AMS 221 Statistical Decision Theory
Homework 2
May 7, 2016
Cheng-Han Yu
1. Problem 1 PRS
Proof.
(i)
(ii)
(iii)
(iv)
u(100) = (0.5)u(−25) + (0.5)u(300) ⇔ 0 = (0.5)u(−25) + 0.5 ⇔ u(−25) = −1
u(300) = (0.5)u(600) + (0.5)u(100) ⇔ 1 = (0.5)u(600) ⇔ u(600) = 2
u(100) = (0.5)u(−100) + (0.5)u(600) ⇔ 0 = (0.5)u(−100) + (0.5)2 ⇔ u(−100) = −2
u(−100) = (0.5)u(−200) + (0.5)u(300) ⇔ −2 = (0.5)u(−200) + (0.5)1 ⇔ u(−200) = −5
Figure 1: Smooth curve of through the six points x = −200, −100, −25, 100, 300 and 600.
(a) What is the CE of a lottery that gives a .5 chance at $300 and a .5 chance at $600?
(0.5)u(300) + (0.5)u(600) = 0.5 + (0.5)2 = 1.5. So we need to find z ∗ such that
u(z ∗ ) = 1.5. According to the smooth curve of the utility function, z ∗ ≈ 428. Note
that CE < EMV (Expected monetary value), i.e, z ∗ < z̄ = 450 because the ulitily
function is concave.
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(b) What is the CE of a lottery that gives a .75 chance at $400 and a .25 chance at -$200?
First, u(400) ≈ 1.4, and so (0.75)u(400)+(0.25)u(−200) = (0.75)(1.4)+(0.25)(−5) =
−0.2. The CE z ∗ such that u(z ∗ ) = −0.2 is z ∗ ≈ 70. Again, z ∗ < z̄ = 250, and
the difference between them is larger than that in (a) since the decision maker is
more risk averse in this range.
(c) You are a compound lottery with a canonical chance at the lottery l of part (b) as one
prize and a complementary chance at no net gain. What would your chance of winning
lottery l have to be before you would accept the offer?
I am confused about this problem but the following is how I interpret this question.
I assume the canonical chance is just 0.5 and winning lottery l means that I get
the lottery and my amount of money becomes 400. Suppose the chance of winning
lottery l is p. Suppose the current amount of money I have is x between −200 and
400. Otherwise, if x > 400, I will not accept the offer no matter how large p is.
Also, I will definitely accept the offer no matter how small p is when x < −200.
Assume that “no net gain” means the current amount of money does not change.
To accept the offer, the current utility without the offer should be less than the
utility before accepting the offer, i.e.,
u(x) < (0.5)[pu(400) + (1 − p)u(−200)] + (0.5)u(x)
⇒ u(x) < p(1.4) + (1 − p)(−5)
⇒ u(x) < p(6.4) − 5
u(x) + 5
. The chance of winning the lottery have to be higher than
6.4
depending how much money I have now.
Hence p >
u(x)+5
6.4
(d) What is the insurance premium of a lottery that gives a 0.5 chance at $0 and a .5 chance
at -$200?
The EMV z̄ = (0.5)($0) + (0.5)($ − 200) = −$100. The CE z ∗ is such that
u(z ∗ ) = (0.5)u(0) + (0.5)u(−200) = (0.5)(−0.7) + (0.5)(−5) = −2.85. And so
the CE z ∗ ≈ −$140. Hence the insurance premium is −(z̄ − z ∗ ) = −(−$100 −
(−$140)) = −$40.
(e) Given that the CE of a lottery is $325, and the lottery gives a π chance at $500 and a
(1 − π) chance at $300, find π.
u(CE) = u(325) = πu(500) + (1 − π)u(300). Given that u(325) = 1.11, u(500) =
1.7 and u(300) = 1, we have
1.11 = π(1.7) + (1 − π),
and hence π ≈ 0.157.
(f) What is the CE of a lottery that ofers a .375 chance at $500, a .125 chance at $600, and
a .5 chance at $0?
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u(z ∗ ) = (0.375)u(500) + (0.125)u(600) + (0.5)u(0). Given that u(500) = 1.7,
u(600) = 2, and u(0) = −0.7, we have
u(z ∗ ) = (0.375)(1.7) + (0.125)2 + (0.5)(−0.7) = 0.5375
and hence z ∗ ≈ 191.8.
(g) You are offered the lottery of part (f) $200; would you buy it? If you were an EMVer
would your choiuce change?
I would not buy it because I am just willing to pay 191.8 to avoid the uncertainty
caused by the lottery. If I were and risk-neutral person, since the EMV is
(0.375)(500) + (0.125)(600) + (0.5)(0) = 262.5,
which is higher than the price of lottery, I would buy it in this case.
