23. "An invariable measure of value"
- So we chose some commodity which we took as our "yard stick" measuring
value. But we have a problem: how do we measure the changes of value
over time?
If we express everything in its exchange-ability expressed in eggs, what happens when the value of an egg changes?
It becomes impossible to determine price-fluctuations...whether it emerges from the commodity or the particular "yard stick's" value changes.
- The "relevant peculiarities" consists only in the inequality in the ratio of labor to means of production in the "successive layers" into which a commodity and the aggregate of its means of production can be analyzed.
- The "balanced" commodity which we considered in §21 would present
no peculiarities we just discussed.
- We admit that as wages fell, such a "balanced commodity" would be as susceptible to change in price (relative to other individual commodities) as anything else could. BUT we should know any such fluctuations would originate in the peculiarities of the production of the compared commodity...the change would not occur on its own.
24. The perfect Composite Commodity
- It's doubtful any single commodity posses the desired properties.
- A mixture of commodities, or a "composite commodity", would do equally well...or even better, since it could be "blended" to suit our requirements.
- The mixture of commodities needs to consists of the same commodities
as its aggregate means of production...i.e., if we take our
concoction, then substitute for each commodity its inputs (means of
productions), then we should have our concoction remain invariant.
- NB: This is a symmetry condition! One could apply representation theory, but that would be overkill...
- Sraffa asks: can such a commodity be constructed? (I'm going to guess "yes"...)
25. Construction of such a commodity: example
- The problem really concerns industries rather than commodities...so we should approach it from that angle.
- Suppose we pick out a subsystem (a "subeconomy" if you will) that forms a "complete miniature system" with some property. Specifically, we want its various commodities represented among its aggregate means of production in the same proportions as they are among its products.
- Consider an example:
90 t iron + 120 t coal + 60 qr wheat + (3/16) labor → 180 t iron
50 t iron + 125 t coal + 150 qr wheat + (5/16) labor → 450 t coal
40 t iron + 40 t coal + 200 qr wheat + (8/16) labor → 480 qr wheat
Notice the columns sum to 180 t iron, 285 t coal, and 410 qr wheat...the labor sums to 1 as usual.
- Pop quiz: What's the national income of this economy?
- Solution: We see iron is completely self-replacing, but the other two sectors have surplus. Thus we see the surplus consists of 165 t coal and 70 qr wheat. This gives us the national income.
- How do we obtain a reduced-scale system?
- We need to reduce the sectors with surplus. Note if we do this, without reducing the iron sector, then automatically the iron sector will have surplus!
- We set up a system of equations, writing "t" for "ton of iron", "c" for "ton of coal", "q" for "quarter of wheat", neglecting labor:
90 t + 120 c + 60 q → 180 t
x (50 t + 125 c + 150 q)→ x (450 c)
y (40 t + 40 c + 200 q)→ y (480 q)
where we are trying to find x and y such that the ratio of the sum of the inputs to the outputs are the same (so the ratio of the sum of the iron inputs across all sectors to iron produced is the same as the coal inputs across all sectors to the coal produced).
So our system of equations may be derived from
(90 + x 50 + y 40)/180 = (120 + x 125 + y 40)/(x 450) = (60 + x 150 + y 200)/(y 480) which works if and only if x = 3/5, and y = 3/4.
- Thus our system becomes
90 t iron + 120 t coal + 60 qr wheat + (3/16) labor → 180 t iron
30 t iron + 75 t coal + 90 qr wheat + (3/16) labor → 270 t coal
30 t iron + 30 t coal + 150 qr wheat + (6/16) labor → 360 qr wheat
- The proportions which the three commodities are produced in the new
system (180 : 270 : 360) are equal to the proportions which they enter
its aggregate means of production (150 : 225 : 300). The composite
commodity sought for is accordingly made up in the proportions
1 t. iron : 1.5 t. coal : 2 qr. wheat.
