184 INTERTEMPORAL CHOICE (Ch. 10)

borrower if c1 > m1, and the interest he has to pay in the second periodwill be r(c1 m1). Of course, he also has to pay back the amount that heborrowed, c1 m1. This means his budget constraint is given by

c2 = m2 r(c1 m1) (c1 m1)

= m2 + (1 + r)(m1 c1),

which is just what we had before. If m1 c1 is positive, then the consumerearns interest on this savings; if m1 c1 is negative, then the consumerpays interest on his borrowings.

If c1 = m1, then necessarily c2 = m2, and the consumer is neither aborrower nor a lender. We might say that this consumption position is thePolonius point.1

We can rearrange the budget constraint for the consumer to get twoalternative forms that are useful:

(1 + r)c1 + c2 = (1 + r)m1 + m2 (10.2)

and

c1 +c2

1 + r= m1 +

m21 + r

. (10.3)

Note that both equations have the form

p1x1 + p2x2 = p1m1 + p2m2.

In equation (10.2), p1 = 1 + r and p2 = 1. In equation (10.3), p1 = 1 andp2 = 1/(1 + r).

We say that equation (10.2) expresses the budget constraint in terms offuture value and that equation (10.3) expresses the budget constraint interms of present value. The reason for this terminology is that the firstbudget constraint makes the price of future consumption equal to 1, whilethe second budget constraint makes the price of present consumption equalto 1. The first budget constraint measures the period-1 price relative tothe period-2 price, while the second equation does the reverse.

The geometric interpretation of present value and future value is given inFigure 10.2. The present value of an endowment of money in two periods isthe amount of money in period 1 that would generate the same budget setas the endowment. This is just the horizontal intercept of the budget line,which gives the maximum amount of first-period consumption possible.

1 Neither a borrower, nor a lender be; For loan oft loses both itself and friend, Andborrowing dulls the edge of husbandry. Hamlet, Act I, scene iii; Polonius givingadvice to his son.