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formal proof of this claim is provided in appendix B. .... R [1 + γEt (φt+2)] Et (MPKt +1) = [1 + γEt (φt+2)] UK + R2λt − γ (τk − RUK)Et (λt+1). −Rµt + γ (1 − δk)Et ...
Financing Constraints, Irreversibility, and Investment Dynamics∗ Andrea Caggese Pompeu Fabra University First version: 6 June 2001 This version: 27 May 2004

Abstract We develop a structural model of an industry with many heterogeneous firms in order to investigate the cyclical behaviour of aggregate fixed investment, variable capital investment and output. In particular, we consider an environment in which the firms cannot borrow unless the debt is secured by collateral and cannot sell fixed capital without liquidating their whole business. We use the model to simulate an industry where firms experience persistent idiosyncratic and aggregate shocks, and we show that the financing and the irreversibility constraint interact and amplify each other. On the one hand the presence of financing imperfections amplifies the effect of the irreversibility constraint on fixed capital investment. On the other hand the presence of the irreversibility constraint amplifies the effects of the financing constraint on variable capital investment. We show that the interplay between the two constraints helps to explain why input inventories and material deliveries of US manufacturing firms are very volatile and procyclical, and why their dynamics are highly asymmetrical across periods of expansion and contraction of output. JEL classification: D21, E22, E32, G31 Keywords: Financing Constraints, Irreversibility, Investment



I am most grateful to Nobu Kiyotaki for his encouragement and valuable feedback on my research. I would also like to thank Kosuke Aoki, Orazio Attanasio, Steven Bond, Heski Bar-Isaac, Vicente Cunat, Aubhik Khan, Francois Ortalo-Magn˙e, Steve Pischke, Michael Reiter and Julia Thomas for their valuable comments and suggestions on earlier versions of this paper, as well as the participants in the ”Capital Markets and the Economy” workshop at the 2003 NBER Summer Institute, in the 2003 EEA meetings in Stockholm, and in the LSE, FMG and UPF seminars. All errors are, of course, my own responsibility. Research support from the Financial Markets Group and from Mediocredito Centrale are gratefully acknowledged. Please, address all correspondence to: [email protected] or Pompeu Fabra University, Department of Economics, Room 1E58, Calle Ramon Trias Fargas 25-27, 08005, Barcelona, Spain

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1

Introduction

This paper has two main motivations. First, there is a body of theoretical literature which has shown that asymmetric information and contract incompleteness may prevent firms from accessing external finance, and thus make them unable to fund profitable investment opportunities1 . However, are these imperfections relevant for the investment decisions of firms? To what extent do they affect aggregate investment and production dynamics over the business cycle? Most of the existing empirical literature2 , following the seminal paper by Fazzari, Hubbard and Petersen (1988), addresses only the first question. It shows that investment is significantly correlated with cash flow, especially for firms likely to face capital markets imperfections3 . Instead much less work has been done on the effects of financing constraints on aggregate investment dynamics4 . Gertler and Gilchrist (1994) and Bernanke Gertler and Gilchrist (1996) show that small firms’ output and inventories are more volatile and procyclical than those of large firms5 . The authors argue that this difference is due to financing constraints, but they do not provide a structural theory of firm behaviour to support such claim. Second, this paper is motivated by the fact that there are two distinct bodies of literature analysing the effect of real constraints6 (adjustment costs) and financing constraints on firm investment. But these constraints have always been analysed in isolation, with very few exceptions (Scaramozzino, 1997 and Holt, 2003). Therefore little is known about the effects of the interactions between irreversibility of fixed capital and financing imperfections on investment dynamics. In this paper we develop a structural model of firm behaviour under financing and irreversibility constraints, and we use the model to address both issues regarding the effects of financing constraints at firm level and the consequences for aggregate investment dynamics. The model has three distinctive features. First, output is produced by a risk neutral profit maximising firm which operates a concave risky technology using two complementary factors of production: fixed and variable capital. Second, new investment in fixed capital takes one period to produce output, while new investment in variable capital is immediately productive. Moreover the fixed capital stock cannot be disinvested unless the whole business is sold, while variable capital is fully reversible. Third, the firm’s only source of external finance is debt secured by collateral asset. We solve the dynamic investment problem and then we study the implications of the contemporaneous presence of both fixed and variable capital and of the irreversibility and the financing constraint for the investment decisions of the firm. We show that not 1

Stiglitz and Weiss (1981), Besanko and Thakor (1986), Milde and Riley (1988), Hart and Moore (1998), Albuquerque and Hopenhayn (2000). 2 See Hubbard (1998) for a review of this literature. 3 But the relevance of this finding has been seriously questioned. Kaplan and Zingales (1997 and 2000) find that the investment-cash flow correlation is stronger for firms which are financially very wealthy and surely not financially constrained. Similar evidence is produced by Cleary (1999), who studies a larger sample of 1317 US firms. 4 Recent works on the aggregate implications of financing constraints are Cooley and Quadrini (2001) and Cooley, Marimon and Quadrini (2002). 5 Bernanke Gertler and Gilchrist (1996) show that one third of aggregate fluctuations in the US manufacturing sector can be accounted for by the difference between small and large firms. 6 See Caballero (1997) for a review.

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only current but also future expected financing constraints affect fixed capital investment decisions. This is because fixed and variable capital are complementary. Therefore if a firm expects that, because of financing problems, future variable capital investment may be constrained to be too low, it will also invest less in fixed capital in the current period. Simulations show that this precautionary behaviour becomes stronger as the gap between current and desired stock of fixed capital increases. Therefore future expected financing constraints reduce the volatility of fixed capital investment in similar way than convex adjustment costs do. It is well know in consumption literature that future expected financing constraints affect the current consumption decisions of a risk averse consumer. It is also easy to show an analogous effect on the investment of risk averse entrepreneurs (see for example Caggese, 2003 and Cagetti and Nardi, 2003). But this paper is, to our knowledge, the first one to show a similar effect also for risk neutral profit maximising firms. The main contribution of the paper is then to show that the irreversibility and the financing constraint interact and amplify each other, and especially that the effects of financing constraints on variable capital are amplified by the irreversibility of fixed capital. This amplification effect is both static and dynamic. The static component is intuitive. If a firm is financially constrained, then it can only invest if it has available internal funds. Therefore a negative exogenous shock that reduces the financial wealth of a firm also reduces its investment capacity. This reduction is absorbed by variable capital investment when fixed capital is irreversible. The dynamic component is less intuitive but not less important. Consider for example a firm who faces a persistent negative productivity shock and expects negative profits for some periods. Fixed capital is inefficiently high and it cannot be liquidated because of the irreversibility constraint. This further reduces profits, the return on capital and the market value of the firm, even in the absence of financing constraints. Consider now a firm that also faces financing imperfections. Negative profits reduce the wealth of the firm, and increase the probability of facing future financing constraints. More importantly, wealth can be reduced up to the point where both constraints are contemporaneously binding. In this case, not only is fixed capital inefficiently high but also variable capital is inefficiently low. As a consequence revenues decrease even more due to the unbalanced use of factors of production. This further increases losses and reduces financial wealth and variable capital investment. When the bad period ends and productivity starts to rise the firm, in case it managed not to default, is very cautious about investing in fixed capital, fearing the consequences of the combination of irreversibility and financing constraints in case of a new negative productivity shock in the future. Therefore the interactions between the financing and the irreversibility constraint at the same time increase the volatility of variable capital during downturns and reduce the volatility of fixed capital during upturns. We quantify the implications of these interactions for aggregate firm dynamics by simulating a partial equilibrium industry with many heterogeneous firms. The crosssectional distribution of net worth and fixed capital among firms is determined by both idiosyncratic and aggregate uncertainty, and it affects the way aggregate output and investment react to aggregate shocks. The simulation results show that the combination 3

of financing and irreversibility constraints help to explain the dynamics of deliveries and input inventories of US manufacturing firms. Empirical evidence on US data and on G7 countries (Ramey and West, 1999) shows that inventories are very volatile and procyclical. This stylised fact has recently received considerable attention, because several papers show that on average the drop in inventories accounts for a large part of the GDP decline in recessions (Ramey and West, 1999; Ramey, 1989; Blinder and Maccini, 1991)7 . Gertler and Gilchrist (1994) and Bernanke Gertler and Gilchrist (1996) argue that financing constraints amplify the contraction in aggregate inventories during recessions. This is because they observe that after an episode of tight monetary policy that led to a recession inventories and output drop considerably more for smaller than for larger firms. If this explanation is correct, then we should observe that inventories movements are asymmetric. That is, inventory investment should be more procyclical during contraction than during expansion periods. In the empirical section of the paper we provide new empirical evidence about this asymmetry. We analyse the dynamics of output, material deliveries and input inventories of several two digits durable goods manufacturing sectors, using quarterly data from 1962 to 1996. We focus on input (materials and work in progress) rather than output inventories, because Ramey (1989), Blinder and Maccini (1991) and Humpreys, Maccini and Schuh (2001) show that input inventories are larger and much more volatile than finished goods inventories. We find that both input inventories and deliveries are significantly more procyclical around recessionary than expansionary periods. In almost all the sectors considered the procyclicality of inventories completely disappears if we only analyse periods in which output is above its trend8 . In order to address this empirical evidence with our model, we calibrate the simulated industry in order to match the long run average investment and output of one two digits manufacturing sector (the Fabricated Metals sectors, SIC code n.34). Since the empirical evidence shows that the dynamics of input inventories are similar to those of deliveries, we do not explicitly model inventory decisions. Rather we calibrate the model so that the gross investment in variable capital can be interpreted as the flow of material deliveries. With the calibrated model we simulate several artificial economies over many periods, with and without financing imperfections and irreversibility of fixed capital. The simulated data show that the interaction between the financing and the irreversibility constraint is essential to account for the high volatility of deliveries relative to fixed capital as well as for their asymmetric behaviour in the different phases of the business cycle. Moreover in the simulated data the asymmetry in the procyclicality of deliveries and input inventories is driven by the smaller firms. Finally, the model implies that investment, conditional on a given productivity shock, is sensitive to internal finance. It also implies 7

Stock and Watson (1998) report that ”Changes in business inventories, which constitute but a small fraction of total GDP, account for one-fourth of the cyclical movements in GDP”. 8 The fact that input deliveries are as procyclical as inventories is already noted by Humpreys, Maccini and Schuh (2001) who show that the high volatility and procyclicality of the stock of input inventories has a direct counterpart in the volatility of the flow of material deliveries. Using annual data about the durable manufacturing sector, they show that the flow of deliveries is more volatile and procyclical than the flow of usage of materials. Therefore input inventories are very volatile and procyclical because deliveries drop more than usage of materials during a downturn and increase more during an upturn.

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that this sensitivity is stronger for inventory investment than for fixed investment, especially for smaller firms. This result is consistent with a study which shows that “inventory investment for small firms absorb from 15% to 40% of cash flow fluctuations” (Carpenter, Fazzari and Petersen, 1998)9 . Three recent papers adopt a similar approach to our paper, and analyse an economy with heterogenous firms where financing constraints are binding for a fraction of them in equilibrium: Cooley and Quadrini (2001), Cooley, Marimon and Quadrini (2003) and Gomes (2001). Cooley and Quadrini (2001) show that financing imperfections in a model of industry dynamics explain a stylised fact regarding growth dynamics of firms which is not explained by models based only on technological shocks. Cooley, Quadrini and Marimon (2002) focus on financing imperfections in the context of long term contracts between firms and banks. They show that imperfect enforceability makes the diffusion of new technologies sluggish and amplifies their impact on aggregate output. Gomes (2001) builds a model with heterogeneous firms and financing constraints that replicates some stylised facts about industry dynamics and shows that cash flow is not significant in reduced form investment regressions when average Q is properly measured. Moreover our paper is also related to the irreversibility literature, and in particular to Bertola and Caballero (1994) and Veracierto (2003). Finally, our paper is related to Holt (2003), who analyses the effects of the interactions between the financing and the irreversibility constraints on the dividend policy and the life cycle pattern of firms. Yet, our paper is substantially different from all those above. We focus on the interactions between financing and irreversibility constraints as well as on business cycle dynamics rather than growth dynamics of firms. Moreover, we analyse a multifactor technology and use the model to explain the empirical evidence of both fixed and variable investment. Thus our paper adds to the existing literature in several important ways: we model theoretically and quantify with simulations the precautionary effect of future expected financing constraints on risky investment. We model theoretically an amplification effect between the irreversibility and the financing constraint and show that such effect is essential for explaining several stylised facts regarding business cycle dynamics of aggregate investment. The paper is organised as follows: section 2 illustrates the theoretical model; section 3 describes the solution method and compares the empirical evidence from two digits durable goods manufacturing sectors with the simulation results; section 4 presents the conclusions.

