salesforce incentives, market information, and production/inventory planning

MANAGEMENT SCIENCEVol. 51, No. 1, January 2005, pp. 6075issn 0025-1909 eissn 1526-5501 05 5101 0060

informs doi 10.1287/mnsc.1040.0217

2005 INFORMS

Salesforce Incentives, Market Information, andProduction/Inventory Planning

Fangruo ChenGraduate School of Business, Columbia University, New York, New York 10027, [email protected]

Salespeople are the eyes and ears of the rms they serve. They possess market knowledge that is critical fora wide range of decisions. A key question is how a rm can provide incentives to its salesforce so that it isin their interest to truthfully disclose their information about the market and to work hard. Many people haveconsidered this question and provided solutions. Perhaps the most well-known solution is due to Gonik (1978),who proposed and implemented a clever scheme designed to elicit market information and encourage hardwork. The purpose of this paper is to study Goniks scheme and compare it with a menu of linear contractsa solution often used in the agency literaturein a model where the market information possessed by thesalesforce is important for the rms production and inventory-planning decisions.

Key words : salesforce; incentives; asymmetric information; screening; production planningHistory : Accepted by Stefanos A. Zenios, special issue editor; received December 31, 2000. This paper waswith the author 7 months for 2 revisions.

1. IntroductionFirms rely on their salespeople to stay in touch withcustomers. Good salespeople know what customersneed and want and the sales prospects of the marketthey serve. This market knowledge is critical for awide range of decisions, such as salesforce compen-sation, new product development, and production-inventory planning. So, how can a rm provideincentives to its salesforce so that it is in their interestto truthfully reveal what they know about the mar-ket and, at the same time, to work hard? We addressthis question in a specic context where the marketinformation is useful in making sound salesforce com-pensation and production/inventory decisions.Inducing salespeople to disclose what they know

about the market and providing incentives for themto work hard can sometimes be conicting goals. Toencourage the salespeople to work hard, the rmmust reward them for their selling effort, which isnot always proportional to the absolute sales volume.For example, when times are bad, a high level ofselling effort may only generate a relatively low vol-ume of sales. Therefore, it is sensible to set a quotathat accurately reects the potential of a sales terri-tory and measure a salespersons performance basedon the percentage of the quota achieved. However,only the salesperson working in the territory reallyknows its potential. If the rm simply asks the sales-person to forecast the sales volume for his/her ter-ritory, the salesperson will try to turn in the lowestpossible forecast. In other words, the rms desire to

motivate hard work leads the salespeople to workaround the system, i.e., forecast poorly.Many people have recognized the above problem

and proposed various remedies. Gonik (1978) re-ported a clever scheme under which it is in the sales-peoples interest to forecast accurately and to workhard. Under his scheme, the rm asks each salesper-son to provide a forecast of the sales volume in hisor her sales territory, and the salespersons compen-sation is determined by the realized sales volume andthe initial forecast. For any submitted forecast, thesalespersons compensation is a nonlinear (more pre-cisely, piecewise linear) function of the realized sales.According to Gonik, this compensation scheme hasworked successfully in IBMs Brazilian operations.1

In agency theory, the above problem is said to com-bine moral hazard (selling effort not observable tothe rm) with adverse selection (the salesperson hassuperior information about the market prior to con-tracting with the rm). A typical solution to this typeof problem is to offer a menu of contracts to the sales-person (see, e.g., Kreps 1990). By observing whichcontract the salesperson chooses, the rm (or princi-pal) learns something about the market. A menu oflinear contracts is often suggested as a solution and issometimes shown to be optimal (see, e.g., Laffont andTirole 1986, Gibbons 1987). The main purpose of thispaper is to compare Goniks solution with a menu oflinear contracts.

1 Variations of the Gonik story are often taught in accounting andmarketing courses in business schools.

60

Chen: Salesforce Incentives, Market Information, and Production/Inventory PlanningManagement Science 51(1), pp. 6075, 2005 INFORMS 61

We consider a model in which a rm sells a sin-gle product through a single sales agent. The marketdemand for the product depends on three factors: themarket condition, the selling effort by the agent, anda random noise. The agent possesses private infor-mation about the market condition, and the sellingeffort is not observable to the rm. The rm mustdesign a compensation scheme for the agent andmake production decisions for the product. Therefore,the agents private information about the market con-dition, if revealed, benets the rm in two ways: Itincreases the accuracy of the rms measurement ofthe agents selling effort, and it reduces the demanduncertainty, enabling the rm to better match sup-ply with demand. The question is how the rm candesign a contract so as to maximize its expected prot,anticipating that a production decision has to be madeafter contract signing, but before demand realization.After describing the model in detail, the paper

presents two benchmarks: an upper and a lowerbound on the rms achievable expected prot. Theupper bound is the rst-best solution: the rms max-imum expected prot if the rm could observe boththe market condition and the agents selling effort.The lower bound is obtained if the rm adopts a sin-gle linear contract, thus forgoing the opportunity tolearn about the market condition. Both of these bench-marks can be obtained in closed form.This paper then proceeds to consider menu con-

