Difficulty: Intermediate | Prerequisites: Market demand, cost functions, profit maximisation basics (Lectures 10-14)
A monopolist faces the entire market demand curve and chooses output where marginal revenue equals marginal cost, just like a competitive firm, but produces less and charges more. The gap between monopoly price and marginal cost is determined by how elastic demand is: the more substitutes consumers have, the smaller the markup. This restriction of output creates a deadweight loss, meaning some trades that would benefit both buyers and sellers never happen.
Monopoly
A market structure in which a single firm is the only seller of a product with no close substitutes. In simple terms, one company controls the entire supply, so it gets to pick the price (subject to what consumers will actually buy).
Market demand
The total quantity of a good that all consumers in a market are willing to purchase at each price level, aggregated from individual preferences. Think of it as the constraint the monopolist faces: they can set the price, but they cannot force people to buy.
Marginal revenue (MR)
The additional revenue earned from selling one more unit of output. For a monopolist, MR is always less than price because selling an extra unit requires lowering the price on all units sold, not just the additional one.
Marginal cost (MC)
The additional cost incurred from producing one more unit of output. In simple terms, it is how much more expensive your total costs become when you make one extra unit.
Price elasticity of demand (epsilon_p)
A measure of how sensitive quantity demanded is to a change in price, defined as the percentage change in quantity demanded divided by the percentage change in price. Think of it as: if I raise my price by 1%, how many percent fewer units will people buy?
Markup pricing
The practice of setting price as a multiplier over marginal cost. The multiplier depends on elasticity: p = [|epsilon_p| / (|epsilon_p| - 1)] x MC. In simple terms, the less elastic demand is, the bigger the gap between what it costs to make and what the firm charges.
Deadweight loss (DWL)
The loss in total surplus (consumer surplus + producer surplus) that occurs because the monopolist produces less than the socially efficient quantity. Think of it as the value of trades that would have happened under competition but do not happen under monopoly.
Natural monopoly
A market structure where one firm can supply the entire market at lower cost than two or more firms could, typically because fixed costs are very large relative to marginal costs. Think of it as industries where the startup cost is so enormous that it only makes sense for one firm to exist (e.g. water utilities, rail networks).
Minimum efficient scale (MES)
The smallest level of output at which a firm minimises its long-run average cost. In simple terms, it is how big you need to get before your per-unit costs stop falling. When MES is large relative to total market demand, the market tends towards monopoly.
Cartel
A group of independent producers who coordinate output or pricing decisions to act collectively as a monopolist. In simple terms, competitors agree to behave like a single seller to push prices up. This is illegal in most developed economies.
Market demand gives the price consumers are willing to pay at each quantity. For a linear demand curve p = a - by, the monopolist can sell one unit at price a - b. Demand is a constraint: the firm picks quantity, and demand determines the resulting price.
The monopolist's profit function substitutes the demand curve into the standard profit equation:
Start with profit: pi = py - c(y)
Substitute demand: pi = (a - by)y - c(y)
Or more generally: pi = r(y) - c(y), where r(y) = p(y) x y is total revenue
To find the profit-maximising output, take the first-order condition:
d(pi)/dy = r'(y) - c'(y) = 0
Which gives: r'(y) = c'(y), or MR = MC
This is the same rule as in perfect competition. The difference is that the monopolist's MR curve differs from its demand curve.
Revenue is r(y) = p(y) x y. Differentiating with the product rule:
MR = y(dp/dy) + p(y)
Factor out p(y): MR = p(y)[1 + (y/p)(dp/dy)]
The term (y/p)(dp/dy) is the inverse of the price elasticity of demand. So:
MR = p(y)[1 + 1/epsilon_p]
Since epsilon_p is negative, rewrite as: MR = p(y)[1 - 1/|epsilon_p|]
Setting MR = MC yields the key condition: p(y)[1 - 1/|epsilon_p|] = MC(y)
Critical implications:
As |epsilon_p| increases (demand becomes more elastic), the monopoly price approaches marginal cost, which is what you see in a competitive market.
A monopolist will never operate on the inelastic portion of the demand curve (where |epsilon_p| < 1), because MR would be negative there. Monopolists always operate where demand is elastic.
Higher elasticity generally means a flatter demand curve, which is usually caused by the availability of close substitutes.
