Research Paper 781 SALESFORCE COMPENSATION PLANS IN ENVIRONMENTS WITH ASYMMETRIC INFORMATION Rajiv Lal* and Richard Staelin**

October, 1984

*

Graduate Shool of Business, Stanford University, Stanford, California 94305 Fuqua School of Business, Duke University, Durham, North Carolina 27706

**

The authors wish to thank Professors Milton Harris, Joseph Hotz, Jim Noel, and Tom Paifrey for their several helpful comments and suggestions.

ABSTRACT In this paper, we present a theory of salesforce compensation plans to explain the type of plans implemented by IBM and St. Regis Paper where the members of the salesforce are provided an opportunity to choose a compensation scheme from a menu of contracts offered by the firm~We model this interaction in an agency-theory framework, as in Basu, Lal, Srinivasan and Staelin (1984), but relax the assumptions of information symmetry and salesforce homogeneity to demonstrate the optimality of such compensation plans. We also show that even when these assumptions are relaxed there are situations where the plans characterized by Basu, Lal, Srinivasan and Staelin (1984) are still optimal.

1. Introduction
Most salesforce compeiisation plans include not only a salary, but also some form of remuneration (e.g., comrni~~sions) based on the output of the salesperson. Recently,

Basu, Lal, Srinivasan, and Staelin (1984), hereafter referred to as BLSS, proposed a model which provided insights into why the proportion of salary to total expected compensation varies across different firms or within a firm over time. However, they did not explore the possibility of a firm offering more than one compensation plan at the same time even though many of the factors identified by BLSS which lead to different contracts may also exist within a firm at any given time. For example, different members of a firm’s salesforce often have different aversion towards risk. Since the risk associated with being compensated primarily by a salary is much less than that associated with a compensation plan based primarily on commissions, it follows that more risk averse salespeople prefer the former contract, while the less risk averse salespeople prefer the latter. Given this difference in preference, a firm might be better off offering a menu of contracts and allowing individual members to select that compensation plan which most closely matches their risk preference. Similarly, members of the salesforce may differ in their beliefs of how likely they can obtain a given level of sales. Those with higher expectation should prefer, all else being equal, a plan based on more commissions and less salary while those with lower expectations should prefer a plan with higher salary and lower commission rates. Given such potential heterogeneity within a salesforce, why do firms offer only one contract? One reason is that by offering only one compensation plan over a period of time, a firm attracts and employs a relatively homogeneous salesforce. Thus with this homogeneity there is no need for the firm to design more than one compensation plan to compensate its salesforce. A second reason is that administering multiple compensation plans can be too complex. Finally, just because the salesforce might like the option of having more than one contract to choose from, it is not obvious that such an arrangement will always lead to higher profits for the firm. We address these issues in more depth in this paper. Interestingly, there are small but growing number of instances where firms are now offering their employees a menu of compensation plans. Probably the most familiar are those situations where employees are allowed to choose between different pension plans

1

and/or medical coverage options. In these instances the firm tries to minimize its cost and at the same time provides a set of plans which meet the needs of the employees more satisfactorily. A second example of multiple contracts occurs when a firm is negotiating with a person to distribute (sell) the firm’s product. Lu some instances the firm offers the interested party the option of managing a retail outlet as an independent business (i.e., being a franchisee) or as a manager of a company owned store (i.e., being an employee).

