Sunday, February 24, 2013

Notes on Sraffa's Production, Chapter 4

(I was planning to schedule this for next week's post, but I botched that up...so I'll take a week off! Or something...)

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.

  • 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
    1. being the constituent material of national income and of the means of production (which is unique to the Standard system); or
    2. 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.
  • 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.

Saturday, February 23, 2013

Notes on Sraffa's Production, Chapter 3

Ch. 3. Proportions of Labor to Means of Production

CAUTION: This entire section appears to be completely abstract reasoning, without manipulating a model at hand. Proceed very slowly!

I found it useful to construct toy economies for my own benefit...which helps me understand the point Sraffa makes in each section.

13. Wages as a Proportion of National Income

  • We now give the wage w successive values ranging from 1 to 0: these represent fractions of the national income (compare §10 and §12).
  • Objective: determine how changes in the wage affects the rate of profits, and the prices of individual commodities...assuming the methods of production remain unchanged.

14. Values when whole National Income goes to Wages

  • When we make w = 1, the whole national income goes to wages and r is eliminated.
  • We thus revert to the systems of equations we began with (in ch. 1)! The difference being the quantities of labor are now shown explicitly instead of being represented by quantities of necessaries for subsistence.
  • The relative values of commodities are in proportion to their labor cost, i.e. the quantity of labor which directly and indirectly gone to produce them. (See Sraffa's Appendix "On Sub-Systems")
  • Sraffa asserts "at no other wage-level do values follow a simple rule".
    • Question: This is fairly cryptic. Does he mean values will not be in proportion to the quantity of labor which directly and indirectly produce the commodities? Or does he mean something else?
    • Answer: What Sraffa means, I believe, is that at no other wage level do we recover the first sort of model we discussed...instead we recover a system where the "relative values of commodities" are not in direct proportion to their labor costs.
    • Remark. It seems this proposition has some bearing on the labor theory of value, although not in the "obvious way"...

15. Variety in the proportions of labor to Means of Production

  • Consider the situation when the wages are reduced (i.e., we don't allocate the national product as wage): a rate of profits will emerge.
  • How do relative prices react to changes in wage? The key lies in the inequality of the proportions in which labor and the means of production are used in the various industries.
    • Remark. This phrasing seems ambiguous to me. What exactly is the "proportion" Sraffa speaks of? Isn't it apples and oranges? Or does he mean the ratio of "the value of the means of production" to the wage?

      It seems, based on reading further text, Sraffa refers to the ratio of the "value of the means of production" to the wage...well, I think he means wage (or else it could be the "value of the labor").

      Sraffa is motivating his "Standard commodity" (the subject of the next chapter!). The ratio, for the moment, is of values...but later we will see it doesn't matter if we use values or actual commodities. Yes it is "apples and oranges", but Sraffa's genius works this out!
  • If the proportion were the same in all industries, no price-changes could ensue regardless of any diversity of the commodity-composition of the means of production in different industries.
  • For in each industry, an equal deduction from the wage would yield just as much as required for paying profits on its means of production at a uniform rate without disturbing existing prices.
    • In these "proportions", the means of production must be measured by their values. But since values may change with a change in the wage, the question emerges: which values?
    • As far as establishing the equality or inequality of the proportions (that's all we're concerned with at the moment)...The answer is: all possible sets of values give the same result.
    • In effect, as we have seen, if the proportions of all the industries are equal, then values (and therefore proportions) do not change with the wage.
    • From this it follows if the proportions are unequal at the set of values corresponding to one wage, they cannot be equal at any other, and so they are unequal at all values.

16. "Deficit-Industries" and "Surplus-Industries"

  • For the same reason, it is impossible for prices to remain unchanged when there is inequality of "proportions".
  • Suppose prices did remain unchanged when the wage was reduced and a rate of profits emerged.
    • Since in any one industry what was saved through the wage-reduction would depend on the number of men employed — while what was necessary for paying profits at a uniform rate would depend on the aggregate value of the means of production used — industries with a sufficiently low proportion of labor to means of production would have a deficit...while industries with a sufficiently high proportion would have a surplus, on their payments for wages and profits.
    • Nothing is assumed at the moment as to what rate of profits correspond to what wage reduction. All we require at this stage is there should be a uniform wage and a uniform rate of profits throughout the system.

