Targeting and salesforce compensation: When sales spill over to unprofitable customers

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.

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 + 2aB − p) $-\left (a+\frac { r\sigma ^{2}}{2}\right )$n_HB² 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 + 2aB = p, which leads to p^S = 2α (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

$$\begin{array}{@{}rcl@{}} \underset{p,A,e}{\max} \pi &=&\left( p-c\right) \left[ n_{H}\left( v_{H}+e-p\right) +n_{L}\left( v_{L}+e-p\right) \right] -A \\ &&\text{subject to\ } Eu\left[ A-\frac{\left( n_{H}+n_{L}\right) e^{2}}{4a} \right] \geq u\left( w_{0}\right) \ \text{(IRC)}. \end{array} $$

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 σ² = 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 + 2aB − p) $-\left (a+\frac {r\sigma ^{2}}{2}\right )$n_HB² 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 $x_{L}^{\ast } \left (e,p\right ) =e-p$ ≤ 0.¹² When α ≥ 1 − z/2, the firm can prevent sales to the lows by ensuring that x_L (B,p) = 0 ⇔ 2aB = p, which leads to p^S = 2αv, and $B^{S}=\frac {v}{1+\frac {r\sigma ^{2}}{2a}}$. The expected sales to the highs is $x_{H}^{\ast } \left (p^{S},B^{S}\right ) =$vn_H and the screening profits are π^S = π (p^S,B^S) = (αv − c)vn_H which can be restated in terms of z as π^S = v² (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

$$\begin{array}{@{}rcl@{}} \underset{p,A,e}{\max} \pi &=&\left( p-c\right) \left[ n_{H}\left( v+e-p\right) +n_{L}\left( e-p\right) \right] -A \\ &&\text{subject to} Eu\left[ A-\frac{\left( n_{H}+n_{L}\right) e^{2}}{4a} \right] \geq u\left( w_{0}\right) \ \text{(IRC)}. \end{array} $$

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 σ² = 0. □