Ine cient Investment Waves

Transcription

1 Ine cient Investment Waves Zigo He University of Cicago Péter Kondor Central Eropean University Janary 0 Abstract We propose a dynamic model of investment and trade in a market of a specialized tecnology sbject to two main frictions. First, agents cannot raise otside capital. Second, a random grop of agents will ave te opportnity to invest in new tecnology and tese opportnities are not contractible. Te rst friction implies te presence of invesment cycles wit abndant invesment and low retrns in booms and little invesment and ig retrns in recessions. Only wen te second friction is present invesment cycles are constrained ine cient. Often te ine ciency is two-sided wit too mc invesment in booms and too little in recessions from a social point of view. Interventions targetting only te nderinvesment in recessions migt make all agents worse o. Also, te two-sided ine ciency typically implies too volatile prices and too freqent realizations of abnormally low prices compared to fndamentals. Key Words: Pecniary externality, overinvestment and nderinvestment, market intervention, Greenspan s pt Introdction Te istory of modern economies is ric in boom and bst patterns. Boom periods wit vast resorces invested in new projects and low expected retrns can abrptly trn into recessions wen long-rn projects are liqidated early, liqid resorces are oarded in safe sort-term assets and tere is little investment in new projects even if expected retrns are ig. Figre, sowing te AAA corporate bond spread and te net percentage of banks tigtening credit conditions and increasing spread on new loans, illstrates tese investment cycles. Te nancial crisis at te end of 000s brogt sc investment cycles into te forefront of te academic and policy debate. Can tese cycles be cased by nancing frictions only? Wen is te investment cycle a sign of ine ciency? If it is, wic part of te cycle is ine cient: is tere Preliminary and Reference Incomplete. address: [email protected], kondorp@ce.. We are gratefl to Arvind Krisnamrty, Gido Lorenzoni, Jon Moore, Balazs Szentes, Jame Ventra, Rob Visny, Ligi Zingales, and nmeros seminar participants.

2 Figre : Credit conditions, corporate spread and recessions in te US. Te solid and dased lines sow te net percentage of senior loan o cers tigtening lending standards and increasing spread compared to bank s cost on comercial and indstrial loans to large and mid-cap rms. (Sorce: Srvey of Senior Loan O cers, Federal Reserve.) Te red crve sows Moody s AAA corporate bond spread over te 0-year treasries. (Sorce: FRED.) Te saded areas are NBER recessions. overinvestment in booms and/or nderinvestment in recession? Relatedly, sold te policy maker intervene in booms, in recessions or bot? In tis paper, we contribte to tis debate as follows. First, wit te elp of a parsimonios dynamic model of investment and trade, we sow tat in an economy were capital to nance risky projects is limited in some states, constrained e cient investment cycles arise natrally. Ts, cycles generated by nancing frictions are not a sign of ine ciency per se. Second, we sow tat ninsrable idiosyncratic socks regarding agents relative valation of available assets may indce a two-sided ine ciency. Tat is, tis friction cases overinvestment in booms wit ig asset prices and nderinvestment in recessions wit low asset prices. As a mirror image, agents store too little liqid resorces in booms and oard too mc of tem in recessions. Tird, we sow tat intervention targeted to raise prices in recessions to avoid nderinvestment typically make overinvestment in booms worse. Wat is more, tis adverse e ect migt be so strong tat te intervention becomes Pareto inferior compared to te case of no intervention at all. We present a simple, stocastic dynamic model of investment and trade were asset prices are endogenos. Tere are two assets; trees and cas. Te economy is poplated by specialists, wo are te only agents wo can operate projects in a representative tecnology. Tese projects are represented by te trees. Specialists can create new trees at a xed cost, or liqidate te trees for a small xed bene t bot in terms of cas. Specialists can also trade trees among eac oter at

3 te eqilibrim price. At a random time eac tree "matres", i.e., pays a one-time dividend and te world ends. Before trees matre, sometimes tey generates interim cas ows, bt oter times tey reqire frter interim investment oterwise tey ave to be sold or liqidated. Te basic friction in or economy is tat specialists cannot raise otside fnds for te interim (or any oter type of) investment. Ts, tey store some cas in order to avoid ine cient liqidation of te project. Te second friction in or economy is tat specialists are sbject to an idiosyncratic sock. Namely, at some point investment opportnity into a new tecnological innovation arrives, bt it is available only for a sbset of te specialists. Tese specialists sell teir trees to te rest and invest all teir cas into te new opportnity. In eqilibrim, te aggregate cas-to-tree ratio serves as or single state variable. It is also or proxy for te level of aggregate liqidity in or economy. Wen interim socks are negative, te cas-to-tree ratio falls, and so does te eqilibrim price of trees raising te expected retrn on bying trees. Wen te price drops to te level of te liqidation bene t, trees are converted back to cas keeping te cas-to-tree ratio above an endogenos lower tresold. We tink of low liqidity states as a recession. Expected retrn is ig in a recession becase specialists ave to be compensated for te increasing probability tat tey will be forced into ine cient liqidation wen no cas will be available for interim investment. We refer to tis as a liqidity premim. As te cas-to-tree ratio rises, tis risk is redced and te price of te trees increases and te premim decreases. Wen te price reaces te cost of creating new trees, specialists create new trees keeping te cas-to-tree ratio below an endogenos pper tresold. We tink of te ig liqidity state wen new projects are created as a boom period. Te focs of or analysis is weter te investment and disinvestment tresolds are at teir e cient levels. In te complete market bencmark, te market soltion and te social planner s coice coincide. In tis case, specialists liqidate trees only wen te cas-to-tree ratio its zero. Tey bild trees wen te cas-to-tree ratio it a positive investment tresold. Tis tresold is determined by a trade-o. On one side, bilding trees is a positive net present vale project. On te oter side storing some cas is necessary to avoid costly liqidation wen interim investment is reqired. Still, expected retrns and economic activities ctate wit te cas-to-tree ratio jst as in te incomplete market eqilibrim. Te investment cycle is not a sign of ine ciencies per se. However, in te market soltion, te investment and disinvestment tresolds are distorted. In particlar, specialists always liqidate trees at a positive cas-to-tree ratio. Also, nder some conditions, tey bild trees at a lower tresold tan te social planner wold. Tat is, tey invest too little (liqidate too mc) in recessions, and overinvest in booms. As a mirror image, tey oard too mc cas in a recession, and old too little cas in a boom. Te intition beind or mecanism is as follows. Te cas-in-te-market pricing implies tat te ex post trading price is ig wen te cas-to-tree ratio is ig, i.e., wen te economy is ooded wit liqidity. Te ex post trading price serves two roles. Te rst role is to move all cas (or trees) to te most e cient ands; te fact tat a ig price wen te cas-to-tree ratio is ig elps acieve tis goal. However, te trading price also a ects te rent distribtion between tree 3

4 olders and cas olders, terefore not flly re ecting te marginal rate of sbstittion from te social perspective. Tis second detrimental role drives te wedge between te private marginal rate of sbstittion between tree and cas and te social planner s, and distorts te individal agent s investment incentives ex ante in eqilibrim. Interestingly, te direction of price distortion depends on te state of te economy. It is so, becase in a boom, te aggregate cas is ig, terefore te price at wic specialists wit te new investment opportnity can sell teir trees is ig. Ts, te private vale of trees is iger tan te social vale of trees in booms. Tis is a pecniary externality indcing overinvestment in trees in booms. As a symmetric argment, in recessions te price of te tree is low, indcing a negative wedge between te private and te social vale of trees. Tis implies even lower prices and nderinvestment in trees in recessions. As we explain, or model can be applied to te boom and bst pattern in constrction and ose prices. Or mecanism sggests tat te volme of real estate development in a boom is ine ciently ig, becase investors bild oses instead of olding liqid nancial assets expecting to be able to sell te real estate for a ig price in case tey nd a new investment opportnity. In recessions, investors old ine ciently ig level of liqid assets expecting to be able to by real estate ceap in case a grop of distressed investors ave to liqidate teir oldings. Te dynamic strctre of or model empasizes tat tere is a two-way interaction between decisions in booms and recessions. Wen a specialists decide to bild a tree, se takes into accont tat tis tree will ave to be liqidated if te state of te economy deteriorates signi cantly. Wen se liqidates a tree in te recession, se similarly takes into accont tat te economy migt revert to a boom. As a reslt of tis interaction, if te policy maker taxes cas-oldings in a recession to raise te price and avoid ine cient liqidation, tis one-sided intervention will typically decrease te investment tresold and make te overinvestment problem worse in te boom. Also, agents expected tility in a recession will natrally respond to te e ect of te intervention in te boom. Ts, intervention in te recession, wile e ective in te recession, often makes all agents worse o. Or paper also gives predictions on te di erent distribtion of prices nder complete and incomplete markets. Before te tree matres, te cas-to-tree ratio follows a niform ergodic distribtion regardless weter te constrained e ciency is acieved. Wen te market incompleteness implies tat te disinvestment tresold is too ig and te investment tresold is too low, te spport of te distribtion of te cas-to-tree ratio is compressed. In contrast, te asset price as a stationary distribtion wit a U-saped density fnction regardless weter te constrained e ciency is acieved. Tat is, wile te distribtion of te cas-to-tree ratio is niform by constrction, te economy spends more time wit very ig and very low prices tan in between. Tese e ects is stronger wit incomplete markets. We also nd tat te price distribtion nder market incompleteness is typically rst-order stocastically dominated by its conterpart in economies wit complete markets. Tese two observations imply tat te conseqences of or One sggestive sign of te ine ciently ig level of real estate development is te freqently observed penomenon of "overbilding" (e.g. Weaton and Torto, 990; Grenadier, 996), tat is, periods of constrction booms in te face of rising vacancies and plmetting demand. 4

5 two-sided ine ciency are volatile prices, more freqent ig realization of te liqidity premim, and larger liqidity premim in average tan nder complete markets. As a metodological contribtion, we propose a novel way to analyze te e ect of aggregate liqidity ctations on asset prices and real activity. Wile or model is flly dynamic, it provides analytical tractability for te fll joint distribtion states and eqilibrim objects. Literatre. To or knowledge, or paper is te rst to sow tat te simple friction of nveri- able idiosyncratic investment opportnities cases overinvestment in booms and nderinvestment in recessions. Or work belongs to a growing literatre analyzing pecniary externalities in incomplete markets. All tis literatre, inclding or paper, bilds on te reslt in Geanakoplos and Polemarcakis (985) tat wen markets are incomplete, te competitive eqilibrim may be constrained inef- cient. In tis setting pecniary externalities can ave a rst order e ect, becase prices fail to eqate te marginal rate of sbstittion of eac agent across all goods. A large stream in tis literatre acieves tis e ect by empasizing a re-sale feed-back loop indced, typically, by a collateral constraint (e.g. Kiyotaki and Moore, 997; Krisnamrty, 003; Gromb and Vayanos, 00; Stein, 0; Jeanne and Korinek, 00; Bianci, 00; Bianci and Mendoza, 0; Lorenzoni, 008; Hart and Zingales, 0). In tese papers agents do not take into accont tat te more tey invest ex ante, te more tey ave to liqidate wen tey it teir constraint wic redces re-sale prices tigtening te constraint and amplifying te e ect. Or paper does not rely in sc an ampli cation mecanism. Instead, or paper follows te tradition of Sleifer and Visny (99), Allen and Gale (994, 004, 005); Caballero and Krisnamrty (00, 003) and Gale and Yorlmazer (0) were an ninsrable sock creates te dispersion in marginal rate of sbstittion of ex-ante identical rms. Or main point of departre is tat in or paper, te Geanakoplos and Polemarcakis (985) mecanism is interacted wit a teory of contercyclical liqidity premim, reslting in distortions of opposite directions in booms and recessions. 3 A grop of recent papers investigating te moral azard problem of incentivizing banks in a macroeconomic context derive related implications to or work. In particlar, or reslt tat onesided interventions can be inferior to no interventions is related to te debate on te pros and cons of asymmetric interest rate policy often referred to as te Greenspan s pt. In teir recent work, Fari and Tirole (0) and Diamond and Rajan (0) arge tat spporting distressed instittions by low interest rates is detrimental to ex ante incentives of nancial intermediaries and encorage teir excessive risk-taking ex ante. As a reslt, ex post intervention to save distressed instittions will be needed more often. Similar to or work, in Gersbac and Rocet (0) banks extend too mc credit in booms and too little in recessions. Teir mecanism relies on te di erence between te private and social soltion of bank s moral azard problem. Namely, if private bene ts of banks are increasing in te size of teir loans, ten it is ceaper to make tem exert e ort by letting See Holmstrom and Tirole (0, cap. 7.) for simpli ed versions and excellent discssion of Sleifer and Visny (99) and Caballero and Krisnamrty (003). 3 See Davila (0) for an excellent comparative analysis of te di erent type of re-sale externalities explored in te literatre. 5

6 tem to increase teir loan size in booms compared to paying tem s cient rent to avoid tis. Tis private contract does not take into accont te price e ect of te reslting procyclicality in aggregate loan size. In contrast to tis literatre, agency frictions and related incentive problems for nancial intermediaries are not central in or argment. Instead, or mecanism is based on te novel observation tat incomplete migt imply tat te price of te prodctive asset is biased in te opposite direction in a boom and in a recession. Ts, watever policy elps in a boom will typically make agents worse o in a recession and vice-versa. Ex ante welfare in any state is te weigted average of tese e ects. From a metodological point of view, as a continos time model wit investment and trade, te closest paper to ors is Brnnermeier and Sannikov (0). As teir focs is balance seet ampli cation, teir model is more complex, bt not analytically tractable. Te strctre of or paper is as follows. Section gives an simple static example to igligt te main intition. In Section 3 we present or model, and analyze te market eqilibrim and te constrained e cient allocations of te social planner. In Section 4 we expose te ine ciencies of te market soltion, and Section 5 considers an alternative more natral sock speci cation. Finally, we conclde. A simple example Before we move on to set p or general model, we rst illstrate te main insigts of or paper by a simple example wit te following -date--good economy. Endowment and goods. At te beginning of date 0 eac agent i of te nit mass of risknetral agents old one nit of apple and c nits of coconts. Transformation tecnology. At date 0, eac agent can adjst teir frit-basket by transforming coconts to apples or te oter way arond. Te tecnology is sc tat eac agent can convert coconts to an apple, or an apple to l coconts were > l. Ts, given te endowment of one nit of apple and c nits of coconts, te individal olding of A i ; C i at te end of date 0 mst satisfy A i + C i + c if A i > ; la i + C i l + c if A i () : Tis bdget constraint re ects te kinked transformation tecnology. Preference and socks. At date 0, eac agent is identical in teir preferences. However, at te beginning of date, alf of te agents trn ot to like only apples wit a marginal tility of R > 0, wile te rest of te agents like only coconts wit a marginal tility of > 0. After tese idiosyncratic socks, agents can trade apples to coconts wit eac oter. As becoming clear later, we assme tat R (l; ). Te market soltion. Recall tat A i ; C i describe te oldings after te adjstment in date 0 bt before te trade in date, and te aggregate conterpart A R A i di and C R C i di. Given te strctre of te idiosyncratic sock, it is clear tat agents are appy to trade in all te frits 6

7 tey dislike for te frits tey like. Ts, in a competitive market, for eac nit of apple eac agent will be able to obtain p CA coconts in date. For tis given price, eac agent solves max A i ;C i J i A i ; C i ; p A i + Ci R + p Ai p + C i sbject to te bdget constraint in (). Given te simple linear strctre, te individal demand fnction is A i c+ ; Ci 0 if p > ; A i ; C i c if l p ; A i 0; C i c + l if p < l: Tis is intitive: individal agents sold old te frit wose relative price is iger tan te marginal rate of transformation, and given te kink in te transformation tecnology inaction may be optimal. We can derive te niqe symmetric market eqilibrim by combining individal demand fnctions () wit te eqilibrim condition () C i A i C p: (3) A It is apparent tat te eqilibrim price p as to lie between te interval [l; ]. We caracterize market eqilibria based on te relative initial cocont endowment c. Case Sppose c > so tat te initial cocont endowment is relatively ig. Ten te market eqilibrim as p, and individal agents invest in apples to reac te oldings of A i + c > ; Ci c c < c: Case Sppose c < l so tat te initial cocont endowment is relatively low. Ten te market eqilibrim as p l, and individal agents invest in coconts to reac te oldings of A i l c l < ; C i c + l c > c: Case 3 Oterwise, wen c [l; ], te market eqilibrim as p c, and individal agents do not invest so tat A i ; C i c: Social planner s problem and ine ciency. Te planner maximizes te sm of tilities of agents. Te only di erence between te planner and te market is tat te market takes prices as given, wile te social planner takes into accont ow individal decisions determine prices. Ts, 7

