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\begin{center}
{\large \textbf{Diseconomies of Scale in Employment Contracts}\\[0pt]
} \vskip 15pt \today  \\[0pt]
\bigskip  Eric Rasmusen and Todd Zenger \\[0pt]
\vskip 1in

Published: Journal of Law, Economics and Organization (June 1990), 6: 65-92 
\textit{Abstract}\\[0pt]
\end{center}

\noindent  We find that small teams can write more efficient incentive
contracts than large teams when agents choose individual effort levels but
the principal observes only the joint output. This result is helpful in
understanding organizational diseconomies of scale and is consistent with
both existing evidence and our own analysis of data from the Current
Population Survey. Our modelling approach, similar to classical hypothesis
testing, is of interest because we need not derive the optimal contract to
show the advantage of small teams.

\vskip .8in {\small \ \noindent  \hspace*{.2in} Rasmusen: John M. Olin
Visiting Assistant Professor, Center for the Study of the Economy and the
State and Graduate School of Business, University of Chicago 60637, and UCLA
AGSM. Phone: (312) 702-9478. Fax: (312) 702-0458. Bitnet: Fac\underline{ }%
ras@gsbacd.uchicago.edu.\newline
\hspace*{.2in} Zenger: School of Business and Management, Pepperdine
University, 3rd floor, 400 Corporate Point, Culver City, Cal. 90230. Phone:
(714) 941-0329. Fax: 213-568-5727. \newline
}

{\small \ \noindent \hspace*{20pt} 2000: Eric Rasmusen, Professor of
Business Economics and Public Policy and Sanjay Subhedar Faculty Fellow,
Indiana University, Kelley School of Business, BU 456,  1309 E 10th Street, 
Bloomington, Indiana, 47405-1701.  Office: (812) 855-9219. Fax:
812-855-3354. Erasmuse@indiana.edu. Php.indiana.edu/$\sim$erasmuse. }

%---------------------------------------------------------------

\newpage

\noindent \textbf{I. Introduction.}

\noindent \textbf{The Problem of Firm Size.}

A central question in industrial organization is what determines the size of
firms. This question is closely linked to a central assumption in
microeconomics generally: that ``managerial diseconomies of scale'' limit
the size of firms. If such diseconomies of scale did not exist, many
industries would be natural monopolies, because average cost declines with
output for any technology with a fixed cost and constant marginal cost. This
was noted by Sraffa (1926) and Kaldor (1934), who concluded that perfect
competition was unrealistic. Modern textbook theory, rejecting the premise
of constant marginal cost, notes that if marginal cost rises with output,
then a large firm is inefficient and a competitive market is feasible. The
combination of rising marginal cost and a fixed cost generates the Vinerian
U-shaped average cost curve, not natural monopoly.

The commonly given reason for increasing long-run marginal cost, a reason
mentioned as far back as Kaldor (1934), is that a larger firm is harder to
manage. Why this should be so is not entirely clear. Indeed, Alfred Marshall
believed that there were managerial \textit{economies} of scale, which would
aggravate the problem noted by Kaldor and Sraffa.$^1$ Schumpeter also seems
to have believed in managerial economies of scale, because ``monopolization
may increase the sphere of influence of the better, and decrease the sphere
of influence of the inferior, brains.''$^2$ Modern casual empiricism, if not
that of Alfred Marshall, suggests that large organizations suffer from
diseconomies. Not only is there no tendency towards monopoly in most
unregulated markets, but in most industries one observes a variety of firm
sizes.$^3$ Coase (1937) has argued that optimal firm size is determined by a
comparative assessment of the costs of internalizing additional transactions
and the costs of market transactions. At some point the cost of organizing
another transaction within the firm becomes greater than the cost of
transacting in the open market. But Coase merely provided a general
framework, not a reason why transactions costs should rise as the firm
becomes larger. As he notes in a 1988 article, why the marginal costs of
internal organization should rise with increased size remains unexplained.

Recently, several theorists have developed arguments to explain
organizational diseconomies of scale. Williamson (1985, Chapter 6) argues
that common ownership of successive stages of production, unlike separate
ownership, creates incentives for managers governed by incentive contracts
to misutilize assets and opportunistically manipulate accounting data.
Consequently, joint ownership of successive production stages requires
costly supplemental monitoring to enforce incentive contracts. These
enforcement costs and residual opportunistic behavior render incentive
contracts inefficient, thereby discouraging joint ownership. Milgrom and
Roberts (1988a,b) and Holmstrom (1988) argue that organizations, unlike
markets, must contend with ``influence costs''--- costs that arise whenever
individuals try to influence decisions to their private benefit and whenever
organizations impose mechanisms to control this behavior. But the above
arguments, although helpful in specifying the costs of internal
organization, do not address why these or other costs of internal
organization would necessarily increase with the size of the firm.

Diseconomies of scale also arise in organizations other than firms. Sugden
(1986, chapter 5) and, more formally, Boyd and Richerson (1988) find that
increasing the size of a group makes evolutionary development of a
convention--- a coordination rule such as driving on the right--- more
difficult. A convention works better when used more widely, and moving
randomly from discoordination to coordination is easier in a smaller group.
Farrell and Lander (1989) use a team model to look at the distinction
between an individual's effort on his own behalf and on the team's behalf
under certainty. They find for a particular contractual form that selfish
effort increases and team effort decreases in team size. More directly
related to firm size, Robert Frank (1985) in \textit{Choosing the Right Pond}
argues that if managers are willing to sacrifice monetary compensation in
exchange for being top man in their firms, then small firms could outcompete
large firms.

Bendor \& Mookherjee (1987) analyze the effect of group size on cooperation
in an infinitely repeated Prisoner's Dilemma. They note that as the group
size becomes larger, the free rider problem increases. For a given discount
rate, cooperation cannot be supported beyond a critical group size that
depends on the discount rate. This has the same flavor as work on the size
of cartels, such as Stigler's 1964 model of a cartel that tries to detect
cheaters using a noisy variable whose mean depends on whether cheating
occurs. In his model, a firm only knows how many customers do not return to
it, so each firm sees a different statistic. As the number of firms
increases, detecting a price-cutter becomes more difficult. The problem has
been taken up again in the ``trigger strategy'' literature, where detection
triggers dissolution of the cartel. Porter (1983) comes to a result similar
to Stigler's: a cartel with more members has a shorter expected lifetime.

\bigskip \noindent \textbf{Our Approach.}

We hope to shed some light on the issue of organizational diseconomies of
scale from a new viewpoint: thinking of the organization as a team of agents
who may differ in effort or ability. We will show why teams incur
diseconomies of scale and why teams can coexist with different management
styles. Our argument will be that the small team can more efficiently
construct incentive contracts to induce high effort and attract talented
workers, an argument that applies to any organization generating a joint
output when the inputs of individual members are difficult to assess.

We will use an agency model similar to Holmstrom (1982) in which total team
output is observable but individual contributions are not. Holmstrom focuses
on the \textit{manager's} role in ensuring optimal effort. Even without a
manager the team faces the difficulty of detecting low effort, but the
manager's presence expands the space of contracts by enforcing drastic
punishment if output is low. In a world of certainty this allows the
first-best to be achieved. The Holmstrom model by itself thus does not
explain diseconomies of scale; on the contrary, one may draw from it the
surprising implication that a properly designed contract can avoid them.

