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Unformatted text preview: in cash. • Suppose the interest amount agreed to by Mary and John is $10. • Then, at the end of the year of the loan, John repays Mary the principal plus the interest, or $110:
– Principal + Interest = $100 + $10 = $110 Example Cont. Example Cont.
• The (simple) interest rate of the loan, denoted r, can be found by solving the following equation for r:
– Principal + Interest = (1 + r) Principal • For this example: • So, we find: $110 = (1 + r) $100 r = 10/100 or 10% • Hence, the interest rate on the loan was 10%. Example Cont. Example Cont.
• Generally, we can find the interest rate by noting that:
– B1 = B0 + r B1 = (1+r) B0 • where B0 = Benefit today, and B1 = Benefit tomorrow The Interest Rate is an Equilibrium of Outcome The Interest Rate is an Equilibrium of Outcome • • C1 = consumption in period 1 C2 = consumption in period 2 • Delay of consumption (saving) in period 1 reduces current utility but increases utility in period 2. The Interest Rate is an Equilibrium of The Interest Rate is an Equilibrium of Outcome Cont. • The intertemporal production possibilities curve (IPP) denotes the between consumption today and consumption in the future. utility. technological possibilities for tradingoff present vs. future consumption. • The curve S, is an indifference curve showing individual preferences • Any point along a particular indifference curve leads to the same level of • Utility maximization occurs at point A, where S is tangent to the IPP. • The interest rate, r, that is implied by this equilibrium outcome, can be found by solving either of the following two equations for r:
– slope of S at point A = (1 + r) – slope of IPP at point A = (1 + r) The Interest Rate is an Equilibrium of The Interest Rate is an Equilibrium of Outcome Cont. • Therefore, if we can determine the slope of either S or IPP at tangency point A, then we can calculate the interest rate, r. This is often done by solving the following individual optimization problem where I is the total income available over the two periods:
Max.{ (C1, C 2 )} U
C1,C 2 æ1ö subject to: I = C1 + ç ÷C 2 è1 + r ø The Interest Rate is an Equilibrium of The Interest Rate is an Equilibrium of Outcome Cont. • which can be written as: 1 L = U (C1, C 2) + l I − C1 + C 2 1+ r FOCS: UC 1 = l UC 1 l⇒ = 1+ r UC 2 = UC 2 1+ r The Indifference Curve The Indifference Curve
• The indifference curve is found by setting: UC 1dC 1 + UC 2 dC 2 = 0 ⇒ dC 2 − UC 1 = = − (1 + r ) dC 1 UC 2 • The indifference curve simply indicates that the equilibrium occurs where an individual cannot improve her intertemporal utility at the margin by changing the amount consumed today and tomorrow, within the constraints of her budget. The Components of Interest Rate The Components of Interest Rate
• Interest rates can be decomposed int...
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 Spring '09
 ZELBERMAN
 Economics, Environmental Economics

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