李嘉图模型中的4个数字

The true meaning of David Ricardo’sfour magic numbersAndrea Maneschi *Department of Economics,Vanderbilt University,Nashville,TN,USAReceived 17October 2002;received in revised form 1November 2002;accepted 20November 2002AbstractThe four numbers in David Ricardo’s example of comparative advantage have been traditionally interpreted as unit labor coefficients in the production of wine and cloth in UK and Portugal.A recent interpretation suggests that they represent instead the labor needed to produce the amounts of wine and cloth actually traded.Ricardo’s four numbers are shown to yield each country’s gains from trade by simply subtracting two of the numbers from the other two.Since the numbers also indicate each country’s comparative advantage,Ricardo established a close connection between comparative advantage and the gains from trade.D 2003Elsevier B.V .All rights reserved.Keywords:David Ricardo;Comparative advantage;Gains from tradeJEL classification:F10;B121.IntroductionDavid Ricardo’s famous paragraphs in his Principles of Political Economy and Taxation of 1817formulating the principle of comparative advantage have often been quoted:The quantity of wine which she [Portugal]shall give in exchange for the cloth of England,is not determined by the respective quantities of labour devoted to the production of each,as it would be,if both commodities were manufactured in England,or both in Portugal.0022-1996/$-see front matter D 2003Elsevier B.V .All rights reserved.doi:10.1016/S0022-1996(03)00008-4*Tel.:+1-615-322-2993;fax:+1-615-343-8495.E-mail address:andrea.maneschi@ (A.Maneschi)./locate/econbaseJournal of International Economics 62(2004)433–443England may be so circumstanced,that to produce the cloth may require the labour of 100men for one year;and if she attempted to make the wine,it might require the labour of 120men for the same time.England would therefore find it her interest to import wine,and to purchase it by the exportation of cloth.To produce the wine in Portugal,might require only the labour of 80men for one year,and to produce the cloth in the same country,might require the labour of 90men for the same time.It would therefore be advantageous for her to export wine in exchange for cloth.This exchange might even take place,notwithstanding that the commodity imported by Portugal could be produced there with less labour than in England.Though she could make the cloth with the labour of 90men,she would import it from a country where it required the labour of 100men to produce it,because it would be advantageous to her rather to employ her capital in the production of wine,for which she would obtain more cloth from England,than she could produce by diverting a portion of her capital from the cultivation of vines to the manufacture of cloth.Thus England would give the produce of the labour of 100men,for the produce of the labour of 80.(Ricardo,1817,pp.134–135)These paragraphs contain what Paul Samuelson (1969)referred to as the ‘four magic numbers’denoting the amounts of labor needed to produce wine and cloth in UK (120,100)and Portugal (80,90).The traditional interpretation of these numbers is that they represent the amounts of labor needed to produce one unit of each commodity in each country,that is,that they are labor input–output coefficients.This interpretation appears in nearly all textbooks of international trade,where the ‘Ricardian’model,presented as the simplest representation of the rationale for trade,is based on fixed labor coefficients.While most authors do not quote from Ricardo’s Principles or use his numbers,others illustrate this model by explicit reference to the four numbers.1With the additional assumption that these coefficients are constant,the textbook Ricardian model yields linear production possibility frontiers and complete specialization in both countries (except that if one of the countries happens to be ‘large’,it remains nonspecialized).Present-day textbooks can cite illustrious precedents for this interpretation of the Ricardian model.For example,Gottfried Haberler (1936,p.128)began his analysis of the theory of comparative cost by asserting that ‘‘In chapter VII of his Principles he [Ricardo]gives the following celebrated example:In England a unit of cloth costs 100and a unit of wine 120units of labour;in Portugal a unit of cloth costs 90and a unit of wine 80units of labour’’.Jacob Viner (1937,p.445)presented a table containing the same four numbers,described as the amounts of ‘labor required for producing a unit’of cloth and wine in UK and Portugal.Basing himself on the same interpretation of Ricardo’s passage,John S.Chipman criticized him for faulty logic:the second paragraph quoted above ‘‘is a non1Caves et al.