Why are marginal wages used when solving for the natural rate of unemployment?

Question

In blanchard's macroeconomics, in analysing the labor market and solving for the natural rate of unemployment, he explains a model where the real wage workers are willing to accept, W/P = F(u,z), where u is unemployment and z captures all other factors that strengthen the bargaining power for workers.

He then sets up an equation for the wage that firms are willing to offer, for the case of a simplified production function Y = N, where N is the number of workers, given a price level P and markup m, and calls this the price determination equation: P = (1+m), or W/P = 1/(1+m)

And the solves for equilibrium.

What is confusing to me, however, is, for the price determination equation, P seems to represent the aggregate or average price level, with m being the aggregate markup and W being the aggregate or average wage. But he emphasizes W being the marginal cost of labor. To quote the text:

The production function Y = N implies that the cost of producing one more unit of output is the cost of employing one more worker, at wage W. Using the terminology introduced in your microeconomics course: The marginal cost of production—the cost of producing one more unit of output—is equal to W.

If there was perfect competition in the goods market, the price of a unit of output would be equal to marginal cost: P would be equal to W. But many goods markets are not competitive, and firms charge a price higher than their marginal cost. A simple way of capturing this fact is to assume that firms set their price according to P = (1 + m) W

I don't understand why the marginal wage is what is relevant in this case?

Answer 1

That follows from profit maximization of a competitive firm. Profits $\Pi$ are given by revenue $P\times Y$ minus costs $W\times N$. Using the production function $Y=N$ we can write this as

$$\Pi=PY-WN=PN-WN.$$

Maximizing profits by choosing the optimal input amount $N$ yields the standard condition that the price needs to be equal to marginal costs: $$P-W=0\Leftrightarrow W=P.$$

Hence, $W$ is the marginal cost of production. Intuitively, it costs $W$ to produce an additional unit of output. If the price of a unit $P$ were lower than that, the firm would make losses on the last unit. If it were higher, you could still make additional profits by increasing production by an additional unit.