Lemons Markets and the Transmission of Aggregate Shocks

Transcription

1 Lemons Markets and the Transmission of Aggregate Shocks Pablo Kurlat Stanford University July 21, 2011 Abstract I study a dynamic economy featuring adverse selection in asset markets. Borrowingconstrained entrereneurs sell ast rojects to finance new investment, but asymmetric information creates a lemons roblem. I show that this friction is equivalent to a tax on financial transactions. The imlicit tax rate resonds to aggregate shocks, amlifying the resonse of the real economy and generating cyclical variation in various measures of liquidity. Financial markets are fragile, volatile and occasionally shut down entirely. The extent to which this instability is related to real economic variables is an imortant oen question for macroeconomics. In this aer I develo a model of financial imerfections to exlain how instability can result from macroeconomic shocks and in turn amlify and roagate them. I focus on one secific financial market imerfection: asymmetric information about the quality of assets. There are several reasons for this focus. First, both in theory (Hart and Moore 1994, Kiyotaki and Moore 1997) and in ractice, assets of various kinds are extensively used to enable intertemoral transactions. If there is asymmetric information about asset qualities, which is a natural assumtion, this has the otential to interfere with a large subset of financial transactions. Second, asymmetric information is a central concern in cororate finance. Following Myers and Majluf (1984), asymmetry of information between firm managers and their outside investors is seen as a key determinant of firms caital structure. Third, I am deely indebted to George-Marios Angeletos, Ricardo Caballero, Bengt Holmström and esecially Iván Werning for their invaluable guidance. I also thank Daron Acemoglu, Sergi Basco, Francisco Buera, Martín Gonzalez-Eiras, Guido Lorenzoni, Monika Piazzesi, Krishna Rao, Julio Rotemberg, Martin Schneider, Jean Tirole, Robert Townsend and articiants at the MIT Macroeconomics Lunch, Theory Lunch and International Breakfast, as well a various seminar articiants. Corresondence: Deartment of Economics, Stanford University. 579 Serra Mall, Stanford, CA, kurlat@stanford.edu 1

2 sometimes financial markets simly cease to function, as documented by Gorton and Metrick (2009) for the reo market in Since Akerlof (1970), it is well known that the comlete breakdown of trade is a theoretical ossibility in economies with asymmetric information. This means that asymmetric information at least has the otential to exlain extreme crises and may shed light on less extreme henomena as well. I embed imerfect financial markets in a simle dynamic macroeconomic model. In the model, entrereneurs hold the economy s stock of caital. Every eriod, they receive idiosyncratic investment oortunities, i.e. oortunities to convert consumtion goods into caital. The only way to obtain financing is to sell existing assets. These are bought by entrereneurs who in the current eriod have oor investment oortunities. Unfortunately, some fraction of existing assets are useless lemons and buyers can t tell them aart from high quality assets (nonlemons). I show that the lemons roblem introduces a wedge between the return on saving and the cost of funding, ersuading some entrereneurs to stay out of the market. This is formally equivalent to introducing a tax on financial transactions. The tax lowers asset rices, the rate of return obtained by uninformed investors and the rate of caital accumulation. Furthermore, the imlicit tax rate deends on the roortions of lemons and nonlemons sold, which resond to aggregate shocks. I consider three tyes of aggregate shocks: a standard roductivity shock, an investment shock which affects all entrereneurs investment oortunities roortionally and an informational shock. I show that for all of these, negative shocks will lead to a worsening of financial market imerfections (as measured by the imlicit tax rate), and vice versa. For informational shocks, the effect is quite direct: an exogenous worsening of information increases frictions. For investment shocks, the effect is due to the fact that investment oortunities are the source of gains from trade; fewer investment oortunities reduce the gains from trade and lead to greater frictions. For roductivity shocks, the mechnism is driven entirely by general equilibrium effects. Negative roductivity shocks lower outut, which reduces the suly of savings; the rate of return on saving rises, asset rices fall and sellers of nonlemons dro out of the market, which worsens the adverse selection roblem. Overall, the endogenous resonse of financial frictions amlifies the effects of shocks on the real economy. Due to general equilibrium forces, financial frictions in the model are sensitive to wealth effects. This is also the case in Kiyotaki and Moore (1997), Bernanke and Gertler (1989), and Carlstrom and Fuerst (1997) among others, who analyze how the wealth of roductive-butconstrained agents affects frictions. In this model, however, wealth effects oerate through a different channel since the decision of entrereneurs with good investment oortunities tokee or sell their nonlemons is indeendent of wealth. Instead, the model highlights the imortance of the balance sheets of less-roductive agents, because they govern the demand for assets. 2

