Impacts of earmarked private donations for disaster fundraising

Faced with large humanitarian emergencies like the earthquakes in Haiti and Japan, aid agencies have to decide how to collect money for their relief work. They can either decide to establish a special fund for the emergency and allow for earmarked donations or they can only allow for unearmarked donations. In this paper, we analyze impacts of this decision on donors, aid agencies, and policy makers. To this end, we compare two prevalent fundraising modes using optimization models: fundraising with the option of earmarking donations and fundraising without an earmarking option. In the earmarked case, we consider a new fundraising challenge, excessive funds raised for certain disaster relief projects. We find that desirable fundraising modes for donors, aid agencies, and policy makers differ depending on levels of several parameters, including an aid agency’s utility of a dollar raised, the fundraising cost factor, and donors’ unit utility of donations. Allowing for earmarking leads to a lower overall fundraising cost percentage. For emergencies with strong media attention and donor interest, allowing for earmarking of donations is likely to reduce fundraising activities of organizations with low fundraising costs, while it is likely to encourage fundraising activities among organizations with high fundraising costs.

1. Following Lus and Muriel (2009), we can transform (1) to show the price and cross-price effect. Letting ϕ _i ≡(a _j A _i −bA _j )/(a _i a _j −b ²), τ _i ≡a _j /(a _i a _j −b ²) and υ≡b/(a _i a _j −b ²), the direct donation supply function can be rewritten as d _i =ϕ _i −τ _i p _i +υp _j . In the resulting donation supply function, ϕ _i is the supply intercept which represents the potential fund size and τ _i and υ are the price and cross-price effects.

This work was supported in full or in part by a grant from the Fogelman College of Business and Economics at the University of Memphis and a start-up grant from School of Administrative Studies at York University.

Authors and Affiliations

1. School of Administrative Studies, York University, 4700 Kelle Street, Toronto, M3J 1P3, ON, Canada

   Fuminori Toyasaki

2. WU (Vienna University of Economics and Business), Nordbergstraße 15, 1090, Vienna, Austria

   Tina Wakolbinger

Corresponding author

Correspondence to Tina Wakolbinger.

Appendix

Prof of Proposition 1

The partial differentiation of \(p^{*}_{e}\) with respect to α ₂ is

(24)

The sign of this equation depends on the second term of the numerator because the first term in the numerator is negative and the denominator is positive. Thus, \(\frac{\partial p^{*}_{e}}{\partial \alpha_{2}} > 0\) if \(\beta\theta\alpha_{1}^{2}(\beta^{2}+\delta_{1}(\beta-2\delta_{2}))+2\theta\alpha_{1}\alpha_{2}\delta_{1}^{2}(-\beta+\delta_{2})+(\beta-\delta_{1})(-\theta\alpha_{2}^{2}\alpha_{1}^{2} +\kappa(\beta^{2}-\delta_{1}\delta_{2})^{2})<0\). Solving this for α ₂, one can obtain \(-\alpha_{1} \frac{\delta_{2} -\beta}{\delta_{1} -\beta}-\frac{\sqrt{\theta(\theta\alpha_{1}^{2}+\kappa(\beta-\delta_{1})^{2})\delta_{1}^{2}(\beta^{2}-\delta_{1}\delta_{2})^{2}}}{\theta(\delta_{1}-\beta)\delta_{1}^{2}}<\alpha_{2}<-\alpha_{1} \frac{\delta_{2} -\beta}{\delta_{1} -\beta}+\frac{\sqrt{\theta(\theta\alpha_{1}^{2}+\kappa(\beta-\delta_{1})^{2})\delta_{1}^{2}(\beta^{2}-\delta_{1}\delta_{2})^{2}}}{\theta(\delta_{1}-\beta)\delta_{1}^{2}}\). From the strict concavity condition of (2), |β|−δ _i <0, \(-\alpha_{1} \frac{\delta_{2} -\beta}{\delta_{1} -\beta}-\frac{\sqrt{\theta(\theta\alpha_{1}^{2}+\kappa(\beta-\delta_{1})^{2})\delta_{1}^{2}(\beta^{2}-\delta_{1}\delta_{2})^{2}}}{\theta(\delta_{1}-\beta)\delta_{1}^{2}}\) is always negative.

