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Dynamic behaviour of a value maximizing firm under personal taxation
European
Economic
Review
DYNAMIC
30 (1986)
BEHAVIOUR UNDER
1043-1062.
North-Holland
OF A VALUE MAXIMIZING PERSONAL TAXATION
FIRM
Geert-Jan C.Th. van SCHIJNDEL* Tilburg
Received
University,
October
5000
1984, final
LE
Tilburg,
version
The
received
Nefherlands
November
1985
In this paper we study the impact of both corporate and personal taxation on the optimal dynamic behaviour of a value maximizing firm by determining simultaneously the optimal linancial, investment and dividend policy over a linite horizon. To do so, an optimal control model is formulated and resolved. We discuss the successive stages in the evolution of the Iirm, derive optimal linancial and investment/dividend decision rules and present sensitivity analysis results that are important for understanding the behaviour of lirms in a dynamic setting. The gap between the personal tax rates on dividend and capital gain respectively, turns out to play an important role in providing the optimal policy string.
1. Introduction
The effect of corporate taxes on the market value of a levered firm continues to be a central issue in recent contributions in finance theory. Following the publication of the Modigliani and Miller (1963) tax correction paper, many writers have sought to reconcile the M&M maximum leverage prediction with observed capita1 structure. In ‘Debt and Taxes’, Miller (1977) argues that personal taxes could offset corporate taxes such that in equilibrium the value of any individual lirm would be independent of its leverage. This approach stimulated several contributions such as Kim, Lewellen and McConnell (1979), DeAngelo and Masulis (1980) and others. These contributions study and extend the Miller-hypothesis that companies following a low leverage strategy will find a market among investors in high tax brackets, while the stock of highly levered firms will be held by investors
*The author would like to thank Prof. dr. P.A. Verheyen (Tilburg University), Prof. dr. P.J.J.M. van Loon (Limburg University, Maastricht), Prof. dr. Ch.S. Tapiero (Hebrew University, Jerusalem) and two anonymous referees for helpful suggestions and comments on this and earlier drafts of the paper. This research was supported by a grant from the Common Research Pool of Tilburg University and Eindhoven University of Technology (Samenwerkingsorgaan KHT-THE) in The Netherlands. 00142921/86/$3.50
0
1986, Elsevier
Science
Publishers
B.V. (North-Holland)
1044
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
with low personal tax rates. Consequently, taxation influences both the policy of the firm and investors choices as a function of their disposable personal income. Therefore, taxes are not ‘neutral’ with respect to the optimal financial policy of the firm. In general, ‘a fiscal regime is defined as neutral in the context of an individual firm if its introduction or modification does not affect the marginal costs of the inputs’ [Moerland (1978), p. 433. King (1977), Stapleton and Burke (1977), Atkinson and Stiglitz (1980) and others have also evaluated several fiscal regimes with respect to their (non)neutrality characteristics. The purpose of this paper is to extend these studies by adding another dimension: dynamics. As the survey of Feichtinger (1982a) and the collections of e.g., Bensoussan, Kleindorfer and Tapiero (1978) and Feichtinger (1982b, 1985) very well show, many recent papers extend the theory of the firm using optimal control to solve dynamic models analytically. Those models provide insight regarding the economic and financial behaviour of firms over time. Accordingly, the first part of this paper is related to the financial models such as those of Leland (1972), Lesourne (1973), Ludwig (1978), Sethi (1978) Van Loon (1983) and others [see also the survey of Lesourne and Leban (1982)], which are dealing with specific problems in finance, dividend and investment optimization. This kind of models has already been extended in many ways: Leban and Lesourne (1980) for example have paid attention to employment policies, whereas Van Loon (1982) has introduced activity analysis to describe the choice among a limited number of production possibilities. Although many authors are aware of the effects of personal taxation their study in a dynamic setting is lacking this subject. Research into this subject has been conducted by Yla-Liedenpohja (1978), but assuming an infinite time horizon and taking debt financing not especially into account a number of interesting topics are left out of consideration. In section 2 we describe a deterministic dynamic model of the firm, which behaves as if it maximizes its value conceived by tax homogeneous shareholders. We distinguish different personal and corporate tax rates and use optimal control theory to provide an analytical solution of the problem, which is indicated in section 3 and proved in the appendix. Sections 4 and 5 present an economic interpretation and a further analysis of our results. Since personal tax rates on capital gain and dividend are distinguished, it turns out that in final stages it could be optimal to retain all earnings in order to finance investments with a net return less than the discount rate of the shareholders. In addition, the firm will at last finance investments with equity only, even in the case of cheap debt financing. Finally we present in section 6 results of a sensitivity analysis concerning parameters that are interesting for economic analysis such as the personal tax rates and the discount rate.
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
1045
2. The model
Within a deterministic setting we consider a value maximizing firm which will at some known planning time z, stop its activities. The shareholders of the firm are assumed to have personal tax rates on dividend ~~ and capital gain TV, such that the ratio (1 -T,J/( 1 -TJ is the same for all shareholders and exceeds one. Let X(T) be the amount of equity and D(T) the level of dividend payments at time 7; then the shareholders have an investment which we value by’ V(O)=
j (l-r,)D(T)e-iTdT+X(z)e-i’-~,(X(z)-X(0))e-iZ, T=O
(1)
where i is the shareholder’s discount rate after personal taxes. The firm has only one production factor, capital goods K(T), and it may attract two kinds of money capital, equity X(T) and debt Y(T). The balance sheet is therefore K(T)=X(T)+
Y(T).
(2)
We assume that the firm is operating under decreasing returns to scale caused by imperfect output markets and/or increasing marginal costs of organizing the production due to the increasing scale of the firm. Therefore, revenue before interest payment and taxation is a concave function of the amount of capital goods. Further, no transaction costs are incurred when borrowing or paying off debt money, and corporate tax is proportional to profit and paid at once. Issues of new shares are not allowed and earnings after corporate tax are used to issue dividend or to increase the level of equity through retained earnings. This leads to
x(T):=g=(l-r,)(O(K)-rY)-D,
where
(3)
O(K) = operating income, i.e., revenue before interest and corporate taxation, O(K)>O, dOldK>O, d=O/dK=
is assumed to be proportional to capital goods. The impact on production capacity is described by the general used
‘Small letters are constants,capitals are
variables.
1046
G.-J.C.Th.
formulation
van
Schijndel.
Dynamic
behaviour
under
personal
taxation
of net investment:
d(T):=~=hK,
where
(4)
I = gross investment, a = deprectiation rate. Next we limit debt by introducing an upper bound in terms of a maximum debt to equity rate h [see Ludwig (1978, p. SS)]: YshX.
(5)
This debt to equity rate, together with the interest rate I, provides a way to deal with uncertainty within the framework of a deterministic model. Since I is an indication of the risk class to which a firm belongs, expression (5) may be conceived as a condition on the financial structure of the firm that must be fulfilled in order to stay in the relevant risk class. To complete the set of assumptions we add: (i)
If
K(T)=0
then
(I-r,)g>max{(l-r,)r,i}.
(6)
The marginal revenue of the first product to be sold exceeds each of the financial costs implying that the firm will consider only those alternatives that are profitable from the start. (ii)
i#(l-rC)i-.
(7)
In this way we avoid degenerated solutions. Moreover, an equity between the discount rate and the net cost of debt would be coincidental. (iii) Finally, we assume that the firm owns a certain initial amount of equity and debt such that Y(0) = hX(O),
X(0) + Y(0) = K(O) > 0.
U-9
3. Solution procedure
Following the discussion of section 2 we may formulate tion problem: max(1 -t,J j D(T)e-‘=dT+(l T=O I. Y.D
-r,JX(z)e-‘“,
the next optimiza-
s.t.
(9)
G.-J.C.Th.
van
Schijndel,
Dynamic
behaviour
under
personal
taxation
1047
X=(1-T,)(O(K)-rY)-D,
(10)
R=I-aK,
(11)
05K=X+Y,
(12)
OsYshX,
(13)
020,
(14)
X(O)+ Y(O)=(l+h)X(O)>O,
X(0)=X,,
K(O)=&.
