Question

Consider a pure exchange economy with 2 goods and 3 agents. Good y is the numeraire, so we set py = 1. • Mr. 1 is Cobb-Douglas of type λ = 0.25 and is endowed with (10, 4). • Mr. 2 is Cobb-Douglas of type λ = 0.5 and is endowed with (5, 9). • Mr. 3 is Cobb-Douglas of type λ = 0.75 and is endowed with (20, 20). In the following steps, you will find the equilibrium of this model economy.

(A) Find the aggregate endowment (or total supply) of good x.

(B) Find Mr. 1’s demand for good x in terms of px. Your answer should be a formula in terms of the letter px.

(C) Similarly, write down both Mr. 2 and Mr. 3’s demand for good x.

(D) Write down the aggregate demand for good x in the economy. Again, your answer should be in terms of the letter px.

(E) Solve for the magic, equilibrium price p ∗ x that makes the supply of good x equal the demand of good x.

(F) Calculate how many units of good x each of the three agents will receive in equilibrium. Check: the three numbers should add up to 35.

(G) Write down all three agents’ demands for good y in terms of px. (H) Calculate how many units of good y each of the three agents will receive in equilibrium.

(I)Check: Walras’ Law says that the supply of good y should automatically equal the demand of good y. Your three answers from part (H) should add up to 33.

(J) Given the equilibrium price p ∗ x , which of the three agents turned out to have the least valuable endowment? (K) Using the utility function u(x, y) = x λy 1−λ , calculate how many units of pleasure each agent receives in equilibrium

Answer 1

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