Hence, (24) is positive when \(\alpha_{2}< -\alpha_{1} \frac{\delta_{2} -\beta}{\delta_{1} -\beta}+\frac{\sqrt{\theta(\theta\alpha_{1}^{2}+\kappa(\beta-\delta_{1})^{2})\delta_{1}^{2}(\beta^{2}-\delta_{1}\delta_{2})^{2}}}{\theta(\delta_{1}-\beta)\delta_{1}^{2}}(\equiv \alpha_{2}^{*})\). □

Prof of Corollary 1

Partial differentiation of the threshold \(\alpha_{2}^{*}\) with respect to κ, \(\frac{\alpha_{2}^{*}}{\kappa}=\frac{(-\beta+\delta_{1})(\beta^{2}-\delta_{1} \delta_{2})^{2}}{2\sqrt{\theta(\theta \alpha_{1}^{2}+\kappa(\beta-\delta_{1})^{2})\delta_{1}^{2}(\beta^{2}-\delta_{1}\delta_{2})^{2}}}\). Partial differentiation of the threshold \(\alpha_{2}^{*}\) with respect to θ, \(\frac{\alpha_{2}^{*}}{\theta}=\frac{\kappa(\beta-\delta_{1})(\beta^{2}-\delta_{1} \delta_{2})^{2}}{2\theta\sqrt{\theta(\theta \alpha_{1}^{2}+\kappa(\beta-\delta_{1})^{2})\delta_{1}^{2}(\beta^{2}-\delta_{1}\delta_{2})^{2}}}\).

From the strict concavity condition of (2), |β|−δ _i <0 and θ≥0, therefore, \(\frac{\alpha_{2}^{*}}{\kappa}>0\) and \(\frac{\alpha_{2}^{*}}{\theta}<0\). □

Proof of Proposition 2

(i) Solving \(d_{e}^{*} - d_{ne}^{*} >0\) for α ₃, one can obtain

from the assumptions related to function (2), that is, α ₁, α ₂, α ₃, δ _i >0, and δ _i >|β| for i=1,2.

(ii) Solving \(U_{e}^{a*} - U_{ne}^{a*} >0\) for \(\alpha_{3}^{2}\), one can obtain

from the assumptions related to function (2), that is, α ₁,α ₂,α ₃,δ _i >0, and δ _i >|β| for i=1,2. □

Proof of Proposition 3

1. (i)

   Partial differentiation of (17) with respect to α ₃, \(\frac{\partial p_{ne}^{*}}{\partial \alpha_{3}}=\frac{1}{2\kappa \delta_{3}}\), is positive because κ>0 and δ ₃>0.

2. (ii)

   The threshold \(\alpha_{2}^{*}\) is derived from Proof of Proposition 1.

 □

Proof of Proposition 4

The ratio of an aid agency’s utility in the non-earmarked case to the earmarked case is: \(\frac{U^{a*}_{ne}}{U^{a*}_{e}}=\frac{\alpha_{3}^{2}(\kappa(\beta^{2}-\delta_{1}\delta_{2})^{2}+\theta(\beta\alpha_{1}-\alpha_{2}\delta_{1})^{2})}{\kappa (\alpha_{2}(\beta-\delta_{1})+\alpha_{1}(\beta-\delta_{2}))^{2}\delta_{3}^{2}}\). Note that the numerator and the denominator of \(\frac{U^{a*}_{ne}}{U^{a*}_{e}}\) are positive. Assuming that \(\frac{U^{a*}_{ne}}{U^{a*}_{e}}<1\), subtracting the denominator from the numerator, and solving it for θ, one can obtain the threshold

