Notes on Sraffa's Production, Chapter 9

66. Quantity of labour embodied in two commodities jointly produced by two processes

• So what results from single-product systems generalize over to joint-product systems?
• One rule we should study: when the rate of profits is zero, the relative value of each commodity is proportional to the quantity of labor which (directly and indirectly) has gone to producing them (§14).
  • For joint-products, there is no obvious criterion apportioning the labor among individual products. It seems doubtful whether it makes any sense to speak of a "separate" quantity of labor as having gone to produce one among many jointly produced commodities.
  • We get no help from the "Reduction" approach, where we sum the various dated labor inputs weighted by the product of rate of profits. (This is further discussed in §68)
• With the system of single-product industries, we had an alternative (if less intuitive) approach using the method of "Sub-systems" (Sraffa discusses this in his Appendix A).

  It was possible to determine — for each of the commodities composing the net product — the share of aggregate labor which could be regarded as directly or indirectly entering its production.

  • This method (with appropriate adaptation) extends to joint-products, so the conclusion about the quantity of labor "contained" in a commodity and its proportionality to value (at zero profits) can be generalized to joint products.
• Consider two commodities jointly produced through each of two processes in different proportions.

  Instead of looking separately at the two processes and their products, lets consider the system as a whole and suppose quantities of both commodities are included in the net product for the system.

• We further assume the system is in a self-replacing state, and whenever the net product is changed...the self-replacing state is preserved (i.e., immediately restored through means of suitable adjustments in the proportions of the processes composing it).
• We also note: it is possible to change (within certain limits) the proportions in which two commodities are produced if we alter the relative sizes of the two processes producing them.
• If we wish to increase the quantity which a commodity enters the net product of the system (while leaving all other components unchanged), we normally must increase the total labor employed by society.

  It's natural to conclude we must increase labor for producing the commodity in question. This may go directly (i.e., directly into the process in question) or indirectly (i.e., producing the means of production).

  • The commodity added will (at the prices corresponding to zero rates of profits) be equal in value to the additional quantity of labor.
• This conclusion seems to hold for commodities jointly produced, as it holds for single-product systems.
  • The conclusion appears to hold even when we change the quantities of the means of production, since any additional labor needed to produce the latter is included as indirect labor in the quantity producing the addition to the net product.
  • Footnote. Since joint-products are present, the contraction for some processes might occur, and thus we might fall into the awkward "negative industries" scenario again...but even then, the adjustments noted include them!

    This can be avoided, provided the initial increase for the commodity in question is supposed to be "sufficiently small", and the net product for the system is assumed to comprise at the start "sufficiently large quantities" of all products...so any necessary contraction may be absorbed by existing processes, without the need for any of them having to receive a negative coefficient.

67. Quantity of labour embodied in two commodities jointly produced by only one process

• Similar reasoning holds for the case when two commodities ('a' and 'b') are jointly produced through only one process...but are used as means of production (in different relative quantities) through two processes, each produes singly the same commodity 'c'.
  • So we have two processes of the form $q_{1,a}a+q_{1,b}b\to <!--\; \to \; -->q_{1,c}c$ and $q_{2,a}a+q_{2,b}b\to <!--\; \to \; -->q_{2,c}c$ where $q_{1,a}/q_{1,b}\ne <!--\; \ne \; -->q_{2,a}/q_{2,b}$, and none of the coefficients vanish.
• We can't change the proportions which 'a' and 'b' appear in the output of their production processes (i.e., the processes producing them). But we can (through altering the relative size of the two processes using them) vary the relative quantities in which they enter as means for producing a given quantity of 'c'.

  We can vary the relative quantities of 'a' and 'b' this way, and this by itself alters the relative quantities in which they enter the net social product. (The relative quantities in which the two enter the gross product are fixed.)

  • Remark. As a childish example, we could have $$\begin{array}{r}a+2b\to <!--\; \to \; -->c\\ 3a+b\to <!--\; \to \; -->c\end{array}$$ So, suppose we have for our toy example $q_a=q_b$ (there is a one-to-one ratio between the quantity of 'a' and 'b' produced).

    The relative quantities of 'a' and 'b' seems like a strange term to me. We could consider enlarging the first process and keep the second process constant: $$\begin{array}{rl}f(\stackrel{⃗<!--\; ⃗\; -->}{x})+L_a& \to <!--\; \to \; -->5a\\ g(\stackrel{⃗<!--\; ⃗\; -->}{y})+L_b& \to <!--\; \to \; -->5b\\ 2(a+2b)& \to <!--\; \to \; -->2c\\ 3a+b& \to <!--\; \to \; -->c\end{array}$$ For simplicity, the production of 'a' and 'b' are blackbox functions which takes "some vector" of inputs. We have combined $5a+5b\to <!--\; \to \; -->3c$. The relative quantities of 'a' and 'b' are, literally, one-to-one. Observe the surplus is 3 c...and we had $L_a+L_b$ contribute.

    But if we change how we produce things, say use only the first process, then we have $$\begin{array}{rl}f(0.4\stackrel{⃗<!--\; ⃗\; -->}{x})+0.4L_a& \to <!--\; \to \; -->2a\\ g(0.8\stackrel{⃗<!--\; ⃗\; -->}{y})+0.8L_b& \to <!--\; \to \; -->4b\\ 2a+4b& \to <!--\; \to \; -->2c\end{array}$$ and hence we have the surplus be 2c. The relative proportion which 'a' and 'b' enter production change; is this what Sraffa means? We varied the size of the processes producing 'a' and 'b', without deforming the processes (i.e., changing the proportions of the coefficients, just reduced the ratio to produce a lesser amount).

    The amount of labor also changed from $L_a+L_b$ to $0.4L_a+0.8L_b$.

• It is thus possible (through an addition to total labor) to arrive at a new self-reproducing state, where a quantity for one of the two products (say 'a') is added to the net product, while all other components of the latter remain unchanged.

