topic 2 decision theory 2015 (1)

8/9/2019 Topic 2 Decision Theory 2015 (1)

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Decision Theory under

Uncertainty (Parts I & II)

Economics of Information

(ECON3016)


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Objectives of this Part

Review how economists model

individuals or firms choices under

uncertainty Review the standard assumptions

about attitudes towards risk

Understand concepts of Paretoefficiency in situations with uncertainty

Application: Optimal Risk Sharing


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Uncertainty

Consumers and firms are frequentlyuncertain about the payoffs from their

choices. Example 1: A farmer chooses to

cultivate either apples or cherries

Uncertainty about profits

Not clear what will (turn out to be) thebetter choice

Profits depend on rain conditions,

plagues, market prices,


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Uncertainty

Example 2: Rolling a fair die

Win 2 if 1, 2, or 3,

Neither win nor lose if 4, or 5 Lose 6 if 6

Example 3: Johns monthlyconsumption:

3600 if he does not get ill

400 if he gets ill (so he cannot work)


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Modelling Uncertainty

Element 1: States of the world 1,2, , ill/healthy, number on the die, weather,

Element 2: Prizes in each state of the world Consumption level, monetary payoff, profit,

Element 3: Probabilities of the states of

the world

Assumption: Probabilities are known!

Assumption:Actors only care about prizes

and their likelihoods, not about states!


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Lotteries

A lottery consists of

outcomes or prizes

, ,

Probability of outcomes : 0 One outcome will be realized:

probabilities add up to one

= ++ 1

Note that we are hiding the states of the

world in this description of uncertainty


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Probability

The probability of a repetitive event is the

relative frequency with which it occurs

Probability of heads when flipping a coin is 0.5(If the coin is fair) More generally, probabilities can be

estimated from data and experience

If available!

Warning: Our conclusions may not carry

over to situations where there is no or

limited information about probabilities.


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Expected Value

Expected value of a lottery with monetary outcomes 1, , and probabilities 1, 2,

++

=

Note: We will only consider lotteries with

monetary prizes

(Or other one-dimensional variables)


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Expected Value of Monthly

Consumption (Example 3)

Example 3: Johns monthly consumption:

1 3600 if he does not get ill 2 400 if he gets ill (so he cannot work) Probability of illness 0.25 Probability of no illness 1 0.75 The expected value is:

+ 0.75 3600 + 0.25 400 2800


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Compare Lotteries with the

same Expected Value

Lottery A: (lottery without risk)

get 2800 for sure (ill or healthy)

() 2800 Lottery B:

get 3600 if healthy ( 0.75) get 400 if ill ( 0.25) expected value also 2800

Which Lottery is better?


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Risk Neutrality

A person is called risk-neutral if she alwayschooses the lottery with the highestexpected value

For a risk-neutral person we have in Ex. 3:

~ The person is indifferent between Lotteriesand because E ()


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Indifference Curves

Lets draw indifference curves for a

risk-neutral person

2 monetary prizes , Fixed probabilities , Indifference Curve

Combinations of , that have thesame expected value


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Risk-Neutral Indifference Curves: combinations

(1, 2)with constant expected value

1 1 2 2 1 1 1 2

1

2 11 1

1 2 2 1

1 1

2 1

1 1

(1 )

1 1

0, ; 0,1

(slope)1

E p x p x E p x p x

pE

x xp p

E Eif x then x if x then x

p p

dx p

dx p


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1

2

/(11)

/1

Decreasing line with slope:

21

11 1


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(Ex. 3)

2800/0.75

45-degree line:lotteries without risk

2800

2800

3600

400

Lottery A

Lottery B

Decreasing line with slope:

21

11 1

0.750.25 3

2

1


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Risk Aversion

Does the expected value criterion (risk-neutrality) provide reasonable predictions?

Would you prefer Lottery or Lottery ? Most individuals would avoid the risk in and choose the safe option

The expected value does not capture the

uncertainty involved in a lottery! BUT: Will see that in some cases, risk-

neutrality is a reasonable assumption.


