Value & Capital, CHAPTER IV, Section 4

In this section Hicks briefly discusses the significance of what has been presented thus far about the systems of equations derived by the methods of Walras.  Hicks considers it "a great achievement to have shown, even so schematically, the interrelation of markets."  But he also acknowledges that many economists have felt that Walras's approach has a certain "sterility" about it.  He suggests that the reason for this sterility is that Walras did not proceed to work out the laws of change for these types of systems.  The theory presented up to this point tells us what conditions must be satisfied by a set of equilibrium prices established for a given system of resources and preferences, but it does not explain what will happen if these preferences or resources change.

Hicks will begin to address these questions in the next chapter.

Housekeeping note:  This will be the final post of 2015.  Thanks to everyone for reading; please continue to read the blog, and send me any feedback you have.  Ideally, I would like to finish the book in 2016, which would mean I would have to pick up the pace significantly:  there are about 240 pages remaining (excluding the mathematical appendix, which I do not intend to cover);  thus I would need to cover about 20 pages per month, instead of the 5 or so that I averaged in 2015.  This will be a challenge, but we'll see how it goes.

Best wishes to all for a happy new year.

Value & Capital, CHAPTER IV, Section 3

In this brief section, Hicks goes into a bit more detail about how we know that the number of prices that must be determined in order to define an equilibrium of exchange is always one less than the number of goods.  His argument runs as follows.

If prices are given, we know (using the methods described earlier) how to determine any individual's demand for each commodity, as well as the quantities of any commodities he already possesses that he will be willing to supply in exchange for these demands.  If we can sum up these demands and supplies for all the consumers, then we can determine the total demand and supply for each commodity.  Then, as Hicks puts it, "If the price-system is such as to make these demands and supplies equal, we have a position of equilibrium.  If not, some prices at least will be bid up or down."

Hicks then goes on to argue that the equation of supply and demand for the standard commodity (that is, the one assumed in the previous section to have some of the qualities of money) follows from the demand-and-supply equations for the rest of the goods.  In Hicks's words:

Once any particular individual has decided how much of each non-standard commodity he will sell or he will buy, he will automatically have decided how much of the standard commodity he will buy or sell.  Thus 

Demand for standard = Receipts from sale of other goods - Expenditure on purchase of others

Supply of standard = Expenditure on purchase of others - Receipts from sale of others 

Therefore for the whole community, 

Demand for - Supply of standard commodity = Total receipt from sale of others

- Total expenditure on purchase of others 

and, once the demand for every non-standard commodity equals the supply, this must = 0.

There are thus n-1 independent equations to determine the n-1 independent prices.

Value & Capital, CHAPTER IV, Section 2

In this section, Hicks begins to outline what he calls a "preliminary survey of the theory of Exchange."  He begins as follows:

Let us suppose that we have to deal with a world where the only objects of exchange are personal services.  The demand for these services will be governed by the laws set out in the preceding chapters;  so will the supplies.  All the complications of production and speculation are eliminated.  If we can get a clear idea of such an economic system, we shall certainly still be a long way from having a realistic model of the actual world; but we shall have a foundation on which to build, and which may be useful in itself for certain limited purposes.

He notes that the economist Léon Walras created the theory of general exchange equilibrium, and explains that he will be summarizing some work of Walras in the same way he previously summarized Pareto's work on the theory of value.

Hicks assumes an elementary case in which there are only two sorts of services and hence only two types of goods to be exchanged.  "Thus every person is either simply a buyer of X and seller of Y, or simply a buyer of Y and seller of X."  If perfect competition is assumed, there is only one price ratio to be determined -- that of X to Y.  The condition that the demand for X must equal the supply of X is available to help determine this price ratio.  Hicks notes that if this condition holds for X, then it must hold for Y as well.  Hicks's earlier investigations explained how the demand and supply for the two goods can be determined for a given price ratio.

Hicks notes that when the argument is extended to more than two goods, the number of prices to be determined will always be one less than the number of goods.  And he explains that if one of the commodities is chosen as a standard of value, then the rate of exchange between any two commodities must always equal the ratio of their prices in terms of the standard commodity.

For if not, one party or the other would always be able to benefit himself by abandoning direct exchange, and splitting the transaction into two parts:  first an exchange of one commodity for the standard, and then an exchange of the standard for the other commodity.

At this stage of the argument Hicks is assuming that the standard commodity has two of the qualities of money -- namely that it is an object of desire and that it is used as a standard of value -- but not any other qualities of money. Thus he is assuming that the standard commodity has "an ordinary place on the scale of preferences of an ordinary individual."  People who come into the market with supplies of the standard commodity may either spend it or reserve some of it as they see fit.

