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Rechts-, Wirtschafts- und
Verwaltungswissenschaftliche
Sektion
Fachbereich
Wirtschaftswissenschaften
Diskussionspapiere der DFG-
Forschergruppe (Nr.: 3468269275):
Heterogene Arbeit: Positive und Normative
Aspekte der Qualifikationsstruktur der Arbeit
Oliver Fabel
Spin-offs of Entrepreneurial Firms :
An O-Ring Approach
Mai 2003
Diskussionspapier Nr. 03/03
http://www.ub.uni-konstanz.de/kops/volltexte/2003/1002/
Diskussionspapier der Forschergruppe (Nr.: 3468269275) “Heterogene Arbeit: Positive und Normative
Aspekte der Qualifikationsstruktur der Arbeit“
Nr. 03/03, Mai 2003
Spin-offs of Entrepreneurial Firms : An O-Ring Approach
Abstract:
The O-Ring theory provides a framework to analyse the emergence of firms organized as partnerships. The owner-
managers of such entrepreneurial firms can benefit from ability matching within their production teams. However, they
must also bear the project risk. Risk-aversion then induces a second-best solution. At the same time, integrated firms
managed on behalf of risk-neutral residual-claimants face information and/or enforcement problems. Hence, they
cannot organize ability-matched teams. It is shown that there exists an equilibrium such that groups of individuals
sharing a superior ability level will found entrepreneurial firms. Low-quality individuals will be employed by managed
firms which hire randomly.
The paper constitutes a significantly revised version of the study formerly entitled “The Emergence of a New Economy:
An O-Ring Approach”, Department of Economics, University of Konstanz, Discussion Paper Series I – 314.
JEL Classification : D2, L2, M2
Keywords : O-Ring Theory; Ability Matching; Entrepreneurial Firms; Managed Firms
Download/Reference : http://www.ub.uni-konstanz.de/kops/volltexte/2003/1002/
Oliver Fabel
Universität Konstanz
Fach D 144
78457 Konstanz
Germany
mail : oliver.fabel@uni-konstanz.de
phone : +49-7531-88-2992
fax : +49-7531-88-4456
1 Motivation
While there exists a variety of de¯nitions of entrepreneurship, Bull and
Willard (1995) note two common features. According to the Knightian
school of thought, the entrepreneur bears the risk associated with an un-
certain business environment. At the same time, the Schumpeterian view
emphasizes the function of the entrepreneur to implement "new combina-
tions". Beginning with Cooper and Bruno (1977) and Cooper (1985), a
number of authors have then already focussed on the "incubator" role of
the place of employment prior to founding a new ¯rm. In this respect, the
recent emergence of the so-called "New Economy" - typically referring to
¯rm foundations in the computer, information and communications, and
bio-technology industries - appears to provide a prime example.
Hence, according to Bhid¶e's (2000, p. 54) survey of the 1989 Inc. 500
companies, 71% of the respective start-ups are founded by individuals who
"replicated or modi¯ed an idea encountered during their previous employ-
ment". Superior technological knowledge can therefore not explain the
emergence of such new ¯rms. Yet, Rajan and Zingales (2000, 2001a) empha-
size that the nature of the "new combinations" has changed. In particular,
the new technologies induce a fundamental shift of power from ¯nancial to
human capital. The innovations originate from human capital rather than
from inanimate assets. Thus, the allocation of the decision rights within the
¯rm becomes the prime issue. Corporate venturing for new opportunities
then results in a subdivision of the corporate physical assets.
Following the same basic argument, Baily and Lawrence (2001) conclude
that the emergence of the "New Economy" re°ects outsourcing decisions of
human capital-intensive productions. Bhid¶e (2000, p. 94) observes that such
spin-o®s are typically led by former high-pro¯le employees. Hence, 83% of
the founders in his survey at least hold a four-year college degree. While
¯nancial innovations have made this development possible, the recent expe-
rience of ¯nancial volatility now adversely a®ects the "New Economy" ¯rms
[Baily and Lawrence (2001)]. Reintegration constitutes a means to overcome
these problems. Moreover, according to Holmstr¿m and Kaplan (2001), the
process of disintegration generally terminates once the critical assets have
been identi¯ed. Reintegration via mergers and acquisitions then follows
again. Thus, even successful spin-o®s are viewed as transitory phenomena.
