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.

CHAPTER II -- THE LAW OF CONSUMER'S DEMAND

[My apologies for the gap in posting.  I hope to get back on a more regular schedule.]

In Section 1 of this chapter Hicks discusses Marshall's deduction of the downward slope of the demand curve from the law of diminishing marginal utility.  A critical step in Marshall's reasoning is apparently his assumption that the marginal utility of money is constant.  This assumption would imply that an individual's demand for any commodity is independent of his income.  Hicks has (in my opinion) a fairly charitable attitude toward this assumption, namely that "it is in fact an ingenious simplification, which is quite harmless for most of the applications Marshall gave it himself.  But it is not harmless for all applications..." and Hicks intends to make things clearer in the coming sections about how demand actually does interact with prices and  income.

This section has an example of something that Hicks is prone to do occasionally -- stating something fairly deep and non-obvious as though it were obvious.  In a footnote to the sentence about Marshall's assumption that the marginal utility of money is constant, he states "This, of course, abolishes any distinction between the diminishing marginal utility of a commodity and the diminishing marginal rate of substitution of that commodity for money."  The reader may be forgiven for thinking, "'of course'?"

CHAPTER I -- Section 9

In this short section -- the final section of Chapter I -- Hicks dispenses with the simplifying assumption that the consumer is choosing between only two possible consumption goods.  Although two-dimensional indifference diagrams are no longer useful for higher dimensions, the mathematical principles illustrated by them still hold.

The marginal rate of substitution can be defined as before, with the added proviso that the quantities consumed of all other commodities (Z...) must remain unchanged.  The consumer is only in full equilibrium if the marginal rate of substitution between any two goods equals their price-ratio.

 The principle of diminishing marginal rate of substitution must be generalized slightly.  In addition to diminishing marginal rate of substitution between each pair of goods,

more complicated substitutions (of some X for some Y and some Z) must be ruled out in the same way. We may express this by saying that the marginal rate of substitution must diminish in every direction.

And this concludes Chapter I!  Thank you for reading along this far.

CHAPTER I -- Section 8

In this section Hicks examines the foundation for the principle of Diminishing Marginal Rate of Substitution.  He reviews the fact that his goal is to deduce laws that deal with the reaction of a consumer to market conditions.  In particular, when conditions change, we expect the consumer to move from one position of equilibrium to another.  The principle of Diminishing Marginal Rate of Substitution must hold at the new position, or else it would not constitute an equilibrium.  Moreover, as Hicks argues, if there were some intermediate point between the two positions of equilibrium where the principle did not hold (and hence there were a "kink" in the indifference curve), then "there will be some systems of prices at which the consumer will be unable to choose between two different ways of spending his income."  The principle of Diminishing Marginal Rate of Substitution is a simple assumption that rules out these kinds of difficulties, and it is consistent with our experience as well.

After some discussion, Hicks argues that "other principles can be discovered whose foundation is exactly similar."  He describes how some of these principles will be worked out in later chapters, and he concludes this section with the statement that, "We are in sight of a unifying principle for the whole of economics."

CHAPTER I -- Section 7

In this section Hicks argues for rejecting the principle of Diminishing Marginal Utility and for replacing it with the principle of Diminishing Marginal Rate of Substitution.  Geometrically, this amounts to the rule that indifference curves must be convex to the axes.  He explains the meaning of Diminishing Marginal Rate of Substitution as follows:

Suppose we start with a given quantity of goods, and then go on increasing the amount of X and diminishing the amount of Y in such a way that the consumer is left neither better off nor worse off on balance; then the amount of Y which has to be subtracted in order to set off a second unit of X will be less than that which has to be subtracted in order to set off the first unit.  In other words, the more X is substituted for Y, the less will be the marginal rate of substitution of X for Y.

Hicks explains the need for this principle by noting that any point where it does not hold cannot be a stable equilibrium.  He notes that this is true even if the marginal rate of substitution equals the price ratio, and he illustrates it by means of a figure that looks somewhat similar to the one below:

The dashed curve doesn't appear in the book;  I've added it to help illustrate his explanation of the figure:

At the point Q on the diagram, the marginal rate of substitution equals the price-ratio, so that the price-line touches the indifference curve through Q.  But the marginal rate of substitution is increasing (the indifference curve is concave to the axes), so that a movement away from Q in either direction along LM would lead the individual on to a higher indifference curve.

The dashed curve is one such higher indifference curve.  Q obviously cannot be a point of equilibrium, because the consumer can move anywhere along the line LM and stay within his budget, therefore he would gain by moving to a point where the higher indifference curve intersects LM.

Hicks concludes this section by raising the question as to the foundation for assuming that Diminishing Marginal Rate of Substitution is a principle that is true in general.  He will deal with this question more in the next section.