Aggregate consequences of limited contractenforceability∗

Thomas CooleyNew York University

Ramon MarimonEuropean University Institute

Vincenzo QuadriniNew York University

February 15, 2001

Abstract

In this paper we develop a general equilibrium model in which entrepreneursfinance investment by signing long-term contracts with a financial intermediary. Be-cause of enforceability problems, financial contracts are constrained efficient. Aftershowing that the micro structure of the model captures some of the observed featuresof the investment policy and dynamics of firms, we show that at the aggregate levellimited enforceability generates serial correlation in output growth and amplifies theresponse of output to shocks.

∗We would like to thank Gianluca Clementi, Mariassunta Giannetti, Urban Jermann and GiuseppeMoscarini for helpful comments, and seminar participants at Atlanta Fed, Arizona State University, CEFmeeting in Barcelona, ES meeting in New Orleans, Econometric World Congress in Seattle, MannheimUniversity, Minnesota Summer Theory Workshop, SED meeting in Costa Rica, Stockholm School ofEconomics, Universita’ di Roma-La Sapienza, Wharton School and Yale University.

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1 Introduction

The ability to attract external financing is crucial for the creation of new firms and theexpansion of existing ones. For that reason the nature of the financial arrangementsbetween lenders and firms has important consequences for the growth of firms. Oneimportant issue in financial contracting is enforceability, that is, the ability of each side torepudiate the contract. This can be an important issue in the financing of firms becauseprojects often involve specific entrepreneurial expertise and might be worth less to investorswithout the services of managers who initiated them. At the same time the developmentof such projects may provide managers with experience that is extremely valuable forstarting new projects. Limited enforceability conditions the kinds of contracts we arelikely to observe and affects the resources that are available for the firm to grow.

Contractual arrangements that are motivated by limited enforceability are most likelyto be important for firms that are small and/or young. Albuquerque and Hopenhayn [1]have shown that these considerations can help to explain some of the growth characteristicsof small and young firms. We know for example that smaller and younger firms are lesslikely to distribute dividends and that, conditional on the initial size, they tend to growfaster. Further, the investment of smaller and younger firms is positively correlated withcash flows and there is at least indirect evidence that they are more likely to be financiallyconstrained.

Even if financial constraints are important at the firm level, it is not obvious thatthey will have important aggregate consequences for the economy. One issue is whetherthe allocation of resources that results from these contracts reduces welfare significantlycompared to a world where contracts are fully enforceable. A related question is whetherthese constraints cause the economy to be more sensitive to aggregate shocks. Thereis a widespread view, associated with the notion of a ”credit channel” that holds thatmarket incompleteness or credit rationing may be an important propagation mechanismfor aggregate shocks. In this paper we show that the financial constraints that arisebecause of limited enforceability not only explain the growth characteristics of firms butthey affect the macro allocation of resources and the propagation of aggregate shocks tothe economy in important ways.

We study a general equilibrium model where entrepreneurs and investors enter intolong-term contractual relationships which are optimal, subject to enforceability constraints.1

Consequently, financial constraints arise endogenously in the model as a feature of the op-timal contract. At the micro level, our approach is closely related to the partial equilibriummodel of Albuquerque and Hopenhayn [1], although our framework differs in importantdetails. One of the differences is the timing of the investment choice. While in Albu-querque and Hopenhayn the investment choice is made after observing the shock, in ourmodel investment is chosen one period in advance, and therefore, before the observation of

1This work is closely related to the existing literature on optimal lending contracts with the possibilityof debt repudiation. Examples are Alvarez and Jermann [2], Atkeson [3], Kehoe and Levine [11] andMarcet and Marimon [14]. Recent papers also related to our work are Monge [17] and Quintin [19].

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the shock. This timing is important for explaining the cash-flow sensitivity of investmentonce we control for the current size and future profitability of the firm. Another specialfeature of our model is that, once the contract has been signed and the project has beeninitiated, the entrepreneur has the ability to leave the firm and start a new project. Be-cause contracts are not fully enforceable, the ability to start a new project affects the valueof repudiating the contract. Aggregate shocks affect the productivity of new projects, andtherefore, the repudiation value. This channel is central for the amplification of aggregateshocks to the economy.

Within this framework we show first that limited enforceability can help to explainsome of the patterns of investment and growth of an individual firm. Secondly, we showthat limited enforceability amplifies the response of the aggregate economy to shocks.Our theory predicts that the extent of contract enforceability matters for the volatility ofoutput: economies in which contracts are more enforceable display less volatility of outputthan economies where enforceability is weak. If we think that the enforcement of contractsin developing countries is weaker than in industrialized countries, then the former shoulddisplay more extreme sensitivity to shocks. This appears to be true in the data as willbe shown in Section 5. In addition, we also show that limited enforceability generatesserial correlation in output growth. A result that cannot be obtained if contracts are fullyenforceable.

There is an extensive literature that studies the importance of financial factors on theinvestment behavior of firms (see, for example, Hubbard [10]) and on the problem of debtrenegotiation (see, for example, Hart and Moore [9]). A large body of the literature isinterested in studying the micro foundation of market incompleteness but abstracts fromthe implications that market incompleteness may have for the macro performance of theeconomy. There are important contributions that embed these approaches in a generalequilibrium analysis to study issues of macroeconomic relevance. Examples are Bernanke,Gertler and Gilchrist [4], Carlstrom and Fuerst [5], DenHaan, Ramey and Watson [7],Kiyotaki and Moore [12], Smith and Wang [20]. However, in most of these attempts,firms’ heterogeneity is either exogenous or it does not play an important role.

In contrast, in our theoretical framework financial frictions induce a non-trivial hetero-geneity of firms which evolves endogenously, and this heterogeneity has important impli-cations for the behavior of the aggregate economy. Because of the financial frictions thatare a feature of the optimal contract, in each period there are two types of firms: thosewho are resource constrained and those who are unconstrained. The constrained firms aremore sensitive to shocks and experience higher rates of growth. For that reason the rela-tive contributions of each type of firm to aggregate output affects the growth of the entireeconomy. After a positive shock, constrained firms increase their investment and expandtheir share in aggregate production because of their higher sensitivity to shocks. Moreover,the share of constrained firms will be higher for several periods (as a consequence of theinitial increase in investment). Because constrained firms grow faster than unconstrainedfirms, the growth rate of aggregate output continues to grow beyond the first period. Thisimplies that the growth rate of output will be serially correlated. Graphically, the response

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of output will be hump-shaped. This pattern is then amplified by the high sensitivity ofconstrained firms to shocks.

The organization of the paper is as follows. In Section 2 we describe the model economyand in Section 3 we derive and characterize the optimal financial contract between theentrepreneur and a financial intermediary. In Section 4 we define the general equilibriumof the economy and in Section 5 we analyze the quantitative properties of the modelwith and without contract enforceability. After parameterizing the model (Section 5.1),we compute the efficiency losses due to limited enforceability (Section 5.2) and study theresponse of the economy to aggregate shocks (Section 5.3). The final Section 6 summarizesand concludes.

2 The model

Preferences and skills: The economy is populated by a continuum of agents of totalmass 1. In each period a mass 1− α of them is replaced by newborn agents and α is thesurvival probability. A fraction e of the newborn agents have the entrepreneurial skills tomanage a firm and become entrepreneurs. The remaining fraction, 1−e, become workers.2

Agents maximize:

E0

∞∑t=0

(α

1 + r

)t (ct − ϕ(lt)

)(1)

where r is the intertemporal discount rate, ct is consumption, lt are working hours, ϕ(lt) isthe disutility from working. Utility flows are discounted by α/(1 + r) as agents survive tothe next period only with probability α. Given the assumption of risk-aversion, r will bethe risk-free interest rate earned on assets deposited in a financial intermediary.3 Theseassets are denoted by a. The function ϕ is strictly convex and satisfies ϕ(0) = 0. Denotingby wt the wage rate, the supply of labor is determined by the condition ϕ′(lt) = wt. Forentrepreneurs, lt = 0 and their utility depends only on consumption.

