Chapter 6
Choice Under Uncertainty
Up until now, we have been concerned with choice under certainty.
A consumer chooses which
commodity bundle to consume.
A producer chooses how much output to produce using which mix
of inputs.
In either case, there is no uncertainty about the outcome of the choice.
We now turn to considering choice under uncertainty, where the objects of choice are not
certainties, but distributions over outcomes.
For example, suppose that you have a choice between
two alternatives.
Under alternative A, you roll a sixsided die.
If the die comes up 1, 2, or 3, you
get $1000.
If it comes up 4, 5, or 6, you lose $300.
Under alternative B, you choose a card from
a standard 52 card deck.
If the card you choose is black, you pay me $200.
If it is a heart, you
get a free trip to Bermuda.
If it is a diamond, you have to shovel the snow o
ff
of my driveway all
winter.
If I were to ask you whether you preferred alternative A or alternative B, you could probably
tell me.
Indeed, if I were to write down any two random situations, call them
L
1
and
L
2
, you could
probably tell me which one you prefer.
And, there is even the possibility that your preferences
would be complete, transitive (i.e., rational), and continuous.
If this is true then I can come up
with a utility function representing your preferences over random situations, call it
U
(
L
)
, such that
L
1
is strictly preferred to
L
2
if and only if
U
(
L
1
)
> U
(
L
2
)
.
Thus, without too much e
ff
ort, we can
extend our standard utility theory to utility under uncertainty.
All we need is for the consumer
to have well de
fi
ned preferences over uncertain alternatives.
Now, recall that I said that much of what we do from a modeling perspective is add structure
to people’s preferences in order to be able to say more about how they behave.
In this situation,
what we would like to be able to do is say that a person’s preferences over uncertain alternatives
158
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Nolan Miller
Notes on Microeconomic Theory: Chapter 6
ver: Aug. 2006
should be able to be expressed in terms of the utility the person would assign to the outcome if
it were certain to occur, and the probability of that outcome occurring.
For example, suppose
we are considering two di
ff
erent uncertain alternatives, each of which o
ff
ers a di
ff
erent distribution
over three outcomes: I buy you a trip to Bermuda, you pay me $500, or you paint my house.
The
probability of each outcome under alternatives A and B are given in the following table:
Bermuda
$500
Paint my house
A
.3
.4
.3
B
.2
.7
.1
What we would like to be able to do is express your utility for these two alternatives in terms
of the utility you assign to each individual outcome and the probability that they occur.
For
example, suppose you assign value
u
B
to the trip to Bermuda,
u
m
to paying me the money, and
u
p
to painting my house.
It would be very nice if we could express your utility for each alternative
by multiplying each of these numbers by the probability of the outcome occurring, and summing.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 wer
 Utility, Nolan Miller

Click to edit the document details