U,=lLn.-hj. (6.37)

Finally, the aggregate demand side of the model is again given by = m - p (equation [6.10]); is again the average of the ,s. In contrast to the Lucas model, the money supply is publicly ob served.

Individual Behavior

Converting the demand equation, qt = y-ri(p, -p), from logs to levels yields Q, = Y(Pi IP) Substituthig this into expression (6.37) gives us

. iP,-W)Y(PiJP) + WLi 1 P

"The absence of an economy-wide labor market is critical to the Lucas model: with such a market, individuals observation of the nominal wage would allow them to deduce the money supply, and would thus make nominal shocks neutral. In contrast, assuming a competitive labor market in the current model is not crucial to the results.

As described in n. 2 and Problem 6.2, when individuals preferences over the different goods give rise to the assumed constant-elasticity demand curves for each product, the appropriate (log) price and output indexes are not exactly equal to the averages of the p, s and the q, s. Problem 6.4 shows, however, that the results of this section are unchanged when the exact indexes are used.

the only input into production. But individuals do not produce their own goods directly; instead there is a competitive labor market where they can both sell their labor and hire workers to produce their goods.

As before, the demand for each good is log-linear; for simplicity, the shocks to the demands for the individual goods (the z,s) are absent. Thus, q, = y - T}(p, - p) (see [6.7]). p is the (log) price level; as in Part A, it is the average of the p, s. To ensure that a profit-maximizing price exists, tj is assumed to be greater than 1. Sellers with market power set price above marginal cost; thus if they cannot adjust their prices, they are willing to produce to satisfy demand in the face of small fluctuations in demand, tn the remainder of the chapter, sellers are therefore assumed not to ration customers.

As in the Lucas model, the utility of a typical individual is Ui = Q-Lj/y (see [6.2]); again C, is the individuals income divided by the price index, P, and I is the amount that he or she works. The production function is the same as before: the output of good / equals the amoimt of labor employed in its production. Individual zs income is the sum of profit income, (P, - W)(li, and labor income, WL,, where Qj is the output of good z and W is the nominal wage. Thus,

Equilibrium

Because of the symmetry of the model, in equilibrium each individual works the same amount and produces the same amount. Equilibrium output is thus equal to the common level of labor supply. We can therefore use (6.41) or (6.42) to express the real wage as a function of output:

j = (6.43)

Substituting this expression into the price equation, (6.40), yields an expression for each producers desired relative price as a function of aggregate output:

= - -\ (6.44)

P TJ - 1

The individual has two choice variables, the price of his or her good (P,) and the amount he or she works (I,). The first-order condition for P, is

r(P, / ) - (P, - W)7,Y(P, /PyHl/P) p

Multiplying this expression by (P, /P)"P, dividing by , and rearranging yields

That is, we get the standard result that a producer with market power sets price as a markup over marginal cost, with the size of the markup determined by the elasticity of demand.

Now consider labor supply. From (6.38), the first-order condition for I,

y-ir=0. (6.41)

I, = (f) . (6.42)

Thus labor supply is an increasing function of the real wage; the elasticity is 1/(y-l).

For future reference, it is useful to write this expression in logs:

(6.45)

= + .

Since producers are symmetric, each charges the same price. The price index P therefore equals this common price. Equilibrium therefore requires that each producer, taking P as given, sets his or her own price equal to P; that is, each producers desired relative price must equal 1. From (6.44), this condition is [-n/ir] - l)]F>"i = 1, or

i/(r-i)

(6.46)

This is the equilibrium level of output.

Finally, we can use the aggregate demand equation, = M/P, to find the equilibnum price level:

M (6.47)

7}- 1

Implications

When producers have market power, they produce less than the socially optimal amount. To see this, note that in a symmetric allocation each individual supplies some amount I of labor, and production of each good and each individuals consumption are equal to that I. Thus the problem of finding the best symmetric allocation reduces to choosing I to maximize I - (lly)V. The solution is simply I = 1. As (6.46) shows, equilibrium output is less than this. Intuitively, the fact that producers face downward-sloping demand curves means that the marginal revenue product of labor is less than its marginal product. As a result, the real wage is less than the marginal product of labor: from (6.40) (and the fact that each P, equals P in equilibrium), the real wage is (tj - l)/??; the marginal product of labor, in contrast, is 1. This reduces the quantity of labor supphed, and thus causes equilibrium output to be less than optimal. From (6.46), equilibrium output is [(tj - 1)/t}]*>"; thus the gap between the equilibrium and optimal levels of output is greater when producers have more market power (that is.