Economie de l'incertain et de l'information

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Economie de l’incertain et de l’information

Christoph Heinzel

To cite this version:

Christoph Heinzel. Economie de l’incertain et de l’information. (Lecture 1: Overview, Basic Concepts I), 2015, 28 p. �hal-02796845�

Lecture 1:

Overview, Basic Concepts I

Course objective:

•

_{Introduction to the economics of risk and time, with an emphasis on the}

measure-ment of relevant preferences, time-related applications, and the theory of insurance.

•

_{Focus on familiarizing with important concepts and economic intuition.}

We will also deal with (but not primary focus)…

- Actual economics of uncertainty (literature on ‘ambiguity’). - Economics of information (only in last lecture).

Unit 1: Basics: Expected Utility, Risk Aversion, and Prudence

(2 lectures)

• _{Distinction of risk and uncertainty}

• _{Lotteries, expected value, expected utility theory: St. Petersburg paradox} • _{Risk aversion and prudence (definition & measurement), comparison of risks}

Unit 2: Measurement, Testing, and Criticism of Expected Utility

(3 lectures)

• _{Experiments on risk attitudes, survey methods, experiments on risk and time preferences} • _{Expected Utility: criticisms, extensions, alternatives}

• _{Numerical analysis of the two-period consumption-saving model}

Unit 3: Discounting a Risky Future

(2 lectures)

• _{Definition of time preferences, discount rate, Ramsey formula for Social Discount Rate} (SDR), determinants of the term structure of the SDR

Unit 4: Theory of Insurance

(2 lectures)

• _{Insurance decisions under symmetric information}

Overview

Literature – as background information:

•

_{Some textbooks on topic:}

Laffont, Jean-Jacques: Economics of Uncertainty and Information. MIT Press, 1989 (French 1985) Hirshleifer, Jack and John G. Riley: The Analytics of Uncertainty and Information. Cambridge

University Press, 1992

Van Zandt, Timothy: Introduction to the Economics of Uncertainty and Information. Oxford University Press, in preparation

Overview

Literature – as background information

(cont.)

:

•

_{Important historic books on topic:}

Knight, Frank: Risk, Uncertainty and Profits. Houghton Mifflin, 1921

Von Neumann, John and Oskar Morgenstern: The Theory of Games and Economic Behavior. Princeton University Press, 1944

Savage, Leonard J.: Foundations of Statistics. Dover Publications, 1954

•

_{Important recent books on topic:}

Gollier, Christian: Economics of Risk and Time. MIT Press, 2001

Gollier, Christian: Pricing the Planet’s Future. The Economics of Discounting in an Uncertain

World. Princeton University Press, 2013

Overview

Literature – used for the course:

• _{Main recommendations:}

§ Eeckhoudt, Louis, Gollier, Christian and Harris Schlesinger: Economic and Financial

Decisions under Risk. Princeton University Press, 2005 (chapters 1, 3, 6, 12)

§ Chavas, Jean-Paul: Risk Analysis in Theory and Practice. Elsevier, 2004 (chapters 1-5)

• _{Other texts, slides, exercises (& eventually solutions) are made available on the course website} on the intranet: ENT > “Identifiez-vous” > Formation - Insertion pro.: Mes cours en ligne

Remarks:

• _{Important for the tests and final exam is the material covered in the lecture !}

• _{Articles on the website are relevant to the extent that they are discussed in class (you do not} need to fully understand them beyond that).

Unit 1: Basic Concepts - Overview

Unit 1A:

Ø Distinction of risk and uncertainty

Ø Lotteries, Expected Value, Expected Utility Theory: § St. Petersburg Paradox

Ø Risk aversion:

§ Definition

§ Coefficients of absolute and relative risk aversion § Risk premium

§ Certainty equivalent

§ Criteria for the shapes of absolute and relative risk aversion

Unit 1: Basic Concepts - Overview

Unit 1B:

• _{Prudence in Temporal Decisions:}

Ø The two-period consumption-saving model (under certainty)

Ø Precautionary saving in the two-period consumption-saving model Ø Behavioral definition and theory of “prudence” (Kimball 1990)

• _{Comparison of Risks:}

Ø Moments of a distribution (e.g., mean, variance, skewness, kurtosis) Ø Increases in risk:

§ “mean-preserving spread” = same mean, increased variance

§ “increase in downside risk” = same mean and variance, but more skewness • _{Prudence as a Higher-Order Risk Attitude:}

Ø Alternative definitions of prudence:

§ Prudence as third-order risk apportionment

Basic Concepts

Literature:

• _Main:

§ Eeckhoudt, Louis, Gollier, Christian and Harris Schlesinger: Economic and Financial

Decisions under Risk. Princeton University Press, 2005, chapter 1

§ Chavas, chapters 2-4, Appendix A

• _Background:

Basic Concepts

Distinction of risk and uncertainty:

• _{Uncertainty in the sense of the title of this course means:}

“a situation in which not everything is known with certainty”.

• _{Knight’s definitions – which I follow in this course – are more precise:}

Ø Under this definition, risk is equivalent to a positive variance of the distribution.

Ø In modern economics, this definition of uncertainty and “ambiguity” are synonymous. Ø We will not deal with what is known as “Knightian uncertainty” in economics: a situation

which may involve unknown outcomes. Definition (Knight 1921):

• _{Under risk, all possible outcomes and their probabilities are known, and the probabilities add} up to 1, but none is 1.

Basic Concepts

Distinction of risk and uncertainty

(cont.)

:

• _{Uncertainty in the sense of the title of this course means:}

“a situation in which not everything is known with certainty”.

• _{Discussion of Chavas’ (2004, chapter 2) position :}

Ø Chavas defines “risk” exactly as uncertainty is meant in the title of this course.

Ø He links risk with time (‘an event is risky if it is not known for sure ahead of time’, p. 6) – which is intuitive, but contradicts the basic idea of risk in risk economics.

Ø He, finally, pragmatically associates both risk and uncertainty with Knight’s stricter risk definition – on the premise that in the studied situations agents will always have at least some subjective probability assessment of the outcomes.

Basic Concepts

Definitions of Lottery and Random Variable:

• _{A “compound” lottery is a lottery over lotteries.} • _{The probabilities may be subjective or objective.}

• _{Cf., for more details, Chavas, chapter 2 and Appendix A.} Definition:

Given the set of possible outcomes S and a probability distribution F (that associates each outcome with a probability), a random variable is a function Z(s) that assigns to each possible outcome ݏ א ܵ either a natural number (discrete RV) or a real number (continuous RV).

• _{A “compound” lottery is a lottery over lotteries.} • _{The probabilities may be subjective or objective}

Definition:

A lottery Z is a collection of outcomes (z₁,…,z_n), measured in the same unit, and associated probabilities ሺ݌_௭భǡ ݌_௭మǡ ǥ ǡ ݌_௭೙ሻ, with Ͳ ൑ ݌_௭೔ ൏ ͳ for all i=1, 2,…, n, which sum up to 1:

Basic Concepts

Concept of Expected Value:

§ The EV corresponds to the arithmetical mean or first moment of a probability distribution. • _{Why is the expected value unsatisfactory as the sole decision criterion?}

Game: A fair coin is flipped until it comes up heads. The number of flips i until it comes up heads for the first time determines the prize, which is equal to € 2i_.

