# Offer Curve or Reciprocal Demand Curve

**DOI:**https://doi.org/10.1057/978-1-349-95189-5_1646

## Abstract

The offer curve made its first appearance in Alfred Marshall’s *Pure Theory of Foreign Trade* (1879), a privately printed paper consisting of the second and third chapters (chosen by Henry Sidgwick) of a four chapter manuscript. Almost 50 years passed before Marshall’s analysis became generally available under his own name as Appendix J to *Money, Credit and Commerce* (1923). Thus, it was mainly through the writings of Edgeworth (1894) and others who had read Marshall’s original contribution (see Whitaker 1975, p. 114n), that the offer curve came to be known.

## Keywords

Adjustment process Community indifference curves Continuity Convexity Duality Excess demand Excess supply External economies Homogeneous programming Income effects Increasing returns Marshall, A. Mill. J. S. Monotonicity Offer curve Reciprocal demand curve Recontracting Stability of equilibrium Substitution effects Tatonnement Trading curve Uniqueness of equilibrium Walras’s Law## JEL Classifications

D0The offer curve made its first appearance in Alfred Marshall’s *Pure Theory of Foreign Trade* (1879), a privately printed paper consisting of the second and third chapters (chosen by Henry Sidgwick) of a four chapter manuscript. Almost 50 years passed before Marshall’s analysis became generally available under his own name as Appendix J to *Money, Credit and Commerce* (1923). Thus, it was mainly through the writings of Edgeworth (1894) and others who had read Marshall’s original contribution (see Whitaker 1975, p. 114n), that the offer curve came to be known.

Newman (1965, p. 104) notes the objections raised by Edgeworth (1924) and Wicksell (1925) to the name *offer curve* which was coined by W.E. Johnson (1913) and used by Bowley (1924). They were concerned that *offer curve* might suggest an asymmetry between supply and demand where, in fact, there was none. The alternative name, *reciprocal demand curve*, or *trading curve* (Newman 1965, pp. 89 ff.), avoids any such suggestion.

supply and demand are but another expression for reciprocal demand; and to say that value will adjust itself so as to equalize demand with supply, is in fact to say that it will adjust itself so as to equalize demand on one side with the demand of the other. (Mill 1852, p. 604)

Mill’s purpose was to close Ricardo’s trade model by finding prices such that ‘demand will be exactly sufficient to carry off the supply’ (Mill 1844, p. 238). Edgeworth, though giving high praise to Mill’s mature statement of his equation of international demand, thought little of Mill’s exact solution; and Marshall commented only ‘that the special example which [Mill] has chosen does not illustrate the general problem in question’ (Whitaker 1975, p. 148). Chipman has argued, however, that Mill, in effect, solved a problem in what would now be called *homogeneous programming.* In claiming Mill’s result to be a ‘genuine and correct proof of the existence of equilibrium … [pre-dating the next such proof] by eighty years’, Chipman remarks of Mill’s law of international value, or *reciprocal demand*, that in ‘its astonishing simplicity, it must stand as one of the great achievements of the human intellect’ (Chipman 1965, Part 1, pp. 491 and 486, respectively).

Modern uses of the offer curve in trade theory and other areas (see Cass et al. 1980; Cass 1980; Grandmont 1985) have a greater affinity with Mill’s analysis of a general equilibrium of supply and demand than with Marshall’s original argument. That argument had three parts. The first is directly relevant to modern theory and concerns what would now be called the *income and substitution effects* of relative price changes. The second part deals with *increasing returns in production*, a phenomenon whose formulation and implications for traditional theory remain controversial. And finally, there is the problem of the *adjustment mechanism*, a part of Marshall’s theory which, though highly regarded (Whitaker 1975, p. 115; Kemp 1964, p. 60), has been almost completely eclipsed by a Walrasian inspired ‘stability’ analysis. What follows is accordingly divided into three sections, following a brief discussion of the formal basis for the offer curve as it exists in modern theory.