3 In addition to roviding an amlification mechanism, the model can account for a number of stylized facts about business cycles. The model redicts that frictions will increase in recessions, which may exlain the rocyclical liquidity of real assets documented by Eisfeldt and Ramini (2006). While different tyes of cyclical frictions could account for this, other redictions are secific to asymmetric information. The model redicts that the decision to sell an asset is a more negative signal about the asset s quality in recessions than in exansions, which is consistent with the evidence in Choe et al. (1993) on seasoned equity issues. Furthermore, unlike other theories of financial frictions that redict that outside financing is either a fixed fraction of firms funding or is even countercyclical, this model redicts that outside financing is rocyclical, consistent with the evidence in Covas and Den Haan (2011). The model further redicts that large negative shocks lead financial markets to shut down entirely. This tye of shar decrease in the volume of financial transactions has been documented for instance by Ivashina and Scharfstein (2010). In the model, entrereneurs investment oortunities are heterogeneous, which imlies that the distribution of investment across entrereneurs matters. Increased frictions in recessions distort the allocation of investment and worsen the average rate of transformation of consumtion goods into caital goods. Justiniano, Primiceri and Tambalotti (2008) have found this tye of effect to lay a large role in aggregate fluctuations and have further found that it correlates with measures of disturbances in financial markets. The structure of the model is close to that develoed by Kiyotaki and Moore (2005, 2008), which also features random arrival of investment oortunities, borrowing constraints and artially illiquid assets. They use a reduced-form model of the limitations on selling caital and investigate whether this may exlain why easier-to-sell assets command a remium. In contrast, I develo an exlicit model of what the sources of these limitations are, which allows me to investigate how they resond to aggregate shocks. Following Stiglitz and Weiss (1981), adverse selection layed an imortant early role in the theory of credit markets. Recent contributions include Bolton et al. (2009) and Malherbe (2009). In macroeconomic settings, the issue has been studied by Mankiw (1986), de Meza and Webb (1987) and House (2006). Hendel and Lizzeri (1999) and Hendel, Lizzeri and Siniscalchi (2005) study the dynamic nature of the lemons roblem, focusing on durable goods market. With durable goods, the gains from trade arise from differences in tastes and random dereciation, whereas in this model trade in assets is merely a instrument for intertemoral trades. Closest to this aer is Eisfeldt (2004). In her model, entrereneurs hold different vintages of rojects and cannot diversify risks. Financial transactions are desirable because they enable entrereneurs to smooth consumtion when they suffer oor realizations of income from revious vintages of risky rojects. Thus her aer is about how asymmetric information interferes with risk-sharing whereas mine is about how it interferes with the financing of investment. 3

4 On a more technical side, one limitation of her aroach is that it requires keeing track of the distribution of ortfolio holdings across different vintages of rojects, for all entrereneurs, which makes it necessary to limit attention to numerical simulations of steady states or simle deterministic cycles, since stochastic simulations are comutationally infeasible. Recently, Bigio (2011) has extended a variant of my model, introducing labour market frictions, and shown that it can quantitatively match some of the features of the recession. 1 The environment Households. There are two kinds of agents in the economy, workers and entrereneurs. There is a continuum of mass L of identical workers, each of whom sulies one unit of labour inelastically; they have no access to financial markets, so they just consume their wage. In addition, there is a continuum of mass one of entrereneurs, indexed by j, who have references with u(c j t) = log(c j t). They do not work. 1 E β t u(c j t) t=0 Technology. Consumtion goods are roduced combining caital and labour. The caital stock consists of rojects owned by entrereneurs. Entrereneur j s holdings of rojects are denoted k j t so the aggregate caital stock is K t = k j t dj. Every eriod a fraction λ of rojects becomes useless or lemons. Each entrereneur s holdings of rojects is sufficiently well diversified that the roortion λ alies at the level of the individual entrereneur as well. Each of the (1 λ) K t rojects that do not become lemons is used for roduction, so that outut is Y t = Y ((1 λ) K t, L; Z t ). Y is a constant-returns-to-scale roduction function that satisfies Inada conditions and Z t is exogenous roductivity; the roerties of the model are such that the stochastic rocess that governs Z t may be left unsecified. The marginal roduct of caital and labour are denoted Y K and Y L resectively. The aggregate resource constraint is Lc w t + (c j t + i j t) dj Y ((1 λ) Kt, L; Z t ) (1) where c w t denotes consumtion er worker, c j t is consumtion by entrereneur j and i j t reresents hysical investment by entrereneur j. Physical investment is undertaken in order to convert consumtion goods into rojects 1 Giving entrereneurs no labour income makes their decisions linear in their caital holdings, which simlifies the characterization of their olicy functions. 4

5 for eriod t + 1. Each entrereneur can transform consumtion goods into rojects using an idiosyncratic linear technology with a stochastic marginal rate of transformation A j t. A j t is iid across time and entrereneurs and is drawn from a continuous distribution F with finite mean and continuous density f. In addition, each nonlemon roject turns into rojects at t + 1, so it is ossible to interret 1 (1 λ) as an average rate of dereciation. Aggregate caital accumulation is given by K t+1 = (1 λ) K t + i j ta j tdj (2) Allocations. The exogenous state of the economy is z t { Z t, Āt}. It includes roductivity Z t and the function Āt, which mas each entrereneur to a realization of A j t. An allocation secifies consumtion and investment for each agent in the economy and aggregate caital after every history: {c w (z t ), c j (z t ), i j (z t ), K (z t )}. Information. At time t each entrereneur knows which of his own rojects have become lemons in the current eriod, but the rest of the agents in the economy do not. Informational asymmetry lasts only one eriod. At t+1, everyone is able to identify the rojects that became lemons at t, so they effectively disaear from the economy, as illustrated in figure 1. This assumtion is made for simlicity as it eliminates the need to kee track of rojects of different vintages. Daley and Green (2009) study the strategic issues that arise when informational asymmetries dissiate gradually over time. The investment oortunity A j t is and remains rivate information to entrereneur j. Period t Period t + 1 Lemon λ roject disaears Lemon 1 roject λ 1 λ Nonlemon rojects enters Y t ( ) 1 λ Nonlemon enters Y t+1 ( ) Figure 1: Information about a roject over time 5