Yet, there exist limits to vertical control associated with (re-)integration.
Prat (2002) shows that °at hierarchies with ability-matched teams are dom-
inant in production environments characterized by positive complementari-
ties between specialized tasks. Moreover, °at hierarchies with "up-or-out"-
promotion schemes for experienced managers provide "incentives [...] to
protect, rather than steal, the source of organizational rents" [Rajan and
Zingales (2001b, p. 805)]. The perspective to become owners themselves
limits the exploitation risk and, therefore, enhances the incentives to spe-
1
cialize for young managers. Then, Bhid¶e (2000, p. 139-140, 367-368) again
reports the particular importance of such team work in corporate ventures.
The two examples provided by Prat (2002) further demonstrate the piv-
otal role of adequate recruiting. In fact, "unusual judgement or perceptive-
ness" in employee selection characterizes the successful entrepreneur [Bhid¶e
(2000, p. 108)]. Case studies show that - during the growth phase following
the immediate start-up period - recruiting experienced managers from es-
tablished ¯rms constitutes a key success factor [Bhid¶e (2000, p. 282-288)].
At the same time, corporate policies in well-established ¯rms to "recruit
individuals who will ¯t their culture and norms to promote cooperation and
team work [...] limit their ability to employ the best individual for a given
task [...]" [Bhid¶e (2000, p. 324)]. Hence, entrepreneurial spin-o®s constitute
a persistent response to the established corporations' failure in organizing
team-work in human capital-intensive industries.
Within the new spin-o®s, stock ownership or stock option plans then
serve as selection devices [Bhid¶e (2000, p. 87, 200)]. Rajan and Zingales
(2001b, p. 832) add that ownership must be wide-spread throughout these
new ¯rms. Concentrated control rights would again increase the threat of ex-
propriation for new team members. Thus, the incentives to specialize would
be reduced. According to Audretsch and Thurik (2001), ownership-like
management incentive schemes characterize "entrepreneurial" ¯rms. Fur-
ther, the change from "managed" to "entrepreneurial" ¯rms constitutes the
single most important characteristic associated with the emergence of the
"New Economy". In fact, the past two decades have witnessed a signi¯cant
increase in managerial stock ownership [Holderness et al. (1999)].
Following the Knightian view of entrepreneurship, Kihlstrom and La®ont
(1979, 1983) already demonstrate the existence of a contract equilibrium.
Less risk-averse individuals become risk-taking entrepreneurs who provide
insurance for their more risk-averse employees. Yet, the recent experience
of "New Economy" spin-o®s sheds doubts on the self-selection of individ-
uals as entrepreneurs, respectively employees according to their degree of
risk-aversion. Thus, the increased necessity to compensate poor employee
stock performance in cash has induced additional ¯nancial problems for the
"New Economy" ¯rms [Zingheim and Schuster (2000)]. Moreover, this de-
velopment gives rise to motivation problems for the manager-owners who
formerly received preferential treatment as high-potential employees [Wein-
berg (2001)]. Hence, partnership-like incentive schemes limit the scope of
spin-o®s. Individuals who - either as high-pro¯le employees in established
corporations, or as potential entrepreneurs - are equally risk-averse will de-
mand a compensating risk-premium in order to join a ¯rm in which incen-
tives are provided via ownership.
The current study focuses on the trade-o® between the bene¯ts of ability-
matching and the costs of partnership-like compensation schemes in entrepre-
neurial spin-o®s. It applies the O-Ring production theory introduced by
2
Kremer (1993)
1
. On ¯rst sight, the O-Ring theory only constitutes a partic-
ular example of positive complementarities in organizing team-work. Yet,
while previous work has focussed on the relationship between asset com-
plementarity and the internal organization of ¯rms, the current study ana-
lyzes entrepreneurial activity in terms of a labor market equilibrium. It is
therefore more closely related to Gromb and Scharfstein (2002) and Landier
(2001). However, both studies analyze informational equilibria. Failed ¯rm
founders are not stigmatized whereas project failure within a ¯rm provides
an informative signal concerning employee quality. In contrast, the O-Ring
framework identi¯es individual abilities with probabilities of failing in task-
performance. Thus, the bene¯ts of ability-matching are directly linked to
reductions in project-risk.