8 we can write te problem of te planner as max A;C A + C C A! R + A CA + C sbject to te aggregate bdget constraint similar to (): max AR + C A;C A + C + c if A > C; la + C l + c if A C: (4) Importantly, te date price does not play any role in social planner s problem. Recall te parameter restriction tat R (l; ), ten te optimal soltion is simply te endowment allocation A ; C c: (5) Intitively, R gives te marginal rate of sbstittion for social welfare. If it lies between te two marginal rates of transformation, ten it is socially wastefl to transform one frit to te oter. However, as sown in te market soltion, individal agents overinvest in apples (nderinvest in coconts) wen te initial endowment of coconts is relatively ig, wile overinvest in coconts (ts nderinvest in apples) wen te initial endowment of coconts is relatively low. Intition and discssion. Let s igligt te main lessons from tis example. In general ine ciency cold be de to ine cient resorce allocations; bt it is not te case ere, as te date trading ensres te e cient resorce allocation among ex post eterogeneos agents. In words, all agents wo like apples eat all te apples and all agents wo like coconts eat all te coconts. In fact, nder bot te planner s soltion and te market one, given te xed pairs of (A; C) te representative agent obtains Z A i + Ci R + p Ai p + C i di AR + C: In fact in or model te ine ciency arises prely de to te divergent ex ante private incentives to transform frits compared to te one of te social planner. To captre divergent investment incentives, we can stdy te marginal rate of sbstittion for bot te social planner and individal agents. Te social planner s vale, given te pair of apple-cocont oldings (A; C), is simply given by J S (A; C) AR + C; and terefore te social planner s marginal rate of sbstittion (MRS S ) between apple and cocont is a constant independent of date market price p. MRS S J A S (A; C) JC S (A; C) R ; (6) 8 In contrast, te private vale of te pair of

9 apple-cocont oldings A i ; C i, given te price p, is J i A i ; C i ; p C R A i + AC A Ci + C A Ai + C i R A i + Ci + p pai + C i : Ts, from te perspective of individal price-taking agents te marginal rate of sbstittion between apple and cocont is MRS i J i A i Tis is exactly te ex post price p! J i C i A i ; C i ; p C A A i ; C i ; p C A (R + p) p: (7) p R + Interestingly, tere is a wedge between te social planner s marginal rate of sbstittion R and tat of individal agents p. Te economic force beind tis wedge is as follows. Altog te ex post (date ) trading garantees te e cient resorce allocation, it introdces te distribtion of economic rents in a way tat in general distorts te individal agent s ex ante (date 0) private marginal rate of sbstittion. To see tis, consider te pair of trading agents so tat one likes cocont bt olds apple, and te oter likes apple bt olds cocont. Te apple in te cocont agent s and delivers im a tility of p, wile te cocont in te apple agent s and delivers Rp (ceck (7)). Becase te social planner s marginal rate of sbstittion R sold not take te rent distribtion into accont, ex post trading price leads to distortion in ex ante incentives and cases a pecniary externality. Moreover, te ex post price p CA depends on te relative abndance of between apples and coconts. Wen te private marginal rate of sbstittion p is iger (lower) tan R, most of te rent from olding coconts (apples) goes to te agents olding apples (coconts), and ts compared to te social planner olding apples (coconts) becomes more attractive tan olding coconts (apples). Finally, note tat in resolving te ine ciency, te social planner does need to identify (or make agents to reveal) wic agent is it by wic idiosyncratic sock, as ex post trading will implement te e cient allocation of frits. ex-ante investment decision. It is s cient for te social planner to control te Te above intition of welfare canging pecniary externalities drives or main reslts in or general model, were we consider te coice between a more prodctive good called tree (instead of apples), and a less prodctive good called cas (instead of coconts). 4 In or fll dynamic stocastic model wit cas ow socks, instead of keeping te combination of (A; C) at a xed ratio, bot te planner and market participants will nd it optimal to coose a (di erent) nondegenerate distribtion of te A to C by transforming one good to te oter only occasionally. Ts, weter te market over- or nderinvests in trees depends on te realized state of te A to C ratio. Tis ratio is determined by te particlar istory of te cas- ow sock of trees, even if te e ciency of te tree remains te same. 4 One important di erence is tat by replacing coconts wit cas, even te agent likes apple ex post sold be able to consme cas wit marginal tility of, wic places te bond of date price p R. 9

10 3 Te Model 3. Assets We model a market for risky projects. Tese projects are represented by trees. Te only oter asset in te economy is frits wic we call cas; bot cas and tree are perfectly storable. All trees matre at te same random time arriving wit Poisson intensity : A matred tree pays a single cas-dividend R or 0 as we specify below. However, before te trees matre, eac tree migt provide some positive cas payot, or it migt reqire maintenance so tat te cas payot is negative. Tis sock is common across trees and driven by dz t, were Z fz t ; F t ; 0 t < g is a standard Brownian-motion on a complete probability space (; F; P). Wen dz t > 0 te tree provide cas; wen dz t < 0; te owner of te tree as to invest tis amont to te tree and oterwise te tree dies. Denote by A t te aggregate qantity of trees. Ten given te aggregate cas sock dz t to eac nit of tree, te evoltion of aggregate cas, witot investment or disinvestment (to be introdced sortly), is 3. Agents and frictions dc t A t dz t : (8) Te market is poplated by a nit mass of risk netral specialists wo tend te trees. We tink of specialists as a pooled set of bankers-entrepreners representing all te agents wo nderstand te risky projects. At eac time instant specialist may decide to plant new trees, trade trees for cas at te eqilibrim price p t, or liqidate te trees. Planting a new tree costs nits of cas, wile liqidating te tree (selling it for rewood) provides l nits of cas were > l. Tis scraping tecnology ensres te limited liability of te asset owner despite te potentially nbonded losses in (8). Specialists can also consme teir cas at any moment for a constant marginal tility of. Becase of linear tecnologies, in general or model featres tresold strategies for (dis)investment in bot market eqilibria and te social planner s problem. Ts, we can simply focs on tresolds to compare di erent (dis)investment strategies. Tere are two major frictions in tis economy. First, tis economy is closed in te sense tat tere are no otsiders wo cold provide cas for specialists at any point nder any arrangement (wit te only exception of te nlimited bying of rewood for l). Tat is, otsiders do not ave te man capital to tend te tree and ftre cas ows are not pledgeable. Tis extreme assmption is a sort-ct for frictions (e.g., informational asymmetries and agency problems) wic in general prevents otside investors to nance all positive NPV projects in te economy. 5 Second, specialists are sbject to non-veri able idiosyncratic socks. Wile specialists are ex ante identical, ex post tey di er in teir skills. Speci cally, wen te tree matres, it comes wit anoter new tecnology. Half of te specialists ave te skill to arvest te tree, bt do not ave te skill to operate te new tecnology. Ts, in teir ands eac tree pays R nit of cas, bt te 5 Were we to partially relax tis assmption by allowing specialists to borrow a limited amont, or reslts wold not cange. 0

11 Figre : Time line. new tecnology provides zero retrn. Te oter alf of te specialists ave te expertise to invest in te new tecnology, bt do not ave te skill to arvest te tree. In te and of a specialist belonging to tis grop, eac tree pays 0 dividend, bt te new tecnology provides > cas retrn for every nit invested. Tis sitation is analogos to te apple-cocont example stdied in Section. Jst as in tat example, we only need tat ex post tere is eterogeneity among agents in teir valations of te available assets. 6 As before, we assme tat te otpt from te new tecnology, i.e.,, are not pledgeable, an assmption we will discss later in Section 4... Trogot we assme tat R > ; (9) wic ensres tat bilding trees is socially e cient wen te economy as s cient cas. Specialists learn wic grop tey belong to only at te moment rigt before te tree matres. We refer to te grop wit te skill to invest in te new tecnology as specialists it by te idiosyncratic sock or skill-sock. After specialists learn weter tey are it by te sock, all specialists ave a last trading opportnity to trade trees for cas. We refer to te potentially in nitely long time interval before te asset matres as ex ante, and refer to te in nitely sort interval wen te asset matres as ex post. We denote te ex post price by p T wic will be determined sortly. Figre smmarizes te time-line of events in or model. Wile te dynamic strctre of or model migt seem nsal, we will arge tat tis strctre ni es te advantages of two period models and in nite period models. In particlar, tis strctre keeps te analysis analytically tractable, bt still gives te opportnity for te analysis of te stationary distribtion of ex ante variables. In Section 5 we consider an alternative speci cation were idiosyncratic skill socks occr over time, and sow tat te main reslts still old. 6 A more intitive assmption wold be to allow for te arrival of te idiosyncratic sock in every instant. Under tis alternative speci cation a given fraction of agents ave to cas in and exit te market wit marignal tility in every instant. Ten, or interpretation tat te idiosyncratic sock is an idea for a new invesment opportnity is more natral. Tere are also oter natral applications. For example, in a osing market context, te idiosyncratic sock is tat an agent as to move to anoter city and sell is ose. In tat case te move provides a marginal tility of : Indeed, we analyze tis alternative speci cation in Section 5.3 sowing tat or mecanism carries trog. However,tat strctre is not analytically tractable. Also, or main speci cation makes or mecanism more transparent.

12 3.3 Agent s problem Consider specialist i wo olds A i t nits of tree and Ct i amont of cas, wose dollar vale of wealt wt i is wt i p t A i t + Ct: i Ten te specialist i is solving te following problem: max d i t 0;Ai t ;Ci t ;dai t max E d i t ;Ai t ;Ci t ;dai t Z E 0 d i t Z Z t e t 0 0 d + A i t + Ci t R + p T Ai tp T + Ct i dt ; (0) were i t is te specialist i s cmlative consmption (so it is non-decreasing wit d i t 0; later we see tat it is zero in eqilibrim), and da i t is te amont of trees tat e liqidates or bild. In te rst line te expectation is formed bot respect to te aggregate sock Z t and te Poisson event of te matrity of te trees. In te second line expectation is formed only wit respect to te aggregate sock Z t : In te second bracket of te second line, we also sed te direct conseqence of or assmptions tat tose it by te skill-sock sell teir trees for p T to tose wo are not it by te sock. For instance, wen te skill sock its, te specialist sells te tree to receive A i tp T, and ten invests tem togeter wit C i t in te new tecnology wit prodctivity. Te problem in (0) is sbject to te dynamics of individal wealt, dw i t d i t da i t + A i t (dp t + dz t ) ; were is te cost of canging te amont of trees so tat 7 ( if da i t > 0 l if da i t < 0 : Also, wealt cannot be negative at any point, i.e., w i t 0 for all t: Combining te investment/disinvestment policy da t, (8) implies tat te dynamics of aggregate cas level in te economy is were dc t A t dz t da t ; () Z A t A i tdi: Te scale-invariance implied by te linear tecnology in or model sggests tat it is s cient to 7 To simplify notation we ingore te possibility tat at any given point in time some agents create trees wile some agents liqidate trees. Hence, te lack of i sperscript of : It will be easy to see tat tis never appens in eqilibrim. i

13 keep track of te dynamics of te cas-to-tree ratio c t C t A t, wic follows dc t dc t A t C t A t da t A t dz t ( + c t ) da t A t : () 3.4 De nition of Eqilibrim Or eqilibrim concept is standard. De nition In an incomplete market eqilibrim. eac specialist cooses d i t; A i t; C i t; da i t to solve (0), and. markets clear in every time instant bot ex ante and ex post, i.e., Z i Z A i tdi A t ; Ctdi i C t ; and i A tp T C t: As we will see, in or framework, te eqilibrim only pins down te aggregate variables: prices, net trade, and net investment and disinvestment. Typically, any combination of individal actions consistent wit te aggregate variables will be an eqilibrim. Ts, often it will be convenient to pick te particlar incomplete market eqilibrim were all specialists follow te same action. We refer to tis case as te symmetric eqilibrim. De nition A symmetric eqilibrim is an incomplete market eqilibrim were d i t d t ; A i t A t ; C i t C t, and da i t da t : In te rest of te paper we omit te time sbscript wenever it does not case any confsion. 3.5 Incomplete Market Eqilibrim We solve for te incomplete market eqilibrim in tis section. It is clear tat in tis economy consmption before matring event is strictly sboptimal, ts d i t d t 0 always Ex post eqilibrim prices Let s start te analysis wit te event wen te trees matre. All specialists wo are it by te skill-sock sell teir trees, becase teir marginal valation of trees drop to zero. As long as te price of te tree is less tan R; all cas olders wo are not it by te sock are appy to cange all teir cas to trees. Ts, appealing to te law of large nmbers, te market clearing condition jst before te asset matres is C Ap T implying p T c: Tis is an eqilibrim price as long as c < R: As we will see, te fll spport of c will be endogenosly determined in or model, as agent will bild (dismantle) trees wenever 3

14 te aggregate cas is s ciently ig (low). For simplicity, we will restrict te parameter space to ensre tat te condition c < R is satis ed for te fll spport of c tat prevails in eqilibrim Ex ante eqilibrim vales, prices, and investment polices Before te matrity event, determining te eqilibrim objects is more sbtle. As we state in te next lemma, or formalization as a nmber of sefl properties. Namely, te only relevant aggregate state variable is te cas-to-tree ratio, te vale fnction is linear in trees and cas. Lemma Let J C; A; A i t; C i t te vale fnction of specialist i: Ten wit aggregate cas-to-tree ratio c CA; tere are fnctions v (c) and q (c) tat, J C; A; A i t; C i t A i t v (c) + C i tq (c) : Tat is, regardless of te specialists portfolio, te vale of every nit of tree is v (c) and te vale of every nit of cas is q (c) ; bot fnctions only depend on te aggregate cas-to-tree ratio and will be determined sortly. Becase of linearity, te eqilibrim price as to adjst in a way tat specialists are indi erent weter to old te tree or te cas. Tat is, te eqilibrim price of tree p (c) mst satisfy tat p (c) v (c) q (c) : Specialists bild trees wenever te price of te asset p reaces te cost ; and liqidate trees wenever te price falls to te liqidation vale l: De ne c (c l ) as te endogenos tresold of te aggregate cas-to-tree ratio were specialists start to bild (liqidate) trees, ten we mst ave v (c ) q c ; and v (c l ) q c l: (3) l As bilding trees redces te cas-to-tree ratio wile liqidating trees increases it, tis implies tat c and c l are re ective bondaries of te process c. Terefore, based on (), te aggregate cas-to-tree ratio c mst belong to te interval [c l ; c ], wit a dynamics of dc dz t du t + dl t ; were du t ( + c ) dat A t re ects c at c from above wile dl t (l + c l ) dat A t re ects c at c l from below. Moreover, te standard properties of re ective bondaries imply te following smoot pasting conditions for or vale fnctions: v 0 (c ) q0 (c ) q0 (c l ) v0 (c l ) 0: (4) 4

15 3.5.3 Caracterizing te incomplete market eqilibrim Now we trn to caracterizing te vale fnction in te range c [c l ; c ] : We give ere a draft and sow te details in te Appendix. Becase of Lemma, we can separately analyze ow te vale of olding a nit of te tree, v (c) ; and te vale of olding a nit of te cas q (c) varies in te range c [c l ; c ] by te following steps. First, we write down te standard Hamiltonian for J C; A; A i t; Ct i : Second, we conjectre and verify tat q (c) > olds, so tat specialists do not consme before te asset matres. Finally, given te indi erence among specialists in te composition of teir portfolios we consider te dynamics of te vale fnction of an agent wo olds only tree and anoter agent wit cas only. Te former gives te ODE for q (c): 0 q00 + ( q (c)) + R c and te latter, given q (c) ; yields te ODE for v (c): q (c) ; (5) 0 q 0 (c) + v00 (c) + (c v (c)) + (R v (c)) : (6) One can interpret tese ODEs as Eler eqations. Tey ensre tat given te dynamics of te state c; agents are indi erent weter to old te cas (or tree). We rst explain te terms witot in bot ODEs. For te cas vale q eqation (5), q00 captres te impact of canging aggregate liqidity; and a similar term sows p in te asset vale v eqation (6). In addition, we ave q 0 (c) in eqation (6). Tis is becase te asset itself generates random cas ows dz t tat are correlated wit te aggregate state c + dz t, and te expected vale of tese cas ows is E t [q (c + dz t ) dz t ] E t q 0 (c) (dz t ) i q 0 (c) dt: Te terms mltiplied by te intensity describe te cange in expected tility if te asset matres. Te rst of tese terms in eqation (5) sows tat, once a specialist olding a nit of cas is it by a skill sock, er vale jmps to from q (c) : If se is not it by te sock, te second term says tat se ses te nit of cas to by p T c nit of tree, so er tility jmps to R c from q (c) : Te interpretation in eqation (6) is similar. One can solve te ODE system in (5)-(6) in closed-form, wic admits te following general form: q (c) + e c K + e c K + R e c Ei ( c) + e c Ei (c) ; (7) and v (c) R + c + ec (K 3 ck ) e c (K 4 + ck ) + cr (e c Ei ( c) e c Ei (c)) ; (8) 5