Ø What is the maximum sum you would be willing to pay to play this game?

Ø Expected value: ʹ ή ଵ_ଶ ൅ Ͷ ή ଵ_ସ ൅ ͺ ή ଵ_଼ ൅ ͳ͸ ή _ଵ଺ଵ ൅ ڮ ൌ ͳ ൅ ͳ ൅ ͳ ൅ ͳ ൅ ڮ

!

infinity.

Ø St. Petersburg “Paradox” (Bernoulli 1738):

Definition:

The expected value (or expectation) of a random variable Z is the sum of its outcomes weighted by their probabilities:

ò

å

×

=

×

=

=

dt

t

f

t

Z

E

continuous

p

z

Z

E

discrete

_z n i zi i

;

:

(

)

:

1

Basic Concepts

Concept of Expected Value

(cont.)

:

Resolution of St. Petersburg “Paradox” (Bernoulli 1738):

• _{Instead of focusing on the expected value, an individual may rather refer to the expected utility} (expected level of satisfaction) of the game.

Ø Focus on the (subjective) utility of the (objective) monetary outcomes; that is, focus on: u(2), u(4), u(8),…, u(2n_{),…, instead of: 2, 4, 8,…, 2}n_,…

• _{Natural assumptions on a utility function:}

Ø The higher the outcome, the higher the utility: ࢛ᇱ ൐ ૙ (“non-satiation”)

Ø The higher the outcome, the less an additional outcome unit adds to utility: ࢛ᇱᇱ ൏ ૙. § Examples of increasing, concave (utility) functions: Natural logarithm, Square-root function. • _{Expected utility of the game arises, then, as sum of the utilities of the outcomes, weighted by}

the probabilities of occurrence of the outcomes:

ݑሺʹሻ ή ͳ_{ʹ ൅ ݑሺͶሻ ή}ͳ_{Ͷ ൅ ݑሺͺሻ ή} ͳ_{ͺ ൅ ڮ  ൌ ෍ ݑ ʹ}௜ ή_ʹͳ_௜൏ λ

ஶ ௜ୀଵ

Basic Concepts

Expected Utility Hypothesis and Theorem:

• _{For a risky prospect Z and a utility function u, the expected utility Eu is for}

a discrete random variable: , and a continuous random variable: . • _{E is the expectation operator.}

• _{For the ‘certain conditions’ (utility axioms), cf. Chavas, chapter 3.}

• _{Note: The utility function provides a mathematical representation of risk attitudes.} Expected Utility Hypothesis:

Individuals evaluate risky prospects such as to maximize the expected level of their utility.

Expected Utility Theorem (von Neumann-Morgenstern 1944):

Under certain conditions, for any two prospects Z₁, Z₂ with the same support there exists a utility function u representing an individual’s risk attitude such that the statement

“Z₁ is preferred to Z₂” is equivalent to Eu(Z₁) " Eu(Z₂). ) ( 1 i n i i p z u ×

å

=

ò

×f (t)dt u(t) _Z

Basic Concepts

Risk Attitudes

(Pratt 1964)

:

• _{Note: Any lottery Z with non-zero expected payoff can be decomposed into its expected}

payoff EZ and a zero-mean risk Z – EZ. (Consider: E(Z – EZ) = EZ – E(EZ) = EZ – EZ = 0.)

• _{Risk neutrality is characterized by:}

Risk love is characterized by:

• _{For the proof, consider Jensen’s inequality:}

Ø A function u(Z) is concave if and only if Eu(Z) # u(EZ); it is linear iff Eu(Z) = u(EZ); and it is convex iff Eu(Z) " u(EZ). [Mistake in Chavas, p. 32!]

Definition:

An agent is risk averse if, at any wealth level w, he or she dislikes every lottery Z with an

expected payoff of zero, EZ = 0, so that:

_Eu

₍

_w

+

_Z

₎

£

_u

₍

_w

₎

_.

,

)

(

)

(

w

Z

u

w

Eu

+

=

.

)

(

)

(

w

Z

u

w

Eu

+

³

Proposition:

An agent with utility function u is risk averse (neutral, loving), i.e., for any lottery Z and wealth w, it holds that

_Eu

₍

_w

+

_Z

₎

£

₍

=

_,

³

₎

_u

₍

_w

+

_EZ

, if and only if u is concave (linear, convex).

₎

Basic Concepts

Risk Attitudes

(Pratt 1964) (cont.)

:

Illustration of relation of risk aversion and utility concavity:

• _{For an initial wealth of 8000, consider a 50-50 lottery Z to gain or lose 4000.}

ܧݑ ݓ ൅ ࢆ ൏ ݑ ܧሺݓ ൅ ࢆሻ ֞ ͲǤͷ ή ݑ ͶͲͲͲ ൅ ͲǤͷ ή ݑ ͳʹͲͲͲ ൏ ݑሺͲǤͷ ή ͶͲͲͲ ൅ ͲǤͷ ή ͳʹͲͲͲሻ

Basic Concepts

Risk Attitudes

(Pratt 1964) (cont.)

:

• _{For an initial wealth of 8000, consider a 50-50 lottery Z to gain or lose 4000:}

ܧݑ ݓ ൅ ܼ ൏ ݑ ܧሺݓ ൅ ܼሻ ֞ ͲǤͷ ή ݑ ͶͲͲͲ ൅ ͲǤͷ ή ݑ ͳʹͲͲͲ ൏ ݑሺͲǤͷ ή ͶͲͲͲ ൅ ͲǤͷ ή ͳʹͲͲͲሻ Examples:

Ø Logarithmic utility: ͲǤͷ ή ͺǤʹͻ ൅ ͲǤͷ ή ͻǤ͵ͻ ൌ ͺǤͺͶ ൏ ͺǤͻͻ

Basic Concepts

Strength & Comparison of Risk Attitudes I

(Pratt 1964)

:

Coefficient of Absolute Risk Aversion:

where u‘ > 0 (non-satiation) and u‘‘ < 0 (concavity).

Ø Technically: rate at which marginal utility u’ decreases when wealth is increased by € 1 (i.e., the coefficient is not unit-free, but measured in units of 1/€).

Ø Interpretation: measure of strength of absolute risk aversion, thus, of strength with which an individual seeks to avoid a risk.

Ø Comparison: Agent 1 (with utility u1) is more risk averse than Agent 2 (with utility u2) iff

for some w.

0

)

(

'

)

(

''

_>

-w

u

w

u

)

(

'

)

(

''

)

(

'

)

(

''

2 2 1 1

w

u

w

u

w

u

w

u

_>

-Basic Concepts

Strength & Comparison of Risk Attitudes I

(Pratt 1964) (cont.)