The remainder of the study is organized as follows. The next section in-
troduces the basic analytical framework. Section 3 investigates the e®ects of
¯rm organization and risk-aversion given exogenous alternative employment
opportunities. It is shown that the ¯rst-best solution can be implemented by
partnerships of risk-neutral individuals. However, partnerships of risk-averse
individuals induce ine±cient input choices. Section 4 then demonstrates the
existence and characteristics of a competitive industry equilibrium with en-
dogenous separation of entrepreneurial and managed ¯rms. Matched groups
of high-ability individuals found partnerships. Low-ability individuals will
seek employment in managed ¯rms which recruit randomly while o®ering
certain income opportunities. The ¯nal section summarizes and discusses
the results with particular reference to the perceived "volatility" of "New
Economy" ¯rms.
2 The basic O-Ring framework
The following theoretical framework modi¯es the basic O-Ring model in-
troduced by Kremer (1993) only marginally. Thus, consider the expected
revenue function
R = pF (k; n)
"
n
Y
i=1
q
i
#
n (1)
where k refers to physical capital input and n denotes the number of
tasks involved in a particular ¯rm's production. For analytic convenience,
the output price p is normalized to equal unity.
Further, q
i
2 [q
L
; q
H
], with 0 < q
L
< q
H
< 1 and i = 1; :::; n, denotes
the ability of the employee, respectively team member assigned to task i.
Ability directly corresponds to the individual probability of perfect task
1
The term "O-Ring theory" refers to the Challenger-accident which is blamed on the
malfunctioning of a simple o-ring. By analogy, the weakest link in a production chain
determines the team's productivity.
3
performance. More precisely, if the individual assigned to task i malper-
forms, the team as a whole cannot produce positive output. This occurs
with probability [1 ¡
Q
n
i=1
q
i
]. Further, the probability q
i
constitutes an
individual characteristic of the particular team member assigned to task i.
Thus, (1) re°ects a typical team production function. The output of
the team depends on the performance of each team member. In fact, given
the O-Ring framework, the productivity of the team is always governed by
the lowest-quality employee hired. Only the fact that output is completely
destroyed upon malperformance of a single team member may be considered
as an extreme assumption. This approach therefore constitutes a particular
variant of the positive complementarities in production discussed by Prat
(2002)
2
.
Intuitively, F (k; n) then de¯nes output per team member given that all
members perform perfectly. It increases with physical capital employed and
the number of tasks involved in production. Hence, increasing n implies a
technological change towards the production of a more sophisticated variant
of the industry's good, or service. For convenience, let F (k; n) = [k]
®
[n]
(1¡®)
in the following.
Spin-o®s typically produce services which replace a formerly integrated
production. Thus, new bio-technology ¯rms often constitute R&D spin-
o®s founded and controlled by former employees of a pharmaceutical ¯rm.
Also, "New Economy" ICT ¯rms customize standard accounting software,
prepare Internet presentations, or optimize server-client networks for "Old
Economy" ¯rms. In principle these services can be - and, in less sophisticated
variants, are still - produced in integrated ¯rms as well.
The analysis therefore considers a particular labor market for profes-
sional specialists. In equilibrium, they are either employees of integrated
¯rms, or partners in entrepreneurial spin-o®s. For simplicity, abilities are
distributed uniformly over the interval [q
L
; q
H
]. Thus, N(q) = ¹n = 0 mem-
bers of the pool of professionals share the ability q. The analysis abstracts
from explicitly considering the labor-leisure trade-o®. Given voluntary par-
ticipation, all individuals supply one unit of labor inelastically.
Throughout the analysis draws on the following concept of a competitive
industry equilibrium:
De¯nition: If the competition of ¯rms for professionals of di®erent qual-
ity induces an allocation such that
a) the residual expected pro¯t in all expected pro¯t-maximizing ¯rms equals
zero,
2
Positive complementarities can be associated with super-modular revenue functions.
