3420A ECON - Lecture 1
Intertemporal Choice
In this chapter, we study how decisions of consumption and investment can be made
in a world of two periods and when there is no uncertainty over future incomes. In
such a setting, the analysis is actually very similar to what we have learned in utility
maximization in basic micro. However, our analysis does produce new insights. In
particular, we see how utility maximization turns out to be equivalent to present value
maximization. This gives rise to an important decision rule in investment, i.e. choose
the investment project that gives the highest present value of future incomes.
Although this rule is derived under the assumption of certainty, we will see later on
that the same rule essentially applies in a world of uncertainty as well.
1. Present Value Rule
Suppose you have $100. You can either consume it now or put it in a bank, which
pays you an interest rate of say, 10%. This means you can withdraw $110 one year
from now. The money market provides you with a choice of instant consumption
of $100 or $110 one year from now. In other words, $ 110 one year from now is
worth only $110/(1+10%) today.
Since a dollar at different future dates has different value, one needs to discount
future cash flows appropriately so that cash flows at different dates can be added
together. One method to do this is the present value method i.e. we discount
each future cash flow to the present before we add them up.
Consider a cashflow stream: -C0, C1, 0, C3. If we invest C0 now (note the negative
sign), then we get C1 at the end of the first year, nothing the second year and C3
the third year. The present value of this stream is:
PV C 0
C3
C1
0
2
(1 r ) (1 r )
(1 r ) 3
where r is the interest rate. If we were given numbers for the stream and a suitable
interest rate, we could calculate PV.
The general rule of investment decision is: If PV turns out to be positive, then the
investment project is profitable at the interest rate of r. Suppose the opportunity
1
cost of investing in the project is some foregone interest that could have been
earned by putting the money in a bank. Then using the bank interest rate as r, a
positive PV means we get a higher return from investing C0 in this project than
from putting it in the bank for three years. Alternatively, if C0 has to be borrowed,
then using the borrowing interest rate as r, a positive PV means we will end up,
after repaying loans, with a surplus in three years. In either case, the investment
decision is positive.
The simple rule of thumb is therefore,
a) The higher is the PV, the more profitable is the project, and so the higher is its
priority (if investment budget is limited).
b) All projects that have positive PV should be undertaken because each of them
can increase wealth.
c) There is always a discount rate r* (could be more than one) that makes the PV
of a cashflow stream equal to 0. This rate is called the (internal) rate of return
of the project generating the stream. The higher is the PV of a project, the
higher also is the rate of return. If the PV of a project is positive at a certain
discount rate r, then the (internal) rate of return of the project is higher than r.
2. Intertemporal Consumption Equilibrium
This model shows how much a consumer chooses to save when there is nowhere
to invest savings.
Suppose a consumer lives for two periods only. In period 1, income is y1 and in
period 2, y2. Assuming that interest rate is r, the (present value) wealth of this
consumer is
W y1
y2
.
(1 r )
Suppose the consumer can borrow or lend money in the market at rate r. If the
consumer chooses to leave nothing to period 2, the maximum amount that can be
consumed in period 1 is simply W. Why? If the consumer chooses not to consume
at all in period 1, then the maximum amount that can be consumed in period 2 is
y1(1+r)+y2 . Why? These two values are the end points of the WW line in the
diagram below.
C2
W
2
C1
The equation of the WW line is
C1
C2
y
y1 2
1 r
1 r
because the present value of consumption must be equal to the present value of
income. The endowment point E represents the original income stream (y1, y2).
Note that if the consumer consumes $1 less in period 1, $(1+r) more can be
consumed in period 2. The slope of the line is therefore (1+r). It is clear that the
consumer can, by borrowing (consuming more than present income) or lending
(consuming less than present income), attain any point on the wealth constraint
WW.
The indifference curves in the diagram represent the consumers preference over
bundles of present and future consumption. As drawn, the consumer lends money
in period 1 to smooth out the consumption stream. If there were no money market
that allows borrowing and lending, consumption amounts would have been very
different in the two periods. Utility would have been lower, at u1 instead of u2.
Note that at the equilibrium point, the slope of WW line is equal to the slope of
the indifference curve u2. The marginal rate of substitution between present and
future consumption is equal to one plus the market interest rate.