:

Coefficient of Relative Risk Aversion:

where u‘ > 0 (non-satiation) and u‘‘ < 0 (concavity).

Ø Technically: rate at which marginal utility u’ decreases when wealth is increased by 1 percent (unit -free); equal, also, to wealth elasticity of marginal utility.

Ø Interpretation: measure of strength of relative risk aversion, thus, of strength with which an individual seeks to avoid a risk.

Ø Relation of coefficients of relative and absolute risk aversion, R(w) and A(w): ܴ ݓ ൌ െ݀ݑԢሺݓሻȀݑ_݀ݓȀݓᇱ ݓ ൌ െݓ ή ݑ_ݑᇱ ᇱᇱ_ݓ ݓ ൌ ݓ ή ܣሺݓሻ

Ø In theoretical and empirical applications both coefficients, A(w) and R(w), are used.

0

)

(

'

)

(

''

_>

×

-w

u

w

u

w

Basic Concepts

Strength & Comparison of Risk Attitudes II

(Pratt 1964)

:

Risk Premium and Certainty Equivalent:

Formally: For a zero-mean risk Z (i.e., EZ = 0), the risk premium ! derives from

Ø The risk premium ! is positive for a risk averse agent; it is zero for a risk neutral agent; and it is negative for a risk loving agent.

Ø Comparison: Agent 1 (with utility u1) is more risk averse than Agent 2 (with utility u2) iff

Definition:

The risk premium $ is the maximum amount of money an agent is ready to pay in order to get rid of a zero-mean risk.

.

Z

w,

all

for

)

,

,

(

)

,

,

(

w

u

₁

Z

p

w

u

₂

Z

p

³

.

)

,

,

(

)]

(

[

)

(

)

(

w

Z

u

w

w

u

1

Eu

w

Z

w

u

Z

Eu

+

=

-

p

Û

p

=

-

-

+

=

p

Basic Concepts

Strength & Comparison of Risk Attitudes II

(Pratt 1964) (cont.)

:

Illustration of risk premium:

• _{For an initial wealth of 8000, consider a 50-50 lottery Z to gain or lose 4000.}

(Source: EGS’05, p. 11)

Basic Concepts

Strength & Comparison of Risk Attitudes II

(Pratt 1964) (cont.)

:

Risk Premium and Certainty Equivalent:

• _{When EZ differs from zero, usually the concept of the certainty equivalent is used.}

Formally: For an arbitrary risk Z, the certainty equivalent e derives from

Ø The certainty equivalent of a risk Z, with Z = EZ + and E = 0, and its risk premium are related as:

(as can be seen by comparing the utility arguments in the equations defining the concepts). Definition:

The certainty equivalent e of a risk Z is the sure increase in wealth that has the same effect on utility as having to bear risk Z.

)

ˆ

,

,

(

)

,

,

(

w

u

Z

EZ

w

u

Z

e

=

-

p

.

)

,

,

(

)]

(

[

)

(

)

(

w

Z

u

w

e

e

u

1

Eu

w

Z

w

e

w

u

Z

Eu

+

=

+

Û

=

-

+

-

=

Basic Concepts

Criteria for the Shapes of Absolute and Relative Risk Aversion:

• _{The strength of risk attitudes is likely to depend on the level of wealth.}

How does (absolute) risk aversion vary with initial wealth?

• Consider a 50-50 lottery to gain or lose 100.

Ø For Agent 1 with initial wealth w = 101, this lottery is potentially life-threatening; for Agent 2 with initial wealth w = 1 000 000, the lottery is essentially trivial.

Ø The former should be ready to pay more for the elimination of risk than the latter.

Indeed, suppose the two agents have a square-root utility function. Then,

§ Risk premium of Agent 1: ! = 43.4 § Risk premium of Agent 2: ! = 0.0025

Basic Concepts

Criteria for the Shapes of Absolute and Relative Risk Aversion

(cont.)

:

Criterion for the Shape of Absolute Risk Aversion (ARA):

• _{Formally: How does the risk premium change with initial wealth?}

Ø Differentiating: ܧݑ ݓ ൅ ܼ ൌ ݑ ݓ െ ߨ ݓ with respect to w yields:

ܧݑ

ᇱ

ݓ ൅ ܼ ൌ ݑ

ᇱ

ݓ െ ߨ ݓ ή ͳ െ ߨ

ᇱ

ݓ ֞ߨ

ᇱ

ݓ ൌ

௨ᇲ ௪ିగ ௪ ିா௨_{௨ᇱ ௪ିగ ௪} ᇲ ௪ା௓



Ø Thus, the risk premium is decreasing (increasing; constant) in w (!‘(w) # ("; =) 0) if and only if ܧݑԢ ݓ ൅ ܼ  ൒ ൑Ǣ ൌ ݑᇱ _{ݓ െ ߨ ݓ Ǥ}

Substitute v = – u’ : ܧݒ ݓ ൅ ܼ ൑ ൒Ǣ ൌ ݒ ݓ െ ߨ ݓ

Ø Note that v (which is increasing) can be interpreted as another utility function, so that :

ߨ

௩

ݓ ൒ ൑Ǣ ൌ ߨ

௨

ݓ ֞ ܣ

௩

ሺݓሻ ൒ ൑Ǣ ൌ ܣ

௨

ሺݓሻ

Ø Re-substituting – u’ = v : െ௨_௨ᇲᇲᇲ_{ᇲᇲ ௪}௪ ൒ ൑Ǣ ൌ ܣ_௨ ݓ

Basic Concepts

Criteria for the Shapes of Absolute and Relative Risk Aversion

(cont.)

:

Criterion for the Shape of Absolute Risk Aversion (ARA) (cont.)

:

Criterion for the Shape of Relative Risk Aversion (RRA):

A similar criterion can be derived for relative risk aversion:

Proposition:

Absolute risk aversion is decreasing (increasing; constant) in wealth if and only if the coefficient of absolute prudence is uniformly larger (smaller than; equal to) absolute risk aversion:

െݑ_ݑᇱᇱᇱ_{ᇱᇱ ݓ ൒ ൑Ǣ ൌ ܣ}ݓ ௨ ݓ Ǥ

Proposition:

Relative risk aversion is decreasing (increasing; constant) in wealth if and only if the coefficient of relative prudence is uniformly larger (smaller than; equal to) relative risk aversion plus one:

Basic Concepts

Exercises:

1. Determine the coefficients of absolute and relative risk aversion for the following classical utility functions. Do they have a constant, increasing or decreasing shape? (Suppose w>0.)

a) Square-root function : b) Natural logarithm : c) Exponential utility (% > 0): d) Power utility :

w

w

u

(

)

=

)

ln(

)

(

w

w

u

=

w

e

w

u

(

)

=

-

-a×

ï

î

ï

í

ì

=

¹

³

-=

-1

for

)

ln(

1

0,

for

1

)

