## Abstract

Targeting selling efforts towards profitable customers is widely known to increase sales and allow firms to charge higher prices. In this paper, we show that targeting of selling efforts may also inadvertently lead to sales spilling over to unprofitable customers when they are not identifiable. Such spillover sales are more likely when the ability of the salesperson and the profitability of target customers are above a threshold. We also show that firms can solve this problem by lowering the sales incentives as well as the price to make their offer unattractive to the unprofitable customers, a strategy commonly referred as screening. When the ability and profitability are both very high, however, the firm is better off allowing sales to unprofitable customers because the cost of preventing sales from spilling over is excessive. This is because the reduction in profits from the target customers that results under screening exceeds the loss from allowing sales to unprofitable customers. Such an accommodation strategy becomes more attractive as the fraction of unprofitable customers in the market decreases. Finally, we show that the spillover problem is even more acute when firms can monitor the selling efforts of a salesperson.

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## Notes

The problem of identification of target customers exists even in retail situations. An incident that gained widespread notoriety, happened in August 2013 when a salesperson at Zurich’s Trois Pommes had reportedly refused to show a $38,000 Hermes handbag to a customer (who was later revealed to be Oprah Winfrey), telling her that she could not afford it. Source: http://www.usatoday.com/story/money/business/2013/08/09/a-38000-handbag-not-unheard-of-in-luxury-market/2635871/(last accessed on 13 November 2014).

All the qualitative results of the paper can be shown to remain valid more generally for

*v*_{L}∈ (0,*c*). Details are available in Appendix A.The utility maximization, given common knowledge of the parameters, is based on rational expectations of selling effort and price using all available information including word of mouth. When the resulting utility is non-positive, we assume that there is neither any interaction between the customer and the salesperson nor any sales. When the resulting utility is positive, note that the customer still needs to experience the salesperson’s effort for the utility to materialize and result in sales.

Note that assuming a Normal Distribution of the error term allows sales to a customer, particularly that of the ‘lows’, to be negative. We interpret negative sales as customer returns. At the level of abstraction where our single (accounting) period model represents one of many repeated stages with the same general characteristics of the model except parameters such as customer valuation, customer returns may include items from inventory carried over from previous accounting periods (Lal and Srinivasan 1993, p.783).

Note that negative aggregate sales may be interpreted as the number of units returned by customers from the previous periods being more than new units purchased by them in the current period. However, Lal and Srinivasan (1993, p. 784) point out that the probability of negative aggregate sales is very small. Here, even in case of the lows, the probability of such negative aggregate sales is lower than 0.5% when

*n*_{L}≥ [3*σ*/ (*e*−*p*)]^{2}.If

*a*= 0 (i.e., the salesperson’s ability is zero), it does not make sense for the firm to hire the salesperson. At the other extreme, when*a*= 1, it can be shown that the firm will be able to induce infinite effort to earn infinite profits. We therefore restrict attention to the realistic case where*a*∈ (0, 1).The particular form of the salesperson’s utility function we use is

*u*(*w*) = 1 − exp [−*r*(*w*−*C*(*T*))].The model assumes, as in the literature (e.g., Inderst and Ottaviani 2009, p. 885), that in equilibrium, the customers are aware of their potential valuation of the information provided by the salesperson. This assumption is based on the commonplace hearsay information about the level of selling efforts that the customers may receive.

As explained in the subsequent paragraphs, in this model any increase in price without a corresponding increase in selling efforts leads to lower sales to the target segment. While the purpose of screening is to prevent expected sales to the lows, its aim is also to preserve sales to the highs. A strategy using a higher price with a lower selling effort reduces sales to the highs more than the alternative strategy using lower effort and a lower price which is the solution to this problem.

It is possible that when customers are heterogeneous in terms of their sensitivities to selling efforts (not in our model) such that \(x_{i}^{\ast } =v_{i}+s_{i}e_{i}-p_{i}\) where

*s*_{L}≠*s*_{H}, and*v*_{L}≥*c*, the firm may want to serve both highs and lows by offering a menu of discriminating contracts.

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## Additional information

The authors are grateful to the Whitman Summer Research Fund for supporting this research.

