Dividends, Safety and Liquidation when Liabilities are Long-term and Stochastic∗ Robin Mason Department of Economics, University of Southampton January 2003
Abstract This paper investigates the optimal management of a firm faced with a long-term liability that occurs at a random date. Three issues are analysed: the optimal dividend policy; optimal expenditure on safety to delay the occurrence of the liability; and the optimal liquidation date of the firm. An owner faced with dynamic unlimited liability never liquidates and therefore accumulates capital to the golden rule level. For longterm liabilities, dividend payments and safety expenditure are non-decreasing over time. The owner protected by limited liability may liquidate the firm in finite time in order to avoid paying the liability. If this is the case, then it accumulates less capital than the dynamic unlimited liability owner; and may decrease dividend payments and safety expenditure over time. The paper shows that a finite liquidation date is more likely to be optimal when the arrival rate of the liability occurrence increases over time.
JEL Classification: C61, D21, H23, K32. Keywords: Liability, Strategic Liquidation, Accumulative Hazards. Address for Correspondence: Robin Mason, Department of Economics, University of Southampton, Highfield, Southampton SO17 1BJ, U.K.. Tel.: +44 (0)23 8059 3268; fax.: +44 (0)23 8059 3858; e-mail:
[email protected]. Filename: LIQHAZ10.tex. ∗
I would like to thank David Myatt, David Newbery, David Ulph, Juuso V¨ alim¨ aki and Chris Wallace. I would also like to thank the two referees and the editor for comments that have improved the paper substantially. The paper has grown out of joint work with Tim Swanson, whose contribution I gratefully acknowledge.
1. Introduction Limited liability rules present firms with the incentive and opportunity for strategic behaviour to avoid paying for liabilities that arise due to their operation. Two factors make such behaviour attractive: size and delay. When the expected liability is large, firms have a strong incentive to undertake actions (even if these actions are costly) to avoid paying the full cost. When there is a long separation between action and realisation of the liability, then the opportunity to avoid the liability also exists. There are many examples of such behaviour. Ringleb and Wiggins (1990) provide evidence that firms in the U.S. have altered capital structure to ensure that assets are not exposed to liability claims. A major contributing factor to the thrifts crisis in the U.S. was the continued payment of large dividends despite the overwhelming liabilities of the savings and loans institutions; see Akerlof and Romer (1993). ‘Orphan’ waste sites, where the firms responsible for environmental damages arising from disposal of toxic waste have long since disappeared, have left a clean-up bill of some U.S.$100 billion; see Menell (1991). The shareholders of a limited liability company have several decisions to make. First, they must decide whether to invest in the (physical) capital of the firm, or to pay out all the proceeds as dividends. Investment will generate higher output in the future; but shareholders may not benefit from this if assets are seized to pay for a liability. Secondly, the shareholders must decide whether to expend on measures to reduce the expected liability. Clearly, doing so reduces current profits, but may increase profits in the long-run. Finally, they must decide when to close the firm. Early closure sacrifices the dividends that shareholders can earn in the future; but it also avoids the payment of any liability that occurs after the firm is liquidated. Mason and Swanson (1996) provide a deterministic model of this situation.1 A firm operates for up to two periods; at the end of the second period, a liability occurs as a result of the firm’s activities. The firm must choose capital levels, safety expenditures (that affect the probability but not timing of the liability occurrence) and the liquidation date. 1
Wiggins and Ringleb (1992) contains a model of long-term liabilities, but they treat the probability of bankruptcy as exogenous. The contribution of Mason and Swanson is to allow bankruptcy to be chosen by firms to avoid the future payment of liabilities.
1
The authors show that ‘looting’—liquidation of the firm at the end of period one to avoid payment of the liability—will be the most attractive strategy for the firm’s shareholders when: (i) the damage is large and long-term; (ii) the firm has low capital productivity, low levels of sunk capital, and low effectiveness of safety expenditure. The firm is run to maximize one-period profits; there is no safety expenditure (so that the liability occurs with maximum probability) and no assets remaining in the second period to meet damages. The two-period model identifies clearly shareholders’ incentives for early liquidation. A key assumption of the model is that the liability can occur only at the end of the second period. This assumption drives several of the model’s detailed results. The limited liability firm makes zero expenditures precisely because it knows that it can avoid the liability with probability 1. Similarly, there are never any assets remaining to cover the liability, because the firm knows when the liability will occur. Finally, the dynamics of investment are particularly simple—initial capital investment is chosen to maximize period-1 output, which is paid out entirely in dividends; then firm is then closed before the liability occurs. But while it may be certain that a liability will occur in the future, the exact date of its occurrence will be uncertain. The robustness of Mason and Swanson’s results to more general, and more realistic, forms of liability occurrence has not been examined. This paper extends the analysis to a continuous-time model in which the time of occurrence of the liability is random. The owner of a firm can delay the occurrence of the liability in a probabilistic sense by expending on safety. If he is protected by limited liability, the owner can avoid paying for the liability by liquidating the firm before the damage has occurred. The paper examines the investment, safety and liquidation policies of two types of owner—one who faces (dynamic) unlimited liability, and one who is protected by limited liability. Three major results are presented. First, if the limited liability owner finds it optimal to liquidate the firm in finite time, then he under-invests in capital. Secondly, when liabilities are long-term, the dynamic unlimited liability owner does not decrease dividend payments or expenditure on safety during any stage of an optimal program; the limited liability owner may (but does not necessarily) reduce both before the firm is liquidated. Finally, it may be that a finite liquidation date is not optimal for the limited liability firm owner: unlimited and limited liability optima may coincide. The limited liability owner are more likely to liquidate in finite time when the arrival rate of the liability occurrence is increasing over
2
time. This result suggests that liquidation to avoid liability (‘looting’) will be a problem for regulators when damages are ‘accumulative’. The analysis suggests a new emphasis for corporate taxation, away from simply raising revenue towards addressing the incentives for looting. The analysis in this paper therefore confirms the general conclusion of the two-period model: that limited liability can encourage early liquidation, or ‘looting’. But the details are quite different. In the fully dynamic analysis, only in exceptional circumstances will the limited liability owner make zero safety expenditures; more generally, some expenditure will be optimal for the owner who is uncertain about when the liability will occur. Secondly, the dynamics of investment are more complicated than the two-period model suggests. Once the certain arrival of the liability is removed, the simple pattern of one-off investment followed by maximum dividend payment (that occurs in the two-period model) can be replaced by paths in which investment and dividend payments are non-monotonic over time. Finally, this model allows a more detailed picture of how the nature of the liability affects a limited liability owner’s behaviour. The two-period model suggests that early liquidation is optimal when the liability is ‘long-term’. The multi-period model extends this condition to liabilities that are accumulative. The rest of the paper is organised as follows. The next section describes the model that will be used to analyse the problem. Section 3 presents the benchmark case of the firm run by an owner with dynamic unlimited liability. This is contrasted in section 4 with the profitmaximising program of a limited liability owner. Section 5 considers the limited liability owner’s decision to liquidate the firm, relating this to the nature of the liability. Section 6 discusses the policy implications of the results, and particularly the role of ex ante taxation of limited liability firms. Section 7 considers the generality of the model and robustness of the results to extensions. Section 8 concludes. The appendix contains all proofs and a numerical example.
2. The Model A firm’s activities give rise to a liability (for example, the firm may produce waste that causes environmental damage). It is not known ex ante when the damage will occur. If it occurs 3
while the firm is still in operation, the owner loses the entire stock of capital. Implicitly, it is assumed that the value of the damage is at least as great as the value of the firm’s capital; this assumption serves to emphasise uncertainty about the timing, rather than the size, of the liability.2 The firm can be liquidated at any time before the liability occurs. To simplify the model results, the possibility of scrapping a fraction of the firm’s capital before the firm is closed is ruled out: either the entire capital stock can be scrapped, or none at all. This assumption means that only one scrapping decision is of importance: the decision to liquidate the firm. It can be justified using the result of Arrow and Kurz (1970): if the production and scrap value functions are strictly concave in capital, the maximised Hamiltonian of the optimal control problem will be strictly concave in capital, and hence it will not be optimal to scrap capital in the interval t ∈ (0, T ), where T is the terminal time of the problem. The owner of the firm receives a scrap value for the firm’s capital. This paper is interested in the effect of limited liability on a firm’s behaviour when faced with a long-term liability which occurs at a random time. Limited liability has a dual nature: static and dynamic. Static limitation means that, at any point in time, the liability of shareholders of a corporation is bounded by the assets of the firm. Dynamic limitation means that only damages that have occurred can be presented for payment from the assets of the injurer. This has two implications: first, the current assets of a firm which is in operation are protected from claims relating to future liabilities; secondly, both the assets and the liabilities of a firm end with its liquidation. The analysis focusses particularly on the effects of dynamic limited liability. Output is generated from capital k according to the production function Φ(k), which is increasing and concave in k.3 D(t) is the percentage dividend payment: if D = 1, then all output is paid out as dividends; if D = 0, all output is invested. Provided the damage has not occurred (which is the case with probability (1 − f (t))), the owner receives dividends 2
In addition, it describes many cases. For example, the asbestosis tort claims against Manville exceeded available cash resources only one year after the setting up of a fund; see Viscusi (1996). The initial demand for US$3 billion damages for the Bhopal disaster exceeded Union Carbide’s entire net worth; see Fischer (1996). 3 State variables are denoted by lower case Latin letters, control variables by upper case. Parameters are denoted by lower case Greek letters. Functions (the production and scrap value functions) are denoted by upper case Greek letters. Unless otherwise stated, all relevant variables are functions of time; the t argument is omitted for brevity.