6 2 Benchmark allocations 2.1 Comlete markets Suose all the information was ublic and there were comlete cometitive markets. X { z, K } be the aggregate state variable, where K mas entrereneurs to caital holdings. Since lemons are useless their rice will be zero. In factor markets, entrereneurs hire workers at a wage of w(x) = Y L (X) and obtain dividends of r(x) = Y K (X) for each nonlemon roject. 2 Consumtion goods are traded for nonlemon rojects, ex-dividend, at a sot rice of NL (X) goods er nonlemon roject. State-contingent claims are traded one eriod ahead: the staterice density is ρ(x, X ). Entrereneurs solve: V (k, b, X) = Let max [u (c) + βe [V c,k,i,d NL,b(X ) (k, b(x ), X ) X]] (3) s.t. c + i + NL (X)d NL + E[ρ(X, X )b(x )] r (X) (1 λ) k + b (4) k = [(1 λ) k + d NL ] + A j (X)i (5) i 0, d NL (1 λ)k (6) Constraint (4) is the entrereneur s budget constraint. The entrereneur s available goods are equal to the dividends from his nonlemons r (1 λ) k lus net state-contingent goods bought the revious eriod b. These are used for consumtion lus hysical investment lus net urchases of nonlemons d NL lus urchases of state-contingent goods for the next eriod. Constraint (5) kees track of the entrereneur s holdings of rojects. k, the total number of rojects next eriod, is equal to the nonlemon rojects he owns at the end of the current eriod, which are (1 λ) k + d NL and have grown at rate, lus the rojects that result from his hysical investment in this eriod, A j (X)i. Constraint (6) states that investment must be nonnegative and sales of nonlemons are limited by the number of nonlemons the entrereneur owns. The first order conditions with resect to i and d NL imly: 3 A j (X), with equality if i > 0 (7) NL (X) Let A max be the highest ossible value of A. At each history there will be an entrereneur 2 As is standard, this could be the result of cometitive firms renting caital from entrereneurs or of entrereneurs oerating the roductive technology themselves. With asymmetric information, the latter interretation avoids the need to analyze adverse selection in the rental market. 3 This uses the fact that constraint d NL (1 λ)k will not bind in equilibrium because a roject can be relicated with contingent securities. 6

7 (the best entrereneur) with A j (X) = A max who can transform each consumtion good into A max rojects at t + 1. Equation (7) then imlies that NL (X) = A max for all X. The best entrereneur is the only one to undertake hysical investment. He finances this investment by selling claims to consumtion goods one eriod ahead (i.e. borrowing) which he then satisfies with the dividends lus roceeds of selling the newly created rojects in the sot market. Since r(x) is stochastic, caital is a risky asset, and the best entrereneur will use state-contingent securities to share this risk with the rest of the entrereneurs. Comlete markets imly that risk-sharing will be erfect. Comlete markets are actually not indisensable for achieving this allocation. Suose there is a market for selling new rojects at the same instant that they are built, when they still haven t slit into lemons and nonlemons. In equilibrium these new rojects will trade at a rice NEW (X) = 1 A max, the marginal cost for the best entrereneur, who will be the only one to build them. Entrereneurs would have no further need to trade contingent securities because (i) their idiosyncratic realizations of A j would have no effect on their budget sets, so they would bear no idiosyncratic risk and (ii) they have identical homothetic references and their exosure to aggregate risk is roortional to their caital, which is their only asset. Hence, they would attain the same allocation as with comlete markets. Proosition 1. With comlete markets, all investment is undertaken by the entrereneur with A j = A max ; all entrereneurs obtain A max rojects er consumtion good saved and bear no idiosyncratic risk. The aggregate economy behaves just like an economy where the rate of transformation of consumtion goods into rojects is fixed at A max, there is a reresentative entrereneur and workers are constrained to live hand-to-mouth. The same allocation is obtained if the only market that exists is for newly-created rojects. 2.2 Borrowing constraints but still no informational asymmetry For various reasons, it may be difficult for an entrereneur to borrow against his future wealth, i.e. to choose negative values of b(x ). For instance, he may be able to run away with his wealth rather than honouring his debts. 4 I what follows, I make the extreme assumtion that borrowing is imossible, which in equilibrium imlies lending is imossible as well. Thus, the entrereneur will solve rogram (3), with the added constraint: b(x ) = 0 (8) As a result, the only market that exists is for ex-dividend sales of existing rojects. This is intended to reresent not just literal sales of used machines but more broadly the use of 4 Alternatively, he could refuse to exert effort if he has ledged his outut, as in Holmström and Tirole (1998). 7

