Optimal Sharing Ratio in a Profit-sharing Arrangement

Optimal Sharing Ratio in a Profit-sharing Arrangement

Muhammad Rashid Basu Sharma University of New' Brunswick

Abstract This paper derives an optimal sharing ratio for a capitalist or non-participating firm in a model where workers get a fixed market-determined wage plus a share of net profit. For the analysis, modern capital murket theoiy and the stock value maximization principle are used. It is shown that the optimal sharing ratio depends on (i) the productivity efferct qf profit sharing, (ii) the corporate tax treatment of the sharing wage, and (iii) the risk preferences of mana,pementlshareholders. A relationship between productivity efjCects from profit sharing und optimal sharing ratio is derived. Assuming normally distributed profits of the firm, the results qf the paper are illustrated numerically.

Rksurnk Cet expose' pre'sente la proportion de partage optimal pour une firme capitaliste ou non-purticipante duns laquelle les employe's recoivent un salaire de'termine' par les ,fin-c.es du marche' avec un pourcentage des profits-net. On utilise pour l'analyse la the'arie du marche' capital moderne et le principe de nmvimalisutinn de la va l~ur rles r-c:ser-ves. I1 est de'montre' que la proportion de partage optimal de'pend de: ( i ) l 'efet que lc partage des prrfits a sur la prodirctixite', (ii) comment les taxes commer~ciales aflectent les salaires partage's et (iii) les pre'je'rences de risque des Kcstionnaires~ac'tionnaires. Un rapport est ktuhli entre les efiets sur la productivite' du partage des projits et la proportion de partage optimal. partir d'une distribution normale des profits d' une ,firme, les re'sultats de cet expose' sont illustre's nume'riyuement.

Profit or revenue or gain sharing has been around for about one and one-half centuries, as it reportedly started in France in the 1840s. However, it is only in recent years that profit sharing has been widely discussed and debated by economists, politicians, journalists, and others. Most of these discussions have focussed mainly on macro-economic implications in terms of employment, inflation and wage rates. For example, see Meade (1986), Mitchell (1987), Nuti (1987), Standing (1988), and Weitzman (1983, 1984, 198.5). The literature on profit sharing at the micro level has centered mainly on whether profit sharing produces "shirking" or, in contrast, a productivity or profitability effect. Some notable authors who have emphasized the possibility of shirking are Jensen and Meckling (1979) and Samuelson (1977). The possibility of a positive productivity effect is pointed out by Fitzroy and Kraft (1986, 1987), Jones and Svejnar (198.5), Meade (1986), Mitchell (1987), Vanek (1963, and several others.

However, the firm's choice of an optimal sharing ratio in a profit-sharing arrangement has been a neglected topic in the literature. The determination of the optimal sharing ratio is addressed by Aoki (1 980, 1982) in a game-theoretic framework but only for a participatory firm. In this paper, the firm's choice of an optimal sharing ratio is derived for a more conventional North American firm which is non-participatory. The

management of this firm makes decisions on behalf of shareholders, based on the share price maximization principle. The paper also derives the implication of tax deductibility provision of sharing wage for the optimal sharing ratio. The effect of risk aversion on the optimal level of the sharing ratio is shown in the framework of the capital asset pricing model. Finally, the paper shows that the optimal sharing ratio rises with the productivity effects from profit sharing, but the optimal sharing ratio increases at a declining rate.