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.
...
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.

Value & Capital, CHAPTER III -- COMPLEMENTARITY

This section begins with the definition of complementary and competitive goods as used by the economists Francis Ysidro Edgeworth and Vilfredo Pareto.

Y is complementary with X in the consumer's budget if an increase in the supply of X (Y constant) raises the marginal utility of Y;  Y is competitive with X (or is a substitute for X) if an increase in the supply of X (Y constant) lowers the marginal utility of Y.

To put this in familiar terms, one could think of hotdog franks and buns as being an example of a pair of complementary goods.  Conversely, one might think of french fries and onion rings as being substitutes.

With the above definition, the complementary-competitive relationship is symmetric or reversible:  If Y is complementary with X, then X is complementary with Y, and similarly for competitive goods.  Also if the marginal utility of money is constant, this definition implies that, for complementary goods, a fall in the price of X, increasing the demand for X, will raise the marginal utility of Y, which will lead to an increase in demand for Y.  Similarly, if X and Y are substitutes, a fall in the price of X will lower the demand for Y.

Hicks then goes on to describe Pareto's difficulties in trying to translate the definitions of complementary and competitive goods into the terms of indifference curves.  Pareto was able to find a connection between the case of complementary goods (according to the definition above) and the case of indifference curves that are highly bent, as in Figure 12.

Similarly, Pareto found a parallelism between the case of X and Y being substitutes and the case of indifference curves that are very flat, as in Figure 13.

But as Hicks explains, Pareto was not able to discover what degree of curvature corresponds to the distinction between complementary and substitute goods.  In addition, Hicks notes that the definition above violates Pareto's principle of not assuming utility to be measurable.  Hicks will show in the next section how these difficulties can be overcome.

Note to Chapter II -- CONSUMER'S SURPLUS

In this section, Hicks uses some of the results reached in this chapter to examine the doctrine of consumer's surplus.  He refers to Alfred Marshall's work on the topic, as well as an earlier paper by Jules Dupuit (that lacks an important qualification supplied by Marshall).  According to Hicks, Dupuit illustrated consumer's surplus using a price-quantity demand diagram, as shown below.

Dupuit claimed that the utility secured by being able to purchase 0n units of a commodity at the price pn is given by the area dpk on the diagram.  According to Hicks, Marshall uses the same diagram and arrives at the same result, but with an important qualification that the marginal utility of money is assumed to be constant.

Hicks recasts the analysis of consumer's surplus using indifference diagrams, as shown in Figure 11.

The consumer's income is given by OM, and the price of good X is given by the slope of the line ML, which touches an indifference curve at P.  Then ON will be the amount of X purchased, and PF will be the amount of money paid for it. (It may be easy to get confused here, as Figures 10 and 11 have slightly different interpretations.  It happens that the quantity pn in Figure 10 is the price paid by the consumer, whereas in Figure 11 PN is the quantity of money retained (not spent on X) by the consumer.)  The point P lies on a higher indifference curve than the point M does.  The consumer, starting with income OM, would be willing to pay RF to consume quantity ON of good X (since he'd be on the same indifference curve, at point R, as when he started).  Because he only has to pay PF instead of RF, consumer's surplus is given by the length of the line RP.

Hicks explains the derivation of Marshall's conclusion as follows:

If the marginal utility of money is constant, the slope of the indifference curve at R must be the same as the slope of the indifference curve at P, that is to say, the same as the slope of the line MP.  A slight movement to the right along the indifference curve MR will therefore increase RF by the same amount as a slight movement along MP will increase PF.  But the increment in PF is the additional amount paid for a small increment in the amount purchased  at the price given by MP, an amount measured by the area pnn'z' in Fig. 10.  The length RF is built up out of a series of such increments, and must therefore be represented on Fig. 10 by the area built up out of increments such as pnn'z'.  This is nothing else than dpno.

RP will therefore be represented on Fig. 10 by dpk -- Marshall's consumer's surplus.

Hicks then goes on to discuss the basis for Marshall's assumption that marginal utility of money is constant.  This assumption neglects the difference between the slopes of the indifference curves at P and R in Figure 11.  This difference will be important if the commodity under consideration is important in the consumer's budget.  Even if this isn't the case, the difference will still be important, according to Hicks, "if RP is large, if the consumer's surplus is large, so that the loss of the opportunity of buying the commodity is equivalent to a large loss of income."

