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.