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