16 p were, Ei (x) is te exponential integral fnction de ned as Ei (x) Z x e t t dt; and te constants K -K 4 are determined from bondary conditions in (4). If te reslting price p (c) v(c) q(c) falls in te range of [l; ] for any c [c ; c l ] ten we ave an eqilibrim. Te following proposition gives s cient conditions for sc an incomplete market eqilibrim to exist and describe te basic properties of tis eqilibrim. We smmarize tis reslt below and give formal proof in te Appendix. Proposition If te di erence between te cost of liqidation, l and te cost of bilding a tree, is relatively small ten an incomplete market eqilibrim wit te following properties exist:. agents do not consme before te tree matres,. eac agent in eac state c [c l ; c ] is indi erent in te composition of er portfolio 3. agents do not bild or liqidate trees wen c (c l ; c ) and, in aggregate, agents spend every positive cas sock to bild trees i c c and nance te negative cas socks by liqidating a s cient fraction of trees i c c l : 4. te vale of olding a nit of cas and te vale of olding a nit of tree are described by v (c) and q (c) and price ex ante is p p (c) v (c) q (c) : 5. Ex post, eac agents it by te sock sells all er trees to te agents wo are not it by te sock for te price p T c; and 6. q (c) is monotonically decreasing, v (c) is monotonically increasing and p (c) is monotonically increasing. Becase all specialists are ex ante indi erent ow mc cas or trees to old at te eqilibrim prices, te properties of or market eqilibrim leave individal portfolios ndetermined. symmetric market eqilibrim picks te market eqilibrim were all individal portfolios are te same. Te Investment waves Te tick, solid lines on te tree panels of Figre 3 illstrate te properties of te market eqilibrim. We call te cas-to-tree ratio c te aggregate liqidity. We tink of te time wit ig (low) aggregate liqidity in wic trees are bilt (liqidated) as a boom (recession). Altog investment takes a simple tresold strategy so tat investment (disinvestment) occrs at c (c l ), we believe it captres te essence of investment waves observed in te data. 6

17 Te economy ctates between tese states becase te aggregate level of liqidity of specialists also ctate. Tis ctation is driven by te interim cas- ow socks. Wen aggregate liqidity is low, te marginal vale of cas increases, te price of te tree falls, and te expected retrn of olding trees rises. Tis is so, becase in tese states te probability tat te economy slips into a recession wen trees ave to be liqidated is large. Ts, specialists old a large amont of low-retrn cas and willing to old trees only for a s ciently large premim. Tis is consistent wit te so-called slow moving capital pzzle. 8 An eqivalent interpretation is tat te marginal vale of cas is ig in tese states becase tey can be trned into ig-expected retrn trees. Wen aggregate liqidity is ig, te marginal vale of cas and expected retrns are small, becase of symmetric reasons. Te constant p parametrizing fnctions v (c) ; q (c) in (8) and (7) as an important role in te following analysis. Intitively, drives te relative importance of te ex post payo s for ex ante decisions. Wen te switcing intensity is ig or a low redces te cance of large interim socks, ex post payo s are important determinants of ex ante decisions. Te following reslts on te investment and disinvestment tresolds are sefl to nderstand te intition beind or reslts. Proposition In te incomplete market eqilibrim. c > ; c l < l. as! ; c! and c l! l: Consider te last reslt. Wen grows witot bond te specialist s ex ante decisions are almost solely determined by ex post payo s. Wen te specialist expects te idiosyncratic sock to be realized immediately, in expectation te specialist is better o trning nits of cas to a tree wenever + R c c + R were te term in te bracket on te left and side is te expected vale of olding a nit of cas, wile te rigt and side is te expected vale of olding a nit of tree wen te asset matres. Clearly, tis ineqality olds for any c > : Tis explains tat c! in tis limit. Away from tis limit, wen creating trees, te specialist considers also te risk of reacing te low liqidity state wen tese trees are liqidated ine ciently. Ts, se decides to bild trees only at a iger tresold, c > : Similarly, te specialist is better o in expectation to trn a tree into l nits of cas wenever + R l c c + R or c < l: Tis gives c l! l: Away from tis limit, wen liqidating trees, te specialist considers 8 See te presidential address of D e (00). 7

18 Figre 3: Panels depict te vale of cas, trees and te price of trees for te baseline model wit incomplete markets (tick solid crves) and te bencmark of complete markets (tin, dased crves). Te solid, vertical line on te rigt of eac grap is at te invesment tresold in complete markets and in social planners soltion, c P ; wile te two dased vertical lines are te disinvesent and invesment tresolds in or baseline case, c l ; c : Te orizontal lines on te bottom panel are at te levels of l and : Parameter vales are R 4:; ; 0:; ; l :8 and : 8

19 also te risk of reacing te ig liqidity state wen tese trees ave to be rebild. Ts, se decides to liqidate trees only at a lower tresold, c l < l: Price distribtions Given te fnction p (c t ) ; we can nd te stationary distribtion of ex ante prices easily. tis, rst note tat te state c follows a niform distribtion on te spport [c l ; c ] by standard argments. 9 Ten te pdf and te cdf of p t is given by respectively. 0 (p t ) c c l (p t ) p (p t ) ; c c l p 0 (p (p t )) ; As bondary conditions (4) imply p 0 () p 0 (l) 0; te vale of te pdf is very ig close to te barriers and l: In fact, te conseqence of te S-saped p (c) is tat pdf is inverse U-saped and te cdf is very steep at te barriers and at in between. We sow an example of te pdf of c and te cdf of p on Panels C and D of Figre 4. Altog te state c is eqally likely to be at any point of its spport, te realized price is most often eiter very ig or very low. In tis sense, te volatility of te price of te tree is mc iger tan te volatility of te nderlying cas- ows. Tis is a form of excess volatility consistent wit te classic observation of Siller (98). For 4 Externalities We stdy pecniary externalities in tis section. As a bencmark, we rst solve for constrained e cient allocation in tis economy. We ten sow tat or model featres a two-sided ine ciency on investment waves, and a one-sided intervention in boosting investment in recession may lead to lower welfare everywere inclding recession. 4. Constrained E cient Allocations In tis part, we discss te constrained e cient soltion of or problem were te planner takes into accont te tecnological constraint. Namely, tat otside capital cannot be injected into te system. Tat is, te aggregate cas as to be kept non-negative by liqidating trees if necessary. We will consider two bencmark economies wic bot prodces te same constrained e cient otcome. 9 Tis is so, becase c t is a Brownian motion wit no drift reglated by re ective barriers. See Dixit (993) pp Clearly, te expressions for te cdf and pdf of p t are meaningfl only if p (c) is monotonically increasing. Altog we do not prove tis property directly, we nd tat it olds for every set of parameters we ave experimented wit. Note tat witot tis tecnological constraint, condition (9) implies tat te planner sold convert any amont of cas to trees. 9

20 First, we discss te social planner s problem wo can dictate te investment policy in tis economy. Compared to te decentralized market eqilibrim, te main di erence is tat in te market eqilibrim specialists take prices as given and tis market price drives teir decision to bild or dismantle trees. In contrast, te social planner directly decides wen to bild or dismantle trees. Second, we consider a decentralized economy were markets are complete so tat eac individal agents ave access to te same investment opportnities (trog contracting), wic allows s to caracterize te asset prices witot ine ciency. 4.. Social planner s problem De ne te social planner s vale fnction as J P (A; C). Te planner can decide wen to bild and liqidate trees. Ts, se optimally reglates te c process sbject to te constraint tat te cas level C mst stay non-negative. Ex post, cas (trees) always ends p in te specialists wit (witot) skill sock at te market clearing price p T c. Terefore, te total vale ex post is AR + C: (9) Ts, given te crrent aggregate state pair (A; C), te problem of te social planner is Z J P (A; C) max E e t (A t R + C t ) dt da 0 (0) sbject to te constraint C t 0 and (). rewriting (0) as Te linearity implies tat we can de ne j P (c) by Z J P (A; C) Aj P (c) max E e t (AR + C) dt : da 0 Becase of te linear tecnology of planting and dismantling trees, reglation wit re ective barriers on c is optimal. Tat is, tere exists low and ig tresolds c P l ; cp, so tat it is optimal to stay inactive wenever c c P l ; cp, and dismantle (bild) jst enog trees to keep c c P l (c c P ) at te lower (pper) tresold. Before trning to te determination of optimal investment/disinvestment tresolds, it is sefl to tink of te social vale given tat c is reglated by any arbitrary re ecting barriers c l ; c : De ne te corresponding (scaled) social vale as j S (c; c l ; c ) so tat Z Aj S (c; c l ; c ) E e t (A t R + C t ) dtjc l ; c : () 0 Clearly, te optimal vale acieved by te social planner is See Dixit (993) for a detailed argment. j P (c) max c l ;c j S (c; c l ; c ) : () 0

21 Using standard reslts in forming expectations on fnctions of reglated Brownian motions, 3 for any c (c l ; c ), j P (c) mst satisfy were we sppressed te argments of j S and j 00 c l ; c te smoot pasting conditions mst old: 0 j00 S + (R + c j S ) ; (3) j c. Also from (), at te re ective [Aj S (c l ; c l ; c [Aj S (c l ; c l ; c )] ; [Aj S (c ; c l ; @ [Aj S (c ; c l ; c )] : We empasize tat tese conditions are not optimality conditions. Tey old for any arbitrarily cosen barriers c l < c as a conseqence of forming expectations on a reglated Brownian motion. Te ODE (3) as a closed from soltion j S (c; c l ; c ) R + c + D (c l ; c ) e c + D (c l ; c ) e c : (5) For any xed c l ; c, te constants D and D are solved by invoking te smoot pasting conditions in (4): R + c + D e c + D e c ( + c ) D e c + D e c ; (6) R + c l + D e c l + D e c l (l + c l ) D e c l + D e c l : (7) Following Dmas (99), to determine te optimal barriers c P l ; cp wic solve (), we ave to add spercontact conditions. For te pper barrier, tis is wic we can rewrite Aj P c P ; c Aj P c P ; c l; ; (8) C j P (c; c l ; c ) cc P D e cp + D e cp : For te lower barrier, we ave to take into accont tat at te optimal coice, te constraint C 0 migt bind. Ts, te spercontact condition Aj P c P l ; c l; c Aj P c P l ; c ; for c P l 0 C wit complementarity. In te next proposition we state tat te lower optimal tresold is always in te corner, i.e., c P l 0, and te pper tresold is te niqe soltion of a simple eqation. Proposition 3 Te social planner liqidates trees wen c reaces 0 and bilds trees wen c reaces 3 See Dixit (993) for a detailed argment.

22 c P > 0: Te tresold cp statics. R R is given by te niqe soltion of l e cp ( + l) ( l) e cp c P + 0: (9) To nderstand te coice of te social planner, it is sefl to consider te following comparative Proposition 4 Te socially optimal investment tresold c P. is converging to 0 as! ; and decreasing in given tat > ^ for a given ^;. decreasing in l and R and increasing in : 3. converging to as R! or! R : Consider te case wen is nbondedly large. Te rst statement sows tat in tis case te social planner does not store any cas, bt converts it to trees immediately. Te idea is as follows. In tis limit eiter is very large or is very small. Bot imply tat te social planner does not ave to worry abot te possible negative socks before te tree matres. It is so, eiter becase te asset matres very fast or becase te interim socks are small. Terefore, se decide to not to store any cas, in line wit condition (9) ensring tat at matrity te payo of a tree is larger tan te marginally tility weigted cost of creating a tree. Wen is not so large te social planner worries abot negative socks. As reinvesting from cas reserves is ceaper tan liqidating trees for reinvestment, se stores some cas and bilds te trees only wen te cas-to-tree ratio reaces te positive tresold c P : At tat point te marginal tility of cas is s ciently small tat trning cas to trees is optimal. Te second statement is also intitive. Wen planting trees is expensive, liqidating tem is very ine cient or teir retrn is low tan reinvesting by liqidating trees as opposed to by sing cas is more painfl. Ts, te social planner olds on to te cas ntil a iger tresold. Finally, wen R is s ciently small, te social planner wold only sligtly prefer to trn cas to trees even absent of te possibility tat a series of bad interim socks indces ine cient liqidation of trees. Ts, te tresold to trn cas to trees increases witot bond. 4.. Market prices nder constrained e cient allocations wit complete market Consider te variant of or decentralized model were markets are complete so tat constrained e cient soltion is acieved. Tere are many di erent ways to model complete markets. In te context of or model wit investment, we simply assme tat te proceeds R and are flly pledgeable so tat individal agents can enjoy te investment opportnities of oters. 4 Ts, all agents know tat tey invest teir cas-oldings to te new tecnology and none of tem loses teir expertise to tend te trees. Te critical point is tat te ex post eterogeneity among agents 4 In te context of ex post perference as in te simple example in Section, complete market reqires Arrow-Debre secrities written on te idiosyncratic preference socks and ts veri able idiosyncratic preference socks.

23 e ectively disappear. We refer to tis variant as te complete market economy or te sbscript cm. By following te same derivation, in tis case q cm (c) and v cm (c) solve te ODE system 0 q00 cm (c) + ( q cm (c)) ; (30) 0 q 0 cm (c) + v00 cm (c) + (R v cm (c)) : (3) Te following statement caracterizes te eqilibrim in tis variant of or model. Proposition 5 In te complete market economy, tere is an eqilibrim for any set of parameters were. agents do not consme before te tree matres,. eac agent in eac state c 0; c P is indi erent in te composition of er portfolio 3. eac agent olding trees se every positive cas sock to bild trees i c c P negative cas socks by liqidating te tree i c 0: and nance te 4. te vale of olding a nit of cas, te vale of olding a nit of tree and te price of te tree is described by were L ; L ; L 3 ; L 4 and c P v cm c P ; q cm q cm (c) + e c L + e c L ; (3) v cm (c) R + c + ec (L 3 cl ) + e c (L 4 cl ) : (33) c P is given by bondary conditions v cm (0) q cm (0) l, v0 cm c P q 0 cm c P v 0 cm (0) 0 (34) and j P (c) v cm (c) + cq cm (c) for all c t: 5. v cm (c) is increasing in c; q cm (c) is decreasing in c and p cm (c) vcm(c) q cm(c) is increasing in c: Te Proposition states tat in tis economy, te market implements te social planner s soltion. Tis economy is constrained e cient, as individal agents ave te same objective as te social planner. It is also important to note tat te qalitative properties of te constrained-e cient economy is qite similar to te market soltion of or baseline economy. In particlar, as te casto-tree ratio decreases, te price fall and te tree trades wit a signi cant liqidity premim. We illstrate tis eqilibrim by te tin, dased crves on Figre 3. Te intition for all tese reslts 3

24 are te same as in te economy wit an idiosyncratic skill-sock. Ts, nderpriced assets, large liqidity premim and slow moving capital is not inconsistent wit a constrained e cient economy. Te state c follows a niform distribtion on te spport 0; c P : Jst as in te baseline model, we can determine te pdf and cdf of prices, cm (p t ) c P p0 cm p cm (p t ) (p t ) p cm (p t ) : Te S-sape of p cm (c) implies an inverse U-saped pdf, jst in te baseline case. Ts, as before, altog te economy is in any state c wit eqal probability, most of te time te price is eiter very ig or very low. c P However, wit complete markets te bondary conditions (34) imply only p 0 cm () 0; bt does not imply p 0 cm (l) 0: As a conseqence, te pdf of te price as a very ig vale only close to te pper barrier. Te tin, dased crves on Panel C and D of Figre 4 sow te pdf of c and cm (p t ) : It is apparent tat cm (p t ) is very steep only arond ; bt not arond l: We will retrn tis isse in te next section, were we discss ow or Second Best bencmarks compare to or baseline, decentralized model wit incomplete markets. 4. Two-sided ine ciency In tis part, we arge tat tere is a large sbset of parameters were te externality imposed by te idiosyncratic sock imply bot overinvestment in prodctive assets in a boom and nderinvestment in prodctive assets in a recession. We refer to tis case as two-sided ine ciency. In particlar, we sow tat nlike te social planner, in te market eqilibrim specialists dismantle trees wen still some cas is arond, c l > 0: Also, specialists create new trees at a lower liqidity level tan te social planner wold do, c < cp : We sow tat in tis case any policy wic raises te pper tresold keeping te lower one constant, or decrease te lower tresold keeping te pper one constant wold nambigosly increase total welfare in or economy. Constraining orselves to te symmetric market eqilibrim, tis is eqivalent to a Pareto improvement bot in te ex ante sense, wile ex post welfare is not e ected for any given realized state. Or case is illstrated on Figres 3 and 4, were te dased vertical lines sow te tresolds, c l ; c of te baseline market eqilibrim, and te solid vertical line sows te investment tresolds in te constrained-e cient economy wit complete market. 4.. Te existence of two-sided ine ciency and intition Te following proposition states tat wile te social planner wold dismantle trees only wen all cas in te economy is gone, te market soltion described by Proposition implies c l > 0 for any parameters. Tat is, in te market eqilibrim agents dismantle trees wen te social planner wold still avoid it. In tis sense tere is nderinvestment in prodctive assets or, eqivalently, 4

