The effect of market structure on the

plausibility of REEMarkets, the more the

better ?

Introduction. Friedman’s argument.

– Speculation is stabilizing. – Speculators

sell when the price is high Buy when it is low They stabilize the market.

– Friedman’s argument has been incredibly influential and is still. The challenge.

– With RE, new markets may create sunspot equilibria. Bowman-Faust

– New markets threaten « eductive » stabillity. Guesnerie-Rochet

– New markets destabilize « evolutive » learning. Brock, Hommes, Wagener.

Bowman-Faust (1997)Bowman-Faust (1997) The model. The model.

– 3 periods, 2 agents, log. Utility..3 periods, 2 agents, log. Utility..– one firm…equity is exchanged..one firm…equity is exchanged..– 0 decision on firm’s investment, reaped at 2. 0 decision on firm’s investment, reaped at 2. – At 1, one of the agent, randomy picked, desires At 1, one of the agent, randomy picked, desires

immediate consumption, (zero value), the other one immediate consumption, (zero value), the other one postponement…postponement…

The equilibrium with one assetThe equilibrium with one asset– Is not P.O : not zero consumption in the bad event..Is not P.O : not zero consumption in the bad event..– No sunspot. No sunspot.

An option :An option :– Completes the market : PO with the option..Completes the market : PO with the option..– Creates sunspotsCreates sunspots

Guesnerie-Rochet. Guesnerie-Rochet. The model :The model :

– A manna of crop at each period w(t), t=1,2A manna of crop at each period w(t), t=1,2– Part on the market, the other goes to inventories. Part on the market, the other goes to inventories. – Inventory Cost : CxInventory Cost : Cx2 2 /2, /2, – P(t)=k{d(t) - w(t) (+/–) S(t)}, S(t), quantity on the P(t)=k{d(t) - w(t) (+/–) S(t)}, S(t), quantity on the

market. market. Profitability of inventories. Profitability of inventories.

– Mean-variance utility : E(Mean-variance utility : E() – (1/2)b(Var ) – (1/2)b(Var ))– ΔΔP(t)=k{P(t)=k{ΔΔ d(t) -2X- d(t) -2X- ΔΔw(t)}, (k=1/B), Var(w(2)) = vw(t)}, (k=1/B), Var(w(2)) = v22

– x() = k(X_ -2X(e))/{bkx() = k(X_ -2X(e))/{bk22vv22+C}+C}– Inventory mass NInventory mass N– X= kN(X_ -2X(e))/{bkX= kN(X_ -2X(e))/{bk22vv22+C}+C}

Equilibrium inventories :Equilibrium inventories :– X* = X_/{2+C/kN + bkvX* = X_/{2+C/kN + bkv22/N}/N}– Plausible…..Plausible…..

The « eductive » The « eductive » process :the inventory process :the inventory

variantvariant An « eductive » story :An « eductive » story :– Expectations X(e), Expectations X(e), – Realizations :Realizations :– -2kNX(e))/{bk-2kNX(e))/{bk22vv22+C}+C}

Results :Results :– N<{bkN<{bk22vv22+C}/2k+C}/2k– BadBad

More tradersMore traders Less risk averseLess risk averse Less uncertaintyLess uncertainty Less costly..Less costly..

-2kN/{bk2v2+C}

X

Inventories with futures Inventories with futures marketsmarkets

M mass of « speculators » : intervene on the market of M mass of « speculators » : intervene on the market of futures, price P(f), one unit of wheat to morrow. futures, price P(f), one unit of wheat to morrow.

Hedging behaviour from primary traders : Hedging behaviour from primary traders : – N[p(f)-p(1)]/C = (N+M)[p(2)-p(1)]/bkN[p(f)-p(1)]/C = (N+M)[p(2)-p(1)]/bk22vv2.2.

– Previously X = X*/{2+C/kN + bkvPreviously X = X*/{2+C/kN + bkv22/N}/N} Now : X = X*/{2+C/kN + bkvNow : X = X*/{2+C/kN + bkv22/(N+M)}/(N+M)}

– Intuition : uncertainty cost born by N+M agents.Intuition : uncertainty cost born by N+M agents.– The variance of prices is decreasedThe variance of prices is decreased

Eductive stability :Eductive stability :– C/kN + bkvC/kN + bkv22/(N+M) >2/(N+M) >2– Intuition.N(M), N decreasing function of M. M>0 is bad.Intuition.N(M), N decreasing function of M. M>0 is bad.

An « evolutive » learning model..

From Brock-Hommes-Wagener : « more hedging instruments may destabilize markets ».

The model (sketch). – Stock p(0,t)q(t+1,s)=p(0,t+1) + y(s), s=1,…S, prob. – N Arrow securities, i pays the vector d(i), i.e pays 1 in

state i=1,..N<S, price p(i,t); – Mean variance utility.