## General Equilibrium

The traditional offer curve arises in the context of a two-country general equilibrium model. Each country has an endowment of resources and a technology for transforming the associated factor service flows into flows of output of two tradable commodities. Resources are owned by the country’s consumers, each of whom has a preference ordering which is continuous, convex, and monotonic. Under constant or decreasing returns to scale, resources and technology generate a *convex* production possibilities set (although some degree of increasing returns to scale is not inconsistent with convexity). The assumptions on preferences guarantee that the set of points ranked ‘at least as good as’ any given point is also *convex*, its boundary defining an indifference curve. If consumers have identical preferences and factor endowments, community indifference curves are simply radial expansions of individual indifference curves (cf. Chipman 1965, part 2, pp. 690–8).

Geometrical derivations of the offer curve utilize techniques introduced by Leontief (1933), Lerner (1932, 1934), and Meade (1952). Implicit in the derivation is the solution to a pair of problems in constrained optimization. At given commodity prices, *P*_{j}, *j* =1,2, outputs, *Y*_{j}^{k}, in each country *k, k* = *A*, *B*, are such that the value of production, *Σ*_{j}*P*_{j}*Y*_{j}^{k} is a maximum, subject to resource constraints which define the production possibilities sets. Simultaneously, for given factor supplies, *F*_{i}^{k}, *i* =1, 2 (assuming two factors for simplicity), rental rates or factor prices, *W*_{i}^{k}, are such that cost of production, *Σ*_{i}*W*_{i}^{k}*K*_{i}^{k}, is a *minimum*, subject to price constraints which state that equilibrium profits are nowhere positive. Duality theory establishes an equality between maximum value and minimum cost. Because consumers own all resources, total cost is equal to total income, and so consumption choices, *X*_{j}^{k} satisfy Walras’s Law: \( {\varSigma}_j{P}_j{X}_j^k={\varSigma}_j{P}_j{Y}_j^k \).

*P*

_{1}and

*P*

_{2}, there may or may not exist a solution or set of solutions, \( \left({Y}_1^k-{X}_1^k,{Y}_2^k-{X}_2^k\right),k=A,B \). If \( {Y}_1^k-{X}_1^k \) is positive (negative), Walras’s Law ensures that \( {Y}_2^k-{X}_2^k \) is negative (positive): country

*k*offers an excess supply of good 1 (good 2) in order to satisfy its excess demand for good 2 (good 1). If both prices are positive, \( {Y}_1^k-{X}_1^k=0 \) implies \( {Y}_2^k-{X}_2^k=0 \); while if one price is zero and satiation is ruled out, the corresponding excess demand will be unbounded. In Fig. 1, the offer curves therefore occupy quadrants II and IV, passing through the origin and approaching the axes asymptotically.

At a given price ratio, measured by the (absolute) slope of a straight line through the origin, the solution for excess supply and excess demand in each country is unique in Fig. 1 and therefore (trivially) convex-valued. The solutions are also upper semicontinuous (Chipman 1965, part 2, p. 717). These two conditions are the basis for the idea of ‘connectedness’ or ‘continuity’ of the offer curve. They follow from the postulates on preferences: continuity, convexity, and monotonicity (where the last can be replaced by the assumption that outputs are strictly positive). The importance of the postulates turns on the fact that when the set of offers by each country, at a given price vector, is closed, convex, and upper semicontinuous, it can be shown that an equilibrium price vector exists. Mill found a *unique* equilibrium for Ricardo’s trade model by assuming (implicitly) unitary price and income elasticities of demand for the two commodities in each country (Chipman 1965, part 1, pp. 483–91). The offer curves in Fig. 1 intersect three times, indicating *three isolated* equilibrium price vectors, *OE*_{1}, *OE*_{2}, and OE_{3}. Perpendiculars from *E*_{1}, *E*_{2}, and E_{3} to the axes mark off the matching reciprocal demands of each country.

