Value & Capital, CHAPTER XV, Section 4

Having sketched a simple model of a dynamic production plan in the previous section, the author turns in this section to the question of which among the various feasible production plans should be the preferred one.  In the static case, the problem was simple:  the entrepreneur would plan production so as to maximize the "surplus of receipts over costs."  For the dynamic case, there is no single instance of receipts and costs; instead, a given production plan will generate a stream of costs and receipts over time.  In the trivial case in which one stream has, at every step, a larger surplus than a second stream, the first stream is obviously preferred over the second.  In general, though, "we need some criterion to enable us to judge whether one stream [of surpluses] is to be reckoned larger than another."

The author makes the assumption (seemingly almost in passing) "that the entrepreneur can lend and borrow freely at given market rates" of interest.  This assumption is key to his being able to conclude that the preferred production plan must be the one that maximizes the capitalized value of the stream of surpluses.  If prices and price-expectations at each time step are known, then the surplus at each step "is determined as soon as the production plan is determined.  And its present value is determined if interest rates and interest-expectations are given."

The author examines a few other considerations of the model, including accounting for costs that the entrepreneur may face due to "contracts entered into in the past."  In this case the costs "are independent of his present decisions [and] cannot be modified by any change in the plan."  Therefore the capitalized value of his receipts, net of these costs, "only differs from the from the capitalized value of his prospective surpluses by a constant, and will be maximized when that is maximized."

He also notes that "any increase in the capital value of his prospective net receipts must always take the entrepreneur to a preferred position."  This is because the increased capital value "will enable him to plan the same expenditures as before ... and still to have something left over."

Finally, the author recalls that "a person's income can be regarded as the level of a standard stream whose present value is the same as the present value of his prospective receipts."  Once the type of standard stream is decided (which, as we saw in Chapter XIV, relates to the definition of income being used), and once "price- and interest-expectations are given," the values of surpluses and expenses are determined, and therefore "any increase in the present value of a stream must raise the level of the standard stream corresponding to it."  The author notes that net profit can be defined as net receipts plus the net effect of appreciation/depreciation (which may be negative).  He concludes this section by noting

We can either say that the entrepreneur maximizes his profits, or that he maximizes the present value of his prospective net receipts, or that he maximizes the present value of his prospective surpluses.  All these tests come to the same thing;  but it is the last of them (what we have called the present value of the plan) which is the most convenient analytically.

Value & Capital, CHAPTER XV, Section 3

In this section the author sketches a simple model of a production plan of the sort that an entrepreneur would seek to determine at some hypothetical date.  The model is as follows:

A0, A1, A2, A3, … , An

B0, B1, B2, B3, … , Bn

·    · ·    · · ·  

X0, X1, X2, X3, … , Xn

Y0, Y1, Y2, Y3, … , Yn

·    · ·    · · ·  

where "A, B, ... are different kinds of inputs, X, Y, ... are different kinds of outputs, and the entrepreneur is supposed to make his plan for a period of n future weeks." Inputs to the production process are simply things that the entrepreneur buys for his enterprise, and outputs are those things that he sells. The author points out that the model is general enough to handle the case in which the entrepreneur plans to shut down the enterprise and sell off all the equipment at some future date. In this case, "the plant he plans to have left over ... [is] a particular kind of output (say Zn), a kind which is only produced in the last week." All outputs are then zero for all time periods after the enterprise is sold.

The general dynamic problem for the enteprise is to select the optimal production plan from among all those that are technically feasible. The author points out the similarity of this problem to the static problem of choosing the set of quantities of factors of production and products. He explains that the technical limitation on production plans (or the "production function") will give the maximum possible quantity of a given output on a given date, if all inputs, and all outputs but the given one, are fixed in magnitude. Similarly, "if all outputs, and all inputs but one, are given in magnitude, [the production function] will give the minimum input necessary on the remaining date." Given this limitation, all changes between production plans reduce to (1) "substituting some amount of one output for some amount of another, (2) ... substituting [some amount of] one input for another," or (3) "increasing or diminishing one input and one output simultaneously" or some combination of these "elementary variations." The author concludes the section by noting that this is "exactly as in statics."

Value & Capital, CHAPTER XV, Section 2

In this section the author notes that "the dynamic theory of production has been the occasion of a great controversy" in economic dynamics.  He identifies the "great name in this department of economics" as being Böhm-Bawerk, whose theory of production he terms the "Austrian theory."

