3.4 Price-Taking Firm's Supply

In the previous section we learned how to compute the price-taking firm’s optimal choice in the short and in the long run. In the example developed in the previous section, the firm’s technology is described by the production function $Q=80\sqrt{L}$ or, equivalently, by the inverse production function $L=Q^2/6400$, and the price of labor is $W=64$. From these data we have computed (as explained in Section 3.2) the marginal cost function, $MC(Q)=Q/50$, and from there, for each possible output price $P$, the firm’s short-run optimal choice: produce the quantity $Q$ solving the equation $MC(Q)=P$. In our example, $Q=50P$.

Supply in the Short Run

Given the firm’s technology and the cost of labor, the function associating each possible output price $P$ to the short-run profit maximizing quantity $Q$ is the firm’s short-run supply function. (In the example recalled above, the firm’s short-run supply function is $Q=50P$.) Graphically, the firm’s short-run supply function coincides with the firm’s marginal cost function.

Assuming that the production function is $Q=A\sqrt{L}$, in the following figure we summarize the construction of the firm’s short-run supply function, and show how the function depends on productivity (parameter $A$) and price of labor, $W$. (As we know, the firm’s short-run optimal choice does not depend on the fixed cost $FC$.) For reference to the underlying cost structure, in the figure we also represent (with dashed thin lines) the average cost (in brown) and the average variable cost (in orange).

Supply in the Long Run

Once we fix not only the firm’s technology and labor cost, but also the firm’s fixed cost, we know how to associate to each price $P$ the long-run profit maximizing quantity $Q$. This mapping is called the long-run supply function of the firm. As we know, in the long run the firm’s behavior is identical to that in the short run, if the market price of output, $P$, is larger than the minimum average cost, $AC_{\text{min}}$. The long-run optimal choice is instead $Q=0$ if $P$ is smaller than $AC_{\text{min}}$. Finally, if $P=AC_{\text{min}}$, the firm has two optimal choices, namely $Q=0$ and $Q=Q^{\text{eff}}$.

Thus, the firm’s long-run supply function graphically coincides with the marginal cost function for prices $P$ larger than minimum average cost, and with the vertical axis (i.e. $Q=0$) for prices $P$ smaller than minimum average cost. When $P=AC_{\text{min}}$, the graph of the long-run supply consists of two points, corresponding to $Q=0$ and $Q=Q^{\text{eff}}$.

Using the same data as in the previous figure, in the following figure we summarize the construction of the long-run supply function of the firm, and show how it depends on productivity (parameter $A$), labor cost $W$, and fixed costo $FC$. Again, to have a reference to the underlying cost structure, in the figure we also represent (with dashed thin lines) average cost (in brown), average variable cost (in orange), and marginal cost (in red).

Producer Surplus

The objective of the firm is to maximize its profit, and in this chapter we have explained how to identify the firm’s corresponding optimal choice in the short and in the long run. Another measure of the firm’s welfare, closely related to profit, is producer surplus, indicated with $PS$, given by the difference between the firm’s revenue and its avoidable cost. As we know, the firm’s fixed cost is sunk in the short run, but avoidable in the long run. Thus, in the short run producer surplus is the difference between revenue and variable cost, $PS=R-VC$, while in the long run producer surplus coincides with profit, $PS=\Pi$.

Maximizing producer surplus is the same as maximizing profit. Indeed, in the short run producer surplus and profit only differ by a constant that is unaffected by the firm’s decisions (the fixed cost, which is sunk in the short run), while in the long run the two notions coincide. Moreover, the concept of producer surplus allows us to express the firm’s profit maximization rule in a very compact fashion. In fact, whether we are in the short or in the long run, we can describe the firm’s optimal choice as follows:

Choose the quantity $Q$ at which $MC(Q) = P$ if, at this quantity, producer surplus is positive or zero (if it is zero, choosing $Q=0$ is also optimal). Otherwise, choose $Q=0$.

It is easy to visualize and compute producer surplus using the firm’s supply function. At any quantity of output $Q$, producer surplus is the area between the horizontal line drawn at the level of the market price $P$ and the firm’s supply function, as we illustrate next.