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General Equilibrium and Welfare in International Trade

Equilibre général et bien-être dans la théorie du commerce international
John S. Chipman
p. 15-33

Résumés

Suivant la tradition de Cournot, Mill et Marshall, on étudie un modèle d’équilibre général s’appliquant à des pays et non à des individus, sous l’hypothèse que les préférences individuelles peuvent être agrégées. Ce modèle couvre à la fois le cas d’équilibre concurrentiel avec libre échange et le cas de commerce restreint par des droits de douanes, analysé par Marshall aussi bien que par Johnson. La théorie de la guerre des droits de douane de Johnson est analysée comme un exemple de théorie du duopole de Cournot, conduisant à un équilibre de Nash par opposition à l’équilibre concurrentiel de libre échange. Pour terminer, on discute le cas d’un équilibre instable, tel que Wan et Zhou l’ont présenté récemment, suggérant ainsi un nouveau concept d’équilibre négocié.

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Notes de la rédaction

The paper has benefited from the comments of two referees.

Texte intégral

1In this paper I consider the general equilibrium of countries in the tradition of Mill (1844; 1852), Mangoldt (1863), Marshall (1879; 1923), Edgeworth (1894), Pareto (1895), and Viner (1937). Attention is not limited to competitive equilibrium, but is also devoted to what is now known as Nash equilibrium, in particular the equilibrium of countries imposing tariffs against each other. The key reference is that of Johnson (1954), whose model turns out to be analytically equivalent to a model of duopoly going back to Cournot (1838).

2I will discuss two different models which can be used to compare the differential effects of free trade and tariff-encumbered trade on the welfares of the citizens of two countries. In each case, in order to simplify, I abstract from processes of production, and assume that in each country there is a constant rate of production of two goods per period. Thus the models are all of simple exchange equilibrium. In each case I examine a tariff-encumbered equilibrium in which the two countries engage in a tariff war along the lines of Johnson’s (1954) seminal contribution. The outcome of the tariff war, which is a Nash equilibrium, is compared with that (or those) of free trade, which are competitive equilibria. A third concept of negotiated equilibrium is suggested.

In section 1, I assume that consumers within and as between countries have the same homothetic (hence aggregable) preferences of the Millian or “Cobb-Douglas’’ type, in which the ratio of expenditure on each of the two commodities to income is constant (in fact, in this application, one-half). While I consider a general model of fixed production, in order to make efficient comparisons I assume that each country produces 1 unit of its import good and Image 10000201000000400000001941FD539207178AB5.png units of its export good per period. In a tariff war between them it turns out that for each country the equilibrium tariff factor is Image 100002010000002300000019386DB2BC611AC34B.png , so that the equilibrium ad valorem tariff rate is Image 1000020100000049000000191C0EFD5F2574A5D1.png . For example, if Image 100002010000003F00000019FF60DF077808CEA3.png , the equilibrium tariff rate will be 41.4%. This equilibrium, based on successive applications of the optimal-tariff reaction function, is approximately achieved in about four stages of the tariff war. Under these assumptions it is shown that both countries are worse off after the tariff war than under free trade (that is, under the Nash equilibrium than under the competitive equilibrium), but better off under the Nash equilibrium than under autarky. It turns out that the welfare levels of the countries at the end of the tariff war are slightly more than midway between those under autarky and those under free trade.

3In section 2, I examine a fascinating recent contribution by Wan and Zhou (2008), in which the free-trade competitive equilibrium is not unique, but consists of two stable equilibria and one unstable equilibrium as in Marshall (1879), the equitable (“fair’’) equilibrium being unstable and the inequitable (“unfair’’) ones stable. This possibility is achieved by assuming a zero endowment (rate of production) of each country’s import good, and a unit endowment of its export good, combined with preferences that are parallel (hence aggregable) with respect to each country’s export good, i.e., exhibiting constant marginal utility of the export good, leading to each country having a strong relative preference for its own import good. What they show is that, for each country, while one of the two stable (unfair) competitive equilibria is better than the (unique) tariff-encumbered equilibrium, the other one is (about equally) worse. The cautious approach for each country is then to choose the Nash equilibrium rather than run the risk of falling into the worse competitive one, or else to negotiate a tariff rate in between.

1. Symmetric endowments and identical Millian preferences

  • 1 Cf. Mill (1852, Vol. II, Book III, Ch. XVIII, 122–151; 1865, 124–153). See also Chipman (1965, 483– (...)

