微观经济学范里安第八版原版第十章

nullChapter TenChapter TenIntertemporal Choice 跨时期选择StructureStructurePresent and future values Intertemporal budget constraint Preferences for intertemporal consumption Intertemporal choice Comparative statics Valuing securitiesIntertemporal ChoiceIntertemporal ChoicePersons 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 ValuesPresent and Future ValuesBegin with some simple financial arithmetic. Take just two periods; 1 and 2. Let r denote the interest rate per period.Future ValueFuture ValueE.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 ValueFuture ValueGiven 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 isPresent Value （现值）Present Value （现值）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 ValuePresent ValueQ: 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) = 1 That is, m = 1/(1+r), the present-value of $1 obtained at the start of next period.Present ValuePresent ValueThe 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 isPresent ValuePresent ValueE.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 isThe Intertemporal Choice ProblemThe Intertemporal Choice ProblemLet 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 ProblemThe Intertemporal Choice ProblemThe 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 ConstraintThe Intertemporal Budget ConstraintTo start, let’s ignore price effects by supposing that p1 = p2 = $1.The Intertemporal Budget ConstraintThe Intertemporal Budget ConstraintSuppose 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 ConstraintThe Intertemporal Budget Constraintc1c2m2m100The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2So (c1, c2) = (m1, m2) is the consumption bundle if the consumer chooses neither to save nor to borrow.m2m100The Intertemporal Budget ConstraintThe Intertemporal Budget ConstraintNow 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 ConstraintThe Intertemporal Budget ConstraintPeriod 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 ConstraintThe Intertemporal Budget ConstraintPeriod 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 ConstraintThe Intertemporal Budget Constraintc1c2m2m100the future-value of the income endowmentThe Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2m100 is the consumption bundle when all period 1 income is saved.The Intertemporal Budget ConstraintThe Intertemporal Budget ConstraintNow 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 ConstraintThe Intertemporal Budget ConstraintOnly $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 ConstraintThe Intertemporal Budget ConstraintOnly $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 ConstraintThe Intertemporal Budget Constraintc1c2m2m100 is the consumption bundle when all period 1 income is saved.the present-value of the income endowmentThe Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2m100is 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.The Intertemporal Budget ConstraintThe Intertemporal Budget ConstraintSuppose that c1 units are consumed in period 1. This costs $c1 and leaves m1- c1 saved. Period 2 consumption will then beThe Intertemporal Budget ConstraintThe Intertemporal Budget ConstraintSuppose that c1 units are consumed in period 1. This costs $c1 and leaves m1- c1 saved. Period 2 consumption will then be which isíïïíîììîslopeinterceptThe Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2m100slope = -(1+r)The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2m100SavingBorrowingslope = -(1+r)The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintis the “future-valued” form of the budget constraint since all terms are in period 2 values. This is equivalent towhich is the “present-valued” form of the constraint since all terms are in period 1 values.As r rises, it is as if period 1 consumption is more expensive or period 2 consumption cheaper.The Intertemporal Budget ConstraintThe Intertemporal Budget ConstraintIf we allow for inflation or deflation, then prices of consumption goods may be different in two periods. Use p1 and p2 to denote prices for consumption in periods 1 and 2. How does this affect the budget constraint?Intertemporal ChoiceIntertemporal ChoiceGiven her endowment (m1,m2) and prices p1, p2 what intertemporal consumption bundle (c1*,c2*) will be chosen by the consumer? Maximum possible expenditure in period 2 is so maximum possible consumption in period 2 isIntertemporal ChoiceIntertemporal ChoiceSimilarly, maximum possible expenditure in period 1 is so maximum possible consumption in period 1 isIntertemporal ChoiceIntertemporal ChoiceFinally, if c1 units are consumed in period 1 then the consumer spends p1c1 in period 1, leaving m1 - p1c1 saved for period 1. Available income in period 2 will then be soIntertemporal ChoiceIntertemporal Choicerearranged isThis is the “future-valued” form of the budget constraint since all terms are expressed in period 2 values. Equivalent to it is the “present-valued” formwhere all terms are expressed in