(15)
This model can be solved analytically by using optimal control theory [see Pontryagin et al. (1962), Kamien and Schwartz (1983)], where the state of the system is described by the amount of equity and capital goods and is controlled by investment, dividend and the amount of debt. The aim of this control is to reach a maximum value of the objective function. In the appendix necessary, and in our case also sufficient, conditions for an optimal solution are obtained by means of the standard maximum principle of Pontryagin. These conditions consist of optimality conditions on control variables, Euler-Lagrange equations, complementary slackness conditions, transversality conditions and some additional conditions concerning continuity properties. Next, we apply the general solution procedure of Van Loon (1983, pp. 115-117) in order to transform the set of necessary conditions into a solution which covers the optimal policy of the firm over the whole planning period. 4. Optimal
policy strings
Applying the solution procedure of Van Loon we may discern five different feasible paths or stages, each characterized by different policies concerning capital structure, growth speed and/or dividend payments. Although we derived these features analytically, it s&ices at this moment to summarize the findings in table 1. An in-depth economic interpretation will be presented when the optimal master trajectories are described. As an example we derive from table 1 that path 1 has excellent possibilities for growth. Under this policy the firm will attract as much debt as possible, retain all earnings and invest both in order to realize a maximum increase of the amount of capital goods. Dependent on the values of the financial and fiscal parameters i, I, rC, TV and TV we get several optimal policy strings. We discuss two of them: two master trajectories both ending with path 3. The first one occurs if i<( 1 -T,)I and is represented by fig. 1. Although
1048
G.-J.C.Th.
van
Schijndel,
Dynamic
behaviour
Table Features
under
taxation
1
of feasible
paths.
Path
Y
D
2
k
K’
1 2 3 4
hX hX
0 0
+ +
+ 0
0 0
5
hX
0 max max
+ 0 0
0’ 0
“K=K;,u(l-s,)(dO/dK)=(l-r,)R, K=Kz++(l-~~)(dO/dK)=i, K = Ktu( 1 - r,)(dO/dK)
personal
Feasible Always Always Always
>fGX =K:
i<(l-s,)r iz(l-r,)r
=K:
= i/( 1 + h) + r( I - r,)h/(
1 + h).
K, Y. D. K 4 I I I I I I I I
I I I I I I I I
I I
I
I I I Q
I I I I I
I I
I
t12
Fig. 1. Master
t23
trajectory
t34
143
z
if i < (1 - c,)r.
debt financing is more expensive than self-financing the firm’s optimal policy is to borrow the maximum amount that is possible given the size of equity. As borrowing increases profit and thus raises the rate of growth of equity, it is profitable to borrow as long as the resulting marginal costs are less than marginal revenue. Accordingly, the firm grows at maximum speed in order to realize a maximum increase of the income stream and amount of cheap equity (path 1). As soon as the amount of capital goods K(T) reaches the level KF,, where (16)
it is profitable
to use retentions to pay back debt money, for
G.-J.C.Th.
uan Schijndel,
Dwamic
behaviour
under
personal
- issuing earnings is valued by the shareholders according rate i < (1 - TJi-, - continuing expansion investment yields a net revenue of
taxation
1049
to their* discount
(1-T,)(dO/dK)<(l-T,)r,
- paying back debt money saves (1 - z,)r rent payments per unit. So the firm stops its expansion in order to use all its revenue to replace debt by equity (path 2). After this period of consolidation it still makes sense to expand the amount of capital goods, because it is financed now by equity only. As no dividend is issued, the firm starts increasing as fast as possible again (path 3). In this way the state of maximum dividend pay-out in a self-financing regime Kg, where K=K+(l-T,)-&,
d0
(17)
is reached as quickly as possible. At this state it is, in first instance, useless to continue expansion investment because additional net costs exceed net revenue (path 4). However, due to the fixed planning horizon and the non-neutrality of the tax regime with respect to dividend and capital gain, it holds for T > t,, that one dollar dividend net of taxes has less worth than a net increase of equity at T = z caused by a retention of one dollar: for
T>t43:(1-Td)e-iT<(l-T~e-i’.-
W4 dX(T)’
Although marginal net revenue (1 -rc)(dO/dK) will fall below the discount rate i, the optimal policy is to stop dividend payment and start expansion investment again, because this loss is counterbalanced by the tax advantage. So, the advantage of the lower tax rate on capital gain (T~( 1 - tJr] it is optimal to borrow the maximum amount that is possible and invest both retained earnings and debt in capital goods in order to realize a maximum growth of the income stream (path 1). It is worth investing at the maximum level till K =KF, because at a level lower than Kg marginal income after taxation exceeds marginal financing
1050
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
K, Y. D.
Fig. 2. Master
trajectory
if i>( 1 -~,)r.
costs in case of maximum debt financing. As soon as the amount of capital goods equals KF, this accelerated growth is cut off abruptly at T=t15, because marginal net revenue equals the weighted sum of net cost of debt and equity: (19) This expression implies an equality of marginal return on equity and the discount rate i. Investment falls down to the replacement level and remaining earnings are issued to the shareholders (path 5). Corresponding with the previous master policy string a moment occurs at which the firm stops dividend payment and starts expansion investment by retaining earnings. Besides this, it is optimal to borrow as much as possible because borrowing still increases profit and raises the rate of growth till K = Kfx (path 1). According to the case of relatively expensive debt, the firm now drops debt to save (1 -xc)r interest payments per unit (path 2) and finally it starts growing in a self-financing regime till the end of the planning period (path 3). The optimal policy string succeeding path 5 depends among others on the spread between the persona1 tax rates, in other words, the tax advantage yielded by shareholders receiving capital gain instead of dividend. A lower value of (1 -Q/(1 -rd) will not only alter the final stage, but also postpone the moment T=t,, and decline the number of stages to pass through [see also section 63. Note, that the firm wants to get rid of debt although it is cheap compared to equity.
G.-J.C.Th.
van
Schijndel,
Dynamic
behaviour
under
personal
taxation
1051
5. A further analysis
Besides the previous presentation of the optimal policy strings, we like to derive two global decision rules, which together constitute the optimal policy of the firm [see Van Lobn (1983)].
(0
Financial decision rule. The financial structure is characterized by the relative amounts of debt and equity. The latter one is restricted by the first. So, the financial structure has two extreme cases: the case that the assets are financed by equity only and the case that the firm is financed by means of the maximum allowed amount of debt. Which of both is the optimal one depends on the marginal return to equity. The firm will try to realize such a financial structure as to maximize marginal return to equity, which implies:
choose for self-financing
if
K>K;,
or
(1 -r,)g
choose for maximum
if
K
or
(1-r,)gz(l-r,)r.
~(1 -Q, (20)
debt
So, personal taxation has no impact on the financial decision rule. (ii) Dividend/investment decision rule. The firm may spend its earnings in two ways: to pay out as dividend or to retain them in order to finance investment and/or to pay back debt money. The last mentioned decision has been discussed implicitly in the previous decision rule: redemption of debt starts as soon as the firm attains the KF,-level on which selffinancing becomes optimal. The second application is certainly preferable as long as marginal return to equity exceeds the discount rate of the shareholders, for this rate expresses the rate of return that shareholders can obtain elsewhere. Accordingly, the optimal policy is, in general, to pay out dividend and invest only on replacement level as soon as marginal return to equity equals the discount rate. So far so good, but what happens if the discount rate exceeds marginal return to equity? In absence of personal taxation one can prove that this situation only occurs if the initial amount of equity and capital goods are too high to make production at that level profitable. Van Loon (1983) argues that the optimal policy is a decrease of the capital stock by issuing all revenues. On top of this statement, however, the introduction of personal taxation brings on a second possible situation, which results in an additional dividend/investment decision rule to point out.
1052
G.-J.C.Th.
van Schijndel.
Dynamic
behaviour
under
personal
taxation
On the stationary dividend stages 4 and 5 the firm has to decide whether it will continue dividend payments or start retaining earnings. If we suppose that the firm holds these earnings only in cash and obtains no additional revenue, shareholders value the first possibility by (1 - rd) emiT and the latter one by (1 -rJe-“. This means that capital gain, on one hand, will be more profitable in view of the tax advantage (r,O. So, the decision to continue dividend payments or retaining earnings and keep cash money depends on the sign of (21)
It is obvious that given the values of tax rates, discount rate and planning horizon only for one value of time, T= tf*, the equality sign of expression (21) will hold. Note, that in absence of personal taxation the decision will always be in favour of a dividend policy and, consequently, t: equals z. In spite of the.decreasing marginal return to equity, however, expansion investment still acquires positive revenue, which can be used again to finance more investment. So, shareholders will not only receive the retained earnings but also the total increase of equity during the time interval [IT;z]. This increase depends on the rate of return on equity during the relevant time interval. The value of time, T =tF, therefore, will be determined by
(l-~d)e-if;=(l-z,)e-i’.
ax(z) -=(l-rJe-“sexp ax(e)
T&M~~ I
d,), (22)
where R,=(l
-rJg
as Y(T)=O,
(23)
as
(24)
Y(T)>O.
> This discussion, therefore, results in the next dividend/investment rule:
decision
- don’t pay out dividend and spend all earnings on investment if
R,>i,
or
R,
(25)
and
T>t:,
(26)
G.-J.C.Th.
