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).