We will assume that contracts are enforceable in that managers will always
impose the punishments stipulated by the contract, and focus instead on team
size under uncertainty, where low output might be due to random noise
instead of shirking. The manager must still deter shirking by the threat of
a teamwide punishment if output is below a chosen threshold. We will examine
the case where the optimal contract is costly: even if inflicting the
punishment incurs deadweight loss, some mistaken punishment occurs because
of the noise, and the first-best cannot be achieved. We will show that the
costs associated with the optimal contract increase with team size. This
result will be obtained in a way that avoids the difficult task of
characterizing the optimal contract. Our argument will apply both to moral
hazard (when effort is variable) and adverse selection (when ability is
variable). The implication is that large teams must be content with low
effort and ability or else use ability testing and effort monitoring instead
of output-based contracts.

We believe that our results on teams are relevant to thinking about
organizational diseconomies of scale. In the case of small firms, our
results apply directly, since the firm may be nothing more than a small team
of production workers. Large firms, however, are commonly comprised not of
one large team, but of many subunits. Despite this, it is relatively
uncommon for any significant element of compensation to be explicitly tied
to subunit output. Why don't large firms simply replicate small-firm
incentives by attaching compensation to subunit performance? Presumably,
these subunits are not teams in our sense, with a clearly observable team
output. Rather, the output of a subunit depends on the inputs of other
subunits and management. Consequently, breaking into small subunits does not
permit the large firm to replicate the contracting advantage of the small
firm.$^4$

Our model also has application to management teams. It may well be that the
individual output of production workers is fully observable, but the board
of directors cannot distinguish the individual contributions of the top five
executives. In this case, our model predicts that firms with fewer top
executives could construct contracts which attract greater talent at the
executive level. If the number of executives limits the size of the firm,
then only relatively small firms will employ high-ability executives and use
incentive contracts.

The teams approach is important because it complements an older idea: the
firm cannot simply replicate divisions, because it can have only one chief
executive, who runs into diminishing returns when used intensively. This is
the idea extended and modelled by the literature starting with Williamson
(1967) and continuing in, for example, Keren and Levhari (1983) and McAfee
and McMillan (1989a). Williamson constructs a model with an exogenous ``span
of control,'' an exogenous ratio of agents to managers at each level of the
firm. Since managers are not directly productive, this assumption makes
average costs increase in output. The interesting questions in Williamson's
paper concern the optimal number of levels in the firm hierarchy, however,
rather than why large firms  are inefficient, since the inefficiency is a
direct assumption about the management technology. The span-of-control
approach asks how to optimize the firm's structure given organizational
diseconomies of scale, but it does not inquire into their origins. Our paper
suggests that one reason for span-of-control problems is the increasing cost
of extending incentive contracts as span increases.

The single-executive explanation for why replication is ineffective in
removing managerial diseconomies of scale is satisfactory only if we take it
for granted that a firm must have only one chief executive. But why not
replace the chief executive with a management team, increasing the number of
executives as the amount of necessary supervision increases? We will show
that using a team is costly, and it is more costly the larger the team.
Hence, making the management team larger by horizontal expansion of the
hiearchy's top level is not a costless substitute for increasing the number
of hierarchical levels by vertical expansion.

The teams approach may also explain why vertical expansion of the hierarchy
with monitoring cannot be replaced by vertical incentive contracts. The
individuals we model as a team need not be at the same hierarchical level.
The reasoning applies even if one of the team members is a boss, whose
ability is twice as high as an ordinary member and who is paid a double
share, but whose effect on team output cannot be untangled from the effect
of his subordinates. The results hold \textit{a fortiori}, since low effort
by the boss is as easy to detect as low effort by two ordinary members with
perfectly correlated errors. All that is required is that the double
importance of the boss is common knowledge. Thus, our model in its broadest
interpretation suggests that as the number of individuals contributing to an
observable output increases, the costs of offering incentive contracts rise
regardless of the individuals' hiearchical location within the organization.

Sections II and III of this article present the model and show why contract
costs increase with team size. Section IV discusses the applicability of the
results to models of adverse selection, in which smaller firms can offer
lower-cost contracts to attract high-ability agents while deterring the
low-ability. Section V lays out the empirical implications of the model and
compares them with evidence from the 1979 Current Population Survey.

%---------------------------------------------------------------
\newpage

\noindent \textbf{II. The Model.}

A principal employs $n$ agents to make up a team. The agents are identical,
with a utility function $U(w, b, e)$ that is increasing in the wage $w$ and
decreasing in the punishment $b$ and the effort $e$. Effort takes one of two
levels, $\overline {q}_h$ or $\overline {q}_l$, where $\overline {q}_h>
\overline {q}_l$. Agent $i$'s contribution to output is the sum of his
effort and a random disturbance: 
\begin{equation}
\label{e1} q_h = \overline {q}_h + \varepsilon_i \;\; \mathrm{or}\;\; q_l =
\overline {q}_l + \varepsilon_i.
\end{equation}
The disturbances $\varepsilon_i$ are identically distributed with mean zero
according to a multivariate normal distribution, and they may be either
independent or positively correlated. If the disturbances are independent,
the random influence is at the level of the individual agent; if they are
perfectly correlated, it is at the level of the team. We use the normal
distribution so that total team output follows the same distribution
regardless of team size. This allows output to range from negative infinity
to positive infinity, so some readers may wish to interpret the variable as
some other measure of performance besides output (e.g. profit).

Competing teams offer contracts to the agents, who choose the contract that
yields the greatest expected utility. After the agents choose contracts and
efforts, nature chooses the values of the disturbances and the team outputs
appear. The principals then pay or penalize the agents according to the
terms of the contracts.

Individual effort and output are prohibitively costly for a principal to
determine, so his contract must rely solely on the level of team output,
denoted $Q_{obs}$. The principal is risk neutral and seeks to maximize
profit, the expected value of $(Q_{obs} - nw)$. The simplest kind of
contract ignores $Q_{obs}$ and pays a fixed wage of $w=\overline{q_l}$,
which elicits low effort and produce an expected profit of zero. If agents
are sufficiently averse to risk and effort, this is the efficient outcome
under asymmetric information--- incentives for high effort are too costly.
We will assume the opposite---that high effort is more efficient, even under
asymmetric information--- for the remainder of this article.

We will also assume that welfare under asymmetric information is not as high
as it would be under full information--- that is, the incentives necesssary
to induce high effort do generate costs. This is important because if
neither large nor small teams incur costs, there is no cost difference
between them. The assumption that incentives are costly can be justified by
various combinations of primitive assumptions. If incentives require
variability, and the agents are risk averse, then any variability in wages
is costly. If the agents are risk neutral then wage variability does not
matter, but if the principal is constrained in the wage contracts he can
write he may be forced to use dissipative penalties. But the agent's utility
function will matter only because of the two requirements that (a) high
effort is efficient under asymmetric information, and (b) the asymmetry of
information has no costless remedy.

The principal wishes to design a contract that supports a Nash equilibrium
in which every agent exerts high effort. Since behavior under the Nash
equilibrium concept presumes that each player calculates whether his
unilateral deviation from equilibrium behavior is profitable, the
willingness of one or more agents to switch would break the equilibrium. The
principal's problem is therefore to make it unprofitable for even a single
agent to choose low effort. If all $n$ agents in a team choose high effort,
output is the random variable 
\begin{equation}  \label{e3}
Q_{n, H} = \sum_{i=1}^n ( \overline{q}_h +\varepsilon_i),
\end{equation}
which is the equilibrium output. If all but one of the $n$ agents choose
high effort, output is the random variable 
\begin{equation}  \label{e2}
Q_{n, L} = \overline{q}_l + \varepsilon_1 + \sum_{i=2}^n ( \overline{q}_h
+\varepsilon_i),
\end{equation}
which is the deviation output.

