1 Introduction

Understanding donor motives can help charity organisations and governments in designing fund-raising campaigns and supporting programmes. The two most prominent motivations discussed in the literature are warm glow and altruism. Whereas pure altruists derive utility from total donations (i.e., their donations and the donations of others are perfect substitutes); pure warm-glow givers only derive utility from their own donation; warm-glow givers are thought to derive utility from the joy of giving (Andreoni, Reference Andreoni1989, Reference Andreoni1990; Null, Reference Null2011). What precisely motivates donors is, however, not easy to identify empirically.

Several techniques have been developed to identify charitable motivations, including Crumpler and Grossman (Reference Crumpler and Grossman2008), Gangadharan et al. (Reference Gangadharan, Grossman, Jones and Leister2018, Reference Gangadharan, Grossman and Xue2023), Fielding et al. (Reference Fielding, Knowles and Peeters2022), Ottoni-Wilhelm et al. (Reference Ottoni-Wilhelm, Vesterlund and Xie2017), and Carpenter (Reference Carpenter2021). Donors are typically classified into two extreme categories: warm-glow givers and pure altruists. However, many experimental participants do not fit either strictly defined category, and are best classified as impure altruists. Critically, impure altruists exhibit great heterogeneity in donation behaviour, which suggests a need for further refinement of this category. Crumpler and Grossman (Reference Crumpler and Grossman2008) measure the strength of warm-glow preferences, while holding constant altruistic preferences. In contrast, Gangadharan et al. (Reference Gangadharan, Grossman and Xue2023) measure the strength of altruistic preferences, while holding constant warm-glow preferences.

We suggest an alternative approach. Whereas Gangadharan et al. (Reference Gangadharan, Grossman, Jones and Leister2018) sort participants into four strict categories (non-donor, warm-glow giver, pure altruist, and impure altruist), we instead identify the relative strength of warm glow and altruistic motives using their same two-stage decision task. To accomplish this, we follow Ottoni-Wilhelm et al. (Reference Ottoni-Wilhelm, Vesterlund and Xie2017) in assuming participants maximise a Cobb–Douglas impure altruism utility function, allowing us to identify structural parameters for giving motives. We thus obtain a novel fluid measure of an individual's strength of altruistic motive, and we validate our measure via a comparison to the survey measure of Carpenter (Reference Carpenter2021).

2 Experimental design

We use the two-stage decision task of Gangadharan et al. (Reference Gangadharan, Grossman, Jones and Leister2018), as implemented by Gandullia et al. (Reference Gandullia, Lezzi and Parciasepe2020). Participants are given 100 points and are asked first to choose an amount, $g_1$ , to donate to their preferred charity.Footnote ¹ They are informed that we (the experimenters) will donate $100-g_1$ points to the charity, such that the charity organisation is guaranteed to receive a total of 100 points, regardless of their donation decision. After making this decision, the participant has $100-g_1$ points left. They are then informed that they have a second (unforeseen) opportunity to donate an amount, $g_2$ , to the charity, knowing that this time no complementary donation is made by us. Thus, the charity organisation will receive in total $100+g_2$ points, while the participant will keep $100-g_1-g_2$ points.Footnote ²

After completing this giving-type task, participants are asked to respond to the question “Think about the last time you gave to charity before today. What was most important to you?” with possible answers being (i) “the total amount given by everyone”, (ii) “the amount that you personally gave”, and (iii) “some other aspect of giving”.Footnote ³ This question was suggested by Carpenter (Reference Carpenter2021) to elicit giving motivations and was validated using the Crumpler and Grossman (Reference Crumpler and Grossman2008) method.

In total, 357 participants completed the experiment/survey on 14 April 2023. These participants (aged 18–65 and fluent in English) were recruited via Prolific and redirected to our Qualtrics survey. Our subject pool includes 213 male and 140 female participants; 3 participants are gender diverse and 1 preferred not to answer. Two participants preferred not to reveal their age; the average age of the participants who revealed their age is 27.26. The median completion time was 3 min and 6 s, and participants earned on average 1.27 GBP including the fixed participation fee of 1.00 GBP.Footnote ⁴ Ethical approval was obtained from the University of Otago Human Ethics Committee (reference code D23/082).

