Say you have $X today and can earn an annual interest rate r by investing it. Let FV denote the future value of your investment and t = time.

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1 Same as with Labor Supply, maximizing utility in the ontext of intertemporal hoies is IDEN- TICAL to what we ve been doing, just with a different budget onstraint. Present and Future Value Say you have $X today and an earn an annual interest rate r by investing it. Let FV denote the future value of your investment and t = time. Then Rearranging terms, we have that X( + r) t = FV. = Future Value Equation X = FV ( + r) t PV = FV ( + r) t = Present Value Equation sine the value of your money today, $X, is the present value of your investment. Example Find the present value of 3 payments of $00 eah if the annual interest rate is %50. Assume the first payment is made today, the seond payment is made year from today, and the third payment is made 2 years from today. Using our present value equation, Budget Constraint P V = 00 ( +.5) ( +.5) + 00 ( +.5) 2 Let = onsumption today, = onsumption year from today, m = inome today in terms of present onsumption, = inome year from today in terms of onsumption year from today, r = nominal interest rate, and π = inflation rate. To understand the derivation of the intertemporal budget onstraint, let s bring bak our old friend p x + p 2 x 2 = m. For this budget onstraint, p is the ost of x, p 2 is the ost of x 2, and m is inome. What s the ost of onsuming today? The interest rate r you miss out on by not investing - an opportunity ost. What s the ost of onsuming year from today? Inflation, sine it eats away the value of your money. Then sine total inome equals inome today + inome year from today, we have that ( + r) + ( + π) = ( + r)m + ( + π). We an write the budget onstraint two ways: in terms of present value and future value. ( + π) + ( + r) ( + π) 2 = m + ( + r) = Present Value Budget Constraint ( + r) ( + π) ( + r) + = ( + π) m + = Future Value Budget Constraint Nominal vs Real Interest Rate The differene is similar to the differene betwewen nominal and real inome. Definition: Nominal Interest Rat The raw interest rate you an earn on an investment. This is the interest rate you will see in the real world on your bank aount, stoks and bonds.

2 Definition: Real Interest Rate The atual, inflation adjusted interest rate you an earn on an investment. You may have seen the Fisher equation: ρ = r π, where ρ is the real interest rate. In fat, this is only an approximation. In truth, + ρ = + r + π. Substituting in + ρ into the budget onstraint, we have that + ( + ρ) = m + ρ = Present Value Budget Constraint ( + ρ) + = ( + ρ)m + = Future Value Budget Constraint I prefer to work with the Future Value Budget Constraint equation sine it looks a bit leaner, but it makes no differene; they re equivalent. In some test questions, they use the budget onstraint ( + r) + ( + π) = ( + r)m +. Note the lak of inflation on. This is beause the problem denotes m and in terms on onsumption, not inome. Read the test questions arefully to know whih to use. From here on, assume that m and are in terms of inome, not onsumption. Graphially, (+r) Slope = (+r) (+π) m m + (+π) (+r) If you end up hoosing (, 2 ) suh that > m, 2 <, we say you re a borrower sine you hoose to spend more in the present than you were initially endowed with, and neessarily must borrow some money from your future inome.

3 Saver (+r) m m + (+π) (+r) Similarly, if you end up hoosing (, 2 ) suh that < m, 2 >, we say you re a saver sine you hoose to spend less in the present than you were initially endowed with, and an invest the money you hose not to spend now, resulting in more future onsumption. Borrower (+r) m m + (+π) (+r) Now, suppose the interest rate r inreases. Then Slope inreases sine r is in the numerator. Thus at a higher interest rate, say r, the budget onstraint looks like this:

4 (+r ) (+r) m + (+π) m (+r ) 2 m m + (+π) (+r) Notie that the graph pivots about the point (, ). This is beause, regardless of the interest rate, you still have m dollars to start with today and dollars to start with year from today; hene, after the interest rate hanges, your budget onstraint must still past through this point.

5 Pratie Problems. A bond is urrently selling for $00. This bond only has one remaining payment to bondholders, a fae value of $X two years from today. If the annual interest rate is 20% for this bond, what is X? 2. r = 200%, π = 50%. Find ρ 3. Interpret MRS in the intertemporal ontext 4. u(, ) = 3 3 2, m = 50, = 60, r = 20%, π = 0%. Find, 2, and determine whether this onsumer is a borrower or a saver. How muh (if any) do they save/borrow? 5. u(, ) = 3 + 2, m = 30, = 30, r = 200%, π = 50%. Find, 2, and determine whether this onsumer is a borrower or a saver. How muh (if any) do they save/borrow?