EconS Sequential Competition

Transcription

1 EconS Sequential Competition Eric Dunaway Washington State University eric.dunaway@wsu.edu Industrial Organization Eric Dunaway (WSU) EconS 425 Industrial Organization 1 / 47

2 A Warmup 1 x i x j (x i ) x i = x j x i (x j ) 0 x k m x l 1 x j Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 47

3 A Warmup Eric Dunaway (WSU) EconS 425 Industrial Organization 3 / 47

4 Introduction Today, we ll talk about a few di erent types of sequential competition. Stackelberg Competition A revisit to Horizontal Di erentiation The Leigh Lecture is being held on March 23rd in CUE 203 at 7:30 PM. Edward Prescott (Nobel Laureate) is coming to discuss the current state of the U.S. economy. Eric Dunaway (WSU) EconS 425 Industrial Organization 4 / 47

5 Stackelberg Competition In 1934, German economist Heinrich Freiherr von Stackelberg re ected on the Cournot model and added his own contribution to it in his work Marktform und Gleichwicht (Translation: Market Structure and Equilibrium). He had noticed that while many rms competed in the duopoly (or oligopoly) context, often one rm would set their output level before the other rms, who would observe the "leader s" output before making their own decisions. The leading rm tended to be the largest rm in the market, and they usually earned the largest pro ts, as well. Stackelberg altered the Cournot model of competition to allow for one rm to move before the others. Eric Dunaway (WSU) EconS 425 Industrial Organization 5 / 47

6 Stackelberg Competition Stackelberg competition is fairly common in the real world. Gerber acts as a Stackelberg leader in the baby food market. Campbell acts as a Stackelberg leader in the US canned soup market. All of the major US car companies have been a Stackelberg leader at one point. Basically, we re moving from a simultaneous game to a sequential one. In our model, we ll assume that rm 1 is the Stackelberg leader and gets to set their output level rst. Eric Dunaway (WSU) EconS 425 Industrial Organization 6 / 47

7 Stackelberg Competition Remember that when we are dealing with a sequential game, we need to use backward induction. We start with the player that moves last, then work our way to the top of the game tree. In this case, we ll start with rm 2 s best response to rm 1 s output level. Then we ll determine rm 1 s optimal output level, given that they know how rm 2 is going to react. Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 47

8 Stackelberg Competition For rm 2, we ll nd that nothing has changed. With an inverse demand function of p(q) where Q = q 1 + q 2 and a constant marginal cost of c, their pro t maximization problem is max q 2 p(q)q 2 cq 2 with rst-order condition with respect to their quantity of π 2 q 2 = p(q) Q Q q 2 q 2 + p(q) c = 0 Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 47

9 Stackelberg Competition p(q) Q Q q 2 q 2 + p(q) c = 0 This is the exact same rst-order condition that we calculated in the Cournot case. Remember that nothing at all has changed for rm 2. They are still responding to a strategy from rm 1, just like before. From this rst-order condition, we can derive a best response function for rm 2 as a function of rm 1 s output level, q 2 (q 1 ). Recall that as rm 1 increases their output, rm 2 will reduce their own, i.e., q 2(q 1 ) q 1 < 0. Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 47

10 Stackelberg Competition Moving up the game tree to rm 1, their pro t maximization problem is max p(q)q 1 cq 1 = max p(q 1 + q 2 )q 1 cq 1 q 1 q 1 but before taking a rst-order condition, recall that rm 1 wants to take rm 2 s reaction into consideration when it makes its output decision. It knows how rm 2 is going to react to its own choice. Thus, we can substitute rm 2 s best response function q 2 (q 1 ) into rm 1 s pro t maximization problem, max q 1 p(q 1 + q 2 (q 1 ))q 1 cq 1 Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 47

