Microeconomics 1. Uncertainty - econ2.uni-bonn.de · Microeconomics - 1. Uncertainty Lotteries...

Microeconomics - 1. Uncertainty

Lotteries Expected Utility Money Lotteries Stochastic Dominance

Microeconomics

1. Uncertainty

Alex Gershkov

http://www.econ2.uni-bonn.de/gershkov/gershkov.htm

11. November 2008

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Lotteries

A decision maker faces a choice among a number of riskyalternatives.

Each alternative can lead to one of a number of possible outcomes.

C is the finite set of all possible outcomes with |C | = N.

Definition: A simple lottery L = (p1, . . . , pN), pi ≥ 0,∑

pi = 1 isa collection of probabilities for the sure outcomes x1, . . . , xN . (Wewrite a lottery as a set {xi : pi}

Ni=1

and denote by L the set of allsimple lotteries over the set of outcomes C .)

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Lotteries

A simple lottery can be represented as a point in simplex.

Definition: The set ∆ = {p ∈ R+N :

∑pi = 1} is called a

N-dimensional simplex.

In its three outcome (=dimensional) case, a simplex can begraphically represented by an equilateral triangle with altitude 1.Each perpendicular then can be interpreted as the probability ofthe outcome at the opposing vertex. Thus every point in thetriangle represents a lottery.

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Lottery

The Lottery Q = {1 : p1, 2 : p2, 3 : p3} with all probabilities equalto one third.

$3

$2 $1

f +

Q

p 1 p 2

p 3

1

½

½

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Lottery

Definition: Given K simple lotteries, Lk = (pk1 , ..., p

kN),

k = 1, ...,K and probabilities αk ≥ 0 with∑

k αk = 1, thecompound lottery (L1, ...,LK ;α1, ..., αK ) is the risky alternativethat yields the simple lottery Lk with probability αk fork = 1, ...,K .

For any compound lottery, the reduced (simple) lottery over finaloutcomes can be specified

pn = α1p1n + ...+ αKpK

n .

or, alternatively,

L = α1L1 + . . .+ αKLK .

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Preferences over Lotteries

We assume that the DM has a rational (complete and transitive)relation on L.

In addition assume:

◮ Reduction axiom. Two compound lotteries are equivalent ifthey yield the same simple lottery.

◮ Continuity. Suppose that L,L′ and L′′ are three simplelotteries such that L ≻ L′ ≻ L′′. Then there existsα, β ∈ (0, 1) such that αL + (1− α)L′′ ≻ L′ ≻ βL + (1− β)L′′

◮ Independence. Suppose that L,L′ and L′′ are three simplelotteries such that L % L′. Then, for any α ∈ (0, 1),αL + (1 − α)L′′ % αL′ + (1 − α)L′′.

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Preferences over LotteriesThese assumptions induce indifference curves in the simplexrepresentation to take the form of straight & parallel lines.

$3

$2$1

$3

$2$1

uu

vv

1/2u + 1/2v

w

1/2u + 1/2w1/2v + 1/2w

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Expected utility example

2 alternatives: A and B

Bermuda -500 0A 0.3 0.4 0.3B 0.2 0.7 0.1

What we would like to be able to do is to express the utility forthese two alternatives in terms of the utility the DM assigns toeach individual outcome and the probability that they occur.

U(A) = 0.3ub + 0.4u−500 + 0.3u0.

U(B) = 0.2ub + 0.7u−500 + 0.1u0.

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Expected utility

If at some alternative, the probabilities of the trip to theBermudas, paying 500 and paying 0 are pB , p−500 and p0,respectively, then the utility of the alternative is

pBub + p−500u−500 + p0u0.

This representation is called the expected utility form.

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Expected utility

Definition. The utility function U : L −→ R has an expected

utility form if there is an assignment of numbers (u1, ..., uN ) to theN outcomes such that, for every simple lotteryL = (p1, ..., pN ) ∈ L, we have

U(L) =∑

i

piui .

A utility function U : L −→ R with the expected utility form iscalled a von Neumann-Morgenstern expected utility function.

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Expected utility

The expected utility form is preserved only by increasing linear

transformations.

If U : L −→ R is an expected utility function for the preferencerelation %, then U : L −→ R is another expected utility functionfor the preference relation % if and only if there are scalars α > 0and β, such that U(L) = αU(L) + β for every L ∈ L.

