Dissertation proposal on financial systems and tax accounting- Havard Style

The Effects of Taxation on the Firm's Investment and Financial Behavior

Author(s): Göran Eriksson

Source: The Scandinavian Journal of Economics , 1980, Vol. 82, No. 3 (1980), pp. 362-377

Published by: Wiley on behalf of The Scandinavian Journal of Economics

Stable URL: https://www.jstor.org/stable/3439747

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THE EFFECTS OF TAXATION ON THE FIRM'S INVESTMENT AND FINANCIAL BEHAVIOR

Goran Eriksson*

Industrial Institute for Economic and Social Research, Stockholm, Sweden

Abstract

The effects of taxation on the behavior of firms are examined in this paper. Earlier studies of this issue have concluded that, given certainty and true economic de- preciation, corporate taxes have no effect on the firm's financial policy. External, e.g. legal, restrictions on the financial options of the firm have to be introduced in order for taxes to have such an effect. Our model turns these conditions around by taking the risk of bankruptcy into account. There then exists an interior finan- cial optimum for the firm. The firm's investment decisions become inseparable from its financial decisions. Furthermore, changes in taxes on profit, personal income and capital gains induce the firm to make smooth adjustments in its optimal financial structure and optimal rate of investment.

I. Introduction

Although considerable attention has been paid to the question of the effects of taxation in the literature, there are very few firm models which incorporate

and can deal with the effects of different taxes and simultaneously take the firm's financial decisions into account in a realistic way. Almost all of the models rest on purely neoclassical assumptions implying, among other things, certainty and perfect capital markets.' Under these assumptions the firm's behavior has no effect on the price of different forms of finance. If there are no additional external restrictions which limit the firm's financial options, this means that the firm will use only one financing source, i.e. that which has the lowest cost after taxes. However, such behavior would imply financing solely through borrowing or by retaining profits. which is inconsistent with empirical evidence.

In this paper we investigate the optimal investment and financial decisions for a growing, profit-maximizing firm and take into account taxes on profits, personal income and capital gains. The basic model used is a somewhat modified

* I am grateful to Charles E. McLure Jr., National Bureau of Economic Research, for valuable criticism and helpful suggestions. 1 See e.g. Stiglitz (1973) and King (1974). In Stiglitz (1972), uncertainty is allowed but in a model without taxes.

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Effects of taxation on the firm's behavior 363

version of one presented in Eriksson (1978).1 This model allows us to drop the

assumption of perfect certainty and introduce the following financial restraints:

the firm's interest rate on borrowed capital is an increasing function of its debt

to equity ratio, and the rate of return required by the stockholders is an in-

creasing function of this ratio and a decreasing function of the proportion of profits paid out.

Due to these financial restraints, we obtain several interesting results which

contrast with earlier studies where taxes have been incorporated. The value of

the firm depends on its debt and dividend policies; an interior optimum exists

with positive volumes of both internal and external finance. A complete inter-

relationship between the firm's borrowing, internal financing and investment

decisions also emerges. Consequently, taxes always have implications for real

magnitudes, i.e. they influence the firm's optimal investments and rate of growth.

II. The Model

11.1. Valuation of the Firm

We assume a linear homogeneous production function for the firm with respect to labor and capital inputs, fixed output and input prices, and no growth costs.2

Given these assumptions and given that the interest rate rises with increased borrowing, it can be shown that the firm's optimal input mix is determined

solely by the price of labor and the marginal value product of labor.3 A profit

tax, personal income tax or capital gains tax will then have no impact on the

proportions in which labor and capital are demanded. This result allows us to

disregard the production activity of the firm. Furthermore, we assume that

depreciation for tax purposes coincides with true economic depreciation and that the depreciation rate is constant. Thus, the rate of return on total capital is assumed exogenously given.

The firm acquires funds for investment only through borrowing and by

retaining profits. For period t, we define the rate of return on total capital (rt)

as profit net of depreciation divided by total assets and the leverage ratio (he)

as debt divided by equity. The rate of return on equity (rEt) is then determined according to the well-known identity

rE:= rt + ht(rt-it), (1)

where it denotes the interest rate.

