Compute the walrasian equilibrium

Question

Assignment:

Suppose there are two goods. Toxic waste, good to, and money, good in. There is a firm which stores toxic waste in a small village called Pleasantville. The firm has constant returns to scale, and it can transform one dollar into five units of storage for toxic waste (it can do so continuously).

There are two consumers, 1 and 2. Consumer 1, who lives outside of Pleas-antville, has utility over money and the amount of toxic waste stored in Pleas-antville is

u₁(w, m₁) = log(w) + m₁

Consumer 2, who lives in Pleasantville, obviously hates the fact that toxic waste is stored in his town, so his utility is

U^R(w,m₂) = 2 log(6 - w) + m₂

Both consumers 1 and 2 have initial endowments of money, e^R_m = eT_m = 10 and own no storage for toxic waste. A competitive (Walrasian) equilibrium is defined by a price of storage equal to the marginal cost of storage, and choices over how much waste to store in Pleasantville.

Q1. Suppose first that 2 cannot pay the firm not to store waste, and that the competitive price for storage is 1. Compute the Walrasian equilibrium and show that it is not Pareto efficient.

Q2. What would be the Pareto efficient amount of waste stored in Pleasantville?

Q3. Now suppose that 2 owns "waste permits" so that if 1 wants to store waste, he has to pay a price equal to the sum of the marginal cost of storage (which equals 1) and the price of a permit, pˆ, and the proceeds from permits go to consumer 2. An equilibrium is a permit price pˆ such that given pˆ, the amount of waste that 1 wants to store is equal to the supply of permits that 2 wants to sell. Compute this Walrasian equilibrium; in particular, find the equilibrium price pˆ, and find how much waste will be produced. Compare this result to parts (1) and (2).

Q4. Now suppose that 1 owns "waste permits" so that if 1 wants to store waste he can do so freely. Prom (a) above we know how much he would choose to store at a marginal price of storage equal to 1. Now assume that 2 can pay a price, p˜ for every unit below the quantity that 1 would produce on his own. that is, there is a market for waste prevention, and 1 can pay for 2's willingness to produce less than his private optimal level. An equilibrium is a permit price p˜ such that given p˜, the amount of prevention that 2 wants to buy is equal to the amount by which 1 is willing to reduce waste. Compute this Walrasian equilibrium; in particular, find the equilibrium price, and find how much waste will be produced.

Q5. Compare your results in parts (c) and (d), and use them to explain the Coase theorem. Also, use them to argue how only efficiency considerations, but not distributional considerations are addressed by it.