(h) Consider the lottery l = {(.2 : $0), (.5 : $150), (.3 : $600)}. For how much would you just
be willing to sell this lottery if you owned it?
CE can be seen as the selling price of a lottery. That is, we would just be willing
to sell the lottery l, if we had it, for the amount CE. So u(z ∗ ) = (0.2)u(0) +
(0.5)u(150) + (0.3)u(600) = (0.2)(0.7) + (0.5)(0.333) + (0.3)(2) = 0.9065. Hence
z ∗ ≈ 280. I will be sell the lottery at price $280 if I owned it.
(i) For how much would you just be willing to buy the lottery of part(h) if you did not own
it?
Again, the amount I am just willing to buy the lottery if I did not own it is the
CE, which is $280 derived in (h). But we need to keep in mind that the amount
adecision maker would just be willing to pay a certain lottery assuming he does
not have it is in general different from the amount he would just be willing to sell
this same lottery for assuming he does have it.
2. Problem 2 PRS
Solution:
I start with u($0) = 0 and u($10000) = 1. After using the method of section 4.3.1 with total
21 points, I got my utility function shown in Figure 2. It is not being smoothed.
3. Problem 4 PRS
Solution:
u(Don’t drill) = u(0) = (0.14)u($100000) + (0.86)u(−$50000) .
u(Keep all) = (0.6)u(−50) + (0.2)u(100) + (0.1)u(200) + (0.07)u(500) + (0.03)u(1000) =
(0.21045)u($100000) + (0.78955)u(−$50000)
u(Sell 1/4) = (0.6)u(−37.5) + (0.2)u(75) + (0.1)u(150) + (0.07)u(375) + (0.03)u(750) =
(0.2088)u($100000) + (0.7912)u(−$50000)
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Figure 2: Utility function of Problem 2 PRS.
u(Drill and sell 1/2) = (0.6)u(−25)+(0.2)u(50)+(0.1)u(100)+(0.07)u(250)+(0.03)u(500) =
(0.19805)u($100000) + (0.80195)u(−$50000)
u(Sell 3/4) = (0.6)u(−12.5) + (0.2)u(25) + (0.1)u(50) + (0.07)u(125) + (0.03)u(250) =
(0.17755)u($100000) + (0.82245)u(−$50000)
Based on this result, the wildcatter should choose keep all because this option gives them
the highest expected utility.
4. Problem 6 PRS
Solution:
I am facing two options:
Option A: In addtional to your regular income you will receive a tax-free gift of z dollars
per year for the rest of my life.
Option B: A single toss of a fair coin will determine whether you get nothing or the fabulous
privilege of an unlimited ability to wrtite checks in any amount you wish for the rest of your
natural life.
Based on my current economic situation, z = $200, 000 satisfies me although I know I will
have one-half probability of getting checks in any amount I want. This kind of explain why
my utility function is bounded from above. If I could get ckecks in any amount I’d like and
my utility function is unbounded, then the expected utility of this game should go to infinity.
But here merely $10000 gives me the same satisfaction as the game.
5. Problem 4.4 PI
Solution:
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Let z be defined in R+ . Bernoulli’s utility function is
u(z) = c log(z) − log(z0 ).
Then
u0 (z) = c/z,
u00 (z) = −c/z 2 < 0 ∀z > 0.
Hence Bernoulli is risk-averse.
Then we compute the absolute risk aversion:
u00 (z)
−c/z 2
=
−
= −1/z,
u0 (z)
c/z
λ0 (z) = 1/z 2 > 0 ∀z > 0,
λ(z) = −
and so λ(z) is increasing in z and hence Bernoulli is not decreaingly risk-averse.
6. Problem 4.6 PI
Solution:
From the textbook corollary 1 of chapter 4, we know that the constantly risk averse utility
function is one of following forms
(
az + b
if λ(z) = 0
u(z) =
−λz
−ae
+ b if λ(z) = λ > 0
where a > 0 and b are constant.
Suppose u(z) = az + b. The certainty equivalent z ∗ = z̄ = $1000(1 − p), which is not equal
to 0.25, 0.6, 0.85 and 0.93 for p = 1/10, 1/3, 2/3, and 9/10. Now suppose u(z) = −ae−λz + b.
Then
∗
−aeλz + b = p(−aeλ0 + b) + (1 − p)(−aeλ1000 + b)
= −pa + b − (1 − p)(−aeλ1000 )
If p = 1/10,
−aeλ0.25 = (−1/10)a − (9/10)aeλ1000
This implies e−0.25 − (9/10)e1000 = 1/10, which cannot be true because e−0.25 − (9/10)e1000 =
0.7788. Hence the utility function cannot be the form of u(z) = −ae−λz + b. So the person’s
utility function is not consistent with constant risk aversion.
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