Version_April_2014.pdf


When warm glow burns: Motivational (mis)allocation
in the non-pro�t sector�

Gani Aldashevy Esteban Jaimovichz Thierry Verdierx

April 16, 2014

Abstract

We build an occupational-choice general-equilibrium model of an economy with the
non-pro�t sector �nanced through private warm-glow donations. Lack of monitoring
on the use of funds implies that an increase of funds of the non-pro�t sector (because
of a higher income in the for-pro�t sector, a stronger preference for giving, or an in�ow
of foreign aid) worsens the motivational composition and performance of the non-pro�t
sector. If motivated donors give more than unmotivated ones, there exist two stable
(motivational) equilibria. Linking donations to the motivational composition of the
non-pro�t sector or a tax-�nanced public funding of non-pro�ts can eliminate the bad
equilibrium.

Keywords: non-pro�torganizations, charitablegiving, altruism, occupational choice,
foreign aid.
JEL codes: L31, D64, J24, D5.

�We thank Emmanuelle Auriol, Roland Benabou, Maitreesh Ghatak, Andy Newman, Cecilia Navarra,
Susana Peralta, Jean-Philippe Platteau, Paul Seabright, Pedro Vicente, and the participants at the N.G.O.
(Non-pro�ts, Governments, Organizations) workshop (London), OSE workshop (Paris), and seminar partic-
ipants at Collegio Carlo Alberto, Nova University of Lisbon, Tinbergen Institute and University of Sussex
for useful suggestions.

yDepartment of Economics and CRED, University of Namur, and ECARES (ULB). Mailing address: De-
partment of Economics, 8 Rempart de la Vierge, 5000 Namur, Belgium. Email: gani.aldashev@fundp.ac.be.

zUniversity of Surrey. Mailing address: School of Economics, Guildford, Surrey, GU2 7XH, UK. Email:
e.jaimovich@surrey.ac.uk.

xCorresponding author: Paris School of Economics and CEPR. Mailing address: PSE, 48 Boulevard
Jourdan, 75014 Paris, France. Email: verdier@pse.ens.fr.

1



1 Introduction

One of the major recent phenomena in both developing and developed countries is the

rising importance of the provision of public goods through private non-pro�t organizations.

In the developing world, the non-governmental organizations (NGOs) play a key role in

provision of health and education services and are fundamental actors of empowerment

of socially disadvantages groups (such as women and minorities) and of monitoring the

adherence by �rms to environmental and labor standards. The number of international

NGOs wordwide increased from less than 5 000 in mid-1970s to more than 28 000 in 2013

(Union of International Associations 2014; see also Werker and Ahmed 2008).

Similarly, in the OECD countries, the role of non-pro�t organizations in providing public

goods is considerable, especially in health, arts, education, and poverty relief (see Bilodeau

and Steinberg 2006, Section 3.2, for a detailed analysis of the scope of the non-pro�t sector).

This sector has a large weight in terms of employment: on average, 7.5% of the economically

active population is employed in the non-pro�t sector, and for some countries (Belgium,

Netherlands, Canada, U.K., Ireland) this share exceeds 10% (Salamon 2010).

One distinctive feature concerning the provision of public goods by non-pro�ts and NGOs

is their �nancing structure: while a part of these organizations� operational cost is covered

by government grants and by user fees, voluntary private donations play a key role in the

�nancing of their budgets. Bilodeau and Steinberg (2006: 1285) report that, on average,

for the 32 countries for which comparable data on non-pro�ts is available, over 30% of their

�nancing comes from voluntary private giving. More than three-quarters of this amount

actually consists of small donations. Given the public-good nature of the services typically

provided by non-pro�ts, and the fact that small donors could hardly expect their contribu-

tions to have any meaningful e¤ect on total provision, this evidence suggests that private

contributions to non-pro�ts must be partially motivated by some strong form of impure

altruism.

Recent research inpublicandexperimental economicshas indeedshownthat rationalizing

empirical regularities about giving requires the acknowledgment of private psychological

bene�ts accruing to the donor from the act of giving. This is the so-called "warm-glow"

motivation, �rst modelled by Andreoni (1989). Using a panel of donations and government

funding from the U.S. to 125 international relief and development organizations, Ribar and

Wilhelm (2002) �nd that only the warm-glow motive is consistent with the observed absence

2



of crowding out of private donations to non-pro�ts.1 In laboratory experiments, Andreoni

and Miller (2002) and Korenok et al. (2013) �nd that voluntary giving tends to respond to

income variations in a way that is more in line with the warm-glow motive than with pure

altruism. A �eld experiment by Tonin and Vlassopoulos (2010) reaches analogous results.

The authors analyze the behavior of student workers who had to exert real e¤ort on a data

entry task, and �nd that in an environment that elicits warm glow altruism workers respond

by increasing e¤ort, while additionally eliciting pure altruism has no further e¤ect on e¤ort.

Non-pro�t organizations rely then heavily on private donors� contributions, which are to

a large extent the result of some intrinsic "joy of giving".2 The prevalence of impure altruism

by donors means that the link between the motivation to give to non-pro�t organizations and

the ultimate provision of public-goods by them is weak. In addition, the very nature of the

goods and services provided by these organizations is such that it is virtually impossible to

write contracts that condition payment or future donations on the output produced by these

entities (see Hansmann, 1996, Chapter 12; and Bilodeau and Slivinski, 2006a, Section 4.1,

for detailed discussions). These features, combined with the fact that individual producers�

intrinsic motivation is private information, imply that the non-pro�t sector is subject to

large scope for funds diversion and rent-seeking by the founders/managers of non-pro�ts.

There is someevidenceofopportunisticbehavior in thenon-pro�t sector. Onetypicalway

through which the society tries to limit the scope for funds diversion is the non-distribution

constraint, which entails that the organization cannot distribute pro�ts but must reinvest

them towards the ful�llment of its mission (Hansmann, 1996: 229-230). However, a clear

downside of this policy is that it lowers the incentives to cut costs. Moreover, given the

di¢culty to control how these costs are calculated, it often spurs in-kind diversion. For

instance, Smillie (1995: 151-153) describes how development-oriented non-pro�ts use in�ated

and hidden overheads to engage in the in-kind diversion of funds. Frumkin and Keating

(2001) analyze the non-pro�t executive pay patterns and conclude that "CEO compensation

is signi�cantly higher in non-pro�t organizations where free cash �ow is present". Malani

1In a recent paper, using an instrumental-variable approach, Andreoni and Payne (2011) document sub-
stantial crowding out of private giving to charities by government grants (about 75%). Virtually all of the
crowding out is caused by non-pro�ts strategically reducing fundraising (rather than donors responding to
grants by consciously decreasing their donations).

2In this paper, we mostly focus of the joy-of-giving (or warm-glow) motive for giving. However, an
additional reason why people might be willing to donate is social-signalling, as modelled by Benabou and
Tirole (2006). Social-signalling motivation would complement and reinforce the joy-of-giving motive that we
focus on in our model.

3



and Choi (2005) exploit the executive compensation data from 2700 nursing homes in the

U.S. and �nd that non-pro�t managers behave as if they cared about pro�ts as much as their

counterparts in for-pro�t �rms. Finally, Fisman and Hubbard (2005) �nd that non-pro�ts

in the U.S. states with weaker oversight have managerial compensation that is more highly

correlated with donation �ows and allocate a smaller percentage of donations to the �rm�s

endowment. One important implication of these facts is that the �nancing of the non-pro�t

sector might have an impact on the composition of the sector, particularly in terms of the

level of intrinsic motivation of its managers.3

Economists have so far analyzed separately the issues of non-contractability and poor

monitoring in the non-pro�t sector, sorting into mission-oriented organizations, and the op-

timal �nancing of non-pro�t organizations. However, we still lack a model that ties all these

key elements together within a tractable general equilibrium framework. Complementing

the previous literature with a general equilibrium analysis is crucial, given that the relative

size of the non-pro�t sector in numerous countries (both developing and developed) is large

enough to imply that policies that in�uence the behavior of non-pro�t managers and entry

into the sector might importantly a¤ect the returns in both the non-pro�t and for-pro�t sec-

tors. As a consequence, partial equilibrium approaches may lead to wrong policy conclusions

(for instance, concerning the desirability of more extensive state �nancing to non-pro�ts or

channeling foreign aid via NGOs).

This paper proposes a tractable occupational-choice model with for-pro�t �rms, non-

pro�t organizations and endogenous private donations. The model relies on four key as-

sumptions. First, private donors give to non-pro�ts essentially because of warm-glow motives

(i.e., with a weak link to the expected public-good output generated by the particular do-

nation). Second, individuals self-select either into the for-pro�t or non-pro�t sectors. Third,

monitoring the behavior and knowing the intrinsic motivation of the non-pro�t managers

is inherently di¢cult. Fourth (and also resulting from the non-measurability of non-pro�ts�

output), private donations to the non-pro�t sector are shared among the existing non-pro�ts

�rms in a manner that is not strictly related to their performance.

The model aims at addressing the following set of questions. What is the equilibrium

3There is substantial narrative evidence several developing countries that generous �nancing by foreign
aid, together with a strong new emphasis on decentralized development, has led to perverse e¤ects by
triggering opportunistic behavior and elite capture in these local NGO projects (see, e.g., Platteau and
Gaspart, 2003; Platteau, 2004; the contributions in Bierschenk et al., 2000; Gueneau and Leconte, 1998, for
Chad; and Bano, 2008, for Pakistan).

4



composition of the non-pro�t and for-pro�t sectors in terms agents� intrinsic motivation?

What are the implications of the external �nancing on the behavior of the non-pro�t sec-

tor? What types of policies can improve the motivational composition of the non-pro�t

sector? What happens when donations respond positively to a better perceived motivational

composition of the non-pro�t sector?

The main mechanism in our model rests on the notion that self-selection into either the

for-pro�t or non-pro�t sectors is altered by the level of donations received by non-pro�t

�rms in equilibrium. Imperfect monitoring of managers in the non-pro�t sector, together

with warm-glow motives by private donors, implies that the scope for rent-seeking in this

sector expands when private giving grows. We show that warm-glow altruism and self-

selection, in a context of asymmetric information about non-pro�ts managers� motivation,

interact sometimes in non-monotonic ways, leading in certain cases to ine¢cient equilibrium

outcomes and allocations. Our model generates the following �ve main results.

First, there exist cases in which rent-seeking motives crowd out altruistic motivation

from the non-pro�t sector. When this occurs, the non-pro�t sector ends up being managed

by intrinsically self-interested agents who exploit the lack of monitoring to divert funds

for their private use. Moreover, since the scope for rent-extraction rises with the level of

donations received by each non-pro�t �rm, this misallocation problem is exacerbated in

richer economies and in economies where private donors give more generously.

Second, foreign aid intermediation through the non-pro�t/NGO sector in a developing

country may entail perverse e¤ects: it may cause the economy to switch from an equilibrium

with a good allocation to one with a bad allocation of pro-social motivation. One implication

of this result is that, in our model, total output of the non-pro�t sector becomes a non-

monotonic function of the amount of foreign aid. At low levels of foreign aid, a small

increase in aid leads to higher total NGO output, since the allocation of motivation in the

non-pro�t sector remains intact, and the motivated managers can produce more with more

funds. However, with larger increases in foreign aid, as soon as the motivational composition

of the sector starts to change (because of the crowding out e¤ect), total non-pro�t output

declines. Such inverted U-shaped relation, in turn, can help explaining the micro-macro

paradox observed by empirical studies of aid e¤ectiveness (i.e. the absence of empirical

positive e¤ect of aid on output at the aggregate level, combined with numerous positive

�ndings at the micro level).

5



Third, if pro-socially motivated donors exhibit a higher propensity to give out of their

private income than unmotivated ones, the model exhibits multiple equilibria. In particular,

for intermediate ranges of private income, the model sustains two very di¤erent types of

equilibria. In one equilibrium there is a high level of pro-social motivation in the non-pro�t

sector, while in the other one the non-pro�t sector is fully managed by unmotivated agents.

The underlying reason for equilibria multiplicity is that when the private sector is rich in

altruistic motivation, a large amount of aggregate donations are given to the non-pro�t

sector, thereby expanding the scope for rent-extraction by non-pro�t managers. Conversely,

when the private sector rich in self-interested agents, only altruistic motivated agents end

up being attracted to the non-pro�t sector in dearth of private donations.

Fourth, if donors� warm-glow motivation somehow responds positively to the expected

productivity of the non-pro�t sector, the low-motivation equilibrium disappears. However,

our model shows that, even in these cases, when the amount of donations becomes su¢ciently

large, unmotivated agents will still end up constituting an important share of the pool of

non-pro�t managers, hurting thus the aggregate provision of public goods.

