Production-ONigepody Cournot Duopoly Suppose there are two firms in an industry. Their strategy spaces are quantities. Their payoffs are profits. Industry demand is given by the inverse demand function, P(Q), where industry production is Q=q1+92.They have identical cost functions c(qi). Profits for each firm are therefore given by: 丌1=P(q1+q2 and 12= P(qi This is a game. The Nash equilibrium occurs when both firms are optimising given the behaviour of the other. Throughout the lecture consider the following linear demand example with constant marginal costs. So: P(Q)=a-Q=a-q1-92 and cq)= action- Oligopoly Profits and Best Respons Profits for each firm are maximised where marginal revenue is equal to marginal cost. Recall firms are interested in nding an optimal level of their quantity for each level their opponent might choose. Suppose firm 2 chooses q q2 acts like a constant. For linear demand, marginal falls at twice the rate of demand Firm 1 sets marginal revenue, a-q2-2q1, equal to marginal cost, c. Hence q1(g2)=-d and q2(qu) These two equations give the best response functions for the two firms. Often called reaction or best reply functions Plotting these yield the reaction or best reply curves
Production — Oligopoly 1 Cournot Duopoly • Suppose there are two firms in an industry. Their strategy spaces are quantities. Their payoffs are profits. • Industry demand is given by the inverse demand function, P(Q), where industry production is Q = q1 + q2. They have identical cost functions c(qi). Profits for each firm are therefore given by: π1 = P(q1 + q2)q1 | {z } Revenue − c(q1) | {z } Cost and π2 = P(q1 + q2)q2 | {z } Revenue − c(q2) | {z } Cost • This is a game. The Nash equilibrium occurs when both firms are optimising given the behaviour of the other. • Throughout the lecture consider the following linear demand example with constant marginal costs. So: P(Q) = a − Q = a − q1 − q2 and c(q) = cq Production — Oligopoly 2 Profits and Best Responses • Profits for each firm are maximised where marginal revenue is equal to marginal cost. Recall firms are interested in finding an optimal level of their quantity for each level their opponent might choose. Suppose firm 2 chooses q2: max q1 π1 = max q1 (a − q1 − q2)q1 − cq1 • q2 acts like a constant. For linear demand, marginal revenue falls at twice the rate of demand. • Firm 1 sets marginal revenue, a − q2 − 2q1, equal to marginal cost, c. Hence: q1(q2) = a − c − q2 2 and q2(q1) = a − c − q1 2 • These two equations give the best response functions for the two firms. Often called reaction or best reply functions. Plotting these yield the reaction or best reply curves
Production-ONigepody Reaction curves Drawing the reaction curves for both firms on the same graph yields the picture below q1(q2) 41 The curve q(qz)yields the optimal level of gn for any given 42. The curve q2(gn)yields the optimal level of qz for any given qr. These curves will cross. Why? a point at which there is no profitable unilateral deviation is a Nash equilibrium. That is a point which is a best response to a best response(and so on)-where the two curves cross, written(qi. 42) action- Oligopoly Nash Equilibriun What is the value of qi and g?? They could be read off from the graph. Alternatively, solve the two equations. Substituting the value of q2 into the equation for qi yields: Symmetrically solving for gives q*=(a-c)/3. With the same linear demand and constant marginal cost assumptions in place but with n firms in the industry it is (mathematically) simple to show that each of the n firms will produce at:
Production — Oligopoly 3 Reaction Curves • Drawing the reaction curves for both firms on the same graph yields the picture below. ................................................................................................................................................................................................................................................................................ . ................................................................................................................................................................................................................................................................................. . . . . ....... ............. ............. ............. 