Suppose demand is p = 100 - (1/2)y and cost is c(y) = (1/2)y^2 + 100
Profit: pi = (100 - (1/2)y)y - (1/2)y^2 - 100
First-order condition: d(pi)/dy = 100 - y - y = 0, so y = 50
Monopoly price: p = 100 - 25 = 75
Monopoly profit: pi = 75 x 50 - (1/2)(50)^2 - 100 = 2400
Competitive equilibrium comparison:
In competition, p = MC = y, so 100 - (1/2)y = y, giving y = 66.67 and p = 66.67
The monopolist produces less (50 vs 66.67) and charges more (75 vs 66.67)
Rearranging the MR = MC condition gives the monopolist's pricing rule:
p(y) = MC(y) / [1 - 1/|epsilon_p|]
Equivalently: p(y) = [|epsilon_p| / (|epsilon_p| - 1)] x MC(y)
The term |epsilon_p| / (|epsilon_p| - 1) is the markup multiplier. Since |epsilon_p| > 1 at the profit-maximising quantity, this multiplier is always greater than 1, confirming the monopolist always charges above marginal cost.
Worked example continued:
At y = 50, p = 75: epsilon_p = (dy/dp)(p/y) = (-2)(75/50) = -3
Markup multiplier: |epsilon_p| / (|epsilon_p| - 1) = 3/2 = 1.5
Check: MC(y) = y = 50, and 50 x 1.5 = 75 = the monopoly price. It holds.
The key assumption underlying single-price monopoly: the monopolist cannot distinguish between consumers. If two people buy two units, the firm cannot tell which one would have paid more, so it must charge both the same (lower) price.
Because the monopolist restricts output below the competitive level, some mutually beneficial trades do not occur. The resulting welfare loss is the deadweight loss (DWL), represented graphically as the area of the triangle between the demand curve and the MC curve, from the monopoly quantity to the competitive quantity (areas B + C in the standard diagram).
Under monopoly, consumers pay a higher price for fewer goods. Part of what was consumer surplus under competition is transferred to the monopolist as profit (area A in the diagram), but the DWL portion is lost entirely, captured by nobody.
A natural monopoly arises when a firm has very large fixed costs but small marginal costs. This produces increasing returns to scale over a wide range of output, so one firm can serve the market more cheaply than two or more firms could.
The natural monopoly creates a regulatory dilemma:
If the firm prices at MC (the efficient price), it produces the right quantity (y_MC) but cannot cover its fixed costs and makes a loss.
If the firm prices at AC (average cost), it covers costs but produces too little relative to the efficient level (y_AC < y_MC).
Minimum efficient scale (MES) is the output level that minimises average cost. Whether a market is competitive or monopolistic depends on MES relative to total demand:
MES small relative to demand (e.g. MES = 100, demand = 100,000): room for many firms, competitive market.
MES large relative to demand (e.g. MES = 10,000, demand = 15,000): room for very few firms, monopoly or oligopoly likely.
A cartel is a coordinated effort among producers to restrict output and raise prices, effectively recreating monopoly behaviour through agreement rather than market structure.
Cartels are illegal in the U.S. and most developed economies under antitrust and competition law.
\text{Linear demand: } p = a - by\text{Profit: } \pi = r(y) - c(y) = p(y) \cdot y - c(y)\text{First-order condition (MR = MC): } r'(y) = c'(y)\text{MR in terms of elasticity: } MR = p(y)\left[1 - \frac{1}{|\epsilon_p|}\right]\text{Monopoly pricing condition: } p(y)\left[1 - \frac{1}{|\epsilon_p|}\right] = MC(y)\text{Markup formula: } p(y) = \frac{|\epsilon_p|}{|\epsilon_p| - 1} \times MC(y)\text{Price elasticity of demand: } \epsilon_p = \frac{\partial y}{\partial p} \cdot \frac{p}{y}Key diagram notes (refer to textbook or lecture slides for the full graphs):
The MR curve has twice the slope of a linear demand curve (slope = -2b vs -b) and the same vertical intercept.
The monopolist's profit-maximising output y* is found where MR crosses MC. The price p* is read off the demand curve at y*.
Deadweight loss is the triangle between demand and MC, from y* (monopoly output) to the competitive output.
The markup pricing formula explains why pharmaceutical companies can charge many times the manufacturing cost for a patented drug: with no close substitutes, demand is relatively inelastic, so the markup multiplier is large. The same logic explains Apple's pricing of iPhones, where the lecture notes the production cost of an iPhone 6 was roughly $200 but the retail price was $649.
Natural monopoly theory is why utilities like water, electricity distribution, and rail infrastructure are typically either government-owned or heavily regulated. The fixed costs are so large that competition would be wasteful, but leaving an unregulated monopolist to set its own price would produce significant deadweight loss.