Finally, we show that a properly designed public �nancing policy of the non-pro�t sec-

tor may improve the motivational composition of the non-pro�t sector and eliminate the

low-motivation equilibrium. This occurs because taxation alters the occupational choice of

individuals in two ways: it reduces the returns in the private sector and increases the aggre-

gate transfers to the non-pro�t sector. In a partial equilibrium setup, both channels would

make the non-pro�t sector relatively more attractive for both motivated and unmotivated

agents. However, in our framework, the implicit general equilibrium re-allocations imply

that if public �nancing is able to increase the aggregate funding of the non-pro�t sector,

while at the same time it su¢ciently increases the number of non-pro�t managers so that

the funding that each non-pro�t �rm obtains from the aggregate pool is lower, this policy

will lead to entry of motivated and exit of unmotivated agents from the non-pro�t sector.

Besides the aforementioned papers by Andreoni (1989) and Benabou and Tirole (2006),

our paper relates to several other key papers that study pro-social motivation and non-pro�t

organizations: Lakdawalla and Philipson (1998), Glaeser and Shleifer (2001), François (2003,

2007), Besley and Ghatak (2005), and Aldashev and Verdier (2010). We contribute to this

line of research by endogenizing the occupational choice decision of individuals and exploring

the general equilibrium implications of the �nancing of the non-pro�t sector.

6



The second related strand of literature is the occupational choice models applied to the

selection into the public sector and politics (e.g., Caselli and Morelli, 2004; Macchiavello,

2008; Delfgaauw and Dur, 2010; Bond and Glode, 2012; Jaimovich and Rud, 2013). We

extend this line of ongoing research by analyzing how the selection mechanisms apply to the

non-pro�t/NGO sector within a context of endogenous voluntary donations.

Finally, there is growing theoretical literature that studies the e¤ects of the modes and

level of foreign aid �nancing on its e¤ectiveness (see, for example, the survey in Bourguignon

and Platteau, 2013a). Among these studies, an early paper by Svensson (2000) underlines

howshort-termincreases inaid�ows maytrigger rent-seeking "wars" amongcompeting elites

in a developing country. Another interesting contribution is a recent paper by Bourguignon

and Platteau (2013b), which concentrates on moral hazard issues related to the increasing

amounts of foreign aid (in particular, the e¤ect of domestic monitoring on the ultimate use

of aid �ows). Our model studies a separate and novel channel, previously unaddressed by the

foreignaid literature: thatofmotivationaladverse selection into the sector that intermediates

foreign aid �ows between outside donors and bene�ciaries.

The rest of the paper is organized as follows. Section 2 builds our baseline model of

occupational choice in the for-pro�tandnon-pro�t sectors; it also introducesandanalyzes the

e¤ects of foreign aid and public �nancing on the e¢ciency of the non-pro�t sector. Section 3

provides an alternative setup with endogenous fundraising e¤ort by non-pro�t organizations,

and shows that our main results remain essentially intact. Section 4 presents two further

key extensions of the baseline model: allowing the donations by private entrepreneurs to be

related to their degree of altruism, and letting donations depend positively on the expected

output of the non-pro�t sector. Section 5 discusses the main premises and modelling choices,

as well as the generalizability of our results to relaxing these assumptions. Section 6 discusses

the main applications of our model, explores several avenues for future work, and concludes.

2 Basic model

We consider an economy populated by a continuum of individuals with unit mass. There

exist two occupational choices available to each agent: she may become either a private

entrepreneur in the for-pro�t sector or a social entrepreneur by founding a �rm in the non-

pro�t sector. Henceforth, we will refer to the two types of �rms as private and non-pro�t

�rms, respectively. For simplicity, we assume that each entrepreneur founds and manages

7



only one �rm. Let N denote the total mass of non-pro�t managers.

All agents are identically skilled. However, they di¤er in their level of pro-social motiva-

tion, denoted by mi. There exist two levels of mi, which we refer to henceforth as types: mH

("motivated") and mL ("unmotivated"), where mH > mL. The type mi is private informa-

tion. For simplicity, we will focus only on the extreme case in which mH = 1 and mL = 0.

In addition, we assume the population is equally split between mH- and mL-types.

2.1 For-pro�t sector

Each private entrepreneur produces an identical amount of output. There are decreasing

returns in the private sector, thus while the aggregate output is increasing in the mass of

private entrepreneurs, 1�N, the output produced by each private entrepreneur is decreasing
in 1�N. More precisely, we assume that each private entrepreneur produces

y =
A

(1�N)1��
, where 0 < � < 1 and A > 0: (1)

Aggregate output is thus given by Y = A(1�N)�. This assumption of decreasing average
output can be justi�ed if, for instance, each �rm is built around some marketable product

idea, and the most productive ideas are discovered �rst; so as the number of private �rms

increases, each additional �rm is built around an ever less productive idea.

Private-sector entrepreneurs derive utility from their consumption of the private good

(c). In addition, they also enjoy warm-glow utility from donating to the non-pro�t sector

(d). In particular, we assume all entrepreneurs have the same Cobb-Douglas type utility

function:4

VP(c;d) = c
1��d�

1

� (1� �)1��
, where 0 < � < 1: (2)

Private-sector entrepreneurs maximize (2) subject to (1). The solution of the maximiza-

tion problem yields c� = (1� �)y and d� = �y, which in turn implies that, at the optimum,
their indirect utility function is

V �P = y: (3)

From the optimization problem of private-sector entrepreneurs, it follows that the total

amount of entrepreneurial donations to the non-pro�t sector is

D = � (1�N)� A: (4)
4In Section 4.1 we relax the assumption that warm-glow donations by private entrepreneurs are indepen-

dent of their level of pro-social motivation by letting � be type-speci�c (�i), with �L = 0 and 0 < �H � 1.

8



As can be readily observed from (4) the total amount of donations increases with the pro-

ductivity of the private sector (A), the number of private �rms (1�N), and the parameter
determining the marginal utility of warm-glow giving (�).

2.2 Non-pro�t sector

The non-pro�t sector is composed by a continuum of non-pro�t �rms with total mass N.

Each non-pro�t �rm is run by a social entrepreneur. We think of each single non-pro�t

�rm as a mission-oriented organization (as, for instance, in the seminal paper by Besley and

Ghatak, 2005) with a narrow mission targeting one particular social problem (e.g., child

malnutrition, air pollution, �ghting malaria, saving whales, etc.).

Each non-pro�t manager i collects an amount of donations �i from the aggregate pool

of donations D. Part of the collected donations �i is used to pay the wage of the non-pro�t

manager wi, while the rest (the undistributed donations) is used as input for the production

of the service towards the organization�s mission. We measure the e¤ectiveness (output)

of each speci�c non-pro�t �rm by gi, which is a function of the undistributed donations

(�i � wi). We assume that the output generated by each speci�c non-pro�t �rm exhibits
decreasing returns with respect to the funds invested into the project, namely:

gi = (�i �wi)
, where 0 < 
 < 1: (5)

A non-pro�t manager derives utility from her own consumption (which equals her wage)

and from her contribution to the solution of the social problem targeted by her organization�s

mission (which is equal to gi). The weight placed oneachof two components of utility is given

by the non-pro�t manager�s level of pro-social motivation mi. More precisely, we assume

that the utility function of a non-pro�t manager with motivation mi is:

Ui(wi;gi) = w
1�mi
i g

mi
i

1

mmii (1�mi)1�mi
, where mi 2 fmH;mLg: (6)

In line with the evidence discussed in the Introduction, we assume that the non-pro�t

sector su¤ers from poor monitoring by donors. For simplicity, we take the extreme assump-

tion that non-pro�t managers enjoy full discretion in setting their own wage (subject to the

feasibility constraint wi � �i). In addition, we assume that the pool of total donations D
is equally shared by all non-pro�t �rms.5 Then, donations collected by each non-pro�t �rm

5In Section 3 we relax this equal-sharing assumption by explicitly modelling fundraising e¤ort by non-
pro�t managers.

9



are:

�i =
D

N
=
�A(1�N)�

N
:

Notice that �i is decreasing in N through two distinct channels: �rstly, because total do-

nations D decrease when the mass of private entrepreneurs (1 � N) is smaller; secondly,
because a rise in the mass of non-pro�t �rms N means that a given total pool of donations

D must be split among a larger mass of non-pro�t �rms.

Given that mH = 1, motivated non-pro�t managers place all the weight in their utility

function on g, and set accordingly w�H = 0. As a result, choosing to become a non-pro�t

manager gives to a motivated agent the indirect utility equal to

U�H =

�
D

N

�
=

�
�A
(1�N)�
N

�
: (7)

Analogously, given that mL = 0, unmotivated non-pro�t managers disregard contributing

to their organizations� mission, and convert all the donations into their wages, w�L = �i. This

implies that choosing to become a non-pro�t manager gives to an unmotivated agent the level

of utility

U�L =
D

N
= �A

(1�N)�
N

: (8)

We can now state the following:

Lemma 1 Let bN denote the level of N at which D( bN) = bN. Then,

U�H R U
�

L if and only if N R bN;

where: (i) �A=(1 + �A) < bN < 1, (ii) bN is strictly increasing in A and � and strictly
decreasing in �, (iii) lim

A!1

bN = 1, (iv) lim
�!0

bN = �A and lim
�!1

bN = �A=(1+ �A).

Proof. U�H R U
�

L i¤ N R bN follows immediately from the expressions in (7) and (8).
The rest of the results follow from noting that �A(1 � bN)�= bN = 1, and di¤erentiating this
expression.

Lemma 1 is a single-crossing result useful for our analysis. It states that a motivated

individual obtains higher utility from becoming a non-pro�t manager, as compared to a

unmotivated individual making the same choice, only when donations per non-pro�t are

small enough, i.e. D=N < 1. Both U�H and U
�

L are strictly increasing in donations per

non-pro�t, D=N. However, when level of donations received by each non-pro�t rises above

10



the threshold level (which here is equal to 1), U�L surpasses U
�

H. The reason for this result

essentially rests on the concavity of gi in (5), combined with the altruism displayed by

motivated non-pro�t managers in (6). These two features translate into a payo¤ function of

motivated non-pro�t managers, U�H, that is concave in D=N. On the contrary, unmotivated

non-pro�t managers exhibit a payo¤ function, U�L, which is linear in D=N. This is because

these agents only care about their private consumption, and hence they exploit the lack of

monitoring in the NGO sector in order to always set wi = D=N.
6

2.3 Equilibrium occupational choice

Let NH and NL denote henceforth the mass of non-pro�t managers of mH- and mL-type,

respectively (the total mass of non-pro�t managers is then N = NH + NL). In equilibrium,

the following two conditions must be simultaneously satis�ed:

1. Given the values of NH and NL, each individual chooses the occupation that yields the

higher level of utility, with some agents possibly indi¤erent between the two occupa-

tions.

2. The allocation (NH;NL) must be feasible: (NH;NL) 2
�
0; 1

2

�
�
�
0; 1

2

�
:

In this basic speci�cation of the model, for a given parametric con�guration, the equi-

librium occupational choice will always be unique (except for one knife-edge case described

in the footnote below). Still, the type of agents (in terms of their pro-social motivation)

who self-select into the non-pro�t sector will depend in an interesting manner on the speci�c

parametric con�guration of the model. In what follows, we describe the main features of the

two broad kinds of equilibria that may take place: an equilibrium where 0 = NH < NL = N

(which we refer to as �dishonest equilibrium�), and an equilibrium where 0 = NL < NH = N

(which we dub as �honest equilibrium�).7

6The result in Lemma 1 does not crucially depend on the extreme assumption that mH = 1, and easily
extends to any situation in which 0 = mL < mH = m � 1: In that case, the mH�type sets w�i =
�i (1�m)=(1�m+
m), which in turn implies that at the optimum

U�m =


m

mm(1�
) (1�m+
m)1�m(1�
)
�
D

N

�1�m(1�
)
= �(m;
)

�
D

N

�1�m(1�
)
:

Therefore, noting that, for any vector (m;
) 2 (0;1]� (0;1), the function �(m;
) satis�es 1 � �(�) � 2, it
follows that whenever D=N R [�(�)]1=m(1�
), then U�L R U�m.

7The above-mentioned two cases exclude the set of parametric con�gurations for which bN = N0, where
N0 is de�ned below in (9). When bN = N0, all individuals in the economy will be indi¤erent in equilibrium

11



Dishonest equilibrium

An equilibrium in which the non-pro�t sector is populated exclusively by unmotivated in-

dividuals arises when all motivated individuals prefer to found private �rms, whereas all

unmotivated ones (weakly) prefer to become social entrepreneurs:

U�H(N) < V
�

P(N) � U�L(N);

where V �P(N) is given by (3), U
�

H(N) by (7), U
�

L(N) by (8), and N = NL � 1=2.
Lemma 1 implies that for U�H(N) < U

�

L(N) to hold the non-pro�t sector should be

su¢ciently small (i.e., N < bN), so that the level of donations received by each non-pro�t
�rm turn out to be su¢ciently high. In addition, the condition V �P(N) � U�L(N) leads to:

N � N0 �
�

1+ �
: (9)

From (9) we may observe that N0 < 1=2. As a result, in a �dishonest equilibrium� it must

necessarily be the case that N = NL = N0, so that the unmotivated agents turn out to be

indi¤erent between the for-pro�t and non-pro�t sectors. Indi¤erence by mL-types leads a

mass 1=2 � N0 of them to become private entrepreneurs, allowing thus "markets" to clear.
Notice, �nally, that U�H(N0) < V

�

P(N0) needs to be satis�ed, hence the crucial parametric

condition leading to a �dishonest equilibrium� boils down to N0 < bN.

