Banking on Collateral: Wealth Distribution, Welfare and the Risk-Free Rate with Collateralized Lending

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1 Banking on Collateral: Wealth Distribution, Welfare and the Risk-Free Rate with Collateralized Lending Geoffrey Dunbar 1 February 25, 2005 Abstract In this paper, I examine the implications of collateral constraints on borrowing for the distribution of wealth, the level of inequality and the risk-free real interest rate. I develop an infinite-horizon model where collateralized lending and borrowing contracts arise. Agents in the model are dynastic and face idiosyncratic productivity shocks. They smooth consumption by investing in either risky and depreciating capital or by borrowing from and lending to banks one-period non-state contingent bonds using their capital as collateral. Numerical simulations demonstrate that collateralized lending exacerbates wealth inequality relative to autarky and that the extent of inequality is linked to the persistence of the productivity shocks. As a result, the market-clearing risk-free interest rate is lower than in an environment without collateral. In addition, the risk-free rate proxies the level of wealth inequality in the collateralized steady-state: conditional on the risk intolerance of an economy, higher levels of inequality are correlated with lower levels of the risk-free rate. The distributions of wealth in a collateralized steady-state are typically bi-modal with a large measure of poor agents. Only if shocks are small and transitory is the distribution of wealth symmetrically distributed around the mean wealth level. Unlike other Bewley-type models, collateral constraints do not imply capital over-accumulation. Capital over-accumulation occurs only where the idiosyncratic shock is both large and persistent. Finally, the model suggests that collateralized intermediation may not be welfare-improving over autarky for all parametrisations of the idiosyncratic shock. JEL classification: E43, E44, G21 Keywords: Banking; Collateral; Wealth Heterogeneity, Risk-Free Rate 1 I thank, without implicating, Gregor Smith, Allen Head, Sumon Majumdar, Huw Lloyd-Ellis, J-F Houde, Thor Koeppl, participants at the MidWest Macro Meetings at Iowa State, participants at the Canadian Economic Association Meetings and PhD seminar participants at Queen s University... This paper follows from joint work with Jason Allen and much of its spirit is due to our many discussions regarding financial intermediation. The author gratefully acknowledges financial support from SSHRC. Corresponding author: Geoffrey Dunbar, Department of Economics, Dunning Hall, Queen s University, Kingston, ON, K7L 3N6, Phone: , dunbarg@qed.econ.queensu.ca

2 1 Introduction That collateral is a central ingredient to the story of financial intermediation is, in most developed economies, a well-known fact. However, the role that collateral plays in the allocation of wealth remains, for the most part, unclear. The main question addressed by this paper is: what is the impact of collateralized lending on wealth inequality? Collateral has two key implications for the distribution of wealth: it is a type of insurance for both borrowers and lenders; and it bars agents with insufficient collateral resources from borrowing. Collateral insures borrowers ex-post by allowing them to exchange a risky asset for a risk-less obligation. In addition, collateral insures lenders ex ante by providing a enforcement mechanism for the repayment of loans. This paper characterizes the wealth distribution in a stationary equilibrium with collateral and shows that collateralized lending increases inequality relative to autarky. The story is the following. I imagine a world populated by agents who own a particular production technology which is identical across all agents. Agents in the model can be thought of as independent producers, i.e. households with a backyard technology. All agents produce an identical consumption good using the production technology. In each period, each agent suffers an idiosyncratic shock which affects the quantity of output she produces. The shock is intended to capture such vagaries as bad weather, crop disease, or illness. The shock provides agents with an explicit consumption-smoothing motive. However, lenders require an enforcement technology to ensure that borrowers will repay their obligations. Collateral suffices to ensure that intermediary contracts are enforceable. Lending and borrowing contracts are therefore two dimensional objects: an amount borrowed (lent) and an amount of capital posted as collateral. The paper is structured as follows. Section 2 describes the relation of this paper to the literature on incomplete markets and limited enforcement of contracts. Section 3 presents the model environment. Section 4 discusses the enforcement technology which arises in equilibrium. Section 6 defines the autarkic equilibrium of the model. Section 7 presents the social planning equilibrium of the model which is, naturally, the complete markets outcome. Section 8 describes the types of contracts which arise as equilibrium behavior. Section 9 defines a competitive equilibrium, where collateralized lending and borrowing contracts are traded in equilibrium. Section 10 discusses the solution concept applied to determine an (approximate) solution to the competitive equilibrium. Section 11 presents the results from different specifications of the model parameters. In particular, the sensitivity of the collateralized equilibrium is discussed. Section 12 concludes. 2

