SOLUTION.
Quantity of Donuts and Carrots that maximizes Donald’s utility.
Utility Function= U (X,Y) =X1/4Y3/4
Income (M) = $120.
Price of Carrots Px = $2
Price of Donuts Py = $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 = PyPx
PyPx is the price ratio of carrots and donuts.
The budget line will be,
Income (M ) = (price of carrots × quantity of carrots) + (price of donuts × quantity of donuts)
M = (Px ×Qy ) + (Py×Qy)
120 = (2×Qx)+(6×Qy)
120=2Qx+6Qy
MRS =marginalutilityofdonutsmarginalutilityofcarrots
MRS=MUyMUx
U=X41Y43
MUx=dxdu=41X4−3Y43
MUy=dydu=43X41Y4−1
MRS=(41X4−3Y)/(43X41Y4−1) simplify the equation
MRS=3XY
MRS=3XY=PxPy=26
Y=9X
Substitute Y=9X into the Budget line equation
120=2Qx+6Qy
120=2Qx+6Q(9x)
120=2Qx+54Qx
120=56Qx
Qx=715
Substitute Qx to the equation to find Qy
120=2(715)+6Qy
120=730+6Qy
Qy=7135
Therefore the Quantities that will maximize Donald’s utility are Qx=715 and Qy=7135
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>PyPx
Therefore, MRS will be greater.
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