University of Toronto

Rotman School of Management

Finance

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

that:

Pa.Na + Po.No = $10

where:

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

or

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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

where

MUa = Marginal Utility of Apples = Change in Utility .

Change in Apples Consumption

And likewise for oranges …

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## Document Summary

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 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: The economic (allocative) role of the interest rate.