Difficulty: Intermediate | Prerequisites: Cost functions, derivatives, basic supply and demand.
A monopolist maximises profit where MR = MC, then reads the price off the demand curve. A perfectly competitive firm is a price taker, so MR = P and the supply curve is the MC curve above the shutdown point. Revenue maximisation (MR = 0) always produces a higher quantity and lower price than profit maximisation (MR = MC with MC > 0).
Market structure determines how firms set prices and quantities. This section covers the two polar cases: monopoly (one seller, price-setter) and perfect competition (many sellers, price-takers). Together they account for a large share of the exam. The monopoly material revolves around manipulating the inverse demand function, while the perfect competition material revolves around cost curves. Both require you to be comfortable taking derivatives and solving simple equations.
Inverse demand function
Demand rewritten with price as a function of quantity: from Q = 300 - 2p, solve for p = 150 - 0.5Q. This form is needed to derive total and marginal revenue.
Marginal Revenue (MR)
The additional revenue from selling one more unit. For a linear inverse demand p = a - bQ, MR = a - 2bQ. It has the same vertical intercept as the demand curve but twice the slope.
Profit maximisation
The firm produces where MR = MC. For a monopolist, MR < P, so the profit-maximising price is above marginal cost.
Revenue maximisation
The firm produces where MR = 0. This ignores costs entirely and always yields a higher quantity (and lower price) than profit maximisation when MC > 0.
Shutdown price
The minimum price at which a competitive firm will produce in the short run. It equals the minimum of Average Variable Cost (AVC). Below this price, the firm loses less by producing nothing.
Average Variable Cost (AVC)
Total variable cost divided by quantity. For C(q) = 2 + 0.5q^2, fixed cost is 2, variable cost is 0.5q^2, and AVC = 0.5q.
Average Cost (AC)
Total cost divided by quantity: AC = C(q)/q. The minimum of AC (where MC = AC) is the break-even price.
Supply function (competitive firm)
The portion of the MC curve above the shutdown price. The firm sets P = MC and solves for q.
Market supply
The horizontal sum of all individual firm supply curves. With n identical firms, Q_S = n · q(P).
Given C(Q) = 250 + 10Q and Q(p) = 300 - 2p:
Step 1: Find the inverse demand. From Q = 300 - 2p, solve for p = 150 - 0.5Q.
Step 2: Derive Total Revenue. TR = p · Q = (150 - 0.5Q)Q = 150Q - 0.5Q^2.
Step 3: Derive Marginal Revenue. MR = dTR/dQ = 150 - Q.
Step 4: Find MC. MC = dC/dQ = 10.
Step 5: Set MR = MC. 150 - Q = 10, so Q = 140.
Step 6: Find the price. p = 150 - 0.5(140) = 80.
The profit-maximising output is Q = 140 at a price of p = 80.
Revenue maximisation sets MR = 0 (costs are irrelevant).
150 - Q = 0, so Q = 150. Price: p = 150 - 0.5(150) = 75.
Compare: profit max gives (Q = 140, p = 80), revenue max gives (Q = 150, p = 75). Revenue maximisation always produces a higher quantity because it ignores the positive marginal cost. A profit maximiser restricts output to the point where MR equals the positive MC, which requires a higher MR and therefore a lower Q.
For any linear inverse demand p = a - bQ, MR = a - 2bQ. The MR curve shares the same vertical intercept (a = 150) but is twice as steep (slope -2b vs. -b). This means MR hits zero at half the quantity where demand hits zero.
A monopolist never produces on the inelastic portion of the demand curve. On that portion, MR is negative, and MC is always positive, so MR = MC cannot hold there.
Given C(q) = 2 + 0.5q^2 with no sunk costs:
Step 1: Find MC. MC = dC/dq = q.
Step 2: Find AC. AC = C(q)/q = 2/q + 0.5q.
Step 3: Find the minimum of AC by setting MC = AC. q = 2/q + 0.5q, so 0.5q = 2/q, giving q^2 = 4 and q = 2.
Step 4: The shutdown price is MC at q = 2, which equals P = 2.
Below P = 2, the firm shuts down. At or above P = 2, the firm produces where P = MC.
The individual supply function comes from P = MC = q, so q(P) = P for P >= 2.
With 200 identical firms, the market supply is Q_S(P) = 200 · P = 200P.
