Chapter 6 - Romer Model

Chapter 6: Growth and Ideas
Ryan W. Herzog
Spring 2021
Ryan W. Herzog (GU) Romer Spring 2021 1 / 48

1 Introduction
2 Economics of Ideas
3 The Romer Model
4 Combining Solow and Romer: Overview
5 Growth Accounting
6 Concluding Our Study of Long-Run Growth
7 Mathematical Solution of Combined Model

Introduction
Learning Objectives
New methods of using existing resources are the key to sustained
long-run growth.
Why “nonrivalry” makes ideas different from other economic goods in
a crucial way.
How the economics of ideas involves increasing returns and leads to
problems with Adam Smith’s invisible hand.
The Romer model of economic growth.
How to combine the Romer and Solow models to get a full theory of
long-run economic performance.

Introduction
Romer Model
Divides the world into objects and ideas
Objects are capital and labor from the Solow Model
Ideas are used in making objects
The distinction between ideas and objects forms the basis for modern
theories of economic growth.

Ideas
The Economics of Ideas
Adam Smith’s invisible hand theorem states that perfectly
competitive markets lead to the best of all possible worlds.
Idea diagram:
Ideas → nonrivalry → increasing
returns → problems with
pure competition

Ideas
Ideas
Ideas in the world are virtually infinite
Objects in the world are finite
Sustained economic growth occurs because of new ideas.

Ideas
Nonrivalry
Objects are rivalrous
One person’s use reduces their inherent usefulness to someone else.
Ideas are nonrivalrous
One person’s use does not reduce their inherent usefulness to someone
else.
Nonrivalry implies we do not need to reinvent ideas for additional use.
Nonrivalry is different from excludability.
Excludability occurs because someone may legally restrict use of a
good.
Ideas may be excludable.

Ideas
Increasing Returns
Firms pay initial fixed costs to create new ideas but don’t need to
reinvent the idea again later.
Increasing returns to scale: A doubling of inputs will result in a more
than doubling of outputs.

Ideas
Increasing Returns

Ideas
Test for Increasing Returns
Multiplying all inputs by two
Increasing returns is present if output is then multiplied by more than
two.
Yt = F(Kt, Lt, At) = AtK
1/3
t L
2/3
t (1)
Double the inputs
Yt = F(2Kt, 2Lt, 2At) = 2A·(2K)1/3
·(2L)2/3
= 2·21/3
·22/3
·AK1/3
L2/3
= 4 · AK1/3
L2/3
= 4 · F(Kt, Lt, At)

Ideas
Problems with Pure Competition: Pareto Optimal
Allocation
There is no way to change an allocation to make someone better off
without making someone else worse off.
Perfect competition results in Pareto optimality because P = MC.

Ideas
Pure Competition and Increasing Returns
Under increasing returns to scale, a firm faces
Initial fixed costs
Marginal costs
If P = MC under increasing returns, no firm will do research to invent
new ideas.
The fixed research costs will never be recovered.
To solve this problem we use patents, government funding, prizes.
However, P > MC results in welfare loss.

Romer Model
The Romer Model
Focuses on the distinction between ideas and objects
Yields four equations
Stipulates that output requires knowledge and labor
The production function of the Romer model
Constant returns to scale in objects alone
Increasing returns to scale in objects and ideas

Romer Model
Assumptions
New ideas depend on
The existence of ideas in the previous period
The number of workers producing ideas
Worker productivity
Unregulated markets traditionally do not provide enough resources to
produce ideas and hence they are underprovided.
The population has workers producing ideas and workers producing
output

Romer Model
Key Equations
Production function for output
Yt = AtLyt (2)
Production function for ideas
∆At+1 = zAtLat (3)
We can use labor to produce output or ideas. The terms Lyt and Lat
represent the shares of labor devoted to producing output and ideas,
respectively.

Romer Model
Key Equations
Resource constraint
Lyt + Lat = L (4)
An exogenous supply of labor is divided into producing output and
ideas.
Allocation of labor
Lat = `L (5)
Lyt = (1 − `)L
where ` represents that share of labor producing ideas.