- Remark. Again, this seems familiar compared with, e.g., Marx's notion of the "total or expanded form of value" discussed in Das Kapital, Ch. 1, § 3.
26. Standard Commodity Defined
- Definition. We shall call this sort of mixture the Standard composite commodity, or Standard commodity for short. The set of equations taken in the proportions producing the standard commodity we call the Standard system.
- In any actual economic system, a miniature Standard system's embedded in it...which can be brought to light by "chipping off" the unwanted bits. (The same way a system not in a self-reproducing state can be transformed into a self-reproducing subsystem.)
- What do we take as the "unit" of the Standard commodity?
- The quantity of the Standard commodity that would form the net
product of a Standard system employing the whole annual labor of the system.
That is to say, the output for a standard system when the labor column sums to 1.
- In our example, the labor column sums to (12/16). We need to
"enlarge" each sector by (1/3). As a result, the system becomes:
120 t iron + 160 t coal + 80 qr wheat + (1/4) labor → 240 t iron
40 t iron + 100 t coal + 120 qr wheat + (1/4) labor → 360 t coal
40 t iron + 40 t coal + 200 qr wheat + (2/4) labor → 480 qr wheat
Observe the surplus in this system is: 40 t iron, 60 t. coal, and 80 qr. wheat. Thus — insofar as I understand this — the unit would consist of 40 t. iron, 60 t. coal, and 80 qr. wheat...or in my notation: 40t+60c+80q.
- The quantity of the Standard commodity that would form the net
product of a Standard system employing the whole annual labor of the system.
- Definition. Such a unit we shall call the Standard net product or Standard national income.
27. Equal Percentage Excess
- The rate which the quantity produced exceeds the quantity used up in production is the same in each sector for a Standard system. Why? Simple: in the Standard system the various commodities produced are in the same proportion as they enter the aggregate means of production.
- In our running example, the rate for each commodity is 20%. (You see, the surplus divided by the input for any commodity is 20%; 40 t. iron divided by 200 t. iron is 40/200 = 1/5 = 20%.)
- Observe for the surplus sectors, when we add the input together then multiply by 120%, we recover the output from the transformed system described in §25.
28. Standard Ratio (R) of Net Product to Means of Production
- The rate which applies to individual commodities is also the rate which the total product of the Standard system exceeds its aggregate means of production, i.e., the ratio of the net product to the means of production of the system. This ratio we call the Standard ratio.
- Note we didn't say the ratio of the values of the net product to the means of production! This is because both collections are made up in the same proportions—because they're quantities of the same composite commodity.
- So if we wrote the standard commodity as σ, for simplicity, then
the ratio would be (xσ)/(yσ). If we used the values,
then we modify σ→σ', and the ratio remains the same.
Hence the ratio of the values of the two aggregates would inevitably always be the ratio of the quantities of their components.
- In the Standard system, the ratio of the net product to means of production would remain the same...regardless of variations in the division of net product between wages and profits, and regardless of consequent price changes.
29. Standard Ratio and Rates of Profits
- If we use a fraction of the net product instead, everything that has been stated holds...why? Because we are working with multiples of a composite commodity! So the ratio of such a fraction to the means of production will remain unaffected by any variations of prices.
- Suppose the Standard net product is divided between wages and profits (taking care that the share of each consists of Standard commodity). The resulting rate of profits would be in the same proportion to the Standard ratio of the system as allotted to profits was to the whole of the system.
- Example. Our running example given above, where the Standard ratio
was 20%. If (3/4) of the Standard national income went to wages, and
(1/4) to profits, then the rate of profits would be 5%...why? Because
(1/4) of 20% is precisely 5%! If half went to each, the rate of
profits would be 10%. And if the whole went to profit, the rate of
profits would reach its maximum level of 20% and coincide with the
Standard ratio.
