EconS 425 - Sequential

Eric Dunaway

Washington State University [email protected]

Industrial Organization

Eric Dunaway (WSU) EconS 425 1 / 47 A Warmup

xi xi = xj 1 xj (xi)

xi (xj)

xj 0 xk m xl 1

Eric Dunaway (WSU) EconS 425 Industrial Organization 2 / 47 A Warmup

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

Today, we’lltalk 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 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: and Equilibrium). He had noticed that while many …rms competed in the duopoly (or ) 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 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 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 Stackelberg Competition

Remember that when we are dealing with a , we need to use . We start with the player that moves last, then work our way to the top of the game tree. In this case, we’llstart with …rm 2’sbest response to …rm 1’soutput level. Then we’ll determine …rm 1’soptimal output level, given that they know how …rm 2 is going to react.

Eric Dunaway (WSU) EconS 425 Industrial Organization 7 / 47 Stackelberg Competition

For …rm 2, we’ll…nd that nothing has changed. With an inverse demand function of p(Q) where Q = q1 + q2 and a constant of c, their pro…t maximization problem is

max p(Q)q2 cq2 q2 with …rst-order condition with respect to their quantity of

∂π2 ∂p(Q) ∂Q = q2 + p(Q) c = 0 ∂q2 ∂Q ∂q2

Eric Dunaway (WSU) EconS 425 Industrial Organization 8 / 47 Stackelberg Competition

∂p(Q) ∂Q q2 + p(Q) c = 0 ∂Q ∂q2

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 from …rm 1, just like before. From this …rst-order condition, we can derive a function for …rm 2 as a function of …rm 1’soutput level, q2(q1). Recall that as …rm 1 increases their output, …rm 2 will reduce their own, i.e., ∂q2 (q1 ) < 0. ∂q1

Eric Dunaway (WSU) EconS 425 Industrial Organization 9 / 47 Stackelberg Competition

Moving up the game tree to …rm 1, their pro…t maximization problem is max p(Q)q1 cq1 = max p(q1 + q2)q1 cq1 q1 q1 but before taking a …rst-order condition, recall that …rm 1 wants to take …rm 2’sreaction 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’sbest response function q2(q1) into …rm 1’spro…t maximization problem,

max p(q1 + q2(q1))q1 cq1 q1

Eric Dunaway (WSU) EconS 425 Industrial Organization 10 / 47 Stackelberg Competition

max p(q1 + q2(q1))q1 cq1 q1 Taking a …rst-order condition with respect to …rm 1’squantity,

∂π1 ∂p(Q) ∂q1 ∂q2(q1) = + q1 + p(Q) c = 0 ∂q1 ∂Q ∂q1 ∂q1   We have a new term within …rm 1’smarginal revenue, ∂q2 (q1 ) , which ∂q1 will alter …rm 1’soptimal decision from . Since ∂q2 (q1 ) < 0, if …rm 1 chose the Cournot output level, marginal ∂q1 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 Stackelberg Competition

Let’slook at a practical example. Suppose the market inverse demand function is

p = a bQ = a b(q1 + q2) and marginal costs remain constant at c. Using backward induction, …rm 2’spro…t maximization problem is,

max (a b(q1 + q2)) q2 cq2 q2 with …rst-order condition,

∂π2 = a bq1 2bq2 c = 0 ∂q2 Solving this expression for q2 gives us …rm 2’sbest response to any output level of …rm 1, a c q1 q2(q1) = 2b 2 which is identical to what was seen in Cournot competition.

Eric Dunaway (WSU) EconS 425 Industrial Organization 12 / 47 Stackelberg Competition

Moving up the game tree to …rm 1, their pro…t maximization problem is, max (a b(q1 + q2)) q1 cq1 q1 factoring this expression and substituting in …rm 2’sbest response function gives us

a c q1 max a c bq1 b q1 q1 2b 2    a c bq1 = max q1 q1 2 2   with …rst-order condition,

∂π1 a c = bq1 = 0 ∂q1 2

Eric Dunaway (WSU) EconS 425 Industrial Organization 13 / 47 Stackelberg Competition

a c bq1 = 0 2

Solving this expression for q1 gives us …rm 1’soptimal output level, a c q = 1 2b which is exactly the level. Plugging this value into …rm 2’sbest response function gives us

a c q1 a c a c a c q = = = 2 2b 2 2b 4b 4b and …rm 2 produces only half of …rm 1’squantity. Moving second appears to be a huge disadvantage.

