Microeconomic Analysis L1D007

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Microeconomic Analysis (L1D007)
School of Economics
Dr. M. Montero

• Choice under Risk
• Expected Utility Theory

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Choice under uncertainty/risk

• So far, the consumer knows the exact consequences
  of his choices
• This is rarely the case in practice
• Examples
• Degree/module choice
• How much will I like it?
• What mark can I expect?
• Investment
• What will be the return?

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• Difference between risk and uncertainty
• In both cases, several outcomes are possible
• Risk Probabilities are determined
• Uncertainty Probabilities are undetermined
• We will only discuss risk. If the objective
  probabilities are not known, we assume
  subjective probabilities.

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Lotteries
A (simple) lottery is a collection of outcomes
with their corresponding probabilities
Outcome
Probability
Outcome
Probability
35,000
50,000
0.5
0.9
0.5
0.1
25,000
15,000
Probabilities are determined (objectively or
subjectively)
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Theories of choice under risk

• Several theories varying in sophistication
• The simplest one assumes that people look
• only at expected monetary payoff
• They choose the option that yields more
• money on average

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Evaluating the lottery Employee
xp(x)
x (outcome)
p(x) (probability)
35,000
17,500
0.5
25,000
0.5
12,500
Total
30,000
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Evaluating the lottery Self-employed
xp(x)
x (outcome)
p(x) (probability)
50,000
45,000
0.9
15,000
0.1
1,500
Total
46,500
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Everybody should be self-employed!

• Is employee really such a bad choice?
• It may give less on average
• . but there is less risk involved!
• It seems reasonable to assume that people care
• also about risk

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Example 1 a fair gamble

• Consider a choice between
• 10,000 pounds for sure
• An even chance of getting 0 or 20,000
• Most people would much prefer the safe option
• A minority will prefer the risky option
• But hardly anybody will be indifferent!
• Nevertheless, the theory predicts indifference
• because on average they both yield 10,000

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Example 2 the sadistic philanthropist(Schotter,
p. 544)
The individual needs 20,000 right now for a
life-saving operation
Outcome
Probability
Outcome
Probability
10,000
0
0.5
0.99
0.5
0.01
15,000
20,000
Expected monetary value 12,500
Expected monetary value 200
Most people would choose gamble B
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Choice under risk so far

• Each choice leads to a probability distribution
  over outcomes
• 2. Choice on basis of average monetary payoff
• 3. Problems with this approach
• Risk is not taken into account
• Utility is ignored (the sadistic philanthropist)
• New theory choice on basis of expected
• ( average) utility

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Expected utility

• 1. People look at average utility
• 2. So we cannot make any predictions without
• knowing the utility function.
• 3. Expected utility allows us to account for
• different risk preferences
• existence of insurance
• 4. It also allows us to have a theory of choice
  when the outcomes are not money

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Utility functions
A utility function assigns a number to each
possible outcome
u(x) is a measure of the satisfaction that the
individual receives from outcome x
If the outcome x is a number (typically an amount
of money) we will usually assume that the
corresponding utility can be computed
systematically by using a formula
Examples u(x) x, u(x) x2, u(x) ?x
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How to calculate expected utility
Probability p(x)
Outcome x
Utility u(x) x2
p(x)u(x)
100
1
0.01
10
1
1
0.25
0.25
0
0
0
0.74
1.25
Expected utility
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We can evaluate all kinds of choices
Outcome x
Probability p(x)
p(x)u(x)
Utility u(x)
DVD player
100
1
0.01
teddy bear
10
2.5
0.25
nothing
0
0
0.74
3.5
Expected utility
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The case of the sadistic philanthropist now
explained!
u(x)
p(x)u(x)
0
0
A
0
0
0
Expected utility
u(x)
p(x)u(x)
0
0
B
1
0.01
0.01
Expected utility
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Different risk attitudes now explained!

• Employee or self-employed?
• Different people may make different choices
• Employee 25,000 with probability 0.5 and
  35,000 with probability 0.5
• Self-employed 50,000 with probability 0.9 and
  0 with probability 0.1
• If u(x) x0.1 we choose Employee
• If u(x) x we choose Self-employed

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A few remarks

• We need more information now u(x)
• The old theory is a special case with u(x) x
• Cardinal and ordinal utility
• If expected utility is going to make sense, we
  need u(x) to be cardinal

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Ordinal and Cardinal Utility

• Preferences are represented by utility functions
• A utility function can be ordinal or cardinal,
• depending on how much information it contains
• An ordinal utility function respects the
  direction of the preferences
• i.e, most preferred options are assigned a higher
  value
• It says nothing about the intensity of preferences

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• Examples of ordinal utility functions

Suppose the consumer has the following
preferences
Utility function u has u(x)10, u(y)9, u(z)1
Utility function v has v(x)100, v(y)1, v(z)0
Both are ordinal utility functions representing
these preferences because they respect the
individuals ranking
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• Cardinal utility functions

Cardinal utility functions contain more
information

• they respect the ranking of the alternatives
• they contain information about the intensity
• of preferences

• In the previous example, if we interpret u and v
  as cardinal, they cannot refer to the same
  individual
• U implies that a and b are quite close
• V implies that a and b are far apart

Ordinal representation almost anything will
do Cardinal representation how constrained are
we?
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Analogy the measurement of temperature
Celsius
Fahrenheit
212
100
100 units
180 units
z
y
32
0
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• Scales may differ in the location of the origin
  or/and the size of the units
• These are the only arbitrary elements knowing
  them is enough to determine temperature
• z 1.8y 32
• utility functions u(x) a v(x) b with a gt
  0
• We can make meaningful statements about
• distances between points

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• Why do we need cardinal utility?

