23  Monopoly

For more information on these topics, see Varian Chapter 24: Monopoly.

23.1 Part A: Monopolist Maximizing Profits with Linear Demand

Until now, we’ve focused on markets with many identical sellers, where each seller is a price taker. This means they can sell as many units as they want at the market price, but if they raise their price above the market level, they won’t sell anything because consumers will choose to buy from their competitors instead.

A monopolist, on the other hand, is the exact opposite. As the only seller in the market, the monopolist is a price maker. They control the price. Like any firm, the monopolist aims to maximize profit. To do this, they set the price strategically to achieve the highest possible profit. Do they set the price infinitely high? No, because no one would buy their product. Instead, they must find the optimal balance, lowering the price enough to maximize overall profit. In this assignment, we’ll derive the monopolist’s price and output under a few different market conditions.

Example

When you want to choose quantity supplied in order to maximize profit, we know that you can write down the profit function, take the derivative with respect to \(Q\), and set it equal to 0 because at a function’s maximum (if it’s something like a downward-facing parabola), the slope of the tangent line is 0.

\[\begin{align} \pi &= TR - TC\\ \frac{d \pi}{dQ} &= 0 = \frac{d TR}{dQ} - \frac{d TC}{dQ}\\ 0 &= MR - MC\\ MR &= MC\\ \end{align}\]

This is the proof you’ve seen before that maximizing profit is the same as setting marginal revenue equal to marginal cost.

Further: if the monopolist faces a demand curve given by \(P = 100 - 2Q\) and the monopolist’s marginal cost is \(MC = 20\),

\[\begin{align} MR &= \frac{d}{dQ} PQ\\ &= \frac{d}{dQ} (100 - 2Q)Q\\ &= \frac{d}{dQ} 100 Q - 2 Q^2\\ &= 100 - 4Q \end{align}\]

This is a result we’ll see many times in the future: when the monopolist faces a linear demand curve, their marginal revenue has the same y-intercept as the demand curve, but twice the slope (so half the x-intercept). Let’s find the monopolist’s profit maximizing \(Q\):

\[\begin{align} MR &= MC\\ 100 - 4Q &= 20\\ 4 Q &= 80\\ Q &= 20 \end{align}\]

The profit-maximizing amount for the monopolist to supply is 20 units. Let’s finish the problem by finding the monopolist’s price (the maximum they can sell those 20 units for), and the monopolist’s profit (assuming zero fixed costs, so \(TC = 20Q\)):

\[\begin{align} P &= 100 - 2Q\\ &= 100 - 2(20)\\ &= 60 \end{align}\]

\[\begin{align} \pi &= TR - TC\\ &= PQ - 20 Q\\ &= 60 (20) - 20^2\\ &= 1200 - 400\\ &= 800 \end{align}\]

So in review: when the monopolist faces the demand curve \(P = 100 - 2Q\) and has a marginal cost of \(20\), they set MR = MC to find their profit maximizing output \(Q = 20\), the maximum price the market can bear is given by the demand curve: \(P = 60\), and the monopolist can make a profit of $800. Here’s what’s happening visually on a graph:

Compare the market under the monopolist to the market if there was instead perfect competition and the price was driven down to marginal cost, like we’ve studied before:

\[\begin{align} P &= MC\\ 100 - 2Q &= 20\\ 2Q &= 80\\ Q &= 40\\\\ P &= 100 - 2(40)\\ P &= 100 - 80\\ P &= 20\\\\ \pi &= PQ - 20Q\\ \pi &= 20 (40) - 20 (40)\\ \pi &= 0 \end{align}\]

So under the monopolist, the price is higher, the quantity exchanged is lower, and profit to firms is higher.

Question 1

  1. If demand is given by \(P = 60 - Q\) and the monopolist’s marginal cost is $20 per unit, find the monopolist’s \(Q\), \(P\), and \(\pi\).

  2. Continuing from part a, find the \(Q\), \(P\), and \(\pi\) under perfect competition.

  3. Compare: under a monopolist, the price is (higher/lower), the quantity exchanged is (higher/lower), and the profit to firms is (higher/lower).

Question 2

  1. If demand is given by \(P = 50 - 0.5 Q\) and the monopolist’s marginal cost is $40 per unit, find the monopolist’s \(Q\), \(P\), and \(\pi\).

  2. Continuing from part a, find the \(Q\), \(P\), and \(\pi\) under perfect competition.

  3. Compare: under a monopolist, the price is (higher/lower), the quantity exchanged is (higher/lower), and the profit to firms is (higher/lower).


23.2 Part B: What Causes a Monopoly

In Part A, we learned that compared to perfect competition, monopolies restrict quantity to drive up prices and maximize their own profit, to the detriment of consumers. So what causes a monopoly? One reason monopolies exist is that they are natural monopolies. A natural monopoly occurs in a market when a firm’s average costs keep declining over the relevant range of demand. For example:

Example

Imagine a city considering how to provide water service. The cost structure might look like:

  • Fixed costs: $10 million for pipes, treatment plants, etc.
  • Marginal costs: $2 for every thousand gallons

That gives us:

  • Total costs: \(TC = 10000000 + 2Q\)
  • Average total costs: \(ATC = 10000000/Q + 2\)

Average total costs only decline as Q increases: that’s a signal there’s a natural monopoly.

Let’s say the city’s demand for water is given by \(P = 100 - 0.00012 Q\) (Q is in thousands of gallons).

Now we can see why this creates challenges:

  1. Perfect Competition Price (\(P = MC\))

    • Setting \(P = MC\) means \(P = 2\)
    • When \(P = 2\), solve for Q to get \(2 = 100 - 0.00012 Q\), which implies \(Q = 816,666\)
    • What’s the firm’s profit at this quantity? \(\pi = TR - TC = 2 (816,666) - 10000000 - 2 (816,666)\), which is negative 10 million dollars of course. If price is kept at marginal cost (either by competition or by government regulation), no firm would expect to make a profit by providing water to this city.

  2. Monopoly Price (Q such that MR = MC; P the highest demand can bear)

    • \(MR = \frac{d}{dQ} (100 - 0.00012 Q)Q = 100 - 0.00024 Q\)
    • \(MR = MC\) gives \(100 - 0.00024 Q = 2\), or \(Q = 408333\): the monopolist would provide 408,333 units of water, which is half as much as in perfect competition
    • The quantity restriction lets the monopolist raise the price to \(100 - 0.00012 (408333) = 51\): 51 dollars per thousand gallons of water
    • Does the monopolist make a profit? \(\pi = TR - TC = 51 (408333) - 10000000 - 2 (408333) = 10008317\): Yes, the monopolist makes over 10 million dollars in profit.

Question 3

A small town is considering how to provide internet service. The cost structure is:

  • Fixed costs = $5 million (for fiber optic cables)
  • Marginal cost = $10 per household
  • Demand: P = 200 - 0.0015Q (where Q is number of households)

  1. Calculate the average total cost function. Does this point to a possible natural monopoly? Why or why not?

  2. Find the quantity and price if the town sets a rule that P = MC. Will this be profitable?

  3. Find the monopoly quantity and price. Is this profitable for the monopolist?


23.3 Multiple-Choice Practice Questions

Question 1: What characterizes a monopolist’s position in the market?






Question 2: When a monopolist faces a linear demand curve, their marginal revenue curve has:






Question 3: Compared to perfect competition, monopolies typically result in:






Question 4: What defines a natural monopoly?






Question 5: In the internet service example, why would setting P = MC be unprofitable?