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2.5.3 Joint products, transformation function, and profit maximization
In more general cases, the technology of a producer is represented by a transformation
function: Fj
(yj
1, . . . , yj
n) = 0, where (yj
1, . . . , yj
n) is called a production plan, if yj
k > 0
(yj
k) then k is an output (input) of j.
Example: a producer produces two outputs, y1 and y2, using one input y3. Its
technology is given by the transformation function (y1)2
+ (y2)2
+ y3 = 0. Its profit
is Π = p1y1 + p2y2 + p3y3. The maximization problem is
max
y1,y2,y3
p1y1 + p2y2 + p3y3 subject to (y1)2
+ (y2)2
+ y3 = 0.
To solve the maximization problem, we can eliminate y3: x = −y3 = (y1)2
+(y2)2
> 0
and
max
y1,y2
p1y1 + p2y2 − p3[(y1)2
+ (y2)2
].
The solution is: y1 = p1/(2p3), y2 = p2/(2p3) (the supply functions of y1 and y2), and
x = −y3 = [p1/(2p3)]2
+ [p1/(2p3)]2
(the input demand function for y3).
2.6 Production function and returns to scale
Production function: Q = f(L, K). MPK =
∂Q
∂L
MPK =
∂Q
∂K
IRTS: f(hL, hK) > hf(L, K). CRTS: f(hL, hK) = hf(L, K).
DRTS: f(hL, hK) < hf(L, K).
Supporting factors:
∂2
Q
∂L∂K
> 0. Substituting factors:
∂2
Q
∂L∂K
< 0.
Example 1: Cobb-Douglas case F(L, K) = ALa
Kb
.
Example 2: CES case F(L, K) = A[aLρ
+ (1 − a)Kρ
]1/ρ
.
2.7 Cost function: C(Q)
Total cost TC = C(Q) Average cost AC =
C(Q)
Q
Marginal cost MC = C (Q).
Example 1: C(Q) = F + cQ
Example 2: C(Q) = F + cQ + bQ2
Example 3: C(Q) = cQa
.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-9-2048.jpg)
![15
4.4 Multiproduct monopoly
Consider a producer who is monopoly (the only seller) in two joint products.
Q1 = D1(P1, P2), Q2 = D2(P1, P2), TC = C(Q1, Q2).
The profit as a function of (P1, P2) is
Π(P1, P2) = P1D1(P1, P2) + P2D2(P1, P2) − C(D1(P1, P2), D2(P1, P2)).
Maximizing Π(P1, P2) w. r. t. P1, we have
D1(P1, P2) + P1
∂D1
∂P1
+ P2
∂D2
∂P1
−
∂C
∂Q1
∂D1
∂P1
−
∂C
∂Q2
∂D2
∂P1
= 0,
⇒
P1 − MC1
P1
=
1
| 11|
+
P2 − MC2
P2
TR2
TR1
21
| 11|
Case 1: 12 0, goods 1 and 2 are substitutes,
P1 − MC1
P1
1
| 11|
.
Case 2: 12 0, goods 1 and 2 are complements,
P1 − MC1
P1
1
| 11|
.
Actually, both P1 and P2 are endogenous and have to be solved simultaneously.
1 −
R2 21
R1| 11|
−R1 12
R2| 22|
1
P1 − C1
P1
P2 − C2
P2
=
1
| 11|
1
| 22|
⇒
P1 − C1
P1
P2 − C2
P2
=
1
11 22 − 12 21
| 22| +
R2
R1
21
| 11| +
R1
R2
12
.
4.4.1 2-period model with goodwill (Tirole EX 1.5)
Assume that Q2 = D2(p2; p1) and
∂D2
∂p1
0, ie., if p1 is cheap, the monopoy gains
goodwill in t = 2.
max
p1,p2
p1D1(p1) − C1(D1(p1)) + δ[p2D2(p2; p1) − C2(D2(p2; p1))].
4.4.2 2-period model with learning by doing (Tirole EX 1.6)
Assume that TC2 = C2(Q2; Q1) and
∂C2
∂Q1
0, ie., if Q1 is higher, the monopoy gains
more experience in t = 2.
max
p1,p2
p1D1(p1) − C1(D1(p1)) + δ[p2D2(p2) − C2(D2(p2), D1(p1))].](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-17-2048.jpg)
![16
Continuous time (Tirole EX 1.7):
max
qt,wt
∞
0
[R(qt) − C(wt)qt]e−rt
dt, wt =
t
0
qτ dτ,
where R(qt) is the revenue at t,R 0, R 0, r is the interest rate, Ct = C(wt) is
the unit production cost at t, C 0, and wt is the experience accumulated by t.
Example: R(q) =
√
q and C(w) = a +
1
w
.
4.5 Durable good monopoly
Flow (perishable) goods: ‡
Durable goods: ˝‹
Coase (1972:) A durable good monopoly is essentially different from a perishable
good monopoly.
Perishable goods: .°v‚ ÒÖ , ©‚·b½h˛.
Durable goods: .°v‚ Ò.Ö , ¥‚˛-‚ÿ..y, ÝBªJž“.
4.5.1 A two-period model
There are 100 potential buyers of a durable good, say cars. The value of the service
of a car to consumer i each period is Ui = 101 − i, i = 1, . . . , 100.
Assume that MC = 0 and 0 δ 1 is the discount rate.
E i, Q
T
Ui, P
rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr
The trace of Ui’s becomes
the demand curve.
ù »j¶: 1. É•.“. PR
1 , PR
2 are the rents in periods 1 and 2.
2. “i. Ps
1 , Ps
2 are the prices of a car in periods 1 and 2.
4.5.2 É•.“
The monopoly faces the same demand function P = 100 − Q in each period. The
monopoly profit maximization implies that MR = 100 − 2Q = 0. Therefore,
PR
1 = PR
2 = 50, πR
1 = πR
2 = 2500, ΠR
= πR
1 + δπR
2 = 2500(1 + δ),
where δ 1 is the discounting factor.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-18-2048.jpg)
![17
4.5.3 “i
We use backward induction method to find the solution to the profit maximization
problem. We first assume that those consumers who buy in period t = 1 do not resale
their used cars to other consumers.
Suppose that qs
1 = ¯q1, ⇒ the demand in period t = 2 becomes
qs
2 = 100 − ¯q1 − Ps
2 , ⇒ Ps
2 = 100 − ¯q1 − q2 and
MR2 = 100 − ¯q1 − 2q2 = 0, qs
2 = 50 − 0.5¯q1 = Ps
2 , πs
2 = (100 − ¯q1)2
/4.
Now we are going to calculate the location of the marginal consumer ¯q1 who is indif-
ferent between buying in t = 1 and buying in t = 2.
(1 + δ)(100 − ¯q1) − Ps
1 = δ[(100 − ¯q1) − Ps
2 ] ⇒ (100 − ¯q1) − Ps
1 = −δPs
2 ,
⇒ Ps
1 = 100 − ¯q1 + δPs
2 = (1 + 0.5δ)(100 − ¯q1).
max
q1
Πs
= πs
1 + δπs
2 = q1Ps
1 + δ(50 − 0.5q1)2
= (1 + 0.5δ)q1(100 − q1) + 0.25(100 − q1)2
,
FOC⇒ qs
1 = 200/(4+δ), Ps
1 = 50(2+δ)2
/(4+δ), Πs
= 2500(2+δ)2
/(4+δ) ΠR
= (1+δ)2500.
When a monopoly firm sells a durable good in t = 1 instead of leasing it, the monopoly
loses some of its monopoly power, that is why Πs
ΠR
.
4.5.4 Coase problem
Sales in t will reduce monopoly power in the future. Therefore, a rational expectation
consumer will wait.
Coase conjecture (1972): In the ∞ horizon case, if δ→1 or ∆t→0, then the monopoly
profit Πs
→0.
The conjecture was proved in different versions by Stokey (1981), Bulow (1982), Gul,
Sonnenschein, and Wilson (1986).
Tirole EX 1.8.
1. A monopoly is the only producer of a durable good in t = 1, 2, 3, . . .. If (q1, q2, q3, . . .)
and (p1, p2, p3, . . .) are the quantity and price sequences for the monopoly product,
the profit is
Π =
∞
t=1
δt
ptqt.
2. There is a continuum of consumers indexed by α ∈ [0, 1], each needs 1 unit of the
durable good.
vα = α + δα + δ2
α + . . . =
α
1 − δ
: The utility of the durable good to consumer α.
If consumer α purchases the good at t, his consumer surplus is
δt
(vα − pt) = δt
(
α
1 − δ
− pt).](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-19-2048.jpg)
![18
3. A linear stationary equilibrium is a pair (λ, µ), 0 λ, µ 1, such that
(a) If vα λpt, then consumer α will buy in t if he does not buy before t.
(b) If at t, all consumers with vα v (vα v) have purchased (not purchased) the
durable good, then the monopoly charges pt = µv.
(c) The purchasing strategy of (a) maximizes consumer α’s consumer surplus, given
the pricing strategy (b).
(d) The pricing strategy of (b) maximizes the monopoly profits, given the purchacing
strategy (a).
The equilibrium is derived in Tirole as
λ =
1
√
1 − δ
, µ = [
√
1 − δ − (1 − δ)]/δ, lim
δ→1
λ = ∞, lim
δ→1
µ = 0.
One way a monopoy of a durable good can avoid Coase problem is price commitment.
By convincing the consumers that the price is not going to be reduced in the future,
it can make the same amount of profit as in the rent case. However, the commitment
equilibrium is not subgame perfect. Another way is to make the product less durable.
4.6 Product Selection, Quality, and Advertising
Tirole, CH2.
Product space, Vertical differentiation, Horizontal differentiation.
Goods-Characteristics Approach, Hedonic prices.
Traditional Consumer-Theory Approach.
4.6.1 Product quality selection, Tirole 2.2.1, pp.100-4.
Inverse Demand: p = P(q, s), where s is the quality of the product.
Total cost: TC = C(q, s), Cq 0, Cs 0.
Social planner’s problem:
max
q,s
W(q, s) =
q
0
P(x, s)ds − C(q, s),
FOC: (1) Wq = P(q, s) − Cq = 0, (2) Ws =
q
0
Ps(x, s)dx − Cs = 0.
(1) P = MC,
(2)
1
q
q
0
Psdx = Cs/q: Average marginal valuation of quality should be equal to the
marginal cost of quality per unit.
Monopoly profit maximization:
max
q,s
Π(q, s) = qP(x, s)−C(q, s), FOC Πq = MR−Cq = 0, Πs = qPs(x, s)−Cs = 0.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-20-2048.jpg)
![19
Ps = Cs/q: Marginal consumer’s marginal valuation of quality should be equal to the
marginal cost of quality per unit.
Example 1: P(q, s) = f(q) + s, C(q, s) = sq, ⇒ Ps = 1, Cs = q, no distortion.
Example 2: There is one unit of consumers indexed by x ∈ [0, ¯x]. U = xs − P, F(x)
is the distribution function of x. ⇒ P(q, s) = sF −1
(1 − q) ⇒
1
q
q
0
Psdx ≥ Ps(q, s),
monopoly underprovides quality.
Example 3: U = x + (α − x)s − P, x ∈ [0, α], F(x) is the distribution function
of x ⇒ P(q, s) = αs + (1 − s)F −1
(1 − q) ⇒
1
q
q
0
Psdx ≤ Ps(q, s), monopoly overpro-
vides quality.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-21-2048.jpg)
![21
payoff functions can be represented by matrices.
infinite – strategy (continuous or discontinuous payoff functions) games like
duopoly games.
3. according to sum of payoffs:
0-sum games – conflict is unavoidable.
non-zero sum games – possibilities for cooperation.
4. according to preplay negotiation possibility:
non-cooperative games – each person makes unilateral decisions.
cooperative games – players form coalitions and decide the redistribution of
aggregate payoffs.
5.2 The extensive form and normal form of a game
Extensive form: The rules of a game can be represented by a game tree.
The ingredients of a game tree are:
1. Players
2. Nodes: they are players’ decision points (personal moves) and randomization
points (nature’s moves).
3. Information sets of player i: each player’s decision points are partitioned into
information sets. An information set consists of decision points that player i can not
distinguish when making decisions.
4. Arcs (choices): Every point in an information set should have the same number of
choices.
5. Randomization probabilities (of arcs following each randomization point).
6. Outcomes (end points)
7. Payoffs: The gains to players assigned to each outcome.
A pure strategy of player i: An instruction that assigns a choice for each information
set of player i.
Total number of pure strategies of player i: the product of the numbers of choices of
all information sets of player i.
Once we identify the pure strategy set of each player, we can represent the game
in normal form (also called strategic form).
1. Strategy sets for each player: S1 = {s1, . . . , sm}, S2 = {σ1, . . . , σn}.
2. Payoff matrices: π1(si, σj) = aij, π2(si, σj) = bij. A = [aij], B = [bij].
Normal form:
II
I d
d
d
σ1 . . . σn
s1 (a11, b11) . . . (a1n, b1n)
...
...
...
...
sm (am1, bm1) . . . (amn, bmn)](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-23-2048.jpg)
![29
Consider the case of a 3-person game. There are 8 subsets of N = {1, 2, 3}, namely,
φ, (1), (2), (3), (12), (13), (23), (123). Therefore, a characteristic form game is deter-
mined by 8 values v(φ), v(1), v(2), v(3), v(12), v(13), v(23), v(123).
Super-additivity: If A ∩ B = φ, then v(A ∪ B) ≥ v(A) + v(B).
An imputation is a payoff distribution (x1, x2, x3).
Individual rationality: xi ≥ v(i).
Group rationality: i∈S xi ≥ v(S).
Core C: the set of imputations that satisfy individual rationality and group rational-
ity for all S.
Marginal contribution of player i in a coalition S ∪ i: v(S ∪ i) − v(S)
Shapley value of player i is an weighted average of all marginal contributions
πi =
S⊂N
|S|!(n − |S| − 1)!
n!
[v(S ∪ i) − v(S)].
Example: v(φ) = v(1) = v(2) = v(3) = 0, v(12) = v(13) = v(23) = 0.5, v(123) = 1.
C = {(x1, x2, x3), xi ≥ 0, xi + xj ≥ 0.5, x1 + x2 + x3 = 1}. Both (0.3, 0.3, 0.4) and
(0.2, 0.4, 0.4) are in C.
The Shapley values are (π1, π2, π3) = (1
3
, 1
3
, 1
3
).
Remark 1: The core of a game can be empty. However, the Shapley values are
uniquely determined.
Remark 2: Another related concept is the von-Neumann Morgenstern solution. See
CH 6 of Intriligator’s Mathematical Optimization and Economic Theory for the mo-
tivations of these concepts.
5.9 The Nash bargaining solution for a nontransferable 2-person cooper-
ative game
In a nontransferable cooperative game, after-play redistributions of payoffs are im-
possible and therefore the concepts of core and Shapley values are not suitable. For
the case of 2-person games, the concept of Nash bargaining solutions are useful.
Let F ⊂ R2
be the feasible set of payoffs if the two players can reach an agreement
and Ti the payoff of player i if the negotiation breaks down. Ti is called the threat
point of player i. The Nash bargaining solution (x∗
1, x∗
2) is defined to be the solution
to the following problem:
E x1
T
x2
T1
T2
x∗
1
x∗
2
max
(x1,x2)∈F
(x1 − T1)(x2 − T2)](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-31-2048.jpg)
![34
6.2 Sequential moves – Stackelberg equilibrium
Consider now that the two firms move sequentially. At t = 1 firm 1 chooses q1. At
t = 2 firm 2 chooses q2.
Firm 1 – leader, Firm 2 – follower.
The consequence is that when choosing q2, firm 2 already knows what q1 is. On
the other hand, in deciding the quantity q1, firm 1 takes into consideration firm 2’s
possible reaction, i.e., firm 1 assumes that q2 = R2(q1). This is the idea of backward
induction and the equilibrium derived is a subgame perfect Nash equilibrium.
At t = 1, firm 1 chooses q1 to maximize the (expected) profit π1 = π1(q1, R2(q1)):
max
q1
[a−b(q1+R2(q1))]q1−c1q1 = [a−b(q1+
a − c2
2b
−0.5q1)]q1−c1q1 = 0.5(a+c2−bq1)q1−c1q1.
The FOC (interior solution) is
dπ1
dq1
=
∂π1
∂q1
+
∂π1
∂q2
dR2
dq1
= 0.5(a − 2c1 + c2 − 2bq1) = 0,
⇒ qs
1 =
a − 2c1 + c2
2b
qc
1, qs
2 =
a + 2c1 − 3c2
4b
qc
2.
Qs
=
3a − 2c1 − c2
4b
Qc
, Ps
=
a + 2c1 + c2
4
Pc
, ((a + c1 + c2)/3 = Pc
c1).
πs
1 =
(a − 2c1 + c2)
8b
, πs
2 =
(a + 2c1 − 3c2)
16b
.
1. πs
1 + πs
2 πc
1 + πc
2 depending on c1, c2.
2. πs
1 πc
1 because πs
1 = maxq1 π1(q1, R2(q1)) π1(qc
1, R2(qc
1)) = πc
1.
3. πs
2 πc
2 since qs
2 qc
2 and Ps
Pc
.
E q1
T
q2
e
e
e
e
e
e
e
e
e
e
R1(q2)
r
rr
rr
rr
r
r
R2(q1)
qc
2
qc
1
qs
2
qs
1
6.2.1 Subgame non-perfect equilibrium
In the above, we derived a subgame perfect equilibrium. On the other hand, the game
has many non-perfect equilibria. Firm 2 can threaten firm 1 that if q1 is too large, he](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-36-2048.jpg)
![35
will chooses a large enough q2 to make market price zero. This is an incredible threat
because firm 2 will hurt himself too. However, if firm 1 believes that the threat will
be executed, there can be all kind of equilibria.
6.2.2 Extension
The model can be extended in many ways. For example, when there are three firms
choosing output quantities sequentially. Or firms 1 and 2 move simultaneously and
then firm 3 follows, etc.
6.3 Conjecture Variation
In Cournot equilibrium, firms move simultaneously and, when making decision, expect
that other firms will not change their quantities. In more general case, firms will form
conjectures about other firms behaviors.
πe
1(q1, qe
2) = (a − bq1 − bqe
2)q1 − c1q1, πe
2(qe
1, q2) = (a − bqe
1 − bq2)q2 − c2q2.
The FOC are
dπe
1
dq1
=
∂πe
1
∂q1
+
∂πe
1
∂qe
2
dqe
2
dq1
= 0,
dπe
2
dq2
=
∂πe
2
∂q2
+
∂πe
2
∂qe
1
dqe
1
dq2
= 0.
Assume that the conjectures are
dqe
2
dq1
= λ1,
dqe
1
dq2
= λ2.
In equilibrium, qe
1 = q1 and qe
2 = q2. The FOCs become
a − 2bq1 − bq2 − bλ1q1 − c1 = 0, a − 2bq2 − bq1 − bλ2q2 − c2 = 0.
In matrix form,
(2 + λ1)b b
b (2 + λ2)b
q1
q2
=
a − c1
a − c2
,
⇒
q1
q2
=
1
[3 + 2(λ1 + λ2) + λ1λ2]b
(a − c1)(2 + λ2) − (a − c2)
(a − c2)(2 + λ1) − (a − c1)
.
If λ1 = λ2 = 0, then it becomes the Cournot equilibrium.
6.3.1 Stackelberg Case: λ2 = 0, λ1 = R2(q1) = 0.5
Assuming that c1 = c2 = 0.
q1
q2
=
1
2b
a
a/2
.
It is the Stackelberg leadership equilibrium with firm 1 as the leader.
Similarly, if λ1 = 0, λ2 = R1(q2) = 0.5, then firm 2 becomes the leader.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-37-2048.jpg)
![38
6.6 A 2-period model
At t = 1, both firms determine quantities q1, q2.
At t = 2, both firms determine prices p1, p2 after seeing q1, q2.
Demand function: P = 10 − Q, MCs: c1 = c2 = 1.
Proposition: If 2qi + qj ≤ 9, then p1 = p2 = 10 − (q1 + q2) = P(Q) is the equi-
librium at t = 2.
Proof: Suppose p2 = 10 − (q1 + q2), then p1 = p2 maximizes firm 1’s profit. The
reasons are as follows.
1. If p1 p2, π1(p1) = q1(p1 − 1) π1(p2) = q1(p2 − 1).
2. If p1 p2, π1(p1) = (10−q2−p1)(p1−1), π1(p1) = 10−q2−2p1+1 = 2q2+q1−9 ≤ 0.
It follows that at t = 1, firms expect that P = 10 − q1 − q2, the reduced profit
functions are
π1(q1, q2) = q1(P(Q)−c1) = q1(9−q1−q2), π2(q1, q2) = q2(P(Q)−c2) = q2(9−q1−q2).
Therefore, it becomes an authentic Cournot quantity competition game.
6.7 Infinite Repeated Game and Self-enforcing Collusion
Another way to resolve Bertrand paradox is to consider the infinite repeated version
of the Bertrand game. To simplify the issue, assume that c1 = c2 = 0 and P =
min{1 − Q, 0} = min{1 − q1 − q2, 0} in each period t. The profit function of firm i at
time t is
πi(t) = πi(q1(t), q2(t)) = qi(t)[1 − q1(t) − q2(t)].
A pure strategy of the repeated game of firm i is a sequence of functions σi,t of
outcome history Ht−1:
σi ≡ (σi,0, σi,1(H0), . . . , σi,t(Ht−1), . . .) .
where Ht−1 is the history of a play of the repeated game up to time t − 1:
Ht−1 = ((q1(0), q2(0)), (q1(1), q2(1)), . . . , (q1(t − 1), q2(t − 1))) ,
and σi,t maps from the space of histories Ht−1 to the space of quantities {qi : 0 ≤
qi ∞}. Given a pair (σ1, σ2), the payoff function of firm i is
Πi(σ1, σ2) =
∞
t=0
δt
πi(σ1,t(Ht−1), σ2,t(Ht−1)) =
∞
t=0
δt
πi(q1(t), q2(t)).
Given the Cournot equilibrium (qc
1, qc
2) = (
1
3
,
1
3
), we can define a Cournot strategy for
the repeated game as follows:
σc
i,t(Ht−1) = qc
i =
1
3
∀Ht−1, σc
i ≡ (σc
i,0, σc
i,1, . . . , σc
i,t, . . .).
It is straightforward to show that the pair (σc
1, σc
2) is a Nash equilibrium for the
repeated game.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-40-2048.jpg)
![41
6.7.5 Folk Theorem of the infinite repeated game
In the above, we consider only the cooperation to divide the monopoly profit evenly.
The same argument works for other kinds of distributions of monopoly profit or even
aggregate profits less than the monopoly profit.
Folk Theorem: When δ→1, every distributions of profits such that the average
payoff per period πi ≥ πc
i can be implemented as a SPNE.
For subgame-non-perfect NEs, the individual profits can be even lower then the
Cournot profit.
6.7.6 Finitely repeated game
If the duopoly game is only repeated finite time, t = 1, 2, . . . , T, we can use backward
induction to find Subgame-perfect Nash equilibria. Since the last period T is the
same as a 1-period duopoly game, both firms will play Cournot equilibrium quantity.
Then, since the last period strategy is sure to be the Cournot quantity, it does not
affect the choice at t = T − 1, therefore, at t = T − 1 both firms also play Cournot
strategy. Similar argument is applied to t = T − 2, t + T − 3, etc., etc.
Therefore, the only possible SPNE is that both firms choose Cournot equilibrium
quantity from the beginning to the end.
However, there may exist subgame-non-perfect NE.
6.7.7 Infinite repeated Bertrand price competition game
In the above, we have considered a repeated game of quantity compeition. We can
define an infinitely repeated price competition game and a trigger strategy similarily:
(1) In the cooperative phase, a firm sets monopoly price and gains one half of the
monopoly profit 0.5πm = 1/8.
(2) In the non-cooperative phase, a firm sets Bertrand competition price pb = 0 and
gains 0 profit.
To deviate from the cooperative phase, a firm obtains the whole monopoly profit
πm = 1/4 instantly. If 1/[8(1 − δ)] 1/4 (δ 0.5), the trigger strategy is a SPNE.
6.8 Duopoly in International Trade
6.8.1 Reciprocal Dumping in International Trade
2 countries, i = 1, 2 each has a firm (also indexed by i) producing the same product.
Assume that MC = 0, but the unit transportation cost is τ.
qh
i , qf
i : quantities produced by country i’s firm and sold in domestic market and for-
eign market, respectively.
Q1 = qh
1 + qf
2 , Q2 = qh
2 + qf
1 : aggregate quantities sold in countries 1 and 2’s market,
respectively.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-43-2048.jpg)
![42
Pi = a − bQi: market demand in country i’s market.
The profits of the international duopoly firms are
Π1 = P1qh
1 + (P2 − τ)qf
1 = [a − b(qh
1 + qf
2 )]qh
1 + [a − (qf
1 + qh
2 ) − τ]qf
1 ,
Π2 = P2qh
2 + (P1 − τ)qf
2 = [a − b(qh
2 + qf
1 )]qh
2 + [a − (qf
2 + qh
1 ) − τ]qf
2 .
Firm i will choose qh
i and qf
i to maximize Πi. The FOC’s are
∂Πi
∂qh
i
= a − 2bqh
i − bqf
j = 0,
∂Πi
∂qf
i
= a − 2bqf
i − bqh
j − τ = 0, i = 1, 2.
In a symmetric equilibrium qh
1 = qh
2 = qh
and qf
1 = qf
2 = qf
. In matrix form, the
FOC’s become:
2b b
b 2b
qh
qf =
a
a − τ
, ⇒
qh
qf =
a + τ
3b
a − 2τ
3b
,
Q = qh
+ qf
=
2a − τ
3b
P =
a + τ
3
.
It seems that there is reciprocal dumping: the FOB price of exports P FOB
is lower
than the domestic price P.
PCIF
= P =
a + τ
3
, PFOB
= PCIF
− τ =
a − 2τ
3
P.
However, since PFOB
MC = 0, there is no dumping in the MC definition of dump-
ing.
The comparative statics with respect to τ is
∂qh
∂τ
0,
∂qf
∂τ
0,
∂Q
∂τ
0,
∂P
∂τ
0.
In this model, it seems that international trade is a waste of transportation costs
and is unnecessary. However, if there is no international competition, each country’s
market would become a monopoly.
Extensions: 1. 2-stage game. 2. comparison with monopoly.
6.8.2 Preferential Trade Agreement, Trade Creation, Trade Diversion
Free Trade Agreement FTA: Free trade among participants.
Customs Union CU: FTA plus uniform tariff rates towards non-participants.
Common Market CM: CU plus free factor mobility.
Consider the apple market in Taiwan.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-44-2048.jpg)
![43
Demand: P = a − Q.
2 export countries: America and Japan, PA PJ .
At t = 0, Taiwan imposes uniform tariff of $ t per unit.
P0 = PA + t, Q0 = a − PA − t, W0 = CS0 + T0 =
Q2
0
2
+ tQ0 =
(a − PA)2
− t2
2
.
At t = 1, Taiwan and Japan form FTA, assume PJ PA + t.
P1 = PJ, Q1 = a − PJ , W1 = CS1 + T1 =
Q2
1
2
=
(a − PJ )2
2
.
W1 − W0 = 0.5[(a − PJ )2
+ t2
− (a − PA)2
] = 0.5[t2
− (2a − PA − PJ)(PJ − PA)].
Given a and PA, FTA is more advantageous the higher a and the lower PJ .
E Q
T
P
d
d
d
d
d
d
d
d
d
d
Q0
P0
PA
PJ T0
CS0
E Q
T
P
d
d
d
d
d
d
d
d
d
d
Q1
P1
CS1
E Q
T
P
d
d
d
d
d
d
d
d
d
d
Q0
P0
PA
Q1
P1
φ
β γ
δ
W1 − W0 = (φ + β + γ) − (φ + β + δ) = γ − δ.
γ: trade creation effect.
δ: trade diversion effect.
W1 − W0 0 if and only if γ δ.
6.9 Duopoly under Asymmetric Information
6.9.1 Incomplete information game
Imperfect information game: Some information sets contain more than one nodes,
i.e., at some stages of the game, a player may be uncertain about the consequences
of his choices.
Incomplete information game: Some players do not completely know the rule of
the game. In particular, a player does not know the payoff functions of other players.
There are more than one type of a player, whose payoff function depends on his type.
The type is known to the player himself but not to other players. There is a prior
probability distribution of the type of a player.
Bayesian equilibrium (of a static game): Each type of a player is regarded as an
independent player.
Bayesian perfect equilibrium (of a dynamic game): In a Bayesian equilibrium, players
will use Bayes’ law to estimate the posterior distribution of the types of other players.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-45-2048.jpg)
![44
6.9.2 Modified chicken game
In the chicken dilemma game, assume that there are 2 types of player 2, 2a and 2b,
2a is as before but 2b is different, who prefers NN to SS, SS to being a chicken:
1
S N
d
d
d
d
¨
©2a
S N S N
5
5
2
10
10
2
0
0
1
S N
d
d
d
d
¨
©2b
S N S N
5
2
2
10
10
0
0
5
Asymmetric information: player 2 knows whether he is 2a or 2b but player 1 does
not. However, player 1 knows that Prob[2a] = Prob[2b] =0.5.
If we regard 2a and 2b as two different players, the game tree becomes:
d
d
d
d
d
d
¡
¡
¡
¡
¡
¡
e
e
e
e
e
e
¡
¡
¡
¡
¡
¡
e
e
e
e
e
e
0
¨
©1
¨
©2a
¨
©2b
5
5
0
2
10
0
10
2
0
0
0
0
5
0
2
2
0
10
10
0
0
0
0
5
1/2 1/2
S N
S N S N
S N
S N S N
In this incomplete information game, player 2b has a dominant strategy, N. Hence,
it can be reduced to a 2-person game. The pure strategy Bayesian equilibria are
SNN and NSN. The mixed strategy equilibrium is different from the ordinary chicken
game.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-46-2048.jpg)
![45
6.9.3 A duopoly model with unknown MC
2 firms, 1 and 2, with market demand p = 2 − q1 − q2.
MC1 = 1, π1 = q1(1 − q1 − q2).
2 types of firm 2, a and b.
MC2a = 1.25, π2a = q2a(0.75 − q1 − q2a).
MC2b = 0.75, π2b = q2b(1.25 − q1 − q2b).
Asymmetric information: firm 2 knows whether he is 2a or 2b but firm 1 does not.
However, firm 1 knows that Prob[2a] = Prob[2b] =0.5.
To find the Bayesian equilibrium, we regard the duopoly as a 3-person game with
payoff functions:
Π1(q1, q2a, q2b) = 0.5q1(1 − q1 − q2a) + 0.5q1(1 − q1 − q2b)
Π2a(q1, q2a, q2b) = 0.5q2a(0.75 − q1 − q2a)
Π2b(q1, q2a, q2b) = 0.5q2b(1.25 − q1 − q2b)
FOC are
0.5(1−2q1−q2a)+0.5(1−2q1−q2b) = 0, 0.5(0.75−q1−2q2a) = 0, 0.5(1.25−q1−2q2b) = 0.
4 1 1
1 2 0
1 0 2
q1
q2a
q2b
=
2
0.75
1.25
.
The Bayesian equilibrium is (q1, q2a, q2b) = (
1
3
,
5
24
,
11
24
).](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-47-2048.jpg)
![49
7.1.4 Comparison between quantity and price games
pc
− pb
=
α
4δ−1 − 1
0, ⇒ qc
− qb
0.
As δ→0, pc
→pb
and when δ = 0, pc
= pb
.
Strategic substitutes vs Strategic complements: In a continuous game, if the
slopes of the reaction functions are negative as in the Cournot quantity game, the
strategic variables (e.g., quantities) are said to be strategic substitutes. If the slopes
are positive as in the Bertrand price game, the strategic variables (e.g., prices) are
said to be strategic complements.
7.1.5 Sequential moves game
The Stackelberg quantity leadership model can be generalized to the differentiated
product case. Here we use an example to illustrate the idea. Consider the following
Bertrand game:
q1 = 168 − 2p1 + p2, q2 = 168 − 2p2 + p1; TC = 0.
⇒ pi = Rb
i (qj) = 42 + 0.25pj, pb
= 56, qb
= 112, πb
= 6272.
In the sequential game version, assume that firm 1 moves first:
π1 = [168 − 2p1 + (42 + 0.25p1)]p1 = [210 − 1.75p1]p1.
FOC: 210 − 3.5p1 = 0, ⇒ p1 = 60, p2 = 57, q1 = 105, q2 = 114, π1 = 6300, π2 =
6498.
⇒ p1 p2, q1 q2, π2 π1 πb
= 6272.
The Cournot game is:
p1 = 168 −
2q1 + q2
3
, p2 = 168 −
2q2 + q1
3
;
⇒ qi = Rc
i (qj) = 126 − 0.25qj, q = pc
= 100.8, pc
= 67.2, πc
= 6774.
In the sequential game version, assume that firm 1 moves first:
π1 = (168 −
2
3
qi −
1
3
qj)qi = (126 −
7
12
qj)qi.
FOC: 126 −
7
6
q1 = 0, ⇒ q1 = 108, q2 = 81, p1 = 69, p2 = 78, π1 = 7452, π2 = 6318.
⇒ p1 p2, q1 q2, π1 πc
π2.
From the above example, we can see that in a quantity game firms prefer to be
the leader whereas in a price game they prefer to be the follower.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-51-2048.jpg)
![52
As N is very large,
∂A
∂pi
≈ 0 and the demand elasticity is approximately |η| =
1
1 − α
.
The cost function of producer i is TCi(qi) = F + cqi.
max
qi
piqi − TC(qi) = A1−α
qα
i − F − cqi.
The monopoly profit maximization pricing rule MC = P(1 − 1
|η|
) means:
p∗
i = c
|η|
|η| − 1
=
c
α
, πi = (p∗
i − c)qi − F =
c(1 − α)
α
qi − F.
In equilibrium, πi = 0, we have (using the consumer’s budget constraint)
q∗
i =
αF
(1 − α)c
, N∗
=
I
p∗q∗
=
(1 − α)I
F
.
The conclusions are
1. p∗
= c/α and Lerner index of each firm is
p − c
p
= 1 − α.
2. Each firm produces q∗
=
αF
(1 − α)c
.
3. N∗
=
(1 − α)I
F
.
4. Variety effect:
U∗
= (N∗
q∗α
)1/α
= N∗1/α
q∗
= [αα
(1 − α)1−α
I]1/α
F1− 1
α c−1
.
The size of the market is measured by I. When I increases, N∗
(the variety)
increases proportionally. However, U∗
increases more than proportionally.
Examples: Restaurants, Profesional Base Ball, etc.
If we approximate the integer number N by a continuous variable, the utility function
and the budget constraint are now
U({q(t)}0≤t≤N ) =
∞
0
[q(t)]α
dt
1/α
,
N
0
p(t)q(t)dt = I.
The result is the same as above.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-54-2048.jpg)
![54
Firm B is located at point L − b, PB is the price of its product.
Each point x ∈ [0, L] represents a consumer x. Each consumer demands a unit
of the product, either purchases from A or B.
Ux =
−PA − τ|x − a| if x buys from A.
−PB − τ|x − (L − b)| if x buys from B.
τ is the transportation cost per unit distance. |x − a| (|x − (L − b)|) is the distance
between x and A (B).
The marginal consumer ˆx is indifferent between buying from A and from B. The
location of ˆx is determined by
−PA − τ|x − a| = −PB − τ|x − (L − b)| ⇒ ˆx =
L − b + a
2
−
PA − PB
2τ
. (1)
The location of ˆx divids the market into two parts: [0, ˆx) is firm A’s market share
and (ˆx, L] is firm B’s market share.
