ECON/MGMT 115 Industrial Organization
1. Cournot Model, reprised 2. Bertrand Model of Oligopoly 3. Cournot & Bertrand
First Hour Reviewing the Cournot Duopoloy Equilibria Cournot vs. competitive markets vs. monopoly Extending Cournot to Multiple Firms Implications of Cournot Introducing the Bertrand Model of Duopoly Second Hour Competing on price vs. quantity Simple Bertrand Model Bertrand Model with quantity constraints Bertrand Model and product differentiation Cournot vs. Bertrand outcomes
We now move to oligopoly models. There are three dominant oligopoly models: Cournot Bertrand Stackelberg They are distinguished by: The decision variable that firms choose The timing of the underlying game Concentrate on the Cournot model in this section 4
The Cournot model begins with a duopoly. Two firms making an identical product (NOTE: Cournot supposed this was spring water); Demand for this product is: P = A - BQ = A - B(q 1 + q 2 ) where q 1 is output of firm 1 and q 2 is output of firm 2 Marginal cost for each firm is constant at c per unit To get the demand curve for one of the firms we treat the output of the other firm as constant So for firm 2, demand is P = (A - Bq 1 ) - Bq 2 5
P = (A - Bq 1 ) - Bq 2 $ The profit-maximizing choice of output by A - Bq 1 firm 2 depends upon the output of firm 1. A - Bq 1 Marginal revenue for firm 2 is: Solve this c MR 2 = (A - Bq 1 ) - 2Bq for output 2 MR 2 MR 2 = MC A - Bq 1-2Bq 2 = c q 2 q* 2 q* 2 = (A - c)/2b - q 1 /2 If the output of firm 1 is increased the demand curve for firm 2 moves to the left Demand MC Quantity 6
q* 2 = (A - c)/2b - q 1 /2 This is the reaction function for firm 2. It gives firm 2 s profit-maximizing choice of output for any choice of output by firm 1. There is also a reaction function for firm 1. By exactly the same argument it can be written:: q* 1 = (A - c)/2b - q 2 /2 Cournot-Nash equilibrium requires that both firms be on their reaction functions. 7
(A-c)/B (A-c)/2B q C 2 q 2 If firm 2 produces The reaction function The Cournot-Nash (A-c)/B then firm for firm 1 is equilibrium is at 1 will choose to q* 1 = (A-c)/2B - q 2 /2 the intersection produce no output If of firm the 2 reaction produces nothing functions then firmthe reaction function 1 will produce the for firm 2 is C monopoly output q* 2 = (A-c)/2B - q 1 /2 (A-c)/2B Firm 1 s reaction function q C 1 (A-c)/2B Firm 2 s reaction function (A-c)/B q 1 KEEP8
(A-c)/B q 2 q* 1 = (A - c)/2b - q* 2 /2 q* 2 = (A - c)/2b - q* 1 /2 Firm 1 s reaction function q* 2 = (A - c)/2b - (A - c)/4b + q* 2 /4 3q* 2 /4 = (A - c)/4b (A-c)/2B (A-c)/3B C Firm 2 s reaction function (A-c)/2B (A-c)/B (A-c)/3B q* 2 = (A - c)/3b q 1 q* 1 = (A - c)/3b 9
In equilibrium each firm produces q * 1 = q * 2 = (A - c)/3b Total output is, therefore, Q* = 2(A - c)/3b Recall that demand is P = A - BQ So the equilibrium price is P* = A - 2(A - c)/3 = (A + 2c)/3 Profit of firm 1 is (P* - c)q * 1 = (A - c) 2 /9 Profit of firm 2 is the same. A monopolist would produce Q M = (A - c)/2b Competition between the firms causes them to produce more product. Price is lower than the monopoly price. But output is less than the competitive output (A - c)/b where price equals marginal cost. 10
To illustrate Cournot, we will use a very linear demand function. Consider the following: Demand: P = 25 Q MC = AC = 5 Competitive Outcome: P = MC = 5 Q = 20 CS = 200, PS = 0 Monopoly outcome: MR = 25-2Q = MC= 5 Q = 10. P = 15 CS = 50, PS = 100, DWL = 50
Now let s find the Cournot Equilibrium: Assume two identical firms (1 and 2). Both set quantity assuming the other firm s quantity is independent of their own choice of output. Equilibrium occurs when each firm does not want to change its output having observed what output the other firm has set. (Nash Equilibrium) 12
