investment and dividends under irreversibility and financial constraints

Journal of Economic Dynamics & Control27 (2003) 467–502

www.elsevier.com/locate/econbase

Investment and dividends under irreversibilityand %nancial constraints

Richard W.P. Holt ∗

Department of Economics, University of Edinburgh, William Robertson Building,50 George Square, Edinburgh EH8 9JY, UK

Abstract

Research %nds that %rms’ investment and dividend policies are distorted by irre-versibility and %nance constraints. Existing work considers the impact of each suchfriction in isolation. Using a real options framework, I explain how the investmentand payout decisions interact over time when irreversibility and %nancial constraintsmay bind jointly. Firms’ behaviour follows a life-cycle pattern: young (small) %rmsinvest for growth; not unless they avoid liquidation and expand through investmentdo they pay dividends. Important rami%cations for empirical work on investment arediscussed. c© 2002 Elsevier Science B.V. All rights reserved.

JEL classi(cation: D92; E22; G31; G33; G35

Keywords: Investment; Dividend policy; Irreversibility; Financial constraints

1. Introduction

The standard approach to modelling economic phenomena is to consider acandidate explanation in isolation. While it can provide a clear insight, suchsimplicity is apt to lead to a plethora of competing explanations for featuresof economic behaviour. Of course it may be possible to distinguish empir-ically between competing explanations. A more serious problem is that thistendency towards simplicity can obscure important interactions between thecandidate explanations, which may cloud empirical results. For example, both

∗ Tel.: +44-131-6508350.E-mail address: [email protected] (R.W.P. Holt).

0165-1889/02/$ - see front matter c© 2002 Elsevier Science B.V. All rights reserved.PII: S0165-1889(01)00057-4

468 R.W.P. Holt / Journal of Economic Dynamics & Control 27 (2003) 467–502

nominal rigidities and real rigidities have been considered separately by re-searchers trying to understand business cycle phenomena, however it is nowrecognised that the existence of real rigidities in goods, labour and %nan-cial markets can increase the eEect of frictions in nominal price adjustment(Romer, 2000). Equally there may be important interactions between diEerentreal rigidities. In this paper, I consider the interaction between irreversibilityand %nancial constraints in determining the %rm’s expansion and distributiondecisions.Recent studies of investment have stressed the importance of frictions

arising from irreversibility and %nancial market imperfections, see Caballero(1999), and Hubbard (1998) for surveys. However, existing work considersthe impact of each such friction in isolation. Both irreversibility and %nancialconstraints can (separately) account for a number of features of investmentbehaviour. 1 This embarras de richesses is potentially problematic. Failure tocontrol for ‘the other’ constraint in existing theoretical and empirical workcasts doubt on (i) our ability to ascribe investment responses to particularcauses, (ii) the magnitudes of estimated eEects and (iii) the validity of theassociated policy recommendations. Also, in the presence of %nancial con-straints, in contrast to the results of Modigliani and Miller (1961), dividendand investment activity are both circumscribed and interdependent. This islikely to be particularly important when %nancial constraints and irreversibil-ity occur together, since funds committed to distribution cannot be used forinvestment, and vice versa. Once again the investment and dividend policyliteratures proceed in splendid isolation. The main focus of the literature on%rms’ dividend policies has been to identify the rationale for and determinantsof dividend policy without directly considering the interaction with investmentdecisions, see Allen and Michaely (1995) for a survey. In the existing liter-ature on investment, dividend policy is typically considered only as a meansof identifying %nancially constrained %rms.Existing results will remain potentially compromised until we understand

%rm behaviour when both constraints may bind simultaneously. Also, withouta theoretical framework which encompasses jointly constrained behaviour itis diHcult to derive testable implications which could discriminate betweenthe two classes of friction. These concerns motivate this paper’s focus onthe interaction of investment, retention and distribution decisions under ir-reversibility and %nancial constraints. I %nd that the %rm behaviour can becharacterised in terms of a life-cycle model: a young (small) %rm will %nd itoptimal to invest; not unless it avoids liquidation and grows will it choose to

1 Irreversibility and %nancial constraints can each explain the history dependence of investmentdecisions; the existence of hurdle rates for investment and periods of inactivity (threshold eEectsand nonlinearities); the dominance of quantity variables over price variables in investmentequations; and the volatility of aggregate investment (both why it is so high and why it islower than that predicted by a frictionless model).

R.W.P. Holt / Journal of Economic Dynamics & Control 27 (2003) 467–502 469

payout. This behaviour results from the complementarity of the irreversibilityand %nancial constraints.The intertemporal nature of these issues drives the choice of modelling

framework. Often in theoretical work on investment and dividend policy,%nancial market imperfections arise through informational asymmetries dueto selection or hidden action problems. Nonetheless, empirical tests of impli-cations of these models adopt an intertemporal framework, in which %nancialconstraints are imposed exogenously, Hubbard (1998). Equally, in the ir-reversible investment literature, informational asymmetry (in the form of alemons problem) is cited as a source of the secondary market imperfection,yet to clarify the dynamic implications of these features an intertemporalframework is used in which the irreversibility constraint is exogenously im-posed on %rms, see for example Dixit and Pindyck (1994). I follow thisapproach and impose constraints exogenously; hence, this paper can mosteasily be interpreted as an extension of the canonical irreversible investmentmodel.The irreversibility constraint is standard, as in Bertola (1998). Financial

constraints, following Milne and Robertson (1996), are captured through theunavailability of external funds combined with the threat of liquidation shouldthe internal funds fall too low. In Milne and Robertson’s model, symmetricconvex costs of investment allow a %rm to manage its physical capital ata limited cost to mitigate the threat of liquidation in the face of low in-ternal funds. Once investment is irreversible, such a behaviour is not fea-sible. Thus, a key diEerence between the current paper and that of Milneand Robertson (1996) is that irreversibility and %nancial imperfections dis-play complementarity, this complementarity underpins the main results of thepaper. 2

In such an environment a %rm is found to exhibit life-cycle type be-haviour: when small (young), it is optimal to invest (for growth); payoutis only optimal if it avoids liquidation and grows, this appears consistentwith the observed behavoiur. These insights arise from the analytic char-acterisation of the investment and dividend thresholds, which are separate,non-linear and concave in capital stock. I link this directly to the concavityof the revenue function in capital stock, in conjunction with the constraintcomplementarity. Moreover, these results seem likely to be robust to alterna-tive (richer) operating environments precisely because the revenue concavityand constraint complementarity properties are likely to be preserved. Another

2 Such complementarity is likely to be robust provided investment is subject to non-convexadjustment technology, such as the costly reversibility discussed in Abel and Eberly (1996).This is because the capital loss involved in selling an equipment to the secondary market willbe more severe under non-convexities than under symmetric convex costs (considered by Milneand Robertson) and will limit the scope for managing physical capital to oEset movements ininternal funds.


important contribution of the paper lies in documenting the consequences forempirical work on investment of the juxtaposition of irreversibility and %-nancial constraints. The discussion highlights the importance of controllingfor one constraint when examining the impact of the other. In particular, theimpact of the complementarity of these constraints suggests that it is betterto work with data where one constraint is unlikely to be present.The next section outlines a model of %rm behaviour under irreversibility

and %nancial constraints. In the following section, properties of the %rm’svalue function and the optimal policy are investigated. Implications for em-pirical work are discussed in Section 4. The robustness of the results toa series of extensions of the %rm’s operating environment is discussed inSection 5. Section 6 contains a conclusion and some suggestions for futureresearch.

2. The model

This section sets out a simple framework for the analysis of investmentand dividend policy under irreversibility and %nancial constraints. The startingpoint for the analysis is the sources and uses of funds identity. A simplecontinuous time version of this relation may be written as

�(t) dt + dS(t) + dB(t) ≡ dX (t) + (P�(t) I(t) +D(t)) dt: (1)

In words, over any time interval, dt, net pro%ts from operations, �(t), plus thenet issue of equity, dS(t), plus the net issue of debt, dB(t), (which togethercomprise the sources of funds), are identically equal to the change in internalfunds, dX (t), plus expenditure on investment, P�(t)I(t), and dividends paidout, D(t) (these comprise the uses of funds).Let net earnings from operations be �(t) ≡ P(t)Q(t), where Q(t) is out-

put and P(t) is output price. Assume that the %rm’s production function iscontinuous and concave in a single factor, capital equipment, in particular letQ(K)=K . If this set-up is adopted in relation to the sources and uses offunds identity, then even if P� is assumed to be time invariant, there will be%ve state variables: B;K; P; S; X and two controls D, I . Such a structure is toocomplex to facilitate analytic results concerning the %rm’s optimal policies.Rather than obtain numeric results for particular parameter values, I use aseries of simplifying assumptions to obtain analytic results. I consider theimplications of relaxing some of these assumptions in Section 5.First, set the output price, P, and the price of capital equipment, P�, as

constant ∀t, and normalise the former to unity. Normalising P to unity simpli-%es the problem somewhat, but it also omits the prime source of uncertaintyin canonical irreversible investment models, which makes evaluation of the


results more diHcult. Instead, uncertainty is allowed to enter pro%ts directly.The Low of net pro%ts over the time interval dt is then

