Sociology Mind
2011. Vol.1, No.2, 33-35
Copyright © 2011 SciRes. DOI:10.4236/sm.2011.12004
Effects of Charity on Social Welfare: A Theoretical Analysis
Madhu S. Mohanty
Department of Economics and Statistics, California State University, Los Angeles, USA.
Received January 24th, 2011; revised January 29th, 2011; accepted January 31st, 2011.
Using a social welfare function approach, this study theoretically demonstrates that charity and selfless service
contribute positively to the welfare of the society without reducing welfare of any individual and thus they in-
variably lead to Pareto-improvement in social welfare. In other words, a society, regardless of its current level of
development, can attain a higher level of subjective wellbeing if the affluent class voluntarily undertakes the
service activities that benefit the needy.
Keywords: Social Wefare Function, Charity, Selfless Giving, Pareto Improvement
Introduction
Giving to the needy and donating to charitable organizations
are common practices in many societies of the world today. In
all religions and cultures, giving is considered as a meritorious
deed. Following the social welfare function approach of Arrow
(1963) and Sen (1970, 1986), this study formally demonstrates
that charitable behavior of the members of the society enhances
the welfare of not only the donor, but also the society as a
whole without reducing the welfare of any individual. The next
section presents a theoretical model showing the positive ef-
fects of charitable donations on the social welfare function.
Section 3 extends this analysis to selfless giving and the final
section summarizes the conclusions and examines their empirical
implications.
A Model of Charity
For the sake of simplicity, assume that the society contains
two groups of people: rich and poor. Rich people have more
than what they need for their consumption and poor have less.
The utility of each group increases as they consume more of
goods and services. Among the rich, there are two groups of
people: those who enjoy giving their surplus wealth to the poor
(donors = D) and those who don’t (non-donors = ND). Defining
group 1 as rich, group 2 as poor and q as quantities of goods
and services (in $), we can write the utility functions of all three
groups just mentioned as follows:


1111 12
,
D
DD
UUqq, (1)

11 1112
,

N
DNDN
UUqqD , (2)


2221 22
,UUqq. (3)
Note that the utility of each group depends on the quantity of
goods and services (qs). The first subscript denotes the group
and the second subscript denotes either consumption or sur-
plus/donation. For example, q11 and q12 in Equations (1) and (2)
denote respectively the dollar values of actual consumption and
surplus for the rich. The superscripts D and ND distinguish rich
donors from rich non-donors. Similarly, q21 and q22 respectively
denote the dollar values of actual consumption and donations
received by the poor from the rich donor. The quantity q22 thus
is related to q12 by the following equation:
22 12
D
qkq, 0 < k < 1, (4)
where k is the fraction of the surplus income of the rich donor
that he/she donates to the poor.
Assume that the utility derived by an individual is positively
related to the quantity of goods and services consumed. Thus,
11 22
21 22 11 11
0,0, 0,0
DND
DND
UU UU
qq qq
 

  
(5)
Assume further that for a donor, utility increases with the in-
crease in donation from his/her surplus income. Since 12
D
q is
fixed during a given time period, this assumption suggests that
the utility of the rich donor rises as the size of 12
D
q becomes
smaller due to donation. Thus, for k < 1,
1
12
0
D
D
U
q

 (6)
From this assumption, it follows that
11121 22
2
12 12
0
DDDD
DD
UUqU q
kk qqk
 


 

 

1 (7)
In other words, the utility of the rich donor rises as the fraction
of his/her surplus income donated to the poor increases. For a
non-donor, however, utility rises with the increase in surplus
hoarding and remains unaffected by whether or not the donor
donates to the poor. Thus,
11
12 12
0, 0
ND ND
ND D
UU
qq




(8)
In this framework of three groups of individuals, the social
welfare function in its simplest form can be written as
1By Equation (6), the first partial derivative after the last equality sign
is negative. By Equation (4), 12 22
20
D
qq
kk





.

M. S. MOHANTY
34






111 121111222112
,,,
D
D DNDNDNDD
WU qqUqqUqkq (9)
To see the effect of an increase in the rate of donation k by
the rich donor on the welfare of the society, it is necessary to
evaluate the sign of Wk. Under the assumption that the
utility function of the rich non-donors remains unaffected by
the changes in k, this partial derivative reduces to
112 222
22 12
1222
12
2
22 12
DD
D
D
D
D
WUqU q
kkq q
UqU
q
q qk






