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27 Part The Diamond Model 2.10 Assumptions We now turn to the Diamond overlappinggenerations model. The central difference between the Diamond model and the RamseyCassKoopmans model is that there is turnover in the population: rather than there being a fixed number of infinitelylived households, new individuals are continually being born, and old individuals are continually dying. With turnover, it tums out to be simpler to assume that time is discrete rather than continuous; that is, the variables of the model are defined for t = 0,1,2,... rather than for aU values of f > 0. To further simplify the analysis, the model assumes that each individual lives for only two periods. It is the general assumption of turnover in the population, however, and not the specific assumptions of discrete time and twoperiod lifetimes, that is crucial to the models results. Ir individuals are born in period t. As before, population grows at rate n; thus Ir = (1 H n)Iri. Since individuals live for two periods, at time t there are Ir individuals in the first period of their lives and Ir i = Ir/(1  n) individuals in their second periods. Each individual supplies one unit of labor when he or she is young and divides the resulting labor income between firstperiod consumption and saving; in the second period, the individual simply consumes the saving and any interest he or she earns. Let Cir and Czt denote the consumption in period t of young and old individuals. Thus the utihty of an individual born at t, denoted U,, depends See Problem 2.14 for a discretetime version of the Solow model. Blanchard (1985) develops a tractable continuoustime model in which the extent of the departure from the infinitehorizon benchmark is governed by a continuous parameter. Weil (1989a) considers a variant of Blanchards model where new households enter the economy but existing households do not leave. He shows that the arrival of new households is sufficient to generate most of the main results of the Diamond and Blanchard models. Finally, Auerbach and Kotlikoff (1987) use simulations to investigate a much more realistic overlappinggenerations model. to begin with that they are difficult to develop further. And models based on ruleofthumb behavior involve sufficiently unconventional assumptions that it is often hard to know how they should be extended. At the same time, it is likely that departures from Ricardian equivalence are quantitatively important. At the very least, the data do not clearly reject the importance of any of the potential sources of failure of Ricardian equivalence we have discussed. Thus despite its logical appeal, there does not appear to be a strong case for using Ricardian equivalence to gauge the likely effects of governments financing decisions in practice.
2.11 Household Behavior 73 on Cif and C2f+i. We again assume constantrelativeriskaversion utility: \s before, this functional form is needed for balanced growth. Because lifetunes are finite, we no longer have to assume p> +{1 ) to ensure that hfetime utihty does not diverge. If p > 0, individuals place greater weight on firstperiod than secondperiod consumption; if p < 0, the situation is reversed. The assumption p > 1 ensures that the weight on secondperiod consumption is positive. Production is described by the same assumptions as before. There are many firms, each with the production function Yt = F(Kt,AtLt). f (•) again has constant returns to scale and satisfies the Inada conditions, and A again grows at exogenous rate g (so Af = [1 i g]Ati). Markets are competitive; Thus labor and capital earn their marginal products, and firms earn zero profits. As in the first part of the chapter, there is no depreciation. The real mterest rate and the wage per unit of effective labor are therefore given as before by n = f(kt) and Wf = f(kt)  ktf(kt). Finally, there is some initial capital stock Kq that is owned equally by all old individuals. Thus, in period 0 the capital owned by the old and the labor supphed o\ the young are combined to produce output. Capital and labor are paid their marginal products. The old consume both their capital income and iieir existing wealth; they then die and exit the model. The young divide their labor income, Wf Af, between consumption and saving. They carry their saving forward to the next period; thus the capital stock in period rl, Kt+i, equals the number of young individuals in period t, U, times each of these individuals saving, WfAt  Cu This capital is combined with iie labor supplied by the next generation of young individuals, and the arocess continues. 2.11 Household Behavior The secondperiod consumption of an individual born at t is Czt+i = (1 + ff+iKWfAf  Cu). . (2.47) Dividing both sides of this expression by 1  rt+i and bringing Cu over to the lefthand side yields the budget constraint: Cu + Czti = AfWf. (2.48) 1 I f+i This condition states that the present value of lifetime consumption equals jiitial wealth (which is zero) plus the present value of lifetime labor income rthich is AtWt).
1 ci 1 i+pie The firstorder conditions are r"  1 . A,w, Zt + l 1 + rt+ Substituting the first equation into the second yields 1 „ . 1 r"  •If > 1 + rt+i 1+P .49) (2.50) (2.51) (2.52) (2.53) This expression is analogous to the Euler equation, (2.19), in our analysis of the infinitehorizon model. It implies that whether an individuals consumption is increasing or decreasing over time depends on whether the real rate of retum is greater than or less than the discount rate, again determines how much individuals consumption varies in response to differences between r and p. If (2.53) fails, the individual can rearrange consumption over his or her lifetime to raise total utility without changing the present value of the consumption stream. We can use (2.53) and the budget constraint, (2.48), to express C\t in terms of labor income and the real interest rate. Specifically, multiplying both sides of (2.53) by and substituting into the budget constraint gives Cu + (1 + rti)iW> (1 +p)i/ = AtWt. (2.54) This imphes One can also derive (2.53) along the lines of the intuitive derivation of the Euler equation in (2.20)(2.21). Specifically, imagine the individual decreasing by a small amount and then using the resulting additional saving and capital income to raise C>t by (1 + rtl)AC. This change has a utility cost of Cf," AC and a unlit) benefit of (1/(1 + p))C2/(l + r,+j)AC. Equating the cost and benefit and rearranging yields (2.53). The individual maximizes utility, (2.46), subject to the budget constraint, (2.48). The Lagrangian is
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