Question

Your utility function is U = 10 X 0.1Y 0.7 and your marginal utility functions are MUx=X^−0.9 Y^ 0.7 MUy=7X^ 0.1 Y^−0.3

Your budget is M and the prices of the two goods are pX and pY .

1. a) Write down the two conditions for utility maximization subject to a budget constraint.

2. b) Derive the demand functions for X and Y .

3. c) Based on your demand functions explain whether:

   • - Y is a normal good or an inferior good and why. (1 mark)

   • - good X satisfies the law of demand. (1 mark)

4. d) Assume you currently have 42 units of Y and 3 units of X . How many units of Y are you

   willing to give up for an additional unit of X while holding your total utility constant? (1 mark)

Answer 1

Given utility function is U = 10X^0.1Y^0.7
Marginal Utility of X = MUx = X^−0.9 Y ^0.7
Marginal utility of Y = MUy = 7X^0.1 Y^−0.3

Also given, price of X, Px and price of Y is Py. Total income is M.
Hence the budget equation is given as,
M = XPx + YPy

a) In two ways utility can be maximised:
Way 1:
MUx / Px = MUy / Py
then by plugging the values in M = XPx + YPy, utility can be maximised.

Way 2:
Maximixe utility : 10X^0.1Y^0.7
subject to budget constraint : M = XPx + YPy

b) Demand function can be derived as follows:

Hence demand function of X = M/8Px
Demand function of Y = 7M / 8Py

c) An inferior good is that whose demand decreases as income increase and vice versa.
A normal good is that whose demand increases as income increase and vice versa.
Here demand for Y = 7M/8Py
Hence as income (M) increases, demand for Y will also increase. Hence Y is normal good.

Demand for X = M/8Px
If price of X (Px) increases, demand for X will decrease and vice versa. Hence demand for X follows law of demand. (LAw of demand states that if price increase, quantity demanded will decrease and vice versa).

d) Given X = 3 units and Y = 42 units.
Utility = 10X^0.1Y^0.7
= 10 * (3)^0.1 * (42)^0.7
= 10 * 1.116 * 13.68
= 152.74
Now if X = 4
U = 152.74
hence , 152.74 = 10 * (4)^0.1 * Y^0.7
or, Y^0.7 = 152.74 / 10 * (4)^0.1
or, Y^0.7 = 152.74 / 11.48
or. Y^0.7 = 13.30
or, Y =( 13.30 )^1/0.7
or, Y = 13.30^1.43
or, Y = 40.46

Difference in Y = 42 - 40.46 = 1.54

Hence to get 1 additional unit of X, around 1.54 units of Y will be given up.