Strategic trading and welfare in a dynamic market. Dimitri Vayanos

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1 LSE Researc Online Article (refereed) Strategic trading and welfare in a dynamic market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral Rigts for te papers on tis site are retained by te individual autors and/or oter copyrigt owners. Users may download and/or print one copy of any article(s) in LSE Researc Online to facilitate teir private study or for non-commercial researc. You may not engage in furter distribution of te material or use it for any profit-making activities or any commercial gain. You may freely distribute te URL (ttp://eprints.lse.ac.uk) of te LSE Researc Online website. Cite tis version: Vayanos, D. (999). Strategic trading and welfare in a dynamic market [online]. London: LSE Researc Online. Available at: ttp://eprints.lse.ac.uk/arcive/449 Tis is an electronic version of an Article publised in te Review of economic studies, 66 (2) pp Blackwell Publising. ttp:// ttp://eprints.lse.ac.uk Contact LSE Researc Online at: [email protected]

2 Strategic Trading and Welfare in a Dynamic Market Dimitri Vayanos April 998 Abstract Tis paper studies a dynamic model of a nancial market wit strategic agents. Agents receive random stock endowments at eac period and trade to sare dividend risk. Endowments are te only private information in te model. We nd tat agents trade slowly even wen te time between trades goes to. In fact, welfare loss due to strategic beavior increases as te time between trades decreases. In te limit wen te time between trades goes to, welfare loss is of order =, and not = 2 as in te static models of te double auctions literature. Te model is very tractable and closed-form solutions are obtained in a special case. MIT Sloan Scool of Management, 5 Memorial Drive E52-437, Cambridge MA , tel , [email protected]. I tank Drew Fudenberg, Pete Kyle, Walter ovaes, Jon Roberts, Lones Smit, Jean Tirole, Jean-Luc Vila, Xavier Vives, Jiang Wang, and Ingrid Werner, seminar participants at Berkeley, Boston University, Cicago, Duke, Harvard, LSE, MIT, ortwestern, Princeton, Stanford, Tel-Aviv University,Toulouse, UAB, UBC, UCLA, UCSD, ULB, Wasington University at St Louis, and Yale, participants at te ESF Summer Symposium at Gerzensee and at te AFA meetings in Boston for very interesting comments. I am especially grateful to Hyun Song Sin (te editor) and two anonymous referees for comments tat greatly improved te paper. Lee-Bat elson, Erik Stuart, and Muamet Yildiz provided very competent researc assistance. JEL os C73, D44, G.

3 Introduction Large traders, suc as dealers, mutual funds, and pension funds, play an increasingly important role in nancial markets. Tese agents' trades exceed te average daily volume of many securities and, according to a number of empirical studies, ave a signicant price impact. Recent studies ave also sown tat large agents strategically reduce te price impact of teir trades by spreading tem over several days. 2 One explanation for te price impact of large trades is tat tey reveal inside information. However, since large traders do not outperform te market, te majority of teir trades cannot be attributed to suc information. 3 In tis paper we study a dynamic model of a nancial market wit large agents wo trade to sare risk. We address te following questions. First, wat trading strategies do large agents employ in order to minimize teir price impact? Second, wat inuences market liquidity? Tird, ow well does te market perform its basic function of matcing supply and demand, i.e. ow small is te fraction of te gains from trade lost because of strategic beavior? We consider a discrete-time, innite-orizon economy wit a consumption good and two investment opportunities. Te rst is a riskless tecnology and te second is a risky stock tat pays a random dividend at eac period. Te only agents in te economy are innitely-lived, risk-averse large traders. For simplicity, no trades are motivated by inside information, i.e. dividend information is public. Instead, agents receive random stock endowments at eac period, and trade to sare dividend risk. Trades ave a price impact because tere is a nite number of risk-averse agents. An agent's endowment is private information. Trade in te stock takes place at eac period and is organized as a Walrasian auction were agents submit demand functions. Our assumptions t particularly well inter-dealer markets, i.e. markets were dealers trade to sare te risk of te inventories tey accumulate from teir customers. Inter-dealer markets are very active for government bonds, foreign excange, and London Stock Excange and asdaq stocks. 4 Dealers are large and generally are te only participants in tese markets. Teir endowments are te trades tey receive from teir customers, and are private information. Finally, in many inter-dealer markets

4 trade is centralized and conducted troug a limit-order book Tis paper is related to tree dierent literatures: te market microstructure literature, te literature on durable goods monopoly, and te literature on double auctions. Te relation to te market microstructure literature is troug te asset trading process. Starting wit Glosten and Milgrom (985) and Kyle (985), te market microstructure literature as mainly focused on ow inside information is revealed troug te trading process. Te literature carefully models insiders' strategic beavior but takes as exogenous te beavior of agents wo trade for reasons oter tan inside information. 5 Tis paper focuses instead on te strategic beavior of tese agents. Our large traders are similar to insiders to some extent, since tey are trying to minimize price impact. However, te \noise" traders, wo are essential for trade in an insider model, are not present in our model. Tis paper is also closely related to te literature on durable goods monopoly. 6 Large traders as well as durable goods monopolists delay trades in order to practice price discrimination. In bot cases tere is a welfare loss due to delaying trades: in tis paper dividend risk is not optimally sared wile in te durable goods monopoly literature future gains are discounted. Te main dierence from most of tis literature, were te monopolist's cost is public information, is tat agents' endowments are private information. 7 Moreover, by assuming tat private information is continuously generated over time (since agents receive endowments at eac period), tis paper diers from previous models wit private information. 8 Finally, tis paper is related to te double auctions literature since it studies ow strategic beavior and welfare loss depend on te size of te market. Most of tis literature considers static models wit risk-neutral agents wo ave - demands. 9 Instead, we consider a more realistic (in a nancial market framework) dynamic model were agents are risk-averse and ave multi-unit demands. Our results are te following. First, agents trade slowly even wen te time between trades,, goes to, i.e. even wen tere are many trading opportunities. Te increase in trading opportunities as a direct and an indirect eect on agents' beavior. To determine te direct eect, we assume tat price impact stays constant. Wit more trading opportunities, agents ave more exibility. Tey break teir 2

5 trades into many small trades and, at te same time, complete teir trades very quickly. Price impact increases, owever, since a trade signals many more trades in te same direction. Te increase in price impact induces agents to trade more slowly, wic furter increases price impact, and so on. Agents trade slowly because of tis indirect eect. Our result is contrary to te Coase conjecture studied in te durable goods monopoly literature. To compare our result to tat literature, we study te case were agents' endowments are public information. Consistent wit te Coase conjecture, agents trade very quickly as goes to. We next study te welfare loss due to strategic beavior. Our second result is tat welfare loss increases as te time between trades,, decreases. Terefore, welfare loss is maximum in te limit wen goes to. Our result implies tat in te presence of private information, dynamic competitive and non-competitive models dier more tan teir static counterparts. We nally study ow quickly te market becomes competitive, i.e. ow quickly welfare loss goes to, as te number of agents,, grows. Our tird result is tat welfare loss is of order = 2 for a xed, but of order = in te limit wen goes to. Terefore, in te presence of private information, dynamic non-competitivemodels become competitive more slowly tan teir static counterparts. Te = 2 result was also obtained in te static models of te double auctions literature. A practical implication of our results is tat aswitc from a discrete call market toacontinuous market may reduce liquidity and not substantially increase welfare. Our results also suggest tat if dealers' trades in te customer market are disclosed immediately, trading in te inter-dealer market will be more ecient. Te rest of te paper is structured as follows. In section 2we present te model. In section 3 we study te bencmark case were agents beave competitively and take prices as given. In sections 4 and 5 we assume tat agents beave strategically. In section 4 we study te case were agents' endowments are private information, and in section 5 we study te case were endowments are public information. Section 6 presents te welfare analysis and section 7 contains some concluding remarks. All proofs are in te appendix. 3

6 2 Te Model Time is continuous and goes from to. Activity takes place at times `, were ` = 2 :: and >. We refer to time ` as period `. Tere is a consumption good and two investment opportunities. Te rst investment opportunity is a riskless tecnology wit a continuously compounded rate of return r. One unit of te consumption good invested in tis tecnology at period ` ; returns e r units at period `. Te second investment opportunity is a risky stock tat pays a dividend d` at period `. We set d = d and d` = d`; + `: (2.) Te dividend sock ` is independent of ` for ` 6= `, i.e.dividends follow a random walk, and is normal wit mean and variance 2. All agents learn ` at period `, i.e. dividend information is public. Tere are innitely-lived agents. Agent i consumes c i ` at period `. His utility over consumption is exponential wit coecient of absolute risk-aversion and discount rate, i.e. ; X `= exp(;c i ` ; `): (2.2) Agent i is endowed wit M units of te consumption good and e sares of te stock at period. At period `, `, e is endowed wit i ` sares of te stock and ;d` ; e ;r i ` units of te consumption good. Te consumption good endowment is te negative of te present value of expected dividends, d`=( ; e ;r ), times te stock endowment, i `. Te stockendowment, i `, is independentof i ` for i 6= i or ` 6= `, independent of `, and normal wit mean and variance 2. We generally assume tat e i ` is private information to agent i and is revealed to im at period `. In section 5 we study te public information case were all agents learn i ` at period `. Trade in te stock takes place at eac period `. Te trading mecanism is a Walrasian auction as in Kyle (989). Agents submit demands tat are continuous functions of te price p`. Te market-clearing price is ten found and all trades take place at tis price. If tere are many market-clearing prices, te price wit minimum 4

7 absolute value is selected (if tere are ties, te positive price is selected). If tere is no market-clearing price, tere is eiter positive excess demand at all prices or negative excess demand at all prices, since demands are continuous. In te former case te price is and all buyers receive negatively innite utility, wile in te latter case te price is ; and all sellers receive negatively innite utility. Te sequence of events at period ` is as follows. First, agents receive teir endowments and learn te dividend sock, `. ext, trade takes place. Ten, te stock pays te dividend, d`, and nally agents consume c i `. We denote by M i ` and e i ` te units of te consumption good and sares of te stock tat agent i olds at period `, after trade takes place and before te dividend is paid. We allow M i ` and e i ` to negative, interpreting tem as sort positions. We denote by x i `(p`) te demand of agent i at period `. We only make explicit te dependence of x i `(p`) on te period ` price, p`. However, x i `(p`) can depend on all oter information available to agent i at period `, i.e. ` for ` `, p` for ` <`, i ` for ` `, and, in te public information case, j ` for j 6= i and ` `. 5

8 3 Te Competitive Case In tis section we study te bencmark case were agents beave competitively and take prices as given. We rst dene candidate demands and deduce market-clearing prices. We ten provide conditions for demands to be optimal given te prices. 3. Candidate Demands and Prices Te demand of agent i at period ` is x i `(p`) =Ad` ; Bp` ; a(e i `; + i `): (3.) Demand, x i `(p`), is a linear function of te dividend, d`, te price, p`, and agent i's stock oldings before trade at period `. Stock oldings are te sum of stock oldings after trade at period ` ;, e i `;, and of te stock endowment at period `, i `. Te parameters A, B, and a are determined in section 3.2. Te market-clearing condition, X i= x i `(p`) = (3.2) and te denition of x i `(p`) imply tat te priceatperiod` is P p` = A B d` ; a i=(e i `; + i `) : (3.3) B Te price, p`, is a linear function of te dividend, d`, and of average stock oldings, P i=(e i `; + i `)=. Te stock oldings of agent i after trade at period ` are e i ` =(e i `; + i `)+x i `(p`) (3.4) i.e. are te sum of stock oldings before trade, e i `; + i `, andof te trade, x i `(p`). (p` now denotes te market-clearing price, 3.3.) To determine x i `(p`), we plug p` back in te demand 3. and get P j=(e j `; + j `) x i `(p`) =a Equations 3.4 and 3.5 imply tat ; (e i `; + i `) : (3.5) e i ` =(; a)(e i `; + i `)+ap j=(e j `; + j `) : (3.6) 6