3 Strength of altruistic motive

Following Gangadharan et al. (Reference Gangadharan, Grossman, Jones and Leister2018), a participant is classified as a non-donor if $g_1=0$ and $g_2=0$ , a warm-glow giver if $g_1>0$ and $g_2=0$ , a pure altruist if $g_1=0$ and $g_2>0$ , and as an impure altruist if $g_1>0$ and $g_2>0$ .Footnote ⁵ Regarding the question of Carpenter (Reference Carpenter2021), participants who gave the first response are categorised as pure altruists and those who gave the second one as high warm-glow altruists. In the following, we construct a more fluid method to measure the extent to which an individual's charitable behaviour is motivated by altruism.Footnote ⁶

Consider an individual with the Cobb–Douglas impure altruism utility function

$\begin{array}{c}(1-\beta -\gamma )\text{ln}\left(x\right)+\beta \text{ln}\left(g\right)+\gamma \text{ln}\left(G\right),\end{array}$

where x is private consumption, g is the amount given by the individual, and G is total funds received by the charity. In the first decision of the Gangadharan et al. (Reference Gangadharan, Grossman, Jones and Leister2018) giving-type task, the individual who seeks to maximise this utility (with $x=100-g_1$ , $g=g_1$ and $G=100$ ) will choose

$\begin{array}{c}{g}_1^{\ast }=\frac{\beta }{1-\gamma }100.\end{array}$

In the second decision, the individual (using $x=100-g_1^{\ast }-g_2$ , $g=g_1^{\ast }$ , and $G=100+g_2$ ) will choose

$\begin{array}{c}{g}_2^{\ast }\left(g_1^{\ast }\right)=\frac{\gamma (100-g_1^{\ast })-(1-\beta -\gamma )100}{1-\beta }=\frac{(1-\beta -\gamma )(2\gamma -1)}{(1-\beta )(1-\gamma )}100.\end{array}$

Thus, choices ${\tilde{g}}_1$ and ${\tilde{g}}_2$ yield the following unique values of $\tilde{\beta }$ and $\tilde{\gamma }$ :

$\begin{array}{c}\tilde{\beta }=\frac{{\tilde{g}}_1(100-{\tilde{g}}_1-{\tilde{g}}_2)}{20,000-{\tilde{g}}_1(200+{\tilde{g}}_2)}\text{and}\tilde{\gamma }=\frac{(100-{\tilde{g}}_1)(100+{\tilde{g}}_2)}{20,000-{\tilde{g}}_1(200+{\tilde{g}}_2)}.\end{array}$

Due to the design of the giving task, $\tilde{\gamma }$ is at least $\frac{1}{2}$ . We normalise the parameters to ensure that they lie within [0, 1] using $\overline{\beta }=2\tilde{\beta }$ and $\overline{\gamma }=2(\tilde{\gamma }-\frac{1}{2})$ . Furthermore, because of the two-stage design of the giving task, we can estimate the parameters directly on the donation choices without the use of statistics.

We note that this estimation procedure aligns well with the derivations laid out by Gangadharan et al. (Reference Gangadharan, Grossman, Jones and Leister2018) for distinguishing different giving motives (see Appendix D, especially equations D7 and D8). An alternative approach might sequentially estimate the $\tilde{\gamma }$ and $\tilde{\beta }$ parameters. However, under such an approach, the individual's preferences change from the first to the second stage, which is inconsistent with the wider literature. Moreover, separating the $\tilde{\gamma }$ and $\tilde{\beta }$ means that the individual never makes a direct trade-off between warm-glow and altruism, which is what we seek to capture. We also note that the rescaling of $\tilde{\gamma }$ and $\tilde{\beta }$ described above is simply a normalisation and does not materially affect the analysis in any way. These monotonic transformations preserve the ordering of parameter values across individuals.

In the spirit of Footnote 9 of Ottoni-Wilhelm et al. (Reference Ottoni-Wilhelm, Vesterlund and Xie2017), we construct a fluid measure of altruism as $\overline{\alpha }=\frac{\overline{\gamma }}{\overline{\beta }+\overline{\gamma }}$ . It captures the amount of generosity that is attributable to the altruistic motive relative to the warm-glow motive. For participants who donated their full endowment during the giving-type task ( ${\tilde{g}}_1+{\tilde{g}}_2=100$ ), values for $\overline{\beta }$ and $\overline{\gamma }$ cannot be properly specified, since their donation decisions meet the constraint and do not follow from the first-order conditions of the maximisation problems. As a consequence, $\overline{\alpha }$ cannot be properly specified either. Likewise, for non-donors ( ${\tilde{g}}_1={\tilde{g}}_2=0$ ), we find $\overline{\beta }=\overline{\gamma }=0$ , so $\overline{\alpha }$ cannot be properly specified for these participants.