Profit Sharing and the Productivity Effect It is controversial whether profit sharing can lead to in-

creased labour productivity. On the one hand, there is the view that under profit sharing, it is rational for each worker to reduce effort because he/she is exposed to only a fraction of output loss derived from his/her own lower effort (Samuelson, 1977). This is termed "shirking" or "leisure-on-the-job.'' A similar ar- gument has been propounded by property-right theorists (Jensen & Meckling, 1979). Alchian and Demsetz ( 1 972) ar- gue that profit sharing is likely to result in shirking by management/shareholders, although they do not deny the possibility of the increased incentives for workers not to shirk. According to them, in the absence of perfectly effective market competition for managers, nianagement/shareholders have the incentive to shirk in checking the performance of workers, unless they are entitled to the residual earnings. Furubotn (1976) has even gone further and has argued that profit sharing may

0 ASAC 1991 259 RCSA I CJAS, 8 (4). 259-266

OPTIMAL SHARING RATIO IN A PROFIT-SHARING ARRANGEMENT RASHID & SHARMA

lead to inefficient allocation of resources and exploitation of capitalists by impatient and opportunistic workers. Standing (1988) argues that under profit sharing, management-worker conflicts might be exacerbated in the face of poor profitability, adverse markets, and investments with long-term profitability at the expense of short-term profitability.

On the other hand, Fitzroy and Kraft ( 1 986, 1987), Jones (1982), Jones and Sevjnar (1983, Meade (1986), Stiglitz (1974), Vanek (1965), and others have pointed out several rea- sons why profit sharing may lead to higher labour productivity: ( 1 ) Workers will be more concerned about the success of their

enterprise, therefore they are likely to show more commitment. This is expected to reduce labour-management conflicts in the workplace, causing output to rise.

(2) There will be greater motivation to improve one's skill and efficiency and less resistance to technological improve- ments.

(3) In order to reduce the possibility of shirking and detect shirking co-workers, peer-group pressure or horizontal monitoring is likely to evolve. Discipline is likely to improve in the workplace.

(4) There will be an improvement in the firm's organizational efficiency. Thus, on theoretical grounds, it is difficult to resolve the is-

sue whether profit sharing reduces productivity by causing shirking, or increases it through the possible favourable factors listed above. Empirical evidence on the productivity effect of profit sharing is also inconclusive. For example, while Cable and Fitzroy (1980 a,b), Fitzroy and Kraft (1986, 1987), Jones (1982), Jones and Backus (1977), and Richardson and Nejad (1985) find apositive productivity effect of various schemes of profit sharing, Livingstone and Henry (1980) and Rosen and Klein (1983) find either no or negative effects of various types of profit sharing on profitability.

In our view, profit sharing may be effective in raising labour productivity in those firms where (i) the cost of detecting shirking is low or peer-group pressure is high, (ii) workers are less mobile, so that cost of their shirking is higher to them, and (iii) the design of work tasks is such that workers' interactive pos- sibilities exist and the contribution of each worker to firm's profitability is discernible. Some of these conditions are discussed in Sharma and Rashid ( 1 987). We shall assume such a firm in our model. However, we are not contending that there will necessarily be a labour productivity effect of profit sharing, but rather that if there is such a productivity effect, there may exist a sharing ratio which maximizes the value of the equity of the firm.

A Model of Profit Sharing and Returns to Equity Holders We shall consider a firm that undertakes an investment out-

lay of $A and hires N workers at the beginning of the period. Capital and labour together will be assumed to produce stochastic revenue of R net of non-depreciation and non-labour costs. The capital assets are assumed to depreciate at a perfectly known rate of @ during the period. Assuming the price of the capital stock to be unity now and unity at the end of the

period, the value of the firm's capital stock at the end of the period will be ( I -cp)A.

Other assumptions which will be made for the model are as follows. Firstly, financing of investment outlay will be assumed to come solely from common stock.' Secondly, workers will be assumed to be homogeneous, so that intra-labour conflicts can be ignored.2 Thirdly, earnings of the firm will be assumed to be taxable at the rate t. Finally, management/shareholders will be initially assumed to be risk neutral.

