Question

Jeanette has the following utility function: U-ain(x) + b*In(y), where a+b=1 a) For a given amount of income I, and prices Px, Py, find Jeanette's Marshallian demand functions for X and Y and her indirect utility function. (6 points)

how to find indirect utility function here?

Answer 1

The utility function would be as $\dpi{80} U= a*ln(x) + (1-a)*ln(y)$ .

(a) The budget constraint would be $\dpi{80} x P_x + y P_y =I$ . The optimal combination of goods would be where $\dpi{80} MRS= \frac{P_x}{P_y}$ or $\dpi{80} \frac{MU_x}{MU_y}= \frac{P_x}{P_y}$ or $\dpi{80} \frac{a/x}{(1-a)/y}= \frac{P_x}{P_y}$ or $\dpi{80} (a/x) P_y = P_x (1-a)/y$ or $\dpi{80} a y P_y = (1-a) P_x x$ or $\dpi{80} y = \frac{(1-a) P_x}{a P_y} x$ .

Putting this in the constraint, we have $\dpi{80} x P_x + \frac{(1-a) P_x}{a P_y} x *P_y = I$ or $\dpi{80} (1 + \frac{(1-a) }{a}) x P_x = I$ or $\dpi{80} \frac{1}{a}* x P_x = I$ or $\dpi{80} x^* = \frac{a * I}{P_x}$ , and since $\dpi{80} y = \frac{(1-a) P_x}{a P_y} x$ , we have $\dpi{80} y^* = \frac{(1-a) P_x}{a P_y} x^*$ or $\dpi{80} y^* = \frac{(1-a) P_x}{a P_y} * \frac{a * I}{P_x}$ or $\dpi{80} y^* = \frac{(1-a)*I}{P_y}$ . These are the required Marshallian demand functions.

For the utility function be U=f(x,y) and the Marshallian demand functions be x^*(Px,I) and y^*(Py,I), the indirect utility function would be U^*=f(x^*,y^*). In this case, the indirect utility function would be as $\dpi{80} U^* = a*ln(x^*) + (1-a)*ln(y^*)$ or $\dpi{80} U^* = a*ln(\frac{a * I}{P_x}) + (1-a)*ln(\frac{(1-a)*I}{P_y})$ or $\dpi{80} U^* = ln(\frac{a^a * I^a}{P_x^a}) + ln(\frac{(1-a)^{(1-a)}*I^{(1-a)}}{P_y^{(1-a)}})$ or $\dpi{80} U^* = ln \left (\frac{a^a * I^a}{P_x^a} * \frac{(1-a)^{(1-a)}*I^{(1-a)}}{P_y^{(1-a)}} \right )$ or $\dpi{80} U^* = ln \left (\frac{a^a (1-a)^{(1-a)} I}{P_x^a P_y^{(1-a)}} \right )$ .