  We can conclude the addition to labor is the quantity which directly and indirectly is required to produce the additional amount of 'a'.

68. Reduction to dated quantities of labour not generally possible

• Sraffa claims there is no equivalent (in the case of joint-products) to the "alternative method", i.e., Reduction to a series of dated labor terms. Sraffa explains the "essence" of Reduction is that each commodity should be (a) produced separately, (b) by only one industry, and (c) the whole operation consists in tracing back the successive stages of a single-track production process.
  • Remark. I am very suspicious of this claim, and I don't follow the reasoning given. After all, consider the system given as $$(1+r)A\stackrel{⃗<!--\; ⃗\; -->}{p}+w\stackrel{⃗<!--\; ⃗\; -->}{L}=\stackrel{⃗<!--\; ⃗\; -->}{p}$$ where A is the input-output matrix, $\stackrel{⃗<!--\; ⃗\; -->}{p}$ is the price-vector, w wage, $\stackrel{⃗<!--\; ⃗\; -->}{L}$ the labor vector, and r the rate of profits. Then we have $$(I-<!--\; -\; -->(1+r)A)\stackrel{⃗<!--\; ⃗\; -->}{p}=w\stackrel{⃗<!--\; ⃗\; -->}{L}$$ where I is the identity matrix. This gives us $$\stackrel{⃗<!--\; ⃗\; -->}{p}=(I-<!--\; -\; -->(1+r)A)\cdot <!--\; \cdot \; -->w\stackrel{⃗<!--\; ⃗\; -->}{L}=\left(\underset{0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}(1+r{)}^n{A}^n\right)w\stackrel{⃗<!--\; ⃗\; -->}{L}$$ Isn't this a Reduction-type equation?

    If so, it could be suitably generalized in the straightforward way for a joint-product. Provided the joint-product system satisfies the conditions Sraffa gives (basically, the general linear algebraic conditions that a solution exists).

  • Remark (Cont'd). Now, we are dealing with a slightly more general situation, specifically: $$(1+r)A\stackrel{⃗<!--\; ⃗\; -->}{p}+w\stackrel{⃗<!--\; ⃗\; -->}{L}=B\stackrel{⃗<!--\; ⃗\; -->}{p}$$ where the matrix B is necessary for joint-products. Without loss of generality, we may assume it is an invertible matrix. Thus we re-write this system as $$(1+r)B^{-<!--\; -\; -->1}A\stackrel{⃗<!--\; ⃗\; -->}{p}+w{B}^{-<!--\; -\; -->1}\stackrel{⃗<!--\; ⃗\; -->}{L}=\stackrel{⃗<!--\; ⃗\; -->}{p}$$ or if we introduce new symbols to stress the similarity to the previous case: $$(1+r)\stackrel{˜<!--\; ˜\; -->}{A}\stackrel{⃗<!--\; ⃗\; -->}{p}+w{\stackrel{⃗<!--\; ⃗\; -->}{L}}^{\prime }=\stackrel{⃗<!--\; ⃗\; -->}{p}.$$ We should observe this becomes the previous situation.
• Sraffa suggests we should have to give a negative coefficient to one of the two joint-production equations and a positive coefficient to the other, thus eliminating one of the products while retaining the other in isolation.

  Some of the terms in the Reduction equation would represent negative quantities of labor, which Sraffa insists "no reasonable interpretation could be suggested."

  • Sraffa insists the series would contain both positive and negative terms, so the "commodity residue" wouldn't necessarily be decreasing at successive stages of approximation. Instead, it might show steady or even widening fluctuations — the series might not converge!
  • Sraffa will investigate this in §79 ("Different depreciation of similar instruments in different uses").
• Reduction could not be attempted if the products were jointly produced by a single process, or by two processes in the same proportions, since the apportioning of the value and of the quantities of labor between the two products would depend entirely on the way the products were used as means of production for other commodities.

69. No certainty that all prices will remain positive as the wage varies

• Sraffa urges us to reconsider another proposition considered earlier: if the prices of all commodities are positive at any one value of the wage between 1 and 0, no price could become negative as a result of varying the wage within those limits (§39).
  • Sraffa denies the possibility we could generalize this proposition to joint-product systems.
• Recall, the premise underpinning this proposition: the price of a commodity could only become negative if the price for some other commodity (one of its means of production) had become negative first — so no commodity could ever be the first to do so.
  • But for joint-products, there is a way around and the price for one of them may become negative...provided the balance was restored by a rise in the price of its companion product sufficient to maintain the aggregate value of the two products above that of their means of production by the requisite margin.

70. Negative quantities of labour

• Sraffa suggests his conclusion is "not in itself very startling". He interprets the situation quite simple. Sraffa notes in fact all prices are positive...but a change in the wages may create a situation which necessarily requires prices to become negative. Since this is unacceptable, those methods giving negative prices would be discarded in favor of those giving positive prices.
• When we consider this with the previous section (concerning the quantity of labor entering a commodity), the combined effect requires some explaining...
  • What's involved is not merely something like "In the remote contingency of the rate of profits falling to zero, the price of such a commodity would (if other things remain equal) have to become negative"...but we conclude in the actual situation, with profits at the perfectly normal rate of (say) 6%, that particular commodity is in fact produced by a negative quantity of labor.
  • Caution: We will work supposing 6% is the "normal rate of profits" throughout this section, so bear that in mind...
• Sraffa says "This looks at first as if it were a freak result of abstraction-mongering that can have no correspondence in reality." He has such a way with words, sometimes!
  • If we apply it to the test employed for the general case in §66, where — under the supposed conditions — the quantity of such a commodity entering the system's net product is increased (the other components remaining constant), we shall find as a result the aggregate quantity of labor society employs has diminished.
• Nevertheless! Since the change in production occurs while the "ruling rate of profits" is 6%, and the system of prices is the one appropriate to that rate, Sraffa argues "nothing abnormal will be noticeable".

