Financial Risk Capacity

Transcription

1 Financial Risk Capacity Saki Bigio Department of Economics New York University Updates October 2, 211 Abstract I study a dynamic macroeconomic model of asymmetric information and financial intermediation. Banks provide intermediation services by dealing with informational asymmetries in capital markets. Intermediation requires banks to take positions on asset pools (collateralized loans, securities) over which they posses disadvantageous information. Banking is risky because the asset quality distribution is uncertain. When the value of their asset pools is surprisingly low, banks incur losses covered by liquidating their own net-worth. A sequence of negative shocks may therefore lead the financial system to lose its capacity to bear further losses. Without this capacity, the volume of intermediation shrinks while exacerbating adverse-selection. Adverse-selection has the effect of reducing the profitability of the financial sector up to a point where banks cannot be recapitalized externally. These episodes are characterized by persistent drops in the growth rate of the economy which are overcome as the financial sector slowly strengthens its balance sheets via retaining earnings only. The model is calibrated to match features of the U.S. economy and used to explore the effects of government policies meant to improve welfare. JEL: E32, G1, G21 Key Words: Financial Crisis, Adverse-Selection, Capacity Constraints, Capital Requirements sbigio@nyu.edu. Homepage: I would like to thank Viral Acharya, Alberto Bisin, V.V. Chari, Douglas Gale, Mark Gertler, Veronica Guerrieri, Urban Jermann, Larry Jones, Jennifer La O, Guido Lorenzoni, Allessandro Lizzeri, Alberto Martin, Cecilia Parlatore and Yongseok Shin for useful comments. I am specially indebted with Ricardo Lagos and Tom Sargent for their advice in this project. 1

2 1 Introduction The financial sector plays a key role for the reallocation of resources in the economy. For this reason, business cycle fluctuations are in part attributed to the financial system s capacity to provide intermediation services. Economic theory also stresses that one of the main functions of banking is to mitigate problems of asymmetric information in capital markets. This paper provides a unified theory to study the dynamics of financial intermediation in capital markets where asymmetric information is present. This theory explains how adverseselection in financial markets can become severe when banks suffer large losses and how adverse-selection can reduce private incentives to recapitalize banks during these episodes. The theory also identifies an externality prescribing a role for regulating financial institutions by trading-off growth against stability. Although there is a large body of theoretical and empirical research on how asymmetric information and financial intermediation may affect the business cycle these aspects have not been studied jointly. There are two reasons as to why this connection is key to understand business cycles and financial regulation. The first is that adverse-selection can occur when banks have low net-worth although the underlying information structure of an economy has not changed. Beginning with the seminal contributions of Stiglitz and Weiss (1981) or Myers and Majluf (1984), we know that asymmetric information can cause credit rationing phenomena that affect economic performance. Yet, this literature abstracts from the fact that most transactions occur through financial institutions (FIs). For example, Gorton (21) argues that The essential function of banking is to create a special kind of debt, debt that is immune to adverse selection by privately informed agents.. Indeed, by transacting with a large number of parties, FIs dilute the risk of transacting under asymmetric information. Nevertheless, FIs cannot dilute risk entirely so financial intermediation remains a risky activity. The capacity to tolerate financial intermediation risk (the financial risk capacity) is tied to the net-worth of FIs and thus lending terms will depend on this capacity. This paper argues that adverse selection can become severe after banks experience large losses. This feature represents a novel transmission mechanism. The paper argues that shocks that adversely affect the financial sector are amplified and spilled-over to the rest of the economy by exacerbating adverse-selection even if the underlying information structure of an economy has not changed. The second motivation is of a dynamic character. The connection between asymmetric information and financial intermediation can also explain why is it that recoveries are particularly slow after a banking crises. A theory that links financial intermediation to the balance sheet of banks must also explain why the financial sector is not quickly recapitalized after such episodes. 1 After all, the intuition behind standard competition arguments suggests that financial intermediation should be more profitable (on the margin) whenever there is less supply of this service. In turn, profitability should ultimately attract funds to the financial sector, reactivate financial intermediation and act as an automatic stabilizer during economic downturns. This paper shows that in presence of informational asymmetries, the profitability of banking may decrease as the aggregate volume of intermediation contracts. In fact, the paper shows that this stabilizer is effective for mild but not for large shocks to the financial industry. This characteristic provides an explanation as to why the private recapitalization of FIs was so weak during the last financial crisis and, consequently, why lending has taken so long to take-off during its aftermath. This matter has been a major concern for policy makers during the crisis. In fact, during his only television interview, FED Chairman, Ben Bernanke, was asked when he would 1 Many recent models of financial intermediation shut this channel by imposing assumptions such that intermediaries are fully invested in their banks. In these models, returns to intermediation are higher when net-worth is low. Since intermediaries are specialists with no additional wealth to recapitalize their firms, equity cannot be injected to replenish their net-worth. This aspect concerns both theorists, practitioners and policy makers (see literature review). Duffie (21) for example describes how financial intermediary equity was particularly slow in adjusting during the recent crisis. 2

