# SUPPLEMENT TO DECENTRALIZED TRADING WITH PRIVATE INFORMATION : ONLINE APPENDIX (Econometrica, Vol. 82, No. 3, May 2014, )

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1 Econometrica Supplementary Material SUPPLEMENT TO DECENTRALIZED TRADING WITH PRIVATE INFORMATION : ONLINE APPENDIX Econometrica, Vol. 82, No. 3, May 2014, ) BY MIKHAIL GOLOSOV, GUIDO LORENZONI, AND ALEH TSYVINSKI S1. PROOF OF LEMMA 5 TO PROVE THE LEMMA, we will find two scalars x and u such that the set X = { x : x 0 x] 2 Ux δ) u for some δ [0 1] } satisfies the desired properties. The proof combines two ideas: use market clearing to put an upper bound on the holdings of the two assets, that is, to show that, with probability close to 1, agents have portfolios in 0 x] 2 ; use optimality to bound their holdings away from zero, by imposing the inequality Ux δ) u. First, let us prove that X is a compact subset of R The following two equalities follow from the fact that Ux δ) is continuous, non-decreasing in δ if x 1 x 2, and non-increasing if x 1 x 2 : { x : Ux δ) u for some δ [0 1] x 1 x 2} = { x : Ux 1) u x 1 x 2} { x : Ux δ) u for some δ [0 1] x 1 x 2} = { x : Ux 0) u x 1 x 2} The sets on the right-hand sides of these equalities are closed sets. Then X can be written as the union of two closed sets, intersected with a bounded set: X = { x : Ux 1) u x 1 x 2} { x : Ux 0) u x 1 x 2}) 0 x] 2 and thus is compact. Notice that x/ X if x j = 0forsomej because of Assumption 2 and u >. Therefore, X is a compact subset of R Next, let us define x and u and the time period T.Givenanyε>0, set x = 4/ε. Goods market clearing implies that 48) P x j t > x s ) ε/4 forall t for j = 1 2 To prove this, notice that 1 = x j tω) dpω s) x j tω) dpω s) x j t ω)>4/ε 4/ε)P x j t > 4/ε s ) 2014 The Econometric Society DOI: /ECTA8911

2 2 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI which gives the desired inequality. Let u be an upper bound for the agents utility function u ) from Assumption 2). Choose a scalar u<u such that u Ux 0 δ 0 ) u u ε 8 for all initial endowments x 0 and initial beliefs δ 0.Suchau exists because Ux 0 δ 0 )>, as initial endowments are strictly positive by Assumption 3, and there is a finite number of types. Then notice that Ux 0 δ 0 ) E[v t h 0 ] for all initial histories h 0, because an agent always has the option to refuse any trade. Moreover, E [ v t h 0] P v t <u h 0) u + P v t u h 0) u Combining these inequalities and rearranging gives P v t <u h 0) u Ux 0 δ 0 ) u u ε 8 Taking unconditional expectations shows that Pv t <u) ε/8. Since Ps) = 1/2, it follows that 49) Pv t <u s) ε/4 forallt for all s Choose T so that 50) P u t v t >u/2 s ) ε/4 forallt T This can be done by Lemma 2, given that almost sure convergence implies convergence in probability. We can then set u = u/2. Finally, we check that Px t X s) 1 ε for all t T, using the following chain of inequalities: Px t X s) P x t 0 x] 2 Ux t δ t ) u s ) P x t 0 x] 2 v t u u t v t u/2 s ) 1 j 1 ε P x j t > x s ) Pv t <u s) P u t v t >u/2 s ) The first inequality follows because Ux t ω) δ t ω)) u implies Ux t ω) δ) u for some δ [0 1]. The second follows because v t ω) u and u t ω) v t ω) u/2 implyu t ω) = Ux t ω) δ t ω)) u/2 = u. The third follows from repeatedly applying Lemma 4. The fourth combines 48), 49), and 50).

3 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 3 S2. PROOF OF LEMMA 6 The idea of the proof is as follows. We construct a Taylor expansion to compute the utility gains for any trade. Then we define the traded amount ζ and the utility gain Δ satisfying 9) and 10). Choose any two portfolios x A x B X and any two beliefs δ A δ B [0 1] such that Mx B δ B ) Mx A δ A )>ε. Pick a price p sufficiently close to the middle of the interval between the marginal rates of substitution: p [ Mx A δ A ) + ε/2 Mx B δ B ) ε/2 ] This price is chosen so that both agents will make positive gains. Consider agent A and a traded amount ζ ζ for some ζ which we will properly choose below). The current utility gain associated with the trade z = ζ p ζ) can be written as a Taylor expansion: 51) Ux A z δ A ) Ux A δ A ) = πδ A )u x 1 A) ζ + 1 πδ A ) ) u x 2 A) p ζ + 1 πδa )u y 1) + 1 πδ A ) ) u y 2) p 2) ζ πδ A ) ) u x 2 A) ε/2) ζ + 1 2[ πδa )u y 1) + 1 πδ A ) ) u y 2) p 2] ζ 2 for some y 1 y 2 ) [x 1 A x1 A ζ] [x 2 + p ζ x2 A A ]. The inequality above follows because p Mx A δ A ) + ε/2. An analogous expansion can be done for agent B. Now we want to bound the last line in 51). To do so, we first define the minimal and the maximal prices for agents with any belief in [0 1] and any portfolio in X: { } p = min Mx δ) + ε/2 x X δ [0 1] { } p = max Mx δ) ε/2 x X δ [0 1] These prices are well defined, as X is a compact subset of R 2 ++ and u ) has continuous first derivative on R 2. Then, choose ++ ζ >0 such that, for all ζ ζ and all p [p p], the trade z = ζ p ζ) satisfies z <θand x + z and x z are in R 2 + for all x X. This means that the trade is small enough. Next, we separately bound from below the two terms in the last line of the Taylor

