Question #80933

9. Given the following marginal utility schedule for good X and good For the individual,and given that the price of X and the price of Y are both $1, and that the individual spends all income of $7 on X and Y, Q 1 2 3 4 5 67 MUX 15 11 9 6 4 3 1 MUY 12 9 6 5 3 2 1 a. indicate how much of X and Y the individual should purchase to maximize utility. b. show that the condition for constrained utility maximization is satisfied when the individual is at his or her optimum. c. determine how much total utility the individual receives when he or she maximizes utility? How much utility would the individual get if he or she spent all income on X or Y?

Expert's answer

Given the following marginal utility schedule for good X and good For the individual,and given that the price of X and the price of Y are both $1, and that the individual spends all income of $7 on X and Y,

Q 1 2 3 4 5 6 7

MUX 15 11 9 6 4 3 1

MUY 12 9 6 5 3 2 1

a. The individual should purchase such amount of X and Y to maximize utility, for which:

MUX/PX = MUY/PY and PX*X + PY*Y = I, so:

For MUX = MUY = 6, X = 4, Y = 1, I = 1*4 * 1*3 = 7, so the utility is maximized.

b. The condition for constrained utility maximization is satisfied, becausr the individual at his or her optimum spends all his income of $7, as we proved in a.

c. Total utility received when he or she maximizes utility is: TUX + TUY = (15 + 11 + 9 + 6) + (12 + 9 + 6) = 68.

If he or she spent all income on X or Y, then:

TUX = 15 + 11 + 9 + 6 + 4 + 3 + 1 = 49,

TUY = 12 + 9 + 6 + 5 + 3 + 2 + 1 = 39.

Q 1 2 3 4 5 6 7

MUX 15 11 9 6 4 3 1

MUY 12 9 6 5 3 2 1

a. The individual should purchase such amount of X and Y to maximize utility, for which:

MUX/PX = MUY/PY and PX*X + PY*Y = I, so:

For MUX = MUY = 6, X = 4, Y = 1, I = 1*4 * 1*3 = 7, so the utility is maximized.

b. The condition for constrained utility maximization is satisfied, becausr the individual at his or her optimum spends all his income of $7, as we proved in a.

c. Total utility received when he or she maximizes utility is: TUX + TUY = (15 + 11 + 9 + 6) + (12 + 9 + 6) = 68.

If he or she spent all income on X or Y, then:

TUX = 15 + 11 + 9 + 6 + 4 + 3 + 1 = 49,

TUY = 12 + 9 + 6 + 5 + 3 + 2 + 1 = 39.

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