The New Palgrave Dictionary of Economics

2018 Edition
| Editors: Macmillan Publishers Ltd

Offer Curve or Reciprocal Demand Curve

  • Harvey Gram
Reference work entry


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.


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


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.

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.

Marshall commented that his ‘International Trade curves … were set to a definite tune, that called by [John Stuart] Mill’ (Pigou 1925, p. 451). It was Mill who had written that

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, Pj, j =1,2, outputs, Yjk, in each country k, k = A, B, are such that the value of production, ΣjPjYjk is a maximum, subject to resource constraints which define the production possibilities sets. Simultaneously, for given factor supplies, Fik, i =1, 2 (assuming two factors for simplicity), rental rates or factor prices, Wik, are such that cost of production, ΣiWikKik, 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, Xjk satisfy Walras’s Law: \( {\varSigma}_j{P}_j{X}_j^k={\varSigma}_j{P}_j{Y}_j^k \).

Given P1 and P2, 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.

Offer Curve or Reciprocal Demand Curve, Fig. 1

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, OE1, OE2, and OE3. Perpendiculars from E1, E2, and E3 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 P1/P2 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 P1/P2 increases, convexity of the production possibilities set ensures that Y1A increases and Y2A 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, X1A decreases and X2A 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 P1/P2 reduces the real purchasing power of country A which is an importer of good 1. If both goods are normal, the reduction in X1A associated with substitution is reinforced while the increase in X2A 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 X1A and reinforcing it in determining X2A. 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 OE1 and for vectors intermediate between OE2 and OE3, and negative for vectors intermediate between OE1 and OE2 and steeper than OE3. 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 OE1, OE2, and OE3 (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).

Modern discussions of the offer curve in the presence of increasing returns are concerned with technological externalities rather than with irreversible economies of large-scale production. A firm’s output may depend on the output of the industry to which it belongs. Output in one industry, or the level of employment of particular factors in that industry, may have external effects on the output of another industry. Theoretical questions then arise concerning the convexity of the production possibilities set, the relationship between opportunity cost and relative price, and whether or not production occurs at a limit point of the feasible set. What the models have is common with Marshall’s discussion is that the associated offer curves are no longer convex-valued functions of the relative price ratio. Marshall indicated this by drawing offer curves with several inflexion points. In modern treatments of external economies in production, offer curves typically have the shape indicated in Fig. 2. Curvature at the origin is opposite to that indicated in Fig. 1, changing abruptly at points of complete specialization. (The latter may or may not correspond to the critical points in Fig. 1 where excess supply reaches a maximum.)

Offer Curve or Reciprocal Demand Curve, Fig. 2

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 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, P1/P2, slightly greater than the (absolute) slope of OE1. Hypothetical exports of good 1 by country A exceed hypothetical imports by country 2, while the opposite is true for good 2. If P1 falls and P2 rises in this situation, P1/P2 falls back towards OE1. The opposite is true for price ratios slightly lower than the (absolute) slope of OE1. Thus, E1 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 OE1, 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 E1 and E3, exporters’ income effects are not strong enough to swamp importer’s income effects plus all substitution effects. At E2, 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 P1/P2, 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).

A variation on the above analysis allows trade to take place out of equilibrium but assumes that demand for imports is always satisfied. A disequilibrium exchange ray, such as 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 P1 then falls and P2 rises, P1/P2 once again falls back toward OE1. 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 versa when the rate of profits in the foreign trade [is] exceptionally low. (Whitaker 1975, p. 151)

This adjustment in the production of exports (imports and the domestic consumption of exports held constant) appears to have been meant by Marshall to reflect a concomitant change in the production of non-traded goods (Marshall 1923, pp. 354–5n). Thus, at points off the offer curves

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.


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


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Authors and Affiliations

  • Harvey Gram
    • 1
  1. 1.