Q. A and B consumes two goods, k and x. The prices of these goods are 1 and consumers have same level of income, I. Each unit of good k provides one unit of consumption to the purchasing consumer and s (0 < s < 1) units of consumption to the other consumer. Consumers A and B have the following utility function:

where kA and kB are the amounts of good k and x4 and xB are the amounts of good x consumed by A and B, respectively.

a. Write down the interpretation of s in few sentences.

b. Assume that x is a private good. Find the Nash equilibrium (each consumer maximizing their own utility given that they are symmetric) levels of consumption for both goods for both consumers.

c. Suppose that the social welfare function is Utilitarian, show that equilibrium is Pareto efficient for s = 0 but inefficient for other values of s.

d. Let's assume that in addition to good k, now each unit of good x provides one unit of consumption to the purchasing consumer and s (0 ≤ S ≤ 1) units of consumption to the other consumer. Find the Nash equilibrium (each consumer maximizing their own utility given they are symmetric) levels of consumption for both goods for both consumers. Show that Nash equilibrium is Pareto-efficient.

e. Explain the results in part d.