3 consider the crisis to be over. He answered When banks start raising capital on their own.. 2 I therefore build a model that formalizes these insights with the purpose of explaining some aspects of financial intermediation: (1) The interaction between risk in financial intermediation and asymmetric information, (2) the implications of these features for the amplification and propagation of real shocks, (3) the externality that arises in this environment and finally, (4), the effects of financial regulation aimed at resolving the related market inefficiencies. On the quantitative side, the theory is capable of replicating several features of financial crises: (1) their frequency of occurrence, (2) the persistence in the response of output, (3) the contractions in volumes of intermediation and financing premia spikes, and (4) the reduction in bank equity returns and stock value which is explains why banks aren t recapitalized. I introduce a banking sector with the features described above into an otherwise standard growth model. The financial sector is modeled as a large competitive sector that provides intermediation services for the reallocation of capital between small agents in two sectors of the economy. As in Kiyotaki and Moore (28), the first sector is composed by producers of capital goods in need of funding to generate new investment projects. The second sector are agents that lack access to investment opportunities but, in contrast to their counterparts, do have the resources to carry out these projects. The fundamental economic problem in this economy is that funds must flow from the latter group to the former, and capital must flow the other way around. The key friction that prevents the efficient flow of resources is that capital goods producers have private information about the quality of their capital units. Banks resolve this problem by offering risk free deposits, transferring funds to capital goods producers and taking existing capital (of imprecise quality) as collateral for these transactions. Thus, in the model, financial intermediation is the engine of economic growth. Without informational asymmetries, the economy runs efficiently without the need of financial intermediation. In fact, the model collapses to a classical stochastic growth economy. When incorporating asymmetric information, FIs serve the purpose of diluting the idiosyncratic risk that small agents face if they trade capital directly while having disadvantageous information. If the financial sector knew the actual distribution of capital quality, financial intermediation would not be risky at all: by trading with a large number of sellers, the financial sector could, in equilibrium, perfectly infer the quality of the traded assets. In fact, without intermediation risk, bank profits would be zero and this economy collapses to other business cycles models driven by shocks that aggravate asymmetric information such as Bigio (21) or Kurlat (29). 3 To make this theory one of banking, it needs two additional ingredients. First, bank net-worth must fluctuate and for this, one needs to introduce risk into intermediation profits. Second, there needs to be a reason in order for net-worth to matter. To introduce risk into financial intermediation I assume that the entire distribution of collateral quality (which is private information) is, a priori, unknown. Mechanically, in equilibrium, banks will hold positions in the lower tail of the asset quality distribution. Since the entire distribution changes from time to time, the value of that tail is risky, which makes bank profits stochastic. 4 Net-worth will matter if, as in the real world, one assumes that banks face limited liability constraints. This means that when the asset quality pool held by banks is surprisingly worse than expected, they must 2 This quote was taken from the March 15, 29 interview titled the The Chairman, 6 Minutes, CBS News. 3 If this is the case, banks can exploit the law of large numbers in a way that allows them to wipe out all financial risk. This happens in Stiglitz and Weiss (1981), for example. Moreover, without any risk, in competitive environment, the law of large numbers takes care of driving profits to and equity is not needed. The conditional expected quality of assets bought at a given price would converge to the actual quality sold in equilibrium. For a formal argument see Glosten and Milgrom (1985) 4 Changes in the distribution of firm revenues have been found to precede recessions (see Bloom (29) and Bloom et al. (29)) for documentation. This feature is a realistic way of modeling financial risk when these shocks stem from private information. Very recently, a number of studies have provide several theories on the propagation of sector or firm specific shocks to the rest of the economy. See for example, Arellano et al. (21) or Gilchrist et al. (21). 3

4 resort to their own funds to finance operational losses. This restriction makes their net-worth a key state variable because today s net-asset position affects financial intermediation tomorrow. This creates a sensitive feedback-loop between asymmetric information and the evolution of net-worth. This aspect of the model is also generates an externality: banks can fail to internalize that by taking more risky positions, on the aggregate, they may suffer large losses that will affect the quality of asset pools held tomorrow by their competitors. The paper argues that this externality is particularly harmful as adverse selection can be exacerbated leading to prolonged downturns. To illustrate the mechanics of the model, let me guide you through a hypothetical simulation. Think about a particular sequence of distributions of capital quality. Assume that these are surprisingly below the financial system s expectations (leading to a sequence of financial losses). Financial losses are financed by liquidating part of the bank s own assets. For a sufficiently adverse sequence, a bank s net-worth is depleted up to the point in which it runs out of capital to finance future losses under the same market conditions. In such states, banks cut-back on lending, requiring a larger discount between the expected value of assets and the actual amounts transferred in exchange. These phenomena exacerbate adverse-selection leading to larger contractions in volumes of intermediation and further increases in financing premia. Depending on the how the quality of collateral responds to changes in prices, the profitability from financial intermediation may decrease in spite of higher premia. Low profitability in the financial industry precludes the external recapitalization of the financial sector. Instead, the financial system is recapitalized only through retained earnings. Yet, this process is slow since volumes of intermediation and expected profits are low to begin with. The paper explains that these u-shaped returns to intermediation are a distinctive feature of the asymmetric information component of the model. Figure 1 suggests that a similar mechanism could have been at play during the financial crisis. The top left panel plots the evolution of tangible net-worth (with and without TARP injections) during the last decade for a group of selected Bank Holding Companies. The figure shows that this variable suffered a deviation away from trend in the quarters prior to the recession (second gray shaded area). The bottom left panel, describes the behavior of the nominal stock of capital in the U.S., which declines during the recession, a symptom of the decline in economic activity. The top right panel shows that return to bank equity (ROE) was very stable during the period prior to the crisis. Not only did bank ROE decline during the recession but it has been persistently below historical averages since. The bottom right panel shows the behavior of bank dividends and equity injections. Although issuance activity per unit of equity was high in comparison to historical records, it remained far from replenishing bank equity. 5 The paper resolves several technical and empirical challenges which may be useful in other applications. On the technical side, the model has features asymmetric information, capacity constraints and aggregate shocks simultaneously. 6 I developed an algorithm that allows the computation of global solution to a model with these characteristics. In spite of these difficulties, it allows one to compute equilibria in a few seconds. On the empirical side, I reconstruct a set of aggregate banking time series to correct for institutional changes and isolate TARP injections. 7 I use the solution method and these reconstructed time series to calibrate the model and provide rough measures of the effects of dividend taxes and capital requirements in the financial system. 5 The figure reports reconstructed time series of banking indicators based on the empirical strategy described in section 6. 6 The model also features heterogeneous agents but it allows for aggregation. 7 Since the beginning of the crisis, many non-bank institutions effectively became Bank Holding Companies. For example, the bank equity time series reported in the Flow of Funds has a spurious jump because Goldman Sachs converted into a Bank Holding Company. 4