4 4 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI expansion 51). Let D A = D A = min x X δ [0 1] 1 πδ) ) u x 2) ε/2 min x X δ [0 1] p [p p] y [x 1 x 1 + ζ] [x 2 p ζ x 2 ] 1[ πδ)u y 1) + 1 πδ) ) u y 2) p 2] 2 Note that D A is positive, D A is negative, but D A ζ 2 is of second order. Then, there exist some ζ A 0 ζ) such that, for all ζ ζ A, D A ζ + D A ζ 2 > 0 and, by construction, Ux A z δ A ) Ux A δ A ) D A ζ + D A ζ 2 Analogously, we can find D B D B,andζ B such that, for all ζ ζ B, the utility gain for agent B is bounded from below: Ux B + z δ B ) Ux A δ B ) D B ζ + D B ζ 2 > 0 We are finally ready to define ζ and Δ.Letζ = min{ζ A ζ B } and Δ = min { D A ζ + D A ζ2 D B ζ + D B ζ2} By construction, Δ and ζ satisfy the inequalities 9) and 10). To prove the last part of the lemma, let λ = 1 2 π1) min x X {u x 1 )} min{d A D B} which, as stated in the lemma, only depends on X and ε. Using a second-order expansion similar to the one above, the utility gain associated to z = ζ pζ), for an agent with portfolio x A and any belief δ [0 1], can be bounded below: Ux A z δ) Ux A δ) π1) min x X { u x 1)} ζ + D A ζ2 Therefore, to ensure that 11) is satisfied, we need to slightly modify the construction above, by choosing ζ so that the following holds: π1) min x X {u x 1 )}ζ + D A ζ2 Δ >λ The definitions of Δ and λ and a continuity argument show that this inequality holds for some positive ζ min{ζ A ζ B }, completing the proof.

5 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 5 S3. PROOF OF LEMMA 7 For all x R 2 and all δ [0 1], define the measure G t as follows: P ) ω : x t ω) = x δ t ω) = δ s 1 G t x δ) P ) ω : x t ω) = x δ t ω) = δ s 2 if δ>1/2 0 if δ 1/2 We first prove that G t is a well-defined measure, and next, we prove properties i) iii). Since P generates a discrete distribution over x and δ for each t, toprove that G t is a well-defined measure we only need to check that Px t = x δ t = δ s 2 ) Px t = x δ t = δ s 1 ) so that G t is nonnegative. Take any δ>1/2. Bayesian rationality implies that a consumer who knows his belief is δ must assign probability δ to s 1 : δ = Ps 1 x t = x δ t = δ) Moreover, Bayes s rule implies that Ps 2 x t = x δ t = δ) Ps 1 x t = x δ t = δ) = Px t = x δ t = δ s 2 )Ps 2 ) Px t = x δ t = δ s 1 )Ps 1 ) Rearranging and using Ps 1 ) = Ps 2 ) and δ>1/2 yields Px t = x δ t = δ s 2 ) Px t = x δ t = δ s 1 ) = 1 δ < 1 δ which gives the desired inequality. Property i) is immediately satisfied by construction. Property ii) follows because Px t = x δ t = 1 s 2 ) = 0forallx, given that δ t = 1 requires that we are at a history which arises with zero probability conditional on s 2. The proof of property iii) is longer and involves the manipulation of market clearing relations and the use of our symmetry assumption. Using the assumption of uniform market clearing, find an M such that 52) x 2 ω) dpω s x 2 t 1) 1 ε for all m M t ω) m Notice that x 1 ω) dpω s x 2 t 1) t ω) m x 1 t ω) dpω s 1) = 1

6 6 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI which, combined with 52), implies that x 2 t ω) m x 1 t ω) x2 t ω)) dpω s 1 ) ε for all m M Decomposing the integral on the left-hand side gives x 1 ) 53) t x2 t dpω s1 ) + x 1 t >x2 t x t [0 m] 2 + x 1 t <x2 t x t [0 m] 2 x 2 t =x1 t x t [0 m] 2 x 1 t x2 t ) dpω s1 ) + x 1 t x2 t ) dpω s1 ) x 1 t >m x 2 t m x 1 ) t x2 t dpω s1 ) ε Let us first focus on the first three terms on the left-hand side of this expression. The second term is zero. Using symmetry to replace the third term, the sum of the first three terms can then be rewritten as 54) x 1 t >x2 t x t [0 m] 2 x 1 t x2 t ) dpω s1 ) + x 1 t >x2 t x t [0 m] 2 x 2 t x1 t ) dpω s2 ) These two integrals are equal to the sums of a finite number of nonzero terms, one for each value of x and δ with positive mass. Summing the corresponding terms in each integral, we have three cases: a) terms with δ t = δ>1/2 and Px t = x δ t = δ s 1 )>Px t = x δ t = δ s 2 ) by Bayes s rule), which can be written as x 1 x 2) Px t = x δ t = δ s 1 ) x 1 x 2) Px t = x δ t = δ s 2 ) = x 1 x 2) G t x); b) terms with δ t = δ = 1/2 andpx t = x δ t = δ s 1 ) = Px t = x δ t = δ s 2 ) by Bayes s rule), which are equal to zero, x 1 x 2) Px t = x δ t = δ s 1 ) x 1 x 2) Px t = x δ t = δ s 2 ) = 0; c) terms with δ t = δ<1/2 andpx t = x δ t = δ s 1 ) = Px t = x δ t = δ s 2 ) once more, by Bayes s rule), which can be rewritten as follows, exploiting symmetry: x 1 x 2) P x t = x 1 x 2) δ t = δ s 1 ) x 1 x 2) P x t = x 1 x 2) δ t = δ s 2 ) = x 1 x 2)[ P x t = x 2 x 1) δ t = 1 δ s 2 )

