Size-dependent firm regulations
in a many-to-one assignment model:
talent misallocation and the return to skill⇤
Jesica Torres-Coronado†
January 2015
Abstract
I combine Garicano and Rossi-Hansberg (2004, 2006) hierarchical model with Lucas (1978)
technology. This general equilibrium model endogenizes the occupational choices and the
assignment into firms of varying sizes of a mass of heterogeneous workers. I use the framework
to study the allocation of talent if only large firms comply with regulations and thus face
higher costs than smaller firms, which I model as a tax on labor for firms of sizes above a
threshold. The analysis suggests that size-dependent policies affect the allocation of talent not
only around the threshold but throughout the economy. The size-dependent tax encourages
employers just above the threshold in the managerial skill distribution to constrain the size of
their firm and effectively avoid the tax, and disincentives their wage-workers to fully exploit
their talent. Employers and employees in the constrained firms turn indifferent to their match
and their positive assortative matching is only one of infinitely many possible outcomes.
The policy also reallocates the marginal worker in each occupation, which in turn changes
the sorting of the infra-marginal workers, and as a result employers coordinate employees less
talented and run smaller firms than in the equilibrium without distortions. The size-dependent
tax unambiguously lowers the return to skill for workers in the constrained firms and at the
top of the skill distribution. Moreover, it increases the earnings of workers at the bottom of
the skill distribution, and decreases the earnings of those at the very top. If complementarities
in production are not strong enough, size-dependent regulations could significantly lower the
average return to skill in the economy.
Comments welcome. This paper was part of my dissertation. I thank Casey Mulligan, Erik Hurst, Chang-Tai
Hsieh and Rafael Lopes de Melo for their helpful feedback and constant encouragement. I have also benefited
from useful discussions with Santiago Levy, Andrés Neumeyer, and Luca Flabbi, and from comments by seminar
participants at various institutions including the 2014 Latin American Meetings of the Econometric Society and the
IADB. I also thank Mexico’s CONACyT (National Council for Science and Technology) and SEP (Secretariat of
Public Education) as well as the Theodore W. and Esther Schultz Economics Fund of the Social Sciences Division
at the University of Chicago for their financial support.
†
Tecnológico de Monterrey School of Government. email address: [email protected]
⇤
Introduction
More educated workers are more likely to form larger firms. In Mexico for example more educated
business owners in Mexico tend to hire more workers.1 Figure 1 shows that approximately 60% of
business owners with 12 or fewer years of schooling do not hire other workers, whereas only 40% of
them are employers. These figures reverse for business owners with 13 or more years of schooling:
42% of them work alone, whereas 58% of them hire others. Years of schooling are thus strongly
correlated with whether a business owner hires other workers. Moreover, more educated business
owners tend to hire more workers, as figure 2 illustrates.
Figure 1: Likelihood of a business owner employing others by years of schooling.
1
Unless I specify otherwise, the figures use data for male workers for 2010 from the Encuesta Nacional de
Ocupación y Empleo (ENOE), a nationally representative cross-section of households for Mexico. I keep those ages
25 to 49 in areas with more than 2,500 inhabitants employed in the private sector, specifically in manufacturing,
construction, retailing and wholesale, and the service sector. I drop workers with 0 years of schooling, who make
up less than 2% of the population.
2
Figure 2: Employers by years of schooling and number of employees.
Years of schooling are strongly correlated with the size of the firm for employees as well. That
is, wage-workers with more years of schooling tend to work for larger firms. Figure 3 displays
this relation. 48% of employees with 6 or fewer years of schooling work for micro-businesses; that
figure is 32% for employees with 7 to 9 years of schooling, 27% for employees with 10 to 12 years
of schooling, and only 12% for employees with 13 or more years of schooling.
In addition, large firms tend to be made up of more educated workers. As I illustrate in figure
4, 30% of workers in micro-firms have 6 or fewer years of schooling, whereas that figure is less than
10% in the largest firms.
3
Figure 3: Employees by years of schooling and size of the firm.
Figure 4: Skill composition of firms by size.
Similar patterns are observed for other countries. Busso, Neumeyer, Spector (2014) document
for OECD and Latin American countries a positive relation between skill and firm size for both
4
employers and employees. Fox (2012) documents evidence for Sweden consistent with hierarchical
matching. Cardiff-Hicks et al. (2014) find that higher quality workers are sorted into large firms
and large establishments in retailing in the US.
If more talented workers form larger firms, then policies that affect firms of different sizes
differently will therefore distort the return to skill. This paper argues in particular that firm
regulations which favor small businesses misallocate talent throughout the entire economy and
lower the return to skill. I first combine Garicano and Rossi-Hansberg (2004, 2006) hierarchical
model with Lucas (1978) technology. This general equilibrium model endogenizes the occupational
choices and the assignment into firms of varying sizes of a mass of heterogeneous workers. In the
model the skill of a worker completely determines his occupation, the quality of his match and the
size of his firm, and the reward for marginally increasing his type is matching with more talented
workers in larger firms. Next I use the framework to study the allocation of talent if only large
firms comply with regulations and thus face higher costs than smaller firms, which I model as a
size-dependent tax on labor.
I find that the size-dependent tax encourages employers just above the threshold in the managerial skill distribution to constrain the size of their firm to effectively avoid the tax, and disincentives
their wage-workers to fully exploit their talent. Employers and employees in the constrained firms
turn indifferent to their match and their positive assortative matching is only one of infinitely many
possible outcomes. The policy misallocates the talent of managers in the unconstrained firms as
well: the size-dependent tax reallocates the marginal worker in each occupation, which in turn
changes the sorting of the infra-marginal workers, and as a result employers coordinate employees
less talented and run smaller firms than in the equilibrium without distortions.
In the model the size-dependent tax also unambiguously lowers the return to skill for workers
in the constrained firms: wage-workers lose the ability to differentiate themselves and therefore
completely lose the marginal reward for increments in skill, and managers lose the additional reward
for pairing with more skilled employees and their earnings are instead linear in skill. Moreover, it
lowers the marginal return for workers at the top of the skill distribution, who now coordinate less
talented employees in smaller firms. The policy increases the earnings of workers at the bottom of
the skill distribution, and decreases the earnings of those at the very top. Low levels of skill turn
therefore more attractive, whereas high levels of skill become less appealing. If complementarities
in production are not strong enough–and therefore a large fraction of managers constrain the size
of their firm–size-dependent regulations could significantly lower the average return to skill in the
economy.
The analysis does not model the firm’s incentives to evade the law (Rauch (1991), Leal (2010)),
nor does it solve for the optimal governmental response (enforce compliance from large firms
only). I instead assume that the government collects taxes from large firms but not small firms,
and study the effects of this policy in the labor market, specifically on the allocation of workers
and equilibrium prices. To the best of my knowledge, this paper is the first one to suggest and
5
formalize the link between size-dependent policies and the return to skill.
Related literature
Rauch (1991)...
There are several instances in the literature that study the effects of policies that distort the
size of firms on aggregate output and TFP. These authors build on the analysis in Restuccia
and Rogerson (2008) and Hsieh and Klenow (2009a) by introducing occupational choices. Guner
et al. (2008) for example develop a growth model where agents choose whether to work for a
wage or run a business and the amount of capital to carry over the next period. In their model
all establishments regardless of their size choose the same capital to labor ratio in equilibrium.
