Neoclassical Models of Endogenous Growth October 2007 () - - PowerPoint PPT Presentation

Neoclassical Models of Endogenous Growth

October 2007

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Motivation

What are the determinants of long run growth? Growth in the "e¤ectiveness of labour" should depend on economic incentives , ! decision makers who make A grow must be rewarded , ! BUT since F(K, AL) exhibits CRS when A is exogenous, it must exhibit IRS when A is a separate factor , ! not all factors can be paid their marginal products , ! inconsistent with perfect competition and, hence, the neoclassical framework.

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Alternative Paradigms of Endogenous Growth

Neoclassical or AK paradigm , ! e¤ectively assumes that raw labour, L, is not a factor of production , ! emphasizes knowledge that is embodied in the work force , ! growth promoting factor (human capital) is a private, rival good with no dynamic externalities Endogenous technological change paradigm , ! incorporates IRS by allowing for imperfect competition in a GE framework , ! emphasizes knowledge that is disembodied , ! growth promoting factor (ideas) is a non–rival, public good with dynamic externalities.

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Basic AK Model

Ramsey model with capital share α = 1 and no technical change: y(t) = Ak(t) Household’s optimal consumption path: ˙ c(t) c(t) = r(t) ρ θ Perfect competition ) r(t) = A δ Consumption growth is then ˙ c(t) c(t) = A δ ρ θ

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Aggregate resource constraint: c(t) + ˙ k(t) + δk(t) = Ak(t). , ! dividing by k(t), we get c(t) k(t) + ˙ k(t) k(t) = A δ. Along a BGP

˙ k(t) k(t) is constant ) c/k must be constant

) ˙ c(t) c(t) = ˙ k(t) k(t) Since y(t) = Ak(t) it follows that ˙ y(t) y(t) = ˙ k(t) k(t) = A δ ρ θ = g

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c k k=0 . Saddlepath

Figure: Phase Diagram for the AK Model

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What about the transversality condition? D(T)k(T) = erT egT k(0) goes to zero as T becomes large if and only if r > g A > A δ ρ θ

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Implications

Simplest possible endogenous growth model ) long–run growth rate depends on level of MP of capital (net of depreciation) relative to discount rate ) growth increases with willingness of households to substitute consumption across time BUT most estimates …nd diminishing returns to physical capital and wages/salaries ' 2/3 of output , ! this simple model does not conform well with basic observations Also implies no conditional convergence

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A One-Sector Model with Physical and Human Capital

Could expand de…nition of "capital" as in augmented Solow model Resource constraint: Y = AK αH1α = C + IK + IH where ˙ K = IK δK and ˙ H = IH δH Implications are very similar to basic AK model (see Barro ch. 5)

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A Two-Sector Model with Physical and Human Capital

Uzawa–Lucas Model

Based on “The Mechanics of Economic Development” (Lucas, 1988) , ! emphasizes the central role of human capital accumulation in driving long-run growth Simpli…ed version: no population growth and no externalities Focus on balanced (steady state) growth path Sectors producing human and physical capital di¤er

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Assumptions

Aggregate output is produced according to Y (t) = AK(t)αH(t)1α, where H(t) = u(t)h(t)L(t). and u(t) = fraction of labour time allocated to working h(t) = human capital per worker In per capita terms: y(t) = Ak(t)α [u(t)h(t)]1α (1) where y(t) = Y (t)/L(t), etc.

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Aggregate Resource constraint c(t) + ˙ k(t) + δk(t) = Ak(t)α [u(t)h(t)]1α (2) Competitive factor markets: r(t) = α u(t)h(t) k(t) 1α δ (3) w(t) = (1 α)

• k(t)

u(t)h(t) α (4) where w(t) = wage per unit of human capital

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Representative household preferences: U =

Z ∞

eρt c(t)1θ 1 θ dt. Dynamic budget constraint ˙ k(t) = r(t)k(t) + w(t)u(t)h(t) c(t) (5) Human capital accumulation ˙ h(t) = B(1 u(t))h(t), (6) Boundary conditions: lim

T !∞ D(T)k(T) 0 and u(t) 2 (0, 1)

Note that there are 2 control variables and 2 state variables

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Optimality conditions if both k and h are accumulated Note that there are 2 control variables and 2 state variables Hamiltonian for household’s optimization problem: J = eρt c1θ 1 θ + λ [rk + wuh c] + µ [B(1 u)h] The Hamiltonian conditions are dJ dc = eρtcθ λ = 0 (7) dJ dk = λr = ˙ λ (8) dJ du = λwh µBh = 0 (9) dJ dh = λwu + µB(1 u) = ˙ µ (10)

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Di¤erentiating (7) w.r.t. time and combining with (8), we get ρ θ ˙ c c = ˙ λ λ = r Di¤erentiating (9) w.r.t. time ˙ λ λ + ˙ w w = ˙ µ µ Substituting out λw in (10) using (9): µBu + µB(1 u) = ˙ µ B = ˙ µ µ

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It follows that r(t) = B + ˙ w(t) w(t) (11) if both h and k are being accumulated by the household, the rates of return must be equal , ! otherwise only the asset with the highest return will be accumulated

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The Balanced Growth Path

, ! situation where all aggregates grow at constant rates (need not be equal) If ˙ h/h is constant ) u(t) = u is constant Let ˙ c/c = g Then from the Euler equation r(t) = r = θg ρ It follows that from (3) that h(t)/k(t) is constant: ˙ h h = ˙ k k Dividing the (2) by k(t) yields c(t) k(t) + ˙ k(t) k(t) + δ = uh(t) k(t) 1α Since ˙ k/k is constant, c(t)/k(t) must be constant ) ˙ c c = ˙ k k = ˙ h h = g

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From (1) it follows that ˙ y y = α ˙ k k + (1 α) ˙ h h = g Since k(t)/h(t) is constant (4) implies ˙ w(t) w(t) = 0. But then from (11) we have r = B It follows that the equilibrium growth rate is g = B ρ θ .

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Implications

Similar expression to basic AK model growth rate BUT growth depends on the productivity of human capital sector, B , ! does not depend on marginal product of physical capital Physical capital accumulation is NOT the “engine of growth" here , ! capital stock adjusts so that r = B in the long run Lucas model generates endogenous growth in a competitive model while preserving diminishing returns to physical capital Transitional dynamics ) conditional convergence

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Extension — Human Capital Externalities

Suppose the production function (in per capita terms) is y(t) = Ak(t)α[uh(t)]1αha(t)γ, where ha = e¤ect of the average human capital not taken into account by …rms , ! perceived marginal product of human capital: w(t) = (1 α)

• k(t)

u(t)h(t) α ha(t)γ , ! but, in equilibrium, ha(t) = h(t)

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