(

1

g

g

g

g

g

w

w

w

u

Basic Concepts

Exercises:

2. An individual with square-root utility ݑ ݓ ൌ ݓ and initial wealth w = 10 faces the lottery ܼ ൌ ሺെ͸ǡ ͲǤͷǢ ͸ǡ ͲǤͷሻ.

• _{Compute this individual’s risk premium and certainty equivalent for this risk.}

Basic Concepts

Exercise:

3. Show that, for a utility function u and initial wealth w, relative risk aversion is decreasing (ܴԢሺݓሻ ൑ Ͳ) if and only if relative prudence is larger than relative risk aversion plus one: 

Lecture 2:

Prudence as a Higher-Order

Preference; Comparison of Risks

Prudence

Unit 1B:

• Prudence in Temporal Decisions I:

Ø The two-period consumption-saving model (under certainty)

Ø Precautionary saving in the two-period consumption-saving model Ø Behavioral definition and theory of “prudence” (Kimball 1990)

• Comparison of Risks:

Ø Moments of a distribution (e.g., mean, variance, skewness, kurtosis) Ø Increases in risk:

§ “mean-preserving spread” = same mean, increased variance

§ “increase in downside risk” = same mean and variance, but more skewness • Prudence as a Higher-Order Risk Attitude:

Ø Prudence as third-order risk apportionment

Literature

Prudence in Temporal Decisions I:

• Main:

§ Eeckhoudt, Louis, Gollier, Christian and Harris Schlesinger: Economic and Financial

Decisions under Risk. Princeton University Press, 2005, subsections 6.2-6.3

• Background:

§ Kimball, Miles: Precautionary Saving in the Small and in the Large. Econometrica, 58(1): 53-73, 1990

§ Eeckhoudt, L. and H. Schlesinger: Higher-Order Risk Attitudes. Draft, 2012, sections 1-5 § Eeckhoudt, Louis and Harris Schlesinger: Putting Risk in its Proper Place. American

Economic Review, 96(1): 280-289, 2006

Prudence

Two-Period Consumption-Saving Model: Motivation

• Intertemporal consumption decisions are important in various areas:

§ Economic growth: saving is a major determinant of the investment level in every period. § Consumption-saving behavior over the life-cycle is studied, e.g., regarding the impact of

income taxes, human-capital investments, social security including health and retirement provisions.

§ Determination of social discount rate (cf. Lectures 6&7).

• Link to two-period consumption-saving model:

Ø Such studies use multi-period or infinite-horizon models of consumption-saving behavior. Ø The two-period consumption-saving model captures much of the intuition that underlies

Prudence

Two-Period Consumption-Saving Model under Certainty:

Utility objective:

where u is instantaneous utility in period t = 0, 1

c₀ is consumption in period 0, c₁ is consumption in period 1 ߚ ؠ _ଵାఋଵ is the utility discount factor, ! is the utility discount rate

)

(

)

(

)

,

(

c

₀

c

₁

u

c

₀

u

c

₁

U

=

+

b

×

Prudence

Two-Period Consumption-Saving Model under Certainty:

where u is instantaneous utility in period t = 0, 1

c₀ is consumption in period 0, c₁ is consumption in period 1 ߚ ؠ _ଵାఋଵ is the utility discount factor, ! is the utility discount rate y₀ is the income in period 0, y₁ is the income in period 1 s₀ is saving in period 0

r is the interest rate (from period 0 to period 1) on the efficient credit market 1 0 1 0 0 0 1 0 1 0 s

r)

(1

s

y

,

s

y

)

(

)

(

)

,

(

max

such that

0

c

c

c

u

c

u

c

c

U

=

+

+

+

=

×

+

=

b

Prudence

Two-Period Consumption-Saving Model under Certainty

(cont.)

:

Equivalently, this model can be written as:

where u is instantaneous utility in period t = 0, 1

" is the utility discount factor

y₀ is the income in period 0, y₁ is the income in period 1 s₀ is saving in period 0

r is the interest rate (from period 0 to period 1) on the efficient credit market

First-Order Condition (FOC):

)

r)

(1

s

y

(

)

(

)

(

max

_{0 0 0 1 0} s₀

U

s

=

u

y

-

s

+

b

×

u

+

+

)

r)

(1

s

y

(

'

)

1

(

)

(

'

0

)

r)

(1

s

y

(

'

)

1

(

)

(

'

0 1 0 0 0 1 0 0

+

+

×

+

×

=

-Û

=

+

+

×

+

×

+

-u

r

s

y

u

u

r

s

y

u

b

b

Prudence

Two-Period Consumption-Saving Model under Certainty

(cont.)

:

First-Order Condition (FOC):

Ø Thus, from FOC :

ܷ

ᇱ

ݏ

_଴

ൌ Ͳ ֜

optimal saving ݏ_଴כ. Ø Interpretation of FOC:

• In equilibrium, the marginal utility from foregoing consumption today has to equal the discounted marginal utility from consuming instead in period 1.

• Intrapersonal ‘saving market’: the left-hand side represents the supply of saving, and the

right-hand side the demand for saving.

)

r)

(1

s

y

(

)

(

)

(

max

_{0 0 0 1 0} s₀

U

s

=

u

y

-

s

+

b

×

u

+

+

)

r)

(1

s

y

(

'

)

1

(

)

(

'

y

₀

-

s

₀

=

×

+

r

×

u

₁

+

₀

+

u

b

Prudence

Two-Period Consumption-Saving Model under Certainty

(cont.)

:

First-Order Condition (FOC):

Second-Order Condition (SOC):

Ø Thus, from FOC :

ܷ

ᇱ

ݏ

_଴

ൌ Ͳ ֜

optimal saving ݏ_଴כ. Ø From SOC : (from utility concavity).

)

r)

(1

s

y

(

)

(

)

(

max

_{0 0 0 1 0} s₀

U

s

=

u

y

-

s

+

b

×

u

+

+

0

)

r)

(1

s

y

(

''

)

1

(

)

(

''

y

₀

-

s

₀

+

×

+

r

2

×

u

₁

+

₀

+

<

u

b

0

)

(

''

s

₀

<

U

)

r)

(1

s

y

(

'

)

1

(

)

(

'

y

₀

-

s

₀

=

×

+

r

×

u

₁

+

₀

+

u

b

Prudence

Two-Period Consumption-Saving Model under Certainty

(cont.)

:

Interpretation of Utility Concavity in Intertemporal Choice under Certainty:

• Consider the model for a utility discount rate " = 1 and a net interest rate r = 0. • Assume that the incomes in periods 0 and 1 are the same, y0 = y1= #.

Then, the FOC reads:

Ø Utility concavity induces consumption smoothing !

Ø In the standard model of intertemporal choice, the coefficient െ௨_௨ᇲᇲ_ᇲ ൐ Ͳ measures the strength of the consumption-smoothing preference, or, put differently, the strength of the

Resistance to Intertemporal Substitution (RIS).