They model distortions as implicit taxes that are applied only to the use of capital above an
exogenously set level. In their exercise some establishments expand, others contract, and new ones
emerge. Distortions always result in an increase in the equilibrium number of establishments and
a decrease in the average establishment size, output per establishment and aggregate output.
Sanchez-Vela and Valero-Gil (2010) use this framework to study the combined effects of the
lower corporate income tax for small firms and the exemption of business owners from the social
security tax in Mexico. They find that these policies alone do not significantly harm the economy,
but in a context of informality and tax evasion their negative macroeconomic impact is significant.
Schivardi and Torrini (????, ????)...
More recently, Garicano et al. (2013)–whose work I was not aware of until the final stages of this
project–develop a general equilibrium model in which a continuum of managers optimally choose
the size of their firm. They use their model to study the effect of size-dependent labor regulations
on the firm size distribution and TFP. As in my model, in the equilibrium with distortions managers
just above the threshold in the managerial skill distribution constrain the size of their firm to avoid
the tax, a result which they support with conclusive evidence for France, where firms which employ
50 or more workers face higher firing costs, and indeed there is a dramatic change in the firm size
distribution at precisely 50 employees.
Hsieh and Klenow (????)....
I describe the model next. Then I examine the effects of the size-dependent tax, solve for the
equilibrium if all firms regardless of size pay the tax, and present my conclusions.
Garicano and Rossi-Hansberg hierarchical model
Garicano and Rossi-Hansberg (2004) assume workers who differ in one trait–knowledge–and must
choose whether to produce alone or in a team with other workers. To produce in this economy
workers must solve one problem in their unit of time and problems vary in their difficulty z
according to some density q (z). A worker with knowledge z can answer all questions of difficulty
6
less than or equal to z. Workers draw a task and if they know the answer, they produce and their
(expected) earnings are therefore the percentage of tasks they can solve. If they do not know the
answer, they do not produce and earn zero.
Agents can also form teams of identical production workers and one manager so that they
ask the team manager to solve the problem whenever they do not know the answer. They work
together then to exploit complementarities in production between the manager and his wageworkers. Production workers in a team do not interact with each other. A team with n workers
draws n problems and therefore the (expected) output in the team is the percentage of tasks that
the manager zm is able to solve in his n units of time:
y = Q (zm ) n
Communicaton in teams is costly. In particular, solving problems for his production workers
substracts from the manager’s time a fraction h per worker. Workers of skill zp will ask the manager
with probability 1 Q (zp ) which limits his span of control:
hn [1
Q (zp )]  1
This constraint is always binding and firm sizes in the model are therefore a function of the
skill of workers in the bottom layer of the firm. Rewritting the production function of the team
yields
y = Q (zm ) n (zp )
which is a hierarchical production function of the form
y = l1 (zm ) l2 [n (zp )]
for some functions l1 and l2 (Garicano and Hubbard (2002) and Garicano and Rossi-Hansberg
(2014)).
In Garicano and Rossi-Hansberg (2006) the skill distribution is also endogenous. Agents now
differ in their cost of acquiring knowledge (cognitive skill) and choose not only their occupation
and their team but also the problems they want to be able to solve. Moreover, workers can now
form firms with more than two layers. The authors theoretically prove that in equilibrium in
this more complete setup the allocation will exhibit positive sorting and scale of operations effects
(Sattinger (1993)) and perfect stratification of workers into occupations (everyone within a segment
enters the same occupation as in Lucas (1978), Rosen (1982), Evans and Jovanovic (1989), and
others): the less skilled become production workers, the next workers by cognitive skill become
first level problem solvers, the next are second level problem solvers, and so on. They also prove
that the earnings profile in equilibrium will be increasing and convex in cognitive ability, that the
equilibrium exists and is unique, and the the equilibrium allocation will be Pareto optimal.
7
Neither their 2004 model nor their 2006 model with an endogenous skill distribution have in
general a closed-form solution. The authors rely instead on numerical examples in their comparative statics. In these examples their propositions hold in equilibrium and in addition firm sizes
follow a power law.
I combine Garicano and Rossi-Hansberg hierarchical model with a Lucas (1978) technology
assuming a fixed skill distribution. In particular, I set l1 (x) = x and l2 (x) = x↵ . Output in teams
in this framework is then
y = zm [n (zp )]↵
Garicano et al. (2013)–whose work I was not aware of until the final stages of my dissertation–
use the same technology but with firm sizes independent of skill exactly as in Lucas (1978).
Instead of specifying a density for the difficulty of tasks and assuming values for the communication cost, I assume that in equilibrium firm sizes follow a Pareto distribution with power 1
and bounded support, which is an empirical regularity (Xabaix (2009)...). I then solve for n (zp )
when finding the equilibrium. Implicit in this approach then is the employee-employer interaction
process developed by Garicano and Rossi-Hansberg.
My framework does not have a closed-form solution either. I rely on numerical examples to
illustrate the comparative statics of the model and work out the effects of size-dependent policies.
The model goes as follows:
Individuals possess one unit of time and an amount z of skill. This amount varies continuously
in the population according to the (given) skill distribution F (z) with support [L, H] and density
f (z). Agents choose the use of their time and talent that maximizes their earnings. They can
either produce alone or in a team (firm or business) with other workers. They work together to
specialize either in managerial activities–running the business–or in production activities–working
as one of the firm’s employees–and thus exploit complementarities in production. Workers therefore
have three options: to produce alone, to run a business hiring others and to work for a wage for
someone else. The formation of firms in the model is endogenous–the entrepreneurs optimally
select the size of their firm and the quality of his employees, whereas the wage-workers optimally
select which of the continuum of teams to join.
A manager can coordinate at most n (zp ) employees of quality zp . This technological constraint
effectively sums up the random task-drawing production process detailed in the original framework.
The model requires n (zp ) > 1 and n0 (zp ) > 0 for all zp . The span of control of the entrepreneur
is therefore limited by the skill of his employees–better employees allow the manager to supervise
a larger number of them. I also assume that n00 < 0.2
2
As I explain in the next section, ↵ needs to be high enough to guarantee that output per worker (average
product) will be increasing in zp , but low enough to guarantee that teams will find it more profitable to assign the
managerial position to the most skilled worker, and the employee positions to the least skilled workers, and not the
other way around. The upper bound on ↵ decreases as n turns less concave and at an extreme, when the function
turns convex, the bounds flip in the wrong order and the equilibrium ceases to exist.
8
The manager applies his human capital to the time of his n identical employees. More precisely,
the match of manager zm and n production workers yields
(1)
z m n↵
units of output, where ↵ > 0. The manager’s problem is to choose the quality of his employees
zp and the size of his firm n to solve
max zm n↵
zp ,n
w (zp ) n
(2)
s.t. n  n (zp )
where w (zp ) denotes the equilibrium wage rate. The inequality constraint will always be
binding: from the manager’s point of view, firm size l is feasible hiring wage-workers of any type
above n 1 (l), but costs are minimal employing workers n 1 (l) who earn the lowest wage, and that
is true for all firm sizes. I therefore rewrite output as
y (zp , zm ) = zm [n (zp )]↵
(3)
and managerial rents are thus
R (zm ) = max zm [n (x)]↵
x
w (x) n (x)
(4)
s (z) denotes the earnings of worker z if he works alone. Workers take wages and managerial
rents as given and choose the occupation that yields the highest earnings given their skill:
max {w (z) , s (z) , R (z)}
(5)
The equilibrium consists of an assignment of individuals into occupations and into firms and
wages and managerial rents such that no agent desires to switch to another occupation or firm.