2

The model

We consider an industry populated by many firms. Each firm chooses investment in fixed and variable capital in order to maximise the expected discounted sum of future dividends. All firms have access to the same risky technology. In this section we will describe and solve the optimal investment problem of one generic firm. kt and lt are respectively the stock of fixed and variable capital of the firm. Variable capital represents variable inputs 9

Even though the authors study total inventories, while we focus on input inventories.

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such as raw materials and work in progress, while fixed capital represents fixed inputs such as plant and equipment10 . Output yt is produced according to a Cobb-Douglas production function: yt = eθt ktα ltβ with α > 0; β > 0; α + β < 1 (1) θt is a stationary autoregressive stochastic process representing the productivity shock. Fixed capital installed at time t takes one period to become productive, while variable capital is immediately productive. Moreover fixed capital is irreversible and can only be disinvested if the firm is liquidated, while variable capital is reversible. Irreversibility of fixed capital is justified by the fact that in many industries plant and equipment do not have a secondary market because they cannot be easily converted to other productions. Yet we allow fixed capital to be used as collateral by assuming that such conversion is easier if the whole of the assets is sold. The assumption that fixed capital is irreversible conditional on the continuation of the activity is consistent with the empirical evidence from a very large sample of US manufacturing plants analysed by Caballero, Engel and Haltiwanger (1995). Therefore, conditional on continuation the firm is subject to the following constraint11 : kt+1 ≥ (1 − δ k ) kt (2) In order to introduce financial markets imperfections in the model we assume that equity finance and risky debt are not available. At time t the firm can borrow (and lend) one period debt, with face value bt+1 , at the market riskless rate r. A positive (negative) bt+1 indicates that E is a net borrower (lender). Lenders only lend secured debt, and the only collateral they accept is the next period residual value of physical capital. Therefore at the time t the amount of borrowing is limited by the following constraint12 : bt ≤ τ k kt + τ l lt

(3)

τ k ≤ (1 − δ k )2

(4)

τ l ≤ 1 − δl

(5)

δ l and δ k are respectively the depreciation rate of variable capital and fixed capital. τ k and τ l are the shares of fixed and variable capital that can be used as collateral. Constraints (4) and (5) holding with equality imply that the end of period t residual capital is fully collateralisable. In the case of fixed capital, at the beginning of period t capital kt has already depreciated at the rate δ k (see equation 8), and therefore at the 10

For simplicity, labour is not considered in the analysis. However the inclusion of an additional factor of production would not affect the results. 11 We assume full irreversibility of fixed capital. Another possibility is to assume partial irreversibility: fixed capital has a resell price lower than its residual value in the firm. As long as the wedge between the selling price and the residual value is not neglegible, both assumptions would generate similar qualitative results. 12 The rationale for this collateral constraint is that the firm can hide the revenues from the production. Being unable to observe such revenues the lenders can only claim, as repayment of the debt, the value of the firm’s physical assets (Hart and Moore, 1998). Therefore the firm can only lend or borrow one period secured debt at the market interest rate r offered by the lenders. We also implicitly assumes that in any default and renegotiation of the debt with the bank the firm has all the bargaining power. Otherwise the bank could use the threat of liquidation of fixed capital to enforce the repayment of uncollateralised debt.

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end of period t, if no positive gross investment has been made (negative gross investment is ruled out by the irreversibility constraint 2), the residual value of fixed capital is equal to (1 − δ)2 kt . Constraints (4) and (5) holding with disequality are equivalent to assume that the firm can ’steal’ a (1 − τ ) fraction of the residual value of capital13 . The timing of the model is the following: the firm inherits from time t − 1 the fixed capital kt and the net financial wealth wt . At the beginning of period t the shock θt is realised, and the firm can either liquidate or continue activity. If the firm is liquidated, its technology becomes useless forever. Its assets are sold and its net financial value wt is distributed as dividends. If the firm continues activity it has to pay a fixed cost F. Then it can borrow (lend) one period debt (credit) with face value bt+1 , receiving the discounted value bt+1 /R. The net worth wt plus the new borrowing bt+1 /R are allocated between dividends, variable capital input and fixed capital: dt + kt+1 + lt = wt +

bt −F R

dt ≥ 0

(6) (7)

dt are the dividends distributed at time t. After producing the firm repays the debt bt at the end of the period. Therefore residual wealth at the end of period t is14 : wt+1 = yt − bt + (1 − δk ) kt+1 + (1 − δ l ) lt

(8)

Liquidation at the beginning of a period can happen for three different reasons: i) exogenous liquidation, with probability 1 − γ. This small probability of exogenous termination is necessary to ensure that the distribution of firms does not degenerate to the point where all firms are very wealthy and no one is financially constrained. ii) Voluntary liquidation: after observing θt , the firm decides to liquidate when the net present value of expected profits are so low that do not compensate the current fixed cost F. iii) Forced liquidation: a firm is forced to liquidate when, even though its business has a positive net present value, it does not have enough funds to pay the fixed cost F. If fixed capital is reversible then forced liquidation never happens in equilibrium, because a loss making firm would voluntarily liquidate before reaching the level at which it cannot pay F. Instead if fixed capital is irreversible forced liquidation of a firm with a positive net present value of the stream of future profits may happen when such firm has too much of its wealth invested in fixed capital, and can only pay F if liquidates it. A formal proof of this claim is provided in appendix B. 13

We believe that it is realistic to allow kt and lt to be used as collateral, but not kt+1 . kt is phisical capital existing at the beginning of time t, and can be observed by the lenders. lt is purchased at the beginning of period t just after the shock. To assume that it is collateralisable it is equivalent to assume that the firm is given short term trade credit by its suppliers, who then discount such credit at a bank, which is willing to assume the liability because this is secured by the value of variable capital. Conversely kt+1 is fixed capital that will be in place only at the end of period t, and can be used as collateral only in period t + 1. Nonetheless to assume that kt+1 can be used as collateral in period t would not change the results of the paper. 14 Equations (1) and (8) imply that one unit of installed fixed capital kt+1 is fully productive in period t + 1, while it’s market value is reduced by the depreciation rate δ k .

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We formulate now the intertemporal maximisation problem. We denote the expected value of the firm at time t, after θt is realised, by Vt (wt , θt , kt ), where wt , θt and kt are the three state variables of the problem: Vt (wt , θt , kt ) = γSt Vtstay (wt , θt , kt ) + (1 − γSt ) wt

(9)

Where St is the following binary variable:

St =

(

h

1 if Vtstay (wt , θt , kt ) > wt and F ≤ wt − (1 − δ k ) − 0 otherwise h

h

i

τk R

i

i

¯ kt ¯¯ ¯ ¯

(10)

The condition F ≤ wt − (1 − δ k ) − τRk kt is necessary to rule out forced exit, as explained in appendix B. Therefore we can define Vtstay , the value of continuing activity, as the following: i 1 h stay (wt+1 , θt+1 , kt+1 ) + (1 − γSt+1 ) wt+1 (11) Vtstay (wt , θt , kt ) = max dt + Et γSt+1 Vt+1 lt ,kt+1 ,bt R

The firm’s problem is defined by equation (11) subject to constraints (2), (3) , (6) and (7). These constraints define a compact and convex feasibility set for lt , kt+1 and bt+1 , and the law of motion of wt+1 conditional on wt , kt and θt is continuous. Therefore, given the assumptions on θt and the concavity of the production function, a unique solution to the problem exists15 . The solution to the problem is obtained using a numerical method, and is illustrated in detail in appendix A. In the following subsections of the paper we provide a description of the first order conditions of the problem in the special case when endogenous exit never happens in equilibrium (St = 1 for any t). This simple case is useful to illustrate the effects of the interactions between financing and irreversibility constraints on investment decisions. We substitute St = St+1 = 1 and Vt = Vtstay in equation (11). By substituting recursively in (11) we obtain: V0 (w0 , θ0 , k0 ) = M AX∞ d0 + E0 {lt, kt+1 ,bt }t=0

Ã∞ ! X γdt + (1 − γ) wt

Rt

t=1

(12)

Let µt λt and φt be the Lagrangian multipliers associated to the constraints (2) , (3) and (7). Substituting dt in (12) using (6) and taking the first order conditions of equation (12) with respect to bt , lt and kt+1 , it is possible to show that the solution of the problem is given by the optimal sequence of {kt+1 , lt , bt , dt, λt , µt , φt | kt , wt , θ t , Θ}∞ t=0 which satisfies equations (2), (13), (14), (15) and (16), plus the standard complementary slackness conditions on λt , µt and φt : ³ ´ φt = Rλt + γEt φt+1 (13) M P Lt = U L + 15

See Lucas and Stokey (1989), chapter 9.2.

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R (R − τ l ) λt ³

1 + γEt φt+1

´

(14)

γ R

h

³

´i

h

³

´i

1 + γEt φt+2 Et (M P Kt+1 ) = 1 + γEt φt+2 U K + R2 λt − γ (τ k − RU K) Et (λt+1 ) ³ ´ h ³ ´ i −Rµt + γ (1 − δ k ) Et µt+1 + Rγ cov M P Kt+1 , γEt+1 φt+2 − 1 (15) µ ¶ τl τk dt + 1 − lt + kt+1 ≤ wt + kt (16) R R

∂yt+1 ∂y M P Kt+1 = ∂k and M P Lt = ∂l are the marginal productivities and UK and U L t t+1 are the user costs of fixed and variable capital respectively:

U K = R − (1 − δ k ) ; U L = R

(17)

Equation (16) combines the budget constraint (6) and the collateral constraint (3) and implies that the downpayment necessary to buy kt+1 , lt and dt must be lower than firm’s net wealth16 . λt is positive when the financing constraint is binding, and is equal to zero otherwise. φt is positive when λt has some probability to bind today or in the future. This can be easily seen by iterating forward equation (13): φt = R

∞ X

γ j Et (λt+j )

(18)

j=0

A positive φt represents the additional value of financial wealth for the firm in terms of its ability to reduce financing constraints. As long as φt > 0 then the return on money invested in the firm is higher than the market return,³ and´ the firm does not distribute dividends. Thus dt = 0 for any t. The term (1 − δ) βEt µt+1 is the cost of future expected irreversibility constraints. µt is positive when the irreversibility constraint is binding, and is equal to zero otherwise. In the remaining part of this section we will describe the main qualitative features of the model. We first analyse the solution without the financing constraint, then we analyse the solution without the irreversibility constraint, and finally we explain how the two constraints interact with each other.

2.1

Solution with the irreversibility constraint only.

In this subsection we rule out current and future expected financing constraints by assuming that w0 , the initial wealth of the firm, is high enough so that the borrowing constraint (3) is never binding. This means that λt = 0 for any t, and the first order conditions (13), (14) and (15) can be simplified to the following equations: φt = 0

(19)

M P Lt = U L

(20)

³ ´ γ R γ Et (M P Kt+1 ) = U K − µt + (1 − δ k ) Et µt+1 R 1−γ 1−γ

(21)

16

The optimal choices of kt+1 , lt and dt determine the optimal level of net financial wealth but do not determine how such wealth is allocated between debt and financial assets. In fact given that lending and borrowing rates are the same, when the borrowing constraint is not binding the firm is indifferent between borrowing up to the limit and investing in financial assets and keeping some spare borrowing capacity.

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Equation (20) determines the optimal level of variable capital according to the simple equation between marginal cost and marginal return. Equations (2) and (21) determine µt and kt+1 . They describe the solution to a version of a well known irreversible investment problem (e.g. see Bertola and Caballero, 1994). Since we allow for a reversible factor of production to be used in conjunction with the irreversible one, the intuitive consequence is that lt+1 , the reversible factor, is more volatile than kt+1 , the irreversible factor, both after a positive and a negative shock17 .

2.2

Solution with the financing constraint only

In this section we rule out current and future expected irreversibility constraints by assuming that both variable capital and fixed capital are ³reversible. Hence the irreversibility ´ constraint (2) no longer applies. Substituting µt = Et µt+1 = 0 in (15) and rearranging it we obtain the following: h

³

´

γ γ (τ k − RU K) Et (λt+1 ) + R2 λt γcov M P Kt+1 , γEt+1 φt+2 − 1 ³ ´ h ³ ´i Et (M P Kt+1 ) = U K− + R 1 + γEt φt+2 R 1 + γEt φt+2 (22)

i

The solution of the problem is in this case defined by the sequence {kt+1 , lt , bt , dt, λt , φt | kt , wt , θt , Θ}∞ t=0 which satisfies equations (2), (13), (14), (16) and (22). If the firm does not have enough resources to invest optimally then constraint (16) is binding with equality and λt > 0. Equations (14) and (22) imply that a positive λt increases both M P Lt and Et (M P Kt+1 ). Instead if λt = 0 then equation (14) simplifies to equation (20). This means that future expected financing constraints do not affect variable capital investment decisions. Regarding fixed capital, if the firm does not expect to be financially constrained in the future (λt = 0 and φt = 0 for any t), then equation (22) can be simplified to the following profits maximising rule: γ Et (M P Kt+1 ) = U K (23) R The more interesting case is when λt = 0 but the financing constraint can be binding in the future (Et (λt+j ) > 0 for some j > 0). In this case, kt+1 is affected by future expected financing conditions. In order to understand why, it is useful to evaluate equation (22) for λt = 0 and rearrange it as follows: Et

Ã

1 1−α kt+1

!