0 q2 q1 q ∗ 2 q ∗ 1 q1(q2) q2(q1) • The curve q1(q2) yields the optimal level of q1 for any given q2. The curve q2(q1) yields the optimal level of q2 for any given q1. These curves will cross. Why? • A point at which there is no profitable unilateral deviation is a Nash equilibrium. That is a point which is a best response to a best response (and so on) — where the two curves cross, written (q ∗ 1 , q ∗ 2 ). Production — Oligopoly 4 Nash Equilibrium • What is the value of q ∗ 1 and q ∗ 2 ? They could be read off from the graph. Alternatively, solve the two equations. • Substituting the value of q2 into the equation for q1 yields: q ∗ 1 = 1 2 ½ a − c − a − c − q ∗ 1 2 ¾ =⇒ q ∗ 1 = a − c 3 • Symmetrically solving for q ∗ 2 gives q ∗ 2 = (a − c)/3. • With the same linear demand and constant marginal cost assumptions in place but with n firms in the industry it is (mathematically) simple to show that each of the n firms will produce at: q ∗ i = a − c n + 1
Production-ONigepody Profits and pri How much do the firms charge and how much profit do they make? Price is simply read off from the demand Equilibrium industry supply is Q2*=9i+4?. Hence equilibrium price is: Profit is given by revenue less cost: Ti=(P-c)ai=(a-c)2/9 just as easy to calculate the price and profit in the case of n firms: P andz Notice that, as the number of firms grows, price gets closer to marginal cost and profits get close to zero. These are the conditions in a perfectly competitive market. An oligopoly with many firms is like action- Oligopoly How does the case of Cournot duopoly differ from monopoly? If the two firms could collude they would act like a monopoly to maximise total profits. Recall a monopolist faces the entire demand curve and sets MR=MC. With linear demand P=a-Q and constant marginal costs c, the optimality condition requires: A monopolist produces less and hence prices higher at Pm=(a+c)/2. Profits are rm=(a-c)2/4 If the two firms could collude they would be able to split the profits in two, each firm getting(a-c-/8 by producing of=(a-c)/4. This is bigger than their Cournot equilibrium profit But they cannot. If one of the firms produced (a-c)/4 the other would not choose to produce the same. The best response function reveals that the firm has a better response where if (for example) firm I produced(a-c)/4 ={--"} This will yield higher profits. How can collusion be explained?
Production — Oligopoly 5 Equilibrium Profits and Prices • How much do the firms charge and how much profit do they make? Price is simply read off from the demand curve. • Equilibrium industry supply is Q∗ = q ∗ 1 + q ∗ 2 . Hence equilibrium price is: P ∗ = P(Q ∗ ) = a − q ∗ 1 − q ∗ 2 = a + 2c 3 • Profit is given by revenue less cost: π ∗ 1 = (P ∗ − c)q ∗ 1 = (a − c) 2/9. • It is just as easy to calculate the price and profit in the case of n firms: P ∗ = a n + 1 + nc n + 1 and π ∗ i = µ a − c n + 1 ¶2 • Notice that, as the number of firms grows, price gets closer to marginal cost and profits get close to zero. These are the conditions in a perfectly competitive market. An oligopoly with many firms is like perfect competition. Production — Oligopoly 6 Collusion • How does the case of Cournot duopoly differ from monopoly? If the two firms could collude they would act like a monopoly to maximise total profits. Recall a monopolist faces the entire demand curve and sets MR = MC. • With linear demand P = a − Q and constant marginal costs c, the optimality condition requires: MR = MC =⇒ a − 2Q m = c =⇒ Q m = a − c 2 • A monopolist produces less and hence prices higher at P m = (a + c)/2. Profits are π m = (a − c) 2/4. • If the two firms could collude they would be able to split the profits in two, each firm getting (a − c) 2/8 by producing q m 1 = (a − c)/4. This is bigger than their Cournot equilibrium profit. • But they cannot. If one of the firms produced (a − c)/4 the other would not choose to produce the same. The best response function reveals that the firm has a better response where if (for example) firm 1 produced (a − c)/4: q2(q1) = a − c − q1 2 = 1 2 ½ a − c − a − c 4 ¾ = 3 8 (a − c) • This will yield higher profits. How can collusion be explained?
