Graduate Microeconomics II Lecture...

What Should be Known Expected Utility Measure of Risk Stochastic dominance Information partitions Common knowledge

Graduate Microeconomics IILecture 1:

Patrick Legros

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Outline

What Should be Known

Expected Utility

Measure of Risk

Stochastic dominance

Information partitions

Common knowledge

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What Should be Known

• Constrained optimization

• Kuhn et Tucker• Dynamic programming• Comparative Statics

• Information economics• Risk aversion• Stochastic dominance• Value of information

• Basics of game theory• Extensive forms• Equilibrium

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Some Reviews

1. Measures of risk

2. Stochastic dominance

3. Information partitions and common knowledge

4. No trade theorems

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Utility for Money

Start from preferences over lotteries L = (p1, · · · , pN).Impose two axioms.

• CONTINUITY :∀L, L′, L′′, the sets α ∈ [0, 1] : αL + (1− α)L′ L′′ andα ∈ [0, 1] : αL + (1− α)L′ L′′ are closed.

• INDEPENDENCE: ∀L, L′, L′′, ∀α ∈ (0, 1),[L < L′]⇔ [αL + (1− α)L′′ < αL′ + (1− α)L′′]

Theorem (von Neuman expected utility)

If < satisfies CONTINUITY and INDEPENDENCE, then it admitsan expected utility representation: ∃u1, · · · , uN s.t.

L < L′ ⇔N∑

i=1

piui >N∑

i=1

p′iui

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Example 1: Allais Paradox

Four (simple) lotteries: A =

(10 0.10 0.9

), B =

(15 0.090 0.91

),

C =

(10 10 0

), D =

(15 0.90 0.1

)

• Do you prefer A to B? Do you prefer C to D?

• Most often: B A and C D

Violation of expected utility

A = 0.1 ∗ C + 0.9 ∗ 0

B = 0.1 ∗ D + 0.9 ∗ 0

B A ⇔ D C

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Example 1: Allais Paradox

Four (simple) lotteries: A =

(10 0.10 0.9

), B =

(15 0.090 0.91

),

C =

(10 10 0

), D =

(15 0.90 0.1

)

• Do you prefer A to B? Do you prefer C to D?

• Most often: B A and C D

Violation of expected utility

A = 0.1 ∗ C + 0.9 ∗ 0

B = 0.1 ∗ D + 0.9 ∗ 0

B A ⇔ D C

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Example 2: Mean-VarianceOr Beware of Seemingly “Natural” Utility Functions• Businessman who never heard of Bernoulli nor von Neumann

Morgenstern nor expected utility. Thinks that it is a good ideato get a high expected profit but he thinks that someadjustment has to be made for risk.

• He may think, if he had some rudimentary statistics in hisyouth, the variance is a good measure of risk. He may thendecide to assign the following utility to a lottery f (x) withmean E and variance V :

W (f ) = E − aV

where a can be taken as a measure of “risk aversion.”

• However, there is no function u(x) independent of f so thatU(f ) =

∑u(x)f (x). This means that the decision rule of our

businessman violates some axioms of expected utility and leadto some “weird” decisions.

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Example 2: Mean-VarianceOr Beware of Seemingly “Natural” Utility Functions• Businessman who never heard of Bernoulli nor von Neumann

Morgenstern nor expected utility. Thinks that it is a good ideato get a high expected profit but he thinks that someadjustment has to be made for risk.

• He may think, if he had some rudimentary statistics in hisyouth, the variance is a good measure of risk. He may thendecide to assign the following utility to a lottery f (x) withmean E and variance V :

W (f ) = E − aV

where a can be taken as a measure of “risk aversion.”• However, there is no function u(x) independent of f so that

U(f ) =∑

u(x)f (x). This means that the decision rule of ourbusinessman violates some axioms of expected utility and leadto some “weird” decisions.