The positive relation between the entrepreneur’s span of control and the quality of his employees, together with the complementarity introduced in the production function between the
manager’s skill and his productive resources–in other words, the positive cross derivative in the
production function–yield an equilibrium that exhibits positive sorting in both quality and quantity,
as in Garicano and Rossi-Hansberg (2004, 2006) and Garicano and Hubbard (2002). Specifically,
the best managers match with the best wage-workers to form the largest teams, the second-best
managers match with the second-best wage-workers to form the second-largest firms, and so on.
The smallest firms in the market are thus the match of the least-skilled entrepreneurs to the
9
least-skilled wage-workers.
The equilibrium also exhibits perfect stratification of workers into occupations based on their
skill, but unlike the equilibrium in the original framework, the equilibrium in this model could
exhibit segregation of workers as in Kremer and Maskin (1996) in which the top agents match
together whereas workers at the bottom produce alone. Whether the equilibrium entails segregation
or not depends on (i ) the slope of s (·), which indicates the sensitivity of earnings to skill in
the outside option, (ii ) the parameters of the skill distribution f (·), and (iii ) the strength of
complementarities in production (the returns to increasing the team size in the production function)
captured by ↵. If s (·) is increasing, then the equilibrium does not exhibit segregation only when
the skill density has thin and short tails (L and H are not too far apart) or ↵ is not very large. If
s (·) equals a constant, the only equilibrium exhibits segregation.
Earnings in the outside option increasing in skill
If ↵ < 1 (production in teams exhibits decreasing returns to increasing the size of the firm) and
s (·) is an increasing (not necessarily linear) function of workers’ talent, then in equilibrium the less
skilled agents become production workers, workers in the middle produce alone, while the most
skilled ones work managing others. Intuitively, if ↵ < 1, then managers tend to demand unskilled
workers to set up small firms (thus preventing the decreasing returns in n from turning significant),
which leaves workers in the middle of the distribution producing alone. The following figure shows
the equilibrium allocation of workers into occupations in this scenario.
Figure 5: Allocation of workers across occupations with ↵ < 1 and s (·) increasing.
To find the equilibrium I follow Garicano and Rossi-Hansberg’s algorithm. I conjecture the
existence of two thresholds z1 and z2 such that agents with skill z  z1 optimally become wageworkers, agents with skill z1 < z < z2 optimally choose to work alone, and agents with skill z z2
optimally choose to manage others. The first step is to find the equilibrium assignment function
zm (zp ), which denotes the manager zm that corresponds to workers of skill zp .
In equilibrium the market for managers clears: for all s in [L, z1 ] it has to be the case that
ˆ
s
L
f (x)
dx = F (zm (s))
n (x)
which by Leibniz rule implies
10
F (z2 )
(6)
f (s)
dzm
= f (zm (s))
n (s)
ds
(7)
In equilibrium it is also the case that, for any s in [L, z1 ], the share of firms with n (s) or fewer
employees equals the share of managers hiring employees of skill s or lower:
G (n (s)) =
F (zm (s)) F (z2 )
1 F (z2 )
(8)
where G (·) denotes the firm-size distribution. Equation 8 implies
dzm
f (zm (s))
dn
ds
g (n (s))
=
ds
1 F (z2 )
(9)
I assume that the market equilibrium generates a Pareto firm-size distribution with power
1 and (known) support [n, n]. Equation 9 therefore provides a differential equation for n (·). I
solve for zm (zp ) and n (zp ) using the system of differential equations in 7 and 9 with zm (L) = z2
(which states that the worst wage-workers are matched to the worst employers) and n (L) = n
(which states that the least-skilled wage-workers group in the smallest teams) as the boundary
conditions. Unlike the original model, n (·) in my framework is not given but solved-for once I
assume a particular firm size distribution in equilibrium.
The second step is to combine the first order condition of the entrepreneur’s problem with the
equilibrium assignment function to find the equilibrium wage function. The firm’s problem is to
select zp to maximize R = zm [n (zp )]↵ w (zp ) n (zp ). The first order condition is
zm ↵ [n (zp )]↵
1
dn
dw
dn
=
n (zp ) + w (zp )
dzp
dzp
dzp
(10)
I plug in zm (zp ) and n (zp ) from step one and solve the differential equation for w (zp ) using
the equilibrium condition w (z1 ) = s (z1 ), which states that the last wage-workers are indifferent
between working for a wage or working alone.
The final step is to pin down the constants of integration in the two previous steps to completely
characterize the assignment function zm (zp ), and the earnings functions w (z) and R (z). To do so
I solve zm (z1 ) = H (which states that the best wage-workers are matched to the best managers
and guarantees that n (z1 ) = n holds as well) and R (z2 ) = s (z2 ) (which states that the worst
managers are indifferent between running a team or working alone).
When a solution to the algorithm exists, it is unique. However, there are several combinations
of parameter values which give rise to the same firm size distribution and no solution exists for all
combinations of parametes.
I display in figure 6 the earnings profiles for an hypothetical economy where ↵ = 0.75, s (z) = z,
and the skill distribution follows a power law with L = 1, H = 31 and exponent equal to 1/20.
Indeed, in that example it is the case that w (z)
s (z) and w (z)
R (z) for all z in [L, z1 ],
11
s (z) w (z) and s (z) R (z) for all z in [z1 , z2 ], and R (z) s (z) and R (z) s (z) for all z in
[z2 , H], and therefore the conjecture is correct and the allocation is an equilibrium.3
Figure 6: Earnings in equilibrium in a numerical exercise.
Unlike equilibrium earnings in Garicano and Rossi-Hansberg (2006), equilibrium earnings in
my framework are convex only at the top of the distribution. The wage function is concave as in
their 2004 paper and thus does not exhibit a superstar effect (Rosen (1981)).
Changing the skill distribution
The equilibrium is not invariant to changes in the skill distribution. A mean-preserving spread
of the skill density could shift the equilibrium to one where workers at the bottom produce alone
and those in the middle become wage-workers. Intuitively, longer tails in the skill density indicate
that workers at the top turn more and more skilled (H is higher), that workers at the bottom
turn more and less skilled (L is lower), and that there is a smaller mass around the mean. If
managers at the top become more and better, they will now optimally demand more employees
with more talent, turning wage-working more attractive than producing alone for owners in the
middle of the distribution. Similarly, if workers at the bottom turn less skilled, they will demand
less talented managers, but their potential managers (in the middle of the distribution) now find it
more profitable to work for a wage, thus leaving them with producing alone as their best alternative.
The best workers then choose to work together in this scenario, leaving the least-skilled producing
alone. This equilibrium entails the same concept of segregation that Kremer and Maskin (1996)
describe.