Eric Dunaway (WSU) EconS 425 Industrial Organization 14 / 47 Stackelberg Competition

Next, we calculate the market price,

a c a c a + 3c p = a b(q + q) = a b + = 1 2 2b 4b 4   and lastly, our equilibrium pro…ts,

a + 3c a c (a c)2 π = (p c)q = c = 1 1 4 2b 8b     a + 3c a c (a c)2 π = (p c)q = c = 2 2 4 4b 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 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’sbest 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 Stackelberg Competition

Why doesn’tthe 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 of our model, it is a possible solution outcome.

Eric Dunaway (WSU) EconS 425 Industrial Organization 17 / 47 Stackelberg Competition

Follower Cournot Stackelberg

(a –c)2 (a –c)2 (a –c)2 (a –c)2 Cournot 9b , 9b 9b , 9b Leader 2 2 2 2 Stackelberg (a –c) (a –c) (a –c) (a –c) 12b , 18b 8b , 16b

Eric Dunaway (WSU) EconS 425 Industrial Organization 18 / 47 Stackelberg Competition

Follower Cournot Stackelberg

(a –c)2 (a –c)2 (a –c)2 (a –c)2 Cournot 9b , 9b 9b , 9b Leader 2 2 2 2 Stackelberg (a –c) (a –c) (a –c) (a –c) 12b , 18b 8b , 16b

Eric Dunaway (WSU) EconS 425 Industrial Organization 19 / 47 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 perfect Nash equilibrium.

Eric Dunaway (WSU) EconS 425 Industrial Organization 20 / 47 Stackelberg Competition

Player 1

Cournot Stackelberg

Player 2

(a –c)2 9b (a –c)2 Cournot Stackelberg 9b

(a –c)2 (a –c)2 12b 8b (a –c)2 (a –c)2 18b 16b

Eric Dunaway (WSU) EconS 425 Industrial Organization 21 / 47 Stackelberg Competition

Player 1

Cournot Stackelberg

Player 2

(a –c)2 9b (a –c)2 Cournot Stackelberg 9b

(a –c)2 (a –c)2 12b 8b (a –c)2 (a –c)2 18b 16b

Eric Dunaway (WSU) EconS 425 Industrial Organization 22 / 47 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 Stackelberg Competition

p

a + c Monopoly 2

a + 2c Cournot 3

a + 3c Stackelberg 4

Bertrand c Q

Eric Dunaway (WSU) EconS 425 Industrial Organization 24 / 47 Product Di¤erentiation

Now that we have established a general framework for competition between …rms, let’sreturn 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, θˆ1 and θˆ2 respectively, and sell their product to consumers. Consumers have unit demand, so they buy from exactly one …rm, or not at all. We’llassume 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 Product Di¤erentiation

0 1

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

1 2 0 1 θ θ

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

To start things out, I am going to say that the …rm locations, θˆ1 and θˆ2, are exogenous (we’llrelax 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 Product Di¤erentiation

Recall that consumers have the following payo¤ function (their surplus), K t(θi , θˆ) 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(θi , θˆ) is the transportation cost that consumer i pays when consuming the product located at θˆ, and p is the price of that good. We’ll assume a linear transportation cost, giving us

K t θi θˆ p

Eric Dunaway (WSU) EconS 425 Industrial Organization 29 / 47 Product Di¤erentiation

Using backward induction, we …rst must determine the optimal strategy for each consumer. While it isn’treally feasible to determine each consumer’sstrategy individually, we can at least …gure out generally what consumers will do.

For this to work, we are going to assume that θˆ1 < θˆ2, i.e., …rm 1’s product is located to the left of …rm 2’sproduct on the Hotelling line.

There will be some consumer θm 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, θˆ1 < θm < θˆ2. Since consumer m is indi¤erent between both …rms, the payo¤ that they receive from each …rm will equal one another,

K t(θm θˆ1) p1 = K t(θˆ2 θm ) p2

Eric Dunaway (WSU) EconS 425 Industrial Organization 30 / 47 Product Di¤erentiation

K t(θm θˆ1) p1 = K t(θˆ2 θm ) p2 We can solve this expression for θm to determine the exact location of the indi¤erent consumer,

t(θm θˆ1) t(θˆ2 θm ) = p2 p1 t(2θm (θˆ1 + θˆ2)) = p2 p1 θˆ1 + θˆ2 p2 p1 θm = + 2 t Notice that the …rst term in this expression is the midpoint between θˆ1 and θˆ2. From there, the indi¤erent consumer’slocation increases or decreases depending on whether p2 or p1 is larger. More importantly, though, we know that all consumers located to the left of θm will purchase from …rm 1, and all consumers located to the right of θm will purchase from …rm 2.