tracts. For a menu of linear contracts, the optimalvalues of the contract parameters can be obtainedin closed form, leading to a simple calculation ofthe rms optimal expected prot. For Goniks (1978)menu of nonlinear contracts (taking his contract struc-ture while ignoring his particular choices of param-eter values), the problem becomes much harder.However, based on several analytical results, we areable to devise an efcient algorithm for computingthe optimal values of the contract parameters and thecorresponding expected prot for the rm. We thencompare these two solutions in a numerical exper-iment. Our results show that the Gonik solution isdominated by a menu of linear contracts. This appearsto be due to the Gonik solutions inability to inducedifferent effort levels for different market conditions.The numerical analysis also compares these solutionswith the above benchmarks and shows how the dif-ferences among these scenarios vary as some of themodel parameters change.The salesforce compensation problem has been

widely studied in the marketing literature; seeCoughlan (1993) for a comprehensive review. Manyof the existing models assume, as is done here, thatthe sales response function has a random noise term;i.e., the total sales generated by a given level of sell-ing effort is random. This, together with the fact that

the selling effort is often unobservable to the rm,leads to the moral hazard problem: the problem ofmotivating the salespeople to work, given that theirreward can only be based on an imperfect signal ofeffort. If, in addition, the salespeople have superiorinformation about the sales response function (theproductivity of selling effort, the sensitivity of cus-tomers to price changes, the sales prospects, etc.),then the rm is at an informational disadvantage interms of the sales environment. This is the adverse-selection problem mentioned earlier. The basic modelof the moral hazard problem has been studied andrened by many in the economics/agency-theory lit-erature; see, e.g., Shavell (1979); Harris and Raviv(1978, 1979); Holmstrom (1979, 1982); and Grossmanand Hart (1983). This machinery was then introducedto the salesforce compensation literature in market-ing by Basu et al. (1985). Many marketing researchershave since extended the basic moral hazard model(one product, one salesperson, symmetric informa-tion) to models with asymmetric information andmultiple sales territories; see, e.g., Lal (1986), Lal andStaelin (1986), Rao (1990), and Raju and Srinivasan(1996). Our model is a combination of moral haz-ard and adverse selection. Closely related to the cur-rent paper is Mantrala and Raman (1990), who haveformalized Goniks idea and focused on analyzingthe sales agents response to a Gonik scheme. Basedon this analysis, they suggested several guidelines asto how the rm can go about choosing the contractparameters under Goniks scheme. One of our contri-butions here is to directly address the problem facingthe rm (the agents response is only a subproblem)and to provide analytical results that lead to an ef-cient algorithm for computing the optimal contractparameters. It is also worth noting that a novel featureof our model as compared to Mantrala and Raman(1990), and for that matter, most models in agencytheory, is that the principal (the rm) has an activityi.e., production/inventory planningthat stands tobenet from the revelation of the agents privateinformation. On the other hand, the operations lit-erature often takes as given the knowledge aboutdemand (e.g., the characterization of the demand pro-cess), whereas here part of the demand informationis unknown to the rm and salesforce incentives arerequired for its revelation. We therefore contribute tothe growing body of research in an area often labeledthe marketing-operations interface. For some ofthis interface research with a specic connection tosalesforce incentives, see Dearden and Lilien (1990),Porteus and Whang (1991), and Chen (2000).The rest of this paper is organized as follows.

Section 2 describes the model and presents some pre-liminary results. Section 3 presents two benchmarksolutions: the rst-best solution and one based on a

Chen: Salesforce Incentives, Market Information, and Production/Inventory Planning62 Management Science 51(1), pp. 6075, 2005 INFORMS

single linear contract. Section 4 shows how an opti-mal menu of linear contracts can be determined inclosed form. Section 5 deals with the Gonik solution.Section 6 presents the numerical results. Concludingremarks are in 7.

2. The ModelA rm sells a single product through a sales agent.The total sales or demand, X, is determined by theagents selling effort (a), the market condition (), anda random noise () via the following additive form:

X = a+ + where a is a nonnegative real number, and and areindependent random variables with Pr = H = and Pr = L = 1 for 0 < < 1 and H > L > 0,and N02. Moreover, assume that L is suf-ciently large that the probability of X being negativeis negligible.The relationship between the rm and the sales-

person is that of a principal and an agent in thesense that the latter sells the product on behalf ofthe former. The principal designs the agents wagecontract and makes production decisions, while theagentendowed with private information about themarket conditiondecides whether or not to accept acontract and, if so, how much selling effort to exert.More specically, we assume the following sequenceof events: (1) The rm (or principal) offers a menu ofwage contracts; (2) the agent privately observes thevalue of ; (3) the agent decides whether or not to par-ticipate (work for the rm) and if so, which contract tosign; (4) under a signed contract, the rm determinesthe production quantity, and the agent makes theeffort decision; and (5) both parties observe the totalsales (i.e., the value of X). The rm cannot directlyobserve the agents effort level, and thus must com-pensate the agent based on the realized value of X.The problem facing the rm is thus a mixture of

moral hazard (postcontractual opportunism associ-ated with the effort decision) and adverse selection(precontractual asymmetric information regarding themarket condition). A typical response to this type ofproblem is to offer a menu of contracts to the agent,whoknowing the market conditionthen choosesone of them to sign. By observing the choice madeby the agent, the rm may learn something about themarket condition (this is often referred to as screeningin economics), and this knowledge may be helpful inmaking production decisions.Consider the agents decisions when offered a

menu of contracts. First, he would consider each con-tract on the menu and determine the maximumexpected utility that could be obtained under the con-tract. Suppose s is the contract being considered.

(Thus, sx is the wage paid to the agent if the totalsales is x.) Assume the agents utility for net income zis Uz=erz with r > 0. (The negative exponentialutility function is widely used in the agency litera-ture.) Note that U is increasing and concave, imply-ing that the agent is risk averse. The net income is thewage received, sX, minus the cost of effort, whichis assumed to be V a = a2/2, an increasing, convexfunction.2 To determine the maximum expected utilityachievable under s, the agent solves the followingoptimization problem:

maxa

EersXV a

Recall that the agent has already observed thevalue of when evaluating the contract. Therefore,the above expectation is with respect to , given theobserved value of . For convenience, we say that theagent is of high type if he has observed = H , andlow type otherwise. The optimal effort decision thusdepends not only on the contract s, but also theagents type. Let as t be the optimal effort decisiongiven contract s and agents type t (= H or L). Letus t be the corresponding expected utility for theagent, i.e., the maximum achievable expected utilityunder s and t. If us t is greater than or equal toU0the agents reservation utility representing thebest outside opportunity for the agentthen s is saidto be acceptable to the agent.3 Among all the contractson the menu, the agent chooses the one with the high-est achievable expected utility and participates if thisutility level exceeds U0. Throughout this paper, wewill assume that the menu offered by the principalcontains at least one contract that is acceptable to thehigh type and, similarly, at least one acceptable to thelow type. This avoids the situation where the princi-pal is left without a salesperson.We now turn to the principals problem. The above

sequence of events implies that the rm must make itsproduction decision before observing the total sales.This is reasonable when the customers demand fastdelivery of their orders and the production lead timeis relatively long. (It is thus impossible to followmake-to-order.) Let Q be the production quantity. Letc be the cost per unit produced. When supply doesnot match demand, additional costs are incurred. IfX Q, the excess supply is salvaged at p per unit.On the other hand, if X >Q, the excess demand mustbe satised via an emergency action such as a specialproduction run at a cost of c per unit. Let the unit

2 Other functional forms for the cost of effort can be used withoutfundamentally changing the analysis.3 It is reasonable to assume that the reservation utility does notdepend on the agents type, because what distinguishes the hightype from the low type is the market condition, something unre-lated to the agents intrinsic quality.


selling price be 1+ c (the prot margin is thus nor-malized to 1). To avoid trivial cases, assume p < c F (4)

where (, ), u, and v are contract parameters chosenby the rm with u > ( > v > 0. Figure 1 depicts twopossible contracts under the above scheme. This menuof piecewise linear (in x) contractswith properlychosen parametersis precisely what Gonik (1978)proposed and implemented at IBM Brazils sales

Figure 1 Goniks Menu of Piecewise Linear Contracts

u

v

F F x

s (x/F )s (x/F )

x+

operations. The purpose for the rest of this sectionis to show how the optimal values of ()u, and vcan be determined to maximize the rms expectedprot.

5.1. The Agents ProblemConsider the problem facing the agent when given amenu of contracts specied in (4). The agent makestwo decisions: the forecast F and the selling effort a.These decisions are made after the agent sees themarket condition (), but before the demand noise is realized. The objective is to maximize the agentsexpected utility, which is

EersXF V a=erV aEersXF Recall that X = + a+ N + a2. Using (4),

we have

EersXF = F

ersF F uFx1,

(x a

)dx

+ F

ersF F +vxF 1,

(x a

)dx

where , is the standard normal density function.Using sF F = (F + ) in the above expression andsimplifying, we have

EersXF = er(F+)[eruz+

12 ru

2.