4
D(t)Φ(k(t)) minus safety expenditure γS(t) (where γ > 0 is the unit cost of safety expenditure), or the scrap value of the firm Ψ(k(T ), T ) if the firm is liquidated. The scrap value function is increasing in capital:
∂Ψ ∂k
> 0; and decreasing with time:
feature of the model is that capital is assumed to be sunk:
∂Ψ ∂k
∂Ψ ∂T
≤ 0.4 An important
< 1, so that the owner cannot
recoup the full value of capital when the firm is liquidated. If the damage has occurred (with probability f (t)), then the entire capital stock of the firm is appropriated by the courts to pay for the liability and the owner receives nothing.5 Investment is irreversible and capital does not depreciate. The probability that the liability will occur by time t is given by the distribution function f (t). If the maximum safety expenditure is made (S = 1), then this probability does not increase (df /dt ≡ f˙ = 0); Rt ˙ ≥ 0 and if no output is spent on safety, then f (t) = 1−exp(− 0 λ(s)ds), where λ(t) > 0, λ(t) ¨ ≤ 0.6 λ is referred to as the arrival rate of the liability. The results of the paper will hold λ(t)
¨ is small. Allowing the arrival for cases where λ is a convex function of time, provided that λ
rate to depend on the capital stock: λ = λ(K) makes little difference to the results (provided the steady state is the appropriate transversality condition for the dynamic unlimited liability program—see later). Proposition 5 will show that a finite optimal liquidation date is less likely when λ˙ < 0; so a time-increasing arrival rate is the more interesting case. The hazard rate equals
f˙(t) 1−f (t)
= (1 − S(t))λ(t).
The risk neutral owner’s problem is to choose its investment/dividend policy, safety expenditure and the liquidation date to maximise the sum of a stream of dividend payments 4 The sign of Ψk is reasonable enough. The sign of Ψ T can be justified by supposing that the liability may attach to the capital, so that the buyer of the firm could be responsible for any damages arising from historic use of the capital; this possibility is referred to as ‘successor liability’. If the original owner’s safety expenditure is not verifiable, then the value of the firm’s capital for re-sale will decline over time. 5 Hence it is assumed that the liability is at least as large as the firm’s capital. This assumption is relaxed in Hartl et al. (1999); but they do not consider the possibility that the firm liquidates. 6 As one referee points out, safety expenditure is modelled as a flow, not a stock. Three observations motivate this choice. First, this assumption suits well certain types of expenditure: for example, continuous effort to prevent leaks of industrial solvents into drinking water supplies. It may be less appropriate for cases involving on-going safety investment, such as capital expenditures at nuclear power plants. Secondly, while the accumulated stock of safety expenditure does not enter the program directly, the history of safety expenditure is important in the way that it affects the state variable f , the distribution function describing liability occurrence. Finally, addition of a third state variable, in the form of accumulated safety expenditures, would complicate the model to such an extent as to make analytical solution unlikely.
5
minus safety costs, discounted at rate ρ: max
{D(t),S(t),T }
Z
T
e−ρt [D(t)Φ(k(t)) − γS(t)][1 − f (t)]dt + e−ρT Ψ(k(T ), T )(1 − f (T ))
0
˙ s.t. k(t) = (1 − D(t))Φ(k(t));
k(0) = k0
f˙(t) = (1 − S(t))(1 − f (t))λ(t); f (0) = 0 D(t), S(t) ∈ [0, 1].
(1) (2) (3)
The firm’s objective functional can be understood by considering the possible profit flows of the firm over the time interval [t, t + dt] before liquidation. There are three possibilities: 1. the liability has already occurred: the probability of this outcome is f (t), and the firm’s payoff is zero; 2. the liability does not occur during the interval: the probability of this outcome is 1 − f (t + dt), and the firm’s payoff is (approximately) (D(t)Φ(k(t)) − γS(t)); 3. the liability occurs at time t: the probability of this outcome is f˙(t), and the firm’s payoff is zero. Letting dt → 0, to give the continuous time limit, yields the firm’s profit functional as above. If the firm liquidates at time T , then it receives the expected discounted present scrap value of its assets. The probability that the liability has not occurred by time t is 1 − f (T ); the appropriate discount factor is e−ρT ; and the scrap value received is Ψ(k(T ), T ). Note that profits are not constrained to be non-negative. The firm is allowed to borrow on a perfect capital market (at interest rate ρ) to make safety expenditures, but not to invest in capital or make dividend payments. This assumption simplifies the dynamics considerably (particularly for the dynamic unlimited liability case) without having a major effect on the qualitative features of the model (as will be shown in section 7). The model captures, in a stylised way, five important features: 1. The owner of the firm faces a trade-off between current and future consumption, where future payouts are threatened by the occurrence of a liability. 6
2. The occurrence of the liability is a random event: the owner of the firm does not know the exact date at which damages will arise. 3. The liability can be delayed (in a probabilistic sense) by expending on safety; and can be prevented completely by maintaining safety expenditure at its maximum. 4. The owner can protect assets from the liability by making dividend payments or by liquidating the firm before the damage occurs. 5. Capital is sunk. The model is similar in many respects to existing work on growth/pollution trade-offs when there is the risk of catastrophe; see Clarke and Reed (1994), and Tahvonen and Withagen (1996) for examples. In turn, these models build on earlier work on machine maintenance and breakdown; see Thompson (1968) and Kamien and Schwartz (1971) for the classic papers. Thompson analyses the optimal maintenance policy and sale date of a machine whose scrap value declines deterministically over time; Kamien and Schwartz examine the optimal maintenance and sale date of a machine subject to stochastic failure. These papers find that (i) maintenance is a non-increasing function of time; and (ii) it makes little difference whether the arrival rate of machine failure is constant or time-dependent. These conclusions cannot be carried over directly to this model. The analytical results of these models are critically dependent on the ability to reduce the problem to one control variable. The authors are able to do this because there is no capital accumulation—all income is spent on consumption or care expenditure; and so for a constant flow of income, one of the controls can be eliminated. In this paper, the value of operating the firm increases over time when the owner accumulates capital. The competing factors of increasing output and rising probability of the damage occurring complicate the dynamics considerably. More importantly, whether the arrival rate is constant or increasing with time will be critically important in this model for determining the optimal liquidation date of the firm.7 7
Clarke and Reed (1994) show how the nature of the arrival rate of the catastrophe is important in determining steady state levels and dynamics. They show this, however, in an infinite horizon model, and so do not consider the optimal liquidation date.
7
Define the current valued Hamiltonian: H(t) = [D(t)Φ(k(t)) − γS(t)][1 − f (t)] + p1 (t)(1 − D(t))Φ(k(t)) + p2 (t)(1 − S(t))(1 − f (t))λ(t). p1 (t) is the adjoint variable attached to the capital stock of the firm, p2 (t) the adjoint variable attached to the probability of occurrence of the liability. The following conditions are necessary and sufficient for a maximum (see theorems 11–13, pp. 143–146 of Seierstad and Sydsaeter (1987)): 0 > (1 − f (t)) ⇒ D(t) = p1 (t) DSI , = 1 < < γ 0 ⇒ S(t) = |p2 (t)| SSI , = λ 1 >
(4)
(5)
p˙1 (t)
=
ρp1 (t) − [D(t)Φ0 (k(t))(1 − f (t)) + p1 (t)(1 − D(t))Φ0 (k(t))] ,
(6)
p˙2 (t)
=
ρp2 (t) + [D(t)Φ(k(t)) − γS(t) + p2 (t)(1 − S(t))λ(t)] .
(7)
DSI and SSI are the value of the controls during singular intervals; Φ0 is the derivative of the production function. Equation (4) states that maximum dividend payment occurs if the expected unit return (1 − f ) exceeds the marginal value of capital p1 ; conversely, maximum investment occurs when the marginal value of capital is high. In the neoclassical growth model (i.e., when either λ or γ equals 0), the expected unit return from capital is 1; this value must be reduced when there is a liability occurring at a random date. Equation (5) states that safety expenditure is zero while the absolute marginal value of the probability of liability occurrence is small; as this marginal value becomes more negative, the optimal policy switches to maximum safety expenditure. The critical value of p2 is − γλ , which is the present value of an infinite stream of expenditures γ, discounted at the liability arrival rate λ. Table 1 summarises the eight possible combinations of controls: four non-singular (NS1–NS4) and four singular (S1–S4).8 8
Lemma A.1 in the appendix shows that these are the only possible singular solutions for this problem.
