FIGURE 2.3 The dynamics of and

arrow points to the left. Finally, at Point E both and are zero; thus there is no movement from this poinl.

Figure 2.3 is drawn with k* (the level of that imphes = 0) less than the golden-rule level of (the value of associated with the peak of the = 0 locus). To see that this must be the case, recall that * is defined by f{k*) = p + eg, and that the golden-rule is defined by /"( ) = n + . Since f"(fc) is negative, k* is less than kcR if and only it p + eg is greater than n + g. This is equivalent to p-n ~{1 - ) > 0, which we have assumed to hold so that lifetime utility does not diverge (see [2.2]). Thus * is to the left of the peak of the = 0 curve.

The Initial Value of

Figure 2.3 shows how and must evolve over time to satisfy households intertemporal optimization condition (equation [2.22]) and the equation relating the change in to output and consumption (equation [2.23]) given initial values of and k. The initial value of is given; but the initial value of must be determined.

There are two other points where and are constant. The first is the origin: if the economy starts with no capital and no consumption, it remains there. The second is the point where the = 0 curve crosses the horizontal axis. Here all of output is being used to hold constant, so = 0 and f(k) = [n + g)k. Since having consumption change from zero to any positive amount violates households intertemporal optimization condition, (2.22), if the economy is at this point it must remain there to satisfy (2.22) and (2.23). As we will see shortly, however, the economy is never al this point.

2.3 The Dynamics of the Economy 49

This issue is addressed in Figure 2.4. For concreteness, k(0) is assumed to be less than k*. The figure shows the trajectory of and for various assumptions concerning the initial level of c. If c(0) is above the = 0 curve, at a point Uke A, is positive and k negative; thus the economy moves continually up and to the left in the diagram. If c(0) is such that is initially zero (Point B), the economy begins by moving directly up in (k,c) space; thereafter is positive and negative, and so the economy again moves up and to the left. If the economy begins slightly below the = 0 locus (Point C), is initially positive but small (since is a continuous function of c), and is again positive. Thus In this case the economy initially moves up and slightly to the right; when it crosses the = 0 locus, however, becomes negative and once again the economy is on a path of rising and falling k.

Point D shows a case of very low initial consumption. Here and are both initially positive. From (2.22), is proportional to c; when is small, is therefore small. Thus remains low, and so the economy eventually crosses the = 0 line. At this point, becomes negative, and remains positive. Thus the economy moves down and to the right.

and are continuous functions of and k. Thus there must be some critical point between Points and D-Point F in the diagram-such that at that level of initial c, the economy converges to the stable point, Point E. For any level of consumption above this critical level, the curve is crossed before the = 0 line is reached, and so the economy ends up on a path of perpetually rising consumption and falling capital. And if consumption is less than the critical level, the = 0 locus is reached first, and so the

c = 0

kiO) k*

FIGURE 2.4 The behavior of and for various initial values of

The Saddle Path

Although all of this discussion has been in terms of a single value of k, the idea is general. For any positive initial level of k, there is a unique initial level of that is consistent with households intertemporal optimization, the dynamics of the capital stock, households budget constraint, and the requirement that cannot be negative. The function giving this initial as a function of is known as the saddle path; it is shown in Figure 2.5. For any starting value for k, the initial must be the value on the saddle path. The economy then moves along the saddle path to Point E.

2.4 Welfare

A natural question is whether the equilibrium of this economy represents a desirable outcome. The answer to this question is simple. The First Welfare theorem, from microeconomics, tells us that, if markets are competitive and complete and there are no externalities (and if the number of agents is finite), the decentralized equilibrium is Pareto-efficient-that is, it is im-

economy embarks on a path of falUng consumption and rising capital. But if consumption is just equal to the critical level, the economy converges to the point where both and are constant.

All of these various trajectories satisfy equations (2.22) and (2.23). But we have not yet imposed the requirement that households satisfy their budget constraint, nor have we imposed the requirement that the economys capital stock cannot be negative. These conditions determine which of the trajectories in fact describes the behavior of the economy.

If the economy starts at some point above F, must eventually become negative for (2.22) and (2.23) to continue to be satisfied. Since this is not possible, we can rule out such paths.

To rule out paths starting below F, we use the budget constraint expressed in terms of the limiting behavior of capital holdings, equation (2.12). If the economy starts at a point like D, eventually exceeds the golden-rule capital stock. After that time, the real interest rate, fik), is less than n+g, so e K(s)g(m-g)s jg rising. Since is also rising, e"**e*""sk(s) diverges. Thus lims-co e"*e<"+s*k(s) is infinity; from the derivation of (2.12), we know that this is equivalent to the statement that the present value of households lifetime income is infinitely larger than the present value of their lifetime consumption. Thus households can attain higher utility, and so such a path cannot be an equilibrium.

Finally, if the economy begins at Point F, converges to k*, and so r converges to fik*) = p + . Thus eventually is falling at rate

p - - (1 - 6)0 = )3 > 0, and so limscc e-<e<"+s)fc(s) js zero. Thus the path beginning at F, and only that path, is possible.