-
van
Schijndel,
invest only on replacement if
Dynamic
hehaviour
under
personal
taxation
level and pay out all remaining
1053
earnings
R, =i,
(27)
- decrease capital stock and pay out all earnings if
R,
and
T
(28)
6. Sensitivity analysis In this section we study the influence of environmental changes on six different features of the optimal policy string. This is a sensitivity analysis concerning parameters that are interesting for economic analysis: interest rate r, discount rate i, corporate tax rate r, and especially personal tax rates rd and rB. Features of the policy string are the level of capital goods on stationary stages K*, speed of growth of equity 8, level of capital gain X(z)-X(O), start and finish of the stationary stage, tt and C? respectively, and leverage Y/X. 6.1. Fiscal parameters
(i) Corporate tax. consequences:
A reduction
of the corporate
tax rate has two direct
- a rise of the net cost of debt due to the tax deductibility, - possibilities to increase (expansion) investments in view of lower tax payments. The first one enables a switch of the sign of ip( 1 -Tc)r resulting in a policy switch of the firm to a low leverage strategy. Contrary to marginal revenue, however, net marginal cost of finance is only partially influenced. Thus, such a reduction results in increasing profit and a larger amount of capital goods at stationary stages. A reduction of the corporate tax rate also increases earnings after tax payment from which (expansion) investment is to be paid. In this way growth accelerates which is demonstrated by - g=(o(K)-rY)sO. E
On top we derive that such a reduction will put forward the stationary stage. No conclusion, however, is possible with respect of dividend stages, because a reduction of the corporate tax rate opposite influences: a rise of the speed of growth and a larger
(29)
end of the to the start causes two amount of
1054
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
capital goods to be reached. Nevertheless, we conclude that a reduction of the corporate tax rate raises the value of the firm and stimulates entrepreneur’s activities. (ii) Personal tax. The impact of changes in the value of the personal tax rates on the features of the firm are obvious: a fall of the dividend tax rate causes decreasing interest in capital gain, which is expressed in a postponement of finishing the stationary stages 4 and 5 respectively [see expressions (21) and (22)]. As the speed of growth does not change, the total amount of net dividend payment rises. Besides this, the final policy may alter due to the postponement of t:. By means of the transversality conditions we select those paths that may be final paths, i.e., paths feasible at T=z (see the appendix). Using the features of the several stages we derive the conditions stated in table 2. Table Conditions Path
Final
1 2 3 5 3
i>(l-r,)r i>(l -7,)r
4
i<(l
i>(l-7,)r i<(l-7,)r i<(l-7,)r
path
=~,)r
2
for linal stages.
if and l<(1-7J/(1-7,J-ci/(l-7T,)r and (1-7J/(l-Ta)=i/(l-T,)r and (l-7J/(l-7J>i/(l-7,)r and (1-73/(1-T,4)=1 and (l-r&)/(1-Q)>l and (1-73/(1-Td)=l
Table 2 distinguishes a financial and a fiscal condition. The first one states the financial setting of the firm and is the same as in models without personal taxes. The latter one indicates the investor’s attitude towards the desired kind of income: the bigger the difference between the personal tax rates the more he prefers capital gain, i.e., a rise of the amount of equity, instead of dividend. The fiscal conditions of table 2 are well understandable if we consider that the return on equity net of all taxes of a dividend paying firm is equal to (l -7Ji. So, a policy of retaining earnings is only optimal at T=z if net return after all taxes, (1 -rJ( 1 -rJ(dO/dK), equals at last (1 -q,)i. Recalling the result of table 1, we see, e.g., that policy 3 is optimal final stage if (1-7J(1-7,)r>(l-7J(l
-r,)$=(l
-Td)i.
So, due to a lower value of the dividend tax rate, net return of a dividend paying policy rises and, accordingly, a policy of retaining earnings remains
G.-J.C.Th.
oan
Schijndel,
Dynamic
behaoiour
under
personal
taxation
1055
an optimal policy only if marginal return rises up to the higher level of (1 -z,Ji. Therefore, path 2 or even path 1 becomes a feasible final policy. Consequently, it turns out that the firm does not always want to get rid of debt money. Finally note that dividend is issued on a final stage only when the personal tax rates equal each other, which shows that similar models without personal taxation [Ludwig (1978), Van Loon (1983)] are in this respect special cases of our model. If we consider, on the other hand, a rise of the dividend tax rate, the possibility occurs that the stationary dividend stage will disappear, i.e., the difference between r,, and rB is such that it is never optimal to pay out dividend [see dividend/investment rule of section 51. The former discussion could also be used to illustrate the results of a switch of the tax regime, e.g., from a classical system to an imputation system. Under the imputation system [see Stapleton and Burke (1977), King (1977) and Moerland (1978)], a lower effective dividend tax rate could be obtained and, consequently, more investors prefer a maximum dividend policy rather than a capital gain policy. A fall or rise of the tax rate on capital gain gives almost the opposite picture of a change in the dividend tax rate. 6.2. Financial
parameters
As earlier research has been done in sensitivity analysis in relation to financial parameters [Van Loon (1983)], we discuss the results only briefly and refer for a comprehensive discussion to the research mentioned. A rise of the interest rate T causes increasing costs of finance implying a drop in the speed of growth in stages with both debt and equity. In the case of expensive debt the termination point of the stationary stage will be postponed. A switch of the sign of i2( 1 -rc)r resulting in a switch of the policy of the firm also belongs to the possibilities. A change in the value of the discount rate has no influence on the speed of growth. The level of capital goods in stationary stages will decline and the end of these paths will be postponed due to the larger revenue for the investor elsewhere. The results of this analysis are summarized in table 3. 7. Summary
In this paper we considered especially the impact of tax systems on the optimal dynamic policy of a value maximizing firm by introducing both corporate and personal taxation. In this way we extended the dynamic theory of the firm. We derived an analytical solution of the deterministic
1056
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
Table 3 Sensitivity results. A fall of Impact on Level stationary stage K* Speed of growth X Capital gain X(z) - X(0) Begin stationary stage t: End stationary stage tf Leverage y/X
+ +oo + ? -
0
0
0 + 0
+ 0 0
+/+o + ?I-IO +
+ + -I-
‘+ =rise of the feature value, - =fall of the feature value, O=no influence on the relevant feature, ?=no conclusion due to opposite influences, ./. =conclusion before separation sign applies only if i<(l -r,)r, after separation sign if i > (1 - 5,)r.
dynamic model by means of optimal control theory. It turns out that the results differ from those of similar dynamic models without personal taxation in three ways: - in final stages no dividend will be issued, - in some situations the firm wants to get rid of debt, even when debt is cheap compared to equity, - due to the difference between the tax rate on dividend and capital gain, it may be optimal to start investments with a net return less than the discount rate. Finally, sensitivity analysis showed the importance of the value of the personal tax rates in providing the optimal (final) policy.
Appendix: A reduced form of the model and its solution In order to simplify this solution procedure, control variable Y: We can rewrite (12) as
we will first eliminate
the
Y=K-X. Substitution of the above expression in (9)-(15) results in the next reduced form of the model:
yy(l-~1
j W?e -iTdT+(l
T=O
-
e-
iz
)
s.t.
(A.2)
(A.31
G.-J.C.Th.
van Schijndel.
Dynamic
hehauiour
under
personal
taxation
1057
R=I-aK,
(A.4)
K-X20,
64.5)
(l+h)X-KZO,
64.6)
Combining expressions (A.5) and (A.6) however makes (A.8) superfluous, so we will leave it out. To avoid jumps in state variables we need a closed control region by putting artificial boundaries to dividend and investment. The proposition of Arrow and Kurz, however, enables us to prove that under our model formulation and assumptions a jump will never be optimal except possibly at the initial time point [Arrow and Kurz (1970, p. 57)]. Therefore, we omit these artificial boundaries. The model in its reduced form is a non-standard-control problem, however, because of the state constraints (A.5) and (A.6). To prevent discontinuous co-state variables we could apply the procedure of Russak (1970) [see also Van Loon (1983, pp. 107-l 13)]. Geerts (1985), however, showed that under our assumptions and model formulation the co-state variables will be continuous. Therefore, we apply the standard maximum principle of Pontryagin et al. (1962) to obtain neccesary conditions for an optimal solution. Let the Hamiltonian be H=(1-T,)De-iT+ICI1[(1-Tz,)(O(K)-r~+r~)-~]+~,[~-a~],
(A.9)
and let the Lagrangean
be
L=H+&(K-X)+&((l+h)X-K)+Q, tij=t,Gj( T)
1, = ,I,( T)
where
(A.lO)
adjoint variables or co-state variables which denote the marginal contribution of the jth level state variable to the performance level, dynamic Lagrange multipliers representing the dynamic ‘shadow price’ or ‘opportunity costs’ of the sth restriction.
