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Department of Economics
Working Paper Series
CONTRACTS AND THE DIVISION OF LABOR
Daron Acemoglu
Pol Antras
Elhanan Helpman
Working Paper 05-1
May 4, 2005
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
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y^'^'^
Contracts and the Division of Labor
Daron Acemoglu
Pol Antras
Department of Economics,
MIT
Department of Economics, Harvard University
Elhanan Helpman
Department of Economics, Harvard University
May
4,
2005
Abstract
We present a tractable framework for the analysis of the relationship between contract incompleteness, technological complementarities
and the
firm decides the division of labor and contracts with
division of labor. In the
its
model economy, a
worker-suppliers on a subset of activities
they have to perform. Worker-suppliers choose their investment levels in the remaining activities
anticipating the ex post bargaining equilibrium.
We
show that greater contract incompleteness
reduces both the division of labor and the equilibrium level of productivity given the division of
labor.
The impact
of contract incompleteness
workers are more complementary.
division of labor
We
is
greater
when the
tasks performed by different
also discuss the effect of imperfect credit markets
and productivity, and
studj' the choice
between the employment relationship
versus an organizational form relying on outside contracting. Finally,
of our framework for productivity differences
JEL
Classification: D2, J2, L2,
Keyw^ords: incomplete
on the
we
derive the implications
and comparative advantage across
countries.
03
contracts, division of labor, productivity
*We thank Gene Grossman, Oliver Hart, Giacomo Ponzetto and Rani Spiegler, as well as participants in the
Canadian Institute for Advanced Research conference and seminar participants at Harvard, MIT, Tel Aviv and
Universitat Pompeu Fabra, for useful comments. We also thank Davin Chor, Alexandre Debs and Ran Melamed for
excellent research assistance. Acemoglu and Helpman thank the National Science Foundation for financial support.
Much of Helpman's work for this paper was done when he was Sackler Visiting Professor at Tel Aviv University.
Introduction
1
This paper develops a simple framework for the analysis of the relationship between contracting
and the division of
(1776) and Allyn
labor.
Young
While a major strand of the economics
and managing organizations consisting of many individuals each performing
implicit) relationships
in accounting
and use
its
workers.
Murphy
(1992),
It is
thus
or
difficult
we observe today without the important advances
of information within organizations.-^ This perspective has
example, by Becker and
different
more comphcated (contractual
and greater coordination between the firm and
to imagine a society reaching the division of labor
for
literature has recognized the role of the costs
Naturally, greater division of labor will necessitate both
tasks.
Smith
(1928), emphasizes the importance of the extent of the market as the
major constraint on the division of labor, the recent
of designing
Adam
literature, following
who
been emphasized,
suggest that the division of labor
is
not limited
by the extent of the market, but by "coordination costs" inside the firm. Nevertheless, they do not
provide a microeconomic model of
This
is
how and why
these costs vary across societies and industries.
one of our objectives in this paper.
Our approach
Grossman and Hart's (1986) and Hart and Moore's (1990) seminal
builds on
papers, which develop an incomplete-contracting model of the internal organization of the firm.
The key
insight
in order to
is
to think of the ownership structure
and the
size of
the firm as chosen partly
encourage investments by managers, suppliers and workers.
Hart and Moore show
how
this
how
the degree of contract incompleteness and the nature of the production technology affect
approach can be used to determine the boundaries of the firm, but do not investigate
the division of labor.
We
extend Hart and Moore's framework by considering an environment
with partially incomplete contracts and varying degrees of technological complementarities
(i.e.,
complementarities between different production tasks). Using this frainework, we investigate how
the degree of contracting incompleteness (for example, determined by the contracting institutions
of the society)
and technological complementarity impact on the equilibrium degree
of division of
labor.
In our baseline model, a firm decides the range of tasks that will be performed and the division
of labor, recognizing that a greater division leads to greater productivity. However, together with
the greater division of labor comes the need to contract with multiple workers (suppliers) that are
undertaking relationship-specific investments.^
A
undertake are contractible, while the
the work by Grossman-Hart-Moore, are nonveri-
fiable
rest, as in
fraction of the activities that workers have to
and noncontractible. The fraction of contractible
activities is
our measure of the quality of
^See Pollard (1965) on the advances in accounting and other information-collection technologies during the 19th
and Yates (1989) on the role of clerks and advances in information processing during the
early 20th century in the reorganization of production, and Chandler (1977) on the rise of the modern corporation and
century, Michaels (2004)
the relationship between this process and increasing informational
demands
of organizations.
A
different perspective,
which we do not pursue in this paper, would build on Marglin (1974) and emphasize the benefits of division of labor
in monitoring workers.
"Throughout, we focus on the relationship between a firm and its "worker-suppliers," and simphfy the terminology
by referring to these as workers. Since our model is sufficiently abstract, it can also be applied to the relationship
between a downstream firm and multiple upstream suppliers.
While workers are contractually obliged to perform
contracting institutions.'^
contractible activities, they are free to choose their investments
their duties in the
and can withhold
their services in
As
noncontractible activities, and this leads to an ex post multilateral bargaining problem.
arid
Moore
we use the Shapley value
(1990),
We
the firm and the workers.
in
Hart
to determine the division of ex post surplus between
derive an exphcit solution for the Shapley value of the firm and the
workers, which enables us to provide a closed-form characterization of the equilibrium.
Workers' expected surplus in this bargaining process determines their willingness to invest in the
noncontractible activities. Since they are not the
full
residual claimants of the productivity gains
derived from their investments, they tend to underinvest.
Our
first
major
result
that greater
is
contracting incompleteness reduces worker investments, and via this channel, limits the profitability
and the equihbrium extent
of the division of labor. Secondly, a greater degree of technological
plementarity limits the division of labor, since
this
is
it
interesting; although greater technological
to each worker,
it
also
makes
their
payments
further discourages investments.
also
show that better contracting
featuring greater complementarity.
less sensitive to their
institutions are
These
The reason
for
complementarity increases equilibrium payments
noncontractible investments,
and reduces investments. These lower investments make a greater division
We
com-
results also
more important
of labor less profitable.
for firms
have natural implications
with technologies
for productivity.
For example, better contracting institutions lead to greater productivity both because the equilibhigher and because, conditional on the division of labor, workers invest
rium division
of labor
more and
more productive.
We
are
is
also characterize the equilibrium
when workers cannot make ex ante
transfers to the firm
to compensate for the ex post rents they receive in the bargaining process.
comparative
static results are the
same
as in the case with ex ante transfers, the relationship
between the equilibrium division of labor and productivity
productivity can
transfers
and
now be
in the case
While the major
is
more complex. In
fact,
higher than both in the case with incomplete contracts and ex ante
with complete contracts. The reason
transfers, the firm uses investments in contractible tasks as
is
that, in the absence of ex ante
an instrument
for extracting surplus
from the workers, potentially inducing overinvestment. Even though overinvestment
it
increases the observed level of productivity.
division of labor
We
may be
also use this
equilibrium
is
inefficient,
Consequently, without ex ante transfers, greater
associated with lower productivity.
framework to discuss when the employment relationship, where various produc-
tion tasks are performed
by employees of the firm rather than by outside contractors, may emerge as
an equilibrium outcome. The key observation
relationship with the firm
when they
is
that agents have greater bargaining power in their
are outside contractors, which
is
similar to non-integrated
suppliers having greater bargaining power in Grossman-Hart-Moore's theory of the firm.
Maskin and Tirole (1999) question whether the presence
argument is not
necessarily lead to incomplete contracts. Their
of nonverifiable actions
We show
and unforeseen contingencies
it assumes the presence
central to our analysis because
of "strong." contacting institutions (which, for example, allow contracts to specify sophisticated mechanisms), while in
our model a fraction of activities may be noncontractible not because of "technological" reasons but because of weak
contracting institutions. In fact, we interpret the fraction of noncontractible activities as a measure of the quality of
contracting institutions.
that with ex ante transfers, the equihbrium always involves outside contractors, because this organizational form increases agents' bargaining power and investment.''
can
is
arise as
an equilibrium arrangement
may
that agents
obtain too
much
in the absence of
The employment
relationship
ex ante transfers, however. The reason
rents as outside contractors,
and the employment relationship
enables the firm to reduce these rents. Interestingly, in this case, the employment relationship also
makes the
division of labor
more
therefore leads to a theory of
While
in his theory, given
employment
A
succinct
way
relationship
employment
and thus encourages a greater
rents
and leads to greater division of labor. Our framework
complementary to that by Marglin (1974).
the employment relationship, the division of labor
rents from workers, in ours, the
ities
profitable,
is
used to extract more
relationship itself enables the firm to obtain greater
division of labor.
of expressing the results in our paper
and ex post bargaining, the (equilibrium)
is
that in the presence of noncontractibil-
profit function of
a firm
is
modified to
AZF{N)-C{N)
where
N
represents the division of labor,
or the scale of the market.
and depends on the degree
Naturally,
C {N) its cost and ^ is a measure of aggregate demand
F {N) is an increasing function. Here Z is a constant
of contract incompleteness
comparative static results follow from the response of
Z
and technological complementarity.
Key
to parameter changes. For example,
Z
is
decreasing in the degree of contract incompleteness and technological complementarity.
This simple form of the equilibrium profit function enables us to embed our model in a general
The
equilibrium framework.
straint,
an improvement
interesting result here
in contracting institutions
is
that,
because of the equilibrium resource con-
does not increase the division of labor in
tors (firms). Instead, the division of labor increases in sectors that are
and declines
in others. In our
all
sec-
more "contract-dependent"
model, sectors with greater technological complementarity are more
contract-dependent and experience an increase in the division of labor in general equilibrium, while
those with a high degree of substitut ability experience a decline in the division of labor. In the
context of an open economy, the same interactions lead to endogenous comparative advantage as
a function of contracting institutions. In particular,
among
countries with identical technologies,
those with better contracting institutions will specialize in sectors with greater complementarities
among
inputs.
While our main focus
is
on the division of labor, the results
in this
paper are also relevant for
the literature on cross-country differences in (total factor) productivity. Although there
is
a broad
consensus that differences in total factor productivity ("efficiency") are a major part of the large
*This result
is
driven by the fact that the firm does not undertake any investments other than
its
technology
choice.
^Our framework can also be used to discuss the trade-off between vertical integration and non-integration with
workers interpreted as suppliers. In this context, Aghion and Tirole (1994) and Legros and Newman (2000) show
how inefficient vertical integration can arise in the presence of imperfect credit markets. The interesting result in our
framework, different from these papers, is that the employment relationship (or vertical integration) can be "efficient"
because of
its
imphcations
for the division of labor.
Klenow and Rodriguez,
cross country differences in living standards (e.g.,
1999, Caselli, 2004), there
is
no agreement on the sources of these productivity
results suggest that differences in contracting institutions
differences.
1997, Hall
Societies facing the
same technological
and Jones,
differences.
Our
can be an important source of productivity
possibilities,
but functioning under different
contracting institutions, will choose technologies corresponding to different degrees of division of
labor and will naturally have different levels of productivity. Furthermore, because of the differences
in workers' investment levels, these societies will also experience different levels of productivity even
A
conditional on the division of labor.
detailed investigation of whether this could be an important
source of (total factor) productivity differences across countries
is
an interesting area for future
research.
The
In addition to the papers mentioned above, our work relates to two large literatures.
investigates the relationship
the work by
such as
between the division of labor and the extent of the market.
Yang and Borland
Romer
The second
and Grossman and Helpman
(1990)
level of (equilibrium)
(1991), as well as the product-variety models of
demand
in the
economy
It
first
includes
endogenous growth,
(1991), that can be interpreted as linking the
to the division of labor.
literature derives implications for the division of labor
from the theory of the
firm.
Important work here includes the Grossman-Hart-Moore papers discussed above as well as their
predecessors, Klein, Crawford
and Alchian (1978) and Williamson (1975, 1985), which emphasize
incomplete contracts and hold-up problems.
most
a
closely related to our work.
number
value.
of workers
Stole
The two papers by
Stole
and Zwiebel (1996a,b) are
and Zwiebel consider a relationship between a firm and
where wages are determined by ex post bargaining according to the Shapley
They show how the firm may overemploy
in order to reduce the bargaining
power
of the
workers and discuss the implications of this framework for a number of organizational design issues.^
Stole and Zwiebel's framework does not incorporate relationship-specific investments, which
is
at
the center of our approach, and they do not discuss the division of labor or the effects of contract
incompleteness and complementarities on the equilibrium division of labor.
Another important strand of the literature stems from the seminal work by Holmstrom and
Milgrom
(1991, 1994),
which emphasizes issues of multi-tasking, job design and complementarity of
tasks as important determinants of the internal organization of the firm. These considerations naturally lead to a theory of the division of labor,
where the firm may want to separate substitutable
tasks in order to prevent severe multitask distortions. Yet another approach, perhaps closer to the
spirit of
the contribution by Becker and
Murphy
cation or information processing within the firm
(1992), explicitly models the costs of
(e.g.,
Sah and
Stiglitz, 1986,
communi-
Geanakoplos and
Milgrom, 1991, Radner 1992, 1993, Radner and Van Zandt, 1992, Bolton and Dewatripont, 1994,
and Garicano, 2000). Another strand
of the literature, e.g., Calvo
and Wellisz (1978) and Qian
(1994), links the size of the firm to other informational problems, such as monitoring or selection.
Finally, a
number
of recent papers,
most notably Levchenko (2003), Costinot (2004), Nunn (2004)
^Bakos and Brynjolfsson (1993) develop a related model to study the
technology on the number of suppliers that firms choose to contract with.
effects of
improvements
in
information
and Antras (2005), apply the insights of these Hteratures to generate endogenous comparative advantage from differences in contracting institutions.
closely related since he also focuses
Among
these papers, Costinot (2004)
most
is
on the division of labor (though using a more reduced-form
approach).
The
rest of the
paper
organized as follows. Section 2 introduces the basic partial equilibrium
is
framework and characterizes the equilibrium with complete contracts. Section
equilibrium with incomplete contracts and contains the
how the
investigates
results
main comparative
3 characterizes the
static results. Section 4
change when there are no ex ante transfers. Section 5 uses our frame-
work to study the choice between employment relationship and outside contracting.
embeds the
partial equilibrium
model
in a general equilibrium
Section 6
framework and discusses how
differ-
ences in contracting institutions generate endogenous comparative advantage. Section 7 concludes,
while Appendix
A
contains the main proofs.
Model
2
In the next four sections,
results,
2.1
and
we
present our model of the firm and derive the main partial equilibrium
later turn to general equilibrium analysis.
Technology
Consider a firm, facing a
demand curve
for its
quantity and p denotes the price, and
demand
economy and
in the
from a constant
is
j3
G
product of the form q
(0,1).
^ >
=
Ap~'^f^^~^\ where q denotes
corresponds to the level of aggregate
taken as given by the firm. This form of demand can be derived
elasticity of substitution preference structure for the
consumers over
many
products
(see Section 6). It implies that the firm in question generates a revenue of
R{q)
from producing a quantity
The
=
A^-l^q^
(1)
q.
firm decides the technology of production which specifies a continuum range of tasks
that will be performed, and the
number
of worker-suppliers that will perform these tasks,
[0,
M
.
N]
For
now, we do not distinguish whether these worker-suppliers are employees of the firm or outside
contractors,
perform e
j
=
1, 2,
and simply
= N/M
...M.
tasks,
refer to
them
and we denote the
The production
Each worker
as workers (see, however, Section 5).
services of tasks performed
by worker j by
technology, which can only be operated by the firm,
X [j)
will
for
is:
l/a
^^
q
=F
[[X [j)]f^^\
M^n) ^
N^+^-'l"
j=i
A number of features of this production function
of substitution
between tasks, 1/
(1
—
a),
is
N
M
,
< Q <
are worth noting. First, since
1,
a >
K
>
0,
0.
