Difficulty: Intermediate to Advanced | Prerequisites: Demand curves, basic calculus (derivatives), surplus concepts
A firm with market power maximises profit by setting the price where marginal revenue equals marginal cost (MR = MC). Elasticity tells you whether the current price is too high or too low by revealing whether an extra unit sold adds or loses revenue. When a firm can charge different prices in different markets (price discrimination), it earns more than a single-price strategy, but the welfare effects for each group of consumers differ.
Price elasticity of demand (E or Ed)
The percentage change in quantity demanded divided by the percentage change in price. It measures how sensitive buyers are to price changes. A negative number (demand slopes down). |E| > 1 is elastic, |E| < 1 is inelastic, |E| = 1 is unit elastic.
Marginal revenue (MR)
The additional revenue earned from selling one more unit. For a price-setting firm, MR < P because lowering the price to sell one more unit also lowers revenue on all previous units.
Marginal cost (MC)
The additional cost of producing one more unit. For the problems in this set, MC is constant (the derivative of a linear total cost function).
Profit maximisation rule: MR = MC
A firm with market power maximises profit by producing the quantity where marginal revenue equals marginal cost, then charging the price consumers will pay for that quantity (read off the demand curve).
Inverse demand function
The demand function rearranged to express price as a function of quantity: P(Q). This form is used to derive the revenue function R(Q) = P(Q) x Q.
Price discrimination (third-degree)
Charging different prices to different groups of consumers (separated by geography, age, or other observable characteristics) for the same product. Each group faces its own demand curve, and the firm sets MR = MC in each market separately.
Market power
The ability of a firm to set its own price rather than taking the market price as given. Typically arises from patents, branding, or limited competition. MelCo has market power through its patent on Xamoff.
MelCo sells Xamoff, an anti-anxiety medicine protected by a patent. Two data points:
At P = $10: Q = 25 million
At P = $11: Q = 22 million
Production cost: $2 per package. MelCo is considering raising the price to $12.
Use the arc elasticity formula:
E = (% change in Q) / (% change in P)
At P = $10:
% change Q = (22m - 25m) / 25m = -12%
% change P = (11 - 10) / 10 = 10%
E at $10 = -12% / 10% = -1.2
At P = $11:
% change Q = (25m - 22m) / 22m = 13.6%
% change P = (10 - 11) / 11 = -9.1%
E at $11 = 13.6% / -9.1% = -1.5
Note that the elasticity estimates differ depending on which price is the starting point. This is a feature of arc elasticity, not an error.
For a firm with market power, marginal revenue relates to price and elasticity:
MR = P x (1 + 1/E)
At P = $10, E = -1.2:
MR = 10 x (1 + 1/(-1.2)) = 10 x (1 - 0.833) = $1.67
Since MR ($1.67) < MC ($2.00), the last unit sold actually loses money at the margin. MelCo should raise the price.
At P = $11, E = -1.5:
MR = 11 x (1 + 1/(-1.5)) = 11 x (1 - 0.667) = 11 x 0.333 = $3.67
Since MR ($3.67) > MC ($2.00), MelCo could still sell additional units profitably. MelCo should lower the price (from $11).
At $10, MR < MC: price too low. At $11, MR > MC: price too high. The profit-maximising price lies between $10 and $11.
Using elasticity at $11 (E = -1.5):
% change P = (12 - 11) / 11 = 9.1%
% change Q = -1.5 x 9.1% = -13.65%
New Q = 22m x (1 - 0.1365) = approximately 18,997,000
New revenue = 18,997,000 x $12 = approximately $227,964,000
Old revenue at $11 = $242,000,000
Revenue falls by about $14 million
Because |E| > 1 at $11 (demand is elastic), a price increase reduces total revenue.
Sal's Streaming serves the US and Canada with demand:
US: Qus = 50 - (1/3)Pus
Canada: Qca = 80 - (2/3)Pca
Q in thousands of subscriptions per year. Total cost: TC = 1000 + 30Q (so MC = 30, fixed cost = 1000).