9 Stock oldings after trade are a weigted average of stock oldings before trade and average stock oldings. Trade tus reduces te dispersion in stock oldings. Te parameter a measures te speed at wic disperse stock oldings become identical, and dividend risk is optimally sared. In section 3.2 we sowtata is equal to. In te competitive casestock oldings become identical after one trading round. 3.2 Demands are Optimal Our candidate demands and prices were dened as to satisfy te market-clearing condition. To sow tat tey constitute a competitive equilibrium, we only need to sow tat demands are optimal given te prices. In tis section we study agents' optimization problem and provide conditions for demands to be optimal. Te conditions are on te parameters A, B, and a. We formulate agent i's optimization problem as a dynamic programming problem. Te \state" at period ` is evaluated after trade takes place and before te stock pays te dividend. Tere are four state variables: te agent's consumption good oldings, M i `, te dividend, d`, te agent's stock oldings, e i `, and te average stock oldings, P e j= j `=. Tere are two control variables cosen between te state at period ` ; and te state at period `: te consumption, c i `;, and te demand, x i `(p`). Te dynamics of M i ` are given by te budget constraint M i ` = e r (M i `; + d`; e i `; ; c i `; ) ; d` ; e ;r i ` ; p`x i `(p`): (3.7) Te agent beaves competitively and takes prices as given. Terefore te price, p`, in te budget constraint is independent of te agent's demand and given by equation 3.3. Te dynamics of d` and e i ` are given by equations 2. and 3.4, respectively. (Equation 3.6 gives te dynamics of e i ` only wen te agent submits is equilibrium demand.) Finally, tedynamics of P j= e j `= are given by P j= e j ` P j=(e j `; + j `) = (3.8) since average stock oldings are equal before and after trade. ote tat te agent does not observe oter agents' stock oldings directly. However e can infer average stock oldings from te price, p`, using equation

10 Summarizing, agent i's optimization problem, (P c ), is subject to sup c i ` x i `(p`) ;E ( X `= exp(;c i ` ; `)) P p` = A B d` ; a i=(e i `; + i `) B M i ` = e r (M i `; + d`; e i `; ; c i `; ) ; d` ; e ;r i ` ; p`x i `(p`) and te transversality condition d` = d`; + ` e i ` =(e i `; + P i `)+x i `(p`) P e j= j ` j=(e j `; + j `) = lim ` E V c M i ` d` e i ` P j= e j ` were V c is te value function. 2 Our candidate value function is P V c M e j= j ` i ` d` e i ` = ;exp(;( ; e;r M i `+d`e i `+F (Q exp(;`)= (3.9) B@ e i ` P j= e j ` In expression 3., F (Q v) = (=2)v t Qv for a matrix Q and a vector v (v t transpose of v), Q is a symmetric 2 2 matrix, and q is a constant. CA)+q)): (3.) is te In proposition A., proven in appendix A, we provide sucient conditions for te demand 3. to solve (P c ), and for te function 3. to be te value function. Te conditions are on te parameters A, B, a, Q, andq, and are derived from te Bellman equation ( sup c i `; x i `(p`) V c M i `; d`; e i `; P j= e j `; ;exp(;c i `; ) + E`; V c M i ` d` e i ` P j= e j ` = ) exp(;) : (3.) Tere are two sets of conditions, te \optimality conditions" and te \Bellman conditions". To derive te optimality conditions, we write tat te demand 3. maximizes te RHS of te Bellman equation. To derive te Bellman conditions, we write tat te value function 3. solves te Bellman equation. 8

11 We now derive euristically te optimality conditions and later use tem to compare te competitive case to te private and public information cases. Tere are 3 optimality conditions. Te rst optimality condition concerns A=B, te sensitivity of price to te dividend. To derive tis condition, we set e i `; + i ` and P j=(e j `; + j `)= to, and d` to. If agent i submits te demand 3., is trade is by equation 3.5. Suppose tat e modies is demand and buys x sares. His oldings of te consumption good decrease by p`x, wic is (A=B)x by equation 3.3. His stock oldings become x, wile average stock oldings remain equal to. Equation 3. implies tat te value function does not cange in te rst-order in x if ; ; e;r A +=: (3.2) B Terefore, A=B, te sensitivity of price to te dividend, is =( ; e ;r ), te present value of expected dividends. Te second optimality condition concerns a=b, te sensitivity of price to average stock oldings. To derive tis condition, we set d` to, and e i `; + i ` and P j=(e j `; + j `)= to. If agent i submits te demand 3., is trade is by equation 3.5. Suppose tat e modies is demand and buys x sares. His oldings of te consumption good decrease by p`x, wic is ;(a=b)x by equation 3.3. His stock oldings become x, wile average stock oldings remain equal to. Te value function does not cange in te rst-order if ; e ;r (Q i j denotes te i j't element of te matrix Q.) a B + Q + Q 2 =: (3.3) Te tird optimality condition concerns te speed of trade, a. To derive tis condition, we setd` and P j=(e j `; + j `)= to, and e i `; + i ` to. Agent i tus needs to sell one sare to te oter agents. If e submits te demand 3., e sells a sares. Suppose tat e modies is demand and sells x fewer sares. His oldings of te consumption good do not cange since p` =. His stock oldings become ; a +x, wile average stock oldings remain equal to. Te value function does not cange in te rst-order if ( ; a)q =: (3.4) 9

12 Equation 3.4 implies tat a =. In te competitive case te agent equates is marginal valuation to te price, and sells te one sare immediately. His stock oldings become equal to average stock oldings, and dividend risk is optimally sared. To derive te Bellman conditions we need to compute te expectation of te value function. Te value function is te exponential of a quadratic function of a normal vector, and its expectation is complicated. Terefore, te Bellman conditions are complicated as well. In appendix A we sow tat te optimality conditions and te Bellman conditions can be reduced to a system of 3 non-linear equations in te 3 elements of te symmetric 2 2 matrix Q. Te Bellman conditions simplify dramatically in te limit wen endowment risk, 2 e, goes to. In proposition A.2, proven in appendix A, we solve te non-linear system in closed-form for 2 e = and use te implicit function teorem to extend te solution for small 2 e. Te Bellman conditions ave avery intuitive interpretation for 2 e =. To illustrate tis interpretation, we determine Q + Q 2 and Q using te Bellman conditions. Te expressions for Q + Q 2 and Q will also be valid in te private and public information cases. In te next two sections we will combine tese expressions wit te optimality conditions, to study market liquidity and te speed of trade. we get Adding up te Bellman conditions for Q and Q 2,i.e. equations A.23 and A.25, Q + Q 2 =(; 2 +(Q + Q 2 ))e ;r : (3.5) Te LHS, Q + Q 2, represents agent i's marginal benet of olding x sares at period `, wen d` = and e i ` = P j= e j ` =. Tis marginal benet is te sum of two terms. Te rst term, ; 2 e ;r, represents te marginal benet of olding x sares between periods ` and ` +. Tis term is in fact a marginal cost, due to dividend risk. Te marginal cost is increasing in te coecient of absolute risk-aversion,, and in te dividend risk, 2. Te second term, (Q + Q 2 )e ;r, represents te marginal benet of olding x sares at period ` +. It is te same as te marginal benet at period ` (except for discounting) because d`+, e i `+, and P e j= j `+ are equal in expectation to teir period ` values. Te Bellman conditions simplify for 2 e = because te marginal benet between periods ` and ` +is only

13 due to dividend risk. For 2 e >, te marginal benet is also due to endowment risk and is a complicated function of Q. Equation 3.5 implies tat Q + Q 2 = ; 2 e ;r : (3.6) ;r ; e Terefore, Q +Q 2 is simply a present value of marginal costs due to dividend risk. Te Bellman equation for Q is equation A.23, i.e. Q =(; 2 +(; a) 2 Q )e ;r : (3.7) Te LHS, Q, represents te marginal benet of olding x sares at period `, wen d` = P j= e j ` =and e i ` =. Tis marginal benet is te sum of two terms. Te rst term represents te marginal benet of olding x sares between periods ` and ` +. Te second term represents te marginal benet of olding ( ; a)x sares at period ` +. 3 Tis marginal benet is obtained from te marginal benet at period ` by discounting and multiplying by ( ; a) 2. Te \rst" ; a comes because e i `+ is ; a instead of (in expectation) and te \second" because we consider ( ; a)x sares instead of x. Equation 3.7 implies tat Terefore, Q 2 e ;r Q = ; : (3.8) ; ( ; a) 2 ;r e is a present value of marginal costs due to dividend risk, exactly as Q + Q. However, te discount rate is iger tan r, and incorporates te parameter a tat measures te speed of trade. Tis is because for Q + Q 2 agent expects to old one sare forever, wile for Q sare over time. te te agent expects to sell te

14 4 Te Private Information Case In tis section we study te case were agents beave strategically. We refer to tis case as te private information case in order to contrast it to te case studied in te next section, were agents beave strategically and were endowments are public information. We rst construct candidate demands, and provide conditions for tese demands to constitute a as equilibrium. We ten study market liquidity and te speed of trade. 4. Demands Te demand of agent i at period ` is x i `(p`) =Ad` ; Bp` ; a(e i `; + i `): (4.) Demand is given by te same expression as in te competitive case. Terefore te price, te trade, and stock oldings after trade are also given by te same expressions. Te parameters A, B, and a will owever be dierent. In particular, te parameter a, tat measures te speed of trade, will be smaller tan. ote tat agent i's demand does not depend on is expectation of oter agents' stock oldings, wic is Pj6=i P P e j `; since stock oldings are j6=i(e j `; + j `). 4 In fact, if we introduce a term in P j6=i e j `; in te demand 4., we will nd tat its coecientis. Tis is surprising: if j6=i e j `; increases, agent i expects larger future sales from te oter agents. His demand at period ` sould tus decrease, olding price constant. Demand does not cange because \olding price constant" P means tat te period ` stock endowments, Pj6=i j `, are suc tat stock oldings, j6=i(e j `; + j `), remain constant. Our candidate demands constitute a as equilibrium if it is optimal for agent i to submit is candidate demand wen all oter agents submit teir candidate demands. 5 We now study agent i's optimization problem and provide conditions for te demand 4. to be optimal. Te only dierence between tis optimization problem and te optimization problem in te competitive case, concerns te price, p`. In te competitive case p` is independent of te agent's demand and given by equation 3.3. In te 2