  In effect the diminution in the expense for labor will be more than balanced by an increased charge for profits, the addition to net output will entail a positive addition to the cost of production.

• So, what happened? In order bring about the required change in the net product, one of the two joint-production processes must be expanded while the other contracted.

  In the case under consideration, the expansion of the former employs (either directly or through "other processes as it carries in its train the ensure full replacement") a quantity of labor which is smaller...but means of production which at the prices appropriate to the given rate of profits are of greater value — and thus attracts a heavier charge for profits — than the contraction of the latter process "under a similar proviso".

• Sraffa concludes "It seems unnecessary to show in detail that what has been said in this section concerning negative quantities of labor can be extended (on the same lines as was done for positive quantities in §67) to the case in which two commodities are jointly produced by only one process, but are used as means of production by two distinct processes both producing a third commodity."

71. Rate of fall of prices no longer limited by rate of fall of wages

• Sraffa has one further proposition about prices which needs reconsideration for the case of joint products.
• We have seen (§49) for single-product industries, when the wage falls in terms of the Standard commodity that no product can fall in price at a higher rate than does the wage.
  • The premise underpinning this: were a product able to do so, it must be owing to one of its means of production falling in price at a still higher rate.

    Since this could not apply to the product that fell at the highest rate of all, that product itself could not fall at a higher rate than wage.

• With one of a group of joint products, there is the alternative possibility the other commodities jointly produced with it should rise in price (or suffer only a "moderate" fall) with the fall of wage so as to make up — in the aggregate product of the industry — for any excessive fall of the first commodity's price.

  To such a rise, there is no limit...and thus there is none to the rate at which one of the several joint products may fall in price.

• But as soon as it is admitted the price of one (out of two or more joint products) can fall at a higher rate than does the wage, it follows even a singly produced commodity can do so...provided it employs — as one of its means of production, and to a sufficient extent — the joint product so falling.

72. Implication of this

• This possibility — price may fall faster than the wage — has some noteworthy consequences...
• First we have an exception to the rule "The fall of wage in any Standard involves a rise in the rate of profits."
• Suppose a 10% fall in the Standard wage entails (at a certain level) a larger proportionate fall — say 11% — in the price of 'a' as measured in the Standard product.
  • This means labor has risen in value by about 1% relative to the commodity 'a'.
    • Remark. I think the ratio would be $90/89\approx <!--\; \approx \; -->1.01123595505$ or the rise of value of labor relative to 'a' is about 1.12%.
  • If we were to express the wage in terms of commodity 'a', a fall in such a wage over the same range would involve a rise in the Standard wage and consequently a fall in the rate of profits.
• Moral. We can't speak of a rise or fall in the wage unless we specify the standard, for what is a rise in one standard may be a fall in another.
• For the same reasons, it becomes possible for the wage-line and price-line of a commodity 'a' to intersect more than once as the rate of profits varies
  • Figure 5: Several intersections are possible in a system of multiple-product industries.
    

• As a result, to any one level of the wage in terms of commodity 'a', there may correspond several alternative rates of profits.
  • In figure 5, the several points intersective the solid black curve — representing the price of 'a' — with the dashed wages curve represent equality in value between a unit of labor and a unit of commodity of 'a'...i.e., the same wage in terms of 'a'.

    Of course, they represent different levels of wage in terms of the Standard commodity.

  • On the other hand, as in the case of the single-products system, to any one level of the rate of profits there can only correspond one wage, whatever the standard in which the wage is expressed.

Notes on Sraffa's Production, Chapter 8

Ch. 8. The Standard System with Joint Products

53. Negative Multipliers: I. Proportions of Production Incompatible with Proportions of use

• When we consider in detail how we construct a Standard system with joint products, it becomes obvious some of the multipliers may be negative.
• Consider two products jointly produced, each through two different methods.
  • The possibility that varying the extent to which one or the other method is used ensures a certain range of variation in the proportions in which the two goods may be produced in aggregate.
  • For each commodity, its two methods limits the range of proportionality. The limits are reached as soon as one or the other method is exclusively employed.
• Now suppose in all cases which two joint products 'a' and 'b' are used as means of production, the proportion in which 'a' is employed relatively to 'b' is invariably higher than the highest of the proportions in which it is produced.
  • In such circumstances we may say some process must enter the Standard system with a negative multiplier: but whether such a multiplier will have to be applied to the low producer or high user of commodity 'a' cannot be determined a priori—it can only be discovered through the solution of the system.

54. Negative Multipliers: II. Basic and non-basic jointly produced

• Non-basic products are "the most fertile ground" for negative multipliers.
  • (NB: non-basic goods needs a new definition under these new circumstances…but we may say that the main class, i.e. products altogether excluded from the means of production, will still be non-basic; see §60)
• Consider again the case of two commodities (jointly produced in different proportions by two processes). One is to be included in the Standard product while the other — not entering the means of production for any industry — must be excluded from the Standard product.
  • This will be effected by giving a negative multiplier to the process which produces relatively more of the second commodity, and a positive one to the other process.

    The two multipliers being so proportioned when the two equations are added up to the two quantities produced of the non-basic exactly cancel out…while a positive balance of its companion product is retained as a component of the Standard commodity.

55. Negative Multipliers: III. Special raw material

• Once negative multipliers have been admitted for some processes, others (which shine with a reflected light) are liable to appear.
• Hence, suppose we have a raw material be directly used in only one process. Suppose that process has a negative multiplier. Then the industry which produces the raw material will itself follow suit and enter the Standard system with a negative multiplier.

