A Tax Game in a Cournot Duopoly. A.Galegov, A.Garnaev. Department of Computer Modelling and Multiprocessor Systems,. Faculty of Applied Mathematics and ...
A Tax Game in a Cournot Duopoly A.Galegov, A.Garnaev Department of Computer Modelling and Multiprocessor Systems, Faculty of Applied Mathematics and Control Processes, St Petersburg State University, Universitetskii prospekt 35, Peterhof, St Petersburg 198504, Russia
Abstract In many countries the tax rate depends on the tax base. In Russia in 2003, so called simpli ed tax system, which consists of two tax rates (6 and 15 percents), was introduced in order to support small business. That is why some rms face up a problem of choosing one of them: either the pure pro t tax rate (when rm pays tax from total revenue minus total cost) or the total revenue tax rate (when rm pays tax from total revenue). The total revue tax rate (6 percent) is less than the pure pro t tax rate (15 percent), because the tax base in the rst case is greater. In the environment of competition, when there are few rms producing a homogeneous good for the market, the problem of choice becomes a game-theoretical problem since every rm should take into consideration behavior of its rival. We will call this situation a tax game. The aim of this paper is to generalize the problem of choice of the tax rate as well as the criteria of this choice for the competition case. In this paper we consider two plots of this game in the Cournot settings. The rst one is a two stage game. On rst stage rms plan their production in the Cournot settings for each combination of possible tax rates, while on the second stage they decide what tax rate will be better to use. The second plot is the one stage game where the rms choose the tax rate by optimal way after setting production plan.
1
Introduction
In many countries the tax rate depends on the tax base. In Russia in 2003, so called simpli ed tax system, which consists of two tax rates (6 and 15 percents), was introduced in order to support small business. That is why some rms face up a problem of choosing one of them: either the pure pro t tax rate (when rm pays tax from total revenue minus total cost) or the total revenue tax rate (when rm pays tax from total revenue). The total revue tax rate (6 percent) is less than the pure pro t tax rate (15 percent), because the tax base in the rst case is greater. In the environment of competition, when there are few rms producing a homogeneous good for the market, the problem of choice becomes a game-theoretical problem since every rm should take into consideration behavior of its rival. We will call this situation a tax game. The aim of this paper is to generalize the problem of choice of the tax rate as well as the criteria of this choice for the competition case. In this paper we consider two plots of this game in the Cournot settings. The rst one is a two stage game. On rst stage rms plan their production in the Cournot settings for each combination of possible tax rates, while on the second stage they decide what tax rate will be better to use. So, the
two stage game will be a combination of the Cournot settings and a bimatrix game as it was done in a R&D in transport and communication game by Lambertini, Mantovani and Rossini ([1], [2]) The second plot is the one stage game where the rms choose the tax rate by optimal way after setting production plan and this plot is a modi cation of the Cournot settings for this particular scenario.
2
The Settings of the Two Stage Game
We analyse a duopoly where two rms (1 and 2) compete non-cooperatively in a two stage framework in the Cournot settings. Both rms produce a homogeneous good. Market competition takes place as a Cournot game where each rm chooses the pro t-maximizing quantity separately. On rst stage rms plan their production in the Cournot settings for each combination of possible tax rates, while on the second stage they decide what tax rate will be better to use. We resort to backward induction to get subgame perfection.
3
The First Stage of the Game: Cournot settings
In this section and in the following four subsections we study the rst stage of the tax game where the rms plan their production in the Cournot settings for each combination of possible tax rates (the total revenue and the pure pro t ones). Let qi be the quantity of the product produced by rm i where i = 1, 2 and p be the price of the product, which depends on its total quantity on the market, and it is given by the following linear model ([3]) p = A − q1 − q2 where A is the maximal possible price of the product accessible by the market. Also we assume that the marginal cost of the product for the both rms is c and A > c because of non-negativity of the marginal revenues.
