CHAPTER II -- Section 7

We're nearing the end of Chapter II of Value and Capital -- after this section, the only remaining material is a technical note connecting the results of this chapter with the topic of consumer's surplus.  In Section 7, Hicks extends his analysis to consider the case of a consumer who comes to the market as both a buyer and a seller of some commodity X.

If we assume the price of X remains fixed, then the previous conclusions of this chapter are unaffected.  The consumer can be assumed to sell at the given price whatever stock of commodity X he brought to the market, then use the proceeds (and whatever other income he had) to purchase a bundle of commodities to maximize his utility.

If the price of X can vary, the situation changes slightly.  The substitution effect works the same as before;  a fall in the price of X will increase the demand for X through consumers substituting X for some of the other purchases they would have made.  The income effect is different, though.  A seller of X is made worse off by a fall in the price of X, so he will decrease his own purchases of X (unless X is for him an inferior good).  Thus, for a seller, income and substitution effects work in opposite directions (except in the unusual case of inferior goods), whereas for buyers the two effects work in the same direction.

Hicks notes that sellers often derive large parts of their income from one particular thing they sell and that in such a case "the income effect is just as powerful as the substitution effect, or is dominant.  We must conclude that a fall in the price of X may either diminish its supply or increase it."  He goes on to argue that this phenomenon is most pronounced in the case of the factors of production.

Thus a fall in wages may sometimes make the wage-earner work less hard, sometimes harder; for, on the one hand, reduced piece-rates make the effort needed for a marginal unit of output seem less worth while, or would do so, if incomes were unchanged; but on the other, his income is reduced, and the urge to work harder in order to make up for the loss in income may counterbalance the first tendency.

Hicks notes that this asymmetry between supply and demand had long been known.  But he regards the explanation of its cause in terms of income and substitution effects "as one of the first-fruits of our new technique."

CHAPTER II -- Section 6

In this section Hicks summarizes the conclusions thus far about the law of demand.  The demand curve (expressing the quantity of a commodity demanded as a function of its price) must always slope downward whenever the commodity is not an inferior good.  Even when the commodity is an inferior good, the demand curve will still slope downward as long as the proportion of income spent on the commodity is small.  And finally, even if neither of the above qualifications apply, the demand curve may still slope downward if substitution effects are large.

Hicks notes that, "Consumers are only likely to spend a large proportion of their incomes on what is for them an inferior good if their standard of living is very low," and he notes that the Giffen case, quoted by Alfred Marshall exactly fits this description.  But cases such as this are clearly rare.

Therefore, Hicks concludes that, "The simple law of demand -- the downward slope of the demand curve -- turns out to be almost infallible in its working.  Exceptions to it are rare and unimportant."

CHAPTER II -- Section 5

In this brief section, Hicks discusses making the transition from analyzing individual demand to analyzing market demand.

He notes that market demand is the sum of individual demands.  Therefore, the change in market demand is the sum of changes in the individual demands.  A change in market demand due to a change in price can be divided into substitution and income effects.  The substitution effect consists of the sum of the individual substitution effects, and the income effect consists of the sum of the individual income effects.  Since all the individuals' substitution effects imply increased consumption of a good whose price falls, the market substitution effect must imply the same.  Individual income effects are not as reliably uniform in direction, therefore the group income effect must be similarly unreliable.  Finally, group income effects will tend to be negligible for any commodity on which the group spends a small proportion of its total income.

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'?"