(z

+ ru

)

+ ervz+ 12 rv2 .(z

+ rv

)] (5)

where z = F a , . is the standard normal cdf,and . = 1 .. Therefore, the agents optimizationproblem can be stated as

maxaz

erV a(aer(r)/z (6)

where

/z = eru(z+ 12 ru2.(z

+ ru

)

+ erv(z+ 12 rv2 .(z

+ rv

)

which also depends on (u, and v, but is independentof ).Note that the agents objective function in (6) is

separable in a and z. The optimal selling effort a

minimizes V a (a. With V a = a2/2, a = (. Itis interesting that the optimal effort level is entirelydetermined by only one contract parameter, (, and itis independent of the agents type. Now let /z beminimized at z. The optimal forecast decision thenis, F = z + a + = z +(+ . Note that z dependson (u, and v, but it is independent of ) and the


agents type. Therefore, the high-type agent forecastsFH = z + (+ H and the low-type forecasts FL = z +(+L. As a result, after receiving the agents forecast,the rm discovers the market condition and makesthe production decision accordingly. This is the sepa-rating case discussed in 2.We pause here to collect several results on /,

which are useful in determining z.

Theorem 1. / is quasi-convex with /z 1 forall z.

Proof. Dene

f1(z = er(z and

f2z = eruz+ 12 ru2.(z

+ ru

)

+ ervz+ 12 rv2 .(z

+ rv

)

Thus, /z = f1(zf2z. Because df1(z/dz =r(f1(z,

/z= r(f2z+ f 2zf1(z (7)Because f1(z > 0 for all z, the sign of /z is thesame as the sign of r(f2z+ f 2z. Note that

f 2z = rueruz+12 ru

2.

(z

+ ru

)

+ eruz+ 12 ru2,(z

+ ru

)1

+ rvervz+ 12 rv2 .(z

+ rv

)

ervz+ 12 rv2,(z

+ rv

)1

= rueruz+ 12 ru2.(z

+ ru

)

+ rvervz+ 12 rv2 .(z

+ rv

)

Therefore,

r(f2z+ f 2z

= ru(eruz+ 12 ru2.(z

+ ru

)

r( vervz+ 12 rv2 .(z

+ rv

)

= rervz[u(eruvz+ 12 ru2.

(z

+ ru

)

( ve 12 rv2 .(z

+ rv

)] (8)

Therefore, the sign of r(f2z + f 2z is the sameas the sign of the term inside the square brackets,

which is equal to an increasing function of z minusa decreasing function of z. Moreover, the term insidethe square brackets is negative as z and posi-tive as z+. Therefore, the sign of r(f2z+f 2zchanges from negative to positive exactly once, imply-ing that / is quasi-convex.To prove that /z 1 for all z, note that the

type- agents expected utility satises the followinginequalities:

EersXF V a erEsXF V a erEsXV a= er(+a+)V a (9)

where sx= (x+) because the utility function is con-cave (for the rst inequality) and sx F sx for allx and F (for the second inequality). As we saw ear-lier, EersXF V a = erV a(aer(r)/z. Com-bining this inequality with (9), we have /z 1 forall z.

Theorem 2. z is increasing in (, but decreasing in uand v.

Proof. From (7) and the quasi convexity of /, weknow z is the solution to

r(f2z+ f 2z= 0As ( increases to (, r(f2z+ f 2z < 0. To makethis inequality an equality, the quasi convexity of /implies that z must be increased.To show that z is decreasing in u, take the deriva-

tive of r(f2z + f 2z with respect to u (using theexpression in (8)):

reruz+12 ru

2.

(z

+ ru

)(10)

+ ru(rz+ r2ueruz+ 12 ru2.(z

+ ru

)(11)

+ ru(eruz+ 12 ru2,(z

+ ru

)r (12)

The term (10) is clearly positive. Now consider thesign of the sum of (11) and (12) (and ignore the com-mon term ru (eruz+ 12 ru2 because it is positive).Note that

rz+ r2u.(z

+ ru

)+,

(z

+ ru

)r

= r((

z

+ ru

).

(z

+ ru

)+,

(z

+ ru

))> 0

where the last inequality follows because w.w +,w > 0 for any w. (To see the latter, just show thatthe function of w is increasing and tends to 0 as


w.) Therefore, as u increases, r(f2z+ f 2zincreases, implying that z is decreasing.The fact that z is decreasing in v can be shown sim-

ilarly. The derivative of r(f2z+ f 2z with respectto v (using the expression in (8)) is

rervz+12 rv

2 .(z

+ rv

)

r( vrz+ r2vervz+ 12 rv2 .(z

+ rv

)

