The transmission of information and the efficient market hypothesis. What is an efficient market ?

The transmission of information and the efficient

market hypothesis. What is an efficient market ?

The logic of information transmission

Information transmission through prices :– Agents decision are correlated to their private information. – Hence the price is correlated to the individuals’ private

information. – Hence the observation of the price allows to extract non

private information. – Conditional on having the right knowledge of the joint

distribution price-information. – Paradoxical limit, the fully revealing equilibrium.

Standard Theory :the RE equilibrium. – A rational expectations equilibrium :– Self-fulfilling correlation between price and information. – Every agent understands what the others do, depending on

their information. – Price observation allows to infer (part) of the agents’private

information.– Partially (/fully) revealing equilibria.

The classical model The model :

– A risky asset. – ”CARA/Gaussian”)– Grossman (1976), Grossman et Stiglitz (1980), Hellwig (1980)

The equilibrium concept– Linear RE equilibrium partially revealing. – A fixed point of the mapping : the expected relationship price-

state of nature actual relationship. The model (next).

– Observed price p, unknown future value – A continuum [0, 1] of agents, (no market manipulation)– Identical but differently informed.

Private information : – s(i)=+(c)+(i) ----N(0,m.), (c)----N(0, b(c)), (i)----N(0, b-b(c)), i.i.d

The model (next) Social information

– S (mean signal)– S=+(c)– The conditional distribution /S is the same as /..s(i)..– s(i)=S+(i).

Noisy traders :– Demand e : N(0,q)

Timing : , e, s*, drawn at random– x(i,s(i),p) auctionneer.– Z(p,s*))=e – P(S,e), partial information on s*.

The model (next) A linear equilibrium :

– P(S,e)=p(0)+cS+de– Then, if true /s(i),p is normal.

Demand (CARA), /s(i),p is normal: – X(i, s(i),p)={E[/s(i),p]-p}/aVar[/s(i),p]– X(i,s(i),p) is linear..– Z is linear – Z= T(.)+T’(.)s*+T’’(.)e– Has T,T’,T’’ a fixed point ?– A mapping from R3 into R3.

Existence : a unique Linear REE.– P(S,e)=p(0)*+c*S+d*e

Eductive stability. Eductive stability : the standard view point

– It is tentatively CK that – the « affine » map that relates equilibrium price to S and

e– Is close to the equilibrium map.– Is it CK knowledge that the equilibrium map takes place.

Remarks. – Linearity is taken for granted. – Local view point (local Strong Rationality)– It amounts to the fact that the mapping T,T’,T’’ is

contracting locally. (eigenvalues of the mapping dT) Results.

– EER locally SR Var[/p]>Var[/s(i)]

Eductive stability : the economic logic.

Results.– EER locally SR Var[/p]>Var[/s(i)]– You trust more your information than the market

information. – If not, you are not inclined to use your information and

then you donot provide it to the market. – Philosophy : the market cannot transmit too much

information. Additional comments.

– If the private signal is better : Var[/s(i)] =x decreases– But Var[/p]=y decreases too, ambiguous. – In fact the equilibrium REE relationship is y=A(x.x)– Stability requires that x large enough, so that individual

signals are not too informative….

Crash in information Crash in information transmission.transmission.

The multiplicity hypothesis..The multiplicity hypothesis..

A model with informed and A model with informed and non informed agents and noisy non informed agents and noisy

supply.supply. The framework : The framework :

– Asset value , H or B. Asset value , H or B. – Proportion a informed.Proportion a informed.– Mean-variance pref. . Mean-variance pref. . – Noisy (noise traders).Noisy (noise traders).

Equilibrium : Equilibrium : Z, BeliefsZ, Beliefs Z(p,I)=ad(I,p)+(1-a)d(NI, p)=Z(p,I)=ad(I,p)+(1-a)d(NI, p)=ee

p(I,e) clears the market. p(I,e) clears the market. – BeliefsBeliefs NI bayesian NI bayesian – d(I,p) Dominant Str.d(I,p) Dominant Str.– If p : If p :

e =- Z(p,H) ou -Z(p,B)e =- Z(p,H) ou -Z(p,B)– Compute Compute

E(H/p) and E(H/p) and E(s/p)= HE(H/p) +B(1-E(H/p))E(s/p)= HE(H/p) +B(1-E(H/p))

– d(NI,p) = E(s/p) –p. d(NI,p) = E(s/p) –p.

d

p

High informed Demand

Non informed Demand

Equilibrium in the noisy model.Equilibrium in the noisy model.

