ps4_solution - Noah Williams Department of Economics...

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Noah Williams Economics 503 Department of Economics Macroeconomic Theory Princeton University Fall 2006 Problem Set 4 Solutions 1. We now consider a recursive version of the neoclassical model with distorting taxes. An infinitely-lived representative household owns a stock of capital k which it rents to firms. The household’s capital stock depreciates at rate δ , and denote aggregate capital by K . Households do not value leisure and are endowed with one unit of time each period with which they can supply labor N to firms. They have standard time additive expected utility preferences with discount factor β and period utility u ( c ). Firms produce output according to the production function zF ( K, N ) where z is the stochastic level of technology which is Markov with transition function P ( z 0 , z ) . There is also a government which levies a proportional tax τ on households’ capital income and labor income. The tax rate is the same for both types of income and constant over time. The government uses the proceeds of the taxes to provide households with a lump sum transfer payment T (which may vary over time), and suppose that in equilibrium the government must balance its budget every period. (a) Define a recursive competitive equilibrium for this economy. Be specific about all of the objects in the equilibrium and the conditions they must satisfy. Solution. A recursive competitive equilibrium with a government consists of pric- ing functions r ( K, z ) , w ( K, z ) , decision rules K 0 = K ( K, z ) , C ( K, z ) , N ( K, z ) , T ( K, z ) , and a value function V ( k, K, z ) , given the Markov transition function P ( z 0 , z ) and the tax rate τ , such that (i) V solves the household’s problem, i.e., it solves V ( k, K, z ) = max c,n,k 0 u ( c ) + β Z V ( k 0 , K 0 , z 0 ) dP ( z 0 , z ) (1) s.t. c + k 0 = (1 - δ ) k + (1 - τ ) [ w ( K, z ) n + r ( K, z ) k ] + T ( K, z ) (2) K 0 = K ( K, z ) (3) P ( z 0 , z ) is exogenously given, hence determining decision rules k 0 = k ( k, K, z ) , c ( k, K, z ) , and n ( k, K, z ) ; (ii) the firm maximizes profits, i.e., it solves max K,N { zF ( K, N ) - rK - wN } , hence determining pricing functions r ( K, z ) and w ( K, z ) ; (iii) the government balances its budget in every period, i.e., T ( K, z ) = τ [ r ( K, z ) K + w ( K, z ) N ( K, z )] , (4) 1
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hence determining T ( K, z ) ; (iv) markets clear, i.e., C ( K, z ) + K ( K, z ) = (1 - δ ) K + zF ( K, N ( K, z )) ; (5) (v) the consistency conditions between individual and aggregate are satisfied, i.e., k ( K, K, z ) = K ( K, z ) (6) c ( K, K, z ) = C ( K, z ) (7) n ( K, K, z ) = N ( K, z ) , (8) i.e., if k = K then the functions k ( · ) , c ( · ) , and n ( · ) should yield the aggregate levels of capital, consumption, and labor, respectively, (vi) since we have a representative-agent setting, we also need c = C and k = K . (b) Characterize the equilibrium by finding a functional (Euler) equation which the house- hold’s optimal capital accumulation policy must satisfy (i.e., k 0 as a function of k , K , and z ). Solution. First notice that since preferences don’t depend on leisure, since the agent’s assets are increasing in labor n , and since the agent takes the wage as given, we optimally set n ( k, K, z ) = 1 , k, K, z . (9) Also, it will be easier to solve the problem if we let the state variable be the agent’s wealth a (1 - δ ) k + (1 - τ ) [ w ( K, z ) + r ( K, z ) k ] + T ( K, z ) , (10) in which case the law of motion is a 0
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