Normally the form of the compensation to the firm’s agent differs substantially under these two arrangements. Moreover the differences in the profits to the firm can be significant (McGuire and Staelin (1983) and Lal (1984)). A somewhat similar situation occurs when a firm uses more than one type of selling arrangement to market its products. For example, Anderson (1984) reports that many firms in her sample used salesforces that consisted of both manufacturing reps and salespeople. For manufacturing reps, compensation was entirely via commissions while the latter usually received a straight salary. Finally IBM (Goni]c, 1978) and St. Regis Paper (National Conference Board, Inc 1968) have reported the usage of salesforce compensation schemes which consist of a menu of contracts for salespeople to choose from. These schemes allow the salesperson to select the particular compensation plan by prespecifying how much they believe they will sell during the period. Figure 1 graphically depicts two such compensation plans as reported by Gonik. As can be seen, the two compensation schemes differ on the possible amount of remuneration a person gets for the same sales output measure; scheme 1 provides more compensation for lower sales than scheme 2 while the converse is true for high sales. Moreover scheme 1 has less variation in possible remuneration, varying from 67 percent to 200 percent above the base level, while scheme 2 varies from zero to 270 percent above the base level. The two significant characteristics of scheme 1, lower variation across the range of possible sales

and higher compensation at low sales levels (compared to scheme 2), should appeal to salespeople who (a) are more risk averse and/or (b) believe that a given level of effort will result in lower sales for the selling period. In this paper we explicitly explore the possible benefits to the firm from offering multiple contracts to a salesforce and characterize when one might expect such contracts to exist. The basic paradigm used herein is similar to that used by BLSS (1984), in that we 2

formulate the interaction between the firm and the salesforce as a principal-agent problem. However, whereas BLSS limit their attention to salesforces assumed to be homogeneous with respect to their attitudes towards risk and their beliefs about the selling environment, we relax these two assumptions to investigate the optimality of multiple contracts and the appropriateness of compensation plans suggested by BLSS. The effects of salesforce heterogeneity have been studied before by Srinivasau (1981) within the context of designing optimal commission rate policies. He examined the effect of salesforce heterogeneity with respect to the sales response function and the salesperson’s reservation price for accepting employment. He limited his analysis to determine the effects of these two factors on the optimal commi~sionrate policy. In our analysis we broaden our attention by considering all possible plans which are a function of sales (e.g., straight salary, a combination of salary and commission etc.). Moreover, we allow the salesforce to be heterogeneous with respect to their attitude towards risk in addition to heterogeneity with respect to the sales response function and the reservation price. We incorporate the sales response function heterogeneity by assuming that each individual in the salesforce has a unique perception of the response function associated with his/her territory. However, the firm is assumed to only have a prior over the range of possible response functions for each of these territories. This kind of information asymmetry between the salesperson and the firm is most likely to exist in new product markets or highly active markets where local conditions are constantly changing. Such differences in the information possessed by the salesperson and the firm may exist even if the sales manager makes a concerted effort to collect information about the selling environment. Heterogeneity with respect to risk characteristics is modeled by allowing the degree of risk aversion to vary across the salesforce. As with the sales response function, the salesperson knows his/her aversion to risk while the firm is assumed to be informed only about the distribution of the degree of risk aversion over the population of potential salespeople. Finally, the reservation price is allowed to vary with the salesperson’s aversion to risk. The general approach taken in this paper is to formulate the problem as a game where the risk-neutral firm declares a set of compensation schemes at the beginning of the selling period; the risk averse salesperson then chooses (if there is a choice) the compensation scheme which maximizes his/her expected utility and decides on the effort level to be

3

devoted to the selling activity. The effort level together with a random exogenous factor, referred to as uncertainty, affects the final sales. At the end of the selling period, sales are realized and are observed by both parties (i.e., the firm and the salesperson); the salesperson is then compensated based on the observed sales in accordance with the chosen plan. Such an approach requires a number of assumptions about the salesperson’s behavior and the control the salesperson has on the selling environment. In the next section we discuss these assumptions and develop our model for determining the optimal contract(s) for any given sales environment. We next use this model to explore when, if ever, it is worthwhile for the firm to offer multiple contracts. We do this first for an environment where the salesperson’s estimate of the possible sales outcomes are different than those of the salesmanager and then for the situation where the salesforce is heterogeneous with respect to their tolerance for risk. We conclude with a discussion of results. 2. Assumptions Many of the assumptions made herein are the same as those made by BLSS (1984). However we deviate in two important ways. First, as already mentioned, we allow for differences between the salesperson’s and the firm’s perceptions of the salesperson’s a) aversion to risk and b) ability to generate sales for a given effort level. Second we do not make any specific distributional assumptions about the conditional distribution of sales for a given level of effort, but instead the optimal contracts are searched over the set of contracts which are non-decreasing in sales, BLSS did not use such an assumption but instead used distributions (i.e., gamma and the binomial) that satisfied the sufficient conditions identified by Holmstrom (1979) to assure non-decreasing optimal compensation plans. More specifically, we assume the following. (a) The salesperson’s utility function is characterized by two separable components, the wealth which s/he receives from selling the firm’s products and the effort required to realize the sales. Mathematically if W~(w, represents the utility function for the ith salesperson t) and w and t denote wealth and effort respectively, then W1(w, t) = U~(w) V(t), where U~(w)and V(t) represents the utility for wealth and effort for the ith salesperson. Thus
—

