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

13.3 The Demand for Labor


Profit as a function of labor

In addition to looking at the competitive firm’s profit-maximizing supply of output, we can also look at its profit-maximizing demands for its inputs. This is particularly important because the firm buying inputs — hiring workers and borrowing capital — form the “demand” side of labor and financial markets.

As a reminder, the firm’s profit is its total revenue minus its total cost. For a competitive firm, this is \(\pi = p \times q - (w \times L + r \times K)\) In solving for the profit-maximizing level of output in the short run, we wrote this as profit as a function of $q$: \(\pi(q) = p \times q - c(q)\) where \(c(q) = w \times L(q) + r \times \overline K\) was the cost of the inputs required to produce $q$ units of output. However, because $q = f(L)$, we could have proceeded differently, by writing \(\pi(L) = p \times f(L) - wL - r\overline K\) If we take the derivative of this and set it equal to zero, we get \(\overbrace{p \times MP_L}^{MRP_L} = w\) The left-hand side of this equation is called the marginal revenue product of labor (or $MRP_L$). It’s the marginal revenue from hiring another unit of labor: hiring that unit of labor increases output by $MP_L$, and those $MP_L$ units are sold for price $p$ each, so the firm earns $p \times MP_L$ additional revenue from hiring the additional unit of labor. The right-hand side is the wage rate, which is of course the marginal cost of hiring an additional unit of labor.

Optimal labor as a function of wage

Let’s look at our example of $f(L,K) = \sqrt{LK}$, with capital fixed at $\overline K = 32$. In this case we have $MP_L = \sqrt{8/L}$, so the profit-maximizing condition equating $MRP_L = w$ is \(\begin{aligned} p \times \sqrt{8 \over L} &= w\\ \sqrt{8 \over L} &= {w \over p}\\ L^\star(w,p) &= {8p^2 \over w^2} \end{aligned}\)

This gives us the demand for labor as a function of the wage rate and the output price. To plot the labor demand, we hold the price of output constant and plot $L$ as a function of $w$ (as usual, plotting the price $w$ on the vertical axis and the quantity $L$ on the horizontal axis). For example, holding $p = 12$ constant gives us $L^\star(w) = 8 \times 12^2/w^2$:

Note that when we analyzed the firm’s output supply problem when $p = 12$ and $w = 8$, we found that its optimal supply was $q^\star = 24$. Here we find that at those prices, its profit-maximizing labor demand is $L = 18$. In fact, if the firm hires $L = 18$ workers, it produces \(f(12) = \sqrt{18 \times 32} = 24\) So this is one and the same decision.

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