How hierarchical structures impact on competition and taxation Текст научной статьи по специальности «Математика»

Alexsandr Galegov and Andrey Garnaev

Saint Petersburg State University, Russia E-mail: agarnaev@rambler.ru

Abstract Stackelberg models for hierarchical oligopolistic markets with a homogenous product were studied by researchers extensively. The goal of this paper is to extend the classical solution in closed form of the Stackelberg model for a general hierarchical structures composed by firms arranged into groups of different hierarchical levels and to find optimal tax rate for this model.

1. Introduction

Stackelberg models for hierarchical oligopolistic markets with a homogenous product were studied by researchers extensively. Mainly two type of the models were considered. One is a hierarchical Stackelberg game in which each firm chooses its output at a stage sequentially. This is formulated as a multi-stage game. The other is a standard two stage game in which multiple leaders choose outputs simultaneously and independently at first, and multiple followers decide outputs simultaneously and independently later, given the leader’s total output.

Several researchers have tackeled to investigate the existence and uniqueness of the hierarchical Stackelberg equilibrium. Under linear demand and cost functions Boyer and Moreax (1986), and Vives (1988) showed the existence of the unique Stackelberg equilibrium of the hierarchical Stackelberg game by directly computing its solution. Robson (1990) established the existence of the Stackelberg equilibrium under general conditions of demand and cost functions. For the Stackelberg models with many leaders and followers researchers tackled questions concerning the existence and uniqueness of the Stackelberg equilibrium. In duopoly case, Okamura, Futagarni, and Ohkawa (1998) proved that there exists a unique Stackelberg equilibrium under general demand and cost functions. The convexity of the follower’s reaction function is essential for uniqueness of the Stackelberg equilibrium. In cases of a single leader and multiple followers. Sherall, Soyster and Murphy (1983) showed the existence and uniqueness of the Stackelberg equilibrium under general demand and cost functions, and also that convexity of the reaction function of the follower’s total output with respect to the leader’s output is crucial for the uniqueness of the Stackelberg equilibrium.

This paper aims to obtain generalization of closed form solution for a general hierarchical structure of firms arranged by leaderships into groups which can be modelled by multi-stage game with perfect information in which sequentially level by level multiple players (firms) of each level choose outputs simultaneously and independently, and multiple followers (firms) of the next (lower) level of hierarchical structure decide outputs simultaneously and independently later, given the players’s of the higher level their total output, and then after all these sequential setting the firms of the highest level assigns simultaneously their outputs. After this will extend received result on different tax rates and find out what tax is preferable.

It is worth to note that in the modern market a lot of hierarchical structures arise. For example, market of operation systems is split mainly between Windows (67.1%) and Linux (22.8%) meanwhile all the rest operations system takes together 10.1% of the market. So, in the operation systems markets presets three level hierarchical structure where the first and second levels are occupied by one OS (Windows and Linux) each meanwhile the third one is shared by all the rest OS. The world market of tobacco (except China) is split into four levels. The first level is shared by Altria (28%) and British American Tobacco (25%). Japan Tobacco holds the second one (16%). The third level is split among Imperial Tobacco (6%) and Altadis (3%). All the rest equal competitors share the forth level.

When one deals with such hierarchical structures as a a first approximation one could consider the produced product as a homogeneous one. Of course, products sold in both mentioned markets are differentiated. Sure, the importance of product differentiation is underscored by smokers brand loyalty in the market for tobacco products and by positive network externalities (stemming from the need of compatibility of an application software with an operating system) in the market for operating systems. But as a first and very rough approximation under very strong assumption about homogeneous nature of the products these markets could be described in frame of Cournot and Stackelberg models. When one starts studying Cournot model even for two firms presented on a market - the first two usual questions one has to answer are to find Cournot-Nash and Stackelberg equilibria and compare them (Gibbons, 1992).

The goal of this paper is to extend the classical solution in closed form of the Stackelberg model for a general hierarchical structures composed by firms arranged into groups of different hierarchical levels acting sequentially level by level and simultaneously inside of a level and to find optimal tax rate for this model.

