Consumer Choice With Uncertainty Part I: Intertemporal Choice

Agenda:

1. Uncertainty & The Universe

2. Sources of Uncertainty Brainstorm

3. Key math: Present and Future Value

4. Intertemporal Choice

5. Economic Modeling & Policy Implications

“The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature…By showing the world how to understand risk, measure it, and weight its consequences, they converted risk-taking into one of the prime catalysts that drives modern Western Society.”

Demand Over Time (“Intertemporal Choice”)

Now!

Later

The Intertemporal Budget Constraint

today

tomorrow

“Present value of lifetime income”

How can we spend more than our income today?

Endowment Bundle: What you get today and tomorrow

Optimal Bundle: Marginal rate of intertemporal substitution=slope of budget

Key math concept: Present Value

A dollar today is not worth the same as a dollar tomorrow.

22( )

1

MPV M

r

The present value is the value divided by 1+ the risk-free interest rate.

Example:

Congratulations! You won $1,000 in the lottery, BUT, you can’t collect it for a year. If the risk-free rate is 10%, how much is your lottery ticket worth today?

22

$1,000( ) $909

1 1.1

MPV M

r

Time Value with Multiple Time Periods

22( )

1

MPV M

r

0

( )1

Ti

T ii i

MPV M

r

The Mi and ri can change each time period!

Example:

Congratulations! You won $1,000 in the lottery. You will be paid $500 today, $300 next year and $200 in two years. The interest rate for one year is 5% and the interest rate for two years is 7%. What is the present value of your winnings?

2

$300 $200$500 $960.40

(1.05) (1.07)PV

Key math concept: Future Value

1 1( : , ) 1t

FV M t r M r

The future value is the present value multiplied by 1+ the risk-free interest rate raised to the power of the number of time periods (compound interest).

Example:

Congratulations! You won $1,000 in the lottery. If you collect the money today and put it in the bank at 10% annual interest, how much will it be worth in one year?

11( ;1,10%) $1000(1.1) $1,100FV M

Read about delayed gratification and success:http://www.newyorker.com/reporting/2009/05/18/090518fa_fact_lehrer

How much will it be worth in five years?

51( ;5,10%) $1000(1.1) $1,610.51FV M

Another Future Value problem…

Congratulations! You won $1,000 in the lottery. If you receive $100 per year for 10 years and save all of your earnings at an interest rate of 10% what is the future value of your winnings in year 10 (right after you get your last payment)?

9

0

0 1 2 3 9

(1 )

100(1.1) 100(1.1) 100(1.1) 100(1.1) ...100(1.1)

100 110 121 133.1 146.41 161.05 177.16 194.87 214.36 235.79

( )1593.74

ii

i

M r

apx

How much of your earnings would you give up (e.g. how much less than $1,000) would you accept to get the lump sum today?

9

1593.74675.90

(1.1)

1000 675.90 324.10( )

PV

rounded

Congratulations! You just got a new job that pays you $500 per week. The interest rate is 3%. Draw a graph & answer the following:

1. What is the present value of your income for two weeks?

2. What is the future value of income for two weeks?

3. What is your endowment bundle?

4. If your marginal rate of intertemporal substitution at your endowment bundle is 1, should you a) live pay check to pay check; b) borrow to spend more today or c) save to spend more tomorrow?

$985.44

$1,015Endowment Bundle

($500, $500)MRIS = 1Optimal Bundle

MRIS = 1.03Save to spend more tomorrow

What if the interest rate rises to 5%?

What if the interest rate drops to 0.5%

Interest Rates and the Intertemporal Budget Constraint

When interest rates go DOWN we have less money tomorrow if we save, but tomorrow’s dollars are worth more today.

When interest rates go UP we have more money tomorrow if we save, and less today if we discount future earnings.

The intertemporal budget constraint PIVOTS around the

original endowment.

BOTH the X and Y intercepts change!

Will you be happier if interest rates go UP or DOWN?

It depends on your time preference and the shape of our indifference curve!

M1=M2 is the original endowment

What happens if interest rates go up?

What happens if interest rates go down?

0

( )T

iT i

i

PV M M

“discount rate”

1

1i

i

ir

Beta – a time consistent discount rate

0

( )1

Ti

T ii i

MPV M

r

Looking ahead…What if we have inconsistent time preferences?

Static optimization One period

Type of Optimization Notational Set-up Form of Answer

A number

Dynamic OptimizationCalculus of Variations Euler Equation

0max ( )

xU f x

00

max ( , )T

tt

xt

U f t x

10

0

max ( , , )T

it t

xt

U f t x x

Optimal pathNot just the change in value, but the change in the curve

Optimal control Maximum Principal 00

0

max ( , ( ), ( ))

. . ( , , ); (0)

T

uF t x t u t dt

s t x g t x u x x

Optimal paths (curves) for state, control & co-state

Dynamic Programming Principal of Optimality

Decision Ruleoptimal control path

A vector of numbers Multi-periodSeries of discrete time problems

( )max ( , ( ), ( ); )

( , ( ), ( ); );

uf s x s u s

V t x t u t s t

Static and Dynamic Economic Models

Policy issue: dynamic scoringhttp://www.nber.org/digest/jul05/w11000.htmlhttp://crfb.org/category/blog-issue-areas/dynamic-scoring

Policy Summary

The higher the risk, the higher the discount rate

1. Times of war2. Risky new businesses3. Individuals with terminal illness – less uncertainty, lower discount!

Monetary Policy

1. Lower rates get people to spend more now!2. Lower rates mean lower hurdles for project investment3. Higher rates take money out of the economy now, increasing

hurdle rate for projects, reducing threat of inflation