1. We can solve this game by backward induction. First, consider firm 2’s problem. If, in stage 1 firm 1 set a price above marginal cost (ie p1 > mc), firm 2 would set a price just below (ie p2  < p1).  Alternatively, if p1  < mc, firm 2 will want to set a price p2  > p1 .  Finally, if p1 = mc, firm 2 is indifferent between setting p2 = mc and some p2  > mc.  The above description will constitute firm 2’s strategy in equilibrium.

Firm 1 will take firm 2’s reaction into account when setting p1 . As in the Bertrand model, there is a (subgame perfect) Nash equilibrium involving p1 = p2 = mc. Note that there are also other equilibria in which firm 1 sets p1 > mc and firm 2 adopts the above strategy. Why can this be a (subgame perfect) Nash equilibrium? Does the leader or the follower have an advantage in this game?

2. Asymmetric capacity constraints.

(a) To find firm 1’s reaction function, first note that, because firm 2 is not capacity con-

strained, if firm 1 relents (raises price), then firm 2 will get the whole market. Hence, firm 1 earns no profits by relenting.  If firm 2’s price is above marginal cost, firm 1 will therefore always undercut.  Hence, we can describe firm 1’s reaction function as follows:

i. if p2 > pM , set p1 = pM (where pM is the monopoly price);

ii. if MC < p2 < pM , undercut (ie set p1 = p2 _ 6);

iii. if p2 < MC, set a price p1 > p2 .

Note that there is a slight complication for Firm 1. Because of Firm 1’s capacity con- straint, they are unable to satisfy demand at the unconstrained monopoly price of 0.65. Instead, the monopoly price for Firm 1 will be 0.66. At this price, they use their entire capacity.

(b) The intuition for firm 2’s reaction function is the same as for the symmetric Edgeworth model (but the details are different). If p1 is high, firm 2 would like to undercut. If p1 is low, firm 2 would like to relent and act as a monopolist on residual demand. We can work out the cut-off price at which firm 2 switches in the same way as in class. If firm

2 undercuts by setting a price slightly lower than p1 , profits are given by eU (p1 ) ≈ (1000 _ 1000p1 )(p1 _ 0.3).

[Note that this step is different to the symmetric case because firm 2 has no capacity constraints.] Alternatively, if firm 2 relents, profits are given by

eR (p2 ) = (660 _ 1000p2 )(p2 _ 0.3),

To calculate relenting profits, we must first solve for the optimal relenting price. This is the solution to the problem maxpeR (p). Find the first order conditions in the usual way:

660 _ 1000p _ 1000(p _ 0.3) = 0

2000p = 960

= 0.48

The optimal relenting price is therefore pR  = 0.48.  Substituting back into the profit function, we obtain eR = 32.4.

To work out the point at which firm 2 switches between undercutting and relenting, equate eU (p1 ) and eR . This yields

(1000 _ 1000p1 )(p1 _ 0.3)   =   32.4

_1000p1(2) +1300p1 _ 300   =   32.4

p1(2) _ 1.3p1 +0.3324   =   0

This is a quadratic equation. The solution is p 1(*) ≈ 0.35. (The largest root is above the

monopoly price, so we can ignore this one).

Hence, firm 2’s reaction function is:

i. if p1 > pM , set p2 = pM ;

ii. if p1(*) < p1 < pM , undercut (ie set p2 = p1 _ 6);

iii. if p1 < p 1(*), set a price p2 = pR = 0.48.

(c) Figure 1 contains the reaction functions.  You will notice that there is again no (pure strategy) Nash equilibrium.  Unlike the symmetric case, firm 2 is the only firm that relents as part of their reaction function.

3. Cartel problem

(a) Firm i solves the problem maxqi e = qi(140 _ 0. 1qi _ 0. 1qj _ 20).

This yields FOCS: 140 _ 0.2qi _ 0. 1qj _ 20 = 0 = qi = 600 _ 0.5qj . This is Firm i’s reaction function.

Given the symmetry of the problem, we will look for a symmetric Nash equilibrium, so let qi = qj = qN . Hence, in the Nash equilibrium, qN = 400, and the market price is p = 60. Each firm earns profits of eN = 400(60 _ 20) = 16, 000.

(b) Because the firms have constant marginal costs, to calculate the cartel’s optimal out- put, we can simply calculate the monopoly output. A monopolist solves the problem

maxQ e = Q(140 _ 0. 1Q _ 20).

This yields FOCS: 140 _ 0.2Q _ 20 = 0 = Q = 600.  For each firm, the cartel out- put is then qC  = 300, and the market price is p = 80.   Each firm earns profits of eC = 300(80 _ 20) = 18, 000.

(c) We have already calculated the reaction function for each firm in part (3a). The defec- tion or cheating output is then qD  = 600 _ 0.5qC  = 450. If Firm 1 were to cheat and produce 450, while firm 2 produced the cartel output of 300, the market price would be p = 65, and the profits to Firm 1 would be eD = 450(65 _ 20) = 20, 250.

(d) If (say) Firm 1 cooperates, they obtain a payoff of VC = eC (1 +8 +82 + . . .).

If Firm 1 defects from the cartel (cheats), they would produce qD in the current period, and obtain payoffs of VD = eD +eN (8 +82 +83 + . . .).

The cartel is then sustainable if  > eD +  .

Hence, eC  > (1 _ 8)eD + 8eN = 8 >  = 8* . So, the cartel is sustainable if 8 > 8* =  =  .

(e) If (say) Firm 1 cooperates, they obtain a payoff of VC = eC (1 +8 +82 + . . .).

If Firm 1 defects from the cartel (cheats), they would produce qD in the current period, and obtain payoffs of VD = eD +8eN +eC (82 +83 + . . .).

The cartel is then sustainable if eC (1 +8) > eD +8eN .