**See Problem 6.11 for a simple version of this result.

discussed above, the Taylor model exhibits price level inertia: the price level adjusts fully to a monetary shock only after a sustained departure of output from its normal level. As a result, it is often claimed that the Taylor model accounts for inflation inertia.

Ball (1994a) demonstrates, however, thai this claim is incorrect. To see why, consider a Taylor economy with steady inflation and output equal to its flexible-price value, and consider two possible changes in policy. In the first, there is a one-time downward adjustment in the path of money; that IS, money growth is low for one period but then returns to its usual value. In the second, the shift to lower money growth is permanent.

Equation (6.80) shows that the prices that individuals set depend on the entire expected future path of money. The permanent fall in money growth leads to much larger reductions in the expected future values of the money stock than does the one-time shift. As a result, the permanent change in money growth has a much larger effect on newly set prices than does the one-time reduction, and hence a much smaller short-run impact on output, m - p.

In fact. Ball, using a continuous-time version of the model, estabUshes the following result. Consider a permanent reduction in money growth achieved by reducing money growth linearly over an interval equal to the length of time between a representative individuals price changes; that IS, if the initial and hnal money growth rates are go and gi, if prices are in effect for intervals of length , and if the reduction begins at time to, then money growth at is 00 - [(f - 1 )/ ]( - gi) for to < t < to + . Ball shows that such a policy, instead of causing a recession, causes output to rise above its normal level. Thus the fact that price (and wage) changes are staggered does not account for the difhculty of reducing inflation.!

6.9 The Caplin-Spulber Model

The Fischer and Taylor models assume that the timing of price changes is determined solely by the passage of time. Although this is a good approx-unation for some prices (such as wages set by union contracts, wages that are adjusted annually, and prices in some catalogues), it is not a good description of others. Many retail stores, for example, can adjust the timing of their price changes fairly freely in response to economic developments. It is therefore natural to analyze the consequences of such state-dependent pricing. Our hnal model of staggered price changes, the Caplin-Spulber model, provides an example of such an analysis.

The model is set in continuous time. Each individuals optimal price at time t, p*(r), is again {1) + ) ( ). Money growth is always positive; as we will see, this causes p* to always be increasing. The key assumption of the model is that price-setters follow an Ss pricing policy. Specifically,

"In addition, tiiis result helps to justify the assumption that the initial distribution of p, - p,* is uniform between s and S. p, - p,* for each price-setter equals each value between s and 5 once during the interval between any two price changes; thus there is no reason to expect a concentration anywhere within the interval. Indeed, Caplin and Spulber show that under simple assumptions, a given price-setters p, - p,* is equally likely to take on any value between s and 5.

whenever a price-setter adjusts his or her price, he or she sets it so that the difference between the actual price and the optimal price at that time. Pi - P*, equals some target level, S. The individual then keeps the nominal price fixed until money growth has raised p* sufficiently that p, - p* has fallen to some trigger level, s. He or she then resets p, - pf to 5, and the process begins anew.

Such an Ss policy is optimal when inflation is steady, aggregate output is constant, and there is a fixed cost of each nominal price change (Barro, 1972; and Sheshinski and Weiss, 1977). In addition, as Caplin and Spulber describe, it is also optimal in some cases where inflation or output is not constant. And even when it is not fully optimal, it provides a simple and tractable example of state-dependent pricing.

Two technical assumptions complete the model. First, to keep prices from overshooting s and to prevent bunching of the distribution of prices across price-setters, m changes continuously. Second, the initial distribution of Pi - pf across price-setters is imiform between s and S. The remaining assumptions are the same as in the Fischer and Taylor models.

Under these assumptions, money is completely neutral in the aggregate despite the price stickiness at the level of the individual price-setters. To see this, consider an increase in m of amount Am < S -s over some period of time. We want to find the resulting changes in the price level and output, and . Since pf = (1 - ) + , the rise in each price-setters optimal price is (1 - ) -i- . Price-setters change their prices if p, - pf falls below s; thus price-setters with initial values of p, - pf that are less than s -i- [(1 - ) + ] change their prices. Since the initial values of p, - pf are distributed uniformly between s and S, this means that the fraction of price-setters who change their prices is [(1 - ) + Am]/{S - s). Each price-setter who changes his or her price does so at the moment when his or her value of p, - pf reaches s; thus each price increase is of amount S ~s. Putting aU of this together gives us

- ) +

S (6.83)

= (1 - ) + .

Equation (6.83) implies that = Am, and thus that = 0. Thus the change in money has no impact on aggregate output.

To understand the intuition for this result, consider the case where = 1, so that p, - pf is just p, - m. Now think of arranging the points in the interval (s, 5] around the circumference of a circle; this is shown in

FIGURE 6.2 model

The effects ofan increase in the money stock in the Caplin-Spulber

Figure 6.2. Initially, price-setters are distributed uniformly around the circle. Now notice that an increase in m of Am moves every price-setter aroimd the circle counterclockwise by a distance Am. To see this, consider Hrst a price-setter, such as the one at Point A, with an initial value of p, - pf that is greater than s + Am. Such a price-setter does not raise his or her price when m rises by m; since p* rises by Am, p; - p* therefore falls by Am. Thus the price-setter moves counterclockwise by amount Am. Now consider a price-setter, such as the one at Point C, with an initial value of p, - p* that is of the form s + k, where is less than Am. For this price-setter, Pi - pf falls until m has risen by k; thus he or she is moving counterclockwise around the circle. At the instant that the increase in m reaches k, pt jumps by 5 - s, and so Pi - pf jumps from s to 5. In terms of the diagram, however, this is just an infinitesimal move around the circle. As m continues to rise, the price-setter does not change his or her price further, and thus continues to travel around the circle. Thus the total distance such a price-setter travels is also Am.

Since the price-setters are initially distributed uniformly around the circle, and since each one moves the same distance, they end up still uniformly distributed. Thus the distribution of p, - m is unchanged. Since p is the average of the Pis, this implies that p - m is also unchanged.

The reason for the sharp difference between the results of this model and those of the Taylor model is the nature of the price-adjustment policies. In the Caplin-Spulber model, the number of price-setters changing their prices at any time is larger when the money supply is increasing more rapidly; given the specific assumptions that Caphn and Spulber make, this