Value & Capital, CHAPTER IX, Section 7

This section provides a short summary of the three fundamental concepts used in the dynamic model, along with brief justifications and descriptions of their use.  Using the notion of a week allows the analysis "to treat a process of change as consisting of a series of temporary equilibria;  this enables us still to use equilibrium analysis in the dynamic field."  In the case of the plans that the firms (and private persons) are assumed to make each week, and that are assumed to unfold over the course of the week,

we find ourselves able to conceive of the situation at the end of the week being different from the situation at the beginning;  thus the new temporary equilibrium which is established in a second week must be different from that which was established in the first;  going on in like manner, we have a process under way.

Using the device of definite expectations enables the use of the same type of analysis as that used in the static case of determining an equilibrium for a private individual or firm.  In the dynamic case, however, we are determining the effects of both current prices and expected prices on the plans that firms and individuals make.

These three fundamental notions enable the concept of market equilibrium in the dynamic case to be explored using the "machinery" of the static case ("without abandoning our model to stationariness").

Value & Capital, CHAPTER IX, Section 6

This section discusses two aspects in which the assumptions about individuals' expectations in the dynamic model are "excessively rigid."  The plans made by producers and consumers in each time period are assumed to depend on "definite" ideas of the prices of all goods of interest in any future time period.  One erroneous aspect of this kind of assumption is that people don't have expectations of specific prices, so much as they have "expectations of market conditions."  Therefore, the text argues, "the assumption of precise price-expectations is really one aspect of the assumption of perfect competition, which we have maintained throughout, and shall continue to maintain here."

The second aspect in which the assumptions are overly rigid occupies the remainder of the discussion in this section, and it relates to the fact that people don't have precise price expectations but instead have a range of prices they consider possible.  Such a price range includes some value considered to be the most probable, "but deviations from this most probable value on either side are considered to be more or less possible."  The text argues that it is sometimes sufficient to look only at the most probable value of a quantity of interest, although "for most purposes the dispersion has a very real importance."

When we think of the choices the individuals must make in determining their plans, we should expect that "a person's readiness to adopt a plan which involves buying or selling" at a given price on a given future date "may be affected, if he becomes less certain about the probability of that price, if the dispersion of possible prices is increased."  In general, even if the most probable price were to remain unchanged, we would expect an increase in price dispersion to make individuals less willing to commit to buying or selling at the most probable price.  For individuals planning to buy, an increase in price dispersion will have the same effect as an increase in the expected price;  for those expecting to sell, "an increased dispersion will have the same effect as a reduction of the expected price."  Therefore, the text argues, "we must not take the most probable price as the representative expected price."  Instead we should use the most probable price, plus or minus an allowance for risk.

This adjustment for risk underlies the analysis of the dynamic model that will follow.  The author acknowledges that this "is not an absolutely satisfactory way of dealing with risk" and expresses the opinion that "there ought to be an Economics of Risk on beyond the Dynamic Economics we shall work out here."  Given that this book was originally written in the 1930's, it is not surprising that the decades since its publication have seen extensive developments in the economic analysis of risk.

The discussion next makes two final points about the analysis of risk.  One is that the allowance for risk depends not only on the degree of uncertainty, but also on the decision maker's preferences -- in particular his willingness to bear risks.  The second point (and the author describes it as "the most serious weakness of our treatment") is that the willingness to bear any particular risk "will be appreciably affected by the riskiness involved in the rest of the plan.  I can do very little about this on present methods," he states, "though some consequences of the interrelations of risks will come to our notice now and then."  (The text does not state this, but a particularly important consideration is the extent to which various risks may be correlated, meaning that the realization of one risk implies that certain other risks are more likely to come to fruition -- as opposed to a situation in which risks are independent.)

The section concludes by restating that the analysis will "formally assume that people expect particular definite prices."  On occasion, though, these expectations will be interpreted as "those particular figures which best represent the uncertain expectations of reality."

Value & Capital, CHAPTER IX, Section 5

This section explores a "second property of the week" in the dynamic model described in preceding sections.  Since the previous section's description of the model made the simplifying assumption that markets were open only one day per week (say Monday), the discussion in the present section makes the further assumption that "Mondays are the planning dates too."  The author notes the fundamental importance of realizing that the planning decisions about buying and selling "nearly always form part of a system of decisions which is not bounded by the present, but has some reference to future events."

The discussion goes on to note some aspects in which the treatment of planning in the model is unrealistic.  For example, it would be more realistic to describe firms as "making plans at irregular intervals."  Furthermore, the assumption "that every firm more or less reconsiders the whole situation every Monday" likely implies greater efficiency than the system would actually possess.  (The author notes that "an inefficient firm will make major plans as rarely as possible, and do all its planning by small adjustments of detail, which take only a few elements of the situation into account, and do not need much thinking.")  The author argues that these assumptions do not matter much for the model.

Thus this section arrives at the assumption that "firms (and private persons) draw up or revise their plans on Mondays in light of the market situation which is disclosing itself;  and that any minor adjustments made during the week can be neglected."  At the close of business on Mondays, then, markets "have reached the fullest equilibrium which is possible on that date;  not only have prices settled down, but every one has made the purchases and sales which seem advantageous to him at those prices."  Plans have been adjusted to these prices as well as they can be, given the imperfect efficiency of the planners.

Value & Capital, CHAPTER IX, Section 4

This section begins to describe how Alfred Marshall's framework for analyzing the dynamic model of a simple one-good economy can be generalized to study "a whole economic system."  Hicks explains that it is not worthwhile to retain Marshall's "tripartite division" of the model into Temporary Equilibrium on the first "Day," "Short Period" equilibrium, and "Long Period" equilibrium.  He bases his choice not to retain the tripartite classification on questions about the actual tendency toward stable equilibrium, as well as concerns about the length of time that adjustments to equilibrium could require.  He states his intent, instead, "to keep the truth it embodies (the time taken in adjustment) clearly in mind."

Hicks chooses to work in terms of a "week" (chosen, somewhat arbitrarily, for illustration and also "to distinguish it from Marshall's Day").  He assumes it to be "that period of time during which variations in prices can be neglected."  For illustration Hicks supposes "that there is only one day (say Monday) when markets are open, so that it is only on Mondays that contracts can be made."  Contracts can be fulfilled during the week (goods can be delivered, payments made, etc.) but any new contracts would have to wait until the following Monday to be drawn up (as would any revisions to existing contracts). In this scenario the prices set on Monday will "rule throughout the week, and they will govern the disposition of resources during the week."

During the week, when markets are not open, there is no opportunity for prices to change, hence they will remain constant.  But Hicks goes on to argue that changes in price are also "negligible" on Mondays "when the market is open and dealers have to fix market prices by higgling and bargaining, trial and error.  This implies that the market (indeed, all markets) proceeds quickly and smoothly to a position of temporary equilibrium -- in Marshall's sense.  Marshall gave certain grounds for supposing this to be a reasonable assumption under the conditions of his model;  I shall examine in the note at the end of this chapter how far these grounds are available to us."

For the sake of the present discussion, Hicks asks the reader to accept his assumption of "an easy passage to equilibrium" as being similar to other common assumptions in economic reasoning (he cites the specific example of assuming "that every one knows the current prices in all those markets which concern him").  He will explore the properties that follow from his assumptions in subsequent sections.

Value & Capital, CHAPTER IX, Section 3

In this section the author, Sir John Hicks, describes his approach to analyzing the dynamic problem as being similar to the approach of Alfred Marshall (as opposed to the Austrian approach that relies on the stationary state model).  One difference between Hick's approach and Marshall's is that Marshall only analyzed the determination of value for a single commodity, whereas Hicks is "concerned with the determination of the whole system of values."

Marshall's analysis considers a supply of goods being brought forward for sale on a particular day (which Hicks calls Day I).  This supply is considered to be completely determined by past expectations (which may or may not match the actual supply and demand conditions that exist on Day I).  The demands, on the other hand, "will be determined by the preferences and income conditions that actually exist on Day I; they may also be affected by the expectations which exist on Day I, particularly if the commodity is durable, and some persons expect an increased demand (or diminished supply) in the future."

