00 / Prerequisite · Berry92

Entry and market structure

Compare the incentives of operating firms and potential entrants.

The entry-equilibrium discussion draws on Berry (1992). The small example below illustrates the incentives rather than reproducing the paper’s airline estimation.

Follow the entry decision

Choices change competition

Entry comes before profit.

StartPotential firms

Stage 1Enter or stay out

Stage 2Compete

CheckSwitching gains

πᵢ(n) = vᵢ − δ log(n)

Illustrative net profit · outside option = 0
vᵢ includes firm-specific cost disadvantages.

Operating firm

πᵢ(n) ≥ 0

Exiting must not improve profit.

Excluded firm

πᵢ(n + 1) ≤ 0

Entry adds a firm. Recheck competition.

Complete information: firms know the profit indices. Levin, pp. 6–7 ↗ · Kawaguchi, Entry and Exit ↗

Four potential firms

Try a candidate set.

v = (1.4, 1.0, 0.9, 0.3)
for A, B, C, D.

Competition penalty δ

Inspect an operating set

Equilibrium count: —
Equilibrium identities: —
Ranked construction selects: —

Gain from changing only this firm’s action.

Firm	Switch to	Profit gain

Loading saved checks…

Positive gain: this candidate set fails the equilibrium check.

Count ≠ identity

Two firms. Two possible sets.

At δ = 1, AB and AC both pass.Ranking profitability chooses AB. It constructs one equilibrium; it does not list them all.

A common crowding penalty makes the number easier to determine.
Rival-specific effects can change that result.
Exact zero-profit ties can admit different counts under weak inequalities.

Count and identity assumptions: Berry (1992), §3.1 ↗.

Where this fits

Profitability needs a model.

AB08: demand, cost and conduct generate operating profits.
02 One Economy: changing the operating set requires solving again.
03 Estimation: model outcomes become moments to match with data.

Scope & the course’s entry routine

Enumeration checks all 16 sets; it scales exponentially.
Ranking relies on the shared, decreasing crowding term.
The course deletes firms and resolves; it does not test every excluded firm’s re-entry.
Its stopping rule can retain loss-making firms. A returned set needs further checks before being called an entry Nash equilibrium.

pure_equilibria([1.4, 1.0, 0.9, 0.3],
                delta=1.0)
# AB and AC both satisfy the deviation tests.

Python example ↓Further exercise ↗

Try it yourself

Check every operating set on your computer.

Download the Python example and run it in the folder where you saved it. It needs Python 3.10 or later with NumPy, and takes under a second. In the course repository, run python lessons/00_prerequisites/berry92.py from the root instead.

python berry92.py

It checks all 16 sets of the four firms at δ = 1, lists the sets that pass the deviation tests and gives the set that ranking by profitability chooses. Add --self-test to run the checks first.

Check your result

Equilibrium: AC; deviation gains [A,B,C,D]: [-0.706853 -0.098612 -0.206853 -0.798612]
Equilibrium: AB; deviation gains [A,B,C,D]: [-0.706853 -0.306853 -0.198612 -0.798612]
Ranked selection: AB

These are the two sets that pass at δ = 1 on this page, with the same deviation gains. Every gain is negative, so no firm wants to enter or to leave. Ranking by profitability chooses AB.

Code on GitHub lessons/00_prerequisites/berry92.py ↗