Hicks's work in this area was first published in 1934 in an article he wrote jointly with R. G. D. Allen (1934a), but it is more comprehensively set out in Chapters 1-3 of Value and Capital (1939a). Their work in this area was a development of earlier work by Edgeworth, Pareto and Slutsky, and they established that ordinal utility (elaborated in terms of indifference curves and budget lines) could derive the same propositions as cardinal utility (elaborated in terms of measurable marginal utilities) but that the former achieved the same results more clearly and more precisely. However, they accepted that the two theories were saying the same thing, e.g. 'Tangency between the price line and an indifference curve is the expression ... of the proportionality between marginal utilities and prices' (Hicks, 1939a, p. 17). This can be demonstrated conveniently as in Figure 2. The slope of indifference curve, I, can be written as dY/dX and measures the marginal rate of substitution (MRS) of X for Y. However, along an indifference curve the total utility of the consumer is constant. Therefore it is possible to demonstrate that the slope of an indifference curve can also be defined as the ratio of the marginal utilities (MU) of X for Y.2 Thus the slope of indifference curve, I = dY/dX = MUx/MUy = MRS of X for Y. However, the slope of the budget line, AB = Px/Py. Therefore at the point of tangency MUx/MUy = Px/Py = MRS of X for Y, i.e. the ordinal and cardinal conditions for a maximum are equivalent. It is also of interest to note the similarities between this analysis and our previous discussion of isoquants. The elasticity of substitution is a property of an isoquant but the same general principle holds for an indifference curve. Specifically the marginal rate of substitution diminishes along a convex indifference curve and the marginal rate of technical substitution (MRTS) diminishes along a convex isoquant. Figure 2 can then be relabelled to demonstrate the cost minimising output level, i.e. measuring capital along the Y axis and labour along the X axis, cost minimisation occurs where PL/Pk = MRTS of L for K. In fact, it was the realisation of this symmetry that led Hicks in the direction of the 1934 Hicks-Allen article.
Apart from considerations of presentation and clarity the major advance which indifference curve analysis offered was its ability to distinguish between the income and substitution effects of a price change. Moreover, Hicks and Allen pointed out that the sign of this income effect could not be predicted from the simple assumption that the consumer seeks to maximise utility. However they demonstrated that in the absence of income effects the substitution effects were remarkably regular. For example, they were able to show that not only is the direct substitution effect of a change in the price of X on the quantity demanded of X always negative, but that a number of secondary substitution theorems could also be deduced - the most important being the proposition that the substitution effect of a change in the price of X on the quantity demanded of Y must be exactly equal to the effect of a change in the price of Y on the quantity demanded of X.
The latter substitution theorem refers to the cross elasticity of demand and its importance derives from the fact that it successfully cleared up the elasticity problem that had initially prompted the Hicks-Allen article. Specifically, Henry Schultz had estimated some cross elasticities of demand (the demand for X against the price of Y and vice versa) and found them to be non-symmetric whereas Marshallian demand theory suggested they should be symmetric. Hicks and Allen were then able to point out that 'Schultz had left out the income effects which for direct elasticities may indeed be negligible, as Marshall (in effect) supposed them to be; but for cross elasticities there is no reason why they should be negligible' (Hicks, 1974b).
This quotation is important in another respect. Basically, Marshall had ignored the income effect by assuming a constant marginal utility of money. The quotation suggests that from the point of view of ordinary demand analysis Hicks accepted Marshall's neglect of the income effect. Indeed, such a neglect is justifiable as long as the good in question represents a small part of the consumer's budget and Marshall was always careful to make this assumption. Thus in Hicks's view the ordinal approach 'was not so clear an advance (on the older marginal utility approach) as is usually supposed' (1976, p. 137). By the same token Hicks has never accepted the extension of his analysis via Samuelson's revealed preference approach as either necessary or desirable. 'Marshall's consumer who decides on his purchase by comparing the marginal utility of what is to be bought with the marginal utility of the money he will have to pay for it is more like an actual consumer' (Hicks, 1976, p. 138). In fact in his Revision of Demand Theory (1956a) Hicks rejects the revealed preference approach even for econometric purposes, stating that for the data to which econometrics is usually applied on ordinal scale of preferences seems the most sensible hypothesis with which to begin the analysis.
However, despite his own reservations, Hicks's ordinal approach in the end met with very little criticism and quickly became a standard part of the economist's tool kit. Moreover, many observers would claim that Hicks has underestimated the step forward that his use of indifference curves entailed, e.g. Blaug explains that European marginal utility doctrine around the First World War proliferated 'in subtle distinctions and metaphysical classifications. [Thus], one is made to realize how much has been swept away by the Hicksian Revolution - all to the good we would say' (Blaug, 1976, p. 388).