## Income and Substitution Effects

The shape of each curve in Fig. 1 reflects the income and substitution effects of relative price changes. Consider country *A.* When *P*_{1}/*P*_{2} is zero output of good 1 is zero and demand is unbounded so that the offer curve shoots off to the right in quadrant IV. As *P*_{1}/*P*_{2} increases, convexity of the production possibilities set ensures that *Y*_{1}^{A} increases and *Y*_{2}^{A} decreases (unless both remain constant at a vertex). Assuming hypothetically that country *A* is confined to a given, convex-to-the-origin community indifference curve, *X*_{1}^{A} decreases and *X*_{2}^{A} increases (unless both remain constant at a ‘corner’ of the curve). These two substitution effects, one in production and one in consumption, reduce excess demand for good 1 (and excess supply of good 2) in quadrant IV, while raising excess supply of good 1 (and excess demand for good 2) in quadrant II. Note, however, that excess supply of good 1 reaches a maximum at *a* in quadrant II, while the same is true for good 2 at a' in quadrant IV. The reason for this is the income effect of the relative price change. In quadrant IV, a higher *P*_{1}/*P*_{2} reduces the real purchasing power of country *A* which is an importer of good 1. If both goods are normal, the reduction in *X*_{1}^{A} associated with substitution is reinforced while the increase in *X*_{2}^{A} is offset. In quadrant II, the reverse is true. Country *A*, as an exporter of good 1, gains from an increase in the relative price of good 1. Now the income effect is pushing against the substitution effect in determining *X*_{1}^{A} and reinforcing it in determining *X*_{2}^{A}. Along the offer curve between *a* and *a*′ substitution effects in production and consumption dominate the income effect. Beyond those critical points, the income effect is dominant in the sense that the excess supply of a commodity is lower when its relative price is higher.

Marshall’s explanation of the critical point *a* is somewhat different. The independent variable in his analysis is the quantity of imports rather than the relative price ratio. In Marshall’s normal class, an increase in imports in the neighbourhood of the origin results in an increase in receipts and, for this reason, the volume of exports which can be produced at normal profits increases. Receipts from imports pay the cost of exports. The slope of the offer curve increases (in absolute value) from the origin to point *a* because demand for imports is elastic. Beyond point *a* import demand turns inelastic, receipts fall off, and so the volume of exports which can be produced at normal profits declines. Marshall referred to this situation as Class I.

A final aspect of the income effect of relative price changes concerns changes in the distribution of purchasing power among consumers within each country, a problem which can only be addressed if consumers have different resource endowments. Assume therefore that country *A* and country *B* in Fig. 1 are, in fact, two groups of consumers within a single country. Aggregate excess demand for good 1 and excess supply of good 2 are *positive* for price vectors flatter than *OE*_{1} and for vectors intermediate between *OE*_{2} and *OE*_{3}, and *negative* for vectors intermediate between *OE*_{1} and *OE*_{2} and steeper than *OE*_{3}. An offer curve defined for the two groups of consumers would therefore pass through the origin three times, tying itself in a bow. Its slope at the origin has three isolated values given by the slopes of *OE*_{1}, *OE*_{2}, and *OE*_{3} (cf. Johnson 1959, 1960). A pair of such curves, constructed for two countries, each composed of two differentiated groups of consumers, would intersect at various points in quadrants II and IV. This indicates the possibility of trade pattern reversals as the relative price ratio takes on different equilibrium values.

The one proposition that Marshall insisted upon as ‘the only law to which the curves must conform under all circumstances’ is violated by offer curves which form loops through the origin. This was his Proposition VI to the effect that country *A*’s offer curve ‘cannot in any case be cut more than once by a horizontal line. Similarly [country B’s curve] cannot in any case be cut more than once by a vertical line’ (Whitaker 1975, p. 140). Marshall’s argument, however, had nothing to do with the income redistribution effects which, upon aggregation of consumer groups (as above), countries (see Chipman 1965, part 2, p. 217), or generations (see Cass et al. (1980) pp. 25–6), can result in offer curves exhibiting the floral patterns first noted by Johnson (1959, 1960). Rather he was concerned with the problem of increasing returns.

