intermediate microeconomic theory intertemporal choice

Intermediate Microeconomic Theory

Intertemporal Choice


So far, we have considered: How an individual will allocate a given amount of money over

different 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 to borrow for consumption in the present).


To think about this, instead of considering how an individual trades off one good for another and vice versa, 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.


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 or borrowing, 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}.

Intertemporal Budget Constraint

Consider an individual has an intertemporal endowment of {m1, m2} and can 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?


What is slope? Hint: How much more

consumption will he have next period if he saves $x this period?

To put another way, how much does consuming an extra $x this period “cost” in terms of consumption next period.

What will intercepts be?

x

c2

m2

m1 c1

?x

?


Intercepts Vertical – What if you saved all of your

period 1 endowment, how much would you have for consumption in period 2?

Horizontal – How 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?


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?

Writing the Intertemporal Budget Constraint Given this framework, we want to write

out the intertemporal budget constraint in the typical form

We know the interest rate r will determine relative prices, but like with goods, we have to determine our “numeraire”.

Writing the Intertemporal Budget Constraint So intertemporal budget constraint can

be written in two equivalent ways:

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

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

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

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

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

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

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?


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?


Similarly, we can solve for each individual’s demand functions for consumption now and consumption in the future, given interest rate (i.e. relative price) and endowment.

c1(r,m1,m2)

c2(r,m1,m2)

So if u(c1, c2) = c1a c2

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?


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

Analog to Buying and Selling

So instead of being endowed with coconut milk and mangos (or time and non-labor income) we can think of being endowed with money now and money in the future.

Moreover, instead of being a buyer of coconut milk by selling mangos, we can think of being a buyer of consumption now (i.e. a borrower) by selling future consumption.

Comparative Statics in Intertemporal Choice Suppose the interest rate decreases.

Will borrowers always remain borrowers?

Will lenders always remain lenders?

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?

Present Value and Discounting

The intertemporal budget constraint reveals that timing of payments matter.

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

1. Scheme 1 - Pay you $200 one year from today.

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

3. Scheme 3 - Pay you $200 today.

Assuming buyers’ word’s are good, which payment scheme should you take? Why? (Hint: think graphically)


This is idea of present value discounting. To compare different streams of payments, we have to have some way

of evaluating them in a meaningful way. So we consider their present value, or the total amount of

consumption each would buy today. Also called discounting.

In terms of previous example, with r = 0.10 the present value of each stream is:

1. PV of Scheme 1 = $200/(1+0.10) = $181.822. PV of Scheme 2 = $100 + $100/(1+0.10) = $190.913. PV of Scheme 3 = $200

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


What about more than two periods? As we saw, if r is interest rate one period ahead, PV of

payment 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 pay it 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?

Interest Rate and Uncertainty

So far, we have assumed there is no uncertainty. Individuals know for sure what payments they will

receive 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?

intermediate microeconomic theory intertemporal choice

Documents