Romer Model
The Romer Model
The model has four endogenous variables (Yt, At, Lyt, Lat)
The model has four equations:
Output production function: Yt = AtLyt
Idea production function: ∆At+1 = zAtLat
Resource constraint: Lyt + Lat = L
Allocation of labor: Lat = `L
Exogenous parameters: z, L, `, A0

Romer Model
Solving the Romer Model
Output per person depends on the total stock of knowledge.
yt ≡
Yt
L
= At(1 − `) (6)
Key results of this model stems from the nonrivalry of ideas.
Ideas spread. Whereas in the Solow model output per worker was
dependent on capital per worker.
Capital per worker is rivalrous, cannot be shared. My use of the
printer prevents someone else from using it.

Romer Model
Solving the Model - Stock of Knowledge
We need to solve for A, in our case the growth rate of A.
∆At+1 = zAtLat
Lat = `L
so...
∆At+1
At
= z`L (7)
The stock of ideas grows at a constant rate g = z`L
The stock of ideas depends on the initial level of knowledge
At = A0(1 + g)t
(8)

Romer Model
Final Solution
Combining Equations:
yt = At(1 − `), and
At = A0(1 + g)t, we have:
yt = A0(1 − `)(1 + g)t
(9)
The level of output per person is now written entirely as a function of
the parameters of the model.
Growth is constant.

Romer Model
Constant Growth

Romer Model
Growth in the Romer Model
The Romer model produces the desired long-run economic growth
whereas the Solow did not.
In the Solow model, capital has diminishing returns. Eventually,
capital and income stop growing.

Romer Model
Growth in the Romer Model
The Romer model does not have diminishing returns to ideas because
they are nonrivalous.
∆At+1 = zAtLat
Look at the exponents on the endogenous terms on the right side.
Labor and ideas have increasing returns together.
Returns to ideas are unrestricted.

Romer Model
Balanced Growth
The Solow model relies on transition dynamics
The Romer model
Does not exhibit transition dynamics
Instead, has balanced growth path.
The growth rates of all endogenous variables are constant.
g = z`L

Romer Model
Experiments in the Romer Model
Focus on the key parameters:
A0: Initial stock of ideas
`: fraction of population doing research
z: Productivity
L: Population
yt = A0(1 − `)(1 + g)t
g = z`L

Romer Model
Changing the Population L
A change in population changes the growth rate of knowledge.
An increase in population will immediately and permanently raise the
growth rate of per capita output.

Romer Model
Changing the Population

Romer Model
Changing the Research Share `
An increase in the fraction of labor making ideas, holding all other
parameters equal, will increase the growth rate of knowledge.
If more people work to produce ideas, less people produce output.
The level of output per capita jumps down initially.
But the growth rate has increased for all future years.
Output per person will be higher in the long run.

Romer Model
Change in the Research Share

Romer Model
Growth Effects versus Level Effects
The exponent on ideas in the production function Determines the
returns to ideas alone
If the exponent on ideas is not equal to 1:
The Romer model will still generate sustained growth.
Growth effects are eliminated if the exponent on ideas is less than 1.
due to diminishing returns

Romer and Solow
Combining Romer and Solow
Nonrivalry of ideas results in long-run growth along a balanced
growth path
Exhibits transition dynamics if economy is not on its balanced growth
path
For short periods of time countries can grow at different rates.
In the long run countries grow at the same rate.

Romer and Solow
Setting up the Combined Model
The model has five endogenous variables (Yt, Kt, At, Lat, Lyt)
The model has five equations:
Production function: Yt = AtK
1/3
t Lyt
2/3
Capital accumulation: ∆Kt+1 = It − dKt, where It = sYt.
Idea production function: ∆A = zAtLat
Resource constraint: Lyt + Lat = L
Allocation of labor: Lat = `L
Exogenous parameters: z, s, d, L, `, K0

Accounting
Growth Accounting
Growth accounting determines the sources of growth in an economy
And how they may change over time
Consider a production function that includes both capital (Kt) and
ideas (At).
Yt = AtK
1/3
t Lyt
2/3
The stock of ideas (At) is referred to as total factor productivity
(TFP).