�(t) dt=K(t) dt + �K(t) dW (t); (2)

where W (t) is a Wiener process.Irreversible investment is frequently motivated by appealing both to as-

set speci%city and to ‘lemons’ problems in the secondary market for capitalequipment. This is used to justify an (exogenous) irreversibility constrainton investment in equipment: researchers do not feel compelled to model theinformational asymmetry directly, but instead impose the constraint and eval-uate its consequences. Equally, %nancial constraints are often seen as theconsequence of informational asymmetries between the %rm and the providersof external %nance, so here I impose an exogenous %nancial constraint, andproceed to evaluate its behavioural consequences.In particular, %nancial constraints can be incorporated into the model by

assuming that the %rm %nds it prohibitively expensive to raise external %-nance. This might occur because of imperfect knowledge amongst prospec-tive equity and debt holders of the current state of and prospects for the %rm.For instance, Hellmann and Stiglitz (2000) outline conditions under which a%rm may be simultaneously credit-rationed and equity-rationed under adverseselection. To be precise, such an assumption restricts new issues of equityand new borrowing. It does not aEect repurchases of equity or redemptionof outstanding debt. To simplify the analysis it is assumed that repurchas-ing and redemption cannot occur; these constraints involve the restrictionsdB(t)=dS(t)=0; ∀t. Under these assumptions, the sources and uses of fundsidentity may be rearranged to give the evolution of internal funds as 3

dX (t)= (K(t) − P�I(t)−D(t)) dt + �K(t) dW (t): (3)

To allow %nancial constraints to bite, assume that the %rm faces automaticliquidation if internal funds, X (t), fall below some threshold. This makesinternal funds a hedge against the threat of liquidation. For simplicity, thethreshold is set at zero, which is consistent with irreversibility of investment,and gives the restriction 4

X (t) ≥ 0: (4)

The concept of the irreversibility of dividend payments is captured throughthe constraint,

D(t) ≥ 0: (5)

3 For simplicity it is assumed that no interest can be earned on internal funds.4 Under these assumptions, since capital has no resale value, the net worth of the %rm equals

its internal funds, X .


Irreversibility of investment is captured by the standard constraint,

I(t) ≥ 0: (6)

The result is that the level of capital stocks can only fall if depreciation ispositive. Suppose, for convenience that the rate of depreciation is zero. Thencapital stock only changes when investment is undertaken, so that the capitalaccumulation equation becomes

dK(t)= I(t) dt: (7)

The analytical advantage of this simpli%cation is that while capital remains astate variable, so that capacity choice matters, the dimension of the diEerentialequation governing V in the continuation region is reduced by one. 5 Inother words, capital stock acts as a scaling variable. This also simpli%es theanalysis of the regions of state space in which the various regimes (inaction,investment, etc.) hold.The %rm’s objective is to maximise the discounted value of a stream of

dividend payments, D, by judicious control of dividend and investment poli-cies. Its value function is given as

V (K(t); X (t)) = max{D(s); I(s)}

Et

[∫ �

te−�(s−t)D(s) ds+ e−�(�−t)V (K(�); 0)

];

(8)

subject to (3)–(7). Here, Et denotes the expectation operator conditional onthe information available at time t; � is the %rst time that the internal fundsfall to zero, so that the %rm enters liquidation and V (K(�); 0)=0 by assump-tion. Note that X (t)=0 forms an absorbing state.The %rm’s behaviour is an optimal resolution of the conLicts between the

desire to pay irreversible dividends to impatient shareholders, the requirementto retain funds within the %rm as a barrier against liquidation, and the desireto boost future pro%ts by undertaking irreversible investment expenditures toincrease future revenues. The impatience of the %rm’s owners is capturedby the assumption that shareholders rate of time preference, �, is strictlynon-negative, �¿0. If this condition does not hold then the %rm would max-imise value by accumulating capital and internal funds into the inde%nitefuture so as to render the threat of liquidation inoperative.

3. Solution to the �rm’s problem

The solution to the %rm’s problem is considered in this section. This in-volves identifying the form of the solution outside and inside the continuation

5 Pindyck (1988) adopts such a procedure to analyse the investment decision of a %nanciallyunconstrained %rm.


region and determining two sets of unknowns: the constants of integration andthe location of the action thresholds (the latter de%nes the shape of the con-tinuation region). The analysis is complicated by the existence of severalrelevant unknown action thresholds associated with payout, investment andjoint action regimes and by the complementarity of irreversibility and %nancialconstraints. First, the Hamilton–Jacobi–Bellman (HJB) equation governing Vis determined. Then the payout and investment thresholds are characterisedunder restrictions on the curvature of the value function. Finally, value func-tion itself is identi%ed, conditional on the thresholds.

3.1. The Hamilton–Jacobi–Bellman equation

The HJB equation satis%ed by the %rm’s value function is obtained by thestandard approach of expanding the integral (8) over an interval [t; t + dt),during which the %rm does not enter liquidation, using the evolution equations(3) and (7), taking the limit as dt → 0 using Ito’s lemma, and dividing bydt to obtain

�V (K(t); X (t))= max{D(t); I(t)}

�2K(t)2

2 VXX (K(t); X (t))

+K(t) VX (K(t); X (t))

−(P�I(t) +D(t))VX (K(t); X (t))

−I(t)VK(K(t); X (t)) +D(t)

: (9)

To proceed further, it is necessary to undertake the maximisation withrespect to I and D in order to determine the optimal policy rules for thesevariables. In addition, it is important to understand the conditions under whichthe optimal policies are feasible. These issues are discussed next.

3.2. The structure of optimal dividend and investment policy

In this sub-section it is shown that, under restrictions on the value functiongiven in Conjecture (1), the investment and payout thresholds are boundedand non-negative.

Conjecture 1. Suppose that the value function V (K; X ) is twice continuouslydi<erentiable with derivatives VK¿0; VKK¡0; VX¿0; VXX¡0 and VX (K; 0)¿1; VK(0; X )¿P�.

These conditions are reasonable. VK¿0; VKK¡0 and VK(0; X )¿P� arestandard in models of irreversible investment under perfect capital markets;VX¿0; VXX¡0 and VX (K; 0)¿1 require the marginal value of internal fundsis positive, decreasing in the level of those funds and, if internal funds arescarce, greater than the marginal value of paying out funds. This condition


holds in Milne and Robertson’s model. 6 Under Conjecture (1) the structureof the optimal policies is summarised in the following statement: 7

Proposition 2. Consider a (rm with value function (8) subject to the evo-lution equations (3) and (7) and constraints (4); (5) and (6). Assume thatthe value function satis(es Conjecture (1); then (a) Conditioning on thelevel of capital stock; the optimal dividend policy takes the form: ∃XD(K)s.t. D=0; ∀X ≤ XD(K); with instantaneous barrier control at X ≥ XD(K);where XD(K)∈ (0;∞). (b) Optimal investment for a given capacity takes theform ∃KI s.t. I =0; ∀K ¿KI ; with instantaneous barrier control at K ≤ KI ;where KI ∈ (0;∞).

Proof. See appendix.

This result indicates that the optimal investment threshold bounds K frombelow, while the payout threshold bounds X from above. Since K is %xed inthe absence of investment activity, it is simpler, from a notational viewpoint,to write the investment threshold as XI ≡ XI (K). The optimal policy is thatthe %rm invests when X¿XI (K). Proposition 2 indicates that state space isdivided into a number of regions which the %rms %nd it optimal to do oneof a number of things: (i) do nothing (X¡XD(K); X¡XI (K)), (ii) invest(X¡XD(K); X ≥ XI (K)), (iii) payout (X ≥ XD(K); X¡XI (K)), (iv) payoutand invest (X ≥ XD(K); X ≥ XI (K)), and (v) enter liquidation X =0. 8 Solu-tion of the %rm’s problem involves both identifying the regions of state spacewhere these regimes hold and determining the rules governing transitions be-tween regimes; so the optimisation problem determines an optimal hierarchyof actions. The novel feature, and the key reason that the threshold rules andthe optimisation problem diEer both from those in the canonical irreversibleinvestment model, Bertola (1998) and from those in Milne and Robertson’s

6 However, it is necessary to verify Conjecture 1. A direct analytic proof is not feasible,since the value function must be computed numerically. Nonetheless, it is a simple matter toverify numerically that the solution to the %rm’s problem outlined in this section has theseproperties throughout the inaction region for various sets of parameter values (on a %nite gridin (K; X )-space).

7 This proposition diEers from Milne and Robertson (1996) in that statement (a) generalisesMilne and Robertson’s statement to allow for the dependence of the dividend threshold onK; ∀K¿0, not just a subset of possible values. Statement (b) has no counterpart in Milne andRobertson (1996), since they do not consider irreversible investment.

8 In the empirical literature examining investment under %nancial constraints, dividend policyis sometimes used as a means of classifying %nancially constrained %rms (low or zero dividend%rms are seen as %nancially constrained). In the context of the model in this paper, any %rmpaying dividends is, de facto, %nancially unconstrained, while %rms are %nancially constrainedto the extent that the ratio of X=K falls short of that which triggers dividend payouts at thatscale of operation.


(1996) threat of liquidation model is that the irreversibility and %nancial con-straints complement each other and hence investment and dividend policy areinterdependent. 9 The full solution requires boundary conditions, which arediscussed in the next sub-section.