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Expected Utility

Standard model for prefs. over lotteries

Individuals evaluate outcomes/prizes using

a strictly increasing utility function () Lottery is preferred over lottery if theexpected utility for is higher than for :

=

> =

is called the Bernoulli utility function

(or von Neumann-Morgenstern utility function)


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Expected Utility

Note that Expected Utility is computed

similar to Expected Value:

The mathematical expectation is taken overutility values rather than monetary prizes.

=

++

Remark: EU criterion can also be applied for

non-monetary lotteries.


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Expected Utility Example 3

Suppose John has Bernoulli utility function Lottery: 1 (2800) 1 2800 53 Lottery

:

0.75 3600 + 0.25 (400) 0.75 3600 + 0.25 400 50 John strictly prefers

over


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Indifference Curves

2 monetary prizes , Fixed probabilities

, Indifference Curve Combinations of , that yield the

same expected utility

for given Bernoulli utility function (.)


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Slope of Indifference Curves

+

+ (1 ) Total differentiation:+ 1 0Rearrange:

1

1 1

< 0


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Indifference Curves

Features of indifference curves:

Decreasing

Slope on 45-degree line is= 11 1

11 1

Same slope as the constant expected value curve!

Individuals are approx. risk-neutral on the 45-degree line.

If is concave, then indifference curves areconvex (see first problem set).

Will see that concavity of the Bernoulli utility function

means that a person is risk-averse.


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Concavity and Convexity

A (twice differentiable) function () is: concave if 0, convex if 0, strictly concave if < 0, strictly convex if > 0.


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Indifference Curves (Ex. 3)

Lotteries without Risk:45-degree line Constant Expected Value:

21 11 1 0.750.25 3

2

1

Indifference Curves:

11 1


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Indifference Curves and Risk Aversion

Convex Indifference Curve:

Among the lotteries with

constant expected value

(red line) the lottery on the45-degree line is most

preferred.

Person chooses certain

outcome ( ) if shedoes not have to reduceexpected value to do this.

2

1


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2800/0.752800

2800

3600

400

Lottery A

Lottery B

2

1

Convex Indifference Curve:

Lottery is preferred toLottery . (no uncertainty) (uncertainty)

We have Indifference

between B and C even

though 2800Lottery C


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Risk Aversion: Formal Definition

Consider decision maker (DM) with Bernoulli utility

function (). Definition: The Certainty Equivalent

CE(X)of a

lottery is the amount of money for which the DMis indifferent between the lottery and gettingCE(X) for sure:

(CE(X))EU(X) Definition: DM is risk averse (risk loving) if forevery lottery that involves risk, the certaintyequivalent is lower (higher) than exp. value of

CE(X))


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Risk Aversion: Formal Definition

Related to the certainty equivalent:

Definition: The risk-premiumR for a lotteryis the difference between the expected value of

and its certainty equivalent:R E(X)CE(X) Easy to see from definitions:

DM is risk averse/risk loving if her risk-premium ispositive/negative for all lotteries that involve some

risk.

One can show:

Risk aversion is equivalent to concavity of ().


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2800/0.752800

2800

3600

400 Lottery B

2

1

Suppose 2.800

EU 50 (see last week) CE EU CE 50 2500 ( )R CE 300

Lottery C


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Examples of commonly used Utility

functions for risk averse individuals

( ) ln( )

( )

( ) 0 1

( ) exp( * ) 0

a

U x x

U x x

U x x where a

U x a x where a


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Risk Neutrality

DM is risk-neutral if Equivalently: For all lotteries:

and 0 When does risk-neutrality arise? Average of many independent draws of the

same lottery equals expected value.

This is similar to diversification. Large companies (e.g. insurance companies)

can diversify risks and can therefore behave

risk-neutral.


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Classification


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Measuring Risk Aversion

Intuitively, risk-aversion becomes stronger

(CE smaller) if is more concave orindifference curves are more convex

Definition:Arrow-Pratt measure of absolute

risk aversion:

For risk averse individuals, () < 0 () is positive for risk averse individuals

"( )

( ) '( )

U Xr X

U X


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Comparing Risk Aversion of different

Decision Makers/Utility Functions

Consider and such that for alllevels of wealth:

> is more risk-averse than :

For every lottery that involves risk: CE < CE and R > R

More risk-averse DM is willing to pay a

higher premium to avoid risk.


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Risk Aversion at different wealth

levels for the same utility function

Consider a lottery with prizes , , . How do R(X) and CE() change if DM gets

wealthier?

