DP005
Rebate versus Matching, Again:
Does Self-selection Matter?

Shusaku Sasaki, Osaka University
Takunori Ishihara, Kyoto University of Advanced Science
Hiroki Kato, Hitotsubashi University

Rebate versus Matching, Again: Does Self-selection Matter?

Shusaku Sasaki1

Takunori Ishihara2

Hiroki Kato3First Draft: May 1st, 2023

Abstract
Standard economic theory predicts that matching and rebate will have the same effect on individuals’
donation behavior when the donation price is equivalent. However, several experimental studies havereported that their donation behaviors are promoted under matching more than under rebate. This study
reveals how treatment effects of matching and rebate change when people can self-select whether touse such schemes or not. Although traditional experimental studies have measured the causal effects
of mandatory policy assignment, real-world policies are often applied to only those who accept them.
We conduct an incentivized nationwide experiment on 2,400 Japanese residents with four treatments,
two 1:1 matching treatments (compulsory / self-selection) and two 50% rebate treatments (compulsory/ self-selection), and provide the following findings: Initially selected amount under the compulsory
matching is smaller than under the compulsory rebate, while total amount donated to the charity under
the former scheme is larger than that under the latter scheme, which is consistent with the existing
theories. The treatment effect on the total donation amount among those who self-select to receive thetreatment is still larger under the matching scheme than under the rebate scheme, but this difference isfurther larger than that in the case when using compulsory treatments. The superiority of matchingover rebate for the total donation amount becomes more pronounced in the case with a self-selection
process.
Keywords: Charitable subsidy, Tax incentive, Public economics, Behavioral economics, Online
experiment
JEL classification: D91 · H20 · C90

Center for Infectious Disease Education and Research (CiDER), Osaka University, Japan (Corresponding
author: ssasaki.econ@cider.osaka-u.ac.jp).
2
Kyoto University of Advanced Science, Japan.
3
Hitotsubashi University, Japan.
1

We obtained ex-ante approval from the ethics committee of Center for Infectious Disease Education andResearch, Osaka University, Japan (2022CRER0120-2). We also registered the experimental design with
the AEA RCT Registry (Sasaki, Ishihara, and Kato, 2023). This research is financially supported by theJapan Society for the Promotion of Science [JSPS Grant Numbers: 19K13722 (S., Sasaki)].

1

1. Introduction
Standard economic theory predicts that, when the donation price and all other factors are equal, there
should be no difference in individuals’ donation behavior under matching and rebate schemes. For
example, a 1:1 matching is equivalent to a 50% rebate. In the former scheme, when one chooses todonate 5000 JPY to a charity, the same amount will be added to this donation, thus making the total
amount donated to the charity 10,000 JPY. In the latter scheme, when one chooses to donate 10,000
JPY to a charity, half of the amount will be refunded, making the actual donation expenditure 5000
JPY. Similarly, a 2:1 matching is equivalent to a 33% rebate, and a 4:1 matching is equivalent to a
20% rebate. However, Eckel and Grossman (2003) experimentally reveal that donation rates andaverage donation expenditures for matching are higher than for rebate. Sasaki, Kurokawa, and Ohtake(2021) use a Japanese nationwide sample and report the findings consistent with Eckel and Grossman(2003).
This study’s purpose is to determine in a randomized controlled trial how treatment effects
of matching and rebate change when people can self-select whether to use such schemes or not. Most
traditional policy research using randomized controlled trials has measured the causal effects ofmandatory policy assignment. However, implementing a policy intervention in a mandatory manneris rare in the real world. This is because mandatory implementation requires a system that enables a
policy to be applied to all individuals involved and monitors their adherence to it. Also, a policy must
be made mandatory by law, and implementation costs tend to be extremely high. In practice, policies
are often applied to only those who choose to accept them, in particular by employing an opt-in scheme,
where a policy is not applied by default, but rather only upon request.
Under conditions with self-selection, overall policy impacts will vary depending on theheterogeneous effects across individuals and which individuals self-select to receive the policy. For
example, if a policy is widely accepted by people for whom a large (or significant) positive policy
effect appears, the overall policy impact will become larger than if the policy intervention was
mandated, and thus the policy function more efficiently due to self-selection. Conversely, if those whoare likely to experience small or negative effects choose to receive a policy intervention, the overall
policy impact will become relatively small, and self-selection will prevent it from functioning
efficiently. To accurately understand the real-world implications of policy interventions, it is essential
to ascertain the influence of self-selection on policy efficiency. Recent field experimental studies havebegun to measure policy intervention effects after considering self-selection in energy saving (Wanget al. 2020; Fowlie et al. 2021; Ito et al. 2021; Ida et al. 2022) and health promotoion (Kurokawa andSasaki, 2023).
This study adds to this emerging literature stream new evidence in the context of charitablegiving by measuring the treatment effects of matching and rebate, while considering self-selection.

