Source: Microeconomic Theory, Texas A&M University
Tags: monopoly, market power, marginal revenue, inverse demand, profit maximisation, deadweight loss, price discrimination, first-degree, second-degree, third-degree, block pricing, quantity discount
A monopolist is the sole seller in a market and faces the downward-sloping market demand curve, meaning its marginal revenue is below price. It maximises profit by producing where MR = MC, then pricing from the demand curve. This results in higher prices, lower output, and a deadweight loss compared to competition. Price discrimination allows the monopolist to capture more surplus by charging different prices to different consumers or for different quantities.
Monopoly
A market with only one seller but many buyers.
Inverse demand function, p(q)
Expresses price as a function of quantity. Derived by solving the demand function Q(p) for p.
Marginal revenue (MR) under monopoly
MR(q) = R'(q). Because the monopolist must lower its price to sell more, MR lies below the demand curve. For a linear inverse demand p(q) = a − bq, revenue is R(q) = aq − bq² and MR(q) = a − 2bq, which has the same intercept as the demand curve but twice the slope.
Price discrimination
Charging different prices to different consumers or for different units of the same good.
First-degree price discrimination
Charging each consumer their exact willingness to pay. CS = 0, DWL = 0, and the monopolist captures all surplus.
Second-degree price discrimination
Charging different per-unit prices for different quantities of the same good, e.g. block pricing, quantity discounts, buy-one-get-one-50%-off offers.
Third-degree price discrimination
Dividing consumers into distinct groups with separate demand curves and setting a different price for each group, e.g. student/senior discounts, weekday vs. weekend hotel rates, economy vs. business class.
In a competitive market, the firm's demand curve is horizontal at the market price, so MR = p.
A monopolist faces the entire market demand curve, which slopes downward. To sell an additional unit, the monopolist must cut the price on all units (assuming a single-price scheme). The revenue gained from the extra unit is therefore less than the price charged, so MR < p.
Step 1: Write the inverse demand function p(q) by solving Q(p) for p
Step 2: Compute revenue R(q) = p(q) × q
Step 3: Differentiate to get MR(q) = R'(q)
Worked Example 1: Q(p) = 100 − 2p.
p(q) = 50 − 0.5q
R(q) = 50q − 0.5q²
MR(q) = 50 − q
Note: MR has the same vertical intercept (50) as p(q), but the slope is −1 rather than −0.5, i.e. twice as steep.
Worked Example 2: Q(p) = 20 − 0.5p.
p(q) = 40 − 2q
R(q) = 40q − 2q²
MR(q) = 40 − 4q
Short run:
Produce q* > 0 if MR(q*) = MC(q*) and p(q*) ≥ AVC(q*). Otherwise shut down.
Long run:
In the long run, FC = 0, so AVC(q) = AC(q). Produce q* > 0 if MR(q*) = MC(q*) and p(q*) ≥ AC(q*). Otherwise shut down.
For simplicity, most problems from Chapter 10 onward are long-run problems where only AC is provided.
Step 1: Find q* where MR(q*) = MC(q*)
Step 2: Read p* from the demand curve at q*
Step 3: Confirm p* ≥ AC(q*)
Step 4: Profit = (p* − AC(q*)) × q*, shown as a rectangle on the graph
Example 1: Q(p) = 400 − 2p, C(q) = 100 + q² (short-run problem, FC = 100).
p(q) = 200 − 0.5q, so MR(q) = 200 − q
MC(q) = 2q, AVC(q) = q
MR = MC: 200 − q = 2q, giving q* = 200/3
p(q*) = 200 − 100/3 = 500/3
Check: p = 500/3 ≥ AVC = 200/3 ✓
Profit = (500/3)(200/3) − [100 + (200/3)²] ≈ 6,566.7
Example 2: Q(p) = 200 − 2p, AC(q) = MC(q) = 10 (constant cost, long run).
p(q) = 100 − 0.5q, so MR(q) = 100 − q
MR = MC: 100 − q = 10, giving q* = 90
p(q*) = 100 − 45 = 55
Check: p = 55 ≥ AC = 10 ✓
Profit = (55 − 10) × 90 = 4,050
Example 3 – Graphical: In a graph with MC, AC, AVC, Demand, and MR curves, the profit-maximising price is found by going from the MR = MC intersection up to the demand curve.
Example 4: Q(p) = 10 − 2p, MC = AC = 2.50.
p(q) = 5 − 0.5q, MR(q) = 5 − q
MR = MC: 5 − q = 2.5, giving q* = 2.5
p(q*) = 5 − 1.25 = 3.75
Profit = (3.75 − 2.50) × 2.5 = 3.125
Compared to competitive equilibrium (where p = MC):
The monopolist charges a higher price (Pm > Pc)
Produces less output (Qm < Qc)
Consumer surplus falls (decreases by A + B)
Producer surplus changes (increases by A, decreases by C, net change is A − C)
A deadweight loss of B + C arises
The monopolist could have produced the competitive quantity but chooses not to, because restricting output and raising price yields higher profit.
