A Market Based Solution to Price Externalities: A Generalized Framework

Transcription

1 A Market Based Solution to Price Externalities: A Generalized Framework Weeracart T. Kilentong University of te Tai Camber of Commerce Robert M. Townsend Massacusetts Institute of Tecnology August 7, 2014 Tee Kilentong is grateful to financial support from te University of te Tai Camber of Commerce UTCC. Robert Townsend is grateful to financial support from te Eunice Kennedy Sriver National Institute of Cild Healt and Human Development NICHD under grant R01 HD027638, te Centre for Economic Policy Researc CEPR and te Department for International Development DFID under grant MRG , te Jon Templeton Foundation, and te CFSP at te University of Cicago troug a grant from te Bill & Melinda Gates Foundation. Te findings and conclusions contained in te report are tose of te autors and do not necessarily represent te views of te funders. 1

2 Abstract Pecuniary externalities ave regained te interest of researcers as tey seek policy interventions and regulations to remedy externality-induced distortions, e.g., balance seet effects, amplifiers and fire sales. We sow ow bundling, exclusivity and additional markets internalize tese pecuniary externalities; ex ante competition can acieve a constrained-efficient allocation. We cover a large variety of example economies wic differ from one anoter in te particular source of te constraint generating te externality. We do not need to identify and quantify some policy intervention. Wit te appropriate ex ante design we can let markets solve te problem. Keywords: price externalities; segregated excanges; Walrasian equilibrium; market-based solution; collateral; exogenous incomplete markets; moral azard wit retrading; idden information wit retrading; liquidity constraints; fire sales. 1 Introduction A large variety of economies can suffer from pecuniary externalities and for a variety of distinct reasons. Tere is long istory of researc into tis subject, wit a solid tradition rigt from te beginnings of general equilibrium teory. Lately, and particularly in te aftermat of te recent financial crisis, models wit pecuniary externalities ave regained te interest of researcers as tey seek policy interventions and regulations to remedy externality-induced distortions, e.g., balance seet effects, amplifiers and fire sales, to name some key words. Here in tis paper we go back to first principles and sow ow to design financial contracts and markets in suc a way tat ex ante competition can acieve a constrained-efficient allocation. Te key as in general equilibrium teory is to extend te commodity space in suc a way tat bundling, exclusivity and additional markets internalize pecuniary externalities. We devise in tis paper a general way of proceeding tat covers as a general case te large variety of example-economies wic differ in te particular in te source of te constraint generating te externality. One class of example economies are tose in wic collateral is used to back promises. Te issuer of a promise in te contract period is required to back te promise wit collateral. 2

3 In te spot market/trading period te issuer can eiter onor te promise or default, anding over te collateral. But te value of collateral is endogenous in te spot market, determined by te forces of supply and demand at tat time. Maximizing agents take prices as given and do not take into account tat teir actions in te contract period, te promises tey issue and teir savings and collateral, determine ex post spot market prices. Tat is, tere is a collateral constraint facing te issuers of promises, wic as in it tose spot market prices. Te problem of aircuts in wic collateral backing loans receives deep discounts wen markets seem to suffer from illiquidity as received, rigtly, muc attention. See, e.g., Begalle et al. 2013; Gorton and Metrick 2012; Krisnamurty et al Our own first paper on collateral-constraint-induced externalities, Kilentong and Townsend 2014b, sows ow to remedy te problem, in tat specific context. Tis is wat got us started along tis more general pat. Likewise, and even simpler in some respects, wen security markets are incomplete for some exogenous reason, i.e. missing an equivalent set of Arrow Debreu securities, te welfare teorems fail and competitive markets do not acieve an ex ante optimal allocation, even wen te comparison of competitive equilibrium allocations to tose acievable by a planner limits te planner to te same set of limited securities. Here spot market prices can move wit initial security positions, someting te planner can take into account, but in competitive markets te consequent wealt effects cannot be covered due to missing markets. See, e.g., Geanakoplos and Polemarcakis Pareto improving taxation as been proposed as a remedy 1 see, e.g., Geanakoplos and Polemarcakis, 2008; Greenwald and Stiglitz, 1986; Herings and Polemarcakis, Our general framework encompasses economies wit information problems and retrading in spot markets. For example in a moral azard insurance economy te incentive to take appropriate action ex ante is induced by vectors of consumption rewards ex post. But if retrading of goods is allowed, as in ex post spot markets, tis can undercut ex ante incentives. Indeed tis is raised as a criticism of te realism of Prescott and Townsend 1984a,b wo sow ow to apply te standard welfare teorems to many limited-information 1 We are reminded of Fari and Werning 2013 toug teir externalities arises from sticky prices and a zero bound on interest rates, wic are different from our general set up. 3

4 environments but wo also do not allow retrading as in more typical dynamic spanning arguments. However, see Acemoglu and Simsek 2012; Kilentong and Townsend 2011 for particular remedies. Tis again is one of our examples tat set us on te general pat. Likewise, te well-known Diamond and Dybvig bank-run environment can, in te norun equilibrium at least, acieve an information-constrained efficient allocation. But tis is undercut wen retrading in a bond market is allowed, a point made forcefully in te seminal work of Jacklin Fari et al use tis environment to argue for portfolio restrictions in te regulation of financial institutions. As already mentioned, tere is a growing literature on fire sales and amplifiers; were private agents undervalue net wort in a period of financial distress because tey fail to internalize tat net wort as positive spill overs on oter agents. Bianci and Mendoza 2012 study te relationsip of tese so-called credit externalities to financial fragility and welfare losses from balance seet effects, for example. Tere is a muc larger literature in te international context, e.g., Caballero and Krisnamurty 2001; Jeanne and Korinek Lorenzoni 2008 is a seminal paper on inefficient credit booms and we feature tat particular environment ere. Oters propose regulation of international capital flows, Korinek 2010, and taxation, Jeanne and Korinek On te flip side, so to speak, Hart and Zingales 2113 study an economy were markets for te forward sale of labor are precluded, due to uman capital considerations. Ironically, tis leads to excess savings, not over borrowing; savings backs liquidity instruments used in trade wen tere are Wicksell absence-of-double-coincidence-of-wants in real goods. Tey sow ow fiscal policy following a large negative sock can increase ex ante welfare. Again we can cast every one of tese environments as an example-economy of a larger, generalized framework. Te key is to be clear -- beyond preferences, endowments, and tecnology and beyond ordinary budget constraints -- is tat tere are extra constraints typically binding on a subset of agent types wic contain market clearing price. Tese can be ex ante security prices or ex post prices, or bot. Indeed te price-containing constraints wen binding are te definition of pecuniary externalities. Examples are, again, collateral constraints, spot market budget constraint in incomplete markets, retrading entering into incentives to take ex ante action or announce ex post socks trutfully, no default conditions, 4