the salesforce is heterogeneous with respect to its utility for wealth and homogeneous with

4

respect to leisure. This assumption of homogeneity with respect to leisure is not restrictive in that heterogeneity with respect to leisure can be easily translated into heterogeneity with respect to the response function by using the results in Section 4 of Appendix A presented in BLSS (1984); hence we do not consider this type of heterogeneity separately. (b) The salesperson’s utility for wealth increases at a decreasing rate, i.e., U(w) > 0, and U111(w) < 0. The salesperson’s disutility for effort increases at an increasing rate; i.e., V’(t) > 0 and V”(t) > 0. (c) The minimum acceptable expected utility (from compensation and leisure) is the same for all members of the salesforce and is ~i. We relax this reservation price assumption in a later section. (d) The firm’s objective is to maximize it’s expected one period profits. Efforts spent on activities such as after-sales service, collecting information, etc. are considered only as they affect sales in the period of analysis. (e) The marginal cost of production, c, is constant. Without loss of generality we assume this cost to be zero. (f) Sales are influenced by the effort devotedby the salesperson, marketing efforts of the firm and a random variable which characterizes the stochastic effects of the market forces such as, unpredictable effects of the marketing mix variables (e.g., advertising, promotion, etc.) and competitive forces. More specifically sales are modeled as:
=

~j(t) + ~j

where x~ the level of sales realized by the salesperson i, ~~(t) is the deterministic response is function for the ith salesperson’s effort t, where ~~(O)denotes the response to the marketing efforts of the firm, and ~ is the random disturbance term. In addition ~(t) is assumed to reflect diminishing marginal returns to scale: i.e., ~(t) > 0 and ~‘(t) 1,

the supply of optimistic salespeople exceeds the demand. In such situations, it is easy
6

to show (and we do show later) that the optimal strategy for the firm is to offer just one contract and obtain a homogeneous salesforce composed of only those salespeople who have optimistic beliefs about the selling environment. Although assumptions (i) and

(j)

seem to be appropriate for many situations, they

may not be reasonable for some special situations. However, we expect these assumptions to hold when a firm already has an established salesforce and is interested in selecting an appropriate plan without conducting an extensive search for salespeople beyond those currently available. Such a situation would occur if search costs, costs of training and/or the opportunity time associated with finding and training new salespeople are too high. Even when a firm is starting a salesforce from scratch, P will be less than 1 if the firm views the problem as finding and keeping the salesforce over time. This is because the costs of identifying q candidates with optimistic beliefs by prescreening from a larger pool of candidates can be substantial. Such costs will be even higher if the expectations of the salespeople are likely to change over time, because, even if the firm were able to identify and hire q salespeople with high sales expectations the management will have to repeat the process each period to replace those salespeople whose expectations changed during the preceding period. Consequently we believe the assumption P < 1 to be reasonable. 3. Analysis We initially limit our attention to the sit’uation where, U1(w) = U(w) V i, i.e., all members of the salesforce have identical tolerance for risk. We also recognize that since there exist only two types of salespeople, i.e., those who are optimistic and those who are pessimistic, the maximum number of contracts that a firm would have to offer is two. Consequently we limit our attention to the following three strategies. Strategy Al: The firm offers one contract such that only optimistic salespeople (i.e., those who believe the conditional distribution of sales to be f0(x I t)) choose to work for the firm. In this way the firm obtains a homogeneous salesforce with optimistic beliefs about the selling environment. Strategy A2: The firm offers one contract such that only pessimistic salespeople choose to work for the firm. Strategy A3: The firm offers a set of contracts (in this case the set consists of two
7