=

RU K ³

β αγEt eθt+1 lt+1

17

´ + Ct

(24)

This is evident from the comparison between equations (20) and (21). Equation (20) implies that variable capital always reacts to both positive and negative productivity shocks. Equation (21) instead implies that fixed capital investment volatility is reduced by the irreversibility constraint. After a negative productivity shock at time t, When constraint (2) is binding, kt+1 cannot be reduced, and as a consequence expected marginal productivity of capital is lower than the user cost U K. This is compensated by a positive µt on the right hand side of equation (21). Instead after a positive productivity shock the firm ¡ wants ¢ to invest more in both factors. Therefore constraint (2) is not binding and µt = 0. Instead (2) can be binding at time Et µt+1 > 0 because, applying the same reasoning made before, ¡ constraint ¢ t + 1 conditional on a future negative shock. The positive Et µt+1 represents the cost associated to future expected irreversibility. Such cost increases the required marginal productivity of fixed capital Et (M P Kt+1 ) , thereby reducing kt+1 .

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Figure 1: Policy functions for a given productivity shock kt Irreversibility constraint is binding Borrowing constraint is binding Both constraints are binding The firm is liquidated

wt

h

³

´

i

³

´

β α−1 γcov M P Kt+1 , γEt+1 φt+2 − 1 − γR (τ k − RU K) Et (λt+1 ) cov eθt+1 lt+1 , kt+1 h ³ ´i ³ ´ ³ ´ Ct = − β β αγ 1 + γEt φt+2 Et eθt+1 lt+1 Et eθt+1 lt+1 (25) Future expected financing constraints affect the term Ct , but this is of second order importance with respect to the first term on the right hand side of equation (24). In order to understand how future expected financing constraints affect fixed capital investment decisions, it is useful to remember that Et (lt+1 ) decreases in Et (λt+1 ) (see equation 14). This means that the more the firm is expected to be financially constrained in the next period, the lower is the expected level of variable capital investment lt+1 . Since the two factors of production are complementary, this also reduces expected profitability of fixed capital, and lowers investment in fixed capital in period t. The consequence is that fixed capital investment can be extremely sensitive to internal finance when the financing constraint is not binding today but it may be binding in the future. The implication is that two firms with identical technology and identical profitability, and both currently not financially constrained, may choose very different fixed capital investment levels depending on their internal finance availability. in the next section we will show that this ”precautionary” effect on investment is quantitatively important, and may reduce the volatility of fixed capital in a way that is similar to the effect of convex adjustment costs.

2.3

Solution with the financing and the irreversibility constraints.

We now consider the solution of the problem with both constraints. Figure 1 summarises the different types of optimal policy functions kt+1 (wt , kt | θt ) and lt (wt , kt | θt ) in the {kt , wt } space, conditional on a certain productivity shock. The black area on the top left corresponds to a situation in which the firm is liquidated. Instead of describing in detail such solution, we focus only on the most interesting feature: the fact that irreversibility and financing constraints interact and amplify each other. When both constraints are

11

binding, µt is determined by equation (15). By substituting recursively we obtain: µt =

1 R

i h ∞ h P γ(1−δk ) j

j=0

³

[ 1 + γEt φt+j+2

R

´i

[RUK − γEt (MP Kt+j+1 )] h

³

´

i

+λt+j − γ (τ k − RU K) Et (λt+j+1 )] + Rγ cov M P Kt+1 , γEt+1 φt+2 − 1

(26)

In order to interpret equation (26) it is useful to evaluate it for the case of no financing constraints (λt = φt = 0 for any t): "

∞ 1 X γ (1 − δ k ) µt = R j=0 R

#j

(1 + γ) [RU K − γEt (M P Kt+j+1 )]

(27)

Equation (27) shows that µt , the shadow cost of the irreversibility of fixed capital, is the expected discounted sum of the marginal loss in revenues caused by the fact that stock of fixed capital is inefficiently high when the irreversibility constraint is binding. Now compare equations (26) and (27). When the firm also faces financing constraints, µt is increased by an additional term, λt+j − γ (τ k − RU K) Et (λt+j+1 ) , which can be easily shown to be positive, and which represents the additional cost of irreversibility imposed by future expected financing constraints. This is because the lower return on capital caused by the irreversibility constraint is going to be a problem when the firm is unable to efficiently access external finance, and must rely on internal finance for new investment. Therefore financing constraints increase the cost of irreversibility µt . But this ³ ´ implies that they also increase the value of Et µt+1 when the irreversibility constraint is not binding, and amplify the cautious investment effect on fixed capital. Importantly, the interactions between irreversibility and financing constraints go both ways. Not only financing constraints increase the cost of future irreversibility constraints, but also the irreversibility of fixed capital amplifies the effects of financing constraints on variable capital investment. The easiest way to see it is by combining equations (3) and (6) and by evaluating them for dt = 0 and kt+1 = (1 − δ)kt . This allows us to define an upper limit on variable capital investment: µ



R wt0 lt ≤ R − τl ·

(28) ¸

τk kt (29) R Equations (28) and (29) show that the amount of resources available to invest in variable capital is reduced by the fact that the firm cannot liquidate the fixed capital. As a result the firm may have both constraints contemporaneously binding: too much fixed capital and too little financial resources to invest in variable capital. The immediate consequence is that variable capital investment may fall sharply after a negative shock. The more long term consequence is that the unbalanced use of factors of production leads to lower wealth accumulation and more expected financing constraints in the future. In the next sections we will show that this amplification effect is quantitatively important, and is essential to explain investment dynamics in material deliveries in the US manufacturing sector. wt0 = wt − (1 − δ k ) −

12

3

Numerical Solution and simulation

3.1

Model’s solution

We solve the intertemporal maximisation problem using a numerical method (see appendix C for details). Adding the subscript i to indicate the generic i − th firm, the solution consists of the optimal policy functions ki,t+1 (wi,t , θi,t , ki,t ) and li,t (wi,t , θi,t , ki,t ) , the associated Lagrange multipliers λi,t (wi,t , θi,t , ki,t ) , µi,t (wi,t , θi,t , ki,t ) and φi,t (wi,t , θi,t , ki,t ) , the value function Vi,t (wi,t , θi,t , ki,t ) and the liquidation rule St (wi,t , θi,t , ki,t ). θ i,t is defined as follow: θi,t = θfi,t εt (30) θfi,t is the idiosyncratic productivity shock, and εt is the industry-wide shock common to all firms. εt is introduced in order to study the implications of the model for the dynamic of investment at the industry level. The model is otherwise partial equilibrium, with constant interest rate and constant relative prices, normalised to 1. Both θ fi,t and εt are first order autoregressive stochastic processes: θfi,t = ρθ θfi,t−1 + ζ θi,t ζ fi,t ∼ iid N (0, σ 2θ )

(31)

εt = ρε εt−1 + ζ εt ζ εt ∼ iid N (0, σ 2ε )

(32)

In all the following simulations θ fi,t and εt are discretised respectively as a two states and eight states symmetric Markov processes: θfi,t {θL , θH } where θL < θH εi,t {ε1 , ε2 , ..., ε8 } where ε1 < ε2 < ... < ε8 The benchmark parameters are illustrated in table 1. We calibrate the model to match the long run averages of output and capital stock for the ”Fabricated Metals Sector” in the US from 1962 to 1995 (source: NBER-CES manufacturing industry database, SIC code 34). Therefore the ability of the model to explain the cyclical fluctuations of aggregate output and investment will be measured by comparing the simulated statistics from the artificial economy with the corresponding empirical data for the SIC 34 sector. This sector has been chosen as a generic representative one, but the theoretical model can be applied to any other sector where productive units use a combination of reversible and irreversible factors of production and can be subject to borrowing constraints. Moreover in the next section we show that the stylised facts that this model aims at explaining are common to almost all the other 2 digits durable US manufacturing sectors. The main advantage of using a single two digits sector, rather than the whole manufacturing industry, is that we have less aggregation problems in the empirical data. Moreover, as table 2 shows, the SIC 34 sector is populated by many firms, 36429 according to the statistics for year 1992, the majority of which were small firms: 40.8% of the total payroll was generated from 13

Table 1: benchmark parameters

α β δl ρa σa θ γ F1 w01 δk σθ ρθ τl τk r

Base calibration Empirical restriction 0.072 kt /lt 0.898 returns to scale (α + β) 0.5 Materials usage / Input invent. 0.9 corr (yt , yt+1 ) 0.0085 st.dev.(salest ) 0.7908 average size of fixed assets 0.97 % firms ≤ 90% avg. size 0.066 annual exit rate 1.93 (fraction of constrained firms) it 0.035 k³t ´ 0.015 std kitt 0.8 0 0.679 0.01

³

´

corr kitt , kit−1 t−1 (imperfect enforceability) debt/assets ratio quarterly real interest rate

Empirical data 1.06 (SIC 34) 0.97 (SIC 34) ≈1 (SIC 34) 0.77 (SIC 34) 0.043 (SIC 34) 58 851 (SIC 34) 71% (SIC 34) 6.4% (SIC 34) N.A. 0.145 (Compustat) 0.139 (Compustat)

Simulated data 1.13 0.97 1 0.73 0.042 58851 70% 6.44% 0.33 0.145 0.083

0.239 (Compustat)

0.140

0.3 (corporate sector) 1%

35% 1%

1: both F and w0 are expressed as fractions of the average size of sales for the simulated firms

Table 2: summary statistics from the 1992 survey of manufacturers % of total payroll SIC N. firms Total payroll from firms smaller code than 50 employees Stone, Clay, Glass Products 32 16254 13113 30.8 Primary Metal 33 6501 22202 7.2 Fabricated metal products 34 36429 38962 40.8 Industrial machinary 35 53956 57231 24 Electronic and other electric equip. 36 16922 8978 4.7 Transportation Equipment 37 11287 62733 3.5 Instruments and related products 38 11354 33067 8.5

14

firms with less than 50 employees. Therefore it is reasonable to assume that a large share of these firms may be affected by financing imperfections, like the firms in the simulated artificial economy. In calibrating the model the set of technological parameters α, β, δ l , θ, ρa and σ a are chosen to match the aggregate statistics about the SIC 34 sector: α and β match the average ratio of fixed over variable capital and a chosen level of return to scale of 0.97, which is consistent with empirical micro studies (Burnside, 1996). δ l is set in order to match the average life of materials, with the following procedure: in the model (1 − δ l )lt is the residual value of variable capital at the end of period t after production takes place. Therefore we can interpret (1 − δ l )lt as the stock of input inventories, and the gross investment in variable capital can be interpreted as deliveries: deliveriest = lt − (1 − δ l )lt−1

(33)

Total variable capital available for production in period t is lt , and δ l lt can be interpreted as the usage of materials in period t. Therefore δ l must satisfy the following equation: (1 − δ l )lt input inventories at the end of year t (34) = δ l lt usage of materials in year t Solving for δ l we obtain: δl =

1 input inventories at the end of year t usage of materials in year t

+1

(35)

inventories at the end of year t is approximately equal to one during the Since the ratio inputusage of materials in year t sample period, δ l is calibrated to be equal to 0.5. θ is set to match the average size of the fixed assets. ρa and σ a are chosen in order to match the volatility and autocorrelation of aggregate output of the SIC 34 sector18 . The parameter F is set to match firms turnover. We choose it so that the share of firms that voluntarily liquidate in the model matches the annual average exit rate of 6.4% in the SIC 34 sector (this value has been calculated as the average rate of plants deaths in the years 1995-2000). The exogenous exit probability γ is introduced in the model in order to provide an upper bound to firms size. Therefore we calibrate it in order to match the empirical evidence on the size distribution of firms. More precisely, we focus on all firms in the SIC 34 sector with less than 500 employees, and we choose γ in order to match the skewness of the size distribution. The value w0 (the initial wealth of every newborn firm) is calibrated so that approximately 30% of firms are on average financially constrained. This is of course arbitrary, because we do not have direct information about the intensity of financing constraints for the empirical data. We believe that 30% is not unreasonable given that we calibrate the model on a sector with high concentration of small firms. Nevertheless in section 3.4 we will analyse the sensitivity of our main findings for higher values of w0 . δ k , σ f and ρf are calibrated to jointly match the average, standard deviation and autocorrelation of the investment rate in fixed capital. Since this information is not 18

Since the simulated data are stationary, we detrend the empirical data using the Hodrick-Prescott filter

