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Adding one risk to another
1 Introduction
A large number of studies examined background risk due to the belief that
background risk impacts the wealth and hence the welfare of the agent (see,
for example Menoncin (2002), among others). However, the impact of back-
ground risk on the agent’s decision variable(s) was not examined by the lit-
erature. Since the agent has control over the decision variable(s) only, it is
much more useful to investigate the impact of background risk on the agent’s
decision variable(s).
The literature focused on the impact of the additive form of background
risk. Examples include Gollier and Pratt (1996), Quiggin (2003) and Machina
(1982). On the other hand, the multiplicative background risk was largely ne-
glected. Exceptions include Franke et al. (2003) and Pratt (1988). Studies deal-
ing with more general forms of background risk are virtually non-existent.
Even the ones that examined the additive or multiplicative form provided re-
strictive models and results. They placed restrictions on the functional form,
probability distributions, and/or the characteristics of the risk such as unde-
sirable risk. For example, Gollier and Pratt (1996) and Franke et al. (2003)
adopted undesirable risk and restricted the functional form of utility.
Another important restriction which is imposed by the previous models is
the statistical independence assumption. That is, the background risk is inde-
pendent of the other risk(s). Moreover, background risk is normally incorpo-
rated in choice models (Quiggin, 2003). Therefore, the impact of background
risk on production decisions is rarely investigated. In sum, all the previous
models are restrictive in multiple aspects.
This paper overcomes all the restrictions and limitations of the previous
models. First, it relaxes the independence assumption. Second, it adopts a
general functional form. Third, it adopts a general type of risk (as opposed to
undesirable risk or mean-zero risk). Fourth, it introduces a new general form
of background risk. Finally, it incorporates background risk into theory of the
firm, as opposed to choice models. In sum, this paper serves as a general
theory of modeling background risk and the impact of adding one risk to
another.
2 The model
Profit is given by π = η($ + py = c(y)) ≡ η($ + Π), where y is output, p
is price, c is the cost function, $ is a random variable representing wealth
(Franke et al., 2003) or additive background risk, η > 0 is a multiplicative
background risk.1 The risk averse firm maximizes the expected utility of the
profit
Max Eu(∏)
y
where u is a von Newmann-Morgestern utility function.
1Franke et al. (2003) used the restriction En ≤ 1, while Gollier and Pratt (1996) used the
restriction E$ ≤ 0
Revista de Economia del Rosario. Vol. 14. No. 1. Enero - Junio 2011. 57-60