5 or te price of liquidity assets used in trade. Wit tis generalized set-up, we establis tat tere is a common way to internalize te externality, by aving markets for rigts to trade at te culprit equilibrium spot and/or security prices, bundled wit te oter traded objects. Tat is, agents pay a market participation fee, or receive a compensation, depending on te excange, te targeted price pair, and rigts to trade. An agent wo cooses to trade in a security excange can act as if se can trade commodities and securities wit te specific prices indicated, in effect indexing te commodity/security trade by te price. Moreover, tese are self-fulfilling in tat security excanges, given te composition of traders attracted to te excange, are in an equilibrium fixed point. Importantly, agents are not allowed to trade wit agents in oter excanges, tat is, our remedy requires a registration/identification system and an exclusivity assumption; tat is, an agent type can trade exclusively witin is excange but not wit agents in oter excanges. But contemporary financial markets currently already ave and utilize tecnologies wic allow tis to appen. We elaborate on tis in some detail in te conclusion of our first paper Kilentong and Townsend 2014b. To reiterate, we create rigts to trade in tese security excanges and in turn tese rigts are priced in competitive markets 2. Tese rigts are te externality-correcting commodities 3. 2 Te rigts to trade or te discrepancy from te fundamental is key to our solution concept. It is related to consumption rigts in Bisin and Gottardi 2006, wic internalize te consumption externality due to adverse selection problem. Te key difference is tat our rigts to trade only requires own type information endowments, preferences, production decision, and financial asset/security positions and te knowledge of te equilibrium prices, wic is a standard Walrasian assumption, wile te determination of te consumption rigts for eac type in an adverse selection environment utilizes information on oter types see Eq. 3.2 in Bisin and Gottardi 2010 wic specifies correct conjectures of wat oter types are doing and te related no-envy conditions in Prescott and Townsend 1984a,b. Anoter related paper is Stein 2012, wo proposed a cap-and-trade approac, in wic banks are granted permits reserves for private money creation and te interest rate is te price at wic teses permits are traded in te market. Te key difference from ours and Bisin and Gottardi 2006 is tat is approac is a government intervention in wic rigts to trade are te policy instrument, so it is not entirely a decentralized, market-based approac. 3 To ensure te consistent execution of eac security excange, te sum of te rigts to trade or te discrepancies witin te excange must, by te definition of consistency, be zero so tat te specified price tat indexed ex ante contracts is te one wic prevails in equilibrium. Tis is like a club constraint in oter literature, e.g., Prescott and Townsend Tis solution concept wit segregated security excanges 5

6 For example, in te collateral equilibrium model in Kilentong and Townsend 2014b, tis object is called te type- discrepancy from te fundamental, tat object wic determines te spot market price, and tis discrepancy depends on type endowments, is coice of collateral allocation, and te specific target fundamental/price. An alternative example is in Kilentong and Townsend 2011, in wic te rigts to trade are implicitly embedded in te incentive comparability constraints. More generally, an agent s required rigts to trade can be defined by excess demand functions for relevant commodities, wic are again functions of is endowments, is coice of trades, and te specific target price. A key take away from our approac is tat we do not need to identify and quantify some policy intervention. Policy makers may accept te arguments in te literature about taxation, or portfolio restrictions, for example, and ten ask ow exactly to implement, i.e., wat is te order of magnitude of te intervention. In contrast in our approac, we let markets determine prices for rigts to trade, determined in te usual manner. Tere is no need for intervention. Te outcome will be constrained-efficient and indeed any constrained efficient allocation can be acieved wit conventional ex ante lump-sum taxes and transfers tat do not impede te operation of markets. As mentioned at te outset, tere are clear antecedents for wat we are doing in te general equilibrium literature. Tis includes Arrow 1969 s early suggestion to expand te commodity space to include te object creating te externality, following Meade 1952 s early treatment of externalities. Te exclusive security excanges we feature in tis paper are founded in te tradition of te geograpic assignment problem of Koopmans and Beckmann 1957, te labor problem of Sattinger 1993, te treatment of te firm as an endogenous object of McKenzie 1959, 1981, te treatment of firms and plants in general equilibrium of Hornstein and Prescott 1993, and te treatment of firms as clubs of Prescott and Townsend Te remaining of te paper proceeds as follows. Section 2 introduces key ingredients of te general model and ten sows ow te ingredients of te main example-economies is also related to te assignment literature e.g., Koopmans and Beckmann, 1957; Prescott and Townsend, 1984a,b. Mortensen and Wrigt 2002 internalizes a searc externality using directed searc into segregated submarkets tat promise different expected waiting times. See also Guerrieri, Simer, and Wrigt

