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Indirect taxation and consumer welfare in an asymmetric Stackelberg oligopoly
Accepted Manuscript Indirect Taxation and Consumer Welfare in an Asymmetric Stackelberg Oligopoly Leonard F.S. Wang, Chenhang Zeng, Qidi Zhang PII: DOI: Article Number: Reference:
S1062-9408(19)30200-1 https://doi.org/10.1016/j.najef.2019.101034 101034 ECOFIN 101034
To appear in:
North American Journal of Economics & Finance
Received Date: Accepted Date:
11 April 2019 23 July 2019
Please cite this article as: L.F.S. Wang, C. Zeng, Q. Zhang, Indirect Taxation and Consumer Welfare in an Asymmetric Stackelberg Oligopoly, North American Journal of Economics & Finance (2019), doi: https://doi.org/ 10.1016/j.najef.2019.101034
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Indirect Taxation and Consumer Welfare in an Asymmetric Stackelberg Oligopoly∗ Leonard F.S. Wang†
Chenhang Zeng‡
Qidi Zhang§
Abstract This paper studies undesirable competition in an asymmetric Stackelberg oligopoly under both unit and ad valorem taxation. We find that i) under unit taxation, a rise in the number of inefficient followers hurts consumers, but the harm to consumer welfare is less severe than that under Cournot competition; ii) under ad valorem taxation, in addition to the entry of inefficient followers, a rise in the number of efficient leaders may also hurt consumers. The harm to consumer welfare could be more severe than that under Cournot competition when the cost difference between leaders and followers is large; iii) unit taxation yields higher consumer welfare in comparison to ad valorem taxation. Our result is important for competition policy.
Key words: ad valorem tax; unit tax; consumer welfare, Stackelberg JEL Classification: D43; H21; L13
∗
We thank the editor, Hamid Beladi, and two anonymous reviewers for their constructive comments. We also thank Kazuhiko Kato, Yoshihiro Tomaru for their insightful comments and suggestions. Financial support from the Fundamental Research Funds for the Central Universities, Zhongnan University of Economics and Law (Grant No. 2722019JCT039), the National Natural Science Foundation of China (Grant no. 71773063 and Grant no. 71603078), and the key program of the National Social Science Foundation of China (Program No. 17ZDA038) are gratefully acknowledged. † Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., Wuhan, China. ‡ Correspondence to: Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., Wuhan 430073, China, cz [email protected], Telephone and Fax: +86-027-88387186. § Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., Wuhan, China.
1
Indirect Taxation and Consumer Welfare in an Asymmetric Stackelberg Oligopoly
Abstract This paper studies undesirable competition in an asymmetric Stackelberg oligopoly under both unit and ad valorem taxation. We find that i) under unit taxation, a rise in the number of inefficient followers hurts consumers, but the harm to consumer welfare is less severe than that under Cournot competition; ii) under ad valorem taxation, in addition to the entry of inefficient followers, a rise in the number of efficient leaders may also hurt consumers. The harm to consumer welfare could be more severe than that under Cournot competition when the cost difference between leaders and followers is large; iii) unit taxation yields higher consumer welfare in comparison to ad valorem taxation. Our result is important for competition policy.
Key words: ad valorem tax; unit tax; consumer welfare, Stackelberg JEL Classification: D43; H21; L13
1
1
Introduction
It is usually believed that greater competition benefits consumers and improves social welfare. Such a belief is often the foundation of policy decisions. However, as argued in the literature, greater competition, either due to cost reductions (Lahiri and Ono, 1988; and Zhao, 2001), or due to entry of a firm (Klemperer, 1988; Mukherjee et al., 2009),1 may reduce social welfare, but consumers are always better off. An important paper by Dinda and Mukherjee (2014) firstly show that a rise in the number of inefficient firms hurts consumer welfare with the consideration of strategic unit taxation, while that of efficient firms benefits consumer welfare as believed. The study of adverse effect of competition on consumers is conducted in an asymmetric Cournot oligopoly with a number of identical efficient firms and symmetric inefficient firms. It is commonly recognized that unit (or specific) taxation and ad valorem taxation may lead to different outcomes under imperfect competition. A recent paper by Wang et al. (2019) reexamine the effect of greater competition on consumer welfare with the consideration of strategic ad valorem taxation. The authors find that a rise in the number of efficient firms may unexpectedly hurt consumer welfare. However, the analyses in these two papers are conducted in the framework of Cournot oligopoly. Neither of them investigate this issue in a Stackelberg oligopoly, even though Stackelberg competition is widely observed in the real world. In this paper, we aim to better understand the competition policy associated with indirect taxation in a different market structure, i.e., Stackelberg oligopoly with efficient leaders and inefficient followers. We mainly investigate the following two issues: (i) the relationship between greater competition and consumer welfare under both unit and ad valorem taxation, and (ii) the comparison between Cournot and Stackelberg on the adverse effect of competition on consumers.2 To the best of our knowledge, this is the first paper to examine how greater competition, i.e., a rise in the number of (efficient and inefficient) firms, affects consumer welfare with the consideration of strategic taxation in a Stackelberg oligopoly. Our main results are summarized as follows. Firstly, under unit taxation, a rise in the number of inefficient followers hurts consumers, which is consistent with the main finding by Dinda and Mukherjee (2014). We further show that the harm to consumer welfare is less severe than that under Cournot competition. Secondly, under ad valorem taxation, in addition to the entry of inefficient followers, a 1 More recently, Mukherjee and Zhao (2017) find that the entry of inefficient firms improves both industry output and profit of the cost efficient incumbent. 2 The market structure is very important for policy implementation. A recent paper by Cato and Matsumura (2019) discusses entry-license tax in a free entry market, and also conducts a comparison of the Cournot and Stackelberg competition. The authors show that the optimal entry tax is decreasing (res. increasing) in the productivity of the incumbent under Stackelberg competition (res. under Cournot competition).
2
rise in the number of efficient leaders may also hurt consumers when the cost difference is large. The main reason lies in that ad valorem taxation creates two counteracting effects: output reduction effect which decreases industry output, and output shift effect which shifts output from inefficient followers to efficient leaders. When the number of efficient leaders increases, the government is more likely to be induced to shift production to efficient leaders by raising the tax rate, which therefore reduces industry output. More surprisingly, the harm to consumer welfare could be more severe than that under Cournot competition under certain conditions. Lastly, we show that unit taxation yields higher consumer welfare in comparison to ad valorem taxation. The intuition for this result is as follows. The additional output shift effect under ad valorem taxation will induce the government to raise the tax rate. This action reduces the industry output under ad valorem taxation. Our findings on the comparison between Cournot results and Stackelberg results are new in the literature, and contains meaningful implications on competition policy. Our paper is also related to a growing literature on the two common forms of commodity taxation, such as Delipalla and Keen (1992), Denicolo and Matteuzzi (2000), Wang and Zhao (2009) and Wang et al. (2018). To examine the superiority between ad valorem and unit taxation, the common approach in the literature is to assume that the two tax regimes yield equal industry output or equal tax revenue. By contrast, we calculate the optimal tax rate under each taxation and conduct the consumer welfare comparison. The rest of the paper is organized as follows. Section 2 introduces the main model. Section 3 presents our analysis and results under unit taxation. In Section 4, we show our main results with the consideration of ad valorem tax. And Section 5 concludes. We organize most of the proofs and claims in the Appendix.
2
The Model
Consider a Stackelberg oligopoly with n ≥ 1 identical leaders and m ≥ 1 symmetric followers. All leaders and followers produce a homogeneous product.3 Assume that the marginal cost of followers is c and the marginal cost of leaders is λc, where 0 ≤ λ ≤ 1, which implies that the leaders are more efficient in production.4 3
Stackelberg model is a very important model in the literature concerning imperfect competition in addition to Cournot model, which may yield different welfare implications. See also in Sherali (1984), Daughety (1990), and Ino and Matsumura (2012, 2016). 4 The assumption of efficient leaders is commonly observed in the literature, such as Ono (1978), and Mukherjee and Wang (2013). However, the opposite, λ > 1, may happen as shown by Matsumura (1997) and Hirata and Matsumura (2011). Our analysis can be easily extended to the case in which λ > 1. When leaders are less efficient in comparison to followers, a rise in the number of leaders always raises consumer welfare under both unit and ad valorem taxation.
