Neoclassical Models of Endogenous Growth
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1 Neoclassical Models of Endogenous Growth October 2007 () Endogenous Growth October / 20
2 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. () Endogenous Growth October / 20
3 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. () Endogenous Growth October / 20
4 Basic AK Model Ramsey model with capital share α = 1 and no technical change: y(t) = Ak(t) Household s optimal consumption path: ċ(t) c(t) = r(t) θ ρ Perfect competition ) Consumption growth is then r(t) = A δ ċ(t) c(t) = A δ ρ θ () Endogenous Growth October / 20
5 Aggregate resource constraint:,! dividing by k(t), we get Along a BGP k(t) k(t) Since y(t) = Ak(t) it follows that c(t) + k(t) + δk(t) = Ak(t). c(t) k(t) + k(t) k(t) = A δ. is constant ) c/k must be constant ) ċ(t) c(t) = k(t) k(t) ẏ(t) y(t) = k(t) k(t) = A δ ρ = g θ () Endogenous Growth October / 20
6 c. k=0 Saddlepath Figure: Phase Diagram for the AK Model k () Endogenous Growth October / 20
7 What about the transversality condition? D(T )k(t ) = e rt e gt k(0) goes to zero as T becomes large if and only if r > g A > A δ ρ θ () Endogenous Growth October / 20
8 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 () Endogenous Growth October / 20
9 A One-Sector Model with Physical and Human Capital Could expand de nition of "capital" as in augmented Solow model Resource constraint: Y = AK α H 1 α = C + I K + I H where K = I K δk and Ḣ = I H δh Implications are very similar to basic AK model (see Barro ch. 5) () Endogenous Growth October / 20
10 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 () Endogenous Growth October / 20
11 Assumptions Aggregate output is produced according to Y (t) = AK (t) α H(t) 1 α, 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)] 1 α (1) where y(t) = Y (t)/l(t), etc. () Endogenous Growth October / 20
12 Aggregate Resource constraint c(t) + k(t) + δk(t) = Ak(t) α [u(t)h(t)] 1 α (2) Competitive factor markets: u(t)h(t) 1 α r(t) = α δ (3) k(t) k(t) α w(t) = (1 α) (4) u(t)h(t) where w(t) = wage per unit of human capital () Endogenous Growth October / 20
13 Representative household preferences: U = Dynamic budget constraint Z 0 e ρt c(t)1 θ 1 θ dt. Human capital accumulation Boundary conditions: k(t) = r(t)k(t) + w(t)u(t)h(t) c(t) (5) ḣ(t) = B(1 u(t))h(t), (6) lim D(T )k(t ) 0 and u(t) 2 (0, 1) T! Note that there are 2 control variables and 2 state variables () Endogenous Growth October / 20
14 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 = e ρt c1 θ The Hamiltonian conditions are + λ [rk + wuh c] + µ [B(1 u)h] 1 θ dj dc dj dk dj du dj dh = e ρt c θ λ = 0 (7) = λr = λ (8) = λwh µbh = 0 (9) = λwu + µb(1 u) = µ (10) () Endogenous Growth October / 20
15 Di erentiating (7) w.r.t. time and combining with (8), we get Di erentiating (9) w.r.t. time ρ θ ċ c = λ λ = r λ λ + ẇ w = µ µ Substituting out λw in (10) using (9): µbu + µb(1 u) = µ B = µ µ () Endogenous Growth October / 20
16 It follows that r(t) = B + ẇ(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 () Endogenous Growth October / 20
17 The Balanced Growth Path,! situation where all aggregates grow at constant rates (need not be equal) If ḣ/h is constant ) u(t) = u is constant Let ċ/c = g Then from the Euler equation r(t) = r = θg It follows that from (3) that h(t)/k(t) is constant: Dividing the (2) by k(t) yields ḣ h = k k c(t) k(t) + k(t) k(t) + δ = ρ uh(t) 1 k(t) Since k/k is constant, c(t)/k(t) must be constant ) ċ c = k k = ḣ h = g α () Endogenous Growth October / 20
18 From (1) it follows that ẏ y = α k k + (1 α)ḣ h = g Since k(t)/h(t) is constant (4) implies But then from (11) we have ẇ(t) w(t) = 0. r = B It follows that the equilibrium growth rate is g = B θ ρ. () Endogenous Growth October / 20
19 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 () Endogenous Growth October / 20
20 Extension Human Capital Externalities Suppose the production function (in per capita terms) is y(t) = Ak(t) α [uh(t)] 1 α h a (t) γ, where h a = e ect of the average human capital not taken into account by rms,! perceived marginal product of human capital: k(t) α w(t) = (1 α) h a (t) γ u(t)h(t),! but, in equilibrium, h a (t) = h(t) () Endogenous Growth October / 20
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