7 map into te generalized framework, including a collateral economy, an exogenous incomplete markets economy, a moral azard wit retrading economy, and a liquidity constrained economy. For expositional purposes, eac key ingredient of te general model is followed by its counterpart for eac example-economies, and in te text we limit ourselves to tese four to control lengt. Section 3 presents te competitive equilibrium wit obstacles to trade and constrained optimality and establises te existence of te externality. To save on space in te main text, and as its already covered wit te notation of te general model, te competitive equilibrium and constrained optimality for eac of te four example-economies are presented in Appendix A. Section 4 formally defines our market-based solution concept, te competitive equilibrium wit segregated excanges, and sows ow to establis te existence and welfare teorems. Te oter two example-economies, including a fire sales economy and a idden information wit retrading economy are presented in entirety in Appendix B. 2 A General Model wit Price Externalities and Its Prototypical Economies Tis section formulates a general model tat captures key features regarding price externalities of prototypical economies including a collateral economy Kilentong and Townsend, 2014b, an exogenous incomplete markets economy Geanakoplos and Polemarcakis, 1986; Greenwald and Stiglitz, 1986, a moral azard wit retrading economy Kilentong and Townsend, 2011, and a liquidity constrained economy Hart and Zingales, Eac subsection presents a key ingredient of te model along wit te relevant part of eac prototypical economy. More prototypical economies, namely a fire sales economy Lorenzoni, 2008 and a idden information wit retrading economy Diamond and Dybvig, 1983; Jacklin, 1987, are presented in Appendix B. 7

8 2.1 Basic Ingredients: Commodity Space, Preferences, Endowments, and Tecnology Tere are L commodities. Tese can be basic underlying commodities and also date and/or state contingent were te date and/or state are public. In order to incorporate private information problems into tis framework, we also allow a subset of commodities to be contingent on recommended but unobserved actions or on reported but unobserved states. For actions, let a be te recommended action and wit te incentive compatibility constraints in place te actually taken action, and a be potentially deviating action. For privately observed states, let a be te reported state and wit incentive compatibility constraints in place te actual state, and let a be some potentially counterfactual report. Let A R + be te set of possible actions/states, i.e., a, a A. Tere is a continuum of agents of measure one. Te agents are divided into H ex-ante types, eac of wic is indexed by = 1, 2,, H. Eac type consists of α [0, 1] fraction of te population suc tat α = 1. In addition, tis model allows for ex-post diversity denoted by ex-post eiter observable or unobservable type ω Ω. More formally, let ζ ω be te fraction of agents of type wose ex-post type is ω. An ex-post type ω may depend on an observed output, an unobserved action, and/or unobserved state of nature as well. Eac agent type is endowed wit an endowment e R L +. Note tat c and e lie in te L-dimensional commodity space. Te preferences of an agent of type are represented by te utility function U c, were c R L is te consumption allocation for an agent of type. Eac agent of type as an access to a production tecnology defined implicitly by F y 0, 1 were y R L is te vector of its inputs and outputs in commodity space L. Tis production tecnology is generally a multidimensional vector of constraints wit dimension O, i.e., F y [ F ] o y O. o=1 8

9 2.1.1 Basic Ingredients for te Collateral Economy Tis is a two-period economy, t = 0, 1. Tere are a finite S states of nature in te second period t = 1, i.e., s = 1, 2,..., S. Let 0 < π s 1 be te objective and commonly assessed probability of state s occurring, were s π s = 1. Tere are two goods, called good 1 and good 2 in eac period. Tese two goods can be traded in eac date and in eac state, and we refer to tose markets as spot markets wit good 1 as te numeraire good in every date and state. Tus, tere are L = S commodities. Tere is no unobserved action or privately observed state. Eac agent of type = 1, 2,..., H is endowed wit good 1 and good 2, e 0 = e 10, e20 in te first period and e s = e 1s, e2s, in eac state s = 1,, S. Let e = e 0,, es be te endowment profile of an agent of type over te first period and all states s in te second period, respectively. Tere is no ex-post diversity in tis economy, and terefore we simply omit all related notation. Te preferences of an agent of type are represented by te utility function u c 1, c2, and te discounted expected utility of is defined by: U c u S c 10, c 20 + β π s u c 1s, c2s, 2 were β is te discount factor. Good 1 is consumable but cannot be stored from t = 0 to t = 1 is completely perisable, wile good 2 is consumable and storable. Te good 2 tat is stored can be collateralizable, i.e., can serve as collateral to back promises. Hencefort, good 2 and te collateral good will be used intercangeably. Eac unit of good 2 stored as input will become R s units of good 2 in state s. As a result, te production function in our general framework can be written as follows: Fs y = y2s + R s y20 = 0, for s = 1,..., S, 3 s=1 were y 20 R and y 2s R +, s = 1, 2,..., S are inputs and outputs, respectively. We use te standard convention under wic an input must be non-positive and an output must be non-negative. Tis economy as O = S production functions. 9

10 2.1.2 Basic Ingredients for te Exogenous Incomplete Markets Economy Consider an economy wit two periods, t = 0, 1. Tere are S possible states of nature in te second period t = 1, i.e., s = 1,..., S, eac of wic occurs wit probability π s suc tat s π s = 1. Tere are 2 goods, labeled good 1 and good 2, in eac date and in eac state. Tus, tere are L = S commodities. Because te endowment profiles are te same as specified in te collateral economy discussed above, we omit te details in tis section for brevity. Te preferences of an agent of type are represented by te utility function u c 1, c 2, and te discounted expected utility of is defined by: U c u S c 10, c 20 + β π s u c 1s, c2s, 4 were β is te discount factor. Tere is no ex-post diversity in tis economy, and endowments and preferences are known ex-ante, and terefore we simply omit all related notation. s=1 For simplicity, we assume tat tere is no production. Tus, F o can be suppressed. As a result, tere would be no externalities if preferences were identically omotetic, as spot prices are determined by ratio of aggregate endowment only, wic no one can influence. So we assume oterwise; tat is, preferences are not identically omotetic Basic Ingredients for te Moral Hazard wit Retrading Economy Tere are two pysical commodities, labeled as good 1 and good 2, in eac states. Tese commodities can be produced using te sole input, called action, a. Let A be te number of possible actions. As in te literature, te random production tecnology is given by fq a, wic is te probability density function of te output vector of good 1 and good 2, q = q 1, q 2, conditional on an action a taken by an agent. In oter words, te probability tat te realized output will be q is fq a wen an agent takes an action a. Te action tat an agent takes is private information. Hence, tere is a moral azard problem. Tere is a continuum of ex ante identical agents of mass 1, i.e., no diversity in types so trivially α 1 = 1. For simplicity, we assume tat eac agent is endowed wit zero units of bot goods. We will now map tis moral azard economy into our general model wit securities trading. Different combinations of outputs q define idiosyncratic states or indexes for 10

