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| A Reproduction Scheme (Based on Biddard 1990) |
One way of reading the Cambridge Capital Controversy (CCC) is an internal exploration of and debate about neoclassical price theory1. Both sides agreed to concentrate on the case of perfect competition, with no principal agent problems, no asymmetric information, etc. The Cambridge-Italian critics thought themselves to have demonstrated that neoclassical economists could not consistently with their theory claim that equilibrium prices were indices of relative scarcity. Such a claim is not well-founded in the theory, and economists should turn away from biotechnological determinism and turn toward developing price theories in which class power matters.
In this post, I want to outline the sophisticated neoclassical response, in the 1970s, to the Cambridge-Italian critics.
2.0 General Equilibrium Models of Intertemporal and Temporary EquilibriumThis neoclassical response asserted that price theory was best expressed in terms of General Equilibrium Theory (GET). Capital theory involves production over time. Models of intertemporal and temporary equilibrium have been developed in GET. And these models, it is claimed, are both logically consistent and unaffected by Cambridge-Italian criticism2.
2.1 The Arrow-Debreu Model of Intertemporal EquilibriumThe Arrow-Debreu model is a model of disaggregated individuals interacting solely through a single centralized market in existence at the beginning of time. Commodities are distinguished by their physical characteristics, where they become available, when they become available, and the state of the world in which they become available. The givens in the Arrow-Debreu model consist of, roughly:
Equilibrium is achieved in some sort of no-time before the beginning of time. The centralized market opens, and the auctioneer informs all agents of the prices of all commodities. These commodities include, for example, a contract to buy an umbrella in New York City on 20 June 2015, given it is raining. The agents inform the auctioneer of which contracts they are willing to enter and offer at the given prices. The auctioneer adjusts prices, depending on mismatches between offers and demands, until all markets clear. No transactions are allowed to take place until equilibrium is achieved3.
In an equilibrium, the plans of all agents are pre-reconciled. For any commodity with a positive (spot or forward) price, the quantity supplied equals the quantity demanded. Some goods are not commodities in equilibrium; they have a price of zero. The supply of these goods exceeds the demand.
Once equilibrium has been established, all agents make their promised trades and carry out their plans throughout time.
2.2 The GET Model of Temporary EquilibriumThe idea that all plans are only made in one big market transaction at one instant of time is hard to swallow. This constraint is relaxed in models of temporary equilibrium, as developed by, for example, J. R. Hicks (1946).
In Hicks' model, a market opens at the start of each successive week. If I recall correctly, with a single exception, all markets are spot. That is, supplies and demands are contracted each Monday to be delivered or taken during the following week, but not for later weeks. If an agent plans, for example, to hire labor two weeks hence, they must agree on the wage on the Monday at the start of that week. No market exists in which a worker can be hired for a period of more than one week.
In this model, the plans of agents only need to be consistent in equilibrium for a single week. Supply and demand of both factors of production and commodities for consumption match in the spot market. But these supplies and demands may be based on plans that entail inconsistencies in future weeks. When agents see spot markets failing to clear, say, next week, they revise their plans. And, once again, these revisions and the haggling in the market are assumed to bring about instantaneous market clearing.
The single exception to the requirement that all markets be spot is a market for something like a bond or an annuity. Forward markets exist for all time periods in which one can offer to pay a unit of the numeraire commodity at the start of this week for delivery of a given quantity of the numeraire quantity at the start of, say, the next week. So a whole complex of interest rates exist for the numeraire. An individual's expectations include expectations of movements of future spot prices of all commodities. That is, each individual has expectations for not only future movements in, say, one week interest rates for the numeraire commodity but beliefs about own-rates of interest for all commodities. Nothing in the model brings about consistency among agents of these expectations and beliefs about the future.
This model continues to impose the requirement that no false trading occur. When the markets open on Monday, no transactions can take place until a market-clearing set of prices is found. Only after equilibrium has been achieved do commodities change hands. And production proceeds in each week during the period in which markets are closed.
These models impose very few restrictions on equilibrium, even with specific assumptions on expectations. Perhaps models of temporary equilibrium are better, within mainstream economics, than the Arrow-Debreu model of exploring the formation of expectations.