15 private information case p` is given by te market-clearing condition X j6=i (Ad` ; Bp` ; a(e j `; + j `)) + x i `(p`) =: (4.2) Te price in te private information case coincides wit te price in te competitive case only wen agent i submits is equilibrium demand. Agent i's optimization problem, (P pr ), is subject to X j6=i sup c i ` x i `(p`) ;E ( X `= exp(;c i ` ; `)) (Ad` ; Bp` ; a(e j `; + j `)) + x i `(p`) = M i ` = e r (M i `; + d`; e i `; ; c i `; ) ; d` ; e ;r i ` ; p`x i `(p`) and te transversality condition d` = d`; + ` e i ` =(e i `; + P i `)+x i `(p`) P e j= j ` j=(e j `; + j `) = lim ` E V pr M i ` d` e i ` P j= e j ` exp(;`)= were V pr is te value function. Our candidate value function is P V pr M e j= j ` i ` d` e i ` = ;exp(;( ; e;r M i `+d`e i `+F (Q B@ e i ` P j= e j ` CA)+q)): (4.3) Te value function is given by te same expression as in te competitive case. Te parameters Q and q will owever be dierent. In proposition B., proven in appendix B, we provide sucient conditions for te demand 4. to solve (P pr ), and for te function 4.3 to be te value function. Tese conditions are te optimality conditions and te Bellman conditions. Te rst two optimality conditions are te same as in te competitive case, namely ; ; e;r A B += (4.4) 3

16 and ; e ;r a B + Q + Q 2 =: (4.5) Te conditions are te same as in te competitive case because tey are derived for e i `; + i ` equal to P j=(e j `; + j `)=, and equal to or. In bot cases agent i does not trade if e submits te demand 4.. Terefore, if e modies is demand, te rstorder cange in te value function will be independent of weter te price canges or not. To derive te tird optimality condition, we set d` and P j=(e j `; + j `)= to, and e i `; + i ` to. Agent i tus needs to sell one sare to te oter agents. If e submits te demand 4., e sells a sares. Suppose tat e modies is demand and sells x fewer sares. Te market-clearing condition 4.2 implies tat p` increases by x=( ; )B, and tus becomes x=( ; )B instead of. Terefore te agent's oldings of te consumption good increase by (x=( ; )B)(a ; x). His stock oldings become ; a + x, wile average stock oldings remain equal to. Te value function does not cange in te rst-order if ; e ;r a ( ; )B +(; a)q =: (4.6) Condition 4.6 diers from condition 3.4 in te competitive case, because of te rst term. Tis term represents te price improvement from trading more slowly. Because of tis term, a will be smaller tan. Te Bellman conditions are te same as in te competitive case. In appendix B we sow tat te optimality conditions and te Bellman conditions can be reduced to a system of 4 non-linear equations in a and in te 3 elements of te symmetric 2 2 matrix Q. In proposition B.2, proven in appendix B, we solve te non-linear system in closed-form for 2 e = and use te implicit function teorem to extend te solution for small 2 e. 4.2 Market Liquidity and te Speed of Trade 4.2. Market Liquidity Before dening market liquidity, we study a=b, te sensitivity of price to average stock oldings. Combining te optimality condition 4.5 wit te Bellman condition 4

17 3.6, we get ; e ;r a B = 2 e ;r : (4.7) ;r ; e To provide some intuition for equation 4.7, we set d` to, and e i `; + i ` and P j=(e j `; + j `)= to. Agent i tus does not trade, and expects to old one sare forever. Since e does not trade, te price is equal to is marginal valuation. Te LHS of equation 4.7 corresponds to te price, wic is ;a=b. Te RHS corresponds to te marginal valuation, wic is ; 2 e ;r =( ; e ;r ), i.e. is a present value of marginal costs due to dividend risk. To dene market liquidity, we assume tat an agent deviates from is equilibrium strategy and sells one more sare. Market liquidity is te inverse of te price impact, i.e. of te cange in price. Te market-clearing condition 4.2 implies tat if agent i sells one more sare, te price decreases by =( ; )B. Terefore, price impact is =( ; )B. ote tat price impact is equal to te sensitivity of price to average stock oldings, a=b, divided by a( ;). Tis is because wen agent i deviates from is equilibrium strategy and sells one more sare, te oter agents incorrectly infer tat e did not deviate but received an endowment sock iger by =(a( ; )). Indeed, tey do not observe agent i's endowment and buy =( ; ) sares in bot cases. In te second case, price impact is te product of te increase in average stock oldings, =(a( ; )), times te price sensitivity, a=b. Equation 4.7 implies tat price impact is ( ; )B = a( ; ) 2 2 e ;r ( ; e ;r ) : (4.8) 2 ote tat price impact is large wen a is small, since a sale of one sare signals tat many more sares will follow. As te time between trades,, goes to, a will go to. Since price sensitivity goes to a strictly positive limit, price impact will go to and market liquidity to Speed of Trade We now study te parameter a tat measures te speed of trade. Combining te optimality condition 4.6 wit te Bellman condition 3.8, we get ; e ;r a ( ; )B =(; a) 2 e ;r : (4.9) ; ( ; a) 2 ;r e 5

18 To provide some intuition for equation 4.9, we set d` and P j=(e j `; + j `)= to and e i `; + i ` to. Agent i tus needs to sell one sare to te oter agents. At period ` e sells a sares. If e sells fewer sares, e trades o an increase in te price at period ` wit an increase in dividend risk at future periods. Te LHS of equation 4.9 represents te price improvement. It is proportional to te price impact, =( ; )B, and te trade, a. Te RHS represents te increase in dividend risk and is a present value of marginal costs. Using equation 4.7 to eliminate B, we get a 2 e ;r ; a(2e ;r +( ; )( ; e ;r )) + ( ; 2)( ; e ;r )=: (4.) Equation 4. gives a as a function of te number of agents,, and te time between trades,, for 2 e =. To study ow a depends on and, we use equation 4. for 2 e =and extend our results by continuity for small 2 e. In proposition 4., proven in appendix B, we study ow a depends on and. Proposition 4. For small 2 e, a increases in and. Te intuition for proposition 4. is te following. If is large, te price improvement obtained by delaying trades is small. If is large, tere are few trading opportunities. Terefore, te increase in dividend risk from delaying trades is large. We nally study a for two limit values of. Te rst value is and corresponds to te bencmark static case. In te static case a is ( ; 2)=( ; ). 6 Te second value is and corresponds to te continuous-time case. In proposition 4.2 we sow tat a is of order and tus goes to as goes to. Proposition 4.2 is proven in appendix B. Proposition 4.2 For small 2 e, a= goes to a > as goes to. To understand te implications of proposition 4.2 for te speed of trade, assume tat at calendar time t agent i olds one sare and average stock oldings are. Te agent tus needs to sell one sare, and sells a sares at time t. At timet + e needs to sell ; a sares, and sells a( ; a) sares. At timet +2 e needs to sell ( ; a) 2 sares, and so on. Terefore, at calendar time t suc tat t ; t is a multiple of, te agent needs to sell ( ; a) t ;t sares. Since a= goes to a > as goes to, ( ; a) t ;t goes to e ;a(t ;t) 2 ( ). Te agent tus sells slowly. 6

19 Te result of proposition 4.2 is somewat surprising since it implies tat agents trade slowly even wen tere are many trading opportunities. To provide some intuition for tis result, we assume tat goes to, and proceed in two steps. First, we assume tat price impact, =( ;)B, stays constant, and determine te direct eect of on a. Second, we take into account te cange in price impact, and determine te indirect eect. Equation 4.9 implies tat if price impact stays constant, a is of order p. Terefore, bot a and ( ; a) t ;t go to as goes to. Since a goes to, agents break teir trades into many small trades. At te same time, since ( ; a) t ;t goes to, agents complete teir trades very quickly. Te intuition is tat wit constant price impact, agents' problem is similar to trading against an exogenous, downward-sloping demand curve. As goes to, agents can \go down" te demand curve very quickly, acieving perfect price-discrimination and bearing little dividend risk. ote tat bot te LHS and te RHS of equation 4.9 are of order p, and tus go to as goes to. Te LHS represents te price improvement obtained by delaying. Since agents break teir trades into many small trades, te price improvement concerns a small trade and goes to. Te RHS represents te increase in dividend risk from delaying. It goes to since agents complete teir trades very quickly and bear little dividend risk. Equation 4.8 implies tat as and a go to, price impact does not stay constant, but goes to. Te intuition is tat a trade signals many more trades in te same direction. Te increase in price impact implies a decrease in a, wic implies a furter increase in price impact, and so on. To determine tis indirect eect of on a, we note tat te LHS of equation 4.9 does not go to as goes to. Indeed, te price improvement obtained by delaying is proportional to te trade, a, wic is small, and te price impact, =( ; )B, wic is large. Te product of te trade and te price impact is in turn proportional to te price sensitivity, a=b, wic goes to a strictly positive limit. Since te LHS does not go to, te RHS does not go to. Te increase in dividend risk from delaying does not go to onlywenagents trade slowly and a is of order. 7

20 Te result of proposition 4.2 is contrary to te Coase conjecture studied in te durable goods monopoly literature. According to te Coase conjecture, a durable goods monopolist sells very quickly as te time between trades goes to. Our result is contrary to te Coase conjecture because endowments are private information. Wit private information, price impact is larger tan wit public information, and agents ave an incentive to trade slowly. Price impact is larger because if an agent sells more sares, te oter agents incorrectly infer tat e received a iger endowment sock. Terefore, tey expect larger sales in te future. In next section we study te case were endowments are public information. We sow tat, as goes to, price impact goes to and agents trade very quickly. 8

21 5 Te Public Information Case In tis section we study te case were agents beave strategically and were endowments are public information. Tis case serves as a useful bencmark, allowing us to compare our results to te durable goods monopoly literature. In te interdealer market interpretation of te model, te public information case corresponds to mandatory disclosure of trades tat te dealers receive from teir customers. We rst construct candidate demands and provide conditions for tese demands to constitute a as equilibrium. We sow tat tere exists a continuum of as equilibria and select one using a trembling-and type renement. We ten study market liquidity and te speed of trade. 5. Demands Te demand of agent i at period ` is x i `(p`) =Ad` ; Bp` ; A e P j=(e j `; + j `) ; a(e i `; + i `): (5.) Te only dierence wit te private information case is tat agent i's demand depends on oter agents' stock oldings. We willgivesomeintuition for tis result later. Te price at period ` is P p` = A B d` ; A e + a i=(e i `; + i `) : (5.2) B Te only dierence wit te private information case is tat te sensitivity of price to average stock oldings is (A e + a)=b and not a=b. Agent i's trade and stock oldings after trade are given by te same expressions as in te private information case. Our candidate demands constitute a as equilibrium if it is optimal for agent i to submit is candidate demand wen all oter agents submit teir candidate demands. We now study agent i's optimization problem and provide conditions for te demand 5. to be optimal. Tere are two dierences between tis optimization problem and te optimization problem in te private information case. First, agent i's demand, x i `(p`), can depend on oter agents' stock oldings, e j `; + j ` for j 6= i, since tese are public information. Second, te market-clearing condition is X j6=i P (Ad` ; Bp` ; A k=(e k `; + k `) e ; a(e j `; + j `)) + x i `(p`) = (5.3) 9

