Financial Frictions, Investment, and Tobin s q

Transcription

1 Financial Friction, Invetment, and Tobin q Dan Cao Georgetown Univerity Guido Lorenzoni Northwetern Univerity Karl Walentin Sverige Rikbank February 6, 2017 Abtract We develop a model of invetment with financial contraint and ue it to invetigate the relation between invetment and Tobin q. A firm i financed partly by inider, who control it aet, and partly by outide invetor. When inider wealth i carce, they earn a rate of return higher than the market rate of return and thu the firm value include a quai-rent on inveted capital. Thi implie that two force drive q: change in the value of inveted capital and change in the value of the inider future rent per unit of capital. Thi weaken the correlation between q and invetment, relative to the frictionle benchmark. We preent a calibrated verion of the model, which, due to thi effect, can generate more realitic correlation between invetment, q, and cah flow. Keyword: Financial contraint, optimal financial contract, invetment, Tobin q, limited enforcement. JEL code: E22, E30, E44, G30.

2 1 Introduction Dynamic model of the firm imply that invetment deciion and the value of the firm hould both repond to expectation about future profitability of capital. In model with contant return to cale and convex adjutment cot thee relation are epecially clean, a invetment and the firm value repond exactly in the ame way to new information about future profitability. Thi i the main prediction of Tobin q theory, which implie that current invetment move one-for-one with q, the ratio of the firm financial market value to it capital tock. Thi prediction, however, i typically rejected in the data, where invetment appear to correlate more trongly with current cah flow than with q. In thi paper, we invetigate the relation between invetment, q, and cah flow in a model with financial friction. The preence of financial friction introduce quai-rent in the market valuation of the firm. Thee quai-rent break the oneto-one link between invetment and q. We tudy how the preence of thee quairent affect the tatitical correlation between invetment, q, and cah flow, and ak whether a model with financial friction can match the correlation in the data. Our main concluion i that the preence of financial friction can bring the model cloer to the data, but that the model implication depend crucially on the hock tructure. The crucial obervation i that in a model with financial friction it i till true that invetment and q repond to future profitability, but the two variable now repond differently to information at different horizon. Invetment i particularly enitive to current profitability, which determine current internal financing, and to near-term financial profitability, which determine collateral value. On the other hand, q i relatively more enitive to profitability farther in the future, which will determine future growth and thu the ize of future quai-rent. Therefore, to break the link between invetment and q, we need the preence of both hort-lived hock which tend to move invetment more and have relatively maller effect on q and long-lived hock which do the oppoite. To develop thee point, we build a tochatic model of invetment ubject to 1

3 limited enforcement, with fully tate-contingent claim. We how that our limited enforcement contraint i equivalent to a tate-contingent collateral contraint, o our model i eentially a tochatic verion of Kiyotaki and Moore 1997) with adjutment cot and tate-contingent claim. 1 We how that the model lead to a wedge between average q which correpond to the q meaured from financial market value and marginal q which capture the marginal incentive to invet and i related one-to-one to invetment. 2 We then analyze two verion of the model and look at their implication for an invetment regreion in which the invetment rate i regreed on average q and cah flow. Firt, we focu on a verion of the model with no adjutment cot, which, under ome implifying aumption, can be linearized and tudied analytically. We conider three different hock tructure. In a cae with a ingle peritent hock, the model ha indeterminate prediction regarding invetment regreion coefficient. Thi imply follow becaue in thi cae q and cah flow are perfectly collinear. In a cae with two hock a temporary hock and a peritent hock the one-to-one relation between q and invetment break down becaue invetment i driven by productivity in period t and t + 1 while q repond to all future value of productivity. Finally, we conider a cae with new hock, that i, we allow agent to oberve J period in advance the realization of productivity hock. In thi cae, we how that increaing the length of the horizon J reduce the coefficient on q and increae the coefficient on cah flow in invetment regreion. Thi i due again to the differential repone of invetment and q to information on productivity at different horizon. The model with no adjutment cot, while analytically tractable, i quantitatively unappealing, a it tend to produce too much hort-run volatility and too little peritence in invetment. Therefore, for a more quantitative evaluation of the model we introduce adjutment cot. We calibrate the model to data moment 1 Related recent tochatic model that combine tate-contingent claim with ome form of collateral contraint include He and Krihnamurthy 2013), Rampini and Viwanathan 2013) and Di Tella 2016). 2 The terminology goe back to Hayahi 1982), who how that the two are equivalent in a canonical model with convex adjutment cot. 2

4 from Computat and analyze it implication both in term of impule repone and in term of invetment regreion. Our baeline calibration i baed on the two hock tructure, with temporary and peritent hock. In thi calibration we how that q repond relatively more trongly to the peritent hock while invetment repond relatively more trongly to the tranitory hock, in line with the intuition from the no-adjutment-cot cae. Thi lead to invetment regreion with a maller coefficient on q and a larger coefficient on cah flow, relative to a model with no financial friction, thu bringing u cloer to empirical coefficient. However, the q coefficient i till larger than in the data and the cah flow coefficient i maller than in the data. When adding the poibility of new hock, the diconnect between q and invetment increae, leading to further reduction in the q coefficient and increae in the cah flow coefficient. Fazzari et al. 1988) tarted a large empirical literature that explore the relation between invetment and q uing firm-level data. The typical finding in thi literature i a mall coefficient on q and a poitive and ignificant coefficient on cah flow. 3 Fazzari et al. 1988), Gilchrit and Himmelberg 1995) and mot of the ubequent literature interpret thee finding a a ymptom of financial friction at work. More recent work by Gome 2001) and Cooper and Ejarque 2003) quetion thi interpretation. The approach taken in thee two paper i to look at the tatitical implication of imulated data generated by a model to undertand the empirical correlation between invetment, q and cah flow. 4 In their imulated economie with financial friction q till explain mot of the variability in invetment, and cah flow doe not provide additional explanatory power. In thi paper, we take a imilar approach but reach different concluion. Thi i due to two main difference. Firt, Gome 2001) and Cooper and Ejarque 2003) model financial friction by introducing a tranaction cot which i a function of the flow of outide finance iued each period, while we introduce a contractual imperfection that impoe an upper bound on the tock of outide liabilitie a a fraction of total aet. Our approach add a tate variable to the problem, namely 3 See Hubbard 1998) for a urvey. 4 An approach that goe back to Sargent 1980). 3