22. Experimental Design
2.1.Overview
We conduct an online economic experiment through MyVoice.com Ltd., which offers online surveys
and experiments. Around 1 million adults living throughout Japan register the company as responsemonitors. We conduct a screening survey and sample 2400 participants from the monitors to match
the proportions of a national representative sample in terms of age, sex, and residential area. Here, thescreening survey includes questions related to sex, age, residential area, knowledge on matching andrebate, calculation problems, willingness to pay for matching and rebate.
The experiment in this study is incentivized using “points,” which can be exchanged forgift cards (nationwide gift cards, Amazon gift cards, App Store & iTunes gift cards, book cards, etc.).
The participants receive basic points by answering the survey and can earn additional points, whichvary depending on their choices in the experiment. Note that the exchange rate is 1 point = 1 JPY.
The structure of this experiment can be divided into three main part. First, we present theparticipants with questions that capture their behavioral economic characteristics, including social,time, and risk preferences, in addition to questions related to donation experience.
Second, we randomly divide them into matching treatments, rebate treatments, or a control,and conduct the economic experiment to capture their donation behavior under each assigned
condition. We construct two groups, respectively for the 1:1 matching treatments (compulsory andself-selection) and the 50% rebate treatments (compulsory and self-selection).
Third, we present participants with questions to capture their socioeconomic attributes,
including marital status, number of children, years of education, household income, place of residence,
etc., and gather their responses.

2.2.Interventions
We randomly divide participants into matching treatments, rebate treatments, or a control, and conduct
the economic experiment to capture their donation behavior under each assigned condition. We
construct two groups, respectively for the 1:1 matching treatments (compulsory and self-selection)and the 50% rebate treatments (compulsory and self-selection).
⚫

1:1 matching (compulsory): All the participants assigned to this group will select their donation
amount under the 1:1 matching, where for every donation amount they pass on to the charity, theexperimenter will match it with an additional equal amount.

⚫

1:1 matching (self-selection): Participants assigned to this group are required to apply in advanceif they wish to use the 1:1 matching.

⚫

50% rebate (compulsory): All the participants assigned to this group will select their donation
amount under the 50% rebate, where for every donation amount participants pass on to the charity,
the experimenter will refund 50% of the amount to you.
3⚫

50% rebate (self-selection): Participants assigned to this group are required to apply in advanceif they wish to use the 50% rebate.

2.3.Outcomes
In this economic experiment, participants are informed that, in addition to the basic reward points foranswering the survey, one in ten will have a chance to earn another reward, and the additional reward
points are worth 1000 JPY. They are then needed to decide how much of the 1000 JPY they are willingto pass on to a social contribution project, assuming they could win and earn this additional reward. Ifthey win and earn 1000 JPY, their donation decision will be carried out as they answer. Our primary
outcomes are initially selected amount and total amount donated to the charity.

2.4.Balance Check
Table 1 reports the means and standard deviations of participants’ sex, family structure, educational
years, household income, and place of residence for each group to check the random allocation ofparticipants between the five groups. We confirm that there is no statistically significant difference inthe means of all the variables in the table between the groups.
[Table 1 is here]43. Hypothesis Setting and Analytical Procedure
3.1.Hypothesis Setting
Based on Hungerman and Ottoni-Wilhelm (2021), the treatment effect of matching on initially selectedamount is always less than that of rebate (please see the Appendix of “Theoretical Background”). We
can decompose the effect of matching on the initially selected amount into two components: 1) an
effect of decreasing the initially selected amount based on the optimal total donated amount and theamount added by the third party’ matching and 2) a positive or negative change in the optimal total
donated amount itself. Since the decreasing effect by the added amount always exceeds the change inthe optimal total donated amount, the treatment effect of matching on initially selected amount isalways less than that of rebate (Hypothesis 1).
Second, the treatment effect of matching on the total amount donated to charity is expected
to differ from that of rebate (Hypothesis 2). The size of the treatment effect for both matching andrebate on the total donated amount is likely to vary, depending on how warm-glow influences
substitution effect. Specifically, if warm-glow weakens the substitution effect, the treatment effect ofmatching is more significant than rebate. On the other hand, if warm-glow strengthens the substitution
effect, the treatment effect of matching is smaller than that of rebate. If donors do not have a warmglow preference, the results of both treatments are equal.
Third, since rebate improves consumer surplus to a greater extent than matching, we can
expect that donors are more likely to prefer receiving rebate than matching (Hypothesis 3-1). It isespecially true for donors who increase their actual donation expenditure as a result of receiving thetreatment. Thus, the treatment effect on the initially selected amount for donors who self-select toreceive rebate will exceed the treatment effect for donors who self-select to receive matching. Next,
the treatment effect is expected to be more significant for donors who voluntarily receive
matching/rebate than for those who receive the treatments in a compulsory manner (Hypothesis 3-2).
Further, the difference in TOT effects on the total donation amount between matching and rebate isvary depending on donors’ warm-glow preference (Hypothesis 3-3).