%---------------------------------------------------------------

\bigskip  \noindent \textbf{Testing for Shirkers.}

Rather than directly attacking the problem of optimal contract design, we
will start with the subproblem of detecting a low-effort agent. Under
classical hypothesis testing, we establish a null hypothesis,

\begin{center}
``$H_0$: All agents chose high effort, ''
\end{center}

and an alternative hypothesis,

\begin{center}
``$H_1$: At least one agent chose low effort.''
\end{center}

$H_1$ is a little more general than is required here, since for Nash
equilibrium all that is necessary is to test for one agent choosing low
effort. A strategy combination is a Nash equilibrium if no player has any
incentive to unilaterally deviate from his strategy. In this case, all
agents will choose high effort in the proposed equilibrium, so we must test
to see whether a single agent can benefit by deviating and choosing low
effort. Whether several players might benefit by forming a coalition and
simultaneously deviating is irrelevant to whether the strategy combination
is a Nash equilibrium. But in this particular model, the Nash equilibrium
concept is not restrictive. A test that detects deviation by a single
low-effort agent would even more easily detect deviation by several agents
who chose low effort simultaneously.

\noindent  The principal is concerned with two types of errors:

\noindent \textbf{Type I Error: } Rejecting $H_0$, when it should be
accepted.\newline
\hspace*{.2in} (false rejection, associated with a low significance level)

\noindent \textbf{Type II Error:} Accepting $H_0$, when it should be
rejected.\newline
\hspace*{.2in} (false acceptance, associated with low power)

The principal wishes to avoid both types of error. Equivalently, he wants
the detection test to have high values for (1) its \textit{power} (the
probability that a low-effort agent will be detected when present) and (2)
its \textit{significance level} (the probability of avoiding false
detection).$^5$ If the power is high, the firm's probability of punishing a
shirking agent is high and the agents fear to shirk. If the significance
level is high, false detection is rare and agents need not be paid a large 
premium as compensation for expected accidental punishments.  Both the power
and the significance level are determined by  the particular test.

Economists are used to trading off the levels of desirable characteristics.
Here, however, the principal just desires a power that deters shirking.
Higher levels of power are no better, so the lexicographic approach of
classical hypothesis testing is appropriate. The classical statistician
chooses the significance level for a test and then maximizes the power given
that significance level. Here, the principal chooses the test's power (to be
high enough to deter shirking), and then maximizes the significance level
(which minimizes the premium for accidental punishment).

We will concentrate on the simple test that uses only the information of
whether total output has exceeded a threshold level $T$. The principal
accepts $(H_0$: All agents chose high effort) if $Q_{obs} \geq T$ and $(H_1$%
: At least one agent chose low effort) if $Q_{obs} < T$. This test is
illustrated by Figure 1. The significance level of the test, $S$, is one
minus the probability that the principal mistakenly believes that a single
agent chose low effort: 
\begin{equation}  \label{e4}
S = 1- \mathrm{Prob} (Q_{n, H} \leq T ).
\end{equation}
The significance level falls as the threshold rises: 
\begin{equation}  \label{e5}
\frac{dS}{dT} < 0.
\end{equation}
The power of the test, $P$, is the probability that a single shirking agent
will be detected when present: 
\begin{equation}  \label{e6}
P = \mathrm{Prob}(Q_{n, L} \leq T) .
\end{equation}
The power rises as the threshold rises: 
\begin{equation}  \label{e7}
\frac{dP}{dT} > 0.
\end{equation}

\begin{center}
\textbf{Figure 1: Power and Significance Level}
\end{center}

The characteristics of this test are well-known, for it is simply a test for
the mean of a normal distribution with known variance against a composite
alternative hypothesis. It is, in fact, a textbook example for power and
significance level, and it provides the \textit{best critical region} for
testing our hypothesis. It is also the \textit{uniformly most powerful}
test: not only can it test for the family of alternative hypotheses in which 
$n$ agents shirk, it is the best such test for all of those alternative
hypotheses.$^6$

%---------------------------------------------------------------

\noindent \textbf{The Compensation Contract.}

A natural contract to associate with the threshold test consists of the
triplet $(T, w, b)$. Each agent receives $w$ if output exceeds the threshold 
$T$ and suffers punishment $b$ if output fails to exceed $T$. We have no
reason to believe that the optimal contract lies within this restricted
contract space, but as will be explained at the end of Section III, this is
not so restrictive as it might seem.

Using the contract $(T, w, b)$, the principal must choose threshold and
punishment values such that if the agent shirks, his expected utility is
lower than if he works. The expected utility of shirking is based on the
probabilities of (a) being caught and rightfully punished (the power), and
(b) successful deception and a wage of $w$. But the agent must compare the
expected utility of shirking not with a fixed utility from working hard, but
with the expected utility of working hard. The expected utility of working
hard is based on the probabilities of (a) being mistakenly punished, and (b)
being paid $w$ (the significance level).

The first requirement for a contract is that it deter shirking. Any of a
wide range of powers can succeed in this, since the punishment can be chosen
appropriately to the power--- a bigger punishment for a smaller power. The
second requirement for a contract is that it minimize the cost of mistaken
punishment. Thus, a high significance level is desirable. But there is a
tradeoff between these two requirements: as equations  (\ref{e5}) and (\ref
{e7}) show, if the contract's threshold increases, the significance  level
declines while the power rises.

We have assumed that the punishment $b$ reduces the agent's utility without
increasing the principal's profit. Since we are interested in comparing the
efficiency of different contracts, it is important that the punishment incur
deadweight loss in this way rather than just being a transfer. An example of
a pure transfer is the forfeiture of a performance bond when both the
principal and the agents are risk neutral and collecting the bond incurs no
administrative costs. In that case, a contract that results in more frequent
bond forfeitures is no less efficient. An agent would only care about
expected income, so he would be indifferent between ($w=\$190$, $b=\$10$, 
\textit{50 percent probability of punishment}) and ($w=\$110$, $b=\$10$, 
\textit{10 percent probability of punishment}). The expected income is \$100
under either contract.

Punishment creates deadweight loss whenever the punishment's disutility to
the agent is greater than its utility to the principal. There are several
reasons why punishments that create deadweight loss are commonly used.$^7$
One reason is that when agents are risk averse even a monetary penalty
introduces riskiness into compensation, which hurts the agent without
correspondingly benefiting the principal (who, in fact, must also bear
risk). A second reason is that any instance of efficient punishment benefits
the employer, which raises the problem of deliberate unwarranted punishment.
A third reason is that monetary penalties which go beyond a wage of zero to
seize part of the agents' assets incur high transactions costs to enforce. A
fourth reason is that bankruptcy protection and other legal constraints
impose limits on monetary penalties, requiring substitution to nonmonetary
penalties such as dismissal or embarassing reprimands. Even if monetary
penalties are bounded, the same effect can be achieved by the uses of
bonuses. Instead of a high ordinary wage and a severe fine if output is low,
the principal pays a low ordinary wage and a large bonus if output is high.
But bonuses are particular vulnerable to cheating on the part of the
principal; he may misreport that output is low, or even deliberately
sabotage output. In addition, the bankruptcy constraint can bite at the
level of the principal as well as of the agent. If the principal must pay
large bonuses to the entire team, he too can go bankrupt.

Punishments take a number of forms in actual employment. Examples include
loss of wage premiums (Becker and Stigler 1974), loss of future wage
increases (Doeringer and Piore 1971), damaged reputation (Klein and Leffler
1981), and job search costs after dismissal (Shapiro and Stiglitz 1984).
Even if the level of the punishment is exogenous, or the firm gains some
advantage from the punishment, the model continues to apply, so long as the
gain to the firm is less than the loss to the agent.