8 existing assets as collateral for borrowing. In articular, selling rojects is the only way to achieve intertemoral trades. 5 Imortantly, only existing rojects, not newly-built ones, may be traded, a distinction also emhasized by Kiyotaki and Moore (2008). As a result of constraint (8), the best entrereneur will not be able to undertake all the investment in the economy. Instead, there will be a cutoff A (X) = NL (X) such that entrereneurs with A j (X) < A (X) will not invest and entrereneurs with A j (X) > A (X) will sell all their existing nonlemons in order to obtain consumtion goods for investment. This equilibrium is inefficient in two related ways. First, the economy does not exclusively use the most efficient technology (A max ) for converting consumtion goods into rojects. Second, entrereneurs are exosed to idiosyncratic risk. If they draw a low value of A j, they must convert their consumtion goods into rojects through the market, which only rovides a return A (X), whereas if they draw a higher value they convert them at a rate A j (X). 3 Asymmetric information Assume now that only the owner of a roject knows whether it is a lemon, and each entrereneur observes only his own A j. Those who urchase rojects have rational exectations about λ M, the roortion of lemons among the rojects that are actually sold in the market. individual rojects is still the only financial transaction allowed. 6 Selling With asymmetric information, entrereneurs do not observe the state of the economy because they do not observe other entrereneurs investment oortunities. Fortunately, the endogenous aggregate variables r, and λ M that are relevant for the entrereneur s decisions deend only on roductivity and caital, which the entrereneur does observe. Therefore I will look for a recursive cometitive equilibrium with X {Z, Γ} as a state variable, where Γ(k, A) is the cumulative distribution of entrereneurs over holdings of caital and investment oortunities. 7 The relevant state variable for entrereneur j s roblem is {k j, A j, X} so (droing 5 The difference between selling a roject and romising b with the roject as collateral is that if the value of the roject at t + 1 is v the buyer/lender obtains v in case of a sale and min{v, b} in case of collateralized borrowing. Given the binary outcomes for rojects, if there was no aggregate risk, both transactions would be identical u to a normalization, aying a fixed ositive amount if the roject is still a nonlemon at t + 1 and zero if it becomes a lemon. Due to aggregate risk, the ayoff in case the roject is still a nonlemon is not fixed, so the two transactions differ slightly. Still, they share the feature that the exected value of an entrereneur s existing rojects determines the extent to which he is able to obtain consumtion goods for new investment. 6 In articular, I do not allow for selling a reresentative ortfolio of rojects, which would have no adverse selection. This could result from the difficulty of determining what the entrereneur s total holdings are or from a cost in bundling the ortfolio, as assumed by Kiyotaki and Moore (2003). I also do not allow for other securities besides outright sales. DeMarzo and Duffie (1999) and Nachman and Noe (1994) analyze security design in related settings. 7 Since A j is iid, then it is indeendent of k j and Γ is just the roduct of F and the distribution of k. The more general formulation could easily accommodate the case where an entrereneur s individual A j has some ersistence, which would create correlation between k j and A j. 8

9 the j suerscrit) he solves the following rogram: V (k, A, X) = s.t. max [u (c) + βe [V c,k,i,s L,s NL,d (k, A, X ) X]] (9) c + i + (X) [d s L s NL ] r (X) (1 λ) k k = [ (1 λ) k + ( 1 λ M (X) ) d s NL ] + Ai i 0, d 0 s L [0, λk], s NL [0, (1 λ) k] Program (9) incororates the borrowing constraint (8) and the fact that the rice (X) alies equally for sales of lemons s L, sales of nonlemons s NL and urchases of rojects of unknown quality d, a roortion λ M (X) of which turn out to be lemons. Denote the solution to this rogram by {c (k, A, X), k (k, A, X), i (k, A, X), s L (k, A, X), s NL (k, A, X), d (k, A, X)} and define the suly of lemons and nonlemons, total suly of rojects and demand of rojects resectively as S L (X) s L (k, A, X) dγ(k, A), S NL (X) snl (k, A, X) dγ(k, A), S (X) S L (X) + S NL (X), D (X) d (k, A, X) dγ(k, A). Definition 1. An equilibrium consists of rices { (X), r (X), w (X)}; market roortions of lemons λ M (X); a law of motion Γ (X) and associated transition density Π (X X); a value function V (k, A, X) and decision rules {c w (X), c (k, A, X), k (k, A, X), i (k, A, X), s L (k, A, X), s NL (k, A, X), d (k, A, X)} such that (i) factor rices equal marginal roducts: w (X) = Y L (X), r (X) = Y K (X); (ii) workers consume their wage c w (X) = w (X); (iii) {c (k, A, X), k (k, A, X), i (k, A, X), s L (k, A, X), s NL (k, A, X), d (k, A, X)} and V (k, A, X) solve rogram (9) taking (X), r (X), λ M (X) and Π (X X) as given; (iv) the market for rojects clears: S (X) D (X), with equality whenever (X) > 0; (v) the market roortion of lemons is consistent with individual selling decisions: λ M (X) = S L(X) and (vi) the law of motion of Γ is consistent S(X) with individual decisions: Γ (k, A)(X) = k dγ( k, Ã) F (A). ( k,ã,x) k 3.1 Equilibrium conditions I solve the entrereneur s roblem and find equilibrium conditions in stes. First I show that all the olicy functions are linear in k, which imlies an aggregation result. Second I show that, given choice of c and k, the choices of d, s L, s NL and i reduce to an arbitrage condition. Third I solve a relaxed roblem, converting the entrereneur s nonlinear budget set into a weakly larger linear one and show that there is a simle static characterization of the consumtion-savings decision. Based on the solution to the relaxed roblem it is ossible to derive suly, demand and a market clearing condition. Finally I show that the equilibrium rice must satisfy the market-clearing condition whether or not the solutions to the two rograms coincide. 9

