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22

c = 0

k = 0

k-

FIGURE 2.7 The linearized phase diagram

Figure 2.7 shows the line along which the economy converges smoothly to (k*,c*) (the saddle path AA in the figure). It also shows the line along which the economy moves directly away from {k*,c*) (the line BB). If the initial values of c(0) and k(0) lay along this line, (2.30) and (2.31) would imply that - c* and k-k* would grow steadily at rate 1x2- Since f"(*) is negative, (2.32) implies that the relation between -c* and k-k* has the opposite sign from fx. Thus the saddle path AA Is positively sloped, and the BB line is negatively sloped.

Thus if we linearize the equations for and , we can characterize the dynamics of the economy in terms of the models parameters. At time 0, must jtunp to c* -t- [f"(k*)c*/{efi\)](k -k*). Thereafter, and converge to their balanced-growth-path values at rate fi\; that is, k{t) = k* + ei4k(0) - k*] and c(f) = c* + ewt[c(o) - c*]}

"Of course, it is not possible for the initial value of {k,c) to lie along the BB line. As we saw in Section 2.3, if it did, either would eventually become negative or households would accumulate infinite wealth.

This analysis can be used to characterize the path of and that would be Implied by (2.28) and (2.29) if the initial value of {k, c) were on neither the AA nor BB lines. (Again, this could not in fact occur in equilibrium; see n. 14.) Consider a point (k - k*,c c*) that can be written as a sum of a point on AA and a point on BB. That is, suppose we can find a ka and a k(, such that

(k(0)-k*.c(0)-c*)= K,-k-, (k,-k*) + kt,-k*, (kfc-k*)

= (ka - *, -c*)-b(kfc-k*,Cfc-c).

(continued)



The Speed of Adjustment

To understand the imphcations of (2.35) for the speed of convergence to the balanced growth path, consider our usual example of Cobb-Douglas production, f(k) = k". This implies f"{k*) = a(a-l)k*"- = l(a-l)/a]r*/ f{k*), where r* = ak*" is the real interest rate on the balanced growth path. Thus in this case we can write the expression for pi as

a

1/2"

(2.36)

where s* = I - [c* / f{k*)] is the saving rate on the balanced growth path.

On the balanced growth path, saving is (n + g)k*; thus s* = iji+g)k*lk*< = a(n-(-0)/r*.Finally,(2.19)implies r* = p+e.Substituting these facts into (2.36) yields

Ml =

2 11- + - a{n + g))

. (2.37)

Equation (2.37) expresses the rate of adjustment in terms of the underlying parameters of the model.

To get a feel for the magnitudes involved, suppose a = 1/3, p = 4%, n =2%, g \%, and e = 1. Using the facts above, these parameter values imply r * = 5% and s * = 20%; in addition, the deftnition of /3 as p - n - (1 - e) implies j3 = 2%. Equation (2.36) or (2.37) then implies pi = -5.4%. Thus adjustment is quite rapid in this case; for comparison, the Solow model with the same values of a, n, and g (and as here, no depreciation) implies an adjustment speed of 2%per year (see equation [1.26]). The reason for the difference is that in this example, the saving rate is greater than s * when is less than *

The first point on the right-hand side is on AA and the second is on BB (see [2.32]). Because (fe(0) - k*. c(0) - c*) is the sum of (K, - k*. cĄ - c*) and ( - *, *), and because (2.28) and (2.29) are linear, the economys dynamics starting at (k{0) - k*,c{0) - c*) are the sum of what they would be starting at (A, - k*, Ca - c*) and what they would be starting at {kt - k", - *). Thus,

k(t) - k* = e(ka - k*) + e(k - k*),

and similarly for c(f) - c*. Because 1 is negative and 2 positive, the first term goes to zero and the second term diverges. Thus asymptoticaUy k(t) - k" and c(f) - c* grow at rate 2, and the economy approaches the BB line. The only way to avoid this outcome is for kj, - k* to be zero (which implies that Cj, - c* is also zero)-that is, for the economy to begin on the saddle path AA.

FinaUy, note that we can write any point in (k-k*,c-c*) space as a sum of a point on AA and a point on BB: the ftrst equation above can be written as two linearly independent equations, one for k(0) - k* and one for c(0) - c*, in two unknowns, - k* and kj, - k*. Thus this approach can be used to characterize the dynamics implied by (2.28) and (2.29) for any assumed initial values of and c.



Adding Government to the Model

Assume that the govemment buys output at rate G(f) per unit of effective labor per unit time. Government purchases are assumed not to affect utility from private consumption; this can occur either if the govemment devotes the goods to some activity that does not affect utihty at all, or if utiUty equals the stun of utiUty from private consumption and utility from government-provided goods. Similarly, the purchases are asstuned not to affect future output; that is, they are devoted to pubhc consumption rather than public investment. The purchases are financed by lump-sum taxes of amount G(f) per unit of effective labor per imit time; thus the government always runs a balanced budget. The next section discusses deficit finance.

Investment is now the difference between output and the stun of private constunption and government pinrchases. Thus the equation of motion for k, (2.23), becomes

m = fm)) - c(t) - Git) - (n + g)k(t).

(2.38)

\ higher value of G shifts the = 0 locus down: the more goods that are purchased by the govemment, the fewer that can be purchased privately if is to be held constant.

By asstunption, households preferences ([2.1]-[2.2] or [2.14]) are unchanged. Since the Euler equation ([2.19] or [2.22]) is derived from households preferences without imposing their lifetime budget constraint, this condition continues to hold as before. The taxes that finance the governments purchases affect households budget constraint, however. Specifically, (2.7) becomes

g-R(f)c(f)e("+9)f rff < :(o) + e-"V(t) - ( )] *"+" dt. (2.39)

and less than s* when is greater than k*;m the Solow model, in contrast, s is constant by asstunption.

2.7 The Effects of Government Purchases

Thus far, we have left government out of our model. Yet modern economies devote their resources not just to investment and private consumption but also to public uses. In the United States, for example, about 20 percent of total output is purchased by the government; in many other countries the figure is considerably higher. It is thus natural to extend our model to include a government sector.



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