2. Cournot model

In Cournot model of oligopoly there are M firms producing the same good. Each firm i, i e {1, ...M} has a constant marginal cost of production c*. Each firm simultaneously and independently sets the quantity q* of the good its is going to produce. Am inverse aggregate demand function of p(q) = max{A — q, 0}, where q = qi + ... + qM, is given. Then, the payoff to firm i (i e {1, ..., M}) is given as follows

M

ni(qU . ..,qM ) = (A — qj )qi — ciq*. (1)

j = 1

Then the following result is a well known (see, for example (1) and we produce it here only for convenience of readers.

Theorem 1. In the Cournot model the equilibrium strategies are given as follows

1 I M \ M

qi = mTT ( A + ^ cA forte (2)

\ j = 1,j=i )

with payoffs

1 - 2 nt = jwrw{A+c~{M + 1)Cl) ■

Aggregate output is given by

M

. , M + 1

i=1

For the case with equal production cost ci = c, i G {1, M} the equilibrium strate-

gies are given as follows

1

= ~n------(A — c)

H M +lK J

with payoff

n' = WTW-{A-cf

M M

Aggregate output is given by

EM , . .

''*= mTT

i=1

Of course, in Theorem 1 we deal only with conception of interior solution which exists under assumption that the parameters of the model are such that all the qi given by (10) are positive, namely, if the following inequalities hold:

M

A + Cj > Mci for i G {1,..., M}.

j=1,j=i

3. Tax extension of Cournout model

In this section we will consider the influence of tax systems used in Russian Federation (Vasin and Morozov, 2005) under condition that firms produce the same good. Videlicet, we will consider total revenue tax, pure profit, excise tax and VAT. In Cournout model profit function ni of firm i will be given as follows:

(a) Total revenue tax:

ni = pt I A — qj I qi — ciqi, i G {1, ..., MK (3)

\ je{1,...,M} J

where pt = 1 — Tt, and Tt - tax rate for total revenue tax.

(b) Excise tax:

ni = I A — qj — t I qi — ciqi, i G {1, ..., M}, (4)

\ je[1,M] J

where t - excise tax rate.

(c) Pure profit tax:

ni = pp I { A — qj 1 qi — ciqi 1 , i G {1, ..., M}, (5)

W je[1,M] J J

where pp = 1 — Tp, and Tp- tax rate for pure profit tax.

(d) VAT:

ni = pn I A — qj — cz I qi — ciqi, i G {1, ...,M}, (6)

\ je[1,M] )

where pn = 1 — Tn, and Tn - VAT tax rate, cz - cost of purchased products and ci

- own firms cost.

Following theorem will generalize theorem 1 on case of payment mentioned above taxes.

Theorem 2. In case of taxation the Cournot model the equilibrium strategies are given as follows

(a) Total revenue tax:

ptA + C - (M + 1)q .

« =---------MMTT)------------ A/} (7)

with profits

Total output

* _ (PtA + C - (M + 1 )cj) i_ Pt{M+lf '

A _ ptMA - C qi ~ pt(M + 1) '

(b) Excise tax:

with profits

Total output

A— t + C — (M + l)ci .

Qi =---------WTl----------- *

^ _ _ M(A-t) — C M +1 ■

i=1

(c) Pure profit tax:

with profits

r„_ Pp (A + (5 — (M +1)ci)2

n*

(M +1)2 Total output

M MA-C

E

qi

M+1

i=1

(d) VAT:

with profits

f3n(A — cz) + C — (M + l)cj .

« =---------------AIM + l)----------------- A/} (10)

Total output

{Pn{A — cz) -\- C — (M + l)cj)

/3n(M+l)2 •

_ (3nM(A — cz) — C hqi~ fUM+i) •

4. Stackelberg model

In this section we consider the strong linear hierarchical structure model Leader-Follower where the number of levels coincides with number of firms. This kind of Stackelberg model can be solved in the sense the subgame perfect Nash equilibrium. Without loss of generality we can assume that the first level leader is firm 1, the second level leader is firm 2 and so on. Thus, firm M is lowest firm in the hierarchical structure. The game is played in M stages. On the first stage firm M chooses its strategy to maximize IIm assuming that all others strategies are fixed. So, the firm sets up its strategy as a root of the equation dnM/dqM = 0 where

M

nM = qj)qM — cMqM■ (11)

j=l

Thus,

_ 11. M— ^ 1

qM ~ 2 1 qj J ~ 2°M'