17. A Watershed Proportion

  • There would be a "critical proportion" of labor to means of production which marked the watershed between "deficit" and "surplus" industries.
  • An industry with such a proportion would show an even balance—the proceeds of the wage-reduction would provide exactly what was required for the payment of profits at the general rate.
  • Whatever the precise value of that "proportion" in any system, it can be said a priori that—in a system with two or more basic industries—the industry with the lowest proportion of labor to means of production would be a "deficit" industry and the one with the highest proportion would be a "surplus" industry.

18. Price-Changes to Redress Balance

  • Thus with a wage-reduction, price-changes would necessary to redress the balance in each of the "deficit" and "surplus" industries.
  • We expect the price-ratio between each product and its means of production "to come into play".
    • Consider the "deficit" industry when wage is reduced. A rise in the price of the produce relatively to the means of production would help to eliminate the deficit, since it would release some of that share of the gross product into the industry which had been going to pay for the replacement of the (now cheapened) means of production.

      This would be added to the quantity available for the distribution as wages or profits.

      The price rise by itself would thus result in an increase in the magnitude (and "not merely in the value") of that part of the product of the industry which is available for distribution, despite the methods of production remaining unchanged.

  • A further effect of the rise in the price of the product (relative to the means of production) would be to help a given quantity of product to go a "longer way" towards achieving the required rate of profit.
  • Independent of this, the steeper the rise in the product's price relative to labor, the smaller the quantity of it absorbed by the wage.
  • Conversely, price-movements in the opposite direction would accomplish the disposal of the surplus which otherwise would appear in an industry using a high "proportion" of labor to the means of production.

19. Price-Ratios of Product ot Means of Production

  • It does not follow that the price of the product of an industry having a low proportion of labor to means of production (and hence a "potential deficit") would necessarily rise, with a wage-reduction, relative to its own means of production.

    "On the contrary," Sraffa writes, "it might possibly fall." The reason for this seeming contradiction: the means of production for an industry are themselves the product of one or more industries which (in turn) may employ a still lower proportion of labor to the means of production (and the same may be said for these commodities' means of production, etc.)

    In this case, the price of the product — although produced by a "deficit" industry — might fall in terms of its means of production. Its deficiency would have to be made good through a particularly steep rise relative to labor.

  • Result: as wages fall, the price of the product for a low-proportion ("deficit") industry may rise or fall, or even alternate in rising and falling, relative to its means of production...while the price of the product of a high-proportion ("surplus") industry may fall or rise, or alternate. What neither can do, as we will see in §§21--22, is remain stable in price relative to its means of production throughout any range (long or short) of the wage-variation.

20. Price-Ratios between Products

  • These considerations dominate the price-relation of a product to its means of production and equally to its relations to any other product.
  • It's the "proportions" of labor to means of production which determines the relative "price" between commodities. NB: this is iterative, so those means of production used up are subject to the same method determining its "relative price".
  • The net result and justification for price-variations from a change in distribution remains a simple one: redressing the balance in each industry.

21. A Recurrent Proportion

  • We can now revert to the "critical proportion" (mentioned in §17) as the border between "deficit" industries and "surplus" ones.
  • Assumption. Suppose we had an industry sector with that "critical proportion" of means of production to labor, and moreover each sector (producing each commodity used as a means of production) are themselves in this "critical proportion" state...and all the sectors involved in producing the means of production used in the production of the means of production are in that critical state, and so on.
  • The commodity produced in such a sector would have its value not be affected when wages rose or fell. This can only happen from a potential deficit or surplus...but we assumed the industry was "in balance"!
    • NB: A commodity of this sort would not change its value relative to other commodities.
  • Two separate conditions have been assumed to attain this result:
    1. The "balancing" proportion is used", and
    2. one and the same proportion recurs in all successive layers of the industry's aggregate means of production without limit.
  • Note the second condition implies the first. This is the subject of the next section...

