Econ 384
Intermediate
Microeconomics II
Instructor: Lorne Priemaza
Lorne.priemaza@ualberta.ca
A. Intertemporal Choice
A.1 Compounding
A.2 Present Value
A.3 Present Value Decisions
A.4 Lifecycle Model
A.1 Compounding
If you invest an amount P for a return r,
After one year:
You will make interest on the amount P
Total amount in the bank = P(1+r) = P + Pr
After another year:
You will make interest on the initial amount P
You will make interest on last year’s interest Pr
Total amount in the bank = P(1+r)2
This is COMPOUND INTEREST. Over time you
make interest on the interest; the interest
compounds.
A.1 Compounding
Investment: $100
Interest rate: 2%
Derived Formula:
S = P (1+r)t
S = value after t years
P = principle amount
r = interest rate
t = years
Year
Calc.
1
Amount
100
100.00
2 100*1.02
102.00
3 100*1.022
104.04
4 100*1.023
106.12
5 100*1.024
108.24
A.1 Compounding Choice
Given two revenues or costs, choose the one
with the greatest value after time t:
A: $100 now B:$115 in two years, r=6%
(find value after 2 years)
S = P (1+r)t
SA =$100 (1.06)2 = $112.36
SB =$115
Choose option B
A.1 Compounding Loss Choice
This calculation also works with losses, or a
combination of gains or loses:
A: -$100 now B: -$120 in two years, r=6%
(find value after 2 years)
S = P (1+r)t
SA =-$100 (1.06)2 = -$112.36
SB =-$120
Choose option A. (You could borrow $100
now for one debt, then owe LESS in 2 years
than waiting)
A.2 Present Value
What is the present value of a given sum of
money in the future?
By rearranging the Compound formula, we
have:
S
PV 
t
(1  r )
PV = present value
S = future sum
r = interest rate
t = years
A.2 Present Value Gain Example
What is the present value of earning $5,000 in
5 years if r=8%?
S
PV 
t
(1  r )
$5,000
PV 
5
(1.08)
PV  $3,403
Earning $5,000 in
five years is the
same as earning
$3,403 now.
PV can also be
calculated for
future losses:
A.2 Present Value Loss Example
You and your spouse just got pregnant, and
will need to pay for university in 20 years. If
university will cost $30,000 in real terms in
20 years, how much should you invest now?
(long term GIC’s pay 5%)
PV = S/[(1+r)t]
= -$30,000/[(1.05)20]
= -$11,307
A.2 Present Value of a Stream of
Gains or Loses
If an investment today yields future returns of
St, where t is the year of the return, then
the present value becomes:
S1
S2
ST
PV  So 

 ... 
2
T
(1  r ) (1  r )
(1  r )
If St is the same every year, a special ANNUITY
formula can be used:
A.2 Annuity Formula
PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]
PV = A[1-xt] / [1-x]
x=1/{1+r}
A = value of annual payment
r = annual interest rate
n = number of annual payments
Note: if specified that the first payment is
delayed until the end of the first year, the
formula becomes
PV = A[1-xt] / r
x=1/{1+r}
A.2 Annuity Comparison
Consider a payment of $100 per year for 5
years, (7% interest)
PV= 100+100/1.07 + 100/1.072 + 100/1.073
+ 100/1.074
= 100 + 93.5 + 87.3 + 81.6 + 76.3 = $438.7
Or
PV = A[1-(1/{1+r})t] / [1- (1/{1+r})]
PV = A[1-xt] / [1-x]
x=1/{1+r}
PV = 100[1-(1/1.07)5]/[1-1/1.07] = $438.72
A.3 Present Value Decisions
When costs and benefits occur over time,
decisions must be made by calculating the
present value of each decision
-If an individual or firm is considering optionX
with costs and benefits Ctx and Btx in year t,
present value is calculated:
X
X
X
X
X
X
(
B

C
)
(
B

C
)
(
B

C
X
X
X
1
1
2
2
T
T )
PV  ( B0  C0 ) 

 ... 
2
(1  r )
(1  r )
(1  r )T
Where r is the interest rate or opportunity cost
of funds.
A.3 PV Decisions Example
A firm can:
1) Invest $5,000 today for a $8,000 payout in
year 4.
2) Invest $1000 a year for four years, with a
$2,500 payout in year 2 and 4
1
1
If r=4%,
(
B

C
)
1
1
1
PV  ( B0  C0 ) 
3
3
3
(1  r )
(8,000)
1
PV  (0  5,000) 
(1.04)3
PV 1  $2,112
A.3 PV Decisions Example
2) Invest $1000 a year for four years, with a
$2,500 payout in year 2 and 4
If r=4%,
1
1
1
1
1
1
(
B

C
(
B

C
)
(
B

C
)
1
1
1
3
3)
1
1
2
2
PV  ( B0  C0 ) 


2
(1  r )
(1  r )
(1  r )3
(2,500  1,000) (1,000) (2,5,00  1,000)
1
PV  (1,000) 


