Unit Tax versus Ad Valorem Tax: A Tax Competition Model with Cross-border Shopping∗
Hiroshi Aiura Oita University [email protected]
Hikaru Ogawa Nagoya University [email protected]
Abstract Within the framework of spatial tax competition with cross-border shopping, we examine the choice of tax method between ad valorem tax and unit (specific) tax. The paper shows that governments endoge- nously choose ad valorem tax not because of a classic welfare reason, but because it is a good strategy in competing for mobile customers. Another key finding is that while governments are committed to the ad valorem tax method, the choice is not efficient; Tax-cutting compe- tition becomes more serious when countries adopt ad valorem tax, and competition in ad valorem tax yields smaller payoffs than competition in unit tax.
Keywords: spatial tax competition, cross-border shopping, unit (specific) tax, ad valorem tax.
JEL Classification Number. H21,H77,L13,R12
∗ Research support from JSPS (no.22330095) is gratefully acknowledged.
1 1 2 3 4 5 6 7 8 9 1 Introduction 10 11 12 More than 50 years have passed since economists formally compared the 13 effects of ad valorem tax and unit (specific) commodity tax on a non- 14 competitive market. Suits and Musgrave’s (1953) path-breaking study pre- 15 sented a formal comparison in monopoly analysis to show that ad valorem 16 tax is welfare superior to unit tax. Prior to their analysis, Cournot (1838, 17 18 1960) had already found that the two tax methods needed to be treated 19 differently in analyzing an imperfect market, and Wicksell (1896, 1959) af- 20 firmed the analogous argument of Suits and Musgrave: for any given tax 21 revenue, prices in a monopoly market will be lower with ad valorem tax, 22 indicating that ad valorem tax causes less distortion and is welfare superior 23 1 24 to unit tax. Researchers have reexamined and substantiated this argument 25 through various intensive studies. Several studies have focused on the effects 26 of unit and ad valorem taxes on the equilibrium characteristics in oligopoly 27 and monopolistic competition, and most of them confirm that the Suits and 28 Musgrave argument still holds, while some indicates the possibility of having 29 a counterview.2 30 31 The purpose of this paper is to provide further insights into this classic 32 argument in a two-country framework. In most of the literature examining 33 the effects of ad valorem and unit commodity taxes, comparisons are drawn 34 within a single country framework, in which consumers are forced to buy a 35 domestic product no matter how high the prices and taxes are. Instead, in 36 37 this paper, we explore the choice of tax method in consideration of cross- 38 border shopping in a two-country model. We argue that each country has 39 incentives to adopt the ad valorem tax method not because of a classic 40 welfare reason, but because ad valorem tax is superior to unit tax from 41 the angle of attracting cross-border consumers. In fact, we first show that 42 43 governments choose the ad valorem tax method as a dominant strategy 44 to compete for mobile consumers. Then, we show further fact that the 45 superiority of ad valorem in attracting cross-border consumers becomes at 46 the root of significant tax-cutting competition. In other words, tax-cutting 47 competition is more severe when a country is committed to ad valorem 48 49 tax. Therefore, competition in ad valorem tax yields lower tax revenue than 50 competition in unit tax. This result suggests a critical implication that 51 governments are plunged into a prisoner’s dilemma when choosing their tax 52 1 53 A recent study by Carbonnier (2011) tests this theoretical hypothesis and finds that 54 in the alcoholic beverages market in France, the shifting of prices of per unit excise taxes was significantly larger than the shifting of ad valorem VAT. 55 2 56 See Keen (1998) for a general review. 