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### Microeconomic Analysis (L1D007)

School of Economics

Dr. M. Montero

Choice under Risk:

Expected Utility Theory

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?

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.

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)

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

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

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

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

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

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

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

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

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

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

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

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

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

A few remarks

- We need more information now: u(x)
- The old theory is a special case withu(x) = x
- Cardinal and ordinal utility

If expected utility is going to make sense, we need u(x) to becardinal

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

Suppose the consumer has the following preferences:

Examples of ordinal utility functions

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 functionsrepresenting these

preferences because they respect the individual’s

ranking

Cardinal utility functions contain more information

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

Cardinal utility functions

- 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?

F

212

100

y

z

32

0

- 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 > 0
- We can make meaningful statements about
- distances between points

Consider three alternatives, x y and z, with

Why do we need cardinal utility?

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 < 9 = u(y)

According to v: 0.5v(x)+0.5v(z)= 50 > 1 = v(y)

No coherent answer to the question!

U(y)

U(z)

Consider three alternatives, x y and z, with

Constructing our own utility function

- 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

The theory of expected utility assumes that people

evaluate gambles according to their average utility

Conclusion

If x is an amount of money, the previous theory of expected monetary value is obtained as a particular case with u(x) = x

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)

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

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

x

u(x)

x

u(x)

x

Can we infer risk attitudes from the shape

of u(x) ?

- Distinguish three cases:
- Decreasingmarginal utility
- Increasingmarginal utility
- Constantmarginal utility

- 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 > loss, then the consumer prefers lottery A

(concave u)

u(x)

u(75)

gain

u(50)

loss

u(25)

0

25

50

75

100

x

The sure outcome is better!

25

50

75

100

Increasing marginal utility

(convex u)

u(x)

u(75)

gain

u(50)

loss

u(25)

x

The sure outcome is worse!

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

(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!

(convex u)

u(x)

u(40)

u(A)

u(20)

u(10)

0

10

20

40

x

Again u(A) = 2/3 u(10) + 1/3 u(40)

As expected, the sure outcome is not preferred

Relation between u and risk attitude

- Risk averse

Concave u

(u’’(.) < 0)

- Risk preferring

(u’’(.) > 0)

Convex u

- Risk neutral

Linear u

(u’’(.) = 0)

The certainty equivalent of a lottery

The certainty equivalent of lotteryA is the amount of moneyxe(A) such that the decision maker is indifferent between receiving xe(A) for sure and receiving A

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

In order to find the certainty equivalent we need the utility function

- Suppose u(x)=√x
- Then the expected utility of lottery C is
- The certainty equivalent is the value x such that

u(x) = U(C), or √x = 2. Thus xe(C) = 4

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

Risk averse consumer: xe(A) < E(A)

- Risk preferring consumer: xe(A) > E(A)

- Risk neutralconsumer:xe(A) = E(A)

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