11 Stackelberg Competition max q 1 p(q 1 + q 2 (q 1 ))q 1 cq 1 Taking a rst-order condition with respect to rm 1 s quantity, π 1 = p(q) q1 + q 2(q 1 ) q 1 + p(q) c = 0 q 1 Q q 1 q 1 We have a new term within rm 1 s marginal revenue, q 2(q 1 ) q 1, which will alter rm 1 s optimal decision from Cournot competition. Since q 2(q 1 ) q 1 < 0, if rm 1 chose the Cournot output level, marginal revenue would actually be more than marginal cost. This implies that rm 1 will set an output level higher than they would under Cournot competition. Eric Dunaway (WSU) EconS 425 Industrial Organization 11 / 47

12 Stackelberg Competition Let s look at a practical example. Suppose the market inverse demand function is p = a bq = a b(q 1 + q 2 ) and marginal costs remain constant at c. Using backward induction, rm 2 s pro t maximization problem is, max q 2 (a b(q 1 + q 2 )) q 2 cq 2 with rst-order condition, π 2 = a q 2 bq 1 2bq 2 c = 0 Solving this expression for q 2 gives us rm 2 s best response to any output level of rm 1, q 2 (q 1 ) = a c q 1 2b 2 which is identical to what was seen in Cournot competition. Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 47

13 Stackelberg Competition Moving up the game tree to rm 1, their pro t maximization problem is, max q 1 (a b(q 1 + q 2 )) q 1 cq 1 factoring this expression and substituting in rm 2 s best response function gives us a c q 1 max a c bq 1 b q 1 q 1 2b 2 a c bq 1 = max q 1 q with rst-order condition, π 1 = a c q 1 2 bq 1 = 0 Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 47

14 Stackelberg Competition a c 2 bq 1 = 0 Solving this expression for q 1 gives us rm 1 s optimal output level, q1 = a c 2b which is exactly the monopoly level. Plugging this value into rm 2 s best response function gives us q2 = a c 2b q1 2 = a c 2b a c 4b = a c 4b and rm 2 produces only half of rm 1 s quantity. Moving second appears to be a huge disadvantage. Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 47

15 Stackelberg Competition Next, we calculate the market price, a c p = a b(q1 + q2 ) = a b 2b + a c 4b and lastly, our equilibrium pro ts, = a + 3c 4 a + 3c π1 = (p c)q1 = 4 a + 3c π2 = (p c)q2 = 4 a c a c c 2b c 4b = = (a c)2 8b (a c)2 16b As we can see, rm 1 receives the same pro t level that they would under a collusive agreement, while rm 2 receives half of that. Eric Dunaway (WSU) EconS 425 Industrial Organization 15 / 47

16 Stackelberg Competition Clearly, we have a case of rst mover advantage in this model. Firm 1 is able to leverage what they know about rm 2 s best response to claim a large share of the market. These theoretical results are very consistent with what is seen in reality. A rm that enters a market before its rivals is typically able to secure a large share of that market, and is able to continue market dominance Eric Dunaway (WSU) EconS 425 Industrial Organization 16 / 47

17 Stackelberg Competition Why doesn t the second mover just produce the Cournot output? The second mover could threaten that no matter what the Stackelberg leader chooses as their output level, they will produce the Cournot level. This would mean that the optimal output level for the leader would also be the Cournot level. Since this is a Nash equilibrium of our model, it is a possible solution outcome. Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 47

18 Stackelberg Competition Cournot Follower Stackelberg Leader Cournot Stackelberg 9b 12b,, 9b 18b 9b 8b,, 9b 16b Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 47

19 Stackelberg Competition Cournot Follower Stackelberg Leader Cournot Stackelberg 9b 12b,, 9b 18b 9b 8b,, 9b 16b Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 47

20 Stackelberg Competition The problem with doing this is that it is not a credible threat. If the leader chooses the Stackelberg output level in the rst stage, it would be irrational for the second mover to pick the Cournot level. This is just like the Battle of the Sexes game, since the leader gets to move rst, they can choose the Nash equilibrium that gives them the best payo. In this case, while both the Stackelberg output level and the Cournot output level are Nash equilibria of our model, only the Stackelberg output level is a subgame perfect Nash equilibrium. Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 47

21 Stackelberg Competition Player 1 Cournot Stackelberg 9b 9b Cournot Player 2 Stackelberg 12b 18b 8b 16b Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 47

