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Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References

We introduce a post-entry liquidity constraint to the standard model of a firm with serially correlated profitability and an irreversible exit decision. We assume that firms with no cash holdings and negative cash flow must either exit or raise new cash at a transaction cost. This creates a precautionary motive for holding cash, which must be traded off against the liquidity cost of holding cash. We characterize the optimal exit and payout policy. The direct effect of financial frictions is to impose inefficient exit, but there is also an indirect effect through higher equilibrium price that leads to inefficient survival.

1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References

We analyze how financial frictions affect the optimal policy and survival prospects of a firm that operates under persistent cash flow uncertainty. The persistence in profitability implies that a firm should exit if the current cash flow falls sufficiently low. Financial frictions imply that a firm may also exit due to insufficient liquidity even when continuation would be economically efficient. Our model captures the interaction of these two dimensions—profitability and liquidity—underlying firm exit and shows how the firm should optimally manage its cash reserves to cope with the liquidity constraint. The solution is a policy for exit and payouts that depends on the current levels of both profitability and cash holdings. We also analyze the associated steady-state distribution of firms in a competitive industry and show how it can involve either too much or too little exit, the latter case being a type of “survival of the fattest.”

Our starting point is a standard real option model of a firm with serially correlated profitability and an irreversible exit decision.1 In this setup the potential for future profits and the irreversibility of exit make it optimal for a firm to continue even when facing expected losses. Cash holdings are irrelevant in the absence of financial constraints, and the optimal policy is simply a negative threshold level of profitability below which the firm exits. The optimal exit policy thus requires the ability to sustain negative cash flows indefinitely. It seems realistic in many contexts that a firm with a long history of losses would find it difficult to keep raising more funds. But as soon as there is a limit to a firm's ability to sustain losses the firm's problem changes in a fundamental way.

In our basic case we model the liquidity constraint as the complete inability to raise new funds. The firm has an initial stock of cash that can only be augmented with retained earnings. A firm without cash and with a negative cash flow is forced to exit immediately regardless of its future prospects, so firms have an incentive to hoard cash to avoid inefficient exit in the future. This precautionary saving is costly due to the liquidity premium: Cash holdings earn interest at a rate below the discount rate. Therefore, if the firm is sufficiently safe from forced exit—with a sufficiently benign combination of cash flow and cash holdings—it is strictly optimal to pay out some of the cash to the owners. Thus, besides affecting the optimal exit policy, the model also generates the optimal dividend policy. At the same time, if the firm is currently unprofitable and the remaining cash holdings are relatively small, it can be optimal to pay out the remaining cash and close down operations instead of running the risk of forced exit later on. We call this feature of the optimal policy “precautionary exit.” We characterize the optimal policy and analyze its dependence on the properties of the cash flow process. Our model leads to a free boundary partial differential equation problem that does not have an analytical solution. Instead of attempting to solve the firm's problem directly we formulate it as a recursive dynamic programming problem and show how it can be easily solved by value function iteration. The solution has an intuitive interpretation, and we illustrate its comparative statics properties graphically. Our numerical results show that even a small liquidity premium has a large impact on optimal firm behavior.

We do not explicitly model the causes behind the liquidity constraint. One natural cause is asymmetric information: It can be difficult for a firm or a manager to credibly convey to investors the potential for profits.2 Aside from the liquidity constraint, our model has no other imperfections such as agency problems.

The literal interpretation of the decision-maker in our basic model is a risk neutral owner–entrepreneur who can increase cash holdings only through retained earnings. Nevertheless, we believe our findings have relevance in the wider context. In an extension, we show that our results are robust to allowing the owners to raise new funds at a transaction cost; in effect the basic model assumes that this cost is prohibitive.

We also analyze the impact of the liquidity constraint at the level of an industry. Our concept of competitive industry equilibrium with entry and exit of firms is essentially that of Hopenhayn (1992), and we assume that the uncertainty faced by individual firms is due to idiosyncratic productivity shocks. In this setup the liquidity constraint causes an obvious overselectivity effect in terms of productivity: Some marginally productive firms that should survive a temporary loss exit due to insufficient funds (or, more accurately, to preempt forced exit). This effect tends to make the remaining industry on average more productive by weeding out marginally productive firms that would need financing to survive. However, the liquidity constraint also induces some formerly productive firms with sufficient cash holdings to stay on even when their productivity falls below the socially efficient exit threshold. This is a type of “survival of the fattest” as coined by Zingales (1998). As a result of these counteracting effects, the liquidity constraint may either reduce or increase the average productivity of firms. In a calibrated example, we find that the former effect dominates as the liquidity constraint increases average productivity by 3.6%. The welfare loss due to the liquidity constraint shows up as a consumer price that is higher by 7.3%.

1.1. Related Literature

Our model builds on elements from the literature on the optimal exercise of options, where the seminal papers are by McDonald and Siegel (1986), who model the optimal timing of investment under uncertain cash flow, and by Dixit (1989), who analyzes the firm's optimal entry and exit decisions in the same framework. A large number of extensions to various directions is summarized by Dixit and Pindyck (1994). Our article extends this line of research to another direction by adding a liquidity constraint that may prevent the firm from covering operating losses.

One paper that addresses the effects of liquidity constraints on the optimal exercise of real options is by Boyle and Guthrie (2003), who analyze the optimal timing of investment when uncertain wealth prior to the investment affects the firm's ability to finance the investment. Our article, by contrast, focuses on post-investment uncertainty and its effects on optimal payouts and exit.

A special case of our model, where we assume away the liquidity premium, bears close resemblance to the problem of a financially constrained firm in Mello and Parsons (2000), who analyze the optimal hedging policy for a firm that faces persistent cash flow risk and cannot raise new funds. Gryglewicz (2011) presents a model of a financially constrained start-up firm, where the mean level of a stochastic cash flow is learned over time. Eventually, as firms mature, they either go bankrupt or their confidence of being high type converges to certainty, in which case they face only i.i.d. risk and their cash holdings increase without limit. In these models the firm has to choose the optimal exit policy, but it has no reason to ever pay out dividends.

It is important to make a clear distinction between our model and an ostensibly similar stream of literature that considers the problem of a liquidity constrained firm under nonpersistent cash flow risk. This other literature models cumulative earnings as a Markovian stochastic process, which leads to independently distributed earnings across periods, whereas we model the level of earnings as the state variable, which results in serially correlated earnings. Milne and Robertson (1996) is a representative model of a firm facing a memoryless profit stream under a financial constraint, where the firm faces exogenous liquidation if cash balance falls below a given threshold. The optimal policy is to accumulate a buffer stock of savings up to a point and pay out as dividends all income above that level. A number of other papers analyze various additional features in a similar framework: Radner and Shepp (1996) and Dutta and Radner (1999) add an operation policy that controls risk-return properties of the earnings process, Décamps and Villeneuve (2007) analyze the optimal exercise of a growth option, Peura and Keppo (2006) introduce a delay time to recapitalization, and Rochet and Villeneuve (2005) allow flexible allocation of reserves in risky and safe alternatives. Décamps et al. (2011) assume costly recapitalization and analyze the implications of such financing frictions on the firm's cash management and stock price dynamics.