5 the tock of exiting liabilitie of the firm a a fraction of aet, thu generating lower dynamic in the gap between internal fund and the deired level of invetment. Second, we explore a variety of hock tructure, which, a we argue below, play an important role in our reult. A related trand of recent literature ha focued on violation of q theory coming from decreaing return or market power, leaving aide financial friction. 5 We ee our effort a complementary to thi literature, ince both financial friction and decreaing return determine the preence of future rent embedded in the value of the firm. Alo in that literature the hock tructure play an important role in the reult. For example, Eberly et al. 2008) how that it i eaier to obtain realitic implication for invetment regreion by auming a Markov proce in which the ditribution from which peritent productivity hock are drawn witche occaionally between two regime. Abel and Eberly 2011) alo how that in model with decreaing return it i poible to obtain intereting dynamic in q with no adjutment cot, imilarly to what we do in Section 3 in a model with contant return to cale and financial contraint. The implet hock that break the link between q and invetment in model with financial contraint i a purely temporary hock to cah flow, which doe not affect capital future productivity. Abent financial friction thi hock hould have no effect on current invetment. Thi idea i the bai of a trand of empirical literature that tet for financial contraint by identifying ome ource of purely temporary hock to cah flow. Thi i the approach taken by Blanchard et al. 1994) and Rauh 2006), which provide reliable evidence of the preence of financial contraint. Our paper build on a imilar intuition, by howing that in general hock affecting profitability at different horizon have differential effect on q and invetment and ak whether, given a realitic mix of hock, a model with financial friction can produce the unconditional correlation oberved in the data. In thi paper we ue the implet poible model with the feature we need: 5 See Schiantarelli and Georgouto 1990), Alti 2003), Moyen 2004), Eberly et al. 2008), Abel and Eberly 2011), Abel and Eberly 2012). 4

6 an occaionally binding financial contraint; a dynamic, tochatic tructure; adjutment cot that can produce realitic invetment dynamic. There i a growing literature that build richer model that are geared more directly to etimation. In particular, Henney and Whited 2007) build a rich tructural model of firm invetment with financial friction, which i etimated by imulated method of moment. They find that the financial contraint play an important role in explaining oberved firm behavior. In their model, due to the complexity of the etimation tak, the financial friction i introduced in a reduced form manner, by auming tranaction cot aociated to the iuance of new equity or debt, a in Gome 2001) or Cooper and Ejarque 2003). 6 We ee our effort a complementary, a we have a more tylized model, but with financial contraint coming from an explicitly modeled contractual imperfection. Latly, relative to thee paper, and the entire invetment regreion literature, we demontrate the importance of flexible hock tructure, beide financial friction, in bringing the model implication cloer to the data. A growing number of paper ue recurive method to characterize optimal dynamic financial contract in environment with different form of contractual friction Atkeon and Cole 2005), Clementi and Hopenhayn 2006), DeMarzo and Sannikov 2006), DeMarzo et al. 2012)). The limited enforcement friction in thi paper make it cloer to the model in Albuquerque and Hopenhayn 2004) and Cooley et al. 2004). Within thi literature Biai et al. 2007) look more cloely at the implication of the theory for aet pricing. In particular, they find a et of ecuritie that implement the optimal contract and then tudy the tochatic behavior of the price of thee ecuritie. Here, our objective i to examine the model implication for q theory, therefore we imply focu on the total value of the firm, which include the value of all the claim held by inider and outider. 7 6 The difference in reult, relative to thee paper, appear due to the fact that Henney and Whited 2007) alo match the behavior of a number of financial variable. 7 Our formulation alo allow for the exploration of flexible hock tructure, which are abent in the aforementioned paper. For example, mot continuou time model aume hock under the form of Brownian motion, thu rule out intereting hock tructure uch a permanent veru temporary hock and new hock conidered in our paper. 5

7 In Section 2 we preent the model. In Section 3, we tudy the cae of no adjutment cot, deriving analytical reult. In Section 4, we tudy the model with adjutment cot, relying on numerical imulation. 2 The Model Conider an infinite horizon economy, in dicrete time, populated by a continuum of entrepreneur who invet in phyical capital and raie fund from rik neutral invetor. The entrepreneur technology i linear: K it unit of capital, intalled at time t 1 by entrepreneur i, yield profit A it K it at time t. We can think of the linear profit function A it K it a coming from a contant return to cale production function in capital and other variable input which can be cotlely adjuted. Therefore, change in A it capture both change in technology and change in input and output price. For brevity, we jut call A it productivity. Productivity i a function of the tate it, A it = A it ), where it i a Markov proce with a finite tate pace S and tranition probability π it it 1 ). There are no aggregate hock, o the cro ectional ditribution of it acro entrepreneur i contant. Invetment i ubject to convex adjutment cot. The cot of changing the intalled capital tock from K it to K it+1 i G K it+1, K it ) unit of conumption good at date t. The function G include both the cot of purchaing capital good and the intallation cot. We aume G i increaing and convex in it firt argument, decreaing in the econd argument, and diplay contant return to cale. For numerical reult, we ue the quadratic functional form G K it+1, K it ) = K it+1 1 δ) K it + ξ K it+1 K it ) 2. 1) 2 K it All agent in the model are rik neutral. The entrepreneur dicount factor i β and the invetor dicount factor i ˆβ, with ˆβ > β. We aume invetor have a large enough endowment of the conumption good each period o that the 6