3
The inverse market demand function is P = a − Q, where P is market price, a represents the size of market demand, and Q = Σni=1 qi + Σm j=1 qj is industry output, where qi and qj denote the output of each leader i, and each follower j, respectively. We consider the case that all firms are taxed at the same rate t per unit of output under unit taxation, and the same rate of τ fraction of gross revenue under ad valorem taxation. We propose a three-stage game with the following timing. In the first stage, the government determines a unit tax rate or an ad valorem tax rate to maximize social welfare. In the second stage, all leaders independently and simultaneously decide about their individual supply, taking the taxation set by the government as given. In the last stage, all followers decide upon their outputs independently and simultaneously after observing the total output supplied by the leaders. As usual, the game is solved by backward induction.
3
Unit Taxation
We first consider the case in which all firms are taxed at the same rate t per unit of output. In the last stage, the profit function for each follower can be written as πj = (a − Σni=1 qi − Σm j=1 qj − t − c)qj ,
j = 1, 2, . . . , m.
By solving the first-order conditions, the equilibrium output of a typical follower, denoted by qf , can be found as qf =
a − t − c − Σni=1 qi . 1+m
(1)
We assume that the market demand (represented by a) is sufficiently large such that all followers are active in production. In the second stage, each leader determines its output to maximize πi = (a − t − λc − Σni=1 qi − mqf )qi ,
i = 1, 2, . . . , n,
where qf is determined in (1). By solving the first-order conditions, the equilibrium output of a typical leader, denoted by ql , can be found as ql =
a − t + c(m − (1 + m)λ) . 1+n
(2)
Incorporating the expression of ql into (1) leads to qf =
a − t − c − nql a − t − c(1 + n(1 + m)(1 − λ)) = . 1+m (1 + m)(1 + n)
(3)
The equilibrium outputs of all firms are positive if a > t + c (1 + n(1 + m)(1 − λ)) , which 4
is assumed to hold. The industry output is obtained as Q=
a(m + n + mn) − (m + n + mn)t − c(m + nλ + mnλ) . (1 + m)(1 + n)
(4)
Lemma 1. The following properties of equilibrium quantities in the second-stage subgame hold: (i) ∂ql /∂t < 0, ∂qf /∂t < 0, and ∂Q/∂t < 0; (ii) ∂Q/∂m > 0, and ∂Q/∂n > 0. Lemma 1(i) indicates that unit taxation reduces the output for both leaders and followers. This is what we call the output reduction effect of unit taxation. Lemma 1(ii) implies that, for any given tax rate t, a rise in the number of firms (regardless of their marginal costs) increases industry output, thus making the consumers better off. In the first stage, the government decides t to maximize social welfare SW = Σni=1 πi + Σm j=1 πj + CS + T,
(5)
where CS = Q2 /2 denotes consumer welfare, and T = tQ denotes tax revenue. Simple calculations yield the equilibrium tax rate as t∗ =
c(m + (1 + m)nλ) − a(m + n + mn) . (m + n + mn)2
(6)
It follows that the equilibrium tax rate is negative, which indicates that the government always subsidizes firms to produce. By (4) and (6), we obtain the equilibrium outputs as ql∗
(1 + m) cm(m + n + mn − 1) + a(m + n + mn) − c m2 + (1 + m)2 n λ = , (m + n + mn)2
2 n2 − (1 + m)n(m + n + mn − 1)λ a(m + n + mn) − c m + m(1 + m)n + (1 + m) qf∗ = , (m + n + mn)2 Q∗ =
a(m + n + mn) − c(m + nλ + mnλ) . m + n + mn
(7)
We next examine how consumer welfare changes with a higher market competition. The following results follow straightforwardly from (7). Proposition 1. In a Stackelberg oligopoly, (i) an increase in the number of leaders, n, raises consumer welfare; but (ii) an increase in the number of followers, m, reduces consumer welfare. Proposition 1 confirms the result of undesirable competition found by Dinda and Mukherjee (2014) in an asymmetric Cournot Oligopoly, which implies that the moves (simultaneously or sequentially) of efficient and inefficient firms do not change this result. 5
We recalculate the equilibrium industry output under Cournot competition when all efficient firms have positive marginal cost λc as in our model. After the usual straightforward calculations mirroring those in Dinda and Mukherjee (2014), We obtain that Q∗c =
a(m + n) − c(m + nλ) , m+n
(8)
where the subscript “c” denotes the case of Cournot oligopoly. Next, it is very interesting to ask i) which market structure yields a higher consumer welfare? and ii) under which market structure an increase in the number of m leads to more severe harm to consumer welfare? We therefore summarize the results as follows. Proposition 2. In comparison to Cournot oligopoly, (i) the industry output is higher in a Stackelberg oligopoly; and (ii) an increase in the number of m leads to less severe harm to consumer welfare in a Stackelberg oligopoly. A unit tax imposed on each firm is equivalent to an equal increase in marginal cost, which does not cause essential distortions on the cost and objective of each firm. Proposition 2(i) confirms the predominance of Stackelberg model over Cournot model on industry output. Thus, the Stackelberg consumer welfare is greater than the Cournot consumer welfare. We have seen that, an increase in m induces the government to reduce subsidy, which therefore leads to a smaller industry output. Proposition 2(ii) indicates that the harm to consumer welfare in a Stackelberg oligopoly is less severe than that in a Cournot oligopoly. The main reason is due to the well-known output expansion effect of Stackelberg models, which weakens the negative impact of m on industry output. This result is new in the literature and thus should be noticed.
4
Ad Valorem Taxation
We next switch to the case in which all firms are taxed at the same rate of τ fraction of gross revenue. In the last stage, the profit function for followers can be written as πj = (1 − τ )(a − Σni=1 qi − Σm j=1 qj )qj − cqj ,
j = 1, 2, . . . , m.