In some circumstances it may be possible to avoid the deadweight loss of
punishments entirely and attain just as high utility under asymmetric
information as under full information. A ``boiling-in-oil'' contract is a
threshold contract under which the principal imposes very severe penalties
if output is so low that such an output level would occur with zero
probability if every agent had exerted high effort. Holmstrom (1982) uses
such a contract to obtain the first-best outcome in a teams model with
hidden effort but no uncertainty in output. Under a boiling-in-oil contract
the significance level is equal to one, since the agents exert high effort
in equilibrium and the punishment is never inflicted. Such a contract is
infeasible here because the support of the normal distribution is the whole
range of output. Whatever effort is chosen by the agent, output might be
very low, so no threshold can guarantee a significance level of one hundred
percent. Holmstrom also shows, using the approach of Mirrlees (1974), that
even with uncertainty the first-best outcome can be approximated by a simple
threshold contract with large penalties infrequently inflicted. This is the
case if there is no bound on penalties and the distribution of output
satisfies assumptions ensuring that the product of the penalty and the
probability of its infliction can be made vanishingly small.$^8$

When boiling-in-oil contracts are infeasible, the principal knows he will
sometimes punish agents even when they exert high effort. He also knows that
in equilibrium the agents always exert high effort, so every instance of
punishment is mistaken. This is a paradox common to every model of costly
punishment. In discussing classical hypothesis testing, our language has
strayed from that of Bayesian games. If the equilibrium is common knowledge
(the standard assumption), the principal should rationally assign zero
probability to the presence of a low-effort agent, but we have spoken as if
the principal actually believes the test when it indicates low effort and he
carries out the punishment. This language is for expositional convenience.
In actuality, it is important that both agents and principal have committed
to follow the contract and carry out the costly punishment, even though
everyone knows that (1) in equilibrium only the innocent are punished and
(2) even if an agent did shirk, by the time output is observed it is useless
to punish him. Such a situation cannot be avoided, because only by
precommitting to carry out punishments can shirking be deterred.

%---------------------------------------------------------------

\newpage  \noindent \textbf{III. The Number of Agents.}

We will now try to discover which is better at providing incentive
contracts, a large team or a small team. We will compare the significance
levels and powers for threshold tests in teams of size $n$ and $n+1$, where
the null hypothesis is that all agents are choosing high effort and the
alternative hypothesis is that one agent is shirking.

\noindent \textbf{Lemma 1:} \textit{If the power of the tests used by teams $%
n$ and $n+1$ is the same, the significance level is higher for team $n$.}

\noindent \textbf{Proof:} See Appendix. \bigskip

In view of Lemma 1, as the number of agents increases, the attractiveness of
the contract diminishes because of the increase in mistaken detection. The
expected wage in equilibrium is always $\overline{q}_h$, since it must yield
zero profits, so contracts differ in their attractiveness based on the
frequency and size of the punishment in equilibrium. If the significance
level of one of the contracts is lower, that contract ends up inflicting a
greater expected punishment, and hence is less attractive to agents for a
given expected wage. This is true, in particular, if $P$ is the power
generated by the optimal contract $(T^*, b^*, w^*)$ for a firm of size $n$.
To increase the size of the team and maintain the same power is costly. This
is stated in Proposition 1, which frames the situation in terms of the cost
of providing a given level of utility to the agent (in equilibrium, this
will be the maximum level that allows an optimizing firm to maintain zero
profits).

\bigskip  \noindent \textbf{Proposition 1:} \textit{If teams $n$ and $n+1$
choose contracts of the form $(T, w, b)$ to maximize their own profits
subject to providing agents with a given level of utility, team $n$ incurs
lower costs of contracting.}  \bigskip

\noindent \textbf{Proof:} A certain level of power $P^*$ is associated with
the contract that maximizes profit for team $n+1$ subject to the reservation
utility constraint. Let us denote this optimal contract by $(T^*, w^*, b^*)$%
. Suppose that team $n$ offers a contract with the same wage $w^*$,
punishment $b^*$, and power $P^*$ ($n$'s threshold will be different to
maintain the same power). By Lemma 1, $S(n, P^*) > S(n+1, P^*)$, so team $n$
will punish the agents less often in equilibrium. Team $n$'s workers will
therefore have higher utility than team $(n+1)$'s. This means that the
reservation utility constraint is not binding and team $n$ can reduce the
wage. This does not depend on whether the agents are risk averse or not.
Moreover, team $n$ could also choose the power and punishment optimal for
itself, instead of using $P^*$ and $b^*$, so the wage might be reduced still
further. Hence team $n$ has lower costs.\newline
Q.E.D.

\bigskip

It might be appropriate here to repeat the assumptions of the model.
Proposition 1 arises from a free-rider problem of sorts, but not a simple
free-rider problem. The simple statement that workers shirk more in larger
firms because they influence output less is not true, because incentive
contracts can sometimes get around the free-rider problem. Holmstrom (1982)
shows that if there is no random noise, or particular kinds of random noise,
shirking can be prevented, and McAfee and McMillan (1989b) shows that even
with general uncertainty, incentive contracts can be found that will deter
shirking. The question is how the cost of shirking differs between firms,
and that is the question addressed by Proposition 1.

So far, we have maintained the restriction that the contract is of the form $%
(T, w, b)$. We can easily relax this restriction and allow contracts
consisting of a finite number of such triplets, as stated in Proposition 2.
Note a caveat absent from Proposition 1: Proposition 2 applies only if the
first-best cannot be achieved; otherwise the question of firm size is
vacuous.

\bigskip  \noindent \textbf{Proposition 2:} \textit{If teams $n$ and $n+1$
choose contracts of the form $(T_i, w_i, b_i), \; i = 1, \ldots, k$ to
maximize their own profits subject to providing agents with a given level of
utility, and if these contracts impose real costs, then team $n$ incurs
lower costs.}

\noindent \textbf{Proof:} Each of the $i$ components of the contract has a
particular power $P_i$ and significance level $S_i$. Lemma 1 says that for
each component, firm $n$ can maintain the power at $P_i$ while increasing
the significance level. Increasing the significance level is desirable for
the reasons discussed in the proof of Proposition 1. (The proof of
Proposition 1 did assume that profits equalled zero, which might not be true
of each component, but the proof can easily be adapted to any fixed level of
profit.) Firm $n$ is therefore superior in each of the $k$ parts, so agents
in firm $n$ can be paid a lower wage than agents in firm $n+1$.\newline
Q.E.D.  \bigskip

Proposition 2 has quite general application because combinations of the $(T,
w, b)$ contracts can be used to build step contracts to approximate any
continuous contract. Thus, although we have limited ourselves to a class of
contracts that might not include the optimal contract, and we can say almost
nothing about its form, our results on team size fit a very wide class of
contracts.

As an example of how to apply Proposition 2, consider the contract
consisting of a flat wage of $\overline{q}_h$ plus a punishment $b^{\prime}$
inflicted if the team output is less than a particular threshold $T^{\prime}$%
. This contract is outside of the space allowed by Proposition 1, but it can
be closely approximated by the two triplets $(T_1= -\alpha, w_1=\overline{q}%
_h, b_1=0), (T_2= T^{\prime}, w_2=0, b_1=b^{\prime})$, where $\alpha$ is an
arbitrarily large number. (This contract is an approximation only because
the wage is not quite flat; it falls to zero if output is below $-\alpha$.)

Propositions 1 and 2 continue to be valid even if one requires that
contracts use only limited penalties and wages, e.g. $b \in [\underline{b}, 
\overline{b}]$ and $w \in [\underline{w}, \overline{w}]$, so long as high
effort continues to be second-best efficient. This is because Lemma 1
concerns detection, rather than punishment, so the smaller team has a lower
level of false punishment for \textit{any} punishment-detection combination,
not just the one optimal without the penalty limitation. Hence, a limit on
the size of penalties does not remove the advantage of the smaller team.