11 contracting purposes. Tere is no loss of generality to assume tat tere are a finite Q states, q Q. Following te mecanism design literature, an optimal consumption of te two goods under moral azard depends on realized output q and recommended action a; tat is, c 1 q, a and c 2 q, a. Accordingly, we define commodity using bot output/state q and recommended action a. In particular, for eac recommended action a, tere are Q states. Tere are two commodities in eac state. In addition, actual action a itself is anoter commodity. Terefore, tere are L = 2QA + 1 commodities in tis model. Eac agent is endowed wit te instantaneous common utility function for te two goods and action, u c 1, c 2, a. Again, let a be recommended action, and a be taken possibly out-of equilibrium action. Te discounted expected utility of an agent wo is reported action a but took action a is define by: U c = q πq a u c 1 q, a, c 2 q, a, a 5 were πq a denote te probability of realizing outputs q given action a actually taken, wic satisfies te following probability constraint: πq a = 1, a. 6 q Ex-post diversity in tis model is determined by actual ex ante action and realized ex post outputs, i.e., ω = q, a. For generality, let δ a be te fraction of agents wo took action a. Recall tat te fraction of agents wo realized outputs q conditional on taking action a is f q a. As a result, te fractions of agents of ex-post type q, a is ζ 1 q, a = f q a δ a. As in te literature, te probability distribution across outputs/states depend on agent s coice of action a. Tis dependency is modeled as a general production function F wose input is actual action a and outputs are q: F q, a = f q a πq a = 0, q, a 7 In words, different actions will lead to different probability distributions. Tere are, as in 7, O = QA production functions. Combining tese production tecnologies wit te probability 11

12 conditions 6 leads to te standard probability constraints of production function fq a: πq a = 1 f q a = 1, a. 8 q q Basic Ingredients for te Liquidity Constrained Economy Consider an economy wit four periods, t = 0, 1, 2, 3. Tere are two types of agents, called doctors and builders, eac of wic consists of α > 0 for all = b, d fraction of te population wit =b,d α = 1. Eac agent = b, d is endowed wit e = e units of weat at period t = 0. Tis is a simplified and deterministic version of Hart and Zingales 2113 in wic we assume tat te doctors will buy building services in period t = 1 first, and te builders will buy doctor services later in period t = 2. As in te collateral model in section 2.1.1, tere is no unobserved action or privately observed state, and terefore we simply omit all related notation. Tere are two commodities in period t = 0, weat w0, and storage f0, were te latter is formally defined below. Tere are tree commodities in period t = 1, storage f1, building services b d and labor supply of te doctors l d. Similarly, tere are tree commodities in period t = 2, storage f2, doctor services d b, and labor supply of te builders l b. Tere is one commodity, weat w3, in te last period t = 3. Terefore, tere are L = 9 commodities in tis model. Te preferences of doctors and builders are represented by U d c = u d w, d, b, l l 2 = w3 + b 2, 9 U b c = u b w b, d b, b b, l b l b 2 = w3 b + d b 2, 10 respectively. Note tat doctors do not consume doctor services, and vice versa for builders. We can write te utility function in a more general from as follows: U c = u w, d, b, l = w3 + δb b + 1 δd l 2 d 2, 11 were δb = 1 if b, and zero oterwise. Tere are two tecnologies or assets available in period t = 0. First, te collateralizeable asset is a storage tecnology, wose return from t = 0 to t = 3 is 1 unit of weat, i.e., saving 12

13 one unit of weat in te first period t = 0 will return 1 unit of weat in te last period t = 3. In addition, te claim on te output of tis tecnology is transferable, and terefore can be used as private money or collateral during periods t = 1 and t = 2. Te second asset is an investment project, wose return from t = 0 to t = 3 is R > 1 units of weat. However, tis asset cannot be used as collateral. For simplicity, we consider only a deterministic return case ere. Let f0 be te amount of weat stored by an agent type = b, d, and accordingly, te agent type invests e f0 units of weat in te investment project. Te production tecnologies are irreversible; tat is, teir outputs will be realized in te last period t = 3 only. Te production function of te storage tecnology denoted by subscript s for an agent type is defined as follows: F s f 2, y31 = y s3 f2 = 0, = b, d, 12 were f 2 is te number of claims on te storage tecnology eld by te agent type at te end of period t = 2, and y s3 is te output in unit of weat in period t = 3 received by te agent type from te storage tecnology. Similarly, te production function for investment project denoted by subscript i is defined by were f 0 Fi e f 0, yi3 = y i3 R e f0 = 0,, 13 is te amount of weat stored by an agent type in period t = 0, and y i3 is te output in unit of weat in period t = 3 received by te agent type from te investment tecnology. In addition, te builders and te doctors produce building and doctor services denoted by subscript o, respectively, using te following simple linear tecnologies: F o y, l = y l = 0, = b, d, 14 wic use labor as te only input. For notational convenience, we also set F 0 y = y = 0, = b, d, 15 were y = yd b, yb d denote building services produced by doctors and vice versa. To sum up, tere are O = 4 production functions. 13