57 58 59 2 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 method, and also that the tax method used for commodity tax competition 10 3 11 is different from the one used in capital tax competition. Accordingly, the 12 endogenous choice of tax method, between ad valorem tax and unit tax, in a 13 commodity tax competition framework is one of the distinctive contributions 14 of this paper. We aim to contribute to the consolidation of the issues on 15 governments’ choices of tax methods and tax competition. 16 17 Many studies on spatial commodity tax competition, which have been 18 carried out since the established work of Kanbur and Keen (1993), have 19 made their presence felt in society.4 For instance, if we look at the reduc- 20 tion of shipping and transportation costs, as a simple example, cross-border 21 activities have strong effects compared to earlier less open economies. Fur- 22 23 thermore, political efforts to create a broad economic union also reflect the 24 importance of cross-border shopping, and globalization has enabled firms to 25 procure funds and buy materials from the world over. The development of 26 global markets has significantly increased the role of cross-border shopping 27 by consumers and cross-border material procurement by producers. Our 28 29 model of spatial tax competition in a two-country framework give a de- 30 scription of actual choice of tax method, and the insights from our analysis 31 suggest the possible inefficiency of ad valorem tax method in the integrated 32 market. 33 The paper is organized as follows. Section 2 introduces the basic model. 34 35 The choice of tax rates is examined in Section 3. We derive the main propo- 36 sitions on choosing tax methods in Section 4. Section 5 presents the discus- 37 sion of the model, which is extended to include the government’s alternative 38 objectives and the market structure. Finally, Section 6 concludes the paper. 39 3 40 The choice between ad valorem tax and unit tax in a capital tax competition model 41 was examined in Akai et al. (2011) and Lockwood (2004), which show that unit tax 42 competition is superior to ad valorem tax competition, and selecting the unit tax method 43 is a dominant strategy for governments. The choice of tax method was also examined in a tariff war model by Jorgensen and Schr¨oder (2005) and Lockwood and Wong (2000). 44 4 45 See Braid (1993, 2000), Cremer and Ghavari (2000), Lucus (2004), Nielsen (2001, 46 2002), Ohsawa (1999, 2003), Ohsawa and Koshizuka (2003), and Wang (1999). See also et al. 47 Leal (2010) for an extensive summary of tax competition analyses of cross-border 48 shopping. Non-spatial models of commodity tax competition such as de Crombrugghe 49 and Tulkens (1990), Haufler (1996), Kimberley (1999), Lockwood (1993), Lucas (2004), 50 Mintz and Tulkens (1986), Moriconi and Sato (2009), Scharf (1999), and Trandel (1994) 51 are similar to this paper in their tax competition structure, but contrast with this paper in that they do not have a stake in the endogenous choice of tax method. 52 53 54 55 56 57 58 59 3 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 2 Model 10 11 12 Our simple Hotelling economy, described in Figure 1, consists of two sym- 13 metric countries, i = 1, 2. The location space of the economy is given by 14 θ [ 1/2, 1/2], divided into two countries at θ = 0, the length of each ∈ − 15 country is therefore 1/2. In each country, there is a single private firm at 16 both ends, x1 = 1/2, x2 = 1/2, where x1 and x2 are the location points 17 5− 18 of firms 1 and 2. The firms are fixed at their locations, and they sell their 19 products at price pi. 20 21 Consumers. Consumers are endowed with a utility function separable in 22 money and the utility derived from a given product, and each one is required 23 24 to buy one product unit. They differ with respect to location and are uni- 25 formly distributed along a unit interval. We represent a consumer’s location 26 as y [ 1/2, 1/2]. The utility of a consumer located at y, having money m, ∈ − 27 and buying a product sold by firm 1 is given by u1 = v + m p1 τ(y x1), 28 − − − where v stands for the utility of the product and τ(y x1) the transportation 29 − cost (τ > 0).6 In a similar way, the utility of a consumer buying a product 30 sold by firm 2 can be given by u = v + m p τ(x y). 