3 2 Research Context This paper intersects three strands of the literature on financial intermediation. In particular, it examines the role of financial intermediation in a similar manner to models which focus on: incomplete markets and wealth inequality; endogeneous borrowing constraints; and limited contract enforcement mechanisms. The primary motivation of this paper is to build on Huggett [17] and Aiyagari [3] by developing a heterogeneous-agents model with collateralized borrowing/lending. Several authors have constructed theoretical models where collateral arises as an optimal debt contract. Lacker [28] constructs a two-period model where output is only observable to the borrower and demonstrates that, when the borrower values collateral more than the lender, collateral ensures repayment. Kocherlakota [24] demonstrates that collateralized debt contracts are optimal in models where the ex-post value of collateral and the ex-post investment return are known only to the borrower. Geanakoplos and Zame [16] construct a two-period general equilibrium model where agents can default at any time and show that financial assets are only traded when backed by collateral. Finally, Andolafatto and Nosal [5] construct a model where agents endogenously circulate claims which are implicitly backed by collateral. The model examined in this paper can be viewed as a simplified version of the type of model examined theoretically by Araujo, Pascoa and Torres- Martinez [6], who demonstrate the existence of stationary equilibria in models with collateral constraints. This paper demonstrates that a collateral requirement restricts the amount of intermediation somewhat like endogenous solvency constraints. That is, the amount an agent may borrow is limited by the assets she may post as collateral. In this respect, the present paper is related to Kehoe and Levine [20] and Alvarez and Jermann [4] who study the effects of solvency constraints. One difference of this paper from [20] and [4] is that I do not assume that agents determine borrowing constraints to ensure that repayment is individually rational for the borrower. Kiyotaki and Moore [21], Kocherlakota [23], Lustig [30], Kubler and Schmedders [27] and Cordoba and Ripoll [11] examine the macroeconomic impacts of collateral constraints explicitly. Kiyotaki and Moore construct an infinite-horizon economy populated by two types of agents, farmers and gathers (where the nature of the economy specifies a fixed rate of interest) and demonstrate the presence of credit cycles resulting from the collateral constraints. Kocherlakota considers an open economy variant of a model with collateral constraints and shows that the amplification of shocks inherent in collateral constraints depends on the parametrisation of the economy. Cordoba 3

4 and Ripoll examine the extent to which collateral acts as an amplification mechanism for exogenous shocks. They find that the amplification effects of collateral are typically small. Lustig examines a model of bankruptcy where agents post shares in Lucas-trees as collateral for state-contingent loans. He demonstrates that the set of equilibria are constrained relative to those of Alvarez and Jermann. Kubler and Schmedders examine a similar model and characterize the wealth distribution when there are two agents. My paper extends the literature by examining the consequences of collateralized lending on the distribution of wealth in a general equilibrium with many agents. Finally, bankruptcy rules have been examined in a number of papers, including Livshits, MacGee and Tertilt [29], Chatterjee, Corbae, Nakajima and Rios-Rull [9] and Zame [36] among others. Although bankruptcy is possible in my model, I abstract from modeling different bankruptcy rules and instead assume that assets offered as collateral are seized by creditors once repayment stops. Once collateral is seized any remaining debt is discharged by the lender. 3 General Model In this section, a simple heterogeneous agent model is developed where collateralized lending arises in equilibrium. Agents are differentiated by a shock process in their production function. Apart from the shock, the production technology is identical across all agents. As a result, considerations of human capital, monopolistic competition, or labor supply are absent. 3.1 Preferences and Technology There exists a continuum of agents who live forever. The population is constant and normalized to unity. Agents have the following preferences: U = E β t u(c i t); (1) t=0 where U is the discounted lifetime utility of agent i, β (0, 1) is the discount factor and u(c i t) is the period utility of an agent of type i in period t with consumption c i t R +. The function u(c i t) is assumed to satisfy u (c i t) > 0, u (c i t) < 0 and u (0) =. Each agent is initially endowed with k0 i units of the capital stock and a production technology: y i t = η i t(k i t) α (2) where y i t R + is the output of the consumption good by agent i in period t, k i t R + is the capital stock used in production by agent i in period t, α (0, 1) is the productivity of capital 4

5 and ηt i N = {η, η}; η > 1 > η, is a mean zero, idiosyncratic, stochastic shock to agent i in period t. The shock, ηt, i follows a Markov process with stationary transition probabilities π(η η) = P rob(ηt+1 i = η ηt i = η) > 0 for η, η N. The shock is intended to capture idiosyncratic elements of production, such as: drought, illness, median age, median experience, time constraints, etc. 1 The shock is assumed to be private information. Agents realize their shock and output simultaneously. I assume the initial distribution of the shock is such that η0 i = η for exactly half of the agents, while the remaining agents have ηi 0 = η. Finally, capital is assumed to depreciate by a fraction δ [0, 1) each period. 3.2 Contracting The idiosyncratic shock implies that intertemporal risk-sharing can be Pareto-improving; hence contracting between agents is advantageous. The assumptions specified in this section are sufficient for collateralized lending and borrowing contracts to arise. Assumption 1 An agent s shock, η, is private information. Assumption 2 Capital stock holdings, kt, i and preferences are common knowledge (or costlessly verifiable). Assumption 3 Contracts which specify a transfer, b i t, from (to) an agent i in period t at a cost q i t 1 bi t in period t 1 may be written costlessly. knowledge and contracts may be breached. Contracts are assumed to be common Assumption 4 Agents may contract once per period through an intermediary (henceforth referred to as the bank). There is assumed to be free entry to banking. Assumption 1 implies that agents cannot be differentiated by observation. In particular, agents with a high shock may consume their additional output without public observation. Moreover, unlike a two-person model, no agent can infer the output (or shock) of any other agent given the assumption of an infinity of agents.assumption 3 characterizes the nature of the bargaining problem. Assumption 4 rules out some contracting equilibria in that each agent can only bargain once per period. 1 In an infinite-horizon model, age and experience are clearly identical across agents. However, infinite-horizon agents may be considered also as dynastic households. Thus, the shock process also can represent the proportion of productive members of the household at any given point in time. 5

