GrowthModel09 - One-Sector Growth Model Notes 1 The...

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One-Sector Growth Model Notes 1 The representative household's problem The representative household enjoys consumption and leisure time in all periods t = 0, 1,..., f . The household has no exogenous income, but it has fixed time endowments in each period, h , and an initial capital endowment, 0 K . Assume that the household's utility function is time-separable, and takes the form 0 1 0 1 0 ( , ,., , ,) ( , ) t t t t U C C L L u C h L E f ± ¦ where t C and t h L ± denote consumption and leisure in period t, u( ·,· ) is a period utility function which is strictly increasing, strictly concave and satifies the Inada Conditions ( 0 lim ( , ) , lim ( , ) t t C t t L t t C L h u C h L u C h L o o ± f ± ±f ; these help ensure that the solutions are interior). (0,1) E is the household's discount factor. 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: 1 (1 ) , 0,1,., t t t K K I t G ² ± ² f where (0,1) G is the depreciation rate, and t I denotes the household's investment in period t. In all periods, the household has capital t K and rents it to the firm at a rental rate t r . After the firm uses the capital, it returns the remaining stock (1 ) t K G ± to the household. Once we include investment t I , the household gets the next period's capital stock 1 t K ² .
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