Effects of Charity on Social Welfare: a Theoretical Analysis

doi:10.4236/sm.2011.12004

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Sociology Mind

2011. Vol.1, No.2, 33-35

Effects of Charity on Social Welfare: A Theoretical Analysis

Madhu S. Mohanty

Department of Economics and Statistics, California State University, Los Angeles, USA.

Email: mmohant@calstatela.edu, [email protected]

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

UUqq, (1)



11 1112



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

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

UU UU

 





 

(5)

Assume further that for a donor, utility increases with the in-

crease in donation from his/her surplus income. Since 12

q is

fixed during a given time period, this assumption suggests that

the utility of the rich donor rises as the size of 12

q becomes

smaller due to donation. Thus, for k < 1,



 (6)

From this assumption, it follows that

11121 22

12 12

DDDD

UUqU q

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,

12 12

0, 0

ND ND

ND D









(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











M. S. MOHANTY













111 121111222112

,,,

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

1222

WUqU q

kkq

UqU













 







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

q is variable which exactly equals the

donation received by the poor. Thus,

12111 1222

DD D

qqqqq  (11)

Due to the variable nature of the surplus income 12

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

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











11121211112

22122 12

D DNDNDND

WU qqqUqq

Uqq q

 

 (14)

and the effect of selfless giving on social welfare can be evalu-

ated from the following equation:

1111 22

1211 121212

112

11 12

DD D

WU qUUq

qqqq q

UUU













 



(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





. 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

anization website

htt

://www.sath

asai.or

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

(Varian, 1993, Chs. 21, 22; Mas-Colell et al., 1995, Chs. 16, 21).

M. S. MOHANTY 35

New York: Wiley.

http://www.sathyasai.org

Mas-Colell, A., Whinston, M. D., & Green. J. R. (1995). Microeco-

nomic theory. New York: Oxford University Press.

Sen, A. (1970). Individual choice and social welfare. San Francisco:

Holden Day.

Sen, A. (1986). Ch. 22: Social choice theory. In Arrow, K. & Intriliga-

tor, M. (Eds.), Handbook of mathematical economics. Amsterdam:

North-Holland.

Varian, H. (1993). Microeconomic analysis (3rd Ed.). New York: Nor-

ton.