15

available for the SIC 34 sector, we match the values for the Compustat sample provided by Gomes (2000). Regarding the financial parameters we set r, the quarterly real interest rate, equal to 1%. τ l , the fraction of variable capital that can be used as collateral, is set equal to zero, while τ k matches the debt assets ratio observed on average in the US corporate sector. This choice of allocation of collateral capacity between fixed and variable capital is based on an imperfect enforceability argument19 , but it is not necessarily consistent with the empirical evidence20 . Therefore in the next section we analyse the sensitivity of the results to different assumptions regarding τ k and τ l . The last two columns of table 1 report the empirical moments matched in the calibration and the corresponding moments in the simulated economy. Given the interaction and nonlinearities of the model, it is very difficult to perfectly match all the empirical facts, especially those regarding firm level investment. In the next sections we compare the simulations of the economy with the financing and irreversibility constraints with three alternative economies, one with only the financing constraints, one with only the irreversibility constraints, and one with no constraints. The benchmark parameters for these economies are illustrated in appendix D. Figures 2 and 3 illustrate the value of the multipliers λt (wt , θ t , kt ) and φt (wt , θt , kt ) and the policy functions lt (wt , θ t , kt ) and kt+1 (wt , θt , kt ) in the space of financial wealth wt , for selected values of kt and θt . θt is chosen so that figures 2 and 3 represent the investment decision of a firm with a negative productivity shock. More precisely, θf = θL and ε = ε5 . Figure 2 illustrates the policy function of a firm that is only subject to the financing constraint, and hence is free to adjust fixed capital, while figure 3 illustrates the policy functions of a firm which is also subject to irreversibility of fixed capital. For both figures kt is chosen at an intermediate level so that, in the case of figure 3, the irreversibility constraint is not binding unless wealth is very low. The most interesting comparison is for values of wealth lower than W ∗ , where the financing constraint is binding and λt is positive. In this case a decrease in wt by one unit must be compensated by a one to one decrease21 in lt + kt+1 . In the case of figure 2, the drop in wealth is mostly absorbed by fixed capital. This is because the financing constraint is binding and hence the value of internally generated money is higher than the market price of it. Since the firm is risk neutral and discounts future dividends at the market risk free interest rate, then φt measures the premium in the value of one additional unit of net worth for the firm in terms of its ability to reduce current and future expected financing constraints. The value of φt (the ”fi” line) can be as high as 7% for very low levels of financial wealth. This means that it is very profitable to generate more output today, and the firm prefers to use its limited financial resources to invest in variable capital, which is immediately productive, and to reduce the investment in fixed capital. In figure 3 the irreversibility constraint is binding for very low levels of wealth. In this case the financing constraint must be absorbed by variable capital investment, which 19

We assume that the lenders cannot enforce the repayment of the debt from the firms by seizing revenues and variable capital. Their only mean of enforcing the repayment of the debt is the treath of liquidation of the firms’ fixed assets. 20 Firms in practice borrow in the very short term trade credit to finance material deliveries. 21 The one to one relationship is caused by the fact that τ l = 0,and hence lt has no collateral value.

16

Figure 2: policy functions. Firms with only the financing constraint 30000

0.7 0.6

25000

0.5 20000

lambda (left scale) fi (left scale) k (right scale) l (right scale)

0.4 0.3

15000

0.2 10000

0.1

5000

0 13744

W*

48103

82463

decreases sharply as wealth decreases. Lower variable capital means that the firm produces less output today, generates less cash flow, and is expected to face even worse financial conditions in the future periods. The interaction of the two constraints implies that variable capital, output and profits drop sharply after a negative shocks. The magnitude of this amplification effect is summarised by the value of φt , which is almost ten times higher in figure 3 than in figure 2, reaching levels as high as 65%. This means that this firm would be willing to pay up to 65% quarterly interest rate on new financing. This extreme level is in practice rarely reached by the simulated firms, while values of φt between 1015% are more frequent among the simulated firms with both constraints binding. This may still seem an unrealistically high premium for the availability of external finance, but actually the annualized interest rate firms implicitly pay when they delay the payment of trade credit is often found to be above 40% (Ng et Al, 1999). One interesting feature of the policy functions, present in both figures 2 and 3, is the fact that fixed capital is sensitive to financial wealth even when the financing constraint is not currently binding. As we mentioned in the previous section, this behaviour takes place because the value of fixed capital investment today depends not only on the future expected productivity shock, but also on future variable capital investment. Even if financial wealth is sufficient to efficiently invest today, it may be insufficient tomorrow conditional on a positive shock that increases the demand for capital of the firm. This reduces the current level of fixed capital investment. This effect is quantified in figure 4, which considers a firm that has a current value of kt = 5633. The black line represents the optimal investment in fixed capital kt+1 . At the right end, for very high levels of wealth, the firm is never constrained (φt is equal to zero), and kt+1 reaches the optimal

17

Figure 3: policy functions. Firm with both the financing and the irreversibility constraint 0.7

30000

0.6

25000

0.5 0.4

20000

lambda (left scale) fi (left scale) k (right scale) l (right scale)

0.3 0.2

15000

10000

0.1 0

5000 13744

W*

48103

82463

∗ 22 unconstrained level, which we call kt+1 . The gray line is the maximum feasible investment in fixed capital if the firm borrows up to the limit and invests all available wealth. When the black line is below the grey line it means that the borrowing constraint is not binding, but still the firm chooses a level of kt+1 lower than the optimal level kt+1 . The distance between the gray line and the black line is the reduction in capital due to future expected ∗ financing constraints. In figure 4 this reduction can amount to 31% of the value of kt+1 . This is because the difference between current capital kt = 5633 and optimal desired ∗ = 13575 is large. Conversely in figure 3 we shoved an already large firm capital kt+1 ∗ (kt = 24501) that wants to expand its size much less (kt+1 = 27115). In this case the effect of future expected financing constraints on fixed capital is much smaller. The comparison of the two figures implies that future expected financing constraints act in a similar way as convex adjustment costs, they reduce the incentive to invest when the gap between current and desired capital is large. In the next sections we will show that this implies that financing constraints may reduce fixed capital investment volatility even in the absence of adjustment costs.

.

3.2

Summary of the empirical evidence

The aim of this section is twofold. First, we will analyse the dynamics of input inventories in the durable manufacturing sector. Ramey and West (1999) show that inventory investment is procyclical, and that even though the stock of inventories is very small rela22

This maximum investment level takes into account the expected irreversibility constraint. That is, the firm would invest even more if fixed capital was reversible.

18

Figure 4: Effect of future expected financing constraints on fixed capital investment 14000

0.2 0.18

12000

0.16 0.14

k (left scale) 10000

l (left scale)

0.12

max investment capacity (left scale)

0.1

fi (right scale)

8000

0.08 0.06

6000

0.04 0.02

4000

0 6398

31991

57583

83176

tive to GDP the reduction in inventories account for a relatively large part of the decline in GDP. One explanation of this evidence is that inventories generally exhibit very high short term volatility (Hornstein, 1998), and there is nothing specific about the behaviour of inventories during recessions with respect to the other phases of the business cycle. Another explanation (Gertler and Gilchrist, 1994 and Bernanke Gertler and Gilchrist, 1996) is that financing constraints determine the reduction in inventories during recessions. If this explanation is correct, then we should observe that inventories movements are asymmetric. That is, inventory investment do not increase as sharply (relative to sales) during expansions as they decrease during contractions. In this section we will show that this is the case for the investment in input inventories. They are significantly more procyclical during downturns than upturns. This is true both for the whole of the manufacturing sector and for single two digits durable manufacturing industries. Second, we will show that the procyclicality and asymmetric behaviour of input inventories is always mirrored by a similar procyclicality and asymmetric behaviour of deliveries. This finding supports our choice, in order to keep the model more simple and tractable, to not explicitly model inventory decisions. Rather we interpret the net flow of material inputs in the model as the flow of deliveries, and in the following sections we show that the interaction between financing and irreversibility constraints generate a behaviour of deliveries and input inventories consistent with the empirical evidence. Table 3 summarizes the empirical evidence about the volatility of deliveries and inventories for the SIC 34 sector which we used to calibrate the model. The standard deviations are computed on the deviations from the Hodrick-Prescott nonlinear trend23 . Since quarterly data about deliveries are not available, we estimate them by assuming that the ratio 23

We choose λ=1600 for quarterly series and λ=100 for yearly series.

19

Table 3: summary statistics, Fabricated Metals Sector (SIC 34) Avg. % drop in recessions Yearly average1 St.deviation2 relative to the drop in Sales Y Y Q Q Sales 129.512 0.0564 0.0429 100 Output 129.611 0.0568 0.0440 109 Mat. deliveries 65975 0.0654 0.0594 143 Mat. inventories 5689 0.0734 0.0556 58 W.i.p.. inventories 4977 0.0527 0.0325 44 Finished g. inv. 4276 0.0350 0.0343 18 Materials usage 65892 0.0584 Employment (n. workers) 1439 0.0515 Employment (hours) 2208 0.0544 Capital stock (Equip) 34087 0.0234 Capital stock (Plant) 24764 0.0110 1.Values are in real terms. 2.standard deviations of percentage deviations from the trend

between usage of materials and output is constant during each year. Following Humpreys, Maccini and Schuh (2001) we define deliveries as follows: deliveriesy,q = usage of materialsy,q + ∆input inventoriesy,q where y denotes the year (from 1962 to 1996) and q denotes the quarter. We estimate the unobservable quarterly usage of materials as follows: usage of materialsy,q =

usage of materialsy ∗ outputy,q outputy

This calculation should be reasonably accurate since the ratio between usage of materials and output is fairly stable in the short term. The statistics reported in table 3 confirm the empirical evidence that input inventories (materials and work in progress) are more volatile than output inventories and that deliveries are more volatile than materials usage, sales and output. This is true both for quarterly and for yearly data. Moreover deliveries drop relatively more than sales during recessions, while at the same time input inventories drop relatively more than finished goods inventories. In the following tables we provide detailed evidence of the volatility, procyclicality, and asymmetric behaviour of material deliveries and input inventories. Tables 4 illustrates the statistics for our benchmark sector and for the other two digits durable manufacturing sectors for which the Bureau of the Census provides detailed historical data. Industry statistics confirm the empirical evidence, documented by Humpreys, Maccini and Schuh (2001) for the total manufacturing sector, that input inventories are more volatile than finished goods inventories. Moreover deliveries are more volatile than sales in all sectors. The last two columns show that both investment in input inventories

20

Sector SIC SIC SIC SIC SIC SIC SIC

32 33 34 35 36 37 38

Table 4: volatility and procyclicality of deliveries and inventories Standard Deviations Correlations ³ ´ ³ ´ Material W.i.p. Final ∆invt delt Sales Deliveries , s , s t t st st inv. inv. g. inv. 0.043 0.056 0.034 0.039 0.032 0.038(0.009) 0.006(0.065) 0.086 0.094 0.050 0.044 0.059 0.038(0.012) 0.068(0.023) 0.043 0.059 0.056 0.033 0.034 0.076(0.026) 0.180(0.060) 0.061 0.080 0.061 0.058 0.051 0.078(0.021) 0.162(0.053) 0.057 0.075 0.075 0.061 0.058 0.118(0.019) 0.155(0.059) 0.066 0.083 0.058 0.066 0.053 0.040(0.016) 0.191(0.035) 0.039 0.073 0.068 0.069 0.059 0.027(0.031) 0.312(0.099)

Standard deviations are calculated on the deviations from the trend computed using the Hodrick-Prescott filter. Standard errors are in parenthesis

Table 5: correlation Tot. Man. SIC 32 SIC 33 SIC 34 SIC 35 SIC 36 SIC 37 SIC 38

³

∆inventoriest , salest salest

´

all sample

below trend

above trend

0.08 (0.014)

0.109 (0.025)

0.038 (0.033)

0.076 (0.021)

0.084 (0.020)

0.038 0.038 0.076 0.078 0.118 0.041 0.027

0.037 0.070 0.101 0.174 0.063 0.071 0.113

(0.017) 0.040 (0.020) (0.023) 0.006 (0.023) (0.044) 0.040 (0.058) (0.042) -0.036 (0.048) (0.037) 0.175 (0.038) (0.027) 0.002 (0.031) (0.052) -0.072 (0.057)

0.041 (0.013) 0.037 (0.017) 0.128 (0.036) 0.104 (0.032) 0.139 (0.029) 0.0486 (0.022) 0.089 (0.046)

0.036 (0.015) 0.042 (0.019) 0.022 (0.037) 0.072 (0.029) 0.102 (0.027) 0.047 (0.026) -0.007 (0.043)

(0.009) (0.012) (0.026) (0.021) (0.019) (0.016) (0.031)

decreasing sales increasing sales

Standard error in parenthesis.

and deliveries are procyclical24 .The correlation coefficient between sales and the deliveries/sales ratio is also positive and significant in all sectors except SIC 32. The next three tables answer to the following question: is such procyclicality uniform in the different phases of the business cycle? Tables (5), (6) and (7) provide a negative answer to this question. Table 5 shows t the estimations of the correlation between ∆inventories and salest separately for different salest subperiods.The positive correlation is significantly stronger in the periods in which sales are below their trend. This is true for the total of the manufacturing sector and for 5 out of 7 two digits sectors. In the periods in which sales are above trend the correlation is not significantly different from zero for 5 sectors. The second part of the table shows that the correlation is on average stronger when sales are decreasing than when sales are increasing. Table 6 provides a similar picture regarding the procyclicality of deliveries. These statistics are consistent with the anecdotal evidence that inventories are very procyclical during recessions. They also show that the asymmetric behaviour of input 24

The coefficients of correlations regarding inventory investment are smaller in absolute values than the similar coefficients estimated by Ramey and West (1999). This is due to the several differences in the way we compute the statistics. We consider durable manufacturing sectors and input inventories while Ramey and West consider the whole of domestic sales and total inventories. We apply the HP filter while Ramey and West apply a linear trend.

