Question

Leon consumes two goods, X and Y. His utility function is given by U(X,Y) = 56XY. The price of X is $14 a unit; the price of Y is $8 a unit; and Leon has $672 to spend on X and Y.

a. Provide the equation for Leon’s budget line. (Your answer for the budget line should be in the form Y = a – bX, with specific numerical values given for a and b.)

b. Provide the numerical value of Leon’s marginal rate of substitution of X for Y when utility is maximized.

c. Provide the numerical value of Leon’s utility-maximizing amount of X.

d. Provide the numerical value of Leon’s utility-maximizing amount of Y.

e. Provide the numerical value of Leon’s marginal utility of income when utility is maximized.

Answer 1

Utility is given by U = 56XY
Price of X, Px = $14
Pprice of Y, Py = $8
Total budget to be spent on X & Y is $672

a) Budget equation of Leon is as follows:
Total budget = Price of X * quantity of X + Price of Y * Quantity of Y
672 = 14X + 8Y

This can be re-written as follows:
8Y + 14X =672
or, 8Y = 672 - 14X
or, Y = (672/8) - (14/8)X
or, Y = 84 - 1.75X

Hence the budget equation in the specific form is Y = 84 - 1.75X

b) Utility maximises where MUx/Px = MUy/Py

Given utility = 56XY
MUx = 56Y
MUy = 56X

Px = 14
Py = 8

At the utility maximising point, 56Y/14 = 56X /8
or, 8Y = 14X
Plugging the above equality in the budget equation,

672 = 14X + 8Y
or, 672 = 14X + 14X

or, 672 = 28X
or, X = 672 / 28 = 24

plugging X = 24 in 8Y = 14X
or, 8Y = 14*24
or, Y = 14*24/8
   = 14 * 3
= 42

Hence at X = 24 and Y = 42, utility will be maximum. We are calculating MRS for these values of X & Y.
Marginal rate of substitution is calculated by the ratio of utilities.
MRSxy = MUx / MUy
= 56Y / 56X
= 56 * 42 / 56 * 24
= 42 / 24  
= 1.75

Hence numerical value of MRSxy = 1.75

c) As calculated in part (b), utility maximising amount of X = 24 units
d) As calculated in part (b), utility maximising amount of Y = 42 units

e) Value of marginal utility of income is calculated by substituting the value of X & Y in the utility function.
Utility = 56XY = 56 * 24 * 42 = 56448
Hence marginal utility of income = 56448