Epargne de précaution et statique comparative avec aversion pour le risque

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Risk Aversion in intertemporal contexts Comparative Risk Aversion Application to choice under uncertainty Summary

Comparative Risk Aversion:

A Formal Approach with Applications to

Saving Behaviors

A. Bommier1, A. Chassagnon2 F. Le Grand3

1_{GREMAQ, Université Toulouse I and TSE,}2_{LEDa-SDFi, Université} Paris-Dauphine and PSE3_{EMLyon Business School}

Séminaire "Economie du Risque", 13 octobre 2009

Chaire de la Fondation du Risque, Dauphine-ENSAE-Groupama: «Les Particuliers Face aux Risques»

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Risk Aversion in intertemporal contexts Comparative Risk Aversion Application to choice under uncertainty Summary

Outline of the talk

1 Risk Aversion in intertemporal contexts

Are there good reasons to focus on risk aversion ? Disentangling risk aversion and intertemporal substitution Comparative statics: certainty equivalents Vs dispersion criteria

2 Comparative Risk Aversion

A simple model

Heads or tails gambles

Comparative Risk Aversion for popular classes of preferences

3 _{Application to choice under uncertainty}

Model free set-up Model free result

Three applications to saving under uncertainty Continuous distributions

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Risk Aversion in intertemporal contexts

Comparative Risk Aversion Application to choice under uncertainty Summary

Robustness of the impact of risk aversion

The idea is that conclusions about the impact of risk aversion radically depend on the setup which is chosen. This is

particularly true, for example, when discussing the relation between risk aversion and precautionary savings.

In a simple two-period model,

working with Kihlstrom and Mirman (1974) ’s preferences or with Quiggin (1982)’s preferences leads to conclude that precautionary savings increase with risk aversion (also, Drèze and Modigliani (1972), Yaari (1987), Bleichrodt and Eeckhoudt, (2005)).

On the contrary the relation is ambiguous when using Epstein and Zin’s (1989) preferences (also Kimball and Weil (2009))

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Risk aversion Vs other indicators

Sometime, ambiguous results have been interpreted as the indication that other indicators matters. In Kimball (1990) precautionary saving relates to a well identified aspect of preferences other than risk aversion. . . and gave birth to other notions like prudence

Gollier (2001, p236):

Definition (Kimball, 1990): An agent is prudent if adding an insurable zero mean risk to his future wealth raises his optimal saving

Gollier (2001, p237)

Local prudence and local risk aversion are independent concepts in the sense that, at least locally, one could be risk averse and prudent, risk averse and imprudent, risk lover and prudent, or, risk-lover and imprudent.

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Risk aversion in multidimensional models

Risk aversion definition is not immediate when uncertainty is over multi-dimensional profiles. In particular, hard to separate “risk aversion” from the other dimensions.

We restrict to preferences over “certain × uncertain pairs (c1, ˜c2).

Kihlstrom and Mirman(1974) convincingly explain that comparing agents’ risk aversions is possible if and only if agents have identical preferences over certain prospects. We shall therefore focus on utility classes that allows us to consider different attitudes towards risk, while keeping preferences over certain consumption paths unchanged.

This means that risk aversion is built upon an ordinal utility function U(C1, C2).

This rules out the standard class of expected utility models assuming additively separable utility functions. Indeed, once additive separability is assumed, it is impossible to change risk preferences, without altering ordinal preferences.

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Some popular classes of preferences

A common approach to discuss the role of risk aversion involves considering particular class of preferences.

Most of the classes are built assuming additively separable utility functions over certain prospects: U(C1, C2) = U1(C1) + U2(C2). A utility function U(c1, ˜c2)is called:

a Kihlstrom and Mirman utility function if there exist increasing u1, u2, k:

UKMk (c1, ˜c2) = k−1(E [k (u1(c1) + u2(˜c2))])

a Selden utility function if there exist increasing u1, u2, v:

UvS(c1, ˜c2) = u1(c1) + u2

“

v−1(E[v(˜c2)])

”

a Quiggin utility function if there exist increasing u1, u2, and φ : [0, 1] → [0, 1]:

UφQ(c1, ˜c2) = u1(c1) + Eφ[u2(˜c2)]

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Epstein and Zin classes of preferences

A very popular specification is obtained when choosing

isoelastic utility function in Selden representation. Indeed, one obtains a class of utility function assuming assuming constant intertemporal elasticity of substitution and homothetic

preferences. Such utility functions are often called “Epstein and Zin utility functions”, though it is imperfectly indicative of what can be found in Epstein and Zin (1989) who considered preferences over infinitely long consumption paths.