22 Stackelberg Competition Player 1 Cournot Stackelberg 9b 9b Cournot Player 2 Stackelberg 12b 18b 8b 16b Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 47

23 Stackelberg Competition Regarding welfare, price is still above marginal cost, so deadweight loss will still exist in our model. However, the price in this case is actually lower than the Cournot price, so there is less of a distortion to the market. Again, to calculate all of these values, simply use the triangle formulas that we have seen in the past. Eric Dunaway (WSU) EconS 425 Industrial Organization 23 / 47

24 Stackelberg Competition p a + c 2 a + 2c 3 Monopoly Cournot a + 3c 4 c Stackelberg Bertrand Q Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 47

25 Product Di erentiation Now that we have established a general framework for competition between rms, let s return to our models of product di erentiation that we looked at before and adapt them to the duopoly setting. This is much more interesting than the monopoly framework. Starting with horizontal di erentiation, suppose that consumer preferences were uniformly distributed from 0 to 1 along a Hotelling line. Each rm is able to select its location along the Hotelling line, ˆθ 1 and ˆθ 2 respectively, and sell their product to consumers. Consumers have unit demand, so they buy from exactly one rm, or not at all. We ll assume that the consumers value the good enough such that everyone buys from one of the rms. Eric Dunaway (WSU) EconS 425 Industrial Organization 25 / 47

26 Product Di erentiation 0 1 Eric Dunaway (WSU) EconS 425 Industrial Organization 26 / 47

27 Product Di erentiation θ 1 θ Eric Dunaway (WSU) EconS 425 Industrial Organization 27 / 47

28 Product Di erentiation To start things out, I am going to say that the rm locations, ˆθ 1 and ˆθ 2, are exogenous (we ll relax that later). The game proceeds as follows: In the rst stage, rms simultaneously choose prices for their product. In the second stage, consumers purchase from the rm that gives them the highest payo. Eric Dunaway (WSU) EconS 425 Industrial Organization 28 / 47

29 Product Di erentiation Recall that consumers have the following payo function (their surplus), K t(θ i, ˆθ) p where K is their inherent valuation of the good, which we assume is very large in this case to guarantee that every consumers buys from one of the rms; t(θ i, ˆθ) is the transportation cost that consumer i pays when consuming the product located at ˆθ, and p is the price of that good. We ll assume a linear transportation cost, giving us K t θ i ˆθ p Eric Dunaway (WSU) EconS 425 Industrial Organization 29 / 47

30 Product Di erentiation Using backward induction, we rst must determine the optimal strategy for each consumer. While it isn t really feasible to determine each consumer s strategy individually, we can at least gure out generally what consumers will do. For this to work, we are going to assume that ˆθ 1 < ˆθ 2, i.e., rm 1 s product is located to the left of rm 2 s product on the Hotelling line. There will be some consumer θ m that will be completely indi erent between purchasing from rm 1 or rm 2. Intuitively, this consumer should be between the location of the two products, ˆθ 1 < θ m < ˆθ 2. Since consumer m is indi erent between both rms, the payo that they receive from each rm will equal one another, K t(θ m ˆθ 1 ) p 1 = K t(ˆθ 2 θ m ) p 2 Eric Dunaway (WSU) EconS 425 Industrial Organization 30 / 47

31 Product Di erentiation K t(θ m ˆθ 1 ) p 1 = K t(ˆθ 2 θ m ) p 2 We can solve this expression for θ m to determine the exact location of the indi erent consumer, t(θ m ˆθ 1 ) t(ˆθ 2 θ m ) = p 2 p 1 t(2θ m (ˆθ 1 + ˆθ 2 )) = p 2 p 1 θ m = ˆθ 1 + ˆθ p 2 p 1 t Notice that the rst term in this expression is the midpoint between ˆθ 1 and ˆθ 2. From there, the indi erent consumer s location increases or decreases depending on whether p 2 or p 1 is larger. More importantly, though, we know that all consumers located to the left of θ m will purchase from rm 1, and all consumers located to the right of θ m will purchase from rm 2. Eric Dunaway (WSU) EconS 425 Industrial Organization 31 / 47