Consider the conventional wage system in which the wage rate per worker, to be denoted by w, is non-stochastic and pre- announced. With N workers and w as the wage rate per worker over the period, the firm's wage bill is

Bf= WN ( 1 ) Under profit sharing, the firm's wage bill depends upon the

nature of profit sharing.' We assume here a profit-sharing scheme under which workers receive a fixed market-determined wage, plus a share of the net profit. Such a scheme is known as full profit sharing (Standing, 1988, p. 10). This is un- like the scheme of Weitzman (1984, 1985, 1986) which involves paying a "core" wage which is below the market-determined wage, plus a share of profits or revenue. Given a high degree of risk aversion of workers, Weitzman's profit-sharing scheme appears to be less pragmatic. Full profit sharing on the other hand does not create any downside risk for the workers, because they are assured of w. The full profit-sharing scheme, to be used in this paper, is also consistent with "gainsharing" plans such as the Halsey, Rucker, Scanlon, and Improshare plans. Fitzroy and Kraft (1986, 1987), among others, have used the full profit-sharing plan in their analyses. Hereafter, we shall refer to the full profit-sharing plan simply as profit sharing.

Our proft-sharing scheme is in line with the major tenets of efficiency wage models. For a review of this literature, see Ak- erlof and Yellen (1986), and Yellen (1984). The efficiency models realistically assume that even though contracts can be made for supply of labour units, workers have discretion over supply of effort. Thus, labour contracts are incomplete on the second count. This motivates employers to introduce incentive rewards in such forms as profit sharing. The issue of the determination of equilibrium wage rate should therefore be addressed in this frame of reference, when extended to a general equilibrium framework for wage determination. The analysis here is based on a partial equilibrium framework.

For simplicity of the analysis, the following definitions will be used:

?i' = the firm's revenue with no profit sharing net of costs

q = cpA + wN (the depreciation and labour cost of the

h= the profit-sharing ratio, with 0 5 h < 1. Revenue under the profit-sharing system is then4,5

where a = the profit-sharing productivity factor. If a = 0, then obviouslyR'=?i'; that is, there is no productivity effect of profit

other than depreciation and labour cost;

firm); and

R ' = ( 1 +h)"R, OIa< 1 (2)

260 RCSA I CJAS. 8 (4). 2.59-266


sharing. From equation ( 2 ) , it is evident that 4r" rises with h, but at a declining rate.

Given the above assumptions and definitions, we define the profit non-sharing operator as:

Using the above notation and assuming that workers share only positive before-tax earnings,' the wage bill of the sharing firm is:

8, = h( 1 - 5)~R' - q~ + M". (4) Afew points may be noted about B,. First, this formulation

assumes that the workers share in the net operating earnings of the firm. Some other forms of profit-sharing arrangements, where h is related to sales, gross operating profit, after-tax in- come, growth of assets, etc., are also possible. However, the choice of the net operating earnings as the profit sharing base is motivated by the fact that it is the most commonly under- stood measure of profitability. Second, according to equation (4), the workers share only operating profit, not operating loss. This is consonant with the existing profit-sharing schemes in the U S . , Canada, Japan, and some European countries. Third, we assume in equations (1) and (4) that the firm is always able to meet its wage bill; otherwise the wage rate, w, will become stochastic, which will add unnecessary complications.

Since = B, when h = 0, subscripts to identify the fixed wage system and the profit sharing will be dropped.

We assume that the firm's profits are taxed at a constant rate t. The firm's profits, which are subject to tax, exclude all labour costs, including any profits shared out to labour. Then the total cash flow to equity holders of the firm under either labour compensation system is:

It may be noted that when h = 0, this equation gives the cashflow to equity holders under the fixed wage system, while if h > 0, it yields the cashflow to equity holders of the sharing

2'= (R'-q)(l - t)ll - (1 - 5)hl + A (5)

firm.

value of C under the fixed wage system is: Introducing E as the expectational operator, the expected

E(C,) = (E(R) - q)( 1 - t ) + A (6) The expected value of C under the profit sharing system,

after some rearrangement of terms, is: E(CJ = [(( 1 + h)" E ( R ) - 4)]( 1 - t)( 1 - h) + A (7)

+ ( 1 - t)h[( 1 + h)" E(R5) - qE(5)l. In equation (7), E(5) is the probability that the firm will not

share profits with workers, as can be seen from:

E(5) = J >(R)dR = F ( 4 ) (8) -m

where f(R) is the density function of R.

