Question

U( X,Y,Z)=10X^2Y^7Z^1

1.What is the own price elasticity of demand for Y?

2.What is the cross price elasticity of demand for Y with respect to the Price of X? With respect to the price of Z?

3.What is the income elasticity of demand for Y?

Answer 1

First, we find the demand function of X, Y, and Z;

Let p_1,p_2 and p_3 be the price of the good X, Y and Z respectively.

Budget constraint

Using Lagrange multiplier method

First-order condition

Differentiating equation (3) w.r.t X, we get

Similarly, differentiating equation (3) w.r.t Y, we get

Again, differentiating equation (3) w.r.t Z, we get

And again, differentiating equation (3) w.r.t λ, we get

Dividing equation (4) by (5), we get

Dividing equation (5) by (6), we get

Put value from (7) and (8) in eq (2), we get

From equation (7), we get the demand function of Y:

Own price elasticity of demand for Y:

the cross-price elasticity of demand for Y with respect to the Price of X is zero

( )

the cross-price elasticity of demand for Y with respect to the Price of Z is zero.

since the demand function of Y is independent of the price of Z

as

Income elasticity of demand Y:

Hence income elasticity of demand Y is 1.