Serving the many or serving the most needy?

Abstract

Free, subsidized, or cost-covering? The decision on how much to charge for a good or service is fundamental in social business planning. The higher the fee paid by the recipient, the more people in need can be served by the additional revenues. However, charging a fee simultaneously excludes the very poor from consumption. This paper argues that the entrepreneur’s trade-off between both effects is governed by her level of poverty aversion, i.e., her preference intensity for the service of needy people with different incomes. Additionally, we account for the possibility of excess demand for the provided good and assume that applicants are rationed by non-price-allocation mechanisms. We therefore contribute to the extensive literature on the pricing and rationing behavior of nonprofit firms. Within our theoretical model, we find ambiguous reactions of the entrepreneur to a cut in donations. Given a sufficiently low level of status-quo donations, entrepreneurs with relatively high poverty aversion tend to increase the project volume, whereas those with relatively low poverty aversion do the opposite.

Appendix

Proof of Proposition 5

Let $D\in \left( {0,\bar{{n}}\left( 0 \right)\cdot c} \right)$ and $\alpha <\hat{{\alpha }}$ with

$$\begin{aligned} \hat{{\alpha }}=\left. {\left[ {{\ln \left( {1-\frac{1}{\bar{{n}}_F \cdot c}} \right)}/{\left( {\ln \bar{{n}}-\ln \underline{n}} \right)}} \right]} \right|_{F=0} . \end{aligned}$$

Then, according to proposition 3, $F^{{*}}>0$. Now, consider Eq. (11). With $f=f^{{*}}>0$, an increase in donations enlarges the entrepreneur’s total income by $c/{\left( {c-f^{{*}}} \right)}$. Consequently, an increase in donations leads to a decrease in the optimal project volume if ${dF^{{*}}}/{dD}<-c/{\left( {c-f^{{*}}} \right)}$. Applying the implicit function theorem to the first-order condition yields

$$\begin{aligned} \frac{dF^{{*}}}{dD}=\left. {-\frac{{\partial ^{2}U\left( {F;D} \right)}/{\partial F\partial D}}{{\partial ^{2}U\left( {F;D} \right)}/{\partial F^{2}}}} \right|_{F=F^{{*}}} <-\frac{c}{c-f^{{*}}}, \end{aligned}$$

which can be rearranged to

$$\begin{aligned} \left. {\left[ {\frac{\partial ^{2}U\left( {F;D} \right)}{\partial F^{2}}-\frac{c-f}{c}\cdot \frac{\partial ^{2}U\left( {F;D} \right)}{\partial F\partial D}} \right]} \right|_{F=F^{{*}}} >0. \end{aligned}$$

${\partial ^{2}U\left( {F;D} \right)}/{\partial F^{2}}$ is given by Eq. (9) and $\left. {\left( {{\partial ^{2}U\left( {F;D} \right)}/{\partial F\partial D}} \right)} \right|_{F=F^{{*}}}$ by Eq. (12). Hence, the optimal project volume decreases if

$$\begin{aligned} \Omega :=\left[ {\left. {\left. {\frac{\partial ^{2}U\left( {F;D} \right)}{\partial F^{2}}-\frac{c-f}{c}\cdot \frac{\partial ^{2}U\left( {F;D} \right)}{\partial F\partial D}} \right]} \right|_{F=F^{{*}}} =} \right[\left( {\bar{{n}}^{\alpha }-\underline{n}^{\alpha }} \right)\cdot \bar{{n}}_{FF} \nonumber \\ \left. {\left. {+\alpha \cdot \bar{{n}}_F^2 \cdot \left[ {\bar{{n}}^{\alpha -1}-\left( {1-\frac{1}{\bar{{n}}_F \cdot c}} \right)\cdot \underline{n}^{\alpha -1}} \right]} \right]} \right|_{F=F^{{*}}} >0 \text{.} \end{aligned}$$

(13)

