代做Microeconomics Theory: Assignment 5代做Prolog

Microeconomics Theory: Assignment 5

Please read the following instructions:

.  Please try to answer all questions.  Your assignment will be graded based on effort rather than accuracy.

.  If you discuss with your classmates to finish the assignment, please list their names on your assignment. You should still hand in your own version of assignment.

.  Please scan or take a picture of your finished assignment and upload it to Canvas before the due time.

.  The due time is 11:59 PM on April 23rd.  Please submit your assignment on Canvas.  Email submission is not accepted.

1.  Consider a pure exchange economy with two goods,x and y; two individuals, A and B, each with utility function Ui(xi, yi) where i ∈ {A, B}. The initial endowments are eA  = (10, 0) and eB  = (0, 10) .

(a)  Assume that the utility functions are Ui(xi, yi) = min{xi, yi} for individuals i ∈ {A, B}.  Find the set of Pareto efficient allocations and the set of Walrasian equilibrium allocations.

(b) Assume that the utility functions are UA (xA , yA ) = x AyA , UB (xB , yB ) = min{xB , yB }. Find the set of Pareto efficient allocations and the set of Walrasian equilibrium allocations.

2.  Consider an economy with two consumers i  ∈  {A, B},  one  firm  and two goods l  ∈  {1, 2}.   The endowments of A and B are eA  = (2/1,2/1), eB  = (2/1,2/1) . The utility functions are

The firm produces good 2 using good 1.  The production function is y2  = √y1.  Consumer B owns the firm (denote π the firm’s profit). Good 2 is the numeraire good, i.e., p2 = 1.

(a)  Determine the demands for good 1 of consumers and the firm.

(b)  Show that there is a unique equilibrium price p1.

(c)  Assume that the production function is now y2  = y1, and thus satisfies constant return to scale. Determine the equilibrium price and allocation.

(d)  Consider the pure exchange economy with only consumer A and B (in other words, eliminate the firm). Determine the equilibrium price and allocation.

3. Consider an economy with two individuals i = {A, B}, each with identical Cobb-Douglas utility function

Ui(x1(i), x2(i)) = x 1(i)x2(i), and the initial endowments are eA  = (200, 100) and eB = (100, 200) .

(a)  Find the Pareto efficient allocations.

(b) Find a Walrasian equilibrium allocation. For simplicity, assume p 1  = p2  = 1.

(c)  Assume that the government sets a tax t on purchases of good 1, which is refunded to the consumers as a lump sum payment, Ti  = tx 1(i). Find the post-tax Walrasian equilibrium allocation, and compare it with your results in (b).