Lecture10_IntertemporalChoice

8/12/2019 Lecture10_IntertemporalChoice

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Intermediate Microeconomic Theory

Intertemporal Choice


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So far, we have considered:

How an individual will allocate a given amount of money overdifferent consumption goods.

How an individual will allocate his time between enjoying leisure

and earning money in the labor market to be used for consuming

goods.

Another thing to consider is how an individual will decide how

much of his money should be consumed now, and how much he

should save for consumption in the future (or how much toborrow for consumption in the present).


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To think about this, instead of considering how an

individual trades off one good for another and viceversa, we can think about how an individual trades

off consumption (of all goods) in the present for

consumption (of all goods) in the future.

i.e. two goods we will consider are:

c1- dollars of consumption (composite good) in the

present period, and

c2- dollars of consumption (composite good) in a

future period.


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So an intertemporal consumption bundle is just a pair {c1, c2}.

E.g. a bundle containing $50K worth of goods this year,and $30K next year is denoted {c1 = 50K, c2= 30K}.

Endowment now describes how many dollars of consumption

an individual would have in each period, without saving orborrowing, denoted {m1, m2}.

For example,

An individual who earns $50K each year in the labor

market {m1 = 50K, m2= 50K}. An individual who earns nothing this year but expects

to inherit $100K next year {m1 = 0, m2= 100K}.


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Intertemporal Budget Constraint

Consider an individual has an

intertemporal endowment of {m1, m2} andcan borrow or lend at an interest rate r.

What will be his intertemporal budget

constraint?

What is one bundle you know will be

available for consumption?

What else can he do?


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What is slope?

Hint: How much moreconsumption will he have

next period if he saves $x this

period?

To put another way, how

much does consuming anextra $x this period cost in

terms of consumption next

period.

What will intercepts be?

x

c2

m2

m1 c1

?

x

?


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Intercepts

VerticalWhat if you saved all of yourperiod 1 endowment, how much would you

have for consumption in period 2?

HorizontalHow much could you borrow

and consume today, if you have to pay it

back next period with interest?

What happens to budget constraint when

interest rate r rises?


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Example:

Suppose person is endowed with $20K/yr

Interest rate r = 0.10

What will graph of BC look like?

What if r falls to 0.05?


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Writing the Intertemporal Budget Constraint

Given this framework, we want to write

out the intertemporal budget constraint inthe typical form

We know the interest rate r will determinerelative prices, but like with goods, we have

to determine our numeraire.


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Writing the Intertemporal Budget Constraint

So intertemporal budget constraint can

be written in two equivalent ways:

Future value: future consumption is numeraire, price ofcurrent consumption is relative to that.

How much does another dollar of current consumptioncost in terms of foregone future consumption?

BC: (1+r)c1+ c2= (1+r)m1+ m2

Present value:present consumption is numeraire, price offuture consumption is relative to that

How much does another dollar of future consumptioncost in terms of foregone current consumption?

BC: c1+ c2 (1/(1+r))= m1+ m2 (1/(1+r))


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Intertemporal Preferences

Do Indifference Curves make sense in this

context?

What does MRS refer to in this context?

Do Indifference Curves with Diminishing

MRS makes sense in this context?

What Utility function might be appropriate

to model decisions in this context?


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We can again think of analyzing optimal

choice graphically.

What does it mean when optimal choice is

a bundle to the left of endowment bundle?

How about to the right of the endowment

bundle?


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Similarly, we can solve for each individuals demand

functions for consumption now and consumption in thefuture, given interest rate (i.e. relative price) and endowment.

c1(r,m1,m2)

c2(r,m1,m2)

So if u(c1, c2) = c1ac2

b, an endowment of (m1,m2) and an

interest rate of r, what would be the demand function for

consumption in the present? In the future?


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As we showed graphically,

If c1(r,m1,m2) > m1

the individual is a borrower

If c1(r,m1,m2) < m1

the individual is a lender

Equivalently,

If c2(r,m1,m2) < m2

the individual is a borrower

If c2(r,m1,m2) > m2

the individual is a lender


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Analog to Buying and Selling

So instead of being endowed with coconut

milk and mangos (or time and non-laborincome) we can think of being endowed

with money now and money in the future.

Moreover, instead of being a buyer ofcoconut milk by selling mangos, we can

think of being a buyer of consumption now

(i.e. a borrower) by selling future

consumption.


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Comparative Statics in Intertemporal Choice

Suppose the interest rate decreases.

Will borrowers always remain borrowers?

Will lenders always remain lenders?


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Comparative Statics in Intertemporal Choice

How does this model inform us about

government interest rate policy?

Why might government lower interest rates?

Raise interest rates?


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Present Value and Discounting

The intertemporal budget constraint

reveals that timing of payments matter.

Suppose you are negotiating a sale and 3buyers offer you 3 different paymentsschemes:

1. Scheme 1 - Pay you $200 one year fromtoday.

2. Scheme 2 - Pay you $100 one year fromnow and $100 today.

3. Scheme 3 - Pay you $200 today.

Assuming buyers words are good, whichpayment scheme should you take? Why?(Hint: think graphically)


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This is idea ofpresent value discounting.

To compare different streams of payments, we have to have some wayof evaluating them in a meaningful way.

So we consider theirpresent value, or the total amount ofconsumption each would buy today.

Also called discounting.

In terms of previous example, with r = 0.10 thepresent valueof eachstream is:

1. PV of Scheme 1 = $200/(1+0.10) = $181.82

2. PV of Scheme 2 = $100 + $100/(1+0.10) = $190.91

3. PV of Scheme 3 = $200

While you certainly might not want to consume the entire paymentstream today, as we just saw, the higher the present value the biggerthe budget set (assuming same interest rate applies to all schemes!)


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What about more than two periods?

As we saw, if r is interest rate one period ahead, PV ofpayment of $x one period from now is $x/(1+r). What is

intuition?

If you were going to be paid $m two years from now,

what is the most you could borrow now if you had to payit back with interest in two years?

So what is general form for present value of a payment of$x n periods from now?

What is form for a stream of payments of $x/yr for the

next n years?


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Interest Rate and Uncertainty

So far, we have assumed there is no uncertainty.

Individuals know for sure what payments they willreceive in the future, both in terms of endowments and

loans given out.

What happens if there is uncertainty regarding whether

you will be paid back the money you lend or will be

able to pay back the money you borrow?

Lecture10_IntertemporalChoice

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