1 For references to other models, see Gordon (1962) and Lerner & Carleton (1966). 2 Since dynamic restraints are covered on the financial side in the form of rising costs of capital (see Section II. 2), an additional restraint on the production side by allowing growth costs in the model would only strengthen the obstacles to expansion due to financial restraints and would work in the same direction with respect to the optimal payout ratio, investment rate and growth rate; see Eriksson (1978), Chapter 5. 3 Op cit., Chap. 5.

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364 G. Erikson

Assume a proportional profit tax rate (8s) and let ut denote the payout ratio. The growth rate of equity will then be

Vt = (1 -Ut) (1 -SV) rEn (2)

If Et is equity, dividends paid by the firm are t(L -sv)rEEt. Letting sI represent a proportional tax rate on personal income, it follows that the

dividends received by the stockholders after taxes will be

Ut = ut(l - s,)(I- sv) rEt E.(3

The stockholders discount rate (k) is determined by what they can earn on

alternative financial investments. A proportion (SI) of these earnings is paid

as income tax. This makes the discount rate after income tax equal to (1 - 8) k.

Furthermore, the increase in the value of stocks (i.e. capital gains) is assumed

to be taxed on an accrual basis.' Let sR be the fraction of the capital gains each

year that is taxed as personal income. As s, is the income tax rate, S1Rs is the annual rate of capital gains taxation. If dP/dt is the increase in the value

of stocks, then the corresponding increase after capital gains tax amounts to

(1 -SIjs)dPldt. We conceive of a market in which the firm's stocks are traded. On this market

all existing and potential owners of the firm have access to equal and costless

information about the actual price of its stocks and other relevant facts about

the firm. The market is cleared when the stockholders' discount rate after in-

come tax is equal to the dividend yield plus the rate of growth in their own

price of all stocks, after taxes, in each period. This market equilibrium con-

dition can be stated

(I - s) kt = Ut + (1 -s 8IS) }p /Pt, (4)

where Pt is the price of the stocks. We now introduce a very simple form of uncertainty. We assume that the

actual rates of return on total assets at any point in time (rt) is normally distributed around the long-run expected mean value (r) with constant vari-

ance. Given a time-invariant leverage ratio and payout ratio, this simple

stochastic profitability function is compatible with steady-state expansion of the firm with short-run fluctuations in all its monetary variables around

given trend values.2 Of course such a dynamic steady state does not exactly

I Of course it would have been desirable to take into account the fact that, in reality, capital gains are taxed at the time of realization. But in order to do this satisfactorily, an analysis of the optimal length of shareholding and consequently, a much more compli- cated model than ours would be required. Therefore we dispense with these complications by assuming quite simply that accrued-and not realized-gains are taxed.

Since the analysis is partial and only considers the effect of taxation on the individual firm, the discount rate may be regarded as unaffected by either profit or capital gains taxes. For a similar treatment of the incidence of these taxes, see Bergstrom & S6dersten (1976). 2 See Lintner (1964).

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Effects of taxation on the firm's behavior 365

hold in reality. But this simplifying assumption is common in the literature and

accepted by many authors as a description of how firms expand over long periods of time.'

When the firm has decided on a certain long-run growth rate v, the growth of its expected dividends in the long run is given by

Ut = UOeVt. (5)

Assuming payouts in an unlimited future, solution of the differential equa- tion (4) then gives2

p=| Ut -ss (t s, -exp (- (1- 8,) k) dt (-, (1ss (6)

where

UO= u(I -.-s) (I -sV)rEEO (7)

v = (1-U) (1 -8V)rE (8)

rE= r+h(r-i). (9)

The amount of internal funds expressed by Eo is predetermined by rationed earnings from earlier periods.

II.2. Financial Restraints

We assumed random fluctuations in the actual rates of return on total assets.