3.1 Both rms choose the pure pro t tax rate In this subsection we assume that both rms choose the pure pro t tax rate. Then their pro t functions in the Cournot settings are given as follows
π1pp = βp ((A − q1 − q2 )q1 − cq1 ), π2pp = βp ((A − q1 − q2 )q2 − cq2 ), where βp = 1 − Tp and Tp is the pure pro t tax rate. Each rm maximizes its pro t taking into consideration the quantity sold on the market. From market stage rst order conditions we get the following equilibrium quantities: pp pp q∗1 = q∗2 =
A−c . 3
By substituting the equilibrium quantities into the pro t functions we obtain the equilibrium total pro ts: βp (A − c)2 pp pp π∗1 = π∗2 = . 9
3.2 Both rms choose the total revenue tax rate In this subsection we assume that both rms choose the total revenue tax rate. Then their pro t functions in the Cournot settings are given as follows
π1tt = βt (A − q1 − q2 )q1 − cq1 , π2tt = βt (A − q1 − q2 )q2 − cq2 , where βt = 1 − Tt and Tt is the total revenue tax rate. Each rm maximizes its pro t taking into consideration the quantity sold on the market. From market stage rst order conditions we get the following equilibrium quantities: tt tt q∗1 = q∗2 =
βt A − c . 3βt
By substituting the equilibrium quantities into the pro t functions we obtain the equilibrium total pro ts: (βt A − c)2 tt tt π∗1 = π∗2 = . 9βt
3.3 Firms choose the dierent tax rates In this subsection we assume that the rms choose the dierent tax rates. Say, the rm 1 chooses the pure pro t tax rate and the rm 2 chooses the total revenue tax rate. Then their pro t functions in the Cournot settings are given as follows
π1pt = βp ((A − q1 − q2 )q1 − cq1 ), π2pt = βt (A − q1 − q2 )q2 − cq2 . Each rm maximizes its pro t taking into consideration the quantity sold on the market. From market stage rst order conditions we get the following equilibrium quantities:
βt A + (1 − 2βt )c 3βt βt A + (βt − 2)c = . 3βt
pt q∗1 = pt q∗2
By substituting the equilibrium quantities into the pro t functions we obtain the equilibrium total pro ts:
βp (βt A + (1 − 2βt )c)2 , 9βt2 (βt A + (βt − 2)c)2 . = 9βt
pt π∗1 = pt π∗2
If the rm 2 chooses the pure pro t tax rate and the rm 1 chooses the total revenue tax rate. Then their pro t functions are given as follows
π1tp = βt (A − q1 − q2 )q1 − cq1 , π2tp = βp ((A − q1 − q2 )q2 − cq2 ). It has the following equilibrium quantities and corresponding total pro ts:
βt A + (βt − 2)c , 3βt βt A + (1 − 2βt )c = , 3βt (βt A + (βt − 2)c)2 = , 9βt βp (βt A + (1 − 2βt )c)2 = . 9βt2
tp q∗1 = tp q∗2 tp π∗1 tp π∗2
4
The Second Stage of the Game: choosing the tax rate
In this section we study the second stage of the game where the rms choose the tax rate to maximize their incomes. So each rm has two pure strategy: to choose the pure pro t tax rate (P ) and to choose the total revenue tax rate (T ). Thus, the second stage of the game can be described by the following bimatrix:
P T P (b11 , b11 ) (b12 , b21 ) T (b21 , b12 ) (b22 , b22 ) where
b11 = b21 = b12 = b22 =
βp (A − c)2 , 9 (βt A + (βt − 2)c)2 , 9βt βp (βt A + (1 − 2βt )c)2 , 9βt2 (βt A − c)2 . 9βt
Since the tax base for the total revenue tax is greater than the one for pure pro t to equalize approximately the tax payos in the real world tax rates Tt < Tp is assigned. So, βt > βp . For example, in Russian Federation rms can use simpli ed tax system, where Tt = 0.06 and Tp = 0.15 ([5]). So, βt = 0.94 and βp = 0.85. We will investigate our game for these particular values, thus our goal is to nd NE of the following matrix game
P T P (a11 , a11 ) (a12 , a21 ) T (a21 , a12 ) (a22 , a22 )
(1)
where
17 2 A − 180 47 2 = A − 450 17 2 = A − 180 47 2 = A − 450
a11 = a21 a12 a22
17 17 2 Ac + c, 90 180 53 2809 2 Ac + c, 225 21150 374 8228 2 Ac + c, 2115 99405 2 50 2 Ac + c. 9 423
Theorem 1. Let 2 160 1 √ ( − 7990) ≈ 1.0016, 47 3 3 2 √ 7 7990 ≈ 1.065, t2 = − 3 141 2 160 1 √ t3 = ( + 7990) ≈ 3.54, 47 3 3 7 2 √ t4 = + 7990 ≈ 3.6. 3 141 t1 =
Then (a) (P ,P ) is the NE if and only if A ∈ [t2 c, t4 c], (b) (T ,T ) is the NE if and only if A ≤ t1 c or A ≥ t3 c, (c) (T ,P ) is the NE if and only if A ∈ [t1 c, t2 c], (d) (P ,T ) is the NE if and only if A ∈ [t1 c, t2 c], (e) (P ,P ) Pareto dominates (T ,T ) if and only if A ∈ [t1∗ c, t2∗ c] where √ √ 5 7990 5 7990 t1∗ = − ≈ 1.0327, t2∗ = + ≈ 2.3006. 3 141 3 141
Proof. (a) follows from (1) and the fact that (P ,P ) is the NE if and only if a11 ≥ a21 .