Risk Neutral Equity Holders and an Optimal Sharing Ratio Initially we are assuming risk neutral stockholders. The ef-

fect of risk aversion on an optimal choice of h will be illustrated in the next section.

With risk neutral stockholders. the market value of the equity of the firm at the beginning of the period when the workers are paid only the fixed wage is given by:

v/= E(C/)/( 1 + 1.) (9)

where r = the risk free rate of interest.

sharing is: Similarly, the value of the equity of the firm under profit

V,\ = E(T,\)/( 1 + 1.) (10) Ifthe objective ofthe film is to maximize the value of its eq-

uity. the firm will shift to profit sharing only if V, > V,. I t will choose that sharing ratio, h, which maximizes Z, the difference between V, and V,. The firm's problem, then, is to choose h such that Z is maximized.

Using equations (9) and (lo), Z can be written: z = [E(C,\) - E(C/ ) ] / ( 1 + 1.). ( 1 1 )

Noting the definitions of E(C,) and E(Z;) from equations (6) and (7) respectively, elaborating some of the terms, and defin- ing E(R) = p, Z becomes:

Z = ( / ( I + h)"p - h((l + h)"p - y)i( 1 - t ) - M"(I - t ) + rcpA + +( I-cp)A+ ( 1 - [)A[( 1 + h)" E(R5) - qE(@ - p( 1 - t )

+ ~ ~ N ( l - t ) - ~ [ ( ~ A - ( l - ( ~ ) A ] / i l + r ) .

Cancelling out terms, factoring out ( 1-t), and collecting terms which involve h linearly, we obtain:

z= [(( 1 + h y p - 1-11 - h

{(( 1 + h)"p - q) - ( 1 + h)"E(RT;) + q E ( 5 ) / ] ( 1 - t ) / ( 1 + r ) . We shall assume that the firm's revenue net of costs other

than labour and depreciation, R, is normal with mean p and standard deviation o. Under this umption, the partial mean, E(R5), becomes':

Using equations (8) and (12), Z becomes: E ( m = - o2A4). (12)

z = [{( 1 + A)" p - p} - hi(( 1 + h)" p - 4 ) - ( 1 + h)"(pF(q) - 02f(q))

+ qF(q),'I( 1 - t ) / ( 1 + 1.1. (13) Before obtaining an optimal value of the sharing ratio, , we note the effect of the productivity parameter, a, on Z. If a = 0, from equation (13), we see that for any h > 0, Z < 0. That is, if there is no productivity effect of profit sharing, then sharing profits with workers, given that the fixed wage rate does not change, leads to a lower value of the equity of the firm. This result continues to hold for low values of a. The reason for this is that in our model workers share all the operating profits, not just incremental profits. Unless labour productivity increases sufficiently to exceed the total sharing wage, it will not be economical for the firm to adopt the profit sharing plan. However, at a given h, dZ / da is:

d-%a = ( 1 - t ) / ( I + r)[p( 1 - h) +h{pF(q) - o2f(q))1(1 + A)" log(1 + A). Since (( 1 - t ) / ( 1 + r))( 1 + h)* log( 1 + h) is positive, the

sign of dZ/da depends on the sign of the terms in the

(14)

26 1 RCSA J CJAS, 8 (4), 259-266


square brackets. Given that A< I , p( 1 - h) is positive, the partial moment y ~ ( 4 ) - (~ l f iq ) is also positive, unless o2 is very high and simultaneously p is low. Even if this partial moment is negative, the first term in the square brackets will dominate. Thus, at a given h, and other things held constant, an increase in the productivity factor raises the market value of the equity of the firm.

