## Oligopoly Theory

### STRATEGIC SITUATION Individuals face a strategic situation if the payoff to each person depends directly on the actions chosen by other persons, and each person is aware of this fact when he chooses his action. Game theory is a formal way of modeling strategic situations. The most widely used solution concept in game theory is the Nash equilibrium. A set of strategies constitute a Nash equilibrium if, holding the strategies of all other players unchanged, no single player can attain a higher payoff by choosing a different strategy. Let Si be the set of all possible strategies for player i. Let si denote an element of Si, and let s-i denote the strategies of all players other than i. The payoff to individual i is represented by yi(s1, ..., sN). The group of strategies (s1*, ..., sN*) constitute a Nash equilibrium if, for each player i, yi(si*, s-i*) >= yi(si,s-i*) for all si in Si. A Nash equilibrium need not be Pareto efficient is "self-enforcing" need not be unique need not exist in pure strategies COURNOT MODEL Assumptions: quantity is the strategic variable (Nash equilibrium in quantities) price clears the market homogeneous goods (not essential) Let P = P(q1,q2) be the inverse demand function. Let C1(q1) and C2(q2) be the cost functions of firms 1 and 2. If (q1*, q2*) is a Nash equilibrium, then q1* maximizes P(q1,q2*)q1 - C1(q1) q2* maximizes P(q1*,q2)q2 - C2(q2) These two conditions can be written in terms of two first order conditions: P + q1*P' - C1' = 0 P + q2*P' - C2' = 0 price exceeds marginal costs marginal costs need not be equalized across firms, i.e., production is inefficient compare the monopoly pricing equation: P + QMP' - C' = 0 Oligopolists only take into partial account the effect of their output expansion on output price. Therefore equilibrium price is lower and equilibrium quantity is higher than the monopoly solution. we can re-arrange the first order conditions to get (P - MCi)/P = ai/e where ai is the market share of firm i and e is the industry's elasticity of demand. DIAGRAMATIC REPRESENTATION The best response function (also known as reaction function) shows the optimal (i.e., profit-maximizing) output q1 as a function of q2. It is implicitly defined by the first order condition for firm 1: P'(q1 + q2)q1 + P(q1 + q2) - C1'(q1) = 0. We can write q1 = BR1(q2). Apply a similar reasoning to firm 2, we can write q2 = BR2(q1). The Nash equilibrium can alternatively be defined as the solution to the two equations q1 = BR1(q2). q2 = BR2(q1). Diagramatically, the Nash equilibrium can be represented by the intersection of the best response curves. In the Cournot model, the best response function is often assumed to be downward sloping because an increase in other firms' output will tend to reduce the incentive for each firm to increase output. (That is, d/dqj(dyi/dqi) tends to be negative.) When best response functions are downward sloping, we call the strategic variables strategic substitutes. We can also draw iso-profit curves for firm 1 that shows combinations of q1 and q2 that give firm 1 equal profits. These curves have the following characteristics: the preference direction is downwards the isoprofit curve has an inverted U shape and it reaches a maximum on the BR1 curve Putting the best response functions and the iso-profit curves together, we can see there is potential for profitable collusion there are enforcement problems associated with collusion EXAMPLES 1. Let D = 1 - P, C1 = c1q1, C2 = c2q2, Firm 1 maximizes (1 - q1 - q2)q1 - c1q1. The first order conditions is (1 - q1 - q2) - q1 - c1 = 0. Rearrange this to q1 = (1 - q2 - c1)/2 =BR1(q2). Note the BR function is downward sloping the BR function will shift out when marginal cost falls Write a similar first order condition for firm 2: (1 - q1 - q2) - q2 - c2 = 0. Solve the two first order equations simultaneously to get the Nash equilibrium output: q1* = (1 - 2c1 + c2)/3. q2* = (1 + c1 - 2c2)/3. Substitute these equilibrium output back into the original profit function to get the equilibrium profits y1* = [(1 - 2c1 + c2)/3]2. y2* = [(1 + c1 - 2c2)/3]2. Notice that firm 1's output and profits depends (positively) on firm 2's cost. 2. Let D = 1 - P n firms, each with the same