10 Linearity of olicy functions. The constraint set in rogram (9) is linear in k and the utility function is homothetic. Hence the olicy functions c (k, A, X), k (k, A, X), i (k, A, X), s L (k, A, X), s NL (k, A, X) and d (k, A, X) are all linear in k. This imlies the following result: Lemma 1. Prices and aggregate quantities do not deend on the distribution of caital holdings, only on total caital K. By Lemma 1, {Z, K} is a sufficient state variable; in order to comute aggregate quantities and rices it is not necessary to know the distribution Γ. Buying, selling and investing decisions. Take the choice of k as given. The entrereneur s roblem then reduces to choosing d, s L, s NL and i to maximize c. This rogram is linear so the entrereneur will generically choose corner solutions. The decision to kee or sell lemons is trivial: as long as > 0 the entrereneur will sell the lemons (s L = λk), since they are worthless to him if ket. The decisions to kee or sell nonlemons and to invest in new rojects or in urchasing rojects deend on A. The return (i.e. the number of t + 1 rojects obtained er consumtion good sent) from buying rojects is A M (1 λ M ). I refer to this as the market rate of return. 8 Conversely, the number of t + 1 nonlemon rojects an entrereneur must give u to obtain one consumtion good is > AM. The return on investing is simly A. This imlies that the otimal choices of d, s NL and i are given by two cutoffs, shown in figure 2. A M (1 λm ) Buyer: Kee nonlemons Buy rojects Keeer: Kee nonlemons Invest Seller: Sell nonlemons Invest A Figure 2: Buying, selling and investing decision as a function of A For A < A M, entrereneurs are Buyers: the return from buying is greater than the return from investing so i 0 and s NL 0 bind and d > 0. For A [A M, ] entrereneurs are Keeers: investing offers a higher return than buying but not higher than the oortunity cost of selling nonlemons at the market rice, so the entrereneur neither buys rojects nor sells nonlemons; d 0 and s NL 0 bind and i > 0. For A > entrereneurs are Sellers: the return from investing is high enough for the entrereneurs to sell nonlemons in order to finance investment; d 0 and s NL (1 λ) k bind and i > 0. 9 Combining these arbitrage conditions 8 Noting, however, that it involves two different goods (rojects and consumtion goods) as well as two different dates. 9 This assumes that k (1 λ) k so the entrereneur wants more caital that just his own nonlemons. If instead k < (1 λ) k (which by lemma 4 below is inconsistent with equilibrium), then Buyers and Keeers would choose i = d = 0 and s NL > 0 while Sellers would choose d = 0, s NL = (1 λ) k and i > 0. 10

11 with the constraint from rogram (9) yields the following lemma: Lemma 2. Given k, the otimal d, s L, s NL and i are given by Buyers: A [ 0, A M] Keeers: A ( ] A M, Sellers: A s L = λk λk λk { } d = max k (1 λ)k, (1 λ M ) { } { } s NL = max (1 λ)k k, 0 max (1 λ)k k, 0 (1 λ) k { } i = 0 max k (1 λ)k k, 0 A A (, ) (10) Consumtion-savings decision under a relaxed budget set. An entrereneur with investment oortunity is A must choose c k k k A[(1 λ)r + ] and k k from his budget set, shown in figure 3. (1 λ) + [(1 λ) r + λ] A (1 λ) + [(1 λ) r + λ] A M (1 λ) Keeer Buyer Seller x True budget sets Relaxed budget sets (1 λ)r +λ (1 λ)r + Figure 3: Budget sets c k Point x reresents an entrereneur who chooses s L = λk and i = s NL = d = 0, an otion available to all entrereneurs. He simly consumes the dividends (1 λ) rk and the roceeds from selling lemons λk, and enters eriod t + 1 with (1 λ) k rojects. Consider first the decision of a Keeer. If he wishes to increase consumtion beyond oint x he must sell nonlemons, which means giving u future rojects for each additional unit of consumtion. If instead he wishes to carry more rojects into t + 1, he invests with rate of transformation A. Hence the budget constraint is kinked: to the right of x the sloe is whereas to the left it is A. Consider next a Buyer. His budget set is the same as for the Keeer excet that the return he obtains from saving beyond oint x is the market return A M, which is higher than his individual return on investment A but lower than that of Keeers. Lastly, a Seller will sell 11

12 all his rojects and his budget constraint is linear with constant sloe A. Define the entrereneur s virtual wealth as W (k, A, X) [ ( { })] λ (X) + (1 λ) r (X) + max (X), k (11) max {A, A M (X)} Virtual wealth corresonds to to the extension of the left half of the budget constraint onto the horizontal axis. It consists of the goods the entrereneur has (dividends lus roceeds of selling lemons) lus the nonlemon rojects, valued at the maximum of either their sale rice or their k relacement cost. The linear budget set max { A, A M (X) } [ ] W (k,a,x) c max{a,a M (X)} k k k is weakly larger than the true kinked budget, so substituting it in rogram (9) leads to the following relaxed rogram: V (k, A, X) = max [u (c) + βe [V (k, A, X ) X]] (12) c,k s.t. k = max { A, A M (X) } [W (k, A, X) c] (13) Lemma 3. Under rogram (12), the entrereneur s consumtion is c (k, A, X) = (1 β) W (k, A, X) Due to log references, entrereneurs will always choose to consume a fraction 1 β of their virtual wealth and save the remaining β, by some combination of keeing their old nonlemons, buying rojects and hysical investment. Note that the entrereneur s decision, while rational and forward looking, does not deend on the transition density Π (X X) or on the stochastic rocess for A. This feature will make it ossible to solve for the equilibrium statically. Notice that the function W is decreasing in A. Different agents have different valuations of rojects but asymmetric information revents them from trading away those differences. In that sense, caital is illiquid. Furthermore, Lemma 3 imlies that high-a agents who value rojects the least also consume less, so roject valuation is negatively correlated with the marginal utility of consumtion. Therefore agents would be willing to save in a risk-free asset with a lower exected return, a remium that would disaear if there was symmetric information. Kiyotaki and Moore (2008) find that a similar remium arises if resaleability constraints revent entrereneurs from reselling a fraction of their rojects. Here instead the difference between the values laced on rojects by entrereneurs with different investment oortunities is derived endogenously as a result of asymmetric information. Suly and demand under the relaxed rogram. By (10), the suly of rojects will include all the lemons lus the nonlemons from Sellers. Hence S () = [ ( λ + (1 λ) 1 F ( ))] K (14) 12

