Intertemporal choiceLecture 5 β Tom Holden
Intermediate Microeconomics Semester 2
http://micro2.tholden.org/
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Objectives for this topic
β’ To understand and use (net) present values.
β’ To appreciate the budget constraint that individuals face when choosing between consumption over time.
β’ To show how choices are affected by individualsβ preferences between consumption today and tomorrow.
β’ To relate this analysis to borrowing and saving behaviour.
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Thinking about intertemporal choice
β’ Intertemporal choice: choice over levels of consumption over time.
β’ E.g.:
β’ Given people usually receive income in monthly βlumpsβ, how should the income be spread over the following month?
β’ How much should we save for retirement?
β’ Is it rational to take out student loans?
β’ When should firms invest in a project that pays off in the future?
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Think about your intertemporal choices
β’ Did you save before you came to university?
β’ Are you borrowing?
β’ At what rate of interest?
β’ Does your borrowing change depending on the rate of interest you face?
β’ To what extent do you think about your future income when making these decisions?
β’ Do you think your consumption decisions will change if you get a definite job/placement offer?
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Present and future values
β’ Suppose period 0 is the present.
β’ π is the interest rate per period.β’ E.g. If π = 0.1 = 10%, then Β£100 saved in period 0 becomes Β£110 in period 1.
β’ More generally, saving π in period 0means you have π 1 + π in period 1.β’ If you keep your money in savings until period π‘, you have π 1 + π π‘.
β’ What is the value today of Β£π in period 1?β’ It must be less than Β£π, as if you took that Β£π now and invested it you would
have Β£π 1 + π in period 1.
β’ Instead the present value (PV) in period 0 of Β£π in period 1 is Β£π
1+π, since if
you invest this amount at 0 you get Β£π at 1.
β’ More generally, the present value in period 0 of Β£π in period π‘ is Β£π
1+π π‘.
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Example present values
β’ Present values reduce quite fast as we look further into the future under reasonable interest rates.
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Rate π = π π = π π = π π = ππ π = ππ π = ππ π = 25 π = 30
0.05 .95 .91 .78 .61 .48 .37 .30 .23
0.10 .91 .83 .62 .39 .39 .15 .09 .06
0.15 .87 .76 .50 .25 .25 .06 .03 .02
0.20 .83 .69 .40 .16 .16 .03 .01 .00
Value of Β£π π years in the future under different interest rates
More complicated present values
β’ Suppose you believe buying a share in a company (in period 0) would give you a stream of dividends given by π1, π2, β¦ where ππ‘ is the dividend you get in period π‘.
β’ What is the present value of this dividend stream?
β’ It is the PV of getting π1 in period 1 plus the PV of getting π2 in period 2, plusβ¦
β’ I.e. π1
1+π 1+π2
1+π 2+β― = π‘=1
β ππ‘
1+π π‘
β’ So at what price should you be prepared to buy the share? What should you do if you couldnβt afford it at this price?
β’ Note that high interest rates reduce present values.
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More on present value (1/4)
β’ Suppose a consumerβs income stream is currently given by π¦0, π¦1, π¦2, β¦ where π¦π‘ is their income at π‘, and period 0 is the present.
β’ If they are offered the chance to switch to an alternative stream π§0, π§1, π§2, β¦ should they take it?
β’ Could be a new job, or a degree, or an investment.
β’ PV of the original stream, π£π¦ is π£π¦ = π¦0 +π¦1
1+π+π¦2
1+π 2+β― = π‘=0
β π¦π‘
1+π π‘.
β’ PV of the alternative stream, π£π§ is π£π§ = π§0 +π§1
1+π+π§2
1+π 2+β― = π‘=0
β π§π‘
1+π π‘.
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More on present value (2/4)
β’ Suppose that π£π§ > π£π¦ and they do take the π§π‘ stream.
β’ Then, by going to the bank they can borrow π§1
1+πagainst their period 1
income. (When period 1 arrives they will have to repay π§1
1+π1 + π = π§1,
which is their period 1 income.) And π§2
1+π 2against period 2 income, etc etc.
β’ So they can borrow π‘=1β π§π‘
1+π π‘= π£π§ β π§0 in total (note sum starts at 1), so if
they wanted to they could spend π£π§ in period 0 and starve from then on.
β’ Suppose they borrow all of this money, then immediately put a total of π‘=1β π¦π‘
1+π π‘= π£π¦ β π¦0 into their saving account, leaving π£π§ β π£π¦ β π¦0 =
π£π§ β π£π¦ + π¦0 in their pocket.
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More on present value (3/4)
β’ As π£π§ β π£π¦ > 0, this enables them to spend more than π¦0 in the first period.