I shall first take up the case of Millian preferences, in which each country acts as though it maximizes a utility function of the form Image 10000201000000D8000000191F4CBE5F55421988.png , where Image 100002010000001B0000001968FE853E8A097B49.png denotes country Image 100002010000001F00000019BA9ADADEC2BCB062.png consumption of commodity Image 1000020100000007000000198047A0AA5CFF7D2F.png .1 Letting the first subscript denote the commodity and the second the country, country Image 100002010000001F00000019BA9ADADEC2BCB062.png demand function for commodity Image 1000020100000007000000198047A0AA5CFF7D2F.png is

Image 100002010000028B0000004B1329A8AA50FF99E4.png

  • 2 Note that in the absence of saving and capital movements, if country 1 imposes a tariff on its impo (...)

which is equal to its disposable income divided by twice the price of commodity Image 1000020100000007000000198047A0AA5CFF7D2F.png on country Image 100002010000001F00000019BA9ADADEC2BCB062.png markets. On the supply side, I shall assume that country Image 100002010000000C00000019B958D326355F743E.png produces constant amounts Image 100002010000002B000000199BBE8F231729E219.png and Image 100002010000002400000019E404260D27C41D78.png of commodities 1 and 2 in each period, so that its earned income is Image 100002010000005D0000001997C8178595B4407F.png Image 100002010000004300000019556929351A1F0D02.png . Its disposable income will be this amount plus the deficit in its balance of payments on current account, Image 100002010000001D000000197D37C0392399A977.png  (denominated in its own prices), which in this application will consist of its tariff revenues.2

Let us now consider a situation of trade between countries 1 and 2, in which the parameters Image 100002010000002B000000199BBE8F231729E219.png are such that country 1 exports Image 10000201000000BA00000019AFBE94FCAD9BD802.png units of commodity 1 to and imports Image 10000201000000A2000000197FDC45FA7BD18B9B.png  units of commodity 2 from country 2, and likewise country 2 exports Image 10000201000000B7000000197A4873BADD1758D9.png units of commodity 2 to and imports Image 10000201000000A400000019A693CD86416E4644.png units of commodity 1 from country 1, where in equilibrium the material-balance conditions Image 1000020100000084000000194F2758D8AD1872F2.png necessarily hold for commodities Image 100002010000004800000019B63452DD0FD02DBD.png .

1.1. Derivation of tariff-modified offer functions

Country 1’s trade-demand function Image 10000201000000180000001982D0923AE914CE97.png for its import good (commodity 2), which is expressed as a function of its domestic prices Image 100002010000004C000000191B15956D065826F1.png and the deficit in its balance of trade Image 100002010000001D000000194FD718108363AEDE.png , is defined in terms of its ordinary demand function Image 1000020100000020000000194CBB7B7747F0CDD3.png by

Image 100002010000028B000000895A7FF263F174124C.png

where Image 100002010000009B000000194ED3B85CBAEB8840.png

For any tariff factor Image 100002010000007B00000019B5A32202C6EFDDDA.png imposed by country 1 on its import of commodity 2 (where Image 1000020100000017000000195FC4510047150BAC.png is the ad valorem tariff rate), country 1’s tariff-modified excess-demand function for this import good (which is expressed as a function of world prices Image 1000020100000036000000196A1232307EDAB258.png ) is defined implicitly by

Image 100002010000028B00000075CED04E838F8F82E9.png

where we have used Image 100002010000013300000019AD1548C050C9F9D0.png for the tariff revenues. Collecting terms, this leads to the explicit formula

Image 100002010000028B000000615CB0FBA160F6F346.png

Normalizing world prices to Image 100002010000003600000019BB7469D482C02252.png where Image 1000020100000076000000191A238DDA1964498F.png country 1’s tariff-modified inverse excess-demand function Image 100002010000005F0000001962645E8892CF1265.png for its import good 2 is now defined as the solution of the equation

Image 100002010000028B0000007A7F1A6091C59D6516.png

wich is
Image 100002010000028B00000061EE7EF5CC3015FC0E.png

4Country 1’s tariff-modified offer function is then defined as

Image 100002010000028B00000062425DB255C979F8FC.png

5An exactly similar formula holds for country 2. Thus we have a system of Marshallian offer functions

Image 100002010000028B0000006B532FD883EBBAE0C9.png

1.2. General equilibrium of the tariff-modified system

We note from (1.7) that this system is linear in the reciprocals of the trades. Then, using the material-balance conditions Image 100002010000008400000019EB93BC17FFCC01C6.png for Image 100002010000004600000019FCC6879A6ED4FDF7.png , (1.7) may be written in the matrix form

Image 100002010000028B000000635F9030376F4DCE1C.png

A sufficient condition for its solution is that Image 10000201000000F9000000196C30D2D0E716246A.png

6The solution to (1.8) is

Image 100002010000028B00000079D151C8267DFB6D3A.png

or equivalently,

Image 100002010000028B00000089341CB3E2BE496997.png

7For the consumption levels, we have for country 1

Image 100002010000028B0000004CFBE4D625E82B6672.png

and

Image 100002010000028B0000004DF5B5F4D65007B9C3.png

with analogous formulas for country 2.

