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University of Toronto
Rotman School of Management
D.J.S. Brean
21 September 2004
The 2-Period Model of Intertemporal Allocation of Consumption
The 2-period model of intertemporal allocation of consumption, together with its extension to a
simple model of investment, provides a useful framework for understanding several important
relationships and decisions in finance.
The 2-periods may be thought of as:
now versus later, or
today versus tomorrow, or
the present generation versus future generations.
The 2-period context captures the idea that one makes decisions today concerning consumption,
savings and investment that have consequences for consumption tomorrow.
The focus is on consumption. The idea is that money is used to consume things - food, clothing,
housing, entertainment, et cetera.
In our course in Finance, the reason for introducing the 2-period model is to provide a conceptual
framework for understanding the economic function of capital markets and the behaviour of
individuals with respect to that market. Among the concepts and issues addressed in the 2-period
model are:
Why people save
Why people lend
Why people invest
Determinants of the interest rate
The economic (allocative) role of the interest rate
Present value
Return on investment
The opportunity cost of investment.
Optimal investment criterion

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Value maximization
Dividend decision
The relevance of finance
Capital market - financial intermediaries
A Quick Review of Consumer Theory
There is a strong parallel between the analytic techniques applied in the problem of 2-period
allocation of consumption and the techniques used in standard consumer theory.
In consumer theory, individuals maximize utility defined over goods. The total amount of
consumption is subject to a budget constraint.
For example, consider the problem of buying apples and oranges with a given amount of money,
say $10.00. Apples cost $0.50 each and oranges cost $1.00 each.
Since the total amount of apples plus oranges that can be purchased is limited to $10.00, we know
Pa.Na + Po.No = $10
Pa = Price of an apple
Na = Number of apples purchased
Po = Price of an orange
No = Number of oranges purchased
How many apples and how many oranges does a person buy? That depends on her preference for
apples and oranges in light of the prices and her budget constraint ($10.00).
In the jargon of consumer theory, a consumer maximizes utility - defined, in this case, over apples
and oranges - subject to a budget constraint. The budget constraint is:
Pa.Na + Po.No = $10
The consumer's utility function represents her preferences, viz.,
Utility is a function of apples and oranges

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Ui = U(A, O)
where the subscript i refers to the ith or the representative consumer. The U on the right-hand-side
is the function.
In this context, the theory of consumer behaviour is premised on a number of points:
1. Utility increases with higher purchases (consumption) of apples.
2. Utility increases with higher purchases (consumption) of oranges.
3. The utility of apples consumption diminishes at the margin. That means that the second
apple provides less utility than the first apple, the third apple provides less utility than the
second apple, and so on.
4. Likewise, the utility of oranges consumption diminishes at the margin.
5. To purchase more apples, one must purchase fewer oranges.
6. To purchase more oranges, one must purchase fewer apples.
7. Since apples and oranges are both desirable things, a consumer will buy some apples
and some oranges.
8. The rate at which a consumer is able to substitute apples for oranges (what she can
do) is determined by the relative price of apples for oranges, which in this case is Pa/Po or
$0.50/$1.00 or 0.5. This means that two apples can always be substituted for one orange,
or one orange can always be substituted for two apples.
9. The consumer's choice can be solved if the utility function is known. Consumers
behave as if each person equates the ratio of marginal utility to price for each good, viz:
MUa / Pa = MUo / Po
MUa = Marginal Utility of Apples = Change in Utility .
Change in Apples Consumption
And likewise for oranges
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