However, the more general concept of super-modularity does not require that the revenue
function is at least twice di®erentiable. Compare the discussion in Prat (2002).
4
b) and all individuals of a given quality q 2 [q
L
; q
H
] obtain an identical
expected utility level either as members of a partnership maximizing
the expected utility of their partners, or being employed by an expected
pro¯t-maximizing ¯rm,
this allocation is said to constitute a competitive industry equilibrium.
Given that the two types of ¯rms - partnerships and expected pro¯t-
maximizing ¯rms - coexist, a separating competitive industry equilibrium
must further satisfy that
c) (i) no current member of a partnership prefers to be employed by an
expected pro¯t-maximizing ¯rm, and (ii) no group of current employees
of expected pro¯t-maximizing ¯rms prefers to found a new partnership.
The next section serves to prepare the equilibrium analysis by investi-
gating the optimal input decisions associated with ¯rm type. Only for this
purpose, let V = 0 then denote an exogenous reservation income associated
with alternative employment. Also, all ¯rms are able to select individual
workers to organize their production teams. Thus, abilities are publicly
observable and the ¯rms' recruitment decisions can be perfectly enforced.
These assumptions allow to isolate the e®ects of ¯rm organization and risk-
aversion on the e±ciency of input choices. They will be removed in section
4. The separating reservation income is then determined endogenously in
labor market equilibrium.
3 Firm organization and risk-aversion
3.1 Expected pro¯t-maximizing ¯rms
In order to construct a benchmark case for further analysis, consider the
standard expected pro¯t-maximizing ¯rm. Its residual claimant solves
Max
(fq
i
g;n;k)
¼(fq
i
g ; k; n) = R (fq
i
g ; n; k) ¡
n
X
i=1
w(q
i
) ¡ rk (2)
where w(q
i
), with i = 1; ::; n, denotes the wage income o®ered to the
employee assigned to task i. The expression fq
i
g then refers to the ability-
pro¯le of the ¯rm. Also, r is the rental rate of capital in a perfectly com-
petitive capital market.
It can now be shown:
Proposition 1 Suppose the reservation wage V is su±ciently low to allow
for positive production. Also, there exist only expected pro¯t-maximizing
¯rms in the industry.
5
(a) Then, with observable individual abilities, there exists a competitive
industry equilibrium such that all ¯rms employ a single ability-type only.
Moreover, the corresponding identical ability level of the team members de-
termines a unique optimal team size and capital input level for each ¯rm.
The optimal team size increases in the team members' identical ability level.
The shares of revenue devoted to repaying capital and rewarding labor are
given by the production elasticities ®, respectively (1 ¡ ®).
(b) The allocation implemented by the competitive industry equilibrium
is e±cient.
Proof. To begin with, note that, given (2), an expected pro¯t-maximizing
¯rm will not be induced to change its current ability pro¯le, if
@¼(fq
i
g ; k; n)
@q
i
= 0 = (3)
F (k; n)
2
4
Y
j6=i
q
j
3
5
n ¡
dw(q
i
)
dq
i
; 8i = 1; ::; n
Now, assume that, in equilibrium, ¯rms employ a single ability-type
only. The expected pro¯t of a ¯rm hiring employees of ability q can then be
obtained as
¼(q) = F (k; n)q
n
n ¡ w(q)n ¡ rk (4)
If an interior optimum exists, it is characterized by the ¯rst-order con-
ditions
®k
(®¡1)
n
(1¡®)
q
n
n = r (5)
and
[2 ¡ ® + log(q)n] k
®
n
(1¡®)
q
n
= w(q) (6)
Given that only expected pro¯t-maximizing ¯rms exist in equilibrium,
condition b) of the de¯nition of the competitive industry equilibrium re-
quires the existence of a wage-schedule which unambiguously assigns a wage-
payment w(q) to all individuals of ability q. If production teams are homo-
geneous, the optimality of the recruiting decisions according to (3) implies
that this schedule must satisfy
F (k; n)q
(n¡1)
n =
dw(q)
dq
(7)
Then, if the ¯rms' equilibrium choices of capital input and team size
actually satisfy the ¯rst-order conditions (5) and (6), rearranging condition
(5) reveals that
k
¤
=
µ
®q
n
¤
r
¶
1
1¡®
(n
¤
)
2¡®
1¡®
(8)
6
with superscripts "¤" indicating optimal values. Inserting from (8) into
(7) yields
dw
¤
(q)
dq
=
³
®
r
´
®
1¡®
(n
¤
)
1
1¡®
q
³
n
¤
1¡®
¡1
´
n
¤
=) w
¤
(q) = (1 ¡ ®)
³
®
r
´
®
1¡®
(n
¤
)
1
1¡®
q
³
n
¤
1¡®
´
+ c (9)
The constant of integration c must equal zero. In the limit q = q
L
! 0, it
re°ects the wage o®ered when organizing teams consisting of professionals
with zero quality. However, such teams would produce zero output with
certainty. Thus, they cannot o®er positive wage-income.
