Value & Capital, CHAPTER XIV, Section 4

This section continues -- seemingly in mid-paragraph -- the discussion of the previous section, by beginning, "For consider what happens, first, if interest rates are expected to change."  In this case, the author argues, "a definition based upon constancy of money capital becomes unsatisfactory."  He then constructs a numerical example in which the interest rate becomes fixed in the second time period at a rate that is twice the rate in the first period (and it remains at the second-period rate thereafter).  In his example, the individual can make the stream of expenditures

 £10, £20, £20, £20, ...  

and maintain his same quantity of capital, whereas that quantity of capital being available at the beginning of the second time period allows the individual to spend the stream

 £20, £20, £20, £20, ...  

The author notes that "It will ordinarily be reasonable to say that a person with the latter prospect is better off than one with the former."

Accordingly, the author arrives at the definition he calls "Income No. 2."  Namely, it is "the maximum amount the individual can spend this week, and still expect to be able to spend the same amount in each ensuing week."  For constant interest rates, this definition is the same as Income No. 1, but not when interest rates are expected to change.  The author concludes, "Income No. 2 is then a closer approximation to the central concept than Income No. 1 is."

Value & Capital, CHAPTER XIV, Section 3

This section considers the first of several approximations to the "central meaning" of the term income.  This first approximation "would make everything depend on the capitalized money value of the individual's prospective receipts."  The author explains by use of an example (in terms of pounds sterling, denoted by £):

Suppose that the stream of receipts expected by an individual at the beginning of the week is the same as that which would be yielded by investing in securities a sum of £M.  Then if he spends nothing in the current week, reinvesting any receipts which he gets ... he can expect that the stream which will be in prospect at the end of the week will be £M plus a week's interest on £M.  But if he spends something, the expected value of his prospect at the end of the week will be less than this.  There will be a certain particular amount of expenditure which will reduce the expected value of his prospect to exactly £M.  On this interpretation, that amount is his income.

It might be easier to understand this explanation by seeing it expressed mathematically.  Let r denote the rate of interest earned by investments; let M (as above) be the initial sum available for investment, and let y denote income.  Based on the author's description of income as an expenditure that will reduce the expected prospect to M, we have the following equation:

(M − y)⋅(1 + r) = M

Performing one step of multiplication, we get

M + r⋅M − y⋅(1 + r) = M

Subtracting M from both sides and rearranging, we get

r⋅M = y⋅(1 + r)

Therefore,

y = M⋅r / (1 + r).

The author notes that this definition makes sense in the case of income derived from property.  He goes on to note that it is "less obviously sensible" in the case of labor income but "is still quite consistent with ordinary practice."  The author refers to this definition as "Income No. 1" and notes that it is "the maximum amount which can be spent during a period if there is to be an expectation of maintaining intact the capital value of prospective receipts (in money terms)."  Looking at the equation above, we can see that y is less than a single period of interest on the original investment M because of the term of 1 + r in the denominator (assuming r is positive).  This makes sense because the expenditure of y would cause interest to be earned on less than the full investment of M.

The section concludes by noting that this definition is likely the one that "most people do implicitly use in their private affairs;  but it is far from being in all circumstances a good approximation to the central concept."

Value & Capital, CHAPTER XIV, Section 2

This section begins to discuss the meaning of the term income, in the context of dynamic economics.  In the stationary state of the dynamic model, as well as in the static case, the meaning is much easier to specify.  For a person who expects to receive “the same amount in every future week as he receives this week, it is reasonable to say that that amount is his income.”  

In the more general dynamic case, complications arise.  The author uses the example of a person being paid more in the current week than in some other weeks (if, for example, “this week’s receipts may include wages for several weeks’ work”);  in such a case “we should not regard the whole of his current receipts as income.” Similarly, for someone paid once every four weeks, if “the present week was one in which his salary was not paid, we should not regard his income this week as being zero.”

If not zero, then what should the income be in this example?  In proceeding to try to answer this question, the author notes that

The purpose of income calculations in practical affairs is to give people an indication of the amount which they can consume without impoverishing themselves.  Following out this idea, it would seem that we ought to define a man’s income as the maximum value which he can consume during a week and still expect to be as well off at the end of the week as he was at the beginning.

He concludes that this definition of income “is what the central meaning must be,” because “the practical purpose of income is to serve as a guide to prudent conduct.”

He closes the section by noting that “business men and economists alike are usually content to employ one or other of a series of approximations to the central meaning.”  The next several sections of the book will look at examples of such approximations.

