Welfare with the maximum‐revenue tariff is compared to free‐trade welfare under Cournot duopoly with differentiated products; under Bertrand duopoly with differentiated products; and under perfect competition in the case of a large country able to affect its terms of trade. Under Cournot duopoly and Bertrand duopoly, assuming linear demands and constant marginal costs, welfare with the maximum‐revenue tariff is always higher than free‐trade welfare. Under perfect competition, assuming linear demand and supply, welfare with the maximum‐revenue tariff will be higher than free‐trade welfare if the country has sufficient market power

Collie, David R.

English

Online Research @ Cardiff

442  |    Scott J Polit Econ. 2020;67:442–447.wileyonlinelibrary.com/journal/sjpe1  | INTRODUC TIONA country may set its import tariff to maximise tariff revenue rather than welfare if it does not have alternative sources of government revenue such as an efficient income tax or a sales tax system. In fact, even though the federal government of the USA has an efficient income tax system, Donald Trump has tweeted that import tariffs are a ‘great revenue producer’.1 Therefore, the use of import tariffs as a source of government revenue has become a matter of  1See Chad P. Bown and Douglas A. Irwin, ‘Is Trump right when he tweets that tariffs bring in government revenue? Here are 5 things you need to know’, The Washington Post, 16th July 2019, https://www.washi ngton post.com/polit ics/2019/07/16/tarif f-reven ue-trump -tweet s-thing s-you-need-know/Accepted: 18 March 2020DOI: 10.1111/sjpe.12245  O R I G I N A L  A R T I C L EMaximum-revenue tariffs versus free tradeDavid R. CollieThis is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.© 2020 The Authors. Scottish Journal of Political Economy published by John Wiley & Sons Ltd on behalf of Scottish Economic SocietyCardiff Business School, Cardiff University, Cardiff, UKCorrespondenceDavid R Collie, Cardiff Business School, Cardiff University, Cardiff, UK.Email: collie@cardiff.ac.ukAbstractWelfare with the maximum-revenue tariff is compared to free-trade welfare under Cournot duopoly with differen-tiated products; under Bertrand duopoly with differenti-ated products; and under perfect competition in the case of a large country able to affect its terms of trade. Under Cournot duopoly and Bertrand duopoly, assuming linear demands and constant marginal costs, welfare with the maximum-revenue tariff is always higher than free-trade welfare. Under perfect competition, assuming linear de-mand and supply, welfare with the maximum-revenue tariff will be higher than free-trade welfare if the country has suf-ficient market power.K E Y W O R D SBertrand oligopoly, Cournot oligopoly, free trade, maximum-revenue tariff, perfect competitionJ E L  C L A S S I F I C A T I O NF11; F12; F13     |  443COLLIEtopical policy debate. Obviously, the welfare of the country with the maximum-revenue tariff is lower than its welfare with the optimum-welfare tariff, but is it possible that its welfare with the maximum-revenue tariff is higher than its free-trade welfare? This intriguing question will be answered in this note for three cases: under Cournot duopoly with differentiated products; under Bertrand duopoly with differentiated products; and under perfect competition with a large country able to influence its terms of trade. Under Cournot duopoly and Bertrand duopoly, assuming linear demands and constant marginal costs, welfare with the maximum-revenue tariff will be shown to be always higher than free-trade welfare. Under perfect competition, assuming linear demand and supply functions, welfare with the maximum-revenue tariff will be shown to higher than free-trade welfare if the country has sufficient market power.Under perfect competition, Johnson (1950) showed that the maximum-revenue tariff always exceeds the op-timum-welfare tariff whereas Collie (1991) and Clarke and Collie (2006) showed that the optimum-welfare tariff may exceed the maximum-revenue tariff under Cournot oligopoly and under Bertrand oligopoly. The latter result is more likely the lower the costs of the home firm relative to the foreign firm and the greater the degree of product