Find the demands for goods X and Y for agent

Question

Agent B's preferences are given by:

U (x,y) = y - ((x)^(-2))/2

Which are quasi-linear preferences

Budget constraint is -> x*p +y = 10

(Since p= price of good x, and Price of good y is normalized to 1, agent has an endowment of 10 units of y).

How do I solve for the Demand of each good?

Answer 1

This is a little Tricky. For an Interior solution we need MRS=Px/Py = Px as Py=1 we will have a corner solution when all money is spent on X. We need to determine that "choke price". First lets figure out what the MRSxy is. The MUx = X^-3 = (1/X³)and MUy = 1 so MRS = (.1/X³). So we need to know what Value of X and px will have

(1/X³) = Px (1)

and

PxX = 10 (2)

If I multiply both sides of 1 by X I get

(1/X²) = PxX = 10 or 1/10 = X² or X = 10^-1/2 = (1/10)^1/2

from (1) we know Px = (1/X)³ =(10^1/2)³ = 10^3/2. so for any price higher than 10^3/2 all 10 dollars of the budget is spent on X so X= 10/Px for Px >= 10^3/2. for any price of x less than 10^3/2 we have an interior solution where MRS = Px. so from (1) we have X³= 1/Px or X=(1/Px)^1/3 which is independent of income. There will be only substitution effects so all of the compensated demand curves will "lie on top" of the ordinary demand curve. With in interior solution, we go back to the budget constraint

10 = PxX + Y = Px(1/Px^1/3) +Y = Px^2/3+ Y or

Y= 10-Px^2/3 for Px<= 10^3/2 and 0 elsewhere

and,

X= 10/Px for Px >= 10^3/2 and X = (1/Px)^1/3 for 0<Px<10^3/2

If you have any questions. feel free to Reach out. I have been doing this stuff for Decades!

Best,

Lenny