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Taxes and Risk SharingAuthor(s): John C. Fellingham and Mark A. WolfsonSource: The Accounting Review, Vol. 60, No. 1 (Jan., 1985), pp. 10-17Published by: American Accounting AssociationStable URL: http://www.jstor.org/stable/246960 .

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Page 2: Taxes and Risk Sharing

THE ACCOUNTING RE VIE W Vol. LX, No. I January 1985

Taxes and Risk Sharing

John C. Fellingham and Mark A. Wolfson

ABSTRACT: Models that characterize Pareto-efficient sharing of joint venture profits or constrained Pareto-efficient sharing of income in principal-agent contracting problems have ignored tax considerations. We extend the theory by showing that the effect of taxes on optimal contracting (both in the face of and in the absence of moral hazard problems) is related to the effect of changes in risk attitudes towards lotteries over pre-tax income. For example, optimal contracts will reflect the tax-induced demand for insurance of a risk-neutral individual who faces a progressive income tax schedule; that is, the risk-neutral individual will not bear all the risk, and in the face of moral hazard on the act selection of a risk-neutral agent, demand for monitoring will be created where none existed in the absence of the progressive tax. We also show that Pareto-optimal risk-sharing contracts do not generally result in expected tax minimization, ever. when taxes are modeled as a deadweight loss to the system.

T HIS paper is concerned with the nature of efficient risk sharing and incentive arrangements in a part-

nership (joint ownership of an income- producing project) when part of each owner's income from the partnership is unavailable for private consumption purposes. The difference between an owner's gross income from the partner- ship and net income (over which prefer- ences are assumed to be directly defined) results from taxes that are a function of gross income. We analyze risk sharing and incentive arrangements between two partners. We assume that both partners are VonNeumann-Morgenstern (VN-M) expected utility maximizers.

The outcome (e.g., cash) available to the partnership, x(sa), is treated as a joint function of an action choice, a, taken by one of the partners (typically termed the agent) and a random state occurrence, s. The action is assumed to be a source of welfare reduction to the agent, i.e., the agent is effort-averse. We will consider two distinct but related settings. If the action choice is unobserv-

able to the passive partner (the principal), a moral hazard setting is said to exist. In such a setting the optimal contract must be structured so as to in- duce the agent to work "hard," since without appropriate incentives shirking will result without fear of detection. In a full information setting, the agent's action choice is observable (or inferra- ble) by the principal and, hence, can be used as a contracting variable. Then the optimal action can be contracted on directly. For example, the agent may be severely penalized unless the contractu-

The comments of Stanley Baiman, Joel Demski, William Kinney, James Patell, Jan Stoeckenius, and an anonymous reviewer are gratefully acknowledged.

John C. Fellingham is Associate Pro- fessor ofAccounting, University of Texas at A ustin, and Mark A. Wolfson is Asso- ciate Professor of Accounting, Stanford University.

Manuscript received January 1982. Revisions received August 1983 and June 1984. Accepted July 1984.

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Page 3: Taxes and Risk Sharing

Fellingham and Wolfson 11

ally specified act is supplied. Hence, without loss of generality we can sup- press effort level in the formulation of the contracting problem in the full in- formation setting. Initially, we consider a full information formulation in which the problem is to characterize the set of Pareto-optimal risk-sharing contracts.' The basic formulation of an optimal risk sharing contract without moral hazard is due to Wilson [1968]. We extend this result to a setting that includes taxation. We will address the moral hazard incentive problem in a taxation environ- ment in a later section of the paper.

TAXES AND PARETO-OPTIMAL RISK SHARING: THE FULL INFORMATION CASE

The problem confronting the partner- ship is to construct a sharing rule that distributes the outcome X to the partners for all possible realizations, x.

Each partner is subject to taxes on partnership income. Let zi(x) be partner i's share of the partnership income, when x is realized. Let Ti(zi) denote after-tax income; that is, Ti is a function (assumed to be continuously differentiable) that maps pre-tax income zi into after-tax income Tl(zi). Note that the tax is equal to zi - (zi).2 Finally, Ui(7T(zi)) denotes the (VN-M) utility function, which is defined over after-tax income.

Pareto-optimal risk-sharing contracts in a full information setting are given by the following variational problem:

Max ful(T (x - z2(x)))f(x)dx Z 2(X )

+ xU2(T2(z2(x)))f(x)dx, (1)

where 2 is a non-negative multiplier on partner 2's expected utility. Since all Pareto-optimal contracts are charac- terized by z1(x)+z2(x)=x for all x, we have expressed the problem in terms of

choice of the sharing function to partner 2. zI(x) is then simply equal to x-z2(x). As noted above, we may suppress the action choice and treat the partnership as endowed with a random variable X with density function f (x).