+ r( vervz+ 12 rv2,(z

+ rv

)r

which can be shown to be positive. (Here one usesthe fact that ,ww.w > 0 for any w. To see this,simply note that the function of w is decreasing andtends to zero as w+.) It is interesting that the agents forecast decision

is analogous to the order quantity decision in anewsvendor problem. Suppose a retailer can buy aproduct for free and sell it for a per-unit price of (.The retailer must determine a purchase quantity Qbefore the demand X is realized. The retailers incen-tives for matching supply with demand come fromthe overage and underage costs: If supply exceedsdemand, the retailer incurs a disposal cost of u(per unit of excess inventory; otherwise, if demandexceeds supply, the retailer must satisfy the excessdemand by making an emergency purchase at a per-unit cost of (v. The retailers prot is (Xu( Q X+ ( vX Q+, which differs fromsX Q only by a constant ). Therefore, our agentsoptimal forecast is the same as the retailers optimalorder quantity as long as the retailer is also risk aversewith the same utility function. Risk-averse newsven-dor problems have been studied by, e.g., Eeckhoudtet al. (1995) and Agrawal and Seshadri (2000). Sen-sitivity results similar to those in Theorem 2 havebeen reported in Eeckhoudt et al. (1995). Note thatthe agents problem is only part of a larger problem,which is the principals choice of contract parameters.Continuing with the above newsvendor analogy, wenote that the values of (, u, and v determine theretailers price/cost parameters. So, one can think ofthe principals problem as one facing a manufacturerwho can control the retailers price/cost parameters.This kind of problem has often been addressed in thecontext of channel/supply chain coordination; see,e.g., Pasternack (1985), with risk neutrality at bothlevels being a common assumption. We now turn tothe principals problem.

5.2. The Principals ProblemThe principals expected prot consists of threecomponents: expected revenue EX, expected wageEsX F , and the expected production/inventory

costs. Recall from the previous subsection that theagents optimal effort is a = ( regardless of his type.Thus, X = + (+ . Because E = 0, EX = + (.Also from the previous subsection, the agents opti-mal forecast depends on his type: A low type leads toa forecast lower than the high types. Therefore, theagents forecast unambiguously signals the marketcondition. Under this separating case, the minimumexpected production/inventory costs are G1 (see 2).It remains only to derive the expected wage paid tothe agent.Given that the agent is type , forecasts F , and

exerts effort a, the expected wage is

EsX F F a= Es+ a+ F = F

sF F uF x 1,

(x a

)dx

+ F

sF F + vx F 1,

(x a

)dx

= (F +)u z/

zy,ydy

+ v z/

y z,ydy

where z= F a and the last equality follows aftera change of variable x= y+ + a. Note that z/

zy,ydy =

(z

.

(z

)+,

(z

))

and z/

y z,ydy =(z

.(z

),

(z

))

Because FH = H + (+ z and FL = L + (+ z, EF =+(+ z. The ex ante expected wage is

(+(+ z+)u(z

.

(z

)+,

(z

))

v(z

.(z

),

(z

))

= (+(+ ( vz +) u v

(z

.

(z

)+,

(z

))

There are two participation constraints, one for eachagent type:

erV a(aer(H+)/zU0and

erV a(aer(L+)/zU0where the agents maximum expected utility for eachtype comes from the agents objective function in (6),


with the decision variables a and z replaced with theiroptimal values. It is clear that the constraint for thehigh type is redundant. Recall that ), the xed salary,does not impact the values of a and z. Therefore, theoptimal ) (from the rms standpoint) binds the par-ticipation constraint for the low type. Because a = (and V a= a2/2, we have the optimal ) (as a functionof the other contract parameters):

)=(2

2(L

lnU0r

+ ln/z

r

Therefore, the minimum wage given (u, and v is

(+( (2

2(L

lnU0r

+ ln/z

r+ ( vz

u v(z

.

(z

)+,

(z

)) (13)

Combining the above revenue and cost terms, wehave the rms expected prot:

+(G1{(+( (

2

2(L

lnU0r

+ ln/z

r+(vz

uv(z

.

(z

)+,

(z

))} (14)

The rm seeks the values of (, u, and v that maximizethe above expression.

5.3. BoundsThis subsection presents bounds that are helpfulin determining a numerical solution to the princi-pals contract-design problem. We rst show that z

is within easily computable bounds. These bounds,together with the quasi convexity of /, can be usedfor a quick calculation of z. We then show that theprincipals objective function is bounded from aboveby a concave quadratic function of (. This leads tobounds on the optimal value of (. This subsection canbe safely skipped in the rst reading.We rst show that / can be bounded below by

a simple function. Dene sux F = (F +)uF xand svx F = (F + )+ vx F for all x. It is clearthat sx F =min!sux F svx F ". Therefore,

EersXF = Eermin!suXF svXF "= Eemax!rsuXF rsvXF "= Emax!ersuXF ersvXF " max!EersuXF EersvXF "

Suppose the agent is type , forecasts F , and exertseffort a. Let z = F a. Because X = + a + ,suX F N(z+ + a+ ) uz u2. Therefore,

the agents certainty equivalent of the random incomesuX F is (z+ + a+ ) uz 12 ru2. Similarly,the certainty equivalent of svX F is (z+ + a+) vz 12 rv2. Consequently,EersXF

max!er(z++a+)uz 12 ru2er(z++a+)vz

12 rv

2"

= er(+a+)max!eru(z+ 12 ru2 erv(z+ 12 rv2"On the other hand, replace F in (5) with z++a, andwe have