6 4 Enforcement In this section, I describe the enforcement mechanism for contracts. That an enforcement technology is required is straightforward to establish. Proposition Private contracts will not arise without an enforcement mechanism for those contracts. Proof: If there is no cost to a borrower to forfeit payment at at time t then she will default since the continuation payoffs are (weakly) greater after an agent defaults a debt than when an agent repays. Any enforcement mechanism used by banks to enforce contracts must be binding in the sense that the enforcement mechanism must be robust to free-entry. That is, no borrower can avoid enforcement costs by banking with an entrant bank. One enforcement mechanism which is robust to free entry is a collateral mechanism. I assume that a collateral mechanism is used by banks to enforcement contracts. Assumption 5 Banks use a collateral mechanism to enforce contracts where a collateral mechanism is defined as the exchange of a claim which transfers ownership of an asset, a i t+1 [0, (1 δ)k i t], from the borrower i to the lender conditional on the breach of a contract in period t + 1. The collateral claim is assumed to be costless to write and the seizure of collateral, conditional on the breach of a contract, is assumed to be costless and immediate. Together with Assumption 2, this implies collateral cannot be privately consumed by the borrower. However, collateral may be used by agent i for production. In addition, only capital currently held by the agent in period t can be pledged as collateral for period t + 1. That is, agents cannot commit to capital investment. 2 I note that collateral constraints are different from other constraints considered in the literature. In particular, [20] and [4] among others, have studied economies where the individual borrowing constraints are determined by individual rationality constraints which are, in effect, censure mechanisms. A censure mechanism in the context of this paper would be a mechanism which excludes 2 The restriction that only capital currently held by the agent can be pledged as collateral prevents a Ponzi style game from arising. If agents could borrow against future but unenforceable capital investment, agents have an incentive to continually promise more capital investment (without necessarily following through) in order to pay-off current period debts while banks have a similar incentive to believe such promises in order to remain solvent. Such an equilibrium would be, in effect, a type of Ponzi scheme. 6

7 an agent from borrowing or lending for a period of time, ν. In relation to the model studied in this paper, censure and collateral have contrasting implications for borrowing. First, no censure contract which specifies a period of censure ν > 0 can be constructed as a collateral contract. Clearly, for censure to impose an enforcement cost then the defaulting agent must have lower expected utility when censured than she would otherwise. However, the agent who invokes censure also suffers a cost in addition to the cost she suffers when the repayment of her loan is forfeit. The additional cost results from the inability to intermediate with the censured agent. With collateral, she does not incur the additional cost and, generally, reaps a positive return from the value of the collected collateral. Second, censure mechanisms imply less (more) borrowing than collateral mechanisms for poor (rich) agents. For example, imagine an agent with very low capital holdings k 0 such that the costs of autarky are relatively large. Then, for this agent, the value of intermediation is relatively high and thus the borrowing limit where she is indifferent between default and repayment is relatively high. However, under a collateral mechanism, she has very low access to intermediation because she has little capital to post as collateral. Hence, the distributional effects of collateral mechanisms will be quite different from those of censure mechanisms. The final assumptions are largely self-explanatory. Assumption 6 The bank proposes (the terms of) contracts. Assumption 7 The bank can disburse received collateral. Without Assumption 7, it may not be feasible for a bank to write collateral contracts since there may be a positive probability that a bank acquires collateral in a given period. Without a mechanism to disburse collateral, the bank is acquiring an asset to which it ascribes no value. Hence, for collateral contracts to be feasible for a bank, there must be a mechanism by which the bank can transform collateral to output or transfer collateral to creditors in lieu of output claims. 4.1 Banking Sector Banks are assumed to be profit-maximizing and the banking sector is assumed to have free entry, although in equilibrium I assume there exists only one bank which earns zero profits. The bank s profit may be generally written as follows. θ t = i b i t + i q i tb i t+1 + i e i t (3) 7

8 where θ t is profit in period t, b i t is the amount borrowed (or lent) to agent i in period t, qt i is the price of borrowing and lending in period t for agent i and e i t is the net profit from collateral in period t from agent i (e.g. the amount of collateral collected in excess of the face value of bonds defaulted). For the moment, let b > be a lower bound on bond holdings, b < be an upper bound and define S as: [b, b] N. Further, let B S be the Borel σ-algebra and define µ as a probability measure defined on (S, B S ) with assumed transition function P : S B S. In a slight abuse of notation, I write P t as the transition function in period t such that the distribution of agents in period t, µ t = S P tdµ t 1. In any stationary equilibrium, P t = P t 1 = P t. In addition, given agent s choices of beginning-of-period bonds and capital stock, b i t and kt, i the collateral requirements a i t+1 and end-of-period bonds and capital stock, b i t+1 and ki t+1, then P t is a known function. Hence, given Assumptions 1-4 and a law of large numbers, P t is a known function for all t. Let ˆθ t be the expected profit in period t: ˆθt = S θ tdµ t. Since there is free entry into banking, a bank cannot make any expected profit ˆθ t > 0. In addition, I assume that banks face a solvency requirement such that ˆθ t 0 t, since with free entry a bank cannot offset a loss with future profit. Hence, whenever ˆθ t < 0 then the bank declares bankruptcy and is replaced by an entrant. Bankruptcy arises because in any period where ˆθ t < 0 the net obligations of the bank are greater than its net assets. I assume that once bankrupt a bank ceases it s operations and simply disburses any remaining assets (loans or their collateral claims) on a pro-rated basis to any remaining creditors (i.e. those agents with b i t > 0). Proposition q i t = q t i. Proof: Given the environment, a bank could write contracts which are contingent on an agent s capital holdings, i.e. qt i = q(kt), i where q(kt) i is some pricing kernel. However, in equilibrium, it must be that qt i = q t, i. To see this, let q represent the price charged to an agent who borrows and q represent the price offered to an agent who lends. 1. Suppose q i t < qj t, Then an entrant bank can offer contracts qi t + ɛi and q j t + ɛj, ɛ j < 0 < ɛ i which both agent i and agent j prefer. That is, if q i t < qj t < 1 then the cost of borrowing for agent i is lower and the return for agent j is higher. If 1 < q i t < qj t then the return to borrowing for agent i is higher and the cost of lending for agent j is lower. Moreover,ɛ i and ɛ j may be chosen such that i bi t+1 + j bj t+1 = 0, i.e. such that market clearing is unaffected. Hence q i t < qj t cannot be an equilibrium. 8