3.3. Optimal policies, action thresholds and boundary conditions

Further progress towards understanding the %rm’s behaviour requires thatthe constraints under which it operates be mapped into boundary conditionsat the edge of the continuation region. 10 There are two classes of boundaries%xed and free, which are analysed in turn.Fixed boundaries, whose locations are known, arise from the existence

conditions for the %rm. At liquidation X =0, then V (K; 0)=0; ∀K¿0, sincecapital has no resale value. This is an absorbing state. Also, since a %rmwith neither physical capital nor internal funds is of no value to its own-ers, the required condition at K =X =0 is V (0; 0)=0. These conditions aresummarised as

V (K; 0)=0; ∀K¿0;

V (0; 0)=0: (10)

The boundary conditions at the free boundaries indicate when and where itbecomes optimal to undertake investment and distribution (rather than reten-tion) under the behavioural constraints, (5) and (6). Along the payout actionthreshold, the condition is VX (K; X )=1 and along the investment thresholdVK(K; X )=P�VX (K; X ). These (necessary) ‘smooth-pasting’ conditions fol-low from the sample path continuity of Brownian motion, but are insuHcientto determine the optimal policies as they do not identify the location of theaction thresholds. SuHcient conditions are given by the second derivatives ofthe value function along the respective thresholds: 11

VX =1;

VXX =0;X ≥ XD(K); (11)

9 If the constraints were independent, then the XD(K) would be horizontal in (K; X )-space,whilst XI (K) would be vertical. The extent of the interdependence is displayed in Fig. 2, andwill be explained in Section 4.10 Once the %rm is in the continuation region, regulation (investment and=or dividend payment),

prevents it from leaving except through liquidation. For the present, assume that the initial(K; X ) pair lies inside the continuation region. The situation where the initial (K; X ) lie outsidethe continuation region is dealt with in Section 3.7.11 Since the form of such conditions is standard within the literature, Dixit and Pindyck (1994),

proofs are omitted.


VK =P�VX ;

VXK =P�VXX ;X ≥ XI (K): (12)

It is worth pausing to interpret these conditions. Consider the payout de-cision. For 0¡X¡XD, an extra unit of operating pro%t is worth more to the%rm held as internal funds, X , to stave oE the threat of liquidation, than dis-tributed as dividends to the owners. That is VX (K; X )¿1. When the marginalvalue of cash balances falls to the marginal value of dividends, VX (K; X ) = 1,and dividends are paid. Next, consider the investment decision. Recall thatmarginal q is given as the ratio of the marginal value product of capital to thereplacement cost of capital, q ≡ VK=P�. In the standard irreversible investmentmodel, the %rm invests when q rises above unity. 12 Here, the %rm investswhen q rises to the marginal value of internal funds, q=VX (K; X ). When0¡X¡XI , the %rm does not invest since its marginal valuation of the extraunit of capital is less than the marginal unit of capital’s value at replacementcost, weighted by the marginal value of internal funds. In other words, itis only worth investing when the capitalised revenue Lows associated withthe marginal unit of capital exceeds the purchase price of new capital equip-ment weighted by a factor measuring the value of (the detrimental eEect ofinvestment on) the incremental unit of internal funds.It is, of course, possible that a %rm can choose to pay dividends and un-

dertake investment simultaneously. Then substituting the boundary conditionsat the payout threshold, Eq. (11), into those at the investment threshold, Eq.(12), three conditions summarise the optimal policy under joint action,

VK =P�;

VKX =0;

VXX =0

(K; X ) | (K; X )∈R2+; X ≥ XI (K)=XD(K)=XID(K): (13)

Having reexpressed inequality constraints (4)–(6) as boundary conditions onthe %rm’s value function, the next step is to determine some general propertiesof the value function.

3.4. Some general properties of the value function

The main result of this section lies in the identi%cation of the form of the%rm’s value function, while subsidiary results determine the elements of thisfunction.

12 This marginal q-based investment rule is the value of the (rm form discussed by Dixitand Pindyck (1994). Although this looks much like the rule used in frictionless models ofinvestment, those models neglect the fact that the marginal value product of capital accountsfor the eEects of exercising the option to invest.


The boundary conditions outlined in Section 3.3 may be substituted backinto (9), giving a diEerential equation in V . To recap: the structure of the%rm’s problem has been simpli%ed by assuming that capital assets do notdepreciate either over time or with use, that internal funds, rather than productdemand, is the source of innovations and that no interest can be earned oninternal funds. The %rst two assumptions allow the problem to be modelledin two dimensions by solving a linear homogeneous second order ODE in X ,Eq. (14), with the other state variable, capital, acting as a scaling factor. The%nal assumption ensures that this ODE is of constant coeHcient form. Overthe continuation (inaction) region, for which constraints (5) and (6) hold asequalities, this diEerential equation is

�2K2

2VXX + K VX − �V =0; (14)

subject to boundary conditions (10)–(12). The general solution of Eq. (14)takes the form

V (K; X )=A1(K)e�1X + A2(K)e�2X ; (15)

where �i; i∈{1; 2}, are the roots of the characteristic polynomial

�2

2K2 �2 + K � − �=0:

That is,

�i=−1±

√1 + 2��2

�2K ; i∈{1; 2}: (16)

Taking �1 as the positive root of the characteristic equation, since �; �; K ≥ 0,it follows that |�2|¿|�1| and �2¡− 1=�2K ¡0¡�1: The characteristic rootscan be written as functions of K; �i ≡ �i(K); i∈{1; 2}.From Eqs. (10) and (15), it follows that A1(K)=− A2(K), thus

V (K; X )=A1(K)[e�1(K)X − e�2(K)X ]: (17)

To determine the solution fully, the constants of integration and locations ofthe action thresholds must be determined under the diEerent regimes. Theaction thresholds are identi%ed in Section 3.5; the constants of integration arediscussed in Section 3.6.

3.5. Identi(cation and characterisation of the action thresholds

The aim of this section is to express XI , and XD as functions of K , anddetermine the slope and curvature of these functions. This information deter-mines the locations of the diEerent regimes. The behaviour of the investmentand payout thresholds are considered separately.


3.5.1. The payout thresholdThe payout threshold is obtained by imposing boundary conditions (11) on

the value function (17). A closed form representation is obtained. The resultsare summarised as follows:

Proposition 3. When the value function of the (rm outlined in Section 2satis(es Eq. (17) subject to Eqs. (10)–(13); the payout threshold is globallyconcave in K; ∀K¿0, and is given by

XD(K)

=�2 ln [(1+(1+2��2)+2

√1+2��2)=(1+(1+2��2)−2

√1+2��2)]

2√1+2��2

K :

(18)


The curvature of the payout threshold plays a key role in determining theregime (payout or investment) under which the %rm operates.

3.5.2. The investment thresholdThe analysis of the investment threshold is more convoluted than that for

the payout threshold because it is only possible to obtain implicit expressionsfor XI (K). One can solve this numerically, but to determine the location ofthe investment threshold directly, extra information is used drawing on the(boundary) conditions which hold under joint action. This allows the locationof the investment threshold to be determined at two points in (K; X )-space.The main result of this section is that for given parameter values, joint actionoccurs only at (0; 0) and a unique non-negative pair (KID; XID).First, an implicit function de%ning the investment threshold is derived.

Given an initial level of capacity K : XI (K)¡XD(K), the free boundary,XI (K) is determined by imposing boundary conditions (12) on value function(17).

Lemma 4. When the value function of the (rm outlined in Section 2 satis-(es Eq. (17) subject to Eqs. (10)–(13); the investment threshold is de(nedimplicitly by

G(K; XI) ≡

�′1(K)e

(�1(K)−�2(K))XI + �′2(K)e(�2(K)−�1(K))XI

+(�1(K)− �2(K))(�′2(K)− �′1(K))XI+(�1(K)− �2(K))2P� − (�′1(K) + �′2(K))

=0: (19)



Fig. 1. The G(K; X ) surface. P� =1; =0:7; �=0:5; �=0:1.

G(K; XI) represents the points for which the general surface G(K; X ) isequal to 0. 13 Noting the explicit functional form for output Q ≡ K ; 0¡ ¡1,write G(K; X ) as

G(K; X )=

[C1e!XK

− K−( +1) + C2e−!XK

− K−( +1)

+!2P�K−2 + !2XK−(2 +1) + C3K−( +1)

]; (20)

where C1 =− [(−1+√1 + 2��2)=�2]¡0; C2 =− [(−1−

√1 + 2��2)=�2]¿0;

C3 = [ − 2 =�2]¡0; !=2√1 + 2��2=�2. Fig. 1 illustrates the behaviour of

G(K; X ) over a region of state space for the baseline case �=0:1; �=0:5; =0:7; P�=1. The behaviour of the surface G(K; X ) determines the exis-tence of the investment threshold. It was not possible to determine a closed

13 The notion of using the boundary conditions to de%ne an implicit function recalls Abeland Eberly’s (1996) study of costly reversible investment. However, in that paper, the implicitfunction de%ned the ratio of the threshold values at acquisition and disposal as a function ofprimary to secondary market price (the degree of irreversibility). Here, the investment thresholdXI is directly, if implicitly, identi%ed by K .


form solution to the equation G(K; X )=0 for 0¡X¡∞. The asymptoticproperties of this surface are considered in the appendix and used to deducethe existence of the investment threshold.Although, full characterisation of the investment threshold is only feasible

numerically, analytical methods may be used to determine the location ofthe investment threshold at the joint action points. That is the points wherethe investment and dividend thresholds intersect. To analyse this situation,simplify expression (20) by multiplying throughout by K +1 to give

H (K; XI) ≡[C1e!XIK

− + C2e−!XIK

−

+!2P�K1− + !2XIK− + C3

]=0: (21)

For XI =0, it follows that K =0, since

C1 + C2 + C3 + !2P�K1− =!2P�K1− =0:

Note that when K = 0 the payout threshold is also zero, XD(K) = 0. So thepayout and investment thresholds intersect at (K; X )= (0; 0). This and otherpoints of intersection of the thresholds XD(K) = XI (K) = XID(KID), can bedetermined by substituting for XI (KID) in Eq. (21) using Eq. (18):

XD(KID)=C4

!K ID (22)

where C4 = ln[(1 + ��2 +√1 + 2��2)=(1 + ��2 −

√1 + 2��2)]. Then

H (KID)≡

C1[(1+��2+

√1+2��2)=(1 + ��2 −

√1 + 2��2)]

+C2[(1 + ��2 −√1 + 2��2)=(1 + ��2 +

√1 + 2��2)]

!2P�K1− ID + !C4 + C3

=0:

This non-linear equation has two solutions KID0 = 0 and

KID1 =

−

C1

[1+��2 +

√1+2��2

1+��2−√

1+2��2

]+C2

[1+��2−

√1+2��2

1+��2+√

1+2��2

]+ !C4 + C3

!2P�

1=1−

:

(23)

It is not possible to determine the sign of this expression analytically, butnumerical analysis showed that KID1¿0; ∀�∈ (0; 0:25); �∈ (0; 1): allowing abroad range of parameter values.