If the rate of interest on debt is not subject to such variations, then according

to (1), a higher leverage ratio brings about higher relative variability over time in the rate of return on equity. This increases the probability that the rate of return on equity will fall below zero, in which case the firm may not

be able to meet its obligations and may be forced to go bankrupt.3 Since bankruptcy involves administrative expenses, outlays for reorganization of the firm and losses due to the adverse effects of financial difficulties on the firm's

1 E.g. Gordon (1962), Marris (1964), Lintner (1964), Solow (1971) and Jakobsson (1976). 2 Note that eq. (6) expresses the value of the firm from the stockholders' point of view as the discounted value of future expected paid out dividends, net of personal taxes on dividends and capital gains. Since this valuation formula is derived from the market equilibrium condition (4) and the discount rate (1 - s) k reflects the individual income tax rate and before-tax yield on alternative investments, the use of (4) only shows that valuation of the stocks at the margin equals the required rate of return. For similar treatment in deriving the market price of stocks, see Lintner (1964), King (1974) and S6dersten (1977).

Note also that (1 -si) k must be greater than (1 -8Isp)v in order for the present value of future dividends (PO) to be finite. 3 These arguments are in line with the traditional view on this subject; see e.g. Schwartz (1959). A more detailed account of how changes in the debt to equity ratio affect the risk of bankruptcy and rate of interest is given in Eriksson (1978), Chapter 4, where cross- sectional estimations also show a significant positive relationship between the interest rate and the leverage ratio.

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366 G. Eriksson

flow of operating earnings, there are indeed real costs associated with excessive

debt. This is an important reason for expecting a positive covariation between

the leverage ratio and the rate of interest demanded by the firm's lenders.'

We formulate the interest rate function

i = i(h), (10)

where Oi/Sh > 0 and aS2i/ah2 > 0.

On purely theoretical grounds, however, it is more difficult to reach un-

ambiguous conclusions concerning the degree to which the interest rate will

rise with higher leverage. Nor have any empirical studies been published on

this subject. Therefore, we accept the common view that increased reliance on

debt should not exert an appreciable influence on the likelihood of bankruptcy

at a low leverage ratio, whereas this influence will be much stronger at a high

leverage ratio.2

Next, we turn to the question of the influence of debt financing on the

stockholders' discount rate. It is clear that the increased probability of in-

solvency of the firm due to a higher leverage ratio also affects stockholders.

This, in turn, establishes the same kind of positive relationship between the

discount rate and the leverage ratio as for the rate of interest on borrowed

capital. It should be emphasized that even if markets were perfect, the opera- tion of the Modigliani-Miller arbitrage process could not make the rate of

discount decline at high levels of debt. This is due to the fact that every debt-

holder is in a preferred position relative to stockholders in their claims on all

cash flows and assets of the firm and to the presumed risk aversion of all in-

vestors.3 There is also empirical evidence which indicates that the discount

rate is a monotonically increasing function of the leverage ratio.4

Uncertainty should also imply that the stockholders' rate of interest rises

as they increase their borrowing on personal account. Thus dividend policy

might affect the discount rate demanded by the stockholders. We now propose

the hypothesis that the discount rate is a falling function of the payout ratio within the relevant interval of values for this ratio (i.e. between zero and one).

One reason is that stockholders' consumption should be paid for partly by current dividends, and increased borrowing costs might prevent them from

fully substituting increased personal borrowing for postponed dividends. Another reason is that investors regard dividends expected in a distant future

as more risky so that such dividends are discounted at a higher rate than those expected in the near future. Assuming risk aversion on the part of the stock- holders, a lower payout ratio, which pushes dividends further into the future,

1 The firm obviously has to declare bankruptcy as soon as the negative rate of return causes the value of its total assets to fall short of the value of its debt. 2 See e.g. Baxter (1969). 3 Robicheck & Myers (1965), Chapter 3. 4 See e.g. Brigham & Gordon (1968) and Bennet, Graham & Tran Van Hoa (1969).

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Effects of taxation on the firm's behavior 367

will raise the average discount rate for the entire flow of dividends.' Further-

more, it seems plausible that firms which try to speed up the rate of growth

by lowering the payout ratio would, at the same time, switch to more risky

activities and devote relatively more resources to research and development

of new products, which involve greater risk.2

Of course, we are aware that these arguments are not uncontroversial. Many

authors have claimed that changes in internal equity financing do not affect,

or at least have a neglible effect, on the cost of capital to the firm.3 But the

Modigliani-Miller conclusion of no effect at all requires, among other things,

that any change in the leverage of the firms can be completely offset by changes

in homemade leverage. This hardly is a realistic assumption. Although reasons

can be found for market imperfections and thus for the existence of dividend

effects, it is another matter to make any precise statement about the way in

which dividend policy influences the discount rate. Whether this rate is an

increasing or decreasing function of the payout ratio should ultimately be an

empirical question.

