A perfectly competitive firm maximises profit by producing where price equals marginal cost (P = MC). Profit is total revenue minus total cost, and the breakeven price is the price at which profit is exactly zero, found where MC equals average total cost.
Difficulty: Intermediate
Prerequisites: basic understanding of cost functions (total cost, variable cost, fixed cost), derivatives for finding marginal cost, and the concept of price-taking behaviour in competitive markets.
This topic sits at the heart of the theory of the firm in a perfectly competitive market. You are learning how a single firm, which has no influence over market price, decides how much to produce. The firm takes the price as given and chooses output to make itself as well off as possible. If you understand cost curves and how to take a derivative, you have what you need. This material leads directly into supply curves, long-run equilibrium, and welfare analysis.
Variable cost (VC)
The portion of total cost that changes with the quantity produced. In this context, VC = 2q². Think of it as the costs you only pay when you actually make something: materials, energy, hourly labour.
Fixed cost (FC)
Costs that do not change regardless of output. Here, FC = $72. In simple terms, these are the bills you pay even if you produce nothing: rent, insurance, equipment leases.
Marginal cost (MC)
The additional cost of producing one more unit. Formally, MC = dTC/dq. For VC = 2q², MC = 4q. Think of it as the answer to "how much more does it cost me to make the next one?"
Total cost (TC)
The sum of variable and fixed costs: TC = VC + FC. Here, TC = 2q² + 72.
Total revenue (TR)
Price multiplied by quantity sold: TR = P × q. In a competitive market the firm sells every unit at the same market price.
Profit (π)
Total revenue minus total cost: π = TR – TC. This is economic profit, which includes the opportunity cost of the owner's resources.
Breakeven price
The lowest price at which the firm earns zero economic profit. Found where MC = ATC. In simple terms, it is the price at which the firm exactly covers all its costs, including fixed costs.
Average total cost (ATC)
Total cost divided by quantity: ATC = TC/q. The breakeven condition is the price at which MC crosses ATC from below.
A perfectly competitive firm is a price taker. It cannot influence the market price.
To maximise profit, the firm produces the quantity where price equals marginal cost.
Given P = 36 and MC = 4q, set 36 = 4q, which gives q* = 9.
Intuition: if P > MC for the next unit, the firm gains by producing it. If P < MC, the firm loses money on that unit. The optimum is where they are equal.
Total revenue: TR = P × q = 36 × 9 = 324.
Total cost: TC = VC + FC = 2(9²) + 72 = 162 + 72 = 234.
Profit: π = TR – TC = 324 – 234 = 90.
A positive profit means the firm earns more than enough to cover all costs, including the opportunity cost of capital.
The breakeven price is the price at which economic profit is zero. This occurs where MC = ATC.
ATC = TC/q = (2q² + 72)/q = 2q + 72/q.
Set MC = ATC: 4q = 2q + 72/q.
Rearrange: 2q = 72/q, so 2q² = 72, giving q = 6.
Plug q = 6 into MC: P = 4(6) = 24.
The breakeven price is $24. At any market price above $24 the firm earns positive profit; at $24 it earns exactly zero.
VC = 2q^2TC = 2q^2 + 72MC = \frac{dTC}{dq} = 4qATC = \frac{TC}{q} = 2q + \frac{72}{q}\pi = TR - TC = Pq - (2q^2 + 72)Profit-maximisation condition: P = MC.
Breakeven condition: MC = ATC, solved for q, then P = MC at that q.
A wheat farmer facing a market price they cannot influence uses exactly this logic: produce more bushels as long as the cost of the next bushel is below the going price. Manufacturing firms with rising marginal costs (due to overtime labour, machine wear, or scarce inputs) follow the same P = MC rule to decide their production runs.
Students often set MR = MC for a competitive firm and then try to derive MR separately. In perfect competition, MR = P, so P = MC is the correct shortcut. No need to differentiate total revenue.
A common error is forgetting to include fixed costs when calculating total cost. The profit-maximising quantity does not depend on FC (because MC does not include it), but profit itself absolutely does.
Some students confuse the breakeven price with the shutdown price. Breakeven is where P = ATC (zero profit). Shutdown is where P = AVC (the firm cannot even cover its variable costs).
When solving for breakeven, students sometimes set P = ATC and stop. You must also confirm that MC = ATC at that quantity, because the breakeven price is at the minimum of the ATC curve.
⚠️ The P = MC rule is tested constantly. Be ready to apply it with any cost function, not just quadratic ones.
⚠️ Expect a question asking you to distinguish breakeven price from shutdown price. Know both formulas cold.
⚠️ Profit calculation is a frequent free-marks question. Show all steps: TR, TC, then subtract. Partial credit comes from showing the components.
⚠️ The breakeven algebra (setting MC = ATC and solving) often appears as a standalone sub-part. Watch for the cross-multiply step when ATC has a fraction.
True or false: A perfectly competitive firm maximises profit where MR = MC, and MR equals the market price. (True.)
Fill in the blank: If VC = 2q² and FC = 72, then TC = ______. (2q² + 72.)
True or false: Fixed costs affect the profit-maximising quantity. (False. FC does not appear in MC.)
Fill in the blank: The breakeven price is found where MC = ______. (ATC.)
True or false: At the breakeven price the firm should shut down. (False. It earns zero economic profit, which means it is covering all costs including opportunity costs. It has no reason to shut down.)
Q: A perfectly competitive firm has VC = 2q² and FC = 72. If the market price is $36, what quantity maximises profit?
A: Set P = MC. MC = 4q, so 36 = 4q, giving q = 9.
Q: Using the same cost structure, what is the firm's profit at q = 9?
A: TR = 36 × 9 = 324. TC = 2(81) + 72 = 234. Profit = 324 – 234 = 90.
Q: What is the breakeven price for this firm?
A: Set MC = ATC. 4q = 2q + 72/q. Multiply through by q: 4q² = 2q² + 72, so 2q² = 72, q = 6. The breakeven price is MC at q = 6: P = 4(6) = 24.
Q: If the market price fell to $20, should the firm produce? Why or why not?
A: Find the shutdown price first. AVC = VC/q = 2q. At P = MC: 20 = 4q, so q = 5. AVC at q = 5 is 2(5) = 10. Since P = 20 > AVC = 10, the firm should produce in the short run (it covers variable costs and contributes to fixed costs), even though it makes a loss because P < breakeven price of 24.
Q: Explain why fixed costs do not affect the profit-maximising output level.
A: The profit-maximising condition is P = MC. Marginal cost is the derivative of total cost with respect to q. Since fixed costs are constant, their derivative is zero, so they drop out of the MC expression entirely.
This material connects directly to the firm's short-run supply curve, which is the MC curve above the shutdown price. Once you understand how a single firm chooses output, you can aggregate across firms to build the market supply curve.
The breakeven and shutdown analysis here also sets up long-run competitive equilibrium, where free entry and exit drive all firms to zero economic profit (P = minimum ATC).
Welfare analysis (consumer surplus and producer surplus in competitive markets) builds on the supply curve you derive from this profit-maximisation framework.
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