Chapter TenIntertemporal ChoicePresent and Future ValuesFuture ValueSlide 5Present ValueSlide 7Slide 8Slide 9The Intertemporal Choice ProblemSlide 11The Intertemporal Budget ConstraintSlide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Price InflationSlide 43Slide 44Slide 45Real Interest RateSlide 47Comparative StaticsSlide 49Slide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56Slide 57Slide 58Valuing SecuritiesSlide 60Slide 61Valuing BondsSlide 63Slide 64Slide 65Valuing ConsolsSlide 67Slide 68Slide 69Chapter TenIntertemporal ChoiceIntertemporal ChoicePersons often receive income in “lumps”; e.g. monthly salary.How is a lump of income spread over the following month (saving now for consumption later)?Or how is consumption financed by borrowing now against income to be received at the end of the month?Present and Future ValuesBegin with some simple financial arithmetic.Take just two periods; 1 and 2.Let r denote the interest rate per period.Future ValueE.g., if r = 0.1 then $100 saved at the start of period 1 becomes $110 at the start of period 2.The value next period of $1 saved now is the future value of that dollar.Future ValueGiven an interest rate r the future value one period from now of $1 is Given an interest rate r the future value one period from now of $m isFV r 1 .FV m r ( ).1Present ValueSuppose you can pay now to obtain $1 at the start of next period.What is the most you should pay?$1?No. If you kept your $1 now and saved it then at the start of next period you would have $(1+r) > $1, so paying $1 now for $1 next period is a bad deal.Present ValueQ: How much money would have to be saved now, in the present, to obtain $1 at the start of the next period?A: $m saved now becomes $m(1+r) at the start of next period, so we want the value of m for which m(1+r) = 1That is, m = 1/(1+r),the present-value of $1 obtained at the start of next period.Present ValueThe present value of $1 available at the start of the next period isAnd the present value of $m available at the start of the next period isPVr11.PVmr1.Present ValueE.g., if r = 0.1 then the most you should pay now for $1 available next period isAnd if r = 0.2 then the most you should pay now for $1 available next period isPV   11 0 191$0 .PV   11 0 283$0 .The Intertemporal Choice ProblemLet m1 and m2 be incomes received in periods 1 and 2.Let c1 and c2 be consumptions in periods 1 and 2.Let p1 and p2 be the prices of consumption in periods 1 and 2.The Intertemporal Choice ProblemThe intertemporal choice problem:Given incomes m1 and m2, and given consumption prices p1 and p2, what is the most preferred intertemporal consumption bundle (c1, c2)?For an answer we need to know:– the intertemporal budget constraint– intertemporal consumption preferences.The Intertemporal Budget ConstraintTo start, let’s ignore price effects by supposing that p1 = p2 = $1.The Intertemporal Budget ConstraintSuppose that the consumer chooses not to save or to borrow.Q: What will be consumed in period 1?A: c1 = m1.Q: What will be consumed in period 2?A: c2 = m2.The Intertemporal Budget Constraintc1c2m2m100The Intertemporal Budget Constraintc1c2So (c1, c2) = (m1, m2) is theconsumption bundle if theconsumer chooses neither to save nor to borrow.m2m100The Intertemporal Budget ConstraintNow suppose that the consumer spends nothing on consumption in period 1; that is, c1 = 0 and the consumer saves s1 = m1.The interest rate is r.What now will be period 2’s consumption level?The Intertemporal Budget ConstraintPeriod 2 income is m2.Savings plus interest from period 1 sum to (1 + r )m1.So total income available in period 2 is m2 + (1 + r )m1.So period 2 consumption expenditure isThe Intertemporal Budget ConstraintPeriod 2 income is m2.Savings plus interest from period 1 sum to (1 + r )m1.So total income available in period 2 is m2 + (1 + r )m1.So period 2 consumption expenditure isc m r m2 2 11  ( )The Intertemporal Budget Constraintc1c2m2m100mr m211( )the future-value of the incomeendowmentThe Intertemporal Budget Constraintc1c2m2m100 is the consumption bundle when all period 1 income is saved. ( , ) , ( )c c m r m1 2 2 10 1  mr m211( )The Intertemporal Budget ConstraintNow suppose that the consumer spends everything possible on consumption in period 1, so c2 = 0.What is the most that the consumer can borrow in period 1 against her period 2 income of $m2?Let b1 denote the amount borrowed in period 1.The Intertemporal Budget ConstraintOnly $m2 will be available in period 2 to pay back $b1 borrowed in period 1.So b1(1 + r ) = m2.That is, b1 = m2 / (1 + r ).So the largest possible period 1 consumption level isThe Intertemporal Budget ConstraintOnly $m2 will be available in period 2 to pay back $b1 borrowed in period 1.So b1(1 + r ) = m2.That is, b1 = m2 / (1 + r ).So the largest possible period 1 consumption level isc mmr1 121 The Intertemporal Budget Constraintc1c2m2m100 is the consumption bundle when all period 1 income is saved. ( , ) , ( )c c m r m1 2 2 10 1  mr m211( )mmr121the present-value ofthe income endowmentThe Intertemporal Budget Constraintc1c2m2m100 ( , ) , ( )c c m r m1 2 2 10 1  ( , ) ,c c mmr1 2 1210 is the consumption bundle when period 1 borrowing is as big as possible. is the consumption bundle when period 1 saving is as large as possible.mr m211( )mmr121The Intertemporal Budget ConstraintSuppose that c1 units are consumed in period 1. This costs $c1 and leaves m1- c1 saved. Period 2 consumption will then bec m r m c2 2 1 11   ( )( )The Intertemporal Budget ConstraintSuppose that c1 units are consumed in period 1. This costs $c1 and leaves m1- c1 saved. Period 2 consumption will then bewhich isc m r m c2 2 1 11   ( )( )c r c m r m2 1 2 11 1    ( ) ( ) .slope interceptThe Intertemporal Budget Constraintc1c2m2m100 ( , ) , ( )c c m r m1 2 2 10