22 instead of 4.2, since demands depend on average stock oldings. Agent i's optimization problem, (P p ), is subject to X j6=i sup c i ` x i `(p`) ;E ( X `= exp(;c i ` ; `)) P (Ad` ; Bp` ; A k=(e k `; + k `) e ; a(e j `; + j `)) + x i `(p`) = M i ` = e r (M i `; + d`; e i `; ; c i `; ) ; d` ; e ;r i ` ; p`x i `(p`) and te transversality condition d` = d`; + ` e i ` =(e i `; + P i `)+x i `(p`) P e j= j ` j=(e j `; + j `) = lim ` E V p M i ` d` e i ` P j= e j ` exp(;`)= were V p is te value function. Our candidate value function is P V p M e j= j ` i ` d` e i ` = ;exp(;( ; e;r M i `+d`e i `+F (Q B@ e i ` P j= e j ` CA)+q)): (5.4) Te value function is given by te same expression as in te private information case. In proposition C., proven in appendix C, we provide sucient conditions for te demand 5. to solve (P p ), and for te function 5.4 to be te value function. Tese conditions are te optimality conditions and te Bellman conditions. Te optimality conditions are and ; e ;r ; e ;r ; ; e;r A += (5.5) B A e + a B + Q + Q 2 = (5.6) a ( ; )B +(; a)q =: (5.7) 2

23 Conditions 5.5 and 5.7 are te same as in te private information case. Condition 5.6 is dierent, since te sensitivity of price to average stock oldings is (A e + a)=b and not a=b. Te Bellman conditions are te same as in te private information case. Since tere are as many conditions as in te private information case, but tere is te additional parameter A e, tere exists a continuum of as equilibria. Te intuition for te indeterminacy is tat in te public information case agents know te market-clearing price, and are indierent as to te demand tey submit for all oter prices. Terefore te slope of te demand function, B, is indeterminate. 7 Tis intuition is te same as in Wilson (979), Klemperer and Meyer (989), and Back and Zender (993). To select one as equilibrium, we use a renement tat as te avor of tremblingand perfection in te agent-strategic form. 8 We consider te following \perturbed" game. Eac consumption and demand coice of agent i is made by a dierent \incarnation" of tis agent. All incarnations maximize te same utility. Tey take te coices of te oter incarnations, and of te incarnations of te oter agents, as given. Consumption is optimally cosen, but demand is x i `(p`)+u i `, were x i `(p`) is te optimal demand and u i ` is a \tremble". For tractability, u i ` is independent of u i ` for i 6= i or ` 6= `. Trembles are tus independent over agents. Trembles are also independent over time, i.e. over incarnations of an agent, and tis is wy we need to consider trembling-and perfection in te agent-strategic form (tat involves incarnations) rater tan trembling-and perfection. For tractability, u i ` is normal wit mean and variance u. 2 Since te normal is a continuous distribution, agent i trembles wit probability. However, beavior as a rational avor since expected demand is x i `(p`), te optimal demand. A as equilibrium of te original game satises our renement if and only if it is te limit of as equilibria of te perturbed games as 2 u goes to. In appendix C we sow tat our renement selects one as equilibrium. Te intuition is tat an agent does not know te market-clearing price, because of te trembles of te oter agents. Terefore, e is not indierent as to te demand e submits for eac price, and te slope of te demand function can be determined. More precisely, our renement implies an additional optimality condition, tat concerns 2

24 te slope of te demand function. To derive euristically tis condition, we set d`, e i `; + i `, and P j=(e j `; + j `)= to. We also assume tat u i ` =, i.e. agent i does not tremble, and tat P j= u j `= =. If agent i submits is equilibrium demand 5., e sells one sare to te oter agents at price =B. Tis follows from te market-clearing condition X j6=i P (Ad` ; Bp` ; A k=(e k `; + k `) e ; a(e j `; + j `)+u j `)+x i `(p`) = (5.8) i.e. condition 5.3 adjusted for te trembles. If agent i modies is demand and sells x fewer sares, te price becomes =B +x=( ; )B. Te agent's oldings of te consumption good increase by (=B +x=( ; )B)( ; x) ; =B. His stock oldings become ;( ; x), wile average stock oldings remain equal to. Te value function does not cange in te rst-order if ; e ;r ; 2 ( ; )B + Q =: (5.9) In appendix C we sow tat te optimality conditions and te Bellman conditions can be reduced to a system of 3 non-linear equations in te 3 elements of te symmetric 2 2 matrix Q. In proposition C.3, proven in appendix C, we solve te non-linear system in closed-form for 2 e = and use te implicit function teorem to extend te solution for small 2 e. In te public information case, agent i's demand depends on oter agents' stock oldings, P j6=i(e j `; + j `). By contrast, in te private information case, agent i's demand does not depend on is expectation of oter agents' stock oldings, P j6=i e j `;. Te intuition for te dierence is te following. Suppose tat in te private information case P j6=i e j `; increases, olding price constant. Holding price constant means tat te period ` stock endowments, P j6=i j `, are suc tat stock oldings, Pj6=i(e j `; + j `), remain constant. Terefore, agent i does not expect larger future sales from te oter agents, and is demand remains constant. By contrast, suppose tat in te public information case P j6=i(e j `; + j `) increases, olding price constant. Holding price constant now means tat te oter agents \trembled" and by mistake sold less at period `. Terefore, agent i still expects larger future sales, and is demand decreases. 22

25 5.2 Market Liquidity and te Speed of Trade 5.2. Market Liquidity Te market-clearing condition 5.8 implies tat if agent i deviates from is equilibrium strategy and sells one more sare, te price decreases by =( ;)B. Terefore, price impact is =( ; )B. Combining te optimality condition 5.9 wit te Bellman condition 3.8, we get ( ; )B = 2 2 e ;r ( ; 2)( ; e ;r )( ; ( ; a) 2 e ;r ) : (5.) Price impact is tus dierent tan in te private information case. In fact, it is easy to ceck tat for any a 2 [ ( ; 2)=( ; )], price impact is smaller tan in te private information case. Te intuition is tat if in te private information case agent i deviates and sells more sares, te oter agents incorrectly infer tat e received a iger endowment sock. Terefore, tey expect larger sales in te future. By contrast, if in te public information case agent i deviates and sells more sares, te oter agents correctly infer tat e deviated (or \trembled"). Terefore, tey expect smaller sales in te future, and te price does not decrease by as muc. ote tat price impact is small wen a is large, i.e. wen dividend risk is sared quickly. To provide some intuition for tis result, we suppose tat agent i deviates and sells more sares. Te oter agents correctly infer tat i deviated and tat total stock oldings remain constant. Te deviation increases teir stock oldings only for a sort period, since dividend risk is sared quickly. Terefore, te price decrease is small. As te time between trades,, goes to, a will go to a strictly positive limit, i.e. agents will trade very quickly. Equation 5. implies tat price impact will go to, and market liquidity to Speed of Trade Combining te optimality condition 5.7 wit te Bellman condition 3.8, we get ; e ;r a ( ; )B =(; a) 2 e ;r : (5.) ; ( ; a) 2 ;r e Equation 5. determines a, and is te same as in te private information case. Te LHS represents te price improvement obtained by delaying. Te RHS represents te 23

26 increase in dividend risk. Using equation 5. to eliminate B, we get Equation 5.2 is valid not only for 2 e directly from equations 5.7 and 5.9. a = ; 2 ; : (5.2) = but for any 2 e, since it can be derived Te main implication of equation 5.2 is tat a is independent of te time between trades,. Terefore, agents trade very quickly as goes to. Indeed, assume tat at calendar time t agent i olds one sare and average stock oldings are. At calendar time t suc tat t ; t is a multiple of, te agent olds ( ; a) t ;t is independent of, ( ; a) t ;t quickly. sares. Since a goes to as goes to. Te agent tus sells very To provide an intuition for wy agents trade very quickly, we assume tat goes to, and proceed as in te private information case. First, we assume tat price impact stays constant, and determine te direct eect of on a. Second, we take into account te cange in price impact, and determine te indirect eect. Equation 5. implies tat if price impact stays constant, a is of order p, and tus agents trade very quickly. Tis direct eect is te same as in te private information case. Price impact does not stay constant, owever. Equation 5. implies tat if a is of order p, price impact goes to as goes to. Te decrease in price impact implies an increase in a, wic implies a furter decrease in price impact, and so on. Tis indirect eect reinforces te result tat agents trade very quickly. ote tat te indirect eect works in te opposite direction relative to te private information case. 24

27 6 Welfare Analysis In te private and public information cases, stock oldings do not become identical after one trading round. Terefore, dividend risk is not optimally sared. Tis entails a welfare loss. In tis section we dene welfare loss and study two questions. First, ow doeswelfare loss depend on te time between trades,? In particular, ow does welfare loss cange wen we move from a static to a dynamic model? Second, ow quickly does te market become competitive aste number of agents,, grows, i.e. ow quickly does welfare loss go to as grows? 6. Denition of Welfare Loss To dene welfare loss, we introduce some notation. We denote te time certainty equivalents of an agent in te competitive, private information, and public information cases by CEQ c, CEQ pr, and CEQ p, respectively. (Since agents are symmetric, certainty equivalents do not depend on i.) We also denote te time certainty equivalent of an agent in te case were trade is not allowed by CEQ n. Te \no-trade" case is studied in appendix D. Welfare loss, L, is in te private information case and L = CEQ c ; CEQ pr CEQ c ; CEQ n (6.) L = CEQ c ; CEQ p CEQ c ; CEQ n (6.2) in te public information case. Te numerators of expressions 6. and 6.2 represent te welfare loss in te private and public information cases, respectively, relative to te competitive case. We normalize tis welfare loss, dividing by te maximum welfare loss, i.e. te welfare loss in te no-trade case relative to te competitive case. Tis denition of welfare loss is te same as in te double auctions literature. Proposition 6. sows tat in te limit wen 2 e goes to, L is given by avery simple expression. Te proposition is proven in appendix E. Proposition 6. In bot te private and public information cases lim L = ( ; a)2 ( ; e ;r ) : (6.3) 2 e ; ( ; a) 2 ;r e 25

28 To provide some intuition for expression 6.3, we rewrite it as (( ; a) 2 +(; a) 4 e ;r + :: +(; a) 2(`+) e ;r` + ::)( ; e ;r ): We also assume tat at time t agent i olds one sare and average stock oldings are. Te agent tus needs to sell one sare, but only sells a sares at time t. Te welfare loss of not selling ( ; a) sares corresponds to te term ( ; a) 2. At time t + te agent needs to sell ( ; a) sares but only sells a( ; a) sares. Te welfare loss of not selling ( ; a) 2 sares, discounted at time t, is ( ; a) 4 e ;r, and so on. ote tat expression 6.3 goes from to asa goes from to. 6.2 Welfare Loss and We now study ow welfare loss, L, depends on te time between trades,. In particular, we study ow welfare loss canges wen we move from a static model ( = ) to a dynamic model. Tis exercise allows us to compare our results to te double auctions literature tat mainly considers static models. Corollary 6. follows easily from proposition 6. and is proven in appendix E. Corollary 6. Suppose tat 2 e in, but in te private information case L decreases in. is small. In te public information case L increases In te public information case, welfare loss decreases as te time between trades,, decreases. Tis result is consistent wit our results on te speed of trade. Indeed, in te public information case, agents trade very quickly as goes to. Terefore, welfare loss goes to as goes to. In te private information case, welfare loss increases as decreases. Tis result is not a simple consequence of our earlier results. Indeed, proposition 4.2 implies tat agents trade slowly as goes to. Tis means only tat welfare loss goes to a strictly positive limit as goes to. To provide some intuition for wy welfare loss increases as decreases, we assume tat at time t agent i olds one sare and average stock oldings are. Equation 4.9, tat we reproduce below, decribes te agent's trade-o. ; e ;r a ( ; )B =(; a) 2 e ;r : (6.4) ; ( ; a) 2 ;r e 26