(1993),Kenen (1994)and Krugman and Obstfeld (2000)are examples of the former approach;Appleyard and Field (2001)and So ¨dersten and Reed (1994)of the latter.Appleyard and Field present Ricardo’s four numbers in a table (p.29)titled ‘‘Ricardian Production Conditions in England and Portugal’’,while So ¨dersten and Reed show a table (p.6)that identifies them as ‘‘labour cost of production (in hours)’’.A.Maneschi /Journal of International Economics 62(2004)433–443434A.Maneschi/Journal of International Economics62(2004)433–443435 sequitur,since nothing so far has been said about Portugal;indeed,the argument could be turned around merely by redefining the units of measurement’’.The first two sentences of the third paragraph are‘‘equally unsatisfactory,except when read in conjunction with the first’’(Chipman,1965,pp.479–80).Other authors,including the present writer(Man-eschi,1998,p.53),have echoed Chipman’s interpretation.A troubling aspect of this criticism is that it implies an illogical(or,at the very least,careless)thought process on the part of the master logician of political economy.Since many international economists believe that comparative advantage was Ricardo’s most significant contribution to classical political economy,it is unfortunate to have to qualify one’s admiration of his formulation of this principle by pointing out its imperfections.Section2sets out a new interpretation by Ruffin(2002)of Ricardo’s numerical example,which absolves him of any such criticism and presents the first clear interpre-tation of the meaning of the four magic numbers.Ruffin,however,did not expand on how Ricardo’s formulation of comparative advantage is intimately related to the gains from trade,as I wish to argue in this article.Ricardo’s measure of the gains from trade is consistent with what Viner named the‘eighteenth-century rule’characterizing such gains. Section3shows that the gains from trade according to this rule are obtained simply by subtracting two of the four numbers from the other two.The concluding Section4 examines the reasons why the new interpretation of Ricardo’s four numbers was over-looked for so long,and how the adoption of the standard interpretation might have affected the evolution of the theory of international trade.2.A new interpretation of Ricardo’s four numbersA recent reinterpretation of the above passage from Ricardo’s Principles has cast a new light on it,and at the same time rescued Ricardo from the charge of inconsistency or carelessness.In fact,it shows to great advantage the revolutionary character of Ricardo’s contribution.Ruffin(2002)convincingly argued that the four labor input numbers presented by Ricardo are not input–output coefficients,but rather the quantities of labor needed to produce the amounts of wine and cloth actually traded by UK and Portugal.This is implied by Ricardo’s usage of the terms‘the cloth’and‘the wine’in the second paragraph quoted above,which refer to the amounts traded that were mentioned in the first paragraph.As Ruffin states,Let X be‘the quantity of wine’that is traded for Y units of cloth.If England requires 120men for one year to make X units of wine and100men to make Y units of cloth,‘‘England would therefore find it her interest to import wine,and to purchase it by the exportation of cloth.’’He[Ricardo]then went on to Portugal,which required80men to produce the wine and90to produce the cloth.Obviously,Portugal would save10men producing X wine and trading it for England’s Y cloth.(Ruffin,2002,pp.741–742) He adds(p.742)that‘‘Ricardo’s proof is elegant,simple,and sublime.It not only uses the separation property of the law of comparative advantage,but the logical structure applies to any number of goods or countries,unlike textbook expositions’’.Further reflection on Ruffin’s novel interpretation of Ricardo’s numerical example has convinced me that it can be developed so as to yield new insights into the relationship between comparative advantage and the gains from trade.Given that X units of Portuguese wine are traded for Y units of English cloth (Ricardo never assigned values to X and Y ),the unit labor coefficients for wine and cloth in UK and Portugal are given by the following table:The terms of trade are Y /X units of cloth per unit of wine.In UK the opportunity cost of wine is 1.2Y /X units