56. Interpretation of negative components of the Standard commodity

• Since no meaning could be attached to "negative industries" which such multipliers entail, it becomes impossible to visualize the Standard system as a conceivable rearrangement of the actual processes.
• We must therefore (in the case of joint-products) be content with the system of abstract equations, transformed by appropriate multipliers, without trying to think of it as having a bodily existence.
  • Remark. I'm sure many marginalist economists howl out in frustration over this, which is amusingly ironic.
• The Standard system's purpose is to provide a Standard commodity. When it has negative components, there is no difficulty interpreting them: they are liabilities or debts. This is analogous to accounting (negative numbers = liabilities/debts; positive numbers = assets).
• Hence a Standard commodity which includes both negative and positive quantities may be adopted as money of account without straining the imagination, provided the unit represents a fraction of each asset and each liability (like a share in a company)…with the liability in the shape of an obligation to deliver without payment certain quantities of particular commodities.

57. Basics and non-basics, new definition required

• We have another difficulty we must tackle before constructing the Standard commodity: the criterion distinguishing basic and non-basic goods fail…since it's ambiguous whether a product entering the means of production for only one industry producing a given commodity should or should not be regarded as entering directly the means of production for that product.
  • Footnote: The trouble lies deeper, and as we shall see presently there would be uncertainty even if the commodity entered directly the means of production of all the processes in the system! See §59.
• And the uncertainty would naturally extend to the question whether it did or did not enter "indirectly" the production of commodities, into which the latter entered as means of production.

58. Three types of non-basics

• All three distinct types of non-basics are met in the single-product system will find their equivalents in the case of multiple-product industries.

  Taking advantage of this circumstance, we begin defining for the latter case the three types of non-basics, each as the extension of the corresponding single-product type (cf. §35).

  1. Products which do not enter the means of production for any industry. This type can be immediately extended to the multiple-product system without modifying anything.
  2. Products each of which enters only its own means of production. The equivalent would be a commodity which enters the means of production for each of the processes by which it is itself produced, and no others — but enters them to such an extent that the ratio of its quantity among the means of production to its quantity among the products is exactly the same in each of the processes concerned.
  3. Products which only enter the means of production for an interconnected group of non-basics; in other words, products which (as a group) behave in the same way as a non-basic of the second type does individually.
• In order to define (in the multiple system of k processes) the type which corresponds to the third case (with the interconnected group consisting of 'a', 'b', and 'c'), we arrange the quantities in which these commodities enter any one process, as means of production, and as products, in a row. We shall thus obtain k rows ordered in 2×3 columns as follows: $$\begin{array}{r}{A}_1{B}_1{C}_1{A}_{(1)}{B}_{(1)}{C}_{(1)}\\ A_2{B}_2{C}_2{A}_{(2)}{B}_{(2)}{C}_{(2)}\\ \dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\\ A_k{B}_k{C}_k{A}_{(k)}{B}_{(k)}{C}_{(k)}\end{array}$$
  • Footnote: Some of these quantities may be zero, of course.
• The condition for the three products being non-basic: not more than three of the rows should be independent, and the others should be a linear combination of those three. (For the general definition, see §60.)

59. Example of the third type

• This third type gives us "curiously intricate patterns". Sraffa demonstrates this with an example.
• Given a system of four processes and four products, two commodities ('b' and 'c') are jointly produced by one process and are produced by no other.

  But while 'b' does not enter the means of production for any process, 'c' enters the means of all four processes.

  Supposing the process producing 'b' and 'c' corresponds to the equation $$(A_1{p}_a+C_1{p}_c+K_1{p}_k)(1+r)+L_{1}w=A_{(1)}{p}_a+B_{(1)}{p}_b+C_{(1)}{p}_c+K_{(1)}{p}_k$$ the "rows" for the two commodities will be $$\begin{array}{cccc}-<!--\; -\; -->& C_1& B_{(1)}& C_{(1)}\\ -<!--\; -\; -->& C_2& -<!--\; -\; -->& -<!--\; -\; -->\\ -<!--\; -\; -->& C_3& -<!--\; -\; -->& -<!--\; -\; -->\\ -<!--\; -\; -->& C_4& -<!--\; -\; -->& -<!--\; -\; -->\end{array}$$ Only the first row and any other are independent, the remaining two rows are linear combinations of the first pair. So both 'b' and 'c' are non-basic.

• If we look at the matter from constructing the Standard system, we see: (a) it's obvious 'b' can't enter the Standard commodity, (b) 'c' looks like it could be a suitable component.
  • However, since 'b' occurs only in one process, the only way to eliminate 'b' is omitting that process altogether.
  • But that process was the exclusive producer of 'c', so it only appears as means of productions…not as a produced commodity. So 'c' cannot possibly enter the Standard commodity, and must be dropped.

60. General Definition

• The formal definition given is not as satisfying as the intuition "entering (or not entering) the means of production for all commodities", which it supercedes. BUT it has the advantage of greater generality.
• Lets observe the first two types of non-basics may be considered as special cases of the third.
• The definition covers the three types of the single-product system.
  • (It is quite general, and as the example in §59 suggests, it includes a final type of non-basic, which is introduced subsequently…namely commodities which enter the means of production but are not produced — a type which land is the outstanding example.)
• We can give this general formulation between the distinction between basic and non-basic goods:
• Critera. In a system of k productive processes and k commodities (no matter whether produced singly or jointly), we say that a commodity — or more generally a group of n linked commodities (where 1≤ n< k) — are "Non-Basic" if:

  of the k rows (formed by the $2n$ quantities in which they appear in each process) not more than n rows are independent, the others being linear combinations of these.

  Or, in linear algebraic terms, the matrix of k rows and $2n$ columns is of rank less than or equal to n.

• All commodities which do not satisfy this condition are "Basic" (Note that, as has been stated in §6, every system is assumed to include at least one basic product.)

61. Elimination of non-basics

• It follows we can (through linear transformations) entirely eliminate non-basic commodities from the system…both on the side of the means of production and the products.
• This operation achieves the same result as we obtained in the single-products system by the much simpler method of crossing out equations of industries producing non-basics (§35).

