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 that tastes are homothetic for both individuals.
(a) Draw the Edgeworth Box for this economy, indicating on each axis the dimensions of the box.
(b) Suppose that the two individuals have identical tastes. Illustrate the contract curve—i.e. the set of all efficient allocations of the two goods.
(c) True or False: Identical tastes in the Edgeworth Box imply that there are no mutually beneficial trades.
(d) Now suppose that the two individuals have different (but still homothetic) tastes. True or
False: The contract curve will lie to one side of the line that connects the lower left and upper right corners of the Edgeworth Box—i.e. it will never cross this line inside the Edgeworth Box.
B: Suppose 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). Some of the questions below are notationally a little easier to keep track off if you also denote E1 = e11 +e2 1 as the economy’s endowment of x1 and E2 = e12 +e22 as the economy’s endowment of x2.
(a) Let x1 denote the allocation of x1 to individual 1, and let x2 denote the allocation of x2 to individual 1. Then use the fact that the remainder of the economy’s endowment is allocated to individual 2 to denote individual 2’s allocation as (E1 − x1) and (E2 − x2) for x1 and x2 respectively. Derive the contract curve in the form x2 = x2(x1)—i.e. with the allocation of x2 to person 1 as a function of the allocation of x1 to person 1.
(b) Simplify your expression under the assumption that tastes are identical — i.e. α = β. What shape and location of the contract curve in the Edgeworth Box does this imply?
(c) Next, suppose that α 6= β. Verify that the contract curve extends from the lower left to the upper right corner of the Edgeworth Box.
(d) Consider the slopes of the contract curve when x1 = 0 and when x1 = E1. How do they compare to the slope of the line connecting the lower left and upper right corners of the Edgeworth Box if α > β? What if α < β?
(e) Using what you have concluded, graph the shape of the contract curve for the case α > β and for the case when α < β?
(f) Suppose that the utility function for the two individuals instead took the more general constant elasticity of substitution form u = (αx1 + (1−α) x2) −1/ρ. If the tastes for the two individuals are identical, does your answer to part (b) change?