8
D
NS1 NS2 NS3 NS4 S1 S2 S3 S4
0 0 1 1 DSI 1 DSI DSI
S 0 1 0 1 =1 0 SSI ∈ (0, 1) SSI =1 1
p1 > (1 − f ) > (1 − f ) < (1 − f ) < (1 − f ) = (1 − f ) < (1 − f ) = (1 − f ) = (1 − f )
p2 > − γλ < − γλ > − γλ < − γλ > − γλ = − γλ = − γλ < − γλ
Table 1: Policy Regimes
Finally, the following assumption will be made to reduce the number of cases that must be considered in sections 3 and 4: Assumption 1: k0 is sufficiently low that all optimal programs involve an initial period of capital accumulation. Although this is an ‘endogenous’ assumption (i.e., it is not a restriction on the primitives of the model),9 it is nevertheless reasonable. When the capital stock is low, the marginal value of capital will be high, and it should be optimal to accumulate capital. This effect will be reinforced at the beginning of the program, when the arrival rate of the liability is at its lowest i.e., the probability of capital loss through occurrence of the liability is small. The assumption has two effects. First, it truncates programs so that (uninteresting) initial phases with no investment are ignored. Secondly, it rules out optimal programs in which the limited liability owner sets up a firm purely to extract dividends, making no investment and expending nothing on safety before liquidation. Note that the assumption means that all programs must start in either of regimes NS1 or NS2. 9
When assumption 1 does not hold, it is straightforward to show for dynamic unlimited liability optimal programs and limited liability optimal programs that involve maximum terminal safety expenditures (S(T ) = 1) that there exists a k such that, if k0 < k, D(0) = 0. The same cannot be shown in general for limited liability programs with S(T ) = 0; in this case, it is possible that a firm will be set up with no investment beyond the initial capital level, and liquidated after a period of dividend payment. Two approaches can be used to rule out this sort of program. First, it could be assumed that the elasticity of the scrap value function with respect to capital is greater, for all values of T , than the elasticity of the production function. This is a very strong assumption, and is not made here. The alternative approach is to use assumption 1.
9
3. The Dynamic Unlimited Liability Benchmark This section examines a benchmark case in which the firm generating a liability is not protected by liquidation: the firm’s assets are seized whenever the liability occurs. But to emphasise the effect of dynamic limited liability, the firm has static limited liability—only the firm’s capital stock is seized to meet the cost of the damage. A firm that has truly unlimited liability would not have its liability capped by its capital stock—the private assets of the owners could also be seized by the court to meet damages. The focus of this paper is on the dynamic effects of limited liability, and so the issue of personal assets will be ignored. (Alternatively, assume that the owners of the firm have negligible assets outside of the firm.) Definition 1: The dynamic unlimited liability (DUL) firm’s liability is capped by the firm’s current capital stock, but does not end with liquidation. The DUL firm therefore continues in operation until the liability occurs. It does not consider the full value of social damages, but only the accumulated capital stock of the firm. This case can therefore be seen as a sort of second-best social optimum. Since the firm has no incentive to liquidate, the time horizon of the owner’s maximisation problem is infinite: max
{D(t),S(t)}
Z
∞
e−ρt [D(t)Φ(k(t)) − γS(t)][1 − f (t)]dt,
0
subject to the constraints on the state and control variables in equations (1)–(3). In addition to the necessary conditions given by equations (4)–(7), a transversality condition determines the terminal state of the system. The choice of the appropriate transversality condition is difficult in infinite horizon control problems; see Pitchford (1977), Michel (1982) and Seierstad and Sydsaeter (1987). The next two assumptions ensure the necessity of the transversality condition that the DUL program reaches a steady state. Assumption 2: Let kGR be the ‘golden rule’ capital level given by Φ0 (kGR ) = ρ. Then Φ(kGR ) ≥ lim γ t→∞
10
ρ + λ(t) . λ(t)
At the steady state, p˙2 = 0, p2 ≤ − γλ (since S = 1), D = 1, and k = kGR (see proposition 1). Hence assumption 2 ensures that the steady state exists.10 The assumption restricts attention to problems where the cost of safety expenditure is not prohibitively high; and where the liability is not such a remote prospect that it is effectively ignored by the firm’s owner. Assumption 3: For any time t, let tγ ( ρ+λ ) ≡ λ
Φ(kγ ( ρ+λ ) ) λ
kGR
dk , k ρ+λ Φ(k) γ( ) λ ρ + λ(t) γ . λ(t) Z
Then
1 Φ(kGR ) > tγ ( ρ+λ ) ∀t. ln ρ+λ(t) λ ρ γ λ(t)
The primary role of the assumption is to rule out a number of possibilities for the DUL program. The model here involves a multi-dimensional linear optimal control problem; as is common in such problems, the control dynamics may be very complicated. (See e.g., Clark et al. (1979) and Seierstad and Sydsaeter (1987).) As proposition 2 shows, the assumption ensures that no singular interval can occur before the steady state is reached. As a consequence, this more complicated model, with an extra state because of the possibility of liability occurrence, is similar to its neoclassical version with no liability, in which capital is accumulated as quickly as possible to the golden rule level. It is not straightforward to interpret the technical condition in assumption 3. All of the model parameters appear in the condition, and do so in non-linear expressions. One useful 10
For example, if the production function is Cobb-Douglas: Φ(k) = k α , 0 < α < 1, and the arrival rate is α 1−α a constant λ, then this parameter restriction is αρ ≥ γ ρ+λ . λ
11
way to view the condition is to consider what values of λ satisfy the assumption. This is done in the next lemma. ¯ Lemma 1: For any values of γ and ρ and for any production function Φ(·), there exists a λ ¯ (which will depend on the parameter values) such that assumption 3 is satisfied for all λ ≤ λ.
Hence assumption 3 requires that the liability be sufficiently long-term i.e., that λ be sufficiently small. Intuitively, bounding λ above ensures that the more complicated model with liability occurrence is sufficiently ‘close’ to its neo-classical version. This is reflected in proposition 2 by the similarity of the dynamics of the DUL and neo-classical optimal programs. The first proposition describes the steady state. Proposition 1: The DUL program must reach the steady state (regime S4 in table 1): D = DSI = 1 and S = 1. The steady state capital stock of the DUL firm is the golden rule level kGR . The steady state values of D and S are easy enough to understand: both are equal to 1, ensuring that there is no growth in capital or the cumulative probability of liability occurrence. Perhaps a surprising result is that the DUL firm chooses the capital level without reference to the liability. There are two ways to understand the result. In the absence of a future damage, the model is a standard neoclassical growth problem and capital will be accumulated to kGR ; see e.g., Barro and Sala-i-Martin (1995). When a future liability exists, the DUL firm sets S = 1 in the steady state; as a result, the cumulative probability of liability occurrence is stationary. The factor (1 − f ) in the objective functional is then just a constant that drops out of consideration in the necessary conditions. Consequently, the DUL firm accumulates capital to kGR . An alternative explanation compares the marginal benefits of consumption and investment. At time t, the marginal value of consuming one more unit is given by (1 − f (t)). The R∞ marginal value of investing that unit is t e−ρs Φ0 (k(s))(1 − f (s))ds. Necessary conditions 12
for a steady state are that f (t) = f and k(t) = k i.e., both state variables are constant. R∞ In turn, this implies that 0 e−ρt Φ0 (k(t))(1 − f (t))dt = Φ0 (k)(1 − f )/ρ. Thus optimality
imposes that (1 − f ) = Φ0 (k)(1 − f )/ρ. This leaves the golden rule: Φ0 (k) = ρ.
The analysis now turns to the dynamics of the DUL program. Proposition 2 summarises the feasible DUL programs. Proposition 2: Given assumptions 1–3, there are two feasible DUL programs, illustrated below. No program involves singular intervals.
D = 0, S = 0
D = 0, S = 1 t1
0
SS1
t
DUL Program 1
D = 0, S = 1 SS2
0
t
DUL Program 2 SSi are the times at which the steady-state is reached in the two programs, so that D = S = 1 for t > SSi . In both programs, an initial period of investment is followed by the steady state of maximum dividend payment. In the first program, the DUL firm starts by making no expenditure on safety; but once the shadow value p2 of the state variable f reaches a critical level, the owner switches to maximum safety expenditure. In the second program, safety is at 13
the maximum level permanently, so that the steady state probability of liability occurrence is zero.11
4. The Limited Liability Owner The owner’s optimal control problem when full limited liability (both static and dynamic) applies is now considered. The terminal time T is treated as finite and fixed in this section; this is for expositional reasons only, and will be relaxed in section 5. This means that the necessary conditions (4)–(7) must be supplemented by the transversality conditions: p1 (T ) = Ψk (k(T ), T )(1 − f (T )),
(8)
p2 (T ) = −Ψ(k(T ), T ).