By solving the first-order conditions, the equilibrium output of a typical follower, denoted by qf , can be found as qf =
a−
c 1−τ
− Σni=1 qi . 1+m
6
(9)
We focus on the case when all firms are active in equilibrium. The profit function for each leader in the second stage can be written as πi = (1 − τ )(a − Σni=1 qi − mqf )qi − λcqi ,
i = 1, 2, . . . , n,
where qf is determined in (9). The first-order conditions yield the equilibrium output of a typical leader as ql =
a(1 − τ ) + c(m(1 − λ) − λ) . (1 + n)(1 − τ )
(10)
As a result, we obtain that qf =
a−
c 1−τ
− nql a(1 − τ ) − c((1 + m)n(1 − λ) + 1) = . 1+m (1 + m)(1 + n)(1 − τ )
(11)
As before, we assume that a is sufficiently large such that all firms are active in equilibrium, i.e., a > c (1 − (1 + m)(1 − λ)n)/(1 − τ ). The industry output is given by Q=
a(m + n + mn)(1 − τ ) − c(m + nλ + mnλ) . (1 + m)(1 + n)(1 − τ )
(12)
Lemma 2. The following properties of equilibrium quantities in the second-stage subgame hold: (i) ∂ql /∂τ > 0 when λ < m/(1 + m) and otherwise ∂ql /∂τ ≤ 0, ∂qf /∂τ < 0, and ∂Q/∂τ < 0; (ii) ∂Q/∂m > 0, and ∂Q/∂n > 0. Similar as Wang et al. (2019), Lemma 2(i) reveals two important effects of ad valorem taxation under Stackelberg competition. On one hand, ad valorem taxation reduces the industry output. This is what we call the output reduction effect. On the other hand, when λ < m/(1 + m), ad valorem taxation penalizes inefficient firms more than efficient firms, which therefore shifts production toward efficient firms as tax increases. We call this effect output shift effect. As in Wang et al. (2019), the two effects play a key role in the following consumer welfare analysis. In the first stage, the government decides τ to maximize social welfare SW = Σni=1 πi + Σm j=1 πj + CS + T, where T = τ P Q denotes tax revenue. Simple calculations yield the equilibrium tax rate as τ
∗∗
a(m + nλ + mnλ) − c m(1 + m)2 n(1 + n)(λ − 1)2 + G = , a(m + (1 + m)nλ) − c(1 + m)(1 + n)G
(13)
where G = m + mn + m2 n − 2mnλ − 2m2 nλ + nλ2 + 2mnλ2 + m2 nλ2 . We then obtain the
7
equilibrium outputs as ql∗∗ = qf∗∗
(1 + m)(aλ(m + (1 + m)nλ) − c(m(λ − 1) + λ)X) , (m + (1 + m)nλ)2
m a − c(1 + n + mn)2 + (1 + m)n(a + 3cm(1 + n + mn))λ + Y = , (m + (1 + m)nλ)2
where X = m(1 + n + mn) − 2m(1 + m)nλ + (1 + m)2 nλ2 , Y = c(1 + m)2 nλ2 (n(m(λ − 3) + λ − 1) − 1). And the industry output is given by ∗∗
Q
a(m + nλ + mnλ) − c m2 n(−1 + λ)2 + nλ2 + m 1 + n − 2nλ + 2nλ2 = m + nλ + mnλ
. (14)
We next examine how consumer welfare changes with a higher market competition. The following results follow straightforwardly from (14). Proposition 3. In a Stackelberg oligopoly, (i) an increase in the number of leaders, n, raises consumer welfare if λ > m/(1 + m); otherwise it reduces consumer welfare; and (ii) an increase in the number of followers, m, reduces consumer welfare. In line with Wang et al. (2019) which analyze how greater competition affects consumer welfare in an asymmetric Cournot oligopoly, we find that an increase in the number of efficient leaders may also reduce consumer welfare when the cost difference between leaders and followers is large, which implies that the adverse effect of competition on consumers is irrespective of market structure. The main reasons are due to the above-mentioned effects of ad valorem taxation. When leaders are sufficiently efficient, an increase in n will induce the government to increase τ so as to shift production toward these efficient leaders (i.e., output shift effect). Then industry output decreases due to the output reduction effect, which hurts consumers. Furthermore, the industry output under Cournot competition is given by Q∗∗ c
a(m + nλ) − c m 1 + n(−1 + λ)2 + nλ2 = . m + nλ
(15)
where the subscript “c” denotes the case of Cournot oligopoly. A straightforward comparison leads to the following results. Proposition 4. In comparison to Cournot oligopoly, (i) the industry output is higher in a Stackelberg oligopoly if λ > m/(1 + m), otherwise the opposite happens; (ii) an increase in the number of m leads to less severe harms to consumer welfare in a Stackelberg oligopoly when λ > λ1 (m, n), where λ1 (m, n) solves 2n2 λ3 − m3 (1 − λ)(1 + nλ) − 2m2 nλ(1 − λ)(2 +
8
nλ) + mnλ2 (2 − n(3 − 4λ)) = 0;5 otherwise the opposite happens; (iii) when n2 < 1 + m, an increase in the number of n always leads to more severe harms to consumer welfare in a Stackelberg oligopoly; when n2 > 1 + m, an increase in the number of n leads to less severe √ harms to consumer welfare in a Stackelberg oligopoly when λ > m/(n 1 + m); otherwise the opposite happens. The result in Proposition 4(i) is in sharp contrast to that in Proposition 2(i) under unit taxation. Under ad valorem taxation, the Stackelberg model may lead to a smaller industry output than the Cournot model does. Such an inferiority of Stackelberg models can be obtained for the following reasons. On one hand, the Stackelberg model yields a higher industry output than the Cournot model without the consideration of ad valorem taxation. This is the well-known output expansion effect associated with Stackelberg models. On the other hand, ad valorem taxation creates another opposite effect for Stackelberg models due to the output shift between inefficient followers and efficient leaders. The output increase of efficient firms will be smaller under Stackelberg competition in comparison to Cournot competition as the tax rate increases. We call this negative effect tax-inducing effect of Stackelberg models. When the leaders are very efficient, the tax-inducing effect may dominate the output expansion effect, and thus leads to a smaller industry output in Stackelberg models. Proposition 4(ii) and (iii) indicate that an increase in the number of either leaders or followers may lead to more severe harm to consumer welfare under Stackelberg competition when the cost difference is large (i.e., λ is small). This result is puzzling. As the number of firms increases, the market competition intensifies with a reduced Herfindahl-Hirschman Index (HHI), which results in a higher industry output and therefore benefits consumers. However, the strategic ad valorem taxation penalizes all the firms with a large cost difference between leaders and followers, and that will mitigate the former positive competitive effect and further drive down the industry output in equilibrium. In other words, ad valorem taxation creates an anti-competitive effect in our model. Especially when the cost difference is sufficiently large (i.e., λ < λ1 (m, n) √ in part (ii) or λ < m/(n 1 + m) in part (iii)), the industry output reduction due to ad valorem taxation in Stackelberg models tends to be larger than that in Cournot models. It is very important to note that λ1 (m, n) in Proposition 4(ii) is approximately equal to one when m ≥ n.6 That is, when the number of inefficient firms is relatively large, an increase in the number of m leads to more severe harms to consumer welfare in a Stackelberg oligopoly. Implied by Proposition 4(iii), a similar result is obtained when the number of m is sufficient large, i.e., m > n2 − 1. These results indicate that a Stackelberg game with a small number 5
It is important to note that λ1 (m, n) is the unique real root of this cubic equation. The other two are imaginary roots which are thus omitted. Furthermore, we have 0 < λ1 (m, n) < 1 (see details in the proof). 6 For example, λ1 (m, n) = 0.9990 when m = 100, n = 10; and λ1 (m, n) = 0.9966 when m = 100, n = 100.
9
of followers has quite different implications from that with a large number of followers. The antitrust authorities should pay more attention to the markets with large number of followers (see also in Ino and Matsumura, 2016). Lastly, we compare the corresponding equilibrium industry outputs under unit and ad valorem taxation. A direct observation reveals that Q∗ > Q∗∗ , which implies that the consumer welfare is worse off under ad valorem taxation. Proposition 5. The consumer welfare is worse off under ad valorem taxation in comparison to unit taxation. Proposition 5 has also been investigated by Wang et al. (2019) in an asymmetric Cournot oligopoly. The industry output is higher under unit taxation in comparison to ad valorem taxation. As we seen in Section 3, unit taxation generates an output reduction effect. In equilibrium, the government always subsidizes firms to encourage production under unit taxation. However, ad valorem taxation creates two counteracting effects: output reduction effect and output shift effect. The additional output shift effect under ad valorem taxation will induce the government to raise the tax rate.7 This action reduces the industry output; consequently, the industry output under ad valorem taxation is lower compared to that under unit taxation.
5
Concluding Remarks
To the best of our knowledge, this is the first paper to examine how the entry of firms (a rise in the number of firms) affects consumer welfare with the consideration of strategic taxation in a Stackelberg oligopoly. Both unit and ad valorem taxation are considered in this paper. We find that i) under unit taxation, a rise in the number of inefficient followers hurts consumers, but the harm to consumer welfare is less severe than that under Cournot competition; ii) under ad valorem taxation, in addition to the entry of inefficient firms, a rise of the number of efficient leaders may also hurt consumers when the cost difference is large. Furthermore, the harm to consumer welfare could be more severe than that under Cournot competition; iii) unit taxation yields higher consumer welfare in comparison to ad valorem taxation. Our findings on the comparison between Cournot results and Stackelberg results are new in the literature, and contains meaningful implications for the government. A number of areas are worthwhile directions for future research based on the present model. Similar with many previous literature, the linear inverse demand function and identical cost functions are assumed for tractability. The first direction is to consider an extension with general demand and cost functions. Another direction is to carry out similar analysis under Stackelberg 7
It can be easily verified that the ad valorem tax rate can be positive in some situations.