Honest equilibrium

This type of equilibrium takes place when all unmotivated individuals prefer to found private

�rms, whereas all motivated ones prefer (weakly) to be social entrepreneurs: U�L(N) <

V �P(N) � U�H(N);where N = NH � 1=2.
Lemma 1 states that for U�H(N) > U

�

L(N) to hold, the non-pro�t sector should be su¢-

ciently large in size: N > bN. The condition U�L(N) < V �P(N) requires that N > N0 (this is
because the unmotivated agents prefer to stay out of the non-pro�t sector when the size of

this sector is too large, as the available rents per manager are too low in that case). Unlike

in the previous case, in the �honest equilibrium� one cannot rule out the possibility of full

sectorial specialization of the two motivational types of agents (i.e., in principle, an �honest

equilibrium� may well feature NL = 0 and NH = 1=2).

across the two available occupations. Moreover, because of that, there is actually equilibrium multiplicity,
and the set equilibria is given by

�
N�H +N

�
L = N0; j0 � N�H � 12;0 � N

�
L � 12

	
. Hereafter, for the sake of

brevity, we skip this knife-edge case:

12



For future reference, we denote with N1 the value of N that makes mH-types indi¤erent

between occupations. From (1) and (7) we observe that:

(1�N1)
1��(1�
)



N1
� A

1�


�
: (10)

Equilibrium characterization

The following proposition characterizes the di¤erent kinds of equilibria that may arise, given

the speci�c parametric con�guration of the model.

Proposition 1 Whenever A(1+ �)
1�� 6= 1, the equilibrium occupational allocation (N�H;N�L)

is unique. The type of agents who manage the non-pro�t sector is determined solely by

whether A(1+ �)
1��

is strictly larger or smaller than one:

1. If A(1+ �)
1��

> 1, in equilibrium there is a mass N� = N�L = N0 of non-pro�t �rms,

all managed by mL-types. The mass of private entrepreneurs equals 1 � N0; a mass 12
of them are motivated, the remaining 1

2
�N0 are unmotivated.

2. If A(1+ �)
1��

< 1, in equilibrium there is a mass N� = N�H = min
�
N1;

1
2

	
of non-

pro�t �rms, all managed by mH-types. Moreover, if N
�

H = N1 (respectively, N
�

H =
1
2
);

the mass of private entrepreneurs equals 1�N1 (respectively, 12). When N
�

H = N1, the

mass of private entrepreneurs consists of a mass 1
2
of unmotivated individual and a

mass 1
2
� N1 of motivated ones. Instead, when N�H = 12, all private entrepreneurs are

unmotivated.

Proof. See Appendix A.

Proposition 1 characterizes the three main types of equilibria that may arise in the model,

depending on the speci�c parametric con�gurations. These three cases are depicted in Figure

1, panels A, B, and C, respectively. This �gure portrays the indirect utilities of motivated

and unmotivated agents in the non-pro�t sector (UH and UL, respectively) and of individuals

in the private sector (y), as a function of the size of the non-pro�t sector, N.

[Insert Figure 1 about here]

13



Consider Figure 1A, and suppose that the non-pro�t sector is initially of size zero. This

situation is not an equilibrium, since both UH and UL lie above y when N = 0, and the

utility di¤erential would attract both types of agents into the non-pro�t sector. As the size

of the non-pro�t sector grows, the utility di¤erential shrinks for both types of agents. At

the intersection of the UH-curve with y-curve, the motivated types are indi¤erent between

the two sectors, but the mL-types still prefer the non-pro�t sector. Therefore, the non-pro�t

sector must still grow further. The equilibrium is only reached when the size of the non-pro�t

sector equals N0, at which point the unmotivated agents are indi¤erent between the pro�t

and non-pro�t sectors while all motivated agents prefer the private sector.

The situation plotted in Figure 1B is analogous, except that the utility di¤erential for

unmotivated types vanishes earlier than it does for the motivated ones, which serves as the

basis for the honest equilibrium (with non-pro�t sector size equal to N1). Finally, in the case

depicted in Figure 1C, once all the existing motivated agents have entered the non-pro�t

sector, the utility di¤erential is positive even for the last entrant. The size of the non-pro�t

sector is thus equal to 1
2
and is rationed by the number of mH-agents.

An interesting implication of Proposition 1 is that more productive economies (i.e., those

with a relatively large A) tend to exhibit a �dishonest equilibrium�. This result rests on the

fact that a larger A entails greater pro�ts to private entrepreneurs. Hence, in equilibrium,

a larger amount of donations to any non-pro�t �rm (�i) are needed to compensate for the

higher opportunity cost of managing a non-pro�t �rm (i.e., the fact of not becoming a

private entrepreneur). This result has interesting implications for an initially poor economy

on a positive growth path. As the productivity parameter A increases from an initial level

below 1=(1+ �)
1��
, some non-pro�t mH-type managers start leaving the non-pro�t sector to

found their private �rms, which are becoming increasingly pro�table. Importantly, while this

process takes place, the level of donations received by each of the remaining non-pro�ts �rms

will also grow. As private productivity keeps rising over time, A will eventually surpass the

threshold 1=(1+ �)
1��
, and the economy will experience a radical transformation in their

non-pro�t sector: all motivated managers leave the non-pro�t sector to found a private �rm,

while a mass N0 of unmotivated agents leave the private sector to found non-pro�t �rms.

A similar intuition applies to the e¤ect of a higher warm-glow utility from giving; that is,

a greater �.8 This yields a larger amount of total donations, D, for a given mass of non-pro�ts

8A rise in � could be caused, for instance, by the e¤ects of stronger social norms of giving, or by an
increase in the social prestige associated with observable giving by private-sector managers.

14



N, making the non-pro�t sector relatively more attractive to unmotivated agents than to

motivated ones. Again, beyond some threshold of �, this in turn will lead to a reshu­ing of

the motivational composition of the non-pro�t sector, analogous to the one described just

above for an increase in A.

In terms of policy implications, the results obtained above imply that the value-added of

betteraccountability for theperformanceof thenon-pro�t sector increaseswiththeaggregate

generosity in the economy. In other words, donations and accountability are complementary

inputs in the aggregate production function of the non-pro�t sector. A larger D �either

resulting from higher aggregate private income or larger �� corresponds to an increase of

only one input into the aggregate production function of the non-pro�t sector. However,

such a rise in D, without an accompanying increase in the other input (i.e., accountability),

may turn out to be actually negative for the functioning of the non-pro�t sector.

2.4 E¤ect of foreign aid on the equilibrium allocation

So far, all donations in our model were generated (endogenously) within the economy. How-

ever, foreign aid is also a crucial source of revenue for non-pro�ts organizations and NGOs in

many developing countries. In fact, a growing share of foreign aid is being channeled through

the NGOs. For instance, data from the United States shows that over 40 per cent of U.S.

overseas development funds �ows through NGOs (Barro and McCleary 2006). International

aid agencies as well have been increasingly preferring NGOs to public-sector channels: e.g.,

whereas between 1973 and 1988, a tiny 6 per cent of World Bank projects went through

NGOs, by 1994 this share exceeded 50 per cent (Hudock 1999). As Kanbur (2006) argues,

the rise of NGOs during the 1980s was one of the key changes in the functioning of the

foreign aid sector.

What would be the e¤ect of a rise in foreign aid on the motivational composition and

performance of the non-pro�t sector of the recipient economy? In this subsection, we ap-

proach this question. To do so, we slightly modify the previous model to allow an injection

of amount � > 0 of foreign aid (outside donations) into the economy.

Foreign aid represents an exogenous increase in the total amount of donations available

to the national non-pro�t sector. Donations collected by a non-pro�t �rm now become:

D

N
=
�A(1�N)� +�

N
: (11)

Asdoneabove inLemma1, we�rstpindownthethreshold bN suchthat, forallN > bN the

15



utility obtained by unmotivated non-pro�t managers dominates that obtained by motivated

non-pro�t managers.

Lemma 2 (i) Whenever 0 � � � 1, there exists a threshold bN � 1 such that U�H(N) R
U�L(N) i¤ N R bN; the threshold bN is strictly increasing in �, and lim�!1 bN = 1. (ii)
Whenever � > 1, U�H(N) < U

�

L(N) for all 0 < N � 1:

Proof. The �rst part follows from noting that bN must solve the following equality: � =
bN ��A(1� bN)� � �( bN), where �0( bN) > 0 , hence @ bN=@� > 0: Also, given that �0( bN) > 0
and �(1) = 1, it follows that, for any 0 � � � 1, the solution of �( bN) = � must necessarily
satisfy bN � 1. The second part follows directly from observing that when � > 1, the
right-hand side of (11) is strictly greater than unity for all 0 < N � 1.

The �rst result in Lemma 2 essentially says that the set of values of N for which the

inequality U�H(N) < U
�

L(N) holds �which is given by the interval (0;
bN)� expands as the

amount of foreign aid � increases. The second result states that when foreign aid is su¢-

ciently large, the dominance relation U�H(N) < U
�

L(N) becomes valid for any feasible value

of N.

The injection of foreign aid thus enlarges the set of parameters under which the economy

featuresanequilibriumwithunmotivatednon-pro�tmanagers (�dishonest equilibrium�). The

proposition below formalizes this perverse e¤ect of foreign aid. For brevity, we restrict the

analysis only to the more interesting case, in which A(1+ �)
1��

< 1.

It is useful to denote by N the level of N for which y(N) in (1) equals one; that is,

N � 1�A
1

1�� : (12)

In addition, in order to disregard situations in which N � 0 fails to exist, we henceforth set
the following upper-bound on A:

Assumption 1 A � 1:

Note that if A > 1, then the condition A(1+ �)
1��

< 1 for an �honest equilibrium� in

Proposition 1 could never hold, and the model would always deliver �by construction� a

�dishonest equilibrium�.9

9Another way to avoid the problem of obtaining a �dishonest equilibrium� by construction is to assume
that the production function of private entrepreneurs is given by y(N), with y0(N) > 0; y00(N) < 0, y(1) = 1
and y(0) = 0. Notice that all these properties are satis�ed by (1), except for y(0) = 0, which in (1) is actually
y(0) = A. Intuitively, what is needed to give room for an �honest equilibrium� is that y(N) � 1 for some
N � 0. Assumption 1 ensures this is always the case.

16



Proposition 2 Let A(1+ �)
1��

< 1 so that when � = 0 the economy features an �honest

equilibrium�. Let also �0 � 1�A
1

1��(1+ �); and note bN = N when � = �0:

1. If 21��A > 1, there exist two thresholds, �A > �0 > 0, such that:

(a) When 0 � � < �0, all non-pro�t �rms are managed by mH-types: 0 < N�H < 12
and N�L = 0, where N

�

H is strictly increasing in �.

(b) When �0 < � � �A, all non-pro�t �rms are managed by mL-types: 0 < N�L � 12
and N�H = 0, where N

�

L is strictly increasing in � whenever N
�

L <
1
2
.

(c) When � > �A, non-pro�t �rms are managed by a mix of types with mL-type

majority, namely: N�L =
1
2
and 0 < N�H <

1
2
, where N�H is strictly increasing in �

and lim�!�A N
�

H = 0.

2. If 21��A < 1, there are two thresholds, �0 > �B > 0, such that:

(a) When 0 � � � �B, all non-pro�t �rms are managed by mH-types: 0 < N�H � 12
and N�L = 0, where N

�

H is strictly increasing in � whenever N
�

H <
1
2
.

(b) When �B < � � �0, non-pro�t �rms are managed by a mix of types with mH-
type majority, namely: N�H =

1
2
and 0 < N�L <

1
2
, where N�L is strictly increasing

in � and lim�!�B N
�

L = 0.

(c) When � > �0, non-pro�t �rms are managed by a mix of types with mL-type

majority, namely: N�L =
1
2
and 0 < N�H <

1
2
, where N�H is strictly increasing in �

and lim�!�0 N
�

H = N � 12 > 0.

Proof. See Appendix A.

