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Intertemporal ChoiceOptimization over time,
Borrowing/Lending
Intertemporal Choice
Receive income in lumps(monthly salary)
o How is lump income spread over following month (save now,
consume later)?
o How is consumption nanced by borrowing now again received at
end of month?
Present and Future Values
2 periods; r! interest rate per period
Future Valueexif r ! "#$
! $"" saved at start of period $ %$$" at start of period 2
value of ne&t period of %$ saved now ! FV of that dollar
' $ period from now of %$
FV " # $ r
' $ period from now of %m
FV " m%# $ r&
Present Value
*aying %$ now for %$ ne&t period ! bad idea; if you save
than %$ will increase
How much % would have to be saved now to obtain %$ at start of
ne&t period?o %m saved now %m($+r) at start of ne&t want
value of m
m($+r) ! $m " #/%#$r& the PVof %$ obtained at the start of
ne&t period
* of %m available at start of ne&t period
PV " m / %# $ r&
'xr ! "#$, most you should pay now for %$ available ne&t
period* ! $ ($ + "#$) ! %"#-$
r ! "#2 * ! $ ($ + "#2) ! %"#./
Intertemporal Choice Pro(lem
0$,m2 ! incomes received in periods $,2
1$,c2 ! consumptions in periods $,2
*$,p2 ! prices of consumption in periods $,2
Given incomes and consumption prices, what is the most preferred
intertemporal consumption
bundle (c1,c2)?o eed to 3now
4ntertemporal budget constraint
4ntertemporal consumption preferences
Intertemporal Budget Constraint
4gnore price e5ect by
*$ ! p2 ! %$
1onsumer chooses not to saveborrow
o 1onsumed in period $ c$ ! m$o 1onsumed in period 2 c2 ! m2
o %c#,c)& " %m#,m)& " consumption (undle
consumer spends nothing on consumption in period $
o c$ ! " saves s$ ! m$
o int rate r6
o period 2 consumption level m2
saving + interest from period $ ! %# $ r&m#
income in period 2 m) $ %#$r&m#
c) " m) $ %#$r&m#
consumer spends everything on consumption period $
o c2 ! "
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o most can borrow in period $ against period 2 income of %m
(# " (orrowed amount in p#
%m2 to pac3 bac3 %b$ (#%#$r&"m) (# " m)%#$r&
largest consumption level in periods $ C# " m# $
m)/%#$r&
c$ consumed in period $ costs %c$, leaves m$7c$ saves
period 2 consumption
c) " m) $ %#$r&%m#!c#&c) " !%#$r&c# $m) $%#$r&m#
7($+r) ! slope m2 +($+r)m$ ! intercept
%# $ r&c# $ c) " %# $ r&m# $ m)o ' for of budget
constraint (all terms in period 2 values
o 89uivalent tto *
C# $ c)/%# $ r& " m# $ m)/%#$r&o * form of constraint
(period $ values)
Adding prices
How does it a5ect budget constraint?
8ndowment (m$,m2) and prices (p$,p2) what intertemporal bundle
(c$:,c2:) 9ill be
chosen?o 0a& possible expenditurein period 2
*) $ %# $ r&m#o 0a& possible consumptionin period 2
C) " +m) $ %#$r&m#/p)o 0a& possible expenditurein period
$
*# $ m)/%#$r&o 0a& possible consumption in period $
C# " +m# $ m)/%#$r&/p#
1$ units consumes in period $ spends p$c$ in period $ leaving m#
- p#c#for period o vailable income in period 2
m) $ %#$r&%m#!p#c#&p)c) " m) $
%#$r&%m#!p#c#&
o Rearranged
%#$r&p#c#" %#$r&m# $ m) ' form of budget constraint
p#c# $ +p)/%#$r&c) " m# $ m)%#$r& *
'uture alue
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Price In.ation 4nflation rate 6 p#%# $ & " p)
*$ ! $ p2 ! $ +
can rewrite budget constraint
p#c# $ +p)/%#$r&c) " m# $ m)%#$r&to
c# $ +%#$&/%#$r&c) " m# $ m)/%#$r&
rearrange
no price in@ation (p$!p2!$)
o slope of budget constraint ! 7($+r)
with price in@ation
o slope of
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Li4e!C5cle *odel
Budget Line
1o ! consumption in period o
0o ! income in period o
i B interet rate
right side of e9uiation ' of income stream
left side' of consumption bundle
slope ! 7($+i) ! opp cost
Co%# $ i& $ C# " *o%# $ i& $ *# FV
Co $ C#/%#$i& " *o $ *#/%#$i& PV
1o ! 0o, 1$ ! 0$ endowment
s i increases budet line pivots on endowment
Intermporal 6llocation o4 Li4etime Income
Eo ma& utilityconsumption bundle on budget line where
o 0RF ! ($+i) or
o 0arginal rate of time preference ! ($+i)
o