7 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 7 P x t = x 2 x 1) δ t = 1 δ s 1 )] = x 2 x 1) G t x 2 x 1) 1 δ ) Combining all these terms, the integral 54)isequalto x 1 x 2) dg t x δ) x 1 >x 2 δ>1/2 x [0 m] 2 + x 1 >x 2 δ<1/2 x [0 m] 2 x 2 x 1) dg t x 2 x 1) 1 δ ) = x 1 x 2) dg t x δ) + x 1 >x 2 δ>1/2 x [0 m] 2 = x 1 x 2) dg t x δ) x [0 m] 2 x 2 >x 1 δ>1/2 x [0 m] 2 x 1 x 2) dg t x δ) where the first equality follows from a change of variables and the second from the fact that G t is zero for all δ 1/2. We can now go back to the integral on the right-hand side of 53), and notice that the integrand x 1 t x 2 t ) in the fourth term is positive, so replacing the measure P with the measure G t,which is smaller than or equal to P, reduces the value of that term. Therefore, the inequality 53), in terms of the measure P, leads to the following inequality in terms of the measure G t : x 1 x 2) dg t ε x 2 m completing the proof of property iii). S4. PROOF OF LEMMA 3 We start with the usual convergence properties. Since the marginal rates of substitution of informed agents converge, by Proposition 1, and there is at least amassα of informed agents, using Lemmas 4 and 5 we can find a compact set X R 2 ++ and a time T such that there is a sufficiently large mass of informed agents with marginal rates of substitution sufficiently close to κ t s) within ε/2) and portfolios in X: P Mx t δ t ) κ t s) < ε/2 δt = δ I s) x t X s ) >3/4)α for all t T and for all s, where ε is defined as in Proposition 2.

8 8 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI Now we provide an important concept. We want to focus on the utility gains that can be achieved by small trades of norm less than θ), by agents with marginal rates of substitution sufficiently different from each other by at least ε/2). Formally, we proceed as follows. Take any θ>0. Using Lemma 6, we can then find a lower bound for the utility gain Δ>0 from trade between two agents with marginal rates of substitution differing by at least ε/2 and with portfolios in X, making trades of norm less than θ. It is important to notice that this is the gain achieved if the agents trade but do not change their beliefs. Therefore, it is also important to bound from below the gains that can be achieved by such trades if beliefs are updated in the most pessimistic way. This bound is also given by Lemma 6, which ensures that λδ is a lower bound for the gains of the agent offering z at any possible ex post belief where λ is a positive scalar independent of θ). Next, we want to restrict attention to agents who are close to their long-run expected utility. Per period utility u t converges to the long-run value ˆv t,by Lemma 2. We can then apply Lemma 4 and find a time period T T such that, for all t T and for all s: 55) and 56) Pu t ˆv t αδ/4 x t X s) > 1 ε/2 P Mx t δ t ) κ t s) < ε/4 ut ˆv t αδ/8 δ t = δ I s) x t X s ) >α/2 Equation 55) states that there are enough agents, both informed and uninformed, close to their long-run utility. Equation 55) states that there are enough informed agents close to both their long-run utility and to their longrun marginal rates of substitution. We are now done with the preliminary steps ensuring proper convergence and can proceed to the body of the argument. Choose any t T. By Proposition 2, two cases are possible: either i) the informed agents long-run marginal rates of substitution are far enough from each other, κ t s 1 ) κ t s 2 ) ε; or ii) they are close to each other, κ t s 1 ) κ t s 2 ) < ε, but there is a large enough mass of uninformed agents with marginal rate of substitution far from that of the informed agents, P Mx t δ t ) κ t s) 2 ε s) ε for all s. In the next two steps, we construct the desired trade z for each of these two cases, and then complete the argument in step 3. Step 1. Consider the first case, in which κ t s 1 ) κ t s 2 ) ε. In this case, an uninformed agent can exploit the difference between the informed agents marginal rates of substitution in states s 1 and s 2,makinganofferatanintermediate price. This offer will be accepted with higher probability in the state

10 10 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI The argument is as follows: with positive probability, we can reach a point where it is possible to separate the marginal rates of substitution of a group of uninformed agents from the marginal rates of substitution of a group of informed agents. This means that the uninformed agents in the first group can make an offer z to the informed agents in the second group and they will accept the offer in both states s 1 and s 2. If the probabilities of acceptance χ t z s 1 ) and χ t z s 2 ) are sufficiently close to each other, this would be a profitable deviation for the uninformed, since their ex post beliefs after the offer is accepted would be close to their ex ante beliefs. In other words, in contrast to the previous case, they would gain utility but not learn from the trade. It follows that the probabilities χ t z s 1 ) and χ t z s 2 ) must be sufficiently different in the two states, which leads to either 4) or to 5). To formalize this argument, consider the expected utility of an uninformed agent with portfolio x t and belief δ t, who offers a trade z and stops trading from then on: u t + δ t χ t z s 1 ) Ux t z 1) Ux t 1) ) + 1 δ t )χ t z s 2 ) Ux t z 0) Ux t 0) ) where u t is the expected utility if the offer is rejected and the following two terms are the expected gains if the offer is accepted, respectively, in states s 1 and s 2. This expected utility can be rewritten as 59) u t + χ t z s 1 ) Ux t z δ t ) Ux t δ t ) ) + 1 δ t ) χ t z s 2 ) χ t z s 1 ) ) Ux t z 0) Ux t 0) ) using the fact that Ux t δ t ) = δ t Ux t 1) + 1 δ t )Ux t 0) by the definition of U). To interpret 59), notice that, if the probability of acceptance was independent of the signal, χ t z s 1 ) = χ t z s 2 ), then the expected gain from making offer z would be equal to the second term: χ t z s 1 )Ux t z δ t ) Ux t δ t )). The third term takes into account that the probability of acceptance may be different in two states, that is, χ t z s 2 ) χ t z s 1 ) may be different from zero. An alternative way of rearranging the same expression yields 60) u t + χ t z s 2 ) Ux t z δ t ) Ux t δ t ) ) + 1 δ t ) χ t z s 1 ) χ t z s 2 ) ) Ux t z 1) Ux 1) ) In the rest of the argument, we will use both 59)and60). Suppose that there exist a trade z and a period t which satisfy the following properties: a) the probability that z is accepted in state 1 is large enough, χ t z s 1 )>α/4