A utility function U(c1, ˜c2)is called an Epstein and Zin utility function if there

exist positive scalars ρ 6= 1 and γ 6= 1 such that:

UγEZ(c1, ˜c2) = c1−ρ₁ 1 − ρ+ 1 1 − ρ “ E h ˜ c1−γ2 i”1−ρ_1−γ

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Lower certainty equivalents representation of RA

One way to quantify and to compare risk aversion is to focus on how agents compare lotteries to certain outcomes.

Preceding class preferences seem well ranked in terms of risk aversion in the sense that “more risk averse agents” always need less certainty equivalents. Formally, if one compare U and UA_(UA_{being more risk averse):}

U(c1, ˜c2) = U(C1, C2) ⇒ UA(c1, ˜c2) < UA(C1, C2).

Following this criteria, agents are more risk averse in Kihlstrom and Mirman, for k2more concave that k1

in Selden utility function, for v2more concave that v1

in Epstein and Zin, for γ2> γ1,

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Aversion for dispersion and RA

Risk aversion represents the aversion for increases in risk. Naturally then, another way to compare risk aversion is to focus on how agents accepts risky profiles.

Preceding class preferences are not necessarily well ranked in terms of risk acceptance in the sense that “more risk averse agents” always accept “riskier profiles”. Formally, if one compare U and UA_(UA_{being more risk averse), for}

any profile (c1, ˜c2)that is “riskier” than the profile (C1, ˜C2):

U(c1, ˜c2) = U(C1, ˜C2) ⇒ UA(c1, ˜c2) < UA(C1, ˜C2).

Following this criteria, when considering spreads in Head and Tails, agents are more risk averse, in Kihlstrom and Mirman, for k2more concave that k1

and in Quiggin, for φ2more convex that φ1

but,

there is no simple definition of “riskier profile” that could be render compatible a ranking of Epstein and Zin preferences following that criteria. a γ2agent

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Comparative Risk Aversion

Application to choice under uncertainty Summary

A simple model Heads or tails gambles

General setup

X is an abstract set of consequences, without any

particular structure (not necessarily convex, etc.)

L(X)the set of lotteries with outcomes in X.

Economy populated by two agents A and B with

preferences A and Bover L(X).

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Acceptance sets and comparative risk aversion

Consider two rational agents A_andB _.

Standard approach (initiated by Yaari, 1969):

1 _{Take as given (for all agents) a partial universal order}

`"riskier than" on M(X).

2 _{For all l ∈ M(X) define agent i}0_{s acceptance set of l by:}

Ωi(l) = {m ∈ M(X)|m i land m `i l}

3 Comparative risk aversion (definition)

Ais more risk averse than B if and only if

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Meaning of "Riskier than"

1 _{Yaari’s minimalist definition (1969): it is assumed that}

l`i

mif and only if m is degenerate (m = δxfor some x)

2 _{This definition is the one used by Epstein and Zin, to argue the}

preferences they consider are ordered in terms of risk aversion.

3 _{In this paper a less partial order (inspired from Jewitt, 1987).}

l ` i

mif and only if there exist an x0∈ X such that

For all x i_dx0 we have Pr(l idx) ≥ Pr(m

i d x) For all x idx0 we have Pr(l idx) ≤ Pr(m id x)

4 The more restrictive `, the stronger the corresponding

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Heads or tails gambles: Definition

First a focus on extremely simple random elements =Heads or

tails gambles:

(xl; xh)is the lottery that gives xl∈ X with probability 0.5

and xh ∈ X with probability 0.5. We suppose xh xl.

H(X) =set of such gambles.

Assume that H(X) is endowed with a relation of preferences .

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A marginal increase in risk

Definition (Gamble spread)

The gamble (xl; xh)is a spread of (yl; yh), which is noted

(xl; xh) ` (yl; yh), if the following relationship holds:

(xl; xh) ` (yl; yh) ⇐⇒ xh yh yl xl

Example, x is a spread of y (x ` y):

x 100 0 y 19 18

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1 u F(u) xl yl yh xh 1 2

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Spread compatibility enough to define a unique

ordering in terms of risk aversion

A partial order “riskier than” R is spread compatible, iff: (xl; xh) ` (yl; yh) ⇒ (xl; xh)R(yl; yh)

Moreover, if (xl; xh) ` (yl; yh)and xh yhor yl xlthen it cannot be the case that (yl; yh)R(xl; xh).

Preferences over gambles i _{fulfills ordinal dominance iff:}

(xh yh and xl yl) ⇒ (xl; xh) i (yl; yh)

Result

If preferences of A and B fulfill ordinal dominance and if A is more risk averse than B with respect to a spread compatible R,

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Preferences over certain × uncertain consumption pairs

1 _{An agent with utility function U}KM

kA is more risk averse than

an agent with utility function UKM

kB with respect to any

spread compatible relation R iff kA is more concave than kB.