With c = 0, (9) then implies
n
¤
w
¤
(q) = (1 ¡ ®)(k
¤
)
®
(n
¤
)
(1¡®)
q
n
¤
n
¤
= (1 ¡ ®)R
¤
(10)
since ®
h
2¡®
1¡®
i
+ (1 ¡ ®) =
1
(1¡®)
. This proves that the total wage-bill
in each ¯rm equals the share (1 ¡ ®) of revenue. Moreover, according to
(5), the rental payment for capital rk
¤
amounts to the share ® of revenue in
each of these ¯rms. Thus, as required by condition a) of the de¯nition of a
competitive industry equilibrium, residual expected pro¯ts equal zero.
Further, inserting from (10) into (6) implies
1
n
¤
= ¡ log(q) (11)
Thus, the optimal team size is increasing in the ability level of the team
members. Interpreting this result, note that the function nq
n
attains a
unique maximum for
1
n
¤
= ¡ log(q).
Next, investigating the second-order conditions for the ¯rm's optimiza-
tion problem, reveals that
@
2
¼
¤
(q)
@k@k
= ® (® ¡ 1) (k
¤
)
(®¡2)
(n
¤
)
(1¡®)
q
n
¤
n
¤
=
¡ (1 ¡ ®) r
k
¤
< 0 (12)
@
2
¼
¤
(q)
@k@n
= (1 ¡ ®) ®(k
¤
)
(®¡1)
(n
¤
)
(1¡®)
q
n
¤
=
(1 ¡ ®)r
n
¤
> 0 (13)
@
2
¼
¤
(q)
@n@n
= ¡(k
¤
)
®
(n
¤
)
¡®
q
n
¤
=
¡rk
¤
(n
¤
)
2
µ
1 + ® (1 ¡ ®)
®
¶
< 0 (14)
upon utilizing (11) and (5). Thus,
¯
¯
¯
¯
@
2
¼
¤
(q)
@k@k
@
2
¼
¤
(q)
@k@n
@
2
¼
¤
(q)
@k@n
@
2
¼
¤
(q)
@n@n
¯
¯
¯
¯
=
r
2
(1 ¡ ®)
(n
¤
)
2
®
> 0 (15)
7
Given the proposed characterization of the industry equilibrium, (n
¤
; k
¤
)
thus constitute unique optimal choices for ¯rms employing homogeneous
teams consisting of type-q individuals.
Production will actually take place in ¯rms characterized by team ability
q such that
w
¤
(q) = V (16)
or some q 2 [q
L
; q
H
]. By virtue of (9) w
¤
(q) is monotonically increasing in
q. Hence, positive production in this industry can be assured by assuming
w
¤
(q
H
) > V = w
¤
(q
L
). Then, q
¤
2 [q
L
; q
H
[ - de¯ned by w
¤
(q
¤
) = V
- characterizes the competitive labor market equilibrium. All professionals
exhibiting abilities q = q
¤
will be employed in the industry, while individuals
of quality q < q
¤
will prefer the alternative employment.
So far, it has still be assumed that, in the competitive industry equi-
librium, the ¯rms select teams which are homogeneous with respect to the
team members' abilities. However, note that the increasing wage-schedule
derived in (9) now also implies that (3) actually characterizes an optimal
choice of the i ¡ th team member's ability level.
