CHAPTER II -- Section 4

In this brief section Hicks describes the extension of the previous section's argument to cases involving a collection of more than two commodities.  The heart of his explanation lies in the following two statements:

...[S]o long as the terms on which money can be converted into other commodities are given, there is no reason why we should not draw up a determinate indifference system between any commodity X and money (that is to say, purchasing power in general).

 and

So long as the prices of other consumption goods are assumed to be given, they can be lumped together into one commodity 'money' or 'purchasing power in general.'

Therefore all the goods other than X can be lumped together into a money commodity, and we can analyze the indifference curves between X and money just as before.

Hicks indicates that this principle has quite general applications, some of which will be pointed out later on.  For the purposes of this section, however, the application is as follows:

For the present, we shall only use this principle to assure ourselves that the classification of the effects of price on demand into income effects and substitution effects, and the law that the substitution effect, at least, always tends to increase demand when price falls, are valid, however the consumer is spending his income.

CHAPTER II -- Section 3

In contrast to the previous section, which examined the effects of changes in income (with prices fixed), this section begins by considering changes in price with income fixed.  As before, Hicks uses an indifference diagram representing a consumer's preferences for two goods, X and Y.  Letting one of the prices vary (the price of X) while holding the other fixed, he represents the consumption possibilities by the diagram in Figure 7.  The different prices of X determine diagonal lines, such as LM and L'M in the figure, defined by the consumer's income.  For each such diagonal, there will be an equilibrium point where the diagonal touches an indifference curve.  The set of all such equilibrium points defines a curve, represented by MPQ in the figure, that Hicks calls the price-consumption curve.
Hicks next compares the price-consumption curve with the income-consumption curve (defined in the previous section), using Figure 8 for illustration.  He notes that the point Q, where indifference curve I2 is tangent to a line through Q and M, lies to the right of P', where the indifference curve is tangent to a line parallel to LM.  He points out that this follows from the convexity of the indifference curves.  To spell that out a bit, let me note that convexity in this context implies that the slope of the indifference curve is increasing (specifically, becoming less negative) as we move from left to right.  The line L"M, where Q is tangent, has a less negative slope than LM (which has the same slope as L'M').  Thus the point where the indifference curve is tangent to L"M must occur to the right of the point where it is tangent to L'M'.

Hicks claims that this proposition is "quite fundamental to a large part of the theory of value" and discusses a few of its implications.  When the price of X falls, the consumer can afford more of it with the same income; thus he moves along the price-consumption curve from equilibrium P to equilibrium Q.  Hicks states that

We now see that this movement from P to Q is equivalent to a movement from P to P' along the income-consumption curve, and a movement from P' to Q along an indifference curve.  We shall find it very instructive to think of the effect of price on demand as falling into these two separate parts.

There are thus two effects of the change in price:  an effect that is similar to an increase in income, and an effect of substitution of the now-cheaper commodity for other commodities.  The total effect is the sum of these two effects.  Hicks notes that the relative importance of these two effects will depend on the proportion of income that the consumer was spending on the commodity whose price has changed.  If the consumer was not buying much of X, then a fall in its price may not gain him much, and the income effect will tend to be swamped by the substitution effect.  Hicks states that this point is the justification of Marshall's assumption of constant marginal utility.

Hicks goes on to note that the substitution effect will always happen and will always cause an increase in demand for a commodity when its price falls.  The income effect is less reliable.  Although it will ordinarily work similarly to the substitution effect, in the case of inferior goods, the income effect of a decrease in price may actually lead to a decrease in demand.

CHAPTER II -- Section 2

In this section Hicks returns to the study of the indifference diagram.  Figure 5, shown below, plays an important role in the discussion in this section.  For a given amount of income, the set of possible consumption choices (assuming income is fully spent) will be defined by the diagonal line (LM in the figure) that connects the two points that are defined by spending all the income on one of the two goods and none on the other.  The consumer will choose a point along this line that touches an indifference curve (this will be the highest-valued indifference curve that the consumer could achieve with that income).

If the consumer's income increases, the diagonal line (which we can think of as the consumer's budget constraint) will move to the right. (The line L'M' in the figure shows one such example.)  As long as the prices do not change, the new budget constraint will be parallel to the old one.

As the consumer's income continues to increase, the budget constraint line moves to the right, and the equilibrium consumption point traces out a curve (labeled as C in the figure).  Hicks calls this the income-consumption curve.  He explains that the income-consumption curve will ordinarily slope upward and to the right, but he shows in Figure 6 two cases where this does not hold.  Below I've tried to redraw Figure 6 as it appears in the text.

It is not obvious why income-consumption curves might look like curves C1 and C2, so I've drawn another graph that attempts to show how this might come about.  In this graph, which I call Figure 6a, I've shown the consumer's income increased to the line L'M' .

We are assuming there could exist cases in which either C1 or C2 intersects L'M'  at an equilibrium point.  These cases correspond to different shapes of the indifference curve.  The dotted curve is an indifference curve that causes C1 to intersect the budget constraint at an equilibrium point.  The dashed curve corresponds to the case where C2 intersects at an equilibrium point.  Note that both of these cases involve one of the goods being significantly more desirable than the other.