Difficulty: Intermediate
A monopoly maximises profit by producing the quantity where marginal revenue equals marginal cost, then reading the price off the demand curve at that quantity. Because a monopolist faces a downward-sloping demand curve (unlike a competitive firm's flat one), its marginal revenue is always below its price, so it restricts output and charges above marginal cost. The size of the markup depends on how elastic demand is, not on the firm's size.
Monopoly
The sole supplier of a good that has no close substitute. A monopoly faces the entire market demand curve, so it can set its own price.
In simple terms: there is only one seller, and buyers have nowhere else to go for this product.
Inverse demand function, p(Q)
The function that gives the price a monopolist receives for selling a given quantity Q. It is the demand curve written with price on the left-hand side: p = f(Q).
Think of it as: "if I want to sell Q units, this is the price the market will pay."
Marginal revenue (MR)
The change in total revenue from selling one additional unit of output. For a monopolist, MR = p(Q) + [dp(Q)/dQ] x Q, which is always less than the price at any positive quantity because the firm must lower its price on all units to sell one more.
In simple terms: the extra money from the next unit, after accounting for the price cut on every unit already being sold.
Profit-maximisation condition (MR = MC)
A monopoly (like any firm) maximises profit by choosing the output where marginal revenue equals marginal cost. This is the necessary first-order condition. The second-order condition requires the MR curve to be steeper (falling faster) than the MC curve at that point.
Think of it as: keep producing as long as the next unit adds more to revenue than to cost. Stop when they are equal.
Price elasticity of demand (epsilon)
The percentage change in quantity demanded divided by the percentage change in price: epsilon = (dQ/dp)(p/Q). It is always negative for a downward-sloping demand curve. A monopoly always operates where demand is elastic (epsilon < -1), never in the inelastic region.
In simple terms: how much buyers cut back when the price goes up. If demand is inelastic, the monopolist can always raise profit by raising price further.
Shutdown rule (monopoly)
In the short run, a monopoly shuts down if the profit-maximising price is below average variable cost. Equivalently, it shuts down if revenue is less than variable cost at the optimal output.
Think of it as: produce only if you can at least cover your running costs.
A monopoly's profit function is pi(Q) = R(Q) - C(Q).
To maximise, set the derivative equal to zero: dR/dQ - dC/dQ = 0, which gives MR(Q*) = MC(Q*).
The second-order condition requires d2R/dQ2 < d2C/dQ2 at Q*. In practice this holds because MR is downward-sloping and MC is flat or upward-sloping.
A monopoly uses Q to denote its output (not lowercase q), because the monopoly's output is the entire market output.
Revenue is R(Q) = p(Q) x Q. By the product rule: MR = p(Q) + [dp/dQ] x Q.
The first term, p(Q), is the price received on the extra unit.
The second term, [dp/dQ] x Q, is negative (because dp/dQ < 0 for a downward-sloping demand curve). It captures the revenue lost by having to cut the price on all Q units already being sold.
So MR < p at every positive quantity. The MR curve sits below the demand curve.
For a competitive firm, dp/dQ = 0 (flat demand), so MR = p. That is why competitive firms are price takers.
If the inverse demand curve is linear, p(Q) = a - bQ, then revenue is R = aQ - bQ2 and MR = a - 2bQ.
The MR curve has the same vertical intercept (a) but twice the slope (-2b vs -b).
It hits the quantity axis at half the distance: MR = 0 at Q = a/(2b), while p = 0 at Q = a/b.
This relationship holds for every linear demand curve.
MR can be rewritten as MR = p(1 + 1/epsilon).
When demand is elastic (epsilon < -1), MR is positive.
When demand is unit-elastic (epsilon = -1), MR = 0.
When demand is inelastic (-1 < epsilon < 0), MR is negative.
A profit-maximising monopoly always operates in the elastic portion of its demand curve, because in the inelastic region MR is negative while MC is positive, so profits can be raised by reducing output.
