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

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Unformatted text preview: University of Minnesota Department of Economics Econ 3102: Intermediate Macroeconomics Handout 6 1 The representative households 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 . Assume that the households utility function is time-separable, and takes the form U ( C ,C 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 util- ity function which is strictly increasing, strictly concave and satifies the Inada Conditions (lim C t u C ( C t , t ) = , lim t u ( C t , t ) = ; these help ensure that the solutions are in- terior). (0 , 1) is the households discount factor. Note that N t = h- t . 1.1 The households 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- ) K t + I t , t = 0 , 1 ,..., where (0 , 1) is the depreciation rate, and I t denotes the households 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- ) K t to the household. Once we include investment I t , the household gets the next periods capital stock K t +1 . The households 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 households utility maximization problem as 1 max { C t , t ,I t } t =0 X t =0 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- ) K t + I t , t = 0 , 1 ,..., (1.3) K given C t , t h where N t = h- t Under our assumptions over u(...
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Handout 6 - University of Minnesota Department of Economics...

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