Several authors, e.g. Brigham and Gordon (1968) and Bennet, Graham and

Tran Van Hoa (1969) have verified empirically that the discount rate rises

with enlarged internal equity financing. Furthermore, calculations using

Swedish data (allowing for the fact that an exogenous change in the discount

rate might, in turn, affect the payout ratio) show a highly significant rela-

tionship between these two variables. This implies that the discount rate rises

at an increasing rae when the payout ratio is decreased.4 These hypotheses can

be formulated as follows:

k = k(h, ),(11)

where Ok/ah >0, O2k/Oh2 >0, O.ck/u < 0 and a2k/aU2 > 0.

Given that every stockholder's valuation of the firm equals the price of

its shares and the fact that the negative impact of uncertainty on the price is

taken into account through a higher rate of discount, we can conclude our

enumeration of the assumptions by stating that the stockholders' objective

is maximization of the share price.

1 See e.g. Gordon (1962) and Walter (1963). 2 See Brems (1976). 3 The most prominent pure-earning theorists are Modigliani and Miller (1958). 4 Eriksson (1978), Chapter 4. Note that our dividend hypothesis is not at variance with the observations that profitable and fast-growing firms which are regarded as safe invest- ments usually retain a relatively large part of their profits, and that the raising or lowering of dividends is often seen as a signal of changes in future prospects. Instead, these implications may be derived from the model. Assume, for instance, an exogenously caused permanent increase in the rate of return on total capital. It can then be shown that according to the model, this increases the retention rate and rate of growth of the firm. If the elasticity of the payout ratio with respect to the rate of return is numerically smaller than one, the volume of dividends paid out is also increased.

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368 G. Eriksson

III. The Firm's Optimal Decisions

We now turn to the optium financial position of the firm. According to (6),

by maximizing the price of the firm's shares (PO) with respect to the decision variables, the leverage ratio (h) and the payout ratio (u), we obtain the neces-

sary first-order conditions:'

_ _ _ _ _ _ _ _ t i rE a i Ah~O ,D~kP-vr-i-h ----k ih]=0 (12) Sh rE(lcI- v) [ h k Ochj

apo PO

au ( 8-,8r8) u(l'-_V)

X [(1 -83) (E- u ak- I--(8V + 8j8.e(1-8V)} rS = ? (13)

The assumption of steady-state and balanced growth implies that these

optimality conditions, which hold for the initial period of time, also hold for all future periods. Condition (12) can be fulfilled only if the bracketed expres-

sion equals zero.2 In other words, the firm will borrow capital to such an

extent that the rate of return on total capital equals the marginal cost of debt.

This marginal cost reflects the effects of an increased leverage ratio on the

interest rate and the stockholders' required rate of return.

Other important findings are apparent from (12). First, since the discount

rate is rising, with the leverage ratio (Okiih >0), debt financing should not be

pushed so far that r = i + hai/Sh, which is the condition for maximization of the rate of return on equity. Second, if not only the discount rate but also the

interest rate were independent of the leverage ratio (akih = ailah = 0), we would have a situation that prevails in the certainty models, implying either zero

leverage or a maximal leverage determined only by outside restraints. Third, the tax parameters do not affect the criteria for optimal debt financing. This

might seem curious at first. However, since equity capital is a predetermined variable, the highest share value must be obtained with respect to the leverage

ratio when the percentage increase in the rate of return on equity, followed

by an extra unit of debt, equals the percentage increase in the discount rate.

Clearly, this happens at the same leverage ratio regardless of whether the discount rate and rate of return are defined before or after taxes.

Condition (13) can also only be fulfilled if the bracketed expression equals zero.3 This implies that the stockholders' required marginal rate of return net of personal income tax equals the rate of return after taxes on ploughed-back

profits. The term sv+s1sR(I -sv) may be seen as the total tax burden on re-

1 Note that k' =(1-sI) kI(1-sI8S). 2 Because k'POIrE(k'-v) cannot be zero. 3 Because Po/u(k' - v) (1- 818R) cannot be zero.