11 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 11 and b) there is a positive mass of uninformed agents with portfolios and beliefs that satisfy 61) 62) 63) u t ˆv t α/4)δ Ux t z δ t ) Ux t δ t ) Δ Ux t z δ) Ux t δ) λδ for all δ [0 1] for some Δ>0 and λ>0. In words, the uninformed agents are sufficiently close to their long-run utility, their gains from trade at fixed beliefs have a positive lower bound Δ, and their gains from trade at arbitrary beliefs have alowerbound λδ. Now we distinguish two cases. Suppose first that χ t z s 2 ) χ t z s 1 ). Then, for the uninformed agents who satisfy 61) 63), the expected utility 59) is greater than or equal to ˆv t α/4)δ + χ t z s 1 )Δ χ t z s 2 ) χ t z s 1 ) ) λδ From individual optimality, this expression cannot be larger than ˆv t,since ˆv t is the maximum expected utility for a proposer in period t. We then obtain the following restriction on the acceptance probabilities χ t z s 1 ) and χ t z s 2 ): χ t z s 1 )1 + λ)δ αδ/4 + χ t z s 2 )λδ Since χ t z s 1 )>α/2andχ t z s 1 ) χ t z s 2 ), it follows that α/4 <1/2) χ t z s 2 ), and we obtain χ t z s 1 )1 + λ) χ t z s 2 )1/2 + λ) which is equivalent to 64) χ t z s 1 ) 1 + λ 1/2 + λ χ tz s 2 ) This shows that the probability of acceptance in state s 1 is larger than the probability of acceptance in state s 2 by a factor 1 + λ)/1/2 + λ) greater than 1. Consider next the case χ t z s 2 )<χ t z s 1 ). Then, for the uninformed agents who satisfy 61) 63), the expected utility 60) is greater than or equal to ˆv t αδ/4 + χ t z s 2 )Δ χ t z s 1 ) χ t z s 2 ) ) λδ An argument similar to the one above shows that optimality requires χ t z s 2 ) 1 + λ 1/2 + λ χ tz s 1 )

12 12 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI Some algebra shows that this inequality and χ t z s 1 )>α/2imply 65) 66) 1 χ t z s 1 )>1 α 1/2 + λ λ 1 α/2)1/2 + λ) 1 χ t z s 1 )> 1 χt z s 2 ) ) 1 α/2)1/2 + λ) α/4 giving us a positive lower bound for the probability of rejection 1 χ t z s 1 ) and showing that 1 χ t z s 1 ) exceeds 1 χ t z s 2 ) by a factor greater than 1. To complete this step, we show that there exist a trade z and a period t which satisfy properties a) and b). Notice that P Mx t δ t ) κ t s 1 ) 2 ε s 1 ) ε requires that either PMx t δ t ) κ t s 1 ) 2 ε s 1 ) ε/2 holds or PMx t δ t ) κ t s 1 ) + 2 ε s 1 ) ε/2. We concentrate on the first case, as the second is treated symmetrically. Set the trading price at p = min{κs 1 ) κs 2 )} ε/2. Lemma 6 implies that there are positive scalars Δ and λ and a trade z = ζ pζ) with z <θ that satisfy the following inequalities: 67) Ux t z δ t ) u t + Δ Ux t z δ) u t λδ for all δ [0 1] if Mx t δ t )<p ε/4 andx t X and 68) Ux t + z δ t ) u t + Δ if Mx t δ t )>p+ ε/4 andx t X Since Mx t δ t ) κ t s 1 ) < ε/4 implies Mx t δ t )>κ t s 1 ) ε/4andκ t s 1 ) ε/4 is larger than p + ε/4 by construction, conditions 56) and68) guarantee that there is a positive mass of informed agents who accept z, ensuring that χ t z s 1 )>α/2, showing that z satisfies property a). Next, we want to prove that there is a positive mass of uninformed agents who gain from making offer z. To do so, notice that κ t s 1 ) κ t s 2 ) < ε implies which implies p ε/4 = min { κ t s 1 ) κ t s 2 ) } 3/4) ε κ t s 1 ) 7/4) ε>κ t s 1 ) 2 ε P Mx t δ t )<p ε/4 s 1 ) P Mxt δ t ) κ t s 1 ) 2 ε s 1 ) ε/2 This, using Lemma 4 and condition 55), implies P Mx t δ t )<p ε/4 u t ˆv t αδ/4 x t X s 1 ) > 0 which, combined with 67), shows that the trade z satisfies property b).

