Question

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

 His utility function is as follows: U(X,Y) = X^1/4Y^3/4. Donald has an income (M) of $120 and the price of carrots (P_X) is $2 while the price of donuts (P_Y) is $6. What quantities of carrots and donuts will maximize Donald's utility? How does MRS_XY change as the firm uses more X, holding utility constant



Answer 1

SOLUTION.

Quantity of Donuts and Carrots that maximizes Donald’s utility.

Utility Function= $U$ $(X,Y)$ =$X^{1/4} Y^{3/4}$

Income $(M)$ = $120.

Price of Carrots $P_{x}$ = $2

Price of Donuts $P_{y}$ = $6

The marginal rate of substitution ($MRS$ ) is the rate, which the consumer is willing to substitute one good for another.

In this case, $MRS$ = $\frac{P_x}{P_y}$

$\frac{P_x}{P_y}$  is the price ratio of carrots and donuts.

The budget line will be,

Income ($M$ ) $=$ (price of carrots $\times$ quantity of carrots) $+$ (price of donuts $\times$ quantity of donuts)

$M$ $=$ ($P_x$ $\times Q_y$ ) $+$ ($P_y \times Q_y)$

$120$ $=$ $( 2 \times Q_x) + (6 \times Q_y)$

$120 = 2Q_x + 6Q_y$

$MRS$ $= \frac{marginal utility of carrots}{marginal utility of donuts}$

$MRS = \frac{MU_x}{MU_y}$

$U = X^\frac{1}{4}Y^\frac{3}{4}$

$MU_x = \frac{du}{dx} = \frac{1}{4}X^\frac{-3}{4}Y^\frac{3}{4}$

$MU_y = \frac{du}{dy} = \frac{3}{4}X^\frac{1}{4}Y^\frac{-1}{4}$

$MRS = (\frac{1}{4}X^\frac{-3}{4}Y) / ( \frac{3}{4}X^\frac{1}{4}Y^\frac{-1}{4} )$ simplify the equation

$MRS = \frac{Y}{3X}$

$MRS = \frac{Y}{3X} = \frac{P_y}{P_x} = \frac{6}{2}$  

          $Y=9X$

Substitute $Y=9X$ into the Budget line equation

$120=2Q_x+6Q_y$

$120=2Q_x+6Q(9x)$

$120=2Q_x+54Q_x$

$120=56Q_x$

$Q_x=\frac{15}{7}$

Substitute $Q_x$ to the equation to find $Q_y$

$120=2(\frac{15}{7})+6Q_y$

$120= \frac{30}{7}+ 6Q_y$

$Q_y=\frac{135}{7}$

Therefore the Quantities that will maximize Donald’s utility are $Q_x=\frac{15}{7}$ and $Q_y=\frac{135}{7}$

How does MRS_xy change as the firm uses more X, holding utility constant?

• Since the firm will use more X and less Y, the $MRS>\frac{P_x}{P_y}$

Therefore, MRS will be greater.

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