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Why should we care about Sonnenschein-Mantel-Debreu?

from Lars Syll

Along with the Arrow-Debreu existence theorem and some results on regular economies, SMD (Sonnenschein-Mantel-Debreu) theory fills in many of the gaps we might have in our understanding of general equilibrium theory …

It is also a deeply negative result. SMD theory means that assumptions guaranteeing good behavior at the microeconomic level do not carry over to the aggregate level or to qualitative features of the equilibrium. It has been difficult to make progress on the elaborations of general equilibrium theory that were put forth in Arrow and Hahn 1971 …

24958274Given how sweeping the changes wrought by SMD theory seem to be, it is understand-able that some very broad statements about the character of general equilibrium theory were made. Fifteen years after General Competitive Analysis, Arrow (1986) stated that the hypothesis of rationality had few implications at the aggregate level. Kirman (1989) held that general equilibrium theory could not generate falsifiable propositions, given that almost any set of data seemed consistent with the theory. These views are widely shared. Bliss (1993, 227) wrote that the “near emptiness of general equilibrium theory is a theorem of the theory.” Mas-Colell, Michael Whinston, and Jerry Green (1995) titled a section of their graduate microeconomics textbook “Anything Goes: The Sonnenschein-Mantel-Debreu Theorem.” There was a realization of a similar gap in the foundations of empirical economics. General equilibrium theory “poses some arduous challenges” as a “paradigm for organizing and synthesizing economic data” so that “a widely accepted empirical counterpart to general equilibrium theory remains to be developed” (Hansen and Heckman 1996). This seems to be the now-accepted view thirty years after the advent of SMD theory …

S. Abu Turab Rizvi

And so what? Why should we care about Sonnenschein-Mantel-Debreu?

Because  Sonnenschein-Mantel-Debreu ultimately explains why New Classical, Real Business Cycles, Dynamic Stochastic General Equilibrium (DSGE) and “New Keynesian” microfounded macromodels are such bad substitutes for real macroeconomic analysis!

These models try to describe and analyze complex and heterogeneous real economies with a single rational-expectations-robot-imitation-representative-agent. That is, with something that has absolutely nothing to do with reality. And — worse still — something that is not even amenable to the kind of general equilibrium analysis that they are thought to give a foundation for, since Hugo Sonnenschein (1972) , Rolf Mantel (1976) and Gerard Debreu (1974) unequivocally showed that there did not exist any condition by which assumptions on individuals would guarantee either stability or uniqueness of the equlibrium solution.

Opting for cloned representative agents that are all identical is of course not a real solutionto the fallacy of composition that the Sonnenschein-Mantel-Debreu theorem points to. Representative agent models are — as I have argued at length here — rather an evasionwhereby issues of distribution, coordination, heterogeneity — everything that really defines macroeconomics — are swept under the rug.

Of course, most macroeconomists know that to use a representative agent is a flagrantly illegitimate method of ignoring real aggregation issues. They keep on with their business, nevertheless, just because it significantly simplifies what they are doing. It reminds — not so little — of the drunkard who has lost his keys in some dark place and deliberately chooses to look for them under a neighbouring street light just because it is easier to see there …

  1. August 4, 2017 at 1:10 am

    Why should we care? Indeed…

    Demonstrating how a complex aggregation of variables might interact with each other (GDP) if a specific fantasized ‘event’—e.g., a tax increase—were to occur is an interesting exercise in abstract mathematical speculation, one which can be rather fun to play with if you are facile with advanced mathematics.

    But most DSGE models are rendered worthless by demand functions corrupted by flawed assumptions re: the impact on market demand of a change in prices or disposable incomes. As usual, fallacies of composition are to blame.

    For example when these models are used to predict what will happen to the economic aggregates (GDP) if taxes are increased, they use demand functions which ignore the fact that the real purchasing power of economic agents depends ultimately upon whether or not EVERYONE’S disposable income changes.

    The impact—on actual purchasing power—of a change in the disposable income of a single household, all else equal, is profoundly different from the impact (on purchasing power) of a change in the disposable income of ALL households.

    An extended elaboration on this point can be found here:


    From my POV, there is no better way to attack the perceived legitimacy of neoclassical DSGE models than to expose the fallacies of composition that are willfully built into their demand functions.

    • August 4, 2017 at 2:21 pm


      It is not that simple. SMD theorem invalidate not only DSGE model but also almost all price theories. Are you ready to deny all concepts that uses demand function? The true meaning of SMD theorem is this.

      Do you believe that economics does not need price theory? If you do not, you should consider of making a very new theory.

      • August 4, 2017 at 4:12 pm


        Looking up SMD I found “the excess-demand function may have more than one root – more than one price vector at which it is zero (the standard definition of equilibrium in this context”.

