Question

Consider the following utility function of 2 goods, x and y: U(x,y)=min{x+2y, y+3x} ; x,y ≥0.

1. a) Carefully draw the indifference curve when utility level is 10. Explain your answer.

2. b) If income is $100 and price of good x and y is $10 and $20 respectively, then find out the consumption bundle that maximizes utility.

Answer 1

a) According to the utility function we have x + 2y = y + 3x

This gives 2x = y or y = 2x.

The indifference curves are L-shaped and the ratio of x and y is x/y = 1/2 implying that the two are used in a fixed proportion.

For U = 10 we have x + 2y = 10 and y + 3x = 10

This gives x + 2*2x = 10 or x = 2 and y = 4 while 2x + 3x = 10 gives x = 2 and y = 4

The kink has x = 2 and y = 4.

b) In the budget line we have M = xpx + ypy

100 = 10x + 20y

100 = 10x + 20*2x

100 = 50x

This gives x = 100/50 = 2 and y = 2*2 = 4

Optimal bundle has x = 2 units and y = 4 units.