An agent with entrepreneurial skills has the ability to implement one of the projectsavailable in that particular period as described below. Entrepreneurial skills fully depreci-ate if the agent remains inactive. This implies that, as long as the value of a new project ispositive, newborn agents with entrepreneurial skills will always undertake a project whenyoung. At the same time, by undertaking a project, an entrepreneur maintains the abilityto start new projects in future periods. This assumption eliminates the possibility thatskilled agents remain inactive and wait for better investment opportunities.

Technology and shocks: In each period there is the arrival of new investment projectscharacterized by a productivity level z ∈ z1, z2. New investment projects are equally

2It is straightforward to make the fraction of new entrepreneurs e endogenous. Because this feature ofthe model is not essential for the results, we take e as exogenous.

3On each unit of assets deposited in a financial intermediary, agents receive (1+r)/α if they survive tothe next period and zero otherwise. Therefore, the financial intermediary acts as a life-insurance companyand the expected return on these deposits is r.

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likely to be of high or low productivity. The productivity of new investment projects actsas an aggregate shock and in this economy business cycle expansions are driven by thearrival of more productive technologies rather than the improvement of existing ones. Inthis sense, the economy has the typical features of a model with vintage capital.4

An investment project with productivity z generates revenues according to:

z · F(

mink , ξ · l, η)

(2)

where k is the input of capital, l is the input of labor, η an idiosyncratic shock and z isthe project-specific level of productivity. Each project is characterized by a particular zwhich remains constant for that project. Different projects, however, may have differentz depending on the vintage of the project. The shock η is idiosyncratic to the projectand changes randomly over time. It is independently and identically distributed in thepositive section of the real line with distribution function G(η). The function F is strictlyincreasing in both arguments, strictly concave with respect to the first argument, andsatisfies F (0, .) = 0 and F (., 0) = 0. As long as the project remains active, capitaldepreciates at rate δ.

Given the Leontif structure of the production function, in equilibrium the capital-labor ratio is equal to the parameter ξ. Therefore, we can write the production functionas zF (k, η). To simplify the notation, we will also redefine w to be the equilibrium wagerate divided by the capital-labor ratio ξ. Using this notation the cost of labor for the firmis wk. Henceforth, we will refer to the variable w as the “wage rate”, although it shouldbe remembered that the rate paid per unit of labor—the actual wage rate—is wξ.

The input of capital is chosen one period in advance, before observing the shock. This isone of the specific features that differentiates our model from Albuquerque and Hopenhayn[1], where the investment choice is made after the observation of the shock. The capitalinvested in a project is specific to that project and it cannot be reallocated to a new one.Consequently, if the firm is liquidated, the liquidation value of capital is zero. On theother hand, if the project remains active, the internal value of capital is (1− δ)k. Undercertain conditions, this assumption implies that it is never optimal to liquidate an activeproject, even if new technologies are more productive. In the rest of the paper we keepthe assumption that the difference between z1 and z2 is sufficiently small that it is neveroptimal to replace an existing project.

The last assumption about the revenue technology is that with probability 1 − φ theproject becomes unproductive. In this case the entrepreneur loses the entrepreneurialskills and become a worker.5

4In general, we can make the shock persistent by assuming that the z of new investment projectsfollows a Markov process. We keep the simpler assumption of i.i.d. shocks because the model alreadygenerates enough persistence without having persistent shocks.

5There are two sources of exogenous liquidation of the firm. The entrepreneur may die with probability1 − α or the project becomes unproductive with probability 1 − φ. The demographic assumption of anexogenous death is introduced for analytical convenience. With this assumption new entrepreneurs arenewborn agents who do not own assets to finance investment. Without this assumption we would have to

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Financial contract and repudiation: An entrepreneur who starts a new project,finances the input of capital by signing a long-term contract with a financial intermediary.The contract is not fully enforceable. As in Albuquerque and Hopenhayn [1], Alvarezand Jermann [2], Kehoe and Levine [11], and Marcet and Marimon [14], enforceabilityproblems arise as the contractual parties cannot commit to future obligations and theycan repudiate the contract at any moment. Because of this, the value of repudiating thecontract is the key object that affects the properties of the financial contract.

In case of repudiation, the intermediary will lose the whole value of the contract.Therefore, the repudiation value for the intermediary is zero. This implies that, as longas the value of the contract is positive, the intermediary will not repudiate the contract.

For the entrepreneur, the derivation of the repudiation value is more complex. Weassume that, if a contract is repudiated, the entrepreneur is able to appropriate (andconsume) the cash flow revenue generated by the production process. In addition toappropriating the current cash flow, the entrepreneur can also start a new investmentproject by entering into a new financial relationship.6 Repudiation, however, also carrieswith it a cost κ for the entrepreneur. This cost can be interpreted as legal punishmentsthat reduce the utility of the entrepreneur. Alternatively, we could assume that, in case ofrepudiation, the intermediary carries over to the next contract a credit κ. In the absenceof such a cost, a financial contract may not exist. Because the repudiation value sets alower bound on the value of the contract for the entrepreneur, when this value is high,the financial intermediary may not break even for low productivity projects. The role ofκ is to reduce this lower bound by reducing the repudiation value. This parameter can beinterpreted as an index of the enforcement system in the economy.

Denote by V 0(s) the value of a new investment project (new contract) for the en-trepreneur, where s are the aggregate states of the economy as will be specified later inthe paper. Then the value of repudiating an active contract is Dz(s, k, η) = zF (k, η) +V 0(s) − κ. The repudiation value is indexed by the subscript z because the cash flowdepends on the productivity of the project currently run by the entrepreneur.

It should be noted that, although it is never efficient to switch from a low productivityproject to a more productive project (given that capital is sunk and given the restrictionson z), this does not mean that the entrepreneur has no incentive to repudiate the contractand start a new investment project. The optimal contract must be structured so that theentrepreneur has no incentive to do so (incentive compatibility).

keep track of the distribution of assets among potential entrepreneurs. The exogenous probability 1 − φis introduced to generate enough turnover in the distribution of firms. This can also be obtained withoutassuming that projects become unproductive if we reduce the survival probability α. The implied deathprobability, however, would be too large.

6Remember that by running the firm the entrepreneur maintains the entrepreneurial skills which allowhim or her to manage a new investment project.

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3 The optimal financial contract

A contract specifies, for each history of the realization of the individual and aggregateshocks, the payments to the intermediary τ , the payments to the entrepreneur d (divi-dends), and the next period capital input k′. The payments to the entrepreneur, d, cannotbe negative. The payments to the intermediary, instead, can be negative. To characterizethe optimal financial contract we use the recursive approach developed in Marcet andMarimon [15]. This approach studies the optimal contract as the solution to a planner’sproblem who attributes different weights to the entrepreneur and the intermediary. Forthe moment we assume that the weights assigned by the planner to the two contractualparties are given. Later we use the fact that there is free entry in the intermediationsector to determine these weights. The planner takes as given the equilibrium prices andthe problem is subject to incentive-compatibility and resource constraints.

To simplify the characterization of the optimal contract, for the moment we ignore thepossibility that the intermediary could repudiate the contract. Therefore, in the analysisthat follows we assume that the intermediary commits to fulfill any obligations (one-sidecommitment). Then, later, we will study the conditions under which the value of theoptimal contract for the intermediary is non-negative in any possible contingency. If theseconditions are satisfied, the incentive compatibility constraint for the intermediary is neverbinding and it will never repudiate the contract. Consequently, the optimal contract withone-side commitment is also the optimal contract for the more general model withoutcommitment.

Consider a contract signed at time t. Define λt the weight assigned to the entrepreneurand normalize to 1 the weight assigned to the intermediary. Under the assumption ofone-side commitment of the intermediary, the planner’s problem takes the form:

maxds,τs,ks+1∞t=0

Et

∞∑s=t

βs−t(λtds + τs) (3)

subject to

Es

∞∑j=s

βj−sdj ≥ Dz(ss, ks, ηs) (4)

ks+1 = zF (ks, ηs) + (1− δ − w(ss))ks − ds − τs (5)

ds ≥ 0, kt = 0 (6)

The objective (3) defines the surplus of the contract for the planner as the expecteddiscounted value of per-period flows. The per-period flow is defined as the weighted sumof the returns for the entrepreneur and the intermediary. Future flows are discounted byβ = αφ/(1 + r)—rather than 1/(1 + r)—because the entrepreneur survives to the nextperiod with probability α and the project remains productive with probability φ. Notice

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that the wage variable w is determined by the clearing condition in the labor market,which depends on the aggregate states of the economy s. The set of aggregate states willbe specified below.