(b) follows from (1) and the fact that (T ,T ) is the NE if and only if a22 ≥ a12 . (c) follows from (1) and the fact that (T ,P ) is the NE if and only if a21 ≥ a11 and a12 ≥ a22 . (d) follows from (1) and the fact that (P ,T ) is the NE if and only if a12 ≥ a22 and a21 ≥ a11 . (e) follows from (1) and the fact that (P ,P ) Pareto dominates (T ,T ) if and only if a11 ≥ a22 .
Theorem 2. The game has a mixed NE if and only if A ∈ (t1 c, t2 c) or A ∈ (t3 c, t4 c). Proof. Let rm 1 and rm 2 employ strategies x = (p, 1 − p) and y = (q, 1 − q) where
p, q ∈ (0, 1). Then
π1 (x, y) = a11 xy + a21 y(1 − x) + a12 x(1 − y) + a22 (1 − x)(1 − y).
Suppose that the strategy y of rm 2 is xed. Then, since rm 1 intends to maximize its pro t which is π1 (x, y). Let W be equal to: 1 6627A2 − 30080Ac + 23480c2 W = . 3 c(282A − 649c) Then for a xed y ∈ [0, 1] we have for q < W, 0 p = any value from [0, 1], for q = W, 1 for q > W. Similarly, since
π2 (x, y) = a11 xy + a21 x(1 − y) + a12 y(1 − x) + a22 (1 − x)(1 − y). The best reply of rm 2 for a xed strategy x of rm 1 is given by for p < W, 0 q = any value from [0, 1], for p = W, 1 for p > W. If p = W , q = W and W ∈ [0, 1] (A ∈ (t1 c, t2 c) or A ∈ (t3 c, t4 c)) we have the NE ((W, 1 − W ), (W, 1 − W )) with the payo vector (π, π), where
π=
5
34 (2209A2 − 4559Ac + 2341c2 )c . 2115 282A − 649c
The Switching Point for two stage game
In this section we consider the switching point from one tax rate to another. First consider the switching without the Cournot settings. Let T R be the total revenue, T C be the total cost. Then pro t is π = T R − T C . So we have that tax payos for both tax rates are equal if the following condition holds ([4])
0.06(T R − T C) = 0.015T R. Thus,
TC = 0.6. (2) TR This relation can be interpreted as follows: if total cost is greater than 60 percent of total revenue then the rm chooses pure pro t tax rate, and if total cost is less than 60 percent of total revenue then the rm chooses total revenue tax rate. Now consider one rm Cournot game and nd the equivalent of the condition (2) when the rm prefers to change the tax rate. Following the considered above plot we assume that rst the rm nds the optimal production plan for each combination of tax rates and
after that it chooses the optimal tax rate. Then payos, the optimal production plans and the corresponding pro ts are given as follows:
π p = βp ((A − q)q − cq), A−c q∗p = , 2 βp (A − c)2 π∗p = , 4 π t = βt (A − q)q − cq, βt A − c , q∗t = 2βt (βt A − c)2 π∗t = . 4βt Comparing π∗p and π∗t we nd that the rm will employ the total revenue tax rate if A > t2∗ c. Pass on to the two rms two stage game. First note (a) if A > t3 c then the only accessible NE for the rms is (T , T ) since for A ∈ [t3 c, t4 c] it dominates the pure NE (P , P ) and using the mixed strategy for A ∈ [t3 c, t4 c] is not reasonable because of the existence of the pure NE. (b) if A ∈ (t2 c, t3 c) then there is the only NE for the rms is (P , P ), (c) The non-negativity of equilibrium pro ts and equilibrium quantities for the total revenue tax rate implies that A > 50c/47. So there is no reason to consider the total revenue tax rate for A ≤ 50c/47. If A ∈ (50c/47, t2 c) then the situation becomes extremely uncertain and competitive with small pro t for the rms. Thus, although in the two stage game there are few NE, only two of them are accessible, namely (T , T ) and (P , P ) and the switching point is t3 c. This switching point is greater than the switching point for one rm two stage game. That means that in the environment of competition the switching point goes up and it guaranties less but more stable pro t.