Hicks then explores the extent to which the price that results on Day I is "determinate," i.e. conclusively determined.  Hicks notes that Marshall uses "an ingenious argument" to show that the price is determinate -- namely that "in the end, supply and demand must be equated." At the price that results, buyers buy the quantity they want to buy at that price, and sellers sell the quantity they want to sell at that price.  Hicks states in a footnote that he will return to this point in a note at the conclusion of this chapter.

Hicks next explores what happens to the supply of goods on some "Day II, or perhaps some 'days' later."  There will begin to be effects from the price that results on Day I, in addition to the continuing influence of decisions made before Day I.  The Day I price will have different effects in the long run than in the short run.  In the short run the supplies of machinery, specialized skills, and other capital, along with "the appropriate industrial organization" have not had sufficient time to adapt to demand; instead, producers must make the best adjustments to demand that they can.  In the long run, these investments in productive capacity "have time to be adjusted to the incomes which are expected to be earned by them."  Given the price on Day I, producers may begin to plan for increasing output, and may make short-term or long-term plans in this direction.

The section concludes by explaining that, for given Day I and Day N, one could "inquire what output producers will plan to produce on Day N, if they expect the price on Day N to be such and such."  The different pairs of expected price and expected output will form a curve.  "Such a curve could be drawn up for each particular future date;  Marshall's short and long period curves are samples taken out of this potentially large collection."

Value & Capital, CHAPTER IX, Section 2

This section discusses the analysis of economic dynamics by exploring the question of whether one could apply static analysis (of the sort presented in the book up to this point) and simply date all the quantities to the same moment in time.  The discussion notes that the main problem with such an approach is that "the adjustments needed to bring about an equilibrium take time."

A rise in the price of a commodity exercises, at once, only a small influence upon the supply of that commodity; but it sets entrepreneurs guessing whether the higher price will continue.  If they decide that it probably will continue, they may start upon the production of a considerably increased supply for a future date.  This decision will affect their current demand for factors;  the current position in the factor markets will thus be governed by the way entrepreneurs interpret the rise in the price of the product.

Similarly, the current supply of a commodity depends not so much upon what the current price is as upon what entrepreneurs have expected it to be in the past.  It will be those past expectations, whether right or wrong, which mainly govern current output;  the actual current price has a relatively small influence.

This is the first main crux of dynamic theory ... 

The author lays out two alternatives for how to proceed:  either incorporate the fact that quantities such as supplies and demands depend as much on expected prices as on current prices, or else "evade the issue by concentrating on the case where these difficulties are at a minimum."  He describes the first alternative as being "the method of Marshall."  He describes the second as
"(broadly speaking) ... the method of the Austrians."

The remainder of this section focuses on this second alternative, and in particular on what some Austrian-school economists called "the stationary state."  Hicks clearly states up front that "it is my firm belief that the stationary state is, in the end, nothing but an evasion" but he believes it to require attention because of the large part it had played in economic thought of the time.  He defines the stationary state as "that special case of a dynamic system where tastes, technique, and resources remain constant through time."  If entrepreneurs expect these constant conditions to remain in effect, then we can expect current prices to be the same as expected prices.  We can conclude that a price system in such a stationary state is the same as the static price-system already studied.  The explanation of this similarity with "the static world" notes the importance of a "'stationary' assumption that capital remains intact."

The discussion then notes that the static theory left out any account of the dependence of input-output (i.e. production) relations on the quantity of capital (in the form of intermediate products).  The author asks how this quantity of capital will be determined, with the answer being "through the rate of interest."  The assumption of a stationary state gives a relation between the (constant) size of the capital stock and the interest rate.  We have a second relation between the interest rate and quantity of capital stock by virtue of the effect of a stationary state on the (constant) level of saving.  This level "depends partly upon the propensities to save of the individuals composing the community, partly upon their real incomes -- and these depend again upon the size of the stock of intermediate products."  Thus there are two equations to determine the two unknowns:  size of capital stock and rate of interest.  While this is "a plausible theory of a stationary state" there are many complications that are left out.

The section closes with a discussion of the various real-world complications that are left out of the stationary state model.  The discussion also highlights the model's questionable assumptions and finishes the section with a withering criticism of the stationary state model.  If one does not assume a stationary state, then it is no longer valid to equate actual prices with expected prices, or interest rates from one period with those of another, or income with product, or money rates of interest with real rates of interest.  While it has always been known that the actual state of a real-world economy is never stationary, "stationary-state theorists naturally regarded reality as 'tending' toward stationariness;  though the existence of such a tendency is more than questionable."  The stationary state theory only "tells us that if we got to a stationary state, then (other things being equal) we should stick;  but it gives us no indication that we are in fact aiming for such a position;  for it can tell us nothing about anything actual at all."

Value & Capital, Part III -- THE FOUNDATIONS OF DYNAMIC ECONOMICS, Chapter IX -- THE METHOD OF ANALYSIS

In the first section of Chapter IX, the author, Sir John Hicks, introduces the subject of economic dynamics.  He describes it as being those parts of economic theory in which every quantity must be time-stamped.  (He actually uses the term "dated" but the concept of something's being time-stamped is probably fairly familiar to us in this age of e-commerce transactions.)  Up to now, the book has been concerned with "economic statics."  So in thinking about quantities of commodities consumed, or quantities of factors used to produce quantities of products, the discussion hasn't addressed questions as to when the consumption happens, or when the factors are available, or when the products will be finished.  In economic dynamics, we do address such questions.

Hicks goes on to explain that his rule of having abstained from looking at time-stamping ("dating") of quantities will have "great advantages" as he proceeds to examine the theory of economic dynamics.  This is because the methods of analysis he used in developing his static theory will be useful in analyzing dynamic problems.  He concludes this section as follows:

It is not obvious that anything like the same methods will do.  Nevertheless, we shall find, as we proceed, that there is a way of reducing the dynamic problem into terms where it becomes formally identical with that of statics.  Thus the results of static theory can be used after all; though almost all of them need drastic reinterpretation.

Value & Capital, Note to Chapter VIII -- CONVENTIONAL OR RIGID PRICES

Chapter VIII concludes with a note that examines what happens when some price is fixed.  Figure 22 shows supply and demand curves (S and D) for a single commodity, with the prices of all other commodities assumed to be fixed.  If the price of this particular commodity were not fixed, then it would move to the level represented by the intersection of the supply and demand curves.

If the price were fixed at a higher level (say at the level OL), then the demand curve implies that only a quantity ON would be sold, even though the supply curve implies that sellers would be willing to sell a quantity LT.  (Note that, by construction, the quantities ON, LP, and MQ are all equal.)  The note then argues that

The situation is therefore identical with that which would have arisen if a price OL had been fixed for buyers only, a price OM for sellers only, the difference between those prices being handed over as a bonus to those sellers who actually do make sales.  (Alternatively, we may suppose that a tax equal to LM per unit is laid upon the commodity, and the proceeds of that tax handed over to the sellers.  A process made familiar to us by the [UK] Ministry of Agriculture!)

The text points out that this construction retains an equilibrium in which supply equals demand, but "we have to sacrifice the rule that there is only one price in the market."  The real price is the assumed fixed price, and the text introduces the term "shadow price" to represent the price determined by equilibrium conditions.  In this context, the shadow price is OM.  The text notes that this shadow price is not the price that the sellers actually receive, so there is no income effect associated with it, but it does govern substitution effects on the supply side.

The text then assumes that demand for the commodity increases (represented by a change in the demand curve from D to D').  If the price remains fixed at OL, the amount bought will increase from LP to LP', and the shadow price will increase from OM to OM'.  The text points out the significance of the fact that the "supply will increase in just the same way (apart from the income effect) as if the actual price had increased from OM to OM'.  That is why the shadow price is important."  Any reactions in the markets for other commodities that are caused by changes in the supply of this commodity "will proceed as if there had been a real change in price; it is only is only reactions on the demand side which are cut off by the price-fixation."