The definition of capitalistic production as time-consuming production; of the amount of capital employed as an indicator of the amount of time employed; of the effect of a fall in interest on the structure of production as consisting in an increase in the amount of time employed; all these ideas give to the subject an apparent clarity which is, at first sight, irresistable.  The theory stands up very well to the more obvious objections which can be made against it; yet, as one goes on, difficulties mount up.

The author notes some of the criticisms that have been made against Böhm-Bawerk's theory by Knight and Kaldor but claims that "the main issue is still left unsettled."  He previews the discussion in upcoming sections by saying "I hope to show, that when we transcend ... artificially simple cases ... the central propositions change their character rather markedly."  In a satisfactory general theory of capital,  Böhm-Bawerk's theory is "valid as a limiting case, though not a very important case.  The general theory differs from Böhm-Bawerk's in some important respects."

Value & Capital, CHAPTER XV -- THE PLANNING OF PRODUCTION

In this section, the first one of the first chapter of Part IV (The Working of the Dynamic System) the author, Sir John Hicks, lays out the "programme" of analysis that he will consider in "this fourth and final part" of the book.  Essentially he will discuss a series of problems that parallel the discussion of the dynamic system, as presented in Part III, when it is put through the same analysis as that of the static system done in Part II.

We have to consider again ... the private individual, and ... the laws of his behaviour;  only we have now more things to take into account.  We have to consider the ways in which his conduct may be affected, not only by present prices, but also by interest rates, and also by price- and interest-expectations;  we have to examine, not only his demand and supplies of commodities, as before, but also his demand or supply of securities (including ... money).  We have to make a similar investigation for the case of the firm.  Then ... we have to bring these laws together to give us laws for the working of the whole price-system.

Although the author perceives it will be difficult to proceed much beyond a temporary equilibrium analysis (such as how the dynamics might work during a particular "week"), he will endeavor "to see what can be said about the laws of development of the price-system through time."

The author notes that "firms probably work out their production plans a good deal more fully than private individuals work out their expenditure plans."  There are advantages of presenting an "analysis of the determination of a plan" in a more realistic case;  therefore, the discussion will consider first the behavior of the individual firm, and will proceed after that with the study of the individual person.

Value & Capital, Notes to Chapter XIV -- B. INTEREST AND THE CALCULATION OF INCOME, part 3

In this section, the author considers a "common sense" analysis of a person expecting to receive funds "derived from the exploitation of a wasting asset, liable to give out at some future date."  In this case, the author explains, "we should say that his receipts are in excess of his income, the difference between them being reckoned as an allowance for depreciation."  To avoid consuming more than his income, such a person must invest part of his receipts (or, in the language of the text, "re-lend" them) so that his income from these investments (or the interest he earns on re-lending) will compensate "for the expected failure of receipts from his wasting asset in the future."  In this case, with receipts expected to decline, income will decrease if the interest rate decreases.  If we change the assumption of a wasting asset and assume that expected receipts will increase (for whatever reason), income will be higher for a lower rate of interest (because the person consuming as much as his income will have to borrow during early periods to have enough funds to consume so much).

The author then recalls the previous section's interpretation of the capital value of a stream of receipts as the weighted average period of the receipts (weighted by discounted values of receipts), and the comparison of this average period of receipts with the period of a standard stream of receipts to test whether a rise in the interest rate would increase or decrease income.  The author asks whether this test can be reinterpreted so as to agree with the common-sense case described above.

The author gives his answer for the case in which prices and interest rates are expected to remain constant.  In this case all three approximations of income give the same results, with the standard stream of receipts having the same constant value in all periods.  If the average period of the given stream of receipts is greater than that of the standard stream, then the given stream must have lower value initially.  But because the two streams must have the same capitalized value, the given stream must catch up later.  (In the language of the text, there must a "crescendo.")  The author concludes that "The average period turns out to be nothing else but an exact method of measuring the crescendo (or diminuendo) of a stream of values."  In the case of a stream of identical quantities, continuing indefinitely, and "discounted throughout by the same rate of interest" the author shows that the average period works out to be the reciprocal of the rate of interest, the calculations being as follows:


Finally, the author gives a formula for the crescendo of a stream of values, with each period's value expanding by the same proportion.  The formula for the crescendo c is

c = i – 1 / P

where i is the interest rate, and P is the average period of the stream.

Value & Capital, Notes to Chapter XIV -- B. INTEREST AND THE CALCULATION OF INCOME, part 2

This section lays out the graphical construct alluded to at the conclusion of part 1 of Note B.  As explained there, the purpose of the construct is to study the relation between interest rates and the present value of actually expected receipts.  It plots capitalized values along the horizontal axis;  along the vertical axis, it plots the discount ratio, which is related to the interest rate.  Namely, if i is the interest rate, then the discount ratio β equals 1 / (1 + i).