Students often think a monopolist can charge any price it wants. It cannot. The monopolist is constrained by the demand curve: raise the price too high and nobody buys.
Students often confuse "where MR = MC" with "where price = MC." In monopoly, MR = MC determines the output, but price is read off the demand curve at that output. Price exceeds MC.
Students sometimes think deadweight loss is money that goes to the monopolist. It is not. DWL is surplus that disappears entirely, captured by no one. The transfer from consumers to the monopolist is a separate rectangle (area A), not the DWL triangle (areas B + C).
Students often assume a monopolist can operate on the inelastic portion of demand. It cannot, because MR would be negative there. A profit-maximising monopolist always sets output in the elastic region.
Deriving the profit-maximising quantity and price from a given demand curve and cost function is a standard exam calculation. Be able to do it quickly: substitute demand into profit, take the FOC, solve for y, plug back into demand for p.
The relationship between MR and elasticity (MR = p[1 - 1/|epsilon_p|]) is frequently tested. Know how to go from the demand function to the elasticity at a given point, and from there to the markup multiplier.
Comparing monopoly vs competitive outcomes (output, price, profit, surplus) for the same demand and cost functions is a classic exam question. The monopolist always produces less and charges more.
Understanding the deadweight loss diagram and being able to identify areas A (transfer), B + C (DWL), and the welfare effects on consumers vs producers.
The natural monopoly pricing dilemma (MC pricing causes losses, AC pricing restricts output) comes up in policy-oriented questions.
True or False: A monopolist maximises profit where price equals marginal cost. (False: where MR = MC.)
Fill in the blank: The MR curve for a linear demand curve has ______ the slope of the demand curve. (Twice.)
True or False: A monopolist can profitably operate where demand is inelastic. (False: MR would be negative.)
Fill in the blank: Deadweight loss from monopoly is the area between the ______ curve and the ______ curve, from monopoly output to competitive output. (Demand, MC.)
True or False: The markup multiplier |epsilon_p| / (|epsilon_p| - 1) is always less than 1. (False: always greater than 1.)
Q: Given demand p = 100 - (1/2)y and cost c(y) = (1/2)y^2 + 100, find the monopolist's profit-maximising output, price and profit.
A: Profit = (100 - (1/2)y)y - (1/2)y^2 - 100. FOC: 100 - y - y = 0, so y = 50. Price: p = 100 - 25 = 75. Profit: 75(50) - (1/2)(2500) - 100 = 3750 - 1250 - 100 = 2400.
Q: For the same demand and cost functions, what is the competitive equilibrium output and price?
A: In competition, p = MC = y. Set demand equal to MC: 100 - (1/2)y = y, so (3/2)y = 100, giving y = 66.67 and p = 66.67.
Q: Why does the monopolist's MR curve lie below the demand curve?
A: Because selling one more unit requires lowering the price on all units, not just the marginal one. MR = p + y(dp/dy), and since dp/dy < 0, the second term is negative, pulling MR below p.
Q: A monopolist faces demand with |epsilon_p| = 5 at its current output. What is the markup multiplier?
A: Markup = |epsilon_p| / (|epsilon_p| - 1) = 5/4 = 1.25. The monopolist charges 25% above marginal cost.
Q: Explain the regulatory dilemma facing a natural monopoly. Why can the government not simply require the firm to price at marginal cost?
A: A natural monopoly has large fixed costs and low marginal costs, so MC is below AC over the relevant output range. Pricing at MC means the firm sells at a loss (price < AC) and cannot survive without a subsidy. Pricing at AC covers costs but restricts output below the efficient level.
Q: Why do monopolists always operate on the elastic portion of the demand curve?
A: On the inelastic portion (|epsilon_p| < 1), marginal revenue is negative. The firm could increase revenue by reducing output. Since reducing output also reduces costs, the firm would keep reducing output until it reaches the elastic portion. A profit maximiser never stops where MR < 0.
This material builds directly on the profit maximisation and cost function content from earlier lectures. The MR = MC condition is the same rule used in perfect competition; the difference is that a monopolist's MR diverges from price because of the downward-sloping demand curve.
Deadweight loss connects to welfare economics and the concept of Pareto efficiency. In later lectures on oligopoly and game theory, you will see how the monopoly outcome serves as one extreme (the other being perfect competition), with oligopoly results falling in between.
The elasticity and markup pricing material ties into price discrimination (likely a future lecture), where a monopolist tries to charge different prices to different consumers based on their willingness to pay, which requires being able to distinguish between consumer types.
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