Finally, the text considers an example in which there is "a minimum price for wheat, combined with jut sufficient restriction of supply to make the minimum price effective."  If there is an increase in demand, our model above shows that even though the fixed price doesn't change, there should be an increase in supply (as described by the supply curve).  Because this expansion in supply may come at the expense of some other crops (i.e. lower quantities of these other crops may be supplied), the prices of these other crops may rise, just as if the price of wheat had risen.

The note concludes that "The significance of this proposition (which is equally valid for maximum prices, when all term are reversed) is self-evident.  Price control can damp down a general rise in prices; but, unless it is absolutely complete, it cannot prevent it altogether."

Value & Capital, CHAPTER VIII, Section 7

This section explores in further detail some of the results of preceding sections by examining the role of "the standard commodity" (and considering what happens in the case where this is money).

Previous sections assumed an increase in the supply of factor A or in the demand for product X and examined the effects on the prices of other commodities, depending on whether these other commodities are substitutes or complements of the given product or factor.  The discussion in the previous section implicitly assumed that

... the amount of A offered at given prices increased, and the suppliers demanded nothing but some of the standard commodity in exchange.  If the standard commodity is money, this implies that they hoard all the income which they derive from the new units they supply.  Similarly, in [Section 5 of this chapter] it is implicitly assumed that the new demand is demand in terms of the standard commodity; so that if the standard commodity is money, the new demand comes from dishoarding, not from economizing on other goods.

If instead of making this implicit assumption we were to assume that the changes mentioned above were accompanied by changes in the demands for products, these latter changes would "produce an effect on general prices which goes in the opposite direction from the primary effect."  In particular, suppose that along with the increased supply of factor A there is an increased demand for products. The text argues that "prices in general" will only decrease if there is hoarding.  In other words, unless suppliers keep all their additional proceeds as money, they will spend some of it on products, which will tend to increase prices.  Conversely, suppose that along with the increased demand for product X there is a decreased demand for other products. Unless the increased demand for X comes from some dishoarding, all of the increased demand must be paid for through reduced spending for other products, thus decreasing their prices.

The discussion in the text also makes reference to the possible difficulties of calculating the net effect on prices of a combination of increased supply of a factor and increased demand for certain commodities.  Sometimes this can be simplified by choosing a representative consumption good as the standard commodity.  In other cases, there are obstacles to using this sort of device.  If there are any prices in the system that are fixed in terms of money, "severe intellectual contortions are needed" unless we choose money as the standard commodity.

Value & Capital, CHAPTER VIII, Section 6

This section examines the effects of an increase in the supply of some factor of production A. When the supply increases, without an increase in demand, the price must fall.  The techniques covered in previous sections can allow the effects on other prices to be worked out as well.

The text calls attention to one type of price effect it calls "particularly interesting."  This is the case of the effect on the price of some other factor of production that is used in the same industry or industries as factor A.  One possibility is that this factor, call it B, is complementary to factor A, and indeed, it has been pointed out previously that complementarity is the most likely relation among factors employed together in production.  The text notes that the direct effect in this case is to raise the price for B;  this makes intuitive sense, as the increased supply of A will lead to an increased quantity being used in production, which increases the demand for B.  But the text also notes that an important indirect effect will work in the opposite direction;  namely, the product produced from A and B is in a sense a substitute for them.  Hence B is a substitute for a substitute of A, which will tend to make its price fall.  Therefore

The net effect on the price of B is thus compounded out of two contrary tendencies, a direct effect tending to raise it, an indirect effect tending to reduce it;  either may be dominant.

If A and B are substitutes, then the combined effects will tend to reduce the price of B.

The section closes by noting that complementary factors are the commodities most likely to increase in price when the supply of a factor increases.  Even here, however, such a price

will only actually rise if the prices of their common products are little affected, that is to say, if the demands for the products are fairly elastic, or the products are good substitutes for other commodities. 

Value & Capital, CHAPTER VIII, Section 5

This section gives several examples of the kinds of analysis made possible by the results of preceding sections, specifically looking at what happens when there is an increased demand for a product.  (The next section will look at what happens when there is an increase in the supply of some factor of production.)

An increased demand for some product X will cause the price of X to rise and, in fact, will have a general tendency to raise prices "throughout the whole system."  Unless X is "a commodity of very great importance," however, the observed price increases may not be significant except for those commodities that are "nearly related" to the product X,   These commodities include factors employed in the production of X.

The increase in demand for X will only cause prices to decline for commodities that are "directly or indirectly complementary with X."  The author divides the complements into three groups as follows:
(1) Commodities complementary with X in consumption -- if the demand for X rises without a corresponding increase in demand for one of these complements, such a complement's price will tend to decrease, although the author notes that, in practice, there may be an increase in demand that masks the tendency toward a price decline.
(2) Products complementary with X in production -- products produced jointly with X will increase in supply as X increases in supply.  Again, absent an increase in demand for the complement, its price will fall.  (The author refers to this as "the familiar text-book case of wool and mutton.")
(3) Factors regressive against X -- As noted earlier in the book, regression is more plausible in the case of joint production than when there is a single product.  Here, the author notes that if any of the joint products are substitutes for X, their production will decrease, and hence the demand for factors needed to produce them may decrease as well.

The indirectly complementary commodities mentioned earlier are "either substitutes of the direct complements, or complements of the direct substitutes (whose prices rise)."  The latter includes such commodities as those that are "complements in consumption of other products whose prices had risen because they needed in their production some of the same factors as were needed for the manufacture of X."  The former includes factors of production for commodities complementary in consumption with X, as well as products for which "production is facilitated" by the decreased prices of these factors.

For those commodities the author calls "remoter indirect complements," the overall prevalence of substitution in a system will tend to "swamp much indirect complementarity."  Hence it is unlikely that their prices will fall.

Value & Capital, CHAPTER VIII, Section 4

This section begins by asserting that the previous discussion of stability is sufficient to conclude that "a perfectly stable system of production equilibrium is a reasonable hypothesis."  The discussion then assumes that such a system exists and proceeds to examine its properties.  The rules derived in Chapter V for a general equilibrium system still apply, with some additional interpretation needed.

With a stable system, an increase in the demand for any commodity must raise its price (in terms of the standard commodity).  Conversely, an increase in the supply of any commodity must lower its price.  These properties hold for both factors and products.  The extent of such a price change depends on the degree of substitutability in the system.  This makes sense, since if it is easy to find a substitute for a commodity for which the demand increases, then some of the demand can be accommodated by the substitute (in other words, the given causes of increased demand might have caused an even greater demand had a substitute not been readily available).  Factors and products are considered to be in a relation of substitution.  "Thus, the more elastic the marginal productivity curve of any factor in terms of its product, the less will the price of any commodity (factor or product) be affected by a change in the demand (or supply) of it."  Again, this makes sense, as we may think of a little bit of the factor as "going a long way" in production of the product when the marginal productivity curve is highly elastic.  To look at it another way, a highly elastic curve has large quantities of product associated with small changes in price;  therefore a small change in demand must be associated with a very small change in price.

The discussion also points out that the effects on prices for various commodities that will result from a change in the supply or demand for some commodity depend "primarily on whether these other commodities are substitutes or complements for the first."  This is elaborated as follows:

To a first approximation, we may say that a rise in the price of a commodity X will be accompanied by a rise in the prices of all those goods which are directly substitutes for X, and a fall in the prices of those goods that are complementary.  But in the second place, we may have to allow for indirect effects through other prices ... . If a good is such that it is at the same time a direct substitute for X, and the complement of a substitute, the direct and indirect effects will pull in opposite directions.

Finally, "in the third place," there may be income effects.  A change in price may make some people richer and others poorer, and the overall effects on supply and demand may not cancel out.  The section closes by noting that

It is very difficult to say anything in general about this income effect;  sometimes its working can be guessed, but very often it can only be treated as a source of random error.

