Value & Capital, CHAPTER IV -- THE GENERAL EQUILIBRIUM OF EXCHANGE, Section 1

With this chapter of Value and Capital, Hicks begins Part II of the book -- GENERAL EQUILIBRIUM.  Hicks begins this first section of the chapter by reviewing what was accomplished in his elaboration of the the theory of consumer's demand.  Among the accomplishments were finding "a precise meaning for the assumption that an individual's 'wants' are given:  it must mean that he has a given scale of preferences."  Then he investigated "how an individual with a given scale of preferences, and given supplies of commodities, will seek to exchange those commodities for others, when the prices of both sets (the commodities he gives up and those that he acquires) are given."  Then he explored how those decisions change as prices change.  Finally he extended these findings to groups of individuals.

Hicks next points out that his analysis applies beyond the obvious example of an "ordinary consumer spending his income on the satisfaction of his immediate personal wants."  The necessary condition for the analysis being applicable is that the objects being bought and sold are objects of desire that can be arranged in an indifference system.  Hicks emphasizes that the indifference system must itself be independent of prices, and he highlights two important cases that are excluded.

One is the demand and supply of goods from producers.  A factor of production, to a producer, is ordinarily not something for which he has a place on his own scale of preferences.  His demand for it is a derived demand, depending on the price of its product.  He intends to sell the product, and then satisfy his wants with the proceeds;  without any information about the price of the product, he cannot tell what it will be worth his while to pay for a unit of the factor.
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The other case which is excluded is the case of speculative demand. ... [A] fall in price may fail to increase demand, or may even diminish it, because it creates an expectation of the price falling farther.

Hicks highlights one important example of speculative demand:  the demand for money.  "There is no demand for money for its own sake," he writes, "but only as a meas of making purchases in the future.  It is therefore always liable to be affected by expectations of the future.  Every theory of money has always had to take account of this fact in one way or another."

Hicks also deals briefly with a further exclusion that he calls "a trifle compared with the important exclusions."  This is the "Veblenesque example beloved of the text-books:  the demand for an object of ostentatious expenditure... ."  Veblen's term conspicuous consumption refers to the buying of luxury goods or services as a public display of economic power or status.  Thus if the price of some luxury good were to fall, the demand for it could fall as well.

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 7

In this final section of the chapter, Hicks treats a proposition, not included in the first edition of the book, that he describes as, "probably the ultimate generalization of the theory of demand."  Hicks considers an arbitrary change in the system of prices confronting a consumer.  He discusses isolating the substitution effect of such a change by considering one that keeps the consumer on the same indifference level, and he says the following:

...[W]e can always say that the new collection of goods purchased must have a higher value in terms of the old prices than the old collection of goods had. ... Similarly the old collection of goods must have a higher value in terms of the new prices than the new collection of goods has.

The following figure (not in the book) illustrates this in the case of two goods X and Y.  A consumer with the given indifference curve, facing the old set of prices, chooses a collection of the goods labelled "old" (where the quantity of each good purchased corresponds to the distance along its axis).  The old and new systems of prices are each characterized by the slope of the (straight) budget constraint lines (or as Hicks has called them, "price-lines").  When the prices change to the new set of prices, the consumer chooses the collection labelled "new."  For each of the systems of prices, a pair of lines is shown  -- one line passing through the old collection of goods (i.e. the collection chosen at the old prices) and the other line passing through the new collection of goods.

The figure illustrates how Hicks's statement above is true:  the new collection costs more than the old at the old prices, and the old collection costs more than the new at the new prices.  This is somewhat reminiscent of the argument made earlier in the book that a consumer's utility will be maximized at a point where an indifference curve is tangent to the price-line.

Hicks goes on to make an argument in words that I found easier to understand by working it out mathematically for the case of two goods.  Let Pxo and Pxn be the old and new prices, respectively, of good X, and let Qxo and Qxn be the quantities of X chosen in the old and new collections, respectively.  Similarly, let Pyo and Pyn be the old and new prices of good Y, and let Qyo and Qyn be the quantities of Y chosen in the old and new collections.  Hicks states that it follows, from the new collection of goods having a higher value in terms of the old prices than the old collection has, that "the sum of the increments in amounts purchased must be positive when valued at the old prices."  We can express this mathematically as follows: The value of the new collection at the old prices is Qxn * Pxo + Qyn * Pyo, and the value of the old collection at the old prices is Qxo * Pxo + Qyo * Pyo. So we have that

Qxn * Pxo + Qyn * Pyo >  Qxo * Pxo + Qyo * Pyo

Subtracting the right-hand side from both sides, and factoring out the prices, we get

( Qxn -  Qxo ) * Pxo + ( Qyn - Qyo ) * Pyo  >  0.

The left-hand side expression is exactly the sum of the increments in amounts purchased when valued at the old prices.

