# The Stackelberg Model

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- Additional Contributors: None
Categories: CategoryModel CategoryMicro CategoryIndustrialOrganisation CategoryImperfectCompetition

Related Models: Cournot_Model

- Created: 2008-05-28
Suggested Citation: See license page.

Contents

## Introduction

The eponymous Stackelberg model takes extends the basic quantity setting approach of the Cournot_Model but replaces simultaneous quantity choice by a sequential approach. Specifically in the Stackleberg model quantities are chosen sequentially with later firms knowing the quantity chosen by earlier firms. In the original model focused on a duopoly but the model is easily generalized to an n-firm setting.

## History

According to http://ideas.repec.org/a/eej/eeconj/v18y1992i2p171-186.html:

- The static equilibrium dominant firm price leadership model is traced to a seminar presentation by Karl Forchheimer in 1906, who seems to have originated the concept of a dominant firm facing competition from fringe rivals maximizing profits on the basis of residual demand--industry demand less quantity supplied by the fringe. Heinrich von Stackelberg completed the model analytically in 1934, although in a duopoly context absent stable equilibrium. George Stigler finally combined von Stackelberg's comparative statics with Forchheimer's price-taking fringe rivals, to articulate (in 1940) the equilibrium model as it has been used in countless intermediate microeconomics texts and classrooms for the half century since.

## Model

### Stackelberg Specific

- N firms
- Firms choice variable is their output: $$q_{i}$$
- Firms 'move' (i.e. choose output quantity) in sequential order.
- Firms' payoffs are as for Cournot: $$ \Pi_{i} = q_{i} \cdot p(Q) - C(q_{i}) $$

## Solving

The standard solution concept is sub-game perfect Nash equilibrium. For Stackelberg this will involve solving for the last firm as a function of the current total committed production quantity. Given this, one can then solve for the Nth-1 firm given the behaviour of the nth firm the total committed production of the previous N-2 firms etc.

As little can be said about the general case we shall proceed straight to some specific examples.

## Examples

All of the following examples are based on the case of:

- Linear demand: $$p = A - BQ$$
- Constant marginal costs: C(q) = cq$$
- Without loss of generality we may take $$c=0$$ (just replace $$A$$ by $$A'=A-c$$).

The key insight in solving this (and other cases) is to appreciate that given a total committed quantity $$Q'$$ at some stage in the game, the remainder of the game (whatever its form) is a function only of that committed quantity and hence has the same solution as if that game were being played on its own with a demand curve: $$A' - BQ$$ where $$A' = A - BQ'$$.

### 2 firms (Classic Stackelberg)

In this case post output choice of $$q_{1}$$ by the first-mover the second firm faces a standard monopoly game with demand curve $$(A-Bq_{1}) - Bq_{2} = A'(q_{1}) - Bq_{2}$$. From the Monopoly_Model we know this has solution with:

$$ \[ p=\frac{A'}{2}, q_{2} = \frac{A'}{2B} \] $$

Hence the first firm solves the reduced problem:

$$ \[ \max_{q_{1}} q_{1}p(q_{1}) = q_{1}\frac{A'}{2} = \frac{(A-Bq_{1})}{2} \] $$

This is just the classic monopoly profit function scaled by a half and thus it has the solution:

$$ \[ q_{1} = \frac{A}{2B} \Rightarrow p = \frac{A}{4} \] $$

For completeness we note that profits are ($$\Pi^{M}$$ are profits under monopoly):

$$ \[ \Pi_{1} = \frac{A^{2}}{8B} = \frac{\Pi^{M}}{2}, \Pi_{2} = \frac{A^{2}}{32B} = \frac{\Pi^{M}}{8} \] $$

Thus the first-mover firm supplies a quantity equal to that supplied by a monopolist, the second-mover supplies half this amount and the price is half what it would be under monopoly (and higher than it is under Cournot).

### N Firms Sequentially

We can proceed by induction:

1. The first firm commits to quantity $$q_{1}$$

2. Given $$q_{1}$$ the subsequent $$N-1$$ firms are playing N-1 Stackelberg in the reduced game with demand $$(A-Bq_{1}) - BQ_{N-1} = A'-BQ_{N-1}$$ ($$Q_{N-1}$$ being the total output of the remaining N-1 firms). Suppose this has solution $$p(q_{1}) = f(A',B)$$. Examination of the demand diagram immediately leads us to conjecture that:

$$ \[ p(q_{1}) = \frac{A'}{2^{N-1}} \] $$

This being firm 1's problem is profits are:

$$ \[ q_{1} p(q_{1}) = \frac{A-Bq_{1}}{2^{N-1}} \] $$

Again this is just the classic monopoly profit function scaled by a constant and thus has solution:

$$ \[ q_{1} = \frac{A}{2B} \Rightarrow p = \frac{A}{2^{N}} \] $$

Thus our inductive hypothesis is confirmed and we have (via induction) that the N-player Stackelberg model has:

$$ \[ p = \frac{A}{2^{N}}, q_{k} = \frac{q_{1}}{2^{k}} = \frac{A}{2^{k}B} \] $$

Finally profits are given by:

$$ \[ \Pi_{k} = \frac{A^{2}}{2^{N}2^{k}B} = \frac{\Pi^{M}}{2^{N-1}2^{k-1}} \] $$

#### N Firms with N-1 Follower Firms Playing Cournot

From the Cournot_Model we know that with N-1 firms, constant marginal costs, and demand curve $$A' - BQ$$ the symmetric equilibrium has:

$$ \[ q_{i} = \frac{A'}{NB} \] $$

$$ \[ p = \frac{A'}{N} \] $$

Thus the first-mover solves:

$$ \[ \max_{q_{1}} q_{1}p(q_{1}) = q_{1}\frac{A'}{N} = \frac{(A-Bq_{1})}{N} \] $$

Again this is just the classic monopoly profit function scaled by $$N$$ and thus it has the solution:

$$ \[ q_{1} = \frac{A}{2B} \Rightarrow p = \frac{A}{2(N)} \] $$

Thus, the first-mover, just as in 2 firm model, supplies a quantity equal to that supplied by a monopolist. For completeness we note that profits for first-mover are ($$\Pi^{M}$$ being profits under monopoly):

$$ \[ \Pi_{1} = \frac{A^{2}}{4B(N+1)} = \frac{\Pi^{M}}{N+1} \] $$

Meanwhile profits of followers are:

$$ \[ \Pi_{i} = \frac{ A^{2} }{ 4BN^{2} } = \frac{\Pi^{M}}{N^{2}) = \frac{\Pi_{1}}{N} \] $$

Thus, profits of the first-mover fall by the reciprocal of $$N$$ while those of followers fall with the reciprocal of $$N^{2}$$. Hence the ratio of first-mover profits to follower profits rise in proportion to $$N$$.