The first order condition for (1) is given by:

U'1 (T1 (X - Z2 (X)))T'(X -Z2 z(X))

= A U'2(T2 (z2(x)))TO(z2(x))2x, (2)

where primes (') denote derivatives. We now get the slope of the optimal risk sharing contract by differentiating (2) with respect to x and substituting U' T1'/U'2 T for A. The slope indicates how a marginal change in the total partnership payoff is allocated to the partners:

Pi =i P Ti- 2 2 (3)

E piTfi-Ti L 1

where pi=- - U/UI) is evaluated at after-tax income and Ti(_ TI/T') is evalu- ated at pre-tax income. Note that pi is the global measure of risk aversion (called absolute risk aversion) developed by Pratt [1964], which he shows to be "twice the risk premium per unit of variance for infinitesimal risks" (p. 125].

Ti can be viewed as a measure of pro- gressivity in the income tax. A progressive income tax is defined as one that is convex in pre-tax income. We assume that margi- nal tax rates are always below 100 per- cent. Hence, a progressive tax corre- sponds to Ti <0, and a regressive tax is equivalent to Ti > 0.3 Note that Ti = (

I A risk-sharing contract is Pareto-optimal if there does not exist another contract that leaves both of the partners at least as well off and one of the partners strictly better off in the sense of a higher expected utility.

2 Taxes are assumed to be deadweight losses to the partners. This is discussed more fully below.

3 Recall that T(z) is after-tax income, and the tax is - T(z). Denote this tax t(z). A progressive tax is one

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Page 4: Taxes and Risk Sharing

12 The Accounting Review, January 1985

when the tax is proportional, a lump-sum, or a combination of lump-sum plus proportional.

In the absence of taxes, (3) simplifies to the familiar result of Wilson [19681:

Zf = I 1 , (4) PI + P2

Each partner's variable share of partner- ship income is equal to one minus the ratio of his risk aversion to the aggregate risk aversion of both partners.4

Note from (4) that if partner 2 is risk- neutral (U'2=0), then P2=0 and z2= In this case, all Pareto-optimal risk- sharing contracts (in the absence of taxes) involve partner 2 bearing all the risk and partner I receiving a fixed pay- ment independent of partnership income. In the face of taxes, however, this risk- sharing arrangement is not generally optimal. For example, a progressive tax induces risk aversion. Even though part- ner 2 is risk-neutral towards after-tax lotteries, he is strictly risk-averse towards pre-tax lotteries. The reason is that a mean-preserving spread in pre-tax in- come increases the expected tax and hence reduces expected after-tax income, since the progressive tax is convex in pre-tax income. If the tax were a regres- sive one, risk-seeking behavior would be induced.'

When both partners are strictly risk- averse, even a lump sum tax will typically affect the set of Pareto-optimal risk- sharing contracts. Even though (3) and (4) look similar in this case, the pi are evaluated at different levels of income.

The problem can be viewed as equiva- lent to a taxless setting in which each partner earns non-partnership-related in- come in addition to his partnership share. When one of the partners experi- ences a change in non-partnership wealth level, his absolute risk aversion (-U"')/(U) will typically change. As can

be seen by examination of (3), this will cause an adjustment to the optimal con- tract. The only exceptions to this are constant absolute risk aversion utility functions. For these functions-linear, negative exponential, and affine transfor- mations thereof-absolute risk aversion is invariant with respect to wealth level. See Pratt [1964]. In these cases, the partnership contracting choice can be decomposed from non-partnership wealth levels, whether from outside in- come or the imposition of a lump sum tax.

TAX MINIMIZATION AND PARETO-OPTIMAL RISK SHARING

Pareto-optimal risk sharing does not, in general, minimize expected taxes. However, the set of Pareto-optimal risk- sharing contracts coincides with the set of expected tax-minimizing contracts for the special case in which both partners are risk-neutral, as we prove below. Note that tax minimization is equivalent to maximization of after-tax income when all partnership income is distributed. Hence, the set of expected tax-minimizing contracts is given by:

that satisfies d2t(z)dz2 >OVZ, or, equivalently, d2T(z)/dz2 <OW. We assume that the marginal tax rate is always less than 100 percent. Hence. dT(z-),d->OV:, and a progressive tax corresponds to r<O. Similarly. a regressive tax corresponds to z > 0.

