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Marshallian Cross Diagrams and Their Uses before Alfred Marshall: The Origins of Supply and Demand Geometry ThonaasM. Humphrey Undoubtedly the simplest. and most frequently used tool of microeconomic analysis is the conventional partial equilibrium demand-and-supply-curve diagram of the textbooks. Economics professors and their students put the diagram to at least six main uses. They use it to depict the equilibrium or market-clearing price and quantity of any particular good or factor input. They employ it to show how (Walrasian) price or (Marshallian) quantity adjustments ensure this equilibrium: the first by eliminating excess supply and demand, the second by eradicating disparities between supply price and demand price. They use it to illustrate how parametric shifts in demand and supply curves induced by changes in tastes, incomes, technology, factor prices, and prices of related goods operate to alter a good’s equilibrium price and quantity. They apply it to show how the shifting and incidence of a tax or tariff on buyers and sellers depends on elasticities of demand and supply. With it they demonstrate that price ceilings and price floors generate shortages and surpluses, respectively. Finally, they employ it to compare the allocative effects of competitive versus monopoly pricing and to indicate the welfare costs of market imperfections. And Karl Rau (184 l), Jules Dupuit (1844), Hans von Mangoldt (1863), and Fleeming Jenkin (1870) thoroughly developed it years before Marshall presented it in his Pzm Theory of Domestic Vafues (1879) and later in his Pnitciples offionomics (1890). Far from merely introducing the diagram, these writers applied it to derive many of the concepts and theories often attributed to Marshall or his followers. The notions of price elasticity of demand and supply, of stability of equilibrium, of the possibility of multiple equilibria, of comparative statics analyses involving shifts in the curves, of consumers’ and producers’ surplus, of constant, increasing and decreasing costs, of pricing of joint and composite products, of potential benefits of price discrimination, of tax incidence analysis, of deadweight-welfare-loss triangles and the allocative inefficiency of monopoly: all find expression in early expositions of the diagram. The diagram’s applications are of course well known. Not so well known, however, are its origins and early history. Economists typically tend to associate the diagram with Alfred Marshall, its most persuasive and influential nineteenth century expositor. So strong is the association that economists have christened the diagram the Marshallian cross or Marshallian scissors after Marshall’s analogy comparing the price-determining properties of a brace of demand and supply curves with the cutting properties of the blades of a pair of scissors. These expositions, however, have not always been fully appreciated. John Maynard Keynes ([ 192.51 1956: 24), for example, dismissed them as vastly inferior to “Marshall’s diagrammatic exercises” which “went so far beyond the ‘bright ideas’ of his predecessors that we may justly claim him as the founder of modern diagrammatic economics.” In the same vein, Michael Parkin (1990: 85) recently has claimed that of early graphical treatments of supply and demand only Marshall’s is sufficiently modern to be recognized by textbook readers today. Such discounting of the pre-Marshall work has contributed to what Joseph Schumpeter (1954: 839 n. 13) complained of as economists’ “uncritical habit of attributing to Marshall what should, in the ‘objective’ sense, be attributed to others (even the ‘Marshallian’ demand curve!).” Seen this way, Marshall’s definitive contribution emerges as the cdmination of the diagram’s development. Economists, Schumpeter thought, misperceived it as the origin. The diagram itself, however, long predates Marshall. Antoine-Augustin Cournot originated it in 1838. In an effort to correct such misperceptions and to give the earlier writers their due, this paper traces FEDERAL RESERVE BANK OF RICHMOND 3 their pioneering contributions so as to counter the notion that the Marshallian cross diagram begins with Marshall. To paraphrase Frederick Lavington’s famous remark that “it’s all in Marshall, if you’ll only take the trouble to dig it out,” (Wright 1927: 504) the following paragraphs attempt to show, with respect to the diagram and its applications, that it’s all in Marshall’s predecessors too. ANTOINE-AUGUSTINCOURNOT COURNOT’S 1 DEMAND CURVE [quantity] 0 (1801-1877) Though hardly the first to state that supply and demand determine price, Antoine-Augustin Cournot, in his 1838 Recherchs sur l’esprincipes mathtfmatiques ak b t&wriedes tihesses (Reseamk into tk Mathematial Principbs of the Thor-y of Wealth), was the first to draw market demand and supply curves for a particular good.’ Of the two curves, Cournot analyzed demand before introducing supply. Figure 1 shows his diagram of the demand function D = F@) where D is quantity sold at different prices p, which Cournot measured along the horizontal rather than the vertical axis as is customary today.z Cournot ([1838] 197 1: 52-3) noted that corresponding to each price on the demand curve ab is a price-quantity rectangle whose area R = pD represents total revenue R associated with that price. He sought to determine the particular price at which revenue is at a maximum. A profit-maximizing monopolist would charge this price if his costs were either zero or a fixed sum independent of quantity produced. To find the revenue-maximizing price, Cournot differentiated the revenue function R = pD = pF@) with respect to price. He obtained the expression pF’(p) + F(p), where F’ is the derivative of the demand function F. Setting this expression equal to zero as required for a maximum and rearranging, he gotp = -F(p)/F’@), which says that the revenuemaximizing price must equal the ratio of the quantity demanded to the slope of the demand curve at that price. To depict this solution diagrammatically, he rewrote the expression as F@)lp = -F’@). ’ That supply and demand determine price must have been known for thousands of vears. But the phrase “sutmlv and demans itself is of more recent vintage; Sir James Steuart originated it in 1767 (Thweatt 1983). Adam Smith (117761 19307:56) in his weo/t/r‘ofNationsused ;he notion of sup$y and demand to explain deviations of actual market price from natural price determined by long-run cost of production. And David Ricardo ([ 18 17-Z 1] 195 1) gave the idea added prominence when he incorporated it into the title “On the Influence of Demand and Supply on Price” of Chapter 30 of his Principlesof Political Economy and Taxation. . . I * On Cournot see Theocharis (1983: 138-46) and Ekelund, Furubotn, and Gramm (1972: 15-19). 