21

Table 6: correlation all sample Tot. Man. SIC 32 SIC 33 SIC 34 SIC 35 SIC 36 SIC 37 SIC 38

³

deliveriest , salest salest

´

below trend

above trend

0.089 (0.017) 0.145 (0.029)

0.091 (0.039)

0.083 (0.024)

0.096 (0.024)

(0.136) -0.029 (0.114) (0.044) 0.028 (0.045) (0.110) 0.030 (0.130) (0.103) -0.133 (0.118) (0.112) 0.317 (0.117) (0.057) 0.129 (0.067) (0.168) 0.017 (0.185)

-0.006 (0.088) 0.099 (0.036) 0.230 (0.090) 0.213 (0.078) 0.108 (0.088) 0.186 (0.046) 0.533 (0.146)

0.021 0.044 0.120 0.155 0.199 0.225 0.169

0.006 0.068 0.180 0.162 0.155 0.190 0.312

(0.065) (0.023) (0.060) (0.053) (0.059) (0.035) (0.099)

0.053 0.108 0.280 0.411 0.003 0.239 0.569

decreasing sales increasing sales

(0.099) (0.032) (0.090) (0.074) (0.081) (0.055) (0.138)

Standard error in parenthesis.

Table 7: elasticity of deliveries to sales :

∆deliveriest salest ∆salest deliveriest

Contractions in Sales Expansions in Sales 1st quarter 2nd quarter 3rd quarter 1st quarter 2nd quarter 3rd quarter Tot. Man. SIC 32 SIC 33 SIC 34 SIC 35 SIC 36 SIC 37 SIC 38

1.23 (0.15)

1.27 (0.12)

1.20 (0.37)

0.93 (0.18)

1.21 (0.19)

1.17 (0.28)

0.70 0.79 1.15 1.41 1.37 1.09 1.19

0.78 0.75 1.66 1.77 0.99 1.21 1.66

0.99 1.18 1.98 2.06 1.30 0.76 1.77

0.82 1.27 1.14 0.67 0.20 0.96 1.98

1.35 1.17 1.18 0.53 1.28 1.00 0.59

1.22 1.18 0.91 1.00 1.34 0.58 1.79

(0.33) (0.07) (0.36) (0.24) (0.31) (0.12) (0.46)

(0.41) (0.11) (0.29) (0.45) (0.34) (0.08) (0.77)

(0.51) (0.12) (0.83) (0.29) (0.40) (0.17) (1.22)

(0.31) (0.09) (0.49) (0.34) (0.46) (0.07) (0.58)

(0.34) (0.11) (0.31) (0.38) (0.42) (0.14) (0.61)

(0.72) (0.17) (0.31) (0.34) (0.34) (0.48) (0.83)

inventories is mirrored by the one of material deliveries. In the next section we will show that the contemporaneous presence of the financing and the irreversibility constraint generates dynamics consistent with this empirical evidence. The explanation provided by our theoretical model is based on the following intuition. At the beginning of a downturn the negative aggregate productivity shock implies that some firms would like to downsize their activity, but they are prevented to do so by the presence of the irreversibility constraint. As the downturn continues and productivity and revenues worsen, a fraction of these firms may also have a binding financing constraint, and hence may be forced to reduce their demand for production inputs. If this explanation is correct, we expect that inventories and deliveries drop more sharply conditional on a sequence of negative shocks. Table 7 confirms this claim. It shows the estimation of the elasticity of deliveries to sales conditional on the number of periods of subsequent reduction or increase in detrended real sales25 . During periods of contraction in sales we observe that in 5 out of 7 sectors the elasticity tends to increase conditional on the contraction lasting longer. This is especially true for sectors 34, 35, and 38. These sectors are also those that in table 5 exhibited the highest degree of asymmetry in the procyclicality of deliveries. The same picture is not present 25

The estimated elasticities conditional on more than three consecutive periods of contraction or expansion are not reported because too few observations do not allow us to estimate significant coefficients.

22

during periods of expansion in sales. Here the elasticity of deliveries is generally lower, and does not exhibit any clear increasing or decreasing pattern. Finally, our theory predicts that financing constraints are an essential factor to generate the asymmetry in the procyclicality of deliveries. If this is true, we should observe that such asymmetry is stronger for smaller rather than larger firms. Unfortunately we do not have the availability of data disaggregated in the size dimension. Nonetheless we illustrated, in table 2, the relative importance of small firms in the aggregate production of the two digits sectors. Apart from sector SIC 32, which does not show any significant procyclicality of deliveries in the first place, we observe that the three sectors with highest density of smaller firms (SIC 34, SIC 35 and SIC 38) show the highest degree of asymmetry in the procyclicality of deliveries and inventory investment. Conversely the only sector in which the asymmetry goes in the opposite direction (SIC 36), is also one of the sectors with lower density of smaller firms.

3.3

Dynamics of aggregate output and investment

In this section we use the solution of the model to simulate the investment and production path of many heterogeneous firms. We will show that the combination of the irreversibility and the financing constraint generates investment dynamics consistent with the empirical evidence illustrated in tables 3-7. In the simulated economy all firms are identical ex ante, but each of them is subject to a different realisation of the idiosyncratic productivity shock θfi,t , which is uncorrelated across firms and serially correlated for each firm. The distribution of {wi,t , ki,t } across firms depends on the set of exogenous parameters and on the history of aggregate shocks {εj }tj=0 . In this section we compare the empirical data from the SIC 34 Sector with the statistics generated by the simulation of an economy of 5000 firms for 2000 periods. In each period a fraction of the firms is liquidated. We assume that an identical number of new firms enters production, so that the total number of firms remain constant. Each newborn firm draws an initial value of θf from a uniform distribution, and has an endowment of w0 and a fixed capital of k0 . w0 , together with F and γ, determines the aggregate distribution of wealth and hence the intensity of financing constraints. For a given F and γ, if w0 is too small we would have that no firm ever manages to expand enough and become unconstrained, and all firm are liquidated after few periods of life. If w0 is too large then all firms can expand to the level at which they are never financially constrained. As we mentioned in the previous section, w0 is not directly calibrated on the empirical evidence. In the benchmark calibration we set a value that corresponds, in the economy with both constraints, to an average share of financially constrained firms around 30%. In the next section we illustrate a sensitivity analysis of the main results of the paper conditional on different values of w0 . In any case it is possible to show that for the chosen values of w0 a stochastic steady state exists such that a fraction of firms is on average financially constrained, and among those a significant fraction is contemporaneously financially and irreversibility constrained. The choice of k0 is also very important, because it influences the level of deliveries and sales of the first period of life of a firm. We assume that the firms that enter in period t have a 23

st.dev2 . Sales st. dev2 . Deliveries st.dev2 . Fixed Capital st.dev.(deliveries) st.dev(f ixed capital)

All periods Below trend Above trend Decrease in sales Increase in sales average % of binding financing constraints average % of binding irreversibility constraints average % of both constraints binding

Table 8: simulated statistics Empirical data Simulated data Both Const. Only f.c. Only Irr. No Const. 0.0429 0.0416 0.0531 0.0454 0.0424 1.38 - 1.161 1.23 1.52 1.26 1.36 1 0.41 0.50 1.15 0.93 1.05 1 2.79 2.44 1.32 1.36 1.29 corr(deliveriest /salest ,salest ) 0.18** 0.21 0.43 0.28 0.33 0.28** 0.24 0.29 0.21 0.29 0.03 0.11 0.42 0.27 0.29 0.23* 0.26 0.26 0.14 0.25 0.12 0.03 0.37 0.26 0.23 N/A

35%

53%

0%

0%

N/A

26%

0%

32%

0%

N/A

5%

0%

0%

0%

1.Based on yearly data. 2. Standard deviations of percentage deviations from the trend. **significant at the 99% confidence level; *significant at the 95% confidence level

level of fixed capital which is ex ante optimal, conditional on w0 and on the information set at time t − 1. Tables 8, 9 and 10 compare output and investment dynamics for the US manufacturing firms in the SIC 34 sector (first column) with the data from four different simulated economies corresponding to the four versions of the model described in the previous section: without any constraint, with one of the two constraints only, and with both constraints. For the empirical data we consider as fixed capital the stock of equipment. The volatility of sales in all the simulated economies are roughly the same as in the empirical data, because we calibrate the aggregate productivity shock in order to match this data. Therefore, in this section, the interaction between the financing and the irreversibility constraints will be measured in terms of the ability of the model to replicate the following empirical evidence: i) the relative volatility of deliveries with respect to fixed capital ii) the procyclicality and asymmetry of deliveries. The last column in table 8 reports the simulated statistics from the economy without the irreversibility and the financing constraint26 . In this case fixed capital stock is approximately as volatile as sales are. Deliveries are more volatile than fixed capital because they have an higher elasticity with respect to the productivity shock27 . The correlation between the delivery-sales ratio and the level of sales is positive and much larger than zero. This is due to the time to build assumption about fixed capital. After a positive 26

Despite simulate data are stationary, we apply the same Hodrick Prescott filter used on the empirical data, in order to ensure that similar frequencies are filtered. 27 Deliveries are measured as gross investment in the variable capital stock (see equation 33). By construction they are more volatile than the stock of variable capital lt , which is roughly as volatile as sales.

24

shock firms can immediately increase variable capital input, while investment in fixed capital takes one period to become productive. Therefore deliveries increase proportionally more than sales after a positive shock28 . The time to build assumption alone can account for the procyclicality of deliveries, but it cannot account for the asymmetric nature of their procyclicality: the procyclicality of deliveries is symmetrical across expansion and contraction phases, as it is expected given the absence of any asymmetric element in the simulated economy. The next column illustrates the statistics of the economy with the irreversibility constraint only. Fixed capital becomes less volatile relative to deliveries and sales. This is because in every period the irreversibility constraint is binding for a fraction of firms in the sample. These firms therefore do not change fixed capital in response to a productivity shock. Despite the irreversibility constraint is on average binding for as much as 32% of the firms, the change in the relative volatilities of variable and fixed capital is rather small. This is because of two reasons: i) the aggregate negative shocks are usually not large enough to change the fraction of firms with a binding irreversibility constraint. Fixed capital is adjusted with a lag. After a negative shock at the beginning of time t, kt is unchanged whether or not fixed capital is reversible. Instead kt+1kt−kt is restricted to be bigger than −δ in the economy with the irreversibility constraint. This restriction is the more likely to bind the larger is the negative aggregate shock. But for the realistic parameters of this simulations the aggregate shock is not large enough to generate a large difference between the economy with only the irreversibility constraint and the economy with no constraints. This finding is consistent with Veracierto’s (2003) general equilibrium result that, in a model without financing imperfections, the irreversibility constraint at the establishment level does not have significant effects on business cycle dynamics. ii) The two factors of production are complementary. Therefore after a negative persistent productivity shock if a firm is unable to reduce fixed capital it has also less incentive to reduce variable capital input. This implies that the lower volatility of fixed capital also causes a lower volatility of deliveries. Regarding the asymmetric behaviour of deliveries in the business cycle, deliveries are more procyclical during upturn phases than during downturns. Therefore the introduction of an element of asymmetry in the model, the presence of the irreversibility constraint, actually generates an asymmetry in deliveries that is opposite to the one observed in reality. The reason is also related to the fact that the two factors of production are complementary. In this economy a downturn begins with a negative aggregate productivity shock that reduces output. As long as the low productivity period persists, aggregate fixed capital is gradually reduced towards the new optimal level. The fact that irreversibility is binding for some firms implies that aggregate fixed capital is inefficiently high, and hence also variable capital stock is higher than otherwise. The inefficiently large capital implies that output drops more than variable capital during a downturn, and hence the deliveries/sales ratio is less procyclical in this phase than during an upturn. The next column shows the simulated data for an economy with the financing constraint only. In the absence of the irreversibility constraint the firms are always able to 28

This can be interpreted as an increase in capacity utilization of the existing fixed capital stock.