Hicks goes on to argue that this weakness in Marshall's argument need not be retained, as the notion of consumer's surplus "is not wanted for its own sake; it is wanted as a means of demonstrating a very important proposition, which was supposed to depend upon it."  Although it isn't clear at this point just what "important proposition" Hicks is talking about, he states a page later that

This is all that is necessary in order to establish the important consequences in the theory of taxation which follow from the consumer's surplus principle.  It shows, for example, why (apart from distributional effects) a tax on commodities lays a greater burden on consumers than an income tax.

So, this is where Hicks is headed.  How does he get there?

He states that consumer's surplus is "the compensating variation in income, whose loss would just offset the fall in price, and leave the consumer no better off than before."  He goes on to show a lower bound on this compensating income, which is all that is needed for his argument.  He illustrates the bound on compensating income by means of the following example:

Suppose the price of oranges is 2d. each, and at this price a person buys 6 oranges.  Now suppose that the price falls to 1d., and at the lower price he buys 10 oranges.  What is the compensating variation in income?  We cannot say exactly, but we can say that it cannot be less than 6d.  For suppose again that, at the same time as the price of oranges fell, his income had been reduced by 6d.  Then, in the new circumstances, he can, if he chooses, buy the same amount of oranges as before, and the same amounts of all other commodities;  what had previously been his most preferred position is still open to him;  so he cannot be worse off.

In a footnote, Hicks says that the "compensating variation can thus be proved to be greater than the area kpzk' on Figure 10."  To see that this is the case, think of Hicks's oranges example as being depicted on Figure 10. Buying 6 oranges at the price 2d. corresponds to buying the quantity 0n at the price pn, and buying 10 at the price 1d. corresponds to buying 0n' at the price p'n' (distances are not to scale).  The area kpzk' equals 6d.  In the footnote, Hicks examines whether the compensating variation can be proved to be less than the area kz'p'k'.  In discussing this question, he explains that

At first sight, one might think so;  but in fact it is not possible to give an equally rigorous proof on this side.  This comes out clearly if we use the indifference diagram (Fig. 11).  The line exhibiting opportunities of purchase, when the price of oranges falls by 1d. and income is reduced by 10d., no longer passes through the original point of equilibrium P.  Thus we have no reliable information about the indifference curve it touches.

And without this indifference curve, we cannot compute the compensating variation for the price change.

This completes Hicks's demonstration that a tax on commodities is more burdensome on consumers than an income tax.  He states that other deductions that have been drawn using the concept of consumer's surplus could be similarly analyzed, and in a footnote he points to a then-recent paper by Harold Hotelling, published in Econometrica in July of 1938, as making a similar argument.

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'?"

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.

CHAPTER I -- Section 6

In this section Hicks takes the first step forward in his effort to build a theory of consumer's demand from concepts not implying a quantitative measure of utility.  He begins with the concept of marginal utility.  He notes that if we fix the quantities of two commodities, then the slope of the indifference curve at that point is equal to the ratio of the marginal utilities of those goods, and the ratio is "independent of the arbitrariness in question."  He calls this quantity the marginal rate of substitution of X for Y, and defines it as "the quantity of Y that would just compensate the consumer for the loss of a marginal unit of X." For given market prices, an individual in equilibrium will have his marginal rate of substitution between any two goods be equal to the ratio of their prices.  If this were not the case, he could gain by reducing his consumption of one of the goods and spending the savings on the other good.  The condition for equilibrium on the market will therefore be written in terms of marginal rates of substitution rather than marginal utilities.  Hicks ends this section by noting that we can say that a commodity's price equals the marginal rate of substitution of that commodity for money.

CHAPTER I -- Section 5

In this short section Hicks sets out the goal of constructing a theory of consumer's demand, starting from the indifference map alone.  He points out that Pareto's work will be of no further help, as Pareto himself did not rework Marshall's conclusions in the light of his significant discovery about indifference maps.  He then notes that an important article on this topic had been published by the Russian economist and statistician E. E. Slutsky, but that it had remained relatively unknown -- probably due its having been published in Italian, as well as the fact that it was published in 1915 (i.e. during war time).  Value and Capital contains "the first systematic exploration of the territory which Slutsky opened up," according to Hicks.

It is interesting to note the role of Hicks's ability to read Italian in advancing his work.  In addition to being able to read Slutsky's article, Hicks was able to read Pareto's work in the original Italian.  In this interview Hicks relates an interesting story of how he came to know about Pareto's work.  A colleague of his at the London School of Economics, Hugh Dalton, advised him that since he read Italian he should read Pareto.  Dalton had first learned about Pareto while spending several months in a hospital in Italy after being wounded toward the end of World War I (he was serving in a British force sent to help the Italians on the Austrian front).  He wasn't seriously wounded, so he had time to learn to speak and read Italian.  Dalton was a Cambridge-trained economist, so he searched for Italian books on economics and found Pareto's Manual of Political Economy.