32 Product Di erentiation θ m θ 1 θ Eric Dunaway (WSU) EconS 425 Industrial Organization 32 / 47

33 Product Di erentiation θ m θ 1 θ 2 0 Firm 1 Firm 2 1 Eric Dunaway (WSU) EconS 425 Industrial Organization 33 / 47

34 Product Di erentiation This is enough information to move back to the rst stage of the game. In the rst stage, rms simultaneously choose their prices, so starting with rm 1 s maximization problem, max p 1 p 1 q 1 cq 1 To keep things easy, we ll assume c = 0. What about rm 1 s quantity, though? We can gure that out from the location of the indi erent consumer. Since everyone to the left of the indi erent consumer purchases from rm 1, we know that from 0 to θ m on the Hotelling line is the proportion of the market that rm 1 serves. Thus, q 1 = N Z θm 0 f (θ)dθ where N is the total number of consumers and f (θ) is the distribution of those consumers along the Hotelling line. Eric Dunaway (WSU) EconS 425 Industrial Organization 34 / 47

35 Product Di erentiation Again, we ll keep things simple and assume that N = 1. Also, we speci ed that consumers are uniformly distributed along the Hotelling line, so f (θ) = 1. Thus, q 1 = Z θm 0 dθ = θ m = ˆθ 1 + ˆθ p 2 p 1 t which should make sense since that is simply rm 1 s market share. Substituting this back into rm 1 s pro t maximization problem, " # ˆθ 1 + ˆθ 2 max p 1 + p 2 p 1 p 1 2 t and taking a rst-order condition with respect to rm 1 s price gives us, π 1 = ˆθ 1 + ˆθ 2 + p 2 p 1 p 1 = 0 p 1 2 t t Eric Dunaway (WSU) EconS 425 Industrial Organization 35 / 47

36 Product Di erentiation ˆθ 1 + ˆθ p 2 p 1 t p 1 t = 0 From here, we solve this expression for p 1 to determine rm 1 s best response to any price chosen by rm 2,! p 1 (p 2 ) = t ˆθ 1 + ˆθ 2 + p Analyzing this expression, rm 1 s optimal price increases as rm 2 s price increases (which makes sense since they would be substitutes). It also increases as the transportation cost, t, increases. This also makes sense. As consumers incur a higher cost of consuming a product that is less like their ideal, rms know they are less likely to switch from one product to another. This reduces competition and thus, increases prices for both rms. Eric Dunaway (WSU) EconS 425 Industrial Organization 36 / 47

37 Product Di erentiation Now let s look at rm 2 s pro t maximization problem, max p 2 p 2 q 2 cq 2 where rm 2 s quantity is all of the consumers above the indi erent consumer, θ m, Z 1 q 2 = N f (θ)dθ θ m Imposing our assumptions of N = 1 and the uniform distribution, we obtain Z 1 ˆθ 1 + ˆθ 2 p 2 p 1 q 2 = dθ = 1 θ m = 1 θ m 2 t and substituting this back into the pro t maximization problem, we have, " # ˆθ 1 + ˆθ 2 p 2 p 1 max p 2 1 p 2 2 t Eric Dunaway (WSU) EconS 425 Industrial Organization 37 / 47

38 Product Di erentiation max p 2 p 2 " 1 ˆθ 1 + ˆθ 2 2 p 2 p 1 t # Taking a rst-order condition with respect to rm 2 s price gives us, π 2 p 2 = 1 ˆθ 1 + ˆθ 2 2 p 2 p 1 t p 2 t = 0 and solving this expression for p 2 gives us rm 2 s best response to any price chosen by rm 1,! p 2 (p 1 ) = t ˆθ 1 + ˆθ p Notice that this best response function is not symmetric to rm 1 s, which should make sense since their products are di erent. Eric Dunaway (WSU) EconS 425 Industrial Organization 38 / 47

39 Product Di erentiation p 1 = t 2 p 2 = t 2 ˆθ 1 + ˆθ 2 2 1! ˆθ 1 + ˆθ p 2 2! + p 1 2 To nd our equilibrium, we simply solve this system of two equations and two unknowns, giving us p 1 = t 6 (2 + ˆθ 1 + ˆθ 2 ) p 2 = t 6 (4 (ˆθ 1 + ˆθ 2 )) Eric Dunaway (WSU) EconS 425 Industrial Organization 39 / 47