10
competition with the introduction of foreign firms. For example, it is very interesting to see how the presence of foreign followers change the results on consumer welfare. Still a third avenue is to introduce product differentiation, as in Wang and Zhao (2009), into the model and study Bertrand competition with domestic and foreign firms.
11
Appendix: Proofs and Derivations Proof of Lemma 1: ∂qf ∂ql ∂Q 1 1 ∂t = − 1+n < 0, ∂t = − (1+m)(1+n) < 0, and ∂t a−t−c(m(λ−1)+λ) a−c−t > 0, and ∂Q > 0. ∂n = (1+m)2 (1+n) (1+m)(1+n)2
By (2)-(4), we have that 0. Furthermore,
∂Q ∂m
=
=
−m−n−mn (1+m)(1+n)
<
Proof of Proposition 1: Straightforward calculations yield that As a result, CS =
(Q∗ )2 /2
∂Q∗ ∂m
=
cn(λ−1) (m+n+mn)2
< 0, and
∂Q∗ ∂n
=
cm(1+m)(1−λ) (m+n+mn)2
> 0.
decreases with m, but increases with n.
Proof of Proposition 2: cm2 n(λ−1) (m+n)(m+n+mn) < 0, which proves part (i). ∗ cn(λ−1) ∂Q∗c cn(λ−1) we have ∂Q ∂m = (m+n+mn)2 < 0, and ∂m = (m+n)2 < 0. ∗ ∂Q∗ c that ∂Q ∂m < ∂m < 0, which implies that an increase in the number
By (7) and (8), Q∗c − Q∗ = For part (ii), straightforwardly
It follows of m leads
to less severe harm to consumer welfare in a Stackelberg oligopoly.
Proof of Lemma 2: For part (i), we obtain that
∂ql ∂τ
= − c(m(λ−1)+λ) by (10), which is positive when λ < (1+n)(τ −1)2
negative otherwise. Furthermore, by (11) and (12), we have that ∂Q ∂τ
and
=
c(m+nλ+mnλ) − (1+m)(1+n)(τ −1)2
=
< 0.
For part (ii), simple calculations yield that a(τ −1)+c(m(λ−1)+λ) (1+m)(1+n)2 (τ −1)
∂qf ∂τ
m 1+m , and c(−1+(1+m)n(λ−1)) < 0, (1+m)(1+n)(τ −1)2
∂Q ∂m
=
c+a(τ −1) (1+m)2 (1+n)(τ −1)
> 0, and
∂Q ∂n
=
> 0.
Proof of Proposition 3: For part (i), we have that
∂Q∗∗ ∂n
=
cm(1+m)(1−λ)(m(λ−1)+λ) , (m+nλ+mnλ)2
which is positive when λ >
m 1+m ,
cn(1−λ)((n(λ−1)−1)λ+2mn(λ−1)λ+m2 (λ−1)(1+nλ)) (m+nλ+mnλ)2
< 0,
and negative otherwise. For part (ii), we have that
∂Q∗∗ ∂m
=
which implies that an increase in m reduces consumer welfare.
Proof of Proposition 4: ∗∗ = By (14) and (15), we have that Q∗∗ c − Q
λ<
m 1+m ;
otherwise,
Q∗∗ c
−
Q∗∗
cm2 n(λ−1)(m(λ−1)+λ) (m+nλ)(m+(1+m)nλ) ,
=< 0. Hence, we have part (i).
12
which is positive when
cn(1−λ)((−1+n(−1+λ))λ+2mn(−1+λ)λ+m2 (−1+λ)(1+nλ)) ∂Q∗∗ < ∂m = (m+nλ+mnλ)2 ∗∗ cn(−1+n(−1+λ))(−1+λ)λ c 0, and ∂Q < 0. We next compare the two values and obtain ∂m = − (m+nλ)2 ∗∗ c 2 3 3 ∂Q ∗∗ cmn(1−λ)(2n λ −m (1−λ)(1+nλ)−2m2 n(1−λ)λ(2+nλ)+mnλ2 (2+n(−3+4λ))) ) that ∂Q . It ∂m − ∂m = (m+nλ)2 (m+(1+m)nλ)2
For part (ii), we have that
follows that an increase in the number of m leads to less severe harms to consumer welfare in a Stackelberg oligopoly when 2n2 λ3 − m3 (1 − λ)(1 + nλ) − 2m2 nλ(1 − λ)(2 + nλ) + mnλ2 (2 − n(3 − 4λ)) > 0, which reduces to λ > λ1 (m, n), where λ1 (m, n) solves g(λ) ≡ 2n2 λ3 − m3 (1 − λ)(1 + nλ) − 2m2 nλ(1 − λ)(2 + nλ) + mnλ2 (2 − n(3 − 4λ)) = 0. Note that the above equation, g(λ) = 0, only has one real root. The other two are imaginary roots which are thus omitted. Furthermore, simple calculations lead to that g(0) = −m3 < 0, and g(1) = 2n2 + mn(2 + n) > 0. Thus, we have 0 < λ1 (m, n) < 1. For part (iii), we focus on the case λ <
m 1+m
∂Q∗∗ c
in which an increase in n reduces consumer cm(1−λ)(m(λ−1)+λ) (m+nλ)2
∗∗
c welfare. Simple calculations lead to that ∂n = < 0. As a result, ∂Q ∂n − cm2 (1−λ)(m(λ−1)+λ)(n2 (1+m)λ2 −m2 ) ∂Q∗∗ = , which is positive when λ < n√m ; and negative ∂n (m+(1+m)nλ)2 (m+nλ)2 1+m
otherwise. As a result, an increase in the number of n leads to more severe harms to consumer √m ; otherwise, the opposite happens. Notice n 1+m m √ . Thus, λ < n√m automatically holds. n 1+m 1+m
welfare in a Stackelberg oligopoly when λ < that when n2 < 1 + m, it holds that
m 1+m
<
Proof of Proposition 5: By (7) and (14), we have that Q∗ − Q∗∗ =
cm(1+m)2 n(1+n)(λ−1)2 (m+n+mn)(m+(1+m)nλ)
13
> 0.