3.2.Analytical Procedure
3.2.1. Basic Analysis Plan
We describe our primary analysis. We consider the following linear regression model.

𝑦𝑖 = 𝛼 +∑𝜏𝑔𝐼𝑇𝑇 𝑍𝑖𝑔 + 𝜖𝑖 ,𝑔∈{𝐶𝑀,𝐶𝑅,𝑆𝑀,𝑆𝑅}𝑦𝑖 represents the total amount donated to the charity or initially selected amount by individual 𝑖. In
addition, 𝑔 represents each group in the RCT, where CM, CR, SM, and SR represent the groups5receiving the compulsory matching treatment, compulsory rebate treatment, self-selection matchingtreatment, and self-selection rebate treatment, respectively. 𝑍𝑖𝑔 is a dummy variable that takes 1 if
individual 𝑖 is assigned to group 𝑔 and 0 otherwise. 𝛼 represents a constant term, and 𝜏𝑔represents the average treatment effect in each group. In particular, for SM and SR, where treatment
is determined by self-selection, 𝜏𝑔 represents the intention-to-treat (ITT) effect since it is thetreatment effect for the treatment group as a whole. Finally, 𝜖𝑖 represents the error term.
We consider the following two-stage regression model to identify the treatment effect for
those who receive treatment (TOT effect: treatment-on-treated effect).

1st stage: 𝐷𝑖 = 𝛽 +∑𝛾𝑔 𝑍𝑖𝑔 + 𝜂𝑖 ,𝑔∈{𝑆𝑀,𝑆𝑅}

2nd stage: 𝑦𝑖 = 𝛼 +∑̂𝑖𝑔 + 𝜖𝑖 ,𝜏𝑔𝑇𝑂𝑇 𝐷𝑔∈{𝑆𝑀,𝑆𝑅}

where 𝐷𝑖 is a self-selection dummy that takes 1 if individual 𝑖 selects to receive the treatment and0 otherwise. 𝛾𝑔 represents the probability that an individual will receive the treatment if assigned togroup 𝑔. Also, 𝜂𝑖 represents the error term in the first-stage estimation. A two-stage estimation
method is used to estimate these unknown parameters.
We first test the following null hypotheses to analyze differences in the treatment effects
of rebate and matching on initially selected amounts and total donation amounts.
𝐼𝑇𝑇
𝐼𝑇𝑇
𝐻0 : 𝜏̂ 𝐶𝑀= 𝜏̂ 𝐶𝑅,

From our hypotheses setting, we determine predict the relationship between these treatment effects
𝐼𝑇𝑇
𝐼𝑇𝑇
when the outcome is the total amount donated, but we would expect 𝜏̂ 𝐶𝑀< 𝜏̂ 𝐶𝑅
for initially selectedamounts.
Next, we test the following null hypotheses for the probability of selecting a treatment.
𝐻0 : 𝛾̂𝑆𝑀 = 𝛾̂𝑆𝑅 ,From the above hypotheses, we can expect 𝛾̂𝑆𝑀 ≤ 𝛾̂𝑆𝑅 , since donors are expected to prefer matchingover rebate.
Finally, we examine the following null hypotheses regarding the effects of self-selected
rebate treatment and matching treatment.
𝑇𝑂𝑇
𝐼𝑇𝑇
𝐻0 : 𝜏̂ 𝐶𝑀= 𝜏̂𝑆𝑀
,

6

𝑇𝑂𝑇
𝐼𝑇𝑇
𝐻0 : 𝜏̂ 𝐶𝑅= 𝜏̂𝑆𝑅,
𝑇𝑂𝑇
𝑇𝑂𝑇
𝐻0 : 𝜏̂𝑆𝑀= 𝜏̂𝑆𝑅
,

𝑇𝑂𝑇
𝑇𝑂𝑇
𝐼𝑇𝑇
𝐼𝑇𝑇
From the hypotheses, we expect 𝜏̂ 𝐶𝑀≤ 𝜏̂𝑆𝑀, 𝜏̂ 𝐶𝑅≤ 𝜏̂𝑆𝑅
because donors with higher treatment

effectiveness are expected to receive treatments when self-selection is used rather than compulsory
𝑇𝑂𝑇
𝑇𝑂𝑇
treatments. We also expect 𝜏̂𝑆𝑀≤ 𝜏̂𝑆𝑅
because donors prefer rebate over matching.