%---------------------------------------------------------------

\newpage

\noindent \textbf{IV. Identical Effort, Different Abilities}

The model so far has been constructed for identical agents who choose effort
(moral hazard), but it could also have been constructed for agents whose
effort is fixed but who differ in ability (adverse selection).$^9$ Suppose
that agents have either high or low ability, where the proportion of
high-ability agents in the economy equals $\theta$, and agents have utility
functions $U(w, b)$ that are increasing in the wage $w$ and decreasing in
the penalty $b$. Agents may be either risk averse or risk neutral in the
wage. Agent $i$'s output, which depends on his ability and random
disturbance, equals 
\begin{equation}
\label{e100} q_h = \overline {q}_h + \varepsilon_i \;\; \mathrm{or}\;\; q_l
= \overline {q}_l + \varepsilon_i,
\end{equation}
where $\overline {q}_h> \overline {q}_l$.  These assumptions parallel those
in the moral hazard model, but adverse selection requires somewhat more care
in modelling because some agents produce high output and some produce low
output even in equilibrium, and the equilibrium contract might be either
pooling or separating. There are various ways to specify how offers and
counteroffers are made in an adverse selection model, and under some
specifications the existence of equilibrium is a problem. We do not need to
discuss those specifications here; for discussions see Riley (1979) or
chapters 8 and 9 of Rasmusen (1989). All that is relevant is whether the
equilibrium is pooling or separating.

If a pooling contract were to be part of equilibrium in this game, it would
pay the same wage for all outputs and never inflict penalties. For profits
to equal zero, the wage would equal the average ability, so the contract
would specify a wage of $\theta \overline{q}_h + (1-\theta)\overline{q}_l$
and a penalty of zero. The size of the team would be irrelevant, since no
attempt would be made to detect low-ability agents.

If a separating contract were to be part of an equilibrium, the reasoning of
Proposition 2 implies that the cost of offering a separating contract to
attract just high-ability agents would increase with team size. Any team
which offered a contract with a larger team size would have to pay a higher
expected wage, and since the smallest teams would earn zero profits under
competition with each other, the larger team would earn negative profits.
The high-ability agents are thus hired by small teams, and the low-ability
agents are hired by teams of any size that use fixed-wage contracts at a
wage of $\overline{q}_l$.

An important difference between the effort and ability versions of the model
is that the ability version has implications not only for the size of teams,
but also for the distribution of talent among them. Only small teams would
be able to offer contracts which attract high-ability agents. Large teams
would have to pay higher wages to attract high-ability agents with a
contract that still deterred low-ability agents. Hence, high-ability agents
would choose small teams that provided contracts which ensured the high
quality of working peers. Large teams could still be composed of low-ability
agents who are paid a fixed wage, and since size is irrelevant to the
efficiency of the fixed-wage contract, some small teams might also be
composed of low-ability agents.

\newpage \noindent \textbf{V. Empirical Evidence}

If we have persuaded the reader that a teams model has something to say
about firm size, our model has a number of empirical implications for
industrial organization. From an organizational point of view, our results
imply that the optimal team size is a single member. A larger team cannot
offer as attractive a contract, because it requires a higher probability of
mistaken punishment, so if teams can take any size, the optimal team has a
single member. This is an implication of any model of managerial
diseconomies of scale. But although single-agent firms are common, they
certainly do not represent the full range of sizes observed. Indeed, in many
industries we observe a wide range of firm sizes at the same time. This
diversity of firm sizes is not incompatible with our results, since
managerial diseconomies of scale are not the only influence on firm size. If
technological economies of scale are present or if external contracting is
particularly costly (Coase, 1937; Williamson, 1975), these elements will be
traded off against managerial diseconomies of scale to determine the optimal
firm size. In addition, our model makes no prediction for the size of firms
that employ low-effort or low-ability agents on fixed-wage contracts. Such
firms can be either large or small. Finally, firms need not rely solely on
the measurement of team output to detect shirking or low ability. They also
have the option of monitoring effort or testing ability. If firms can detect
shirking or low ability for a fixed cost per agent, then as firm size
increases and the cost of incentive contracts rise, testing or monitoring
become cheaper than incentive contracts. If there are economies of scale to
monitoring and testing, then large firms using those methods might coexist
with small firms using incentive contracts.

Our model suggests that although large firms and small firms can exist in
the same industry, they will differ in their management styles and
employment contracts. Large firms will offer fixed-wage contracts and make
heavier use of monitoring, testing, and easily observable employee
characteristics to control productivity. Small firms will link pay and
performance more closely, extract greater effort, and hire the most talented
employees (conditioning on observable variables). Small firms may also be
willing to employ those low-quality individuals rejected by the screening of
large firms, because the small firms can pay them an appropriately low wage
using output-based contracts.

We will compare these empirical implications with evidence from earlier
studies and data from the 1979 Current Population Survey. The CPS is
designed to be representative of the entire U.S. labor force. Employees were
surveyed, and they estimated the size of their firm by choosing between five
size categories: 1) 1-24 employees, 2) 25-99 employees, 3) 100-499
employees, 4) 500-999 employees, and 5) 1000 or more employees. We use wage
regressions to examine the predicted differences between large and small
firms in employment contracts. We use regressions of self-reported hours
worked to examine the predicted differences in effort.

\bigskip \noindent \textbf{A. Use of Observable Employee Characteristics in
Wage-Determination}

A first prediction is that large firms will rely more heavily than small
firms on directly observable worker characteristics such as education or
seniority, as a substitute for performance-based contracts. Small firms,
which are more efficient at detecting low effort and ability, will more
closely link pay and performance. The implications of this can be looked at
in two ways: (1) individual variables such as tenure will explain wages
better for large firms, and (2) the set of such variables will explain wages
better for large firms.

Table 1 presents descriptive information for individuals in each of the five
size categories. Table 2 presents five equations (for the five firm sizes)
using the CPS data to regress the log of hourly wages on tenure with the
firm, work experience, education, and various dummy control variables such
as industry, occupation, location, and union membership.$^{10}$

\begin{center}
Table 1: DESCRIPTIVE STATISTICS

Table 2: LOG WAGE EQUATIONS: THE EFFECT OF TENURE
\end{center}

Tenure has a significant effect on compensation in all size categories.
Consistent with our hypothesis, the effect of tenure is greater in large
firms than in small for nearly the entire range of tenure values (up to 41
years). The estimated coefficients suggest, for example, that an employee
with two years tenure receives 1.30\% in additional income for remaining an
additional year at a small firm, but 1.84\% at a large firm.$^{11}$ At the
entire sample's mean tenure of 8.25 years, an additional year yields 1.26\%
additional compensation in a small firm, but 1.59\% additional compensation
in a large firm (1000+ employees).$^{12}$

There are many specific reasons why tenure might matter more in large firms.
Large firms might have more complex bureaucracies, or rely more heavily on
deferred compensation, or attach more importance to learning about ability
and effort over time. But all of these specifics are subheadings of the
general reason we propose: that compensation in large firms relies less on
current output and more on other factors than does compensation in small
firms.

The other regression variables we expected to be important were experience
and education. The coefficients for ``Other Experience'' are significant,
but roughly one third the size of tenure's, and without important variation
across firm size, except for relative unimportance in the largest category.
Education also shows no clear size-related pattern. To the extent that
previous experience and education are collinear with ability, we would not
expect much variation in these parameters across firm sizes, since these
attributes are easily observed.

\bigskip

A second way to interpret the wage equations is to look at how well wages
are explained by the right-hand-side variables in aggregate. The model
predicts greater residual variance in the small-firm regression, because
small firms link pay to performance instead of to the right-hand-side
variables. The easiest way to check for residual variance is to look at the $%
R^2$ values in Table 2. The $R^2$ for small firms is $.358$, whereas the
values for the four larger categories are $.402$, $.447$, $.450$, and $.421$%
. These results are generally consistent with our prediction, although the
small value for the largest firms is anomalous.

