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Econ 384 Intermediate Microeconomics II

Instructor Lorne Priemaza Lorne.priemaza_at_ualbert

a.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(1r) P Pr
- After another year
- You will make interest on the initial amount P
- You will make interest on last years interest Pr
- Total amount in the bank P(1r)2
- This is COMPOUND INTEREST. Over time you make

interest on the interest the interest compounds.

A.1 Compounding

- Investment 100
- Interest rate 2

Year Calc. Amount

1 100 100.00

2 1001.02 102.00

3 1001.022 104.04

4 1001.023 106.12

5 1001.024 108.24

Derived Formula S P (1r)t S value after

t years P principle amount r interest rate t

years

A.1 Compounding Choice

- Given two revenues or costs, choose the one with

the greatest value after time t - A 100 now B115 in two years, r6
- (find value after 2 years)
- S P (1r)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, r6
- (find value after 2 years)
- S P (1r)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 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 r8?

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

GICs pay 5) PV S/(1r)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

If St is the same every year, a special ANNUITY

formula can be used

A.2 Annuity Formula

PV A1-(1/1r)t / 1- (1/1r) PV

A1-xt / 1-x x1/1r 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

A1-xt / r x1/1r

A.2 Annuity Comparison

Consider a payment of 100 per year for 5 years,

(7 interest) PV 100100/1.07 100/1.072

100/1.073 100/1.074 100 93.5 87.3

81.6 76.3 438.7 Or PV A1-(1/1r)t /

1- (1/1r) PV A1-xt / 1-x

x1/1r PV 1001-(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

Where r is the interest rate or opportunity cost

of funds.

A.3 PV Decisions Example

- A firm can
- Invest 5,000 today for a 8,000 payout in year

4. - Invest 1000 a year for four years, with a 2,500

payout in year 2 and 4 - If r4,

A.3 PV Decisions Example

2) Invest 1000 a year for four years, with a

2,500 payout in year 2 and 4 If r4,

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 (1young and 2old), 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,
- c2i2(1r)i1
- If you only consumed when young
- c1i1i2 /(1r)

Lifecycle Budget Constraint

The slope of this constraint is (1r). Often

point E is referred to as the endowment point.

i2(1r)i1

Old Consumption

i2

E

O

i1

i1i2 /(1r)

Young Consumption

A.4 Lifecycle Budget Constraint

- Assuming a constant r, the lifecycle budget

constraint is

Note that if there is no borrowing or lending,

consumption is at E where c1i1, therefore

A.4 Lifetime Utility

- In the lifecycle model, an individuals lifetime

utility is a function of the consumption in each

time period - Uf(c1,c2)
- If the consumer assumptions of consumer theory

hold across time (completeness, transitivity,

non-satiation) , this produces well-behaved

intertemporal indifference curves

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)

Intertemporal Indifference Curves

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

U2

1

Uv2

0

c1

1

2

4

Marginal Rate of Intertemporal Substitution (MRIS)

- Utility is constant along the intertemporal

indifference curve - An individual is willing to SUBSTITUTE one

periods consumption for another, yet keep

lifetime utility even - ie) In the above example, if someone starts with

consumption of 2 each time period, theyd be

willing to give up 1 consumption in the future to

gain 3 consumption now - Obviously this is unlikely to be possible

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)

A.4 MRIS Example

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

Maximizing Intertemporal Utility

c2

Point A affordable, doesnt maximize

utility Point B unaffordable Point C affordable

(with income left over) but doesnt maximize

utility Point D affordable, maximizes utility

IBL

D

B

C

IIC2

A

IIC1

c1

0

A.4 Maximization Example

A.4 Maximization Example 2

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

Maximizing IntertemporalUtility

c2

Utility is always maximized at the tangent to the

indifference curve

U817,316

c1

0

A. Conclusion

- Streams of intertemporal costs and benefits can

be compared by comparing present values - To examine consumption timing, one can use the

LIFECYCLE MODEL - An intertemporal budget line has a slope of (1r)
- The slope of the intertemporal indifference curve

is the Marginal Rate of Intertemporal

Substitution (MRIS) - Equating these allows us to Maximize