A monopolist with a Leontief (fixed-proportions) production function derives a linear total cost, then sets marginal revenue equal to marginal cost to find its profit-maximising quantity and price. The monopoly price exceeds the competitive price, output is lower, and the gap creates deadweight loss.
Difficulty: Intermediate to Advanced
Prerequisites: production functions (especially Leontief/fixed-proportions), cost minimisation with input prices, the concept of marginal revenue, and basic welfare analysis (consumer surplus, producer surplus).
This topic brings together production theory, cost theory, and market power in one extended problem. You start from a production function, derive the cost function from input prices, then use the demand curve to find the monopolist's optimal output. The welfare analysis at the end (consumer surplus, producer surplus, deadweight loss) shows precisely why monopoly is inefficient compared to perfect competition. This is a capstone exercise that ties several weeks of material together.
Leontief production function (fixed-proportions)
A production function of the form q = min(aL, bK). Inputs must be used in a fixed ratio; extra units of one input without the other add nothing to output. Think of it as a recipe: you need exactly 2 cups of flour per egg, and extra flour without eggs is wasted.
Total cost function (TC)
The minimum cost of producing q units, derived from the production function and input prices. For q = min(2L, 5K) with w = r = 10, TC = 7q.
Marginal cost (MC)
The cost of one more unit. When TC = 7q, MC = 7 (constant).
Market demand / inverse demand
The relationship between price and quantity demanded in the entire market. Here, P = 247 – 4Q. The monopolist faces the entire market demand curve.
Total revenue (TR)
Price times quantity. TR = P × Q = (247 – 4Q)Q = 247Q – 4Q².
Marginal revenue (MR)
The additional revenue from selling one more unit. MR = dTR/dQ = 247 – 8Q. For a linear demand curve P = a – bQ, MR = a – 2bQ. In simple terms, marginal revenue falls twice as fast as the demand curve because selling one more unit requires lowering the price on all units.
Consumer surplus (CS)
The area between the demand curve and the price, up to the quantity sold. It measures how much more consumers would have been willing to pay beyond what they actually paid.
Producer surplus (PS)
The area between the price the firm receives and its marginal cost, over the quantity sold. For a constant-MC monopolist, PS = (P^M – MC) × Q^M.
Deadweight loss (DWL)
The value of mutually beneficial trades that do not happen because the monopolist restricts output. It is the triangle between the demand curve and MC, from Q^M to Q^C.
Competitive quantity (Q^C) and competitive price (P^C)
The quantity and price that would prevail under perfect competition, where P = MC. Used as the benchmark for measuring how much welfare monopoly destroys.
Production function: q = min(2L, 5K). Inputs w = 10, r = 10.
For efficiency, both arguments of the min function must be equal: 2L = 5K = q.
Conditional factor demands: L* = q/2, K* = q/5.
Total cost: TC = wL* + rK* = 10(q/2) + 10(q/5) = 5q + 2q = 7q.
Because TC is linear in q, the marginal cost is constant: MC = 7.
The monopolist faces market demand P = 247 – 4Q.
Total revenue: TR = PQ = (247 – 4Q)Q = 247Q – 4Q².
Marginal revenue: MR = dTR/dQ = 247 – 8Q.
Set MR = MC: 247 – 8Q = 7, so 8Q = 240, giving Q^M = 30.
Plug Q^M = 30 into the demand curve: P^M = 247 – 4(30) = 247 – 120 = 127.
L* = Q^M / 2 = 30/2 = 15.
K* = Q^M / 5 = 30/5 = 6.
Under perfect competition, P = MC: 247 – 4Q = 7, so Q^C = 60 and P^C = 7.
The competitive outcome serves as the baseline for welfare comparisons.
CS is the triangle above the price and below the demand curve.
CS = ½ × (P_max – P^M) × Q^M = ½ × (247 – 127) × 30 = ½ × 120 × 30 = 1,800.
PS is the rectangle between price and MC, over the monopoly quantity.
PS = (P^M – MC) × Q^M = (127 – 7) × 30 = 120 × 30 = 3,600.
DWL is the triangle of lost surplus between Q^M and Q^C.
DWL = ½ × (P^M – MC) × (Q^C – Q^M) = ½ × (127 – 7) × (60 – 30) = ½ × 120 × 30 = 1,800.
q = \min(2L,\, 5K)L^* = \frac{q}{2}, \quad K^* = \frac{q}{5}TC = wL^* + rK^* = 7qMC = 7P = 247 - 4QTR = 247Q - 4Q^2MR = 247 - 8QCS = \frac{1}{2}(P_{\max} - P^M) \cdot Q^MPS = (P^M - MC) \cdot Q^MDWL = \frac{1}{2}(P^M - MC)(Q^C - Q^M)Pharmaceutical companies with patent protection behave as monopolists: they set prices well above marginal cost, generating large producer surplus and deadweight loss. The welfare analysis you learn here is exactly the framework regulators use when deciding whether to intervene in markets with dominant firms.