Equation (4) defines the intertemporal participation constraint: the value of contin-uing the contract for the entrepreneur, at each point in time s, must always be greaterthan or equal to the value of repudiating it. The repudiation value is equal to the cashflow generated by the firm, plus the value of starting a new investment project, that is,Dz(s, k, η) = zF (k, η) + V 0(s) − κ. The contract will be structured so that the incentivecompatibility constraint is never violated and repudiation never arises in equilibrium.

The initial weight λt affects the value of the contract for the two contractual parties.This weight is determined by the relative contractual power of the two parties and it is thesame for all contracts signed in the same period t. However, contracts signed in differentperiods will have different weights. The determination of λt will be specified in Section 4.

After writing this problem in Lagrangian form, with γs the Lagrange multiplier at times associated with the incentive compatibility constraint (4), the planner’s problem takesthe following saddle-point formulation:

minµs+1∞s=t

maxds,τs,ks+1∞s=t

Et

∞∑s=t

βs−t[µs+1ds + τs − (µs+1 − µs)Dz(ss, ks, ηs)

](7)

subject to

ks+1 = zF (ks, ηs) + (1− δ − w(ss))ks − ds − τs (8)

µs+1 = µs + γs (9)

ds ≥ 0, kt = 0, µt = λt (10)

By Theorem 1 in Marcet and Marimon [15], a solution to the saddle point problem is asolution to the original planner’s problem.7 Of particular interest is the co-state variableµ that evolves according to µs+1 = µs + γs. That is, µ increases when the Lagrangemultiplier γs is positive, which happens when the intertemporal participation constraint(4) is binding. The saddle point formulation shows how the planner assigns variableweights to the entrepreneur and the intermediary along an accumulation path: it startswith µt = λt and increases the weight when the enforceability constraint is binding. Thisproperty has a very simple intuition. The weight used by the planner determines the valueof the contract for the entrepreneur: the larger is this weight, the higher is the value of thecontract for the entrepreneur. When the enforcement condition is binding, the value of thecontract for the entrepreneur is smaller than the repudiation value. In this circumstance,

7Theorem 1 in Marcet and Marimon [15] is a sufficiency theorem and our model clearly satisfies therequired assumptions. However, it does not satisfy the convexity assumptions needed to guarantee thatall solutions of the original planner’s problem can be obtained as solutions to the corresponding saddlepoint problem: the function −Dz(s, ·, η) fails to be quasiconcave.

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to prevent repudiation, the promised value must increase. The way to increase this valueis by attributing a larger weight to the entrepreneur, that is, by increasing µs+1.

From the saddle-point formulation, we can rewrite the problem recursively as follows:

Wz(s, k, µ, η) = minµ′

maxd,τ,k′

µ′d+ τ − (µ′ − µ)Dz(s, k, η) + βEWz(s

′, k′, µ′, η′)

(11)

subject to

k′ = zF (k, η) + (1− δ − w(s))k − d− τ (12)

d ≥ 0, µ′ ≥ µ (13)

s′ ∼ H(s) (14)

where the prime denotes the next period variable and the function H is the distributionfunction for the next period aggregate states (law of motion), given the current states.The aggregate states are given by the distribution (measure) of firms over the variables z,k and µ, which we denote by M , and by the productivity of new investment projects Z(we use the capital letter to distinguish it from the productivity of an existing project z).Therefore, s = (Z,M).

Before proceeding, we observe that for λt > 1, problem (3) is not well defined. This isbecause the planner would attribute more weight to the entrepreneur and always prefers toshift resources from the intermediary to the entrepreneur (remember that τ is unboundedbelow). So it will be optimal to ask for an infinite amount of transfers to the entrepreneur.This also implies that µ in the recursive formulation cannot be larger than 1.

3.1 Characterization of the optimal contract

Conditional on the survival of the firm, the optimization solution is characterized by thefollowing first order conditions:

µ′ − 1 ≤ 0, (= if d > 0) (15)

Dz(s, k, η)− β∂EWz(s

′, k′, µ′, η′)

∂µ′− d ≤ 0, (= if µ′ > µ) (16)

βE

[z∂F (k′, η′)

∂k′+ 1− δ − w(s′)− (µ′′ − µ′)

∂Dz(s′, k′, η′)

∂k′

]− 1 = 0 (17)

Patterns of firms’ growth: Our underlying benchmark model has a very simple pat-tern of growth when contracts can be fully enforced. In this economy enforceability con-straints are never binding, i.e., the enforcement technology makes the repudiation value

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sufficiently low. Conditions (15)-(17), provide an immediate characterization of such aneconomy with µ′′ = µ′ = µ = λt. In particular, condition (16) plays no role in determiningthe process of capital accumulation and projects are financed to achieve their optimalinput of capital as determined by equation (17). This optimal input of capital will bedenoted by kz(s).

8 Moreover, (15) shows that, unless λt = 1, the intermediary receives allthe rents. Of course, competition in the intermediation sector guarantees that, in equilib-rium, intermediaries and entrepreneurs are equally weighted by the planner, i.e., λt = 1.In this case the distribution of dividends is undetermined, meaning that the division ofthe surplus between the two contractual parties can be obtained with a multiplicity ofschemes.

In contrast, an economy in which contracts are not fully enforceable experiences avery different pattern of growth. The fact that enforceability constraints are likely tobe binding in the future means—by condition (17)—that the entrepreneur cannot startthe contract with the optimal level of capital, as in the economy with fully enforceablecontracts. Furthermore, in those periods in which the enforceability constraint is binding,condition (16) is satisfied with equality (and zero dividends, unless the unconstrainedstatus is reached that period). The pattern of growth will then be determined by thiscondition. In other words, whenever enforceability constraints are binding, the relevanttechnology is the “outside option technology”, Dz(s, k, η) = zF (k, η) + V 0(s) − κ, notthe “firm’s technology”, zF (k, η) + (1 − δ − w(s))k. Furthermore, as we will show inthe next section, whenever the enforceability constraint is not binding, firms do not growon average. Once the variable µ reaches the value of 1, the structure of the contract issimilar to a contract with full enforceability. This is the state in which the firm becomesunconstrained.

In summary, the pattern of firms’ growth is markedly different when there are enforce-ability problems. In particular, there is a process of accumulation, not just a jump to theunconstrained level of capital. This accumulation process captures two features that werementioned in the introduction: conditional on the initial size, small firms grow faster thanlarge firms and small firms are the ones that are financially constrained. As we will seelater, this pattern of growth will be important for the propagation of aggregate shocks tothe economy.

Dividend policy and binding constraints: Condition (15) tells us that if some divi-dend is paid to the entrepreneur, then µ′ must be set to 1. Condition (16) imposes a limitto the firm’s growth. When the enforceability condition is binding (that is, Dz(s, k, η)is greater than the expected discounted value of dividends), then µ′ > µ and the nextperiod stock of capital grows at the rate that satisfies equation (16) with equality. Whenthe enforceability condition is not binding, µ′ = µ and (16) can be satisfied with the in-equality sign. Because the repudiation value is increasing in the value of the idiosyncraticshock, the enforcement condition is binding only for values of the shock above a certain

8The optimal input of capital still depends on the aggregate states because the wage variable affectsthe marginal profit with respect to the input of capital and the wage is affected by the aggregate states.

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threshold. This threshold depends on the aggregate and individual states and we denoteit by η(s, k, µ). Therefore, for η ≤ η(s, k, µ) the enforcement condition is not binding andµ′ = µ. In summary, surviving firms experience a growth pattern characterized by bind-ing enforcement constraints—when shocks are sufficiently high—and along such paths nodividends are paid to the entrepreneur. A feature which is a first approximation to theobserved fact that small firms are less likely to distribute dividends than large firms. Asimilar result is also obtained in Albuquerque and Hopenhayn [1], Cooley and Quadrini[6] and Quadrini [18].

Investment policy: An important property of this model is that the investment ofconstrained firms is sensitive to cash flows. Before showing this result, however, we firststate a property of the model that will be used in characterizing the investment policy.

