Teamwork Efficiency and Company Size

1.1 Literature Review

The paper contributes to two strands of the literature. The moral hazard in teams literature was introduced by Holmstrom (1982), who showed that the provision of effort in teams will be generally suboptimal due to externalities in effort levels and the impossibility of monitoring individual efforts perfectly. Legros and Matthews (1993) showed that the problem of deviation from efficient level effort may be effectively mitigated if the sharing rules are well-designed. ^[2]Kandel and Lazear (1992) suggest peer pressure to mitigate the 1/N effect: the increase in the number of workers lowers the marginal payoff from higher effort. When the firm gets larger, the output is divided between a larger quantity of workers, while they bear the same individual costs. Hence, the effort of each worker should grow less as firms grow larger, and the peer pressure should compensate for this decline. ^[3]

Adams (2006) showed that the 1/N effect may not occur if the efforts of workers are complementary enough. Because he uses a CES production function with constant returns to scale, the determinant of sufficient complementarity is the value of the elasticity of substitution. McGinty (2014) extends this argument to power production functions. In this framework, two outcomes are generic: either to always increase, or always to reduce the firm size. By generalizing, we obtain a nontrivial optimal company size. This allows us to contribute to the firm size literature too. Theories of firm boundaries are classified as technological, organizational and institutional (see Kumar et al. 1999). The technological theories explain the firm size by the productive inputs and the ways in which the valuable output is produced. Basically, five technological factors are taken into account in describing the firm size: market size, gains from specialization, management control constraints, limited workers’ skills, and loss of coordination. For example, Adam Smith defined the firm size by benefits from specialization limited by the market size. By his logic, workers can specialize and invest in a narrower range of skills, hence economizing on the costs of skills. Becker and Murphy (1992) focus on the tradeoff between specialization and coordination costs. The larger the firm, the larger the costs of management to put them together to produce the valuable output.

Williamson (1971), Calvo and Wellisz (1978) and Rosen (1982) use loss of control to explain the firm size. Williamson points out that the size of a hierarchical organization may be limited by loss of control, assuming that the intentions of managers are not fully transmitted downwards from layer to layer. Calvo and Wellisz (1978) show that the effect of the problem largely depends on the structure of monitoring. If the workers do not know when the monitoring occurs, the loss of control doesn’t hinder the firm size, but it may do so if the monitoring is scheduled. Rosen (1982) highlights the tradeoff between increasing returns to scale in management and the loss of control. Because highly qualified managers foster the productivity of their workers, able managers should have larger firms. However, the attention of managers is limited, hence having too many workers results in loss of control and substantially reduces the productivity of their team. The optimal firm size in this model is reached when the value produced by the new worker is less than the losses due to attention being diverted from his teammates.

In this literature, Kremer (1993) is the paper closest to ours, because this is one paper that obtains the optimal size of the firm based solely on the firm’s production function. This paper focuses on the tradeoff between specialization and the probability of failure associated with low skill of workers. He assumes that the the value of output is directly proportional to the number of tasks needed to produce it. A larger number of workers – and hence tasks tackled – allows for the production of more valuable output, but each additional worker is a source of the risk of spoiling the whole product. Hence, the size of the firm is explained by the probability of failure by the workers, which correlates with the worker’s skill.

Acemoglu and Jensen (2013) analyze a problem similar to ours. Agents pariticipate in an aggregative game, where the payoff of each agent is a function only of the agent himself and of the aggregate of the actions of all agents, and they establish existence and comparative statics results for games of this type. Nti (1997) offers a similar analysis for contests. We allow general interactions, but under certain assumptions we can summarize these interactions in a similar way, which does not depend on additive separability. In addition, Acemoglu and Jensen (2013) and Nti (1997) study comparative statics for this general class of games with respect to the number of players, whereas we go a step beyond, looking at the optimal number of players from the perspectives of different managerial objectives. Jensen (2010) establishes the existence of pure strategy Nash equilibrium in aggregative games, but does not explore the symmetry of the equilibrium or the comparative statics.

2.1 Effort Choice in a Team: Equilibrium Outcome

The equilibrium is a collection of the efforts of agents {e1∗,..eN∗} such that each worker i solves his problem (2) treating the efforts of the other peers as given:

where e−i∗ denotes the values of {e1∗,..,eN∗} omitting ei∗.