Hicks also states that it follows, from the old collection of goods having a higher value in terms of the new prices than the new collection has, that "the sum of the same increments, valued at the new prices, must be negative."   Proceeding similarly, we note that the value of the old collection at the new prices is Qxo * Pxn + Qyo * Pyn, and the value of the new collection at the new prices is Qxn * Pxn + Qyn * Pyn. So we have that

Qxo * Pxn + Qyo * Pyn >  Qxn * Pxn + Qyn * Pyn

Subtracting the left-hand side from both sides, and factoring out the prices, we get

0  >  ( Qxn -  Qxo ) * Pxn + ( Qyn - Qyo ) * Pyn.

The right-hand side expression is exactly the sum of the increments in amounts purchased when valued at the new prices.

Hicks writes that the two statements about the value of the increments purchased "can only be consistent with one another if the sum of the increments, valued at the increment of the corresponding price in each case, is negative."  Note that the two inequalities above, taken together, imply that

( Qxn -  Qxo ) * Pxo + ( Qyn - Qyo ) * Pyo  >  ( Qxn -  Qxo ) * Pxn + ( Qyn - Qyo ) * Pyn.

Subtracting the left-hand side from both sides, and factoring out the increments of the quantities, we get

0  >  ( Qxn -  Qxo ) * ( Pxn -  Pxo ) + ( Qyn - Qyo ) * ( Pyn - Pyo ).

The right-hand side expression represents the sum of the increments in amounts purchased when valued at the increments of the corresponding prices.  Hicks states that, "This is the sense in which the most generalized change in prices must set up a change in demands in the opposite direction."  Another way to think about this is that positive changes in price will drive negative changes in demand, and negative changes in price will drive positive changes in demand, so the product of a good's increments in demand and price will be negative.

This is the end of Chapter III, and Part I of the book.  Thanks for reading this far.

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 6

In this section Hicks deals with a couple of additional points about the effect of a change in the price of one good on the demands for other goods.

The first point is that the principles he has been discussing in preceding sections are "just as applicable to market demand as to the demand of the individual consumer."  Thus, two goods X and Y can be regarded as complementary for a group of consumers (or substitutes, for the group as a whole), depending on whether the total substitution effect causes the total demand for Y to increase when the price of X falls (or, in the case of substitutes, the total demand for Y falls when the price of X falls).

Another important principle is that "when the relative prices of a group of commodities can be assumed to remain unchanged, they can be treated as a single commodity."  This means that substitution effects can exist between the group as a whole and a single commodity X that is not in the group.  Thus a fall in the price of X relative to the prices of the other goods gives rise to a substitution in favor of X and away from the other commodities (although Hicks reminds the reader that the expenditure on these other commodities may be rearranged such that the expenditure on some of them is increased).  Similarly, if the price of X remains fixed and the prices of the commodities in the group all fall in the same proportion, this must cause a substitution in favor of the group as a whole.

Hicks gives us a hint of things to come by noting that, "We shall find, as we go on, that this proposition is a distinctly useful one."  But he closes this section by clarifying the limits of the proposition.  He notes that

It does not mean that there must be a substitution effect in favour of each commodity in the group taken separately, so that (apart from income effects) the demand for each commodity separately must increase.  It is always possible that the demands for some goods in the group may diminish, since they are substituted by other goods in the group.

Finally, regarding income effects, Hicks notes that when the group is large, so that the consumer spends a significant fraction of his income on it, the income effect will be large.  But negative income effects for a large group are not likely;  in other words, "it is unlikely that the consumer will spend less money upon a whole large group of goods when his income increases."  Therefore, regarding the demand for the group of goods, he concludes that "we should expect the income effect to pull in the same direction as the substitution effect."

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 5

In this section Hicks sums up his conclusions about the effect that a change in price of a commodity has on a consumer's expenditure.  Noting that the fall in price of some good X affects both the demand for X and the
demand for other commodities through an income effect and a substitution effect, Hicks discusses four cases in detail:

(1) A good Y may be highly complementary with X.  In this case the substitution effect will likely be large enough to drown out any income effect, so the demand for Y will definitely increase.

(2) Y may be mildly complementary with X.  In this case the income effect becomes important.  This will usually mean that both effects cause an increase in demand for Y, unless Y is an inferior good, in which case the strength of the income effect will determine whether demand for Y increases, decreases, or remains unchanged.

(3) A good Y may be mildly substitutable for X.  As Hicks notes, the income and substitution effects work in opposite directions in this (very common) case.  Thus the net effect on the demand for Y would tend to be small and could go either way.  (If Y is an inferior good, however, its demand will definitely decrease in this case.)

(4) A good Y may be highly substitutable for X.  In this case, Hicks notes, "the substitution effect will be decidedly dominant, and the demand for Y must diminish."

