1. Horizontal differentiation

Locations

i. To find the demand for each firm, we need to find the fraction of consumers buying from each firm.  So, we need to work out the location of the consumer who is indifferent between each firm’s product. Call the location of this consumer x. It must therefore be that

u −t(x−a)2 − p1 = u −t(x−b)2 − p2 p1 +t(x−a)2 = t(x−b)2 + p2

t(x2 −2ax+a2 −x2 +2b−b2 ) = p2 − p1

2tx(b−a) = p2− p1 +t(b2 −a2 )

x =  +

= 2t(b−a) +    2

b − a      p2 − p1 

2       2t(b−a)

Hence, the demand for each firm is given by

q1 = x = a+  + 

q2 = 1 − x = 1 −b+  + 

Consider Firm 1’s demand.  Effectively, Firm 1 has a consumers to the left of her who prefer her product in the absence of price considerations.  Half of the consumers between Firm 1 and Firm 2 also prefer Firm 1’s product. The third term describes the effect of relative price. Notice that relative prices are less important if travel costs are higher and if the firms are located further apart.

ii. To find the reaction function for each firm, we need to first work out firm profits, then solve for the profit maximising problem for each firm. First, profits for Firm

1 are given by

π 1    =   q1 (p1−c)

=   (p1−c) (a+  + )

The first order conditions for firm 1’s profit maximising problem are then

b − a      p2 − p1        p1 − c  

p1              a+b        p2 + c  

t(b−a) =    2    + 2t(b−a)

p1 = t(b−a) + 

=  +t ≡ R1(p2 )

This is Firm 1’s reaction function.

Similarly, for Firm 2

p2 =  +t ≡ R2(p1 )

The term b2 − a2 measures the squared distance between Firm 1 and Firm 2. The distance between the firms thus plays a key strategic role in price setting.  The closer firms are together, the more intensely they compete on price. The effect of distance is amplified by travel costs. The greater consumers dislike traveling (i.e. higher t), the greater the impact of firm distance on pricing.

(b) Recall from class that firms face a trade off when deciding where to locate. The closer they locate to their rival, the more customers they can‘steal’from their competitor. However, the closer the firms are, the greater the intensity of price competition. In this question, there is no price competition, so we might expect the first effect to dominate, and the firms to locate near each other.

Let’s argue this a bit more carefully. First, notice that if there is no price competition, all consumers will just buy from the firm that is closer to them. In a Nash equilibrium, neither firm will have an incentive to change their location given the location of their rival.

Suppose that Firm 1 locates on the right half of the city (ie 0.5 < x1 ≤ 1).  Firm 2’s optimal location would then be just a tiny bit to the left of Firm 1. This way, Firm 2 will capture all the consumers to the left of Firm 1. However, if Firm 2 locates just to the left of Firm 1, Firm 1’s optimal location is then just to the left of Firm 2. Consequently, we cannot have a Nash equilibrium in which Firm 1 locates on the right half of the city. By the same reasoning, Firm 1 cannot locate on the left half of the city in a Nash equilibrium (Firm 2 would then wish to locate just to the right, and Firm 1 would then wish to change it’s location to just right of Firm 2).

The same arguments hold for Firm 2. There can be no Nash equilibrium in which Firm 2 locates either in the right side of the city, or the left side.

Now, if Firm 1 locates exactly in the centre, Firm 2’s optimal location is also in the centre. This way, they get half of the market, rather than slightly less than half if they locate either side. So, both firms have no incentive to move if they are located in the centre of the city. Hence, the unique Nash equilibrium to this game involves both firms locating in the centre of the city.

2. Collusion and obfuscation

(a) If Firm 1 plays grim-trigger, she receives payoffs of

VC = 7(1 +δ+δ2 + ... ) =

Firm 1’s most profitable deviation is to marginally undercut pm . This yields payoffs VD ≈ 14 +0(δ+δ2 + ... )

Collusion is sustainable if

VC ≥ VD   ⇐⇒  ≥ 14

⇐⇒ 2(1 −δ) ≤ 1

⇐⇒ δ ≥ 1/2

That is, collusion is sustainable if firms are sufficiently patient.  Note that this is the same result as the repeated Bertrand model we considered in class.

(b) If Firm 1 plays grim-trigger, in odd periods she receives continuation value

Vod(C)d =

Firm 1’s most profitable deviation is to marginally undercut pm = 14. This yields value Vod(D)d ≈ 14α = 8

There is no incentive to deviate in odd periods if

Vod(C)d  ≥ Vod(D)d   ⇐⇒  ≥ 8

⇐⇒ 7 +2δ ≥ 8 −8δ2

⇐⇒ 8δ2 +2δ − 1 ≥ 0

⇐⇒ δ ≥ 1/4