A competitive firm is a price taker, so MR = P. Profit maximisation (MR = MC) therefore becomes P = MC. This holds in both the short run and the long run.
Concept | Formula |
|---|---|
Inverse demand (linear) | p = a - bQ |
Total Revenue | TR = aQ - bQ^2 |
Marginal Revenue (linear) | MR = a - 2bQ |
Profit max condition | MR = MC |
Revenue max condition | MR = 0, so Q = a / 2b |
MC (from C = F + cq^2) | MC = 2cq |
AVC (from C = F + cq^2) | AVC = cq |
AC (from C = F + cq^2) | AC = F/q + cq |
Shutdown price | min(AVC), or min(AC) when no sunk costs |
Individual supply | q(P) = P / (2c) for P >= shutdown price |
Market supply (n firms) | Q_S = n · q(P) |
Monopoly pricing explains why pharmaceutical companies with patent protection charge prices far above marginal cost. The gap between P and MC is the source of monopoly profit, and it is precisely what generic competition eliminates.
The shutdown decision explains why restaurants sometimes stay open during slow hours despite not covering full costs: they are covering variable costs and contributing to fixed costs, which they would still owe if closed.
Students often set P = MC for a monopolist. That is the competitive condition. A monopolist sets MR = MC and then reads the price off the demand curve, which is above MC.
A common confusion is between revenue maximisation (MR = 0) and profit maximisation (MR = MC). Revenue maximisation ignores costs, so it always gives a higher Q.
Students sometimes think the shutdown price is where MC = 0 or where profit = 0. The shutdown price is the minimum of AVC (short run) or AC (when there are no sunk costs).
A frequent error is thinking a monopolist might produce on the inelastic portion of the demand curve. MR is negative there, so it can never equal a positive MC.
⚠️ Converting demand to inverse demand is the first step in every monopoly question. Practise this conversion until it is automatic.
⚠️ The relationship between MR and the demand curve (same intercept, twice the slope) is tested directly in True/False questions.
⚠️ Profit max vs. revenue max comparison is a 10-mark long-answer question. Know both conditions, compute both outcomes, and explain the intuition.
⚠️ Finding the shutdown price requires MC = AC (no sunk costs) or MC = AVC (sunk costs present). Read the question carefully.
⚠️ Market supply = n × individual supply is a straightforward but commonly missed step in multiple-choice.
Fill in the blank: A monopolist maximises profit where ____ = ____. (MR, MC)
True or False: The MR curve for a linear demand has the same intercept as the demand curve. (True)
True or False: A monopolist can produce on the inelastic portion of the demand curve. (False)
Fill in the blank: The shutdown price equals the minimum of ____. (AVC)
True or False: In perfect competition, P = MC because firms are price takers. (True)
Q: For C(Q) = 250 + 10Q and Q(p) = 300 - 2p, find the profit-maximising Q and p.
A: Inverse demand: p = 150 - 0.5Q. MR = 150 - Q. MC = 10. Set MR = MC: Q = 140, p = 80.
Q: What is the Total Revenue function in terms of Q?
A: TR = (150 - 0.5Q)Q = 150Q - 0.5Q^2.
Q: Find the revenue-maximising Q and p.
A: MR = 0: 150 - Q = 0, Q = 150. p = 150 - 0.5(150) = 75.
Q: For C(q) = 2 + 0.5q^2 with no sunk costs, what is the minimum price for positive supply?
A: MC = q, AC = 2/q + 0.5q. Set MC = AC: q = 2/q + 0.5q, so q = 2 and P = MC(2) = 2.
Q: With 200 identical photographers each with supply q = P, what is market supply?
A: Q_S(P) = 200P.
Monopoly pricing connects to welfare analysis and deadweight loss, which may appear in later chapters. The gap between the monopoly price and MC represents the inefficiency.
Perfect competition supply connects to cost minimisation (the previous topic). The cost function that determines MC comes from the firm's optimal input choices.
The comparison between profit maximisation and revenue maximisation illustrates why a firm's objective matters. A manager who maximises revenue (perhaps to grow market share) produces more and charges less than one who maximises profit.
monopoly, monopolist, marginal revenue, inverse demand, profit maximisation, revenue maximisation, MR = MC, MR = 0, perfect competition, price taker, supply function, shutdown price, average variable cost, AVC, average cost, market supply, deadweight loss, ECN 312, intermediate microeconomics, ASU, market structure, linear demand