Value & Capital, CHAPTER VIII, Section 3

This section continues the analysis of the equilibrium of production, examining the stability of the equilibrium.  Because this analysis concerns the stability of markets, much of the book's earlier investigations can be applied here.

The previous chapter examined the effects of a change in price on the behavior of a single firm;  here we are interested in the effect on a group of firms.  As the author points out, "For the most part, this effect can be got by aggregating the effects on single firms, as we found we could aggregate the effects on private individuals; so far the group must obey the same laws as the single firm."  The complicating factor occurs when "the change in prices has the effect of altering the number of firms producing a particular commodity."  Hicks calls this "a notoriously tricky matter" and proceeds by considering two cases:  one in which the price change stimulates a new firm to begin production of a commodity X by using entrepreneurial resources that had not been used before, and a second in which such a firm takes entrepreneurial resources that previously had been used to make other products and transfers them to the production of X.  In the first case the production creates a new source of supply of X and a new source of demand for the factors used to produce X.  In the second case, the shift to production of X means that the supply of certain other products may diminish;  similarly, the demand for factors used to produce those other products may also diminish.  Hicks concludes that "in direction of change, though not perhaps in extent, the complications due to new firms are similar in character to those we have already covered."

Hicks argues that the only possible source of instability in the equilibrium of production is, as with exchange equilibrium, the presence of strong asymmetry in income effects.  This would imply, for example, (using the language of Chapter V) that "the sellers of X will have to be much more anxious to consume more X when they become better off than the buyers of X are."  In considering how likely it is for such an asymmetry to cause instability, Hicks notes that (as seen in the previous chapter) supply and demand from firms are not subject to income effects.  Therefore he proceeds by separately considering four different markets:
(1) The markets for products.  Here a fall in price makes consumers better off and entrepreneurs worse off;  therefore income effects exist on both sides here.  As with pure exchange, instability is only possible if the product in question is an inferior good, or if it is consumed to significant extent by the entrepreneurs.  Even if these conditions exist, however, the market will only be unstable if these effects dominate the substitution effects.  In the present case we have the substitution effects from consumers choosing between the given product and other commodities, and we also have the effect on production, which, as we saw in a previous section acts as a substitution effect.  Both of these effects work toward stability.
(2) The markets for factors.  Here a fall in price makes the suppliers of the factor worse off, while making the entrepreneurs who purchase the factor better off.  Hicks argues that the specifics of this case are likely to cause an income effect that could tend toward instability;  again, however, there are the stabilizing effects of both substitution by individual consumers (between leisure and consumption, for example) as well as the effect from production.
(3) The markets for direct services.  Here there is no production, so these markets work exactly as described for exchange.
(4) Markets for intermediate products.  In this case both the supply and demand come from firms, so there is no income effect;  hence these markets are necessarily stable.

Summarizing all these considerations, Hicks concludes that the situation is similar to that of the equilibrium of exchange, but in the present situation the absence of income effects leads toward stability.  Any danger of instability is concentrated in the markets for factors.  The section closes by examining the question of how likely it is that an instability in the factor markets could cause instability in the system as a whole.  His answer is as follows:

It would seem that it is not at all likely.  For we must always remember that the predominant relation on the technical side between factors and products reckons as a relation of substitution, and that it is usually a strong relation.  The possibility of considerable changes in the rate of conversion of factors into products as a result of quite small changes in relative prices is a strong stabilizing element.  It is this more than anything else which gives us ground for supposing that the general equilibrium of production will be stable in most ordinary circumstances.

Value & Capital, CHAPTER VIII, Section 2

To examine the workings of an economic system with both private individuals and firms, this section starts from the points of view of the private individuals and of the individual entrepreneurs who run the firms.  Every individual is assumed to have resources of one or both of the following two types:  (1) factors of production, which can be bought and sold on the market, and (2) entrepreneurial resources, which cannot be traded, but which can be used, in combination with the various factors, to produce marketable products.  Just because individuals have these entrepreneurial resources, however, does not mean they will necessarily use them;  it must be the case that using them will generate a positive surplus, given the market prices for the factors and products.  If so, such an individual will become an entrepreneur and use his resources to maximize the surplus.  Doing so will determine the individual's demand for factors and supply of products.  The surplus thus generated is then available for the entrepreneur to use (along with his other income) for consumption as a private individual.

A private individual, who either does not possess entrepreneurial resources or does not find it worthwhile to use them, has to decide how much of his supply of each factor he will sell and how much of each commodity he will purchase.  Again, the system of prices determines these decisions.

Hicks summarizes the workings of this system as follows:

Taking entrepreneurs and private individuals together, the demands and supplies of all sorts of commodities are determined, once the system of prices is given.  Strictly speaking, we have to distinguish four kinds of markets:  (1) the markets for products, where demand comes from private accounts (of private individuals and entrepreneurs), supply comes from the business accounts of entrepreneurs (that is to say, from firms); (2) markets for factors, where demand comes from firms, supply from private accounts; (3) markets for direct services, where supply and demand both come from private accounts; (4) markets for intermediate products, which are products for one firm and factors for another, so that supply and demand both come from firms.  In all kinds of markets, however, supply and demand are determined, once the price-system is given.

Hicks closes the section by noting briefly that, as in the theory of exchange, we take one of the commodities as a standard, and, if the number of commodities is n, we have n - 1 equations to determine the prices of the other commodities in terms of the standard.

Value & Capital, CHAPTER VIII -- THE GENERAL EQUILIBRIUM OF PRODUCTION, Section 1

The opening paragraph of this section summarizes what has been covered in the book up to this point.  Chapters I-III explore "what determines the equilibrium of the private individual, and how he may be expected to react to changes in prices."  Chapters IV and V use the insights from the previous chapters "to elucidate the working of an economic system" consisting only of private individuals and in which the exchange of existing goods and services is the only economic activity. Chapters VI and VII introduce "a new kind of economic unit, the firm" and describe how a firm will conduct itself in the market.  At this point the stage is set "to examine the working of an economic system containing both kinds of units, private individuals and firms; so that the price-system does not only regulate exchange, but also regulates production."

Not surprisingly, the author, Sir John Hicks, calls the General Equilibrium of Production "an hypothesis of much wider applicability than the General Equilibrium of Exchange."  There are "quite a number" of economic problems where it can be applied safely, although it is possible to misuse it. In fact, Hicks claims that "the misuse of this system is one of the most fruitful sources of error in economic theory."  The reasons for this have to do with the areas of the economy that it abstracts away.

The section closes by enumerating the three "main deficiencies" of the system of the General Equilibrium of Production.  The first is that it leaves out the possibility of monopoly and imperfect competition.  The second is that "it abstracts from the economic activity of the State."  The third is that "it abstracts from capital and interest, saving and investment, and all that complex of activities ... earlier ... called 'speculation.'"  The book will treat this final deficiency in later chapters.

Value & Capital, CHAPTER VII, Section 6

Having treated some special cases in earlier sections, Chapter VII concludes in this section by summarizing what is known about the general case of a firm employing any number of factors to produce any number of products.  The summary still assumes that, "the factors ... co-operate with a fixed productive opportunity of limited capacity, so that the condition of increasing marginal cost is satisfied."  The discussion looks at what happens when a price of either a factor or a product changes (with all other prices being unchanged).

If there is a fall in price for some factor A, then the demand for A must increase.  If this happens, it must be balanced somehow -- either by the supply of some products increasing, or the demand for some other factors decreasing, or both.  Hicks goes on to explain:

The typical result of a fall in the price of a factor is then this:  that the supplies of products will expand, and the demand for other factors will expand too.  But to each of these general rules a limited amount of exception is possible, when the fixed resources are influential enough;  some factors may be substitutes for the first factor, some products may be regressive against it; the demands for substitute factors, and the supplies of regressive products, will decline.

