A NOTE ON THE PROPERTIES OF PRODUCTS OF RANDOM VARIABLES WITH REFERENCE TO ECONOMIC APPLICATIONS*
| DOI | http://doi.org/10.1111/j.1468-0084.1980.mp42004003.x |
| Published date | 01 November 1980 |
| Date | 01 November 1980 |
| Author | ROBERT W. BACON |
A NOTE ON THE PROPERTIES OF
PRODUCTS OF RANDOM VARIABLES WITH
REFERENCE TO ECONOMIC APPLICATIONS*
By ROBERT W. BACON
INTRODUCTION
The recent growth of interest in the role of uncertainty in economics in both
theoretical and applied fields has increased the need to have available a wider range
of statistical results on the properties of random variables. One problem, which is
now appearing in many fields of economics, is related to the properties of products
of random variables. Because only very special and limited results concerning the
properties of random variables have been used in the economic literàture and
because even in the theoretical statistical literature the results are not widely
referred to, the purpose of this note is to draw attention to a series of statistical
results which may be of considerable use in economics. The first part of the note
reviews some areas of economics where considerations of products of random
variables arise, while the second part indicates the main relevant statistical results
and gives a limited bibliography relating to the subject.
ECONOMIC MODELS INVOLVING PRODUCTS OF RANDOM VARIABLES
One of the main areas of economics where uncertainty has been introduced is
the theory of the firm. Many of the recent developments in the theory of the firm
under uncertainty have been discussed by Young while the general literature has
been reviewed by Hey. Traditionally theoretical models have concentrated on
demand uncertaintythe firm either presets output (via its known production
function) and then faces an uncertain market clearing price, or it presets price and
then, via the uncertain quantity demanded, faces an uncertain change in stocks.
Such firms have usually been assumed to be risk neutral and to maximize the
expected value of profits, although higher moments have been used to model
different attitudes to risk.
In the econometric literature on production functions a different source of
uncertainty has been investigated, notably by Zehner, Kmenta and Drèze. Their
firms face certain output prices (and quantity demanded) but the production
function itself (as seen by the firm) is uncertain. Again firms are assumed to
maximize expected profits or, as in the work of Gander, a weighted average of the
mean and variance of profits. Gander generalizes further by allowing both demand
and supply to be uncertain.
In the field of agricultural supply functions the explicit introduction of
uncertainty has been particularly common. Early emphasis was placed on the
randomness of prices and on the resulting effects on farmers planting decisions.
* I am grateful to Leslie Young for helpful comments on an earlier draft of this note.
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