Lemma 3.1 The next period capital k′ is fully determined by the variables (s, k, η) andthere is a map µ′ = ψ(s, k′) that satisfies conditions (15)-(17), with ∂ψ(s, k′)/∂k′ > 0.

Proof 3.1 See the appendix.

According to this lemma, the capital stock and the idiosyncratic shock (along with theaggregate states) are sufficient statistics for the characterization of the contract. Moreover,for given states s, there exists an increasing function ψ that uniquely relates µ′ to k′. Atypical shape of the function ψ is plotted in Figure 1. The function is constructed for aparticular value of z and for given aggregate states s. This result is important because itallows us to reduce the individual states of an optimal contract to k or µ instead of (k, µ)(once we know the function ψ). We would like to point out, however, that this propertycannot be generalized to other models because it derives from the linearity of preferencesleading to the result that equation (16) does not depend on µ.

A corollary to the above lemma states the dependence of investment on cash flows forconstrained firms.

Corollary 3.1 Controlling for the aggregate states, the next period stock of capital forconstrained firms is increasing in zF (k, η) if the enforceability constraint is binding andremains constant if the enforceability constraint is not binding.

Proof 3.1 In lemma 3.1 we have seen that the next period stock of capital depends only on(s, k, η). Moreover, when the enforceability constraint is not binding, µ′ = µ and k′ = k, ifwe control for s. If the enforceability constraint is binding, instead, the next period capitalis determined by equation (16) after setting d = 0, until k′ reaches kz(s). It is then clearthat k′ is increasing in zF (k, η). Q.E.D.

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1

kz(s)

6

k′

µ′

•

Figure 1: Relation between the next period state µ′ and capital k′, given project-specificproductivity z and aggregate states s.

The cash flows sensitivity of investment has a simple intuition. The increase in revenuesinduced by the increase in the stock of capital, translates into an increase in the value ofthe entrepreneur (through the increase in the value of repudiating the contract) and adecrease in the value for the intermediary. Because the planner is giving a larger weightto the intermediary than to the entrepreneur, it prefers not to expand the stock of capital,unless it must do so to prevent repudiation. If the firm survives enough periods, µ′

converges to 1. At this point the firm is unconstrained and the input of capital is alwayskept at the optimal level kz(s). This implies that the investment of unconstrained firmsis no longer sensitive to cash flows.

The cash flows sensitivity of investment derives from the assumption that capital ischosen one period in advance, before observing the shock. With respect to this feature, ourmodel differs from Albuquerque and Hopenhayn [1] who instead assume that investmentis chosen in the current period after the observation of the shock. In their model, oncewe control for the current size of the firm and their future profitability (by assuming i.i.d.shocks, for example), investment is no longer sensitive to cash flows.

Life-cycle of the financial contract: Once the firm reaches the unconstrained status,the structure of the contract becomes undetermined. One possible way to structure thecontract at this stage is as follows: Define Rz(s−1, s, η) = zF (kz(s−1), η)−(δ+w(s))kz(s−1)to be the net profits of the firm. Then the payments to the intermediary are:

τ =

τ , if Rz(s−1, s, η) > τRz(s−1, s, η), if Rz(s−1, s, η) < τ

(18)

Basically, the intermediary will receive a constant flow of payments if the firm is able

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to make these payments. It will receive a smaller payment or even a loss if the firm isunable to make the payment τ . This structure of the contract resembles a debt contractin which the lender receives a constant flow of interest payments on the loan unless thefirm is unable to fulfill its obligations. Of course, the payment τ is such that the lenderwill expect a certain return from the loan. It also resembles the dividend payments tothe shareholders of large public companies. Despite very high volatile profits, these firmsprefer to distribute a sufficiently smooth flow of dividends to their shareholders. Therefore,the optimal financial contract follows a pattern which is typical of the life-cycle of firmsfinanced through venture capital. These firms start with limited funds. These funds areincreased subsequently if the firms are successful, until they go public. At this point theyfinance part of the capital with new issues of shares and with debt.

Another possible structure of the contract once the firm reaches the unconstrained sizeis to continue repaying the financial intermediary until the entrepreneur has a sufficientlylarge credit in the intermediary. At this point the intermediary will pay a constant flow ofinterests in every period to the firm. These payments are sufficient to cover any possiblelosses that the firm can realize. This contract, however, would not be feasible if theintermediary is unable to commit to future obligations. In fact, under these circumstances,the value of the contract for the intermediary is negative and it would have an incentiveto repudiate the contract.

4 Value of a new firm and general equilibrium

The analysis conducted in the previous section takes as given the initial weight λt. Asanticipated, this weight affects the distribution of the surplus of the contract between thetwo contractual parties. Fixing λt is equivalent to fixing the initial values of the contractfor the entrepreneur and the intermediary. Define V E(s, k, µ, η) the value of the contractfor the entrepreneur given the states s, the capital k, co-state µ and realization of theidiosyncratic shock η. Similarly, define V I(s, k, µ, η) to be the value of the contract for theintermediary. The following lemma defines these functions.

Lemma 4.1 Assume that W (s, k, µ, η) is well defined and let η(s, k, µ) be the value of theshock below which the enforceability constraint is not binding. Then for given (s, k, µ, η),the values of the contract for the entrepreneur and the intermediary are:

V Ez (s, k, µ, η) =

Dz(s, k, η(s, k, µ)) if η ≤ η(s, k, µ)

Dz(s, k, η) if η > η(s, k, µ)(19)

V Iz (s, k, µ, η) = Wz(s, k, µ, η)− µV E

z (s, k, µ, η) (20)

Proof 4.1 See the appendix.

13

After defining the value of a contract for the entrepreneur and the intermediary forgiven (s, k, µ, η), we can now define the current expected values of a new contract signedat time t, with initial kt+1 and µt+1. By lemma 3.1, we know that there is a uniquecorrespondence between µt+1 and kt+1, that is, µt+1 = ψ(st, kt+1). Therefore the value ofa new contract can be written as:

V E(st, kt+1) = βEV Ez (st+1, kt+1, ψ(st, kt+1), ηt+1) (21)

V I(st, kt+1) = βEV Iz (st+1, kt+1, ψ(st, kt+1), ηt+1) (22)

Notice that V I is the value of the contract for the intermediary after the anticipationof the initial capital. Before anticipating this capital, the value of a new contract isV I(st, kt+1)− kt+1.

The initial value of kt+1 depends on the contractual power of the two parties. Byassuming that financial markets are competitive, kt+1 solves the problem:

V 0(st) = maxkt+1

V E(st, kt+1) (23)

s.t.

V I(st, kt+1)− kt+1 ≥ 0 (24)

The next proposition states the uniqueness of the solution.

Proposition 4.1 If an optimal contract exists, the solution to (23) is unique and satisfiesV I(st, kt+1)− kt+1 = 0.

Proof 4.1 It is enough to show that the function V E is strictly increasing for all kt+1 <kz(st). From the definition of V E

z (s, k, µ, η) of lemma 4.1, we know that this function isweakly increasing in k for each value of η and strictly increasing for some η. This impliesthat the expected value of V E, with respect to η, is strictly increasing in k. Q.E.D.

The determination of the initial value of kt+1, is shown in the first panel of Figure2. The figure plots the values of V E(st, kt+1) and V I(st, kt+1) − kt+1 as a function of theinitial input of capital, for given aggregate states. The second function has been assumedto be decreasing for all values of k. This, however, does not have to be the case. Theinitial input of capital is given by the point in which the lender’s value of the contractcrosses the horizontal axis. This is the point that maximizes the value of the contract forthe entrepreneur, without violating the non-negativity of the value of the contract for theintermediary. This value is denoted by k0

t+1.The figure also shows another interesting feature of the model. The initial input of

capital k0t+1 can be interpreted as the maximum value that the entrepreneur can initially

borrow. However, if the entrepreneur had some funds to finance the initial investment,then the contract could have been started with a higher value of capital. To increase the

14

0.0

Entrepreneur’svalue

Lender’svalue

kt+1k0t+1

PPPPPPPPPPPPP

r -

6

Entrepreneur’sassets

Capitalfinancedby lender

-

6

a) Initial capital b) Borrowing limit

Figure 2: Initial input of capital and borrowing limit for a new contract.

initial input of capital, the entrepreneur needs to contribute for the difference betweenthe horizontal axis and the value of the contract for the intermediary. Therefore, we canidentify a relation between the capital financed by the intermediary and the initial assetsof the entrepreneur. This relation, plotted in the second panel of Figure 2, defines theinitial borrowing limit for the entrepreneur.