0 L
d d
a L − b
r
ˆx
A’s share B’s share' E' E
Therefore, given (PA, PB), the demand functions of firms A and B are
DA(PA, PB) = ˆx =
L − b + a
2
−
PA − PB
2τ
, DB(PA, PB) = L−ˆx =
L − a + b
2
+
PA − PB
2τ
.
Assume that firms A and B engage in price competition and that the marginal costs
are zero. The payoff functions are
ΠA(PA, PB) = PA
L − b + a
2
−
PA − PB
2τ
, ΠB(PA, PB) = PB
L − a + b
2
+
PA − PB
2τ
.
The FOCs are (the SOCs are satisfied)
L − b + a
2
−
2PA − PB
2τ
= 0,
L − a + b
2
+
PA − 2PB
2τ
= 0. (2)
The equilibrium is given by
PA =
τ(3L − b + a)
3
, PB =
τ(3L − a + b)
3
, QA = ˆx =
3L − b + a
6
, QB = L−ˆx =
3L − a + b
6
.
ΠA =
τ(3L − b + a)2
18
=
τ(2L + d + 2a)2
18
, ΠB =
τ(3L − a + b)2
18
=
τ(2L + d + 2b)2
18
,
where d ≡ L − b − a is the distance between the locations of A and B. the degree of
product differentiation is measured by dτ. When d or τ increases, the products are
more differentiated, the competition is less intensive, equilibrium prices are higher
and firms are making more profits.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-56-2048.jpg)
![56
0 L
d d
a L − b
Consumers in [0, a] move together.
Consumers in [L − b, L] move together.
E PA
T
ˆx
L
r
rr
rr
rr
r
0
PB − (L − a − b)τ PB + (L − a − b)τ
L − b
a
The true profit function of firm A is
ΠA(PA, PB) =
PA
L − b + a
2
−
PA − PB
2τ
a ˆx L − b
0 ˆx ≤ a
PAL L − b ≥ ˆx
=
PAL PA ≤ PB − dτ
PA
L − b + a
2
−
PA − PB
2τ
PB − dτ PA PB + dτ
0 PA ≥ PB + dτ,
where d ≡ L − a − b. Consider the case a = b, d = L − 2a. The reaction function of
firm A is (firm B’s is similar):
PA = RA(PB) =
0.5(τL + PB) (τL+PB)2
8τ[PB−dτ]
≤ L or PB
τ
/∈ (3L − 4
√
La, 3L + 4
√
La)
PB − (L − 2a)τ (τL+PB)2
8τ[PB−dτ]
≥ L or PB
τ
∈ [3L − 4
√
La, 3L + 4
√
La]
For a ≤ L/4, the reaction functions intersect at PA = PB = τL. When a L/4, the
reaction functions do not intersect.
E PA
T
ΠA
¡
¡
¡
¡
¡
¡
PB − dτ PB + dτ
E PA
T
PB
¨¨
¨¨
¨¨
¨
¡
¡
¡
¡
¡
¡
¡
Case: a ≤ L/4, P ∗ = τL,
RA
RB
E PA
T
PB
¨¨
¨¨
¨
¡
¡
¡
¡
¡
Case: a L/4, no equilibrium.
RA
RB
7.4.4 Quadratic transportation costs
Suppose now that the transportation cost is proportional to the square of the distance.
Ux =
−PA − τ(x − a)2
if x buys from A.
−PB − τ[x − (L − b)]2
if x buys from A.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-58-2048.jpg)
![57
The marginal consumer ˆx is defined by
−PA − τ(x − a)2
= −PB − τ[x − (L − b)]2
⇒ ˆx =
L − b + a
2
−
PA − PB
2τ(L − a − b)
. (3)
The demand functions of firms A and B are
DA(PA, PB) = ˆx =
L − b + a
2
−
PA − PB
2τ(L − a − b)
, DB(PA, PB) = L−ˆx =
L − a + b
2
+
PA − PB
2τ(L − a − b)
.
The payoff functions are
ΠA(PA, PB) = PA
L − b + a
2
−
PA − PB
2τ(L − a − b)
, ΠB(PA, PB) = PB
L − a + b
2
+
PA − PB
2τ(L − a − b)
.
The FOCs are (the SOCs are satisfied)
L − b + a
2
−
2PA − PB
2τ(L − a − b)
= 0,
L − a + b
2
+
PA − 2PB
2τ(L − a − b)
= 0. (4)
The equilibrium is given by
PA =
τ(3L − b + a)(L − a − b)
3
, QA = ˆx =
3L − b + a
6
, ΠA =
τ(3L − b + a)2
(L − a − b)
18
,
PB =
τ(3L − a + b)(L − a − b)
3
, QB = L−ˆx =
3L − a + b
6
, ΠB =
τ(3L − a + b)2
(L − a − b)
18
.
∂ΠA
∂a
0,
∂ΠB
∂b
0.
Therefore, both firms will choose to maximize their distance. The result is opposite
to the linear transportation case.
Two effects of increasing distance:
1. Increase the differentiation and reduce competition. Π ↑.
2. Reduce a firm’s turf. Π ↓.
In the linear transportation case, the 2nd effect dominates. In the quadratic trans-
portation case, the 1st effect dominates.
Also, ˆx is differentiable in the quadratic case and the interior solution to the profit
maximization problem is the global maximum.
Welfare comparison:
Equilibrium transportation cost curve
0 L
d d
A B
ˆx = L/2 0 Lˆx = L/2
d d
A B
L/4 3L/4
Social optimum transportation cost curve](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-59-2048.jpg)
![59
Since each consumer needs 1 unit, TVC = c is a constant. The total cost to the
society is the sum of aggregate transportation cost, TVC, and fixed cost.
SC(N) = T(N)+c+NF =
τ
4N
+c+NF, FOC: −
τ
4N2
+F = 0, ⇒ Ns
= 0.5
τ
F
.
The conclusion is that the equilibrium number of firms is twice the social optimum
number.
Remarks: 1. AC is decreasing (DRTS), less firms will save production cost.
2. T(N) is decreasing with N, more firms will save consumers’ transportation cost.
3. The monopolistic competition equilibrium ends up with too many firms.
7.6 Sequential entry in the linear city model
If firm 1 chooses his location before firm 2, then clearly firm 1 will choose li = L/2 and
firm 2 will choose l2 = L/2+
. If there are more than two firms, the Nash equilibrium
location choices become much more complicated.
Assume that there are 3 firms and they enter the market (select locations) sequen-
tially. That is, firm 1 chooses x1, then firm 2 chooses x2, and finally firm 3 chooses
x3. To simplify, we assume that firms charge the same price p = 1 and that x1 = 1/4.
We want to find the equilibrium locations x2 and x3.
1. If firm 2 chooses x2 ∈ [0, x1) = [0, 1/4), then firm 3 will choose x3 = x1 + .
π2 = (x2 − x1)/2 1/4.
0 x1 = 1/4 1
rd
x2 d
x3
π2 = (x2 + x1)/2 1/4.
2. If firm 2 chooses x2 ∈ (x1, 3/4) = (1/4, 3/4), then firm 3 will choose x3 = x2 + .
π2 = x2 − (x2 + x1)/2 = (x2 − x1)/2 1/4.
3/40 1/4 1
dx1 dx2 dx3
π2 = (x2 − x1)/2 1/4.
3. If firm 2 chooses x2 ∈ [3/4, 1], then firm 3 will choose x3 = (x2 + x1)/2.
π2 = 1 − 0.5(x2 + x3)/2 = (15 − 12x2)/16.
3/40 1/4 1
d
x1 d
x2d
x3
π2 = (15 − 12x2)/16.
The subgame perfect Nash equilibrium is x2 = 3/4, x3 = 1/2 with π1 = π2 = 3/8 and
π3 = 1/4.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-61-2048.jpg)
![69
If
(1 − k1)2
4
−E 0, firm 2 will choose not to enter. Therefore, firm 2’s true reaction
function is
k2 = R2(k1, E) =
1 − k1
2
if k1 1 − 2
√
E
0 if k1 1 − 2
√
E.
E k1
T
k2
r
rr
rr
1
2
1 − 2
√
E
k2 = R2(k1; E)
In t = 1, firm 1 takes into consideration firm 2’s discontinuous reaction function.
If 1 − 2
√
E ≤
1
2
(⇒ E ≥
1
16
), then firm 1 will choose monopoly output k1 =
1
2
and
firm 2 will stay out. This is the case of entry blockaded.
Next, we consider the case E
1
16
. If firm 1 chooses monopoly output, firm 2 will
enter. Firm 1 is considering whether to choose k1 ≥ 1 − 2
√
E to force firm 2 to give
up or to choose k1 1 − 2
√
E and maximizes duopoly profit.
1. entry deterrence: If firm 2 stays out (k1 ≥ 1 − 2
√
E and k2 = 0), firm 1 is a
monopoly by deterrence:
πd
1(E) = max
k1≥1−2
√
E
k1(1 − k1) = kE(1 − kE), where kE ≡ 1 − 2
√
E.
2. entry accommodate: If firm 2 enters (k1 1 − 2
√
E and k2 0), firm 1 is the
leader of the Stackelberg game:
πs
1(E) = max
k11−2
√
E
k1(1−k1−k2) =
1
2
k1(1−k1), ⇒ πs
1 =
1
2
ks
(1−ks
) =
1
8
where ks
≡
1
2
.
E k1
T
πd
1
πs
1 =
1
8
r
kE
r
r
ks
E 0.00536
Entry accommodate
E k1
T
πd
1
πs
1
r
kE
r
r
ks
0.00536 E
1
16
Entry deterred
E k1
T
πd
1
πs
1
r
kE
r
r
ks
=km
E
1
16
Entry blockaded
πd
1 (E) = (1 − 2
√
E)[1 − (1 − 2
√
E)] πs
1(E) =
1
8
if E
(1 − 1/2)2
16
≈ 0.00536.
In summary, if E 0.0536, firm 1 will accommodate firm 2’s entry; if 0.00536 E
1
16
, firm 1 will choose k1 = kE to deter firm 2; if E F116, firm 1 is a monopoly and
firm 2’s entry is blockaded.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-71-2048.jpg)
![72
E ¯k
T
π1
d
d
d
qc
¯qqs
1
c 1
5
E ¯k
T
π1
d
d
d
d
d
d
qc
¯q qs
1
c 1
5
In Stackelberg model, firm 1 can choose any point on firm 2’s reaction curve R2(q1)
to maximize π1. In Dixit model, firm 1’s choice is restricted to the section of R2(q1)
such that ¯k ∈ [0, ¯q]. When ¯q ≥ qs
1, the result is the same as Stackelberg model. When
¯q qs
1, firm 1 can only choose ¯k = ¯q.
In both cases q1 = ¯k. Therefore, firm 1 does not over-invest (choose ¯k q1) to
threaten firm 2’s entry.
8.4.6 Capital replacement model of Eaton/Lipsey (1980)
It is also possible that an incumbent will replace its capital before the capital is com-
plete depreciated as a commitment to discourage the entry of a potential entrant.
t = −1, 0, 1, 2, 3, . . ..
In each period t, if only one firm has capital, the firm earns monopoly profit H.
If both firms have capital, each earns duopoly profit L.
Each firm can make investment in each period t by paying F.
Denote by Ri
t (Ci
t ) the profit (cost) of firm i in period t.
Πi =
∞
t=0
ρt
(Ri
t − Ci
t ), Ri
t =
0 no capital
L duopoly
H monopoly
Ci
t =
0 no invest (NI)
F invest (INV).
Assumption 1: An investment can be used for 2 periods with no residual value left.
Assumption 2: 2L F H.
Assumption 3: F/H ρ (F − L)/(H − L).
Firm i’s strategy in period t is ai
t ∈ {NI, INV}. Assume that a1
−1 = INV. We
consider only Markov stationary equilibrium.
If firm 2 does not exist, firm 1 will choose to invest (INV) in t = 1, 3, 5, . . .. Given the
threat of firm 2’s possible entry, in a subgame-perfect equilibrium, firm 1 will invest
in every period and firm 2 will not invest forever.
The symmetrical SPE strategy is such that an incumbent firm invests and a potential](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-74-2048.jpg)
![75
An industry configuration: (pI
, qI
).
Feasibility: (1) pI
= a − qI
. (2) πI
= pI
qI
− (F + cqI
) ≥ 0.
Sustainability: ∃(pe
, qe
) such that pe
≤ pI
, qe
≤ a − pe
, πe
= pe
qe
− (F + cqe
) 0.
A contestable-market equilibrium: A feasible, sustainable configuration.
E q
T
p
d
d
d
d
d
d
d
d
d
d
pI r
qI
p = a − q
ATC =
F
q
− c
Extension: 1. More incumbent firms. 2. More than 1 products.
Comments: 1. If there are sunk costs, the conclusions would be reversed. See Stiglitz
(1987) discussed before.
2. If incumbents can respond to hit-and-run entries, they can still make some positive
profits.
8.6 A Taxonomy (}é¶) of Business Strategies
Bulow, Geanakoplos, and Klemperer (1985), “Multimarket Oligopoly: Strategic Sub-
stitutes and Complements,” JPE.
A 2-period model: Firm 1 chooses K1 at t = 1 and firms 1 and 2 choose x1, x2
simultaneously at t = 2.
Π1
= Π1
(K1, x1, x2), Π2
= Π2
(K1, x1, x2).
The reduced payoff functions at t = 1 are
Π1
= Π1
(K1, x∗
1(K1), x∗
2(K1)), Π2 = Π2(K1, x∗
1(K1), x∗
2(K1),
where x∗
1(K1) and x∗
2(K1) are the NE at t = 2, given K1,
∂Π1
dx1
=
∂Π2
dx2
= 0.
top dog (»−, _5T‘): be big or strong to look tough or aggressive.
puppy dog ({ü, Qm‘): be small or weak to look soft or inoffensive.
lean and hungry look (_|Ï.ñ÷.§, ü−): be small or weak to look tough or
aggressive.
fat cat (]ý×j): be big or strong to look soft or inoffensive.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-77-2048.jpg)
![76
8.6.1 Deterence case
If firm 1 decides to deter firm 2, firm 1 will choose K1 to make Π2
= Π2
(K1, x∗
1(K1), x∗
2(K1)) =
0.
dΠ2
dK1
=
∂Π2
∂K1
+
∂Π2
∂x1
dx∗
1
dK1
.
Assume that ∂Π2
/∂K1 = 0 so that only strategic effect exists.
dΠ2
/dK1 =
∂Π2
∂x1
dx∗
1
dK1
0: The investment makes firm 1 tough.
dΠ2
/dK1 =
∂Π2
∂x1
dx∗
1
dK1
0: The investment makes firm 1 soft.
To deter entry, firm 1 will overinvest in case of tough investment (top dog strat-
egy _5T‘íõÆ%, entrant øªVÿbç.)
and underinvest in case of soft investment (lean and hungry look _|Ï.ñ÷.§
íšä, U entrant JÑÌ‚ªÇ).
8.6.2 Accommodation case
If firm 1 decides to accommodate firm 2, firm 1 considers maximizing Π1
= Π1
(K1, x∗
1(K1), x∗
2(K1)).
dΠ1
dK1
=
∂Π1
∂K1
+
∂Π1
∂x2
dx∗
2
dK1
=
∂Π1
∂K1
+
∂Π1
∂x2
dx∗
2
dx1
dx∗
1
dK1
.
Assume that ∂Π1
/∂x2 and ∂Π2
/∂x1 have the same sign and that ∂Π2
/∂K1 = 0.
sign
∂Π1
∂x2
dx∗
2
dx1
dx∗
1
dK1
= sign
∂Π2
∂x1
dx∗
1
dK1
× sign(R2).
There are 4 cases:
1. Tough investment
∂Π2
∂x1
dx∗
1
dK1
0 with negative R2: “top dog” strategy, be big
or strong to look tough or aggressive. _5T‘J9„ entrant −‘.
2. Tough investment with positive R2: “puppy dog” strategy, be small or weak to
look soft or inoffensive. Qm‘, .bK@ entrant Jnù–¬Á.
3. Soft investment
∂Π2
∂x1
dx∗
1
dK1
0 with negative R2: “lean and hungry look” strat-
egy, be small or weak to look tough or aggressive. _|Ï.ñ÷.§íšäJH
U entrant =−.
4. Soft investment with positive R2: “fat cat” strategy, be big or strong to look
soft or inoffensive. ]ý×jJî¸ entrant ı/.
top dog: ı%v9Ê,Þí%, Xº6.
puppy: AŠí*üä, IAnÀíÄ/A.
fat cat: (ö5) ½bí’ŒA, À3. ‹ ‘5×A. A )ícA.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-78-2048.jpg)
![78
t = 2 is 9 − 1 = 8 whereas the lose at t = 1 due to imitation is 9 − 5.83(4.17 − 4) ≈ 8.
The separating equilbrium is as follows. At t = 1,
q1(c1 = 0) = 5.83, p1(c1 = 0) = 4.17, π1(c1 = 0) = 24.31;
q1(c1 = 4) = 3, p1(c1 = 4) = 7, π1(c1 = 0) = 9.
At t = 2, firm 2 will enter only if p1 4.17.
8.7.5 Pooling equilibrium
If the standard distribution of c1 is smaller, a separating equilibrium will not exist.
Instead, the high cost firm 1 will immitate the low cost firm 1 and firm 2 will not
enter.
For example: Prob[c1 = 0] = 0.8 and Prob[c1 = 4] = 0.2.
8.7.6 A finite version
Assume that p1
1 ∈ {7, 5, 4}, the game tree is
d
d
d$$$$$$$$$$$$
$$$$$$$$$$$$
ˆˆˆˆˆˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆˆˆˆˆˆ
¡
¡
¡
e
e
e
¡
¡
¡
e
e
e
¡
¡
¡
e
e
e
¡
¡
¡
e
e
e
¡
¡
¡
e
e
e
¡
¡
¡
e
e
e
0
1a
1b
¨
©2
¨
©2
¨
©2
p 1-p
p1
1 = 7 p1
1 = 5 p1
1 = 4
E N E N E N E N E N E N
0
10
7
0
18
0
34
0
−1.9
46
0
0
0
6
7
0
14
0
38
0
−1.9
50
0
0
0
1
7
0
9
0
37
0
−1.9
49
0
0
There is a separating equilbrium p1
1a = 4, p1
1b = 7 and firm 2 chooses not to en-
ter (N) if p1
1 = 4 and enter (E) if p1
1 ∈ {7, 5}.
For p 7/8.9, there is also a pooling equilbrium p1
1a = p1
1b = 5 and firm 2 chooses
not to enter (N) if p1
1 ∈ {5, 4} and enter (E) if p1
1 = 7.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-80-2048.jpg)
![80
T = 1: It is easily shown that the NE, {µ∗
(I), µ∗
(E)}, is
µ∗
(I) = a, µ∗
(E) =
e if p0
b
1 + b
≡ ¯p
s if p0
¯p.
T = 2: There are three cases.
(1) q0
aδ − 1
aδ
≡ ¯q: It is not worthwhile for I to fight to deter entry:
µ∗
(I1) = µ∗
(I2) = a, µ∗
(E1) = µ∗
(E2s) =
e if p0
¯p
s if p0
¯p,
µ∗
(E2e) =
e if I accommodates at t = 1
s if I fights at t = 1.
(2) q0
¯q and p0
¯p: Fighting will deter entry and µ∗
(I1) = F.
(3) q0
¯q and p0
¯p: Both fighting and accommodating are not equilibrium. I
will randomize. Let β ≡ Prob[F]. β is such that E2’s posterior probability that I is
“tough” equals ¯q:
Prob[“tough”|F] =
p0
p0 + β(1 − p0)
= ¯q ⇒ β =
p0
(1 − p0)b
.
The total probability of fighting at t = 1 for E1 is
p0
+ (1 − p0
)β = p0
(b + 1)/b.
Therefore, E1 will enter if p0
¯p2
and stay out otherwise.
T = 3: (a) p0
¯p2
, I will fight and E1 will stay out at t = 1.
(b) ¯p3
p0
¯p2
, I will randomize at t = 1.
(c) p0
¯p3
, I will accommodate and E1 will enter t = 1.
T 3: Entrants will stay out until t = k such that p0
¯pk
.
When δ = 1: (a) q0
a/(1 + a), I will accommodate at first entry and reveal
its type. Hence, limT →∞ π/T = 0.
(b) q0
a/(1 + a), there exists an n(p0
) such that I will fight until there are no more
than n(p0
) entrants remaining. Hence, limT →∞ π/T = (1 − q0
)a − q0
.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-82-2048.jpg)
![81
9 Research and Development (RD)
RD«àß“ÞG|—Ë
1. hÞßj¶Z‰ ¼A…!Z, Uß“ Ò2 0½h|c
2. h߹Ljhä
3. RD …™Ñ ¼¬ÁµI
4. xXØà½æ
5. RD, _, D%Èê
6. RD D Merger activities 5É[
RD2Гç5ª0 (OECD 1980):
NØ 23%, lÂœ 18%, Úä 10%, »“ 9%, ë¹, ˘“, ˆ, ]E, ×- 1%
Production and cost functions are black boxes created by economists. Investigat-
ing RD processes helps us to open the boxes.
Process innovation: An innovation that reduces the production cost of a prod-
uct.
Product innovation: An innovation that creats a new product.
The distinction is not essential. A process innovation can be treated as the creation
of new intermediate products that reduce the production costs. On the other hand,
a product innovation can be regarded as an innovation that reduces the production
cost of a product from infinity to a finite value.
9.1 Classification of Process Innovations
Consider a Bertrand competition industry.
Inverse demand function: P = a − Q.
In the beginning, all firms have the same technology and P0 = C0. Suppose that an
inventor innovates a new production procedure so that the marginal production cost
reduces to c c0.
E Q
T
P
d
d
d
d
d
d
d
d
d
r
P0 = c0
Q0
c
E Q
T
P
d
d
d
d
d
d
d
d
d
r
c
Pm(c)
Qm](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-83-2048.jpg)
![84
E α
T
I/V
Eπs
(1) = 0 or
I
V
= α
Eπs
(1) = Eπs
(2) or
I
V
= α(1 − α)
1 firm
2 firms E α
T
I/V
(I)
(II)
(III)
(I): Social optimal is 1 firm, the same as market equilibrium number of firms.
(II): Social optimal is 1 firm, market equilibrium has 2 firms.
(III): Social optimal is 2 firms, the same as market equilibrium number of firms.
Area (II) represents the market inefficient area.
9.2.3 Expected date of discovery
Suppose that the RD race will continue until the discovery.
ET(n): Expected date of discovery if there are n firms.
ET(1) = α + (1 − α)α2 + (1 − α)2
α3 + . . . = α
∞
t=0
(1 − α)t−1
t =
α
[1 − (1 − α)]2
=
1
α
.
ET(2) = α(2−α)+(1−α)2
α(2−α)2+. . . = α(2−α)
∞
t=0
(1−α)2(t−1)
t =
1
α(2 − α)
ET(1).
9.3 Cooperation in RD
Firms’ cooperation in price setting is against anti-trust law. Cooperation in RD
activities usually is not illegal. Therefore, firms might use cooperation in RD as a
substitute for cooperation in price setting.
In this subsection we investigate the effects of firms’ cooperation in RD on social
welfare.
A 2-stage duopoly game with RD
t = 1: Both firms decide RD levels, x1 and x2, simultaneously.
t = 2: Both firms engage in Cournot quantity competition.
Market demend is P = 100 − Q.
Firm i’s RD cost: TCi(xi) = x2
i /2.
Firm i’s unit production cost: ci(xi, xj) = 50 − xi − βxj.
β 0; if β 0, it represents the spillover effect of RD; if β 0, it is the interference
effect.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-86-2048.jpg)
![85
9.3.1 Noncooperative RD equilibrium
When firms do not cooperate in RD, they decide the RD levels independent of
each other. We solve the model backward.
At t = 2, x1 and x2 are determined. The Cournot equilibrium is such that
Πi(ci, cj) =
(100 − 2ci + cj)2
9
− TCi(xi).
Substituting the unit cost functions, we obtain the reduced profit function of t = 1:
Πi(xi, xj) =
[100 − 2(50 − xi − βxj) + (50 − 2xj − βxi)]2
9
−
x2
i
2
=
[50 + (2 − β)xi + (2β − 1)xj]2
9
−
x2
i
2
.
At t = 1, firm i chooses xi to maximize Πi(xi, xj). FOC is
∂Πi
∂xi
= 0 =
2(2 − β)[50 + (2 − β)xi + (2β − 1)xj]
9
− xi.
In a symmetric equilibrium, xi = xj = xnc
:
x1 = x2 = xnc
=
50(2 − β)
4.5 − (2 − β)(1 + β)
, c1 = c2 =
50[4.5 − 2(2 − β)(1 + β)]
4.5 − (2 − β)(1 + β)
,
Pnc
− cnc
= Qnc
=
75
4.5 − (2 − β)(1 + β)
, Π1 = Π2 = Πnc
=
252
[9 − 2(2 − β)]
[4.5 − (2 − β)(1 + β)]2
.
9.3.2 Cooperative RD equilibrium
When firms cooperate in RD, they choose x1 = x2 = x so that
Πi = Πj = Π(x) =
[50 + (1 + β)x]2
9
−
x2
2
.
Then they decide the level of x to maximize Π(x). FOC is
∂Π
∂x
= 0 =
2(1 + β)[50 + (1 + β)x]
9
− x.
Denote by xc
the optimal level of x,
x1 = x2 = xc
=
50(1 + β)
4.5 − (1 + β)2
, c1 = c2 =
50[4.5 − 2(1 + β)2
]
4.5 − (1 + β)2
,
Pc
− cc
= Qc
=
75
4.5 − (1 + β)2
, Π1 = Π2 = Πc
=
252
[9 − 2(1 + β)2
]
[4.5 − (1 + β)2]2
.
Conclusions:](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-87-2048.jpg)
![90
where [h(xi)]−1
is expected discovery time of firm i.
E(Ti) = [h(xi)]−1
, h(xi) 0, h (xi) 0, h (xi) 0.
Industry discovery time: ˆT ≡ min{T1, T2} ∼ 1 − e−[h(x1)+h(x2)] ˆT
because
Prob{ ˆT T} = Prob{T1 T, T2 T} = e−h(x1)T
e−h(x2)T
= e−[h(x1)+h(x2)]T
.
Firm 1’s winning probability: Prob[T1 = T, T2 T | ˆT = T] =
h(x1)
h(x1) + h(x2)
:
Prob[T1, ˆT ∈ (T, T + dt)]
Prob[ ˆT ∈ (T, T + dt)]
≈
h(x1)e−h(x1)T
dt[1 − (1 − e−h(x2)T
)]
[h(x1) + h(x2)]e−[h(x1)+h(x2)]T dt
=
h(x1)
h(x1) + h(x2)
.
E T1
T
T2
T +dt
T
A
BC
T +dtT
{ ˆT ∈ (T, T + dt)} = A ∪ B ∪ C
A = {T1 ∈ (T, T + dt), T2 ≥ T + dt}
B = {T2 ∈ (T, T + dt), T1 ≥ T + dt}
C = {T1, T2 ∈ (T, T + dt)}, Prob[C] ≈ 0.
Given (x1, x2), the expected payoff of firm 1, Π1(x1, x2) is (Π2 is similar)
∞
0
h(x1)
h(x1) + h(x2)
∞
ˆT
e−rt
V dt −
ˆT
0
e−rt
x1dt [h(x1) + h(x2)]e−[h(x1)+h(x2)] ˆT
d ˆT
=
∞
0
h(x1)
h(x1) + h(x2)
e−r ˆT
V
r
−
(1 − e−r ˆT
)x1
r
[h(x1) + h(x2)]e−[h(x1)+h(x2)] ˆT
d ˆT
=
h(x1)V/r
h(x1) + h(x2) + r
−
x1
r
+
[h(x1) + h(x2)]x1/r
h(x1) + h(x2) + r
=
h(x1)V − rx1
r[h(x1) + h(x2) + r]
.
FOC for symmetric NE with x1 = x2 = x:
[2h(x)+r][h (x)V −r]−[h(x)V −rx]h (x) = h(x)h (x)V +rh (x)(x+V )−2rh(x)−r2
= 0.
Social welfare when x1 = x2 = x:
W(x) =
∞
0
∞
ˆT
e−rt
V dt −
ˆT
0
e−rt
2xdt 2h(x)e−2h(x) ˆT
d ˆT
=
1
r
∞
0
e−r ˆT
V − (1 − e−r ˆT
)2x 2h(x)e−2h(x) ˆT
d ˆT
=
2h(x)V/r
2h(x) + r
−
2x
r
+
2h(x)x/r
2h(x) + r
=
2[h(x)V − rx]
r[2h(x) + r]
.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-92-2048.jpg)
![91
FOC for social optimal:
[2h(x) + r][h (x)V − r] − 2[h(x)V − rx]h (x) = rh (x)(2x + V ) − 2rh(x) − r2
= 0.
Example: h(x) = 2
√
x, h (x) = 1/
√
x.
FOC for NE:
2V +
r(x + V )
√
x
− 4r
√
x − r2
= 0, ⇒
√
xn =
2V − r2
+ (2V − r2)2 + 12r2V
6r
.
FOC for social optimal:
r(2x + V )
√
x
− 4r
√
x − r2
= 0, ⇒
√
xs =
−r2
+
√
r4 + 8r2V
4r
.
Derivation: Let zn ≡
√
xn, zs ≡
√
xs, fn(z) ≡ 3rz2
+ (r2
− 2V )z − rV , and fs(z) ≡
2rz2
+r2
z−rV . fn(zn) = 0 and fs(zs) = 0. Direct computation shows that fn(zs) 0,
limz→∞ fn(z) = ∞ 0. Hence zn zs and therefore
√
xn
√
xs
=
2
3
k − 1 +
√
k2 + 4k + 1
−1 +
√
1 + 4k
1, k =
2V
r2
.
Therefore, the equilibrium RD level is greater than the social optimal level.
Extensions: 1. h(x) = λ¯h(x/λ) = λ1−a
xa
.
2. When there are n 2 firms.
3. n is endogenouse and optimal x and n.
9.7.2 RD race between an incumbent and a potential entrant
2 firms compete in RD to win the patent on a new procedure with unit cost c.
Firm 1: Incumbent with initial unit production cost ¯c c.
Firm 2: A potential entrant.
Πm
(¯c): Firm 1’s monopoly profit before the discovery of the new procedure.
Πm
(c): Firm 1’s monopoly profit if firm 1 wins.
Πd
1(¯c, c): Firm 1’s duopoly profit if firm 2 wins.
Πd
2(¯c, c): Firm 2’s duopoly profit if firm 2 wins.
Assumption 1: Πm
(c) ≥ Πd
2(¯c, c) + Πd
1(¯c, c).
Assumption 2: The patent length is ∞.
Using the same derivation as basic model,
V1(x1, x2) =
h(x1)Πm
(c) + h(x2)Πd
1(¯c, c) + r[Πm
(¯c) − x1]
r[h(x1) + h(x2) + r]
,
V2(x1, x2) =
h(x2)Πd
2(¯c, c) − rx2
r[h(x1) + h(x2) + r]
.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-93-2048.jpg)
![94
10.0.4 A standardization game
Firm A (USA) and firm B (Canada) are choosing a standard for their product.
α-standard (L„d, CÔ¬iW , etc.) β-standard (t„d, CÔ˝iW , etc.)
A B α β
α (a, b) (c, d)
β (d, c) (b, a)
1. If a, b max{c, d} (battle of the sexes), then (α, α) and (β, β) are both NE. They
choose the same standard ( ™Ä“).
2. If c, d max{a, b}, then (α, β) and (β, α) are both NE. They choose the dif-
ferent standards (®Wwu).
10.1 Network Externalities
10.1.1 Rohlfs phone company model
Rohlfs 1974, “A Theory of Interdependent Demand for a Communication Service,”
Bell Journal of Economics.
Consumers are distributed uniformly along a line, x ∈ [0, 1].
0 1
r
x
Consumers indexed by a low x are those who have high willingness to pay to subscribe
to a phone system.
Ux
=
n(1 − x) − p if x subscribes to the phone system
0 if x does not,
n, 0 n 1: the total number of consumers who actually subscribe.
p: the price of subscribing.
ˆx: the marginal consumer who is indifference between subscribing and not.
0 1
t
ˆxsubscribers
In a rational expectation equilibrium, ˆx = n and U ˆx
= n(1 − ˆx) − p = 0, ⇒
Inverse demand function: p = ˆx(1 − ˆx).
Demand function: ˆx =
1 ±
√
1 − 4p
2
.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-96-2048.jpg)
![96
10.1.2 The standardization-variety tradeoff ™Ä“CÖj“?
Consumers are distributed uniformly along a line, x ∈ [0, 1].
0 1aprefer A prefer B
2 brands/standards, A (à¬G) and B (à˝G).
a 0 consumers prefer A-standard.
b = 1 − a 0 consumers prefer B-standard.
xA: number of consumers using A-standard.
xB: number of consumers using B-standard.
δ: the disutility of using a less prefered standard.
UA
=
xA use A-standard
xB − δ use B-standard,
UB
=
xA − δ use A-standard
xB use B-standard.
Consumer distribution: (xA, xB) such that xA, xB ≥ 0 and xA + xB = 1.
A-standard distribution: A distribution such that (xA, xB) = (1, 0).
B-standard distribution: A distribution such that (xA, xB) = (0, 1).
Incompatible AB-standards distribution A distribution with xA, xB 0.
Equilibrium: (xA, xB) such that none wants to switch to a different brand/standard.
Proposition 10.3: If δ 1, then both A-standard and B-standard are equilibrium. If
δ 1, then both A-standard and B-standard are not equilibrium.
Proof: If δ 1 and every one chooses the same brand (either A or B), then none
wants to switch to a different brand. If δ 1, then the cost of switching to a preferred
brand is less than the benefit and therefore a single standard equilibrium cannot exist.
Proposition 10.4: If a, b
1 − δ
2
, then (xA, xB) = (a, b) is an equilibrium.
Proof: Given the distribution (xA, xB) = (a, b), the utility levels are
UA
=
a use A br.
b − δ = 1 − a − δ a use B br.
UB
=
a − δ = 1 − b − δ b use A br.
b use B br.
Therefore, none will switch to a different brand.
E a
T
b
d
d
d
d
d
d
d
d
d
d
d
dd
2-standard equilibrium
range
1−δ
2
1−δ
2
r
r
1−δ
1−δ](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-98-2048.jpg)
![97
Social welfare: W(xA, xB) = aUa
+ bUb
, where Ua
(Ub
) is the utility level of A-
prefered consumers (B-prefered consumers).
W(A) = W(1, 0) = a + b(1 − δ) = 1 − bδ.
W(B) = W(0, 1) = a(1 − δ) + b = 1 − aδ.
W(AB) = W(a, b) = a2
+ b2
= (1 − b)2
+ b2
= 1 − b − b(1 − 2b) = 1 − 2ab.
Proposition 10.5: If a b, then W(A) W(B).
Proof: If W(A) − W(B) = a + b(1 − δ) − [a(1 − δ) − b] = (a − b)δ 0.
Proposition 10.6: 1. If δ 1, then W(AB) max{W(A), W(B)}.
2. If δ 1 and
δ
2
max{a, b}, then W(AB) max{W(A), W(B)}.
Proof: Assume that a b (or 1 − 2b 0). (case b a is similar.)
1. If δ 1, then max{W(A), W(B)} = W(A) = 1 − bδ 1 − b 1 − b − b(1 − 2b) =
W(AB).
2. If δ 2a 1, then W(AB) − W(A) = b(δ − 2a) 0.
Proposition 10.7: If δ 1, then market failure can happen.
Remark: When a b and δ 1, the social optimal is A-standard. However, both
the incompatible standards and B-standard can also be equilibrium.
