# 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) = X^{1/4}Y^{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

**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 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.