## Increasing Returns

Marshall put increasing returns under the heading of ‘problems of Exceptional Class II’ (Whitaker 1975, p. 144). Where an increase in the production of exports leads to the introduction of extensive economies, a reduction in the volume of exports to a level previously experienced would not require as large a volume of imports to cover their costs of production as had previously been the case (assuming implicitly an elastic demand for imports). Thus, a movement along the offer curve would simultaneously shift the curve *towards* the export axis. Moreover, ‘if time was allowed for the development of economies of production on a large scale, time ought to be allowed for the general increase of demand’ (Pigou 1925, p. 49). In that event, a given volume of imports would yield higher receipts thereby shifting the offer curve *away* from the import axis.

Marshall’s long period offer curve does not show any maximum level of exports, as does every static curve constructed on the basis of given resources and technology. Moreover, the slope of the long period curve can *decrease* (in absolute value) as a consequence of technological change. It was in this context that Marshall denied that any given volume of imports would cover the costs of more than one volume of exports. His Proposition VI claimed that economies of scale would never be sufficient to lower the total cost of a larger volume of exports below that of a smaller volume previously produced. Marshall had made the same assumption in the first edition of his *Principles*, but dropped it subsequently (Whitaker 1975, p. 116).

The curves in Fig. 2 have three intersections indicating three equilibrium trades. (These may be reduced to two by drawing curves which are mutually tangent at the origin indicating equal pre-trade price ratios which, in Fig. 1, would be sufficient to rule out an equilibrium with positive trade.) There is nothing in principle to prevent all three points from falling along a single ray. The resulting indeterminacy in the volume and direction of trade is the main distinguishing feature of trade models with external economies in production. Chacholiades (1978, pp. 197–9) has considered the problem in some detail, arguing that, in general, a country benefits from specialization in the production of the commodity subject to external economies. If this is the same commodity in each country, then, depending on the pattern of demand, one country may lose from trade. This suggests an even sharper conflict of interest than is evident in Fig. 1 where a country is clearly better off in that equilibrium in which its exports are smallest and its imports are largest.

## Stability

Stability of equilibrium is defined in relationship to a process of adjustment which determines the movement of prices and/or quantities when the system is out of equilibrium. A distinction has been drawn between processes which focus on price changes and those which focus on quantity changes. A frequently considered case is that in which prices respond to differences between *hypothetical* supply and demand (those quantities which would prevail on each side of the market if the current price were an equilibrium price). Transactions, however, only take place in equilibrium. This is a *recontracting process* and its convergence to an equilibrium of supply and demand is often referred to as the Walrasian *tâtonnement* or ‘groping’ process. Walras, in fact, referred to *tâtonnement* in connection with the problem of bringing a set of interrelated markets into equilibrium *sequentially*, and, as such, it was problematical since ‘few prices will lie quiet at equilibrium while others are brought to heel, and the whole thing may turn out to be like the labour of Sisyphus’ (Newman 1965, p. 103).

The offer curves in Fig. 1 can be used to illustrate the stability of a recontracting process. Consider a price ratio, *P*_{1}/*P*_{2}, slightly greater than the (absolute) slope of *OE*_{1}. Hypothetical exports of good 1 by country *A* exceed hypothetical imports by country 2, while the opposite is true for good 2. If *P*_{1} falls and *P*_{2} rises in this situation, *P*_{1}/*P*_{2} falls back towards *OE*_{1}. The opposite is true for price ratios slightly lower than the (absolute) slope of *OE*_{1}. Thus, *E*_{1} is a stable point. Note that, as prices move, the four substitution effects (in production and consumption in both countries) contribute towards reducing the initial divergence between supply and demand for each good. The same is true of part of the income effect. As the price ratio falls towards *OE*_{1}, for example, country *A* is made worse off and country *B* is made better off. As *importers*, country *A* demands less of good 2 and country *B* demands more of good 1 (assuming that the goods are normal), and this reinforces the substitution effects. But, as *exporters*, country *A* demands less of good 1 while country *B* demands more of good 2, thereby exacerbating the initial excess supply (of good 1) and excess demand (for good 2). At stable points, such as *E*_{1} and *E*_{3}, exporters’ income effects are not strong enough to swamp importer’s income effects plus all substitution effects. At E_{2}, however, a slight increase in the relative price of good 1 would be associated with a hypothetical excess demand for good 1 and excess supply of good 2. The recontracting hypothesis would therefore result in a further increase in *P*_{1}/*P*_{2}, reflecting the fact that the initial increase has generated exporters’ income effects which swamp all other income and substitution effects (cf. Caves and Jones 1985, pp. 492–4).