EersXF = er(+a+)/zConsequently,

/z /lbzdef= max{eru(z+ 12 ru2 erv(z+ 12 rv2}4 (15)

i.e., / is bounded below by the maximum of twosimple convex functions of z, one increasing in z andthe other decreasing.The immediate benets of the above lower-bound

function are easily computable bounds around z. Lety0 be the value of /z for any given z. For example,y0 = /0 = e 12 ru2.ru + e 12 rv2 .rv. Let z =max!z5 /lbz y0" and z =min!z5 /lbz y0". From(15), z z z. To obtain z and z, simply solve

eru(z+12 ru

2 = y0 and erv(z+ 12 rv2 = y0to arrive at

z= ln y012 ru

2

ru( and z=ln y0 12 rv2

rv(

The minimum point of /z, z, can then be obtainedby, e.g., a bisection search over z z z.Next, we derive an upper bound on the rms

expected prot. First, x ( and consider the rmsexpected prot as a function of u and v. (Recall that) is always set at its minimum value, which bindsthe participation constraint for the low-type agent. Inother words, ) has been replaced by a function ofthe other contract parameters.) Note that the rmsexpected revenue and expected operations costs areindependent of u and v, and only the agents expectedwage depends on u and v. Consider the expectedwage in (13) after removing all the terms that are notaffected by u and v:

ln/zr

+(vzuv(z

.

(z

)+,

(z

))

(16)


(Note that z depends on u and v.) To derive a lowerbound on the above expression (and thus eventuallyan upper bound on the rms objective function), wereplace / with /lb and allow the rm to choosethe value of z. That is, the solution to the followingproblem provides a lower bound on (16):

minz

ln/lbzr

+ ( vz

u v(z

.

(z

)+,

(z

)) (17)

To solve the problem in (17), recall that /lb isthe maximum of two convex functions, one increasingand the other decreasing. The two convex functionsintersect at z0, which is the solution to

ru(z+ 12ru2 = rv(z+ 1

2rv2

Thus, z0 = 12 r2u+v. For z z0, the objective func-tion in (17) can be written as

u(z+ 12ru2+(vz

uv(z

.

(z

)+,

(z

))

= uvzuv(z

.

(z

)+,

(z

))+ 12ru2

because /lbz= eru(z+ 12 ru2 for z z0. It is easy toverify that the above function is increasing in z. Sim-ilarly, the objective function is decreasing for z < z0.Therefore, the solution to (17) is z= z0, with the min-imum objective function value

u vz0 u v(z0.

(z0

)+,

(z0

))+ 12ru2

= 12rv2 u v

( 12 r2u+ v

.

( 12 r2u+ v

)

+,( 12 r2u+ v

)) (18)

Now consider the problem of minimizing (18) overu and v. If we keep u + v constant and decreasev (thus increase u), the above expression decreasesbecause w.w+,w > 0 for all w. Setting v= 0, wehave a new lower bound:

u( 12 ru.( 12 ru)+,( 12 ru))Letting w = 12 ru, the above lower bound can bewritten as

2rww.w+,w (19)

Lemma 1. Dene f w=ww.w+,w for all w.Then, f is quasi-convex. Moreover, f w is minimizedat w06 with f 0601.

Proof. Note that

f w= 2w.w+,w= 2,w(w.w

,w+ 12

)

To examine the sign of f w, one only needs to lookat the sign of the term inside the brackets. Note that(

w.w

,w

)= .w+w,w+w

2.w

,w> 0

because .w+w,w+w2.w > 0 for all w. (To seethis, simply note that .w + w,w + w2.w =2,w + 2w.w > 0, a fact noted earlier, and thatlimw.w + w,w + w2.w = 0.) Also, onecan easily nd a w value with f w < 0 and a wvalue with f w > 0 (such as w = 0). Therefore f changes from negative to positive exactly once, imply-ing that f is quasi-convex. Numerical optimiza-tion shows that f w is minimized at w 06 withf 0601. The above lemma implies that a lower bound on (19)

is 02/r . Combining this lower bound on (16) withthose terms in the expected wage expression (13) thatwere removed because they are independent of u andv, we have a lower bound on the expected wage (13):

(+( (2

2(L

lnU0r

02r

Combining the above expression with the rmsexpected revenue and expected production/inventorycosts, we have an upper bound on the rms protfunction:

6(def= +(G1

{(+( (

2

2(L

lnU0r

02r

}

= 1(+(G1+(2

2+(L+

lnU0r

+ 02r

This upper bound is a concave, quadratic func-tion of (. Let 60 be a feasible value of the rmsexpected prot. Dene (=min!(5 6(60" and (=max!(5 6(60". The following result is immediate.Theorem 3. ( ( (, where ( is the optimal value

of (.

