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Economics of Antitrust Handout on Oligopoly Models This handout is intended to give you a basic feel for how oligopoly models work. It will cover both Cournot and Bertrand oligopolies. If there is demand for it, I’ll provide further handouts with either extensions of these two models or details on other oligopoly models. Counot (1838) Oligopoly Consider N firms, with the profits of the ith firm being given by the following expression: [1] πi = Pqi – TC(qi) Price is determined by market demand, which is a function of the total quantity produced by all of the firms, Q = ∑qi. The market demand can be described by an inverse demand equation. [2] P = P(Q) Before picking specific function for TC and P, it’s worth considering the general approach to solving this model. Plug [2] into [1], take the derivative with respect to qi, and set the result equal to zero. This is known as the first order condition (FOC) for firm i, and must be true for any optimal choice of qi (assuming an interior solution – if it is optimal for the firm to produce no output, the foc may not be satisfied). You can solve the first-order condition for qi, in which case you will have firm i’s reaction function, which gives its optimal qi as a function of the q’s chosen by each of the other firms. Since we have N of these reaction functions (one for each firm) and N variables (one q for each firm), we have N equations in N unknowns, which we can solve for all N of the q’s. Once we have the optimal Q’s, it is a simple matter to plug them into [2] to get the market clearing price, and into [1] to get the profits for each firm. There is one trick that is sometimes useful in solving this system of equations. If the N firms are identical, and we are at an interior solution (i.e., all of the firms are producing a positive amount), then each firm will produce the same quantity. Thus, we can substitute qi = q for all i, and reduce the system of equations to N identical equations, with one variable. Note: when using this trick it is very important not to make this substitution until after taking the derivative. If you make the substitution before taking the derivative, you are assuming that the firm assumes that all of its competitors will match its quantity choice, i.e., that the firm is picking one quantity for all of the firms, and not just its own quantity. If you do this, you will get q equal to 1/N of the monopoly Q, leading the monopoly pricing (which is wrong). This may seem confusing, so let’s move away from the general case, and consider a specific example that we can actually solve. Let’s start with some simplifying assumptions: A1: All of the firms have the identical cost function TC(qi) = MC*qi. A2: The inverse demand function is P = AQ1/ε, where b<0 (A > 0 and ε < 0 are constants). First consider A2. Note that if you calculate the price elasticity of demand, you get that it is constant and equals ε, the constant with a cleverly chosen name. [You can check this as an exercise.] Profits for firm 1 can then be written as follows: [3] π1 = A(∑qi)1/ε*q1 – MC*q1 Taking the derivative with respect to q1 gives us firm 1’s fist order condition: [4] q1*(A/ε)*(∑qi)-1+1/ε + A(∑qi)1/ε – MC = 0 Rather than solving for firm 1’s reaction function, I’ll apply the trick, setting qi equal to q for all i (including i = 1). [5] q*(A/ε)*(Nq)-1+1/ε + A(Nq)1/ε – MC = 0 (Nq)1/ε*[q*(A/ε)*(Nq)-1 + A – MC*(Nq)-1/ε] = 0 q*(A/ε)*(Nq)-1 + A – MC*(Nq)-1/ε = 0 A/(Nε) + A = MC*(Nq)-1/ε 1 A * 1 N q MC * 1 N [A digression on HHI] Economists some measure how concentrated an industry is using something called the Herfindahl-Hirschman Index (HHI). HHI is defined to be the sum of the squares of the market shares of the firms in the industry. Example: Suppose there are four firms in an industry, with market shares of 45%, 40%, 10% and 5%, respectively. Then the industry’s HHI would be 0.452 + 0.42 + 0.12 + 0.052 = 0.3750. Alternatively, if market shares are measured as whole numbers, the industry’s HHI would be 452 + 402 + 102 + 52 = 3750. Note: Both methods of measuring market shares are used. It should be fairly clear from context which method is being used. If shares are measured as fractions, the HHI will a number between zero (perfect competition) and one (monopoly). If shares are measured as whole numbers, the HHI will be a number between zero (perfect competition) and 10,000 (monopoly). Note that HHI = 1/N if the N firms all have equal size, and with HHI > 1/N if there are N firms with unequal size. [End of digression.] In this case, we can substitute HHI for 1/N into equation [5] to get [6] HHI A * 1 q MC * HHI Thus, we have found the equilibrium quantities as a function of marginal costs, the HHI measure of concentration, and marginal costs. From here it is pretty easy to find the equilibrium price and profits per firm. [7] P = AQ1/ε P = A(Nq)1/ε 1 A * 1 1 N P A* N * * MC N [8] P MC 1 1 N P MC HHI 1 πi = P*q – MC*q πi = (P– MC)*q 1/ HHI MC A1 i MC * MC HHI 1 HHI 1 * HHI 1 HHI * MC 1 HHI * A1 Now we can analyze the solution. Unless otherwise stated, this analysis assumes that demand is not “too” inelastic, specifically that ε < -HHI. Equations [6] to [8] then tell us the following things: 1. P > MC, but P < PM (the monopoly price). Q is greater than the monopoly quantity, but less then the competitive level. 2. As HHI goes to zero (i.e., N goes to infinity), P goes to MC, qi goes to zero, Q goes to P(MC), πi goes to zero, and ∑πi goes to zero. In other words, as the number of firms increases, the market outcomes approach the perfectly competitive outcomes. 3. As MC increases, P increases, qi decreases, and Q decreases. 4. Suppose demand becomes more elastic. Then P goes to MC, qi goes to zero, Q goes to P(MC), πi goes to zero, and ∑πi goes to zero. Just as with increasing the number of firms, making demand more elastic causes the equilibrium to approach the competitive level. 