CHAPTER I - Section 2

In Section 2 Hicks summarizes the utility theory writings of Vilfredo Pareto.  After starting off with ideas similar to those developed by Marshall, Pareto turned his attention to analyzing goods that are related -- in the sense of being complementary or of being (at least partial) substitutes.  Pareto used a geometric device used earlier by the Irish-English economist Francis Ysidro Edgeworth -- the indifference curve.  For two commodities X and Y we can construct a map like that shown in the figure below (an imitation of Hicks's Figure 2).

The points in this X-Y plane correspond to quantities of the goods X and Y that are consumed -- these quantities of consumption are understood to provide utility to the consumer.  The curves shown in the figure are indifference curves:  namely, curves along which all points have the same total utility.  We can think of these curves as being similar to contour lines on a map -- they show points that are at the same "elevation" in three dimensions.  Referring to this figure, Hicks goes on to explain:

If P and P' are on the same indifference curve, that means that the total utility derived from having PM and PN [these distances correspond to quantities of the goods X and Y, respectively] is the same as that derived from having P'M' and P'N'.  If P" is on a higher indifference curve than P (the curves will have to be numbered so as to distinguish higher from lower), then P"M" and P"N" will give a higher total utility than PM and PN. 

Hicks explains that, assuming positive marginal utility (each increment increases utility, at least a little), the indifference curves must slope downwards from left to right.  One way to understand this is to realize that increasing X without decreasing Y should lead one to a higher indifference curve.  Put another way, if we increase the amount of X consumed, the only way to keep the same utility is to decrease the amount of Y.

As Hicks explains, the slope of an indifference curve at any point has a definite and important meaning, "It is the amount of Y which is needed by the individual in order to compensate him for the loss of a small unit of X." Assuming quantities are small, the decrease in utility from losing the small unit of X is given by the product (amount of X lost) x (marginal utility of X).  The gain in utility from increasing the amount of Y is given by the product (amount of Y gained) x (marginal utility of Y).  We're assuming these utilities exactly balance each other, so (amount of Y gained) x (marginal utility of Y) = (amount of X lost) x (marginal utility of X).  A small amount of algebra then gives us [apologies -- I need learn a better way to paste equations in]:

(amount of Y gained) / (amount of X lost) = (marginal utility of X) / (marginal utility of Y)

But the left-hand side of this equation is the slope, so the slope of the indifference curve equals the ratio of the marginal utilities.  (There are two things to keep in mind here -- one is that the slope of the indifference curve is changing as we consider different points along the curve;  the other is to notice which of the marginal utilities is in the numerator and the denominator on the right-hand side.)

Hicks concludes this section by examining the question of whether diminishing marginal utility implies that the indifference curves must be convex to the origin.  (One way to think about convexity here is to think of it as implying that, for any two points on the curve, the curve in between these two points must lie below a straight line connecting the two points.)  Hicks argues that, at first sight, this would seem to be true, but that it does not necessarily follow.  A counterexample is given by the case of related goods in which "the increase in X lowers the marginal utility of Y [and vice versa, and]... these cross-effects are considerable." 

Thanks for reading this far.  In the next section (a short one), Hicks notes "a really remarkable thing" about indifference curves.