We can also test for the difference in explanatory power more formally.
Under our model the wage equations are misspecified for small firms, since
performance is a relevant and omitted variable. But we can take as our null
hypothesis that the wage equations are correctly specified, that large firms
are identical to small ones in their use of the explanatory variables, and
that the random disturbances follow the same normal distribution for all
firms. Under this null hypothesis, the variance of the residual is identical
for the five categories, and the ratio of the squared standard errors from
any two of the five regressions follows the F-distribution, with degrees of
freedom equalling the sample size for each regression. Table 2 shows the
F-statistics for the differences between the standard errors of the smallest
firms and each of the four other categories. The regression for the smallest
firms has a significantly greater standard error than for any of the other
size categories, rejecting the null of no difference.$^{13}$

\bigskip \noindent \textbf{B. Self-Reported Hours of Work.}

A second prediction is that as a result of these size-related differences in
employment contracts, effort will be lower in large firms than in small
firms. CPS respondents reported the number of hours they worked during the
week preceding the survey, and we can use their self-reported hours as an
indication of effort. Hours worked is clearly not a perfect measure of
overall effort, since hours worked measures only the duration of effort and
not its intensity. Indeed, if hours worked were a perfectly accurate measure
of effort for all employees, then all employees would presumably be paid by
the hour. But our tolerance for measurement error can be considerably
greater than the tolerance of managers, since managers, unlike researchers,
must directly compensate for the uncertainty imposed by errors in
measurement. Hence, for our purposes, a substantial correlation between the
duration of effort and overall effort, as seems reasonable, is sufficient.
The likelihood of such a correlation between true effort and hours worked is
partly contingent on managers not using hours worked as a measure of effort.
If all employees in the large firm were paid by the hour, the correlation
between hours and effort would decline as employees adjusted their behavior
toward longer, but less intensive effort. The measurement of hours worked by
government statisticians does not have this same behavior-altering effect on
employees.

Table 3 presents separate regressions for hourly and non-hourly employees of
hours worked per week on firm size, work experience, education, and various
dummies including industry, union membership, occupation, and location. Only
the coefficients for the dummy size categories are displayed, with very
large size (1000+ employees) as the excluded category. The results indicate
that full-time, non-hourly workers employed in very small firms (1-24
employees) on average work 2.4 hours per week more than non-hourly workers
employed in very large firms. Those employed in small firms (25-99
employees) on average work 1.4 hours per week more than non-hourly workers
in very large firms. Note also that the relationship between firm size and
hours worked appears to be non-linear since hours worked do not differ
significantly among those employees in the three large firm size categories.

\begin{center}
Table 3: FIRM SIZE AND HOURS WORKED
\end{center}

Our model suggests that since time is a directly monitored input for hourly
employees, they should should work the same number of hours in large and
small firms. Table 3 shows that even for hourly employees there is a
significant negative relationship between size and hours, but not as strong
or as significant as for non-hourly employees. Hourly employees of very
small firms work .85 hours more per week than hourly employees of very large
firms. Hourly employees in small and medium-sized firms (100-499 employees)
work on average just over $.5$ hour more per week than in very large firms.
Reasons for these size-related differences must be found outside our model,
but the results help to calibrate the extent to which the coefficient on
firm size in the non-hourly regressions is due to omitted variables, and
show it to be small.

Cross-industry empirical work of this kind, while useful for finding whether
an effect is widespread, is also subject to the criticism that it might be
driven by omitted industry variables, however many control variables are
included in the regressions. Another approach is to examine firms within a
single industry. An example is Zenger (1989), which compares contracts at
large and small firms using survey responses from a sample of engineers who
had left two large high-technology firms. The findings suggest that
contracts at smaller firms involve greater equity ownership, link firm
performance more closely to pay, involve less formal monitoring, and impose
greater employment risk. Moreover, among engineers departing the two firms,
those of higher ability left for smaller firms and those of lower ability
left for larger firms. Finally, the engineers in small firms worked more
hours per week, consistent with the CPS regressions of Table 3.

\bigskip \noindent \textbf{C. Previous Work on Firm Size and Employment
Contracts}

Various investigators have found a relationship between firm size and the
employment contract. Garen (1985) examines wage models from the National
Longitudinal Survey and finds a marginally significant relationship between
wages and the interaction of ability (measured by test scores) and firm size
(measured by the percentage of the industry's labor force employed in firms
with more than 500 employees). Bishop (1987) similarly finds that
productivity has an important positive effect on wages in small, non-union
establishments, but almost no effect in large unionized establishments.
Medoff and Abraham (1980) examine the compensation practices of two large
firms and confirm a weak link between pay and performance. These results
support the conclusion that ability and compensation are more closely
associated in small firms than large firms.

Our model may be particularly applicable to R\&D settings, where individual
outputs are difficult to discern and teamwork is essential. Also consistent
with our reasoning and these results is the common, although not undisputed
empirical finding that R\&D is more efficiently performed by small and
medium-sized than by large firms (see Chapter 3 of Kamien and Schwartz,
1982). Our model predicts that large firms cannot efficiently offer
contracts that induce high effort and attract high ability. The survey data
from Zenger (1989) supports this view.

Another prediction is that earnings at small firms should vary more than at
large firms. High-ability employees should all be attracted to small firms,
while low-ability employees might work at large or small firms. It is well
known that average earnings are higher in large firms than in small firms,
contrary to our model's prediction, though it is not clear why this is so,
since the effect persists even after controlling for observable indicators
of worker quality. In fact, the thorough study of Brown and Medoff (1989)
finds that the effect is just as strong for \textit{piece-rate} workers at
large firms. This may be the result of large firms employing testing or
monitoring procedures that weed out workers who are low-quality in terms of
either observables or unobservables. Then a more complete prediction of our
model would be that small firms will include some firms with incentive
contracts employing high-output workers and some firms with fixed-wage
contracts employing very-low-ability workers who are rejected by the large
firms. In addition to our findings, Brown and Medoff, Garen (1985), and
Stigler (1962) have found greater variability in earnings among employees of
small firms than among employees of large firms.

Our results linking firm size and effort are also consistent with
experimental studies in psychology examining the effects of group size.
These studies confirm negative relationships between group size (ranging
from 2 to 8 members) and individual effort in rope-pulling, brainstorming,
hand-clapping, shouting, and use of an air pump.$^{14}$ They also find a
curvilinear relationship consistent with the regression results of Table 3:
the marginal effect of the Nth person on the efforts of group members is
less than the marginal effect of the (N-1)th individual.

%---------------------------------------------------------------

\newpage \bigskip \noindent \textbf{VI. Concluding Remarks}

This paper develops a model of the relationship between team size and the
efficiency with which contracts based on team output can resolve problems of
moral hazard and adverse selection. The model implies that contracts based
on team output are more efficient in identifying and deterring low effort
and low ability for small teams than for large teams. We argue that these
conclusions are also relevant to firm size. Consequently, large firms are
more likely than small firms to avoid contracts that base workers'
compensation on firm output. Instead, large firms offer fixed-wage contracts
that do not closely link compensation to ability or effort, but use
seniority or other observable criteria to determine compensation. In
addition, large firms will aggressively test for low ability. Small firms
will identify and reward ability and effort by linking pay and firm output.
As a consequence of these contractual differences, we predict that small
firms will induce higher effort and employ low-ability workers rejected by
large firms as well as high-ability workers attracted by incentive
contracts. Empirical results are consistent with these predictions.