        So a price is a number – a one-dimensional vector – but in two or more dimensions derivatives are complex and have two roots. Price thus corresponds to the radius (varying with time) of a circle whose continuous circumference may or may not represent an infinity of directions. As I’ve said recently of growth, a Price has to be a price OF something, but in this case it is also a price IN something, i.e. money, whose relative value can be externally manipulated. In short, prices are arbitrary.

        To get such a heresy published it may have helped to dress it up in complicated mathematics, but if prices are arbitrary, it is desirable to establish them by some workable convention, this taking the place of your “new theory”. My convention has socially affordable credit limits rather than money, costed replacement of the used rather than prediction of usage, and penalties on credit use creating excess demand. To minimise distribution costs it works best with ideas globalised, mass-production regionalised and computerised/artisan production localised.

      • August 4, 2017 at 6:09 pm

        “Do you believe that economics does not need price theory?”

        Let me put it this way: I believe MACRO economics does not need any price theory which over-emphasizes the imagined importance of a representative agent’s decision-making process. Individuals base their purchasing decisions on certain “guesses” they make: their perceptions of their current and anticipated future economic circumstances.

        Those guesses are quite often flawed, sometimes based on wishful thinking, sometimes based on irrational fears (e.g., of inflation), sometimes based on their simple ignorance of the very predictable consequences of EVERYBODY experiencing the same gain/loss of disposable dollars.

        If I wanted to attempt to accurately model these consumer choice guesses for use in a general equilibrium theory, I’d consider it necessary to model at least one set of assumptions wherein all those affected by some exogenous change are fully knowledgeable re: the predictable impact it will have on their purchasing power.

        Under such assumptions, for example, rich people would not panic re: a big increase in their marginal income tax rates for they’d know that it would not diminish their purchasing power in the marketplace in any way, in spite of the fact that their disposable incomes had been reduced substantially.

        Price theory on the level of the Marshallian Cross is very useful, I believe, but attempts to draw macroeconomic conclusions from profoundly inadequate characterizations of the “demand functions” of individual agents is generally a waste of everyone’s time, IMO.

      • Craig
        August 5, 2017 at 7:37 pm

        You’re right we need an entire new theory and understanding of how to address pricing.

        The economy being utterly embedded in the temporal universe must reflect its process of starting, changing and stopping if it is to attain and maintain a continuous flow.

        DSGE is flawed in that it assumes general equilibrium when the reality of modern economies’ most basic problem is a scarcity ratio of total individual incomes to total costs/prices. Merely equating that ratio does not resolve the problem because it only statistically/momentarily creates an equilibrium and so violates the process/free flowingness nature of the temporal universe.

        What is required is to take adult, responsible and ethical control of the pricing system so that the scarcity ratio above….is inverted….so that the continuous reality is what I call “the higher disequilibrium” of total individual incomes exceeding total costs/prices. This will enable the economy to attain and maintain free flowingness and hence align with the nature of the temporal universe.

  2. August 5, 2017 at 11:22 am

    Dear Dave Taylor,

    your convention seems to be a set of institutional restrictions (or policies) which are preferable to be implemented and enforced in a way or other. I do not argue if your concrete convention is workable and effective. You have a good chance that you have a good policy idea. However, I do not think that a set of policies can substitute a theory.

    A theory (in this case the core part of economics) is a tool to understand economic mechanisms. Do you think that you can get a good policy without knowing how the economy works? It is possible that you get one by pure chance. But you will not be so lucky that you can find a series of good workable policies without having a good understanding.

    • August 6, 2017 at 9:26 am

      Yoshinori, my theory is that price determination is two-dimensional, so the mechanism is not like a tug of war between supply and demand, it is more like Locke’s balance of forces between legislature, executive and judiciary, these between them not determining but setting bounds to what is permitted in the system. My theory, further, is that people within the system are not physically constrained by the boundary, but are informed of it by law and custom, constrain themselves by internalising knowledge of it and choosing to respect it.

  3. August 5, 2017 at 2:11 pm

    Dear James Kroeger,

    I have read your article:
    Why Rich People Should Insist On A Full-Employment Economy
    New Analysis Challenges Key Institutional Assumptions

    You are trying to persuade rich people. That is good. It may be necessary things to do. I have no intention to disturb your efforts. But there are many other things to do. Theory is one of them. As Keynes put it, we are often victims of an economic theory. That is the reason why we should always be watchful for the state of economics. If the economics is bad, it is necessary to replace it. This is no easy work. It is just like making a Copernican revolution in astrology.

    The present mainstream economics became a highly “refined” theory but stands on a false conception of the world. The mainstream economics is comparable to Ptolemaic system. It was extremely complex and boasted high accuracy of prediction. Before the arrival of Kepler, Copernican system was less accurate than modified Ptolemaic system. But the latter was fundamentally wrong. A similar state is now happening in economics. This is why we need a new economics, a real-world economics.