References [1] Cato, S. and T. Matsumura (2019): “Entry license tax: Stackelberg versus Cournot.” Journal of Institutional and Theoretical Economics, 175, 258-271. [2] Daughety, A.F. (1990): “Beneficial concentration.” American Economic Review, 80, 12311237. [3] Delipalla, S. and M. Keen (1992): “The comparison between ad valorem and specific taxation under imperfect competition,” Journal of Public Economics, 49, 351-367. [4] Denicolo, V. and M. Matteuzzi (2000): “Specific and ad valorem taxation in asymmetric Cournot oligopolies,” International Tax and Public Finance, 7, 335-342. [5] Dinda, S. and A. Mukherjee (2014): “A note on the adverse effect of competition on consumers,”Journal of Public Economic Theory, 16, 157-163. [6] Hirata, D. and T. Matsumura (2011) : “Price leadership in a homogeneous product market,” Journal of Economics, 104(3), 199-217. [7] Ino, H. and T. Matsumura (2012) : “How many firms should be leaders? beneficial concentration revisited,” International Economic Review, 53(4), 1323-1340. [8] Ino, H. and T. Matsumura (2016) : “Welfare-improving effect of a small number of followers in a Stackelberg model,” The B.E. Journal of Theoretical Economics, 16(1), 243-265. [9] Klemperer, P. (1988): “Welfare effects of entry into markets with switching costs,” Journal of Industrial Economics, 37, 159-165. [10] Lahiri, S. and Y. Ono (1988): “Helping minor firms reduces welfare,” Economic Journal, 98, 1199-1202. [11] Matsumura, T. (1997): “A two-stage Cournot duopoly with inventory costs,” Japanese Economic Review, 48, 81-89. [12] Mukherjee, A., U. Broll and S. Mukherjee (2009): “The welfare effects of entry: the role of the input market,”Journal of Economics, 98, 189-201. [13] Mukherjee, A. and L.F.S. Wang (2013): “Labour union, entry and consumer welfare,” Economics Letters, 120, 603-605. [14] Mukherjee, A. and L.X. Zhao (2017): “Profit raising entry,”Journal of Industrial Economics, 65, 214-219. 14
[15] Ono, Y. (1978): “The equilibrium of duopoly in a market of homogeneous goods,” Economica, 45, 287-295. [16] Sherali, H.D. (1984): “A multiple leader Stackelberg model and analysis,” Operations Research, 32, 390-404. [17] Wang A.K., P.Y. Chou and W.J. Liang (2018): “Specific versus ad valorem taxes in the presence of cost and quality differences,” International Tax and Public Finance, 25, 11971214. [18] Wang, L.F.S., C.H. Zeng and Q.D. Zhang (2019): “Indirect taxation and undesirable competition,” Economics Letters, 181, 104-106. [19] Wang, X.H. and J.G. Zhao (2009): “On the efficiency of indirect taxes in differentiated oligopolies with asymmetric costs,” Journal of Economics, 96, 223-239. [20] Zhao, J.G. (2001): “A characterization for the negative welfare effects of cost reduction in Cournot oligopoly,” International Journal of Industrial Organization, 19, 455-469.
15
S1062-9408(19)30200-1 https://doi.org/10.1016/j.najef.2019.101034 101034 ECOFIN 101034
To appear in:
North American Journal of Economics & Finance
Received Date: Accepted Date:
11 April 2019 23 July 2019
Please cite this article as: L.F.S. Wang, C. Zeng, Q. Zhang, Indirect Taxation and Consumer Welfare in an Asymmetric Stackelberg Oligopoly, North American Journal of Economics & Finance (2019), doi: https://doi.org/ 10.1016/j.najef.2019.101034
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Indirect Taxation and Consumer Welfare in an Asymmetric Stackelberg Oligopoly∗ Leonard F.S. Wang†
Chenhang Zeng‡
Qidi Zhang§
Abstract This paper studies undesirable competition in an asymmetric Stackelberg oligopoly under both unit and ad valorem taxation. We find that i) under unit taxation, a rise in the number of inefficient followers hurts consumers, but the harm to consumer welfare is less severe than that under Cournot competition; ii) under ad valorem taxation, in addition to the entry of inefficient followers, a rise in the number of efficient leaders may also hurt consumers. The harm to consumer welfare could be more severe than that under Cournot competition when the cost difference between leaders and followers is large; iii) unit taxation yields higher consumer welfare in comparison to ad valorem taxation. Our result is important for competition policy.
Key words: ad valorem tax; unit tax; consumer welfare, Stackelberg JEL Classification: D43; H21; L13
∗
We thank the editor, Hamid Beladi, and two anonymous reviewers for their constructive comments. We also thank Kazuhiko Kato, Yoshihiro Tomaru for their insightful comments and suggestions. Financial support from the Fundamental Research Funds for the Central Universities, Zhongnan University of Economics and Law (Grant No. 2722019JCT039), the National Natural Science Foundation of China (Grant no. 71773063 and Grant no. 71603078), and the key program of the National Social Science Foundation of China (Program No. 17ZDA038) are gratefully acknowledged. † Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., Wuhan, China. ‡ Correspondence to: Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., Wuhan 430073, China, cz [email protected], Telephone and Fax: +86-027-88387186. § Wenlan School of Business, Zhongnan University of Economics and Law, 182 Nanhu Ave., Wuhan, China.
1
Indirect Taxation and Consumer Welfare in an Asymmetric Stackelberg Oligopoly
Abstract This paper studies undesirable competition in an asymmetric Stackelberg oligopoly under both unit and ad valorem taxation. We find that i) under unit taxation, a rise in the number of inefficient followers hurts consumers, but the harm to consumer welfare is less severe than that under Cournot competition; ii) under ad valorem taxation, in addition to the entry of inefficient followers, a rise in the number of efficient leaders may also hurt consumers. The harm to consumer welfare could be more severe than that under Cournot competition when the cost difference between leaders and followers is large; iii) unit taxation yields higher consumer welfare in comparison to ad valorem taxation. Our result is important for competition policy.
Key words: ad valorem tax; unit tax; consumer welfare, Stackelberg JEL Classification: D43; H21; L13
1
1
Introduction
It is usually believed that greater competition benefits consumers and improves social welfare. Such a belief is often the foundation of policy decisions. However, as argued in the literature, greater competition, either due to cost reductions (Lahiri and Ono, 1988; and Zhao, 2001), or due to entry of a firm (Klemperer, 1988; Mukherjee et al., 2009),1 may reduce social welfare, but consumers are always better off. An important paper by Dinda and Mukherjee (2014) firstly show that a rise in the number of inefficient firms hurts consumer welfare with the consideration of strategic unit taxation, while that of efficient firms benefits consumer welfare as believed. The study of adverse effect of competition on consumers is conducted in an asymmetric Cournot oligopoly with a number of identical efficient firms and symmetric inefficient firms. It is commonly recognized that unit (or specific) taxation and ad valorem taxation may lead to different outcomes under imperfect competition. A recent paper by Wang et al. (2019) reexamine the effect of greater competition on consumer welfare with the consideration of strategic ad valorem taxation. The authors find that a rise in the number of efficient firms may unexpectedly hurt consumer welfare. However, the analyses in these two papers are conducted in the framework of Cournot oligopoly. Neither of them investigate this issue in a Stackelberg oligopoly, even though Stackelberg competition is widely observed in the real world. In this paper, we aim to better understand the competition policy associated with indirect taxation in a different market structure, i.e., Stackelberg oligopoly with efficient leaders and inefficient followers. We mainly investigate the following two issues: (i) the relationship between greater competition and consumer welfare under both unit and ad valorem taxation, and (ii) the comparison between Cournot and Stackelberg on the adverse effect of competition on consumers.2 To the best of our knowledge, this is the first paper to examine how greater competition, i.e., a rise in the number of (efficient and inefficient) firms, affects consumer welfare with the consideration of strategic taxation in a Stackelberg oligopoly. Our main results are summarized as follows. Firstly, under unit taxation, a rise in the number of inefficient followers hurts consumers, which is consistent with the main finding by Dinda and Mukherjee (2014). We further show that the harm to consumer welfare is less severe than that under Cournot competition. Secondly, under ad valorem taxation, in addition to the entry of inefficient followers, a 1 More recently, Mukherjee and Zhao (2017) find that the entry of inefficient firms improves both industry output and profit of the cost efficient incumbent. 2 The market structure is very important for policy implementation. A recent paper by Cato and Matsumura (2019) discusses entry-license tax in a free entry market, and also conducts a comparison of the Cournot and Stackelberg competition. The authors show that the optimal entry tax is decreasing (res. increasing) in the productivity of the incumbent under Stackelberg competition (res. under Cournot competition).