P =25 - (q 1 +q 2 ) P = 25 q 1 q 2 Treat (25 q 1 ) as a constant. Total revenue for Firm 2 is: Pq 2 = (25 q 1 )q 2 q 2 2 MR = (25 q 1 ) 2q 2 MR = MC; (25 q 1 ) 2q 2 = 5 2q 2 = - 20 + q 1 This gives the reaction function for Firm 2: q 2 =(20-q 1 )/2 Repeat the procedure for Firm 2. 13
q 1 =(20-q 2 )/2 and q 2 =(20-q 1 )/2 Plug q 2 into the q 1 equation and solve for q 1 : q 1 = 20/3. Through symmetry, q 2 =20/3 P=35/3 p 1 = p 2 = 400/9; PS=800/9 CS=800/9 CS+PS=1600/9=177.78 DWL = 22.22 14
Perfect Competition Monopoly Cournot Oligopoly (2 firms) PRICE 5 15 11 2/3 (35/3) QUANTITY 20 10 13 1/3 (40/3) SURPLUS CS = 200; PS = 0 CS = 50; PS = 100 CS = 89; PS = 89 DEADWEIGHT LOSS 0 50 22 15
What if there are more than two firms? Much the same approach. Say that there are N identical firms producing identical products This denotes output Total output Q = q 1 + q 2 + of + every q N firm other than firm 1 Demand is P = A - BQ = A - B(q 1 + q 2 + + q N ) Consider Firm 1. It s demand curve can be written: P = A - B(q 2 + + q N ) - Bq 1 Use a simplifying notation: Q -1 = q 2 + q 3 + + q N So demand for Firm 1 is P = (A - BQ -1 ) - Bq 1 16
P = (A - BQ -1 ) - Bq 1 $ The profit-maximizing A - BQ choice of output by Firm -1 1 depends upon the output of the other firms. A - BQ -1 Marginal revenue for Firm 1 is: MR 1 = (A - BQ -1 ) - 2Bq 1 MR 1 = MC A - BQ -1-2Bq 1 = c Solve this c for output q 1 q* 1 q* 1 = (A - c)/2b - Q -1 /2 If the output of the other firms is increased the demand curve for firm 1 moves to the left Demand MC MR 1 Quantity 17
How do we solve this for q* 1? q* 1 = (A - c)/2b - Q -1 /2 As the number of firms increases output of each firm falls Q* -1 = (N - 1)q* 1 The firms are identical. q* 1 = (A - c)/2b - (N - 1)q* 1 /2 So in equilibrium they (1 + (N - 1)/2)q* 1 = (A - c)/2b will have identical q* outputs 1 (N + 1)/2 = (A - c)/2b As the number of q* 1 = (A - c)/(n + 1)B firms increases Q* = N(A - c)/(n + 1)B aggregate output P* = A - BQ* = (A + Nc)/(N + 1) increases Profit of firm 1 is P* 1 = (P* - c)q* 1 = (A - c) 2 /(N + 1) 2 B KEEP18
q* 1 = (A - c)/2b - Q -1 /2 Q* -1 = (N - 1)q* 1 How do we solve this As the The firms for number q* are 1? of identical. q* 1 = (A - c)/2b - (N - 1)q* 1 /2firms increases As the number output of So in equilibrium they of each firms increases falls (1 + (N - 1)/2)q* 1 = (A - c)/2b will aggregate have identical As the output number of q* outputs 1 (N + 1)/2 = (A - c)/2b firms increases increases price q* 1 = (A - c)/(n + 1)B tends to marginal cost Q* = N(A - c)/(n + 1)B P* = A - BQ* = (A + Nc)/(N + 1) Profit of firm 1 is P* 1 = (P* - c)q* 1 = (A - c) 2 /(N + 1) 2 B KEEP19
q* 1 = (A - c)/2b - Q -1 /2 Q* -1 = (N - 1)q* 1 How do we solve this As the The firms for number q* are 1? of identical. q* 1 = (A - c)/2b - (N - 1)q* 1 /2firms increases As the number output of So in equilibrium they of each firms increases falls (1 + (N - 1)/2)q* 1 = (A - c)/2b will aggregate have identical As the output number of q* outputs 1 (N + 1)/2 = (A - c)/2b firms As increases the increases number price of q* 1 = (A - c)/(n + 1)B firms tends increases to marginal profit cost of each firm falls Q* = N(A - c)/(n + 1)B P* = A - BQ* = (A + Nc)/(N + 1) Profit of firm 1 is P* 1 = (P* - c)q* 1 = (A - c) 2 /(N + 1) 2 B KEEP20
What if the firms do not have identical costs? Much the same analysis can be used. Marginal costs of Firm 1 are c 1 and of Firm 2 are c 2. Demand is P = A - BQ = A - B(q 1 + q 2 ). We have marginal revenue for Firm 1 as before. MR 1 = (A - Bq 2 ) - 2Bq 1 Equate to marginal cost: (A - Bq 2 ) - 2Bq 1 = c 1 Solve this for output q 1 q* 1 = (A - c 1 )/2B - q 2 /2 q* 2 = (A - c 2 )/2B - q 1 /2 A symmetric result holds for output of firm 2 21
