Consider an economy of three-period-lived people in overlapping generations. Each person is endowed with y goods when young and nothing when middle-aged and old. The population of each generation born in period t is Nt , where Nt = n Nt1.

The young can save a part of their endowment for capital which yields return after 2 periods. The rate of return on capital is x in 2 periods time and it depreciates at the rate d every period. Loans can be exchanged between the young and the middle-aged individuals, which the borrower has to pay back to the lender in the next period.

Additional Information if needed:

• Borrowers are endowed with nothing when young and with y

when old.

• Lenders are endowed with y when young and with nothing when old.
• For simplicity there is no at money. We'll start with the case of no capital either.

... In the 1st period, lenders divide the endowment between consumption (c1,L) and loans to borrowers (l)

... When old, they just consume (c2,L) the repayment on their loans (rl)

...To simplify, let's focus only on stationary equilibrium.

Budget constraints are: c1,L+l=y (14)

• c2,L=rl (15)

Combined they become:

c1,L + (c2,L)/r = y

Borrower's problem

When young, the borrower consumes (c1,B) only what he borrows (b).

When old, he consumes (c2,B) what's left of the endowment after repaying the loan.

Budget constraints are: c1,B=b (17)

c2,B=y - rb (18)

combined they become:

c1,B + (c2,B)/r y/r

Private Debt and Capital

Now, let's introduce capital.

For simplicity, assume capital pays a fixed gross rate of return:x

Now lenders have to choose between 2 assets: capital and loans.

We further assume there is no risk in either asset.

How to choose one's portfolio?

Suppose x

Now suppose that x r*

Lenders will want to give loans only if the interest rate is equal to x.

This means that the interest rate will exceed r*

QUESTIONS:

a.Explain how credit can be used to provide for consumption when middle-aged. Explain your answer in terms of the use and source of funds for these loans.

b.Point out who lends to whom and write the condition for the equality of supply and demand for loans in period t.

c.Write the budget constraints for the young, the middle-aged and the old with clear and fully explained/ well defined notations.

d.Find an equation that represents the set of feasible stationary allocations in presence of a central planner and explain it in words.