MR = 24 - 2Q. MC = 2Q. Setting MR = MC: 24 - 2Q = 2Q, so Q* = 6.
Price: p = 24 - 6 = 18. Average cost: AC = (36 + 12)/6 = 8.
Profit: pi = (18 - 8) x 6 = 60.
Revenue is maximised at Q = 12 (where MR = 0), but profit is maximised at Q = 6 (where MR = MC). Profit-maximising output is always less than revenue-maximising output.
Elasticity at Q* = 6: epsilon = 1 - 24/6 = -3. This is in the elastic region, as expected.
A monopoly can equivalently set price or quantity; the demand curve links the two, so the profit-maximising outcome is the same either way.
The monopoly cannot choose a price-quantity pair above the demand curve. It is constrained to points on the curve.
Convention: we usually model the monopoly as choosing quantity, then reading price off the demand curve.
In a competitive market, shifting demand traces out equilibria along the supply (MC) curve, so each quantity maps to one price.
A monopoly has no supply curve. A demand shift can change price alone, quantity alone, or both, depending on how the new demand and MR curves intersect the MC curve.
Knowing the monopoly's MC curve is not enough to predict what it will sell at any given price. You also need the demand curve.
Profit function
\pi(Q) = R(Q) - C(Q)Profit-maximisation first-order condition (Eq. 11.1)
\frac{dR(Q^*)}{dQ} - \frac{dC(Q^*)}{dQ} = 0 \quad \Rightarrow \quad MR(Q^*) = MC(Q^*)Second-order (sufficient) condition (Eq. 11.2)
\frac{d^2R(Q^*)}{dQ^2} - \frac{d^2C(Q^*)}{dQ^2} < 0Equivalently: the slope of MR must be less than the slope of MC at Q*.
Marginal revenue via the product rule (Eq. 11.3)
MR(Q) = p(Q) + \frac{dp(Q)}{dQ} \cdot QFirst term = price received on the extra unit. Second term = revenue lost on existing units from the price cut.
Marginal revenue in terms of elasticity (Eq. 11.4)
MR = p\left(1 + \frac{1}{\varepsilon}\right)Where epsilon is the price elasticity of demand (always negative).
Linear demand shortcut
If p(Q) = a - bQ, then MR(Q) = a - 2bQ. Same intercept, twice the slope.
Pharmaceutical patents. Plavix cost roughly 3 cents per pill to produce but sold for about $7, because patent protection gave the manufacturer a monopoly. Harvoni (hepatitis C) sold for $1,350 per pill. These are textbook examples of MR = MC at a price far above marginal cost, sustained by legal barriers to entry.
Amazon Prime pricing. In 2018, Amazon raised the annual Prime price from $99 to $119 (a 20% increase). Analysts estimated a price elasticity of about -0.5, meaning Amazon expected to lose roughly 10% of its members. The revenue gain from higher prices on remaining members ($1,620 million) exceeded the revenue lost from departing members ($891 million), making the increase profitable. This illustrates MR reasoning with real numbers.
Apple iPad. When the iPad launched in 2010, Apple held 87% of the tablet market, close to a monopoly. With an estimated inverse demand of p = 770 - 11Q (millions of units), MC = $220 (constant), and F = $2 billion, the profit-maximising output was 25 million units at $495 each, yielding roughly $4.875 billion in profit. Within a year, competitors entered and Apple's share fell to 29% by 2018.
Taylor Swift concert pricing. Swift's 2018 tour had empty seats at some venues. This is profit-maximising behaviour: selling out requires cutting the price until MR falls below MC, which destroys profit. Higher prices with some empty seats yield area A + B in profit, compared to a smaller area B + C at the sellout price.
"A monopoly can charge any price it wants." It cannot. The monopoly is constrained by the demand curve. It picks the best point on the curve, but it cannot set both price and quantity independently. If it raises price, it sells fewer units.