31 2 − 2 − 2 − 32 Utility maximization means picking the minimum of p + τ(y x ) and 1 − 1 33 p + τ(x y). As x = 1/2 and x = 1/2, the utility that a consumer 2 2 − 1 − 2 34 residing aty ˆ derives from buying a product of either of the two firms is the 35 same, wherey ˆ = 0.5(p p )/τ. Hence, the demand function that firm i 36 2 − 1 37 has to meet, Di, can be expressed as 38 39 1 p2 p1 1 p2 p1 40 D1(p1,p2)= + − and D2(p1,p2)= − . (1) 41 2 2τ 2 − 2τ 42 43 Governments. In each country, there is a single revenue-maximizing govern- 44 ment, which raises revenue only through commodity taxes; the government 45 7 46 can choose either the unit tax or ad valorem tax method. If the govern- 47 5As long as the location is symmetric and exogenous, the assumption that the firms 48 locate at both ends is not crucial. For instance, if the firms locate at the center of each 49 country, x1 = −1/4 and x2 = 1/4, the firms would simply compete for customers locating 50 at somewhat short intervals [−1/4, 1/4]. 51 6A linear transport cost makes the analysis tractable, which is familiar in the literature 52 of spatial tax competition. See Kanbur and Keen (1993), Ohsawa (1999), and Wang 53 (1999), among others. 54 7Following Kanbur and Keen (1993), Ohsawa (1999), and Wang (1999), we begin by 55 describing the government’s objective in its simplest form, deferring discussions on gener- 56 alizations until later. 57 58 59 4 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 ment adopts the unit tax method, taxes will be imposed on the number units 10 11 sold, and if it selects the ad valorem tax method, taxes will be imposed on 12 the amount of sales. If the government in country i employs the unit tax 13 method, the tax revenue will be obtained by 14 15 R = T D , (2) 16 i i i 17 where Ti denotes the unit tax rate. On the other hand, if the government 18 imposes ad valorem tax, the tax revenue of country i will become 19 20 21 Ri = tipiDi, (3) 22 where t (< 1) denotes the ad valorem tax rate. 23 i 24 With two countries, the governments can employ four possible tax method 25 combinations: in case (i), both the countries will compete for mobile con- 26 sumers in ad valorem tax; in case (ii), both the countries will compete in 27 unit tax; in case (iii), country 1 will compete in unit tax and country 2 will 28 compete in ad valorem tax; and in case (iv), country 1 will compete in ad 29 30 valorem tax and country 2 will compete in unit tax. In Section 3, we analyze 31 the four cases by one and clarify the possible equilibria properties. 32 33 Firms. Each firm tries to maximize its profits, indicated by 34 35 36 (pi c)Di TiDi when country i employs the unit tax method, 37 πi = − − , (pi c)Di tipiDi when country i employs the ad valorem tax method, 38 − − 39 and where c> 0 is the constant unit cost of the product. 40 41 42 3 Choice of tax rates: Second-stage outcome 43 44 45 In this section, we consider a simple three-stage tax competition model and 46 address the issue of governments choosing their tax method endogenously. 47 In our three-stage game model, the governments choose either the unit tax 48 or ad valorem tax method in the first stage. In the second stage, they choose 49 the tax rate of the tax instrument selected in the previous stage. In the final 50 stage, the firms choose their price. 51 52 53 3.1 Unit tax competition by both governments 54 55 We refer to the case in which both governments employ the unit tax method 56 as UU. In the third stage, given Ti, firm i maximizes πi = (pi c)Di TiDi, 57 − − 58 59 5 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 where D is given by (1). The first-order condition is ∂π /∂p = (τ + c + 10 i i i T + p 2p )/τ = 0, where i = j and i, j = 1, 2, leading to 11 i j − i 6 12 2T1 + T2 T1 + 2T2 13 p1 = τ + c + and p2 = τ + c + . (4) 14 3 3 15 Substituting (4) into (2), the tax revenue of country i in the second stage 16 can be obtained from 17 18 1 T T 19 R = T + j − i , i = j, i, j = 1, 2. (5) i i 2 6τ 6 20 21 Country i’s second-stage problem is to maximize its tax revenue with 22 respect to T . The first-order condition is obtained by ∂R /∂T = (3τ + T 23 i i i j − 24 2Ti)/τ = 