CHAPTER I -- Section 4

In this brief section Hicks points out that the process of working out marginal utility theory in terms of indifference curves achieves the remarkable accomplishment of arriving at the same results while leaving behind some of the original information.  Specifically, Marshall's theory assumed we know a consumer's utility surface.  The indifference maps of Pareto's theory contain less information.  They are somewhat analogous to contour lines on a map without any key that tells us how much change in elevation occurs between adjacent lines.  It turns out that this extra information was not necessary to explain market phenomena.  And Hicks argues that, "on the principle of Occam's razor, it is better to do without it."

CHAPTER I -- Section 3

The remarkable thing about indifference curves that Hicks explains in this section is the fact that they allow us to draw detailed conclusions about a consumer's optimal consumption choices without knowing details about the utility derived from that consumption.  Here he uses a simple example:

Suppose that we have a consumer with a given money income, who is spending the whole of that income on the commodities X and Y, no others entering into the picture.  Suppose that the prices of those commodities are given on the market.  Then we can read off the amounts that he will buy directly from his indifference map, without any information about the amounts of utility he derives from the goods.

He illustrates this with the following figure:

The line LM is constructed as follows:  the length OL represents the quantity of good X that the consumer could buy if he spent all his income on it.  Similarly, OM represents the quantity of Y he could afford if he spent everything on Y.  Any point along the line LM corresponds to a pair of quantities of the two commodities that would use up all his income.  The slope of the line LM corresponds to the ratio of the prices of the two goods (since it is the rate at which one commodity could be evenly exchanged for the other).  Hicks notes that the consumer's utility will be maximized at a point where an indifference curve is tangent to the price-line, "For at a point of tangency, the consumer will get on to a lower indifference curve if he moves in either direction."

Hicks concludes this section by relating the tangency between an indifference curve and price line to the principle (mentioned in an earlier post) of proportionality between marginal utilities and prices.  He does this as one simple assertion, which I had intended to discuss with mathematical expressions.  Unfortunately my attempts to paste equations into this blogging tool have been unsuccessful so far, so I'll have to rely simply on words for now.  Suffice it to say that at the point of tangency, the derivative of the indifference curve is equal to the slope of the price line.  The latter is proportional to the ratio of prices, whereas the former is proportional to the ratio of marginal utilities.  We can discuss in comments if anyone's interested.

Thanks again for reading.

CHAPTER I - Section 2

In Section 2 Hicks summarizes the utility theory writings of Vilfredo Pareto.  After starting off with ideas similar to those developed by Marshall, Pareto turned his attention to analyzing goods that are related -- in the sense of being complementary or of being (at least partial) substitutes.  Pareto used a geometric device used earlier by the Irish-English economist Francis Ysidro Edgeworth -- the indifference curve.  For two commodities X and Y we can construct a map like that shown in the figure below (an imitation of Hicks's Figure 2).

The points in this X-Y plane correspond to quantities of the goods X and Y that are consumed -- these quantities of consumption are understood to provide utility to the consumer.  The curves shown in the figure are indifference curves:  namely, curves along which all points have the same total utility.  We can think of these curves as being similar to contour lines on a map -- they show points that are at the same "elevation" in three dimensions.  Referring to this figure, Hicks goes on to explain:

If P and P' are on the same indifference curve, that means that the total utility derived from having PM and PN [these distances correspond to quantities of the goods X and Y, respectively] is the same as that derived from having P'M' and P'N'.  If P" is on a higher indifference curve than P (the curves will have to be numbered so as to distinguish higher from lower), then P"M" and P"N" will give a higher total utility than PM and PN. 

Hicks explains that, assuming positive marginal utility (each increment increases utility, at least a little), the indifference curves must slope downwards from left to right.  One way to understand this is to realize that increasing X without decreasing Y should lead one to a higher indifference curve.  Put another way, if we increase the amount of X consumed, the only way to keep the same utility is to decrease the amount of Y.

As Hicks explains, the slope of an indifference curve at any point has a definite and important meaning, "It is the amount of Y which is needed by the individual in order to compensate him for the loss of a small unit of X." Assuming quantities are small, the decrease in utility from losing the small unit of X is given by the product (amount of X lost) x (marginal utility of X).  The gain in utility from increasing the amount of Y is given by the product (amount of Y gained) x (marginal utility of Y).  We're assuming these utilities exactly balance each other, so (amount of Y gained) x (marginal utility of Y) = (amount of X lost) x (marginal utility of X).  A small amount of algebra then gives us [apologies -- I need learn a better way to paste equations in]:

(amount of Y gained) / (amount of X lost) = (marginal utility of X) / (marginal utility of Y)

But the left-hand side of this equation is the slope, so the slope of the indifference curve equals the ratio of the marginal utilities.  (There are two things to keep in mind here -- one is that the slope of the indifference curve is changing as we consider different points along the curve;  the other is to notice which of the marginal utilities is in the numerator and the denominator on the right-hand side.)