Deadweight loss example: With q* = 90, p* = 55, and MC = 10, the competitive quantity would be 180 (where p = MC on the demand curve).
DWL = ½ × (180 − 90) × (55 − 10) = ½ × 90 × 45 = 2,025
Under a single-price scheme, the monopolist leaves two sources of value on the table:
Some consumers have a willingness to pay above the price, so their surplus is not fully extracted
Some consumers who value the good above marginal cost are priced out entirely
Price discrimination addresses both.
The firm charges each consumer their exact willingness to pay.
PS = the entire area between the demand curve and the MC curve (triangle ABC)
CS = 0
DWL = 0
Output is the efficient (competitive) level
This is typically impractical because the firm would need to know every consumer's valuation. Imperfect examples: personalised car discounts, personalised tax service pricing.
Worked example: With demand intercept 100, MC = 10, and competitive quantity = 180:
PS without discrimination = ½ × 90 × 90 = 4,050 (the monopoly profit from the earlier example)
PS with first-degree discrimination = ½ × 180 × 90 = 8,100
Different per-unit prices for different quantities of the same good.
Block pricing: the more you consume, the higher your average price. Common in utility pricing, and it can promote conservation.
Quantity discounts: lower per-unit price for larger purchases. This works because consumers typically have diminishing marginal utility and therefore diminishing willingness to pay for additional units.
Examples: buy one get one 50% off, loyalty cards, family-size snacks being cheaper per unit.
Consumers are divided into groups with separate demand curves, and each group is charged a different price.
Examples: hotel rates (weekdays vs. weekends), economy vs. business class, student/senior pricing, coupons, gender-based pricing (pink tax).
Worked Example 1: A hotel offering a negotiated group rate for bulk bookings.
This is second-degree price discrimination (quantity discount: buying more rooms leads to a different per-unit price).
Worked Example 2: The Metropolitan Museum of Art charges $25 for adults, $17 for seniors, $12 for students.
This is third-degree price discrimination (consumers are divided into demographic groups, each facing a different price).
Inverse demand: solve Q(p) for p to get p(q)
Revenue: R(q) = p(q) × q
Marginal revenue: MR(q) = R'(q)
For linear demand p(q) = a − bq: MR(q) = a − 2bq
Monopoly output: set MR(q*) = MC(q*)
Monopoly price: p* = p(q*) from the demand curve
Monopoly profit: Π = (p* − AC(q*)) × q*
Deadweight loss: DWL = ½ × (Qc − Qm) × (Pm − MC(Qm)), the triangle between MC and demand from Qm to Qc
First-degree PS: entire area between demand and MC
⚠️ For linear demand, MR has the same intercept as the inverse demand curve but twice the slope. This is the fastest way to write MR in exam conditions.
⚠️ The monopolist's price is read from the demand curve at q*, not from the MR curve. A common mistake is to use the MR value as the price.
⚠️ The shutdown condition for a monopolist uses AVC in the short run and AC in the long run, just like a competitive firm.
⚠️ Deadweight loss under monopoly is the triangle between demand and MC, from Qm to Qc. Be ready to compute it.
⚠️ Know the three degrees of price discrimination and be able to classify real-world examples. Group rates are second-degree (quantity discount). Demographic pricing is third-degree.
⚠️ Under first-degree price discrimination, output is efficient (no DWL), but all surplus goes to the producer.
Q: A monopolist faces Q(p) = 60 − p. What is MR(q)?
A: Inverse demand: p(q) = 60 − q. R(q) = 60q − q². MR(q) = 60 − 2q.
Q: If MC = 20 and MR(q) = 60 − 2q, what is the profit-maximising quantity and price?
A: 60 − 2q = 20, so q* = 20. Price: p* = 60 − 20 = 40.
Q: Why is a monopolist's marginal revenue below the price?
A: To sell one more unit, the monopolist must lower the price. Under a single-price scheme, the lower price applies to all units, not just the marginal one, so the gain from the extra unit is partly offset by the revenue lost on all previous units.
Q: A "buy 3 get 1 free" promotion at a bakery is an example of which degree of price discrimination?
A: Second-degree. Different quantities carry different effective per-unit prices.
Q: Under first-degree price discrimination, what happens to consumer surplus and deadweight loss?
A: Both become zero. The monopolist extracts every consumer's willingness to pay, capturing the entire surplus, and produces the efficient output level.
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