(2)
the elasticity
always greater than one and determines the degree
Second, we follow Benassy (1998) in introducing the term
of complementarity of the technology.
jYK+i-i/a^ which allows us to disentangle the elasticity of substitution between tasks from the
elasticity of
output with respect to the
In this case,
for. all j.
productivity,
To
level of technology.
when technology
is
N,
=
q
see this,
N'^'^'^X; so greater
and the parameter k determines the relationship between
In the text,
we
suppose that
N
A''
X {j)
=
X
translates into greater
and productivity.^
simplify the analysis further by considering the case in which
M -^
and work
cx),
with the production function
1/q
g
= F^
{[X
0')]f.o)
iV-+i-V«
=
assigning tasks to workers, and
To
contracts.
is
we
M^
(3)
is
N.^ In Appendix B, we show that with complete
cx)
as long as there are "diseconomies of scope" in
where the only "technology" choice of the firm
contracts the equilibrium will feature
^l\x Ur dj
under which
M —^ oo with incomplete
useful to suppose,
from now on, that there
also provide conditions
simplify the discussion further,
a continuum of workers and that every task
it
is
performed by a separate worker.
is
Under these
circumstances -N stands for the measure of tasks as well as the measure of workers, and we refer to
A''
as the division of labor of the technology.^
Each worker assigned
measure of (symmetric)
question
to a task needs to undertake relationship-specific investments in a unit
activities,
each entailing a marginal cost
Cx-
The
services of the task in
then a Cobb-Douglas aggregate,
is
1
X (j) = exp
where x
(i,
j)
denotes the
level of
Inx
{i,j) di
investment in activity
(4)
performed by the worker.
i
We
adopt
this
formulation to allow for a tractable parameterization of contract incompleteness, whereby a subset
of the investments necessary for production will
also
assume that the
Section
cost of investments
is
be nonverifiable and thus noncontractible.
non-pecuniary
(e.g.,
"effort" cost, see
We
equation (38) in
6).
Finally,
we denote the
costs of division of labor
by C{N), which could include potential
(ex-
ogenous) coordination costs, costs of investment in technology, and upfront payments to cover the
outside options of the workers (when these are positive, see Section
"^In
(i.e.,
contrast, with the standard specification of the
K =
1/q — 1),
total output would be g
= N^'^X,
CES
6).
We
assume
production function, without the term ]\['^+''-~^'° in front
elasticities are governed by the same parameter,
and these two
a.
^Equation
(3) is
obtained as the limit of equation
(2)
when
M—
>
oo.
With
a slight abuse of notation,
we
use j to
index workers in both the discrete and continuum cases.
^In other words, we suppose that the measure of workers performing the tasks in any interval [a, b] is b — a. Note
also that, alternatively, we could have started with the production function (3) and the assumption that every task
is
performed by a separate worker.
Assumption
For
(i)
all
1
N>
0,
>
C (iV)
is
twice continuously differentiable, with C" (A^)
NC" (N) /C (N) >
(ii)
For
all
The
first
part of this assumption
tions
A^
0,
and to ensure
is
[/3
(k
+
1)
-
1]
/ (1
The second
standard.
-
and C" {N) >
>
0.
/3).
part
is
necessary for second-order condi-
interior solutions.
Payoffs
2.2
The
firm and the workers maximize expected returns. ^*^
j consists of
two
Suppose that the payment to worker
an ex ante payment, t{j), and a payment after the x{i,j)s have been
parts:
delivered, s {j). Then, the payoff to worker j
TTx (j)
=E
is
(j)+s(j)-
/
C:,x{i,j)
Jo
where
E
denotes the expectations operator, which applies
if
there
uncertainty regarding the ex
is
we assume
post payment, s{j), which will be the case in the bargaining game. As stated above,
that
if
a worker does not participate in the relationship with the firm, then she has an outside
option of zero. Similarly, the payoff to the firm
is
N
7r
=E
^1-/3_^/3(k+1-1/q)
0/^
(^exp(^y^
lnx(i,J)d^Jj
rN
-
djj
which follows by substituting
(3)
and
(4) into (1),
[r [j)
j^
j^
+
s [j)] dj
- C [N)
and by subtracting the payments to workers and
the costs of division of labor.
Division of Labor with Complete Contracts
2.3
As
a benchmark, consider the case of complete-contracts (the "first best" from the viewpoint of
the firm), which corresponds to the case in which the firm has
and pays each worker her outside option.
control over
full
all
investments
In analogy to our treatment of the division of labor
with incomplete contracts below, consider a game form where the firm chooses a technology with
division of labor
N
and makes a contract
of potential workers.
If
We
(z,
J)}^^\q
a worker accepts this contract, she
stipulated in her contract.
^^
offer, {.x
The subgame
^i
is
,gfo
m
,
{s (j)
i
t (j)},g[o
obliged to supply
perfect equilibrium of this
{a; (j,
tvJ'
*° ^ ^^*
j)}jg[o
game with complete
ii
a^
contracts
use risk neutrality only in the calculation of Shapley values to express the payoff of a player as the expected
(average) value of his or her marginal contributions to
all
possible coalition structures.
corresponds to the solution to the following maximization problem:
/"^
^i-/3^/3(«+i-i/c«)
max
("e^p
(exP
[
I
(
f' lnx{ij)dij
/
(
dj
j
+ sU)]dj-CiN)
lT{j)
10
subject to
s
U)
+
r (j)
-c. f X
(i,
j)
di>Q
for all j
G
N]
[0,
.
Jo
Since the firm has no reason to provide rents to the workers in this case, the constraints bind, and
substituting for these to eliminate t (j) and s(j), which are perfect substitutes in this case, the
program
simplifies to
r-N
Ai-^iV^('^+i-i/°)
max
N,{x{t,j)},^
Put
/
L./0
/
/
lnx(i,j)dzjj
i'^^^i
differently, the firm's profits equal
\ \ °
/•!
activities in all tasks
all
is
equally costly, the firm will choose the
and we can simplify the program
first-order conditions of this
(3{1
+
problem
yield a unique solution {N*,x*), which
(6)
solution for
and
A''*,
(7)
It is also useful to
tracts.
Since
A''
When
all
is
_
c^x
-
C {N) =
- c^N =
level
x
for all
(5)
0,
0,
implicitly defined by:
=
—
^
=
c' {N*)
(7), yields
(6)
,
-.
(7)
can be solved recursively. Given Assumption
which, together with
same investment
{i,j)
to
^K/3V(i-/5)c;Mi-/3)
x*
Equations
x{i,j)didj-C{N).
are:^^
k) ^i-z^x'^AT^l^+i)-!
(N*)^^"^^
/
-c^Nx-C{N).
l3A'-^x''-'N^^--^'^
and
/!
-c^
a convex problem in the investment levels x
maDcA^-^N/^^^+^hl^
The
rN
'^Z"
revenue minus the cost of investments of workers and the
cost of division of labor. Moreover, since this
and these investments are
1
dj
1,
equation
(6) yields a
unique
a unique solution for x*.
derive the productivity implications of the equilibrium with complete con-
the investment levels are identical and equal to x, output equals q
=
N'^'^^x.
workers are taking part in the production process, a natural definition of productivity
'^Appendix
A
shows that the second-order conditions are
satisfied
under Assumption
1.
is
output divided by N,
i.e.,
P=
N'^x. In the case of complete contracts this productivity level
P* = {N*fx*.
We
compare
will
The following
and N*
>
A
workers and the associated productivity
level of
for details):
1 Suppose that
Assumption
Then program
1 holds.
(5)
has a unique solution x*
characterized by (6) and (7), and an associated productivity level
Furthermore, this solution
dN*
dx*
da
da
In the case with complete contracts, consistent with
Adam
^
dx*
dP*
^
oA
oA
^
oA
market as parameterized by the demand
how
P* >
given by
>
(8).
satisfies:
dN*
illustrates
with incomplete contracts. ^^
proposition summarizes the above discussion and highlights the effects of some key
Appendix
Proposition
(8)
this productivity level to the level of productivity
parameters on the division of labor, the investment
levels (see
is
level
A
dP*
da
„
Smith's emphasis, the size of the
Our model
affects the division of labor.
further
a larger market size tends to be associated with greater investments by workers and
higher productivity.-'^
The other noteworthy
division of labor
1/ (1
—
implication of this proposition
and productivity are independent of the
a). This will contrast with the equilibrium
is
that under complete contracts the
elasticity of substitution
between
under incomplete contracts, where the
tasks,
elasticity
of substitution influences bargaining.
Division of Labor with Incomplete Contracts
3
3.1
Contractual Structure
We now
consider a world with incomplete contracting, where each worker has control over the
investment levels in the various
which equates
'^Notice that
A''
if
x
activities. In
the text,
with the division of labor. Appendix
{i,j)s
we work with the production function
B shows
(3),
that the firm will indeed prefer to
= N^. Our assumption
an environment where these investments are not well
in fact measured in practice, especially using aggregate
were adequately measured, the index of productivity would be P2
that the x{i,j)s correspond to "effort" naturally
maps
into
measured. Moreover, P appears to be much closer to what is
data, than P2.
Note also that P measures "price-adjusted" productivity, because it is in terms of output rather than revenue.
An alternative concept, often used in practice because of data limitations, would be revenue divided by N, i.e..
Pi = A^'^Af'^'""'"'^'"^!''. Throughout, none of our conclusions depend on which of these two measures of productivity
are used.
'^It is also straightforward to
show that the
division of labor, investment levels,
and productivity
the marginal cost of investment, Cx- In addition, as long as parameter values are such that A''*
(and thus P*) are increasing in the elasticity of output with respect to the division of labor, k.
these comparative statics to save space.
>
1,
We
all
decline with
both
N* and
i*
do not emphasize
have the maximal division of labor given A^ as long as
/9
be interpreted either as treating the production function
production function
We
[0,1]
<
Therefore, the results here
satisfied.^^
model the imperfection of the contracting institutions by assuming that there
such that, for every task
investments in activities
j,
G
i
A
contract stipulates investment
the remaining
— ^
1
nonverifiable as in
Grossman and Hart,
and the workers.
We
follow the incomplete contracts literature
we assume that
•
The
{'i',j)]i=o
'
'''
is
•
Workers
The
threat point
To
way around,
are possible.
as follows:
0)}' f°r every task
G
i
upfront transfer to worker
The
this below).
parties have access to perfect credit markets, so ex ante transfers
all
in contractible activities
•
and assume that the ex post
game (more on
firm adopts a technology (a division of labor) with
by {[xc
weakness of
not to provide the services for the noncontractible activities.
is
firm, or the other
of events
or because of the
governed by multilateral bargaining, and we adopt the Shapley value as
is
of each worker in bargaining
The timing
fJ,,
ex post distribution of revenue between the firm
the solution concept for the multilateral bargaining
from workers to the
[0,/^].
Under these circumstances workers choose the investment
activities in anticipation of the
start with,
6
i
which can be enforced by a court. '•^ For
and Moore, 1990,
1986, Hart
in this society ).^^
on noncontractible
distribution of revenue
<
G
/z
ex ante contracts are not possible (either because the x {i,j)s are
activities,
contracting institutions
i
exists
are observable and verifiable.
[O./j]
Consequently, complete contracts can be written for the investment levels for activities
levels x{i,j) for
may
a primitive, or as derived from the
(3) as
with these additional requirements
(2)
a.^''
[0,/u.],
j
€
[0, A''],
A''
tasks,
where the Xc
and
(i,
denoted
offers a contract,
j)s are the investment levels
which have to be performed by worker
j,
and r
(j) is
an
j.
firm chooses TV workers from a pool of applicants, one for each task j.
G
j
[0, A'']
simultaneously choose investment levels x{i,j) for
contractible activities
•
The workers and the
•
Output
is
i
G
[0, fi]
they invest x
{i,j)
=
Xc
(i,
j) for
every
all i
G
[0, 1].
In the
j.
firm bargain over the division of revenue.
produced and
sold,
and the revenue
R (g)
is
distributed according to the bargaining
agreement.
We will characterize
a symmetric
the bargaining outcomes in
all
subgame
perfect equilibrium
(SSPE
subgames are determined by Shapley
for short) of this
game, where
values.
^''Provided that a mild regularity condition on diseconomies of scope
is satisfied. This regularity condition will be
example, when there are no economies of scope, or when there are sufficient diseconomies of scope.
'^In the general equilibrium version of the model developed in Section 6, the elasticity of substitution between
satisfied, for
varieties of final
goods
will
be 1/ (1-/3). The inequality
/3
<q
thus corresponds to the case in which tasks are more
substitutable than final goods.
^^For example, the court can impose a large penalty
if the employee does not deliver the contractually-specified
do not introduce these penalties explicitly to simplify the exposition.
For similar reasons, we assume that revenue-sharing agreements between the firm and the workers are not
level of services.
We
enforceable.
10
Equilibrium Formulation
3.2
The
which
in
A''
represents the division of labor, Xc
investment in noncontractible activities and f
every j €
i
G
SSPE can be
along-the-equilibrium path behavior in the
[0,yu]
the upfront payment
0,A^
and x
{i,j)
=
x„
for
6
i
(/-/.,
1].
workers have deep pockets), so f
(i.e.,
The SSPE
=
f
,
and the investment
is
distributed
among
Importantly, at this point of the
the Shapley values of this
ation in which
activities.
game
<
is
{N,
xe,
x„)
+ Ns^
game
To
activities,
Xc
and x„ are given.
in the next section
all
and
is
in
We
(N'^'^^XcXn
'^ )
discuss the computation of
and x„
iir
situ-
noncontractible
Sq
{N,Xc,Xn)
These values naturally exhaust the entire revenue,
x„ Xn) = ^^"^ (a^-^+^x^x^-^)
They
are,
i.e.,
.
At that stage
all
workers
know
the
however, free to choose the investment levels in the noncontractible
when the technology
j,
x„ (— j)
is
choice
is A'',
Xc
let
is
.s^;
[A', Xc,
x„ (— j) x„
,
Sx
be the Shapley
all
the workers other
same, since we are looking for a symmetric equilibrium), and x„
which was introduced above,
=
[j)]
the required investment in the contractible
the investment level in the noncontractible activities by
j (these are all the
Sx {N,Xc,Xn)
If
non-
in
Appendix A. For now, consider a
the investment level in every noncontractible activity by worker j.^^
i.e.,
Xc for
the game.
investment
is
R = A^~^
contractible activities
characterize the symmetric equilibrium,
value of worker
for
N and have to invest Xc in the contractible activities, as they have committed to
in their contract.
activities.
A'',
to one stage before the bargaining takes place.
technology choice
than
{N,
=
{i,j)
last stage of
Let s^ {N,Xc,Xn) denote the Shapley value of a representative worker and
Now move
do so
x
is
the workers and the firm according to their Shapley values.
denote the Shapley value of the firm.
Sg
levels are
is,
>
possible.
workers have invested Xc in
all
f
For now, we assume that there are perfect credit markets
contractible activities, then (l)-(4) imply that the available revenue
This revenue
TV, x^, x„,
the upfront transfer to every worker. That
is
investment in contractible activities and x„
is
<
the investment in contractible activities, Xn
is
be characterized by backward induction. Consider the
will
the level of technology, Xc
A'' is
t (j)
is
represented as a tuple
is
The
function Sx
(A'',
(j) is
Xc,x„),
the along-the-equilibrium-path value of s^ [N,Xc,Xn {—j) ,Xn
{N,Xc,Xn,Xn)-
We
(j)],
develop an explicit formula for this function in the next
subsection.
At
this stage of the
game, worker
x„
j chooses
(j) to
maximize
Sx
[A",
Xc,
x^ (— j) Xn {])] As
,
implied by the concept of equilibrium, the firm expects the workers to choose their best response
investment levels in noncontractible
activities:
x„ € arg max s^ [N,
This
is
x^, x„,
x„
(j)]
-
(1
-
p) CxXn {j)
the workers' "incentive compatibility constraint," and also imposes the
(9)
symmetry
require-
^^More generally, we would need to consider a distribution of investment levels, {xn (i,i)},gro jj for worker j/, where
some of the activities may receive more investment than others. It is straightforward to show that the best deviation
for a worker is to choose the same level of investment in all noncontractible activities, and to save on notation, we
restrict attention to
such deviations.
11
ment
(in
that x„
is
the best response to x„).^^
Next move one step backward
in the
from a pool of applicants. This pool
game
is
tree, to
empty
the stage in which the firm chooses
workers expect to receive
if
option. Therefore, for production to take place, the final
yields a net
reward
Sx
This
j
€
[0,
at least as large as the workers' outside option, that
(A'',
Xc, Xn, Xn)
+T>
+
fic^Xc
—
(1
/i)
c^Xn for x„ that
given
the workers' "participation constraint":
is
good producer has
to offer a contract that
is:
satisfies (9).
