The Romer Model

Prof. Lutz Hendricks

Econ520

February 7, 2017

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Issues

I We study models where intentional innovation drivesproductivity growth.

I Romer model:I The standard model of R&D goes back to Romer (1990).I Innovations are produced like any other good using R&D labor

as input.

I Policy effectsI Policies, such as R&D subsidies, can change the rate at which

innovations are produced.I Surprisingly, it turns out that policies have no effect onlong-run growth.

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Learning Objectives

In this section you will learn:

1. how to analyze the Romer model2. why R&D policies do not change the long-run growth rate of

the economy

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The Romer modelSolow block

I Production of goods works exactly like in the Solow ModelI Aggregate production function:

Yt = Kαt (AtLYt)

1−α (1)

I Capital accumulation as in the Solow model

Kt = sKYt−δKt (2)

I Labor input grows at a constant rate

g(L) = n (3)

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Solow BlockWhat has changed?

Final goods production function has:

I constant returns to rival inputs: K and LY .I 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

I Ideas are produced just like other goods.I The input is labor (LAt)

I not much changes if capital is an input, too.

I The output is a number of new ideas.I At is the number of ideas that have been invented up to t.I At is the number of ideas discovered today (or the rate at

which they are discovered).

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R&D Block

I The ideas production function:

At = BLλAt (4)

I λ determines returns to scale.I B is a productivity parameter.

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Ideas are inputs to innovation

I How easy it is to produce a new idea depends on how muchhas already been discovered.

B = B Aφ (5)

I If ideas help produce new ideas: φ > 0: A ↑=⇒ B ↑.I If there is "fishing out:" φ < 0.I Assume φ ≤ 1. (If φ > 1 odd things happen...).I The ideas production function is then

A = B LλA Aφ (6)

g(A)=B LλA Aφ−1 (7)

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Ideas production function

A

g(A)

Even though ideas foster innovation (φ > 0), more ideas implyslower g(A).

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Ideas production function

Note how similar this is to the law of motion for capital in theSolow model

Model Productivity “Capital” Labor DepreciationSolow Kt = sA1−α Kα

t L1−αt −δKt

Romer At = B Aφ

t LλAt −0

It follows that there cannot be long-run growth in A/L whenλ + φ < 1 (details follow).But we still can get long-run growth in Y/L.

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The Romer modelBehavior

So far we have described technologies.To describe behavior, we make a Solow assumption:

I A constant saving rate

S/Y = I/Y = sK

I A constant labor allocation:

LA = sAL (8)LY = (1− sA) L (9)

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Model summaryThe Solow block:

Y = Kα (A LY)1−α (10)

K = sK Y−δ K (11)Lt = L0 ent (12)

Production of ideas:A = B Lλ

A Aφ (13)

Constant behavior:

LY = sY L; LA = sA L (14)

The growth rate of ideas:

g(A) = B (sA L)λ Aφ−1 (15)

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Model summary

I This looks complicated, but isn’t.I We have tricked the model such that Y and K don’t matter for

how A evolves.A = B Lλ

A Aφ (16)

I This would change, if we let A depend on KI but that would not affect the resultsI only the algebra would be more complicated (see Romer 2011)

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Does the Model Make Sense?

I The production functions are arbitrary.I But what matters are certain qualitative features, not the

exact functional form.I We wil get back to this.

I There is only one input. Only one good.I All of this can be relaxed without changing anything too

important.

I Where are the households, consumption, population growth ...I We can add those - it does not make any difference.

I The labor allocation is fixed.I This is important.I The literature does not make this assumption. It can talk

about patents, policy, ...

I 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) = BLλ

AA1−φ

(18)

BGP: g(A) is constant =⇒ LλAAφ−1 is constant

Take growth rates of that

g(g(A)) = λg(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) (20)

g(A)=λ n

1−φ(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 growthBehavior does not matter: sK and sA do not appear in (22).

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Intuition

I Consider the case φ = 0.I Ideas production is then

A = B LλA (23)

I If the population is constant, LA is constant.I In each period, the economy produces a constant number of

ideas.I The growth rate of ideas, g(A) = B LA/A, falls to zero over

time.I A fixed number of people cannot produce a growing stream of

ideas.

Population growth is necessary for sustained innovation (at aconstant rate).

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How growth is sustained

A

g(A)

g(A) = BAφ−1LλA

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Special Case: Phi = 1

With φ = 1, idea production becomes

g(A) = B LλA (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.I But we are still producing new ideas all the time.I How can we reconcile this?

2. What if the population shrinks over time?I Is the long-run growth rate negative?

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Reading

I Jones (2013b), ch. 5.

Optional:

I Romer (2011), ch. 3.1-3.4I Jones (2013a), ch. 6

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Advanced Reading

I Jones (2005) talks in some detail about the economics of ideas.I 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 economicgrowth, 1, 1063–1111.

——— (2013a): Macroeconomics, W W Norton, 3rd ed.

Jones, Charles; Vollrath, D. (2013b): Introduction To EconomicGrowth, W W Norton, 3rd ed.

Lucas, R. E. (2009): “Trade and the Diffusion of the IndustrialRevolution,” American Economic Journal: Macroeconomics,1–25.

McGrattan, E. R. and E. C. Prescott (2009): “Openness,technology capital, and development,” Journal of EconomicTheory, 144, 2454–2476.

Romer, D. (2011): Advanced macroeconomics, McGraw-Hill/Irwin.

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