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17 households total consiunption at t, C(t)Ht)/H, equals consumption per unit of effective labor, c{f), times the households quantity of effective labor, A{t)Lit)/H. Similarly, its initial capital holdings are k(0), capital per unit of effective labor at time zero, times A(0)L(0)/H. Thus we can rewrite (2.5) as R(t) c(t) A(mt) dt < k(0) ( ) ) .rWdt. (2.6) H A{t)L{t) equals A(0)I(0)e<"+3f. Substituting this fact into (2.6) and dividing both sides by (0)1(0)/ yields e «(fc(f)e("+9f dt < + e«(fw(f)e<"+3" dt. (2.7) In many cases, it is difficult to find the integrals in (2.7). Fortunately, we can express the budget constraint in terms of the limiting behavior of the households capital holdings; even when it is not possible to compute the integrals in (2.7), it is often possible to describe the limiting behavior of the economy. To see how the budget constraint can be rewritten in this way, first bring all of the terms of (2.6) over to the same side and combine the two integrals; this gives us K{0) e«(f[w(f)  c{t)]A(t) t=o dt > 0, (2.8) where we have used the fact that k(0)A(0)I(0) = K(0). We can write the integral from f = Otof = ooasa limit. Thus (2.8) is equivalent to KjO) H e«(f)[w(f)  c(t)]A(t) dt t=o H >0. Now note that the households capital holdings at time s are ,«()«(0[(f) C(f)]idt. (2.9) (2.10) To understand (2.10), note that eK(0)/H is the contribution of the households initial wealth to its wealth at s. The households saving at t is [w(f) c(t)]A{t)Ut)/H (which may be negative); e(s)«(f shows how the value of that saving changes from f to s. The expression in (2.10) is e times the expression in brackets in (2.9). Thus we can write the budget constraint simply as Soo H (2.11) Expressed in this form, the budget constraint states that the present value
of the households asset holdings cannot be negative in the limit. Since K(s) IS proportional to k(5)e<"+ we can also write this as lime<e<""3k(s)>0. (2.12) Finally, we can also rewrite the households objective function, (2.1)2.2), m terms of consumption per unit of effective labor. C(f), consumption per worker, equals A(t)c(t). Thus, C(f)i« lA(t)c{t)V 1  [A(0)e*?n"*c(f)i< (2.13) = A(0)e<i>3f c(f) Substituting (2.13) and the fact that L{t) = ) " into the households objective function yields pfC(f)i«I(f) 1 (0)1 (1 )9 c(r)i» 1 l(0)e"f dt jt=o i  e (2.14) e/3t rff = )!" p  „  (1  e)0. f=o 1  e From (2.2), )3 is assumed to be positive. Household Behavior The households problem is to choose the path of c(f) to maximize lifetime utility subject to the budget constraint. Although this involves choosing at each instant of time (rather than choosing a finite set of variables, as in standard maximization problems), conventional maximization techniques can be used. Since the marginal utility of consumption is always positive, the household satisfies its budget constraint with equality. We can therefore use the objective function, (2.14), and the budget constraint, (2.7), to set up
i~fi £ = B " • (2.15. kiO) The household chooses at each point in time; that is, it chooses infinitely many c(f )s. The firstorder condition for an individual c(f) is BePc(t) = Ae*fe"+3)f. (2.I6) The households behavior is characterized by (2.16) and the budget constraint, (2.7). To see what (2.16) implies for the behavior of consumption, first take logs of both sides: InB  pt elnc(f) = InA  R(t) + in + g)t. (2.17) Now note that since the two sides of (2.17) are equal for every t, the derivatives of the two sides with respect to t must be the same. This condition is )3  =r(f) + (n + 0), (2.18) where we have used the definition of R(t) as J/q ( ) to find dR(t)ldt. Solving (2.18) for c(f)/c(f) yields c(f) nt) n  g~p c(f) (2.19) r(f)   where the second line uses the definition of )3 as p  n  (1  ) . To interpret (2.19), note that since Cit) (consumption per worker, rather than consumption per unit of effective labor) equals c{t)A{t), the growth For an introduction to maximization subject to equality constraints, see Dixit (1990, Chapter 2), Simon and Blume (1994, Chapters 1819), or Chiang (1984, Chapter 12). For the case of inequality constraints, see Dixit (Chapter 3), Simon and Blume (Chapter 18), Chiang (Chapter 21), or Kreps (1990, Appendix 1). This step is slightly informal; the difficulty is that the terms in (2.16) are of order dt in (2.15); that is, they make an infinitesimal contribution to the Lagrangian. There are various ways of addressing this issue more formally than simply "canceling" the dts (which is what we do in [2.16]). For example, we can model the household as choosing consumption over the finite intervals [0, At), [At,2At), [2At,3At),with its consumption required to be constant within each interval, and then take the limit as approaches zero. This also yields (2.16). Another possibility is to use the calculus of variations (see n. 11, below). the Lagrangian:
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