contracts). Each salesperson chooses a contract at the beginning of the selling period based on his/her beliefs about the sales response function and is paid in accordance with the chosen contract at the end of the selling period after sales are realized. In this situation both types of salespeople are employed by the firm although the compensation plan chosen by each type of salespeople is different. One might also envision a fourth strategy, A4, where the firm offers one contract which is constructed to appeal to both types of salespeople. However, such a strategy can be viewed as a special case of A3, where the two optimal contracts offered are identical. As will become more evident, the constraints involved in characterizing the optimal contracts for strategy A4 include all those needed for strategy A3 plus the additional constraint that both contracts are identical. Given this additional constraint, it follows that the profits from A3 weakly dominate those from A4. Hence we do not consider A4 as a separate strategy. In order to choose the optimal strategy, it is necessary to identify the conditions under which the firm finds each of these different strategies to be most profitable. We do this by formulating the optimization problem for each of these strategies and then comparing the firm’s profits from these different strategies. The optimal contract for Al can be derived by recognizing that the firm wants to maximize profits by offering only one compensation plan, s0(x), which appeals only to those potential salespeople who believe the sales response function to be represented by f0(z I t). Such a plan must ensure that potential salespeople with optimistic beliefs will accept the contract and those with pessimistic beliefs will not. This is done by guaranteeing an expected utility greater to or equal to that available from an alternative source of employment for those salespeople who have optimistic expectations about the conditional distribution of sales, while ensuring that the expected utility is lower than that available from an alternative source for those with pessimistic expectations. Finally the plan must acknowledge that each salesperson who accepts the contract will set his or her effort level so as to optimally trade off the expected value of the compensation with that derived from leisure. More precisely, define the maximization problem MP1 (which characterizes Strategy Al) as follows.1 The firm wants to design a compensation plan, s0(z), which maximizes ~ All contracts in this paper are chosen from a set of contracts which are non-decreasing 8

the firm’s expected profits, i.e.,

MP1: subject to the following constraints:

max

f

qP(z

—

s0(x))f0(z I t0)dz

(1.0)

(a) the expected utility for the salespeople with optimistic expectations is greater than or equal to that available from an alternative job (fli), i.e.,

f f

U(a~,(x))f~,(z 0)dx It

—

V(t0)

~,

(1.1)

(b) the expected utility for the salespeople with pessimistic expectations is less than that available from an alternative job (wi), i.e.,

U(s0(z))f~,(x t~)dz V(t~) ~i. But such a
MP1. Thus
— —

f U(s(x))f~(x I

condition contradicts the assumption that s (x) is the optimal interior solution to program MP2 since, as shown by Holmstrom(1979),

f U(s~(z))f~,(x t)da I

—

V(t)

=

~i, where

t satisfies equation (1.3). Thus s(z) is the optimal solution to MP1 (i.e., strategy Al). Finally, note that since the objective function in MP2 is linear in P, this function increases linearly as P increases. We use this fact in subsequent analyses. The framework needed to derive an optimal contract for strategy A2 is similar to that used above. The optimal contract for this strategy can be obtained by solving the maximization problem MP3. MP3: such that max

f

q(1

—

P)(x

—

a~(x))f~(z I t~)dz

(3.0)

J J

U(s~,(x))f~,(x I t~,)dz V(t~) ~i,
—

(3.1) (3.2)

t~, argmaxJ U(s~(z))f~(xt~)dz V(t~), E
—

I

U(s,,(x))f0(z I 10)dx

—

V(10)