29 Te LHS represents te price improvement obtained by selling fewer sares at time t, wile te RHS represents te increase in dividend risk. Te increase in dividend risk corresponds to te marginal welfare loss. Total welfare loss is proportional to marginal welfare loss and to ; a, te number of sares tat agent i does not sell at time t. As decreases, a decreases. If marginal welfare loss does not cange, total welfare loss increases, since agent i sells fewer sares at time t. Te cange in marginal welfare loss is equal to te cange in price improvement. Price improvement canges only sligtly, since te increase in price impact, =( ; )B, osets te decrease in te trade, a. Corollary 6. does not imply tat welfare in te private information case, CEQ pr, decreases as decreases. Indeed, as decreases, tere are more trading opportunities, and welfare in te competitive case, CEQ c, increases. Terefore, CEQ pr may increase. 9 Corollary 6. rater implies tat in te presence of private information, dynamic competitive and non-competitive models dier more tan teir static counteparts. 6.3 Convergence to a Competitive Market Finally, we study ow quickly te market becomes competitive, i.e. ow quickly welfare loss goes to, as te number of agents,, grows. Corollary 6.2 answers tis question in te limit wen 2 e goes to. Te corollary follows easily from proposition 6. and is proven in appendix E. Corollary 6.2 In te public information case, lim L = e In te private information case lim L = e ; e ;r ( ; ) 2 ; ;r: (6.5) (;)2 e ( ; ) 2 ( ; e ;r ) + o( 2 ) (6.6) and lim lim e L = ; : (6.7) 27

30 In te public information case, welfare loss is of order = 2. In te private information case, welfare loss is of order = 2, for a xed. However, in te limit wen goes to, welfare loss is of order =. Te two results are consistent: for a xed small, welfare loss is \of order" =, except wen is very large in wic case welfare loss is of order = 2. Te = 2 result was also obtained in te static models of te double auctions literature. Corollary 6.2 implies tat in te presence of private information, dynamic noncompetitive models become competitive more slowly tan teir static counterparts. To provide some intuition for tis result, we assume tat at time t agent i olds one sare and average stock oldings are. Equation 6.4 implies tat price improvement is of order =. Terefore, marginal welfare loss is also of order =. Total welfare loss is proportional to marginal welfare loss and to ; a, te number of sares tat agent i does not sell at time t. In te static case, agent i keeps te ;a sares forever. Terefore, marginal welfare loss is proportional to ; a. Since marginal welfare loss is of order =, ; a is of order =, and total welfare loss is of order = 2. (In fact, ; a ==( ; ) since in te static case a =( ; 2)=( ; ).) As goes to, a goes to and ; a goes to. Terefore, total welfare loss is of order =. For a xed small, ; a is close to, except wen is very large. Wen is very large, te benet of delaying is small and ; a is of order = as in te static case. 28

31 7 Concluding Remarks Tis paper studies a dynamic model of a nancial market wit strategic agents. Agents receive random stock endowments at eac period and trade to sare dividend risk. Endowments are te only private information in te model. We nd tat agents trade slowly even wen te time between trades goes to. In fact, welfare loss due to strategic beavior increases as te time between trades decreases. In te limit wen te time between trades goes to, welfare loss is of order =, and not = 2 as in te static models of te double auctions literature. Te model is very tractable and closed-form solutions are obtained in a special case. We made two important simplifying assumptions. Te rst assumption is tat te only agents in te market are large and strategic. Tis assumption implies tat te equilibrium price fully reveals teir endowments. If small \noise" traders were present in te market, price would not be fully revealing and large agents would not be identied immediately. However, if we introduce noise traders in tis model, we run into te innite regress problem, i.e. into an innite number of state variables. Te intuition is tat since price is not fully revealing, an agent uses is expectation of oter agents' stock oldings wen forming is demand. To form tis expectation, e uses past prices and is own past stock oldings. Since oter agents are forming teir expectations in te same way, te agent needs to form expectations of oter agents' past stock oldings, and so on. Vayanos (998) studies a model wit one large trader wo receives random stock endowments, a competitive risk-averse market-maker, and noise traders. Since te large trader as better information tan te market-maker, tere is no innite regress problem. Wen te time between trades goes to, te trading process consists of two pases. During te rst pase, te large trader sells very quickly a fraction of is endowment and is identied by te market-maker. He completes slowly is trades during te second pase. If tere are many noise traders, te large trader may \manipulate" te market, overselling during te rst pase and buying during te second pase. Te second assumption is tat dividend information is public. Tis assumption rules out trades motivated by inside information. Cau (998) studies a model wit one large trader wo receives bot inside information and random stock endowments, 29

32 a competitive market-maker, and noise traders. He also nds tat te large trader may manipulate te market. 3

33 Appendix A Te Competitive Case We rst provide conditions for te demand 3. to solve (P c ), in proposition A.. We ten sow tat te conditions can be reduced to a system of 3 non-linear equations. Finally, we solve te system in closed-form for 2 e = and extend te solution for small e, 2 in proposition A.2. A. Te Conditions To state proposition A., we dene te following symmetric 2 2 matrices. First, we dene Q by and Q 2 = ; ; e;r Q 2 2 = ; e;r Q =(; a) 2 Q a 2 B + a( ; a)q +(; a)q 2 2a 2 B + a2 Q +2aQ 2 + Q 2 2 : (A.) (A.2) (A.3) ext, we dene 2, te variance-covariance matrix of te vector ( i ` P j= j `=). Tis matrix is Finally, we dene R, R, and P by 2 = 2 e B@ CA (A.4) R = I + Q 2 (A.5) and R = Q 2 R ; Q P =(Q ; R ; 2 ;)e ;r : (A.6) (A.7) In equation A.7, ; is a 2 2 matrix wose elements are for (i j) = ( ) and oterwise. 3

34 Proposition A. Te demand 3. solves (P c ) and te function 3. is te value function, if te following conditions old. First, te optimality conditions 3.2, 3.3, 3.4, and te inequality Q <. Second, te Bellman conditions Q = P (A.8) and q = log(jrj)e;r 2( ; e ;r ) + (e;r ; r) ( ; e ;r ) ; log( e r ; ): In equation A.9, jrj denotes te determinant of te matrix R. (A.9) Proof: Te demand 3. solves (P c ) and te function 3. is te value function, if te following are true. First, te function 3. solves te Bellman equation 3., for te demand 3. and te optimal consumption. Second, te demand 3. and te optimal consumption satisfy te transversality condition 3.9. We will sow tat conditions 3.2, 3.3, 3.4, Q equation and te transversality condition. Bellman Equation <, A.8, and A.9 are sucient for te Bellman We proceed in 3 steps. First, we sow tat te optimality conditions 3.2, 3.3, 3.4, and Q < are sucient for te demand 3. to maximize te RHS of te Bellman equation. Second, we compute te expectation of te RHS w.r.t. period ` ; information. Finally, we sow tat te Bellman conditions A.8 and A.9 are sucient for te function 3. to satisfy te Bellman equation. Step : Optimal Demand Before trade at period `, agent i knows ` and i `, but not j ` for j 6= i. He tus cooses is demand, x i `(p`), to maximize ;E j ` j6=iexp(;( ; e;r ;d` +d`(e i `; + i ` + x i `(p`)) + F (Q ; e ;r i ` ; p`x i `(p`) B@ e i `; + i ` + x i `(p`) P j= (e j `;+ j `) CA))) were te price, p`, is given by equation 3.3. Te agent can infer P j6=i j ` from te price. Terefore is problem is te same as knowing P j6=i j ` and coosing a trade, 32

35 x, to maximize ; ; e;r A B d` ; a B +d`(e i `; + i ` + x)+f (Q P j=(e j `; + j `) x ; d` i ` B@ e i `; + i ` + x P j= (e j `;+ j `) If te trade 3.5 is optimal, ten te demand 3. is optimal, since it produces tis trade. Setting x = a P j=(e j `; + j `) CA): ; (e i `; + i `) +x te demand 3. is optimal if x = maximizes P ; ; e;r A B d` ; a j=(e j `; + j `) P j=(e j `; + j `) (a B ;d` i ` + d` ( ; a)(e i `; + i `)+ap j=(e j `; + j `) +x +F (Q B@ ( ; a)(e i `; + i `)+a P j= (e j `;+ j `) P j= (e j `;+ j `) +x ; (e i `; + i `) +x) CA): (A.) Conditions 3.2, 3.3, and 3.4 ensure tat te rst-order condition is satised for x =. Condition Q < ensures tat expression A. is concave. Using te denition of te matrix Q,we can write expression A. for x =as d`e i `; + F (Q Step 2: Computing te Expectation B@ e i `; + i ` P j= (e j `;+ j `) CA): (A.) We ave to compute te expectation of te period ` value function w.r.t. period `; information, i.e. w.r.t. ` and j `. Using te budget constraint 3.7 and expression A., we ave to compute E`; exp(;( ; e;r e r (M i `; + d`; e i `; ; c i `; )+(d`; + `)e i `; +F (Q B@ e i `; + i ` P j= (e j `;+ j `) 33 CA)+q) ; ): (A.2)

36 Computing expectations w.r.t ` is straigtforward. We get E ; j `exp(;( e;r e r (M i `; ; c i `; )+e r d`; e i `; ; 2 2 e 2 i `; +F (Q B@ e i `; + i ` P j= (e j `;+ j `) CA)+q) ; ): To compute expectations w.r.t. j `, we use te formula (A.3) E(exp(;(a+b t x+ 2 xt cx))) = exp(;(a; 2 bt 2 (I +c 2 ) ; b+ 2 logji +c2 j)) (A.4) were x is a n normal vector wit mean and variance-covariance matrix 2, I te n n identity matrix, a a number, b an n vector, and c an n n symmetric matrix. (Formula A.4 gives simply te moment generating function of te normal distribution for c =. We can always assume c = by also assuming tat x is a normal vector wit mean andvariance-covariance matrix 2 (I + c 2 ) ;.) We set x = B@ 2 te matrix dened by equation A.4, i ` P j= j ` CA a = ; e;r e r (M i `; ; c i `; )+e r d`; e i `; ; 2 2 e 2 i `; +F (Q B@ e i `; P j= e j `; b = Q B@ e i `; P j= e j `; CA)+q + and c = Q. Using te denitions of R and R,we can write expression A.2 as CA exp(;( ; e;r e r (M i `; ; c i `; )+e r d`; e i `; ; 2 2 e 2 i `; +F (Q ; R B@ e i `; P j= e j `; CA)+ logjrj + q) ; ): 2 Finally, using te denition of P, we can rewrite tis expression as exp(;( ; e;r e r (M i `; ; c i `; )+e r d`; e i `; 34