The attraction of modeling the level of profits as a memoryless process is that it results in one-dimensional state space, which yields analytical solutions. The drawback is that the liquidity constraint is then the only reason why the firm would ever exit, because the future always looks equally profitable. This is reasonable for a firm that consists of financial assets whose prices react to news in an efficient market but is less suited as a model of a firm facing uncertainty over real (nonfinancial) operations. In our setup, the firm's profitability (the level of expected profit flow) fluctuates, making entry and exit natural features of the economy irrespective of whether there are liquidity constraints or not. Having a first-best benchmark that involves firm exit allows us to analyze how the liquidity constraint affects firm survival and how, at the industry level, it impacts firm selection.

There are also a few papers on the macroeconomic effects of financial frictions that are related to ours. Cooley and Quadrini (2001), Gomes (2001), and Jones (2003) use as building blocks models of firm dynamics with serially correlated productivity. In Gomes's and Jones's papers firms also face an exit decision, and in the latter paper the financial constraint may force the firm to exit in states where it would be socially efficient to continue. However, due to different focuses, none of these papers characterize the joint exit-payout policy of the firm.

Our setup is also related to the models of precautionary saving. The seminal papers on precautionary saving by Zeldes (1989) and Deaton (1991) analyze the problem of optimal lifetime consumption. Under serially correlated income shocks the state space is two-dimensional (savings and expected income) as in our model; the key difference is that consumers do not face an exit decision. For consumers, precautionary saving results from the convexity of marginal utility, whereas in our model it results from the threat of forced exit.

Next we characterize the problem of the firm. We begin with the basic model, where the firm cannot raise outside funds, and then add the possibility of raising cash at a transaction cost. In Section 'SOLUTION METHOD', we solve the firm's optimal policy. In Section 'INDUSTRY EQUILIBRIUM', we analyze the implications of the liquidity constraint for a competitive industry and present a calibrated example.

2. THE PROBLEM OF THE FIRM

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References

The firm faces a stochastic revenue flow inline image that follows geometric Brownian motion

  • display math(1)

where inline image is the increment of a standardized Wiener process (i.e., with mean zero and variance inline image). The firm earns a profit flow inline image where the fixed cost c is a positive constant. Exit is irreversible and without an additional exit cost or scrap value. (The entry decision will only show up in industry equilibrium.) The objective is to maximize the expected present value of the income to the owners, discounted at rate inline image.

There are two fundamentally different cases. An unconstrained firm can accumulate negative profits indefinitely if needed. The problem of an unconstrained firm is described by the standard real option model of optimal exit. The sole decision is to choose the exit threshold for inline image, so there is no meaningful decision for when (if at all) to retain cash or pay dividends.

A constrained firm has to worry about its ability to cover negative profits using its existing cash reserves. The optimal exit policy depends both on revenue inline image and cash holdings inline image. The firm's cash holdings are augmented by the profit flow and by the interest earned on the cash holdings at an exogenous rate inline image. The difference inline image is the liquidity premium. If inline image then the cash held inside the firm incurs a cost to the owners, so they face a meaningful decision of how to pay dividends.3 The downside of payouts is that reduced cash holdings lower the capability to cover any future losses. We start by assuming that the liquidity constraint is very stark in the sense that it is not possible to inject more cash into the firm. In this case the firm is forced to exit if it has no cash while facing a negative cash flow. We later extend the model to the case where new funds may be raised at some transaction cost; the basic version can be thought of as a special case in which such transaction costs are prohibitive.

2.1. Unconstrained Firm

The unconstrained firm will exit if the cash flow becomes too negative. The value function inline image gives the expected discounted future cash flows for a firm with current revenue level inline image, and it is defined by the familiar differential equation

  • display math(2)

(see, e.g., Dixit and Pindyck 1994, chapter 7) with the constraints that inline image be continuous (“smooth pasting”) and have a finite limit. This ODE has a well-known closed-form solution. The firm exits when inline image falls to inline image given by

  • display math(3)
  • display math(4)

The unconstrained value function is

  • display math(5)

2.2. Fully Constrained Firm

The constrained firm has an initial cash balance s0 that is exogenous to the problem. Cash earns interest at rate inline image. At any moment t , the firm can run down its cash balance by paying dividends. Paying dividends is costless and instantaneous. The objective of the firm is to maximize the expected discounted stream of dividend payments. We denote by inline image the cumulative dividend process. The restrictions that we impose on this process are the following: First, since we allow only positive dividend payments, the process must be increasing. Second, the dividend payment inline image at time t can only be conditioned on the past history of cash flows. Stated in technical terms, D must be adapted to the filtration generated by the Brownian motion inline image. Third, we assume that D is right continuous (upward jumps in D represent lumpy dividend payments). Finally, D must satisfy the liquidity constraint, which requires that inline image for all t, where the dynamics of the cash balance inline image are given by:

  • display math(6)

The firm is forced to exit if inline image and inline image, so the exit time τ is given by

  • display math(7)

The objective of the firm is to choose a dividend process to maximize

  • display math(8)

subject to (6), (7), and inline image for all inline image. Note that this formulation allows voluntary exit when inline image and inline image by paying out the remaining cash as the liquidation value: inline image.4

The firm's problem becomes much more intuitive once recast as a Markovian control problem with suitably chosen state variables. Note that the history at time t consists of past cash flows inline image, past dividends inline image, and the initial cash balance s0. Since the cash flow process is Markovian, the part of the history that defines the probability distribution for future incomes is summarized in the current cash flow level inline image. Similarly, the part of the history that defines the firm's capacity to satisfy the liquidity constraint is summarized as the current cash holdings inline image, as derived from past cash flows and dividend payments through Equation (6). Therefore, the pair inline image summarizes the history part that is payoff relevant for the future and is sufficient for deciding the optimal policy at t by the Bellman's Principle of Optimality. Consequently, we may denote by inline image the value of the firm that solves (8) starting from an arbitrary state point inline image.

In effect, the problem of the firm is to choose between three policy options at each point of the state space. First, the firm may exit, which is irreversible, and results in the exit value inline image. Second, the firm may pay a positive dividend inline image to the owners, which shifts the firm in the state space to cash balance level inline image. Third, the firm can continue without paying dividends, in which case the cash balance evolves according to

  • display math(9)

The solution to the firm's problem is a division of the inline image-space into regions in each of which one of the three policy options is optimal. The following proposition characterizes the solution in the case where inline image (the special case inline image will be discussed later). For illustration, see Figure 1.

Proposition 1. (Optimal Policy when inline image) There are constants inline image, inline image, and inline image such that the optimal policy has the following features:

  1. If inline image, it is optimal to exit immediately irrespective of inline image.
  2. If inline image, then there is a cut-off value inline image such that it is optimal to exit if and only if inline image. inline image is decreasing in inline image, and inline image.
  3. If inline image, it is optimal not to exit, irrespective of inline image.
  4. If inline image or inline image, it is strictly optimal to pay out some dividends.
image

Figure 1. OPTIMAL POLICY REGIONS OF A LIQUIDITY CONSTRAINED FIRM, WITH EXIT BOUNDARY inline image AND DIVIDEND BOUNDARY inline image IN THE STATE SPACE OF REVENUE x AND CASH HOLDINGS s

Note: The steep dashed line ending at the zero profit-zero cash point (c, 0) shows where inline image. The small gray region inside the continuation region is a transitory region.