13 This imlies a market roortion of lemons of λ M () = λ ( ( λ + (1 λ) 1 F )) (15) and a market rate of return of: 10 ( ( )) A M () = ( 1 λ M () ) = (1 λ) 1 F ( ( )) (16) λ + (1 λ) 1 F Demand for rojects will come from Buyers. By Lemma 3, under the relaxed rogram they choose k =βa M W (k, A, X). By Lemma 2, they each demand k (1 λ)k (1 λ M ) rojects. Using (11) and adding over all Buyers, demand for rojects will be: D () = Market clearing requires ( [ β λ + (1 λ) r ] ) (1 β) (1 λ) F ( A M () ) K (17) 1 λ M () S() D() with equality whenever > 0 (18) Equilibrium conditions under the true rogram. Lemma 4. D > 0 only if the solutions to rograms (9) and (12) coincide for all entrereneurs The solutions to rograms (9) and (12) will not coincide whenever in the relaxed rogram, some entrereneurs wish to choose oints to the right of x. Lemma 4 states that if this is the case there will be no demand for rojects. Therefore equilibrium rices can be found simly by solving (14)-(18). If a ositive solution is found, then the relaxed rogram has yielded the correct solution; otherwise the equilibrium rice is zero. Omitting the deendence on the state X, I simly denote the equilibrium rice by. Once is determined, it is straightforward to solve, also statically, for the rest of the equilibrium objects. λ M and A M follow from (15) and (16). If > 0 then virtual wealth and, by Lemma 3, consumtion for each entrereneur can be found using (11) and s L, s NL, d and i are given by (10). If instead the only solution to (18) is = 0, it is still ossible to solve the relaxed roblem (12), which results in k = β (1 λ) (Ar + ) k 10 Define A M (0) 0. 13

14 This satisfies k (1 λ) k iff A Ā (1 β). Hence for entrereneurs with A Ā, r β consumtion and investment can be comuted in the same way as when > 0 whereas entrereneurs with A < Ā chose c = (1 λ) rk and k = (1 λ) k. Aggregate caital accumulation is found by relacing the equilibrium values of i into the law of motion of caital (2), yielding K K = (1 λ) + + A M [βa [λ + (1 λ) r] (1 β) (1 λ) ] df (A) (19) βa [ + (1 λ) r] df (A) In general, the market return A M () can be either increasing or decreasing in. An increase in the rice has a direct effect of lowering returns by making rojects more exensive and an indirect effect of imroving returns by increasing the roortion of entrereneurs who choose to sell their nonlemons. This imlies that there could be more than one solution to (18). I assume that the equilibrium rice is given by the highest solution. Furthermore, there could exist a rice > such that A M ( ) > A M ( ) even when is the highest solution to (18). Stiglitz and Weiss (1981) argue that when this is the case the equilibrium concet used above is not reasonable and it would be more sensible to assume that Buyers set a rice above that maximizes their return and ration the excess suly. For some of the results below, it will simlify the analysis to just assume that arameters are such that the issue does not arise: 11 Assumtion 1. A M () is decreasing 3.2 Equivalence with an economy with taxes As shown in figure 2, asymmetric information introduces a wedge between the return obtained by Buyers, A M, and the return given u by Sellers,. This wedge is isomorhic to the wedge that would be introduced by imosing a tax on the sales of rojects. Consider the economy with borrowing constraints and symmetric information of section 2.2, but now assume that the government imoses an ad-valorem tax of τ (X) on urchases of rojects. The total revenue T (X) = τ(x)(x)s((x)) collected from this tax is rebated to ( ) [ ( ( ))] 11 Assumtion 1 holds if h λ λ 1 F, where h is the hazard function of A. If the hazard at A = is too high, a small decrease in the rice will lead a large fraction of the Sellers to become Keeers, increasing λ M and lowering market returns. Many common distributions fail to satisfy Assumtion 1 at their right tail, i.e. for very low rices; for instance with λ = 0.1, a standard lognormal fails Assumtion 1 above its 91 st ercentile. None of the results below require that Assumtion 1 hold globally, and those that require that it hold locally are noted. When Assumtion 1 fails, the main results still hold but need to be restated in terms of changes in the degree of rationing rather than changes in rices. 14

15 entrereneurs in roortion to their caital holdings. Entrereneurs solve: s.t. V (k, A, X) = max [u (c) + βe [V c,k,i,s NL,d (k, A, X ) X]] (20) c + i + (X) [d (1 + τ (X)) s NL ] r (X) (1 λ) k + T (X) k = [(1 λ) k + d s NL ] + Ai i 0, d 0 s NL [0, (1 λ) k] k K(X) This roblem can be solved by the same stes used to solve rogram (9). equilibrium conditions leads to the following equivalence result. Solving for the Proosition 2. Suose τ (X) = λm (X), where 1 λ M (X) λm (X) is the equilibrium value of the asymmetric information economy. Then rices and allocations of the symmetric-informationwith-taxes and the asymmetric information economies are identical. In the asymmetric information economy, Buyers cannot avoid aying for useless lemons whenever they buy rojects, which is like having to ay a tax on their urchases. Moreover, asymmetric information gives all entrereneurs the ossibility of earning λk from selling lemons to others, which is the counterart of the redistribution of the government s revenue. Overall, the effect of having a roortion λ M of lemons in the market is exactly equivalent to the one that would result from a tax at the rate τ = λm. The magnitude of this distortion is endogenous 1 λ M and, as discussed in section 4 below, it resonds to aggregate shocks in secific ways. The fact that the distortions are equivalent to taxes suggests that erhas they could be undone by government olicy. In rincile, this is indeed ossible. Suose the government were to subsidize Buyers, aying for a fraction σ(x) of the urchase rice of any rojects that they buy, and financed this subsidy with a tax on caital holdings. By the same reasoning that leads to Proosition 2, it can be shown that if σ(x) = λ M ( SI (X)) (where SI (X) is the rice that would revail under symmetric information and no taxes), then the allocation in the asymmetric information economy with subsidies would coincide with the one in the symmetric information economy and no taxes. The transaction subsidy would have to be greater in those states where the roortion of lemons is endogenously higher. There are, however, both theoretical and ractical objections to undertaking such a olicy. First, since the olicy involves a subsidy on transactions, the government would need a way to revent the same eole from trading the same roject several times and collecting the subsidy each time. This would require, for instance, keeing track of who owned each roject at the beginning of the eriod and only subsidizing urchases from the original owner. The government would also need to know entrereneurs caital holdings in order to imose the 15