0, giving the equilibrium unit tax rates in the UU case as 25 26 UU UU T1 = T2 = 3τ. (6) 27 28 Substituting (6) into (5), we obtain the equilibrium tax revenue: 29 30 3 RUU = RUU = τ. (7) 31 1 2 2 32 33 3.2 Ad Valorem tax competition by both governments 34 35 We refer to the case in which both countries adopt the ad valorem tax 36 method as AA. In the third stage, given ti, firm i maximizes πi = (pi 37 − c)D t p D . The first-order condition for the optimization problem is as 38 i − i i i follows: ∂π /∂p = [c + (1 t )(τ + (p 2p ))]/τ = 0, leading to 39 i i − i j − i 40 41 42 c 2 1 c 1 2 p1 = τ + + and p2 = τ + + . (8) 43 3 1 t1 1 t2 3 1 t1 1 t2 44 − − − − 45 Substituting (8) into (3), we obtain the following tax revenue: 46 47 48 t c 2 1 c 1 1 R = i τ + + τ . (9) 49 i 2τ 3 1 t 1 t − 3 1 t − 1 t 50 − i − j − i − j 51 Country i’s second-stage problem is to maximize its tax revenue given 52 53 by (9) with respect to ti. The first-order condition is given by 54 55 56 57 58 59 6 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 10 11 ∂Ri 1 c 2 1 c 1 1 12 = τ + + τ ∂t τ 3 1 t 1 t − 3 1 t − 1 t 13 i − i − j − i − j 14 ct c 4 1 + i τ = 0. 15 2 3τ(1 ti) − 3 1 ti − 1 tj 16 − − − 17 In the symmetric equilibrium, the tax rate in the AA case, t = t = tAA, 18 1 2 19 satisfies 20 21 22 3(1 tAA)3 + k(1 tAA)(3 2tAA) k2tAA = 0, (10) 23 − − − − 24 where k c/τ > 0. Solving (10) with respect to k, we obtain 25 ≡ 26 AA AA 1 t AA 27 k(t )= − 3 2t + 9 8(tAA)2 . (11) 2tAA − − 28 q 29 AA AA k(t ) is a monotone decreasing function, and limtAA→+0 k(t ) = and 30 AA AA∞ limtAA→1 k(t ) = 0; thus, the equilibrium ad valorem tax rate, t , is an 31 AA 32 inverse function of k. Conversely, t (k) is a monotone decreasing function, AA AA 33 and limk→+0 t (k) = 1 and limk→∞ t (k) = 0. Figure 2 depicts the graph 34 of tAA(k) for reference. Substituting (11) into (9), we obtain the tax revenue 35 under the given equilibrium tax rate in the AA case: 36 37 AA 2 AA AA 3+ 9 8(t (k)) 38 R1 = R2 = − τ. (12) 39 p 4 40 41 3.3 Preliminary results 42 43 At this stage, we obtain some remarkable results. 44 45 Lemma 1. The price of products in unit tax competition is higher than 46 that in ad valorem tax competition, pUU >pAA. 47 UU AA k 48 Proof. From (4) and (6), p = τ(4+k). From (8), p = τ(1+ − AA ). 49 1 t (k) Using (11), pUU pAA = 0.5τ(2tAA(k) + 3 9 8(tAA(k))2). Since 50 − − − 51 3 9 8(tAA(k))2 > 0 tAA(k) [0, 1), pUU >pAA. (Q.E.D.) p 52 − − ∀ ∈ 53 Lemmap 2. The tax rate in both unit tax competition and ad valorem tax 54 competition has a positive relationship with transportation costs. 55 56 57 58 59 7 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 Proof. From (6), ∂T UU /∂τ > 0. From (11), we can find that tAA(k) is 10 a monotone decreasing function. As k c/τ, tAA increases with τ. 11 ≡ 12 (Q.E.D.) 13 14 The first lemma confirms that the classic argument presented by Suits 15 and Musgrave (1953, p.603) in a single country model still preserves in the 16 two-country tax competition model: consumers have to pay a higher price 17 18 when their countries use the unit tax method. The second lemma also 19 presents a common view: as the mobility of consumers increase, represented 20 by a decrease in τ, governments are likely to engage in tax-cutting compe- 21 tition. 22 Next, we discuss equivalent translation between ad valorem tax and unit 23 8 24 tax. Assume that country i adopts an ad valorem tax method ti. We define 25 the effective unit tax rate for country i as 26 27 Ti = tipi, (13) 28 29 by replacing ad valorem tax with unit tax. Using the equivalence tax rate, 30 we obtain the following proposition. 31 32 Proposition 1. Compared with ad valorem tax, tax competition is less 33 AA UU 34 severe when countries adopt the unit tax method: T < T . 35 Proof AA k 36 . Substituting p = τ(1 + 1−tAA(k) ) and (11) into (13), under 37 symmetric equilibrium, we obtain 38 39 3+ 9 8tAA(k) 40 T AA = − τ. (14) 41 p 2 ! 42 43 Comparing (6) with (14), for tAA(k) [0, 1), we obtain T AA T UU = ∈ − 44 ( 9 8(tAA(k))2 3)/2 < 0. (Q.E.D.) 45 − − 46 p Furthermore, we obtain the following result, which is directly linked to 47 48 Proposition 1. 