    Sonnenschein-Mantel-Debreu theorem is of great significance because it made clear that general equilibrium theory cannot be a good microfoundation of a macroeconomics. Simply speaking, SMD theorem showed that (1) uniqueness of equilibrium states and (2) stability or local convergence to equilibrium are not generally assured. These are two of many requirements that a theory could be a good microfoundation. Thus, DSGE and almost all other macroeconomics which rely on GE framework lost their foundations. This means that those macroeconomics are simply castles in the air.

    However, this is only one Implication of SMD theorem. General equilibrium model like Arrow-Debreu competitive equilibrium was proposed in order to avoid theoretical inconsistency which lied in the Marshallian framework. Marshallian cross was a part of it. Please note that SMD theorem tells that aggregate excess demand function can take any form provided that it is continuous and satisfies Walrass law. Marshallina cross ordinarily assumes that demand curve is decreasing but it can take all other possible shapes. Stability and uniqueness are also violated in general.

    SMD theorem overthrew not only Walrassian-type general equilibrium theory but also Marshallian-type partial equilibrium theory.

    You may not be able to imagine a price theory that does not contain Marshallian cross in its core. It exists. See my paper:

    The Revival of Classical Theory of Values

    This is now published as Chapter 8, pp.151-172 in Nobuharu Yokokawa, Kiichiro Yagi, Hiroyasu Uemura and Richard Westra (Eds.) The Rejuvenation of Political Economy, May 2016, Oxon and New York: Routledge.

    • August 6, 2017 at 8:56 pm

      I had some time to read over your paper, Yoshinori, and wanted to give you this quick response.

      Under perfect competition, perfect knowledge, and constant [money-expressed] Demand assumptions, I fully agree with you that sellers’ cost-of-production becomes the ultimate determinant of market prices, since sellers are forced in such an environment to sell at only a price that will generate a “normal” profit.

      But since these assumptions almost never exist in the Real World, I must tell you that I have a difficult time simply assuming the validity of any mathematical conclusions upon which they are based.

      Yoshinori: “Do you believe that economics does not need price theory? If you do not, you should consider…making a very new theory.”

      Let me then share with you a quite different analytical perspective re: the topic of price determination, one that I believe far more accurately reflects what we observe in the Real World.

      My price theory is based almost entirely on the assumption that sellers always charge the highest price that the market will bear, period.

      Yes, in the short term, they may fall short of realizing that “highest price”, but they will continually adjust the prices they “set” until they arrive at the highest price at which they are able to sell all of the product they have to offer.

      What this means, of course, is that seller costs actually have nothing, or little, to do with market prices. If a seller brings product to market and finds that she can only get a price that is less than her unit costs, then that is the price she is going to sell at. Many marginal firms do precisely that.

      But if she finds that demand is such that she can sell her products at a high multiple of her unit costs, then that is the price at which she will sell.

      So in this sense, all sellers are ultimately “price-takers” in that they will only be able to maintain a price that the demand side will accept (has the means + desire to accept).

      At the same time, they are also price-makers in that they have complete freedom to set prices as high or low as they want…but that freedom means only that they will have the opportunity to continually adjust their prices until they find that highest price at which the public is willing to buy all that they have to sell.

      If this fundamental assumption is recognized as being largely true, then we are led by logic to the conclusion that prices are therefore ultimately determined by Demand, or rather, the distributions of disposable dollars which constitute money-expressed Demand.

      The seller simply finds out what the demand side is going to allow her get away with.

      Cost is certainly an important variable in determining changes in prices over time (assuming competition), but I don’t think it is a good idea to elevate its level of importance to that of Ultimate Determinant of prices.

      I’ll confess to being curious as to how you might be able to illustrate some of these considerations in your mathematical model, Yoshinori.

      I might add that, while I appreciate the fact that you took the time to read the beginning and end of my article, I think you would have found some very useful analysis related to your project if you had spent a bit more time on the sections, “How The Marketplace Determines Prices” and “Something You Never Knew About Inflation.”

    • August 7, 2017 at 1:39 am

      James Kroeger,

      I have to thank you for reading my paper. Have you read the paper titled The Revival of Classical Theory of Values? In this paper or others, I use “prefect competition,” “perfect knowledge” only in critical context and never use them as my own concepts. I am even critical to the terminology like “imperfect competition,” because this assumes perfect competition as state of reference.

      In my Revival paper, I used the word “perfect,” “imperfect,” “perfectly”, and “imperfectly” eight times in total. In no place I used these notions as my own concepts.

      The word “perfect” first appears in page 161 (in published edition, just before heading “How are markup rates determined?”). I have written there:

      >>Britain was in a state of nearly perfect competition and prices fluctuated freely and frequently until the turn of the century when imperfect competition intruded in the economy. I doubt that this image is accurate. It is not conceivable that all prices changed every week or so.

      The second appearance is in the same page (just after the above same heading):

      >>The Oxford Economists’ Research Group investigation revealed observations that presented an “inconvenient fact” to economics, as the full-cost principle ran counter to both perfect and imperfect competition theories. Perfect competition theory assumed that all firms are price-takers; the Research Group investigation revealed that most firms are price-setters.