3
The positive sorting of workers is guaranteed as long as the second-order condition holds. I do not prove the
uniqueness of the equilibrium (which Garicano and Rossi-Hansberg (2006) do for the particular functions they use),
but in my numerical exercises equilibria with different stratifications do not exist.
12
Figure 7: Change in the stratification of workers across occupations with a mean-preserving spread
in the skill distribution.
Indeed, in my numerical exercises I find that as I spread out the original skill distribution while
preserving its mean, the share of owners producing alone in equilibrium decreases–better managers
demand better employees, which induces the least-skilled entrepreneurs currently working alone to
become wage-workers, whereas less talented wage-workers demand less talented managers, which
induces the most-skilled entrepreneurs currently working alone to become employers. For example,
the equilibrium which figure 6 depicts in which 73.32% of individuals work for a wage, 6.89% work
alone, and 19.78% hire others turns into one in which these shares amount to 74.86%, 0.79%, and
24.35%, respectively. For a large enough spread in the distribution, the equilibrium stratification
exhibits segregation of workers–now those in the middle become wage-workers whereas those at the
bottom produce alone. The earnings in equilibrium in this example are displayed in the following
figure, where I split the graph into two in order to appreciate the earnings of those workers in the
left-hand tail of the skill distribution. There 11.37% of workers work alone, except this time these
are the least-skilled workers in the market.
13
Figure 8: Earnings profiles in equilibrium after a mean-preserving spread of the distribution in
figure 6.
Changing the returns to increasing the firm size
The strength of complementarities in production also determine whether the equilibrium exhibits segregation or not. Note first that smaller returns to increasing the size of the firm evidently
lead to more workers producing alone in equilibrium–if complementarities between the manager
and his productive resources become weaker, workers are less likely to benefit from producing together. When I decrease ↵ in my model from 0.75 to 0.65, for example, the allocation of workers
changes from 93.11% in team production and 6.89% working alone to 84.12% producing together
and 15.88% producing alone. There is, however, a lower bound on ↵. In equilibrium output per
worker in teams must be increasing in p (the employees’ skill)–were that not the case, wages would
be decreasing in talent. The largest possible share of workers producing alone in equilibrium when
s (·) is increasing is therefore attained when ↵ equals its lower bound.
Conversely, stronger complementarities in production lead to fewer workers producing alone
in equilibrium. Indeed, there is a value of ↵ which guarantees no segregation and a mass zero of
workers producing alone, as figure 9 illustrates. If ↵ increases beyond that point, the equilibrium
stratification of workers turns into one with segregation, as I illustrate in figure 10. Intuitively, as
↵ increases, managers employ ever more skilled workers to set up bigger firms, thus exploiting the
larger returns. Larger values of ↵ therefore lower the demand for the least-skilled workers, who
at some point may find producing alone more attractive. There is, however, an upper bound on
the size of ↵–in equilibrium it should always be optimal for the most skilled workers in a team
to work as managers while the least skilled ones work as employees, lest the equilibrium cease to
exist. There is no need, however, for this upper bound to be less than 1.
14
Figure 9: Equilibrium with a mass zero of workers producing alone when s (·) is increasing.
Figure 10: Equilibrium stratification of workers with s (·) increasing and ↵ large.
To sum up, when s (z) is increasing there are two possible equilibria, depending on the skill
distribution and the strength of complementarities in production. If the skill density has thin and
short tails (L and H are not too far apart) or ↵ is not very large, the assignment of workers into
15
occupations in equilibrium does not exhibit segregation–the least skilled ones work for a wage,
workers in the middle produce alone, and the most skilled ones work managing others. If the
density has fat and long tails (L and H are far apart) or ↵ is large enough, the top workers choose
to work together, leaving the least-skilled to work alone.
Earnings in the outside option independent of talent
I now consider the equilibrium when workers producing alone earn an amount z regardless of their
skill level: s (z) = z. In this scenario there is no equilibrium in which workers at the bottom enter
wage-working: for that to be the case, the wage function w (·) would need to be above s (·) at
low values of z, which either implies a discontinuity in the earnings profile if the wage function
is increasing (which in turn means that the last wage-worker can never be indifferent between
wage-working or producing alone), or requires a decreasing wage function (which cannot be). The
only other possibility is an equilibrium where workers at the bottom produce alone and those in
the middle work for managers at the top.
For an equilibrium with segregation to exist if s (·) equals a constant, the model needs a value of
↵ high enough (although not necessarily above 1). That way, managers are encouraged to exploit
the strong returns by employing high-skill workers, thus lowering the demand for the least-skilled
ones, who in turn find producing alone more attractive. Figure 11 illustrates the equilibrium
stratification of workers in this scenario, while figure 12 displays the earnings in equilibrium for
an hypothetical economy. To solve the model I follow the same steps the I describe the previous
section.
Figure 11: Stratification of workers in equilibrium when s (·) equals a constant.
16
Figure 12: Earnings in equilibrium in the numerical exercise of figure 6 when ↵ = 4/3 and s (·)
does not vary with talent.
The equilibrium stratification in this case does not change with a mean-preserving spread of
the distribution nor with increases in ↵. The largest value of ↵ that guarantees that an equilibrium
exists is that which yields zero workers producing alone, as I illustrate in the following figure. The
smallest value of ↵ for an equilibrium to exist is that which guarantees that output per worker in
teams is increasing in employees’ skill.
Figure 13: Equilibrium with a mass zero of workers producing alone when s (·) does not vary with
talent.
17
Equilibrium under size-dependent firm regulations with perfect enforcement
I use the model to study the effects of the following policy: firms which hire more than N workers
pay ⌧ % of their wage rate. This policy is perfectly enforced. I find that the size-dependent tax
reduces the average firm size in the economy–managers just above the threshold in the managerial
skill distribution constrain the size of their firm to N and the largest firm sizes disappear. The
magnitude of this effect depends on the strength of complementarities in production relative to
the size of the tax rate.
The size-dependent tax only reallocates workers in the middle of the skill distribution into
another occupation. It does not distort the occupational choices of the least and the most talented
individuals in the market but lowers the average quality of those who work for a wage. Moreover,
if the equilibrium does not exhibit segregation, the size-dependent tax unambiguously increases
the fraction of workers in small firms (firms with N or fewer employees).
The tax breaks the positive assortative matching of wage-workers and managers in the constrained firms. Wage-workers in this segment of the market are identical from the point of view of
the managers, and employees are indifferent to the skill of their manager because in equilibrium
they earn the same wage regardless of their firm. Positive sorting is therefore only one of infinitely
many possible outcomes. The tax does not break the positive sorting of workers in unconstrained
firms but it misallocates managerial talent in this segment of the market–the tax distorts the selection into occupations, which alters the rank of each worker within each occupation, which in
turn distorts the equilibrium assignment of managers to wage-workers.