Proposition 2 describes the e¤ects of changes in the amount of foreign aid � on the

equilibrium allocation of an economy which, in the absence of any foreign donations, would

display an �honest equilibrium�. The most interesting results arise when A(1+ �)
1��

< 1 <

21��A. In this case, when foreign aid is not too large (0 � � < �0), the non-pro�t sector
continues to be managed only by motivated agents. However, when the level of donations

surpasses the threshold �0, unmotivated agents start being attracted into the non-pro�t

sector due to the greater scope for rent extraction. Interestingly, for any �0 < � � �A,

17



the economy experiences a complete reversal in the equilibrium occupational choice: all mH-

types choose the private sector, while the non-pro�t sector becomes entirely managed by

mL-types. Finally, when � > �A, foreign aid becomes so large that the non-pro�t sector

starts attracting back some of the mH-types in order to equalize the returns of motivated

agents in the for-pro�t and non-pro�t sectors. Notice, however, that when � > �A the mass

of non-pro�ts run by unmotivated agents is still larger than the mass of non-pro�ts managed

by mH-types.

[Insert Figure 2 about here]

Figure 2 depicts the above-mentioned results when A(1+ �)
1��

< 1 < 21��A. The

solid lines represent U�H(N) and U
�

L(N) when � = 0 , the dashed lines shows non-pro�t

managers� payo¤s when �0 < � � �A, and the dotted lines plots those payo¤s when
� > �A. A gradual injection of foreign aid from � = 0 to � = �0 initially has no e¤ect

on the motivational composition of the non-pro�t sector, given that the utility di¤erential

between the two sectors remains negative for the unmotivated types. Beyond the amount of

aid � = �0, this utility di¤erential becomes positive for the unmotivated types, whereas it

turns negative for the motivated ones. At that point, the motivational composition of the

non-pro�t sector is completely reversed. Further increases in foreign aid have no e¤ect on the

non-pro�t sector�s output, up to the point � = �A. There, all the unmotivated agents have

moved into the non-pro�t sector and thus its size equals 1
2
. From then on, further injections

of aid (beyond �A) start to attract back some motivated agents into the non-pro�t sector,

and the motivational composition of the sector therefore improves.

A key corollary that stems from Proposition 2 refers to the total output of the non-pro�t

sector, G, at di¤erent values of �. Bearing in mind that only motivated non-pro�t managers

use donations to produce the mission-oriented output gi, an implication of Proposition 2 is

thatG(�) is non-monotonic in�. Inparticular, in thecasewhereA(1+ �)
1��

< 1 < 21��A,

non-pro�t output grows initially with the amount of foreign aid, up to the level when � = �0

when it reaches G(�0) = N; this is the enhancing e¤ect of foreign donations when the non-

pro�ts are managed by motivated managers. However, for �0 < � � �A, the motivation in
the non-pro�t sector gets completely "polluted" by the presence of unmotivated managers,

and G(�) drops suddenly to zero. Finally, when foreign donations rise beyond �A, non-

pro�t output begins to grow again (starting o¤ from G = 0), as some of the donations will

18



end up in the hands of mH-types. This non-monotonicity is illustrated by Figure 3, which

depicts the total output of the non-pro�t sector as a function of foreign aid in�ow.

[Insert Figure 3 about here]

Similar results are obtained when 21��A < 1. In this case, G(�) increases monotonically

with � for all � < �0, reaching G(�0) =
1
2
. However, as soon as � rises above �0, aggregate

non-pro�t output falls discretely to N � 1
2
. Thereafter, for all � > �0, G(�) grows again

monotonically with �, starting from G(�) = N � 1
2
.

Our analysis con�rms some of the concerns raised by critiques of foreign aid, by pointing

out at one precise mechanism through which the negative e¤ect of aid operates: the encour-

agement of unmotivated agents to replace motivated ones in the NGO sector. For instance,

Dambisa Moyo writes in her book entitled Dead Aid (Moyo 2009):

"Donors, development agencies and policymakers have, by and large, chosen

to ignore the blatant alarm signals, and have continued to pursue the aid-based

model even when it had become apparent that aid, under whatever guise, is

not working... Foreign aid does not strengthen social capital - it weakens it.

By [...] encouraging rent-seeking behavior, siphoning o¤ scarce talent from the

employment pool [...] aid guarantees that in most aid-dependent regimes social

capital remains weak and the countries themselves poor" (pp. 27, 59)

Note that our mechanism is quite di¤erent from the several arguments previously raised

concerning the perverse e¤ects of foreign aid on the functioning of the public sector (for

example, due to higher corruption, break-up of accountability mechanisms of elected o¢-

cials, triggering ethnic-based rent-seeking; see Svensson 2000). Our model shows that even

when foreign aid is channeled through the NGO sector (therefore, by-passing the public bu-

reaucracy) perverse e¤ects might still arise, since more massive aid in�ows may lead to a

worsening of motivational composition of the NGO sector in the recipient country.

In that regard, our results also help shedding light on the so-called micro-macro paradox

found in the empirical foreign aid literature (Mosley 1986). This paradox refers to the

fact that, at the microeconomic level, there are numerous studies that �nd the positive

e¤ect of foreign-aid �nanced projects on measures of welfare of bene�ciaries, while at the

aggregate level most studies actually fail to �nd a signi�cant positive e¤ect. Our model

19



explains this paradox as follows: when aid in�ows are small (or, alternatively, when you hold

the motivational composition of the NGO sector constant) the general equilibrium e¤ect

described in our model becomes negligible (or, alternatively, disappears altogether). Under

such circumstances, empirically, one �nds a positive e¤ect of aid projects. However, when aid

in�ows are su¢ciently large (e.g. when the well-functioning micro-level projects are scaled

up), the general equilibrium e¤ects kick in, and the motivational adverse selection e¤ect may

neutralize the positive e¤ect found at the micro level.

2.5 Taxes and public �nancing of non-pro�ts

In most economies, an important part of non-pro�ts� revenues comes from public grants

�nanced by taxes. This raises two questions: What is the e¤ect of partial public �nancing

on the motivational composition and size of the non-pro�t sector? Can public �nancing

generate an improvement on the composition of the non-pro�t sector, as compared to the

decentralized equilibrium, and if so, how should such �nancing be designed? In this section,

we address these questions by adding a set of public policy variables into our basic model.

Let the government impose a proportional tax on income in the for-pro�t sector and use

its proceeds as (unconditional) grants to non-pro�ts. Thus, the payo¤s of individuals in the

private sector becomes:

V �P = (1� t)y; (13)

where y is as stated in (1). The level of donations collected by each non-pro�t in this case

are given by:

�i =
D

N
=

private donationsz }| {
� (1� t)(1�N)y +

public grantz }| {
t(1�N)y

N
: (14)

Public �nancing via such a tax/grant system alters occupational choices of individuals

via two distinct channels. On the one hand, we can see in (13) that taxation lowers returns

in the private sector. On the other hand, as the public sector donates back all the taxes

it collects while the private sector only gives a fraction � of its net income, �i in (14)

increases with the tax rate t. Both channels, ceteris paribus, turn the non-pro�t sector

more attractive to all individuals. However, within our general equilibrium framework, the

key issue is whether public �nancing increases the attractiveness of the non-pro�t sector

relatively more for altruistic or for self-interested individuals.

20



To study the more interesting case, let us focus on a setting where our basic economy

(without public �nancing) would give rise to a �dishonest equilibrium�: A(1+ �)
1��

> 1:

[Insert Figure 4 about here]

Consider now an increase in taxes, with the transfer of all the proceeds to nonpro�ts as

grants. For such policy to induce a motivational improvement in the nonpro�t sector, it is

crucial that, in the new equilibrium (after taxes), the dishonest individuals who were initially

managing the nonpro�t sector switch occupations and move to the private sector. This will

occur only if the policy attracts enough motivated agents from the private sector into the

nonpro�t sector, such that this entry su¢ciently dilutes the amount of funds per non-pro�t

�rm, even after taking into account the larger total funding of the non-pro�t sector as a

whole. The proposition below formally proves that such a tax/grant policy exists.

Proposition 3 For A(1+ �)
1��

= 1 + �, where 0 < � < �, there exist a feasible range of

tax rates [t;t], where t > 0 and t � (1 � �)=(2 � �), such that when t 2 [t;t] an �honest
equilibrium� arises.

Figure 4 (Panel A) plots the equilibrium regions for di¤erent combinations of values of

A and t (see Appendix A for the derivation of the equilibrium regions). There are four

di¤erent regions. For combinations of relatively low values of A and t, the model features an

�honest equilibrium� where the non-pro�t sector is fully managed by motivated agents. On

the other hand, given a certain level of t, for su¢ciently high levels of A we have a �dishonest

equilibrium�. Notice that when t = 0, the boundary between these two regions is given by

A = 1=(1 + �)1��, as previously stated in Proposition 1. In addition, with public �nancing,

two new equilibrium regions arise: one with a mixed-type equilibrium with a fraction of

motivated agents in the non-pro�t sector larger than one half (f > 0:5), and one with a

mixed-type equilibrium with f < 0:5. These two types of equilibria occur when the tax rate

is su¢ciently large, while the former also requires that A is su¢ciently small and the latter

that A takes intermediate values.

A crucial feature of Figure 4 is that threshold of A splitting the �honest� and �dishonest�

equilibrium regions is increasing in t (up to the point in which t = t). As a consequence,

there exist situations in which introducing public funding of non-pro�ts via (higher) taxes on

private incomes can make the economy switch from a �dishonest� to an �honest� equilibrium.

21



This is depicted in Figure 4 (Panel B) by the dashed line arrow. This result rests on a subtle

general equilibrium interaction. Consider an economy with no taxes that is on the �dishonest

equilibrium� region, located, for example, at point Z. At Z, all mH-types prefer the private

sector, while mL-types are indi¤erent between the non-pro�t sector.
10 Since a higher tax

rate makes the non-pro�t sector more attractive, by su¢ciently raising t we can make all

mH-types prefer non-pro�t sector as well. However, when all motivated agents switch to

the non-pro�t sector, the value of N will rise and the returns in this sector will accordingly

decrease. When t lies within the interval [t;t], the new equilibrium allocation induced by the

increase in t leads to an increase in total funding of the non-pro�t sector but reducing the

value of per-organization funding (�i) strongly enough such that only motivated agents are

attracted to the non-pro�t sector.11

It is important to note that this motivational reshu­ing will not occur if the public

�nancing policy is too small. In particular, a mild increase in taxes will actually make

things even worse for the non-pro�t sector, as this would only raise the total funding of

the non-pro�t sector without altering its motivational composition. Graphically, this would

corresponds to any increase in taxes below the level t in Figure 4 (Panel B).

A well-designed tax/grant public policy will then to increase the variety (number) of

non-pro�t �rms enough so as to simultaneously reduce the per-nonpro�t �nancing (made

of voluntary donations and the grant). What are the implications of this insight for public

policies towards the non-pro�t sector? In our setting, exactly like the donors, the policy-

maker is subject to the same asymmetric information regarding the motivational type of

each speci�c agent. However, the policy-maker can change the relative returns in the two

sectors so as to induce the motivational "cleansing" of the nonpro�t sector by scaling-up

funding through expanding the extensive margin (i.e., inducing a greater number of non-

pro�t organizations), while simultaneously shrinking the intensive margin (i.e., reducing

per-organization funding level). In other words, in our setting "small is indeed beautiful":

starting from a dishonest equilibrium, the policy-maker should make sure that the funding

received by each non-pro�t �rms decreases. In our general equilibrium framework, this is

10Hence, in the equilibrium at Z, a part of the mL-types will choose the private sector and the other part
will found non-pro�t �rms.

11Notice thatall this implies that, in thenewequilibrium, the totalmassofnon-pro�t�rmsmustnecessarily
be larger than in Z, since from (14) it follows that �i will grow with t for a given level of N. In other words,
after t is raised to a level within [t;t], a mass N�L of unmotivated non-pro�t managers will be replaced by a
mass N�H of motivated non-pro�t managers, where N

�
H > N

�
L.

22



achieved by inducing a massive entry of new non-pro�t managers.

In terms of actual implementation, our result imply that it may be advisable to give

starting grants to new nonpro�ts, possibly even at the expense of cutting the �nancing

to the existing larger ones. For instance, consider the recent proposals to do "philanthropy

through privatization" (see Salamon 2013), which consists in returning part of proceeds from

the privatization of public sector assets to foundations and charities. Our analysis suggests

that this policy would work correctly only if the way these proceeds are used is such that

they are scattered through a multitude of small organizations, rather than concentrating

them on a few large nonpro�ts: the latter risks worsening the motivational composition of

the sector by attracting unmotivated agents, whereas the former ensures that the returns in

the non-pro�t sector remain low enough to attract only highly motivated managers.