10.2 Supporting Services and Network Effects
Network effects can occur even there is no network externalities. For example, when
there is a complementary supporting industry exhibiting increasing returns to scale
such as PC industry.
10.2.1 Basic model
Chou/Shy (1990), “Network effects without network externalities,” International Jour-
nal of Industrial Organization.
A PC industry with 2 brands, A and B and prices PA and PB.
Consumers are distributed uniformly along a line, δ ∈ [0, 1].
E δ
0 1δprefer A prefer B
Let NA and NB be the numbers of software pieces available to computers A and
B, respectively.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-99-2048.jpg)
![102
11 Advertising
Advertising is defined as a form of providing information about prices, quality, and
location of goods and services.
2% of GNP in developed countries.
vegetables, etc., 2% of sales.
cosmetics, detergent, etc., 20-60 % of sales.
In 1990, GM spent $63 per car, Ford $130 per car, Chrysler $113 per car.
What determines advertising in different industries or different firms of the same
industry?
Economy of scale, advertising elasticity of demand, etc.
Kaldor (1950), “The Economic Aspects of Advertising,” Review of Economic Studies.
Advertising is manipulative and reduces competition.
1. Wrong information about product differentiations ⇒ increases cost.
2. An entry-deterring mechanism ⇒ reduces competition.
Telser (1964), “Advertising and Competition,” JPE
Nelson (1970),“Information and Consumer Behavior,” JPE
Nelson (1974),“Advertising as Information,” JPE
Demsetz (1979), “Accounting for Advertising as a Barrier to Entry,” J. of Business.
Positive sides of advertising: It provides produt information.
Nelson:
Search goods: Quality can be identified when purchasing. ⇒ .Û µ
Experience goods: Quality cannot be identified until consuming. ⇒ µª«àÛb
Persuasive advertising: Intends to enhance consumer tastes, eg diamond.
Informative advertising: Provides basic information about the product.
11.1 Persuasive Advertising
Q(P, A) = βA a
P p
, β 0, 0 a 1, p −1.
A: expenditure on advertising.
a: Advertising elasticity of demand.
p: Price elasticity of demand.
c: Unit production cost.
max
P,A
Π = PQ − cQ − A = (P − c)βA a
P p
− A.
FOC with respect to P:
∂Π
∂P
= βA a
[( p + 1)P p
− c pP p
] = 0, ⇒ Pm
=
c p
p + 1
,
Pm
− c
Pm
=
p
−1
.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-104-2048.jpg)
![104
11.2.1 A simple model of informative advertising
1 consumer wants to buy 1 unit of a product.
p: the price. m: its value.
U =
m − p
0 .
If the consumer does not receive any advertisement, he will not purchase.
If he receives an advertisement from a firm, he will purchase from the firm.
If he receives 2 advertisements from 2 firms, he will randomly choose one to buy.
2 firms, unit production cost is 0, informative advertising cost is A.
Each chooses either to advertise its product or not to advertise it.
πi =
p − A if only firm i’s ad is received.
p
2
− A if both firms’ ad are received.
−A if firm i’s ad is not received.
0 if firm i chooses not to advertise.
Let δ be the probability that an advertisement is received by the consumer.
Eπi =
δ(1 − δ)(p − A) + δ2
(
p
2
− A) − (1 − δ)A ≡ π(2) if both choose to advertise.
δ(p − A) − (1 − δ)A ≡ π(1) if only firm i chooses to advertise.
0 if firm i chooses not to advertise.
If p/A 1/δ, ⇒ π(1) 0, ⇒ at least one firm will choose to advertise.
If p/A 2/[δ(2 − δ)], ⇒ π(2) 0, ⇒ both firms will choose to advertise.
E
δ
T
p/A
π(2) 0,
2 firms in equilibrium.
1
firm
π(1) 0, 0 firm q
1
1/δ
2/[δ(2 − δ)]
Welfare comparison:
EW =
δ(2 − δ)m − 2A ≡ W(2) if 2 firms advertise.
δm − A ≡ W(1) if only one firm advertises.
0 if no firm advertises.
If
m
A
1
δ
, ⇒ W(1) 0, ⇒ social optimal is at least one firm advertises.
If
m
A
1
δ(1 − δ)
, ⇒ W(2) W(1), ⇒ social optimal is both firms advertise.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-106-2048.jpg)
![105
E
δ
T
m/A
q
1
1/[δ(1 − δ)]
W(2) W(1)
W(1) 0
W(1) 0
E
δ
T
p/A = m/A
q
1
market failure
µxXTòU δ Ó‹ When δ → 1, one firm would be enough. However, if m/A 1,
both firms will advertise.
11.3 Targeted Advertising ‡ú4 µ
(1) Consumers are heterogeneous with different tastes. (2) Large scale advertising is
costly. (3) Intensive advertising will result in price competition.
⇒ ̶nßF í¾‘6, .° ¼Sà.°4”5‡ú4 µ, ‡ú.°5¾‘6íˇ
11.3.1 The model
2 firms, i = 1, 2, producing differentiated products.
2 groups of consumers: E experienced consumers and N inexperienced consumers.
θE of experienced consumers are brand 1 oriented, 0 θ 1.
(1 − θ)E of experienced consumers are brand 2 oriented.
2 advertising methods: P (persuasive) and I (informative).
Each firm can choose only one method.
AS1: Persuasive advertising attracts only inexperienced consumers. If only firm i
chooses P, then all N inexperienced consumers will purchase brand i. If both
firms 1 and 2 choose P, each will have N/2 inexperienced consumers.
AS2: Informative advertising attracts only the experienced consumers who are ori-
ented toward the advertised brand, i.e., if firm 1 (firm 2) chooses I, θE ((1−θ)E)
experienced consumers will purchase brand 1 (brand 2).
AS3: A firm earns $1 from each customer.
From the assumptions we derive the following duopoly advertising game:
firm 1 firm 2 P I
P (N/2, N/2) (N, (1 − θ)E)
I (θE, N) (θE, (1 − θ)E)](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-107-2048.jpg)
![109
12 Quality
12.1 Vertical Differentiation in Hotelling Model
Quality is a vertical differentiation character.
2-period game:
At t = 1, firms A and B choose the quality levels, 0 ≤ a b ≤ 1, for their products.
At t = 2, they engage in price competition.
Consumers in a market are distributed uniformly along a line of unit length.
0 1
r
i
x
Each point x ∈ [0, 1] represents a consumer x.
Ux =
ax − PA if x buys from A.
bx − PB if x buys from B.
The marginal consumer ˆx is indifferent between buying from A and from B. The
location of ˆx is determined by
aˆx − PA = bˆx − PB ⇒ ˆx =
PB − PA
b − a
. (7)
The location of ˆx divids the market into two parts: [0, ˆx) is firm A’s market share
and (ˆx, 1] is firm B’s market share.
0 1
r
ˆx
A’s share B’s share' E' E
Assume that the marginal costs are zero. The payoff functions are
ΠA(PA, PB; a, b) = PA ˆx = PA
PB − PA
b − a
, ΠB(PA, PB; a, b) = PB(1−ˆx) = PB 1 −
PB − PA
b − a
.
The FOCs are (the SOCs are satisfied)
∂ΠA
∂PA
=
PB − 2PA
b − a
= 0,
∂ΠB
∂PB
= 1 −
2PB − PA
b − a
= 0. (8)
The equilibrium is given by
PA =
b − a
3
, PB =
2(b − a)
3
, ⇒ ˆx =
1
3
.
The reduced profit functions at t = 1 are
ΠA =
(b − a)
9
, ΠB =
4(b − a)
9
.](https://image.slidesharecdn.com/fch-160330173657/75/F-ch-111-2048.jpg)
Uploaded byWilfredo Bacilio Alarcón
F ch
This document provides lecture notes on industrial organization. It covers topics such as market structure, conduct, and performance. Key concepts discussed include utility maximization, demand curves, profit maximization, supply curves, and competitive market equilibrium. In a competitive market, firms are price takers in the short run. In long run equilibrium with free entry and exit, there are no barriers to entry or profits. The market reaches equilibrium when quantity demanded equals quantity supplied at the market clearing price.
F ch
- 1. 1 Lecture Notes on IndustrialOrganization (I) Chien-Fu CHOU January 2004
- 2. 2 Contents Lecture 1 Introduction1 Lecture 2 Two Sides of a Market 3 Lecture 3 Competitive Market 8 Lecture 4 Monopoly 11 Lecture 5 Basis of Game Theory 20 Lecture 6 Duopoly and Oligopoly – Homogeneous products 32 Lecture 7 Differentiated Products Markets 46 Lecture 8 Concentration, Mergers, and Entry Barriers 62 Lecture 9 Research and Development (R&D) 81 Lecture 10 Network Effects, Compatibility, and Standards 93 Lecture 11 Advertising 102 Lecture 12 Quality 109 Lecture 13 Pricing Tactics 112 Lecture 14 Marketing Tactics: Bundling, Upgrading, and Dealerships 114
- 3. 1 1 Introduction 1.1 Classificationof industries and products 2M¬Å¼¹™Ä}é, 2M¬ÅW“™Ä}é; «%Íß%’eé. 1.2 A model of industrial organization analysis: (FS Ch1) Structuralist: 1. The inclusion of conduct variables is not essential to the development of an operational theory of industrial organization. 2. a priori theory based upon structure-conduct and conduct-performance links yields ambiguous predictions. 3. Even if a priori stucture-conduct-performance hypotheses could be formulated, attempting to test those hypotheses would encounter serious obstacles. Behaviorist: We can do still better with a richer model that includes intermediate behavioral links. 1.3 Law and Economics Antitrust law, tÃ>q¶ Patent and Intellectual Property protection ù‚D N‹ßžˆ Cyber law or Internet Law 昶 1.4 Industrial Organization and International Trade
- 4. 2 Basic Conditions Supply Demand Rawmaterials Price elasticity Technology Substitutes Unionization Rate of growth Product durability Cyclical and seasonal characterValue/weight Purchase methodBusiness attitudes Marketing typePublic polices c Market Structure Number of sellers and buyers Product differentiation Barriers to entry Cost structures Vertical integration Conglomerateness c Conduct Pricing behavior Product strategy and advertising Research and innovation Plant investment Legal tactics c Performance Production and allocative efficiency Progress Full employment Equity
- 5. 3 2 Two Sidesof a Market 2.1 Comparative Static Analysis Assume that there are n endogenous variables and m exogenous variables. Endogenous variables: x1, x2, . . . , xn Exogenous variables: y1, y2, . . . , ym. There should be n equations so that the model can be solved. F1(x1, x2, . . . , xn; y1, y2, . . . , ym) = 0 F2(x1, x2, . . . , xn; y1, y2, . . . , ym) = 0 ... Fn(x1, x2, . . . , xn; y1, y2, . . . , ym) = 0. Some of the equations are behavioral, some are equilibrium conditions, and some are definitions. In principle, given the values of the exogenous variables, we solve to find the endogenous variables as functions of the exogenous variables: x1 = x1(y1, y2, . . . , ym) x2 = x2(y1, y2, . . . , ym) ... xn = xn(y1, y2, . . . , ym). We use comparative statics method to find the differential relationships between xi and yj: ∂xi/∂yj. Then we check the sign of ∂xi/∂yj to investigate the causality relationship between xi and yj.
- 6. 4 2.2 Utility Maximizationand Demand Function 2.2.1 Single product case A consumer wants to maximize his/her utility function U = u(Q) + M = u(Q) + (Y − PQ). FOC: ∂U ∂Q = u (Q) − P = 0, ⇒ u (Qd) = P (inverse demand function) ⇒ Qd = D(P) (demand function, a behavioral equation) ∂2 U ∂Q∂P = UP Q = −1 ⇒ dQd dP = D (P) < 0, the demand function is a decreasing function of price. 2.2.2 Multi-product case A consumer wants to maximize his utility function subject to his budget constraint: max U(x1, . . . , xn) subj. to p1x1 + · · · + pnxn = I. Endogenous variables: x1, . . . , xn Exogenous variables: p1, . . . , pn, I (the consumer is a price taker) Solution is the demand functions xk = Dk(p1, . . . , pn, I), k = 1, . . . , n Example: max U(x1, x2) = a ln x1 + b ln x2 subject to p1x1 + p2x2 = m. L = a ln x1 + b ln x2 + λ(m − p1x1 − p2x2). FOC: L1 = a x1 − λp1 = 0, L2 = b x2 − λp2 = 0 and Lλ = m − p1x1 − p2x2 = 0. ⇒ a b x2 x1 = p1 p2 ⇒ x1 = am (a + b)p1 , x2 = bm (a + b)p2 SOC: 0 −p1 −p2 −p1 −a x2 1 0 −p2 0 −b x2 2 = ap2 2 x2 1 + bp2 1 x2 2 > 0. ⇒ x1 = am (a + b)p1 , x2 = bm (a + b)p2 is a local maximum. 2.3 Indivisibility, Reservation Price, and Demand Function In many applications the product is indivisible and every consumer needs at most one unit. Reservation price: the value of one unit to a consumer. If we rank consumers according to their reservation prices, we can derive the market demand function. Example: Ui = 31 − i, i = 1, 2, · · ·, 30.
- 7. 5 E i, Q T Ui,P rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr The trace of Ui’s becomes the demand curve. 2.4 Demand Function and Consumer surplus Demand Function: Q = D(p). Inverse demand function: p = P(Q). Demand elasticity: ηD ≡ p Q dQ dp = pD (p) D(p) = P(Q) QP(Q) . Total Revenue: TR(Q) = QP(Q) = pD(p). Average Revenue: AR(Q) = TR(Q) Q = P(Q) = pD(p) D(p) . Marginal Revenue: MR(Q) = dTR(Q) dQ or MR(Q) = P(Q) + QP (Q) = P(Q) 1 + P (Q)Q P(Q) = P(Q) 1 + 1 η . Consumer surplus: CS(p) ≡ ∞ p D(p)dp. 2.4.1 Linear demand function: Q = D(p) = A b − 1 b p or P(Q) = A − bQ TR = AQ − bQ2 , AR = A − bQ, MR = A − 2bQ, η = 1 − a bQ , CS(p) = A p D(p)dp = (A − p)p 2b . 2.4.2 Const. elast. demand function: Q = D(p) = apη or P(Q) = AQ1/η TR = AQ1+ 1 η , AR = AQ1/η ,and MR = 1 + η η AQ1/η . 2.4.3 Quasi-linear utility function: U(Q) = f(Q) + m ⇒ P(Q) = f (Q) TR = Qf (Q), AR = f (Q), MR = f (Q) + Qf (Q), CS(p) = f(Q) − pQ = f(Q) − Qf (Q).
- 8. 6 2.5 Profit maximizationand supply function 2.5.1 From cost function to supply function Consider first the profit maximization problem of a competitive producer: max Q Π = PQ − C(Q), FOC ⇒ ∂Π ∂Q = P − C (Q) = 0. The FOC is the inverse supply function (a behavioral equation) of the producer: P = C (Q) = MC. Remember that Q is endogenous and P is exogenous here. To find the comparative statics dQ dP , we use the total differential method discussed in the last chapter: dP = C (Q)dQ, ⇒ dQ dP = 1 C (Q) . To determine the sign of dQ dP , we need the SOC, which is ∂2 Π ∂Q2 = −C (Q) < 0. Therefore, dQs dP > 0. 2.5.2 From production function to cost function A producer’s production technology can be represented by a production function q = f(x1, . . . , xn). Given the prices, the producer maximizes his profits: max Π(x1, . . . , xn; p, p1, . . . , pn) = pf(x1, . . . , xn) − p1x1 − · · · − pnxn Exogenous variables: p, p1, . . . , pn (the producer is a price taker) Solution is the supply function q = S(p, p1, . . . , pn) and the input demand functions, xk = Xk(p, p1, . . . , pn) k = 1, . . . , n Example: q = f(x1, x2) = 2 √ x1 + 2 √ x2 and Π(x1, x2; p, p1, p2) = p(2 √ x1 + 2 √ x2) − p1x1 − p2x2, max x1.x2 p(2 √ x1 + 2 √ x2) − p1x1 − p2x2 FOC: ∂Π ∂x1 = p √ x1 − p1 = 0 and ∂Π ∂x2 = p √ x2 − p2 = 0. ⇒ x1 = (p/p1)2 , x2 = (p/p2)2 (input demand functions) and q = 2(p/p1) + 2(p/p2) = 2p( 1 p1 + 1 p2 ) (the supply function) Π = p2 ( 1 p1 + 1 p2 ) SOC: ∂2 Π ∂x2 1 ∂2 Π ∂x1∂x2 ∂2 Π ∂x1∂x2 ∂2 Π ∂x2 1 = −p 2x −3/2 1 0 0 −p 2x −3/2 2 is negative definite.
- 9. 7 2.5.3 Joint products,transformation function, and profit maximization In more general cases, the technology of a producer is represented by a transformation function: Fj (yj 1, . . . , yj n) = 0, where (yj 1, . . . , yj n) is called a production plan, if yj k > 0 (yj k) then k is an output (input) of j. Example: a producer produces two outputs, y1 and y2, using one input y3. Its technology is given by the transformation function (y1)2 + (y2)2 + y3 = 0. Its profit is Π = p1y1 + p2y2 + p3y3. The maximization problem is max y1,y2,y3 p1y1 + p2y2 + p3y3 subject to (y1)2 + (y2)2 + y3 = 0. To solve the maximization problem, we can eliminate y3: x = −y3 = (y1)2 +(y2)2 > 0 and max y1,y2 p1y1 + p2y2 − p3[(y1)2 + (y2)2 ]. The solution is: y1 = p1/(2p3), y2 = p2/(2p3) (the supply functions of y1 and y2), and x = −y3 = [p1/(2p3)]2 + [p1/(2p3)]2 (the input demand function for y3). 2.6 Production function and returns to scale Production function: Q = f(L, K). MPK = ∂Q ∂L MPK = ∂Q ∂K IRTS: f(hL, hK) > hf(L, K). CRTS: f(hL, hK) = hf(L, K). DRTS: f(hL, hK) < hf(L, K). Supporting factors: ∂2 Q ∂L∂K > 0. Substituting factors: ∂2 Q ∂L∂K < 0. Example 1: Cobb-Douglas case F(L, K) = ALa Kb . Example 2: CES case F(L, K) = A[aLρ + (1 − a)Kρ ]1/ρ . 2.7 Cost function: C(Q) Total cost TC = C(Q) Average cost AC = C(Q) Q Marginal cost MC = C (Q). Example 1: C(Q) = F + cQ Example 2: C(Q) = F + cQ + bQ2 Example 3: C(Q) = cQa .
- 10. 8 3 Competitive Market Industry(Market) structure: Short Run: Number of firms, distribution of market shares, competition decision vari- ables, reactions to other firms. Long Run: R&D, entry and exit barriers. Competition: In the SR, firms and consumers are price takers. In the LR, there is no barriers to entry and exit ⇒ 0-profit. 3.1 SR market equilibrium 3.1.1 An individual firm’s supply function A producer i in a competitive market is a price taker. It chooses its quantity to maximize its profit: max Qi pQi − Ci(Qi) ⇒ p = Ci(Qi) ⇒ Qi = Si(p). 3.1.2 Market supply function Market supply is the sum of individual supply function S(p) = i Si(p). On the Q-p diagram, it is the horizontal sum of individual supply curves. E Q T p S1 S2 S E Q T p S1 S2 Sd d d d d d d d dD p∗ Q∗ Q∗ 1 Q∗ 2 3.2 Market equilibrium Market equilibrium is determined by the intersection of the supply and demand as in the diagram. Formally, suppose there are n firms. A state of the market is a vector (p, Q1, Q2, . . . , Qn). An equilibrium is a state (p∗ , Q∗ 1, Q∗ 2, . . . , Q∗ n) such that: 1. D(p∗ ) = S(p∗ ). 2. Each Q∗ i maximizes Πi(Qi) = p∗ Qi − Ci(Qi), i = 1, . . . , n. 3. Πi(Q∗ i ) = p∗ Q∗ i − ci(Q∗ i ) ≥ 0.
- 11. 9 3.2.1 Example 1:C1(Q1) = Q2 1, C2(Q2) = 2Q2 2, D = 12 − p 4 p = C1(Q1) = 2Q1, p = C2(Q2) = 4Q2, ⇒ S1 = p 2 , S2 = p 4 , S(P) = S1+S2 = 3p 4 . D(p∗ ) = S(p∗ ) ⇒ 12− p∗ 4 = 3p∗ 4 ⇒ p∗ = 12, Q∗ = S(p∗ ) = 9, Q∗ 1 = S1(p∗ ) = 6, Q∗ 2 = S2(p∗ ) = 3. E Q T p ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢¢ S1 S2 Sd d d d d d d d d D 12 9 E Q T p d d d d d d d d d D Sc Q∗ 3.2.2 Example 2: C(Q) = cQ (CRTS) and D(p) = max{A − bp, 0} If production technology is CRTS, then the equilibrium market price is determined by the AC and the equilibrium quantity is determined by the market demand. p∗ = c, Q∗ = D(p∗ ) = max{A − bp∗ , 0}. If A b ≤ c then Q∗ = 0. If A b c then Q∗ = A − bc 0. 3.2.3 Example 3: 2 firms, C1(Q1) = c1Q1, C2(Q2) = c2Q2, c1 c2 p∗ = c1, Q∗ = Q∗ 1 = D(p∗ ) = D(c1), Q∗ 2 = 0. 3.2.4 Example 4: C(Q) = F + cQ or C (Q) 0 (IRTS), no equilibrium If C (Q) 0, then the profit maximization problem has no solution. If C(Q) = F + cQ, then p∗ = c cannot be and equilibrium because Π(Q) = cQ − (F + cQ) = −F 0.
- 12. 10 3.3 General competitiveequilibrium Commodity space: Assume that there are n commodities. The commodity space is Rn + = {(x1, . . . , xn); xk ≥ 0} Economy: There are I consumers, J producers, with initial endowments of com- modities ω = (ω1, . . . , ωn). Consumer i has a utility function Ui (xi 1, . . . , xi n), i = 1, . . . , I. Producer j has a production transformation function F j (yj 1, . . . , yj n) = 0, A price system: (p1, . . . , pn). A private ownership economy: Endowments and firms (producers) are owned by consumers. Consumer i’s endowment is ωi = (ωi 1, . . . , ωi n), I i=1 ωi = ω. Consumer i’s share of firm j is θij ≥ 0, I i=1 θij = 1. An allocation: xi = (xi 1, . . . , xi n), i = 1, . . . , I, and yj = (yj 1, . . . , yj n), j = 1, . . . , J. A competitive equilibrium: A combination of a price system ¯p = (¯p1, . . . , ¯pn) and an allocation ({¯xi }i=1,...,I, {¯yj }j=1,...,J ) such that 1. i ¯xi = ω + j ¯yj (feasibility condition). 2. ¯yj maximizes Πj , j = 1, . . . , J and ¯xi maximizes Ui , subject to i’s budget con- straint p1xi 1 + . . . + pnxi n = p1ω1 1 + . . . + pnωi n + θi1Π1 + . . . + θiJ ΠJ . Existence Theorem: Suppose that the utility functions are all quasi-concave and the production transfor- mation functions satisfy some theoretic conditions, then a competitive equilibrium exists. Welfare Theorems: A competitive equilibrium is efficient and an efficient allocation can be achieved as a competitive equilibrium through certain income transfers. Constant returns to scale economies and non-substitution theorem: Suppose there is only one nonproduced input, this input is indispensable to produc- tion, there is no joint production, and the production functions exhibits constant returns to scale. Then the competitive equilibrium price system is determined by the production side only.
- 13. 11 4 Monopoly A monopolyindustry consists of one single producer who is a price setter (aware of its monopoly power to control market price). 4.1 Monopoly profit maximization Let the market demand of a monopoly be Q = D(P) with inverse function P = f(Q). Its total cost is TC = C(Q). The profit maximization problem is max Q≥0 π(Q) = PQ−TC = f(Q)Q−C(Q) ⇒ f (Q)Q+f(Q) = MR(Q) = MC(Q) = C (Q) ⇒ QM . The SOC is d2 π dQ2 = MR (Q) − MC (Q) 0. Long-run existence condition: π(Qm) ≥ 0. Example: TC(Q) = F + cQ2 , f(Q) = a − bQ, ⇒ MC = 2cQ, MR = a − 2bQ. ⇒ Qm = a 2(b + c) , Pm = a(b + 2c) 2(b + c) , ⇒ π(Qm) = a2 4(b + c) − F. When a2 4(b + c) F, the true solution is Qm = 0 and the market does not exist. E Q T P d d d d d d d d dD e e e e e e e e eMR MC Qm MC Pm 4.1.1 Lerner index The maximization can be solved using P as independent variable: max P ≥0 π(P) = PQ − TC = PD(P) − C(D(P)) ⇒ D(P) + PD (P) = C (D(P))D (P) ⇒ Pm − C Pm = − D(P) D (P)P = 1 | | . Lerner index: Pm − C Pm . It can be calculated from real data for a firm (not necessarily monopoly) or an industry. It measures the profit per dollar sale of a firm (or an industry).
- 14. 12 4.1.2 Monopoly andsocial welfare E Q T P d d d d d d d d dD e e e e e e e e eMR MC Qm MC Pm E Q T P d d d d d d d d dD MC=S Q∗ P∗ E Q T P d d d d d d d d dD e e e e e e e e eMR MC Q∗ Qm 4.1.3 Rent seeking (¬•) activities RD, Bribes, Persuasive advertising, Excess capacity to discourage entry, Lobby expense, Over doing RD, etc are means taken by firms to secure and/or maintain their monopoly profits. They are called rent seeking activities because monopoly profit is similar to land rent. They are in many cases regarded as wastes because they don’t contribute to improving productivities. 4.2 Monopoly price discrimination Indiscriminate Pricing: The same price is charged for every unit of a product sold to any consumer. Third degree price discrimination: Different prices are set for different consumers, but the same price is charged for every unit sold to the same consumer (linear pricing). Second degree price discrimination: Different price is charged for different units sold to the same consumer (nonlinear pricing). But the same price schedule is set for different consumers. First degree price discrimination: Different price is charged for different units sold to the same consumer (nonlinear pricing). In addition, different price schedules are set for different consumers. 4.2.1 Third degree price discrimination Assume that a monopoly sells its product in two separable markets. Cost function: C(Q) = C(q1 + q2) Inverse market demands: p1 = f1(q1) and p2 = f2(q2) Profit function: Π(q1, q2) = p1q1 + p2q2 − C(q1 + q2) = q1f1(q1) + q2f2(q2) − C(q1 + q2) FOC: Π1 = f1(q1)+q1f1(q1)−C (q1 +q2) = 0, Π2 = f2(q2)+q2f2(q2)−C (q1 +q2) = 0; or MR1 = MR2 = MC. SOC: Π11 = 2f1 + q1f1 − C 0, 2f1 + q1f1 − C −C −C 2f2 + q2f2 − C ≡ ∆ 0. Example: f1 = a − bq1, f2 = α − βq2, and C(Q) = 0.5Q2 = 0.5(q1 + q2)2 . f1 = −b, f2 = −β, f1 = f2 = 0, C = Q = q1 + q2, and C = 1.
- 15. 13 FOC: a −2bq1 = q1 + q2 = α − 2βq2 ⇒ 1 + 2b 1 1 1 + 2β q1 q2 = a α ⇒ q1 q2 = 1 (1 + 2b)(1 + 2β) − 1 a(1 + 2β) − α α(1 + 2b) − a . SOC: −2b − 1 0 and ∆ = (1 + 2b)(1 + 2β) − 1 0. E q T p MC d d d d d d d d d d MR1 MR2 ——————————————— MR1+2 Qmq∗ 2q∗ 1 MC∗ MC = MR1+2 ⇒ Qm, MC∗ MC∗ = MR1 ⇒ q∗ 1 MC∗ = MR2 ⇒ q∗ 2 4.2.2 First Degree Each consumer is charged according to his total utility, i.e., TR = PQ = U(Q). The total profit to the monopoly is Π(Q) = U(Q) − C(Q). The FOC is U (Q) = C (Q), i.e., the monopoly regards a consumer’s MU (U (Q)) curve as its MR curve and maximizes its profit. max Q Π = U(Q) − C(Q) ⇒ U (Q) = C (Q). The profit maximizing quantity is the same as the competition case, Qm1 = Q∗ . How- ever, the price is much higher, Pm1 = U(Q∗ ) Q∗ = AU P∗ = U (Q∗ ). There is no inefficiency. But there is social justice problem. E Q T P d d d d d d d d d e e e e e e e e e r rr rr rr rr C (Q) Q∗ Qm r AU P∗ Pm1 Q∗ = Qm1 D = MUMR 4.2.3 Second degree discrimination See Varian Ch14 or Ch25.3 (under).
- 16. 14 E Q T P d d d d d D2 D1 A C B EQ T P d d d d d D2 D1 ' E Q T P d d d d d D2 D1 By self selection principle, P1Q1 = A, P2Q2 = A+C, Π = 2A+C is maximized when Q1 is such that the hight of D2 is twice that of D1. 4.3 Multiplant Monopoly and Cartel Now consider the case that a monopoly has two plants. Cost functions: TC1 = C1(q1) and TC2 = C2(q2) Inverse market demand: P = D(Q) = D(q1 + q2) Profit function: Π(q1, q2) = P(q1 + q2) − C1(q1) − C2(q2) = D(q1 + q2)(q1 + q2) − C1(q1) − C2(q2) FOC: Π1 = D (Q)Q + D(Q) − C1(q1) = 0, Π2 = D (Q)Q + D(Q) − C2(q2) = 0; orMR = MC1 = MC2. SOC: Π11 = 2D (Q) + D (Q)Q − C1 0, 2D (Q) + D (Q)Q − C1 2D (Q) + D (Q)Q 2D (Q) + D (Q)Q 2D (Q) + D (Q)Q − C1 ≡ ∆ 0. Example: D(Q) = A − Q, C1(q1) = q2 1, and C2(q2) = 2q2 2. FOC: MR = A − 2(q1 + q2) = MC1 = 2q1 = MC2 = 4q2. 4 2 2 6 q1 q2 = A A ⇒ q1 = 0.2A, q2 = 0.1A, Pm = 0.7A E q T p d d d d d d d d d d e e e e e e e e e e MR D¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ MC1 £ £ £ £ £ £ £ £ £ £ MC2 MC1+2 Qmq∗ 1q∗ 2 MR∗ pm MR = MC1+2 ⇒ Qm, MR∗ MR∗ = MC1 ⇒ q∗ 1 MR∗ = MC2 ⇒ q∗ 2
- 17. 15 4.4 Multiproduct monopoly Considera producer who is monopoly (the only seller) in two joint products. Q1 = D1(P1, P2), Q2 = D2(P1, P2), TC = C(Q1, Q2). The profit as a function of (P1, P2) is Π(P1, P2) = P1D1(P1, P2) + P2D2(P1, P2) − C(D1(P1, P2), D2(P1, P2)). Maximizing Π(P1, P2) w. r. t. P1, we have D1(P1, P2) + P1 ∂D1 ∂P1 + P2 ∂D2 ∂P1 − ∂C ∂Q1 ∂D1 ∂P1 − ∂C ∂Q2 ∂D2 ∂P1 = 0, ⇒ P1 − MC1 P1 = 1 | 11| + P2 − MC2 P2 TR2 TR1 21 | 11| Case 1: 12 0, goods 1 and 2 are substitutes, P1 − MC1 P1 1 | 11| . Case 2: 12 0, goods 1 and 2 are complements, P1 − MC1 P1 1 | 11| . Actually, both P1 and P2 are endogenous and have to be solved simultaneously. 1 − R2 21 R1| 11| −R1 12 R2| 22| 1 P1 − C1 P1 P2 − C2 P2 = 1 | 11| 1 | 22| ⇒ P1 − C1 P1 P2 − C2 P2 = 1 11 22 − 12 21 | 22| + R2 R1 21 | 11| + R1 R2 12 . 4.4.1 2-period model with goodwill (Tirole EX 1.5) Assume that Q2 = D2(p2; p1) and ∂D2 ∂p1 0, ie., if p1 is cheap, the monopoy gains goodwill in t = 2. max p1,p2 p1D1(p1) − C1(D1(p1)) + δ[p2D2(p2; p1) − C2(D2(p2; p1))]. 4.4.2 2-period model with learning by doing (Tirole EX 1.6) Assume that TC2 = C2(Q2; Q1) and ∂C2 ∂Q1 0, ie., if Q1 is higher, the monopoy gains more experience in t = 2. max p1,p2 p1D1(p1) − C1(D1(p1)) + δ[p2D2(p2) − C2(D2(p2), D1(p1))].
- 18. 16 Continuous time (TiroleEX 1.7): max qt,wt ∞ 0 [R(qt) − C(wt)qt]e−rt dt, wt = t 0 qτ dτ, where R(qt) is the revenue at t,R 0, R 0, r is the interest rate, Ct = C(wt) is the unit production cost at t, C 0, and wt is the experience accumulated by t. Example: R(q) = √ q and C(w) = a + 1 w . 4.5 Durable good monopoly Flow (perishable) goods: ‡ Durable goods: ˝‹ Coase (1972:) A durable good monopoly is essentially different from a perishable good monopoly. Perishable goods: .°v‚ ÒÖ , ©‚·b½h˛. Durable goods: .°v‚ Ò.Ö , ¥‚˛-‚ÿ..y, ÝBªJž“. 4.5.1 A two-period model There are 100 potential buyers of a durable good, say cars. The value of the service of a car to consumer i each period is Ui = 101 − i, i = 1, . . . , 100. Assume that MC = 0 and 0 δ 1 is the discount rate. E i, Q T Ui, P rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr The trace of Ui’s becomes the demand curve. ù »j¶: 1. É•.“. PR 1 , PR 2 are the rents in periods 1 and 2. 2. “i. Ps 1 , Ps 2 are the prices of a car in periods 1 and 2. 4.5.2 É•.“ The monopoly faces the same demand function P = 100 − Q in each period. The monopoly profit maximization implies that MR = 100 − 2Q = 0. Therefore, PR 1 = PR 2 = 50, πR 1 = πR 2 = 2500, ΠR = πR 1 + δπR 2 = 2500(1 + δ), where δ 1 is the discounting factor.
- 19. 17 4.5.3 “i We usebackward induction method to find the solution to the profit maximization problem. We first assume that those consumers who buy in period t = 1 do not resale their used cars to other consumers. Suppose that qs 1 = ¯q1, ⇒ the demand in period t = 2 becomes qs 2 = 100 − ¯q1 − Ps 2 , ⇒ Ps 2 = 100 − ¯q1 − q2 and MR2 = 100 − ¯q1 − 2q2 = 0, qs 2 = 50 − 0.5¯q1 = Ps 2 , πs 2 = (100 − ¯q1)2 /4. Now we are going to calculate the location of the marginal consumer ¯q1 who is indif- ferent between buying in t = 1 and buying in t = 2. (1 + δ)(100 − ¯q1) − Ps 1 = δ[(100 − ¯q1) − Ps 2 ] ⇒ (100 − ¯q1) − Ps 1 = −δPs 2 , ⇒ Ps 1 = 100 − ¯q1 + δPs 2 = (1 + 0.5δ)(100 − ¯q1). max q1 Πs = πs 1 + δπs 2 = q1Ps 1 + δ(50 − 0.5q1)2 = (1 + 0.5δ)q1(100 − q1) + 0.25(100 − q1)2 , FOC⇒ qs 1 = 200/(4+δ), Ps 1 = 50(2+δ)2 /(4+δ), Πs = 2500(2+δ)2 /(4+δ) ΠR = (1+δ)2500. When a monopoly firm sells a durable good in t = 1 instead of leasing it, the monopoly loses some of its monopoly power, that is why Πs ΠR . 4.5.4 Coase problem Sales in t will reduce monopoly power in the future. Therefore, a rational expectation consumer will wait. Coase conjecture (1972): In the ∞ horizon case, if δ→1 or ∆t→0, then the monopoly profit Πs →0. The conjecture was proved in different versions by Stokey (1981), Bulow (1982), Gul, Sonnenschein, and Wilson (1986). Tirole EX 1.8. 1. A monopoly is the only producer of a durable good in t = 1, 2, 3, . . .. If (q1, q2, q3, . . .) and (p1, p2, p3, . . .) are the quantity and price sequences for the monopoly product, the profit is Π = ∞ t=1 δt ptqt. 2. There is a continuum of consumers indexed by α ∈ [0, 1], each needs 1 unit of the durable good. vα = α + δα + δ2 α + . . . = α 1 − δ : The utility of the durable good to consumer α. If consumer α purchases the good at t, his consumer surplus is δt (vα − pt) = δt ( α 1 − δ − pt).