4 Figure Source: Cournot ([11338] 1971: Fig. 1 at end of book) The left-hand side he represented, for any point n on the demand curve, by the slope qn/Oq of the ray On (see Figure I). The right-hand side he portrayed by the slope qnlqt of the line segment nt. He noted that these slopes are equal only when Oq equals qt, which uniquely determines the revenue-maximizing price Oq as one-half of Ot. Price Elasticity of Demand Cournot (53-4) also anticipated Marshall’s concept of price elasticity of demand, defined as the percentage change in quantity demanded divided by percentage change in price: (dD/D)/(dp/p) or pdD/Ddp. True, William Whewell had foreseen the idea before him in 1829. But Whewell, whose work was unknown to Cournot, did not use diagrams or draw a demand curve. Cournot, in drawing the curve, argued that one can determine the effect of a small change in price Ap on total revenuepF@) by comparing the curve’s slope AD/Ap with the ratio of quantity demanded to price D/p. A small rise in price, he claimed, will cause total revenue to rise, fall, or remain unchanged at its peak level as ADlAp is less than, greater than, or equal to D/p. Setting ADlAp = D/p and dividing both sides by D/p yields (p/D) (AD/Ap) = 1. Now the left-hand side of this expression is Marshall’s measure of elasticity. Hence Cournot’s statement that total revenue ECONOMIC REVIEW. MARCtiIAPRIL 1992 Figure achieves its stationary maximum when ALI/& equals D/p is equivalent to saying that it does so at the point of unitary elasticity on the demand curve. Similarly, his contention that revenue rises or falls with a rise in price as ADlAp is less than or greater than D/p is tantamount to declaring that it does so as elasticity is above or below one in value. Supply Curve COURNOT’S TAX SCISSORS INCIDENCE 2 DIAGRAM AND ANALYSIS [price] 2D Introduced After specifying the elasticity-revenue relationship, Cournot incorporated supply curves into his diagram. Assuming a regime of perfect competition, he (90-Z) argued that profit-maximizing firms equate price with marginal cost. Since marginal cost after a certain point rises with the level of output, each firm produces along a schedule showing output as an increasing function of price. Summing over the individual firms’ supply functions, Cournot obtained the total market supply function S@) expressing a schedule of quantities offered at all possible prices. Finally, he equated market supply and demand S@) = F@) to determine equilibrium price and output. All this he depicted in Figure ‘2 in which the downward-sloping demand curve MN is combined with the upwardsloping supply curve PQ to yield the equilibrium price-quantity combination S.3 T I 1r- -7 [quantity] Source: Cournot ([1838] 1971: Fig. 6 at end of book). Effects of a Tax Having depicted supply-demand equilibrium, Cournot (92-3) next used his curves to show the effect of a per-unit tax of amount VS’ levied on a particular good. He argued that the tax, by adding L’S’ francs to the cost of supplying each unit of output, would shift the supply curve upward by that same amount. The result is a rise in the equilibrium price from OT to OT’ and a fall in the equilibrium quantity from TS to T’S’. He noted, however, that with an upward-sloping supply curve the price increase m’ is always less that the tax VS’ provided the demand curve is not vertical. Indeed, if the demand curve were horizontal the tax would not increase price at all. Cournot was thus the first to show that given a positively sloped supply curve the portion of an excise tax shifted to buyers in the form of higher prices varies inversely with demand elasticity, being nil at infinite and complete at zero values of that parameter. Cournot, not Marshall, invented geometrical tax-incidence analysis. This is Cournot’s Figure 6 with the axes switched to conform to current practice. 3 KARL HEINRICH RAU (1792-1870) Cournot did not use his scissors diagram to expound the stability of market-clearing equilibrium. Although he employed stability analysis in his famous reaction-function model of duopoly, he never did so in his competitive supply and demand model. In 184 1, three years after the publication of Cournot’s Researc/res but with no knowledge of its contents, the German economist Karl Heinrich Rau rectified this oversight (Hennings 1979).4 In so doing, he became the first to employ a Marshallian cross diagram to investigate the stability of market equilibrium and to indicate the forces that restore that equilibrium once it is disturbed. Rau’s diagram, which owed nothing to Cournot’s and indeed differed from it by putting price on the vertical axis and quantity on the horizontal, first 4 Hennings (1979: 6) argues that Rau never saw a copy of Cournot’s Researchesand thus was ignorant of its diagrams. He points out that neither Rau’s library nor that of his university possessed Cournot’s book. FEDERAL RESERVE BANK OF RICHMOND 5 appeared in a note in the 1841 volume of the Bultin de C’Acadhie des Sciences et des Bekdettres de Bmxefhs.~ He published a more elaborate version in the appendix to section 154 of the fourth (1841) and all subsequent editions of his textbook &wz~s&ze der lWh-witihscha~slehm (see Figure 3). There he argued that, given the linear demand curve g/l and the vertical supply curve a& a below-equilibrium price ec would produce an excess demand cb. Suppliers would take advantage of the resulting shortage of the good to raise price until the shortage disappeared and equilibrium m was restored.6 Rau thought this result important enough to state algebraically. His formula, cm = (ab -ac) tan tw, expresses the price rise cm as the product of the excess demand (ab -ac) times the slope of the demand curve tan w where w is the angle formed by lines cb and bm. This trigonometric expression, of course, holds only for the linear demand-vertical supply case. Alternatively, if the demand curve were instead the concave schedule j, the same excess demand would induce sellers to raise price top. And if the supply schedule were the positively sloped curve ek, then excess demand would spur sellers to boost price to I or n depending on which demand curve they faced. Conversely, an above-equilibrium price would generate an excess supply that would put downward pressure on price until equilibrium was restored. Rau’s diagram seems to have influenced only one of his contemporaries, Hans von Mangoldt, who used it as the starting point for his own demand-andsupply-curve analysis in 1863, twenty-two years after Rau first drew it.’ In the meantime, other writers, including the French engineer Jules Dupuit who published demand-curve diagrams in 1844, either ignored it or were unaware of its existence. 5 On Rau’s diagram and its applications see Hennings (1979) who provides an English translation of the relevant passages. 6 Of course demanders too could take the initiative and bid up the price. But Rau did not mention this possibility. Figure 3 RAU’S DIAGRAM [price] Despite their originality, neither Cournot nor Rau derived welfare propositions from their diagrams. Because they saw demand curves as empirical sales schedules rather than as replicas of theoretical marginal utility functions, they said nothing about the welfare implications of monopoly pricing, public utility rate setting, discriminatory pricing, or commodity taxation.8 Jules Dupuit, however, was not so hampered. Explicitly identifying demand curves with marginal utility schedules, he became, in 1844, the first to derive welfare theorems with the aid of a Marshallian diagram. True, he drew no supply curves. He merely assumed a constant supply price or one that varies independently of the level of output. But he made path-breaking use of the demand curve to define such Marshallian concepts as total utility, consumer surplus, and deadweight-welfare-loss triangles, not to mention Laffer-curve relationships between tax rates and revenues. In so doing he advanced demand theory far beyond Cournot and Rau, whose work was unfamiliar to him.9 B 45 a# 40--o 33 'OH 30 - 25-a” -a’ .20 - I5 - -a 10 _ ’ Hennings (1980: 670) notes than Mangoldt, in advertising his book, explicitly cited Rau as a precedent for employing diagrams. h e Source: JULESDLJPUIT(1804-1866) Rau (1841b: 527). x [quantity] 8 On Cournot’s and Rau’s view of demand functions as purely empirical schedules, see Ekelund, Furubotn, and Gramm (1972: 17) and Hennings (1979: 2). 