So, after substituting qM into (1) for i G {1, ■■■, M — 1} we obtain that the payoff to

firm i is given as follows:

Hi = ~ ^4 — qi — (ci — — gm^ qi, i G {1, ..., M — 1}. (12)

On the second stage firm M — 1 chooses its strategy as a root of the equation

dnM—i/dqM-1 = 0. Thus,

1 ( M—2 \ 1

qM-1 = 2 I ^ — ^ I ~ 2 ~ Cm) ■

After substituting qM—1 into (12) for i G {1, ■■■, M — 2} we obtain that the payoff to firm i is given as follows:

iTj = — f A — qjj qi — — —(2cm-i + cm)| qi, * € {1, ..., M — 2}.

Thus,

1 ( M—3 \ 1

qM-2 = 2 i _ E qi I _ 2^CM_2 - ^cm-i - cm).

and so on. Then, step by step firm M — k, k G {1, ■■■M — 2} recursively sets its

strategy as a root of the equation dnM—k/dqM—k = 0. Thus,

1 / m—k—i\ 1 k-1

qM-k = 9 I ^ _ I _ o(2kcM-k ~ Y#cM-i)

\ j=1 J j=0

and payoffs on step k +1 are given as follows:

1 / m—k—1\f 1 k

= 2fc+T ( ^ _ E qi) qi~ ( Ci ~ 2cM-j ) qi, * G {1,..., M—k—1}.

So,

1 ( M—2.

qi = - I A - 2M 1c1 + ^ 2jcM-

V j=0

and so, moving backward we have that on the level i, k G {1, ■■■, M} the firm have the following optimal strategy:

1 / M—1

qi = — I A + 2■’cm-j ~ 2MCi ] , i G {1, ..., M}

Thus, we proved the following result:

Theorem 3. In the Stackelberg model the equilibrium strategies are given as follows

M — 1

I A 42»

qi — 7^1 ( ^ + — 2MCj ] , i G {1,..., M} (13)

j=0

with payoffs

1 I m—1

Hi = ^ U - 2MCi + J2 2jcM-j | for * G {1,..., M}.

Aggregate output is given by

M / 1 \ 1 M—1-J2qi= \1~2m)A~2mY 23cM-j-

i=1 V ' j=0

Of course, in Theorem 3 we deal only with conception of interior solution which exists under assumption that the parameters of the model are such that all the qi given by (13) are positive, namely, if the following inequalities hold:

M — 1

A + 2jcM—j > 2Mci for i G {1, ■■■, M}■

j=0

It is clear that a firm increases own production if production cost of its rival is increasing and it reduces own production if its own production cost arises. Namely, q.i is increasing on each Cj where j = i and q. is decreasing on each c..

For a particular case with equal production cost c. = c, i G {1,^,M} from Theorem 3 we have the following result.

Theorem 4. For the case with equal production cost c. = c, i G {1,„;M} the equilibrium strategies are given as follows

with payoffs

— 2 M+i _ c)2’ *G{1,...,M}.

Aggregate output is given by

^ qi - ( A - c) (l - ) ■

i=1

If the number of firms with equal production cost c increases then the aggregate output tends to A — c.

5. General case

As a general case we consider a hierarchical structure composed by M firms arranged into N groups of firms r1,... rN of different hierarchical level such that the groups r composes ith level and consists of M. firms. Let F. = Uj=1F, i G {1, ■■■N} and Mi = ^ij=1 M. is the number of firms which are in levels from 1 to i. Then MN = M. Also, let M0 = 0. Thus, the payoff of firm i in new notations is given as follows:

n. = IA — 53 qj I q.— ^q^ i G rN■ (14)

\ jerN J

Let start stage by stage, level by level from the level N (first stage) which is the lowest one and it is composed by firms of group rN. Since d2ni/dq2 = —2 these firms set up their strategies as a solution of the system of equations dn./dq. = 0, i G rN or

—2q. + A — 53 qj— ci = 0 i G rN ■

je Fn Vi.}

Thus,

® = m^T~i{A~ ? ' (Cj' M^Tl) ’ iefN' (15)