For each market, derive the inverse demand, then the revenue function, then set MR = MC.
US market:
Inverse demand: Pus = 150 - 3Qus
Revenue: R = 150Qus - 3(Qus)^2
MR = 150 - 6Qus
MR = MC: 30 = 150 - 6Qus, so Qus = 20 (thousand)
Pus = 150 - 3(20) = $90
Canadian market:
Inverse demand: Pca = 120 - 1.5Qca
Revenue: R = 120Qca - 1.5(Qca)^2
MR = 120 - 3Qca
MR = MC: 30 = 120 - 3Qca, so Qca = 30 (thousand)
Pca = 120 - 1.5(30) = $75
When consumers can access either country's stream via VPN, Sal must charge one price. Add the demand curves:
Q = Qus + Qca = [50 - (1/3)P] + [80 - (2/3)P] = 130 - P
Inverse demand: P = 130 - Q
Revenue: R = 130Q - Q^2
MR = 130 - 2Q
MR = MC: 30 = 130 - 2Q, so Q = 50 (thousand)
P = 130 - 50 = $80
Two-market (price discrimination):
Revenue: (20 x $90) + (30 x $75) = $1,800 + $2,250 = $4,050 (thousands)
Cost: 1000 + 30(50) = $2,500 (thousands)
Profit: $4,050 - $2,500 = $1,550 (thousands)
Single market:
Revenue: 50 x $80 = $4,000 (thousands)
Cost: 1000 + 30(50) = $2,500 (thousands)
Profit: $4,000 - $2,500 = $1,500 (thousands)
Sal earns $50,000 more with price discrimination. Total quantity is the same (50,000 subscriptions), so the cost is identical. The difference is pure revenue extraction from charging each market closer to its willingness to pay.
Canadian consumers prefer the two-market scenario. Under single pricing they pay $80 instead of $75: a price increase of $5. Some are priced out entirely, and those who stay pay more. Their consumer surplus falls.
US consumers prefer single pricing. Their price drops from $90 to $80, and more consumers enter the market. Their consumer surplus rises.
The intuition: price discrimination extracts more from the less elastic market (US, higher price) and gives a break to the more elastic market (Canada, lower price). A uniform price averages these out.
Arc elasticity
E = (% change in Q) / (% change in P) = [(Q2 - Q1) / Q1] / [(P2 - P1) / P1]
Marginal revenue for a firm with market power
MR = P x (1 + 1/E)
This lets you test whether a given price is optimal without knowing the full demand curve.
From demand to MR (calculus approach)
Rearrange demand Q(P) into inverse demand P(Q)
Revenue: R(Q) = P(Q) x Q
MR = dR/dQ
For a linear inverse demand P = a - bQ: MR = a - 2bQ (same intercept, double the slope)
Profit
Profit = Total revenue - Total cost = P x Q - TC(Q)
Decision rule
If MR > MC: the firm can increase profit by lowering the price and selling more.
If MR < MC: the firm can increase profit by raising the price and selling less.
If MR = MC: the firm is at the profit-maximising output.
Airlines, cinemas, and software companies all practise third-degree price discrimination: student discounts, regional pricing, early-bird fares. The logic is identical to Sal's problem: separate markets with different demand elasticities, charge more where demand is less elastic.
The MR = MC rule applies to any firm with pricing power, from a local monopoly to a pharmaceutical company deciding what to charge for a patented drug.
Students often confuse "elastic" with "steep." A steep demand curve is inelastic (quantity barely changes with price). A flat demand curve is elastic. The visual shortcut: flat = elastic, steep = inelastic.
A common error in the MelCo problem is using the same base for elasticity in both directions. Elasticity at $10 uses $10 and 25m as the base; elasticity at $11 uses $11 and 22m. This is why you get different numbers.
Students sometimes think MR = MC means "set the price equal to MC." That is the competitive-market result. For a firm with market power, P > MC at the optimum; MR = MC determines the quantity, and the price is read off the demand curve at that quantity.