(A-c 1 )/B (A-c 2 )/2B q 2 Industrial Organization The equilibrium output of firm 2 q* 1 = (A - c 1 )/2B - q* 2 /2 increases and If the of marginal firm 1 falls q* 2 = (A - c 2 )/2B - q* 1 /2 R 1 cost of firm 2 q* 2 = (A - c 2 )/2B - (A - c 1 )/4B falls its reaction + q* 2 /4 curve shifts to 3q* 2 /4 = (A - 2c 2 + c 1 )/4B the right R 2 C q* 2 = (A - 2c 2 + c 1 )/3B q* 1 = (A - 2c 1 + c 2 )/3B (A-c 1 )/2B (A-c 2 )/B q 1 What happens to this equilibrium when costs change? 22
In equilibrium, the firms produce: q * 1 = (A - 2c 1 + c 2 )/3B; q * 2 = (A - 2c 2 + c 1 )/3B Total output is, therefore, Q* = (2A - c 1 - c 2 )/3B Recall that demand is P = A - BQ So price is P* = A - (2A - c 1 - c 2 )/3 = (A + c 1 +c 2 )/3 Profit of Firm 1 is (P* - c 1 )q * 1 = (A - 2c 1 + c 2 ) 2 /9 Profit of Firm 2 is (P* - c 2 )q * 2 = (A - 2c 2 + c 1 ) 2 /9 Equilibrium output is less than the competitive level. Output is produced inefficiently: the low-cost firm should produce all the output. 23
Assume there are N firms with different marginal costs We can use the N-firm analysis with a simple change Recall that demand for Firm 1 is P = (A - BQ -1 ) - Bq 1 But then demand for firm i is P = (A - BQ -i ) - Bq i Equate this to marginal cost c i A - BQ -i - 2Bq i = c i This can be reorganized to give the equilibrium condition: A - B(Q* -i + q* i ) - Bq* i - c i = 0 P* - Bq* i - c i = 0 P* - c i = Bq* i But Q* -i + q* i = Q* and A - BQ* = P* 24
P* - c i = Bq* i The price-cost margin Divide by P* and multiply the right-hand side by Q*/Q* P* - c i P* = BQ* for each firm is q* i determined by its P* Q* market share and But BQ*/P* = 1/ and q* i /Q* = s i P* - c so: i = s i P* Extending this we have: P* - c P* = H demand elasticity Average price-cost margin is determined by industry concentration 25
In a wide variety of markets, firms compete in prices: Internet access Restaurants Consultants Financial services With monopoly, setting price or quantity first makes no difference. In oligopoly it matters a great deal. nature of price competition is much more aggressive then quantity competition. 26
In Cournot prices are set by market mechanisms. An alternative approach is to assume that firms compete in prices. This is the approach taken by Bertrand. This leads to dramatically different results. Take a simple example: Two firms producing an identical product... choose the prices at which they sell their products. Each firm has constant marginal cost of c Demand is P = A BQ In terms of Q = a bp with a = A/B and b= 1/B 27
We need the derived demand for each firm: demand conditional upon the price charged by the other firm. For Firm 2, if Firm 1 sets a price =p 1 then: If Firm 2 sets a price > p 1 it sells nothing. if Firm 2 sets a price < p 1 it gets the whole market. If Firm 2 sets a price = p 1 consumers are indifferent between 1 and 2, so market is split. So we have the derived demand for Firm 2: q 2 = 0 if p 2 > p 1 q 2 = (a bp 2 )/2 if p 2 = p 1 q 2 = a bp 2 if p 2 < p 1 28
This can be illustrated by the graph to the right. Demand is discontinuous. The discontinuity in demand carries over to profit. p 1 p 2 (a - bp 1 )/2 There is a jump at p 2 = p 1 a - bp 1 a q 2 29
Firm 2 s profit is: p 2 (p 1,, p 2 ) = 0 if p 2 > p 1 p 2 (p 1,, p 2 ) = (p 2 - c)(a - bp 2 ) if p 2 < p 1 p 2 (p 1,, p 2 ) = (p 2 - c)(a - bp 2 )/2 if p 2 = p 1 Clearly this depends on p 1. For whatever reason! Now suppose Firm 1 sets a very high price: greater than the monopoly price of p M = (a +c)/2b 30
What price should firm 2 With p 1 > (a + c)/2b, Firm 2 s profit set? looks like this: Firm 2 will only earn a positive profit by cutting its So firm Firm 2 2 s Profit The monopoly price should to (a just + c)/2b or less undercut p price 1 a bit and At p2 = p1 get almost all the firm 2 gets half of the p 2 < monopoly profit monopoly profit What if firm p 1 1 prices at (a + c)/2b? p 2 = p 1 p 2 > p 1 c (a+c)/2 b p 1 Firm 2 s Price 31