9 Ekelund and Htbert (1983: 260) note that Dupuit was ignorant of Cournot’s work even though both writers once lived and worked in Paris at the same time. ECONOMIC REVIEW, MARCH/APRIL 1992 Laws of Demand prices and responding to lower prices, give the curve its characteristically convex shape. Figures 4, 5 and 6 depict Dupuit’s diagrams as presented in the appendix to his 1844 article “On the Measurement of the Utility of Public Works” in the Annales des Ponts et Chausstfes.‘OThe diagrams as shown illustrate Dupuit’s two “laws” of demand.” Law number one says that demand curvesexpressed by Dupuit as y = f(x), where y denotes quantity demanded and x price-slope downward because of diminishing marginal utility: extra quantities of a good or service add less and less to total satisfaction and thus command lower demand prices. Law number two says that a given fall in price induces larger increases in quantity demanded the lower the price at which it occurs. This law Dupuit attributed to the pyramidal distribution of income; each price decrement activates the demands of a new group of buyers larger and poorer than the group above it on the income scale. The resulting new demands, added to those already existing at higher 10These are Dupuit’s Figures switched to facilitate inspection. Marginal Utility, Total Utility; Consumers’ Surplus (Figure 4) Having deduced the shape of the demand curve, Dupuit used his first diagram to refute J. B. Say’s contention that a good’s market price measures the utility of each unit consumed. Not so, said Dupuit ([1844] 1969: 280-l). Price Up measures the (marginal) utility nr of the last unit purchased only. Preceding or inframarginal units such as r’ and T” yield higher marginal utilities as indicated by the higher demand prices p ’and p ” buyers would pay for those units rather than go without. Summing over these successive demand prices as one moves up the demand curve gives a measure of the total utility of the entire quantity consumed. This measure is represented by the roughly trapezoidal area OPnr under the demand curve and not, as Say implied, by the price-times-quantity or total-expenditure rectangle Opnr. Say’s measure understates utility by the amount of the consumers’ surplus triangle pPncalled utifitb relative by Dupuit. Taken to its extreme, Say’s analysis erroneously implies that total 1, 3 and 4 with the axes 11On Dupuit’s two laws of demand, see Houghton (1958: 50) and Ekelund and Thornton (1991). Figure 4 DUPUIT MARGINAL ON TOTAL UTILITY, UTILITY, AND CONSUMERS’ SURPLUS [price] r Source: Dupuit ([1844] 1969: 280). FEDERAL RESERVE BANK OF RICHMOND N [quantity] utility is zero when price is zero. In fact, total utility would then be at its maximum equal to the entire area under the demand curve. As for the triangular area Nm in the lower righthand corner of the diagram, Dupuit defined it as the utility lost when a positive market price Cp constrains consumption short of the satiation point N. l2 Under competitive conditions this loss is the natural result of resource scarcity and cannot be avoided. Here price measures the satisfaction forgone on other goods whose production falls so that resources can be freed to produce an extra unit of the good in question. That the last rN units of this good possess marginal utilities falling below price indicates that they are not worth producing; their opportunity cost exceeds the extra satisfaction they would bring. Thus units r through N are forgone so that resources can be put to higher-valued uses elsewhere. Under monopoly pricing, however, the loss may be due to contrived restriction of output and is a true measure of social harm. A costless monopoly charging price Op, for example, needlessly deprives consumers of satisfaction equivalent to the area Nm. Here is the first diagram in the history of economic thought to depict a deadweight-loss-from-monopoly triangle and a total utility trapezoid under the demand curve. And it is the first to partition the trapezoid into a price-quantity rectangle showing buyer expenditure on the good and a consumers’ surplus triangle showing the excess of what consumers would pay over what they actually do pay. In short, Dupuit, not Marshall, was the first to demonstrate diagrammatically that consumers get more utility than they pay for when they buy a good or service at a single market price. He was also the first to extend this insight to the evaluation of public works. In particular, he noted that the potential benefits of such projects cannot be measured by their costs. One needs to estimate the area under the demand curve. Dupuit’s Tax Theorems (Figure 5) Armed with the utility concepts developed in his first diagram, Dupuit (281-2) used them in his second diagram to derive key propositions concerning the welfare effects of commodity taxes. His first proposition states that the imposition of a tax results in a loss in consumers’ surplus that exceeds the yield of the levy. His diagram shows how a per unit tax 12On Dupuit’s concept of lost utility (uhfh!pe&e) (1970: 271-3). 8 see Ekelund Figure DUPUIT’S 5 TAX THEOREMS [price] P K M p” P” P’ P C N [quantity] Source: Oupuit (I18441 1969: 282) of qn ’raises price by pp ‘, thus reducing purchases top ‘n ’and consumers’ surplus top ‘Pn ‘. On thep ‘n ’ units still bought, consumers pay a total tax ofpp ‘n ‘q to the government, which Dupuit assumed puts it to socially productive uses. But, on the qn units no longer bought, buyers lose consumers’ surplus qnn ’ with no corresponding gain to the government. Hence the tax causes a loss of consumers’ surplus that exceeds the tax yield by the roughly triangular area n ‘qn: the deadweight loss of the tax. This loss, consisting of the tax-induced distortion of relative prices and consumption patterns, persists even if the government returns the proceeds to the taxpayers. Dupuit’s second theorem states that the deadweight loss is proportional to the square of the tax rate. As mentioned above, the welfare loss AU is the area of the triangle n ‘qn whose height is the ECONOMIC REVIEW. MARCH/APRIL 1992 tax rate t and whose base is the reduction in quantity bought AQ caused by the duty. Since the area of a triangle is half its height times its base, one sees that the net loss in utility is half the tax rate times the fall in amount purchased or AU = %tAQ. Now AQ, the change in amount bought, can by definition be expressed as AQ = kt, where k is the inverse AQ/t of the slope of the triangle’s hypotenuse. Substituting this expression into its predecessor yields Dupuit’s taxation theorem: AU = ‘/zkt2 or, in his (28 1) words, “the loss of utility increases as the square of the tax.