Intermediate Microeconomics

PROBLEM 2: Theory of Demand

Randy chooses between two goods, x and y, with prices p_x and p_y, respectively. He has an income I and his preferences are represented by the utility function U (x, y) =x + y.

1. Assuming that an interior solution exists to the constrained utility maximization problem,

derive Randy's Marshallian demand function for each of the two goods. Are both goods

normal? Explain

2. Find the indirect utility function, V (p_x, p_y, I).

3. Derive Randy's Hicksian demand function for each of the two goods and the expenditure function. Compare the Marshallian demand for good x and the Hicksian demand for good x. Are these different functions? If so, why? If not, why not?

4. Suppose that I = 100, p_x = 1 and p_y = 2. How much of good x and good y will Randy

optimally choose?

5. Now the price of good x rises to p_x = 2, while income (I = 100) and the price of good y,

p_y= 2, remain unchanged. What quantities does Randy buy and what is her resulting utility?

Illustrate graphically

6. Find the income and substitution effect for good x due to this price change. That is, which change in the consumption of x is due to pure substitution and which change in the consumption of x is due to the income effect? Illustrate your response graphically.

7. If the government wants to compensate Randy for the price increase of good x

from two to four dollars, by giving her some extra income, how much extra income would be needed to bring her back to the old utility level? In other words, find the compensating variation for the price change. Show all your calculations and illustrate graphically.