In the model firm sizes, the selection of workers into occupations, and the quality of the
equilibrium assignments affect the return to skill. Size-dependent regulations lower the return to
skill for both employees and employers in the constrained firms. If complementarities in production
are not strong enough–and therefore a large fraction of managers constrain the size of their firm–the
tax could significantly lower the average return to skill in the market. Size-dependent regulations
also lower the return to skill for managers at the top of the skill distribution. In general they also
lower the slope and the level of the profit function for managers other than the best, but if the
reallocation of workers across occupations is only minor, then the marginal return and the rents of
managers in the middle of the (managerial) skill distribution are larger than before the tax. The
effect of the tax on the slope of the wage function for workers in firms other than the constrained
firms is unclear. The tax, however, unambiguously increases the wage rate of employees at the
bottom and decrease the wage rate of employees at the top.
I now describe how to solve for the equilibrium in the scenario without segregation. The
equilibria with distortions in the other two scenarios are obtained in a similar way.
Define z1 ⌘ n 1 (N ) (the employees’ skill that correspond to team size N ) using the (now
known) technology which groups workers into teams–n (·)–in the equilibrium without distortions.
18
Consider manager zm . His optimization problem with the size-dependent tax is:
max zm [n (zp )]↵
zp
8
<w (z ) n (z )
if n (zp )  N or zp  z1
p
p
:w (z ) n (z ) (1 + ⌧ ) if n (z ) > N or z > z
p
p
p
p
1
(11)
The optimality condition varies with the skill of the manager: there is an interior solution for
managers of low and high skill, and a corner solution at z1 for managers in the middle. More
clearly, note that the extra benefit of increasing zp is
zm ↵ [n (zp )]
↵ 1
✓
dn (zp )
dzp
◆
(12)
which is increasing in zm . The additional cost is independent of zm , but the tax introduces a
discontinuity at z1 . If n (zp )  N , the marginal cost is
✓
dw (zp )
dzp
◆
n (zp ) + w (zp )
✓
◆
(13)
(1 + ⌧ )
(14)
dn (zp )
dzp
If n (zp ) > N , it is instead
✓
dw (zp )
dzp
◆
n (zp ) + w (zp )
✓
dn (zp )
dzp
◆
The following figure depicts the optimality condition for different values of zm , assuming that
the marginal benefit is decreasing in zp and the marginal cost is increasing in zp , which does not
necessarily have to be the case as long as the second-order condition holds.
19
Figure 14: Marginal benefit and marginal cost of increasing zp for different values of managerial
talent zm .
I use z5 to stand for the last manager zm for whom
zm ↵ [n (zp )]
↵ 1
✓
dn (zp )
dzp
◆
=
✓
dw (zp )
dzp
◆
n (zp ) + w (zp )
✓
dn (zp )
dzp
◆
(15)
z5 is thus the manager who hires N workers and does not pay the tax. Managers of skill zm  z5
hire n  N workers by selecting zp according to this equation.
Similarly, I use z6 to denote the first manager for whom
zm ↵ [n (zp )]
↵ 1
✓
dn (zp )
dzp
◆
=
✓
dw (zp )
dzp
◆
n (zp ) + w (zp )
✓
dn (zp )
dzp
◆
(1 + ⌧ )
(16)
z6 is thus the first manager to pay the tax. Managers of skill zm z6 choose n N by selecting
zp based on this relation.
Managers of skill zm in [z5 , z6 ] optimally choose N as the size of their firm–for them the marginal
benefit of choosing N workers exceeds its marginal cost, whereas the marginal cost of setting n > N
is above its marginal benefit.
I argue that it cannot be the case that z5 = z6 , which would require the marginal cost in Figure
14 to be continuous in zp but with a kink at z1 . Assume on the contrary that z5 = z6 = z ⇤ , that
is, there is a manager z ⇤ who hires N workers and is indifferent between paying the tax or not
paying the tax. That would require z ⇤ to earn the same rent whether he pays the tax or not. In
other words, z ⇤ N ↵ w (z1 ) N = z ⇤ N ↵ w (z1 ) N (1 + ⌧ ) must hold. But that equality cannot hold
unless ⌧ = 0. The set [z5 , z6 ] is therefore not a singleton.
20
Managers in [z5 , z6 ] therefore optimally demand wage-workers of skill z1 to set up firms of size
N . Note that the employees’ skill affects the manager’s profits through (i ) the size of the firm
and (ii ) the wage rate. If the size of the firm for all managers in [z5 , z6 ] is the same at N , they
would optimally employ workers in [z1 , z2 ] for some z2 > z1 if all these wage-workers earned the
same wage rate (workers of skill below z1 cannot form teams of size N , whereas for workers of
skill above z1 , N is always a feasible firm size). This way managers [z5 , z6 ] would be indifferent
to whom they hire and wage-workers [z1 , z2 ] would be indifferent to the quality of their manager
because they would earn the same wage regardless of their match. In the only equilibrium under
the size-dependent policy there is therefore a group of wage-workers of skill above z1 that will be
allocated to managers [z5 , z6 ] in equilibrium for a constant wage rate, with the endogenous cutoff
z2 as the last of these wage-workers and N ⌘ n (z2 ).4 z6 is the manager indifferent between not
paying the tax with a firm with N employees and paying the tax running a larger firm with N
employees. The figure below shows the relation between z1 , z2 , z5 , and z6 in equilibrium for the
hypothetical example introduced in figure 14.
Figure 15: Cutoffs z1 , z2 , z5 and z6 in equilibrium.
Thus in the new equilibrium there are cutoffs z1 through z6 in the skill distribution such that:
• workers in [L, z3 ] become employees
– those in [L, z1 ] work in firms with less than N workers
– those in [z1 , z2 ] work in firms with exactly N workers for an endogenous wage rate which
does not vary with skill
4
More formally, z1 = z2 would require allocating a mass zero of wage-workers to managers in [z5 , z6 ], which
cannot be. My code is flexible enough to allow for the possibility of z1 = z2 , but I do not find that to be the case.
21
– those in [z2 , z3 ] work in firms with more than N workers
• workers in [z3 , z4 ] produce alone
• workers in [z4 , H] become employers
– those in [z4 , z5 ] run firms with less than N workers and select their firm size according
to equation 15 above
– those in [z5 , z6 ] hire exactly N workers and their profits are therefore linear
– those in [z6 , H] work in teams with more than N workers and select their firm size
according to equation 16 above
Motivated by the terminology in Garicano et al. (2013), I call firms with exactly N workers the
constrained firms as they do.
The assignment functions for each segment of wage-workers in the unconstrained firms are
obtained equating the supply and demand for managers using the equivalent of equation 7. The
assignment of wage-workers in [z1 , z2 ] to managers in [z5 , z6 ] is indeterminate, as I explain below.
The only equilibrium condition for this segment of workers is that the demand for managers for
the constrained firms must equal its market supply:
F (z2 )
F (z1 )
= F (z6 ) F (z5 )
(17)
N
The wage function for the first segment of wage-workers is obtained combining the equilibrium
assignment function and the first order condition in equation 15; the wage function for the second
segment is flat and set equal to w (z1 ); the wage function for the third segment is found combining
the equilibrium assignment function and the first order condition in equation 16. The rents for the
business-owners are defined as production minus labor costs at an optimum, and for the second
segment of business-owners these are linear in their skill. I adjust the equations for each set of
marginal workers accordingly.