      If you understand that I am supporting OERG investigation, this part implies that I am opposing both “perfect” and “imperfect” competition theories.

      In page 164 (under Price setting and its twinned behavior of firms), I explain how the notion of perfect competition is absurd. After I have given a textbook-level explanation of the perfect and imperfect competition, I made this remark:

      >>Prices can be determined in this way. All right! However, the story does not end here. What happens in the market, and how do firms behave if the prediction included in the demand function is not correct? This is the real problem we should examine.

      I wonder why you have misunderstood me so completely.

      • August 7, 2017 at 2:14 am

        Actually, in my statement Yoshinori, I was not saying that you placed such conditions on your model, but only that >>I<< am able to logically agree with its foundational assumption (that seller costs are the primary determinant of prices in the macro economy) ONLY if I first embrace the condition of perfect competition.

        I'm sorry I did not make that more clear in what I wrote.

        Only in a perfect competition environment can I see clearly that prices in every market will be driven down by competition to where the prices they charge are equal to costs they incur (including normal profit). Without that assumption, I'm afraid I need more convincing…

    • August 7, 2017 at 9:36 am

      It’s OK if you apologize in that way. We have much more important a question.

      Is it a good custom to think market situation in terms of perfect or imperfect competition?

      Perfect competitiveness is a sheer product of imagination. It cannot even have a status of IDEAL STATE in describing market transactions. We have to refute such concepts. So let us stop thinking in terms of these concepts.

      You wrote:

      [your] price theory is based almost entirely on the assumption that sellers always charge the highest price that the market will bear.

      As you continue

      [sellers] may fall short of realizing that “highest price”, but they will continually adjust the prices they “set” …,

      there is no great difference between what I call full-cost principle. The only difference lies in your assertion that

      they arrive at the highest price at which they are able to sell all of the product they have to offer.

      When you say “all of the product they have to offer,” are you supposing that producers have already fixed their volume of production? This kind of situation may happen in agriculture for example but in a modern industrial production firms produce continuously and can quickly adjust their speed of production with rather cheap cost. This is the typical producers in modern industry. Those firms set their prices and produce as much as their product sells.

      To sell all the fixed amount of product is rather a rare and, by consequence, exceptional case. In a modern economy, prices may remain constant while quantities change every day or week. In this sense, the prices and quantities are primarily independent with each other.

      Another important fact is that the profit of a firm is not uniquely determined by the product price. If the firm is producing at constant cost, the profit is proportional to the sales volume. (If it incurs fixed cost, you have to reduce this amount.) It may change by the change of business cycle phase, economic growth, seasonal fluctuations, and others.

      You referred to highest price. But, what firms pursue is not the highest price but higher profits. Your price theory lacks these details.

      • August 8, 2017 at 8:31 am

        Re: the aggregate supply-function…

        “In a modern economy, prices may remain constant while quantities change every day or week.”

        Ultimately, any time a seller “sets a price”, it is nothing more than a guess. That guess will be amended over a subsequent period depending on the response the seller gets from the demand side of the market. Firms usually allow some time before making a responsive change in the price, but >change they willgeneralization< which I believe describes actual market outcomes more accurately than models which assume that prices will be largely determined by the seller’s costs.

        The entire purpose of economic theorizing, IMO, is to identify key economic “variables”, try to isolate their impact on other economic variables holding all other variables constant, in order to identify essential relationships between these key variables that policy-makers can make good use of.

        Remember that the “Law of Demand” does not hold in any and all markets at all times. Still, it is a generalization which most economic observers find very useful. It is “largely true” most of the time, enough so that we are able to make predictions with a high degree of accuracy based on them, even though the foundational assumption does not hold up against the most severe scrutiny.

        It is in that sense that I advocate a “supply-function” which relies heavily on the simplifying assumption that sellers always charge the highest price they believe the market will bear. That IS their intention and I believe they mostly succeed most of the time and that predicted outcomes based on that assumption are more often fulfilled than not.

        Any mathematical model which incorporates this basic supply-function assumption will, I believe, produce more accurate predictions than those which do not.

      • August 9, 2017 at 12:26 am

        “but >change they willgeneralization< which"

        Is there any derangement around here?

  4. August 7, 2017 at 9:56 am

    The most significant implication of SMD theorem is that demand function (aggregated for each product) has no real meaning. To assume some properties on demand function is to introduce some arbitrary assumptions and this implies that general equilibrium is not a general theory as it was often supposed.

    The concept of demand function appeared in England in 1860’s. Most of economic theories afterwards were based on demand function. This is the most simple reason why we should return before Jevons and Marshall.

    • August 8, 2017 at 8:44 am

      Re: the aggregate demand-function:

      “To assume some properties on demand function is to introduce some arbitrary assumptions and this implies that general equilibrium is not a general theory as it was often supposed.”