25 Figre 4: Te total vale of te representative agent, te ratio of vale fnctions, te probability density of te state c and te cmlative density of te price of te tree, p; for or baseline model wit incomplete markets (tick solid crves) and te bencmark of complete markets (tin, dased crves). On Panels A and C, te solid, vertical line on te rigt is at te invesment tresold in complete markets and in social planners soltion, c P ; wile te two dased vertical lines are te disinvesent and invesment tresolds in or baseline case, c l ; c : Parameter vales are R 4:; ; 0:; ; l :8 and : over oarding of liqidity in a recession. As we explain below, te market eqilibrim imply over or nderinvestment in prodctive assets in booms, i.e., c? cp Proposition 6 depending on te parameter vales.. For any parameters, c l > 0; so te market soltion implies nderinvestment in trees and over oarding of liqidity in recessions.. Keeping ; l; ; R xed, tere is a tresold ^ tat if > ^; c > cp ; so te market soltion implies nderinvestment in trees and over oarding of liqidity in booms as well. 3. Keeping ; l; xed, tere are tresold ^; and fnction R () tat if > ^ and R R () ten c < cp ; so te market soltion implies overinvestment in trees and nderinvestment in liqidity in booms. Also, ^R () is decreasing in for > ^ and as! ; ^R ()! : 5

26 Te general intition beind or mecanism is essentially given in te simple example in Section. Te ex post market clearing price not only moves resorces to te most e cient ands bt also allocates te rent among di erent agents, and tis distorted price canges te investment and disinvestment tresolds. Importantly, te direction of price distortion depends on te state of te economy as illstrated in te bottom panel of Figre 3. Becase te representative specialist will sell te tree in te market given a skill-sock, te private (expected) ex post vale of a tree s p T + R; wile from te social perspective, te ex post vale of a tree is always R. Terefore, weter te representative agent overvales te trees compared to te planner crcially depends on weter p T > R : Tat is, weter te private marginal rate of sbstittion is larger tan te social marginal rate of sbstittion as calclated in Section. Given tat p T c ctates in te interval [c l ; c ] we sold expect overinvestment in booms and nderinvestment in recessions wenever c l < R < c : (35) Consistent wit Proposition 6, te rst ineqality in (35) is always satis ed becase c l < l < < R, wereas te second ineqality migt or migt not old, depending on weter c > R.5 An intitive application of or model is te boom and bst pattern in real estate development and ose prices. 6 Or mecanism sggests tat te volme of constrction in a boom is inef- ciently ig, becase banks and investors invest in real estate developments instead of olding liqid nancial assets expecting to be able to sell te real estate for a ig price in case tey nd a new investment opportnity. 7 One sggestive sign of tis ine ciency is te freqently observed 5 Tere is an analogos argment by comparing of te social and private vales of a nit of cas. Tis argement leads to te same ineqalities. 6 Siller (007) illstrates tis pattern by te cyclicality of te residential investment to GDP ratio. He points ot tat cycles in tis ratio correspond closely to te recessions after 950, typically peaking few qarters before te start of te recession. Tis pattern was not observed before te recession bt was observed again before te recession. 7 Related argments were made in connection to te development of Japan. It took most Japanese banks years to wittle down te tens of billions of dollars in nrecoverable loans left on teir books after te collapse of a real estate bbble in Japan s overeated 980 s. Tey nally scceeded in te last two or tree years [...]Bt analysts criticize most banks for failing to nd new, more pro table and less risky ways of doing bsiness. Instead, analysts say many ave gone back to lending eavily to real estate development companies and investment fnds, as te rebonding economy as toced o a constrction boom in Tokyo. If te economy stalled, Japanese banks wold ave a bad loan problem all over again, said Naoko Nemoto, an analyst for Standard & Poor s in Tokyo. Ms. Nemoto estimates tat banks loaned.6 trillion yen ($4 billion) to real estate developers in te six monts tat ended last September alf of all new bank lending in tat period." (Te New York Times, Janary 7, 006, pg.4) 6

27 penomenon of "overbilding," tat is, periods of constrction booms in te face of rising vacancies and plmmeting demand. 8 On te oter and, in recessions, or model sggests tat banks and investors old ine ciently ig level of liqid assets expecting to be able to by real estate ceap in case a grop of distressed investors ave to liqidate teir oldings. Proposition 6 translates te above intition in terms of te deep parameters of te model. We constrct tese restrictions by combining reslts in Proposition 4 and Proposition. Vagely speaking, Proposition 6 sggests tat tere is overinvestment in booms and nderinvestment in recessions, if R is small, i.e., te pro tability of te existing tree tecnology is close to tat of te new investment opportnity. As pointing ot te two-sided ine ciency is te major novelty in or paper, we focs mostly on tis case in te rest of te paper. 4.. Wat market failres drive te ine ciency? To be completed. First, te pledgeability of ftre project payo s in ex post period de nitely elp resolve te distorted ex ante investment incentives. Here, te pledgeability means tat we can write contract on te proceeds from bot projects (bot and R). Ts, te individal agent wit skill-sock can ire te agents witot skill sock to arvest te tree on bealf of te agents wit skill sock, and still vale te fll marginal retrn of R from tree. Similarly, te agent do not ave investment opportnity can lend teir cas to te agent wit skill sock and receive te investment bene t. Tis way, all agents are essentially facing te same investment opportnities, and ts ine ciency disappears. Second, te pledgeability does not apply in or example were R and is te agent s preference. Tere, completing te market by introdcing te Arrow-Debre secrities elp. We can sow tat if individal states are contractible, ten date zero trading of tese secrities will restore te investment incentives. 4.3 Social welfare Now we trn into te explicit comparison of welfare nder te second-best acieved by te social planner and te market eqilibrim wit incomplete markets. symmetric eqilibrim were all specialists old te same portfolio ex ante. For simplicity, we focs on te As empasized earlier in Section, in or model trading leads to ex post e cient resorce allocation. Under bot te social planner s soltion and in or incomplete market soltion, te representative specialist it by te skill-sock gets (p T A + C) (ca + C) C; 8 See Weaton and Torto (990) and Grenadier (996) for alternative explanations of overbilding. Overbidling was also observed before te recession in te sense tat rental vacancies peaked in 004, before te peak of te contstrction boom. (See ttp:// 7

28 wile, if se is not a ected, se gets RA + R p T C RA + R c C AR: Tat is, given te state (A; C) te social planner does not cange te welfare of te representative agent ex post. Tis is intitive, as trading after te sock moves te assets (cas) to te ands wit te igest pro tability. Instead, by canging te tresolds c and c l ; te social planner can in ence te ftre distribtion of c (or, eqivalently, te joint distribtion of (A; C)), wic a ects te representative agents ex ante welfare. To be more speci c, following te argment in Section 4., given any tresolds (c l ; c ) wic reglate te process c; total welfare is given by Aj S (c; c l ; c ) wit an explicit soltion determined by (5)-(7). Note tat in a symmetric eqilibrim, Aj S (c; c l ; c ) is also te ex ante vale of te representative agent. If a policy increase total welfare wit respect to te market eqilibrim, it is an ex ante Pareto improvement wit respect to te symmetric market eqilibrim. Given tat te total welfare is state dependent, we can make a distinction between policies wic improve total welfare at some states, e.g. in recessions only, and policies wic improve welfare everywere. 9 In te next proposition we sow tat increasing te lower tresold or decreasing te pper tresold compared to te social planners soltion nambigosly decreases welfare everywere. Proposition 7 For any c < c P and c l > 0 and S (c; c l ; c l < S (c; c l ; c < 0: Te proposition ensres tat wenever c < cp ; specialists nderinvest in recessions and overinvest in booms in te prodctive asset in or market eqilibrim in a well de ned sense. Indeed, a lower disinvestment tresold or a iger investment tresold wold increase total welfare in a market eqilibrim and wold lead to a Pareto improvement in te symmetric market eqilibrim. Te comparison of te solid crves and dased crves on Panel A and B on Figre 4 illstrate tis point. It is apparent tat in an economy wit two-sided externalities, te social planner raises ex ante welfare at every state. At a basic level, or mecanism is in line wit te welfare e ects of pecniary externalities identi ed by Geanakoplos and Polemarcakis (985). Tat seminal paper sows tat wen markets 9 Note tat in or strctre for any crrent c t; te total welfare fnction factors in te e ect of te policy in eac oter state. Te idea beind a policy tat improves welfare, e.g., only in a recession is tat te probability of arriving in a given state depends on te crrent c t; i.e.,in a recession, a boom looks less likely ten a contining recession. 8

29 are incomplete and, conseqently, prices do not eqate marginal rate of sbstittion of agents, ten pecniary externalities migt ave rst-order e ects on welfare. Or mecanism works by te same logic. Te ex post price p T cannot eqate marginal rates of sbstittions becase agents wo are not it by te sock cannot pay more tan c for te asset, even if teir valation is R; becase tis is all te cas tey ave. In a market eqilibrim, for a given p T agents beave optimally wen tey bild trees at c and liqidate teir trees at c l : However, tey fail to take into accont tat, becase of te missing market, price p T does not serve its Wallrasian fnction of signalling relative social vale of di erent goods. Tis makes specialists decisions socially ine cient. Or main contribtion relative to Geanakoplos and Polemarcakis (985) and te sbseqent literatre is to point ot tat te distortion implied by te pecniary externality is likely to cange sign wit te state of te economy, becase te distortion in te price canges sign. Or complete market bencmark elps to see ow te two-sided ine ciency canges te ex ante distribtion of prices. Panel D in Figre 4 compares te cdfs in te complete market case and te incomplete market case. Te rst ting to note tat te price distribtion in te complete market case rst-order stocastically dominates te incomplete market case. Tat is, two-sided ine ciency implies tat lower liqidity premim states appen wit iger probability compared to te second-best. We also nd tat te nconditional volatility of prices is iger in or baseline model and te distribtion is more positively skewed. 0 All tese are consistent wit or previos observation tat wile in or baseline model te pdf of p t approaces in nity bot at te extremes and l; it is only te case at te ig-end in te complete market case. Tis asymmetry is implied by te di erence in bondary conditions (4) and (34). An important advantage of te dynamic strctre of or model is tat in any ex ante state c agents decisions are a ected by teir expectation of economic conditions in all oter states. Sppose tat te economy is in a recession and a policy is introdced wit te promise tat it will be abandoned as soon as te economy recovers. Tis policy will necessarily in ence agents coices in te boom, wic agents foresee. Ts, in trn, tis e ect will in ence teir crrent reaction to te policy. Tis feedback e ect is in te centre of or analysis of possible government interventions in te next section. 4.4 One-sided interventions In tis part, we will analyze sboptimal policies of a class we call one-sided interventions. Te idea is tat at a state close c l te policy maker migt realize tat te price falls dangerosly close to te disinvestment tresold l and migt decide to intervene to raise prices and to avoid ine cient liqidation of prodctive assets. We do not allow te policy maker to reglate prices directly. Instead, te tool we give to te policy maker is any combination of ex ante taxes and sbsidies to te cas olders and asset olders sbject to a balanced-bdget condition. Te policy maker can e ect te eqilibrim prices and te eqilibrim investment/disinvestment tresolds trog 0 We do not ave analytical proofs for tese statements, de to te lack of a closed form expression for fnctions p (p t) and pcm (p t) : Still, we nd tese reslts robst across all set of parameters we experimented wit. 9

30 tese taxes and sbsidies. Since te policy maker migt realize tat raising prices by taxes and sbsidies is nnecessary in a boom, se migt make te policy conditional on being in a s ciently low c state Tax-sbsidy sceme A one sided intervention lowers te disinvestment tresold by de nition. We know from Proposition (7) tat if te investment tresold, c ; remained constant, tis policy wold improve welfare. Wile we sow tat typically a one-sided intervention redces te investment tresold, c ; implying a negative e ect of welfare, tis is not or main reslt. Te main reslt of tis section is tat tis negative e ect can be so strong tat it migt imply tat te policy redces welfare everywere! Tat is, even if te policy redces ine cient liqidation in te recession, agents welfare migt redce even in te recession, becase tey expect tat te crrent policy will make overinvestment in a ftre boom mc worse. Before stating tese reslts formally, we de ne one-sided intervention and te corresponding intervention eqilibrim. We distingis eqilibrim objects nder intervention from teir conterpart nder no intervention wit te index as in fc l ; c ; p (c) ; v (c) ; q (c)g : De nition 3 A one-sided intervention is a tax-sbsidy sceme (c) and an intervention-tresold c 0 sc tat. for eac nit of cas eld in state c specialists pay (c),. for eac nit of tree eld in state c specialists receive c (c), 3. (c) 0 for any c > c 0 ; 4. te disinvestment tresold is redced by te intervention, c l < c l 5. te eqilibrim price is increased at te intervention tresold, p (c 0 ) > p (c 0 ). De nition 4 An intervention eqilibrim is an incomplete market eqilibrim nder te a taxsbsidy sceme de ned by a one-sided intervention. Note tat wat makes one-sided interventions trly asymmetric is not te arbitrarily large intervention-tresold c 0, bt te reqirement tat te price at tat tresold mst be raised by te intervention. Intitively, we tink of one-sided interventions as policies wic raise prices for every c [c l ; c 0] compared to te market eqilibrim by increasing te marginal vale of te asset v (c) and/or decreasing te marginal vale of cas q (c) for every c [c l ; c 0] : However, for or reslts we need less. Ts, we do not restrict te sign of (c) and impose only te weaker reqirement on te e ect on prices in part 5 of te de nition. In te next proposition, we sow tat a one-sided intervention typically decreases te investment tresold, c < c ; wic in tis sense makes overinvestment in te boom worse. Te condition of te proposition is weak in te sense tat we wold expect a one sided intervention wic is designed to raise prices p to te point c 0 to decrease te marginal vale of cas at tat point. 30

31 Proposition 8 Any one-sided intervention ( (c) ; c 0 ) for wic te vale of cas decreases at c 0 ; q (c 0 ) q (c 0 ) ; redces te investment tresold, c < c : Proposition 8 is qite intitive. After all, te vale of te tree in one state is natrally positively related to its vale in every oter state. Ts, wen intervention raises te price of te tree in low states, its price tend to increase also in ig states. However, tis implies tat c as to decrease to make sre tat te condition p (c ) is not violated. Ts, an intervention focsing on improving nderinvestment in te recession will typically make overinvestment worse in te boom An example wit a Pareto-dominated one-sided intervention Te intitive reslt in Proposition 8 opens an interesting qestion. Te price-boosting one-sided intervention in te recession alleviates te nderinvestment problem in recession; owever, becase it leads to iger prices in te boom, tis one-sided intervention necessarily reslts in more severe overinvestment in te boom. Is it possible tat te latter negative eqilibrim e ect dominates te earlier positive e ect in every state, even in te recession were te one-sided intervention is designed for? We provide an a rmative answer to tis qestion, by constrcting an example were a one-sided intervention is Pareto inferior to no-intervention at all in every state. Te simplest example of a one sided intervention is wen te tax-sbsidy is constant, i.e., (c) for every c < c 0 : Following or derivation of te market eqilibrim, vale fnctions in te intervention eqilibrim are de ned by te ODEs 0 q00 + Rc c<c0 + 0 v00 + q 0 + c<c0 c + q c + R were is te indicator fnction, sbject to te bondary conditions v (36) (37) v (c ) ; v (c l ) l (38) q c q c l v 0 (c ) q0 (c ) q0 (c l ) v0 (c l ) 0: (39) We also ave to make sre tat eac fnction is smoot at c 0 : It is simple to ceck tat te following general soltion satis es te system q (c) c<c0 + + e c ( c>c0 M + c<c0 M 5 ) + e c ( c>c0 M + c<c0 M 6 ) + + R e c Ei ( c) + e c Ei (c) 3

32 Figre 5: Te marginal vale of cas, te marginal vale of a tree, te price of te tree and te ratio of vale fnctions for or baseline model wit incomplete markets (tick solid crves) and a particlar one-sided intervention (tin, dased crves). On Panels A, B and C, te solid, vertical lines in te midle are te tresolds for intervention, c 0 ; and te post-intervention investment tresold, c :wile te two dased vertical lines close to te edges of eac Panel are te disinvesent and invesment tresolds in or baseline case, c l ; c : Parameter vales are R 4:; ; 0:; ; l :8 and and c 0 c 0:5 and 0:05: v (c) c<c0 c + R + c + ec (( c>c0 M 3 + c<c0 M 7 ) c ( c>c0 M + c<c0 M 5 )) e c (( c>c0 M 4 + c<c0 M 8 ) + c ( c>c0 M + c<c0 M 6 )) + cr (e c Ei ( c) e c Ei (c)) (40) were te constants M ; :::; M 8 are given by (38)-(39) and te smoot-pasting conditions at c 0 for v (c) and q (c) : We plot one particlar example on Figre 5. It is apparent tat wile te policy raises prices, decreases bot te investment and disinvestment tresolds, c l < c l ; c < c ; it also redces welfare at every point. Ts, te depicted one-sided intervention is Pareto inferior compared to te symmetric market eqilibrim wit no-intervention. It is instrctive to connect tis reslt to te crrent debate on "Greenspan s pt", i.e., te 3