(9)
Subscripts denote partial derivatives, so Ψk ≡
∂Ψ . ∂k
The next proposition determines the values of dividends and capital when the firm liquidates. Proposition 3: If the optimal liquidation date T of the firm is finite, capital is sunk, and assumption 1 holds, then D(T ) = 1 (i.e., the limited liability program always terminates with a period of maximum dividend payment); and the terminal capital stock k(T ) is strictly less 11
When Φ(k) = k α , 0 < α < 1, and λ(t) = λ, a constant, then program 1 holds when " !# α 1 1−α 1−α α −ρ α ρ+λ > exp − k01−α γ . λ ρ 1−α ρ
Program 2 holds when γ
ρ+λ λ
" α 1−α α −ρ ≤ exp ρ 1−α
!# 1 1−α α 1−α − k0 . ρ
Both require assumption (2) to hold. The parameter values under which the programs hold can be determined by solving the two explicitly for the time t ∗∗ at which S switches from 0 to 1. A general expression for t ∗∗ is given in the proof of proposition 2 in the appendix; it is straightforward to evaluate this expression in the special case of the Cobb-Douglas production function. If the time t ∗∗ is greater than zero, program 1 holds; the stated condition ensures this. Otherwise program 2 holds. The second program occurs (obviously enough) when γ is low or λ is high (the effect of other parameters is less clear).
14
than the golden rule level. This is the first difference between operating DUL and limited liability firms: the latter will under-invest in capital. Comparison of the proofs of propositions 1 and 3 show that two factors drive this result: capital sunkness and the transversality conditions (8)–(9). The DUL firm invests to the golden rule capital level because the system must reach a steady state. The limited liability owner pays out maximum dividends before closing the firm (since Ψk < 1); and if liquidation occurs in finite time, this implies that capital must be less than the golden rule level. The fact that the full value of capital cannot be recovered means that owner who chooses to liquidate will invest less in the firm.12 It should be noted that proposition 3 is not a straightforward under-investment result. The comparison between propositions 1 and 3 takes the operation of the DUL and limited liability firms as given. But there will be situations in which DUL firm will choose not to operate (have a negative value), while the limited liability firm finds it profitable to be in business. There are therefore two potential investment inefficiencies. First, limited liability may encourage firms to operate when it is not socially beneficial for them to do so; this causes over-investment. Secondly, conditional on a firm’s operation, limited liability leads to too low a level of long-run capital i.e., under-investment. The analysis turns now to the dynamics of limited liability programs. Proposition 4: There is at least one optimal program with limited liability and a finite liquidation date in which both dividend payments and safety expenditure are non-monotonic over time. The proposition identifies a key difference between DUL and limited liability optimal programs. The former must involve non-decreasing controls; the latter can involve controls that vary non-monotonically over time. In the DUL program, there is, in principle, a choice between going as quickly as possible to the steady state, or spending some time in a regime that involves dividend payments, followed by more investment to move on to the steady 12
A possible objection to this result is that it has nothing to do with the occurrence of a liability, but only with capital sunkness. Note, however, that the limited liability owner would not wish to liquidate if there were no liability.
15
state. The latter scheme is likely to be an attractive option when the probability of liability occurrence is high. But with a sufficiently remote liability—as is assured by assumption 3—it is optimal to accumulate capital at the maximum rate to reach the steady state, golden rule level as quickly as possible. In the LL program with a finite liquidation date, on the other hand, capital accumulation is less valuable to the firm. (This can be seen clearly by comparing the terminal level of the adjoint variable p1 i.e., the shadow value of capital in the two programs. For the DUL program, p1 = 1 − f ; for the LL program, p1 = Ψk (1 − f ) < 1 − f .) The LL firm therefore has a greater incentive than the DUL firm to spend time in a regime that involves dividend payments, before moving to the terminal state. The proof of proposition 4 constructs an example of such a program. In this program, a most rapid approach path is taken to the only feasible singular solution, in which some dividends are paid. The system stays in this regime for as long as possible, before taking the most rapid approach path to terminal regime. This involves more investment to increase the scrap value received on liquidating the firm. (Note that other programs may be optimal, depending on parameter values; and some of these may be monotonic.)13
5. Optimal Liquidation The analysis so far has treated the liquidation date T of the firm as fixed and finite. In this section, the optimal choice of T is investigated. Necessary conditions for this choice, for any particular capital level k, are given by the transversality equations: Φ(k)
= (ρ + λ(T ))Ψ(k, T ) − ΨT (k, T ) if S(T ) = 0,
Φ(k(T )) − γ = ρΨ(k, T ) − ΨT (k, T )
if S(T ) = 1.
(10)
Equations (10) show that the optimal liquidation date T is such that, at the time of liquidation, the value to keeping the firm in operation for one moment longer should equal the loss in scrap value. The former is the total amount paid in dividends (if S(T ) = 0), or dividends 13
The technical reason for the difference lies in the transversality conditions in the DUL and limited liability problems. The proof of proposition 2 in the appendix shows that the combination of the necessity of reaching a steady state and assumption 3 ensures that no singular interval is possible in an optimal DUL program. As a consequence, dividends and safety expenditure cannot decrease over any portion of the DUL program. In contrast, optimal programs with a finite liquidation date do not reach the steady state; and hence singular intervals are possible. This possibility allows the controls to be non-monotonic.
16
minus expenditure on safety (if S(T ) = 1). The latter has three components: (i) the interest foregone on the scrapped value: ρΨ; (ii) the probability of the liability occurring, and hence all scrap value being lost (if S(T ) = 0): λΨ; and (iii) the time depreciation of the scrap value: ΨT . Inspection of the necessary conditions indicates the parameter values that favour a finite liquidation date for the firm: low productivity of capital, low levels of sunk capital, high cost of safety (if S(T ) = 1), a rapidly time-depreciating scrap value (|ΨT | large), a high discount rate ρ and a large arrival rate (if S(T ) = 0). Equations (10) are only necessary conditions; in order for them to yield a finite optimal liquidation date, the sufficient conditions must be satisfied: ˙ )Ψ(k, T ) ≤ 0 if S(T ) = 0 ΨT T (k, T ) − (ρ + λ(T ))ΨT (k, T ) − λ(T ΨT T (k, T ) − ρΨT (k, T )
≤ 0 if S(T ) = 1
(11)
for any particular capital level k. If these conditions are not satisfied, then the limited liability owner will not liquidate the firm in finite time. In this case, the limited liability and DUL optima coincide. The next proposition examines the effect of the arrival rate λ on the sufficient conditions. Proposition 5: Suppose that a finite liquidation date is optimal when S = 1 i.e., suppose that for any k ≥ 0, there exists a T such that ΨT T (k, T ) − ρΨT (k, T ) < 0. There exists ˙ ζ ≥ 0 such that if the time elasticity of the arrival rate, λ(t)/tλ(t) ≥ ζ for all t, then a finite liquidation date is optimal. A numerical example helps to illustrate the proposition. Suppose that the production function is Cobb-Douglas: = k α for 0 < α < 1. Suppose also that the scrap value 2Φ(k) 2 function is Ψ(k, T ) = k β T −T , where 0 < β ≤ 1 and T > 0. If λ(t) is a constant (equal 2 T
to λ), the sufficient condition for a finite optimal liquidation date cannot be satisfied with these functional forms. For example, if γ is large, then the limited liability optimum with a finite liquidation date is likely to involve S(T ) = 0. In this case, equation (10) gives the following expression for the optimal liquidation date T : 1 T = + ρ+λ
s
2
1 T 1 + (1 − k α−β ) > 2 (ρ + λ) ρ+λ ρ+λ 17
where k is the capital stock accumulated in the limited liability program. The sufficient condition from equation (11) is: T ≤
1 . ρ+λ
So a finite liquidation date cannot be optimal when λ is constant. The DUL and limited liability optima coincide. (A similar argument can be used when γ is low so that S(T ) = 1 for a finite T ; see the appendix.) When λ(t) is increasing over time, however, a finite liquidation date may be optimal for the limited liability owner of the firm. Suppose that λ(t) = λt, so that f (t) = Rt 1 − exp −λ 0 (1 − S(τ ))τ dτ , for t ≤ T ; otherwise, λ(t) = λT .14 A numerical solution
is reported in table 2; the details of the solution are presented in the appendix. The pa-
rameter values chosen, while fictitious, are intended to represent a case such as hazardous waste disposal business in the U.S. as it was operated prior to the adoption of the Resource Conservation and Recovery Act (Rcra) in 1976. At that time it was the norm (due to the lack of regulation) to operate waste disposal sites with little or no treatment or care. The typical operator of a dump site depended upon the waste containers and local geology to ensure that any liability arising from the dumping of waste occurred well in the future. After years of operation, and before detection of any leakage, these firm dissolved, leaving the liability for clean-up to be met by the new owners of the sites.15 The sector was therefore characterised by (i) low capital productivity; (ii) expensive safety measures (relative to the return on capital); (iii) a moderate hazard rate (so that the expected period until liability occurrence is sufficient for the firms to earn returns; but not as long as for e.g., asbestosis, which can have a clinical latency of up to 25 years); (iv) rapidly depreciating scrap value of capital. For the parameter values used, program 1 holds for the DUL firm; since γ is high, the following will be used as the optimal limited liability program when the liquidation date is 14
Assumptions 1–3 are satisfied for the parameter values used. Details of orphan sites in the U.S. can be obtained from the Environmental Protection Agency’s web site, at the URL http://www.epa.gov/superfund/. See in particular the case of Binghamton Equipment Company in Vestal, Broome County, New York, reported at http://www.epa.gov/oerrpage/superfnd/web/sites/query/rods/r0289086.htm. 15
18
finite:16
D = 0, S = 0
D = 1, S = 0 t1
0
T
t
Limited Liability Program The numerical results show that, when the arrival rate is constant, the limited liability and DUL firms’ optimal strategy is to operate the firm, accumulating capital for the first 18 years and paying maximum dividends thereafter, until it is ‘closed’ by the occurrence of the liability.17 There is no expenditure on safety before t = 13; after this point, S = 1. This solution is shown in figure 2, which has 4 panels. The left-hand panels show the level of the firm’s capital over time (the upper panel), and its dividend payments (the lower panel). The right-hand panels show the probability distribution function f (t) over time (the upper panel) and safety expenditure (lower panel). When the arrival rate increases over time, however, the limited liability owner’s optimal strategy is to accumulate capital for one year and to liquidate the firm after 7.4 years; no expenditure is made on safety. The owner’s choices of dividends and safety expenditure, and the resulting levels of capital and the probability distribution of the liability over time, are shown in figure 3. Again, the DUL optimal program requires accumulation of capital for the first 18 years of the firm’s operation; maximum safety expenditure commences after 6 years in this case. 16
It is more difficult to determine which limited liability program is optimal, given the multiplicity of programs identified in proposition 4. 17 This is not inconsistent with proposition 3, which states if the liquidation date T is finite, then terminal capital stock of the limited liability firm is strictly less than the golden rule level. The optimal liquidation date is not finite in this case when the arrival rate is a constant.
