Difficulty: Intermediate | Prerequisites: Cost curves (ATC, AVC, MC), basic derivatives.
This topic builds on cost curves to answer the central question in producer theory: how much should a price-taking firm produce? It covers profit maximisation via the MC = P rule, revenue and profit calculations, and how to build a firm's short-run supply curve from its cost function. If you are not comfortable deriving MC from a total cost function, review that material first.
A perfectly competitive firm maximises profit by producing where marginal cost equals the market price (MC = P). Its short-run supply curve is the portion of its MC curve that lies at or above the minimum of AVC. Below that price, the firm shuts down and produces nothing.
Price taker
A firm that has no influence over the market price. It can sell as much or as little as it wants at the going price. In simple terms, the firm faces a perfectly horizontal demand curve.
Profit-maximisation rule (MC = P)
A competitive firm maximises profit by choosing the output where marginal cost equals the market price. Think of it as: keep producing as long as the next unit adds more to revenue than to cost.
Total revenue (TR)
Price multiplied by quantity: TR = P x q. For a price taker, this is a straight line through the origin.
Economic profit
Total revenue minus total cost: π = TR - TC. This includes all opportunity costs, not just accounting costs. Zero economic profit means the firm earns a normal return.
Short-run supply curve
The portion of the MC curve at or above the minimum of AVC. Below min AVC, the firm supplies zero. Think of it as a piecewise function: q = 0 if P < min AVC, and q solves MC(q) = P if P ≥ min AVC.
Shutdown price
The minimum of AVC. At any market price below this, the firm is better off producing nothing and simply absorbing its fixed costs as a loss.
Set MC = P and solve for q. For a quadratic cost function like C(q) = 3 + 2q + 2q², MC = 2 + 4q. Setting 2 + 4q = 10 gives q = 2.
Always check the second-order condition: MC should be increasing at the solution (the MC curve should cross the price line from below). With MC = 2 + 4q, the slope is 4 > 0, so the condition holds.
For cubic cost functions like TC(q) = 5 + 10q - 5q² + 3q³, MC = 10 - 10q + 9q². Setting MC = P gives a quadratic in q. Solve using the quadratic formula and take the root where MC is increasing.
TR = P x q. At the profit-maximising output, simply multiply price by the optimal quantity.
Profit = TR - TC. Substitute the optimal q back into both TR and TC.
Example: With C(q) = 3 + 2q + 2q² and P = 10, optimal q = 2. TR = 10 x 2 = 20. TC = 3 + 4 + 8 = 15. Profit = 20 - 15 = 5.
Find min AVC by taking the derivative of AVC with respect to q, setting it to zero, and solving.
For C(q) = 3 + 2q + 2q²: VC = 2q + 2q², so AVC = 2 + 2q. This is linear and increasing, so min AVC = 2 (at q = 0, or more precisely, as q approaches zero). The firm produces at any P ≥ 2.
For TC(q) = 5 + 10q - 5q² + 3q³: VC = 10q - 5q² + 3q³, so AVC = 10 - 5q + 3q². Take the derivative: dAVC/dq = -5 + 6q = 0, giving q = 5/6. Plug back in: min AVC = 10 - 5(5/6) + 3(25/36) = 10 - 25/6 + 75/36 = 10 - 25/6 + 25/12 = 10 - 25/12 = 95/12 ≈ 7.92.
The supply curve is piecewise: q = 0 when P < min AVC, and q solves MC(q) = P when P ≥ min AVC.
For the cubic case, MC = 10 - 10q + 9q². Setting MC = P and solving for q (using the quadratic formula) gives the supply function. Only the root on the upward-sloping portion of MC is valid.
At P = 100 with MC = 10 - 10q + 9q²: solve 9q² - 10q + 10 = 100, which gives 9q² - 10q - 90 = 0. Applying the quadratic formula: q = (10 ± √(100 + 3240)) / 18 = (10 ± √3340) / 18. Taking the positive root: q ≈ (10 + 57.79) / 18 ≈ 3.77.