Given the monotonicity of the function ψ, choosing k0t+1 is equivalent to the choice of

µ0t+1. The relation between the initial co-state µ0

t+1 and the weight assigned by the plannerto the entrepreneur in problem (3) is now clear: the planner’s weight λt is simply equal toµ0

t+1.9

To derive V 0(s), we have used functions that depend on V 0(s). Therefore, to solve forthe equilibrium we have to solve for a non-trivial fixed point problem. In general we canthink of this fixed point as the solution to a mapping T that maps a set of functions V (s)into itself, that is:

Vs+1(s) = T (Vs)(s) (25)

This mapping is based on the assumption that after starting a project, agents believethat the value of repudiating the contract and starting a new investment project is given bythe function Vs(s). Given these beliefs, the value of starting a project for new entrepreneursis Vs+1. This mapping embeds all the general equilibrium properties of the model, that is,the clearing conditions in the labor and financial markets. The fixed point of this mappingdefines the equilibrium of this economy.

9In writing the saddle-point formulation (7), we imposed the constraint µt = λt. However, if the initialweight is such that the enforceability constraint is not initially binding, then µt+1 = µt. If instead λt issuch that the enforceability constraint is initially binding, then µt+1 = µt + γt. However, the Lagrangemultiplier γt is perfectly known at time t (there is no production and the repudiation value does notdepend on η). Consequently, setting a smaller λt is equivalent to setting the initial weight to λt + γt fromthe beginning. Therefore, we can write λt = µ0

t+1.

15

Definition 4.1 (Recursive equilibrium) A recursive competitive equilibrium is definedas a set of functions for (i) labor supply l(s, a) and consumption c(s, a) from workers; (ii)dividend (consumption) rule d = d(s, k, µ, η), investment rule k′ = k(s, k, µ, η) and func-tion µ′ = ψ(s, k′); (iii) new firm’s value V 0(s); (iv) wage w(s); (v) aggregate demand oflabor from firms and aggregate supply from workers; (vi) aggregate investment from firmsand aggregate savings from workers and entrepreneurs (intermediated by financial inter-mediaries); (vii) distribution function (law of motion) for the next period states s′ ∼ H(s).Such that: (i) the household’s decisions are optimal; (ii) the dividend and investment rulesand the function ψ satisfy the optimality conditions of the financial contract (conditions(15)-(17)); (iii) the value of a new firm is the fixed point of (25); (iv) the wage is theequilibrium clearing price in the labor market; (v) the capital market clears (investmentequals savings); (vi) the distribution function for the next period states is consistent withthe solution of the optimal contract and the stochastic process for the aggregate shock.

4.1 Steady state equilibrium

Proving the existence of an equilibrium is equivalent to proving the existence of a fixedpoint of (25). This is a difficult task because V 0(s) is a function of the whole distribution offirms. In this section we prove the existence and uniqueness of a steady state equilibriumwhich is characterized by an invariant distribution of firms and by constant values of V 0(s)and w(s). Before this, however, we first state two lemmas that will be used below in theproof of the existence of the steady state equilibrium.

Lemma 4.2 Assume that the wage w is constant and z takes only one value. Then themapping T defined in (25) has a unique fixed point V 0. Moreover, for κ ∈ [κ, κ], V 0 isstrictly positive.

Proof 4.2 See the appendix.

The existence of a unique fixed point is guaranteed by the fact that higher values ofV 0 expected for the future make the enforceability constraint tighter. This will reduce thevalue of a new contract for the entrepreneur and therefore T (V 0). The continuity of Tthen guarantees the existence and uniqueness of the fixed point.

Lemma 4.3 Given a constant w, there exists a unique invariant distribution of firms M .

Proof 4.3 It is sufficient to show that the transition function satisfies the conditions ofTheorem 12.12 in Stokey and Lucas [21] (monotonicity and mixing condition). Q.E.D.

We then have the following proposition.

16

Proposition 4.2 There exists a unique steady-state equilibrium.

Proof 4.2 According to lemma 4.3, for each w there exists a unique invariant distributionof firms with associated aggregate demand of labor. If we increase w the demand of laborassociated with the new invariant distribution decreases. On the other hand, the supply oflabor is implicitly defined by ϕ′(l) = wξ, and therefore, is increasing in w. This impliesthat there exists a unique value of w that clears the labor market and defines the uniquesteady state equilibrium. Q.E.D.

We can prove the existence of an equilibrium for the economy with aggregate shocks inthe special case in which the function ϕ is linear. In this case the equilibrium wage rate willbe constant and the function V 0 only depends on Z, that is, V 0(Z). The distribution offirms is only important for the demands of labor and capital. Given the simple structure ofthe utility function, labor and capital are demand determined (at the equilibrium constantprices w and r).

Proposition 4.3 With elastic labor supply the mapping (25) has a unique fixed pointV 0(Z), and therefore, a unique equilibrium.

Proof 4.3 The constancy of w implies that we can neglect the distribution of firms incharacterizing an optimal contract. Then the proof follows the same steps of the proof oflemma 4.2. Q.E.D.

4.2 Lender’s renegotiation

In the analysis conducted up to this point we assumed that the intermediary is able tocommit to future obligations, and therefore, it will not repudiate the contract even ifits value becomes negative. We now study the conditions under which the value of thecontract for the intermediary will never be negative. As for the proof of the existence of anequilibrium with aggregate shocks, it is difficult to find these conditions for any possiblerealization of the aggregate shock. Therefore, in this section we will concentrate on thesteady state equilibrium. The results, however, should hold for small deviations from thesteady state.

Remember that, according to corollary 3.1, in a steady state the stock of capital neverdecreases. This implies that in case the firm realizes losses, these losses have to be coveredby the intermediary. In deciding whether to cover these losses or repudiate the contract,the intermediary compares the current losses with the discounted values of future paymentsit expects to receive. The value of the contract for the intermediary can be written as:

τ(k, η) + V I(k′) (26)

17

where the function V I was defined in (22). Given that η ≥ 0, the minimum value that τcan take is −(δ + w)k or, equivalently, the maximum losses that the intermediary has tocover in the current period are (δ + w)k. Therefore, the condition that guarantees thatthe intermediary will never repudiate the contract is

(δ + w)k ≤ V I(k) (27)

Notice that in the function V I we have set k′ = k because when η = 0, the enforceabilityconstraint is not binding in the optimal contract. Then, according to corollary 3.1, k′ = k.

The next proposition states a condition that guarantees that the intermediary willnever repudiate the contract.

Proposition 4.4 If β(k+κ)− (1+β)(δ+w)k > 0, then condition (27) is always satisfiedand the intermediary will never repudiate the contract.

Proof 4.4 See the appendix.

Notice that this is a sufficient condition. Therefore, even if β(k + κ) − (1 + β)(δ +w)k < 0, the intermediary may still not have an incentive to repudiate the contract.For all numerical exercises conducted in this paper, the non-repudiation condition for theintermediary will always be satisfied.

5 Contrasting economies with and without contract enforceability

In the previous sections we have characterized some of the analytical properties of theeconomy with and without contract enforceability. In this section we further study theproperties of these two versions of the economy numerically. In Section 5.1 we parameterizethe model. In Section 5.2 we study the steady state properties and evaluate the welfarelosses associated with limited enforceability. In Section 5.3 we study the impulse responsesto aggregate shocks.

5.1 Parameterization

The period in the economy is one year and the intertemporal discount rate (equal to theinterest rate), is set to r = 0.04. The survival probability is α = 0.99. The disutility fromworking takes the form ϕ(l) = π ·lν . The parameter ν affects the size of the aggregate econ-omy with and without financial frictions, which is important for the welfare computations.This parameter is also important for the size response of output to shocks: the smallerthe value of ν (the more elastic is the supply of labor) and the larger is the response ofoutput. However, the shape of the impulse response to shocks is not affected significantlyby this parameter. In the baseline model we set ν = 1.1 and in the welfare calculations wewill conduct a sensitivity analysis. After fixing ν, the parameter π is chosen so that one

18

third of available time is spent working. The mapping from π to the working time will bedescribed below.

