The Romer Model
Prof. Lutz Hendricks
Econ520
February 7, 2017
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Issues
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We study models where intentional innovation drives
productivity growth.
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Romer model:
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The standard model of R&D goes back to Romer (1990).
Innovations are produced like any other good using R&D labor
as input.
Policy effects
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Policies, such as R&D subsidies, can change the rate at which
innovations are produced.
Surprisingly, it turns out that policies have no effect on
long-run growth.
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Learning Objectives
In this section you will learn:
1. how to analyze the Romer model
2. why R&D policies do not change the long-run growth rate of
the economy
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The Romer model
Solow block
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Production of goods works exactly like in the Solow Model
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Aggregate production function:
Yt = Ktα (At LYt )1−α
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Capital accumulation as in the Solow model
K̇t = sK Yt − δ Kt
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(1)
(2)
Labor input grows at a constant rate
g (L) = n
(3)
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Solow Block
What has changed?
Final goods production function has:
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constant returns to rival inputs: K and LY .
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has increasing returns to all inputs (including A)
Labor is divided into production (LY ) and R&D (LA ).
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R&D Block
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Ideas are produced just like other goods.
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The input is labor (LAt )
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not much changes if capital is an input, too.
The output is a number of new ideas.
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At is the number of ideas that have been invented up to t.
Ȧt is the number of ideas discovered today (or the rate at
which they are discovered).
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R&D Block
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The ideas production function:
λ
Ȧt = B̄LAt
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λ determines returns to scale.
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B̄ is a productivity parameter.
(4)
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Ideas are inputs to innovation
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How easy it is to produce a new idea depends on how much
has already been discovered.
B̄ = B Aφ
(5)
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If ideas help produce new ideas: φ > 0: A ↑ =⇒ B̄ ↑.
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If there is "fishing out:" φ < 0.
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Assume φ ≤ 1. (If φ > 1 odd things happen...).
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The ideas production function is then
Ȧ = B LAλ Aφ
g (A)=B
LλA
φ −1
A
(6)
(7)
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g(A)
Ideas production function
A
Even though ideas foster innovation (φ > 0), more ideas imply
slower g (A).
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Ideas production function
Note how similar this is to the law of motion for capital in the
Solow model
Model
Solow
Romer
K̇t
Ȧt
=
=
Productivity
sA1−α
B
“Capital”
Ktα
φ
At
Labor
Lt1−α
λ
LAt
Depreciation
−δ Kt
−0
It follows that there cannot be long-run growth in A/L when
λ + φ < 1 (details follow).
But we still can get long-run growth in Y/L.
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The Romer model
Behavior
So far we have described technologies.
To describe behavior, we make a Solow assumption:
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A constant saving rate
S/Y = I/Y = sK
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A constant labor allocation:
LA = sA L
(8)
LY = (1 − sA ) L
(9)
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Model summary
The Solow block:
Y = K α (A LY )1−α
(10)
K̇ = sK Y − δ K
(11)
nt
(12)
Ȧ = B LAλ Aφ
(13)
LY = sY L; LA = sA L
(14)
Lt = L0 e
Production of ideas:
Constant behavior:
The growth rate of ideas:
g (A) = B (sA L)λ Aφ −1
(15)
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Model summary
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This looks complicated, but isn’t.
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We have tricked the model such that Y and K don’t matter for
how A evolves.
Ȧ = B LAλ Aφ
(16)
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This would change, if we let Ȧ depend on K
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but that would not affect the results
only the algebra would be more complicated (see Romer 2011)
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Does the Model Make Sense?
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The production functions are arbitrary.
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There is only one input. Only one good.
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We can add those - it does not make any difference.
The labor allocation is fixed.
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All of this can be relaxed without changing anything too
important.
Where are the households, consumption, population growth ...
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But what matters are certain qualitative features, not the
exact functional form.
We wil get back to this.
This is important.
The literature does not make this assumption. It can talk
about patents, policy, ...
Ideas are produced like goods.
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Balanced growth path
Definition
A BGP is a path along which all variables grow at constant rates.
Why might this be interesting?
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Balanced growth path
At what rates do the endogenous objects grow on the BGP?
Result 1: g (k) = g (y)
Proof:
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Balanced growth path
Result 2: g (y) = g (A)
Proof:
Result
All long-run growth is due to R&D.
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Growth rate of ideas
g (A) =
λ n
1−φ
(17)
Proof:
Ideas production:
g (A) = B
LAλ
A1−φ
(18)
BGP: g (A) is constant =⇒ LAλ Aφ −1 is constant
Take growth rates of that
g (g (A)) = λ g (LA ) − (1 − φ ) g (A) = 0
(19)
With constant time allocation, sA : g (LA ) = n.
Solve for g (A). Done.
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Summary: Balanced growth
Balanced growth in the Romer model is characterized by:
g (y) = g (k) = g (A)
λ n
g (A)=
1−φ
(20)
(21)
All growth is due to innovation.
Why is this true?
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Why is all growth due to innovation?
Solow model:
Romer model:
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Balanced growth: Intuition
g (A) =
λ n
1−φ
(22)
Growth is simply a multiple of population growth
Behavior does not matter: sK and sA do not appear in (22).
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Intuition
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Consider the case φ = 0.
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Ideas production is then
Ȧ = B LAλ
(23)
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If the population is constant, LA is constant.
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In each period, the economy produces a constant number of
ideas.
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The growth rate of ideas, g (A) = B LA /A, falls to zero over
time.
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A fixed number of people cannot produce a growing stream of
ideas.
Population growth is necessary for sustained innovation (at a
constant rate).
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g(A)
How growth is sustained
A
g (A) = BAφ −1 LAλ
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Special Case: Phi = 1
With φ = 1, idea production becomes
g (A) = B LAλ
(24)
This is the case studied by Romer (1990).
The model has exploding growth, unless the population is constant.
This is clearly contradicted by post-war data: LA rose dramatically,
while g (y) was at best constant.
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Reality check
1. The model says: constant population - no growth.
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But we are still producing new ideas all the time.
How can we reconcile this?
2. What if the population shrinks over time?
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Is the long-run growth rate negative?
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Reading
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Jones (2013b), ch. 5.
Optional:
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Romer (2011), ch. 3.1-3.4
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Jones (2013a), ch. 6
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Advanced Reading
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Jones (2005) talks in some detail about the economics of ideas.
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Lucas (2009) and McGrattan and Prescott (2009) on openness
and growth
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References I
Jones, C. I. (2005): “Growth and ideas,” Handbook of economic
growth, 1, 1063–1111.
——— (2013a): Macroeconomics, W W Norton, 3rd ed.
Jones, Charles; Vollrath, D. (2013b): Introduction To Economic
Growth, W W Norton, 3rd ed.
Lucas, R. E. (2009): “Trade and the Diffusion of the Industrial
Revolution,” American Economic Journal: Macroeconomics,
1–25.
McGrattan, E. R. and E. C. Prescott (2009): “Openness,
technology capital, and development,” Journal of Economic
Theory, 144, 2454–2476.
Romer, D. (2011): Advanced macroeconomics, McGraw-Hill/Irwin.
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