Consider a 2-person/2-good exchange economy in which person 1 is endowed with (e11, e12) and person 2 is endowed with (e21, e22) of the goods x1 and x2.
A: Suppose again that tastes are homothetic, and assume throughout that tastes are also identical.
(a) Draw the Edgeworth Box and place the endowment point to one side of the line connecting the lower left and upper right corners of the box.
(b) Illustrate the contract curve (i.e. the set of efficient allocations) you derived in exercise 16.1 Then illustrate the set ofmutually beneficial trades as well as the set of core allocations.
(c) Why would we expect these two individuals to arrive at an allocation in the core by trading with one another?
(d)Where does the competitive equilibriumlie in this case? Illustrate this by drawing the budget line that arises from equilibrium prices.
(e) Does the equilibriumlie in the core?
(f) Why would your prediction when the two individuals have different bargaining skills differ from this?
B: Suppose, as in exercise 16.1, that the tastes for individuals 1 and 2 can be described by the utility functions u1 = xα1 x2(1−α) and u2 = xβ1 x2(1−β) (where α and β both lie between 0 and 1). (a) Derive the demands for x1 and x2 by each of the two individuals as a function of prices p1 and p2 (and as a function of their individual endowments).
(b) Let p∗1 and p∗2 denote equilibrium prices. Derive the ratio p∗2 /p∗1.
(c) Derive the equilibrium allocation in the economy—i.e. derive the amount of x1 and x2 that each individual will consume in the competitive equilibrium (as a function of their endowments).
(d) Now suppose that α = β—i.e. tastes are the same for the two individuals. From your answer in (c), derive the equilibrium allocation to person 1.
(e) Does your answer to (d) satisfy the condition you derived in exercise 16.1B(b) for pareto efficient allocations (i.e. allocations on the contract curve)?1