5 Figure 1: U.S. Banking Indicators and Capital Accumulation. The rest of the paper is organized as follows. The following section discusses how the paper relates to the literature. Section 2 introduces the model. Section 3 provides banking theory interpretation of the model. Section 4 characterizes equilibria. Section 5 presents two examples in closed form solution with the purpose of underscoring the role of asymmetric information in this economy. Section 6 describes the empirical strategy used to calibrate the model and presents the results to the basic quantitative exercise. Section 7 introduces several policy extensions. I conclude in section 9. The computational algorithm, proofs, and data definitions are relegated to the appendix. 1.1 Relation to the literature Banker net-worth. As portrayed by Figure 1, the behavior of the balance-sheet of the financial sector seems to have played a major role as a driving factor during the Great Recession. The literature explains the relevance of the financial system s net asset position for the efficient allocation of resources srom several perspectives. This paper is closely related to other theories in which the financial system s net-worth reduces agency costs or relaxes leverage constraints that interrupt in financial intermediation. 8 Holmstrom and Tirole (1997) are 8 Following Diamond and Dybvig (1983), another stream of research has emphasized the role of the financial sector s capital composition for the occurrence of bank runs caused by maturity mismatches. In these models, financial crises arise when an 5

6 among the first to introduce agency costs into financial intermediation intermediaries. The financial sector s net-worth reduces agency costs because bankers have more incentives to increase the return on loans when they have more to loose if they don t. Several recent studies have introduced a financial sector with these features into quantitative business cycle model to shed some light about the impact of several policies that are being executed throughout the world. For example, Martnez-Miera and Suarez (211) incorporate this feature into a macro model where bank net-worth affects risk and correlation among across loans. This model is used to study capital requirements. Other important references are Gertler and Karadi (29) or Gertler and Kiyotaki (21). There, bank capital is important because banks face a limited enforcement problem in the spirit of the earlier financial accelerator models in Bernanke and Gertler (1989) and Bernanke et al. (1996). These models are used to study the benefits of government equity injections that intend to recapitalize FIs. As here, the intermediaries in Brunnermeier and Sannikov (29) and He and Krishnamurthy (28) also face a limited liability constraint also (which can be obtained endogenously from a contracting problem) and their models are used to explain the impact of financial shocks are amplified by fire-sale spirals. Fire-sales occur when banks must meet obligations liquidating capital whose price may fall rapidly if large quantities of it are liquidated at the same time. 9, 1 This paper and the literature share in common that financial system s net-worth is also key to allocate resources efficiently. The novelty here is that systematic risk faced by the financial system is exacerbated by asymmetric information. In terms of the incentives to recapitalize banks, asymmetric information yields the opposite implication than moral-hazard. With moral hazard only, as net-worth is lost, agency costs increase so the marginal value of bank equity is greater in times of crisis. If in those models one were to allow inside equity injections, one could expect these to be a counterbalancing stabilizing force in times of crisis. Here, in contrast, adverse selection reduces the profitability of financial equity which is why, ultimately, the system is not recapitalized. The limited liability constraint introduced in this paper is also responsible of causing an externality. In that sense, it is also similar to Lorenzoni (28). That paper combines some features of the literature of bank runs with limited commitment in common to this paper. That paper identifies a fire-sale externalities that stems from excessive risky intermediation (with relation to a social optimal). The externality here stems from the bankers not internalizing that by leveraging excessively, they may run out of capital in future periods while causing severe adverse selection effects. Asymmetric information. This paper also follows a recent line of general equilibrium models that introduce adverse selection into capital markets. Carlstrom and Fuerst (1997) introduced private information in the return to investment in the spirit of Stiglitz and Weiss (1981) into general equilibrium to explain credit rationing during the cycle. 11 Eisfeldt (24) studies a general equilibrium model where, like here, the quality exogenous shocks to the demand and supply of short run funds affects financial institutions. This literature underscores that the availability of funds (for the whole financial system) determines the effectiveness of internal insurance mechanisms against idiosyncratic shocks. From this perspective, the financial system s asset position because it can discourage bank-run phenomena caused by fear of reduction on bank solvency. Other seminal contributions in this area include Holmstrom and Tirole (1998) or Allen and Gale (1998). Bolton et al. (21) relate this literature with asymmetric information. See Sargent (211) for a recent survey. 9 Similar feedback loops from financial intermediary capital to the value of capital occur in models such as Gromb and Vayanos (22) and Brunnermeier and Pedersen (29) because they introduce features that make assets less valuable when there is less intermediation. Vayanos and Wang (211) introduce asymmetric information into this framework 1 Fire-sales were first studied by Shleifer and Vishny (1992). Diamond and Rajan study how strategic behavior by banks may cause lead them to reduce lending to exploit future potential fire-sales by competing institutions. 11 Martin (29) compares pooling with separating equilibria in a similar context in which the quantity of collateral interest rates are used as screening devices. 6