{xc,t)^ every worker
and noncontractible
activities
(when both she and other workers invest x„,
in every noncontractible activity). If the firm were to choose a division of labor TV
that did not satisfy this participation constraint,
The
of
its
(10)
N] expects her Shapley value plus the upfront payment to cover the cost of her investments
in contractible
{xc, t)
workers
than their outside
less
and the contract
A''
A''
firm chooses
A'"
and designs a contract
(x^, r) to
Shapley value ininus the upfront payments
the cost of
A''.
Consequently,
max
Sq
it
would not be able to
it
maximize
its profits.
(or plus the transfers
and a contract
hire
These
as in (9),
any workers.
profits consist
from the workers) and minus
solves the following problem:
{N,Xc,Xn)
- Nt - C{N)
subject to (9) and (10).
(11)
N,Xc,Xn,T
The presence
of perfect credit markets implies that the participation constraint (10) can never
be slack (otherwise, the firm would reduce
We
r).
can now solve r from this constraint, substitute
the solution into the firm's objective function, and arrive at the following simpler maximization
problem:
max Sq{N,Xc,Xn) + N[sx{N,Xc,Xn,Xn) -
fLC^^Xc-
{l-
iJ.)c^Xn]
- C{N)
subjcct to
(12)
(9).
N,Xc,Xn
The SSPE
tuple
<
N,Xc,Xn
>
and the corresponding upfront payment,
solves this problem,
f, sat-
isfies
f
=
HCxXc
+ {I-
IJ.)C:cin-Sx (N,Xc,Xn,Xn]
(13)
,
With a
3.3
slight
abuse of terminology, we refer to
<
A", Xc,
x„
\
as the
SSPE.
Bargaining
Before we provide a more detailed characterization of the equilibrium,
in this
game
(see Shapley, 1953, or
our solution as follows (see the proof of
'"In general, (9)
may have
derive the Shapley values
Osborne and Rubinstein, 1994). Since we have a continuum
players and the original Shapley value applies to
Xc by the firm.
below).
we
Lemma
games with a
1 in
Appendix
finite
A
number
of players,
for details).
we
of
derive
Divide the interval
N
multiple fixed points, and there can be multiple equilibria given the choice of
and
ensure uniqueness (see equation (18)
(3) and the assumption that a >
Our production function
12
[0,
N] into
M equally spaced subintervals with
by a single worker
(as in the discussion of equation (2) above).
game between
the bargaining
simplifies the solution
M+
the
Then
M workers and the
players;
1
performed
solve the Shapley values of
The key
firm.
that a coalition that does not include the firm, which
is
N/M
the tasks in each one of length
all
insight that
the only agent
is
that can operate the technology, has a value of zero and a coalition that includes the firm has a
value that equals the revenue from the final output
is
the limit of the solution of this
game with a
player's payoff
{5
...,
is
the average of her contribution to
,
...,
More
3 (M)} be a permutation of
M are the workers, and
the permutation
g.
We denote
—
let z^
{j'
G
by
So
Lemma
1
M—
is
Suppose that
all
cx)
>
(see
Aumann
Shapley values in a
>
(j)
{j')}
g
number
M+
1
of players each
coalitions that consist of players ordered
1,2, ...,M,
0,
a finite
game with
where player
be the
M+
is
players.
1
players, let g
=
the firm and players
set of players ordered
the set of feasible permutations and hy v
:
G
-^
Then the Shapley value
M.
below j
in
the value of
of player j
is
^[.(4uj)-t'(4)]
=
(^^
following result
game with
explicitly, in a
g
\
the coalition consisting of any subset of the
The
when
for a similar derivation of
in a bargaining
in all feasible permutations.
(0) ,5 (1)
1, 2,
of players
a continuum of players).
According to the Shapley value,
below her
number
finite
and Shapley, 1974, and Stole and Zwiebel, 1996b,
game with
can produce. The solution that we use below
it
+
1)'..G
proved in Appendix A:
M -^
and worker
00,
j invests x„ (j) in her noncontractible activities,
all
the other workers invest x„ (— j) in their noncontractible activities, every worker invests Xc in her
contractible activities, and the division of labor
N. Then the Shapley value
is
of worker j
is
(1-p)q
So:
[N, Xc, X„ (-j)
,
Xn
=
(j)]
(1
-
^fM^„ (_j)W-M) jv/'(-+i)-\
A 1-/3
7)
(14)
^n (-j)
where
1
A
number
of features of (14) are
noncontractible activities, x„
s. (AT,
x„ x„) =
(j)
-s.
=
a
x„
=
First,
fraction
is
if all
workers invest equally
in all the
Xn, then
x„, Xn)
=
(1
-
In this event, the joint Shapley value of the workers,
of the revenue (which
(15)
a+P
worth noting.
Xn{—j)
{N,
=
7) A^-^xf^x^(i-^)
A^'Sj, (A'',
equal to A^'/^x^^x^^^'^^ N'^^'^+'^^).
It
Xc, x^), is
A^^('^+i)-i.
equal to a fraction
(16)
1
-7
then follows that the firm receives a
7 of the revenue, or
s, (iV, .x„
X.)
= 7^^-^x^^^^\--) A^^^'^+i)
13
(17)
=
Second, the derived parameter 7
a and
rising in
declining in
That
/3.
+ /3)
a/ (a
represents the bargaining power of the firm;
a higher elasticity of substitution between tasks increases
is,
the competition between workers at the bargaining stage (since each worker
On
production) and reduces their bargaining power.^°
demand
in the price elasticity of
Finally,
the parameter
Si [N, Xc, Xn {-j)
is
,
Xn
(j)],
that in the limit as
for the final
is less
the other hand, an increase in
good) has the opposite
effect.
(3
essential for
(an increase
^^
also determines the concavity of the Shapley value of a worker,
a
with respect to noncontractible activities Xn
M^
it is
00, each
worker has an infinitesimal
The
(j)-
intuition for this
on total output, so she
effect
does not perceive the concavity in revenues coming from the elasticity of
demand
(related to
/3).
Instead, the perceived concavity for each worker simply depends on the substitution between her
task and the tasks performed by other workers, which
substitutable are the tasks, the less concave
on concavity
s^
(•)
regulated by the parameter a.
The more
The impact
in each worker's investment.
of
a
an important role in the results below.
Equilibrium
3.4
To
will play
is
is
characterize a
SSPE, we
max (1-7)A^"^
first
use (14) to solve x„ in
(9).
This
is
the solution
to:
^'-^^"
Xn
'
(j)
xf ^n (-jf^'-'^^ iV/3(-+i)-i _
e, (1
_
^) ^^ (j)
[-J)\
Relative to the producer's first-best, characterized above,
(1
—
7) implies that the worker
is
not the
full
we
see
residual claimant of
two
all
differences.
First, the
term
the investments she undertakes,
Second, as discussed above, multilateral bargaining distorts the
so she will tend to underinvest.
perceived concavity of the private return function.
The
first-order condition for this optimization problem, evaluated at .t„ (j)
=
Xn {—j)
=
^n,
q on 7 more precisely, recall that the Shapley value of the worker is an average of her
The impact of a on the marginal contribution of a worker depends on the
size of the coalition. Consider the marginal contribution of a worker to a coalition of n workers who cooperate with the
firm when the technology choice is N. We show in Appendix A that this marginal contribution equals (/S/a) (n/N)
R/n, where R = A^~ N^'^'^^' x^ Xn ^ is the total revenue of the firm. This marginal contribution is increasing in
a for small coalitions, i.e., when n/N < exp(— a/^S), and decreasing in a for large coalitions, i.e., n/N > exp(— a/^).
^"To understand the
effect of
contributions to coalitions of various sizes.
'^
Intuitively, with only a few other workers in the coalition, a technology that requires
smaller marginal revenues than one that requires inputs that are
more
complementary inputs generates
most workers
substitutable. Conversely, with
the coalition, the marginal contribution of a new worker is highest when technology features relatively high
complementarity. Since the Shapley value equals a weighted average of these marginal contributions, with the weight
being larger for larger coalitions (because there are more permutations leading to large coalitions containing the firm),
in
the impact of
a on the Shapley
value
is
also a weighted average of the
a worker to coalitions of different size and the negative impact of
worker,
—
=
impact of a on the marginal contributions of
receives
more
weight,
making the share
of each
+
jS) N, decreasing in q.
13/ {a
7) /N
intuition behind the negative effect of ^ on
(1
a
7 is similar. An increase in P increases the price elasticity of
and makes revenues less concave in output. This decreases the marginal contribution of a worker
to a coahtion of relatively small size and increases her marginal contribution to a coalition of relatively large size. In
particular, this marginal contribution is decreasing in /3 whenever n/N < exp(— q//3), and increasing in /3 whenever
n/N > exp (— q//3). As before, the Shapley value assigns larger weights to coahtions of larger size, so that the positive
impact of /3 dominates and 1 — 7 = /3/ (q -(- /8) is increasing in (3.
"'The
demand
for the firm
14
yields a unique solution:
Xn
=
= a (1
Xn {N,Xc)
7)(c,)-'xfMi-^iV^(-+i)-i
This choice of investment in noncontractible activities
rises
tractible activities, because different activities performed
marginal productivity of an activity
rises
(18)
is
(18)
with the investment
increasing in a. Economically,
it is
that investments in noncontractible activities
payments to workers
bargaining power.
and
this effect
is
a
(1
—
7)
=
a/3/ (a
the outcome of two offsetting forces. As noted above,
which enters the Shapley value of each worker,
is
the
(i.e.,
^^
activities).
are increasing in a. Mathematically, this simply follows from the fact that
is
level in the con-
by the worker are complements
with investments in other
Another important implication of equation
1/[1-/3(1-m)]
(1
+ /3)
—
7),
decreasing in a. This implies that the level of
,
decreasing in a, because greater substitution between workers reduces their
However, the
effect of
a on
the concavity of Sx () makes x„ increasing in a,
dominates the impact of a through
the level of payments to workers,
also
it
(1
—
7). In essence,
makes these payments more
though a greater a reduces
sensitive to their investments,
encouraging greater equilibrium investments.
Now
using (16), (17) and (18), the firm's optimization problem (12) boils
max A^-^ X^Xn
iV/5('=+l)
{N, Xc)^-^]
-
CxNilXc
- CxN
(1
-
^J.)
down
Xn {N, X,)
to
- C (iV)
(19)
,
N,Xa
where Xn {N,
and
Xc,^^
(NjXc)
Xc)
is
defined in (18). Substituting for x„
we obtain two
to (19) (see
first-order conditions,
Appendix
A
(A'",
Xc)
and
differentiating with respect to
N
which can be manipulated to yield a unique solution
for details):
1-)3(1-m)
A''
1-3
Ak/Ji-^Cj
1-/3
l-a(l-7)(l-M)
/3(l-A^)
c
'
1
-
/3
(1
-
[n)
,
(20)
m)
C N
(21)
K,Cx
The unique SSPE
is
then given by
<^
N, Xc, x„
>
such that
a (1-7) [1-/9(1-/.)]
/?[l-a(l-7)(l-M)]
^^
N on the level of
C'(-)
(22)
i^c.
ambiguous, however. In particular, investment in noncontractible activities
/3 (k + 1) < 1 and rises with the division of labor in the opposite case.
Intuitively, an increase in A'' has opposite effects on the incentives to invest for workers. On the one hand, because
technology features a "love for variety," a higher measure of tasks increases the marginal product of noncontractible
investments. On the other hand, the bargaining share of a worker, (1 - 7) /N, is decreasing in A''. When k — 0, the
necessarily reduces the investment Xnfirst effect disappears and an increase in
^'We show in Appendix A that the second-order conditions of this problem are satisfied under Assumption 1.
The
effect of
declines with the division of labor
Xr, is
when
N
15
Comparing
(7) to (21),
in contractible activities
we can
see that for a given level of
under incomplete contracts,
A'',
the implied level of investment
the investment level with
Xc, is identical to
complete contracts, x*. This highhghts that distortions in the investments
In fact, comparing (6) to (20),
are related to the distortion in the division of labor.
N
and N*
differ
in contractible activities
we
see that
only because of the two bracketed terms on the left-hand side of (20).
represent the distortions created by bargaining between the firm and
its
These
workers. Intuitively, the
choice of the division of labor will be distorted because incomplete contracts and bargaining distort
the level of investments in contractible and noncontractible activities, which are complementary
to the division of labor.
Notice that as ^
have (N,Xc] —^ {N*,x*).
That
is,
as the
zero, the incomplete contracts equilibrium
^
1,
and we
becomes
close to
both of these bracketed terms tend to
1,
measure of noncontractible
(N,Xc] converges
activities
to the complete contracts equilibrium
(A^*,x*).24
To examine the
is
now equal
effect of
incomplete contracting on productivity, notice also that productivity
to:
P=
The next
main comparative
Appendix A):
Proposition 2 Suppose that Assumption
Furthermore, (N,Xc,Xn,P)
Then
1 holds.
N,Xc,Xn >
given by (20), (21) and (22), with
(23).
(23)
proposition summarizes the key features of the equilibrium and the
statics (proof in
by
iV'^-i^4-^
0,
there exists a unique
SSPE In,Xc,x
and an associated productivity
level
P>
given
satisfies
Xr
Xfi <^
and
9N
die
^
Ofl
OfJ.
da
and
din
r.
'
(9q
r.
dP
r.
OjJ.
OfJ.
Qq
'
'
'
5q;
also
<0.
dadfj.
We therefore find that workers always invest less in noncontractible activities than in contractible
activities,
and that the division of
labor, investment levels,
-''Interestingly, however, it can be verified that a.s ^
noncontractible activities does not disappear as |z —» 1.
^
16
1,
Xn
and productivity are
^
x'
,
because the
all
increasing in the
effect of distortions
on the
market, the fraction of contractible activities and the elasticity of substitution between
size of the
tasks.
The
first
from dividing equation
result follows
(22)
by equation
(21) to obtain
x„_ a(l-7)[l-/3(l-M)]
where the inequahty follows from a
the
full
in the
(1
—
=
7)
a/3/ (a
+ /3) <
/3
(recall (15)). Intuitively,
residual claimant of the investments in contractible activities
the firm
is
and dictates these investments
terms of the contract. In contrast, investments in noncontractible activities are decided by
the workers, and as highlighted by (16), they are not the
full residual
claimants of the revenues
generated by these investments.
The
analysis of comparative statics
by Assumption
ditions (which are ensured
side of (20) will also increase
side of (20)
Appendix
is
A
N. As
1),
any change
in the case of
shows that the left-hand side of
when
is
Both
results are intuitive.
greater
An
there
increase in
of this parameter
is
and
in A,
ju.
or
a that
increases the left-hand
(20)
is
also increasing in
/j,
and
a, so that the division of
contract incompleteness and less technological complementarity.
less
Incomplete contracts imply that workers underinvest in noncontractible
/i
increases the degree of contractibility and thus the profitability of
on the concavity
increases workers' investments,
The
fact that, given the second-order con-
complete contracts, we find that the left-hand
further investments in technology (division of labor).
a
by the
increasing in A, and thus the division of labor again rises with the size of the market.
labor
activities.
facilitated
is
The
positive effect of
a stems from the
effect
of the Shapley value of workers, s^ (), discussed above. Greater
making
division of labor
more
profitable for the firm.
other comparative static results in Proposition 2 then follow from eciuations (21), (23),
(24). In particular,
from equation
(23), equilibrium productivity
of the market, the degree of contractibility
division of labor
and
is
also increasing in the size
and the substitutability of
— given the division of labor — the
levels of
tasks, because
both the
investment in contractible and
noncontractible activities are increasing in these variables. ^^
It is
noteworthy that the
effect of
a on productivity
is
of the opposite sign as its effect in
standard Dixit-Stiglitz formulations of technology that do not include the Benassy (1998) term.
In these formulations, the larger
is
the complementarity in production, the larger
of productivity to an increase in the division of labor.
neutralizes this effect
and helps
isolate a
Our
is
the response
specification of technology in (3)
new channel by which
technological complementarity
affects productivity.
Another
result,
which
will play
an important role in Section
6, is
d'^N/dad/j,
that better contracting institutions have a greater effect on the division of labor
greater technological complementarities.