14 2.2 Market Structure: Security and Spot Markets Tere are J securities. Let θ j R denote te amount of security j acquired negative if sold by an agent of type, and D j = [D jl ] L l=1 RL + denote its payoff vector. Note securities ave payoffs of goods in te L-dimensional space of underlying commodities. Notationally, let D = [D j ] J j=1 be te payoff matrix of all securities. Let P RJ + be te price vector of all securities, tat is, P j 0 for j = 1,..., J. In addition, agents can trade in eac of S restricted ex-post spot markets of subsets of commodities. Let L s be te subset of commodities tat can be traded in an s restricted spot markets suc tat S s=1l s L. Note tat ex-post trades are in proper subsets of commodity space L as some dates or states are by ten realized or known. Wit abuse of notation, let L s denote te number of commodities in L s. Tese markets are assumed to be mutually exclusive, i.e., L s L s = for any s s. Let τ denote te set of trades in tese markets wit τls ω denoting te amount of te l t good in market L s acquired negative if surrendered by an agent of ex-ante type and ex-post type ω. Note again tat tese spot trades τls ω are restricted to be traded wit commodities in L s only. Let p s [ p ls ] l L s R Ls + be te price vector of commodities in L s. Te relationsip between consumption, endowments, securities, spot trades, and outputs for an agent of type is defined implicitly by g c, e, θ, τ, y = Tese will be obvious identities or accounting formulas in te examples wic follow. Tis condition is generally multidimensional vector wit dimension N, i.e., g g n c, e, θ, τ, y N n= Market Structure for te Collateral Economy Let θ ks denote securities paying in good k = 1, 2 in state s or net transfers of good k = 1, 2 in state s acquired by an agent type. If tis is negative, it is a promise to pay. Also θ k0 is spot purcase of good k at t = 0 but for convenience we refer to tis as a security trade. Tus tere are J = S securities. Let P l0 and P ls denote te security spot price of 14

15 good l at period t = 0 and te price of a security paying in good l in state s, respectively. We take good l = 1 as te numeraire. Let τks denote spot trade amount of good k = 1, 2 in spot markets Ls in state s acquired by an agent of ex-ante type. Wit abuse of notation, let τk0 denote spot trade amount of good k in spot markets L 0 in period t = 0 acquired by an agent of ex-ante type. Eac spot market as two commodities, namely good 1 and good 2, i.e., L s = 2 for all s = 0, 1,..., S. Tere are S + 1 spot markets ere. We set te spot-market-clearing price of good 1 equal to one te numeraire good, and let p s denote te spot-market-clearing price of good 2 in eac spot market L s. Te consumption-relationsip constraints in tis case are defined as follows: gks c, e, θ, τ, y = e ks + yks + θks + τks c ks = 0, for k = 1, 2; s = 0, 1,..., S, 17 were we set y10 = 0 and y1s = 0 to represent te fact tat good 1 cannot be stored. Tere are N = S consumption-relationsip constraints. As proved in Kilentong and Townsend 2014b, wit complete collateralized contracts, tere is no need for restricted/spot trades τ in tis case. All trades can be accomplised in ex-ante security markets. As a result, te consumption-relationsip constraints can be rewritten as follows: gks c, e, θ, y = e ks + yks + θks c ks = 0, for k = 1, 2; s = 0, 1,..., S. 18 Neverteless, we can define wat te ex-post spot price p s would be tat would clear tese markets witout active trade Market Structure for te Exogenous Incomplete Markets Economy Tere are J < S securities available for purcase or sell in te first period t = 0. Let D = [D js ] be te payoff matrix of tose assets were D js be te payoff of asset j in unit of good 1 te numeraire good in state s in te second period t = 1, s = 1, 2,..., S. Let θj denote te amount of te j t security acquired by an agent of type at t = 0, and P j denote te price of security j. An exogenous incomplete markets assumption specifies tat D is not full rank; tat is again, J < S. Tis is crucial. Let τks denote spot trade amount of good k = 1, 2 in spot markets Ls in state s acquired by an agent of ex-ante type. Wit abuse of notation, let τk0 denote spot trade amount 15

16 of good k = 1, 2 in spot markets L 0 in period t = 0 acquired by an agent of ex-ante type. Eac spot market as two commodities, namely good 1 and good 2, i.e., L s = 2 for all s = 0, 1,..., S. Tere are S + 1 spot markets ere. We set te spot-market-clearing price of good 1 equal to one te numeraire good, and let p 0 and p s denote te spot-market-clearing price of good 2 in spot market L 0 in period t = 0, and te spot-market-clearing price of good 2 in spot market L s at state s in period t = 1, respectively. Te consumption-relationsip functions in te first period t = 0 is defined as follows: g k c, e, θ = e k0 + τ k0 c k0 = 0, for k = 1, Te consumption-relationsip function for good 1 and good 2, respectively, in te state s in te second period is defined as follows: g 1+2s c, e, θ, τ = e 1s + j D js θ j + τ 1s c 1s = 0, for s = 1,..., S, 20 g 2+2s c, e, θ, τ = e 2s + τ 2s c 2s = 0, for s = 1,..., S. 21 Note ere tere will be active spot market trades τ to support te equilibrium allocation. To sum up, tere are N = 21 + S consumption-relationsip constraints Market Structure te Moral Hazard wit Retrading Economy To be consistent wit te general model, one can imagine tat tere are state-contingent securities paying in good k = 1, 2 wen state/output is q and te recommended action is a, namely θ k q, a. Tat is, security j is indexed by q, k, and a. Even toug tere are J = 2QA securities available to trade, eac agent can trade only 2Q securities depending on is recommended action a only. In particular, an agent recommended action a will be able to trade only securities θ a [θ k q, a] k,q. Let P k q, a denote te price of a security paying in good k conditional on output q and recommended action a. Recall tat actual action a = a is an input of te production tecnology, and tere is no loss of generality to assume tat it is non-tradable. Terefore, tere is no price for tat commodity action. Note also tat as in te literature, tese equilibrium securities prices are fair prices. Tere is te possibility of retrade in ex post spot markets. One can tink of two subperiods: te first wit te application of inputs, securities, and production; te second for 16