Thus, if θ is less than the threshold θ ^∗, then one can conclude that \(U^{a*}_{e} >U^{a*}_{ne}\). □

Proof of Corollary 2

Since the denominator of θ ^∗ is always positive, θ ^∗ becomes always negative when its numerator is negative. Solving \(-\alpha_{3}^{2}(\beta^{2} -\delta_{1}\delta_{2})^{2}+(\alpha_{2}(\beta-\delta_{1})+\alpha_{1}(\beta-\delta_{2}))^{2}\delta_{3}^{2}) <0\) for \(\alpha_{3}^{2}\) and considering assumptions of α ₁,α ₂,α ₃,δ _i >0, and δ _i >|β| for i=1,2, one can obtain \(\alpha_{3} > \frac{(\alpha_{2}(\delta_{1} -\beta)+ \alpha_{1} (\delta_{2} - \beta) ) \delta_{3}}{\delta_{1}\delta_{2}-\beta^{2}} (\equiv \alpha_{3}^{*})\). □

Proof of Corollary 3

Since θ ^∗>0, the numerator of θ ^∗ is positive. Thus, \(\frac{\partial \theta^{*}}{\partial \kappa} >0\). □

Proof of Corollary 4

\(\frac{\partial \theta^{*}}{\partial \alpha_{3}} =\frac{-2\kappa(\alpha_{2}(\beta-\delta_{1})+\alpha_{1}(\beta-\delta_{2}))^{2} \delta_{3}^{2}}{\alpha_{3}^{3}(\beta\alpha_{1}-\alpha_{2}\delta_{1})^{2}}<0\).

Recall that \(\theta^{*}\equiv \frac{\kappa(-\alpha_{3}^{2}(\beta^{2} -\delta_{1}\delta_{2})^{2}+(\alpha_{2}(\beta-\delta_{1})+\alpha_{1}(\beta-\delta_{2}))^{2}\delta_{3}^{2})}{(\beta\alpha_{1}-\alpha_{2}\delta_{1})^{2}\alpha_{3}^{2}}>0\). From the strict concavity condition of (2), |β|−δ _i <0, (α ₂(β−δ ₁)+α ₁(β−δ ₂))² increases with an increase in α ₁ and the numerator of θ ^∗ increases. From the assumption of non-negative donation amounts, βα ₁−δ ₁ α ₂≤0, \(\alpha_{3}^{3}(\beta\alpha_{1}-\alpha_{2}\delta_{1})^{2}\) decreases with an increase in α ₁ if β>0. In summary, for β>0, an increase in α ₁ leads to an increase in the numerator and a decrease in the denominator and \(\frac{\partial \theta^{*}}{\partial \alpha_{1}} >0\). □

Proof of Corollary 5

Taking a partial derivative of θ ^∗ with respect to α ₂, one can obtain \(\frac{\partial \theta^{*}}{\partial \alpha_{2}}=\frac{2\kappa(\delta_{1}\delta_{2}-\beta^{2})(\alpha_{3}^{2}\delta_{1}(\beta^{2}-\delta_{1}\delta_{2})+\alpha_{1}(\alpha_{2}(-\beta+\delta_{1})+\alpha_{1}(-\beta+\delta_{2}))\delta_{3}^{2})}{\alpha_{3}^{2}(\beta\alpha_{1}-\alpha_{2}\delta_{1})^{3}}\). Term 2κ(δ ₁ δ ₂−β ²) in the numerator is positive from δ _i >|β| and the denominator is negative from one of the non-negative donation amount conditions, βα ₁−δ ₁ α ₂≤0. Thus, when one solves the term \(\alpha_{3}^{2}\delta_{1}(\beta^{2}-\delta_{1}\delta_{2})+\alpha_{1}(\alpha_{2}(-\beta+\delta_{1})+\alpha_{1}(-\beta+\delta_{2}))\delta_{3}^{2} <0\) for α ₂, the threshold of Corollary 5 can be obtained. □