      I have long had a problem with the aggregate demand functions used in DSGE models because I believe they are fundamentally corrupted by various flawed assumptions re: the composition of aggregate demand.

      Simply summing up the “demand-functions” of all economic agents to supposedly capture an aggregation of all the micro economic elements of demand is a modeling practice which I believe serves no useful purpose to interested observers or to policy makers.

      Why not simply say that all of an economy’s participants will normally seek to obtain more of the scarcest goods/services/experiences that are brought to market, to the extent that their disposable dollar accumulations allow them to do so?

      Why doesn’t that fully describe the ‘demand-function’ of every participant in the economy, both individually and collectively?

      What is generally missing from the presumed demand functions of representative agents is the many misperceptions they are prone to due to money illusion and various fallacies of composition they are frequently guilty of.

      These misperceptions can have a big impact on buying and saving decisions but are not represented in most models.

      Ultimately, the only somewhat justified reason for constructing general equilibrium models is the desire to draw some useful conclusions re: the phenomenon of price inflation. Unfortunately, the models taught in Ph.D. programs fail miserably at this task.

      At the heart of the problem is the profoundly flawed notion of The Price Level that supposedly applies to any economy at any given point in time. As an “economy-wide phenomenon” it is nothing more than a theoretical fantasy which is utterly discredited by the observable fact that one income group can experience robust levels of price inflation at the same time that another income group is experiencing disinflation, or even deflation. (See my article)

      This very important reality is completely hidden from analysts when they settle for an index number which represents an “averaging” of all the price changes that occur in an economy. Both within the Real Economy, and on a conceptual/theoretical level, there is no “Price Level” [that accurately represents what is happening to prices across the whole of the economy].

      If DSGE models cannot be revised to illustrate the significance of these variations in inflation levels experienced by different income groups, they may as well be cast into the fire, for they offer nothing of value.

  5. August 9, 2017 at 1:10 am


    “but >change they willgeneralization< which"
    Is there any derangement around here?

    Yes, an unfortunate error…

    It should read something like this:

    Ultimately, any time a seller “sets a price”, it is nothing more than a guess. That guess will be amended over a subsequent period depending on the response the seller gets from the demand side of the market. Firms usually allow some time before making a responsive change in the price, but change, they will, if they believe they can raise prices further without it costing them anything in terms of sales or in a competitive context.

    Note that when I state that sellers always sell at the highest price they think the market will bear, I am making a generalization, which I believe describes actual market outcomes more accurately than models which assume that prices will be largely determined by the seller’s costs.

    • August 9, 2017 at 10:15 am

      James Kroeger

      You are philosopher. Let us discuss with Cartesian spirit. We must be clear and distinct in our arguments.

      It seems we have two different questions:

      A. The question concerning the rules of price determination.

      B. The question which concerns whether we should refute the concepts such as supply function and demand function.

      Let us argue first question A. Question B will be discussed in a later post.

      You claim and believe that

      (K.1) Sellers always sell at the highest price they think the market will bear.

      (K.2) The above rule describes actual market outcomes more accurately than models which assume that prices will be largely determined by the seller’s costs.

      When you say that your theory is better than other, what you imagine as the other theory is the theory that prices will be largely determined by seller’s costs. If you understand that this is my theory, you have too quickly flipped my paper:
      The revival of the classical theory of value

      I warned that

      “The classical theory of value also takes many versions, from the labor theory of value to the subjective theory of value. … What I consider is the cost-of-production theory of value.” (p.4 in my RG draft: p.153 in the published edition; I cite the published version when there are any text differences)

      “This is what Ricardo wanted to formulate and yet could not arrive at. … A satisfactorily exact formula appeared many years after when Sraffa (1960) published his small book.” (p.4; pp.153-154)

      “Although he could not present a precise formula, Ricardo knew what it was not. Ricardo starts chapter 30 of his Principles with this remark:

      It is the cost of production which must ultimately regulate the price of commodities, and not, as has been often said, the proportion between the supply and demand. (Ricardo 1951, p.382)

      It is also important to note that Ricardo included profits in the cost of production. In his footnote at the end of Section 6, Chapter 1, he wrote:

      Mr. Malthus appears to think that it is a part of my doctrine, that the cost and value of a thing should be the same: — it is, if he means by cost, “cost of production” including profits. (Ricardo, 1951, p.47)

      It is plausible to name Ricardo’s theory of value cost of production theory of value, although this term was not employed explicitly by him.” (p.4; p.154)

      You are completely missed this emphasis. Cost-of-production theory of value assumes that the selling price is set as cost + profit margin. This is what is often called full-cost principle.

      After this, we have two points to argue.

      (A.1) How is the profit margin determined?

      (A.2) How is the cost of production determined when costs depend on prices?

      Argument on the point (A.1) is long. You should read two sections (headings):

      =Cost of production theory of value (2): Oxford Economists’ Research Group

      =How are markup rates determined?