5. If ε > -HHI, then 1+HHI/ε is negative, and both price and quantity will be negative. This obvious absurdity means that the model is invalid for values of HHI and ε that make this happen. If you allow for entry and exit, the Cournot model can be extended so as to be valid in these cases (basically entry will make the HHI will fall enough to make ε < -HHI). Bertrand (1883) Oligopoly Bertrand oligopoly is a little different. Instead of each firm simultaneously choosing its quantity, with the price being set by the market so that supply equals demand, firms simultaneously choose their prices, with the market demanding an appropriate quantity from each firm. Let Q = Q(P) be the market demand function. Let N be the number of firms, and let Ni be the number of firms that choose the same price as firm. All firms have zero fixed costs and constant marginal costs. The demand for firm i’s output is qi = Q(Pi)/Ni if Pi <= Pj qi = 0 otherwise In other words, if firm i charges no more than any other firm in the market, it will share the market equally with all of the firms that had the lowest price. If it charges more, then it will sell nothing (thus earning zero profits). Now consider equilibrium costs. Suppose that the equilibrium price was above marginal costs, so profits are positive for each firm charging the minimum price. Each firm charging such a price gets 1/Ni of the total profits. If a firm were to undercut this price by a single penny, the total industry profits would change only by a very small amount (since P changes very little, total Q changes very little, which implies that total costs change very little). But the firm that undercut the price would earn all of the industry profits, and not just a share of them. For sufficiently small price costs, the effect of going from a market share of 1/Ni to 100% will outweigh any small change in total industry profits, so the firm will want to undercut the price. But this argument holds for any price above marginal cost. Thus, the equilibrium price can be no greater than the marginal cost. But the equilibrium price cannot be below marginal cost, since that would mean that all firms would earn a negative profit. Thus, the only equilibrium is for all firms to charge Pi = MC. Comparing Cournot and Bertrand So, when modeling an oligopoly, which model should you use? One approach is to look at the industry in question, and to ask whether firms pick a price, and then sell as much as the market wants at that price, or do they pick a quantity, and then try to find the price at which exactly that quantity is sold? The problem with that approach is that most firms fiddle with both price and quantity. For example, they have only an imperfect estimate of demand, and the try and pick a point on their demand curve. When the market starts running, they will find themselves either accumulating or running down their inventory, and so they will adjust both their price and their quantity in response. Another approach is to attempt to reconcile the models, usually by making Bertrand look more like Cournot. There are several ways to do this. For instance, if firms are capactity constrained so that no one firm can serve the entire market, then a firm will receive a positive market share even if its price is greater than the lowest price in the market. This leads to equilibrium prices that are higher than marginal costs. In fact, Kreps and Scheinkman (1983) showed that if firms first choose their capacity constraints, and then play the Bertrand game with those capacity constraints, they will choose capacity constraints such that the outcome of the capacity-constrained Bertrand game is exactly the same as the outcome of the Cournot game. Other ways to modify the Bertrand game include adding consumer search costs to the model (so that consumers do not always find the firm with the lowest price), or addinf product differentiation. The key to all of these modifications is they imply that a firm can price slightly above the price of the low-price firm, without loosing all of its sales. This blunts price competition, leading to prices above marginal costs and positive profits. Since the Bertrand model is not robust to these kinds of changes (i.e., even small search cots, capacity constraints, or product differentiation leads to big changes in the predicted outcomes), and since the Cournot model corresponds to our intuition that having only two firms is not enough to get the competitive outcome, the Cournot model is sometimes preferred by economists, at least in markets with homogenous goods. However, there are cases in which the Bertrand model seems to fit the industry quite well. Other Oligopoly Models There several other models of oligopoly, so Cournot and Bertrand are not the end of the story. There are also many extensions to the basic Bertrand and Cournot models, such as adding product differentiation, capacity constraints, consumer search costs, consumer switching costs, entry/exit, advertising, cost-reducing or quality-enhancing investments, and/or incomplete or asymmetric information. In general, economists try to use the simplest possible model that captures the relevant features of the world. Which features of the word are relevant will of course depend on the question the model is being used to answer. References Bertrand, J. 1883. “Theorie Mathematique de la Richesse Sociale,” Journal des Savants, pp. 499-588. Translated by James W. Friedman in Andrew F. Daughety, ed., Cournot Oligopoly, Cambridge, United Kingdom: Cambridge University Press, 1988, pp. 73-81. This is the original source of the Bertrand model. Carleton, D. and J. Perloff. 2000. Modern Industrial Organization (New York: Addison Wesley Longman). This intermediate textbook includes analysis of many models of oligopoly. Cournot, A. 1838. Recherches sur les Principes Mathematiques de la Theorie des Richesses. English edition (ed. N. Bacon): Researches into the Mathematical Principles of the Theory of Wealth (New York: Macmillan, 1897). This is the original source of the Cournot model. Kreps, D. and J. Scheinkman. 1983. “Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes,” Bell Journal of Economics, v. 14, pp. 326337. This article relates the two models. The title pretty much gives away the conclusion. Tirole, J. 1988. The Theory of Industrial Organization (Cambridge, Mass.: MIT Press). This is a good graduate level textbook that includes analysis of many oligopoly models. For you, it should be very challenging, but comprehensible. Viscusi, W., J. Vernon, and J. Harrington. 2000. Economics of Regulation and Antitrust, 3rd edition (Cambridge, Mass.: MIT Press). This is an undergraduate textbook that includes a very nice summary of oligopoly models.