of cloth,while in Portugal it is 0.89Y /X ,so that Portugal has a comparative advantage in wine.Instead of following this procedure in terms of the unknown variables X and Y ,economists have tended to infer the pattern of comparative advantage,and hence the direction of trade,by interpreting the ‘four magic numbers’as unit labor coefficients.They used them to derive the autarky price ratios for the two countries,and offered reasons that explain why the terms of trade usually lie between them.Ricardo is often mildly reprimanded for announcing the terms of trade in the fourth paragraph of the above passage without explaining their determination,a task later undertaken by John Stuart Mill.With the new interpretation of the four numbers,we see that Ricardo’s method was much more direct.By stipulating from the start that certain quantities of wine and cloth exchange for each other in the trade equilibrium,he essentially began with the terms of trade,and went on to specify domestic exchange ratios in the two countries that are consistent with them and lie on either side of them.2The fact that UK uses 100men to produce the cloth she exports,whereas she would need 120men to produce the wine she imports,immediately establishes her comparative advantage in cloth without requiring any knowledge of Portugal’s labor inputs.Likewise,Portugal’s comparative advantage in wine is established by her requiring 80men to produce enough wine to pay for the cloth which she would otherwise produce with the labor of 90men.Thus,in the same breath,Ricardo informed his readers not only about the pattern of comparative advantage,but about each country’s gains from trade.In the case of UK,these gains are given by the difference between the two numbers he cited for wine and cloth,120À100=20,while in the case of Portugal they amount to 90À80=10.They express the labor that each country saves whenWineCloth UK 120/X100/Y Portugal80/X 90/Y 2Ruffin (2002)argues persuasively that Ricardo’s primary concern (as expressed in the first paragraph of the passage from the Principles quoted above)was not how the commodity terms of trade are determined,but the fact that the double factoral terms of trade between two trading countries are usually not equal to unity.While it is theoretically possible (though unlikely)that they equal unity when each country has an absolute advantage in its export commodity,this is impossible when (as in Ricardo’s numerical example)Portugal has an absolute advantage in both commodities.Hence the labor theory of value cannot be relied on as a guide for the determination of international prices.A.Maneschi /Journal of International Economics 62(2004)433–443436A.Maneschi/Journal of International Economics62(2004)433–443437 it trades with the other instead of remaining self-sufficient.These gains,of course,can be realized only if it specializes according to its comparative advantage.Gains from trade accrue to each country as long as the amount of labor embodied in its exports falls short of the labor it would need to produce the commodities it imports.This guarantees that the terms of trade fall inside the cone spanned by the two autarky price ratios.The economy of thought and the richness of results to be drawn from Ricardo’s example are truly astounding.The way in which Ricardo expressed the gains from trade is consistent with a long tradition that preceded him,denoted by Viner as the‘eighteenth-century rule’for the gains from trade.It states that‘‘it pays to import commodities from abroad whenever they can be obtained in exchange for exports at a smaller real cost than their production at home would entail’’(Viner,1937,p.440).It therefore reflects the benefits of trade viewed as an indirect method of production.Consider this in the context of Ricardo’s two-good English economy,where good C(cloth)is exported,good W(wine)is imported,and labor is the only factor of production.Let the superscript t represent the trade equilibrium.If a i is the amount of labor needed to produce one unit of good i,one unit of labor can produce either1=a W units of wine or1=a C units of cloth,which can be exchanged forð1=a CÞðp C=p WÞt units of wine at the terms of tradeðp C=p WÞt:This trade is beneficial at the margin as long asð1=a CÞðp C=p WÞt>1=a W;ð1Þorðp C=p WÞt>a C=a W¼MRT;ð1aÞsince the marginal rate of transformation(MRT)equals the ratio of the marginal costs of producing cloth and wine.The economy produces more cloth untilðp C=p WÞt¼MRT;so that the terms of trade line is tangent to the transformation