2 _{An agent with utility function U}Q

φA is more risk averse than

another one with utility U_φQB with respect to any spread

compatible relation R iif φA₍1 2) ≤ φ

B₍1 2).

Proposition (Epstein and Zin utility functions and risk aversion)

Consider two agents A and B with respective utility functions U_γEZ

A and U

EZ

γB, with γA> γB. There does not exist any spread

compatible relation R, such that A is more risk averse than B with respect to R.

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Elements of proof

We build two heads or tails gambles Ga_{and G}b_{with G}a_{being a spread of G}b_,

such that the agent A is indifferent between both gambles, and the agent B strictly prefers Ga_{. With 0 < ε << 1, c}

1, c2> 0, Gaand Gbare defined as

follows: Ga = (ca; 31−ρ1 _{(1 − ε)}_{( 1} 2), 3 1 1−ρ_{(1 + ε)}_{( 1} 2)) Gb = (cb; 1 − 2ε( 1 2), 1 + 2ε( 12)) where: c1−ρ_a − c1−ρ_b = (1 − 2ε) 1− γA _{+ (1 + 2ε)}1−γA 2 1−ρ 1−γA − 3 (1 − ε) 1−γA _{+ (1 + ε)}1−γA 2 _1−γA1−ρ

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Risk Aversion in intertemporal contexts Comparative Risk Aversion

Application to choice under uncertainty

Summary

A model free set-up

Agents chose an action t ∈ I ⊂ R modifying the payoff of a gamble Gt ≡ (xl(t), xh(t)) ∈ Z.

Typical example: saving choice. A couple assumptions over lotteries:

non constant: If for s = h, l, xs(t1) ∼ xs(t2)then t1 = t2.

actions do not modify outcome orders: if xh(t) xl(t)for all

t∈ I.

lotteries are single-peaked. For s = h, l:

∃ts∈ I, ∀t ∈ I, xs(ts) xs(t)

t1≤ t2≤ ts≤ t3≤ t4(∈ I),



xs(ts) xs(t2) xs(t1)

xs(ts) xs(t3) xs(t4) Best actions are ordered as states of the world: th≥ tl.

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Summary

A model free result

Proposition

We consider two agents A and B choosing an action t providing them a gamble satisfying preceding assumptions. The

preferences of agents A and B are supposed to fulfill the ordinal

dominance and to define respective single optimal actions tA

and tB_{. Then:}

If A is more comparative risk averse than B, then tA ≤ tB_.

Comment.General result. Few assumptions. Direct relationship between risk aversion and optimal choice.

Intuition.tl≤ th. A larger action increases the (ex post) welfare in the good

state and diminishes it in the bad one. Decreasing t = redistribution from good states of the world to bad ones to diminish the ex-post utility dispersion.

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Summary

Idea of the proof

Proof. Suppose that tA_{> t}B_.

tl≤ tB_{< t}A _{≤ th. Otherwise if (for example) t}B _{< tl}_{< th,} then xl(tl) xl(tB)and xh(tl) xh(tB)(single peakedness) with one strict relation. Then from ordinal dominance: B strictly prefers G(tl)to G(tB₎

Single peakedness implies xl(tA_{) xl(t}B_{) xh(t}B_{) xh(t}A_), with one strict inequality. G(tA₎_{is a spread of G(t}B₎

⇒ A prefers G(tA₎_{to G(t}B_{), which is a contradiction with A}

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Summary

Three applications

Three types of uncertainty:

(i) the second period income is random.

(ii) the saving interest rate is uncertain

(iii) the agent faces a mortality risk, i.e. a risk of dying at the end of the first period.

=⇒We use our model free result to draw unambiguous conclusions about the role of risk aversion in these three frameworks.

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Summary

Precautionary saving

Precautionary saving = saving when the 2nd period income is uncertain. Addressed in many papers: Drèze an Modigliani (1972), Kimball (1990), Kimball and Weil (2009). Notion of prudence whose link with risk aversion could not be clarified.

X is the set of 2 period consumption profiles, endowed with

a preference relation : (c1, c2) ∈ X.

H(X)is the set of lotteries and Y ⊂ H(X) is the set of

gambles with deterministic first consumption: (c1; ch2, c

l 2) ∈ Y.

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Summary

Precautionary saving

Two assumptions on ordinal preferences :

is convex on X.

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Summary

Precautionary saving

Proposition (Precautionary saving)

A and B have to chose the first period consumption c1providing

them (c1,ey2+ (1 + R)(y1− c1)), where y1> 0, andey2 = y h 2, y

l 2. fulfills both previous assumptions and that risk preferences verify the ordinal dominance and define respective optimal first

period consumptions cA

1 and c

B

1 for both agents.

agent A is more risk averse than agent B =⇒ cA₁ ≤ cB₁

Intuition. cl 1≤ c

A

1 ≤ cB1 ≤ ch1(normality of c1). Due to the the single

peakedness, the closer cA

1 to cl1, the larger the welfare when the bad state

realizes and the closer cA

1 to ch1, the larger the welfare when the good state

realizes. Increasing cA

1 increases the welafre in h and decreases it in l ⇒

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Summary

Precautionary saving when the interest rate is

uncertain

Same framework as the preceding one, except: y2: certain eR= Rh, Rl: random.

s(y1, y2, R) = arg maxs(y1− s, y2+ (1 + R)s).

