Uncertainty and Risk - DARP

Microeconomics CHAPTER 8. UNCERTAINTY AND RISK
Exercise 8.13 Consider a competitive, price-taking …rm that confronts one of
the following two situations:
“uncertainty”: price p is a random variable with expectation p.
“certainty”: price is …xed at p.
It has a cost function C(q) where q is output and it seeks to maximise the
expected utility of pro…t.
1. Suppose that the …rm must choose the level of output before the particular
realisation of p is announced. Set up the …rm’s optimisation problem and
derive the …rst- and second-order conditions for a maximum. Show that,
if the …rm is risk averse, then increasing marginal cost is not a necessary
condition for a maximum, and that it strictly prefers “certainty” to “uncertainty”.
Show that if the …rm is risk neutral then the …rm is indi¤erent
as between “certainty” and “uncertainty”.
2. Now suppose that the …rm can select q after the realisation of p is announced,
and that marginal cost is strictly increasing. Using the …rm’s
competitive supply function write down pro…t as a function of p and show
that this pro…t function is convex. Hence show that a risk-neutral …rm
would strictly prefer “uncertainty” to “certainty”.
Outline Answer:
1. Pro…t is given by
:= pq
C(q)
where p is a random variable. Maximising expected utility of pro…t Eu()
by choice of q requires the FOC
E(u ()p) E(u ())C q = 0
where u () is the …rst derivative of u(). This will represent a maximum
if
d 2 Eu
dq 2 < 0:
We …nd that this implies
E(u [p C q ] 2 ) E(u )C qq < 0:
Notice that since the …rst term is negative for a risk-averse …rm then the
condition can be satis…ed not only if C qq > 0 but also if C qq < 0 and jC qq j
is not too large. Now consider transforming p to bp thus: bp = (1 )p + p
then bp has the same mean as p but is less dispersed. Maximised utility for
the random variable bp is
Eu([(1 )p + p]q C(q ))
where q is the output satisfying the …rst-order conditions for a maximum.
Di¤erentiate this expected utility with respect to
@Eu
@ = [E(u [p p])]q + [E(u [bp C q ])] @q
@
cFrank Cowell 2006 132