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The proof is in Appendix A.1. Figure 1 illustrates the optimal policy. This is the key figure of our article. The life span of a firm is a stochastic path in the inline image-space. While the firm stays inside the continuation region its law of motion is given by Equations (1) and (9). The gray area inside the continuation region is a transitory region: After leaving it, a firm that follows the optimal policy cannot return there. The firm never ventures inside the dividend region, because payouts (which move the firm down along s-axis) keep it from crossing the boundary of that region. When revenue x is sufficiently high, the dividend region reaches all the way to the inline image line, where the firm operates with zero cash holdings and continually pays out all of the profit flow as dividends. The firm's life span ends when it hits the boundary of the exit region for the first time.

We will next explain the intuition for why the optimal policy takes the form depicted in Figure 1.

2.2.1. Continuation region

The point of accumulating cash is to use it as a buffer that prevents inefficient exit. To see this, consider a situation where the firm's current cash holding inline image is small but strictly positive and where the profit flow is exactly zero, that is, inline image. The firm is not currently making losses and there is a positive option value associated with future profits, so it cannot be optimal to exit. Neither can it be optimal to pay out inline image as dividends, because this would cause the firm to immediately move down to the point inline image, which means that the firm is forced to exit within the “next instant,” thus losing the option value. Therefore, there must be a nonempty continuation region, where it is optimal to retain cash inside the firm despite the difference between the discount rate and the rate of return on cash holdings.

Now let us consider the properties of the value function in the continuation region. Define the value of the constrained firm inline image as gross of the cash holdings, so the value at the time of exit is inline image. Using Ito's lemma, we can write the differential inline image as

  • display math(10)

Taking the expectation and letting inline image be small yields:

  • display math

where inline image is from (9). The Bellman equation is inline image, which can be solved for inline image, leading to the following PDE:

  • display math(11)

Note that this PDE does not contain a cash flow term. The reason is that, in the continuation region, the cash flow between the firm and its owners is zero: Positive cash flow adds to the cash balance and negative flow subtracts from it.

The PDE (11) does not have a closed-form solution. Further, it is valid only in the continuation region, the boundaries of which must be optimally chosen as part of the solution. We will next discuss the properties of these boundaries, which constitute the optimal exit and dividend policies. The numerical solution of the problem is discussed in Section 'SOLUTION METHOD'.

2.2.2. Exit policy

The liquidity constraint can only reduce the continuation value of the firm, so the constrained firm should certainly exit whenever the unconstrained would, that is, when inline image. In addition, the firm is forced to exit when it has no cash to cover the current loss, that is, when inline image. This gives a fixed boundary for the value of the firm:

  • display math(12)

The firm should clearly never exit while current profits are positive (inline image). Now consider a firm with a very small inline image and with inline image. This firm is depleting its cash but could in principle still continue. However, it is very likely to be forced to exit in the near future. For any inline image, and for sufficiently small inline image, the firm is so unlikely to bounce back to a positive cash flow before s hits zero that the owners are better off exiting immediately and just taking the remaining inline image.5 Thus, there must be a boundary between exit and continuation regions that lies strictly above inline image for inline image. We call exiting when inline image and inline image precautionary exit.

We denote the exit threshold by inline image, defined in inline image where inline image is, in practical terms, the lowest revenue at which the firm ever operates. The lower is inline image, the less valuable the continuation value of the firm, and thus the higher the s required for continuation to be optimal, so inline image in inline image.

Inside the continuation region the value of the firm must exceed the exit value s. At the exit boundary the firm is indifferent between taking the exit value and the continuation value, so

  • display math(13)

Inside the continuation region the marginal value of cash must be at least unity, else the owners would be better off by paying out cash. Smooth pasting at the exit boundary requires

  • display math(14)
  • display math(15)

It may seem unintuitive that the exit boundary is strictly above zero for all inline image. To see this point more formally, suppose, by contrast, that the continuation region in fact reached all the way down to inline image for some interval inline image, where inline image. Since a cashless firm is forced to exit at inline image, the boundary value inline image is fixed for all inline image. This implies that the first and second derivatives with respect to x must also be zero within this interval: inline image for all inline image. Substituting these into the PDE (11) that holds in the continuation region yields inline image within this interval. But this leads to a contradiction, because inline image must hold in the continuation region or else cash would be more valuable outside than inside the firm. It follows that the continuation region cannot reach down to inline image for inline image.

The only way in which a firm following the optimal policy can extinguish all funds is to hit exactly the zero-flow-zero-stock point for cash, inline image. Thus the constrained firm will experience a forced exit with probability 0.6 Practically, all exit by liquidity constrained firms is precautionary.

We assume that the scrap value of the firm is zero, so the exit value of the firm is simply equal to its cash holdings. In the unconstrained case, adding a positive scrap value would be equivalent to adding the rental opportunity cost of the scrap value to the flow cost. However, in the presence of a liquidity constraint an opportunity cost is not equivalent to an operating cost as only the latter requires liquidity. In the extreme, having a scrap value so high that the optimal exit threshold of an unconstrained firm is positive, the firm cannot face negative cash flows during its lifetime so the liquidity constraint is redundant. Apart from this extreme case, the problem would not be qualitatively changed by a positive scrap value.

2.2.3. Dividend policy

When inline image, holding cash is costly. The benefit of holding cash is that it may allow the firm to avoid a forced exit in the future when the option value of continuation would still be positive. This benefit is bounded above by inline image, the unconstrained continuation value at the zero profit flow. Since the cost of holding cash increases without bound in s, there exists, for any x, some s high enough such that it is better to stop accumulating cash. This threshold value, denoted inline image, defines the boundary between the continuation region and the dividend region. It can be interpreted as a target level of cash holdings that depends on current profitability. The firm makes payouts to owners to make sure it does not hold more than the target level of cash; hence we call inline image the dividend threshold. The value of the firm above the dividend threshold must be

  • display math

For sufficiently high x the possibility of forced exit is so remote that it is not worth holding on to any cash. We denote the threshold above which it is optimal to not hold any cash by inline image. In the limit inline image, the prospect of forced exit becomes irrelevant, and thus the value of the firm must converge to the value of the unconstrained:

  • display math(16)

At the dividend threshold, cash is equally valuable inside as it is outside the firm, where one dollar is of course worth one dollar. Thus, the value matching condition

  • display math(17)

must hold at the dividend threshold. The associated smooth-pasting condition requires7

  • display math(18)
  • display math(19)

The firm is constrained at the margin only in the continuation region; there having a dollar more would increase the value of the firm by more than a dollar: inline image.