16 caital tax. But if it were ossible for the government to know who owns what, some even simler olicies would also effective, such as mandating that every entrereneur sell their entire ortfolio every eriod, thus eliminating the adverse selection roblem. Second, even if it were feasible, the olicy would not necessarily be desirable. Imagine imlementing, for a single eriod, the tax-subsidy scheme that restores the symmetric information allocation. This olicy always makes some entrereneurs worse off. Consider an entrereneur whose investment oortunity equals the symmetric-information threshold A (X) so that with symmetric information he would choose to neither buy nor sell. With asymmetric information, he can still neither buy nor sell nonlemons but receives extra revenue from selling lemons so, abstracting from general-equilibrium effects on future factor rices, he is strictly better off. 12 General equilibrium effects would make him even better off, since asymmetric information lowers caital accumulation (see Lemma 5 below), which raises the future marginal roduct of caital. Thus, a one-time intervention that restored the symmetric information allocation would not bring about a Pareto imrovement, although erhas it would if it were imlemented every eriod and entrereneurs were sufficiently atient. 4 Aggregate shocks In order to analyze how the economy resonds to shocks I first establish a reliminary result regarding how the isomorhic economy with symmetric information and taxes would resond to an exogenous increase in tax rates. This will be useful for determining what effects are due to the endogenous resonse of λ M to aggregate shocks. Lemma 5. Suose that in the economy with symmetric information and taxes there is an exogenous increase in the tax rate τ. Then 1. the rice of rojects decreases, 2. the market returns for Buyers decrease and 3. caital accumulation decreases if τ is close to zero. An increase in taxes increases the wedge between A M and. Parts 1 and 2 of Lemma 5 establish that this increase in the wedge manifests itself through both lower returns for Buyers and lower rices for Sellers. Both of these effects tend to lower caital accumulation. In addition, taxes have the effect of redistributing resources from Buyers and Sellers to all entrereneurs, 12 The fact that this olicy does not bring about a Pareto imrovement is just a manifestation of the fact that allocations in the static Akerlof model are interim constrained efficient, as roved by Myerson (1983) and Bigelow (1990). 16

17 including Keeers. As with any tax, the relative incidence on Buyers and Sellers deends on elasticities. For small enough τ, the elasticities of suly and demand are mechanically linked, as the density of marginal Buyers, f ( A M) ( ), aroaches that of marginal Sellers, f. Part 3 of Lemma 5 establishes that in this case the redistributive effect always goes against the higher-a agents, reinforcing the effect of lower caital accumulation The resonse to aggregate shocks Thanks to assuming that entrereneurs have log references, no labour income and a single asset to invest in, the equilibrium conditions are static. This imlies that shocks will have the same effects no matter what is the stochastic rocess by which they are generated. Thus by answering the comarative statics question how would the features of the model change if a arameter were different? one also answers the imulse resonse question how would the economy resond to a shock? Consider first the effects of roductivity shocks, which can be of two kinds. Shocks to Z affect the caacity of the economy to roduce goods from the current caital stock while shocks to the distribution of A affect the caacity of the economy to transform goods into caital. I refer to the first simly as a roductivity shock and the second as an investment shock, and consider them in turn. Suose first that there is a roortional roductivity shock, lasting one eriod. Proosition 3. If in equilibrium > 0 then a ositive roductivity shock leads to 1. a higher rice for rojects, 2. lower market returns for Buyers (if Assumtion 1 holds), 3. a lower orortion of lemons in the market and 4. higher caital accumulation (if Assumtion 1 holds) Productivity shocks matter through their effect on the marginal roduct of caital r = Y K. Higher r means that entrereneurs receive higher current dividends. Other things being equal, they want to save a fraction β of the additional dividends. Sellers and Keeers do so through hysical investment but Buyers attemt to buy more rojects, bidding u the rice (art 1) and lowering returns (art 2) to restore market-clearing. Higher rices then ersuade marginal ( ) 13 For τ away from zero, it is ossible to construct counterexamles where f is much higher than f ( A M), so suly is much more elastic than demand. In this case it is ossible for Sellers to be net beneficiaries of redistribution, so taxes can conceivably increase caital accumulation. However, the direct distortion increases more than roortionately with τ so in most examles increasing τ from a high starting oint deresses caital accumulation even more than from a low starting oint. 17