3.2.2. Further Analysis Plan (considering the upper limit of donation)[Figure 1 is here]
Those who receive the rebate treatment have a different upper limit of donation than the matching orcontrol group. Figure 1 shows the difference in this upper limit. We assume that the donor is given
1,000 JPY, from which she decides how much to donate. If the donor is assigned to the control groupwhen she initially selects 𝑥 JPY for the donation, the total amount donated to charity is 𝑥 JPY, andthe actual donation expenditure is 1,000 − 𝑥 JPY. Therefore, the maximum actual donation
expenditure and the total amount donated is 1,000 JPY.
On the other hand, if the donor receives a 1:1 matching treatment, when she initially selects
𝑥 JPY for the donation, the total amount donated is 2𝑥 JPY, and the actual donation expenditure is
1,000 − 𝑥 JPY. Since the initially selected amount is 𝑥 JPY, the upper limit remains 1,000 JPY, but
the total amount donated is 2𝑥 JPY, so the upper limit becomes 2,000 JPY. Also, suppose a donor
receives the 50% rebate treatment. In that case, she initially selects 𝑥 JPY for the donation, the total
amount donated is 𝑥 JPY, and the actual donation expenditure is 1,000 − 𝑥 JPY. However, if thedonor receives rebate treatment, actual donation expenditure is deducted, so actual donation
expenditure is 1/2 𝑥 JPY. Therefore, the maximum actual donation expenditure is 500 JPY. Thus,
the maximum actual donation expenditure and the total amount donated vary depending on the rebate
or matching treatment.
Although the rebate and matching are identical in changing the price of donations, they
have different upper limits on actual donation expenditure and the total amount donated to the charity.
This difference is problematic when comparing the treatment effects of rebate and matching. If wecompare these treatment effects without considering this difference, the impact of the rebate may beunderestimated compared to the treatment effect of matching. Therefore, we design our experimentthat accounts for this difference in the upper limit of donation.
[Table 2 is here]7

To address this difference, we set the donor's budget at 1,000 JPY and establish two upper
limits of donation for the same donor: one with the maximum total amount donation of 2,000 JPY(maximum actual donation expenditure of 1,000 JPY) and the other with the maximum total amountdonation limited to 1,000 JPY (maximum actual donation expenditure of 500 JPY) (Table 2). We also
allow donors in the self-selection condition to decide whether to opt-in or not before determining theinitially selected amount under each upper limit. Then, we conduct our analysis after equalizing theupper limit of donation for each donor. Table 2 shows where each group donates 80% of theirmaximum donation to the donor under their respective upper limit.
Under this experimental design, we compare the effects of matching and rebate using theareas highlighted in orange in Table 2. Specifically, we compare matching versus rebate using a
condition with a maximum total amount donated of 2,000 JPY for the control and compulsory rebategroups and a maximum total amount donated of 1,000 JPY for compulsory matching. For the selfselected group, we use an upper limit of 2,000 JPY for rebates. On the other hand, if the same donation
limit is used for matching, the upper limit will be different for those who opt-in and those who do not
opt-in. Therefore, we will use the upper limit of 1,000 JPY for those who opt-in and the upper limit of
2,000 JPY for those who do not.

