Suppose you and I have the same homothetic tastes over x1 and x2, and our endowments of the two goods are EM = (eM1, eM2) for me and EY = (eY1, eY2) for you.
A (a) Suppose throughout that, when x1 = x2, our MRS is equal to ˆ’1. (a) Assume that eM1 = eM2 = eY1 = eY2. Draw the Edgeworth box for this case and indicate where the endowment point E = (EM, EY) lies.
(b) Draw the indifference curves for both of us through E. Is the endowment allocation efficient?
(c) Normalize the price of x2 to 1 and let p be the price of x1. What is the equilibrium price pˆ—?
(d)Where in the Edgeworth Box is the set of all efficient allocations?
(e) Pick another efficient allocation and demonstrate a possible way to re-allocate the endowment among us such that the new efficient allocation becomes an equilibrium allocation supported by an equilibrium price. Is this equilibrium price the same as pˆ— calculated in (c)?
B: Suppose our tastes can be represented by the CES utility function u(x1, x2)

(a) Let p be defined as in A (c). Write down my and your budget constraint (assuming again endowments EM = (eM1, eM2) for me and EY = (eY1, eY2).)
(b)Write down my optimization problem and derive my demand for x1 and x2.
(c) Similarly, derive your demand for x1 and x2.
(d) Derive the equilibrium price. What is that price if, as in part A, eM1 = eM2 = eY1 = eY2?
(e) Derive the set of pare to efficient allocations assuming eM1 = eM2 = eY1 = eY2. Can you see why, regardless of how we might redistribute endowments, the equilibrium price will always be p = 1?

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