Chapter 3: Individual Choice Under Uncertainty - Advanced ...niedermayer.vwl.uni-mannheim.de/uploads/media/beamer_chapter3_01… · Lotteries and Expected Utility Lotteries as Contingent

Lotteries and Expected Utility Lotteries as Contingent Plans Measures of Risk and Risk Aversion

Chapter 3: Individual Choice UnderUncertainty

Advanced Microeconomics I

Andras Niedermayer1

1Department of Economics, University of Mannheim

Fall 2009

Chapter 3: Individual Choice Under Uncertainty Fall 2009 1 / 76


so far: individual choices had completely predictableconsequences

often choices where consumption involves uncertainty

Examples

goods of uncertain qualitysavings decisionsinvestment portfolioscareer movesinsurance policiesenvironmental policy choices

Question: how does a rational individual evaluate andcompare risky choices?



Choices over baskets of goods under uncertainty arechoices of probability distributions over R

H+

standard theory: different uncertain prospects andprobability laws that they obey exogenously given to theindividual decision maker (von Neumann and Morgenstern(1944))



Definition: A lottery L is a probability distribution over RH (or

RH+, if one cannot lose).

Important special cases:

“Simple lotteries”: Lotteries over a fixed, finite subset ofR

H+, {x1, ..., xS}:

L = (π1, ..., πS) ∈ SS−1 = {π ∈ RS+|

S∑

s=1

πs = 1}.

(S − 1-dimensional simplex)“Money lotteries”: Lotteries over R (“money”): assume theexistence of a given price system → reduce the goodsspace from R

H+ to R.

“Simple state lotteries”: for a fixed set of “states of nature”with probabilities π1, ..., πS choose bundles in R

H+ for each

state of nature: L = (x1, ..., xS) ∈ RHS+ . (See Section 3.2)



simple lotteries with S = 3 important for experiments andexposition → Marshak-Machina triangle



Example: Three possible monetary outcomes ( all in Euro, say):x1 = 0x2 = 500,000x3 = 2,500,000

Here are two lotteries:Lottery L1 : (0,1,0)

L2 : (0.01,0.89,0.1)Question: Which one do you prefer? (As an orientation: theexpected values of these lotteries are 500,000 for the first and695,000 for the second)



Here are two other lotteries (defined over the same outcomesx1 = 0; x2 = 500,000; x3 = 2,500,000):

L3 : (0.89,0.11,0)L4 : (0.9,0,0.1)

Question: Which one of these two do you prefer? (Theexpected values are 55,000 for L3 and 250,000 for L4) Wereturn to the answer of this equation further below.



we look at choice under uncertainty using preferencetheory

L: the set of lotteries

%: a complete and transitive preference relationship on Ltypically more structure, e.g. for simple lotteriesL = SS−1 = {π ∈ R

S+;∑S

s=1 πs = 1}, the unit simplex

Definition: The preference relation % is continuous, if for allL ∈ L the sets {L′ ∈ L; L′ % L} and {L′ ∈ L; L′ - L} are closed(in the relevant topology). By Proposition 2.1, % can thereforebe represented by a continuous utility function U : L → R. Thismeans:

L % L′ ∈ L ⇔ U(L) ≥ U(L′).



One can even use more structure of the space of all probabilitydistributions, “linearity”:

Definition: Given α1, α2 ≥ 0, α1 +α2 = 1, the mixture of lotteryL1 and L2, α1L1 + α2L2, is the risky prospect which yields Li

with probability αi , i = 1,2.

Example simple lotteries: If Li = (πi1, ..., π

iS), then

α1L1 + α2L2 = (α1π11 + α2π2

1, ..., α1π1

S + α2π2S).

Example money lotteries: If Li is given by c.d.f. F i , thenα1L1 + α2L2 given by c.d.f. α1F 1 + α2F 2.



Important (typical) restriction on preferences over L:

Definition: A utility function U : L → R is called a vonNeumann-Morgenstern utility function, if it is linear:

U(α1L1 + α2L2) = α1U(L1) + α2U(L2)

for all α1, α2 ≥ 0, α1 + α2 = 1, and Li ∈ L.



