A consumer maximizes utility u(c, c) by choosing c and c, where c is the amount of consumption in the current period, and c is the amount of consumption in the future period. The consumers income in the current period is y, and income in the future period is y. The consumer pays lump-sum taxes T in the current period, and T in the future period. The real interest rate at which the consumer can save is r>0.

The consumer is given two options. First, he can borrow at the interest rate r but can only borrow up to a maximum amount x, where

x = we-y+T, where we = lifetime wealth.

Second, he can borrow an unlimited amount at the interest rate r > r.

This problem contrasts two alternative forms of credit market imperfections. As one possibility, consumers may either borrow or lend at the same real interest rate r, but face a maximum amount of borrowing. The alternative possibility allows unlimited borrowing, but the interest rate paid on borrowing exceeds the interest rate earned from lending, r > r.

Clearly, consumers who choose to be lenders are unaffected by such constraints. We therefore only need to be concerned about the behavior of borrowers.

Use diagrams to determine which option a borrower would choose, and explain your results.