The two terms of condition (13) characterize the change in the replacement effect resulting from an increase in user-fee revenues. The first term is positive by definition and the second term is nonnegative for all $\alpha \ge {\alpha }^{\prime }$, with

$$\begin{aligned} {\alpha }^{\prime }=\left. {1+\left[ {{\ln \left( {1-\frac{1}{\bar{{n}}_F \cdot c}} \right)}/{\left( {\ln \bar{{n}}-\ln \underline{n}} \right)}} \right]} \right|_{F=F^{{*}}} . \end{aligned}$$

Next, we show that a unique $\mathop {\alpha }\limits ^{\smile } \in \left( {0,{\alpha }^{\prime }} \right)$ exists for which $\Omega$ is zero. Hence, $\Omega$ is positive for all $\alpha >\mathop {\alpha }\limits ^{\smile }$ and negative for all $\alpha <\mathop {\alpha }\limits ^{\smile }$.

Rearranging Eq. (13) yields

$$\begin{aligned} \tilde{\Omega }:=y\left( {\alpha ,\underline{n}} \right)-z\left( {\alpha ,\underline{n}} \right), \end{aligned}$$

(14)

with

$$\begin{aligned} y\left( {\alpha ,\underline{n}} \right):=\left. {\left[ {\left( {\bar{{n}}_{FF} +\alpha \cdot \bar{{n}}_F^2 \cdot \frac{1}{\bar{{n}}}} \right)\cdot \left( {\frac{\bar{{n}}}{\underline{n}}} \right)^{\alpha }} \right]} \right|_{F=F^{{*}}} \end{aligned}$$

and

$$\begin{aligned} z\left( {\alpha ,\underline{n}} \right):=\left. {\left[ {\bar{{n}}_{FF} +\alpha \cdot \bar{{n}}_F^2 \cdot \left( {1-\frac{1}{\bar{{n}}_F \cdot c}} \right)\cdot \frac{1}{\underline{n}}} \right]} \right|_{F=F^{{*}}} . \end{aligned}$$

With $\bar{{n}}>\underline{n}$, $\bar{{n}}_F <0$ and $\bar{{n}}_{FF} >0$, $y\left( {\alpha ,\underline{n}} \right)$ is the product of a linear and a convex increasing function of $\alpha$. Hence, $y\left( {\alpha ,\underline{n}} \right)$ is also increasing and convex in $\alpha$. On the other hand, $z\left( {\alpha ,\underline{n}} \right)$ is linearly increasing in $\alpha$. Consequently, the difference of both terms, $\tilde{\Omega }$, has maximally two roots. Apparently, one is given for $\alpha =0$.¹⁵ There exists a second root for $\alpha =\mathop {\alpha }\limits ^{\smile } >0$ if and only if $\left. {\left( {{dy\left( {\alpha ,\underline{n}} \right)}/{d\alpha }} \right)} \right|_{\alpha =0} <\left. {\left( {{dz\left( {\alpha ,\underline{n}} \right)}/{d\alpha }} \right)} \right|_{\alpha =0}$, i.e.,

$$\begin{aligned} y_\alpha :=\left. {\left( {\frac{dy\left( {\alpha ,\underline{n}} \right)}{d\alpha }} \right)} \right|_{\alpha =0}&= \bar{{n}}_F^2 \cdot \frac{1}{\bar{{n}}}+\bar{{n}}_{FF} \cdot \ln \left( {\frac{\bar{{n}}}{\underline{n}}} \right)\\&<\bar{{n}}_F^2 \cdot \left( {1-\frac{1}{\bar{{n}}_F \cdot c}} \right)\cdot \frac{1}{\underline{n}}=\left. {\left( {\frac{dz\left( {\alpha ,\underline{n}} \right)}{d\alpha }} \right)} \right|_{\alpha =0} =:z_\alpha . \end{aligned}$$