13 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 13 Step 3. Here we put together the bounds established above and define the scalars β and ρ in the lemma s statement. Consider the case treated in Step 1. In this case, we can find a trade z such that the probability of acceptance conditional on each signal satisfies χ t z s 1 )>α/2andχ t z s 2 )<α/4. Therefore, in this case, condition 4) is true as long as β and ρ satisfy β α/2 and ρ 2 Consider the case treated in Step 2. In this case, we can find a trade z such that either χ t z s 1 )>α/2and64) holdor65) and66) hold. This implies that either condition 4) or condition 5) holds, as long as β and ρ satisfy β 1 α 2 1/2 + λ 1 + λ ρ 1 + λ 1/2 + λ ρ 1 α/2)1/2 + λ) 1 α/2)1/2 + λ) α/4 Setting { β = min α/2 1 α 2 { ρ = min 2 } 1/2 + λ > λ 1 + λ 1/2 + λ 1 α/2)1/2 + λ) 1 α/2)1/2 + λ) α/4 } > 1 ensures that all the conditions above are satisfied, completing the proof. S5. PROOF OF LEMMA 8 Since δ t ω) are equilibrium beliefs, Bayesian rationality requires Ps 1 δ t < ε/1 + ε)) < ε/1 + ε) for all ε>0. The latter condition implies Ps 2 δ t < ε/1 + ε)) > 1 ε/1 + ε) and thus Ps 1 δ t <ε/1 + ε)) Ps 2 δ t <ε/1 + ε)) <ε for all ε>0. Bayes s rule implies that Ps 1 δ t <ε/1 + ε)) Ps 2 δ t <ε/1 + ε)) = Pδ t <ε/1 + ε) s 1 )Ps 1 ) Pδ t <ε/1 + ε) s 2 )Ps 2 ) Combining the last two equations and using Ps 1 ) = Ps 2 ) = 1/2 yields P δ t <ε/1 + ε) s 1 ) <εp δt <ε/1 + ε) s 2 ) ε which gives the desired inequality.

14 14 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI S6. PROOF OF LEMMA 9 Let us begin from the first part of the lemma. Suppose, by contradiction, that κ t+j s 1 ) κ t s 1 ) >ε for some ε>0 for infinitely many periods. Then, at some date t, an informed agent with marginal rate of substitution close to κ t s) can find a profitable deviation by holding on to his portfolio x t for J periods and then trade with other informed agents at t + J. Let us formalize this argument. Suppose, without loss of generality, that κ t+j s 1 )>κ t s 1 ) + ε for infinitely many periods the other case is treated in a symmetric way). Next, using our usual steps and Proposition 1, it is possible to find a compact set X, atimet, and a utility gain Δ>0 such that the following two properties are satisfied: i) in all periods t T, there is at least a measure α/2 of informed agents with marginal rate of substitution sufficiently close to κ t s), utility close to its long-run level, and portfolio x t in X, that is, 69) P Mx t δ t ) κ t s) <ε/3 ut ˆv t γ J αδ/2 x t X δ t = 1 s ) >α/2 and ii) in all periods t T in which κ t+j s) > κ t s) + ε, there is a trade z such that 70) Ux z 1)>Ux 1) + Δ if Mx 1)<κ t s) + ε/3andx X and 71) Ux+ z 1)>Ux 1) + Δ if Mx 1)>κ t+j s) ε/3andx X Pick a time t T in which κ t+j s) > κ t s) + ε and consider the following deviation. Whenever an informed agent reaches time t and his portfolio x t satisfies Mx t 1) <κ t s) + ε/3 andx t X, he stops trading for J periods and then makes an offer z that satisfies 70) and71). If the offer is rejected, he stops trading from then on. The probability that this offer is accepted at time t + J must satisfy χ t+j z s 1 )>α/2, because of conditions 69) and71). Therefore, the expected utility from this strategy, from the point of view of time t,is u t + γ J χ t+j z s 1 ) Ux t z 1) u t ) >ut + γ J αδ/2 ˆv t so this strategy is a profitable deviation and we have a contradiction. The second part of the lemma follows from the first part, using Proposition 1 and the triangle inequality.

16 16 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI examples in Sections S7.1 and S7.2, and present some more technical derivations behind the examples in Sections S7.3 and S7.4. S7.1. Example 1: An Equilibrium With No Learning For this example, we introduce an additional modification to our baseline model, assuming that, in period t = 1, all informed agents get to be proposers with probability 1. In particular, in t = 1, informed agents are only matched with uninformed agents and the informed agent is always selected as the proposer. 11 If two uninformed agents meet at t = 1, each is selected as the proposer with probability 1/2. From period t = 2 onward, the matching and the selection of the proposer are as in the baseline model. That is, each agent has the same probability of meeting an informed or uninformed partner and each agent has probability 1/2 of being selected as the proposer. As pointed out above, the changes made in period t = 1 do not affect the long-run properties of the game and the general results of the previous sections still hold. For a given scalar η 0 1), to be defined below, our aim is to construct an equilibrium in which strategies and beliefs satisfy: S1. In t = 1, all proposers of type i offer z i E = η η) x i 0. All responders accept. S2. In t = 2 3, all proposers offer zero trade. S3. In t = 2 3, all uninformed responders reject any offer z that satisfies min { ϕz ϕ)z 2 1 ϕ)z 1 + ϕz 2} < 0; all informed responders reject any offer z that satisfies ϕz ϕ)z 2 < 0 if s = s 1 or 1 ϕ)z 1 + ϕz 2 < 0 if s = s 2 B1. Uninformed responders keep their beliefs unchanged after offer z i E in period t = 1 and after offer 0 in period t 2. B2. In t = 1, after an offer z z i E, uninformed responders adjust their belief to δ = 1 if they are of type 1 and to δ = 0 if they are of type 2. B3. In t = 2 3, after an offer z 0, uninformed responders adjust their belief to δ = 1ifϕz ϕ)z 2 <1 ϕ)z 1 + ϕz 2 and to δ = 0ifϕz ϕ)z 2 >1 ϕ)z 1 + ϕz 2. Notice that, in equilibrium, all agents reach endowments on the 45 degree line after one round of trading and remain there from then on. S1 S3 and B1 B3 describe strategies and beliefs in equilibrium and along a subset of off-the-equilibrium-path histories. This is sufficient to show that we have an equilibrium, since we can prove that, if other agents strategies satisfy S1 S3, 11 This requires assuming α<1/2.