When the firm hits the dividend threshold from inside it pays out just enough cash to not cross the boundary. However, if the firm were to start at inline image inline image, then it would immediately pay out the excess inline imageinline image as a lump sum dividend. (For a new firm this means that the owners have more than enough funds to endow the firm with the optimal level of precautionary cash holdings; the “lump dividend” at the start is then the cash that owners retain for themselves.) A lump sum dividend is also paid out as the liquidation value upon precautionary exit. Note that if a firm that enters the industry at revenue level x0 can choose its initial cash holdings, then inline image inline image is the optimal choice.

2.2.4. Comparative statics

We next illustrate how the firm's optimal policy depends on the parameters of the stochastic process. To do this, we solve the optimal policy numerically in a manner to be explained in Section 'SOLUTION METHOD'. We vary one parameter at a time from a set of baseline parameters. The results are depicted in Figure 2. The solid lines mark the borders of the continuation region in the liquidity constrained case, and the vertical dashed lines mark the optimal exit threshold in the unconstrained case. The left panel shows the relation of the optimal policy and σ, the volatility of the cash flow process. As is well known, the unconstrained exit threshold inline image is decreasing in σ. Here the increased option value shows up as an enlarged continuation region. As in the unconstrained case, higher volatility makes it optimal to accept bigger losses because it increases the upside potential while the downside is still protected by the exit option. Furthermore, the dividend boundary shifts out to the right also because, at any given x, higher volatility increases the risk of facing forced exit within any given period of time.

image

Figure 2. COMPARATIVE STATICS OF THE OPTIMAL POLICY AROUND BASELINE PARAMETER VALUES inline image, inline image, inline image, inline image, inline image

Note: Left panel: volatility: inline image, from smallest to largest region. Right panel: drift: inline image, from smallest largest region.

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The right panel of Figure 2 shows the effect of varying μ, the percentage drift of the cash flow process. Higher μ increases the option value at any given level of losses, as the firm is more likely to bounce back to positive profits within any given period of time. However, higher μ also makes the firm safer at any given point—by making it less likely that forced exit would threaten it within any given time—so it is not obvious that a higher μ should also shift out the dividend boundary. However, we have found no examples of the opposite.

2.3. Special Case: No Liquidity Premium (inline image)

Consider now the special case in which there is no liquidity premium: inline image. Hoarding cash is now costless, so it can never be strictly optimal to pay dividends. The optimal policy is thus defined by dividing the inline image-space between the exit region and the continuation region. The qualitative properties of the exit region and the exit threshold inline image are the same as with inline image.

Holding cash inside the firm can be strictly optimal only when there is a positive probability of being forced to exit in the future. Of course, no matter how high inline image, falling below inline image always remains possible. However, the worst-case cash flow under which the owners would ever want to continue is the cash flow at the unconstrained exit policy, inline image. The firm becomes irreversibly unconstrained if it accumulates so much cash that the interest income from its cash holdings could be used to cover the worst-case losses forever. This defines the escape level of cash as

  • display math(20)

This means that a fixed boundary condition

  • display math(21)

now replaces the free boundary inline image seen in the inline image case. For inline image, the firm is indifferent between paying dividends or not and inline image. Above the escape level of cash, the firm can no longer run out of cash before first becoming so unprofitable that it would want to exit even in the absence of a liquidity constraint. As the firm is then in effect unconstrained, its exit policy is the same as that of an unconstrained firm: exit if and only if inline image. We summarize these results in the following proposition:

Proposition 2. (Optimal Policy when inline image) If inline image and inline image, it is strictly optimal to refrain from paying dividends. If inline image and inline image, the owners are indifferent between paying dividends and continuing without paying dividends. The optimal exit policy is qualitatively the same as when inline image (see Proposition 1).

The special case without a liquidity premium is quite similar to the setup of a financially constrained firm in Mello and Parsons (2000). They study optimal hedging, namely, how firms should use futures contracts on an asset that is correlated with their profits to reduce the risk of inefficient exit. They do not take into account that the firm becomes permanently safe from inefficient exit at a finite level of cash holdings, but instead assume that the constrained firm's value reaches that of the unconstrained case only in the limit of infinite cash holdings. The environment faced by the agent in DeMarzo and Sannikov (2008) also features serially correlated cash flow and saving is possible without liquidity costs; there precautionary exit does not arise because expected cash flow is assumed to be always positive (due to a parameter restriction that implies that the exit threshold is always positive). There is also no liquidity premium in the model of Gryglewicz (2011), so it is never strictly optimal to pay out dividends. To generate predictions about the dividend policy he assumes that the firm pays dividends at the indifference boundary, which is equivalent to our inline image (but is changing over time due to learning).

2.3.1. Small liquidity cost

It turns out that the optimal policy, in particular the dividend policy, is very sensitive to small liquidity costs. Figure 3 shows the impact of varying the return on firm's cash holdings, r. As r gets closer to ρ it becomes less costly to hold cash, so continuation is everywhere more attractive and the continuation region expands. The limiting case results in the escape level of cash inline image, from (20), that is much higher than the highest cash holdings that the firm would ever keep even at a liquidity premium of just one basis point. The limiting case is qualitatively different, because there is no trade-off between the liquidity cost of the cash holdings and the expected benefit of preventing exit.8 However, although the optimal payout policy is very sensitive to r near ρ, the value of the firm is not. When r is very close to ρ the liquidity cost is negligible, and there is a large region in state space where the firm is almost indifferent between retaining and paying out cash. (There the marginal value of cash, inline image, is only very slightly above unity). The high sensitivity of optimal policy to r near ρ means that, even if the liquidity premium were close to zero, the optimal behavior of firms would not be well approximated with a model where the liquidity cost is completely assumed away.

image

Figure 3. OPTIMAL POLICY AT VARYING LEVELS OF THE LIQUIDITY PREMIUM inline image

Note: Discount rate is held fixed at inline image; other parameters are held at inline image inline image, inline image, and inline image from smallest to largest regions. In the limiting case inline image there is no actual dividend region, but above the escape level of cash inline image the owners are indifferent between hoarding and paying out cash.

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image

Figure 4. OPTIMAL POLICY WHEN CASH CAN BE RAISED AT A TRANSACTION COST

Note: Amount inline image of new cash is raised when inline image and inline image.

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2.4. Generalization: New Cash Injections

The assumption that the firm cannot raise new cash is quite stark and is made to identify the effects of the liquidity constraint in its most transparent form. Here we generalize the model to the case where owners can increase the firm's cash holdings at some transaction cost. Specifically, suppose that they can, at any point in time, inject any amount s of cash at cost inline image, where ξ is the fixed and γ the marginal transaction cost. The injection of cash causes the firm to jump directly upwards in the state space (inline image).

Paying the transaction cost can only be optimal when the firm would otherwise face immediate forced exit (inline image and inline image) because otherwise the cost could still be postponed and, with luck, even avoided. If the firm decides to incur the transaction cost, then its target level of cash is

  • display math(22)

The target level inline image equalizes the marginal cost of new cash and its marginal value at the firm, inline image. Transaction costs are independent of current revenue x, so raising cash must be more desirable the higher the current x (as long as inline image). Therefore, if raising cash is ever optimal, then there must be some inline image such that cash is raised on an interval inline image and nowhere else. The lowest x where the firm replenishes its cash holdings, inline image, is the point where the value of exit (which is zero on the inline image line) is just equal to the value of continuing from inline image net of the transaction cost of moving there:

  • display math(23)

If transaction costs are sufficiently high then it is never optimal for the firm to raise new cash. This is the case when inline image, that is, the firm's value net the transaction cost would be negative. Notice that “prohibitive” transaction costs are finite, because the benefit (from some probability of inefficient exit prevented) is necessarily finite.