9 2. Suppose q i t > qj t. Then an entrant bank can offer contracts qi t +ɛi and q j t + ɛj, ɛ j < 0 < ɛ i which both agent i and agent j prefer. That is, the borrowing agent i receives more in period t and the lending agent j is promised more in period t + 1. With free-entry and price-competition, then the sequence of prices diverge such that: q i t and qj t 0. It is trivial that market clearing in period t cannot hold and hence a stationary equilibrium cannot be sustained. Hence q i t = qj t = q t i, t.. In the next proposition, I demonstrate that the zero expected profit on all contracts implies zero expected profit on each contract. Proposition ˆθ h = 0 i bi h = i q hb i h+1 = i ei h = 0 h t. Proof: In period 0, by construction, i q 0b i 1 = 0. To see this consider i q 0b i 1 0. i q 0b i 1 < 0 implies that the bank is adding aggregate resources to the economy which is not possible. Conversely, i q 0b i 1 > 0 implies that the bank is making a profit on the sales of period 1 bonds which cannot happen with free-entry in banking. In addition, i q 0b i 1 > 0 implies that the bank will incur a loss on those contracts in the following period when it must repay its obligations. Thus, the bank would need to offer future contracts which are net profitable, i q 1b i 2 > 0 = i q 0b i 1, in order to maintain solvency. However, an entrant could offer marginally less profitable contracts which are preferred by all agents. Hence i q 0b i 1 = 0, which implies i bi 1 = 0 since q 0 is the same across all agents. Thus, in period 1 for a bank to make positive expected profit, it must be that either i q 1b i 2 > 0 or i ei 1 > 0. i q 1b i 2 > 0, by the logic above, cannot occur and hence i ei 1 0 given the bank s payoff. If i ei 1 > 0 then a direct implication for zero expected profit to be sustained is that i q 1b i 2 < 0. However, no bank has an incentive to offer such contracts since the best an entrant bank can offer is i q 1b i 2 = 0. Hence, no bank for which i ei 1 > 0 will offer a contract i q 1b i 2 < 0 since they can offer a contract i q 1b i 2 = 0 and earn profit. As a result, in period 0 no bank will offer a contract where i ei 1 > 0. As a result, the only zero expected profit contracts which are feasible are i ei 1 = 0 and i q 1b i 2 = 0. By the same logic, there is no profitable deviation for a bank in any period and hence ˆθ t = 0 t. In addition, when there is no uncertainty in the net profit from enforcement contracts then a direct implication is that e i t = 0; t i. As a result of proposition 4.1.1, I examine economies with one-period uncontingent bonds. A direct implication of Proposition is that any unexpected loss suffered by a bank will be sufficient to cause the bank to become insolvent since the bank would be required to make a positive profit during the current or a future period in order for its net present value to be positive. 9

10 5 Timing The timing of the model is as follows. In each period, the following sequence of events occurs: 1. Agents produce using their available capital stocks. 2. Agents learn their shock and receive their output. 3. Agents contract with the bank. I.e. agents settle or default their previous-period contracts, if such contracts exist, and re-contract (or not) with a bank. 4. Agents choose next-period capital holdings and consume. 5. Depreciation occurs. The nature of the timing is not innocuous. In particular, the timing rules out a potential hold-up problem on the part of the lender. Second, the nature of the timing implies that depreciation affects the lender and the borrower in the same manner. 3 6 Autarky Under autarky, agents face a trivial problem inasmuch as they cannot exchange and therefore they either consume their production or invest in capital every period. The period budget constraint for an agent of type i is: c i t + ki t+1 1 δ ki t = η i t(k i t) α (4) Let Λ represents the state vector under autarky, Λ = (k i, η i ); Λ and k i represent the state vector and capital holdings in the t + 1 th period respectively. The agent s problem can be represented as (subscripts are omitted): V (Λ) = max c i,k i u i (c i ) + β η V (Λ )π(η η) (5) Let µ(λ) represent the measure of agents in state Λ. A stationary autarkic equilibrium may be defined as follows: A stationary recursive autarkic equilibrium is a set of functions V (Λ), c(λ), k(λ) and µ(λ) such that given δ, α, and π(η η): 3 The timing of the depreciation also simplifies the computational analysis in later sections. The difficulty with depreciation immediately following production is that it implies that lenders must discount collateral twice before they can consume it while borrowers only discount collateral once. For this reason, I assume that depreciation occurs at the end (or equivalently the beginning) of a period so that borrowers and lenders have the same discount period over collateral. 10