3.5.3. Action threshold curvature and the optimal regimeThis section demonstrates that the optimal policy partitions state space into

two regions, on the basis of the scale of operation, K , in which diEerent policyregimes are relevant. Investment is the preferred action (whenever action isoptimal) ∀K ∈ (0; KID1), and payout is the preferred action (whenever action isdeemed optimal) ∀K ∈ (KID1 ;∞), while joint action occurs only at K =KID1 .This result is established by comparing the elasticities with respect to K ofthe investment threshold and the dividend threshold at the joint action point(KID1 ; XID1).

Proposition 5. For K ∈ (0; KID); XI (K)¡XD(K); that is investment is thepreferred action; and for K ∈ (KID;∞); XD(K)¡XI (K); that is payouts arethe preferred action.

Proof. Since the dividend and investment thresholds intersect only at (K; X )=(0; 0) and (K; X )= (KID1 ; XID1), it must be the case that at (KID1 ; XID1) eitherthe investment threshold cuts the dividend threshold from below or vice versa.It follows that a change of regime occurs at K =KID1 . The relative slopes ofthe investment and dividend thresholds at (KID1 ; XID1) determine which regimedominates for K ? KID1 . These relative slopes are captured in the elasticitiesof the thresholds with respect to K at K =KID1 .The elasticity, dXD(K)

dKK

XD(K)of the payout threshold, XD(K), with respect to

K is

#D=dXD(K)dK

KXD(K)

∣∣∣∣(K;X )=(KID1 ;XID1 )

= ; ∀K¿0:

For the investment threshold, apply the implicit function theorem to Eq. (21)to obtain

dXI (K)dK

≡ !C1XIK−1e!XIK− − !C2XIK−1e−!XIK

− + 2!2XIK−1 − !2(1− )P�

!C1e!XIK− − !C2e−!XIK

− + !2:

(24)

Then, since at the joint action point XI (KID1)=XID1 (KID1)=XD(KID1), theelasticity of the investment threshold at the joint action point can be writtenas

dXI (K)dK

KXI (K)|(K;XI )=(KID1 ;XID1 )

=dXI (K)dK

KXD(K)|(K;X )=(KID1 ;XID1 )

≡ #I :


So,

#I =

!C1XD(KID1)K

−1ID1

e!XD(KID1 )K− ID1

− !C−12 XD(KID1)K

−1ID1

e−!XD(KID1 )K− ID1

+ 2!2XD(KID1)K−1ID1

− !2(1− )P�

!C1e!XD(KID1 )K

− ID1 − !C2e

−!XD(KID1 )K− ID1 + !2

KID1

XD(KID1): (25)

Substituting in Eq. (25) for XD(KID1) and KID1 using Eqs. (22) and (23) andrearranging gives

#I = +(1− )

[C1

[1+��2+

√1+2��2

1+��2−√

1+2��2

]+ C2

[1+��2−

√1+2��2

1+��2+√

1+2��2

]+ !C4 + C3

]C1eC4 − C2e−C4 + !

:

(26)

So, the diEerence between the elasticities of the investment and payout thresh-olds at the joint action point is

#I − #D

=(1− )

[C1

[1+��2+

√1+2��2

1+��2−√

1+2��2

]+ C2

[1+��2−

√1+2��2

1+��2+√

1+2��2

]+ !C4 + C3

]C1eC4 − C2e−C4 + !

≡ ND:

It follows from the discussion surrounding Eq. (23) that the numerator, N ,of this expression is negative (at least ∀�∈ (0; 0:25); �∈ (0; 1)). Moreover,the denominator is also negative. To observe this, substitute for C1; C2 and! and express the denominator, D, directly as a function of ; �; �:

D=

− [−1+

√1+2��2

�2

] [1+��2+

√1+2��2

1+��2−√

1+2��2

]

− [−1−

√1+2��2

�2

] [1+��2−

√1+2��2

1+��2+√

1+2��2

]

+ 2√

1+2��2

�2

:

Some algebraic manipulation gives the result

D=− 4√1 + 2��2

�2=− 2!¡0:

This establishes that #I − #D¿0.

So, at the positive crossing point for the payout and investment thresholds,the latter cuts the former from below; for K ∈ (0; KID); XI (K)¡XD(K)—investment is the preferred action, and for K ∈ (KID;∞); XD(K)¡XI (K)—payouts are the preferred action. This is illustrated in Fig. 2.


Fig. 2. Intersection of the dividend and investment thresholds. P� =1; =0:7; �=0:5; �=0:1.

3.6. The value function in the continuation region

The %rm’s value function over the continuation region follows the generalform of Eq. (15). Its precise form is determined by whether XD R XI atany particular value of K . The two solutions are tied together at K =KIDby the matching conditions at the joint action point XID(KID). The analysisdepends crucially on whether the endogenous boundary to the continuationregion is the investment threshold or the payout threshold. That is whether,for a given K ∈ [0;∞); XD R XI . Note that once the capacity is %xed at anylevel it can only stay unchanged or grow. This growth occurs intermittently atthe points in time at which investment occurs. With the capacity constant, the%rm moves up and down a vertical line (K; X ) |X ∈ (0;min{XI (K); XD(K)}) asX (t) evolves according to evolution (3). First, the case XD¡XI isconsidered.

3.6.1. The value function when payouts would be chosenUsing Eq. (18) substitute for XD in the smooth pasting condition (11) and

rearrange to give the following expression for the constant of


Fig. 3. The value function, V; and its components as functions of X . P� =1; =0:7;�=0:5; �=0:1; K =500.

integration, A1(K),

A1(K)=(�1(K)�2(K)

)2 e−�2(K)=(�1(K)−�2(K))�1(K)e− �2(K) ¿0: (27)

For a concave production function, A1(K) is also concave in K . SubstitutingEq. (27) in Eq. (17) gives an analytic expression for the value function:

V (K; X )=(�1(K)�2(K)

)2 e−�2(K)=(�1(K)−�2(K))�1(K)e− �2(K) [e�1(K)X − e�2(K)X ]; (28)

∀(K; X ):K ∈ [KID1 ;∞); 0 ≤ X¡XD(K)¡XI (K). The typical shape of thevalue function V (K; X ), and its components, as a function of X is displayedin Fig. 3. 14 The dashed line is the complementary function in the positivecharacteristic root. The term below the K-axis is that in the negative char-acteristic root. The value function (the sum of these two) lies in betweenthem and is always non-negative. V (K; X ) is de%ned in this way for K ≥KID; X ∈ (0; XD(K)); the baseline parameter case is illustrated in Fig. 4.

14 The parameter values used in this exercise are �=0:1; �=0:5; =0:7; P� =1 and K =500.


Fig. 4. Value function for K¡KID (investment) and K¿KID (payout). P� =1; =0:7;�=0:5; �=0:1.

3.6.2. The value function when investment would be chosenBoundary conditions (A.4) and (A.5) form a system of (non-linear) equa-

tions allowing both the investment threshold and the constant of integra-tion to be determined implicitly. Eq. (A.6) forms a linear homogeneous (rstorder ODE for the constant of integration A1 in K of the form A′

1(K) +m(K)A1(K)=0. This is reproduced below:

A′1(K) +

[�′1(K)XIe

�1(K)XI − �′2(K)XIe�2(K)XI

−P�(�1(K)e�1(K)XI − �2(K)e�2(K)XI )

][e�1(K)XI − e�2(K)XI

] A1(K)=0: (29)

Eq. (29) holds over the region (K; X )∈ (0; KID1 ]×(0; XI (K)] subject to,

A1(KID1)=(�1(KID1)�2(KID1)

)2 e−�2(KID1 )=(�1(KID1 )−�2(KID1 ))�1(KID1)e− �2(KID1)

; (30)

which ensures the continuity of the value function at the joint action point.Eq. (29) can be solved directly by multiplying throughout by a suitableintegrating factor and integrating out. The integrating factor is de%ned as&(K)= exp{∫ m(K) dK}. Simplifying m(K) in (29), &(K) can be


expressed as

&(K)= exp

∫ �′1(K)XI (K)− P��1(K)

−[ !XI (K)K +1 + !P�K ]

1e!XI (K)K− −1

dK

: (31)

The solution A1(K) is then given as C=&(K), where C is a constant determinedby condition (30). This allows V (K; X ) to be determined implicitly over theinvestment threshold as

V (K; X )=A1(K)(e�1X − e�2X ):

This is computed directly. First, choose ; �; � and solve for both KID1 usingEq. (23), and XI (K) using Eq. (21). Then obtain values for A1(K) by solvingEq. (29) subject to the initial condition (31). This is achieved using a RungeKutta scheme adapted from Press et al. (1992). The resulting value functionholds over the range K ∈ (0; KID); X ∈ (0; XI (K)); V (K; X ) is illustrated forthe baseline parameter value case in Fig. 4.