Finally Hicks asks which cases can have a fall in the price of X that results in no influence on the demand of Y.  This can happen if both the income and substitution effects are negligible or else if they are not negligible but tend to cancel each other out.  Hicks concludes this section by speculating that a fair number of commodities that economists have usually treated as 'independent' of a particular commodity are actually cases of non-negligible effects cancelling each other out.  In other words, "one feels that a good deal of mild substitutability must be present which is prevented from showing itself by being offset by income effects."

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 4

In this section Hicks elaborates on a few details about substitution effects, noting, to begin with, that the substitution being discussed in the context of complementary and competitive goods is exactly the same thing as the substitution discussed in earlier chapters.

When the consumer is choosing consumption amounts of two (and only two) goods, then the goods must necessarily be substitutes.  It is only when there are more than two goods involved that other kinds of relations among them become possible.  Hicks notes that this explains why complementarity cannot be represented on an indifference diagram for two goods, "for X and Y can only be complementary if there is some third thing at whose expense substitution in favor of both X and Y can take place."  A complementary group of commodities requires something outside the group for them to be substituted against.

So with multiple goods it is theoretically possible, in an extreme case, that all but one good could form a complementary group, with each good in the group being a substitute for the one good outside the group.  At the other extreme, there may be no complementary goods at all.  Hicks notes that it will usually be the case that a good will have a relatively small "knot" of other goods that are complementary with it, but "its most probable relation with any other good taken at random will be one of (doubtless mild) substitutability."

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 3

This section examines the operation of the income and substitution effects on complementary and substitute goods.  Hicks begins by noting that indifference diagrams are of little use in this context;  the problem is that the two-dimensional indifference diagrams cannot easily represent the relevant interactions of quantities of the two related goods along with money.  Hicks refers the reader to an algebraic version of the theory in the book's Appendix.  Here he describes the theory in words.

The case of the income effect is relatively straightforward.  As Hicks puts it, "A fall in the price of X acts like a rise in income, and thus tends to increase the demand for every good consumed, excepting inferior goods."  Hicks also notes that these effects will tend to be small if the consumer's spending on X is a small proportion of income.

The substitution effect is somewhat more complicated.  Substitution effects, as Hicks put it, "must involve a substitution in favor of X; and therefore against something other than X."  If we were to lump all other goods into a single composite commodity, then the substitution effect would cause the demand for this "commodity" to decrease with a fall in the price of X.  But it need not be the case that the demand decreases for every one of the commodities making up the composite one.  If Y is one of these commodities and if it is complementary with X, then the increased demand for X will tend to lead to an increased demand for Y.  Hicks gives a detailed explanation of this in terms of marginal rate of substitution for money.  To spell it out in slightly different terms, let me note that the definition of complementary goods (given in the previous section) states that when X is substituted for money, the marginal rate of substitution of a complementary good Y for money is increased.  But we have not assumed the price of Y to have changed, so there is now a mismatch between the price of Y and its marginal rate of substitution for money, which we know from Chapter I Section 6 means the individual cannot be in equilibrium.  The marginal rate of substitution of Y for money would have to decrease to restore equilibrium, which by the principle of Diminishing Marginal Rate of Substitution discussed in Chapter I Section 7, means the substitution of Y for money (i.e. the demand for Y) would have to increase.

By a similar process, a fall in the price of X would encourage a substitution of money against the good Y if Y were a substitute for X.  As Hicks states, "It is our definition of complementarity which draws the exact line between these two situations."

Value & Capital, CHAPTER III -- COMPLEMENTARITY, Section 2

In this section Hicks explains how to overcome the difficulties described in the previous section regarding the definitions of complementary and competitive (i.e. substitute) goods.  The key step is to replace the use of marginal utility in the definitions with "marginal rate of substitution for money."  The definition of a substitute good then becomes:

Y is a substitute for X if the marginal rate of substitution of Y for money is diminished when X is substituted for money in such a way as to leave the consumer no better off than before.

Similarly, Y is complementary with X if the above substitution of X for money results in an increase in the marginal rate of substitution of Y for money.  Hicks motivates the specific nature of the reduction of money in the substitution of X by noting that the definition of a substitute good should make it "absolutely certain that an extra unit of the same physical commodity is a substitute for preceding units."  And we can only be certain of this when the extra unit of X is substituted for money in a way that leaves the consumer no better off than before; then the result is guaranteed by the principle of diminishing marginal rate of substitution.

As Hicks notes, the resulting definition is free from any dependence on a quantitative measure of utility.  In addition, the symmetry properties described in the previous section hold (namely, if Y is a substitute for X, then X is a substitute for Y, and similarly for complements).  Also this definition reduces to the Edgeworth-Pareto definition if the marginal utility of money is assumed constant, while being directly applicable in cases where the assumption does not hold.