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Unformatted text preview: Lecture 9: Asset Markets c & 2008 Je/rey A. Miron Outline: 1. Present and Future Value 2. Bonds 3. Taxes 4. Applications 1 Present and Future Value In the discussion of the twoperiod model with borrowing and lending, we showed that the budget constraint can be written as c 1 + c 2 (1 + & ) = ! 1 + ! 2 (1 + & ) where the ! &s are endowments of the consumption good at time 1 and 2. A simple extension of this framework says that if a consumer receives income in each of two periods, then the budget constraint is c 1 + c 2 (1 + & ) = m 1 + m 2 (1 + & ) where the m &s are income. This equation is an example of a present value formula. In words, it says that the present value of consumption equals the present value of income. It is useful to review and extend the idea of present value. The key insight is that a dollar today is not equivalent to a dollar tomorrow. The exact tradeo/ depends on the interest rate, assuming the consumer can borrow and lend at this rate. A few additional comments are useful reminders or extensions. 1 The concept of present value is related to the concept of future value. The FV in one period of a dollar today is FV (1) = (1 + r ) because you could lend the dollar and earn interest between now and next period. Similarly, the future value of a dollar T periods from now is FV ( T ) = (1 + r ) T The present value of a dollar one period from now is PV (1) = 1 = (1 + r ) because if you had 1 = (1+ r ) and loaned it out you would have one dollar next period. The PV of a dollar T periods from now is PV ( T ) = 1 = (1 + r ) T These formulas apply in cases with multiple time periods and multiple payments. For example, the PV of receiving m t in period t where t goes from, say, 1 to T is PV = P T t =1 m t (1 + r ) t We could also have the summation go to in&nity; this particular case is useful in many settings. Finally, the concept of present or future value does not require the interest rate to be constant; one simply has to use a more general formula to account for this possibility. For example, the PV of a dollar three periods from now would be PV = 1 (1 + r 1 )(1 + r 2 )(1 + r 3 ) where the r i s are the interest rates in the three di/erent periods. This formula gets used frequently in macroeconomics, &nance, and other settings. The crucial economic insight related to present value in the context of budget constraints is that, assuming the consumer can borrow and lend at the market in terest rate, the consumer should be indi/erent about the timing of when income or endowments arise, assuming one is holding the present value of the endowment 2 stream constant. Any change in income or an endowment that raises the present value allows the consumer more choices about what to consume by shifting the bud get constraint out. This applies even if the consumer has &oddpreferences, since, unless they are very odd, the consumer always prefers more to less. This point will arise later in discussions of behavioral economics....
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 Fall '09
 JeffreyA.Miron

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