## Appendices

### Appendix A: The case where 0 < *v*
_{L}< *c*

### The targeting contract

Solving the necessary and sufficient first order conditions for the firm’s objective function, \(\underset {p,B}{\max }{\kern 3pt}n_{H}[\left (p-c\right ) \left (v_{H}+ 2aB-p\right ) -\left (a+\frac {r\sigma ^{2}}{2}\right ) B^{2}]+n_{L}[\left (p-c\right ) \left (v_{L}+ 2aB-p\right ) -\left (a+\frac {r\sigma ^{2}}{2}\right ) B^{2}]\) which is jointly concave in price and sales commissions, and defining \(Z=\frac {v_{H}-c}{ v_{H}-v_{L}}\) we get \(p^{\ast } =c+\frac {\left (v_{H}-v_{L}\right ) \left (Z-\phi \right )} {2\left (1-\alpha \right )} \), and \(B^{\ast } =\frac {\alpha } {2a}\frac {\left (v_{H}-v_{L}\right ) \left (Z-\phi \right )} {1-\alpha } \) ⇒ \(e^{\ast } =\alpha \frac {\left (v_{H}-v_{L}\right ) \left (Z-\phi \right )} {1-\alpha } \). Note that *p*^{∗}≥ *c*, *B*^{∗}≥ 0 only if *Z* ≥ *ϕ*. The resulting total profit is *π* (*p*^{∗},*B*^{∗}) = \(\frac {n_{H}+n_{L}}{4}\frac {\left (v_{H}-v_{L}\right )^{2}\left (Z-\phi \right )^{2}}{1-\alpha } \) which can be rewritten as \(\pi \left (p^{\ast } ,B^{\ast } \right ) =\frac {n_{H}}{4}\frac {\left (v_{H}-v_{L}\right )^{2}Z^{2}\left (1-\phi \right )} {1-\alpha } \)\(-\frac {n_{L}} {4}\frac {\left (v_{H}-v_{L}\right ) \left (c-v_{L}\right ) \left (Z-\phi +\left (1-\phi \right ) Z\right )} {1-\alpha } \) where the profits earned from the highs is obtained by substituting *n*_{L} = 0, i.e., *π*_{H} (*p*^{∗},*B*^{∗}) \(=\frac {n_{H}}{4}\frac {\left (v_{H}-v_{L}\right )^{2}Z^{2}\left (1-\phi \right )} {1-\alpha } \). The remainder of profit which comes from the lows is *π*_{L} (*p*^{∗},*B*^{∗}) = \(-\frac {n_{L}}{4}\frac {\left (v_{H}-v_{L}\right ) \left (c-v_{L}\right ) \left (Z-\phi +\left (1-\phi \right ) Z\right )} { 1-\alpha } <0\) given *Z* ≥ *ϕ* from the condition of *p*^{∗}≥ *c*, *B*^{∗}≥ 0 above. The lows are therefore unprofitable.

Now consider the firm decision when it targets the highs by not selling to the lows, \(\underset {p,B}{\max } \)*π* = (*p* − *c*) *E* (*x*_{H}) − *E* [*w* (*x*_{H})] = (*p* − *c*) *n*_{H} (*v*_{H} + 2*a**B* − *p*) \(-\left (a+\frac { r\sigma ^{2}}{2}\right ) \)*n*_{H}*B*^{2} which is concave in *p* and *B*. The first order conditions lead to \(p^{T}=c+\frac {1}{2}\frac {\left (v_{H}-v_{L}\right ) Z}{1-\alpha } \), \(B^{T}=\frac {1}{2}\frac { \left (v_{H}-v_{L}\right ) Z}{1+\frac {r\sigma ^{2}}{2a}}-a\)\(\Rightarrow e^{T}=\frac {a\left (v_{H}-v_{L}\right ) Z}{1+\frac {r\sigma ^{2}}{2a}-a}\) resulting in profits \(\pi ^{T}=\pi \left (p^{T},B^{T}\right ) =\frac {n_{H}}{4} \frac {\left (v_{H}-v_{L}\right )^{2}Z^{2}}{1-\alpha } \). We can see that *π*^{T} > *π* (*p*^{∗},*B*^{∗}).

### Spillover sales occur

From the above, we can see that when the firm chooses (*p*^{T},*B*^{T}), \(x_{L}^{\ast } \left (e^{T},p^{T}\right ) =v_{L}+ 2aB^{T}-p^{T}=\frac {n_{L}\left (v_{H}-v_{L}\right )} {2}\frac {2\alpha -2+Z}{2(1-\alpha )}\geq 0\) if *α* ≥ 1 − *Z*/2. The expected demand from the lows will, therefore, be positive which leads to the failure of the contract based on demand from only the highs serving whom is more productive for the salesperson.

