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Homework 3 - Partial Answers Jonathan Heathcote Due in Class on Tuesday February 28th In class we outlined two versions of the stochastic growth model: a planner’s problem, and an Arrow-Debreu competitive equilibrium. We were working to- wards showing that allocations in the two setups would be identical. 1. (a) Complete the proof that the sets of equations that characterize (i) the solution to the planner’s problem, and (ii) the competitive equi- librium are identical, and thus that one can solve for equilibrium allocations by solving the planner’s problem. This is pretty straightforward following the class notes, and the text- book. (b) Now consider the following twist on the economy we described in class. Income (from both labor and capital) is taxed at rate τ t , where 0 t < 1 . There is no allowance for depreciation: thus the typical consumer’s budget constraint (in the sequence of markets formulation, without state-contingent claims) is c t + k t +1 =(1 τ t )( r t k t + w t n t )+(1 δ ) k t Revenues are used for non-valued government purchases G .Con s ide r (i) a planner’s problem in which the planner has to set aside a con- stant amount G of output each period for government purchases, and (ii) a competitive equilibrium in which the tax rate τ t is such that at each date equilibrium revenue is equal to the same amount G . i. Describe the planner’s problem and the competitive equilibrium, and the two sets of equations characterizing (i) the planner’s solution and (ii) the equilibrium. The planner’s inter-temporal f rst order condition reduces to something like u c ( c t ,l t )= βE t [ u c ( c t +1 t +1 )(1 δ + MPK t +1 )] The corresponding condition from the consumer’s problem in the competitive equilibrium is u c ( c t t t [ u c ( c t +1 t +1 δ + r t +1 (1 τ t +1 ))] 1

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From the f rm’s problem, r
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