common. For example, the two studies of price adjustment that examine the broadest ranges of goods, Carhons and Blinders, find that intervals between price changes are typically about a year. And price changes for some goods are much less frequent. Cecchetti, for example, finds that the newsstand prices of magazines are changed on average only every three years.

The second theme of these studies is that the price adjustments do not follow any simple pattem. The behavior of L.L. Bean catalog prices, documented by Kashyap, is representative. As in the other studies, the frequency of price changes is low. on average, the price of a good is changed only after inflation has eroded its real price by about 10%. Only an extremely large cost of price adjustment, or an extremely small cost of failing to charge the price that is optimal in the absence of adjustment costs, can reconcile this finding with a menu-cost view. In addition, although Bean issues over 20 catalogs a year, prices are changed in only two of the catalogs (faU and spring). Even in these catalogs, most prices are usually not changed. Neither fact supports the view that the barrier to price adjustment is the cost of printing and posting a new price. In addition, the spacing of the changes is highly irregular: thus the results are not at all consistent with the assumption of the Fischer and Taylor models that there is a fixed interval between changes. Finally, the size of changes varies tremendously, and small changes are as likely as large changes to be followed quickly by an additional change; if the barrier to price adjustment is some kind of fixed cost, then under reasonable assumptions the changes would be fairly uniform in size, and the firm would make a relatively small change only if it expected the new price to be m effect for a relatively long time. In sum, the microeconomic evidence on price stickiness is puzzling.

6.14 Coordination-Failure Models and Real Non-Walrasian Theories

Coordination-Failure Models

All the models of fluctuations we have examined imply that when prices are flexible, the economy has a unique equilibrium. Thus fluctuations arise only from changes in the flexible-price equilibrium (as in real-business-cycle models) or from departures of the economy from that equilibrium (as in models with nominal stickiness). If more than one level of output is a flexible-price equilibrium, however, fluctuations can also arise from movements of the economy among different equilibria.

Cooper and John (1988) present a simple framework for analyzing multiple equilibria in aggregate activity. The economy consists of many identical agents. Each agent chooses the value of some variable, which we call output for concreteness, taking others choices as given. Let U, = V(y,,y) be

FIGURE 6.7 A reaction function that implies a unique equilibrium

agent zs payoff when he or she chooses output y, and all others choose y. (We will consider only symmetric equilibria; thus we do not need to specify what happens when others choices are heterogeneous.) Let y*(y) denote the representative agents optimal choice of y, given y. Assume that V(») is sufhciently weU behaved that y,*(y) is uniquely defined for any y, is continuous, and is always between zero and some upper bound y. y*{y) is referred to as the reaction function.

Equilibrium occurs when y*(y) = y. In such a situation, if each agent believes that other agents will produce y, each agent in fact chooses to produce y.

Figure 6.7 shows an economy without multiple equilibria. The figure plots the reaction function, y*(y). Equilibrium occurs when the reaction function crosses the 45-degree line. Since there is only one crossing, the equilibrium is unique.

Figure 6.8 shows a case with multiple equilibria. Since y*(y) is bounded between zero and y, it must begin above the 45-degree line and end up below. And since it is continuous, it must cross the 45-degree line an odd number of times (if we ignore the possibility of tangencies). The figure shows a case with three crossings, and thus three equilibrium levels of output. Under plausible assumptions, the equilibrium at Point A is unstable. If, for example, agents expect output to be slightly above the level at A, they produce slightly more than they expect others to produce. With natural assumptions about dynamics, this causes the economy to move away from A. The equilibria at and C, however, are stable.

With multiple equilibria, fundamentals do not fully determine outcomes. If a nts e>. Jt "b. 2C- -.or.iy to be at . ends u, tnere; ii they expect il to

FIGURE 6.8 A reaction function that implies multiple equilibria

be at B, it ends up there instead. Thus animal spirits, self-fulfilling prophecies, and sunspots can affect aggregate outcomes.-

II is plausible that V(y,, y) is increasing in y-that is, that a typical individual is better off when aggregate output is higher. In the model of Section 6.6, for example, higher aggregate output shifts the demand curve that the representative firm faces out, and thus increases the real price the firm obtains for a given level of its output. If this condition holds, equilibria with higher output involve higher welfare. To see this, consider two equilibrium levels of output, yi and , with > yi- Since V(y,,y) is increasing in y, Viyi.yz) is greater than V(yi,yi). And since is an equilibrium, y, = 2 maximizes V(yi,y) given = y, and so ( 2, 2) exceeds V(yi,y2). Thus the representative agent is better off at the higher-output equilibrium.-

A sunspot equilibrium occurs when some variable that has no inherent effect on the economy matters because agents believe that it does. Any model with multiple equilibria has the potential for sunspots: if agents believe that the economy will be at one equilibrium when the extraneous variable takes on a high value and at another when it takes on a low value, they behave in ways that validate this belief. For more on these issues, see Cass and Shell (1983); Woodford (1990, 1991); and Farmer (1993).

is no necessary connection between the slope of the reaction function and the welfare properties of equilibria. If agents maximization problem has an interior solution, y,*(y) is defined by Vi(y,*(y),y) = 0, where subscripts denote partial derivatives. Differentiating this condition with respect to yields ,*( ) = Vn /(~ Vu). Since Vu must be negative for y,*(y) to be an interior maximum, the sign of y,*(y) is given by the sign of V12. Relative welfare in different equilibria, on the other hand, is determined by V2. Thus the issues of whether there are multiple equilibria and whether a high-output equilibrium is preferable to a low-output one are distinct.