Question

Suppose a consumer’s utility function is given by U(X,Y) = X*Y. Also, the consumer has $360 to spend, and the price of X, PX = 9, and the price of Y, PY = 1.

a) (4 points) How much X and Y should the consumer purchase in order to maximize her utility?

b) (2 points) How much total utility does the consumer receive?

c) (4 points) Now suppose PX decreases to 4. What is the new bundle of X and Y that the consumer will demand?

d) (6 points) How much money would the consumer need in order to have the same utility level after the price change as before the price change?

e) (6 points) Of the total change in the quantity demanded of X, how much is due to the substitution effect and how much is due to the income effect?

Answer 1

U = XY

Budget line: M = X.PX + Y.PY

(a) Budget line: 360 = 9X + Y

Utility is maximized when MUX / MUY = PX / PY = 9/1 = 9

MUX = U / X = Y

MUY = U / Y = X

MUX / MUY = Y / X = 9

Y = 9X

Substituting in budget line,

360 = 9X + 9X = 18X

X = 20

Y = 9 x 20 = 180

(b) U = 20 x 180 = 3600

(c) When PX = 4, Budget line: 360 = 4X + Y

MUX / MUY = Y / X = 4/1 = 4

Y = 4X

Substituting in budget line,

360 = 4X + 4X = 8X

X = 45

Y = 4 x 45 = 180

(d) Consumption after price change, priced at original PX = 45 x 9 + 180 x 1 = 405 + 180 = 585

Additional money needed = 585 - 360 = 225

(e) Total effect (TE) = 45 - 20 = 25

To find Substitution Effect (SE), we keep utility unchanged at original level, but substitute the new (X / Y) Ratio:

U = 3600 = X.4X = 4X²

X² = 900

X = 30

SE = 30 - 20 = 10

Income effect = 45 - 30 = 15