8

4. Results[Table 3 is here]
First, we set the initially selected amount as one outcome variable and test Hypothesis 1. The
estimation results in column 1 of Table 3 show that the initially selected amount in the compulsory
matching group is 62.71 JPY lower than in the control group (p<.01), and that in the compulsory rebategroup is 164.20 JPY higher than in the control group (p<.01). A direct comparison of the two treatmentgroups shows that the initially selected amount in the compulsory matching group is 226.91 JPY lower
than in the compulsory rebate group (p<.01), which supports Hypothesis 1.
Second, we set the total donation amount as another outcome variable and test Hypothesis
2. The estimation results in column 3 of Table 3 show that the total donation amount in the compulsory
matching group is 223.50 JPY higher than in the control group (p<.01), and that in the compulsory
rebate group is 164.20 JPY higher than in the control group (p<.01). A direct comparison of the twotreatment groups shows that the total donation amount in the compulsory matching group is 59.30 JPY
higher than in the compulsory rebate group (p<.05), which supports one mechanism of Hypothesis 2,
by which warm glow moderates the substitution effect.
[Table 4 is here]
Third, we focus on the self-selection groups and conduct multiple tests of Hypothesis 3.
When we compare the take-up rate for matching (48.75%) to that for rebate (50.42%), the difference
between the two is not statistically significant and does not support the hypothesis that donors preferrebate over matching. However, the treatment effect on the initially selected amount among those who
self-select to receive the treatment is -99.57 JPY (p<.05) for matching and 165.70 JPY (p<.01) for
rebate (column 1 of Table 4). This TOT effect is still smaller for matching than for rebate (p<.01).
Further, the treatment effect on the total donation amount among those who self-select toreceive the treatment is 516.70 JPY (p<.01) for matching and 165.70 JPY (p<.01) for rebate (column
3 of Table 4). The hypothesis that this TOT effect will become larger than the ITT of the compulsory
treatment is supported for matching (p<.01), but not supported for rebate (p=0.97). As a consequence,
the TOT effect on the total donation amount is larger with matching than with rebate (p<.01). While
this tendency itself is the same as in the case of the compulsory treatments, the difference between thematching and rebate treatment effects on the total donation amounts is larger in the case with a selfselection process (351.0 JPY) than in the case when using compulsory treatments (59.30 JPY). Thatis, the superiority of matching over rebate for the total donation amount is more pronounced in thecase of a self-selection process. These results are stably observed after controlling for covariates.

95. Conclusions
Standard economic theory predicts that matching and rebates will have the same effect on individuals’
donation behavior when the donation price is equivalent. However, several experimental studies havereported that their donation behaviors are promoted under matching more than under rebate. This study
reveals how treatment effects of matching and rebate change when people can self-select whether touse such schemes or not. Although most traditional policy research has measured the causal effects ofmandatory policy assignment, policies in the real world are often applied to only those who accept
them. We conduct an incentivized and nationwide experiment on 2,400 Japanese residents with four
treatments, the 1:1 matching treatments (compulsory and self-selection) and the 50% rebate treatments(compulsory and self-selection), and provide the following findings: Initially selected amount under
the compulsory matching is smaller than under the compulsory rebate, while total amount donated tothe charity under the former scheme is larger than that under the latter scheme, which is consistent
with the existing theories (Hungerman and Ottoni-Wilhelm, 2021). The treatment effect on the total
donation amount among people who self-select to receive the treatment is larger under the matching
scheme than under the rebate scheme, and this difference is further larger than that in the case whenusing compulsory treatments. That is, the superiority of matching over rebate for the total donation
amount gets more pronounced in the case of a self-selection process.

10

Reference
Eckel, C.C., and Grossman, P.J. (2003). Rebate versus matching: does how we subsidize charitable
contributions matter?. Journal of Public Economics, 87(3-4), 681-701.
Fowlie, M., Wolfram, C., Baylis, P., Spurlock, C.A., Todd-Blick, A., and Cappers, P. (2021). Default
effects and follow-on behaviour: evidence from an electricity pricing program. The Review of
Economic Studies, 88(6), 2886–2934.
Hungerman, D.M., and Ottoni-Wilhelm, M. (2021). Impure impact giving: Theory and evidence.
Journal of Political Economy, 129(5), 1553-1614.
Ida, T., Ishihara, T., Ito, K., Kido, D., Kitagawa, T., Sakaguchi, S., and Sasaki, S. (2022). Choosing
who chooses: selection-driven targeting in energy rebate programs. National Bureau of Economic
Research. (No. w30469).
Ito, K., Ida, T., and Tanaka, M. (2021). Selection on welfare gains: experimental evidence fromelectricity plan choice. National Bureau of Economic Research. (No. w28413).
Kurokawa, H., and Sasaki, S. (2023). How Does Opt-in Work? A Field Experiment on Financial
Incentives for Physical Activity. Osaka University Discussion Papers In Economics And Business,
No.23-01.
Sasaki, S., Ishihara, T., and Kato, H. (2023). Rebate versus matching, again: Does opt-in matter?. AEA
RCT Registry. https://doi.org/10.1257/rct.10943-1.1
Sasaki, S., Kurokawa, H., and Ohtake, F. (2022). An experimental comparison of rebate and matchingin charitable giving: The case of Japan. The Japanese Economic Review, 73(1), 147-177.
Wang, W., Ida, T., and Shimada, H. (2020). Default effect versus active decision: evidence from a field
experiment in Los Alamos. European Economic Review, 128, 103498.

11

Tables and Figure
Table 1. Balance Check

FemaleMarriedHaving children
Age
Educational years
Household income
No income information
Living in urban areas

Number of observations

(1)

(2)

(3)

(4)

(5)

Control

1 to 1 matching
Compulsory

1 to 1 matching
Self-selection

50% rebate
Compulsory

50% rebate
Self-selection

p-val.