period 1 values.The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2/p2m1/p100The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2/p2m1/p100The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2/p2m1/p100The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2/p2m1/p100Slope = The Intertemporal Budget ConstraintThe Intertemporal Budget Constraintc1c2m2/p2m1/p100SavingBorrowingSlope = Intertemporal PreferencesIntertemporal PreferencesExtreme cases Perfect substitute Perfect complement Intermediate case of well-behaved preferences Intertemporal PreferencesIntertemporal Preferencesc1c2Intertemporal ChoiceIntertemporal Choicec2Comparative StaticsComparative StaticsInterest rate rises or falls Price inflationComparative StaticsComparative StaticsThe slope of the budget constraint is The constraint becomes flatter if the interest rate r falls. The constraint becomes steeper if the interest rate r rises.Comparative StaticsComparative Staticsc1c2m2/p2m1/p1slope =Comparative StaticsComparative Staticsc1c2m2/p2m1/p100slope =The consumer saves.Comparative StaticsComparative Staticsc1c2m2/p2m1/p100slope =A decrease in the interest rate “flattens” the budget constraint.The consumer saves.Comparative StaticsComparative Staticsc1c2m2/p2m1/p100slope =If the consumer choose to remain a lender then her welfare is reduced by a lower interest rate. Sign on saving is ambiguous.The consumer initially saves.Comparative StaticsComparative Staticsc2c1m2/p2m1/p100slope =The consumer borrows.Comparative StaticsComparative Staticsc1c2m2/p2m1/p100slope =A fall in the interest rate “flattens” the budget constraint.The consumer borrows.Comparative StaticsComparative Staticsc1c2m2/p2m1/p100slope =The consumer will remain a borrower. Welfare is increased by a lower interest rate. Will borrow more.The consumer borrows.Borrow or Lend after a Change in Interest Rate?Borrow or Lend after a Change in Interest Rate?Use WARP. Different effects for rising interest rates (see textbook for analysis of rising interest rates).Effect on Consumption (Saving)? Slutsky EquationEffect on Consumption (Saving)? Slutsky Equation (?) (-) (?) (+) If borrower, (m-c)<0, total effect is negative. If lender, (m-c)>0, total effect is ambiguous.Price InflationPrice InflationDefine the inflation rate by p where For example, p = 0.2 means 20% inflation, and p = 1.0 means 100% inflation.Price InflationPrice InflationWe lose nothing by setting p1=1 so that p2 = 1+ p . Then we can rewrite the budget constraint asPrice InflationPrice Inflationrearranges toso the slope of the intertemporal budget constraint isPrice InflationPrice InflationWhen there was no price inflation (p1=p2=1) the slope of the budget constraint was -(1+r). Now, with price inflation, the slope of the budget constraint is -(1+r)/(1+ p). This can be written as r is known as the real interest rate （实际利率）. r is called the nominal interest rate （名义利率）.Real Interest RateReal Interest RategivesFor low inflation rates (p » 0), r » r - p . For higher inflation rates this approximation becomes poor.Real Interest RateReal Interest RateComparative StaticsComparative StaticsThe slope of the budget constraint is The constraint becomes flatter if the inflation rate p rises. The effect is the same as falling (nominal) interest rate. Both have the effect of decreasing the real rate of interest. Earlier comparative static analyses apply.Valuing Securities （证券）Valuing Securities （证券）A financial security （金融证券）is a financial instrument that promises to deliver an income stream. E.g.; a security that pays $m1 at the end of year 1, $m2 at the end of year 2, and $m3 at the end of year 3. What is the most that should be paid now for this security?Valuing SecuritiesValuing SecuritiesThe security is equivalent to the sum of three securities; the first pays only $m1 at the end of year 1, the second pays only $m2 at the end of year 2, and the third pays only $m3 at the end of year 3.Valuing SecuritiesValuing SecuritiesThe PV of $m1 paid 1 year from now is The PV of $m2 paid 2 years from now is The PV of $m3 paid 3 years from now is The PV of the security is thereforeValuing Bonds （债券）Valuing Bonds （债券）A bond is a special type of security that pays a fixed amount $x for T years (its maturity date) and then pays its face value （面值） $F. What is the most that should now be paid for such a bond?Valuing BondsValuing BondsValuing BondsValuing BondsSuppose you win a State lottery. The prize is $1,000,000 but it is paid over 10 years in equal installments of $100,000 each. What is the prize actually worth?Valuing BondsValuing Bondsis the actual (present) value of the prize.Valuing ConsolsValuing ConsolsA consol is a bond which never terminates, paying $x per period forever. What is a consol’s present-value?Valuing ConsolsValuing ConsolsValuing Consols(统一公债）Valuing Consols(统一公债）Solving for PV givesValuing ConsolsValuing ConsolsE.g. if r = 0.1 now and forever then the most that should be paid now for a console that provides $1000 per year is