33 doctrine tat it is s cient if monetary policy intervenes in a recession, bt stays inactive wen te economy is recovered. In or abstract model we can interpret or taxes-and-sbsidies scemes as vage representations of an expansionary monetary policy. An interest rate ct decreases incentives to save cas and increases incentives to invest in prodctive assets, jst as or simple one-sided intervention does. Or reslt sows tat sc interventions migt be armfl even at te recession. Recently, several papers proposed argments against te Greenspan s pt inclding Fari and Tirole (0) and Diamond and Rajan (0). However, teir argment is di erent In Diamond and Rajan (0) te main friction is tat banks can provide only non-state contingent demand deposit contract to oseolds. In sc a world, ex post ine cient bank-rns serve as a disciplining device for banks to onor tese contracts. Anticipated interest rate cts in bad times elps insolvent banks ex post, bt weakens tis disciplining device ex ante. As reslt, banks take on too mc leverage ex ante, and sbject to rns ex post too often. Fari and Tirole (0) makes a similar argment sowing tat tere is strategic complementarity in te coice of increasing leverage ex ante, and, conseqently, needing a more freqent non-directed bail-ot in te form of low interest rates ex post. In bot of tese papers incentive problems related to te agency friction inerent in nancial intermediation play te central role. In contrast, te interaction of a pecniary externality and incomplete markets is in te center of or mecanism. Until now we ave pt little empasis on te parameters wic imply a one-sided ine ciency. Tat is, on te case implying nderinvestment in trees bot in te boom and in te recession. It is sefl to note, owever, tat in tis case, a price increasing one-sided intervention (at least if it is s ciently small) improves welfare by psing te economy closer to te second-best bot in a recession and in a boom. Ts, an alternative reading of or reslts is tat te pros and cons of an asymmetric interest rate policy depend on te natre of te externality we face. In or case, it depends on weter te tecnology represented by trees are mc more prodctive tan te idiosyncratic investment opportnity or not. Only in te latter case a one-sided intervention tends to be armfl. 5 Robstness: an alternative speci cation In tis section, we arge tat in an abstract level, or mecanism is based on tree main ingredients:. Two assets of wic relative spply is a ected by a stocastic process.. A grop of agents wo can transform eac asset to te oter one by a linear tecnology, bt wit some loss in te process. 3. An idiosyncratic sock wic canges some agents relative valation of te assets compared to oter agents. In particlar, we present an alternative speci cation wic also incorporate tis tree ingredients bt a di erent dynamic strctre. Tis variant generates te same main reslts. We empasize 33

34 tat te speci c strctre in or baseline model tat te idiosyncratic sock is connected to te matrity event is immaterial for or reslts. 5. Te Setting In tis variant, we make two major canges. First, te Poisson event tat te sock matres and te idiosyncratic sock is separated. In particlar, te asset matres wit Poisson intensity : Wen it does, all agents wo are still in te market and old te tree arvest te R frit or cas dividend. However, in eac point of time dt fraction of te agents are it by te skill sock. Tat is, tey do not ave te skill to arvest te tree, bt ave te skill to invest in a new opportnity wit a marginal retrn of > : As a reslt, in eac instant, a grop of agents wit measre dt sell all teir trees to te rest of te specialists and exit te market. Te second cange is abot timing of (dis)investment opportnities. Instead of letting te agents to invest and disinvest at any point, we assme tat tey can do so only irreglarly. In particlar, wit intensity ; a Poisson event realizes wen all specialists on te market are allowed to bild trees at cost or liqidate trees at cost l in any amont tey wis. Given tat in tis variant tere is no garantee tat te aggregate cas level is kept away from zero by disinvesting if necessary, we also assme an in nite pool of otside capital wic can inject nit of cas to tis market for a total cost of >. Tat is, otside investors can acqire te knowledge of specialists, bt it is costly. We tink of to be s ciently ig. In a certain sense, in tis variant, te dynamic strctre is te mirror image of tat of or baseline model. Wile in or baseline model specialists can invest and disinvest in any instant before te tree matres, tey learn te realization of te idiosyncratic sock only at matrity at wic tey cannot invest or disinvest frter. In tis variant, in eac instant a grop of agents learn te realization of te idiosyncratic sock, specialists can invest and disinvest only in discrete time periods. 5. HJB Eqations Following or logic before, te vale of tree v (c) and te vale of cas q (c) in te market soltion ave to satisfy te following ODEs as HJB eqations: 0 v 0 (c + p) + q 0 (c t ) + v00 + (p v) + (R v) + (( c>c v (c ) + c<cl v (c l ) + c >c>c l v (c)) v (c)) ; (4) 0 q 0 (c + p) + q00 + ( q) + ( q) + (( c>c q (c ) + c<cl q (c l ) + c >c>c l q (c)) q (c)) : (4) wit p (c) v (c) q (c) ; 34

35 and te following six bondary conditions v 0 (0) 0 (43) q (0) (44) p (c l ) v (c l) q (c l ) l (45) p (c ) v (c ) q (c ) (46) lim c! v0 (c) 0 (47) lim c! q0 (c) 0: (48) In te ODEs, tere are two main canges compared to or baseline model. First, te rst term in eac eqation is de to te fact tat tis variant introdces a drift term into te dynamics of c, as agents wit idiosyncratic socks are leaving wit cas at any instant of time. In particlar, wenever tere is no investment opportnities, we ave dc (c + p) dt + dz t : Te drift is becase dt fraction of agents leave wit te cas wit te normalized vale of teir portfolio c + p. More speci cally, tey leave wit teir cas olding c, and also sell teir tree oldings at a price of p and leave te market wit tese proceeds. Second, te last bracketed term in eac eqation, i.e., (4) and (4), is de to te fact tat wen specialists ave te opportnity to invest or disinvest, tey will do so, only if te price level makes te decision optimal. In particlar, if p > ; tey bild new trees ntil te point were te cas level falls to c were p (c ) : Similarly, if p < l; tey liqidate trees ntil te point were te cas level is raised to c l were p (c l ) l: De to tis adjstment, wenever te opportnity to cange te measre of trees arrives, te vale of a tree, v (c) ; and te vale of cas, q (c) ; jmps to v (c ) and q (c ) if c > c ; and to v (c l ) and q (c l ) if c < c l ; and remains ncanged in every oter case. Trning to te bondary conditions, condition (43) olds becase c 0 is a re ective barrier. Condition (44) as to old, becase otside capital is injected wenever te vale of cas is larger tan : Conditions (45)-(46) are determined by te adjstment of te measre of trees explained above. Te last two conditions ave to old to ensre tat te vale of cas and trees does not increase or decrease witot bond even wen te cas level is very ig. Unlike in te case of te baseline model, no analytical soltion of tis system is available. Ts, we ave to rely on nmerical soltions. Panels A,B and C on Figre 6 sow te fnctions q (c) ; v (c) and p (c) in a particlar example. It is apparent, tat jst as before, q (c) is a decreasing fnction wile p (c) is an increasing fnction. Intitively, tis variant also generates investment waves were te economy ctates between boom periods caracterized by investment and low expected retrns and bst periods caracterized by disinvestment and ig expected retrns. Te 35

36 di erence is tat in or baseline model wenever c it te lower or pper tresold disinvestment or investment began and contined as long as a contrarian sock does not it te system. Under or alternative speci cation, investment and disinvestment is lmpy. Wenever c is below or above te corresponding tresolds and te Poisson even its, a large amont of investment or disinvestment occrs in one instant. Wen te Poisson event does not it, tere is no cange in te nmber of trees regardless of te vale of c. 5.3 Two-Sided Ine ciency and Government Intervention Does tis economy constrained e cient? Jst as before, we asses weter a social planner cold improve welfare by only canging te investment/disinvestment tresolds c ; c l instead of leaving teir determination to te market. We caracterizes te ODE for te eqilibrim vale fnctions v and q in te Appendix, and solve tem nmerically. On Panel D of Figre 6 we compare te market eqilibrim wit tree economies wit sboptimal policies. In te rst one, we marginally decrease c l compared to te market soltion c l : In te second one, we marginally increase c compared to c ; in te tird one we do bot. Under eac scenarios, te vale increases compared to te decentralized otcome in every state. Tis illstrates tat jst as in or baseline model, tere is a two-sided ine ciency in tis variant as well. Te intition is similar to wat we ave illstrated in te baseline model. As in te baseline model, agents migt s er idiosyncratic skill socks wic force tem to sell te tree. And, wen te aggregate cas level is low (ig), te eqilibrim price is low (ig), exactly becase of te cas-in-te-market setting tat we are considering. Terefore, as in or baseline model, tese prices sold a ect te agents ex ante investment incentives wen te investment/disinvestment opportnities arrive occasionally. Take te example wen te crrent aggregate cas is low and agents now can disinvest to convert trees to cas. Becase agents worry tat before te next opportnity comes tey migt be it by a skill sock and terefore sell teir trees to oter agents, wile te social planner sold ignore tis transfer isses, agents will disinvest excessively so tat tey old more cas more tan te social planner wants. Te same idea applies to te state wit abndant aggregate cas. Tere, agents will invest more tan te social planner wants, becase tey will take into accont te fact tat in te near ftre, once it by liqidity socks, te tree can be sold at a ig eqilibrim price. As a reslt, tey will invest cas to obtain te tree, altog te social vale of te cas, i.e.,, can be iger tan v (c ) q (c ). Tese distorted ex ante investment/disinvestment incentives are no longer tere if te investment tecnologies are always available. In te rst case c c + 0:; in te second case c l c l 0:0; wile in te tird case c c + 0:05 and c l c l 0:0: Were we to set c c + 0: and c l c l 0:0 in te tird case, te dotted crve wold be indistingisable from te pper envelope of te solid and dased crves. 36

37 Figre 6: Panel A, B and C sows te vale of a nit of cas, q (c) ; te vale of a nit of asset, v (c) ; and te price of te asset, p (c) ; in te decentralized eqilibrim nder or alternative speci cation. Panel D sows te relative cange in te vale wen te lower and pper tresolds are canged as follows. Te dased crve corresponds to c c + 0:; te solid crve corresponds to c l c l 0:0; wile te dotted crve corresponds to c c + 0:05 and c l c l 0:0: Figres of fnctions v (c) ; q (c) and p (c) nder tese scenarios are indistingisable from te baseline case. Vertical lines in all panels correspond to c l ; c ; wile te orizontal lines on Panel C correspond to and l: Parameter vales are R 3:5; 0:3; 0:; :5; l 0:55 ; 3:8; 0:8; 5; 0:: 37

38 6 Conclsion In tis paper, we bilt an analytically tractable, dynamic stocastic model of investment and trade of a specialized asset. We arge tat if specialized tecnologies are a determining part of te economy, investment cycles arise as a dominant pattern. Tat is, boom periods wit abndant investment and low retrns on tis tecnology will intercange wit bst periods wit low investment and ig retrns. We sowed tat tese cycles migt or migt not be constrained e cient. In particlar, wile nder complete markets, tere are constrained e cient investment cycles, in te presence of nveri able idiosyncratic investment opportnities a two-sided ine ciency can arise. Tat is, tere are two mc investment in te tecnology and too low b er in cas in booms, and tere are too little investment and too mc cas oldings in recessions. We sow tat in tis case a one-sided policy targeting only te nderinvestment in te recession period migt be ex ante Pareto inferior to no intervention in every state. Apart from analyzing two-sided ine ciencies, we also presented a novel way of modelling problems of investment and trade. Tis metod provides analytical tractability in a dynamic stocastic framework for te fll joint distribtion of states and eqilibrim objects. To explore its potential, we se tis framework to analyze te role of sovereign wealt fnds in nancial crises by introdcing grops of specialists wit di erent level of skills in a parallel project. References Allen, Franklin, and Doglas Gale Limited Market Participation and Volatility of Asset Prices. American Economic Review, 84(4): Allen, Franklin, and Doglas Gale Financial Intermediaries and Markets. Econometrica, 7(4): Allen, Franklin, and Doglas Gale From Cas-in-te-Market Pricing to Financial Fragility. Jornal of te Eropean Economic Association, 3(-3): Bianci, Javier. 00. Credit Externalities: Macroeconomic E ects and Policy Implications. American Economic Review, 00(): Bianci, Javier, and Enriqe G. Mendoza. 0. Overborrowing, Financial Crises and Macro-prdential Policy. International Monetary Fnd IMF Working Papers /4. Brnnermeier, Marks K., and Yliy Sannikov. 0. A macroeconomic model wit a nancial sector. ttp://scolar.princeton.ed/marks/pblications/macroeconomic-model- nancialsector. Caballero, Ricardo J., and Arvind Krisnamrty. 00. International and domestic collateral constraints in a model of emerging market crises. Jornal of Monetary Economics, 48(3):

39 Caballero, Ricardo J., and Arvind Krisnamrty Excessive Dollar Debt: Financial Development and Underinsrance. Jornal of Finance, 58(): Davila, Edardo. 0. Dissecting Fire Sales Externalities. ttp:// edavila/papers/dissecting.pdf. Diamond, Doglas W., and Ragram Rajan. 0. Illiqid Banks, Financial Stability, and Interest Rate Policy. NBER Working Papers Dixit, A.K Te art of smoot pasting. Vol., Rotledge. D e, Darrell. 00. Presidential Address: Asset Price Dynamics wit Slow-Moving Capital. Jornal of Finance, 65(4): Dmas, Bernard. 99. Sper contact and related optimality conditions. Jornal of Economic Dynamics and Control, 5(4): Fari, Emnanel, and Jean Tirole. 0. Collective Moral Hazard, Matrity Mismatc, and Systemic Bailots. American Econonomic Review,,: fortcoming. Gale, Doglas, and Tanj Yorlmazer. 0. Liqidity oarding. Federal Reserve Bank of New York Sta Reports 488. Geanakoplos, Jon, and Heracles M. Polemarcakis Existence, Reglarity, and Constrained Sboptimality of Competitive Allocations Wen te Asset Market Is Incomplete. Cowles Fondation for Researc in Economics Discssion Papers 764. Gersbac, Hans, and Jean-Carles Rocet. 0. Aggregate investment externalities and macroprdential reglation. mimeo. Grenadier, Steven R Te Strategic Exercise of Options: Development Cascades and Overbilding in Real Estate Markets. Te Jornal of Finance, 5(5): pp Gromb, Denis, and Dimitri Vayanos. 00. Eqilibrim and welfare in markets wit nancially constrained arbitragers. Jornal of Financial Economics, 66(-3): Hart, Oliver D., and Ligi Zingales. 0. Ine cient Provision of Liqidity. NBER Working Papers 799. Holmstrom, Bengt, and Jean Tirole. 0. Inside and Otside Liqidity. MIT Press. Jeanne, Olivier, and Anton Korinek. 00. Managing Credit Booms and Bsts: A Pigovian Taxation Approac. Peterson Institte for International Economics WP0-. Kiyotaki, Nobiro, and Jon Moore Credit Cycles. Jornal of Political Economy, 05():

40 Krisnamrty, Arvind Collateral constraints and te ampli cation mecanism. Jornal of Economic Teory, (): Lorenzoni, Gido Ine cient Credit Booms. Review of Economic Stdies, 75(3): Siller, Robert J Understanding Recent Trends in Hose Prices and Home Ownersip. National Brea of Economic Researc, Inc NBER Working Papers Sleifer, Andrei, and Robert W Visny. 99. Liqidation Vales and Debt Capacity: A Market Eqilibrim Approac. Jornal of Finance, 47(4): Stein, Jeremy C. 0. Monetary Policy as Financial-Stability Reglation. NBER Working Papers Weaton, William C., and Raymond G. Torto An Investment Model of te Demand and Spply For Indstrial Real Estate. Real Estate Economics, 8(4):

41 A Appendix A. Proof of Lemma and Proposition We constrct te proof in steps. In particlar, we separate Proposition into te following for Lemmas. Lemma A. If te eqation system (7)-(8), (3)-(4) as a soltion were c < R and v (c) is increasing and q (c) is increasing in te range c [c ; c l ] ten Proposition old. Lemma A. Te system (7)-(8), (3)-(4) always ave at least one soltion. Lemma A.3 If l is s ciently small, c < R: Lemma A.4 If increasing in te range c [c ; c l ] : l is s ciently small q (c) is monotonically decreasing and v (c) is monotonically Clearly, te for lemmas are s cient to prove Proposition. A.. Step : Proof of Lemma and Lemma A. Denote te dollar sare of tree in te specialist s portfolio by i t, so tat i t A i tp t w i t. According to or conjectre, te vale fnction can be written as (recall te aggregate cas-to-tree ratio c CA) J A t ; C t ; A i t; Ct i w i t i t q (ct ) + is linear in w t. Note tat tis is eqivalent to te statement J C; A; A i t; C i t A i t v (c) + C i tq (c) i t v (c) J A t ; C t ; w p t i ; t stated in te Lemma. Also, we ave te wealt dynamics, expressed in terms of portfolio coice i t, as dwt i d i t da i t + i twt i (dp t + dz t ) : p t Te HJB of problem (0) can be eqivalently written as te agent is coosing consmption d i t and portfolio sare i t, and te tree to bild or liqidate da i t 0 max d i d i t ; i t + J C E t (dc t ) + J CCdCt + J w E t (dw t ) + JAdA 0 i t + J w;c E t (dw t dc t ) : t ;dai t Let also conjectre tat q (c t ) > for every c t, ts specialists do not consme before te tree matres. Denote te endogenos drift and volatility of prices as dp t p 00 (c) dt + p 0 (c) dz t + dl p t du p t 4