For an assumed given stream of receipts, there is a capital-value curve RR (plotted as a solid curve in the figure below) that shows the capitalized value of that stream for a given discount ratio.  Also, as the author explains

Corresponding to any particular level of income, we have a capital-value curve (dotted in the diagram) which shows the present value of the standard stream corresponding to that particular level of income (according to the definition of income we are using).

The figure follows the usual convention in economics, of putting the independent variable on the vertical axis.

Therefore, "If the discount ratio is OH, the present value of the prospective receipts is HA, and the level of income is that represented by the dotted curve SS, which passes through A."  A change in the discount ratio will move the point A along the curve RR.  Whether a rise in the discount ratio means a rise in the level of income depends on whether SS is steeper than RR as it is drawn here (meaning that SS is less elastic than RR). 

The author then proceeds to discuss elasticity of income with respect to the discount ratio.  If the expected stream of receipts in the various time periods is (x0, x1, ... , xv), then the capital value of this stream is

x0 + β x1 + β2 x2 + … + βv xv.  

Since mathematically the x-elasticity of y is the product of the derivative of y with respect to x, and the ratio x / y, it is clear that the elasticity of the capital value with respect to the discount ratio is 

The author then goes on to explain that

... when we look at the form of this elasticity we see that it may be very properly described as  the Average Period of the stream;  for it is the average length of time for which the various payments are deferred from the present, when the times of deferent are weighted by the discounted values of the payments.

This is clear from looking at the individual terms of the numerator:  the numerical coefficients are the numbers of time periods for which the payment is deferred, and the rest of each term represents the discounted value of the payment; the (common) denominator scales each payment as a fraction of the whole capital value.

The author concludes by noting that a comparison of the above average period of the stream of receipts with the average period of the standard stream will determine whether a fall in the rate of interest will increase income.  If the above average period is greater than the standard, it will raise income;  if not, a rise in the interest rate will raise income.

Value & Capital, Notes to Chapter XIV -- B. INTEREST AND THE CALCULATION OF INCOME, part 1

This section introduces a discussion that will lead to studying graphically the relationship between the rate of interest and income.  The author begins by noting that for each of the three approximations for income studied earlier, "the calculation of income consists in finding some sort of standard stream of values whose present capitalized value equals the present value of the stream of receipts which is actually in prospect."

In each of the approximations to income, the standard stream "maintains some sort of constancy" instead of fluctuating as could happen in reality.  The three approximations make three different choices for what is assumed to remain constant.  Income No. 2 arithmetically "imputes identically the same sum of money value to each successive week."  Income No. 3 assumes constant purchasing power of each week's receipts, therefore "the money values imputed to successive weeks will vary as the price level is expected to vary."  For Income No. 1, the capitalized money value of all future receipts (in the standard stream) will be held constant from week to week, so the actual money values may vary with both prices and the rate of interest.

The author goes on to note that

...in each case we are broadly doing the same thing.  We are replacing the actual expected stream of receipts by a standard stream, whose distribution over time has some definite standard shape.  We ask, not how much a person actually does receive in the current week, but how much he would be receiving if he were getting a standard stream of the same present value as his actual receipts.  That amount is his income.

If the expected future receipts increase, then the equivalent income will increase by raising the equivalent standard stream of receipts, while maintaining "its old standard shape."

The author also notes that variable interest rates complicate matters.  The present values of both the actual expected stream of receipts and the "old standard stream" will change.  "In order to discover the effect on income we have to find which of these two present values is affected the more."  The succeeding discussion will study this effect in the case where the rate of interest is the same for all durations of loans (which the author asserts is "often or usually legitimate" to assume).

Value & Capital, Notes to Chapter XIV -- A. SAVING AND INVESTMENT

Regarding saving and investment, the author states in the preliminary paragraph of his notes on Chapter XIV that "I think the reader has a right to demand some expression of opinion on that controversial topic."  He then begins his note on the topic by explaining that the principal difficulty "evidently arises from the multiplicity of ways in which the terms can be defined."  He observes that saving can be defined in a way that corresponds to each of the definitions of income discussed in the preceding chapter (each of which would, in turn, give rise to a corresponding definition of investment).  He also states that saving can be defined ex ante or ex post.  He argues that the choice of which definition of income to use is not important to the analysis, but he states that "the ex ante - ex post distinction is of course very important."  (One wonders whether the author thought this should be obvious.)

