Download MPP 801 Tutorial Expected Utility

MPP 801 Tutorial
Expected Utility
Kevin Wainwright
October 16, 2006
Uncertainty and expected value
Suppose there are two states of nature (good day, bad day) and that a person’s wealth W , depends on which
state. The probability of each state is given by pi (i = 1; 2) and the wealth in each state is Wi (i = 1; 2).
Together, the two probabilities and the two values of wealth are referred to as a Risky Prospect.
The expected monetary value of a risky prospect (EV ) is given by
EV = p1 W1 + p2 W2
Since p1 + p2 = 1 then p1 = (1
p2 ): We can substitute to eliminate p1
EV
=
=
(1
(1
p2 )W1 + p2 W2
p)W1 + pW2
With only one probability left, we can suppress the subscript (p2 = p)
Expected Utility
Now suppose that a level ofpwealth gives a person a certain level of utility such that U = U (W ): Examples
of utility functions are U = W (risk averse), or U = W 2 (risk lover), or U = 2W (risk neutral). Each type
are illustrated in …gure 1
Expected utility (EU ) is found the same way we found expected value
EU = (1
p)U (W1 ) + pU (W2 )
For expected utility, you …rst …nd the utility in each state and then calculate the expected value of the
Utilities!
U
RISK
AVERSE
U
RISK
LOVER
W
U
W
Figure 1: Types Risk Attitudes
1
RISK
NEUTRAL
W
U
G
10
U(87.2) = 9.338
EU= 9.2
E
B
6
The BG line is
the “expected
utility”function
36
87.2
100
W
Figure 2: The numerical example
Example of EV and EU
p
Suppose a person has the utility function U = W .(or = W 1=2 ) On good days they earn $100 but on bad
days they only earn $36. Bad days occur 20% of the time. Therefore, their expected wealth (or income) is
good
:
Their expected utility is
Bad
z }| { z}|{
EV = (1 p)W1 + pW2 = 0:8($100) + 0:2($36) = $87:2
EU
=
=
(1 p)U (W1 ) + pU (W2 )
0:8(100)1=2 + 0:2(36)1=2
0:8(10) + 0:8(6)
= 9:2
Now, suppose you were to o¤er the person a "Safe" opportunity where earnings NEVER ‡uctuated. Further,
the pay would be $87.2 every day (guaranteed) which is equal to the expected value of the "Risky" job. Their
utility would be
U = (87:2)1=2 = 9:338
Since the two jobs have the same expected value but the less risky job gives a higher utility, this person is
risk averse (9:338 > 9:2). This example is illustrated in …gure 2.
Suppose we asked the question: what wealth, with no risk, would be as good as the risky prospect? Since
the risky prospected gives an (expected) utility of 9.2, we can work back from the utility function to answer
this question. Find WC such that
U (WC ) = 9:2
p
WC = 9:2
WC = (9:2)2 = 84:64
therefore $84. 64 with NO risk gives the same utility number as the risky prospect (Think about what this
means in context of indi¤ erence curves) Where would you put Wc in Figure 2???
Insurance Market
The typical set-up for insurance markets is as follows:
2
Buyers of insurance are individuals with a wealth of W who may experience a loss of L with a probability
of p. Using our "states of nature" approach from above, we have their uninsured situation as
Good
Bad
W1 = W
W2 = W
L
The expected value of being uninsured is
EV
= (1
= W
p)W1 + pW2 = (1
pL
p)W + p(W
L)
The expected utility of being uninsured is
EU = (1
p)U (W ) + pU (W
L)
When comparing people of di¤erent risk classes, they can di¤er either by probability, by loss, or by both.
The most common is to assume all types of people will have the same loss but will di¤er by the probability
of incurring a loss.
Sellers of insurance o¤er a premium, or price, (a1 ) in return for a payout (a2 ) in the event of a loss (bad
state).If insurance markets are competitive, then they will o¤er insurance at fair odds. "Fair odds" insurance
means that the premium is equal to the expected value of the loss (L). i.e.
a1 = pL
If the person can buy partial insurance (units) such that they decide on the amount of a1 they are willing
to pay, then the amount they would receive in the bad state is a2 such that
a1 = pa2
if fully insuring, a2 = L: Note that a1 is paid in both states. i.e.
W1
W2
= W a1
= (W a1 )
L + a2
Therefore, fully insured people (a2 = L) will have a wealth of W
of insurance is
EU
= (1
= (1
p)U (W1 ) + pU (W2 )
p)U (W a1 ) + pU (W
a1 in both states. The expected utility
a1
L + a2 )
It is assumed that people maximize utility. Therefore, they will only insure if EU (insured) EU (uninsured).
This, of course, is ignoring the case of compulsary insurance imposed by government regulation (That is the
natural extension of this story...)
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Tutorial Questions
1. The e¤ort of carrying my umbrella reduces my utility by 1/2 a unit. If it rains and I have no umbrella,
my utility falls by 3 units, while it only falls by 1 unit if I do have an umbrella. I consider that the
probability it will rain is 1/2. Therefore I carry an umbrella.(is this True or False?)
p
2. Skippy has the following utility function U = W , where W is her wealth. Her initial wealth is $2500.
She is going on a trip where she has a 50% chance of losing $1600.
(a) What is Skippy’s
1. expected value of the trip?
2. expected utility of the trip?
3. What is the certainty equivalent wealth (look this up in your text)?
(b) If insurance is o¤ered at fair odds show that Skippy will fully insure. (If you feel clever, you can
use calculus to do this by substituting all the given information into the expected utility function,
…nd dEU=da1 , set equal to zero and solve for a1 )
(c) Carefully graph your results from (a) and (b)
3. Myrtle is the same as Skippy in every way except she has a 30% chance of losing $1600. However the
insurance company cannot tell the two apart, so they o¤er fair insurance based on either
(i) the average probability (0.4), or
(ii) only Skippy’s probability (the highest risk person)
(a) Find Myrtle’s expected utility of the trip with no insurance.
(b) Find the expected utility for both Skippy and Myrtle when the insurance company uses the
average probability
(c) Find Myrtle’s expected utility when her premium is based on Skippy’s probability of loss? Will
she still buy insurance?
(d) Graph your results
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