Hicks explains in a footnote that regression seems to be more plausible in the multiple-product case than the single-product case.  If factor A plays an important role in the production of product X, then we may expect the output of X to increase when employment of A increases.  "But if the entrepreneur's fixed resources are devoted more to the production of X, they will be less available for the production of Y.  Thus A and Y may be regressive."

If there is a rise in the price of some product X, then the supply of X must increase.  Again, this effect must be balanced, and in this case it will be by an increase in employment of factors, a decreased output of some other products, or both.  Hicks also explains that

There are essentially the same reasons for expecting complementarity to be dominant among products as for expecting it to be dominant among factors (all the products must be complementary if the contribution to production of the entrepreneur's fixed resources is negligible).  Thus, though exceptions are possible, it is likely that the outputs of most of the other products will tend to rise.

Thus we would typically expect that an increased price of one product will cause an increased supply of other products and an increased demand for the factors.  "Substitute products and regressive factors will only be possible to a limited extent."

Although not stated explicitly, one can conclude that a price change for either a factor or product in the opposite direction of those assumed in the text will drive results in the opposite directions of those noted.

Hicks concludes by noting that these principles for the market conduct of a firm differ from those governing the behavior of an individual in two important ways:  (1) the income effect is absent, and (2) the general tendency will be for factors employed by the same firm to be complementary and for products jointly produced by a firm to be complementary.

This concludes Chapter VII.  Thank you for reading.

Noted in passing:  I am writing this post on the morning after Britain's vote to leave the European Union.  I wonder what Sir John Hicks would have made of this situation.  On the one hand, I tend to believe that he would generally support free trade and the economic integration facilitated by the EU.  On the other hand, some things I've read make me believe he had a tendency at times toward nationalist views, and I do think this vote was motivated largely by nationalism.

Value & Capital, CHAPTER VII, Section 5

This section further discusses the possible relationships among product and factors described in the previous section, where two factors A and B are used to produce one product X.  As the opening paragraph explains, "It is most likely that A and B will be complements, next most likely that no complementarity will be present and no regression, least likely of all that there will be regression.  The reasons for this all hang together."

The discussion then explores a limiting case in which complementarity must necessarily exist. This case assumes that there is no effect on marginal cost from the "fixed 'productive opportunity' of the enterprise" -- no economies of scale that result in cost savings for expanded production, and also no increase in marginal costs with output.  Hicks makes the following argument for A and B being complementary:

Since marginal cost is constant, the increase in product due to a simultaneous proportionate increase in both factors (the marginal product of the two factors taken together) must be constant.  But this joint marginal product is made up of four parts:

1. the marginal product of A with B constant;
2. the increment (or decrement) of this marginal product due to the simultaneous increase in B.  It will be an increment if A and B are complementary, a decrement if they are substitutes;
3. the marginal product of B with A constant;
4. the similar increment (or decrement) due to the increase in A.  To this the same rule applies.

He then claims that as the quantities of factors employed expand, the first and third of these parts must decline (this is because marginal product must be diminishing for an equilibrium to exist). But by assumption the whole does not decline.  Therefore the decline in 1 and 3 must be offset by increments in 2 and 4. Therefore A and B must be complementary.

So in this special case, in which the entrepreneur's "fixed opportunity" does not have a  limiting effect on the scale of production, the two factors must be complementary.  As soon as the fixed opportunity actually does something to limit expansion, the situation changes and the two factors are not necessarily complements.  (They still could be, if their join marginal product declines slowly.)

When might two factors employed to make a single product be substitutes?  For the case where output is variable, Hicks says this can only happen if
(1) "the fixed resources of the entrepreneur must make an appreciable contribution to production,"
(2) "the factors must be such that they would be close substitutes in the production of a given output."
He doesn't define "close" substitutes in the text, but I infer this to mean that the marginal rate of substitution is relatively high. (It's also worth noting that a footnote here says, "Thus in the case of constant costs and two factors, the two factors are necessarily complements in the production of a variable output, and necessarily substitutes in the production of a constant output.  This is a paradoxical situation, which may easily lead to misunderstandings unless we are careful about it."  He then states that it is more convenient not to regard the case of constant costs as the standard case, but as a limiting case in which the effect of the entreprenurial resources vanishes.  In the variable output case, a pair of factors employed by a single firm will ordinarily tend to be complementary.)

At this point, Hicks is able to provide an interpretation of the regression relationship in the current context:  If factor A and product X are regressive, then factors A and B must be substitutes.  From the preceding discussion, it follows that when A and X are regressive, the fixed resources of the entrepreneur must be playing a significant role in limiting production.  He then argues that this limiting effect, together with the regression relationship, imply that factor A must be especially suited to small-scale production, and factor B must be suited to production on a larger scale.  In this case, a decline in the price of A can lead to more employment of A, thereby leading to smaller-scale production and hence a decline in output.  Hicks then concludes, "Regression turns out to be a phenomenon of increasing returns;  one which is just consistent with perfect competition if the fixed entrepreneurial resources are important enough.  Still, it does not yet appear to be a possibility of which we need take much account."

Value & Capital, CHAPTER VII, Section 4

This section continues the "disentangling" of the possible substitution and complementarity relationships that might exist among products or factors of production.  It focuses on the case in which there are variations in the quantities of both factors and products.  

If the firm produces one product X, using two factors A and B, then, as before, a fall in the price of A will cause an increase in the demand for A.  But what happens with X and with B?  Section 1 and Section 2 of this chapter looked at each of these, respectively, in isolation.  Figure 20 indicated that the supply of X must increase, and Figure 21 indicated that the demand for B would decrease, but these arguments did not account for the possibility of complementarity.

When Hicks brings complementarity into the picture, he concludes that there would appear to be three ways in which to balance an increased demand for A:

(1) The supply of the product X may be increased, and the demand for the other factor B may be reduced (here no complementarity is present).
(2) The supply of X may be increased, but the demand for B may increase as well (here the factors A and B are complementary).
(3) The demand for the factor B may be reduced, but the supply of the product may be reduced too.  Here there is a queer sort of inverted complementarity between factor and product.

From figures 20 and 21 it is fairly clear that the typical relationship between factor and product -- in which more of the former will result in more of the latter -- is similar to the substitute relationship between two commodities, or between two factors, or between two products.  Given this similarity, it is natural to ask whether there is something that would be similar to complementarity, and Hicks identifies case (3) as that very thing.  He calls it "regression."  If factor A and product X are regressive, then substituting A for B will decrease the marginal product of B in terms of X.  This in turn will decrease the supply of X (given the prices of B and X).

Hicks closes this section with an amusing bit of sympathy for the reader:

I have a feeling that at this point the reader will rub his eyes, and declare that something must have gone wrong with the argument.  Regression is such a peculiar relation that it is hard to reconcile it with common sense.  Something, it would seem, must have been left out, which either excludes regression, or at least limits its possibility very drastically.  Let us see what that can be.

Hicks will address this question in the next section.

Value & Capital, CHAPTER VII, Section 3

This brief section begins the discussion of production in cases more complex than the simple cases treated in the previous two sections.  Those sections derived results about the necessary effects resulting from a factor or product price change in the one factor, one product case and in the fixed output, two factor case.

This section opens by discussing an analogy with utility theory, and how similar necessary results were obtained in simple cases. Thus the expectation is stated that we are getting these necessary results for the simple cases in production because we are working with only two variables -- one factor and one product, or two factors.  In more complex cases we may expect this "definiteness" to disappear.

This section considers the case of a firm producing a fixed output, using three factors A, B, and C.  Suppose the price of factor A falls; then, because the ratio of the prices of B and C stays the same, they can actually be considered as a single factor.  So we can conclude that the price drop for A will cause an increase in demand for A, and the demand for the combined factor of B and C must decrease.  As Hicks puts it, "There must be a substitution in favour of A at the expense of the other factors taken together."

Things change in the presence of complementarity. If B is complementary with A, the increased demand for A will cause an expansion in demand for B as well and therefore a substitution in favor of A and B, and against C.  Hicks explains that, as in utility theory, A and B are considered complementary when a substitution of A for C (B remaining unchanged) moves the marginal rate of substitution of B for C in favor of B.  Thus for a constant output, if we consider only substitutions among factors, the same rules emerge as for substitutions in a consumer's budget.

