ECON 581. Endogenous growth

models: Second generation

Instructor: Dmytro Hryshko

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Product-variety Models. Key elements

Productivity growth comes from an expanding varietyof specialized intermediate products.

For each new product there’s a sunk cost of productinnovation that must be incurred once, when theproduct is first introduced (e.g., costs of research).

Technological knowledge consists of a list of blueprints,each describing how to produce a different(intermediate) product.

Fixed costs make intermediate-product marketsmonopolistically competitive; imperfect competitioncreates positive profits, which act as a reward for thecreation of new products.

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There’s a fixed number L of people, each living forever andsupplying one unit of labor inelastically used inmanufacturing.

Utility function is u(c) = c1−θ

1−θ . The growth rate of percapita consumption, from the Euler equation, is

g =r − ρθ

.

Final output is produced under perfect competition; theaggregate technology is

Y (t) = L1−α∫ M(t)

0

xi(t)αdi,

where M(t) is a measure of product variety; intermediateinputs are indexed by i ∈ [0,M(t)]; α ∈ [0, 1].

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Each intermediate product is produced using the final goodas input, one for one. Let X(t) be the total amount of finalgood used in producing intermediate products. Thus,

X(t) =

∫ M(t)

0

xidi.

Suppose that xi(t) = x; then x = X(t)M(t)

. Plugging into theproduction function

Y (t) = L1−α∫ M(t)

0

(X(t)/M(t))α di = M(t)1−αL1−αX(t)α.

∂Y (t)∂M(t)

= (1− α) Y (t)M(t)

> 0. The economy’s GDP is defined as

GDP(t) = Y (t)−X(t).

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Each intermediate product is produced by a monopolistwhose objective is to maximize the flow of profit, measuredin units of final good:

Πi = pixi − xi,

where pi is good i’s price in units of final good. Theperfectly competitive firm takes the price pi as given andchooses xi at the amount when

pi =∂Y

∂xi= αL1−αxα−1i .

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The monopolist’s profit is then

Πi = αL1−αxαi − xi.

The FOC is

α2L1−αxα−1i − 1 = 0.

it follows that the equilibrium quantity is the same in everysector i:

x = α2

1−αL.

Plugging it into the profit function, gives the flow of profit

Π =1− αα

α2

1−αL.

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Since Y (t) = M(t)1−αL1−αX(t)α and X(t) = xM(t),

Y (t) = M(t)L1−αxα

GDP(t) = M(t)(L1−αxα − x

).

Thus,

g =M(t)

M(t).

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Research sector

Assume that

M(t) = λR(t),

where R(t) is the amount of final good used in research,and λ is the productivity of research sector. Researchsector is perfectly competitive. Each blueprint is worth Π/rto the researcher. The flow of profit in research sector is

Π

rλR(t)−R(t) = 0,

which is satisfied when

r = λΠ.

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Using the Euler equation and the expression for Π,

g =λΠ− ρ

θ=λ1−α

αα

21−αL− ρθ

.

Thus, the growth

increases with the productivity of research, λ;

increases with the size of the economy, L;

decreases with the rate of time preference, ρ.

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Romer’s model with labor as R&D inputLet

L = L1 + L2,

where L1 is labor used for final-good production and L2 islabor used in research. Final good is produced as

Y (t) = L1−α1

∫ M(t)

0

xαi di.

As before,

x = α2

1−αL1

Π =1− αα

α2

1−αL1.

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Assume

M(t) = λM(t)L2.

The flow of profit in research sector is

Π

rλM(t)L2 − w(t)L2 = 0,

which is satisfied when

r = λM(t)Π/w(t).

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As before, Y (t) = L1−α1 M(t)xα. Wages should be the same

in the final-goods and research sectors.

w(t) = FL1 = (1− α)L−α1 M(t)xα = (1− α)α2α1−αM(t).

Using our results for Π and w(t),

r = αλL1.

g =M(t)

M(t)= λL2 =

αλL1 − ρθ

=αλL− α

=g︷︸︸︷λL2 −ρ

θ,

g =αλL− ρα + θ

.

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Summary of Romer’s modelGrowth is driven by innovations that lead to introductionof new input varieties.

Productivity growth is driven both by the increasedspecialization of labor working with an increasing numberof intermediate inputs and by the research spilloverswhereby each innovator benefits from the whole existingstock of innovations.

Ideas are nonrival but excludable (each innovation isrewarded by monopoly rents).

A limitation of the model is that it doesn’t capture therole of exit in the growth process; exit is detrimental in themodel as it reduces input specialization. Recent empiricalwork finds a strong correlation between productivitygrowth and exit or turnover of firms and inputs.

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The Schumpeterian model

The model embodies the force of “creativedestruction”—innovations that drive growth by creatingnew technologies also destroy the results of previousinnovations by making them obsolete.

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A one-sector model. The basics

Time is discrete: t = 1, 2, . . ..

In each period, there’re L individuals who live for oneperiod and endowed with one unit of labor, suppliedinelastically.

The final good is produced by perfectly competitivefirms using labor and a single intermediate product:Yt = (AtL)1−αxαt , α ∈ (0, 1).

The intermediate product is produced by a monopolist,using the final good as all input, one for one.

GDPt = Yt − xt.

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Production and profits

The monopolist mazimizes

Πt = ptxt − xt,

where pt is the price of the intermediate product in termsof the final good. The final-goods producer’s demand forgood xt should satisfy:

pt =∂Yt∂xt

= α (AtL)1−α xα−1t .

Thus,

Πt = α(AtL)1−αxαt − xt.

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The FOC implies

xt = α2

1−αAtL,

and an equilibrium profit

Πt = πAtL, π ≡ (1− α)α1+α1−α , (P)

both proportional to the effective labor supply AtL.Plugging xt into the production function,

Yt = α2α1−αAtL.

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InnovationEach period, there’s one entrepreneur who attempts aninnovation; if successful the innovation creates a newversion of the intermediate product, more productive thanprevious ones.

At = γAt−1, γ > 1 if successful

At = At−1 if failure.

The probability µt that an innovation occurs at t dependson the amount of Rt of final good spent on research:

µt = φ(Rt/A∗t︸ ︷︷ ︸

=nt

) = λnσ, σ ∈ (0, 1), A∗t = γAt−1.

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Research arbitrage

Expected net benefit to an innovator with the desiredprofit, Π∗t , is

φ(Rt/A∗t )Π

∗t −Rt.

FOC wrt Rt:

φ′(Rt/A∗t )Π

∗t/A

∗t − 1 = 0.

Using (P),

φ′(nt)πL︸ ︷︷ ︸=MB

= 1︸︷︷︸=MC

. (R)

(R) implies that nt and, therefore, φ(nt) are constant.

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Using Cobb-Douglas function,

n = (σλπL)1

1−σ µ = λ1

1−σ (σπL)σ

1−σ .

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GrowthThe growth in per capita GDP is random and equals

gt =At − At−1At−1

= µ× γAt−1 − At−1At−1

+ (1− µ)× 0.

By LLN,

gt = E(gt) = µ(γ − 1).

In the LR, the economy’s average growth rate equals thefrequency of innovations times the size of innovations.

Using the definition of µ:

g = λ1

1−σ (σπL)σ

1−σ (γ − 1). (G)

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Comparative statics

g = λ1

1−σ (σπL)σ

1−σ (γ − 1). (G)

Growth increases with:

the productivity of innovations, λ;

the size of innovations, γ (the productivityimprovement factor);

an increase in the size of population.

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