25

maintain the optimal ratio between fixed and variable capital. This explains why the relative volatility of the two inputs is very similar to their relative volatility in the economy with no constraints. But the presence of the borrowing constraint implies that both deliveries and fixed capital are more volatile relative to output. This is because of two main reasons: i) in the simulated economy on average 53% of the firms have a binding financing constraint. For these firms the level of fixed and variable capital is inefficiently low, and hence their marginal productivity is high, leading to an higher sensitivity of capital to the productivity shock; ii) a binding financing constraint means that investment is sensitive to changes in net worth also when expected productivity does not change. This effect influences the volatility of capital more than the volatility of sales. For the same reason deliveries are more procyclical in this economy than in the previous two economies without financing imperfections. But also this simulated economy is not consistent with the observed asymmetric behaviour of deliveries. Financing constraints increase the procyclicality of deliveries especially during expansion phases, as it was the case for the economy with only the irreversibility constraint. The next column shows the results of the simulation of the economy with both the financing and the irreversibility constraints. In this economy the volatility of deliveries increases substantially with respect to the volatility of fixed capital, so that the ratio of volatilities becomes very close to the one observed in the data. It is quite surprising to find that no additional convex or concave adjustment cost is needed to reach this result. This happens because of the interactions between the two constraints. On the one hand the financing constraint amplifies the cautious behaviour in fixed capital investment: the firm is less willing to increase the size of the fixed capital stock because a future negative shock could make it inefficiently large. Conditional on such negative shock too much fixed capital lowers profits and cash flow, and this is particularly damaging when the firm also faces borrowing constraints. As a consequence the firm is even more cautious in investing in fixed capital when it faces both irreversibility and borrowing constraints. On the other hand the irreversibility of fixed capital amplifies the effect of financing constraints on variable capital when both constraints are binding. One way to quantify this amplification effect is to notice that if we add the irreversibility constraint to the economy with no frictions, the ratio of the volatility of deliveries versus fixed capital increases only by 5%. Instead, if we add the irreversibility constraint on top of the financing constraint, we obtain an increase by 85%. More importantly, the model with both constraints is consistent with the asymmetric behaviour of deliveries in expansion and contraction phases. Deliveries are more volatile than output during periods in which output is below trend, in a way that is consistent with the empirical evidence. The intuition for this result is that in this economy a fraction of firms has both constraints binding. This implies that, during a downturn, the reduction in wealth and the increase in the intensity of financing constraints is entirely absorbed by a fall in deliveries. This effect more than counterbalances the fact that the more wealthy firms in the sample only have the irreversibility constraint binding and hence do not reduce deliveries with the same intensity. Less intuitive is the reason why deliveries are not procyclical during upturns. In section 3.1 figure 3 illustrated the fact that when 26

Table 9: Dynamics of deliveries, small and large firms Smaller firms1 Both const. only f.c. only irr. no const. corr(delt+1 /salest ,salest ) 0.08 0.43 0.22 0.30 (all periods) corr(delt+1 /salest ,salest ) 0.14 0.29 0.17 0.26 (below trend) corr(delt+1 /salest ,salest ) -0.02 0.39 0.20 0.26 (above trend) Larger firms corr(delt+1 /salest ,salest ) 0.34 0.25 0.28 0.33 (all periods) corr(delt+1 /salest ,salest ) 0.33 0.21 0.21 0.28 (below trend) corr(delt+1 /salest ,salest ) 0.26 0.23 0.28 0.28 (above trend) 1.Smaller firms are the 50% of firms with smaller output.

both constraints are binding the shadow cost of money for the firm reaches very high levels. The implication of this is that the marginal productivity of variable capital for firms with both constraints binding is very high. Therefore when a positive aggregate shock hits and positive profits are realised, a small increase in deliveries generates large increases in output, and hence deliveries increase less than output for these firms. The ability of the model to match the observed behaviour of deliveries is important, because the asymmetry of deliveries is empirically closely related to the asymmetric dynamics of input inventories, as it was shown in tables 5 and 6. Even though we do not explicitly model inventory decisions in the model, we can interpret the residual value of capital at the end of period t, (1 − δ l )lt , as input inventories. Therefore (1 − δ l )(lt − lt−1 ) can be interpreted as inventory investment. Table 11 reports the procyclicality of inventory investment for the empirical data and for the simulated economies. Also in this case the economy with both constraint is the only one that reproduces an asymmetry of inventory dynamics consistent with the empirical data. Our model therefore provides a theoretical justification of the stylised fact that inventories are very procyclical especially during recessionary periods. Furthermore, it is consistent with the finding of Gertler and Gilchrist (1994) and Bernanke, Gertler and Gilchrist (1996) who observe that the dynamics of inventories around recessions are very different between small and large firms. Table 9 shows that the asymmetric behaviour of deliveries in the economy with both constraints is entirely driven by the smaller firms in the sample. The asymmetry of deliveries in the economy with both constraints is also reflected by the elasticity of deliveries to sales. Table 10 shows the estimated elasticity of deliveries to sales conditional on the number of consecutive periods of decreasing and increasing sales. The elasticity of deliveries to sales is very large in the first period of both expansion and contraction phases because capital does not initially adjust to the new productivity shock. Hence the sensitivity of output to the shock is less strong than the sensitivity of 27

Table 10: sales elasticity of deliveries Empirical Data Simulated Data Increase in sales Both Constraints Only irr. Only f.c. ∂delt+1 st st (1 quarter) 1.14 2.39 2.88 2.60 ∂st delt+1 ∂delt+1 st nd (2 q.) 1.18 0.66 1.44 1.59 ∂st delt+1 ∂delt+1 st rd (3 q.) 0.91 0.82 0.62 1.30 ∂st delt+1 ∂delt+1 st th (4 q.) 0.56 0.78 0.5 ∂st delt+1 Decrease in sales ∂delt+1 st (1st quarter) 1.15 2.23 2.93 3.26 ∂st delt+1 ∂delt+1 st nd (2 q.) 1.66 0.87 1.10 0.99 ∂st delt+1 ∂delt+1 st rd (3 q.) 1.98 0.89 0.76 0.50 ∂st delt+1 ∂delt+1 st th (4 q.) 0.93 0.69 0.30 ∂st delt+1 Decrease/increase 1st quarter 1.01 0.93 1.01 1.25 2nd q. 1.41 1.31 0.76 0.62 3rd q. 2.17 1.09 1.23 0.39 4th q. 1.67 0.89 0.60

No Const 2.86 1.25 0.58 1.01

variable capital. This effect is amplified by the fact that we consider deliveries, which are the gross investment in variable output. In order to abstract from this effect we compare, in the bottom part of table 10, the ratio between the elasticity of deliveries to sales in contraction and expansion periods. The ratio tends to increase in the economy with both constraints, as it does in the data, while it generally decreases or it is constant in the other simulated economies.

3.4

Sensitivity analysis

We present now a sensitivity analysis of the main results obtained in table 8. In the economies where financing constraints are present, the simulated statistics depend on the value of w0 , which determines the aggregate distribution of wealth and the average fraction of firms that have a binding financing constraint. In the next figures 5-11 we analyse the effect of a gradual increase in w0 , and hence of a reduction in the intensity of financing constraints, on the volatility of deliveries relative to fixed capital and on the behaviour of deliveries across expansion and contraction phases. In the previous section we claimed that the interactions between the financing and the irreversibility constraint are necessary to generate the high volatility of deliveries relative to fixed capital and their asymmetric dynamics. Here we confirm the claim by showing that these features of the simulated economy gradually disappear when w0 increases. Figures 5-8 illustrate the sensitivity analysis for the economy with both constraints. Figures 9-11 for the economy with only the financing constraint. On the x axis it is reported the ratio w0 /k, where k is the average optimal fixed capital for a firm that does not face current or future expected financing constraints. The smallest value on the x axis corresponds to the value of w0 chosen in the simulations in the previous section. 28

3.01 1.23 0.56 0.73 1.05 0.98 0.98 0.67

h

Table 11: Correlation between sales and inventory investment: corr Empirical data All firms All periods Below trend Above trend Decrease in sales Increase in sales Smaller firms Below trend Above trend Larger firms Below trend Above trend

∆inventoriest , salest salest

i

.

Simulated data Both. Const. Only F.C. Only Irr. No constr.

0.076** 0.101* 0.04 0.128** 0.02

0.20 0.23 0.11 0.27 0.02

0.50 0.33 0.50 0.30 0.45

0.29 0.21 0.28 0.14 0.27

0.33 0.30 0.27 0.29 0.17

N.A. N.A.

0.13 0.01

0.35 0.45

0.17 0.22

0.30 0.27

N.A. N.A.

0.33 0.25

0.27 0.28

0.22 0.28

0.26 0.25

**significant at the 99% confidence level; *significant at the 95% confidence level

Figure 5: Economy with the financing and the irreversibility constraint: fractions of firms with a binding constraint. 40

75

35

70 65

30

60

Avg % of firms with both constraints binding

25

55

20

Avg % of firms with a binding borrowing constraint

15

Avg % of firms with a binding irr. constraint

50 45 40

10

35

5

30 25

0 0.26

0.41

0.58

0.75

0.91

1.08

29

1.25

1.41

1.58

1.75

1.92

In figure 5 we illustrate the relationship between initial wealth and fraction of firms with a binding constraint. As w0 increases, the fraction of firms with a binding financing constraint drops. The fraction of firms with both constraints binding is initially increasing and then rapidly decreasing. Therefore only firms with very low wealth may experience both constraints binding at the same time. Conversely the fraction of firms with the irreversibility constraint binding increases with w0 , because more firms are able to expand the size of their assets, and hence more firms experience a binding irreversibility constraint after a negative shock29 . Figure 6 shows that the volatility of deliveries relative to the volatility of fixed capital first increases with w0 and then slowly decreases, reaching a constant level when the fraction of constrained firms declines to zero. The initial increase is due to the fact that, as w0 increases, more firms are able to invest and expand the scale of their activity. The distribution of firms changes and a larger fraction of firms do not have a binding financing constraint. Still, most of these firms have a level of wealth very close to the region where the financing constraint is binding. From our analysis in section 3.1 we know that expected financing constraints negatively affect fixed capital decisions of these firms. This is especially true in the economy with both constraints. Therefore these firms are very cautious in investing in fixed capital, because if they increase kt+1 too much they risk that a future negative productivity shock could push them in the region where both constraint are binding. But since these firms are not currently constrained in investing in variable capital the relative volatility of deliveries increases considerably. When w0 further increases, the ratio of the volatility between deliveries and fixed capital slowly decreases30 . This is because the distribution of wealth changes so that more firms become very wealthy and with low probability of facing future financing constraints. For these firms the probability to face both constraints binding in the future is negligible. Therefore the observed correlation between the fraction of firms with a binding financing constraint and the excess volatilities of deliveries relative to fixed capital is driven by the fact that, when the fraction of constrained firms decreases, also the interactions between the financing and the irreversibility constraints become less intense. This is why we do not observe a similar correlation in the economy with only the financing constraint (figure 9). In this case the relative volatility of deliveries with respect to fixed capital is more weakly correlated with the fraction of constrained firms. It sharply increases and then decreases for low values of w0 , and then remains roughly constant as the fraction of constrained firms decreases from 30% to 0. The ”spike” in the relative volatility of deliveries for the values of w0 /k around 0.4 is also caused by the effect of future expected 29

For high levels of w0 this fraction can become as high as 75%. The reason is that financing constraints on average reduce the volatility of firm level investment, therefore in the calibrated economy it is necessary an high volatility of the idiosyncratic shock to match the empirical data. When financing constraints are reduced, then firm level investment becomes much more volatile, and a larger fraction of firms can experience a binding irreversibility constraint at every period. 30 The ratio between the volatilities of deliveries and fixed capital decreases from a maximum of 3.4 to 1.8. This is less than the difference between the calibrated economies with both constraints and with the irreversibility constraint only (see table 8), because figure 6 shows an economy with an higher volatility of the idiosyncratic shock with respect to the one of the economy with only the irreversibility constraint analysed in the previous section. When w0 increases and financing constraints disappear, this high volatility generates and unrealistically high volatility of firm level investment, which amplifies the effects of the irreversibility constraint on aggregate investment dynamics.