Value & Capital, CHAPTER XIV -- INCOME

In this section, the first of the chapter, the author, Sir John Hicks, notes that his discussion of interest has concluded, and that he has thereby "concluded all that it is absolutely necessary to say about the foundations of dynamic economics."  He further notes his intention, ultimately, "to analyse the working of the dynamic system, proceeding on parallel lines to those on which we analysed the working of the static system in Part II."  Meanwhile, he notes a possible objection to the discussion so far, namely that it has not addressed topics such as Income, Saving, Depreciation, or "Investment (with a capital I)."

He explains that he deliberately omitted these concepts from the previous five chapters.  He finds "far too much equivocation in their meaning" and believes essentially that "they are not logical categories at all;  they are rough approximations, used by the businessman to steer himself through the bewildering changes of situation which confront him."

He states that "eminent authorities" have confused "each other and even themselves, by adopting different definitions of saving and income, none quite consistent, none quite satisfactory."  He concludes the section by noting the need to bring out the reason for such confusion.  He will spend the remainder of this chapter discussing various definitions of income.  In supplementary notes at the end of the chapter he will discuss the relation between Saving and Investment, as well as the effect of interest rate changes on the calculation of Depreciation, and hence Income.

Value & Capital, CHAPTER XIII, Section 5

In this section, the final one of the chapter, the author summarizes the chapter as having given a "preliminary survey of the relation between money and interest."  The author notes that "The fact that money and securities are close substitutes is absolutely fundamental to dynamic economics."  And he goes on to note that "This close substitutability is much the most important property of actual money which we shall need in our further inquiries."

Money serves as a standard of value, although from the point of view of economic analysis, it is straightforward to consider some other standard commodity "selected from the rest to serve as standard of value."  The author notes that actual money also has two other important properties, "considered for the first time in this chapter," namely "the familiar properties of being a 'medium of exchange' and a 'store of value.'"  He concludes the section, and the chapter, by stating that the "important consequence" of these properties of money "for the working of the price system is simply this:  they explain why there is such a close relation of substitution between money and securities, that is to say, they explain the phenomenon of interest -- money interest."

Value & Capital, CHAPTER XIII, Section 4

This section looks at the various sorts of securities discussed in this chapter, including money, as being substitutes for one another "in very much the same sort of way as a chain of substitute commodities, say different qualities of wheat or sugar.  Money is naturally the highest grade, and that is why other grades ordinarily stand at a discount relatively to money."  As noted earlier in the chapter, it is the lack of general acceptability of investment bills as a method of payment that causes them to stand at a discount.

The discussion notes that in early societies, "some sort of durable material commodity" tended to be used as money.  Although the discussion here does not bring out this point, such durable commodities would have alternate uses, which motivates the following passage.

[I]t was not easy to distinguish the demand for the commodity as money from the demand for it as durable consumption good -- or even to see what the demand for it as money could mean.  But when some sorts of promises to pay money began to be so generally acceptable as to become perfect substitutes for the original money -- and thus to stand with the original money in the highest grade -- it became clear that the pure monetary demand had acquired an independent existence.  Money had left its chrysalis stage of durable consumption good, and had developed into pure money -- which is nothing else but the most perfect type of security.

The discussion goes on to address the next highest grade of security, "bills of short maturity."  These are "not quite perfect money, but still very close substitutes for it."  The argument for this claim is that the fluctuating prices for three-month bills tend to remain much closer to the face value of such bills than the differences between pairs of material commodities regarded as being very good substitutes.

The text notes that longer term securities are a lower grade of security.  Their greater fluctuations in value make them "much less of a perfect substitute" for money, and they sell at a greater discount.  Despite this greater imperfection, the discussion observes that "substitution between money and long-term securities does take place," and it proceeds to describe several forms of such substitution.

The text argues that small investors, who buy securities "in order to live on the interest from them" typically choose when to convert their money into securities based more on when they can afford the cost of making the investment, rather than on a change in interest rates.  More speculative investors that do not have ready access to short-term issues, the text argues, "will use the long-term security market as a repository for funds only temporarily idle."  For these types of investors, "the margin between money and securities is a very sensitive margin; the more conscious they are of the importance of capital losses, the more easily they will switch about when the rate of interest varies."  These investors still have access to "at least one form of short term security" -- namely, they can deposit funds in a bank.  This observation leads into the discussion of a third class of investor -- banks themselves, along with "financial houses, public institutions, large industrial and commercial firms."   These have access to "a whole gamut of securities of different maturity."