Proposition 3.1: (i)In the case of simple lotteries, utilityfunctions are linear if and only if there are S numbers u1, ...uS

such that for every L = (π1, ..., πS) ∈ L

U(L) =S∑

s=1

usπs. (1)

(ii)In the case of money lotteries, utility functions are linear ifand only if there is a function u : R → R such that for everyc.d.f. F ∈ L

U(F ) =

∫

R

u(x)dF (x). (2)

Intuition: If utility is linear, it can be patched together linearlyfrom the “corner utilities” over certain events (Es).



Proof.

for the finite case (i):1. (“if”, i.e. linear utility function is implied) Suppose that thereare u1, ...us such that U can be represented as (1). Clearly,then U is linear.2. (“only if”, implied that there are u1, ...us) For any L ∈ L, letL =

∑sπsEs, where Es = (0, ..,1, ..,0) is the degenerate lottery

with unit weight at xs....



Proof.Then,

U(L) = U

(∑

s

πsEs

)

= U

(π1E1 + (1 − π1)

S∑

s=2

πs

1 − π1Es

)(note:

S∑

s=2

πs

1 − π1= 1)

= π1U(E1) + (1 − π1)U

(S∑

s=2

πs

1 − π1Es

)

= ...

= π1U(E1) + ...+ πSU(ES)

and the conclusion follows by setting us = U(Es) for all s (whichdoes not depend on the specific L).Case (ii) homework.



Common assumptions:

u monotonically increasing

u continuous (for money lotteries)



Graphically, for linear preferences

u1π1 + u2π2 + u3π3 = const (substitute for π2 = 1 − π1 − π3)

⇐⇒ (u1 − u2)π1 + (u3 − u2)π3 + u2 = const

⇐⇒ π1 =const − u2

u1 − u2+

u2 − u3

u1 − u2︸︷︷︸≷0

π3



The previous figure shows that linearity of preferences (overlotteries) is quite a strong assumption, much more than whatwe have assumed for the deterministic case. In the latter,preferences U(x1, x2) = ax1 + bx2 would yield only extremechoices (see Figure 3.3).



Proposition 3.2 : Suppose that the preference relation - over Lis complete, transitive and continuous. Then it can berepresented by a linear utility function iff

for all L,L′,L′′ ∈ L and α ∈ [0,1] : (IA)

L % L′ ⇐⇒ αL + (1 − α)L′′ % αL′ + (1 − α)L′′ (3)

Proof.

“⇒” trivial, “⇐” homework (not easy; easy for simple lotteries,see MWG and Gollier)

“Independence Axiom” (IA): if one mixes each of two lotteries(L,L′) with a third (L′′), then the ordering of the resultingmixtures is independent of L′′



Intuition: Choice between αL + (1− α)L′′ and αL′ + (1 −α)L′′ isthe same as

flip a coin, probability 1 − α tails → you get L′′, α head →either L or L′

choose between L and L′ before knowing head or tails

if tails: choice didn’t matter

if head: back to original choice between L and L′



Proposition 3.2 : Suppose that U : L → R is a vonNeumann-Morgenstern utility function for the preferencerelation % on the set of simple lotteries L. Then U : L → R isanother von Neumann-Morgenstern utility function for % if andonly if there are scalars β > 0 and γ such thatU(L) = βU(L) + γ for every L ∈ L.

Proof.

Choose two lotteries L and L with L % L % L for all L ∈ L. IfL ∼ L every utility function must be constant. The result thenfollows immediately. Next we look at L ≻ L....



Proof.

Write L as L =∑sπsEs, where Es = (0, ..,1, ..,0) is the

degenerate lottery with unit weight at xs. U is a vonNeumann-Morgenstern utility function and U(L) = βU(L) + γ.To be shown: U is also a von Neumann-Morgenstern utilityfunction:...



Proof.

U(L) = U(∑

s

πsEs) = βU(∑

s

πsEs) + γ

= β

[∑

s

πsU(Es)

]+ γ

=∑

s

πs(βU(Es)) +∑

s

πsγ

=∑

s

πs(βU(Es) + γ)

=∑

s

πsU(Es)

Hence U has a von-Neumann-Morgenstern utilityrepresentation.Further, U represents the same preferences asU.



Proof.

Opposite direction of the proof: U and U are vonNeumann-Morgenstern utility functions representing � ⇒U(L) = βU(L) + γ for all L ∈ L. (to be shown)Take any lottery L ∈ L and define αL by

U(L) = αLU(L) + (1 − αL)U(L),

⇔αL =

U(L) − U(L)

U(L) − U(L).

