Lecture Notes on Separable Preferences:
Ted Bergstrom UCSB Econ 210A
When applied economists want to focus attention on a single commodity or
on one commodity group, they often find it convenient to work with a two
commodity model, where the two commodities are the one that they plan
to study and a composite commodity called ”other goods”.
For example,
someone interested in the economics of nutrition may wish to work with
a model where one commodity is an aggregate commodity “food” and the
other is “money left over for other goods.” To do so, they need to determine
which goods are foods and which are not and then define the quantity of the
aggregate commodity, food, as some function of the quantities of each of the
food goods.
In the standard economic model of intertemporal choice model of in
tertemporal choice, commodities are distinguished not only by their physical
attributes, but also by the date at which they are consumed. In this model,
if there are
T
time periods, and
n
undated commodities, then the total
number of dated commodities is
nT
. Macroeconomic studies that focus on
savings and investment decisions often assume that there is just one “ag
gregate good” consumed in each time period and that the only nontrivial
consumer decisions concern the time path of consumption of this single good.
In these examples, and in many other applications of economics, the tactic
of reducing the number of commodities by aggregation can make difficult
problems much more manageable. In general, such simplifications can only
be purchased at the cost of realism.
Here we examine the “separability
conditions” that must hold if this aggregation is legitimate.
To pursue the food example, suppose that the set of
n
commodities can
be partitioned into two groups, foods and nonfoods. Let there be
m
food
commodities. We will write commodity vectors in the form
x
= (
x
F
, x
∼
F
)
where
x
F
is a vector listing quantities of each of the food goods and
x
∼
F
is
a vector quantities of each of the nonfood goods. If we are going to be able
to aggregate, it must be that consumers have preferences representable by
utility functions
u
of the form
u
(
x
F
, x
∼
F
) =
U
*
(
f
(
x
F
)
, x
∼
F
)
f
is a realvalued function of
m
variables and where
U
*
is a strictly increasing
function of its first argument. The function
f
is then a measure of the amount
1
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of the aggregate commodity food. Notice that the function
u
is a function
of
n
variables, while
U
*
is an function of only
n

m
+ 1 variables, which
denote quantities of each of the nonfood variables and a quantity of the food
aggregate
f
.
Economists’ standard general model of intertemporal choice, is based on
the use of dated commodities. Thus if there are
n
“undated commodities,”
and
T
time periods, we define
x
it
to be the quantity of commodity
i
con
sumed in period
t
. We define
x
t
to be the
n
vector of commodities consumed
in period
t
and we define
x
= (
x
1
, . . . , x
T
) to be the
nT
vector listing con
sumption of each good in each period. This is sometimes called a time profile
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 Spring '10
 fallahi
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