Kata Bognar
[email protected]
Economics 142
Probabilistic Microeconomics
UCLA
Fall 2009
5
Decisions in the Presence of Risk
Readings.
Chapter 6.
Concepts.
Gambling problem, investment problem, insurance problem. Fair, favor
able and unfavorable gamble / insurance. First and secondorder conditions. Arbitrage.
Local risk neutrality. State prices.
5.1
Example
•
Previously, we have characterized optimal choice under uncertainty with only a
few available options, i.e. hire the applicant or not, go to restaurant A or B, etc.
You also know a lot about optimal choice with a continuum of available options,
you have learned a lot about that in Economics 11 and 101. Here we put these
two together and talk about individual optimization under uncertainty if there are
many available options
to choose from.
•
EXAMPLE: Exercise 5.1
–
Step 1: Define the elementary events and their respective likelihood.
The
elementary events are (i) theft happens with probability 1/4 and (ii) theft
does not happen with probability 3/4.
–
Step 2: Understand that the person faces a risky act and think about how to
represent this act on a graph. The person has $10
,
000 with probability 3/4
and $5
,
000 with probability 1/4.
We can use the graph introduced earlier
with
z
1
being the payoff if theft does not happen,
z
2
being the payoff if theft
happens. Then we can plot the point (10000
,
5000)
–
Step 3: Understand what buying insurance means for the decision maker. It
refers to a transaction between the decision maker and the insurance company
such that (i) the decision maker pays a given fee to the company and in return
(ii) she gets a money transfer in case a theft or accident, etc. happens. Since
the fee is certain, it decreases the payoffs in each states, you may think about
it as an amount needs to be paid in advance.
–
We use the following notations:
γ
is the fee for a dollar coverage  it is 25 cents
in the example,
K
is the amount of coverage bought and
π
is the probability
of the accident  it is 1/4 in the example.
–
Step 4: Calculate the payoffs in case of full coverage.
The decision maker
buys full coverage when she insures up to the total possible loss. In this case,
her payoff will be the same in both states. The possible loss in the example is
$5
,
000 so to fully insure the decision maker had to buy
K
= 5000 coverage.
This level of insurance costs 0
.
25
×
5
,
000 = 1
,
250 therefore the payoffs will
be 10
,
000

1
,
250 = 8
,
750 in both case. This riskfree act is represented by
the point (8750
,
8750) on the graph.
1
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–
Step 5: Find the set of available acts.
The decision maker can choose any
level of coverage.
1
Any level of coverage defines a pair of payoffs, (
z
1
, z
2
). If
the decision maker buys
K
coverage, then
z
1
= 10
,
000

0
.
25
K
and
z
2
=
5
,
000 +
K

0
.
25
K
= 5
,
000 + 0
.
75
K
are the implied payoffs. A line with a
slope of

0
.
75
0
.
25
will represent these pairs. Notice that by buying insurance the
decision maker shifts money form the good state to the bad state.
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 Fall '09
 Bognar
 Microeconomics, theft, decision maker

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