Hicks concludes this section by examining the question of whether diminishing marginal utility implies that the indifference curves must be convex to the origin.  (One way to think about convexity here is to think of it as implying that, for any two points on the curve, the curve in between these two points must lie below a straight line connecting the two points.)  Hicks argues that, at first sight, this would seem to be true, but that it does not necessarily follow.  A counterexample is given by the case of related goods in which "the increase in X lowers the marginal utility of Y [and vice versa, and]... these cross-effects are considerable." 

Thanks for reading this far.  In the next section (a short one), Hicks notes "a really remarkable thing" about indifference curves.

CHAPTER I UTILITY AND PREFERENCE

Chapter I has nine sections, the first of which I'll discuss in this post.  Section 1 summarizes the main assumptions and conclusions of Alfred Marshall's theory of demand.  The point of that theory is to express mathematically how a consumer chooses to divide his expenditures among several consumption goods, assuming the prices of these goods are already determined.  For mathematical convenience the theory assumes the goods can be purchased in very small units.  A footnote explains:

This convenient assumption of continuity does, of course, falsify the situation a little (or sometimes more than a little) as far as the individual consumer is concerned.  But if our study of the individual consumer is only a step towards the study of a group of consumers on the market, these falsifications can be trusted to disappear when the individual demands are aggregated.

The theory makes several assumptions: that the individual spends his income so as to maximize his satisfaction (or as Marshall, Hicks and other writers have come to call it, "utility");  that utility depends on the quantities of goods purchased;  and that as one increases consumption of some good, each additional unit brings less satisfaction than the previous one -- this is called "the principle of diminishing marginal utility."

With these fairly reasonable (even obvious) assumptions, the conclusion is that "utility will be maximized when the marginal unit of expenditure in each direction brings in the same increment of utility."  In other words, the consumer will choose quantities of the various goods in a way that causes the last (tiny) unit of each good to bring the same additional amount of utility.

This conclusion seems reasonable if one considers a situation where the condition doesn't hold -- that is, in some planned set of purchases, one good has a strictly greater marginal utility than another good.  This clearly can't be the set of purchases that maximizes utility, since the consumer could increase his utility by purchasing (at least a tiny bit) less of the good that has lower marginal utility and using the extra money to buy more of  the good with the higher marginal utility.  With the ability to purchase small units, we have the additional conclusion that the marginal utilities of the various goods are proportional to their prices.  Again, thinking about this conclusion by assuming that the condition doesn't hold, we can see that shifting the planned purchases slightly in the direction of a good with higher marginal utility per unit of price would increase total utility.

In section 2, Hicks will begin to ask some probing questions about the nature of "this 'utility' which the consumer maximizes."

Interesting background on Value and Capital

Nobel Laureate Kenneth Arrow describes Value and Capital as having had a significant influence on him -- and learning about it somewhat by chance:

 I learned about general equilibrium, not from any course, but from the fact that my desk, like those of all the other graduate students, was located in the library.  I was close to where the economics books were, in the stacks, so I would simply go to the stacks, starting flipping the books around, and see if there was anything interesting.  I found a book by the English economist, John R. Hicks, titled, Value and Capital.  Nobody at the Columbia Economics Department knew anything about this book, but as you talk to people educated elsewhere, say in England, many of the economists of my generation were transformed by this book.
     That's why when many years later I was awarded the Nobel Memorial Prize in Economic Science jointly with Hicks, it was an especially great honor to be joined with one whose work I admired so much, who was so influential on me.

And we're off!

My original intent was to read "Value and Capital" by John Hicks and to carry on an email correspondence about it with my first and dearest economics teacher, Jozell Brister.  After giving it some more thought (and getting some encouraging feedback from Jozell on the idea), I decided to try blogging my way through the book.  After some more thought, I'm hoping later on to blog about other well-known books and papers in the economics literature -- hence the name of the blog.

If you want to read the book yourself and follow along, I'll be using the second edition, which was first published in 1946.  It looks like my copy might have been from the fifth printing of that edition, in 1957.  (I suspect other editions will be similar enough that it won't be confusing to follow along.)

I won't write much for the next few days, so you definitely have time to look around for a copy.