# EIRP Proceedings, Vol 10 (2015)

**The
Stackelberg Model for a Leader **

**of
Production and Many Satellites**

**Catalin
Angelo Ioan**^{1}**,
Gina Ioan**^{2}

**Abstract**:
Oligopoly
is a market situation where there are a small number of bidders (at
least two) of a good non-substituent and a sufficient number of
consumers. The paper analyses the Stackelberg model for a leader of
production and many satellites. There are obtained the equilibrium
productions, maximum profits and sales price where one of the company
is the leader of quantity, and other satellites. There are also
survey the situations where the firm based on its marginal cost of
production can effectively take the lead of production.

**Keywords:**
oligopoly; Stackelberg;
equilibrium

**JEL
Classification:**
D43

**1 Introduction**

Oligopoly is a market situation where there are a small number of bidders (at least two) of a good non-substituent and a sufficient number of consumers. Oligopoly composed of two producers called duopoly.

Considering any number of competing firms producing the same normal good, is interesting the analysis of each activity in response to the activity of other companies.

Each of them when it sets the production and the sale price will be considered the productions and prices of other firms. If one of the companies will settle the price or the quantity produced, the other adjusting after it, then the price will be called leader of price or leader of production, the others called price satellites or production satellites.

**2 ****The
Stackelberg Model for a Leader of Production and Many Satellites**

Consider now m
firms F_{i},
i=
,
a function of price: p(Q)=a-bQ,
a,b0
and
the total cost of production TC_{i}=_{i}Q,
i=
,
where _{i}
is the marginal cost of firm F_{i}.

Consider now
that the company F_{s}
is a leader of production, where s=
is fixed. It sets at a given time a production Q_{s}.

Considering a
certain satellite F_{k},
it will establish a production that will seek to maximize its profit
but at the same time it will take into account the production of the
leader which will influence the selling price of the product.
Therefore, let the firm F_{k},
k=
,
ks
reaction function:

Q_{k}=
,
i=
,
is

The leader’s
profit
F_{s}
is through price, a function of both its production and those of
satellites:

=

Taking into
account the reaction functions of satellites: Q_{i}=
,
i=
,
is
follows:

=

The profit of
a satellite F_{k},
ks
is:

= , ks

The condition
of profit maximization of the leader F_{s}
is:

=

and for
satellite F_{k}:

= =0, k= , ks

In conditions
that the production of the company F_{s}
is given, we have the system of equations:

, k= , ks

Adding the resulting m-1 relationships: from where: .

Substituting in the above formula, we find that:

- the reaction
function of F_{k}
to F_{s},
k=
,
ks

Substituting in the condition of profit maximization of the leader, follows:

the satellite
production of F_{k}
being:

, k= , ks

But now, the
condition that F_{s}
be really production leader returns to:
,
k=
,
ks
that is:

, k= , ks

which finally leads to:

Therefore, the
marginal cost of F_{s}
will be higher limited to:

sups=

If _{i}c
- constant, i=
,
is
resulting
from the above:
.

Following
these considerations, in order that the company F_{s}
to maintain or to assume leadership role it is necessary that its
marginal cost is higher limited by sups.

The maximum profits of the firms will be:

= =

=
=
,
k=
,
ks

The
condition that the profit leader be higher than that of any satellite
returns to:

from where:

Therefore, the marginal cost of the leader must satisfy the condition imposed by the upper limit sups and the expression above, otherwise the company is not leading amount or if in affirmative case doesn’t have a higher maximum profit than the one of the satellites.

For two
companies F_{s}
and F_{k}
(m=2) the condition that the maximum profit of the leader be higher
than that of the satellite becomes:

The
requirement that
is equivalent with
.

Therefore,
if:
we have:
and if:
follows that:
.

Returning to the case of the m firms the selling price is:

p^{*}=
=
0

From here, it follows:

p^{*}=

Therefore, a change in the marginal cost of the leader with an amount , will lead to a change in price and a change of all others will change by an amount .

Returning to
the situation of the two companies F_{s}
and F_{k},
the condition that F_{s}
be leader was:

It is possible that none of the companies can not be a leader? It should, in this case that and that is:

Considering the straight lines: and we get the situation in Figure 1:

**Figure
1.**

Therefore, when a firm's marginal cost is sufficiently large relative to the other, it can not take the lead. If the two firms have marginal costs comparable to each other, but large enough, none can assume leadership, the reverse situation being when the marginal costs are low enough, the two companies being able to assume each leadership role.

**3**** References**

Ioan,
C.A. & Ioan, G. (2012), *Matheconomics*.
Galati: Zigotto.

Ioan,
C.A. & Ioan, G. (2012). General Considerations on the Oligopoly.
*Acta
Universitatis Danubius. Oeconomica*,
Vol. 8, No. 3, pp. 115-137.

Spence,
M. (1984). Cost reduction, competition, and industry performance.
*Econometrica*,Vol.
52, No.1, pp. 101-121.

Van
Damme, E. & S. Hurkens (1999). Endogenous Stackelberg leadership.
*Games
and Economic Behavior*,
Vol. 28, No.1, pp. 105-129.

1 Associate Professor, PhD, “Danubius” University of Galati, Romania, Address: 3 Galati Boulevard, 800654 Galati, Romania, Tel.: +40.372.361.102, fax: +40.372.361.290, Corresponding author: catalin_angelo_ioan@univ-danubius.ro.

2 Assistant professor, PhD, “Danubius” University of Galati, Romania, Address: 3 Galati Boulevard, 800654 Galati, Romania, Tel.: +40.372.361.102, fax: +40.372.361.290, E-mail: ginaioan@univ-danubius.ro.

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