- 20. 18 3. A linearstationary equilibrium is a pair (λ, µ), 0 λ, µ 1, such that (a) If vα λpt, then consumer α will buy in t if he does not buy before t. (b) If at t, all consumers with vα v (vα v) have purchased (not purchased) the durable good, then the monopoly charges pt = µv. (c) The purchasing strategy of (a) maximizes consumer α’s consumer surplus, given the pricing strategy (b). (d) The pricing strategy of (b) maximizes the monopoly profits, given the purchacing strategy (a). The equilibrium is derived in Tirole as λ = 1 √ 1 − δ , µ = [ √ 1 − δ − (1 − δ)]/δ, lim δ→1 λ = ∞, lim δ→1 µ = 0. One way a monopoy of a durable good can avoid Coase problem is price commitment. By convincing the consumers that the price is not going to be reduced in the future, it can make the same amount of profit as in the rent case. However, the commitment equilibrium is not subgame perfect. Another way is to make the product less durable. 4.6 Product Selection, Quality, and Advertising Tirole, CH2. Product space, Vertical differentiation, Horizontal differentiation. Goods-Characteristics Approach, Hedonic prices. Traditional Consumer-Theory Approach. 4.6.1 Product quality selection, Tirole 2.2.1, pp.100-4. Inverse Demand: p = P(q, s), where s is the quality of the product. Total cost: TC = C(q, s), Cq 0, Cs 0. Social planner’s problem: max q,s W(q, s) = q 0 P(x, s)ds − C(q, s), FOC: (1) Wq = P(q, s) − Cq = 0, (2) Ws = q 0 Ps(x, s)dx − Cs = 0. (1) P = MC, (2) 1 q q 0 Psdx = Cs/q: Average marginal valuation of quality should be equal to the marginal cost of quality per unit. Monopoly profit maximization: max q,s Π(q, s) = qP(x, s)−C(q, s), FOC Πq = MR−Cq = 0, Πs = qPs(x, s)−Cs = 0.
- 21. 19 Ps = Cs/q:Marginal consumer’s marginal valuation of quality should be equal to the marginal cost of quality per unit. Example 1: P(q, s) = f(q) + s, C(q, s) = sq, ⇒ Ps = 1, Cs = q, no distortion. Example 2: There is one unit of consumers indexed by x ∈ [0, ¯x]. U = xs − P, F(x) is the distribution function of x. ⇒ P(q, s) = sF −1 (1 − q) ⇒ 1 q q 0 Psdx ≥ Ps(q, s), monopoly underprovides quality. Example 3: U = x + (α − x)s − P, x ∈ [0, α], F(x) is the distribution function of x ⇒ P(q, s) = αs + (1 − s)F −1 (1 − q) ⇒ 1 q q 0 Psdx ≤ Ps(q, s), monopoly overpro- vides quality.
- 22. 20 5 Basis ofGame Theory In this part, we consider the situation when there are n 1 persons with different objective (utility) functions; that is, different persons have different preferences over possible outcomes. There are two cases: 1. Game theory: The outcome depends on the behavior of all the persons involved. Each person has some control over the outcome; that is, each person controls certain strategic variables. Each one’s utility depends on the decisions of all persons. We want to study how persons make decisions. 2. Public Choice: Persons have to make decision collectively, eg., by voting. We consider only game theory here. Game theory: the study of conflict and cooperation between persons with differ- ent objective functions. Example (a 3-person game): The accuracy of shooting of A, B, C is 1/3, 2/3, 1, respectively. Each person wants to kill the other two to become the only survivor. They shoot in turn starting A. Question: What is the best strategy for A? 5.1 Ingredients and classifications of games A game is a collection of rules known to all players which determine what players may do and the outcomes and payoffs resulting from their choices. The ingredients of a game: 1. Players: Persons having some influences upon possible income (decision mak- ers). 2. Moves: decision points in the game at which players must make choices between alternatives (personal moves) and randomization points (called nature’s moves). 3. A play: A complete record of the choices made at moves by the players and realizations of randomization. 4. Outcomes and payoffs: a play results in an outcome, which in turn determines the rewords to players. Classifications of games: 1. according to number of players: 2-person games – conflict and cooperation possibilities. n-person games – coalition formation (¯ó©d) possibilities in addition. infinite-players’ games – corresponding to perfect competition in economics. 2. according to number of strategies: finite – strategy (matrix) games, each person has a finite number of strategies,
- 23. 21 payoff functions canbe represented by matrices. infinite – strategy (continuous or discontinuous payoff functions) games like duopoly games. 3. according to sum of payoffs: 0-sum games – conflict is unavoidable. non-zero sum games – possibilities for cooperation. 4. according to preplay negotiation possibility: non-cooperative games – each person makes unilateral decisions. cooperative games – players form coalitions and decide the redistribution of aggregate payoffs. 5.2 The extensive form and normal form of a game Extensive form: The rules of a game can be represented by a game tree. The ingredients of a game tree are: 1. Players 2. Nodes: they are players’ decision points (personal moves) and randomization points (nature’s moves). 3. Information sets of player i: each player’s decision points are partitioned into information sets. An information set consists of decision points that player i can not distinguish when making decisions. 4. Arcs (choices): Every point in an information set should have the same number of choices. 5. Randomization probabilities (of arcs following each randomization point). 6. Outcomes (end points) 7. Payoffs: The gains to players assigned to each outcome. A pure strategy of player i: An instruction that assigns a choice for each information set of player i. Total number of pure strategies of player i: the product of the numbers of choices of all information sets of player i. Once we identify the pure strategy set of each player, we can represent the game in normal form (also called strategic form). 1. Strategy sets for each player: S1 = {s1, . . . , sm}, S2 = {σ1, . . . , σn}. 2. Payoff matrices: π1(si, σj) = aij, π2(si, σj) = bij. A = [aij], B = [bij]. Normal form: II I d d d σ1 . . . σn s1 (a11, b11) . . . (a1n, b1n) ... ... ... ... sm (am1, bm1) . . . (amn, bmn)
- 24. 22 5.3 Examples Example 1:A perfect information game 1 L R d d 2 l r d d 2 L R 1 9 9 6 3 7 8 2 S1 = { L, R }, S2 = { Ll, Lr, Rl, Rr }. II I d d d Ll Rl Lr Rr L (1,9) (1,9) (9,6) (9,6) R (3,7)* (8,2) (3,7) (8,2) Example 2: Prisoners’ dilemma game 1 L R d d d d ¨ ©2 L R L R 4 4 0 5 5 0 1 1 S1 = { L, R }, S2 = { L, R }. II I d d d L R L (4,4) (0,5) R (5,0) (1,1)* Example 3: Hijack game 1 L R d d 2 L R −1 2 2 −2 −10 −10 S1 = { L, R }, S2 = { L, R }. II I d d d L R L (-1,2) (-1,2)* R (2,-2)* (-10,-10) Example 4: A simplified stock price manipulation game ¨¨ ¨ ¨ rr r r 0 1/2 1/2 e e 1 L R ¡ ¡ d d d 1 l r ¡ ¡ e e ¡ ¡ e e ¨ ©2 L R L R 4 2 7 5 5 7 4 5 4 2 3 7 S1 = { Ll, Lr, Rl, Rr }, S2 = { L, R }. II I d d d L R Ll (4, 3.5) (4, 2) Lr (3.5, 4.5) (3.5, 4.5) Rl (5.5, 5)* (4.5, 4.5) Rr (5,6) (4,7) Remark: Each extensive form game corresponds a normal form game. However, different extensive form games may have the same normal form.
- 25. 23 5.4 Strategy pairand pure strategy Nash equilibrium 1. A Strategy Pair: (si, σj). Given a strategy pair, there corresponds a payoff pair (aij, bij). 2. A Nash equilibrium: A strategy pair (si∗, σj∗) such that ai∗j∗ ≥ aij∗ and bi∗j∗ ≥ bi∗j for all (i, j). Therefore, there is no incentives for each player to deviate from the equilibrium strategy. ai∗j∗ and bi∗j∗ are called the equilibrium payoff. The equilibrium payoffs of the examples are marked each with a star in the normal form. Remark 1: It is possible that a game does no have a pure strategy Nash equilib- rium. Also, a game can have more than one Nash equilibria. Remark 2: Notice that the concept of a Nash equilibrium is defined for a normal form game. For a game in extensive form (a game tree), we have to find the normal form before we can find the Nash equilibria. 5.5 Subgames and subgame perfect Nash equilibria 1. Subgame: A subgame in a game tree is a part of the tree consisting of all the nodes and arcs following a node that form a game by itself. 2. Within an extensive form game, we can identify some subgames. 3. Also, each pure strategy of a player induces a pure strategy for every subgame. 4. Subgame perfect Nash equilibrium: A Nash equilibrium is called subgame perfect if it induces a Nash equilibrium strategy pair for every subgame. 5. Backward induction: To find a subgame perfect equilibrium, usually we work backward. We find Nash equilibria for lowest level (smallest) subgames and replace the subgames by its Nash equilibrium payoffs. In this way, the size of the game is reduced step by step until we end up with the equilibrium payoffs. All the equilibria, except the equilibrium strategy pair (L,R) in the hijack game, are subgame perfect. Remark: The concept of a subgame perfect Nash equilibrium is defined only for an extensive form game. 5.5.1 Perfect information game and Zemelo’s Theorem An extensive form game is called perfect information if every information set consists only one node. Every perfect information game has a pure strategy subgame perfect Nash Equilibrium.
- 26. 24 5.5.2 Perfect recallgame and Kuhn’s Theorem A local strategy at an information set u ∈ Ui: A probability distribution over the choice set at Uij. A behavior strategy: A function which assigns a local strategy for each u ∈ Ui. The set of behavior strategies is a subset of the set of mixed strategies. Kuhn’s Theorem: In every extensive game with perfect recall, a strategically equiva- lent behavior strategy can be found for every mixed strategy. However, in a non-perfect recall game, a mixed strategy may do better than be- havior strategies because in a behavior strategy the local strategies are independent whereas they can be correlated in a mixed strategy. ¨¨ ¨¨ ¨¨ rr rr rr d d d d d d ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 0 u11 2 ¨ ©u12 1/2 1/2 a b A B c d c d1 −1 −1 1 2 −2 0 0 −2 2 0 0 A 2-person 0-sum non-perfect recall game. NE is (µ∗ 1, µ∗ 2) = ( 1 2 ac ⊕ 1 2 bd, 1 2 A ⊕ 1 2 B). µ∗ 1 is not a behavioral strategy. 5.5.3 Reduction of a game Redundant strategy: A pure strategy is redundant if it is strategically identical to another strategy. Reduced normal form: The normal form without redundant strategies. Equivalent normal form: Two normal forms are equivalent if they have the same reduced normal form. Equivalent extensive form: Two extensive forms are equivalent if their normal forms are equivalent. Equivalent transformation: (1) Inflation-Deflation; ¨ ¨¨ ¨¨ ¨ e e e e e e r d d d ¨ ©1 ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 2 1 ¨ ¨¨ ¨¨ ¨ e e e e e e r d d d ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 2 ¨ ©1
- 27. 25 (2) Addition ofsuperfluous move; ¨¨ ¨¨ ¨¨ e e e e e e r d d d ¨ ©1 ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 2 2 ¨¨ ¨¨ ¨¨ rr rr rrr d d d ¨ ©1 ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e d d d ¨ ©2 ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 ¨ ©2 (3) Coalesing of moves; ¨¨ ¨¨ ¨¨ e e e e e e r d d d ¨ ©1¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 2 1 ¨¨ ¨¨ ¨¨ i i i i i i i i ii e e e e e e e e e r d d d ¨ ©1¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 2 (4) Interchange of moves. ¨ ¨¨ ¨¨ ¨ e e e e e e r d d d ¨ ©1 ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 2 1 ¨ ¨¨ ¨¨ ¨ e e e e e e r d d d ¨ ©2 ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 1 1 1 5.6 Continuous games and the duopoly game In many applications, S1 and S2 are infinite subsets of Rm and Rn Player 1 controls m variables and player 2 controls n variables (however, each player has infinite many strtategies). The normal form of a game is represented by two functions Π1 = Π1 (x; y) and Π2 = Π2 (x; y), where x ∈ S1 ⊂ Rm and y ∈ S2 ⊂ Rn . To simplify the presentation, assume that m = n = 1. A strategic pair is (x, y) ∈ S1 × S2. A Nash equilibrium is a pair (x∗ , y∗ ) such that Π1 (x∗ , y∗ ) ≥ Π1 (x, y∗ ) and Π2 (x∗ , y∗ ) ≥ Π2 (x∗ , y) for all x ∈ S1 y ∈ S2. Consider the case when Πi are continuously differentiable and Π1 is strictly concave in x and Π2 strictly concave in y (so that we do not have to worry about the SOC’s).
- 28. 26 Reaction functions andNash equilibrium: To player 1, x is his endogenous variable and y is his exogenous variable. For each y chosen by player 2, player 1 will choose a x ∈ S1 to maximize his objective function Π1 . This relationship defines a behavioral equation x = R1 (y) which can be obtained by solving the FOC for player 1, Π1 x(x; y) = 0. Similarly, player 2 regards y as en- dogenous and x exogenous and wants to maximize Π2 for a given x chosen by player 1. Player 2’s reaction function (behavioral equation) y = R2 (x) is obtained by solving Π2 y(x; y) = 0. A Nash equilibrium is an intersection of the two reaction functions. The FOC for a Nash equilibrium is given by Π1 x(x∗ ; y∗ ) = 0 and Π2 y(x∗ ; y∗ ) = 0. Duopoly game: There are two sellers (firm 1 and firm 2) of a product. The (inverse) market demand function is P = a − Q. The marginal production costs are c1 and c2, respectively. Assume that each firm regards the other firm’s output as given (not affected by his output quantity). The situation defines a 2-person game as follows: Each firm i controls his own output quantity qi. (q1, q2) together determine the market price P = a − (q1 + q2) which in turn determines the profit of each firm: Π1 (q1, q2) = (P−c1)q1 = (a−c1−q1−q2)q1 and Π2 (q1, q2) = (P−c2)q2 = (a−c2−q1−q2)q2 The FOC are ∂Π1 /∂q1 = a − c1 − q2 − 2q1 = 0 and ∂Π2 /∂q2 = a − c2 − q1 − 2q2 = 0. The reaction functions are q1 = 0.5(a − c1 − q2) and q2 = 0.5(a − c2 − q1). The Cournot Nash equilibrium is (q∗ 1, q∗ 2) = ((a − 2c1 + c2)/3, (a − 2c2 + c1)/3) with P∗ = (a + c1 + c2)/3. (We have to assume that a − 2c1 + c2, a − 2c2 + c1 ≥ 0.) 5.7 2-person 0-sum game 1. B = −A so that aij + bij = 0. 2. Maxmin strategy: If player 1 plays si, then the minimum he will have is minj aij, called the security level of strategy si. A possible guideline for player 1 is to choose a strategy such that the security level is maximized: Player 1 chooses si∗ so that minj ai∗j ≥ minj aij for all i. Similarly, since bij = −aij, Player 2 chooses σj∗ so that maxi aij∗ ≤ maxi aij for all j. 3. Saddle point: If ai∗j∗ = maxi minj aij = minj maxi aij, then (si∗, σj∗) is called a saddle point. If a saddle point exists, then it is a Nash equilibrium. A1 = 2 1 4 −1 0 6 A2 = 1 0 0 1 In example A1, maxi minj aij = minj maxi aij = 1 (s1, σ2) is a saddle point and hence a Nash equilibrium. In A2, maxi minj aij = 0 = minj maxi aij = 1 and no saddle point exists. If there is no saddle points, then there is no pure strategy equilibrium.
- 29. 27 4. Mixed strategyfor player i: A probability distribution over Si. p = (p1, . . . , pm), q = (q1, . . . , qn) . (p, q) is a mixed strategy pair. Given (p, q), the expected payoff of player 1 is pAq. A mixed strategy Nash equilibrium (p∗ , q∗ ) is such that p∗ Aq∗ ≥ pAq∗ and p∗ Aq∗ ≤ p∗ Aq for all p and all q. 5. Security level of a mixed strategy: Given player 1’s strategy p, there is a pure strategy of player 2 so that the expected payoff to player 1 is minimized, just as in the case of a pure strategy of player 1. t(p) ≡ min j { i piai1, . . . , i piain}. The problem of finding the maxmin mixed strategy (to find p∗ to maximize t(p)) can be stated as max p t subj. to i piai1 ≥ t, . . . , i piain ≥ t, i pi = 1. 6. Linear programming problem: The above problem can be transformed into a linear programming problem as follows: (a) Add a positive constant to each element of A to insure that t(p) 0 for all p. (b) Define yi ≡ pi/t(p) and replace the problem of max t(p) with the problem of min 1/t(p) = i yi. The constraints become i yiai1 ≥ 1, . . . , i yiain ≥ 1. min y1,...,ym≥0 y1 + . . . + ym subj. to i yiai1 ≥ 1, . . . , i yiain ≥ 1 7. Duality: It turns out that player 2’s minmax problem can be transformed sim- ilarly and becomes the dual of player 1’s linear programming problem. The existence of a mixed strategy Nash equilibrium is then proved by using the duality theorem in linear programming. Example (tossing coin game): A = 1 0 0 1 . To find player 2’s equilibrium mixed strategy, we solve the linear programming prob- lem: max x1,x2≥0 x1 + x2 subj. to x1 ≤ 1 x2 ≤ 1. The solution is x1 = x2 = 1 and therefore the equilibrium strategy for player 2 is q∗ 1 = q∗ 2 = 0.5. E x1 T x2 d d d d d d d d 1 1 r E y1 T y2 d d d d d d d d 1 1 r
- 30. 28 Player 1’s equilibriummixed strategy is obtained by solving the dual to the linear programming problem: min y1,y2≥0 y1 + y2 subj. to y1 ≥ 1 y2 ≥ 1. The solution is p∗ 1 = p∗ 2 = 0.5. mixed strategy equilibria for non-zero sum games The idea of a mixed strategy equilibrium is also applicable to a non-zero sum game. Similar to the simplex algorism for the 0-sum games, there is a Lemke algorism. Example (Game of Chicken) 1 S N d d d d ¨ ©2 S N S N 0 0 −3 3 3 −3 −9 −9 S1 = { S, N }, S2 = { S, N }. II I d d d Swerve Don’t Swerve (0,0) (-3,3)* Don’t (3,-3)* (-9,-9) There are two pure strategy NE: (S, N) and (N, S). There is also a mixed strategy NE. Suppose player 2 plays a mixed strategy (q, 1−q). If player 1 plays S, his expected payoff is Π1 (S) = 0q + (−3)(1 − q). If he plays N, his expected payoff is Π1 (N) = 3q + (−9)(1 − q). For a mixed strategy NE, Π1 (S) = Π1 (N), therefore, q = 2 3 . The mixed strategy is symmetrical: (p∗ 1, p∗ 2) = (q∗ 1, q∗ 2) = (2 3 , 1 3 ). 5.8 Cooperative Game and Characteristic form 2-person 0-sum games are strictly competitive. If player 1 gains $ 1, player 2 will loss $ 1 and therefore no cooperation is possible. For other games, usually some coopera- tion is possible. The concept of a Nash equilibrium is defined for the situation when no explicit cooperation is allowed. In general, a Nash equilibrium is not efficient (not Pareto optimal). When binding agreements on strategies chosen can be contracted before the play of the game and transfers of payoffs among players after a play of the game is possible, players will negotiate to coordinate their strategies and redistribute the payoffs to achieve better results. In such a situation, the determination of strate- gies is not the key issue. The problem becomes the formation of coalitions and the distribution of payoffs. Characteristic form of a game: The player set: N = {1, 2, . . . , n}. A coalition is a subset of N: S ⊂ N. A characteristic function v specifies the maximum total payoff of each coalition.
- 31. 29 Consider the caseof a 3-person game. There are 8 subsets of N = {1, 2, 3}, namely, φ, (1), (2), (3), (12), (13), (23), (123). Therefore, a characteristic form game is deter- mined by 8 values v(φ), v(1), v(2), v(3), v(12), v(13), v(23), v(123). Super-additivity: If A ∩ B = φ, then v(A ∪ B) ≥ v(A) + v(B). An imputation is a payoff distribution (x1, x2, x3). Individual rationality: xi ≥ v(i). Group rationality: i∈S xi ≥ v(S). Core C: the set of imputations that satisfy individual rationality and group rational- ity for all S. Marginal contribution of player i in a coalition S ∪ i: v(S ∪ i) − v(S) Shapley value of player i is an weighted average of all marginal contributions πi = S⊂N |S|!(n − |S| − 1)! n! [v(S ∪ i) − v(S)]. Example: v(φ) = v(1) = v(2) = v(3) = 0, v(12) = v(13) = v(23) = 0.5, v(123) = 1. C = {(x1, x2, x3), xi ≥ 0, xi + xj ≥ 0.5, x1 + x2 + x3 = 1}. Both (0.3, 0.3, 0.4) and (0.2, 0.4, 0.4) are in C. The Shapley values are (π1, π2, π3) = (1 3 , 1 3 , 1 3 ). Remark 1: The core of a game can be empty. However, the Shapley values are uniquely determined. Remark 2: Another related concept is the von-Neumann Morgenstern solution. See CH 6 of Intriligator’s Mathematical Optimization and Economic Theory for the mo- tivations of these concepts. 5.9 The Nash bargaining solution for a nontransferable 2-person cooper- ative game In a nontransferable cooperative game, after-play redistributions of payoffs are im- possible and therefore the concepts of core and Shapley values are not suitable. For the case of 2-person games, the concept of Nash bargaining solutions are useful. Let F ⊂ R2 be the feasible set of payoffs if the two players can reach an agreement and Ti the payoff of player i if the negotiation breaks down. Ti is called the threat point of player i. The Nash bargaining solution (x∗ 1, x∗ 2) is defined to be the solution to the following problem: E x1 T x2 T1 T2 x∗ 1 x∗ 2 max (x1,x2)∈F (x1 − T1)(x2 − T2)
- 32. 30 See CH 6of Intriligator’s book for the motivations of the solution concept. 5.10 Problems 1. Consider the following two-person 0-sum game: I II σ1 σ2 σ3 s1 4 3 -2 s2 3 4 10 s3 7 6 8 (a) Find the max min strategy of player I smax min and the min max strategy of player II σmin max. (b) Is the strategy pair (smax min, σmin max) a Nash equilibrium of the game? (c) What are the equilibrium payoffs? 2. Find the maxmin strategy (smax min) and the minmax strategy (σmin max) of the following two-person 0-sum game: I II σ1 σ2 s1 -3 6 s2 8 -2 s3 6 3 Is the strategy pair (smax min, σmin max) a Nash equilibrium? If not, use simplex method to find the mixed strategy Nash equilibrium. 3. Find the (mixed strategy) Nash Equilibrium of the following two-person game: I II H T H (-2, 2) (2, -1) T (2, -2) (-1,2) 4. Suppose that two firms producing a homogenous product face a linear demand curve P = a−bQ = a−b(q1 +q2) and that both have the same constant marginal costs c. For a given quantity pair (q1, q2), the profits are Πi = qi(P − c) = qi(a − bq1 − bq2 − c), i = 1, 2. Find the Cournot Nash equilibrium output of each firm. 5. Suppose that in a two-person cooperative game without side payments, if the two players reach an agreement, they can get (Π1, Π2) such that Π2 1 + Π2 = 47 and if no agreement is reached, player 1 will get T1 = 3 and player 2 will get T2 = 2. (a) Find the Nash solution of the game. (b) Do the same for the case when side payments are possible. Also answer how the side payments should be done?
- 33. 31 6. A singer(player 1), a pianist (player 2), and a drummer (player 3) are offered $ 1,000 to play together by a night club owner. The owner would alternatively pay $ 800 the singer-piano duo, $ 650 the piano drums duo, and $ 300 the piano alone. The night club is not interested in any other combination. Howeover, the singer-drums duo makes $ 500 and the singer alone gets $ 200 a night in a restaurant. The drums alone can make no profit. (a) Write down the characteristic form of the cooperative game with side pay- ments. (b) Find the Shapley values of the game. (c) Characterize the core.
- 34. 32 6 Duopoly andOligopoly–Homogeneous products 6.1 Cournot Market Structure 2 Sellers producing a homogenous product. TCi(qi) = ciqi, i = 1, 2. P(Q) = a − bQ, a, b 0, a maxi ci, Q = q1 + q2. Simultaneous move: both firms choose (q1, q2) simultaneously. π1(q1, q2) = P(Q)q1 − c1q1 = (a − bq1 − bq2)q1 − c1q1, π2(q1, q2) = P(Q)q2 − c2q2 = (a − bq1 − bq2)q2 − c2q2. Definition of a Cournot equilibrium: {P c , qc 1, qc 2} such that Pc = P(Qc ) = a−b(qc 1 +qc 2) and π1(qc 1, qc 2) ≥ π1(q1, qc 2), π2(qc 1, qc 2) ≥ π2(qc 1, q2), ∀(q1, q2). The first order conditions (FOC) are ∂π1 ∂q1 = a − 2bq1 − bq2 − c1 = 0, ∂π2 ∂q2 = a − bq1 − 2bq2 − c2 = 0. In matrix form, 2b b b 2b q1 q2 = a − c1 a − c2 , ⇒ qc 1 qc 2 = 1 3b a − 2c1 + c2 a − 2c2 + c1 . Qc = 2a − c1 − c2 3b , Pc = a + c1 + c2 3 , πc 1 πc 2 = 1 9b (a − 2c1 + c2)2 (a − 2c2 + c1)2 . If c1 ↓ (say, due to RD), then qc 1 ↑, qc 2 ↓, Qc ↑, Pc ↓, πc 1 ↑, πc 2 ↓. 6.1.1 Reaction function and diagrammatic solution From FOC, we can derive the reaction functions: q1 = a − c1 2b − 0.5q2 ≡ R1(q2), q2 = a − c2 2b − 0.5q1 ≡ R2(q1). E q1 T q2 e e e e e e e e e e R1(q2) rr r rr rr rr R2(q1) qc 2 qc 1
- 35. 33 6.1.2 N-seller case Nsellers, MCi = ci, i = 1, . . . , N, P = P(Q) = a − bQ = a − b N j=1 qj. πi(q1, . . . , qN ) = P(Q)qi − ciqi = a − b N j=1 qj qi − ciqi. FOC is ∂πi ∂qi = a − b j qj − bqi − ci = P − bqi − ci = 0, ⇒ Na − (N + 1)b j qj − j cj = 0, ⇒ Qc = j qc j = Na − j cj (N + 1)b , Pc = a + j cj N + 1 , qc i = P − ci b = a + j cj − (N + 1)ci b(N + 1) . Symmetric case ci = c: qc i = a − c (N + 1)b , Pc = a + Nc N + 1 , Qc = N N + 1 a − c b . When N = 1, it is the monopoly case. As N→ ∞, (Pc , Qc ) → (c, a − c b ), the competition case. 6.1.3 Welfare analysis for the symmetric case Consumer surplus CS = (a − P)Q 2 , Social welfare W = CS + j πj. For the symmetric case, CS as functions of N are CSc (N) = 1 2 a − a + Nc N + 1 N N + 1 a − c b = 1 2b N N + 1 2 (a − c)2 . The sum of profits is j πj = j (Pc − c)qj = a − c N + 1 N N + 1 a − c b = N (N + 1)2 (a − c)2 b . Wc (N) = CSc (N) + j πj = (a − c)2 b N + 0.5N2 (N + 1)2 . lim N→∞ Wc (N) = lim N→∞ CSc (N) = (a − c)2 2b , lim N→∞ j πj = 0. E Q, qi T P d d d d d d d d d d CS c Pc Qc π1 π2 π3 q1
- 36. 34 6.2 Sequential moves– Stackelberg equilibrium Consider now that the two firms move sequentially. At t = 1 firm 1 chooses q1. At t = 2 firm 2 chooses q2. Firm 1 – leader, Firm 2 – follower. The consequence is that when choosing q2, firm 2 already knows what q1 is. On the other hand, in deciding the quantity q1, firm 1 takes into consideration firm 2’s possible reaction, i.e., firm 1 assumes that q2 = R2(q1). This is the idea of backward induction and the equilibrium derived is a subgame perfect Nash equilibrium. At t = 1, firm 1 chooses q1 to maximize the (expected) profit π1 = π1(q1, R2(q1)): max q1 [a−b(q1+R2(q1))]q1−c1q1 = [a−b(q1+ a − c2 2b −0.5q1)]q1−c1q1 = 0.5(a+c2−bq1)q1−c1q1. The FOC (interior solution) is dπ1 dq1 = ∂π1 ∂q1 + ∂π1 ∂q2 dR2 dq1 = 0.5(a − 2c1 + c2 − 2bq1) = 0, ⇒ qs 1 = a − 2c1 + c2 2b qc 1, qs 2 = a + 2c1 − 3c2 4b qc 2. Qs = 3a − 2c1 − c2 4b Qc , Ps = a + 2c1 + c2 4 Pc , ((a + c1 + c2)/3 = Pc c1). πs 1 = (a − 2c1 + c2) 8b , πs 2 = (a + 2c1 − 3c2) 16b . 1. πs 1 + πs 2 πc 1 + πc 2 depending on c1, c2. 2. πs 1 πc 1 because πs 1 = maxq1 π1(q1, R2(q1)) π1(qc 1, R2(qc 1)) = πc 1. 3. πs 2 πc 2 since qs 2 qc 2 and Ps Pc . E q1 T q2 e e e e e e e e e e R1(q2) r rr rr rr r r R2(q1) qc 2 qc 1 qs 2 qs 1 6.2.1 Subgame non-perfect equilibrium In the above, we derived a subgame perfect equilibrium. On the other hand, the game has many non-perfect equilibria. Firm 2 can threaten firm 1 that if q1 is too large, he
- 37. 35 will chooses alarge enough q2 to make market price zero. This is an incredible threat because firm 2 will hurt himself too. However, if firm 1 believes that the threat will be executed, there can be all kind of equilibria. 6.2.2 Extension The model can be extended in many ways. For example, when there are three firms choosing output quantities sequentially. Or firms 1 and 2 move simultaneously and then firm 3 follows, etc. 6.3 Conjecture Variation In Cournot equilibrium, firms move simultaneously and, when making decision, expect that other firms will not change their quantities. In more general case, firms will form conjectures about other firms behaviors. πe 1(q1, qe 2) = (a − bq1 − bqe 2)q1 − c1q1, πe 2(qe 1, q2) = (a − bqe 1 − bq2)q2 − c2q2. The FOC are dπe 1 dq1 = ∂πe 1 ∂q1 + ∂πe 1 ∂qe 2 dqe 2 dq1 = 0, dπe 2 dq2 = ∂πe 2 ∂q2 + ∂πe 2 ∂qe 1 dqe 1 dq2 = 0. Assume that the conjectures are dqe 2 dq1 = λ1, dqe 1 dq2 = λ2. In equilibrium, qe 1 = q1 and qe 2 = q2. The FOCs become a − 2bq1 − bq2 − bλ1q1 − c1 = 0, a − 2bq2 − bq1 − bλ2q2 − c2 = 0. In matrix form, (2 + λ1)b b b (2 + λ2)b q1 q2 = a − c1 a − c2 , ⇒ q1 q2 = 1 [3 + 2(λ1 + λ2) + λ1λ2]b (a − c1)(2 + λ2) − (a − c2) (a − c2)(2 + λ1) − (a − c1) . If λ1 = λ2 = 0, then it becomes the Cournot equilibrium. 6.3.1 Stackelberg Case: λ2 = 0, λ1 = R2(q1) = 0.5 Assuming that c1 = c2 = 0. q1 q2 = 1 2b a a/2 . It is the Stackelberg leadership equilibrium with firm 1 as the leader. Similarly, if λ1 = 0, λ2 = R1(q2) = 0.5, then firm 2 becomes the leader.
- 38. 36 6.3.2 Collusion case:λ1 = q2/q1, λ2 = q1/q2 Assume that MC1 = c1q1 and MC2 = c2q2. The FOCs become a − 2b(q1 + q2) − c1q1 = 0, a − 2b(q1 + q2) − c2q2 = 0. That is, MR = MC1 = MC2, the collusion solution. The idea can be generalized to N-firm case. 6.4 Bertrand Price Competition It is easier to change prices than to change quantities. Therefore, firms’ strategic variables are more likely to be prices. Bertrand model: firms determine prices simultaneously. Question: In a homogeneous product market, given (p1, p2), how market demand is going to divided between firm 1 and firm 2? Assumption: Consumers always choose to buy from the firm charging lower price. When two firms charge the same price, the market demand divided equally between them. Let Q = D(P) be the market demand. q1 = D1(p1, p2) = 0 p1 p2 0.5D(p1) p1 = p2 D(p1) p1 p2, q2 = D2(p1, p2) = D(p2) p1 p2 0.5D(p2) p1 = p2 0 p1 p2. E Q T P d d d d d d d d D(P) E q1 T p1 d d d d D1(p1, p2) rp2 E q2 T p2 d d d d D2(p1, p2) rp1 Notice that individual firms’ demand functions are discontinuous. Bertrand game: π1(p1, p2) = (p1 − c1)D1(p1, p2), π2(p1, p2) = (p2 − c2)D2(p1, p2) Bertrand equilibrium: {pb 1, pb 2, qb 1, qb 2} such that qb 1 = D1(pb 1, pb 2), qb 2 = D2(pb 1, pb 2), and π1(pb 1, pb 2) ≥ π1(p1, pb 2), π2(pb 1, pb 2) ≥ π2(pb 1, p2) ∀ (p1, p2). We cannot use FOCs to find the reaction functions and the equilibrium as in the Cournot quantity competition case because the profit functions are not continuous.
- 39. 37 6.4.1 If c1= c2 = c, then pb 1 = pb 2 = c, qb 1 = qb 2 = 0.5D(c) Proof: 1. pb i ≥ c. 2. Both p1 p2 c and p2 p1 c cannot be equilibrium since the firm with a higher price will reduce its price. 3. p1 = p2 c cannot be equilibrium since every firm will reduce its price to gain the whole market. 4. p1 p2 = c and p2 p1 = c cannot be equilibrium because the firm with p = c will raise its price to earn positive profit. 5. What left is p1 = p2 = c, where none has an incentive to change. Bertrand paradox: 1. πb 1 = πb 2 = 0. 2. Why firms bother to enter the market if they know that πb 1 = πb 2 = 0. 6.4.2 If ci cj and cj ≤ Pm(ci), then no equilibrium exists However, if cj Pm(ci), where Pm(ci) is the monopoly price corresponding to marginal cost ci, then firm i becomes a monopoly. Or if there is a smallest money unit e (say, e is one cent), cj −ci e, then pb i = c2 −e, pb j = cj, qi = D(c2 − e), qb 2 = 0 is an equilibrium because none has an incentive to change. Approximately, we say that pb i = pb j = cj and qb i = D(cj), qb j = 0. 6.5 Price competition, capacity constraint, and Edgeworth Cycle One way to resolve Bertrand paradox is to consider DRTS or increasing MC. In Bertrand price competition, if the marginal cost is increasing (TCi (qi) 0), then we have to consider the possibility of mixed strategy equilibria. See Tirole’s Supplement to Chapter 5. Here we discuss capacity constraint and Edgeworth cycle. 6.5.1 Edgeworth model Assume that both firms has a capacity constraint qi ≤ ¯qi = 1 and that the product is indivisible. c1 = c2 = 0. There are 3 consumers. Consumer i is willing to pay 4 − i dollars for one unit of the product, i = 1, 2, 3. E Q T P t t t D(P) 1 2 3 1 2 3 Edgeworth Cycle of (p1, p2): (2, 2)→(3, 2)→(3, 2.9)→(2.8, 2.9)→(2.8, 2.7)→ · · · →(2, 2) Therefore, the is no equilibrium but repetitions of similar cycles.