The production function is specified as F (k, η) = ηkθ. The parameter θ is assigned thevalue of 0.975. The shock is distributed according to an exponential density function, thatis, G(dη) = e−η/ε

ε. The choice of this function is made only for its analytical simplicity. The

production technology becomes unproductive with probability 1 − φ = 0.04. Associatedwith the 1 percent probability that the entrepreneur dies, the exit probability of firms isabout 5 percent.

We would like the steady state of the economy to have a capital-output ratio of 2.8and a labor income share of 0.6. These indices are complicated functions of the wholedistribution of firms. However, because most of the aggregate output is produced byunconstrained firms, we can choose the parameter values so that these numbers are re-produced by unconstrained firms. More specifically we impose that k/E(η)(k)θ = 2.8 andwk/E(η)(k)θ = 0.6. After normalizing the capital stock of unconstrained firms to k = 1,a value of 2.8 for the capital-output ratio implies E(η) = 0.4. This condition pins downthe parameter ε in the distribution function of the shock.

The capital input of unconstrained firms is given by the following expression:

k =

(βθE(η)

1− β(1− δ − w)

) 11−θ

= 1

which is equal to 1 because we have normalized k = 1. After observing that the wagevariable is equal to the ratio between the labor share and the capital-output ratio, thatis, w = 0.6/2.8, this expression determines the parameter δ. The value found is 0.037.Because in each period about 5% of firms exit the market and the capital of these firmsfully depreciates, the depreciation rate for the aggregate stock of capital is about 0.085.

Given the parameterization of the production sector, and the implied wage variable w,the model generates a stationary distribution of firms and an aggregate demand of labor.The parameter κ affects the size of new firms. We set κ so that the initial stock of capitalfor new firms is about 20 times smaller than unconstrained firms. Then the capital-laborratio ξ and the utility parameter π are determined so that in the steady state equilibriumeach worker spends 1/3 of available time working and unconstrained firms employ 1,000workers. This implies ξ = 1/(1, 000 · 0.33) = 0.0033. The number of workers employedby unconstrained firms is not important. The results would not change if we choose adifferent number. To pin down the parameter π, remember that the actual wage rate(per unit of labor) is wξ. The supply of labor is determined by the worker’s first ordercondition νπlν−1 = wξ. Given w, ξ and ν, the condition l = 0.33 will then pin down π.Finally, the mass of new firms (newborn agents with entrepreneurial skills, e), is such thatthe aggregate supply of labor is equal to the aggregate demand. The full set of parametervalues are in table 1.

19

Table 1: Parameter values.

Intertemporal discount rate r = 0.040Disutility from working ϕ(l) ≡ π · lν ν = 1.100

π = 0.001Survival probability of agents α = 0.990Survival probability of projects φ = 0.960Production technology F (k, η) ≡ η · (mink, ξ · l)θ, η ∼ e−η/ε

ε ξ = 0.003θ = 0.975ε = 0.357

Depreciation rate δ = 0.037Cost of repudiation κ = 0.026

5.2 Steady state equilibrium and the efficiency loss due to limited enforce-ability

A steady state is obtained when the productivity level z takes only one value. In thissection, however, we follow a different strategy. We keep the assumption that z takes twovalues and we assume that in each period one half of the new firms have high productivityand the other half low productivity.

Figure 3 plots the distribution of firms in the economy with full enforceability. In thiseconomy the optimal contract determines the input of capital that maximizes the surplusof the firm for each value of z, independently of the distribution of this surplus betweenthe entrepreneur and the intermediary. Consequently, the distribution is characterizedby a mass of firms concentrated in the optimal input of capital associated with the lowproductivity level (panel a) and the high productivity level (panel b).

Figure 3: Steady state distribution of firms in the economy with full enforceability.

In the economy with limited enforceability, we still have that half of the firms have lowproductivity and the other half have high productivity. However, within the class of firmswith the same level of productivity, firms employ different inputs of capital. The steady

20

state distribution is plotted in Figure 4. Panel a plots the distribution of low productivityfirms and panel b the distribution of high productivity firms. If the wage rate in theeconomy with limited enforceability was the same as the wage rate in the economy withfull enforceability, the optimal input of capital employed by unconstrained firms wouldbe equal to the capital employed by the firms operating in the full enforcement economy.However, due to the fact that some of the firms are constrained, the demand of capitaland labor will be smaller in the economy with limited enforceability. This implies that thewage rate is smaller and the optimal size of the firms larger. The average size of firms,however, is smaller in the economy with limited enforceability.

Figure 4: Steady state distribution of firms in the economy with limited enforceability.

Figure 5 plots the values of a new contract for the entrepreneur and the intermediaryfor the low (panel a) and high productivity (panel b). These are the functions V E(z, k)and V I(z, k) − k defined in (21) and (22). As can be seen from the figure, the valueof the contract is increasing in the initial stock of capital for the entrepreneur but it isdecreasing for the intermediary. The assumption of competitive financial markets impliesthat in equilibrium the value of the contract for the intermediary is zero (zero profitcondition).

Figure 5: Values of a new contract as a function of the initial stock of capital.

21

In order to consider the possibility that the intermediary repudiates the contract,Figure 6 plots the two components of the non-repudiation condition (27). The continuationvalue of the contract is V I(z, k) and the maximum current payment is −τ = (δ +w)k. Indeciding whether to renegotiate the contract, the intermediary will compare the currentpayments (if τ is negative) with the value of continuing the contract. As can be seen fromthe figure, for any possible size of the firm, the maximum payments for the intermediaryare smaller than the value of continuing the contract. Therefore, the intermediary willnever repudiate the contract.

Figure 6: Maximum payments from the intermediary and continuation value.

The first panel of Figure 7 plots the ratio of the value of the contract for the interme-diary (after the anticipation of the capital and before observing the shock) and the stockof capital. If we interpret the value of the contract for the intermediary as the liabilitiesof the firm, this ratio is an indicator of the firm’s leverage. This leverage decreases as thefirm grows. The second panel of Figure 7 plots the expected growth rate of capital forfirms of different size. The growth rate is decreasing in the size of the firm and more ingeneral it is bigger for constrained (small) firms. This feature of the optimal contract willbe important in shaping the response of the economy to aggregate shocks as we will seein the next section.

Table 2 reports the welfare gains from making contracts fully enforceable along withsome key statistics of the steady state competitive allocations in the economies with lim-ited and full enforceability. Starting from the steady state of the economy with limitedenforceability, we assume that contracts become fully enforceable. This unanticipatedchange brings the economy to a new equilibrium in which all firms with the same produc-tivity employ the same input of capital. The transition dynamics takes only one period.The welfare gains are computed as the increase in consumption every period necessaryto make all agents indifferent between staying in the economy with limited enforceability(but with the consumption increase) and in the transition to the steady state with fullenforceability.

Different values of the parameter ν are considered: the larger the elasticity of labor is(smaller the value of ν), the larger is the effect of contract enforceability for the macro

22

Figure 7: Leverage ratio and growth rate of firms as function of the initial stock of capital.

allocation of the economy. This is because, when labor is elastic, the increase in thesupply of labor allows previously constrained firms to expand production which in turnallows the increase in consumption. On the other hand, when labor is not very elastic,the expansion in the production scale of previously constrained firms is prevented by theincrease in the wage rate. In this case, the increase in production and consumption is notlarge. Independently of the value of ν, the welfare gains are between 0.5 and 1 percentof consumption. The table also reports the comparison between the steady state welfarelevels (which neglects the transition). Looking only at the steady state welfare, the gainsfrom contract enforceability are larger and they increase with the elasticity of the laborsupply. The reason the welfare numbers are smaller when the transition is considered, isbecause the shift to the new equilibrium requires a large initial investment which initiallyreduces consumption.10

5.3 Contract enforceability and the transmission of shocks

In this section we show how limited enforceability affects the propagation of aggregateshocks to the economy. In particular, we show that limited enforceability generates serialcorrelation in output growth and amplifies the response of the economy to aggregateshocks. Our theory predicts that countries with higher degrees of contracts enforceabilitydisplay less volatility of output than countries where enforceability is weaker. This appearsto have some support in the data as shown in Figure 8. This figure plots the position of 66countries with respect to two variables: the “standard deviation of GDP growth” and the“efficiency of the judiciary system”. The efficiency of the judiciary system, which proxiesfor the enforcement of contracts, is compiled by Business International and it is taken fromMauro [16]. It takes values between 1 and 10 and the values plotted are for the period1980-83. The data for GDP growth is from the Penn World Table for the period 1960-92.As can be seen from the figure, there is a negative association between the volatility of

10Quintin [19] also studies how contract enforceability affects the size distribution of firms and welfarewith special application to Latin America countries.