37 +e r F (P B@ e i `; P j= e j `; CA)+ logjrj + q) ; ): 2 (A.5) Step 3: Bellman Equation To compute te RHSofteBellman equation, we ave to maximize w.r.t. c i `; ;exp(;c i `; ) ; exp(;( ; e;r e r (M i `; ; c i `; )+e r d`; e i `; +e r F (P B@ e i `; P j= e j `; Simple calculations sow tat te maximum is ;exp(;( ; e;r CA)+ logjrj + q) ; ): 2 (A.6) M i `; + d`; e i `; + F (P + 2 logjrje;r + qe ;r + (e;r ; r) B@ e i `; P j= e j `; CA) ; ( ; e;r )log( e r ; ))): Tis is equal to te period ` ; value function if conditions A.8 and A.9 old. Transversality condition (A.7) It is easy to ceck tat, by substituting te optimal c i `; in te second term of expression A.6, we nd expression A.7 times e ;r. Terefore te expectation of te period ` value function at period ` ;, is te period ` ; value function times e ;r. Recursive use of tis equation implies equation 3.9. Q.E.D. A.2 Te System Te system will follow from te Bellman condition A.8. In tis condition, te matrix P is indirectly a function of Q, wic is in turn a function of B, a, and Q. To ave Q as te only unknown, we eliminate rst B and ten a in te denition of Q. Using condition 3.3, we can write Q as a function of a and Q, as follows Q =(; a) 2 Q Q 2 = a(2 ; a)q + Q 2 (A.8) (A.9) 35

38 and Q 2 2 = ;a(2 ; a)q + Q 2 2 : (A.2) Since a = from condition 3.4, Q becomes a function of Q only. Terefore, te system is te Bellman condition A.8, were P, R,andR are given by equations A.7, A.6, and A.5, respectively, Q by equations A.8, A.9, and A.2, and a =. Having solved tis system, we can solve fora, B, andq, using equations 3.2, 3.3, and A.9, respectively. We also ave to ceck tat inequality Q < issatised. A.3 Te Solution Proposition A.2 Te system as a solution for small 2 e. For 2 e = te solution is Q = ; 2 e ;r < Q 2 = ;Q 2 2 = ; 2 e ;2r : (A.2) ;r ; e Proof: We rst solve te system for 2 e =. We ten use te implicit function teorem to extend te solution for small e. 2 Te Solution for e= 2 For 2 e =,R =and Q = P =(Q ; 2 ;)e ;r : (A.22) To solve te system, we use equations A.8 to A.2 and A.22. We treat a as a parameter in equations A.8 to A.2, obtain Q as a function of a, and ten set a = to obtain te Q of te proposition. We treat a as a parameter because te expression of Q as a function of a will also be valid in te private and public information cases. Te equation for Q is Q = (( ; a) 2 Q ; 2 )e ;r (A.23) and is satised for Te equation for Q 2 is 2 e ;r Q = ; : (A.24) ; ( ; a) 2 ;r e Q 2 =(a(2 ; a)q + Q 2 )e ;r (A.25) 36

39 and is satised for Finally, teequation for Q 2 2 is Q 2 = a(2 ; a)q e ;r ; e ;r : (A.26) Q 2 2 =(;a(2 ; a)q + Q 2 2 )e ;r (A.27) and is satised for Setting a =,we obtain te Q of te proposition. Q 2 2 = ; a(2 ; a)q e ;r ; e ;r : (A.28) Small 2 e To extend te solution for small 2 e, we consider te function G(Q Q 2 Q e )= B@ Q ; P Q 2 ; P 2 CA : Q 2 2 ; P 2 2 Te matrices P, R, R, and Q are dened by equations A.7, A.6, A.5, A.8, A.9, and A.2, for a =and Q = Q. We use Q instead of Q to deal wit te case =. We apply te implicit function teorem to G at te point A, were 2 e =, Q = ; 2 e ;r, and Q 2 and Q 2 2 are given by expressions A.2 for Q = Q. can be, in wic casewe extend G, Q 2, and Q 2 2 bycontinuity. Te function G is equal to at A, since Q solves te system. To apply te implicit function teorem, we only need to sow tat te Jacobian matrix of G w.r.t. Q, Q 2, and Q 2 2 is invertible. Using equation A.22 (wic is valid since we compute partial derivatives for 2 e = ) it is easy to ceck te following. First, te partial derivatives of G w.r.t. Q, G 2 w.r.t. Q 2, and G 2 2 w.r.t. Q 2 2, are strictly positive. Second, te partial derivatives of G w.r.t. Q 2, Q 2 2, and G 2 w.r.t. Q 2 2 are. Terefore, te Jacobian matrix is invertible. Since Q < at A, Q < and Q < for small e. 2 Q.E.D. 37

40 B Te Private Information Case We rst provide conditions for te demand 4. to solve (P pr ), in proposition B.. We ten sow tat te conditions can be reduced to a system of 4 non-linear equations. We solve te system in closed-form for 2 e =and extend te solution for small e, 2 in proposition B.2. Finally, we prove propositions 4. and 4.2. B. Te Conditions To state proposition B., we use te matrices Q, 2, R, R, and P, dened in appendix A. Proposition B. Te demand 4. solves (P pr ) and te function 4.3 is te value function, if te following conditions old. First, te optimality conditions 4.4, 4.5, 4.6, and te inequality ; ; e;r 2 ( ; )B + Q < : (B.) Second, te Bellman conditions Q = P (B.2) and q = log(jrj)e;r 2( ; e ;r ) + (e;r ; r) ( ; e ;r ) ; log( e r ; ): (B.3) Proof: We will sow tat te optimality conditions 4.4, 4.5, 4.6, and B. are sucient for te demand 4. to maximize te RHS of te Bellman equation. Te rest of te proof is as in te competitive case. Before trade at period `, agent i knows ` and i `, but not j ` for j 6= i. He tus cooses is demand, x i `(p`), to maximize ;E j ` j6=iexp(;( ; e;r ;d` +d`(e i `; + i ` + x i `(p`)) + F (Q ; e ;r i ` ; p`x i `(p`) B@ e i `; + i ` + x i `(p`) P j= (e j `;+ j `) CA))) were te price, p`, is given by equation 4.2 and depends on x i `(p`). Instead of solving tis problem, we proceed as in Kyle (989) and solve a simpler problem. 38

41 We assume tat te agent knows te residual demand of te oter agents (i.e. knows Pj6=i j `) and simply cooses a trade, x, on tis residual demand. Te market-clearing condition 4.2 implies tat te price, p`, is A B d` ; a Pj6=i(e j `; + j `) x + B ; ( ; )B : Terefore, te agent's problem is to maximize w.r.t. x ; ; e;r A B d` ; a Pj6=i(e j `; + j `) + B ; +d`(e i `; + i ` + x)+f (Q x ( ; )B B@ e i `; + i ` + x P j= (e j `;+ j `) x ; d` i ` If te trade 3.5 solves tis problem, ten te demand 4. is optimal since it produces tis trade for all values of P j6=i j `. demand 4. is optimal if x = maximizes P ; ; e;r A B d` ; a j=(e j `; + j `) + B P j=(e j `; + j `) (a ; (e i `; + i `) CA): Dening x as in te competitive case, te x ( ; )B +x) ;d` i ` + d` ( ; a)(e i `; + i `)+ap j=(e j `; + j `) +F (Q B@ ( ; a)(e i `; + i `)+a P j= (e j `;+ j `) P j= (e j `;+ j `) +x +x CA): (B.4) Conditions 4.4, 4.5, and 4.6 ensure tat te rst-order condition is satised for x =. Condition B. ensures tat expression B.4 is concave. Proceeding as in te competitive case, we can rewrite expression B.4 for x = as d`e i `; + F (Q B@ e i `; + i ` P j= (e j `;+ j `) CA): Q.E.D. 39

42 B.2 Te System and te Solution Te rst 3 equations follow from te Bellman condition B.2, were P, R, and R are given by equations A.7, A.6, and A.5, respectively, and Q by equations A.8, A.9, and A.2. Te last equation is ; (Q + Q 2 )=(; a)q : (B.5) Tis equation follows from conditions 4.5 and 4.6 by eliminating B. Proposition B.2 Te system as a solution for small 2 e. For 2 e =te matrix Q is given by 2 e ;r Q = ; ; ( ; a) 2 e ;r Q 2 = ;Q 2 2 = a(2 ; a)q e ;r (B.6) ; e ;r and a is te unique solution in [ ( ; 2)=( ; )] of a 2 e ;r ; a(2e ;r +( ; )( ; e ;r )) + ( ; 2)( ; e ;r )=: (B.7) Proof: We rst solve te system for 2 e =. To obtain Q as a function of a, we proceed exactly as in te competitive case. To obtain equation B.7, we eliminate Q and Q 2 in equation B.5. Finally, we ceck tat inequality B. is satised. Using equation 4.5, we can write inequality B. as 2 Q + Q 2 + Q < : ; a Since <a<, Q <. Since in addition Q + Q 2 = ; 2 e ;r =( ; e ;r ) <, te inequality issatised. To extend te solution for small 2 e, we consider te function Q ; P G(Q Q 2 Q 2 2 a 2 2 e )= B@ Q 2 ; P 2 Q 2 2 ; P 2 2 (Q ; + Q 2 ) ; ( ; a)q CA : Te matrices P, R, R, and Q are dened by equations A.7, A.6, A.5, A.8, A.9, and A.2, for a = a. We use a instead of a to deal wit te case =. We 4

43 apply te implicit function teorem to G at te point A, were 2 e =, Q is given by expressions B.6 of te proposition for a = a, and a is te unique solution in ( ( ; 2)=( ; )) of (a) 2 e ;r ; a(2e ;r +( ; )( ; e ;r )) + ( ; 2)( ; e ;r )= (B.8) i.e. of equation B.7 for a =a. can be, in wic case we extend G, Q, and a by continuity. Te function G is equal to at A, since Q and a =a solve te system. To apply te implicit function teorem, we only need to sow tat te Jacobian matrix of G w.r.t. te Q i j 's and a is invertible. It is easy to ceck tat te partial derivative of a component ofg, oter tan G 2 2,w.r.t. Q 2 2 is, and tat te partial derivative ofg 2 2 w.r.t. Q 2 2 is strictly positive. Terefore, te Jacobian matrix of G is invertible if and only if te Jacobian matrix of G, G 2,andG a w.r.t. Q, Q 2, and a is invertible. Tis matrix is B@ ;(;a) 2 e ;r 2Q ( ; a)e ;r ;a(2 ; a)e ;r a ; ;2 ; ;e;r ;2Q ( ; a)e ;r Q ; To compute te determinant, we add te rst row to te second and factor out Q ( ; e ;r )=. We get CA : Q ; e ;r ;(;a) 2 e ;r 2( ; a)e ;r a ; ;2 ; ; : Subtracting te second column from te rst and expanding te determinant we get ; e ;r ; ( ; a) 2 e ;r Q + 2( ; a) 2 e ;r wic is strictly negative. Q.E.D. We now prove proposition 4.. Proof: Equation B.8 implies tat a = a is increasing in and decreasing in e ;r, i.e. increasing in. By continuity it is increasing in and for 2 e small. Q.E.D. 4

44 Finally, we prove proposition 4.2. Proof: Proposition B.2 implies tat a = a= is a continuous function of and e. 2 Equation B.8 implies tat a =( ; 2)r=2 for =and 2 e =. By continuity a > for = and 2 e small. Q.E.D. 42