Let e∗(N) be the function that solves

Homogeneity of degree 1 for g(⋅) helps us to study the behavior of e∗(N). Define

This function represents the efficiency of coworking. Observe that

that is, h(N) measures how much more efficient is the team of agents that the efforts of a single person, holding effort level unchanged. Henceforth we will call this the teamwork efficiency function. For instance, if it is linear, the working team is as efficient as its members applying the same effort separately. By Euler’s rule and the symmetry of g(⋅),

Therefore, (7) can be rewritten as

Equation (8) is the incentive constraint that defines e∗(N) as a function of N.

2.2 Effort Choice in a Firm: First Best

Following Holmstrom (1982), we assume that the residual claimant provides the employees with contracts that implement the first-best choice of effort.

The residual claimant would choose the effort size eP(N) to implement by maximizing

which, assuming a symmetric outcome, leads to the first-order condition

The solution of (9), eP(N), is greater than the solution of (8), e∗(N), as long as N>1. The reason is that in equilibrium, the marginal payoff for the individual effort does not take into account the complementarities provided to other workers. Even if the product f(⋅) were not split N ways, but instead were non-rivalrous, ^[9] the additional 1/N in the marginal benefit of the team worker would persist.

2.3 Second-Order Conditions and Uniqueness

Equation (6), the second-order condition of (8), in the equilibrium can be rewritten as

This is because εg(e∗(N),..,e∗(N)|N)=(h(N)/N)e∗(N)e∗(N)h(N)=1N. Let

hold; then (10) is satisfied automatically. If c(x) is more convex than f(y) at every x≥y, this condition is satisfied. Similar math is used to compare the risk-aversity of individuals: for every u(x), εu′(x) is just the negative of Arrow-Pratt measure of relative risk aversion.

The second-order condition for (9) is

which, after dividing by the first-order condition, can be rewritten as

Observe that it is very similar to (11): but the effort level in the argument is different. One would be sure that both (11) and (12) hold if one were sure that c(⋅) is at every point “convexer” than f(⋅)at every point above: εf′(y)<εc′(x)∀y>x. This can be simpler to verify if additional assumptions are imposed on εf′ or εc′:

Second-order conditions hold at maxima automatically, but if they hold everywhere, the solution of the corresponding FOC has to be unique. Result 1 thus provides sufficient conditions for the uniqueness of the pure strategy outcome.

εf′(x) being decreasing has the following interpretation. When εf′(x) is constant and equal to α, it means that f′(x)=Kxα, which makes f(x) a power function, where K is an integration constant (unless α=−1, in which case f′(x)=Klnx). The decreasing εf′(x) implies the “lower power”, or “less convexity” of f(⋅) in larger arguments.

3.1 Team Size that Maximizes Effort

This may be a concern in industries where learning-by-doing is important, and therefore the decisionmakers would like to increase efforts even though this might hurt their immediate profits. Workers may be willing to participate in teams of a size that maximizes their effort to combat their long-term/short-term decisionmaking inconsistency issues. This subsection is crucial to understanding the further analysis. We have therefore sought to keep the analysis in this part very explicit. Other problems will be dealt with in a similar fashion, therefore we relocate the repetitive parts to the Appendix.

From (8) one can deduce e∗(N), well-defined and differentiable over N∈R+.

Take elasticities with respect to N on both sides of (8) to get:

Solve this to obtain

From (13) one can immediately see that the N that maximizes e∗(N) has to satisfy

The denominator of (13) is positive: it is a second-order condition of the effort choice problem, (11). Therefore, whenever εh(N)εf′(e∗(N)h(N))+1>2, e∗(N) is increasing in N, and otherwise it is decreasing in N.

In the space of (x,y)=(εh(⋅),εf′(⋅)), Equation (14) simplifies to:

Solving out the equilibrium will produce a function e∗(N), and therefore a sequence of values of (εh(N),εf′(e∗(N)h(N)). We depict an example of this path in Figure 1a. Denote

Figure 1:

The choice of N to maximize effort in a team; and the Result 2 logic.

For the sequence depicted in the Figure 1, one can observe that e∗(N) is increasing at N≤3, and decreasing for N≥4. Therefore, the optimal “continuous” N (denote it N1) is between 3 and 4, and the integer N that delivers the maximum effort is either 3 or 4.