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Expected utility theorem

Suppose that the rational preference relation % on the space oflotteries satisfies the reduction axiom, continuity andindependence. Then, there exists a function of the expected utilityform that represents %. That is, we can assign a number un toeach outcome n = 1, ...,N such that for any two lotteriesL = (p1, ..., pN ) and L′ = (p′

1, ..., p′

N ) we have

L % L′ if and only if

N∑

n=1

unpn ≥

N∑

n=1

unp′

n.

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Allais paradox

L1 {$ 1: 1, $ 0: 0}

L2 {$ 5: 0.1, $ 1: 0.89, $ 0: 0.01 }

L3 {$ 5: 0.1, $ 0: 0.9}

L4 {$ 1: 0.11, $ 0: 0.89}

(Amounts are in millions.)

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Expected utilityTypically L1 ≻ L2 and L3 ≻ L4—but no parallel, straight level setcan represent both L1 ≻ L2 and L3 ≻ L4!

£5 Mio

£1 Mio £0

£5 Mio

£1 Mio £0

L2 L3

L4

f +

L1

L2 L3

L4

f +

L1

Hence, the typical choice behaviour described by Allais’ paradoxcannot be represented by expected utility theory.

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Money lotteries

Let x be a continuous variable (amounts of money).

We can describe a monetary lottery by means of a cumulativedistribution function F : R −→ [0, 1].

Denote by f the associated density function.

Denote by u(·) the utility function defined on sure amounts ofmoney (Bernoulli utility function) and by U(·), a vN/M utilityfunction.

The expected utility from a lottery F (·) is given by

U(F ) =

∫u(x)dF (x).

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Risk Aversion

Definition: A DM is called risk averse (or said to exhibit risk

aversion) if, for any lottery F (·), the degenerate lottery that yieldsthe amount

∫xdF (x) with certainty is at least as good as the

lottery F (·) itself.

If for any F (·) the DM is indifferent between these two lotteries,we say that he is said to be risk neutral.

We say that he is strictly risk averse if indifference holds only whenthe two lotteries are the same (when F (·) is degenerate).

Opposite relations refer to risk loving.

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Risk aversion

Bernoulli utility fns of

◮ risk averse agents in [x , u(x)] space are concave;the more concave they are, the more risk-averse is the agent

◮ risk loving agents are strictly convex.

This is a direct consequence of Jensen’s inequality; for a realcontinuous concave fn u(·) we have

∫+∞

−∞

u(x)dF (x) ≤ u

(∫+∞

−∞

xdF (x)

).

Hence the expected utility is smaller than the utility of an expectedvalue.

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Risk aversionBernoulli utility functions representing risk averse and risk lovingattitudes.

u(x)u(x)

u(3)

u(3)

u(2)

u(2)

1

2u(1) + 1

2u(3)

1

2u(1) + 1

2u(3)

u(1)

u(1)

$$11 22 33

u(x)u(x)

Since the shape of u(·) matters, expected utility functions can bedefined up to separate increasing linear transformations.

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Risk aversion

A useful concept for the analysis of risk aversion:

Definition: Given a Bernoulli utility function u(·), the certainty

equivalent of F (·), denoted c(F , u) is the amount of money forwhich the individual is indifferent between the gamble F (·) and thecertain amount c(F , u); that is,

u(c(F , u)) =

∫+∞

−∞

u(x)dF (x).

The DM is risk averse if and only if

c(F , u) ≤

∫xdF (x) for all F (·).

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Insurance

◮ strictly risk averse DM has an initial wealth of w

◮ he runs a risk of a loss of D dollars

◮ the probability of the loss is π

◮ one unit of insurance costs q dollars and pays 1 dollar if theloss occurs.

The expected utility of the DM if he acquires α units of insurance

(1 − π)u(w − αq) + πu(w − αq − D + α).

The DM has to choose how much insurance to acquire (α).

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Insurance

If α∗ is an optimum, it must satisfy the first order condition

−q(1 − π)u′(w − α∗q) + π(1 − q)u′(w − α∗q − D + α∗) ≤ 0.

If the insurance is actuarially fair (q = π), the FOC requires that

u′(w − α∗π − D + α∗) − u′(w − α∗q) ≤ 0.