6
One Step Tax Cournot Game
In this section we analyse a duopoly where the rm chooses the tax rate by optimal way after setting production plan. First consider the one rm game. After comparing pro ts for pure pro t and total revenue tax rates the rm pro t is given as follows: (a) if A > 5c/3 then 85 100 ((A − q)q − cq) for q > A − 5c/3, π(q) = 94 for q ≤ A − 5c/3, 100 (A − q)q − cq (b) if A ≤ 5c/3 then
π(q) =
85 ((A − q)q − cq). 100
The optimal quantities for pure pro t and total revenue tax rates are
A 25 A c − , q∗t = − c. 2 2 2 47 t p It is clear that q∗ < q∗ . Three cases have to be considered: q∗p =
(i) Let A − 5c/3 < q∗t < q∗p . Then A ≤ 320 c/141 and the rm chooses the pure pro t tax rate.
(ii) Let q∗t < A − 5c/3 < q∗p . Then 320 c/141 < A ≤ 7 c/3 and √
(a) if 320c/141 < A ≤ (5/3 + 7990/141)c then the rm chooses the pure pro t tax rate, √ (b) if (5/3 + 7990/141)c < A ≤ 7c/3 then the rm chooses the total revenue tax rate,
(iii) Let q∗t < q∗p < A − 5c/3. Then A > 7c/3 and the rm chooses the total revenue tax rate.
Thus, the following theorem is proved for the one rm game.
Theorem 3. In the √ one rm tax game
(a) If A ≤ (5/3 + 7990/141)c then the rm chooses the pure pro t tax rate, the optimal 17 quantity is q∗p = A2 − 2c and the corresponding pro t is 80 (A − c)2 . √ (b) If A > (5/3+ 7990/141)c then the rm chooses the total revenue tax rate, the optimal 47 quantity is q∗t = A2 − 25 c and the corresponding pro t is 200 A2 + 25 c2 − 12 Ac. 47 94 Pass on to the two rms Cournot game. Let the rms produce q1 and q2 quantity then their pro t functions are given as follows: if q2 < A − 5c/3 then 85 100 ((A − q1 − q2 )q1 − cq1 ) for q1 > A − q2 − 5c/3, π1 (q1 , q2 ) = 94 for q1 ≤ A − q2 − 5c/3, 100 (A − q1 − q2 )q1 − cq1 if q2 ≥ A − 5c/3 then
π1 (q1 , q2 ) =
85 ((A − q1 − q2 )q1 − cq1 ), 100
if q1 < A − 5c/3 then
π2 (q1 , q2 ) =
85 100 ((A − q1 − q2 )q2 − cq2 ) for q2 > A − q1 − 5c/3, 94 100 (A − q1 − q2 )q2 − cq2
for q2 ≤ A − q1 − 5c/3,
if q1 ≥ A − 5c/3 then
π2 (q1 , q2 ) =
85 ((A − q1 − q2 )q2 − cq2 ). 100
Analogously to Theorem 3 we have the following theorem.
Theorem 4. In the two rms tax game
(a) If A < 135c/47 then the rm chooses the pure pro t tax rate, the optimal quantity is p p 17 q1∗ = q2∗ = (A − c)/3 and the corresponding pro t is 180 (A − c)2 . (b) If A ≥ 135c/47 then the rm chooses the total revenue tax rate, the optimal quantity 47 50 2 t t = q2∗ = (47A − 50c)/141 and the corresponding pro t is 450 is q1∗ A2 − 29 Ac + 423 c.
7
Conclusion
First note that for one rm one stage game as well as for two stage game the switching points coincide and they are equal to t2∗ c. For two rms games the situation changes. It was shown that the switching point from pure pro t tax rate to total revenue tax rate for the two stage tax game is t3 c and for the one stage game is 135c/47 and t3 c > 135c/47. It could be explained by existence in a two stage plot some extra uncertainty in comparison with the one stage plot. This uncertainty brings in uence on the rms behavior making them agree to get less pro t in order to get more stable position on the market.
References [1] Lambertini L., Mantovani A. Price vs Quantity in a Duopoly with Technological Spillovers: A Welfare Re-Appraisal, Keio Economic Studies, Vol. 38 (2001), pp. 4152. [2] Lambertini L., Mantovani A., Rossini G. R&D in Transport and Communication in a Cournot Duopoly, Rivista Internazionale di Scienze Economiche e Commerciali, Vol. 50, issue 2 (2003), pp. 185198. [3] Petrosyan L.A., Zenkevich N.A. Game Theory. World Scienti c, London, 1996. [4] Naregnii V. Management of small business: Is it pro table to use simpli ed tax system from 2003?: Financial newspaper, N 10 (2002), pp. 1518 (in Russian). [5] Tax Code of Russian Federation, clause N 346.20 (in Russian).