The
detrimental effect on the division of labor
intuition
when
there
is
is
^°Also, as noted in footnote 13, the division of labor, investment
and increasing in k. as long as N' > 1.
17
<
0.
when
It implies
there are
that contract incompleteness has a
more
greater complementarity between tasks.
levels,
and productivity are decreasing with
c^
because the investment distortions are larger
Finally,
it is
useful,
in this case.
both to further illustrate the intuition
and the
to note that combining (18), (19),
of the firm can be expressed as (see
TT
for the results
first-order condition
and
with respect to
for future reference,
Xc, the profit function
Appendix A):
= AZ {a, 1^) N^+^^^^^ ~ C (N)
(25)
where
l-g(l-M)
l-a(l-7)(l-A^)
iS(i-t^)
;3m
1-/3
/3
(c,)-i^
(26)
1-/3(1-/.)
and as before, 7
= a/{a + /3).
Z{a,fi)
is
thus a term that captures "distortions" arising from
incomplete contracting.
The comparative
static results regarding the effect of
follow from the fact that
profit function will
Z (a, /i)
become
is
a and
^u
on the division of labor simply
increasing in both variables. This (equihbrium) reduced-form
useful in our general equilibrium analysis. It also
makes other potential
applications of this framework quite tractable.
The above
analysis also enables us to derive a straightforward comparison of equilibria with
and without complete contracts. Recall that the incomplete contract equihbrium converges on the
complete contract equilibrium when
/i
—+
1.
Proposition 3 Suppose that Assumption
Together with Proposition
2,
this
imphes
Let <N,Xc,Xn> be the unique
1 holds.
incomplete contracts and {N*,x*} be the equilibrium with complete contracts. Let
the productivity levels associated with each of these equilibria.
N < N*,
Xn
< Xc<
X*,
and
P<
SSPE
P
with
and P* be
Then
P*
This proposition implies that, with incomplete contracts, the equilibrium division of labor
is
always suboptimal.^^ Furthermore, investments in both contractible and noncontractible activities
are also lower than those under complete contracts.
productivity with incomplete contracts
is
As
a result of these effects, the level of
also lower than that with complete contracts.
Before concluding this section, and in order to facilitate the transition to the analysis of imperfect credit
markets, consider the implications of the equilibrium for the ex ante transfer
positive, the firm has to
make an upfront payment
to every worker,
and
if it is
f.
If
r
is
negative, the firm
^^It is useful to contrast this result with the overemployment result in Stole and Zwiebel 1996a, b). There are two
important differences between our model and theirs. First, in our model, there are investments in noncontractible
activities, which are absent in Stole and Zwiebel. Second, Stole and Zwiebel assume that if a worker is not in the
coalition in the bargaining game, then she receives no payment to compensate for her outside option. Since in our
model whether the worker provides the services in the noncontractible tasks is not verifiable, whether or not she is
(
in the coalition
payments
cannot
affect her
wage except through bargaining, and consequently she
for her contractible investments.
result in Stole
The treatment
of the outside option
and Zwiebel.
18
is
will
continue to receive the
essential for the
overemployment
We
ought to receive an upfront payment from every worker.
transfer in the
use (13) to calculate the size of this
SSPE.
First note that s^ fiV, x^,
equilibrium revenue,
i.e.,
5„,5„)
= {l-'y)R/N,
R =
where
every worker receives an ex post payment equal to the share (1
of the revenue (her Shapley value).
Second note that
the
A'^'I^N^'^'^+^'^i^^x?,^'^'^^ is
7)
= a(l — j)R/N
c^Xn
(18) implies
—
/N
as a
Combining these
result of the worker's optimal choice of investment in noncontractible activities.
observations with (13) and (24) implies that
/5-a(l-7)
a—
+,^
;
1
(1-7)5.
N
(1-7)[1-/3(1-m)]
When 7 < I— /3,
compensate
to the firm to
7
>
1
— ^,
r
this transfer
is
is
0) for all
no longer necessarily negative.
more complete than
It
i.e.,
/j.
<
f
>
0.
this,
e
/^
[0, 1]
and workers make upfront payments
they earn in the bargaining game. However, when
for the ex post rents
contracts are sufficiently incomplete,
contracts are
<
negative (f
{1
can be shown that
—
a) {1
—
/3)
it
/ {j3/ (1
will
—
be negative
7)
—
more
likely
—
fxCxXn
make an upfront payment
case the firm has to
is
(•)
when
/3)
+
and only
a/3).
if
When
This result stems from the fact that investments in
contractible activities are chosen by the firm and set at a level greater than
have chosen themselves. As a result s^
+
(a
if
—
—
(1
m) CxXc can
what workers would
become
negative, in which
in order to secure workers' participation. This case
a greater fraction of activities are contractible.
Imperfect Credit Markets
4
The above
analysis
presumed that workers and firms had access
same weak contracting
institutions that
make
to ex ante transfers. However, the
certain activities performed
by the worker-suppliers
noncontractible also create credit market frictions and other capital market imperfections, which
may
prevent such transfers (or "bonding" arrangements).
We now
analyze the same niodel with severe credit market imperfections that
transfers from workers to the producer of the final
Assumption
2
Ex
good
is
ante
impossible:"'^
ante transfers from workers to firms are not possible,
This extreme form of credit market imperfections
make ex
useful to focus
i.e.,
r
>
0.
on the interaction between
incomplete contracts and imperfect credit markets. Since we assume that the costs of investments
Xn and Xc are non-pecuniary, credit market imperfections do not directly prevent or
investments." This assumption
is
therefore in line with our interpretation of the x
(z,
restrict these
j)s as potentially
nonverifiable "efforts"
Recall that at the end of the Section 3
f
<
0.
If this
condition
is
we provided
a necessary and sufficient condition for
not satisfied, then workers are not required to
make ex ante
Then, as long as the firm has deep pockets, the equihbrium from the previous section
^'
^*
Given Assumption 3 below, whether or not the firm has deep pockets is not relevant.
And if the firm cannot make ex ante transfers either, then the equilibrium characterized
19
transfers.
prevails.
in this section applies.
We now
proceed to analyze the equilibrium with binding credit constraints. This requires
Assumption 3
^
Given Assumptions
difference
r
>
0,
is
we again look
1-3,
for the
(l-a)(l-/3)
symmetric subgame perfect equilibrium. The main
that in terms of the optimization problem (11),
which
is
under Assumption
(only) binding
Sx {N, Xc, Xn, Xn)
>
jJ-CxXc
+
(1
—
/i)
3.
we now impose the additional
Therefore, the participation constraint becomes
The maximization problem
CxX^.
constraint
of the firm then
becomes
max Sq{N,Xc,Xn) — C {N)
subject to (9) and S^ [N, Xc, Xn, Xn)
The problem can be
be binding. To see
further simplified
jJ-CxXc
+
—
(1
/i)
CxXn-
by noting that the participation constraints
note that Sq {N,Xc.Xn)
this,
>
is
strictly increasing in Xc, so if the participation
constraint were slack, the firm could increase Xc and thus Sq{N,Xc,Xn)-
and
(17)
(18)
will always
and imposing the participation constraint
as an equality,
Therefore, using (16),
we obtain the
simpler
maximization problem
max TTl^-'^xf^x^^i-^) AT^^'^+i) - C {N)
subject to
(27)
N,x,
a{l-j) Ai-^xf^x^(i-'^)iV^('^+i)-i =
(1
The two
- 7) Ai- V^a:^(i-^)iV^(-+i)-i =
f,cxx,
+
c,x„,
(1
-
m) c,x„.
constraints of this problem can be used to solve the investment levels in contractible and
noncontractible activities as functions of the technology choice
Xn
1
— c-'a{l-^)A'-^N^^^+'^-'
—Q+
A''.
These investment
ajj.
Pn
levels are
1/(1-/3)
(28)
ajj.
1/(1-/3)
c-'a
An immediate
(1
-
implication
7)
is
^i-/3iV^(-+i)-i
(Ll^_+^
Xr
are, respectively,
activities. Interestingly, this ratio
as
jj.
—>
0,
1
—a+
Q'/-i
(29)
that
a/j,
X^n
where Xc and x„
/3ij.'
1
(30)
— a + a/x
the equilibrium investments in contractible and noncontractible
does not depend on
/3
and approaches zero
as
/^
^
0. Intuitively,
there are only a few activities that the firm can use to extract the surplus from the
workers, and there will be greater overinvestment in these activities.
Now substituting
(28)
and
(29) into the firm's objective function in (27),
20
we obtain the
following
maximization problem:
m^X7
The
[c-^a
(1
-
first-order condition of this
^-^
7)]
AiV^^^+^ - C (iV)
(
^,
(31)
J
problem characterizes the equilibrium technology
N?^
It is
given
by
Using
this
.
j3(^+i)-i
^
"'
Rk^A
/I
l^V
-
n-
+
0H
rv/;A i-/3
.
1
,-,-^
,
w-n
.
[c-'a{l-j)]^-^=C'{N).
^^J
^
(32)
equation and the equations for the investment levels (28) and (29), the equilibrium
investment levels in the credit constrained equilibrium can be represented by
C (N)
a(l-'y)
,,.;i(Lz2)(i„g)( '-° +
Comparing these equations with
Xc
>
Xc
and x„ >
That Xc > x^
x„.
(21)
and
(22)
in this case
is
extracts the surplus from the workers by forcing
when
°>'
)g:W.
(34)
N = N,we note that Assumption 3 impHes
again a consequence of the fact that the firm
them
now
^"^
to "overinvest" in contractible activities.
Naturally, however, the division of labor differs in the two equilibria, and this will further affect
the equilibrium investment levels.
Analogously to before, we define equilibrium productivity (now without ex ante transfers) as
P=
We
have (proof
in
N^xi^xl-^.
(35)
Appendix A):
Proposition 4 Suppose that Assumptions
1-3 hold.
Then
there exists a unique
ex ante transfers, {N,Xc,Xn}, given by (32), (33) and (34), with N,Xc,Xn
productivity level
P>
'"Moreover,
<v,
and an associated
Xqj
again ensures that the second-order condition is satisfied.
no longer the case that as /^ —> 1, £c -^ a^c, since the firm will use the contractible activities to
1
it is
extract surplus for
without
given by (35). Furthermore, [N,Xc,Xn,P) satisfies
XtJ
'^Assumption
>
SSPE
all yu
<
1.
21
and
^
dA
>
^'
'
'
Ojl
Ojl
Ojl
OjJ.
and
^>0 ^>0
^>0
dA- dA - dA
also
«'*
<
0.
dadfi
As
than
in the case of perfect capital
in contractible activities.
2 above,
workers are not the
their investments.
markets, workers always invest less in noncontractible activities
The reason
full
for this
is
two-fold. First, as
was the case
in Proposition
residual claimants of the increase in productivity generated
by
Second, in the presence of credit constraints, contractible investments are an
instrument of surplus extraction by the firm, which forces workers to overinvest in these activities.
The comparative
statics in Proposition 4 related to the division of labor
noncontractible activities are identical to those derived in Proposition
2.
and to the choice
In particular,
we
of
find that
the division of labor and the investment in noncontractible activities are increasing in the size of
the market, the quality of the contractual environment, and the elasticity of substitution between
tasks.
Furthermore, a higher degree of contractibility and lower technological complementarities
also lead to a lower
gap between the investment
levels in contractible
and noncontractible
Moreover, better contracting institutions have a larger impact on the division of labor
is
activities.
when
there
greater technological complementarity.
Notice, however, that contrary to Proposition
now need not be
activities Xc
tutability of tasks).
2,
equilibrium investment levels in contractible
increasing in the quality of contracting institutions (or in the substi-
The reason
for this
is
again the dual role played by contractible activities; in
addition to their productivity enhancing role, they are also instruments for surplus extraction by
the firm.
When
the degree of contractibihty
is
low, there are fewer activities in
which the firm can
may
increase. Similarly,
extract surplus, so overinvestment in the remaining contractible activities
when
there
is
(recall that
7
The
is
is
increasing in a) and the incentive to set a relatively high Xc
potential for overinvestment also has implications for productivity.
productivity
it
limited substitution between tasks, the share of revenue accruing to workers
is
is
is
higher
greater.
Although, as before,
increasing in the size of the market and the quality of the contracting institutions,
no longer monotonic
in a.
This
is
because, as noted in the previous paragraph, a higher
a
reduces the share of revenue accruing to workers and thus the amount of surplus to be extracted
through the choice of
Xc-
reduction in productivity.
Other things equal,
this lower incentive to overinvest in Xc leads to a
The
a on
overall effect of
P
is
thus ambiguous.
More
interestingly, the
next proposition shows that productivity can be higher with imperfect credit markets than both
22
the case with incomplete contracts and perfect credit markets and the case with complete contracts
Appendix A):
(proof in
Proposition 5 Suppose that Assmnptions
contracts problem,
be the unique
let
N,
<
Xc,
SSPE without
Xn
\
1-3 hold. Let {A''*,x*}
SSPE
be the unique
ex ante transfers. Let P*
be the solution to the complete
with ex ante transfers, and
P, and
,
P
[N,
let
Xc,
x„}
be the productivity attained
in
—
^,
each of these three environments. Then
and
N* >
Suppose further that k <
there exists a unique
Jl
G
1
—
Then
/3.
(0, 1),
N > N.
P>P
{[)
such that
fJ,
for all
(0, 1);
P<
implies
<J1
€
/j,
and moreover
P* and
fi
>
Jl
(ii)
if
7 <
P>
implies
1
P*.
Credit market problems therefore increase the spread between investments in noncontractible
and contractible
and further reduce the division
activities
of labor.
Intuitively, the firm
is
now
obtaining part of the returns by inefficiently extracting the surplus from the workers by means of
high levels of Xc (and this
is
diminishing returns to Xc). This naturally
inefficient, since there are
N
> N.
that P > P
reduces the "productivity" of further increases in the division of labor. Therefore,
More
when
A''
of labor,
interestingly, because Xc
< N.
may be
higher than Xc,
it
now becomes
possible
even
In other words, although credit constraints always depress the equilibrium division
measured productivity can be greater because
of overinvestment in contractible activities.
In fact, the latter will necessarily be the case whenever the productivity gains from division of
labor, represented
by
k, are relatively
for productivity in the equilibrium
low
<
{k.
1
—
Moreover, in this case,
/?).
it is
also possible
with imperfect capital markets to exceed that of the
first
(i.e.,
P>P*).3i
5
The Employment Relationship Versus Outside Contracting
We now
use our framework to discuss the choice between the employment relationship and an
organizational form relying on outside contracting.
Our
analysis so far
assumed that the threat
point of each worker-supplier was not to deliver the noncontractible activities (which
not delivering the entire
of organizational form,
fraction 1
— 6 of their X
X (j)
Conversely,
may dominate.
given the Cobb-Douglas structure in (4)).
we assume
[j]
contractor (in the spirit of
'''^
best
depends on whether she
Grossman and Hart,
when k
is
In fact,
when k approaches
relatively large (k
P' > P > P. Notice also that
makes P > P* possible.
5
in
>
1
— /3),
1986,
is
equivalent to
To endogenize the
that the threat point of a particular agent
and suppose that
is
is
choice
not to deliver a
an employee or an outside
and Hart and Moore, 1990). Our analysis
the productivity loss associated with a limited division of labor
upper bound dictated by Assumption 1, it is necessarily the case that
the absence of perfect credit markets, we have limf,_i P ^ P* and this is what
its
,
23
=
above therefore corresponds to the case where 6
Lemma
3)
demonstrate that
the results so far hold for any value of
all
of outside (arms-length) contracting,
=
relationship, 5
but the results in this section
0,
we assume
also denote the choice of organizational
E
outside contracting and
organizational form, O,
following generalization of
2
activities,
Denote Sq
Lemma
=
Lemma
=
0-
O
form by
G {C,E}, with
1
C
We
corresponding to
assume that the
A'"."^'^
The
rest of the
needs to be modified to account for the fact that
game now depend on the
choice of O.
proved in Appendix A:
1 is
5 and suppose that
M ^ oo, worker j invests x„
(j) in
her noncontractible
the other workers invest Xn {—j) in their noncontractible activities, every worker
all
invests Xc in her contractible activities,
worker j
and with an emplo3rment
chosen concurrently with the choice of technology
is
the Shapley values in the ex post bargaining
Lemma
[0,1),
corresponding to an employment relationship.
analysis continues to apply except that
The
Sc &
In particular, in the case
6.