β’ The next period thanks to interest they now have 1 + π π‘=1β π¦π‘
1+π π‘= π¦1 +
π‘=2β π¦π‘
1+π π‘β1in the bank.
β’ If they spend π¦1 they still have π‘=2β π¦π‘
1+π π‘β1in savings.
β’ The period after this has become 1 + π π‘=2β π¦π‘
1+π π‘β1= π¦2 + π‘=3
β π¦π‘
1+π π‘β2.
β’ Continuing in this way, the consumer may spend π¦π‘ in every period after 0.
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More on present value (4/4)
β’ But they spent more in 0, thus consumption is higher in one period, and the same in all others, so the consumer must be better-off overall, as long as their preferences are strictly increasing.
β’ So when π£π§ > π£π¦ the consumer is strictly better off taking the π§π‘stream, independent of preferences.
β’ As Varian says:β’ βPresent value is the only correct way to convert a stream of payments into
todayβs dollarsβ¦ If a consumer can freely borrow and lend at a constant rate of interest, then the consumer will always prefer a pattern of income with a higher present value to a pattern with a lower present value.β
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PV examples: Comparing two simple income streams
β’ Investment A pays Β£100 now and Β£200 next year.
β’ Investment B pays Β£0 now and Β£310 next year.
β’ With a zero interest rate we just add up the payments Β£310 > Β£300 so investment B is better.
β’ But with a sufficiently high interest rate investment A is preferred.
β’ For example if π = 0.2 then PVπ΄ = 100 +200
1.2= 266.67 & PVπ΅ = 0 +
310
1.2=
258.33.
β’ The fact that A pays more money earlier on means that it will have a higher present value if the interest rate is high enough.
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PV examples: Perpetuities
β’ Buying a perpetuity guarantees the holder of the perpetuity to a payment of π in all future periods. What is the PV of such a perpetuity?
β’ Using the formula from earlier, the present value of the perpetuity, π£,
satisfies π£ = π‘=1β π
1+π π‘=π
1+π+π
1+π 2+π
1+π 3+β―.
β’ Thus 1 + π π£ = π +π
1+π+π
1+π 2+β― = π + π£
β’ So ππ£ = π, i.e. π£ =π
π.
β’ If you invest an amount π£ at an interest rate π, then you get ππ£ each period.
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PV examples: Bonds
β’ Buying a π-period bond guarantees the holder to:β’ A payment of π in periods 1,2,β¦ , π β 1. (This is called βthe couponβ.)
β’ A payment of πΉ in period π. (This is called βthe face valueβ.)
β’ Thus the present value of the bond, π£, satisfies:
π£ =
π‘=1
πβ1π
1 + π π‘+πΉ
1 + π π=π
1 + π+π
1 + π 2+β―+
π
1 + π πβ1+πΉ
1 + π π
β’ Thus:
1 + π π£ = π +π
1 + π+π
1 + π 2+β―+
π
1 + π πβ2+
πΉ
1 + π πβ1
= π +π
1 + π+π
1 + π 2+β―+
π
1 + π πβ2+
π
1 + π πβ1+πΉ
1 + π π+πΉ β π
1 + π πβ1βπΉ
1 + π π
= π + π£ +πΉ β π
1 + π πβ1βπΉ
1 + π π
β’ So ππ£ = π +πΉβπ
1+π πβ1βπΉ
1+π π= π +
ππΉβ 1+π π
1+π π, i.e. π£ =
π
π+πΉβ 1+π
π
π
1+π π.
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PV examples: Investment with maintenance payments
β’ Suppose I buy a machine at a price π0 today, but in each future period π‘ it needs maintenance which costs ππ‘.
β’ The machine generates output worth π¦π‘ in each period π‘.
β’ Then my net income in period π‘ is π¦π‘ β ππ‘.
β’ And the net present value (NPV) of the investment is the PV of the net incomes, i.e. π‘=0
β π¦π‘βππ‘
1+π π‘.
β’ The investment is a good idea if the NPV is positive.
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PV examples: Cost benefit analysis
β’ Cost benefit analysis (CBA) works on exactly the same principles as the investment analysis we have just seen.
β’ If for some policy the net present value to society is positive, then the policy is worth pursuing.
β’ But here all the benefits and costs are included, and an attempt is made to value them as those experiencing them would value them (so market prices are not always used if externalities etc. are important).
β’ π is consdered more broadly as the βsocial rate of time preferenceβ.β’ May differ quite substantially from the interest rate.
β’ E.g. π in Stern report was 2%. Much lower than businesses would use.
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PV examples: Intertemporal budget constraints (2 period case)
β’ πΆ0 is consumption in period 0, πΆ1 is consumption in period 1.
β’ π0 is the price of the consumption good in period 0, π1 is its price in period 1.
β’ π0 is income in period 0, π1 is income in period 1.
