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.