Question

a. In words, interpret the income elasticity of demand for cereal being equal to 1.5. (Hint: your answer should include a %).

b. For the utility function U(x,y) = 8ln(x)+2y, solve for individual demand for goods x and y.

c. Using your result from Question B, assume Px = 1 and Py = 1. What is the income elasticity of demand for good x?

d. The demand for good x is given by x∗ = 60−4Px +2M +Py, where Px is the price of good x, Py is the price of good y, and M is income. Find the income elasticity of demand when Px = 20, Py = 20, and M = 100. Is x a normal or inferior good? Explain.

Answer 1

a)

The income elasticity of demand for cereal is 1.5 which means a 1% increase in income would result in a 1.5% increase in demand for cereal.

b)

U(x,y) = 8lnx + 2y

Budget constraint is

P_xx + P_yy = I

U/x = MUx = 8/x  

U/y = MUy = 2  

MRS = MUx/MUy

= (8/x)/2

= 8/2x

= 4/x

The optimal choice is

MRS = P_x/P_y  

4/x =   P_x/P_y  

x/4 = Py/Px  

x = 4Py/Px

Put  x = 4Py/Px in budget constaint

P_xx + P_yy = I

P_x(4Py/Px )  + P_yy = I

4P_y + P_yy = I  

4Py+ P_yy   = I  

P_yy = I - 4Py

y = I/Py - 4

The demand functions for good x and y are

x = 4Py/Px

y = I/Py - 4

c)

x =   4Py/Px

x/I = 0  

Income elasticity of demand for good x

e_{x , I} = (x/I)(I/x)

= (0)(I/x)

= 0  

Income elasticity of demand for good x is e_{x , I} = 0

d)

The demand for good x is given by

x∗ = 60 − 4Px +2M +Py  

x* = 60 - 4(20) + 2(100) + 20

= 60 - 80 + 200 + 20

= 200

Income elasticity of demand for good x

e_{x* , M} = (x*/M)(M/x*)

x∗ = 60 − 4Px +2M +Py  

(x*/M) = 2

e_{x* , M} = (2)(100/200)

= 200/200

= 1

since e_{x* , M} = 1 > 0 thus x is a normal ggod.