7 of existing assets is private information. In this environment, assets are sold under asymmetric information for risk-sharing motives. That paper shows how asymmetric information limits risk-sharing. Bigio (21) and Kurlat (29) study models in which assets are sold under asymmetric information with the objective of relaxing financial constraints. These papers explain how shocks that exacerbate adverse selection can generate recessions. In the present paper, crisis are persistent because the financial sector may take time to recover. Other models that study lemons markets, such as Hendel and Lizzeri (1999), Kurlat (29) or Daley and Green, also obtain persistent effects through an endogenous mechanism but in all of these cases, this generated by a learning-by-holding dynamics. A formal model of the interaction between financial intermediary capital and asymmetric information in general equilibrium in a contribution of this paper. As a concept, this idea is new for the macroeconomic literature, but there has been several decades of research on how limited liability constraints affect insurance markets. Liability and catastrophe insurance, for example, share a common striking feature with financial markets: crises in this sector are recurrent, are characterized by large swings in insurance premia and by falls in the volume of issued policies. In parallel to the crisis, these events typically followed episodes of large negative profits for insurance firms, suggesting the hypothesis that limited liability constraints where relevant. 12 Gorton and Metrick (21) documents a similar pattern in the collapse of asset backed security markets. The relationship between financial and insurance market crises is not surprising if one thinks of the institutions in either sector as a large clearing houses that pool risks together (funding risky loans, or offering insurance against risky events) and funds itself by issuing liabilities backed by these pools (deposits in the case of banks, and bonds in the case of insurance companies). The modeling choices combine elements from different papers. The balance sheet of banks studied in this paper resemble those in O Hara (1983) (or, actually, the insurance companies studied by Winter (1991a)). The need for trade is follows from Kiyotaki and Moore (28) because not all agents have access to investment technologies. Asymetric information in asset qualities is introduced in a similar way as in Bigio (21). Empirical Literature. In fact, during the recent crisis, authors such as Brunnermeier (29), Krishnamurthy (29) or Gorton and Metrick (21) suggest that both asymmetric information and bank losses played a major role in the transmission of the crisis. Ivashina and Scharfstein (21) provide evidence on the decline in volumes of financial intermediation and increases premia during the crisis. Acharya et al. (21) provide a detailed description of bank equity injections and dividends during the crisis that suggests that banks were not being recapitalized substantially during the crisis. 2 Model 2.1 Environment The model is formulated in discrete time with an infinite horizon. There are two goods: a perishable consumption good (the numeraire) and capital. Every period is divided into two stages, s {1, 2}. There are two aggregate shocks, a TFP shock A t A and a shock φ t Φ {φ 1, φ 2,..., φ N } that determines the distribution of capital quality. (A t, φ t ) form a joint Markov process that evolves according to a transition probability χ : (A Φ) (A Φ) [, 1] with the standard assumptions. 12 See Winter (1991b) for a survey on the liability insurance market. Gron (1994a) and Gron (1994b) provide empirical investigations that tested the tested the implications of the insurance sector s net worth on insurance premiums and volumes for this sector of the insurance industry. Duffie (21) and Saunders and Cornett (21) discuss this behavior in catastrophe insurance markets. 7

