The supply curve of a perfectly competitive firm and its shut-down condition

The setting: a price-taking firm. Under perfect competition a single firm is too small to influence the market price, so it takes the price p as given and can sell as much as it wishes at that price. Its demand (price) line is horizontal, and since every extra unit is sold at p, average revenue and marginal revenue both equal the price: MR = AR = p. A firm's supply curve shows how much output it chooses to produce at each market price, with technology and input prices held constant. Deriving it therefore means asking: at each possible price, which output gives this firm the greatest profit?

Conditions for maximum profit. Profit is total revenue minus total cost. A positive output q0 maximises profit only if three conditions hold there; we take them in the order a firm would ask them, whether to produce and then how much:

1. Price covers average variable cost in the short run (and average cost in the long run). This settles whether to produce at all: otherwise the firm does better by producing nothing.
2. Price equals marginal cost. While MR exceeds MC, one more unit adds more to revenue than to cost and profit rises; while MR is below MC, profit falls. Profit is therefore greatest where MR = MC, which for a price taker means p = MC.
3. Marginal cost is not falling at that output. If MC were falling where it equals p, the units just before that output would have cost more than p, so a slightly smaller output would earn more; the point cannot be a maximum. Only the rising part of the MC curve qualifies.

Deriving the short-run supply curve. Take the firm's short-run marginal cost (SMC) and average variable cost (AVC) curves, and let the price vary.

• Case 1: price at or above minimum AVC. Set the price equal to SMC on its rising part; the output found there also has AVC no higher than the price, so all three conditions hold and that output is what the firm supplies. As the price rises, the point of equality moves up the rising SMC curve, so the quantity supplied rises.
• Case 2: price below minimum AVC. Every positive output now has AVC above the price, which breaks the first condition, so the firm produces nothing.

Putting the two cases together: the firm's short-run supply curve is the rising part of its SMC curve from the minimum of AVC upwards, together with zero output at every price below minimum AVC. The supply curve is upward sloping because marginal cost rises along it.

A worked schedule (our own figures). A small brick kiln in a competitive market has a fixed cost of ₹50 thousand a week (kiln rent, licence) and the following variable costs, output in lakh bricks and costs in ₹ thousand:

MC falls to 16 and then rises; it meets AVC at its minimum of 22 (output 4). The firm produces each unit on the rising part of MC whose marginal cost does not exceed the price, provided the price covers AVC. Its supply schedule follows:

At ₹20 every positive output loses more than the fixed cost (the best, 3 units, loses 56), so the kiln shuts down. At ₹22 producing 4 units just pays the variable cost, so the loss equals the fixed cost either way. At ₹26 the kiln still makes a loss but, by covering its variable cost of 88 with revenue of 104, it cuts its loss from 50 to 34. Above the minimum average total cost (₹34 at output 5) it earns a profit. As the price rises from 26 to 36 to 46, output climbs one step at a time up the rising stretch of the MC column (22, 32, 42), as the derivation predicts.

The shut-down condition.

• Short run. Fixed cost has to be paid whether or not the firm produces, so shutting down leaves a loss equal to total fixed cost. Producing at a price below AVC loses that fixed cost plus part of the variable cost, which is worse than shutting down. The firm therefore keeps producing, even at a loss, as long as the price is at least equal to minimum AVC, and shuts down when the price falls below it.
• Shut-down point. Moving down the supply curve, the last price at which the firm still produces is minimum AVC, the point where the SMC curve cuts the AVC curve; below it there is no production. This is the short-run shut-down point. In the schedule it is the price of ₹22.
• Break-even point. The point of minimum average (total) cost, where the supply curve cuts the average cost curve, is the break-even point, at which the firm earns only normal profit. Between the shut-down and break-even prices the firm produces at a loss in the short run.

The long run. In the long run a firm that shuts down earns zero profit, so it produces only if the price covers long-run average cost. Repeating the derivation with the long-run cost curves gives the long-run supply curve: the rising part of the long-run marginal cost curve from minimum long-run average cost upwards, with zero output at lower prices. The long-run shut-down point is therefore the minimum of the long-run average cost curve.

Conclusion. A competitive firm supplies the output at which price equals a rising marginal cost, so its short-run supply curve is the part of its SMC curve that lies above minimum AVC. Its shut-down condition is that price must at least cover minimum average variable cost in the short run, and minimum average cost in the long run; below that, producing nothing minimises the loss.