8 equilibrium interet rate i 1 + r = 1/ ˆβ. Each period an entrepreneur retire with probability γ and i replaced by a new entrepreneur with an endowment of 1 unit of capital. When an entrepreneur retire, productivity A it i zero from next period on. The retirement hock i embedded in the proce it by auming that there i an aborbing tate r with A r ) = 0 and the probability of tranitioning to r from any other tate i γ. Each period, entrepreneur i can iue one-period tate contingent liabilitie, ubject to limited enforcement. The entrepreneur control the firm capital K it and, at the beginning of each period, can default on hi liabilitie and divert a fraction 1 θ of the firm capital. If he doe o, he re-enter the financial market a a new entrepreneur, with capital 1 θ) K it and no liabilitie. That i, the punihment for a defaulting entrepreneur i the lo of a fraction θ of the firm aet. 2.1 Optimal invetment We formulate the optimization problem of the individual entrepreneur in recurive form, dropping the ubcript i and t. Let V K, B, ) be the expected utility of an entrepreneur in tate, who enter the period with capital tock K and current liabilitie B. For now, we imply aume that the problem parameter are uch that the entrepreneur optimization problem i well defined. In the following ection, we provide condition that enure that thi i the cae. 8 The function V atifie the Bellman equation ubject to V K, B, ) = max C + βe [ V K, B ), ) ], 2) C 0,K 0,{B )} C + G K, K ) A)K B + ˆβE [ B ) ], 3) V K, B ), ) V 1 θ) K, 0, ),, 4) 8 In the Online Appendix we provide a general exitence reult. 7

9 where C i current conumption, K i next period capital tock, and B ) are next period liabilitie contingent on. Contraint 3) i the budget contraint and ˆβE [B ) ] are the fund raied by elling the tate contingent claim {B )} to the invetor. Contraint 4) i the enforcement contraint that require the continuation value under repayment to be greater than or equal to the continuation value under default. The aumption of contant return to cale implie that the value function take the form V K, B, ) = v b, ) K for ome function v, where b = B/K i the ratio of current liabilitie to the capital tock. We can then rewrite the Bellman equation a ubject to v b, ) K = max C 0,K 0 {b )} C + βe [ v b ), ) ] K, 5) C + G K, K ) A)K bk + ˆβE [ b ) ] K, 6) v b ), ) 1 θ) v 0, ),. 7) It i eay to how that v i trictly decreaing in b. We can then find tatecontingent borrowing limit b ) uch that the enforcement contraint can be written a b ) b ),. 8) So the enforcement contraint i equivalent to a tate contingent upper bound on the ratio of the firm liabilitie to capital. Relative to exiting model with collateral contraint, two ditinguihing feature of our model are that we allow for tate-contingent claim and we derive the tate-contingent bound endogenouly from limited enforcement. 9 9 Other recent model that allow for tate-contingent claim include He and Krihnamurthy 2013) and Rampini and Viwanathan 2013). Cao 2013) develop a general model with an explicit tochatic tructure that tudie collateral contraint with non-tate-contingent debt. 8

10 2.2 Average and Marginal q To characterize the olution to the entrepreneur problem let u tart from the firt order condition for K : λg 1 K, K ) = λ ˆβE [ b ] + βe [ v ], 9) where λ i the Lagrange multiplier on the budget contraint 6), or the marginal value of wealth for the entrepreneur. The expreion E [b ] and E [v ] are horthand for E [b ) ] and E [v b ), ) ]. Optimality for conumption implie that λ 1 and the non-negativity contraint on conumption i binding if λ > 1. To interpret condition 9) rewrite it a: λ = βe [v ] G 1 K, K) ˆβE [b ] 1. 10) When the inequality i trict the entrepreneur trictly prefer reducing current conumption to invet in new unit of capital. If C wa poitive the entrepreneur could reduce it and ue the additional fund to increae the capital tock. The marginal cot of an extra unit of capital i G 1 K, K) but the extra unit of capital increae collateral and allow the entrepreneur to borrow ˆβE [b ] more from the conumer. So a unit reduction in conumption lead to a levered increae in capital inveted of 1/G 1 ˆβE [b ]). Since capital tomorrow increae future utility by βe [v ], we obtain 10). Condition 9) can be ued to derive our main reult on average and marginal q. The value of all the claim on the firm future earning, held by invetor and by the entrepreneur at the end of the period, i ˆβE [ B ) ] + βe [ V K, B ), ) ]. 9

11 Dividing by total capital inveted give u average q: q a ˆβE [ b ] + βe [ v ]. Marginal q, on the other hand, i jut the marginal cot of one unit of new capital, q m G 1 K, K). We can then rearrange equation 9) and expre it in term of q a and q m a: q a = q m + λ 1 λ βe [ v ]. 11) Since λ > 1 if only if the non-negativity contraint on conumption i binding, we have proved the following reult. Propoition 1. Average q i greater than or equal to marginal q, with trict equality if and only if the non-negativity contraint on conumption i binding. The difference between average and marginal q i larger if either the Lagrange multiplier λ i larger or the future value of entrepreneurial equity E [v ] i larger, a we can ee from equation 11). A we hall ee in the numerical part of the paper, an increae in indebtedne b increae λ but reduce the future value of entrepreneurial equity, o in general the relation between b and q a q m can be non-monotone. There i a cutoff for b uch that λ = 1 below the cutoff and λ > 1 above the cutoff, o we know the relation i increaing in ome region. The fact that the only Lagrange multiplier appearing in 11) i λ, doe not mean that the collateral contraint i not relevant in determining the gap between average and marginal q. Conider the firt order condition for b ˆβλ + βv b b ), ) = µ ), where µ ) i the Lagrange multiplier on the enforcement contraint 8) expreed a a ratio of π )K for convenience). Uing the envelope condition for b to ubtitute for v b and uing time ubcript we can then write λ t = βˆβ λ t+1 + 1ˆβ µ t+1. 12) 10