*Oe*in Fig. 1, cuts the two offer curves in distinct points. Perpendiculars to the axes from these points indicate excess supply of good 1, which causes inventories to rise, and excess demand for good 2, which causes inventories to fall. If

*P*

_{1}then falls and

*P*

_{2}rises,

*P*

_{1}/

*P*

_{2}once again falls back toward

*OE*

_{1}. During the process, however, country

*A*must be selling assets to country

*B*in order for the trade flow to be financed. If the consequence of this is to alter the position and shape of the offer curves, a more complete and undoubtedly more complex analysis of the convergence to equilibrium would be required (cf. Jones 1961, p. 203). Marshall’s discussion of the adjustment mechanism is concerned neither with a recontracting process nor with inconsistent trades ‘financed’ by changes in inventories. In this theory, profits in export industries are abnormally high at points between a country’s offer curve and the axis measuring its imports. On the other side of the curve, profits in exports are abnormally low. Marshall’s adjustment mechanism is summed up as follows:

when the terms on which a country’s foreign trade is conducted are such as to afford a rate of profits higher than the rate current in other industries, the competition of traders to obtain these higher profits will lead to an increase in the exportation of her wares: and

vice versawhen the rate of profits in the foreign trade [is] exceptionally low. (Whitaker 1975, p. 151)

production is changing in both countries, [and so] the dimensions of the Edgeworth box must be changing, as are also the shapes of the offer curves. The extreme subtlety of the Marshallian conception becomes more apparent the further one probes into it. (Chipman 1965, Part 2, p. 723)

One can only conclude that efforts to formalize Marshall’s ‘dynamics’ (Samuelson 1947, pp. 266–8; Kemp 1964, pp. 66–9; Amano 1968, pp. 326–39) are but valiant attempts to come to terms with an approach to equilibrium which itself moves in an unspecified manner as a consequence of not being attained initially.

## Conclusion

Not surprisingly, that part of Marshall’s *Pure Theory of Foreign Trade* which is most evident in modern discussions of the offer curve concerns the income and substitution effects which are central to the theory of supply and demand equilibrium. His treatment of increasing returns and his discussion of the adjustment process raise dynamic considerations associated with changes in technology and with changes in the structure of productive capacity. It is just such changes which present the equilibrium theory of supply and demand with some of its greatest difficulties.

