Question

Q:

Suppose there are two consumers, A and B.  

The utility functions of each consumer are given by:

U_A(X,Y) = XY³
U_B(X,Y) = X*Y

Therefore:

• For consumer A: MU_X = Y³; MU_Y = 3XY²
• For consumer B: MU_X = Y; MU_Y = X

The initial endowments are:  

A: X = 16; Y = 28
B: X = 54; Y = 12

a) Suppose the price of Y, P_Y = 1. Calculate the price of X, P_X that will lead to a competitive equilibrium.

b) How much of each good does each consumer demand in equilibrium?

Consumer A's Demand for X:
Consumer A's Demand for Y
Consumer B's demand for X
Consumer B's demand for Y

c) What is the marginal rate of substitution for consumer A at the competitive equilibrium?

Answer 1

Solution: The utility functions of each consumer we given by: UA (x) = xy3 Ug(x,y) = **Y A: X-16 Y = 28 B: x=54) ya 12 U = xy3, 42 = xy, W. = (16,28), W2 = (54,12) consumer 1's problem: max x43 sao that Px + y = 162 +28 At equilibrium , marginal rate of subetitution is equal to the price satio: y Yap -> y = 3px p 32y2 = substituting the value of y in the budget constrai- - 16P+28 cnt: PX - 3px 16p +28X up consumer's 28 problem: pe+y = 5up +12 max ny show that -> = P-> y = px -> 2P4 = 54412 CS Scanned with CamScanner ?? x= Sup +12 a supt2 2P

Demande bir t. ond y by both constang, so (1,4,) = ( nooit te voeg ), (12., 9,) = ( Suptr2, 5up ?) 2P total erdasmont igs equal to total At equilibrium demand : 160+28 up SUP + 12 & 2P = 70 16P +28+108P + 24 = 980 p Hence, equilibrium priceswe: - (P2, Py) = (-3,1) domard & we . (b) At those prices, consumer's Gary,) = (25, 25), (12,Y) = (45,15) A at the Competitive (C) MRS for equilibrium consumer ip: Scanned with CamScanner