Question

Maurice has the following utility function: u (X, Y) = 50X + 60Y - 5X² - 2Y², where X is his consumption of CDs with price of $ 5 and Y is his consumption of movie videos, with a rental price of $ 6. He have plans to spend $ 460 on both forms of entertainment. Determine the number of CDs and video rentals that will maximize Maurice's utility. What is Maurice's maximum utility equal to? (Hint 1: MU, = 50 - 10X, MU, = 60 - 4Y) (Hint 2: MU, / P, = MU, / P;)

Answer 1

Answer: 20 CDs ; 60 Movie Videos

Maurice's utility function is as follows;

U(X, Y) = 50X + 60Y - 5X² - 2Y²..........(1), where,

X - consumption of CDs,

Y - consumption of movie videos,

PX = $5

PY = $6

Maurice's total budget is $460 , that he plans to spend on both X, and Y.

X * ($5) + Y * ($6) = $460

Or, $5X + $6Y = $460.......(2)

The utility of two goods (X, Y) is maximized, when the following condition is satisfied,

MUX / PX = MUY / PY

Differentiating equation(1) with respect to X, we get,

dU / dX = 50 - 10X

Or, MUX = 50 - 10X

[ MUX = marginal utility of X = Change in total utility for the change in an additional unit of the consumption of X, keeping Y constant.]

Differentiating equation(1) with respect to Y, we get,

dU / dY = 60 - 4Y

Or, MUY = 60 - 4Y

[ MUY = marginal utility of Y = Change in total utility for the change in an additional unit of the consumption of Y, keeping X constant.]

Now, putting the values of MUX , MUY , PX and, PY in Maurice's utility maximization condition, we get the following,

(50 - 10X) /$5 = (60 - 4Y) / $6

Or, $6 * (50 - 10X) = $5 * (60 - 4Y)

Or, $300 - $60X = $300 - $20Y

Or, - $60X + $20Y = $300 - $300

Or, $20( - 3X + Y) = 0

Or, - 3X + Y = 0

Or, Y = 3X

Now, putting the value of Y in equation(2), we get,

$5X + $6 * (3X) = $460

Or, $5X + $18X = $460

Or, $23X = $460

Or, X = $460 / $23

Or, X = 20

We get, that Y = 3X

Y = 3 * 20

Or, Y = 60

The number of CDs and video rentals that will maximize Maurice's utility are as follows;

CDs = 20 CDs ; Movie Videos = 60 Movie Videos

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Maurice's maximum utility is achieved when,

MUX / PX = MUY / PY

So, Maurice's maximum utility is as follows;

(50 - 10X) / $5 = (60 - 4Y) / $6

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