substitutability. Assuming that welfare is concave in the tariff, if the optimum-welfare tariff exceeds the maximum-revenue tariff, then welfare with the maximum-revenue tariff will be higher than free-trade welfare. If the maximum-revenue tariff exceeds the optimum-welfare tariff, as is always the case under perfect competition, then it is not immediately obvious whether welfare with the maximum-revenue tariff is higher or lower than free-trade welfare. The larger the maximum-revenue tariff relative to the optimum-welfare tariff then the more likely that welfare with the maximum-revenue tariff will be lower than free-trade welfare.The analysis of Collie (1991) and Clarke and Collie (2006) comparing the optimum-welfare and maximum-rev-enue tariff under oligopoly has been extended in a number of directions. For example, Larue and Gervais (2002) consider the case with a number of domestic and foreign firms while Wang, Lee, and Huang (2012) consider the effect of privatisation in a mixed duopoly. Wang and Lee (2012) consider the effect of domestic entry while Wang and Lee (2014) consider the effect of vertical integration.2  | COURNOT OLIGOPOLY WITH DIFFERENTIATED PRODUC TSConsider a Cournot duopoly in the home country consisting of a domestic firm and a foreign firm producing differ-entiated products. The domestic firm has marginal cost c1 and produces output x1 for its domestic market, which sells at price p1, while the foreign firm has marginal cost c2 and produces output x2 for export to the home country, which sells at price p2. The home country imposes a specific import tariff t on imports from the foreign country. Preferences of the representative consumer are derived from a quadratic, quasi-linear utility function:where z is consumption of a numeraire good produced by a perfectly competitive industry using constant returns to scale technology, α> c1, α> c2, β>0 and ϕ∈[0,1] is the degree of product substitutability that is equal to one when products are perfect substitutes and equal to zero when products are independent. Utility maximisation by the con-sumer yields the inverse demand functions facing the domestic firm and foreign firm respectively:Hence, the profits of the domestic firm are π1=(p1−c1)x1, and the profits of the foreign firm are π2=(p2−c2− t)x2. The welfare of the home country is the sum of consumer surplus, profits of the domestic firm and tariff revenue:(1)U=α(x1+x2)−β2(x21+x22+2ϕx1x2)+z(2)p1=α−β(x1+ϕx2)p2=α−β(ϕx1+x2)(3)W=β2(x21+x22+2ϕx1x2)+π1+ tx2444  |     COLLIEIt is straightforward to solve for the Cournot equilibrium outputs of the domestic and foreign firms, which yield:where A1≡2(𝛼−c1)−ϕ(𝛼−c2)>0 and A2≡2(𝛼−c2)−ϕ(𝛼−c1)>0 are both positive if there is an interior solution, where both firms sell positive quantities, under free trade, t=0. Substituting the Cournot equilibrium outputs into the inverse demand functions yields the prices of the domestic and foreign firms:Setting the import tariff equal to zero, t=0, in (4) and (5) then substituting outputs and prices into the expres-sion for welfare (3) yields free-trade welfare:If the country sets its import tariff to maximise tariff revenue, R= tx2, then the maximum-revenue tariff is obtained by solving: R/t=0, which yields tR=A2/4>0. Collie (1991) has shown that the optimum-welfare tariff may exceed the maximum-revenue tariff under Cournot duopoly. Substituting the maximum-revenue tariff into the expression for welfare yields welfare with the maximum-revenue tariff:Subtracting welfare under free trade (6) from welfare with the maximum-revenue tariff (7) yields:This is unambiguously positive, which leads to the following proposition:Proposition 1 Under Cournot duopoly with differentiated products, assuming linear demand and constant marginal cost, welfare with the maximum-revenue tariff is higher than welfare under free trade.3  | BERTR AND DUOPOLY WITH DIFFERENTIATED PRODUC TSNow consider the situation when the two firms in Section 2 compete in prices rather than quantities so that there is a Bertrand duopoly rather than a Cournot duopoly. Inverting the inverse demand functions (2) yields the de-mand functions facing the domestic and foreign firm:Assuming an interior solution, the