Competition and supply

A firm with a cost function still has no reason to prefer one output to another until something is said about what the output sells for.

The simplest possible assumption is that the firm cannot affect the price at all, and this lesson works out its consequences in full. The assumption is false of almost every real firm, which is exactly why it is worth doing first: it produces a benchmark against which market power can be measured, and the next three lessons are all measurements against it.

What price taking means

A firm is a price taker when its own output decision leaves the market price unchanged, so the demand curve facing it is horizontal at the going price. Sell one unit or a thousand, the price is the same, and try to charge a penny more and sales go to zero.

Four conditions are conventionally listed as sufficient for this. There are many buyers and many sellers, each small relative to the market. The product is homogeneous, so nobody has a reason to prefer one seller. Buyers and sellers know the prices on offer. And entry and exit are free in the long run, which does no work in the short run but everything in the long.

Note what these conditions actually deliver. Homogeneity plus information is what makes the individual demand curve horizontal, because a seller charging more is undercut by an identical product. Smallness is what makes the firm's output too little to move the market price. And note the honest point about the elasticity from three lessons ago: market demand for wheat is highly inelastic, while the demand curve facing one wheat farm is effectively perfectly elastic. Those are consistent, and confusing them is a standard error.

Nothing meets all four conditions exactly. Wheat, foreign exchange and standardised financial contracts come close. George Stigler pointed out in 1957 that the concept only became precise long after economists started using it, and its value is as a limiting case rather than a description.

Profit maximisation

Profit is revenue minus cost, $π (q) = R (q) - C (q)$ . Differentiating and setting to zero,

MR (q) = MC (q)

Marginal revenue equals marginal cost. This is the general condition for any firm, competitive or not, and it is the marginal rule from the first lesson with revenue as the benefit. For a price taker, revenue is $R = p q$ with $p$ constant, so $MR = p$ and the condition becomes

p = MC (q)

The second-order condition matters here and is usually skipped. A maximum requires $π'' < 0$ , which for a price taker means $MC' (q) > 0$ : marginal cost must be rising at the chosen output. Where marginal cost is falling, $p = MC$ locates a profit minimum, and the firm should move away from it in either direction. That is why only the upward-sloping part of the marginal cost curve is ever a supply curve, and it is also the first hint that a firm with continuously falling costs cannot be a price taker at all.

Example. A firm has $C (q) = 500 + 5 q + 0.05 q^{2}$ and the market price is £25. What does it produce, and what profit does it make?

Marginal cost is $5 + 0.1 q$ . Setting that equal to 25 gives $q = 200$ , and marginal cost is rising there, so it is a maximum. Revenue is $25 \times 200 = £ 5000$ and total cost is $500 + 1000 + 2000 = £ 3500$ , so profit is £1500. Equivalently, average total cost at 200 units is $3500 / 200 = £ 17.50$ , and profit is the margin times the quantity, $(25 - 17.50) \times 200 = £ 1500$ .

Now you. The same firm faces a price of £15. What does it produce, and what profit does it make?

Answer

$5 + 0.1 q = 15$ gives $q = 100$ . Revenue is £1500 and total cost is $500 + 500 + 500 = £ 1500$ , so profit is exactly zero. That is not a coincidence: the previous lesson found that this firm's average total cost is minimised at $q = 100$ with a value of £15, and marginal cost equals average cost at that point. A price equal to minimum average cost gives zero profit, which the long-run section below turns into a prediction.

When to keep operating at a loss

Suppose the price falls below average total cost, so the firm loses money. Shutting down is not automatically right, because in the short run the fixed cost is paid either way.

Compare the two options. Operating gives profit $p q - V (q) - F$ . Shutting down gives $- F$ . Operating is better when $p q - V (q) > 0$ , that is when $p > AVC$ . The shutdown condition is therefore a comparison with average variable cost, not average total cost, and the fixed cost drops out of it entirely, exactly as a cost common to both branches must.

Example. The same firm faces a price of £8. Should it produce, and what happens to its profit?

Setting $5 + 0.1 q = 8$ gives $q = 30$ . Revenue is £240, total cost is $500 + 150 + 45 = £ 695$ , so the loss is £455. Average variable cost at 30 units is $195 / 30 = £ 6.50$ , which is below the price of £8. So operating is right: shutting down would lose the full £500 of fixed cost, and producing loses only £455. The £45 difference is the contribution the output makes towards the fixed cost.

Now you. The price falls to £6. What should the firm do?

