Question

16. Suppose John has an income of $300 and spends his income to purchase two goods (X and Y ). Price of Y is $5 and the price of X is $10. Furthermore, John always consumes 2 units of Y with 1 unit of X. (a) How many units of X and Y should John consume in order to maximize his utility? (b) Suppose that the price of X goes up to $15 (income and price of Y are the same). How many units of X and Y should John consume in order to maximize his utility? (c) Plot the budget constraints and indi§erence curves for parts (a) and (b) on the same graph. Clearly label the equilibrium points as "A" for the equilibrium in part (a) and "B" for the equilibrium in part (b). Discuss the results in terms of income and substitution effects.

Answer 1

Given;

John's income, M = $300
Price of x, P_x = $10
Price of y, P_y = $5

John always consumes 2 units of y with 1 unit of x. This means that x and y are complement goods. These are those type of goods which are consumed together. The utility function for John will be;

U (x,y) = min {2x,y}

(a) The utility maximizing level for John will be acheived when given the utility function the condition is satisfied;

2x = y

Putting this in the budget constraint;

P_x x + P_y y = M
10x + 5y = 300
2x + y = 60

2x + 2x = 60
4x = 60
x = 15

2*15 = y
y = 30

The utility maximizing bundle is : (15,30)

(b) Now, price of x goes up to $15 (income and price of Y are the same)

New price of x, P'_x = $15

So, the new budget constraint will be;

P'_x x + P_y y = M
15x + 5y = 300
3x + y = 60

The utility maximising condition;

2x = y

Putting this in new budget constraint;

3x + 2x = 60
5x = 60
x' = 12

2x = y
y = 2*12
y' = 24

The utility maximizing bundle is : (12,24)

c) The old budget constraint; 2x + y = 60
The new budget constraint; 3x + y = 60

The utility function;

U (x,y) = min {2x,y}

Old optimal bundle (15,30)

U = min {2*15,30}
U = 30

New optimal bundle (12,24)

U = min {2*12,24}
U = 24

BL represents the original budget line and IC is the indifference curve which will be L-shaped where utility is 30 with x = 15 and y = 30 at the optimal level at point A.

Now, price of good x increases and budget line shifts leftwards through x-axis to BL'. The new indifference curve will be IC' where utility is 24 and x = 12 and y = 24 at the optimal level at point B.

d) As this is the case for complement goods, there will be no substitution effect. This means that;

Substitution effect (S.E) = 0

Thus, the total effect will be the due to the income effect.

The change in income will be;

= 15 (15-10)
= 15 * 5
= 75

Thus, the new income will be;

= M' - M
75 = M' - 300
M' = 375

Income effect will be;

x (P'_x , M) = m / P_x+2P_y
= 300 / 10+2(5)
= 300/20
x (P'_x , M) = 15

x*(P'_x , M') = m' / P'_x+2P_y
= 375 / 15+2(5)
= 375/25
x*(P'_x , M') = 15

Therefore, there will only be income effect as we can see in the figure,