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• For instance, consider the two lotteries, lottery f1: get 1 withproba 0.1 and get −99 with proba 0.9lottery f2: get 1 with proba 0.2 and −99 with proba 0.8suppose a = 0.1, then U(f1) = −179 > U(f2) = −239

• If instead, we start from the utility for money, u(x) = x − ax2

where 0 ≤ a ≤ 1/(2M) and −∞ ≤ x ≤ M then,

U(f ) =∑

xf (x)− a∑

x2f (x)

=∑

xf (x)− a(∑

xf (x))2 − a∑

(x −∑

xf (x))2f (x)

= E − aE 2 − aV

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Example 3: “Bernoulli Utility”Utility for money u(x + s) where s is initial wealth. If preferenceordering over lotteries is independent of s, then

u(x + s) = a(s)u(x) + b(s)

then

u′(x + s) = au′(x)

u′(x + s) = a′u(x) + b′

⇒a′u(x)− au′(x) + b′ = 0

Then,

• a′ = 0⇒ u(x) = b′

a x + C1 (risk neutral)

• a′ 6= 0⇒ u(x)− aa′u′(x) + b′

a′ = 0 ⇒ u(x) = C2e−a′ax− b′

a′

or u(x) = C3e−αx

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Risk aversion

The Bernoulli utility ui incorporates the preferences for risk. Let xbe a random payoff. The certainty equivalent is the sure payoff cx

such that Eu(x) = u(cx)

• Risk loving: u is convex; hence cx > E x

• Risk averse: u is concave; hence cx < E x

• Risk premium: ρx = E x − cx

Intuition: a risk averse agent would be ready to pay for insurance

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Deriving risk premium for small risks

y is a small risk with E y = 0,E y 2 = 1. Let σ be a scaling factor;

• Write the absolute risk premium p when initial wealth is xwhen face the risk σy as u(x − p) = Eu(x + σy). Taylorseries expansion:

u(x)− pu′(x) = E [u(x) + σu′(x)y + (1/2)σ2u′′(x)y 2]

= u(x) + (1/2)σ2u′′(x)

So

p ≈ −u′′(x)

u′(x)

σ2

2

• Relative risk premium: u(x − ρx) = Eu(x + σyx). Seriesexpansion yields

ρ = −u”(x)x

u′(x)

σ2

2

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Measure of Risk Aversion

The coefficient of absolute risk aversion is −u”(x)u′(x) .

The coefficient of relative risk aversion is −u”(x)xu′(x) Usually (stylized

fact?) we think of the absolute risk aversion as declining in wealth(condition on the third derivative on u). Evidence?

• Constant relative risk aversion - u(x) = x1−ρ

1−ρ (CES), ρ ≥ 0.

Note that ρ = −u”(x)xu′(x) .

• Special cases: risk neutrality ( ρ = 0); log utility (ρ = 1)

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How risk averse?

Equity premium puzzle

• (Mehra and Prescott 1985): equity premium ρ ≈ var [y ][Ey ]2 R

where R = −[E y ][u”(E y)]/u′(E y) is the index of relative riskaversion evaluated at the mean return. In data,var [y ]/[E y ]2 ≈ 0.056% .

• Historical data for 1889-1978 (Shiller 1989, Kocherlakota1996), average return S&P is 7% while short-term risk freerate is 1%. Fro 1963-1995, figures are resp. 7,72% and3,14%, hence ρ approx4, 5%, requiring R ≥ 40

In the lab

• Often risk aversion over small amount of money

• Rabin (2000): risk aversion in the small leads to impossibleresults in the large

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“Suppose we knew a risk-averse person turns down 50-50 lose$100/gain $105 for any lifetime wealth level less than $350,000,but we knew nothing about the degree of his risk aversion forwealth levels above $350,000. Then we know that from an initialwealth level of $340,000 the person will turn down a 50-50 bet oflosing $4,000 and gaining $635,670.”

If mu is the marginal utility then sinceu(w + 105)− u(w) < u(w)− u(w − 100),

mu(w + 105) < [u(w + 105− u(w)]/105

<100

105[u(w)− u(w − 100)]/100

<100

105mu(w − 100)

Hence mu falls at a faster rate than that of a geometrical sequence.

• Rubinstein (Nobel symposium 2001): assumes that there is asingle preference relation over set of lotteries; but could havepreferences depend on w .

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Stochastic dominance

How can we say that a lottery is riskier than another? Statementmust be true for all or only some utility functions?

• (Rotschild-Stiglitz) Take mass from middle and spread it atthe tails; mean-preserving risk.