U(s,,(z))f~,(zI t~,)dx V(t9) >
—

J f

U(s~(z))f0(zI 10)dz

v(10)

10,

(4.5)

U(s0(x))f~(xI 1~,)dz V(i~) for all
—

1~. (4.6)

The constraints (4.1) and (4.2) are the minimum utility constraints which ensure that the salesperson will work for the firm. The next two constraints, (4.3) and (4.4), are the action choice constraints which determine the level of effort that will be devoted by the salesperson to maximize his/her expected utility; and the last two constraints, (4.5 and 4.6) are called the self-selection constraints and ensure that the two groups of salespersons are best off selecting the plans s~(z)and s0(z) respectively. We next establish the set of properties (discussed above) which allows us to compare profits from strategy A3 (formalized in MP4) to those from strategy Al (formalized in MP1) for different values of P. These properties are: 11

Property a: The value of the objective function in MP4 evaluated at the optimum is continuous in P. Proof: Suppose it is not (i.e., the objective function evaluated at the optimum is discontinuous in P, as shown in Figure 2). Let (s(x),a(x)) be the solution to this maximization problem and ~a be the value of the objective function at the optimum for P
= Fe,

Also, let 11b be the value of the objective function at the optimum for P=P6 ,where Pa < Pb and the expected profits to the firm at P~1 be greater than at Pb Now, since the set of
.

points that satisfy the constraints to the optimization problem MP4 (i.e., the set of feasible points) are independent of the probability P, (s(x), s (x)) is also a feasible solution for the maximization problem if the probability P=P~ As shown in Figure 2, the value of the
.

objective function evaluated at this feasible point will be H°,since the objective function is linear in P. But fl°is higher than the function value ~b, Since this is a contradiction to the assumed optimum corresponding to the function value I1~’,this function can not be discontinuous as shown in Figure 2. Almost analogous reasoning can be used to show the function can not be discontinuous with the the discontinuity being an increase to the right. Property b: The value of the objective function in MP4 evaluated at the optimum, as P approaches 0 is in the limit given by MP5: subject to constraints (4.l)-(4.6). Proof: Since the objective function in MP4 is continuous in P, as P approaches 0, (4.0) in the limit becomes (5.0). Also, the constraints in the two programs MP4 and MP5 are the same. Therefore the solutions to MP4 when P approaches 0 and MP5 are identical. Property
C:

max

fq(z_sp(z))fp(zItp)dx

(5.0)

If ~(~)is the solution to the problem given below, MP6:

maxfq(x_sp(z))fp(zItp)dx

(6.0)

subject to constraints (4.2) and (4.4) and fl, is the value of the objective function at then the value of the objective function in the maximization problem MP5 evaluated at the optimum is also 11;. Proof: We note that s~(x)= s0(z) = 8(x) is a feasible solution for the maximization problem MP5. This is because, the self selection, the minimum utility, and the action 12

choice constraints are satisfied (the salesperson can always choose an expected utility greater than
~.

t~,= to

which provides

Moreover, since MP5 has more constraints than MP6,

the value of the objective function in program MP5 is always less than or equal to that in the maximization problem MP6. Thus, this feasible solution must also be an optimal solution for the program MP5. This implies that optimal solution to MP4 when P
=

0

is identical to that in MP6. Moreover, this solution is better than not employing any salesperson, since assumption (h) ensures that H; > 0. Property d: The value of the objective function for MP4 evaluated at the optimum as P approaches 1, is in the limit is given by MP7: subject to constraints (4.1)-(4.6). Proof: The proof uses the same general logic as that used for property b. Property e: If s(z) is the solution to the problem, MP8: max max