(^c, 1) because the assets the agent uses are the property of the firm.'^^
6e ^
Moreover, to simplify the exposition we normalize 5c
We
=
that 6
(in particular,
and the division of labor
N. Then the Shapley value
is
of
is
^'
S^ [N, Xc,
Xn i-j) Xn
,
=
{j)]
(1
-
''^%fM^^(_j-).e(i-M);V/3(K+i)-i^
t) A^'^
.Xn{-3)_
where
^- (.o)(i-i°)
Imposing symmetry
straightforward to show that 7 again represents the fraction of revenue
it is
accruing to the firm in the bargaining, with
of the
(1
—
7)
/N
representing the fraction assigned to each
N workers.
Several features of this result are worth mentioning. Notice
values (and thus the
When
(5
>
0,
7
and obtains a
7
is
SSPE
of the
1,
greater than a/ (a
and the firm has
following result provides a
Lemma
3 For
increasing in
all
S
G
[0, 1),
a and a
(1
—
7
is
7)
first
that
game) are identical to those described
+ /3),
so the firm
is
full
—
0,
the Shapley
in the bargaining
1,
game
the bargaining share
bargaining power.
increasing in
is
S
in the previous sections.
where 6 goes to
more complete characterization
/Pj
when
now more powerful
larger fraction of the revenue. In the limit
also goes to
The
is
<^^'
-
a and
nondecreasing
5
of the bargaining share 7:
and decreasing
in
/3.
Furthermore, a
(1
—
7)
in a.
To express these issues more formally, we could introduce X^ {j) as the efficiency units of services provided
by worker j. Then, in the case of disagreement with an outside contractor, X^ {j) = 5cX (j), and in the case of
disagreement when there is an employment relationship, X^ [j) = 5eX (j).
For notational simplicity, we assume that the firm chooses either to have an employment relationship for all of
the tasks or for none (given the symmetry of our setup this is without loss of generality).
24
Using
lemma,
this
respect to
and a
/.i
a and a
(1
—
through
continue to hold. In par-
5)
simply depended on the following properties: 7 and a
7) //37
positive values of
the results regarding the comparative statics with
all
in the previous sections (Propositions 2
ticular, these results
in
can be shown that
it
is
Lemma
nondecreasing in a.
(1
—
7) are increasing
3 implies that these properties hold for all
Consequently, conditional on the equilibrium choice of organizational form.
6.
Proposition 2 through 5 continue to apply.
We
next turn to discussing the equilibrium organizational form and
of labor.
exists a
of the
The
Consider
SSPE
unique
SSPE
<
iV, Xc,
in
simplified since the Envelope
is
7 and therefore
declining in
increasing in x„, which
is
We
given by (20), (21) and (22).
[
to include the choice of organizational form,
analysis
Remember
the model with perfect credit markets.
first
Theorem
i.e.,
its effect
<
on the division
that in this case there
can now expand the definition
O, N,
Xc,
Xn
>,
where
O
G {C, E}.
implies that the firm's profits in (19) are
in 6 (to see this, recall that since
x„
<
x*, the level of profits
decreasing in 7). This immediately implies that the firm prefers
(5
=
is
to
any positive S and establishes:
Proposition 6 Suppose that Assumption
holds.
1
Then
in the
unique
SSPE
with incomplete
contracts and no credit market imperfections, the equilibrium organizational form involves outside
contracting,
i.e.
O=
C.
In the absence of credit constraints,
it
never optimal for the firm to hire the agents as
is
This stems from the fact that increasing the bargaining share of the firm reduces
employees.
the incentives of workers to invest in the noncontractible activities, which in turn reduces the
marginal product of investing
in contractible activities
and
in the division of labor."^^
In other
words, because worker-suppliers are the only agents undertaking noncontractible investments, the
efficient allocation of
power
entails giving
them
as
much ex
perfect capital markets this efficient organizational form
an employment relationship
is
is
attained with outside contracting, and
never optimal.^^
Let us next turn to the case with credit constraints. In this case, though 5
revenue, the firm
is
increase this share.
The employment
To show
this,
From problem
is
made
to
^^Formally,
relationship
from the workers
we proceed
(31)
(in
addition to Xc).
in a
manner analogous
and the Envelope Theorem,
maximize 7
we
maximizes
find that
(1
—
~
7)
dN/dS <
0,
-
it is
Because 7
dxc/dd <
0,
is
We
depends on the
fact that the firm
may want
to
find that in this case the possibility
division of labor.
to that in the case with perfect credit markets.
clear that the choice of organizational form
monotonically increasing
and dxn/dd <
0.
in 6, a
O
higher level of
This can be estabUshed by noting that
the only effect of 5 on the equilibrium value of TV works through 7, which is in turn increasing in
left-hand-side of equation (20) and Xn/xc are increasing in q (1 — 7), and thus decreasing in 5.
•^^This result
sales
then potentially useful as another instrument
is
employment relationship may indeed enhance the
to create an
=
constrained to obtain only a fraction 7 of the revenue ex ante and
for surplus extraction
With
post bargaining power as possible.
5,
and
(ii)
(i)
the
does not make any relationship-specific investments. Otherwise, a
greater share of surplus for the workers would distort the firm's investment incentives (see, for example, Hart and
Moore, 1990, Antras, 2003, or Antras and Helpman, 2004).
25
6 lowers this
then 6
=
term whenever 7 >
maximizes
1
and
profits
—
inequahty holds for
If this
/5.
O = C
6
e
1) (i.e.,
[0,
Therefore,
the best strategy.
is
all
a+P >
1),
—
the
when 7 >
1
,5,
equilibrium organizational form involves outside contracting.
More
interesting results obtain
>
case there exists a unique ^
when
all
when 7 <
such that
1
if
—
/3
for
ds €
some
(O, S)
,
e
5
[0,
1) (i.e.,
a
+ /3 <
In this
1).
then the firm's profits are maximized
agents are hired as employees. Under these circumstances the equilibrium organizational
form involves the employment relationship between the firm and the workers. Furthermore, the
threshold 5
is
by
implicitly defined
^a4-/3
1-
1-r
This threshold
is
decreasing in
a
=
Appendix
(see
Proposition 7 Suppose that Assumptions
transfers, the equilibrium organizational
O = C when a + > 1.
(ii) If a +
< 1, then there
(i)
form
O
Then
0.
(37)
This discussion leads
for details).
1-3 hold.
>
in the
unique
SSPE
to:
without ex ante
satisfies:
/3
/?
for
A
for 5
1
6e >
The
6.
Furthermore, 6
exists a
(i.e.,
a+
may
such cases, the firm
(3
(0, 1)
such that
O =
£" for
6e <
6
=C
and
decreasing in a.
is
intuition behind this result
complementarity
unique 5 E
<
1),
is
as follows.
When
there
a high degree of technological
is
the Shapley value of the firm tends to be relatively low too. In
be able to increase
its profits
(and
its
incentive to invest in the division
of labor) by hiring the agents as employees. In doing so, the firm trades off a larger fraction of the
revenue
for a smaller level of
revenue (due to the reduced investment incentives of workers).
A similar logic explains the last statement in Proposition 7:
parameter space
when
there
is
in
which the employment relationship emerges as an equilibrium outcome
is
larger
greater complementarity between tasks (because, with greater complementarity, the
firm captures an even smaller fraction of sale revenues
more
other things equal, the region of the
likely to prefer
O=
This result relates
to,
Our
dealing with outside contractors,
it is
E).
but
is
different from, Marglin's (1974)
relationship encourages the division of labor as a
power of workers.
when
result implies that the
of reducing the bargaining
way
argument that the employment
of reducing the
autonomy and bargaining
employment relationship
power of the worker-suppliers, and
this
itself
emerges as a way
encourages further division of
labor .^^
is also related to Aghion and Tirole (1994) and Legros and Newman (2000), who develop propertymodels of the firm in which parties have a Hmited ability to transfer surplus ex-ante. An important distinction
between our approach and theirs stems from partial contractibility. In particular, the presence of partial contractibility implies that, even though vertical integration (the employment relationship) is a way of transferring resources
from worker-suppliers to the firm, it emerges in equilibrium only when it is constrained-efficient and increases the
equilibrium division of labor. This is different from existing results in the literature, where credit constraints lead to
^^This result
rights
inefficient integration decisions.
26
General Equilibrium
6
The theory
of the division of labor that
and
applications,
illustrate a
we have developed
We
in particular in applications that require general equilibrium interactions.
number
of such applications in
the case with perfect credit markets, and
contracts.
simple enough to be used in various
is
Thus throughout the
section
what
To save
follows.
we hmit the
space,
we do not present the benchmark
we have
<
/x
<
1
and
5
=
analysis to
results with
complete
Under these circumstances
0.
the firm solves problem (19). Recall from Section 3 that equihbrium profits take the simple form
IT
=
AZ{a,iJ.)N'^^
representation to
embody our
(see (25)), with Z(a,/i) given
model
partial equilibrium
by
(26).
We now
in a general equilibrium
exploit this
framework.
Basics
6.1
Assume
the
- C{N)
'i-'S
that there exists a continuum of final goods q{j), with j E [0,Q], where
number (measure)
Consumer
of final goods.
Q
represents
preferences are given by the quasi-linear utility
function:
/
fQ
u=i
where
e is
between
N
V^
q{zfdz\
<
-c^e,
/?
<
(38)
1,
the total effort exerted by this individual as a worker,^^ and the elasticity of substitution
final goods,
1/(1-/3),
is
greater than
so that final goods are gross substitutes in
1,
consumption.
As
is
where p
is
well
[z) is the price of
good
the ideal price index, which
ences imply the
demand
function, where
A=
We
{z)
=
and p^
=
known, these preferences imply the demand function q
z,
S
we
function
is
the aggregate spending
A (p {z))~
'
~^' that
p^
=
f^
p
{i)~^'^ ~^' di
1.
In other words, these prefer-
we have used
in the derivation of the profit
will take as numeraire,
'
level,
S/p^
[p (z) /p^]
i.e.,
S.
assume that firms
(sectors) differ according to their technological complementarity, a,
and
denote the cumulative distribution function of a by iJ (a) with support as a connected subset of
(0, 1).
This implies that
produced with
The key
w
and
let
Q
products are available
come
general equilibrium interaction will
Assume
that there
is
determined
consumption, a fraction
for
(1
—
in general equilibrium
with
them
of
are
competition between sectors (firms) for
in fixed
rate in terms of real
all
H (a)
a).
one scarce input, labor,
up and manning organizations. The wage
is
for
than 1/
elasticities of substitution smaller
scarce resources.
setting
if
supply
L
that
used for
is
denoted by
More
specifically,
consumption
other endogenous variables.
is
us assume that
C [N] = wCl
where Cl {N)
is
the amount of labor needed to construct and operate a production facility with
the division of labor
''^In particular,
(iV)
A'^.
For example,
if all
that a firm needs to do
is
to
make
sure that
N
agents
the cost of effort Cx corresponds to the constant marginal rate of substitution between effort and
consumption.
27
when other
highly nonlinear
Assumption
'
Using
now
1
Cl
needed to
set
a
Bk
1-/3
{a,
a fraction
jj.
But
up the organization
this function
can be
of production. Clearly
(N).
this notation, the first-order condition in the
^
A''
activities are
applies to
stitution parameter
where
= N.
perform the productive tasks, then Ci (N)
act as workers to
maximization of (25)
for a firm
with sub-
is
AZ {a,
jj.)
N {a,
j3(>c
^i)
+ l)-l
i-s
= wC"l[N (a,
/.l)]
for all
,
a €
(39)
(0, 1)
denotes the division of labor of the producer with complementarity given by a
fi)
when
of tasks are contractible.
This equation shows that the key price that needs to adjust
=
follows to define a
A/w, which can be thought
w/A.
is
It is
convenient for what
of as an adjusted extent of the market,
and rewrite
(39) as:
Rif
-aZ
Although a
is
{a,
endogenous
ii)
/3(k+1)-1
N [a,
^-P
j^)
=
C'^ [A^ (a,
yit)]
for all
,
in general equilibrium, each firm takes
a G
(0,1).
as given. Proposition 2 implies
it
that firms with higher elasticities of substitution choose finer divisions of labor.
Therefore this
equation implies a positive correlation between the degree of substitutability of tasks and the
division of labor in the cross-section of firms.
Given each firm's desired
level of division of labor,
A''
(a,
This implies that market clearing for labor can be expressed
demand
left-hand side describes the
This market clearing equation
Proposition 2 implies that
A'"
fi) is
to increase the division of labor for
not
all A'' (a, fi)s
three alternative models.
6.2
Ci{[N
(a,
/i)].
L.
(40)
for labor while the right-hand side describes its supply.
rising in
all
We now
so
//,
we may expect
better contracting institutions
producers. But the resource constraint, (40), implies that
can increase simultaneously.
adjust to clear the market.
is
generate the main general equilibrium results in this section.
will
(a,
for labor
as'^^
Q f Cl[N (as /i)] dH (a) =
Jo
The
demand
their
/i),
Consequently, the endogenous variable,
examine the implications
a,
has to
of the competition for labor using
^^
Exogenous Number of Products
In this example
produced by a
of labor
More
is
we assume that the number
different firm.
The
increasing in the extent of the market a
wage
will
is
labor market clearing condition
formally, (40) should hold with
see that the
of products, Q,
given,
is
and that every product
is
given by (40). Since the division
and the cost function Cl {N)
complementary slackness together with
w >0,
but
is
it is
increasing in
A^",
straightforward to
always be positive.
''We have also worked out an endogenous growth model with expanding product variety, in which new entrants
draw an a from a known distribution. In this model, the long-run rate of growth depends on the degree of contract
incompleteness. We do not discuss this model in order to save space.
28
demand
the
for labor,
a unique solution for
the left-hand side of (40),
i.e.,
Using this value of a
a.
is
an increasing function of
a,
and
(40) yields
then determines the cross-sectional variation
in (39)
in the division of labor.
To
and
illustrate general equilibrium interactions,
in particular, the implications of differences
compare two countries that are
in contracting institutions,
identical, except for their
/xs.
Differen-
tiating the first-order condition (39) yields
+
a
C^(Q,/z)/i
=
A[A^(a,^)]iV(a,/.),
where y represents the proportional rate of change of variable
Z (a,
[/3(/.
with respect to
i.l)
+ l)-l]/(l-^),
and A
/j,
(iV)
A
1
that (^ {a,
>
fj.)
is
the elasticity of
C£ (N) minus
i.e.,
implies that A
i.l)
(^ (a,
the elasticity of the marginal cost curve
is
C"{N)N
.f.r,-
Assumption
y,
(41)
and ^^
>
(A'")
(a,
//)
Moreover, we show in the proof of Proposition 2 in Appendix
0.
is
I3{K+1)-1
declining in a.
Obtaining the relationship between
jj,
and a
straightforward. Differentiating (40) yields
is
(a) iV (a, m) diJ (a)
f ai
=
(42)
0,
Jo
where ai
tween
jj,
(a)
and
=
C[_
[N
{a,
jj,)]
N {a,
Substituting (41) into (42) gives a simple relationship be-
fi).
a,
.
establishing that a
is
^
i;
declining in
<JL (g)
/z.
C^ (g,
y-)
A [N
In other words,
(g, ^x)]-^
when
dH {a)
,
contracting institutions improve, wages
increase relative to expenditure and a will decline.
Equation (42) also shows that
N (a,
jj,)
(in
response to
/i)
can only be positive for some as and
has to be negative for others, because the resource constraint implies that not
their division of labor.
The question
low a firms. The answer
for high a's
is
and positive
for
more contract dependent,
high
i.e.,
a
firms.
This
clear
is
from
low
i.e.,
is
firms can improve
therefore whether the division of labor improves in high or
(41): since the left-hand side
is
decreasing in a,
it is
negative
Therefore, the division of labor improves in firms that are
a's.
low a firms, and worsens in firms that are
a reflection of the negative cross partial d(^
there exists a critical value a^ such that
a >
all
N (a,
/n)
>
for all
less
{o',fJ')
a < q^ and
contract dependent,
/da. More precisely,
N (a,
/i)
<
for all
a^.
We
summarize
this result as:
Proposition 8 Suppose Assumption
equilibrium
economy with
Q
1
holds.