Now suppose Firm 1 sets a price less than (a + c)/2b Firm 2 s profit looks like this: Firm 2 s Profit As What long price as p 1 > c, Of course, firm 1 Firm should 2 should firm aim 2 just will then undercut to undercut set now? firm 1 Then firm 2 and should so pon also 2 < price at c. Cutting price pbelow 1 cost gains the whole market but loses What money if firm on every 1 customer prices at c? p 2 = p 1 c p 1 (a+c)/2 b p 2 > p 1 Firm 2 s Price 32
We now have Firm 2 s best response to any Firm 1 price: p* 2 = (a + c)/2b if p 1 > (a + c)/2b p* 2 = p 1 - something small if c < p 1 < (a + c)/2b p* 2 = c if p 1 < c We have a symmetric best response for Firm 1 too: p* 1 = (a + c)/2b if p 2 > (a + c)/2b p* 1 = p 2 - something small if c < p 2 < (a + c)/2b p* 1 = c if p 2 < c 33
These best response function for functions look like this: p 2 The best response firm 1 R 1 R 2 The best response function for firm 2 (a + c)/2b c The Bertrand The equilibrium equilibrium has is both with firms both charging firms pricing marginal at cost c c (a + c)/2b p 1 34
The Bertrand model makes clear that competing on price is very different from competition in quantities. COURNOT: P monopoly > P cournot > P competitive BERTRAND: P monopoly > P bertrand = P competitive Since many firms seem to set prices (not quantities), this is a challenge to the Cournot approach. But is this positive outcome likely? We will now consider two extensions of the Bertrand model that will provide a more nuanced approach to oligopolies: the impact of capacity constraints; and product differentiation 35
p = c equilibrium requires both firms have sufficient capacity to meet all demand at p = c However, if there is insufficient capacity, capacity constraints may affect the equilibrium. Consider an example: daily demand for skiing is Q = 6,000 60P Q is number of lift tickets; P is the ticket price. Two resorts: Pepall and Richards have fixed daily capacities: Pepall = 1,000 and Richards =1,400. marginal cost of lift services for both is $10. 36
Is a price P = c = $10 a Nash equilibrium? total demand is 5,400 > 2,400 capacity. Assume Firm 1 sets its price at c. From Firm 2 s perspective: At p = c there is insufficient capacity to serve the entire market. Why not set p 2 > c? Firm 2 loses some customers. But it retains others, from whom it now earns a profit. Therefore, P = c = $10 is not a Nash equilibrium. 37
Assume that at any price where demand at a resort is greater than capacity there is efficient rationing. serves skiers with the highest willingness to pay Then we can derive residual demand: If P = $60, total demand = 2,400 = total capacity. Pepall gets 1,000 skiers. Residual demand to Richards with efficient rationing is Q = 5000 60P or P = 83.33 Q/60 in inverse form. marginal revenue is then MR = 83.33 Q/30 38
Residual demand and MR: Suppose that Richards sets P = $60. Does it want to change? since MR > MC Richards does not want to raise price and lose skiers since Q R = 1,400 Richards is at capacity and does not want to reduce price $83.33 $60 $36.66 Price MR Demand $10 MC 1,400 Same logic applies to Pepall so P = $60 is a Nash equilibrium for this game. Quantity 39
The logic of this example can be generalized: firms are unlikely to expand capacity to serve the whole market when p = mc since they get only a fraction in equilibrium; capacity of each firm is less than needed to serve the whole market; therefore there is no incentive to cut price to marginal cost. Efficiency property of Bertrand equilibrium breaks down when firms are capacity constrained. 40
We now turn to the Bertrand Model when product differentiation is present. The original analysis assumes that firms offer homogeneous products. This creates incentives for firms to differentiate their products to... generate consumer loyalty; and not lose all demand when they price above rivals. 41