8 Notation. I use the notation x t,s to refer to the value of a variable x in period t stage s when the variable changes values between stages. Otherwise, if the variable remains constant throughout the period, I use the time subscript only. Demography. There are two populations of agents: entrepreneurs and bankers. Each population has a unit mass but bankers are assumed to be bigger in a sense to be clear below. bankers face an exogenous constant probability of exit. When an banker exits, he is immediately replaced by a newborn banker. The purpose of introducing stochastic survival is to obtain analytic examples but bankers are treated as long-lived in the numerical exercises. Entrepreneurs. Entrepreneurs are identified with a number z [, 1] and carry their capital stock k t (z) as their individual state variable. At the beginning of the first stage, entrepreneurs are randomly segmented into two groups: capital goods producers and consumption goods producers. I also refer to these types as k-producers and c-producers. Entrepreneurs become capital good producers with a probability π independent of time and z. As a consequence, every period, there are masses π of k-producers and 1 π of c-producers. Capital goods producers have access to an investment technology that allows them to create new capital units using consumption goods but cannot use their capital stock for the production of consumption goods. In contrast, consumption goods producers can use capital to produce consumption goods, but lack the possibility of augmenting their capital stock building capital directly. This segmentation induces a need for trade: k- producers have access to the investment technology but lack the input to operate it. c-producers produce consumption goods but lack access to an investment technology that allow them to accumulate capital. Bankers provide intermediation services between these two groups. Randomizing across activities is introduced for tractability since, otherwise, the relative wealth of each group of entrepreneur would become a state variable. Entrepreneurs have log-preferences over consumption streams and evaluate these according to an expected utility criterion: E β t log (c t ) t where c t is their consumption. Bankers. Bankers are identified by some j [, 1]. At the beginning of every period, they receive a large exogenous endowment of consumption goods ē t (j). In addition, they carry a stock of consumption goods n t,1 (j) stored within banks of their property. n t,1 (j) is the banker s individual state variable which is interpreted as the bank s net-worth. During the first stage, bankers can alter the composition of their financial wealth by injecting equity from their personal endowment to their banks or do the opposite paying dividends. After equity injections and dividends, their banks net-worth evolves from n t,1 (j) to n t,2 (j). Bankers participate in capital markets purchasing capital units from k-producers in the first stage and reselling them to c-producers during the second stage. They purchase capital by issuing tradeable riskless IOUs that entitle the holder to a unit of consumption in the second stage. 13 For now, I implicitly assume that managerial incentives cannot be met, so bankers face no strategic decisions between holding on to capital purchases or selling immediately As discussed in a later section, this institutional environment can be adapted to resemble standard bank practices. Introducing them here as dealers is done for pedagogical reasons. 14 The model can be extended in this direction. The outcome involves a non-trivial fire-sale behavior for financial firms which in turn may have implications for the behavior of the financial system. Fire-sales may induce a different type of externality than the one caused by adverse-selection which is the focus of this paper. Diamond-Rajan, for example. Note: Douglas Gale suggested 8

9 Since φ t arrives between stages, the value of the pool of purchased capital is random. This randomness makes financial intermediation risky. In particular, the banker may suffer losses if his purchase cost (issued IOUs) exceeds the value of his capital purchases. When the banker experiences financial losses, he is forced to draw funds from his bank s equity in order to settle these claims. In principle, financial losses could be financed via the bankers personal endowment. Instead, I assume, as in real world banks, that financial intermediation is subject to a limited liability constraint (LLC) such that the bankers personal endowment is not liable to his banks losses. 15 This condition implies that losses from financial intermediation cannot exceed their bank s net-worth, n t,2 (j). As a consequence of the LLC, the bank s net-worth will affect the bankers capacity to engage in more or less transactions. In that sense, a bank s net-worth acts as a cushion to absorb potential losses. For this reason, there is a distinction between the banker s personal endowment and his bank s equity: net-worth relaxes the LLC constraint whereas the personal endowment does not. The LLC can be obtained endogenously as a result of a commitment problem on the side of Bankers. If bankers cannot be forced to inject equity into their financial firms to cover losses, this constraint would show up as an ex-post incentive compatibility condition. 16 Otherwise, one can take the LLC as an institutional constraint. Upon an exit, bankers sell their bank to an entrant banker. The presence of dividend taxes implies that entrant bankers will rather buy a bank from a an exiting banker than start a new bank on their own. The exit probability, ρ [, 1], is constant. When, ρ =, bankers are characterized by an infinite horizon problem. When ρ = 1, one can solve the model analytically. Bankers have linear preferences over consumption streams and evaluate these according to an expected utility criterion: E t (β f ) t c t where c t is their consumption. Technology. The investment technology transforms one unit of consumption into an efficiency unit of capital available for production the following period. A c-producer that holds a capital stock of k t (z), will produce consumption goods according to a linear technology A t k t (z). In addition, bankers have access to a storage technology that transforms 1 unit of consumption into R b units of consumption. In principle, one can think of this technology as a risk-less government bond financed through lump-sum taxes but I leave the interpretation open. I assume, R b is smaller than the expected return to capital. This assumption implies that it is never efficient to have capital in hands of the financial sector and in a frictionless economy, the financial sector would disappear. I also assume that β f R b < 1 so that bankers would liquidate their banks if they are not compensated with positive profits. Capital. At the beginning of every period, capital held by each entrepreneur is divisible into a continuum of units. Each unit is identified by a quality ω [, 1]. There is a deterministic increasing differentiable function λ(ω) : [, 1] R + that determines the corresponding efficiency units that will evolve from an ω-quality unit. λ(ω) can interpreted as a quality dependent depreciation rate (but they are not restricted to be less than 1). In addition, there is a set of measures over qualities that depends on the realization of φ t. In particular, exploring this direction. 15 Scotland..banking era. 16 Otherwise, this constrained can be obtained as a high penalty for defaulting in IOU s. Like loss in reputation for example. 9