23

Table 2: Properties of the economy with limited and full enforceability.

Limited FullEnforceability Enforceability

ν = 1.05Average size (capital) 0.69 0.81Working time 0.33 0.39Steady state output loss 15.85%Steady state welfare loss 3.30%Welfare gain from transition 0.83%

ν = 1.10Average size (capital) 0.69 0.77Working time 0.33 0.37Steady state output loss 10.69%Steady state welfare loss 2.44%Welfare gain from dynamics 0.79%

ν = 1.50Average size (capital) 0.69 0.71Working time 0.33 0.34Steady state output loss 3.28%Steady state welfare loss 1.20%Welfare gain from transition 0.74%

output growth and the quality of the judiciary system. In particular, all the countrieswith a value of 10 (the highest efficiency) experienced moderate volatility in GDP growth.

To show how contract enforceability affects the response of the economy to aggregateshocks, we conduct the following experiment. Starting from the steady state equilibriumas previously described, we consider a temporary shock in which all new projects have thehigh productivity. The shock is not persistent, which implies that from the next period ononly half of the new projects will have high productivity. Three versions of the economyare considered: (a) full enforceability; (b) limited enforceability with market exclusion; (c)limited enforceability without market exclusion.

These three versions of the economy corresponds to three different degrees of contractenforceability. In the second economy, the entrepreneur has the ability to repudiate thecontract and consume the current cash flows. However, he or she cannot enter into anew contractual relationship once he or she has repudiated the contract. Therefore, therepudiation value is simply equal to D(s, k, η) = zF (k, η). If we interpret the cost κ as theclaim that the previous lender has on the new financial contract in case of repudiation, thenthe market exclusion can be obtained by assuming a sufficiently large κ. In the third versionof the economy an entrepreneur who repudiates the contract is not excluded from thefinancial market. Consequently, as long as the value of a new financial contract is positive,

24

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 2 4 6 8 10 12

Efficiency judiciary system

Sta

ndar

d de

viat

ion

GD

P g

row

th

Source: The Penn World Table 1960-1992 for GDP; Mauro (1995) for Judiciary System ranking.

Figure 8: Efficiency of judiciary system and volatility in GDP growth.

the repudiation value is D(s, k, η) = zF (k, η) + V 0(s) − κ. Broadly speaking, we mayinterpret the second version of the economy as representative of the enforcement system ofmore developed countries and the third version as representative of the enforcement systemof developing countries. The responses of output in the three versions of the economy areplotted in Figure 9.

The response of output in the economy with full enforceability increases at impact andthen returns monotonically to the steady state. This pattern is driven by the entranceof new firms which have higher productivity. After their entry, the proportion of highproductivity firms increases. Due to exit, however, this proportion slowly returns tothe steady state value of 0.5. In this economy aggregate output is closely related tothe proportion of high productivity firms and follows a similar path. This implies thataggregate output does not continue to grow beyond the first period.

The impulse responses change dramatically in the economies with limited enforceabil-ity. Independently of whether repudiating entrepreneurs are excluded or not from thefinancial market, the response of aggregate output is hump-shaped and its growth rate isserially correlated. Within the economies with limited enforceability, market exclusion isimportant for the amplification of the shock. As can be seen from the figure, the responseof output in the economy without market exclusion is much larger than the response of theeconomy with market exclusion. Therefore, lower is the degree of contract enforceability,and larger is the macro impact of the aggregate shock.

The mechanism that generates the serial correlation in output growth derives from thedynamics of new firms. After the shock new firms are more productive and this higherproductivity increases output. However, in contrast to the economy with enforceable

25

Figure 9: Impulse response of output to a temporary positive shock.

contracts, the impact on output will be gradual. This is because new firms will start witha smaller input of capital. This capital, however, will grow in subsequent periods andthis will allow for further growth in aggregate output. In contrast, in the economy withenforceable contracts, new firms start from the beginning with the optimal input of capitaland the largest impact on output will be in the first period of the shock.

To describe the mechanism that generates amplification, let’s compare the economywith full enforceability and the economy with limited enforceability and without exclusion(economies (a) and (c)). In the full enforcement economy, the expansion of aggregateoutput only derives from the higher productivity of new firms. New firms are also moreproductive in the economy with limited enforceability. In addition to this, however, theeconomy with limited enforceability expands because the repudiation value of constrainedentrepreneurs increases. To see this, consider condition (16). For those firms with bindingenforceability constraints and µ′ < 1, this condition becomes:

zF (k, η) + V 0(s)− κ = βEzF (k′, η′) + V 0(s′)− κ

(28)

The shock will increase V 0(s) but, because temporary, it will have only a small effecton EV 0(s′). This implies that the left-hand-side increases more than the right-hand-side.To re-establish the equality, k′ must increase. Therefore, the impact of the shock is equiv-alent to lessening the financial constraints of some entrepreneurs (the constrained ones)which allow them to invest more. This mechanism is absent in the economy with fullenforceability because all firms are unconstrained. This mechanism is also absent in theeconomy with limited enforceability and market exclusion. In this economy, the produc-tivity of new projects does not affect directly the repudiation value of the entrepreneur

26

because entrepreneurs are not allowed (or do not find convenient) to start a new investmentproject.

To compare the performance of new and incumbent firms, Figure 10 plots the patternof the firm value for these two categories of firms after a shock. The value of the firm isthe total value of the contract, that is, the sum of the value for the entrepreneur and theintermediary, and the plotted numbers are relative to the steady state path. As can beseen from the figure, the value of new firms increases while the value of incumbent firmsfalls below the steady state values, with the exception of the first few periods. The initialincrease in the value of incumbent firms comes from the response of the incumbent firmsthat are constrained. As we have seen above, the shock induces an expansion of constrainedfirms and this expansion increases their value. The increase in the value of constrainedfirms is larger than the decrease in the value of unconstrained firms, and the total valueof incumbent firms initially increases. This effect, however, lasts only for few periods afterwhich is the decline in the value of unconstrained firms that dominates. The value ofunconstrained firms declines because the entrance of more productive firms increases thedemand of labor and drives the wage up. For incumbent firms the wage increase reducesprofits which in turn reduces the value of the firm. This pattern is consistent with theinterpretation of Greenwood and Jovanovic [8] about the IT revolution. According to theirinterpretation new firms had better technologies while incumbent firms were tied to lessefficient technologies. This allowed new firms to out-perform incumbent firms.

Figure 10: Total value of new and incumbent firms after a temporary shock.

To summarize, limited enforceability generates a hump-shaped response of output toaggregate shocks. In addition, if repudiating entrepreneurs are not excluded from partic-ipating in financial markets, contract enforceability amplifies the shock. Therefore, lowerthe degree of contract enforceability is, and larger the impact of shocks to the economy.As long as contracts are less enforceable in developing countries, this may contribute togenerate the larger volatility of output in these countries as observed in the data.

27

6 Conclusion

We have studied an economy in which entrepreneurs finance investment by entering intolong-term relationships with financial intermediaries. Contracts are not fully enforceableand the incentive compatibility requirement makes investment dependent on the repudi-ation value of the entrepreneur, which is only binding for small and fast growing firms.In addition, the incentive compatibility requirement makes investment dependent on therealization of profits and the model generates the cash-flows sensitivity of investment. Inthis way the model captures patterns of industry evolution (small firms grow faster, tendto be financially constrained and distribute fewer dividends) and investment (the cash flowsensitivity) observed in empirical studies.

Limited enforceability is important for the macroeconomy for two reasons. On theone hand it impairs the efficient allocation of resources. On the other it generates serialcorrelation in output growth and amplifies the response of the economy to aggregateshocks. Firm heterogeneity, induced by market frictions, is key to these results.