1.3. Optimal tariffs and reaction functions in a leading special case

At this point, in order to simplify I introduce the assumption that Image 100002010000009A00000019566A0912746EF06C.png and Image 10000201000000A4000000192B0519C031170864.png i.e., each country produces Image 10000201000000140000001958243EADBE2B2FC4.png units of its export good and 1 unit of its import-competing good, where Image 1000020100000041000000196FA8BE02D2C988A5.png .

Now if Image 100002010000001800000019DCBF7C39A26A2144.png (country 2’s tariff factor on its import of commodity 1) is given initially, country 1’s problem is to maximize its utility Image 10000201000000CA0000001944EFAE1D4FD9F67F.png with respect to its tariff factor Image 100002010000001A00000019652F7D495603732E.png  (subject to the given value of country 2’s tariff factor Image 100002010000001800000019DCBF7C39A26A2144.png ). From (1.11) and (1.12) the problem is then to maximize

Image 100002010000028B0000004FCD43EE1B60D498CF.png

with respect to Image 100002010000001900000019BB734BA89BBFBAE1.png . Differentiating (1.13) with respect to Image 100002010000001900000019BB734BA89BBFBAE1.png and equating it to zero we obtain

Image 100002010000028B0000004ABC112910F02E3D34.png

8After some elaborate calculations this reduces to the equation

Image 100002010000028B0000003A1CC1A30068D8F72E.png

  • 3 The idea of a reaction function was originated by Cournot (1838, Ch. VII, 90 and Fig. 2; 1897, 81). (...)

9from which we obtain country 1’s reaction function3

Image 100002010000028B0000007EAC7C01F02F993247.png

  • 4 Following Torrens (1844, esp. 331–372), the first systematic discussion of the benefit to a country (...)

expressing its tariff factor Image 100002010000001A00000019652F7D495603732E.png on commodity 2 as a function Image 1000020100000018000000191A73BC8D0B149010.png of country 2’s tariff factor Image 100002010000001800000019DCBF7C39A26A2144.png on commodity 1. Because of the complete symmetry of the situation, by interchanging Image 100002010000001800000019DCBF7C39A26A2144.png and Image 100002010000001A00000019652F7D495603732E.png in (1.16) we obtain country 2’s reaction function Image 100002010000007A00000019CE04AA8F7124A734.png . In the special case in which country 2 does not impose a tariff on its import of commodity 1, i.e., Image 1000020100000047000000197D5A7010B68BE908.png , the formula for country 1’s “optimal tariff’’ factor on its import of commodity 2 reduces to Image 10000201000000CE00000019FEDF7AE9ACEB6984.png 4

Figure 1: Tariff-war reaction functions and dynamic approach to equilibrium. The case ω = 2

Image 1000020100000317000002D8B392A60A514C4D37.png

These two reaction functions are displayed in Figure 1 for the case Image 100002010000003F00000019835D476FB31C03C2.png , and may be compared with those depicted by Cournot (1838) in his Figure 2. They are hyperbolae.

  • 5 A slightly different approach is followed by Gibbons (1992, 75–79).
  • 6 Cf. Nash (1951). See also Gibbons (1992, 14–21). The concept was derived independently by Johnson ( (...)

We may now proceed with Johnson’s (1954) dynamic approach to equilibrium.5 Starting from free trade, if country 1 begins the tariff war, it imposes a tariff factor Image 100002010000001A00000019652F7D495603732E.png given by (1.16) for Image 1000020100000046000000191EE0A5C0CAA18A7A.png . Given this tariff factor, country 2 retaliates by following the same formula (with Image 100002010000001800000019DCBF7C39A26A2144.png and Image 100002010000001A00000019652F7D495603732E.png interchanged) for its new tariff factor Image 100002010000004700000019FFE54C696D5E49E1.png . We now show that there is an equilibrium in this game; in fact it is a Nash equilibrium.6 If there is an equilibrium, then by the symmetry of the model both countries must impose the same tariff factor. Then, setting Image 1000020100000088000000193A7846EB72DCF89E.png in (1.16) and squaring both sides, we obtain the polynomial equation

Image 100002010000028B00000039492FCAB98A28B83E.png

which has one positive root Image 100002010000002300000019386DB2BC611AC34B.png and three negative roots Image 100002010000007300000019D07B8507D3957FAA.png Image 100002010000002300000019386DB2BC611AC34B.png . Thus an equilibrium exists, and the equilibrium tariff factor is Image 100002010000005300000019C5D39A1A7AE67A72.png .

