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16 °° Tit) ePuiCmdt. (2.1) f=o H C(f) is the consumption of each member of the household at time t. u(*) is the instantaneous utility function, which gives each members utility at a given date. I(t) is the total population of the economy; L(t)/H is therefore the number of members of the household. Thus u(C{t))Ut)/H is the households total instantaneous utility at t. Finally, p is the discount rate; the greater is p, the less the household values future consumption relative to current consumption. One could also write utility as J,"(, e<u(C{t))dt, where p = pn. Since I(t) = L(0)e", fhis expression equals the expression In equation (2.1) divided by )/ , and thus has the >ame implications for behavior. Part A The RamseyCassKoopmans Model 2.1 Assumptions Firms There is a large nimiber of identical firms. Each has access to the production function Y = F{K,AL), which satisfies the same assumptions as in Chapter 1. The firrns hire workers and rent capital in competitive factor markets, and sell their output in a competitive output market. Firms take A as given; 3S in the Solow model, A grows exogenously at rate g. The firms maximize profits. They are owned by the households, so any profits they earn accrue to the households. Households There is also a large number of identical households. The size of each household grows at rate n. Each member of the household supplies one unit of labor at every point in time, bi addition, the household rents whatever capital it owns to firms. It has initial capital holdings of K{0)/H, where K(0) is the initial amount of capital in the economy and H is the number of households. For simplicity, in this chapter we assume that there is no depreciation. The household divides its income (from the labor and capital it supplies and, potentially, from the profits it receives from firms) at each point in time between consumption and saving so as to maximize lifetime atility. The households utility function takes the form
2.2 The Behavior of Households and Firms Firms Firms behavior is relatively simple. At each point in time they employ the stocks of labor and capital, pay them their marginal products, and sell the 2See Problem 2.2. To see this, first subtract I /(1  ) from the utility function; since this simply changes utility by a constant, it does not affect behavior. Then take the limit as e approaches 1; this requires using rHopitals rule. The result is In . "Phelps (1966a) discusses how growth models can be analyzed when households can obtain infinite utility. The instantaneous utihty function takes the form C(t)i " w(C(t)) =f. e>0, pn~ae)g >0. (2.2) I  V This utihty function is known as constantrelativeriskaversion (or CRRA) utility. The reason for the name is that the coefficient of relative risk aversion (which is defined as Cu"{C)/u(C)) for this utihty function is , and thus is independent of . Since there is no uncertainty in this model, the households attitude toward risk is not directly relevant. But also determines the households willingness to shift consumption between different periods: the smaller is e, the more slowly marginal utility faUs as consumption rises, and so the more willing the household is to allow its consumption to vary over time. If is close to zero, for example, utility is almost hnear in , and so the household is willing to accept large swings in its consumption to take advantage of small differences between its discount rate and the rate of return it gets on its saving. Specifically, one can show that the elasticity of substitution between consumption at any two points in time is 1/e. Three additional features of the instantaneous utility function are worth mentioning. First, is increasing in if e < 1 but decreasing if e > 1; dividing C" by I  e thus ensures that the marginal utihty of consumption is positive regardless of the value of . Second, in the special case of e 1, the instantaneous utility function simplifies to In ; this is often a useful case to consider. And third, the assumption that pn (1 ~ ) > 0 ensures that lifetime utility does not diverge: if this condition does not hold, the household can attain infinite lifetime utility, and its maximization problem does not have a welldefined solution.*
Ht) = fikit)). (2.3) The marginal product of effective labor is dF{K,AL)/dAL. In terms of *), this is f(k)  kfik)."" Thus the real wage per unit of effective labor is w(f) = fm))  k{t)fm)). (2.4) Since the marginal product of labor (as opposed to effective labor) is AdF{K,AL)/dAL, a workers labor income at time f is A{t)w(t). Households Maximization Problem The representative household takes the paths of r and w as given. Its budget constraint is that the present value of its lifetime consumption cannot exceed its initial wealth plus the present value of its lifetime labor income. To write the budget constraint formally, we need to account for the fact that r may vary over time. To do this, define i(f) as jq ( ) . One unit of the output good invested at time 0 yields e" units of the good at f; equivalently, the value of one unit of output at time I in terms of output at time 0 is e". For example, if r is constant at some level F, R{t) is simply rt and the present value of one unit of output at t is e. More generally, e* shows the effects of continuously compounding interest over the period (0,f]. Since the household has L(t)/H members, its labor income at t is A(t)w{t)Ut)/H, and its consumption expenditures are C(t)Ut)/H. The households budget constraint is therefore r«<"A(t)w(r) dt. (2.5) As in the Solow model, it is easier to work with variables normalized by the quantity of effective labor. To do this, we need to express the budget constraint in terms of consumption and labor income per unit of effective labor. Define c(f) to be consumption per unit of effective labor. The See Problem 1.7, in Chapter 1. resulting output. Because the production function has constant returns and the economy is competitive, firms earn zero profits. As described in Chapter 1, the marginal product of capital, dF(K, AL)/dK, is fik), where fi) is the intensive form of the production function. Because markets are competitive, capital earns its marginal product. And because there is no depreciation, the real rate of return on capital equals its earnings per unit time. Thus the real interest rate at time t is
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