Endogenous Technological Change October 2007 () Endogenous - - PowerPoint PPT Presentation

Endogenous Technological Change

October 2007

() Endogenous Techological Change October 2007 1 / 19

Basic idea

Economic growth ! increased market size ! higher pro…ts ! justi…es the …xed costs of knowledge creation ! knowledge growth ! higher productivity of labour/capital ! economic growth Two classes of model: (1) Growth sustained by increased specialization of labour across an increasing variety of activities (A. Smith, 1776, Romer, 1990) (2) Growth through creative destruction (Schumpeter, 1935, Aghion and Howitt, 1992)

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A baseline GE model with a variety of products

Utility maximizing households , ! …xed population L with CES preferences , ! assets are claims to pro…ts of …rms (no capital) Competitive …nal goods producers i combine labour Li and intermediates to produce output Yi Monopolistically competitive intermediate producers j 2 f1, Ng protected by patents R&D …rms allocate resources to invention, then sell ideas to producers

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Producers of Final Ouput

Production function: Yi = AL1α

i N

∑

j=1

X α

ij

0 < α < 1 where Xij = quantity of intermediate j used by …nal output producer i All …nal output is assumed identical with price = 1 (numeraire) Pro…t: Yi wLi

N

∑

j=1

PjXij FOCs: ∂Yi ∂Li = (1 α)Yi Li = w ∂Yi ∂Xij = αAL1α

i

X α1

ij

= Pj 8 j

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Demand function for Xij : Xij = Li Aα Pj

• 1

1α

Aggregate demand for labour L = ∑

i

Li = 1 α w

∑

i

Yi = (1 α) Y (t) w(t)

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Intermediate Good Producers

Simpli…cation: perpetual monopoly rights conferred by patent Present value of pro…ts: V (t) =

Z ∞

t

D(t, s)πj(s)ds where πj(s) is ‡ow of pro…ts at time s and D(t, s) = e R s

t r(ω)dω () Endogenous Techological Change October 2007 6 / 19

Note that dV dt =

Z ∞

t

dD(t, s) dt πj(s)ds D(t, t)πj(t) ˙ V (t) = r(t)

Z ∞

t

D(t, s)πj(s)ds πj(t) ˙ V (t) = r(t)V (t) πj(t) This can be expressed as r(t) = πj(t) V (t) + ˙ V (t) V (t) , ! rate of return on claims to pro…ts = dividend yield plus capital gains

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Cost of producing one unit of intermediate = one unit of output , ! pro…t is then πj = [Pj 1] Xj where Xj = ∑

i

Xij = Aα Pj

• 1

1α

∑

i

Li = Aα Pj

• 1

1α

L Pro…t-maximization ) max

Pj

πj = [Pj 1] Aα Pj

• 1

1α

L =

• P

1

1 1α

j

P

• 1

1α

j

• (Aα)

1 1α L () Endogenous Techological Change October 2007 8 / 19

FOC is dπj dPj =

• 1

1 1 α

• P
• 1

1α

j

+

• 1

1 α

• P
• 1

1α 1

j

• (Aα)

1 1α L = 0

, ! this simpli…es to

• 1

1 1 α

• +
• 1

1 α

• P1

j

= 0 , ! multiply through by 1 α and solve for Pj Pj = 1 α > 1 , ! monopoly price is a constant mark-up over marginal cost Quantity of each intermediate j used in …nal goods industry i: Xij =

• Aα2

1 1α Li () Endogenous Techological Change October 2007 9 / 19

Final output in industry i Yi = AL1α

i

N

• Aα2

α 1α Lα

i

= A

1 1α α 2α 1α NLi

Aggregate output: Y (t) = A

1 1α α 2α 1α LN(t)

, ! output must grow at the same rate as N Pro…ts are πj = π = 1 α α Aα2

1 1α L

Present discounted value of pro…ts V (t) = 1 α α Aα2

1 1α L

Z ∞

t

D(s)ds

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R&D Sector

The cost of producing one new idea = η units of output Free Entry condition: V (t) η If V (t) < η ) no resources allocated to R&D. We focus on case where V (t) = η. From our asset pricing equation r(t) = π V (t) + ˙ V (t) V (t) , ! it follows that r(t) = r = 1 η 1 α α Aα2

1 1α L () Endogenous Techological Change October 2007 11 / 19

Households

As usual, their optimal decision is characterized by ˙ c(t) c(t) = ˙ C(t) C(t) = r ρ θ = γ where C(t) = c(t)L. Dynamic budget constraint: ˙ z(t) = w(t) + rz(t) c(t) where z(t) = assets.

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In aggregate ˙ Z = wL + rZ C Total assets = market value of …rms Z(t) = V (t)N(t) = ηN(t) It follows that ˙ Z = η ˙ N and so η ˙ N = wL + rηN C , ! divide through by N and note that w(t)L = (1 α)Y (t) η ˙ N N = (1 α)Y (t) N(t) + rη C(t) N(t)

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Along a BGP, ˙ N/N is constant. Since Y /N and r are constant: ˙ Y Y = ˙ N N = ˙ C C = g It follows that output growth is given by γ = 1 θ 1 η 1 α α Aα2

1 1α L ρ

• Note that the transversality condition requires that r > γ

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Implications

A, θ and ρ a¤ect growth in the same way as in the AK model A decrease in the unit cost of R&D, η, raises the rate of return r and thereby raises the growth rate Scale e¤ect: larger the labour endowment ) higher growth rate , ! demand: L " ) larger “market” for ideas ) more incentive to invent , ! supply: L " ) opportunity cost of invention is lower Cross-country, post WWII data ) no scale e¤ect (Jones, 1999) Long term, regional data ) evidence of scale e¤ect (Kremer, 1993) More recent extensions ) endogenous technical change without scale e¤ects

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Comparison to the Social Optimum

Because of imperfect competition, the decentralized equilibrium is not Pareto optimal. Consider a a hypothetical social planner that chooses aggregates to maximize household utility. Output is either consumed, used to produce intermediates or used in R&D: Y = AL1αN1αX α = C + X + η ˙ N and so ˙ N = 1 η

• AL1αN1αX α cL X
• The Hamiltonian for the planner’s problem is

J = eρt c1θ 1 θ + λ 1 η

• AL1αN1αX α cL X
• ,

! here there are 2 control variables, c and X and 1 state variable, N.

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The Hamiltonian conditions are dJ dc = eρtcθ λ η L = 0 (1) dJ dX = λ 1 η

• αAL1αN1αX α1 1

= 0 (2) dJ dN = λ 1 η (1 α)AL1αNαX α = ˙ λ (3) , ! as usual, from (1) we have ρ θ ˙ c c = ˙ λ λ , ! condition (2) implies X N = (αA)

1 1α L () Endogenous Techological Change October 2007 17 / 19

It follows that (3) can be written 1 η (1 α)A (αA)

α 1α =

˙ λ λ , ! substituting out ˙ λ/λ we get ρ + θ ˙ c c = 1 η (1 α)A

1 1α α α 1α

It follows that the optimal growth rate of consumption is γP = 1 θ 1 η 1 α α

• (Aα)

1 1α L ρ

• Recall that

γ = 1 θ 1 η 1 α α Aα2

1 1α L ρ

• ,

! since α < 1, γP > γ

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Implications

The decentralized growth rate is too low relative to the social

• ptimum

, ! output of monopolists is too low ) reduces return to innovation Note that α # ) monopoly mark-up 1/α " and

• γP γ "

) monopoly pro…ts are necessary for growth, but greater monopoly power is not optimal Could achieve social optimum by subsidizing purchases of intermediates

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