“r3 It follows that the government minimizes the welfare burden of a given total tax collection by charging a low rate on a great many goods rather than a high rate on a few. For the welfare loss, which shrinks as the square of a lowered rate, approaches zero as the tax becomes general or diffused and its rate correspondingly small. Dupuit’s third theorem posits an inverted U-shaped or Laffer-curve relationship between tax rates and tax revenue. Like Arthur Laffer in the 198Os, Dupuit in 1844 saw tax revenues rising from zero with small increases in the rate, reaching a maximum pMTQ at rate PM, then falling with further rate increases, and eventually returning to zero when the rate becomes prohibitive. This rate-revenue relationship together with his second tax theorem led him (282) to conclude “that the yield of a tax is no measure of the loss which it causes society to suffer.” For the same yield can be obtained from two different rates entailing markedly different deadweight losses. RatespK andpp “, for example, yield the same revenue; yet the first rate’s welfare-loss triangle is more than ten times the size of the seconds. Likewise, zero and prohibitive tax rates both yield zero revenue. The zero rate, however, produces no welfare loss while the prohibitive rate produces a loss equal to the whole area under the demand curve. Pricing Policies (Figure 6) would be larger and deadweight loss smaller by the amounts pMTn and 71(m, respectively. Dupuit also analyzed price discrimination-the practice of charging separate customers different prices for the same product-with the aid of his third diagram.r4 He argued that discriminatory pricing could render profitable a firm that would suffer losses if it charged a single price. Dupuit examined the case of a monopolist whose fixed costs exceed his receipts OMTR at the revenue-maximizing price UM. The monopolist, by dividing his market MT into two groups, one paying price Op ’for quantity p ‘n ’and the other price OM for quantity q’T, could expand I4 On Dupuit’s analysis of price discrimination (1970: 271-S). see Ekelund Figure 6 DUPUIT ON PRICING PRICE POLICIES AND DISCRIMINATION [price] P and Price Discrimination p’ Dupuit (282-3) employed his last diagram to specify appropriate pricing policies for private and public monopolies having identical fixed costs Upnr. A private monopoly would charge the price OM that maximizes its receipts OMTR and, with costs given and independent of output, its profits too. By contrast, a public utility would charge the lowest price Op that maximizes consumer satisfaction subject to meeting the cost constraint. Consumers’ surplus M P 0 [quantity] 13 See Htbert and Ekelund (1984: 6’2) for an alternative derivation of this formula. Source: Dupuit ([18441 1969: 283). FEDERAL RESERVEBANK OF RICHMOND 9 his receipts by an amount Mp ‘n ‘q’sufficient to defray his costs. Further discrimination would yield even larger receipts. For example, were the monopolist able to charge the maximum price for each successive unit of the MT quantity sold, he could effectively redistribute consumers’ surplus to himself and capture revenue equal to the entire area OP7R. Figure MANGOLDT’S 7 CROSS [price] To Dupuit, however, price discrimination could accomplish more than merely redistributing a fixed sum of welfare from buyers to sellers. It could increase total welfare if it led to increased output. Let a monopolist initially charging price OM on output OR find it profitable to sell extra output Rr at discriminatory price Op. Then total utility, Dupuit claimed, would rise by RTnr at the expense of a corresponding shrinkage in deadweight loss to nrN. Discriminatory pricing, in other words, would yield a net social benefit. Later, in the 1920s and 193Os, Marshall’s students A.C. Pigou and Joan Robinson would echo Dupuit’s declaration of the welfare superiority of output-increasing price discrimination over simple monopoly pricing. [quantity] Source: HANS VON MANGOLDT Taking his cue from Karl Rau, whose work he cited, Mangoldt began his chapter on “The Exchange Ratio of Goods” with a scissors diagram similar to Rau’s (Figure 7). Like Rau, he identified the marketclearing equilibrium price P and described the adjustment process that restores it once it is disturbed. His stability analysis, like Rau’s, highlights the price-equilibrating role of excess demand or supply. Let price fall below equilibrium, he ([ 18631 1962: 32) said, and the resulting excess of demand over supply bids it back to equilibrium. Likewise, an above-equilibrium price activates an excess of supply over demand that puts downward pressure on price until it returns to equilibrium. Curve (Figure 8) Following his stability analysis, Mangoldt (33-S) proceeded to examine the demand curve in great detail. He argued that (1) the height of each point on the curve represents the marginal utility of the 10 Mangoldt ([1863] 1962: 33). (1824-1868) Dupuit had shown how much one could accomplish working with the demand curve alone. It was time, however, to reintroduce the supply curve into the diagram and to examine the role of cost in price determination. Hans von Mangoldt took this step in his 1863 Ghndtis der Voikswbm-hajislehre (Outline of Pol’itical Economy). Demand DIAGRAM corresponding quantity, (2) the curve slopes downward because of diminishing utility of additional units and the resulting reduction in prices buyers are willing to pay, and (3) a rise in price induces buyers to cut back their purchases until the marginal utility of the last unit bought rises to match the higher price. The demand curve cuts the vertical axis, he said, at a price which just exceeds the marginal utility of the good’s first unit (point Dm on Figure 8a). Conversely, demand reaches its satiation point D on the quantity axis when price is zero. As for shifts in the curve, Mangoldt attributed them to population growth, to changes in tastes and knowledge, and to economic development and the resulting rise in income and wealth. Unlike Dupuit, who believed that demand curves must be of convex shape, Mangoldt held that they could be either convex or concave (see Figure 8b) depending on the type of good (luxuries or necessities), on the degree of inequality of income distribution, and on the availability of close substitutes. Finally, he noted certain exceptions to the law of demand. Demand curves, he argued, could possess upward-sloping segments (see Figure 8c) if tastes for conspicuous consumption cvanity”) or expectations of further price hikes (“fear”) motivated consumers ECONOMIC REVIEW, MARCH/APRIL 1992 Figure 8 MANGOLDT’S DEMAND CURVES (b) (a) [price] (c) [price] [price] L7m i I,,,,,, I,,,,,,,,, IIll n [quantity] Source: [quantity] [quantity] Mangoldt (118631 1962: 34-35) to buy larger quantities at higher prices. And rising demand curves, he realized, could intersect supply curves more than once, giving rise to the possibility of multiple equilibria. Supply Curves (Figure 9) Turning his attention to supply curves, Mangoldt (3.57) made his most enduring contribution. He was the first to draw such curves with different shapes depending on the behavior of costs of production. Constant unit costs yield a horizontal or perfectly elastic curve (Figure 9a). Constant costs up to the limit of a rigidly fixed capacity yield a reverse L-shaped curve possessing horizontal and vertical segments (Figure 9b). Constant costs that give way to increasing costs and then to rigidly fared limits yield a curve with perfectly elastic, relatively elastic, and perfectly inelastic components (Figure SC) . Finally, decreasing costs owing to economies of scale over a certain range of output followed by increasing costs due to diseconomies of scale yield a roughly U-shaped curve that falls before it subsequently rises (Figure 9d). As for outward secular shifts in supply curves, Mangoldt ascribed them to technological progress and resource discovery-forces tending to lower the cost of producing any level of output. Comparative Statics Exercises (Figure 10) Today Marshall’s name is associated with the partial equilibrium, comparative statics method. But it was Mangoldt, not Marshall, who pioneered the technique. Having presented curves of demand and supply, Mangoldt (38-40) put them through a series of exercises designed to show how shifts in the curves affect equilibrium price and quantity. Rightward shifts of the demand curve along a perfectly elastic supply curve raise quantity but not price (Figure lOa). Price, that is, is supply-determined in the constant cost case. The same demand shifts occurring along a vertical or perfectly inelastic segment of the supply curve raise price but not quantity (Figure lob). Price is demanddetermined in this