4 Equivalently. z2 = p2) 1'/ 17 + P 2 - ?, where p7 is

defined by Wilson [1968] as the risk tolerance: "half the tolerable variance per unit of compensating risk premium for infinitesimal risks" [p. 121]. Each partner's variable share of partnership income is equal to the ratio of his risk tolerance to the aggregate risk tolerance of both partners.

I If the tax is regressive, the first-order condition for the maximization problem in (1) may not be sufficient. For example, if both partners are risk-neutral and the tax is strictly concave in pre-tax income, Pareto-optimal sharing rules do not exist without further restrictions on the set of available contracts. Henceforth, we restrict attention to nonregressive taxes.

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Fellingham and Wolfson 13

Max F1(X - z2(x )) f (x )dx Z2(X) J

+iT2(z2(x))f(x)dx, (5)

where p is a positive constant correspond- ing to a particular expected after-tax income for partner 2. Following the approach used to derive equation (3) from equation (1), the risk-sharing con- tract that minimizes expected taxes re- sults in the following sharing of risks:

Z2 T 1 ( 1 +?2) (6)

Note that (3) simplifies to (6) for the spe- cial case in which both partners are risk- neutral, since Pi = P2 = in this case. This completes the proof.

The set of Pareto-optimal risk-sharing contracts also coincides with the set of tax-minimizing contracts when one of the partners is strictly risk-averse and is sub- ject to a strictly progressive income tax, while the other partner is risk-neutral and is subject to a lump sum, proportional, or combination of lump sum plus propor- tional tax. It is routine to verify that (3) and (5) coincide in this case; the risk- neutral partner optimally bears all the risk. This situation might approximate the case of an individual contracting with

a wealthy corporate partner. In the more general case, Pareto-

optimal risk sharing departs from tax minimization. To see this, suppose that one partner is subject to a strictly pro- gressive tax (z 1 < 0), and the other partner is subject to a lump sum plus propor- tional tax (z2=0). The tax minimizing contract specified by (6) requires that partner 2 bear all the risk. However, if partner 2 is risk-averse, the Pareto- optimal risk-sharing contract given by equation (3) requires that both partners bear some risk.

A simple numerical example follows. Let partner 1 be risk-neutral and subject to a progressive tax so that after-tax wealth is of this form: T1 (z) = . Similarly, let partner 2 be risk-averse but with a proportional tax, T1(z)=.8z. Let partner 2's VN - M utility function in after-tax wealth be U(w)= 1000 * (1 -e-w/1000). Since partner l's after-tax wealth is a concave function of the part- nership share, a contract which mazi- mizes expected after-tax wealth to the partnership requires that partner 1 re- ceive a fixed fee. For concreteness, let the pre-tax cash inflow to the partnership be one of two equiprobable amounts: $1000 or $100. An expected tax-minimizing contract is:

Pre-tax Share Tax

Outcome Partner 1 Partner 2 Partner 1 Partner 2

1000 100 900.00 16.82 180.00 100 100 0.00 16.82 0.00

The expected aggregate tax to the partnership is $106.82. The expected utility to the two partners is:

Partner 1: 83.18 Partner 2: 256.62

Notice that all the risk is borne by partner 2 who is risk-averse; opportunities for Pareto improvement are clear. Consider the following contract

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Page 6: Taxes and Risk Sharing

14 The Accounting Review, January 1985

Pre-tax Share Tax

Outcome Partner I Partner 2 Partner I Partner 2

1000 205.86 794.14 39.51 158.83 100 0.00 100.00 0.00 20.00

The expected aggregate tax to the partnership is $109.17 versus $106.82 in the previous contract. The expected utility to each partner is:

Partner 1: 83.18 Partner 2: 273.56

Although the expected tax has increased, a welfare improvement is obtained by improving the risk-sharing characteristics of the contract.6

TAXES AND INCENTIVE CONTRACTING UNDER CONDITIONS OF MORAL HAZARD

Thus far, we have restricted attention to the effect of taxes on risk sharing in a full information setting. These effects were shown to be related to induced changes in risk attitudes towards lotteries over pretax income. In this section we maintain the basic partnership structure but abandon the full information assumption. Instead we assume that the productive act is both unobservable to the passive partner (principal) and costly to the active partner (agent). In this way we formulate a standard agency problem and examine the effects of taxes on incentives.

In the familiar principal-agent prob- lem"'8 a demand for a monitoring tech- nology may arise when the agent's act is unobservable to the principal and, hence, cannot be used directly as a con- tracting variable. This demand is a mani- festation of the social loss that results from excessive risk (relative to the Pareto-optimal amount) being forced upon a risk-averse agent in order to motivate the desired level of effort.

