A GRAPHICAL REPRESENTATION FOR FORCE-USING OLIGOPOLISTS IN NONCOOPERATIVE EQUILIBRIA

Johnnie B. Linn IIIConcord University

This Paper

• extends previous work on the behavior of force-using competitive or monopolistically competitive firms to oligopolistic firms.

• A new geometric means of finding the equilibrium is introduced.

Assumptions

• Two populations.• Labor is mobile within populations but not between

populations.• No joint production technologies of output and force.• Ratio rule.• Firms fight over pooled output.• Nash equilibrium.• Firm employees share residuals.

Setting Up the Geometric Solution

• PPF’s of the two firms are placed origin to origin.• Y axes are horizontal and F axes are vertical.• A cursor partitions output according to ratio rule.• Additional firms can be included.

The Geometric Solution

Y1

Y2

F1

F2

O AB P

S1

S2

Return to Change in Force

W

F

Y2 Y1

F1

F2

1 1 2| ( / )( / )YW p F F D Y D

Return to Change in Input

1 2| ( ')(1 / )FW p Y Yp F D

W

Y

Y2 Y1

F1

F2

Marginal Rate of Transformation for Output and Force

1 1 2( / ){ /[ (1 )]} /Y D F Y

The Nash Equilibrium

ii

jj

FF

1 2( )i iW p Y Y

2 11

2 1 1

/( / )

FW pF

Y Y W p

Nash Equilibrium Illustrated

F1

F2

Y2

Y1P

W1

W2

B

AO

Characteristics of the Nash Equilibrium

• No output-only corner solutions.• Force-only corner solution is possible.• If there is not a force-only corner solution, there must be

at least one stable interior solution.• True for all technologies.

The Paradox of Power

• If a population has a sufficiently small endowment, it will employ a force-only corner solution against the other population (Hirschleifer, 1991).

Paradox of Power Illustrated

F1

F2

Y2

Y1B PW1

W2

Introducing Criminal Enterprises

• If smaller population is in a force-only corner solution, it would like to recruit more population, but can’t.

• Individuals in the larger population may go free-lance and set up a force-using firm of their own.

• Result is three entities, two in the larger population and one in the smaller.

Rise of a Criminal Enterprise

F1

F2

Y2

Y1BP1

W1

W2

P2

The Role of Outliers

• Outliers are force-using individuals who do not use force collectively on the margin.

• Outliers are not a firm, they are not regarded collectively as a player in the Nash equilibrium.

• Firms can hire outliers if outlier per-capita winnings are less than the per-capita earnings of firms.

• Outliers will be present only when combined winnings of firms do not exhaust output.

Test for the Presence of Outliers

*1 1 2 1 2 1 1 1 1( ) ( ) ( )p Y Y Y Y wL wG U U

1

1

0U pYU

One Firm in the Presence of Outliers

1 1 1 1[1 (1 ) (1 )] 0p Y

1 11

1

(1 ) 1

or

Two Firms in the Presence of Outliers

1 1 2 21 2

1 2

(1 ) 1 (1 )(1 ) 1

Two Firms in the Presence of Outliers, Identical Technologies

1 2

(1 ) 2 2

(1 ) 2a

or

Non-Cooperative Equilibria in Two Populations, Output-Producing Firms, Absence of Other Entities

2

1

1/(1-)0

DUOPOLY,OUTLIERS PRESENT

DUOPOLY, OUTLIERS ABSENT

Non-Cooperative Equilibria in Two Populations, General Case

2

1

1/(1-)0

DUOPOLY, OUTLIERS INBOTH POPULATIONS, ORTHREE ENTITIES, OUTLIERS PRESENT IN LARGER POPULATION

TWO ENTITIES, OUTLIERS PRESENT IN LARGER POPULATION, ORTHREE ENTITIES, OUTLIERS ABSENT

Equilibrium in Isolated Population

2

11/(1-)0

TWO ENTITIES, OUTLIERS ABSENT, OR MONOPOLY, OUTLIERS PRESENT

TWOENTITIES,OUTLIERS PRESENT

1.11

PURE COMPETITION, MONOPOLISTIC COMPETITION, OUTLIERS PRESENT, OR DUOPOLY ORMONOPOLY AND CRIMINAL ENTERPRISE, OUTLIERS ABSENT