Borrowers and savers have utility function u (C^b_1, C^b₂)= log(C₁^b)+B_blog(C^b₂), u (C^s_1, C^s₂)= log(C₁^s)+B_blog(C^s₂). with b < s. The representative borrower is subject to the period-by-period budget constraints:

C₁^b= Y₁^b q K + D₁^b, C₂^b= Y₂^b+ q K ( 1 + r ) D₁^b,

whereD1bis debt,Ytbis income in periodt, 1+ris the gross real interest rate, andKis a durable asset in fixed supply. The borrower is also subject to the following borrowing constraint:

D1bqK. (1) The representative saver is subject to the period-by-period budget constraints:

C₁^s=Y₁^s+D₁^s,

C₂^s=Y₂^s(1+r)D₁^s.

Explain the economic meaning of constraint (1).

For each agent, derive the intertemporal budget constraint. Explain its meaning.

For each agent, derive the Euler equation. Explain.

Derive a representation of the equilibrium in terms of DB curve (demand of debt) and DS curve (supply of debt), in the space real interest rate-debt. Plot the DB-DS system, including the borrowing constraint. Comment on the difference between constrained and unconstrained equilibrium. (Hint: remember that, in equilibrium, D₁^b=D₁^sD.)

5. Suppose the intersection between the DB and the DS in part (4) occurs on thedecreas- ing partof the DB curve. Suppose now that savers become more patient, that is,sin- creases. Show the new equilibrium in the DB-DS diagram (consider all possible cases).

6. Characterize the effects of the change inson the equilibrium quantity ofdebt, on thebsb s

real interest rate, and onC1,C1,C2, andC2. If it applies, consider both(i)the case where the new intersection between the DB and the DS occurs on the decreasing part of the DB curve and(ii)the case where the new intersection happens on the vertical part of the DB curve.

Is it possible that the equilibrium interest rate becomes negative? Explain.

How would your answer in part (7) change if, in addition to the change ins, the

banking sector would also reduce the maximum LTV ratio?