17 output and possible retrading wit final consumption. Witout aggregate uncertainty, tere is only one set of spot markets S = 1 for good 1 and good 2 L s = 2, in wic everyone participates. Let τ k q, a be spot trade of an agent of ex-post type q, a wen a is bot te recommended and taken action. We set te spot-market-clearing price of good 1 equal to one te numeraire good, and let p denote te spot-market-clearing price of good 2, wic depends on agents action a recommended and taken and securities θ a = [θ k q, a] k,q as a function of recommended action a as if te markets can be partitioned by action a; tat is, p = p θ a, a. As in Kilentong and Townsend 2011 and te collateral example in Section 2.2.1, te spot markets are redundant wit complete contracts, owever. Anyting tat can be done wit spot markets can be done witout tem wit altered security oldings. Terefore, we can omit spot trades, encefort, toug tere is still an implicit sadow spot price. Te consumption-relationsip in tis case is defined as follows: g kqa = q k + θ k q, a c k q, a = 0, q, a; k = 1, Tere are N = 2QA consumption-relationsip constraints Market Structure te Liquidity Constrained Economy To be consistent wit te general model, tere is no security in tis model; tat is, J = 0. All trades occur in te spot markets. Tere are 2 sets of spot markets in period t = 1 and t = 2; tat is, S = 2. Agents can trade storage claim τ f1 and building services τ b in te spot markets in period t = 1 at price p b ; tat is, tere are two commodities in te spot markets in period t = 1 L 1 = 2. Similarly, agents can trade storage claim τ f2 τ d and doctor services in te spot markets in period t = 2 at price p d; tat is, tere are two commodities in te spot markets in period t = 2 L 2 = 2. Te consumption-relationsip constraints are as follows: g b b, yb, τb = b yb + τb = 0,, 23 g d d, yd, τd = d yd + τd = 0,, 24 gw w, yi3, ys3 = w 3 yi3 ys3 = 0,, 25 gft f t, ft 1, τft = f t ft 1 τft = 0, ; t = 1,

18 To sum up, tere are N = 6 consumption-relationsip constraints. 2.3 Trade Frictions: Obstacle-to-Trade Constraints Price externalities in tis model come from te fact tat agents face obstacles to trade tat depend on prices. Tese obstacles are formulated as constraints and are called obstacleto-trade constraints. Eac set of obstacle-to-trade constraints depends on te prices of a particular subset of securities denoted P i or te prices of a particular subset of commodities p i, or bot. Let J i J be te dimensions of P i, and L i L be te dimensions of p i. Tere are I sets of obstacle-to-trade constraints 4. Tat is, tere are I sets of prices p i, P i relevant to obstacle-to-trade constraints. Tese obstacle-to-trade constraints could be in te form of collateral constraints, retrading in exogenous incomplete-market constraints, incentive compatibility constraints under moral azard wit retrading, incentive compatibility constraints under idden information wit retrading, liquidity constraints, and no-default constraints. Eac set of obstacle-to-trade constraints Ci [ ] Ci,a,a A for an agent of type consists a,a =1 of A 2 obstacle-to-trade constraints, eac of wic depends on te same set of prices p i, P i. Tese put restriction on securities, spot trades, and output and are defined as follows: C i,a,a c, θ, τ, y, p i, P i 0, for i = 1,..., I; a A; a A. 27 Te total number of obstacle-to-trade constraints is M = IA 2. Note tat an action as in a moral azard model, or privately observed state indexes te commodities, and terefore is included in c. Te dependency on market-clearing prices of tese obstacle-to-trade constraints is te source of price externalities in tis paper. Most of te literature focuses only on te dependency on te restricted/spot prices. Tis paper explicitly puts security prices into te constraints in order to empasize tat price externalities could arise even wen we sut 4 Tere is no loss of generality in setting te dimensionality of te obstacle-to-trade constraints identical for all agent s types. Consider a model were different types face different numbers of obstacle-to-trade constraints. Let I be te number of obstacle-to-trade constraints faced by an agent of type. Set I = max I. For an agent of type wose I < I, we ten define C i = for I < i I. 18

19 down te spot markets. In oter words, te spot markets/prices are not fundamental to te externality problem. It is an obstacle to trade itself, wic can not be removed, tat is key to te problem. As sown in te collateral economy below, one can get rid of te spot markets tere since tey are redundant. Te collateral constraints te need to back promises by collateral ten depend on security prices only, but te price externality still occurs Trade Frictions for te Collateral Economy As in Kilentong and Townsend 2014b, te collateral constraints or obstacle-to-trade constraints state tat te value of collateral y2s must weakly exceed value of promises to pay θ 1s, θ2s : p s y 2s p s θ 2s + θ 1s, for s = 1,..., S, 28 wic can be rewritten as follow: p s y 2s + θ 2s + θ 1s 0, for s = 1,..., S, 29 were again p s is te spot price of good 2 in units of good 1 in state s. But as mentioned earlier, tese collateral constraints can be rewritten in terms of security prices as following: C s θ, y, P s = P 2s y 2s + θ2s + P1s θ1s 0, for s = 1,..., S, 30 wic results from te fact tat, wit complete state contingent contracts at t = 0 and te possibility of retrading, te spot price ratio p s equals to te ratio of security prices P 2s P 1s. Tis formulation empasizes tat we can sut down te spot markets, but te collateral constraints still depend on security prices, wic still generate externalities. In oter words, te spot markets/prices are not fundamental to te externality problem. It is an obstacle to trade itself, wic can not be removed, tat is key to te problem. Eac agent of type faces I = S sets of obstacle-to-trade constraints, eac of wic contains only one constraint, i.e., tecnically A = 1. Terefore, tere are M = S obstacleto-trade constraints in total. 19

