The Household Problem for T = 2 (the Fisher model)

Joe Six-pack just learned about the two period household maximization model and fell so deeply in love with it that he decides that he wants to live (i.e. consume) according to it. He is in the first period of his life and he spends it in college. In college he works in the cafeteria where he makes $20,000 after taxes. He knows for sure that he is going to get a job at McKinsey (he is an economics major) and will make $110,000 after taxes in the second period of his life (sadly enough he only lives for two periods). He has no initial wealth and he can borrow and lend at an interest rate of 10%. His utility function is

U(c₁, c₂) = log(c₁) + 0.5 log(c₂)

1. Determine Joe's optimal consumption in both periods of his life. Is Joe a saver or a borrower? Determine his optimal level of savings (assets to be brought from period 1 to period 2). 

2. How would Joe's optimal consumption and saving decision change if he had a utility function of the form

U(c₁,c₂)=c₁ ∗c₂^0.5

Justify your answer (note that this question can be answered without any calculations, but you have to remember certain facts about utility functions from your micro course). 

3. Back to Joe's original utility function for the rest of the question. Suppose the interest rate increases to 16%. Now what is Joe's optimal consumption and saving/borrowing plan? 

4. Suppose the interest rate is back at 10%. The government wants to cut taxes to stimulate the economy. Suppose the government cuts the taxes that Joe has to pay in the first period by $2,000, so that Joe's after tax income increases from $20,000 to $22,000. In order to finance this tax cut the government has to increase taxes in the second period by $2000 ∗ (1 + r) = $2, 200. Hence Joe's income in the second period goes down from $110, 000 to $107, 800. What is Joe's new optimal consumption and saving plan. Compare to your answer in Part 1. 

5. Suppose more generally that Joe's utility is given by U(c₁,c₂) = [(c₁^1-σ -1)/(1-σ)] + [β (c₂^1-σ -1)/(1-σ)] where σ > 0 is a fixed parameter. He has no initial wealth, and only income y₁ in period 1. The interest rate is r. What can you say about the effect of a change in the interest rate on consumption c₁, ∂c_1/∂(1+r)? How does this depend on the parameter σ?