3.7. Behaviour outwith the continuation region

To complete the solution to the %rm’s problem it is necessary to considerthe behaviour outside the continuation region. For instance, recalling that theinvestment threshold occurs at a strictly positive K (Proposition 2b), imaginethat a %rm initially has no capacity, K =0, but high internal funds X¿0. Itcould potentially be better oE by investing a fraction (to improve its futuredividend paying capacity) and so increase shareholder value above X .In general, outside the continuation region the %rm solves the problem:

V (K; X )= maxRX;RK

[RX + V (X −RX − P�RK;K +RK)];

where (K; X ) are outside the continuation, RX is the amount of dividendspaid, RK is the amount of investment undertaken and the value functionon the right-hand side is on either of the action thresholds. Let KID denotethe scale of operation at the intersection between the investment and payoutbarriers, then if K¿KID payout is optimal and this brings (K; X ) immediatelyto (K; X − RX ) which is the payout barrier, while if K ≤ KID the optimalpolicy could be invest only, or investment plus payout depending on thevalue of X . The complete description of all the actions the %rm can take isillustrated in Fig. 2.

4. Interpretation and implications of the model

The main results of the model, and the mechanisms underlying these resultsare considered in this section along with implications arising for empiricalwork.


The main theoretical results outlined above concern the separation of theaction thresholds for investment and dividend activity. In particular, for K¡KID1 , investment is the preferred action, whenever activity occurs; for K =KID1 ,the %rm is indiEerent between investment and dividends so that if actionoccurs investment and dividend payments occur jointly; %nally, if K¿KID1

dividend payment is the preferred activity should activity occur. These re-sults clarify the role of expansion in determining when it is optimal to switchregimes. They are consistent with a life-cycle interpretation of %rm behaviour,in which young, small %rms prefer to take advantage of growth opportunitieswith the prospect of higher future dividends, while mature, %nancially secure,%rms pay dividends.The results explain both the endogenous hierarchy with diEerent regimes

(combinations of investment, payout and inaction) which characterise the be-haviour of the %rm in diEerent regions of state space and the rules governingtransitions between those regimes. This hierarchy is based on the marginalcosts of these actions in relation to each other and reLects the price of capitalequipment, the value of dividends and the marginal valuation of internal fundsand of capital. In this respect, the optimal dividend and investment decisionmodelled here bear some resemblance to the hierarchy of activities observedin models of multifactor demand in the presence of non-convex adjustmentcosts, see Eberly and Van Mieghem (1997).These results depend ultimately on the concavity of the revenue function in

capital stock, and on the complementarity of the irreversibility and %nancialconstraints. This complementarity can be straightforwardly understood. If a%rm faces symmetric convex costs of adjusting physical capital stocks, thenit can manage these prudently to ward oE the threat of liquidation when in-ternal funds are low. This is what happens in Milne and Robertson’s (1996)model. By contrast, if capital investment is irreversible, such prudent man-agement is not possible once the capital is in place, leading to greater cautionin the investment decision. 15 The results show that the juxtaposition of ir-reversibility and %nancial constraints yields a greater impact on investmentactivity than either constraint individually. Moreover, %nancial constraints ofthe form considered here may well be more prevalent when the %rm under-takes irreversible investments: a %rm should be less able to raise external%nance if the projects it undertakes require irreversible investment expen-ditures, since potential providers of external %nance will be less willing tosupply funds when the project is irreversible.

15 It is likely that this would carry over to models of costly reversibility based on non-convexadjustment technology, as in Abel and Eberly (1996). This is because the non-convexity willlead to a capital loss if resale to the secondary market is undertaken, and this engenderscaution in the initial investment decision. The analysis of Section 3 suggests that such an eEectis accentuated by %nancial constraints.


The model has implications for empirical work. The possibility raised here,that irreversibility and %nancial constraints may bind simultaneously, suggeststhat in acurately assessing the impact of the constraint in question on %rmbehaviour empirical studies need to %lter out the impact of ‘the other’ con-straint. 16 Otherwise, since both constraints tend to reduce the incentive toinvest, the magnitudes of the eEect of each ‘pure’ constraint may be over-stated, leading to incorrect policy recommendations.For example, one of the tests researchers have devised for the eEects of

irreversibilities on investment activity is via the sign of the investment un-certainty relationship. 17 In the model in Section 3, variations in uncertaintyaEect the incentive to invest through diEerent channels. 18

First, there is a direct eEect of irreversible investment (which many re-searchers expect to be negative if irreversibilities are present). Second, thereis a direct eEect through %nancial constraints, via the precautionary motive,which induces the %rm to display prudence to ward oE the threat of liq-uidation (i.e., prudent behaviour could occur even without irreversibility, asin Milne and Robertson (1996), but would not lead to investment inaction).Finally, there is the indirect eEect of irreversibility, through its complemen-tarity with %nancial constraints, which again reduces the incentive to invest asuncertainty increases. Thus, a negative relationship between variation in un-certainty and investment activity could reLect the pure precautionary motiveor the interaction eEect, and need not reLect the pure irreversibility eEect towhich it is ascribed in existing studies. This suggests that when examining the

16 Researchers who are dubious of the relevance of irreversibilities might point to the evi-dence documented by Hubbard (1998), that linear investment equations (derived from modelswith convex costs of adjustment) are not rejected for those %rms classi%ed as %nancially un-constrained and argue that non-convex adjustment cannot be an important issue. However, asHamermesh and Pfann (1996) discuss, space and time aggregation smooths (and linearises)investment observations, even at the level of the %rm. This may explain the success of linearinvestment equations despite the relevance of non-convexities for investment decisions.17 Direct tests of the impact of irreversibility are diHcult since the main implications of the

model are not for investment itself but for the location of the threshold at which investmentoccurs. Empirical work has focussed on the sign of the relationship between uncertainty andinvestment, with a negative sign being taken as consistent with the option-to-wait eEect underirreversibility and ongoing uncertainty. See Carruth et al. (2000) for a survey of early empiricalwork. Such an approach is problematic since theory both suggests that increases in uncertaintywiden the inaction band by increasing the value of the option-to-wait, and that the increasedvolatility of the underlying process may raise the probability of investment occurring, see Sarkar(2000) for a recent treatment. Potentially, irreversibility may lead increased uncertainty to reduceinvestment in the short-run, in that, if uncertainty increases, the shift in the threshold (wideningof the inaction region) can lead, at least initially, fewer %rms to be in the neighbourhood ofthe (new) investment threshold.18 Sensitivity analysis, omitted from this version of the paper, but available from the author

on request, found that the investment and dividend action thresholds of the model in Section 3both shifted outwards in response to a rise in the level of uncertainty.


implications of irreversibility for investment activity, one should control thepresence of %nance constraints. 19 Similarly, when examining the impact of%nancial constraints on investment activity, one should control the degree ofirreversibility. 20

Finally, consider what testable implications emerge directly from the modeland how one might proceed to examine them. Two key features of modelsbased on non-convex adjustment technologies is the existence of an inac-tion region and of threshold-type behaviour. Due to time-aggregation andspace-aggregation issues, even trying to observe investment zeroes can bediHcult. 21 This appears to make the direct tests of the %ne structure of themodel, such as the level of X=K at the investment threshold at diEerent scalesof operation infeasible, at least using the standard econometric approaches andmost available datasets. An alternative approach which exploits the economicstructure of the model to get a measure of the threshold, without estimating afull dynamic structural model, has been developed by Pindyck and Solimano(1993). The basic idea is to exploit assumptions (about production technol-ogy, market structure, etc.) made in developing the theoretical model, in orderto construct a measure of the marginal value product of capital for a givenunit (over time). 22 The highest values of this measure (over time) give anestimate of the investment threshold for a given unit; one can then examinethe impact of variations in uncertainty (across units) on the threshold value(across units). To date, this approach appears only to have been applied toaggregate and sectoral data, but it could provide a useful approach at the %rmlevel. In the context of the model in Section 3, one would expect to observethat (i) the estimated threshold value of the marginal value product of cap-ital and (ii) the magnitude of the (negative) relationship between (changesin) uncertainty and investment, would both be higher (a) for %rms identi-%ed as %nancially constrained (through their dividend policy) than for thosewhich were %nancially unconstrained, and (b) for those %rms whose invest-ments were primarily irreversible, as opposed to those for which investments

19 Only two recent empirical studies formally control the interaction of the two constraints:Scaramozzino (1997) shows, for a panel of UK %rms, that q-theory holds only for the subset of%rms for which both irreversibility and %nancial constraints are unlikely to be present. Guiso andParigi (1999), using cross-section data on Italian manufacturing %rms, %nd that the impact ofuncertainty on investment is consistent with irreversibility, controlling the existence of %nancialconstraints.20 A straightforward way to do this is to use data on the diEerence between primary and

secondary market prices for the relevant capital equipment’, the author is unaware of anyempirical studies which do so.21 In any case, as the credit rationing literature demonstrates, investment zeroes are not the

sole preserve of %rms with non-convex investment technologies.22 This would be the marginal value of assets in place using the terminology of Dixit and

Pindyck (1994).


were more reversible. 23 This suggests how one might identify the behaviourconsistent with the model, while taking the theoretical structure seriously.The implications of the model suggest a re-appraisal of existing empiri-

cal studies is required, but before accepting such a view, it is important toknow whether these implications are robust to variations of the economicenvironment. This is the subject of the next section.