62. The system of Basic equations

• If the number of basic products is j, the system thus obtained will consist of j equations: these may be described as Basic equations.
• Supposing the j basic commodities are a, b, …, j we shall denote the net quantities in which they appear using the "barred-quantities" $\stackrel{ˉ<!--\; ˉ\; -->}{A}$, $\stackrel{ˉ<!--\; ˉ\; -->}{B}$, …, $\stackrel{ˉ<!--\; ˉ\; -->}{J}$ to distinguish them from the quantities in the original processes.
• The Basic equations will accordingly be as follows:

$$\begin{array}{rl}({\stackrel{ˉ<!--\; ˉ\; -->}{A}}_1{p}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_1{p}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_1{p}_j)(1+r)+{\stackrel{ˉ<!--\; ˉ\; -->}{L}}_{1}w& ={\stackrel{ˉ<!--\; ˉ\; -->}{A}}_{(1)}{p}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_{(1)}{p}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_{(1)}{p}_j\\ ({\stackrel{ˉ<!--\; ˉ\; -->}{A}}_2{p}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_2{p}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_2{p}_j)(1+r)+{\stackrel{ˉ<!--\; ˉ\; -->}{L}}_{2}w& ={\stackrel{ˉ<!--\; ˉ\; -->}{A}}_{(2)}{p}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_{(2)}{p}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_{(2)}{p}_j\\ \dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->& \dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\\ ({\stackrel{ˉ<!--\; ˉ\; -->}{A}}_j{p}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_j{p}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_j{p}_j)(1+r)+{\stackrel{ˉ<!--\; ˉ\; -->}{L}}_{j}w& ={\stackrel{ˉ<!--\; ˉ\; -->}{A}}_{(j)}{p}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_{(j)}{p}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_{(j)}{p}_j\end{array}$$

• This system is equivalent to the original one inasmuch as the values it determines for R and the prices will also be solutions of that system.
• It differs from the original system (aside from obviously excluding non-basics):

  (a) A basic equation does not represent a productive process — it merely is the result of combining the equations of a number of processes.

  (b) It may contain negative quantities as well as positive ones.

63. Construction of the Standard system

• The basic equations are designed for the construction of the Standard product.
  • Footnote. It would be possible to construct the Standard product directly from the original equations, and the final result would have been the same. Why it has seemed simpler to go through the intermediate step of the Basic equations, well, Sraffa explains it in Appendix C.
• The multipliers $q_1$, $q_2$, …, $q_j$ which applied to the $j$ Basic equations give the Standard system are determined by the following equations:

$$\begin{array}{rl}({\stackrel{ˉ<!--\; ˉ\; -->}{A}}_1{q}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_1{q}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_1{q}_j)(1+r)& ={\stackrel{ˉ<!--\; ˉ\; -->}{A}}_{(1)}{q}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_{(1)}{q}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_{(1)}{q}_j\\ ({\stackrel{ˉ<!--\; ˉ\; -->}{A}}_2{q}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_2{q}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_2{q}_j)(1+r)& ={\stackrel{ˉ<!--\; ˉ\; -->}{A}}_{(2)}{q}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_{(2)}{q}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_{(2)}{q}_j\\ \dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->& \dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\dots <!--\; \dots \; -->\\ ({\stackrel{ˉ<!--\; ˉ\; -->}{A}}_j{q}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_j{q}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_j{q}_j)(1+r)& ={\stackrel{ˉ<!--\; ˉ\; -->}{A}}_{(j)}{q}_a+{\stackrel{ˉ<!--\; ˉ\; -->}{B}}_{(j)}{q}_b+\dots <!--\; \dots \; -->+{\stackrel{ˉ<!--\; ˉ\; -->}{J}}_{(j)}{q}_j\end{array}$$

• The equations give an equation for R of the j-th degree, so there may be up to j possible values of R and corresponding sets of values of the q's. Each set will represent a Standard commodity of different composition.

64. Only the lowest value of R economically significant

• When deciding which (among the j possible sets of values) is the relevant one, we can not rely on there being a value of R which corresponds to an all-positive Standard commodity. Why? Because in a system with joint-production all possible Standard commodities may include negative quantities among their components.
• If we reconsider the matter from the perspective of single-product systems, we find while an all-positive Standard makes sense, its superiority is due to it corresponding to the lowest possible value of R (as we shown in §42).
  • We shall see the possession of this last property is — by itself — sufficient to make the Standard net product, endowed with it, the one eligible for adoption as unit of wages and prices.

    This is regardless of whether the Standard commodity with this crucial property consists of all positive quantities or otherwise.

• Suppose that $R^{\prime }$ being the lowest possible value of $R$, we adopted as unit the Standard product corresponding to another value (say $R^{\prime \prime }>R^{\prime }$).
  • As the wage $w$ measured in this Standard was gradually reduced from 1 it would (before vanishing) arrive at a level $w^{\prime \prime }$ such that $$R^{\prime \prime }(1-<!--\; -\; -->w^{\prime })=R^{\prime }$$ when the rate of profits would be equal to $R^{\prime \prime }$.
  • If at such a level of $w$, we suppose on the basis of $R^{\prime }$, then the wage must be zero (since the rate of profits is at its maximum). While on the basis of $R^{\prime \prime }$ the wage must be positive since the rate of profits is below its maximum.
  • We reconcile this through the wage $w^{\prime }$ be a positive quantity of a composite commodity who's exchange value is zero. This is because (as we shown in §41) the exchange value for a Standard commodity the composition corresponds to one solution of $R$ (in our case $R^{\prime \prime }$) at the prices that correspond to another solution of $R$ (for us, $R^{\prime }$) is zero.
  • This implies, under these circumstances, the prices of all commodities would — in terms of the chosen Standard — be infinite(!).

    Economically, such a result is meaningless.

    This anomaly, however, can be avoided if we adopt as unit the Standard net product corresponding to the lowest value of $R$.

    This is the only Standard product in terms of which, at all the levels of wage from 1 to 0 (and so at all the levels of the rate of profits from 0 to its maximum), it is possible for the prices of commodities to be finite.

