Homework 1: Preferences, Lotteries, Expected
Value and Expected Utility
Solution Guide
1. Study your own preferences
(a) In each case, indicate if you prefer one of the goods over the other or
if you are indifferent:
i.
ii.
iii.
iv.
v.
vi.
vii.
viii.
ix.
cheese and eggs
bread and butter
milk and sugar
cheese and potatoes
ham and cheese
ham and eggs
potatoes and eggs
bread and milk
sugar and cheese
(b) Check if your preferences are transitive.
[SOLUTION]
(a) The answer to this question depends on each one’s preferences. Typically, different people will produce different answers.
(b) Based on your answers on (a) you must check that your preferences
are transitive. That is, you must check that whenever the alternative
A is preferred to the alternative B, and alternative B is preferred to
alternative C, then A should also be preferred to C (for whatever
alternatives A, B, and C in cheese, eggs, bread, butter, milk, etc.)
For instance, if you find that cheese is preferred to eggs, eggs preferred
to ham, and ham preferred to cheese, then your preferences are NOT
transitive. Preferences are transitive whenever such “cycles” do not
occur.
2. Represent the following situation. I am considering going for a vacation
during the Easter break, but I am not quite sure which is the best choice.
Usually it rains (with probability 0.7), in which case going to the mountains might be a good choice (it gives me utility of 100), but if the weather
is sunny I will not enjoy the mountains so much (my utility will be 30). If
contrary to the tradition the weather is sunny and hot (with probability
0.3), the best choice will be the beach (utility of 120), but spending my
holiday on the beach when it is raining is horrible (utility -20).
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[SOLUTION]
My actions (the alternatives I have to choose from) are : Mountain and
Beach. That is, I must decide whether go to the mountains or to the
beach. These determine the “action branches” at the beginning of the
three. Then, after I have made my choice (Mountain or Beach), the
weather may be Rainy (70% chances) or Sunny (30% chances). These
define the “chance branches” after each one of the “choice branches”
The “sequence of the tree” is important. The tree must represent the real
problem: I have to decide were to go before knowing the weather.
The problem is represented in the following tree:
Rainy
(0.7)
100
Mountain
Sunny
(0.3)
30
Rainy
(0.7)
-20
Sunny
(0.3)
120
Beach
3. There are 3 cards on the table, you are unable to see the colors, but you
know that there is one blue, one yellow and one red. If you take the
blue one you have a probability of 70% of winning a prize of $100, if you
take the yellow one the probability is 50%, and the red one gives you a
probability of 30%.
(a) Represent the lottery you are playing. (Note: The probability to
withdraw any of the cards is 1/3)
(b) What is the set of final outcomes?
(c) Represent the reduced form of this lottery.
(d) Are you willing to play such gamble?
2
[SOLUTION]
(a) In this case there are no “choice branches”, you have nothing to choose
(well, at the end you must decide if you want to play the gamble or
not, but this is not part of the tree). The gamble proposed is just a
lottery where everything is random. The sequence of the gamble is
that first you choose a card (without knowing its color) and then you
play a lottery whose probabilities of winning depend on the color of
the card you chose in the first place
The problem is represented in the following tree
(b) The set of final outcomes in this case is 0 and 100. These are the
only possible results. Thus, what matters at the end is what are the
probabilities of obtaining each of these two outcomes. We find them
in the “reduced lottery” in the next item
(c) We compute now the probabilities of earning 100 and of earning 0
To earn 100, the possible events are: (blue card and win) or (yellow
card and win) or (red card and win)
In terms of probabilities:
3
p(100) =
=
p(blue) · p(win) + p(yellow) · p(win) + p(red) · p(win) =
1
1
1
(0.7) + (0.5) + (0.3) = 0.5
3
3
3
Thus, we have that p(100) = 0.5 and therefore, p(0) = 0.5 as in the
following tree
(d) I would play the gamble since I have nothing to lose ! (But different
people might have a different view) A different question would be:
How much are you willing to pay to play the gamble ? The answer
to that question would depend on each one’s risk attitude
4. John Smith has in mind 3 investment opportunities: to invest in the wood
industry, water energy, and solar energy. The outcome of each of this
options depends on the energetic policy of his country. If the government adopts a sustainable growth economic policy (this can happened
with probability of 0.5) the payoffs will be different compared to the case
of an open market policy, as in the table below:
Wood
Water
Solar
Ecologist
25
49
64
Open Market
59
36
36
(a) Represent his decision problem.
(b) Assume that he is risk neutral (that is, he values lotteries according
to their expected value). What is his choice?
(c) Assume
now that he is risk averse and his utility function is u(x) =
√
x
10 . What is his choice now?
(d) Assume now that he is risk lover and his utility function is u(x) =
10x2 . What is his choice?
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[SOLUTION]
(a) My actions in this case (the alternatives I have to choose from) are
: Wood, Water and Solar. These define the “action branches”
at the beginning of the three. Then, after I have made my choice,
the energetic policy may be Ecological (50% chances) or Open
Market (50% chances). These define the “chance branches” after
each one of the “choice branches”
Again, the “sequence of the tree” is important. The tree must represent the
real problem: I have to decide my investment before knowing the energetic
policy.
The problem is represented in the following tree
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(b) If John Smith is risk neutral we have that u(x) = x. Hence,
E(W ood)
E(W ater)
E(Solar)
= (0.5) · 25 + (0.5) · 59 = 42
= (0.5) · 49 + (0.5) · 36 = 42.5
= (0.5) · 64 + (0.5) · 36 = 50
Hence, a risk neutral John Smith would choose Solar as it provides
the highest Expected Utility
(c) If John Smith is risk averse we have that u(x) =
E(W ood)
E(W ater)
E(Solar)
√
x
10 .
Hence,
√
√
25
59
= (0.5) ·
+ (0.5) ·
= 0.634
10
10
√
√
49
36
= (0.5) ·
+ (0.5) ·
= 0.65
√10
√10
64
36
= (0.5) ·
+ (0.5) ·
= 0.7
10
10
Hence, a risk averse John Smith would also choose Solar as it provides
the highest Expected Utility
(d) If John Smith is risk lover we have that u(x) = 10x2 . Hence,
E(W ood)
E(W ater)
E(Solar)
= (0.5) · 10(25)2 + (0.5) · 10(59)2 = 20530
= (0.5) · 10(49)2 + (0.5) · 10(36)2 = 18485
= (0.5) · 10(64)2 + (0.5) · 10(36)2 = 26960
Hence, a risk lover John Smith would also choose Solar as it provides
the highest Expected Utility
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