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21 kou) kNEW FIGURE 2.6 The effects of a fall in the discount rate locus is affected. Recall equation (2.22): c(f)/c(f) = [f{k(t))~  ]/e.Thus a fall in p means that, for a given k,c/cis lower than before. Since f"(k) is negative, the needed for to equal zero therefore rises. Thus the = 0 line shifts to the right. This is shown in Figure 2.6. At the time of the change in p, the value of kthe stock of capital per unit of effective laboris given by the history of the economy, and it cannot change discontinuously. In particular, at the time of the change equals the * on the old balanced growth path. In contrast, cthe rate at which households are consumingcan jump at the time of the shock. Given our analysis of the dynamics of the economy, it is clear what occurs: at the instant of the change, jumps down so that the economy is on the new saddle path (Point A in Figure 2.6). Thereafter, and rise gradually to their new balancedgrowthpath values; these are higher than their values on the original balanced growth path. Thus the effects of a fall in the discount rate are similar to the effects of a rise in the saving rate in the Solow model with a capital stock below the goldenrule level. In both cases, rises gradually to a new higher level, and in both initially falls but then rises to a level above the one it started at. Thus, just as with a permanent rise in the saving rate in the Solow model, the permanent fall in the discount rate produces temporary increases in the growth rates of capital per worker and output per worker. The only We are assuming that the change is unexpected. Thus the discontinuous change in does not imply that households are not optimizing. Their original behavior is optimal given their beliefs (which includes the belief that p will not change): the faU in is the optimal response to the new information that p is lower. (See Section 2.7 and Problems 2.9 and 2.10 for examples of how to analyze anticipated changes.)
For a more formal introduction to the analysis of systems of differential equations (such as [2.24][2.25]), see Simon and Blume (1994, Chapter 25). difference between the two experiments is that, in the case of the fall in p, in general the fraction of output that is saved is not constant during the adjustment process. The Rate of Adjustment and the Slope of the Saddle Path Equations (2.22) and (2.23) describe c(f) and k{t) as fimctions of kit) and c(t). A fruitful way to analyze their quantitative implications for the dynamics of the economy is to replace these nonlinear equations with linear approximations aroimd the balanced growth path. Thus we begin by takmg firstorder Taylor approximations to (2.22) and (2.23) around = *, = *. That is, we write clkk*]lcc*], (2.24) k[kk*] + [cc*], • (2.25) where /sk, / , / , and / are all evaluated atk = k*, = c*. Our strategy will be to treat (2.24) and (2.25) as exact and analyze the dynamics of the resulting system. Since c* is a constant, equals  c* (that is, dc(t)/dt equals d[c{t)  c*]jdt). Similarly, equals kk*.We can therefore rewrite (2.24) and (2.25) as cc* ~[kk*] + lcc*]. (2.26) kk* \kk*] + lcc*]. (2.27) (Again, the derivatives are all evaluated atk = k*,c = c*.) Using (2.22) and (2.23) to compute these derivatives yields cc* C.I [kk*], (2.28) kk* [f(k*)  (n + g)][k  k*] Ic c*] = 1(P + eg) {n+ g)]lk  k*] Ic c*] (2.29) = mk*]lcc*],
kk* n From (2.31), the condition that (k  k*)/(k  k*) equal (c  c*)/(c  c*) is thus fJ = P. (2.33) p 2 prOQ , (2.34) This is a quadratic equation in p. The solutions are l3±W4f"ik*)cVeV/ ,3, p =. (2.35) Let p,i and p2 denote these two values of p,. If p is positive, then c(f)  c* and k(f)  k* are growing; that is, instead of moving along a straight line toward (k*, c*), the economy is moving on a straight line away from (k*,c*). Thus if the economy is to converge to (k*, c*), p must be negative. Inspection of (2.35) shows that only one of the p,s, namely {p  [p  4f"(k*)c*/eV}/2, is negative. Let pi denote this value of p,. Equation (2.32) (with p = p,i) then tells us how cc* must be related to  k* for both to be falling at rate p,i. where the second line of (2.29) uses the fact that (2.22) implies that f(k*) = + and the third line uses the deftnition of j3 as p  n  (1  e)g. Dividing both sides of (2.28) by ~ c* and both sides of (2.29) by  k* yields expressions for the growth rates of c* and  k*: cc* f"(k*)c* kk* Equations (2.30) and (2.31) imply that the growth rates of  c* and kk* depend only on the ratio of cc* to kk*. Given this, consider what happens if the values of and are such that  c* and  k* are falling at the same rate (that is, if they imply (c  c*)/(c  c*) = (k  k*)/(k  k*)). This implies that the ratio of  c* to * is not changing, and thus that their growth rates are also not changing. Thus  c* and kk* continue to fall at equal rates. In terms of the diagram, from a point where  c* and kk* are falling at equal rates, the economy moves along a straight line to (k*, c*), with the distance from (k*, c*) falling at a constant rate. Let p, denote (c  c*)/(c  c*). Equation (2.30) implies cc* ( *) *1 32
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