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# ps3sol - Noah Williams Department of Economics University...

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Noah Williams Economics 714 Department of Economics Macroeconomic Theory University of Wisconsin Spring 2010 P ROBLEM S ET 3 Solutions by Tim Lee 1. (a) Although households take prices, taxes and transfers as given, it must be able to project next period’s prices and taxes in order to solve its problem. So we need to introduce a perceived law of motion , which we do by assuming rational expections. To do this, define the aggregate state as the vector X = [ K z ] . D EFINITION 1 A recursive competitive equilibrium (RCE) with a government is a value function V ( k ; X ) , a set of decision rules { c ( k ; X ) , k 0 ( k ; X ) , n ( k ; X ) } , a set of prices { r ( X ) , w ( X ) } , transfers { T ( X ) } , and a perceived law of motion for capital K 0 ( X ) , given the tax rate τ and transistion function P ( z 0 , z ) and N = 1 , such that i. Given prices, taxes, transfers and the law of motion, the decision rules solve the household’s problem: V ( k ; X ) = max c , k 0 u ( c ) + β Z z 0 V ( k 0 ; K 0 ( X ) , z 0 ) dP ( z , z 0 ) c + k 0 = ( 1 - τ )( w ( X ) n + r ( X ) k ) + ( 1 - δ ) k + T ( X ) ii. For all X = [ K z ] , ( K , N ) solves the representative firm’s problem given prices: zF K ( K , N ) = r ( X ) zF N ( K , N ) = w ( X ) iii. For all X, the government balances budget: T ( X ) = τ ( w ( X ) N + r ( X ) K ) iv. Markets clear, i.e. k 0 ( k ; X ) = K 0 ( X ) and n ( k ; X ) = N = 1 for all [ k ; X ] , v. k = K, i.e. aggregate state equals individual state. This is required due to the representative agent setting. vi. The law of motion is induced by K 0 ( X ) = zF ( K , N ) + ( 1 - δ ) K - c ( K ; X ) = k 0 ( K ; X ) , 1

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(b) Given today’s aggregate state X and the law of motion K 0 ( X ) , the household rationally projects next period’s aggregate state as X 0 = [ K 0 ( X ) , z 0 ] . Since there is no preferences for leisure, n ( k ; X ) = 1 for all k , X . To write a FE in terms of k 0 ( k ; X ) , it will be easier to let the agent’s state variable be his wealth, which in turn is a function of [ k X ] : a ( k ; X ) ( 1 - τ )[ w ( X ) + r ( X ) k ] + ( 1 - δ ) k + T ( X ) , so from the perspective of the agent who chooses k 0 but forecasts X 0 , it faces the individual law of motion: a ( k 0 ; X 0 ) = ( 1 - τ )[ w ( X 0 ) + r ( X 0 ) k 0 ] + ( 1 - δ ) k 0 + T ( X 0 ) . (1) Now let V ( a ( k ; X ) , z ) V ( k ; X ) , the Bellman equation for the HH is V ( a ( k ; X ) , z ) = max k 0 u ( a ( k ; X ) - k 0 ) + β Z z 0 V ( a ( k 0 ; X 0 ) , z 0 ) dP ( z , z 0 ) subject to the law of motion ( 1 ). The f.o.c. at the solution is u 0 ( a ( k ; X ) - k 0 ( k ; X )) = β Z z 0 ( 1 - τ ) r ( G ( X ) , z 0 ) + 1 - δ · V a ( a ( k 0 ( k ; X ) ; X 0 ) , z 0 ) dP ( z , z 0 ) and combining with the envelope condition V a ( a ( k ; X ) ; X ) = u 0 ( a ( k ; X ) - k 0 ( k ; X )) we get the Euler equation u 0 ( a ( k ; X ) - k 0 ( k ; X )) = β Z z 0 ( 1 - τ ) r ( X 0 ) + 1 - δ · u 0 a ( k 0 ( k ; X ) ; X 0 ) - k 0 ( k 0 ( k ; X ) ; X 0 ) dP ( z , z 0 ) . Now being explicit about rational expectations and the individual’s state vari- ables, we rewrite u 0 ( a ( k ; K , z ) - k 0 ( k ; K , z )) = β Z z 0 ( 1 - τ ) r ( K 0 ( K , z ) , z 0 ) + 1 - δ × (2) u 0 a ( k 0 ( k ; K , z ) ; K 0 ( K , z ) , z 0 ) - k 0 ( k 0 ( k ; K , z ) ; K 0 ( K , z ) , z 0 ) dP ( z , z 0 ) .
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ps3sol - Noah Williams Department of Economics University...

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