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Unformatted text preview: Econ 3102: Intermediate Macroeconomics Homework #4 Answer Key 1 Problems (55 points) Exercise 1 (35 points) Consider a in&nite horizon setup. The representative household (consumer) enjoys consumption and leisure time in all periods t = 0, 1,.... The household has no exogenous income, but it has &xed time endowments in each period & h , and an initial capital endowment, k . Assume that the households utility function is time-separable, and takes the form U ( C ; C 1 ; :::; ; 1 ; ::: ) = 1 X t =0 & t u ( C t ; t ) where C t and t denote consumption and leisure in period t , u ( & ; & ) is a period utility function which is strictly increasing, strictly concave, twice-di/erentiable and sati&es the Inada conditions ( lim C t ! u C ( C t ; t ) = 1 ; lim t ! u ( C t ; t ) = 1 ; these help ensure that the solutions are interior). & 2 (0 ; 1) is the households discount factor. Note that n t = & h t is the households labor supply. The household owns the capital stock k t and rents it to the &rm at a rental rate r t at every period t . The household makes investment decisions to increase the capital stock over time. The law of motion for the capital stock is given by: k t +1 = (1 ) k t + i t ; t = 0 ; 1 ; ::: where 2 (0 ; 1] is the depreciation rate, and i t denotes the households investment in period t ....
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- Summer '10