Question

A consumer has the utility function over goods X and Y,

U(X; Y) = X1/3Y1/2

Let the price of good x be given by Px, let the price of good y be given by Py, and let income be given by I.

(a) Derive the consumer’s generalized demand function for good X.

(b) Solve for the Marshallian Demand for X and Y using Px, and Py (there are no numbers—use the notation).

(b) Is good Y normal or inferior? Explain precisely.

Answer 1

A) optimal bundle condition,

MUx/px=MUy/py

√y/3*x^2/3*px=x^1/3/2*√y*py

Solve for x,

2*y*py=3*x*px

X=2*y*py/3px { x generalized demand function}

B) budget constraint,

I=px*x+py*y

I=2y*py/3px)*px+py*y=2y*py/3+py*y=5py*y/3

Solve for y,

Y=3I/5py { marshallian demand of y}

2*y*py=3*px*x

Y=3*px*x/2*py

I=3*px*x/2py)*py+px*x=3*px*x/2+px*x=5px*x/2

Solve for x,

X=2I/5px{ marshallian demand of x}

b) Y=3I/5py

DERIVATIVE of demand of y with respect to I and check it is positive or negitive. Positive DERIVATIVE means good is normal and negitive DERIVATIVE means. Good is inferior.

∆Y/∆I=3/5py,

So DERIVATIVE is positive,so good Y is normal good