case. Similarly, leftward shifts in demand produce only price falls when supply is perfectly inelastic (vertical curve PS,,, in Figure 10~) and only quantity reductions when supply is perfectly elastic (curve S&J. In the typical case of relatively elastic supply, however, demand-curve shifts change both price and quantity (Figure 1Od). Finally, simultaneous rightward shifts in both demand and supply curves can cause equilibrium price to rise, fall, or remain unchanged depending upon which shift, if either, predominates (Figure 1Oe). FEDERAL RESERVEBANK OF RICHMOND 11 Figure 9 MANCOLDT’S [price] SUPPLY CURVES [price] (4 (b) [quantity] [price] f Sm Si Source: Mangoldt ([18631 1962:36-37) These exercises alone are sufficient to ensure Mangoldt’s place in the history of economic thought. But he went beyond them to describe a three-step adjustment process by which price and quantity move from one equilibrium position to another. Years before Marshall he (51) posited a Marshallian mechanism. First comes an outward shift in either the demand curve or supply curve. This produces a positive gap between demand price and supply price at the existing level of output. The resulting rise in profits induces producers to expand output until the price differential is eliminated and the new equilibrium is attained. All this, Mangoldt noted, refers to unanticipated shifts in the curves. Should the shifts be anticipated, price immediately jumps to its new 12 equilibrium, thus avoiding the sequential adjustment process. Multiple Equilibria Anticipating Marshall, Mangoldt (SO) discussed the possibility of multiple equilibria of demand and supply. He noted that such phenomena cannot occur when the two curves slope in opposite directions and so intersect no more than once. But they can occur when both curves slope in the same direction. Here Mangoldt cited demand curves that rise with price because of desires for conspicuous consumption or expectations of even higher future prices. Likewise he cited supply curves that fall because of ECONOMIC REVIEW, MARCH/APRIL 1992 Figure 10 COMPARATIVE [price] T STATICS EXERCISES b-4 [quantity] [price] [quantity] [price] (cl (d) .Sm Sm n [quantity] [price] f-+Dm,------ [quantity] (e) --__ Sm I J I [quantity] Source: Mangoldt ([l&53] 1962: 37-38, 40). FEDERAL RESERVE BANK OF RICHMOND 13 increasing returns to scale. In such cases, multiple intersections are possible and there may be several equilibrium prices. Having said this, however, Mangoldt said nothing about which equilibria are stable and which unstable. He failed to apply the analysis he had used before in the case of a single unique equilibrium. With respect to multiple equilibria, he did not recognize the stability problems involved. Pricing of Joint Products (Figure 11) Nevertheless, Mangoldt’s analysis of intersecting supply and demand curves must be judged an outstanding performance that in many respects exceeded those of his predecessors. Equally impressive was his application of the diagram to the problem of price determination when goods are jointly demanded or supplied in fixed proportions. True, John Stuart Mill had briefly discussed this problem in his 1848 Pritzc$des of PohicaL Economy. But Mangoldt’s geometric and algebraic analysis eclipsed Mill’s purely verbal treatment and was not superseded until Marshall’s Principl’es. A “great achievement” and “Mangoldt’s most significant contribution to price theory,” Eric Schneider (1960: 380, 384) called it. A “brilliant theoretical contribution” concurred Jiirg Niehans (1990: 128). What follows sketches the geometric part of Mangoldt’s contribution. Readers can find treatments of the algebraic portion in the appendix to this article as well as in Schneider (1960) and Creedy (1992: 38-46). Mangoldt first examined the “joint demand” case of a pair of goods A and B purchased in fixed proportions under a given spending constraint.r5 He (41-4) showed how demand and supply determine the equilibrium price and quantity of both goods consistent with the constraint. He also showed that a fall in A’s supply price-i.e., a rightward shift in its supply curve (curvefl; in Figure 1 la)-raises B’s price and explained why. The cheapening of A induces buyers to take more of it and, because of fixed proportions, to demand more B too. The resulting upward shift in the demand curve for B (curve ggs) raises its price. In this way, a cost reduction that increases the supply of A raises the price of B. Turning to the “joint supply” case of a pair of goods such as beef and cowhides produced in fixed proportions subject to a given cost constraint, Mangoldt showed how equilibrium is established in both I5Examples include (1) scotch and soda, and (2) copper and zinc used in making brass. 14 markets. He (46-8) also showed that a rise in the demand for beef must lower the price of hides and gave the rationale. With fixed proportions, the increased demand for beef leads to a rise in its output as well as that of hides and so, given the demand for hides, to a fall in their price. In this way an upward shift in the demand curve for beef (curve ee, in Figure 1 lb) produces a downward shift in the supply curve of hides (curve dd,) that lowers their price. Having examined joint demand and supply, Mangoldt for completeness considered composite demand and supply. Composite demand refers to the case where two competing uses (e.g., furniture and firewood) vie for one fixed input (timber). Here Mangoldt (48-50) showed that a fall in the demand for furniture would, by making more timber available for firewood, increase the latter’s supply and lower its price. Composite supply refers to the case where two substitute goods (e.g., flax and cotton) satisfy a single need (for cloth). Here he (44-6) showed that a rise in supply and hence fall in the price of flax lowers the demand for and so the price of cotton. These topics were not further developed until Marshall took them up in his Principles. Mangoldt’s Influence Mangoldt’s diagrammatic analysis should have become common property to all economists by the 1870s. That it did not is attributable to one Friedrich Kleinwachter who, upon publishing a reprint of Mangoldt’s book in 187 1 shortly after his death, deleted the diagrams on the grounds that “it is utterly inconceivable to me that graphs or mathematical formulae could facilitate the understanding of economic laws” (Creedy 1992: 46; see also Schneider 1960: 392). Mangoldt’s contribution fell into oblivion for twenty-three years until Francis Edgeworth ([ 18941 1925: 53) rediscovered it in 1894 and proclaimed its author “one of the independent discoverers of the mathematical theory of Demand and Supply.” Edgeworth might well have said the same thing about Henry Charles Fleeming Jenkin, the distinguished electrical engineer and inventor, who, with no formal training in economics and no acquaintance with the work of Mangoldt or his predecessors, introduced demand and supply curves-indeed the technique of diagrammatic analysis-into the English economic literature circa 1870.16 I6 On these points see Brownlie and Lloyd Prichard (1963: ‘211, 2 16) who note that Jenkin had read little economics other than J. S. Mill’s Principftx ECONOMIC REVIEW. MARCH/APRIL 1992 Figure 11 PRICING OF JOINT PRODUCTS (a) [price] __ ______ / / +-Ii [quantity] (b) \ J’ .,es I I 1 t- C [quantity] Source: Mangoldt ((18631 1962: 43, 47). FEDERAL RESERVE BANK OF RICHMOND 15 FLEEMINGJENKIN (1833-1885) Jenkin presented his analysis in three papers: his 1868 Nmh British Rfx.zkv article “Trade-Unions: How Far Legitimate ?,” his 1870 Recess Stzdies piece “The Graphic Representation of the Laws of Supply and Demand, and