Question

9) (10 points) Consider an exchange economy composed of two individuals A and B and two goods x1 and X2. A's utility function is given bỵ U,-2X1 + X2. Individual B's utility function is givenby u = xx2. In the economy, the total endowment of xņš 2 and the total endowment of x2 is 1. Normalize p2 to 1. We know that, in this economy eauilibrium price is given bypi-1. a. (6 points) Find the equilibrium allocation b. (4 points, write a possible endowment for A and B. That is, with the endowment- have written the eauilibrium price should be 1. (Note that there might be multiple values for endowments that would lead to an equilibrium price of 1. You are only required to find one of these

Answer 1

Solution:

Two individuals: A and B; Two goods: x1 and x2

Utility functions are given as follows: UA = 2*x1 + x2 and UB = x1*x2

Total endowments for two goods are: x1 = 2 and x2 = 1; Equilibrium price set (p1, p2) = (1,1)

a) Finding the equilibrium allocation:

Denoting equilibrium allocation by (x1A, x2A) for individual A and (x1B, x2B) for individual B.

For an equilibrium allocation, two conditions to be satisfied are the utility maximizing condition and feasibility condition.

Utility maximizing: From basic economic theory, we know that utility maximizing condition requires that Marginal rate of substitution (MRS) must equal the price-ratio.

Price-ratio = p1/p2 = 1/1 = 1

MRS_x1,x2 = Marginal utility of x1 (MU_x1)/Marginal utility of x2 (MU_x2)

MU_x1 = and MU_x2 =

Feasibility condition: The equilibrium allocation of each good to both consumers should equal the total allocation of each good in the economy.

Then, feasibility condition says that x1A + x1B = total x1 in economy, and similar for good x2.

Thus, x1A + x1B = 2 and x2A + x2B = 1

Seeing the utility functions, individual A treats good x1 and x2 as perfect substitutes while individual B has the Cobb-Douglas utility form. In case of perfect substitutes, if MRS_x1,x2 > price-ratio, consumer consumes only good x1, MRS_x1,x2 < price-ratio then only good x2 and if the two are equal, then consumer consumes anywhere on the budget line.

For individual A, MU_x1 = 2, MU_x2 = 1, so, MRS^A_x1,x2 = 2/1 = 2 > 1 = price-ratio. Thus, individual consumes only good x1, and 0 units of good x2. Then, using feasibility condition:

x2A + x2B = 1

0 + x2B = 1, that is, individual B consumes 1 unit (total endowment) of good x2.

For individual B, MU_x1 = x2, MU_x2 = x1, so, MRS^B_x1,x2 = x2/x1

By utility maximizing condition then, x2/x1 = 1 implying x1 = x2 (at optimality for individual B) or x1B = x2B

We already established above that x2B= 1, so x1B = 1

Using the feasibility condition for good x1, x1A + x1B = 2

x1A + 1 = 2

x1A = 1

Thus, the equilibrium allocation is:

for individual A, (x1A, x2A) = (1, 0)

for individual B, (x1B, x2B) = (1, 1)

b) Assuming the initial goods endowment with individual A and B as (e1A, e2A) and (e1B, e2B) respectively, budget line of individual A becomes:

p1*x1A + p2*x2A = p1*e1A + p2*e2A which, given the prices, equals x1A + x2A = e1A + e2A

and that of individual B becomes: x1B + x2B = e1B + e2B

Using Budget line of individual A, 1+ 0= e1A + e2A

e1A + e2A = 1... (1)

Using Budget line of individual B, 1 + 1 = e1B + e2B

e1B + e2B = 2 ... (2)

Also, in this framework of no production, given the endowment definition, we must have

For goodx1, e1A + e1B = x1 = 2 ... (3)

For good x2, e2A + e2B = x2 = 1 ... (4)

Now, we have system of equations the above 4 system of equations, solving which we shall find e1A, e2A, e1B, and e2B

Any values for these unknowns satisfying the above 4 equations shall satisfy our solution. By hit and trial, one such combination could be:

(e1A, e2A, e1B, e2B) = (0, 1, 2, 0) (You shall verify!)

That is endowment for A: (e1A, e2A) = (0, 1)

Endowment for B: (e1B, e2B) = (2, 0)