Stackelberg_Model - Atlas of Economic Models
 

The Stackelberg Model

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 $$N$$.