Their Application to Labour,” and his 1872 contribution to the Pmeed’ngs of th Royal Society of Edinbz@ 1871-Z “On the Principles Which Regulate the Incidence of Taxes.” In his 1870 paper he ([ 193 11: 77) drew intersecting curves representing equations which he (I 193 11: 17-8) had stated in his 1868 piece, namely b-= f(A+ I/x) and S = F(B+x) where D and S denote quantities demanded and supplied, respectively, x denotes price, and A and B denote shift parameters that determine the location or height of the curves on the [price] (a) diagram. equals the stock on hand times the fraction offered for sale. This fraction varies directly with the differential between actual market price and traders’ reservation prices, i.e., prices below which stocks are held for future sale and above which they are marketed immediately. Different traders possess different reservation prices stemming from their expectations of future prices (99, 109). Those expecting low future prices have low reservation prices. Those expecting high future prices have high reservation prices. A rising market price surpasses a growing number of reservation prices, thus enlarging the fraction of the Figure 12 JENKIN PRICE ON MARKET-PERIOD DETERMINATION [price] (b) SM 4s He (17-8) noted that the curves’ intersection point depicts the equilibrium price and quantity that solve the market-clearing equation D =S or f(A+ I/x) = F(B +x). To ensure stability of equilibrium he relied on excess supply or demand triggered by price deviations from equilibrium. These excess supplies or demands, he said, act immediately to restore price to its marketclearing level. Market-Period Price Determination (Figures 12 and 13) Anticipating Marshall’s assumption of separate operational time periods, Jenkin (78, 89) conceived two hypothetical intervalsmarket period and long run-to which his analysis applied. In the market period the stock of goods is fured and cost of production plays no role in price determination. Quantity supplied 16 90 80 70 60 50 40 SO 20 IO 0 Source: Quarters Quarters [quantity] [quantity] Jenkin ([1870] 1931: 77, 79). ECONOMIC REVIEW, MARCH/APRIL 1992 stock marketed. Accordingly, quantity supplied rises with price and the supply curve slopes upward until it turns vertical when the entire stock or “whole supply” (Figure 12a) is marketed.17 As for market-period demand curves, Jenkin drew them with a negative slope indicating that lower prices are required to compensate for diminishing marginal utility of additional units bought. Intersection of demand and supply curves yields an equilibrium (Figure 12a) characterized by zero excess supply or demand as the market clears (Figure 12b). This equilibrium, however, is extremely volatile. It changes with every event, real or imagined, that shifts the curves. Demand curves shift with variations in buyers’ whims, desires, and expectations (Figure 13a) as well as with changes in incomes (Figure 13b). Supply curves shift with variations in the size of stocks on hand (Figure 13~) and with changes in traders’ expectations and thus the reservation prices they set (Figure 13d). Figure SHIFTS IN MARKET-PERIOD [price] Shillings 13 CURVES [price] (a) Shillings (b) I Qualtsrs [quantity] Quarters [quantity] [price] [price] Shlllin~s h-0 Long-Run Price Determination (Figure 14) Turning to the long period when output can vary, Jenkin (89-93) showed that the latter adjusts to equilibrate demand and supply prices. Long-period supply price consists of average cost of production, which includes the sum of the costs of factor inputs per unit of output I7 Jenkin’s diagrams which, like Cournot’s and Dupuit’s, measured price horizontally and quantity vertically are shown here with their axes transposed. Quarters [quantity] [quantity] Source: Jenkin ([1870] 1931: 80-82) FEDERAL RESERVE BANK OF RICHMOND 17 Figure LONG-PERIOD PRICE 14 DETERMINATION [price] production approximately determines price while demand approximately determines output. Here is Jenkin’s version of the Marshallian cases of long-run increasing and constant cost industries, respectively. Application Shillings to Labor Unions (Figure 15) As is evident from the titles of his papers, Jenkin developed his supply and demand diagrams for two main purposes: to examine the impact of trade unions on wage determination and to elucidate the welfare effects of taxes. With respect to trade unions, he (94-106) argued that they could, by accumulating a strike fund to support workers during walkouts, raise equilibrium wage rates in both the market period and the long run. In the market period the labor force is given and, if non-unionized, will work for whatever wage it can get. Labor’s supply curve is a vertical line whose intersection with employers’ demand-for-labor curve determines the equilibrium wage (Figure 15a). Enter the trade union. With its strike fund the union allows its members to enjoy a reservation wage below which they withhold labor from employers rather than selling it for what it will fetch. The resulting labor supply curve, instead of being a vertical straight line, becomes a right-angled or reverse Lshaped curve at the reservation wage set by the union (Figure 1Sa). The labor demand curve will cut the supply curve in its horizontal segment such that equilibrium wages will be higher and equilibrium employment lower than in the non-union case. Unions raise wages at the expense of employment. Millions of Bales Source: [quantity] Jenkin ([11370] 1931: 90). plus normal profits of producers. Unit or average cost generally rises with output. The reason: productive factors are in limited supply and must be paid higher prices to bid them away in greater quantities from alternative uses. Consequently, if a good requires specialized inputs that are extremely scarce and thus increasingly costly to obtain, unit costs rise rapidly with output and the supply curve is steeply sloped (curve 2 of Figure 14). Conversely, if inputs are so plentiful that their prices are virtually invariant to increases in the demand for them, unit costs will be nearly constant and the supply curve relatively flat (curve 1 of Figure 14). In this latter case, cost of 18 So much for the market period. In the long run the workforce is variable and labor’s supply curve horizontal. Labor is produced at constant cost consisting of the expense of rearing and maintaining workers at some expected standard of comfort. Trade unions, by setting a reservation wage and so raising the standard of comfort, act to raise labor’s cost of production. The resulting upward shift in the labor supply curve causes it to cut the demand curve at a higher equilibrium wage and a lower equilibrium labor force (Figure 15b). Unions, by raising the standard of comfort, influence the equilibrium size of the population. Here is Jenkin’s most original contribution: his extension of the scissors diagram to the analysis of the labor market. Welfare Effects of Taxes (Figure 16) Jenkin also employed his diagrams to examine the welfare effects of excise taxes, exhibiting originality in conception if not establishing temporal priority in ECONOMIC REVIEW, MARCH/APRIL 1992 Figure 15 JENKIN ON WAGE DETERMINATION (a) supply +dow- I QuadiCy oj labour (b) Wa 8 rciL Quantity 9 labour Source: Jenkin (I18701 1931: 83) publication in doing so. First, without having seen Cournot’s tax incidence analysis, he showed that who bears the tax depends on the slopes of the demand and supply curves. According to Jenkin (114), the steeper the demand curve or the flatter the supply curve the greater the share of the tax shifted to demanders. Conversely, the steeper the supply curve and the flatter the