c = 0

FIGURE 2.8 The effects of a permanent increase in government purchases

Reasoning parallel to that used before shows that this imphes the same expression as before for the limiting behavior of (equation [2.12]).

The Effects of Permanent and Temporary Changes in Government Purchases

To see the imphcations of the model, suppose that the economy is on a balanced growth path with G(t) constant at some level Gl, and that there is an unexpected, permanent increase in G to Gh. From (2.38), the = 0 locus shifts down by the amount of the increase in G. Since government purchases do not affect the Etder equation, the = 0 locus is unaffected. This is shown in Figure 2.8.

We know that in response to such a change, must jump so that the economy is on its new saddle path. If not, then as before, either capital would become negative at some point or households would accumulate infinite wealth. In this case, the adjustment takes a simple form: faUs by the amoimt of the increase in G, and the economy is immediately on its new balanced growth path. Intuitively, the permanent increases in government purchases and taxes reduce households lifetime wealth. Thus consumption falls immediately, and the capital stock and the real interest rate are unaffected.

A more interesting case is provided by an unanticipated increase in G that is expected to be temporary; for simplicity, assume that the terminal date is known with certainty. In this case, does not fall by the full amount of the increase in G, Gh - Gl- To see this, note that if it did, consumption would jump up discontinuously at the time that government purchases re-

Empirical Application: Wars and Real Interest Rates

This analysis suggests that temporarily high government purchases cause real interest rates to rise, whereas permanently high purchases do not. Intuitively, when the governments purchases are high only temporarily, households expect their consumption to be greater in the future than it is in the present. To make them willing to accept this, the real interest rate must be high. When the governments purchases are permanently high, on the other hand, households current consumption is low, and they expect it to remain low. Thus in this case, no movement in real interest rates is needed for households to accept their current low consumption.

As in the example in the previous section, because the initial change in G is unexpected, the discontinuities in consumption and marginal utility at that point do not mean that households are not behaving optimally. See n. 12.

turned to Gl, thus marginal utility would fall discontinuously. But since the return of G to Gl is anticipated, the discontinuity in marginal utility would also be anticipated, which cannot be optimal for households.

During the period of time that government purchases are high. A: is determined by the capital-accumulation equation, (2.38), with G = Gh; after G returns to Gl, it is governed by (2.38) with G = Gl- The Euler equation, (2.22), determines the dynamics of throughout, and cannot change discontinuously at the time that G returns to Gl- These facts determine what happens at the time of the increase in G: must jump to the value such that the dynamics imphed by (2.38) with G = Gh (and by [2.22]) bring the economy to the old saddle path at the time that G returns to its initial level. Thereafter, the economy moves along that saddle path to the old balanced growth path.i

This is depicted in Figure 2.9. Panel (a) shows a case where the increase in G is relatively long-lasting. In this case falls by most of the amount of the increase in G. As the time of the return of G to Gl approaches, however, households increase their consumption and decrease their capital holdings in anticipation of the fall in G.

Since r = f{k), we can deduce the behavior of r from the behavior of k. Thus r rises gradually during the period that government spending is high and then slowly returns to its initial level. This is shown in Panel (b); fo denotes the time of the increase in G, and h the time of its return to its initial value.

Finally, Panel (c) shows the case of a short-lived rise in G. Here households change their consumption relatively little, choosing instead to pay for most of the temporarily higher taxes out of their savings. Because government purchases are high for only a short period, the effects on the capital stock and the real interest rate are small.

+

FIGURE 2.9 The effects of a temporary increase in government purchases

A natural example of a period of temporarily high government purchases is a war. Thus our analysis predicts that real interest rates are high during wars. Barro (1987) tests this prediction by examining military spending and interest rates in the United Kingdom from 1729 to 1918. The most significant complication he faces is that, instead of having data on short-term real interest rates, he has data only on long-term nominal interest rates. Long-term interest rates should be, loosely speaking, a weighted average of expected short-term interest rates. Thus, since our analysis imphes that temporary increases in government purchases raise the short-term rate over an extended period, it also imphes that they raise the long-term rate. Simi-

iSee Section 9.3.