Source: ECN 312 Intermediate Microeconomic Theory, Arizona State University
Difficulty: Intermediate | Prerequisites: Understanding of supply and demand, basic calculus (derivatives), cost functions, and market structures overview.
This material sits at the heart of the monopoly unit in intermediate microeconomics. A monopolist, unlike a perfectly competitive firm, faces the entire market demand curve and therefore chooses both quantity and price. The two central questions here are: what quantity maximises total revenue (ignoring costs), and what quantity maximises profit (accounting for costs)? You need to be comfortable with inverse demand, total and marginal revenue, and the MR = MC condition before moving on to price discrimination, welfare loss, or regulation.
A revenue-maximising monopolist sets marginal revenue equal to zero, while a profit-maximising monopolist sets marginal revenue equal to marginal cost. Profit maximisation always produces a lower quantity and higher price than revenue maximisation (when MC > 0). The mechanics boil down to: invert the demand curve, build the revenue function, differentiate, set the derivative equal to zero (revenue max) or MC (profit max), and solve.
Inverse demand function
The demand curve rewritten to express price as a function of quantity, p(Q), rather than quantity as a function of price. In simple terms, it tells you the highest price consumers will pay for any given quantity.
Total revenue, R(Q)
Price times quantity, R(Q) = p(Q) × Q. Because a monopolist must lower price to sell more, total revenue first rises and then falls as quantity increases.
Marginal revenue, MR(Q)
The additional revenue earned from selling one more unit; the derivative of total revenue with respect to quantity. Think of it as: how much does my revenue change if I push one more unit out the door?
Marginal cost, MC
The additional cost of producing one more unit; the derivative of the total cost function with respect to quantity. For a linear cost function C(Q) = F + cQ, marginal cost is simply the constant c.
Revenue maximisation
Choosing quantity where MR = 0. The firm ignores costs entirely and picks the output level at which total revenue peaks. In simple terms, this is the top of the revenue hill.
Profit maximisation
Choosing quantity where MR = MC. This is what firms are assumed to do in standard economic theory. The firm balances the extra revenue from one more unit against its extra cost.
Fixed cost
A cost that does not vary with output (the constant term in the cost function, e.g. 250 in C(Q) = 250 + 10Q). Fixed costs affect total profit but never affect the profit-maximising quantity, because they vanish from the marginal cost.
Linear demand
A demand curve of the form Q = a - bP (or equivalently P = a/b - Q/b). With linear demand, the MR curve has the same vertical intercept as the inverse demand curve but twice the slope.
The revenue-maximising quantity is found by setting MR = 0. This ignores costs entirely and finds the peak of the total revenue curve.
Step-by-step procedure
Start with market demand: Q(p) = a - bp
Rearrange to get inverse demand: p = (a/b) - (Q/b)
Build the total revenue function: R(Q) = p × Q
Differentiate R(Q) to get MR(Q)
Set MR(Q) = 0 and solve for Q*
Plug Q* back into inverse demand to find p*
Worked Example (Problem 1a)
Given: C(Q) = 250 + 10Q and Q(p) = 300 - 2p
Inverse demand: p = 150 - Q/2
Revenue: R(Q) = 150Q - Q²/2
Marginal revenue: MR(Q) = 150 - Q
Set MR = 0: 150 - Q = 0, so Q* = 150
Price: p* = 150 - 150/2 = 75
Total revenue at the optimum: R = 75 × 150 = 11,250
Note that MR = 0 occurs at exactly half the horizontal intercept of the demand curve (Q = 300). This is always the case with linear demand.
Graphical interpretation
On a price-quantity diagram, the revenue-maximising point sits at the midpoint of the linear demand curve. The MR curve crosses zero at Q* = 150. On a separate total revenue diagram, this corresponds to the peak of the inverted parabola at R = 11,250.
The profit-maximising quantity is found by setting MR = MC. This is the standard optimisation rule for any firm with market power.
Step-by-step procedure
Derive MR(Q) from the revenue function (same steps as revenue maximisation)
Find MC by differentiating the cost function C(Q)
Set MR = MC and solve for Q*
Plug Q* back into inverse demand to find p*
Compute profit: π = R(Q*) - C(Q*), or equivalently π = (p* - ATC) × Q*
Worked Example 1 (Problem 1c): With fixed costs
Given: C(Q) = 250 + 10Q, Q(p) = 300 - 2p
MC = dC/dQ = 10 (constant)
MR = 150 - Q (from the revenue maximisation section)
Set MR = MC: 150 - Q = 10, so Q* = 140
Price: p* = 150 - 140/2 = 80
Profit: π = (80 × 140) - (250 + 10 × 140) = 11,200 - 1,650 = 9,550
Compare with revenue maximisation: the profit-maximising firm produces fewer units (140 vs 150) and charges a higher price (80 vs 75). This is the expected result whenever MC > 0.
Worked Example 2 (Problem 2a): No fixed costs
Given: C(Q) = 80Q, Q(p) = 5000 - 50p
Inverse demand: p = 100 - Q/50
Revenue: R(Q) = 100Q - Q²/50
MR = 100 - Q/25
MC = 80 (constant)
Set MR = MC: 100 - Q/25 = 80, so 20 = Q/25, so Q* = 500
Price: p* = 100 - 500/50 = 90
Profit: π = (90 × 500) - (80 × 500) = 45,000 - 40,000 = 5,000
Notice that with no fixed costs, the firm earns positive economic profit on every unit sold (the markup is p* - MC = 90 - 80 = 10 per unit).
