ECON 582: The Neoclassical Growth Model (Chapter 8, Acemoglu)
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1 ECON 582: The Neoclassical Growth Model (Chapter 8, Acemoglu) Instructor: Dmytro Hryshko 1 / 21
2 Consider the neoclassical economy without population growth and technological progress. The optimal growth problem is then Z 1 s.t. max [k(t);c(t)] 1 t=0 0 exp( t)u(c(t))dt _k(t) = f(k(t)) k(t) c(t); k(0) > 0, k(t) 0 for all t. The current-value Hamiltonian can be written as ^H(k; c; ) = u(c(t)) + (t) [f(k(t)) k(t) c(t)] ; with state variable k, control variable c, and current-value costate variable. 2 / 21
3 Theorem 7.13 Suppose the problem of maximizing (7.60) subject to (7.61) and (7.62). Let ^H(t; x; y; ) be the current-value Hamiltonian given by (7.64). Then except at points of discontinuity of ^y(t), the optimal control pair (^y(t); ^x(t)) satises the following necessary conditions: ^H y (t; ^x(t); ^y(t); (t)) = 0 for all t 2 R + (7.65) (t) _(t) = ^H x (t; ^x(t); ^y(t); (t)) for all t 2 R + (7.66) _x(t) = ^H (t; ^x(t); ^y(t); (t)) for all t 2 R + ; x(0) = x 0 ; lim t!1 b(t)x(t) x 1 (7.67) lim [exp( t)(t)^x(t)] = 0: (7.69) t!1 where (7.69) is the transversality condition. 3 / 21
4 The economy consists of a set of identical households; population within each household grows at the rate n, L(0) = 1: L(t) = exp(nt): Labor supply is inelastic. Per-capita utility u(c(t)); household utility at t is L(t)u(c(t)) = exp(nt)u(c(t)). Then the objective function of each household at time t = 0 can be written as Z 1 0 exp( ( n)t)u(c(t))dt (8.3) 4 / 21
5 Assumption 3 (Neoclassical preferences): u(c(t)) is strictly increasing, concave, and twice dierentiable. Assumption 4 0 (Discounting): > n. (Ensures that, in the model without growth, discounted utility is nite.) Assume the economy without technological progress; production function Y (t) = F (K(t); L(t)). y(t) Y (t) L(t) = F K(t) L(t) ; 1 f(k(t)): Competitive factor markets imply R(t) = f 0 (k(t)) (8.5) w(t) = f(k(t)) k(t)f 0 (k(t)): (8.6) 5 / 21
6 The law of motion for the total household assets is _ A(t) = r(t)a(t) + w(t)l(t) c(t)l(t): Dene per-capita assets as a(t) A(t). Note that L(t) _a(t) = _ A(t)L(t) A(t) _ L(t) L(t)2 = _ A(t) L(t) a(t)n. Thus, Market-clearing implies _a(t) = (r(t) n)a(t) + w(t) c(t): (8.8) a(t) = k(t): (8.9) The no-ponzi condition: Z t a(t) exp (r(s) n)ds) = 0: (8.16) lim t!1 0 6 / 21
7 Denition of equilibrium Denition 8.1 A competitive equilibrium of the neoclassical growth model consists of paths of consumption, capital stock, wage rates, and rental rates of capital, [C(t); K(t); w(t); R(t)] 1 t=0, such that the representative household maximizes its utility given initial asset holdings (capital stock) K(0) > 0 and taking the time path of factor prices [w(t); R(t)] 1 t=0 as given; rms maximize prots taking the time path of factor prices as given; and factor prices are such that all markets clear. 7 / 21
8 Household maximization The current-value Hamiltonian: ^H(t; a; c; ) = u(c(t)) + (t) [w(t) + (r(t) n)a(t) c(t)] : Applying Theorem 7.13, ^H c = u 0 (c(t)) (t) = 0 (8.17) ^H a = (t)(r(t) n) = ( n)(t) _(t) (8.18) lim [exp ( ( n)t) (t)a(t)] = 0: (8.19) t!1 8 / 21
9 (8.18) implies: _(t) = (t) (r(t) ): (8.20) Dierentiating (8.17) wrt t, and dividing by (t), u 00 (c(t)) _c(t) = _(t); Using (8.20), u 00 (c(t))c(t) u 0 (c(t)) {z } "u(c(t)) _c(t) c(t) = _(t) (t) : _c(t) c(t) = 1 " u (c(t)) (r(t) ): (8.22) 9 / 21
10 From (8.22), consumption grows if the discount rate is below the rate of return on assets; the speed of growth is related to the elasticity of marginal utility of consumption. (8.20) implies (t) = (0) exp = u 0 (c(0)) exp Z t 0 Z t (r(s) 0 (r(s) )ds )ds : (8.24) The transversality condition (8.19) can be expressed as lim t!1 lim t!1 2 {z } R t = 0 (n )ds Z t 64exp ( ( n)t) a(t)u 0 (c(0)) exp a(t) exp (r(s) n)ds 0 Z t (r(s) )ds = 0 0 = 0: (8.25) 10 / 21
