1. Introduction
Becker’s Irrational Consumer paper ([1]) is a classic. Becker considered a consumer choosing randomly from their budget set, according to a uniform probability distribution. He then demonstrated that on average both compensated and uncompensated demand curves are downward sloping; put differently, mean choices (in a two good framework) satisfy the weak axiom of revealed preference. Becker’s demonstration was a geometric argument that made particular use of the assumption of a uniform distribution. Becker’s conclusion can be easily generalized. Here we demonstrate that if individuals randomly choose consumption bundles on their budget sets, then on average, both the compensated and uncompensated demand curves are downward sloping, irrespective of the underlying probability distribution of the random choices. The proof is surprisingly simple.
We also note that the Becker framework is formally similar to that of Hildenbrand ([2])1 but the interpretations differ. Hildenbrand characterizes average demand when there are a large number of heterogeneous infinitesimal agents while the Becker paper considers the average behavior of a single individual making a large number of repeated random choices from their budget line. One can also think of the Becker framework as a characterizing mean consumption of a large population of identical individuals each making random choices from their budget line. The Hildenbrand framework, however, is much more general; distribution of population characteristics is arbitrary and the number of goods, while finite, is not restricted to just two. Becker’s framework can easily be generalized along these lines.
2. Downward Sloping Demand in the Two Good Model
There are two goods, x and y with prices 
and
, income is I. 
is a random variable distributed on [0, 1] with mean
. An individual randomly chooses a point on budget line
We suppose that this choice occurs as a realization of
, so that
and
.
Taking expectations, average or mean demand is given by
and
Clearly, mean demands 
and 
both slope downward for 
strictly between 0 and 1. Since
, the mean demand curves 
and 
corresponds to a utility maximizing individual with Cobb-Douglas preferences. Since the mean demand curve arises from utility maximization, mean consumption conforms to WARP. It is clear, however, that there are no restrictions on the underlying distribution of
, with support [0, 1], other than the existence of a mean
. The case Becker considered corresponds to 
distributed uniformly on the unit interval. For this case, the mean of the distribution 
is 1/2. This argument also generalizes to more than 2 commodities, where the support of the random variable is the unit simplex, which is the set of non-negative vectors whose components add up to one.
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