11 1. Agents in state Λ choose c(λ), k(λ) to maximize V (Λ). 2. There exists an invariant probability measure P Λ defined over the ergodic set of equilibrium distributions, Λ. 7 Social Planner Given the characterization of the environment, a mechanism which facilitates exchange between agents of each type may be Pareto-improving. Agents have risk averse preferences and thus prefer, ceteris paribus, to smooth consumption across periods for feasible parametrisations of the intertemporal discount factor. Moreover, all agents, prior to the realization of their productivity shock, prefer to insure against the realization of a bad shock. Consider the social planner s problem. In the current environment, in every period an agent may find herself in one of two possible realizations of the idiosyncratic shock, η i t = {η, η}. Since the social planner is constrained by aggregate resources then, clearly, the feasible set of autarkic allocations is a subset of the feasible set of social planning allocations. Let µ 1 and µ 2 refer to the measures of agents receiving shocks η and η, respectively, in a given period and let ˆK t be the aggregate capital stock in period t. The social planner can transfer capital and the consumable output to solve the following program: W (ˆΛ) = max c 1,c 2 ( λµ 1 u 1 (c 1 t ) ) ( + (1 λ)µ 2 u 2 (c 2 t ) ) + βw (ˆΛ ) s.t. 2 2 µ i [c i t + ki t+1 1 δ ki t] = µ i f(kt) i i=1 i=1 c 1 t, c 2 t 0 2 µ i kt i = ˆK t (6) i=1 where ˆΛ = ( ˆK, µ 1, µ 2 ) refers to the state vector under the planner s problem and λ reflects the planner s weight on the utility of an agent of type 1. The social planner will transfer capital to agents who receive a good shock (to maximize the aggregate production) and subsequently distribute the consumable output. The planner s problem gives some intuition for the role of collateral. The planner redistributes capital to (efficiently) maximize total output. Hence, one possibly overlooked role for collateral contracts is simply that they allow the possibility of a redistribution of capital. Under highly persistent idiosyncratic shocks such a redistribution may be desirable. 11

12 8 Collateralized Contracts For a competitive equilibrium to differ from autarky, agents must have access to intermediation. In this section, I describe a type of intermediary contracts which arise in equilibrium. In order to describe the contracting mechanism, let Ξ i t = A i t B i t represent the space of feasible collateral contracts in period t for agent i where B i t = [b i t, b i t] is the feasible set for credit balances (i.e. borrowing and lending) and A i t = [0, (1 δ) k i t] is the feasible set for collateral. b i t = (1 δ) k i t is the lower bound on borrowing and b i t is the upper bound on lending which is given by the period t budget constraint of agent i. b i t+1 (ai t+1 ; q t) Ξ i t; is a feasible enforceable collateral contract where b i t+1 Bi t+1 denotes a consumption good transfer (payment) in period t + 1 to (by) agent i, at a net cost (revenue) of q t b i t+1 in period t and ai t+1 Ai t+1 denotes a capital obligation contingent on b i t+1 being forfeited in period t + 1 by agent i. Let c i t represent the period t consumption of agent i in autarky and let u i t( c i t q t b i t+1 ) represent the period t utility of an agent i with a prospective contract b i t+1 (ai t+1 ; q t). For the remainder of the paper, I suppress the arguments of b i t+1 (ai t+1 ; q t) and simply write b i t+1.4 Further, let Vt+1 i (ki t+1, bi t+1, ηi t+1 ) be the discounted expected utility value at time t + 1 for an agent of type i with capital stock k i t+1, shock ηi t+1, and existing collateralized contract bi t+1. V i,aut t Finally, = u i t( c i t) + β η t+1 V i t+1 (ki t+1, 0, ηi t+1 )π(η t+1 η t ), is the expected utility value to agent i of foregoing intermediation in period t. This case includes the possibility of postponing intermediation one period. Any contract b i t+1 (ai t+1 ; q t), is individually rational for agent i at time t where: u i t( c i t q t b i t+1) + β η t+1 V i t+1(k i t+1, b i t+1, η i t+1)π(η t+1 η t ) V i,aut t (7) Next, I note that the autarky problem at time t can be written, for an agent i, as: max c,b,k ui t( c i t q t b i t+1) + β Vt+1(k i t+1, i b i t+1, ηt+1)π(η i t+1 η t ) (8) η t+1 subject to: Ξ i t = (9) Thus, whenever Ξ i t is non-empty, then there exists at least one contract in Ξ i t which satisfies (7) for an agent i. A contract b i t+1 does not, however, necessarily imply that loans will be repaid. (7) implies both of the following for a lending agent i: agent i is weakly better off lending and receiving repayment than she would be under autarky 4 Recall that throughout this paper, contracts b i t+1 are assumed to be feasible to honor. That is, agent i has either (both) sufficient output or (and) sufficient capital to satisfy the terms of the contract b i t+1. 12

