Question

Suppose Alex only consumes 3 units of x1 with 8 units of x2. That is, if he is consuming more x1 or x2 in a different ratio, it does not increase his utility

a) Write down Alex’s utility function. What kind of utility function does he have?

b) Suppose Alex wants to have a utility 48. If he desires to make the best use of his money, based on your utility function in a) how many units of x1 and x2 should he consume? When his utility is 48, draw an indifference curve for Alex (with x1 on the horizontal axis). Label two points on this indifference curve.

c) Suppose Alex’s income is m. If price of x1 is $8 (p1 = 8) and price of x2 is $3 (p2 = 3), using the utility function that you chose in a), draw Alex’s Engel curve for x1 (Hint: the Engel curve should have x1 on the horizontal axis and income m on the vertical axis). Make sure to label the axes and indicate the slope of the line

d) Now, suppose Alex’s income is $320. If price of x1 is $16 (p1 = 16) and price of x2 is $6 (p2 = 6), what is

Alex’s optimal consumption bundle? What is Alex’s utility at this bundle?

Answer 1

A) as fixed ratio between x1 & x2

X1/x2 = 3/8

8x1 = 3x2

so, U(x1, x2) = Min(8x1, 3x2)

Utility function is perfectly Complements type

B) U = 48,

48= 8x1= 3x2

x1*= 48/8 = 6

x2*= 48/3= 16

c) from BC , P1X1 + P2X2 = m

8X1 + 3X2= m

at eqm, from utility function

8x1 = 3x2

so, 8x1 + 8x1= m

m = 16x1 : Engel curve

d) m = 320, p1= 16 , p2= 6

Optimal bundle

BC: 16x1 + 6x2= 320

put, 8x1= 3x2

16x1 = 6x2

32x1= 320

x1"= 10, x2"= 80/3

U= Min(80, 3*80/3)

= 80

Graphs