We describe in Section 'SOLUTION METHOD' how the optimal policy with new cash injections can be solved numerically. Figure 4 depicts the optimal policy for a firm that faces positive but not prohibitive transaction costs. The qualitative difference to the optimal policy in the basic model (recall Figure 1) is the segment of horizontal axis where cash is raised and the associated target curve inline image directly above. For sufficiently low cash flow the firm still finds it optimal to exit with positive cash holdings instead of incurring the transaction cost.

Holding cash is costly when inline image, so without a fixed transaction cost firms would raise cash continuously only to offset a contemporaneous negative cash flow. The fixed cost makes it optimal to raise new cash in lumps to postpone the prospects of having to incur it again. In the absence of a marginal transaction cost it would be optimal to “jump” all the way to the dividend boundary. Any transaction costs reduce the value of continuation and shift the exit boundary to the right.

The unconstrained case, with the simple exit threshold inline image in (3), is the limiting case where both the fixed and the marginal transaction costs are zero. The constrained case, where the firm never raises new cash, is equivalent to assuming that the cost parameters are prohibitively high. Hence this setup encompasses both the constrained and unconstrained cases of the basic model.

The setup with cash injections allows various interpretations. One literal interpretation is that of a risk-neutral owner–entrepreneur who allocates her wealth between two assets, one liquid asset that can be used to pay off possible losses, and another illiquid asset that yields a higher rate of return but can only be turned into liquid form at a transaction cost. The entrepreneur has deep pockets in terms of the illiquid asset, but the transaction cost makes it desirable to hold some liquid assets as well and, in some circumstances, let the firm fold rather than incur another transaction cost.

A broad interpretation of the extended model analyzed in this section is a firm that can raise new equity at a transaction cost. This interpretation is similar to Décamps et al. (2011), who analyze the case of nonpersistent cash flow risk. Assuming that there is a fixed cost associated with raising equity, the firm delays the recapitalization until it has used up its liquid assets. Then, upon hitting inline image, it will raise new equity to increase its cash balance to level inline image that equalizes the marginal value of internal cash with the marginal cost of raising equity. The new owners supply the firm with cash and are compensated with an equally valuable stake in the firm. Our calibrated example of industry equilibrium in Subsection 'A Calibrated Example' will adhere to this interpretation of the firm's problem.

2.4.1. Comparative statics

In Figure 5, we illustrate how the level of transaction costs affects the optimal policy. In addition to the case with prohibitive transaction parameters (seen already in Figure 1), we solve the policy for a case of low and high transaction costs. In each case the exit boundary is further left than under prohibitive costs, as the threat of forced exit is not as grave with the possibility to raise new capital. The lower the transaction costs, the further the exit boundary shifts toward the unconstrained exit threshold. With low transaction costs it is cheap to add cash whenever necessary, so it is possible to reduce the liquidity cost and never hold very much cash, so the continuation region becomes smaller. In the limiting case the firm holds no cash; it pays out profits as they come in and raises cash as it makes losses.

image

Figure 5. OPTIMAL POLICY IS UNDER HIGH (LEFT PANEL) AND LOW (RIGHT PANEL) TRANSACTION COST PARAMETERS: inline image IN THE CASE OF “LOW” AND inline image IN THE CASE OF “HIGH” COSTS

Note: The case with prohibitively high transaction costs (i.e., the fully constrained firm) is depicted in dashed curves for comparison. Other parameters are as in Figure 2.

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3. SOLUTION METHOD

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References

The PDE defined by (11) and the various free boundary conditions cannot be solved analytically. To solve the firm's problem we turn to a discrete-time approximation of the problem and solve it numerically.9 In the binomial process approximation of geometric Brownian motion the evolution of x is governed by

  • display math(24)

where Δ, the length of the time period, can be set arbitrarily small.10 The evolution of the cash balance is now

  • display math(25)

where inline image is the dividend paid at time t. The dividend cannot be so high as to make the cash holdings negative at any point in time, so the maximum feasible dividend is restricted by inline image, where inline image inline image.

The value function of the firm, stated in recursive form, is

  • display math(26)

where inline image is from (25).

The recursion in (26) satisfies Blackwell's sufficient conditions so it is a contraction mapping. Thus it can be solved by iterating backwards in time: Starting from an arbitrary inline image, the value function converges to a unique solution that approximates inline image.11

Augmenting the value function with the additional option of raising more cash is straightforward. When solving for the optimal policy, (26) is replaced with

  • display math(27)

where inline image is from (25). The numerical solution method is otherwise unchanged.

4. INDUSTRY EQUILIBRIUM

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References

We saw in Section 'THE PROBLEM OF THE FIRM' how a liquidity constraint causes firms to exit at higher levels of current revenue compared to unconstrained firms. It might therefore seem obvious that, at the level of an entire industry, the liquidity constraint would cause there to be fewer but on average more productive firms. However, as we next show, this firm-level reasoning is misleading, because it does not take into account the impact that the liquidity constraint has on output price in competitive equilibrium.

To analyze the impact of the liquidity constraint on a competitive industry, we use the definition of industry equilibrium similar to Hopenhayn (1992) and Dixit and Pindyck (1994, chapter 8.4).12 There is a continuum of firms. We assume that for each firm the revenue x depends on firm-specific output or “productivity” z and an endogenous industry-specific output price p, so that

  • display math(28)

We assume that productivity z follows geometric Brownian motion

  • display math(29)

with the shocks inline image independent across firms. New firms of known productivity z0 can be established by paying an entry cost ϕ. Entering firms choose the initial level of cash holdings to maximize value, so that inline image, where inline image is from (22). (In the fully constrained case s0 would have to be an exogenous parameter.) To guarantee the existence of steady state, we assume an exogenous “death rate” inline image at which firms are forced to exit with their cash holdings as the exit value (see Appendix A.2 for details).13 In steady state, both the dying and the endogenously exiting firms must be balanced by an equal inflow of new firms of type inline image.

The industry faces a demand curve inline image for its output. We assume that the demand curve is everywhere strictly downward sloping. The equilibrating variables are price of output p and mass of firms m. Firms are atomistic, so there is no aggregate uncertainty in steady state. As p is constant, the revenue of individual firms (28) follows the same process (1) that we assumed earlier in Section 'THE PROBLEM OF THE FIRM'. All firms follow the same optimal policy, which in turn results in a stationary distribution of z. In steady state, m and p must satisfy market clearing

  • display math(30)

where inline image denotes the cross-sectional average output of firms in steady state (inline image depends on p because the exit policy in terms of z depends on p). Entry is endogenous, so equilibrium must also satisfy the zero-profit condition for entering firms

  • display math(31)

where the total entry cost includes financing costs for both the “physical” entry cost ϕ and the initial level of cash holdings inline image.14

Equilibrium price is fully determined by the entry condition (31): p must adjust to eliminate expected rents to entrants. (If entry were profitable, then more firms would enter and m would increase, and if entry resulted in expected loss, then no one would enter and m would decrease.) Since the value function V is increasing in revenue, p is uniquely determined by (31) and V is obtained numerically as described in the previous section. In the unconstrained case, the entry condition (31) is replaced by inline image, where inline image has the closed form seen in (5).

