Lecture 10: Oligopoly

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Quantity Competition: The Cournot Model

One type of competition among oligopolistic firms is quantity competition, in which each firm chooses a quantity and the market price is determined by their (joint) choice.

We’ll look at one particular example of this over the next few weeks: we’ll keep the basic market demand and cost functions the same, and see how different assumptions on market structures, time, and information lead to dramatically different outcomes.

So here’s the setup: suppose demand for a particular good is given by $Q(P) = 14-P$ If we rearrange this to solve for P as a function of Q, we call the inverse demand: $P(Q) = 14 - Q$ One way of thinking about this is that once $Q$ units of the good have been produced, supply is perfectly inelastic, and the price is therefore determined by the total quantity supplied, $Q$:

See interactive graph online here.

Further, let’s assume that any firm can produce this good with no fixed costs, and at a constant marginal cost of 2; that is, for an individual firm producing $q$ units of output, $c(q) = 2q$.

Finally, let’s assume that any firm is going to try to maximize its profits, which are total revenues (price times quantity) minus total cost of production: $\pi(q) = \underbrace{P \times q}_\text{revenue} - \underbrace{c(q)}_\text{cost}$

Baseline Case: Monopoly

For a baseline example, let’s suppose that a single firm (monopoly) served this market; in this case the monopolist is choosing the total market quantity Q. That monopolist’s profit would be written as $\begin{aligned} \pi(Q) &= P(Q) \times Q - c(Q)\\ &= (14− Q)Q− 2Q\\ &= \underbrace{14Q− Q^2}_\text{revenue} - \underbrace{2Q}_\text{cost} \end{aligned}$ To find the quantity at which profit is maximized, we take the derivative⊕ As we’ve established many times, just taking the derivative and setting it equal to zero won’t always find the solution…watch out for corner solutions (i.e. times when this process would find you a negative number) and other times when this doesn’t work! But as a general rule, this is how we’ll at least start to solve these problems. and set it equal to zero, finding the quantity at which marginal revenue equals marginal cost:

$\begin{aligned} \pi^\prime(Q) = \underbrace{14− 2Q}_\text{marginal revenue} - \underbrace{2}_\text{marginal cost} &= 0\\ 2Q &= 12\\ Q^\star & = 6 \end{aligned}$ Plugging $Q = 6$ into the inverse demand function gives us the price of $P(6) = 14 - 6 = 8$ so the monopolist’s profit is $\pi(6) = 8 \times 6 - 2 \times 6 = 36$ Graphically, we see the profit as the area with height ($AR−AC$) and width $Q$:

See interactive graph online here.

Simultaneous Quantity Choice: Cournot Duopoly

Now suppose there are two firms in the market, and that each firm $i$ chooses its quantity $q_i$ (that is, firm 1 chooses $q_1$ and firm 2 chooses $q_2$). Therefore the total market quantity is $Q = q_1 + q_2$ Both firms choose their quantities simultaneously (or at least without knowing what the other firm is doing). Once they choose, the market price is determined by the total quantity produced: $P(q_1,q_2) = 14 - \underbrace{(q1 + q2)}_Q$ Again, we can think of the market supply curve as being vertical once each firm has produced its quantity. We can see this in the following graph, which is the same as the monopolist’s graph except that each firm is independently choosing $q_1$ and $q_2$:

(add graph with just the market demand and q1 and q2, with sliders to control each of those)

Now let’s think of the problem faced by firm 1. Suppose it knows, or believes, or anticipates, that firm 2 is going to choose quantity $q_2$. We could rewrite the market price, just as a function of $q_1$, as $P(q_1) = (14 - q_2) - q_1$ That looks just like the market demand curve facing the monopolist, except that instead of a vertical intercept of 14, the vertical intercept is $(14 − q_2)$. In other words, we can think of firm 1 as facing a residual demand curve given firm 2’s decision: if firm 2 produces zero, then firm 1 faces the entire market demand, but the more firm 2 produces, the lower the demand for good 1 is.