49 50 Proposition 2. The tax revenue in unit tax competition is larger than the AA UU 51 revenue in ad valorem tax competition: Ri < Ri . 52 53 8See, for instance, Lockwood (2004), Salani´e(2002, p.21), and Suits and Musgrave 54 (1953, p.601). 55 56 57 58 59 8 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 Proof. Comparing (7) with (12), we obtain 10 11 12 3+ 9 8(tAA(k))2 3 RAA = − τ < τ = RUU . (Q.E.D.) 13 i 4 2 i 14 p 15 16 Propositions 1 and 2 show that unit tax competition is superior to ad 17 valorem tax competition in terms of raising tax revenue. This is simply 18 19 because ad valorem tax competition induces more severe competition for 20 customers; prices and tax rates are relatively lower in ad valorem tax com- 21 petition.9 Intensified tax competition simply produces significant battle to 22 attract cross-border consumers among regions, resulting in inefficiently lower 23 levels of tax rates and smaller amounts of tax revenue. Hence, selecting the 24 unit tax method is desirable for both countries in terms of raising tax rev- 25 26 enue. Proposition 2, however, leaves the possibility open for the stage 1 game 27 to choose tax methods that have a prisoner’s dilemma structure, where the 28 ad valorem tax is a dominant strategy even though it leaves both countries 29 worse off in equilibrium. Thus, in the next subsection, we characterize the 30 mixed equilibria in which one country sets ad valorem tax, and the other 31 32 sets unit tax. 33 34 35 3.4 Ad valorem tax versus unit tax 36 37 We derive the equilibrium outcome when countries employ different tax 38 methods. We denote the case in which country 1 adopts the ad valorem 39 40 (unit) tax method and country 2 selects the unit (ad valorem) tax method 41 as AU(UA). However, as the two cases, AU and UA, are symmetric, we 42 examine only the AU case. 43 The objective function of firms 1 and 2 in the third stage are, respectively, 44 given by π = (p c)D t p D and π = (p c)D T D . The first- 45 1 1 − 1 − 1 1 1 2 2 − 2 − 2 2 46 order conditions of the optimization problem give 47 48 ∂π c 1 t (1 t )(p 2p ) 49 1 = + − 1 + − 1 2 − 1 = 0, 50 ∂p1 2τ 2 2τ 51 ∂π 1 c + T 2p p 2 = + 2 2 − 1 = 0, 52 ∂p 2 2τ − 2τ 53 2 54 9The reason ad valorem tax induces more severe competition for customers is that a 55 tax-cut incentive is stronger in ad valorem tax than in unit tax, which is explained in 56 Section 4. 57 58 59 9 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 leading to 10 11 12 1 2c 1 c 13 p = τ + + (c + T ) and p = τ + + 2(c + T ) . (15) 1 3 1 t 2 2 3 1 t 2 14 − 1 − 1 15 16 Using (2), (3), and (15), tax revenues are obtained as follows. 17 18 t 1 2c 1 c 19 R = 1 τ + + (c + T ) τ (c + T ) , (16) 1 2τ 3 1 t 2 − 3 1 t − 2 20 − 1 − 1 21 T 1 c R = 2 τ + (c + T ) . (17) 22 2 2τ 3 1 t − 2 23 − 1 24 Given the tax rate of the other country, each country maximizes its tax 25 revenue. The first-order conditions are given by 26 27 28 ∂R 1 1 2c 1 c 29 1 = τ + + (c + T ) τ (c + T ) ∂t τ 3 1 t 2 − 3 1 t − 2 30 1 − 1 − 1 31 ct 1 4c + 1 τ (c + T ) = 0, (18) 32 3τ(1 t )2 − 3 1 t − 2 33 − 1 − 1 34 ∂R2 1 c = 1+ (c + 2T2) = 0. (19) 35 ∂T2 3τ 1 t1 − 36 − 37 From (19), we obtain 38 39 40 3 ct1 T2 = τ + . (20) 41 2 2(1 t1) 42 − 43 Substituting (20) into (18), we obtain country 1’s ad valorem tax rate in the 44 AU case, tAU which satisfies 45 46 47 81(1 tAU )3 + 18k(1 tAU )(3 3tAU + (tAU )2) 48 − 2 AU − AU− AU 2 49 k t (18 5t + (t ) ) = 0, (21) 50 − − 51 where k c/τ > 0. Solving (21) with respect to k( c/τ), we obtain 52 ≡ ≡ 53 54 9(1 tAU ) 3 3tAU + (tAU )2 + 9 8(tAU )2 55 k(tAU )= − − − . (22) 56 htAU (18 5tAU + (tAUp)2) i 57 − 58 59 10 60 61 62 63 64 65 1 2 3 4 5 6 7 8 9 AU AU k(t ) is a monotone decreasing function, and limtAU →+0 k(t ) = and 10 AU ∞AU 11 limtAU →1 k(t ) = 0; thus, country 1’s tax rate in the equilibrium, t , is AU 12 an inverse function of k. Viewed from the opposite side, t (k) is a mono- AU AU 13 tone decreasing function, and limk→+0 t (k) = 1 and limk→∞ t (k) = 0. 14 Figure 3 depicts the graph of tAU (k) for reference. From (11) and (22), we 15 find tAA(k)
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