Equilibrium : Equilibrium : Z, BeliefsZ, Beliefs Z(p,I)=ad(I,p)+(1-a)d(NI, p)=Z(p,I)=ad(I,p)+(1-a)d(NI, p)=ee

p(I,e) clears the market. p(I,e) clears the market. – BeliefsBeliefs NI bayesian NI bayesian – d(I,p) Dominant Str.d(I,p) Dominant Str.– If p : If p :

e =- Z(p,H) ou -Z(p,B)e =- Z(p,H) ou -Z(p,B)– Compute Compute

E(H/p) and E(H/p) and E(s/p)= HE(H/p) +B(1-E(H/p))E(s/p)= HE(H/p) +B(1-E(H/p))

– d(NI,p) = E(s/p) –p. d(NI,p) = E(s/p) –p. PropertiesProperties

– Total demand is decreasing.Total demand is decreasing.– But not necessarily NI But not necessarily NI

demand. demand. – Function / noise précision. Function / noise précision.

Equilibrium is unique.Equilibrium is unique.

Z totale/si B

p

Getting multiple equilibria. Getting multiple equilibria.

The 1987 crash according to The 1987 crash according to Genotte-Leland. Genotte-Leland. – Add informatics programs into Add informatics programs into

the picturethe picture:: Automatic sale whenever Automatic sale whenever

the asset price decreases.the asset price decreases.– Again, Again,

Static framework…Static framework…– Adding : Adding :

A positively sloped A positively sloped curve….curve….

Consequence :Consequence :– Total deamand is no longer Total deamand is no longer

decreasing.decreasing.– Multiplicity.Multiplicity.– The crash : passage The crash : passage

From a high equilibrium From a high equilibrium to a low one. to a low one.

p

d

Or crash of Or crash of expectational expectational coordination ?coordination ?

« Eductive stability » of « Eductive stability » of equilibria.equilibria.

A reminder of Desgranges A reminder of Desgranges The setting : The setting :

– Information transmission à la Grossman-Stiglitz. Information transmission à la Grossman-Stiglitz. – Each small agent receive a piece of noisy information. Each small agent receive a piece of noisy information. – Noise traders. Noise traders. – Aggregate equilibrium excess demand reflects the Aggregate equilibrium excess demand reflects the

average information of the society…. generates average information of the society…. generates individual and aggregate…individual and aggregate…

The analysis. The analysis. – There exists a unique equilibrium.There exists a unique equilibrium.– But not necessarily strongly rational. But not necessarily strongly rational. – Contradiction between the confidence in the market Contradiction between the confidence in the market

transmission and the amount of information transmitted.transmission and the amount of information transmitted.

A model with informed and A model with informed and non informed agents and noisy non informed agents and noisy

supply.supply. The framework : The framework :

– Asset value , H or B. Asset value , H or B. – Proportion a informed.Proportion a informed.– Mean-variance pref. . Mean-variance pref. . – Noisy (noise traders).Noisy (noise traders).

Equilibrium : Equilibrium : Z, BeliefsZ, Beliefs Z(p,I)=ad(I,p)+(1-a)d(NI, p)=Z(p,I)=ad(I,p)+(1-a)d(NI, p)=ee

p(I,e) clears the market. p(I,e) clears the market. – BeliefsBeliefs NI bayesian NI bayesian – d(I,p) Dominant Str.d(I,p) Dominant Str.– If p : If p :

e =- Z(p,H) ou -Z(p,B)e =- Z(p,H) ou -Z(p,B)– Compute Compute

E(H/p) and E(H/p) and E(s/p)= HE(H/p) +B(1-E(H/p))E(s/p)= HE(H/p) +B(1-E(H/p))

– d(NI,p) = E(s/p) –p. d(NI,p) = E(s/p) –p.

d

p

High informed Demand

Non informed Demand

Eductive coordination in the noisy Eductive coordination in the noisy model. model.

A first answer : A first answer : – The equilibrium is eductively stable iif The equilibrium is eductively stable iif ( normal noise):( normal noise):

(1-(1-))22<4<422 – With With , prop.informed, , prop.informed, ,gap, ,gap, 22 variance of noise. variance of noise.– Product of an amplification effect and a sensitivity effect. Product of an amplification effect and a sensitivity effect. – Comments.Comments.

A second answer. A second answer. – The equilibrium is eductively stable iif The equilibrium is eductively stable iif – Aggregate equilibrium demand is enough decreasing. Aggregate equilibrium demand is enough decreasing. – With few informed agents, a Necessary condition is that With few informed agents, a Necessary condition is that

non informed demand is decreasing. non informed demand is decreasing. – Comments.Comments.

Next on financial markets

Sunspots; Market structure.

Where are we ? Financial markets :

– Volatility, bubbles Followers, herd behaviour, rationality of riding the bubble

– Transmission of information through prices Multiplicity, Too much information kills information.

– The market structure. Remains to be seen.

