Question

Questions 11-13 relate to the following consumer utility maximization problem: maximize U(x,y)=x^0.75y^.25 subject to the budget constraint 10x+5y=40.

A) What is the optimal amount of x in the consumer's bundle? (Note that the 10 in the budget constraint is interpreted as the price of good x, 5 is the price of good y, and 40 is the consumer's income.)

B) What is the optimal amount of y in the consumer's bundle?

C) What is the consumer's maximized utility?

Answer 1

(a) U = x^0.75 y^0.25

Marginal utility of x: MUx = dU/dx

=> MUx = 0.75 x ^{0.75 -1} y^0.25

=> MUx = 0.75 x^-0.25 y^0.25

=> MUx = 0.75 (y / x)^0.25

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Marginal utility of y: MUy = dU/dy

=> MUy = 0.25x^0.75y^0.25-1

=> MUy = 0.25x^0.75y^-0.75

=> MUy = 0.25(x/y)^0.75

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Budget constraint: 10x + 5y = 40

=> Price of good x (Px) = 10

=> Price of good y (Py) = 5

=> Income (M) = 40

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At utility maximization point following condition holds:

(MUx / MUy) = (Px / Py)

=>[ 0.75 * (y/x)^0.25 ] / [0.25*(x/y)^0.75] = (10 / 5)

=> 3 (y/x)^0.25 * (y/x)^0.75 = 2

=> 3(y/x) = 2

=> y = (2x /3) -------------------- eq(1)

Put eq(1) in budget constraint,

10x + 5y = 50

=> 10x + 5(2x /3) = 50

=> [10x*3 + 5*2x]/3 = 50

=> 30x + 10x = 50 * 3

=> 40x = 150

=> x = (150/40)

=> x = 3.75

Optimal amount of x in optimal consumer's bundle is 3.75 units.

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(b) Put optimal amount of x in eq (1)

i.e., y = (2x/3)

=> y = (2 * 3.75) / 3

=> y = 2.5

optimal amount of y in the consumer's bundle is 2.5 units.

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(c) optimal amount of x = 3.75

optimal amount of y = 2.5

U = x^0.75 y^0.25

put x = 3.75 and y = 2.5

=> U = (3.75)^0.75 * (2.5)^0.25

=> U = 3.38

Consumer maximized utility is 3.38

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