Do Matches Really Outperform Rebates? New Evidence from a Novel Experiment

The authors gratefully acknowledge support from the National Science Foundation (Award Number: 2315706), the University of Maryland (Faculty-Student Research Award), and the Department of Agricultural and Resource Economics (The Bruce and Mary Ann Gardner Dissertation Enhancement Award).

Abstract: This paper challenges the well-established result among existing experimental studies that donations are significantly more responsive to matches than to rebates. In previous experimental studies the budget sets available to subjects under rebates are constrained relative to those available under matches. We design and run a novel experiment that removes the constraint on rebates, producing equal budget sets for price-equivalent rebates and matches. Contrary to previous studies, we find no statistical difference between the estimated rebate- and match-price elasticities in our experiment. Furthermore, we show that the constraint under rebates in previous studies affects the entire distribution of observed behavior, not only the behavior of individuals for whom the constraint is binding. That is, experimental subjects are observed to be highly sensitive to changes in their available budget sets, and previous elasticity estimates are therefore significantly biased. Finally, to reconcile extant models of giving with our results, we propose an extension of extant models which relaxes an assumption implicitly made in previous work, allowing the model to correctly predict our experimental results.

Subsidies for giving are often used to incentivize donations to charitable organizations. The two most common forms of subsidies used are rebates and matches. Both rebates and matches can be used to lower the price of giving, which is the cost to the donor of providing the charity with $1. If a rebate and match produce the same price of giving, they are price-equivalent.

What are rebates, matches, and the price of giving?

Rebates provide donors with a rebate of \(r\) on each dollar they donate. Thus, the price of giving faced by donors will be equal to \(p=1-r\). For example, if there is a 50% rebate offered (\(r=0.5\)), the donor will face a price of giving of 0.5 (\(p=1-0.5\)), and for every dollar they donate they will receive a rebate of 50 cents. So, if they provide a $10 donation, they will receive a $5 rebate. The most common example of a rebate subsidy for giving is tax policy. In the US, charitable donations are often tax exempt, effectively providing a rebate equal to the donor's marginal tax rate. For example, if a donor faces a marginal tax rate of 30% (\(t=0.3\)), each dollar donated to charity will reduce their tax liability by 30 cents. Thus, they face a price of giving equal to \(p=1-t=0.7\).

Matches increase the amount received by the charity by matching a donor's donation at some rate \(m\). For each dollar a donor donates, the charity ends up receiving \(1+m\) dollars. Under a match, the price of giving becomes \(p=\frac{1}{1+m}\). For example, if a match is offered at rate \(m=1\) and a donor wants to provide the charity with a total of $10, they must provide a $5 donation. With the match, the total received by the charity becomes \(5*(1+m)=10\), and the price of giving is \(p=\frac{1}{1+m}=0.5\).

Both rebates and matches can be used to decrease the price of giving. Furthermore, a rebate and match will produce the same price of giving as long as the rebate rate \(r\) and the match rate \(m\) are such that \(1-r = \frac{1}{1+m}\). In this case, we say that the rebate and match are price-equivalent.

One might expect that price-equivalent rebates and matches should have the same effect on donations. However, previous experimental studies have repeatedly found that this is not the case. Instead, matches are found to result in significantly greater donations than their price-equivalent rebate counterparts. This result has important implications for charities, tax policy, and theoretical models of giving, but researchers have struggled to provide an explanation.

One possible explanation has to do with a technical aspect of the experimental setting in which rebates and matches are compared in previous work. To compare rebates and matches, researchers have experimental subjects play a variation of the standard dictator game in which subjects are provided with an endowment and asked to decide how much of their endowment to pass to a charity.

What is a dictator game?

Dictator games are frequently used in experimental economics. Dictator games involve providing a subject with an endowment (e.g., $10) and having them decide how to allocate the endowment between themself and some other party (e.g., another subject, an anonymous individual, a charity, etc.). Because the other party has no say over the final allocation of the endowment, the subject essentially serves as "dictator," giving the game its name.

A similar game that is also frequently used is the ultimatum game. Just as in the dictator game, in the ultimatum game subjects are provided with an endowment and asked to decide how to allocate it between themselves and some other party. However, unlike the dictator game, in the ultimatum game the other party can choose to either accept the proposed allocation or refuse it, in which case both parties receive nothing. Thus, the first party essentially presents the second party with an ultimatum ("either accept my proposed allocation or receive nothing"), giving the game its name.

Subjects are typically required to make decisions for a menu of decision problems. Some problems will provide subjects with a rebate on any donations they pass to the charity, some will provide a match, and some won't provide any subsidy for giving. By varying the endowment, subsidy type, and price of giving across problems, researchers are able to compare the effectiveness of rebates and matches at increasing donations.

The issue with comparing rebates and matches in this way is that the budget sets faced by subjects are not equal for price-equivalent rebates and matches. Instead, the budget sets under rebates are strict subsets of those under matches. This is most easily seen by considering a simple example:

Example: Suppose a subject is provided with a $10 endowment and either (i) a rebate subsidy at rate \(r=0.5\) or (ii) a match subsidy at rate \(m=1\). (Note that the rebate and match are price-equivalent, with \(p=0.5\) in both cases.) Suppose further that the subject wishes to provide the charity with largest donation possible. In this case, the best the subject can do is to pass their entire endowment. However, the total amount received by the charity (and the subject's earnings) will differ depending on whether the subsidy is a rebate or a match.

In the case of a rebate, the subject will pass all $10 of their endowment, resulting in the charity receiving a total donation of $10 and the subject earning $5 (because of the rebate they receive).

In the case of a match, the subject will again pass all $10 of their endowment, but now the charity will receive a total donation of $20 (because of the match) and the subject will earn $0.

Because subjects can never pass more than their endowment, rebates do not allow subjects to provide as large of donations as what is possible under price-equivalent matches. That is, subjects' budget sets are constrained under rebates relative to matches.

While one might assume that the constraint under rebates should only affect subjects for whom the constraint is binding, this is not necessarily the case. Previous research has found that expanding the budget set available to subjects can affect the entire distribution of behavior, not just the behavior of subjects who are previously constrained. Because of this, the disparity in budget sets may be responsible for the finding that rebates are less effective than matches at increasing donations.

To test this, we design a novel experiment in which price-equivalent rebates and matches provide equal budget sets, and we find that the gap between rebates and matches disappears. That is, rebates and matches are found to be equally effective at increasing donations, contrary to previous studies. To verify that our results are not unique to our subject pool, we also replicate previous experiments. In our replication experiment, we find a large and significant gap between the estimated rebate- and match-price elasticities of giving, consistent with previous studies. Thus, we conclude that previous results are indeed driven by the disparity in budget sets present in previous studies, and donors may in fact view rebates and matches as equivalent.

Furthermore, when comparing the results of our novel experiment with the results of our replication experiment, we find that the entire distribution of observed behavior is affected by the constraint on budget sets present in previous studies, just as predicted. This finding reinforces the importance of carefully designing experimental studies and highlights the extent to which the behavior of subjects in laboratory settings can be highly sensitive to changes in their available budget sets.