## See Also

## Bibliography

- Amano, A. 1968. Stability conditions in the pure theory of international trade – Rehabilitation of the Marshallian approach.
*Quarterly Journal of Economics*82: 326–339.CrossRefGoogle Scholar - Bowley, A. 1924.
*The mathematical groundwork of economics*. New York: Oxford University Press.Google Scholar - Cass, D. 1980. Money in consumption loan type models: An addendum. In
*Models of monetary economics*, ed. J.H. Kareken and N. Wallace. Minneapolis: Federal Reserve Bank of Minneapolis.Google Scholar - Cass, D., M. Okuno, and I. Zilcha. 1980. The role of money in supporting the Pareto optimality of competitive equilibrium in consumption loan type models. In
*Models of monetary economies*, ed. J.H. Kareken and N. Wallace, 13–48. Minneapolis: Federal Reserve Bank of Minneapolis.Google Scholar - Caves, R.E., and R.W. Jones. 1985.
*World trade and payments, an introduction*, 4th ed. Boston: Little, Brown.Google Scholar - Chacholiades, M. 1978.
*International trade theory and policy*. New York: McGraw-Hill.Google Scholar - Chipman, J.S. 1965. A survey of the theory of international trade.
*Econometrica*33, Pt. I, 477–519; Pt. II, 685–760.Google Scholar - Edgeworth, F.Y. 1894. The theory of international values.
*Economic Journal*4: 35–50, 424–443, 606–638. Reprinted in F.Y. Edgeworth,*Papers relating to political economy*[1925], vol. 2. New York: Burt Franklin, 1970.Google Scholar - Edgeworth, F.Y. 1924. Review.
*Economic Journal*34: 430.CrossRefGoogle Scholar - Grandmont, J.-M. 1985. On endogenous competitive business cycles.
*Econometrica*53: 995–1045.CrossRefGoogle Scholar - Johnson, H.G. 1959. International trade, income distribution, and the offer curve.
*Manchester School of Economic and Social Studies*27: 241–260.CrossRefGoogle Scholar - Johnson, H.G. 1960. Income distribution, the offer curve, and the effects of tariffs.
*Manchester School of Economic and Social Studies*8: 215–242.CrossRefGoogle Scholar - Johnson, W.E. 1913. The pure theory of utility curves.
*Economic Journal*23: 483–513.CrossRefGoogle Scholar - Jones, R.W. 1961. Stability conditions in international trade: A general equilibrium analysis.
*International Economic Review*2: 199–209.CrossRefGoogle Scholar - Kemp, M.C. 1964.
*The pure theory of international trade*. Englewood Cliffs: Prentice-Hall.Google Scholar - Leontief, W.W. 1933. The use of indifference curves in the analysis of foreign trade.
*Quarterly Journal of Economics*47: 493–503.CrossRefGoogle Scholar - Lerner, A.P. 1932. The diagrammatical representation of cost conditions in international trade.
*Economica*12: 346–356.CrossRefGoogle Scholar - Lerner, A.P. 1934. The diagrammatical representation of demand conditions in international trade.
*Economica*, NS 1: 319–334.Google Scholar - Marshall, A. 1879.
*The pure theory of foreign trade.*Reprinted in Whitaker (1975).Google Scholar - Marshall, A. 1923.
*Money, credit and commerce*. New York: Augustus Kelley, 1965.Google Scholar - Meade, J.E. 1952.
*A geometry of international trade.*Reprinted. New York: Augustus Kelley, 1971.Google Scholar - Mill, J.S. 1844. On the laws of interchange between nations; and the distribution of the gains of commerce among countries of the commercial world. In
*Essays on some unsettled questions of political economy.*Reprinted in*Collected works of John Stuart Mill*, vol. 4. Toronto: University of Toronto Press, 1967.Google Scholar - Mill, J.S. 1852.
*Principles of political economy.*3rd ed. Reprinted in*Collected works of John Stuart Mill*, vol. 3. Toronto: University of Toronto Press, 1965.Google Scholar - Newman, P. 1965.
*The theory of exchange*. Englewood Cliffs: Prentice-Hall.Google Scholar - Pigou, A.C. (ed.). 1925.
*Memorials of Alfred Marshall*. London: Macmillan.Google Scholar - Samuelson, P.A. 1947.
*Foundations of economic analysis*. Cambridge, MA: Harvard University Press.Google Scholar - Whitaker, J.K. (ed.). 1975.
*The early writings of Alfred Marshall, 1867–1890*, vol. 2. New York: Free Press.Google Scholar - Wicksell, K. 1925. Matematisk nationalekonomi.
*Ekonomisk Tidskrift*27: 103–125. Trans. in*Knut Wicksell, Selected Papers on Economic Theory*, ed. E. Lindahl. Reprinted, New York: Augustus Kelley, 1969.CrossRefGoogle Scholar