      Argument on (A.2) is given in the section

      =Cost of production theory of value (1): Sraffa’s formulation

      You should understand all this when you criticize the cost-of-production theory of value. I have clearly defined the concept and warned against possible misinterpretations.

      If you reflect al these, you will see that cost-of-production theory of value is what you think as your theory in a vague, inexact expression that you claim by proposition (K.1).

      • August 9, 2017 at 10:40 am

        It seems to me Yoshinori is missing the point that a % markup creates an exponential and hence inflation whereas earned wages do not. A genuine full cost pricing system would have the employer pay himself a wage for the work he does, and allow explicitly for capital investment and maintenance rather than hiding lack of it in his profits.

  6. August 9, 2017 at 2:28 pm

    Dave Taylor

    please read the section =Cost of production theory of value (1): Sraffa’s formulation=.
    (pp.6-8: pp.156-158)

    If the material input coefficient matrix A is irreducible and primitive , a positive number R is determined (don’t worry too much on this condition. If this is not satisfied, we only must appeal to a bit more complicated explanation). It is the Frobenius eigenvalue of matrix A. (In the paper it is called “maximal rate of profit”)

    If each markup rate m of every industry is smaller than R, the markup pricing converges to a finite price vector p for any given wage rate w. Note that p satisfies equation (1).

    Continuous inflation process can happen if many of markup rates exceed R. A sufficient condition for that this:

    (1+ M){w a0 + A p} > p (1 bis)

    for some positive price vector p.

    The inflation processes which occurs in this reason were studied under the title of cost push inflation in 1970’s. I have also written two papers in relation to this theme.

    Non-Simultaneous Mark-up Pricing Processes

    Asymptotic Behavior of Perturbed Iterations, the Case of Non-negative Irreducible and Primitive Matrices.

  7. August 9, 2017 at 5:36 pm

    I note that the primitives of A are rates, with Frobenius enabling the generalisation of a varying yearly markup into a continuous rate. Taking w as 1 so p is a relative price, the antiderivative is log p so as I said, p is an exponential. Workers are not industries, but the mark-up R’ for them considered as such is in general less than R so (as you say) tends to converge to a finite level w (i.e. subsistence level). Distribution markups on top of production markups and financial markups on top of that compound the problem. The rich get richer and the poor get poorer.

    This seems to me more about inflating costs than increasing value, which rather rules it out as a theory or measure of value. Having read Ricardo on wages I hope you are not trying to defend him. In practice James Kroeger seems nearer the mark: “My price theory is based almost entirely on the assumption that sellers always charge the highest price that the market will bear, period.” What I will suggest is that this applies chiefly to young traders mimicking what they hear from “fly-by-night” traders now proliferating on the internet. In the old days in country markets the wise could safely by produce they could see and compare, but when it came to products and being tempted by a bargain, the proverbial warning was “buyer beware”.

  8. August 9, 2017 at 11:44 pm

    N.B This is not a reply to davetaylor1’s post dated August 9, 2017 at 5:36 pm, but a correction to my post dated August 9, 2017 at 2:28 pm.

    Dave Taylor and dear all readers,

    I am sorry! I was totally confused in my explanation of cost-push inflation process. I studied the topic many years ago. As you see by the date of my working papers, I worked on them in 1970’s. Some 40 years ago!

    The core idea of cost-push inflation process is the contradiction between firms’ demand for higher markup rates and workers’ demand for warranted level of real wage. In the previous post, I only talked about level of markups. In a purely imaginative economy, it is possible that many of (1 + markup rates) exceed the Frobenius eigenvalue (1 + R), and inflation occurs between some range of commodities even if wage rate is negligible. Of course, this is a purely imaginative state. In the above post, I was also making a confusion between R and 1+R.

    My explanation around the equation (1 bis) is also a pure nonsense. If labor input coefficient vector a0 is positive (meaning that labor is necessary in the production of all product, including intermediate goods), then for any fixed vector p the inequality (1 bis) is always satisfied when w is sufficiently large.

    I apologize all these confusions.

    Now, here is more correct explanation. To make the explanation more plausible, it is necessary to consider workers real wage demand. The most simple way is to assume that workers ask the real wage that they can buy a fixed basket of goods for a unit time of labor. Let it be

    d = (d1, d2, … , dN).

    Then we can define a square matrix D by posing

    D = (a0i ・dj ) = a0 ・d.

    The right side expresses the matrix product of N-column vector by a N-row vector.

    Let M be a diagonal matrix of N-dimension with diagonal elements mj the markup rate for industry j. (I have not explained this in the previous post.)

    If the demand of workers and firms (or capitalists) in the following situation is in a following situation for a positive vector p#:

    (I + M )(D + A) p# > p# (1 ters).

    then there is no positive fixed wage-price couple (w, p) that satisfies equation (1), i.e.

    (I + M )(w a0 + A) p = p (1),

    and real wage requirement:

    〈d, p〉 ≦ w (RW).  