45 C Te Public Information Case We rst provide conditions for te demand 5. to solve (P p ), in proposition C.. We ten study te perturbed game, and sow tat our renement implies an additional optimality condition. We next sow tat te optimality conditions and te Bellman conditions can be reduced to a system of 3 non-linear equations. Finally, we solve te system in closed-form for 2 e = and extend te solution for small e, 2 in proposition C.3. C. Te Conditions for (P p ) To state proposition C., we use te matrices Q, 2, R, R, and P, dened in appendix A. However, we cange te denitions of Q 2 and Q 2 2 to and Q 2 = ; ; e;r Q 2 2 = ; e;r a(a e + a) B 2a(A e + a) B + a( ; a)q +(; a)q 2 (C.) + a 2 Q +2aQ 2 + Q 2 2 : (C.2) Proposition C. Te demand 5. solves (P p ) and te function 5.4 is te value function, if te following conditions old. First, te optimality conditions 5.5, 5.6, 5.7, and te inequality ; ; e;r Second, te Bellman conditions 2 ( ; )B + Q < : (C.3) Q = P (C.4) and q = log(jrj)e;r 2( ; e ;r ) + (e;r ; r) ( ; e ;r ) ; log( e r ; ): (C.5) Proof: We will sow tat te optimality conditions 5.5, 5.6, 5.7, and C.3 are sucient for te demand 5. to maximize te RHS of te Bellman equation. Te rest of te proof is as in te competitive case. Before trade at period `, agent i knows ` and j `, 8j. He tus cooses is demand, x i `(p`), to maximize ;exp(;( ; e;r ;d` ; e ;r i ` ; p`x i `(p`) 43

46 +d`(e i `; + i ` + x i `(p`)) + F (Q B@ e i `; + i ` + x i `(p`) P j= (e j `;+ j `) CA))) were te price, p`, is given by equation 5.3 and depends on x i `(p`). Te agent knows te residual demand of te oter agents. Terefore is problem is te same as coosing a trade, x, on tis residual demand. Te market-clearing condition 5.3 implies tat te agent's problem is to maximize w.r.t. x P ; ; e;r A B d` ; A e k=(e k `; + k `) ; a Pj6=i(e j `; + j `) + B B ; ;d` i ` + d`(e i `; + i ` + x)+f (Q B@ e i `; + i ` + x P j= (e j `;+ j `) x x ( ; )B If te trade 3.5 solves tis problem, ten te demand 5. is optimal since it produces tis trade. x = maximizes CA): Dening x as in te competitive case, te demand 4. is optimal if ; ; e;r (a P A B d` ; A e + a j=(e j `; + j `) x + B ( ; )B P j=(e j `; + j `) ; (e i `; + i `) +x) ;d` i ` + d` ( ; a)(e i `; + i `)+ap j=(e j `; + j `) +F (Q B@ ( ; a)(e i `; + i `)+a P j= (e j `;+ j `) P j= (e j `;+ j `) +x +x CA): (C.6) Conditions 5.5, 5.6, and 5.7 ensure tat te rst-order condition is satised for x =. Condition C.3 ensures tat expression C.6 is concave. Proceeding as in te competitive case, we can rewrite expression C.6 for x = as d`e i `; + F (Q B@ e i `; + i ` P j= (e j `;+ j `) CA): Q.E.D. 44

47 C.2 Te Perturbed Game We rst provide conditions for te demands 5. to constitute a as equilibrium of te perturbed game. We ten study te limit of tese conditions as 2 u goes to, and sow tat our renement implies an additional optimality condition. Te demands 5. constitute a as equilibrium of te perturbed game if tere exist consumption coices, c i `, suc tat te following are true. First, te demand 5. maximizes te utility of te incarnation of agent i tat cooses demand at period ` (te \(i `) demand" incarnation). Second, te consumption c i ` maximizes te utility of te incarnation of agent i tat cooses consumption at period ` (te \(i `) consumption" incarnation). Bot incarnations take te coices of te oter incarnations as given, i.e. assume tat te (j `) demand incarnation cooses te demand 5. plus te noise u j `, and tat te (j `) consumption incarnation cooses c j `. In proposition C.2 we provide sucient conditions for te demands 5. to constitute a as equilibrium of te perturbed game. Tese conditions are on te parameters A, B, A e, a, Q, andq. Te parameters Q and q correspond to te value function, wic is still given by expression 5.4. To state proposition C.2, we dene te following matrices. First, te symmetric 4 4 matrix ^Q. Te top left 2 2 submatrix of ^Q is te matrix Q dened at te beginning of appendix C. Te remaining elements of ^Q are ^Q 2 3 = ; e;r ^Q 3 3 = Q ^Q 3 =(; a)q ^Q 4 = ; e;r a B ; ( ; a)q A e + a B + aq + Q 2 ^Q2 4 = ; ; e;r A e +2a ; aq ; Q 2 B ^Q3 4 = ; ; e;r B ; Q ^Q4 4 = ; e;r 2 B + Q : P We next dene ^ 2, te variance-covariance matrix of te vector ( i ` j= j `= u P i ` u j= j `), and ^R and ^R by ^R = I + ^Q ^ 2 and ^R = ^Q ^ 2 ^R; ^Q : Finally,wedeneP by equation A.7, were R now denotes te top left 22 submatrix of ^R. 45

48 Proposition C.2 Te demands 5. constitute a as equilibrium of te perturbed game, if te following conditions old. First, te optimality conditions 5.5, 5.6, 5.7, 5.9, and te inequality C.3. Second, te Bellman conditions Q = P (C.7) and q = log(j ^Rj)e ;r 2( ; e ;r ) + (e;r ; r) ( ; e ;r ) ; log( e r ; ): (C.8) Proof: We assume tat te value function after trade at period ` is given by expression 5.4. We rst sow tat te demand 5. maximizes te utility of te (i `) demand incarnation. We ten study te optimization problem of te (i ` ; ) consumption incarnation and determine c i `;. We also sow tat te function 5.4 satises te Bellman equation, i.e. tat te value function after trade at period ` ; is given by expression 5.4. Te transversality condition follows as in te competitive case. Terefore, te value function is indeed given by expression 5.4. Step : Optimal Demand Te optimal demand maximizes ;E uj ` j6=iexp(;( ; e;r ;d` +d`(e i `; + i ` + x i `(p`)) + F (Q ; e ;r i ` ; p`x i `(p`) B@ e i `; + i ` + x i `(p`) P j= (e j `;+ j `) CA))) were te price, p`, is given by equation 5.8 and depends on x i `(p`). Instead of solving tis problem, we proceed as in te private information case, and solve a simpler problem. We assume tat te residual demand of te incarnations of te oter agents is known (i.e. P j6=i u j ` is known) and coose a trade, x, on tis residual demand. Te market-clearing condition 5.3 implies tat x maximizes P ; ; e;r A B d` ; A e k=(e k `; + k `) ; a Pj6=i(e j `; + j `) + B B ; B x + ( ; )B x ; d` i ` + d`(e i `; + i ` + x)+f (Q 46 B@ e i `; + i ` + x P j= (e j `;+ j `) P j6=i u j ` ; CA):

49 If te trade a P j=(e j `; + j `) P j6=i u j ` ; (e i `; + i `) ; solves tis problem, ten te demand 5. is optimal since it produces tis trade for P all values of j6=i u j `. Setting P j=(e j `; + j `) x = a P j6=i u j ` ; (e i `; + i `) ; +x te demand 5. is optimal if x = maximizes P ; ; e;r A B d` ; A e + a j=(e j `; + j `) + B B P j=(e j `; + j `) (a ; (e i `; + i `) ; ;d` i ` + d` ( ; a)(e i `; + i `)+ap j=(e j `; + j `) +F (Q B@ ( ; a)(e i `; + i `)+a P P j= (e j `;+ j `) ; j= (e j `;+ j `) P j6=i u j ` + P j6=i u j ` +x) ; Pj6=i u j ` x ( ; )B P j6=i u j ` +x +x CA): (C.9) Conditions 5.5, 5.6, 5.7, and 5.9 ensure tat te rst-order condition is satised for x =. Condition C.3 ensures tat expression C.9 is concave. Expression C.9, evaluated for x =, corresponds to te case were te (i `) demand incarnation submits te demand 5.. Since it submits te demand 5. plus te noise u i `, expression C.9 becomes ; ; e;r (a A B d` ; A e + a B P j=(e j `; + j `) P j=(e j `; + j `) ; (e i `; + i `) ;d` i ` + d` ( ; a)(e i `; + i `)+ap j=(e j `; + j `) +F (Q B@ ( ; a)(e i `; + i `)+a P j= (e j `;+ j `) P j= (e j `;+ j `) P + u j= j ` B P j= + u i ` ; u j ` ) + u i ` ; + u i ` ; P j= u j ` P j= u j ` CA): 47

50 Proceeding as in te competitive case, we can rewrite tis expression as d`e i `; + F ( ^Q B@ (e i `; + i `) P j= (e j `;+ j `) u i ` P j= u j ` CA ): (C.) Step 2: Bellman Equation We rst compute te expectation of te period ` value function w.r.t. period ` ; information, i.e. w.r.t. `, j `, and u j `. Using te budget constraint 3.7 and expression C., we ave to compute E`; exp(;( ; e;r e r (M i `; + d`; e i `; ; c i `; )+(d`; + `)e i `; +F ( ^Q B@ (e i `; + i `) P j= (e j `;+ j `) u i ` P j= u j ` Proceeding as in te competitive case, we get )+q) ; ): CA exp(;( ; e;r e r (M i `; ; c i `; )+e r d`; e i `; ; 2 2 e 2 i `; +F ( ^Q ; ^R B@ e i `; P j= e j `; CA )+ 2 logj ^Rj + q) ; ): Finally, using te denition of P, we can rewrite tis expression as exp(;( ; e;r e r (M i `; ; c i `; )+e r d`; e i `; e r F (P CA)+ 2 logj ^Rj + q) ; ): B@ e i `; P j= e j `; (C.) Te (i ` ; ) consumption incarnation cooses consumption, c i `;, to maximize ;exp(;c i `; ) ; exp(;( ; e;r e r (M i `; ; c i `; )+e r d`; e i `; 48

51 +e r F (P B@ e i `; P j= e j `; As in te competitive case, te maximum is ;exp(;( ; e;r CA)+ 2 logj ^Rj + q) ; ): M i `; + d`; e i `; + F (P + 2 logj ;r ^Rje + qe ;r + (e;r ; r) B@ e i `; P j= e j `; CA) ; ( ; e;r )log( e r ; ))): Tis is equal to te period `; value function if conditions C.7 and C.8 old. Q.E.D. Te optimality conditions of proposition C.2 are tose of proposition C., plus condition 5.9. Te Bellman conditions are dierent tan in proposition C., since te matrices R and P are dierent, and j ^Rj 6= jrj. However, it is easy to ceck tat for 2 u = te Bellman conditions are te same. Terefore, our renement implies te additional optimality condition 5.9. C.3 Te System and te Solution Te system is te Bellman condition C.4, were P, R,andR are given by equations A.7, A.6, and A.5, respectively, Q by equations A.8, A.9, and A.2, and a = ( ; 2)=( ; ). In te public information case, equations A.9 and A.2 follow from equations C., C.2, and 5.6. Te fact tat a = ( ; 2)=( ; ) follows from equations 5.7 and 5.9. Proposition C.3 Te system as a solution for small 2 e. For 2 e =te matrix Q is given by Q = ; 2 e ;r ; (;)2 e Q ;r 2 = ;Q 2 2 = (;2) (;)2 Q e ;r ; e ;r : (C.2) Proof: We proceed exactly as in te competitive case, but wit a =( ;2)=( ; ) instead of a =. Q.E.D. 49