19
Figure 1: Constant arrival rate
20
Figure 2: Increasing arrival rate
21
Constant Arrival Time-increasing Rate λ(t) = λ Arrival Rate λ(t) = λt Limited Liability Optimum Time at which dividend 18 1.0473 payment starts (t1 ) Terminal capital level (k(T )) 100 2.3215 Time at which safety 13 — expenditure starts Terminal liability probability 0.6896 0.9163 Optimal liquidation date (T ) ∞ 7.4240 DUL Optimum Time at which dividend 18 18 payment starts Terminal capital level 100 100 Time at which safety 13 6 expenditure starts Terminal liability probability 0.6896 0.8021 Notes: (i) The DUL and limited liability optima coincide when λ(t) = λ. (ii) The parameter values used are: α = 0.5, β = 1, γ = 5, ρ = 0.05, k0 = 1, λ = 0.09, T = 20. Table 2: Numerical Results
22
6. Policy Implications In this section, the policy implications of the analysis are discussed briefly. There are several possible approaches: modify limited liability rules, including abolishing it outright (see, for example, Hansmann and Kraakman (1991) and Pitchford (1995)); impose an ex ante tax on incorporated firms; require the posting of an ‘environmental bond’ (see e.g., Bovenberg and Heijdra (1998)); mandate safety technology standards (e.g., license for operation only waste sites with approved procedures for disposal); and so on. The first two options have been used extensively, particularly in the operation of the ‘Superfund’ (the common name for the Comprehensive Environmental Response Compensation and Liability Act) in the United States.18 Liability for the costs of cleaning up contaminated sites under the Superfund is strict, joint and several between potentially responsible parties (PRP). The U.S. Congress initially confined PRPs to current owners and operators of offending sites, generators of the waste deposited at the site, and transporters of waste to that site. Problems with the insolvency and non-existence (i.e., early liquidation) of PRPs have lead to an increasing tendency to extend liability towards related firms, particularly those with ‘deep pockets’ such as banks. See Greenburg and Shaw (1992). The lengthy legal battles that have ensued have limited the effectiveness of this approach. The average site takes 10 years to be cleaned up, at a cost of US$24-25 million; a third of expenditures are transaction costs, paid mostly to lawyers. In 1990, less than 6% of the Superfund’s revenues were raised from actions against PRPs. Ex ante measures, particularly corporate taxation, seem more promising in the light of this experience. For example, the Superfund imposes three forms of tax. Imported petroleum products and crude oil are subject to a per barrel tax; this accounted for around 37% of Superfund revenues in 1990. Specified chemicals also are taxed, raising 18% of total 1990 revenues for the Superfund. Finally, a general corporation tax generates the remainder: 40% of total revenues in 1990, worth US$1.3 billion. This extensive taxation of particular products and the corporate sector in general appears to have several advantages. Revenues are raised while firms are in operation in order to cover the costs of future damages. The taxation 18
Details on the Superfund can be obtained from the Environmental Protection Agency’s web site, at the URL http://www.epa.gov/superfund/.
23
of products that may be directly harmful, or have harmful by-products, may go some way towards encouraging appropriate levels of their use. And the reduction of corporate profits may discourage operation when the institution of limited liability would, by itself, lead to a socially excessive number of firms.19 This paper emphasises a different role for a corporate tax. Four potential inefficiencies have been identified that may arise when firms are protected by limited liability and generate long-term damages: (i) early liquidation; (ii) excessive entry to the corporate sector; (iii) under-investment by incorporated firms; and (iv) inefficient levels of safety expenditure. The analysis suggests that policy measures for long-term liabilities should be designed to address these four problems. For example, the emphasis of corporate taxation should be shifted from simple revenue raising (which accounts for a large part of the Superfund’s taxation policy) to elimination of the incentives firms face for liquidation. An illustrative proposal is that firms in specific sectors be charged an ‘exit tax’: on liquidation, they must pay over a proportion of asset value. Such a policy would diminish the incentives for liquidation, and hence the inefficiencies arising from dynamic limited liability.
7. Extensions Analysis of the behaviour of a firm faced with a stochastic liability is complicated by the presence of two state variables (the stock of capital and the probability of liability occurrence). The set-up of the model used in this paper reduces the problem, at least for some values of the control variables, to one state variable—the state f drops out, and singular solutions are defined in terms of k only. This allows features of the optimal program(s) to be derived analytically; see Pitchford (1977). Three assumptions are important for this solution method: (i) linearity of the objective functional and state variable differential equations in the control variables D and S; (ii) a zero hazard rate when safety expenditure is at a maximum; (iii) borrowing is allowed only for safety expenditure, not capital investment. The first assumption is relatively harmless. For example, the unit cost of safety expen19 Taxation can take other forms, such as compulsory insurance. See Winter (1991) for a discussion of the liability insurance market.