For the optimal value of h, we require that dZ/& = 0 and a2Z/ah' < 0. These conditions are:

az/ah=((1 - L ) / ( I + r ) ) ~ a ( ~ +h)a-1 {P( 1 - + W Y q ) - 02f(q))} - { (( 1 + h)"P - 4) - (1 + h)a(PF(4) - oY(4)) + qF(q)]I

a2z/ax2 = ((1 - t)/( I + r.))[a(a - I )( 1 + h)a - 2

{ P( 1 - h) + h(PF(4) - &/))/

(15)

- 2 4 1 + h ) a - l ( p ( l - ~ ( 9 ) ) + 02f(4)1. (16)

d2Z/dh' is negative because a<l, F(q)<l and p( 1 - h) + yF(q) - &(4) is generally positive. With this, if such a sharing ratio exists at which d Z / d h = 0, then Z is an inverted U-shaped function of h. However, an explicit solution of h at which dZ/dh = 0 is not possible because equation (15) is highly nonlinear in h. One way out is to solve for optimal h through a numerical simulation, after postulating a set of reasonably plausible assumptions about h, 0, A, cp, w, N, t and r. This is the approach adopted in the subsequent analysis.

We shall assume the following quantification of parameters and variables in the model.

p = $2.0 million o = $0.7071 million A = $ I .6 million

w = $0.035 million N =40 workers t =0.25 r = 0.045. These data broadly reflect the operating characteristics of a

set of Canadian firms which are using (or have used) some form of profit sharing arrangements. The corporate tax rate of 25 percent is the average combined federal and provincial tax rate applied to small businesses in Canada. The risk free interest rate of 4.5 percent is the historical average of the long-term Government of Canada bond yields over the 1950-83 period (Hatch & White, 1985, p. 58).

In addition, we assume the profit sharing productivity factor, a. to be equal to 0.3. This value of is consistent with empirical findings of Fitzroy and Kraft [ 1986, 19871. With these data, q = wN+cpA = 1.88, F(q) = 0.4364, and f(q) = 0.3939. Replacing these and other numerical values in equation ( 13), Z becomes:

cp =0.3

z= [ ! ( I +h)0.'2-2 -h,l.32415(1 +A)().'- 1.05957)]0.7177.

Searching for the value of h which maximizes Z produces Table 1 and Figure 1.

The inverted U-shaped nature of Z as a function of h can be explained as follows. At a = 0.3 when h > 0, there are incremental returns to shareholders due to productivity effects from profit sharing. At low levels of h there is, however, a signifi- cant partial offset because workers share all operating profits, not just incremental profits. As h rises, the productivity effect from profit sharing increases, thereby further improving shareholders' incremental returns. But simultaneously, a rising sharing ratio increases productivity at a declining rate, because a < 1 .0 while the workers' share of profits rises at the rate of a. The two opposing forces balance at a point which yields an optimal sharing ratio, h*, which is 0.33. With an increase in h beyond h*, a rapid increase in the workers' share of profits, together with increasing productivity at a declining rate, reduces Z and eventually makes it zero at h = 0.72.

The Non-Tax- Deductibility of the Sharing Wage and the Sharing Ratio

If the sharing wage is considered by government to be the distribution of profit rather than a component of operating costs, then it may not be granted tax-deductibility. In this situ- ation, the terms in the second curly brackets of equation ( 13) will not be multiplied with (I-t). Therefore,

Z = [1(1 +h)".'2-2}0.75 -h(l.32415(1 1.05957~]/1.045.

Results are presented in Table 2 and graphed in Figure 2. Comparing Table 2 with Table 1 (or Figure 2 with Figure I ) , it is evident that when the sharing wage is not tax deductible, Z is maximized only at a lower h, because profit sharing now becomes less attractive to the equity holders of the firm.