    In this case, if firms price their products by the conventional full-cost pricing and workers demand and realize their real wage requirement, then a cost-push inflation process is inevitable. Details are given in my Non-simultaneous Markup Pricing Processes paper.

  9. August 10, 2017 at 7:46 am


    I am not defending all of Ricardo’s observations and theses. I do not support his thesis that real wage level is fixed at a subsistence level. I do not support his wage fund theory. All these thesis may have seemed true at the time of Ricardo, i.e. in the first quarter of the 19th century. Up until 1850’s, English real wage remained approximately constant since the 14th century, if we measure it by the wage of London carpenters and the prices of bread or wheat.

    However, as you know very well, the real wage rate began to rise continuously in England in 1860(s and we observe similar trend in many other countries with some delays.

    What I support Ricardo is his theory of value. He is almost unique theorist who really
    understood that it was the cost of production including profit which determines the value for almost all industrial products and clearly refused the law of demand and supply.

    Of course, this does not mean that the classical theory of value ceased to develop since then. On this point, I have written a short account on the development after Sraffa. As it is a short paper which counts only six pages, please read it.

    Sraffa’s research program is advancing

    • August 10, 2017 at 8:51 am


      I see what you mean. Perhaps the issue is including rent due to the relative monopoly of wholesalers and financiers in profit.

      • August 10, 2017 at 9:52 am

        … the rent being paid by labour in the form of excess work. The relatively few employers have been able to form wage cartels in private while making chartered craft guilds redundant and legislating against worker unions in parliament. Feudal serfs at least had the safety net of their own share of arable allotments of land, whereas after Ricardo the Malthusian mob imported mass-produced food, replacing even Speenhamland supplements with near-starvation in Bentham’s penal poor houses. Ugh!

  10. August 10, 2017 at 11:56 pm


    Please read §6. “How are markup rates are determined?” of my paper cited above:

    The revival of the classical theory of value

    I have explained how the markup rates are determined. In the latter half of the section, I refuted the traditional kinked demand curve explanation and introduced a concept of share function. With the latter we can determine the most profitable selling price for producers and retailers. If the share function is only dependent of the price ratio between competing sellers, the most profitable price setting is given by the full-cost principle with a constant markup rate.

    You may called this mechanism “rent due to the relative monopoly” of sellers.

    One important thing here is that sellers set their selling price but they cannot raise them arbitrarily. Even in a duopoly case (which I studied mainly), if the price is too high, customers go to competitors. There are many other complicated stories about markup rates but they cannot be raised above certain bounds whenever the competition is maintained at the same level. By the concept “competition” you may imagine the price sensibility of customers.

    So, let us assume that markup rates are bounded from above. Then, the real wage rate increases when the productivity increases. This explains why workers life level was continuously raised after 1860’s. Ricardo could not foresee this but provided the core of the right value theory.

  11. August 11, 2017 at 3:28 pm

    Yoshinori, I admire your persistence, but this is going to have to be a bit hard. I got so far as reading the abstract of your paper, but it completely misrepresented Keynes, and this from the latest blog on the real life Zika vaccine is sufficient to refute its theory of markup pricing:

    “The prescription drug industry ranks as one of the most dysfunctional sectors of the US economy. Martin Shkreli has made himself a celebrity as the “pharma bro”: a hedge fund boy who jacked up prices on drugs produced by a company he controlled by more than 5,000 percent. Unfortunately, this sort of pricing is typical of the industry, even if most companies tend to be a bit less flamboyant in profiting off patients than Mr. Shkreli.”

    You say in your abstract: “In my opinion, the adoption of equilibrium framework was the fundamental reason of Keynes’s failure. I do not deny that the General Theory contained a revolutionary idea i.e. that of effective demand. But Keynes or other economists after him could not arrive to formulate the concept in a proper way. One of the results of this fact is the disappearance of the concept of effective demand from New Keynesian economics”.

    Keynes did not adopt the equilibrium framework. H started by pointing out that this was what mainstream economists were using, but most of the General Theory (which incidentally is about Employment, Interest and Money, not pricing) builds on the concept of effective demand to avoid equilibrium’s long-term failings. It is not reasonable to expect an innovator to formulate his concept in what in retrospect seems to be the “proper way”, any more than it would have been to expect Orville Wright to design a jumbo jet. It was about 1964 before what Keynes achieved could be classified in terms of the theory of PID servos (effectively the cybernetic theory of navigation). In terms of that, what Keynes did was not to eliminate steering in the short term, but to add long term course adjustment by occasionally checking and correcting for positional drift. We can more justly blame succeeding economists with pecuniary interests in the status quo and undermining control theory in the name of freedom, who succeeded in making effective demand disappear from so-called Keynesian economics.