β’ The budget constraint says the PV of expenditure must equal the PV of consumption.β’ I cannot die with debts, and I do not want to leave any assets behind when I die.
β’ I.e. π0πΆ0 +π1πΆ1
1+π= π0 +
π1
1+π.
β’ Increasing interest rates make next periodβs consumption cheaper. (I have to save less today to pay for it).
β’ Increasing interest rates make next periodβs income less valuable. (I can borrow less today using it).
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The intertemporal budget constraint
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When π0 = π1, the slope of the budget constraint is β (1 + π) for each Β£1 given up now, we get Β£ 1 + π in future. For each extra Β£1 consumed now, we give up Β£ 1 + π in future.
πΌπ‘ is income (i.e. what we were calling ππ‘.)
Preferences
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A person who was totally indifferent between consumption today and in the future would have indifference curves with a slope of β1. People are likely to prefer a mix of consumption in present and future, so ICs are convex.Because impatient individuals prefer to consume today, on the 45Β° line we need more than Β£1tomorrow to give up Β£1 today.Steeper indifference curves imply more impatience.
Example
β’ Suppose:β’ π πΆ0, πΆ1 = πΆ0πΆ1,β’ π0 = π1 = 1 (i.e. we are measuring in units of the consumption good).β’ the consumer has income of Β£100 in the first period and Β£121 in the second,β’ the interest rate is 10%.
β’ MRS is ππ
ππΆ0
ππ
ππΆ1=πΆ1
πΆ0.
β’ The price of consumption in period 0 in units of consumption in period 1 is 1.1.β’ This is minus the slope of the budget constraint.
β’ Thus πΆ1
πΆ0= 1.1 at an optimum, i.e. πΆ1 = 1.1πΆ0.
β’ The budget constraint says πΆ0 +πΆ1
1.1= 100 +
121
1.1.
β’ Thus 2πΆ0 = 100 + 110, so πΆ0 = 105 and πΆ1 = 115.5.
β’ So the consumer borrows Β£5 in the first period and pays it back in the second.
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General case (1/2)
β’ The Lagrangian for the consumerβs optimisation problem is:
β = π πΆ0, πΆ1 β π π0πΆ0 +π1πΆ11 + πβ π0 β
π11 + π
β’ FOC πΆ0: 0 =ππ
ππΆ0β ππ0, so π =
1
π0
ππ
ππΆ0
β’ FOC πΆ1: 0 =ππ
ππΆ1βππ1
1+π, so:
β’ 1 + ππ0
π1= ππ
ππΆ0
ππ
ππΆ1= MRS
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General case (2/2)
β’ Suppose that π1 = 1 + π π0, where π is the inflation rate of all prices in the economy.β’ I.e. the real price of the good has stayed the same.
β’ Then the condition from the last slide says: 1+π
1+π= MRS.
β’ This suggests that 1+π
1+πis the real cost of substituting between period 0 and 1, i.e.
it is one plus the βrealβ interest rate.
β’ So lets define the real interest rate π by π =1+π
1+πβ 1 =
1+π
1+πβ1+π
1+π=πβπ
1+πβ π β π.
β’ Then the condition says: 1 + π = MRS.
β’ So if the real price of a good is constant, at the optimum minus the slope of the indifference curve must equal 1 + π.
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Class exercise: Time-separable utility
β’ Suppose that π πΆ0, πΆ1 = π’ πΆ0 +1
1+ππ’ πΆ1 .
β’ π is the discount rate. Large πmeans more impatience.
β’ Derive the FOC.
β’ When are consumers neither savers or borrowers?
β’ What happens when π’ πΆ = log πΆ?
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Applications of intertemporal decision making: human capital investment
β’ If education leads to higher wages later in life then studying can be regarded as an investment.
β’ As before, there are a stream of benefits and a stream of costs.
β’ By being here you have implicitly calculated that the NPV of this investment is positive(?!?).
β’ It is clear that a number of factors should influence this decision.β’ The cost of fees.β’ Available borrowing arrangements (students currently pay below the market
rate of interest).β’ Income foregone when studying.β’ Expected income on graduation.
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A representation of the investment decision with no credit markets
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Notice that here the that human capital options are entirely continuous. This is probably not the case in reality.
Human capital investment with borrowing and lending
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In this example the decision over investment is entirely separate from preferences about consumption over time. Is this plausible?
Summary
β’ In a world with perfect credit markets consumers can choose between consumption today and in the future.β’ This is relevant for their saving and borrowing behaviour, and the approach
to investments.
β’ However, all these decisions will be affected by the relative price of consumption in different periods as indicated by the interest rate.
β’ NPV is an important concept which enables individuals to evaluate the worth of income and payments at different times.β’ When making decisions we should be choosing those with the highest net
present value.
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