Questions -
Answer all parts of two of the following three questions. Label all graphs and show all calculations.
Question 1 - Suppose an individual's preferences are characterized by the utility function U = X0.25Y0.75.
Does utility increase or decrease as consumption of each good increases? That is, confirm whether the first partial derivatives of the utility function are positive or negative.
Does the law of diminishing marginal utility hold in both of the goods at all quantities? That is, confirm whether the second partial derivatives are negative or positive.
Derive and graph the indifference function associated with the bundle X = 2 and Y = 4 and calculate the margin of substitution (the absolute value of the slope of the indifference curve) at this bundle. Show all steps for credit.
Question 2 - Suppose Fran's preferences are characterized by the Cobb-Douglass utility function U(X, Y) = XaYb. Fran has a period income equal to I, and the per-unit prices of good X and good Y are, respectively, equal to PX and PY. Of integral is characterizing Fran's demand for good X and good Y, the associated price elasticities, and her indirect utility functions. Specifically:
Specify the Lagrangian function that will be maximized:
Derive the 3 first order conditions (FOC's):
Using the FOC's derived in 'B', derive Fran's demand functions for good X and good Y. Show all steps.
Question 3 - Assume an individual receives a per-period income equal to I and exhausts this on goods X and Y, whose prices are respectively, equal to PX and PY. Show the following using the individual's indirect utility function U(I, PX, PY).
A. The equivalent variation (EV) associated with a reduction in the price of good X from PX0 to PX1.
B. The compensating variation (CV) associated with a reduction in the price of good X from PX0 to PX1.
C. The equivalent variation (EV) associated with an increase in the price of good X from PX0 to PX1.
D. The compensating variation (CV) associated with an increase in the price of good X from PX0 to PX1.