For an optimal control history (D*,!*) of the problem formulated by (A.2) through (A.7) and the resulting state trajectory (K*, X*) to be optimal it is
1058
G.-J.C.Th.
uan Srhijndel.
Dynamic
behaoiour
under
personal
taxation
r~~e.ssrr~~~ that there are functions tij( T) and ,I,( T) such that Hoptimal: = H(K*,
X*, D*, I*, tij, A,, T)
=max{H(K*,X*,D,I,~j,~,,T)} D.1
for all
7: OsTsz
and, except at points of discontinuity of (D*, I*), optimality
of‘ control
aL
~=(l-r,)e-“-I//,+I,=O,
(A.1 1)
aL
c3/=*~=o:
Euler-Lagrange
(A.12)
equations
aL
-=-I),
=+bl(l -5,)r+(f
ax
+h)R,-A,,
(A.13)
(A.14)
complementary
slackness
conditions
A,(K-X)=0,
(A.15)
&((l+h)X-K)=O,
(A.16)
A,D=O;
(A.17)
transversality
conditions
$l(z)=e-i’(l
-5J,
*2(z) = 0; additional
(A.18) (A.19)
conditions
*j(T)
are continuous with piecewise continuous derivatives,
(A.20)
G.-J.C.Th.
A( 7’)
wm Schijnde/,
Dynamic
are non-negative {D, I>.
hehiour
under
and continuous
personal
1059
fuxation
on intervals
of continuity of (A.21)
is a concave ASHoptimal
function of (K,X) these conditions are also sufficient. From these conditions we derive that I/,(T) is a continuous function if and only if AJT) is continuous. As $JT) equals always zero, it is also continuous [see expression (A.1 l), (A.12) and (A.19)]. We now apply the general solution procedure of Van Loon (1983, pp. 115117) in order to transform the set of necessary conditions into a solution which covers the optimal policy of the firm over the whole planning period. This procedure consists of four steps. Step 1. Based on the complementary slackness conditions (A.1 5)+A.17) we may discern 2j = 8 different paths or stages, each characterized by active and inactive states of our restrictions. This number will be reduced, however, to five by assumptions (6)-(8). The properties of these stages are presented in table 1. As an example we derive the features of path 1. Path 1: &=0,&>0,1,>0. From (A.15): ,I, =O=xK>X * Y=hX. From (A.16): &>O=K=(l +h)X I From (A.17): A3 >O+D=O. Combining (A.12), (A.14) and ,I, =0 gives ljr(l-r,)
g-r
(
)
As $1=(1-r,)e-iT+J3>0 (1 -7,)$(l After substitution
-&=O. and ,I,>0
--J
or
this implies
K*(T)
of Y = K - X = hX and D = 0 we rewrite
(A.2) as
x=(1-7,)
As O(K) is a concave function of K and dO/dK >I O(K) > rK > h-K/( 1 + h) and therefore X > 0 and K > 0.
this
implies
Step 2. The second step is the selection of those paths that may be final paths, i.e., paths feasible at T =z. A feasible final path satisfies the transversality condirions (A.18) and (A.19). Using the features of the stages we
G.-J.C.Th.
1060
van
derive the conditions of path 5.
Schijndel,
Dynamic
hehaviour
under
personal
taxation
as presented by table 2. We now prove the conditions
Path 5: 1,=1,=0,&>0. Substitution of (A.18) and (A.19) in (A.ll)
for T=z
gives
(l-rd)e-iz-(l-r,)e-iz=O, which means that path 5 can be final stage only if rd = TV Step 3. To couple a feasible preceding path with a final path we use the continuity properties of relevant variables. Therefore we test each stage
whether coupling with the final stage will or will not violate the (necessary) continuity properties of the state variables K and X and the co-state variables $1 and rj2. If the set of feasible preceding stages appears to be empty, then the relevant final stage is the description of the optimal policy of the firm for the whole planning period, supposing that the initial state constraints (15) are fulfilled (step 4). If the set is not empty we apply the testing procedure again. Ih this way we find a still longer string of stages, constituting an optimal policy string. On which conditions will path 5 precede the string path 2+path 3 in the case i>(l-r,)r? To prove the continuity of $, we consider especially the continuity property of I, as this variable is on path 1 and equal to zero on path 5. So, 1, can only be continuous
Example.
lim ,I,=0 or A,>0 Tl’,, From (A.1 l)-(A.14) we derive
when
A,=0
l-path have to positive if
on path 1.
>I -i
-A,(1 -TJ(g
- &+l
+h).
So, 1, > 0 for 1, = 0 if (1 + h)( 1 - r,)(dO/dK) < h( 1 - T,)r + i. Because i > (1 - z,)r and (1 -rJ(dO/dK) > (1 - TJr on path 1 this condition could be fulfilled. Which path could precede the string path l-+path 2
No
K discontinuous
3
No
K -X
4
No
i<(l
5
Yes
discontinuous -TJr
2-+path 3?
Step 4. If the set of feasible preceding stages is empty we stop the iterative procedure of step 3 and, finally, check whether the initial state constraints (15) are fulfilled. Both optimal policy strings start with path 1 on which Y(T)=hX(T)+K(T)=(l+h)X(T).
References Arrow, K.J. and M. Kurz. 1970. Public investment. the rate of return and optimal hscal policy (Johns Hopkins University Press. Baltimore. MD). Atkinson A.B. and J. Stiglitz 1980, Lectures on public economics (McGraw-Hill. New York), Bensoussan, A., P.R. Kleindorfer and ChS. Tapiero. 1978, Applied optimal control (NorthHolland, Amsterdam). DeAngelo. H. and R. Masulis. 1980, Leverage and dividend irrelevancy under corporate and personal taxation, Journal of Finance 35, 453467. Feichtinger, Cl., 1982a, Anwendungen des Maximum-Prinzips im Operations Research, Teil I und 2 (Applications of the maximum principle in operations research, Vols. I and 2), OR Spektrum 4, 17l-IYO and 195-212. Feichtinger, G.. 1982b. Optimal control theory and economic analysis (North-Holland, Amsterdam). Feichtinger, G.. 1985, Optimal control theory and economic anaiysis. Vol. 2, (North-Holland, Amsterdam). Geerts, A., 1985. Optimality conditions for solutions of singular and state constrained optimal control problems in economics. Master’s thesis (Eindhoven University of Technology, Eindhoven). Kamien, M.I. and N. Schwartz. 1983, Dynamic optimization, the calculus of variation and optimal control in economics and management. 2nd reprint (North-Holland, New York). Kim, E.H., W.G. Lewellen and J.J. McConnell. 1979, Financial leverage clienteles, Journal of Financial Economics 7. X3-109. King, M., 1977. Public policy and the corporation, (Chapmann and Hall, London). Leban, R. and J. Lesourne. 1980, The lirm’s investment and employment policy through a business cycle, European Economic Review 13. 43380. Leland, H.E., 1972, The dynamics of a revenue maximizing firm, International Economic Review 13. 376385. Lesourne, J., 1973, Modeles de Croissance des Entreprises (Growth models of the firm), (Dunod, Paris). Lesourne, J. and R. Leban, 1982, Control theory and the dynamics of the lirm, OR Spektrum 4, l-14. Ludwig, Th.. 1978. Optimale Expansionpfade der unternehmung (Optimal growth stages or the firm), (Gabler, Wiesbaden). Miller, M.H., 1977, Debt and taxes, Journal or Finance 32, 261-275. Modigliani, F. and M.H. Miller, 1963, Corporate income taxes and the cost of capital: A correction, American Economic Review 53, 433-l43. Moerland, P.W., 1978, Optimal lit-m behaviour under different liscal regimes, European Economic Review 1 I, 37-58. Pontryagin, L.S., V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mischenko, 1962, The mathematical theory of optimal processes (Wiley, New York). Russak, B.I., 1970, On general problems with bounded state variables, Journal of Optimization Theory and Applications 6, 424-451. Sethi, S.P., 1978, Optimal equity and linancing model of Krouse and Lee: Corrections and extensions, Journal of Financial and Quantitative Analysis 13, 487505.
1062
G.-J.C.Th.
wn
Schijndel,
D~wunic
behaviour
under
personal
taration
Stapleton, R.C. and C.M. Burke, 1977, European tax systems and the neutrality of corporate linancing policy, Journal of Banking and Finance 1, 55-70. Van Loon, P.J.J.M., 1982, Empolyment in a monopolistic Turn, European Economic Review 19, 305-327. Van Loon, P.J.J.M., 1983, A dynamic theory of the firm: production, linance and investment, Lecture Notes in Economics and Mathematical Systems 218 (Springer, Berlin). Yli-Liedenpohja, J., 1978, Taxes, dividends and the capital gains in the adjustments cost model of the firm, Scandinavian Journal of Economics 80, 3999410.