24
diture γ could be an increasing function of S, rather than a constant. The effect of this would be to remove the ‘bang-bang’ nature of the current model; propositions 1 and 3 (comparing terminal capital levels) would not change, but the dynamics of the programs would alter. Nevertheless, the contrast between the dynamics of the DUL and limited liability programs—the monotonicity of the former’s controls, the possibility of decreasing dividends and safety expenditure in the latter—would remain. The second assumption, that maximum safety expenditure reduces the hazard rate to zero (see equation (2)), is critical in allowing an unambiguous comparison of the terminal capital levels in DUL and limited liability optima. Although the problem is non-autonomous when the arrival rate λ is a function of time, nevertheless steady state analysis can be performed in the DUL case, since the probability of liability occurrence is constant when S = 1. This means that the DUL level of terminal capital can be identified exactly as the golden rule level. The limited liability firm, in contrast, accumulates capital to below the golden rule level. Hence under-investment from limited liability is clear. Other assumptions are likely to mean that the DUL does not accumulate capital to the golden rule level, and so will make this comparison more difficult. The third assumption (restrictions on borrowing) has two effects. First, it simplifies the dynamics of the DUL program by ensuring that the critical values of the adjoint variables (1−f (t) for p1 , the shadow value of capital, and − γλ for p2 , the shadow value of the probability of liability occurrence) are not dependent on the controls D and S. Secondly, it means that the analysis ignores the (interesting) question of the effect of the firm’s capital structure on its behaviour towards the liability. The assumption could be relaxed in two ways: either by introducing a constraint that flow profits be non-negative; or by allowing the firm to borrow both for safety expenditure and capital investment. The first approach has little effect on the qualitative features of the model. In the period immediately before the steady state is reached, the DUL firm pays out no dividends and makes maximum safety expenditure (see proposition 2). With a constraint on non-negative profits, this is no longer feasible. Instead, some dividend payment have to be made, and safety expenditure may be less than the maximum level. This delays, but does not prevent, the
25
reaching of the steady state;20 therefore the DUL firm still accumulates capital to the golden rule level. The monotonicity of safety expenditures is preserved, but dividend payments may decrease over the period.21 A non-negativity constraint affects limited liability programs in a similar way: propositions 3–5 are therefore unaltered. The second approach raises questions which are beyond the scope of this paper, and are important areas for further research. Preliminary analysis suggests that the capital structure of the firm will have some quantitative effect on its behaviour. It appears, however, that the qualitative features of the limited liability firm’s behaviour hold even when the firm is able to borrow extensively. Regardless of the level of debt, (dynamic) limited liability allows a firm to avoid the payment of large damages. The incentive to liquidate may be increased or mitigated by borrowing; but in either event, it does not disappear altogether. As a result, the various inefficiencies identified in the previous sections remain in a model which includes borrowing. An example will help to illustrate this point. Consider the following limited liability program:
D = 0, S = 0 0
D = 0, S = 1 t1
D = 1, S = 1 t2
T
t
Figure 3 shows the value of this program for different levels of debt, using the time-increasing hazard rate λ(t) = λt and other parameters values from section 5.22 The firm’s net profits 20
Assumption 2 ensures that the non-negativity constraint is not binding at the steady state. γ ˙ To see this, note that, during the period that the constraint is binding, D˙ = Φ S − D(1 − D)Φ0 . S˙ > 0, since in the absence of the constraint, S would be at its maximum level. Both terms are therefore positive, and so D˙ is ambiguously signed. 22 The shadow value of capital is highest at the start of the program, so the firm is assumed to borrow a sum at t = 0 to invest immediately in capital. Debt is assumed to be riskless, and so the interest rate charged is ρ. This assumption can be motivated in two ways. First, the objective of this exercise is to show that the firm will liquidate in finite time even when it is able to borrow for capital investment. It is therefore an acceptable research tactic to make borrowing as attractive (i.e., cheap) to the firm as possible. Secondly, 21
26
Figure 3: Debt in a Limited Liability Program
are maximised at a debt level of 36. The firm makes maximum safety expenditure from t = 1.1172 onwards. It invests until t = 1.5612, when its capital stock reaches 47.1055 (less than half the golden rule level). It liquidates at t = 1.9897. This is in contrast to the DUL program which, irrespective of the initial debt level incurred, accumulates capital to the golden rule level. (In fact, it can be shown that it is not optimal to borrow at t = 0 in the DUL program, and so the solution is as in section 5.) This example illustrates that the issues highlighted in this paper, relating to the incentives for limited liability firms to liquidate, remain relevant even when the firm can borrow to any debt issued could use the firm’s capital as collateral. Provided that the scrap value of the firm’s assets exceed the amount of the loan, the debt is then riskless, since in the event of the liability occurring, it has higher priority (as secured debt) in bankruptcy proceedings than the liability claim. See Campbell (1992). Note, however, that the practices of successor and lender liability may mean that even secured debt is risky. The value of the program is Φ(k(t2 )) −ρt2 γ −ρt1 t21 −ρT −ρT −ρT −λ 2 −e + −e +e Ψ(k(t2 ), T ) − (k − k0 )eρ , − e e V =e ρ ρ where k is the level of initial borrowing. The details of how to determine the times t 1 , t2 and T can be obtained from the author on request.
27
invest. The analysis is clearly not exhaustive; the question of the interaction of capital structure and firm behaviour in the presence of long-term liabilities is an interesting topic for further work. As a final note, the size of the liability has, effectively, been ignored by assuming that it exceeds that assets of the firm. A model in which both the size and the timing of the liability are uncertain should be a next step for research in this area.
8. Conclusions This paper has extended the discrete-time analysis of Mason and Swanson (1996) to the situation where the timing of the occurrence of a long-term liability is uncertain. The extent to which the scrap value decreases over time, and, most importantly, the nature of the liability that is created, have been shown to be critical factors in determining whether owners will liquidate their firm to avoid future, uncertain liabilities. Liabilities that are both long-term and accumulative (in the sense that the arrival rate of the liability increases over time) are most likely to be avoided by the shareholders through the looting strategy. In other words, industries exposed to ‘accidents’ (e.g., Union Carbide and Bhopal) are unlikely to use liquidation as a means of avoiding liability; but industries where hazards accumulate—such as toxic waste disposal—will find liquidation an attractive option. The analysis suggests a shift in emphasis for corporate taxation away from revenue raising to meet the costs of damages, towards addressing the specific incentives of firms faced by long-term liabilities.
28
R kGR k
γ
(
ρ+λ λ
)
dk Φ(k) 1 ρ
ln
Φ(kGR ) γ ( ρ+λ λ )
replacements
¯ λ
λ
Figure A.1: Assumption 3—long-term liability
APPENDIX This appendix contains the proofs of all lemmas and propositions and the details of the numerical example used in section 5.
A.1. Proof of Lemma 1 Assumption 3 requires that
1 Φ(kGR ) > tγ ( ρ+λ ) . ln λ ρ γ ρ+λ λ
(A1)
Both sides of equation (A1) are increasing functions of λ. It is straightforward to show that if ρ Φ0 kγ ( ρ+λ ) < λ λ
(A2)
then the left-hand side is less steeply sloped. Note that lim λ → 0kγ ( ρ+λ ) = +∞. Hence, due to the concavity λ ˆ such that equation (A2) is satisfied for all λ ≤ λ. ˆ For λ ≤ λ, ˆ therefore, there exists of Φ(·), there exists a λ ¯ such that equation (A1) is satisfied for all λ ≤ λ. ¯ The argument is illustrated in figure A.1. aλ
29
A.2. Proof of Proposition 1 The necessity of the steady state follows from theorem 16, p. 244 of Seierstad and Sydsaeter (1987). In the steady state, k˙ = f˙ = 0; hence D = S = 1 (from equations (1) and (2)). Therefore p 1 ≤ (1 − f ), from equation (4). p˙1 = 0 in the steady state, so that p 1 =
Φ0 (k) ρ (1
− f ), from equation (6). Since (1 − f ) > 0,
this implies that Φ0 (k) ≤ ρ. From assumptions 1 and 2, the program must start with an interval in which D = 0 and end with an interval in which D = 1. At the time that D switches from 0 to 1, p 1 must be weakly decreasing (this is clear from the first-order condition (4) and from the dynamics of f (t) in equation (2)). But for p˙1 ≤ 0 at this point, it must be that (ρ−Φ0 (k))(1−f ) ≤ 0 (substitute into equation (6) p1 = (1−f )). Hence Φ0 (k) ≥ ρ. Together with the earlier condition, this implies that Φ 0 (k) = ρ i.e., k = kGR . In order to prove other lemmas and propositions, general features of the system are first identified. The next lemma states the singular solutions of the system.
Lemma A.1: When λ(t) is a constant λ, there are 4 possible singular intervals: S1 D = DSI = 1, S = 0 and k = kρ+λ ; S2 D = 1, S = SSI ∈ [0, 1] and k = kγ ( ρ+λ ) ; λ
S3 D = DSI ∈ (0, 1), S = SSI ∈ [0, 1] and k > kγ ( ρ+λ ) ; λ
S4 D = DSI = 1, S = 1 and k = kGR (the steady state). ˙ When λ(t) > 0, there are 2 possible singular intervals: S3 D = DSI ∈ (0, 1), S = SSI ∈ [0, 1] and k > kγ ( ρ+λ(t) ) ; λ(t)
S4 D = DSI = 1, S = 1 and k = kGR (the steady state). kρ+λ and kγ ( ρ+λ(t) ) are defined as follows: λ(t) Φ0 (kρ+λ )
ρ + λ, ρ+λ γ . λ
Φ(kγ ( ρ+λ ) ) λ
˙ Proof: Suppose 0 that λ = 0. When D is singular, p 1 (t) = 1 − f (t). From equations (2) and (6), this means that S = 1 − Φ (k(t))−ρ . So when D = DSI , S may equal 0; in this case, k = k ρ+λ and so DSI = 1. This λ is case S1 above. Or S can equal 1 when k = k GR , which requires DSI = 1; this is the steady state of case γ γ S4. When S is singular (i.e., p2 = − λ ), equation (7) means that D = Φ(k(t)) ρ+λ . So when S = SSI , D λ
30
must be greater than zero. If D = 1, then k = k γ ( ρ+λ ) ; this is case S2. If D = D SI < 1, then k > kγ ( ρ+λ ) ; λ
λ
this is case S3. When λ˙ > 0, cases S1 and S2 are ruled out by the assumption that investment is irreversible. Case S1 ˙ ˙ requires that Φ0 (k(t)) = ρ + λ(t); and so k(t) = Φ00λ(k) < 0 since Φ(k) is concave. Case S2 requires that ˙ 2 ˙ ˙ ¨ ¨ ≤ 0). ˙ + γλλ2 ; therefore k(t) < 0 (since λ Φ(k(t)) = γ ρ+λ(t) = Φ01(k) − γλλ2 + γλλ2 − 2γ(λλ) 3 λ(t) The next lemma identifies the dynamics of the system.
