Answer to Question #180641 in Microeconomics for MURIITHI MUCHEMI

Question #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




1
Expert's answer
2021-04-18T19:29:42-0400

SOLUTION.

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


Utility Function= UU (X,Y)(X,Y) =X1/4Y3/4X^{1/4} Y^{3/4}


Income (M)(M) = $120.


Price of Carrots PxP_{x} = $2

Price of Donuts PyP_{y} = $6


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

In this case, MRSMRS = PxPy\frac{P_x}{P_y}

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


The budget line will be,

Income (MM ) == (price of carrots ×\times quantity of carrots) ++ (price of donuts ×\times quantity of donuts)

MM == (PxP_x ×Qy\times Q_y ) ++ (Py×Qy)P_y \times Q_y)

120120 == (2×Qx)+(6×Qy)( 2 \times Q_x) + (6 \times Q_y)

120=2Qx+6Qy120 = 2Q_x + 6Q_y


MRSMRS =marginalutilityofcarrotsmarginalutilityofdonuts= \frac{marginal utility of carrots}{marginal utility of donuts}


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


U=X14Y34U = X^\frac{1}{4}Y^\frac{3}{4}


MUx=dudx=14X34Y34MU_x = \frac{du}{dx} = \frac{1}{4}X^\frac{-3}{4}Y^\frac{3}{4}


MUy=dudy=34X14Y14MU_y = \frac{du}{dy} = \frac{3}{4}X^\frac{1}{4}Y^\frac{-1}{4}


MRS=(14X34Y)/(34X14Y14)MRS = (\frac{1}{4}X^\frac{-3}{4}Y) / ( \frac{3}{4}X^\frac{1}{4}Y^\frac{-1}{4} ) simplify the equation


MRS=Y3XMRS = \frac{Y}{3X}

MRS=Y3X=PyPx=62MRS = \frac{Y}{3X} = \frac{P_y}{P_x} = \frac{6}{2}  


          Y=9XY=9X


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

120=2Qx+6Qy120=2Q_x+6Q_y

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

120=2Qx+54Qx120=2Q_x+54Q_x

120=56Qx120=56Q_x

Qx=157Q_x=\frac{15}{7}


Substitute QxQ_x to the equation to find QyQ_y

120=2(157)+6Qy120=2(\frac{15}{7})+6Q_y

120=307+6Qy120= \frac{30}{7}+ 6Q_y

Qy=1357Q_y=\frac{135}{7}


Therefore the Quantities that will maximize Donald’s utility are Qx=157Q_x=\frac{15}{7} and Qy=1357Q_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>PxPyMRS>\frac{P_x}{P_y}

Therefore, MRS will be greater.



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