When adding demand curves for the single-market price, some students add the inverse demand functions (prices) instead of the quantity functions. You must add quantities at each price, not prices at each quantity.
The MR = P(1 + 1/E) formula is a favourite exam shortcut. Memorise it. It lets you check whether a price is optimal using only the price and the elasticity, without computing the full MR function.
"Is the price too high or too low?" questions always compare MR to MC. MR > MC means too high (lower the price to sell more). MR < MC means too low (raise the price).
Price discrimination problems typically ask you to compare profits and consumer surplus across scenarios. Calculate both carefully: revenue minus cost for profit, and the triangle areas for surplus.
When demand is elastic (|E| > 1), a price increase reduces revenue. When inelastic (|E| < 1), a price increase raises revenue. A profit-maximising monopolist always operates on the elastic portion of the demand curve (because MR > 0 only when |E| > 1).
True or false: If MR > MC, the firm should raise its price. ___
Fill in the blank: For a linear inverse demand P = a - bQ, the MR function is MR = a - ___Q.
True or false: A profit-maximising monopolist always operates where demand is inelastic. ___
Fill in the blank: To add two demand curves for a combined market, add the ___ at each price.
True or false: Price discrimination always makes all consumers worse off. ___
Answers: 1. False (MR > MC means it should lower the price and sell more). 2. 2b. 3. False (always on the elastic portion). 4. quantities. 5. False (the market getting the lower discriminatory price is better off).
Q: A monopolist sells 1,000 units at $50 and 900 units at $55. Estimate the elasticity of demand at P = $50.
A: % change Q = (900 - 1000) / 1000 = -10%. % change P = (55 - 50) / 50 = 10%. E = -10% / 10% = -1.0.
Q: Using the elasticity from above, what is MR at P = $50?
A: MR = 50 x (1 + 1/(-1.0)) = 50 x 0 = $0. At unit elasticity, marginal revenue is zero.
Q: If MC = $5, should the monopolist raise or lower the price from $50?
A: MR ($0) < MC ($5). The last unit sold costs more to make than the revenue it generates at the margin. The firm should raise the price.
Q: A firm faces demand Qus = 100 - 2Pus in the US and Qeu = 60 - Peu in Europe. MC = $10. What is the profit-maximising price in each market?
A: US inverse demand: P = 50 - 0.5Q. MR = 50 - Q. Set MR = 10: Q = 40, P = $30. Europe inverse demand: P = 60 - Q. MR = 60 - 2Q. Set MR = 10: Q = 25, P = $35.
Q: In the question above, if the firm must charge a single price (combined market), what is it?
A: Total demand: Q = 100 - 2P + 60 - P = 160 - 3P. Inverse: P = (160 - Q)/3. R = (160Q - Q^2)/3. MR = (160 - 2Q)/3. Set MR = 10: 160 - 2Q = 30, Q = 65. P = (160 - 65)/3 = $31.67.
Q: Why does a price-discriminating firm earn more than a single-price firm when total quantity is the same?
A: Price discrimination lets the firm charge higher prices in the less elastic market and lower prices in the more elastic market, extracting more revenue from each group's willingness to pay. A single price averages out these differences and leaves revenue on the table.
The elasticity and MR = MC framework connects to welfare analysis (Problem 2): a monopolist restricting output below the competitive level creates deadweight loss, and you can calculate it using the same surplus triangles. It also connects to cost analysis (Problem 1) because the MC in these pricing problems is the same marginal cost concept used to evaluate forward integration.
The price discrimination material connects to market segmentation and antitrust policy: regulators care about price discrimination because it can either increase or decrease total welfare depending on whether it brings new consumers into the market.
price elasticity of demand, arc elasticity, marginal revenue, marginal cost, MR = MC, profit maximisation, inverse demand function, revenue function, price discrimination, third-degree price discrimination, monopoly pricing, market power, patent, single-price monopoly, consumer surplus under monopoly, ECON 101, microeconomics, NYU, firms and markets