Graphical interpretation
On the price-quantity diagram, the profit-maximising quantity is where the MR curve intersects the MC curve. The price is then read off the demand curve at that quantity (not off the MR curve). The area of the rectangle (p* - ATC) × Q* represents total profit.
General linear demand and its derivatives
For demand Q = a - bp:
p = \frac{a}{b} - \frac{Q}{b}R(Q) = \frac{a}{b}Q - \frac{Q^2}{b}MR(Q) = \frac{a}{b} - \frac{2Q}{b}Notice MR has the same intercept as inverse demand but twice the slope coefficient on Q. This is a standard result for any linear demand curve.
Revenue maximisation condition
MR(Q) = 0Profit maximisation condition
MR(Q) = MC(Q)Profit calculation
\pi = R(Q^*) - C(Q^*) = p^* \times Q^* - C(Q^*)Quick reference for the two problems
Problem 1 | Problem 2 | |
|---|---|---|
Demand | Q = 300 - 2p | Q = 5000 - 50p |
Cost | C = 250 + 10Q | C = 80Q |
MC | 10 | 80 |
Rev-max Q*, p* | 150, 75 | n/a |
Profit-max Q*, p* | 140, 80 | 500, 90 |
Students often think the monopolist reads price off the MR curve. The price is always found on the demand curve at the chosen quantity. The MR curve tells you the optimal quantity, the demand curve tells you the price.
Students often confuse revenue maximisation with profit maximisation. Revenue maximisation ignores costs (set MR = 0). Profit maximisation accounts for costs (set MR = MC). They give different answers whenever MC is not zero.
Students sometimes believe that fixed costs affect the profit-maximising quantity. They do not. Fixed costs shift total profit up or down but have no effect on marginal cost, so Q* and p* are unchanged.
Students sometimes set MR equal to the demand curve or equal to price. MR is always set equal to zero (for revenue max) or equal to MC (for profit max), never equal to price or demand.
⚠️ You will almost certainly be asked to compute both revenue-maximising and profit-maximising Q* and p* for a given demand and cost function. Be able to do both from scratch in under five minutes.
⚠️ Graph questions are common: know how to draw demand, MR, and MC on the same axes, mark the optimal Q* on the horizontal axis, and trace up to the demand curve (not MR) for the price.
⚠️ Expect a question that asks you to compare the revenue-maximising outcome with the profit-maximising outcome and explain why they differ. The answer: when MC > 0, the profit-maximising firm restricts output further (below the revenue-max level) and charges a higher price.
⚠️ The "twice the slope" property of MR under linear demand is a quick shortcut that saves time on exams. If inverse demand is p = a - bQ, then MR = a - 2bQ.
True or False: A revenue-maximising monopolist sets MR = MC. (False. MR = 0.)
True or False: With linear demand, the MR curve has the same intercept as inverse demand but twice the slope. (True.)
Fill in the blank: To find the monopolist's price, plug Q* into the ______ curve. (Inverse demand.)
True or False: Fixed costs change the profit-maximising quantity. (False. They affect profit level but not Q*.)
Fill in the blank: Profit-maximising output is always ______ than revenue-maximising output when MC > 0. (Lower.)
Q: A monopolist faces demand Q = 200 - 4p and cost C(Q) = 5Q. What is the profit-maximising price and quantity?
A: Inverse demand: p = 50 - Q/4. Revenue: R = 50Q - Q²/4. MR = 50 - Q/2. MC = 5. Set MR = MC: 50 - Q/2 = 5, so Q* = 90. Price: p* = 50 - 90/4 = 27.50.
Q: Using the same demand, what is the revenue-maximising price and quantity?
A: Set MR = 0: 50 - Q/2 = 0, so Q* = 100. Price: p* = 50 - 100/4 = 25. Revenue-max output is higher and price is lower, as expected.
Q: A monopolist has C(Q) = 1000 + 20Q and faces Q = 400 - 5p. Does removing the fixed cost of 1000 change the profit-maximising output?
A: No. MC = 20 regardless of the fixed cost. Inverse demand: p = 80 - Q/5. MR = 80 - 2Q/5. Set MR = MC: 80 - 2Q/5 = 20, so Q* = 150 and p* = 50, whether the fixed cost is 1000 or zero. The fixed cost changes total profit (by 1000) but not the optimal Q* or p*.
Q: Why does the profit-maximising monopolist produce less than the revenue-maximising monopolist?
A: Because producing the last few units towards the revenue-max quantity adds revenue (MR > 0) but also adds cost (MC > 0). When MR is positive but less than MC, those units cost more to make than they bring in. The firm stops producing where MR = MC, which is before MR reaches zero.
Q: On a graph, how do you find the monopolist's price?
A: Find Q* where MR = MC. Then go straight up from Q* to the demand curve (not the MR curve). The height of the demand curve at Q* is the price p*.
This connects to price discrimination because once you understand single-price monopoly, the next question is whether the monopolist can charge different prices to different consumers and capture more surplus.
This connects to welfare analysis and deadweight loss. The gap between the monopoly quantity and the competitive quantity (where P = MC) creates a deadweight loss triangle, which is the central argument for regulating monopolies.
This connects to perfect competition as a benchmark. In a competitive market, firms are price takers and produce where P = MC. The monopolist's MR = MC condition and the resulting higher price, lower quantity contrast is what makes monopoly a distinct market structure.
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