28

A Appendix: Analytical proofs

Proof of lemma 3.1 First we show that the choice of (k′, µ′) only depends on the state variables(s, k, η), that is, µ is irrelevant once we know (s, k, η). In the case in which the enforceabilityconstraint is binding, equation (16) is satisfied with equality and this equation does not dependon µ. Therefore, in this case k′ is only a function of (s, k, η). A similar result holds if theenforceability condition is not binding. In this case µ′ = µ and k′ only depends on s. To seethis, denote by h(s, k′, µ′) = 0 equation (17) and assume that s = s−1 (the aggregate states donot change from the previous period). If this equation was satisfied in the previous period, thatis, h(s−1, k, µ) = 0, and µ′ = µ, then h(s = s−1, k

′, µ′ = µ) = 0 if k′ = k (and only if providedthe choice of k′ is uniquely determined). If s 6= s−1, then k′ 6= k. However, the change in nextperiod capital is only a function of the change in s. Therefore, independently of whether theenforceability constraint is binding or not, k′ only depends on (s, k, η). Then, given k′, equation(17) will determine µ′. Because (17) does not depend on µ and η, is then clear that there isa unique correspondence between k′ and µ′ which depends on s. Finally, the positive relationbetween k′ and µ′ is clear from inspecting (17), once we take into consideration that µ′ ≤ µ′′ ≤ 1and ∂F (k′, η′)/∂k′ is decreasing in k′. Q.E.D.

Proof of lemma 4.1 Let’s start with the proof of (19). From the original formulation (3),the value of an optimal contract for the entrepreneur must always be greater or equal to therepudiation value D(s, k, η). If µ < 1, it is optimal for the planner to minimize the payments tothe entrepreneur. This implies that the planner will never attribute to the entrepreneur a valuethat is greater than D(s, k, η) when the enforceability constraint is binding, that is, η > η(s, k, µ).On the other hand, we have seen in lemma 3.1 that, conditional on (s, k, µ), the values of k′

and µ′ are independent of the current realization of the shock for η ≤ η(s, k, µ). This impliesthat the conditions of the contract remain the same independently of whether η = η(s, k, µ) (theenforceability constraint is binding) or η < η(s, k, µ) (the enforceability constraint is not binding).Because the value of the entrepreneur is D(s, k, η(s, k, µ)) when η = η(s, k, µ), this must also bethe entrepreneur’s value when η < η(s, k, µ).

To prove (20), consider the recursive formulation (11). After summing and subtractingµV E(s, k, µ, η) and µ′βV E(s′, k′, µ′, η′), (11) can be rewritten as:

Wz(s, k, µ, η)− µV Ez (s, k, µ, η) = τ + µ′d− (µ′ − µ)D(s, k, η)− µV E(s, k, µ, η) +µ′βEV E(s′, k′, µ′, η′) + βE(Wz(s′, k′, µ′, η′)− µ′V E

z (s′, k′, µ′, η′))

For all values of µ′, this reduces to:

Wz(s, k, µ, η)− µV Ez (s, k, µ, η) = τ + βE(Wz(s′, k′, µ′, η′)− µ′V E

z (s′, k′, µ′, η′))

To see this consider first the case in which µ′ = µ. Under this case the result can be verified oncewe consider that V E

z (s, k, µ, η) = d + βEV Ez (s′, k′, µ′, η′). Consider now the case µ′ > µ. When

the enforceability constraint is binding we have already proved that V Ez (s, k, µ, η) = Dz(s, k, η),

or equivalently, d + βEV Ez (s′, k′, µ′, η′) = Dz(s, k, η). The substitution of these two expressions

will give the result.The last expression is a recursive formulation with current return given by the payment to

the intermediary τ . Therefore, Wz(s, k, µ, η)−µV Ez (s, k, µ, η) is the value of the contract for the

intermediary. Q.E.D.

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Proof of lemma 4.2 Consider V 01 < V 0

2 . The optimal solution with V 0 = V 02 is also feasible

(although not optimal) when V 0 = V 01 . In fact, constraint (4) will not be violated if we replace

V 02 with V 0

2 . Therefore, the initial value of the contract for the entrepreneur under V 01 must be

at least as big as the value under V 02 . Because there is some contingency in which the solution is

binding when V 0 = V 02 but it is not binding if we replace V 0

2 with V 01 , then we can find a deviation

that is feasible under V 0 = V 01 and increases the value of the entrepreneur without changing the

value of the intermediary. Q.E.D.

Proof of proposition 4.4 Let’s prove first that if β(k + κ) − (1 + β)(δ + w)k > 0, thenfor unconstrained firms the non-repudiation condition of the intermediary is satisfied, that is,V I(k)− (δ + w)k > 0.

Consider the total surplus of continuing a contract when k = k. This is equal to β(EF (k, η)−(δ +w)k)/(1−β). The entrepreneur’s surplus of continuing the contract is β(EF (k, η)+V 0−κ)while the continuation value for the intermediary is the total surplus minus the entrepreneur’ssurplus, that is, β(EF (k, η) − (δ + w)k)/(1 − β) − β(EF (k, η) + V 0 − κ). Under the contractscheme outlined in (18), when the shock takes the minimum value η = 0, the entrepreneur getsno payment and the intermediary pays (δ +w)k. This implies that the non-repudiation conditionfor the intermediary is:

−(δ + w)k + β(EF (k, η)− (δ + w)k)/(1− β)− β(EF (k, η) + V 0 − κ) > 0 (29)

Now observe that the value of V 0 cannot be smaller than β(EF (k, η) − (δ + w)k)/(1 − β) − k.Therefore, this condition is also satisfied if

β(k + κ)− (1 + β)(δ + w)k > 0 (30)

We now show that, if the non-renegotiation condition is satisfied at k = k, it will also besatisfied for all value of k. Define Ω(k) = V I(k)− k. This can be written as:

Ω(k) = βEF (k, η) + β(1− δ − w)k − k + βEΩ(g(k, η)) (31)

where the function g(k, η) determines the next period capital in an optimal contract. This func-tion, characterized by conditions (15)-(17), is concave in k for each η. This implies that Ω(k) isconcave. Because Ω(k) = V I(k)− k, then obviously V I(k)− (δ + w)k is also concave.

Now remember that V I(k0) − k0 = 0, and V I(k) − k < 0, where k0 and k are the minimumand maximum values of capital. If V I(k)− (δ + w)k > 0, then of course V I(k0)− (δ + w)k0 > 0.The concavity of V I(k) − (δ + w)k will then guarantee that this non-repudiation condition issatisfied for all k. Q.E.D.

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B Appendix: Numerical procedure

Steady state: The steps to solve for the steady state equilibrium are as follows:

1. We guess the wage variable w.

2. We guess the value of a new project V 0.

3. Given w and V 0, we solve the contract on a grid of points for µ, for each z. Given corollary3.1, we use a backward procedure starting from µ = 1. Grid points are joined with step-wiselinear functions.

4. Using the zero profit condition for the intermediary we find the value of a new contractfor the entrepreneur. If this value is different from the initial guess V 0, we restart theprocedure from step 2 until convergence.

5. After solving for the optimal contract and finding the investment rules, we iterate on thedistribution of firms until we find the invariant distribution.

6. Given the invariant distribution of firms, we check the equilibrium in the labor market. Ifthe demand of labor is different from the supply we restart the procedure from step 1 untilconvergence.

Aggregate shocks: In computing the equilibrium with aggregate shocks, we assume that ineach period a fraction p of new projects have low productivity and 1 − p have high productivity.The variable p is drawn from a uniform distribution in the interval [0, 1]. The steps to solve forthe equilibrium are as follows:

1. We parameterize the value of a new project V 0(s) and, for each grid point of µ and for eachz, the values E(µ′′ − µ) and EW . These values are parameterized with linear functionsof the following variables: (a) the fraction of new projects with low productivity; (b) thefraction of all firms with low productivity; (c) the capital used by constrained firms; (d) thecapital used by unconstrained firms. The last three variables are proxies for the distributionof firms.

2. We guess the coefficients of the parameterized functions.

3. Given the parameterized functions, we solve and simulate the model.

4. We estimate the coefficients of the parameterized functions using the simulated data as inKrusell and Smith [13]. These estimates are used as new guesses for these functions andthe procedure is restarted from the previous step until convergence.

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