65. Tax on non-basic product leaves rate of profits and prices of other products unaffected

• The distinction between Basics and Non-Basics has become so abstract in the multiple-product system, we may wonder if it has become void of meaning.
• The chief economic implication of the distinction was the basics have an essential part in determining prices and the rate of profits, while Non-Basics have none. And this remains true under the new definition.
• For single-product systems this implies: if an improvement took place in the method of production for a basic commodity, then the result would necessitate a change in the rate of profits and the prices of all commodities.

  Whereas a similar improvement for a Non-Basic would affect only that particular Non-Basic's price.

  • This cannot be directly extended to a system with multiple products, where both basics and non-basics may be produced through the same process.
  • We can find an equivalent in a tax (or subsidy) on the production for a particular commodity.

    Such a tax is best conceived as a tithe, which can be defined independent of prices and has the same effect as a fall in the output for the commodity in question all other things (viz., the quantities of its mean of production and its companion products) remaining unchanged.

• A tax on a basic product will affect all prices and cause a fall in the rate of profits corresponding to a given wage, while if imposed on a Non-Basic…it will have no effect beyond the price of the taxed commodity, and those other Non-Basics linked with it.
  • Footnote. The effect which tax has on the price for Non-basics will vary with the type of Non-Basics.

    If it does not enter any of the means of production, its price will rise by the amount of the tax.

    If it enters its own means of production, its price will change to the extent required to maintain the original ratio of the value for the aggregate product of the process (after deduction of the wage and tax) to the value of its aggregate means of production.

    If it belongs to a group of interconnected non-basics, the prices of all or some of the components of the group will change s oas to maintain that ratio.

    (In the example of §59 if the production of commodity 'c' were taxed, the price of 'c' itself would be unaffected and the brunt would be borne by the price of 'b' which would have to rise to the necessary extent.)

• This is obvious if we consider the transformed system of Basic equations (which by itself determines the rate of profits and prices of basic products) cannot be affected by changes in the quantity or prices of Non-basics which are not part of the system.

Notes on Sraffa's Production, Chapter 7

50. Two methods of production for two joint products: or, one method for producing them and two methods for using them in the production of a third commodity

• So far we have worked with industries, each producing a single commodity. But we may now ask "What happens if a single industry produces multiple products?" For example, we have something like:

  150 q. wheat + 12 t. iron + 3 units Labor → 3 pigs + 50 q. wheat
  25 q. wheat + 3 t. iron + 25 units Labor → 4 pigs + 30 q. wheat
  

  Note we have two processes producing pigs and wheat. Having joint products usually has multiple different processes producing the joint-products.

• The conditions for production can no longer determine the prices. There would be "more prices to be ascertained than there are processes" (and hence equations) to determine them.
  • The system of equations thus becomes under-determined, as the kids would say nowadays in linear algebra courses...
  • Remark. I realize now if we have a joint-process producing k different "species" of commodities, we need k distinct methods of production for the system to (mathematically) have a solution.

    So for my example above, we need two distinct processes (i.e., two processes which are linearly independent) for a solution to exist. But for the example given, we need an iron sector before we can solve it.

• Sraffa suggests there will be a second, parallel process which will produce the two commodities by a different method...and in different proportions (Sraffa hints this may change later).
  • This is mathematically necessary to solve the system of equations.
  • Sraffa takes a step further and assumes (in such cases) a second process or industry exists.
    • Footnote. Incidentally, considering the proprtions which the two commodities are produced by any one method will (in general) differ from those required for use, the existence of two methods of producing them in different proportions will be necessary for obtaining the required proportion of the two producets through an appropriate combination of the two methods.
• Problem: In every case, will there be a second (or third or n), distinct method of production?

  This is not immediately obvious to me.

  • Sraffa notes "this may appear an unreasonable assumption to make", implying for every process there exists a second, distinct, process which is neither more nor less productive.
  • But no such condition as to equal productiveness is implied! Nor would it have any meaning before prices were determined.

    With different proportions of products, a set of prices can generally be found where the two different methods are equally profitable.

  • Remark. This does not seem satisfactory, to me at least. Is there any reason why we should expect there to be multiple distinct methods of production for joint-products?

    How many different ways are there to raise sheep (which would then be turned into wool and mutton)?

• Thus any other method of producing the two commodities will be compatible with the first, subject only to the general requirement: the resulting equations are mutually independent and have at least one system of real solutions.

  This rules of, e.g., proportionality of both products and means of production in the two processes.

  • The only economic restriction: while the equations may be formally satisfied by negative solutions for the unknowns, only those methods are practicable which do not involve other than positive prices in the conditions actually prevailing (i.e., at the given wage or rate of profits).
• The same result could be achieved through the commodities being used as means of production in different proportions in various processes.
  • Remark. This is an important point that should not be overlooked. If we use the joint-products as means of production in other sectors (or production processes), then we can achieve the same result.

    I am having difficulty grasping this: is Sraffa trying to set up a system resembling:

    (1 + r) Ap + wL = (I + B)p

    where I is the identity matrix, and B is the "deformation" to take into account joint-production?

• It could be achieved even if the two commodities were jointly produced by only one process, provided they were used as means of production to produce a third commodity by two distinct processses...and more generally provided that the number of independent processes in the system was equal to the number of commodities produced.
  • The assumption previously made of the existence of "a second process" could now be replaced by the more general assumption the number of processes should be equal to the number of commodities.

51. A System of Universal Joint Products

• The possibility for an industry having more than one product makes it necessary to reconstruct – to some extent – the equations devised for the case of exclusively single-product industries.

  To do so in a perfectly general way we shall instead of regarding joint products as the exception, assume them to be universal and to apply all processes and all products.