- 40. 38 6.6 A 2-periodmodel At t = 1, both firms determine quantities q1, q2. At t = 2, both firms determine prices p1, p2 after seeing q1, q2. Demand function: P = 10 − Q, MCs: c1 = c2 = 1. Proposition: If 2qi + qj ≤ 9, then p1 = p2 = 10 − (q1 + q2) = P(Q) is the equi- librium at t = 2. Proof: Suppose p2 = 10 − (q1 + q2), then p1 = p2 maximizes firm 1’s profit. The reasons are as follows. 1. If p1 p2, π1(p1) = q1(p1 − 1) π1(p2) = q1(p2 − 1). 2. If p1 p2, π1(p1) = (10−q2−p1)(p1−1), π1(p1) = 10−q2−2p1+1 = 2q2+q1−9 ≤ 0. It follows that at t = 1, firms expect that P = 10 − q1 − q2, the reduced profit functions are π1(q1, q2) = q1(P(Q)−c1) = q1(9−q1−q2), π2(q1, q2) = q2(P(Q)−c2) = q2(9−q1−q2). Therefore, it becomes an authentic Cournot quantity competition game. 6.7 Infinite Repeated Game and Self-enforcing Collusion Another way to resolve Bertrand paradox is to consider the infinite repeated version of the Bertrand game. To simplify the issue, assume that c1 = c2 = 0 and P = min{1 − Q, 0} = min{1 − q1 − q2, 0} in each period t. The profit function of firm i at time t is πi(t) = πi(q1(t), q2(t)) = qi(t)[1 − q1(t) − q2(t)]. A pure strategy of the repeated game of firm i is a sequence of functions σi,t of outcome history Ht−1: σi ≡ (σi,0, σi,1(H0), . . . , σi,t(Ht−1), . . .) . where Ht−1 is the history of a play of the repeated game up to time t − 1: Ht−1 = ((q1(0), q2(0)), (q1(1), q2(1)), . . . , (q1(t − 1), q2(t − 1))) , and σi,t maps from the space of histories Ht−1 to the space of quantities {qi : 0 ≤ qi ∞}. Given a pair (σ1, σ2), the payoff function of firm i is Πi(σ1, σ2) = ∞ t=0 δt πi(σ1,t(Ht−1), σ2,t(Ht−1)) = ∞ t=0 δt πi(q1(t), q2(t)). Given the Cournot equilibrium (qc 1, qc 2) = ( 1 3 , 1 3 ), we can define a Cournot strategy for the repeated game as follows: σc i,t(Ht−1) = qc i = 1 3 ∀Ht−1, σc i ≡ (σc i,0, σc i,1, . . . , σc i,t, . . .). It is straightforward to show that the pair (σc 1, σc 2) is a Nash equilibrium for the repeated game.
- 41. 39 6.7.1 Trigger strategyand tacit collusive equilibrium A trigger strategy σT for firm i has a cooperative phase and a non-cooperative phase: Cooperative phase ¯T Þ– If both firms cooperate (choose collusion quantity qi = 0.5Qm = 0.25) up to period t − 1, then firm i will cooperate at period t. Non-cooperative phase .¯T Þ– Once the cooperation phase breaks down (someone has chosen a different quantity), then firm i will choose the Cournot equi- librium quantity qc i . If both firms choose the same strigger strategy, then they will cooperate forever. If neither one could benefit from changing to a different strategy, then (σT , σT ) is a subgame-perfect Nash equilbrium. Formally, a trigger strategy for firm i is σT = (σT 0 , σT 1 , . . . , σT t , . . .), σT 0 = 0.5Qm, σT t = 0.5Qm if qj(τ) = 0.5Qm ∀ 0 ≤ τ t, j = 1, 2 qc i otherwise. 6.7.2 (σT , σT ) is a SPNE for δ 9 17 Proof: 1. At every period t in the cooperative phase, if the opponent does not violate the cooperation, then firm i’s gain to continue cooperation is Π∗ = 0.5πm + δ0.5πm + δ2 0.5πm + · · · = 0.5πm(1 + δ + δ2 + · · ·) = 1 8(1 − δ) . If firm i chooses to stop the cooperative phase, he will set qt = 3 8 (the profit max- imization output when qj = 0.25) and then trigger the non-cooperative phase and gains the Cournot profit of 1/9 per period. Firm i’s gain will be Πv = 9 64 + δπc i + δ2 0.5πc i + · · · = 9 64 + πc i (δ + δ2 + · · ·) = 9 64 + δ 9(1 − δ) . Π∗ − Πv = 1 576 (17δ − 9) 0. Therefore, during the cooperative phase, the best strategy is to continue cooperation. 2. In the non-cooperative phase, the Cournot quantity is the Nash equilibrium quan- tity in each period. The cooperative phase is the equilibrium realization path. The non-cooperative phase is called off-equilibrium subgames.
- 42. 40 6.7.3 Retaliation triggerstrategy A retaliation trigger strategy σRT for firm i has a cooperative phase and a retaliation phase: Cooperative phase– the same as a trigger strategy. Retaliation phase– Once the cooperation phase breaks down, then firm i will choose the retaliation quantity qr i = 1 to make sure P = 0. If both firms choose the same trigger strategy, then they will cooperate forever. If neither one could benefit from changing to a different strategy, then (σRT , σRT ) is a subgame-non-perfect Nash equilbrium. It is not a perfect equilibrium because retali- ation will hurt oneself and is not a credible threat. Formally, a retaliation trigger strategy for firm i is σRT = (σRT 0 , σRT 1 , . . . , σRT t , . . .), σRT 0 = 0.5Qm, σRT t = 0.5Qm if qj(τ) = 0.5Qm ∀ 0 ≤ τ t, j = 1, 2 1 otherwise. 6.7.4 (σRT , σRT ) is a NE for δ 1 9 Proof: At every period t in the cooperative phase, if the opponent does not violate the cooperation, then firm i’s gain to continue cooperation is (same as the trigger strategy case) Π∗ = 1 8(1 − δ) . If firm i chooses to stop the cooperative phase, he will set qt = 3 8 (same as the trigger strategy case) and then trigger the retaliation phase, making 0 profit per period. Firm i’s gain will be Πv = 9 64 ⇒ Π∗ − Πv = 1 8(1 − δ) − 9 64 = 9δ − 1 64(1 − δ) 0. Therefore, during the cooperative phase, the best strategy is to continue cooperation. The retaliation phase is off-equilibrium subgames and never reached. Since the retal- iation strategy is not optimal, the Nash equilibrium is not subgame-perfect. In summary, if 1 δ 9 17 , then the duopoly firms will collude in a SPNE; if 9 17 δ 1 9 , then the duopoly firms will collusion in a non-perfect NE. If δ 1 9 , then collusion is impossible.
- 43. 41 6.7.5 Folk Theoremof the infinite repeated game In the above, we consider only the cooperation to divide the monopoly profit evenly. The same argument works for other kinds of distributions of monopoly profit or even aggregate profits less than the monopoly profit. Folk Theorem: When δ→1, every distributions of profits such that the average payoff per period πi ≥ πc i can be implemented as a SPNE. For subgame-non-perfect NEs, the individual profits can be even lower then the Cournot profit. 6.7.6 Finitely repeated game If the duopoly game is only repeated finite time, t = 1, 2, . . . , T, we can use backward induction to find Subgame-perfect Nash equilibria. Since the last period T is the same as a 1-period duopoly game, both firms will play Cournot equilibrium quantity. Then, since the last period strategy is sure to be the Cournot quantity, it does not affect the choice at t = T − 1, therefore, at t = T − 1 both firms also play Cournot strategy. Similar argument is applied to t = T − 2, t + T − 3, etc., etc. Therefore, the only possible SPNE is that both firms choose Cournot equilibrium quantity from the beginning to the end. However, there may exist subgame-non-perfect NE. 6.7.7 Infinite repeated Bertrand price competition game In the above, we have considered a repeated game of quantity compeition. We can define an infinitely repeated price competition game and a trigger strategy similarily: (1) In the cooperative phase, a firm sets monopoly price and gains one half of the monopoly profit 0.5πm = 1/8. (2) In the non-cooperative phase, a firm sets Bertrand competition price pb = 0 and gains 0 profit. To deviate from the cooperative phase, a firm obtains the whole monopoly profit πm = 1/4 instantly. If 1/[8(1 − δ)] 1/4 (δ 0.5), the trigger strategy is a SPNE. 6.8 Duopoly in International Trade 6.8.1 Reciprocal Dumping in International Trade 2 countries, i = 1, 2 each has a firm (also indexed by i) producing the same product. Assume that MC = 0, but the unit transportation cost is τ. qh i , qf i : quantities produced by country i’s firm and sold in domestic market and for- eign market, respectively. Q1 = qh 1 + qf 2 , Q2 = qh 2 + qf 1 : aggregate quantities sold in countries 1 and 2’s market, respectively.
- 44. 42 Pi = a− bQi: market demand in country i’s market. The profits of the international duopoly firms are Π1 = P1qh 1 + (P2 − τ)qf 1 = [a − b(qh 1 + qf 2 )]qh 1 + [a − (qf 1 + qh 2 ) − τ]qf 1 , Π2 = P2qh 2 + (P1 − τ)qf 2 = [a − b(qh 2 + qf 1 )]qh 2 + [a − (qf 2 + qh 1 ) − τ]qf 2 . Firm i will choose qh i and qf i to maximize Πi. The FOC’s are ∂Πi ∂qh i = a − 2bqh i − bqf j = 0, ∂Πi ∂qf i = a − 2bqf i − bqh j − τ = 0, i = 1, 2. In a symmetric equilibrium qh 1 = qh 2 = qh and qf 1 = qf 2 = qf . In matrix form, the FOC’s become: 2b b b 2b qh qf = a a − τ , ⇒ qh qf = a + τ 3b a − 2τ 3b , Q = qh + qf = 2a − τ 3b P = a + τ 3 . It seems that there is reciprocal dumping: the FOB price of exports P FOB is lower than the domestic price P. PCIF = P = a + τ 3 , PFOB = PCIF − τ = a − 2τ 3 P. However, since PFOB MC = 0, there is no dumping in the MC definition of dump- ing. The comparative statics with respect to τ is ∂qh ∂τ 0, ∂qf ∂τ 0, ∂Q ∂τ 0, ∂P ∂τ 0. In this model, it seems that international trade is a waste of transportation costs and is unnecessary. However, if there is no international competition, each country’s market would become a monopoly. Extensions: 1. 2-stage game. 2. comparison with monopoly. 6.8.2 Preferential Trade Agreement, Trade Creation, Trade Diversion Free Trade Agreement FTA: Free trade among participants. Customs Union CU: FTA plus uniform tariff rates towards non-participants. Common Market CM: CU plus free factor mobility. Consider the apple market in Taiwan.
- 45. 43 Demand: P =a − Q. 2 export countries: America and Japan, PA PJ . At t = 0, Taiwan imposes uniform tariff of $ t per unit. P0 = PA + t, Q0 = a − PA − t, W0 = CS0 + T0 = Q2 0 2 + tQ0 = (a − PA)2 − t2 2 . At t = 1, Taiwan and Japan form FTA, assume PJ PA + t. P1 = PJ, Q1 = a − PJ , W1 = CS1 + T1 = Q2 1 2 = (a − PJ )2 2 . W1 − W0 = 0.5[(a − PJ )2 + t2 − (a − PA)2 ] = 0.5[t2 − (2a − PA − PJ)(PJ − PA)]. Given a and PA, FTA is more advantageous the higher a and the lower PJ . E Q T P d d d d d d d d d d Q0 P0 PA PJ T0 CS0 E Q T P d d d d d d d d d d Q1 P1 CS1 E Q T P d d d d d d d d d d Q0 P0 PA Q1 P1 φ β γ δ W1 − W0 = (φ + β + γ) − (φ + β + δ) = γ − δ. γ: trade creation effect. δ: trade diversion effect. W1 − W0 0 if and only if γ δ. 6.9 Duopoly under Asymmetric Information 6.9.1 Incomplete information game Imperfect information game: Some information sets contain more than one nodes, i.e., at some stages of the game, a player may be uncertain about the consequences of his choices. Incomplete information game: Some players do not completely know the rule of the game. In particular, a player does not know the payoff functions of other players. There are more than one type of a player, whose payoff function depends on his type. The type is known to the player himself but not to other players. There is a prior probability distribution of the type of a player. Bayesian equilibrium (of a static game): Each type of a player is regarded as an independent player. Bayesian perfect equilibrium (of a dynamic game): In a Bayesian equilibrium, players will use Bayes’ law to estimate the posterior distribution of the types of other players.
- 46. 44 6.9.2 Modified chickengame In the chicken dilemma game, assume that there are 2 types of player 2, 2a and 2b, 2a is as before but 2b is different, who prefers NN to SS, SS to being a chicken: 1 S N d d d d ¨ ©2a S N S N 5 5 2 10 10 2 0 0 1 S N d d d d ¨ ©2b S N S N 5 2 2 10 10 0 0 5 Asymmetric information: player 2 knows whether he is 2a or 2b but player 1 does not. However, player 1 knows that Prob[2a] = Prob[2b] =0.5. If we regard 2a and 2b as two different players, the game tree becomes: d d d d d d ¡ ¡ ¡ ¡ ¡ ¡ e e e e e e ¡ ¡ ¡ ¡ ¡ ¡ e e e e e e 0 ¨ ©1 ¨ ©2a ¨ ©2b 5 5 0 2 10 0 10 2 0 0 0 0 5 0 2 2 0 10 10 0 0 0 0 5 1/2 1/2 S N S N S N S N S N S N In this incomplete information game, player 2b has a dominant strategy, N. Hence, it can be reduced to a 2-person game. The pure strategy Bayesian equilibria are SNN and NSN. The mixed strategy equilibrium is different from the ordinary chicken game.
- 47. 45 6.9.3 A duopolymodel with unknown MC 2 firms, 1 and 2, with market demand p = 2 − q1 − q2. MC1 = 1, π1 = q1(1 − q1 − q2). 2 types of firm 2, a and b. MC2a = 1.25, π2a = q2a(0.75 − q1 − q2a). MC2b = 0.75, π2b = q2b(1.25 − q1 − q2b). Asymmetric information: firm 2 knows whether he is 2a or 2b but firm 1 does not. However, firm 1 knows that Prob[2a] = Prob[2b] =0.5. To find the Bayesian equilibrium, we regard the duopoly as a 3-person game with payoff functions: Π1(q1, q2a, q2b) = 0.5q1(1 − q1 − q2a) + 0.5q1(1 − q1 − q2b) Π2a(q1, q2a, q2b) = 0.5q2a(0.75 − q1 − q2a) Π2b(q1, q2a, q2b) = 0.5q2b(1.25 − q1 − q2b) FOC are 0.5(1−2q1−q2a)+0.5(1−2q1−q2b) = 0, 0.5(0.75−q1−2q2a) = 0, 0.5(1.25−q1−2q2b) = 0. 4 1 1 1 2 0 1 0 2 q1 q2a q2b = 2 0.75 1.25 . The Bayesian equilibrium is (q1, q2a, q2b) = ( 1 3 , 5 24 , 11 24 ).
- 48. 46 7 Differentiated ProductsMarkets 7.1 2-Differentiated Products Duopoly 2 Sellers producing differentiated products. p1 = α − βq1 − γq2, p2 = α − βq2 − γq1, β 0, β2 γ2 . In matrix form, p1 p2 = α α − β γ γ β q1 q2 , ⇒ q1 q2 = β γ γ β −1 α α − p1 p2 , = 1 β2 − γ2 α(β − γ) α(β − γ) − β −γ −γ β p1 p2 ≡ a a − b −c −c b p1 p2 , where a ≡ α β + γ , b ≡ β β2 − γ2 , c ≡ γ β2 − γ2 . If γ = 0 (c = 0), the firms are independent monopolists. If 0 γ β (0 c b) the products are substitutes. When β = γ ⇒ p1 = p2, the products are perfect substitutable (homogenous). E γ T β d d d d d d d d d Complemnts Substitutes hom ogeneous Define δ ≡ γ2 β2 , degree of differentiation. If δ (hence γ, c) → 0, products are highly differentiated. If δ → β (hence γ → c), products are highly homogenous. Assume that TCi(qi) = ciqi, i = 1, 2. In a Cournot quantity competition duopoly game, the payoffs are represented as functions of (q1, q2): πc 1(q1, q2) = (p1−c1)q1 = (α−βq1−γq2−c1)q1, πc 2(q1, q2) = (p2−c2)q2 = (α−βq1−γq2−c2)q2. In a Bertrand price competition duopoly game, the payoffs are represented as func- tions of (p1, p2): πb 1(p1, p2) = (p1−c1)q1 = (p1−c1)(a−bp1+cp2), πb 2(p1, p2) = (p2−c2)q2 = (p2−c2)(a−bp2+cp1). It seems that Cournot game and Bertrand game are just a change of variables of each other. However, the Nash equilibrium is totally different. A change of variables of a game also changes its Nash equilibrium.
- 49. 47 7.1.1 Change ofvariables of a game Suppose we have a game in (x1, x2): πa 1 (x1, x2), πa 2 (x1, x2). FOCs of the x-game: ∂πa 1 ∂x1 = 0, ∂πa 2 ∂x2 = 0, ⇒ (xa 1, xa 2). Change of variables: x1 = F(y1, y2), x2 = G(y1, y2). The payoff functions for the y-game is πb 1(y1, y2) = πa 1 (F(y1, y2), G(y1, y2)), πb 2(y1, y2) = πa 2 (F(y1, y2), G(y1, y2)). FOCs of the y-game: ∂πb 1 ∂y1 = 0 = ∂πa 1 ∂x1 ∂F ∂y1 + ∂πa 1 ∂x2 ∂G ∂y1 , ∂πb 2 ∂y2 = 0 = ∂πa 2 ∂x1 ∂F ∂y2 + ∂πa 2 ∂x2 ∂G ∂y2 , ⇒ (yb 1, yb 2). Because the FOCs for the x-game is different from that of the y-game, xa 1 = F(yb 1, yb 2) and xa 2 = G(yb 1, yb 2) in general. Only when ∂F ∂y2 = ∂G ∂y1 = 0 will xa 1 = F(yb 1, yb 2) and xa 2 = G(yb 1, yb 2). In case of the duopoly game, a Bertrand equilibrium is different from a Cournot equilibrium in general. Only when δ = γ = c = 0 will the two equilibria the same. That is, if the two products are independent, the firms are independent monopolies and it does not matter whether we use prices or quantities as the strategical variables. 7.1.2 Quantity Game Assume that c1 = c2 = 0. The payoff functions are πc 1(q1, q2) = (α − βq1 − γq2)q1, πc 2(q1, q2) = (α − βq1 − γq2)q2. The FOC of firm i and its reaction function are α−2βqi −γqj = 0, ⇒ qi = Rc i (qj) = α − γqj 2β = α 2β − γ 2β qj dqi dqj Rc i = − γ 2β = −0.5 √ δ. E qj T qi rr rr rr rr rr Rc i (qj) dqi dqj Rc i = − γ 2β = −0.5 √ δ The larger δ, the steeper the reaction curve. If products are independent, δ = 0, qi is independent of qj.
- 50. 48 In a symmetricequilibrium, q1 = q2 = qc , p1 = p2 = pc , qc = α 2β + γ = α β(2 + √ δ) , pc = α − (β + γ)qc = αβ 2β + γ = α 2 + √ δ , and π1 = π2 = πc , πc = α2 β (2β + γ)2 = α2 β(2 + √ δ)2 . ∂qc ∂δ 0, ∂pc ∂δ 0, ∂πc ∂δ 0. Therefore, when the degree of differentiation increases, qc , pc , and πc will be increased. When δ = 1, it reduces to the homogeneous case. 7.1.3 Price Game Also assume that c1 = c2 = 0. The payoff functions are πb 1(p1, p2) = (a − bp1 + cp2)p1, πb 2(p1, p2) = (a − bp2 + cp1)p2. The FOC of firm i and its reaction function are a − 2bpi + cpj = 0, ⇒ pi = Rb i (pj) = a + cpj 2b = a 2b + c 2b pj dpi dpj Rb i = c 2b = 0.5 √ δ. E pj T pi ¨ ¨¨ ¨¨ ¨¨ ¨¨ ¨ Rb i (pj) dpi dpj Rb i = 0.5 √ δ The larger δ, the steeper the reaction curve. If products are independent, δ = 0, pi is independent of pj. In a symmetric equilibrium, p1 = p2 = pb , q1 = q2 = qb , pb = a 2b + c = a b(2 + √ δ) = α(β − γ) 2β − γ , qb = a − (b − c)pb = ab 2b − c = a 2 − √ δ , and π1 = π2 = πb , πb = a2 b (2b − c)2 = a2 b(2 − √ δ)2 = α2 (β − γ)β (2β − γ)2(β + γ) = α2 (1 − √ δ) β(2 − √ δ)2(1 + √ δ) . ∂pb ∂δ 0, ∂qb ∂δ 0, ∂πb ∂δ 0. Therefore, when the degree of differentiation increases, pb , qb , and πb will be increased. When δ = 1, it reduces to the homogeneous case and pb →0.
- 51. 49 7.1.4 Comparison betweenquantity and price games pc − pb = α 4δ−1 − 1 0, ⇒ qc − qb 0. As δ→0, pc →pb and when δ = 0, pc = pb . Strategic substitutes vs Strategic complements: In a continuous game, if the slopes of the reaction functions are negative as in the Cournot quantity game, the strategic variables (e.g., quantities) are said to be strategic substitutes. If the slopes are positive as in the Bertrand price game, the strategic variables (e.g., prices) are said to be strategic complements. 7.1.5 Sequential moves game The Stackelberg quantity leadership model can be generalized to the differentiated product case. Here we use an example to illustrate the idea. Consider the following Bertrand game: q1 = 168 − 2p1 + p2, q2 = 168 − 2p2 + p1; TC = 0. ⇒ pi = Rb i (qj) = 42 + 0.25pj, pb = 56, qb = 112, πb = 6272. In the sequential game version, assume that firm 1 moves first: π1 = [168 − 2p1 + (42 + 0.25p1)]p1 = [210 − 1.75p1]p1. FOC: 210 − 3.5p1 = 0, ⇒ p1 = 60, p2 = 57, q1 = 105, q2 = 114, π1 = 6300, π2 = 6498. ⇒ p1 p2, q1 q2, π2 π1 πb = 6272. The Cournot game is: p1 = 168 − 2q1 + q2 3 , p2 = 168 − 2q2 + q1 3 ; ⇒ qi = Rc i (qj) = 126 − 0.25qj, q = pc = 100.8, pc = 67.2, πc = 6774. In the sequential game version, assume that firm 1 moves first: π1 = (168 − 2 3 qi − 1 3 qj)qi = (126 − 7 12 qj)qi. FOC: 126 − 7 6 q1 = 0, ⇒ q1 = 108, q2 = 81, p1 = 69, p2 = 78, π1 = 7452, π2 = 6318. ⇒ p1 p2, q1 q2, π1 πc π2. From the above example, we can see that in a quantity game firms prefer to be the leader whereas in a price game they prefer to be the follower.
- 52. 50 7.2 Free entry/exitand LR equilibrium number of firms So far we have assumed that the number of firms is fixed and entry/exit of new/existing firms is impossible. Now let us relax this assumption and consider the case when new/existing firms will enter/exit the industry if they can make a positive/negative profit. The number of firms is now an endogenous variable. Assume that every existing and potential producer produces identical product and has the same cost function. TCi(qi) = F. P = A − Q Assume that there are n firms in the industry. The n-firm oligopoly quantity compe- tition equilibrium is (see 6.1.2) qi = A n + 1 = P, ⇒ πi = A n + 1 2 − F ≡ Π(n). If Π(n + 1) 0, then at least a new firm will enter the industry. If Π(n) 0, then the some of the existing firms will exit. In LR equilibrium, the number of firms n∗ will be such that Π(n∗ ) ≥ 0 ≥ Π(n∗ + 1). Π(n∗ ) = A n + 1 2 − F ≥ 0 ⇒ n∗ = A √ F − 1. E n T Π n∗ n∗ +1 Π(n∗ ) 0 and Π(n∗ + 1) 0. The model above can be modified to consider the case when firms produce differ- entiated products: pi = A−qi−δ j=i qj, πi = A − qi − δ j=i qj qi−F, ⇒ FOC: A−2qi−δ j=i qj = 0, where 0 δ 1. If there are N firms, in Cournot equilibrium, qi = q∗ for all i, and q∗ = p∗ = A 2 + (N − 1)δ , π∗ = A 2 + (N − 1)δ 2 − F. The equilibrium number of firms is N∗ = A δ √ F + 1 − 2 δ = 1 δ A √ F − 1 + 1 − 1 δ .
- 53. 51 When δ→1, itreduces to the homogenous product case. When δ decreases, N∗ increases. 7.3 Monopolistic Competition in Differentiated Products The free entry/exit model of last section assumes that firms compete in a oligopoly market, firms in the market are aware of the co-existent relationship. Now we consider a monopolistic competition market in which each firm regards itself as a monopoly firm. 7.3.1 Chamberlin Model There are many small firms each produces a differentiated product and has a nega- tively sloped demand curve. The demand curve of a typical firm is affected by the number of firms in the industry. If existing firms are making positive profits Π 0, new firms will enter making the demand curve shifts down. Conversely, if existing firms are negative profits Π 0, some of them will exit and the curve shifts up. In the industry equilibrium, firms are making 0-profits, there is no incentive for entry or exit and the number of firms does not change. E q T p AC(Q) d d d d d d d d D Π 0, new firms are entering, D is shifting down. © E q T p AC(Q)d d d d d d D Π 0, some firms are exiting, D is shifting up. E q T p AC(Q) d d d d d d d D In equilibrium, Π = 0, D is tangent to AC. p∗ q∗ 7.3.2 Dixit-Stiglitz Model Dixit-Stiglitz (AER 1977) formulates Chamberlin model. There is a representative consumer who has I dollars to spend on all the brands available. The consumer has a CES utility function U(q1, q2, . . .) = ∞ i=1 (qi)α 1/α , 0 α 1. The consumer’s optimization problem is max N i=1 (qi)α , subject to N i=1 piqi = I, ⇒ L = N i=1 (qi)α + λ(I − N i=1 piqi). FOC is ∂L ∂qi = α(qi)α−1 − λpi = 0 ⇒ qi = λpi α 1 α−1 = I j p −α 1−α j p −1 1−α i = Ap −1 1−α i .
- 54. 52 As N isvery large, ∂A ∂pi ≈ 0 and the demand elasticity is approximately |η| = 1 1 − α . The cost function of producer i is TCi(qi) = F + cqi. max qi piqi − TC(qi) = A1−α qα i − F − cqi. The monopoly profit maximization pricing rule MC = P(1 − 1 |η| ) means: p∗ i = c |η| |η| − 1 = c α , πi = (p∗ i − c)qi − F = c(1 − α) α qi − F. In equilibrium, πi = 0, we have (using the consumer’s budget constraint) q∗ i = αF (1 − α)c , N∗ = I p∗q∗ = (1 − α)I F . The conclusions are 1. p∗ = c/α and Lerner index of each firm is p − c p = 1 − α. 2. Each firm produces q∗ = αF (1 − α)c . 3. N∗ = (1 − α)I F . 4. Variety effect: U∗ = (N∗ q∗α )1/α = N∗1/α q∗ = [αα (1 − α)1−α I]1/α F1− 1 α c−1 . The size of the market is measured by I. When I increases, N∗ (the variety) increases proportionally. However, U∗ increases more than proportionally. Examples: Restaurants, Profesional Base Ball, etc. If we approximate the integer number N by a continuous variable, the utility function and the budget constraint are now U({q(t)}0≤t≤N ) = ∞ 0 [q(t)]α dt 1/α , N 0 p(t)q(t)dt = I. The result is the same as above.
- 55. 53 7.3.3 Intra-industry trade Heckscher-Ohlintrade theory: A country exports products it has comparative ad- vantage over other countries. Intra-industry trade: In real world, we see countries export and import the same products, eg., cars, wines, etc. It seems contradictory to Heckscher-Ohlin’s compar- ative advantage theory. Krugman (JIE 1979): If consumers have preferences for varieties, as in Dixit-Stiglitz monopolistic competition model, intra-industry trade is beneficiary to trading coun- tries. No trade: p∗ = c/α, q∗ = (1 − α)F/(αc), N∗ = (1 − α)I F , U∗ ∝ I1/α . Trade: p∗ = c/α, q∗ = (1 − α)F/(αc), N∗ = (1 − α)2I F , U∗ ∝ (2I)1/α . Gros (1987), with tariff. Chou/Shy (1991), with non-tradable good sector. 7.4 Location Models The models discussed so far assume that the product differentiation is exogenously determined. Location models provide a way to endogenize product differentiation. 7.4.1 Product characteristics ß¹Ô Different products are characterized by different characters. P0, ¯ , æH, ¹”. Firms choose different product characters to differentiated each other. Vertical differentiation: Consumers’ preferences are consistent w.r.t. the differen- tiated character, eg.,. ¹”. (Ch12) Horizontal differentiation: Different consumers have different tastes w.r.t. the dif- ferentiated character, eg., P0, ¯ . To determine the characteristics of a product is to locate the product on the space of product characteristics. ß¹ìP 7.4.2 Hotelling linear city model Assume that consumers in a market are distributed uniformly along a line of length L. 0 L d d A B a L − b r i x Firm A is located at point a, PA is the price of its product.
- 56. 54 Firm B islocated at point L − b, PB is the price of its product. Each point x ∈ [0, L] represents a consumer x. Each consumer demands a unit of the product, either purchases from A or B. Ux = −PA − τ|x − a| if x buys from A. −PB − τ|x − (L − b)| if x buys from B. τ is the transportation cost per unit distance. |x − a| (|x − (L − b)|) is the distance between x and A (B). The marginal consumer ˆx is indifferent between buying from A and from B. The location of ˆx is determined by −PA − τ|x − a| = −PB − τ|x − (L − b)| ⇒ ˆx = L − b + a 2 − PA − PB 2τ . (1) The location of ˆx divids the market into two parts: [0, ˆx) is firm A’s market share and (ˆx, L] is firm B’s market share. 0 L d d a L − b r ˆx A’s share B’s share' E' E Therefore, given (PA, PB), the demand functions of firms A and B are DA(PA, PB) = ˆx = L − b + a 2 − PA − PB 2τ , DB(PA, PB) = L−ˆx = L − a + b 2 + PA − PB 2τ . Assume that firms A and B engage in price competition and that the marginal costs are zero. The payoff functions are ΠA(PA, PB) = PA L − b + a 2 − PA − PB 2τ , ΠB(PA, PB) = PB L − a + b 2 + PA − PB 2τ . The FOCs are (the SOCs are satisfied) L − b + a 2 − 2PA − PB 2τ = 0, L − a + b 2 + PA − 2PB 2τ = 0. (2) The equilibrium is given by PA = τ(3L − b + a) 3 , PB = τ(3L − a + b) 3 , QA = ˆx = 3L − b + a 6 , QB = L−ˆx = 3L − a + b 6 . ΠA = τ(3L − b + a)2 18 = τ(2L + d + 2a)2 18 , ΠB = τ(3L − a + b)2 18 = τ(2L + d + 2b)2 18 , where d ≡ L − b − a is the distance between the locations of A and B. the degree of product differentiation is measured by dτ. When d or τ increases, the products are more differentiated, the competition is less intensive, equilibrium prices are higher and firms are making more profits.
- 57. 55 ¡«Ç¦Uhmú D SOGO¬Áy—Ë. ∂ΠA ∂a 0, ∂ΠB ∂b 0. Moving towards the other firm will increase one’s profits. In approximation, the two firms will end up locating at the mid-point L 2 . In this model the differentiation is minimized in equilibrium. Note: 1. In this model, we can not invert the demand function to define a quan- tity competition game because the Jocobian is singular. 2. The degree of homogeneity δ = γ β is not definable either. Here we use the distance between the locations of A and B as a measure of differentiation. 3. The result that firms in the Hotelling model will choose to minimize product differentiation is so far only an approximation because the location of the marginal consumer ˆx in (7) is not exactly described. It is actually an upper-semi continuous correspondence of (PA, PB). The reaction functions are discontinuous and the price competition equilibrium does not exist when the two firms are too close to each other. See Oz Shy’s Appendix 7.5. Welfare index: aggregate transportation costs Equilibrium transportation cost curve 0 L dd A B ˆx = L/2 d d d d d d d d d d 0 Lˆx = L/2 d d A B L/4 3L/4 d d d d d d d d d d Social optimum transportation cost curve 7.4.3 Digression: the exact profit function In deriving the market demand, we regard the location of the marginal consumer ˆx as the market dividing point. This is correct only when a ˆx L − b. When ˆx ≤ a, all the consumers buy from B, QA = 0 and QB = L. The reason is, if the consumer located at a (the location of firm A) prefers to buy from firm B, since the transportation cost is linear, all consumers to the left of a would also prefer to buy from B. Similarily, when ˆx ≥ L − b, all the consumers buy from A, QA = L and QB = 0.
- 58. 56 0 L d d aL − b Consumers in [0, a] move together. Consumers in [L − b, L] move together. E PA T ˆx L r rr rr rr r 0 PB − (L − a − b)τ PB + (L − a − b)τ L − b a The true profit function of firm A is ΠA(PA, PB) = PA L − b + a 2 − PA − PB 2τ a ˆx L − b 0 ˆx ≤ a PAL L − b ≥ ˆx = PAL PA ≤ PB − dτ PA L − b + a 2 − PA − PB 2τ PB − dτ PA PB + dτ 0 PA ≥ PB + dτ, where d ≡ L − a − b. Consider the case a = b, d = L − 2a. The reaction function of firm A is (firm B’s is similar): PA = RA(PB) = 0.5(τL + PB) (τL+PB)2 8τ[PB−dτ] ≤ L or PB τ /∈ (3L − 4 √ La, 3L + 4 √ La) PB − (L − 2a)τ (τL+PB)2 8τ[PB−dτ] ≥ L or PB τ ∈ [3L − 4 √ La, 3L + 4 √ La] For a ≤ L/4, the reaction functions intersect at PA = PB = τL. When a L/4, the reaction functions do not intersect. E PA T ΠA ¡ ¡ ¡ ¡ ¡ ¡ PB − dτ PB + dτ E PA T PB ¨¨ ¨¨ ¨¨ ¨ ¡ ¡ ¡ ¡ ¡ ¡ ¡ Case: a ≤ L/4, P ∗ = τL, RA RB E PA T PB ¨¨ ¨¨ ¨ ¡ ¡ ¡ ¡ ¡ Case: a L/4, no equilibrium. RA RB 7.4.4 Quadratic transportation costs Suppose now that the transportation cost is proportional to the square of the distance. Ux = −PA − τ(x − a)2 if x buys from A. −PB − τ[x − (L − b)]2 if x buys from A.