Answer

$5 + 0.1 q = 6$ gives $q = 10$ . Revenue is £60 and total cost is $500 + 50 + 5 = £ 555$ , a loss of £495. Average variable cost is $55 / 10 = £ 5.50$ , still below the price, so operating is still marginally better than the £500 loss from shutting. Below £5 the firm's average variable cost can never be covered at any output, so £5 is its shutdown price.

The long run differs because there are no fixed costs in it: every commitment can be ended. So the long-run exit condition compares price with average total cost, and a firm that cannot cover all its costs leaves.

From the firm to the industry

The firm's supply curve is therefore its marginal cost curve above average variable cost, and nothing at all below that. Rearranging $p = 5 + 0.1 q$ gives $q = 10 (p - 5)$ for $p \geq 5$ .

Market supply is the horizontal sum of the individual curves: at each price, add the quantities. Horizontal, not vertical, because the price is common and the quantities add. With $n$ identical firms, $Q = 10 n (p - 5)$ .

Example. One hundred such firms face market demand $Q = 25000 - 500 p$ . Find the short-run equilibrium price, the industry quantity, each firm's output and its profit.

Industry supply is $Q = 1000 (p - 5) = 1000 p - 5000$ . Setting supply equal to demand, $1000 p - 5000 = 25000 - 500 p$ , so $1500 p = 30000$ and $p = £ 20$ . Industry output is 15,000 units and each firm makes 150. Checking against the firm: marginal cost at 150 is $5 + 15 = £ 20$ , matching the price. Each firm's total cost is £2375, so profit is $3000 - 2375 = £ 625$ .

Now you. Suppose 200 such firms are in the industry, facing the same demand curve. Find the equilibrium and each firm's profit.

Answer

Supply is $2000 (p - 5)$ , and $2000 p - 10000 = 25000 - 500 p$ gives $2500 p = 35000$ , so $p = £ 14$ . Industry output is 18,000 and each firm makes 90, at an average total cost of $1355 / 90 = £ 15.06$ . Each firm loses about £95. Two hundred firms is too many for this market at this demand.

Entry, exit, and the zero-profit result

Those two answers bracket the long run. With 100 firms there are profits, which attract entry; with 200 there are losses, which force exit. Entry shifts the industry supply curve right and drives the price down; exit does the reverse. Neither stops until economic profit is zero.

Zero profit requires $p = ATC$ . Profit maximisation requires $p = MC$ . Both hold only where $MC = ATC$ , which the previous lesson established happens exactly at the minimum of average total cost. So the long-run competitive equilibrium has

p = MC = \min ATC

For this firm that price is £15 and that output is 100 units. At £15 the market demands $25000 - 7500 = 17500$ units, so the industry settles at 175 firms each making 100 units. That number was not assumed anywhere; it fell out of the technology, which fixes the efficient scale, and the demand, which fixes how many such firms fit.

Three things are worth extracting from this. Each firm ends up at its own minimum average cost, so the industry's output is produced as cheaply as the technology allows. Price equals marginal cost, which the next lesson shows is the condition for efficiency. And economic profit is zero, meaning owners earn exactly what their resources would earn elsewhere, which is a statement about opportunity cost rather than about hardship.

The long-run industry supply curve is horizontal at £15 if input prices do not change as the industry expands, which is the constant cost case. If expansion bids up the price of a specialised input, minimum average cost rises with industry output and long-run supply slopes upward: that is the increasing cost case, and it is what makes long-run supply of anything using land or a scarce skill less than perfectly elastic.

What the model gets right and what it assumes away

Entry and exit are not theoretical. In the United States, roughly a tenth of employer firms are born and a comparable share die in a typical year, with the churn concentrated in retail, food service and construction, which are the industries closest to the assumptions here. The mechanism the model describes is visibly at work even though no market meets its conditions exactly.

Two assumptions do most of the load-bearing and deserve to be named. Free entry means entry at the same costs as incumbents, with no patent, licence, sunk investment or established brand in the way. William Baumol, John Panzar and Robert Willig showed in 1982 that this matters more than the number of firms: a market with two firms but genuinely free entry and costless exit behaves competitively, because the threat of entry disciplines price. Contestability, not headcount, is what the zero-profit argument needs.

And the whole apparatus assumes the firm has one product, one price, and no ability to be preferred by any buyer. Drop the last of those and the firm faces a downward-sloping demand curve of its own, which is the situation of nearly every real business. Before that, though, the competitive outcome needs to be evaluated rather than merely described: the next lesson asks what is good about $p = MC$ , and gets a sharper answer than might be expected.