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18 "Formally, (2.19) implies c(f) = c(0)ei*<""+"i», which implies ee"*>c(t) = c(0)ei<i«R(t)+(e"p)ti/e. c{0) is thus determined by the fact that c(0)Jf!(,eii«>*«>*<"rfi«dt must equal the righthand side of the budget constraint, (2.7). rate of equals the growth rate of plus the growth rate of A. That is, (2.19) jnpUes that consumption per worker is growing at rate [r(t)  ]/ . Thus, 2 19) states that consumption per worker is rising if the real return exceeds ne rate at which the household discounts future consumption, and is falling : the reverse holds. The smaller is 6the less marginal utility changes as  jnsumption changesthe larger are the changes in consumption in response to differences between the real interest rate and the discount rate. Equation (2.19) is known as the Euler equation for this maximization problem. A more intuitive way of deriving (2.19) is to think of the households consumption at two consecutive moments in time. Specifically, imagine the household reducing at some date f by a small (formally, mfinitesirnal) amount , investing this additional saving for a short (again, infinitesimal) period of time Af, and then consuming the proceeds at time h Af; assume that when it does this, the household leaves consumption and capital holdings at all times other than t and f  unchanged. If the household is optimizing, the marginal impact of this change on hfetime utility must be zero. From (2.14), the marginal utility of c(f) is S«?c(f) Thus the change has a utility cost of Bec(t)" . Since the instantaneous rate of return is r(t), at time f  can be increased by glr(t)ng]it Similarly, since is growing at rate c(f)/c(f), we can HTite c(t + ) as c(t)e<, thus the marginal utility of c(f + M) is Be <f+f)c(f + )« = Be / ( + 0[ ( ) 1 (0/ ( )] ] jus for the path of consumption to be utilitymaximizing, it must satisfy Be fcit) Ac=Be Wt)[c{t)etMlc(t)]M] e[m ngw (2.20) Di\iding by Be~c(f)" and taking logs yields:   eAt+ lr(t)n  g]At = 0. (2.21) Finally, dividing by and rearranging yields the Euler equation in (2.19). Intuitively, the Euler equation describes how must behave over time given c(0): if does not evolve according to (2.19), the household can rearrange its consumption in a way that raises lifetime utility v«thout changing the present value of its lifetime spending. The choice of c(0) is then determined by the requirement that the present value of lifetime consumption o\ er the resulting path equals initial wealth plus the present value of future earnings. When c(0) is chosen too low, consumption spending along the path satisfying (2.19) does not exhaust lifetime wealth, and so a higher path is possible; when c(0) is set too high, consumption spending more than uses up lifetime wealth, and so the path is not feasible.**
The Dynamics of Since all households are the same, equation (2.19) describes the evolution of not just for a smgle household but for the economy as a whole. Since r(f) = fikit)}, we can rewrite (2.19) as m fikmpeg cit) (2.22) Thus is zero when f(k) equals p + eg. Let k* denote this level of k. When exceeds k*, f(k) is less than p + , and so is negative; when is less than *, is positive. This information is summarized in Figure 2.1. The arrows show the direction of motion of c. Thus is rising if < k* and falling if > *. The = 0 line at = * indicates that is constant for this value of k. The Dynamics of As in the Solow model, equals actual investment minus breakeven investment. Since we are assuming that there is no depreciation, breakeven c = 0 (c>0) (c<0) FIGURE 2.1 The dynamics of 2.3 The Dynamics of the Economy The most convenient way to describe the behavior of the economy is in terms of the evolution of and k.
2.3 The Dynamics of the Economy 47 investment is (n + g)k. Actual investment is output minus consumption, f(k)  c. Thus: kt) = fm))  c(f)  ( + g)k(t). (2.23) For a given k, the level of that implies = 0 is given by f{k) (n + g)k; in terms of Figure 1.6 (Chapter 1), is zero when consumption equals the difference between the actual output and breakeven investment lines. This value of is increasing in until f{k) = n + g (the goldenrule level of k) and then decreasing. When exceeds the level that yields = 0, is falling; when is less than this level, is rising. For sufficiently large, breakeven mvestmeht exceeds total output, and so is negative for all positive values of c. This information is summarized in Figure 2.2; the arrows show the direction of motion of k. The Phase Diagram Figure 2.3 combines the information in Figures 2.1 and 2.2. The arrows now show the directions of motion of both and k. To the left of the = 0 locus and above the = 0 locus, for example, is positive and negative. Thus IS rising and falling, and so the arrows point up and to the left. The arrows in the other sections of the diagram are based on similar reasoning. On the = 0 and = 0 curves, only one of and is changing. On the = 0 line above the = 0 locus, for example, is constant and is falling; thus the (k<0)
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