demand curve the greater the share borne by suppliers. In the limiting case of a perfectly vertical demand curve or perfectly horizontal supply curve, all of the tax is shifted to demanders. But when supply is perfectly inelastic or demand perfectly elastic, the entire burden falls on suppliers. Second, Jenkin, in his tax analysis, derived the Marshallian concepts of consumers’ and producers’ surplus. Being unaware of Dupuit’s invention of the former idea, he (110) thought both were novel. Only the latter concept, however, was new with him. Like Marshall, he (109) defined it as the excess of sellers’ actual receipts from supplying a good over the minimum necessary to induce them to do so. And, like Marshall, he measured it by the roughly triangular area lying between the price line and the supply curve (area 2 of Figure 16a). He (110) also pointed out that consumers’ and producers’ surplus triangles 2 and 2 are larger the more steeply sloped the demand and supply curves, respectively. Finally, Jenkin (113-4) used these triangles to show that (1) the welfare losses of consumers and producers always exceed the tax they pay and (2) these losses rise with the rate of the tax. Thus, starting from pre-tax equilibrium D in Figure 16b, a tax of C’C or MM’ per unit of output drives a wedge between sellers’ supply price OM and buyers’ demand price OM’thus reducing output by C’D. The government obtains tax revenue equal to the sum of rectangular areas 2 and 2, of which buyers pay area 2 and sellers area 2. But buyers lose consumers’ surplus equal to the sum of areas 2 and 3, this loss exceeds their tax payment by the amount of area 3. Similarly, sellers lose producers’ surplus equal to the sum of areas 2 and 4, which exceeds their tax liability by the amount of area 4. These excess-burden triangles form constituent parts of the deadweight-loss triangle 3 + 4 whose size increases with the tax wedge. At very high tax rates the deadweight-loss triangle dwarfs the tax-yield rectangle and is a powerful argument for keeping rates low. Influence and Recognition Jenkin’s work, especially his 1870 paper, fully foreshadowed Marshall’s. Nobody saw this more clearly than Marshall himself. Marshall had lectured on demand and supply curves since 1868, but, at the time of Jenkin’s writings, had published nothing. He realized that Jenkin had “scooped” him, as the following passage (Whitaker 1975: 45) from H. S. FEDERAL RESERVE BANK OF RICHMOND 19 Figure 16 CONSUMERS’ AND PRODUCERS’ SURPLUS THE WELFARE EFFECTS OF A TAX AND [price] [price] (b) (4 A A4 P F v 0 Y [quantity] Source: Jenkin ([18721 1931: 108, 113) Foxwell’s letter of 24 April 1925 to J. M. Keynes confirms: “I happened to come across Uenkin’s 1870 article] in the Easter vacation of 1870, when I was attending Marshall’s lectures on diagrammatic economics, & I shall never forget his chagrin as he glanced through the article when I showed it to him. There was nothing in Cournot which so closely agreed with Marshall’s general approach to the Theory of Value & particularly to his statement of the equation of supply & demand.” Nevertheless, Marshall continued to insist that he, partly under the tutelage of Cournot, had invented the scissors diagram and pioneered its applications independently of Jenkin. William Stanley Jevons 20 [quantity] likewise dismissed Jenkin’s contribution with the claim that he Uevons) had used intersecting curves to depict price determination in lectures at Owens College as early as 1863, seven years before Jenkin published his diagrams. Such disparaging comments, together with Jenkin’s lack of formal training in economics, caused his innovations to go unnoticed (Brownlie and Lloyd Prichard, 1963: 21.5-6). Even today his name is unfamiliar to most economists who instinctively think of Marshall when supply-anddemand analysis is mentioned.18 18Thus Blaug and Sturges (1983: 186) remark that Jenkin’s work “was little noticed . . . and had little effect on the subsequent course of economic thought despite its striking quality and originality.” ECONOMIC REVIEW. MARCH/APRIL 1992 CONCLUSION extended it to the labor market to explain wage determination. Cournot formulated the elasticity concept and Dupuit analyzed price discrimination. There is little in Marshall’s use of the diagram that was not anticipated by his predecessors. Economists typically consider Alfred Marshall the father of the Marshallian cross or scissors diagram. The pre-Marshall literature, however, reveals a somewhat different picture. In particular: 1. The Marshallian cross diagram did not originate with Alfred Marshall. At least five economists-Cournot, Rau, Dupuit, Mangoldt, and Jenkin-employed it in print before Marshall published it. And the first four did so before Marshall began his career as an economist. 2. Of the diagram’s five originators, all but Mangoldt, who knew of Rau’s contribution, were ignorant of the work of the others. The diagram was a multiple independent discovery. 3. Besides conceiving the diagram itself, its originators supplied all its components and pioneered most of its applications. Cournot contributed the original curves. Rau and Mangoldt provided the stability analysis. Mangoldt furnished the comparative statics exercises which Jenkin applied to the analysis of price determination in the market period and long run, respectively. The tax incidence analysis stems from Cournot, Dupuit, and Jenkin. The consumers’ surplus and deadweight-loss triangles are Dupuit’s and Jenkin’s. Jenkin devised the idea of producers’ surplus. Mangoldt applied the diagram to the problem of the pricing of joint and composite goods. Jenkin 4. The diagram thus illustrates Stigler’s (1980) Law of Eponymy according to which no scientific discovery is named for its original discoverer. The Marshallian cross diagram bears Marshall’s name because he gave it its most complete, systematic, and persuasive statement, not because he was the first to invent it. His account was definitive, not pathbreaking. For this he received-and deserved-credit. 5. Later economists could have obtained their demand-and-supply analysis from Marshall’s predecessors as well as from his Prim-tjdes. Had they done so, today’s microeconomics textbooks probably would be little changed. Given the existence of a well-formulated geometry of supply and demand before Marshall, it follows that his contribution was a sufficient but hardly a necessary condition for the diagram’s subsequent dissemination. Had he never published, later economists probably would have discovered the work of his predecessors or invented the diagram themselves. REFERENCES Blaug, M. and Sturges, P. 1983. %40’s Who in Economics. Cambridge: MIT Press. Brownlie, A.D. and Lloyd Prichard, M.F. 1963. Professor Fleeming Jenkin, 1833-1885, pioneer in engineering and political economy. OxfordEconomicPapers, NS 15: 204-16. Cournot, A.A. 1838. ResearchesInto the MathematikalPrinc@es of the Theoryof Weak/r. Trans. by N.T. Bacon, New York: Macmillan, 1929; reprinted, New York: Augustus Kelley, 1971. Ekelund, R.B., Jr. 1970. Price discrimination and product differentiation in economic theory: an early analysis. QuarterZyJournal of Economics84, May, 268-78. Ekelund, R.B., Jr., Furubotn, E.G., and Gramm, W.P. 1972. The evolution and state of contemporary demand theory. In Evolution of Modenr Demand Theory, ed. R.B. Ekelund, Jr., E.G. Furubotn, and W.P. Gramm, Lexington: D.C. Heath. Creedy, J. 1992. Demand and Exchange in Economic Analysti. Aldershot: Edward Elgar. Ekelund, R.B., Jr. and Htbert, R.F. 1983. A Historyof Economic Theoryand Method. 