Risk Aver-se Equity Holders und an Optimul Shur-in!: Rmtio For this section, we shall assume that shares ofthe common

stock of the firm are traded and all the assumptions of the capital asset pricing model are valid. Then, the market value of equity when the firm is under the conventional wage system is given by:

1'= [E(Z;) - 8Cfn(Ct ,Qj / ( 1 + 1.)

where r,,l 8

oil Cov(Z;, <,,) = covariance between C, and <,,.

(17)

- = rate of return on the market portfolio, = [E(FY,,) - r ] / o i , is the market price per unit of risk,

= variance of <,,, and

Similarly, the market value of the equity of the firm under the profit sharing plan, assuming once again the tax-deductibility of the sharing wage, is:

V', = IE(C,) - 8CO\~(C,. F?,J]/( 1 + 1 . ) . (18) Noting the expected values of C/ and 2*\ from equations (6) and (7) respectively, the difference between V'\ and V', , which will arise if the firm shifts from the conventional wage system to the profit-sharing arrangement is given by:

262 RCSA ICJAS, 8 (4). 259-266

OPTIMAL SHARING RATIO IN A PROFIT-SHARING ARRANGEMENT RASHID XC SHARMA

z' = v', - v'+= z - ( 1 - t)e[coLq?, F ~ , ) ! ( I + A)"( I - h) - I I + h( 1 + l JYho5 ,<i l ) - hyCo1(5, FJI / ( 1 + 1.) (19)

where Z i s the same as in equation (13) above. The last term expresses the effect of risk aversion on the differential market valuation of the equity of the firm, arising from adopting the profit-sharing arrangement. Equation ( 1 9) can be further sim- plified by noting that' :

and:

co~(R5, Frt1) = COL(R, F n I ) [ ~ ( 4 ) - Yf(q)]

Cov(5, Tn1) = - Cov(R, F,lllf(4)

Consequently, Z' becomes:

~~

Table 1 Optimal Sharing Ratio with Risk Neutrality and Tax Non- Deductible Sharing Wage.

h Z ($ Million) 0.1 0.01989 0.2 0.3 0.33 0.4 0.5 0.6 0.7 0.8

0.03204 0.03723 0.03753 0.036 12 0.02928 0.01707 0.00003

-0.02172

2 ($ Millions) 0 04 ,

0.03

0.02

0.01

0

-0.01

-0.02

Figure 1: Optimal Sharing Ratio Under Risk Neutrality

Z' = z - e( 1 - r)Cm~(R,Fr,,)[ (( 1 + h)? 1 - h) - I )

+ h(1 + h)"IF(4) - C l f ( 4 + hd(~/ ) l / ( l + 1.1. ( 20 1

In addition t o the data used to derive results in Tables 1 and 2. we require quantification of 8, Cov(R, <,, ), E(F,i, ) and $,. From Hatch and White (1985, p. 5 8 ) , E(OI ) and oiil are 12.95 percent and 18.17 percent respectively. These values, together with r = 0.045, yield 8 = 2.58. In addition, assuming a .20 cor- relation between the firm's R and Fnl, Cov(R, <,,) becomes 0.0257. With these values and noting that F(q) - qf(q) = 304 1, 2' is:

Z'=Z-0.0476[((1 +h)'"(l - A ) - I }

- 0.3041h(I + + 0.7405hl.

Table 2 Optimal Sharing Ratio Under Risk Neutrality and Tax Non- Deductible Sharing Wage.

h Z ($ Million) 0.1 0.0 1264 0.2 0.01 5814 0.3 0.0 1046 0.33 0.0073 0.4 -0.00265 0.5 -0.0227 0.6 -0.04968 0.7 -0.0826 0.8 -0.0 121 22

Z ($ Millions)

Tax-deductible Sharing Wage

0

Tax Non-deductible -0.04 * Sharing Wage

-0.06

-0.08

-O" I I I * -OI2 -0 14 0 '- 0.2 0 33

0 7 Sharing Ratio h

Figure 2: Optimal Sharing Ratio, Risk Neutrality and Tax Treatment of Sharing Wage.

