A theory of dynamic contracting with financial constraints

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Abstract

Financial constraints preclude many surplus producing economic transactions, and inhibit the growth of many others. This paper models financial constraints as the interaction of two forces: the agent has persistent private information and is strapped for cash. The wedge between the optimal and efficient allocation, termed distortion, increases over time with each successive “bad shock” and decreases with each “good shock”. At any point in the contract, an endogenous number of “good shocks” are required for the principal to provide some liquidity and then eventually for the contract to become efficient. Efficiency is reached almost surely. The average rate at which contract becomes efficient is decreasing in persistence of shocks; in particular, the iid model predicts a quick dissolution of financial constraints. This speaks to the relevance of modeling persistence in dynamic models of agency. The problem is solved recursively, and building on the literature, a technical tool of finding the minimal subset of the recursive domain that houses the optimal contract is further developed.

Introduction

Long term economic transactions are often marred by financial constraints. A sizeable body of empirical work documents the wide prevalence of financial constraints, their micro impact on firms size and growth, and macro impact on the misallocation of capital in an economy.1 With an aim to provide theoretical constructs to these empirical regularities, Kiyotaki (2012), in an elegant note, advocates “a mechanism design approach to illustrate how different environments of private information and limited commitment generate different financial frictions.” In the spirit of the said agenda, this paper posits financial constraints as a product of the interaction between (i) persistent private information, and (ii) limitations on the ability of agents to generate timed cash flows.

We study a dynamic screening problem with Markovian shocks where the principal offers history dependent allocations and transfers to the agent. If the agent has a cash reserve or pledgeable assets, the principal will ask the agent to post a bond or deposit collateral. Broadly, optimal distortions are then frontloaded, going from maximal to zero, and optimal payments are backloaded, maximally delayed to the extent possible. However, in many real situations, e.g. in supply contracts, managerial compensation, provision of public goods and regulation, the agent may not have the requisite cash to post a bond or collateralize existing assets. This has implication for both the optimal structure of distortions, and the sequential breakup of payments.2

Taking inspiration from the literature on financial contracting, we model the aforementioned situation by restricting the stage (or per-period) utility to be positive. The idea being that the agent requires, in the least, the amount of cash that covers the consumption/production decisions in every period. The economic force generated by the interaction of this stronger feasibility restriction and private information is termed as the financial constraint. This is because if there is no private information, the efficient allocation is implementable, and in the presence of a bond or collateral, efficiency is achievable much more easily through maximal backloading of payoffs. So, it is the interaction of the two forces together that produces financial constraints. Further, we show that persistence in private information, an empirically relevant feature of the model, makes this interaction even richer in terms of constraining the optimal allocation.3

The big picture question is: when do these financial constraints bind and when they do, what dynamic distortions do they generate? In asking and then trying to answer this question, we provide a deeper understanding of the role of financial constraints in dynamic mechanism design, and a deeper understanding of the role of persistence in agency frictions in dynamic financial contracting.4

The rest of this introduction is divided into three parts. First is the structure of the optimal contract and a plausible mechanism that implements it. Second is unpacking the economic content of the novel elements – (i) cash versus no cash constraints, (ii) interpretation of positivity of stage utility, and (iii) the role of persistence in agency frictions in generating financial constraints. Third, is an overview of the literature. We expound upon each after briefly describing the model.

The formal model is as follows. A big firm (principal) repeatedly producing a final good contracts with a smaller firm (agent) that supplies an important input. Each period, the small firm privately observes either a low (“good shock”) or high (“bad shock”) marginal cost. After being drawn from a prior, costs evolve according to an exogenous two state Markov process. Preferences are quasi-linear. The small firm requires a constant cash flow to cover its costs of production, hence the stage utility must be positive: we say that the agent is thus strapped for cash. The big firm is tasked with designing a contract which sets supply of inputs by the small firm, and payments for its production. Both parties can commit to a dynamic contract.