18 Keeers to sell their nonlemons, reducing the severity of the lemons roblem, as measured by the equivalent tax rate τ = λm. Note that roductivity does not matter er se but rather 1 λ M for its effect on current dividends. A similar effect would result, for instance, if there was a helicoter dro of consumtion goods on the economy. This shows that, as in other macroeconomic models with financial constraints, balance sheet effects matter. However, unlike most of the transmission channels anaylzed in this literature, the effects do not work through the wealth of roductive-but-constrained high-a entrereneurs; for given rices, the threshold A for becoming a Seller rather than a Keeer is just and does not resond to current dividends. Instead, what matters is the amount of consumtion goods held by the low-a entrereneurs, because this determines the demand for assets and therefore asset rices and frictions. Focusing exclusively on how shocks affect the balance sheets of financially constrained agents would miss these otentially imortant general-equilibrium effects. Turn now to an investment shock, reresented as a roortional change in the investment oortunity of every entrereneur, from A to ϕa. Proosition 4. A ositive investment shock leads to 1. an ambiguous effect on the rice of rojects 2. higher market returns for Buyers (if Assumtion 1 holds) 3. a lower roortion of lemons in the market and 4. higher caital accumulation (if Assumtion 1 holds) Proosition 4 imlies that higher roductivity in the roject-roducing sector also alleviates the lemons roblem. Because hysical investment has become more attractive, marginal Keeers decide to sell their nonlemons, imroving the mix of rojects. In this case, the effect on asset rices is ambiguous: the direct effect of the shock is to lower asset rices (because the shock lowers the roduction cost of new rojects), but if the selection effect is strong enough, then rices could increase. Instead, the effect on returns for Buyers is unambiguously favourable, since the direct and selection effects go in the same direction. In addition to roductivity and investment shocks, it is also ossible to analyze urely informational shocks. A negative information shock that increases informational asymmetry can be modeled as a one-eriod increase in λ, comensated by an increase in K such that (1 λ) K remains unchanged. This shock has no effect on the roduction ossibility frontier of the economy and, with symmetric information, would have no effect on allocations. Proosition 5. A negative information shock leads to 1. a lower rice for rojects, 18

19 2. lower market returns for Buyers, 3. a higher orortion of lemons in the market and 4. lower caital accumulation for λ small enough. The increase in λ M that results from this tye of shock is equivalent to an exogenous increase in taxes, so Lemma 5 can be alied directly. One interretation of this tye shock may be the following. Suose every eriod entrereneurs receive an endowment of K useless lemons, so the total number of lemons is (λ + ) K rather than λk. However, in ordinary times it is ossible to tell aart the endowment-lemons from the nonlemons, so their existence is irrelevant. A shock to λ of the kind described above is equivalent to entrereneurs losing the ability to detect endowment-lemons, a form of deterioration of information. Business cycles may be generated by any combination of roductivity, investment and erhas information shocks. Proositions 3-5 jointly show that no matter what is the relative imortance of either tye of shock, financial market distortions (as measured by the imlict tax) will be countercyclical or, to ut it differently, liquidity will be rocyclical. For negative shocks, the adverse selection effect can be sufficiently strong that the rice falls to zero and no nonlemons are traded, an outcome that I label a market shutdown. 14 Proosition 6. A sufficiently negative roductivity, investment or information shock leads to a market shutdown. When roductivity is low, dividends are low so entrereneurs have few consumtion goods. For roject demand to be ositive, the market return must be high enough to ersuade Buyers to give u their scarce consumtion goods in return for additional rojects (i.e. to choose k above the kink in figure 3). In a world with symmetric information, the rice would dro until the return from buying rojects was sufficiently attractive to clear the market. With asymmetric information, the adverse selection effect laces an uer bound on the A M () return function. If dividends are sufficiently low, the return required for demand to be ositive exceeds this uer bound and there is no demand at any ositive rice. Similarly, when investment-roductivity is low, the measure of entrereneurs who are willing to sell nonlemons is low, which raises the fraction of lemons in the market and lowers returns. For large negative shocks, it does so to the oint where demand is zero at any ositive rice. The rediction of a market shutdown requires that the return function A M () have an uer bound. This feature is sensitive to the assumtions regarding the distribution of roject qualities. For instance, suose that instead of being worthless, lemons had low but ositive 14 At = 0, demand is indeterminate, so a market shutdown could equally mean no trade at all or just trade of lemons at a zero rice. 19

20 value. Then A M () would be unbounded because for sufficently low rices returns from buying would be arbitrarily large even if only lemons were being sold. Therefore lemons would always be traded at ositive rices in equilibrium and there would be no market shutdowns. More generally, with many ossible qualities, a necessary condition for A M () to be bounded is that the lower bound of the suort of the distribution of roject quality be zero The role of endogenous changes in distortions A way to isolate the role that asymmetric information lays in the transmission of aggregate shocks is to analyze how the resonse to shocks of this economy differs from that of an otherwise identical economy with symmetric information. In order to establish the right benchmark for comarison, I use the result from Proosition 2 and assume that in the symmetric information economy there are (fixed) taxes on transactions at a rate such that, absent the shock, rices and allocations in both economies would be exactly the same. Proosition 7. In resonse to a ositive roductivity shock 1. the rice of rojects increases more, 2. the market returns for Buyers fall less and 3. caital accumulation increases more (for λ small enough) while in resonse to a ositive investment shock 4. the rice of rojects falls less, 5. the market returns for Buyers increase more and 6. caital accumulation increases more (for λ small enough) in the asymmetric information economy than in the benchmark symmetric information economy. One of the longstanding questions in macroeconomics is why the economy exhibits large fluctuations, given that it is hard to locate large exogenous shocks driving them. Proosition 7 shows that asymmetric information about asset qualities may be art of the answer, since it amlifies the effects of either tye of shock on the real economy. Hence it can be one of the sources of the so-called financial accelerator. The role of asset markets in generating amlification has been noted before, for instance by Kiyotaki and Moore (1997). What is newer to this model is that, rather than being a fixed arameter as in Kiyotaki and Moore 15 The same is true in Akerlof s model, as shown by Wilson (1980). 20