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Effect of taxation on the firm's behavior 369

tained profits.' Due to the falling discount rate with respect to the payout

ratio (ak/au<o), financing by retained profits should not be increase to the point where the discount rate equals the rate of return on equity. If the

discount rate were not affected by the payout ratio (ak/au=0), which would

be the case if there were no uncertainty regarding the future flow of dividends

from the firm, a corner solution arises where the entire profit is either retained

or paid out, depending on whether the rate of return on equity is higher or

lower than the discount rate net of taxes. The fact that the rate of discount is a

falling function of the payout ratio also explains why firms pay dividends even

when the personal tax rate is higher than the tax rate on ploughed-back

profits.2

Let us now compare our results with those of Stiglitz and King, who also

assume perfect competition in the output and input markets and that the firm

maximizes the wealth of its stockholders. Given true economic depreciation, they found that the firm's real capital should be adjusted in each short time

period so that the marginal value product of capital equals the gross rate of

interest, that is:

paQ/aK =pK(a + i), (14)

where p and P, are the product price and price of capital goods, respectively,

aQ/lK is the marginal product of capital, a is the depreciation rate and i is a constant market rate of interest. This is consistent with our model in the

following respects. First, if we assume, as Stiglitz and King do, that the

interest rate and the discount rate are equal to a constant market rate (i),

we obtain an optimality condition with the same meaning as (14) with respect

to borrowing.3 Second, the fact that there are no tax parameters in (14) also

conforms to our borrowing optimality condition. However, because of the

separation between investment and dividend decisions4 in the models of

Stiglitz and King, taxation does not interfere with the firm's investment

policy, as it does in our model.

It should be pointed out that this conclusion is modified when legal con-

straints prevent the firm from using only the cheapest source of finance. Such

constraints, along with taxes that discriminate between different ways of

obtaining funds, imply that the firm, in fact, faces a stepwise rising capital cost

l S6dersten (1977), p. 5. 2 Assume that the rate of return on equity equals the discount rate at zero retention be- fore taxes. It is then obvious according to (13) that only when sk/au < 0, relatively heavier taxation on personal income need not induce the firm to give up dividend payments entirely.

3 Defining the return on equity as VE -=PQ -PLL -(a + i) (1 + h) E, where h is the leverage ratio and E is equity, it is clear from maximization of PO with respect to h, that condition (14) is obtained. Note the identity pKK = (1 + h) E and that E is assumed fixed. 4 These decisions are separated due to their conception that there is no difference between debt and equity and that the firm can always obtain the desired amount of funds at constant cost.

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370 C. Eriksson

curve. The transition from one step to the next (higher) step indicates the rise in

the cost level, as the firm is forced to switch to a more expensive financing

method. Here, taxation may influence the firm's behavior. But this only

happens when the curve for the return on marginal investment intersects the

capital cost curve in the horizontal segment.

Returning to our model, we may note that it can rather easily be extended

to deal with external equity finance. If funds are aquired through issues of

new shares in a volume which is proportionate to profits in each period and

if no transaction costs are involved, the present value of all future net dividends

(the payouts minus the inflow of funds from new issues) at point in time t =0

will be

[(1 - s1) U - c] (1- SV) rEEO Pno (I - sl) k-[(1 - ) (I -ss.) +c] (I - s)rE ( where c is the new issue rate. The best financial policy for the owners of the

firm would then be found by setting values of u and c which maximize Pno.

If the discount rate k were a function only of the net payout ratio (u - c), the

fact that dividends are more heavily taxed than capital gains, s8 < 1, would

imply that the firm's net worth can always be increased by reducing both u

and c, leaving the net paid-out dividends unchanged. On the other hand, in

the case of no tax subsidy on capital gains, 8R =1, net worth will be inde-

pendent of such changes between internal and external equity financing and

no interior optimal solution exists. However, a much more reasonable assump-

tion seems to be that old and new stockholders have different risk and liquidity

preferences, which means that k is affected differently by equal changes in

u and c. This might explain why firms use both retained profits and new issues

sources of finance.