Varying R ≡ varying the price of c2(c1=numeraire).

Two effects: (i) substitution effect and (ii) income effect.

⇒ smay increase or decrease as a function of R depending on which effect dominates (equivalent to EIS < or > 1).

Proposition (Optimal saving when interest rate is uncertain)

Under analogous assumptions, the following implication holds: agent A is more risk averse than agent B =⇒

8 < : sA1 ≤ sB1 if R 7→ s(y1, y2, R)is increasing or sB1 ≤ s A 1 if R 7→ s(y1, y2, R)is decreasing

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Summary

Optimal saving when the lifetime is uncertain

Impact of risk aversion on savings with lifetime uncertainty addressed without assuming expected utility and allowing for bequest.

Initial endowment W.

May live for one or two periods

ca₂= (1 + Ra)(W − c1)is the second period consumption

when surviving and cd

2= (1 + Rd)(W − c1)is the bequest.

We assume that Ra> −1 and Rd ≥ −1 but make no

assumption on how Raand Rdcompare. If no annuity and

no tax on bequest, Ra = Rd, while with perfect annuities,

Ra > Rd= −1.

X _{= (R}+)2× {a, d} and we note consumption profiles

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Summary

Optimal saving when the lifetime is uncertain

We make a set of assumptions regarding ordinal preferences:

The agent is always better off when surviving:

∀c₁, (c1, (1 + Ra)(W − c1))a (c1, (1 + Rd)(W − c1))d. The saving when surviving is larger than when dying: sa> sd.

Ordinal preferences are convex on both (R+)2× {a} and

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Summary

Optimal saving when the lifetime is uncertain

The following proposition holds:

Proposition (Optimal saving with uncertain lifetime)

Agents A and B face an (identical) exogenous risk of dying after the first period. They chose a saving s providing them a

consumption profile (W − s, (1 + Ra)s)_aif they survive and a

consumption/bequest profile (W − s, (1 + Rd)s)_d when not.

Agents A and B preferences fulfill ordinal dominance and define optimal saving sA_{and s}B_{. If ordinal preferences fulfill previous} set of assumptions then:

agent A is more risk averse than agent B =⇒ sA ≤ sB

Positive correlation between risk aversion and impatience. Strong assumption: identical mortality risk for both agents.

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Summary

Continuous distributions

Our previous results hold for continuous distributions, as long as we consider a proper definition for “riskier than”.

Previous results rely on an undisputable notion of “riskier than” for gambles.

However to gain generality, one needs to consider a broader class of lotteries.

No consensus regarding the notion of “riskier than” (Chateauneuf, Cohen, and Meilijson (2004)).

⇒ we chose a notion of spread, which generalizes the gamble case.

We suppose that the ordinal preference relationship over X can be represented by a function U : X → R.

The cdf for L ∈ L(X) is noted FLand defined over R.

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Summary

Comparative riskiness

Definition (p−Spread)

Let p ∈]0, 1[. L is said to be a p−spread of L0 (L `pL0), if it exists

u0∈ R such that: 1 _FL(u 0) = p 2 _{for all u ≤ u} 0, FL(u) ≥ FL0(u), 3 _{for all u ≥ u} 0, FL(u) ≤ FL0(u). Remarks:

The p−spread is an ordinal concept and not a cardinal one. States of the world can be split into “bad states" (measure 1 − p) and “bad states" (measure p), such that:

better outcomes for both lotteries in good states.

Conditional on being good, L first order dominates L0, and oppositely.

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Summary

Three applications to saving under uncertainty Continuous distributions 1 u F(u) L0 L u0 p Figure:p−spread L `pL0

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Risk Aversion in intertemporal contexts Comparative Risk Aversion Application to choice under uncertainty

Summary

Conclusions

We develop a notion of risk aversion that reflects the willingness to reduce risks (marginally), and not only the willingness to avoid all sources of risk by choosing a certain outcome.

We focus on heads or tails gambles and prove that KM and Q utility functions are well ordered in terms of risk aversion. But this is not the case for Selden (while well ordered in terms of the willingness to opt for certain outcomes). We develop a model free approach to derive unambiguous results about the impact of risk aversion without assuming a particular form of rationality.

3 applications regarding precautionary saving,

precautionary saving when the interest rate is uncertain and optimal saving when the lifetime is uncertain.