The name is absent

Alghalith

Lemma 1. In the presence of a multiplicative background risk only (i.e., $ = 0)
output is equal to its certainty-equivalent level.²

Proof. The first order condition is

(p - c )(y*)Eu (∏*)η = 0=(p - c (y*))                 (1)

since Eu⁰(π*)η > 0, clearly the certainty equivalent output y satisfies the con-
dition p = c⁰(y) and thus y = y*.                                            □

Theorem 2. A. Starting with wealth risk, adding a multiplicative background risk
does not affect the optimal output.

B. Starting with a multiplicative background risk, adding a wealth risk does not
affect the optimal output

Proof. A. The first-order condition in the presence of both risks

(p - c')(y**)Eu'(∏**)η = 0=(p - c'(y**))                (2)

therefore, using the lemma, y = y* = y**.

The proof of B is similar and thus omitted.                            □
3 General background risk

A general form of gackground risk is specified as π = g($ + Π, η). The first
order condition is

( p - c⁰ )( y** ) Eu' ( ∏) g⁰ ( ∙ ) =0 = ( p - c' ( y ))                  (3)

since Eu⁰(π)g'(∙) = 0.

Thus the results of the theorem hold under a general background risk. That
is, a general background risk does not impact the optimal output. Clearly,
under the multiplicative form g'(∙) = η and g'(∙ ) = 1 under the additive form.
An example of another form of background risk is π = ()^η.

An important implication of this result is that even though, the addition of
the background risk may affect the welfare of the agent, it does not influence
his decision if the price of the asset is certain.

References

Franke, G., H. Schlesinger, and R. C. Stapleton (2003, May). Multiplicative
background risk. (52), 146-153.

Gollier, C. and J. W. Pratt (1996). Risk vulnerability and the tempering effect
of background risk. Econometrica 64(5), 1109-1123.

Machina, M. J. (1982). Expected utility analysis without the independence
axiom. Econometrica 50(2), 277-323.

2

Revista de Economla del Rosario. Vol. 14. No. 1. Enero - Junio 2011. 57 - 60

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