Lemma A.2: The set of necessary conditions for switches between policy regimes to take place are given in table A.1. (A switch out of regime S4 is not shown, since this is the steady state.)
Proof: The proof will be given only for the top row (switches from regime NS1 to other regimes)—other rows are calculated similarly. For the switch NS1 → NS2 to occur, it must be that p˙2 < before the switch (i.e., p2 must decrease from being greater than
− γλ
˙ γλ λ2
immediately
to being less than this value). But in
regime NS1, p˙2 = (ρ + λ)p2 < 0; so the switching condition is satisfied for all levels of capital. For the switch NS1 → NS3 to occur, it must be that p˙1 < −f˙ i.e., p1 must fall from above (1 − f ) to below this value. Since p˙1 = (ρ − Φ0 (k))p1 and −f˙ = −(1 − f )λ in regime NS1, this implies that Φ 0 (k) > ρ + λ i.e., k < kρ+λ . ˙ For the switch NS1 → NS4 to occur, both p˙1 < −f˙ and p˙2 < γλλ2 must hold before the switch. From the arguments above, the latter imposes no conditions on the level of capital; the former requires that k < k ρ+λ . The switch NS1 → S1 requires that p˙1 < −f˙ i.e., k < kρ+λ , once again. The switches NS1 → S2, S3 both ˙ require that p˙1 < −f˙ and p˙2 < γ λ2 . As above, this implies that k < kρ+λ . Finally, the switch NS1 → S4 λ
˙ γλ appears to require only that p˙1 < −f˙ and p˙2 < λ2 . But this switch can be ruled out because S4, the steady )−γ state, requires that p 2 = − Φ(kGR . Generically, this steady state value is not equal to the switching ρ
value − γλ .23 The continuity of adjoint variables means that p 2 cannot jump from − λγ to the steady state value; and so the switch NS1 → S4 is infeasible. Table A.1 will be used heavily in the subsequent proofs.
A.3. Proof of Proposition 2 The proposition is proved by showing that given assumptions 1–3, the optimal controls of the DUL program have the following features: (i) there are no singular intervals before the steady state; 23
Equality of the steady state and switching values of p 2 would require that kGR = γ When λ˙ > 0, the two values cannot be equal.
31
ρ+λ λ
when λ˙ = 0.
NS2
NS3
NS4
S1
S2
S3
S4
NS1
—
No restriction
k < kρ+λ
k < kρ+λ
k < kρ+λ
k < kρ+λ
k < kρ+λ
×
NS2
×
—
×
k < kGR
×
×
×
k < kGR
NS3
k > kρ+λ
×
—
k > kρ+λ
×
NS4
×
×
k > kγ ( ρ+λ )
—
S1
×
×
×
S2
i h k > max kρ+λ , kγ ( ρ+λ )
× ×
32
NS1
S3
λ
h k > max kρ+λ , Φ−1 γ ρ+λ + λ
˙ γλ λ
i
Φ(k) < γ
ρ+λ λ
+
˙ γλ λ2
×
Φ(k) < γ
ρ+λ λ
+
˙ γλ λ2
×
×
×
×
k < kγ ( ρ+λ )
—
×
×
×
×
×
×
—
k > kρ+λ
×
×
×
×
×
—
×
λ
λ
The table shows conditions that must apply immediately before a switch from the regime in a row to the regime in a column.
Table A.1: Policy Switches
(ii) D(t) is non-decreasing over time; (iii) S(t) is non-decreasing over time; (iv) D(t) cannot switch from 0 to 1 while S(t) = 0. (i) There are no singular intervals before the steady state. Singular solutions S1 and S2 can be ruled out purely from the dynamics of the system and the necessary condition that the steady state is reached. To see this, consider each of the singular solutions in turn. S1 From the dynamics in table A.1, the system cannot reach the steady state from this regime; so the system must leave this singular solution. It must switch to regime NS4 (again, from the dynamics in table A.1); this requires that k < kγ ( ρ+λ ) . But the system cannot reach the steady state from regime ρ
NS4; and it cannot leave NS4 unless k > k γ ( ρ+λ ) . Since there is no capital accumulation in regime ρ NS4 (D = 1), this is a contradiction. Hence this singular solution cannot occur in a DUL program. S2 The system cannot reach the steady state directly from this regime, and so a switch must occur, either to regime NS1 or S3; both switches require that k > k ρ+λ . But S2 can be reached only from regime NS1 or NS3. If the former, then k < kρ+λ for the switch to take place. If the latter, then again it must be that k < kρ+λ : NS3 cannot be the initial regime of the system (by assumption 1); it must be reached from regime NS1, which requires k < k ρ+λ .24 Since there is no capital accumulation during regime S2, this must mean that the necessary condition for the switch from S2 cannot hold. S3 Consider the following program: the system starts in regime NS1. It switches to regime S3 (which requires that k < kρ+λ ). It then switches back to regime NS1 once capital has accumulated to a high enough level to satisfy the switch requirements. From NS1, the system switches to NS2 (there is no restriction on the level of capital for this switch to take place); from NS2, the system moves to the steady state. This program satisfies all of the necessary conditions for optimality. Therefore only singular solution S3 can occur before the steady state is reached. The time taken for capiR kGR dk k ρ+λ∗ Φ(k) , λ γ( λ∗ ) where λ∗ is the level of the liability arrival rate when the system first reaches singular solution S3. When
tal to accumulate from its singular level to the steady state level is no greater than t γ ( ρ+λ∗ ∗ ) ≡
D = 0, S = 1 (i.e., the interval before the steady state), p˙2 = ρp2 − γ < 0. Integrating the differential equation for p2 , it takes
t∗∗
=
1 Φ(kGR ) , ln ∗∗ ρ γ ρ+λ λ∗∗
24
There is an alternative route to regime NS3: NS2 → NS4 → NS3. The first switch requires that k < k GR ; the second that k > kγ ( ρ+λ ) . If this route is taken, the system cannot switch from NS3 to S2, which is the ρ switch that is of interest here. The switch NS3 → S2 requires that k < k γ ( ρ+λ ) , since S2 is feasible only if ρ λ˙ = 0.
33
for p2 to fall from − λγ∗∗ to its steady state value of − when S switches from 0 to 1; λ∗∗ > λ∗ , so that
t∗∗ > t∗ =
Φ(kGR )−γ ρ
. λ∗∗ is the level of the liability arrival rate
1 Φ(kGR ) . ln ∗ ρ γ ρ+λ ∗ λ
Assumption 3 ensures that tγ ( ρ+λ∗ ∗ ) < t∗∗ , so that singular solution S3 is infeasible for a DUL program. λ (ii) D(t) is non-decreasing over time. This part of the lemma follows directly from part (i) and the optimality of most rapid approach paths (MRAPs) in linear optimal control problems (see Clark (1990)). Since there are no singular intervals before the steady state is reached, the system takes a MRAP to the steady state. Therefore, from assumption 1 and proposition 1, D is monotonic, being 0 before the steady state is reached, and switching to 1 once capital reaches the golden rule level. (iii) S(t) is non-decreasing over time. When D = 0 and S = 1, equation (7) gives p˙2 = ρp2 − γ < 0. So p2 is monotonically decreasing in any period during which D = 0, S = 1; and so S must be non-decreasing, from the first-order condition (5). When D = 1 and S = 1, p˙2 = ρp2 − γ + Φ(k). Since there is no subsequent period of capital accumulation (D is non-decreasing), stock must equal k GR . If S decreases, S : 1 → 0, then at the switch point, the ∗capital ˙ ∗) λ(t ρ+λ(t ) ∗ is the time at which the system switches from it must be that −γ λ(t∗ ) + Φ(kGR ) > γλ(t ∗ )2 , where t S = 1 to S = for to 1 (as is required for the system to reach a steady state), it must 0. But S to switch back ˙ ∗∗ ) ρ+λ(t∗∗ ) γ λ(t be that −γ λ(t∗∗ ) + Φ(kGR ) < λ(t∗∗ )2 , where t∗∗ is the time at which the system switches S : 0 → 1. ¨ ≤ 0; and so S cannot decrease when D = 1. This is a contradiction, since λ˙ > 0 and λ (iv) D cannot switch from 0 to 1 while S = 0. Suppose not i.e., suppose that the system can switch from NS1 to NS3. Then table A.1 shows that k < k ρ+λ . The monotonicity of D means that there is no subsequent period of capital accumulation. This contradicts proposition 1.
A.4. Proof of Proposition 3 Since p1 (T ) = Ψk (k(T ), T )(1 − f (T )) and Ψk < 1, p1 (T ) < (1 − f (T )) and D(T ) = 1 by equation (4). By assumption 1, D(0) = 0. Let τ be such that D(t) = 1 over the interval τ ≤ t ≤ T . Then p˙1 (τ ) must be strictly negative (since p 1 (T ) < (1 − f (T ))). Equation (6) states that p˙1 (τ ) = (ρ − Φ0 (k)(τ ))(1 − f (τ )) at τ ; and so Φ0 (k(τ )) > ρ. But since D(t) = 1 for t > τ , k(T ) = k(τ ). Hence k(T ) < kGR .
