Question 2 (40 points) - Consumer and Producer Theory

Assume that consumers derive utility from consuming public transporta- tion and a household basket (which includes food, clothes, etc.). The utility for each consumer is given by:

u(x1, x2) = x1(1 + q) + log x2,

where x1 denotes the consumption of public transportation and x2 denotes the consumption units of the household basket. Consumers get higher utility as the quality of public transport, q, increases. Consumers only choose the amount of x1 and x2 that they consume, since they cannot directly affect the quality of public transportation. The price of public transportation is p1 and the price of the household basket is p2. Each individual has a wealth of wi.

(a) Derive the Marshallian demand for public transportation for each consumer x1(p, wi, q).

(b) Compute the consumer's indirect utility function and write an implicit equaiton to solve for the expenditure function e(p, u).

(c) Can the aggregate demand for public transportation be expressed as a function of the prince p and aggregate wealth i wi? Why or why not?

Assume that there are N consumers in this economy and that wi = w for each consumer i (they all have the same wealth), so the aggregate demand for good x1 is N x1(p, w, q). Public transportation is provided by one firm. The market price is set by the government at p1 (the firm is regulated by the government to charge p1 per unit sold), and the firm faces a constant marginal cost c of producing public transportation. The firm can provide

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public transport with "zero" quality at no cost, and then each additional unit of quality that is added costs k to the firm. That is, by incurring an additional cost of k q, the firm can provide public transport with quality q to all of its riders.

(d) Write down the profit maximization problem for the firm.

(e) The government is considering raising the price of public trans- port. How will a firm's optimal choice of q change if the market price of public transportation increases?