- 59. 57 The marginal consumerˆx is defined by −PA − τ(x − a)2 = −PB − τ[x − (L − b)]2 ⇒ ˆx = L − b + a 2 − PA − PB 2τ(L − a − b) . (3) The demand functions of firms A and B are DA(PA, PB) = ˆx = L − b + a 2 − PA − PB 2τ(L − a − b) , DB(PA, PB) = L−ˆx = L − a + b 2 + PA − PB 2τ(L − a − b) . The payoff functions are ΠA(PA, PB) = PA L − b + a 2 − PA − PB 2τ(L − a − b) , ΠB(PA, PB) = PB L − a + b 2 + PA − PB 2τ(L − a − b) . The FOCs are (the SOCs are satisfied) L − b + a 2 − 2PA − PB 2τ(L − a − b) = 0, L − a + b 2 + PA − 2PB 2τ(L − a − b) = 0. (4) The equilibrium is given by PA = τ(3L − b + a)(L − a − b) 3 , QA = ˆx = 3L − b + a 6 , ΠA = τ(3L − b + a)2 (L − a − b) 18 , PB = τ(3L − a + b)(L − a − b) 3 , QB = L−ˆx = 3L − a + b 6 , ΠB = τ(3L − a + b)2 (L − a − b) 18 . ∂ΠA ∂a 0, ∂ΠB ∂b 0. Therefore, both firms will choose to maximize their distance. The result is opposite to the linear transportation case. Two effects of increasing distance: 1. Increase the differentiation and reduce competition. Π ↑. 2. Reduce a firm’s turf. Π ↓. In the linear transportation case, the 2nd effect dominates. In the quadratic trans- portation case, the 1st effect dominates. Also, ˆx is differentiable in the quadratic case and the interior solution to the profit maximization problem is the global maximum. Welfare comparison: Equilibrium transportation cost curve 0 L d d A B ˆx = L/2 0 Lˆx = L/2 d d A B L/4 3L/4 Social optimum transportation cost curve
- 60. 58 7.5 Circular MarketModel It is very difficult to generalize the linear city model to more than 2 firms. Alterna- tively, we can assume that consumers are uniformly distributed on the circumference of a round lake. We assume further that the length of the circumference is L. First, we assume that there are N firms their locations are also uniformly distributed along the circumference. Then we will find the equilibrium number of firms if entry/exit is allowed. Firm i faces two competing neighbers, firms i − 1 and i + 1. TCi(Qi) = F + cQi. d d dr −L/N 0 L/N i − 1 i i + 1 x As in the linear city model, each consumer needs 1 unit of the product. The utility of consumer x, if he buys from firm j, is U(x) = −Pj − τ|x − lj|, j = i − 1, i, i + 1, where lj is the location of firm j. To simplify, let L = 1. Given the prices Pi and Pi−1 = Pi+1 = P, there are two marginal consumers ˆx and −ˆx with ˆx = P − Pi 2τ + 1 2N , ⇒ Qi = 2ˆx = P − Pi τ + 1 N . d d dr r −1/N 0 1/N i − 1 i i + 1 −ˆx ˆx Πi = (Pi − c) P − Pi τ + 1 N − F, ⇒ FOC: c + P − 2Pi τ + 1 N = 0. In equilibrium, Pi = P, P = c + τ N , Π = τ N2 − F. In a free entry/exit long run equilibrium, N is such that Π(N) ≥ 0, Π(N + 1) ≤ 0 ⇒ N∗ = τ F , P∗ = c + √ τF, Q∗ = 1 N . Equilibrium transportation cost curve −1/2N 0 1/2N d d d d d d d d d d Aggregate transportation costs: T(N) = N( 1 2N τ 2N ) = τ 4N . Aggregate Fixed Costs = NF.
- 61. 59 Since each consumerneeds 1 unit, TVC = c is a constant. The total cost to the society is the sum of aggregate transportation cost, TVC, and fixed cost. SC(N) = T(N)+c+NF = τ 4N +c+NF, FOC: − τ 4N2 +F = 0, ⇒ Ns = 0.5 τ F . The conclusion is that the equilibrium number of firms is twice the social optimum number. Remarks: 1. AC is decreasing (DRTS), less firms will save production cost. 2. T(N) is decreasing with N, more firms will save consumers’ transportation cost. 3. The monopolistic competition equilibrium ends up with too many firms. 7.6 Sequential entry in the linear city model If firm 1 chooses his location before firm 2, then clearly firm 1 will choose li = L/2 and firm 2 will choose l2 = L/2+ . If there are more than two firms, the Nash equilibrium location choices become much more complicated. Assume that there are 3 firms and they enter the market (select locations) sequen- tially. That is, firm 1 chooses x1, then firm 2 chooses x2, and finally firm 3 chooses x3. To simplify, we assume that firms charge the same price p = 1 and that x1 = 1/4. We want to find the equilibrium locations x2 and x3. 1. If firm 2 chooses x2 ∈ [0, x1) = [0, 1/4), then firm 3 will choose x3 = x1 + . π2 = (x2 − x1)/2 1/4. 0 x1 = 1/4 1 rd x2 d x3 π2 = (x2 + x1)/2 1/4. 2. If firm 2 chooses x2 ∈ (x1, 3/4) = (1/4, 3/4), then firm 3 will choose x3 = x2 + . π2 = x2 − (x2 + x1)/2 = (x2 − x1)/2 1/4. 3/40 1/4 1 dx1 dx2 dx3 π2 = (x2 − x1)/2 1/4. 3. If firm 2 chooses x2 ∈ [3/4, 1], then firm 3 will choose x3 = (x2 + x1)/2. π2 = 1 − 0.5(x2 + x3)/2 = (15 − 12x2)/16. 3/40 1/4 1 d x1 d x2d x3 π2 = (15 − 12x2)/16. The subgame perfect Nash equilibrium is x2 = 3/4, x3 = 1/2 with π1 = π2 = 3/8 and π3 = 1/4.
- 62. 60 3/41/20 1/4 1 d x1d x2d x3 SPE: π1 = π2 = 3/8, π3 = 1/4. 7.7 Discrete location model Consider now that the consumers are concentrated on two points: 87 96 N0 firm A 87 96 NL firm B transportation cost is T N0 consumers live in city 0 where firm A is located. NL consumers live in city L where firm B is located. The round trip transportation cost from city 0 to city L is T. Given the prices (PA, PB), the utility of a consumer is U0 = −PA −PB − T UL = −PA − T −PB. nA (nB) is the number of firm A’s (firm B’s) consumers. nA = 0 PA PB + T N0 PB − T PA PB + T N0 + NL PA PB + T nB = 0 PB PA + T NL PA − T PB PA + T N0 + NL PB PA + T. Nash Equilibrium: (Pn A, Pn B) such that Pn A maximizes ΠA = PAnA and Pn B maximizes ΠB = PBnB. Proposition: There does not exist a Nash equilibrium. Proof: 1. If Pn A − Pn B T, then ΠA = 0, firm A will reduce PA. Similarily for Pn B − Pn A T. 2. If |Pn A − Pn B| T, then firm A will increase PA. 3. If |Pn A − Pn B| = T, then both firms will reduce their prices. Undercut proof equilibrium (UE):(P u A, Pu B, nu A, nu B) such that 1. Pn A maximizes ΠA subject to ΠB = Pu Bnu B ≥ (N0 + NL)(Pu A − T). 2. Pn B maximizes ΠB subject to ΠA = Pu Anu A ≥ (N0 + NL)(Pu B − T). In choosing PA, firm A believes that if PA is too high, firm B will undercut its price to grab A’s consumers and vice versa. There is a undercut proof equilibrium: nu A = N0, nu B = NL,and (Pu A, Pu B) satisfies Pu BNL = (N0 + NL)(Pu A − T), Pu AN0 = (N0 + NL)(Pu B − T).
- 63. 61 ∆P = Pu B− Pu A = (N0 + NL)(N0 − NL)T N2 0 + N2 L + N0NL , ∆P 0 if N0 NL. In the symmetric case, N0 = NL and Pu A = Pu B = 2T. E PB T PA ¨¨ ¨¨ ¨¨ ¨¨ ¨¨ RA(PB)firm B will undercut firm B will not undercut E PB T PA ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ RB(PA) firm A will not undercut firm A will undercut E PB T PA ¨¨ ¨¨ ¨¨ ¨¨ ¨¨ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ t UE
- 64. 62 8 Concentration, Mergers,and Entry Barriers ß“ÞG: h ‹p; H ¢|; ¼¯9; ¼}j hEß“, A ß“, A¹ß“ Õ2 D‚â5É[; úÕ2 íÄ; 1. òQßã: #„}j, öŽ ¼¯9 2. Š¢H S¦ªÒ®×G¨ 8.1 Concentration Measures We want to define some measures of the degree of concentration of an industry in order to compare different industries or a similar industry in different countries. i = 1, 2, . . . , N, Q = q1 + q2 + · · · + qN . Market shares: si ≡ qi Q × 100%, s1 ≥ s2 ≥ s3 ≥ · · · ≥ sN . I4 ≡ s1 + s2 + s3 + s4, I8 ≡ 8 i=1 si. Herfindahl-Hirshman Index: IHH ≡ N i=1 s2 i . Gini coefficient: G ≡ 1 N N j=1 j i=1 si = 1 N N 1 (N − i)si. Entropy: IE ≡ N 1 si ln si. 8.1.1 Relationship between IHH and Lerner index Consider a quantity competition oligopoly industry. P(Q) = P(q1 + q2 + · · · + qN ) is the market demand. λi is the conjecture variation of firm i, i.e., when firm i increases 1 unit of output, he expects that all other firms together will response by increasing λi units of output. πi = P(Q)qi − ci(qi), Q−i ≡ j=i qj, λi ≡ dQe −i dqi . FOC: ∂Πe i ∂qi = ∂πi dqi + ∂πi dQ−i dQe −i dqi or P + qiP (1 + λi) = ci. Lerner index of firm i, Li: Li ≡ P − ci P = −qiP (1 + λi) P = −QP P qi Q (1 + λi) = 1 QP si(1 + λi). Industry average lerner index L: L ≡ N i=1 siLi = 1 QP N i=1 s2 i (1 + λi) = IHH + i s2 i λi QP .
- 65. 63 If λi =λ for all i, then L = IHH (1 + λ) P Q . The case of collusion λi = Q−i qi : FOC is MR = MCi and Li = L = 1 P Q . 8.2 Mergers Mergers, takeovers, acquisitions, integration. 3 types: Horizontal mergers: between the same industry Vertical mergers: between upstream industry firms and down stream industry firms Conglomerate mergers: other cases. In US economic history there were 4 active periods: 1901: mostly horizontal and vertical 1920, 1968, 1980: other types, mostly influenced by changes in Anti-trust Law. Purpose: (1) reduce competition, (2) IRTS, (3) differences in the prospective of firms between sellers and buyers, (4) managers’ intension to enlarge their own careers, (5) the insterests of the promoters. 8.2.1 Horizontal merger ¯9 → concentration ratio ↑ → competition ↓ → Welfare ↓? Not necessary. If high cost (inefficient) firms are taken-over, efficiency increases. Example: In Cournot duopoly model, assume c1 = 1, c2 = 4, and P = 10 − Q. qc 1 = 4, qc 2 = 1, Pc = 5, πc 1 = 16, πc 2 = 1, CS = 12.5, Wc = 29.5. If firms 1 and 2 merge to become a monopoly, the monopoly would shut down the production of firm 2 so that the MC of the monopoly is c = 1. Qm = 4.5, Pm = 5.5, πm = 20.25, CS = 10.125, W m = 30.375 Wc . Comparison of Welfare: Wm Wc , it is the trade-off between the production effi- ciency and monopoly inefficiency. Comparison of IHH: Ic HH = 6, 800, Im HH = 10, 000. According to Anti-trust law, such a merger is prohibited, but it is good to the society. However, if the market is originally in Bertrand price competition, the conclusion is totally different. In a Bertrand competition market, the industry is efficient.
- 66. 64 8.2.2 Vertical merger AB 1 2 c c e e e e… ¡ ¡ ¡ ¡ ¯9 ⇒ A1 B 2 c If both upstream and downstream industries produce homogeneous products and are Bertrand price competition markets, the merger does not affect anything. Assumption: The upstream was originally in Bertrand competition and the down- stream was in Cournot competition. Downsteam market demand: P = α − q1 − q2. MC1 = c1, MC2 = c2. ⇒ qi = α − 2ci + cj 3 , πi = (α − 2ci + cj)2 9 ; Q = 2α − c1 − c2 3 , P = α−Q = α + c1 + c2 3 . Upstream: Assume MCA = MCB = 0. Bertrand equilibrium: pA = pB = c1 = c2 = 0. Pre-merge Equilibrium: q1 = q2 = α 3 , π1 = π2 = α2 9 , πA = πB = 0; P = α 3 , Q = 2α 3 . Post-merge: Assume that A1 does not sell raw material to 2. B becomes an upstream monopoly. We ignore the fact that 2 is also a downstream monopsony. πB = c2q2 = c2(α − 2c2 + c1) 3 = pB(α − 2pB) 3 , max pB πB ⇒ pB = c2 = α 4 . Post-merge Equilibrium: qA1 = 5α 12 , q2 = α 6 , P = 5α 12 , Q = 7α 12 , πA1 = PqA1 = 25α2 144 , = π2 = α2 36 , πB = PBq2 = α2 24 . The effects of merge: P ↑, q1 ↑, q2 ↓, π2 ↓, πB ↑, πA + π1 ↑, πB + π2 ↓. Firm 2 and consumers are the losers. 8.2.3 merger of firms producing complementary goods Firm X produces PCs and Firm Y produces monitors. A system is S = X + Y. Ps = Px + Py.
- 67. 65 Market demand: Qs= α − Ps = α − Px − Py, Qx = Qy = Qs. Pre-merge: Πx = PxQx = Px(α − Px − Py), Πy = PyQy = Py(α − Px − Py). FOC: α − 2Px − Py = 0, α − Px − 2Py = 0. ⇒ Px = Py = α 3 , Qs = Qx = Qy = α 3 , Πx = Πy = α2 9 . Post-merge: Πs = PsQs = Ps(α − Ps), ⇒ Ps = Qs = α 2 , Πs = α2 4 . The effects of merger: Ps = α 2 Px + Py = 2α 3 , Qs = α 2 Qx = Qy = α 3 , Πs = α2 4 Πx + Πy = 2α2 9 . Therefore, one monopoly is better than two monopolies. Remarks: 1. It is similar to the joint product monopoly situation. When a monopoly produces two complementary goods, the profit percentage should be lower than in- dividual Lerner indices. In this case, a higher Px will reduce the demand for Y and vice versa. After merger, the new firm internalizes these effects. 2. The model here is isomorphic to the Cournot duopoly model with price variables and quantity variables interchanged. 3. If there are 3 products, X, Y, and Z, and only X and Y merge, the welfare is not necessarily improving. 8.3 Entry Barriers h ªpí®‰ Û ¼ (Incumbent) íi‘ == h ªpí®‰ == Äç‚â 1. Economy of scale, large fixed cost 2. Production differentiation advantages (reputation, good will) 3. Consumer loyalty, network externalities 4. Absolute cost advantages (learning experiences) 5. Location advantage (sequential entry) 6. Other advantages Incumbents may also take entry deterrence (ªÒß×) strategies.
- 68. 66 8.3.1 Fixed costand IHH In Dixit-Stiglitz monopolistic competition model, N = (1 − α)I F : IHH = N 100 N 2 = 10, 000 N = F (1 − α) 10, 000, ∂IHH ∂F 0. In quantity competition with free entry/exit model, N ≈ (A − c)2 √ bF : IHH = 10, 000 N = √ bF A − c 10, 000, ∂IHH ∂F 0. In the circular city model, N = τ F , IHH = F τ 10, 000. 8.3.2 Sunk cost Sunk costs: Costs that cannot be reversed. ÇŸ‘, µ‘, _|, firm specific equip- ments, etc. Sunk costs B×, h B.ߪҬÁ Stiglitz (1987): Éb øõ Sunk Cost æÊ, ¹ªUh ¼ (Potential Entrants) ú— .‡, UÛ ¼ (Incumberts) ?./:ƒÖ´‚â A: An Incumbent, B: A Potential entrant. ΠA = Πm − if no entry − B enters, ΠB = 0 do not enter − B enters, where Πm is the monopoly profit and is the sunk cost. d d d B Enter Stay out − − Πm − 0 ∗ Proposition: As long as 0 Πm , there exists only one subgame perfect equilib- rium, i.e., B stays out. Conditions: 1. A and B produce homogeneous product with identical marginal cost. 2. Post-entry market is a Bertrand duopoly. 3. A cannot retreat. If B can invest in product differentiation to avoid homogeneous product Bertrand competition or the post-entry market is a Cournot duopoly, then the proposition is not valid any more.
- 69. 67 If A canresale some of its investments, say, recover φ 0. The game becomes d d d B Enter Stay out d d d A Stay in Exit − − φ − Πm − ∗ Πm − 0 The only subgame-perfect equilibrium is that B enters and A exits. However, the result is just a new monopoly replacing an old one. The sunk cost can be regarded as the entry barrier as before. Notice that there is a non-perfect equilibrium in which A chooses the incredible threat strategy of Stay in and B chooses Stay out. 8.4 Entry Deterrence 8.4.1 Burning one’s bridge strategy (Tirole CH8) Two countries wishing to occupy an island located between their countries and con- nected by a bridge to both. Each army prefers letting its opponent have the island to fighting. Army 1 occupies the island and burns the bridge behind it. This is the paradox of commitment. Army 1 Army 2 87 96 island 8.4.2 Simultaneous vs. Sequential Games Consider a 2-person game: Π1(x1, x2, y1, y2), Π2(x1, x2, y1, y2), where (x1, y1) is firm 1’s strategy variables and (x2, y2) is firm 2’s strategy variables. Simultaneous game: Both firms choose (x, y) simultaneously. FOC: ∂Π1 ∂x1 = ∂Π1 ∂y1 = 0, ∂Π2 ∂x2 = ∂Π2 ∂y2 = 0. Sequential game: In t = 1 both firms choose x1 and x2 simultaneously and then in t = 2 both firms choose y1 and y2 simultaneously. To find a subgame perfect equilibrium, we solve backward:
- 70. 68 In t =2, x1 and x2 are given, the FOC are ∂Π1 ∂y1 = 0 and ∂Π2 ∂y2 = 0, ⇒ y1 = f(x1, x2) and y2 = g(x1, x2). In t = 1, the reduced game is: π1(x1, x2) = Π1(x1, x2, f(x1, x2), g(x1, x2)), π2(x1, x2) = Π2(x1, x2, f(x1, x2), g(x1, x2)). The FOC is ∂π1 ∂x1 = ∂Π1 ∂x1 + ∂Π1 ∂y1 ∂f ∂x1 + ∂Π1 ∂y2 ∂g ∂x1 = 0, and ∂π2 ∂x2 = ∂Π2 ∂x2 + ∂Π2 ∂y1 ∂f ∂x2 + ∂Π2 ∂y2 ∂g ∂x2 = 0. Since ∂Π1 ∂y1 = ∂Π2 ∂y2 = 0, the FOC becomes ∂π1 ∂x1 = ∂Π1 ∂x1 + ∂Π1 ∂y2 ∂g ∂x1 = 0, and ∂π2 ∂x2 = ∂Π2 ∂x2 + ∂Π2 ∂y1 ∂f ∂x2 = 0. Compare the FOC of the sequential game with that of the simultaneous game, we can see that the equilibria are not the same. In t = 1, both firms try to influence the t = 2 decisions of the other firms. For the simultaneous game, there is no such considerations. ∂Π1 ∂y2 ∂g ∂x1 and ∂Π2 ∂y1 ∂f ∂x2 are the strategic consideration terms. Entry deterrence application: In t = 1, only firm 1 exists: Π1(x1, y1, y2), Π2(x1, y1, y2). Firm 1 is the incumbent and tries to influence the entry decision of firm 2. 8.4.3 Spence (1977) entry deterrence model In this model the incumbent attempts to use strategic capacity investment to deter the entry of a potential entrant. t = 1: Firm 1 decides its capacity-output level k1. t = 2: Firm 2 decides its capacity-output level k2. If k2 0, firm 2 enters. If k2 = 0, firm 2 stays out. π1(k1, k2) = k1(1 − k1 − k2), π2(k1, k2) = k2(1 − k1 − k2) − E enter 0 stay out, where E is the fixed cost if firm 2 enters. Back induction: In t = 2, k1 is given. If firm 2 enters, its profit maximization FOC is ∂π2 ∂k2 = 1−k1−2k2 = 0, ⇒ k2 = 1 − k1 2 , π2 = 1 − k1 2 1 − k1 − 1 − k1 2 −E = (1 − k1)2 4 −E.
- 71. 69 If (1 − k1)2 4 −E 0, firm 2 will choose not to enter. Therefore, firm 2’s true reaction function is k2 = R2(k1, E) = 1 − k1 2 if k1 1 − 2 √ E 0 if k1 1 − 2 √ E. E k1 T k2 r rr rr 1 2 1 − 2 √ E k2 = R2(k1; E) In t = 1, firm 1 takes into consideration firm 2’s discontinuous reaction function. If 1 − 2 √ E ≤ 1 2 (⇒ E ≥ 1 16 ), then firm 1 will choose monopoly output k1 = 1 2 and firm 2 will stay out. This is the case of entry blockaded. Next, we consider the case E 1 16 . If firm 1 chooses monopoly output, firm 2 will enter. Firm 1 is considering whether to choose k1 ≥ 1 − 2 √ E to force firm 2 to give up or to choose k1 1 − 2 √ E and maximizes duopoly profit. 1. entry deterrence: If firm 2 stays out (k1 ≥ 1 − 2 √ E and k2 = 0), firm 1 is a monopoly by deterrence: πd 1(E) = max k1≥1−2 √ E k1(1 − k1) = kE(1 − kE), where kE ≡ 1 − 2 √ E. 2. entry accommodate: If firm 2 enters (k1 1 − 2 √ E and k2 0), firm 1 is the leader of the Stackelberg game: πs 1(E) = max k11−2 √ E k1(1−k1−k2) = 1 2 k1(1−k1), ⇒ πs 1 = 1 2 ks (1−ks ) = 1 8 where ks ≡ 1 2 . E k1 T πd 1 πs 1 = 1 8 r kE r r ks E 0.00536 Entry accommodate E k1 T πd 1 πs 1 r kE r r ks 0.00536 E 1 16 Entry deterred E k1 T πd 1 πs 1 r kE r r ks =km E 1 16 Entry blockaded πd 1 (E) = (1 − 2 √ E)[1 − (1 − 2 √ E)] πs 1(E) = 1 8 if E (1 − 1/2)2 16 ≈ 0.00536. In summary, if E 0.0536, firm 1 will accommodate firm 2’s entry; if 0.00536 E 1 16 , firm 1 will choose k1 = kE to deter firm 2; if E F116, firm 1 is a monopoly and firm 2’s entry is blockaded.
- 72. 70 Spence model isbuilt on the so called Bain-Sylos style assumptions: 1. Firm 2 (entrant) believes that firm 1 (incumbent) will produce q1 = k1 after firm 2’s entry is deterred. However, it is not optimal for firm 1 to do so. Therefore, the equilibrium is not subgame perfect. 2. Firm 2 has sunk costs but not firm 1. The model is not symmetrical. 8.4.4 Friedman and Dixit’s criticism 1. Incumbent’s pre-entry investment should have no effects on the post-entry market competition. The post-entry equilibrium should be determined by post-entry market structure. 2. Therefore, firm 2 should not be deterred. 3. Firm 1’s commitment of q1 = k1 is not reliable. Also firm 2 can make commitment to threat firm 1. The first-mover advantage does not necessarily belong to incumbent. 8.4.5 Dixit 1980 If firm 2 is not convinced that k1 = q1 if firm 2 enters, then firm 1 cannot choose kE to deter firm 2’s entry. Consider a 2-period model: t = 1: Firm 1 chooses ¯k. t = 2 (Cournot competition): Firms 1 and 2 determine q1, q2 simultaneously. Firm 1’s MC curve in t = 2 is E q1 T MC1 c ¯k MC1 = 0 if q1 ≤ ¯k c if q1 c In t = 2, Firm 1’s FOC is 1 − 2q1 − q2 = MC1. Its reaction function is q1 = R1(q2) = 1 − q2 2 if q1 ≤ ¯k 1 − c − q2 2 if q1 ¯k = (1 − q2)/2 if q2 ≥ 1 − 2¯k ¯k if 1 − c − 2¯k q2 1 − 2¯k (1 − c − q2)/2 if q2 1 − c − 2¯k. Firm 2’s reaction function R2(q1) = 1 − c − q1 2 has nothing to do with ¯k. Firm 1’s reaction function is affected by ¯k. Therefore, the Cournot equilibrium in t = 2 is affected by ¯k.
- 73. 71 E q2 T q1 d d d d d d d d 1 1−c 2 ¯k è™ú|⇒ E q1 T q2 d d d d d d d d R1(q2)1 1−c 2 ¯k E q1 T q2 d d d d d d d d R1(q2) ˆˆˆˆˆˆˆˆˆˆ R2(q1) r 1 1−c 2 1−c 2 ¯k In t = 1, firm 1 will choose ¯k to affect the Cournot equilibrium in t = 2. There are 3 cases: (1) ¯k ≤ qc ≡ 1 − c 3 : q1 = q2 = qc , π∗ 1 = πc ≡ (1 − c)2 9 , i.e., Cournot equilibrium. (2) qc ¯k ¯q ≡ 1 + c 3 : q1 = ¯k, q2 = 1 − c − ¯k 2 , p = 1 + c − ¯k 2 , π1 = (p − c)q1 = (1 − c − ¯k)¯k 2 . (3) ¯q ≤ ¯k: q1 = ¯q, q2 = 1 − c − ¯q 2 , p = 1 + c − ¯q 2 , π∗ 1 = p¯q − c¯k = (1 − c − ¯q)¯q 2 − c(¯k − ¯q). E q1 T q2 d d d d d d d dd R1(q2) ˆˆˆˆˆˆˆˆˆˆ R2(q1) r 1 1−c 2 1−c 2 ¯k 1−c 3 1−c 3 E q1 T q2 d d d d d d d d R1(q2) ˆˆˆˆˆˆˆˆˆˆ R2(q1) r 1 1−c 2 1−c 2 ¯k E q1 T q2 d d d d d d d d R1(q2) ˆˆˆˆˆˆˆˆˆˆ R2(q1)r 1 1−c−¯q 2 ¯k¯q The reduced profit function of firm 1 is π1(¯k) = (1 − c)2 9 ¯k ≤ qc (1 − c − ¯k)¯k 2 qc ¯k ¯q (1 − c − ¯q)¯q 2 − c(¯k − ¯q) ¯q ≤ ¯k. Let qs 1 be the Stackelberg leadership quantity of firm 1: qs 1 ≡ 1 − c 2 . 1. If c 1 5 , then qs 1 ¯q and firm 1 will choose ¯k∗ = qs 1. 2. If c 1 5 , then qs 1 ¯q and firm 1 will choose ¯k∗ = ¯q.
- 74. 72 E ¯k T π1 d d d qc ¯qqs 1 c 1 5 E ¯k T π1 d d d d d d qc ¯q qs 1 c 1 5 In Stackelberg model, firm 1 can choose any point on firm 2’s reaction curve R2(q1) to maximize π1. In Dixit model, firm 1’s choice is restricted to the section of R2(q1) such that ¯k ∈ [0, ¯q]. When ¯q ≥ qs 1, the result is the same as Stackelberg model. When ¯q qs 1, firm 1 can only choose ¯k = ¯q. In both cases q1 = ¯k. Therefore, firm 1 does not over-invest (choose ¯k q1) to threaten firm 2’s entry. 8.4.6 Capital replacement model of Eaton/Lipsey (1980) It is also possible that an incumbent will replace its capital before the capital is com- plete depreciated as a commitment to discourage the entry of a potential entrant. t = −1, 0, 1, 2, 3, . . .. In each period t, if only one firm has capital, the firm earns monopoly profit H. If both firms have capital, each earns duopoly profit L. Each firm can make investment in each period t by paying F. Denote by Ri t (Ci t ) the profit (cost) of firm i in period t. Πi = ∞ t=0 ρt (Ri t − Ci t ), Ri t = 0 no capital L duopoly H monopoly Ci t = 0 no invest (NI) F invest (INV). Assumption 1: An investment can be used for 2 periods with no residual value left. Assumption 2: 2L F H. Assumption 3: F/H ρ (F − L)/(H − L). Firm i’s strategy in period t is ai t ∈ {NI, INV}. Assume that a1 −1 = INV. We consider only Markov stationary equilibrium. If firm 2 does not exist, firm 1 will choose to invest (INV) in t = 1, 3, 5, . . .. Given the threat of firm 2’s possible entry, in a subgame-perfect equilibrium, firm 1 will invest in every period and firm 2 will not invest forever. The symmetrical SPE strategy is such that an incumbent firm invests and a potential
- 75. 73 entrant does not: ai t= INV if aj t−1 = NI NI otherwise. Proof that the above strategy is optimal if the oppoent plays the same strategy: Π1 = H − F 1 − ρ , Π2 = 0. If firm 1 deviates and chooses a1 0 = NI, Π1 becomes H (H − F)/(1 − ρ) (by Assumption 3), because firm 2 will invest and become the monopoly. If firm 2 deviates and chooses a2 0 = INV, then Π2 = (1 + ρ)(L − F) + ρ2 (H − F) 1 − ρ 0 (also by Assumption 3). 8.4.7 Judo economics ü ªÒé™ ·5− The (inverse) market demand is P = 100 − Q. t = 1: Firm 2 (entrant) determines whether to enter and if enters, its capacity level k and price pe . t = 2: Firm 1 (incumbent) determines its price pI . Assume that firm 1 has unlimited capacity and, if pI = pe , all consumers will purchase from firm 1. qI = 100 − pI pI ≤ pe 100 − k − pI pI pe qe = k pe pI 0 pe ≥ pI Backward induction: At t = 2, (k, pe ) is given. 1. If firm 1 decides to deter entry, he chooses pI = pe and πI D = pe (100 − pe ). 2. If firm 1 decides to accommodate firm 2, he chooses to maximize πI A = pI (100 − k − pI ). max pI pe pI (100−k−pI ), ⇒ FOC: 0 = 100−k−2pI , ⇒ pI A = 100 − k 2 = qI A ⇒ πI A = (100 − k)2 4 . Firm 1 will accommodate firm 2 if πI A πI D, or if (100 − k)2 4 ≥ pe (100 − pe ), whence πe = pe k 0. If firm 2 chooses a small k and a large enough pe , firm 1 will accommodate. E k T πI πI A πI D pe (100 − pe ) ' EAccommodate
- 76. 74 At t =1, firm 2 will choose (k, pe ) such that max pe k subject to (100 − k)2 4 ≥ pe (100 − pe ). 8.4.8 Credible spatial preemption #WªÒ2 Suppose that the incumbent is a monopoly in two markets selling substitute products j = 1, 2. If an entrant enters into one (say, product 1) of the two markets, the in- cumbent will give up the market in order to protect the monopoly profit of the other market (product 2). Reason: If the incumbent stays in market 1, the Bertrand competition will force p1 down to its marginal cost. As discussed in the monopoly chapter, the demand of product 2 will be reduced. Example: Suppose that firm 1 has a Chinese restaurant C and a Japanese restaurant J in a small town, both are monopoly. There are two consumers (assumed to be price takers), c and j. Uc = β − PC dine at C β − λ − PJ dine at J, Uj = β − λ − PC dine at C β − PJ dine at J, β λ 0, β is the utility of dinner and λ is the disutility if one goes to a less preferred restau- rant. Monopoly equilibrium: PC = PJ = β, π1 = 2β. Suppose now that firm 2 opens a Chinese restaurant in the same town. 1. If firm 1 does not close its Chinese restaurant, the equilibrium will be P C = 0, PJ = λ, π1 = λ, π2 = 0. 2. If firm 1 closes its Chinese restaurant, the equilibrium will be P C = β = PJ , π1 = β = π2. The conclusion is that firm 1 will close its Chinese restaurant. 8.5 Contestable Market of Baumol/Panzar/Willig (1982) Deregulation trend in US in later 1970s. Airline industries: Each line is an individual industry. Contestable market: In certain industries entry does not require any sunk cost. In- cumbent firms are constantly faced by threats of hit-and-run entry and hence behave like competitive firms (making normal profits). Assumption: Potential entrants and an incumbent produce a homogenous product and have the same cost function TC(qi) = F + cqi. The market demand is p = a − Q.
- 77. 75 An industry configuration:(pI , qI ). Feasibility: (1) pI = a − qI . (2) πI = pI qI − (F + cqI ) ≥ 0. Sustainability: ∃(pe , qe ) such that pe ≤ pI , qe ≤ a − pe , πe = pe qe − (F + cqe ) 0. A contestable-market equilibrium: A feasible, sustainable configuration. E q T p d d d d d d d d d d pI r qI p = a − q ATC = F q − c Extension: 1. More incumbent firms. 2. More than 1 products. Comments: 1. If there are sunk costs, the conclusions would be reversed. See Stiglitz (1987) discussed before. 2. If incumbents can respond to hit-and-run entries, they can still make some positive profits. 8.6 A Taxonomy (}é¶) of Business Strategies Bulow, Geanakoplos, and Klemperer (1985), “Multimarket Oligopoly: Strategic Sub- stitutes and Complements,” JPE. A 2-period model: Firm 1 chooses K1 at t = 1 and firms 1 and 2 choose x1, x2 simultaneously at t = 2. Π1 = Π1 (K1, x1, x2), Π2 = Π2 (K1, x1, x2). The reduced payoff functions at t = 1 are Π1 = Π1 (K1, x∗ 1(K1), x∗ 2(K1)), Π2 = Π2(K1, x∗ 1(K1), x∗ 2(K1), where x∗ 1(K1) and x∗ 2(K1) are the NE at t = 2, given K1, ∂Π1 dx1 = ∂Π2 dx2 = 0. top dog (»−, _5T‘): be big or strong to look tough or aggressive. puppy dog ({ü, Qm‘): be small or weak to look soft or inoffensive. lean and hungry look (_|Ï.ñ÷.§, ü−): be small or weak to look tough or aggressive. fat cat (]ý×j): be big or strong to look soft or inoffensive.
- 78. 76 8.6.1 Deterence case Iffirm 1 decides to deter firm 2, firm 1 will choose K1 to make Π2 = Π2 (K1, x∗ 1(K1), x∗ 2(K1)) = 0. dΠ2 dK1 = ∂Π2 ∂K1 + ∂Π2 ∂x1 dx∗ 1 dK1 . Assume that ∂Π2 /∂K1 = 0 so that only strategic effect exists. dΠ2 /dK1 = ∂Π2 ∂x1 dx∗ 1 dK1 0: The investment makes firm 1 tough. dΠ2 /dK1 = ∂Π2 ∂x1 dx∗ 1 dK1 0: The investment makes firm 1 soft. To deter entry, firm 1 will overinvest in case of tough investment (top dog strat- egy _5T‘íõÆ%, entrant øªVÿbç.) and underinvest in case of soft investment (lean and hungry look _|Ï.ñ÷.§ íšä, U entrant JÑÌ‚ªÇ). 8.6.2 Accommodation case If firm 1 decides to accommodate firm 2, firm 1 considers maximizing Π1 = Π1 (K1, x∗ 1(K1), x∗ 2(K1)). dΠ1 dK1 = ∂Π1 ∂K1 + ∂Π1 ∂x2 dx∗ 2 dK1 = ∂Π1 ∂K1 + ∂Π1 ∂x2 dx∗ 2 dx1 dx∗ 1 dK1 . Assume that ∂Π1 /∂x2 and ∂Π2 /∂x1 have the same sign and that ∂Π2 /∂K1 = 0. sign ∂Π1 ∂x2 dx∗ 2 dx1 dx∗ 1 dK1 = sign ∂Π2 ∂x1 dx∗ 1 dK1 × sign(R2). There are 4 cases: 1. Tough investment ∂Π2 ∂x1 dx∗ 1 dK1 0 with negative R2: “top dog” strategy, be big or strong to look tough or aggressive. _5T‘J9„ entrant −‘. 2. Tough investment with positive R2: “puppy dog” strategy, be small or weak to look soft or inoffensive. Qm‘, .bK@ entrant Jnù–¬Á. 3. Soft investment ∂Π2 ∂x1 dx∗ 1 dK1 0 with negative R2: “lean and hungry look” strat- egy, be small or weak to look tough or aggressive. _|Ï.ñ÷.§íšäJH U entrant =−. 4. Soft investment with positive R2: “fat cat” strategy, be big or strong to look soft or inoffensive. ]ý×jJî¸ entrant ı/. top dog: ı%v9Ê,Þí%, Xº6. puppy: AŠí*üä, IAnÀíÄ/A. fat cat: (ö5) ½bí’ŒA, À3. ‹ ‘5×A. A )ícA.