This condition holds since $\alpha \rightarrow 0$ implies that $F^{{*}}\rightarrow F_{\max }$ and $\underline{n}\rightarrow 0$.¹⁶ Although the limits of $y_\alpha$ and $z_\alpha$ are infinity for $\underline{n}\rightarrow 0$, the application of l’Hôpital’s rule shows that $y_\alpha$ and $z_\alpha$ diverge and $z_\alpha >y_\alpha$ results:

$$\begin{aligned} \mathop {\lim }\limits _{\underline{n}\rightarrow 0} \frac{y_\alpha }{z_\alpha }=\mathop {\lim }\limits _{\underline{n}\rightarrow 0} \left. {\left[ {\frac{\left( {{\partial ^{2}y\left( {\alpha ,\underline{n}} \right)}/{\partial \alpha \partial \underline{n}}} \right)}{\left( {{\partial ^{2}z\left( {\alpha ,\underline{n}} \right)}/{\partial \alpha \partial \underline{n}}} \right)}} \right]} \right|_{\alpha =0} =\frac{\bar{{n}}_{FF} \cdot \underline{n}}{\bar{{n}}_F^2 \cdot \left[ {1-\left( {1/{\left( {\bar{{n}}_F \cdot c} \right)}} \right)} \right]}=0. \end{aligned}$$

Consequently, there exists a unique $\mathop {\alpha }\limits ^{\smile } \in \left( {0,{\alpha }^{\prime }} \right)$ for which the value of $\tilde{\Omega }$, or respectively $\Omega$, is zero.

Yet, we assumed that $\alpha <\hat{{\alpha }}$ and derived the requirement that $\alpha >\mathop {\alpha }\limits ^{\smile }$. Consequently, an increase in donations leads to a reduction in the optimal project volume if $\mathop {\alpha }\limits ^{\smile } <\hat{{\alpha }}$ and $\alpha \in \left( \mathop {\alpha }\limits ^{\smile } ,\hat{{\alpha }} \right)$. However, $\mathop {\alpha }\limits ^{\smile } <\hat{{\alpha }}$ requires a sufficiently low level of donations. For $D\rightarrow \bar{{n}}\left( 0 \right)\cdot c$, $\ln \bar{{n}}-\ln \underline{n}$, which determines $\hat{{\alpha }}$ and ${\alpha }^{\prime }$, is infinitely large, such that $\hat{{\alpha }}\rightarrow 0$ and ${\alpha }^{\prime }\rightarrow 1$. Since $\mathop {\alpha }\limits ^{\smile } <{\alpha }^{\prime }$, it must hold that $\mathop {\alpha }\limits ^{\smile } \in \left( {0,1} \right)$ and, consequently, $\mathop {\alpha }\limits ^{\smile } >\hat{{\alpha }}$. In other words, given that the amount of donations is relatively high, all entrepreneurs react by enlarging their project volumes when they experience an increase in donations. In contrast, for $D\rightarrow 0$, $\left. {\left( {\ln \bar{{n}}-\ln \underline{n}} \right)} \right|_{F=0} \rightarrow 0$ and, hence, $\hat{{\alpha }}\rightarrow \infty$. According to proposition 3, all entrepreneurs with $\alpha <\hat{{\alpha }}$ choose $F^{{*}}>$0. Consequently, $\left. {\left( {\ln \bar{{n}}-\ln \underline{n}} \right)} \right|_{F=F^{{*}}} >0$ and ${\alpha }^{\prime }\in \left( {1,\infty } \right)$. Since $\mathop {\alpha }\limits ^{\smile } <{\alpha }^{\prime }$, it holds that $\mathop {\alpha }\limits ^{\smile } <\hat{{\alpha }}$. As a result, there exists a specific level of donations $D^{\prime }$ such that $\mathop {\alpha }\limits ^{\smile } =\hat{{\alpha }}$, if $D={D}^{\prime }$, and $\mathop {\alpha }\limits ^{\smile } <\hat{{\alpha }}$, if $D\in \left( {0,D^{\prime }} \right)$. Hence, for all $D\in \left( {0,D^{\prime }} \right)$, an increase in donations leads to a reduction in the optimal project volume $F^{{*}}+D$, if $\alpha \in \left( {\mathop {\alpha }\limits ^{\smile } ,\hat{{\alpha }}} \right)$. $\square$