αLU(L) + (1 − αL)U(L) = U(αLL + (1 − αL)L) impliesL ∼ αLL + (1 − αL)L.



Proof.

U represents same preferences as U ⇒

U(L) = U(αLL + (1 − αL)L)

= αLU(L) + (1 − αL)U(L)

= αL[U(L) − U(L)] + U(L).

Substituting for αL we obtain U(L) = βU(L) + γ where

β =U(L) − U(L)

U(L) − U(L), and

γ = U(L) − U(L)U(L) − U(L)

U(L) − U(L)



Independence Axiom (⇔ linear utility) controversial, because ofexperimental evidence, e.g. Battaglio, Kagel, Jiranyakul (J. RiskUncertainty, 1990):Three pairs of simple lotteries on {0,12,20}

S1 = (0,0.4,0.6), R1 = (0.16,0,0.84)

S2 = (0,1,0), R2 = (0.4,0,0.6)

S3 = (0.8,0.2,0), R3 = (0.88,0,0.12).

Graphically, in Marschak-Machina triangle:



indifference curves are straight lines ⇒ either Si � R i for all i orR i � Si for all iHowever, in experiments non-homogeneous choices (not onlyS, not only R)



Example: Allais Paradox: Take the lotteries we have looked atbefore: L1 = (0,1,0),L2 = (0.01,0.89,0.1), andL3 = (0.89,0.11,0),L4 = (0.9,0,0.1) on payments x1 = 0;x2 = 500,000; x3 = 2,500,000, graphically:



With expected utility, graphically:

preferences are either relatively steep (see figure), → L1

and L3 will be preferred.

Or: preferences are relatively flat → L2 and L4 arepreferred (of course, individuals may be indifferent).



Mathematically:Take LA = (0,1,0), LB = ( 1

11 ,0,1011)

L1 = 0.89(0,1,0) + 0.11LA L2 = 0.89(0,1,0) + 0.11LB

L3 = 0.89(1,0,0) + 0.11LA L4 = 0.89(1,0,0) + 0.11LB

In experiments: large part (often majority) L1 � L2 and L4

� L3 → indifference curves “fan out”



In most applications, one still works with linear utility, because

it is conceptually simple,

people have always done it,

it allows simple derivations of concepts such as riskaversion, etc.

choice under uncertainty is indeed different from choiceunder certainty: the linear combination of lotteriesconsidered in the Independence Axiom is not actuallyconsumed (at least when you view it as a two-stagelottery): the comparison is rather between preferring L to L′

ex-ante or ex-post. In the context of choice under certainty,linear combinations of bundles are consumed ascombinations.



theory of “state lotteries” or of lotteries as contingent plans(similar to theory of choice under certainty as discussed inChapter 2)

here: for the finite case (“simple state lotteries”)

fix a set of probabilities π1, ..., πS ≥ 0,∑sπs = 1,

vary the consumption bundles xs ∈ RH+ that the individual

can get with these probabilities.Formally, the set of all these lotteries is identical to (RH

+)S′

,the set of all x = (x1, ..., xS) with xs ∈ R

H+.

Interpretation: S “states of nature”, with probabilities πs.Choice today: “contingent consumption bundle” (a plan forstate-dependent consumption).



complete, transitive, and continuous preferences % overthese plans ⇒ continuous utility function U representing %

Note: L = RHS is a natural linear space, (for any x , y ∈ L

and α, β ∈ R+, αx + βy ∈ L)

interpretation is natural (the lottery αx + βy yieldsαxs + βys in each state), but different from the generaldefinition of linear combinations given earlier

earlier definition: different form of linear combination:mixing two state lotteries x and y with random binaryvariable (probabilities α and 1 − α) → binary lotteryinvolving xs and ys in each state s



Using this notion (linearity in probabilities instead oflinearity in quantities), allows us to retrace the approach ofsimple lotteries taken earlier. Independence Axiom:

Definition: A utility function over contingent plans satisfies theIndependence Axiom if, for any x , x ′ ∈ R

HS+ , ys ∈ R

H+, zs ∈ R

H+,

U(x−sys) ≥ U(x ′

−sys) if and only if U(x−szs) ≥ U(x ′

−szs)

Here, x−sys is the contingent plan x with xs replaced by ys.

Analog to Prop. 3.2Proposition 3.4: (Additivity) Assume that there are at leastthree states. If and only if % satisfies the Independence Axiom,there exist state-dependent “elementary utility functions”us : R

H+ → R such that

U(x) =∑

s

us(xs) for all x .