17 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 17 the payoff from any deviating strategy is bounded above by the equilibrium payoff. An important element of our construction is that following off-theequilibrium-path offers at t 2, uninformed agents hold pessimistic beliefs, meaning that they expect the state s to be the one for which the present value of the offer received is smaller. This, together with the fact that all agents are on the 45 degree line starting at date 2, implies that deviating agents have limited opportunities to trade after period 1. The argument to prove that an equilibrium with these properties exists is in two steps. First, we show that zero trade from period t = 2 on is a continuation equilibrium. Second, we go back to period t = 1 and show that making and accepting the offers z i E is optimal at t = 1. To show that no trade is an equilibrium after t = 2, we use property S3. Let Vx δ)denote the continuation utility of an agent with endowment x and belief δ [0 1] at any time t This value function is independent of t since the environment is stationary after t = 2. Since other agents strategies satisfy S3, the endowment process of a deviating agent who starts at x δ) satisfies the following property: if the state is s 1, any endowment x reached with positive probability at future dates must satisfy ϕ x ϕ) x 2 ϕx ϕ)x 2 This property holds because, in s 1, neither informed nor uninformed agents will accept trades that increase the expected value of the proposer s endowment computed using the probabilities ϕ and 1 ϕ. A similar property holds in s 2, reversing the roles of the probabilities ϕ and 1 ϕ. These properties, together with concavity of the utility function, imply that the continuation utility Vx δ) is bounded as follows: 72) Vx δ) δu ϕx ϕ)x 2) + 1 δ)u 1 ϕ)x 1 + ϕx 2) In the continuation equilibrium, all agents start from a perfectly diversified endowment with x 1 = x 2. So an agent can achieve the upper bound in 72) by not trading. This rules out any deviation by proposers on the equilibrium path. A similar argument shows that the off-the-equilibrium-path responses in S3 are optimal. 13 So we have an equilibrium for t 2. The argument for no trade in periods t 2 is closely related to the no-trade theorem of Milgrom and Stokey 1982). Turning to period t = 1, consider a rich informed proposer with endowment x 1 0 in state s 1. If he deviates and offers z z 1 E, the uninformed responder s belief goes to δ = 0. Then the offer will only be accepted if Vx z 0) Vx 1 0 0) and the payoff of the proposer, if the offer is accepted, would be 12 The value function is defined at the beginning of the period, before knowing whether the game ends or there is another round of trading. 13 See Proposition 6 in Section S7.3.

18 18 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI Vx 1 0 z 1). In Section S7.3, we derive an upper bound on this payoff. We can then find parametric examples and choose η so that offering z 1 E yields a higher payoff. The reason why this is possible is that, if the proposer offers z 1 E, the uninformed responder s belief remains at δ = 1/2, so the informed proposer is able to trade at better terms in period 1. Consider next a poor informed proposer with endowment x 2 0 in state s 1.If he deviates, the uninformed responder adjusts his belief to δ = 1. Since the poor informed proposer also holds belief δ = 1, we can show that the best deviation by a poor informed proposer is to make an offer that reaches the 45 degree line for both. However, the responder s outside option is higher at δ = 1 than at δ = 1/2, because he holds a larger endowment of asset 1. This makes the participation constraint of the responder tighter than at the equilibrium offer z 2 E and allows us to construct parametric examples in which the poor informed proposer prefers not to deviate. Having shown that both the rich informed proposer and the poor informed proposer prefer not to deviate, we can then show that an uninformed proposer prefers not to deviate either, using the argument that the payoff of an uninformed agent under a deviation is weakly dominated by the average of the payoffs of an informed agent with the same endowment. In Section S7.3, we derive sufficient conditions that rule out deviations in period t = 1 and show how to construct parametric examples that satisfy these conditions. The following proposition contains such an example. PROPOSITION 4: Suppose the utility function is uc) = c 1 σ /1 σ) and the parameters σ ω ϕ) are in a neighborhood of 4 9/ / )). There is an η 0 1) and a cutoff ᾱ>0 for the fraction of informed agents in the game, such that S1 S3 and B1 B3 form an equilibrium if 0 α<ᾱ. An important ingredient in the construction of the example in the proposition is to assume that the fraction of informed agents α is sufficiently small. This puts a bound on the utility from trading in periods t 2, because it implies that an agent only gets a chance to trade with informed agents with a small probability. In particular, property S3 means that an agent who starts at x at t = 2 and only trades with uninformed agents before the end of the game can only reach endowments x that satisfy both and ϕ x ϕ) x 2 ϕx ϕ)x 2 1 ϕ) x 1 + ϕ x 2 1 ϕ)x 1 + ϕx 2 This restriction is crucial in constructing upper bounds on the continuation utility of deviating agents at date t = 1. The intuition is that the trades z i E at date t = 1 are proposed and accepted in equilibrium because the outside