For his explanation the author chooses to use Income No. 1 as his basis for defining saving and investment;  he notes that if he chose another definition of income, "the whole argurment would be exactly duplicated."  Using this definition of income (and the same kind of one-week model he has used in earlier analyses) he defines a person's savings ex ante to be "the difference between his actual consumption during the week and that level of consumption which would leave the money value of the prospect he can expect to have at the end of the week the same as it actually was at the beginning."  If the week is assumed to be a short enough time that "the accretion of interest" is negligible , then saving can also be said to be "the increment in the money value of his prospect planned to accrue during the week."  If any changes in his earning power are ruled out, then "his saving may also be written as the planed increment in the value of his property.  All this is saving ex ante; saving ex post will be the realized increment in the value of his property."

Regarding saving and investment ex post, the author gives an argument for the equality of their aggregate amounts that is so straightforward that I prefer to reproduce it in full:

Savings ex post may be aggregated for all members of the community.  Their sum total will equal the total increment in the money value of all persons' property which accrues during the week.  Now property has three forms:  it may consist of physical goods (real capital), or securities, or money.  But money, as we have seen, is either a physical good, like gold, or a security, like notes or bank deposits.  Our three categories thus reduce to two.  Further, securities are simply debts of various sorts from one person (or concern) to another; and therefore, when all property is aggregated, they cancel out.  Total savings ex post therefore reduce to nothing else but the increment in the value of physical capital; which is what seems to be meant by investment -- of course investment ex post.

While the author notes that this equality is a "mere truism," indicating only that "all the capital goods in the economy belong to somebody," the relationship between saving and investment ex ante is more interesting.  For the simple one-week model that the author uses, in which all demands and corresponding supplies are assumed to be equal during the week, it is indeed true that savings equal investment ex ante, but this is a property of the one-week model, and it will not hold for a longer time period.  As the author explains, "The ex post magnitudes will be equal whatever period we take, but the ex ante magnitudes will only be necessarily equal if plans are consistent."  Over a longer time period, planned saving could exceed planned investment.  "If an attempt is made to carry through the plans without readjustment, supplies of commodities will begin to exceed demands, and (so far as we can see at present) prices will tend to fall.  Similarly, if planned investment exceeds planned saving, there will be a tendency for prices to rise."

Toward the end of this note, the author exclaims "What a tricky business this all is!"  He then goes on to note four statements that John Maynard Keynes made regarding savings and investment:
(1) In his Treatise on Money, that they are only equal in conditions of equilibrium, and
(2) that an excess of investment over saving means rising prices, and vice versa;
(3) in the General Theory, that savings and investment are always equal, and
(4) that this is a mere identity or truism, without significance for the determination of prices.

"As far as I can make out," Hicks notes, "there are relevant and important senses in which all four of these statements are each of them right and each of them wrong."

Value & Capital, CHAPTER XIV, Section 8

This section begins by noting a dilemma that confronts anyone wanting to calculate social income:  "The income he can calculate is not the true income he seeks;  the income he seeks cannot be calculated."  The text explains that the only real way out of this dilemma is to start with the objective quantity, Social Income ex post, and to adjust it "for those changes in capital values which look as if they have had the character of windfalls."  The author argues that such an estimation procedure is perfectly reasonable, but he emphasizes that the result is only a statistical estimate, not a true measurement of an economic quantity.

In discussing the problem of estimating income, the author proposes one method, which is

to take the actual capital goods existing at the end of the period, and to value them at the prices which any similar goods would have had at the beginning;  any accumulation of capital which survives this test will be an accumulation in real terms.  By adding the amount of consumption during the period, we get at least one sense of income ex post;  by then correcting for windfalls we get a useful measure of real social income.

In closing this section, the author expresses the hope that this chapter has made clear the important influence of income calculations on economic conduct, both at the individual level and the social level.  At the same time, he expresses the hope that the chapter has made clear that "income is a very dangerous term, and it can be avoided."  Finally, he previews future discussions by noting that "a whole general theory of economic dynamics can be worked out" without using income.  Even when a need arises "at a very late stage in our investigations," an exact definition will not prove necessary;  "something quite rough" will be sufficient.

Value & Capital, CHAPTER XIV, Section 7

This section begins with an issue that emerges when considering social income, i.e. the aggregate of individual incomes.  As seen earlier, the concept of individual income depends on the expectations of the particular individual.  Furthermore, as explained in the discussion of equilibrium and disequilibrium, the expectations of different individuals may not be consistent.  As the author notes, "one of the main causes of disequilibrium in the economic system is a lack of consistency in expectations and plans."  So it is obviously problematic to be considering an aggregate based on these possibly inconsistent expectations.  But if this is not what social income is, the author asks, what is it?