Firm sizes
In equilibrium the size-dependent tax splits both the set of wage-workers and the set of managers
into three, as I illustrate in figure 16 for the three possible equilibria: within each occupation,
individuals at the bottom work in firms with less than N employees, those just above the threshold
iwork in firms with exactly N employees, and those at the top work in firms with at least N
employees, for some endogenous size N > N . Workers do not set up firms of sizes N, N ; they
instead constrain the size of their firm to N as I illustrate in figure 17.
22
Figure 16: Equilibrium allocation of workers under the policy for each type of stratification.
Figure 17: Technology n (·) and observed firm sizes under size-dependent regulations.
Size-dependent regulations lower the average firm size in the economy, not only because a
portion of workers constrain their firm sizes to N , but also because as I explain below, the tax
23
lowers the skill of both the best employee–which in turn lowers the largest firm size in the market–
and the worst employee (in the equilibrium with segregation)–which in turn lowers the smallest
firm size.
The following figure illustrates the change to the firm size density with a size-dependent tax
if there is no segregation of workers in equilibrium, that is, if the smallest firm size is still n (L)
(if the equilibrium exhibits segregation, the only difference is that the smallest firm size does not
stay constant but is lower with the tax). As the graph shows, there is a mass zero of firms of size
N, N and a mass point at N . Firms of very large sizes disappear.
Figure 18: Firm size density with and without size-dependent taxes.
The fraction of managers that constrain the size of their firm depends on the strength of
complementarities in production relative to the size of the tax rate. When ↵ is relatively high,
only few managers constrain the size of their firm and more teams optimally choose to pay the
tax. The weaker the complementarities in production relative to the tax rate, the larger the
share of firms that constrain their size. Indeed, when the tax rate is high enough relative to ↵,
no team pays the tax in equilibrium. That is, if the tax rate is high enough (or alternatively,
if the complementarities in production are weak enough) the segment of both wage-workers and
employers splits into only two: those who set up firms with less than N employees and those who
choose exactly N as their firm size. The equilibrium has only four cutoffs in that case. I display
these cutoffs in the figure below. Indeed, when I consider ↵ = 0.65 in my numerical exercises, the
24
only equilibrium under the policy has this shape, even at low tax rates.
Figure 19: Allocation of workers in markets with size-dependent tax rates high relative to the
strength of complementarities in production.
Occupational choices
I discuss the effect of the size-dependent tax on occupational choices in each of the three equilibria.
The policy alters the occupational choices of individuals at the margin. When s (z) is increasing
and the equilibrium does not exhibit segregation, the tax induces the best wage-workers and the
worst employers to produce alone, as I illustrate in figure 20 below. Intuitively, with the policy
workers have now incentives to form smaller teams, which can only be accomplished by lowering
the quality of the wage-workers by sending the best wage-workers into business-ownership (who
produce without hiring others). A smaller mass of wage-workers optimally demands a smaller
mass of managers, and the worst managers in the economy are induced to produce alone–the top
managers in the market outbid the bottom managers in the competition for wage-workers, and the
bottom managers are left to produce alone. The share of workers producing alone unambiguously
increases.
25
Figure 20: Reallocation of workers across occupations in the equilibrium without segregation.
Consider next the equilibrium when s (z) is flat. There the tax induces the best owners working
alone to become wage-workers (thus lowering the quality of the worst wage-worker in the market)
and the best wage-workers to become employers, as I illustrate in figure 21. Again, to set up
smaller firms, the market attracts the best workers producing alone into team production, thus
lowering the quality of the economy’s wage-workers. In turn, the new lower-skill wage-workers
require lower-skill managers, and thus the best wage-workers enter entrepreneurship. This last
effect is absent in the equilibrium without segregation–there the worst wage-worker is also the
most unskilled worker in the market (L). The share of workers producing alone decreases and the
share of employers in the market increases.
Figure 21: Reallocation of workers across occupations in the equilibrium with s (·) independent of
talent.
The most interesting reallocation arises when s (z) is increasing and the equilibrium exhibits
segregation, which I display in figure 22. Size-dependent regulations could reallocate some of the
workers at the top of the distribution into working alone. Once more, the policy lowers the quality
of both the worst and the best wage-workers in the market in order to lower firm sizes, and the
best wage-workers are induced to become entrepreneurs (producing alone or hiring others). With
earnings in the outside option increasing, some of these top wage-workers may now find producing
alone more attractive than hiring others. There are therefore three cases:
1. All of the wage-workers who leave the sector become employers;
2. Some of the wage-workers who leave the sector choose to produce alone and the rest become
employers;
3. All of the wage-workers who leave the sector and some of the employers at the bottom choose
to produce alone.
26
Figure 22: Reallocation of workers across occupations in the equilibrium with s (·) increasing and
segregation.
Intuitively, wage-workers of low quality now enter the sector. These workers demand employers
of lower quality and this demand induces the top wage-workers who leave the sector to become
employers. At the same time, the top wage-workers leaving the sector could reduce the mass of
wage-workers in the market, and because the top managers outbid those at the bottom in the
competition for wage-workers, some of the currently bottom managers could be left to produce
alone. It is therefore not clear the effect of size-dependent regulations on the share of workers in
each occupation in this scenario.
The effect of the policy on the distribution of workers across firm sizes depends on the equilibrium we consider. When s (·) is increasing and the equilibrium does not exhibit segregation, the
tax definitely lowers the fraction of individuals working for someone else (see figure 20). A larger
fraction of them set up firms with N employees or less and the fraction of employees in small firms,
therefore, increases. In the other two scenarios that conclusion is not as straightforward, since it
is not clear what happens to the amount of workers in each occupation. It is clear, however, that
in these two cases fewer workers produce alone (see figures 21 and 22).
Equilibrium assignments
The size-dependent tax breaks the positive assortative matching of wage-workers and managers in
the constrained firms. Wage-workers in this segment of the market are identical from the point
of view of the managers–employers prefer one type of wage-workers over another if they group in
larger teams, but if all wage-workers in the constrained segment of the market group in teams of
the same size, then no type is more attractive than the next. Managers in these firms are thus
indifferent to whom they hire. Employees are also indifferent to their type of manager because they
earn the same wage regardless of their firm. This segment of the assignment function is therefore
indeterminate. Managers could potentially mix employees from different types in a single firm,
27
and positive sorting is only one of infinitely many possible outcomes.
The policy does not break the positive sorting of employees and employers in the unconstrained
firms. However, it distorts the selection of workers into occupations, which in turn misallocates
managerial talent in this segment of the market. More precisely, the tax changes the marginal
worker in each occupation, which changes the rank of each individual within each occupation,
which in turn distorts the equilibrium assignment of managers to wage-workers.
In figures 23 and 24 I illustrate the change to the assignment function in the market without
segregation. There size-dependent regulations decrease the quality of the best employee and increase that of the worst employer in the market (see figure 20). Note that without distortions
the assignment function is concave–more talented employees need fewer managers to coordinate
their work, and therefore the rate at which wage-workers are matched to managers decreases with
their skill. If the reallocation of individuals across occupations is such that the new marginal
wage-worker and the new marginal employer are close enough to the initial ones, so that the new
assignment function cuts the old one as I display in figure 23, then size-dependent regulations
match the bottom and top wage-workers in the unconstrained firms to managers of higher quality, and the middle wage-workers to employers less talented than before. If the distance between
the old and the new marginal wage-workers and the old and the new marginal employers is large
enough enough, as I display in figure 24, then under size-dependent regulations all wage-workers
in the unconstrained firms are matched to managers more talented than before.