- 79. 77 8.7 Limit Pricingas Cost Signaling, Milgrom/Roberts (1982) Incumbent àQgµIV[ýAÐÑQA…, ò^05 ¼, ñíu ® a potential entrant. 8.7.1 Assumptions of the model t = 1, 2. Each period’s demand is P = 10 − Q. Firm 1 is the incumbent, a monopoly in t = 1. Firm 2 decides whether to enter in t = 2. If firm 2 enters, the market becomes Cournot competition. c2 = 1, F2 = 9, i.e., TC2(q2) = 9 + q2 if q2 0 c1 = 0 with 50% probability and c1 = 4 with 50% probability. Firm 1 knows whether c1 = 0 or c1 = 4 but firm 2 does not. It is an incomplete information game: Firm 1 knows both its and firm 2’s payoff functions but firm 2 knows only its own payoff functions. 8.7.2 Complete information case If there is no uncertainty, firm 1 will choose the monopoly quantity at t = 1, i.e., q1(c1 = 0) = 5, p1(c1 = 0) = 5, π1(c1 = 0) = 25; q1(c1 = 4) = 3, p1(c1 = 4) = 7, π1(c1 = 0) = 9. At t = 2, the duopoly equilibrium for firm 1 with c1 = 0 and firm 2 (c2 = 1) is p = 11/3, q1(c1 = 0) = 11/3, q2 = 8/3, π1(c1 = 0) = 121/9, π2 = 64 9 − 9 = −17/9. The duopoly equilibrium for firm 1 with c1 = 4 and firm 2 (c2 = 1) is p = 5, q1(c1 = 4) = 1, q2 = 4, π1(c1 = 4) = 1, π2 = 16 − 9 = 7. Therefore, firm 2 will enter if c1 = 4 and not enter if c1 = 0. The high cost firm 1 has incentives to confuse firm 2. 8.7.3 The case when firm 2 has no information If firm 2 does not know the type of firm 1, firm 2’s expected profit when enters is 0.5(7) + 0.5(−17/9) = 23/9 0. Therefore firm 2 will choose to enter. Firm 1 has incentives to let firm 2 know firm 1’s type. 8.7.4 Separating equilibrium Firm 2 will use the monopoy price at t = 1 to make inference about the type of firm 1, i.e., firm 2 will not enter if p = 5 will enter if p = 7. However, this is not an equilibrium. If the low cost firm 1 chooses the monopoly quantity q1(c1 = 0) = 5, the high cost firm 1 will have incentives to imitate. To avoid being imitated, the low cost firm 1 will choose a lower price p = 4.17 (q1 = 5.83). In such case, the high cost firm 1 cannot gain by imitating because the gain at
- 80. 78 t = 2is 9 − 1 = 8 whereas the lose at t = 1 due to imitation is 9 − 5.83(4.17 − 4) ≈ 8. The separating equilbrium is as follows. At t = 1, q1(c1 = 0) = 5.83, p1(c1 = 0) = 4.17, π1(c1 = 0) = 24.31; q1(c1 = 4) = 3, p1(c1 = 4) = 7, π1(c1 = 0) = 9. At t = 2, firm 2 will enter only if p1 4.17. 8.7.5 Pooling equilibrium If the standard distribution of c1 is smaller, a separating equilibrium will not exist. Instead, the high cost firm 1 will immitate the low cost firm 1 and firm 2 will not enter. For example: Prob[c1 = 0] = 0.8 and Prob[c1 = 4] = 0.2. 8.7.6 A finite version Assume that p1 1 ∈ {7, 5, 4}, the game tree is d d d$$$$$$$$$$$$ $$$$$$$$$$$$ ˆˆˆˆˆˆˆˆˆˆˆˆ ˆˆˆˆˆˆˆˆˆˆˆˆ ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e ¡ ¡ ¡ e e e 0 1a 1b ¨ ©2 ¨ ©2 ¨ ©2 p 1-p p1 1 = 7 p1 1 = 5 p1 1 = 4 E N E N E N E N E N E N 0 10 7 0 18 0 34 0 −1.9 46 0 0 0 6 7 0 14 0 38 0 −1.9 50 0 0 0 1 7 0 9 0 37 0 −1.9 49 0 0 There is a separating equilbrium p1 1a = 4, p1 1b = 7 and firm 2 chooses not to en- ter (N) if p1 1 = 4 and enter (E) if p1 1 ∈ {7, 5}. For p 7/8.9, there is also a pooling equilbrium p1 1a = p1 1b = 5 and firm 2 chooses not to enter (N) if p1 1 ∈ {5, 4} and enter (E) if p1 1 = 7.
- 81. 79 8.8 Chain-Store Game Selton(1978), “The chain-store paradox,” Theory and Decision. A single long-run incumbent firm (I) faces potential entry by a series of short-run firms, each of which plays only once but observes all previous play. Each period, a potentail entrant (E) decides whether to enter (e) or stay out (s) of a single market. If s, I enjoys a monopoly in that market; if e, I must choose whether to fight (F) or to accommodate (A). d d d E e s d d d I F A −1 −1 0 b a 0 Single period game In the 1-period game, the only SPNE is {A, e}. In the T-period game, using backward induction, the only SPNE is also {A, e} in each period. In the ∞-period game, there is a SPNE such that {F, s} is chosen in each period. ¨¨ ¨¨ ¨¨ ¨¨ ¨¨ ¨ r rr rr r rr rr rr rr rr r d d d d d d d d d d d d d d d d d d E1 E2F E2s E2A I1 I2F I2A I2s e s F A e s e s e s F A F A F A −2 −1 −1 −1 −1 b −1 b −1 0 b b a−1 0 −1 a 0 b a−1 −1 0 a b 0 2a 0 0 2-period game Paradox: When T is large, the incumbent is tempered to fight to try to deter entry. The SPNE above is counterintuitive. 8.8.1 Incomplete information and reputation Kreps and Wilson (1982), “Reputation and imperfect information,” JET. Milgrom and Roberts (1982), “Predation, reputation and entry deterrence,” JET. With probability p0 the incumbent is “tough”, ie., will fight for sure. With probability q0 an entrant is “tough”, ie., will enter for sure. The events that entrants being “taugh” are independent. δ: the discount factor. Since the strategies of both “tough” type I and E are given, we need only analyze the equilibrium strategies of “normal” type.
- 82. 80 T = 1:It is easily shown that the NE, {µ∗ (I), µ∗ (E)}, is µ∗ (I) = a, µ∗ (E) = e if p0 b 1 + b ≡ ¯p s if p0 ¯p. T = 2: There are three cases. (1) q0 aδ − 1 aδ ≡ ¯q: It is not worthwhile for I to fight to deter entry: µ∗ (I1) = µ∗ (I2) = a, µ∗ (E1) = µ∗ (E2s) = e if p0 ¯p s if p0 ¯p, µ∗ (E2e) = e if I accommodates at t = 1 s if I fights at t = 1. (2) q0 ¯q and p0 ¯p: Fighting will deter entry and µ∗ (I1) = F. (3) q0 ¯q and p0 ¯p: Both fighting and accommodating are not equilibrium. I will randomize. Let β ≡ Prob[F]. β is such that E2’s posterior probability that I is “tough” equals ¯q: Prob[“tough”|F] = p0 p0 + β(1 − p0) = ¯q ⇒ β = p0 (1 − p0)b . The total probability of fighting at t = 1 for E1 is p0 + (1 − p0 )β = p0 (b + 1)/b. Therefore, E1 will enter if p0 ¯p2 and stay out otherwise. T = 3: (a) p0 ¯p2 , I will fight and E1 will stay out at t = 1. (b) ¯p3 p0 ¯p2 , I will randomize at t = 1. (c) p0 ¯p3 , I will accommodate and E1 will enter t = 1. T 3: Entrants will stay out until t = k such that p0 ¯pk . When δ = 1: (a) q0 a/(1 + a), I will accommodate at first entry and reveal its type. Hence, limT →∞ π/T = 0. (b) q0 a/(1 + a), there exists an n(p0 ) such that I will fight until there are no more than n(p0 ) entrants remaining. Hence, limT →∞ π/T = (1 − q0 )a − q0 .
- 83. 81 9 Research andDevelopment (RD) RD«àß“ÞG|—Ë 1. hÞßj¶Z‰ ¼A…!Z, Uß“ Ò2 0½h|c 2. h߹Ljhä 3. RD …™Ñ ¼¬ÁµI 4. xXØà½æ 5. RD, _, D%Èê 6. RD D Merger activities 5É[ RD2Гç5ª0 (OECD 1980): NØ 23%, lÂœ 18%, Úä 10%, »“ 9%, ë¹, ˘“, ˆ, ]E, ×- 1% Production and cost functions are black boxes created by economists. Investigat- ing RD processes helps us to open the boxes. Process innovation: An innovation that reduces the production cost of a prod- uct. Product innovation: An innovation that creats a new product. The distinction is not essential. A process innovation can be treated as the creation of new intermediate products that reduce the production costs. On the other hand, a product innovation can be regarded as an innovation that reduces the production cost of a product from infinity to a finite value. 9.1 Classification of Process Innovations Consider a Bertrand competition industry. Inverse demand function: P = a − Q. In the beginning, all firms have the same technology and P0 = C0. Suppose that an inventor innovates a new production procedure so that the marginal production cost reduces to c c0. E Q T P d d d d d d d d d r P0 = c0 Q0 c E Q T P d d d d d d d d d r c Pm(c) Qm
- 84. 82 2 Cases: Drastic innovation(large or major innovation): If Pm(c) ≡ a + c 2 c0 = P0, then the innovator will become a monopoly. Non-drastic innovation (small or minor innovation): If Pm(c) ≡ a + c 2 c0 = P0, then the innovator cannot charge monopoly price and has to set P = c0 − . E Q T P d d d d d d d d d r c P = Pm(c) Qm c0 r π Drastic innovation E Q T P d d d d d d d d d r c Pm(c) Q P = c0 r π Non-drastic innovation A drastic innovation will reduce the market price. A non-drastic innovation will not change the Bertrand equilibrium price. In both cases the innovator makes positive profits. 9.2 Innovation Race êp (innovation): …V.æÊí¼¹, %â RD z…“¨|V _ (immitation): y¹Aíêp; ¦Â%âL²«˙ (backward engineering) ½µAíêp (duplication): .ø85-, %â RD ½µAíêp Patent right (ù‚ž): á¤#êpð5Öðù“ž; . _C.ø8½µ, ·uù ‚ž ù‚}é: hß¹, hj¶, hA}, hql ¦)ù‚5‘K: à (Usefulness), µÆ4 (non-triviality), hJ4 (novelty) Oubçt .?¦)ù‚ ù‚cf: ÑfnAíù‚, êpð.âS¦híûê˜( Innovation race (ù‚¬ˇ): ÖPêpð¬óûêÁ¦/ø¼¹5ù‚ž |lêp6¦) ù‚, Or(6ªS¦ù‚cfG¨ûê Hß¹ .¬|lêp6¦Â} œÖ5¾‘6 y (Consumer Loyalty) ½æ: u´}¨A¬ ¬Á, ¨‘ØÖ5ûê’Ä? Assumptions: 1. 2 firms compete to innovate a product. 2. The value of the patent right to the product is $ V.
- 85. 83 3. To compete,each firm has to spend $ I to establish a research lab. 4. The probability of firm i innovating the product is α. The events of firms being successful is independent. 5. If only one firm successes, the firm gains $ V. If both success, each gains 0.5 V. If a firm fails, it gains 0. 6. The entry is sequential. Firm 1 decides first and then firm 2 makes decision. 9.2.1 Market equilibrium Eπk(n): The expected profit of firm k if there are n firms competing. ik: The investment expenditure of firm k, ik ∈ {0, I}. n = 1: Eπ1(1) = αV − I ⇒ i1 = I if αV ≥ I 0 if αV I. n = 2: Eπ2(2) = α(2 − α) 2 V − I ⇒ i2 = I if α(2 − α) 2 V ≥ I 0 if α(2 − α) 2 V I. E α T I/V Eπ1(1) = 0 or I V = α Eπ2(2) = 0 or I V = α(2 − α) 2 1 firm 2 firms 9.2.2 Social Optimal If there are more firms, the probability of success is higher; On the other hand the investment expenditure will be also higher. Eπs (n): The expected social welfare if there are n firms attempting. Eπs (1) = Eπ1(1) = αV − I Eπs (2) = 2α(1 − α)V + α2 V − 2I, Eπs (2) ≥ Eπs (1) if and only if α(1 − α) ≥ I.
- 86. 84 E α T I/V Eπs (1) =0 or I V = α Eπs (1) = Eπs (2) or I V = α(1 − α) 1 firm 2 firms E α T I/V (I) (II) (III) (I): Social optimal is 1 firm, the same as market equilibrium number of firms. (II): Social optimal is 1 firm, market equilibrium has 2 firms. (III): Social optimal is 2 firms, the same as market equilibrium number of firms. Area (II) represents the market inefficient area. 9.2.3 Expected date of discovery Suppose that the RD race will continue until the discovery. ET(n): Expected date of discovery if there are n firms. ET(1) = α + (1 − α)α2 + (1 − α)2 α3 + . . . = α ∞ t=0 (1 − α)t−1 t = α [1 − (1 − α)]2 = 1 α . ET(2) = α(2−α)+(1−α)2 α(2−α)2+. . . = α(2−α) ∞ t=0 (1−α)2(t−1) t = 1 α(2 − α) ET(1). 9.3 Cooperation in RD Firms’ cooperation in price setting is against anti-trust law. Cooperation in RD activities usually is not illegal. Therefore, firms might use cooperation in RD as a substitute for cooperation in price setting. In this subsection we investigate the effects of firms’ cooperation in RD on social welfare. A 2-stage duopoly game with RD t = 1: Both firms decide RD levels, x1 and x2, simultaneously. t = 2: Both firms engage in Cournot quantity competition. Market demend is P = 100 − Q. Firm i’s RD cost: TCi(xi) = x2 i /2. Firm i’s unit production cost: ci(xi, xj) = 50 − xi − βxj. β 0; if β 0, it represents the spillover effect of RD; if β 0, it is the interference effect.
- 87. 85 9.3.1 Noncooperative RDequilibrium When firms do not cooperate in RD, they decide the RD levels independent of each other. We solve the model backward. At t = 2, x1 and x2 are determined. The Cournot equilibrium is such that Πi(ci, cj) = (100 − 2ci + cj)2 9 − TCi(xi). Substituting the unit cost functions, we obtain the reduced profit function of t = 1: Πi(xi, xj) = [100 − 2(50 − xi − βxj) + (50 − 2xj − βxi)]2 9 − x2 i 2 = [50 + (2 − β)xi + (2β − 1)xj]2 9 − x2 i 2 . At t = 1, firm i chooses xi to maximize Πi(xi, xj). FOC is ∂Πi ∂xi = 0 = 2(2 − β)[50 + (2 − β)xi + (2β − 1)xj] 9 − xi. In a symmetric equilibrium, xi = xj = xnc : x1 = x2 = xnc = 50(2 − β) 4.5 − (2 − β)(1 + β) , c1 = c2 = 50[4.5 − 2(2 − β)(1 + β)] 4.5 − (2 − β)(1 + β) , Pnc − cnc = Qnc = 75 4.5 − (2 − β)(1 + β) , Π1 = Π2 = Πnc = 252 [9 − 2(2 − β)] [4.5 − (2 − β)(1 + β)]2 . 9.3.2 Cooperative RD equilibrium When firms cooperate in RD, they choose x1 = x2 = x so that Πi = Πj = Π(x) = [50 + (1 + β)x]2 9 − x2 2 . Then they decide the level of x to maximize Π(x). FOC is ∂Π ∂x = 0 = 2(1 + β)[50 + (1 + β)x] 9 − x. Denote by xc the optimal level of x, x1 = x2 = xc = 50(1 + β) 4.5 − (1 + β)2 , c1 = c2 = 50[4.5 − 2(1 + β)2 ] 4.5 − (1 + β)2 , Pc − cc = Qc = 75 4.5 − (1 + β)2 , Π1 = Π2 = Πc = 252 [9 − 2(1 + β)2 ] [4.5 − (1 + β)2]2 . Conclusions:
- 88. 86 1. Πc Πnc . 2.If β 0.5 then xc xnc and Qc Qnc . 3. If β 0.5 then xc xnc and Qc Qnc . When β 0.5, consumers will be better off to allow RD cooperation; social welfare will definitely increase. When β 0.5, consumers will be worse off to allow RD cooperation; but the social welfare also depends on the change in firms’ profits. 9.4 Patents êp5òQgM: “¨Þß6‚⣾‘6”ì êp5ÈQgM: óêhêp, ªœ.q©¾ ù‚ž: þ}#êpð5Ñ{, àJ2¥“h ù‚¨AÖ´, Ou³ ù‚„ †³ —DÓÄV2¥êp Êù‚„ |Û5‡, êpðÉ?àòíj¶VˆAÐíž‚ 9õ,, ÛHíêpð? àòíj¶V¦)ªù‚yÅíÖ´‚â Wà, Stradivarius Violin, Coca Cola da Vinci †Ñ_AE 7êp ù‚žÅ : 1Å 17 Ä, r¹ 20 Ä, «É 20 Ä žDù‚žÅ .° bçt .?C~ù‚, Oªò Ú7,ñ˘k ž ½æ: ù‚žÅ bÖýn?Ê2¥“hDÖ´’Ä…Ã5Ȧ)|_~¬? 9.4.1 Nordhous 1969 partial equilibrium model P = a − Q: Demand function of a Bertrand competition market. c: Unit production cost before RD. x: RD magnitude. TC(x) = x2 /2: RD expenditure. c − x: Unit production cost of the innovator after RD. Assume that the innovation is non-drastic. E Q T P d d d d d d d d d d d c c−x a−c a−c+x M(x) DL(x) M(x) = x(a − c) DL(x) = x2 /2 T: Patent length (duration). M(x): Innovator’s expected profit per period during periods T = 1, 2, . . . , T. DL(x): Deadweight Loss due to monopoly (Bertrand competition). ρ = 1/(1 + r): discount factor. (r is market interest rate.)
- 89. 87 Innovator’s problem: max x π(x :T) = T t=1 ρt−1 M(x)−TC(x) = 1 − ρT 1 − ρ M(x)−TC(x) = 1 − ρT 1 − ρ (a−c)x− x2 2 . FOC: 1 − ρT 1 − ρ (a − c) − x ⇒ x = 1 − ρT 1 − ρ (a − c). Comparative statics: ∂x ∂T 0, ∂x ∂a 0, ∂x ∂c 0, ∂x ∂ρ 0. Social optimal duration of patents: W(T) = T t=1 ρt−1 M(x) + ∞ t=T +1 ρt−1 DL(x) − x2 2 = (a − c)x 1 − ρ − 1 − ρT 1 − ρ x2 2 . max x,T (a − c)x 1 − ρ − 1 − ρ2 1 − ρ x2 2 subject to x = 1 − ρT 1 − ρ (a − c). Eliminating x: max T (a − c) 1 − ρ 1 − ρT 1 − ρ (a−c)− 1 − ρ2 1 − ρ 1 2 1 − ρT 1 − ρ (a − c) 2 = a − c 1 − ρ 2 1 − ρT − (1 − ρT )3 2(1 − ρ) . Make change of variable z ≡ 1 − ρT , or T = ln(1 − z) ln ρ . The problem becomes max z z − z3 2(1 − ρ) FOC: 1 − 3z2 2(1 − ρ) = 0, ⇒ z∗ = 2(1 − ρ)/3 ⇒ T∗ = ln(1 − 2(1 − ρ)/3) ln ρ . 9.4.2 General equilibrium models K. Judd (1985) “On performance of patent,” Econometrica is a general equilibrium model. His conclusion is that T∗ = ∞: 1. All products are monopoly priced with the same mark-up ratio and therefore there is no price distortion. 2. The RD costs of an innovation should be paid by all consumers benefited from it to avoid intertemporal allocation distortion. Therefore, infinite duration of patents is needed. C. Chou and O. Shy (1991) “Optimal duration of patents,” Southern Economic Jour- nal: If RD has DRTS, optimal duration of patents may be finite. There are also many nonsymmetrical factors, eg., some products are competitively priced, demand elasticities are different, etc.
- 90. 88 9.5 Licencing ù‚¤ž Morethat 80% of innovators licence their patents to other firms to collect licencing fees rather than produce products and make monopoly profits. Kamien 1992 Consider a Cournot duopoly market with demand P = a − Q. Firm 1 invents a new procedure to reduce the unit production cost from c to c1 = c−x. Firm 2’s unit cost is c2 = c if no licencing. 9.5.1 Equilibrium without licencing q1 = a − c + 2x 3 , q2 = a − c − x 3 , P = a + 2c − x 3 , π1 = (a − c + 2x)2 9 , π2 = ¯π2 = (a − c − x)2 9 . 9.5.2 Equilibrium with per-unit fee licencing Firm 1 can make more profit by licencing the new procedure to firm 2 and changing per-unit fee for every unit sold by firm 2. The maximum fee is φ = c2 − c1 = x. Firm 2’s total cost per unit is still c2 (= c1 + x). Therefore, the equilibrium is the same as without licencing except that firm 1 now collects x dollars per unit of q2: πφ 1 = (a − c + 2x)2 9 + (a − c − x)x 3 , π2 = ¯π2 = (a − c − x)2 9 . 9.5.3 Equilibrium with fixed-fee licencing Firm 1 can also choose to charge firm 2 a fixed amount of money F, independent of q2. Firm 2’s total cost per unit becomes c2 = c1 = c − x. Therefore, the equilibrium is now qF 1 = qF 2 = a − c + x 3 , PF = a + 2c − 2x 3 , πF 1 = (a − c + x)2 9 +F, πF 2 = (a − c + x)2 9 −F. The (maximum) F is such that πF 2 = ¯π2. Therefore F = (a − c + x)2 9 − (a − c − x)2 9 = (a − c)4x 9 , ⇒ πF 1 = (a − c + x)2 9 + (a − c)4x 9 . 9.5.4 Comparison between πφ 1 and πF 1 9 × (πφ 1 − πF 1 ) = (a − c)x 0. Therefore, firm 1 will prefer per-unit fee licencing. The reason is: In the case of fixed-fee licencing, q1 + q2 ↑ and P ↓ and therefore firm 1’s total profit is smaller than that of per-unit fee licencing.
- 91. 89 9.6 Governments andInternational RD Race 9.6.1 Subsidizing new product development Sometimes governmental subsidies can have very substantial strategical effects. Krugman (1986), Strategical Trade Policy and the New International Economics. Boeing (I, a US firm) and Airbus (II, an EU firm) are considering whether to develop super-large airliners. Without intervention, the game is: I II Produce Don’t Produce Produce (-10, -10) (50, 0) Don’t Produce (0, 50) (0, 0) There are two equilibria: (Produce, Don’t) and (Don’t, Produce). If EU subsidizes 15 to Airbus to produce, the game becomes: I II Produce Don’t Produce Produce (-10, 5) (50, 0) Don’t Produce (0, 65) (0, 0) There is only one equilibrium: (Don’t, Produce). In this case, by subsidizing product development, a governmental can secure the world dominance of the domestic firm. 9.6.2 Subsidizing process innovation If we regard the RD levels x1 and x2 in the RD cooperation model as the amount of RD sponsored by governments 1 and 2, it becomes a model of government subsidy competition. 9.7 Dynamic Patent Races Tirole section 10.2. Reinganum (1982) “A dynamic game of RD,” Econometrica. 9.7.1 Basic model 2 firms compete in RD to win the patent of a new product. xi: the size of RD lab established (incurring a continuous cost of xi per unit of time) by firm i, i = 1, 2. V : the value of the patent per unit of time. r: interest rate. Ti: firm i’s discovery time. Assumption: T1 and T2 are independent exponential random variable: Ti ∼ 1 − e−h(xi)Ti , density function: h(xi)e−h(xi)Ti ,
- 92. 90 where [h(xi)]−1 is expecteddiscovery time of firm i. E(Ti) = [h(xi)]−1 , h(xi) 0, h (xi) 0, h (xi) 0. Industry discovery time: ˆT ≡ min{T1, T2} ∼ 1 − e−[h(x1)+h(x2)] ˆT because Prob{ ˆT T} = Prob{T1 T, T2 T} = e−h(x1)T e−h(x2)T = e−[h(x1)+h(x2)]T . Firm 1’s winning probability: Prob[T1 = T, T2 T | ˆT = T] = h(x1) h(x1) + h(x2) : Prob[T1, ˆT ∈ (T, T + dt)] Prob[ ˆT ∈ (T, T + dt)] ≈ h(x1)e−h(x1)T dt[1 − (1 − e−h(x2)T )] [h(x1) + h(x2)]e−[h(x1)+h(x2)]T dt = h(x1) h(x1) + h(x2) . E T1 T T2 T +dt T A BC T +dtT { ˆT ∈ (T, T + dt)} = A ∪ B ∪ C A = {T1 ∈ (T, T + dt), T2 ≥ T + dt} B = {T2 ∈ (T, T + dt), T1 ≥ T + dt} C = {T1, T2 ∈ (T, T + dt)}, Prob[C] ≈ 0. Given (x1, x2), the expected payoff of firm 1, Π1(x1, x2) is (Π2 is similar) ∞ 0 h(x1) h(x1) + h(x2) ∞ ˆT e−rt V dt − ˆT 0 e−rt x1dt [h(x1) + h(x2)]e−[h(x1)+h(x2)] ˆT d ˆT = ∞ 0 h(x1) h(x1) + h(x2) e−r ˆT V r − (1 − e−r ˆT )x1 r [h(x1) + h(x2)]e−[h(x1)+h(x2)] ˆT d ˆT = h(x1)V/r h(x1) + h(x2) + r − x1 r + [h(x1) + h(x2)]x1/r h(x1) + h(x2) + r = h(x1)V − rx1 r[h(x1) + h(x2) + r] . FOC for symmetric NE with x1 = x2 = x: [2h(x)+r][h (x)V −r]−[h(x)V −rx]h (x) = h(x)h (x)V +rh (x)(x+V )−2rh(x)−r2 = 0. Social welfare when x1 = x2 = x: W(x) = ∞ 0 ∞ ˆT e−rt V dt − ˆT 0 e−rt 2xdt 2h(x)e−2h(x) ˆT d ˆT = 1 r ∞ 0 e−r ˆT V − (1 − e−r ˆT )2x 2h(x)e−2h(x) ˆT d ˆT = 2h(x)V/r 2h(x) + r − 2x r + 2h(x)x/r 2h(x) + r = 2[h(x)V − rx] r[2h(x) + r] .
- 93. 91 FOC for socialoptimal: [2h(x) + r][h (x)V − r] − 2[h(x)V − rx]h (x) = rh (x)(2x + V ) − 2rh(x) − r2 = 0. Example: h(x) = 2 √ x, h (x) = 1/ √ x. FOC for NE: 2V + r(x + V ) √ x − 4r √ x − r2 = 0, ⇒ √ xn = 2V − r2 + (2V − r2)2 + 12r2V 6r . FOC for social optimal: r(2x + V ) √ x − 4r √ x − r2 = 0, ⇒ √ xs = −r2 + √ r4 + 8r2V 4r . Derivation: Let zn ≡ √ xn, zs ≡ √ xs, fn(z) ≡ 3rz2 + (r2 − 2V )z − rV , and fs(z) ≡ 2rz2 +r2 z−rV . fn(zn) = 0 and fs(zs) = 0. Direct computation shows that fn(zs) 0, limz→∞ fn(z) = ∞ 0. Hence zn zs and therefore √ xn √ xs = 2 3 k − 1 + √ k2 + 4k + 1 −1 + √ 1 + 4k 1, k = 2V r2 . Therefore, the equilibrium RD level is greater than the social optimal level. Extensions: 1. h(x) = λ¯h(x/λ) = λ1−a xa . 2. When there are n 2 firms. 3. n is endogenouse and optimal x and n. 9.7.2 RD race between an incumbent and a potential entrant 2 firms compete in RD to win the patent on a new procedure with unit cost c. Firm 1: Incumbent with initial unit production cost ¯c c. Firm 2: A potential entrant. Πm (¯c): Firm 1’s monopoly profit before the discovery of the new procedure. Πm (c): Firm 1’s monopoly profit if firm 1 wins. Πd 1(¯c, c): Firm 1’s duopoly profit if firm 2 wins. Πd 2(¯c, c): Firm 2’s duopoly profit if firm 2 wins. Assumption 1: Πm (c) ≥ Πd 2(¯c, c) + Πd 1(¯c, c). Assumption 2: The patent length is ∞. Using the same derivation as basic model, V1(x1, x2) = h(x1)Πm (c) + h(x2)Πd 1(¯c, c) + r[Πm (¯c) − x1] r[h(x1) + h(x2) + r] , V2(x1, x2) = h(x2)Πd 2(¯c, c) − rx2 r[h(x1) + h(x2) + r] .
- 94. 92 Comparing the payofffunctions reveals that firm 2’s payoff function is essentially the same as that of the basic model. Further comparison between firm 1 and firm 2’s payoff functions reveals that firm 1’s incentives are different in two ways: Efficiency effect: Πm (c) − Πd 1(¯c, c) ≥ Πd 2(¯c, c), firm 1 has more incentives to win the race. and therefore x1 tends to be greater than x2 in this aspect. Replacement effect: If firm 1 wins, he replaces himself with a new monopoly. Therefore, firm 1 tends to delay the discovery date. ∂2 V1 ∂Πm(¯c)∂x1 0 tends to make x1 smaller. The net effect depends on which one dominates. Following are two extreme cases: Drastic innovation: Πd 1(¯c, c) = 0 and Πd 2(¯c, c) = Πm (c). No efficiency effect and x1 x2. Almost linear h(x) case: h = λh(x/λ), λ → ∞, and h(x) ≈ h (0)x. In this case h(x1) and h(x2) are very large and firm 1 is more concerned with his win- ning the race rather than replacing itself. Therefore, replacement effect dominates and x1 x2.
- 95. 93 10 Network Effects,Compatibility, and Standards æ˜^‹, ß¹1ñ4, ß¹d™Ä Aéuþ}4Ó . ÞßC¾‘· Õ¶4 (externalities) ÞßÞ: }«¯T, Ãóß× ¾‘Þ: ˇ , þ}%, ²m7, Ãów° Ui = Ui (xi ) ⇒ Ui = Ui (xi , x−i , y), Fj (yj ) = 0 ⇒ Fj (yj , y−j , x) = 0. Arrow-Debreu general equilibrium model assumes no externalities. With production and/or consumption externalities, we have to modify Arrow-Debreu model. Þß, ¾‘Õ¶4Dæ˜^‹, ß¹1ñ4 É %ÈWÑA.âÃóº¯n?êµ%È^0, T¯Þº¹” Wà xkdå, œb§ (qwerty,dvorak) ¾© (L„, t„) ¦d† (Ô¬i, Ô˝i, š5, ˜ ) ¶ „ …PŒÞ (q2 , p×ÀP) —áY$ø2ÅQO6$ø¥d†, ?¹™Ä“ (standardization), êr1ñ4 ½æ: Aéʪ¥, ø….iÊÚ«, hß¹.i|Û, ج#|™Ä“}®×ê 0ä%È V r7“ (globalization), 1ñ4D™Ä“ V Aѽb{æ ®Å ¼·ı AÐíß¹d?AÑ0ä™Ä, J¦)1ñ4, êµ|×5æ˜^‹, Á¦|×5 Ò2 0 Wà: òj Úe HDTV, Ú7£w¶iß¹, ® æ˜ß¹ 10.0.3 3 concepts 1. Compatibility (1ñ4): .° ÉKªJòUà ( øu°™Ä) Standardization (™Ä“): F ÉK·ªJòUà 2. Downward-compatibility: hß¹ªJ HHß¹, OHß¹.øì ªJ Hhß¹ Wà Pentium III vs 486 3. Network externalities (æ˜Õ¶4): ¾‘6^àÓ° UàAb Ó‹7Ó‹ ª1ñß¹5W: ßà ¯ .ª1ñß¹5W: ÎóœœñDŸå ¯ 8”, ª1ñß¹$Aø__üÕÈ, ÕÈqѵI:_ ¼ VHS vs β, òœ, CD player, DVD, MO, etc. ASCII Ñ 7-bit ™Ä“å{ Extended ASCII Ñ 8-bit ³ ™Ä“å{ ( rÖ.°™Ä)
- 96. 94 10.0.4 A standardizationgame Firm A (USA) and firm B (Canada) are choosing a standard for their product. α-standard (L„d, CÔ¬iW , etc.) β-standard (t„d, CÔ˝iW , etc.) A B α β α (a, b) (c, d) β (d, c) (b, a) 1. If a, b max{c, d} (battle of the sexes), then (α, α) and (β, β) are both NE. They choose the same standard ( ™Ä“). 2. If c, d max{a, b}, then (α, β) and (β, α) are both NE. They choose the dif- ferent standards (®Wwu). 10.1 Network Externalities 10.1.1 Rohlfs phone company model Rohlfs 1974, “A Theory of Interdependent Demand for a Communication Service,” Bell Journal of Economics. Consumers are distributed uniformly along a line, x ∈ [0, 1]. 0 1 r x Consumers indexed by a low x are those who have high willingness to pay to subscribe to a phone system. Ux = n(1 − x) − p if x subscribes to the phone system 0 if x does not, n, 0 n 1: the total number of consumers who actually subscribe. p: the price of subscribing. ˆx: the marginal consumer who is indifference between subscribing and not. 0 1 t ˆxsubscribers In a rational expectation equilibrium, ˆx = n and U ˆx = n(1 − ˆx) − p = 0, ⇒ Inverse demand function: p = ˆx(1 − ˆx). Demand function: ˆx = 1 ± √ 1 − 4p 2 .