Now we show that this Nash equilibrium is stable. If country 1 starts the tariff war under conditions of free trade Image 100002010000005700000019FACD960C4A47E465.png , it will clearly set its tariff factor Image 10000201000000460000001980E4FA2ED147C330.png . If country 2 had started the tariff war, it would of course have done the same. But now that country 1’s tariff factor Image 100002010000001A00000019652F7D495603732E.png has increased, country 2 will choose a lower tariff factor Image 100002010000001800000019DCBF7C39A26A2144.png than it would have done if it had started the tariff war, contradicting the intuition that a worse attack warrants a greater retaliation. This is because, from (1.16),

Image 100002010000028B0000005038C5C305346E0884.png

Now that country 2 has increased its tariff factor Image 100002010000001800000019DCBF7C39A26A2144.png , country 1 will reduce its tariff factor Image 100002010000001A00000019652F7D495603732E.png in the next round. And then country 2 will increase it tariff factor Image 100002010000001800000019DCBF7C39A26A2144.png . The two tariff factors will thus converge.

An illustration is given in the table below, where Image 100002010000003F00000019476F0DE24A17980D.png , hence the optimal tariff factor is Image 10000201000000B70000001952DF1E40DD73E27D.png . The dynamic process is illustrated in Figure 1, exhibiting the successive pairs (1,1),(1,1.581),(1.365,1.581),(1.365,1.431), (1.409,1.431), etc., showing that the equilibrium is virtually reached in four iterations. Note that in seven iterations the equilibrium tariff is reached with an accuracy of six decimal points.

T1

T2

1

1.58113883008

1.36453054220

1.43051123097

1.40903150706

1.41587780352

1.41368079121

1.41438429260

1.41415886865

1.41423108545

1.41420794843

1.41421536096

1.41421298615

1.41421374698

1.4. Welfare properties of the free-trade, optimal-tariff, and autarky equilibria

If Image 100002010000008700000019A2CDBCCDC2F6D636.png , the above formula (1.13) provides an indicator of each country’s welfare in an equilibrium with a common tariff factor Image 100002010000001000000019EDDFBBE40A1244FC.png , which we may denote

Image 100002010000028B00000067764FC046D8CB898A.png

It is of interest to compare this welfare level under three situations: free trade Image 100002010000004E00000019E4328834A7F62862.png , a common Cournot-Johnson-Nash equilibrium tariff Image 100002010000006000000019CD00A16C42767588.png , and autarky Image 100002010000005900000019CE1B7A842000B9E5.png . It is not hard to see that the optimal equilibrium with equal tariff factors Image 100002010000007900000019AD15BF8E38DD0A86.png (which includes subsidies for the case Image 100002010000006B000000191A70C4822AF7168D.png ) is the free-trade equilibrium Image 100002010000003D0000001900A12ADA78AD92C3.png . This is seen by differentiating Image 100002010000003300000019DB6AC4D1D7E31626.png with respect to Image 100002010000001000000019EDDFBBE40A1244FC.png , leading to the cubic equation

Image 100002010000026200000042F8C40207DBE0E199.png

which has the unique positive root Image 100002010000003D0000001900A12ADA78AD92C3.png .

The following table provides the welfare level of each country for three values of Image 10000201000000140000001958243EADBE2B2FC4.png , under the three alternative equilibria.

10It may be noted that in each case, the welfare level under the optimal-tariff equilibrium is slightly more than midway between those of autarky and free trade.

Image 100002010000028B0000004BA0759C14B4981620.png

Image 10000201000002F40000042A7C20DEE7391363F9.png

An illustration is provided in Figure 2 displaying the Marshallian offer curves for the case Image 10000201000000400000001960D9E8AC9B123C62.png , where from (1.7) the offer functions have the form

Image 100002010000028B00000062C84317D078E5F9CA.png

under free trade, and (as indicated by the dashed curves)

Image 100002010000028B0000005B4D04869AFB85DB1D.png

under the common Nash-equilibrium tariff factor Image 1000020100000049000000199757F01129304714.png . A tradeindifference curve for country 1 is drawn through the competitiveequilibrium point (.5,.5); this passes above the Nash-equilibrium point (.24,.24) (where from (1.10) we have Image 10000201000000FE000000196495B1F60AE6E095.png Image 1000020100000047000000196FAA61952D990350.png ), showing that country 1 is better off under free trade than under the Cournot-Johnson-Nash optimal-tariff equilibrium (and similarly for country 2). It should be emphasized that this conclusion follows from the strong symmetry assumptions we have made as between the countries; otherwise, as Johnson (1954) showed, one of the countries could be better off under the tariff-war equilibrium than under free trade.