Proof of Proposition 5

In the case of earmarking, the optimal size of the solicited population, \(p_{e}^{u*}\) can be expressed as

The fundraising cost percentage, \(\mathit{FCP}_{e}^{u*}\), can be expressed as

Comparing \(p_{e}^{u*}\) with \(p_{e}^{*}\), one can easily obtain that \(p_{e}^{u*}-p_{e}^{*}=\frac{-\theta\sigma_{2}^{2}(\beta\alpha_{1} -\alpha_{2}\delta_{1})(\beta^{2}-\delta_{1}\delta_{2})}{\kappa(\beta^{2} -\delta_{1}\delta_{2})^{2} +\theta(\beta\alpha_{1}-\alpha_{2}\delta_{1})^{2}} <0\) from δ _i >|β| for i=1,2 and one of the non-negative donation amount conditions, βα ₁−δ ₁ α ₂≤0.

Similarly, comparing \(\mathit{FCP}_{e}^{u*}\) with \(\mathit{FCP}_{e}^{*}\),

from δ _i >|β| for i=1,2 and one of the non-negative donation amount conditions, βα ₁−δ ₁ α ₂≤0.

Regardless of the uncertainty, donation amounts per donor given to the general fund and the special fund remain the same. Taking into account the result of \(p_{e}^{u*}-p_{e}^{*}<0\), the expected total donation amount in the uncertain case is always smaller than the total donation amount in the certain case. □

Proof of Corollary 6

and

from δ _i >|β| and one of the non-negative donation amount conditions, βα ₁−δ ₁ α ₂≤0. As highlighted in the previous proof, donation amounts per donor given to the general fund and the special fund remain the same, regardless of the level of uncertainty. Hence, the expected total donation amount decreases as \(\sigma_{2}^{2}\) increases. □

Proof of Proposition 6

The ratio of the solicited population in the non-earmarking case to the earmarking case is: \(\frac{p_{ne}^{*}}{p_{e}^{*}}=\frac{\alpha_{3} (\kappa(\beta^{2}-\delta_{1}\delta_{2})^{2}+\theta(\beta\alpha_{1}-\alpha_{2}\delta_{1})^{2})}{\kappa(\alpha_{2}(\beta-\delta_{1})+\alpha_{1}(\beta-\delta_{2}))(\beta^{2}-\delta_{1}\delta_{2})\delta_{3}}\). Note that regardless of the sign of β the numerator and the denominator of \(\frac{p^{*}_{ne}}{p^{*}_{e}}\) are positive from δ _i >|β|. Solving the quadratic equation of α ₂ that is derived from \(\frac{p_{ne}^{*}}{p_{e}^{*}}<1\), one can obtain:

(31)

and

(32)

such that \(\hat{\alpha}_{2 low}<\alpha_{2}<\hat{\alpha}_{2 \mathit{high}}\). Thus, if α ₂ is within the range, then one can conclude that \(p_{e}^{*} \geq p_{ne}^{*}\).

Now, we derive the condition that guarantees that \(\hat{\alpha}_{2 \mathit{low}}\) and \(\hat{\alpha}_{2 \mathit{high}}\) are real numbers. From the discriminant of the quadratic equation of α ₂ shown above, the condition that the two real roots of the quadratic equation are real numbers is

The argument of Proposition 6 is limited to this case. □

Proof of Corollary 7

Differentiating \(\hat{\alpha}_{2 \mathit{low}}\) and \(\hat{\alpha}_{2 \mathit{high}}\) partially with respect to κ, one can obtain the following results from condition \(\frac{\alpha_{1}}{\delta_{1}}\ge\frac{\alpha_{3}}{\delta_{3}}\) and δ _i >|β|,

and

 □

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