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28 Equation (2.55) shows that the interest rate determines the fraction of income the individual consumes in the first period. Letting s(r) denote the fraction of income saved, (2.55) implies (1 + rfieye = (l+p)i/e+(i + r)(i)/e (2.56) We can therefore rewrite (2.55) as Cir = [1  s(rM)]AtWt. (2.57) Equation (2.56) implies that young individuals saving is increasing in r if and only if (1 + r)i"* is increasing in r. The derivative of (1 + rfs/ with respect to r is [(1  )/ ]{1 + rf~2e)ie xhus 5 is increasing in r if e is less than 1, and decreasing if is greater than 1. Intuitively, a rise in r has both an income and a substitution effect. The fact that the tradeoff between consumption in the two periods has become more favorable for secondperiod consumption tends to increase saving (the substitution effect), but the fact that a given amount of saving yields more secondperiod consumption tends to decrease saving (the income effect). When individuals are very willing to substitute consumption between the two periods to take advantage of rateofreturn incentives (that is, when is low), the substitution effect dominates. When individuals have strong preferences for similar levels of consumption in the two periods (that is, when is high), the income effect dominates. And in the special case of e = 1 (logarithmic utility), the two effects balance, and young individuals saving rate is independent of r. 2.12 The Dynamics of the Economy The Equation of Motion of As in the infinitehorizon model, we can aggregate individuals behavior to characterize the dynamics of the economy. As described above, the capital stock in period t  1 is the amount saved by young individuals in penod t. Thus, Kt+j = s(r,+i)LtAtWt. (2.58) Note that because saving in period f depends on labor income in that period and on the return on capital that savers expect in the next period, it is w in period f and r in period f  1 that enter the expression for the capital stock in period f b 1.
The Evolution of Equation (2.60) implicitly defines k,+i as a function of kt. (It defines kt+i only implicitly because kt+i appears on the righthand side as vrell as the lefthand side.) It therefore determines how evolves over time given its initial value. A value of kt such that kj+i = ki satisfies (2.60) is an equilibrium value of k: once reaches that value, it remains there. We therefore want to know whether there is an equilibrium value (or values) of k, and whether converges to such a value if it does not begin at one. To answer these questions, we need to describe how kti depends on kf. Unfortunately, we can say relatively little about this for the general case. We therefore begin by considering the case of logarithmic utility and CobbDouglas production. With these assumptions, (2.60) takes a particularly simple form. We then briefly discuss what occurs when these assimiptions are relaxed. Logarithmic Utility and CobbDouglas Production When e is 1, the fraction of labor income saved is 1/(2 + p) (see equation [2.56]). And when production is CobbDouglas, f(k) is k" and w is (1  a)k". Equation (2.60) therefore becomes  = (l + n)(l + ,)7b"" Figure 2.11 shows kt+i as a function of kt. A point where the kt+i function intersects the 45degree line is a point where fct+i equals kt. In the special case we are considering, kt+i equals kt at kt = 0; it rises above kt when kf is small; and it then crosses the 45degree line and remains below. There is thus a unique equilibriiun level of (aside from = 0), which is denoted k*. k* is globally stable: wherever starts (other than at 0), it converges to k*. Suppose, for example, that the initial value of k, , is greater than k*. Because kt+i is less than kf when kt exceeds k*, ki is less than . And Dividing both sides of (2.58) by Lt+iAt+i gives us an expression for Kt+i/At+iLt+], capital per unit of effective labor: We can then substitute for n+i and Wt to obtain kti = JTf+jfftMkt)  ktfih)]. (2.60)
2.12 The Dynamics of the Economy 77 FIGURE 2.11 The dynamics of k because kj exceeds k* and kt+i is increasing m kt, k] is larger than k*. Thus ki IS between k* and . moves part way toward k* This process is repeated each period, and so converges smoothly to k* A similar analysis applies when is less than k* These dynamics are shown by the arrows in Figure 2 11 Given , the height of the k,+i function shows ki on the vertical axis To find , we first need to find ki on the horizontal axis, to do this, we move across to the 45 degree line The height of the kt+i function at this point then shows kz, and so on The properties of the economy once it has converged to its balanced growth path are the same as those of the Solow and Ramsey economies on their balanced growth paths the saving rate is constant, output per worker is growmg at rate g, the capital output ratio is constant, and so on To see how the economy responds to shocks, consider our usual example of a fall in the discount rate, p, when the economy is initially on its balanced growth path The fall in the discount rate causes the young to save a greater traction of their labor income Thus the kf+i function shifts up This is depicted in Figure 2 12 The upward shift of the kf+i function increases k*, the value of on the balanced growth path. As the figure shows, rises monotonically from the old value of k* to the new one Thus the effects of a fall m the discount rate in the Diamond model m the case we are considering are similar to its effects m the RamseyCassKoopmans model, and to the effects of a rise in the saving rate in the Solow model The change shifts the paths over tune of output and capital per worker permanently up, but it leads only to temporary increases m the growth rates of these variables
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