Monopoly

A monopolist is defined as being a sole seller in a market, but the general model of monopoly is central to the analysis of antitrust issues because it can be applied more widely. For example, suppose that all sellers in a market are somehow able to collude and set a price that maximizes the total profit, which is then divided among them. In this case, the monopoly model would be relevant, either for providing a prediction of price and quantity, or as a benchmark from which to measure the success of the cartel.

The monopoly model is also applied to the case of one large firm and a number of small “fringe” firms that behave competitively (expanding output as long as the price that they can get is above the cost of each additional unit of output). The behavior of the firms in the competitive fringe can be represented by a supply function, which shows the quantity provided by these firms in total as a function of the price. Whether or not a group of small firms can be counted upon to behave competitively is a behavioral issue that might be investigated with an experiment. Let this fringe supply function be represented by S(P), which is increasing in price if marginal costs are increasing for the fringe firms. Then the residual demand is defined to be the market demand, D(P), minus the fringe supply, so the residual demand is R(P) = D(P) – S(P). This residual demand

function indicates a relationship between price and the sales quantity that is not taken by the fringe firms; this is the quantity that can be sold by the large dominant firm. In this situation, it may be appropriate to treat the dominant firm as a monopolist facing a demand function Q = R(P).

From the monopolist’s point of view, the demand (or residual demand) function reveals the amount that can be sold for each possible price, with high prices generally resulting in lower sales quantities. It is useful to invert this demand relationship and think of price as a function of quantity, i.e. selling a larger quantity will reduce price. For example, a linear inverse demand function would have the form: P = A – BQ, where A is the vertical intercept of demand in a graph with price on the vertical axis, and – B is the slope, with B > 0. The experiments to be discussed in this chapter all have a linear inverse demand, which for simplicity, will be referred to as the demand function.

The left side of Figure 13.1 shows the results of a laboratory experiment in which each person had the role of a monopoly seller in a market with a constant cost of $1 per unit and a linear demand curve: P = 13 – Q, where P is price and Q is the quantity selected by the monopolist. Since the slope is minus one, each additional unit of output raises the cost by $1 and reduces the price by $1. The vertical axis in the figure is the average of the quantity choices made by the participants in the experiment. It is apparent that the participants quickly settle on a quantity of about 6, which is the profit-maximizing choice, as will be verified next.

The demand curve used in the experiment is also shown in the top two rows of Table 13.1. Notice that as price in the second row decreases from 12 to 11 to 10, the sales quantity increases from 1 to 2 to 3. The total revenue, PQ, is shown in the third row, and the total cost, which equals quantity, is given in the fourth row. Please take a minute to fill in the missing elements in these rows, and to subtract cost from revenue to obtain the profit numbers that should be entered in the fifth row. Doing this, you should be able to verify that profit is maximized with a quantity of 6 and a price of 7.

Table 13.1 Monopoly with Linear Demand and Constant Cost

Quantity 1 2 3 4 5 6 7 8 9 10 11 12 Price 12 11 10 9 8 7 6 5 4 3 2 1

TR 12 22 30

TC 1 2 3

Profit 11 20 27

MR 12 10 8

MC 1 1 1 Even though the profit calculations are straightforward, it is instructive to consider the monopolist’s decision as quantity is increased from 1 to 2, and then to 3, while keeping an eye on the effects of these increases on revenue and cost at the margin. The first unit of output produced yields a revenue of 12, so the marginal revenue of this unit is 12, as shown in the MR row at the left. An increase from Q = 1 to Q = 2 raises total revenue from 12 to 22, which is an increase of 10, as shown in the MR row. These additional revenues for the first and second units are greater than the cost increases of $1 per unit, so the increases were justified. Now consider an increase to an output of 3. This raises revenue from 22 to 30, an increase of 8, and this marginal revenue is again greater than the marginal cost of 1. Notice that profit is going up as long as the marginal revenue is greater than the marginal cost, a process that continues until the output reaches the optimal level of 6, as you can verify.

