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Chapter 13 / Output Supply and Input Demands for a Competitive Firm

13.4 Response to Price and Wage Changes


In the previous two sections we showed that the firm’s optimal output was \(q^\star(w,p) = {16p \over w}\) and that its optimal amount of labor was \(L^\star(w,p) = {8p^2 \over w^2}\) Note that these are, in fact, one and the same result: if we plug this optimal amount of labor into the production function, we get \(q^\star(w,p) = f(L^\star(w,p)) = \sqrt{\frac{8p^2}{w^2} \times 32} = {16p \over w}\) This makes sense, because we’ve established that there is a 1:1 correspondence between the firm’s choice of inputs (in this case, the variable amount $L$ and the fixed amount $\overline K$) and its choice of output.

In plotting the output supply and labor demand curves, though, we drew a diagram for the quantity of one thing as a function of its own price, holding the other price constant. Let’s now consider what happens to the firm’s supply of output when the wage rate changes, and to the firm’s demand for labor when the price changes.

The following diagram shows both the firm’s output supply (in the left diagram) and its labor demand (in the right diagram). The initial condition it shows is the one we’ve been analyzing, with $p = 12$ and $w = 8$. Try adjusting the price and/or the wage, and see what happens to the diagrams.

As you can see:

In each case, both $L^\star$ and $q^\star$ move in the same direction; and indeed, the relationship $q = f(L)$ must always hold.

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