Practically the same thing would happen if the quantities of factors were kept constant and the firm varied its production of various products in response to changes in prices.  The only difference is that a rise in price of product X would lead to a substitution in favor of product X, as opposed to a price rise in a factor leading to a substitution against that factor.

Value & Capital, CHAPTER VII, Section 2

This section begins the "disentangling" (mentioned at the end of the previous section) of the possible substitution and complementarity relationships that might exist among commodities that could be products or be factors used in production of other products.  The first step in the analysis is to construct a simple case in which the firm will produce a fixed amount of output and, to do so, it will employ two factors, A and B.  The goal for the firm is to choose the quantities of the factors so as to minimize the cost of production.  Figure 21 illustrates the possible choices.

We assume the production curve is concave up.  This corresponds to the assumption of diminishing marginal rate of substitution between factors.  The line PK represents possible tradeoffs between quantities of the factors A and B, where each pair of quantities on the line has the same total cost, for the given factor prices.  The point P, where PK is tangent to the production curve, represents a position of equilibrium when the ratio of the prices of A and B is MK to PM.

Suppose the price of A were to fall.  Then, the amount of B having an equal value to the quantity ON of A would also fall, from MK to, say, MK1, and the total cost of production (valued in terms of factor B) falls from OK to OK1.  But since PK1 is not tangent to the production curve, the production costs can be reduced by moving to the point P' which is where the line PK2 (parallel to PK1) is tangent to the production curve.

At this new equilibrium, the production costs have been reduced to OK2; less of factor B is employed, and an additional quantity of factor A has been substituted for it.  These results follow just as necessarily as did the expansion of supply of the product when the factor price fell, in the case of one factor and one product.

Value & Capital, CHAPTER VII -- TECHNICAL COMPLEMENTARITY AND TECHNICAL SUBSTITUTION, Section 1

This chapter begins right where Chapter VI leaves off, by asking "what happens when a firm which has been at equilibrium at certain prices of products, and prices of factors, experiences a change in those prices."  How will those price changes affect the quantities of input factors it uses and the quantities of products it produces?  Hicks notes the similarity of the question to those addressed in Chapters II and III for the private individual.

Considering the simplest case, discussed in the last chapter, of an entrepreneur employing a single factor to produce a single product, the equilibrium is as shown in Figure 19 in the last chapter and can be seen in Figure 20 below -- in both figures denoted by P.  If the price of the factor falls, the most immediate effect (before any change is made in production) is that the entrepreneur's surplus increases from OK to OK1.  The reason for this is that the dashed line that represents the exchange of product for factor after the price change, will not decrease as much in moving from point P back to the vertical axis as did the prior exchange line PK;  this is because the quantity ON of factor consumed in production is not as costly in terms of product as before the price change.)  But the line PK1 is not tangent to the production curve, so OK1 is not the maximum surplus that the entrepreneur can achieve under the new conditions.  He will be better off at the new equilibrium P' on the production curve where the tangent P'K2 has the same slope as PK1.

We assume the production curve is concave down, so "the point P', where the tangent slopes upwards less steeply than at P, must lie to the right of P."  Therefore the fall in the price of the factor results in an increased use of the factor as an input to production and an increased output of the product.  As Hicks notes, a rise in the price of the product will also cause a decreased slope of the tangent, with the same effects.

In comparing these results with the earlier results for the private individual, Hicks notes that here the change in price leads to a new point where the tangent line touches the same (production) curve as before the price change, rather than a different curve.

Therefore, in the case of production, we do not have anything similar to the income effects which gave us so much trouble in utility theory. The only 'production effect' is something similar in character to the substitution effect; it is a movement along the curve (in this case a production curve, as in that case an indifference curve), the curve whose properties we know from the stability conditions.

Hicks notes another complexity within the production effect, however:  that of complementarity.  It turns out that complementarity is more complicated in production theory than in utility theory, because we have to consider the relations between two kinds of commodities -- the factors and the products.  Hicks closes with a brief glimpse of upcoming sections, saying "Their mutual relations and their cross-relations will take a little disentangling."

Value & Capital, Chapter VI, Section 5

This section, the final one in Chapter VI, returns to the case of perfect competition and spells out the conditions for equilibrium in the general case of a firm converting multiple input factors into multiple products.

As in the simple case of a single factor and a single output, we have a relation between the quantities of factors used as inputs and the quantities of products resulting from production.  In this case we can think of the relation as a surface in multiple dimensions.  It will be useful to think of the elevation of such a surface as reflecting a single quantity, so Hicks explains how, for example, "given all the quantities of factors, and all quantities of products but one, the maximum producible amount of the remaining product can be deduced.  Similarly, given all the quantities of products, and all quantities of factors save one, the minimum amount needed of the remaining factor can be deduced."  In a footnote Hicks points out that such a relation will not be defined everywhere, as there will be cases for which "no amount of a remaining factor will be sufficient to produce the given collection of products."

Starting from a set of factor quantities, and the quantities of products that result from using the factors in production, Hicks notes that variations in production can happen in many ways, but they can all be reduced to some combination of three types of variations:

(1) "One product may be increased at the expense of another, i.e. substituted for another at the margin."
(2) "One factor may be substituted for another."
(3) "One factor and one product may be simultaneously increased (or diminished)."

In a footnote, Hicks states that the first two types can actually be reduced to the third.

We naturally assume that the enterprise will seek to maximize its surplus (the value of products it produces minus the costs of the factor quantities required in producing those quantities of products).  This leads to three conditions of equilibrium corresponding to the condition that price must equal marginal cost:

(1) "The price-ratio between any two products must equal the marginal rate of substitution between the two products."  Hicks calls this a "technical rate of substitution" (as it reflects the technology of production rather than happening according to the preferences of a consumer).
(2) "The price-ratio between any two factors must equal their marginal rate of substitution."
(3) "The price-ratio between any factor and any product must equal the marginal rate of transformation between the factor and product (that is to say, the marginal product of the factor in terms of this particular product)."

Next, the conditions for an equilibrium to be stable are as follows.  For stability in the process of transforming a factor into a product, the condition is that of diminishing marginal rate of transformation (or diminishing marginal product);  this carries over directly from the one-factor one-product case.  For substituting one product for another the stability condition is that of increasing marginal rate of substitution, or as Hicks explains, "increasing marginal cost in terms of the other product (marginal opportunity cost)."  For stability in substituting one factor for another, the condition is diminishing marginal rate of substitution.  Hicks explains in a footnote the intuition behind the opposite direction of the product and factor substitution conditions.

Increasing marginal rate of substitution for products, because the total value of products secured has to be maximized;  diminishing marginal rate of substitution for factors, because the total value of factors used has to be minimized.  These conditions are easily verified graphically, if the amounts of other factors and products are assumed given, and the two products (or factors) in question are measured along two axes.

Hicks explains that the stability conditions must hold for a one-for-one substitution or transformation (one factor or product for one other factor or product) but also for group substitutions or transformations.  Also

The marginal rate of substitution between any pair of groups of products must increase, and between any pair of groups of factors must diminish; the marginal rate of transformation between any group of factors and group of products must diminish.

Finally, Hicks discusses the conditions related to the existence of positive surplus.  Instead of a single condition, there are now multiple conditions.  Namely, it must not pay to abandon production of any subset of the set of all products.

Therefore the average cost of producing each product must be rising, and the average cost of producing each group of products must be rising, including the whole group that includes all the products.

Having laid out the conditions for equilibrium in the general case, Hicks will proceed as in part I of the book.  He will assume that the stability conditions and the conditions for positive surplus hold in the neighborhood of an equilibrium point, and he will then derive laws of market conduct for the firm.