In the standard agency setting, the effort level of the agent affects the solu- tion in two ways. Effort is an argument of the agent's preference function which is typically represented as additive in in-

come and effort:

U(:, a)-U2(Z)-V(a), where a is effort level.

Furthermore, the effort level of the agent is modeled as a parameter of the sto- chastic production function that is repre- sented by the conditional density func- tion J(x a). The typical assumption is that increasing the effort level shifts the outcome density function to the right in the sense of first-order stochastic domi- nance.

If the effort is unobservable and, hence, unavailable for contracting purposes, the agent will choose an effort level so as to satisfy

Max | U2(T2(z2(x)))-f (xla)dx- V(a). a.)

6 Actually, further improvements are theoretically possible by shifting more of the risk to partner 1, but this would require assigning a negative income to partner 1 half the time, over which, in this example, the tax function is not defined.

I For a recent survey of this literature, see Baiman [1982].

8 The basic features of the problem are: (1) the principal and agent jointly share the realization from a stochastic outcome (production) distribution; (2) there are (at least) two factors of production: nature (the random factor) and effort (the choice of which is delegated to the agent); (3) the agent derives disutility from expending effort; and (4) effort is not directly observable by the principal.

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Page 7: Taxes and Risk Sharing

Fellingham and Wolfson 15

The first-order condition is

T U2(T2(z2(x)))fa(xla)dx= V'(a). (7)

The socially preferred (Pareto-optimal) effort level solves the following problem:

Max {Ui(Tl(x-z2(x)))f(xla)dx a f

+ ju {U2(T2(z2(x)))f (xla)dx - pV(a).

The first-order condition is

Tu1 (T1 (x - Z2(x)))fa(xla)dx

+ U2(T2(z2(x)))fa(xIa)dx= u V'(a). (8)

For both (7) and (8) to hold requires

fU1(TI(X -Z2(X))) fa (x I) = ?. (9)

To satisfy (9), in general, x - z2(x) must be independent of x or z2(x) = x - c, where c is any constant. This means that the agent internalizes fully the social effect of his act selection. That is, he or she derives all the benefits from the act in terms of increased productivity, as well as bearing the entire cost, V(a). However, the agent also assumes all the risk of the uncertain outcome, and optimal risk sharing as embodied in (3) may be sacri- ficed. We say that there is an "agency problem" if there does not exist a con- tracting arrangement that simultaneously satisfies (8) and (3). Risk sharing and act selection considerations must be traded off against each other. When an "agency problem" exists, there arises a demand for a monitoring technology that renders the agent's act observable, thereby making it accessible for contracting pur- poses.

The demand for a monitoring tech-

nology is affected by the imposition of a a tax in a way similar to the effect of a change in the agent's risk aversion. For example, a progressive tax could create a demand for monitoring where none existed in the absence of the tax. To see this, suppose that the agent is risk- neutral. In this case it is optimal (in a Pareto-efficient sense) for the agent to bear all the risk with the principal re- ceiving a "fixed fee."9 Since the agent is risk-neutral there is no efficiency loss from a risk-sharing standpoint. Further- more, with the agent bearing all the risk, the externality problem is now solved in the sense that the agent will now select that effort level which equates the expected utility of exactly 100 percent of his marginal productivity with the margi- nal cost of achieving the additional out- put in terms of effort disutility.

The contract that satisfies (9) is z2(x) = x- c. This amounts to selling the firm to the agent for an amount c. From equation (3) it follows that this contract also satisfies optimal risk sharing if the agent is risk-neutral (P2= 0) and the tax is linear (E2 = 0). For P2 = 2 = 0, equation (3) implies that z'2 = 1.

If a progressive tax is imposed on the agent, we have ?2 >0. Then both (9) and (3) cannot hold simultaneously and an agency problem is created. To motivate the agent to exert the proper level of effort, the agent must bear more risk than the Pareto-optimal amount. Sub-optimal risk sharing in this case results entirely from the agent's income tax-induced risk aversion.

A simple numerical example illustrates the point. Let the partnership be en- dowed with the stochastic productive opportunities embodied in the following act-state matrix with equiprobable states,

I See Harris and Raviv [19791 or Grossman and Hart [19831.

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Page 8: Taxes and Risk Sharing

16 The Accounting Review, January 1985

where all entries are dollar outcomes (cash) to the partnership.

Act State 01 02

a, 1000 100 a2 100 100

Act a, is more costly to the agent than a2:

V(aJ)= 100 V(a2) = O

Initially, suppose that both partners (principal and agent) are subject to a 20 percent proportional tax. In a full information setting (when the agent's act is observable to the principal), one efficient contract is to pay the agent $250 to supply a, and give the principal the residual. The expected utility payoffs to the risk-neutral partners are

Agent: 100 (= 250 x 80o ?- V(a1))

Principal: 240 (-80 0 of the expected residual).