Structure of the optimal contract. A Pareto-optimal contract chooses allocations and transfers that satisfy incentive compatibility and cash-strapped constraints to maximize the profit of the big firm while ensuring a minimum ex ante payoff for the small firm. Fig. 1a depicts a typical sequence of technology shocks. For a history of cost realizations ht and current cost θi, let q(θi|ht) and U(θi|ht) be the allocation and expected utility of the small firm. At this point, if the marginal cost of incentive provision is zero, then q(θi|ht)=qe(θi), that is the (statically) efficient quantity is supplied. If it is positive, then q(θi|ht)=qe(θi)d(θi|ht) where d measures the history dependent optimal distortion. As is standard, the low cost type always supplies the efficient quantity: q(θL|ht)=qe(θL).5 On the other hand, each “bad shock” increases optimal distortions: q(θH|ht,θH)<q(θH|ht)<qe(θH).6 In addition, the realization of a “good shock” decreases the optimal distortion: q(θH|ht)<q(θH|ht,θL). An endogenous number of consecutive “good shocks”, say n(ht), is required for the optimal distortion to reach zero. For every additional “bad shock”, as distortions increase, this number increases: n(ht,θH)n(ht). Once the optimal distortion reaches zero it stays at zero, that is, efficiency is an absorbing state. In the long run, the efficient contract is supplied almost surely.7

With reference to Fig. 1a, the expected utilities of both the low and high cost types go up after a “good shock” and go down after a “bad shock”. That is, as long as the contract is inefficient: (U(θL|ht,θH),U(θH|ht,θH))(U(θL|ht),U(θH|ht))(U(θL|ht,θL),U(θH|ht,θL)). Two thresholds on the vector of expected utilities divide the evolution of the optimal contract into three regions - illiquidity, liquidity and efficiency; see Fig. 1b. The contract typically starts in the illiquid region – both incentive and cash-strapped constraints bind. A low cost type either keeps the contract in illiquidity or can transition it to liquidity. A high cost type decreases the expected utility of the small firm which keeps it illiquid. After an endogenous number of low cost realizations, the expected utility of the small firm reaches a critical threshold at which the cash-strapped constraint becomes slack. This is called the liquid region. Liquidity is not an absorbing state, a high cost realization can push the small firm back into illiquidity. The liquid region forms a penultimate zone towards efficiency. Once liquid, the realization of one more low cost pushes expected utility of the small firm beyond the second threshold into the absorbing state of efficiency.

At a technical level, we use the recursive approach to characterize the optimal contract. More specifically, we construct a “shell”, the minimal subset of the recursive domain which houses the optimal constrained contract. The recursive domain is too large to make crisp predictions about the exact structure of dynamic distortions. We show that as long as the optimal contract is inefficient, the expected utility of the agent must always lie in this shell. It allows us to show all the aforementioned monotonicity properties of the evolution of the optimal contract. We also provide a simple price-theoretic explanation of the construction of the shell.

A combination of working capital and eventual take-over implements the optimal contract. In the illiquid region, the cash-strapped constraint binds and the big firm only provides working capital to the small firm. Through a sequence of consecutive low cost realizations, the small firm has to earn its way into liquidity. In the liquid region, the big firm promises to take over the small firm on the realization of one more low cost type for a determinable strike price. Thereafter, the small firm operates in-house, producing the efficient quantity.8

The role of financial constraints and persistence of private information. Allowing for a long-term contract helps mitigate the problem of agency frictions by backloading payoffs. Financial constraints, though, restrict the extent of backloading. Dynamic distortions in our framework are an additive sum of two effects: backloading of payoffs and illiquidity due to financial constraints; the latter increases with each “bad shock”, overturning the standard result of decreasing distortions in dynamic mechanism design. Efficiency is still a certainty, though the path towards it is much more constrained in comparison to the model sans financial constraints.

We also reconsider the interpretation of the positivity of stage utility as a limited liability constraint for small businesses. It is clear that the cash strapped constraint is welfare reducing from the perspective of total welfare (or surplus), but is it “beneficial” for the agent? Consider the principal profit maximizing contract on the Pareto frontier in which the big firm has all the bargaining power. The ex ante expected utility of the small firm from the contract is determined endogenously as part of the optimum. We show that in the iid limit the ex ante expected utility of the agent is higher in our model than in the benchmark, and in the perfect persistently limit the ranking can reverse for certain parameters. This points to a cautious interpretation of the positivity of the stage utility as a limited liability constraint, which is the standard in the literature.

