Theoretical Economics Letters, 2013, 3, 283-287
http://dx.doi.org/10.4236/tel.2013.35047 Published Online October 2013 (http://www.scirp.org/journal/tel)
Matched Charitable Contributions: Comparative
Statics and Equilibrium
Daniel K. Biederman
Department of Economics, University of North Dakota, Grand Forks, USA
Email: daniel.biederman@business.und.edu
Received August 8, 2013; revised September 6, 2013; accepted September 13, 2013
Copyright © 2013 Daniel K. Biederman. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
A small optimizing agent max imizes his utility by allocati ng his inco me to private consumption an d to a contribution to
a particular charity. The agent’s contribution may be matched, at a particular rate, by a large ag ent. We provid e a com-
prehensive comparative-statics analysis of the agent’s prob lem, allowing for change s in the agent’s income, the agent’s
conjecture about other agents’ contributions, and the match rate. A Nash equilibrium among n such agents is shown to
exist if private consumption is a normal good for all agents. The equilibrium is unique if private consumption and
charitable giving are normal goods for all agents.
Keywords: Charitable Contributions; Match Rate; Monotonic Transformation; Nash Equilibrium
1. Introduction
Fundraising efforts by charities sometimes feature an
offer to match individual contributions at a particular rate,
commonly called the match rate. Solicitations may in-
form potential contributors that a certain benefactor has
offered to contribute, say, 50 cents or one dollar or two
dollars for every dollar contributed by others. To the ex-
tent, match offers and other fundraising techniques are
successful in generating higher levels of contributions
than otherwise would be observed, they enhance the sup-
ply of certain privately-provided public goods, thereby
mitigating th e undersupp ly of such goo ds. How ever, only
in recent years, economists began to devote considerable
efforts to the study of fundraising methods. Karlan and
List [1, p. 1774] note that fundraisers are “typically long
on rules of thumb and short on hard scientific evidence,
divided as to the most efficient means to attract ... dol-
lars.”
We are concerned, in this paper, with matched contri-
butions.1 We formulate a model in which each of
“small,” purely-altruistic, optimizing agents allocates his
income to private consumption and a contribution to a
particular charitable organization. The agent is aware that
his contribution, as well as the contributions of others,
will be matched at a particular rate. They formulate a
Nash conjecture about the total ch aritable contribution of
the other
n
1n
small agents. Given values of his in-
come, his Nash conjecture, and the match rate, the agent
maximizes his utility subject to his budget constraint.
The purpose of the paper is twofold. First, we aim to
fill a gap in the existing literature by providing a com-
prehensive comparative-statics analysis of the problem
faced by an agent whose charitable contribution is
matched. We derive relatively simple, elasticity-based
expressions for the partial derivatives of the agent’s
charitable contribution with respect to a change in his
income, a change in his conjecture about other agents’
contributions, and a change in the match rate. The signs
of these partial derivatives are directly related to the elas-
ticities of the agent’s marginal rate of substitution with
respect to changes in the agent’s private consumption
and in the total contribution to the charity.
Second, we apply our comparative-statics results to
establish conditions that ensure existence and uniqueness
of Nash equilibrium among the agents. These condi-
tions overlap, in some measure, the conditions obtained
by Bergstrom, Blume, and Varian [9] for the existence
and uniqueness of Nash equilibriu m for a class of models
in which a public go od is privately-provided . Our results
generalize the Bergstrom et al. resu lts in th e sen se th at, i f
we set the net match rate equal to zero in our model, we
obtain, essentially, the Bergstrom et al. model. Further-
n
1Previous theoretical and empirical studies of matched contributions
include Andreoni [2], Eckel and Grossman [3], Gong and Grundy [4],
Huck and Rasul [5], Karlan, List, and Shafir [6], Meier [7], and Ron-
deau and List [8].
C
opyright © 2013 SciRes. TEL
D. K. BIEDERMAN
284
more, Bergstrom et al. offer a single set of conditions
from which existence and uniqueness follow, whereas
we distinguish between conditions that ensure existence
and conditions that ensure existence and uniqueness.
We present the basic model and our results on indi-
vidual contributions in Section 2. Nash equilibrium is the
subject of Section 3. An example follows in Section 4,
and we conclude with Section 5.
2. The Model and Comparative Statics
There are “small” agents, with small age nt i allocating
his income, i, to expenditure on a composite private
good, i
ny
x
, and a contribution, i, to a particular charita-
ble organization.2 The organization produces a public
good from the private good via a linear technology.
Normalizing the price of the private good at unity, agent
’s budget constraint is simply
c
i
. (1)
ii
yxc
i
Apart from the small agents, there exists a bene-
factor (or “large” agent) who offers to match the
small agents’ contributions such that the total contribu-
tion to the charity, , including the match, amounts to
n
C
n
1.
n
i
i
CK c

