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Handout 6 - University of Minnesota Department of Economics...

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University of Minnesota Department of Economics Econ 3102: Intermediate Macroeconomics Handout 6 1 The representative household’s problem The representative household enjoys consumption and leisure time in all periods t = 0, 1,..., . The household has no exogenous income, but it has fixed time endowments in each period, ¯ h , and an initial capital endowment, K 0 . Assume that the household’s utility function is time-separable, and takes the form U ( C 0 , C 1 , ..., ‘ 0 , ‘ 1 , ... ) = X t =0 β t u ( C t , ‘ t ) where C t and t denote consumption and leisure in period t, u( · , · ) is a period util- ity function which is strictly increasing, strictly concave and satifies the Inada Conditions (lim C t 0 u C ( C t , ‘ t ) = , lim t 0 u ( C t , ‘ t ) = ; these help ensure that the solutions are in- terior). β (0 , 1) is the household’s discount factor. Note that N t = ¯ h - t . 1.1 The household’s investment decision and its budget constraints The household makes investment decisions to increase its initial capital stock over time. The law of motion for capital is given by: K t +1 = (1 - δ ) K t + I t , t = 0 , 1 , ..., where δ (0 , 1) is the depreciation rate, and I t denotes the household’s investment in period t. In all periods, the household has capital K t and rents it to the firm at a rental rate r t . After the firm uses the capital, it returns the remaining stock (1 - δ ) K t to the household. Once we include investment I t , the household gets the next period’s capital stock K t +1 . The household’s budget constraint is given by C t + I t = w t ( ¯ h - t ) + r t K t t = 0 , 1 , ..., We can then summarize the household’s utility maximization problem as 1
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max { C t ,‘ t ,I t } t =0 X t =0 β t u ( C t , ‘ t ) (1.1) s.t. C t + I t = w t ( ¯ h - t ) + r t K t t = 0 , 1 , ..., (1.2) K t +1 = (1 - δ ) K t + I t , t = 0 , 1 , ..., (1.3) K 0 given C t 0 , 0 t ¯ h where N t = ¯ h - t Under our assumptions over u( · , · ), one can prove a solution exists. (Technically, we need a transversality condition. Instead we use an assumption that after some T,
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