growth

# growth - 1 The Neoclassical Growth Model The neoclassical...

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1 The Neoclassical Growth Model The neoclassical growth model, which originated with the work of Solow and Swan, consists of the following relationships: a production function, y t = f ( h t ;k t ) ,wh e r e y t is output, h t is labor, and k t is capital at date t =0 ; 1 ; 2 ::: , plus a law of motion for the capital stock, k t +1 =(1 ¡ ± ) k t + ¾y t , where ± 2 (0 ; 1) is the depreciation rate and ¾ 2 (0 ; 1) is the savings rate. We assume that f is homogeneous of degree 1, increasing, concave, and twice continuously di¤erentiable. We assume for now that h t is constant and normalize h t =1 (this assumption is not very accurate, but we will soon endogenize h t by letting agents choose how much time to spend working and how much to spend in other activities). Let F ( k )= f (1 ;k ) .T h e n F 0 > 0 , F 00 < 0 . We further assume that F (0) = 0 , F 0 (0) = 1 and F 0 ( 1 )=0 .Th e initial capital stock, k 0 , is given exogenously. Note that behavior is exogenous here: the representative individual in this economy simply works a …xed number of hours h t =1 ,savesorinvests i t = ¾y t , and consumes c t =(1 ¡ ¾ ) y t , each period. Substituting the production function into the law of motion for capital yields a …rst order di¤erence equation in k t , k t +1 =(1 ¡ ± ) k t + ¾F ( k t ) ´ g ( k t ) : (1) Together with the initial condition k 0 , (1) completely determines the entire time path of the capital stock. Given this path we can compute the paths for y t , c t ,etc .A steady state of the system is a solution to k = g ( k ) . 1 1 One often sees the model in continuous time, in which case the law of motion for capital is written _ k t = ¾F ( k t ) ¡ ±k t (see below for an explicit derivation of the continuous time as the limit of discrete time models). Also, one often sees the model augmented to include population growth, with the number of agents at t given by N t = e nt .I n this case, one needs to distinguish between total and per capital variables. Thus, let Y t = f ( N t ;K t )= e nt F ( K t =N t ) be the production function; then in per capita terms we have y t = F ( k t ) . Also, let _ K

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Figure 1: Neolclassical Growth Model We graph g ( k ) versus k in Figure 1. Our assumptions imply that, as shown, g (0) = 0 , g 0 (0) > 1 , and there is a unique k ¤ > 0 such that k ¤ = g ( k ¤ ) . Hence, the model has two steady states, k =0 and k = k ¤ .Mo reov e r ,fo ra l l k 0 > 0 , k t ! k ¤ (monotonically). Hence, as t !1 , y t ! y ¤ , c t ! c ¤ ,etc.At k ¤ ,wehave ¾F ( k ¤ )= ±k ¤ , which implies that savings just o¤sets depreciation and the capital-output ratio is k y = ¾ ± , and also that c ¤ = y ¤ ¡ ±k ¤ . Clearly, k ¤ is increasing in ¾ ± .Mo r eov e r , c ¤
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growth - 1 The Neoclassical Growth Model The neoclassical...

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