(2)
In (2), 1
K
is the gross match rate. In the absence
of a match offer, 1
K
and we have the ordinary case
of a privately-prov ided public good. If 2
K
, there is a
one-to-one match; if , there is a two-to-one match;
etc. The match rate is not restricted to integer-values.
3K
We will not take up, in this paper, the problem faced
by the large agent, that is, the benefactor who matches
others’ contributions. Instead, we regard the match rate,
K
, as exogenous. We assume small agents are motivated
to contribute to the charity by “pure altruism” in the sens e
of Andreoni [11]. That is, agent experiences no “warm
glow” from contributing; rather, his utility depends only
upon his private consumption, i
i
x
, and upon the total
contribution, . Agent ’s utility function,
C i
,
ii
uxC

>0, >0
ii
xc
,
is twice continuously differentiable and strictly quasi-
concave, with positive marginal utilities. We confine the
analysis to the case of interior solutions .
Agent formulates a fixed, Nash-type conjecture
about the total contributions of the other
i1n
small
agents.3 Denoting agent ’s conjecture about the con- i
tributions of others by ij
j
i
C
c
, it follows that his
ex ante conjecture about the total contribution, , is
i
C
.
ii
CKcC
 i
Accordingly, agent ’s problem is to choose
ii
x
and
to
i
c

Max ,,
iii i
uxKc C

subject to (1). Employing the notatio n
,
iii i
uuxCx

and

,
Ciii
uuxCC i
,
the first-order conditions imply

,
,.
x
ii Cii
uxC KuxC (4)
At his optimum, the agent’s marginal rate of substitu-
tion
MRS
x
C
uu is equal to the gross match rate,
and his optimal choices of i
x
and satisfy (1) and
(4). i
c
2.1. Comparative Statics
We are interested in how the agent responds to changes
in ,
i
yC
i
, and . The total differentials of (1) and (3)
are K
ddd
ii
yx i
c
(5)
and

ddd d
ii ii
CcCKCK
 ,K (6)
respectively. Employing th e notation

222
,, ,
x
xiiii xCiiiii
uuxCxuuxCxC
,
etc., the total differential of (4) is
dddd d
xxixCiCxi CCiC
uxu CKuxuCuK .
(7)
We wish to rewrite (7) in elasticity form. First, we will
identify four elasticities of marginal utility. Sub sequently,
we will relate those four elasticities to two elasticities of
the MRS. We use
x
x
and
x
C
to denote the elastic-
ities of
x
u with respect to i
x
and , respectively:
i
C
and
.
xi ixx
xx ix x
xi ixC
xC ix x
ux xu
xuu
uC Cu
Cu u


(8)
We use Cx
and CC
to denote the elasticities of
with respect to
C
ui
x
and , respectively:
i
C
and
.
Ci iCx
Cx iC C
Ci iCC
CC iCC
ux xu
xu u
uC Cu
Cu u


(9)
2Some models allow for multiple (rival) charitable organizations; see,
for example, Apinunmahakul and Barham [10].
3The Nash--conjecture assumption is standard in the analysis of pri-
vately--provided public goods in general, and in the analysis of charita-
b
le contributions in particular; see, for example, Bergstrom, Blume,
and Varian [9], Andreoni [2,11,12], and Gong and Grundy [4]. The elasticity of the agent’s MRS with respect to i
x
,
Copyright © 2013 SciRes. TEL
D. K. BIEDERMAN 285
which we denote by
x
, is equal to
x
xCx
:

2.
xC i
xixC
CxxxCx iC
x
xCx
xC
uu x
xuu
uuuu xu
uu
C


(10)
Similarly, the elasticity of the agent’s MRS with re-
spect to , which we denote by
i
C
, is equal to
x
CCC
:

2.
xC i
CixC
CxCxCC iC
x
CCC
xC
uu C
Cuu
uuuu Cu
uu


(11)
Using (4) and (8)-(11), we can r ewrite (7) as follows:
dd
d.
ii
xC
ii
xC
K
x
CK