.

0

)

(

'

)

(

'

* 1 * 0 * 0 0 0

c

c

s

s

y

u

s

y

u

=

Û

=

Þ

+

=

-)

r)

(1

s

y

(

)

(

)

(

max

_{0 0 0 1 0} s₀

U

s

=

u

y

-

s

+

b

×

u

+

+

Prudence

Two-Period Consumption-Saving Model under Certainty

(cont.)

:

Interpretation of Utility Concavity in Intertemporal Choice under Certainty:

• The most common measure of consumption-smoothing preference (in macroeconomics and finance) is the Elasticity of Intertemporal Substitution (EIS):

Ø Measure of the willingness to allow for intertemporal substitution of consumption.

• Under univariate, additive utility, the Relative Resistance to Intertemporal Substitution (RRIS) turns out to be the inverse of the Elasticity of Intertemporal Substitution: ܴܴܫܵ ൌ ͳȀܧܫܵ . Ø In this model, the coefficient of RRIS is formally identical to coefficient of relative risk aversion

in the case with risk under Expected Utility.

Ø Consumption-smoothing and risk preferences cannot be distinguished in this model !

)

r)

(1

s

y

(

)

(

)

(

max

_{0 0 0 1 0} s₀

U

s

=

u

y

-

s

+

b

×

u

+

+

) y fy f pp ( ) Definition:

Elasticity of intertemporal substitution is the elasticity of the ratio of future over current

Prudence

Precautionary Saving in the Two-Period Consumption-Saving Model:

For example, add risk on period-1 income:

where u is instantaneous utility in period t = 0, 1

E is the expectation operator

y₀ is the income in period 0, Y₁ is the risky income in period 1

s₀ is saving in period 0

r is the interest rate (from period 0 to period 1) on the efficient credit market

)

r)

(1

s

(

)

(

)

(

max

_{0 0 0 1 0} s₀

V

s

=

u

y

-

s

+

b

×

E

u

Y

+

+

Definition:

Precautionary saving is the saving which arises in a consumption-saving model in response to the

Prudence

Precautionary Saving in the Two-Period Consumption-Saving Model:

Model with Risk on Period-1 Income:

where u is instantaneous utility in period t = 0, 1

E is the expectation operator

y₀ is the income in period 0, Y₁ is the risky income in period 1

s₀ is saving in period 0

r is the interest rate (from period 0 to period 1) on the efficient credit market

First-Order Condition (FOC):

Ø In equilibrium, the marginal utility from foregoing consumption today has to equal the discounted expected marginal utility from consuming instead in period 1.

)

r)

(1

s

(

)

(

)

(

max

_{0 0 0 1 0} s₀

V

s

=

u

y

-

s

+

b

×

E

u

Y

+

+

)

r)

(1

s

(

'

)

1

(

)

(

'

y

₀

-

s

₀

=

×

+

r

×

Eu

Y

₁

+

₀

+

u

b

Prudence

Precautionary Saving in the Two-Period Consumption-Saving Model

(cont.)

:

Is Saving under Income Risk Different from Saving under Certainty?

Ø Notation: ݏ_଴ǡ௖כ optimal saving under certainty, ݏ_{଴ǡ௥௜௦௞}כ optimal saving under income risk. Thus, precautionary saving: ݏ_{଴ǡ௥௜௦௞}כ െݏ_଴ǡ௖כ .

Ø Note: because ܷᇱ ݏ_଴ǡ௖כ ൌ Ͳ and ܷᇱᇱ ݏ_଴ ൏ Ͳ , it holds for the first derivative of the model under certainty evaluated at ݏ_{଴ǡ௥௜௦௞}כ that:

Ø Consider the model under certainty for ݕ_ଵ ൌ ܧࢅ_ଵ:

Ø Strategy: Search for a criterion for which the first derivative of the above problem with ݕଵ ൌ ܧࢅଵ, evaluated at ݏ଴ǡ௥௜௦௞כ , is negative (zero, positive) !

* , 0 * , 0 * , 0

)

(

,

)

0

(

,

)

(

'

s

_risk

s

_risk

s

_c

U

<

=

>

Û

>

=

<

)

r)

(1

s

(

)

(

)

(

max

_{0 0 0 1 0} s₀

U

s

=

u

y

-

s

+

b

×

u

EY

+

+

Prudence

Precautionary Saving in the Two-Period Consumption-Saving Model

(cont.)

:

Is Saving under Income Risk Different from Saving under Certainty?

• Strategy: Search for a criterion for which the first derivative of the model under certainty with

ݕଵ ൌ ܧࢅଵ, evaluated at ݏ଴ǡ௥௜௦௞כ , is negative (zero, positive):

Ø Consider the model under certainty for ݕ_ଵ ൌ ܧࢅ_ଵ:

Ø Evaluate the first derivative of this problem at optimal saving under income risk: ܷᇱ ݏ_{଴ǡ௥௜௦௞}כ ൌ െݑᇱ ݕ଴ െ ݏ଴ǡ௥௜௦௞כ ൅ ߚ ͳ ൅ ݎ ݑᇱሺܧࢅଵ ൅ ݏ଴ǡ௥௜௦௞כ ሺͳ ൅ ݎሻሻ

Ø Substitute using FOC under income risk ݑᇱ ݕ_଴ െ ݏ_଴ ൌ ߚ ͳ ൅ ݎ ܧݑԢሺࢅ_ଵ ൅ ݏ_଴ሺͳ ൅ ݎሻሻ as: ܷᇱ ݏ_{଴ǡ௥௜௦௞}כ ൌ ߚ ͳ ൅ ݎ ሾݑᇱ ܧࢅ_ଵ ൅ ݏ_{଴ǡ௥௜௦௞}כ ͳ ൅ ݎ െ ܧݑԢሺࢅ_ଵ ൅ ݏ_{଴ǡ௥௜௦௞}כ ሺͳ ൅ ݎሻሻሿ

)

r)

(1

s

(

)

(

)

(

max

_{0 0 0 1 0} s₀

U

s

=

u

y

-

s

+

b

×

u

EY

+

+

Prudence

Precautionary Saving in the Two-Period Consumption-Saving Model

(cont.)

:

Is Saving under Income Risk Different from Saving under Certainty?

§ Remember:

and .

Ø Note:

Substitute v = –u ’ : Ø Recall:

Ø Hence:

v = –u’ is concave, thus: u’ is convex, so that u’’’ $ 0 .

Ø u’’’ ! 0 is a necessary and sufficient condition for saving under income risk to exceed saving

under certainty, and, thus, for precautionary saving to arise!

)

r)

(1

s

(

))

1

(

(

)

r)

(1

s

(

'

))

1

(

(

'

0

)

(

'

* , 0 1 * , 0 1 * , 0 1 * , 0 1 * , 0

+

+

³

+

+

Û

+

+

£

+

+

Û

£

risk risk risk risk risk

Y

Ev

r

s

EY

v

Y

Eu

r

s

EY

u

s

U

Proposition:

An agent with utility function u is risk averse, that is, for any lottery Z and any wealth level w, it holds that ܧݑሺݓ ൅ ܼሻ ൑ ݑሺݓ ൅ ܧܼሻ , if and only if u is concave.