Economic
Review
DYNAMIC
30 (1986)
BEHAVIOUR UNDER
1043-1062.
North-Holland
OF A VALUE MAXIMIZING PERSONAL TAXATION
FIRM
Geert-Jan C.Th. van SCHIJNDEL* Tilburg
Received
University,
October
5000
1984, final
LE
Tilburg,
version
The
received
Nefherlands
November
1985
In this paper we study the impact of both corporate and personal taxation on the optimal dynamic behaviour of a value maximizing firm by determining simultaneously the optimal linancial, investment and dividend policy over a linite horizon. To do so, an optimal control model is formulated and resolved. We discuss the successive stages in the evolution of the Iirm, derive optimal linancial and investment/dividend decision rules and present sensitivity analysis results that are important for understanding the behaviour of lirms in a dynamic setting. The gap between the personal tax rates on dividend and capital gain respectively, turns out to play an important role in providing the optimal policy string.
1. Introduction
The effect of corporate taxes on the market value of a levered firm continues to be a central issue in recent contributions in finance theory. Following the publication of the Modigliani and Miller (1963) tax correction paper, many writers have sought to reconcile the M&M maximum leverage prediction with observed capita1 structure. In ‘Debt and Taxes’, Miller (1977) argues that personal taxes could offset corporate taxes such that in equilibrium the value of any individual lirm would be independent of its leverage. This approach stimulated several contributions such as Kim, Lewellen and McConnell (1979), DeAngelo and Masulis (1980) and others. These contributions study and extend the Miller-hypothesis that companies following a low leverage strategy will find a market among investors in high tax brackets, while the stock of highly levered firms will be held by investors
*The author would like to thank Prof. dr. P.A. Verheyen (Tilburg University), Prof. dr. P.J.J.M. van Loon (Limburg University, Maastricht), Prof. dr. Ch.S. Tapiero (Hebrew University, Jerusalem) and two anonymous referees for helpful suggestions and comments on this and earlier drafts of the paper. This research was supported by a grant from the Common Research Pool of Tilburg University and Eindhoven University of Technology (Samenwerkingsorgaan KHT-THE) in The Netherlands. 00142921/86/$3.50
0
1986, Elsevier
Science
Publishers
B.V. (North-Holland)
1044
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
with low personal tax rates. Consequently, taxation influences both the policy of the firm and investors choices as a function of their disposable personal income. Therefore, taxes are not ‘neutral’ with respect to the optimal financial policy of the firm. In general, ‘a fiscal regime is defined as neutral in the context of an individual firm if its introduction or modification does not affect the marginal costs of the inputs’ [Moerland (1978), p. 433. King (1977), Stapleton and Burke (1977), Atkinson and Stiglitz (1980) and others have also evaluated several fiscal regimes with respect to their (non)neutrality characteristics. The purpose of this paper is to extend these studies by adding another dimension: dynamics. As the survey of Feichtinger (1982a) and the collections of e.g., Bensoussan, Kleindorfer and Tapiero (1978) and Feichtinger (1982b, 1985) very well show, many recent papers extend the theory of the firm using optimal control to solve dynamic models analytically. Those models provide insight regarding the economic and financial behaviour of firms over time. Accordingly, the first part of this paper is related to the financial models such as those of Leland (1972), Lesourne (1973), Ludwig (1978), Sethi (1978) Van Loon (1983) and others [see also the survey of Lesourne and Leban (1982)], which are dealing with specific problems in finance, dividend and investment optimization. This kind of models has already been extended in many ways: Leban and Lesourne (1980) for example have paid attention to employment policies, whereas Van Loon (1982) has introduced activity analysis to describe the choice among a limited number of production possibilities. Although many authors are aware of the effects of personal taxation their study in a dynamic setting is lacking this subject. Research into this subject has been conducted by Yla-Liedenpohja (1978), but assuming an infinite time horizon and taking debt financing not especially into account a number of interesting topics are left out of consideration. In section 2 we describe a deterministic dynamic model of the firm, which behaves as if it maximizes its value conceived by tax homogeneous shareholders. We distinguish different personal and corporate tax rates and use optimal control theory to provide an analytical solution of the problem, which is indicated in section 3 and proved in the appendix. Sections 4 and 5 present an economic interpretation and a further analysis of our results. Since personal tax rates on capital gain and dividend are distinguished, it turns out that in final stages it could be optimal to retain all earnings in order to finance investments with a net return less than the discount rate of the shareholders. In addition, the firm will at last finance investments with equity only, even in the case of cheap debt financing. Finally we present in section 6 results of a sensitivity analysis concerning parameters that are interesting for economic analysis such as the personal tax rates and the discount rate.
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
1045
2. The model
Within a deterministic setting we consider a value maximizing firm which will at some known planning time z, stop its activities. The shareholders of the firm are assumed to have personal tax rates on dividend ~~ and capital gain TV, such that the ratio (1 -T,J/( 1 -TJ is the same for all shareholders and exceeds one. Let X(T) be the amount of equity and D(T) the level of dividend payments at time 7; then the shareholders have an investment which we value by’ V(O)=
j (l-r,)D(T)e-iTdT+X(z)e-i’-~,(X(z)-X(0))e-iZ, T=O
(1)
where i is the shareholder’s discount rate after personal taxes. The firm has only one production factor, capital goods K(T), and it may attract two kinds of money capital, equity X(T) and debt Y(T). The balance sheet is therefore K(T)=X(T)+
Y(T).
(2)
We assume that the firm is operating under decreasing returns to scale caused by imperfect output markets and/or increasing marginal costs of organizing the production due to the increasing scale of the firm. Therefore, revenue before interest payment and taxation is a concave function of the amount of capital goods. Further, no transaction costs are incurred when borrowing or paying off debt money, and corporate tax is proportional to profit and paid at once. Issues of new shares are not allowed and earnings after corporate tax are used to issue dividend or to increase the level of equity through retained earnings. This leads to
x(T):=g=(l-r,)(O(K)-rY)-D,
where
(3)
O(K) = operating income, i.e., revenue before interest and corporate taxation, O(K)>O, dOldK>O, d=O/dK=
is assumed to be proportional to capital goods. The impact on production capacity is described by the general used
‘Small letters are constants,capitals are
variables.
1046
G.-J.C.Th.
formulation
van
Schijndel.
Dynamic
behaviour
under
personal
taxation
of net investment:
d(T):=~=hK,
where
(4)
I = gross investment, a = deprectiation rate. Next we limit debt by introducing an upper bound in terms of a maximum debt to equity rate h [see Ludwig (1978, p. SS)]: YshX.
(5)
This debt to equity rate, together with the interest rate I, provides a way to deal with uncertainty within the framework of a deterministic model. Since I is an indication of the risk class to which a firm belongs, expression (5) may be conceived as a condition on the financial structure of the firm that must be fulfilled in order to stay in the relevant risk class. To complete the set of assumptions we add: (i)
If
K(T)=0
then
(I-r,)g>max{(l-r,)r,i}.
(6)
The marginal revenue of the first product to be sold exceeds each of the financial costs implying that the firm will consider only those alternatives that are profitable from the start. (ii)
i#(l-rC)i-.
(7)
In this way we avoid degenerated solutions. Moreover, an equity between the discount rate and the net cost of debt would be coincidental. (iii) Finally, we assume that the firm owns a certain initial amount of equity and debt such that Y(0) = hX(O),
X(0) + Y(0) = K(O) > 0.
U-9
3. Solution procedure
Following the discussion of section 2 we may formulate tion problem: max(1 -t,J j D(T)e-‘=dT+(l T=O I. Y.D
-r,JX(z)e-‘“,
the next optimiza-
s.t.
(9)
G.-J.C.Th.
van
Schijndel,
Dynamic
behaviour
under
personal
taxation
1047
X=(1-T,)(O(K)-rY)-D,
(10)
R=I-aK,
(11)
05K=X+Y,
(12)
OsYshX,
(13)
020,
(14)
X(O)+ Y(O)=(l+h)X(O)>O,
X(0)=X,,
K(O)=&.