34
A.5. Proof of Proposition 4 The proposition will be proved first be establishing the general properties of optimal limited liability programs; and then by an example.
Lemma A.3: Given assumptions 1–3, an optimal limited liability program with a finite liquidation date has the following features: (i) singular solutions are feasible; (ii) an optimal program must terminate in a non-singular interval with D = 1; it cannot start with a singular solution; (iii) singular solutions S1 and S2 are infeasible.
Proof: The lemma has three parts: (i) Singular solutions are feasible. Singular solutions other than the steady state are ruled out in DUL programs by the requirements that: (i) capital accumulates to the golden rule level (this eliminates singular solution S1); (ii) the system leaves the singular solution (ruling out solution S2); (iii) assumption 3 holds (ruling out solution S3). Clearly, requirement (i) does not hold in limited liability optimal programs (see proposition 3), and so singular solution S1 cannot be ruled out. In addition, assumption 3 does not rule out singular solutions in limited liability cases. To understand why, note that the assumption is such thatthe time taken for the adjoint Φ(kGR )−γ γ is greater than the variable p2 to fall from its switching value − λ to its steady state level − ρ
time taken for capital to accumulate from the singular level k γ ( ρ+λ ) to the golden rule level kGR . In the λ limited liability program, p 2 (T ) = −Ψ(k(T ), T ), fromthe transversality condition (9). But from the optimal stopping conditions (10), this means that p 2 (T ) > −
Φ(k(T ))−γ ρ
where k(T ) < kGR . Therefore assumption
3 does not rule out this singular solution.
(ii) An optimal program must terminate in a non-singular interval with D = 1; it cannot start with a singular solution. D must be non-singular and equal 1 at the liquidation date T (assuming that this is finite), since the transversality condition shows that p 1 (T ) < (1 − f (T )). This leaves only one possibility for the program to terminate with a singular solution: that is S2: S = S SI , D = 1, k = kγ ( ρ+λ ) if λ is a constant. The optimal λ
stopping condition in this case is: Φ(kγ ( ρ+λ ) ) − γSSI λ
h γi = − ΨT (kγ ( ρ+λ ) , T ) − (ρ + λ(1 − SSI )) λ λ
35
where the fact that k(T ) = kγ ( ρ+λ ) and the transversality condition λ equation leads to a contradiction:
γ λ
= Ψ(k(T ), T ) have been used. This
0 = −ΨT (kγ ( ρ+λ ) , T ) > 0. λ
Therefore this singular solution cannot be terminal. Hence the system must terminate in either regime NS3 or NS4. By assumption 1, D(0) = 0; so the system cannot start with D singular, since any singular solution ρ+λ(0) γ requires D > 0 (see lemma A.1). But if S(0) is to be singular, then D(0) ≥ Φ(k(0)) > 0. The λ(0)
system cannot, therefore, start with a singular solution; and so any singular solution must be in the interior of a limited liability program. (iii) Singular solutions S1 and S2 are infeasible.
Consider first S1. The program cannot terminate in this regime (see part (ii) above), and so the system must switch from S1. Only the switch S1 → NS4 is feasible; this requires k < k γ ( ρ+λ ) . Once in regime λ NS4, no further switch is possible (the only switch that could occur would be NS4 → NS3; but this requires k > kγ ( ρ+λ ) , contrary to the previous inequality). So the program terminates with S(T ) = 1. From the λ optimal stopping condition (10): Φ(k(T )) = γ + ρΨ(k(T ), T ) − ΨT (k(T ), T ) > γ + ρΨ(k(T ), T ) ρ+λ > γ , λ where the last inequality follows from the terminal conditions that p 2 (T ) = −Ψ(k(T ), T ) (transversality) and p2 (T ) < − λγ (S(T ) = 1). This implies that k > kγ ( ρ+λ ) , which is a contradiction. λ
The system cannot reach regimes NS3 or NS4 directly from S2, and so a switch must occur. Only switches to regimes NS1 or S3 are feasible; both require that k > k ρ+λ . But S2 can be reached only from regime NS1 or NS3. If the former, then k < k ρ+λ for the switch to take place. If the latter, then again it must be that k < kρ+λ : NS3 cannot be the initial regime of the system (by assumption 1); it must be reached from regime NS1, which requires k < k ρ+λ .25 Since there is no capital accumulation during regime S2, this must mean that the necessary condition for the switch from S2 cannot hold. Consider the following program, which takes a MRAP to (the only feasible) singular solution, h →˙ i S3: NS1 ρ+λ −1 S3 → NS1 → NS2 → NS4. The first switch requires k < k ρ+λ ; the second, k > max kρ+λ , Φ γ λ + γλλ ;
there is no restriction on capital for the third switch to take place; and the last requires k < k GR . So this program satisfies all of the properties in lemma A.3, the necessary conditions in table A.1, is consistent with assumption 1, the necessary and sufficient conditions equations (4)–(7), the transversality condition 25
As before, the alternative route to regime NS3: NS2 → NS4 → NS3, can be ruled out.
36
(8) (which implies D(T ) = 1), and the optimal stopping condition (10) (which implies, for S(T ) = 1, that k(T ) > kγ ( ρ+λ ) ; but this is assured when the system switches S3 → NS1). λ
D is non-monotonic in this program, first increasing above 0 (in the switch NS1 → S3) and then back to 0. S also is non-monotonic. To see this, note that k > k ρ+λ for the switch S3 → NS1. During the regime S3, p1 = (1 − f ); this implies that p˙1 ⇒ Φ0 (k)
= (ρ − Φ0 (k))(1 − f ) = −(1 − SSI )(1 − f )λ; = ρ + (1 − SSI )λ.
Combining this equality with the necessary condition for the switch S3 → NS1 implies that S SI > 0 before the switch. S is therefore non-monotonic, starting at 0 (in regime NS1), increasing above zero during the singular interval S3, and returning to zero after the switch S3 → NS1.
A.6. Proof of Proposition 5 A finite liquidation date is optimal if both the necessary conditions (10) and sufficient conditions (11) are satisfied. By hypothesis, Ψ T T (k, T ) < ρΨT (k, T ), so that the necessary and sufficient conditions when ˙ S = 1 are satisfied. When S = 0, the sufficient condition requires that Ψ T T ≤ ρΨT + λΨT − λΨ. Since ΨT T (k, T ) < ρΨT (k, T ), this sufficient condition is certainly satisfied if ˙ λ(t) −ΨT (k, T ) ≡ζ ≥ max k,T tλ(t) T Ψ(k, T )
(A3)
for all t. Finally, note that condition (A3) requires that λ(t) be non-decreasing over time, since ζ ≥ 0. When λ˙ > 0, then the switching value − λγ also increases over time. Hence it is more likely that the optimal program will verify S = 1, and hence satisfy the necessary and sufficient conditions by hypothesis.
A.7. Numerical Example in Section 5 This section of the appendix gives the analytical expressions for the solution to the numerical example in section 5, with the following functional forms: Φ(k) = k α , α ∈ (0, 1);
Ψ(k) = k β
2
T −T 2 2 T
, β ∈ (0, 1].
When the arrival rate is a constant λ, then the two non-linear simultaneous equations that yield t 1 (the date at which capital accumulation stops and maximum dividend payments start) and T (the liquidation date of
37
the firm) are: 2
βe
−(ρ+λ)T
k
−(1−β)
T − T2 T
2
!
− e−(ρ+λ)t1
=
T
=
k
=
where
αk −(1−α) −(ρ+λ)T e − e−(ρ+λ)t1 , ρ+λ s 2 1 1 T + + (1 − k α−β ), ρ+λ (ρ + λ)2 ρ+λ 1 k01−α + (1 − α)t1 1−α .
When the arrival rate increases over time, so that λ(t) = λt, then the two non-linear simultaneous equations in t1 (the time at which capital accumulation stops) and T , when S(T ) = 0 (γ is large) are: 2
βe
2 −ρT − λ −(1−β) 2T
e
k
T − T2 T
2
!
λ 2
− e−ρt1 e− 2 t1
= −αk −(1−α)
Z
T
e−s(ρ+ 2 s) ds, λ
t1
k
(α−β)
=
2T T
k
=
2
+ (ρ + λT )
2
T − T2
k01−α + (1 − α)t1
T
2
1 1−α
!
,
.
These equations can be solved numerically (subject to the constraints that T ≤ T and t 1 < T ) to give the results in section 5. The DUL case 1 accumulates capital until SS 1 , given by SS1 =
1 1−α
α − k01−α . ρ
There is no expenditure on safety until t 1 , given by α
1 t1 = SS1 − ln ρ
λ(t1 ) αρ 1−α γ(ρ + λ(t1 ))
!
.
When λ(t) = λ, a constant, this equation gives an analytical expression for t 1 ; otherwise the resulting non-linear equation must be solved numerically.
38
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