5. Generalisations

The results on the separation of the action thresholds, on their curvatureand on the endogenously determined hierarchy of actions (established in Sec-tion 3) are obtained in the context of a speci%c restricted model. Naturally,it is important to know how robust these results are likely to be once theserestrictions are relaxed. Here, extensions of the model to allow for depre-ciation of capital, relaxation of the %nancial constraint, explicit treatment ofdemand uncertainty, and non-residual dividend policy are discussed. As ex-plained in the last section, the results of the model appear to be driven bythe assumptions about the concavity of the revenue function in capital andthe complementarity of the two constraints. While each generalisation en-riches the model, the basic results remain intact because the concavity andcomplementarity features are preserved.

5.1. Depreciation

Depreciation would enrich the set of feasible sample paths by allowingmovement between various regimes which, in the restricted model, were in-accessible once other regimes had been entered. For example, with deprecia-tion, capital stocks decline in the absence of investment, so %rms may cyclearound the joint action point, allowing joint action to become a more frequentoccurrence.Also, since investment decisions are eEectively less reversible once depre-

ciation is admitted, investment may occur at lower X for given K (this wouldbe consistent with the literature on irreversible investment without %nancialconstraints), while the increased requirement for investment (to replace de-preciated capital) may lead the dividend threshold to shift outwards. TheseeEects would lead joint action to occur at a higher K , but there is no reasonto expect the investment threshold to become linear or convex in K . Thefact that the concavity of the revenue function and the complementarity of

23 Note that the implications of the model are consistent with the methodology of empiricalwork on %nancial constraints and investment activity, which suggests that the dividend policy,size and age, should be used to classify %rms as %nancially constrained, Hubbard (1998).


the constraints are preserved suggests that the basic separation and curva-ture properties of the thresholds will be maintained—although it is no longerpossible to guarantee that there will only be one joint action point.

5.2. Financial constraints

An important question is whether the results continue to hold on relaxingthe restriction that the %rm has access only to internal funds and faces liq-uidation if these fall too low. Clearly, if external %nance was available atno premium, the %rm’s problem would return to that of the canonical irre-versible investment model. However, suppose that there was an upper bound(low enough to make the problem interesting) on the availability of exter-nal funds—this could be incorporated without introducing extra state vari-ables simply by adding these external funds to the available internal funds,X . This suggests that the problem is a little changed since the constraintcomplementarity and revenue function concavity properties, which drive theresults, remain intact. For such a relaxation to be interesting there must besome reason for a (meaningful) limit to be imposed on the availability ofexternal %nance. One good reason for such a feature is that %rms with highlyirreversible prospective investments are precisely the sort of %rms that thelenders would be reluctant to lend to, because of the losses that the lenderswould face in a bad state of the world. Of course this is just another illustra-tion of the complementarity of the irreversibility and %nancial constraints. 24

5.3. Demand uncertainty

In the model of Sections 2 and 3, demand uncertainty is captured in areduced form through Eq. (2), but this makes direct comparison with the re-sults of the literature on irreversible investment diHcult. For instance, without%nancial constraints, under suitable assumptions about the production and de-mand functions, it is possible to obtain a closed form solution in which thelevel of demand, Z , at which investment is triggered is a linear increasingfunction of scale of operation, K , whereas the investment threshold in Sec-tion 3 determines XI as an increasing concave function of K . It would beinteresting to know whether either or both of these features is preserved ifdemand uncertainty is introduced directly.Incorporating demand uncertainty directly (perhaps with the evolution of

demand modelled as a geometric Brownian motion) introduces a number of

24 If there was no upper bound on the availability of external %nance, but the latter requireda premium over internal %nance, then the equivalence to the problem in Section 3 may notfollow. Nonethless, constraint complementarity would remain important to the extent that %rmsundertaking more irreversible investments face a higher premium on external %nance than %rmswhose undertaking is more reversible.


interesting possibilities. For instance, given the concavity of its revenue func-tion, a %rm with high demand, high internal funds and low capital stock willhave an incentive to invest in order to take advantage of high demand oppor-tunities. 25 On the other hand, a %rm with low demand, high internal fundsand high capital stock, recognising the low marginal value of its growth op-portunities, may be inclined to payout. The presence of high demand (andpro%ts) could allow the %rm to be less concerned about the threat of liq-uidation. These possibilities give some insight into the richness generatedby modelling demand uncertainty directly. On the other hand, this richnesscomes at the cost of analytical tractability—both the investment and payoutthresholds will be surfaces in (K; X; Z)-space, and there is no clear guide as tohow they intersect. Nonetheless, the constraint complementarity and concav-ity features that generated the results in Section 3 are still present, so there iscertainly no reason to expect the investment and dividend thresholds to be su-perposed. The investment threshold, XI (K), holding demand, Z , constant maywell continue to vary non-linearly in the scale of operation. Moreover as thedemand increases, this investment threshold XI (K) could be expected to fallat each value of K; as the higher demand increases the incentive to invest.From a diEerent perspective, holding internal funds, X , constant, the level ofdemand which triggers investment, ZI (K) in the %nancially constrained casewould be expected to lie above that which triggers investment in the %nan-cially unconstrained case in (K; Z)-space, and as the level of internal fundsrises, the investment threshold would be expected to converge towards thatin the %nancially unconstrained case. 26 In short, adding demand uncertaintyenriches the problem, but does not remove the key features that drive theresults.

5.4. Non-residual dividend policy

Some readers might take exception to the characterisation of dividend pol-icy as a residual in %rm’s decisions. 27 Although the dividend policy outlinedabove appears to be consistent with a life-cycle view of the %rm, there areother features of observed payout behaviour that would be interesting to cap-ture. In particular, %rms appear to smooth dividends, and dividends appear to

25 Presumably, the incentive to invest will be attenuated if the process governing demand ismean reverting rather than exhibiting the persistence of a geometric Brownian motion.26 This intuition could be checked by obtaining the numerical solution to equivalent problems

for a %nancially constrained %rm and unconstrained %rm, and plotting the investment threshold(level of demand as a function of scale of operation), both for the %nancially unconstrainedcase and at a variety of diEerent levels of internal funds for the %nancially constrained case.27 In fact, this approach is used in the empirical work on the role of %nancial constraints in

the investment activity, in which dividend policy is used to identify those %rms facing %nancialconstraints, Hubbard (1998), pp. 202–203.


have some information content for equity-holders. One popular explanationfor this is that dividends play a signalling role conveying information aboutthe level and variability of expected earnings, Miller and Rock (1985). In par-ticular, dividend initiations and increases convey good news to the relativelyuninformed equity-holders, while dividend omissions and reductions conveybad news. Moreover, there is some evidence that dividend initiations=increasesconvey information regarding the lower risk of the %rm, Dyl and Weigand(1998). Might not a non-residual dividend policy alter the nature of the in-vestment threshold obtained in Section 3?To begin to capture such a phenomena, one could consider the dividend

initiation decision as irreversible. That is, the %rm either pays no dividends,or commits irreversibly to pay an exogenous fraction (∈ (0; 1) of expectedearnings. In return for this commitment, the %rm, by signalling lower risk,can oEer a lower rate of return, �1 ∈ (0; �0), where �0 is the required rate ofreturn which applies prior to dividend initiation. So in the place of Eq. (5)

D(t)=

{0; �=�0;

(K ; �=�1;

while other Eqs. (2)–(7) remain as before. 28

Signalling models distinguish between the informed %rm and its uninformedowners, and this can be handled by rewriting the problem from the viewpointof the %rm, which now maximises the discounted present value of expectedfuture pro%ts, where, in Sections 2 and 3, it maximised the discounted presentvalue of expected future dividends.The dividend initiation aspect of this problem can be solved in a similar

fashion to an irreversible start up decision for a discrete project (of the formdiscussed in Dixit and Pindyck (1994), Chapter 5). That is, %rst determinethe form of the solution for D=0 (call the value function V 0). Next, deter-mine the form of the solution for D= (K ¿0 (call the value function V 1).Finally, determine the optimal dividend initiation point using value match-ing and smooth pasting conditions (V 0 =V 1, V 0

X =V1X ) to tie the solutions

together. 29 The twist is that within the D=0 and D¿0 regions the %rmsolves an incremental irreversible investment problem with boundary condi-tions given by Eq. (12). The fact that the constraint complementarity and

28 Note that, once dividends are initiated, they are much smoother than the realised net earningsand may increase with scale of operation. Of course, ( should be determined endogenously,but this complication is ignored for simplicity.29 Note that the %rm does not incur a sunk cost to initiate dividends, it undertakes to pay the

dividend. For this reason, such a model is a rather weak characterisation of this commitment.The %rm is committed to paying non-zero dividends, even in the face of liquidation, becausewe assume it to be so. Certainly such a model oEers only a loose resemblance to a signallingmodel. Nonetheless, this conveys the Lavour of a non-residual based dividend policy and allowsdiscussion of the impact of a non-residual dividend policy on investment activity.