Û

³

₀*_, * , 0 risk

s

c

s

)]

r)

(1

s

(

'

))

1

(

(

'

[

)

1

(

)

(

'

s

₀*_,risk

=

+

r

×

u

EY

₁

+

s

₀*_,risk

+

r

-

Eu

Y

₁

+

*_0,risk

+

U

b

* , 0 * , 0 * , 0

)

(

,

)

0

(

,

)

(

'

s

_risk

s

_risk

s

_c

U

<

=

>

Û

>

=

<

Prudence

Behavioral Definition and Theory of “Prudence”

(Kimball 1990)

:

• The positive 3rd _{derivative, u’’’>0, is associated with the notion of “prudence”:}

Ø In the consumption-saving model, u’’’>0 is necessary and sufficient for precautionary saving.

• Coefficient of Absolute Prudence

(analogous to coefficient of absolute risk aversion):

Ø Measure of the intensity of the reaction of an optimal choice to risk.

Ø In consumption-saving model: measure of the intensity of the precautionary saving motive.

0

''

''

'

>

-u

u

Definition:

An agent is prudent if, at any wealth level w, adding an uninsurable zero-mean risk Z (with EZ = 0) to his or her wealth raises the agent’s optimal saving, due to: ܧݑԢሺݓ ൅ ܼሻ ൒ ݑԢሺݓሻ.

Prudence

Behavioral Definition and Theory of “Prudence”

(Kimball 1990) (cont.)

:

• Precautionary Premium:

Formally: For a zero-mean risk Z (i.e., EZ = 0), the precautionary premium ! derives from

Ø The precautionary premium ! is positive for a prudent agent, and is zero or negative for an

imprudent agent.

Ø Comparison: Agent 1 (with utility u1) is more prudent than Agent 2 (with utility u2) iff

Ø Comparing the precautionary premium ! to the risk premium ", it holds that: ɗ ݓǡ ݑǡ ܼ ൌ ߨሺݓǡ െݑᇱǡ ܼሻ

Definition:

The precautionary premium % is the sure reduction in wealth w that has the same effect on the optimal decision variable as the addition of the zero-mean risk Z.

.

)

,

,

(

)]

(

'

[

'

)

(

'

)

(

'

w

Z

u

w

w

u

1

Eu

w

Z

w

u

Z

Eu

+

=

-

y

Û

y

=

-

-

+

=

y

.

Z

w,

all

for

)

,

,

(

)

,

,

(

w

u

₁

Z

y

w

u

₂

Z

y

³

Prudence

Behavioral Definition and Theory of “Prudence”

(Kimball 1990) (cont.)

:

• Precautionary Equivalent:

Formally: For an arbitrary risk Z, the precautionary equivalent e_p derives from

Ø The precautionary equivalent of a risk Z, with Z = EZ + " and E" = 0, and its precautionary

premium are related as:

(as can be seen by comparing the utility arguments in the equations defining the concepts). Definition:

The precautionary equivalent e_p is the sure increase in wealth w that has the same effect on the optimal decision variable as having to bear risk Z.

.

)

,

,

(

)]

(

'

[

'

)

(

'

)

(

'

w

Z

u

w

e

e

u

1

Eu

w

Z

w

e

w

u

Z

Eu

+

=

+

_p

Û

_p

=

-

+

-

=

_p

)

ˆ

,

,

(

)

,

,

(

w

u

Z

EZ

w

u

Z

e

_p

=

-

y

Comparison of Risks

Unit 1B:

• Prudence in Temporal Decisions I:

Ø The two-period consumption-saving model (under certainty)

Ø Precautionary saving in the two-period consumption-saving model Ø Behavioral definition and theory of “prudence” (Kimball 1990)

• Comparison of Risks:

Ø Moments of a distribution (e.g., mean, variance, skewness, kurtosis)

Ø Increases in risk:

§ “mean-preserving spread” = same mean, increased variance

§ “increase in downside risk” = same mean and variance, but more skewness • Prudence as a Higher-Order Risk Attitude:

Ø Prudence as third-order risk apportionment

Literature

Comparison of Risks:

• Background:

§ Rothschild, Michael and Joseph Stiglitz: Increasing Risk I: A Definition. Journal of

Economic Theory, 2(3): 225-243, 1970

§ Menezes, Carmen, Geiss, Charles and John Tressler: Increasing Downside Risk. American

Economic Review, 70(5): 921-932, 1980

Comparison of Risks

Increases in Risk: General Remarks

Ø Risks can be compared by comparing characteristics of the associated distribution functions, for example, their moments or their stochastic dominance relationship.

• Moments of a distribution:

Ø Specific moments are:

§ Mean (1st _{raw moment):} Ɋ ؠ m

1 ൌ ܧܺ ൌ ׬ ݔ݂ ݔ ݀ݔ ஶ

ିஶ

§ Variance (2nd _{central moment):} ɐଶ ؠ Var ܺ ൌ ܧ ܺ െ ߤ ʹ_{, where & standard deviation}

§ Skewness (3rd _{standardized moment): Skew} ܺ ൌ ܧ ௑ିఓ

ఙ ͵

§ (Excess) kurtosis (4th _{standardized moment minus 3): Kurt} ܺ ൌ ܧ ௑ିఓ

ఙ Ͷ െ ͵

Ø Specific moments are:

§ Mean (1st _{raw moment):} Ɋ ؠ m

1 ൌ ܧܺ ൌ ׬ ݔ ݂ ݔ ݀ݔ

ஶ ି

׬׬ _ஶ

§ Variance (2nd_{central moment):} ɐଶ ؠ Var ܺ ൌ ܧ ܺ െ ߤ ʹ_{, where & standard d}

§ Skewness (3rd _{standardized moment):}

for example, their moments or their stochastic dominance relationship. • Moments of a distribution:

Skew ܺ ൌ ܧ ௑ିఓ ͵

Definition:

The kth _{(raw) moment m}

k of a (continuous) distribution F with density f is defined as: 

m_k ؠ ܧܺ݇ ൌ න ݔ݇ ή ݂ ݔ ݀ݔ

ஶ ିஶ

Comparison of Risks

Increases in Risk: Mean-Preserving Spreads – Definition

• Consider two cumulative distribution functions (CDF), F and G, with common bounded support in the open interval (a,b), thus: F(a) = G(a) = 0, F(b) = G(b) = 1.

• Mean-preserving spread (Rothschild-Stiglitz 1970):

Ø Note: ׬ ܩ ݕ ݀ݕ_௔௫ ൒ ׬ ܨ ݕ ݀ݕ_௔௫ for all a ' x ' b implies ߪீ ൒ ߪி. Ø Basic notion of a risk increase in economics.