So far we allowed the preferences of an individual to depend onstate s. Disallowing this we get the full analog ofU(L) =

∑Ss=1 usπs shown previously:

Proposition 3.5: (Linearity) If, in addition, U does not dependon s directly, but only on the probability distribution of the xs

(i.e. on the πs), then there is a u : RH+ → R such that

U(x) =∑

s

πsu(xs) for all x .



state independence of preferences is a strong assumption(e.g. insurance/accidents)

but: with larger space of consumption bundle →preferences depending on state directly

alternatively: parameterize us by simple parameter("endowment loss") (especially for monetary model H = 1)



relationship between two concepts "simple lottery" and "statelottery":

state lottery: choose a lottery (x1, ..., xS ;π1, ..., πS), choiceis among (x1, ..., xS) ∈ R

HS+ , the πs are fixed

simple lottery model: choose a lottery [x1, ..., xS ;π1, ..., πS ],choice is among the (π1, ..., πS) ∈ SS−1, the xs are fixed

Focus

empirical literature: mostly simple lottery, replacingExpected Utility hypothesis

theoretical General Equilibrium literature: state-lotterymodel



Application 1: Demand for Insurance

H = 1 (monetary world)

S = 2, where s = 1 ↔ accident, s = 2 ↔ no accident

state-contingent endowments e1 < e2

before endowments materialize: market in state-contingentclaims (monetary payments)

p: price of state-1 consumption in terms of state-2consumption (i.e. p = p1/p2: “how many units of state-2consumption must be given for 1 unit of state-1consumption”)

utility function

U(x1, x2) = u1(x1) + u2(x2)



optimal allocation of endowments:

maxx1,x2

u1(x1) + u2(x2)

s.t. px1 + x2 ≤ pe1 + e2

with monotinicity "≤" → "="

with differentiability and u′

i (0) = ∞: → u′

1(x1) = pu′

2(x2)(marginal rate of substitution = price):

u′

1(x1)

u′

2(x2)= p

→optimal choice of insurance (however, you may havex2 > e2, x1 < e1)



Additional restriction: linearity. This yields

ui(xi ) = πiu(xi), i = 1,2

=⇒ π1

π2

u′(x1)

u′(x2)= p

πi : subjective/objective probabilities for state i .p = π1

π2(“actuarially fair insurance”)⇒ x1 = x2 (if u′

monotone)p 6= π1

π2⇒ demand depends on shape of u

if u concave

U(x1, x2) = π1u(x1) + π2u(x2)

< u(π1x1 + π2x2)

= U(π1x1 + π2x2, π1x1 + π2x2)

decision maker equalizes over statesfor p > π1

π2(“actuarially unfair insurance”) we have x1 < x2

(under-insurance)



Portfolio Choice

Assumptions

Monetary model: H = 1

S states of nature, probabilities πs.

U does not depend on state directly.

K assets (securities) that can be purchased before thestate of nature is realized. Asset k pays off vk

s ∈ R in states.

Asset prices q ∈ RK

budget b to spend on assets, no endowments in thedifferent states



Individual’s problem: choose asset holdings z ∈ RK to

maximize expected utility subject to budget constraint:

maxx,z

U(x) =∑

s

πsu(xs)

s.t. xs =∑

k

zk vks

∑

k

qkzk = b

negative consumption or negative asset holdings ("shortsales") allowed



Matrix notation: Let

V = (v1, ..., vK ) =

v11 ... vK

1...

. . .v1

S ... vKS

be the payoff matrix of the K assets. Then the problem is

maxx

∑

s

πsu(xs)

s.t. x = Vz (4)

q · z = b



Special cases:

Arrow securities: There are K = S assets, and asset kpays 1 unit in state k and 0 in all other states:

V =

1 ... 0...

. . .0 ... 1

Redundant securities: A security k is redundant if there aresecurities k1, ..., km 6= k and numbers αi such thatvk =

∑mi=1 αivki .



Complete asset market: The set of securities is complete(constitutes a complete asset market) if the payoff matrix Vhas rank S: for each x ∈ R

S there is a z such that x = Vz.Eliminate redundant securities (not unique) → V is full rankS × S matrix, and the utility maximization problem can bewritten as

maxx

∑

s

πsu(xs)

s.t. q · V−1x = b

The risk-free asset: An asset that has the same payoff ineach state: vk

s = v . (Note: why “the" risk-free asset?)