(15)
This model can be solved analytically by using optimal control theory [see Pontryagin et al. (1962), Kamien and Schwartz (1983)], where the state of the system is described by the amount of equity and capital goods and is controlled by investment, dividend and the amount of debt. The aim of this control is to reach a maximum value of the objective function. In the appendix necessary, and in our case also sufficient, conditions for an optimal solution are obtained by means of the standard maximum principle of Pontryagin. These conditions consist of optimality conditions on control variables, Euler-Lagrange equations, complementary slackness conditions, transversality conditions and some additional conditions concerning continuity properties. Next, we apply the general solution procedure of Van Loon (1983, pp. 115-117) in order to transform the set of necessary conditions into a solution which covers the optimal policy of the firm over the whole planning period. 4. Optimal
policy strings
Applying the solution procedure of Van Loon we may discern five different feasible paths or stages, each characterized by different policies concerning capital structure, growth speed and/or dividend payments. Although we derived these features analytically, it s&ices at this moment to summarize the findings in table 1. An in-depth economic interpretation will be presented when the optimal master trajectories are described. As an example we derive from table 1 that path 1 has excellent possibilities for growth. Under this policy the firm will attract as much debt as possible, retain all earnings and invest both in order to realize a maximum increase of the amount of capital goods. Dependent on the values of the financial and fiscal parameters i, I, rC, TV and TV we get several optimal policy strings. We discuss two of them: two master trajectories both ending with path 3. The first one occurs if i<( 1 -T,)I and is represented by fig. 1. Although
1048
G.-J.C.Th.
van
Schijndel,
Dynamic
behaviour
Table Features
under
taxation
1
of feasible
paths.
Path
Y
D
2
k
K’
1 2 3 4
hX hX
0 0
+ +
+ 0
0 0
5
hX
0 max max
+ 0 0
0’ 0
“K=K;,u(l-s,)(dO/dK)=(l-r,)R, K=Kz++(l-~~)(dO/dK)=i, K = Ktu( 1 - r,)(dO/dK)
personal
Feasible Always Always Always
>fGX =K:
i<(l-s,)r iz(l-r,)r
=K:
= i/( 1 + h) + r( I - r,)h/(
1 + h).
K, Y. D. K 4 I I I I I I I I
I I I I I I I I
I I
I
I I I Q
I I I I I
I I
I
t12
Fig. 1. Master
t23
trajectory
t34
143
z
if i < (1 - c,)r.
debt financing is more expensive than self-financing the firm’s optimal policy is to borrow the maximum amount that is possible given the size of equity. As borrowing increases profit and thus raises the rate of growth of equity, it is profitable to borrow as long as the resulting marginal costs are less than marginal revenue. Accordingly, the firm grows at maximum speed in order to realize a maximum increase of the income stream and amount of cheap equity (path 1). As soon as the amount of capital goods K(T) reaches the level KF,, where (16)
it is profitable
to use retentions to pay back debt money, for
G.-J.C.Th.
uan Schijndel,
Dwamic
behaviour
under
personal
- issuing earnings is valued by the shareholders according rate i < (1 - TJi-, - continuing expansion investment yields a net revenue of
taxation
1049
to their* discount
(1-T,)(dO/dK)<(l-T,)r,
- paying back debt money saves (1 - z,)r rent payments per unit. So the firm stops its expansion in order to use all its revenue to replace debt by equity (path 2). After this period of consolidation it still makes sense to expand the amount of capital goods, because it is financed now by equity only. As no dividend is issued, the firm starts increasing as fast as possible again (path 3). In this way the state of maximum dividend pay-out in a self-financing regime Kg, where K=K+(l-T,)-&,
d0
(17)
is reached as quickly as possible. At this state it is, in first instance, useless to continue expansion investment because additional net costs exceed net revenue (path 4). However, due to the fixed planning horizon and the non-neutrality of the tax regime with respect to dividend and capital gain, it holds for T > t,, that one dollar dividend net of taxes has less worth than a net increase of equity at T = z caused by a retention of one dollar: for
T>t43:(1-Td)e-iT<(l-T~e-i’.-
W4 dX(T)’
Although marginal net revenue (1 -rc)(dO/dK) will fall below the discount rate i, the optimal policy is to stop dividend payment and start expansion investment again, because this loss is counterbalanced by the tax advantage. So, the advantage of the lower tax rate on capital gain (T~
1050
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
K, Y. D.
Fig. 2. Master
trajectory
if i>( 1 -~,)r.
costs in case of maximum debt financing. As soon as the amount of capital goods equals KF, this accelerated growth is cut off abruptly at T=t15, because marginal net revenue equals the weighted sum of net cost of debt and equity: (19) This expression implies an equality of marginal return on equity and the discount rate i. Investment falls down to the replacement level and remaining earnings are issued to the shareholders (path 5). Corresponding with the previous master policy string a moment occurs at which the firm stops dividend payment and starts expansion investment by retaining earnings. Besides this, it is optimal to borrow as much as possible because borrowing still increases profit and raises the rate of growth till K = Kfx (path 1). According to the case of relatively expensive debt, the firm now drops debt to save (1 -xc)r interest payments per unit (path 2) and finally it starts growing in a self-financing regime till the end of the planning period (path 3). The optimal policy string succeeding path 5 depends among others on the spread between the persona1 tax rates, in other words, the tax advantage yielded by shareholders receiving capital gain instead of dividend. A lower value of (1 -Q/(1 -rd) will not only alter the final stage, but also postpone the moment T=t,, and decline the number of stages to pass through [see also section 63. Note, that the firm wants to get rid of debt although it is cheap compared to equity.
G.-J.C.Th.
van
Schijndel,
Dynamic
behaviour
under
personal
taxation
1051
5. A further analysis
Besides the previous presentation of the optimal policy strings, we like to derive two global decision rules, which together constitute the optimal policy of the firm [see Van Lobn (1983)].
(0
Financial decision rule. The financial structure is characterized by the relative amounts of debt and equity. The latter one is restricted by the first. So, the financial structure has two extreme cases: the case that the assets are financed by equity only and the case that the firm is financed by means of the maximum allowed amount of debt. Which of both is the optimal one depends on the marginal return to equity. The firm will try to realize such a financial structure as to maximize marginal return to equity, which implies:
choose for self-financing
if
K>K;,
or
(1 -r,)g
choose for maximum
if
K
or
(1-r,)gz(l-r,)r.
~(1 -Q, (20)
debt
So, personal taxation has no impact on the financial decision rule. (ii) Dividend/investment decision rule. The firm may spend its earnings in two ways: to pay out as dividend or to retain them in order to finance investment and/or to pay back debt money. The last mentioned decision has been discussed implicitly in the previous decision rule: redemption of debt starts as soon as the firm attains the KF,-level on which selffinancing becomes optimal. The second application is certainly preferable as long as marginal return to equity exceeds the discount rate of the shareholders, for this rate expresses the rate of return that shareholders can obtain elsewhere. Accordingly, the optimal policy is, in general, to pay out dividend and invest only on replacement level as soon as marginal return to equity equals the discount rate. So far so good, but what happens if the discount rate exceeds marginal return to equity? In absence of personal taxation one can prove that this situation only occurs if the initial amount of equity and capital goods are too high to make production at that level profitable. Van Loon (1983) argues that the optimal policy is a decrease of the capital stock by issuing all revenues. On top of this statement, however, the introduction of personal taxation brings on a second possible situation, which results in an additional dividend/investment decision rule to point out.
1052
G.-J.C.Th.
van Schijndel.
Dynamic
behaviour
under
personal
taxation
On the stationary dividend stages 4 and 5 the firm has to decide whether it will continue dividend payments or start retaining earnings. If we suppose that the firm holds these earnings only in cash and obtains no additional revenue, shareholders value the first possibility by (1 - rd) emiT and the latter one by (1 -rJe-“. This means that capital gain, on one hand, will be more profitable in view of the tax advantage (r,
It is obvious that given the values of tax rates, discount rate and planning horizon only for one value of time, T= tf*, the equality sign of expression (21) will hold. Note, that in absence of personal taxation the decision will always be in favour of a dividend policy and, consequently, t: equals z. In spite of the.decreasing marginal return to equity, however, expansion investment still acquires positive revenue, which can be used again to finance more investment. So, shareholders will not only receive the retained earnings but also the total increase of equity during the time interval [IT;z]. This increase depends on the rate of return on equity during the relevant time interval. The value of time, T =tF, therefore, will be determined by
(l-~d)e-if;=(l-z,)e-i’.
ax(z) -=(l-rJe-“sexp ax(e)
T&M~~ I
d,), (22)
where R,=(l
-rJg
as Y(T)=O,
(23)
as
(24)
Y(T)>O.
> This discussion, therefore, results in the next dividend/investment rule:
decision
- don’t pay out dividend and spend all earnings on investment if
R,>i,
or
R,
(25)
and
T>t:,
(26)
G.-J.C.Th.
