The result that equilibrium log output is zero implies that the equilibrium level of output is 1. This results from the 1/y term multiplying in the utility function, (6.2).

"If the Individual knew others prices as a result of making purchases, he or she could deduce p, and hence r,. This can be ruled out in several ways. One approach Is to assume that the household consists of two individuals, a "producer" and a "shopper," and that communication between them is limited. In Lucass original model, the problem is avoided b\ assuming an overlapping-generations structure where individuals produce in the first period of their lives and make purchases in the second.

Averaging the p, s and using the fact that the average of the z, s is zero, we obtain

. p-yry+p. (6.13)

Equation (6.13) implies that the equilibrium value of is simply

= 0. (6.14)

Finally, (6.14) and (6.10) imply

m = p. (6.15)

Not surprisingly, money is neutral in this version of the model: an increase in m leads to an equal increase in all p, s, and hence in the overall price index, p. No real variables are affected.

6.3 The Case of Imperfect Information

We now consider the more interesting case where producers observe the prices of their own goods but not the aggregate price level.

♦

Producer Behavior

Defining the relative price of good i by r, = p, - p, we can write

= ,- ) (ggj

= p + rt.

Thus, in logs, the variable that the individual observes-the price of his or her good-equals the sum of the aggregate price level and the goods relative price.

The individual would like to base his or her production decision on r, alone (see [6.6]). The individual does not observe r,, but must estimate it given the observation of pi At this point, Lucas makes two simplifying assumptions. Fust, he assumes that the individual finds the expectation of Yi given Pi, and then produces as much as he or she would if this estimate were certain. Thus (6.6) becomes

£, =- [ , I p,]. (6.17)

As Problem 6.1 shows, this certainty-equivalence behavior is not identical to maximizing expected utility: m general, the utility-maximizing choice of f, depends not just on the individuars estimate of r,, but also on his or her uncertainty about r,. The assumption that individuals use certamty equivalence, however, simplifies the analysis and has no effect on the central messages of the model.

Second, and very importantly, Lucas assumes that the producer finds the expectahon of r, given p, rationally. That is, E[r, \ p, ] is assumed to be the true expectation of r, given p, and given the actual joint distribution of the two variables. Today, this assumption of rational expectations seems no more pecuUar than the assumption that individuals maximize utility. When Lucas introduced Muths (1960, 1961) idea of rational expectations into macroeconomics, however, it was highly controversial. As we will see. It is one source-but by no means the only one-of the strong implications of Lucass model.

To make the computation of E[r, \ p,] tractable, the monetary shock (m) and the shocks to the demands for the individual goods (the 2,s) are assumed to be normally distributed, m has a mean of £ [m 1 and a variance of Vm The z, s have a mean of zero and a variance of V., and are independent of m. We will see that these assumptions imply that p and r, are normal and independent. Since p, equals p - r,, this means that it is also normal; its mean is the sum of the means of p and r,, and its variance is the sum of their variances. As we will see, the means of p and ri,E[p\ and E[r], are equal to E\m] and zero, respectively; and their variances, Vp and VV, are comphcated functions of Vm and Vz and of the other parameters of the model.

The individuals problem is to find the expectation of r, given p,. An important result in statistics is that when two variables are jointly normally distnbuted (as with r, and p, here), the expectation of one is a linear function of the observation of the other (see, for example. Mood, Graybili, and Boes, 1974, pp. 167-168, or some other introductory statistics textbook). Thus E{ri I p,] takes the form

£[r, I p,] = a + /3p,. (6.18)

In this partiailar case, where p, equals r, plus an independent variable, (6.18) takes the specific form:

(6.19)

y-lVr + Vp (6.20)

= b(pi - E[p]).

Averaging (6.20) across producers (and using the definitions of and p) gives us an expression for overall output:

= b(p - £[p]). (6.21)

Equation (6.21) is the Lucas supply curve. It states that the departure of output from its normal level (which is zero in the model) is an increasing function of the surprise in the price level.

The Lucas supply curve is essentially the same as the expectations-augmented Phillips curve of Chapter 5 with core inflation replaced by expected inflation (see equation [5.38]). Both state that, if we neglect dis turbances to supply, output is above normal only to the extent that inflation (and hence the price level) is greater than expected. Thus the Lucas model provides microeconomic foundations for this view of aggregate supply.

Equilibrium

Combining the Lucas supply curve, (6.21), with the aggregate demand equation, (6.10), and solving for p and yields

This conditional-expectations problem is referred to as signal extraction. The variable that the individual observes, p,, equals the signal, r,, plus noise, p. Equation (6.19) shows how the individual can best extract an estimate of the signal from the observation of p,. The ratio of Vr to Vp is referred to as the signal-to-noise ratio.

Equation (6.19) is intuitive. First, it implies that if p, equals its mean, the expectation of rj equals its mean (which is zero). Second, it states that the expectation of r, exceeds its mean if p, exceeds its mean, and is less than its mean if p, is less than its mean. Third, it tells us that the fraction of the departure of p, from its mean that is estimated to be due to the departure of r, from its mean is VrKVy + Vp); this is the fraction of the overall variance of p, (Vr + Vp) that is due to the variance of r, (V,-). If, for example, Vp is zero, all of the variation in p, is due to r,, and so ffr, p, J is Pi - E[m]. If Vr and Vp are equal, half of the variance in p, is due to r,, and so £[r, I Pi] = (pi - Elm])/2. And so on.""

Substituting (6.19) into (6.17) yields the indixaduals labor supply:

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