Since α∗ > 0,

u′(w − α∗π − D + α∗) = u′(w − α∗q).

u′′ < 0, therefore, w − D + α∗(1 − π) = w − α∗q or α = D.

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Insurance

ωN

ωA

ω

ω − D

1−qq

certainty line The slopes of the indifference cur-ves are

dωA

dωN

|dEu=0 = −(1 − Π)u′(ωN)

Πu′(ωN).

How much insurance will the DM acquire?

(1 − Π)u′(ω − αq)

Πu′(ω − αq − D + α)=

1 − q

q.

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Measurement of risk aversion

Definition: Assume u(·) is C2 and increasing. Then theArrow-Pratt coefficient of absolute risk aversion r(·) of a Bernoulliutility function defined over outcomes xi is

r(x) = −u′′(x)

u′(x).

The following figure should make this idea clearer: at x , both u1(·)and u2(·) give the same utility u(x), but the lotteries {x − ε, x + ε}are evaluated differently by the two agents. Hence the shape of theutility function gives an indication of the degree of risk aversion.

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Measurement of risk aversion

xx − ε x + ε

u(x)

x

c(F , u1)c(F , u2)

u1(·)

u2(·)

Since c(F , u) is bigger for the utility fn with the higher curvature,risk aversion increases with curvature.

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Comparisons across individuals

Assume two Bernoulli utility functions u1(·) and u2(·). When canwe say that u2(·) is more risk averse than u1(·)?

The following statements are equivalent

1. r(x , u2) ≥ r(x , u2) for every x

2. there exists an increasing concave function ψ : R −→ R suchthat u2(x) = ψ(u1(x)) for all x

3. c(F , u2) ≤ c(F , u1) fore any F (·).

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Comparisons across wealth levels

Definition: The Bernoulli utility function u(·) exhibits decreasing

absolute risk aversion (DARA) if r(x , u) is a decreasing function ofx .

Definition: The Bernoulli utility function u(·) exhibits constant

absolute risk aversion (CARA) if r(x , u) is a constant function ofx . For example if u(x) = −e−αx for α > 0, then r(x , u) = α.

Definition: The Bernoulli utility function u(·) exhibits increasing

absolute risk aversion (IARA) if r(x , u) is an increasing function ofx .

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Stochastic dominance

We are interested in knowing whether one distribution offers higherreturns then another.

2 possible criteria:

1. Every DM prefers F (·) to G (·).

2. For every amount of money x , the probability of getting at leastx is higher under F (·) than under G (·).

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Stochastic dominanceDefinition The probability distribution of F first-order

stochastically dominates (fosd) that of G if for non-decreasing u(·)

∫+∞

−∞

u(x)dF (x) ≥

∫+∞

−∞

u(x)dG (x).

cumulative density

x

1

G(x)

cumulative density

x£1 £2 £3

0

F(x)

1

G(x)

0

F(x)

.5

E[F(x)]E[G(x)]

p3F

p2F

p1F

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DefinitionLet two probability distributions F ,G share the same mean (i.e.∫

xdF (x) =∫

xdG (x)). Then distribution F second-order

stochastically dominates (sosd) distribution G if, for concave u(·)

∫u(x)dF (x) ≥

∫u(x)dG (x).

Sosd is about distributions with the same expectation but differentrisks. An example is a mean preserving spread of a givendistribution. If F sosd’s G , for all x0 we have

∫ x0

−∞

G (t)dt ≥

∫ x0

−∞

F (t)dt.

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The idea of Sosd is captured by a mean-preserving spread of agiven distribution F (·).

Example: Let

◮ F (·) be an even probability dist’n between 2$ and 3$

◮ now ‘spread’ the 2$ outcome to an even probability between 1and 2$, and the 3$ outcome to an even probability between 3and 4$ (i.e. giving probability 1

4to all outcomes)

◮ call the resulting distribution G (·)

◮ notice that our probability manipulations do not change themean of the distributions E[F (·)] = E[G (·)]

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1 2 3 4xx

1

4

1

2

1

2

3

4

11

PrPr

F (·)F (·)

G (·)

G (·)

x x ′

A

B

Sosd and mean-preserving spreads of some dist’n F (·).

Saying that G (·) is a mean-preserving spread of F (·) is equivalentto saying that F (·) second order stochastically dominates G (·).

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