Remark: complete asset market ⇒ there is always a risk-freeasset (portfolio with constant payoff):

z = V−1

1...1

Often: label this portfolio “asset 1", and ignore one of theoriginal assets.(This can also work for incomplete asset market)



A reformulation of the portfolio problem: Let ak = qkzk

denote the amount of wealth invested in asset k . Letrk = vk/qk be the (gross) return of asset k . Then problem (4)can be written as

maxa

Eu(

K∑

k=1

ak rk ) (5)

s.t.K∑

k=1

ak = b



Measures of Risk and Risk Aversion

here: H = 1 (monetary allocations).

Interpretation: U over lotteries is an indirect utility functionfor a given set of prices

⇒von Neumann-Morgenstern utility is defined over“income” x ∈ R

set of all possible "states": R ⇒“state model” and the“lottery model” are identical:

in the “lottery model” one chooses a cumulative distributionfunction (c.d.f.) F ,in the “state model” one has a c.d.f. G over R (the states)and choose h : R → R (states → incomes), which yields ac.d.f. over R (income), as well.



Following the discussion of the last section, we will consider adecision maker who has preferences over the space

L = {F :R → [0,1] ; F (−∞) = 0,F (∞) = 1,

F continuous from the right}

and whose preferences can be represented by a linear utilityfunction U:

U(F ) =

∫

R

u(x)dF (x).

u is continuous and increasing (⇔ monotonicity in the certaincase). → “money-utility function”



Digression: The St.-Petersburg-Paradox:

Suppose an individual has vNM utility with unboundedmoney-utility. Let xn be defined by u(xn) = 2n,n ≥ 1.

(e.g. u(x) =√

x =⇒ xn = 4n, and x10 = 1,048,576)

Following lottery: Coin tossed repeatedly, until “heads”comes up. If it comes up at the n-th toss, the individualgets paid xn. Individuals willingness to pay for this lottery:

U(F ) =

∞∑

n=1

12n u(xn) =

∑1 = ∞

Question: Is this reasonable?Answer: Probably not. This phenomenon does not occur if u isbounded above. (But note that in applications these types ofunbounded gambles never occur)



Risk Aversion

Definition: For an individual with money-utility u, the certaintyequivalent of a monetary lottery F , c(F ,u), is defined as

u(c(F ,u)) =

∫

R

u(x)dF (x)

c(F ,u) is the amount of money which the individual, if obtainedfor sure, values as equivalent to the lottery F .



Definition:An individual is risk averse if c(F ,u) <

∫xdF (x) for all F ,

An individual is risk neutral if c(F ,u) =∫

xdF (x) for all F ,An individual is risk loving if c(F ,u) >

∫xdF (x) for all F .



→“global” risk attitude: for all wealth levels and all lotteries.alternatively: an individual attitude towards small risks around agiven wealth level:

Definition: Let ε be a random variable with mean 0 and c.d.f.φ. The risk premium (demanded by a given individual) forgamble ε at wealth level x , ρ(x , ε), is given by

u(x − ρ(x , ε)) =

∫

R

u(x + ε)dφ(ε).

In other words: x − ρ(x , ε) is the certainty equivalence of x + ε.



Proposition 3.6: For an individual with vNM utility the following

statements are equivalent:(i) The individual is risk averse.(ii) ρ(x , ε) > 0 for all x , ε with E ε = 0.(iii) u is strictly concave.



Proof.

(i)=⇒(ii): Take any x , ε with E ε = 0. Since x − ρ(x , ε) is thecertainty equivalent of x + ε, (i) implies

x − ρ(x , ε) <

∫(x + ε)dφ(ε)

= x + E ε

= x

=⇒ ρ(x , ε) > 0

...



Proof.

(ii)=⇒(iii): Take any y , z ∈ R and α ∈ [0,1]. Letx = αy + (1 − α)z.Define the gamble

ε =

{y − x with proba αz − x with proba 1 − α

Then ∫u(x + ε)dφ(ε) = αu(y) + (1 − α)u(z).

...



Proof.

By the definition of ρ,

u(x − ρ(x , ε)) =

∫u(x + ε)dφ(ε)

Hence, u(x − ρ(x , ε)) = αu(y) + (1 − α)u(z).Since u is strictly increasing, (ii) implies

u(x) > αu(y) + (1 − α)u(z).