0.502

0.502

0.502

0.502

0.502

1.000

(0.023)(0.023)(0.023)(0.023)(0.023)

0.550

0.492

0.517

0.546

0.517

(0.023)(0.023)(0.023)(0.023)(0.023)

0.248

0.237

0.258

0.283

0.246

(0.020)(0.019)(0.020)(0.021)(0.020)46.39846.36945.94646.10846.144(0.595)(0.601)(0.606)(0.603)(0.600)14.49314.74814.63414.69214.760(0.097)(0.096)(0.097)(0.094)(0.093)587.961

566.012581.190619.613609.509(16.838)(15.179)(15.092)(17.060)(16.418)

0.198

0.165

0.169

0.171

0.185

(0.018)(0.017)(0.017)(0.017)(0.018)

0.554

0.556

0.563

0.560

0.571

(0.023)(0.023)(0.023)(0.023)(0.023)

480

480

480

480

480

0.350
0.547
0.983
0.274
0.124
0.655
0.988

Note: Some participants did not answered annual household income. We imputed the average amout of the income for such respondents while consideringthat they did not answer it by using the variable of no income information.
12

Table 2. Experimental Design
Control

1 to 1 matching

1 to 1 matching

50% rebate

50% rebate

Compulsory

Self-selection

Compulsory

Self-selection

opt-inBudget constraint (JPY)no

opt-inno1,000
2000（Upper limit of actual donation expenditure: 1,000）

Upper limit of total amount donated to the charity (JPY)(i) Initially selected amount (JPY)

800

800

800

800

800

800

800

(ii) Actual donation expenditure (JPY)

800

800

800

800

400

400

800

(iii) Reward to self (JPY)

200

200

200

200

600

600

200

(iv) Total amount donated to the charity (JPY)

800

1,600

1,600

800

800

800

800

Budget constraint (JPY)

1,000
1,000（Upper limit of actual donation expenditure: 500）

Upper limit of total amount donated to the charity (JPY)(i) Initially selected amount (JPY)

400

400

400

400

400

400

400

(ii) Actual donation expenditure (JPY)

400

400

400

400

200

200

400

(iii) Reward to self (JPY)

600

600

600

600

800

800

600

(iv) Total amount donated to the charity (JPY)

400

800

800

400

400

400

400

13

Table 3. Intention-to-Treat (ITT) Effects(1)
Dependent variable =(2)

Initially selected amount

(3)

(4)

Total amountdonated to the charity

CM: 1 to 1 matching-62.71***-62.91***223.5***

221.9***(Compulsory)(17.56)(17.17)(23.93)(23.33)

OM: 1 to 1 matching-48.54***-47.12***251.9***

253.0***(Self-selection)(18.74)(18.19)(27.28)(26.44)164.2***

162.1***

164.2***

160.9***(22.40)(21.59)(22.40)(21.59)83.54***

81.99***

83.54***

81.13***(22.44)(21.86)(22.44)(21.98)

CR: 50% rebate(Compulsory)
OR: 50% rebate(Self-selection)
FemaleMarriedHaving children
Age
Educational years
Household income
No income information
Living in urban areas

Constant

53.36***

67.00***(12.58)(16.23)

8.04

2.75

(16.74)(21.21)(13.71)(6.36)(17.25)(21.90)5.045***

6.834***(0.53)(0.68)

3.25

6.600*(3.06)(4.00)

0.0715***0.0801***(0.02)(0.02)-41.86**-60.03***(16.90)(21.38)8.27(0.94)(12.46)(16.05)349.0***

1.672

349.0***-132.0*(14.84)(54.27)(14.84)(70.19)Number of observations

2,400

2,400

2,400

2,400

R-squared

0.071

0.132

0.051

0.116

CM v.s. CR

0.000

0.000

0.018

0.012

OM v.s. OR

0.000

0.000

0.000

0.000

CM v.s. OM

0.339

0.273

0.339

0.277

CR v.s. OR

0.001

0.001

0.001

0.001

Notes: *** p<0.01, ** p<0.05, * p<0.1. The numbers in the bottom part show P-values of hypothesis
testings.
14

Table 4. Treatment-on-Treated (ITT) Effects(1)
Dependent variable =TOT of matching(2)

Initially selected amount

(3)

(4)

Total amountdonated to the charity-99.57**-96.57**516.7***

520.4***(40.80)(39.62)(51.97)(50.71)165.7***

162.7***

165.7***

163.5***(39.45)(38.40)(50.25)(49.15)

Covariates

OUT

IN

OUT

IN

Number of observations

2,400

2,400

2,400

2,400

TOT of M = TOT of R

0.000

0.000

0.000

0.000

CM = TOT of M

0.297

0.325

0.000

0.000

CR = TOT of R

0.964

0.986

0.972

0.969

TOT of rebate

Notes: *** p<0.01, ** p<0.05, * p<0.1. The numbers in the bottom part show P-values of hypothesis
testings.

