1. (10 total points) Suppose a consumer\'s utility function is given by U(X,Y)-X

Question

1. (10 total points) Suppose a consumer's utility function is given by U(X,Y)-X12*y2. Also, the consumer has $32 to spend, and the price of X, Px S1, and the price of Y, Py-$4. a) (2 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) (2 points) Now suppose Px increases to S4. What is the new bundle of X and Y that the consumer will demand? d) (2 points) How much additional money would the consumer need in order to have the same utility level after the price change as before the price change? (Note: this amount of additional money is called the Compensating Variation.) e) (2 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? (Note: since there is an increase in the price of Good X, these values will be negative).

Explanation / Answer

Initial budget line: 32 = X + 4Y

MRS = Y/X = 1/4

X = 4Y

Substituting in budget line,

32 = X + X = 2X

X = 16

Y = X/4 = 16/4 = 4

Utility = (16)1/2(4)1/2 = 4 x 2 = 8

After price change, budget line: 32 = 4X + 4Y, or 8 = X + Y

MRS = Y/X = 4/4 = 1

X = Y

Substituting in budget line,

8 = X + X = 2X

X = 4

Y = X = 4

Therefore,

(e)

Total effect (TE) of increase in price of X = 4 - 16 = - 12

To find substitution effect (SE), we keep utility unchanged at 8 & substitute X = Y in utility function:

U = X1/2X1/2 = X

X = 8

Therefore, SE = 4 - 8 = - 4

Income effect = TE - SE = - 12 - (- 4) = - 12 + 4 = - 8

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