"Monopolies maximise revenue." They maximise profit, not revenue. Revenue is maximised where MR = 0 (unit elasticity). Profit is maximised where MR = MC, which occurs at a smaller output because MC > 0. In the textbook example, revenue peaks at Q = 12 but profit peaks at Q = 6.
"A monopoly has a supply curve." It does not. A supply curve maps each price to a single quantity. For a monopoly, the same quantity can correspond to different prices depending on the shape of the demand curve. Knowing MC alone is not enough to predict the monopoly's output.
"Bigger monopolies have more market power." Size does not determine market power. What matters is the elasticity of demand at the profit-maximising quantity. A small monopoly selling a drug with no substitutes can have far more pricing power than a large firm facing elastic demand.
True or False: A monopoly's marginal revenue curve lies above its demand curve. (False. MR lies below the demand curve at every positive quantity.)
Fill in the blank: For a linear inverse demand curve p = a - bQ, the marginal revenue curve is MR = ____. (a - 2bQ)
True or False: A profit-maximising monopoly will always operate where demand is inelastic. (False. It always operates in the elastic region, where epsilon < -1.)
Fill in the blank: Profit is maximised at a ____ output than revenue. (smaller)
True or False: If a monopoly's demand curve shifts, the new equilibrium must lie on the same MC curve at a different point, just like a competitive firm. (False. A monopoly has no supply curve, so demand shifts can produce various price-quantity combinations that do not trace out the MC curve.)
Q: A monopoly faces the inverse demand curve p(Q) = 100 - 5Q and has a cost function C(Q) = 2Q2. Find the profit-maximising quantity, price, and profit.
A: MR = 100 - 10Q. MC = 4Q. Set MR = MC: 100 - 10Q = 4Q, so 14Q = 100, Q* = 100/14 = 50/7 (approx. 7.14). Price: p = 100 - 5(50/7) = 100 - 250/7 = 450/7 (approx. 64.29). Profit: pi = pQ - C = (450/7)(50/7) - 2(50/7)2 = 22,500/49 - 5,000/49 = 17,500/49 = 2,500/7 (approx. 357.14).
Q: Explain, using the MR formula MR = p(1 + 1/epsilon), why a profit-maximising monopoly never operates where demand is inelastic.
A: In the inelastic region, -1 < epsilon < 0, so 1/epsilon < -1, and (1 + 1/epsilon) < 0. This makes MR negative. Since MC is always non-negative, MR < 0 < MC, so the firm could raise profit by reducing output. The monopoly will keep reducing output until it reaches the elastic portion of the demand curve, where MR is positive and can equal MC.
Q: A monopoly has constant marginal cost of $10. Its linear inverse demand function is p = 50 - 2Q. What is the Lerner Index at the profit-maximising output?
A: MR = 50 - 4Q. Set MR = MC: 50 - 4Q = 10, so Q* = 10. Price: p = 50 - 20 = 30. Lerner Index = (p - MC)/p = (30 - 10)/30 = 20/30 = 2/3 (approx. 0.667).
Q: Why does a monopoly not have a supply curve?
A: A supply curve maps each price to a unique quantity supplied. For a competitive firm, this mapping comes from its MC curve. For a monopoly, the profit-maximising output depends on both the MC curve and the shape of the demand curve. If the demand curve shifts, the monopoly may produce the same quantity at a different price, or a different quantity at the same price. There is no one-to-one relationship between price and quantity.
This material builds directly on the competitive firm's profit maximisation from Chapter 8 (same MR = MC rule, but MR = p for competitive firms). The cost concepts (MC, AVC, AC, fixed cost) come from Chapter 7. Elasticity of demand, introduced in Chapter 2, is central here because it determines the gap between price and MR. In Part 2 of these notes, we use the MR = p(1 + 1/epsilon) result to derive the Lerner Index and analyse deadweight loss, connecting to the welfare analysis from Chapter 9.
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