IV. The Effects of Taxation

We now examine how the firm reacts to changes in tax parameters. When a

tax parameter is changed the new value is expected to last forever and the

optimal ratios of the firm's endogenous variables, determined on the basis of

this value, also persist indefinitely. Since there are no growth costs in our

model, the firm immediately switches from one set of optimal ratios and an

optimal rate of steady-state growth to another.

By means of total differentiation of the optimality conditions (12) and (13)

-see Appendix-we get the following effects on the firm's optimal leverage

ratio (h*) and payout ratio (u*).

The second column in Table 1 clearly shows that increased profit or capital

gains taxation affects the firm's propensity to invest negatively (raises the

payout ratio), while higher income taxation has the opposite effect. These ef-

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Effects of taxation on the firm's behavior 371

Table 1. Changes in h* and u* associated with increases in tax parameters

h* ,*

Tax parameter

8v ? + Si ?_ SR ? +

fects, which arise due to changes in the tax parameters, raise the stockholders'

marginal required rate of return net of personal income tax MCu = (1 -s8)(k -uXak/Su), relative to the rate of return after taxes on ploughed-

back profits, MRu=[1-(SV+SISR(1-sV)]rE at given values of u. Since a2k/Ou2 >0, the optimality condition (13)' can then be restored by raising u.

The first column in Table 1 indicates that we cannot draw unambigous con-

clusions about the effects on the optimal leverage ratio. The effects of tax

parameters on h* are indirect through their influence on u*, and a rise in u*

can induce either an increase or a decrease in h*. The form of the discount rate

function determines which of these changes will occur. It can be shown that a

higher elasticity of substitution between h and u in the discount rate function

raises the probability that an increase in sv or sR decreases the optimal relative indebtedness (h*).l

However, the way in which the firm's optimal leverage ratio is affected can

always be ascertained according to the following three well-known formula-

tions, where the k-function is of (a) an additive type, with h and u perfectly

exchangeable; (b) a Cobb-Douglas type; and (c) a Leontief type, with a fixed

relation between h and u.

Higher profit or capital gains taxation or lower personal income taxation

implies a decrease, no change at all or an increase in the firm's optimal leverage

ratio according to a, b and c, respectively.2

Let us now examine the effects of taxation on the growth and market value

of the firm. Higher profit or capital gains tax or lower personal income tax

have been found to increase the optimal payout ratio (u*) and decrease the

optimal leverage ratio (h*). Assuming an elasticity of substitution between

h and u in the discount rate function greater than one, it follows from equa-

tions (8) and (9) that these changes in h* and u* bring about a decrease in the

optimal rate of growth (v*). In the case where the elasticity of substitution

1 These tax changes raise u*. The rise in u* makes (aklsh)lk shift upwards when x = a[(9k/9h)/k]/su = {k s2k/sh su - (skl/sh) (skisu) }/k2 > 0, which disrupts the debt financing condition (12). Due to sil/h >0, O.2/sh2 > 0, sk/sh > 0 and 92k/9h2 > 0, the r.h.s. of (12) is an increasing function of h. This condition can be restored by lowering h. Lower negative values of 92k/9h su mean higher elasticity of substitution in the discount rate function. Obviously the following must hold: higher elasticity of substitution implies a higher x, which in turn indicates a greater probability that an increase in u* will induce a fall in h*. 2 The additative, Cobb-Douglas and Leontief-type k-function imply x > 0, x = 0 and x < 0, respectively.

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372 G. Eriksson

Table 2. Changes in v* and P* associated with increases in tax parameters

v* Pa*

Tax parameter

.9v - - 8i + 8R

is less than one, things become more tricky because both h* and u* are in-

creased. Will the negative effets on v* due to an increase in u* still be so strong

that they override the positive effects of the higher h*? The answer to this

question seems to be yes. If v* were to be increased, the rise in h* must be

considerably larger than the rise in u*.1 This, in turn, means not only that the

elasticity of substitution between h and u in the k-function has to be very low

(which is necessary for the partial derivatives ak/lch in the debt financing con- dition (12) to be strongly affected by the increased u*), but also that the

interest rate and the discount rate have to be very insensitive to variations

in h (necessary in order for the optimal leverage ratio h* to be materially

affected by a change in ek/Sh).