2. Free trade and optimal tariffs under symmetrically disparate preferences and multiple equilibrium

  • 7 Also the unpublished paper by Yinggang Zhou, “A Tariff Reduces Unpredictability: A Solution to the (...)

11In this section I discuss an ingenious paper by Wan and Zhou (2008).7 They assume that there are aggregate preferences in each of two countries which can be represented by the quasi-linear, quadratic, and mirror-image utility functions

Image 100002010000028B0000005B994890A6F2DF7779.png

  • 8 If one replaces x11,x21, and Y1 in (2.2) below by the demands (...)

where Image 100002010000001F000000197DE84BF0647103F2.png is country Image 100002010000001D000000199EF79BF727E9E554.png consumption of its own export good and Image 100002010000001700000019FA34717FFD63A2C4.png its consumption of its import good Image 1000020100000036000000193C5A5EF1BA9366FE.png . Note that for marginal utilities to be positive, these functions are well-defined only for Image 100002010000003B000000190CF08E88D57428D5.png Image 100002010000000C00000019F5488AD529F34BFF.png and Image 100002010000004E00000019D2DAA2AD4CE339B1.png , which we hereby assume. Note also that they are each linear in the country’s export good. Thus, individual preferences are “parallel’’ with respect to the country’s export good (which plays the role of money: cf. Boulding 1945, 857) permitting their aggregation into preferences for the respective countries (but of course not for the world).8

Wan and Zhou assume further that country Image 100002010000000D000000192662C17CABF073EC.png produces exactly one unit of its export good (commodity Image 100002010000000D000000192662C17CABF073EC.png ), and none of its import good (commodity Image 10000201000000370000001994B2CC061E6D045F.png ). In the notation of section 1 we have Image 1000020100000050000000197E0F8BE0E511CD38.png and Image 100002010000004E000000196C701683C71B0FCD.png for Image 100002010000003500000019FAA0C61CC0ABDA6E.png .

Maximizing country 1’s Wan-Zhou utility function U1 in (2.1) subject to the budget constraint Image 10000201000000D70000001947184DA39CBA6723.png , we obtain the demand functions

Image 100002010000028B000000518100AB8C96077820.png

where Image 100002010000001A00000019682A942A44E01C42.png is country 1’s disposable national income, equal to the sum of its domestic product, which is simply the domestic (and world) price Image 1000020100000016000000190722FE94357CDCF4.png of its export good (commodity 1), and its tariff revenues Image 100002010000008600000019417417F60C434E53.png , where Image 100002010000001700000019AD9DF07E9C8DD28E.png is the world price of its import good (commodity 2) and Image 100002010000007500000019A3999271EB80320C.png is the negative of its import of good 2. Since from (2.2) its demand for imports is independent of its income, its excess-demand function for its import good, expressed as a function of world prices, is simply

Image 100002010000028B0000004C1C70B50FBEC9C764.png

Proceeding somewhat as in section 1, we combine (2.3) with the balanced-trade condition Image 10000201000000B9000000192005EB6DF4B976E3.png and the material-balance condition Image 1000020100000076000000196FC2E08441B963CD.png to get

Image 100002010000028B00000035BFF84BAC1D39799F.png

(expressed in world prices). Substituting (2.4) into (2.3) we obtain

Image 100002010000028B0000004A2608893FCDF446E5.png

whence country 1’s tariff-inclusive offer function is

Image 100002010000028B00000033CD350E52A1EAE4E1.png

By symmetry, a similar formula obtains for country 2:

Image 100002010000028B00000035CC589D131F098FE9.png

Wan and Zhou compute the optimal tariff factors to be Image 100002010000009B000000191E3A683F7192D342.png Under free trade Image 100002010000008F00000019AE2F44CC84FC35FA.png , there are three solutions to (2.5) and (2.6), namely

Image 100002010000028B0000006130370C107FEEADE0.png

12which compute to

Image 100002010000028B00000035B59147E6D8162858.png

Figure 3: Optimal tariffs versus free trade under Wan-Zhou preferences

Image 10000201000002AF0000029108C89AF9CE1F6E10.png

  • 9 The possibility of multiple equilibrium in economics seems to have been first noticed by Mill ([185 (...)
  • 10 For a phase diagram of this case, see Figure 12 in Chipman (1987, 936; 2008, 2016).