The graph below shows how this works. Try moving $q_2$ to be zero; then firm 1 faces the entire market demand. But now slowly increase $q_2$, and you can see that the residual demand curve facing firm 1 shifts in:

(add graph with three panels)

Now, facing this residual demand curve, firm 1 will choose the quantity which maximizes its profit: $\begin{aligned} \pi_1(q_1\ |\ q_2) &= P(q_1,q_2) \times q_1 - c_1(q_1)\\ &= [(14 - q_2) - q_1)q_1 - 2q_1 &= \underbrace{14q_1 - q_1q_2 - q_1^2}_\text{revenue} - \underbrace{2q_1}_\text{cost} \end{aligned}$ Taking the derivative and setting it equal to zero, we find that firm 1 will maximize its profits where its marginal revenue equals marginal cost: $\begin{aligned} \pi_1^\prime(q_1\ |\ q_2) = 14 - q_2 - 2q_1 - 2 &= 0\\ 2q_1 &= 12 - q_2\\ q_1^\star(q_2) &= 6 - \tfrac{1}{2}q_2 \end{aligned}$ Now technically (and this can be important on a homework or test question!) we should specify that the firm will never produce less than zero units of output. Since this expression for $q_1^\star$ would be negative if $q_2 > 12$, therefore, we should really write $q_1^\star(q_2) = \begin{cases} 6 - \tfrac{1}{2}q_2 & \text{ if }q_2 \le 12\\ 0 & \text{ if }q_2 \ge 12 \end{cases}$ Note that firm 1’s optimal choice depends on $q_2$, but they don’t observe $q_2$: rather, this tells us formulaically what its “best response” to firm 2’s choice of $q_2$ is. One way of thinking about this is if we were to write out the payoff matrix of this game, it would look (in part) like the following:

(game table)

If we circle firm 1’s best responses to each of firm 2’s strategies, we can see that 5 is a best response to 2, 4 is a best response to 4, and 3 is a best response to 6. But of course, we can’t list out a matrix with all real-number possibilities of quantities chosen by each firm – that would be an infinite matrix! So we express the best response of firm 1 to the quantity chosen by firm 2 as a best response function.

OK, so we’ve found firm 1’s best response to firm 2. Given that firm 2 faces the same problem (it has the same cost function, and the price is determined in the same way), then its best response function will be $q_2^\star(q_1) = 6 - \tfrac{1}{2}q_1$ To solve for equilibrium, we look for the point where each firm is best-responding to the other: that is, where $q_1 = q_1^\star(q_2)$ and $q_2 = q_2^\star(q_1)$. Putting these two conditions together gives us two equations in two unknowns: $\begin{aligned} q_1 &= 6 - \tfrac{1}{2}q_2\\ q_2 &= 6 - \tfrac{1}{2}q_1 \end{aligned}$ If we substitute the second equation into the first, we get $\begin{aligned} q_1 &= 6 - \tfrac{1}{2}\left(6 - \tfrac{1}{2}q_1\right)\\ q_1 &= 6 - 3 + \tfrac{1}{4}q_1\\ \tfrac{3}{4}q_1 &= 3\\ q_1^* &= 4 \end{aligned}$ By symmetry or substitution, we can see that $q_2 = 4$ as well. Visually, if we plot the two best-response functions against one another, we can see that the equilibrium occurs at their intersection:

(graph of best response functions)

Therefore in this Cournot equilibrium, $q_1 = q_2 = 4$, and the market price is $P=14-q_1-q_2=6$; so each firm earns a profit of $\pi = 6 \times 4 - 2 \times 4 = 16$.

