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."