1The Slutsky equation

For each of the following utility functions, i) U (x₁, x₂) = ax^b + x₂, where a > 0 and0 < b < 1 and ii) U (x₁, x₂) = x₁x₂, do the following:

a).derive the(ordinary) demand and compensated demand function for good 1,

x^c (p₁, p₂, u₀),

b).verify the Slutsky equation:

dx₁dp₁

1 1

d x^c=x

dp₁

dx₁dI

c).for simplicity, let's assume that I = 10 and p₂ = 1; find

c.1)the maximum utility u₁ the consumer gets when p₁ = 1,

c.2)the maximum utility u₂ the consumer gets when p₂ = 2,

d).in the same price-quantity diagram, draw the following curves: i) the demand for good 1, ii) the compensated demand for good 1 with the utility level fixed at u₁ (that is, u₀ = u₁ in the compensated demand function), and iii) the compensated demand for good 1 with the utility level fixed at u₂,

e).from the curves that you have drawn in part d), indicate CV, CS and EV for the price changes from p₁ = 1 to p₁ = 2.

2Labor Supply

Consider a consumer who chooses an amount of consumption c and leisure time h to maximize the utility function given by

U (c, h) = c h

subject to the constraint c = w(24 h), where w > 0 is the wage rate.

a).Suppose a 25% tax rate is imposed on the consumer's labor income. Find the consumer's demand functions.

b).Suppose a lump-sum tax of size 3w is imposed on the consumer. Find the consumer's demand functions.

c).From your answers to the above two parts, which of the two tax policies would you recommend to the government? Explain.

3Savings and borrowings

Consider a person who chooses consumption c₁ and c₂ in the current (c₁) and future (c₂) period to maximize the utility function U (c₁, c₂) = c²c₂. This person has incomeI₁> 0 in the first period and I₂> 0 in the second.

a).Formulate the relevant utility-maximization problem, and find the demands for

c₁ and c₂.

b).Let I₁ = I₂ = 100 and r = 0.1. Does this person borrow or save money, or neither?

Now suppose the interest rate increases. How are current and future consumption aected? What happens to the amount this person borrows or saves?