15

Reward to self

Reward to self

1,000 JPY

1,000 JPY
50% rebate

50% rebate

Control

Control
1:1 matching

1:1 matching

0

500 JPY 1,000 JPY

Actual donation
expenditure0(a)1,000 JPY(b)

Figure 1. Budget Constraint Lines

16

2,000 JPY

Donated amountto the charity

Theoretical Background of“Rebate versus matching, again: Does opt-in matter?”
Shusaku Sasaki 1 , Takunori Ishihara 2 , and Hiroki Kato 3
1

2

Center for Infectious Disease Education and Research (CiDER), Osaka University
Institute for International Academic Research, Kyoto University of Advanced Science3
Graduate School of Economics, Osaka University
First Draft: February 12, 2023

1

Theoretical Predictions

1.1

Setup and Optimal Giving

To derive theoretical predictions, we use the impure impact giving model (Hungerman and OttoniWilhelm, 2021). Suppose that a donor obtains utility from private consumption (c), the out-ofpocket gift to the charity (g), and the charity’s amount received (G), that is, U (c, g, G). The donor
tries to maximize utility subject to two constraints. The first constraint is the budget constraint:c + g = y + tg, where t is a rebate rate. The second constraint is the relationship between outof-pocket gifts and the amount received: G = (1 + m)g, where m is a match rate. Using two
constraints, we can produce a constraint in terms of the amount received as follows:c + pt pm G = y,(1)

where the rebate price is pt = 1 − t and the match price is pm = 1/(1 + m). Substituting equation(1) into utility, we have

U (y − pt pm G, pm G, G).
Note that g = G/(1 + m) = pm G.

1(2)

We determine the amount received G to maximize the utility (2). The first-order condition is−pt pm Uc (y − pt pm G, pm G, G) + pm Ug (·, ·, ·) + UG (·, ·, ·) = 0.

(3)This equation depends on three exogenous factors: own income y, the first price term pt pm , andthe second price term pm . Thus, the optimal amount received is a function of these arguments:G∗ = q(y, pt pm , pm ).

(4)

Also, the optimal out-of-pocket gift isg ∗ = pm q(y, pt pm , pm ).

(5)

Under the rebate incentive system, the optimal out-of-pocket gift is equivalent to the amount received. However, under the matching inventive system, the optimal out-of-pocket gift is different
from the amount received.

1.2

Price Effects

First, we predict the rebate-price effect and the match-price effect. The rebate-price effect is∂G∗−pm Uc ∂G∗=−
pm G∗ ,
∂pt
∆
∂y

(6)

where ∆ is the negative of the second-order condition, that is, ∆ > 0. Assuming the amountreceived is normal goods so that the income effect ∂G∗ /∂y is positive, we predict that the rebateprice effect is always negative.
We can decompose the rebate-price effect into the conventional Sltsuky equation:∂G∗ pt∂pt G∗−pm pt Uc ∂G∗
=
−
pt pm
G∗ ∆
∂y
 ∗


−pm pt Uc
∂G y
G∗
=
−
pt pm
G∗ ∆
∂y G∗
y

et =

= eH
t − ey bG ,(7)

where eHt is compensated rebate-price elasticity (negative), ey is income effect (positive), and bG2

is the amount received as a proportion of income.
Next, the match-price effect is∂G∗ −pt Uc ∂G∗
=
−
pt G∗
∂pm
∆
∂y
pm pt Ugc − pm Ugg − UGg ∗
Ug
+
−
G .
∆
∆

(8)

The third and fourth term comes from the warm-glow utility (Ug ). We have the Sltsuky decomposition of the match-price effect as follows:∂G∗ pm∂pm G∗
pm pt Ugc − pm Ugg − UGg
p m Ug−pm pt Uc ∂G∗=−pm pt + ∗ −
pm
∗
G ∆
∂y
G ∆
∆


pm pt Ugc − pm Ugg − UGg−pm pt Uc pm Ug∂G∗ Ug
=
+ ∗ −
pm −
pm
G∗ ∆
G ∆
∆
∂y Uc
∂G∗
∂G∗ Ug−pm pt +
pm
∂y
∂y U
 ∗
 c