Intuitively, since the LHS of (3) is monotonically increasing in
h
Q
j6=i
q
j
i
,
¯rms which have hired the highest-quality employees for the ¯rst (n ¡ 1)
tasks, will always bid most in order to ¯ll the n-th position in the team. This
implies that a ¯rm which has decided to recruit the top-quality employee
for one task will recruit only such top-qualities for all tasks.
Similarly, a ¯rm which has decided to begin hiring by recruiting some
medium-quality professional cannot successfully compete for higher-quality
individuals when ¯lling other positions. However, it will succeed in attract-
ing other employees of the same quality when competing with ¯rms which
have started hiring lower-quality employees.
Thus, given the wage schedule w
¤
(q) provided in (9), the ¯rms' optimal
recruitment decisions described by (3) imply that each ¯rm will employ a
single ability type only. Then, (n
¤
(q); k
¤
(q)) as characterized by (8) and (11)
constitute the corresponding unique expected pro¯t-maximizing choices in
¯rms with teams consisting of individuals with identical ability q. Hence,
the competitive industry equilibrium introduced in part a) of Proposition 1
exists.
For expositional reasons, the proof that this equilibrium implements an
e±cient allocation is relegated to the Appendix.
3
3
Recall that, by assumption, condition c) de¯ning a separating competitive industry
equilibrium does not apply to the case made in Proposition 1. While the proof of the
proposition is then constructive, it has also been particularly organized to allow for the
subsequent use of the ¯rst-order conditions for the expected pro¯t-maximing ¯rm's opti-
mization problem. Thus, there may be other, and certainly more rigorous ways to prove
Proposition 1.
8
Obviously, the e±ciency-property of this competitive equilibrium moti-
vates its further use as a benchmark case. When referring to this solution,
the input choices (k
¤
(q); n
¤
(q)) will be denoted ¯rst-best in the following.
In addition to the characteristic features noted in Proposition 1, also
recall that larger teams of superior ability produce more sophisticated vari-
ants of the industry's commodity or service. Hence, there exists a multitude
of ¯rms o®ering services of di®erent sophistication in competitive industry
equilibrium. According to (9), small di®erences in team abilities then yield
rather large income di®erentials across ¯rms. At the same time, there is no
wage-di®erentiation within the ¯rm.
3.2 Risk-neutral partnerships
Contrasting with the assumption of pro¯t maximization utilized above, spin-
o®s typically constitute entrepreneurial ¯rms. If not organized as formal
partnerships, they distribute a large fraction of their economic pro¯t among
their employees via stock or stock option plans. Hence, it is appropriate to
assume that such ¯rms are self-managed by the members of the production
team. Then, given that individuals are risk-neutral, they rather maximize
surplus per team member. Thus, they solve
Max
(fq
i
g;n;k)
R (fq
i
g ; n; k) ¡ rk
n
(17)
Note, however, that
d
2
[R (fq
i
g ; n; k) =n]
dq
i
d
h
Q
j6=i
q
j
i
> 0 (18)
as well. Hence, consider two professionals each founding such a part-
nership ¯rm and one characterized by higher ability than the other. The
superior ability founder will always be able to o®er a more attractive part-
nership for other high-ability professionals. This remains to be true as the
production teams grow by attracting even more partners. In labor mar-
ket equilibrium entrepreneurial ¯rms will therefore also consist of partners
sharing an identical ability level.
Replacing [
Q
n
i=1
q
i
] by q
n
when solving (17), di®erentiating with respect
to capital k restates (5). The ¯rst-order condition with respect to the num-
ber of tasks n further reveals
(1 ¡ ®)k
®
n
¡®
q
n
+ k
®
n
(1¡®)
q
n
log(q) +
rk
n
2
= 0
=) [1 + nlog(q)] = 0 (19)
9
upon substituting from (5). Again, the optimal choice of technology
yields team size n
¤
(q). Obviously, this also implies that the capital de-
manded by the ¯rms is given by k
¤
(q). The second-order su±cient conditions
can be obtained as shown in (12) to (15) above.