30

Figure 6: Economy with the financing and the irreversibility constraint: volatility of deliveries relative to fixed capital. 3.4

35 SCALE ON THE LEFT: ratio between the volatility of deliveries and the volatility of fixed capital

3.2 3

30 25

SCALE ON THE RIGHT: % of firms with a binding financing constraint

2.8

20

2.6 15

2.4

10

2.2 2

5

1.8

0 0.26

0.41

0.58

0.75

0.91

1.08

1.25

1.41

1.58

1.75

1.92

financing constraints on fixed capital investment . But this effect quickly disappears as w0 further increases, despite there still a large fraction of constrained firms. This is because in this economy there is no interaction between the financing and the irreversibility constraint. Following a negative shock, the firms can quickly sell fixed capital and readjust it to the optimal level. This makes them much less cautious in investing in fixed capital, as soon as their wealth increases above the constrained region. In figure 7 we analyse the asymmetry in the procyclicality of deliveries as a function of w0 . As w0 increases, the observed asymmetry between the procyclicality of deliveries around contraction and expansion phases disappears. Figure 8 shows that the difference in the procyclicality of deliveries below and above the trend in output is correlated with the intensity of financing constraints. Both series drop by almost 50% as w0 /k increases from 0.26 to 0.45. This is the range of values of w0 for which a non negligible share of firms have both constraints contemporaneously binding. Deliveries of these firms are extremely sensitive to financial wealth, and a small upward shift in the wealth distribution, caused by a small increase in w0 , has large consequences on investment dynamics. For levels of w0 /k above 0.45 both series are still negatively correlated to wealth, but with a much smaller sensitivity. The fact that the interactions between the financing and the irreversibility constraint drive this result is confirmed by the fact that the picture looks different in the economy with only the financing constraint (figures 10 and 11). Here deliveries are initially much more procyclical when output is above than below its trend, but this asymmetry disappears as w0 increases above the initial level. Further increases in w0 and declines in the intensity of financing constraints have a small effect on the asymmetry in the procyclicality of deliveries. Finally, table 12 reports the sensitivity analysis of the results in table 8 for different 31

Figure 7: Economy with the financing and the irreversibility constraint: correlation bet tween deliveries and salest , economy with the finncing and the irreversibility constraints salest 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.18

Observations above the trend

0.16

Observations below the trend

0.14 0.12 0.1 0.26

0.41

0.58

0.75

0.91

1.08

1.25

1.41

1.58

1.75

1.92

Figure 8: Economy with the financing and the irreversibility constraint: difference in t and sales correlation between deliveries salest SCALE ON THE LEFT: corr(deliveries/sales,sales): difference between observations below and observations above trend. SCALE ON THE RIGHT: % of firms with a binding financing constraint

0.12 0.1

SCALE ON THE RIGHT: % of firms with both constraints binding

35 30 25

0.08 20 0.06

15

0.04

10

0.02

5

0

0 0.26

0.41

0.58

0.75

0.91

1.08

32

1.25

1.41

1.58

1.75

1.92

Figure 9: Economy with the financing constraint only: volatility of deliveries relative to fixed capital.

2.45

50

SCALE ON THE LEFT: ratio between the volatility of deliveries and the volatility of fixed capital

2.25

40

SCALE ON THE RIGHT: % of firms with a binding financing constraint

2.05

30

1.85 20 1.65 10

1.45 1.25

0 0.26

0.41

0.58

0.75

0.91

1.08

1.25

1.41

1.58

1.75

Figure 10: Economy with the financing constraint only: correlation between salest , economy with the finncing and the irreversibility constraints

1.92

deliveriest salest

0.42 0.4 0.38

Observations above the trend

0.36

Observations below the trend

0.34 0.32 0.3 0.28 0.26 0.24 0.22 0.2 0.26

0.41

0.58

0.75

0.91

1.08

33

1.25

1.41

1.58

1.75

1.92

and

Figure 11: Economy with the financing constraint only: difference in correlation between deliveriest and sales salest

0.05

50

40

0

-0.05

SCALE ON THE LEFT: corr(deliveries/sales,sales): difference between observations below and observations above trend.

30

SCALE ON THE RIGHT: % of firms with a binding financing constraint

20

-0.1 10

0

-0.15 0.26

0.41

0.58

0.75

0.91

1.08

1.25

1.41

1.58

1.75

1.92

values of τ k and τ l . The first column reports the empirical data, and the second column the simulated data in the economy with both constraints for the benchmark values τ k = 0.679 and τ l = 0. The last three columns show the simulated statistics for increasingly larger values of τ l . At the same time τ k is reduced in order to keep the average collateral capacity of the firms approximately constant. Comparing the simulated data in columns 2-5 it is clear that the main results of the simulations are not very sensitive to the assumptions regarding τ k and τ l . This finding can be explained by looking back at the analysis in section 2.3. In equation (29) the term wt0 represents the amount of wealth that can be invested in variable capital. From equation (28) it follows that an increase in τ l reduces the downpayment necessary to buy variable capital and increases the maximum borrowing capacity for a given level of wt0 . But at the same the reduction in τ k reduces the ability to borrow upfront the irreversible fixed capital. This reduces the level of wt0 . The two effects more or less compensate each other in the simulated economies in columns 2-5 in table 12. An exception is the simulated economy in column 6. Here deliveries are not more procyclical during contraction than during expansion phases. But this is mainly due to the fact that, for τ k as low as 0.2, more firms find optimal to liquidate the assets when a bad productivity shock hits and the irreversibility constraint binds. This reduces the average fraction of firms with both constraints binding and therefore the degree of interactions between the two constraints that generates the asymmetric dynamics of deliveries. We believe that this outcome is not realistic and therefore it does not reduce the validity of the results obtained before, for two reasons: i) the outcome in column 6 is based on a share of voluntary liquidations of nearly 17% every year, which is much higher than the empirical value. ii) it is reasonable to assume that variable capital has some collateral value, but 34

Table 12: Simulated statistics, economies with different values of τ k and τ l Empirical data Simulated economy with both constraints τ k = 0.679 τ k = 0.5 τ k = 0.35 τ k = 0.2 τl = 0 τ l = 0.179 τ l = 0.329 τ l = 0.479 st.dev. Sales 0.0429 0.0416 0.0413 0.0394 0.0393 1 st. dev. Deliveries 1.38 - 1.16 1.23 1.25 1.30 1.39 st.dev. Fixed Capital 0.411 0.50 0.49 0.51 0.70 st.dev.(deliveries) 1 2.79 2.44 2.56 2.54 1.99 st.dev(f ixed capital) corr(deliveriest /salest ,salest ) All periods 0.18** 0.21 0.21 0.22 0.22 Below trend 0.28** 0.24 0.24 0.25 0.20 Above trend 0.03 0.11 0.12 0.14 0.18 Decrease in sales 0.23* 0.26 0.24 0.21 0.19 Increase in sales 0.12 0.03 0.06 0.11 0.13 average % of binding N/A 35% 25% 29% 32% financing constraints average % of binding N/A 26% 23% 17% 11% irreversibility constraints average % of both N/A 5% 3.7% 3.1% 2.4% constraints binding % of endogenous exits 6.4% 6.44% 8.32% 10.84% 16.76% **significant at the 99% confidence level; *significant at the 95% confidence level

the assumption of τ l = 0.479 is not realistic because it implies that nearly all the value of input inventories can be pledged as collateral to obtain credit at the benchmark interest rate. In reality the value of inventories is the implicit collateral for trade credit, which usually is implicitly priced well above the market interest rate (Ng et al, 1999).

3.5

Irreversibility and aggregate output volatility

In this section we investigate on the effects of the financing and irreversibility constraint on aggregate output volatility. Veracierto (2003), in a general equilibrium real business cycle model with heterogeneous firms, shows that the presence of the irreversibility constraint at firm level has negligible effects on aggregate investment and output dynamics. In the previous sections of this paper the simulations of partial equilibrium economies with many heterogeneous firms showed that the effect of the irreversibility constraint on aggregate investment dynamics changes dramatically depending on whether or not firms are also subject to financing imperfections. in this section we will show that the irreversibility of fixed capital has a significant effect on the volatility of output of the simulated industry when financing imperfections are also present. Even though this result is subject to all the limitations of the partial equilibrium analysis approach, it suggests that the irrelevance result obtained by Veracierto (2003) should change if financing constraints were to be introduced in a simulated general equilibrium economy31 . This is because 31

Even though we consider a model of an industry rather than of the whole economy, our setting is

35

of the type of interaction between the two constraints. As Veracierto shows, the irreversibility constraint does not affect aggregate output volatility because, for an economy calibrated with realistic parameters, the aggregate shocks is not large enough to significantly change the fraction of firms that have a binding irreversibility constraint. But if financing constraints are also present, then the cross sectional distribution of wealth and fixed capital across firms implies that some firms have, or may have conditional on a negative shock, both constraints binding. For these firms a negative aggregate shock, even if it is small, reduces the wealth available to invest in variable inputs (labour or deliveries). This increases the distortion in the use of factors of production and the implicit cost of the irreversibility constraint. The implication is that by eliminating the irreversibility of fixed capital in such an economy we should have a significant effect on aggregate investment and output volatility even when aggregate shocks are relatively small. In Table 13 we simulate four economies with the two sets of parameters used respectively to calibrate the economy with only the irreversibility constraint and the economy with both constraints. In the first two columns we consider the benchmark parameters we used in section 3.3 to estimate the model with both constraints, and we illustrate the results of the simulations with both constraints in the first column and with only the financing constraint in the second column. In the second part of the table we consider the benchmark parameters we used in the previous section to estimate the model with only the irreversibility constraint, and we illustrate the results of its simulation in the fourth column and of the simulation of the economy with no constraints in the fifth column. Therefore the comparison of columns one and two is an estimation of the partial equilibrium effect on aggregate output of eliminating the irreversibility constraint in an economy with financing imperfections. The comparison between columns four and five is the same exercise on an economy without financing imperfections. This latter case is the partial equilibrium equivalent of the Veracierto (2003) analysis32 . Eliminating the irreversibility constraint in an industry without financing imperfections has the effect of increasing output volatility by 20%. This is much higher than then 5% estimated by Veracierto (2003), but still quite small, considering the partial equilibrium nature of our exercise. Also the relative volatilities of fixed capital and deliveries do not significantly change when the irreversibility constraint is eliminated. The set of parameters 1 case yields very different results. If we eliminate the irreversibility constraint from an economy where firms face financing imperfections, then the volatility of fixed capital relative to output increases by 66%. This large increase is due to the fact that by eliminating the irreversibility constraint we also eliminate the interactions with the financing constraint, which induced a cautious behaviour in fixed capital investment. The result is that the volatility of sales increase by as much as 91%. This large partial equilibrium effect would probably be much reduced in a general equilibrium setting where the interest rate and the relative price if the inputs are endogenous. Nonetheless it is reasonable to think that the increase in aggregate output volatility would still be non negligible. similar to Veracierto’s (2003). In both cases output is produced by a combination of an irreversible factor (capital) that is installed one period in advance and a reversible factor (labour in Veracierto (2003), variable capital in our model) that is immediately productive. 32 Columns three and four correspond to the q = 0 and q = 1 cases in table 1 in Veracierto (2003).

36

Table 13: Irreversibility and output volatility Set of parameters 1 Set of parameters 2 Both Only fin. Only No ∆% ∆% const. constr. Irr. const. st.dev. Sales 0.0416 0.0776 87% 0.0454 0.0546 20% st.dev. Deliveries 1.23 1.35 9.7% 1.26 1.33 5.5% st.dev. Fixed Capital 0.50 0.90 80% 0.93 1.04 8.9% st.dev.(deliveries) 2.44 1.49 -39% 1.36 1.28 -5.9% st.dev.(f ixed capital) Avg. % of binding 35% 33% -5.7% 0% 0% 0% financing constr. Avg. % of binding 26% 0% -100% 32% 0% -100% irrever. constr. Aaverage % of both 5% 0% -100% 0% 0% 0% constraints binding Set of parameters 1: parameters used in the benchmark calibration of the economy with both constraints. Set of parameters 2: parameters used in the benchmark calibration of the economy with only the irreversibility constraints.

4

Conclusions

In this paper we illustrated a structural model of a profit maximising firm subject to both borrowing constraints and irreversibility of fixed capital. The solution of the optimal investment problem shows that not only expected productivity but also current and expected financing problems affect investment decisions. In particular, despite the firm is risk neutral, future expected financing constraints may reduce current investment in fixed capital. This ”precautionary” reduction in investment may be substantial. Consider two firms identical in everything except than in their endowment of financial wealth. Our simulations show that the poorer firm may invest up to 80% less in the risky technology than the richer one, and the difference is entirely due to future expected financing constraints. Importantly, we showed that the irreversibility and the financing constraint are complementary. In particular, the irreversibility of fixed capital amplifies the effects of the financing constraint on variable investment. By simulating an artificial economy with many heterogeneous firms we showed that this amplification effect helps to explain why aggregate investment in input inventories and deliveries of US durable manufacturing firms are very volatile (relative to capital) and procyclical, and why such procyclicality is highly asymmetrical, so that it disappears in periods in which aggregate output is above its trend. Our model is also consistent with the stylised fact that fixed and especially inventory investment are sensitive to the net worth, even when marginal productivity of capital is taken into account. Although we calibrate the model to match one specific US two digits durable manufacturing sector, we show that a similar behaviour of inventories and deliveries is present also in most other durable good sectors. More generally, the implications of the model could be useful in understanding firm dynamics in any productive sector that satisfies the following assumptions: i) both financing and irreversibility constraints are binding for a non negligible share of firms in equilibrium; ii) firms produce output using a combination 37

of reversible and irreversible inputs.