5.4. OptimizationThe above results suggest the following algorithm fordetermining the values of the contract parameters (,), u, and v that maximize the rms expected prot.As noted earlier, the optimal value of ) is uniquelydetermined by the other three contract parametersthrough the binding participation constraint of thelow-type agent. It remains only to determine theoptimal values of (, u, and v. To this end, rst compute


the bounds ( and ( on the optimal (. The algorithmhas two main loops. The outer loop iterates througha nite number of values of ( in the interval ( (,determined by a given step size. For each value of(, the inner loop performs a grid search over uv.For each given pair uv inside the inner loop, rstdetermine z and then use this value to determine theoptimal corresponding value of ), as well as the rmsexpected prot. The optimal values of (, u, and v arethe ones that return the maximum value of (14), therms expected prot. Note that G1 can be computedoutside of the two loops, because it does not affect theoptimal values of the contract parameters.

6. Numerical ExamplesHere we numerically compare the Gonik solutionwith a menu of linear contracts, using as benchmarksthe rst-best solution and the pooling solution undera single linear contract. We xed some parameters

Figure 2 Comparisons of Firm Prots

From Gonik Solution to a Menu of Linear Contracts

0

5

10

15

20

25

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

(Menu Gonik)/Menu

Num

ber o

f Exa

mpl

es

From a Menu of Linear Contracts to First Best

0

5

10

15

20

25

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

(FB Menu)/FB

Num

ber o

f Exa

mpl

es

From a Single Linear Contract to Gonik Solution

0

5

10

15

20

25

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

(Gonik Linear)/Gonik

Num

ber o

f Exa

mpl

es

and varied the others to arrive at 81 examples: = 05;U0 = 1; p = 01; c = 1; L H = 23 34 25; = 020508; r = 0512; and c = 020508. Thethree pairs of L H are indexed by 1, 2, and 3 inFigure 3.For each example, we computed the rms max-

imum expected prots under the rst-best solution,a single linear contract, a menu of linear contracts,and the Gonik scheme. The relative differences amongthese prots are tabulated and reported in Figure 2.Figure 3 then examines these differences more closely,seeking relationships, if any, between the quality of asolution and model parameters. The numerical resultslead to the following observations.First, the Gonik solution dominates the pooling

solution, and the average percentage prot differenceis 3.97%. This shows that the rm benets from amenu contract that leads to the separation of theagent types. (Put differently, the market information


Figure 3 Prot Gaps and System Parameters

0%

2%

4%

6%

8%

10%

12%

1theta_combination

0%

2%

4%

6%

8%

10%

12%

0.2 0.5 0.8sigma

0%

2%

4%

6%

8%

10%

12%

0.5 1.0 2.0risk aversion

Gonik vs. Single Linear Menu Linear vs. Gonik First Best vs. Menu Linear

0%

2%

4%

6%

8%

10%

12%

0.2 0.5 0.8unit production cost

2 3

is valuable.) The gap increases when there is greatervariability in the market conditions (reected in alarger H L), the demand variance decreases, or theunit production cost is close to the middle between pand c. The gap is insensitive to the agents degree ofrisk aversion.Second, a menu of linear contracts dominates the

Gonik solution, with an average percentage prot dif-ference of 4.27%. The gap between these two solu-tions increases when (a) there is greater variability inthe market conditions, (b) the variance of the randomnoise term decreases, and (c) the agent becomes lessrisk averse. Recall that under Goniks scheme, bothtypes of agent exert the same level of effort, whereasa menu of linear contracts calls for a higher level ofeffort for the high type. It is perhaps this exibilitythat makes a menu of linear contracts a better solu-tion. As H L increases, the abovementioned exi-bility becomes more important, explaining (a).Third, the gap between a menu of linear contracts

and the rst-best solution is signicant, with an aver-age percentage difference of 8.82%. The gap increaseswhen (i) the market contracts (reected in a smaller value), (ii) the variance of the random noise termincreases, and (iii) the agent becomes more riskaverse. Note that (ii) and (iii) are consistent withthe basic moral hazard model: Larger means morenoise in the measurement of the agents effort, andmore risk aversion means the provision of incentivesbecomes more costly for the principal; both of thesepoint to a more serious moral hazard problem. Thereason for (i) seems to be that as decreases, therms prot decreases under both solutions, amplify-ing their relative gap.

Finally, note that the unit production cost c (andfor that matter, the salvage value p and the over-time cost c) does not affect the rms optimizationproblem for choosing the optimal contract parametersunder the rst-best solution, a menu of linear con-tracts, and the Gonik scheme. Therefore, a change in cwill cause the rms prots under these solutions tochange by the same amount. The sensitivity analysispresented in Figure 3 over the unit production costshows that the percentage prot differences (amongthe three solutions) become higher for c = 05. Thereason is that it is often true that if the values of cpand c c become closer, while keeping their total con-stant, the minimum inventory cost increases (this iseasy to verify in the EOQ model with backorders).Note that one can arbitrarily increase (or decrease) thepercentage prot differences between two contractingapproaches by increasing (or decreasing) the operat-ing costs.