\ je Fn-1 )

where

Ck =Y cj , k G{1,:M }■

je rk

So, after substituting (15) into (14) for i G rN—1 we obtain that the payoff to firm i is given as follows:

ni = MaT+T (A ~ E n) « “ (16)

\ je rN-1 )

Pass on to the next level (the second stage), namely, to level N — 1 composed by firms from group rN—1. Since d2ni/dq2 = —2/(MN + 1) these firms set up their strategies as a solution of the system of equations dn./dq. = 0 , i G rN—1 where

II. are given by (16). Then

— 2qi + A — 53 qj — (MN + 1)ci + CN = 0 i G rN — 1■

je Fn-i\{i}

Thus,

* = ASTTTT I -4 “ S n

Mn—1 + 1

je rN-2 J (17)

(PN — 1ci — PN CN — 1 — CN) for i G rN — 1,

where

and

Prs = n(Mk + 1) for 1 < s < r < N

s

k=r

PS = 1 for S > r^

Thus, substituting q. from (17) into (16) we obtain the payoffs of the firms from group FN—2 given as follows:

= ~pN ( A - 53 I qi

N — 1 ' jerN-2

j— (PnCn-i + Cn) ) Qi, i G Pn—2• P NN 1

Now, let us pass on to the level M — k composed by firms of group rN—k. Since d2n.i/dq2 = —2/PN_ 1 these firms set up their strategies as a solution of the system of equations dn./dq. = 0 , i G rN—k. Thus,

1

Mn-Ie + 1

« = . 11 \A~ 53 qi

jeI^N-k-1 k

1 ' PN-kci ~YjPN-0 + 1Cn-3 I forie^

Mn—k + 1 T N—ki H N

— j=0

1

c

and

— pn ( A 53 qi I qi

PN — k' jePN-k-1

1k

) qi f°r * e -Hv-fc-1-

“ pJV N-j+l^N-j

PN — k j=o

So, for the highest (the first) level firms we have the following optimal strategy

1 ( N — 1\ PN

qi ~ Ml + 1 + E PN-j+lCN-jJ - Ml + 1Ci

and the joint goods produced by firm of the first level is

S* = Mrh('4+Sp«^|--p"c-

ie 1 1 j=1

Moving backward we have that on the level k, k G {1, ■■■N} the firms have the following optimal strategies:

1 / N — 1_\ pn

qi = J5k ( A + 53 PN-j+lCN-j\ ~ Mkk+1Ci’ i e Fk

and the joint goods produced by firm of k-th level is

£^qi - ~j^ [A + 53Pi+lC’j ) “ Pk+lCk-

ie rk 1 V j = 1

Thus, we proved the following result

Theorem 5. In the Stackelberg model with N groups of firms the equilibrium strategies are given as follows:

<n = jk ^+53Pj+iQ - piNc*). * g a (18)

with payoffs

(a + Ef=1 PJ++A — PN c.)2

n* = ~------------OT--------------->

Pk pn

Aggregate output is given by

M f 1 \ 1 N

53*= f1 _ iw) A~ pw!2pi+ici-

i=1 P1 P1 i=1

Of course, in Theorem 5 we deal only with conception of interior solution which exists under assumption that the parameters of the model are such that all the qi given by (18) are positive, namely, if the following inequalities hold:

N

A + ^pN+1Cj > PNCi for i e rk, k e{1,...,N}.

j=i

For a particular case with equal marginal cost ci = c, i e{1, ...,M} from Theorem 3 we have the following result.

Theorem 6. For the case with equal production cost ci = c, i e {1,...,M} in the Stackelberg model with N group of firms the equilibrium strategies are given as follows

Qi = -fjj: (A — c), i e A,

P1

with payoffs

TT _(^-c)2

11 j - -----j----TT— , % KZ 1 h .

i Pk PN k

1 P1

Aggregate output is given by

M f 1 \

= y-~ ~]WJ (A - c)-

6. Tax extension of Stackelberg model

In this section we consider influence of tax systems used in Russian Federation (Vasin and Morozov, 2005) on Stackelberg equilibrium. Profit functions will be the same with profit functions from section 3.

Following theorem will generalize theorem 5 on case of taxes payment.

