Each household obtains utility from consumption at each period of time and from

Each household obtains utility from consumption at

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Each household obtains utility from consumption at each period of time and from the number of children when she is an adult. Preferences of an indi- vidual born in period t are represented by the following utility function: ln c 1, t + β ln c 2, t + 1 + β 2 ln ( c 3, t + 2 + e ) + μ ln n t + 1 (3) where β ( 0, 1 ) is the subjective discount factor, μ ( 0, 1 ) measures the love for children and n t + 1 is the number of children. At each age, the household can consume the market good produced by firms. Hence, c j , t amounts for con- sumption of this good when young ( j = 1), adult ( j = 2) and old ( j = 3). We also assume that when old, the household has a consumption of a home- produced good e > 0, which is, to simplify, a perfect substitute of the market good. 5 As explained by Aguiar and Hurst (2005), Hurst (2008) and Schwerdt (2005), home production at the retirement age is quite a realistic feature since it seems to solve the consumption puzzle when households are retired. 6 While each household uses her time endowment for home production when old, she supplies labor to firms when young and adult. When young, this labor supply is one unit. In contrast, labor supply is endogenous when adult. In- deed, each household has n t + 1 children at the end of the first period and faces a rearing cost ψ w t + 1 per child in terms of the consumption good in middle-age, where 0 < ψ < 1 measures the fraction of time an adult spends for each child. Having children takes time when adult. Such a specification is often used in the literature on fertility (de la Croix and Doepke (2003), Galor (2005)). When young, the household invests an amount a t + 1 in a productive asset. When it corresponds to an investment in human capital, it represents training, education or acquisition of new skills (PhD or Executive MBA for instance). It provides some adding returns q t + 1 a t + 1 at the period where the agent is work- ing for the market good sector, i.e. when she is adult, but no return when she 4 We denote by F i ( ., . ) the derivative with respect to the i th argument of the function. 5 For home-produced good, the reader can refer to Gronau (1986) for a survey and Benhabib et al. (1991) for a macro-model with home production. 6 This puzzle is based on the observation that consumption expenditures are lower when indi- viduals are retired, but the amount of consumption of goods is not lower. 5
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is retired, i.e. when old. 7 This means that while the wage is the only income when young, the return of productive investment is a source of income com- plementary to the wage in the middle-age. 8 Households may also invest in the speculative asset when young ( b 1, t ) and adult ( b 2, t + 1 ). This asset is supplied in one unit and b i , t represents the value of the share of this asset bought ( b i , t > 0) or sold ( b i , t < 0) by a young or an adult household. Since this asset has no fundamental value, it is a bubble if it has a strictly positive price. Given that the supply is in one unit, the price b B t is given by the sum of the values of the shares hold by households living at the same time, b B t = N t b 1, t + N t - 1 b 2, t . When b B t = 0 and b 1, t = b 2, t = 0, the price of the speculative asset is zero and there is no bubble.
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  • Spring '10
  • Economics, Capital accumulation

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