### Screening

Since the lows are unprofitable as shown above, a screening contract that prevents sales to lows requires the conditions \(x_{H}^{\ast } \left (e,p\right ) =v_{H}+e-p\geq 0\) and \(x_{L}^{\ast } \left (e,p\right ) =v_{L}+e-p\)≤ 0. When *α* ≥ 1 − *Z*/2,the firm can prevent sales to the lows by ensuring that *x*_{L} (*B*,*p*) = 0 ⇔ *v*_{L} + 2*a**B* = *p*, which leads to *p*^{S} = 2*α* (*v*_{H} − *v*_{L}) + *v*_{L} and \(B^{S}=\frac {v_{H}-v_{L}}{1+\frac {r\sigma ^{2}}{2a}}\). The expected sales to the highs is \(x_{H}^{\ast } \left (p^{S},B^{S}\right ) =\)*n*_{H} (*v*_{H} − *v*_{L}) and the screening profits are *π*^{S} = *n*_{H} (*v*_{H} − *v*_{L})^{2} (*α* + *Z* − 1).

### Accommodation

Now considering the alternative strategy of accommodating the lows by solving the problem without the constraint \(x_{L}^{\ast } \left (p,B\right ) = 0 \) , we get \(p^{A}=c+\frac {1}{2}\frac {\left (v_{H}-v_{L}\right ) \left (Z-\phi \right )} {1-\alpha } \) and \(B^{A}=\frac {\alpha } {2a}\frac {\left (v_{H}-v_{L}\right ) \left (Z-\phi \right )} {1-\alpha } \) which lead to profits \(\pi ^{A}=\frac {n_{H}+n_{L}}{4}\frac {\left (v_{H}-v_{L}\right )^{2}\left (Z-\phi \right )^{2}}{1-\alpha } \) . Comparing with the profits under screening, we get *π*^{A} − *π*^{S} ≥ 0 if \(\frac {1}{4}\frac {\left (Z-\phi \right )^{2}}{1-\alpha } \geq \left (1-\phi \right ) \left (\alpha -1+Z\right ) \) . Using the positive root of *α* under the condition of sales spillover to lows, the condition above reduces to \(\alpha \geq 1-Z/2+ \frac {1}{2}\sqrt {\frac {\phi } {1-\phi } \left [ Z\left (2-Z\right ) -\phi \right ]}\).

### Monitoring

Considering the firm decisions under observable efforts we have

Solving, we get \(p^{M}=c+\frac {\left (v_{H}-v_{L}\right ) \left (Z-\phi \right )} {2\left (1-a\right )} \) and \(A^{M}=\frac {\left (n_{H}+n_{L}\right ) e^{M2}}{4a}\) where \(e^{M}=a\frac {\left (v_{H}-v_{L}\right ) \left (Z-\phi \right )} {1-a}\). The resulting profit \(\pi ^{M}=\frac {n_{H}+n_{L}}{4}\frac { \left (v_{H}-v_{L}\right )^{2}\left (Z-\phi \right )^{2}}{1-a}\) which can be rewritten as \(\frac {n_{H}}{4}\frac {\left (v_{H}-v_{L}\right )^{2}Z^{2}\left (1-\phi \right )} {1-a}\)\(-\frac {n_{L}}{4}\frac {\left (v_{H}-v_{L}\right ) \left (c-v_{L}\right ) \left (Z-\phi +\left (1-\phi \right ) Z\right )} {1-a}\) where the profit from sales to the highs is obtained by substituting *n*_{L} = 0, i.e., *π*_{H} (*p*^{M},*B*^{M}) \(=\frac {n_{H}}{4}\frac { \left (v_{H}-v_{L}\right )^{2}Z^{2}\left (1-\phi \right )} {1-a}\), the remainder being the profit from the lows: *π*_{L} (*p*^{M},*B*^{M}) \(=-\frac {n_{L}}{4}\frac {\left (v_{H}-v_{L}\right ) \left (c-v_{L}\right ) \left (Z-\phi +\left (1-\phi \right ) Z\right )} {1-a}<0\) implying that lows are unprofitable. The firm therefore earns a higher profit by targeting only the highs by offering \(A^{T}=\frac {n_{H}a}{4}\left (\frac {\left (v_{H}-v_{L}\right ) Z}{1-a}\right )^{2}\) but charging the price (\(p^{T}=c+ \frac {\left (v_{H}-v_{L}\right ) Z}{2\left (1-a\right )} \) ) and requiring effort (\(e^{T}=a\frac {\left (v_{H}-v_{L}\right ) Z}{1-a}\)). This leads to the profit *π*^{T} = HCode \(\frac {{n_{H}^{2}}\left (v_{H}-v_{L}\right )^{2}Z^{2}}{ 4\left (1-a\right )} \). As in the case of unobservable effort, sales spill over to the lows if *v*_{L} + *e*^{T} > *p*^{T} which reduces to *a* ≥ 1 − *Z*/2. All other results can be obtained from those stated under The Targeting Contract by substituting *r* = 0 or *σ*^{2} = 0.