52 D Te o-trade Case We rst study agents' optimization problem and, in proposition D., provide a set of sucient conditions. Te conditions can be reduced to a non-linear equation. We solve te equation in closed-form for 2 e =and extend te solution for small e, 2 in proposition D.2. D. Te Optimization Problem We formulate agent i's optimization problem as a dynamic programming problem. Te state variables are te consumption good oldings, M i `, te dividend, d`, te stock oldings, e i `, and te average stock oldings, P j= e j `=. Average stock oldings will, of course, be irrelevant. We include tem to facilitate te comparison of te no-trade case to te competitive, private information, and public information cases. Te only control variable is te consumption, c i `;. Te dynamics of M i `, d`, and P e j= j `= are given by equations 3.7, 2., and 3.8, respectively. Te dynamics of e i ` are simply Agent i's optimization problem, (P n ), is subject to sup c i ` ;E ( e i ` = e i `; + i `: X `= exp(;c i ` ; `)) M i ` = e r (M i `; + d`; e i `; ; c i `; ) ; d` ; e i ` ;r P j= e j ` and te transversality condition lim ` E V n M i ` d` e i ` d` = d`; + ` e i ` =(e i `; + i `) P j=(e j `; + j `) = P j= e j ` exp(;`)= 5

53 were V n is te value function. Our candidate value function is P V n M e j= j ` i ` d` e i ` = ;exp(;( ; e;r M i `+d`e i `+F (Q B@ e i ` P j= e j ` CA)+q)): (D.) In proposition D. we provide sucient conditions for te function D. to be te value function. To state te proposition, we set Q = Q, and use te matrices 2, R, R, and P, dened in appendix A. Proposition D. Te function D. is te value function, if te Bellman conditions Q = P (D.2) and old. q = log(jrj)e;r 2( ; e ;r ) + (e;r ; r) ( ; e ;r ) ; log( e r ; ) (D.3) Proof: To sow te Bellman equation, we proceed in two steps. First, we compute te expectation of te RHS w.r.t. period ` ; information. Second, we sow tat te Bellman conditions D.2 and D.3 are sucient for te function D. to satisfy te Bellman equation. Te proof, and te proof of te transversality condition, are as in te competitive case. Q.E.D. D.2 Te Equation and te Solution Te equation follows from te Bellman condition D.2, were P, R, and R are given by equations A.7, A.6, and A.5, respectively, andq = Q. Setting Q 2 = Q 2 2 =,it is easy to ceck tat te equations corresponding to Q 2 and Q 2 2 are satised. We are left wit te equation corresponding to Q. Proposition D.2 Te equation as a solution for small 2 e. For 2 e =te solution is Q = ; 2 e ;r : (D.4) ;r ; e 5

54 Proof: To solve te equation for 2 e = we proceed as in te competitive case. To extend te solution for small 2 e, we apply te implicit function teorem to te function G(Q 2 2 )=Q ; P e at te point A, were 2 e =and Q is given by expression D.4 of te proposition. It is easy to ceck tat te partial derivative of G w.r.t. Q Q.E.D. is strictly positive. 52

55 E Welfare Analysis We rst prove proposition 6.. Proof: oting tat d = d, M i = M, and e i = e, eac CEQ is ; e ;r M + de +(Q +2Q 2 + Q 2 2 )e 2 + q: Te 4 CEQ's are equal for 2 e =. Indeed, propositions A.2, B.2, C.3, and D.2 imply tat Terefore, Q +2Q 2 + Q 2 2 Q + Q 2 = ; 2 e ;r ; e ;r Q 2 + Q 2 2 =: (E.) is te same in te competitive, private information, public information, and no-trade cases. To sow tat q is also te same, we need to sow tat R is te same. Indeed, in all 4 cases R = I. Since te 4 CEQ's are equal, we need to use l'hospital's rule and replace tem by teir partial derivatives w.r.t. 2 e at 2 e =. We rst sow tat te partial derivative of Q +2Q 2 + Q 2 2 is te same in all 4 cases. We ten compute te partial derivative ofq. Partial Derivative of Q +2Q 2 + Q 2 2 We dierentiate implicitly te Bellman condition Q = P w.r.t. 2 e at 2 e =. oting tat 2 = and R = I, we get dq = dp =(dq ; Q d 2 Q )e ;r : From tis matrix equation, we \extract" one scalar equation (equation S), multiplying from te left by x t = ( ) and from te rigt by x. To sow tat te partial derivative of Q +2Q 2 + Q 2 2 is te same in te competitive, private information, public information, and no-trade cases, we willsow tat (i) equation S is te same in all 4 cases, and (ii) te terms in dq i j in equation S are only in d(q +2Q 2 + Q 2 2 ). Since te terms obtained from Q d 2 Q are in Q + Q 2 and Q 2 + Q 2 2, and since, by equations A.8, A.9, A.2, and E., tese are te same in all 4 cases, equation S is te same in all 4 cases. Moreover since, by equations A.8, A.9, and A.2, d(q +2Q 2 + Q 2 2) =d(q +2Q 2 + Q 2 2 ) te terms in dq i j in equation S are only in d(q +2Q 2 + Q 2 2 ). 53

56 Partial Derivative of q We dierentiate te Bellman condition q = log(jrj)e;r 2( ; e ;r ) + (e;r ; r) ( ; e ;r ) ; log( e r ; ) w.r.t. 2 e at 2 e =. oting tat djrj = dr + dr 2 2 and dr = Q d 2 e = e ;r 2( ; e ;r ) (Q + 2Q 2 + Q 2 2 ): Since Q + Q 2 and Q 2 + Q 2 2 are te same in all 4 cases, l'hospital's rule implies tat in te private information case and L = (Q ) c ; (Q ) pr (Q ) c ; (Q ) n L = (Q ) c ; (Q ) p (Q ) c ; (Q ) n in te public information case, were (Q ) c, (Q ) pr, (Q ) p, and (Q ) n denote Q in te competitive, private information, public information, and no-trade cases, respectively. Using equation A.8 and propositions A.2, B.2, C.3, and D.2, we get equation 6.3. We now prove corollary 6.. Q.E.D. Proof: We use equation 6.3 for 2 e =.We ten use continuityof@l=@ to extend our results for small 2 e. In te public information case, we set a =( ; 2)=( ; ) in equation 6.3. In te private information case, we combine equations 4.9 and 6.3 into L =(; a) ( ; e;r ) e ;r Using equation 4.8, we can write tis equation as a ( ; )B : Since a increases in, L decreases in. L = ; a ; : (E.2) Q.E.D. We nally prove corollary

57 Proof: In te public information case, we set a = ( ; 2)=( ; ) in equation 6.3. In te private information case, equation 6.6 follows from equation E.2 and a =; ( ; )( ; e ;r ) + o( ; ): To prove tis fact, we set x ==( ; ), write equation 4. as xa 2 e ;r ; a(2xe ;r +(; e ;r )) ; (x ; )( ; e ;r )= and dierentiate implicitly w.r.t. x at x =, a =. Equation 6.7 follows from equation E.2 and te fact tat a goes to as goes to. Q.E.D. 55

58 otes See, for instance, Kraus and Stoll (972), Holtausen, Leftwic and Mayers (987, 99), Hausman, Lo, and McKinlay (992), Can and Lakonisok (993), and Keim and Madavan (996). 2 See, for instance, Can and Lakonisok (995) and Keim and Madavan (995,996,998). 3 For instance, te growt and growt and income funds in te 994 Morningstar CD turn over 76.8% of teir portfolios every year. However, tey underperform te market by.5%. See Cevalier and Ellison (998). 4 In te foreign excange market, inter-dealer trading is 8% of total volume (Lyons (996)). In te London Stock Excange and te asdaq te numbers are 35% and 5% (Reiss and Werner (995) and Gould and Kleidon (994)). 5 For surveys of te market microstructure literature, see Admati (99) and O'Hara (995). Admati and Peiderer (988) endogenize te beavior of uninformed agents in a limited way. Bertsimas and Lo (998) study te beavior of a large agent wo may or may notbeinformed. However, tey do not endogenize prices. 6 See, for instance, Stokey (98), Bulow (982), and Gul, Sonnenscein, and Wilson (986). 7 Anoter dierence is tat tis paper introduces risk aversion and a dierent trading mecanism. Te closest paper in tat respect is DeMarzo and Bizer (998) wic allows for increasing marginal costs and a similar trading mecanism. 8 Cramton (984), Co (99), and Ausubel and Deneckere (992) assume tat te monopolist's cost is private information. 9 See, for instance Gresik and Sattertwaite (989), Sattertwaite and Williams (989), and Rusticini, Sattertwaite, and Williams (994). Wilson (986) considers a dynamic model wit risk-neutral agents wo ave - demands. Economides and Scwartz (995) suggest tis as one of te reasons wy te YSE 56

59 sould incorprorate a call market into its continuous trading system. We introduce te consumption good endowment for tractability. Wit tis endowment, te risk represented by te stock endowment is independent of te dividend level. Te model witout te consumption good endowment is somewat more complicated but produces te same results. Te consumption good endowment isconsis- tent wit te inter-dealer market interpretation of te model. If a dealer receives a positive endowment sock, i.e. buys stock from a customer, e pays te customer in return. 2 Te transversality condition 3.9 is standard for optimal consumption-investment problems. See, for instance, Merton (969) and Wang (994). 3 We assume tat te agent sells ax sares at period ` +, wile for Q + Q 2 we assumed tat te agent keeps all x sares. By te envelope teorem, te two assumptions are equivalent. In te private and public information cases, it is easier to determine Q making te rst assumption. Indeed, if te agent sells ax sares, p`+ is independent of x and equal to. If te agent keeps all x sares, we ave to take into account te cange in p`+. For Q + Q 2 te cange does not matter because te agent does P not trade in expectation at period ` +. 4 Agent i can infer j6=i e j `; from te period ` ; price, p`;. 5 ote tat tere may exist oter as equilibria in wic demands are non-linear or depend on more variables tan p`, d`, and e i `; + i ` (suc as \trigger-strategy" equilibria). Studying tese equilibria is beyond te scope of tis paper. 6 For 2 e =, te result follows from equation 4.. For 2 e small) te proof is available upon request. > (and not necessarily 7 It migt seem tat we get indeterminacy only because we allow agent i's demand to depend on oter agents' stock oldings, tus introducing te additional parameter A e. Tis is incorrect. If in te private information case we allow agent i's demand to depend on is expectations of oter agents' stock oldings, we will obtain an additional optimality condition. Tis condition will imply tat A e =. 57

60 8 See Selten (975) and capter 8 in Fudenberg and Tirole (99). 9 We do not study ow CEQ pr depends on because wen canges, te dividend process and preferences cange. Tis problem can be avoided by considering a \continuous" model were dividends follow a Brownian motion and preferences are over consumption ow. We do not present te continuous model because it is complicated. (Te Bellman conditions are dierential rater tan algebraic equations.) We sould note tat te continuous model produces te same welfare loss, L, as te discrete model. 58

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