Consider three alternatives, x y and z, with
How does y compare to an equal chance of x and
z?
u(x)10, u(y)9, u(z)1
v(x)100, u(y)1, u(z)0 Both u and v are
ordinal representations According to u
0.5u(x)0.5u(z) 5.5 lt 9 u(y) According to v
0.5v(x)0.5v(z) 50 gt 1 v(y)
No coherent answer to the question!
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• Constructing our own utility function

Consider three alternatives, x y and z, with

• Suppose you can choose between
• y for sure
• x with probability p, z with probability 1-p

Which value of p makes you indifferent? Suppose
this is p 1/3 Then, if we set u(x) 1 and u(z)
0, U(y)1/3u(x)2/3u(z)1/3
See also Schotter, p.549
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• Conclusion

The theory of expected utility assumes that
people evaluate gambles according to their
average utility
If x is an amount of money, the previous theory
of expected monetary value is obtained as a
particular case with u(x) x
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Choice under uncertainty so far

• 1. Each choice leads to a probability
  distribution
• 2. Choice on basis of average monetary payoff
• 3. Problems with this approach
• Risk is not taken into account
• Utility is ignored (the sadistic philanthropist)

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Alternative expected utility
1. People look at average utility 2. Addresses
the two shortcomings 3. Intensity of preferences
becomes important 4. Next how the shape of the
utility function determines the attitude
towards risk
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Different attitudes toward risk
Suppose we have two alternative lotteries A. 75
with probability 1/2, otherwise 25 B. 50 for sure

• Note B is the average money from A for sure
• Risk-neutral indifferent between A and B
• Risk-averse strictly prefers B
• Risk-preferring strictly prefers A

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Can we infer risk attitudes from the shape of
u(x) ?

• Distinguish three cases
• Decreasing marginal utility
• Increasing marginal utility
• Constant marginal utility

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A way to view the lotteries

• We can see lottery A as a situation in which
• we start from 50 (lottery B!) and then
• With probability 1/2 we win 25
• This generates a utility gain
• With probability 1/2 we lose 25
• This generates a utility loss

If gain gt loss, then the consumer prefers lottery
A
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Decreasing marginal utility (concave u)
u(x)
u(75)
gain
u(50)
loss
u(25)
0
25
50
100
75
x
The sure outcome is better!
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Increasing marginal utility (convex u)
u(x)
u(75)
gain
u(50)
loss
u(25)
x
The sure outcome is worse!
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Constant marginal utility (linear u)
u(x)
u(75)
gain
u(50)
loss
u(25)
x
It doesnt matter!
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The previous demonstration is only valid when we
are choosing between a sure outcome and an equal
chance of two other outcomes
What if the choice was the following A. 40 with
probability 1/3, otherwise 10 B. 20 for sure
There are still utility gains and losses but we
cannot compare them directly because they occur
with different probability
We can still check that risk attitude is
determined by the shape of u
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Decreasing marginal utility (concave u)
u(x)
u(40)
u(20)
u(A)
u(10)
0
10
20
40
x
u(A) 2/3 u(10) 1/3 u(40) As expected, the
sure outcome is better!
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Increasing marginal utility (convex u)
u(x)
u(40)
u(A)
u(20)
u(10)
0
10
40
20
x
Again u(A) 2/3 u(10) 1/3 u(40) As expected,
the sure outcome is not preferred
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Relation between u and risk attitude

• Risk averse

Concave u
(u(.) lt 0)

• Risk preferring

(u(.) gt 0)
Convex u

• Risk neutral

Linear u
(u(.) 0)
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The certainty equivalent of a lottery
The certainty equivalent of lottery A is the
amount of money xe(A) such that the decision
maker is indifferent between receiving xe(A) for
sure and receiving A
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Example (and notation)

• Lottery C 25 with probability 0.4
• 0 with probability 0.6

• We can denote C as (25, 0.4 0, 0.6)

• The expected monetary value of C, denoted by
  E(C), is found as

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• In order to find the certainty equivalent we need
  the utility function
• Suppose u(x)vx
• Then the expected utility of lottery C is
• The certainty equivalent is the value x such that
  
• u(x) U(C), or vx 2. Thus xe(C) 4

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Lottery A (40,1/310,2/3)
u(x)
u(40)
u(A)
u(10)
0
10
15
20
40
x
Expected monetary value of A is 20
The certainty equivalent of A is 15
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• Risk averse consumer xe(A) lt E(A)

• Risk preferring consumer xe(A) gt E(A)

• Risk neutral consumer xe(A) E(A)