Then
there exists a^ £
constant, an increase in
29
ji
(0, 1)
such that in the general
raises the division of labor N{a,i-t) in all
firms with
a < a^ and reduces
it
in all firms witii
a >
a^j,.
Comparative Advantage
6.3
We now
extend the analysis to a world with two trading countries, indexed by
=
^
We
implications for comparative advantage as a function of contracting institutions."^^
assume that there are
fixed
products produced in
its
numbers
of products,
own country but
also
and every product
1,2, to derive
continue to
distinct not only
is
from products produced
from other
in the foreign country. All
products can be freely traded between the two countries.
We focus
on the impact of contract incompleteness on comparative advantage. For
=
suppose that the two countries are identical, in particular, Vfor all
a G
Without
(0,1),
but the fraction of activities
we assume that
loss of generality,
fj}
Q^
,
= Q^
that are contractible
/i^
>
L'^
ij?
and H^
this
{a.)
purpose
=
H"^ (a)
across countries.
diff'ers
so that country 1 has better contracting
,
institutions.
The equilibrium
wage
condition for the division of labor, (39), holds in both countries, with different
rates, w^^ for the
diture, since all
two countries [A
goods are traded
freely).
is
the same for both countries and equal to world expen-
= A/w^
Defining a^
and N^
=
TV (a,
a G
(0, 1)
(a)
/i^),
this implies
that
^'^
-a^Z (a,
1-/3
A N^
(a)^'"-'^"'
=
C'^
\n^ (a)!
for all
,
(43)
.
In addition, the labor market clearing condition, (40), holds in both countries. This immediately
implies that the country with better contracting institutions,
i.e.,
country
1,
has higher wages, thus
lower a^.^^
The pattern
value of
a
Proposition
of trade can
now be determined by comparing
2)
shows that revenue of a producer with
= AZ
(a,
elasticity
A N' {a)^P
\~ ^
(1-/3)
Revenues are increasing
in total
world expenditure, A,
include an additional correction which
R^
Appendix
(technological complementarity) in the two countries.
R'{a)
If
the revenues of firms with the
{a) /i?2
(q,)
>
jj
Ri- (q,)
is
in
a
in
^^
~
country
A
£
is
same
(see the proof of
given
by:"*^
^'^
^.
(44)
[1-/3(1-/)^
Z [a, /x),
in the division of labor, but also
the last term.''^
dH (a) /
[q R'^ (q)
dH (a),
then country
1 is
a net exporter of
goods with substitution parameter a. Consequently, we simply need to determine the distribution
The link between contracting institutions and endogenous comparative advantage was previously discussed by
Levchenko (2003), Costinot (2004), Nunn (2004) and Antras (2005).
""Suppose not, then from (43), A^"" (a) > A''^ (a) for all a (since N (q) is increasing in fi for given a), and this would
violate market clearing.
Firms use their revenue to pay for labor costs and compensation of worker-suppliers, with the rest being profits.
This last term does not appear in the profit function, since it corresponds to the part of the revenue paid out to
workers to compensate for their effort.
"*
30
of
R^
/R~ (a) across
(q)
R^{a)
From
different as.
_
R^a)
(44),
Z{a,f,']
l-/3(l-/.2)
Z(q,/.2)
Proposition 8 implies that there exists an
and
>
(a)
iV-^
A'"^
(a) for
> Z
(a, //')
a <
all
1
N (a,
decreases further with
exists
also export
a when
a^ e
(0, 1)
such that R^ (a)
/ R-
the opposite inequality holding for
all
a net exporter in low
institutions,
is
summarized
in the next proposition:
is
(a)
a >
a
Proposition 9 Suppose Assumption
The
< a^ and
declining with
>
j^ Pj (a)
implication of this result
is
in
all
a > a^
C^
(A'') is
(A^)
not constant,
a constant,
is
dH (a) / J^ R~
and a net importer
holds and A
(A'") is
1
is
neg-
equation (45). Consequently, there
a^. In this event country
sectors
1
a
a net importer of products with
nous comparative advantage
that are
(a) for
which has the better contract-
1,
marginal cost
elasticity of
such that in the two-country world equilibrium, country
products with a
A''^
higher (because the cross partial d(^^ (a, ^) /da
/z is
This implies that R^ {a) /R^ (a)
ative).
country
<
some high-a products. On the other hand, when A
country
jj.)
such that AT^ (a)
result,
when the
(q, /i^)). Nevertheless,
may
As a
a^.
q^,
^^^^
1-/3(1-^1)^-
some 1ow-q products and imports some high-a products (remember that
ing institutions, exports
Z
we have
1,
dH (a)
/j.^
a
Then
>
for all
a < a^ and
with the better contracting
in high
constant.
with
(a)
/x^
sectors. This result
there exists a^ e
is
is
(0, 1)
a net exporter of the
a > a^.
that differences in contracting institutions have created endoge-
for the
country with better contracting institutions towards sectors
more "contract dependent," which,
com-
in this case, are those with greater technological
plementarity.
Free Entry
6.4
We
have so
far
assumed that the number
of products
Q
is
fixed in every country.
of general equilibrium interactions coines from endogenizing
briefly discuss these issues,
the
number
of products
is
suppose that there
is
Q
through a
free entry into the
is
not
production of
final
the amount of labor required for entry. This cost
addition to the cost of division of labor. Moreover, in the spirit of
While
free entry condition.
goods and
Hopenhayn
is
born in
(1992) and Melitz
assume that an entrant does not know a key parameter of the technology prior to
in their
know a
models the entrant does not know
(but knows that
the relationship between
of productivity levels
is
To
endogenously determined. In particular, suppose that an entrant faces
a fixed cost of entry wf, where /
(2003),
Another source
a
is
its
productivity,
we assume instead that
entry.
it
does
drawn from the cumulative distribution function H{a)). Since
a and productivity
is
determined
in general equilibrium, the distribution
endogenous.
After entry a firm learns
its
a and maximizes
the profit function (25). This leads to the
first-
order condition (39), which determines the division of labor as a function of the extent of the market
31
a,
the degree of contract incompleteness
n (q,
with
Z
[a,
fx)
we do not
fi,
a)
given by (26), and
and
fj,
its
a. Let us define
= aZ {a, ^) N (a, ^f^^-^^^^ - Cl [N (a, ^)]
A''
{a,
fj.)
as the profit-maximizing choice of division of labor (which
condition on a to simplify notation as before). In other words, U{a,fi,a)
rect profit function, with profits
measured
in units of labor.
11 (a,
/j,
a)
is
is
increasing in
the indiall
three
arguments.
Free entry implies that expected profits must equal the entry cost wf,
[ U{a,iJ,,a)dHia) =
(46)
f.
Jo
Since the expected profits are increasing in the adjusted extent of the market,
condition uniquely determines the equilibrium a
(i.e.,
a, this free
entry
without reference to the labor market clearing
condition, (40))
Now, given the equilibrium value
of a, the first-order condition (39) determines the cross-
sectional variation in the division of labor. Firms with larger as choose a finer division of labor
and
they are more productive. Consequently, the distribution of a induces a distribution of productivity
and firm
size.''^
Finally,
Since labor
we can
is
use the labor market clearing condition to determine the
now used
clearing condition (40)
is
to cover both the costs for entry
A''
(a,
/i)
+
f
=
of labor, the
market
L.
that were determined by (39) and (46),
market clearing condition to solve
7
of entrants.
replaced by
CL{[N{a,^)]dH{a)
Using the values of
and the division
number
for the
number
of entrants
we can use
this
modified labor
Q^^
Conclusion
In this paper,
we develop a tractable framework
for the analysis of the
impact of the degree of
contract incompleteness (or more generally contracting institutions) and technological complementarities
on the equilibrium division of labor. In our model, a firm decides the division of labor and
contracts with a set of worker-suppliers on a subset of activities they have to perform. Investments
The degree
of dispersion of productivity
depends on the degree of contract incompleteness.
In particular, in
countries with better contracting institutions there will be less productivity and size dispersion.
'Another interesting implication of this analysis is that the overall supply of labor L is not a source of comparative
If two countries that differ only in L freely trade with each other, their wages are equalized, their division
of labor is the same in every industry a, and they have the same distribution of productivity and firm size. The only
difference is that the larger country has proportionately more final good producers. This contrasts with the case with
exogenous number of products, where differences in L would generate comparative advantage.
advantage.
32
remaining activities are noncontractible, so workers choose them anticipating the ex post
in the
bargaining equilibrium.
We model the ex post
bargaining problem using the Shapley value and provide a characterization
of the division of surplus. Using this characterization,
we show
that greater contract incompleteness
reduces both the division of labor and the equilibrium level of productivity given the division of
The impact
labor.
of contract incompleteness
between the tasks performed by
when
greater
is
different workers.
there
is
greater complementarities
Similar result hold
when
credit
markets are
imperfect, so that ex ante transfers are not possible. Interestingly, however, imperfect credit markets
may
increase (measured) productivity, because the firm
may
force its workers to overinvest in
contractible activities in order to extract the surplus they capture in the bargaining game.
show how the firm may want
also
bargaining game, and
how
to hire worker-suppliers as employees in order to change the
this interacts
Finally, in the last section,
We
with the equihbrium division of labor.
we embody the
partial equilibrium
model
in a series of general
equilibrium setups and derive implications for the cross-sectional distribution of division of labor
and the impact of contracting institutions on comparative advantage. The most interesting
here
is
result
that societies with better contracting institutions have comparative advantage in sectors
with greater technological complementarities (which are the more "contract-dependent" sectors,
since contractual problems create
The
more
analysis of general equilibrium
inefficiency in those sectors).
is
tractable thanks to the reduced-form (equilibrium) profit
function derived in partial equilibrium, where the degree of contract incompleteness and technological
complementarities affect a single derived parameter. This property makes
to use our
A
framework
number
it
relatively simple
in various applications.
of areas are left for future research.
These include, but are not limited
to,
the
following:
• It
would be interesting to derive the implications of
this
model
for the relationship
between
(contracting) institutions and productivity across countries. This would be an important area
for future research partly because, despite increasing evidence that differences in (total factor)
productivity are an important element of cross-country differences in income per capita,
far
from a theory of productivity
in the "technology" of
The model assumes
that
is
are symmetric.
more general production
differently
are
leads to both endogenous differences
in the efficiency
used.
all activities
similar results hold with a
some workers/tasks
Our model
production (as measured by the division of labor) and
with which a given technology
•
differences.
we
than others,
for
An important
function,
extension
is
to see
whether
where the firm may wish to treat
example, depending on how essential they are
for production.
•
Another area
for future
study
is
a more systematic investigation (beyond what
we have
presented in Appendix B) of the joint determination of the range of tasks and the number
33
of workers performing these tasks.
forces analyzed in this
Finally,
it is
paper and those emphasized by Marglin (1974)
in a single
framework.
important to investigate whether the relationship between contracting institu-
tions, technological
when we
Such an analysis would enable us to incorporate both the
complementarities and the division of labor
is
fundamentally different
use different approaches to the theory of the firm, for example, as in the work by
Holmstrom and Milgrom
(1991).
34
Appendix A: Proofs of the Main Results
Second-Order Conditions
n=
Let
^i-/3iV/3(«+i)a;/3
-
in the
C (TV).
C:,Nx -
second-order conditions can be expressed
a2n/ax2
/
d'^nldxdN ^
yd^u/dNdx
d^n/dN^~
The second-order
-
(^
second diagonal element
is
its
(1
-
^Ncl (C')~'
/3)
if
if
this
and only
C"N
positive
is
if
[^(l
the
first
{I
+
k)
<
and only
inequality. Therefore
Assumption
+ «)]
+
1) «-ic' - c"
«)](«
negative definite, which requires
The
diagonal element
first
is
its
^
'
j
diagonal
negative, the
+
Ac)
+
-1](k-
1)
if
^
1 is
+ k) -
(1
^
both these conditions are
1
is
+
K
C
ii /S
the matrix of the
(7),
if
C"N
Note that
-tv-^ [i-/?(i
matrix
C
and the determinant
and
-c, [1-/3(1
+ «)]
determinant to be positive.
negative
(6)
as:
conditions are satisfied
elements to be negative and
Using the first-order conditions
-c,[i -/?(!
[
J
Complete-Contracting Case
1
1-/3
•
and
satisfied,
if /3 (1
-I-
k)
>
1
the second inequality implies
necessary and sufficient for the second-order conditions to be
satisfied.
Proof of Proposition
The
first
1
part of the proposition
is
a direct implication of Assumption
from the implicit function theorem by noting that, except
a parameter
response of
if
N*
and only
if
for P,
N*
1.
The comparative
statics of
follow
A''*
increases in response to an increase in
this parameter raises the left-hand side of
(6).
Using the results concerning the
to changes in parameters together with (7) then implies the responses of x* to parameter
changes.
Proof of
Let there be
all
Lemma
1
and Related Results
M workers each one controlling a range £ = N/M of the continuum of
workers provide an amount Xc of contractible
activities.
a situation in which a worker j supplies an amount x„
{j)
As
tasks.
Due
to
symmetry,
for the noncontractible activities, consider
per non-contractible activity, while the
M—
1
remaining workers supply the same amount Xn {-j) (note that we are again appealing to symmetry).
To compute the Shapley value for
this particular
of this worker to a given coalition of agents.
F,,v (n,
when
the worker j
N-
is
e)
in
=
A
worker j, we need to determine the marginal contribution
coalition of
^i-S;v''{K+:-i/a)^^M
[(„
_
i)
n workers and the firm
,^^ (-jf-^^^
+ ex^
yields a sale revenue of
^''^
(j)''"^)"]
,
(Al)
the coalition, and a sale revenue
Foc/T(n,iV;e)
=
Ai-''Ar/3('^+i-i/-)x^M [n£.T„ (-j)'^"'^)"]"^"
35
(A2)
when worker
j
is
when n < N, the term
not in the coahtion. Notice that even
because
it
represents a feature of
brackets
is
lower.
tlie
N^'''^'^^~^^°''>
remains in
technology, though productivity suffers because the term in square
Following the notation in the main text, the Shapley value of player j
is
l-^[.(-u,)-^(^i)].
(M +
The
V
(2;i)
is
which g
fraction of permutations in
=
0,
= FiM {l,N;e),
1/M
value of V (2^)
is
[i
=
is
and
i is
1/
(M +
1) for
=
{j)
-^Fj^
is
1,
U
- l.N;e).
:;)
It
,
given g
follows
(j)
=
i,
=
N;e).
{I,
By
{i,
If
i.
g
If
j.
=
(j)
g
then v
=
(j)
- 1/M.
after j with probability 1
j^Fim
is
every
necessarily ordered after
1
(x:^
U j)
=
then the firm
In the former case
Therefore the conditional expected
0.
similar reasoning, the conditional expected
jjFqut {0,N;e). Repeating the same argument
pected value of V (z^
j^FouT
{j)
while in the latter case t;(z^Uj)
value of t;(z^Uj), given g
(A3)
1)
because in this event the firm
ordered before j with probability
v(z^gUj)
front,
N; e), and
for
g
{j)
=
i,
i
>
I;
the conditional ex-
the conditional expected value of v (z^)
is
from (A3) that
M
-—
(M+1) Yl
:
^
e)
- Four
[i
-
1,
N- e)]
Af; e)
- Fqut
{i
-
1,
N;
[Fin {h N-
M
1
Y^ is [F/jv
^
'
(z,
e)] e.
1=1
Substituting for (Al) and (A2),
^
^l-/3^/3(K+l-l/a)^/3M
y^Jeliexn{-j)
{N + £)N
(l-M)a
+e
Xn
(l-^)c
v(1-m)o
/3/«
I
(j)
i=l
/3/a
For e small enough, the first-order Taylor expansion gives
(Ar
^(ze)[z£x„(-j)^^-^^"]
+ £)A^
or
^l-/3^/3(.+ l-l/a)
^l''"'"-^-.(-if^^-''^ ^
(A^ + e) Af
=
(^/^)
i
Now
taking the limit as
M
—
>
00,
and therefore
e
=
NjM
-^
0,
the
IE
,^/«
e.
l
sum on the
right
hand side
equation becomes a Riemann integral:
lim
M--00
p
\
£
^l-/3;V/3(«+l-l/a)
(^/^,)
^l''"'"-^''-"(-^-f^^-^'
A'^''/"dz.
