An individual consumes products X and Y and spends 25 per ti

An individual consumes products X and Y and spends $25 per time period. The prices of the two goods are $3 per unit for X and $2 per unit for Y. The consumer in this case has a utility function expressed as: U(X,Y) = 0.5XY MU_X = 0.5Y MU_Y = 0.5X.

a. Express the budget equation mathematically.

b. Determine the values of X and Y that will maximize utility in the consumption of X and Y.

c. Determine the total utility that will be generated per unit of time for this individual.

Solution

(a) In the mathematical form, budget equation for given case will be expressed as follows -

P_X * X + P_Y * Y = I

3 * X + 2 * Y = 25

3X + 2Y = 25

The budget equation is 3X + 2Y = 25

(b) Utility is maximized when,

MRS = P_X/P_Y

MRS = MU_X/MU_Y

MU_X = 0.5Y

MU_Y = 0.5X

MRS = 0.5Y/0.5X

Putting value of MRS in utility maximizing condition,

MRS = P_X/P_Y

0.5Y/0.5X = 3/2

Y = 1.5X

Putting value of Y in budget equation,

3X + 2Y = 25

3X + 2*1.5X = 25

6X = 25

X = 4.17

Y = 1.5X = 1.5 * 4.17 = 6.25

Thus, the value of X and Y that will maximize utility in consumption of X and Y is

X = 4.17 units

Y = 6.25 units

(c) Calculate Total Utility -

TU = 0.5XY = 0.5 * 4.17 * 6.25 = 13.03

The total utility that will be generated per unit of time for this individual is 13.03.