Theorem 7. For tax extension of Stackelberg model with N groups equilibrium strategies are given as follows:

(a) Total revenue tax:

qi = JhPf \l3tA + Y P?+& ~ ) ’ ie rk

with profit

j=1

htA + ZN=1 PN+iCj - PN ci)

IIi = ---------------T-Hv---------—, *erfc.

PtP^P^ ’

Total output:

M f 1 \ 1 N

J2^={1-jf)A-^fJ2p^N+ic^

i=1 v 1 ' 11 i=1

(b) Excise tax:

with profit

Total output

qi — -pk + E 3V* _ P\ ci ) , * € Tk

A - t + EN=1 PN+1Cj - PN ci)2

n* = -------------j*pN-------------

M f 1 \ 1 N

Yqi= f1 “ ~5n ) (A _t) -

i=1 \ P1 / P1 i=1

(c) Pure profit tax:

qi= pk iA + YPJN+iCj - PiNcA , ie rk

with profit

Pp (a + Ef=1 PN+1Cj - PN ci)2 Hi = ~--------------ierk.

M f 1 \ 1 N

Yqi= f1 ~ -pw) A~

i=1 \ P1 J P1 i=1

Total output:

(d) VAT:

qi = /3 Pk ^n<yA ~ Cz+53^+!^' _ PN°i ] , i & rk

with profit

. 2

^ -n(A - cz) + E^Li Pj+iQ ~ PNc<) _

774 “ ’ *Grfc-

Total output

M / 1 N 1 N

Yqi= f1 “ ) (A~cz) - B pN YPi+ lCi'

i=1 V P1 ' PnP1 i=1

7. Conclusions

In this work we considered the hierarchical structures in general form in the frame of Cournout-Stackelberg model and constructed the optimal strategies in closed form. We can apply this closed form solutions to estimate which impact they produce on the market. As a criteria of such impact we can consider the market price joint p or the quantity of the goods (Q = A — p) produced by all the firms. Then Q is given as follows:

(a) In the case of the absent of the hierarchial structures among M firms:

(b) In the case of the linear hierarchial structure where each of the M firms occupies per level:

(c) In the general case where the hierarchical structure is composed by M firms arranged into N groups:

(d) In the general case with total revenue tax where the hierarchical structure is composed by M firms arranged into N groups:

(e) In the general case with excise tax where the hierarchical structure is composed by M firms arranged into N groups:

(f) In the general case with pure profit tax where the hierarchical structure is composed by M firms arranged into N groups:

(g) In the general case with VAT where the hierarchical structure is composed by M firms arranged into N groups:

M

Q{1},{2},...,{M }

N

Q{1,...,Mi},{Mi + 1,...,M2},...,{Mn-i + 1,...,Mn }

Q{1,...,Mi},{Mi + 1,...,M2},...,{Mn-i + 1,...,Mn }T

Q{1,...,M1},{M1 + 1,...,M2},...,{Mn-i + 1,...,Mn }P

We would like to mention that only pure profit tax does not reduce the firms output and does not increase price of the product, which is optimal for customers.

For example if there are three firms (M = 3) with marginal cost of production Ci, i = 1, 2, 3 equals 1, 2 and 3, A = 10, cz = 0.5 and t = 0.4. Also lets consider tax rates from Russian Federation: f3p =1 — 0.15 = 0.85, f3t = 1 — 0.06 = 0.94 and f3n = 1 — 0.18 = 0.82, Then, Q{i,2,3} = 6, Q{i,2},{3} = 6.833, Q{i},{2,3} = 7 and Q{1}i{2},{3} = 7.375 and the market prices are P{1j2j3} = 4, Q{1j2},{3} = 3.167, P{1},{2,3} = 3 and P{1}i{2}i{3} = 2.625. In this case we see that most preferable for customers is case with 1 firm at each hierarchy level, because in this case we have maximal total output. Lets consider taxation:: (a) Q{1}i{2}i{3} = 7.2872 and P{i},{2},{3} = 2.7128 for total revenue tax, (b) Q{i},{2},{3} = 7.375 andP{i},{2},{3} = 2.625 for pure profit tax, (c) Q{1}i{2}i{3} = 7.025 and P{1}i{2}i{3} = 2.975 for excise tax, (d) Q{ 1}j{2}j{3} = 6.6357 and P{1}i{2}i{3} = 3.3644 for vAt.

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