Lecture 10 The Bertrand Model

1. ECON 4100: Industrial Organization Lecture 10 The Bertrand Model

2. Oligopoly Models • There are three dominant oligopoly models • Cournot • Bertrand • Stackelberg • Now we will consider the Bertrand Model

3. Price Competition: Bertrand • In the Cournot model price is set by some market clearing mechanism, firms seem relatively passive • An alternative approach is to assume that firms compete in prices: => Bertrand • This leads to dramatically different results • Take a simple example • two firms producing an identical product • firms choose the prices at which they sell their water • each firm has constant marginal cost of $10 • market demand is Q = 100 - 2P Check that with this demand and these costs the monopoly price is $30 and quantity is 40 units

4. Bertrand competition (cont.) • We need the derived demand for each firm • demand conditional upon the price charged by the other firm • Take Firm 2. Assume that Firm 1 has set a price of $25 • if Firm 2 sets a price greater than $25 she will sell nothing • if Firm 2 sets a price less than $25 she gets the whole market • if Firm 2 sets a price of exactly $25 consumers are indifferent between the two firms • the market is shared, presumably 50:50 • So we have the derived demand for Firm 2 • q2 = 0 if p2 > p1 = $25 • q2 = 100 - 2p2 if p2 < p1 = $25 • q2 = 0.5(100 - 50) = 25 if p2 = p1 = $25

5. Bertrand competition (cont.) Demand is not continuous. There is a jump at p2 = p1 • More generally: • Suppose Firm 1 sets price p1 p2 • Demand to Firm 2 is: q2 = 0 if p2 > p1 p1 q2 = 100 - 2p2 if p2 < p1 q2 = 50 - p1 if p2 = p1 • The discontinuity in demand carries over to profit 100 - 2p1 100 q2 50 - p1

6. Bertrand competition (cont.) Firm 2’s profit is: p2(p1,, p2) = 0 if p2 > p1 p2(p1,, p2) = (p2 - 10)(100 - 2p2) if p2 < p1 For whatever reason! p2(p1,, p2) = (p2 - 10)(50 - p2) if p2 = p1 Clearly this depends on p1. Suppose first that Firm 1 sets a “very high” price: greater than the monopoly price of $30

7. If p1 = $30, then Firm 2 will only earn a positive profit by cutting its price to $30 or less What price should Firm 2 set? At p2 = p1 = $30, Firm 2 gets half of the monopoly profit Bertrand competition (cont.) So, if p1 falls to $30, Firm 2 should just undercut p1 a bit and get almost all the monopoly profit With p1 > $30, Firm 2’s profit looks like this: What if Firm 1 prices at $30? The monopoly price of $30 Firm 2’s Profit p2 < p1 p2 = p1 p2 > p1 p1 $10 $30 Firm 2’s Price

8. Bertrand competition (cont.) Now suppose that Firm 1 sets a price less than $30 As long as p1 > c = $10, Firm 2 should aim just to undercut Firm 1 Firm 2’s profit looks like this: What price should Firm 2 set now? Of course, Firm 1 will then undercut Firm 2 and so on Firm 2’s Profit p2 < p1 Then Firm 2 should also price at $10. Cutting price below costgains the whole market but loses money on every customer What if Firm 1 prices at $10? p2 = p1 p2 > p1 p1 $10 $30 Firm 2’s Price

9. Bertrand competition (cont.) • We now have Firm 2’s best response to any price set by Firm 1: • p*2 = $30 if p1 > $30 • p*2 = p1 - “something small” if $10 < p1< $30 • p*2 = $10 if p1< $10 • We have a symmetric best response for Firm 1 • p*1 = $30 if p2 > $30 • p*1 = p2 - “something small” if $10 < p2< $30 • p*1 = $10 if p2< $10

10. Bertrand competition (cont.) The best response function for Firm 1 The best response function for Firm 2 These best response functions look like this p2 R1 The Bertrand equilibrium has both firms charging marginal cost R2 $30 The equilibrium is with both firms pricing at $10 $10 p1 $10 $30