Coming back on volatility– Multiplicity issue : sunspots

Going into the market structure problem. – Bowman Faust, – Guesnerie Rochet– Brock-Hommes-

Back on volatility

Sunspot Equilibria.

Erratic fluctuations with short Erratic fluctuations with short

lived agentslived agents.. Standard modelling Standard modelling

– Generations model : each agent « lives » for 2 periods.Generations model : each agent « lives » for 2 periods.– « lives », short planning horizon : « lives », short planning horizon : – Monetary theory.Monetary theory.

The model reinterpreted.The model reinterpreted.– One good.One good.– A production process (« manna ») A production process (« manna »)

Gives d per period : Gives d per period : Other resources the endowments of the « young » AOther resources the endowments of the « young » A

– Asset held at t by generation t-1, consume d/sell the Asset held at t by generation t-1, consume d/sell the asset. asset.

– Asset price p(t) and p(t+1)Asset price p(t) and p(t+1) The optimisation problem.The optimisation problem.

– Max U(c(t))+V(c(t+1)) Max U(c(t))+V(c(t+1)) – c(t)= A - y(t)p(t), c(t+1)=dy(t)+p(t+1)y(t)c(t)= A - y(t)p(t), c(t+1)=dy(t)+p(t+1)y(t)– c(t)+ (p(t)/(p(t+1)+d))c(t+1) = Ac(t)+ (p(t)/(p(t+1)+d))c(t+1) = A– Solution D(p(t)/(p(t+1)+d), .)Solution D(p(t)/(p(t+1)+d), .)

p(t+1)

p(t)

c(t)

c(t+1)

AA-p(t)

A(d+p(t+1))/p(t)

p*p’ p’’

D(p(t),p(t+1)=1

•Income effect dominates the Income effect dominates the

substitution effect. substitution effect.

Equilibria: Equilibria:

definitions and analysisdefinitions and analysis.. Equilibrium: p(t), Equilibrium: p(t),

(p(t+1)(p(t+1) – D(p(t)/(p(t+1)+d))=1D(p(t)/(p(t+1)+d))=1– c(t), c(t+1)…..c(t), c(t+1)…..

Stationary equilibrium :Stationary equilibrium : p*, c*(1)=A-p*,c*(2)= d+p*p*, c*(1)=A-p*,c*(2)= d+p*

– D(p*/(p*+d)=1D(p*/(p*+d)=1

(p*+d)/p*=U’(A-p*)/V’(d+p*)(p*+d)/p*=U’(A-p*)/V’(d+p*) r =d/p* r =d/p* Comment : Comment :

– Price constant… Price constant… – Relation fv ? :Relation fv ? :– p*=? d /(1+(d/p*)) p*=? d /(1+(d/p*))

+d /(1+(d/p*))+d /(1+(d/p*))22 +…+…

Cyclical equilibriumCyclical equilibrium (order (order 2)2)– p’, p’’p’, p’’– D(p’/ p’’+d) =1D(p’/ p’’+d) =1– D(p ’’/p’+d) =1D(p ’’/p’+d) =1– c’(1)= A – p’, c’(2)=p’’+d.c’(1)= A – p’, c’(2)=p’’+d.– c’’(1)=A-p’’,c’’(2) = p’+ d.c’’(1)=A-p’’,c’’(2) = p’+ d.

There exists equilibriaThere exists equilibria– StationaryStationary– Cycles of order 2Cycles of order 2

Intuition Intuition – Low price to-day because Low price to-day because

high to-morrow. high to-morrow. – Vice versa. Vice versa.

Price equal FV ?Price equal FV ?– In which sense ?In which sense ?– Volatility.Volatility.– No « bubble » ?.No « bubble » ?.

Equilibria : NextEquilibria : Next..

Sunspot Equilibria.Sunspot Equilibria.– Sunspots s, ns.Sunspots s, ns.– If s/prob s,ns depends of s If s/prob s,ns depends of s – ( Markovian)( Markovian)

D(p(s),{D(p(s),{(s), p(s), p(ns)}=1(s), p(s), p(ns)}=1 D(p(ns){D(p(ns){(ns) p(s),p(ns)}=1.(ns) p(s),p(ns)}=1. Markovian of order 2 sunspot Markovian of order 2 sunspot

equilibria. equilibria. – Cycle of order 2 limit of SSE.Cycle of order 2 limit of SSE.– Ssi Ssi cycle of order 2 cycle of order 2

(Azariadis-Guesnerie) (Azariadis-Guesnerie) Price fluctuations des prix not Price fluctuations des prix not

based only on fondamentals based only on fondamentals but on exogenous but on exogenous phenomenaVolatilité.phenomenaVolatilité.

Stochastic.Stochastic. No ?(avec les proba risque-No ?(avec les proba risque-

neutres).neutres).p(t+1)

p(t)