Accordingly, a great deal suggests that the firm's rate of growth should be

negatively affected by higher profit and capital gains taxation, but positively

by higher income taxation. In addition, a higher profit tax rate reduces earning

capacity, i.e. profits after tax are reduced. Even without any induced effects

on the firm's debt and dividend policy, increased profit taxation would never-

theless decrease the growth rate due to this capacity effect.

The way in which tax parameters affect the optimum (maximum) share

value PO is found by applying the fact that PO is the firm's maximand. Assume

first that h and u are held constant. When sv or s, or SR is reduced, it is obvious from (6)-(9) that PO must rise. Now, allow h and u to take on new optimal values with respect to the altered tax parameters. Such an adjustment can only

bring about a further rise in PO. Thus, we conclude that the firm's maximum share value will be negatively influenced by higher profit, personal income

and capital gains taxation, respectively. The results regarding the effects on

v* and P* are summarized in Table 2.

Finally, it is interesting to note that the implications of taxation on the

1 Through total differentiation of (8) and (9) and setting dv = 0, we get some idea about how large the rise in h must be relative to the rise in u so as not to cause any change at all in v. We then obtain

dh = r+ (r-i)h (1 - u) {r - i-h gzl ah}

Even if silah = 0, it is clear from the above equation that dh has to be substantially larger then 1 in order for the variable to have reasonable values. Assume, for instance, r = 0O.10, i = 0.05, h = 1.0 and u = 0.5, which gives dh/du = 0.15/0.025 = 6.

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Effects of taxation on the firm's behavior 373

firm's behavior have recently been derived by Solow (1971) and Jakobsson

(1976). In their models, dynamic restraints are introduced in the form of

growth costs. Otherwise, they make the same basic assumptions as we do.

They assume given prices, maximization of the share value and balanced

of the firm. Not surprisingly, they also find that decreased taxes on profits or

capital gains and increased tax on personal income have a favorable influence

on the firm's optimal rate of growth.

Our analysis has been carried out under the assumption that changes in tax

parameters always alter the relation between the stockholders' marginal re-

quired rate of return and the firm's rate of return on ploughed-back profits at

given levels of the leverage ratio and payout ratio. But there are situations when

this does not occur. Consider, for example, the case where the total tax burden

on ploughed-back profits equals the personal income tax rate. It is then clear

from the internal financing condition (13) that changes in any of the tax

parameters will leave the optimal payout ratio unchanged. At the same time,

this means that there is no incentive for the firm to alter the optimal leverage

ratio. Another example is when capital gains are not given preferential treat-

ment vis-a-vis personal incomes. According to (13), the payout ratio and

leverage ratio will then be the same, regardless of the size of the personal in-

come tax rate.

V. Summary

Distinguishing features of the models used to analyze the effects of taxes on

the firm's behavior are the assumptions of certainty and perfect capital

markets. In this situation, the firm's cost for each financing method is con-

stant, given true economic depreciation and full deductability of interest costs.

As shown by Stiglitz (1973) and King (1974), under these conditions taxes will

not influence the optimal investment of the firm as long as legal constraints do

not limit its financial choice. Even in the presence of such constraints, which would force the firm to use more expensive financing methods when it in-

creases its demand for funds, changes in taxes do not always induce any reac-

tion from the firm.

A dynamic steady-state model which allows for taxes has been presented

for a value maximizing firm. As compared to its predecessors, our model is more realistic insofar as a simple form of uncertainty is taken into account.

By assuming short-run random fluctuations in the flow of gross earnings from

the firm, we have demonstrated that the firm's cost of capital rises with an

increased rate of borrowing or retention due to a greater risk of bankruptcy.

The introduction of these "internal" financial restraints in our model has pro-

duced some intuitively understandable results concerning the criteria according

to which investment and financial decisions are made and how the firm reacts to changes in tax laws.