3 Endogenous fundraising e¤ort

In the basic model in Section 2, we have assumed that total donations are split (quite me-

chanically) between all non-pro�t �rms. It is well known, however, that non-pro�ts compete

for donations and engage actively in fundraising. For instance, in his analysis of the hu-

manitarian relief NGOs, De Waal (1997) describes the so-called Gresham�s Law of the NGO

sector:

"[An organization that is] most determined to get the highest media pro-

�le obtains the most funds [...] In doing so it prioritizes the requirements of

fundraising: it follows the TV cameras, [...] engages in picturesque and emotive

programmes (food and medicine, best of all for children), it abandons scruples

about when to go in and when to leave, and it forsakes cooperation with its peers

for advertising its brand name."

Similarly, in his poignant account of the development aid industry, Hancock (1989) de-

scribes the example of World Vision (a large U.S.-based NGO), aggressively competing for

donors in the Australian market with local religious organizations:

"On 21 December 1984, unable to resist the allure of Ethiopian famine pic-

tures, World Vision ran an Australia-wide Christmas Special television show

calling on the public in that country to give it funds. In so doing it broke an

23



explicit understanding with the Australian Council of Churches that it would

not run such television spectaculars in competition with the ACC�s traditional

Christmas Bowl appeal. Such ruthless treatment of �rivals� pays, however: the

American charity is, today, the largest voluntary agency in Australia."

In this section, we relax the assumption of �xed division of donations by incorporating

the endogenous fundraising choice by non-pro�ts. In terms of the private sector, we keep the

same structure described in Section 2.1. The main di¤erence is that now non-pro�t managers

can in�uence the share of funds they obtain from the pool of total donations by exerting

fundraising e¤ort. More precisely, we assume that each non-pro�t manager i is endowed

with one unit of time, which she may split between fundraising and working towards the

mission of her non-pro�t organization (project implementation). Fundraising e¤ort allows

the non-pro�t manager to attract a larger share of donations (from the pool of aggregate

donations) to her own non-pro�t, while implementation e¤ort is required in order to make

those donations e¤ective in addressing the non-pro�t�s mission. We denote henceforth by

ei � 0 the e¤ort exerted in fundraising and by &i � 0 the implementation e¤ort. The time
constraint implies that ei + &i 2 [0;1].
As before, the non-pro�t manager collects an amount of donations �i from the aggregate

pool of donations D. One part of �i is used to pay the wage of non-pro�t manager wi,

while �i �wi is used as input for the non-pro�t�s production. In this section, in the sake of
algebraic simplicity, we assume that the output of a non-pro�t �rm is linear in undistributed

donations, namely:

gi = 2(�i �wi)&i: (15)

Notice, however, that (15) implies that undistributed donations (�i�wi) and implementation
e¤ort (&i) are complements in the production function of the non-pro�t.

We assume that aggregate fundraising e¤ort does not alter the total pool of donations

channeledto thenon-pro�t sector, D. However, the fundraisinge¤ort exertedbyeachspeci�c

non-pro�t manager does a¤ect how a given D is divided among the mass of non-pro�t �rms,

N. In other words, we model fundraising as a zero-sum game over the division of a given D.

Formally, we assume that

�i =
D

N
� ei
e
=
�A(1�N)�

N
� ei
e
; (16)

where e denotes the average fundraising e¤ort in the non-pro�t sector as a whole.

24



Again, non-pro�t managers derive utility from their own consumption and from their

contribution towards their mission, with weights on each of two sources of utility determined

by the agent�s level of pro-social motivation, mi. In addition, we assume the total e¤ort

exerted by non-pro�t managers entails a level of disutility which depends on the agent�s

intrinsic pro-social motivation:

Ui(wi;gi) =
w1�mii g

mi
i

mmii (1�mi)1�mi
� (1�mi)(ei + &i) , where mi 2 fmH;mLg:

Since mH = 1, in the optimum, motivated non-pro�t managers will always set w
�

H = 0

and e�H+&
�

H = 1. The exact values of e
�

H and &
�

H are determined by the following optimization

problem

e�H � argmax
ei2[0;1]

: gi = 2
D

N

ei
e
(1�ei) ;

with &�H = 1�e�H. The above problem yields,

e�H = &
�

H =
1

2
; (17)

which in turn implies that an mH-type non-pro�t manager obtains a level of utility given by

U�H =
1

2e

D

N
=
1

2e

�A(1�N)�
N

: (18)

With regards to unmotivated non-pro�t managers, again, they will always set w�L = �i.

In addition, since unmotivated agents care only about their private consumption and &i

is only instrumental in producing non-pro�t output, in the optimum, they will always set

&�i = 0. As a consequence, the level of e
�

L will be determined by the solution of the following

maximization problem

e�L � argmax
ei2[0;1]

: wi =
D

N

ei
e
�ei;

which, given the linearity of both the bene�t and the cost of e¤ort, trivially yields

e�L =

(
0, if e�1D=N < 1;

1, if e�1D=N � 1:
(19)

As a result, the utility that an unmotivated agent obtains from becoming a non-pro�t

manager is

U�L = max

�
D

N

1

e
�1;0

�
: (20)

Note that the indirect utility of the unmotivated agent decreases, as before, with the

size of the non-pro�t sector; however, it reaches zero at an interior value, whereas in the

25



basic model it reached zero only when N = 1. This is because now donations are not

simply "manna from heaven" but must be obtained through exerting costly e¤ort. For a

su¢ciently large size of the non-pro�t sector, the level donations per non-pro�t �rm that

can be obtained through fundraising e¤ort is just too small to justify the necessary e¤ort

cost. This means that an unmotivated agent will choose to stop competing for donations

if the number of non-pro�ts �rms N reaches a certain critical level. (Beyond such critical

level of N unmotivated managers would optimally choose to exert no e¤ort and collect zero

donations, which accordingly yields U�L = 0).

Honest equilibrium

In an honest equilibrium all non-pro�t managers are of mH-type and set e
�

H = 0:5. Denoting

by N�H the equilibrium mass of non-pro�t managers in an honest equilibrium, this implies

that they will end up raising

��H =
�A(1�N�H)

�

N�H
: (21)

Recalling (3), (18) and (20), we can observe that an honest equilibrium exists if and only

if ��H � 1 when motivated agents are indi¤erent between the non-pro�t and the for-pro�t
sectors. Hence, an honest equilibrium exists if and only if

�A(1�N�H)
�

N�H
� 1;

where N�H solves U
�

H(N = N
�

H;e = 0:5) = V
�

P(N = N
�

H). Proposition 4, presented below,

shows that the necessary and su¢cient parametric condition for an honest equilibrium to

exist is that A � 1=(1+ �)1��, and that this equilibrium is unique.

Dishonest equilibrium

In a dishonest equilibrium all non-pro�t managers are of mL-type and set e
�

L = 1. Denoting

now by N�L the equilibrium mass of non-pro�t managers in a dishonest equilibrium, this

implies that they will end up raising

��L =
�A(1�N�L)

�

N�L
: (22)

Using again (3), (18) and (20), it follows that a dishonest equilibrium exists if and only if

��L > 2 when unmotivated agents are indi¤erent between sectors. Therefore, a dishonest

equilibrium exists if and only if
�A(1�N�L)

�

N�L
� 2;

26



where N�L solves U
�

L(N = N
�

L;e = 1) = V
�

P(N = N
�

L). Proposition 4 shows that the nec-

essary and su¢cient parametric condition for the existence of a dishonest equilibrium is

A � [2=(2+ �)]1��, and that this equilibrium is unique.

Mixed-type equilibrium

In a mixed-type equilibrium all agents are indi¤erent across occupations and the non-pro�t

sector is managed by a mix of mH and mL types. That is, a mixed-type equilibrium is

characterizedbyU�H(N
�) = U�L(N

�) = V �P(N
�), whereN� = N�L+N

�

H and0 < N
�

L;N
�

H � 1=2.
Equality among (18) and (20) requires that average fundraising e¤ort satis�es emixed =

0:5 � (D=N), which in turn means that U�H(N�) = U�L(N�) = 1. The returns in the private
sectormust thenalsobeequal toone, which, using(3), implies that inmixed-typeequilibrium

the total mass of non-pro�ts must be equal to N� = 1 � A
1

1�� : In addition, since e�H = 0

while e�L = 1, then the fact that emixed = 0:5 � (D=N) together with N� = 1 � A
1

1�� pin

down the exact values of N�L and N
�

H, so as to ensure indi¤erence across the two occupations

by all agents. Proposition 4 shows that the necessary and su¢cient parametric condition for

the existence of a mixed-type equilibrium is 1=(1+ �)
1��

< A < [2=(2+ �)]
1��
, and that

this equilibrium is unique.

Equilibrium characterization with fundraising e¤ort

The following proposition characterizes the type of equilibrium that arises, given the speci�c

parametric con�guration of the model with fundraising e¤ort.

Proposition 4 The type of equilibrium allocation that arises is always unique and depends

of the speci�c parametric con�guration of the model:

1. If A � 1=(1+ �)1��, the economy exhibits an �honest equilibrium� with N� = N�H =
�=(1 + �). All non-pro�t managers exert the same level of fundraising and project

implementation e¤ort: e�H = &
�

H = 0:5.

2. If A � [2=(2+ �)]1��, the economy exhibits a �dishonest equilibrium� with N� = N�L,
where �=(2 + �) < N�L < �=(1 + �). All non-pro�t managers exert the same level of

fundraising and project implementation e¤ort: e�L = 1 and &
�

L = 0.

27



3. If 1=(1+ �)
1��

< A < [2=(2+ �)]
1��
, the economy exhibits a mixed-type equilibrium

with a mass of non-pro�t �rms equal to N�mixed = 1�A
1

1��, where

N�H = 2
h
1�A

1
1�� (1+ �=2)

i
; and N�L = A

1
1�� (1+ �)�1: (23)

Motivated non-pro�t managers set e�H = &
�

H = 0:5, while unmotivated agents set e
�

L = 1

and &�L = 0: The average level of fundraising e¤ort is then:

emixed =
1

2

�A
1

1��

1�A
1

1��

: (24)

Proof. See Appendix A.

[Insert Figure 5 about here]

The result of an �honest equilibrium� when A � 1=(1+ �)1�� is the analogous to that
one previously obtained in the basic model (as shown graphically in Figure 5, Panel A).

Similarly, when A � [2=(2+ �)]1�� the model features a pure �dishonest equilibrium� (see
Figure 5, Panel B). However, in this alternative setup, we can observe the set of parameters

under which such an equilibrium arises is actually smaller than in the basic model in Section

2. Moreover, a novelty of this alternative setup is that for the intermediate range of A there

exists a "mixed-type" equilibrium (one under which the non-pro�t sector is populated by

both types of agents). Intuitively, the necessity of competition for donations reduces the

utility of the unmotivated agents. As a consequence, this creates parameter con�gurations

under which, in the absence of fundraising competition the non-pro�t sector would be popu-

lated only by unmotivated agents, whereas in the presence of competition a fraction of them

moves into the private sector (and are in turn replaced by a fraction of motivated agents).

It is interesting to compare the �ndings of this model to those of Aldashev and Verdier

(2010), where more intense competition for funds leads to higher diversion of donations by

non-pro�tmanagers. Thisoccursbecausewhenagentshavetospendmoretimeraising funds,

less time is then left to be devoted to working towards the non-pro�t mission, and thus the

opportunity cost of diverting money for private consumption falls. In that model, all agents

are intrinsically identical, and thus the issue of more intense competition lies in aggravating

a moral hazard problem. Here, instead, the existence of motivationally heterogeneous types

implies that the main problem is one of adverse selection, and, interestingly, a more intense

competition for funds mitigates the severity of this adverse selection problem.

28



4 Extensions

The basic model of the previous section made two particularly strong assumptions. The �rst

� a behavioral one � is that donations by private entrepreneurs were unrelated to their degree

of altruism. The second � an institutional one �, that donors were completely unaware of

the motivational problems in the non-pro�t sector and enjoyed giving independently of who

is actually managing the non-pro�t sector. In this section, we present two extensions of the

model that relax these assumptions.

4.1 Extension 1: Pure and impure altruism

The model presented in Section 2 assumes that all private entrepreneurs (regardless of their

pro-social motivation) donate an identical fraction of their income to the non-pro�t sector.

However, if warm glow giving is actually the result of some sort of altruistic behavior, it

seems more reasonable to expect the propensity to donate out of income to be increasing in

the degree of pro-social motivation. Here, we modify the utility function in (2) by letting

the propensity to donate be type-speci�c (�i) and increasing in mi. In particular, we now

assume that �i = �H 2 (0;1] when mi = mH, whereas �i = �L = 0 when mi = mL.12

The key di¤erence that arises when �i is an increasing function of mi is that, for a

given value of 1 � N, the total level of donations will depend positively on the ratio (1 �
NH)=(1 � N). Intuitively, the fraction of entrepreneurial income donated to the non-pro�t
sector will rise with the (average) level of warm-glow motivation displayed by the pool of

private entrepreneurs.