13 agent i is weakly better off lending and receiving the collateral payment than she would be under autarky For a borrowing agent i who has the default choice, it implies either: agent i is weakly better off borrowing and repaying than she would be under autarky agent i is weakly better off borrowing and defaulting the collateral obligation than she would be under autarky The types of collateral contracts, b i t+1, which may be written in an economy are limited only by feasibility. Contracts are feasible iff: and b i t+1 = 0 i a i t+1 (1 δ)k i t i. The first condition is standard and the second implies that the amount pledged as collateral is less than or equal to the capital stock for each agent i. The literature on incomplete markets and borrowing constraints typically focuses on imposing a solvency constraint where it is never individually rational for the borrower to default. This exposition makes the point that such contracts may be overly tight in the sense that they impose greater restrictions than may be socially optimal or even privately necessary in order to permit exchange. For instance, such contracts imply, by necessity, that the borrowing agent always prefers to repay. In their essence, collateral contracts permit a form of intertemporal trade which is conducted through the collateral constraint. 5 Let τt i represents an agent s choice over default or repayment, a i t represent the collateral required for a contract b i t, q t be the price of capital and b i t+1 be the constraint on borrowing owing to collateral. Then, agents face the following budget constraint: c i t + ki t+1 1 δ (ki t (1 τ i t )a i t) + q t b i t+1 = η i t(k i t) α + τ i t b i t (10) b i t+1 b i t+1 (11) τ i t = {0, 1} (12) 5 Dubey, Geanakoplos and Shubik [14] make a similar point. 13

14 Table 1: Notation Definition c i t kt i δ b i t+1 q t b i t+1 τt i = 0 τt i = 1 a i t α η i t consumption by agent i in period t capital stock of agent i in period t depreciation rate on capital bond savings at time t by agent i the price of a bond at time t the endogenous collateral constraint on borrowing agent i defaults in period t agent i does not default in period t collateral demanded for a loan b i t productivity parameter for capital idiosyncratic productivity shock of agent i at time t The endogenous collateral constraint on borrowing, b i t+1 and the savings technology which gives rise to q t are discussed in Section 9.1. Under default agents are assumed to forgo repayment of a debt b i t and forfeit their collateral. Hence the remaining capital stock of agent i after default is simply the difference between the capital stock at the beginning of the period and the amount claimed by the bank. 6 A comment on the specific role of the bank in determining the collateral requirement is deserved. P1 defines the bargaining problem implicit in collateralized lending and borrowing. Indeed, collateral contracts may be written where default is certain but which are incentive compatible for both the borrower and the lender. In such cases, collateral contracts act as mechanisms facilitating inter-temporal trade in the collateral good. As a consequence, the amount a required for a given loan b, is difficult to characterize in a decentralized environment where agents contract individually since collateral may act as either a enforcement mechanism or as a means to inter-temporal trade. The value of collateral is potentially different in each case. A borrower will default whenever V (k a, 0, η) V (k, b, η). A lender prefers default whenever V (k + a, 0, η) V (k, b, η). Hence, depending on the curvature of the value function, it can be that lenders will accept repayment of an amount b but would prefer sure default of an amount a < b while a borrower would prefer to repay. As a result, most of the literature (c.f. [27] and [28]), assumes an exogenous menu of collateral requirements. In this paper, the bank obviates the need for an exogenous menu of collateral requirements. 6 Default in the model is a binary choice variable. That is, agents cannot default on a fraction of their debt. This assumption is by construction restrictive but it serves the purpose of restricting renegotiation. However, the assumption of a binary choice over default seems justified since only one-period debt contracts are considered. Were this model extended to include multiple-period debt, then it would seem plausible to allow agents to default on their debt at some maturities and not others. I leave this analysis to extensions of the current paper. 14

15 9 Equilibrium In this section, I define a stationary, recursive, competitive equilibrium. 9.1 Equilibrium Collateral Contracts The first requirement in determining the structure of equilibrium contracts is to pin down the collateral requirement for a given contract. The following proposition describes the collateral requirement. Proposition Free-entry to banking implies a profit-maximizing bank will set the collateral requirement for a loan b i t+1 such that it is indifferent between receiving the loan or the collateral. Proof: Essentially, this is a Bertrand result. If the bank sets the collateral requirement higher than the face value of the loan then a competing bank may enter and offer a marginally lower collateral requirement without any concommitent loss of profits (no agent would default if a i t+1 > bi t+1 ). Since collateral constrains consumption smoothing, any relaxation of the collateral requirement will attract all the agents to the new bank. In addition, no bank will set the amount of collateral lower than the face value of the loan. To see this, consider the arguments of Rothschild and Stiglitz [32]. Suppose the bank solves the individual problem of both high shock and low shock agents with given capital stocks, k i t. The enforcement value of collateral to an agent is the expected future period, t + 1, marginal value of capital to that agent. Suppose the bank sets the borrowing constraint for all agents with k i t at the level of borrowing at which the agent with the highest future period marginal value of capital is indifferent. Call this agent the low type. By the arguments of [32] this cannot be a pooling equilibrium since all high types would borrow to the level of the low type and then default. Moreover, when a low type contract is offered, all high types have an incentive to act as a low type. Thus, there can be no separating equilibrium. The only other option is a pooling equilibrium where the borrowing constraint is set at the level of the high type. However, by [32] this can not be a pooling equilibrium in a competitive market since there exists a profitable deviation (contract) for a low type agent who has a higher future period marginal value of capital (i.e. relaxing the borrowing constraint). Hence, only high type borrowing constraints cannot be an equilibrium. Thus, the only equilibrium which exists is where a i t+1 = bi t+1. Hence, the bank determines a i t+1 such that the expected value of the collateral ai t+1 to the expected value of the bond repayment, b i t+1. is equal This implicitly determines the endogenous 15