42 were dl p t (du p t ) re ects p at p (c l ) l (p (c ) ). Tis is becase as we explained in te main text, in any market eqilibrim specialists will create trees if p t and destroy trees if p t l oterwise tey wold se te market to adjst te nmber of teir trees. We derived te bondary conditions in te main text. Also, by risk netrality and ex ante omogeneity of agents, before te tree matres te price of te tree as to make specialists indi erent weter to old trees or cas. Oterwise markets cold not clear. We also explained tat p T c t. Ts, for te range of p (l; ) we can rewrite te Hamiltonian as (we drop i from now on) 8 < 0 max : t ; B t w tq 00 c + q (c t ) t w t +w t p t dt + q 0 (c t ) p 00 (c) t R p t + ( t ) R c t + tw t t p t c t + ( t ) p t ( + p 0 ) dt q (c) Note tat tis is indeed linear in w t : It is also linear in t ; so specialists are indi erent in teir coice of t : Ts, we can separate te problem for calclating te dynamics of te cas vale by coosing t 0 and dynamics of tree vale by coosing : Te rst coice directly implies (5). As long as ; R c cas get eiter or R c 9 ; : > ; or conjectre tat q (c) > mst also old as specialists olding only at matrity and tey do not discont te ftre. Coosing t gives 0 q00 (c) + q (c) p 00 (c) + q 0 (c) p t p t Rearrange, we ave + p 0 + p t (R + c) q (c) p t : 0 p (c) q00 +q (c) p 00 (c) + R +q 0 (c t ) + p 0 + (c q (c) p t) + (R q (c) p t) Since v (c) p (c) q (c) and v 0 q 0 p + p 0 q, we can rewrite tis 0 p (c) q00 (c) + q (c) p 00 (c) + R + q 0 (c t ) + p 0 + (c v) + (R v) sing tat v 00 q 00 p + p 0 q 0 + p 00 q gives (6). Given tat te ODEs for v (c) and q (c) were derived by sbstitting in t and t 0; it is easy to see tat tese fnctions can be interpreted as te vale of a tree and tat of a nit of cas. Tis implies tat J C; A; wt i wt i i t q (c) + i t w i p tv (c) q (c) w t t verifying bot Lemma and or conjectre on te form of J C; A; w i t. Also, te de nition of v (c) is eqivalent to a no-arbitrage condition ensring tat specialists are indi erent weter to old te tree or cas of eqivalent vale. One can ceck tat (8) and (7) are indeed soltions of or system of ODEs. 4

43 A.. Step : Proof of Lemma A. First, note tat for any arbitrary c and c l from (4), we can express K ; K ; K 3 and K 4 in (7)-(8) as a fnction of c and c l only. Sbstitting back to (7)-(8) we get or fnctions parameterized by c and c l wic we denote as v (c; c l ; c ) and q (c; c l ; c ) : Evalating tese fnctions at c c l and c c ; we get te following expressions. Tese notations elp s rewrite or critical eqations as q (c l ; c l ; c ) + R e c (Ei[c ] Ei[c l ]) + e c (Ei[ c ] Ei[ c l ]) e (c c l ) e (c c l ) + R f l (c l ; c ) gl (c l ; c ) v (c l ; c l ; c ) R + c l + m (c l; c ) + R c l f l (c l ; c ) q (c ; c l ; c ) + e c K + e c K + R ( e c Ei [ c ] + e c Ei [c ]) p + R e cl (Ei[c ] Ei[c l ]) + e cl (Ei [ c ] Ei [ c l ]) e (c c l ) e (c c l ) + R f (c l ; c ) v (c ) R + c + ec K 3 e c K 4 c q (c ) R + c + e(c cl) + e (c cl) e (c c l ) e (c c l ) + R e cl (Ei[c ] Ei[c l ]) + e cl (Ei [ c l ] Ei [ c ]) e (c c l ) e (c c l ) c R + c m (c l; c ) + R q (c ) g (c l ; c ) c f (c l ; c ) were we sed te de nitions f l (c l ; c ) e c (Ei[c ] Ei[c l ]) + e c (Ei[ c ] Ei[ c l ]) e (c c l ) e (c c l ) g l (c l ; c ) e c (Ei[c ] Ei[c l ]) + e c (Ei [ c l ] Ei [ c ]) e (c c l ) e (c c l ) f (c l ; c ) e c l (Ei[c ] Ei[c l ]) + e c l (Ei [ c ] Ei [ c l ]) e (c c l ) e (c c l ) g (c l ; c ) e c l (Ei[c ] Ei[c l ]) + e c l (Ei [ c l ] Ei [ c ]) e (c c l ) e (c c l ) m (c l ; c ) e(c c l ) + e (c c l ) (0; ) 43

44 De ne te fnction c l H (c ) by wile c l L (c ) is de ned implicitly by p (c ; c l ; c ) v (c ; c l ; c ) q (c ; c l ; c ) p (c l ; c l ; c ) v (c l; c l ; c ) q (c l ; c l ; c ) l: It is easy to see tat if tere is a c tat H (c ) L (c ) ten tis particlar c and te corresponding c l H (c ) is a soltion of (3)-(4), (7)-(8). To sow tat tis intercept exists, we rst establis properties of L (c ) ten we proceed to te properties of H (c ). Properties of L (c ) It is sefl to observe l ec + e l (e c e c l ) c f l c l e (c c l ) e (c l c l + ec + e l c l (e c e c g l l l e (c c l ) e (c l c ) lim f l ; c l!c c lim g l 0; c l!c lim m 0: c l!c. We sow tat f l (c ; c l ) is monotonically decreasing in c l :Its slope in c l l ec + e cl (e c e c l ) f l (c ; c l ) < 0 c l second derivative f 4e c e cl e c + e c! l c l (e c e c l ) (e c e c ec + e cl l ) (e c e c l ) c l e c + e c l (e c e c f l l (c ; c l ) + ec + e c l ) c l (e c e c l ) c l c l f l (c ; c Ts, tis fnction can ave only minima, bt no maxima. At te limit! e c + e cl lim c l!c c l (e c e c (c l l f l (c ; c l ) ) c ) < 0 c 44

45 Ts, f l (c ; c l ) is decreasing at c c l : tis implies tat it as to be monotonically decreasing over te wole range of c l < c : (sppose tat tere is a ^c l < c were te rst derivative is zero, so f is minimal. as we decrease c l from c f is increasing as long as ^c l < c l < c ; bt it is a contradiction wit ^c l being a minimm point). We sow tat te fnction g l(c ;c l ) c l f l (c ; c l ) is monotonically increasing in c l :(all te derivatives in tis part are wit respect to c l ) gl 0 c l f l + ec + e cl e c + e c! l c l (e c e c g l l (c l ; c ) ) (e c e c (c l l f l (c ; c l ) ) + f l (c l ; c ) ) + ec + e cl gl (c l ; c ) c l (e c e c c l l f l (c ; c l ) + ec + e c! l ) (e c e c f l l (c l ; c ) ) if te rst derivative is zero tan at tat point gl c l f l f l (c l ; c ) c l (ec +e c l ) (e c e c l ) we also know tat gl lim c l!c lim c l!c gl c l f l 0 0 cf 00 3c < 0 so for any xed c ; c l c is a local maximm. In general gl c l f l 00 c l 4e c + ec + e c l gl (e c e c l ) e c l gl (e c e c l ) l c l f l l c l f l + Ts, if tere were a ^c l tat te rst derivative were zero and c derivative were c l e c l 0 f 0 (c l ; c ) + 4e c B (e c e c l f l c l (ec +e c l ) (e c e c l ) c l > ^c l ten te second + C A ec e c l e c + e c l f l c l 45

46 can tis be non-negative? for tat we wold need f l (c l ; c ) < c l c l (ec e c l) e c +e c l (ec e c l ) e c +e c l as f (c l ; c ) is decreasing in c l and at c l lim f l (c l ; c ) c l!c c l (ec e c l ) e c +e c l (ec e c l ) e c +e c l c l < c l tis cold be non-negative only at a c l > c : So te second derivative is always negative at a point were te rst derivative is zero, wic implies it does not ave anoter extreme point 0 so cf > 0 for any cl < c : g 3. Given tat f l is decreasing c l ; q (c l ; c l ; c ) is also decreasing in c l for any c l < c : Given tat gl c l f l 0 > 0 c l + e (c c l) +e (c c l ) e (c c l) e (c c l exp ( c + c l ) + (e c +c l + ) > 0 v (c l ; c l ; c ) is increasing in c l : Ts, p (c l ; c l ; c ) is increasing in c l for any c l < c : Also and Ts, as long as lim p (c l ; c l ; c ) c l!c lim p (c l; c l ; c ) tan (c ) < 0 c l!0 R + c + R c c + R c R + c tere is a niqe soltion c l for any c of R + R c p (c l ; c l ; c ) l: l R + c R + R : c Ts, L (c ) exist. From te monotonicity in c l, and continity of p (c l ; c l ; c ) we also know tat L (c ) is continos. Properties of H (c ) and H (c ) [0; c ] : rst we sow tat for any xed c [l; R] ; H (c ) is a continos fnction 46

47 It will be sefl to know tat and in l. we ave g f l (c ;c l f l (c ; c l ) c l e (c c l ) e (c c l (g l (c ; c l l e (c c l ) e (c c l ec + e (e c e c f l (c ; c l ) ) e c + e c (e c e c g l (c l ; c ) ) c l lim f ; c l!c c lim g 0 c l!c e (c c l) e (c c l) < 0 from previos reslts. f c g f c g (c ; c l ) c f l (c ; c l ) + l e (c c l ) e (c c l ) g 0 (c ; c l ) c f 0 (c ; c l ) c c l e (c c l ) e (c c l ) + +e (c c l ) e( (c c l )) + e (c c l ) c l g l (c ; c l ) c f l (c ; c l ) + c cl if te rst derivative is zero at a point c > c l ;ten te second derivative is c l + (ec +e c l ) g (e c e c l ) l (c l ; c ) c f l (c ; c l ) + c cl c c c l e (c c l ) e c l (c c l ) c l e (c c l ) e (c c l ) < 0: for any c > c l ; wic implies tat it can ave no minimm in tat range. lim c lim c l!c g g f c l f c l 3c 47

48 so c l c mst be te niqe maximm in te range c c l : Ts, for c l < c g f l > 0 3. Conseqently, q (c ; c ; c l ) is monotonically decreasing and v (c ; c ; c l ) is monotonically increasing in c l : Ts, p (c ; c ; c l ) is monotonically increasing in c l : 4. Also Rc + c v (c l ; c ) lim c l!c q (c l ; c ) R c c + R c + (R ) c R c + (R ) c R > 0 > 0 > 0 R + c + R c c + R c Rc + c c + R R > olds if c > : wile lim p (c ; c l ; c ) c l!0 Ts, for any c tere is a niqe c l [0; c ] wic solves p (c ; c l ; c ) : From te monotonicity of p (c ; c ; c l ) in c l and te continity in c ; te reslting fnction H (c ) is continos in c : c Intercept of H (c ) and L (c ). Here we sow tat H () > L () :.We know tat H () as lim v (c l ; ) c l! q (c l ; ) R + + R + R R + + R + R owever as v l (c l ; ) lim c l! q l (c l ; ) R + + R + R R + + R + R and v l(c l ;) q l (c l ;) is increasing in c l; and L () is de ned by v l(l();) q l (L();) l < ; L () < 48

49 mst old.. Here we sow tat lim c! H (c ) 0 < lim c! L (c ) : Observe, tat lim f e c (Ei[c ] Ei[c l ]) + (Ei[ c ] Ei[ c l ]) l lim c! c! e ( c l) e (c c l ) lim g l lim c! c! Ei[ c l] e ( c l) e c (Ei [ c l ] Ei [ c ]) + e c (Ei[c ] Ei[c l ]) e (c c l ) e (c c l ) Ei [ c l] e ( c l) lim f e c l (Ei[c ] Ei[c l ]) + e c l (Ei [ c ] Ei [ c l ]) lim c! c! e (c c l ) e (c c l ) lim g e cl (Ei[c ] Ei[c l ]) + e cl (Ei [ c l ] Ei [ c ]) c! e (c c l ) e (c c l ) e c le c (Ei[c ] Ei[c l ]) + e c le c (Ei [ c l ] Ei [ c ]) e ( c l) e (c c l ) 0 0 e ( c l) e (c c l ) 0 Ts, v (c l ; c l ; c ) lim c! q (c l ; c l ; c ) R + c l lim c! R + c l + + R + m (c l; c ) + R Ei[ c l ] gl (c l ;c ) + R f l (c l ; c ) Ei[ c e ( c l ] c l) l e ( c l) Ei[ c l ] e ( c l) c l f l (c l ; c ) and lim c! L (c ) is te nite positive soltion of R + c l + + R Ei[ c l ] Ei[ c e ( c l ] c l) l e ( c l) Ei[ c l ] e ( c l) l: In contrast, v (c ; c l ; c ) lim c! q (c ; c l ; c ) R + c lim c! lim c! lim c! R c + + R g (c l ;c ) c m (c l; c ) + R g (c l ;c ) c f (c l ; c ) + R f (c l ; c ) c + R g (c l ;c ) c f (c l ; c ) c + R f (c l ;c ) c ; R f (c l ;c ) c As v(c ;c l ;c ) q(c ;c l ;c ) grows witot bond for any xed c l and v(c ;c l ;c ) q(c ;c l ;c ) is monotonically increasing in c l ; In order to ave a soltion of lim c! v(c ;c l ;c ) q(c ;c l ;c ) l; c l as to go to zero, implying lim c! H (c ) 0: 49

50 Te two reslts imply tat tere is always an intercept c (; ) tat H (c ) L (c ) : Tis concldes te step proving tat (3)-(4), (7)-(8) as a soltion A..3 Step 3: Proof of Lemma A.3 We ave sown tat H () : Note also tat if c c l ten v q v l q l : Tis, and te continity of H () and L () in l; implies tat at te limit l! ; tere is a soltion of te system (3)-(4), (7)-(8) tat c l c! 0 and c ; c l! : Ten, te statement comes from < < R; A..4 Step 3: Proof of Lemma A.4 First we sow tat q (c) is always deceasing, and tere exists a critical vale bc (c l ; c ) so tat q 00 (c) < 0 for c (c l ; bc) and q 00 (c) > 0 for c (bc; c ). Moreover, for c (c l ; bc) were q 00 (c) < 0, we ave tat q 000 (c) > 0. We rst sow tat q 0 < 0. From te ODE 0 q R c De to bondary conditions, we ave q satis ed by q, we ave 0 R q000 c q 0 : (A.) q000 R c > 0 at bot ends c l and c. De ne F (c) q0 (c); ten F (c l ) F (c ) 0 and F 00 (c l ) F 00 (c ) > 0. Sppose F (bc) > 0 for some points; ten tere mst exist a point bc so tat F (bc) > 0 and F 00 (bc) 0 (oterwise F (c) is convex always and never comes back to zero). Bt becase F 00 (bc) R + F (bc) > 0; bc contradiction. Tis proves tat q 0 < 0. We know tat q 00 (c l ) < 0 and q 00 (c ) > 0, terefore tere exists bc so tat q 00 (bc) 0: We sow tis point is niqe. Becase 0 q R c q, we ave 0 R q000 c q 0 (A.) 0 q R c 3 q 00 (A.3) Sppose we ave mltiple soltions for q 00 (bc) 0. Clearly, it is impossible to ave te possibility tat q 00 (bc) 0 bt q 00 (bc ) > 0 and q 00 (bc+) > 0; oterwise q 0000 (bc) > 0 wic contradicts wit (A.3). 50

51 Ten tere mst exist two points c > bc and c > c > bc tat q 00 (c ) 0, q 00 (c ) < 0 and q 0000 (c ) > 0; and q 00 (c) < 0 for c (c ; c ) Terefore q0000 (c ) R c 3 + q 00 (c ) < 0: As a reslt, tere exists anoter point c 3 (c ; c ) so tat q 0000 (c 3 ) 0 wit q 00 (c 3 ) < 0. Bt tis contradicts wit (A.3). Now we sow tat for c (c l ; bc) so tat q 00 (c) < 0, we ave q 000 (c) > 0, i.e., q 00 (c) is increasing. Sppose not. Since q 000 (c l ) > 0 so tat q 00 (c) is increasing at te beginning, tere mst exist some re ecting point c 4 so tat q 0000 (c 4 ) 0. Bt becase q 00 (c 4 ) < 0, it contradicts wit (A.3). Now we sow tat if v 00 (c l ) > 0; ten v (c) is increasing. Let F (c) v 0 (c), so tat 0 q 00 + F 00 + F wit bondary conditions tat F (c l ) F (c ) 0. Since F 0 (c l ) > 0 we know tat if tere are some points wit F 0 (c) < 0 ten it mst exists two points c and c (a maximm and a minimm) so tat c < c bt F 00 (c ) < 0 F 00 (c ) > 0, F 0 (c ) F 0 (c ) 0 and F (c ) > 0 > F (c ). From ODE 0 q 00 (c ) + F 00 (c ) + F (c ) 0 q 00 (c ) + F 00 (c ) + F (c ) It is easy to sow tat q 00 (c ) < 0, wic implies tat c < c < bc. eqations also imply tat q 00 (c ) > > q00 (c ) However, te above two contradiction wit te previos lemma wic sows tat q 00 is increasing over [c l ; bc] : Finally we sow tat if l is s ciently close to 0, ten v 00 (c l ) > 0: From or ODE, v 00 (c l ) (cl + R) v (c l ) R + (c l; c ) + R gl (c l ; c ) c l f l (c l ; c ) : We know tat as l! 0; c c l! 0: We will prove te statement by sowing tat () 5