revenue function concavity properties are still present, that value functiontakes a similar form (with the addition of a particular solution of the formK =�i, where i∈{0; 1}) and that the boundary conditions are identical to thosein Sections 2 and 3 strongly suggests that the investment threshold will beconcave and increasing in K as in Eq. (19).One unresolved issue is whether, at a given scale of operation, the %rm

would be more or less willing to invest when D¿0 as opposed to when D=0.This is the same as asking whether in (K; X )-space the investment thresholdwhen D¿0 lies above the equivalent threshold when D=0. There are twoeEects to take into account. First, when D¿0, the required rate of return islower: �1¡�0. This would give the %rm a greater incentive to invest, andtend to reduce the investment trigger value, XI (K) ∀K , once the %rm startsto pay dividends (compared with when it pays no dividends). Against this,dividend initiation commits a fraction of expected pro%ts to a particular use,which increases the threat of liquidation, and will tend to make the %rm morewary of committing to irreversible investment. Commitment tends to raise thetrigger value, XI (K) ∀K , when the %rm pays dividends compared with when itdoes not. The magnitude of these eEects depends on the particular parametervalues, but suppose that for expositional purposes the latter eEect dominates,so that the investment threshold, when dividends are paid, lies above thatwhich is obtained prior to the initiation of dividends.In general, at each level of capacity, K , there is some X =XI (K) at which

investments are triggered, and some X =XD(K) at which dividends are initi-ated. In the D=0 region, these thresholds are such that XD(K)¿XI (K) andthe %rm solves the incremental investment problem for D=0. Since internalfunds can grow more quickly when there is no commitment to distributing afraction of earnings, the %rm will, in the absence of liquidation, reach a pointat which internal funds are suHciently high, and the marginal value productof capital suHciently low, such that the %rm can initiate dividend payments.At this point, XD(K)=XI (K)=XID(KID). Following dividend initiation, the%rm solves the incremental investment problem with D¿0, so there maybe a discontinuity in the operational investment threshold at the joint actionpoint. 30

This discussion suggests that, even in the face of a non-residual dividendpolicy the investment threshold will continue to be a concave increasingfunction of K—although it may now exhibit a discontinuity around the div-idend initiation point. Dividend policy will continue to have a life-cycle in-terpretation. The robustness of these features of the model to variations in

30 Unfortunately, it is less easy to obtain analytic results on the joint action point for thedividend initiation problem, because one has to use information on two implicit functions(de%ning the investment threshold (i) without payouts and (ii) with payouts) and the matchingand smooth pasting conditions for dividend initiation, whereas in Section 3 the closed form forthe dividend threshold facilitates analytic results.


the structure of dividend policy once again stems from the complementarityof the irreversibility and %nancial constraints and the concavity of revenuesin K .

6. Conclusion

This paper has considered investment and dividend policy under the com-bined eEect of %nancial constraints and irreversibility. Given the growingempirical evidence for the pervasiveness of each of these constraints sepa-rately the current paper takes the obvious next step and provides a uni%edtheoretical analysis of the dynamic consequences of the constraints and shouldprovide a point of departure for future empirical work. The main lesson todraw from this paper concerns the nature of the interdependency of invest-ment and dividend policy under irreversibility and %nancial constraints. Com-plementarity of these constraints has important consequences for empiricalwork.The key theoretical result is that of the separation of the thresholds. This

arises directly from the concavity of the production function in K in conjunc-tion with the complementarity of the irreversibility and %nancial constraints.These features ultimately generate the hierarchy of activities thresholds whichare endogenously determined, conditional on the location of the %rm withinstate space, on the basis of the costs of each of the activities. As a result ofcomplementarity and concavity of the production function, the optimal policydepends on the scale of operation in a manner consistent with a life-cycleview of the %rm in which small (young) %rms with good growth opportu-nities (high marginal value product of capital) retain funds to expand, whilelarge, secure (mature) %rms, those with low marginal value product of capitaland low marginal value of internal funds choose to pay dividends.The model has some important implications for empirical work. First, the

mere possibility of the joint presence of irreversibility and %nancial con-straints, which is basically ignored in the empirical literature, indicates thatthe magnitude of the eEects ascribed to a single constraint may be mismea-sured, since both constraints would aEect the incentive to invest in a similarmanner. Due to the complementarity of the two constraints, measurement ofthe ‘pure’ eEect of a single constraint is best done in the case of %nancialconstraints by working with data in which investments are as reversible aspossible, and in the case of irreversibilities, by working with %rms that are%nancially unconstrained.The analytic results on the separation of the thresholds were derived un-

der speci%c functional forms and strong versions of irreversibility and %nan-cial constraints. However, the general properties of threshold separation andnon-linearity of the thresholds in K seem likely to be robust to a range of


more complex environments, such as allowing for capital depreciation, relax-ing the %nancial constraint and explicit consideration of demand uncertainty.The robustness seems to follow precisely because in all the generalisationsconsidered, complementarity of the constraints and concavity of the produc-tion function in capital stock are preserved.The results suggest a number of directions for future work. First, as dis-

cussed in Section 4, empirical work on investment should proceed by allowingand controlling for both irreversibility and %nancial constraints and existingresults should be reassessed. Second, the generalisations suggested in Section5 (and others) could be examined more formally using numerical methods.Third, the nature and consequences of the constraint complementarity eEectfor lenders and borrowers decisions could be examined using an asymmetricinformation approach in order to obtain a diEerent perspective.

Acknowledgements

This paper is a substantially revised version of Chapter 5 of my PhDthesis. The research was funded in part by the Royal Economic Society.I would like to thank the anonymous referees, Derek Arthur, Simon Burgess,Robert Chirinko, John DriHll, Stuart Sayer, Noel Smyth, Ian Tonks, DavidWinter, participants at the Conference on Investment Under Uncertainty,Hamburg and The European Economic Association Meeting, Santiago diCompostela for helpful comments and discussions. Any errors remain myresponsibility.

Appendix A.

This appendix contains proofs of various claims in the text.

A.1. Proof of Propositions 2

Proof. (a) Since the costs of control are linear in the extent of dividend pay-ment, the %rm’s optimal dividend policy will take a bang-bang form, (instan-taneous barrier control). Dividend policy will be either to pay no dividendsor to pay at the maximum possible rate. Since the value function is increasingand concave in K and X , there will be a single (continuation) region overwhich it is optimal to retain earnings and pay no dividends. The boundaryof this region will be characterised by some threshold level of internal funds,XD, such that for X¿XD it is optimal for the %rm to distribute internal fundsas dividends. This is because, given the curvature of the value function, themarginal value of internal funds falls as X rises; above the threshold level,


X¿XD, the marginal value of internal funds is less than its value paid outto shareholders. Unless VKX (XD; K)=0; ∀K , the payout threshold, XD is afunction of capacity XD ≡ XD(K).

Next, consider the bounds on the location of the trigger threshold in thepayout region of state-space. Suppose that the %rm sets XD=∞. Then div-idends are not paid out before the end of the planning horizon. In thatcase, since 0¡�, the value of the %rm to the shareholders is V (K; X ) ≤limT→∞ e−�T

∫ T0 K(s)

ds=0. Therefore, the %rm must always do better bysetting XD¡∞, provided it has strictly positive internal funds and capac-ity. Suppose instead that the %rm chooses XD ≤ 0, then it enters liquidationimmediately. It gains nothing from its existing non-negative holdings of phys-ical capital since these have zero resale value, nor is it compensated for theearnings associated with this capacity lost through liquidation. Therefore, adividend policy with XD ≤ 0 is sub-optimal. From Conjecture (1), sinceVX (K; 0)¿1, the initial increment to internal funds is more valuable than dis-tribution, therefore, the %rm chooses XD ∈ (0;∞). This establishes statement(a).(b) The linearity of the value function in the control variables ensures

that the %rm’s investment policy is either to invest nothing or invest at themaximum possible rate (instantaneous barrier control). Since the %rm’s valuefunction is continuous and concave in capacity and internal funds throughoutthe continuation region, there will be a single region in (K; X ) space overwhich inaction is the optimal investment policy and a single region overwhich it is optimal to invest, see Dixit and Pindyck (1994). The boundarybetween the two regions, the threshold KI , will be such that for K¡KI ,it is optimal for the %rm to invest. This follows from the continuity andcurvature of the value function. Investment reduces the marginal value productof capital, so it is only worthwhile investing when capacity is low enough thatthe value of the marginal unit of cash in hand invested in capacity exceedsthe value of retaining funds internally.Next, consider the bounds on the investment trigger. Suppose that KI ≡

K(XI)=∞, then, assuming P� to be strictly non-negative, the %rm inevitablygoes into liquidation trying to meet the cost of the investment expendituresinvolved in attaining the desired threshold level of capital stock. Clearly, thisis not optimal since, on liquidation, the secondhand value of the installedphysical capital is zero. Instead, suppose that KI ≡ K(XI)=0. Then the %rm’svalue is bounded below by the value of its internal funds V (0; X ) ≥ X .Continued operation yields no pro%t, since K =0; ∀t ≥ 0. In this case, giventhe shareholders’ impatience, the %rm’s optimal dividend policy is immediatedistribution. Again, as a consequence of immediate payout, this policy isuninteresting. Provided that VK(0; X )¿P�, given the continuity and curvatureof the value function, it follows that KI ∈ (0;∞). This establishes statement(b).


A.2. Proof of Proposition 3

Proof. First, impose the boundary conditions at the trigger point XD.