marginal cost = c Consider a typical firm i. Firm i maximizes (1 - q1 - ... - qn)qi - cqi. The first order condition is (1 - q1 - ... - qn) - qi - c = 0. In a symmetric equilibrium, q1 = ... = qn = q. We can then rewrite the first order condition in the form: 1 - nq - q - c = 0. (Note it is important to impose symmetry only after deriving the first order condition.) Solve the above to get q = (1 - c)/(n + 1). Q = nq = n(1 - c)/(n + 1) P = 1 - Q = (1 + nc)/(n + 1) Note dP/dn < 0 P approaches c when n approaches infinity STACKELBERG MODEL Consider a two-stage game where firm 1 (the leader) chooses its output level q1 in the first stage. In stage 2, firm 2 (the follower) chooses its output level q2, knowing what q1 is. The Stackelberg leader is implicitly assumed to have some ability to commit to its chosen output level q1. It will not change q1 after knowing what q2 is, even though such a change might be profitable ex post. To solve the Stackelberg game, we proceed backwards. In stage 2, firm 2 maximizes profits taking q1 as given. The optimal q2 is given by q2 = BR2(q1). This BR function is just like the one in the Cournot model. In stage 1, firm 1 chooses q1 to maximize its profits, taking into account firm 2's optimal response: maximize P(q1 + BR2(q1))q1 - C1(q1). The first order condition is (1 + BR'2)P'q1 + P = MC1. Compare with the Cournot outcome, where P'q1 + P = MC1, we can see that q1S > q1C. Moreover, since BR'2 < 0, q2S = BR2(q1S) < BR2(q1C) = q2C. EXAMPLE D = 1 - P C1 = c1q1 C2 = c2q2 As in Example 1 in the section on Cournot model, Firm 2's best response function is BR2(q1 = (1 - q1 - c2)/2. Firm 1 maximizes (1 - q1 - (1 - q1 - c2)/2)q1 - c1q1. The solution gives q1S = (1 + c2 - 2c1)/2. Substitute this back into the best response function for firm 2, we get q2S = (1 - 3c2 + 2c1)/4. BERTRAND MODEL The Bertrand model can be interpreted as a game in which prices are the strategic variables. Example 1. Assume homogeneous products constant marginal costs with no capacity constraint Let D(p) be the market demand function. Then the demand function for firm 1 is D1(p1,p2)= D(p1) if p1 < p2; D(p1)/2 if p1 = p2; 0 if p1 > p2. Notice that this demand function is discontinuous. price = c1 (minus a very small number) if c1 > c2 with more than 2 firms, price is equal to the second lowest marginal cost if firms have the same marginal cost, price = marginal cost. pure strategies equilibrium may not exist if marginal costs are not horizontal Example 2. Assume differentiated products; e.g., D1(p1,p2) is the demand curve for firm 1. D1 is decreasing in the first argument and increasing in the second cost functions are C1(q1) and C2(q2) Suppose the demand function takes the form D1 = a - bp1 + dp2. Then profits for firm 1 is y1 = (a - bp1 + dp2)p1 - C1(a - bp1 + dp2) We have dy1/dp1 = (a - 2bp1 + dp2) + bC'1(a - bp1 + dp2) d/dp2(dy1/dp1) = d + bdC''1(a - bp1 + dp2) The incentive the raise p1 is higher when p2 is higher. This suggests that the best response function is upward sloping. When the best response function is upward sloping, we call the strategic variables strategic complements. In a Bertrand model with differentiated goods we usually assume that the best response functions are upward sloping the isoprofit curves for firm 1 has the following characteristics the preference direction is upwards it has a U shape and it reaches its lowest point on the BR1 function CONJECTURAL VARIATIONS MODEL The conjectural variations model cannot be interpreted as the Nash equilibrium of a game. Yet the model is useful for describing the degree of monopoly power in oligopolistic markets. Consider a duopoly with homogeneous goods. Let P = P(q1 + q2). Define a parameter Lij to be i's conjecture of j's output response when qi increases. I.e., Lij = dqj/dqi. Firm i maximizes P(qi + qj)qi - Ci(qi). The first order condition is: P'qi(1 + Lij) + P - C'i = 0. if L = -1, then P = C' (perfect competition) if L = 0, then P'qi + P = C' (Cournot model) if L = 1/s - 1 (s is market share), then P'Q + P = C' (monopoly) Another possibility is to impose the condition that Lij has to be equal to the equilibrium dqj/dqi. This is known as the consistent conjectural variations model. Lecture Notes Klemperer Fat cats