The size distribution which emerges is a solution to the problem: allocate production over firms to minimize total cost.
It goes without saying that this is counterfactual for firm size.
Here is the actual firm size distribution (Axtell, Science 2001).

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458906643050496

Lucas aims to implement Henry G. Manne (JPE 1965) suggestion that the market assigns more resources to better managers: "the observed size distribution is a solution to the problem: allocate productive factors over managers of different ability so as to maximize output."

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458909562183680

Lucas starts from an economy with a given amount of capital and labor.

All agents have a level of managerial talent. Talent raises the productivity of the supervised bundle of labor and capital

A firm is one manager, together with the capital and labor under her control.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458912057884673

The key implication is the "scale of operations effect" : more talented managers can run larger firms more efficiently and earn higher profits, which leads to a positive correlation between firm size and earnings.

Earnings grow more than proportionally with talent.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458914737946624

Now think dynamically about firm growth

Gibrat law: A well-known feature of observed patterns in firm growth, however, is the independence of firm growth and size.

Lucas: to obey Gibrat's law a necessary and sufficient condition is for the span function to be a power funct.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458916818345984

The model "predicts" the full size distribution of firms, but only given a distribution of managerial talent.
Lucas proceeds to test it empirically and finds that
as the model predicts there is a systematic effect of GNP on firm size (first row)

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458919280476167

Lucas argues his model has two main holes: homogenous workers, only one layer.

Sherwin Rosen (1982), another (sadly deceased too early) @uchicago economist (and inspiration of much of my work), set to solve these problems.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458922262646784

Rosen had invented in 1981 the idea of superstar markets. Think of Messi, Ronaldo, Taylor Swift, Le Bron James:
1) close connection between personal rewards and the size of one's market;
2) both market size and reward are skewed toward the most talented in the activity.
Why?

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458924464586754

The key input to explain these facts is the limited substitution between the talent of different artists, doctors etc. "hearing a succession of mediocre singers does not add up to a single outstanding performance. "

The reward function raises with the size of the market-- fast!

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458927346053120

In 1982, Rosen turns to superstar firms by expanding on Lucas' model. Crucially, he includes multiple layers of hierarchy. It is a multiplicative technology, where the talent of one layer increases the value of those in the layer below.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458929636122624

Two factors:
- Management involves discrete and indivisible choices (such as which goods to produce, in what varieties and volume)--a strong scale economy
- a supervisory activity that congests management scale economies and produces determinate firm size

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458932131774466

These two factors explain the puzzle:
- The scale economy of management requires that the most able personnel be assigned to top level positions in large firms.
- The diseconomy of direct supervision binds firm size, that can be relaxed by subordinating many hierarchical layers

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458934379921408

So now we have in firms something like the superstar markets:
- Equilibrium assignment of more talented people to top control positions in largest firms
- Multiplicative productivity interactions mean distribution of control and income more skewed than the talent distribution.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458937307611136

Hence talent of workers matters, and hierarchies have many layers.
But Rosen takes a short cut. Workers and managers are linear substitutes in "efficiency units" (talent is proportional).

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458939618664449

Rosen argues that it is necessary to introduce true complementarities between worker and manager talent.
@HansbergRossi and I do that in our QJE (2006) paper

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458942168813568

Workers benefit from having good managers (they solve more problems and make their own time more productive), and managers benefit from having good workers (they need less help and allow managers to have a larger team).

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458944807022593

Hence unlike in all previous literature, there is a one to one "marriage" between workers and managers.
Our framework allows us to have predictions on communication technology (makes organizations larger, deeper, deskills workers), and IT(shorter organizations, empowers workers.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458947629686787

.@xgabaix and @augustinlandier take Lucas' theory in a different direction. They calibrate it to study whether recent changes in CEO Pay may be related, as the model predicts, to firm size changes.

🐦🔗: https://n.respublicae.eu/lugaricano/status/1660458950163144704