VX (K; XD)=A1(K)[�1(K)e�1(K)XD − �2(K)e�2(K)XD ]= 1 (A.1)

VXX (K; XD)=A1(K)[�1(K)2e�1(K)XD − �2(K)2e�2(K)XD ]= 0 (A.2)

Rearranging (A.2) gives a closed form expression for XD as a function ofthe characteristic roots,

XD(�1; �2)=ln((�2=�1)2)�1 − �2 ¿0: (A.3)

Substituting for �i(K), i∈{1; 2}, using (16), yields after some algebra,

XD(K)

=�2 ln[(1+(1+2��2)+2

√1+2��2)=(1+(1+2��2)−2

√1+2��2)]

2√1+2��2

K :

That the payout function is globally concave results from elementary calculus:

dXDdK

= XDK¿0;

d2XDdK2 =

( − 1)XDK2 ¡0;

∀K¿0:

A.3. Proof of Lemma 4

Proof. Write the free boundary conditions (12) at the investment thresholdin full using Eq. (17):

(VK(K; XI−)P�VX (K; XI)

)=

A1(K)

�′1(K)XIe�1(K)XI

−�′2(K)XIe�2(K)XI

−P�(�1(K)e�1(K)XI

−�2(K)e�2(K)XI

)

+A′1(K) [e

�1(K)XI − e�2(K)XI ]

=0; (A.4)


(VKX (K; XI)

−P�VXX (K; XI)

)=

A1(K)

�′1(K)XIe�1(K)XI

−�′2(K)XIe�2(K)XI

+�1(K)�′1(K)XIe�1(K)XI

−�2(K)�′2(K)XIe�2(K)XI

−P�(�21(K)e

�1(K)XI

−�22(K)e�2(K)XI

)

+A′1(K)

[�1(K)e�1(K)XI

−�2(K)e�2(K)XI

]

=0:

(A.5)

Rearranging (A.4), it follows that

A′1(K)=

−A1(K)

[�′1(K)XIe

�1(K)XI − �′2(K)XIe�2(K)XI

−P�(�1(K)e�1(K)XI − �2(K)e�2(K)XI )

]

[e�1(K)XI − e�2(K)XI ]: (A.6)

Substituting for A′1(K) in (A.5), dividing throughout by A1(K); multiplying

throughout by e�1(K)XI − e�2(K)XI gives Eq. (A.7)

0 =

[e�1(K)XI − e�2(K)XI ] ·

�′1(K)e�1(K)XI − �′2(K)e�2(K)XI

+�1(K)�′1(K)XIe�1(K)XI

−�2(K)�′2(K)XIe�2(K)XI

−P�(�21(K)e�1(K)XI − �22(K)e�2(K)XI )

−

[�1(K)e�1(K)XI

−�2(K)e�2(K)XI

]·[�′1(K)XIe

�1(K)XI − �′2(K)XIe�2(K)XI

−P�(�1(K)e�1(K)XI − �2(K)e�2(K)XI )

]

:

(A.7)

Now expanding the expression on the right-hand side of (A.7), collectingterms in e2�1(K)XI ; e2�2(K)XI and e(�1(K)+�2(K))XI ; multiplying throughout bye−(�1(K)+�2(K))XI gives the desired result.

A.4. Asymptotic properties of G(K; X ) and the existence of XI (K)

To provide some insight into the behaviour of G(K; X ) consider its asymp-totic properties.


Lemma 6. The limiting behaviour of the function G(K; X ) de(ned by Eq.(20) is as follows(a) limK→0G(K; X )=−∞; ∀X ∈ (0;∞).(b) limK→∞G(K; X )=0+; ∀X ∈ (0;∞).(c) lim

K;X→∞|XK =*0( XK )=∞; *0 a real; non-negative constant.

(d) limK;X→∞| XK =*1

G(K; X )=0+; *1 a real; non-negative constant.

Proof. (a) Assume that 0¡X¡∞. Consider the asymptotic behaviour asK → 0+. The term C2e−!XK

− K−( +1) tends to zero, as the exponential com-

ponent declines more quickly than the component in K−( +1) expands. Allthe other terms become unbounded asymptotically, but the dominant contri-bution arises from the term in C1e!XK

− K−( +1) which is negative and expands

exponentially. Thus, limK→0G(K; X )=−∞.(b) Take limits in Eq. (20) as K → ∞. All terms in Eq. (20) tend to

zero asymptotically (the one in e−!XK−

decays exponentially). The termsC1e!XK

− K−( +1) and C3K−( +1) are negative, whilst the other terms are pos-

itive. The dominant contribution comes from the term in K−2 ; which isuniformly positive, so limK→∞G(K; X )=0+.(c) Again, suppose that K; X → ∞ such that X=K =*0; is an arbitrary pos-

itive constant. Then X=K =(X=K)K1− =*0K1− . Clearly, limK;X→∞|X=K=*0

(X=K )=∞; therefore, the %rst exponential term in Eq. (20) becomes un-bounded, whilst the other terms tend to zero (at least geometrically) in thelimit. So limK;X→∞|(X=K)=*0G(K; X )=−∞.(d) The %nal asymptotic property of G(K; X ) to be considered here is the

limiting behaviour as K; X → ∞, such that X=K =*1; is an arbitrary positiveconstant. The elements of Eq. (20) containing the exponential terms e!XK

−

and e−!XK−

are then constant and G(K; X ) can be written as

G(K; X )=

[C1e!*1K−( +1) + C2e−!*1K−( +1)

+!2P�K−2 + !2*1K−( +1) + C3K−( +1)

]:

In the limit as K; X → ∞; the dominant term is that in K−2 ; which isuniformly positive. It follows that limK;X→∞|(X=K )=*1G(K; X )=0+.The foregoing analytic arguments provide some insight into the asymptotic

properties of the function G(K; X ); and therefore into its overall shape. Thearguments are also consistent with the particular parameterisation used inFig. 1.

Proposition 7. ∀K¿0; a solution; XI (K); to the equation G(K; X )=0 exists;where G(K; X ) is de(ned by Eq. (20).


Proof. Since G(K; X ) is clearly continuous in K and X on (K; X )∈ (0;∞)×(0;∞); it follows from Lemma 6 that a solution to G(K; X )=0 exists. 31

References

Abel, A.B., Eberly, J.C., 1996. Optimal investment with costly reversibility. Review ofEconomic Studies 63, 581–953.

Allen, F., Michaely, R., 1995. Dividend policy. In: Jarrow, R.A., Maksimovic, V., Ziemba,W.T. (Eds.), Handbook of Operations Research and Management Science: Finance, Vol. 9.North-Holland, Amsterdam, pp. 793–838.

Bertola, G., 1998. Irreversible investment. Research in Economics 52, 3–37.Caballero, R.J., 1999. Aggregate investment. In: Taylor, J.B., Woodford, M. (Eds.), Handbook

of Macroeconomics, Vol. 1. North-Holland, Amsterdam, pp. 813–862.Carruth, A., Dickerson, A., Henley, A., 2000. What do we know about investment under

uncertainty. Journal of Economic Surveys 14, 119–154.Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press,

Princeton, NJ.Dyl, E., Weigand, R., 1998. The information content of dividend initiations: additional evidence.

Financial Management 27, 27–35.Eberly, J.C., Van Mieghem, J.A., 1997. Multi-factor dynamic investment under uncertainty.

Journal of Economic Theory 75, 345–387.Guiso, L., Parigi, G., 1999. Investment and demand uncertainty. Quarterly Journal of Economics

114, 185–227.Hamermesh, D., Pfann, G., 1996. Adjustment costs in factor demand. Journal of Economic

Literature 34, 1264–1292.Hellmann, T., Stiglitz, J., 2000. Credit and equity rationing in markets with adverse selection.

European Economic Review 44, 281–304.Hubbard, R.G., 1998. Capital market imperfections and investment. Journal of Economic

Literature 36, 193–225.Miller, M., Rock, K., 1985. Dividend policy under asymmetric information. Journal of Finance

40, 1031–1051.Milne, A., Robertson, D., 1996. Firm behaviour under the threat of liquidation. Journal of

Economic Dynamics and Control 20, 1427–1450.Modigliani, F., Miller, M.H., 1961. Dividend policy, growth and the valuation of shares. Journal

of Business 34, 411–433.Pindyck, R.S., 1988. Irreversible investment, capacity choice and the value of the %rm. American

Economic Review 78, 163–178.Pindyck, R.S., Solimano, A., 1993. Economic instability and aggregate investment. NBER

Macroeconomics Annual 8, 259–303.Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P., 1992. Numerical Recipes in

FORTRAN: The Art of Scienti%c Computing, 2nd Edition. Cambridge University Press,Cambridge.

31 This analysis does not establish whether a solution to the equation G(K; X )= 0 is unique.Uniqueness cannot be addressed using numerical analysis of the solution for particular parametervalues, since one can only establish uniqueness within a %nite region of (K; X )-space, forparticular (%nite) parameter values, and global results cannot be obtained using these techniques,and in order to proceed implicit expressions for XI (K) and A1(K) must be found.


Romer, D., 2000. Advanced Macroeconomics, 2nd Edition. McGraw Hill, London.Sarkar, S., 2000. On the investment-uncertainty relationship in a real options model. Journal of

Economic Dynamics and Control 24, 219–225.Scaramozzino, P., 1997. Investment irreversibility and %nancial constraints. Oxford Bulletin of

Economics and Statistics 59, 89–108.

investment and dividends under irreversibility and financial constraints

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