John lives three periods (t=0, 1, 2) and consumes non-durable goods (c) and housing (H). For t=0, John's utility is Uo = 0. At the end of t=0, John receives an endowment Ao into a savings account and a house Ho. Between t and t+1, each dollar earns interest of r dollars.

In t=1,2 John splits between non-durables (ct) and investing in housing (ht). Ct costs pt dollars and ht costs vt dollars. Housing depreciates by a rate of ? (i.e., Ht=(1-?)Ht-1 + ht. John receives wages of y1 dollars in t=1 and y2 dollars in t=2.

A1 is savings account after t=1. A2=0 (John's account is shut down since he is dead.) John can borrow/save freely and invest/disinvest in his house, i.e., A1/h1/h2 can be

John's lifetime utility function is

U(c1, c2, H1, H2)=u(c1, H1) + B u(c2, H2)

for t = 1,2, sub utility is:

u(ct, Ht) = a log (ct) + (1-a) log (Ht) John's canonical lifetime utility problem can be written as:

maxc1,c2,H1,H2 U(c1,c2,H1,H2) s.t. ?1c1 + ?2c2 + ??1H1 + ??2H2 = S.

9. Solve John's utility maximization problem. What are his optimal choices (cf, c3, H i\" , H 3)? Also, nd the expressions for At as well as ( 'f, h;). 10. Discuss, in words, what would happen to Hf if '01 increases? Make sure you ad dress income effect, substitution effect, and endowment effect in your answer. A full mathematical derivation is not required here. 11. How would John's problem change if he could not disinvest in housing