Finally, we take this model as representative of firm dynamics in an economy with financial constraints and numerically show how persistence in agency frictions makes a marked difference to the substantive predictions of the model. We make three broad points. The fraction of financially constrained firms in the short-run is monotonically increasing in the persistence of technology shocks. The average rate at which firms converge to the state of being unconstrained is decreasing in persistence; in particular, the iid model predicts a quick dissolution of financial constraints. And, variance in the total value of both constrained and unconstrained firms is larger with persistence. The standard dynamic financial contracting literature that operates in the iid world would miss all these, empirically important, comparative statics.9

Related literature. This paper sits at the intersection of at least two strands of theoretical models: dynamic mechanism design with serially correlated information (see surveys by Vohra (2012), Krähmer and Strausz (2015a), Pavan (2016), and Bergemann and Välimäki (2019)) and dynamic financial contracting with iid technologies (see surveys by Biais et al. (2013), and Sannikov (2013). Three ingredients interact to determine the structure of dynamic inefficiencies: correlation in agency frictions, strength of feasibility restrictions, and permissibility of termination. The overarching role of each combination of ingredients is to create frictions in dynamic contracting that lead to realistic qualitative predictions.

Table 1, Table 2 enlist the most closely related papers; Table 1 features screening and Table 2 features cash flow diversion as the underlying agency friction. Within each table, papers are classified along inclusion/exclusion of the three aforementioned modeling ingredients. In terms of long-term predictions, once the recursive problem is appropriately set up, it can be shown that in each of the papers the optimal contract converges to the efficient allocation in the absence of the termination clause, and it converges either to efficiency or termination in the presence of the termination clause.10 The key economic force that leads to this result is backloading of information rents to the extent possible.11

At a high level, ours is the first paper to precisely characterize the short-run predictions in terms of the monotonic nature of dynamic distortions: “good shocks” monotonically push the allocation towards efficiency and “bad shocks” take it away from it. As noted in Table 1, it is the first paper to analyze a dynamic screening model (as opposed to a cash flow diversion or moral hazard problem) with both persistence in private information and financial constraints, nudging the literature on dynamic mechanism design to explicitly incorporate financial constraints. Further it (i) identifies the minimal subset of the recursive domain that houses the optimal contract; (ii) clarifies the connection between limited liability and being strapped for cash; (iii) provides an explicit characterization of the optimal contract in the perfectly persistent limit which shows “good shocks” have a stronger effect on distortions than “bad shocks”, this fact underlies the long-term efficiency result; (iv) solves for the optimal contract in continuous time, which seeks to unify the literatures on cash flow diversion and screening, since the models converge to the same limit in continuous time; and (v) explores the implications of persistence in agency frictions on firm dynamics.

The two most closely related papers are Battaglini (2005) and Krishna et al. (2013). Battaglini (2005) studies a similar screening model, but where the agent has cash to post a bond. More specifically, only the total expected utility over time is required to be positive in every period. The structure of short-run distortions are thus quite different – the contract becomes efficient forever as soon as the agent assumes a “good shock”, and it has decreasing distortions along the history of constant “bad shocks”. In a departure from that paper, and more generally the literature on dynamic mechanism design, our paper explores the implications of cash constraints for the agent with persistent private information.

Krishna et al. (2013) study the same model as ours, repeated screening with the cash-strapped constraint, but where the agent's types follow an iid process. Since theirs is a special case of our model, all our results also hold in their setup. However, the focus of the paper is on long-term efficiency. We build upon their work in at least three ways. First, the monotonicity of allocation rule, even for the iid model is novel to our paper. Second, the Markov model is technically much harder to solve, as has already been noted in simpler dynamic mechanism design models without financial constraints.12 Third, persistence adds greater empirical relevance to the analysis, as is evident from the applications of standard dynamic mechanism design models to public finance (see Stantcheva (2020)).13

Clementi and Hopenhayn (2006) and Fu and Krishna (2019) both study the problem of cash flow diversion by the agent in a repeated setting, the former looks at an iid technology and the latter at a Markovian one. A simple way to map their framework into ours would be to change the time structure: At the start of every period the agent commits to a production plan after which his cost type is realized. The type is reported, agreed upon input quantity is supplied, and the agent is compensated for by the principal. The interpretation here is that agent does not know whether his cost would be low or high when he makes the production decision. Despite being a low cost type, he can misreport to be a high cost type, supply some portion of the produced quantity and sell the rest in the black market – a diversion of the economic surplus. While these models produce similar long-term predictions, the short-run structure of the optimal contract here is quite different than the screening literature.14