He answers this question by noting that the various "definitions of income that we have hitherto discussed are ex ante definitions -- they are concerned with what a person can consume during a week and still expect to be as well off as he was.  Nothing is said about the realization of this expectation."  If the value of the individual's prospect at the end of the week is greater than expected, he is said to have experienced a "windfall" profit for that week (conversely, if less, then a windfall loss).  If the value of the windfall profit is added to the income ex ante (or the windfall loss is subtracted from it), the result is defined as the income ex post.  As the author notes, if this is applied to the concept he earlier termed Income No. 1, then "it equals Consumption plus Capital Accumulation."

The author goes on to note that this particular income concept "has one supremely important property" -- namely that "it is almost completely objective."  As long as the capital accumulation deals with income from property and not with changes in "human capital" then income ex post can be measured objectively.  The author notes that "this is a very convenient property" but argues that it does not justify extensive use of the income ex post concept in economic theory.

The income ex post of any particular week cannot be calculated until the end of the week, and then it involves a comparison between present values and values which belong wholly to the past.  On the general principle of 'bygones are bygones', it can have no relevance to present decisions.

Value & Capital, CHAPTER XIV, Section 6

This section reaches an important and somewhat surprising conclusion about the definition of income.  Recalling that the "central criterion" is "that a person's income is what he can consume during the week and still expect to be as well off at the end of the week as he was at the beginning," the author notes that the various approximations to this criterion indicate "how unattractive it looks when subjected to detailed analysis."  Indeed, he goes on to explain why the approximations raise doubt as to whether the central criterion "does, in the last resort, stand up to analysis at all."

His explanation compares two "prospects" of an individual possessing a certain stock of consumption goods at the beginning of a week and expecting "a stream of receipts that will enable him to acquire in the future other consumption goods, perishable or durable."  The two prospects, which he calls Prospect I and Prospect II, are related in that Prospect II is the new prospect that emerges, one week after having started with Prospect I.  Prospect II "will have a new first week which is the old second week, a new second week which is the old third week, and so on."  The author argues that if both prospects were available at the beginning of the same week, the individual would know which one he preferred.  "But to inquire whether I on the first Monday is preferred to II on the second Monday is a nonsense question; the choice between them could never be actual at all."

He then concludes the section with the following summation of the key insight of this chapter:

This point ... has the same sort of significance as the point we made at a much earlier stage of our investigations, about the immeasurability of utility.  In order to get clear-cut results in economic theory, we must work with concepts which are directly dependent on the individual's scale of preferences, not on any vaguer properties of his psychology.  By eschewing utility we were able to sharpen the edge of our conclusions in economic statics;  for the same reason, we shall be well advised to eschew income and saving in economic dynamics.  They are bad tools, which break in our hands.

Value & Capital, CHAPTER XIV, Section 5

This section continues the discussion of the definition of income.  Whereas the previous section examined the definition in the context of interest rates that are expected to change, the present section considers what happens when we expect prices to change.  In this case, the author introduces a new definition (the third one of the chapter):

Income No. 3 must be defined as the maximum amount of money which the individual can spend this week, and still expect to be able to spend the same amount in real terms in each ensuing week.

The author considers an individual who plans to spend £10 each week.  If prices are expected to rise each week, then the individual must expect to be less well off as time proceeds (because the rising prices imply that he is getting less for his money each week).  If prices can fluctuate up or down, and £10 is to be the individual's weekly income (in the sense of Income No. 3 as defined above), then "he will have to expect to be able to spend in each future week, not £10 but a sum greater or less than £10 by the extent to which prices have risen or fallen in that week above or below their level in the first week."

As the author notes, this sort of correction is "obviously desirable," but in general there is no completely satisfactory solution.  One could take a set of planned expenditures and expected prices, to compare with a given income to see "whether it is such that the planner is living within his income," but unless expenditures exactly equalled income, it would be unclear "exactly how much his income is."

As the author notes, this indeterminateness is not the only difficulty with Income No. 3; there is also the matter of durable consumption goods.  By definition, a given amount of expenditure on durable goods does not constitute that amount of consumption of such goods.  The definition of income should really refer to an amount that can be consumed (not spent) during a period of time while expecting to be as well off at the end of the time period as before.  "It is only if ... the acquisition of new consumption goods just matches the using up of old ones, that we can equate consumption to spending, and proceed as before."  If these things do not match, if the individual is drawing down his stock of durable goods, he must take other steps so that his planned consumption leaves him as well off at the end of the planning period as at the beginning.