16 borrowing constraint for an agent i such that b i t+1 (1 δ)(ki t (1 τ i t )a i t) b i t+1 b i t+1 < 0. Since no agent will refuse (default on) the repayment of a bond, then a i t+1 = 0 bi t+1 0. The price of capital is trivially determined in the model. Equation (10) implies that the price of capital is simply the numeraire price, 1, t. The bank is assumed to sell off the received collateral so that all agents receive an identical share as long as they have sufficient resources to purchase capital. 7 However, in equilibrium, it must be that: kt+1 i 1 δ (ki t (1 τt i )a i t) a i t (13) τt i=0 i If equation (13) is not satisfied then the total amount of collateral to be sold is greater than the total amount of net investment and hence the price of capital must adjust. Proposition In equilibrium, no agent defaults, i.e. τ i = 1 i when Equation (13) holds. Proof: Since the price of capital and the face value price of bonds are identical, then there is no effect on an agent s current period budget constraint of exchanging capital for bonds. Hence, no agent can be made better off by defaulting when the price of capital is 1. In addition, agents who face borrowing constraints are strictly better off by having more capital and thus prefer to pay off their bond debts using current period income and, if need be, some of their capital stock. By defaulting, they pay off their bond debts using just their capital stocks which means that the amount of capital available to collateralize loans would be less than if they used some current period income. Hence, default will not arise in equilibrium.. Remark: Proposition implies that for default to arise in equilibrium in models similar to the one presented in this paper either (or both) of the following must be true: 1. The price of capital the realized face-value price of bonds (e.g. if bonds were statecontingent). 2. Capital investment is irreversible In both instances, default acts as means of portfolio re-balancing which is otherwise not feasible for agents to undertake. Finally, the bank determines q t in the model so that lending and borrowing are expected to be in zero net supply. In equilibrium it must be that: b i t = 0 s.t. (11) (14) i 7 Alternative disbursement programs won t affect the aggregate amount of net investment and thus the sale format has no impact on the equilibrium in this case. 16

17 9.2 Agent s Problem The agent s problem can be formulated as a dynamic program. Formally, given the state S = (kt, i b i t, ηt) i and economy prices, q t, and collateral constraint, a i t+1, agent i chooses (ci t, kt+1 i, bi t, τt i ) to satisfy: V (S) = max u(c i t) + β V (S )π(η η) (15) c t,k t+1,b t+1,τ t η s.t. (10), (11), (12) The choice over future period states reflects the future effects of default. S denotes the next period state. Finally, µ(s) refers to the measure of agents in state (S). 9.3 Equilibrium Definition A stationary recursive competitive equilibrium in the model is defined as a: set of functions, V (S), b(s), k(s), c(s), f(s), τ(s), µ(s), and a bond price q, such that given u, δ, α, β, η and π(η η): 1. The agent chooses b(s), k(s), c(s), τ(s) to maximize her dynamic problem V (S) given by equation (15). 2. The agent s output is given by f(s). 3. The bank determines q to satisfy equation (14). 4. Aggregates result from individual behavior, K = µ(s)k(s) and A = µ(s)a(s). 5. There exists an invariant probability measure P defined over the ergodic set of equilibrium distributions. 10 Computation In this section, I parametrise an infinite-horizon model and numerically simulate the stationary equilibrium. I use a CRRA utility function which lends itself well to considerations of risk-aversion and time discounting. The specification of the utility function is: U i = t=0 β t (ci t) 1 γ 1 γ (16) 17

18 where c i t is consumption at time t by agent i, β is a time invariant discount factor and γ is a constant parameter of relative risk aversion. γ ω 1 where ω measures an agent s willingness to smooth consumption through time Computational Issues In any stationary steady-state equilibrium it must be the case that the bank accrues zero profits. As a result, in a stationary equilibrium the bank makes no forecasting errors of the price p t+1. Moreover, this restriction also implies that q t and the steady-state distribution of agents µ(s) must satisfy equation (14) and that the amount of received collateral is not greater than the amount of capital demanded, equation (13). In a stationary equilibrium, the bank s program must be satisfied. For the autarky model, a grid for capital is assumed and value-function iteration is used to determine agent s policy functions. Natural cubic spline interpolation is used to determine choices that are not on the grid. A rough grid of 200 points is used for the value function iteration. A convergence tolerance of is chosen for the norm of the distribution of agents, µ. For the social planning problem, a value-function collocation iteration method is used. The grid points for capital are the Cheybshev nodes. A grid of 20 points was chosen for the social planning problem as larger grids did not materially increase the accuracy of the interpolation. For collateral, a state space grid for capital and bonds is assumed for the model. The collateral constraint on borrowing implies that the maximum borrowing is a fraction, (1 δ), of the maximum grid point of capital. The computational complexity of the problem is due largely to the need for a two dimensional grid. That is, capital is not a sufficient statistic to determine an agent s borrowing decision. In equilibrium, an agent s behavior is restricted by the amount of capital with which she enters the period. Her behavior is also affected by the amount of debt (savings) which she has upon entering the period. As a result, for capital to be a sufficient statistic, it must be that an agent s capital stock at the beginning of a period uniquely identifies her debt (savings) at that point. However, it does not since collateral constraints imply that one agent may have faced an unconstrained optimization in the previous period while the second agent may have faced a constrained optimization. As a result, a second statistic - bond holdings - is necessary to identify an agent s equilibrium behavior. The grid for capital is set to be the same as the autarky grid, roughly 41 points, while the grid for bonds is set at roughly 45 points. Hence the total number of possible state realizations for an agent is approximately The exact number depends on the parametrisation. 18