22. Balancing Ratio and Maximum Rate of Profits

  • It will be convenient to replace the "proportion" (quantity of labor to means of production) with one of the corresponding "pure" ratios between homogeneous quantities.

    There are two such ratios:

    1. the quantity-ratio of direct to indirect labor employed; and
    2. the value-ratio of net product to means of production.

    These two ratios coincide when the value-ratio is calculated at the values for w = 1.

    Sraffa uses the latter ratio here.

  • The rate of profits is uniform in all industries (and depends only on the wage), the value-ratio of the net product to the means of production is in general different for each industry and mainly depends on its particular circumstances of production.
  • Exception: When we make the wage zero (i.e., w = 0) and the whole net product goes to profits, in each industry the value-ratio of the net product to means of production necessarily comes to coincide with the general rate of profits r. At this level the "value ratios" of all industries are equal, regardless of how different the "value ratios" may have been at other wage-levels.
  • The only "value-ratio" which can be invariant to changes in wage (and thus capable of being "recurrent" in the sense defined in §21) is the one that is equal to the rate of profits corresponding with zero wage. And that is the "balancing" ratio.
  • Definition. The Maximum Rate of Profits is the rate of profits as it would be if the whole national income went to profits, and we denote it by R.

Monday, February 18, 2013

Notes on Sraffa's Production, Chapter 2

Ch. 2. Production with a Surplus

4. The rate of Profits

  • If the economy produces more than the bare minimum necessary for replacement, i.e., we have some surplus to be distributed, then our model becomes self-contradictory. Why? Because the "Gross National Product" (the right hand side) will contain the sum of the columns on the left hand side plus bonus parts. We cannot use the basic linear algebra one might naively use.
  • We allot the surplus simultaneously as when the prices are determined.
    • We cannot allot the surplus before the prices are determined. The surplus must be distributed in proportion to the means of production advanced in each industry. Such a proportion between two aggregates of heterogeneous goods ("the rate of profits") cannot be determined before we know the prices of goods.
    • OTOH, we cannot defer the allotment of the surplus till after the prices are known, since the prices cannot be determined before knowing the rate of profits.
    • The distribution of the surplus must be determined through the same mechanism and at the same time as the prices of commodities.
  • We add the rate of profits ("which must be uniform for all industries") as an unknown, r, and the system becomes
    (Aapa + Bapb + ... + Kapk)(1 + r) = Apa
    (Abpa + Bbpb + ... + Kbpk)(1 + r) = Bpb
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
    (Akpa + Bkpb + ... + Kkpk)(1 + r) = Kpk

    where, since we have a self-replacing state, we need Aa + Ab + ... + AkA, Ba + Bb + ... + BkB, and so on. In other words: the quantity produced of each commodity is at least equal to the quantity of it used in other sectors' production together.

  • The system has k independent equations, which determines the k - 1 unknowns plus the rate of profit.

5. Example of Rate of Profits

  • Lets revise our example from §1 to have a surplus:
    280 qr wheat + 12 t iron → 575 qr. wheat
    120 qr wheat + 8 t iron → 20 t. iron
    The answer is 15 qr wheat = 1 t. iron will restore the initial condition, and the rate of profit is r = 25%
    • The trick is to rewrite this as producing one unit output in each sector:
      (56/115) qr wheat + 3/5 t iron → 1 qr. wheat
      (24/115) qr wheat + 2/5 t iron → 1 t. iron
      Then we let pw be the price of 1 qr wheat, pi be the price of 1 t iron, r the rate of profits.
    • We have an eigen-problem of the form: Ax = λx where λ = 1/(1+r), A is the matrix we deduced, and x is the vector (pw, pi).
    • Note that the matrix has eigenvalues λ = 4/5, 2/23.
    • The rate of profit r = λ-1 - 1 could be either 1/4 or 23/2. Since the rate of profit must satisfy 0 ≤ r ≤ 1, we see r = 1/4 is the solution.