References [1] Albuquerque, R. and H.A. Hopenhayn (2000), ”Optimal Dynamic Lending Contracts with Imperfect Enforceability”, mimeo, University of Rochester. [2] Bernanke, B., Gertler, M. and S. Gilchrist (1996), ”The Financial Accelerator and the Flight to Quality”, The Review of Economics and Statistics, 78, 1-15. [3] Bertola, G. and R. Caballero (1994), ”Irreversibility and Aggregate Investment”, Review of Economics Studies, 61, 223-246. [4] Besanko,D. and A.V. Thakor (1986), ”Collateral and Rationing: Sorting Equilibria in Monopolistic and Competitive Credit Markets”, International Economic Review, 28, 671-89. [5] Blinder, A.S. and L. Maccini (1991), ”Taking Stock: A Critical Assessment of Recent Research on Inventories”, Journal of Economic Perspectives, 5, 73-96. [6] Burnside, C. (1996), ”Production Function Regressions, Returns to Scale, and Externalities”, Journal of Monetary Economics, 37, 177-201. [7] Caballero, R.J. (1997), ”Aggregate Investment”, NBER WP #6264 [8] Caballero, R.J., Engel, E. and J. Haltiwanger (1995), ”Plant-Level Adjustment and Aggregate Investment Dynamics”, Brookings Papers on Economic Activity, 1-54. [9] Cagetti, M. and M. De Nardi (2003), ”Entrepreneurship, Frictions and Wealth”, Mimeo, University of Virginia. [10] Caggese, A. (2003), ”Financing Constraints, Irreversibility and Firm Dynamics”, Financial Markets Group W.P. n.440. [11] Carpenter,R.E., Fazzari S.M. and B. Petersen (1998), ”Financing Constraints and Inventory Investment: A Comparative Study with High-Frequency Panel Data”, The Review of Economics and Statistics, 80, 513-519. [12] Carroll, C.D. (2001), ”A Theory of the Consumption Function, with and without Liquidity Constraints”, Journal of Economic Perspectives, 15, 23-45. [13] Cleary, S. (1999), ”The Relationship between Firm Investment and Financial Status” Journal of Finance, 54, 673-692. [14] Cooley, T., Marimon, R. and V. Quadrini (2003), ”Aggregate Consequences of Limited Contract Enforceability”, NBER w.p. n.10132. [15] Cooley, T. and V. Quadrini (2001), ”Financial Markets and Firm Dynamics”, American Economic Review, 91, 1286-1310. 38

[16] Erickson, T. and T. Whited (2000), ”Measurement Error and the Relationship between Investment and q”, Journal of Political Economy, 108, 1027-57 [17] Fazzari, S.M, Hubbard, G.R. and B.C. Petersen (1988), ”Financing Constraints and Corporate Investment”, Brooking Papers on Economic Activity, 141-195. [18] Gentry, W.M. and R.G. Hubbard (2000), ”Entrepreneurship and Household Saving”, Mimeo, Columbia University. [19] Gertler, M. and S. Gilchrist (1994), ”Monetary Policy, Business Cycle, and the Behaviour of Small Manufacturing Firms”, Quarterly Journal of Economics, 109, 309340. [20] Gomes, J. (2001), ”Financing Investment”, American Economic Review, 91, 12631285. [21] Heaton, J. and D. Lucas (2000), ”Portfolio Choices and Asset Prices: The Importance of Entrepreneurial Risk”, Journal of Finance, 55, 1163-1198. [22] Hamilton, B.H., (2000), ”Does Entrepreneurship Pay? An Empirical Analysis of the Returns of Self-Employment”, Journal of Political Economy, 108, 604-631. [23] Hart,O and J.H. Moore (1998), ”Default and Renegotiation: a Dynamic Model of Debt”, Quarterly Journal of Economics, 113, 1-41. [24] Holt, R.W.P. (2003), ”Investment and Dividends Under Irreversibility and Financial Constraints”, Journal of Economic Dynamics and Control, 27, 467-502. [25] Hornstein, A (1998), ”Inventory Investment and the Business Cycle”, Federal Reserve Bank of Richmond Economic Quarterly, 84, 49-71 [26] Hubbard, G.R. (1998), ”Capital-Market Imperfections and Investment”, Journal of Economic Literature, 36, 193-225. [27] Kaplan, S.N. and L. Zingales (1997), ”Do Investment-Cash Flow Sensitivities Provide Useful Measures of Financing Constraints?”, Quarterly Journal of Economics, 112, 169-215. [28] Kaplan, S and L. Zingales (2000), ”Investment-Cash Flow Sensitivities Are Not Valid Measures Of Financing Constraints”, The Quarterly Journal of Economics, 115, 707712. [29] Kashap, A., Lamont, O and J. Stein (1994), ”Credit Conditions and the Cyclical Behaviour of Inventories”, Quarterly Journal of Economics, 109, 565-592. [30] Kiyotaki, N. and J.H. Moore (2002), ”Liquidity and Asset Prices”, Mimeo, London School of Economics. [31] Lucas, R.E. and N. Stokey (1989), Recursive Methods in Economic Dynamics, Harvard University Press. 39

[32] Milde, H. and J.G. Riley (1988), ”Signalling in Credit Markets”, Quarterly Journal of Economics, 103, 101-129. [33] Ng, C.K., Smith J.K. and R.L.Smith (1999) “Evidence on the Determinants of Credit Terms Used in Interfirm Trade” Journal of Finance, 54, 1109-29. [34] Oliner,S.D. and G.D.Rudebusch (1996),” Is There a Broad Credit Channel for Monetary Policy?”, Federal Reserve Bank of San Francisco Economic Review, 1, 3-13. [35] Ramey, V.A. (1989), ”Inventories as Factors of Production and Economic Fluctuations”, American Economic Review, 79, 338-54. [36] Ramey, V. and K. West (1999), ”Inventories”, Handbook of Macroeconomics, North Holland. [37] Scaramozzino, P (1997), ”Investment Irreversibility and Finance Constraints”, Oxford Bulletin of Economics and Statistics, 59, 89-108 [38] Stiglitz, J. and A.Weiss (1981), ”Credit Rationing in Markets with Imperfect Information”, American Economic Review 71, 912-927. [39] Stock, J. H. and M. Watson (1998), ”Business Cycle Fluctuations in U.S. Macroeconomic Time Series”, N.B.E.R. Working Paper n.6528. [40] Zeldes, S.P. (1989), ”Consumption and Liquidity Constraints: An Empirical Investigation”, Journal of Political Economy, 97, 305-46.

Appendix A: solution of the problem with endogenous exit: In section 2 of the paper we illustrated the first order conditions of the firm’s problem for the simplified case in which endogenous exit does not happen in equilibrium. Here we stay illustrate the full solution of the problem. If we substitute recursively Vt+1 (wt+1 , θt+1 , kt+1 ) in equation (11) and we add constraints (2), (3) and (6) at times t and t + 1, with the associated Lagrangian multipliers µ, λ and φ, we can represent the problem in the following way: ³

´

Vtstay (wt , θ t , kt ) = (1 + φt ) wt + bRt − kt+1 − lt + µt [kt+1 − (1 − δ k ) kt ] + λt [τ k kt + τ l lt − bt ] + ³ ´³ ´ + R1 Et {(1 − γSt+1 ) wt+1 + γSt+1 { 1 + φt+1 wt+1 + bt+1 − k − l t+2 t+1 + R +µt+1 n [kt+2 n − (1h− δ k ) kt+1 ] + λt+1 [τ k kt+1 + τ l lt+1 − Rbt+1 ]}}+ ioo stay γ (wt+2 , θt+2 , kt+2 ) + (1 − γSt+2 ) wt+2 + R2 Et St+1 Et+1 γSt+2 Vt+2 (36) The first order conditions of the problem are the following: ³

´

1 + φt = Rλt + Et 1 + γSt+1 φt+1 + γEt (Γt+1 ) M P Lt = UL +

R (R − τ l ) λt

³

Et 1 + γSt+1 φt+1 + γΓt+1 40

´

(37) (38)

R3 λt +RUKEt (1+γSt+1 φt+1 +γΓt+1 )−R2 µt Et (M P Kt+1 ) = + γ 2 Et (St+1 Ψt+1 ) γREt {St+1 [λt+1 τ k −µt+1 (1−δ k )]}+γREt [Ωt+1 ]+γ 2 covart+1 − γ 2 Et (St+1 Ψt+1 )

(39)

Where: Γt+1

∂St+1 ≡ ∂wt+1

(

γ (1 − γ) Et+1 [Vt+2 (wt+2 , θt+2 , kt+2 )] + wt+2 R γ

(

"

#)

Ωt+1

∂St+1 ≡ ∂kt+1

(

γ (1 − γ) Et+1 [Vt+2 (wt+2 , θt+2 , kt+2 )] + wt+2 R γ

(

"

#)

Ψt+1 = St+1

"

#

³ ´ 1 + Et+1 St+2 φt+2 γ

covart+1 = cov (M P Kt+1 , St+1 Ψt+1 )

− wt+1

)

(40)

− wt+1

)

(41) (42) (43)

Equations (13), (14) and (15) are a special case of (37), (38) and (39) for St = 1 for any t. The terms Γt+1 and Ωt+1 can be shown to be always equal to zero if forced exit never happens in equilibrium.

Appendix B: forced liquidation Let’s combine equations (3) and (6): µ



τl τk dt + 1 − lt + kt+1 ≤ wt + kt − F R R

(44)

The right hand side of equation (44) represents the investment capacity of the firm. If wealth is very low, in order to satisfy constraint (44) the firm can set both lt and dt equal to zero, while kt+1 cannot be set below (1 − δ k ) kt . By substituting dt = lt = 0 and kt+1 = (1 − δ k ) kt in equation (44) we get: ·

¸

τk F ≤ wt − (1 − δ k ) − kt R

(45)

Condition (45) may not hold if wt is very low. In order to see it, we substitute wt in condition (45) using equation (8), and we consider the “worst case scenario” in which the firm was lacking the funds to invest in variable capital in the previous period: yt−1 = 0, lt−1 = 0, bt−1 = τ k kt−1 and kt = (1 − δ k ) kt−1 . In this case condition (45) becomes: F ≤ τk

·

¸

1 (1 − δ k ) − 1 kt−1 R

(46)

Condition (46) cannot be satisfied even if F = 0. But with F = 0 any firm, no matter how small its scale of activity is, has a positive net present value of the discounted stream of future profits. Therefore equation (46) implies that in the worst case scenario a firm may be forced to retire for being unable to repay its debt without liquidating its fixed assets, even though it would be profitable to continue. Equation (45) holding with equality also implies that, for a given value of kt and θt , the minimum wealth level wtf r below which the firm is forced to retire increases one to one with the increase in F : ³

∂ wtf r | kt , θt δF

´

=1

But it is also possible to show that wtvr , the level of wealth below which the firm voluntary liquidates, is more sensitive to the increase in F than wtf r : ∂ (wtvr | kt , θt ) >1 δF 41

Our simulations show that, for the values of F employed in the paper, wtvr is uniformly higher than wtf r for all possible values of (kt , θt ) , implying that firms are never forced to liquidate their activity.

Appendix C: numerical solution In order to obtain a numerical solution of the dynamic nonlinear system of equations defined by equations (2), (37), (38), (39) and (16), plus the standard complementary slackness conditions on λt , µt and φt , we discretise the state space as follows: each one of ki,t and wi,t are discretised in a 30 points grid, while θt is discretised in 16 elements, which correspond to the eight states of the aggregate shock and the two states of the idiosyncratic shock. The solution of the problem is simplified by the fact that, for all the set of parameters chosen in the paper, forced exit never happens in equilibrium, so that we can calculate the solution for the special case in which the terms Γt+1 and Ωt+1 are always equal to zero. In order the solution, first we formulate ³ to compute ´ ³ an initial ´ guess of the forward terms Et St+1 φt+1 , Et (St+1 Ψt+1 ) , Et (St+1 λt+1 ) , Et St+1 µt+1 and covart+1 . Second, we solve the static optimisation problem conditional on this guess, for each discrete value of the state variables wi,t , ki,t and θi,t . This static optimisation problem is nonlinear as well. Conditional on the value of θi,t the solution falls into four possible categories, depending on the values of the couple {wi,t , ki,t } . These³ categories ´ correspond to the four areas in figure 1. Third, we update the guess of Et St+1 φt+1 , Et (St+1 Ψt+1 ) , Et (St+1 λt+1 ) , ³

´

Et St+1 µt+1 and covart+1 . We repeat these steps until the value function converges.

Appendix D: calibration (incomplete).

The main difference with the benchmark parameters for the economy with both constraints is the difference in the volatility and persistency of the productivity shock. In the economy with only the irreversibility constraint it is necessary a very high persistency of the idiosyncratic shock to generate some autocorrelation in the fixed capital investment rate. ρθ is equal to 0.98 against ρθ = 0.8 in the economy with both constraints. In the economy with only the financing constraint and with no constraints it is not possible to match the autocorrelation of the fixed capital investment rate, no matter how close to 1 ρθ is. Therefore we chose to set ρθ = 0.8 in the economy with only the financing constraint (the same as for the economy with both constraints). and ρθ = 0.98 for the economy with no constraints (the same as for the economy with only the irreversibility constraint).

42