Illustration:

(Source: MacMinn 1997)

• Mean-preserving spreadd (Rothschild-Stiglitz 1970):

Ø Note: ׬ ܩ ݕ ݀ݕ׬׬_௔௫ ൒ ׬ ܨ ݕ ݀ݕ׬׬_௔௫ for all a ' x ' b implies ߪீ ൒ ߪி. Ø B i i f i k i i i

Definition:

CDF G is a mean-preserving spread of CDF F if ݉_ଵீ ൌ ݉_ଵி and ׬ ܩ ݕ ݀ݕ௫

௔ ൒ ׬ ܨ ݕ ݀ݕ

௫

௔ for all

Comparison of Risks

Increases in Risk: Mean-Preserving Spreads – Example

• Be w, k₁, k₂ positive constants, defined so that all outcomes are strictly positive. Consider the following 50-50 lotteries, A₂ and B₂:

A₂ : w – k₁ – k₂ B₂: w – k₁ w w – k₂ § Means: ܧ ܣ_ଶ ൌ ͲǤͷ ݓ െ ݇_ଵ െ ݇_ଶ ൅ ͲǤͷݓ ൌ ݓ െ ͲǤͷሺ݇_ଵ ൅ ݇_ଶሻ ܧ ܤଶ ൌ ͲǤͷ ݓ െ ݇ଵ ൅ ͲǤͷሺݓ െ ݇ଶሻ ൌ ݓ െ ͲǤͷሺ݇ଵ ൅ ݇ଶሻ § Variances: ܸܽݎ ܣ_ଶ ൌ ͲǤʹͷሺ݇_ଵ ൅ ݇_ଶሻଶ, ܸܽݎ ܤ_ଶ ൌ ͲǤʹͷሺ݇_ଵ െ ݇_ଶሻଶ Ø ܧ ܣ_ଶ ൌ ܧ ܤ_ଶ and ܸܽݎ ܣ_ଶ ൐ ܸܽݎ ܤ_ଶ .

Comparison of Risks

Increases in Risk: Increases in Downside Risk – Definition

• Consider two cumulative distribution functions (CDF), F and G, with common bounded support in the open interval (a,b), thus: F(a) = G(a) = 0, F(b) = G(b) = 1.

• Increase in downside risk (Menezes-Geiss-Tressler 1980):

Ø Note: ׬ ׬ ܩ ݕ_௔௫ _௔௕ ሺ݀ݕሻଶ ൒ ׬ ׬ ܨ ݕ_௔௫ _௔௕ ሺ݀ݕሻଶ for all a ' x ' b implies ݉_ଷீ ൏ ݉_ଷி, or that G is

more skewed to the left than F.

Illustration of a left-skewed density function:

(Source: Abbe Lefkowitz on maassmedia.com, 2014)

Ø Note: ׬׬׬ ׬ ܩ ݕ_௔௫׬׬_௔௕ ሺ݀ݕሻଶ ൒ ׬ ׬ ܨ ݕ׬׬_௔௫ ׬׬_௔௕ ሺ݀ݕሻଶ for all a ' x ' b implies ݉_ଷீ ൏ ݉_ଷி, or that G is

more skewed to the left than F.

Definition:

CDF G has more downside risk than CDF F if ݉_ଵீ ൌ ݉_ଵி, ɐீ ൌ ɐிand, for all a ' x ' b, ׬ ׬ ܩ ݕ_௔௫ _௔௕ ሺ݀ݕሻଶ ൒ ׬ ׬ ܨ ݕ_௔௫ _௔௕ ሺ݀ݕሻଶ.

Comparison of Risks

Increases in Risk: Increases in Downside Risk – Example

• For positive constants w and k, a zero-mean random variable ߝǁ, and w – k – * > 0 for all realizations of ߝǁ, consider the following 50-50 lotteries, A₃ and B₃:

A₃: w – k + ߝǁ B₃: w – k

w w + ߝǁ

• To verify the moment conditions, consider the full compound lotteries, with ߝǁ as a 50-50 lottery with outcomes c or – c (for c > 0):

c w – k + c A₃: w – k _{à Reduced A3}: w – k – c – c w w w – k w – k B₃: c _{à Reduced B3}: w + c w w – c 1/2 1/4 1/4 1/2 1/4 1/4

Comparison of Risks

Increases in Risk: Increases in Downside Risk – Example

w – k + c w – k Reduced A₃: w – k – c Reduced B₃: w + c w w – c § Means: ܧ ܣ_ଷ ൌ ݓ െ ଵ_ଶ݇, ܧ ܤ_ଷ ൌ ݓ െ ଵ_ଶ݇ ֜ ܧ ܣ_ଷ ൌ ܧ ܤ_ଷ § Variances: ܸܽݎ ܣ_ଷ ൌ ௞_଼మ ൅ _ସଵሾ ܿ ൅ ௞_ଶ ଶ ൅ ܿ െ ௞_ଶ ଶሿ, ܸܽݎ ܤ_ଷ ൌ ௞_଼మ ൅ ଵ_ସሾ ܿ ൅ ௞_ଶ ଶ ൅ ܿ െ ௞_ଶ ଶሿ ֜ ܸܽݎ ܣଷ ൌ ܸܽݎ ܤଷ § 3rd _moments: ݉ ଷ஺య ൌ ܧሺܣଷሻଷൌ ݓଷ െ ଷ_ଶݓଶ݇ െ ௞ య ଶ ൅ ଷ ଶݓሺ݇ଶ ൅ ܿଶሻ െ ଷ ଶ݇ܿଶ ݉_ଷ஻య ൌ ܧሺܤ_ଷሻଷൌ ݓଷ െ ଷ ଶݓଶ݇ െ ௞య ଶ ൅ ଷ ଶݓ ݇ଶ ൅ ܿଶ ֜ ݉_ଷ஺య _{െ ݉} ଷ ஻_{య ൌ െଷ} ଶ݇ܿଶ ൏ Ͳ ֜݉ଷ஺య ൏ ݉ଷ஻య

Ø Lottery A3 has a more downside risk than lottery B3.

1/2 1/4 1/4 1/2 1/4 1/4

Literature

Prudence as a Higher-Order Risk Attitude:

• Main:

§ Eeckhoudt, L. and H. Schlesinger: Higher-Order Risk Attitudes. Draft, 2012, sections 1-5

• Background:

§ Eeckhoudt, Louis and Harris Schlesinger: Putting Risk in its Proper Place. American

Economic Review, 96(1): 280-289, 2006

§ Friedman, Milton and L.J. Savage: The Utility Analysis of Choices Involving Risk. Journal of

Prudence

Prudence as a Higher-Order Risk Attitude:

• The above definitions of risk increases as well as the illustrating binary lotteries are related to definitions of risk aversion and prudence as static risk preferences.

(I) Prudence as Downside Risk Aversion:

§ Aversion to mean-preserving spreads is equivalent to all agents being risk averse (u’’ < 0)

(Rothschild-Stiglitz 1970).

