Question

(10 Question 1: marks) Given is the Total Utility Function along with Budget Constraint: Utility Function: U (X, Y) = X°.270.3 Budget Constraint: I = XP, + YP, a. What is the consumer's marginal utility for X and for Y? b. Suppose the price of X is equal to 4 and the price of Y equal to 6. What is the utility maximizing proportion of X and Y in his consumption? {construct the budget constraint) c. If the total amount of money he is willing to spend on the two goods is equal to I= 60, how much of each will he consume? d. Calculate the optimal consumption of X and Y if the consumer has income I.

Answer 1

1. (a) The marginal utility for X is
MUX au ax
or $MU_X = \frac{\partial }{\partial X}(X^{0.2} Y^{0.3})$ or
MUX = 0.2x-0.8y0,3
. The marginal utility for Y is $MU_Y = \frac{\partial U}{\partial Y}$ or
a MUY = (X0-203) ay
or .

(b) The budget constraint is . The slope of the budget constraint $Y = - \frac{2 X}{3} + \frac{I}{6}$ would be $\frac{\mathrm{d} Y}{\mathrm{d} X} = - \frac{2}{3}$ , ie 2/3. The utility maximizing combination of goods would be where MRS is equal to the slope of budget constraint, which would be where $MRS = \frac{2}{3}$ or $\frac{MU_X}{MU_Y} = \frac{2}{3}$ or $\frac{0.2 X^{- 0.8} Y^{0.3}}{0.3 X^{0.2} Y^{- 0.7}} = \frac{2}{3}$ or $\frac{0.2 Y}{0.3 X} = \frac{2}{3}$ or $\frac{Y}{X} = 1$ or .

(c) The budget constraint for the given income would be as . Putting the utility maximizing combination of goods in it, we have or , and since , we have or . These are the required demand for goods.

(d) For the budget constraint with an arbitrary income value, we have . Again, putting in the budget constraint, we have or $X^* = \frac{I}{10}$ , and since , we have $Y^* = \frac{I}{10}$ . These are the required optimal consumption values for arbitrary income (the equations basically represents the Engel's curve for X and Y).