RCSA I CJAS, 8 (4). 259-266 263


The following table and graph indicate h* and compare the result with that of the risk neutrality case.

It is evident from Table 3 and Figure 3 that with risk aversion of shareholders, profit sharing becomes more attractive to the firm. This is due to the fact that risk of operating profits is partially shifted to workers under profit sharing, relative to the conventional wage system. This has been recognized in the literature by Mukhopadhyay and Pendse (1983) and Stiglitz (l974), among others. Our analysis suggests that when stockholders are risk averse, the firm is better off with profit sharing at each value of the sharing ratio. The optimal sharing ratio rises from 33 percent under the case of risk neutrality to 37 percent under the assumption of risk aversion.

The Relationship Between the Productivity Factor; , and the Optimal Sharing Ratio, h*

From equation (IS), which yields h* when dZ/& = 0, we see that h* is a function of a, among other things. Since a closed form solution of ah/& is not possible because equation (15) is highly nonlinear in h, we use the same data as used above to derive this relationship.

To simplify, we assume risk neutrality and a tax-deductible sharing wage. With these assumptions, Z from equation (1 3) becomes: Z = [{( 1 + h)"2 - 21 - h{ l.32415( 1 + h)" - 1.05957]]0.7177.

obtained in the same fashion as in Table 1 and Table 2. The results are presented in Table 4 and Figure 4.

We see that h* rises with a, but at a declining rate. In other words, h* is a concave function of a. This shape of the function can be explained as follows. We know that at a given a, h* is obtained at a point where Z is maximized. From that level of a, if a increases, there is an exponential increase in the productivity effect from profit sharing. This suggests that the new optimal sharing ratio will now be higher. However, with suc- cessive increases in a, h* does not rise as much because rising h increases the sharing wage at the expense of the shareholder$ after-tax earnings. For example, h close to unity, implies that all profits accrue mostly to workers, and therefore that the market value of equity must decline. As Table 4 and Figure 4 indicate, for in the 0.7 - 0.9 range, h" is about 60% and not higher.

For selected values of a in the range of 0.1 - 0.9, optimal h is

Conclusions

This paper has addressed the following issue, which has been neglected in the existing literature: if a non-participatory finn wants to adopt a profit-sharing plan to replace the existing conventional wage system. which sharing ratio should it choose'? Using the share price maximization principle and capital market theory, an optimal sharing ratio was obtained. To the best of our knowledge. this paper is the first attempt to link the determination ofthe sharing ratio to the modem capital market theory.

We found that i f workers get a fixed market-determined wage, plus a share of net profit. the optimal sharing ratio is Lero

~ ~~

Table 3 Optimal Sharing Ratio with Risk Aversion.

'i I

0.1 0.0 199 0.2 0.0320 0.3 0.0372 0.33 0.0375 0.37 0.037 1 0.4 0.036 1 0.5 0.0293 0.6 0.0171 0.7 0.0000 0.8 -0.02 17 Obtained from Table I .

Z ($ Millions) 0.05

** 0.04

0.03

0.02

0.01

0

-0.01

-0.02

-0.03

0.0214 0.0353 0.0429 0.0439 0.0446 0.0444 0.0406 0.03 16 0.02 13 0.0002

With Risk Aversion T

' 0 0.33 0 37 0.7

Sharing Ratio h

Figure 3: Risk Aversion and Optimal Sharing Ratio

if there is no productivity effect of profit sharing. In fact, with a small productivity effect, the optimal sharing ratio is still zero, because workers share all the profits, not just the incremental profits. With a discernible productivity effect, we showed that the optimal sharing ratio exists in the range of zero to unity. The optimal sharing ratio was shown to depend cru- cially upon ( i ) whether or not the share wage is tax deductible to the firm, and ( i i ) whether the equity holders of the firm are risk neutral or risk averse. Both the tax deductibility of the sharing wage and risk aversion of shareholders make profit sharing more attractive to the finn.