### Appendix B: Proofs

### Proof 1 (Proof of Proposition 1)

The firm’s objective function, \(\underset {p,B}{\max } n_{H}[\left (p-c\right ) \left (v + 2aB-\right .\)\(\left .p\right )-\left (a+\frac {r\sigma ^{2}}{2}\right ) B^{2}]+n_{L}[\left (p-c\right ) \left (2aB-p\right ) -\left (a+\frac {r\sigma ^{2}}{2}\right ) B^{2}]\) (12) is jointly concave in price and sales commissions. Solving the necessary and sufficient first order conditions assuming \(\alpha =\frac {a}{1+\frac {r\sigma ^{2}}{2a}}\), \(z=\frac {v-c}{v}\) and \(\phi =\frac { n_{L}}{n_{H}+n_{L}}\), we get \(p^{\ast } =c+\frac {v\left (z-\phi \right )} { 2\left (1-\alpha \right )} \) , and \(B^{\ast } =\frac {\alpha } {2a}\frac {v\left (z-\phi \right )} {1-\alpha } \)⇒ \(e^{\ast } =\alpha \frac {v\left (z-\phi \right )} {1-\alpha } \). Note that *p*^{∗}≥ *c*, *B*^{∗}≥ 0 only if *z* ≥ *ϕ*. The resulting total profit is *π* (*p*^{∗},*B*^{∗}) = \(\frac {n_{H}}{4}\)\(\frac {v^{2}\left (z^{2}-\phi ^{2}\right )} {4\left (1-\alpha \right )} -\)\(\frac {n_{L}}{4}\frac {v^{2}\left (z-\phi \right ) \left (2-z-\phi \right )} {1-\alpha } \). The profit earned from sales to the lows is given by *π*_{L} (*p*^{∗},*B*^{∗}) = − \(n_{L}\frac {v^{2}\left (z-\phi \right ) \left (2-z-\phi \right )} {4\left (1-\alpha \right )} \) < 0 since *z* ≥ *ϕ* which is a necessary condition for *π* (*p*^{∗},*B*^{∗}) > 0. The lows are therefore unprofitable.

Now considering the firm decision when it targets the highs by not selling to the lows (*B*_{L} = 0) restated (from Eq. 12) as \( \underset {p,B}{\max } \)*π* = (*p* − *c*) *E* (*x*_{H}) − *E* [*w* (*x*_{H})] = (*p* − *c*) *n*_{H} (*v* + 2*a**B* − *p*) \(-\left (a+\frac {r\sigma ^{2}}{2}\right ) \)*n*_{H}*B*^{2} is concave in p and B. The first order conditions lead to \( p^{T}=c+\frac {1}{2}\frac {vz}{1-\alpha } \) , \(B^{T}=\frac {1}{2}\frac {vz}{1+\frac {r\sigma ^{2}}{2a}}-a\)\(\Rightarrow e^{T}=\frac {avz}{1+ \frac {r\sigma ^{2}}{2a}-a}\) resulting in profits \(\pi ^{T}=\pi \left (p^{T},B^{T}\right ) =\frac {n_{H}}{4}\frac {v^{2}z^{2}}{1-\alpha } >\pi \left (p^{\ast } ,B^{\ast } \right ) \). □

### Proof 2 (Proof of Proposition 2)

From the proof of proposition 1, when the firm chooses (*p*^{T},*B*^{T}), \(x_{L}^{\ast } \left (e^{T},p^{T}\right ) = 2aB^{T}-p^{T} =\frac {vz}{2}\frac {1}{1-\alpha } -v\geq 0\) if *α* ≥ 1 − *z*/2. The expected demand from the lows will, therefore, be positive which leads to the failure of the contract based on demand from serving only the highs. □