12 Thi condition how that λ t i a forward looking variable determined by current and future value of µ t+1. Poitive value of thi Lagrange multiplier in the future induce the entrepreneur to reduce conumption today to increae internal fund available. The forward looking nature of λ t will be ueful to interpret ome of our numerical reult about new hock. If β = ˆβ, condition 12) implie that if, at ome date t, the entrepreneur conumption i poitive and λ t = 1, then the non-negativity contraint and the collateral contraint can not be binding at any future date. In other word, once the entrepreneur i uncontrained he can never go back to being contrained. Thi i due to the aumption of complete tate contingent market. Auming β < ˆβ enure that entrepreneur can alternate between poitive and zero conumption. We conclude thi ection by introducing ome aet pricing relation that will be ued to characterize the equilibrium. We ue the notation G 1,t and G 2,t a horthand for G 1 K t+1, K t ) and G 2 K t+1, K t ). Propoition 2. The following condition hold in equilibrium and λ t = βe t [ λ t+1 A t+1 G 2,t+1 b t+1 G 1,t ˆβE t b t+1 [ ] [ At+1 G 2,t+1 βλt+1 ˆβE t 1 E t G 1,t λ t ], 13) ] A t+1 G 2,t+1. 14) G 1,t The lat two condition hold with trict inequality if the collateral contraint i binding with poitive probability. Notice that A t+1 G 2,t+1 b t+1 G 1,t ˆβE t b t+1 repreent the levered rate of return on capital. Condition 13) further illutrate the forward-looking nature of λ t. In particular, it how that λ t i a geometric cumulate of all future levered return on capital. Condition 13) can alo be interpreted a a tandard aet pricing condition, dividing both ide by λ t and oberving that βλ t+1 /λ t i the tochatic dicount factor of the entrepreneur. 11

13 The expreion A t+1 G 2,t+1 G 1,t i the unlevered return on capital. When the collateral contraint i binding the firt inequality in 14) i trict and thi implie that the expected rate of return on capital i higher than the interet rate 1 + r. Thi implie that the levered return on capital i higher than the unlevered return. The entrepreneur will borrow up to the point at which the dicounted levered rate of return i 1, by condition 13). At that point the dicounted unlevered return will be maller than 1, by the econd inequality in 14). Thi econd inequality can alo be interpreted a capturing the fact that inveting in phyical capital ha the additional benefit of relaxing the collateral contraint. Define the finance premium a the difference between the expected return on entrepreneurial capital and the interet rate which i equal to 1/ ˆβ): [ ] At+1 G f p t 2,t+1 E t 1 + r). 15) G 1,t The firt inequality in 14) how that the finance premium i poitive whenever the collateral contraint i binding. We will ue thi definition of the finance premium in Section Model with No Adjutment Cot: Analytical Reult We now conider the cae of no adjutment cot, which arie when G K t+1, K t ) = K t+1 1 δ) K t. In thi cae, we can derive ome analytical reult that help build the intuition for the numerical reult in the following ection. For thi ection we aume a trict 12

14 inequality between the dicount factor of entrepreneur and invetor, β < ˆβ, o that we can focu on cae in which the collateral contraint i alway binding. Abent adjutment cot, the value function take the linear form V K, B, ) = Λ ) [R ) K B], 16) where R i the gro return on capital defined by R ) A ) + 1 δ. Notice that R ) K B i the total net worth of the entrepreneur at the beginning of the period, the total value of the capital tock minu the entrepreneur liabilitie. With a linear value function the borrowing limit are b) = θr ), 17) and they have a natural interpretation: the entrepreneur can pledge a fraction θ of the firm gro return. We now make aumption that enure that the problem i well defined and that the collateral contraint i alway binding in equilibrium. Aume the following three inequalitie hold for all : βe [ R ) ] > 1, 18) θ ˆβE [ R ) ] < 1, 19) 1 γ) 1 θ) βe [R ), = r ] 1 θ ˆβE [R ) ] < ζ, 20) for ome ζ < 1. Condition 18) implie that the expected rate of return on capital i greater than the invere dicount factor of the entrepreneur, o the entrepreneur prefer invetment to conumption. Condition 19) implie that pledgeable return are inufficient to finance the purchae of one unit of capital, i.e., invetment cannot be fully financed with outide fund. Thi condition enure that 13