Our model and empirical analysis provide a partial explanation for the
managerial or organizational diseconomies of scale assumed in price theory
and transactions-cost economics. The costs of organizing rise with firm size
because larger firms are less efficient than smaller firms in offering
contracts that induce high effort and attract high-ability workers.

%---------------------------------------------------------------

\newpage

\begin{center}
{\Large APPENDIX: Proof of Lemma 1}
\end{center}

If just one agent chooses low effort, the outputs for teams $n$ and $n+1$
are 
\begin{equation}  \label{e18}
Q_{n, L} = \overline{q}_l + \varepsilon_1 + \sum_{i=2}^n ( \overline{q}_h
+\varepsilon_i)
\end{equation}
and 
\begin{equation}  \label{e19}
Q_{n+1, L} = \overline{q}_l + \varepsilon_1 + \sum_{i=2}^{n+1} ( \overline{q}%
_h +\varepsilon_i).
\end{equation}
The two variables $Q_{n, L}$ and $Q_{n+1, L}$ both have normal
distributions, because they are the sums of normally distributed random
variables. (This is true whether the random variables are independent or
not.) Their expected values are 
\begin{equation}  \label{e20}
\mu_{n, L} = \overline{q}_l + (n-1)\overline{q}_h
\end{equation}
and 
\begin{equation}  \label{e21}
\mu_{n+1, L} = \overline{q}_l + n \overline{q}_h.
\end{equation}
The variance of output depends on the team size and the correlation between
the errors, but not on whether agents shirk. If the errors are independent,
then 
\begin{equation}  \label{e38a}
\sigma^2_n = n \sigma^2
\end{equation}
and 
\begin{equation}  \label{e38b}
\sigma^2_{n+1} = (n+1) \sigma^2.
\end{equation}
If the errors are perfectly correlated, then 
\begin{equation}  \label{e38c}
\sigma^2_n = n^2 \sigma^2
\end{equation}
and 
\begin{equation}  \label{e38d}
\sigma^2_{n+1} = (n+1)^2 \sigma^2.
\end{equation}
In either case, or for any positive degree of correlation between errors (or
even a sufficiently small negative correlation), 
\begin{equation}  \label{e38e}
\sigma_{n+1} > \sigma_{n}.
\end{equation}
If the power equals $P$ for either size team, then 
\begin{equation}  \label{e24}
P = Prob ( Q_{n, L} \leq T_n) = Prob ( Q_{n+1, L} \leq T_{n+1}).
\end{equation}
Using normality, 
\begin{equation}  \label{e25}
P= Prob ( Q_{n, L} \leq T_n) = \Phi \left( \frac{ T_n - \mu_{n, L}}{
\sigma_n } \right) = \Phi \left( \frac{ T_n - \overline{q}_l - (n-1)%
\overline{q}_h} { \sigma_n } \right)
\end{equation}
and 
\begin{equation}  \label{e26}
P= Prob ( Q_{n+1, L} \leq T_{n+1}) = \Phi \left( \frac{ T_{n+1} - \mu_{n+1,
L}}{ \sigma_{n+1} } \right) = \Phi \left( \frac{ T_{n+1} - \overline{q}_l -
n \overline{q}_h}{\sigma_{n+1} } \right).
\end{equation}
Let us define 
\begin{equation}  \label{e26a}
A_1 \equiv \frac{ T_n - \overline{q}_l - (n-1)\overline{q}_h}{ \sigma_n }
\end{equation}
and 
\begin{equation}  \label{e26b}
A_2 \equiv \frac{ T_{n+1} - \overline{q}_l - n \overline{q}_h}{ \sigma_{n+1} 
}.
\end{equation}
From the fact that (\ref{e25}) and (\ref{e26}) equal the same $P$, we can
conclude that $A_1= A_2$.

If all the agents choose high effort, the outputs are 
\begin{equation}  \label{e28}
Q_{n, H} = \sum_{i=1}^n ( \overline{q}_h +\varepsilon_i)
\end{equation}
and 
\begin{equation}  \label{e29}
Q_{n+1, H} = \sum_{i=1}^{n+1} ( \overline{q}_h +\varepsilon_i).
\end{equation}
These two variables also have normal distributions. \noindent  The
significance levels for the given power are 
\begin{equation}  \label{e31}
\begin{array}{ll}
S(n, P) & = Prob ( Q_{n, H} \geq T_n) \\ 
& = 1- Prob ( Q_{n, H} \leq T_n)
\end{array}
\end{equation}
and 
\begin{equation}  \label{e32}
\begin{array}{ll}
S(n+1, P) & = Prob ( Q_{n+1, H} \geq T_{n+1}) \\ 
& = 1- Prob ( Q_{n+1, H} \leq T_{n+1}).
\end{array}
\end{equation}
These two significance levels are not necessarily equal. We can rewrite them
using the normality assumption and the definitions of $A_1$ and $A_2$.
Equation (\ref{e31}) becomes 
\begin{equation}  \label{e33}
\begin{array}{ll}
S(n, P) & = 1- \Phi \left( \frac{T_n - \mu_{n, H}}{\sigma_n} \right) \\ 
& = 1- \Phi \left( \frac{T_n - n\overline{q}_h}{\sigma_n} \right) \\ 
& = 1- \Phi \left( \frac{ T_n - \overline{q}_l - (n-1)\overline{q}_h - 
\overline{q}_h + \overline{q}_l } {\sigma_n} \right) \\ 
& = 1- \Phi( A_1 - \frac{ ( \overline{q}_h - \overline{q}_l) } { \sigma_n} ).
\end{array}
\end{equation}
In the same way, equation (\ref{e32}) becomes 
\begin{equation}  \label{e34}
\begin{array}{ll}
S(n+1, P) & = 1- \Phi \left( \frac{T_{n+1} - \mu_{n+1, H}}{\sigma_{n+1}}
\right) \\ 
& = 1- \Phi \left( \frac{T_{n+1} - (n+1) \overline{q}_h}{\sigma_{n+1} }
\right) \\ 
& = 1- \Phi( \frac{T_{n+1} - \overline{q}_l - n\overline{q}_h - \overline{q}%
_h + \overline{q}_l } {\sigma_{n+1}} ) \\ 
& = 1- \Phi( A_2 - \frac{( \overline{q}_h - \overline{q}_l)} {\sigma_{n+1} }
).
\end{array}
\end{equation}
By equation (\ref{e38e}), $\sigma_{n+1} > \sigma_{n}$, so 
\begin{equation}  \label{e37}
\frac{ ( \overline{q}_h - \overline{q}_l) } {\sigma_n} > \frac{( \overline{q}%
_h - \overline{q}_l) } {\sigma_{n+1}}.
\end{equation}
It follows from (\ref{e37}) and the fact that $A_1 = A_2$, that 
\begin{equation}  \label{e38}
\Phi \left( A_1 - \frac{( \overline{q}_h - \overline{q}_l) } {\sigma_n}
\right) < \Phi \left( A_2 - \frac{( \overline{q}_h - \overline{q}_l) } {
\sigma_{n+1}} \right),
\end{equation}
so by equations (\ref{e33}) and (\ref{e34}) it is true that $S(n, P) >
S(n+1, P)$.\newline
Q.E.D.

%---------------------------------------------------------------

\newpage

\begin{center}
{\Large REFERENCES.}
\end{center}

\begin{list}{ }{\leftmargin .5in\itemindent -
                        .5in\listparindent .5in}


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\item[]
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                        \item[]
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 Holmstrom,  Bengt. 1982.  ``Moral Hazard in Teams.'' 13 {\it Bell
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 Lucas,  Robert. 1978. ``On the Size Distribution of Business Firms.''
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                        \end{list}

%---------------------------------------------------------------

\newpage  \noindent \textbf{Footnotes.}

We would like to thank Steven Lippman, Ivan Png, Emmanuel Petrakis, Steven
Postrel, Robert Topel, and Sang Tran for helpful comments. The data was made
available in part by the Inter-university Consortium for Political and
Social Research.