      Correction: my reasoning for r = 1/4 is incorrect here: it's because the exchange corresponding to r = 23/2 gives us a negative quantity of iron may be exchanged for a positive quantity of wheat, which is absurd. Hence we throw it away...and that is the correct reasoning here.
    • Now that we have our solution for r, we plug it into one of the sectors at random and solve for the prices. Since 1 + r = 5/4, we see:
      (280 pw + 12 pi)(5/4) = 575 pw
      implies 15 pi = 225 pw, or equivalently 1 t iron may be traded for 15 qr wheat.

6. Basic and Non-Basic Products

  • Definition. Notice before, without surplus, all commodities produced must be used in the production of other commodities. But now, with surplus, we may have commodities which are not needed in the production process. These commodities are called luxury goods.
    • You should really convince yourself this must be the case, since we affirmed before the sum of the columns for the production matrix must be equal to the output. That is to say: the total inputs must be equal to the total outputs. Otherwise, by definition, there is surplus.
  • Notice also the luxury goods do not affect the rate of profits.
  • If the production of a given luxury good doubled with constant inputs, the price per unit of given luxury good would halve. The price relations of all other goods would remain the same, however.
  • The price of a luxury good is not an unknown we are trying to solve for. But the prices of non-luxury goods are unknowns which we need to determine. With the non-luxury good prices determined, we may deduce the luxury goods prices.
  • Definition. The criteria is: does a commodity enter (either directly or indirectly) the production of all commodities. Those that do, we shall call basic and those which do not are non-basic commodities.

7. Terminological Note

  • Why do we call the ratios satisfying conditions of production "values" or "prices" rather than "costs of production"?
  • The latter would be adequate so far as non-basic products were concerned, since their exchange ratio is merely a reflection of what must be paid for their means of production, labour, and profits in order to produce—there is no mutual dependence.
  • Basic products have another dimension
    • Its exchange-ratio depends on its use in the production of other basic commodities, as much as on the extent to which those commodities enter its own production.
    • One might be tempted to say "it depends as much on the Demand side as on the Supply side", but one would be wrong
  • The price of non-basic products depends on the prices of its means of production, but these (the prices of its means of production) do not depend on it
  • A basic product has the prices of its means of production depend on its own price no less than the latter depends on them
  • Sraffa argues a "less one-sided description than cost of production seems therefore required".
    • Classical terms meeting the case include:
      1. "necessary price" (e.g., the physiocrats;
        Thomas Hodgskin's Popular Political Economy, IX.1 "...the natural and necessary price of money being determined...by the quantity of labour required to produce it"),
      2. "natural price" (e.g., Adam Smith's Inquiry Ch. 4, Ch. 7 "The natural price...is...the prices of all commodities are continually gravitating";
        David Ricardo's Principles Ch. 5;
        J.S. Mill's Principles III.4 "value...proportional to its cost of production, [is] its Natural Value (or its Natural Price)";
        NB: John Locke appears to be the first(?) English economist to use the terms "natural price" and "market price" in Some Considerations), or
      3. "price of production" (e.g., Marx?)
    • But value and price have been preferred, because (a) it's shorter; and (b) in the present context — which has no reference to "market prices" — it's no more ambiguous.
  • In general Sraffa avoids the term "cost of production", as well as the term "capital" (in its quantitative connotation), even at the expense of tiresome circumlocution
    • These terms have become inseparably linked with the supposition they stand for quantities which can be measured independently of — and prior to — the determination of the prices of the products.
    • Consider the "real costs" of Marshall ("But now we have to take account of the fact that the production of a commodity generally requires many different kinds of labour and the use of capital in many forms. The exertions of all the different kinds of labour that are directly or indirectly involved in making it; together with the abstinences or rather the waitings required for saving the capital used in making it: all these efforts and sacrifices together will be called the real cost of production of the commodity." Alfred Marshall's Principles of Economics, Book 5, Chapter 3) and the "quantity of capital" which is implied in the marginal productivity theory.
    • Sraffa avoids suppositions which such terms connote, since he's trying to critique the marginalist paradigm.