It is these professional investors, operating upon the whole gamut, and paying close attention to small differences in rates, who provide most of the logic of the interest system (just as it is the professional arbitrageurs who provide most of the logic of the system of foreign exchange rates).... The whole working of the system of interest rates is an example of the working of the general rule of substitution:  if two commodities are close substitutes for an important section of a market, they will behave as close substitutes for the market as a whole.

Value & Capital, CHAPTER XIII, Section 3

This section summarizes the conclusions to be drawn from the preceding sections regarding the relationship between money and interest.  When someone purchases a "bill" to be paid back with interest after a maturity period, there is a possibility that the purchaser will want the money back prior to the maturity period.  If this happens, "he would have to rediscount his bill."  What this means is that he would have to find someone to buy the bill from him.  But if interest rates have gone up since the bill was purchased, any potential buyer would prefer to buy bills bearing the higher interest rate.  To attract a buyer, the original purchaser would have to discount the price of his bill, so that the total amount of money it repays (in principal and interest) would correspond to having received the new interest rate.

The longer the time before the maturity of the bill, the more serious this ... risk is likely to be; and thus, as we saw in our previous discussions of the long-term rate of interest, the long rate is normally likely to exceed the short rate by a risk-premium, whose function it is to compensate for the risk of an adverse movement of interest rates.

Conversely, of course, the shorter the maturity period of the bill, the less risk there will be of needing to rediscount the bill on unfavorable terms.  Lenders will still have to be compensated for the trouble of holding securities that are "imperfectly 'money'" and this compensation for the marginal lender (the lender who is just indifferent between lending and not lending) will correspond to the interest rate.  For short-term interest rates, the risk of having to rediscount on unfavorable terms is only likely to be important "essentially in conditions of great strain -- more or less crisis conditions."

Value & Capital, CHAPTER XIII, Section 2

This section continues the exploration -- hinted at toward the end of the previous section -- into the nature of interest.  It begins by arguing that the way to "get nearest to the true nature of interest" is by "[considering] the relation between money and that type of security which comes nearest to being money, without quite being money."  It then goes on to state that this type of security is "the very short bill, a bill payable in the very near future, when that bill is regarded as perfectly safe from risk of default."  If there is some reason that such a security should sell at a discount (or "stand at less than its face value"), then this will be "a reason for the existence of pure interest."

The discussion then makes use of the model discussed earlier for analyzing dynamic economics in general and interest rates in particular.  Suppose, in this model, that markets are only open on Mondays, and suppose further that "the shortest currency of any bill is from one Monday to the next."  The natural question to ask is "Is it possible for such a bill to stand at a discount relatively to money?"  If there is nothing to stop an individual from converting all his surplus money into such bills and holding them during the week to earn interest, "then money has no superiority over bills, and therefore cannot stand at a premium relatively to bills."  (Or looked at another way, no one would have an incentive to sell such bills.)  "The rate of interest must be nil."

The discussion goes on to explore the incentive for holding money in such a model system, which is that converting money into bills requires going to the trouble of making a separate transaction.  The discussion goes on to note that the rate of interest on the given investment must reflect the inconvenience of the conversion transaction to the marginal lender (in other words, to the lender who causes the supply of loanable money to equal the demand).  This explanation is still incomplete, however.  For there to be some inconvenience in converting money to bills, it must be necessary to make the conversion transaction.  But making such a conversion is only necessary if people are not already holding bills.  This implies, for example, that people are not being paid in bills for things they sell, but instead are paid in money.  Conversely, some individuals to whom a borrower must make payments may not accept bills and instead insist on being paid in cash (again, perhaps requiring a conversion transaction).  Thus even though the bills may be regarded as having no default risk by those individuals who trade in them, there may still be an interest rate that is required to compensate lenders for the cost of investment.  The section finishes with this conclusion:

Thus the imperfect 'moneyness' of those bills which are not money is due to their lack of general acceptability;  it is this lack of general acceptability which causes the trouble of investing in them, and that causes them to stand at a discount.

Value & Capital, CHAPTER XIII -- INTEREST AND MONEY

In this section, the first one of the chapter, the author, Sir John Hicks, notes that "there are certain kinds of promissory documents, not usually reckoned as securities, but included as types of money."  Among these are bank deposits, which are essentially promises to pay money in the future, as well as bank notes.