J J

q(x

—

a0(z))f0(x

I t0)dx

(7.0)

q(x

—

se(x))fe(x

I t0)dz

(8.0)

subject to constraints (4.1) and (4.3) and l1~is the value of the objective function evaluated at s(z), then the value of the objective function evaluated at the optimum in the maximization problem MP4 will always be less than 11. Property e implies that the profit from MP4 (strategy A3) will be less than the profit from MP1 (strategy Al) at P=1, since MP8 is identical to MP2 when P=l and the solution to MP2 is identical to that in MP1. Proof: First note that fl is the profit from MP2 when P=1. Also, since MP8 has less constraints than MP7, the value of the objective function (evaluated at the optimum) in MP7 is always less than or equal to that in problem MP8 formulated above. For the values of the two objective functions to be equal, it is necessary that there exist a such that (a(z)), s~,(x)is a feasible solution to problem MP7. However, for s~(z)to be feasible in MP7 it must satisfy the minimum utility constraint (4.2) and the self selection constraint (4.5). We know from constraint (4.1) that

f U(s(x))f0(z I t)dz—V(t)
13

= ~

since

8(x)

is a solution to MP8. Substituting this equality into the left hand side of (4.5) implies that

if (~(~), is feasible, then s~(x)should satisfy the constraint, s~(x))

/
Moreover
8~(X) must

U(s~(x))f0(xI 10)dz

—

v(10) 0 (guaranteed

n; ( greater than zero,

see assumption (h)) when P=0. Finally profits from strategy Al when P=0 are 0. Since the profits from strategy A3 are continuous in P, are lower than those from Al when P=1 and are greater than when P=0, there must exist at least one point of intersection. Let the first point of intersection from the right hand side be P~. hus profits from strategy A3 are T lower than from Al for P for P
= P~, fl~ = =

1 to P

= P~.Also

let

(8,8) =

be the solution to program MP4

qf(x

—

s~(z))f0(xI t~)dx, nd, fl a

qf(x

—

s~(x))f~(x t)dx. Then I
—

the value of the objective function in program MP4 can be rewritten as, PJ1~ (1 + Therefore, at the point of intersection, P0H+(1-P~)fl=P0rI, where the right hand side is the profit from Al. Consequently, 11
=

P)fl.

P~rI:~1~:) next We

show that fl > fl~so that 11 > 0. Note that s~(x)can not be the same as s~(z),since 14

we have already shown in the proof of property e that there does not exist any s~(z)such that (s(x), s~(z))is a feasible solution to MP4. Thus fl > fl, and 11 > 0. As shown in Figure 3, we already know that profits from strategy A3 are lower than those from Al over the range P
=

1 to P

=

P~. oreover, the solution (s~(x),s(x)) is a M

feasible solution for all values of P. Thus the straight line joining the point of intersection and ri; comprises a set of achievable values for the objective function. Hence the optimal value of the objective function for any value P < P0 will always lie on or above this line; i.e., there will be no other points of intersection with the plot of the values of the objective function in Al. This completes our characterization of the conditions under whichprofits from strategy A3 can be greater or less than those from the only other feasible strategy Al. 4. Discussion of the Solution We have shown that A2 is infeasible and that A4 is weakly dominated by A3. Moreover when there exist only two types of salespeople (one with optimistic expectations and the other with pessimistic expectations) we have shown that it is more profitable for the firm to offer two contracts whenever the sales managers believes that the proportion of the available salespeople who have optimistic expectations is less than sales manager’s priors are that
P> P.

Conversely if the

P then it is more profitable for the firm to offer only

one contract which is designed to attract optimistic salespeople only. This will result in a homogeneous salesforce with some territories remaining vacant. In this situation (i.e.,

