Span of Control¶
Models following the work of Lucas (1978) and Hopenhayn (1992) have competitive firms that operate technologies with decreasing returns to scale and whose owners (or managers) differ in their "managerial talent." This is captured by differences in firms' productivity \(\left(z_{i}\right)\). More "talented" managers have a larger "span of control," that is, their optimal firm size is larger. The decreasing returns to scale determine how differences in talent (productivity) translate into differences in size, but they also imply that firms make profits in equilibrium. However, these profits are not the result of market power but of the returns to the "managerial input." These are rents to the fixed input that the manager owns.
While we will not cover these models in detail, we a brief description of the problem is useful.
Note
This description follows the lecture notes by Chris Edmond: http://www.chrisedmond.net/phd2014/90065_lecture1.pdf.
I focus on Lucas (1978). Lucas' model is static and is concerned with explaining the distribution of firms. Consider a firm that produces in two stages, first it combines capital and labor with a standard constant-returns-to-scale technology, \(y=F\left(k,n\right)\), then that output is mixed with managerial talent to produce output, so that total output is
where \(g\) is strictly increasing and strictly concave. The managerial talent \(z_{i}\) is the fixed input and the concavity of \(g\) implies that there are decreasing returns to scale to the firm. The practical implication of the decreasing returns is that (unlike with constant returns to scale) the best manager (highest \(z\)) cannot use all the resources.
We can simplify the problem by using the homogeneity of \(F\) to write output as
where \(\kappa=\frac{k}{n}\) is capital per-worker and \(f\left(x\right)=F\left(x,1\right)\). This turns to be convenient because we isolate the labor demand of firms as being affected directly by \(g\). This matters because the objective of this theory is to explain the distribution of firm size, with size measured by employment.
The problem of a firm manager is then to choose labor and capital-per-worker:
(note that without \(g\) this problem would be linear in \(n\), leading to corner solution in scale)
The first order conditions are
We can use these conditions to solve for the optimal capital-per-worker. Dividing them gets us
Recall that firms face competitive input markets, so prices \(w\) and \(r\) are common across firms. They also operate the same technology. The only difference is in managerial talent \(\left(z\right)\). So, this equations implicitly defines the optimal \(\kappa\) for all firms. Lets call that level \(\kappa^{\star}\).
The scale of the firm is then pinned down by solving for \(y^{\star}\left(z\right)\) out of
once we have \(y^{\star}\left(z\right)\) we can also get \(n^{\star}\left(z\right)=\frac{y^{\star}\left(z\right)}{f\left(\kappa^{\star}\right)}\).
Lucas then uses this solution to ask about the properties of \(g\) that are needed for the distribution of firms to satisfy the patterns observed in the data. The most salient feature is known as Gibrat's law and it captures the fact that firm growth is independent of firm size. In particular, as factor prices change, firms (employment) will grow (or shrink). This growth should be independent of their (employment) size.
The size of the firm is captured by \(n^{\star}\left(z_{i},w,r\right)\). Different firms have different \(z\) and hence different sizes, but they all face the same prices. The total differential of the function \(\ln n^{\star}\) as factor prices change gives us the growth rate of firm size,
For this to be independent of size it must be that
for that to always be the case it must be that
These conditions impose restrictions over the shape of \(g\). Lucas solves these differential equations and finds that the only function that satisfies them is \(g\left(x\right)=Ax^{\alpha}\) for some constant \(A>0\) and a power \(\alpha\in\left(0,1\right)\). We can verify this directly. The optimal (inner) output size is
and so the optimal labor demand is
Even though this expression looks daunting, it delivers the result we want immediately, because it implies that the labor (size) of the firm is log-separable in \(z\) and the terms that depend on prices \(\left(w,r,\kappa^{\star}\right)\).
which deliver s the result required by Gibrat's law.
So we have a single parameter \(\alpha\) that determines the (decreasing) returns to scale of the firms. The scale parameter \(A\) plays no real role in these results and so we normalize to 1.
We can further model firm entry as a reflection of the managers' occupational choice. People have different managerial talent, but they all have the same skills as a worker. If a person can supply their (full) time as either a worker or a manager then they will choose whichever delivers them the highest income (absent any preferences or amenities, as is the case in this model). The income as a worker is given by the wage, \(w\), and is constant across people. The income as a manager is given by the managerial profits, \(\pi\left(z\right)\), and depends on talent. There is then a cutoff for the managerial talent, above which people become managers. That is, a \(z^{\star}\) such that
This is a zero profit condition for the marginal manager that takes into account their opportunity cost of not being a worker. That constitutes the fixed cost of setting up a firm.
The work of Hopenhayn (1992) and the literature that it generated extends these ideas to a dynamic setting. It also takes into account the role of fixed set-up costs in determining firm entry.
A special case
We can further simply the problem if we assume that the firm operates only with labor, so that \(y=n\). Then the managerial technology is \(z_{i}n^{\alpha}\). Because there is no capital, the first order condition is now \(zg^{\prime}\left(n\right)=w\). Solving it gives us a closed-form expression for the size of the firm as a function of managerial talent:
If we further assume that managerial talent is distributed Pareto with a CDF \(1-z^{-\xi}\), the distribution of firm size is also Pareto, but with a Pareto parameter \(\xi\left(1-\alpha\right)\).
To see this recall that the defining feature of the Pareto distribution is that its counter-CDF has the form \(Bn^{-p}\), with \(p\) the Pareto parameter and \(B\) a constant. so we want to know \(p\) from
the last equality says that for a firm to have more than \(n\) employees \(\left(\tilde{n}>n\right)\) it must have managerial talent \(\tilde{z}\) above \(\frac{w}{\alpha}n^{1-\alpha}\). We know the share of firms satisfying that condition:
this gives us our result. The employment of firms is distributed Pareto with power \(p=\xi\left(1-\alpha\right)\).
This result gives us insight over the role of the market (in this case through the returns to managerial talent) in amplifying differences across firms (although similar insights apply to differences in income or wealth across individuals). First, note that all profits in this environment are the return to the fixed (managerial) factor. The technology \(z_{i}n^{\alpha}\) makes these returns to be \(1-\alpha\) of total output (just as in the general Cobb-Douglas case, just define \(x_{i}=z_{i}^{\frac{1}{1-\alpha}}\) to write output as \(x^{1-\alpha}n^{\alpha}\)). Then, note that the returns to scale amplify differences in scale as \(\alpha\to1\). That is, the more scalable the technology, the larger the differences between firms:
We should therefore expect larger differences in size (thicker right tails) in industries with more scalable technologies (higher \(\alpha\)).