  (12)
We use (5), (6), and (12) to obtain

d
1d
ii
KxC
K
dd
,
ixiiC
Cii
cCy
Cx K
 
 
 (13)
where
.
Ci xi
K
xC

 (14)
Using (13), we easily obtain the partial derivatives that
are central to our comparative-statics analysis:

,
iii x
cy C
 (15)

,
Cii i
cC Kx (16)
 

1.
Ciii
cK CxK (17)

The second-order sufficient condition for a maximum
is >0
, and that condition is ensured by the assump-
tion of strictly-quasiconcave utility.4
In view of (15), >0
ii
cy if and only if (“iff”)
<0
x
. This is the case when the agent’s MRS, at the
optimum, decreases with his private consumption. Fur-
thermore, using (14) and (16), we can determine that
<0
x
iff >1
ii
cC
 . Thus, to say that an agent’s
charitable contribution, i, is normal is to say that when
that agent conjectures an increase in the contributions of
other agents, his best reply is such that his conjecture
about the total contribution, , increases.
c
i
C
Noting that 1
iiii
x
yc y , it follows from (14)
and (15) that >0
ii
xy iff C>0.
This is the case
when the agent’s MRS, at the optimum, increases with
his conjecture about the total contribution. Using (16), it
follows as well that <0
ii
cC
iff >0.
C
To say
that an agent’s private consumption is normal, then, is to
say that an increase in his conjecture about others’ con-
tributio n s prompts him to decreas e h i s own contribution.
From (17) it is clear that the agent will increase his
contribution in response to an increase in the match rate
iff his MRS is inelastic with respect to his conjecture
about the total contribution (that is, iff <1
C
). We
showed directly above that >0
C
is necessary and
sufficient for normality of private consumption and for a
decreasing best reply. However, we have no a priori
knowledge about the sign of 1C
c
. Indeed, it seems
entirely reasonable to believe that, depending on an
agent’s income and preferences, i may increase, de-
crease, or remain unchanged in response to an increase in
the match rate.5
We can summarize our comparative-statics results as
follows. For agent , we have
i
0< <1,0< <1
ii ii
xy cy
, and 1< <0
ii
cC

iff <0
x
and >0
C
. It is worth noting that these last
two inequalities also ensure satisfaction of the second-
order sufficient condition for the agent’s problem.
2.2. ALEP Complementarity
For a particular agent, private consumption and the total
charitable contribution are said to be Auspitz-Lieben-
Edgeworth-Pareto (ALEP) complements if , and
weak ALEP complements if . It has been
widely-acknowledged that the primary limitation of this
notion of complementarity, which originated with Aus-
pitz and Lieben [13], is that, in general, the signs of the
second-order partial derivatives of a utility function are
not invariant with respect to monotonic transformations.
Nevertheless, in many problems the assumption of ALEP
complementarity (or weak ALEP complementarity) al-
lows for stronger comparative-statics results than are
obtainable without the assumption; see, for example,
Chipma n [14] and Weber [15].
>0
xC
u
0
xC
u
In our model, if we assume that marginal utilities are
diminishing ( and ) and that ,
then <0
xx
u<0
CC
u0
xC
u
<0
x
and >0
C
and, consequently, all of the
desirable” comparative-statics results—normality of i
x
and of , etc.—are ensured. However, the assumption
that is too strong. One can find strictly-quasi-
concave utility functions for which
and for which
i
C0
xC
u
,uxC
<0
xC
u
<0
x
and >0
C
.
Moreover, the elasticities
x
and C
are invariant
to monotonic transformations. Suppose we start with a
utility function
,uxC
x
for which CC ,
and , so that <0, <0uu
xx
0
xC
u<0
and >0
C
. A monotonic
4Ordinarily, the quasiconcavity condition for the problem at hand
would be written as . It can be shown that
2
2
xxxC CC
uKuKu 5Without probing the matter in detail, Gong and Grundy [4] note that
individual contributions can either increase or decrease with the match
rate. We provide an illustrative example in Section 4 below.
>0


2
2xC CC
Ku Ku
ii xxx
Cx uu
 .
Copyright © 2013 SciRes. TEL
D. K. BIEDERMAN
286
transformation

