Intertemporal Choice

1
Intertemporal Choice
2
Intertemporal 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?

3
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.
• e.g., if r 0.1 (10) then 100 saved at the
  start of period 1 becomes 110 at the start of
  period 2.

4
Future Value

• The value next period of 1 saved now is the
  future value of that dollar.
• 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 is

5
Present Value

• Suppose you can pay now to obtain 1 at the start
  of next period.
• What is the most you should pay?
• Would you pay 1?
• No. If you kept your 1 now and saved it then at
  the start of next period you would have (1r) gt
  1, so paying 1 now for 1 next period is a bad
  deal.

6
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(1r) at the start of
  next period, so we want the value of m for which
  m(1r) 1That is, m
  1/(1r),the present-value of 1 obtained at the
  start of next period.

7
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 is
• E.g., if r 0.1 then the most you should pay now
  for 1 available next period is 0.91

8
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.

9
The Intertemporal Choice Problem

• The intertemporal choice problemGiven 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.

10
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/p1.
• Q What will be consumed in period 2?
• A c2 m2/p2

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The Intertemporal Budget Constraint
c2
So (c1, c2) (m1/p1, m2/p2) is the consumption
bundle if theconsumer chooses neither to save
nor to borrow.
m2/p2
0
c1
m1/p1
0
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Intertemporal Choice

• Suppose c1 0, expenditure in period 2 is at its
  maximum at
• since the maximum we can save in period 1 is m1
  which yields (1r)m1 in period 2
• so maximum possible consumption in period 2 is

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Intertemporal Choice

• Conversely, suppose c2 0, maximum possible
  expenditure in period 1 is
• since in period 2, we have m2 to pay back loan,
  the maximum we can borrow in period 1 is m2/(1r)
• so maximum possible consumption in period 1 is

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The Intertemporal Budget Constraint
c2
m2/p2
0
c1
m1/p1
0
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Intertemporal Choice

• Finally, if both c1 and c2 are greater than 0.
  Then the consumer spends p1c1 in period 1, and
  save m1 - p1c1. Available income in period 2
  will then beso

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Intertemporal Choice

• Rearrange to get the future-value form of the
  budget constraint
• since all terms are expressed in period 2
  values.
• Rearrange to get the present-value form of the
  budget constraint

where all terms are expressed in period 1values.
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The Intertemporal Budget Constraint

• Rearrange again to get c2 as a function of other
  variables

intercept
slope
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The Intertemporal Budget Constraint
c2
m2/p2
0
c1
m1/p1
0
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The Intertemporal Budget Constraint

• Suppose p1 p2 1, the future-value constraint
  becomes
• Rearranging, we get

20
The Intertemporal Budget Constraint

• If p1 p2 1 then,

c2
slope (1 r)
m2
0
c1
m1
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Slutskys Equation Revisited

• Recall that Slutskys equation is
• ?xi ?xis (?i x i) ?xim
  ?pi ?pi ?m

• An increase in r acts like an increase in the
  price of c1. If p1 p2 1, ?1 m1 and x1 c1.
  In this case, we write Slutskys equation as
• ?c1 ?c1s (m1 c1) ?c1m
• ?r ?r ?m

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Slutskys Equation Revisited

• ?c1 ?c1s (m1 c1) ?c1m
• ?r ?r ?m
• If r decreases, substitution effect leads to an
  .. in c1
• Assuming that c1 is a normal good then
• if the consumer is a saver m1 c1 gt 0 then
  income effects leads to a ... in c1 and total
  effect is ...
• if the consumer is a borrower m1 c1 lt 0 then
  income effects leads to a .. in c1 and total
  effect must be .

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Slutskys Equation RevisitedA fall in interest
rate r for a saver
c2
Þ
Pure substitution effect
Þ
Income effect
m2
m1
c1
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Price Inflation

• Define the inflation rate by p where
• For example,p 0.2 means 20 inflation, andp
  1.0 means 100 inflation.

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Price Inflation

• We lose nothing by setting p11 so that p2 1 p
  
• Then we can rewrite the future-value budget
  constraintas
• And rewrite the present-value constraint as

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Price Inflation
rearranges to
intercept
slope
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Price Inflation

• When there was no price inflation (p1p21) the
  slope of the budget constraint was -(1r).
• Now, with price inflation, the slope of the
  budget constraint is -(1r)/(1 p). This can be
  written asr is known as the real interest rate.

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Real Interest Rate
gives
For low inflation rates (p 0), r r - p .For
higher inflation rates thisapproximation becomes
poor.
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Real Interest Rate
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Budget Constraint
c2
slope
m2/p2
0
c1
m1/p1
0
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Budget Constraint

• The slope of the budget constraint is
• The constraint becomes flatter if the interest
  rate r falls or the inflation rate p rises (both
  decrease the real rate of interest).

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Comparative Statics

• Using revealed preference, we can show that
• If a saver continue to save after a decrease in
  real interest rate , then he will be worse off
  
• A borrower must continue to borrow after a
  decrease in real interest rate , and he must
  be better off

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Comparative Statics A fall in real interest
rate for a saver
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Comparative Statics A fall in real interest
rate for a saver
c2
An increase in the inflation rate or a
decrease in the interest rate ..
the budget constraint.
m2/p2
0
c1
m1/p1
35
Comparative Statics A fall in real interest
rate for a saver
c2
If the consumer still saves then saving and
welfare are .. by a lower interest
rate or a higher inflation rate.
m2/p2
0
c1
m1/p1
0
36
Comparative Statics A fall in real interest
rate for a borrower
37
Comparative Statics A fall in real interest
rate for a borrower
c2
An increase in the inflation rate or a
decrease in the interest rate ..
the budget constraint.
m2/p2
0
c1
m1/p1
0
38
Comparative Statics A fall in real interest
rate for a borrower
c2
The consumer must continue to borrow
Borrowing and welfare are .. by a lower
interest rate or a higher inflation rate.
m2/p2
0
c1
m1/p1
0
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Valuing Securities

• A financial security 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?

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Valuing Securities

• The 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.

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Valuing Securities

• The 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 therefore

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Valuing Bonds

• 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 F.
• What is the most that should now be paid for such
  a bond?

43
Valuing Bonds
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Valuing Bonds

• Suppose 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?

45
Valuing Bonds
is the actual (present) value of the prize.
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Valuing Consols

• A consol is a bond which never terminates, paying
  x per period forever.
• What is a consols present-value?

47
Valuing Consols
48
Valuing Consols
Solving for PV gives
49
Valuing Consols
E.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