P > P*) the cost to the firm of not filling all sales territories by offering only one contract that attracts only the high expectation salespeople is less than the costs of generating more sales by adding extra salespeople who need to be compensated by means of a separate contract. Said differently, it is better for the firm to leave some territories open and offer only one contract rather than hire both the low and the high expectation salespeople by offering multiple contracts which provide expected utility levels that are greater than or equal to those available from alternative jobs. Although the above implies that when P > P the salesforce compensation plans as suggested by BLSS (1984) are optimal, we have also shown that the salesforce compensation plans as suggested by BLSS(1984) are dominated by a multiple contract solution when P < In this way, we have demonstrated the importance of the sales manager’s information 15

about the available supply of salespeople in determining the best strategy for designing compensation schemes. 5. The case of a salesforce with heterogeneous preferences We next explore the implications of having a potential pool of salespeople who are heterogeneous with respect to their risk aversion characteristics. We do this by limiting our attention to this dimension of heterogeneity by assuming that the sales response function is identical for all members of the salesforce, but that each member of the salesforce can ‘be categorized as either highly risk-averse (h) or less risk-averse (t). The risk aversion is measured in terms of the differences in certainty equivalents for the same gamble across the two groups. We initially assume that minimum acceptable expected utility across the two groups is the same and is ~i. The last assumption may be restrictive under

certain conditions since one could argue that the minimum utility desired by the less riskaverse members may be higher than that by the more risk-averse members, i.e., m~ >

mj~.

Although we relax this last assumption subsequently, however, we first explore the solution under this assumption since, often the level of minimum acceptable utility depends on the demand and supply conditions in the economy and not the salesperson’s (arbitrary) desires. Finally, it should be noted that risk-aversion may be a more enduring characteristic than a salesperson’s expectations of the sales response function in a sales territory. Thus, if the degree of risk-aversion is the only characteristic which differentiates a pool of candidates, there may be more justification for a firm to incur the high cost of pre-screening a large group of candidates along this characteristic and then hiring q salespeople with low aversion to risk. However, given our interest in the form of contracts in situations where the salesforce is heterogeneous with respect to their risk characteristics, we assume that the firm can not wait to find the q qualified salespeople who have the desired risk characteristics; i.e., we still assume that P

1, where P is now defined as, the number of less risk

averse salespeople available divided by the number of people needed by the firm.

Analysis
The set of strategies available to the firm are almost identical to those in the previous analysis. Thus a firm will choose one of the following strategies:

Strategy Bl: offer one contract,

SL(z),

such that only the less risk-averse members

16

of the salesforce choose to work for the firm; Strategy B2: offer one contract, s,1(z), such that only the highly risk averse members of the salesforce choose to work for the firm; Strategy B3: offer a menu of contracts ~ in this case the menu consists of two contracts) and the salespeople self-select the compensation schemes that will. be used to compensate them at the end of the selling, period; as in MP4, the solution to this strategy can be determined by solving the mathematical program MP9 given below: MP9: subject to, max a,. (x),., (x)

/

q(l

—

P)(x

—

8~(z))f(z

I t,,~)dz+

/

qP(x

—

BL(z))f(z I t~)dz (9.0)

/ /
—

Uh(sh(x))f(z I t,~)dx V(tk)
—

~,

(9.1) (9.2) (9.3) (9.4) for all

U~(aL(x))f(zI t~)dx V(tL)
—

~i,
—

th E argmax

/

Uh(sh(z))f(z I tk)dz

V(t, ), 1 V(t ), 1
—

t~ argmaxJ Uj(sj(z))f(~ E

I t~)dz

—

/ /

Uh(sh(z))f(x I th)dx Ug(st(z))f(x I tt)dz

V(th)> V(tL) >

—

/ /

Uk(8~(z))f(zI Ih)dz Uj(ah(x))f(z I IL)dz

V(Ik)

1,~, (9.5) i~. (9.6)

—

V(I~) for all

Strategy B4: offers one contract and all members of the salesforce choose to work for the firm. It can be easily shown (as in Appendix) that the same two strategies B2 and B4 are suboptimal. In fact the proofs are almost identical to those in the previous case. The properties of the objective function for the other two strategies are also exactly the same as discussed earlier. Therefore this analysis leads to the same conclusions as in the previous case. That is, ifthe flrms-priors~ about-the proportion of less risk-averse salespeople available are such that P

P, then the firm should offer a single contract and thereby employ only

the less risk-averse salespeople. On the other hand if the proportion of less risk averse salespeople is believed to be less than Pa, then the firm should offer two contracts and employ both the more and less risk averse salespeople. 17