• We consider a system of k distinct processes each of which produces, in various proportions, the same k products.
• This does not eliminate the possibility that some of the products have a zero coefficient (i.e., are not produced) in some of the processes: just as it has been admitted throughout it is not necessary for each of the basic products to be used directly as means of production by all the industries.
• The system of single-product industries is thus subsumed as an extreme case in which each of the products, while having a positive coefficient in one of the processes, has a zero coefficient in all others.
• An industry or production-process is characterized now by the proportions in which it uses and the proportions in which it produces the various commodities.
• Notation Change! In the present chapter, and the next, processes will be distinguished by arbitrarily assigned numbers 1, 2, …, k (instead of their products 'a', 'b', …, 'k').
  • Thus A₁, B₁, …, K₁ denote the quantities of the various goods 'a', 'b', …, 'k' which are used as means of production in the first process; A₂, B₂, …, K₂ those in the second; etc.
  • The quantities produced will be distinguished with their indices in parenthetics: $A_{(1)}$, $B_{(1)}$, …, $K_{(1)}$ being the products of the first process; $A_{(2)}$, $B_{(2)}$, …, $K_{(2)}$ the products of the second process; etc.
  • In this notation, we have the joint-production equations:

$$\begin{array}{rl}(A_1{p}_a+B_1{p}_b+\dots <!--\; \dots \; -->+K_1{p}_k)(1+r)+L_{1}w& =A_{(1)}{p}_a+B_{(1)}{p}_b+\dots <!--\; \dots \; -->+K_{(1)}{p}_k\\ (A_2{p}_a+B_2{p}_b+\dots <!--\; \dots \; -->+K_2{p}_k)(1+r)+L_{2}w& =A_{(2)}{p}_a+B_{(2)}{p}_b+\dots <!--\; \dots \; -->+K_{(2)}{p}_k\\ \dots <!--\; \dots \; -->& =\dots <!--\; \dots \; -->\\ (A_k{p}_a+B_k{p}_b+\dots <!--\; \dots \; -->+K_k{p}_k)(1+r)+L_{k}w& =A_{(k)}{p}_a+B_{(k)}{p}_b+\dots <!--\; \dots \; -->+K_{(k)}{p}_k\end{array}$$

52. Complications in constructing the Standard system

• We can also construct the Standard system in the same way as was done in the case of exclusively single-product industries (§33). How?
  
  Namely by finding a set of multipliers which — applied to the k production equations — will result in the quantity of each commodity in the aggregate means of production for the system bearing to the quantity of the same commodity in the aggregate product a ration which is equal for all commodities.
• Before proceeding to do so, however, it is necessary to remove certain difficulties.

  These arise from the greater complexity of the interrelations, which results in the creeping in of negative quantities on the one hand, and the disappearance of the one-to-one relation between products and industries on the other.

Notes on Sraffa's Production, Chapter 6

45. Cost of production aspect

• Sraffa considers prices from their "cost of production" aspect, and examines the way they "resolve themselves" into wages and profits.
• Sraffa would have introduced the argument earlier "had it not been for the necessity of following one line of argument at a time".

46. "Reduction" defined

• Definition. We call Reduction to Dated Quantities of Labor (or "Reduction" for short) an operation where the equation for a commodity, the different means of production used are replaced with a series of quantities of labor, each with its appropriate "date".
• Consider the equation representing the production for commodity 'a' (where wage and prices are expressed in terms of the Standard commodity): $$(A_a{p}_a+B_a{p}_b+...+K_a{p}_k)(1+r)+L_{a}w=A{p}_a$$
  • We start with replacing the commodities forming the means of production for A with their own means of production and quantities of Labor.

    In other words: we replace them with the commodities and labor which (as appears from their own respective equations) must be employed to reproduce those means of production; and they, having been expended a year earlier (§9), will be multiplied by a profit factor at a compound rate for the appropriate period…namely, the means of production by (1 + r)² and labor by (1 + r).

  • It may be noted that A_a — the quantity of commodity a itself used in the production of A — is to be treated like any other means of production…i.e., replaced by its own means of production and labor.
  • Remark. Here we are "almost dynamic" but "still quite static"! We are taking into account time, kind of, but we are really…not.
• We next replace these latter means of production with their own means of production and labor, and to these will be applied a profit factor for one more year. Or to the means of production (1 + r)³ and to the labor (1 + r)².
• We can carry this operation on as far as we like. If next to the direct labor L_a we place the successive aggregate quantities of labor which we collect at each step and we call respectively $L_{a_1}$, $L_{a_2}$, …, $L_{a_n}$, …, we shall obtain the Reduction Equation for the product in the form of an infinite series $$L_{a}w+L_{a_1}w(1+r)+...+L_{a_n}w(1+r{)}^n+...=A{p}_a.$$
• How far reduction needs to be pushed in order to obtain a given degree of approximation depends on the level of the rate of profits: the nearer the latter is to its maximum, the further must the reduction be carried.
• Beside the labor terms, there will always be a "commodity residue" consisting of minute fractions of every basic production; but it is always possible, by carrying the reduction sufficiently far, to render the residue so small as to have a negligible effect on price (at any prefixed rate of profits short of R).
  • Remark. I object to this supposition. If we carry this operation "infinitely far back", then we carry it back to a time predating humans. From a strictly historical perspective, humans began with labor alone and constructed simple tools…then constructed complex tools. Sraffa, I believe, errs suggesting "things were as they are" — a common sin among economists!
• Sraffa notes only at r = R the residue becomes all-important as the sole determinant of the price of the product.
  • Mathematically, this makes sense since w = 0 when r = R. Hence the infinite series sums infinitely many zeroes.

