Adding one risk to another: generalizing the unavoidable (background) risk.

• Author: Alghalith, Moawia
• Position: Report
• Pages: 57(4)

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 background risk on the agent's decision variable(s) was not examined by the literature. 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 neglected. Exceptions include Franke et al. (2003) and Pratt (1988). Studies dealing with more general forms of background risk are virtually non-existent. Even the ones that examined the additive or multiplicative form provided restrictive models and results. They placed restrictions on the functional form, probability distributions, and/or the characteristics of the risk such as undesirable 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 independent of the other risk(s). Moreover, background risk is normally incorporated 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], 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, [??] > 0 is a multiplicative...