The supply function of a profit-maximizing price-taking firm

A firm's profit is its revenue minus its cost. If the price p at which the firm can sell its output is not significantly affected by the size of its output, it is reasonable to model the firm as taking the price as given. In this case, its total revenue is

TR(y) = py,

where y is its output. Thus the firm's profit function is

(y) = TR(y) TC(y) = py TC(y),

where TC is either the firm's short run cost function or its long run cost function, depending on whether we are interested in short run or long run supply.

Theory: The firm chooses its output y to maximize its profit (y), taking price as given.

If we solve the maximization problem for all values of p, we get a function y(p). This function is the firm's supply function.

Differentiating (y) with respect to y we obtain

p TC'(y) = 0,

or, recalling that the derivative to TC is MC,

p = MC(y*).

Further, at a maximum (rather than a minimum) the second derivative of profit at y* must be negative, or

MC'(y*) < 0,

MC'(y*) > 0:

the marginal cost curve must be upward sloping at a profit-maximizing output.

Short run supply

In the short run the firm may face a fixed cost even if it produces no output, and we need to check whether it would be better off producing no output rather than y*. If it produces no output it makes a loss equal to FC. Thus the firm's optimal decision is to produce nothing if its best positive output y* yields a loss greater than FC, and otherwise to produce y*. Put differently, the optimal decision is to produce no output if the price is less than the minimum of the firm's average variable cost (in which case for every unit the firm sells it makes a loss).

In summary: A firm's short run supply function is given as follows.

If price is less than the minimum of the firm's AVC then the optimal output is zero.
If the price exceeds the minimum of the firm's AVC then the optimal output y* satisfies the conditions that p = SMC(y*) and SMC is increasing at y*.

In words, a firm's short-run supply function is the increasing part of its short run marginal cost curve above the minimum of its average variable cost.

The short run supply function of a firm with "typical" cost curves is shown in the figure.

Note: At the output it chooses, the firm may make a loss. The loss must be less than its fixed cost (otherwise it would be better for the firm to produce no output), but it definitely may be positive.

At the output it chooses when the price is p, the firm's profit is

y*(p SAC(y*)).

This profit is the red rectangle (length y*, height p

SAC(y*)) in the following figure.

Examples and exercises on short-run profit maximization

Long run supply

In the long run the firm pays nothing if it does not operate. Thus its supply function is given by the part of its marginal cost function above its long run average cost function. (If its maximal profit it positive it wants to operate; if its maximal profit it negative it does not want to operate.)

In summary: A firm's long run supply function is given as follows.

If price is less than the minimum of the firm's LAC then the optimal output is zero.
If the price exceeds the minimum of the firm's LAC then the optimal output y* satisfies the conditions that p = LMC(y*) and LMC is increasing at y*.

In words, a firm's long-run supply function is the increasing part of its long run marginal cost curve above the minimum of its long run average cost.