47. Pattern of movement of individual terms with changes in distribution

• As the rate of profits rises, the value for each of the labor terms is pulled in the opposite direction by the rate of profits and by the wage…and it moves up or down as the one or the other prevails.
• The relative weight of these two factors varies at different levels of distribution. Besides, it varies differently in the case of terms of different "date", as we shall see.
• We have seen (§30) that — if wage is expressed in terms of the Standard net product — when the rate of profits r changes, the wage w moves as $$w=1-<!--\; -\; -->\frac{r}{R}$$ where R is the maximum rate of profits.
• Substituting this expression for the wage in each term in the reduction equation, the general form of any n^th labor term becomes $$L_{a_n}\left(1-<!--\; -\; -->\frac{r}{R}\right)(1+r{)}^n$$
• Consider the values (for this expression) as r moves from 0 to its maximum R.
  • At r = 0, the value for a labor term depends exclusively on its size regardless of date.
  • With a rise in the rate of profits, terms fall into two groups:
    1. those that correspond to labor done in more recent past (which begin at once to fall in value and fall steadily throughout);
    2. those representing labor more remote in time (which rise at first, then as each of them reaches its maximum value, turn and begin downward movement).
  • In the end, at r = R, the wage vanishes and with it vanishes the value of each labor term.
  • This is best shown by a selection of curves, representing terms of widely different dates n and different quantities of labor. Lets doodle this when R is 25%.

    Variation in value of "Reduction terms" of different periods $[L_{n}w(1+r{)}^n]$ relative to the Standard commodity as the rate of profits varies between zero and R (assumed to be 25%).

    The quantities of labor ($L_n$) in various "terms", which have been chosen so as to keep the curves within the page, are as follows: $L_0=1.04$ (dashed black line); $L_4=1$ (orange line); $L_8=.76$ (red line); $L_{15}=.29$ (green line); $L_{25}=.0525$ (blue line); $L_{50}=0.0004$ (solid black line).
• It is as if the rate of profits (when moving from 0 to R) generate a wave along the row of labor terms, the crest formed by successive terms, as one after another reach their maximum value.
  • At any value of the rate of profits, the term which reaches its maximum has the "date" $$n=\frac{1+r}{R-<!--\; -\; -->r}$$
  • Conversely, the rate of profits at which any term of date n is at its maximum when $$r=R-<!--\; -\; -->\frac{1+R}{n+1}$$
  • Accordingly, all terms for which n ≤ R^-1 have their maximum at r = 0 and thus form the group of "recent dates" mentioned above as falling in value for increasing r.

48. Movement of an aggregate of terms

• The labor terms may be regarded as the constituent elements of the price of a commodity, the combination of which may (with variation in the rate of profits) give rise to complicated patterns of price-movement, with several ups and downs.
• The simplest case, the "balanced commodity" (§21) or its equivalent, where the Standard commodity taken as an aggregate: its Reduction would result in a regular series, the quantity of labor for any term being (1 + R) times the quantity in the term immediately preceding it in date.
• Consider a complicated example: we suppose two products which differ in three of their labor terms, while being identical in all others.
  • One of them, a, has an excess of 20 units labor applied 8 years before, whereas the excess of the other (b) consists of 19 units employed in the current year and 1 unit bestowed 25 years prior.
  • (They are thus not unlike the familiar instances, respectively, of the wine aged in the cellars and of the old oak made into a chest.)
  • The difference between their Standard prices at various rates of profit, i.e. $$p_a-<!--\; -\; -->p_b=20w(1+r{)}^8-<!--\; -\; -->(19w+w(1+r{)}^{25})$$ is represented in the following figure:
• The price of "old wine" rises relative to the "oak chest" when the rate of profits move from 0 to 9%, then falls between 9% and 22% to rise again from 22% to 25%.
• (The reduction to dated labor has some bearing on the attempts to find in the "period of production" an independent measure of the quantity of capital which could be used — without circular reasoning — for determining prices and the shares in distribution.
  
  (But the case just considered seems conclusive showing the impossibility of aggregating the "periods" belonging to several quantity of labor into a single magnitude which could be regarded as representing the quantity of capital.
  
  (The reversals in direction of the movement of relative prices, in the face of unchanged methods of production, cannot be reconciled with any notion of capital as measurable quantity independent of distribution and prices.)
  • In this parenthetic remark, Sraffa just decimated the Neoclassical theory of production.

49. Rate of fall of prices cannot exceed rate of fall of wages

• Something restricts the movement of any product's price: if (as a result of a rise in the rate of profits) the price falls, its rate of fall cannot exceed the rate of fall of the wage.
• So, if we draw two lines showing how the price for product a and the wage (both expressed in terms of the Standard commodity) vary with the rise of the rate of profit, we see the price line cannot cut the wage line more than once…and even then, only in one direction: such that the price (from being lower) becomes higher than the wage with the rise of the rate of profits.
• How to see this? We may look at the Reduction series or the original production equation for a. Sraffa considers the former.
  • The only variables (besides the price for a) are the wage and rate of profits, which rises with the fall of the wage…the combined effect of the two can never fall in the price more than in proportion to that of the wage.
  • Sraffa next considers the production equation for commodity a. The prices for the means of production might upset the proposition if they were themselves capable of falling at a greater rate.

    But to see this is impossible, it is sufficient to turn our attention to the product whose rate of fall exceeds that of all others: this product (since it cannot have means of production capable of falling at a greater rate than it does) must itself fall less than wage.

• The conclusion is not affected if we take as measure of wages and prices any arbitrarily chosen product (instead of the Standard commodity), since what we are concerned with is the price-relation between labor and the given product…a relation which is independent of the medium adopted.
• It follows if wage is cut in terms of any commodity (no matter whether it is one that will rise or fall relative to the Standard commodity) the rate of profits will rise…and vice-versa for an increase of the wage.
• It also follows, if wage is cut in terms of one commodity, it is thereby cut in terms of all…and similarly for an increase. The direction of change is the same in relation to all commodities, however different may be the extent.

Addendum (5 March 2016): I realize I mistakenly wrote the equation $$p_a-<!--\; -\; -->p_b=20w(1+r{)}^3-<!--\; -\; -->(19w+w(1+r{)}^{25})$$ instead the exponent in the first term should not be a 3 but an 8, i.e., it should be: $$p_a-<!--\; -\; -->p_b=20w(1+r{)}^8-<!--\; -\; -->(19w+w(1+r{)}^{25})$$ This correction should produce the correct plot (with w = 1 - r/25%, of course).