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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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