The assumption that g(⋅) is CES makes εh(N) constant; the assumption that f′(⋅) is a power function makes εf′(⋅) constant. Example 1 predicts that whether e∗(N) is increasing or decreasing everywhere depends upon the elasticity of substitution of g(⋅) precisely because, in the world of Example 1, f(x)=xα and g(⋅) is CES. Γ1 is a single point in these assumptions. Therefore, in order to have a nontrivial prediction about the optimal effort size, one needs either a decreasing εh(N), or a decreasing εf′(⋅), or both. Obtaining values in the general case in inherently complicated, but one can make comparative statics predictions without knowing the precise specification of relevant functions.

The purpose of this Result is to illustrate that the effort choice comparative statics are governed by the variation in εf′. This illustrates that a simplifying assumption, such as constant elasticity, for the production function is not innocuous. Even assumptions such as the concavity of f can restrict the economically important behavior:

and, therefore, for every N, εh(N),εf′(e∗(N)h(N))<(1,1), no matter whatc(⋅)is. The individual effort decreases withN for every N.

3.2 Firm Size that Maximizes Effort

As in the previous part, this problem occurs in industries where learning-by-doing is important, and long term planning may motivate to increase workers’ effort by manipulating the number of workers. We assume that when the firm designs a contract, it tries to implement the first-best, which takes into account the agents’ complementarities in g(⋅). If the social planner were choosing the effort for the agents, his FOC would suggest a higher effort for a given N (see the discussion of the 1/N effect on p. 12). Since c′(⋅) is increasing, this immediately implies that eP(N)≥e∗(N), with equality at N=1, and therefore the effort-maximizing sizes of a firm and a team do not have to coincide.

The first-order condition ^[12] becomes

Again, if the left-hand side is larger than the right-hand side, the effort is increasing in N, and the reverse holds when the left-hand side is smaller than 1. The change of the managerial objective affects multiple components of the optimal size problem:

1. The threshold that governs when the firm is big enough, Φ1, is now replaced by

The reason why 2 in the definition of Φ1 is replaced by 1 in the definition of Ψ1 is exactly because the marginal 1/N effect, which appeared because the individual marginal benefit did not include the benefits provided to the other participants, went away.

1. Since eP(N)>e∗(N) for almost every level of N, the values of εf′(eP(N)h(N))≠εf′(e∗(N)h(N)), unless f(⋅) is a power function in the relevant domain.

Figure 2:

Choosing N to maximize effort, the firm case.

3.3 Team Size that Maximizes Utility

Would team members invite more members to join the team? If this increases the utility of each team member, yes. Thus, the team size that maximizes the utility of a member of the team is the team size that would emerge if teams were free to invite or expel members.

N3 should solve the following first-order condition:

Again, at values of N where the left-hand side is larger (smaller) than 1, the utility is increasing (decreasing) in N. Let Φ2 be the set of locations where (16) holds with equality. This line, evaluated at N=N1, is plotted over Γ1 and Φ1 on Figure 3.

Figure 3:

Choosing N to maximize individual utility.

One can immediately see that:

1. There is a unique intersection of Φ1 and Φ2, which happens at εˉh=1/εf(e(N1)h(N1)).

2. The path of Γ1 intersects Φ1 above Φ1⋂Φ2 if and only if N1<N3. In general, when two different maximands are used, different answers are to be expected, but our result makes issues clearer: the only thing necessary to establish whether N1<N3 is the value of εh(N1) and of εf(e∗(N1)h∗(N1)).

Therefore, if the elasticity of f(⋅) at the size of the team chosen by team members N3 is too small, it is likely that the team will be too large to implement high efforts (N3>N1).

Observe that the local monotonicity of εf(x) is informative about the comparison between εf′(x)+1 and εf(x):

In particular, f(x)>0 implies εf(x)′>0⇔εf′(x)+1>εf(x), and the condition in Result 4 means that the elasticity of f(⋅) is either not decreasing too fast, or that it is decreasing quite quickly. Since adding and subtracting constants to the production function does not change εf′(x), but does change εf(x), both cases (N1<N3 and N1>N3) are generic.

