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A. Intertemporal Choice
A.1 Compounding
A.2 Present Value
A.3 Present Value Decisions
A.4 Lifecycle Model
A.1 CompoundingIf 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 PYou will make interest on last year’s interest PrTotal 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 CompoundingInvestment: $100 Interest rate: 2%
Year Calc. Amount
1 100 100.00
2 100*1.02 102.00
3 100*1.022 104.04
4 100*1.023 106.12
5 100*1.024 108.24
Derived Formula:
S = P (1+r)t
S = value after t years
P = principle amount
r = interest ratet = years
A.1 Compounding ChoiceGiven 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 ChoiceThis 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:
PV = present valueS = future sumr = interest ratet = years
tr
SPV
)1(
A.2 Present Value Gain Example
What is the present value of earning $5,000 in 5 years if r=8%?
403,3$
)08.1(
000,5$
)1(
5
PV
PV
r
SPV
t
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:
TT
o r
S
r
S
r
SSPV
)1(...
)1()1( 221
If St is the same every year, a special ANNUITY formula can be used:
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 paymentr = annual interest raten = 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 Formula
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
OrPV = 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.2 Annuity Comparison
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 Ct
x and Bt
x in year t, present value is calculated:
T
XT
XT
XXXXXXX
r
CB
r
CB
r
CBCBPV
)1(
)(...
)1(
)(
)1(
)()(
22211
00
Where r is the interest rate or opportunity cost of funds.
A.3 PV Decisions ExampleA 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 $2,500 payout in year 2 and 4If r=4%,
112,2$
)04.1(
)000,8()000,50(
)1(
)()(
1
31
3
13
131
010
1
PV
PV
r
CBCBPV
A.3 PV Decisions Example2) Invest $1000 a year for four years,
with a $2,500 payout in year 2 and 4If r=4%,
850$
333,1925442,1000,1
)04.1(
)000,100,5,2(
)04.1(
)000,1(
)04.1(
)000,1500,2()000,1(
)1(
)(
)1(
)(
)1(
)()(
1
1
321
3
13
13
2
12
12
11
111
010
1
PV
PV
PV
r
CB
r
CB
r
CBCBPV
Option 1 is best.
A.4 Lifecycle ModelAlternately, 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)
18
Lifecycle Budget Constraint
Young Consumption
Old
Con
sum
ptio
n
O i1+i2 /(1+r)
The slope of this constraint is (1+r).
Often point E is referred to as the endowment point.
i1
i2
i2+(1+r)i1
E
A.4 Lifecycle Budget Constraint
Assuming a constant r, the lifecycle budget constraint is:
1122
12
r)c(1-r)i(1i
lope)c(-
c
sInterceptc
Note that if there is no borrowing or lending, consumption is at E where c1=i1, therefore:
22
1122
i
r)c(1-r)c(1i
c
c
20
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 well-behaved intertemporal indifference curves:
21
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)
22
U=√2
U=2
c1
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.
•
•
•
•
0 1 2 4
1
2
23
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
24
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)
1
2constantutility
1
2
c
c
c
cMRIS
25
A.4 MRIS Example
)()(
)(
122
121
21
1
2
12
2
21
cccccMRIS
cUc
cMRIS
cUc
ccU
26
A.4 Maximizing the Lifecycle Model
• 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:
1122
21),(
r)c(1-r)i(1i ..
),(max21
cts
ccUcc
27
c2
c1
•IIC2
IBL
0
C •• B
• IIC1
A
D
Point A: affordable, doesn’t maximize utilityPoint B: unaffordablePoint C: affordable (with income left over) but doesn’t maximize utilityPoint D: affordable, maximizes utility
28
A.4 Maximization Example
)1(
)1(
)1(:, ,
21
12
1
212
1
221
rcc
ccr
rc
cIBCcc
c
cMRISccU
2))1((
)1(
)1()1(
122
2122
1122
iric
ciric
criric
2)1(
)1(2))1((
)1(
12
1
121
21
iri
c
riric
rcc
29
A.4 Maximization Example 25%r $100,000,i ,$1,500,000 , 2121 iccU
500,837$
2)000,500,1)$05.1(000,100$(
2))1((
2
2
122
c
c
iric
619,797$2
000,500,1$)05.1(
000,100$
2)1(
1
1
12
1
c
c
iri
c
316,817
)619,797)(500,837($
)619,797)(500,837($
21
U
U
U
ccU
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
31
c2
c1
•
0
U=817,316
12
21),(
r)c(1-r)$100,000(1000,500,1$ ..
),(max21
cts
ccUcc
12 r)c(1-r)$100,000(1000,500,1$ c
500,837$2 c619,797$1 c
32
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
