Class 21 (answers) Decision under uncertainty

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Microeconomics II 
Undergraduate degree in Economics 
 
Class nr. 21 
 
Subject: 3. Decision under uncertainty (practice class) 
 
 
Exercise 12.2. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, p. 159-160 
Willy owns a small chocolate factory, located close to a river that occasionally floods in 
the spring, with disastrous consequences. Next summer, Willy plans to sell the factory 
and retire. The only income he will have is the proceeds of the sale of his factory. If 
there is no flood, the factory will be worth $500,000. If there is a flood, then what is left 
of the factory will be worth only $50,000. Willy can buy flood insurance at a cost of 
$0.10 for each $1 worth of coverage. Willy thinks that the probability that there will be 
a flood this spring is 1/10. Let cF denote the contingent commodity dollars if there is a 
flood and cNF denote dollars if there is no flood. Willy’s von Neumann-Morgenstern 
(expected) utility function is ( ), 0.1 0.9F NF F NFU c c c c= + 
 
a) If he buys no insurance, what is Willy’s contingent commodity bundle (his 
consumption in each contingence or state of nature)? 
b) What is Willy’s contingent commodity bundle if he buys an insurance that pays 
him $x in the event of a flood? 
c) Write Willy’s budget constraint equation. 
d) Determine Willy’s marginal rate of substitution between the two contingent 
commodities, that is, in both states of nature. 
e) Determine Willy’s optimal contingent commodity bundle. How much will he get 
of compensation in case of flood? How much is the amount of insurance 
premium he will have to pay? 
 
Answers: 
a) ( ) ( ), 50,000; 500,000F NFc c = 
b) ( ) ( ), 50,000 0.9 ;500,000 0.1F NFc c x x= + − 
 
 
 
 
 
 
 
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Undergraduate degree in Economics 
 
c) 
 
 
 
 
 
 
 
 
 
g ≡ ratio between the insurance premium and the amount of coverage 
0.1 0.1
1
γ = = 
Slope of the budget constraint: 
γ
γ
−− 1 
Value of the budget constraint on the axis: cNF: 
( )0.1500,000 50,000 500,000 50,000 505,555. 5
1 1 0.1
γ
γ
⎛ ⎞− − × = + × =⎜ ⎟− −⎝ ⎠ 
Value of the budget constraint on the axis cF: 
1 1 0.150,000 500,000 50,000 500,000 4,550,000
0.1
γ
γ
⎛ ⎞− −− − × = + × =⎜ ⎟⎝ ⎠ 
 
 
 
 
 
 
 
 
 
 
Budget constraint equation: 
( )505,555. 5
1NF F
c cγ γ= − − ; ( ) ( )
0,1505.555, 5 505,555. 5
0,9 9
F
NF F NF
cc c c= − ⇔ = − 
cF 
cNF 
Choice 
Endowment €500,000 
€500,000 - gx 
€50,000 €50,000 + x - gx 
cF 
cNF 
Choice 
Endowment €500,000 
€500,000 - gx 
€50,000 €50,000 + x - gx 
€505,555.(5) 
€4,500,000 
Microeconomics II 
Undergraduate degree in Economics 
 
d) 
( ) ( ) ( )1 12 2, 0.1 0.9 0.1 0.9F NF F NF F NFU c c c c c c= + = + 
( )
( )
( )
( )
1 1
2 2
, 1 1
2 2
0.1 0.5 0.1 1
90.9 0.5 0.9
F
F NF
NF
c F NFNF NFF
c c
F c F
NF F
NF
U
MU c cc ccMRS Uc MU cc c
c
−
−
∂
× × ×∂ ∂≡ = − = − = − = − = −∂∂ × × ×∂
 
f) Optimum condition: Equality between the marginal rate of substitution (slope of 
the indifference curve) and the slope of the budget constraint. 
FNF
F
NF
F
NF
F
NF
F
NF
cc
cc
c
c
c
c
c
c
c
c
MRS
NFF
=⇔
⇔=⇔=⇔−=−⇔−=−⇔−−= 119
1
9
1
9,0
1,0
9
1
1, γ
γ
 
By using the optimum condition and the budget constraint: 
0.1 4,500,000 1500,000 50,000 50,000
1 0.1 9 9 9 9
455,000
455,0009 4,500,000 50,000 10 4,550,000
NF F NF F
F NF
NF NF
NF F NF F F
NFNF NF NF
c c c c
c cc c
c c c c c
cc c c
=⎧ =⎧⎪ ⎪⇔ ⇔⎨ ⎨⎛ ⎞= + × − = + × −⎜ ⎟⎪ ⎪− ⎩⎝ ⎠⎩
= = =⎧ ⎧ ⎧⇔ ⇔ ⇔⎨ ⎨ ⎨ =× = + − × = ⎩⎩ ⎩
 
 
 
 
 
 
 
 
 
 
 
 
Amount of insurance coverage underwritten by Willy: 
45,000500,000 455,000 500,000 0.1x 455,000 45,000 0.1x
0.1
450,000
x x
x
γ− = ⇔ − = ⇔ = ⇔ = ⇔
⇔ =
 
cF 
cNF 
Choice 
Endowment €500,000 
€455,000 
€50,000 
€505,555.(5) 
€4,500,000 €455,000 
Microeconomics II 
Undergraduate degree in Economics 
 