Eric Dunaway (WSU) EconS 425 Industrial Organization 31 / 47 Product Di¤erentiation

m

1 θ 2 0 1 θ θ

Eric Dunaway (WSU) EconS 425 Industrial Organization 32 / 47 Product Di¤erentiation

m

1 θ 2 0 Firm 1 Firm 2 1 θ θ

Eric Dunaway (WSU) EconS 425 Industrial Organization 33 / 47 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’smaximization problem,

max p1q1 cq1 p1 To keep things easy, we’llassume 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 θm on the Hotelling line is the proportion of the market that …rm 1 serves. Thus,

θm q1 = N f (θ)dθ Z0 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 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,

θ m θˆ1 + θˆ2 p2 p1 q1 = dθ = θm = + 2 t Z0 which should make sense since that is simply …rm 1’smarket share. Substituting this back into …rm 1’spro…t maximization problem,

θˆ1 + θˆ2 p2 p1 max p1 + p1 " 2 t # and taking a …rst-order condition with respect to …rm 1’sprice gives us, ∂π1 θˆ1 + θˆ2 p2 p1 p1 = + = 0 ∂p1 2 t t

Eric Dunaway (WSU) EconS 425 Industrial Organization 35 / 47 Product Di¤erentiation

θˆ1 + θˆ2 p2 p1 p1 + = 0 2 t t

From here, we solve this expression for p1 to determine …rm 1’sbest response to any price chosen by …rm 2,

t θˆ1 + θˆ2 p2 p1(p2) = + 2 2 ! 2 Analyzing this expression, …rm 1’soptimal 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 Product Di¤erentiation

Now let’slook at …rm 2’spro…t maximization problem,

max p2q2 cq2 p2 where …rm 2’squantity is all of the consumers above the indi¤erent consumer, θm, 1 q2 = N f (θ)dθ Zθm Imposing our assumptions of N = 1 and the uniform distribution, we obtain 1 θˆ1 + θˆ2 p2 p1 q2 = dθ = 1 θm = 1 2 t Zθm and substituting this back into the pro…t maximization problem, we have, θˆ1 + θˆ2 p2 p1 max p2 1 p2 " 2 t #

Eric Dunaway (WSU) EconS 425 Industrial Organization 37 / 47 Product Di¤erentiation

θˆ1 + θˆ2 p2 p1 max p2 1 p2 " 2 t #

Taking a …rst-order condition with respect to …rm 2’sprice gives us,

∂π2 θˆ1 + θˆ2 p2 p1 p2 = 1 = 0 ∂p2 2 t t

and solving this expression for p2 gives us …rm 2’sbest response to any price chosen by …rm 1,

t θˆ1 + θˆ2 p1 p2(p1) = 1 + 2 2 ! 2 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 Product Di¤erentiation

t θˆ1 + θˆ2 p2 p1 = + 2 2 ! 2

t θˆ1 + θˆ2 p1 p2 = 1 + 2 2 ! 2

To …nd our equilibrium, we simply solve this system of two equations and two unknowns, giving us t p = (2 + θˆ + θˆ ) 1 6 1 2 t p = (4 (θˆ1 + θˆ2)) 2 6

Eric Dunaway (WSU) EconS 425 Industrial Organization 39 / 47 Product Di¤erentiation

t p = (2 + θˆ + θˆ ) 1 6 1 2 t p = (4 (θˆ1 + θˆ2)) 2 6

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 θˆ1 or θˆ2 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 Product Di¤erentiation

Originally, we just let the value of θˆ1 and θˆ2 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 θˆ1 and θˆ2 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 θˆ1 < θˆ2. 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 θˆ1 = θˆ2 = 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 Product Di¤erentiation

If both …rms position at the middle of the market, prices are t t p = (2 + 0.5 + 0.5) = 1 6 2 t t p = (4 (0.5 + 0.5)) = 2 6 2 which are both positive and above the marginal cost of 0. The products aren’tdi¤erentiated at all though. This is just . 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 Product Di¤erentiation

d’Aspremont, Gabszewicz and Thisse discovered this problem in 1979 in their review of Hotelling’swork "On Hotelling’s"Stability in Competition"." Basicaly, Hotelling’sequilibrium 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’tlike consuming things they don’tprefer.

Eric Dunaway (WSU) EconS 425 Industrial Organization 43 / 47 Product Di¤erentiation

Adding in a quadratic transportation cost changes the results completely. Firm 1 now positions at θˆ1 = 0 and …rm 2 at θˆ2 = 1. Rather than meeting in the middle, …rms now prefer to di¤erentiate their products as much as possible. Since the consumers really don’twant 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 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 Next Time

Entry Deterrence. Reading: 9.1-9.2. Midterm is a week from today!

Eric Dunaway (WSU) EconS 425 Industrial Organization 46 / 47 Homework 4-3

Using the basic Stackelberg model of competition we covered, suppose now that each …rm has di¤erent marginal costs, c1 and c2, respectively. Assume that c2 < c1 < 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