40 Product Di erentiation p 1 = t 6 (2 + ˆθ 1 + ˆθ 2 ) p 2 = t 6 (4 (ˆθ 1 + ˆθ 2 )) Like we noticed in the best response function, as the transportation cost, t, increases, both rms are able to charge a higher price to consumers since they are less willing to consume a produce di erent from their ideal. Likewise, as either ˆθ 1 or ˆθ 2 increase, rm 1 receives more favorable market conditions as more consumers are naturally closer to their product. This allows them to raise their price while rm 2 must lower their price. Eric Dunaway (WSU) EconS 425 Industrial Organization 40 / 47

41 Product Di erentiation Originally, we just let the value of ˆθ 1 and ˆθ 2 be taken as given. What if we had the rms choose their location, as well? If we did this, the rst stage of the game would now have the rms simultaneously choose values for ˆθ 1 and ˆθ 2 that maximize their pro ts. Choosing prices would now be stage 2, and consumers purchasing would now be stage 3. The calculus is complicated, and relies heavily on the assumption we made that ˆθ 1 < ˆθ 2. Basically, rm 1 is always going to want to move closer to rm 2 to claim market share from rm 2, and rm 2 is going to want to do the same to rm 1. This is the classic conclusion of the Hotelling model, where both rms will choose ˆθ 1 = ˆθ 2 = 0.5 and position at the middle of the market. Do you see a problem with this arrangement? Eric Dunaway (WSU) EconS 425 Industrial Organization 41 / 47

42 Product Di erentiation If both rms position at the middle of the market, prices are p 1 = t 6 ( ) = t 2 p 2 = t 6 (4 ( )) = t 2 which are both positive and above the marginal cost of 0. The products aren t di erentiated at all though. This is just Bertrand competition. In this situation, the rms could undercut one another and ght over the consumers, driving the price down to marginal cost. Eric Dunaway (WSU) EconS 425 Industrial Organization 42 / 47

43 Product Di erentiation d Aspremont, Gabszewicz and Thisse discovered this problem in 1979 in their review of Hotelling s work "On Hotelling s "Stability in Competition"." Basicaly, Hotelling s equilibrium predictions were incomplete since they would just collapse into Bertrand competition. They also proposed a solution to this problem: quadratic transportation costs. Linear transportation costs make sense in the spatial model, since physically moving from one location to another would have a linear relationship. For product di erentiation, quadratic costs would make more sense since we are talking about consumer preferences. People don t like consuming things they don t prefer. Eric Dunaway (WSU) EconS 425 Industrial Organization 43 / 47

44 Product Di erentiation Adding in a quadratic transportation cost changes the results completely. Firm 1 now positions at ˆθ 1 = 0 and rm 2 at ˆθ 2 = 1. Rather than meeting in the middle, rms now prefer to di erentiate their products as much as possible. Since the consumers really don t want to consume a product di erent from their ideal, rms take advantage of this by di erentiating completely. This allows them both to charge prices that are quite high. Eric Dunaway (WSU) EconS 425 Industrial Organization 44 / 47

45 Summary The Stackelberg model shows us what happens when rms compete sequentially. The rst mover has a very large advantage of the second mover. Eric Dunaway (WSU) EconS 425 Industrial Organization 45 / 47

46 Next Time Entry Deterrence. Reading: Midterm is a week from today! Eric Dunaway (WSU) EconS 425 Industrial Organization 46 / 47

47 Homework 4-3 Using the basic Stackelberg model of competition we covered, suppose now that each rm has di erent marginal costs, c 1 and c 2, respectively. Assume that c 2 < c 1 < a. 1. Find the equilibrium quantities produced by both rms (assume rm 1 is the Stackelberg leader). 2. Is it possible for rm 2 to produce a higher quantity than rm 1? If so, what condition must hold? Eric Dunaway (WSU) EconS 425 Industrial Organization 47 / 47