The paper also derived a relationship between the productivity factor and the optimal sharing ratio. The optimal sharing

264 RCSA I CJAS. 8 (4). 259-266

OPTIMAL SHARING RATIO IN A PROFIT-SHARING ARRANGEMENT KASHII) Kr SHARMA

Table 4 Relationship between the Productivity Factor, a, and Optimal Sharing Ratio, h

N 1% 0. I 0.0 0.15 0.05 0.2 0.18 0.3 0.33 0.4 0.43 0.5 0.49 0.6 0.53 0.7 0.57 0.8 0.60 0.9 0.63

Optimal Sharing Ratio h 0 7

0.6

0.5

0.4

0.3

0.2

0.01

n

/

I

i i

L, 0 0.2 0.4 0.6

Productivity Factor 0.8 1

Figure 4: Relationship between the Productivity Factor and Optimal Sharing Ratio

ratio rises as the productivity factor rises, but the increase in the optimal sharing ratio is at a declining rate.

Several useful extensions of the model of this paper are possible. Firstly, the productivity effect of profit sharing can be made endogeneous and then estimated. Secondly, debt financing can be introduced, with the possibility that the firm's earnings may not cover interest expense in some states of nature. This will introduce bankruptcy considerations into the model.

1 .

2.

3.

4.

5.

6.

7.

8.

Endnotes The introduction of other corporate securities such as pre- ferred stock and bonds will not change the results of the model, provided they are assumed to remain constant in the analysis. However, this neglect is non-trivial if the optimal capital structure of the firm is affected by changing labour compensation arrangements. Since we are abstracting from the optimal capital structure problem of the firm in this paper, zero fixed commitment obligations are assumed for simplicity. Aoki ( 1982), in a shareholder-employee compensation game model, shows that the seniority principle ofemploy- ment leads to lower employment, higher wage rates, and rationing new jobs. There are several different types of profit-sharing schemes. A profit-sharing program can be top-hat or broadbased. A top-hat plan is only for managerial staff while a broadbase plan permits every worker to participate. At the same time, a plan may be a current payment plan or a deferred payment plan. It is also possible to have a combination of both. Current payment plans are further distinguished on the basis of whether net profit or revenue or only incremental gains are shared, and whether payment to workers is made in terms of cash or company shares. Similarly, deferred distributions plans are also of different variety. Jones and Svejnar (1985) show that the productivity effect of various profit-sharing schemes may operate in both embodied and disembodied form. Fitzroy and Kraft (1987) have estimated the effect of profit sharing on productivity of the firm in the disembodiment form. Our equation (2) assumes the productivity effect in a disembodied form, although results of this paper will not change qualitatively if the productivity effect is assumed in the embodied form. In equation (2), we have assumed a productivity effect, provided 0. A useful extension of this paper arises if an endogeneous productivity effect from some underlying production technology of the firm is introduced. In such an extension, the focus of research should be the estimation of the productivity effect of profit sharing, rather than the determination of the optimal sharing ratio. The literature on the employment effect of profit sharing abounds. For the review of this literature, see Nuti (1987) and Standing (1988). In this paper, we are abstracting from this controversial effect in order to focus on the determination of optimal sharing ratio under the share price maximization principle. This assumption is necessitated by a single-period nature of our model. In a multi-period framework, the operating loss carry-backward and carry-forward provision can be easily incorporated. For the derivation of equation (12), see Winkler, Roodman, and Britney (1972, p. 294). For the derivation of these equations, see Mood and Gray- bill (1963, p. 202).

265 RCSA / CJAS, 8 (4). 259-266


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Optimal Sharing Ratio in a Profit-sharing Arrangement