In teaching, many lecturers assign home assignments for group work. Some lecturers use fixed group sizes, other lecturers allow students to form groups of their own choosing. If higher effort is desirable (for instance, because effort in the classroom is valuable on the labor market, which is not fully understood by students), it may be a good idea to restrict the group size, notwithstanding the complaints of students. If the elasticity of f(⋅) at N1 is greater than 12(εf′(⋅)+1) at the same N1, students will yearn for an increase of the size of the group, and they will complain that the required group size is too large otherwise. ^[13] Instead of assigning the group sizes, a teacher who wants to implement teamwork projects can manipulate the group’s payoff implied by the project design, to make sure the maximal effort group size is close to the maximal utility group size.

3.4 Firm Size that Maximizes Utility

When the principal extracts all surplus from the workers, maximizing the payoff per worker translates to maximizing profit per worker. The principal maximizes the surplus per worker, not the total surplus, because the principal can own more than one firm, as fast food franchisers do.

At N4, the following holds (see Appendix for derivation):

When εf(eP(N)h(N))εh(N)>1, the utility of each member of the firm increases with the size of the firm, and the utility is reduced otherwise.

One can see the difference between (15) and (17); they have to be equal only when ∀x,εf(x)=εf′(x)+1, which implies that f(x) is the power function.

This Result helps to establish why people do not work efficiently in different environments. The problem is not so much in the returns to scale of the production function; the relevant threshold is the comparison of the first and second derivatives of the production function, which is known if it is known that the elasticity of the production function is locally increasing or decreasing. Those employee-owned companies whose employees feel that they would be more motivated and would work harder had they had more collaborators have εf(eP(N)h(N))<εf′(eP(N)h(N))+1. The curvature of their production function is increasing.

This Result shows that the issue of which companies are bigger, teams or firms, boils down to the properties of the production function, and the only limitations for the rest of the fundamentals (such as the cost function and effort aggregation function) is to guarantee that assumptions hold. The precise shape of h(⋅) determines the value of N3 and N4, but is not always needed to establish which one is bigger. Obviously, there’s plenty of f(⋅) whose elasticities are not monotone, but (a) the part that is harder to observe, the teamwork efficiency function, may not require estimation, and (b) the monotonicity is only important locally, for company sizes near N3 and N4.

Results for other managerial objectives can be obtained in a similar fashion: for instance, a residual claimant that collects a fixed proportion of the total surplus of the firm will employ more than N4 workers as long as (12) holds. We reserve these for future research.

3.5 The Quagmire of Constant Elasticities

The previous analysis showed that at least one of two elasticities cannot be constant in order to obtain a well-defined optimal company size. However, even holding one of two elasticities constant can mislead. In the following example, we assume that εh(N) is decreasing from a large enough value to 0, and the production function is a power function.

The effort size e∗(N) chosen by the members of the team satisfies

Let us order firm sizes chosen with different managerial objectives. When εh(N) is decreasing,

1. N1, the team size that maximizes the effort when the effort level is chosen simultaneously and independently, satisfies εh(N1)=2/α;

2. N2, the firm size that maximizes the effort when the effort level is chosen according to the first best, satisfies εh(N2)=1/α;

3. N3, the team size that maximizes the team member’s utility when the effort level is chosen simultaneously and independently, solves εh(N)=1α+N−1Nβ−α, the right-hand side of which is monotone and converges to 1α+1β from below;

4. N4, the firm size that maximizes the utility per worker ^[14] when the effort level is chosen according to the first best, satisfies εh(N4)=1/α.

Example 3 supplies the following intuition for different maximands (see Figure 4):

• 1 & 2 The effort-maximizing size of the firm is greater than the effort-maximizing size of the team. This is a consequence of f(⋅) being a power function (see Result 3), and need not hold in general.

• 1 & 3 The company size chosen by the team when the decision to hire is in the hands of the team members is greater than the company size chosen to maximize the effort size. This is not a general result, but a consequence of a close connection between εf(⋅)=α and εf′(⋅)=α−1. Compare (14) and (16): when N is such that (8) is satisfied, (16) suggests that the utility of each participant increases with the size of the team.

• 2 & 4 The size of the firm that maximizes employees’ utilities is maximizing their effort as well. This is not a general result, but a direct consequence of f(x)=xα: conditions (15) and (17) coincide algebraically.

• 3 & 4 When a self-organized team becomes incorporated, it will become larger. This, however, is not a general result, but a consequence of a power production function.

Figure 4:

Ordering solutions from Example 3.

This exercise demonstrates many spurious findings arising simply from the desire for closed form solutions. Some of the strong predictions are generalizable, but most are a consequence of the power function assumptions.