2nd edn., New York: McGrawHill. Dupuit, J. 1844. On the measurement of the utility of public works. Trans. by R.H. Barback in Readings in Welfare Economics,ed. K.J. Arrow and T. Scitovsky. Homewood, IL: Richard D. Irwin, 1969, 255-83. Ekelund, R.B., Jr. and Thornton, M. 1991. Geometric analogies and market demand estimation: Dupuit and the French connection. History of Political Economy 23: 397-418. Edgeworth, F.Y. 1894. The pure theory of international values. Economic Journal 4. Reprinted in his Papers Relating to PoliticalEconomy, Vol. 2, London: Macmillan, 1925. Htbert, R.F. and Ekelund, R.B., Jr. 1984. Welfare economics. In EconomicAnalysisin Historicaol Penpective, ed. J. Creedy and D.P. O’Brien, London: Butterworths. FEDERAL RESERVEBANK OF RICHMOND 21 Hennings, K. 1979. Karl Heinrich Rau and the graphic representation of supply and demand. Diskussionspapier Serie C, Nr. 35, Fachbereich Wirtschaftswissenschaften, Universitat Hannover. Hennings, K. 1980. The transition from classical to neoclassical economic theory: Hans von Mangoldt. Kyklos33: 658-81. Houghton, R.W. 1958. A note on the early history of consumer’s surplus. Economic0 2.5: 49-57. Jenkin, F. 1868. Trade-unions: how far legitimate? Norr/r Britirh Rex&o, March. In Papen, Literaty, Scient& 6c, ed. SC. Colvin and J.A. Ewing, London: Longmans, Green & Co., vol 2, 1877; reprinted, London: The London School of Economics, 1931. Jenkin, F. 1870. The graphic representation of the laws of supply and demand, and their application to labour. In Recess Studies, ed. A. Grant, Edinburgh. Reprinted in Papers, Literary, Scientific, 6c, ed. SC. Colvin and J.A. Ewing, London: Longmans, Green & Co., vol 2, 1877; reprinted, London: The London School of Economics, 1931. Jenkin, F. 1872. On the principles which regulate the incidence of taxes. Prvceedingsof the Royal SocietyofEdinburgh 18722. In Papetx, Litmary, scientific, &c, ed. S.C. Colvin and J.A. Ewing, London: Longmans, Green & Co., vol 2, 1877; reprinted, London: The London School of Economics, 1931. Keynes, J.M. 1925. Alfred Marshall, 1842-1924. In Memotials of A&FedMa&a& ed. A.C. Pigou, London: Macmillan; reprinted, New York: Kelley and Millman, 19.56. Mangoldt, H. von. 1863. The exchange ratio of goods. Trans. by E. Henderson from GNndriss der Voh%wirtschufifrs/e/lre,Stuttgart: Engelhorn. In IntenaatibnalEconomic Papers, No. 11, 32-59, London: Macmillan, 1962. Marshall, A. 1879. The Purz Thory of Domestic Values. Privately printed; reprinted, London: London School of Economics, Scarce Works in Political Economy No. 1, 1930; and in amplified form in Whitaker, 1975. Marshall, A. 1890. Ptinca>ks of Economics, Vol. I. London: Macmillan. Niehans, J. 1990. A Historyof Economic Theory. Baltimore: The Johns Hopkins University Press. Parkin, M. 1990. Economics. Reading, Mass.: Addison-Wesley. Rau, K.H. 1841a. (Communications:) Economic politique. Bulletinde l’Acadhie des scienceset Be//es-L.&% de Bruxebs viii, pt. I: 148-52. Rau, K.H. 1841b. Gtundsiitzeder Volkswirthschaj?slehm. 4th ed, Heidelberg. Ricardo, D. 18 17-Z 1. On t/e Principlesof Politico/Economyand Taxation. Ed. P. Sraffa, Cambridge: Cambridge University Press, 195 1. Schneider, E. 1960. Hans von Mangoldt on price theory: a contribution to the history of mathematical economics. Economehico 28, 380-92. Schumpeter, J.A. 1954. HistoryofEconomic Analysti. London: George Allen & Unwin. Smith, A. 1776. Th wea/h of Nations. Ed. E. Cannan, New York: Modern Library, 1937. Stigler, S.M. 1980. Stigleis law of eponymy. Transactionsof tke NenmYo& Academy of Sciences Series II, 39: 147-58. Theocharis, R.D. 1983. Eurly Developmentsin Mathematiol Economics. 2nd edn. Philadelphia: Porcupine Press. Thweatt, W.O. 1983. Origins of the terminology, supply and demand. &ottbh Jouma/ ofPoliticalEconomy30, November, 287-94. Whitaker, J.K. (ed.) 1975. The E&y Economic Whitings of Alfi-edMa&all, 2867-1890. Vol. 1. London: Macmillan. Wright, H. 1927. Frederick Lavington. Economic JoumaL 37, 503-S. APPENDIX Mangoldt’s Algebraic Model of Joint Demand As stated in the article, Mangoldt ([ 18631 1962: 43-50) analyzed price determination of interrelated goods algebraically as well as graphically. His first algebraic model refers to two goods A and B jointly demanded in fixed proportions under a given expenditure constraint. Joint Demand Model Letting D denote quantity demanded, S quantity supplied, p price, E total expenditure, f a functional relationship between two variables, 1zthe fixed ratio in which the two goods (identified by subscript) are jointly demanded, Mangoldt’s first model consists of the following equations: 22 and Supply (1) (2) (3) DA = nDj3 E = PADA + PBDB PA = fA(s,l (4) PB (5) (6) = SA De = SB. = fB(sB) DA Equation (1) states the fixed-proportion assumption and equation (2) the expenditure or budget constraint. Equations (3) and (4) are supply functions, while equations (5) and (6) are market-clearing conditions. (1) To solve the system, Mangoldt first substituted into (2) and solved for pB to obtain ECONOMIC REVIEW, MARCH/APRIL 1992 (7) PB = E/D, - np~. This expression he interpreted as good B’s demand function containing good A’s price as a shift parameter. He then eliminatedpA from the expression by using (3), (5), and (1) to obtain (8) PB = IX& - ny-ah&~, which he equated with supply function (4)to determine the market-clearing price and quantity of good B and thus equilibrium of the system as a whole. Finally, from equation (7) he demonstrated that a fall in good A’s supply pricepA raises the demand price% of good B. He explained why: As good A becomes less expensive, buyers take more of it. Because the two goods are consumed in fiied proportions, however, buyers necessarily take more of good B too. The resulting increased demand for B bids up its price. In short, a movement down the demand curve for A is accompanied by an outward shift in the demand curve for B. In this way cost reductions that increase the supply of A raise the price of B. Equation (9) states the fixed-proportions assumption that the goods are supplied in the ratio tl. Equation (10) says that revenues must cover total cost. Equation (11) defines total cost K as the product of the unit cost c of a complex unit of output SA +& = (2 +n)SA times the number of units produced, or K = c(2 +n)SA = kSA where k = ~(2 +n). Equations (12) and (13) constitute the goods’ demand functions, while equations (14) and (15) state the marketclearing condition that supplies equal demands. To solve this system, Mangoldt substituted equations (9), (ll), (13), (14), and (15) into (10) to obtain an expression for B’s price in terms of A’s: (16) Further (17) pB = (k-PA/n. substitution yielded the expression SA = (Z/n)f/(k-PA/n/ which Mangoldt interpreted as A’s supply function given equilibrium in the market for good B. Equating (17) with (12) allowed him to solve for A’s marketclearing price and quantity and thus for equilibrium of the system as a whole. Joint Supply Model Mangoldt’s second model refers to two goods A and B jointly supplied in fixed proportions under a given cost constraint. Letting K and R denote total and unit costs, respectively, and using the same variables defined above, his model appears as follows: (9) SB ? nSA (10) PADA (11) K + (12) DA (13) (14) (15) De = -L&t,) SB = DB SA = DA. = PBDB = He concluded that a rise in the demand for A reduces B’s price and explained why. As A’s output rises to match the increased demand so too, via the fixed-proportions assumption, does B’s output. With the demand for B given, however, the extra output of that good constitutes an excess supply that puts downward pressure on its price. The price of B varies inversely with the demand for A. K k& = j-t&d FEDERAL RESERVE BANK OF RICHMOND 23