We can examine product differentiation and Bertrand competition using the Hotelling spatial model, with the following assumptions: a Main Street where consumers are evenly distributed; supplied by two competing shops located at opposite ends of the street; where each consumer buys exactly one unit of the good provided its full price is less than V, from the shop offering the lower full price; and where consumers incur transport costs of t per unit of distance in traveled to a shop. 42
Recall the broader interpretation: metaphorical, representing why products are differentiated. What prices will the two shops charge? 43
Price p 1 p 1 Industrial Organization x m marks the location of What Assume if shop that 1 the shop raises marginal 1 sets buyer one Price price its p 1 price? and shop 2 who setsis indifferent price p 2 between buying either firm s good p 2 Shop 1 x x m All consumers m to x the And all consumers left of x m buy from m moves to the left: some consumers to the right buy Shop from shop 1 switch to shop shop 2 2 2 44
p 1 + tx m = p 2 + t(1 - x m ) 2tx m = p 2 - p 1 + t How is x x m (p m 1, p 2 ) = (p 2 - p 1 + t)/2t This determined? is the fraction There are N consumers in total of consumers who So demand to firm 1 is D 1 = N(p 2 - pbuy 1 + t)/2t from firm 1 Price Price p 2 p 1 x m Shop 1 Shop 2 45
Profit to firm 1 is p 1 = (p 1 - c)d 1 = N(p 1 - c)(p 2 - p 1 + t)/2t p 1 = N(p 2 p 1 - p 12 + tp 1 + cp 1 - This cp 2 -ct)/2t is the best response function Solve this Differentiate with respect to p 1 for firm 1for p 1 p 1 / p 1 = N 2t (p 2-2p 1 + t + c) = 0 p* 1 = (p 2 + t + c)/2 This is the best What about firm 2? response By symmetry, function it has a similar best response function. for firm 2 p* 2 = (p 1 + t + c)/2 46
p* 1 = (p 2 + t + c)/2 p* 2 = (p 1 + t + c)/2 p 2 R 1 2p* 2 = p 1 + t + c = p 2 /2 + 3(t + c)/2 c + t p* 2 = t + c (c + t)/2 p* 1 = t + c Profit per unit to each firm is t (c + t)/2 Aggregate profit to each firm is Nt/2 c + t R 2 p 1 47
Two final points on this analysis. 1. t measures transport costs. It is also a measure of the value consumers place on getting their most preferred variety. when t is large competition is softened and profit is increased when t is small competition is tougher and profit is decreased 2. Locations have been taken as fixed. if product design can be set by the firms, firms must balance temptation to be close (business stealing) against desire to be separate (softer competition). 48
Industrial Organizaton Best response functions are very different with Cournot and Bertrand they have opposite slopes, reflecting different forms of competition firms react differently e.g. to an increase in costs q 2 p 2 Firm 1 Firm 2 Firm 1 Firm 2 Cournot q 1 Bertrand p 1 49
Suppose Firm 2 s costs increase. This causes Firm 2 s Cournot best response function to fall. at any output for firm 1, firm 2 now wants to produce less Firm 1 s output increases and Firm 2 s falls Firm 2 s Bertrand best response function rises. at any price for firm 1, firm 2 now wants to raise its price Firm 1 s price increases as does Firm 2 s. q 2 p 2 Firm 1 passive response by firm 1 aggressive response by firm 1 Firm 2 Firm 1 Firm 2 q 1 p 1 Cournot Bertrand 50
When best response functions are upward sloping (e.g. Bertrand), we have strategic complements. passive action induces passive response When best response functions are downward sloping (e.g. Cournot), we have strategic substitutes. passive actions induces aggressive response Difficult to determine strategic choice variable: price or quantity output in advance of sale probably quantity production schedules easily changed and intense competition for customers probably price 51
Read Chapter 11. 52