20 2.3.2 Trade Frictions for te Exogenous Incomplete Markets Economy Te obstacle-to-trade or spot-budget constraint for an agent of type in eac state s is simply te budget constraint in tat state: C s τ s, p s = τ 1s + p s τ 2s = 0, for s = 1,..., S, 31 Note tat te spot price p s is determined by pre-trade position of endowments and securities were endowments are exogenous but securities are endogenous. Eac agent of type faces I = S sets of obstacle-to-trade constraints, eac of wic contains only one constraint, i.e., A = 1. Terefore, tere are M = S obstacle-to-trade constraints in total Trade Frictions for te Moral Hazard wit Retrading Economy Te possibility of retrade in ex post spot markets creates obstacle to trade in tis model. Wit te possibility of retrade, te ex-post utility maximization problem of an agent wo was recommended action a receiving compensation c 1 q, a, c 2 q, a, but took action a wen te spot market price is p is as follows: v c 1 q, a, c 2 q, a, a, p = max τ 1,τ 2 u c 1 q, a + τ 1, c 2 q, a + τ 2, a 32 subject to te budget constraint: taking spot-market-clearing price p as given. τ 1 + pτ 2 = 0, 33 As in Kilentong and Townsend 2011, te possibility of retrade in ex post spot markets and te moral azard problem imply tat te incentive compatibility constraints IC are as following: a, a, C 1,a,a c, p = q u c 1 q, a, c 2 q, a, a fq a 34 q v c 1 q, a, c 2 q, a, a, p fq a 0, 35 Here te agent takes te recommended action a and so a = a. Tere is only one set of obstacle-to-trade constraints, I = 1, and tere are A 2 constraints for tis one i. Terefore, tere are M = A 2 incentive compatibility constraints in total. 20

21 2.3.4 Trade Frictions for te Liquidity Constrained Economy Te obstacle-to-trade or spot market constraints for an agent type = b, d in period t are as follows: C1 C2 τ f1, τ b, p b = τ f1 + p b τ b = 0, = b, d, 36 τ f2, τ d, p d = τ f2 + p d τ d = 0, = b, d, 37 were p b and p d are te spot-market-clearing prices of building and doctor services in period t = 1 and t = 2, respectively; tat is, p b is suc tat =b,d α τ b = 0, and vice versa. Note tat te spot price p b and p d are determined by storage positions of all agents wic are endogenous. Eac agent of type faces I = 2 sets of obstacle-to-trade constraints, eac of wic contains only one constraint, i.e., A = 1. Terefore, tere are M = 2 obstacle-to-trade constraints in total. 3 Price Externalities Tere is an externality because te consumption feasibility set of an agent type depends on oter agents coices troug prices. To reiterate, tis dependency results from te obstacleto-trade constraints. If tere were no obstacle-to-trade constraints, ten eac agent s consumption feasibility set would be independent of oter agents coices and terefore tere would be no externality. Intuitively, an infinitesimal agent as no influence on aggregate resource allocation, wic determines prices. On te oter and, a constrained planner knows se can influence prices troug agents coices collectively. Te asymmetry between te influence of te planner versus agents generates an inefficiency wen any one of obstacle-totrade constraints, wic contain prices, is binding for any agent type. We now present te formal statement below. For simplicity, we focus on identical allocations for eac type. 3.1 Competitive Equilibrium wit Obstacles to Trade An agent of type maximizes is utility: 21

22 Program 1. max U c 38 c,θ,τ,y subject to te budget, consumption-relationsip, tecnology, and te obstacle-to-trade constraints, respectively: taking prices p i, P i, and P as given. L s l=1 J P j θj 0, 39 j=1 p ls τ ls ω 0, s, ω, 40 gn c, e, θ, τ, y = 0, n, 41 y = 0, o, 42 F o C i,a,a c, θ, τ, y, p i, P i 0, i, a, a, 43 In order to deal wit ex post diversity in private information problems, we define some participation mecanism. Let ξ j, called eligibility weigt, denote te mass of agents of type wo are eligible to trade security j, and ence ξ j = α ξ j be te total mass of agents of all types wo are eligible to trade security j, adding up over ex-ante types. Note tat ξ j can depend on observable actions or unobserved states in A. In addition, we also need to introduce financial intermediaries below in order to generally represent economies wit collateral requirements, incomplete markets, private information, liquidity constraints, and fire sales in te same general framework. It is worty of empasis tat te existence of financial intermediaries is not te cause of price externalities in tis model. Intermediaries simply bundle commodities and facilitate trade. Te representative financial intermediary supplies securities ψ = ψ j j, j = 1, 2,..., J per unit of eligible agents to maximize its profit taking prices as given. Te profit maximization problem of te representative financial intermediary is as follows: J max P j ξ j ψ j 44 ψ j=1 subject to feasibility constraints potentially multi-dimensional J Ψ mj ξ j ψ j = 0, for m = 1,..., M, 45 j=1 22

23 taking prices P as given, were Ψ = [Ψ mj ] m,j is te matrix of security weigts for te j t security in te m t feasibility constraint. Note tere are constant returns to scale so we act as if tere will be one price-taking intermediary. Te market clearing conditions for securities are as follows: α θj = ψ j, j 46 Tese conditions simply state tat te net demand for securities equates tat supplied by te intermediary sector. Note tat combining te feasibility constraints 45 and te market clearing constraints for securities 46 gives te resource constraints 49, written below. Te market clearing conditions for every spot market L s are as follows: α ζ ω τls ω = 0, l L s ; s S. 47 ω Te market clearing constraints for restricted/spot trades 47 ensure tat spot prices p i are consistent. Tat is, spot prices p i are suc tat 47 is satisfied and so eac restricted/spot market clears. Definition 1. A competitive equilibrium wit obstacles to trade is a specification of allocation c, θ, τ, y for ouseolds and ψ for intermediaries, and prices p i, P i, P suc tat i for eac type, c, θ, τ, y solves 38 subject to 39-43, taking prices as given; p i, P i, P ii for te financial intermediary, ψ solves 44 subject to 45, taking prices P as given; iii markets for securities and for spot trades clear; 46 and 47 old. For expositional simplicity and witout loss of generality, we consider only equal-treatment for eac type and interior solutions, and assume tat all functions are differentiable. A necessary optimal condition for a competitive equilibrium allocation wit respect to y l is N n=1 γ g n g n y l + O o=1 γ F o F o y l + I A A γc i,a,a i=1 a=1 a =1 C i,a,a y l = 0,, l 48 23

