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.
[ See interactive graph online at https://www.econgraphs.org/graphs/firm/input_demands/output_supply_and_labor_demand ]
As you can see:
- an increase in the market price results in an upward movement along the supply curve, and an outward shift of the labor demand curve. That’s because an increase in the output price makes each unit of labor more valuable (it increases its $MRP_L$), so at any given wage rate the firm is willing to hire more people.
- an increase in the market wage rate results in an upward movement along the labor demand curve, and an inward shift of the supply curve. That’s because an increase in the wage rate raises the marginal cost of producing each unit of output, so at any given market price the firm will produce fewer units.
In each case, both $L^\star$ and $q^\star$ move in the same direction; and indeed, the relationship $q = f(L)$ must always hold.