The difference between securities that are "types of money" and those that are not is that money securities do not pay interest.  In other words, "their present value equals their face value."  Hicks goes on to observe that

Looked at in this way, money appears simply as the most perfect type of security; other securities are less perfect, and command a lower price because of their imperfection.  The rate of interest on these securities is a measure of their imperfection -- of their imperfect 'moneyness'. The nature of money and the nature of interest are therefore very nearly the same problem.  When we have decided what it is which makes people give more for those securities which are reckoned as money than for those securities which are not, we shall have discovered also why interest is paid.

Hicks notes that his earlier chapter on interest identified two elements, both relating to risk, that comprise parts of the interest paid on securities.  These elements are the risk of default and the risk premium due to uncertainty over future interest rates.  Hicks observes that Keynes in his writings "appears to reduce all interest into terms of these two risk factors."  But having noted the existence of a rate of interest on perfectly safe securities, Hicks suggests that to say this interest rate is determined by uncertainty over future interest rates would be a somewhat circular argument.  His conclusion is that "one feels an obstinate conviction that there must be more in it than that.  Let us try to discover what that something more can be."

Value & Capital, CHAPTER XII, Section 6

In this section, the final one of the chapter, the author relates his discussion of the determination of the rate of interest to the discussion contained in John Maynard Keynes's article "The General Theory of Employment."  The author explains that simply identifying the difficulties with computing the rate of interest is not sufficient "to decide how it is best to regard the determination of the rate of interest."  On the one hand, eliminating the money equation and determining the prices of commodities based on their supplies and demands, while determining the rate of interest by the supply and demand of loan funds, seems to be "the most natural course to pursue."  On the other hand, "we can follow Mr. Keynes in eliminating the other equation which stands out from the rest as being peculiar -- the equation of borrowing and lending, or purchase and sale of securities."

While the author goes into some detail about the advantages and disadvantages of the two methods, he states that "either of these methods is perfectly legitimate; the choice between them is purely a matter of convenience." And he concludes that "all these advantages and disadvantages are matters of opinion; there is no reason why we should commit ourselves to the regular use of one method or the other.  It is indeed very useful to have two methods to serve as a check."

The author closes the section (and the chapter) by noting that the advantage of Keynes's method is that it helps in "stressing the closeness of the connexion between money and interest."  This is the topic to which the next chapter will turn.

Value & Capital, CHAPTER XII, Section 5

This section explores the significance of eliminating the extra equation from the systems of equations discussed earlier in this chapter.  The author explains the meaning of this elimination as follows:

It means that if a system of prices is established which equates the demand and supply for each of the n - 1 goods and services, and equates the demand and supply for securities (or loans), then the demand and supply for money must be equal, so that that equation has nothing further to tell us. 

Mathematically what we have is simply a system of equations and unknowns, and any one of the equations could be chosen for elimination.  Because one equation is superfluous, the same values of the unknowns would result from solving the remaining system of equations, no matter which equation is the one eliminated.

The author focuses on two special cases for the choice of which equation to eliminate.
In the first case, the money equation is eliminated, and prices and interest are determined through the markets for goods and services, along with the market for loans.  In this case the author describes the money equation as "otiose," which, according to the dictionary, can mean "lacking use or effect."

When the money equation is included in the system of equations, it can determine the money values of the goods and services, but some other, independent means of determining their relative values will be needed.  As the author explains, "it is impossible to determine even relative prices except in terms of some standard.  Thus the prices of goods and services must first be fixed in terms of some auxiliary standard commodity."  He goes on to note that writers of "the classics" used unskilled labor as their auxiliary standard, whereas more modern writers used a representative consumption good.  This leads to the other special case the author highlights:  the elimination of the equation for supply and demand of the auxiliary standard commodity.

As the author notes, "this is a perfectly legitimate line of approach," but he emphatically warns that it is subject to a "great danger."  This danger has to do with confusion about "what happens to the rate of interest."  An economist who is not careful "will find himself determining, not the true rate of interest," but instead "a rate indicating the value of future deliveries of the auxiliary standard commodity in terms of current deliveries of the same auxiliary standard."  There is no reason why this rate of interest in terms of the auxiliary standard commodity should be the same as the money interest rate.  We saw in the previous chapter that the two rates were the same only if the futures price of the commodity is the same as the spot market price.  This condition will be satisfied if the money price of the standard commodity is expected to remain constant - and if there is no risk that it will not.  Although the author concedes that the difficulties with using this approach could be overcome, he concludes the section by noting that, "It looks as if it will be better to eliminate a different equation."

