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54 (itr The representative household maximizes the expected value of = Xe""u(c,,l~t). (4.5) f=0 " u{») is the instantaneous utility function of the representative member of the household, and p is the discount rate. Nf is population and is the number of households; thus N,/ is the number of members of the household. Population grows exogenously at rate n: In N, = N+ nt, n < p. (4.6) Thus the level of N, is given by Nf = e". • The instantaneous utihty function, u(»), has two arguments. The first is consumption per member of the household, c. The second is leisure per member, which is the difference between the time endowment per member (normahzed to 1 for simphcity) and the amount each member works, £. Since all households are the same, = IN and f = L/N. For simplicity, u(») is loglinear in the two arguments: Uf = lnct + foln(lf), b>0. (4.7) The final assumptions of the model concern the behavior of the two driving variables, technology and government purchases. Consider technology "Section 4.9 briefly discusses realbusinesscycle models with distortionary taxes. The usual way to express discounting in a discretetime model is as 1/(1 + p) rather than as e But because of the loglinear structure of this model, the exponential formulaUon is more natural here. There is no important difference between the two approaches, however; specifically, if we define p = e  1, then e f" = 1/(1 + p). Kt+i =Kt + It~ 8K, (4.2) = Kt + YtCtGt SK,. The governments purchases are financed by lumpsum taxes.Because households are infinitelylived and there are no capitalmarket imperfections, the precise timing of the tax levies does not affect household behaviorthat is, Ricardian equivalence holds. Labor and capital are paid their marginal products. Thus the real wage and the real interest rate in period t are Wt = (1  a)K,4A,L,rAt (4.3)
Intertemporal Substitution in Labor Supply To see what the utility function implies for labor supply, consider first the case where the household lives only for one period and has no initial wealth. In addition, assume for simplicity that the household has only one member. In this case, the households objective function is simply Inc i foln(l  £), and its budget constraint is simply = w£. first. To capture trend growth, the model assumes that in the absence of any shocks, In Af would be A i gt, where g is the rate of technological progress. But technology is also subject to random disturbances. Thus, InAf = A + 0f + Af, (4.8) where A reflects the effects of the shocks. A is assumed to follow a firstorder autoregressive process. That is, At = pAAti + aA.t, 1< <1, (4.9) where the £a,fs are whitenoise disturbancesa series of meanzero shocks that are uncorrelated with one another. Equation (4.9) states that the random component of In Af, Af, equals fraction pa of its previous periods value plus a random term. If pA is positive, this means that the effects of a shock to technology disappear gradually over time. We make similar assumptions about government purchases. The trend growth rate of per capita government purchases equals the trend growth rate of technology; if this were not the case, over time government purchases would become arbitrarily large or arbitrarily small relative to the economy. Thus, InGt =~G+in + g)t + Gt, (4.10) Gt = pcGti + sg.t, I < pg <l, (4.11) where the f gs are whitenoise disturbances that are uncorrelated with the f as. This completes the description of the model. 4.4 Household Behavior The two most important differences between this model and the Ramsey model are the inclusion of leisure in the utility function and the introduction of randomness in technology and government purchases. Before analyzing the models general properties, this section therefore discusses these features implications for households behavior.
Since the budget constraint requires = w£, (4.13) implies A = 1/w. Substituting this into (4.14) yields + = 0. (4.15) The wage does not enter (4.15). Thus labor supply (the value of £ that satisfies [4.15]) is independent of the wage. Intuitively, because utility is logarithmic in consumption and the household has no initial wealth, the income and substitution effects of a change in the wage offset each other. The fact that the level of the wage does not affect labor supply in the static case does not mean that variations in the wage do not affect labor supply when the households horizon is more than one period. This can be seen most easily when the household lives for two periods. Continue to assume that it has no initial wealth and that it has only one member; in addition, assume that there is no uncertainty about the interest rate or the secondperiod wage. The households lifetime budget constraint is now Cl + C2 = Wii + W22, (4.16) where r is the real interest rate. The Lagrangian is £ = Incn bind  A) + e"[lnC2 + foln(l  Wi£i + YVV22  Cl  YC2j . (4.17) The households choice variables are Ci,C2,i, and fz Only the firstorder conditions for £i and £2 are needed, however, to show the effect of the relative wage in the two periods on relative labor supply. These firstorder conditions are = Awi, (4.18) The Lagrangian for the households maximization problem is £ = Inc + bind) + A(wc). (4.12) The firstorder conditions for and f, respectively, are A = 0, (4.13) +Xw = 0. (4.14)
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