To keep the analysis simple, we abstract from fundraising e¤ort, and assume again that

the mass of total donations are equally split by the mass of non-pro�ts. In addition, we let

the payo¤ functions by motivated and unmotivated non-pro�t entrepreneurs be given again

by (7) and (8), respectively. Donations collected by a non-pro�t is given by:

D

N
=

�H A
�
1
2
�NH

�

(1�NH �NL)1�� (NH +NL)
: (25)

When the total amount of donations to the non-pro�t sector depends positively on the

fraction of pro-socially motivated private entrepreneurs, the model exhibits multiple equi-

12Notice that, in the speci�c case in which �H = 1, the utility functions in the private sector and the
non-pro�t sector would display the same structure for both mH- and mL-types: for the former, all the utility
weight is being placed on pro-social actions (either warm-glow giving or producing gi); for the latter, all the
utility weight is being placed on private consumption.

29



libria. The main reason for equilibrium multiplicity is that, when �i is increasing in mi, the

ratio between U�H and U
�

L does not depend only on the level of N � as it was the case with

(7) and (8) in Section 2 � but, looking at (25), it follows that it also depends on how N

breaks down between NH and NL. Such dependence on the ratio NH=NL generates a positive

interaction between the incentives by mL-types to self-select into the non-pro�t sector and

the self-selection of mH-types into the private sector. The next proposition deals with this

issue in further detail.

Proposition 5 Let �i = �H 2 (0;1] for mi = mH and �i = �L = 0 for mi = mL. Then,

1. Unique �honest equilibrium�: If A < (1� �H=2)1��, the equilibrium in the economy is
unique, and characterized by �H=(2+2�H) < N

�

H <
1
2
and N�L = 0:

2. Unique �dishonest equilibrium�: If A > [(2+ �H)=(2+2�H)]
1��
, the equilibrium in the

economy is unique, and characterized by N�L = �H=2 and N
�

H = 0:

3. Multiple equilibria: If (1� �H=2)1�� < A < [(2+ �H)=(2+2�H)]1��, there exist three
equilibria in the economy,13

a) an �honest equilibrium� where �H=(2+2�H) < N
�

H <
1
2
and N�L = 0;

b) a �dishonest equilibrium� where N�L = �H=2 and N
�

H = 0;

c) a �mixed-type equilibrium� where N�H =
1
2
� 1�A1=(1��)

�H
and N�L =

[1�A1=(1��)](1+�H)
�H

� 1
2
.

Proof. See Appendix A.

Proposition 5 shows that for A su¢ciently small the economy will exhibit an �honest

equilibrium�, whereas when A is su¢ciently large the economy will fall in a �dishonest equi-

librium�. These two results are in line with those previously presented in Proposition 1.

However, Proposition5alsoshowsthat thereexistsan intermediate range, (1� �H=2)1�� <
A < [1� �H=(2+2�H)]1��, in which the economy displays multiple equilibria. For those
intermediate values of A, the exact type of equilibrium that takes place will depend on how

agents� expectations coordinate. If agents expect a large mass of mH-types to choose the

non-pro�t sector (case a above), then the total mass of private donations (for a given N)

will be relatively small, sti�ing the incentives of mL-types to become non-pro�t managers.

13In the speci�c cases where A = (1��H=2)1�� or A = [1� �H=(2+2�H)]1��, the �mixed-type equilib-
rium� described below disappears, while the other two equilibria remain.

30



Conversely, if individuals expect a large mass of mH-types to become private entrepreneurs

(case b above), the value of D (for a given N) will turn out to be large, which will enhance the

incentives of mL-types to enter into the non-pro�t sector more than it does so for mH-types.

Notice that the range of productivity A for which multiple equilibria occur increases with

the (relative) generosity of the motivated individuals, �H. This is depicted in Figure 6: the

range of values of A subject to multiple equilibria vanishes as �H approaches zero.

[Insert Figure 6 about here]

Finally, there is also the possibility of intermediate consistent expectations (case c above),

in which both motivated and unmotivated agents are indi¤erent across occupations, and a

mix of mL- and mH-types share the non-pro�t sector.

4.2 Extension 2: Conditional warm glow giving

So far, we have assumed that mH-type private entrepreneurs donate a fraction �H of their

income simply because they enjoy the act of giving. This is the essence of warm glow giving

and impure altruism. However, if these agents were actually motivated by pure altruism,

then motivated entrepreneurs would not be willing to donate money to non-pro�ts managed

by mL types, and a �dishonest equilibrium� could never arise in our model.

In this subsection, we relax to some extent the assumption of impure altruism, although

we do not go all the way to assuming pure altruism by private entrepreneurs with rational

expectations.14 More precisely, we extend our model in Section 4.1 to allow �H to rise

with the fraction of motivated non-pro�t managers, by postulating that mH-type private

entrepreneurs have the following utility function:

VH(c;d) =

�
e�
e�H

H (1�e�H)1�
e�H

�
�1

c1�
e�H d

e�H, where e�H = f �H and f �
NH

NH +NL
: (26)

The utility function (26) displays conditional warm glow altruism, in the sense that the

intensity of the warm glow giving parameter (e�H) is linked to the likelihood that the donation
ends up in the hands of a motivated non-pro�t manager.

14We must stress that our desire to maintain some impure altruism component is not just due to mod-
elling convenience, but also for consistency: Andreoni (1988) shows that under pure altruism, voluntary
contributions to public good provision would vanish when the number of donors is su¢ciently large.

31



When pro-socially motivated private entrepreneurs are characterized by (26), the level of

donations obtained by a non-pro�t �rm will be given by:

D

N
=

�H A
�
1
2
�NH

�
NH

(1�NH �NL)1�� (NH +NL)2
: (27)

Proposition 6 Let the propensity to donate be given by e�i = f �i; where �H 2 (0;1], �L = 0
and f � NH=(NH +NL). Then, de�ning � � [(2+ �H)=(2+2�H)]1�� :

1. If A � �, in equilibrium, N�H = �H(A) and N�L = 0, where: @�H=@A < 0, and
limA!� �H (A) = �H=(2+2�H).

2. If � < A � 1, in equilibrium, 0 < N�H < 12 and 0 < N
�

L <
1
2
, with N�H + N

�

L =�
1�A1=(1��)

�
. In particular, N�H = nH(A) and N

�

L = nL(A), where:

nH(A) =
1

4
�

s
1

16
�
�
1�A1=(1��)

�2

�H
,

nL(A) =
�
1�A1=(1��)

�
�nH:

Moreover, when � < A � 1, the fraction of pro-socially motivated non-pro�t managers
is strictly decreasing in A; that is, @f=@A < 0.

Proof. See Appendix A.

Proposition 6 states that when warm glow weights depend on the fraction of motivated

agents within the pool of non-pro�t managers, the possibility of multiplicity of equilibria

disappears. The responsiveness of e�H to f in (26) counterbalances the e¤ect that a larger
mass of mH-type entrepreneurs has on total donations in (25), and thus neutralizes the

source of interaction that leads to multiple equilibria in Proposition 5. In addition, condi-

tional warm glow altruism removes the possibility that the non-pro�t sector is managed fully

by unmotivated agents, since in those cases motivated private entrepreneurs would refrain

from donating any of their income. Nevertheless, conditional warm glow altruism does not

preclude the fact that the non-pro�t sector may end up being partly managed by mL-types.

This occurs when A is su¢ciently large, which is in line again with the results of the baseline

model in Proposition 1. Furthermore, Proposition 6 shows that the fraction of dishonest

non-pro�t managers is monotonically increasing in A.

32



5 Discussion

In this section, we discuss several key assumptions and modelling choices on which our

analysis is built, as well as the robustness of our results to varying them.

5.1 Decreasing returns in the non-pro�t sector

One key assumption of the model is the decreasing returns in the non-pro�t sector (0 < 
 <

1). It underlies the single-crossing result (Lemma 1).

The nature of the functioning of the non-pro�t sector organizations indicates that this

assumption is appropriate. As non-pro�t organizations are de�ned by their missions, the

fundamental scarce resource of these organizations is motivated labor, i.e. individuals who

believe into (or aligned with) the mission of a particular non-pro�t. The practitioners of the

sector underline that �nding such people and expanding the sta¤ of the organization is often

extremely di¢cult, mainly because of the existing variety of missions and organizations (this

has also been highlighted by the matching-to-mission model of Besley and Ghatak, 2005).

A fundamental di¤erence of this sector with respect to for-pro�t �rms is that money cannot

easily buy time (but time can buy money, through fundraising activities). Thus, when the

funding of a non-pro�t expands, while its motivated labor remains �xed, the diminishing

marginal product of funds guarantee that the returns are decreasing. For instance, Robin-

son (1992) notes, concerning development non-pro�t working in rural areas, that "ambitious

attempts to expand or replicate successful projects can founder on the paucity of appropri-

ately trained personnel who are experienced in community development" (p. 38). Similarly,

Hodson (1992) states that

"Upgrading the management capability [of a development non-pro�t] usually

implies new talent. Unfortunately, the story-book scenario under which the orig-

inal team continues to develop its management capability at a rate su¢cient to

cope with rapid growth rarely comes true..." (p. 132)

In addition, beyond a certain scale, the successful projects of non-pro�ts have to rely

on public infrastructure and employees (for instance, at a national level). As underlined by

EdwardsandHulme(1992), this immediately clasheswith theusual government ine¢ciencies

of developing countries:

33



"E¤ective development work on a sustainable and signi�cant scale is a goal

which has eluded [development non-pro�ts, because of] the failure to make the

right linkages between their work at micro-level and the wider systems and struc-

tures of which they form a small part. For example, village co-operatives are

undermined by de�ciencies in national agricultural extension and marketing sys-

tems; �social-action groups� can be overwhelmed by more powerful political inter-

ests within the state or local economic elites; successful experiments in primary

health care cannot be replicated because government structures lack the ability

or willingness to adopt new ideas..." (p. 15)

Secondly, the type of tasks that a non-pro�t organization typically carries out, especially

in developing countries, changes along its expansion path. The �rst activities usually con-

centrate on some form of emergency: saving individuals from imminent physical danger or

starvation, helping to avoid some irreversible health problem, etc. In this sense, the mar-

ginal returns are extremely high at the beginning. Unfortunately, there is no shortage of

such problems to solve, and very often, the observation of similar severe problems is exactly

what motivates numerous motivated individuals to establish a non-governmental organiza-

tion that targets it. However, the next activities of the non-pro�t�s project involve tasks

which are less emergency-driven and more oriented towards making the livelihoods of ben-

e�ciaries sustainable (e.g. putting children to school, providing economic activities so that

bene�ciaries can earn their living). This is typically the stage of "teaching how to �sh rather

than providing �sh". Smillie (1995) argues that this second type of tasks is much harder to

accomplish successfully and involves a much longer period of time before results can be ob-

served. Such long-run perspective also implies that many organizations prefer to concentrate

on the emergencies; however, the resulting competition among them for "saving lives" limits

their expansion, as has been underlined by observers of large-scale humanitarian emergencies

such as the 2004 tsunami (Mattei 2005). In our case, this implies that, for a given non-pro�t

organization, graphically, the slope of g is fairly steep at low levels of funding (the emergency

activities), and becomes �atter beyond certain level (sustainable development activities).

5.2 Informational asymmetries in the non-pro�t sector

We have assumed (except in the extension with conditional warm-glow giving) that motives

for giving are disconnected from the performance of the non-pro�t sector. This assumption

34



also implies that non-pro�ts are unable to signal their (motivational) type to donors. In

theory, such signalling would be possible by allowing non-pro�t managers to "burn money"

(in such case, in a separating equilibrium, the altruistic types would engage in burning money

whereas sel�sh types would not). However, in practice it is di¢cult to imagine an easy way

of doing so. One possibility is to allow for self-imposed restrictions on overheads; but, to be

credible, such a scheme would require a third-party certi�cation of such restrictions (e.g. by

the government). Assuming away such credibility problems, the possibility of self-imposed

restrictions would not destroy our main mechanism, but is likely to reduce the range in which

multiple equilibria occur.

Another form of signalling is possible if conditionally warm-glow donors di¤er in size (e.g.

a few largeandmanysmalldonors), and largedonors canobtain (even imperfect) information

about the non-pro�t managers� types at a reasonable cost.15 Again, this would reduce the

range of parameters in which the bad equilibrium exists (both in the unique-equilibrium and

the multiple-equilibria cases).