12.4 Comparing Monopoly, Oligopoly, and Competitive Outcomes We found before that a monopolist would have set Q = 6, resulting in P = 8, and earning π = 36 in profit; in Cournot duopoly, we found that the overall market quantity was Q = 8, resulting in P = 6, and that each firm earned π = 16 in profit (for an overall profit of 32). If there were free entry, firms would set P= M C; since M C = 2, it follows that the price would drop to P = 2, with Q = 10. This would obviously be best for consumers! The graphs below show the equilibrium price and total quantity (green dot), profit (green area), and consumers’ surplus (blue area) from these three market structures, holding the demand structure and cost structure constant: 14 P (Q) 12 10 8 P CS 6 π 4 2 0 M C 14 P (q1 + q2) 12 10 8 6 P CS 4 π1 π2 2 0 M C 0 2 4 6 8 10 12 14 Q Monopoly 0 2 4 6 8 10 12 14 Q Cournot Duopoly 14 D 12 10 8 6 4 2 P CS 0 0 2 4 6 8 10 12 14 Perfect Competition S Q In fact, we can think of monopoly and perfect competition as being special cases of Cournot with different numbers of players. Suppose that instead of a duopoly, we had N identical firms competing in Cournot-style quantity competition, with c(q) = 2q for each firm and P (q1, q2, …, qN ) = 14− i qi. In this case it turns out that Firm 1’s best response function is BR1(q2, q3, …qN ) = 6− 1 2 N i=2 qi Let’s look for a symmetric equilibrium in which all firms produce someˆ q, and Firm 1 best responds to this by producingˆ q. In this case we’d have ˆ q = 6− 1 2 × (N− 1)ˆ q which solves to ˆ q= 12 N + 1 12N So the market quantity and price are 12N Q= Nˆ q= P = 14− Q = 14− N + 1 N + 1 We can see that in the monopoly case where N = 1, ˆ q= Q = 6 and P = 8, and in the duopoly case where N = 2, ˆ q = 4, Q = 8, and P = 6. Furthermore, as N gets large, each firm becomes very small compared to the market (ˆ q becomes very small), so each individual firm actually affects the market price very little; Q approaches 12; and P approaches 2 – i.e., the competitive outcome! So we can in fact think of “perfect competition” as an extreme example of an oligopoly model in which lots of firms have an infinitesimally small amount of market power. 5 12.5 Hotelling Duopoly The Cournot model deals with undifferentiated products: it’s a good way of thinking about things like oil. Many products (food, movies, etc) are differentiated: they appeal to people with different tastes, and each one tries to carve out its “niche.” One way of modeling differentiated products is to think about spatially differentiated products. This might be literal – where stores are located – or it might be some quality of the good being sold. In 1929, Stanford economist Harold Hotelling posited such a kind of competition in his paper Stability in Competition, which you can download here if you’re interested. He suggested that if it was costly for consumers to travel from where they were to the location of the seller, that they would go to the nearest seller if prices were equal. If prices weren’t equal, they would go to whichever seller offered them the best value – that is, utility from the product, minus price, minus travel costs. He presents the following example: there is a town of length l blocks. Firm A is located a blocks from one end of town; firm B is set up b blocks from the other end of town. Suppose that the customer who’s indifferent between the two firms is located x blocks from A and y blocks from B: a b x y A B Suppose that the two firms sell identical goods, and that the cost of traveling distance d to the store is cd, so c is the cost per block. Then for the person above to be indifferent, it must be that pA + cx = pB + cy It’s also true that a + b + x + y= l Solving for x and y we have pA− pB 2c 1 x = 2 (l− a− b)− 2c 1 y= 2 (l− a− b) + pA− pB 2c Note that 1 2 (l− a− b) is the midpoint between the two. So if pA = pB , then the indifferent person is equidistant. If pA > pB , then x < y so the indifferent person is closer to Firm A (i.e., Firm A loses market share), and vice versa. Since firm A’s market share is a + x, this means that firm A faces the demand function qA(pA, pB ) = a + 1 2 (l− a− b)− pA− pB 2c = 1 2 (l + a− b)− 1 pA + 1 pB 2c Likewise, firm B’s demand function is qB (pA, pB ) = b + 1 2 (l− a− b) + pA− pB 2c = 1 2 (l− a + b)− pB + 2c pA 2c Note that each firm’s quantity is decreasing in their own price, but increasing in their competitor’s: that is, they sell more units if their price is low or if the other firm’s price is high. This seems quite natural! From here on in we can solve it just like we solved the Cournot: set up the profit functions, differentiate with respect to the firm’s strategy (in this case their price), set the derivative equal to zero, solve for the best response functions, and find the equilibrium prices. Hotelling does this in his paper, so you can see the derivation there; it’s easier when there are numbers involved, as you’ll see in section and on the homework.

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