2
rise in the number of efficient leaders may also hurt consumers when the cost difference is large. The main reason lies in that ad valorem taxation creates two counteracting effects: output reduction effect which decreases industry output, and output shift effect which shifts output from inefficient followers to efficient leaders. When the number of efficient leaders increases, the government is more likely to be induced to shift production to efficient leaders by raising the tax rate, which therefore reduces industry output. More surprisingly, the harm to consumer welfare could be more severe than that under Cournot competition under certain conditions. Lastly, we show that unit taxation yields higher consumer welfare in comparison to ad valorem taxation. The intuition for this result is as follows. The additional output shift effect under ad valorem taxation will induce the government to raise the tax rate. This action reduces the industry output under ad valorem taxation. Our findings on the comparison between Cournot results and Stackelberg results are new in the literature, and contains meaningful implications on competition policy. Our paper is also related to a growing literature on the two common forms of commodity taxation, such as Delipalla and Keen (1992), Denicolo and Matteuzzi (2000), Wang and Zhao (2009) and Wang et al. (2018). To examine the superiority between ad valorem and unit taxation, the common approach in the literature is to assume that the two tax regimes yield equal industry output or equal tax revenue. By contrast, we calculate the optimal tax rate under each taxation and conduct the consumer welfare comparison. The rest of the paper is organized as follows. Section 2 introduces the main model. Section 3 presents our analysis and results under unit taxation. In Section 4, we show our main results with the consideration of ad valorem tax. And Section 5 concludes. We organize most of the proofs and claims in the Appendix.
2
The Model
Consider a Stackelberg oligopoly with n ≥ 1 identical leaders and m ≥ 1 symmetric followers. All leaders and followers produce a homogeneous product.3 Assume that the marginal cost of followers is c and the marginal cost of leaders is λc, where 0 ≤ λ ≤ 1, which implies that the leaders are more efficient in production.4 3
Stackelberg model is a very important model in the literature concerning imperfect competition in addition to Cournot model, which may yield different welfare implications. See also in Sherali (1984), Daughety (1990), and Ino and Matsumura (2012, 2016). 4 The assumption of efficient leaders is commonly observed in the literature, such as Ono (1978), and Mukherjee and Wang (2013). However, the opposite, λ > 1, may happen as shown by Matsumura (1997) and Hirata and Matsumura (2011). Our analysis can be easily extended to the case in which λ > 1. When leaders are less efficient in comparison to followers, a rise in the number of leaders always raises consumer welfare under both unit and ad valorem taxation.
3
The inverse market demand function is P = a − Q, where P is market price, a represents the size of market demand, and Q = Σni=1 qi + Σm j=1 qj is industry output, where qi and qj denote the output of each leader i, and each follower j, respectively. We consider the case that all firms are taxed at the same rate t per unit of output under unit taxation, and the same rate of τ fraction of gross revenue under ad valorem taxation. We propose a three-stage game with the following timing. In the first stage, the government determines a unit tax rate or an ad valorem tax rate to maximize social welfare. In the second stage, all leaders independently and simultaneously decide about their individual supply, taking the taxation set by the government as given. In the last stage, all followers decide upon their outputs independently and simultaneously after observing the total output supplied by the leaders. As usual, the game is solved by backward induction.
3
Unit Taxation
We first consider the case in which all firms are taxed at the same rate t per unit of output. In the last stage, the profit function for each follower can be written as πj = (a − Σni=1 qi − Σm j=1 qj − t − c)qj ,
j = 1, 2, . . . , m.
By solving the first-order conditions, the equilibrium output of a typical follower, denoted by qf , can be found as qf =
a − t − c − Σni=1 qi . 1+m
(1)
We assume that the market demand (represented by a) is sufficiently large such that all followers are active in production. In the second stage, each leader determines its output to maximize πi = (a − t − λc − Σni=1 qi − mqf )qi ,
i = 1, 2, . . . , n,
where qf is determined in (1). By solving the first-order conditions, the equilibrium output of a typical leader, denoted by ql , can be found as ql =
a − t + c(m − (1 + m)λ) . 1+n
(2)
Incorporating the expression of ql into (1) leads to qf =
a − t − c − nql a − t − c(1 + n(1 + m)(1 − λ)) = . 1+m (1 + m)(1 + n)
(3)
The equilibrium outputs of all firms are positive if a > t + c (1 + n(1 + m)(1 − λ)) , which 4
is assumed to hold. The industry output is obtained as Q=
a(m + n + mn) − (m + n + mn)t − c(m + nλ + mnλ) . (1 + m)(1 + n)
(4)
Lemma 1. The following properties of equilibrium quantities in the second-stage subgame hold: (i) ∂ql /∂t < 0, ∂qf /∂t < 0, and ∂Q/∂t < 0; (ii) ∂Q/∂m > 0, and ∂Q/∂n > 0. Lemma 1(i) indicates that unit taxation reduces the output for both leaders and followers. This is what we call the output reduction effect of unit taxation. Lemma 1(ii) implies that, for any given tax rate t, a rise in the number of firms (regardless of their marginal costs) increases industry output, thus making the consumers better off. In the first stage, the government decides t to maximize social welfare SW = Σni=1 πi + Σm j=1 πj + CS + T,
(5)
where CS = Q2 /2 denotes consumer welfare, and T = tQ denotes tax revenue. Simple calculations yield the equilibrium tax rate as t∗ =
c(m + (1 + m)nλ) − a(m + n + mn) . (m + n + mn)2
(6)
It follows that the equilibrium tax rate is negative, which indicates that the government always subsidizes firms to produce. By (4) and (6), we obtain the equilibrium outputs as ql∗
(1 + m) cm(m + n + mn − 1) + a(m + n + mn) − c m2 + (1 + m)2 n λ = , (m + n + mn)2
2 n2 − (1 + m)n(m + n + mn − 1)λ a(m + n + mn) − c m + m(1 + m)n + (1 + m) qf∗ = , (m + n + mn)2 Q∗ =
a(m + n + mn) − c(m + nλ + mnλ) . m + n + mn
(7)
We next examine how consumer welfare changes with a higher market competition. The following results follow straightforwardly from (7). Proposition 1. In a Stackelberg oligopoly, (i) an increase in the number of leaders, n, raises consumer welfare; but (ii) an increase in the number of followers, m, reduces consumer welfare. Proposition 1 confirms the result of undesirable competition found by Dinda and Mukherjee (2014) in an asymmetric Cournot Oligopoly, which implies that the moves (simultaneously or sequentially) of efficient and inefficient firms do not change this result. 5
We recalculate the equilibrium industry output under Cournot competition when all efficient firms have positive marginal cost λc as in our model. After the usual straightforward calculations mirroring those in Dinda and Mukherjee (2014), We obtain that Q∗c =
a(m + n) − c(m + nλ) , m+n
(8)
where the subscript “c” denotes the case of Cournot oligopoly. Next, it is very interesting to ask i) which market structure yields a higher consumer welfare? and ii) under which market structure an increase in the number of m leads to more severe harm to consumer welfare? We therefore summarize the results as follows. Proposition 2. In comparison to Cournot oligopoly, (i) the industry output is higher in a Stackelberg oligopoly; and (ii) an increase in the number of m leads to less severe harm to consumer welfare in a Stackelberg oligopoly. A unit tax imposed on each firm is equivalent to an equal increase in marginal cost, which does not cause essential distortions on the cost and objective of each firm. Proposition 2(i) confirms the predominance of Stackelberg model over Cournot model on industry output. Thus, the Stackelberg consumer welfare is greater than the Cournot consumer welfare. We have seen that, an increase in m induces the government to reduce subsidy, which therefore leads to a smaller industry output. Proposition 2(ii) indicates that the harm to consumer welfare in a Stackelberg oligopoly is less severe than that in a Cournot oligopoly. The main reason is due to the well-known output expansion effect of Stackelberg models, which weakens the negative impact of m on industry output. This result is new in the literature and thus should be noticed.
4
Ad Valorem Taxation
We next switch to the case in which all firms are taxed at the same rate of τ fraction of gross revenue. In the last stage, the profit function for followers can be written as πj = (1 − τ )(a − Σni=1 qi − Σm j=1 qj )qj − cqj ,
j = 1, 2, . . . , m.
