Public Finance, 2nd Edition

ALTRUISM, FREE RIDING, AND CROWDING OUT OF

PRIVATE CHARITY

The notion of pareto-optimal redistribution has practical as well as theoret-

ical diYculties when private charity and public assistance exist side by side.

Gain±gain redistributions motivated by altruism have two very strong prop-

erties. One is the powerful incentive for donors to free ride on the gifts of

other donors. The other is that public assistance crowds out (reduces) private

charity dollar for dollar under altruism. Neither property is even roughly

consistent with the facts in the United States.

James Andreoni has developed a simple endowment model with altruism

to illustrate the eVects of these two properties for large economies. He begins

with the case of only private charity and explores the propensity to free ride

on the gifts of others.

9

Do People Free Ride?

Assume a nation of N people in which everyone has the same tastes, with

utility deWned over a composite commodity good y and the total amount of

charitable giving, G:

U

i

 U

i

y

i

,G i  1, ...,N

P

y

 1, the numeraire, and a unit of G is $1, a cash grant. Each person i has

an endowment w

i

.

DeWne g

i

as person i's own charitable contribution, and G

i

as the total

charitable contributions of everyone except person i. Assume a Nash envir-

onment in which person i takes G

i

as given. Under the Nash assumption

each person i solves the problem:

max

y

i

; g

i



U

i

y

i

,G

equivalently

max

y

i

; G

U

i

y

i

,G

s:t: y

i

 g

i

 w

i

g

i

 0s:t: y

i

 G  w

i

 G

i

G  G

i

Using the equivalent formulation on the right, the demand for G can be

written as

the potential pitfalls of simple linear utility representations of altruism of the form

V

i

 b

ii

U

i



P

j6i

b

ij

U

j

, b

ij

 0. To give one example, he considers the case in which utility is a

function of one private composite good and one public good and shows that all pareto-optimal

allocations in the altruistic economy are pareto-optimal allocations in the egoistic economy in

which utility is a function only of one's own consumption. The linear representation of altruism

does not buy anything.

9

J. Andreoni, ``Privately Provided Public Goods in a Large Economy: The Limits of

Altruism,'' Journal of Public Economics, February 1988.

322 ALTRUISM, FREE RIDING, AND CROWDING OUT OF PRIVATE CHARITY

G  maxfg w

i

 G

i

,G

i

g i  1, ..., N (10:9)

where g ( ) is i's Engel curve for charitable giving, identical for all individuals.

Assume y and G are both normal goods, so that 0 < g

0

 a < 1. If person i is

at an interior solution, then:

G  g w

i

 G

1

 (10:10)

Invert g and then add g

i

to both sides to obtain:

g

1

Gw

i

 G

1

(10:11)

and

g

i

 w

i

 G  g

1

Gw

i

 fG (10:12)

with

fGg

1

GG (10:13)

Note for future reference that f

0

 1/a  1, and f

1

0

 a/1  a < 1.

Let w*  the amount of endowment at which the individual is just

indiVerent between giving and not giving. From Eq. (10.12),

g

i

 0  w*  fG (10:14)

or

w*  fG (10:15)

Therefore, also from Eq. (10.12),

g

i

 w

i

 w*, for w

i

> w*

g

i

 0, for w

i

 w*

(10:16)

and

G 

P

w

i

>w*

w

i

 w* (10:17)

But, G  f

1

w* from Eq. (10.15). Therefore,

f

1

w*

P

w

i

>w*

w

i

 w* (10:18)

Next consider the average amount of charity per person, H

N

, equal to

H

N

 f

1

w*/N  1/N

P

w

i

>w*

w

i

 w* (10:19)

and ask what happens to the average as N becomes large.

Note, Wrst, that the level of wealth at which an individual is just indiVer-

ent to giving varies with N. Thus, the general expression for the average

amount of charity per person, H

N

,is

10. THE FIRST-BEST THEORY OF TAXATION 323