### Proof 3 (Proof of Proposition 3)

Since the lows are unprofitable (proposition 1), a screening contract that prevents sales to lows requires the conditions \(x_{H}^{\ast } \left (e,p\right ) =v-p+e\geq 0\) and \(x_{L}^{\ast } \left (e,p\right ) =e-p\) ≤ 0.^{Footnote 12} When *α* ≥ 1 − *z*/2, the firm can prevent sales to the lows by ensuring that *x*_{L} (*B*,*p*) = 0 ⇔ 2*a**B* = *p*, which leads to *p*^{S} = 2*α**v*, and \(B^{S}=\frac {v}{1+\frac {r\sigma ^{2}}{2a}}\). The expected sales to the highs is \(x_{H}^{\ast } \left (p^{S},B^{S}\right ) =\)*v**n*_{H} and the screening profits are *π*^{S} = *π* (*p*^{S},*B*^{S}) = (*α**v* − *c*)*v**n*_{H} which can be restated in terms of *z* as *π*^{S} = *v*^{2} (*z* + *α* − 1) *n*_{H}. □

### Proof 4 (Proof of Proposition 4)

Now considering the alternative strategy of accommodating the lows by solving the problem without the constraint \(x_{L}^{\ast } \left (p,B\right ) = 0 \) , we get \(p^{A}=c+\frac {1}{2}\frac {v\left (z-\phi \right )} {1-\alpha } \) and \(B^{A}=\frac {\alpha } {2a}\frac {v\left (z-\phi \right )} {1-\alpha } \) which lead to profits \(\pi ^{A}=\frac {n_{H}+n_{L}}{4}\frac {v^{2}\left (z-\phi \right )^{2}}{1-\alpha } \). Comparing with the profits under screening (see Proof of Proposition 3), we get *π*^{A} − *π*^{S} ≥ 0 if \(\frac {\left (z-\phi \right )^{2}}{4\left (1-\phi \right )} \geq \left (1-\alpha \right ) \left [ z-\left (1-\alpha \right ) \right ] \). Using the positive root of *α* under the condition of sales spillover to lows (*α* ≥ 1 − *z*/2 from proposition 2), the condition above reduces to \(\alpha \geq 1-z/2+\frac {1}{2}\sqrt {\phi \left (1-\frac {\left (1-z\right )^{2}}{1-\phi } \right )} \). □

### Proof 5 (Proof of Proposition 5)

Considering the firm decisions under observable efforts we have

Solving, we get \(p^{M}=c+\frac {vz}{2\left (1-a\right )} \) and \(A^{M}=\frac { \left (n_{H}+n_{L}\right ) e^{M2}}{4a}\) where \(e^{M}=\frac {avz}{1-a}\). The resulting profit *π*^{M} = HCode \(\frac {n_{H}v^{2}z^{2}}{4\left (1-a\right )} \)\( -\frac {n_{L}}{4}\frac {v^{2}z\left (2-z\right )} {1-a}\) where the profits from the lows (as in the Proof of Proposition 1) \({\pi _{L}^{M}}=\)\(-\frac { n_{L}}{4}\frac {v^{2}z\left (2-z\right )} {1-a}<0\). The firm therefore earns a higher profit by targeting only the highs by offering \(A^{T}=\frac {n_{H}a}{4} \left (\frac {vz}{1-a}\right )^{2}\) but setting the same price (*p*^{T} = *p*^{M} ) and effort (*e*^{T} = *e*^{M}). This leads to the profit *π*^{T} = HCode \(\frac { n_{H}\left (vz\right )^{2}}{4\left (1-a\right )} \). As in proposition 2, sales spill over to the lows if *e*^{T} > *p*^{T} which reduces to *a* > 1 − *z*/2. All other results can be obtained from those stated under Proof of Proposition 1 by substituting *r* = 0 or *σ*^{2} = 0. □

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### Cite this article

Banerjee, S., Thevaranjan, A.P. Targeting and salesforce compensation: When sales spill over to unprofitable customers.
*Quant Mark Econ* **17**, 81–104 (2019). https://doi.org/10.1007/s11129-018-9208-2

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DOI: https://doi.org/10.1007/s11129-018-9208-2

### Keywords

- Salesforce compensation
- Targeting
- Principal-agent models
- Agency theory

### JEL Classification

- M31
- M52
- D86
- D82
- L14