The same result can be achieved in a setting of moral hazard where the act (and state) is unobservable to the princi- pal, but the outcome is observable; for example, pay the agent $375 if the out- come is $1000 and $125 if the outcome is $100. Although the principal cannot detect shirking directly, the agent is still willing to choose the personally more costly act, a,, because of a commensurate increase in expected payoff.10 Although the agent assumes more risk under this contract, there is no efficiency loss to the partnership because the agent is risk- neutral.

Now consider an identical setting as above, except that the agent is subject to a progressive tax (recall that T(z) is after-tax income):

T(z) = za

An efficient contract in a full information

setting is for the principal to pay the agent $250 to choose a1. The expected payoffs to the-partners are

Agent: 43.93 Principal: 240.00

This result cannot be replicated in a moral hazard setting. It remains optimal in this example to induce the agent to choose a,. The minimum amount of risk required to do this, keeping the agent's expected utility equal to what it was in the full information case, is given by the following contract :11

Partnership Payment Outcome to Agent

$1000 $449.28 100 66.88

The expected utility payoffs are

Agent: 43.93 Principal: 233.54

The efficiency loss to the partnership is represented by the decline in expected payoff to the principal of $240.00- $233.54 = $6.46. This is due to inefficient risk sharing caused by the agent's tax- induced risk aversion. In the moral haz- ard setting there is a demand for a monitoring technology (such as auditing) in order to render the act selection ob- servable and, hence, available for con- tracting. In this example, the principal is willing to pay $6.46/0.8 = $8.08 for perfect monitoring, assuming that the cost of the technology is tax-deductible at the principal's tax rate of 20 percent.

'0 We make the usual assumption that if the agent is indifferent between two action choices, the one pre- ferred by the principal will be chosen.

Note that if in the full information setting it is optimal for the agent to supply the lower effort level (a9, then the introduction of the progressive tax would not introduce an incentive problem (i.e., no risk would have to be imposed on the agent to induce the supply of the desired effort.

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Page 9: Taxes and Risk Sharing

Fellingham and Wolfson 17

CONCLUSIONS AND LIMITATIONS

This paper may be viewed as an analysis of optimal contracting behavior in the face of taxes. Our focus has been on contracts written for the efficient sharing of tax-induced risk aversion. We have demonstrated that changes in the tax law may induce changes in economic behavior, including changes in the demand for a monitoring technology. Some of the tax-induced contracting effects may be non-intuitive. For ex- ample, not all recontracting behavior in response to tax changes is for the pur- pose of tax reduction. Some contracting changes may actually increase the ex- pected aggregate partnership tax liability as more efficient (tax-induced) risk- sharing arrangements are sought.

Throughout the paper, we have viewed the difference between the part- ner's gross income from the partnership and net income (over which preferences are assumed to be directly defined) as re- sulting from taxes that are a function of gross income. However, other interpre- tations are possible. For example, one could view this paper as an analysis of

efficient risk-sharing contracting in the face of contract enforcement costs or in the face of franchising fees that are a function of the partners' gross shares from the partnership.

Finally, taxes have been modeled in this study as if they are deadweight costs to the partners. While this may be true of contract enforcement costs or franchise fees, there is certainly another side to taxes, viz., receipt of government benefits (B). If we were to assume that preferences over income and government benefits admit to an additive and separable utility representation, i.e., U(z, B) = F(z) + G(B), and if we assume that the distribution function of B is independent of the sharing rules adopted, then all of the earlier analysis holds more generally. But these are highly unrealistic assumptions. In- deed, one might think of B as representing income redistribution, in which case U(z, B) = F [T(z) + B( Y)], where Y repre- sents aggregate income. Even if we were to assume that Bi is independent of zi, risk tolerance will still generally depend upon Bi, unless there is constant risk aversion.

REFERENCES

Baiman, S., "Agency Research in Managerial Accounting: A Survey," Journal of Accounting Literature (Spring 1982), pp. 154-213.

Grossman, S. and 0. Hart, "An Analysis of the Principal-Agent Problem," Econometrica (January 1983), pp. 7-46.

Harris, M. and A. Raviv, "Optimal Incentive Contracts with Imperfect Information," Journal of Economic Theory (April 1979), pp. 231-259.

Pratt, J., "Risk Aversion in the Small and in the Large," Econometrica (January-April 1964), pp. 122-136.

Wilson, R., "The Theory of Syndicates," Econometrica (January 1968), pp. 119-132.

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