first-order conditions for a Bertrand-Nash equilibrium are:(4)x1=A1+ϕt(4−ϕ2) x2=A2−2tβ(4−ϕ2)(5)p1= c1+A1+ϕt4−ϕ2p2= c2+ t+A2−2t4−ϕ2(6)WF=3A21+A22+2ϕA1A24(4−ϕ2)2(7)WR=20A22+(12A1+ϕA2) (4A1+3ϕA2)32(4−ϕ2)2(8)ΔW=WR−WF=A28ϕA1+(4+3ϕ2)A232𝛽(4−ϕ2)2>0(9)x1=(1−ϕ)−p1+ϕp2(1−ϕ2) x2=(1−ϕ)+ϕp1−p2(1−ϕ2)     |  445COLLIESolving for the Bertrand equilibrium prices of the domestic and foreign firms yields:where B1=(2−ϕ2) (𝛼−c1)−ϕ(𝛼−c2)>0 and B2=(2−ϕ2) (𝛼−c2)−ϕ(𝛼−c1)>0 are both positive if there is an in-terior solution where both firms sell positive quantities under free trade, t=0. Substituting the prices into the demand functions (9) yields the sales of the domestic and foreign firms:where Δ=β(1−ϕ2) (4−ϕ2)>0 is clearly positive. Setting t=0 in (11) and substituting the sales of the two firms into the expression for welfare (3) yields free-trade welfare:If the country sets its import tariff to maximise tariff revenue, R= tx2, then the maximum-revenue tariff is obtained by solving: R/t=0, which yields tR=B2/(2(2−ϕ2))>0. Clarke and Collie (2006) have shown that the optimum-welfare tariff may exceed the maximum-revenue tariff under Bertrand duopoly. Substituting the maximum-revenue tariff into the expression for welfare yields welfare with the maxi-mum-revenue tariff:Subtracting free-trade welfare (13) from welfare with the maximum-revenue tariff (14) yields:This is unambiguously positive, which leads to the following proposition:Proposition 2 Under Bertrand duopoly with differentiated products, assuming linear demand and constant marginal cost, welfare with the maximum-revenue tariff is higher than welfare under free trade.The result under Bertrand duopoly is the same as the result under Cournot duopoly: welfare with the max-imum-revenue tariff is always higher than free-trade welfare. One might conjecture that the result would be ambiguous under oligopoly with many domestic and foreign firms.(10)π1p1=α(1−ϕ)−2p1+ϕp2+c1β(1−ϕ2) =0π2p2=(1−ϕ)+ϕp1−2p2+c2+ tβ(1−ϕ2) =0(11)p1= c1+B1+ϕt(4−ϕ2) p2= c2+ t+B2−(2−ϕ2)t(4−ϕ2)(12)x1=B1+ϕtΔx2=B2−(2−ϕ2)tΔ(13)WF=β(3−2ϕ2)B21+2ϕB1B2+B22Δ2(14)WR=8(2−ϕ2)2Δ2[4(2−ϕ2)2 (3−2ϕ2)B21+4(2−ϕ2)(5−3ϕ2)B1B2+(20−25ϕ2+11ϕ4−2ϕ6)B22](15)ΔW=WR−WF=B24ϕ(2−ϕ2)B1+(2(1−ϕ2)2+(2−ϕ2))B28(2−ϕ2)2 (4−ϕ2)Δ>0446  |     COLLIE4  | PERFEC T COMPETITION WITH A L ARGE COUNTRYThe literature on trade policy under oligopoly is well known for yielding results that are at odds with well-estab-lished results under perfect competition. For example, as shown by Collie (1991) and Clarke and Collie (2006), the optimum-welfare tariff may exceed the maximum-revenue tariff under oligopoly, but this is never the case under perfect competition, as shown by Johnson (1950). Therefore, it seems desirable to assess the robustness of the result that welfare with the maximum-revenue tariff exceeds free-trade welfare to the form of market structure by considering the case of perfect competition. Of course, the motive for trade policy intervention under perfect competition is the terms of trade argument rather than the profit-shifting argument under oligopoly.Consider a product produced by a perfectly competitive industry in the home country that also imports the product from the foreign country. Preferences are assumed to be quasi-linear so there is no income effect and, for simplicity, demand and supply functions are assumed to be linear. In the home country, demand for imports is: D(pD)=a−bpD where pD is the domestic price and the parameters are positive, a,b>0. The supply of imports from the foreign country is S(pW)=dpW−e where pW is the world price and the parameters are positive, d,e>0. To ensure that the quantity of imports is positive under free trade, it is assumed that ad−be>0. Assuming that the foreign country pursues a policy of free trade, if the home country imposes a specific import tariff t, then tariff revenue is R= tX, where X is the quantity of imports, while welfare in the home country is given by the sum of surplus from imports and tariff revenue: W=X2/2b+ tX. With a specific tariff, the domestic price is equal to the world price plus the tariff, pD=pW+ t, and equating demand and supply for imports yields the equilibrium world price and the quantity of imports:The import tariff