    You also say workers’ conditions gradually improved from 1860. To the extent that is true here it wasn’t due to industrial policy but to the success of religiously motivated (e.g. Quaker) industrialists and social Liberals inspired by humanitarian J S Mill rather than Ricardo. Marx and Ruskin too were motivated by how bad things still were. Catholic leaders had lived comfortably in the Vatican for hundreds of years, but in 1891, when briefed by the English Cardinal Manning about families starving because the port authorities had locked out dockers striking against the equivalent of today’s zero hours contracts not providing families with a livelihood, the then Pope, Leo XIII intervened by publishing “Rerum Novarum”, a brief survey of what was happening in this “new age”, and what ought to be happening. As a Catholic, that’s where my interest started.

    Right now, as a UK West Midlands citizen, I’m seeing strikes over refuse collection being mechanised in ways the public finds unacceptable, to “save money” by down-sizing wages. I’m seeing over-used motorways choking themselves (and us) near what have become essential repairs, just as restoration of the railway alternative is being abandoned because of incompetent management’s loss of control of fund-raising and financial cost-effectiveness. Conditions here may still be better than they were in 1860, but they are getting worse fast.

  12. August 12, 2017 at 9:10 am


    As I am going into summer vacation for more than one week, I cannot reply to your post for about one or two week time. I have to be short in this reply too.

    As for price theory, do not try to refute it by pointing an exception. A general theory like price theory has to explain first of all majority of transactions.

    As for Keynes, have you ever reflected why his theory and reputation have so quickly fallen down? Don’t you think that some internal theoretical problem has worked in this decline together with other social and political factors? Keynes was great but we can learn from him only a few numbers of ideas. We should discard many of his ideas. I am questioning how to do this sorting.

    As for workers’ improved conditions, I am only saying that this long continued trend can be explained by my theory of prices. What I pointed is that my theory can explain why real wage rate was improved. Am I am contending that their condition was improved by an industrial policy? Not at all! Instead talking about real wage rate, you are talking that working conditions were improved. You are talking totally different thing. I do not deny that that kind of social pressures had contributed to make work conditions better. But that does not imply that classical theory of value is invalid.

    You raised several episodes that some people were staving and so on. Here also you are committing an error of trying to refute a general theory by a small number of exceptions. The real wage rate is a social entity that you can grasp by statistics. To refute an established observation, you should start to reorganize the statistics. It is no easy task. I suppose you do not dare to do that.

    You have the right to worry about any kind of people’s conditions but it is very difficult to relate all of them to an economic theory. Economics is mainly concerned with mass phenomena like statistics.

  13. August 12, 2017 at 12:42 pm

    Yoshinori, have a good holiday, and be sure – despite my disagreements with you – my best wishes go with you. It would be nicer to try and resolve our cross-purposes across a table, but airing them in public does at least allow others to learn something too.

    Here, you say very helpfully that [for you] a price theory has to explain first of all the majority of transactions. My aspirations are higher: I want a price theory to explain ALL transactions.
    That takes us into the logical problem of Russell’s Paradox (is the set of ALL sets a member of itself?), which was resolved by accepting that there are different types of transactions [or more generally propositions] requiring different explanations.

    Let me explain that though I got into economics through Catholic social theory, I got into maths through a mathematician explaining the difference between graphs of linear and quadratic equations, using economics as an example. Representing the different interests of buyer and seller by different dimensions, he showed how the calculus could be used to find an optimum price: not in the majority of cases but in ALL cases. It doesn’t follow that actual prices will be the optimum, for in each transaction one side may be kindly and the other greedy, keeping down or putting up actual prices (i.e. what turn up in such cases as are accounted for in statistics). The contrary doctrines of increasing returns to scale and diminishing returns likewise follow immediately, with undemanded over-production now disrupting supply by lowering prices.

    The basic mathematics of economics thus led me via computer electronics into Algol68 computer logic and communication. Whereas earlier computer languages had provided for more or less of the same thing, Algol68 distinguished static things from active processes, so the logic had to deal not just with parts and wholes but with four types of object in the language (much like the adjectives, nouns, adverbs and verbs of English grammar). The computer logic, too, had to deal with finding and correcting errors in the data input or its processing (see Shannon’s “The Mathematical Theory of Communication,1948), so the equivalent of economic auditing was built in.

    For me, therefore, a price theory has to account not only for all transactions but for whether the processing of them works (the fallacy of composition ruling out one-dimensional aggregation of prices) and whether the auditing system suffices (which it doesn’t, if doesn’t account for the past and the foreseeable future as well as the present). The insights of Adam Smith and Keynes deserving recognition are thus to my mind “the invisible hand” correcting errors audited in the present and Keynes’s unemployment correcting errors in employment accumulating in the past (i.e. in the buying and selling of labour processes rather than good things). The intuition of entrepreneurial minds acts quickly enough when they become aware of problems or opportunities approaching in the future!

    As for the undermining of Keynes by economists unable or unwilling to see or even look for the baby and lots of interesting playthings in the bathwater, more fool them for throwing it and them out, IMHO.

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