One thing to notice about the MR row of table 13.1 is that each unit increase in quantity, which reduces price by 1, will reduce marginal revenue by 2, since marginal revenue goes from 12 to 10 to 8, etc. This fact is illustrated in Figure 13.2, where the demand line is the outer thick line, and the marginal revenue line is the thick dashed line. The marginal revenue line has a slope that is twice as negative as the demand line. The MR line intersects the horizontal marginal cost line at a quantity of 6. Thus the graph illustrates what you will see when you fill out the table, i.e. that marginal revenue is greater than marginal cost

for each additional unit, the 1st, 2nd, 3rd, 4th, 5th, and 6th, but the marginal revenue of the 7th unit is below marginal cost, so that unit should not be added.

(The marginal revenues in the table will not match the numbers on the graph exactly, since the marginal revenues in the table are for going from one unit up to the next. For example, the marginal revenue in the sixth column of the table will be 2, which is the increase in revenue from going from 5 to 6 units. Think of this as 5.5, and the marginal revenue for 5.5 in the figure is exactly 2.)

In addition to the table and the graph, it is useful to redo the same derivation of the monopoly quantity using simple calculus. (A brief review of the needed calculus formulas is provided in the appendix to this chapter.) Since demand is: P = 13 – Q, it follows that total revenue, PQ, is (13 – Q)Q , which is a quadratic function of output: 13Q – Q². Marginal revenue is the derivative of total revenue, which is 13 – 2Q, which is a line that starts at a value of 13 when Q

= 0 and declines by 2 for each unit increase in quantity, as shown in Figure 13.2.

Since marginal cost is 1, it follows that marginal revenue equals marginal cost when 1 = 13 – 2Q, ie. when Q = 6, which is the monopoly output for this market.

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0 1 2 3 4 5 6 7 8 9 10 11 12 13

quantity

price Demand

MR MC

Figure 13.2. Monopoly Profit Maximization

as the area under the demand curve, and the net loss is obtained by looking at the area below demand, to the right of 6 and above the marginal cost line in the figure. This triangular area, bounded by light gray lines, is a measure of the welfare cost of monopoly, as compared with the competitive outcome where price is equal to marginal cost ($1) and the quantity is 12.

It is important to mention that the convergence of the quantity to the monopoly prediction on the left side of Figure 13.1 was observed in a context where the demand side of the market was simulated. This kind of experiment is probably appropriate if one is thinking of a market with a very large number of consumers, none of whom have any significant size or power to bargain for reductions from the monopoly price levels.

II. Cournot Duopoly

Next consider what would happen if a second firm were to enter the market that was considered in the previous section. In particular, suppose that both firms have constant marginal costs of $1, and that they each select an output quantity, with the price then being determined by the sum of their quantities, using the top two rows of Table 13.1. This duopoly structure was the basis for the second part of the experiment summarized in Figure 13.1. Notice that the quantity per seller starts out at the monopoly level of 6 in round 9, which results in a total quantity of 12 and forces price down to $1 (= 13 - 12). When price is

$1, which equals the cost per unit, it follows that earnings are zero. The average quantity is observed to fall in round 10; which is not surprising following a round with zero profit. The incentive to cut output can be seen from the graph in Figure 13.2. Suppose that one seller (the entrant) knew the other would produce a quantity of 6. If the entrant were to produce 0, the price would stay at the monopoly level of $7. If the entrant were to produce 1, the price would fall to $6, etc. These price/quantity points for the entrant are shown as the large dots labeled

“Residual Demand” in Figure 13.2. The marginal revenue for this residual demand curve has a slope that is twice as steep, as shown by the thin dotted line that crosses MC at a quantity of 9. This crossing determines an output of 3 for the entrant, since the incumbent seller is producing 6. In summary, when one firm produces 6, the best response of the other is to choose a quantity of 3. This suggests why the quantities, which start at an average of 6 for each firm in period 9, begin to decline in subsequent periods, as shown on the right side of Figure 13.1. The outputs fall to an average of 4 for each seller, which suggests that this is the equilibrium, in the sense that if one seller is choosing 4, the best response of the other is to choose 4 also.