A/-2
36
of this
Solving the integral delivers
lim
(s,/£)
^n
= (1-7)^1-^
(1-m)c
(i)
xl^x^{-jf^^-^^ Nf'^'-^^^-\
X-3)
=
with 7
aj (q
+ /3).
This corresponds to equation (14) in the main text, and completes the proof of the
lemma.
In addition, imposing symmetry,
so
i.e., a;„
(j)
—
Xn (-j), the
= A'-^N^^^+'^x^.'^xJ^'-^^ -
=
Nsj
firm's payoff
is
7Ai-''xf^x^(i-^)iV''('=+i),
as stated in equation (17) in the text.
We
choice
can also compute the marginal contribution of a worker to a coalition of size n when the technology
is
N. Imposing symmetry,
contribution
i.e.,
Xn
=
{j)
x„ (-j),
if
the worker controls a range e of tasks, her marginal
is:
Ve
which taking the limit
=
e
A'-^N'^^'^+'-'f^^x^^'^
—
\
\{n
+ s) xi'-^^'-
^4l-/3^^(K+l)^^M^,5(l-M)
R=
A^
Al-^)c
B/a
delivers
»
/ jl
limf^
e— V £
where
10 1 a
a
\P
v"/"
("/"'(iv)
As stated
^N^^'^'^^'>x^'^Xn
a when n/N < exp {—a/P) and decreasing
in
a
in the
for
main
n/N >
text, this
R
j-j
/
n
s
a/a
=^W»)(^)
,
marginal contribution
increasing in
is
exp (—a/P).
Proof of Proposition 2
First,
we
verify that the second-order conditions are again satisfied
that the problem in (9)
expression into (19)
convex and delivers a unique
is
=
a;„
under Assumption
S„ {N,Xc), as given in
1.
To
(18).
see this, note
Plugging this
we obtain
Tr
= A i-fX'-*^) i)N^+ i-/3(i-m) a;^M/li-/3(i-M)l _c^x^N^i- C {N)
<i!
=
(A4)
where
The second-order
1/3(1-m)/[1-/3(1-A')1
(C:,)
[1-q.(1-m)(1-7)]
conditions can be checked, analogously to the case with complete contracts, by computing
the Jacobian and checking that
to illustrate
^a(l -7)
how the reduced-form
for a given level of
negative definite.
it is
present here an alternative proof which also serves
profit function (25) in the
N, the problem
of choosing Xc
solution
^-1^
Xq
We
is
main
convex (since
text
/3/x
in
N
(A4) then delivers
TT
derived. In particular, notice that
1
—
(3
+ Pfj.)
and
delivers a unique
^[l-W-M)l/(l-/3)^^^^
-'T-
.1-/3(1-m)J
Plugging this solution
is
<
= AZN^+
^-^
37
-
C {N)
'-I3
.
(A5)
where
-1
/3m/(1-/3)
Pa
l-/3(l-/i)J
as in equation (26) in
< if and
= AZN^~^
tlie text.
d'^H/dN'^
only
c^XcNfj.
^^^
if
1-/3
^[l-/3(l-p)]/(l-/3)
,1-/3(1-m),
This reduced-form expression of the profit function immediately imphes that
the second part of Assumption
Pfi/ (1
—
1
which combined with
/?),
from equation (A5) we also have
holds. Finally,
(15)
and
(24) implies that the firm's revenues
are:
R =
TT
+ C{N)
+C^XaNlJ.
g(>^+i)-i
+ C^XnN{l-
l-/3(l-/i)
/
1-^(1-/.) + P
(1-/3)
which
will
be used
in Section 6 (see
IJ.)
Oc
equation (44)).
Next, the comparative static results follow from the implicit function theorem as in the proof of Proposition
1.
First,
dN/dA >
follows immediately.
To show that dN/dce >
and
dN /djj, >
0, let
us take
logarithms of both sides of (20), to obtain
+y
1-
^"^
-
i IniV
+ In (akBt^c'^^
where
f (,,„ = i^iiifio
1-/3
and
i?
=
Q'(l
—
,„
f
=
F(i?,M)
lnC'(7V),
^^
i^ii^ V
vi-/3(i-M)y
1
" ("O
,
7) <,0is monotonically increasing in a. Simple differentiation delivers
(i_/,)(/3-^)
dF{^,fi)
(1-/3)^(1-^(1-m))^
d-d
which implies dF/da
>
0,
and establishes that
dN /da >
0.
Furthermore,
^-/3 + /3(l-^(l-M))(ln(i5|g^)+ln(f)
dF{-d ,;.)
(l-i?(l-M))(l-/3)
and
- ^f
<0.
(l-^(l-/.))^l-/3(l-M))(l-/3)
d^F{'d,ij)
(/3
9^2
Thus,
dF (i?,
ij.)
/9/i reaches its
minimum
over the set
t?
-^
fj.
^
=
/3.
We
thus have shown that
(3
dF id,
+
fj.)
pin
/d/j.
[0, 1]
-h /3 In (/3/t?)
1-/3
where the inequality follows from d —
£
[P/'d)
>
at
/x
=
1, in
which case
>0,
being decreasing in
i9
for all n, so that dN/dfj.
For the cross-partial derivative in the Proposition, simply note that
(/3-1?)
(1-^(1-A.))-79(1-^)
38
equals
it
<0,
for
>
i?
0.
<
/3,
and equalhng
at
<
which impKes d'^N/dadjj.
Incidentall}',
in
equation (41)
dF {-d, fi) /9/^ >
note that
is
0.
and d'^N/dad/j, <
both positive and decreasing
imply that the
and d {xn/xc) /d^ >
N, where
0.
And,
it
and 5„ are nondecreasing
follows that productivity
P
A,
in
C"
()
a and
>
^,
/dA =
a and
equation (21), the effects of A,
the inequalities becomes strict whenever
Finally, given that Xc
parameters,
in light of
C^ (a, aO defined
in a.
Next, note that straightforward differentiation of (24) delivers d{xn/xc)
those on
elasticity
/x
0,
d{xn/xc) /da
>
on Xc follow directly from
0.
and given that A^
increasing in these
is
also necessarily increasing in these parameters.
is
Proof of Proposition 3
The proof
P
and
follows from Proposition
are
all
increasing in
/i
2,
since
<N,Xc,P\
converge to {N* ,x*,P*} as
(nondecreasing in ^, in the case of the investment
/i
-^
1,
and
TV",
Xc,
C7 x,
'^C7
levels).
Proof of Proposition 4
The proof
1
is
similar to that of Proposition
2.
First,
ensures that the second-order conditions are met.
it
is
This
straightforward to establish that Assumption
is
easily verified
by noting that equation (31)
implicitly defines a reduced-form profit function analogous to that in (25), with the
As
in
previous proofs, to show that
the left-hand side of (32)
is
dN /dA >
dN /da >
0,
0,
dN /dji >
and
increasing in these three parameters. This
the market A. For the other two parameters,
let
is
same exponent on N.
we need only show that
evident for the case of the size of
us take logarithms of both sides of (32), to obtain
ln(^c;-)+G(a,,) = lnC'(iV),
^i^^til^ln7V +
where
G(a,M)-ln7+^ln(i^^)+^ln(a(l-,)).
It is
straightforward to show that dN/dfi
>
0.
Note simply that because 7
is
independent of
/U,
we have
that
dG{a,^)
((l-a + a^)ln(^^^^^f^)-(l-a))/3
5m
(l-a(l-M))(l-/3)
and
g-g(^.M)
{1-
diida
Thus,
this
dG {a, 11) /dfi
reaches
we can conclude
partial derivative
To show that
its
that for
minimum
q <
1,
dN /da >
da
0, let
P)
,Q
{I- aila £
over the set
dG {a,^) /djj. >
above also implies d'^N /dadji <
dG (a, /i)
[0, 1]
i^)f
a
a
=
at
1, in
and thus dN/dfj. >
which case
0.
it
equals
0.
From
Notice also that the cross
0.
us write:
d\nj
da
1
/
this expression
is
P ain(l-7)
1-/3
da
(l-Q)(l-/i)/3
a{l - a{l - 11)) [I - p)
y'a7
(l-a)(l-M)/3
a(l-Q(l-;u))(l-/3)-
/37
1-/3-7;
7V
When 1 — /3 — 7 >
(l-a)/3
^
da
positive, because
7 increases
39
in a.
Next consider the case
1
— /3 — 7 <
0.
Assumption 3 implies
—
{l-a){l-P)
l-M>l-7:
TI^
=
..
1-g-^
(l-a)(l-/3)-i(^
T^ffT^
i^_(l_a)(l-/3)
Therefore
aG(a,/i)^l/^
5a
It follows
that as long
^" 57
/57
1-/5-7/ 5<^
7 \
asl-/3-7<0we
have (9G (q, ^) jdo.
1
The
inequahty follows since a
last
and thus
dN/da >
(1
-
7)
=
Q
07
—7
9(1
a/?/ (q
<
+ /?)
(1-/3-7)
ci'7 (1 — /9)
>
if
1.
is
increasing in a, proving that
Let us next turn to the comparative statics related to the investment
of (30) delivers d (Xn/xc)
the effects of
is
a.
ft
/9q!
>
jdA =
0,
9 {x^jx^ /da >
on x„ follow directly from those on
and d [xn/xc)
N
(since
7
is
note that differentiation
levels. First,
ld\i
>
0.
Next, in light of equation (34),
independent of
fi).
On
the other hand, x„
dN /da > (shown above) and that 0(1-7) /Pi is nondecreasing in
when 7 = q/ (a + /3), in which case q (1 — 7) /Pj = 1. Lemma 3 states that
nondecreasing in a provided that
The
this
is
latter
is
clearly the case
also true for the general definition of
Finally, consider the
directly
in section
7
5.
comparative statics related to productivity. The positive
from noting that Xc and Xn are nondecreasing
on productivity note that, using
(32),
P = TV^.t^x^"'^
in
which depends on
ji
5
Consider
1
As
first
for the
the inequality
> x^/xc >
-
,
C
only through the term
Proof of Proposition
A, while
(iV)
x„/xc-
N
is
can be rewritten
/
3.
SG (a, yu)
0.
N^~^
The
,
s
i-0\
^
on
P
follows
increasing in A. For the effect of
/./,
as:
I/''
thus establishing
first
effect of
inequality
is
dP /dji >
0.
a direct implication of Proposition
second inequality, straightforward manipulation of (24) and (30) indicates that Xn/xc
> Xn/xc
provided that Assumption 3 holds.
Next, we note that the inequality
N > N.
Using equations (20) and
this will
be the case as long as
A''*
(32),
>
N
was established in Proposition
and the arguments
'-gC--^)
pT^
l-Of(l-7)(l-^x)
Thus we only show that
it
should be clear that
«(!-,.)
[r'o(i-7)
1-/3(1-m)
3.
in previous propositions,
1-5
>
1.
Taking logarithms on both sides of the inequality, and after some manipulation, we can write this condition
40
l-a + a^^
\
^A
,,..^^(1-7),,
Pi
\
J
ai^
"
1-/3(1 -^0,„/^(r^ -"/?(1-M)
1-/3
l-/?(l-/i)
1^
"^'J
^
l-a + a^J^
Simple differentiation delivers
jj,(
Q
^
.
(1
-
- g) (1 - ^) - M (/?/ (1 - 7) - (g + /3) +
(l-Q(l-7)(l-^))(l-a. + QM)(l-/3)
7) ((1
1-/3
1-Q + Qfi
1-/3 (1-Ai)
I
Q^))
which can be rewritten as
1
— /3\i — Q4-Q/i
y
where
(l-7)((l-Q)(l-^)-/^(^/(l-7)-(a + /3) +
(l-a(l-7)(l-M))^M
The
fact that
tp is
positive
implied by Assumption
is
1-a + QM
> a (1 —
— In (1 + ip)
(where the inequality follows from
We
next use the fact that
bilj
''
is
Notice also that
3.
which implies (24)
decreasing in
H'
its
minimum
at this
H (m)
upper bound,
>
if ((1
which completes the proof that
To obtain the ranking
-
which case
in
a) (1
-
/3)
in the text.
whenever
tp
6 (1
+
7/^)
<
to conclude that
1,
<
{n)
whenever Assumption 3 holds. But note that Assumption
reaches
„
,^
/3(1-q(1-7)(1-m))
7)),
/?
Q^))
it
-
/ (/3/ (1
3 places
equals
-
7)
an upper bound on
Mathematically,
0.
(a
+ /3) +
ap))
=
/j.,
and thus
we have
H
[ji)
that
0,
N > N.
of productivitj' levels, combine (21), (22), (33), (34),
and the
definitions of
P
and
P, to obtain
P
P
/7VV^^(l-^)('"^)'c'(Ar)
\NJ
/'
a(l-7)[l-/3(l-M)]
V
'"
C'(n)'
\l3[l-a{l—,){l-n)])
Using equations (20) and (32) to eliminate the terms
P
p
Part
(i)
fN\^'^
/a
\nJ
V
-
(1
7) (1
-
For part
3,
the second term in brackets
(ii)
of the proposition,
let
us
is
/?
J
(AM and C" {N), and
(1
-
/x)) (1
simplifying deli\
- a (1 - m))\
^
aM(l-a(l-7)(l-M))
P
of the proposition then simply follows from
Assumption
C
K
N
<
N
(as
shown above) and the
fact that,
under
greater than one.
first
show that P/P*
41
is
increasing in
/x.
Following a similar
procedure as
in
the case of
P/P
- but this time using (6) and
N-C'{N)
a(l-7)
[N')''C'{N*)
/37
P _
P*
Notice that this ratio
increasing in
fi.
To
necessarily increasing in
is
yL
see the latter, take logarithms
/
,
- we obtain:
l-g + g^
afi
y
)
0,
C" (N) >
'^'y
^
because dN/dfj.
>
'
0,
and the
last
term
is
and notice that
2 f
1„ f l-a+afj.
d'[^^-['=^^))
,,
\
(7)
a^i-aa
Q,
-il-a)
(1 _ Q + ail)
and thus the partial derivative
(l-a + a^)ln(l^g^)-(l-a)
9(/.ln(l=g^))
^
dy.
attains a
minimum
d [P/P')
ld[i
>
1
a £
over the set
where the inequality follows from
Next, plugging
(6)
and
this ratio
Notice finally that
P<
Proof of
The proof
when k <
P* but
,
Lemma
is
0.
This establishes that
terms
C (N') and C"
(7V),
and simplifying
delivers
becomes
- /?
1
/ iV
if /x
>
/I,
V-"
<
then
^
\N* J
1
—
P>
/?
/1-7\T^
\
then [N/N*)^^^^~
This establishes that provided that k
p, then
equals
N ^-W(i^f±^)"a(l-7)y"
P*
<
it
(
V7V*
P
/i
which case
M
{N*fC'{N*)
(32) to eliminate the
~
P*
1,
in
N < N*.
P _
^
1,
goes to
0, this ratio
P*
fi
=
a/j.
0.
Next note that when ^ -^
When
a
at
[0, 1]
—a+
and 7
P
> 1, which combined with 7 < 1 - /? implies P > P*.
< 1 — /3, there exists a unique Jl S (0, 1), such that if
P*
2
similar to that of
Lemma
1.
A
coalition of
n workers and the firm
yields a sale revenue of
-,0/a
FjM
(n,7V;e)
= A^'^ N^^^+'-'^^^x^,^
[(n
- l)ex„ {-jf-^^^+eXr, {jf'^^'^ + {N -
ne)5'^Xr,
{-jf~^^
(A6)
42
when
the worker j
Four
(n,
N;
e)
=
is
and a
in the coahtion,
sale revenue
A^-^N^^^+'-'^'^^x^^ [nex„ (-j)*'"''^"
+ e^x^
{jf''^^"
+
(at
_
(n
+
1) e)
5"x„ (-j)''"^'^"
0/0
(A7)
when worker
As
j
not in the coaUtion.
is
in the case
with 5
=
0,
the Shapley value of worker j can be written as
M
——
[N + e)N ^
ie
[FjN
N;
{i,
s)
- Fqut
{i
-
1,
N;
e)] e.