20 20 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI This assumption allows us to characterize analytically the value function Vx δ)for δ = 0andδ = 1. Notice that CARA preferences do not satisfy the property lim c 0 uc) =, which was assumed in Section 2 Assumption 2). However, the only purpose of that property was to ensure that endowments stay in a compact set with probability close to 1 in equilibrium. Here, we can check directly that endowments remain in a compact set in equilibrium. So all our general results still apply. The analysis of this example proceeds in two steps. First, we characterize the equilibrium focusing on the behavior of informed agents which can be done, given that uninformed agents are in zero mass. Second, we look at the uninformed agent s problem at date t = 1 and derive conditions that ensure that his optimal strategy is to experiment, making an offer that perfectly reveals the state s. For the first step, we need to derive four equilibrium offers z i E s) which depend on the proposer type i and on the state s. In equilibrium, all proposers of type i make offer z i E s) in state s, all responders accept, and both proposer and responder reach a point on the 45 degree line. The offers z i E s) are found maximizing Vx i 0 z δ) subject to Vx i 0 + z δ) Vx i 0 δ) with δ = 1ifs 1 and δ = 0ifs 2. Since the two agents share the same beliefs, it is not difficult to show that the solution to this problem yields an allocation on the 45 degree line for both agents and that they stop trading from period t = 2 onward. For this argument, it is sufficient to use the upper bound 72), which was used for our Example 1 and also holds here. We then obtain the following proposition. 14 PROPOSITION 5: If all agents are informed, there is an equilibrium in which all agents reach the 45 degree line in the first round of trading and stop trading from then on. Before turning to our second step, however, we need to derive explicitly the form of the V function and the offers z i E s). These steps are more technical and are presented in Section S7.4. The assumption of CARA utility helps greatly in these derivations, as it allows us to show that the value function takes the form Vx)= exp{ ρx 1 }fx 2 x 1 ) for some decreasing function f and that the function f can be obtained as the solution of an appropriate functional equation. We can then turn to our second step and consider uninformed agents in period t = 1. The case of uninformed responders is easy. Since they meet informed proposers with probability 1 and these proposers make different offers 14 A detailed proof is in Section S7.4.

21 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 21 in the two states, they acquire perfect information on s and respond like informed agents, accepting the offer and reaching the 45 degree line. The case of uninformed proposers is harder. We want to show that an uninformed proposer with endowment i makes offer z i E s 1 ) if i = 1 and offer z i E s 2 ) if i = 2, where z i E s) are the offers derived above for informed agents. In other words, uninformed proposers mimic the behavior of rich informed proposers. By making these offers, uninformed proposers get to learn exactly the state s, because their offer is accepted with probability 1 in one state and rejected with probability 1 in the other. While this implies that they become informed from period t = 2 onward, it also implies that, with probability 1/2, they do not reach the 45 degree line in the first round of trading. Our characterization of the V function in Section S7.4 allows us to compute their payoff in this case and to characterize their trading in periods t = 2 3. In particular, the optimal behavior of uninformed agents who fail to trade in period t = 1is to make an infinite sequence of trades in all periods t 2 in which they are selected as proposers. The difference x 1 x 2 is reduced by a factor of 1/2each time they get to trade, so that they asymptotically reach the 45 degree line. Let us show that the offers described above are optimal for the uninformed proposer. We focus on type i = 1, as the case of i = 2 is symmetric. To prove that offering z 1 E s 1 ) is optimal at time t = 1, we need to check that: 1. offer z 1 E s 1 ) is rejected by informed agents in s 2 ; 2. offer z 1 E s 1 ) dominates any other offer accepted by informed agents only in s 1 ; 3. offer z 1 E s 1 ) dominates any offer accepted by informed agents only in state s 2 ; 4. offer z 1 E s 1 ) dominates any offer accepted by informed agents in both states. Conditions 1 to 3 are proved in Section S7.4 andholdforanychoiceofparameters. The most interesting condition is the last one, which ensures that the uninformed agent prefers to learn even though it entails not trading with probability 1/2 in the first period. In Section S7.4, we derive an upper bound for the expected utility from any offer accepted by informed responders in both s 1 and s 2. Making the following assumptions on parameters: ρ = 1 ϕ= 2/3 ω= 1 we can compute the expected utility from offering z 1 E s 1 ) and the upper bound just discussed. The values we obtain are plotted in Figure S1 for different values of γ. As the figure shows, there is a range of γ for which experimenting dominates non-experimenting, so uninformed agents offer z 1 E s 1 ). Notice that as γ goes to 1, the expected utility of the uninformed agent converges to the expected utility of the informed agent and both converge to the expected utility in a Walrasian rational expectations equilibrium. So unlike in Example 1, in this

22 22 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI FIGURE S1. Example 2. case, as the frequency of trading increases, the equilibrium payoffs converge to those of a perfectly competitive rational expectations equilibrium. As a final remark, notice that, in this example, some agents the uninformed proposers whose offer is rejected in the first round only reach an efficient allocation asymptotically. However, we can characterize the speed at which this convergence occurs. To measure distance from efficiency, let us use the distance d t x 1 t x 2 t. Since this distance is reduced by 1/2 every time the agent gets to make an offer and trade, after n trades in which the agent is selected as the proposer, the distance is reduced to 2 n d 0. It is then possible to show that, for every ɛ, there is a γ large enough that the probability of d t <ɛis largerthan 1 ɛ. That is, with γ close to 1, the allocation approaches efficiency also for uninformed first-round proposers. S7.3. Example 1: Proofs PROPOSITION 6: In the economy of Example 1, individual strategies are optimal for t 2. PROOF: Consider first a proposer, informed or uninformed, who has not deviated up to time t, so he holds an endowment x on the 45 degree line, either η η) or 1 η 1 η),andbeliefsδ [0 1].Frominequality72), the expected payoff from any deviating strategy is bounded above by δu ϕx ϕ)x 2) + 1 δ)u 1 ϕ)x 1 + ϕx 2) = u x 1)