The Leontief production function models assembly-line processes where inputs must be combined in exact proportions, such as one engine per car body or one CPU per motherboard.
Students often set P = MC for the monopolist. That is the competitive rule. A monopolist sets MR = MC, then reads the price off the demand curve.
A frequent error is doubling the slope of MR incorrectly. For P = a – bQ, MR = a – 2bQ. The intercept stays the same; only the slope doubles.
When computing DWL, students sometimes use the monopoly price minus the competitive price as the height of the triangle. The correct height is P^M – MC, and the base is Q^C – Q^M.
Students confuse profit with producer surplus. PS = (P – MC) × Q for constant MC, while profit = TR – TC. When there are no fixed costs (as here, TC = 7q), they happen to be equal. With fixed costs they differ.
⚠️ Deriving the cost function from a Leontief production function is a very common exam question. Practise setting the two arguments equal and solving for L* and K*.
⚠️ The MR = MC step is the single most-tested monopoly concept. Know the formula for MR given a linear demand.
⚠️ Welfare calculations (CS, PS, DWL) frequently appear as a block of sub-questions worth significant marks. Draw the graph first, label everything, then compute the areas.
⚠️ Finding the competitive benchmark (P = MC) to compute DWL is a step students skip or do incorrectly. The competitive quantity here is Q^C = 60, not 30.
True or false: For q = min(2L, 5K), the efficient input ratio is L/K = 5/2. (True. Setting 2L = 5K gives L/K = 5/2.)
Fill in the blank: If P = 247 – 4Q, then MR = ______. (247 – 8Q.)
True or false: A monopolist charges a price equal to marginal cost. (False. The monopolist charges a price read off the demand curve at the MR = MC quantity.)
Fill in the blank: DWL measures the value of ______ that do not occur under monopoly. (Mutually beneficial trades.)
True or false: Consumer surplus under monopoly is larger than under perfect competition. (False. Monopoly raises the price and reduces quantity, shrinking CS.)
Q: Given q = min(2L, 5K) with w = 10 and r = 10, derive the total cost function.
A: Set 2L = 5K = q. Then L = q/2 and K = q/5. TC = 10(q/2) + 10(q/5) = 5q + 2q = 7q.
Q: What is the monopolist's profit-maximising quantity and price if demand is P = 247 – 4Q?
A: TR = 247Q – 4Q². MR = 247 – 8Q. Set MR = MC: 247 – 8Q = 7, so Q^M = 30. P^M = 247 – 4(30) = 127.
Q: How many units of labour and capital does the monopolist hire?
A: L* = 30/2 = 15. K* = 30/5 = 6.
Q: Calculate consumer surplus, producer surplus, and deadweight loss under monopoly.
A: CS = ½(247 – 127)(30) = 1,800. PS = (127 – 7)(30) = 3,600. For DWL, first find Q^C from P = MC: 247 – 4Q = 7, so Q^C = 60. DWL = ½(127 – 7)(60 – 30) = 1,800.
Q: Why is the deadweight loss exactly equal to consumer surplus in this example?
A: It is a coincidence of the numbers. The DWL triangle and the CS triangle happen to have the same base (30 units) and the same height (120). In general, CS and DWL are computed from different regions and will differ.
Q: If the government imposed a price ceiling equal to MC (P = 7), what would happen to output?
A: Output would rise to Q^C = 60, the competitive quantity. The monopolist would produce more because it can no longer restrict output to raise price. DWL would fall to zero.
This material connects to price discrimination: once you understand single-price monopoly, the next step is asking what happens when the monopolist can charge different prices to different consumers (first-, second-, and third-degree discrimination).
The welfare analysis here underpins antitrust economics and regulation. Understanding DWL is essential for evaluating mergers, patents, and natural monopoly regulation.
The Leontief production function connects back to isoquant analysis (L-shaped isoquants) and forward to general equilibrium, where production functions determine economy-wide factor demands.
monopoly, monopolist pricing, MR = MC, marginal revenue, Leontief production function, fixed-proportions, min function, total cost derivation, constant marginal cost, consumer surplus, producer surplus, deadweight loss, welfare analysis, competitive benchmark, P = MC, inverse demand, market power, allocative inefficiency, intermediate microeconomics, ECON 302, Penn State