52 (cl v(c lim (cl +R) l ) cl!c v (c l ) 0 and () lim l < 0: Tese two statements imply tat wen c c l is s ciently small ten v 00 (c l ) > lim c l!c v 00 (c l ) 0: Ts, te cain of eqalities (cl + R) lim c l!c v (c l ) lim c l!c R + R R (c l; c ) + R gl (c l ; c ) 0 0 c l f l (c l ; c ) Proof. (cl +R) v (c l ) lim c l prove te lim c l!c (c l; c ) + R gl (c l ;c ) c l f l (c l ; c l e (c c l ) lim c l!c e (c c l ) + + R + ec + e cl e c + e cl c l (e c e c g l l ) (e c e c (c l l f l ) ( + ) + R c c c 4 < 0 A. Proof of Proposition Te reslt c > is a conseqence of te fact tat we de ned H (c ) as te niqe c l solving v (c l ;c ) q (c l ;c ) wen c > : (see part 4 in section A...) For te reslt c l l; consider te possibility tat c l > l: Te following Lemma states tat in tis case p 00 (c l ) < 0: Tis implies tat tis is not an eqilibrim as p0 (c l ) 0 by te bondary conditions v 0 (c l ) q0 (c l ) 0; ts p00 (c l ) < 0 wold imply tat p (c) < l for a c s ciently close to c l : Lemma A.5 Te sign of p 00 (c l ) is te same as te sign of (l c l ) : 5

53 Proof. v p 00 (c 0 l ) q q 0 v 0 q (v00 q + v 0 q 0 (q 00 v + v 0 q 0 )) q q 3 v 0 q q 0 v v00 q q 00 v (c l + R) + lq (c l ) q q (c l + R) + c l q (c l ) v c l q (c l + R) + lq (c l ) + c l q (c l ) c l q (c l ) q (c l + R) + c l q (c l ) v c l q (c l + R) + c l q (c l ) q (l c l ) c l q v c + (l c l ) q (c l ) q l (c l + R) c l q (c l ) + q (c l ) q wic gives te Lemma by realizing tat or observation tat q is decreasing in c and te bondary q 0 (c l ) 0 implies tat and q > 0: (c l + R) + c l q (c l ) / q00 < 0 Te tird statement is a conseqence of te following Lemma. Lemma A.6 We ave te following limiting reslts: lim f l ; lim! c f, lim l! c g 0,! lim g l 0, lim!! c, lim! c l l: Proof. Using L Hopital rle repeatedly, we ave lim f (Ei[ c ] Ei[ c l ]) Ei[ c ] Ei[ c l ] l lim!! e ( c lim l)! e( c l) lim! e c e c l e ( cl) + ( c l) e lim ( c l)! e c l ( c l ) e ( c l) c l lim f lim e cl (Ei[c ] Ei[c l ]) + e cl (Ei [ c ] Ei [ c l ])!! e (c c l ) e (c c l ) lim e cl (Ei[c ] Ei[c l ])! e (c lim c l )! lim! e c e c l e (c ) + c e c 53 lim! e c c e c (Ei[c ] c ec Ei[c l ])

54 were we sed tat lim! ec l (Ei [ c ] Ei [ c l ]) (Ei [ c ] Ei [ c l ]) lim! e c lim l! Also, and e c e c l ( c l ) e ( c l) lim! lim g e c l (Ei[c ] Ei[c l ]) + e c l (Ei [ c l ] Ei [ c ]) lim!! e (c c l ) e (c c l ) lim! (Ei[c ] e c Ei[c l ]) lim! e c c e c e c l lim 0! c lim g e c (Ei[c ] Ei[c l ]) + e c (Ei [ c l ] Ei [ c ]) l lim!! e (c c l ) e (c c l ) e (Ei[ c ] Ei[ c l ]) c lim! e ( c lim l)! e c l ( c l ) e ( c l) e cl lim! ( c l ) e ( c l) 0: ( c l ) 0: Tis implies tat v R + c lim lim! q! R + c R + R c m (c l; c ) + R g (c l ;c ) c f (c l ; c ) + R f (c l ; c ) Ts, in te limit te soltion of v q wic gives lim! c : Similarly, v R + c l l lim lim! q! R + c l + R + R c l implying tat te soltion of v l q l wic gives lim! c l l: is te soltion of R + c R + R c + m (c l; c ) + R in tis limit as to solve R + c l R + R c l gl (c l ;c ) c l f l (c l ; c ) + R f l (c l ; c ) l 54

55 A.3 Proof of Proposition 3 From (6)-(8) and or conjectre tat c P l 0; we ave R + D + D l ( D + D ) (A.4) R + c P + D e cp + D e cp + c P D e cp + D e cp (A.5) D e cp + D e cp 0 (A.6) Te rst and te last eqation give D (R l) e cp, D l + ( + l) e cp R l l + ( + l) e cp wic if we sbstitte in to te second eqation we get (9). To validate tat c P l ceck tat jp 00 (0) < 0: As 0; we ave to j 00 P (0) D + D (R l) e cp e cp + l e cp + < 0 tis is indeed te case. wit Now we sow tat nder certain conditions te soltion exists and niqe. Let Now de ne Since G (c) R R l ec ( + l) ( l) e c (c + ) G (0) R l < 0; and G (). R l (R ) g (c) G 0 (c) (R l) (l + ) e c + e c ( l) : g (0) R l R l < 0: and g (c) canges sign only once. Conseqently, tere is a niqe ^c tat g (^c) 0: Tis implies tat G (c) is decreasing for any c < ^c and increasing for any c > ^c: As G (0) < 0 and G () ; tere mst be a niqe c P tat G cp 0: Te following series of reslts caracterize te property of social planner s vale fnction j P (c), wic satis es wit bondary conditions 0 j00 P (c) + (R + c j P (c)) (A.7) j P (0) ljp 0 (0) ; j P c P t + c P j 0 P c P ; and j 00 P c P 0: 55

56 Note tat te bondary conditions imply tat j P c P R + c P. Lemma A.7 Te social vale fnction j P (c) is concave and increasing over [0; c ] ; and j P (c) < R + c. Proof. First of all, from smoot pasting condition we ave jp 0 c P j P c P t + c P R + c P t + c P t R t + c P Second, taking derivative again on (A.7) and evalate at te optimal policy point c P, we ave < 0: Combining bot reslts, we ave 0 j000 P c P + j 0 P c P : j 000 P c P R t t + c P > 0; and as a reslt jp 00 cp < 0. Sppose tat j P fails to be globally concave over [0; c ]. Ten tere exists some point jp 00 > 0, and pick te largest one bc so tat j00 P is concave over [bc; c ] wit j 00 P (bc) 0 and j 000 P (bc) < 0 bt since j 00 P is concave over [bc; c ], j 0 P (bc) > j0 P (c ) >, terefore j000 P c P j 0 P c P > 0; contradiction. Terefore j P is globally concave over 0; c P. Finally, since j 0 P c P >, we mst ave jp 0 (c) > > 0 always. Combining wit te fact tat j P c P R + c P, we ave j P (c) < R + c. QED. We can frter extend j P (c) otside c P by recognizing te optimal policy is investing. Sppose tat C > Ac ; ten immediately te economy sold bild x trees so tat C xt A + x c ) x C c A + c A c c + c ; and te total vale is Aj (c) (A + x) j P c P A + c cp + c P j c P + c A + c P j c P 56

57 Terefore te envelope of vale is ( j P (c) j P (c) +c j +c P P c P 0; c P c > c P : A.4 Proof of Proposition 4 Let Note tat limit of c P as G (c) R R l ec ( + l) ( l) e c (c + ) : lim! as! as to satisfy R R l (ec ( + l) ( l) e c ) c lim! R R l (ec ( + l) ( l) e c ) c + j c cc p 0,c as to converge to 0: Tis is te rst part of te rst statement. R R l cec ( + l) + le c c (l ) e c + le c (c + ) wic is clearly positive for s ciently large : Finally, from te proof of Proposition 3 we know tat Ts, for s ciently large p Tis concldes te rst part. (c) cc P > 0: R l (cec ( + l) + le c c (l ) e c + le c ) (c j cc p < (c) (c) R R R l ec ( + l) ( l) e c < 0 ( c) (R ) e c + R + Re e c (R l) > 0 l (R l) > 0 57

58 wic conclde te second part. Finally, lim R! G (c) lim! R G (c) no nite soltion in tis limit. Instead, lim R! R R l (ec ( + l) ( l) e c ) c + j c cc p 0 (c + ) ; so tere is as to old, wic implies tat lim R! R R l (e c (+l) ( l)e c ) c Tis is possible only if c p!. Tis concldes te proposition. as to converge to a constant. A.5 Proof of Proposition 5 Te argment tat we can separate te vale of cas and tat of a nit of asset in te vale fnction of agents and te derivation of (30)-(3) are analogos to te incomplete market case, so it is omitted. It is easy to ceck tat te general forms (3)-(33) are indeed soltions of (30)-(3). Now we sow tat eqations (34) ave a soltion L ; L ; L 3 ; L 4 ; c cm were c cm c P ; and tat j P (c) v cm (c) + cq cm (c) : (A.8) For tis, rst observe tat rst tat v cm (c) + cq cm (c) R + c + e c L 3 + e c L 4 : (A.9) Also, (34) can be written as R + L 3 + L 4 + L l (A.0) + L + L 3 L L 4 L 0 (A.) e ccm L + e ccm L 0 (A.) + eccm (L 3 c cm L ) L e ccm e ccm (L 4 c cm L ) L e ccm 0 (A.3) R + c cm + eccm L 3 + e ccm L 4 + e ccm L + e ccm + c cm L :(A.4) Adding c cm times (A.) to (A.3) gives + eccm (L 3 L ) e ccm (L 4 + L ) 0: (A.5) Togeter wit (A.) tis implies L 3 L ; L L 4 : (A.6) 58

59 Sbstitting tis into (A.) gives e ccm L 4 + e ccm L 3 0: (A.7) Also,expressing (L + L ) from (A.) and plgging into (A.0) gives and by (A.6), (A.4) is eqivalent to R + c cm + eccm R + L 3 + L 4 l ( + L 3 L 4 ) (A.8) L 3 + e ccm L 4 ( + c cm ) L 4e ccm + L 3 e ccm : (A.9) Observe tat te system (A.7)-(A.9) is eqivalent wit te system (A.4)-(A.6), ts L 3 D, L 4 D and c cm c P : Given (A.9) and te fact tat (A.6), we proved te statement. Finally, we sow tat vcm 0 (c) > 0 and qcm 0 (c) < 0 for every c (0; c cm ) wic proves tat te price is monotonically increasing. For q 0 cm (c) < 0 observe tat For v 0 cm (c) > 0 observe tat q 0 cm (c) e c L + e c L e c L 4 + e c L 3 e c D + e c D (R l) e c e (cp c) < 0: e cp + l e cp + v 0 cm (c) + ec (L 3 cl ) L e c e c (L 4 cl ) L e c c D e c c D e c + c (R l) e c e (cp c) e cp + l e cp + > 0: A.6 Proof of Proposition 6 Te rst statement comes from te constrction of te Proof of Proposition??. In particlar, from te fact tat c and c l are constrcted as te intercept of continos fnctions H (c ) ; L (c ) wic map [; )! R ++ and H () > L () > 0 and 0 < lim c! L (c ) lim c! H (c ) 0 < lim c! L (c ) <. Ts, bot c (; ) and c l (0; c ) : Te second statement is te conseqence of Lemma A.6 and te rst reslt in Proposition 4. For te last statement, we provide a constrctive proof. By te tird statement in Proposition 4 for any parameters, we can pick an R s ciently close to tat c P > : Call tis ^cp : By te 59

60 fact tat c P solves R R l e cp ( + l) ( l) e cp c P + 0 for any tere is an R () de ned as R () e^cp ( + l) e ^cp i ( l) e^cp ( + l) ( l i ) e ^cp l ^c P + ^c P + tan for te pair ; R () c P ^cp : Also, for s ciently large ; c (; R ())! ; c l (; R ())! l by an analogos argment to Lemma A.6. Ts, tere is a s ciently large ^ tat for any > ^ and R () we ave c < cp. (R ^c P e ^cp ( l) ^c P + l e ^cp c e ^cp + l + e ( ^cp ) le ( ^cp ) + ^c P e ^cp + e ^cp wic is negative if > l q ^c P ^c P + 4l : Ts, to complete te proof, let s pick ^ max ; l s ^c P ^c P + 4l!! : A.7 Proof of Proposition 7 Sppose tat we are given te policy (c l ; c ) pair, so tat c l > 0 and c < c P were cp satis es te sper-contact condition j 00 P cp ; 0; cp 0: For simplicity, we denote te social vale (wic is j S (c; c l ; c ) in te main text) as j (c; c l ; c ), and we need to sow (c; c l ; c l < 0 (c; c ; c l > 0. If tis olds for any pair of (c l ; c ) tat lies in te interval of 0; c P, ten we know tat for 0 < c l < c l < c < c < cp, we ave j c; c l ; c < j c; c l ; c : Before proceed, let s sow tat given (c l ; c ), we ave te failre of sper contact condition on bot ends c and c l : j 00 (c ; c l ; c ) < 0 and j 00 (c l ; c l ; c ) < 0: 60

61 To see tis, notice tat given (c l ; c ) as non-optimal policies, it mst be tat j (c; c l ; c ) < j P (c) R + c were te last ineqality comes from Lemma A.7. Ten 0 j00 (c) + (R + c j (c)) implies tat te vale fnction is strictly concave at bot ends. And, for all c l and c we mst ave j (c ; c l ; c ) (c + ) j 0 (c ; c l ; c ) 0; (A.0) j (c l ; c l ; c ) (c l + l) j 0 (c l ; c l ; c ) 0: (A.) Tese two conditions old for any policy (inclding te market soltion, te social planner s soltion, or any soltion.) First, we focs on te top policy c. De ne F (c; c l ; c ) wic is te impact of policy on vale. j (c; c l ; c ) 0 j00 (c; c l ; c ) + (R + c j (c; c l ; c )) ; (A.) j 00 (c; c l ; j (c; c l ; c ) 0, or F 00 (c; c l; c ) F (c; c l ; c ) 0: (A.3) Moreover, take te total derivative wit respect to c on te eqality (A.0), i.e., take derivative tat a ects bot te policy c and te state c, j (c ; c l ; c ) + j 0 (c ; c l ; c ) j 0 (c ; c l ; c ) + (c + ) j 0 (c ; c l ; c ) + j 00 (c ; c l ; @c j (c ; c l ; c ) Since j 00 (c ; c l ; c ) < 0, we ave (c j 0 (c ; c l ; c ) (c + ) j 00 (c ; c l ; c ) < 0: F (c ; c l ; c ) (c + ) F 0 (c ; c l ; c ) < 0; (A.4) wic gives te bondary condition of F at c. At c l we can take total derivative wit respect to c on te eqality (A.), j (c l ; c l ; c ) (c l j 0 (c l ; c l ; c ) ) F (c l ; c l ; c ) (c l + l) F 0 (c l; c l ; c ) 0 (A.5) Based on (A.3) we can sow tat tese two bondary conditions imply F as to be positive 6

62 always. As a reslt, F (c; c l ; c ) lower policy c l mst be positive. Lemma A.8 We ave F (c) > 0 for c [c l ; j (c; c l ; c ) as te impact of raising c for any state and any Proof. First we sow tat F (c) cannot cange sign over [c l ; c ]. Sppose tat F (c l ) > 0; ten from (A.5) we know tat F 0 (c l ) > 0. Te ODE (A.3) implies tat F 00 > 0 always, so it is convex and always positive. If F (c l ) < 0 ten te similar argment implies tat F is concave and negative always. Now sppose tat F (c l ) 0 bt F canges sign at some point. Witot loss of generality, tere mst exist some point bc so tat F 0 (bc) 0, F (bc) > 0 and F 00 (bc) < 0; bt tis contradicts wit te ODE (A.3). Now let G (c) F (c) (l + c) F 0 (c) so tat G0 (c) (l + c) F 00 (c) (l+c) F (c). As a reslt, G 0 (c) cannot cange sign. Since frter tat G (c l ) 0, G (c) 0 cannot cange sign eiter. Sppose conterfactally tat F (c) < 0 so tat G 0 (c) > 0 and G (c) > 0. Bt we ten ave G (c ) F (c ) (l + c) F 0 (c ) F (c ) ( + c) F 0 (c ) + ( l) F 0 (c ) F (c ) ( + c) F 0 (c ) + l l + c (F G ) < 0 were we ave sed (A.4). Terefore F (c) < 0. Te argment for te e ect of c l is similar. De ne F (c; c l ; c ) impact of lower-end policy. Ten since wic implies tat 0 j00 (c; c l ; c ) + (R + c j (c; c l ; c )) ; F 00 (c; c l ; c ) F (c; c l ; c ) 0 Using (A.0) and (A.), and take total derivatives, we can sow tat F (c ; c l ; c ) (c + ) F 0 (c ; c l ; c ) 0 F (c l ; c l ; c ) (c l + l) F 0 (c l ; c l ; c ) l j (c; c l ; c ) wic is te Based on (A.3) we can sow tat tese two bondary conditions imply F as to be negative always. 6