1. See page 265 of Marshall (1920): ``In other words, we say broadly that
while the part which nature plays in production shows a tendency to
diminishing return, the part which man plays shows a tendency to increasing
return. The \textit{law of increasing return} may be worded thus:--- An
increase of labour and capital leads generally to improved organization,
which increase the efficiency of the work of labour and capital.''

2. Schumpeter (1950), p. 101. More recent work along these lines includes
Lucas (1978) and Calvo and Wellisz (1980), who argue that high-ability
managers will go to large firms where their greater capacity to manage can
be put to better use. This is a valid point, but our model will assume that
the individual contribution of an agent to the team's output is the same
regardless of the team's size. We will try to isolate just one effect, and,
like technological economies of scale, the increasing sphere for talent
could swamp the disincentive effect we find for large teams.

3. See, e.g., Stigler's 1958 article on the Survivor Principle.

4. See Williamson (1985, Chapter 6) for a more complete discussion of the
constraints faced by large firms in replicating small firm incentives
through multiple subunits.

5. Strictly speaking, the probability of avoiding a Type I error is the 
\textit{size} of the test, and a test of size 0.95 is a test of \textit{%
significance level} 0.95, 0.94, 0.93, and so forth. Since the term ``size''
is somewhat obscure (it is omitted from the index of many statistics texts),
we will use ``significance level'' here, with the understanding that we mean
the test's highest significance level.

6. One textbook that uses this test as an example in discussing these
statistical points is Bickel \& Docksum (1977). See Chapters 5 and 6, and
especially pages 168-71, 192, and 198.

7. The question of why costly punishments are used is also a lively question
in the economics of crime. Shavell (1985) is a recent reference.

8. Our output distribution satisfies those assumptions, which means that any
size team can achieve ``almost'' the first-best if penalties are unbounded.

9. A third possibility is that both ability \textit{and} effort vary between
agents. We do not address that here; for a discussion, see McAfee and
McMillan (1989b).

10. Some of these control variables may depend on whether it is efficient
for the firm to use incentive contracts. Since unions frown on incentive
pay, for example, it might be that large firms, for which a flat wage might
be more efficient, would resist unionization less strongly. If that is true,
then by controlling for unionization we underestimate the effect of firm
size.

11. These values were determined by calculating the effect of tenure at 3
years and then subtracting the effect of tenure at 2 years. For instance,
the value 1.84\% was calculated: $3(.0194)+ 3^2(-.002) - 2(.0194)+
2^2(-.0002)$.

12. We have also performed the regressions for a subsample of CPS data
covering just professional and technical employees, whose output is
particularly hard to measure. The results are similar. Similar regressions
were also performed for a subsample that included only non-union males.
These results were also similar and, indeed, stronger than the results in
Table 2.

13. F(120,120) = 1.35 at the 5 percent level, 1.53 at the 1 percent level
(Maddala 1977, pp. 510-11). The lowest F-statistic in Table 2 is 1.494, and
the lowest sample size is 755, so every test is clearly significant.

14. For surveys of this literature, see Albanese and Fleet (1985) and Latane
(1981).

% ---------------------------------------------------------------

\newpage

\begin{center}
\FRAME{itbpF}{4.7573in}{2.3393in}{0in}{}{}{%
90.jleo.fig1.power.and.significance.level.jpg}{\special{language "Scientific
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\textbf{Figure 1: Power and Significance Level}
\end{center}

\newpage

\begin{center}
Table 1: DESCRIPTIVE STATISTICS
\end{center}

\newpage

\begin{center}
Table 2: LOG WAGE EQUATIONS: THE EFFECT OF TENURE
\end{center}

\newpage

\begin{center}
Table 3: MEAN HOURS WORKED PER WEEK BY FIRM SIZE
\end{center}

%---------------------------------------------------------------

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\[
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\vskip-\lastskip\vskip -2pt \vskip -4pt \noindent Source: A matched file of
the May and June 1979 Current Population Survey. Part }

\textsc{\noindent $\dag$ Numbers in parentheses are standard deviations. }

\textsc{\noindent Note: Tenure is defined as the number of years employed in
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\textsc{%____________________________________
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\textsc{\noindent Source: Matched file of May and June 1979 Current
Population Survey. Part-time e }

\textsc{\noindent Note: Standard errors are in parentheses. }

\textsc{\noindent $\dag$ The omitted size category is \textit{Very Large}
1000+ employees. }

\textsc{\noindent $\ast$ Equations additionally include the variables,
education, tenure, and othe transportation and public utilities, wholesale
trade, retail trade, finance insu }

\textsc{%\noalign {\vskip -3pt}
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\textsc{%______________________________________________
%
%
%This is for a table. October 19 1989.
}

\textsc{\hoffset .92in \voffset .5in \parskip=12pt }

\begin{center}
\textsc{{TABLE 1}\\[0pt]
{DESCRIPTIVE STATISTICS} }

\textsc{
\begin{tabular}{rcccccc}
\hline\hline
& \multicolumn{5}{c}{Means by Firm Size Category} &  \\ \hline
& 1-24 & 25-99 & 100-499 & 500-999 & 1000+ &  \\ \hline\hline
Weekly-earnings\hfil & 224.36 & 256.65 & 276.24 & 282.86 & 321.98 &  \\ 
& (133.94)$\dag$ & (143.33) & (152.91) & (150.20) & (160.10) &  \\ 
Tenure (years) \hfil & 5.67 & 6.92 & 7.63 & 8.35 & 10.02 &  \\ 
Other Experience (years)\hfil & 11.23 & 11.16 & 10.11 & 8.62 & 8.07 &  \\ 
Education (years) \hfil & 13.26 & 13.50 & 14.08 & 14.45 & 14.30 &  \\ 
College (percent)\hfil & 14.79 & 19.63 & 26.86 & 30.42 & 26.39 &  \\ 
Selected Occupations: &  &  &  &  &  &  \\ 
Professional (Percent)\hfil & 9.94 & 13.65 & 22.62 & 29.37 & 22.46 &  \\ 
Manager (Percent) & 10.90 & 13.92 & 12.99 & 11.24 & 10.99 &  \\ 
Craftsmen (Percent) & 20.71 & 17.20 & 12.00 & 10.98 & 15.31 &  \\ 
Selected Industries: &  &  &  &  &  &  \\ 
Retail (Percent) & 21.29 & 15.71 & 7.46 & 6.35 & 7.90 &  \\ 
Durable Goods (Percent) & 5.30 & 13.28 & 16.59 & 16.40 & 26.06 &  \\ 
Non-durable Goods (Percent) & 3.85 & 9.95 & 11.41 & 12.43 & 12.31 &  \\ 
Female (Percent) & 36.15 & 36.40 & 41.78 & 44.44 & 33.15 &  \\ 
Union (Percent) & 8.36 & 22.28 & 30.37 & 34.66 & 41.74 &  \\ 
Hours worked per week & 45.01 & 43.91 & 43.28 & 42.83 & 43.74 &  \\ 
& (9.32) & (7.79) & (6.91) & (6.94) & (6.57) &  \\ \hline
\end{tabular}
}
\end{center}

\textsc{\noindent  Source: A matched file of the May and June 1979 Current
Population Survey. Part time employees (those working less than 36 hours per
week, are not included).\newline
$\dag$ Numbers in parentheses are standard deviations.\newline
Note: Tenure is defined as the number of years employed in current job.
Other experience is defined as the number of years employed in current job. }

\end{document}