1=1
Plugging the new formulas for F/at
(n,
A'';
e)
and Fqut
(n,
N;
e) in
(A6) and (A7) then delivers
0/c
(Ar
+ e)Af
+Nd"Xr, (-j)
(i-m)q
{N + e)N
,S/q
i=l
For e small enough, the first-order Taylor expansion gives
^l-Sjy/3(K+l-l/a)
(^/^)
(;^
_
ja^
11
(.V
(:-/i)a
Xnjj)
^r.(-i)
Y,
+ £) N
i
n+
TV^"]^"-"'/"
/3Mt4^^n
/'_o^/'(1-A')
i-j)
'
(l
"^
z[z{l-6°')
Ar2
Finally, integrating
+
j_ N5A/-A"i('3-")/",
by parts delivers
(1-A.)a
lim
M—»oo
(sj/e)
= (l-7)Ai-^
3..5m^„(_j)/3(1-m)a^/3(.+1)-1^
where
1
(a
as claimed in the
Proof of
-dca+P
+ /3)(l-d")'
Lemma.
Lemma
£.
-|(l-M)a
.0)
Ir.(-j)
'
-
M ^ oo, now yields
Taking the limit as
lim
(«£) [(^e) (1
=l
3
Let
A = ln7 =
,
,
In
(
a
—
-^
Q+
\
/3
43
)
+
,
In
I
1-5'a+0"
1-r
(AS)
From
dA/dd >
inspection of (A8), in order to prove that
d^
-
(In (1
_
d^))
-X
((5^
dddx
for all
X e
S
G
>
But
[0, 1).
x G (0,2) and 5 G
in 5 for
d-y/dS
and
(0, 2)
this inequality follows
Notice in addition that using I'Hopital
0.
To prove that
dA/dP <
rule,
r)A
1
^
>
(A9)
Hence, for 6 G
1.
(In
[0, 1),
easily verified that
—
(5)
1 is
decreasing
dA/dd >
lim^^i 7
=
and thus
1.
>
We
0.
+
-^
r.
-^
In 5)
a+P
dp
by
5
2
from the fact that 5^ — x
it is
n
-
using I'Hopital rule to
1)
show that
note that
0,
while (A9) implies d'^A/dPdS
-
(Ind)
5 {1-5'')
and equals zero when ^
[0, 1),
sufficient to
it is
thus need only show that 5A/9/3
<
when
5
=
We
1.
establish this
show that
/(5Mn<5\
,
1
lim
s^i\lfrom which
57/5^ <
We
follows that
it
dA/dP =
dA/da >
letting
x
=a+
/?,
(1-5^)
a{a +
and thus
It suffices to
the set P G
show that d'^A/dadp
[0, 1]
at
/3
=
0, in
((^"-2-x^(ln^<5))5"
_
~
X
<
2.
>
5)
=
52-
_
(0)
=
c,'
(a;)
= -dMn (d)
=
{1
25^
we prove
0,
-
this
((1
+ x (In 5)) +
+
1
0.
x'-
+
We
dA/da would
(In^ d)
-
(In 5))
25^^ is
reach
its
minimum
over
thus need only show that
+
(5"
1
V
1
>
g' (x)
>
- 25^) >
0,
by showing that
X
l
5^)2 x2
positive, because in such case
is
Given that g
positive because h{x)
that h' (x)
(1
-
which case dA/da equals
g (x)
<
5- {In
we can write
a^A
for
x
>
increasing in x and equals
In particular,
0.
at
x
=
0.
To
see
note that
/i'(x)
and we showed above that
We
dA/dP <
fi_s°^+l3\(^i_go.y
P)
dadp
is
implies that for 5 G [0,1),
Note that
0.
P
da
which
J
0.
next turn to prove that
for all
X
when 5=1. This
dA_
But
6
next show that 0(1
1
+
—
x
7)
(In 5)
/Pi
—
is
5^
=
<
2In(d)(xln(d)-(5^
0.
+
l),
This completes the proof that
nondecreasing in a. Notice that this
44
is
dA/da >
0.
equivalent to showing that
is
nonnegative. But this follows from the fact that
5"^
-{ln5)x-l >
G
for all 6
which was established
[0, 1),
above.
Because
increasing in a,
is
/?7
can conclude that a
—
(1
should be clear that, from
increcising in a.
is
7)
it
0(1-7) /Pi
being nondecreasing in a, we
This completes the proof of the Lemma.
Proof of Proposition 7
That
O = C when
a+P >
We thus
has been proved in the main text.
1
employment relationship dominates outside contracting only
case, the
focus on the case
if
7
(1
—
^yy/i'^-'^)
a+P <
In such
I.
under an
jg ];iigher
employment contract than outside contracting. Defining the function
afl- d"+^
^
(Q + /3)(l-r)'
9(5)(where clearly 7
=
(S)), this
g
g {5e)
condition can be written as
-
[1
g
{Se)]^ >g{0)[l-g (0)]^ =
Notice that the left-hand side of this inequality
when
equals
—
^f;
>
1 (see
that the left-hand side of the inequality
>
gi^s)
1
—
Lemma
the proof of
is
Coupled with the
/?.
fact that
g'
when g
maximum
then g (0) <
when a +
/?
<
1
(Se)
>
1
^^i g
{5 e)
=
—
/3,
6e G
and
0,
it
— P and
'i-
decreasing in 5e
[1
—
when
strictly
is
5(<^e)]'"''
[0, 1].
and thus
for sufficiently small Se, the left-hand side of
(i.e.,
O—
>
there exists a unique 6 € (0,1) such that, for 5e
1,
Notice also that the threshold 5
iP,„).ln,(J) +
(i.e.,
O=
E). In such case, and given that
S,
the above inequality
reversed
is
C).
implicitly defined by:
is
^,„(l-.P))-l„(^)-^l„(;^)=0.
theorem to show that
finally use the implicit function
necessarily decreasing in 5 since, at
On
=
necessarily increasing in 5e- This implies that the above inequality (AlO) will hold for sufficiently
and outside contracting becomes optimal again
We
<
{5e)
implies that g{5E)
0, this
in the set
small values of 5e making an employment relationship optimal
Irnis
(AlO)
.
d
Importantly,
is
1-/9
)
Furthermore, straightforward differentiation indicates
increasing in 5b
concave in 5e and attains a unique
(AlO)
{
equal to the right-hand side whenever 5e
is
3).
-^ -^
5,
the function g {5)
[1
5
~
is
g
decreasing in a. Notice
{5)\
^^^
is
decreasing in
first
5, (i.e.,
that
g
[6)
J
(5,
>
I
a)
—
is
P).
the other hand,
dJ
(5,
a)
_
d\ng
>
g (S)
>
1-/3,
and thus 86 /da <
dg
dln{l-g(5)) _
^ _ J__
(S)
/
l-P-g{5)
[{1
-
/da >
da
1-/3
da
da
Since 1
(S)
p) {I
-g
{!))
/3
(1
a (a
/da _
\ dg
(6)
J
g{5)
45
/3
-
-h /3) (1
q)
-
/3)
/3(l-^-a)
a{a + P){l-P)-
(from the proof of Lemma 3) and a-|-^
0.
-
<
1,
we have 5 J
(6,
a) /da
<
0,
Appendix B: The Division of Labor and the Scope of Firms
This Appendix analyzes the simultaneous choice of division of labor and scope of firms,
M and
i.e.,
iV,
using
the production function (2) under complete and incomplete contracts. Recah that in this case production
requires the combination of
performs e
= N/M
tasks).
tasks and these tasks are divided equally
A''
We
assume,
has to incur a cost V {N/M), where F
{N/M) =
limM-+oo AfF
that
more
it is
The
training.
Now
tion
This
0.
is
in addition to
M workers
among
a nonnegative, increasing and convex function, with F (0)
captures "diseconomies of scope" at the individual
.cost
(so that each
the assumptions in the main text, that each worker
costly for a worker to deal with
more
tasks, for example, because
it
level,
=
and
and implies
requires greater effort or
requirement implies that these economies of this scope are sufficiently powerful.
last
an argument similar to that in the text implies that, with complete contracts, the profit maximiza-
problem of the firm
is
l/a
M
EM
^i-/3a^,8(k+i-i/-)
max
N
N,M,{x(i,j)},^.
Inx
exp
t""
M
{i,j) di
x{i,j)di-C{N)
-MT
3=1
limM^oo
GiA^en this formulation,
as
M—
complete contracts.
We
MT {N/M) =
implies that the
maximum of this program will be reached
maximum division of labor under
In other words, sufficient diseconomies of scope leads to
oo.*''
»
It is also
straightforward to see that the rest of the analysis applies in this case.
next turn to study the choice of division of labor under incomplete contracts. Here
M influences ex
post bargaining and through this channel has a direct effect on the investments x„. Consequently, the profit-
maximizing choice of
and the
cost
a function of
To
M, Xn
illustrate
will
fact, in
As
M). This
is
equivalent to maximizing the ex post bargaining power of workers.
these considerations shape the choice of
number
of workers equal to
M. Suppose
for the noncontractible activities, consider
^n{j) per non-contractible activity, while the
To compute the Shapley value
(n,
M) =
M-
for this particular
1
M, we now compute
the worker j
is
worker
j,
coalition of
a1-^7V'3(-+i-i/")2;^a'
N
in the coalition,
Four
when worker
(n,
M) =
and a
we need
to determine the marginal contribution
n workers and the
,(l-/i)Q
firm generates a revenue of
j
is
not in the coalition.
''^To see this,
it
suffices that the solution to this
implies that this term
is
-^0/a
N
(l-fx)c
(Bl)
U)
sale revenue
N
a'-^N^^^+^'^^^'^x^,''
problem
M comes from the
minimized as M —
implies that the only dependence on
when
a situation in which a worker j supplies an amount
m''
when
the Shapley value
that each worker invests Xc in contractible
remaining workers supply the same amount Xn {-j)-
A
of this worker to a given coalition of agents.
Fin
activities
in
{xc,
how
there are a finite
activities.
M not simply minimize MT{N/M). In
the absence of credit constraints
M would be chosen to maximize workers' investment noncontractible
as
F {N/M),
will
46
•n(1-m)c
be symmetric with x{i,j) = x for
The assumption that limM^oo
last term.
oo.
1/3/a
I
(B2)
all i,j,
and
this
MT {N/M) =
Following the notation in the main text, the Shapley value of player j
(M +
l)!
is
geG
M
1
^
(M +1)M
'
^^'^
^^ "
^^'
-^^^^
(^
-
1.
^)1
we obtain
Substituting for (Bl) and (B2),
/3/a
(l-M)a
M
^1-/3^/3(k+1)2./3m
M+
l
M
B/a
EM
2
At a symmetric equilibrium, with x„ (-j)
=
x„
(z-l)-ia;„(-j)(i-'^)"
(j)
=
x„, this yields the following payoff for each worker
R
s,=(P (M, a/P)
where, as in the main text,
R = A^
The
function
/3/a,
</)(•)
M
M+
goes to the firm.
also that as
M -^
It
(i
—
1) /M
follows that the effect of
oo,
V
Now
in the
sum
a on these
is
all
the workers combined.
smaller than
fractions
is
1.
The
It rises
with
—
<p{-),
residual fraction, 1
the same as in the main text. Notice
<^
^
+
(B4)
l3
and therefore
4>{M,a/P)
as defined in equation (15) in the
consider the impact of
maximize
M^ M
l
I
1=1
lim
is
M ._^^0/o
\^ —
—
M ^ \M
M—.oo
where 7
revenue, and
we obtain
m'-*oo
as before,
is
M'
describes the fraction of the revenue allocated to
declines with q, because
i.e., it
'''"
^N^^'^'^^^x^^Xn
>{M,a/l3)
(B3)
Sj
— Cx^
{1
—
l-i)
x„
main
=
^a+p
text.
M and Xc on investment
(j).
1-7,
in noncontractible activities,
worker j seeks to
Using (B3), the first-order condition, evaluated at Xn {—j)
=
x^
{j)
=
Xn,
is:
M
Xn{Xc,M)
This
is
1/[1-/3{1-m)]
/3/aN
(c.)-x?M-^iV^^(-)-^^^g(^)
(B5)
the same expression as (18) in the text, except that here jji^iY^i^i {jj)
Evidently, here too investment
is
rising in
a (because the i/M terms
in the
sum
is
declining in a. Moreover, from (B4)
we have
to infinity.
47
j^ Z)i=i iji)
a/{a +
are smaller than
obtain once again that investments in noncontractible activities are increasing in
to the workers
replaces
"
1).
a and the share
— a/ (a + /?)
>
as
P).
So we
accruing
M goes
As argued above,
M on the fixed costs associated with the diseconomies of scope,
ignoring the effect of
the profit-maximizing choice of
M wih seek to maximize (B5), or simply
^
1
The
Lemma
following
lemma
M
Proof. The two
for /?/a
>
is
monotonic
in
M
S
(1)
=
Furthermore,
1.
claims were shown above, so
last
,•
\/3/"
characterizes the properties of this function:
4 The function S (M)
decreasing in
/
In particular,
.
1/2 and liniM^co
S {M) > S (1) = 1/2 for /3/a < 1 and 5 (M) < 5 (1) =
(3 /a < 1 (the case /3/q > 1 is symmetric), this follows from
S{M)
increasing in
is
S (M) = a/
M
M + 1^\m)
^
1/2 for
M + 1^[m )
(3
/a >
I
1
and
+ /?).
(a
whenever
M>
for
2
<
for /3/q-
we focus here on proving the monotonicity
note, that
case
it
of
S (M).
M>
2.
First
In the
2.
Next note that using
M
5(M+1)
(M + 2)(M +
we can
0/a
+ {M+l)
.1=1
write
1
S{M+l)-SiM)
Combining
P/a <
1)''/'^
Y^i'^^°'
M+
S (M)
2
with the inequality
this expression
S (M) >
1/2 for
p/a <
we can estabhsh
1,
that for
1,
5 (M +
1)
-
5"
(M)
>
(M +
1
1)M'5/"
M + 2 + (M + 2)(M+1)''/"
M 7m + i\'-^/" (M +
M
V
1
2
which establishes that
The
case with
S (PI)
P/a >
1
is
monotonically increasing whenever P/a
can be analyzed similarly. Using
S{M + 1)-S {M)
<
M+2 +
M
2{M +
and hence S (M)
is
:
S [M) <
Consider the implications
/3
<
for this
Lemma
for the
<
1/2,
1.
we can
M+
M
V
/
l
\^-^/°
+ p).
<0,
m
profit-maximizing choice of
a, even in the absence of diseconomies of scope,
48
establish that
(M + 1) M'^/°'
{M + 2){M+lf^"
monotonically decreasing from 1/2 to a/ {a
implies that as long as
>0,
2)
i.e.,
for
M.
First, this
F [N/M) =
0,
Lemma
the firm
prefers
M -^
investments.
because this choice maximizes the ex post bargaining power of workers and therefore their
oo,
The presence
of diseconomies of scope only reinforces this effect. This implies:
Proposition 10 Suppose that there are incomplete contracts and no
Then
0,
for all nonnegative, increasing
the profit-maximizing choice of the firm
This result
treated the
justifies
same
(i.e.,
credit constraints,
and convex cost functions F [N/M) that
is
M—
>
and that P <
satisfy hmjv/^oo .^^r
a.
(N/M) =
oo.
our focus in the text on the case where the range of tasks and the division of labor are
N
is
equated with the division of labor). Moreover, given
M -^
oo, the equilibrium
derived in the text applies exactly.
The assumption limM^oo -^r {N/M)
M'T [N/M') for all M' > 1. Essentially,
that the function
is
satisfied, for
MF {N/M)
In contrast,
when
/3
The exact equihbrium
is
is
large
{x)
=
can be relaxed to \m\M^oa
MT (N/M)
<
the results requires that there are no "economies of scope" and
strict
minimum
for
some M' >
1.
This assumption
for all x.
the choice of M
from a
between minimizing diseconomies
>
M —* oo) and maj^imizing ex ante incentives to invest Xn (which dictates M =
will result
a,
trade-oflf
in
in this case will
the main text (which imposes
scope
in this proposition
does not have an interior
example, when F
of scope (which dictates
=
M
—<'
depend on the shape
of the
F
function.
When
1).
/3
>a
our analysis in
oo) can be justified by assuming that the effect of the diseconomies of
enough to dominate the outcome. Mathematically,
rising sufficiently fast.
49
this
corresponds to assuming that
r{N/M)
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