2
(1.04)
(1.04)
(1.04)3
PV 1  1,000  1,442  925  1,333
PV 1  $850
Option 1 is best.
A.4 Lifecycle Model
Alternately, often an individual needs to
decide WHEN to consume over a lifetime
To examine this, one can sue a
LIFECYCLE MODEL*:
*Note: There are alternate terms for the Lifecycle
Model and the curves and calculations seen in
this section
A.4 Lifecycle Budget Constraint
Assume 2 time periods (1=young and 2=old),
each with income and consumption (c1, c2, i1,
i2) and interest rate r for borrowing or lending
between ages
If you only consumed when old,
c2=i2+(1+r)i1
If you only consumed when young:
c1=i1+i2 /(1+r)
Lifecycle Budget Constraint
The slope of this constraint
is (1+r).
Old Consumption
i2+(1+r)i1
Often point E is referred to
as the endowment point.
i2
O
E
i1
i1+i2 /(1+r)
Young Consumption
18
A.4 Lifecycle Budget Constraint
Assuming a constant r, the lifecycle budget
constraint is:
c2  Intercept - ( slope)c 1
c2  i 2  (1  r)i1 - (1  r)c1
Note that if there is no borrowing or lending,
consumption is at E where c1=i1, therefore:
c2  i 2  (1  r)c1 - (1  r)c1
c2  i 2
A.4 Lifetime Utility
• In the lifecycle model, an individual’s lifetime
utility is a function of the consumption in each
time period:
U=f(c1,c2)
• If the consumer assumptions of consumer
theory hold across time (completeness,
transitivity, non-satiation) , this produces wellbehaved intertemporal indifference curves:
20
A.4 Intertemporal Indifference
Curves
• Each INDIFFERENCE CURVE plots all
the goods combinations that yield the
same utility; that a person is indifferent
between
• These indifference curves have similar
properties to typical consumer indifference
curves (completeness, transitivity,
negative slope, thin curves)
21
c2
Consider the utility function U=(c1c2)1/2.
Each indifference curve below shows
all the baskets of a given utility level.
Consumers are indifferent between
intertemporal baskets along the same
curve.
2
•
•
•
1
•
U=2
U=√2
0
1
2
4
c1
22
Marginal Rate of Intertemporal
Substitution (MRIS)
• Utility is constant along the intertemporal indifference
curve
• An individual is willing to SUBSTITUTE one period’s
consumption for another, yet keep lifetime utility even
– ie) In the above example, if someone starts with
consumption of 2 each time period, they’d be willing
to give up 1 consumption in the future to gain 3
consumption now
• Obviously this is unlikely to be possible
23
A.4 MRIS
• The marginal rate of substitution (MRIS) is the gain
(loss) in future consumption needed to offset the loss
(gain) in current consumption
• The MRS is equal to the SLOPE of the indifference
curve (slope of the tangent to the indifference curve)
 c2
MRIS 
c1
c2
utilityconstant 
c1
24
A.4 MRIS Example
U  c1c2
c2  U c1
2
c2
2
MRIS 
  (U c1 )
c1
MRIS   ( c1c2 c1 )   (c2 c1 )
2
25
A.4 Maximizing the Lifecycle Model
max U (c1 , c2 )
( c1 ,c2 )
s.t. c2  i 2  (1  r)i1 - (1  r)c1
• Maximize lifetime utility (which depends on c1
and c2) by choosing c1 and c2 ….
• Subject to the intertemporal budget constraint
– In the simple case, people spend everything,
so the constraint is an equality
• This occurs where the MRIS is equal to the
slope of the intertemporal indifference curve:
26
c2
IBL
Point A: affordable, doesn’t maximize utility
Point B: unaffordable
Point C: affordable (with income left over) but
doesn’t maximize utility
Point D: affordable, maximizes utility
•
C •
D
•
•
0
B
A
IIC2
IIC1
c1
27
A.4 Maximization Example
c2
c2
U  c1c2 , MRIS 
  c2 c1 , IBC :
 (1  r )
c1
c1
 (1  r )   c2 c1
c1  c2 (1  r )
c2  i2  (1  r )i1  (1  r )c1
c2  i2  (1  r )i1  c2
c2  (i2  (1  r )i1 ) 2
c1  c2 (1  r )
c1  (i2  (1  r )i1 ) 2(1  r )
 i2


 i1 
(1  r )

c1  
2
28
A.4 Maximization Example 2
U  c1c2 , i1  $1,500,000, i 2  $100,000, r  5%
c2  (i2  (1  r )i1 ) 2
c2  ($100,000  (1.05)$1,500,000) 2
c2  $837,500
 i2


 i1 
(1  r )

c1  
U  c1c2
U  ($837,500)(797,619)
2
 $100,000


 $1,500,000 
(1.05)

c1  
U  ($837,500)(797,619)
U  817,316
2
c1  $797,619
29
A.4 Maximization Conclusion
Lifetime utility is maximized at 817,316
when $797,619 is consumed when
young and $837,500 is consumed
when old.
*Always include a conclusion
c2
max U (c1 , c2 )
( c1 ,c2 )
s.t. c2  $1,500,000  (1  r)$100,000 - (1  r)c1
c2  $1,500,000  (1  r)$100,000 - (1  r)c1
•
c1  $797,619
c2  $837,500
U=817,316
0
c1
31
A. Conclusion
1) Streams of intertemporal costs and benefits can be
compared by comparing present values
2) To examine consumption timing, one can use the
LIFECYCLE MODEL:
a) An intertemporal budget line has a slope of
(1+r)
b) The slope of the intertemporal indifference
curve is the Marginal Rate of Intertemporal
Substitution (MRIS)
c) Equating these allows us to Maximize 32