A German manufacturer, Bosch, supplies electric motors to Tern, a Taiwanese e-bikes manufacturer. Bosch can supply high-quality motors, or it can cut corners and make low-quality ones. Tern must decide whether to buy 10,000 or 20,000 motors from Bosch's current production run. All motors in a batch are of the same quality. Tern cannot tell the quality of the motors when it decides how many to buy, but it does learn the quality once the shipment arrives and is opened. (The nonrefundable payment must be made before the motors are shipped.)

If Tern buys 20,000 units, its profits are $30 million if quality is high, and zero if quality is low. When it buys 10,000 units, its profits are $20 million if quality is high, and $10 million if quality is low. If Bosch sells 20,000 units, then it earns a profit of $40 million if it makes low-quality motors, but $15 million if it supplies high-quality ones. If Bosch sells 10,000 units, its profits are $10 million if it makes low-quality motors, but zero if it makes high-quality ones.

(a) Suppose the two companies interact only once, and they make their decisions simultaneously (i.e., Bosch decides on quality before knowing how large an order it will receive, and Tern must decide how many units to order before learning their quality). Describe the game in matrix form and find the Nash equilibrium.

(b) What outcome is collectively preferred to the above-described equilibrium outcome? Explain why this better outcome cannot be achieved in a one-shot simultaneous move game. In particular, who has an incentive to deviate from that outcome?

Now suppose that Bosch and Tern are in a long-run ongoing business relationship. We can model such an ongoing relationship as a repeated game in which the two firms play the one-shot game of part (a) in every period. We aim to study the possibility of a cooperative equilibrium in which Bosch makes high- quality motors in every period, and Tern buys 20,000 units in every period.

(c) Describe clearly (in words) possible strategies of each player that could sustain such a cooperative arrangement. (In particular, specify how the two companies should agree to play the game, and what each would do in case anyone cheats.)

Suppose that there is a production run every month, and that the monthly discount rate is .

(d) Find the conditions on under which Bosch wants to sustain the cooperative agreement. Interpret the economic intuition of these conditions.

(e) Find the conditions on under which Tern wants to sustain the cooperative agreement. Interpret the economic intuition of these conditions.