For any p, the mass of firms is determined from (30) as inline image. The role of m is merely to close the model. We are not interested in the number of firms but rather on the cross-sectional distribution of productivity, which is independent of m and of the shape of the demand curve because the model has, at industry-level, constant returns to scale.15 Thus m and D will not feature in our analysis.16

4.1. Survival of the Fattest

The model assumes perfect competition, so the only component of welfare that can be affected by the liquidity constraint is consumer surplus, which varies in the opposite direction as p. Maximum welfare is, of course, attained in the unconstrained case, so the liquidity constraint can only increase p. In real terms, there are potentially three different components to the price distortion: higher aggregate entry cost (due to higher turnover), lower average productivity, and higher liquidity costs. As it turns out, turnover and productivity can move to either direction.

To understand why the impact of the liquidity constraint on mean productivity is ambiguous, consider, for simplicity, the fully constrained case where there are no cash injections. Suppose that the entering firms have no cash holdings (inline image). The position of firms in (inline image)-space is illustrated in Figure 6. Entry level z0 is at the point to the right of the zero-profit level (inline image) where the continuation value matches the entry cost. As price is distorted upwards, the lowest type to ever continue (inline image) is below the unconstrained exit threshold (inline image), even though the associated revenue level is higher (recall inline image in Figure 1). The higher price makes it optimal for firms with sufficient cash reserves to continue at productivity levels that would trigger exit in the unconstrained world. The light shaded region (inefficient survival) covers firms that would exit in the unconstrained solution but stay in under the liquidity constraint. The dark region (inefficient exit) covers firms that are more productive than the unconstrained exit threshold inline image but exit due to the liquidity constraint. Whether mean productivity is increased or decreased by a liquidity constraint depends on which of these two effects dominates.17 In the working paper version of this article we analyze numerically how the steady-state outcomes vary in z0 and s0 and show that mean productivity is decreased when the entry cost is sufficiently low (see Murto and Terviö, 2010). Here we instead move to a calibrated example with cash injections.

image

Figure 6. LIQUIDITY CONSTRAINT AND AVERAGE PRODUCTIVITY IN INDUSTRY EQUILIBRIUM

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4.2. A Calibrated Example

We now present a quantitative example based on the general model with cash injections, using the transaction cost parameters estimated by Hennessy and Whited (2007) with U.S. data. They estimated that (financial companies excluded) the marginal cost of raising new equity is inline image and the fixed cost of recapitalization is inline image.18 We take most parameters from Miao (2010), who also models the firm revenue process as a geometric Brownian motion. Specifically, we set the drift inline image volatility inline image the exogenous firm death rate inline image, and the discount rate inline image. We set the liquidity premium at inline image, the same as used by Décamps et al. (2011), which results in inline image

We are left with the fixed operating cost c and entry cost ϕ. We aim to choose these parameters so as to match a firm turnover rate of 0.07 and the recapitalization rate of 0.175. We take the turnover rate from Miao (2010), and the recapitalization rate is estimated by Hennessy and Whited (2007). We aim to choose the fixed operating cost c and entry cost ϕ to match the firm turnover and recapitalization rates with the targets. To do this, we calculate the steady-state firm distributions for a wide range of combinations c and ϕ. This is done by first solving numerically the optimal firm policy (as explained in Section 'SOLUTION METHOD'). The steady-state distribution is then obtained by iterating the firm distribution according to this policy until the distribution converges (see Appendix A.2.1 for more details). Various statistics including turnover rate and recapitalization rate are then readily computed from the steady-state distribution.

Figure 7 depicts the contours of our calibration targets in the space of endogenously determined parameters (entry cost ϕ, operation cost c). Holding fixed c, equilibrium turnover (lighter contours) is decreasing in ϕ; this is natural as ϕ is from the economy's point of view a cost of replacing underperforming firms with new ones. Less obviously, the recapitalization rate (black contours) is also decreasing in ϕ, even though a higher entry cost makes recapitalization more attractive relative to “replacing” the entire firm. However, higher entry cost also implies a higher equilibrium price of output, so entering firms (whose productivity is exogenous) are more profitable and thus less likely to need a recapitalization.

image

Figure 7. LEVEL CURVES OF CALIBRATION TARGETS IN THE SPACE OF ENDOGENOUSLY DETERMINED PARAMETERS

Note: Dashed black curves depict the recapitalization rate, and the lighter dashed curves depict the turnover rate. Target levels are depicted as solid curves.

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The shapes of the target curves make it clear that there is a unique match inline image. To understand why this is so, consider for a moment scaling up all cost parameters (ϕ and c, as well as transaction costs γ and ξ). Such a change would amount to scaling of monetary units, so equilibrium price would also be scaled by the same factor. In (inline image)-space space such scaling amounts to moving along a ray from the origin, so turnover rate would be constant along any ray. However, in the calibration, transaction cost parameters are fixed, so while moving out on a ray from the origin, recapitalization becomes relatively cheaper. This explains why contours of recapitalization rate bend downwards so that the recapitalization rate is higher further out from the origin on any ray. Recapitalizing an existing firm and replacing it with a new firm (turnover) are substitutes from a welfare point of view. Turnover contours bend in upwards, so the turnover rate is decreasing along a fixed ray from the origin. As seen in the figure, this implies exactly one point where both targets are matched at the same time.

The unique match of the calibration targets is at inline image, inline image. This implies a fixed one-time entry cost of $480, 000, and a yearly fixed cost of $120, 000. In interpreting these figures, note that we do not explicitly model inputs such as labor and physical capital. The revenue is best interpreted as describing revenue net of adjustable inputs under a constant returns to scale production technology.

Having calibrated the model, we calculate other steady-state outcomes at the calibrated parameter values. To quantify the impact of financial frictions, we also calculate the corresponding distributions in the absence of transactions costs (see Appendix A.2. for details). The top row of Table 1 reports the key figures. Equilibrium output price is 7.2% higher than in the absence of transaction costs. Resulting average productivity is 3.6% higher than in the absence of transaction costs. Thus, at the calibrated parameters, the productivity-enhancing effect of excessive “weeding out” is stronger than the “survival of the fattest.”19 As the model has perfect competition, the increase in output price paid by the consumers represents the welfare loss from financial frictions.