28
Figure 23: Change in the assignment function in the equilibrium without segregation and without
significant reallocation of workers across occupations.
29
Figure 24: Change in the assignment function in the equilibrium without segregation and with
significant reallocation of workers across occupations.
In the following figure I illustrate the change in the assignment function when s (·) equals
a constant. In that scenario size-dependent regulations lower the quality of the best and the
worst wage-worker and of the worst manager in the market, and therefore all wage-workers in the
unconstrained firms are coordinated by more skilled managers than before.
30
Figure 25: Change in the assignment function in the equilibrium with s (·) independent of talent.
As I argued in the previous section, in the equilibrium with s (·) increasing and segregation
there are three possibilities for the reallocation of workers across occupations. If all of the wageworkers who leave the sector become employers (case 1) or if some of them choose to produce alone
(case 2), then the change to the assignment function follows the pattern displayed in figure 25–sizedependent regulations lower the quality of the best and the worst wage-worker and of the worst
manager in the market, and therefore all wage-workers in the unconstrained firms are coordinated
by more skilled managers than before. If not only the wage-workers who leave the sector but also
some of the employers at the bottom choose to produce alone (case 3), then the change to the
assignment function is similar to one of the two displayed in figures 23 and 24, and which of the
two depends on the extent of the reallocation of workers across occupations (the distance between
the new and the old marginal workers in each occupation).
Earnings
To obtain the effect of size-dependent regulations on earnings I first examine the change in the
slope of each segment of the earnings function. The marginal return to skill for the employers
follows
31
R0 (zm ) = [n (zp (zm ))]↵
(18)
which is derived from the Envelope Theorem and where zp (zm ) denotes the skill of the wageworkers assigned to manager zm in equilibrium.5 The change in the slope of the profit function is
therefore determined by the change in the equilibrium assignment function. If under size-dependent
regulations a manager is assigned more resources in equilibrium, the return to increments in his
skill is higher. If to the contrary he runs a smaller firm, the return to increments in his skill is
lower. The profits of managers in the constrained firms turn linear in their skill.
The return to incremental skill for the wage-workers in the equilibrium without distortions is
⇥
w0 (zp ) = zm (zp ) ↵ [n (zp )]↵
1
w (zp )
⇤ n0 (zp )
n (zp )
(19)
which is simply a rewrite of the first-order-condition of the entrepreneur’s problem. Under
size-dependent regulations the marginal return follows equation 19 for employees in small firms,
changes to zero for employees in the constrained firms, and follows
"
zm (zp ) ↵ [n (zp )]↵
w (zp ) =
1+⌧
0
1
w (zp )
#
n0 (zp )
n (zp )
(20)
for employees in firms which pay the tax. The change in the slope of the wage function for
workers in small firms therefore depends on the change in the assignment function–if an employee
is matched to a better manager, the return to incremental skill is larger–and the change in the
wage rate–if the employee earns a higher wage, the return to increments in his skill is lower. For
workers in firms which choose to pay the tax the return to incremental skill depends also on the
tax rate–ceteris paribus, the tax lowers the employee’s marginal product, which in turn lowers his
marginal return.
The change in the marginal worker in each occupation pins down the effect of size-dependent
regulations on the level of earnings. For instance, if the profit function has now a higher slope
and the marginal employer is now of less talent, the earnings of managers are higher under sizedependent regulations compared to the equilibrium without distortions. If the marginal employer
is more skilled than before, the earnings of managers at the bottom of the (managerial) skill
distribution are lower than before, while the earnings of managers at the top are higher compared
to the equilibrium without distortions. The change in the level of profits coupled with the change
in output allows me next to pin down the change in the level of wages.
Figure 26 shows the change in the earnings function in the equilibrium without segregation
when the new marginal workers in each occupation are close enough to the initial ones, that is,
when the assignment function moves according to figure 23. Note first that the worker indifferent
´x
For all x z2 f (i) n (zp (i)) di = F [zp (x)] holds in equilibrium. Differentiating with respect to x yields (the
inverse) assignment function.
5
32
between working alone and higher others is of higher skill. Managers at the bottom and the top
are matched to workers of talent lower than before, and therefore the profit function for them turns
flatter. Managers in the middle run firms larger than in the equilibrium without distortions, and
therefore their profit function is steeper. It has to be the case therefore that the profit function
in the equilibrium with distortions intersects twice the profit function in the equilibrium without
distortions–managers at the bottom and the top earn less than before while managers in the middle
receive higher earnings.
Figure 26: Change in the earnings profile in the equilibrium without segregation and without a
major reallocation of workers across occupations.
The agent indifferent between working alone and working for someone else is of lower skill than
before. The wage function for the top wage-workers is therefore below the original one. These
employees are matched to better managers, yet the tax decreases their marginal product. The
change in the slope of the wage function is unclear. Wage-workers at the bottom are matched
to more talented managers. Their teams therefore produce more than before but their managers
receive a lower rent. Their wages therefore have to be higher than in the equilibrium without
distortions. Once more, the effect on their marginal wage rate is unclear–on the one hand, these
workers match with better managers, which pushes upward the slope of the wage function; on the
other hand, they receive a higher wage, which pushes the slope in the opposite direction. Because
employees at the top earn a lower wage rate, the wage function with distortions intersects the wage
function without size-dependent regulations, as I show in the figure.
33
Figure 27 shows the change to equilibrium earnings when there is no segregation and all managers run smaller firms, that is, when the assignment function changes as in figure 24. The profit
function turns flatter and all managers earn lower profits compared to the equilibrium without
distortions.
Figure 27: Change in the earnings profile in the equilibrium without segregation and with a
significant reallocation of workers across occupations.
Once more, the wage function for the top wage-workers is below the initial one–the wage-worker
indifferent between working alone and working for someone else is of lower skill. These employees
are matched to better managers, yet the tax decreases their marginal product–the change in the
slope of their segment of the wage function is unclear. Wage-workers at the bottom are matched
to more talented managers (their teams produce more than before) but their managers receive a
lower rent–their wages are higher than in the equilibrium without distortions. The effect on the
marginal wage rate is unclear.
Figure 28 illustrates the change in equilibrium earnings when earnings in the outside option
are independent of talent. In the new equilibrium all managers run smaller firms (see figure 25)
and therefore their profit function turns flatter. The worker indifferent between working for a wage
and running a business hiring others is of lower talent, and therefore the rents of all the managers
and the wages of the top employees are lower than before. The change in the slope of the wage
function is once more unclear.
34
Figure 28: Change in the earnings profile when s (·) equals a constant.
Figures 29 through 32 show the changes in equilibrium earnings when s (·) is increasing and the
equilibrium exhibits segregation. The first of these figures illustrates the changes consistent with
the equilibrium in which all wage-workers who leave the sector become employers (case 1). The
second one illustrates the changes in equilibrium earnings consistent with the equilibrium in which
some of the wage-workers who switch occupation choose to work alone (case 2). In both cases the
managers and the top wage-workers earn lower rents while the bottom wage-workers earn a higher
rate. The slope of the profit function is lower while the effect of size-dependent regulations on the
slope of the wage function is unclear.