curve,or until it specializes fully in cloth.The assumption of increasing opportunity costs in the transformation of wine into cloth is used and justified in Section3.It is more general than the textbook version of the Ricardian model,where the transformation curve is linear and each trading economy becomes fully specialized.3.Ricardo’s measure of the gains from tradeRather than basing the gains from trade on a unit of labor shifted from the importable to the exportable sector,as implied by(1),Ricardo expressed these gains with reference to a country’s total imports and exports.To express his measure algebraically,assume that m W units of wine,rather than produced directly,are imported in exchange for exports x C of cloth.These amounts are shown in Fig.1,where the autarky equilibrium is at A and the terms of trade are given by the slope of PC.If consumption occurs at point C,the trade triangle is CBP,its sides measuring exports BP of cloth and imports CB of wine.If y i is the output of commodity i and S i the amount of labor used in sector i,let y C¼fðS CÞand y W¼gðS WÞbe the production functions of cloth and wine in UK.Assume that f V ;g V >0;f W ;g W V 0;and either f W or g W <0;so that there are diminishing returns to labor in the production of at least one good.If L is the economy’s fully employed labor force,L ¼S C þS W ¼f À1ðy C Þþg À1ðy W Þ,where f À1and g À1are the inverse functions of f and g .This equation yields the economy’s concave transformation curve shown in Fig.1.3If d t i represents the demand for commodity i in UK when it trades with Portugal,theamount of labor required to produce x C ¼y t C Àd t C cloth exports is f À1ðy t C ÞÀf À1ðd t C Þ:The labor that would be required to produce m W ¼d t W Ày t W wine imports is g À1ðd t W ÞÀg À1ðy t W ).According to Ricardo,the amount of labor saved (LS)through trade is thereforeLS ¼g À1ðd t WÞÀg À1ðy t W ÞÀf À1ðy t C Þþf À1ðd t C Þ¼120À100¼20men :ð2Þ3Findlay (1974)obtained a concave transformation curve when he assumed constant returns to labor in the manufacturing sector ðf W ¼0Þand diminishing returns in the agricultural sector ðg W <0Þ:The technological assumptions made in this section are analogous to those underlying the Ricardo–Viner (sometimes called ‘specific-factors’)model,which assumes that each sector of the economy uses a factor of production specific to it as well as a ‘mobile’factor,usually taken to be labor,common to all sectors.Krugman and Obstfeld (2000)discuss this model in the chapter immediately following that devoted to the ‘Ricardian’model.In a box titled ‘Specific Factors and the Beginnings of Trade Theory’,they state that ‘‘Yet almost surely the British economy of 1817was better described by a specific factors model than by the one-factor model Ricardo presented’’(p.58).They implicitly assume that the agricultural (corn)sector uses land as its specific factor,that capital is specific to the manufacturing sector,and that both sectors share the mobile factor,labor.They go on to discuss the distributional implications of the Corn Laws for landlords and capitalists,and the reasons for Ricardo’s opposition to theselaws.parative advantage and Ricardo’s measure of the gains from trade.A.Maneschi /Journal of International Economics 62(2004)433–443438A.Maneschi/Journal of International Economics62(2004)433–443439 Since L¼fÀ1ðy t CÞþgÀ1ðy t WÞ;(2)can be rewritten asLS¼fÀ1ðd t CÞþgÀ1ðd t WÞÀL:ð2aÞA measure of the gains from trade often used in neoclassical trade theory is the Hicksian equivalent variation(EV),defined as the amount of income that must be added to national income evaluated at autarky prices at point A in order to attain the level of welfare realized at point C.If the‘equivalence’underlying the equivalent variation is of the Slutsky type,EV S,the community is able to buy the free-trade consumption bundle itself at the autarky prices,so thatEV S¼p aðd tÀd aÞ;ð3Þwhere p and d are the price and demand vectors,and the superscript a represents the autarky equilibrium.4If cloth is the numeraire and CH is drawn parallel to AG,EV S is measured in Fig.1by the difference GH between the values of the consumption bundles at C and A at the autarky prices given by the slope of the line AG.5An alternative measure of the gains from trade is the compensating variation(CV),defined as the income that must be taken away when the community consumes bundle C at free trade prices in order to reduce welfare to its autarky level.A measure that is analogous to both the EV and the CV is the difference