-
van
Schijndel,
invest only on replacement if
Dynamic
hehaviour
under
personal
taxation
level and pay out all remaining
1053
earnings
R, =i,
(27)
- decrease capital stock and pay out all earnings if
R,
and
T
(28)
6. Sensitivity analysis In this section we study the influence of environmental changes on six different features of the optimal policy string. This is a sensitivity analysis concerning parameters that are interesting for economic analysis: interest rate r, discount rate i, corporate tax rate r, and especially personal tax rates rd and rB. Features of the policy string are the level of capital goods on stationary stages K*, speed of growth of equity 8, level of capital gain X(z)-X(O), start and finish of the stationary stage, tt and C? respectively, and leverage Y/X. 6.1. Fiscal parameters
(i) Corporate tax. consequences:
A reduction
of the corporate
tax rate has two direct
- a rise of the net cost of debt due to the tax deductibility, - possibilities to increase (expansion) investments in view of lower tax payments. The first one enables a switch of the sign of ip( 1 -Tc)r resulting in a policy switch of the firm to a low leverage strategy. Contrary to marginal revenue, however, net marginal cost of finance is only partially influenced. Thus, such a reduction results in increasing profit and a larger amount of capital goods at stationary stages. A reduction of the corporate tax rate also increases earnings after tax payment from which (expansion) investment is to be paid. In this way growth accelerates which is demonstrated by - g=(o(K)-rY)sO. E
On top we derive that such a reduction will put forward the stationary stage. No conclusion, however, is possible with respect of dividend stages, because a reduction of the corporate tax rate opposite influences: a rise of the speed of growth and a larger
(29)
end of the to the start causes two amount of
1054
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
capital goods to be reached. Nevertheless, we conclude that a reduction of the corporate tax rate raises the value of the firm and stimulates entrepreneur’s activities. (ii) Personal tax. The impact of changes in the value of the personal tax rates on the features of the firm are obvious: a fall of the dividend tax rate causes decreasing interest in capital gain, which is expressed in a postponement of finishing the stationary stages 4 and 5 respectively [see expressions (21) and (22)]. As the speed of growth does not change, the total amount of net dividend payment rises. Besides this, the final policy may alter due to the postponement of t:. By means of the transversality conditions we select those paths that may be final paths, i.e., paths feasible at T=z (see the appendix). Using the features of the several stages we derive the conditions stated in table 2. Table Conditions Path
Final
1 2 3 5 3
i>(l-r,)r i>(l -7,)r
4
i<(l
i>(l-7,)r i<(l-7,)r i<(l-7,)r
path
=~,)r
2
for linal stages.
if and l<(1-7J/(1-7,J-ci/(l-7T,)r and (1-7J/(l-Ta)=i/(l-T,)r and (l-7J/(l-7J>i/(l-7,)r and (1-73/(1-T,4)=1 and (l-r&)/(1-Q)>l and (1-73/(1-Td)=l
Table 2 distinguishes a financial and a fiscal condition. The first one states the financial setting of the firm and is the same as in models without personal taxes. The latter one indicates the investor’s attitude towards the desired kind of income: the bigger the difference between the personal tax rates the more he prefers capital gain, i.e., a rise of the amount of equity, instead of dividend. The fiscal conditions of table 2 are well understandable if we consider that the return on equity net of all taxes of a dividend paying firm is equal to (l -7Ji. So, a policy of retaining earnings is only optimal at T=z if net return after all taxes, (1 -rJ( 1 -rJ(dO/dK), equals at last (1 -q,)i. Recalling the result of table 1, we see, e.g., that policy 3 is optimal final stage if (1-7J(1-7,)r>(l-7J(l
-r,)$=(l
-Td)i.
So, due to a lower value of the dividend tax rate, net return of a dividend paying policy rises and, accordingly, a policy of retaining earnings remains
G.-J.C.Th.
oan
Schijndel,
Dynamic
behaoiour
under
personal
taxation
1055
an optimal policy only if marginal return rises up to the higher level of (1 -z,Ji. Therefore, path 2 or even path 1 becomes a feasible final policy. Consequently, it turns out that the firm does not always want to get rid of debt money. Finally note that dividend is issued on a final stage only when the personal tax rates equal each other, which shows that similar models without personal taxation [Ludwig (1978), Van Loon (1983)] are in this respect special cases of our model. If we consider, on the other hand, a rise of the dividend tax rate, the possibility occurs that the stationary dividend stage will disappear, i.e., the difference between r,, and rB is such that it is never optimal to pay out dividend [see dividend/investment rule of section 51. The former discussion could also be used to illustrate the results of a switch of the tax regime, e.g., from a classical system to an imputation system. Under the imputation system [see Stapleton and Burke (1977), King (1977) and Moerland (1978)], a lower effective dividend tax rate could be obtained and, consequently, more investors prefer a maximum dividend policy rather than a capital gain policy. A fall or rise of the tax rate on capital gain gives almost the opposite picture of a change in the dividend tax rate. 6.2. Financial
parameters
As earlier research has been done in sensitivity analysis in relation to financial parameters [Van Loon (1983)], we discuss the results only briefly and refer for a comprehensive discussion to the research mentioned. A rise of the interest rate T causes increasing costs of finance implying a drop in the speed of growth in stages with both debt and equity. In the case of expensive debt the termination point of the stationary stage will be postponed. A switch of the sign of i2( 1 -rc)r resulting in a switch of the policy of the firm also belongs to the possibilities. A change in the value of the discount rate has no influence on the speed of growth. The level of capital goods in stationary stages will decline and the end of these paths will be postponed due to the larger revenue for the investor elsewhere. The results of this analysis are summarized in table 3. 7. Summary
In this paper we considered especially the impact of tax systems on the optimal dynamic policy of a value maximizing firm by introducing both corporate and personal taxation. In this way we extended the dynamic theory of the firm. We derived an analytical solution of the deterministic
1056
G.-J.C.Th.
van Schijndel,
Dynamic
behaviour
under
personal
taxation
Table 3 Sensitivity results. A fall of Impact on Level stationary stage K* Speed of growth X Capital gain X(z) - X(0) Begin stationary stage t: End stationary stage tf Leverage y/X
+ +oo + ? -
0
0
0 + 0
+ 0 0
+/+o + ?I-IO +
+ + -I-
‘+ =rise of the feature value, - =fall of the feature value, O=no influence on the relevant feature, ?=no conclusion due to opposite influences, ./. =conclusion before separation sign applies only if i<(l -r,)r, after separation sign if i > (1 - 5,)r.
dynamic model by means of optimal control theory. It turns out that the results differ from those of similar dynamic models without personal taxation in three ways: - in final stages no dividend will be issued, - in some situations the firm wants to get rid of debt, even when debt is cheap compared to equity, - due to the difference between the tax rate on dividend and capital gain, it may be optimal to start investments with a net return less than the discount rate. Finally, sensitivity analysis showed the importance of the value of the personal tax rates in providing the optimal (final) policy.
Appendix: A reduced form of the model and its solution In order to simplify this solution procedure, control variable Y: We can rewrite (12) as
we will first eliminate
the
Y=K-X. Substitution of the above expression in (9)-(15) results in the next reduced form of the model:
yy(l-~1
j W?e -iTdT+(l
T=O
-
e-
iz
)
s.t.
(A.2)
(A.31
G.-J.C.Th.
van Schijndel.
Dynamic
hehauiour
under
personal
taxation
1057
R=I-aK,
(A.4)
K-X20,
64.5)
(l+h)X-KZO,
64.6)
Combining expressions (A.5) and (A.6) however makes (A.8) superfluous, so we will leave it out. To avoid jumps in state variables we need a closed control region by putting artificial boundaries to dividend and investment. The proposition of Arrow and Kurz, however, enables us to prove that under our model formulation and assumptions a jump will never be optimal except possibly at the initial time point [Arrow and Kurz (1970, p. 57)]. Therefore, we omit these artificial boundaries. The model in its reduced form is a non-standard-control problem, however, because of the state constraints (A.5) and (A.6). To prevent discontinuous co-state variables we could apply the procedure of Russak (1970) [see also Van Loon (1983, pp. 107-l 13)]. Geerts (1985), however, showed that under our assumptions and model formulation the co-state variables will be continuous. Therefore, we apply the standard maximum principle of Pontryagin et al. (1962) to obtain neccesary conditions for an optimal solution. Let the Hamiltonian be H=(1-T,)De-iT+ICI1[(1-Tz,)(O(K)-r~+r~)-~]+~,[~-a~],
(A.9)
and let the Lagrangean
be
L=H+&(K-X)+&((l+h)X-K)+Q, tij=t,Gj( T)
1, = ,I,( T)
where
(A.lO)
adjoint variables or co-state variables which denote the marginal contribution of the jth level state variable to the performance level, dynamic Lagrange multipliers representing the dynamic ‘shadow price’ or ‘opportunity costs’ of the sth restriction.