Production-ONigepody tackelberg Leadership Suppose that firm 1(the leader)is already in the market. Firm 2(the follower) is about to enter. This situation is known as Stackelberg leadership. It is a different game and has a different equilibriun Firm 1 chooses Frm2 chooses q=(1,丌2) ced qi firm 2 simply chooses ise their profits. Using the best response function Working backwards: What should firm 1 do? Firm I now knows which quantity firm 2 will choose and how it depends upon their own choice. Therefore. including this information in their profit function, firm I optimises action- Oligopoly Profits and Best Resp Suppose firm I chooses q1. Using the same example, firm 2 will choose gz to maximise *2=(a-q1-92)q2-cq2. The reaction es the solution to this problem. Hence, firm 2 chooses 2 Firm 1 knows this and uses this information to when maximising their profit. Firm l has profit equal to 丌1=5{(a-q1)q1-cq} This is exactly half the profit a monopolist would receive. Hence firm 1, setting MR= MC, operates just like a monopolist-and produces the monopoly output i=(a-c)/2-and receives half the profit(a-c)2/8 Firm 2's output is worked out from the best response function yielding: ga=(a-c)/4
Production — Oligopoly 7 Stackelberg Leadership • Suppose that firm 1 (the leader) is already in the market. Firm 2 (the follower) is about to enter. • This situation is known as Stackelberg leadership. It is a different game and has a different equilibrium. ................................................................................................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................................................................................................ . ................................................................................................................................................................................................................................................................................................................ . . . . . . . . . . . . . . . . . . . . . • Firm 1 chooses q1 • Firm 2 chooses q2 =⇒ (π1, π2) • Firm 2 has a relatively simple problem to solve. Knowing that firm 1 has already produced q1 firm 2 simply chooses their quantity to maximise their profits. Using the best response function reveals the action firm 2 will take. • Working backwards: What should firm 1 do? Firm 1 now knows which quantity firm 2 will choose and how it depends upon their own choice. Therefore, including this information in their profit function, firm 1 optimises. Production — Oligopoly 8 Profits and Best Responses • Suppose firm 1 chooses q1. Using the same example, firm 2 will choose q2 to maximise π2 = (a − q1 − q2)q2 − cq2. The reaction curve gives the solution to this problem. Hence, firm 2 chooses: q s 2 (q1) = a − c − q1 2 • Firm 1 knows this and uses this information to when maximising their profit. Firm 1 has profit equal to: π1 = µ a − q1 − a − c − q1 2 ¶ q1 | {z } Revenue − cq1 |{z} Cost =⇒ π1 = 1 2 {(a − q1)q1 − cq1} • This is exactly half the profit a monopolist would receive. Hence firm 1, setting MR = MC, operates just like a monopolist — and produces the monopoly output q s 1 = (a − c)/2 — and receives half the profit (a − c) 2/8. • Firm 2’s output is worked out from the best response function yielding: q s 2 = (a − c)/4
Production-ONigepody Equilibrium The equilibrium is characterised by firm one producing the monopoly output(a-c)/2 and firm 2 playing a best response(a-c)/4. Equilibrium price is therefore Ps=a-q1 Finally, equilibrium profits are xi=(a-cr2/8 and 2=(a-c)2/16 Compare this with Cournot. The leader does better. The follower does worse. Price is lower. Total output is larger Compare this with collusion. The leader does as well. The follower does worse. Price is lower. Output is larger. However, like Cournot, comparing this to perfect competition reveals both firms do better than perfectly ompetitive firms -who get zero profit. Prices are higher and output is lower. action- Oligopoly Bertrand Duopoly Suppose instead firms choose prices - does this make a difference? Firm I and firm 2 choose P1 and P2 simultaneously. The firm that charges the lower price serves the entire market If both firms charge the same price, they both serve half the market. Profits are payoffs. Writing down profits, suppose that market e is Q()where P is market price(the lower the two prices, Pi and p2). Suppose again that marginal oosts are constant, MC=c PiQ(P1)-cQ(p1) if Pi P2 i iPr(p1)-cQ(p1)) if Pi=P2 ifPi> To illustrate the best reply functions consider what firm 1s optimal response is if firm 2 sets a price p2. If p2 is greater than marginal cost then firm I will wish to undercut firm 2 by a small amount If py is equal to marginal cost, firm I would be willing to set any price greater than or equal to p2- all such prices will result profits. Firm I will never set a price below P2 if p2 is less than c since this results in losses. In this last case, firm I would set any price strictly greater than p2 and get zero profits