63 A.8 Proof of Proposition 8 Consider te fnctions ~q (c; q 0 ; v 0 ; c ) and ~v (c; q 0 ; v 0 ; c ) of c parameterized by q 0 ; v 0 ; and c de ned by te system and te bondary conditions 0 ~q00 (c) + ( ~q (c)) + R c ~q (c) (A.6) 0 ~q 0 (c) + ~v00 (c) + (c ~v (c)) + (R ~v (c)) : (A.7) De ne te fnction c (q 0 ; v 0 ) implicitly by We are interested in te derivatives ~v 0 (c ) ~q 0 (c ) 0 (A.8) ~q (c 0 ) q 0 ; ~v (c 0 ) v 0 : (A.9) ~v (c ; q 0 ; v 0 ; c ) ~q (c ; q 0 ; v 0 ; c @v 0 ~v q 0 0 ~q q 0 0 ~v c 0 ~q c 0 ~v v 0 0 ~q v 0 0 ~v c 0 ~q c 0 For tis, consider te following Lemmas. Lemma ~q(c ;q 0 ;v 0 ;c 0 e c e c 0 +e c e c 0 > 0 Proof. we know tat q (c 0 ) q 0 and q 0 (c ) 0: We can rewrite te earlier as e c 0 K + e c 0 K + l q q 0 wic implies Rewrite te latter as l q e c 0 K + q 0 e c 0 K : e c K + e c K + s q 0 wic implies K e c K s q e c e c lq ec0 K +q 0 e c 0 e c s q e c0 + e c ( l q + q 0 ) e c0 + e c : 0 s q 63

64 or Ts, and implying tat s q K e c lq+q 0 e c 0 ( + e c e c 0 ) 0 e c e c e c 0 + e c e 0 e c 0 e c 0 e c e c e c 0 + e c e c 0 e c e c e c 0 + e c e c (c 0 e c e c 0 + e c e c 0 + e c e c 0 + e c e c 0 e c e c 0 + e c e c 0 > 0 Lemma ;q 0 ;v 0 ;c 0 e (c c 0) +e (c c 0 ) > 0 Proof. Te general soltion is v (c) R + c + ec (K 3 ck ) e c (K 4 + ck ) + cr p e c Ei ( c) e c Ei (c) v 0 (c) + R ( e c Ei[c] + e c Ei[ c]) p + Rc (e c Ei[c] + e c Ei[ c]) p + + e c (( c ) K + K 3 ) + e c ((c ) K + K 4 ) or bondaries are v (c 0 ) v 0 and v 0 (c ) 0 from te rst e c 0 K 3 e c 0 K 4 + l v v 0 were l v does not depend on K 3 andk 4 or v 0 and we rewrite te second bondary as e c 0 K 3 v 0 + e c 0 K 4 l v K 3 e c 0 v 0 + e c 0 K 4 e c 0 l v s v + e c K 3 + e c K 4 0 s v + e c e c 0 v 0 + e c 0 K 4 e c 0 l v + e c K 4 0 K 4 s v e c (e c 0 v 0 e c 0 l v ) (e c e c 0 + e c ) 64

65 @v (c 4 e c e c 0 e c e c0 + e 3 e c0 + e c 0 e c e 0 e c e c0 + e c e c e c0 + e c 0 e c e c 0 e c e c0 + e c e c e c e c 0 e c e c0 + e c e (c c 0 ) + e (c c 0 ) > 0 Lemma ;q 0 ;v 0 ;c 0 e(c c0) e (c c0) (c c 0 ) e (c c 0) +e (c c 0 ) (e c 0 e c +e c 0 e c < 0 ;q 0 ;v 0 ;c ) ) e (c c 0) +e (c c 0 0 e(c c0) e (c c0) (c + c 0 ) (e c 0 e c +e c 0 e c < 0: ) Proof. We rewrite v (c 0 ) and v 0 (c ) as v (c 0 ) e c0 (K 3 c 0 K ) e c0 (K 4 + c 0 K ) + l vq v 0 (c ) s vq + e c (( c ) K + K 3 ) + e c ((c ) K + K 4 ) Ts, te bondaries v (c 0 ) v 0 imply K 3 c 0 K + e c0 v 0 e c0 l vq + e c0 (K 4 + c 0 K ) and v 0 (c ) 0 K 4 ( e c (c c 0 + )) K + e c (c ) + c 0 e c 0 e c K + (e c 0 e c ) v 0 + (s qv e c 0 e c l qv ) e c + e c 0e c 0 e c (e c 0 e c (c ) + c 0 e c 0 e c ) e c e c0 + e c e c 0 e c 0 e c + e c 0e c ( e c 0 e c (c c 0 + )) e c e c 0 e c + e c 0e c e c e c0 + e c e c 0 were we sed te previos 0 : e c0 c 0 c 0 4 q 0 q 0 q 0 q 0 e c 0 e c ec 0 e c c 0 (e c 0 e c + e c 0 e c ) (e c 0 e c + e c 0e c ) e c e c e c0 + e c e c e c0 e c c e c e c0 + e c e c c 0 + e c ec0 e c c 0 (e c0 e c + e c0 e c) 0 (e c 0 e c + e c 0e c ) e c e c 0 e c + c 0 (e c 0 e c + e c 0 e c ) (e c 0 e c + e c 0e c ) 65

66 (c 0 e 3 q 0 e 4 q 0 (c 0 e c 0 e c + c 0 (e c 0 e c + e c 0 e c ) (e c 0 e c + e c 0e c ) e c 0 e c c 0 (e c 0 e c + e c 0 e c ) (e c 0 e c + e c 0e c ) (c 0 e(c c 0 ) e (c c 0 ) + c 0 (e c 0 e c + e c 0 e c ) (e c 0 e c + e c 0e c ) (c 0 te ineqality comes from te fact tat te fnction e(c c 0 ) e (c c 0 ) (c c 0 ) e (c c 0 ) + e (c c 0 ) (e c 0 e c + e c 0e c ) < 0: e x e x x e x + e x is negative and monotonically decreasing for all x > 0: Te second statement comes directly from te expression 0 : Lemma A. if v 0 q 0 < ; ten ~v c ~q c > 0: Proof. Using... we write q (c) + e c K + e c K + R F (c) v (c) R + c + ec (K 3 ck ) e c (K 4 + ck ) cr F (c) and F 0 (c) G (c) were F (c) e c Ei [ c] + e c Ei [c] G (c) e c Ei[c] e c Ei[ c]: We want to sow tat te fnction ~v (c; q 0 ; v 0 ; c) ~q (c; q 0 ; v 0 ; c) is negative at c c 0 and positive at c! : Terefore, tere is a c c were tis fnction is zero (satisfying te de nition of c) and were te slope of tis fnction is positive. To get ~v (c; q 0 ; v 0 ; c) ~q (c; q 0 ; v 0 ; c) we ave to solve (A.6)-(A.7) wit te bondary conditions ~v 0 (c) ~q 0 (c) 0 (A.30) ~q (c 0 ) q 0 ; ~v (c 0 ) v 0 : (A.3) Ts, indeed, ~v (c 0 ; q 0 ; v 0 ; c 0 ) ~q (c 0 ; q 0 ; v 0 ; c 0 ) v 0 q 0 < 0 by te condition of te proposition. 66

67 Now we sow tat lim c! ~q (c; q 0 ; v 0 ; c) : For tis note tat solving for K and K from (A.6)-(A.7), (A.30)-(A.3) we get Using lim c! F (c) lim c! G (c) 0 gives implying te reslt. e c + e(c c 0) R G(c) + R F (c 0 ) q 0 e (c c0) + e c e c K 0 e c K + R lim c! ec K lim e c K 0: c! G (c) e c K : Now we sow tat lim c! ~v (c; q 0 ; v 0 ; c), (A.6)-(A.7), (A.30)-(A.3) gives e c K 4 e c K 3 e (c 0 c) (R + c 0 v 0 c 0 q (c 0 )) + e (c 0 c) + e ( c+c 0) RcG (c) + + q (c) e c K 4 + RcG(c) + q (c) sing tat lim c! RcG(c) 0 gives implying te reslt. Ts, lim c! ~v (c; q 0 ; v 0 ; c) Ptting togeter te tree Lemma gives lim e c K 4 0; lim e c K 3 c! c! ~q (c; q 0 ; v 0 ; c) wic proves @v 0 ~v 0 q 0 ~q 0 q 0 ~v 0 c ~q 0 c < 0 ~v 0 v 0 ~q 0 v 0 ~v 0 c ~q 0 c > 0 Tis implies tat c < c wenever q (c 0 ) q (c 0 ) and v (c 0 ) v (c 0 ) : Te only remaining part of te Lemma is to sow tat even if v 0 and q 0 decrease proportionally so v 0 q 0 remains constant ten v(c ) q(c ) wold decrease. We increase q 0 to q 0 q 0 + " were " is very small. To make sre tat v 0 q0 v 0 q 0 ; we need tat v 0 v 0 + a" were a v 0 q 0 : Let s refer to all te objects after te cange wit te bar. Using te rst two Lemmas above, we ave q (c ) q (c ) + " e c e c0 + e c e c 0 v (c ) v (c ) + " e(c c 0 ) e (c c 0 ) (c c 0 ) e (c c 0 ) + e (c c 0 ) (e c 0 e c + e c 0e c ) + v 0 " q 0 e c e c0 + e c e c 0 67

68 we sow we want to sow tat v (c ) + " e(c c 0) e (c c 0) (c c 0 ) e (c c 0) +e (c c 0 ) (e c 0 e c +e c 0 e c + v 0 ) q 0 " e c e c 0 +e c e c 0 q (c ) + " e c e c 0 +e c e c 0 < v (c ) q (c ) wic is eqivalent to e (c c 0 ) e (c c 0 ) (c c 0 ) e (c c 0 ) + e (c c 0 ) (e c 0 e c + e c 0e c ) < v (c ) q (c ) v 0 q 0 wic olds, becase te left and side is negative as e x e x x (e x + e x ) < 0 for every x. A.9 Appendix for Section 5.3 Te following Proposition gives te system of ODEs determining te social welfare for an arbitrarily given c and c l in or alternative setting. Proposition 9 Te total welfare in te alternative speci cation for arbitrarily given tresolds c l < c is were v S and q S is given by te system Aj S (c; c l ; c ) A (v S (c) + q S (c) c) 0 qs 0 (c + p S ) + q00 S + ( q S ) + ( q S ) c c + c>c (q S (c ) v S (c )) + q S (c ) q S (c) c + cc + c<cl (q S (c l ) q S (c)) ; 0 qs 0 (c t ) vs 0 (c + p S ) + v00 + (p S v S ) + (R v) cl c + c>c (v S (c ) v S (c)) + c<cl (v S (c l ) lq S (c l )) + v S (c l ) v S (c) : l + c l wit te bondary conditions (43)-(44) and (47)-(48). Proof. For social welfare j (c) so tat c > c g, once investment opportnity arrives, immediately te economy sold bild x trees so tat C x A + x cg ) x C + c g cg A A c cg + c g ; and te total vale is Aj (c) (A + x) j c g A + c cg + c g j c g + c A + c g j c g 68

69 If instead tat C < Ac g l, ten te economy sold dismantle x trees so tat C + xl A x cg l ) x cg l A C l + c g l A cg l c l + c g ; l and te total vale is Aj (c) (A x) j c g l A c g l l + c g l c j c g l + c l A l + c g j c g l l So essentially we ave to evalate 00 0 j j 0 (c + p) + (p + c) + (R + c j) + + c + B (c) j (B (c)) j (c) However we need to know p and terefore we need to solve for v and q. Take te pper as example. Sppose tat social planner bild trees trog taxing cas and do not toc tree. we need to bild amont of trees, wic need A c cg t + c g A c cg + c g amont of cas. Since existing cas is Ac, per nit of cas te taxation is c c g + c g c in te meantime eac gets A c cg +c g trees tat tey can easily sell to te market to get A c cg v (c ) + c g q (c ) As a reslt, te net taxation per nit of cas is c c g + c g c c c g v (c ) + c g q (c ) c c cg c + cc g t v (c ) c cg q (c ) c + cc g ( p (c )) If p (c ) > ten cas is getting positive taxes. Terefore for q eqation, we ave (we need to mltiply above te by q c g to get back to tilities) 0 q 0 (c + p) + c c g q00 + ( q)+ ( q)+ c + cc g q c g v c g + q c g q (c) for c > c g : 69

70 Similarly, wen c is low, social planner wants to dismantle amont of trees x A cg l c l + c g l wic brings cas of c l + c g l l A cg l So for eac tree, investor as to send to te social planner of cg l c g l c l+c g l l+c g l c amont of tree, and getting back l amont of cas. Becase tey can immediately sell tese trees to te market, e ectively eac tree is taxed at c g l l + c g l c v c g l lq c g l : Wen p c g l < l ten trees are getting tax sbsidy, and we ave 0 q 0 (c t ) v 0 (c + p) + v00 + (p v)+ (R v)+ c g l l + c g l c p c g l l + v c g l v (c) for c < c g l 70

71 B Appendix (online only): An alternative eqilibrim In te main text, we sowed tat an eqilibrim exist wen l is s ciently small. Wile or condition is only s cient, and not necessary, it is possible tat te type of eqilibrim we present does not exist. In tis Appendix, we provide some insigts on te type of eqilibrim wic arises instead. We arge tat te main properties of tis alternative eqilibrim are very similar to te one presented. Wile te eqation system (7)-(8), (3)-(4) always ave a soltion, for some parameters tis soltion implies tat for a c s ciently close to c l ; te price is below te tresold l: Tis obviosly cannot be an eqilibrim. Tis is so, becase agents wold liqidate te rst instant wen te price drops below te liqidation vale. For tis set of parameters, te eqilibrim is canged. Te main di erence is tat tere is a c x (c l ; c ) tat for every c [c l ; c x] p (c) v (c) q (c) l and an endogenos fraction of trees are liqidated at every instant. Tat is, in tis range te price is constant in c and specialists liqidate an increasing fraction of teir trees as c drops frter from c x. Te following Proposition describes tis eqilibrim. Proposition B. Sppose tat tere is a c < R; c x (l; c ) ; q 0; K ; K ; K 3 ; K 4 solving (7)-(8), (3) + Rcx (l c x ) q 0 (c x ) l + Rcx (l c x ) v 0 (c x ) v (c x ) q (c x ) l; v (c ) q c v 0 (c ) q 0 (c ) 0: Ten tere is an incomplete market eqilibrim wit partial liqidation were. agents do not consme before te tree matres,. eac agent in eac state c [l; c ] is indi erent in te composition of er portfolio 3. agents do not bild or liqidate trees wen c (c x ; c ) and, in aggregate, agents spend every positive cas sock to bild trees i c c and nance an endogenos fraction of te negative cas socks by liqidating a fraction of trees i c [l; c x ] : Wen c l; agents nance te every negative cas sock by liqidating trees. 7

72 4. te vale of olding a nit of cas and te vale of olding a nit of tree are described by q (c) and v (c) and te ex ante price is p v(c) q(c) wen c [c x; c ] and by q m (c) q 0 + (l R) (c l) v m (c) lq m (c) and te ex ante price is p l wen c [l; c x ]. c l i + lr (ln c ln l) 5. Ex post, eac agents it by te sock sells all er trees to te agents wo are not it by te sock for te price p T c: Proof. Under te conditions of te Proposition, agents start to disinvest wenever p (c) l and along te way Here, if te disinvestment rate is y dc x (c) dt + dz t : da A, ten x (c) dc A C da A A lda A C da (l + c) y: A A Te following mst old as agents are always indi erent. Ten we mst ave 0 x (c) q 0 (c) + q00 (c) + + R c q (c) 0 x (c) v 0 (c) + q 0 (c) + v00 (c) + (c + R) v (c) v (c) lq (c) 0 x (c) lq 0 (c) + q 0 (c) + lq00 (c) + (c + R) lq (c) 0 x (c) lq 0 (c) + lq00 (c) + l + R lq (c) c so tat or, q 0 (c) + l (c + R) q 0 (c) + R 0 c + R (l c) 0: c 7

73 As q 0 (c l ) 0 as to old, c l l: Te closed-form soltion is q (c) q 0 + (l R) (c l) c l i + lr (ln c ln l) And, q 00 (c) + lrc < 0: We know tat for c [l; c x ] we ave v (c) lq (c) wic gives x (c) q00 (c) + R c + q (c) q 0 (c) For c > c x we ave te ODE as sal. We ten searc for te c x ; c pair tat satis es te conditions of te proposition. Plotting v, q and p give very similar graps to Figre 3 wit te main di erence tat at te range c [l; c x ] te price is at at te level l: In te same range q (c) is decreasing implying tat v (c) lq (c) is also decreasing. 73