...



Proof.

(iii)=⇒(i): Remember Jensen’s inequality: If u is strictlyconcave,

u(∫

xdF (x)

)>

∫u(x)dF (x) for all non-trivial F .

Suppose that∫

xdF (x) ≤ c(F ,u) for one F (i.e. no riskaversion). Since u is increasing and strictly concave

∫u(x)dF (x) <

Jensenu(∫

xdF (x)

)≤

u increasingu(c(F ,u))

which is a contradiction to the definition of c.



Measuring Risk Aversion

Two main questions:

1 Can one compare the risk aversion of different individuals?In particular, can one give a sense to the statement “i ismore risk averse than j”?

2 Can one define a “measure of risk aversion”, which wouldnot only allow to compare different utility functions, but alsoto give sense to statements such as “i has very high riskaversion”, “j has risk aversion of around 4”?



Question 1: Can one compare the risk aversion of differentindividuals? In particular, can one give a sense to thestatement “i is more risk averse than j”?Two obvious candidates for answers are

“ui is more concave than uj ”

“at all wealth levels and for all gambles ε, i demands ahigher risk premium”

Proposition 3.7: Consider two money-utility functions u1 andu2. The following two statements are equivalent:(i) There exists a strictly concave function ψ such thatu1(x) = ψ(u2(x)) for all x(ii) ρ1(x , ε) > ρ2(x , ε) for all x and ε with E ε = 0.



Proof.

(i)=⇒(ii): By the definition of the risk premium,ui(x − ρi(x , ε)) =

∫ui(x + ε)dφ(ε). Since ui is strictly increasing

and thus can be inverted, we have

ρi(x , ε) = x − u−1i

(∫ui(x + ε)dφ(ε)

)



Proof.Hence,

ρ1(x , ε) − ρ2(x , ε) = u−12

(∫u2(x + ε)dφ(ε)

)

−u−11

(∫u1(x + ε)dφ(ε)

)

= u−12

(∫u2(x + ε)dφ(ε)

)

−u−11

(∫ψ(u2(x + ε))dφ(ε)

)



Proof.

By Jensen’s inequality (applied to the strictly concave functionψ, after change of variable η = u2(x + ε), note: u′

2 > 0):∫ψ(u2(x + ε))dφ(ε) < ψ(

∫u2(x + ε)dφ(ε))

Hence, because u−11 is increasing,

ρ1(x , ε) − ρ2(x , ε) > u−12

(∫u2(x + ε)dφ(ε)

)

−u−11

(ψ(

∫u2(x + ε)dφ(ε))

)



Proof.

Sinceu1(x) = ψ(u2(x)) ⇐⇒ u−1

2 (u) = u−11 ψ(u)

the right hand side of the last inequality is 0. This completesthe first part of the proof.(ii)=⇒(i): homework.



Question 2 (more general): Can one define a “measure ofrisk aversion”, which would not only allow to comparedifferent utility functions, but also to give sense tostatements such as “i has very high risk aversion”, “j hasrisk aversion of around 4”?

natural question: "what is the risk premium per unit ofvariance of a gamble that a risk averse individualdemands?"



consider a “small gamble” (all mass close to 0) ε.

How much is the individual ready to pay in order to avoidhaving the random wealth x + ε?

By Taylor expansion

u(x + ε) ≈ u(x) + εu′(x) +ε2

2u′′(x) for small ε ∈ R (6)

=⇒∫

u(x + ε)dφ(ε) ≈ u(x) +12

u′′(x)var(ε)



On the other hand, also by Taylor expansion (since for ε small,ρ(x , ε) is small),

u(x − ρ(x , ε)) ≈ u(x) − ρ(x , ε)u′(x)

By the definition of ρ :

u(x) − ρ(x , ε)u′(x) ≈ u(x) +12

u′′(x)var(ε)

⇐⇒ ρ(x , ε)var(ε)

≈ −12

u′′(x)

u′(x)

Hence, for “small gambles around x”, the risk premium per unitof variance demanded by a risk averse individual isapproximately −1

2u′′(x)u′(x) .



Definition: If the money-utility function of a utility maximisingindividual is twice differentiable,

rA(x) = −u′′(x)

u′(x)

is called the coefficient of absolute risk aversion at x .