15 invetment i finite. Finally, condition 20) enure that the entrepreneur utility i bounded. The lat condition allow u to ue the contraction mapping theorem to fully characterize the equilibrium marginal value of wealth Λ ) in the following propoition. The proof of thi lemma and of the following reult in thi ection are in the appendix. Lemma 1. If condition 18)-20) hold there i a unique function Λ : S [1, ) that atifie the recurion and Λ ) = 1 for = r. Λ ) = β 1 θ) E [Λ ) R ) ] 1 θ ˆβE [R, for all = r, 21) ) ] Notice that 21) i a pecial cae of condition 13), in which the contraint i alway binding. The following propoition characterize an equilibrium. Propoition 3. If condition 18)-20) hold and Λ ) atifie Λ ) > βˆβ Λ ), 22) for all, S, then the collateral contraint i binding in all tate, conumption i zero until the retirement hock, invetment in all period before retirement i given by K 1 δ) K K = 1 θ) R ) 1 θ ˆβE [R 1 δ), 23) ) ] and average q i q a = E [ 1 θ) βλ ) + θ ˆβ ) R ) ]. 24) Condition 22) enure that entrepreneur never delay invetment. Namely, it implie that they alway prefer to invet in phyical capital today rather than buying a tate-contingent ecurity that pay in ome future tate. The entrepreneur problem can be analyzed under weaker verion of 18)- 22), but then the contraint will be non-binding in ome tate. It i ueful to 14

16 remark that we could embed our model in a general equilibrium environment with a contant return to cale production function in capital and labor and a fixed upply of labor. In thi general equilibrium model A ) i replaced by the endogenou value of the marginal product of capital. It i then poible to derive condition 18)-22) endogenouly if hock are mall and the non-tochatic teady tate feature a binding collateral contraint. We now aume condition 18)-22) hold and analyze the model auming that there are mall hock to A around the level Ā and linearizing the equilibrium condition 23)-24) around the non-tochatic teady tate. The invetment rate i defined a invetment over aet and i denoted by IK t K t+1 1 δ) K t K t. We will ue a bar to denote teady tate value and a tilde to denote deviation from the teady tate. In teady tate equation 21) yield Λ = and the invetment rate i β 1 θ) γ R 1 θ ˆβ + 1 θ) 1 γ) β ) R. IK = 1 θ) R 1 δ). 1 θ ˆβ R The following propoition charaterize the dynamic of invetment and Tobin Q around the teady tate. Propoition 4. If the economy atifie 18)-22) a linear approximation give the follow- 15

17 ing expreion for invetment and average q: IK t = 1 [ θ Ã t + θ ˆβ R 1 θ ˆβ R 1 θ ˆβ R E ] ] t [Ãt+1, 25) q a t = [ β 1 θ) γ + 1 γ) Λ) + θ ˆβ ] E t [Ãt+1 ] + + β 1 θ) 1 γ) RE t [ Λ t+1 ], 26) where Λ t = Λ/ R 1 θ ˆβR ) j 1 γ) Λ ] E t [Ãt+j, 27) j=0 γ + 1 γ) Λ conditional on t = r. Equation 25)-26) expre invetment and average q in term of current and future expected value of productivity. Since A t i equal to profit over capital, we match it to cah flow over aet in the empirical literature. Given aumption about the proce for A t, equation 25) and 26) give u all the information about the variance-covariance matrix of IK t, q a t, Ã t ) and thu about invetment regreion coefficient. The crucial obervation i that average q i affected by the marginal value of entrepreneurial net worth, which i a forward looking variable that reflect expectation about all future exce return on entrepreneurial capital. 10 Through thi channel, average q repond to information about future value of A t at all horizon. At the ame time, invetment i only driven by the current and next period value of A t. The current value determine internal fund, the next period value determine collateral value. Putting thee fact together implie that hock that affect profitability differentially at different horizon will break the link between average q and invetment. We now turn to a few example that how how different hock tructure lead to different implication for the variance-covariance matrix of invetment, average q and cah flow and thu for invetment regreion. 10 See the dicuion following Propoition 2. 16

18 Example 1. Productivity à t follow the AR1) proce: à t = ρã t 1 + ε t, where ε t i an i.i.d. hock. ] In thi example, we have E t [Ãt+j = ρ j à t o all future expected value of à t are proportional to the current value. Subtituting in 25)-26), it i eay to how that both q t a and IK t are linear function of à t. Therefore, in thi cae cah flow and average q are both, eparately, ufficient tatitic for invetment. Thi i true even though there i a financial contraint alway binding, imply due to the fact that a ingle hock i driving both variable. In thi example, the coefficient of a regreion of invetment on average q and cah flow are indeterminate due to perfect collinearity, but adding cah flow to a univariate regreion of invetment on average q alone doe not increae the regreion explanatory power. Example 2. Productivity à t ha a peritent component x t and a temporary component η t : à t = x t + η t with x t = ρx t 1 + ε t. In thi example, we have E t [Ãt+j ] = ρ j x t, and ubtituting in 25)-26), we arrive at: IK t = 1 θ) 1 1 ρ) Rθ ˆβ ) 1 θ ˆβ R ) x 2 t + 1 θ 1 θ ˆβ R η t, [ β q t a = 1 θ) γ + 1 γ) Λ) + θ ˆβ ) ρ + β 1 θ) 1 γ) γ + 1 γ) ] Λ) 1 θ ˆβ R ) Λρ γ + 1 γ) 1 ρ) Λ) If we now run a regreion of invetment on average q and cah flow, cah flow i the only variable that can capture variation in η t, o the coefficient on cah flow x t. 17

19 will be poitive and equal to 1 θ 1 θ ˆβ R, and cah flow improve the explanatory power of the invetment regreion. The coefficient on cah flow here i bigger than 1, but that clearly due to the abence of adjutment cot. In the next ection we will build on the logic of thi example, to analyze quantatively the effect of financial contraint on invetment regreion. Notice that in thi example, invetment, q and cah flow are fully determined by the two random variable x t and η t and the coefficient are independent of the variance parameter. Thi implie that, given all the other parameter, the coefficient of the invetment regreion are independent of the value of the variance σ 2 ε and σ 2 η, a long a both are poitive. A we hall ee, thi reult doe not extend to the general model with adjutment cot. A an aide, notice that in thi example, the coefficient on cah flow i higher for firm with larger value of θ, i.e., for firm that can finance a larger fraction of invetment with external fund. Thee firm repond more becaue they can lever more any temporary increae in internal fund. Thi i reminicent of the obervation in Kaplan and Zingale 1997) that the coefficient on cah flow in an invetment regreion hould not be ued a meaure of the tightne of the financial contraint. We now turn to our lat example, in which we introduce new hock. Example 3. The productivity proce i a in Example 2 but the value of the permanent component x t i known J period in advance, with J 1. In the appendix, we how that in thi example invetment and q dynamic are given by β 1 θ) γ + 1 γ) Λ) + θ ˆβ q t a = β1 θ)1 γ) Λ + ) x t+1 + ε t 28) 1 θ ˆβR) 1 1 γ) Λρ γ+1 γ) Λ 18