Table 1. CALIBRATION RESULTS
Caseϕcinline image (%)inline image (%)
NOTE
  1. Matched entry cost ϕ, fixed operating cost c, and the implied percentage impact on steady-state price p and average productivity inline image

Baseline0.490.127.23.6
Gomes (2001) parameters0.740.184.83.8
Low liquidity premium0.550.146.63.4
High liquidity premium0.400.098.94.7

Finally, we look at the robustness of the industry outcomes with respect to our key parameters. First, we replace the transaction cost parameters taken from Hennessy and Whited (2007) with the corresponding parameter values taken from Gomes (2001), namely inline image and inline image. Note that Gomes estimates the fixed transaction cost to be considerably higher and the marginal cost lower than Hennessy and Whited (2007). The results are reported in the second row of Table 1. We see that these parameters imply somewhat higher entry and operating costs, whereas the price impact is lower, 4.8%. Second, we investigate how changing the liquidity premium inline image affects the results by repeating the calibration exercise with inline image (low liquidity premium) and inline image (high liquidity premium). The results are reported in last two rows of Table 1. Both the entry and operating costs are increased (decreased) from baseline case as the liquidity premium is decreased (increased). Making financial frictions more severe by increasing the liquidity premium increases the market price as expected.

5. CONCLUSION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References

We have analyzed the problem of a liquidity constrained firm that faces persistent cash flow uncertainty. We view our model as a natural framework for analyzing how profitability and liquidity jointly affect the firm's exit and payout policies and how liquidity constraint distorts production at the industry level. We abstract away from some important aspects of firm dynamics such as growth options, agency issues, and choice of financing structure. We are hopeful that our model framework proves useful for future work along these lines.

APPENDIX

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References

A.1. Proof of Proposition 1

A.1.1. Preliminaries

We begin by three lemmas that collect together the key properties of inline image utilized in the proof. The first one merely records properties of inline image that are discussed in more detail in Subsection 'Fully Constrained Firm' of the main text.

Lemma A.1. inline image is continuous and increasing in both arguments, and inline image for all inline image. Depending on the optimal policy at inline image,

  • If it is optimal to exit, then inline image.
  • If it is optimal to continue without paying dividends, then inline image, inline image, and the following partial differential equation holds locally at inline image:
    • display math(A.1)
  • If it is optimal to pay dividends and continue thereafter, then inline image and inline image.

Proof. Choosing dividend inline image and exiting immediately thereafter is a feasible policy at every point in state space and gives value inline image. It follows immediately that inline image for all inline image. In particular inline image whenever it is optimal to exit and inline image whenever it is strictly optimal to continue. The application of Bellman's principle and Ito's lemma imply that if it is optimal to continue without paying dividends, then the value function must satisfy the Hamilton–Jacobi–Bellman Equation (A.1) locally at inline image, and inline image (see Sub-section 'Fully Constrained Firm' in the main text). Finally, if it is optimal to pay a positive dividend inline image and continue thereafter, the principle of dynamic programming gives inline image, which implies that inline image for all inline image. Continuity and monotonicity of inline image follow from the properties of state transition dynamics and monotonicity of cash flow with respect to x. ▪

Lemma A.2 establishes lower and upper bounds for inline image:

Lemma A.2. For all inline image, we have

  • display math(A.2)

where

  • display math(A.3)

and where inline image is given by (5) and β is given by (4) in the main text.

Proof. Consider the following policy: Pay out immediately any positive cash reserves and thereafter keep cash holdings at inline image by immediately paying out any incoming cash. This leads to forced exit as soon as inline image. The unique value function that satisfies the appropriate differential equation (Equation (2)) together with the boundary condition inline image is given by (A.3). Since this policy is feasible, it gives a lower bound for the value of the optimally managed firm. On the other hand, the net value of a firm that faces no liquidity constraint is inline image, and this must be an upper bound for the liquidity constrained firm. ▪

Finally, Lemma A.3 states that a firm that is at the edge of being profitable (inline image) is more valuable to its owners than its cash holdings. This lemma guarantees that positive cash holdings are optimal at least under some conditions.

Lemma A.3. inline image for all inline image.

Proof. The key to this result is the kink in the value function inline image at inline image. Take an arbitrary inline image and let inline image, inline image. Take a sequence inline image such that inline image and inline image for each n. Denote by inline image the expected payoff of a feasible (but suboptimal) policy, according to which the firm continues without paying dividends for a period of length inline image and thereafter pays out all incoming cash:20

  • display math

Since inline image is a geometric Brownian motion, we have

  • display math

Standard properties of Normal distribution imply

  • display math

Since Normal distribution is symmetric around its mean, we have

  • display math

so that

  • display math

where inline image denotes terms that go to zero at least linearly in inline image. Denoting by ζ the derivative from right of inline image at the kink:

  • display math

and noting that

  • display math

we have

  • display math

Therefore, for n large enough, inline image. But since the optimal policy is at least weakly better than this strategy, we have inline image for any n, and it follows that

  • display math

Proof of Proposition 1. Part 1: We want to show that there is some inline image such that stopping is optimal for all inline image, inline image. Suppose the contrary. Then we can find a sequence inline image with inline image for all n, inline image and inline image, such that all points inline image are within the continuation region so that (A.1) holds by Lemma A.1.21 Since inline image, it follows from Lemma A.2 that inline image for all s. Therefore

  • display math

as inline image. By the smooth-pasting condition of the unconstrained firm, we have inline image, and therefore we must have

  • display math

(Otherwise we would have either inline image or inline image for n large enough, hence violating Lemma A.2.)

Since (A.1) must hold at all points in the sequence inline image, we have

  • display math

On the other hand, from the corresponding Hamilton–Jacobi–Bellman equation of the unconstrained firm (Equation (2)) we have

  • display math

and therefore

  • display math

But since inline image and inline image, this implies that inline image for n large enough. This is a contradiction with Lemma A.2. We can conclude that inline image for all s for some inline image. We let

  • display math(A.4)

Part 2: By Lemma A.3, we have inline image for all inline image. It follows from continuity of the value function that inline image for some inline image, so that inline image defined in (A.4) satisfies inline image.

Next, we show that for all inline image, there is some inline image such that inline image for all inline image . Suppose, by contrast, that there is some inline image such that inline image for all inline image. Since inline image is increasing in x, this implies that inline image for all inline image, inline image. Therefore, there is a continuation region that reaches all the way down to inline image for the interval inline image, and by Lemma A.1, (A.1) must hold for all s sufficiently small. However, since a cashless firm is forced to exit at inline image for inline image, the boundary condition inline image must hold for the whole interval, and therefore also inline image for all inline image. Substituting these into (A.1) yields inline image for inline image. But since inline image for all inline image by Lemma A.1, this is a contradiction. It follows that inline image for all inline image and for all inline image for some inline image. Define for all inline image:

  • display math

It remains to show that inline image is decreasing in inline image and inline image. The former property follows from the monotonicity of inline image in x: Suppose on the contrary that inline image for some inline image. But then, inline image which violates the property that inline image is increasing in x. The latter property follows from the continuity of inline image: suppose that there is some inline image such that inline image for all x in some open neighborhood of c. But this means that inline image for all inline image when x is arbitrarily close to c, and this is in contradiction with continuity of inline image and our previous finding that inline image for all inline image.

Part 3: One available (nonoptimal) policy is to pay-out all incoming cash and keep cash balance at inline image. When inline image, this policy gives value inline image, so it cannot be optimal to exit.

