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55 ie? i + r AW2. (4.19) To see the implications of (4.18)(4.19), divide both sides of (4.18) by wi and both sides of (4.19) by W2/(l i r), and equate the two resulting expressions for A. This yields 1 (4.20) 12 W2 1  V?l Wi 1 W2 12 ep{l + r)wi (4.21) Equation (4.21) implies that relative labor supply in the two periods responds to the relative wage. If, for example, Wi rises relative to W2, the household decreases firstperiod leisure relative to secondperiod leisure; that is, it increases firstperiod labor supply relative to secondperiod supply. Because of the logarithmic functional form, the elasticity of substitution between leisure in the two periods is 1. Equation (4.21) also implies that a rise in the interest rate raises firstperiod labor supply relative to secondperiod supply. Intuitively, a rise in r increases the attractiveness of working today and saving relative to working tomorrow. As we will see, this effect of the interest rate on labor supply is crucial to employment fluctuations in realbusinesscycle models. These responses of labor supply to the relative wage and the interest rate are known as intertemporal substitution in labor supply (Lucas and Rapping, 1969). Household Optimization under Uncertainty The second way that the households optimization problem differs from its problem in the Ramsey model is that it faces uncertainty about rates of return and future wages. Because of this uncertainty, the household does not choose deterministic paths for consumption and labor supply, instead, its choices of and f at any date potentially depend on all of the shocks to technology and government purchases up to that date. This makes a complete description of the households behavior quite complicated. Fortunately, we can describe key features of its behavior without fully solving its optimization problem. Recall that in the Ramsey model, we were able to derive an equation relating present consumption to the interest rate and consumption a short time later (the Euler equation) before imposing the budget constraint and determining the level of consumption. With uncertainty, the analogous equation relates consumption in the current period to expectations concern ePb
prA = Et <!)%.(1 H n,i) Cf+] . (4.22) Since e +1( / +1/ ) "" is not uncertain and since Nf+i = Nfg", this simplifies to  = e"£f[(1 + r,i)l (4.23) Cf Cf+i This is the analogue of equation (2.19) in the Ramsey model. Note that the expression on the righthand side of (4.23) is nof the same as e"p£f[l/Cf+i]£f[l i ff+i]. That is, the tradeoff between present and future consumption depends not just on the expectations of future marginal utility and the rate of return, but also on their interaction. Specifically, the expectation of the product of two variables equals the product of their expectations plus their covariance. Thus (4.23) implies  = e~P f£f[]£f[1 + ff+i] + Cov(, 1 r ff+i)) , (4.24) Cf V Cf+i Cf+i / where Cov(l/Cf+bl + Vt+i) denotes the covariance of 1/Cf+i and 1 i ,+\. Suppose, for example, that when r,+\ is high, Cf+i is also high. In this case, Cov(l /Cf+i, 1 + rt+i) is negativethat is, the return to saving is high in The households problem can be analyzed more formally using dynamic programming (see Section 10.4, below; Dixit, 1990, Chapter 11; or Kreps, 1990, Appendix 2). This also yields (4.23), below. ing interest rates and consumption in the next period. We will derive this equation using the informal approach we used in equations (2.20)(2.21) to derive the Euler equation.! Consider the household in period t. Suppose it reduces current consumption per member by a small amount Ac and then uses the resulting greater wealth to increase consumption per member in the next period above what it otherwise would have been. If the household is behaving optimally, a marginal change of this type must leave expected utility unchanged. Equations (4.5) and (4.7) imply that the marginal utihty of consumption per member in period t is ep{Nt/H){l/Ct). Thus the utility cost of this change is ePNtIH)(AcICt). Since the household has e" times as many members in period fi1 as in period t, the increase in consumption per member in period f  1 is e " (1 + rt+\)Ac. The marginal utility of periodf +1 consumption per member is e"p*"i>(Nf4 IH)(\ICti). Thus the expected utility benefit as of period f is £t[ep<f+i>(Nf+i/H)e"(l i rf+i)/Cf+i]Ac, where denotes expectations conditional on what the household knows in period t (that is, given the history of the economy up through period t). Equating the costs and expected benefits implies
The Tradeoff between Consumption and Labor Supply The household chooses not only consumption at each date, but also labor supply. Thus a second hrstorder condition for the households optimization problem relates its current consumption and labor supply. Specifically, imagine the household increasing its labor supply per member in period t by a small amount and using the resulting income to increase its consumption in that period. Again if the household is behaving optimally, a marginal change of this type must leave expected utility unchanged. From equations (4.5) and (4.7), the marginal disutility of working in period t is e 4Nt/H)[fo/(l  f)]. Thus the change has a utility cost of e"PHNt/H)[b/{1  t)]A. And since the change raises consumption by Wf , it has a utility benefit of e~P(Nt/H){l/Ct)WtM. Equating the cost and benefit gives us (4.26) l£t b Equation (4.26) relates current leisure and consumption given the wage. Because it involves current variables, which are known, uncertainty does not enter. Equations (4.23) and (4.26) are the key equations describing households behavior. 4.5 A Special Case of the Model Simplifying Assumptions The model of Section 4.3 cannot be solved analytically. The basic problem, as Campbell (1994) emphasizes, is that it contains a mixture of ingredients that are linearsuch as depreciation and the division of output into consumption, investment, and govemment purchasesand ones that are loglinearsuch as the production function and preferences. In this section, we therefore investigate a simplified version of the model. the times when the margmal utility of consumption is low. This makes savmg less attractive than it is if 1/Ct+i and rt+i are uncorrelated, and thus tends to raise current consumption. Chapter 7 discusses the impact of uncertainty on optimal consumption further.
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