20 where 11 ε t = J 1 j=1 ) j β 1 θ) 1 γ) Λ 1 γ) 1 θ ˆβR ) Λ ) 1 1 γ) ε Λρ t+1+j, γ + 1 γ) Λ γ+1 γ) Λ and IK t = 1 θ 1 θ ˆβR x t + η t ) + 1 θ) Rθ ˆβ 1 θ ˆβR ) 2 x t+1. We can then how that increaing J affect the coefficient and the R 2 of the invetment regreion a follow. Propoition 5. In the economy of Example 3, all ele equal, increaing the horizon J at which hock are anticipated decreae the coefficient on average q, increae the coefficient on cah flow, and reduce the R 2 of the invetment regreion. The proof of thi reult i in the appendix. Invetment, a in the previou example, i jut a linear function of productivity at time t and t + 1, which fully determine current cah flow and collateral value. On the other hand, q i a function of all future value of A t and, given the preence of new, thee value are driven by anticipated future hock which have no effect on invetment. Thi weaken the relation between q and invetment. Moreover, ince q i the only ource of information about x t+1, and, with new hock, it become a noiier ource of information, thi alo reduce the joint explanatory power of q and cah flow. Notice that new hock here are acting very much like meaurement error in q, by adding a hock to it that i unrelated to the hock driving invetment. However, financial friction are eential in introducing thi ource of error. Abent financial friction future value of productivity hould not affect q, and it i only becaue q include future quai-rent that the relation arie. In the next ection, we will ee that the force identified in thee three example carry over to a more general model with adjutment cot. 11 When J = 1, ε t = 0. 19

21 4 Model with Adjutment Cot: Quantitative Analyi We now turn to the full model with adjutment cot and analyze it implication uing numerical imulation. While the no adjutment cot model analyzed above i ueful to build intuition, it ha a number of unrealitic implication in particular for the inertial behavior of invetment. The full model with adjutment cot, on the other hand, can be calibrated to match ome moment of the oberved procee for profit and invetment, o that we can look at it quantitative implication. We tart by decribing our choice of parameter and characterize the equilibrium in term of policy function and impule repone. We then run invetment regreion on the imulated output and explore the model ability to replicate empirical invetment regreion. 4.1 Calibration The time period in the model i one year. The baeline parameter value are ummarized in Table 1. The firt three parameter are pre-et, the remaining parameter are calibrated on Computat data. We now decribe their choice in detail. The invetor dicount factor ˆβ i choen o that the implied interet rate i 8.7%. A argued by Abel and Eberly 2011) the interet rate ued in thi type of exercie hould correpond to a rik-adjuted expected return. The number we chooe i in the range of rate of return ued in the literature. 12 The entrepreneur dicount factor β ha effect imilar to the parameter γ which govern their exit rate. In particular, both affect the incentive of entrepreneur to accumulate wealth and become financially uncontrained and both affect the forward looking component of q. Therefore, we fix β at a level lower than ˆβ and 12 Abel and Eberly 2011) and DeMarzo et al. 2012) chooe number near 10%, while Moyen 2004) and Gome 2001) ue r = 6.5%. 20

22 Table 1: Parameter Preet β ˆβ θ Calibrated to cah flow moment µ a ρ x σ ε σ η Calibrated to invetment and q moment δ ξ γ calibrate γ. 13 Regarding the fraction of non-divertible aet θ, there i only indirect empirical evidence, and exiting imulation in the literature have ued a wide range of value. Here we chooe θ = 0.3 in line with evidence in Fazzari et al. 1988) and Nezafat and Slavik 2013). In particular, Fazzari et al. 1988) report that 30% of manufacturing invetment i financed externally. Nezafat and Slavik 2013) ue US Flow of Fund data for non-financial firm to etimate the ratio of fund raied in the market to fixed invetment, and find a mean value of The parameter in the econd line of Table 1 are calibrated to match moment of the firm-level cah flow time erie in Computat. We aume that profit per unit of capital A t are the um of a peritent and a temporary component. Namely, A it = x it + η it x it = 1 ρ x )µ a + ρ x x it 1 + ε it where η it and ε it are i.i.d. Gauian hock with variance σ 2 η and σ 2 ε. We identify profit per unit of capital in the model, A it, with cah flow per unit of capital in the data, denoted by CFK it. 14 The parameter µ a i et equal to average cah flow per unit of capital in the data. The value of ρ x, σ ε and σ η are choen to match the firt and econd order autocorrelation and the tandard deviation of cah flow 13 Changing the choen value of β in a reaonable range doe not affect the reult ignificantly. 14 Cah flow i equal to net income before extraordinary item plu depreciation. 21