improves the terms of trade of the home country, since 𝜕pW∕𝜕t<0, and reduces the quantity of imports, since 𝜕X/𝜕t<0. Under free trade, the import tariff is equal to zero, so substituting t=0 into (16) and then deriving free-trade welfare yields:If the country sets its import tariff to maximise tariff revenue, then the maximum-revenue tariff is obtained by solving: R/t=0, which yields tR=(ad−be)/(2bd)>0. Under perfect competition, Johnson (1950) showed that the maximum-revenue tariff is larger than the optimum-welfare tariff. Substituting the maximum-revenue tariff into the expression for welfare yields welfare with the maximum-revenue tariff:Subtracting welfare under free trade (17) from welfare with the maximum-revenue tariff (18) yields:This is positive if 2b>d, which leads to the following proposition:Proposition 3 Under perfect competition, assuming linear demand and supply functions, welfare with the maxi-mum-revenue tariff is higher than welfare under free trade if 2b>d.(16)pW=a+e−btb+dX=ad−be−bdtb+d(17)WF= (ad−be)22b (b+d)2(18)WR=(2b+3d)(ad−be)28bd (b+d)2(19)ΔW=WR−WF=(2b−d)(ad−be)28bd (b+d)2     |  447COLLIEThe greater the market power of the home country relative to the foreign country then the more likely that welfare with the maximum-revenue tariff is higher than free-trade welfare. Obviously, if the country is small, then welfare with the maximum-revenue tariff must be lower than free-trade welfare as any tariff reduces welfare.5  | CONCLUSIONSUnder oligopoly, welfare with the maximum-revenue tariff is always higher than free-trade welfare whether there is Cournot or Bertrand competition. Under perfect competition, welfare with the maximum-revenue tariff may be higher than free-trade welfare if the country has sufficient market power. Despite this result, a country would be better off using the optimum-welfare tariff combined with domestic taxes to raise government revenue rather than using the maximum-revenue tariff. Also, this analysis assumes that the foreign country passively pursues a policy of free trade while the home country unilaterally sets a tariff. If the foreign country retaliates to the tariff set by the home country, then the most likely outcome is that there will be welfare losses for both countries. Therefore, countries are better off pursuing multilateral free trade through international agreements such as the World Trade Organisation and using domestic taxes to raise government revenue rather than unilaterally using import tariffs to raise government revenue.ORCIDDavid R. Collie  https://orcid.org/0000-0002-3132-648X R E FE R E N C E SClarke, R., & Collie, D. R. (2006). Optimum-welfare and maximum-revenue tariffs under bertrand duopoly. Scottish Journal of Political Economy, 53(3), 398–408. https://doi.org/10.1111/j.1467-9485.2006.00386.xCollie, D. (1991). Optimum welfare and maximum revenue tariffs under oligopoly. Scottish Journal of Political Economy, 38(4), 398–401. https://doi.org/10.1111/j.1467-9485.1991.tb003 27.xJohnson, H. G. (1950). Optimum welfare and maximum revenue tariffs. The Review of Economic Studies, 19(1), 28–35. https://doi.org/10.2307/2296268Larue, B., & Gervais, J.-P. (2002). Welfare–maximizing and revenue–maximizing tariffs with a few domestic firms. Canadian Journal of Economics/Revue Canadienne D'économique, 35(4), 786–804. https://doi.org/10.1111/1540-5982.00154Wang, L. F. S., & Lee, J.-Y. (2012). Domestic entry, optimum-welfare and maximum-revenue tariffs. Research in Economics, 66(1), 106–109. https://doi.org/10.1016/j.rie.2011.07.001Wang, L. F. S., & Lee, J.-Y. (2014). Ranking the optimum tariff and the maximum revenue tariff in vertically related mar-kets. Research in Economics, 68(3), 222–229. https://doi.org/10.1016/j.rie.2014.03.001Wang, L. F. S., Lee, J.-Y., & Huang, C.-S. (2012). Maximum-revenue and optimum-welfare tariffs in international mixed duopoly: Does the order of firms’ move matter? Journal of Industry, Competition and Trade, 12(3), 273–283. https://doi.org/10.1007/s1084 2-011-0099-0How to cite this article: Collie DR. Maximum-revenue tariffs versus free trade. Scott J Polit Econ. 2020;67:442–447. https://doi.org/10.1111/sjpe.12245

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