 



k
(10)
Note that by Equations (5) and (6), both terms after the second
equality are positive, and consequently, 0Wk. This con-
firms that with a rise in the rate of donation k, the utility of rich
donors as well as the utility of the poor receiving charitable help
rise, and consequently, the social welfare rises without reducing
the welfare of rich non-donors. Clearly, this is a Pareto-improve-
ment which further justifies charitable donation as a means of
enhancing social welfare.2
Selfless Giving
The charity model outlined above can be easily extended to a
model of selfless giving. In a model of charity, the donation is
given out of the surplus income q12 only, whereas in a model of
selfless giving, the donor goes a step further to donate not only
the entire surplus income (i.e., k = 1), but also a part of the in-
come kept aside for his/her own consumption q11 if necessary.
The desire to sacrifice in this model is so strong that the donor
does not hesitate even to forego a part of his/her own consump-
tion. In this model, the total income of the rich donor is fixed at
, but unlike the charity model, the part of the income kept
aside for donation 12
0
1
q
D
q is variable which exactly equals the
donation received by the poor. Thus,
0
12111 1222
,
DD D
qqqqq  (11)
Due to the variable nature of the surplus income 12
D
q ear-
marked for donation and more intense desire of the donor to
help the poor, the partial derivatives of the donor’s utility func-
tion, unlike those in charity model, assume the following signs:
112
0
DD
Uq (12)
and




1 111 12 12111 120
DDDD DDDD
UqUqqqUq
(13)
by Equations (11) and (12).
The social welfare function in this model can be rewritten as





0
11121211112
22122 12
,,
,
D
D DNDNDND
D
WU qqqUqq
Uqq q
 
 (14)
and the effect of selfless giving on social welfare can be evalu-
ated from the following equation:
1111 22
22 1211 121212
112
22 11 12
.
DD D
2
D
DD D
DD
DD
WU qUUq
q qqqq q
UUU
q qq
D







 

(15)
The three terms after the last equality in Equation (15) are
positive respectively by Equations (13), (12) and (5), and con
sequently, 12 0
D
Wq

 . Thus in a model of selfless giving,
where the donor does not hesitate even to cut down his/her own
consumption to help the poor, the social welfare rises as the
amount of donation by the rich donors increases because it
increases the welfare of not only the poor (indicated by the
positive 3rd term after the last equality in Equation (15)), but
also the rich donor (first and second terms after the last equality
in Equation (15) are positive) without reducing the welfare of
the rich non-donors. Selfless giving thus leads to a Pareto im-
provement in social welfare.
Summary and Discussion
Following a social welfare function approach and assuming a
positive relationship between charitable donation and donor’s
utility, this study demonstrates theoretically that giving to the
needy from one’s surplus income increases the welfare of not
only those who receive the donaion, but also those who donate,
leading thus to an increase in the aggregate social welfare. Ex-
tending this analysis to selfless giving, the study further shows
that helping the needy at the cost of one’s own consumption
also increases social welfare without reducing the welfare of
any individual. Charitable donation and selfless giving thus
lead invariably to a Pareto improvement in social welfare.
This study has important empirical implications, especially
when the government fails to provide adequate public goods
necessary for enhancing the welfare of its citizens. In every
society, there are large numbers of rich citizens with abundant
surplus income along with numerous poor people lacking even
the basic necessities of life. If the rich voluntarily comes for-
ward to help the poor, it will not only solve numerous eco-
nomic problems facing the society, but also augment the ag-
gregate social welfare, leading to a more prosperous and hap-
pier society. This is not a new finding. The generous efforts by
numerous service organizations, such as Red Cross, Habitat for
Humanity, Feed the Children, Salvation Army, Sathya Sai Ser-
vice Organization etc. all over the world, provide ample testi-
mony to the beneficial effects of giving for worthy causes.3 The
study simply reinforces the fact that even an economically de-
veloped society can attain a still higher level of subjective
wellbeing if the affluent class voluntarily undertakes the service
activities that benefit the needy.
References
Arrow, K. J. (1963). Social choice and individual values (2nd Ed.).
3While providing free super-specialty medical care to the needy, free
education even at the university level to all students, and free drinking
water to numerous villages and cities of India, the Sathya Sai Service
Organization has recently extended its operation to several countries
outside India. See “Sai Institutions & Service Projects” under the in-
ternational Sai Or
g
anization website
,
htt
p
://www.sath
y
asai.or
g
.
2Pareto optimality in standard welfare economics refers to a state in
which no one can be made better off without making someone worse
off. A Pareto improvement, on the other hands, indicates a scenario
where at least one individual is better off when no one else is worse of
f
(Varian, 1993, Chs. 21, 22; Mas-Colell et al., 1995, Chs. 16, 21).

M. S. MOHANTY 35
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http://www.sathyasai.org
Mas-Colell, A., Whinston, M. D., & Green. J. R. (1995). Microeco-
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Sen, A. (1970). Individual choice and social welfare. San Francisco:
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tor, M. (Eds.), Handbook of mathematical economics. Amsterdam:
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Varian, H. (1993). Microeconomic analysis (3rd Ed.). New York: Nor-
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