between the amount of labor needed to produce the consumption bundle under trade(d t C;d t W)and that needed to produce the bundle(d a C;d a W)consumed under autarky.It is the additional labor force,over and above L;that would be needed in autarky to produce the consumption bundle available under free trade.It is here denoted as the equivalent variation in terms of labor,or EV L.It differs from both the EV and CV since it is expressed in terms of the labor input rather than of expenditure.Since in autarky d a C¼y a C;d a W¼y a W;and fÀ1ðy a CÞþgÀ1ðy a WÞ¼L;this measure is given byEV L¼fÀ1ðd t CÞþgÀ1ðd t WÞÀfÀ1ðy a CÞÀgÀ1ðy a WÞ¼fÀ1ðd t CÞþgÀ1ðd t WÞÀL:ð4ÞThis is identical to the value of LS in(2a),and therefore to the measure proposed by Ricardo.The beauty and appeal of Ricardo’s measure is that it coincides with EV L,and is thus analogous to the neoclassical measures of the gains from trade,the equivalent and compensating variations.The transformation curve shown in Fig.1,characterized by increasing opportunity costs in transforming one commodity into the other and incomplete specialization,is4On the difference between Hicks and Slutsky compensations,see Varian(1992,pp.135–137).While not shown in Fig.1,it is clear that the Slutsky EV usually exceeds the Hicksian EV.5Note that the Slutsky equivalent variation is smaller than p a m;the value of net imports at autarky prices, where m is the economy’s net import vector.The value of p a m is known to be positive thanks to the generalization of the concept of comparative advantage by Deardorff(1980)and Dixit and Norman(1980).In terms of cloth,it is given in Fig.1byÀx Cþðp W=p CÞa m W¼ÀBPþðp W=p CÞa CB¼ÀBPþðBH=CBÞCB¼PH;which exceeds the value of the equivalent variation given by GH.consistent with Ricardo’s statement that ‘‘a country possessing very considerable advan-tages in machinery and skill,and which may therefore be enabled to manufacture commodities with much less labour than her neighbours,may,in return for such commodities,import a portion of the corn required for its consumption’’(Ricardo,1817,p.136;emphasis added).This implies that some of the corn consumed is produced domestically.The passage from Ricardo’s Principles quoted at the beginning of this article nowhere states or implies that specialization is complete after trade in either UK or Portugal.A concave transformation curve is also ideally suited to portray both the static and the dynamic gains from trade.If the cloth and wine in Ricardo’s example are relabeled as ‘manufactures’and ‘corn’(a term used by the classical economists to denote wheat and other grains),it is easy to show that when the country with a comparative advantage in manufactures imports corn,whose production is subject to diminishing returns,it enjoys an increase in the profit rate caused by a rise in the marginal product of labor in the corn sector (Findlay,1974,1984;Maneschi,1983).A ‘‘rise in the rate of profits’’is mentioned by Ricardo (1817,p.132)as a gain from trade on a par with the static gain discussed above.Since a higher profit rate is associated with a higher rate of capital accumulation,the economy enjoys dynamic gains from trade by moving to a trajectory marked by a higher growth rate.6Another consequence of assuming a concave transformation curve and incomplete specialization is that the trade triangle is much smaller than that associated with a linear transformation curve leading to complete specialization.The smaller volume of trade may explain the relatively small numbers of workers associated with the production of exports and imports in UK (100,120)and Portugal (80,90)in Ricardo’s example.4.Concluding remarksThe gains from trade outlined above,in terms of the amount of labor saved by trading exports for importable goods instead of producing the latter directly,represent the static benefits from exploiting trade as an indirect method of production.In Ricardo’s example,Portugal’s comparative advantage can be deduced from the fact that her relative productivity advantage in wine is 33%,while that in cloth is 10%.Commentators have often speculated on why Ricardo picked his four numbers so that Portugal holds an absolute advantage over UK in producing both commodities.These numbers strike one as especially peculiar with regard to cloth,given that the industrial revolution occurred in UK rather than Portugal.A possible explanation for Ricardo’s choice is that he wished to make the rhetorical point that,even if she were inefficient in producing both commodities,UK