- Exercise. It seems difficult for me to grasp that this transformed matrix would produce, from this procedure, the desired eigenvalue. One should probably rigorously prove this...and by "one", I mean "I"...
- The rate of profits in the Standard system therefore appears as a ratio between quantities of commodities irrespective of their prices.
30. Relation between wage and rate of profits in Standard System
- Let us re-capitulate what has been determined:
If R is the Standard ratio or Maximum rate of profits, and w is the proportion of the net product that goes to wages, the rate of profit is
r = R(1 - w).
Thus as wages gradually reduce from 1 to 0, the rate of profits increase in direct proportion. The relationship is a straight line plotted on the axes (r, w).
31. Relation Extended to any system
- Now, here we should note we've been working with a very peculiar "Standard system"...but does our results hold for any arbitrary economic system? (C.f., my exercise in §29.)
- The question is equivalent to determining whether the decisive role
the Standard commodity plays lies in its
- being the constituent material of national income and of the means of production (which is unique to the Standard system); or
- in its supplying the medium in which wages are estimated?
For the latter is a function which the appropriate Standard commodity can fulfil in any case, regardless whether the system in in Standard proportions or not.
- The second alternative appears wrong. So lets look at it in some
more detail...
- In the Standard system, the wage is paid out in proportion to the
Standard commodity. This draws its special significance from the
fact the "left overs" from profit will be a quantity of the Standard
commodity. Moreover, it will be similar in composition to the means
of production.
The result: the rate of profits (being the ratio of two homogeneous quantities) can be seen to rise in direct proportion to any reduction in wages.
- Consider an "actual system". When the equivalent of the same quantity of the Standard commodity has been paid for wages, there is no reason to believe the value of what is left over for profits should stand in the same ratio to the value of the means of production...unlike the corresponding quantities do in the Standard system.
- In the Standard system, the wage is paid out in proportion to the
Standard commodity. This draws its special significance from the
fact the "left overs" from profit will be a quantity of the Standard
commodity. Moreover, it will be similar in composition to the means
of production.
- The actual system consists of the same basic equations as the Standard system...just in different proportions. Once the wage is given, the rate of profits is determined for both systems regardless of the proportions of the equations in either of them.
- Particular proportions (e.g., the Standard ones) may give transparency
to a system, and render visible what was hidden...but they cannot
change its mathematical properties.
- Remark. I think what has happened with the Standard system: we took our equations of production, expressed it as a matrix, examined the "Basic (commodities) subspace", projected the matrix obtaining a submatrix, then obtained an equivalent matrix. We've determined various properties of this equivalent matrix. The conclusion Sraffa reaches: equivalent matrices have equivalent rates of profits.
- The straight-line relation between wage and rate of profits therefore
hold in all cases...provided only the wage is expressed in terms of
the Standard product.
- The same rate of profits (which in the Standard system is obtained as a ratio between quantities of commodities) will in the actual system result from the ratio of aggregate values.
32. Example
- Working with our running example, if in the actual system (as outlined in §25, with R = 20%) the wage is fixed in terms of the Standard net product, to w = 3/4 there will correspond r = 5%.
- While the share of wages will be 3/4 of the Standard national income,
it does not follow the share of profits will be the remaining 1/4 of
the Standard income.
The share of profits will consist of whatever is left of the actual national income after deducting from it the equivalent 3/4 of the Standard national income for wages.
- Prices must be such as to make the value of what goes to profits equal to 5% of the value of the actual means of production.
33. Construction of the Standard commodity: the q-system
- We will restate our results in general terms.
- The problem constructing a Standard commodity amounts to finding a set of k suitable multipliers, which may be called qa, qb, ..., qk to be applied (respectively) to the production equations of commodities a, b, ..., k.
- The multipliers satisfy the property that the resulting quantities of
of the various commodities will bear the same proportions to one another on
the right-hand sides of the equations (as products) as they do on the
aggregate of the left-hand sides (as means of productions).