8. Subsistence-Wage and Surplus-Wage

  • We have regarded wages as consisting of the necessary subsistence of the workers, and enters the system on equal footing as fuel for engines or feed for cattle.
  • Sraffa takes into account the "other aspect of wages" since, besides the ever-present element of subsistence, they may include a share of the "surplus product".
  • We separate the wages into two components: one is the subsistence, which we keep as inputs on equal footing as fuel or feed; the other is the "division of the surplus", which we should as variable.
    • Working with tradition, we will refrain from parting with tradition, and shall follow the usual practice treating the whole wage as variable.
  • Drawback: This approach relegates the necessaries of consumption to the "limbo" of non-basic products.
    • This is because the necessaries of consumption no longer appear alongside the other means of production, i.e., they don't appear on the left hand side of the equations.
    • An improvement in the methods of production for necessaries of life will no longer directly affect rates of profits and the prices of other products.
    • Necessaries are essentially basic, and if they are prevented from exerting influence on prices and profits under that label, they do so in devious ways (Sraffa suggests, e.g., "by setting a limit below which the wage cannot fall", a limit which would itself fall with any improvement in the methods of production for necessaries, "carrying with it a rise in the rates of profits and a change in the prices of other products".)
  • The discussion Sraffa entertains can "easily be adapted to the more appropriate, if unconventional, interpretation of the wage suggested above".

9. Wages paid out of the product

  • We shall hereafter assume the wage is paid post factum as a share of the annual product. Thus we abandon the classical economists' idea of a wage "advanced" from capital.
  • We retain the supposition of an annual cycle of production with an annual market.

10. Quantity and Quality of Labor

  • The quantity of labor employed in each industry should now be represented explicitly, taking the place of the corresponding quantities of subsistence.
  • We suppose labor to be uniform in quality or (what amounts to the same thing) we assume any difference in quality to have been previously reduced to equivalent differences in quantity, so each unit of labor receives the same wage.
  • We call La, Lb, ..., Lk the annual quantities of labor respectively employed in the industries producing a, b, ..., k and we define them as fractions of the total annual labor of society, which we take as unity. So:
    La + Lb + ... + Lk = 1
    • Remark. It seems labor is treated differently than other commodities. For example, it doesn't have its own "sector" (equation). And it's already normalized!
  • We call w the wage per unit of labor, which like the prices will be expressed in terms of the chosen standard
    • (See also, on the choice of a standard, in §12)

11. Equations of Production

  • So, with these additional assumptions, the equations take the form:
    (Aapa + Bapb + ... + Kapk)(1 + r) + Law = Apa
    (Abpa + Bbpb + ... + Kbpk)(1 + r) + Lbw = Bpb
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
    (Akpa + Bkpb + ... + Kkpk)(1 + r) + Lkw = Kpk
  • We assume, as before, the system is in a self-replacing state, so Aa + Ab + ... + Ak ≤ A, Ba + Bb + ... + BkB, etc.

12. National Income in a Self-Replacing System

  • Definition. The National Income of a system in a self-replacing state consists of the set of commodities which are "left over" after the articles replacing the means of production are used up.
    • In other words, if we denote ΔA = A - (Aa + Ab + ... + Ak) and so on for all other industries, we have A)pa + (ΔB)pb + ... + (ΔK)pk be the national income.
  • The value of this set of commodities, or "composite commodities" as it may be called, which forms the national income...we set to 1.
  • Thus the national income becomes the standard in terms of which the wage and k prices are expressed (taking the place of the arbitrarily chosen single commodity in terms of which the k - 1 prices, besides the wage, were expressed).
  • We have the additional equation:
    A)pa + (ΔB)pb + ... + (ΔK)pk = 1.
  • It is impossible for the aggregate quantity of any commodity represented in this expression to be negative, otherwise we contradict the assumption the economy is in a self-replacing state!
  • This gives k + 1 equations as compared to k + 2 variables (the k prices, the wage w, and the rate of profits r).
  • The result of adding the wage as one of the variables is that the number of these now exceeds the number of equations by one. The system has "one degree of freedom". If one of the variables is fixed, the others will be too.

Tuesday, February 12, 2013

Notes on Sraffa's Production, Chapter 1

Sraffa wrote his Production of Commodities in a very "mathematician's style"...i.e., it was a grocery list of propositions chunked together. Consequently, I analyze each "chunk" in my series of notes. The section titles are Sraffa's.