∂G y
G∗ Ug
G∗
H
=em −
pt pm
−
pm
∂y G∗
y
Uc
y

em =

=eH
m − ey b̃G(9)where eH
m is the compensated match-price elasticity. To compare with the rebate-price elasticity,
we rewrite the Sltsuky equation of match-price as follows:pm pt Ugc − pm Ugg − UGg
p m Ug
∂G∗
−
p
−
pm pt
m
G∗ ∆
∆
∂y


pm pt Ugc − pm Ugg − UGg
pm Ug
= eH
−
pm − ey bG .
t +
∗
G ∆
∆

em = eHt +(10)

The match-price elasticity is different from the rebate-price one due to the second term. This termcomes from the warm-glow part of compensated match-price elasticity eH
m . Thus, a change ofsubstitution effect due to warm-glow preferences creates the difference between em and et . We
summarize the price effect on the amount received as follows:
Prediction 1 (Effect on Amount Received). (a) If warm-glow preferences weaken the conventional
substitution effect, then em > et . (b) On the other hand, if warm-glow preferences strengthen theconventional substitution effect, then et > em . (c) If a donor has pure impact preferences (nowarm-glow preferences), then et = em .
The rebate-price elasticity of the out-of-pocket gift is also et because the optimal out-of-pocket3

is same as the optimal amount received for any preference. However, the match-price elasticityof the out-of-pocket gift is different from em . By g ∗ = pm G∗ , the match-price effect on theout-of-pocket gift is∂g ∗∂G∗= G ∗ + pm
.
∂pm
∂pm

(11)

The match-price elasticity of the out-of-pocket gift, em,o ispm ∂G∗
pm
+
pm ∗
∗
g
∂pm
g
∗
∂G pm
=1+
∂pm G∗

em,o = G∗= 1 + em .

(12)

Using the equation (10), we rewrite em,o as follows:
em,o = eHt +pm pt Ugc − pm Ugg − UGg
pm Ug
1+ ∗ −
pm − ey bG .
G ∆
∆

(13)

the expression in brackets is always positive because (pm pt Ugc − pm Ugg − UGg )/∆ is less thanone. Thus, we have em,o > et .
As an extreme case, consider the match-price elasticity of the out-of-pocket gift for the purewarm-glow donor (UG = 0 for any G). By the first-order condition, we have
pm Ug
G∗ ∆
−pm pt Uc pm Ug
=
+ ∗
G∗ ∆
G ∆
eH
t +

(14)

=0.
Moreover, by the second-order condition, we havepm pt Ugc − pm Ugg − UGg
pm + ey bG∆pm pt Ugc − pm Ugg − UGg
∂G∗
=
pm +
pt pm
∆
∂y
pm pt Ugc − pm Ugg − UGg−pm pt Ucc + pm Ugc
=
pm +
pt pm
∆
∆
=1.
Thus, em,o = 0 for the pure warm-glow donor.

4

(15)

Finally, we summarize the price effect on the out-of-pocket gift as follows:
Prediction 2 (Effect on Out-of-pocket Gift). For any preference, em,o > et . That is to say, thematching incentive is less effective than the rebate incentive in terms of the out-of-pocket gift.
Especially, the matching incentive does not affect the out-of-pocket gift for the pure warm-glow
donor.

1.3

Opt-in Decision

We predict whether donors use the matching or rebate incentive, using the indirect utility. Substituting the optimal amount received (4) into the direct utility (2), we have indirect utility as follows:
V (y, pm , pt )(16)

=U (y − pt pm G(y, pm pt , pm ), pm G(y, pm pt , pm ), G(y, pm pt , pm )).
Using the envelope theorem, the effect of the rebate-price on indirect utility is∂V= −pm Uc G∗ .
∂pt

(17)

The effect of the rebate-price on indirect utility is negative, that is, introducing the rebate incentive
scheme increases indirect utility. Since the term −pm Uc strengthens the rebate-price effect (6), thestronger this effect is, the more sensitive the optimal amount received is.
Also, the effect of the match-price on indirect utility is∂V= (−pt Uc + Ug )G∗ .
∂pm

(18)

This effect is weaker than the effect of the rebate-price on indirect utility. Especially, for the purewarm-glow donors, this effect is zero because −pt Uc + Ug = 0. Since −pt Uc + Ug strengthens thematch-price effect (8), the stronger this effect is, the more sensitive the optimal amount received
is.
∂V
∂V
Prediction 3 (Opt-in Decision). (a) Let pm = pt . Then, ∂p≥ ∂p. That is to say, the rebatemtincentive scheme strictly prefer or indifferent to the matching incentive scheme for any preferences.
In particular, the pure impact donor is indifferent between the two incentives. Also, the matching
incentive scheme does not improve the utility of the pure warm-glow donor. (b) The more donors
who increase their donations by offering incentives, the more they will use the incentives.

5