could end up garnering the lion’s share (20/30or two-thirds)of the worldwide gains from trade.The new interpretation of Ricardo’s four numbers is consistent with diminishing returns in production,so that a trading economy need not fully specialize in its export good.The6This dynamic interpretation of the gains from trade,consistent with Ricardo’s insistence throughout his Principles of Political Economy that Britain’s corn trade should be liberalized,is stressed in Maneschi (1992).A.Maneschi /Journal of International Economics 62(2004)433–443440A.Maneschi/Journal of International Economics62(2004)433–443441 corollary is that there is a dynamic as well as a static component of the gains from trade. This conforms with Ricardo’s opinion,forcefully expressed in Parliament as well as in his writings,that the Corn Laws led to a decline in Britain’s rate of profit.Their repeal would reverse this trend,since cheaper corn imports would boost the profit rate and restore dynamism to the British economy.If the above interpretation of Ricardo’s four numbers is correct,why did most economists,including some very distinguished ones,7for almost two centuries interpret them instead to be unit labor coefficients in the production of wine and cloth in the two countries?A second and equally challenging question is whether this interpretation distorted in any way the profession’s understanding of the nature of Ricardian comparative advantage.In answering the first question,recall that John Stuart Mill(1844,1848) completed Ricardo’s theory by developing the principle of reciprocal demand as a determinant of the terms of trade.In both his books,he cited paragraphs from his father’s Elements of Political Economy(James Mill,1844)that contained examples of Ricardian comparative advantage relating to English cloth being traded for Polish corn.Just as in Ricardo’s case,James Mill’s four numbers represented the labor needed to produce the amounts actually traded by the two countries.But John l also inserted examples of English broad cloth being exchanged for German linen that were expressed in terms of the domestic and international terms of trade,not in terms of the amounts of labor embodied in the commodities traded.This had the effect of focusing attention on the two trading economies’autarky price ratios as limits within which the terms of trade must lie.It was natural for subsequent economists to assume(in line with Ricardo’s own labor theory of value)that these autarky price ratios were given by labor inputs per unit of output,and that Ricardo’s four numbers represent the four labor coefficients per unit of output needed to calculate them.Ricardo’s failure to speculate on the determination of the terms of trade was viewed as a defect which l’s theory was designed to correct.Given the intellectual prestige carried by Mill’s interpretation of the Ricardian model,subsequent generations of economists followed the same interpretation without bothering to read closely Ricardo’s original wording of the principle of comparative advantage.8A second and related question is whether the standard interpretation of Ricardo’s principle led economists to form a distorted view of Ricardian comparative advantage.As argued above,Ricardo began his example by postulating the equilibrium terms of trade, and went on to assume that the two trading countries’opportunity cost ratios lie on either side of them,without explaining what those ratios depend on.The textbook versions of the Ricardian model invert Ricardo’s own order of presentation by first theorizing about the 7As Ruffin(2002,p.742)points out,Sraffa is an exception to this generalization.He correctly interpreted Ricardo’s paragraphs on comparative costs when he stated that via trade UK‘gains the labour of20Englishmen’, while‘‘Portugal gains the labour of10Portuguese’’(Sraffa,1930,p.541).As noted below,James Mill is another exception.The history of economic thought contains many examples of insights gained by earlier economists that were subsequently lost.Sraffa’s insight of1930(unrelated to his subsequent larger reinterpretation of Ricardo’s work)was ignored for over70years until Roy Ruffin noticed it and brought it to public attention.Perhaps the new view of Ricardo’s four numbers as the amounts of labor embodied in trade flows should be referred to as the Sraffa–Ruffin interpretation.l’s role in diverting attention away from Ricardo’s formulation of the labor embodied in the bundle of traded goods is also underscored by Ruffin(2002,pp.742–743).。