For an optimal control history (D*,!*) of the problem formulated by (A.2) through (A.7) and the resulting state trajectory (K*, X*) to be optimal it is
1058
G.-J.C.Th.
uan Srhijndel.
Dynamic
behaoiour
under
personal
taxation
r~~e.ssrr~~~ that there are functions tij( T) and ,I,( T) such that Hoptimal: = H(K*,
X*, D*, I*, tij, A,, T)
=max{H(K*,X*,D,I,~j,~,,T)} D.1
for all
7: OsTsz
and, except at points of discontinuity of (D*, I*), optimality
of‘ control
aL
~=(l-r,)e-“-I//,+I,=O,
(A.1 1)
aL
c3/=*~=o:
Euler-Lagrange
(A.12)
equations
aL
-=-I),
=+bl(l -5,)r+(f
ax
+h)R,-A,,
(A.13)
(A.14)
complementary
slackness
conditions
A,(K-X)=0,
(A.15)
&((l+h)X-K)=O,
(A.16)
A,D=O;
(A.17)
transversality
conditions
$l(z)=e-i’(l
-5J,
*2(z) = 0; additional
(A.18) (A.19)
conditions
*j(T)
are continuous with piecewise continuous derivatives,
(A.20)
G.-J.C.Th.
A( 7’)
wm Schijnde/,
Dynamic
are non-negative {D, I>.
hehiour
under
and continuous
personal
1059
fuxation
on intervals
of continuity of (A.21)
is a concave ASHoptimal
function of (K,X) these conditions are also sufficient. From these conditions we derive that I/,(T) is a continuous function if and only if AJT) is continuous. As $JT) equals always zero, it is also continuous [see expression (A.1 l), (A.12) and (A.19)]. We now apply the general solution procedure of Van Loon (1983, pp. 115117) in order to transform the set of necessary conditions into a solution which covers the optimal policy of the firm over the whole planning period. This procedure consists of four steps. Step 1. Based on the complementary slackness conditions (A.1 5)+A.17) we may discern 2j = 8 different paths or stages, each characterized by active and inactive states of our restrictions. This number will be reduced, however, to five by assumptions (6)-(8). The properties of these stages are presented in table 1. As an example we derive the features of path 1. Path 1: &=0,&>0,1,>0. From (A.15): ,I, =O=xK>X * Y=hX. From (A.16): &>O=K=(l +h)X I From (A.17): A3 >O+D=O. Combining (A.12), (A.14) and ,I, =0 gives ljr(l-r,)
g-r
(
)
As $1=(1-r,)e-iT+J3>0 (1 -7,)$(l After substitution
-&=O. and ,I,>0
--J
or
this implies
K*(T)
of Y = K - X = hX and D = 0 we rewrite
(A.2) as
x=(1-7,)
As O(K) is a concave function of K and dO/dK >I O(K) > rK > h-K/( 1 + h) and therefore X > 0 and K > 0.
this
implies
Step 2. The second step is the selection of those paths that may be final paths, i.e., paths feasible at T =z. A feasible final path satisfies the transversality condirions (A.18) and (A.19). Using the features of the stages we
G.-J.C.Th.
1060
van
derive the conditions of path 5.
Schijndel,
Dynamic
hehaviour
under
personal
taxation
as presented by table 2. We now prove the conditions
Path 5: 1,=1,=0,&>0. Substitution of (A.18) and (A.19) in (A.ll)
for T=z
gives
(l-rd)e-iz-(l-r,)e-iz=O, which means that path 5 can be final stage only if rd = TV Step 3. To couple a feasible preceding path with a final path we use the continuity properties of relevant variables. Therefore we test each stage
whether coupling with the final stage will or will not violate the (necessary) continuity properties of the state variables K and X and the co-state variables $1 and rj2. If the set of feasible preceding stages appears to be empty, then the relevant final stage is the description of the optimal policy of the firm for the whole planning period, supposing that the initial state constraints (15) are fulfilled (step 4). If the set is not empty we apply the testing procedure again. Ih this way we find a still longer string of stages, constituting an optimal policy string. On which conditions will path 5 precede the string path 2+path 3 in the case i>(l-r,)r? To prove the continuity of $, we consider especially the continuity property of I, as this variable is on path 1 and equal to zero on path 5. So, 1, can only be continuous
Example.
lim ,I,=0 or A,>0 Tl’,, From (A.1 l)-(A.14) we derive
when
A,=0
l-path have to positive if
on path 1.
>I -i
-A,(1 -TJ(g
- &+l
+h).
So, 1, > 0 for 1, = 0 if (1 + h)( 1 - r,)(dO/dK) < h( 1 - T,)r + i. Because i > (1 - z,)r and (1 -rJ(dO/dK) > (1 - TJr on path 1 this condition could be fulfilled. Which path could precede the string path l-+path 2
No
K discontinuous
3
No
K -X
4
No
i<(l
5
Yes
discontinuous -TJr
2-+path 3?
Step 4. If the set of feasible preceding stages is empty we stop the iterative procedure of step 3 and, finally, check whether the initial state constraints (15) are fulfilled. Both optimal policy strings start with path 1 on which Y(T)=hX(T)+K(T)=(l+h)X(T).
References Arrow, K.J. and M. Kurz. 1970. Public investment. the rate of return and optimal hscal policy (Johns Hopkins University Press. Baltimore. MD). Atkinson A.B. and J. Stiglitz 1980, Lectures on public economics (McGraw-Hill. New York), Bensoussan, A., P.R. Kleindorfer and ChS. Tapiero. 1978, Applied optimal control (NorthHolland, Amsterdam). DeAngelo. H. and R. Masulis. 1980, Leverage and dividend irrelevancy under corporate and personal taxation, Journal of Finance 35, 453467. Feichtinger, Cl., 1982a, Anwendungen des Maximum-Prinzips im Operations Research, Teil I und 2 (Applications of the maximum principle in operations research, Vols. I and 2), OR Spektrum 4, 17l-IYO and 195-212. Feichtinger, G.. 1982b. Optimal control theory and economic analysis (North-Holland, Amsterdam). Feichtinger, G.. 1985, Optimal control theory and economic anaiysis. Vol. 2, (North-Holland, Amsterdam). Geerts, A., 1985. Optimality conditions for solutions of singular and state constrained optimal control problems in economics. Master’s thesis (Eindhoven University of Technology, Eindhoven). Kamien, M.I. and N. Schwartz. 1983, Dynamic optimization, the calculus of variation and optimal control in economics and management. 2nd reprint (North-Holland, New York). Kim, E.H., W.G. Lewellen and J.J. McConnell. 1979, Financial leverage clienteles, Journal of Financial Economics 7. X3-109. King, M., 1977. Public policy and the corporation, (Chapmann and Hall, London). Leban, R. and J. Lesourne. 1980, The lirm’s investment and employment policy through a business cycle, European Economic Review 13. 43380. Leland, H.E., 1972, The dynamics of a revenue maximizing firm, International Economic Review 13. 376385. Lesourne, J., 1973, Modeles de Croissance des Entreprises (Growth models of the firm), (Dunod, Paris). Lesourne, J. and R. Leban, 1982, Control theory and the dynamics of the lirm, OR Spektrum 4, l-14. Ludwig, Th.. 1978. Optimale Expansionpfade der unternehmung (Optimal growth stages or the firm), (Gabler, Wiesbaden). Miller, M.H., 1977, Debt and taxes, Journal or Finance 32, 261-275. Modigliani, F. and M.H. Miller, 1963, Corporate income taxes and the cost of capital: A correction, American Economic Review 53, 433-l43. Moerland, P.W., 1978, Optimal lit-m behaviour under different liscal regimes, European Economic Review 1 I, 37-58. Pontryagin, L.S., V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mischenko, 1962, The mathematical theory of optimal processes (Wiley, New York). Russak, B.I., 1970, On general problems with bounded state variables, Journal of Optimization Theory and Applications 6, 424-451. Sethi, S.P., 1978, Optimal equity and linancing model of Krouse and Lee: Corrections and extensions, Journal of Financial and Quantitative Analysis 13, 487505.
1062
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wn
Schijndel,
D~wunic
behaviour
under
personal
taration
Stapleton, R.C. and C.M. Burke, 1977, European tax systems and the neutrality of corporate linancing policy, Journal of Banking and Finance 1, 55-70. Van Loon, P.J.J.M., 1982, Empolyment in a monopolistic Turn, European Economic Review 19, 305-327. Van Loon, P.J.J.M., 1983, A dynamic theory of the firm: production, linance and investment, Lecture Notes in Economics and Mathematical Systems 218 (Springer, Berlin). Yli-Liedenpohja, J., 1978, Taxes, dividends and the capital gains in the adjustments cost model of the firm, Scandinavian Journal of Economics 80, 3999410.