Value & Capital, Chapter VI, Section 4

In this section Hicks discusses some of "the above difficulties" -- apparently referring to difficulties of satisfying the conditions of equilibrium when there are economies of scale.  One way of proceeding in our analysis is by "sacrificing the assumption of perfect competition."  When a firm is to some extent a monopolist, it can set a price above its marginal cost.  This may be a necessary condition of profitability, because average cost could sometimes be greater than marginal cost; but the problem with extending the assumption of monopoly too far is that, "Under monopoly the stability conditions become indeterminate; and the basis on which economic laws can be constructed is shorn away."

His conclusion, essentially, is that the only way out of the situation ("this wreck" as he calls it) is to assume that most of the markets that we will analyze are not significantly different from perfectly competitive markets.  Thus if prices exceed marginal costs by some percentage, we will suppose that these percentages are "neither very large nor very variable."  We will also suppose that diminishing marginal costs are rare, and therefore that marginal costs generally increase with output at the equilibrium point.

Hicks acknowledges that this assumption is a "dangerous step," that may restrict "to a serious extent" the problems that our analysis will be able to address.  He is doubtful, though, that the problems thus excluded are even "capable of much useful analysis by the methods of economic theory."

Value & Capital, Chapter VI, Section 3

This section consists of a discussion of the validity of the assumption, made in the previous section, that production has decreasing returns to scale, namely increasing marginal and average cost, and diminishing marginal and average product, as the scale of production increases.

Hicks lists two considerations that sometimes lead to criticism of the assumed conditions.  One consideration is "the frequent conviction of entrepreneurs themselves" that they have decreasing average costs.  The other consideration is that of indivisibility of certain types of investment in factors of production.

Hicks explains that for short-run problems, the existence of "fixed equipment or plant of the firm, which has been built up in the past, and is likely to be to some extent unique" can cause a situation of a factor of production being combined with resources that the firm cannot purchase on the open market.  He argues that this kind of situation can tend to cause diminishing returns, or increasing costs.

For long-run problems, the argument for increasing marginal cost follows from "the increasing difficulty of controlling an enterprise, as its scale of production grows."

Hicks devotes the final paragraph of the section to discussing the implications of having conditions on both marginal cost and average cost.  As he notes, "Marginal costs must rise as the firm expands, in order to ensure that its expansion stops somewhere."  But this condition alone is not enough to specify where the expansion stops.  The firm can be expected to sell at a price equal to marginal cost, but this marginal cost must not be too close to its minimum;  otherwise marginal cost would be below average cost, and the firm would be selling at a loss.

Value & Capital, Chapter VI, Section 2

This section presents an analysis that derives the equilibrium conditions for a firm operating in a perfectly competitive market.  In the simplest case, the firm is able to take advantage of technical opportunities for converting a single factor A into a single output X.  When the firm faces prices for A and X that are given by the market, it will "embark upon production, so long as the total value of the product secured is greater than the total value of factor employed.  Further, it will be to its advantage to produce that quantity of product which will make the excess as large as possible."

Figure 18 illustrates this graphically.  The horizontal axis measures quantities of the input A, the vertical axis measures quantities of the output X, and the curved line illustrates the set of production possibilities (input-output combinations) available to the firm.

Suppose the distance ON corresponds to the quantity of factor being employed as input.  The curve shows that the distance PN gives the quantity of product that results.  Suppose the line PK has slope equal to the ratio of the unit price of factor to the unit price of product.  Then the distance MK represents the quantity of product having the same market value as ON, the quantity of factor consumed in production. (This interpretation of MK may be easier to grasp by keeping in mind the familiar definition of slope as "rise over run.")  Then the distance OK corresponds to the surplus product that the firm generates.  The market value of OK is the net value of receipts minus costs.  As Hicks notes, "The conditions of equilibrium are that OK should be a maximum, and should be positive."

As the text points out, OK is not maximized in Figure 18.  The line PK could be raised (thus increasing OK) until PK is tangent to the production curve, as in Figure 19.

The text goes on to set out the conditions of equilibrium as follows:

(1)  To maximize the excess value of production, the line PK must be tangent to the production curve.  Thus the slope of the production curve at equilibrium must equal the ratio of the price of the factor to the price of the product.  The slope of the production curve also represents the marginal product.  Therefore, as Hicks notes, this condition for equilibrium "can be put in either of two familiar forms:  the price of the factor equals the value of its marginal product, or the price of the product equals its marginal cost."  If this condition did not hold, the firm would find it more profitable to undertake the production process with some other level of input.

(2)  Because OK must be a maximum rather than a minimum, the production curve must be "convex upwards" at the point of tangency (or as one likely heard such a curve described in introductory calculus, "concave down").  This implies that the slope (and therefore the marginal product) must be decreasing at the equilibrium point.

Hicks then discusses the similarity between these two conditions and those derived earlier for the theory of subjective value.

The production curve, as we have drawn it, is remarkably similar in its properties to an indifference curve.  Where we had equality between a price-ratio and a marginal rate of substitution, we now have equality between a price-ratio and a marginal product -- which may be looked on, if we choose, as a marginal rate of transformation.  As for the stability condition, diminishing marginal rate of substitution is replaced by diminishing marginal product.  These two conditions are therefore substantially identical, and by their means we shall be able to construct a theory of the conduct of the firm closely similar to our theory of the conduct of the private individual.

Hicks then discusses a third condition of equilibrium, for which he notes that there is no correspondence in the theory of subjective value.

(3)  The surplus OK must be positive.  This can only be the case if the slope of OP is greater than that of PK.  Since the production curve lies at or below PK, this implies that the slope of OP must be diminishing as P moves to the right.  Since the slope of OP corresponds to the average product, this means that average product must be diminishing.  In a footnote, Hicks derives the equivalent expression of this condition in terms of average cost:

Alternatively, we may argue in the following way.  If there is a positive surplus, price must be greater than average cost.  But price equals marginal cost.  Therefore marginal cost must be greater than average cost.  Therefore the production of an additional unit must raise average cost.  Therefore average cost must be increasing.

Hicks concludes the section by summarizing the equilibrium conditions in the following lists:

1. Price of factor = value of marginal product.           1. Price of product = marginal cost.

2. Marginal product diminishing.                             2. Marginal cost increasing.

3. Average product diminishing.                              3. Average cost increasing.

Value & Capital, CHAPTER VI -- THE EQUILIBRIUM OF THE FIRM, Section 1

The author, Sir John Hicks, sets out in this opening section his goal for the chapter, which is "to bring out a certain parallelism which exists between the case of the firm and that of the private person."  By doing so, he intends "to extend the theory of exchange set out in the last chapter to take account of production as well."

In the previous chapter Hicks assumed that an individual consumer, having come into the market with some collection of commodities or services, could only obtain other ones by means of exchange.  Now he will allow the possibility that sometimes they can generate new commodities through production -- a process of technical transformation of a set of inputs into a set of outputs.  An opportunity to undertake such a transformation will be chosen only if the set of goods generated has a higher value than the set consumed in the transformation.  Changes in market conditions can result in changes in the set of production opportunities that are profitable.  In addition, the set of individuals who can take advantage of these opportunities may change.

In general, an entrepreneur will acquire services that are factors of production and will do so "not because he has any direct desire for them, but because he needs them for the full exploitation of his productive opportunities."  The production that is enabled by these factors will determine the amounts of factors employed.  The enterprise that converts the factors into products may be regarded as an economic unit, operating completely separately from the private account of the entrepreneur.  "It acquires factors and sells products;  its aim is to maximize the difference between their value."

Value & Capital, CHAPTER V, Section 8

In this brief section, the last one of the chapter, Hicks summarizes by saying that he doubts much more can be said about the theory of exchange, at least at a similar level of generality.  He states that it would be possible to proceed to dealing with applications -- especially given how much of the traditional theory of international trade was developed by studying the simple exchange of two goods -- but he is choosing not to proceed in that direction in this book.

He notes, however, that having spent so much time on the theory of exchange will prove useful in the upcoming chapters.  With production and with dynamic problems, "almost exactly the same questions" as those examined in this chapter will appear.  Although these questions will seem slightly more complicated at first, Hicks claims they can be cast in familiar forms, "and so it will turn out that we know the answers already."