5.3 Lack of contractibility of non-pro�t output

Third, we have assumed severe contractual problems on non-pro�ts� output; in particular,

we have imposed that it is completely unobservable or unveri�able. The existence of these

contractual problems has a double implication for the model: motivation serves as a substi-

tute for contracts; however, it is exactly this non-contractibility that attracts low-motivation

individuals into the non-pro�t sector. Clearly, making the non-pro�ts� output measurable

would ease the problem of adverse selection. However, in the sectors where output is well

measurable, the role of non-pro�ts is less important (at the extreme, if the output is perfectly

measurable, the production can be fully taken care of by for-pro�t �rms), as has been argued

by Glaeser and Shleifer (2001). Thus, the strong assumption that we impose is justi�ed by

the scope of the applications of our analysis.

5.4 Absence of non-pecuniary incentives

Finally, we have assumed away other (not strictly pecuniary) forms of incentives, that have

been studied in the organizational economic literature (see, for instance, Besley and Ghatak

2008 and Bradler et al. 2013). It is possible that such incentives are asymmetrically valued

15The models by Vesterlund (2003) and Andreoni (2006), where obtaining a large leadership donation
serves as a credible signal of quality, can serve as a microfoundation for this type of analysis.

35



by motivated and unmotivated types. If, for instance, the prestige associated with working

in the non-pro�t sector, independent from the level of output, is valued relatively more by

motivated types, this shifts the UH curve upwards and thus increases the range of parameters

with honest equilibrium. On the contrary, if prestige is valued more by unmotivated types

(e.g. because of the indirect pecuniary bene�ts that such prestige can deliver), the range of

honest-equilibrium parameters would shrink.

6 Conclusion

In this paper, we have built a theory of private provision of public goods via voluntary

contributions to organizations in the non-pro�t sector, in a general-equilibrium occupational-

choice framework. The main applications of this theory lie in two domains.

The �rst is foreign aid intermediation by NGOs. Aid is being increasingly channelled

via NGOs, essentially driven by increasing emphasis of project ownership, decentralization,

and participatory development. This emphasis is mostly driven by the disillusionment in

government-to-government project aid, which is often considered to be politicized and/or

easily corruptible (see, for instance, empirical evidence by Alesina and Dollar 2000 and

Kuziemko and Werker 2006). However, little analysis so far has been made concerning the

implication of massive channelling of aid via NGOs (with the exception of the few papers

mentioned earlier and the recent review study by Mansuri and Rao, 2013). The application

of our theory to foreign aid allows to explore these implications, in particular, the two e¤ects

of aid in�ows on the functioning of the NGO sector: dilution (increase in N) and selection

(unmotivated agents� entry into the NGO sector). The key implication of our results is that

as the NGO channel of aid expands, the investment into better accountability in the NGO

sector (e.g. restrictions on diversion of funds for private perks) is fundamental, so as to

prevent the appearance of the dishonest equilibrium. Optimal aid delivery through NGOs

requires harder controls accompanying the scaling-up of aid e¤orts.

The second application, instead, pertains to the recent debates on the accountability,

value-for-money, and performance-based pay in the non-pro�t sector in developed countries.

Existing literature recognizes that �rms in the non-pro�t sector, because of the inherent

di¢culty of measuring their performance, are prone to asymmetric information and agency

problems. Understanding the conditions under which these problems are most salient is

an open issue in public economics literature. Our analysis contributes to this debate by

36



indicating that the role of (endogenously determined) relative outside options of unmotivated

and motivated individuals inside the non-pro�t sector is crucial. In particular, what matters

is the type of individuals (i.e. motivated or unmotivated ones) that exit more intensively the

non-pro�t sector, when incomes in the private sector (and thus donations to the non-pro�t

sector) decrease. If, as in our model, unmotivated agents exit more intensively, the recession

can have a cleansing e¤ect, in terms of motivational composition of the non-pro�t sector.

This is, in our view, an interesting hypothesis that can be tested empirically in future work.

Two further promising avenues for future research are worth mentioning. The �rst is

the role of speci�c public policy instruments towards the non-pro�t sector. Several recent

studies on the economic of charities and non-pro�ts have explored the e¤ectiveness of direct

versus matching grants (Andreoni and Payne 2003, 2011; Karlan et al. 2011). Our analysis

in Section 3 indicates that matching grants might have an additional e¤ect that operates

through motivational composition of the non-pro�t sector: such �nancing induces non-pro�ts

to engage more actively in fundraising (and thus to reduce their internal resources devoted to

working on their projects), and this might induce the motivated individuals to quit the non-

pro�t sector. A more complete analysis of the e¤ectiveness of matching grants as compared

to direct ones, that takes into account these various e¤ects, looks very promising.

The second possibility relates to the key speci�city of the non-pro�t/NGO sector: the

disconnection between who �nances and who bene�ts from the activity of this sector. The

resulting monitoring problems create the need in coordinating scaling up of �nancing with

investment into better monitoring of the sector. As suggested by Ruben (2012), evaluation

of aid e¤ectiveness can generate social bene�ts even when one can learn relatively little

from the evaluation exercise, because the very fact of being evaluated makes rent extraction

more di¢cult and therefore might improve the motivational composition on non-pro�t/NGO

sector. The framework developed in this paper might allow to build an analysis of these

indirect e¤ects of evaluation of development projects.

37



Appendix A: Omitted Proofs

Proof of Proposition 1. Part (i). First of all, notice that by replacing N = N0

into (8), it follows that A(1+ �)
1��

> 1 implies U�L(N0) > 1: Hence, since U
�

L(
bN) = 1,

it must necessarily be the case that N0 < bN. Because of Lemma 1, this also means that
U�L(N0) > U

�

H(N0): Now, since U
�

L(N0) = y(N0), then y(N) < U
�

L(N0) for any N < N0,

meaning that whenever N < N0 the mass of non-pro�t managers must at least be equal

to 0:5 (the total mass of mL-types). But this contradicts the fact that N0 < 0:5; hence an

equilibrium with N < N0 cannot exist. Moreover, an equilibrium with N > N0 cannot exist

either, because whenever N > N0 holds, y(N) > U
�

H(N) and y(N) > U
�

L(N), contradicting

the fact that there is a mass of individuals equal to N > 0 choosing to become non-pro�t

managers. As a result, when A(1+ �)
1��

> 1, an allocation with N� = N�L = N0 represents

the unique equilibrium. Since U�H(N0) < U
�

L(N0) = y(N0), in the equilibrium, all mH-type

become private entrepreneurs, and a mass 0:5 � N0 of mL-type agents (who are indi¤erent
between the two occupations) also become private entrepreneurs.

Part (ii). Since A(1+ �)
1��

< 1 implies U�L(N0) < 1, when the former inequality

holds, N0 > bN. Moreover, notice that an equilibrium with N � N0 cannot be exist, as it
would contradict the fact that N0 < 0:5. In turn, because the equilibrium must necessarily

verify N > N0 > bN, only motivated agents will become non-pro�t managers, while all
unmotivated agents will self-select into the for-pro�t sector. Now, by the de�nition of N1

in (10), it follows that if N1 � 0:5, then N� = N�H = N1 represents the unique equilibrium
allocation. (Notice that A(1+ �)

1��
< 1 ensures N1 > N0:) In that situation, the mH-types

are indi¤erent across occupations (and there is a mass 0:5�N1 of them in the private sector),
while when N < N1 all motivated agents wish to become non-pro�t managers contradicting

N < 0:5, andwhenN > N1 nobodywouldactually choose the non-pro�t sector contradicting

N > 0. With a similar reasoning, it is straightforward to prove that when N1 > 0:5, the

unique equilibrium allocation is given by N� = N�H = 0:5, as in that case the condition

U�L
�
1
2

�
< y

�
1
2

�
< U�H

�
1
2

�
holds, whereas for N < 0:5 all mH-types intend to become non-

pro�t managers, and when N > 0:5 there is either nobody or only a mass one-half of agents

who wish to go the non-pro�t sector.

Proof of Proposition 2. Part (i). (a) First of all, recalling (12), notice 21��A > 1

implies N < 1
2
. Using the results in Proposition 1, it then follows that when A(1+ �)

1��
<

38



1 < 21��A and � = 0, in equilibrium, N� = N�H = N1, where recall that N1 is implicitly

de�ned by (10). Let now NH be implicitly de�ned by the following condition:

N�
H [�A(1�NH)
�
+�]


(1�NH)1�� � A; (28)

in raw words, NH denotes the level of N that equalizes (1) and the utility obtained by a
motivated non-pro�t manager when D=N is given by (11). From (28), it is easy to observe

that when � = 0, NH = N1. In addition, di¤erentiating (28) with respect to NH and �, we
obtain that @NH=@� > 0. Let now

�0 � 1�A
1

1��(1+ �); (29)

and, using (12), notice that [�A(1�N)� +�0]=N = 1; hence NH(�0) = N. As a con-
sequence of all this, when A(1+ �)

1��
< 1 < 21��A, for all 0 � � < �0, in equilibrium,

N� = N�H = NH(�), where @NH=@� > 0, and NH (�) : [0;�0) ! [N1;N).
(b) Using again the fact that [�A(1�N)� +�0]=N = 1, from (11) it follows that, for

all � > �0, the utility achieved as non-pro�t managers by mL-types must be strictly larger

than that obtained by mH-types. Let now

�A � 2��A
h�
21��A

�1�

 � �

i
: (30)

Using (1) and (11), notice that when N = 1
2
and � = �A, the utility obtained by motivated

non-pro�t managers is equal to y
�
1
2

�
. All this implies that, when A(1+ �)

1��
< 1 < 21��A,

for all �0 � � < �A, in equilibrium, N� = N�L = NL(�) � 12, where NL(�) is non-
decreasing in�: Inparticular, forall�0 � � � 2��A(1� �) the functionNL(�) is implicitly
de�ned by �

�A(1�NL)� +�
NL

�
(1�NL)1�� � A; (31)

while for all 2��A(1� �) < � < �A, NL(�) = 12. Lastly, when � = 2
��A(1� �), the

expression in (31) implies NL = 12, proving that NL(�) : (�0;�A] !
�
N; 1

2

�
is continuous

and weakly increasing.

(c) First, note that when � > �A, the expression in (28) delivers a value of NH > 12.
As a result, motivated agents must necessarily be indi¤erent in equilibrium between the

two occupations, since some of them must choose to actually work as non-pro�t managers to

allow NH > 12. In addition, since by de�nition of �A in (30), �A [(1�N)
�
+�A]=N > y(N)

39



when N = 1
2
, all unmotivated agents must be choosing the non-pro�t sector when � > �A.

Let thus NLH be implicitly de�ned by the following condition:

N�
LH [�A(1�NLH)
�
+�]


(1�NLH)1�� � A: (32)

Di¤erentiating (32) with respect to NLH and �, we can observe that @NLH=@� > 0. From
(32), we can also observe that lim�!�A NLH = 12 and lim�!1 NLH = 1. As a result, we may
write NLH(�) : (�A;1) !

�
1
2
;1
�
, with @NLH=@� > 0. Moreover, since N�L = 12;8� > �A,

it must be the case that in equilibrium N�H = NLH(�)� 12.

Part (ii). (a) Because of Proposition 1, when � = 0, in equilibrium, N�H � 12 and
N�L = 0: Next, let �B � 2��A(1� �), and note that:

2
�
�A
�
1
2

��
+�B

�
= 21��A; (33)

and note that the right-hand side of (33) equals y(1
2
), while its left-hand side equals D=N

when N = 1
2
and � = �B. Furthermore, notice that 2[�A

�
1
2

��
+ �] is strictly increasing

in �: As a consequence, it follows that in equilibrium, N�L = 0 for any 0 � � � �B. In
addition, denoting byNH (�) = minf12;�g, where � is the solutionof [�A(1��)

�
+�]=� =

A=(1��)1��, the result, N�H = NH (�) for any 0 � � � �B obtains.
(b) This part of the proof follows from the de�nition of �0 in (29), together with the

fact that 2[�A
�
1
2

��
+�] > 21��A, for all � > �B. As a result, we may implicitly de�ne the

function NHL(�) by

�
�A(1�NHL)� +�

NHL

�
(1�NHL)1�� � A;

and observe that @NHL=@� > 0. Noting that, whenever N = NHL(�), mL-types are

indi¤erent across occupations completes the proof of this part.

(c) This part of the proof follows again from the de�nition of �0 in (29), which implies

that for all � > �0, the expression in (11) yields D=N > 1 when N = N. For this reason,

whenever � > �0, the mH-types must be indi¤erent across occupations in equilibrium,

while all mL-types will strictly prefer the non-pro�t sector. We can then implicitly de�ne

the function NLH(�) by

N
�
LH [�A(1�NLH)

�
+�]

�
(1�NLH)1�� � A;

and observe that @NLH=@� > 0 to complete the proof.

40



Proof of Proposition 4. Part (i). First, recall that in an honest equilibrium e = 1
2
.