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374 G. Ericsson

First, we found that the firm acts in the interest of its stockholders when it borrows up to the point where the marginal cost of debt-due to rising interest

and discount rates-equals the rate of return on total capital. The firm retains

profits to such an extent that the marginal cost of retention-due to a rising

discount rate, net of personal income tax-equals the rate of return on ploughed-back profits after all taxes. Second, these optimality conditions show

that there indeed exist an optimal debt to equity ratio and an optimal payout

ratio. This interior financial solution provides theoretical justification for the

observation that firms not only use debt and retained profits simultaneously

as sources of finance, but also pay dividends despite discriminatory treatment

of paid-out funds in the form of favorable tax treatment of debt.

Through the effect of taxes on the above concepts of marginal capital costs

and rates of return, we established their impact on the firm's behavior. The main results are:

(a) The imposition of taxes on profit, personal income and capital gains

affects the firm's financing and investment decisions. These taxes thus have

effects on resource allocation; this conclusion does not coincide with the findings from certainty models. However, a proper combination of taxes

might very well be nondistortionary. This is the case when the total tax

burden on ploughed-back profits equals the personal income tax rate.

(b) A lower profit or capital gains tax or a higher personal income tax

increases the firm's optimal retention rate and rate of growth due to the effects of the taxes on the marginal cost of capital in relation to the rate of return on

ploughed-back profits. These tax changes also increase the firm's optimal indebtedness when the elasticity of substitution between the leverage ratio and

payout ratio is high in the stockholders' discount rate function. When substi-

tutability is low, the impact on indebtednes is reversed. These findings con-

tradict the common view that profit taxes always favor debt financing owing

to the deductability of interest costs.

Appendix

Determination of the directional effects on optimal values of h and u

Comparative dynamic analysis is used in this appendix to determine the changes in the optimal leverage ratio (h) and payout ratio (u) brought about by increases in tax parameters.

We begin by totally differentiating the first-order conditions (12) and (13) and obtain the following equations written in matrix form:

Oah a 2p [_ 1- ___ A'S |=H1 Oh 8ps (Al)

Ou a32P

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Effects of taxation on the /irm'8 behavior 375

where H-1 is the inverse of the Hessian matrix

- 8p 82p -

H Oh2 ahau H I a2p (A2)

LauOh Ju2

and

8 = [8V, 81, 8]. (A3)

Next we ascertain the signs of the elements of (a) the vectors [S2P/aha8,

a2P/s@8] and (b) the inverse matrix H-1.

(a) Keeping h and u fixed, partial derivation of (12) and (13) with respect to

8 V sr and 88 gives

[ 1 ]1 [1] [ bP 1 [01 ( ah~ ahVA1 88 and a811 I and~ h8 (A 4) a2p a2P aUs2P

OU08J LJ a a8J L+ Lau 8,J L I

The signs in (A4) follow immediately since there are no tax parameters in the

second term of (12) and the second term of (13) is

[(1 -8s) (k -uak/au) -(1- 8v) (1 -8188) rE]/(l -818R). (b) From (12) and (13) we obtain

a2 B{k J2- s a2k} (A 5)

a =B rE1 2k (A6)

where

B= U(1 81)2(18-v)E0 > (A 7) [( 1- 8,) k - ( 1U) (1 - 8. SPA) ( 1 -s8) rE

Due to the assumptions a2rE/ah2 <0, S2ka/h2 >0 and S2k/1u2 > 0, it follows that a2P/ah2 < 0, and b2PIau2 < 0.

Furthermore we obtain

2p= 82p Bra - ak __k 2k) ~~h~u~u~h kk3a sa (A 8)

assuming S2k/Shau = S2k/auah. According to (A8) S2Pf/hau ? 0 when x = [ka2k/Shau - (ak/Sh) (8k/lu)] t 0.

25-804817 Scand. J. of Economic. 1980

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376 G. Eriklson

The second-order condition for a maximum requires that det. H >O. This

condition, combined with 02P/Ih2< 0 and a2p/au2 <0, give the elements in

H-1 the following sign:

[H+-] (A 9)

when x >0, or

[--= ](A 10)

when x <0.

Finally, by inserting (A4) and (A9) in (Al) and inserting (A4) and (A10)

in (A 1), we obtain:

OhI@s v <0, ch/Ss, >0 and chlasR <0 when x >0,

ch/asv>O, h/las1<O and ahIas>O when x<O,

as well as

aulasv > 0. cu/as, < 0 and at/I8R > 0 regardless of the sign of x.

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