In summary, we have shown that in an environment with asymmetric information, depending on the sales manager’s priors, it is optimal for the firm to offer either one contract aimed at attracting only those salespeople with the sought after characteristics or offer more than one contract letting each type of salesperson choose the plan which they think is the best for themselves. In this way, we have (a) shown that the contracts proposed by BLSS (1984) are not always optimal and (b) provided a plausible explanation for the kind of compensation plans introduced and implemented by IBM and St. Regis Paper. In addition our results are compatible with the practice of manufacturers using both company owned and franchised owned outlets to distribute their products. Although one reason why manufacturers seem to use different channels of distribution may be the cost considerations in reaching different market segments, another argument may well be that by offering multiple channels, the manufacturers allow the potential applicants to decide whether they want to operate an independent store and be paid on some combination of salary and commissions or work for the manufacturer and be paid by salary only. In addition, we have shown that even if the potential salesforce is heterogeneous with respect to their expectations about the sales response function or risk aversion, there are situations where it is optimal for a firm to offer a single contract which is similar to that described by BLSS (1984). 6. Extension to the case where the minimum utility desired by the more and less risk averse salespeople may not be the same. What happens if the minimum utility available (wi) is allowed to be different for salespeople with different risk aversion characteristics. It is easy to see that the solution will not change if the more risk averse salespeople required a higher minimum utility, i.e., m~ me. > However, this is not likely to be the case. Instead one might conjecture that in such risk sharing situations, people with higher tolerance for risk are more sought after and thus are able to command a higher reservation price. In such situations me > m,~.If this is the case, then it is possible that strategy B2 which was always infeasible in the previous analysis may become feasible. In fact, it is also possible that strategy Bi may become infeasible while strategy B2 is still feasible. Hence in general, depending on the values of the minimum acceptable utility for the more and less risk-averse salespeople there are four possible scenarios. They are 18

Scenario 1: is feasible2. Scenario 2: feasible. Scenario 3: feasible. Scenario 4: feasible.

Strategy Bi is infeasible, Strategy B2 is infeasible and Strategy B3

Strategy Bi is infeasible, Strategy B2 is feasible and Strategy B3 is

Strategy Bi is feasible, Strategy B2 is infeasible and Strategy B3 is

Strategy Bi is feasible, Strategy B2 is feasible and Strategy B3 is

Next we discuss each of these scenarios and characterize the solution to the firm’s problem by comparing the profits to the firm from the feasible strategies as a function of the proportion of less risk averse salespeople (P) as described earlier. Scenario 1: Since Strategy B3 is the only feasible strategy, the firm will always offer a set of contracts and let the salespeople choose the one that they believe is best for themselves. Scenario 2: We explore what a firm will do under such a scenario by looking at the profits associated with employing only the highly risk averse salespeople (strategy B2) and the profits from the strategy which attracts both the more and less risk averse salespeople (strategy B3).
-.

We first note that as P approaches 0, the profits to the firm from employing both kinds of salespeople can never be greater than those from employing only the more risk averse salespeople. This is because, as P approaches 0, the objective function for the two strategies becomes identical, but the optimization problem where the firm wishes to employ both kinds of salespeople has more constraints. Hence we next investigate if and when the profits from the two strategies will be equal. To explore this, let .~~(z) the optimal contract for strategy B2; i.e., .~,,(x)is the be interior solution to the mathematical program MP1O. MP1O: max q(l P)(z s,1(x))f(z I th)dx
— —

/

(10.0)

a,.(z)

subject to

/

Uh(8,~(z))f(x th)dZ I

—

V(th)

~

(10.1)

2

We do not consider strategy B4 since it is a special case of strategy B3. 19

ti,, E arg maxf U,~(8h(z))f(x I th)dx

—

V(t,~),and

(10.2) (10.3)

/

Uj(sh(z))f(z

I

1~)dx V(It)
—