Proposition 3.8: If the two money-utility functions u1 and u2

are twice differentiable, then the following two statements areequivalent:(i) There exists a strictly concave function ψ such thatu1(x) = ψ(u2(x)) for all x(ii) rA

1 (x) > rA2 (x) for all x .



Proof.

We can always write u1(x) = ψ(u2(x)).Since u1 and u2 strictly increasing, ψ must be so as well.Differentiating:

u′

1(x) = ψ′(u2(x))u′

2(x)

u′′

1(x) = ψ′′(u2(x))(u′

2(x))2 + ψ′(u2(x))u′′

2(x)

Divide the second by the first equation:

rA1 (x) = −ψ

′′(u2(x))

ψ′(u2(x))u′

2(x) − u′′

2(x)

u′

2(x)

= −ψ′′(u2(x))

ψ′(u2(x))u′

2(x) + rA2 (x)

Hence, rA1 > rA

2 ⇐⇒ ψ′′ < 0.



The most important comparative statics property of rA

concerns the question how the attitude towards risk isinfluenced by wealth:Proposition 3.9: An individual’s money-utility function exhibitsdecreasing absolute risk aversion (i.e., rA ց in x) if and only ifher risk premium is decreasing in wealth (ρ(x , ε) ց in x) forevery gamble ε.Proof : Exercise.



instead of additive gamble x + ε in (6) → multiplicative gamble(1 + ε)x (similar approximation as above):maximum share of her wealth that a risk averse individual isready to pay in order not to lose or gain a random share ε ofher wealth is approximately

−12

var(ε)xu′′(x)

u′(x).

→ relative risk premium.



Definition: If the money-utility function of a utility maximizingindividual is twice differentiable,

rR(x) = −xu′′(x)

u′(x)

is called the coefficient of relative risk aversion at x .Note:

coefficient of relative risk aversion: no dimension

coefficient of absolute risk aversion: dimension of 1/moneyunit



For computational reasons, the following utility functions areoften used:

CARA u(x) = −e−ax ,a > 0. This function has constantabsolute risk aversion with coefficient a.

CRRA u(x) =

{ 11−γ x1−γ if γ 6= 1, γ > 0

ln x if γ = 1This function

has constant relative risk aversion with coefficientγ.

LRT u(x) = 1γ−1(α+ γx)1−1/γ , for α+ γx > 0. This

function has linear (better: affine) risk tolerance(which means that the inverse of its absolute riskaversion is linear in wealth). They are also calledHARA utility functions (hyperbolic absolute riskaversion - because absolute risk aversion as afunction of wealth describes a hyperbola).



Proposition 3.10: Consider the portfolio choice problem of

Section 3.2, assume that there are K assets, and that asset 1is risk-free with return r . If an agent has utility with linear risktolerance (LRT), then the solution a of his portfolio choiceproblem (5) is linear in wealth. This means: there existnumbers λk , µk ∈ R such that

ak = λk + µkb, k = 1, ...,K .

Proof : Exercise.



Stochastic Dominance

introduced to economics by Rothschild and Stiglitz (JET 1970)Denote F (x) and G(x) two c.d.f. (F and G have to beupper-semicontinuous); both are money lotteries.

Definition: The distibution function F first-order stochasticallydominates distribution function G if

∫∞

0u(x)dF (x) ≥

∫∞

0u(x)dG(x) (7)

for every nondecreasing function u : R+ → R.The distibutionfunction F second-order stochastically dominates distributionfunction G if EF x = EGx and (7) holds for every concavefunction u : R+ → R.



Interpretation

first-order stochastic dominance: no expected utilitymaximizer will prefer G to F .

second-order stochastic dominance: consider two lotterieswith the same mean. No risk-averse expected utilitymaximizer prefers G to F .



Because of the following proposition stochastic dominance canbe defined without having to explicitly specify utilities.Proposition 3.11 : (i) The distribution function F first-orderstochastically dominates distribution function G if and only if, forall x ≥ 0, F (x) ≤ G(x).(ii) The distribution function F second-order stochasticallydominates distribution function G if and only if they have thesame mean and the Lorenz curve of G does not lie below theLorenz curve of F ,

∫ x

0F (y)dy ≤

∫ x

0G(y)dy

for all x ≥ 0.The proof is a bit lengthy.


Chapter 3: Individual Choice Under Uncertainty - Advanced ...niedermayer.vwl.uni-mannheim.de/uploads/media/beamer_chapter3_01… · Lotteries and Expected Utility Lotteries as Contingent

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