Endow DM with additional certain wealth . New lottery + with outcomes+ , , +

Result: If is is decreasing(increasing/const.) in , then CE + is increasing (decr./const.) in

R

+ is decreasing (incr./const.) in


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Risk Aversion and Wealth

Example 1: () ln()where > 0 Arrow-Pratt measure:

1

Risk aversion decreases as wealth

increases


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Risk Aversion and Wealth

Example 2:

exp()where > 0. Arrow-Pratt measure:

Risk aversion is constant as wealth

increases


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Willingness to Pay for Insurance

Consider a person with a current wealth of

100,000 who faces a 25% chance of losing

his automobile worth 20,000 Lottery Suppose the Bernoulli utility function is

() ln ()


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Willingness to Pay for

Insurance

The expected utility isEU 0.75 100 + 0.25 80

EU 0.75 ln 100 + 0.25 ln (80) 11.46 Certainty equivalent:

ln CE EU CE . 94,574 Risk-Premium:R CE

95 000 94 574 426


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Willingness to Pay for Full

Insurance

Pay premium for full insurance: Certain final wealth of 100,000 (Warning:

is not the risk premium)

DM is willing to for full insurance if: < 100,000 < 100,000

< 100,000 < 100,000 + () 5,426 Result: Willingness to pay is equal to

Expected Loss plus Risk Premium.


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Actuarially Fair Premium

Definition: Insurance policy with

actuarially fair premium yields zero

expected profits for insurancecompany

(Note: marketing and administration

expenses are not included in thecomputation of the actuarially fair

premium)


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Actuarially Fair Premium

Previous example: Expected profit0.75 + 0.25 20,000 0 Result:Actuarially fair premium is:

5,000 Notice: the actuarially fair premium is smaller thanthe maximum premium that the individual is willingto pay (5426). There is room for the insurance company and the individual to

trade and improve their profits/welfare. (Even with administrativecosts.)

This happens if (and only if) an individual is risk averse.


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Summary (so far)

The expected value is an adequate criterion to choose among

lotteries if DM is risk neutral, but not if DM dislikes risk.

If DM prefers to receive B rather than a lottery with expected

value greater than B, then we say the DM is risk averse. If U(x) is the utility function, we always assume U(x)>0

If an individual is risk averse, then U(x)


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Efficiency and Risk Sharing

Will analyse situations in which two

parties can allocate risk

Allocation: Specifies outcome for eachperson in each state of the world.

Definition: Ex-ante (Pareto) efficiency

An allocation is efficient if it is notpossible to increase the expected

utility of one person without decreasing

EU of someone else.


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Efficiency and Risk Sharing: Example

Two states of the world 1,2 Two people:

, Allocation: (, , , ) Feasibility constraint:+ < and + <

where is the total amount of money availablein state . The allocation defines lotteries and.

Bernoulli Utilities:

and

.


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Example: Finding PE allocations

Fix EU of one individual: EU EU Maximise EU of the other one:

max EU s.t. EU EUand feasibility Insert everything:

max + (1 ) s.t. + (1 ) EU


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-- Finding PE allocations

Set up Lagrangean:

Frist-order Conditions yield:

Make sure you can compute this condition.

PE allocations: Indifference curves touch!

Careful: Sometimes the optimal solution is at the

boundary! Then the we have inequalities from Kuhn-

Tucker conditions and indifference curves may intersect.


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Suppose (no aggregate uncertainty)and individuals are risk-averse.

> implies < Risk-aversion (concavity of ) implies:

< 1


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Suppose > (aggregate uncertainty) A is risk-averse and B is risk-neutral.

B risk-neutral implies 1

Hence: PE-efficient allocations are on the 45-linethrough the origin for A. (Plus corner solutions!)

Risk averse A is fully insured and risk-neutral B

takes all the risk.


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Summary (Risk Sharing)

Efficient risk-sharing: Ratio of marginal

utilities are equal for different DMs (except

at corner solutions)

No aggregate risk & risk aversion:

PE requires full insurance for both parties

Aggregate risk & one risk-neutral DM

PE implies full insurance for risk-averse DM

More generally: Less risk averse DM bears

more risk.

topic 2 decision theory 2015 (1)

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