11. Bertrand Equilibrium: modifications • Bertrand: competition in prices is very different from competition in quantities • Since many firms seem to set prices (and not quantities) this is a challenge to the Cournot approach • But the Bertrand model has problems too • for the p = marginal-cost equilibrium to arise, both firms need enough capacity to fill all demand at price = MC • but when both firms set p = c they each get only half the market • So, at the p = marginal cost equilibrium, there is huge excess capacity • This calls attention to the choice of capacity • Note: choosing capacity is a lot like choosing output which brings us back to the Cournot model • The intensity of price competition when products are identical that the Bertrand model reveals also gives a motivation for Product differentiation

12. An Example of Product Differentiation Coke and Pepsi are nearly identical but not quite. As a result, the lowest priced product does not win the entire market. QC = 63.42 - 3.98PC + 2.25PP MCC = $4.96 QP = 49.52 - 5.48PP + 1.40PC MCP = $3.96 There are at least two methods for solving this for PC and PP

13. Bertrand and Product Differentiation Method 1: Calculus Profit of Coke: pC = (PC - 4.96)(63.42 - 3.98PC + 2.25PP) Profit of Pepsi: pP = (PP - 3.96)(49.52 - 5.48PP + 1.40PC) Differentiate with respect to PC and PP respectively Method 2: MR = MC Reorganize the demand functions PC = (15.93 + 0.57PP) - 0.25QC PP = (9.04 + 0.26PC) - 0.18QP Calculate marginal revenue, equate to marginal cost, solve for QC and QP and substitute in the demand functions

14. Bertrand competition and product differentiation Both methods give the best response functions: The Bertrand equilibrium is at their intersection PC = 10.44 + 0.2826PP PP Note that these are upward sloping RC PP = 6.49 + 0.1277PC These can be solved for the equilibrium prices as indicated RP $8.11 B $6.49 PC $10.44 $12.72

15. Bertrand Competition and the Spatial Model • An alternative approach is to use the spatial model from Chapter 4 • a Main Street over which consumers are distributed • supplied by two shops located at opposite ends of the street • but now the shops are competitors • each consumer buys exactly one unit of the good provided that its full price is less than V • a consumer buys from the shop offering the lower full price • consumers incur transport costs of t per unit distance in travelling to a shop • What prices will the two shops charge?

16. Xm marks the location of the marginal buyer—one who is indifferent between buying either firm’s good Bertrand and the spatial model What if shop 1 raises its price? Assume that shop 1 sets price p1 and shop 2 sets price p2 Price Price p’1 p2 p1 xm x’m Shop 1 All consumers to the left of xm buy from shop 1 xm moves to the left: some consumers switch to shop 2 Shop 2 And all consumers to the right buy from shop 2

17. This is the fraction of consumers who buy from Firm 1 Bertrand and the spatial model p1 + txm = p2 + t(1 - xm) How is xm determined? => 2txm = p2 - p1 + t => xm(p1, p2) = (p2 - p1 + t)/2t There are N consumers in total So demand to Firm 1 is D1 = N(p2 - p1 + t)/2t Price Price p2 p1 xm Shop 1 Shop 2

18. Bertrand equilibrium Profit to Firm 1 is p1 = (p1 - c)D1 = N(p1 - c)(p2 - p1 + t)/2t p1 = N(p2p1 - p12 + tp1 + cp1 - cp2 -ct)/2t Solve this for p1 This is the best response function for Firm 1 Differentiate with respect to p1 N p1/ p1 = (p2 - 2p1 + t + c) = 0 2t p*1 = (p2 + t + c)/2 This is the best response function for Firm 2 What about Firm 2? By symmetry, it has a similar best response function. p*2 = (p1 + t + c)/2

19. Bertrand and Demand p2 p*1 = (p2 + t + c)/2 R1 p*2 = (p1 + t + c)/2 2p*2 = p1 + t + c R2 = p2/2 + 3(t + c)/2 c + t => p*2 = t + c (c + t)/2 => p*1 = t + c p1 (c + t)/2 c + t

20. Next: Stackelberg • Firms choose outputs sequentially • leader sets output first, and visibly • follower then sets output • The firm moving first has a leadership advantage • can anticipate the follower’s actions • can therefore manipulate the follower