The Budget Constraint
Introduction
Economics is “all over” optimization subject to constraint. No better example
exists than for the consumer. The version of consumer theory that we talk about
in Econ. 380 is static in its orientation. That is, we shall not index consumption
by time. Thus, we don’t talk about saving or even the purchase of durables except
as a service or rental. In 382, saving is addressed. However, it is clear that most of
the time income, I, constrains consumption. For a model of labor supply,
however, we model a conscious decision about how much income to have to
allocate to consumption rather than treating it as fixed. Further, the question is
often asked: “who’s constraint?” Typically, we talk about an individual or a
household that behaves like an individual and discuss little about intra-household
bargaining. With these caveats, we embark together on understanding how to
construct budget constraints and what they mean. Throughout much of Nicholson
(and Snyder), two goods are considered, x and y. However, in economic theory, it
would be more likely to see notation like an N vector x = (x
1
, x
2
,…,x
N
) where the
subscript represents a particular good (e.g., x
1
=the quantity of apples, x
2
=bananas,
x
3
gasoline and so on). This is occasionally mentioned in the text. I think the
ideas are usually best developed with the simple two good model.
Linear Pricing
Most of the time economists think of a budget set, B which is a subset of
2
R
. It is
the set of all possible consumption bundles available to an individual. If a
consumer has income I, it is the set
(1)
:{( , ):
}
xy
B
xy px py I
, where x and y are two consumption goods
(apples and bananas) and
x
p
and
y
p
the unit prices of x and y respectively.