Accounting
Using our rules for growth rates we can “linearize” the production
function:
gyt = gAt +
1
3
gKt +
2
3
gLyt
where:
gyt: growth rate of output
gAt: growth rate of knowledge
gKt: growth contribution from capital
gLyt: growth contribution from workers

Accounting
Controlling for Labor Hours
Need to subtract gLt from all terms (divide by Lt in production
function)
gyt − gLt = gAt +
1
3
(gKt − gLt) +
2
3
(gLyt − gLt)
where:
gyt − gLt: growth of Y /L
gAt: TFP growth
gKt − gLt: contribution from K/L
gLyt − gLt: labor composition.

Accounting
From 1973 − 95
Output in the United States grew half as fast as from 1948 − 73.
This slower era of growth is known as the productivity slowdown.

Accounting
From 1995 − 2002
Output grew nearly as rapidly as before the productivity slowdown.
This recent era is known as the new economy.

Accounting
Contributions to Growth
1948-14 1948-73 1973-95 1995-07 2007-14
Output per hour, Y /L 2.4 3.3 1.6 2.8 1.4
Contribution of K/L 0.9 1.0 0.8 1.1 0.6
Contribution of labor 0.2 0.2 0.2 0.2 0.3
Contribution of TFP 1.3 2.1 0.6 1.5 0.5

Conclusion
Concluding Long-Run Growth
Institutions (property rights, laws) play an important role in economic
growth.
The Solow and Romer models
Provide a basis for analyzing differences in growth across countries.
Do not answer why investment rates and TFP differ across countries.

Solow and Romer
Looking at Growth Rates
Using our rules for growth rates we can “linearize” the production
function:
gyt = gAt +
1
3
gKt +
2
3
gLyt
where:
gyt ≡ ∆Yt+1
Yt
gAt = ∆At+1
At
= zLat = z`L
gKt = ∆Kt+1
Kt
= s Yt
Kt
− d

Solow and Romer
Growth Rate of Capital and Output
In the Solow model we saw: Yt
Kt
= d
s
This implies the growth rates of capital and output will be constant.
So gK = gY .
Given a fixed supply of labor implies gL = 0.

Solow and Romer
Reduced form
Plugging in these results:
gyt = gAt +
1
3
gKt +
2
3
gLyt
g∗
y = g +
1
3
g∗
+
2
3
· 0
g∗
y =
3
2
g =
3
2
z`L
For the long-run combined model, this equation pins down the growth
rate of output and output per person

Solow and Romer
Key Findings
The growth rate of output is even larger in the combined model than
in the Romer model.
Output is higher in this model because ideas have a direct and an
indirect effect.
Increasing productivity raises output because
productivity has increased
higher productivity results in a higher capital stock.

Solow and Romer
Output Per Person
The equation for the capital stock can be solved for the
capital-output ratio along a balanced growth path.
The capital to output ratio is proportional to the investment rate
along a balanced growth path.
g∗
y = g∗
k =
K∗
t
Y ∗
t
=
s
g∗
y + d

Solow and Romer
Output Per Person
This solution for the capital-output ratio can be substituted back into
the production function and solved to get:
y∗
t =
Y ∗
t
L
=
s
g∗
y + d
1/2
A
∗3
2
t (1 − `)
Growth in At leads to sustained growth in output per person along a
balanced growth path
Output yt depends on the square root of the investment rate
A higher investment rate raises the level of output per person along
the balanced growth path.

Solow and Romer
Output Per Person
A permanent increase in the investment rate in the combined model
implies:
The balanced growth path of income is higher (parallel shift).
Current income is unchanged.
The economy is now below the new balanced growth path
The growth rate of income per capita is immediately higher.
The slope of the output path is steeper than the balanced growth path

Solow and Romer
Questions for Review
Why does the nonrivalry of ideas make growth possible?
What role does population play in helping us understand long-run
growth?
Are there likely too many or too few resources devoted to discovering
new ideas? Why?
What considerations affect the future of economic growth?

Solow and Romer

Chapter 6 - Romer Model

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