7. Concluding RemarksGonik (1978) said it well when he described the objec-tives all managers are shooting for in a complex salesenvironment:

First, they want to pay salesmen for their absolute salesvolume. Second, they want to pay them for their effort,even if they are in tough areas where they will sellless. Third, they want good and fresh eld informationon market potential for planning and control purposes.(p. 118)

A clever method to achieve these goals, the Gonikscheme is widely quoted in both the marketing and


the economics/agency literature. The contribution ofthis paper is, then, (1) to devise a model that blendsthe above rst and second challenges articulated byGonik with a concrete benet of good and fresh eldinformation in production and inventory planning,(2) to provide a careful analysis of the Gonik schemein the context of this model, and (3) to compare theGonik scheme with a menu of linear contracts, a con-temporary solution often suggested in the agency lit-erature. Our analysis shows that the Gonik scheme isdominated by a menu of linear contracts.As mentioned earlier, the Gonik scheme induces the

same level of effort for both agent types, whereas amenu of linear contracts can cause the agent to exertdifferent levels of effort under different market con-ditions. One can modify the Gonik scheme so as toinduce different levels of effort from different agenttypes. Consider sF F = f0 + f1F + f2F 2 for someconstants f0 f1 f2. (The original Gonik scheme hasf2 = 0.) Using this new form of sF F in (4), wehave a new menu of nonlinear contracts. Call thisthe new Gonik scheme. A full-blown analysis of thiscontract form is complex. However, if we assumethat u= v= sF F = f1+ 2f2F , then the analysis isfairly straightforward. In this case, the rm offersa menu of linear contracts, and if the agent sub-mits a forecast F , he has effectively chosen a linearcontract sx F = (F x + )F , where (F = f1 +2f2F and )F = sF F (F F = f0 f2F 2. For thisnew Gonik scheme, it can be shown that the differentagent types will indeed choose different linear con-tracts and exert different levels of effort. An immedi-ate question is whether or not this new scheme willoutperform the menu of linear contracts consideredin 4. The answer is no. This is simply because thetwo linear contracts chosen by the two agent typesunder the new Gonik scheme can be offered directlyto the agent at the outset (i.e., without going throughthe sF F calculations). The question is how muchof the gap between the original Gonik scheme anda menu of linear contracts (as considered in 4) canbe closed by the new Gonik schemeaccording topreliminary numerical analysis, not much. The rea-son appears to be that the new Gonik scheme offersan innite number of linear contracts, and these con-tracts are not unrelated because they must be tan-gents of the same quadratic function. This gives theagent much exibility at the expense of the principal.However, this disadvantage of the new Gonik schemeis likely to diminish as the number of possible mar-ket conditions increases. Of course, the Gonik ideamay still prevail with more general functional forms,and/or with u and v chosen optimally; we will leavethis for future research. (The analysis of the quadraticGonik scheme can be obtained from the author uponrequest.)

Although our numerical comparison shows thatthe Gonik scheme is dominated by a menu of linearcontracts, there are other aspects that need to beconsidered before a denitive conclusion can bedrawn about these two solutions. This paper hasassumed a single sales agent serving a single sales ter-ritory that has two possible market conditions (highor low). Reality is always much more complicated.As the sales environment gets more complicated, itseems that the Gonik scheme has the ability to keepthe contract simple, and thus easily implementable, ascompared with a menu of linear contracts. For exam-ple, with each additional sales territory, the numberof contracting parameters increases by one under theGonik scheme (the rm needs to provide a quota Oifor territory i and compensates agent i by the amountsXi/Oi Fi/Oi, where Xi is the actual sales in terri-tory i and Fi is the forecast by agent i), whereas undera menu of linear contracts, the number of contract-ing parameters (the slope and intercept of each lin-ear function on the menu) increases by two times thenumber of different possible market conditions in theadded territory. By this count, the Gonik scheme issimpler than a menu of linear contracts. This advan-tage of the Gonik scheme must be considered togetherwith the rms prot performance to say which ofthese solutions is truly better in a multiterritory salesenvironment.

AcknowledgmentsThe author would like to acknowledge nancial supportfrom the Columbia Business School, the National ScienceFoundation, and the Stanford Global Supply Chain Man-agement Forum. Thanks to Bob Gibbons of MassachusettsInstitute of Technology, two anonymous referees, and theseminar participants at Columbia University, the INFORMS2000 meeting in Salt Lake City, the Stanford Workshop onIncentives and Coordination in Operations Management,the University of British Columbia, and the University ofCalifornia at Berkeley for helpful suggestions and com-ments. Aradhna Krishna of the University of Michigan rstbrought the Gonik article in Harvard Business Review to theauthors attention. The rst draft of this paper was writ-ten while the author was on sabbatical leave at StanfordUniversity. Thanks to the Operations Information and Tech-nology group of the Stanford Business School for their hos-pitality.

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