(see Figure 3).9 As has been well known since the time of Marshall (1879), the first and third of these equilibria are dynamically stable, but the second is unstable.10 However, under the optimal tariffs Image 100002010000003E000000196DB081BC918F05AE.png Image 100002010000006800000019648DCDBAA9C6DD78.png , there is a single equilibrium

Image 100002010000028B00000037C3E6AE300E89BBA5.png

which is stable (again see Figure 3).

Figure 3 is the Marshallian diagram (Marshall 1879, Figure 8; 1923, 353, Figure 20)—still current in the literature on international trade—which is similar to the “Edgeworth box’’ but with country 1’s origin at the southeast corner Image 10000201000000A8000000191FFEA90D8656397C.png where Image 10000201000000910000001967B61AB0F09EEBF1.png Image 100002010000004C000000193347344701E35A48.png and Image 100002010000006000000019C9E84CC324C560A0.png hence Image 10000201000000A800000019E2679A1343B5267D.png ; thus country 1’s direction of preference is northwest. Likewise, country 2’s origin is at the northwest corner (0,1), and its preference direction is southeast. It is clear from the diagram that the stable free-trade equilibrium (.905, .345) is inferior for country 1 to the optimal-tariff equilibrium (.375, .375), being dominated in both components. Of course, the latter is in turn also inferior for country 1 to (because dominated by) the other stable equilibrium (.345, .905).

The following table provides information about the welfare levels of country 1 and country 2 at the three stable equilibrium points shown in Figure 3, where Image 100002010000013B000000196B6132820F036885.png and Image 1000020100000141000000190B299C1D4712789E.png

Image 10000201000004D40000015FE8D2079C1A4FE3AB.png

Starting from the stable optimal-tariff equilibrium (.375, .375), which affords welfare levels of 1.84375 for both countries, and faced with the option of free trade, policy makers in country 1 are confronted with the risk of moving to the stable free-trade equilibrium (.90451, .34549) with an inferior welfare level of 1.23872, or moving to the other stable free-trade equilibrium (.34549, .90451) with a superior welfare level of 2.63627. If these policy-makers are minimaxers, they will choose to remain in their Johnson-Nash equilibrium (.375, .375) (see Figure 3). If they are Bayesians, and if they consider the two stable free-trade equlibria to be equally likely a priori, then since the expected utility is 1.937495 they will take a chance on free trade, provided the given numerical utility function correctly reflects their risk preferences. However, if instead one replaces Image 1000020100000078000000190BFB1CADD29847B7.png by its (natural) logarithm, the expected utility of free trade will be less than the utility of protection. And it is not as if the policy-makers could choose free trade one year and optimal tariffs the next; they must consider a succession of years under the same free-trade equilibrium versus a succession of years under the same optimal-tariff equilibrium.

13An interesting policy proposal made by Wan and Zhou is for the countries to negotiate a common tariff that is less than that of the Cournot-Johnson-Nash equilibrium (and thus closer to free trade), but still large enough to rule out multiple equilibria.

14All in all, this contribution may be considered as a fascinating and impressive achievement. It would provide an ideal basis for international negotiation of tariff reduction, provided the model fits the data precisely. But this question of its realism and relevance requires some discussion.

  • 11 Of course, this assumes that either transport costs or trade impediments are present to cause these (...)

15I start with the question of the basis for international trade, which has long been considered to be comparative advantage in some sense. That doctrine is based on the premise that the main basis for trade is relative differences in resources and factor endowments, as opposed to differences in tastes. Houthakker (1957), in examining the reasons for differing consumption patterns in different countries, came to the conclusion that these differences could be explained largely by differences in relative prices, i.e., the higher consumption of pork relative to beef in Germany in comparison with France could be explained by differences in their relative prices in the two countries in that period, rather than by differences in tastes.11

16On the other hand, both the Ricardian and Heckscher-Ohlin explanations of trade patterns are based entirely on differences either in “climate’’ or in resource endowments. True, for the HeckscherOhlin theorem to hold, one must make stringent assumptions about preferences, namely that they are identical and homothetic within and between countries (Chipman 1987, 938; 2008, 218). These assumptions are undoubtedly unrealistic; indeed, homotheticity of preferences violates Engel’s law. And there are convincing examples of changes in trade patterns that reflect differences in relative demands resulting from different income levels and Engel’s law (Minabe 1966, 1205).

