An economy I know has 1,000 identical consumers and two goods. The two goods are phiffle, a nonnarcotic stimulant, and manna, a basic foodstuff. Each consumer has a utility for phiffle and manna of the form u(x,z) = w(x) + z, where x is the amount of phiffle consumed and z the amount of manna. Assume that w is strictly concave and continuously differentiable. The function w is concave and satisfies w' (0) = 100 and (100)= 1. The price of manna is always $1, and each consumer has wealth exceeding $10,000 which is spent entirely on phiffle and manna.
(a) What is the shape of the market demand curve for phiffle, for prices between $1 and $100? Two firms sell phiffle. For some reason, one firm is willing to sell at a price of $2, while the other insists on a price of $3. Moreover, the first firm is willing to sell only 50,000 units of phiffle. The second firm is willing to sell to all comers. Assume that w' (90) = 2 and w' (75) = 3.
(b) Suppose that the following mechanism is used for the distribution of phiffle. The first firm asks each consumer how much phiffle he wishes to buy at a price of $2. If total orders are less than 50,000, each consumer is given whatever he asks for. Otherwise, phiffle from firm 1 is rationed equally to each person who wants some, up to the amount that the person wants. That is, we divide the 50,000 units equally among all the consumers and ask whether any consumer is getting more than he asked for. If so, those consuill.ers who asked for less are given what they asked for, and the remainder of their share is divided equally among all the consumers who are still rationed, and so on.) After this rationing scheme, consumers can go to the second firm, if they wish, and buy as much (more) phiffle as they would like for $3. What will be the result of this scheme?
(c) Suppose that phiffle is distributed as follows. The first firm asks each consumer how much phiffle she wishes and then distributes the phiffle as follows: A consumer is chosen at random and is given whatever she asks for, up to 50,000 units. Then, if anything remains of the 50,000 units, a second consumer is chosen at random, and so on, until either all consumers have been served or the 50,000 units are exhausted. After this distribution, consumers can purchase phiffle from the other firm, at $3 per unit. What will be the outcome of this distribution system if phiffle cannot be resold?
(d) What will be the outcome of the distribution system in part (c) if phiffle can be resold? (Assume that consumers in this economy are very savvy folks.)
(e) If you like a challenge, think about these three scenarios with the added complication that the economy is populated with a large number of hetemgenous consumers. You will need to formulate some of the problem for yourself.