In order to show that the Cournot equilibrium is in fact 4 units per seller, we need to run through some other best response calculations, which are shown in Table 13.3. This table shows a firm’s profit for the example under consideration

for each if its own output decisions (increasing from bottom to top) and for each output decision of the other firm (increasing from left to right). First consider the column labeled “0” on the left side, i.e. the column that is relevant when the other firm’s output is 0. If the other firm produces nothing, then this is the monopoly case, and the profits in this column are just copied from the monopoly profit row of Table 13.1: a profit of 11 for an output of 1, 20 for an output of 2, etc. Note that the highest profit in this column is 36 in the upper left corner, at the monopoly output of 6. This profit has been highlighted by putting a dark border around the box. To be sure that you understand the table, you should fill in the two missing numbers in the right column. In particular, if both have outputs of 6, then the total output is 12, the price is ____, the total revenue for the firm is ___, the total cost is ____, and then the profit is 0 (please verify). Some of the other best response payoffs are also indicated by boxes with dark borders. Recall that the best response to another firm’s output of 6 is to produce 3 (as seen in Figure 13.2), and this is why the box with a payoff of 9 in the far-right column has a dark border.

Table 13.3 A Seller’s Own Profit Matrix and Best Responses Key: Columns for Other Seller’s Outputs, and Rows for Own Outputs

0 1 2 3 4 5 6

6 36 30 24 18 12 6

5 35 30 25 20 15 10

4 32 28 24 20 16** 12 8

3 27 24 21 18* 15 12 9

2 20 18 16 14 12 10 8 1 11 10 9 8 7 6 5

A Nash equilibrium in this duopoly market is a pair of outputs, such that each seller’s output is the best response to that of the other. Even though 3 is a best response to 6, the pair (6 for one, 3 for the other) is not a Nash equilibrium (problem 1). The payoff of 16, marked with a double asterisk in the table, is the location of a Nash equilibrium, since it indicates that an output of 4 is a best response to an output of 4, so if each firm were to produce at this level, there would be no incentive for either to change unilaterally. Of course, if they could

The fact that the Nash equilibrium does not maximize joint profit raises an interesting behavioral question, i.e. why couldn’t the subjects in the experiment somehow coordinate on quantity restrictions to raise their joint earnings? The answer is given in the legend on the right side of Figure 13.1, which indicates that the matchings were random for the duopoly phase. Thus each seller was matched with a randomly selected other seller in each round, and this switching would make it difficult to coordinate quantity restrictions. In experiments with fixed matchings, such coordination is often observed, particularly with only two sellers.

Holt (1985) reported patterns where sellers sometimes “walked” the quantity down in unison, e.g. both duopolists reducing quantity from 7 to 6 in one period, and then to 5 in the next, etc. This kind of “tacit collusion” occurred even though sellers could not communicate explicitly. There were also cases where one seller produced a very large quantity, driving price to 0, followed by a large quantity reduction in an effort to send a threat and then a conciliatory message and thereby induce the other seller to cooperate. Such tacit collusion is less common with more than 2 sellers. Part of the problem in these quantity-choice models is that when one seller cuts output, the other has a unilateral incentive to expand output, so when one shows restraint, the other has greater temptation.

The Nash equilibrium just identified is also called a Cournot equilibirum, after the French mathematician who provided an equilibrium analysis of duopoly and oligopoly models in 1838. Although the Cournot equilibrium is symmetric in this case, there can be asymmetric equilibria as well. A further analysis of Table 13.3 indicates that there is at least one other Nash equilibrium, also with a total quantity of 8, and hence an average of 4. Can you find this asymmetric equilibrium (problem 3)?

As was the case for monopoly, it is useful to illustrate the duopoly equilibrium with a graph. If one firm is producing an output of 4, then the other can produce an output of 1 (total quantity = 5) and obtain a price of 8 (= 13 - 5), as shown by one of the residual demand dots in Figure 13.3. Think of the vertical axis as having shifted to the right at the other firm’s quantity of 4, as indicated by the vertical dotted line, and the residual demand dots yield a demand curve with a slope of minus 1. As before, marginal revenue will have a slope that is twice as negative, as indicated by the heavy dashed line in the figure. This line crosses marginal cost just above the quantity of 8, which represents a quantity of 4 for this firm, since the other is already producing 4. Thus the quantity of 4 is a best response to the other’s quantity of 4. As seen in the figure, the price in this duopoly equilibrium is $5, which is lower than the monopoly price of $7 for this market.

01 23 45 67 89 1011 1213

0 1 2 3 4 5 6 7 8 9 10 11 12 13 quantity

price Demand

Residual MR MC

Residual Demand

Figure 13.3. Cournot Duopoly