  • 12 This contrasts with the situation under CES preferences (cf. Chipman 2010) in which multiple equili (...)

But in the Wan-Zhou model, one must have both different endowments and different preferences, acting in concert. Given the demand functions (2.2), at unitary relative prices we have Image 100002010000006900000019156A21D61C2263F9.png and Image 1000020100000069000000194D944BFD29ECE0C7.png in country 1, and likewise Image 100002010000006900000019444A4A2DF9C28E82.png and Image 100002010000006900000019B7883ABA9AC3B16D.png in country 2, indicating very different consumption patterns, with each country exhibiting a strong relative preference for its import good.12

17And the model has each country producing only its export good and no import-competing goods. This appears to remove the main political force behind tariff protection. This latter assumption has been somewhat relaxed in Zhou (2006) (see footnote 7 above), but the combined force of the two assumptions remains.

  • 13 An alternative but hard-to-interpret sufficient condition has been proposed by Arrow and Hahn (1971 (...)

In the literature on international trade, there has been much attention devoted to comparison of free trade and autarky, but very little until recently to comparison of free trade and restricted or tariff-encumbered trade. The importance of multiple equilibrium to the latter has now been demonstrated by Wan and Zhou. But remarkably, in the literature on general equilibrium there has been very little attention paid to it. The standard result that one finds in the literature is that of Arrow, Block, and Hurwicz (1959, 89–90), who proved that a sufficient condition for general exchange equilibrium to be unique is that commodities be gross substitutes in consumption (Metzler 1945): that is, that a rise in the price of one commodity should lead to a rise in the excess demands for all the other commodities.13 This result followed from the proof of uniqueness in Wald (1936, 652–6; 1961, 383–7), which I discussed in Chipman (1965, 725–7; 2008, 86–8) in relation to the application of Wald’s condition to the case in which utility functions have the CES (constant elasticity of substitution) form Image 100002010000007B00000019B1C61070F1FEE838.png , where Image 1000020100000041000000191F867D9B17115B00.png is the constant elasticity of substitution. Wald’s sufficient condition for uniqueness in this case reduces to Image 100002010000003E00000019087C4366BAB76B61.png . This is a very stringent condition! But it is by no means necessary, since it turns out that in a two-agent mirror-image pure-exchange economy in which both agents have CES utility functions with Image 100002010000002E00000019B1B797B867A09B90.png , competitive equilibrium is necessarily unique; and even if Image 100002010000002C0000001948C70EC01D344B0A.png , multiple equilibrium is extremely rare, requiring consumers in each country to have a very high relative preference for its export good. I discuss this in detail in Chipman (2010, Theorem 2, 136).

There are many ways in which the model presented here could be improved. In particular, it would be desirable to allow for variable production as represented by supply (“Rybczynski’’) functions Image 100002010000002D00000019976BC05F458A780E.png Image 100002010000007900000019D92547D3FBCD96FE.png  expressing the output of commodity Image 1000020100000007000000192EB492AF44848179.png in country Image 100002010000000A00000019BE970400EF72B959.png as a function of the world prices and the endowments of labor Image 100002010000001200000019D42AD977B00E5C1B.png and capital Image 10000201000000140000001950AE6DF8560E36FC.png in country Image 100002010000000A00000019BE970400EF72B959.png ; cf. Rybczynski (1955), Chipman (1987, 931–2; 2008, 203–6). Still a further relaxation of assumptions could be provided by allowing separate preferences for labor and capital in each country, as in Johnson (1959; 1960). Such a model would undoubtedly provide wide scope for multiple world equilibrium, yet still retain enough structure to give rise to interesting results.

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Notes

1 Cf. Mill (1852, Vol. II, Book III, Ch. XVIII, 122–151; 1865, 124–153). See also Chipman (1965, 483–491; 1979), Melvin (1969), Harwitz (1972), and Appleyard and Ingram (1979). The given utility functions constitute a special case of identical homothetic preferences, allowing one to aggregate preferences within each country (cf. Chipman 1974).

2 Note that in the absence of saving and capital movements, if country 1 imposes a tariff on its import of commodity 2, it will have a deficit in its balance of trade when reckoned in its own domestic prices, even though its trade will be balanced when reckoned in world prices. This is because the tariff revenues (which we assume to be distributed in lump-sum fashion by the government to the population) bring about and thus correspond to an excess of domestic consumption over domestic production at domestic prices, hence they constitute a deficit in country 1’s balance of trade when expressed in its domestic prices; thus, country 1’s budget (balance-oftrade) equation
Image 100002010000018E00000019E12F9EF93D1330A7.png
which states that its balance of trade denominated in domestic prices is equal to its tariff revenues, immediately implies that p11z11 + p21z21 = 0, i.e., that its trade is balanced when expressed in external prices.