 
Amount of insurance premium paid by Willy: 
0.1 450,000 45,000xγ = × = 
 
Exercise 12.12. from Bergstrom and Varian’s book (2006) “Workouts in Intermediate 
Microeconomics”, pp. 168-169 
Dan Partridge is a risk averter who tries to maximize the expected value of c , where c 
is his wealth. Dan has $50,000 in safe assets and he also owns a house that is located in 
an area where there are many forest fires. If his house burns down, the remains of his 
house and the lot it is built on would be worth only $40,000, giving him a total wealth 
of $90,000. If his house doesn’t burn, it will be worth $200,000 and his total wealth will 
be $250,000. The probability that his home will burn down is 0.01. 
 
a) Calculate his expected utility if he doesn’t buy fire insurance 
b) Calculate the certainty equivalent of the lottery he faces if he doesn’t buy fire 
insurance. (Technical note: The certainty equivalent of a lottery is the amount of 
money you would have to be given with certainty to be just as well-off with that 
lottery). 
c) Suppose that he can buy insurance at a price of $1 per $100 of insurance. For 
example if he buys $100,000 worth of insurance, he will pay $1,000 to the 
company no matter what happens, but if his house burns, he will also receive 
$100,000 from the company. What is the value of Dan's wealth if he decides to 
do an insurance coverage of € 160,000? Is this value sufficient for him to have 
his wealth fully insured? 
d) Calculate the certainty equivalent of his wealth for the amount of insurance 
coverage contracted, and his expected utility. 
 
Answers: 
a) 
Let us call c1 to Dan Partridge’s wealth if his house burns down, and c2 if it doesn’t. 
( ) ( ) ( )1 1 2 2 0.01 50,000 40,000 0.99 50,000 200,000E U c U c U cπ π⎡ ⎤ = + = × + + × + =⎣ ⎦ 
4984953 =+= 
 
 
 
 
 
 
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Undergraduate degree in Economics 
 
b) 
 
 
 
 
 
 
 
 
 
 
The certainty equivalent (c *) of a risky situation is the amount of wealth that gives Dan 
the same utility as the expected utility of an uncertain situation. 
( ) ( ) 2* * 498 * 498 * 248,004U c E U c c c c⎡ ⎤= ⇔ = ⇔ = ⇔ =⎣ ⎦ 
 
 
 
 
 
 
 
 
 
 
 
Point L represents the situation in which the home burns down and it wasn't insured. 
Point H represents the situation in which the house doesn't burn down, and it wasn't 
insured. 
Point U represents the risk before we know if the house will burn or not and it wasn't 
insured. 
Point C represents the certainty situation equivalent to the risk before we know if the 
house will burn or not. 
 
 
 
c 
U(c) 
c2 c1 E[c] 
U(c*) = E[U(c)] 
Utility function of wealth: ( ) ccU = 
L 
H 
U 
c* 
C 
c 
U(c) 
90,000 
E[c] = 
= 0.01*90,000+0.99*250,000 = 
=248,400 
498 
Utility function of wealth: ( ) ccU = 
L 
H 
U 
248,004 
C 
250,000 
Microeconomics II 
Undergraduate degree in Economics 
 
c) 
x ≡ amount of the coverage underwritten. 
gx ≡ value of the premium paid 
g ≡ ratio between the premium paid and insurance coverage underwritten (= 0,01 on this 
case) 
[ ] ( ) ( )1 1 2 2
160,000 160,0000.01 90,000 160,000 0.99 250,000
100 100
E c c x x c xπ γ π γ= + − + − =
⎛ ⎞ ⎛ ⎞= × + − + × − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
 
2, 484 245,916 248,400= + = 
Yes, this amount assures him total coverage of his wealth ($250.000), but he has to pay 
the insurance premium ($1.600), in sum he gets a net wealth of $248.400 in both states 
of nature (either if the house burns down or if it doesn’t). 
Amount of Dan’s wealthif the house burns down: 
1
160,00090,000 160,000 248,400
100
c x xγ+ − = + − = 
Amount of Dan’s wealth if the house doesn’t burn down: 
2
160,000250,000 248,400
100
c xγ− = − = 
The insured amount corresponds to the loss suffered by Dan if the house burns down (€ 
200,000 - € 40,000). The loss is totally covered. 
d) 
Expected utility: 
( ) ( ) ( )1 1 2 2 0.01 248,400 0.99 248,400 248,400 498.4E U c U c U cπ π⎡ ⎤ = + = × + × = ≈⎣ ⎦
 
Certainty equivalent: 
( ) ( ) ( )2* * 248,400 * 248,400 * 248,400U c E U c c c c⎡ ⎤= ⇔ = ⇔ = ⇔ =⎣ ⎦ 
 
Note, on the following chart that the uncertainty is eliminated by taking out the 
insurance; therefore the certainty equivalent is equal to the situation where, although 
there is a risk of the house burns down, the total risk was insured. 
 
 
 
 
 
 
Microeconomics II 
Undergraduate degree in Economics 
 
 
 
 
 
 
 
 
 
 
 
 
Now, the wealth (ci’) is always 248,400 in any state of nature. 
 
c 
U(c) 
E[c] = 
= 0.01*(90,000+160,000-1,600)+0.99*(250,000-1,600) = 
=248,400 = c1’ = c2’ = c* 
498.4 
Wealth utility function: ( ) ccU = C’ ≡ U’ ≡ L ≡ H