The Consumer Problem
Consumer theory is concerned with how a rational consumer would make consumption
decisions. What makes this problem worthy of separate study, apart from the
general problem of choice theory, is its particular structure that allows us to derive
economically meaningful results. The structure arises because the consumer’s
choice sets sets are assumed to be defined by certain prices and the consumer’s
income or wealth. With this in mind, we define the consumer problem (CP) as:
max
x∈Rn
+
u(x)
s.t. p · x ≤ w
The idea is that the consumer chooses a vector of goods x = (x1, ..., xn) tomaximize
her utility subject to a budget constraint that says she cannot spend more than
her total wealth.
What exactly is a “good”? The answer lies in the eye of the modeler. Depending
on the problem to be analyzed, goods might be very specific, like tickets to different
world series games, or very aggregated like food and shelter, or consumption and
leisure. The components of x might refer to quantities of different goods, as if
all consumption takes place at a moment in time, or they might refer to average
rates of consumption of each good over time. If we want to emphasize the roles of
quality, time and place, the description of a good could be something like “Number ........
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