Production — Oligopoly 9 Equilibrium • The equilibrium is characterised by firm one producing the monopoly output (a − c)/2 and firm 2 playing a best response (a − c)/4. Equilibrium price is therefore P s = a − q1 − q2 = (a + 3c)/4. • Finally, equilibrium profits are π1 = (a − c) 2/8 and π2 = (a − c) 2/16. • Compare this with Cournot. The leader does better. The follower does worse. Price is lower. Total output is larger. • Compare this with collusion. The leader does as well. The follower does worse. Price is lower. Output is larger. • However, like Cournot, comparing this to perfect competition reveals both firms do better than perfectly competitive firms — who get zero profit. Prices are higher and output is lower. Production — Oligopoly 10 Bertrand Duopoly • Suppose instead firms choose prices — does this make a difference? • Firm 1 and firm 2 choose p1 and p2 simultaneously. The firm that charges the lower price serves the entire market. If both firms charge the same price, they both serve half the market. • Profits are payoffs. Writing down profits, suppose that market demand is Q(P) where P is market price (the lower of the two prices, p1 and p2). Suppose again that marginal costs are constant, MC = c. π1 = p1Q (p1) − cQ (p1) if p1 p2 • To illustrate the best reply functions consider what firm 1’s optimal response is if firm 2 sets a price p2. If p2 is greater than marginal cost then firm 1 will wish to undercut firm 2 by a small amount. • If p2 is equal to marginal cost, firm 1 would be willing to set any price greater than or equal to p2 — all such prices will result in zero profits. Firm 1 will never set a price below p2 if p2 is less than c since this results in losses. In this last case, firm 1 would set any price strictly greater than p2 and get zero profits
Drawing this argument in an informal way gives the below "best response functions P(P2) 0 Nash equilibrium (P1,P2)=(c,c The firms price at marginal cost the efficient outcome. It makes no difference how many firms are in the market. The basic idea is that the firms will continue to undercut one another until they reach marginal cost. They will go no lower as this would involve making a loss. This is very different to Cournot. However, the Cournot equilibrium can be recovered with capacity constraints. action- Oligopoly A Price Leadership Game Consider the following price leadership game- note the similarity with Stackelberg leadership. Note the difference between this and Varians rather odd game, where profits accrue to firm I during the game. Firm I chooses Pl Firm 2 chooses P2 Again, working from the back: Firm 2 will choose to undercut firm 1s initial price in order to gain the whole market Firm 1. unable to make positive profits, can choose any price above marginal cost. Either firm 2 will undercut and make positive profits or, if firm I chooses to price at marginal cost both firms make zero profit In particular (P1, P2)=(c e)is still an equilibrium- another difference between price and quantity competition. Again, it seems collusion could result in higher profits. But firms are unable to collude successfully. Why
Production — Oligopoly 11 Reaction Curves and Nash Equilibrium • Drawing this argument in an informal way gives the below “best response functions”. ................................................................................................................................................................................................................................................................................ .......................................................................................... . . . . . . . . 0 p2 p1 c c p1(p2) p2(p1) • The only place where the two curves cross — the Nash equilibrium — is at (p1, p2) = (c, c). • The firms price at marginal cost — the efficient outcome. It makes no difference how many firms are in the market. • The basic idea is that the firms will continue to undercut one another until they reach marginal cost. They will go no lower as this would involve making a loss. • This is very different to Cournot. However, the Cournot equilibrium can be recovered with capacity constraints. Production — Oligopoly 12 A Price Leadership Game • Consider the following price leadership game — note the similarity with Stackelberg leadership. Note the difference between this and Varian’s rather odd game, where profits accrue to firm 1 during the game. ................................................................................................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................................................................................................ ................................................................................................................................................................................................................................................................................................................ . ................................................................................................................................................................................................................................................................................................................ . . . . . . . . . . . . . . . . . . . . . • Firm 1 chooses p1 • Firm 2 chooses p2 =⇒ (π1, π2) • Again, working from the back: Firm 2 will choose to undercut firm 1’s initial price in order to gain the whole market. • Firm 1, unable to make positive profits, can choose any price above marginal cost. Either firm 2 will undercut and make positive profits or, if firm 1 chooses to price at marginal cost both firms make zero profit. • In particular (p1, p2) = (c, c) is still an equilibrium — another difference between price and quantity competition. • Again, it seems collusion could result in higher profits. But firms are unable to collude successfully. Why?