,uxC uxC
, (where >0
)
may or may not pr eserve the inequality , but it is
certain to preserve the values of the MRS elasticities. In
turn, it follows that the transformation preserves the
agent’s demand func t ions.
0
xC
u
C
3. Nash Equilibrium
In this section we apply the comparative-statics results
derived in Section 2 to the existence and uniqueness of
Nash equilibrium among the agents.
n
cPropo sition 1. Suppose i is continuous in i
for all
. Then, for given values of
1,2, ,in

2, ,1,
i
y
in
and :
K
(A) a Nash equilibrium exists if private consumption is
a normal good for all agents;
(B) a unique Nash equilibrium exists if private con-
sumption and the charitable contribution are normal
goods for all agents.
Proof. We apply Theorems 2.7 and 2.8 from Vives
[16]. In addition to the continuity requirement, it is re-
quired that agents’ best replies depend only on the ag-
gregate actions of other agents. This requirement is
clearly met. A further requirement is that agents’ strategy
spaces must be compact. This requirement is met since
i
for all i. Then Vives’ Theorem 2.7 asserts
existence of an equilibrium if best replies are strictly de-
creasing, as is the case when i
0i
cy
x
is normal. This proves
(A). Vives’ Theorem 2.8 asserts existence and unique-
ness of equilibrium, provided 1<
i
<cC
0
ii
for all
, as is the case when
i
x
and are normal. This
proves (B). Q.E.D. i
c
Note that Proposition 1 was established without regard
to the sign or magnitude of i
cK
. However, we can
establish existence of equilibrium for cases in which
agents respond to an increase in the match rate by de-
creasing their individual contributions or by leaving them
unchanged.
Proposition 2. Suppose is continuous in ii
c C
and
0
i
cK for all . Then, for given values
1, ,i
2,n
of
1,2, ,
i
y
in and , a Nash equilibrium exists.
K
Proof. It follows from (16) and (17) that if
0
i
cK, then <0
ii
cC
 . Applying Vives’ [16]
Theorem 2.7, we then have existence of an equilibrium.
Q.E.D.
As we noted in Section 1, our results on Nash equilib-
rium are related to the results obtained by Bergstrom,
Blume, and Varian [9] for the case of a privately-pro-
vided good. Again, however, the Bergstrom et al. model
does not focus specifically on charitable contributions,
and as such there is no match rate in that model. Fur-
thermore, Bergstrom et al. offered a single set of condi-
tions that ensures existence and uniqueness of equilib-
rium, whereas we have disentangled that set of condi-
tions.
4. An Example
Consider the constant-elasticity (CE) utility function:


11
1 if 01 or if 1
,1
lnln if 1
i
i
i
xC
uxC
xC






where >0
. Let 1
. Using (1) and (4), agent
’s optimal contribution is given by
i
1
1.
i
i
yKC
cK


i
(18)
4.1. Comparative Statics
It is straightforward to show, for the CE function, that
x
and C
. Expressions for the partial de-
rivatives ,
iii i
cycC
 , and i
cK can be ob-
tained either through direct differentiation of (18) or by
the application of (14) - (17). We present the partials in
both forms:
1,
i
ii ii
C
cy
K
xC
K


(19)
1
1i
ii ii
K
x
K
cC
K
xC
K

 
(20)



2
1
11.
ii ii
iii
yC Cx
cK
K
Kx C
KK
 



It is clear that
0< <1
ii
cy
and 1< <0
ii
cC

for all values of
, and that 0
i
cK as
(that is, as ).
1
1
4.2. ALEP Complementarity
The CE utility function is additively separable, with u
0
xC
u
. Accordingly, we refer to
x
and as ALEP
independents in this case. Now let C

1
uuu

. This
increasing (and concave) transformation results in

3
2
xCi i
uuxC

0


, so x and are now ALEP C
substitutes. Nevertheless, the MRS elasticities are un-
changed (xx


and CC


), leaving the
agent’s demand functions u nchange d as well.
4.3. Nash Equilibrium
If all agents’ utility functions are of the CE type, then
private consumption and the charitable contribution are
Copyright © 2013 SciRes. TEL
D. K. BIEDERMAN
Copyright © 2013 SciRes. TEL
287
[3] C. C. Eckel and P. J. Grossman, “Subsidizing Charitable
Contributions: A Natural Field Experiment Comparing
Matching and Rebate Subsidies,” Experimental Econom-
ics, Vol. 11, No. 3, 2008, pp. 234-252.
http://dx.doi.org/10.1007/s10683-008-9198-0
normal for all agents, and Proposition 1, Part B asserts
existence and uniqueness of Nash equilibrium. Suppose,
for instance, that the agents are identical in that they
all have the same CE utility function and the same level
of income, . Setting
n
y
1
i
Cn
i
in (18), we ob-
tain the Nash-equilibrium contribution, , for each
agent:
c*
c[4] N. Gong and B. D. Grundy, “Charitable Fund Raising:
Matching Grants or Seed Money,” Unpublished Manu-
script, 2011.

