Exchange economy

1. Suppose that, in a simple, two-good, exchange economy, the two individuals, A and B have the following utility functions U_A=X^1/2  Y^1/2 and U_B=X^1/3Y^2/3. A is initially endowed with 100 units of good x and 100 units of good y, and B also with 100 units of good x and 100 units of good y.

(i) Find the marginal rate of substitution for both individuals.
(ii) Is the initial allocation pareto efficient.
(iii) With the opportunity to exchange, would a reallocation occur? Who would trade what for what?
(iv) Given the assumption that the price of y is 1, find each individual's demand for each good. (note that the MRS is different for each individual.)
(v) Given these demands, find the competitive equilibrium price of x, and the accompanying allocation of goods across the two individuals.
(vi) Draw the edgeworth box showing the initial allocation, utility curves and any reallocation.

Market structure

2. Consider a market for a homogeneous good that can be described using an inverse demand function of P=150-q, where P is the price of the good, and q is the quantity of the good demanded. Any firm that produces the good faces a fixed cost, F, and a per-unit cost of 20.

(i) If there are many firms in this market, operating under perfect competition, find the price, quantity and profit each firm will make. What level of fixed costs can this market structure sustain?

(ii) Draw a graph of the market, showing consumer and producer surplus. Calculate values for the surpluses.

(iii) Suppose the market is just served by one firm, a monopoly. Find the price, quantity and profit levels.

(iv) Graph the result, show and calculate the consumer surplus, producer surplus and dead weight loss. Who wins and who losses under monopoly?

(v) If there are two firms operating under cournot competition, find the reaction functions and solve for price quantity and profits.

(vi) For what range of F will there be at least two firms serving this market in equilibrium?

Externalities

3. Consider an industry that produces a homogenous good but the production process also produces pollution. The inverse demand for the good is P = 5 - 0.5Q.

The private cost (direct costs payable by the firm) of producing the good is c^P= 2Q. The total social cost is the private cost plus the environmental cost and is equal to c^P+E = 3Q.

(i) In each case find the marginal cost.
(ii) Under perfect competition, find the output, price and profit if only the private cost is taken account of.
(iii) Under perfect competition, find the output, price and profit if both the private cost and the externality are taken account of.
(iv) If this firm was a monopoly, would society as a whole be better or worse off? (compared to perfect competition where only the private cost is taken account of.)
(v) Show all 3 scenarios on a graph, calculating the consumer, producer surpluses, any dead weight losses and externality costs.