11 Thus, the transversality condition implies the no-ponzi condition (8.16). Since in equilibrium a(t) = k(t), it can be written as lim t!1 k(t) exp Z t 0 (r(s) n)ds = 0: (8.26) Thus, the discounted market value of capital in the very far future should equal zero. Any pair (^k(t); ^c(t)) that satises (8.22) and (8.25) corresponds to a competitive equilibrium. Since r(t) = R(t), (8.22) implies _c(t) c(t) = 1 " u (c(t)) (f 0 (k(t)) ): (8.28) 11 / 21
12 Steady-state equilibrium Requires that _c(t) = 0. From (8.28), f 0 (k ) = + : (8.35) (8.35) corresponds to the modied golden rule. Together with Assumption 4 0 this implies SS consumption is r = f 0 (k ) = > n: (8.36) c = f(k ) (n + )k : 12 / 21
13 Redene the production function as f(k) = A ~ f(k), where A is a shift parameter. Additional comparative statics results (A; ; n; < (A; ; n; < (A; ; n; = (A; ; n; < 0: A lower discount rate implies greater patience and thus greater savings. In the model with no technological progress, the savings rate in SS is s = (n + )k f(k : (8.38) ) 13 / 21
14 Proposition 8.4 In the neoclassical growth model with no technological progress, with Assumptions 1{4 0, there exists a unique equilibrium path starting from any k(0) > 0 and converging monotonically to the unique steady-state (k ; c ) with k given by (8.35). This equilibrium path is identical to the unique optimal growth path. 14 / 21
15 Technological change and the canonical neoclassical model The production function takes the form Y (t) = F (K(t); A(t)L(t)); A(t) = A(0) exp(gt). Kaldor facts: constant capital-output ratio, constant growth in output and constant capital share in income. Since K (t) = R(t)K(t), this implies that R(t) Y (t) and therefore r(t) should be constant. For consumption to grow at a constant rate, " u (c(t)) should be constant as well. 15 / 21
16 Balanced growth in the neoclassical model requires that asymptotically all technological change is purely labor-augmenting and the elasticity of intertemporal substitution tends to a constant " u. 16 / 21
17 It can be shown that the intertemporal elasticity of substitution is the inverse of the elasticity of marginal utility. The Arrow-Pratt coecient of relative risk aversion for a twice dierentiable concave utility function u(c) is dened as < = u 00 (c)c u 0 (c) : The Constant Relative Risk Aversion utility function (CRRA) satises the property that < is constant: u(c) = c1 1 if 6= 1; 0 1 = log c if = 1: 17 / 21
18 The Euler equation of the representative household is _c(t) c(t) = 1 (r(t) ) : (8.50) Consumption per capita will be growing; dene ~c(t) = C(t) A(t)L(t) = c(t) A(t) ; which is constant along the BGP. Then, d~c(t)=dt ~c(t) = 1 (r(t) g) : 18 / 21
19 The law of motion of k(t) = K(t) A(t)L(t) : _k(t) = f(k(t)) ~c(t) (n + g + )k(t): (8.51) The transversality condition: Z t k(t) exp lim t!1 0 [f 0 (k(s)) g n]ds Since ~c(t) remains constant on a BGP, = 0: (8.52) f 0 (k ) = + + g: (8.53) The equation uniquely determines the SS value of k. 19 / 21
20 The neoclassical growth model, like the Solow model, endogenizes the capital-labor ratio, but not the growth rate of the economy. The advantage of the neoclassical growth model is that the capital-labor ratio, normalized output and consumption are determined by the preferences of the individuals rather than by an exogenously given saving rate. 20 / 21 The transversality condition can be written as Z t k(t) exp [ (1 )g n]ds lim t!1 This is ensured if the following holds: 0 = 0; Assumption 4 (discounting with technological progress): n > (1 )g. This is equivalent to r > n + g.
21 The role of policy Suppose the returns on capital net of depreciation are taxed at the rate and the proceeds are redistributed lump-sum back to households. The relevant equations are _k(t) = f(k(t)) ~c(t) (n + g + )k(t) d~c(t)=dt ~c(t) = 1 (r(t) g) = 1 ((1 )(f 0 (k(t)) ) g) On a BGP, f 0 (k ) = + + g 1 : (8.57) Thus, a higher tax rate reduces k and income per capita. 21 / 21
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