The demand :– Z(t)=IV(N)[-R(p(0,t)+E{q(t+1)}, -Rp(t)+E{d}]t

– V(N) prop. cov (q(t+1)/d)– Steady state.

p(0,*)= (y*-aQ)/(R-1), variance of the stock, a coef risk aversion, Q total amount of the stock.

p(s,*)=(1/R)( -abQ), cov. Vect. q,d

An « evolutive » learning model..

The learning process :– Heterogenous expectations away from the RE benchmark. – Deviations f(h,t) depends on the type of the traders. – Remarks.

Along the path, Arrow securities are correctly priced in the REE If beliefs are homogenous, no trade on Arrow securities

– The fraction of agents following a given strategy depends on its fitness : average profit of the previous period corrected by riskiness

– The speed of adjustment measured by c>0, – c small, the fundamental equilibrium is stable– When c becomes large, the fundamental equilibrium is

destabilzed Results.

– With one more Arrow security, the fundamental equil. Is destabilized earlier for a smaller c !

– The mechanism : Optimists (resp. pessimists) buy (resp. sell) the stock and hedge with Arrow securities…

An « evolutive » learning model..

Results. – With one more Arrow security, the fundamental equil. Is

destabilized earlier for a smaller b !– The mechanism.– With more insurance possibility more hedging and risk

taking, and profit if you are on the right side of the opinion, and these strategies through reinforcement mechanisms will attract more followers,

– And vice versa… – More movement of opinion and of prices….

Other results. – « Rational » agents may or may not stabilize the

market…– Dubious…

Excess confidence. Excess confidence. A cognitive bias A cognitive bias

– Well established ?Well established ? Modelling : Investors. Modelling : Investors.

– Random variable Random variable vv , mean 0, 2 signals t(1)=v+e(1), , mean 0, 2 signals t(1)=v+e(1), t(2)=v+e(2),t(2)=v+e(2),

– e(1),e(2)e(1),e(2) mean zero, precision ( mean zero, precision (e(i)e(i))= )= ρρ– 2 categories of investors A and B 2 categories of investors A and B – A, (resp.B) over estimates precision t(1),cA, (resp.B) over estimates precision t(1),cρρ,(resp.t(2),c,(resp.t(2),cρρ), c>1), c>1

Modelling : the firrm. Modelling : the firrm. – Long term value Long term value w = u+v+e’, uw = u+v+e’, u mean mean uu >0 >0– signal signal ss, on u, centered on , on u, centered on uu, precision , precision ρρ(s). (s). – u,v,e’, s, u,v,e’, s, normal precision denoted normal precision denoted ρρ(.)(.)– Without cognitive bias : E(Without cognitive bias : E(w/w/s,t)= s,t)= – uu + [ + [ρρ(s)/((s)/(ρρ(u)+ (u)+ ρρ(s))][s - (s))][s - u]u] + + ii[[ρρ/(/(ρρ(v)+2(v)+2ρρ)][t(i)])][t(i)]– ++ii[1/([1/(ρρ*+2)][t(i)], with *+2)][t(i)], with ρρ*= *= ρρ(v)/((v)/(ρρ).).

Excess confidenceExcess confidence Modelling : the firm Modelling : the firm

– Long term value Long term value w =u+v+e’, uw =u+v+e’, u mean mean uu >0 >0– signal signal ss, on , mean , on , mean uu, precision , precision ρρ(s). (s). – U,v,e’,e(i),s, U,v,e’,e(i),s, normal precision normal precision ρρ(.)(.)

Cognitive bias : Cognitive bias : – With bias : E(With bias : E(w/w/s,t)= s,t)= – uu + [ + [ρρ(s)/((s)/(ρρ(u)+ (u)+ ρρ(s))][s- (s))][s- u]u] + + – For A : [cFor A : [cρρ/(/(ρρ(v)+(v)+ρρ(1+c)][t(1)] + [(1+c)][t(1)] + [ρρ/(/(ρρ(v)+(v)+ρρ(1+c)][t(2)](1+c)][t(2)]– The difference between the a posteriori belief of A and B rewrites : The difference between the a posteriori belief of A and B rewrites : – [(c-1)/([(c-1)/(ρρ*+1+c)][t(1)-t(2)]*+1+c)][t(1)-t(2)]– Exchange of shares after observation of t (with or without s)Exchange of shares after observation of t (with or without s)

Si Si tt22 > > tt11 B too optimistic /w : B buys shares to B too optimistic /w : B buys shares to A at his valuationA at his valuation Si Si tt11 > > tt22 then A too optimistic, but cannot buy to then A too optimistic, but cannot buy to B.B.

Value of the firm ex ante :Value of the firm ex ante :– V(0) = V(0) = uu+ [(c-1)/(+ [(c-1)/(ρρ*+1+c)][*+1+c)][ [ [ρρ*(c+1)/2c*(c+1)/2c](écart type (](écart type (vv))))– Hint : increasing the standard deviation of v a value for initial owners. Hint : increasing the standard deviation of v a value for initial owners.