econ11_09_lecture3 - Utility Maximisation Problem Simon...

Utility Maximisation Problem
Simon Board
*
This Version: September 20, 2009
First Version: October, 2008.
The utility maximisation problem (UMP) considers an agent with income m who wishes to
maximise her utility. Among others, we are interested in the following questions:
• How do we determine an agent’s optimal bundle of goods?
• How do we derive an agent’s demand curve for a particular good?
• What is the effect of an increase in income on an agent’s consumption?
1 Model
We make several assumptions:
1. There are N goods. For much of the analysis we assume N = 2, but nothing depends on
this.
2. The agent takes prices as exogenous. We normally assume prices are linear and denote
them by { p
1
, . . . , p
N
} .
3. Preferences satisfy completeness, transitivity and continuity. As a result, a utility func-
tion exists. We normally assume preferences also satisfy monotonicity (so indifference
curves are well behaved) and convexity (so the optima can be characterised by tangency
conditions).
*
Department of Economics, UCLA. http://www.econ.ucla.edu/sboard/. Please email suggestions and typos
to [email protected]
1

Eco11, Fall 2009 Simon Board
4. The consumer is endowed with income m .
The utility maximisation problem is:
max
x
1
,...,x
N
u ( x
1
, . . . , x
N
) subject to
N
X
i =1
p
i
x
i
≤ m (1.1)
x
i
≥ 0 for all i
The idea is that the agent is trying to spend her income in order to maximise her utility. The
solution to this problem is called the Marshallian demand or uncompensated demand. It is
denoted by
x
*
i
( p
1
, . . . , p
N
, m )
The most utility the agent can attain is given by her indirect utility function . It is defined
by
v ( p
1
, . . . , p
N
, m ) = max
x
1
,...,x
N
u ( x
1
, . . . , x
N
) subject to
N
X
i =1
p
i
x
i
≤ m (1.2)
x
i
≥ 0 for all i
Equivalently, the indirect utility function equals the utility the agent gains from her optimal
bundle,
v ( p
1
, . . . , p
N
, m ) = u ( x
*
1
, . . . , x
*
N
) .
1.1 Example: One Good
To illustrate the problem, suppose N = 1. For example, the agent has income m and is choosing
how many cookies to consume. The agent’s utilities are given by table 1.
In general, we solve the problem in two steps. First, we determine which bundles of goods are
affordable. The collection of these bundles is called the budget set . Second, we find which
bundle in the budget set the agent most prefers. That is, which bundle gives the agent most
utility.
Suppose the price of the good is p
1
= 1 and the agent has income m = 4. Then the agent can
afford up to 4 units of x
1
. Given this budget set, the agent’s utility is maximised by choosing
x
*
1
= 4, yielding utility v = 28.
2

Eco11, Fall 2009 Simon Board
Units of x
1
Utility
1 10
2 18
3 24
4 28
5 30
6 29
7 26
8 21
Table 1:
Utilities from different bundles. Observe that this agent is satiated at 5 units.
Next, suppose the price of the good is p
1
= 1 and the agent has income m = 8. Then the
agent can afford up to 8 units of x
1
. Given this budget set, the agent’s utility is maximised by
choosing x
*
1
= 5, yielding utility v = 30. In this example, the consumer can afford 8 units but
chooses to consume 5. If the agent’s preferences are monotone, then she will always spend her
entire budget.