23 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 23 given that x 1 = x 2. Making a zero offer today and not trading in all future periods achieves the upper bound ux 1 ), so the agent cannot gain by deviating. Turning to a responder, consider first an uninformed responder. Suppose he receives an off-the-equilibrium-path offer z 0att 2, with min { ϕz ϕ)z 2 1 ϕ)z 1 + ϕz 2} < 0 Suppose, without loss of generality, that ϕz ϕ)z 2 is smaller than 1 ϕ)z 1 +ϕz 2, and thus smaller than zero. From B3, his belief after receiving offer z is δ = 1. Using 72), his expected utility after accepting the offer is bounded above by u ϕ x 1 + z 1) + 1 ϕ) x 2 + z 2)) <u ϕx ϕ)x 2) = u x 1) Rejecting the offer yields the payoff ux 1 ) and strictly dominates accepting the offer. A similar argument shows optimality for informed responders. Q.E.D. The following lemma provides an upper bound on the continuation utility which will be used below. Define the function 73) Wx δ) max Ux z δ) z s.t. ϕz ϕ)z ϕ)z 1 + ϕz 2 0 and the quantity Ξα) 1 γ 1 γ + αγ/2 which is the probability of trading only with uninformed agents between any period t 2 and the end of the game. LEMMA 10: The following is an upper bound on the value function Vx δ): Vx δ) Ξα) [ δw x 1) + 1 δ)w x 0) ] + 1 Ξα) ) u for δ {0 1} PROOF: Consider first an informed agent with δ = 1. Consider a deviating strategy starting at x 1) at time t, and consider any history h t+j along which the agent only meets uninformed agents. Let xh t+j ) = x + z n be his endowment at that history, where z n are all the successful trades made along the history. Each trade z n must satisfy ϕz 1 n + 1 ϕ)z2 n 0and1 ϕ)z1 n + ϕz2 n 0,

24 24 M. GOLOSOV, G. LORENZONI, AND A. TSYVINSKI and so z = z n must satisfy the same inequalities. By the definition of W, the expected utility at history h t+j then satisfies U x h t+j) 1 ) Wx 1) Following any history in which the agent meets informed agent, the utility is bounded by the upper bound u. Taking expectation over all future histories yields the bound ΞW x 1) + 1 Ξ)u. An uninformed agent cannot do better than receiving perfect information on the state s and then re-optimize, which yields the bound in the lemma. Q.E.D. For the following result, we define 74) w max R Wx 1 0 z 1) z s.t. Wx z 0) Ux 2 0 0) PROPOSITION 7: If η satisfies 75) u1 η) > w R 76) uη) < Ux 2 0 0) 77) uη) > 1 2 Wx 2 0 0) Wx 2 0 1) there is an ˆα 0 1) such that, if α< ˆα, the strategies in S1 S3 are individually optimal. PROOF: Proposition6 shows optimality in all periods t 2, so we can restrict attention to time t = 1. Consider first the behavior of responders. We focus on responders with x 2 0 ; the case of responders with x 1 0 is symmetric. All responders are uninformed and they accept the equilibrium offer if 78) uη) 1 2 Vx 2 0 1) Vx 2 0 0) Using Lemma 10, a sufficient condition for 78)is [ 1 uη) Ξα) 2 Wx 2 0 0) + 1 ] 2 Wx 2 0 1) + 1 Ξα) ) u Assumption 77), together with lim α 0 Ξα) = 1, ensures that this condition holds for α 0.

25 DECENTRALIZED TRADING WITH PRIVATE INFORMATION 25 Consider next informed proposers. We focus on proposers with endowment x 1 0 ; the case of proposers with x 2 0 is symmetric. Suppose the proposer deviates by offering z z 1 E and the responder s belief goes to δ = 0. The z offer will be rejected if Ξα)W x z 0) + 1 Ξα) ) u<ux 2 0 0) given that the left-hand side is an upper bound on the continuation utility after accepting the offer and the right-hand side is a lower bound on the continuation utility after rejecting the offer. If the state is s 1, the proposer is a rich informed agent and his continuation utility is bounded above by Wx 1 0 z 1). The utility from deviating is then bounded above by w R α) = max Ξα)W x 1 0 z 1) + 1 Ξα) ) u z s.t. Ξα)W x z 0) + 1 Ξα) ) u Ux 2 0 0) The function w R α) is continuous in α and w R 0) = w R, so assumption 75) ensures that u1 η) > w R α) as α 0. If the state is s 2, the proposer is a poor informed agent and the utility from deviating is bounded above by w P α) = max Ξα)W x 1 0 z 0) + 1 Ξα) ) u z s.t. Ξα)W x z 0) + 1 Ξα) ) u Ux 2 0 0) When α = 0, the solution to this problem is given by perfect risk sharing with x z1 = x z2 = u 1 Ux 2 0 0)). The proposer s payoff is then w P 0) = u 1 u 1 Ux 2 0 0) )) This payoff is strictly dominated by offering z 1 E if 1 η>1 u 1 Ux 2 0 0) ) which is equivalent to assumption 76). A continuity argument ensures that u1 η) > w P α) for α 0. Finally, consider uninformed proposers. Since informed proposers can condition their strategy on the realization of s, an uninformed proposer cannot do better than the expected gain of the informed proposer s deviations. Since this gain is negative under both values of s, the expected gain is negative and the uninformed proposer strictly prefers not to deviate. Q.E.D. S Proof of Proposition 4 Given Proposition 7, we need to find parameters that satisfy conditions 75) 77). The following lemma simplifies this construction.

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