Recall that the total wage bill in the pro¯t maximizing case always equals
the share (1 ¡ ®) of expected revenue. With risk-neutral team members,
the wage for employees in such ¯rms is thus equal to the expected surplus
net of capital rental payments. Obviously, this exactly coincides with the
individual expected income generated in an entrepreneurial ¯rm. Hence,
given the results above, the ability level q
¤
also satis¯es
(1 ¡ ®) [k
¤
(q
¤
)=n
¤
(q
¤
)]
®
h
(q
¤
)
n
¤
(q
¤
)
n
¤
(q
¤
)
i
= V (20)
Thus, without further proof, it follows:
Proposition 2 Again, assume that individual abilities are observable. How-
ever, suppose that there exist only ¯rms organized as partnerships of risk-
neutral individuals in the competitive industry equilibrium. Then, this equi-
librium can be characterized as derived in Proposition 1 for the case when
there exist only expected pro¯t-maximizing ¯rms.
Given the particular O-Ring team production function (1), this equiva-
lence result should be obvious. Rewarding factor inputs according to their
marginal revenue implies that expected residual pro¯ts equal zero. With
risk-neutral individuals, the institutional structure of the ¯rm only deter-
mines the means to distribute income. It does not a®ect the realized income
distribution.
Further, recalling that the distribution of abilities over the pool of pro-
fessionals is uniform, there exist
R
1
q
¤
h
¹n
n
¤
(q)
i
dq of such ¯rms in industry equi-
librium. Due to (11) and (19) ¯rm size increases with team quality. Thus,
the number of ¯rms characterized by a particular team quality increases
with decreasing team quality. It follows that small increases in the reserva-
tion income V induce the closing of a rather large number of ¯rms in the
industry
4
.
3.3 Partnerships of risk-averse individuals
Let the members of the industry's pools of professionals now be risk-averse.
Hence, they maximize their expected utility. Instantaneous preferences are
4
Alternatively, suppose that the pdf of abilities is single-peaked. Given normally or
log-normally distributed abilities - as often assumed - this result is even reinforced as long
as the industry attracts professionals with above-average abilities. Moreover, in this case
a small increase of V also induces a rather signi¯cant migration of professionals from the
industry into alternative occupations.
10
characterized by a utility function U(y), with U
0
(y) > 0 and U
00
(y) < 0 for
incomes y > 0. Of course, assuming that there only exist managed ¯rms
which maximize expected pro¯ts, all results of section 3.1 can be retained.
However, the manager-owners of entrepreneurial ¯rms will maximize
EU =
"
n
Y
i=1
q
i
#
U(Y + F (k; n) ¡
rk
n
) +
Ã
1 ¡
"
n
Y
i=1
q
i
#!
U(Y ¡
rk
n
) (21)
Introducing exogenous income Y > 0 in (21), the analysis will exclusively
focus on interior solutions. Such solutions can be ensured by assuming that
U(y) satis¯es the usual Inada-conditions.
If it is ever bene¯cial to found such ¯rms, they will consist of teams of
individuals characterized by identical abilities again. This follows from the
fact that @EU=@
h
Q
j6=i
q
j
i
> 0 for F(k; n) > 0. Thus, high-ability individu-
als will always ¯nd it more attractive to join partnerships already consisting
of higher-quality team members in the ¯rst (n¡1) tasks. Replacing [
Q
n
i=1
q
i
]
by q
n
in (21) then yields the ¯rst-order conditions
q
~n
U
0
(Y + F (
~
k; ~n) ¡
r
~
k
~n
)
h
®
~
k
(®¡1)
~n
(1¡®)
¡
r
~n
i
= (1 ¡ q
~n
)U
0
(Y ¡
r
~
k
~n
)
r
~n
(22)
and
¡q
~n
log(q)
"
U(Y + F (
~
k; ~n) ¡
r
~
k
~n
) ¡ U(Y ¡
r
~
k
~n
)
#
= q
~n
U
0
(Y + F (
~
k; ~n) ¡
r
~
k
~n
)
"
(1 ¡ ®)
~
k
®
~n
¡®
+
r
~
k
~n
2
#
(23)
+(1 ¡ q
~n
)U
0
(Y ¡
r
~
k
~n
)
r
~
k
~n
2
where (
~
k; ~n) denote the respective optimal choices in this case.
