Assume the household has preferences given by lnc + lnc0 and (0, 1]. The household is given income y in period 1 and net income y 0 in period 2. Assume there is no government in this problem and taxes are therefore equal to zero in every period.

1. Solve for optimal c

2. Suppose the household wins a lottery in thefirst period. Show how consumption changes with respect to a change in temporary income y

3. Now suppose that y = y 0 (assume no lottery win for part (c)). Suppose instead that households can only borrow against a fraction of their discounted income in the second period, i.e. let the borrowing constraint be given as:

s y 0 1 + r

where (0, 1) and < 1(1+r) 1+ .

Solve for the household's consumption when he is faced with this borrow constraint.

4. Suppose again the household wins a lottery in the first period. Show how consumption changes with respect to a change in a temporary increase in y when the household is borrowing constrained.