Question

Donald derives utility from only two goods, carrots (X) and donuts (Y). His utility

function is as follows: U(X,Y) =X0.1 Y0.9 . Donald has an income (M) of $900 and the price of carrots (PX) and donuts (PY) are $45 and $90 respectively. Based on this information, What quantities of carrots and donuts will maximize Donald's utility?

Answer 1

Solution:

U (_X,Y) = X^0.1Y^0.9

Budget constraint: M = PxX + PyY

900 = 45X + 90Y

Derive MRTS: $\frac{MUx}{MUy} = \frac{Px}{Py}$

MUx = $\frac{\partial U} {\partial X}$ = 0.1X^-0.9Y^0.9

MUy = $\frac{\partial U} {\partial Y}$ = 0.9X^0.1Y^-0.1

0.1X^-0.9Y^0.9 $\div$ 0.9X^0.1Y^-0.1 = $\frac{45}{90}$ 45/90

Y = 4.5X

Substitute in the budget constraint:

900 = 45X + 90Y

900 = 45X + 90(4.5X) = 45X + 405X = 450X

900 = 450X

X = 2

Y = 4.5X = 4.5(2) = 9

Carrots (X) = 2

Donuts (Y) = 9

The quantities of carrots and donuts that will maximize Donald's utility are = 2 and 9