Ch. 1. Production for Subsistence

1. Two Commodities

  • We consider an economy with two commodities, producing just enough to maintain itself.
  • Commodities are produced by separate industries and are exchanged at a market held after the harvest.
  • Wheat and iron are produced, representing sustenance for the workers and the means of production (respectively).
  • We suppose that 280 quarters of wheat and 12 tons of iron are used to produce 400 quarters of wheat, whereas 120 quarters of wheat and 8 tons of iron are used to produce 20 tons of iron. We represent a year's operations as:
    280 qr. wheat + 12 t. iron → 400 qr. wheat
    120 qr. wheat + 8 t. iron → 20 t. iron.
    • Note that when we add up the columns, it's equal to the output. This is what we mean when we say "There is no surplus."
    • These relations Sraffa calls "the methods of production and productive consumption" or the methods of production.

2. Three or More

  • The same applies to any number of commodities. For example, if we add a third sector (pigs):
    240 qr. wheat + 12 t. iron + 18 pigs → 450 qr. wheat
    90 qr. wheat + 6 t. iron + 12 pigs → 21 t. iron
    120 qr. wheat + 3 t. iron + 30 pigs → 60 pigs
  • The exchange values which ensure replacement all around are 10 qr wheat = 1 t iron = 2 pigs.
  • The general procedure for calculating this is to normalize the output, so we consider
    (8/15) qr. wheat + (4/7) t. iron + (3/10) pigs → 1 qr. wheat
    (3/15) qr. wheat + (2/7) t. iron + (2/10) pigs → 1 t. iron
    (4/15) qr. wheat + (1/7) t. iron + (5/10) pigs → 1 pig
  • Then we just use linear algebra to solve this as usual.

3. General Case

  • We have commodities of different "species" (e.g., iron, wheat, etc.). Lower case variables will keep track of each species a, b, c, ..., k. NB: the term "species" is mine, not Sraffa's.
    • Caution: Sraffa abuses notation and mixes up A and a when referring to the species.
    • Abuse of Notation: We will write k to indicate the number of sectors, as well as its proper meaning. It will depend on the context, but it should be clear when used.
  • We denote A the quantity annually produced of a, B the quantity annually produced of b, etc.
  • We will call Aa, Ba, ..., Ka the quantities of a, b, ..., k annually used in the industry producing a, and so on.
  • The prices are the unknowns pa, pb, ..., pk, which represent the values of 1 unit of the commodities a, b, ..., k (respectively) necessary to restore the initial position.
  • We can now cast the conditions of production as:
    Aapa + Bapb + ... + Kapk = Apa
    Abpa + Bbpb + ... + Kbpk = Bpb
    . . . . . . . . . . . . . . . . . . . . . . . . . .
    Akpa + Bkpb + ... + Kkpk = Kpk
  • Note that we assume the system to be in a self-replacing state, so
    Aa + Ab + ... + Ak = A,
    Ba + Bb + ... + Bk = B,
    and so on. That is to say, the sum of the first column equals the output of the first row, etc.
    • Again, the approach one should take is to work with modified quantities A'a = Aa/A and so on, working with primed quantities as inputs and output quantities (NOT PRICES) set to 1.
  • It is unnecessary to assume every commodity enters directly into the production of every other. We could have pigs enter into production of wheat but not iron, and no iron enter into the production of pigs.
  • One commodity is taken as the standard measure of value, and its price is set to 1. This leaves k - 1 unknowns. It doesn't really matter which commodity we pick, since the relative exchange-value doesn't change. This leaves k - 1 independent linear equations which uniquely determine the k - 1 prices. (This is similar to Marx's "general form of value" discussed in Das Kapital, Vol. 1, Ch. 1.)

Monday, February 4, 2013

History of Economic Thought Website

The New School of Economics used to have a great website on the history of economic thought, which has been down for a while.

For my own reference, it's available on the internet archive.

Now...off to work!

: The entire website was saved as a pdf available online.

Someone may want to try to reconstruct the website based off of this pdf...