Second, using (21) and (3) when N = N�H, we have that

�A(1�N�H)
�

N�H
=

A

(1�N�H)
1��

, N�H =
�

1+ �
<
1

2
:

Therefore, an honest equilibrium must necessarily feature N�H = �=(1+ �), with mH types

indi¤erent across the two occupations. In such an equilibrium, they obtain a level of utility

equal to A(1+�)1��. Third, from (19) it follows that this solution is a Nash equilibrium, as

the best response by mL-type non-pro�t managers would be eL = 0 when 2A(1+�)
1�� < 1,

while eL = 1 otherwise. In both cases, A(1 + �)
1�� � 1 implies that unmotivated agents

should prefer the private sector to the non-pro�t sector. Moreover, this must be the unique

Nash equilibrium solution, since the incentives for an mL-type agent to start a non-pro�t will

decline with the average level of e, which in equilibrium will never be below 0:5 as implied

by (17).

Part (ii). Preliminarily, let �rst de�ne eN � �=(2+�). Note then that, when e = 1; the
payo¤ functions (18) and (3) are equalized when N = eN; namely, U�H( eN) = V �( eN). Next,
notice that, for a given e, both (18) and (20) are strictly decreasing in N, while they grow

to in�nity as N goes to zero. Hence, to prove that a dishonest equilibrium exists, it su¢ces

to show that the condition A � [2=(2+ �)]1�� implies U�H( eN) � U�L( eN). To prove that
the dishonest equilibrium is the unique equilibrium, notice �rst that an honest equilibrium

is incompatible with A � [2=(2+ �)]1��. Therefore, the only other alternative would be a
mixed-type equilibrium with all agents indi¤erent between the private and non-pro�t sector.

Yet, for (18) and (20) to be equal, it must be that D=N = 2e. This equality in turn implies

that all activities must yield a payo¤ equal to 1, however, when A � [2=(2+ �)]1��, this
would be inconsistent with e < 1, therefore a mixed-type equilibrium cannot exist either.

Part (iii). First of all, following the argument in the proof of part (i) of the proposition,

notice that an honest equilibrium cannot exist, since when A(1 + �)1�� > 1 unmotivated

agents would like to deviate to the non-pro�t sector and set eL = 1. Secondly, notice that

a necessary condition for a dishonest equilibrium to exist is that U�H > 1 when N =
eN and

e = 1, but replacing N = eN and e = 1 into (18) yields a value strictly smaller than 1 when
A < [2=(2+ �)]

1��
. Asaresult, whenA(1+�)1�� < A < [2=(2+ �)]

1��
theequilibriummust

necessarily be of mixed-type, with all agents indi¤erent across occupations. This requires

that U�H(N
�) = U�L(N

�) = V �P(N
�) = 1. >From (3) we obtain that V �P(N

�) = 1 implies

41



N�mixed = 1 � A
1

1�� : In addition, U�H(N
�) = U�L(N

�) requires that 2emixed = D=N, which

using N�mixed = 1�A
1

1�� leads to (24). Therefore, using the facts that e�H = 0:5 and e
�

L = 1,

the levels of N�H and N
�

L in (23) immediately obtain. Lastly, to prove that this equilibrium

is unique, notice that e�mixed in (24) lies between 0:5 and 1, thus there must exist only one

speci�c combination of N�H and N
�

L consistent with a mixed-type equilibrium.

Proof of Proposition 5. First of all, notice that NH = 0:5 cannot hold in equilibrium,

as (25) implies that in that case D=N = 0, an no agent would then choose the non-pro�t

sector. We can then focus on three equilibrium cases: (i) N�L = 0 and 0 < N
�

H < 0:5, with

mL-types strictly preferring the private sector (ii) N
�

L � 0:5 and N�H = 0, with mH-types
strictly preferring the private sector (iii) 0 � N�L � 0:5 and 0 � N�H < 0:5, will all types
indi¤erent across occupations.

Case (i). For this case to hold in equilibrium, the following condition must be veri�ed:

�HA
�
1
2
�NH

�

(1�NH)1�� NH| {z }
U�
L
(NH;0)

<
A

(1�NH)1��| {z }
y(NH;0)

=

"
�HA

�
1
2
�NH

�

(1�NH)1�� NH

#

| {z }
U�
H
(NH;0)

: (34)

For U�L(NH;0) < y(NH;0) in (34) to hold, NH > �H=(2 + 2�H) must be true. Next, since

U�L(NH;0) < U
�

H(NH;0) , U�L(NH;0) < 1, and y(NH;0) is strictly increasing in NH while
U�H(NH;0) is strictly decreasing in it and U

�

H(
1
2
;0) = 0, a su¢cient condition for (34) to hold

in equilibrium is that

�HA
�
1
2
�NH

�

(1�NH)1�� NH
< 1 when NH =

�H
2+2�H

;

which in turn leads to the condition A < [(2+ �H)=(2+2�H)]
1��
.

Case (ii). The case takes place when the following condition holds:

� 1
2
�HA

(1�NL)1�� NL

�

| {z }
U�
H
(0;NL)

<
A

(1�NL)1��| {z }
y(0;NL)

�
1
2
�HA

(1�NL)1�� NL| {z }
U�
L
(0;NL)

: (35)

Using the expressions in (35), notice that for U�L(0;NL) > y(0;NL) to hold, NL < �H=2.

But, since 0 < �H � 1, NL < �H=2 and U�L(0;NL) > y(0;NL) cannot possibly hold together.
As a consequence, in equilibrium, U�L(0;NL) = y(0;NL) must necessarily prevail, implying

in turn that NL = �H=2. Next, since U
�

L(NH;0) > U
�

H(NH;0) , U�L(NH;0) > 1, a su¢cient

42



condition for (35) to hold in equilibrium is that

1
2
�HA

(1�NL)1�� NL
> 1 when NL =

�H
2
;

which in turn leads to the condition A > (1� �H=2)1��.
Case (iii). Keeping in mind that U�L(NH;0) = U

�

H(NH;0) , U�L(NH;0) = 1, this case
will arise when the following equalities hold:

A

(1�NH �NL)1��| {z }
y(NH;NL)

=
�HA

�
1
2
�NH

�

(1�NH �NL)1�� (NL +NH)| {z }
= 1

U�
L
(NH;NL)

: (36)

Recallingthede�nitionofN in (12), U�L(NH;NL) = 1 leads to [�H (0:5�NH)]=
�
1�A1=(1��)

�
=

1, from where we obtain:

NH =
1

2
� 1�A

1
1��

�H
: (37)

Next, using again the de�nition of N in (12), we may obtain NL =
�
1�A1=(1��)

�
� NH,

which using (37) yields:

NL =
�
1�A

1
1��

� 1+ �H
�H

� 1
2
: (38)

Lastly, (37) implies that NH > 0 , A > (1� �H=2)1�� ; while (38) means that NL > 0 ,
A < [(2+ �H)=(2+2�H)]

1��
, completing the proof.

Proof of Proposition 6. First of all, from (27), it is straightforward to observe that

neither NH = 0:5, nor 0 = NH < NL can possibly hold in equilibrium, as both situations

would imply D=N = 0, an no agent would thus choose the non-pro�t sector.

Second, set NL = 0 into (27), and take the limit of the resulting expression as NH

approaches zero, to obtain

lim
NH!0

D

N

����
NL=0

=
�H A

2

NH

(NH)
2
= 1:

The above result in turn implies that 0 = NH = NL cannot hold in equilibrium either, as in

that case the non-pro�t would become in�nitely appealing to mH-types.

Third, suppose 0 < NH < NL =
1
2
. Using (1) and (27), for this to be an equilibrium, it

must necessarily be the case that

�H A
�
1
2
�NH

�
NH

�
1
2
�NH

�1�� �1
2
+NH

�2 �
A

�
1
2
�NH

�1�� : (39)

43



However, the condition (39) cannot possibly hold, since it would require �H (0:5�NH)NH �
(0:5+NH)

2
, which can never be true.

Because of the previous three results, the only possible equilibrium combinations are: (i)

N�L = 0 and 0 < N
�

H < 0:5, (ii) 0 � N�L � 0:5 and 0 < N�H < 0:5, will all types indi¤erent
across occupations.

Case (i). For this case to hold in equilibrium, condition (34) must be veri�ed, which

following the same reasoning as before in the Proof of Proposition 5 leads to the condition

A < [(2+ �H)=(2+2�H)]
1��
.

Case (ii). For this case to hold in equilibrium, the following equalities must all simulta-

neously hold:

D

N
=

�H A
�
1
2
�NH

�
NH

(1�NH �NL)1�� (NH +NL)2
= y(N) =

A

(1�NH �NL)1��
= 1: (40)

Taking into account the de�nition of N in (12), it follows that y(N) = 1 requires NH +NL =

1�A
1

1�� . As a result, (40) boils down to the following condition:

�H
�
1
2
�NH

�
NH �

�
1�A

1
1��

�2
= 0 (41)

The expression in (41) yields real-valued roots if and only if

A �
�
1�

p
�H=4

�1��
: (42)

When (42) is satis�ed, the solution of (41) is given by:

NH =

8
>>>><
>>>>:

r0 �
1

4
�

s
1

16
�
�
1�A1=(1��)

�2

�H
;

r1 �
1

4
+

s
1

16
�
�
1�A1=(1��)

�2

�H
:

(43)

Note now that the roots r0 and r1 are not necessarily equilibrium solutions for NH. More

precisely, since NL = [1�A
1

1�� ]�NH, then NL � 0 , NH � [1�A
1

1�� ]. As a consequence,

for NH = r1 in (43) to actually be an equilibrium solution, it must then be the case that

r1 � 1�A
1

1�� . But this inequality is true only in the speci�c case when A =
�
1�

p
�H=4

�1��

and
p
�H = 1, which in turn also implies that r1 = r0 in (43). Without any loss of generality,

we may thus fully disregard r1, and check under which conditions r0 � 1�A
1

1�� .

Using (43), and letting x � 1�A
1

1�� , an equilibrium with NL � 0 when NH = r0 requires
the following condition to hold:

	(x) � 1
4
�
s
1

16
� x

2

�H
� x; (44)

44



Now, notice 	(x) = x when A = [(2+ �H)=(2+2�H)]
1��
. In addition, noting that 	0(x) >

0 and 	00(x) > 0, it then follows that: i) 	(x) < x, for all A > [(2+ �H)=(2+2�H)]
1��
;

while 	(x) > x, for all (1�
p
�H=4)

1�� < A < [(2+ �H)=(2+2�H)]
1��
. Consequently, when

A � [(2+ �H)=(2+2�H)]1��, there is anequilibriumwithNH = r0 andNL = [1�A
1

1�� ]�r0.
Lastly, to prove that @f=@A < 0, note that f = 	(x)=x, hence

@f

@A
=

1

4x2
@x

@A
� 1
16x3

�
1

16
� x

2

�H

�
�
1
2 @x

@A
;

from where @f=@A < 0 stems from noting that @x=@A < 0 and that

1� 1
4x

�
1

16
� x

2

�H

�
�
1
2

> 0;

because of (43).

Derivation of Equilibrium Regions in Figure 4. i) Honest Equilibrium Region: This

type of equilibrium arises when �i < 1 < V
�

p for any 0 � N � 12, where V
�

p is given by (13)

and �i by (14). For �i < V
�

p to hold for any 0 � N � 12 it su¢ces to pin down when it holds
for N = 1

2
, which in turn leads to

t < t � (1� �)=(2� �) : (45)

Next, for �i < V
�

p we need that

N <
� (1� t)+ t
1+ � (1� t): (46)

Therefore, plugging the RHS of (46) into (14), leads to the condition that �i < 1 whenever

A <
1

(1� t)� [1+ � (1� t)]1��
: (47)

As a result, the region bounded by (45) and (47) features an �honest equilibrium�.

ii) Dishonest Equilibrium Region: This type of equilibrium needs, �rst, that condition (47)

fails to hold. Second, it also needs that (�i)

< V �p holds, so that mH-types choose the

private sector. For (�i)

< V �p to obtain, it must be that

A >
[t+ � (1� t)]


1�

21�� (1� t)
1

1�

: (48)

Notice now that the RHS of (47) is equal to the RHS of (48) when t = t; while the former lies

above (below) the latter when t < t (when t > t). As a consequence, the region exhibiting

45



a �dishonest equilibrium� is given by A > (1� t)�� [1+ � (1� t)]��1 whenever t � t and by
(48) whenever t > t.

iii) Mixed-type Equilibrium Region with f > 1
2
: From the previous results it follows that

when (47) holds and t > t, we must necessarily have an equilibrium in which all mH-types

choose the non-pro�t sector, while mL-types lie indi¤erent between the two sectors, and a

fraction of them choose the non-pro�t sector as well.

iv) Mixed-type Equilibrium Region with f < 1
2
: From the previous results it also follows that

when both (47) and (48) fail to hold and t > t, we must necessarily have an equilibrium in

which mL-types choose the non-pro�t sector, while mH-types lie indi¤erent between the two

sectors, and a fraction of them choose the non-pro�t sector as well.

46



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