3 The idea of a reaction function was originated by Cournot (1838, Ch. VII, 90 and Fig. 2; 1897, 81). The terminology appears to be due to Fellner (1949, p. 59), who provided an excellent exposition. He also referred to it there (pp. 64, 68n) as a “Cournot function’’.

4 Following Torrens (1844, esp. 331–372), the first systematic discussion of the benefit to a country of a tariff appears to be that of Mill (1844), who held (p. 27) that it “almost always falls in part upon the foreigners who consume our goods.’’ The first formula for an optimal tariff rate appearing in the literature appears to be that of Launhardt (1885, §17, 85; 1993, Ch. 17, 86), which is based on a quadratic utility function. The next is one due to Auspitz and Lieben (1889, Book VI, Ch. 80–82, 408–429; 1890; 1914, 267–280), who provided a geometric treatment on the assumption that the marginal utility of the country’s export good is constant, followed (1889, 442; 1914, 288) by a formula x · V(x) for the optimal tariff rate (see Niehans and Jäggi 1995, 382), where V(x) is their curve (Fig. 77) of the “cost of exports and benefit of imports’’. The third is that of Bickerdike (1907), followed by Edgeworth (1908), who provided formulas based on the assumption of vanishing cross-elasticities (for a discussion see Chipman 1993). The final and definitive one was that of Johnson (1951, 29), who acknowledged the assistance of J. de V. Graaff.

5 A slightly different approach is followed by Gibbons (1992, 75–79).

6 Cf. Nash (1951). See also Gibbons (1992, 14–21). The concept was derived independently by Johnson (1954), and of course goes back to Cournot (1838, Ch. VII, 88–100; 1897, 79–89, and Figures 2 and 3). However, Johnson (1954, 148–9) showed that in general there need not be an equilibrium; there could instead be a limit cycle.

7 Also the unpublished paper by Yinggang Zhou, “A Tariff Reduces Unpredictability: A Solution to the Tariff Reduction Paradox’’ (2006), Cornell University, Ithaca, NY.

8 If one replaces x11,x21, and Y1 in (2.2) below by the demands Image 100002010000002D000000149CFD6F8998848686.png and income Image 100002010000001200000014372878C6D6F890AF.png of the νth individual in country 1, and then averages the two demands over the N individuals, one obtains exactly the same formulas, where xi1 and Y1 are respectively replaced by the averages Image 1000020100000095000000140F05088D38EBD8CB.png and Image 100002010000002900000014EEBF5751A15C0A3D.png Image 100002010000004A0000001439FB2C6FB2A90320.png  This shows that the “parallel preferences’’ (with respect to the export good) can be aggregated over the whole economy, and thus generated by utility functions (2.1) expressed in terms of these averages.

9 The possibility of multiple equilibrium in economics seems to have been first noticed by Mill ([1852], 1865: Vol. II, Book III, Ch. XVIII, §6), in a passage referred to by Marshall (1879, 15) who pointed out that Mill “has seen that under certain curcumstances there may be several different positions of equilibrium of trade’’; but Marshall objected that “it appears to me that the special example which he has chosen does not illustrate the general problem in question.’’ (Mill’s analysis in his Principles built upon his own earlier 1844 work, from which he quoted.) Marshall proceeded to use his own diagrammatical analysis to illustrate the problem and demonstrate the possibility of multiple equilibrium.

10 For a phase diagram of this case, see Figure 12 in Chipman (1987, 936; 2008, 2016).

11 Of course, this assumes that either transport costs or trade impediments are present to cause these differences in relative prices, contrary to the model being presented here.

12 This contrasts with the situation under CES preferences (cf. Chipman 2010) in which multiple equilibrium requires each country to have a relative preference for its own export good. The Wan-Zhou result is also an exception to Mas-Colell’s rule (1991, 285) that “uniqueness is more likely in models that generate a large volume of trade.’’

13 An alternative but hard-to-interpret sufficient condition has been proposed by Arrow and Hahn (1971, 211).

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John S. Chipman, « General Equilibrium and Welfare in International Trade »Œconomia [En ligne], 2-1 | 2012, mis en ligne le 01 juin 2015, consulté le 29 mars 2024. URL : http://journals.openedition.org/oeconomia/1641 ; DOI : https://doi.org/10.4000/oeconomia.1641

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John S. Chipman

University of Minnesota - jchipman@umn.edu

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