Part 4: Fix inline image, and suppose that it is not optimal to pay dividends even at high values of x so that (A.1) holds for all x. Let inline image. From Equation (5) in the main text and (A.3), inline image, and therefore it follows from Lemma A.2 that inline image. This means that inline image, inline image, and inline image, so that

  • display math

But then, combining (A.1) and Equation (2) in the main text,

  • display math

which is a contradiction because we have inline image and inline image. It follows that the continuation region must be bounded from the right: It is optimal to pay dividends for high enough x. We let

  • display math

Finally, fix inline image and suppose that it is not optimal to pay dividends even at high values of s. But then, as inline image, it follows from (A.1) that

  • display math

But this is in contradiction with (A.2) holding for all x and s, and the fact that inline image, inline image, inline image, and inline image are all bounded and independent of s. We can therefore conclude that the continuation region must be bounded from above: It is optimal to pay dividends for high enough s. We let

  • display math

A.2 Stationary Distributions

A.2.1. Unconstrained case

In the unconstrained case, the steady-state firm distribution and its properties reported in Section 'INDUSTRY EQUILIBRIUM' can be derived analytically as follows: Denote inline image. The exit threshold is inline image and new firms are born at inline image. Taking a discrete time approximation, y follows the binomial process

  • display math

where Δ is the length of a period, inline image, and inline image. The steady-state condition gives a difference equation for the mass of firms located at an arbitrary state point y,

  • display math

where inline image is the mass of all firms and inline image is the mass of newborn firms at state point y. Taking the limit inline image leads to a differential equation for the stationary firm density:22

  • display math(A.5)

with inline image and inline image as boundary conditions. In our setup inline image is positive at y0 and zero elsewhere. The point y0 splices the differential equation into two regions, with the inline image as a boundary condition in the middle (f is finite but not differentiable at y0). The value of f0 can be solved from the condition that total probability density integrates to one. Combining the boundary conditions with (A.5) yields the closed-form solution:

  • display math(A.6)

where inline image inline image, and

  • display math(A.7)

There is no economically sensible steady state unless inline image has a finite mean. Here inline image is a necessary and a sufficient condition for the finite mean. Taking out the terms that are independent of y in (A.6), the finite mean requirement becomes

  • display math(A.8)

This holds if inline image, which simplifies to inline image.

A.2.2. Constrained case

The stationarity proof in the unconstrained case is sufficient for the stationarity of the distribution of z in the constrained process. As s is endogenously bounded by the optimal dividend policy and, firm by firm, depends deterministically on the history of z, the fact that z has a stationary distribution suffices for the stationarity of the joint distribution inline image. However, now the optimal policy has no closed-form solution, so the steady-state distribution must be computed numerically. In the discrete time approximation the life span of each individual firm is a Markov chain in the discretized state space. Therefore, the steady-state distribution is obtained by first computing the optimal policy of an individual firm and then, starting from some initial firm distribution, iterating the firm distribution according to the state transition equations associated with the policy (where a constant mass of new firms are established at the birth point within each iteration) until the firm distribution converges to the steady state.

  1. 1

    See, for example, chapter 7 in Dixit and Pindyck (1994).

  2. 2

    For evidence on the importance of liquidity constraints for firms, see, for example, Evans and Jovanovic (1989), Holtz-Eakin et al. (1994), and Zingales (1998). There is also a literature on endogenous borrowing constraints, for example, Albuquerque and Hopenhayn (2004) and DeMarzo and Sannikov (2006). Holmström and Tirole (2011, especially chapters 1 and 2) discuss why agency problems may cause a firm to face a liquidity constraint.

  3. 3

    Another literal interpretation is that the owner–manager is risk neutral but discounts consumption by more than the return on cash. Decamps et al. (2008) interpret inline image as a reduced form of an agency cost, caused by the manager engaging in wasteful activities with the firm's liquid assets.

  4. 4

    We allow voluntary exit when inline image, but this would never be optimal.

  5. 5

    It can be shown that the probability with which the firm bounces up to positive profits before running out of cash goes to zero at a rate faster than s.

  6. 6

    The firm's position in (inline image)-space cannot evolve along the boundaries of the continuation region because, if inline image and inline image then inline image, and if inline image then the firm exits if it hits the boundary inline image.

  7. 7

    In terms of Dumas (1991), the dividend is “an infinitesimal requlator” (although exit is a discrete regulator) so there must be “supercontact” at inline image.

  8. 8

    It can be shown that with optimal policy converges to the limiting case as inline image , in the sense that, for every point (inline image) in the continuation region of the limiting case inline image, there exists inline image for which (inline image) is in the continuation region.

  9. 9

    The program for solving the optimal policy is available at http://www.hse-econ.fi/murto.

  10. 10

    This way of discretizing geometric Brownian motion was inspired by Cox et al. (1979).

  11. 11

    A natural starting point for the backward induction is inline image This means that the problem is turned into a finite-horizon problem with forced exit in the last period. By increasing T the value function at inline image converges to that of the infinite horizon problem.

  12. 12

    Liquidity constraints are introduced to a similar steady-state setting by Gomes (2001) to study the relation of cash flow and investment and by Cooley and Quadrini (2001) to study the age-conditional relation of growth and firm size. Jones (2003) averages over simulated time series of individual firms to study the impact of liquidity constraints on the propagation of aggregate shocks.

  13. 13

    The risk of exogenous exit changes the firm's optimal policy slightly compared to Section 'THE PROBLEM OF THE FIRM': The firms discount the future at rate inline image instead of ρ and the Bellman equation of the constrained firm includes a term inline image on the right-hand side of (11).

  14. 14

    Note that entry, unlike recapitalization, may take place at a positive level of revenue.

  15. 15

    A doubling of entry flow doubles the steady-state industry output.

  16. 16

    For a more detailed exposition of this industry equilibrium concept, see Miao (2005), who studies capital structure (in the absence of liquidity constraints).

  17. 17

    If s0 is sufficiently high and ϕ not too high, then inline image and the picture is more complicated, as some of inefficiently exiting firms are replaced by less productive firms.

  18. 18

    Hennessy and Whited estimate the costs separately for small and large firms, as well as for the total population of nonfinancial firms. In our model, small firms can become large and vice versa, so we use their results for the total population.

  19. 19

    We find that this is reversed if entry cost is set sufficiently low, but this implies turnover rates that are too high to be consistent with this data.

  20. 20

    Note that inline image and inline image, so that the firm is not under threat of immediate forced exit. Therefore, as we consider short intervals inline image, we can safely ignore the possibility that inline image for some inline image.

  21. 21

    Part 2 of the Proposition 1, which we will prove later, states that it is optimal to exit whenever s is small enough for all x < c, and therefore we can assume a limit point inline image

  22. 22

    See Dixit and Pindyck (1993, chapter 8, section 4.c) for more details.

References

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. THE PROBLEM OF THE FIRM
  5. 3. SOLUTION METHOD
  6. 4. INDUSTRY EQUILIBRIUM
  7. 5. CONCLUSION
  8. APPENDIX
  9. References