Our paper is also related to the recent work by Guo and Hörner (2018): They consider a dynamic principal-agent model with persistent private information where preferences are perfectly aligned, transfers are not allowed and the principal wants to maximize efficiency. The optimal contract converges almost surely either to permanent allocation (efficiency) or permanent non-allocation (immiseration), driven by the fact that both front-loading and backloading of payoffs can occur at the optimum. In our framework, preferences are misaligned, the expected utility is continuously backloaded, and the optimal contract always converges to efficiency. A technical aspect we share with Guo and Hörner (2018) is the characterization of a subset of the recursive domain that houses the optimal contract, which allows us to make precise statements about the short and long run properties.15,16

Financial constraints have also been explored in the sequential screening literature pioneered by Courty and Li (2000). For example, Krähmer and Strausz (2015b) consider a sequential screening model with ex post participation constraints. They show that with these additional constraints the optimal contract is static and does not illicit the agent's information sequentially. One way to map their framework into ours would be to consider the two period version of our model, and require the first period allocation to be (exogenously) zero. Then, the cash-strapped constraints require that no payments can be charged in the first period. As a consequence the optimal contract replicates the “static optimum”. In contrast our model highlights that multi-period interactions can extract private information in an incentive compatible fashion, even with stronger feasibility restrictions.17

Finally, the cash-strapped constraint breaks the linearity of transfers across time. The spirit of this exercise is shared by other related works: Amador et al. (2006), and Halac and Yared (2014) study models of delegation. Thomas and Worrall (1990), Garrett and Pavan (2015), Luz (2015), and Arve and Martimort (2016) consider dynamic models of private information where the agent is risk averse. Krasikov et al. (2019) analyze a dynamic screening model with individual rationality, but where the principal is more patient than the agent. In all these papers, there is a cost to moving transfers or payments across time.

Section snippets

Model

The key economic forces in dynamic contracting with persistent private information can be formulated through various related models. We choose the repeated version of the marginal cost screening model, based on Laffont and Martimort (2002). A big firm (principal) specializing in a final good requires a non-durable input that is produced by a smaller firm (agent) every period at a cost θq, where θ is the small firm's private information.18

Optimal contract

Define s(θ,q)=V(q)θq to be the static surplus, succinctly expressed as s(θ)=V(q(θ))θq(θ) for the direct mechanism. It is straightforward to note that the efficient quantity that maximizes the surplus is given by V(qe(θ))=θ. Moreover, let S=t=1Tδt1E[s(θ˜t)] be the (ex ante) expected surplus. The principal's problem, (P), can be stated as:(P)maxU,qS[μLU(θL)+μHU(θH)] subject to q0,(PK):μLU(θL)+μHU(θH)v0, andICL(ht1),ICH(ht1),CL(ht1),CH(ht1)ht1Ht1t, where (PK) is the

Role of financial constraints and persistence in private information

There are at least three conceptual points that emanate from studying this dynamic screening model with persistent private information and cash-strapped constraint: (i) the interaction of incentive constraint with stronger feasibility restrictions generates novel dynamic distortions, (ii) a foundation for when positivity of stage utility can be interpreted as a limited liability restriction, and (iii) the impact of persistence in agency frictions on the evolution of the optimal contract and

Final remarks

This paper motivates the study of financial constraints in dynamic contracting through the interaction between persistent private information and cash or liquidity constraints. The agent has access to a viable technology marred by agency frictions, and is strapped for cash. The paper situates itself in between the literatures on dynamic mechanism design and dynamic financial contracting.

In the appendix, we discuss a number of extensions and other results not covered in the main text. First, we

Appendix

We divide the appendix into twelve subsections - the benchmark model, two period model, the sequential approach, followed by the recursive approach, proof of the main theorem, optimal limit contract, conceptual interpretation of cash-strapped as limited liability, dynamics of payments, general IID model, sufficiency conditions, introducing termination, and the model in continuous time. Throughout we will invoke the general model where f(θL|θi)=αi for i=L,H. We shall assume the following, their

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    We are indebted to Nageeb Ali, Felix Bierbrauer, Aislinn Bohren, Jinwen Wang, Nima Haghpanah, Vijay Krishna, George Mailath, Bruno Strulovici, and to the seminar participants at Northwestern University, University of Pittsburgh, Pennsylvania economic theory conference, Vanderbilt mechanism design conference, Midwest economic theory conference at University of Michigan and SED conference in Edinburgh for their comments. We would also like to thank the editor, Alessandro Pavan, the associate editor, and anonymous referees for detailed suggestions that helped improve the paper. We dedicate this paper to our beloved friend and colleague Edward James Green. His paper Green (1987) was a pioneering contribution in dynamic contracts. Ed passed away on October 26, 2019.

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