By solving the first-order conditions, the equilibrium output of a typical follower, denoted by qf , can be found as qf =
a−
c 1−τ
− Σni=1 qi . 1+m
6
(9)
We focus on the case when all firms are active in equilibrium. The profit function for each leader in the second stage can be written as πi = (1 − τ )(a − Σni=1 qi − mqf )qi − λcqi ,
i = 1, 2, . . . , n,
where qf is determined in (9). The first-order conditions yield the equilibrium output of a typical leader as ql =
a(1 − τ ) + c(m(1 − λ) − λ) . (1 + n)(1 − τ )
(10)
As a result, we obtain that qf =
a−
c 1−τ
− nql a(1 − τ ) − c((1 + m)n(1 − λ) + 1) = . 1+m (1 + m)(1 + n)(1 − τ )
(11)
As before, we assume that a is sufficiently large such that all firms are active in equilibrium, i.e., a > c (1 − (1 + m)(1 − λ)n)/(1 − τ ). The industry output is given by Q=
a(m + n + mn)(1 − τ ) − c(m + nλ + mnλ) . (1 + m)(1 + n)(1 − τ )
(12)
Lemma 2. The following properties of equilibrium quantities in the second-stage subgame hold: (i) ∂ql /∂τ > 0 when λ < m/(1 + m) and otherwise ∂ql /∂τ ≤ 0, ∂qf /∂τ < 0, and ∂Q/∂τ < 0; (ii) ∂Q/∂m > 0, and ∂Q/∂n > 0. Similar as Wang et al. (2019), Lemma 2(i) reveals two important effects of ad valorem taxation under Stackelberg competition. On one hand, ad valorem taxation reduces the industry output. This is what we call the output reduction effect. On the other hand, when λ < m/(1 + m), ad valorem taxation penalizes inefficient firms more than efficient firms, which therefore shifts production toward efficient firms as tax increases. We call this effect output shift effect. As in Wang et al. (2019), the two effects play a key role in the following consumer welfare analysis. In the first stage, the government decides τ to maximize social welfare SW = Σni=1 πi + Σm j=1 πj + CS + T, where T = τ P Q denotes tax revenue. Simple calculations yield the equilibrium tax rate as τ
∗∗
a(m + nλ + mnλ) − c m(1 + m)2 n(1 + n)(λ − 1)2 + G = , a(m + (1 + m)nλ) − c(1 + m)(1 + n)G
(13)
where G = m + mn + m2 n − 2mnλ − 2m2 nλ + nλ2 + 2mnλ2 + m2 nλ2 . We then obtain the
7
equilibrium outputs as ql∗∗ = qf∗∗
(1 + m)(aλ(m + (1 + m)nλ) − c(m(λ − 1) + λ)X) , (m + (1 + m)nλ)2
m a − c(1 + n + mn)2 + (1 + m)n(a + 3cm(1 + n + mn))λ + Y = , (m + (1 + m)nλ)2
where X = m(1 + n + mn) − 2m(1 + m)nλ + (1 + m)2 nλ2 , Y = c(1 + m)2 nλ2 (n(m(λ − 3) + λ − 1) − 1). And the industry output is given by ∗∗
Q
a(m + nλ + mnλ) − c m2 n(−1 + λ)2 + nλ2 + m 1 + n − 2nλ + 2nλ2 = m + nλ + mnλ
. (14)
We next examine how consumer welfare changes with a higher market competition. The following results follow straightforwardly from (14). Proposition 3. In a Stackelberg oligopoly, (i) an increase in the number of leaders, n, raises consumer welfare if λ > m/(1 + m); otherwise it reduces consumer welfare; and (ii) an increase in the number of followers, m, reduces consumer welfare. In line with Wang et al. (2019) which analyze how greater competition affects consumer welfare in an asymmetric Cournot oligopoly, we find that an increase in the number of efficient leaders may also reduce consumer welfare when the cost difference between leaders and followers is large, which implies that the adverse effect of competition on consumers is irrespective of market structure. The main reasons are due to the above-mentioned effects of ad valorem taxation. When leaders are sufficiently efficient, an increase in n will induce the government to increase τ so as to shift production toward these efficient leaders (i.e., output shift effect). Then industry output decreases due to the output reduction effect, which hurts consumers. Furthermore, the industry output under Cournot competition is given by Q∗∗ c
a(m + nλ) − c m 1 + n(−1 + λ)2 + nλ2 = . m + nλ
(15)
where the subscript “c” denotes the case of Cournot oligopoly. A straightforward comparison leads to the following results. Proposition 4. In comparison to Cournot oligopoly, (i) the industry output is higher in a Stackelberg oligopoly if λ > m/(1 + m), otherwise the opposite happens; (ii) an increase in the number of m leads to less severe harms to consumer welfare in a Stackelberg oligopoly when λ > λ1 (m, n), where λ1 (m, n) solves 2n2 λ3 − m3 (1 − λ)(1 + nλ) − 2m2 nλ(1 − λ)(2 +
8
nλ) + mnλ2 (2 − n(3 − 4λ)) = 0;5 otherwise the opposite happens; (iii) when n2 < 1 + m, an increase in the number of n always leads to more severe harms to consumer welfare in a Stackelberg oligopoly; when n2 > 1 + m, an increase in the number of n leads to less severe √ harms to consumer welfare in a Stackelberg oligopoly when λ > m/(n 1 + m); otherwise the opposite happens. The result in Proposition 4(i) is in sharp contrast to that in Proposition 2(i) under unit taxation. Under ad valorem taxation, the Stackelberg model may lead to a smaller industry output than the Cournot model does. Such an inferiority of Stackelberg models can be obtained for the following reasons. On one hand, the Stackelberg model yields a higher industry output than the Cournot model without the consideration of ad valorem taxation. This is the well-known output expansion effect associated with Stackelberg models. On the other hand, ad valorem taxation creates another opposite effect for Stackelberg models due to the output shift between inefficient followers and efficient leaders. The output increase of efficient firms will be smaller under Stackelberg competition in comparison to Cournot competition as the tax rate increases. We call this negative effect tax-inducing effect of Stackelberg models. When the leaders are very efficient, the tax-inducing effect may dominate the output expansion effect, and thus leads to a smaller industry output in Stackelberg models. Proposition 4(ii) and (iii) indicate that an increase in the number of either leaders or followers may lead to more severe harm to consumer welfare under Stackelberg competition when the cost difference is large (i.e., λ is small). This result is puzzling. As the number of firms increases, the market competition intensifies with a reduced Herfindahl-Hirschman Index (HHI), which results in a higher industry output and therefore benefits consumers. However, the strategic ad valorem taxation penalizes all the firms with a large cost difference between leaders and followers, and that will mitigate the former positive competitive effect and further drive down the industry output in equilibrium. In other words, ad valorem taxation creates an anti-competitive effect in our model. Especially when the cost difference is sufficiently large (i.e., λ < λ1 (m, n) √ in part (ii) or λ < m/(n 1 + m) in part (iii)), the industry output reduction due to ad valorem taxation in Stackelberg models tends to be larger than that in Cournot models. It is very important to note that λ1 (m, n) in Proposition 4(ii) is approximately equal to one when m ≥ n.6 That is, when the number of inefficient firms is relatively large, an increase in the number of m leads to more severe harms to consumer welfare in a Stackelberg oligopoly. Implied by Proposition 4(iii), a similar result is obtained when the number of m is sufficient large, i.e., m > n2 − 1. These results indicate that a Stackelberg game with a small number 5
It is important to note that λ1 (m, n) is the unique real root of this cubic equation. The other two are imaginary roots which are thus omitted. Furthermore, we have 0 < λ1 (m, n) < 1 (see details in the proof). 6 For example, λ1 (m, n) = 0.9990 when m = 100, n = 10; and λ1 (m, n) = 0.9966 when m = 100, n = 100.