\text{Profit-max condition: } MC(q) = PTR = P \times q\pi = TR - TC = P \cdot q - TC(q)\text{Supply curve: } q = \begin{cases} 0 & \text{if } P < \min AVC \\ MC^{-1}(P) & \text{if } P \geq \min AVC \end{cases}\text{Shutdown price} = \min_{q} AVC(q)To find min AVC: set dAVC/dq = 0, solve for q, then evaluate AVC at that q.
Students often set MC = ATC to find the profit-maximising output. That gives the break-even point, not the profit-max quantity. The rule is MC = P.
When the MC curve is U-shaped (from a cubic TC), there may be two quantities where MC = P. Students sometimes pick the wrong one. Always choose the quantity on the upward-sloping portion of MC (where the second-order condition holds).
Students regularly forget the shutdown condition and write the supply curve as MC = P for all prices. The supply curve only applies for P ≥ min AVC. Below that, supply is zero.
A common algebra error: forgetting to exclude fixed costs when computing AVC. AVC = VC/q, not TC/q. The fixed-cost term (the constant in the cost function) belongs only in ATC.
⚠️ Deriving the profit-maximising output from a given cost function and price is the single most likely calculation on this topic. Be able to do it quickly.
⚠️ Expect to be asked for the shutdown price and to write the supply curve as a piecewise function.
⚠️ Problems with cubic cost functions (like TC = 5 + 10q - 5q² + 3q³) test whether you can handle the quadratic formula and pick the correct root.
⚠️ Be prepared to compute profit at a specific price and explain whether the firm should operate or shut down.
True or false: A perfectly competitive firm maximises profit where MC = ATC. (False: it maximises profit where MC = P.)
Fill in the blank: The short-run supply curve is the portion of the MC curve above ________. (Minimum AVC)
True or false: If a firm's profit is negative, it should always shut down in the short run. (False: it should shut down only if P < min AVC.)
Fill in the blank: Total revenue for a price taker equals ________. (P x q)
True or false: When solving MC = P with a cubic cost function, both roots of the quadratic are valid supply points. (False: only the root on the upward-sloping portion of MC is valid.)
Q: A firm has C(q) = 3 + 2q + 2q² and faces a market price of $10. What is the profit-maximising output?
A: MC = 2 + 4q. Set 2 + 4q = 10, so 4q = 8, giving q = 2.
Q: What is this firm's total revenue and profit at the profit-maximising output?
A: TR = 10 x 2 = 20. TC = 3 + 2(2) + 2(4) = 3 + 4 + 8 = 15. Profit = 20 - 15 = 5.
Q: What is the minimum price at which this firm will produce a positive level of output in the short run?
A: VC = 2q + 2q², so AVC = 2 + 2q. This is increasing in q, so its minimum is at q approaching 0, giving min AVC = 2. The firm produces as long as P ≥ 2.
Q: Given TC(q) = 5 + 10q - 5q² + 3q³, describe the short-run supply curve and identify the shutdown point.
A: MC = 10 - 10q + 9q². AVC = 10 - 5q + 3q². Min AVC occurs where dAVC/dq = -5 + 6q = 0, so q = 5/6. Min AVC = 10 - 25/6 + 25/12 = 95/12 ≈ 7.92. The supply curve is: q = 0 if P < 95/12; otherwise q solves 9q² - 10q + (10 - P) = 0, taking the larger root.
Q: How much does this firm produce at P = 100?
A: Solve 9q² - 10q + 10 = 100, i.e. 9q² - 10q - 90 = 0. By the quadratic formula, q = (10 + √(100 + 3240)) / 18 = (10 + √3340) / 18 ≈ (10 + 57.79) / 18 ≈ 3.77 units.
This connects to cost curves: the MC, AVC, and ATC curves from the previous topic are exactly what you need to build the supply curve and identify shutdown and break-even prices.
It also feeds directly into market-level supply. The market supply curve is the horizontal sum of all individual firms' supply curves. Understanding how one firm's supply is derived is the prerequisite for policy analysis involving taxes and quantity restrictions.
In the long run, free entry and exit drive economic profit to zero, which means the market price converges to the minimum of long-run ATC. The short-run profit calculations here show why firms enter (positive profit) or exit (persistent losses).
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