10 at t, these measures are defined by an absolutely continuous function f φt : [, 1] R +. which, in turn, is a function of φ t. Between periods, each piece is transformed into future efficiency units by scaling pieces by λ (ω) f φ (ω). Thus, λ (ω) f φ (ω) k (ω) efficiency units remain from an ω quality next period. To simplify the analysis, I assume that these measures constant across entrepreneurs but change through time depending on the realizations of φ t. Once capital units are scaled by their corresponding efficiency units, pieces are merged back into a single homogeneous capital unit. This unit will be divided again in the same way in the following period and the process is repeated indefinitely. Thus, by the end of the second stage, the effective capital stock that remains from an original stock k is k λ (ω) f φ (ω) dω. Distinguishing between (λ (ω) and f φ (ω)) only provides an interpretation to these shocks. For allocations, only their product matters. Producers won t necessarily hold on to all of their capital units. On the contrary, they may choose to sell particular units during the first stage to obtain consumption goods. These decisions are summarized by an indicator function I (ω) : [, 1] {, 1} that takes a value of 1 when units of quality ω are sold. Because each quality has measure, the restriction to all-or-nothing sales is without loss of generality. When choosing I(ω), the entrepreneur transfers k I (ω) dω units of capital to a bank. These units evolve into k λ (ω) I (ω) f φt (ω) dω efficiency units of t+1 capital. Simple accounting shows that the efficiency units that remain with the entrepreneur are k λ (ω) [1 I (ω)] f φ (ω) dω. Taking into account possible investments and purchases of capital, the k-producer s capital stock evolves according to: k = i + k b + k λ (ω) [1 I (ω)] f φ (ω) dω, (1) where i are the capital units created by exploiting the investment technology (when available) and k b are purchases of efficiency units of t+1 capital. I impose some structure on the quality distributions {f φ }: Assumption 1 The set {f φ } φ Φ satisfies that E φ [λ (ω) ω < ω ] I [ω<ω ]λ (ω) f φ (ω) dω is weakly decreasing in φ for any ω. This condition states that the average efficiency unit that remains from qualities under some ω is a decreasing function of φ. That is, for any threshold quality, the average quality under that threshold is worse as φ is larger. In equilibrium, bankers will always purchase qualities under some threshold, so this assumption implies that the amount of effective capital held by bankers decreases with φ regardless of the amounts of capital purchased. The assumption can be interpreted as conditional expectations satisfying first order dominance condition relation and provides an ordering to the shocks. 17 Let λ (X) E φ [λ (ω)] be the unconditional effective depreciation in state X. The setup is general to accommodate to two polar cases of particular interest. The first case is when all qualities are the same conditional on a shock, λ(ω) = λ, but efficiency units are decreasing in φ (f φ integrates to smaller number as φ is larger). 18 In this case, φ has the effects of permanent capital depreciation shock that induces risk into financial intermediation. This polar case is used to study risky financial intermediation without asymmetric information Note that Assumption 1 is neither a general nor a particular case of first or second order stochastic dominance. The standard definitions of stochastic dominance are related to properties of the distribution of qualities, f φ. Instead, here, the condition refers to the properties of functions that are the conditional expectation under a threshold quality (where the threshold is the argument of the function). The condition in Assumption 1 implies that conditional expectations (and not distributions) satisfy first order dominance. 18 For this polar cases, one can allow f φ to be improper distributions (so they don t necessarily integrate to 1). 19 This is the same shock studied in Brunnermeier and Sannikov (29) or Gertler and Karadi (29). 1

11 The second case of interest is when λ (ω) f φ (ω) dω = λ (X) for any φ Φ. This condition states that φ is mean preserving (MPS). This implies that the production possibility of the economy is independent of φ. Under the information structure described below, the second case corresponds to an environment with ex-ante adverse selection. This feature has the connotation that if φ has any effect on allocations, it is because these shocks affect equilibrium but not the feasible set of allocations, as explained later. Information. There are two endogenous aggregate states, K t = k t (z) dz and N t, s = n t, s (j) dj, the capital stock held by entrepreneurs and the net-worth of the financial sector respectively. It will be shown that in order to characterize equilibria, it is only necessary to keep track of their ratio κ t,s N t,s /K t as a unique state variable. Thus, the state of this economy is X t,1 = {A t, φ t 1, κ t,1 } X A Φ K and X t,2 = {A t, φ t, κ t,2 } X A Φ K. X t,s is common knowledge. On the other hand, the ω quality behind a capital unit is only known to its owner. This means that financial firms can observe only the volume of capital units purchased k I (ω) dω but ignore, k λ (ω) I (ω) f φt (ω)dω, the efficiency units that remain from this purchase. In contrast, k-producers also face uncertainty about the realization of φ t but know exactly which ω units are being sold. Thus, if he knew φ t, the producer could know the value of λ (ω) I (ω) f φt (ω)dω whereas the banker could only infer this value. For tractability, I assume that the entrepreneur s type is also observable. This assumption is enough to ensure that, in equilibrium, c-producers are excluded from selling capital. Bankers are informed about their exit, at the beginning of the second stage. Markets. Markets are incomplete. There exists no insurance against type-risk and k-producers cannot sell claims against new investment projects. Instead, the only possible transaction are purchases and sales of capital from or to bankers. This assumption is not modelled formaly but is motivated by the assumption that financial firms are larger than producers. In this context, being larger means that bankers can engage in many more transactions and, therefore, have the advantage of exploiting the law of large numbers to dilute the idiosyncratic risk faced when participating in a small number of transactions (under asymmetric information). Thus, implicitly, I assume that banks fully diversify financial contracts (or mutually insure against idiosyncratic risk). Consequently, profits/losses from financial intermediation are perfectly correlated across bankers. There are two markets for capital. The first is the market for capital units sold by k-producers and bought by banks and where units are sold under asymmetric information. This market opens during the first stage and satisfies the following assumption: Assumption 2 Banks are competitive and capital markets are anonymous and non-exclusive. Without additional instruments to screen, anonymity is key to guarantee that the market in the first stage is a pooling market. Without anonymity, bankers could pay a different price depending on the capital traded by the entrepreneur. With exclusivity, bankers could use dynamic incentives as screening devices. Assumption 2, therefore, implies that because quantity cannot be used as a screening device.[guerrieri-shimer.] Hence, there will be a unique pooling price p t in the first stage market for capital. I refer to the market in the first stage as the pooling market. The second market is the one in which bankers sell back the purchased units while revealing the actual efficiency units behind capital. In essence, this is a market for t+1 capital units traded in period t. This market opens during the second stage and clears at a price q t. I refer to the market in the second stage as the resale market. 11