- In other words, after dilating both sides by these multipliers, the ratio of the sum of the ath column to the output qaA is the same as the sum of the bth column to the output qbB or any other such ratio.
- Definition.
This implies the percentage which the output of a commodity exceeds
the quantity of it entering the aggregate means of production is the
same for all commodities.
This percentage we have called the Standard ratio and we have
denoted it by the letter R.
- Remark. This is an abuse of notation, since R has already been used for the maximum rate of profits. Since these two quantities are equivalent, this abuse really isn't terrible.
- As good mathematicians know, such properties take the form of
equations. What's our equations?
We have a system of equations, arranged in a different order which looks like:
(Aaqa + Abqb + ... + Akqk)(1 + R) = Aqa
(Baqa + Bbqb + ... + Bkqk)(1 + R) = Bqk
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
(Kaqa + Kbqb + ... + Kkqk)(1 + R) = Kqk
We refer to this system of equations as the q-system.
- This system of equations is under-determined. To finish it up, we must
define the unit the multipliers are to be expressed...and since we
wish the quantity of labor employed in the Standard system to be the
same as in the actual system (as discussed in §26), we define the
unit by an additional equation reflecting that condition:
Laqa + Lbqb + ... + Lkqk = 1.
We thus have k + 1 equations which determine the k multipliers and R.
34. Standard national income as unit
- When we solve our system of equations, we find qa, qb, ...,
qk.
- NB: Sraffa refers to the solutions as q'a, q'b, ..., q'k. This is his notation for fixed values of q*.
- We apply these to the equations of the production system §11 and
thus transform it into a Standard system as follows:
q'a [(Aapa + Bapb + ... + Kapk)(1 + r) + Law] = q'aApa
q'b [(Abpa + Bbpb + ... + Kbpk)(1 + r) + Lbw] = q'bBpb
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
q'k [(Akpa + Bkpb + ... + Kkpk)(1 + r) + Lkw] = q'kKpk
- From this we derive the Standard national income which we adopt as the unit of wages and prices in the original system of production.
- The unit equation of §12 is therefore replaced by the following,
where the q's stand for known numbers while the p's are variables:
[q'aA - (q'aAa + q'bAb + ... + q'kAk)] pa + [q'bB - (q'aBa + q'bBb + ... + q'kBk)] pb + ... + [q'kK - (q'aKa + q'bKb + ... + q'kKk)] pk = 1
This composite commodity is the Standard of wages and prices we have been seeking from §23.
35. Non-Basics excluded
- We excluded non-basic products from the system, so it's impossible
they could influence...anything. The multiplier appropriate for their
equations can only be zero.
- The same is true for non-basics which, while not entering the means of production for commodities in general, but are used in producing non-basics...including themselves (e.g., special raw materials for luxury goods; luxury animals reproduce themselves; etc.).
- Insofar as a commodity of this kind entered the production of non-basic products of this type, it follows the latter's fate having zero for its multiplier.
- NB: the ratio of its quantity as a product to its quantity as means
of production would be exclusively determined through its own
production equation. Therefore it would in general be unrelated to
R and be incompatible with the Standard system.
The multiplier appropriate to it would therefore also be zero.
Sraffa has a footnote stating: "Strictly speaking the multiplier would be zero for every possible value of R except the one that was equal to the ratio of the quantity of that non-basic in the net product to its quantity in the means of production. This is a freak case of the type referred to in Appendix B: at that particular value of R all prices would be zero in terms of the non-basic in question."
- We may simplify the discussion by assuming all non-basic equations are eliminated at the outset so only basic industries come under consideration.
- NB: the absence of non-basic industries from the Standard system does not prevent the latter from being equivalent in its effects to the original system since (as we have seen in §6) their presence or absence makes no difference to the determination of prices and of the rate of profits.
Addendum (). Fixed a typo in the subscripts appearing in the definition of the q-system in section 33.
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