This completes the discussion of the first five chapters; thank you for reading this far.

Value & Capital, CHAPTER V, Section 7

This section adds one final proposition to the discussion of the laws of exchange.  Hicks had previously shown that when demand for some good X rises, the price of X must rise.  In this section he addresses the question of what governs the extent of the rise in price.  His answer (again ignoring income effects) is the following:

It can be shown that a given rise in demand will affect the price of X less, the more substitutability or the less complementarity there is between any pair of commodities in the system.

It makes intuitive sense that if there are a large number of substitutes for X, then the increased demand for X will cause some consumers to shift to a substitute rather than pay the higher price for X.  Hicks's explanation is similar, but he adds that the substitutes themselves will also tend to rise in price, although the rise will be spread broadly over the whole set of substitutes and will therefore affect each one very little.

When there is less complementarity in the system, one could think of this as a situation in which it is relatively painless to shift demand from one good to another;  making do with less of a commodity does not tend to diminish the utility gained from consuming other commodities.  Hicks comes at the question from the other direction, assuming a large group of complements; then, an increase in the demand for X will lead suppliers of the increased volume of X to tend to dispose of goods complementary with X.  If the demand for these complementary goods had not increased, this would tend to drive down their prices, which would then lead to increased demand.  Since X is complementary with them, demand for X would increase as well, thus further increasing its price.

Hicks notes that we can apply these same concepts "at a second remove" to the substitutes and complements themselves.  If such a good (either a substitute or complement for X) has good substitutes, the effect on its price from the change in demand for X will be less pronounced.  If on the other hand it is a member of a set of complements, the effect on its price will be increased.

Hicks concludes the section by noting that

Complementarity, like imperfect substitutability, is therefore to be regarded as an element of rigidity in the system, which diminishes the elasticity of supply of any particular good.  Similarly, of course, if we had begun with an increase in supply of X, we should have found the same factors diminishing the elasticity of demand.

Value & Capital, CHAPTER V, Section 6

In this section Hicks begins to lay out his explanation of how economic laws can be derived from the stability conditions for a system of exchange.  He starts by supposing that some of the persons trading experience an increased desire for one of the commodities.  He explains that they are prepared to satisfy this desire by "increasing their supply (or diminishing their demand) for the standard commodity" while leaving their demands and supplies for all other commodities unaffected.  (To be clear, he's talking about increasing their supply of the standard commodity to the market, not increasing their own reserves of the commodity.)  He then asks what changes in prices should result.  To get at this answer he notes that the changes must cause an increase in supply from the other traders that would be sufficient to match the increase in demand from the first group.  The stability conditions have already specified what changes in prices will lead to an excess supply:  to increase the supply of a good X, its price must be raised.

What about the effects on other prices?  If we ignore income effects, and if we can assume that market reactions only take place for one other good Y, then the effect of the increased demand for X on the price of Y follows from the same analysis as shown in section 4:  the price will increase if X and Y are substitutes and fall if they are complementary.  Only this kind of change will maintain equilibrium in the Y-market.

Things get more interesting if more than one other price is affected.  Hicks runs through several possibilities, including the following:  "If Z is a substitute for X, the price of Z will be raised [by the increased demand for X]; and if Y is also a substitute for Z, this in its turn will raise the price of Y."  Also, "if Z is complementary with X and a substitute for Y, the effect through the Z-market will be to lower the price of Y."  He summarizes these cases as follows:

Indirect effects through third markets thus obey the rule that an increased demand for X will raise the prices of those goods which are substitutes of substitutes, or complements of complements, for X; it will lower the prices of those goods which are complements of substitutes, or substitutes of complements.

In cases where there are more commodities and prices involved, multiple indirect effects may be significant.  "Sometimes, perhaps often, they will all go in the same direction," Hicks states, giving the example where X and Y are part of a group of goods that are all substitutes for each other.  When X and Y are members of a group of goods that are all complements, however, things are more complicated;  the direct effect would be to lower the price of Y, but due to indirect effects, the net effect on the price of Y could go in either direction.

According to Hicks, "A system of multiple exchange in which no complementarity was present at all would obey a simple rule.  However many indirect effects were allowed for, they would all go in the same direction."  In addition, an increase in the demand for X would raise the price of X;  it would also raise the prices of all the other goods, but proportionately less than the price of X.  Hicks states that "Complete absence of complementarity, in this manner, is of course not at all a probable condition," but he places an interesting footnote here, in which he points out that the case of international currency exchange is an example where this property may be realized approximately.  "To the foreign-exchange dealers, bills in various currencies are probably all substitutes for one another."

Hicks spends the next-to-last paragraph of this section arguing that many actual situations can be expected to approximate the situation of complete absence of complementarity.  He states that in taking two goods at random, we would more likely expect them to be substitutes than complements.  In addition, indirect effects of complementarity tend to neutralize the direct effects.

Hicks's conclusion to this section's discussion is that "it does appear that an increase in demand for a particular good (or group of goods) is most likely to have an upward effect upon prices in general."  Due to indirect effects, it's possible that some goods that are directly or indirectly complementary with the one whose demand increased could have their prices fall, but these would be the exception.  In addition, the general upward effect on prices will not be widespread unless the good or goods whose price increased were "of considerable importance."

Value & Capital, CHAPTER V, Section 5

Hicks opens this section by stating his conclusion about the first of the two questions raised in the previous section.  His "tentatively negative answer" is as follows:  "If the market for a commodity X is stable, taken by itself, it is not likely to be rendered unstable by reactions through other markets."  He turns next to his second question -- that of whether a market for X that is unstable when considered by itself can likely be made stable by the reactions that happen through other markets.

In order for the market for X to be unstable when considered by itself, a rise in the price of X (with other prices given) will raise the excess demand for X.  (Looking back at the excess demand curve given in section 2 of this chapter, we can see that a rising excess demand curve already puts us in very strange territory.  Demand would have to outpace supply -- and increasingly so -- as the price for X rises.)  Reactions in other markets can only stabilize the market for X if they cause a lower excess demand for X, and Hicks argues that this is very unlikely.  Consider some other commodity Y, and assume for now that there are no income effects.  If Y is a substitute for X, a rise in the price of X should increase the excess demand for Y, thereby raising the price of Y;  this in turn should increase the excess demand for X.  If Y is complementary with X, a rise in the price of X should lower the excess demand for Y, thereby lowering the price of Y;  but because of the complementarity, this should increase the excess demand for X.  So in both cases the indirect reactions should increase the excess demand for X.  Thus, Hicks concludes, a market for X that is unstable, when taken alone, must be even more unstable when indirect effects are considered.

Hicks then goes on to acknowledge that this argument is not conclusive, because of potential complications when more than one other market is considered as well as the potential of income effects.  Hicks notes that the X-market will only be unstable to begin with, taken by itself, when income effects are large.  And if income effects tended to increase the demand for X when the price of X goes up, a similar effect could be possible when the price of Y changes.  As a result, the Y-market could exercise a stabilizing influence on an X-market that is unstable when taken by itself.  Hicks downplays the importance of this possibility but notes it as a possible exception to the rules he intends to set out in the next section of this chapter.

Hicks summarizes his conclusions about stability as follows:

There is no doubt that the existence of stable systems of multiple exchange is entirely consistent with the laws of demand.  It cannot, indeed, be proved a priori that a system of multiple exchange is necessarily stable. But the conditions of stability are quite easy conditions, so that it is quite reasonable to assume that they will be satisfied in almost any system with which we are likely to be concerned.  The only possible ultimate source of instability is strong asymmetry in the income effects. A moderate degree of substitutability among the bulk of commodities will be sufficient to prevent this cause being effective.

Finally, noting that if a system of exchange is stable at all it is likely to be perfectly stable, Hicks considers it "quite justifiable" to proceed to investigate how a perfectly stable system of multiple exchange reacts to changes in prices.