*
1.
y
cnK

[5] S. Huck and I. Rasul, “Matched Fundraising: Evidence
from a Natural Fie ld Expe riment,” Journal of Public Eco-
nomics, Vol. 95, No. 5-6, 2011, pp. 351-362.
http://dx.doi.org/10.1016/j.jpubeco.2010.10.005
It can be verified that the equilibrium total contribu-
tion, , increases with , irrespective of the
value of
*
CKnc[6] D. Karlan, J. A. List and E. Shafir, “Small Matches and
Charitable Giving: Evidence from a Natural Field Expe-
riment,” Journal of Public Economics, Vol. 95, No. 5-6,
2011, pp. 344-350.
http://dx.doi.org/10.1016/j.jpubeco.2010.11.024
*K
.
5. Summary and Conclusions
[7] S. Meier, “Do Subsidies Increase Charitable Giving in the
Long Run? Matching Contributions in a Field Experi-
ment,” Journal of the European Economic Association,
Vol. 5, No. 6, 2007, pp. 1203-1222.
http://dx.doi.org/10.1162/JEEA.2007.5.6.1203
We have focused on purely-altruistic, utility-maximizing
agents who allocate their incomes to private consumption
and to charitable contributions that are matched at an
exogenous rate. W e showed that the nature of an agent’s
response to a change in his income is inexorably linked
to the nature of his best reply to a change in the contribu-
tions of other agents. We also derived a general charac-
terization of an agen t’s response to a chang e in the match
rate in terms of the elasticity of the agent’s MRS with
respect to a change in his conjecture about the total con-
tribution. These comparative-statics results are useful in
establishing conditions that ensure existence and unique-
ness of Nash equilibrium for the case of agents.
n
[8] D. Rondeau and J. A. List, “Matching and Challenge
Gifts to Charity: Evidence from Laboratory and Natural
Field Experiments,” Experimental Economics, Vol. 11,
No. 3, 2008, pp. 253-257.
http://dx.doi.org/10.1007/s10683-007-9190-0
[9] T. Bergstrom, L. Blume and H. Varian, “On the Private
Provision of Public Goods,” Journal of Public Economics,
Vol. 29, 1986, pp. 25-49.
http://dx.doi.org/10.1016/0047-2727(86)90024-1
[10] A. Apinunmahakul and V. Barham, “Strategic Interaction
and Charitable Fundraising,” Modern Economy, Vol. 3,
2012, pp. 338-345.
http://dx.doi.org/10.4236/me.2012.33044
A more general model of matched contributions might
allow for an endogenous match rate. For example, the
match rate might be chosen by a utility-maximizing
benefactor who is cognizant of how small contributors
respond to changes in that rate. In addition, agents may
be driven to contribute by motives other than pure altru-
ism. Finally, in an environment that features heteroge-
neous agents, so me of those ag ents may find it op timal to
refrain altogether from contributing at certain match rates.
Consideration of these factors makes for ongoing re-
search.
[11] J. Andreoni, “Giving with Impure Altruism: Applications
to Charity and Ricardian Equivalence,” Journal of Politi-
cal Economy , Vol. 97, No. 6, 1989, pp. 1447-1458.
http://dx.doi.org/10.1086/261662
[12] J. Andreoni, “Toward a Theory of Charitable Fund-
Raising,” Journal of Political Economy, Vol. 106, No. 6,
1998, pp. 1186-1213. http://dx.doi.org/10.1086/250044
[13] R. Auspitz and R. Lieben, “Untersuchungen über die
Theorie des Preises,” Duncker & Humblot, Leipzig, 1889.
[14] J. S. Chipman, “An Empirical Implication of Auspitz-
Lieben-Edgeworth-Pareto Complementarity,” Journal of
Economic Theory, Vol. 14, No. 1, 1977, pp. 228-231.
http://dx.doi.org/10.1016/0022-0531(77)90096-5
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