3. Equilibrium with Investment Opportunity (no borrowing or lending)
We now consider a different scenario, with one investment but no
borrowing/lending. Consider the situation where projects of different profitability
(i.e. different rates of return) are available to the consumer.
C2
F
K
u2
3
E
u1
In the above diagram, the consumer has an endowment at E. If current
consumption is reduced from E, more consumption can be obtained in the next
period. The substitution possibility between present and future consumption is
represented by the frontier FF. The shape of FF shows that as investment
increases, projects of lower and lower profitability have to be undertaken. How
much the consumer ends up investing depends again on preference. As drawn, the
equilibrium point is at K, and I (horizontal distance between E and K) is the
amount of investment. Utility increases from u1 to u2.
At the equilibrium point, the indifference curve is tangent to the investment
frontier. So the marginal rate of substitution between present and future
consumption is equal to one plus the rate of return of the last project taken.
4. Equilibrium with Both Investment and Borrowing/Lending
We now construct a more complete model by combining the two models above.
C2 w
F
Q
K
u3
u2
w
E
B
u1
F
C2
I
Again endowment is at E. Given investment opportunities, each point on FF is
attainable. Making use of the money market (the slope of WW is given and
constant), the consumer maximizes utility at K. The consumer invests amount I
and borrows amount B in period 1. Given both money and investment markets, the
4
highest utility that can be attained is u3.
There is another way to look at the model. The agent maximizes utility in two
steps.
In the first step, the agent maximizes wealth and chooses Q. In the second step,
the agent maximizes utility by consumption smoothing and chooses K. The
maximization problem of the first step is
max y1
y2
subject to frontier FF.
(1 r )
The solution of this problem is Q. Note that at Q, FF and WW are tangent whereas
to the right of Q, FF is steeper than WW. It follows that at Q, all projects with
rates of return higher than the market interest r are accepted. It also means that,
using 1/(1+r) to do the discounting, all the projects accepted have a PV that is
positive. This confirms the investment decision rule set out in Section 1.
It is important to note that our investment decision rule is applicable to all agents
regardless of their preferences. The first step of utility maximization does not
involve indifference curves. This observation leads to the Separation Theorem
which says that investment decision and consumption decision can be dealt with
separately.
The second step is basically a consumption smoothing problem. Given the income
stream (after undertaking investment) at Q, the consumer maximizes utility by
borrowing the amount B.
Reference:
Hirshleifer, Jack, Price Theory and Application, Prentice Hall
5
Example
Consider a project which requires capital injection of $100 up front and $30 at the end
of two years, and yields incomes of $50 at the end of one year, $ 80 at the end of three
years, and $200 at the end of four years.
How much is this project worth today if
1. Money is diverted from another project which yields a rate of return of 10%
2. Money is borrowed from a bank which charges a fixed rate of interest of 10%
3. Suppose the project costs $117.4 in the market, and you need to borrow all funding
required, how much money do you expect to collect at the end of the project. Assume
the interest rate charged is 10%.
(Ans. Project worth is $117.4, being the PV of the cashflow stream.
Q3: If the market is efficient, information is perfect and competition among investors
bids up the price of the project until no profit can be made.)
6
Discussion Questions
1. Suppose you are certain you will receive the income stream C0 in the current
period, C1 one period from now, C2 two periods from now, C3 three periods from
now, where C0, C1, C2 and C3 could be positive or negative, what is the present
value of this income stream if the interest rate is r%?
2. What is the meaning of r? If r is larger, would PV be smaller? Relate r to the rate of
interest that you can get from other investments. Interpret r as the opportunity cost of
investing your money in the project being considered.
3. Anything that allows you to receive future incomes is actually an asset to you.
Whether it is physical (like a house that gives you rental income, a bond that gives
you interest payments, or a stock that gives you dividend) or intangible (like skills and
knowledge that allow you to earn higher wages) makes no difference. Discuss.
4. What is the meaning of wealth? If Mr. A has more money than Ms. B in their bank
accounts, does it follow that Mr. A is wealthier than Ms. B? What else do you need to
know to make a comparison of their wealth?
5. Since 2009, interest rates around the world have been kept at a low level, largely
because governments increase money supply in the hope of stimulating investment
and reducing financing cost of companies. However, what worries people is that
interest rates cannot fall further and in the event it goes up, investors in bonds and
stocks will suffer losses, causing another financial crisis. Discuss.
7