§ Aversion to increases in downside risk (“downside risk aversion”) is equivalent to all agents being prudent (u’’’ > 0) (Menezes-Geiss-Tressler 1980).

(II) Prudence as 3rd_{-Order Risk Apportionment (Eeckhoudt-Schlesinger 2006):}

§ Preference for the disaggregation of harms within binary lotteries

§ Preference for a particular location of a zero-mean risk between two binary lotteries can be shown to correspond to u’’’ > 0 under Expected Utility.

Prudence

Prudence as a Higher-Order Risk Attitude: 3

rd

_{-Order Risk Apportionment}

Risk Apportionment Preferences (Eeckhoudt-Schlesinger 2006, 2012): General Remarks

• A combination of sure losses or (independent) zero-mean risks, if added to one branch of a

binary 50-50 lottery, can be seen as “mutually aggravating”.

Ø Risk apportionment is a preference for disaggregating harms (within binary 50-50 lotteries). § Example: 2nd_{-order risk apportionment}

• Assume an agent has wealth w > 0 and prefers more to less wealth. Let k₁ > 0 and k₂ > 0 be positive constants. Moreover, all variables be defined so as to maintain strictly positive wealth.

A₂ : w – k₁ – k₂ B₂: w – k₁

w w – k₂

Ø Where would a risk-averter prefer to locate –k2 (i.e., which lottery preferred)? Why?

Prudence

Prudence as a Higher-Order Risk Attitude: 3

rd

_{-Order Risk Apportionment}

_(cont.)

Risk Apportionment Preferences (Eeckhoudt-Schlesinger 2006, 2012): General Remarks

§ Example: 3rd_{-order risk apportionment}

• Assume an agent has wealth w > 0 and prefers more to less wealth. Let k > 0 be positive constants and ߝǁ be a zero-mean random variable. Be w – k – * > 0 for all realizations of ߝǁ. Consider the following two binary lotteries, A₃, B₃, with equally probable branches:

A₃: w – k + ߝǁ B₃: w – k w w + ߝǁ

Ø Where would a prudent agent prefer to locate ߝǁ (i.e., which lottery preferred)? Why?

Answer: Lottery B₃, because sure loss and zero-mean risk are distributed between the branches. Moreover, lottery A₃ has more downside risk over lottery B₃.

Prudence

Prudence as a Higher-Order Risk Attitude: 3

rd

_{-Order Risk Apportionment}

_(cont.)

How Does Risk Apportionment Relate to Expected Utility?

• Consider the utility premium of a risk (Friedman-Savage 1948):

Ø The utility premium measures the “pain”, in terms of utility, involved in adding a risk.

§ For a risk averter, utility concavity (u’’ < 0) and Jensen’s inequality imply: ܷܲ ݓ ؠ ܧݑ ݓ ൅ ߝǁ െ ݑ ݓ ൏ ͲǤ

Definition:

The utility premium UP(.) of a zero-mean risk ߝǁ is the difference between the expected utility of initial wealth w plus this risk and the utility of initial wealth w: ܷܲ ݓ ؠ ܧݑ ݓ ൅ ߝǁ െ ݑ ݓ .

Prudence

Prudence as a Higher-Order Risk Attitude: 3

rd

_{-Order Risk Apportionment}

_(cont.)

Risk Apportionment Preferences and Expected Utility (Eeckhoudt-Schlesinger 2006, 2012)

Comparison with risk premium.. (cf also following figure)

Prudence

Prudence as a Higher-Order Risk Attitude: 3

rd

_{-Order Risk Apportionment}

_(cont.)

Risk Apportionment Preferences and Expected Utility (Eeckhoudt-Schlesinger 2006, 2012)

§ Compare the two lotteries A3 and B3 based on the involved utility premia:

A₃: w – k + ߝǁ B₃: w – k

w w + ߝǁ

Ø The utility premium compares expected utility and utility at a certain wealth level for a zero-mean risk added: ܷܲ ݓ െ ݇ ൌ ܧݑ ݓ െ ݇ ൅ ߝǁ െ ݑ ݓ െ ݇ , ܷܲ ݓ ൌ ܧݑ ݓ ൅ ߝǁ െ ݑ ݓ Ø Intuitively, the utility premium should decrease absolutely with initial wealth wǣܷܲᇱ ݓ ൐ Ͳ Ø With a similar reasoning as for ܷܲ ݓ ൏ Ͳ under risk aversion, it follows that, if marginal

utility is convex, u’’’ > 0 : ܷܲᇱ ݓ ൌ ܧݑᇱ ݓ ൅ ߝǁ െ ݑᇱ ݓ ൐ Ͳ.

Ø Thus, for a (absolutely) decreasing utility premium (ܷܲԢ ݓ ൐ Ͳ), it holds for all k > 0: ܧݑ ݓ െ ݇ ൅ ߝǁ െ ݑ ݓ െ ݇ ൏ ܧݑ ݓ ൅ ߝǁ െ ݑ ݓ Ǥ

Prudence

Conclusion:

Prudence, as associated with convex marginal utility (u’’’ > 0), can be defined

§ behaviorally, in consumption-saving model: if, at any wealth level w, adding an uninsurable zero-mean risk Z (with EZ = 0) to his or her wealth raises the agents optimal saving.

In static contexts:

§ as 3rd_{-order risk apportionment: between two 50-50 lotteries the one where the sure loss k > 0}

and the zero-mean risk are disaggregated between the branches is preferred (B₃ over A₃) A₃ : w – k +ߝǁ B₃: w – k

w w – ߝǁ

§ as downside risk aversion: CDF G has more downside risk than CDF F if and only if all prudent agents prefer F to G.

Ø Similar relationships hold for risk aversion, as associated with utility concavity (u’’ < 0), 2nd

Notes on Test 1

You should..

• know the basic definitions,

• be able to explain what the coefficients of risk aversion and prudence measure, • master the exercises.

Lectures 3:

Measurement of Risk and Time

Preferences

Literature

Measurement of Risk Attitudes:

Main:

§ Holt, Charles A. and Susan K. Laury: Risk Aversion and Incentive Effects. American

Measurement of Risk and Time Preferences

Overview:

• Risk-Preference Elicitation à la Holt-Laury (2002):

Ø Methodological approach

Ø Results: incentive and stakes effects, shapes of absolute and relative risk aversion, predictions under expo-power utility

Ø Caveats on experimental risk-preference estimates à la Holt-Laury • Eliciting Risk and Time Preferences:

Ø Approaches using survey data (Barsky et al. 1997)

Ø Experiments on higher-order risk preferences (Noussair et al. 2015) Ø Experiments combining the risk and time dimensions I:

§ Joint elicitation of risk aversion and utility discount rate (Andersen et al. 2008)

Measurement of Risk and Time Preferences

Motivation:

Measurement of Risk Preferences

Holt-Laury (2002): Methodological Approach

(cont.)