9
of followers has quite different implications from that with a large number of followers. The antitrust authorities should pay more attention to the markets with large number of followers (see also in Ino and Matsumura, 2016). Lastly, we compare the corresponding equilibrium industry outputs under unit and ad valorem taxation. A direct observation reveals that Q∗ > Q∗∗ , which implies that the consumer welfare is worse off under ad valorem taxation. Proposition 5. The consumer welfare is worse off under ad valorem taxation in comparison to unit taxation. Proposition 5 has also been investigated by Wang et al. (2019) in an asymmetric Cournot oligopoly. The industry output is higher under unit taxation in comparison to ad valorem taxation. As we seen in Section 3, unit taxation generates an output reduction effect. In equilibrium, the government always subsidizes firms to encourage production under unit taxation. However, ad valorem taxation creates two counteracting effects: output reduction effect and output shift effect. The additional output shift effect under ad valorem taxation will induce the government to raise the tax rate.7 This action reduces the industry output; consequently, the industry output under ad valorem taxation is lower compared to that under unit taxation.
5
Concluding Remarks
To the best of our knowledge, this is the first paper to examine how the entry of firms (a rise in the number of firms) affects consumer welfare with the consideration of strategic taxation in a Stackelberg oligopoly. Both unit and ad valorem taxation are considered in this paper. We find that i) under unit taxation, a rise in the number of inefficient followers hurts consumers, but the harm to consumer welfare is less severe than that under Cournot competition; ii) under ad valorem taxation, in addition to the entry of inefficient firms, a rise of the number of efficient leaders may also hurt consumers when the cost difference is large. Furthermore, the harm to consumer welfare could be more severe than that under Cournot competition; iii) unit taxation yields higher consumer welfare in comparison to ad valorem taxation. Our findings on the comparison between Cournot results and Stackelberg results are new in the literature, and contains meaningful implications for the government. A number of areas are worthwhile directions for future research based on the present model. Similar with many previous literature, the linear inverse demand function and identical cost functions are assumed for tractability. The first direction is to consider an extension with general demand and cost functions. Another direction is to carry out similar analysis under Stackelberg 7
It can be easily verified that the ad valorem tax rate can be positive in some situations.
10
competition with the introduction of foreign firms. For example, it is very interesting to see how the presence of foreign followers change the results on consumer welfare. Still a third avenue is to introduce product differentiation, as in Wang and Zhao (2009), into the model and study Bertrand competition with domestic and foreign firms.
11
Appendix: Proofs and Derivations Proof of Lemma 1: ∂qf ∂ql ∂Q 1 1 ∂t = − 1+n < 0, ∂t = − (1+m)(1+n) < 0, and ∂t a−t−c(m(λ−1)+λ) a−c−t > 0, and ∂Q > 0. ∂n = (1+m)2 (1+n) (1+m)(1+n)2
By (2)-(4), we have that 0. Furthermore,
∂Q ∂m
=
=
−m−n−mn (1+m)(1+n)
<
Proof of Proposition 1: Straightforward calculations yield that As a result, CS =
(Q∗ )2 /2
∂Q∗ ∂m
=
cn(λ−1) (m+n+mn)2
< 0, and
∂Q∗ ∂n
=
cm(1+m)(1−λ) (m+n+mn)2
> 0.
decreases with m, but increases with n.
Proof of Proposition 2: cm2 n(λ−1) (m+n)(m+n+mn) < 0, which proves part (i). ∗ cn(λ−1) ∂Q∗c cn(λ−1) we have ∂Q ∂m = (m+n+mn)2 < 0, and ∂m = (m+n)2 < 0. ∗ ∂Q∗ c that ∂Q ∂m < ∂m < 0, which implies that an increase in the number
By (7) and (8), Q∗c − Q∗ = For part (ii), straightforwardly
It follows of m leads
to less severe harm to consumer welfare in a Stackelberg oligopoly.
Proof of Lemma 2: For part (i), we obtain that
∂ql ∂τ
= − c(m(λ−1)+λ) by (10), which is positive when λ < (1+n)(τ −1)2
negative otherwise. Furthermore, by (11) and (12), we have that ∂Q ∂τ
and
=
c(m+nλ+mnλ) − (1+m)(1+n)(τ −1)2
=
< 0.
For part (ii), simple calculations yield that a(τ −1)+c(m(λ−1)+λ) (1+m)(1+n)2 (τ −1)
∂qf ∂τ
m 1+m , and c(−1+(1+m)n(λ−1)) < 0, (1+m)(1+n)(τ −1)2
∂Q ∂m
=
c+a(τ −1) (1+m)2 (1+n)(τ −1)
> 0, and
∂Q ∂n
=
> 0.
Proof of Proposition 3: For part (i), we have that
∂Q∗∗ ∂n
=
cm(1+m)(1−λ)(m(λ−1)+λ) , (m+nλ+mnλ)2
which is positive when λ >
m 1+m ,
cn(1−λ)((n(λ−1)−1)λ+2mn(λ−1)λ+m2 (λ−1)(1+nλ)) (m+nλ+mnλ)2
< 0,
and negative otherwise. For part (ii), we have that
∂Q∗∗ ∂m
=
which implies that an increase in m reduces consumer welfare.
Proof of Proposition 4: ∗∗ = By (14) and (15), we have that Q∗∗ c − Q
λ<
m 1+m ;
otherwise,
Q∗∗ c
−
Q∗∗
cm2 n(λ−1)(m(λ−1)+λ) (m+nλ)(m+(1+m)nλ) ,
=< 0. Hence, we have part (i).
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which is positive when
cn(1−λ)((−1+n(−1+λ))λ+2mn(−1+λ)λ+m2 (−1+λ)(1+nλ)) ∂Q∗∗ < ∂m = (m+nλ+mnλ)2 ∗∗ cn(−1+n(−1+λ))(−1+λ)λ c 0, and ∂Q < 0. We next compare the two values and obtain ∂m = − (m+nλ)2 ∗∗ c 2 3 3 ∂Q ∗∗ cmn(1−λ)(2n λ −m (1−λ)(1+nλ)−2m2 n(1−λ)λ(2+nλ)+mnλ2 (2+n(−3+4λ))) ) that ∂Q . It ∂m − ∂m = (m+nλ)2 (m+(1+m)nλ)2
For part (ii), we have that
follows that an increase in the number of m leads to less severe harms to consumer welfare in a Stackelberg oligopoly when 2n2 λ3 − m3 (1 − λ)(1 + nλ) − 2m2 nλ(1 − λ)(2 + nλ) + mnλ2 (2 − n(3 − 4λ)) > 0, which reduces to λ > λ1 (m, n), where λ1 (m, n) solves g(λ) ≡ 2n2 λ3 − m3 (1 − λ)(1 + nλ) − 2m2 nλ(1 − λ)(2 + nλ) + mnλ2 (2 − n(3 − 4λ)) = 0. Note that the above equation, g(λ) = 0, only has one real root. The other two are imaginary roots which are thus omitted. Furthermore, simple calculations lead to that g(0) = −m3 < 0, and g(1) = 2n2 + mn(2 + n) > 0. Thus, we have 0 < λ1 (m, n) < 1. For part (iii), we focus on the case λ <
m 1+m
∂Q∗∗ c
in which an increase in n reduces consumer cm(1−λ)(m(λ−1)+λ) (m+nλ)2
∗∗
c welfare. Simple calculations lead to that ∂n = < 0. As a result, ∂Q ∂n − cm2 (1−λ)(m(λ−1)+λ)(n2 (1+m)λ2 −m2 ) ∂Q∗∗ = , which is positive when λ < n√m ; and negative ∂n (m+(1+m)nλ)2 (m+nλ)2 1+m
otherwise. As a result, an increase in the number of n leads to more severe harms to consumer √m ; otherwise, the opposite happens. Notice n 1+m m √ . Thus, λ < n√m automatically holds. n 1+m 1+m
welfare in a Stackelberg oligopoly when λ < that when n2 < 1 + m, it holds that
m 1+m
<
Proof of Proposition 5: By (7) and (14), we have that Q∗ − Q∗∗ =
cm(1+m)2 n(1+n)(λ−1)2 (m+n+mn)(m+(1+m)nλ)
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> 0.
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