Value & Capital, CHAPTER XII, Section 4

Following the previous section, which examined the simplified model of an economy with only "short lending," this section examines the case in which all lending is "long lending," i.e. with securities that pay a perpetual stream of payments where each payment's amount corresponds to the interest rate at the time of the payment.  There are, as before, n different prices, where one "price" is for the interest rate, and the other n - 1 prices are for the goods and services (and one of the goods plays the role of money).
As before, there are  n + 1 equations that match supply and demand for the  n - 1 goods and services, while also determining the supply and demand for money and determining the interest rate.  Again, as before, one of these equations is redundant and can be eliminated.  Because the process is somewhat different than in the previous model, the text spells out the details.  Where the discussion in the previous section  considered the possibility of an individual lending out money, the current model instead describes an individual as acquiring a security (which then pays the individual indefinitely).  Therefore the following equation holds for each private individual:

Acquisition of cash = Receipts (including interest from securities owned) 

- Expenditures - Value of securities acquired

For firms, instead of considering the possibility of repayment of loans, the long lending model considers the payment of interest on debts.  Thus for a firm we have the following:

Acquisition of cash = Value of output - Value of Input

- Interest on debts - Dividends 

 +Value of securities issued (or sold)

As before, adding these two equations will imply that net acquisition of cash = 0, by virtue of the following facts:

(1)  If demand equals supply in the output market, then

Net Expenditures by private persons = Value of net output.

(2)  If demand equals supply in the input market, then 

Net Receipts by private persons = Value of net input + Dividends + Interest payments

(3)  If demand equals supply in the securities market, then 

Value of securities bought = Value of securities sold

Therefore, for the community as a whole, net acquisition of cash by trading equals 0.  "As before, the system is determined with n unknowns and n independent equations."

Value & Capital, CHAPTER XII, Section 3

This somewhat lengthy section examines the simplified model of an economy with only "short lending" and describes the determination of the system of spot prices of all goods and services, as well as the single-period "short rate" of interest.  The prices in this system must be those that equate the supply and the demand for all goods and services.  If we assume there are n different kinds of exchangeable goods and services, then there must be n different prices to be determined.  As in previous models, we assume one of the goods is "taken as a standard of value (money)."  Then each of the n - 1 other goods must have a price in terms of the standard;  with the interest rate on single period loans, this makes a total of n prices to be determined.  To determine these prices there are n - 1 equations for the supply and demand of the goods and services, plus one equation for the supply and demand for money, plus one for the supply and demand for loans.  It would appear, then, that there are n + 1 equations.  But, as the discussion notes, "one of these n + 1 equations follows from the rest."  Therefore there are the same number of equations as unknowns.  "The system is neither over- nor under-determined."

The text goes on to explain (at length) how one of the n + 1 equations can be eliminated.  An individual cannot spend more than he receives from trading, unless he borrows.  Conversely, an individual who receives more from trading than he spends must end up with more cash on hand (unless he lends out more than his profits from trading).  Thus we have the following equation for any private individual:

Acquisition of cash by trading = Receipts - Expenditures - Lending

(keeping in mind that lending may be negative).  With firms, the situation is slightly more complicated.  In addition to the effects of borrowing and paying back loans, a firm "will reduce its cash balance by any acquisition it makes of factors of production, increase it by any sales of products.  Finally, it will diminish its cash balance by any dividend it pays out to entrepreneurs."  Thus the following equation holds for any firm:

Acquisition of cash by trading = Value of output - Value of input - Repayment of loans + New borrowing - Dividends

In considering what happens over the entire community of individuals and firms, taken in aggregate, the supply and demand equations for unfinished goods can be taken as canceled out.  "The input to be reckoned is simply the input of labour and material property provided by private persons;  the output is simply the output of finished goods sold to private persons."  If demand equals supply in the input markets, the net receipts of private persons equal the value of  (labor) inputs to firms, plus dividends, plus loan repayments.  If demand equals supply in the output markets, the value of the output of production equals the net expenditure of private individuals.  Also, borrowing will equal lending if supply equals demand in the loan market.
 
Therefore, the following equation holds for the community as a whole:
Net Acquisition of cash by trading
= (Value of Output - Net expenditures by private persons)
+ (Net receipts by private persons - Value of Input - Dividends - Repayment of loans)
+ (Borrowing - Lending)
= 0

Therefore, if there is equilibrium in the markets for all goods and services, and if there is equilibrium in the market for loans, then there must also be equilibrium in the market for money.  Thus there are only n independent equations to determine the n unknown prices.