• (FOSD) unambiguous higher returns;F 1 G ⇔ ∀x ,F (x) ≤ G (x) (< on a set of positive measure)FOSD ⇔ ∀φ increasing,EFφ ≥ EGφ

• (SOSD) unambiguous lowerrisk;F 2 G ⇔ ∀y ,

∫ y0 F (x)dx) ≤

∫ y0 G (x)dx (< on a set of

positive measure.SOSD ⇔ ∀u concave,EF u ≥ EG u

• (Lorentz) Inequality measure; φ(F (y)) =R y

0 xf (x)dx

EF x . AssumingEF x = EG x , F is more unequal than G ifF (y1) = G (y2)⇒ φ(F (y1)) ≥ φ(G (y2))

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Blackwell theorema ∈ A action, θ ∈ Θ states, p dist. on θ, u(a, θ) utility function.An environment is t = (u,A).Optimization: maxa∈A Epu(a, θ)⇒ a(p, t)Signal x1 or x2, πi (x |θ) conditional of signal x given θ. Posterior is

pi (θ|x i ) =πi (x |θ) · p(θ)

πi (x i )

Optimal action as a function of signal received and indirect utilityS i (x i ; t)

• x1 is more informative than x2 if S1(x1; t) ≥ S2(x2; t) for allt.

• x1 is sufficient for x2 if there exists a matrix B such that

1.∑

x2 b(x2, x1) = 1 for all x1

2. π2(·|θ) = B · π1(·|θ); hence π2(·|θ) =∑

x1 b(x2, x1)π1(x1|θ)

Theorem (Blackwell, Cremer) x1 is more informative than x2 if and onlyif x1 is sufficient for x2

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Information partitions

(Hintikka 1962) Information structure is a pair (Ω,P), Ω set ofstates and P : Ω→ 2Ω describes the individual information (ifstate is ω agent can exclude all states not in P(ω).Basic Properties

P-1 ω ∈ P(ω)

P-2 If ω′ ∈ P(ω)⇒ P(ω′) ⊆ P(ω)

P-3 If ω′ ∈ P(ω)⇒ P(ω′) ⊇ P(ω)

Ex (awareness) Identify first digit but notice second digit only if itis the same; P(33) = 33 but P(23) = 13, 23, 33 violates P-3Ex (measurement error) States from 00 to 99, mistake of 1 atmost. P(n) = n − 1, n, n + 1, violates P-2.

Proposition An information structure is partitional if and only if itsatisfies P-1,P-2,P-3.

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Knowledge

E is known at ω if P(ω) ⊆ E“Knowledge” of E at state ω induces a set-theoretical definition of“to know” E .

K (E ) = ω|P(ω) ⊆ E

Note that K (E ) is an event, and can define K (K (E )),−K (E ) = Ω− K (E )etc.Definition implies

K-0 E ⊆ F ⇒ K (E ) ⊆ K (F ): K (E ∩ F ) = K (E ) ∩ K (F );K (Ω) = Ω

K-1, axiom of knowledge K (E ) ⊆ E

K-2, axiom of transparency K (E ) ⊆ K (K (E ))

K-3, axiom of wisdom −K (E ) ⊆ K (−K (E ))

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Common knowledge

Let K1,K2 be two knowledge operators for agents 1 and 2. Then,

Common Knowledge E is common knowledge between 1 and 2 inthe state ω if ω is a member of all sets of the typeK1(E ),K2(E ),K1(K2(E )),K2(K1(E )), · · ·

P1 = ω1, ω2, ω3, ω4, ω5, ω6, ω7, ω8P2 = ω1, ω2, ω3, ω4, ω5, ω6, ω7ω8

• E = ω1, ω2, ω3, ω4. Then E is not common knowledge atany ω.

• F = ω1, ω2, ω3, ω4, ω5. Then F is common knowledge inany state ωi , i = 1, · · · , 5

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No trade theorem

Consider two agents who:

• have the same prior beliefs on Ω

• have different information structures P1 and P2

• have a mutual understanding of the information structures

Question Is it possible that they have CK at some ω that agent 1believes that the expectation of a lottery L is strictly above α andthat agent 2 believes that the expectation of L is strictly below α?

If yes: exchange is acceptable to both agents. If no, no tradepossible. No if partitional information structures

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