35
Figure 29: Change in the earnings profile with segregation and s (·) increasing. All wage-workers
who leave the sector become employers.
Figure 30: Change in the earnings profile with segregation and s (·) increasing. Some wage-workers
who leave the sector become employers and the rest work alone.
36
The last two figures show the changes consistent with the equilibria in which some of the
employers at the bottom of the managerial skill distribution switch to producing alone. Figure
31 shows the change in the profit function if the new assignment function cuts the old one, as in
figure 23, while figure 32 shows the change in the profit function if the new assignment function is
above the old one, as figure 24 illustrates. The changes in the level and the slope of the earnings
function correspond to the changes in the equilibrium without segregation that I describe above.
Figure 31: Change in the earnings profile with segregation and s (·) increasing. All employers who
leave the sector become employers. New assignment function close to the original one.
37
Figure 32: Change in the earnings profile with segregation and s (·) increasing. All employers who
leave the sector become employers. New assignment function not close to the original one.
Thus, regardless of the equilibrium that I consider, size-dependent regulations lower the return to skill for both employees and employers in the constrained firms. If complementarities in
production are not strong enough–and therefore a large fraction of managers constrain the size
of their firm–the tax could significantly lower the average return to skill. The size-dependent tax
also lowers the return to skill for managers at the top of the skill distribution. In general it also
lowers the slope and the level of the profit function for managers other than the best, but if the
reallocation of workers across occupations is only minor (so that the new assignment function cuts
the old one), then the marginal return and the rents of managers in the middle of the (managerial)
skill distribution are larger than before the tax. The effect of size-dependent regulations on the
slope of the wage function for workers in firms other than the constrained firms is unclear. They,
however, unambiguously increase the wage rate of employees at the bottom and decrease the wage
rate of employees at the top.
Numerical exercises
Figures 33 through 35 below display the changes in the equilibrium earnings function when I
introduce the policy in my numerical examples. The first plot illustrates the changes when s (z) = z
and L and H are close enough. The tax rate is 10%. In figure 34 I use the same assumptions
and increase the tax rate to 20% and find that no team sets up firms with more than N workers.
38
Figure 35 illustrates the changes in the equilibrium earnings profile when s (z) is a constant.
Figure 33: Change in earnings with ⌧ = 10% and s (·) is increasing.
Figure 34: Change in earnings with ⌧ = 20% and s (·) is increasing.
39
Figure 35: Change in earnings with ⌧ = 10% and s (·) independent of skill.
Equilibrium when all firms in the market pay the tax
In this section I examine the equilibrium if the tax is not size-dependent. I assume that s (·) is
increasing in talent and ↵ < 1. The equilibrium therefore does not exhibit segregation.
As in the equilibrium without distortions, there are two cutoffs x1 and x2 such that workers
in [L, x1 ] work for a wage, workers in [x1 , x2 ] produce alone, and workers in [x2 , H] produce hiring
others. The assignment function solves equation 7 above. The difference is that the wage function
now solves the following differential equation for all zp in [L, x1 ]:
zm (zp ) ↵ [n (zp )]
↵ 1
✓
dn (zp )
dzp
◆
=
and the rents for the managers follow
✓
dw (zp )
dzp
R (zm ) = zm [n (zp (zm ))]↵
◆
n (zp ) + w (zp )
✓
dn (zp )
dzp
w (zp (zm )) n (zp (zm )) (1 + ⌧ )
◆
(1 + ⌧ )
(21)
(22)
for all zm in [x2 , H].
In figure 36 I present the equilibrium earnings without distortions, the equilibrium earnings
when all firms in the market pay the tax, and the equilibrium earnings with a size-dependent tax.
I assume that the reallocation of workers across occupations is such that the new and the old
assignment functions do not cross.
40
Figure 36: Earnings profile without distortions, with a size-dependent tax, and when all firms in
the market pay the tax.
When the tax is size-dependent there are more workers producing alone compared to the equilibrium without distortions, but fewer than in the equilibrium where all firms pay the tax. Intuitively,
if the tax becomes size-dependent, then workers have incentives to form small firms, and workers
who would produce alone instead run their own firm. These managers are optimally allocated the
least productive wage-workers, therefore inducing the least talented individuals producing alone
to become employees and work for the most skilled employers.
The earnings for those who choose to work for someone else are unambiguously higher when
the policy is size-dependent. The same is true for managers at the bottom of the managerial skill
distribution. To the contrary, managers at the top earn lower rents compared to their earnings
when the tax is independent of the size of the firm. By exempting small firms from the tax,
authorities therefore transfer resources away from employers at the top and toward wage-workers
and managers at the bottom.
This analysis assumes that the size-dependent tax and the tax on firms of all size impose
the same rate. A revenue-neutral change from a size-dependent tax to a tax on all firms would
presumably involve a smaller rate, with less reallocation of workers across occupations as a result.
41
Concluding remarks
In the model size-dependent regulations significantly affect the allocation of talent in the market.
The size-dependent tax encourages employers just above the threshold in the managerial skill
distribution to choose sub-optimal firm sizes and disincentives their wage-workers to fully exploit
their skill. Employers and employees in the constrained firms turn indifferent to their match
and their positive sorting in equilibrium is no longer certain. The tax misallocates the talent of
managers in the unconstrained firms as well. It reallocates both the marginal wage-worker and
the marginal employer, which changes the sorting of workers within each occupation, and as a
result employers coordinate employees less talented and run smaller firms than in the equilibrium
without distortions.
The other major effect of size-dependent regulations in the model is on the return to skill. The
tax unambiguously lowers the return to skill for workers in the constrained firms. Wage-workers
and managers in these firms lower the size of their firm and effectively avoid the tax, but by
doing so wage-workers lose the ability to differentiate themselves and therefore lose the reward for
increments in skill, whereas managers lose the reward for pairing with ever more skilled employees
(their earnings change from convex to linear in skill). In addition, the tax increases the earnings of
workers at the bottom of the skill distribution, and decreases the earnings of those at the very top.
Low levels of skill turn therefore more attractive, whereas high levels of skill become less appealing.
Workers in the middle of the skill distribution are also worse-off, although some managers in the
middle of the managerial skill distribution may benefit with the policy in certain scenarios.
The major drawback of my analysis is that some results directly follow from a step tax policy,
specifically the absence of positive sorting of wage-workers and managers in a segment of the
market, and a zero or constant return to incremental skill for these individuals. If the policy were
instead a convex function of firm sizes, positive assortative matching would be guaranteed. Indeed,
if the same tax rate is imposed on firms of all sizes, then no firm artificially constrains their size
and the equilibrium exhibits positive assortative matching of wage-workers and employers. The
effect of a convex tax policy on the return to skill is less clear, although, as in the case of a constant
tax rate for all firms, it would not imply a zero return in the wage function or a constant return
in the profit function for those in the constrained firms. My conjecture is that under a convex tax
policy both the wage function and the profit function would be flatter everywhere relative to the
profiles without distortions–entailing a lower return to skill for all workers–and the earnings of the
wage-workers at the bottom of the skill distribution would be higher as in the case of a step tax
policy, but this conclusion requires further analysis.
42
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