23 Table 2: Target moment and model value Moment ρ 1 CFK) ρ 2 CFK) σcfk) µik) σik) µq a ) Target value Model value per unit of capital in the data, denoted, repectively, by ρ 1 CFK), ρ 2 CFK) and σcfk). Thee moment are etimated uing the approach of Arellano and Bond 1991) and Arellano and Bover 1995) and are reported in Table Notice that imply computing raw autocorrelation in the data a ometime done in the literature would lead to biaed etimate, given the hort ample length. 16 In term of ample, we ue the ame ub-ample of Computat ued in Gilchrit and Himmelberg 1995) o that we can compare our imulated regreion to their reult. 17 The next three parameter in Table 1, δ, ξ, and γ, are choen to match three moment from the Computat ample: the mean and tandard deviation of the invetment rate, µik) and σik), and the mean of average q, µq a ). The reaon why δ and ξ help determine the level and volatility of the invetment rate i intuitive, a thee two parameter determine the depreciation rate and the lope of the adjutment cot function. The parameter γ control the peed at which entrepreneur exit, o it affect the dicounted preent value of the quai-rent they expect to 15 We etimate the firm-pecific variation in cah-flow by firt taking out the aggregate mean for each year and then applying the function xtabond2 in STATA. Thi implement the GMM approach of Arellano and Bover 1995). Thi approach avoid etimating individual fixed effect affecting both the dependent variable cah flow) and one of the independent variable lagged cah flow), by firt-differencing the law of motion for cah flow, and then uing both lagged difference and lagged level a intrument. We ue the firt three available non-autocorrelated) lag in difference a intrument, with lag choen eparately for the 1t and 2nd order autocorrelation etimation. One lagged level i alo ued a an intrument. 16 Thi type of bia wa firt documented in Nickell 1981). The bia i non-negligible in our ample. For the firt-order autocorrelation, the Arellano and Bond 1991) approach give ρ 1 CFK) = 0.60, while the raw autocorrelation in the data i In particular, we retrict attention to the ample period and ue the ame 428 lited firm ued in their paper. 22

24 receive in the future and thu average q. However, the three parameter interact, o we chooe them jointly by a grid earch in order to minimize the average quared percentage deviation between the three model-generated moment and their target. The target moment from the data and the model generated moment are reported in Table Notice that there i a tenion between hitting the target for µik) and σik). Increaing any of the parameter, δ, ξ, γ reduce µik), bringing it cloer to it target value, but alo decreae σik), bringing it farther from it target. Notice alo that it i important for our purpoe that the model generate a realitic level of volatility in the invetment rate, given that IK i the dependent variable in the regreion we will preent in Section 4.3 below. Our calibration alo determine the average ize of the wedge between average and marginal q. In particular, µq a ) = 2.5 i the mean value of average q while ξ and µik) determine the mean value of marginal q, which i 1 + ξµik) δ) = Therefore, the average wedge between average and marginal q i Since the preence of the wedge i what break the ufficient tatitic property of q it i ueful that our calibration impoe ome dicipline on the wedge ize. All the imulation aume that entrepreneur enter the economy with a unit endowment of capital and zero financial wealth i.e., zero current profit and zero debt). Since the entrepreneur problem i invariant to the capital tock and all our empirical target are normalized by total aet, the choice of the initial capital endowment i jut a normalization. We have experimented with different initial condition for financial wealth, but they have mall effect on our reult given that with our parameter the tate variable b converge quickly to it tationary ditribution. It i ueful to compare our reult to thoe of a benchmark model with no financial friction. To make the parametrization of the two model comparable, we re-calibrate the parameter δ, ξ and γ for the frictionle cae. The moment and aociated parameter are reported in Table 3. Notice that the frictionle model generate a low value of µq a ). For given IK, increaing ξ would increae 18 The target tandard deviation σik) i a pooled etimate. 23

25 Table 3: Calibration of frictionle model Parameter δ ξ γ Moment µik) µq a ) σ IK) Target value Model value marginal and average q which are the ame in the frictionle cae), but it would reduce the volatility of invetment. In Section 4.5 we conider an alternative calibration approach, that target the average finance premium, a defined in equation 15). 4.2 Model dynamic We now characterize the optimal olution to the entrepreneur problem, firt decribing optimal choice and value a function of the tate variable and next howing what thi behavior implie for the repone of endogenou variable to different hock Characterization To illutrate the model behavior, it help intuition to ue a tate variable A and n, where n i defined a n A + 1 δ b, 29) rather than uing A and b. The variable n i a meaure of net worth over aet. Net worth excluding adjutment cot i AK + 1 δ)k B. Dividing by K lead to 29). 19 On each row of Figure 1 we plot, repectively, the value function per unit of 19 An alternative i to evaluate intalled capital at it hadow value, thu getting net worth equal to AK G 2 K, K)K B. The figure are imilar. 24

26 Figure 1: Characterization of equilibrium 3.5 Low x 3.5 Average x 3.5 High x v K'/K λ wedge n n n Note: The three column correpond to the 20th, 50th, and 80th percentile of the peritent component of productivity x. The range for the net worth variable n i between the 10th and 90th percentile of the ditribution of n conditional on x. capital) v, the optimal invetment ratio K /K, the Lagrange multiplier λ on the entrepreneur budget contraint, and the wedge between average q and marginal q. Each column correpond to different value of peritent component of productivity x. In particular, we report three value correponding to the the 20th, 50th and 80th percentile of the unconditional ditribution of x. On the horizontal axi we have n, but the domain differ between column a we plot value between the 10th to 90th percentile of the conditional ditribution of n, conditional on the reported value of x The joint ditribution of n, x) i computed numerically a the invariant joint ditribution gen- 25