19 Value function iteration is used to determine agents policy function choices. In addition, the solution algorithm must iterate on the bond price q in order to clear the bond market. Since the bond price q is a function of the measure of agents, µ then the solution algorithm must also update µ after solving for q and then check to see if the bond price q is consistent with the new distribution. The exact algorithm used is: 1. Choose a grid for capital and for bonds. 2. Choose an initial distribution of agents µ. 3. Iterate on the value function defined over the grid to determine the optimal policy functions for k, b. 4. Solve for the market clearing bond price q using a bisection algorithm. 5. Update the distribution of agents µ using a transition matrix defined over the optimal policy functions. 6. Repeat steps 3-5 until the distribution of agents µ converges to a stationary distribution, ˆµ. 7. Conditioning on ˆµ iterate on the value function defined over the grid according to the following procedure: (a) Choose the optimal policy function given the grid values. (b) Fix the choice of capital, k, and use a natural cubic spline to interpolate over choices of b. Choose the optimal b using a golden section bisection approach. (c) Fix the choice of bonds, b, and use a natural cubic spline to interpolate over the choices of k. Choose the optimal k using a golden section bisection approach. (d) Choose highest value among points (k, b ), (k, b ), (k, b ). 8. Solve for the market clearing bond price q. 9. Update the distribution of agents µ using a transition matrix defined over the optimal policy functions. For choices k and b which are not on the grid, allocate a fraction of agents to the nearest two grid points. 10. Repeat steps 7(a) - 9 until the distribution of agents converges. 19

20 Convergence tolerances for the collateral program are set as follows: 1. The convergence tolerances for the value function iteration and the bond price iteration are set at The convergence tolerance for bond market clearing is required to be less than The convergence tolerance for the distribution of agents, µ is set to The measure of convergence chosen is µ new µ old. The convergence tolerance for the stationary distribution is, admittedly, not very strict in an absolute sense. However, given the grid size employed in the simulations it appears to reflect the level of convergence attainable. It should also be mentioned that the results do not appear to qualitatively change for small perturbations of the grid or for modest increases in the size of the grid. Experiments with 80 grid points for bonds did not qualitatively change the results and resulted in a significantly slower simulations. 11 Numerical Examples In this section, I present the results of numerical examples illustrating the macro-economic effects of collateral constraints under a number of different parametrisations of the model. The results suggest that endogenous collateral borrowing constraints have a significant effect on the distribution of wealth in the economy. By limiting the opportunities for risk-sharing, collateral constraints distort the market clearing prices of bonds and capital which, in turn, tend to distort the distribution of wealth in the economy. In particular, under most parametrisations, the steady-state distributions of wealth are skewed towards the poor, with some parametrisations exhibiting bi-modal characteristics. In the following subsections, I examine the sensitivity of the wealth distribution to the size and persistence of the productivity shock and the the sensitivity to depreciation and risk aversion. In particular, I consider: High Risk: {η, η} = {1.25, 0.75} Low Risk: {η, η} = {1.05, 0.95} and: High Persistence: π(η η) =

21 Low Persistence: π(η η) = 0.5 Unless otherwise specified, β = 0.96, α = 0.4, γ = 2 and δ = 0.1 for all examples. The general flavor of the results is as follows. Collateral constraints typically cause bi-modal distributions of wealth as agents transit at a high rate from the middle class. There are two reasons for this result. Firstly, under some parametrisations, significant numbers of the poor face binding borrowing constraints and cannot borrow, despite the fact that they benefit the most since their marginal return to capital investment is the highest. Thus, the poor receive little from financial intermediation. The middle class are the net beneficiaries. Since the wealthy still prefer to smooth their income, they end up lending to the middle class debtors, albeit at a lower return than they could obtain from the poor. 8 Secondly, the distortions from the collateral constraints tend to depress the real interest rate which means that agents have a marginally greater incentive to accumulate capital than under models with a higher real interest rate. The net effect is that, at the margin, the middle class hold higher allocations of risky capital and thus face marginally riskier incomes hence increasing their rates of transition among wealth states. Figures 1-4 in Appendix A illustrate the steady-state distributions of wealth in the collateral equilibria compared to those in the autarkic equilibria across the main examples considered. In addition, Figures 5-8 illustrate the steady-state comparisons of the consumption distributions. 9 With the exception of the example with Low Risk and Low Persistence where the distribution is symmetrically distributed around the mean wealth level, all of the distributions exhibit a bi-modal wealth distribution with most agents having low wealth. Figures 9 and 10 in Appendix A illustrate the bond versus capital decompositions of wealth holdings for two of the parametrisations computed. In particular, they are illustrative of the bimodal nature of wealth inequality and demonstrate that the source of the inequality is primarily the size of bond holdings. Figure 9 depicts High Risk and High Persistence with δ = 0.1, γ = 2 while Figure 10 depicts High Risk and Low Persistence with δ = 0.1, γ = 2. In the first, when the idiosyncratic shock is persistent, agents tend to choose one of two levels of capital depending on the realization of their shock. In the second, the expected marginal return to capital is independent of the realization of the shock and hence agents tend to cluster around one level of capital. However, in both Figures 9 and 10 the tendency of the bond distributions to be bi-modal is evident. The 8 One suspects therefore, that the rich have an incentive to create financial intermediation between themselves and the poor. 9 The distributions presented are kernel-smoothed estimates of the actual distributions. The kernel estimator chosen was the Nadarya-Watson estimator with bandwidth chosen to roughly match the typical grid distance. 21