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84

When the price of the producers good increases, there is some chance that the increase reflects a rise in the price level, and some chance that it reflects a rise in the goods relative price. The rational response for the producer is to attribute part of the change to an increase in the price level and part to an increase in the relative price, and therefore to increase output somewhat. This implies that the aggregate supply airve slopes up: when the aggregate price level rises, all producers see increases in the prices of their goods, and (not knowing that the increases reflect a rise in the price level) thus raise their output.

The next two sections develop this idea in a model where individuals produce goods using their own labor, sell their output in competitive markets, and use the proceeds to buy other producers output. The model has two types of shocks. First, there are random shifts in preferences that change the relative demands for different goods. These shocks lead to changes in relative prices and m the relative production of different goods. Second, there are disturbances to the money supply, or more generally, to aggregate demand. When these shocks are observed, they change only the aggregate price level and have no real effects. But when they are unobserved, they change both the price level and aggregate output.

As a preliminary. Section 6.2 considers the case where the money stock is publicly observed; in this situation, money is neutral. Section 6.3 then turns to the case where the money stock is not observed.

6.2 The Case of Perfect Information

Producer Behavior

There are many different goods in the economy. Consider a representative producer of a typical good, good i. The individuals production function is simply

Q, =1,, (6.1)

where L, is the amount that the individual works and Q , the amount he or she produces. The individuals consumption, C,, equals his or her real income; this equals revenue, P, Q,, divided by the pnce of the market basket of goods, P. P is an index of the prices of all goods (see equation [6.9], below).

Utility depends positively on consumption and negatively on the amount worked. For simplicity, it takes the form

= C, - -Ll 7 > 1. (6.2)

Thus there is constant marginal utility of consumption and increasing marginal disutility of work.



Demand

Producers behavior determines the supply curves of the various goods. Determining the equilibrium in each market requires specifying the demand curves as weh. The demand for a given good is assumed to depend on three factors: real income, the goods relative price, and a random disturbance to preferences. For tractability, demand is log-linear. Specifically, the demand for good is

Qi = y + Zi - r]{p, -p), r]> 0, (6.7)

where is log aggregate real income, z, is the shock to the demand for good i, and tj is the elasticity of demand for each good, is the demand per producer of good i} The z, s have a mean of zero across goods; thus they are purely relative demand shocks, is assumed to equal the average across goods of the q, s, and p is the average of the p, s:

y = qt, (6.8)

That is, the total (log) demand for good i is InN + + z, - rjlp, - p), where N is the number of producers of each good.

When the aggregate price level P is known, the individuals maximization problem is simple. Substituting C, = P, Q, / and Q, = L, into (6.2), we can rewrite utility as

ui = -hj. (6.3)

Since markets are assumed to be competitive, the individual chooses I,- to maximize utility taking P, and P as given. The first-order condition is

j - - = 0, (6.4)

I,-=(P,/P)i/<-i>. (6.5)

Letting lowercase letters denote the logarithms of the corresponding uppercase variables, we can rewrite this condition as

£i = -{Pt - P). (6.6)

Thus the individuals labor supply and production are increasing in the relative price of his or her product.



Equilibrium

Equilibrium in the market for good z" requires that demand per producer equal supply. From (6.6) and (6.7), this requires

(Pi - p) = -b z, - 7( , - p). (6.11)

Solving this expression for p, yields

Pi = , (y + z,) + p. (6.12)

Although (6.7)-(6.9) are intuitive, deriving these exact ftmctional forms from individuals preferences over the various goods requires some approximations. The difficulty is that if preferences are such that demand for each good takes the constant-elasticity form in (6.7), the corresponding (log) price index is exactly equal to the average of the individual p, s only in the special case ot rj = 1. See Problem 6.2. This issue has no effect on the basic messages of the model

p = P,. (6.9)

Intuitively, (6.7)-(6.9) state that the demand for a good is higher when total production (and thus total income) is higher, when its price is low relative to other prices, and when individuals have stronger preferences for it. Finally, the aggregate demand side of the model is

= m - p. (6.10)

There are various interpretations of (6.10). The simplest, and most appropriate for our purposes, is that it is just a shortcut approach to modeling aggregate demand. Equation (6.10) implies an inverse relationship between the price level and output, which is the essential feature of aggregate demand. Since our foais is on aggregate supply, there is little point in modeling aggregate demand more fully. Under this interpretation, M should be thought of as a generic variable affecting aggregate demand rather than as money.

It is also possible to derive (6.10) from models with more complete monetary specifications. Blanchard and Kiyotaki (1987), for example, replace C, in the utility function, (6.2), with a Cobb-Douglas combination of C, and the individuals real money balances, M, / . With an appropriate specification of how money enters the budget constraint, this gives rise to (6.10). Rotemberg (1987) derives (6.10) from a cash-in-advance constraint. Under Blanchard and Kiyotakis and Rotembergs interpretations of (6.10), it is natural to think of m as literally money; in this case the right-hand side should be modified to be m -i- v - p, where v captures aggregate demand disturbances other than shifts in money supply.



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