24 were γ g n, γ F o and γ C i,a,a are te Lagrange multipliers wit respect to te n t consumptionrelationsip constraint, 41; te o t tecnology constraint of agent, 42; and obstacle-totrade constraint i, a, a of an agent of type, 43, respectively. 3.2 Constrained Optimality Attainable allocations are tose tat can be acieved by excanges of commodities and by production subject to bot resource constraints and obstacle-to-trade constraints. In addition, it also requires tat te allocation of relevant commodities under te i t obstacleto-trade constraints must lead to market-clearing prices p i and P i. More formally, tis consistency condition can be written as follows p i = p i c, θ, y, e and P i = P i c, θ, y, e were c, θ, y, e c, θ, y, e. Definition 2. An allocation c, θ, τ, y is attainable if i it satisfies te resource constraints: J Ψ mj ξ j α θj = 0, m = 1,..., M, 49 j=1 α ζ ω τls ω = 0, s = 1,..., S; l = 1,... L s, 50 ω ii for eac, it satisfies te consumption-relationsip function, gn c, e, θ, τ, y = 0, n ; iii for eac, y follows te production function, F o y = 0, o; iv for eac, it satisfies te obstacle-to-trade constraints: C i,a,a c, θ, τ, y, p i, P i 0, i, a, a ; 51 v all consistency conditions are satisfied, i.e., p i = p i c, θ, y, e and P i = P i c, θ, y, e for all i, i.e., prices are suc tat 46 and 47 are satisfied. Te nonlinearity of obstacle-to-trade constraints could cause te attainable set to be non-convex. Tis non-convexity implies tat randomization, a lottery, could be potentially 24

25 useful. However, for expositional reasons, tis section will not use lotteries now. Note tat te existence of an externality sown is still true wit and witout lotteries Kilentong and Townsend, We do generalize and formalize wit lotteries in Section 4.1 below. A constrained optimal allocation identical witin types but tis is witout loss of generality is caracterized using te following planner s problem, wic maximizes te utility of type 1 subject to fixed utility of oter types, as follows: Program 2. subject to max U 1 c 1 52 c,θ,τ,y U c U, for = 2,..., H, 53 gn c, e, θ, τ, y = 0, n,, 54 J Ψ mj ξ j α θj = 0, m, 55 j=1 α ζ ω τls ω = 0, l, s, 56 ω F o C i,a,a c, θ, τ, y, p i, P i were U is a promised utility level for an agent of type. n=1 y = 0, o,, 57 0, i, a, a,, 58 p i = p i c, θ, y, e, i, 59 P i = P i c, θ, y, e, i, 60 A necessary optimal condition for a Pareto optimal allocation wit respect to yl is5 N µ g O n g n + µ F I C o i,a,a yl F o + yl yl + o=1 H =1 I i=1 were µ g n, µ F o, and µ C i,a,a a i=1 a a µ Ci,a,a a µ C i,a,a L i l=1 C i,a,a p i l p i l y l + J i j=1 C i,a,a P i j P j i = 0,, l61 are te Lagrange multipliers wit respect to te n t consumptionrelationsip, 54, te o t tecnology constraint, 57, and i, a, a obstacle-to-trade constraint of agent of type, 58, respectively. 5 A similar conclusion can be drawn from a necessary optimal condition wit respect to c l and θ j. y l 25

26 Of special interest, te last term depends not only on te bindingness of obstacle-to-trade constraints for as we vary y l but also te bindingness of oter agents obstacle-to-trade constraints. Tis implies tat if any type of agent s obstacle-to-trade constraint is binding, it will impact everyone troug prices. Tis is te source of te price externalities. Note tat an infinitesimal agent takes prices p and P, including p i and P i, as invariant. But to te contrary, te constrained planner can influence subsets of prices p i and P i troug aggregate allocation, c, θ, y, e. Tis key influence is te term in pi l and P j i in te planner yl yl problem. Te difference between te impact of te planner and tat of te agents creates price externalities and causes an inefficiency. Tecnically, if te last term in 61 were zero, ten condition 48 would ave been exactly te same as 61. Te last term in 61 could always be zero if eiter µ Ci = 0 for all and all i, or pi l = 0 and P j i = 0, for all l, all i, all l, all j, and all. Generically, tose prices yl yl vary wit te market fundamental wic is determined by c, θ, y, e. As a result, te last term in 61 will not be zero as long as an obstacle-to-trade constraint of an agent of type binds. Wit tis non-zero term, a competitive equilibrium wit obstacles to trade will be constrained inefficient. It is te interaction between te bindingness of obstacle-to-trade constraints and equilibrium prices tat is te key to te existence of price externalities. 4 Market-Based Solution: Endogenous Security Excanges Tis section proposes a market-based solution to price externality problems. Te solution is to create markets for te rigts to trade in a particular security excange, indexed by p, P 6. Te p, P are to be te equilibrium prices for commodities and securities witin te specified security excange, but from te agent stand point all possible price pairs across excanges are available. Agents pay a market participation fee, or receive a compensation, as fixed fee depending on te excange, te price pair and rigts indexed by i. An agent wo cooses to trade in a security excange p, P can act as if se can trade commodities 6 Recall tat a subset of tese prices enters into te obstacle-to-trade constraint i, i = 1, 2,..., I. 26