12 Resale Market Pooling Market Trade X t+1,1 updated X t,1 (A t, φ t 1, κ t,1 ) dividend/equity φ t realized consumption/investment X t,2 (A t, φ t, κ t,2 ) Figure 2: Timing Timing. (1) At the beginning of the period, X t,1 is realized and k-producers choose I (ω). Bankers choose an amount of capital purchases, equity injections and dividend pay-outs. (2) During the second stage, φ t is realized and X t,2 is updated. Bankers learn the average quality of the pool purchased capital and resell these pools as homogeneous units of t + 1 capital. k-producers and c-producers simulateneously choose over consumption and purchases of capital and k-producers make their investment decisions. By the end of the period, bankers redeem all issued IOUs and realize profits from intermediation. The timing of the model is summarized by Figure 2. The following sections describes the problem faced by the agents in this economy and the corresponding market clearing conditions that define equilibria. This economy has a recursive representation so from now on I drop time subscripts. I use x to denote the value of a variable x in the subsequent stage. 2.2 First Stage Problems k-producer s first stage. During the first stage, k-producer s enter the period with a capital stock k. At this stage, they only choose which capital units to sell in exchange for consumption goods: Problem 1 (k-producer s s=1 problem) The k-producer s problem at the first stage is: s.t. x = pk [ V1 k (k, X) = max E V k ( 2 k ( φ ), x, X ) X] I(ω) {,1} 1 I (ω) dω and k ( φ ) = k λ (ω) [1 I (ω)] f φ (ω) dω The first equation is the producer s budget constraint were x represents the quantity of consumption goods available during the second stage. x is obtained by selling capital in the pooling market. The second equation accounts for the capital kept by the entrepreneur which depends on the qualities sold and φ. This term corresponds is the last term in equation (1). The solution to this problem determines a supply schedule for capital units in the pooling market. c-producer s first stage. Since c-producers are excluded from capital markets during the first stage, their value function is the expected value of their second stage value function: 12

13 Problem 2 (c-producer s s=1 problem) The c-producer s problem at the first stage is: 1 (k, X) = E [ V2 c ( k ( φ ), x, X ) X ] V c s.t. x = Ak and k ( φ ) = k λ (ω) f φ (ω) dω Bankers first stage. A banker enters the period with n consumption goods stored in his bank and ē as personal endowment. Bankers will choose an amount of his endowment e as equity injections to his bank. He can do the opposite transferring d consumption units as dividend payoffs to consume in the current period after paying a linear dividend tax τ. 2 Equity injections and dividends are limited by their sources: e ([, ē]) and d [, n]. The banker s consumption during the period is c = (ē e)+(1 τ) d and his bank s net-worth in the next stage is n = n + e d. The presence of a dividend tax is important because it affect the dynamics of the net-worth of financial institutions. Let Q be the quantity of capital purchased by the banker. He purchases this amount by issuing pq in marketable IOUs (inside liquidity). During the second stage, the value his capital purchases is qe φ [λ (ω) X]. E φ [λ (ω) X], the value of his the expected depreciation rate under the distribution f φ (consistent with the sales decisions I (ω)). The LLC states that the amount of issued IOUs cannot exceed the bank s net-worth plus the value of its capital: pq qe φ [λ (ω) X] Q + n for any ( X, X ) X X Let Π (X, X ) qe φ [λ (ω) X] p be the bankers profits per unit of capital purchased. Π (X, X ) is a function of X and X since, as we shall see, q is a function of both X and X. The bankers problem is summarized by, Problem 3 The Bankers problem at the first stage is: [ V f 1 f (n, X) = max c + E V f 2 Q,e [,ē],d [,n] (n + Π ( X, X ) ] Q, X ) X s.t. Π ( X, X ) Q n, X (2) c = (ē e) + (1 τ) d n = n + e d The first constraint of this problem is the LLC, the second is the bankers budget constraint and the last is the evolution of his bank s balance net-worth. 2.3 Second Stage Problems Capital good producer s second stage. During the first stage, k-producers sell part of their capital stock in exchange for x consumption goods brought into the following stage. Their individual state vector is given by x and k, the capital units ready for production the following. Given this state, they solve,, 2 A distinction between cost of equity injections and dividend taxes is common in the dynamic corporate finance literature. See for example Hennessy and Whited (25) or Palazzo (21) among others. In this environment, only their ratio of the cost of equity and dividend taxes matters, so I normalize the tax rate to account for this differences. 13

