there are 51 market participants: 25 of them are producers, where producer i can produce at most one unit of a good at cost i $; 25 of them are consumers, where consumer j is willing to buy at most 1 unit of the good for at most j $; the 51st market participant is a market maker, or an intermediary who runs the market.

(a) What are Pareto-ecient outcomes of this allocation problem?

(b) Suppose all trade must be conducted through the market maker, and he can set two dierent prices: a bid price (at which he passes goods on to consumers), and an ask price (a price at which he acquires the goods) from the producers. Assume also that at these prices he must deliver all goods that consumers want to buy, and that he must purchase all goods that producers want to sell. What bid and ask price would he want to set? Would that be Pareto-ecient? What about if he were only allowed to set one price, what price would he then set then, and would that be Pareto-ecient?

(c) Suppose now that there are again many periods, all market participants discount the future at  < 1. In each period a producer can either choose to trade through the intermediary or if the producer chooses otherwise, she is randomly matched with one of the consumers who is not trading through the intermediary (chosen with equal likelihood among all those not trading through the intermediary). The intermediary sets prices after he sees which producers and consumers are trading at his desk. A producer and consumer who are randomly matched split the surplus equally, if there is any, and they do not trade if there is no surplus. All market participants who have traded exit the market, the others proceed to the next period. Describe what happens in a SPNE of this market game.