Substituting from (22) into (23) implies
¡log(q)
U(Y + F (
~
k; ~n) ¡
r
~
k
~n
) ¡ U(Y ¡
r
~
k
~n
)
U
0
(Y + F (
~
k; ~n) ¡
r
~
k
~n
)
=
~
k
®
~n
¡®
(24)
Concavity then implies that
¡log(q)~n < 1 (25)
Thus, partnerships managed by risk-averse team members are ceteris
paribus smaller than risk-neutral partnerships. Rearranging (22) it also
follows
11
~
k
(®¡1)
~n
(1¡®)
~nq
~n
=
r
®
"
q
~n
+ (1 ¡ q
~n
)
U
0
(Y ¡
r
~
k
~n
)
U
0
(Y + F (
~
k; ~n) ¡
r
~
k
~n
)
#
(26)
Substituting for r from (5) above,
~
k
(®¡1)
~n
(1¡®)
~nq
~n
= (k
¤
)
(®¡1)
(n
¤
)
(1¡®)
n
¤
q
n
¤
"
q
~n
+ (1 ¡ q
~n
)
U
0
(Y ¡
r
~
k
~n
)
U
0
(Y + F (
~
k; ~n) ¡
r
~
k
~n
)
#
(27)
Recall from Proposition 1 that, for a given team ability level q, nq
n
is
maximized by setting n = n
¤
. Thus, n
¤
q
n
¤
> ~nq
~n
, 8q 2 [q
L
; q
H
]. Risk-averse
partners not only require an expected income su±cient to cover their share
of capital costs. In addition, they must be compensated for risk associated
with paying the capital rental costs even if production fails. Consequently,
the cost of attracting partners is higher than in the risk-neutral case. This
implies that the size of the production team falls short of maximizing the
expected team output.
Also, due to decreasing marginal utilities, the last term in the RHS of
(27) is greater than one. Thus, n
¤
> ~n implies (k
¤
=n
¤
) > (
~
k=~n) and k
¤
>
~
k.
Let ~y(q) then denote the certainty equivalent income of such risk-averse part-
ners self-managing a ¯rm of team quality q. Hence, ~y(q) = U
¡1
(EU(q; ~n;
~
k)).
De¯ning ~w(q) = ~y(q) ¡ Y as the respective certainty equivalent return to
participating in the partnership, it is also immediately clear that
~w(q) <
~
k
®
~n
(1¡®)
q
~n
¡
r
~
k
~n
(28)
< (k
¤
)
®
(n
¤
)
(1¡®)
q
n
¤
¡
rk
¤
n
¤
= w
¤
(q)
for all q 2 [q
L
; q
H
]. The second inequality follows from the allocative
distortions associated with maximizing expected utility.
Finally, note that
@EU(q; ~n;
~
k)
@q
= ~nq
(~n¡1)
"
U(Y + F (
~
k; ~n) ¡
r
~
k
~n
) ¡ U(Y ¡
r
~
k
~n
)
#
> 0 (29)
for all q > 0. Again, assume that there exists q = ~q 2 [q
L
; q
H
[ such that
EU(~q; ~n(~q);
~
k(~q)) = U(Y + V ). Then, production will take place. However,
only individuals of ability q = ~q will actually found entrepreneurial ¯rms.
The inequalities (28) then yield ~q > q
¤
.
Summarizing the analysis, it therefore follows:
12
Proposition 3 Ceteris paribus, risk-aversion induces less ¯rms founded as
partnerships in competitive industry equilibrium. The ¯rms actually founded
produce less sophisticated services with ine±ciently small teams. Moreover,
capital input and capital per team member is ine±ciently low.
Else, the equilibrium with risk-averse partnerships shares the proper-
ties derived above already. Small variations in team quality again induce
large expected income di®erentials between the partnerships. Also, a small
increase in the relative attractiveness of alternative jobs induces a rather
large reduction of the number of ¯rms in the industry. Due to smaller ¯rm
sizes, the latter e®ect is even reinforced by introducing risk-aversion.
4 Endogenous separation of managed and entre-
preneurial ¯