14 Problem 4 (i-entrepreneur s s=2 problem) The k-producer s problem at the second stage is: [ V2 k (k, x, X) = max log (c) + βe V k ( c,i,k b 1 k, X ) X], j {i, p} c + i + qk b = x and k = k b + i + k The budget constraint says that the k-producer uses x to consume c, invest i, or purchase k b capital at price q. The capital accumulation equation is consistent with (1) since k already incorporates sales and depreciation (accounted for in the previous stage). Consumption good producer s second stage. except that he is restricted to set i : The c-producer s problem is identical to the k-producer s Problem 5 (p-entrepreneurs s=2 problem) The c-producer s problem at the second stage is: [ V p 2 (k, x, X) = max log (c) + βe V j ( c,i,k b 1 k, X ) X], j {i, p} c + i + qk b = x and k = k b + i + k Bankers second stage. A bankers only action during the second stage consists on reselling all the capital units purchased during the first [ stage while revealing ] their actual depreciation. Thus, their value function in this stage is V f 2 (n, X) = βf E V f 1 (Rb n, X ) X if they remain in the industry or V f 2 = (1 τ) βf R b n if they exit. 2.4 Market Clearing Conditions and Equilibrium Notation. I append terms like j (k, X) to variables that indicate the policy function of an entrepreneur of type j in state (k, X). I use I(ω, k, X) to refer to a k-producer s decision to sell an ω quality when his state is (k, X). Aggregation. In every period and stage, there are measures over capital holdings across the population of k and c-producers. I denote these measures by Γ k and Γ c respectively. By independence, the measures satisfy: Γ k (dk) = πk and Γ c (dk) = (1 π) K. (3) Their evolution is consistent with individual decisions and type process. In addition, there is also a distribution Λ of bankers net-worth. These measures are not state variables because the economy admits aggregation. First stage. In the first stage, market clearing requires that the demand for capital by bankers to equal the supply of capital by k-producers. This condition is given by: Q (n, X) Λ (dn) = k 1 I (ω, k, X) dωγ k (dk) Second stage. In the second stage, market clearing requires that the supply for capital by bankers be equal to the demand of purchases of efficiency units by all entrepreneurs. The demand for efficiency units by c and k-producers are respectively, D c ( X, X ) k b (x(k, X), k 14 1 λ (ω) f φ (ω) dω, X ) Γ c (dk)

15 and D k ( X, X ) k b (x (k, X), k 1 The supply of capital by bankers is inelastic and given by: S ( X, X ) E φ [λ (ω) I (ω, k, X)] λ (ω) [1 I (ω, k, X)] f φ (ω) dω, X ) Γ k (dk). Q (n, X) Λ (dn) where E φ [λ (ω) I (ω, k, X)] stands for the expected quality of capital given the policy functions of k-producers in the first stage. Market clearing requires S (X, X) = D c (X, X) + D k (X, X). Recursive competitive equilibria is defined in the following way: Definition 1 (Recursive Competitive Equilibrium) A recursive competitive equilibrium (RCE) is (1) a set of price functions, {q (X, X), p (X)}, (2) a set of policy functions for c-producers c c (x, k, X), k b,c (x, k, X), i c (x, k, X), a set of policy functions for k-producers c k (x, k, X), k b,k (x, k, X), I k (ω, k, X) and a set of policy functions for Bankers Q (n, X), e (n, X), d (n, X), (3) sets of value functions, { V1 l (k, X)} l=c,k, { V2 l { } (x, k, X)} l=c,k and Vs f (n, X), (4) a law of motion for the aggregate state X, such that for any distributions s=1,2 Γc, Γ k and Λ satisfying the consistency condition (3), the following hold: (I) The entrepreneurs policy functions are solutions to their problems taking q (X, X), p (X) and the law of motion for X as given. (II) Q (n, X), e (n.x), d (n.x) are the solutions to the Bankers problem taking as given q (X, X), p (X) and the law of motion for X as given. (III) Capital markets clear at the first stage. (IV) Capital markets clear at the second stage. (V) The law of motion X is consistent with policy functions and the transition function χ. All expectations are consistent with the law of motion and agent s policies. The definition of equilibria does not depend on the measure over asset holdings because this economy admits aggregation. There is an important detail. Nothing in the definition of equilibrium precludes multiplicity of prices. In particular, we may find two price function p (X) consistent with the definition of equilibria. In particular, in a given state, there could be two quantities ω consistent with equilibria. As prices increase, both the average quality of capital sold and the quantity increase so profits for financial firms may potentially be non-monotonic. Thus, the LLC could be satisfied with equality for multiple values of ω while yielding positive expected returns to banks. Multiplicity is a common feature in models with asymmetric information like Stiglitz and Weiss (1981), for example and although this multiplicity is interesting in itself, it is not the focus of this paper. Thus, for the rest of the paper, I introduce an equilibrium refinement: Definition 2 (Pareto un-improvable Equilibrium) A RCE is Pareto un-improvable if given the law of motion for X, X, there does not exist any p o > p (X), such that p o satisfies market clearing in the first stage, and induces a second stage market clearing price q (p, X ) that is consistent with the agents policy functions and the LLC. This refinement selects the RCE where the volume of intermediation is the largest possible. A later section shows that such equilibria, indeed, cannot be Pareto improved upon. We need to show some intermediate results first. Before proceeding to the characterization, I provide a description of alternative interpretations of the LLC constraint and financial intermediation. 15