27 A higher level of n lead to a higher value v and a higher level of invetment K /K. Moreover, the value function i concave in n. The Lagrange multiplier λ i equal to the derivative of the value function and therefore i decreaing in n. The fact that λ i decreaing in n reflect the fact that a higher ratio of net worth to capital allow firm to invet more, leading to a higher hadow cot of capital G 1 and thu to a lower expected return on invetment. Eventually, for very high value of n we reach λ = 1. However, a the figure how thi doe not happen for the range of n value more frequently viited in equilibrium. The bottom row document how the wedge varie with the level of net worth n and with the peritent component of productivity x. Let u firt look at the effect of n. Even though λ i decreaing in n, the wedge, q a q m, doe not vary much with n for a given value of x. Our analytical derivation in Section 2 help explain thi outcome. Recall from equation 11) that the wedge i equal to λ 1 λ βe [ v ]. When we reach the uncontrained olution and λ = 1 the wedge diappear. However, for lower level of n, for which the contraint i binding, the relation i in general non-monotone. An increae in n reduce the marginal gain from an extra unit of net worth. However, at the ame time it increae the future growth rate of firm capital tock and o it increae the bae to which thi marginal quai-rent i applied. Thi econd effect i captured by the expreion E[v ], becaue the value per unit of capital v embed the future growth of the firm and i increaing in n. The plot in the bottom row of Figure 1 how that in the relevant range of n thee two effect roughly cancel. On the other hand, comparing the value of the wedge acro column, how that peritent component of productivity x ha large effect on the wedge and that the wedge i increaing in x. The reaon i that higher value of x lead both to higher value of λ, a the marginal benefit of extra internal fund increae with productivity, and to higher value of K /K and v, becaue higher productiverated by the optimal policie. 26

28 ity allow the firm to raie more external fund and grow fater. Therefore both element of the wedge increae with higher value of x Impule repone function We now preent impule repone function that illutrate the model dynamic following the two hock. To contruct thee impule repone function, we take a firm tarting at the median value of the tate variable n and x. We then ubject the firm to a hock at time t, imulate 10 6 path following the hock, and report the difference between the average imulated path, with and without the initial hock. Given the non-linearity of the model, the initial condition for n and x in general affect the repone. However, in our imulation thee non-linear effect are relatively mall, o the plot below are repreentative. In the top panel of Figure 2 we plot the repone of marginal and average q, and cah flow per unit of capital to a 1-tandard-deviation peritent hock ε. 21 Following a peritent hock all variable increae and return gradually to trend. The repone of average q i larger than that of marginal q, thu producing an increae in the wedge. In the bottom panel of Figure 2 we plot the repone of the ame variable to a 1-tandard-deviation temporary hock η. Alo in thi cae all three variable repond poitively, but the repone i more hort-lived. Moreover, now the repone of average q i lightly maller than the repone of marginal q, o the wedge how a mall decreae after the hock. Notice that average q i a forward-looking variable that incorporate the quairent that the entrepreneur i expected to receive in the future. It i not urpriing that thee quai-rent are only marginally affected by a temporary hock. In the model with no adjutment cot, the effect i zero, a hown in Section 3 above. Here, becaue of adjutment cot, there i a light poitive effect, due to the fact that the invetment repone diplay a mall but poitive degree of peritence 21 The repone of invetment K /K i alway proportional to the repone of marginal q and i thu omitted. 27

29 Figure 2: Impule repone function Repone to peritent hock q a q m cah flow Time Repone to temporary hock q a q m cah flow Time Note: Average path following a hock at time 1, in level) deviation from average path following no hock. Cah flow i cah flow per unit of capital. and high invetment in the future increae the future value of intalled capital. But thi effect i mall. In the cae of a peritent hock, intead, future quai-rent are directly affected by higher future productivity, which i going to lead to fater growth a hown in Figure 1), thu explaining the large increae in q a in the top panel of Figure 2. The dicuion following Figure 1, help to explain the repone of the wedge q a q m. A temporary hock, by increaing A temporarily, lead to a pure increae in net worth per unit of capital, a n = A b. A we argued when preenting Figure 1, the effect of uch an increae on the wedge i in general ambiguou and, with our parameter choice, cloe to zero. In the cae of a peritent hock, intead, the effect i unambiguouly to increae the wedge, a the increae in x lead to a 28

30 higher λ and to a higher E[v ]. The relative repone of cah flow and marginal q are alo different acro the two hock. In particular, we have a larger repone of marginal q relative to the cah flow repone in the cae of a peritent hock. The reaon i that in the cae of a peritent hock the collateral value of capital increae, thu amplifying the effect on invetment. 4.3 Invetment regreion We now turn to invetment regreion, and ak whether the model can replicate the coefficient on q and cah flow oberved in the data. In particular, we ak to what extent doe the preence of a financial friction help in obtaining a maller coefficient on q and a poitive and large coefficient on cah flow. To anwer thi quetion, we generate imulated data from our model and run invetment regreion on it. In line with the empirical literature, we generate a balanced panel of 500 firm for 20 period, and run the following invetment regreion: 22 IK it = a i0 + a 1 q a it + a 2CFK it + e it, 30) where we allow for firm-level fixed effect. All reported reult are the mean value for 50 imulated panel. The regreion coefficient for the baeline model are preented in the firt row of Table 4. A reference point, in the econd row we report the coefficient that arie in the model without financial friction and in the lat row the empirical etimate in Gilchrit and Himmelberg 1995), which are repreentative of the order of magnitude obtained in empirical tudie. 23 We alo report coefficient of univariate regreion of invetment on average q and cah-flow eparately. The reult for the frictionle benchmark are reported in the econd line of Ta- 22 The model feature random exit, o to generate a balanced panel we only keep firm for which exit doe not occur for 20 period. 23 We do not report tandard error, but they are mall le than 0.04) for both coefficient in our imulated data. They are alo mall in the empirical etimate of Gilchrit and Himmelberg 1995). 29