Source: Chapters 9 & 11 Practice Questions, Microeconomic Theory (Texas A&M University)
Tags: monopoly, profit maximisation, marginal revenue, marginal cost, MR = MC, inverse demand curve, monopoly pricing, monopoly output, market power, deadweight loss monopoly, single-price monopolist
A monopolist maximises profit by producing where marginal revenue equals marginal cost (MR = MC). Because a monopolist faces the entire market demand curve, its marginal revenue is always less than price. The key skill is deriving MR from the inverse demand function, setting MR = MC, solving for quantity, then plugging back into demand to find the profit-maximising price.
Monopoly
A market structure with a single seller. The firm is the market, so it faces the downward-sloping market demand curve directly.
Marginal revenue (MR)
The additional revenue from selling one more unit. For a monopolist with a linear inverse demand curve p = a - bQ, marginal revenue is MR = a - 2bQ. It has the same intercept as demand but twice the slope.
Marginal cost (MC)
The additional cost of producing one more unit. Profit maximisation requires the firm to produce up to the point where the revenue from the last unit equals its cost.
MR = MC rule
The profit-maximising condition for any firm with market power. Produce where MR = MC, then read the price off the demand curve at that quantity. This is the single most important rule for monopoly problems.
Inverse demand curve
The demand curve written as price as a function of quantity: p = f(Q). This form is used to derive marginal revenue directly.
If MR > MC at the current output, the firm gains more revenue from the next unit than it costs to produce. It should produce more.
If MR < MC at the current output, the last unit costs more to produce than it brings in. The firm should produce less.
Profit is maximised only when MR = MC.
Example: MC = 16 and MR = 10. Since MC > MR, the firm is overproducing. It should produce less.
Correct answer to Q6: C) should produce less.
Step-by-step method for linear demand:
Start with the inverse demand curve: p = a - bQ.
Derive total revenue: TR = p × Q = aQ - bQ².
Derive marginal revenue: MR = dTR/dQ = a - 2bQ.
Set MR = MC and solve for Q*.
Plug Q* back into the demand curve to find the profit-maximising price p*.
Example: p = 100 - 2Q, MC = 16.
MR = 100 - 4Q (same intercept, twice the slope).
Set MR = MC: 100 - 4Q = 16.
Solve: 4Q = 84, so Q* = 21.
Price: p* = 100 - 2(21) = 100 - 42 = 58.
The monopolist produces 21 units and charges $58.
Correct answer to Q7: B) is achieved when 21 units are produced.
Common mistake in the answer options: option C says "setting price equal to 21," which confuses the quantity with the price. The quantity is 21; the price is 58.
A perfectly competitive firm can sell any quantity at the market price, so MR = P.
A monopolist must lower the price on all units to sell one more, not just the marginal unit. This means MR < P for every unit after the first.
For linear demand p = a - bQ:
The demand curve has slope -b.
The MR curve has slope -2b (twice as steep).
MR hits zero at half the quantity where demand hits zero.
Even after finding MR = MC, verify the firm earns enough to stay open.
Short run: produce if price covers average variable cost (P ≥ AVC). Otherwise, shut down.
Long run: produce if price covers average total cost (P ≥ ATC). Otherwise, exit.
In the example above, with P = 58 and MC = 16 (constant, so AVC = ATC = 16), the firm is highly profitable. No shutdown concern.
Marginal revenue from linear inverse demand:
If p = a - bQ, then MR = a - 2bQ.
Profit-maximising quantity:
Set a - 2bQ = MC, solve for Q* = (a - MC) / 2b.
Profit-maximising price:
p* = a - bQ* = a - b × (a - MC) / 2b = (a + MC) / 2.
For the example (a = 100, b = 2, MC = 16):
Q* = (100 - 16) / (2 × 2) = 84 / 4 = 21
p* = (100 + 16) / 2 = 58
Monopoly profit (with constant MC):
π = (p* - MC) × Q* = (58 - 16) × 21 = 42 × 21 = $882.
⚠️ The MR = MC rule tells you the quantity. The price comes from plugging that quantity into the demand curve, not from setting P = MC. Monopolists do not price at marginal cost.
⚠️ For linear demand p = a - bQ, always double the slope coefficient to get MR. The intercept stays the same.
⚠️ If MC > MR, the firm should cut output. If MC < MR, the firm should expand. Direction of adjustment is a common multiple-choice trap.
⚠️ Do not confuse price and quantity in the final answer. In Q7, the answer is Q = 21, but the price is 58. Option C tries to trick you by putting 21 in as the price.
⚠️ The shortcut formula p* = (a + MC) / 2 only works when demand is linear and MC is constant. Know the general method (set MR = MC, solve, plug back in) for any other case.
Q: A firm has MC = 16 and MR = 10 at its current output. What should it do?
A: Produce less. MC exceeds MR, meaning the last unit costs more than it earns. Reducing output will raise profit.
Q: A monopolist faces p = 100 - 2Q and has MC = 16. What quantity maximises profit?
A: MR = 100 - 4Q. Set 100 - 4Q = 16, giving Q = 21.
Q: In the same problem, what price does the monopolist charge?
A: p = 100 - 2(21) = 58. The price is read off the demand curve at Q = 21.
Q: Why is marginal revenue less than price for a monopolist?
A: To sell an additional unit, the monopolist must lower the price on all units, not just the marginal one. The revenue gained from the extra unit is offset by the revenue lost on all previous units.
Q: For linear inverse demand p = a - bQ, what is the formula for marginal revenue?
A: MR = a - 2bQ. Same vertical intercept as demand, but twice the slope.
Q: What is the quick formula for the monopoly profit-maximising price with linear demand and constant MC?
A: p* = (a + MC) / 2, where p = a - bQ is the inverse demand. This only works for the linear, constant-MC case.
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