16  Monopolies

For more information on these topics, see Allen, Doherty, Weigelt, and Mansfield Chapter 8: Monopoly and Monopolistic Competition.

No reading required.

16.1 Classwork 16: The Monopolist’s Output Decision

A monopoly exists when there is only one seller in a market with no close substitutes for its product. Unlike firms in perfect competition, a monopolist is not a price-taker. They can restrict quantity (Q) to drive price (P) up as much as they want. When they profit maximize, they choose the point on the demand curve that maximizes their profit.

  1. Monopolist’s Profit Maximization: Suppose a monopolist faces the demand curve \(P = 100 - Q\) and has a total cost function \(TC = 20 + 2Q\).

    1. Show that the monopolist’s total revenue function is \(TR = 100 Q - Q^2\).

    2. Find the profit-maximizing quantity \(Q^*\). Hint: profit is maximized where \(\frac{d \pi}{d Q} = 0\). Since profit is TR - TC, we also have \(\frac{dTR}{dQ} - \frac{dTC}{dQ} = 0\), or in other terms, \(MR - MC = 0\). Move MC to the righthand side to get: \(MR = MC\). Setting MR equal to MC should help you find \(Q^*\).

    3. When the monopolist sets output to be the \(Q^*\) you found in part b, how much will consumers pay (according to the demand function)? That is, what \(P\) will the good sell for?

    4. Calculate the monopolist’s profit.

    5. Calculate the consumer surplus, producer surplus, and total surplus. Recall that CS is the area under the demand curve and above what the consumer pays, PS is the area above the supply curve (marginal cost) and below the price the monopolist receives, and TS is CS + PS.

  2. Key properties: As you work through problem 1, you’ll discover some key properties of the monopolist’s decision. Fill in the blanks to complete:

    1. For a linear demand curve \(P = a - bQ\), the Marginal Revenue (MR) curve is \(\underline{\hspace{2cm}}\). Plot the demand curve and the MR curve from problem 1. Note: for any quantity of output, the price consumers are willing to pay always is (greater than/less than/equal to) the producer’s marginal revenue. This means that \(P\) will always be (greater than/less than/equal to) \(MR\), which is our first sign that the monopolist’s decision is different from a firm in perfect competition, where \(P = MR = MC\).

    2. The monopolist’s profit maximizing \(Q\) is where MR = \(\underline{\hspace{2cm}}\). For that \(Q\) (and every \(Q\)), \(P > MR\).

  3. Comparing Monopoly to Perfect Competition: Using the same demand and cost functions from question 1, calculate the equilibrium price, quantity, and total surplus under perfect competition. Assume the supply curve is given by the marginal cost curve.

    1. Under perfect competition where \(MC = P\), in equilibrium, P = \(\underline{\hspace{2cm}}\) and Q = \(\underline{\hspace{2cm}}\). Under the monopolist, P = \(\underline{\hspace{2cm}}\) and Q = \(\underline{\hspace{2cm}}\). So we learned that monopolists charge (lower/higher) prices and produce (less/more) output.

    2. Under perfect competition, CS = \(\underline{\hspace{2cm}}\), PS = \(\underline{\hspace{2cm}}\), and TS = \(\underline{\hspace{2cm}}\). Under a monopolist, CS = \(\underline{\hspace{2cm}}\), PS = \(\underline{\hspace{2cm}}\), and TS = \(\underline{\hspace{2cm}}\). Monopolists create (less/more) total surplus compared to perfect competition.

  4. Marginal Revenue and Elasticity

    1. Suppose the price elasticity of demand is -2. Will a rise in price raise total revenue or lower it?

    2. In CW6, we showed that across a linear demand curve, the elasticity goes from \(- \infty\) when \(Q = 0\), to \(-1\) at the midpoint, all the way to \(0\) at the x-intercept. That is, low amounts of output correspond to elastic demand, and high amounts of output mean inelastic demand. So as you increase \(Q\), the monopolist’s total revenue increases while you’re on the elastic portion of the demand curve, and then starts to decrease when you hit the inelastic portion. So marginal revenue is positive on the elastic portion, then becomes negative on the inelastic portion. What part of the demand curve will the monopolist always be on? Verify your answer by checking whether the monopolist in problem 1 targeted the demand curve on the upper or lower half. :::