# Answer in Microeconomics for MURIITHI MUCHEMI #180641

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

His utility function is as follows: U(X,Y) = X1/4Y3/4. Donald has an income (M) of \$120 and the price of carrots (PX) is \$2 while the price of donuts (PY) is \$6. What quantities of carrots and donuts will maximize Donald’s utility? How does MRSXY change as the firm uses more Xholding utility constant

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SOLUTION.

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

Utility Function= “U” “(X,Y)” =“X^{1/4}nY^{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 MRSxy 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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