- 97. 95 E x T p 2 9 1/3 2/301 ˜ ˜ 'E' U ˆx0 ˆx ↓ U ˆx0 ˆx ↓ U ˆx0 ˆx ↑ If 0 p 1 4 , then there are two possible marginal consumers, ˆx = 1 ± √ 1 − 4p 2 . The smaller is unstable. The diagram uses p = 2 9 to illustrate. The phone company maximizes its profits: max ˆx pˆx = ˆx2 (1 − ˆx), FOC: 2ˆx − ˆx2 = 0, ⇒ ˆx∗ = 2 3 , p∗ = 2 9 . Dynamic model and critical mass (@ä!…îEb): Assumption 1: The phone company sets p = p∗ = 2 9 . Assumption 2: At t, consumers expect that nt = ˆxt−1. nt(1 − ˆxt) = p = 2 9 , ⇒ nt(1 − nt+1) = p = 2 9 , ⇒ nt+1 = 1 − 2 9nt ≡ f(nt). There are two equilibria: n∗∗ = 1 3 and n∗ = 2 3 . n∗∗ is unstable and n∗ is stable: f (n) = 2 9n2 f(n∗∗ ) = 2 1, f(n∗ ) = 1 2 1. If the initial subscription is n0 1/3, nt → 0. If the initial subscription is n0 1/3, nt → 2/3. 1/3 is the critical mass. E nt T nt+1 f(nt) 45 ◦ r unstable rstable n∗∗ n∗
- 98. 96 10.1.2 The standardization-varietytradeoff ™Ä“CÖj“? Consumers are distributed uniformly along a line, x ∈ [0, 1]. 0 1aprefer A prefer B 2 brands/standards, A (à¬G) and B (à˝G). a 0 consumers prefer A-standard. b = 1 − a 0 consumers prefer B-standard. xA: number of consumers using A-standard. xB: number of consumers using B-standard. δ: the disutility of using a less prefered standard. UA = xA use A-standard xB − δ use B-standard, UB = xA − δ use A-standard xB use B-standard. Consumer distribution: (xA, xB) such that xA, xB ≥ 0 and xA + xB = 1. A-standard distribution: A distribution such that (xA, xB) = (1, 0). B-standard distribution: A distribution such that (xA, xB) = (0, 1). Incompatible AB-standards distribution A distribution with xA, xB 0. Equilibrium: (xA, xB) such that none wants to switch to a different brand/standard. Proposition 10.3: If δ 1, then both A-standard and B-standard are equilibrium. If δ 1, then both A-standard and B-standard are not equilibrium. Proof: If δ 1 and every one chooses the same brand (either A or B), then none wants to switch to a different brand. If δ 1, then the cost of switching to a preferred brand is less than the benefit and therefore a single standard equilibrium cannot exist. Proposition 10.4: If a, b 1 − δ 2 , then (xA, xB) = (a, b) is an equilibrium. Proof: Given the distribution (xA, xB) = (a, b), the utility levels are UA = a use A br. b − δ = 1 − a − δ a use B br. UB = a − δ = 1 − b − δ b use A br. b use B br. Therefore, none will switch to a different brand. E a T b d d d d d d d d d d d dd 2-standard equilibrium range 1−δ 2 1−δ 2 r r 1−δ 1−δ
- 99. 97 Social welfare: W(xA,xB) = aUa + bUb , where Ua (Ub ) is the utility level of A- prefered consumers (B-prefered consumers). W(A) = W(1, 0) = a + b(1 − δ) = 1 − bδ. W(B) = W(0, 1) = a(1 − δ) + b = 1 − aδ. W(AB) = W(a, b) = a2 + b2 = (1 − b)2 + b2 = 1 − b − b(1 − 2b) = 1 − 2ab. Proposition 10.5: If a b, then W(A) W(B). Proof: If W(A) − W(B) = a + b(1 − δ) − [a(1 − δ) − b] = (a − b)δ 0. Proposition 10.6: 1. If δ 1, then W(AB) max{W(A), W(B)}. 2. If δ 1 and δ 2 max{a, b}, then W(AB) max{W(A), W(B)}. Proof: Assume that a b (or 1 − 2b 0). (case b a is similar.) 1. If δ 1, then max{W(A), W(B)} = W(A) = 1 − bδ 1 − b 1 − b − b(1 − 2b) = W(AB). 2. If δ 2a 1, then W(AB) − W(A) = b(δ − 2a) 0. Proposition 10.7: If δ 1, then market failure can happen. Remark: When a b and δ 1, the social optimal is A-standard. However, both the incompatible standards and B-standard can also be equilibrium. 10.2 Supporting Services and Network Effects Network effects can occur even there is no network externalities. For example, when there is a complementary supporting industry exhibiting increasing returns to scale such as PC industry. 10.2.1 Basic model Chou/Shy (1990), “Network effects without network externalities,” International Jour- nal of Industrial Organization. A PC industry with 2 brands, A and B and prices PA and PB. Consumers are distributed uniformly along a line, δ ∈ [0, 1]. E δ 0 1δprefer A prefer B Let NA and NB be the numbers of software pieces available to computers A and B, respectively.
- 100. 98 The utility ofconsumer δ is Uδ = (1 − δ) √ NA if δ buys A-system δ √ NB if δ buys B-system. In the above, √ Ni can easily be generalized to Nα i . Marginal consumer ˆδ: U ˆδ (A) = (1 − ˆδ) NA = U ˆδ (B) = ˆδ NB, ⇒ ˆδ = √ NA √ NA + √ NB . Market shares: δA = ˆδ and δB = 1 − ˆδ. If NA increases (or NB decreases), ˆδ will decrease, A’s market share will increase and B’s market share will decrease. δA = ˆδ = √ NA √ NA + √ NB , δB = 1 − ˆδ = √ NB √ NA + √ NB , δB δA = 1 − ˆδ ˆδ = √ NB √ NA . In this model, there are two monopolistic competition software industries, A-software and B-software. Assume that each consumer has Y dollars to spend on a computer system. If a consumer chooses i-stytem, he has Ei ≡ Y − Pi to spend on software. There is a variety effect in each software industry and the number of software pieces is propotional to the aggregate expenditure spent on them: NA = kδAEA = kˆδ(Y − PA), NB = kδBEB = k(1 − ˆδ)(Y − PB). 1 − ˆδ ˆδ = √ NB √ NA = 1 − ˆδ ˆδ Y − PB Y − PA , ⇒ 1 − ˆδ ˆδ = Y − PB Y − PA = EB EA . Therefore, the equilibrium market shares are δA = ˆδ = EA EA + EB = Y − PA 2Y − PA − PB , δB = 1 − ˆδ = EB EA + EB = Y − PB 2Y − PA − PB . Network effects: When ˆδ goes down, δA goes down (δB goes up), which in turn will reduce NA (increase NB). Finally, A-users’ utility levels will decrease (B-users’ utility levels will increase). The network effect here is the same as the variety effects in Dixit/Stiglitz monopolis- tic competition model. Duopoly price competition: The profit functions of firms A and B are ΠA(PA, PB) = δAPA = PA(Y − PA) 2Y − PA − PB , ΠB(PA, PB) = δBPB = PB(Y − PB) 2Y − PA − PB . The price competition equilibrium is derived in Chou/Shy (1990).
- 101. 99 10.2.2 Partial compatibility Chou/Shy(1993) “Partial compatibility and supporting services”, Economic Letters. In the basic model, A-computers and B-computers are incompatible in the sence that A-computers use only A-software and B-computers use only B-software. The model can be generalized to the case when computer firms design their machines in such a way that some fraction of B-software can be used in A-machines and vice versa. Let ρA (ρB) be the proportion of B-software (A-software) that can be run on A- computers (B-computers). Incompatibility: ρA = ρB = 0. Mutual compatibility: ρA = ρB = 1. One-way compatibility: ρA = 1, ρB = 0 or ρA = 0, ρB = 1. nA: number of software pieces written for A-computers. nB: number of software pieces written for B-computers. NA = nA + ρAnB, NB = nB + ρBnA, ⇒ nA = NA − ρANB 1 − ρAρB nB = NB − ρBNA 1 − ρAρB . (5) δiEi = δi(Y − Pi): Aggregate expenditure on software from i-computer users. ni Ni δiEi + ρjnj Nj δjEj: Aggregate expenditure on i-software. As in the basic model, the number of i-software, ni, is proportional to the aggre- gate expenditure on i-software: nA = k nA NA δAEA + ρBnA NB δBEB , nB = k ρAnB NA δAEA + nB NB δBEB , ⇒ NA = (1 − ρAρB)δAEA k(1 − ρB) , NB = (1 − ρAρB)δBEB k(1 − ρA) . (6) As in the basic model, 1 − ˆδ ˆδ = √ NB √ NA = 1 − ˆδ ˆδ (1 − ρB)EB (1 − ρA)EA , ⇒ 1 − ˆδ ˆδ = (1 − ρB)EB (1 − ρA)EA = (1 − ρB)(Y − PB) (1 − ρA)(Y − PA) . Other things being equal, if firm A increases the degree of compatibility ρA, the number of software pieces run on A-computers will decrease and hence its market share δA = ˆδ will also decrease.
- 102. 100 10.3 The ComponentsModel Matutes/Regibeau (1988), “Mix and Match: Product Compatibility Without Net- work Externalities,” RAND Journal of Economics. Economides (1989), “Desirability of Compatibility in the Absence of Network Exter- nalities,” American Economic Review. AS1 2 firms, A and B, producing XA, YA, XB, YB. AS2 Marginal costs are 0. AS3 X and Y are completely complementary. AS4 3 consumers: AA, AB, BB. You need an X and a Y to form a system S. 2 situations: 1. Incompatibility: A and B’s products are not compatible. You have to buy XAYA or XBYB. 2. Compatibility: A and B’s products are compatible. There are 4 possible systems: XAYA, XAYB, XBYA, and XBYB. Consumer ij’s utility, ij = AA, AB, BB, is Uij = 2λ − (Px i + Py j ) i j = ij, i.e., X, Y ·¯ λ − (Px i + Py j ) i = i or j = j but i j = ij, i.e., X, Y øᯠ−(Px i + Py j ) i = i and j = j, i.e., X, Y ·.¯ 0 .¾‘ 10.3.1 Incompatibility There are only 2 systems: A-system (XAYA) and B-system (XBYB). PA = Px A + Py A, PB = Px B + Py B: Price of system A and system B, respectively. Equilibrium: (PI A, PI B; qI A, qI B) such that 1. PI i maximizes Πi(Pi, PI j ). 2. (qI A, qI B) are the aggregate demand of the consumers at price (P I A, PI B). Lemma 10.1. In an equilibrium, consumer AA (consumer BB) purchases A-system (B-system). Proof: If in an equilibrium consumer AA purchases B-system, it must be PB = 0. In that case, firm A can set 0 PA 2λ to attract consumer AA. Proposition 10.13. There are 3 different equilibria: 1. (PI A, PI B; qI A, qI B) = (λ, 2λ; 2, 1). AA and AB purchase A-system and BB pur- chases B-system, ΠA = ΠB = 2λ, CS = λ, social welfare is 5λ.
- 103. 101 2. (PI A, PI B;qI A, qI B) = (2λ, λ; 1, 2). AA purchases A-system and BB and AB pur- chase B-system. ΠA = ΠB = 2λ, CS = λ, social welfare is 5λ. 3. (PI A, PI B; qI A, qI B) = (2λ, 2λ; 1, 1). AA purchases A-system and BB purchases B- system. AB chooses to do without. ΠA = ΠB = 2λ, CS = 0, social welfare is 4λ. 10.3.2 Compatibility 4 systems: AA-system (XAYA), AB-system (XAYB), BA-system (XBYA), and BB- system (XBYB). Equilibrium: (Pc Ax, Pc Ay, Pc Bx, Pc By; qc Ax, qc Ay, qc Bx, qc By) such that 1. (Pc ix, Pc iy) maximizes Πi(Pix, Piy; Pc jx, Pc jy). 2. (qc Ax, qc Ay, qc Bx, qc By) are the aggregate demand of the consumers at price (P c Ax, Pc Ay, Pc Bx, Pc By). Proposition 10.14. There exists an equilibrium such that P c Ax = Pc Ay = Pc Bx = Pc By = λ, qc Ax = qc By = 2, qc Ay = qc Bx = 1, Πc A = Πc B = 3λ, UAA = UAB = UBB = 0, and social welfare is 6λ. 10.3.3 Comparison 1. Consumers are worse off under compatibility. 2. Firms are better off under compatibility. 3. Social welfare is higher under compatibility. Extension to a 2-stage game: If at t = 1 firms determine whether to design compatible components and at t = 2 they engage in price competition, then they will choose compatibility.
- 104. 102 11 Advertising Advertising isdefined as a form of providing information about prices, quality, and location of goods and services. 2% of GNP in developed countries. vegetables, etc., 2% of sales. cosmetics, detergent, etc., 20-60 % of sales. In 1990, GM spent $63 per car, Ford $130 per car, Chrysler $113 per car. What determines advertising in different industries or different firms of the same industry? Economy of scale, advertising elasticity of demand, etc. Kaldor (1950), “The Economic Aspects of Advertising,” Review of Economic Studies. Advertising is manipulative and reduces competition. 1. Wrong information about product differentiations ⇒ increases cost. 2. An entry-deterring mechanism ⇒ reduces competition. Telser (1964), “Advertising and Competition,” JPE Nelson (1970),“Information and Consumer Behavior,” JPE Nelson (1974),“Advertising as Information,” JPE Demsetz (1979), “Accounting for Advertising as a Barrier to Entry,” J. of Business. Positive sides of advertising: It provides produt information. Nelson: Search goods: Quality can be identified when purchasing. ⇒ .Û µ Experience goods: Quality cannot be identified until consuming. ⇒ µª«àÛb Persuasive advertising: Intends to enhance consumer tastes, eg diamond. Informative advertising: Provides basic information about the product. 11.1 Persuasive Advertising Q(P, A) = βA a P p , β 0, 0 a 1, p −1. A: expenditure on advertising. a: Advertising elasticity of demand. p: Price elasticity of demand. c: Unit production cost. max P,A Π = PQ − cQ − A = (P − c)βA a P p − A. FOC with respect to P: ∂Π ∂P = βA a [( p + 1)P p − c pP p ] = 0, ⇒ Pm = c p p + 1 , Pm − c Pm = p −1 .
- 105. 103 FOC with respectto A: ∂Π ∂A = aβA a−1 P p (P − c) − 1 = 0, ⇒ Pm − c Pm = A PQ 1 a , a p = A PQ . Proposition: The propotion of advertising expenditure to total sales is equal to the ratio of advertising elasticity to price elasticity. 11.1.1 Example: β = 64, a = 0.5, p = −2, c = 1 Q = 64 √ AP−2 , P = 8A1/4 Q−1/2 , ⇒ Pm = 2, Qm = 16 √ A, Am = 64, Π = 16 √ A, ⇒ CS(A) = 16 √ A 0 P(Q)dQ − Pm Qm = 16 √ A 0 8A1/4 Q−1/2 dQ − 32 √ A = 32 √ A. Social welfare: W(A) ≡ CS(A) + Π(A) − A = 48 √ A − A. Social optimal: W (A) = 0 = 24 √ A − 1, ⇒ A = 242 = 576 Am = 64. Remark: 1. Does CS(A) represent consumers’ welfare? If it is informative adver- tising, consumers’ utility may increase when A increases. However, if consumers’ are just persuased to make unnecessary purchases, the demand curve does not really re- flect consumers’ marginal utility. 2. Crowding-out effect: Consumption for other goods will decrease. 3. A can be interpreted as other utility enhancing factors. 11.2 Informative Advertising Benham (1972), “The effects of Advertising on the Price of Eye-glasses,” J. of Law and Economics. Ê1Å, Š¢iŸ µí˚iŸg¦Âœò Consumers often rely on information for their purchases. The problem is whether there is too little or too much informative advertising. Butters (1977), “Equilibrium Distributions of Sales and Advertising Prices,” Review of Economic Studies. Informative Advertising level under monopolistic competition equilibrium is social optimal. Grossman/Shapiro (1984), “Informative Advertising with Differentiated Prdoducts,” Review of Economic Studies. In a circular market, informative advertising level is too excessive. Meurer/Stahl (1994), “Informative Advertising and Product Match,” IJIO. In the case of 2 differentiated products, the result is uncertain.
- 106. 104 11.2.1 A simplemodel of informative advertising 1 consumer wants to buy 1 unit of a product. p: the price. m: its value. U = m − p 0 . If the consumer does not receive any advertisement, he will not purchase. If he receives an advertisement from a firm, he will purchase from the firm. If he receives 2 advertisements from 2 firms, he will randomly choose one to buy. 2 firms, unit production cost is 0, informative advertising cost is A. Each chooses either to advertise its product or not to advertise it. πi = p − A if only firm i’s ad is received. p 2 − A if both firms’ ad are received. −A if firm i’s ad is not received. 0 if firm i chooses not to advertise. Let δ be the probability that an advertisement is received by the consumer. Eπi = δ(1 − δ)(p − A) + δ2 ( p 2 − A) − (1 − δ)A ≡ π(2) if both choose to advertise. δ(p − A) − (1 − δ)A ≡ π(1) if only firm i chooses to advertise. 0 if firm i chooses not to advertise. If p/A 1/δ, ⇒ π(1) 0, ⇒ at least one firm will choose to advertise. If p/A 2/[δ(2 − δ)], ⇒ π(2) 0, ⇒ both firms will choose to advertise. E δ T p/A π(2) 0, 2 firms in equilibrium. 1 firm π(1) 0, 0 firm q 1 1/δ 2/[δ(2 − δ)] Welfare comparison: EW = δ(2 − δ)m − 2A ≡ W(2) if 2 firms advertise. δm − A ≡ W(1) if only one firm advertises. 0 if no firm advertises. If m A 1 δ , ⇒ W(1) 0, ⇒ social optimal is at least one firm advertises. If m A 1 δ(1 − δ) , ⇒ W(2) W(1), ⇒ social optimal is both firms advertise.
- 107. 105 E δ T m/A q 1 1/[δ(1 − δ)] W(2) W(1) W(1) 0 W(1) 0 E δ T p/A = m/A q 1 market failure µxXTòU δ Ó‹ When δ → 1, one firm would be enough. However, if m/A 1, both firms will advertise. 11.3 Targeted Advertising ‡ú4 µ (1) Consumers are heterogeneous with different tastes. (2) Large scale advertising is costly. (3) Intensive advertising will result in price competition. ⇒ ̶nßF í¾‘6, .° ¼Sà.°4”5‡ú4 µ, ‡ú.°5¾‘6íˇ 11.3.1 The model 2 firms, i = 1, 2, producing differentiated products. 2 groups of consumers: E experienced consumers and N inexperienced consumers. θE of experienced consumers are brand 1 oriented, 0 θ 1. (1 − θ)E of experienced consumers are brand 2 oriented. 2 advertising methods: P (persuasive) and I (informative). Each firm can choose only one method. AS1: Persuasive advertising attracts only inexperienced consumers. If only firm i chooses P, then all N inexperienced consumers will purchase brand i. If both firms 1 and 2 choose P, each will have N/2 inexperienced consumers. AS2: Informative advertising attracts only the experienced consumers who are ori- ented toward the advertised brand, i.e., if firm 1 (firm 2) chooses I, θE ((1−θ)E) experienced consumers will purchase brand 1 (brand 2). AS3: A firm earns $1 from each customer. From the assumptions we derive the following duopoly advertising game: firm 1 firm 2 P I P (N/2, N/2) (N, (1 − θ)E) I (θE, N) (θE, (1 − θ)E)
- 108. 106 11.3.2 Proposition 11.5 1.(P, P) is a NE if only if N/2 ≥ θE and N/2 ≥ (1 − θ)E or 1 − N 2E ≤ θ ≤ N 2E . (If strict inequality holds, the NE is unique.) 2. (I, I) is a NE if only if N ≤ θE and N ≤ (1 − θ)E or N E ≤ θ ≤ 1 − N E . (If strict inequality holds, the NE is unique.) 3. (P, I) is a NE if only if N/2 ≤ (1 − θ)E and N ≥ θE or θ ≤ min{1 − N 2E , N E }. 4. (I, P) is a NE if only if N/2 ≤ θE and N ≥ (1 − θ)E or θ ≥ max{1 − N E , N 2E }. E N/E T θ ¨¨ ¨¨ ¨¨ ¨¨ ¨¨ ¨¨ ¨¨ ¨ ¨ ¨ d d d d d d d d d rr rr rr rr rr rr rr r r r (I, I) (P, P) (I, P) (P, I) (I,P)(P,I) 1 1 2 N/E 1 − N/E N/2E 1 − N/2E 11.4 Comparison Advertising Comparison advertising: The advertised brand and its characteristics are compared with those of the competing brand. It became popular in the printed media and broadcast media in the early 1970s. EEC Legal conditions: Material (xñ) and verifiable (ª„õ) details, no misleading (³ Ïû), no unfair (t£). Advantages of comparison ads: 1. Provide consumers with low-cost means of evaluating available products. 2. Makes consumers more conscious of comparison before buying. 3. Forces the manufacturers to build into the products attributes consumers want. Negative points: 1. Lack of objectivity. 2. Deception and consumer confusion due to information overload. Muehling/Stoltman/Grossbart (1990 J of Advertising): 40% of ads are comparison. Pechmann/Stewart (1990 J of Consumer Research): Majority of ads (60%) are indi- rect comparison; 20% are direct comparison.
- 109. 107 11.4.1 Application ofthe targeted ad model to comparison ad Plain ad. = Persuasive ad, aiming at inexperienced consumers. Comparison ad. = Targeted ad, aiming at experienced consumers. Applying Proposition 11.5 and the diagram, we the following results: 1. Both firms will use comparison ad only if E 2N. 2. If 2E N, both firms will use plain ad. 3. Comparison ad is used by the popular firm and plain ad is used by the less popular firm in other cases in general. 11.5 Other Issues 11.5.1 Can information be transmitted via advertising? Search goods: False advertising is unlikely. Experience goods: Producers will develop persuasive methods to get consumers to try their products. Facts: 1. Due to assymmetry of information about quality, consumers can not simply rely on ads. 2. High-quality experience products buyers are mostly experienced consumers. Schmalensee (1978), “A Model of Advertising and Product Quality,” JPE. Low-quality brands are more frequently purchased and firms producing low-quality brands advertise more intensively. ⇒ There is a negative correlation between adver- tising and the quality of advertised products. Kihlstrom/Riordan (1984), “Advertising as a Signal,” JPE. High-quality firms have an incentive to advertise in order to trap repeated buyers. ⇒ the correlation between ad and quality is positive. Milgrom/Roberts (1986), “Price and Advertising Signals of Product Quality,” JPE. A signalling game model with ad as a signal sent by high-quality firms. Bagwee (1994), “Advertising and Coordination,” Review of Economic Studies. and Bagwell/Ramey (1994), “Coordination Economics, Advertising, and Search Behavior in Retail Markets,” AER. Efficient firms with IRTS tend to spend large amount on advertising to convince buyers that large sales will end up with lower prices. ⇒ ad is a signal to reveal low cost. 11.5.2 Advertising and concentration Is there a positive correlation between advertising and concentration ratio? Perfect competition industry: Individual firms have no incentives to advertise their products due to free rider effect. Collectively the industry demand can be increased
- 110. 108 by advertising. However,there is the problem of free rider. Monopoly industry: Due to scale economy, monopoly firms may have more incentives to advertise. Kaldor (1950): In an industry, big firms advertise more. Telser (1964) “Advertising and Competition,” JPE. Very little empirical support for an inverse relationship between advertising and competition. Orenstein (1976), “The Advertising - Concentration Controversy,” Southern Eco- nomic Journal, showed very little evidence that there is increasing returns in adver- tising. Sutton (1974), “Advertising Concentration, Competition,” Economic Journal. The relationship between scale and advertising is not monotonic. Both perfect competi- tion and monopoly firms do not have to advertise but oligopoly firms have to. E concentration ratio T ad 11.5.3 A simple model of dvertising and prices Cost: TC(Q) = cHQ if Q ≤ Q∗ cLQ if Q Q∗ , Demand: P = a1 − Q if advertising a0 − Q if not advertising. No Ad equilibrium: Q0 = (a0 − cH)/2, P0 = (a0 + cH)/2. Ad equilibrium: Q1 = (a1 − cL)/2, P1 = (a1 + cL)/2. If cH − cL a1 − a0, then P1 P0, i.e., advertising reduces the monopoly price.
- 111. 109 12 Quality 12.1 VerticalDifferentiation in Hotelling Model Quality is a vertical differentiation character. 2-period game: At t = 1, firms A and B choose the quality levels, 0 ≤ a b ≤ 1, for their products. At t = 2, they engage in price competition. Consumers in a market are distributed uniformly along a line of unit length. 0 1 r i x Each point x ∈ [0, 1] represents a consumer x. Ux = ax − PA if x buys from A. bx − PB if x buys from B. The marginal consumer ˆx is indifferent between buying from A and from B. The location of ˆx is determined by aˆx − PA = bˆx − PB ⇒ ˆx = PB − PA b − a . (7) The location of ˆx divids the market into two parts: [0, ˆx) is firm A’s market share and (ˆx, 1] is firm B’s market share. 0 1 r ˆx A’s share B’s share' E' E Assume that the marginal costs are zero. The payoff functions are ΠA(PA, PB; a, b) = PA ˆx = PA PB − PA b − a , ΠB(PA, PB; a, b) = PB(1−ˆx) = PB 1 − PB − PA b − a . The FOCs are (the SOCs are satisfied) ∂ΠA ∂PA = PB − 2PA b − a = 0, ∂ΠB ∂PB = 1 − 2PB − PA b − a = 0. (8) The equilibrium is given by PA = b − a 3 , PB = 2(b − a) 3 , ⇒ ˆx = 1 3 . The reduced profit functions at t = 1 are ΠA = (b − a) 9 , ΠB = 4(b − a) 9 .
- 112. 110 Both profit functionsincrease with b − a. Moving away from each other will increase both firm’s profits. The two firms will end up with maximum product differentiation a = 0 and b = 1 in equilibrium. Modifications: 1. High quality products are associated with high unit production cost. 2. Consumer distribution is not uniform. 12.2 Quality-Signalling Games ̾êW There are one unit of identical consumers each with utility function U = H − P à‹ƒò¹”ß¹ L − P à‹ƒQ¹”ß¹ 0 . CH CL ≥ 0: Unit production costs of producing high- and low-quality product. AS1 The monopolist is a high-quality producer. AS2 H L CH. Signalling equilibrium (̾êW): Pm = H and Qm = L − CL H − CL . Proof: 1. For a low-quality monopolist, to imitate the high-quality monopolist is not worthwhile: ΠL(Pm , Qm ) = (Pm − CL)Qm = (H − CL) L − CL H − CL = L − CL = ΠL(L, 1). 2. The high-quality monopolist has no incentives to imitate a low-quality monopolist: ΠH(Pm , Qm ) = (Pm − CH)Qm = (H − CH) L − CL H − CL ΠH (L, 1) = L − CH. The last inequality is obtained by cross multiplying. The high-quality monopolist has to reduce its quantity to convince consumers that the quality is high. If the information is perfect, he does not have to reduce quantity. The quantity reduction is needed for signalling purpose. 12.3 Warranties ¹”„z Spence (1977), “Consumer Misperceptions, Product Failure, and Producer Liability,” Review of Economic Studies. Higher-quality firms offer a larger warranty than do low-quality firms. Grossman (1980), “The Role of Warranties and Private Disclosure about Product Quality,” Journal of Law and Economics. A comprehensive analysis of a monopoly that can offer a warranty for its product.
- 113. 111 12.3.1 Symmetric informationmodel ρ: The probability that the product is operative. V : The value to the consumer if the product is operative. Symmetric information: ρ is known to both the seller and the buyer. P: Price. C: unit production cost. Assumption: ρV C. U = V − P ¹”„z ρV − P ̹”„z 0 .. Monopoy equilibrium without warranty: P nw = ρV and Πnw = ρV − C. Expected cost of a unit with warranty: Cw = C + (1 − ρ)C + (1 − ρ)2 C + · · · = C ρ . (On average, 1/ρ units will end up with one operative unit.) Monopoy equilibrium with warranty: P w = V and Πw = V − C ρ . It seems that the monopoly makes a higher profit by selling the product with a warranty. However, if the consumer purchases 1/ρ units to obtain an operative unit, the result is the same. (Oz Shy’s statement is not accurate unless the consumer is risk averse.) 12.3.2 Asymmetric information with warranty as a quality signal ρL: The operative probability of a low-quality product. ρH ρL: The operative probability of a high-quality product. Asymmetric information: The quality of a product is known only to the seller. Bertrand equilibrium without warranty: P nw = C and Πnw i = 0, i = H, L. Bertrand equilibrium with warranty: P w = C/ρL, QH = 1, QL = 0, Πw H = Pw − C ρH 0, Πw L = Pw − C ρL = 0, U = V − Pw 0. The high-quality firm has a lower unit production cost of the warranty product. In the market for warranty product, the low-quality firm cannot survive.
- 114. 112 13 Pricing Tactics 13.1Two-Part Tariff Oi (1971), “A Disneyland Dilemma: Two-Part Tariffs for a Mickey Mouse,” QJE. In addition to the per unit price, a monopoly firm (amusement parks Y—Ò, sports clubs U™E—¶) can set a second pricing instrument (membership dues Ä‘) in order to be able to extract more consumer surplus. P: price, φ: membership dues, m: consumption of other goods. Budget contraint: m + φ + PQ = I. Utility function: U = m + 2 √ Q. max Q U = I − φ − PQ + 2 Q ⇒ demand function: P = 1 Qd , Qd = 1 P2 . 13.1.1 No club annual membership dues Club capacity: K. Club profit: Π(Q) = PQ = √ Q maxQ Π(Q) ⇒ Qm = K, Pm = 1 √ K , Πm = √ K. 13.1.2 Annual membership dues max φ Πa(φ) = φ subject to I − φ + 2 √ K I = I0, ⇒ φ∗ = 2 √ K = Πa Πm. Using annual membership dues, the monopoly extracts all the consumer surplus, like the 1st degree price discrimination. E Q T m U1 U0 rI ———————————— m + PmQ = I K r rc T c T φ∗ Πm E Q T m U1 U0 U4 rI ————————————— m − φ3 + P4(Q − Q3) = I © Q4 = KQ3 r r c T φ3
- 115. 113 13.1.3 Two-part tariff Thereare two problems with membership dues: 1. It is difficult to estimate consumers’ utility function and the profit maximizing φ∗ . If the monopoly sets φ too high, the demand would be 0. 2. Consumers are heterogeneous. Therefore, the monopoly offers a “package” of Q3 K and annual fee φ3 φ∗ . In addition, the monopoly offers an option to purchase additional quantity for a price P4. 13.2 Peak-Load Pricing ª¼, ׼Ϧg High- and Low-Seasonal Demand Structure: P H = AH − QH , PL = AL − QL , AH AL 0. Cost Structure: TC(QH , QL , K) = c(QH + QL ) + rK for 0 ≤ QL , QH ≤ K. c: unit variable cost, K: capacity, r: unit capacity cost. max QH,QL,K Π = PH QH + PL QL − c(QH + QL ) − rK subject to 0 ≤ QL , QH ≤ K. FOC: MRH (QH ) = c + r, MRL (QL ) = c, QL QH = K. ⇒ PH = AH + c + r 2 PL = AL + c 2 . Regulation for efficiency: P H = c + r PL = c. n-period case: MRH (QH ) = c + r n , MRL (QL ) = c. Modification: Substitutability between high- and low-seasonal demand.
- 116. 114 14 Marketing Tactics:Bundling, Upgrading, and Dealer- ships 14.1 Bundling (¾ù) and Tying (»») Bundling: Firms offer for sale packages containing more than one unit of the product. It is a form of nonlinear pricing (2nd degree price discrimination). Tying: Firms offer for sale packages containing at least two different (usually com- plementary) products. Examples: Car and car radio, PC and software, Book and T-shirt. 14.1.1 How can bundling be profitable? Monopoly demand: Q(P) = 4 − P, MC = 0. Monopoly profit maximization: P m = 2 = Qm , Πm = 4. Bundling 4-unit package for $8: (1) The consumer will have no choice but buying the package. (2) The monopoly profit becomes Π = 8 Πm . The monopoly in this case uses bundling tactics to extract all the consumer surplus. 14.1.2 How can tying be profitable? A monopoly sells goods X and Y. 2 consumers, i = 1, 2 who have different valuations of X and Y. Valuations: V 1 x = H, V 1 y = L; V 2 x = L, V 2 y = H, H L 0. Assume that consumers do not trade with each other. Equilibrium without tying: Pnt x = Pnt y = H if H 2L L if H 2L and Πnt = 2H if H 2L 4L if H 2L. Equilibrium with tying, PT = Pxy, QT = Qxy: Pt T = H + L, and Πt = 2(H + L) Πnt . 14.1.3 Mixed tying Adams/Yellen (1976), “Commodity Bundling and the Burden of Monopoly,” QJE. A monopoly sells goods X and Y. 3 consumers, i = 1, 2, 3 who have different valuations of X and Y. Valuations: V 1 x = 4, V 1 y = 0; V 2 x = 3, V 2 y = 3, V 3 x = 0, V 3 y = 4.
- 117. 115 Assume that consumersdo not trade with each other. Equilibrium without tying: (1) Px = Py = 3, Qx = Qy = 2, Π(1) = 12. (2) Px = Py = 4, Qx = Qy = 1, Π(2) = 8 Π(1). Therefore, Pnt x = Pnt y = 3, Qny x = Qnt y = 2, Πnt = 12. Equilibrium with pure tying, PT = Pxy, QT = Qxy: (1) PT = 4, QT = 3, Π(1) = 12. (2) PT = 6, QT = 1, Π(2) = 6 Π(1). Therefore, Pt T = 4, Qt T = 3, Πt T = 12. Equilibrium with mixed tying: Pmt x = Pmt y = 4, Pmt T = 6, Qmt x = Qmt y = 1, Qmt T = 1, Πmt = 14 Πt = 12. But mixed tying is not always as profitable as pure tying. 14.1.4 Tying and foreclosure (‡½) US antitrust laws prohibit bundling or tying behavior whenever it leads to a reduced competition. What is the connection between tying and reduced competition? 2 computer firms, X and Y, and a monitor firm Z (compatible with X and Y). 2 consumers i = 1, 2 with utility functions U1 = 3 − Px − Pz buys X and Z 1 − Py − Pz buys Y and Z 0 buys nothing, U2 = 1 − Px − Pz buys X and Z 3 − Py − Pz buys Y and Z 0 otherwise, Bertrand equilibrium with 3 independent firms: (1) Px = Py = 2, Pz = 1, Qx = Qy = 1, Qz = 2, Πx = Πy = Πz = 2. (2) Other equilibria: (Px, Py, Pz) = (1, 1, 2) = (0, 0, 3) = (3, 3, 0). Assume that firm X buys firm Z and sells X and Z tied in a single package. Total foreclosure equilibrium: Ptf xz = 3, Qtf xz = 1, Qtf y = 0, Πtf xz = 3 Πx + Πz = 4, Πtf y = 0. Py does not matter. Consumer 2 is not served. The industry aggregate profit is lower under total foreclosure. -foreclosure equilibrium: Pxz = 3 − , Qxz = 2, Py = , Qy = 1, Πxz = 2(3 − ), Πy = . Consumer 2 buys one XZ and one Y and discards X.
- 118. 116 14.1.5 Tying andInternational markets segmentation Government trade restrictions like tarriffs, quotas, etc., help firms to engage in price discrimination across international boundaries. A two countries, k = 1, 2, with one consumer in each country. A world-monopoly producer sells X. It can sell directly to the consumer in each country or open a dealership in each country selling the product tied with service to the consumer. The utility of the consumer in each county (also denoted by k = 1, 2) is U1 = B1 + σ − Ps 1 if 1 buys X service B1 − Pns 1 if 1 buys X only 0 if 1 does not buy, U2 = B2 + σ − Ps 2 if 2 buys X service B2 − Pns 2 if 2 buys X only 0 if 2 does not buy, where Ps k (Pns k ) are the price with service (without service) in country k, k = 1, 2, and σ 0 is the additional value due to service. AS1 B1 B2. AS2 Marginal production cost is 0. AS3 Unit cost of service provided by the dealership is w ≥ 0. No attempts to segment the market: Pns = B2 if B1 2B2 B1 if B1 2B2 Πns = 2B2 if B1 2B2 B1 if B1 2B2 Segmenting the market: Ps k = Bk + σ, Πs = B1 + B2 + 2(σ − w). Nonarbitrage condition: B1 − B2 σ.