3.0 Acceptance of Cambridge-Italian CriticismsThese models of general equilibrium may be internally consistent. Both sides of the CCC, however, came to recognize they did not support the beliefs about causal properties still relied on to this day in mainstream applied theory. The faulty and unfounded idea is something like this: compare two equilibria, in which the exogenous (given) data is identical, except the quantity of some given endowment varies across the two cases. Then, if one abstracts from violations of the assumptions of pure competition and is ignorant of price theory, one might expect the price of that endowment to be higher in the case where it is more scarce. Similarly, such an ignoramus would expect the price of commodities produced more intensively with the more scarce endowment to be higher. Despite the short run nature of the models outlined above, such beliefs are unfounded in GET. Here is Christopher Bliss forthrightly acknowledging such:
"Even people who have made no study of economic theory are familiar with the idea that when something is more plentiful its price will be lower, and introductory courses on economic theory reinforce this common presumption with various examples. However, there is no support from the theory of general equilibrium for the proposition that an input to production will be cheaper in an economy where more of it is available." -- Christopher Bliss (1975).3.1 The Meaning of Quantities, Prices, and Commodities in GET
One issue with GET is remaining clear on the meaning of prices, commodities, and endowments. As with Bliss, I have stated the claim about endowments, prices, and scarcity indices in a timeless, static equilibrium. However, the two disaggregated models outlined above are set in logical time. For example, consider the endowment of a natural resource, such as oil. Only the endowment at the beginning of time is taken as data; the quantities of oil available at the start of the second, third, etc., weeks are different commodities. And these quantities are found by solving the model; they are not taken as given. Likewise, a grade of gasoline produced from oil is a different commodity, depending on which week in which it is produced. Christian Bidard is clear about this distinction between commodities that may be physically identical, but available at different times:
"Intertemporal general equilibrium prices associated with a finite or infinite path of consumption and accumulation have no general properties: the reason being that goods called corn at date t, iron at date t, corn at date t + 1, iron at date t + 1 are formally considered as four distinct commodities of an atemporal economy." -- Christian Bidard (1990).
You might find in GET a statement about the price of one commodity and the quantity of one commodity that goes into the production of that commodity at some more-or-less distant time beforehand. But this is not a statement about the whole vector of prices of physically identical commodities distinguished by the time of their availability. In general, GET theory does not provide simple intuitive propositions about prices and quantities of multiple commodities.
3.2 Steady StatesThe possibility of a third model might be thought to provide a work-around. Consider the limit, as time increases without bound in the models, of relative quantities and relative prices all referring to that single point in time at infinity. This is a model of a steady-state4. Does this background help explain the following from Frank Hahn?:
"It is possible that the outputs produced in an Arrow-Debreu economy in the far distant future are independent of its initial endowments. That would mean that in such an economy the relative scarcities prevailing now would have no influence on the relative prices and rentals in the distant future. This should be enough to persuade the critics that the theory is not committed to a relative scarcity theory of distribution, though they seem to believe it is and that often motivates them in their attacks." -- Frank Hahn (1981).
Christopher Bidard also considers such a limiting process:
"The subscripts refer to the date, the horizontal arrow indicates production by means of inputs and labor and the data in italics are neoclassical theory calls the 'endowments' of the economy. The interesting feature of this scheme is that all inputs at, except for the very initial ones a0, are obtained as the result of previous production. If we admit that the influence of the primitive inputs a0 vanishes in the long run, we have a pure reproduction process (a mechanical endogenization of labour, identified with a given wage basket, would be useful for a comparison with von Neumann's theory, but the operation is not necessary here)." -- Christian Bidard (1990).
In a simple model of a steady state, one might as well drop time indices off state variables for relative quantities and relative prices. They are invariant over time in such a model. But even so, one cannot apply theorems of static equilibrium to such a model. The logic of steady states is not one of allocating scarce resources. Inputs into production, insofar as they are produced, are not exogenous givens. Rather, they are found as part of the model solution. And the kind of relationships that Bliss says above are without foundation in GET are equally unfounded in models of steady-states.
4.0 Does a Logically Consistent Vulgar Neoclassical Theory Exist?I consider the beliefs that economics is solely about the allocation of scarce resources and that equilibrium prices are indices of relative scarcity to be vulgar neoclassical doctrines. Beliefs about properties of equilibrium, if you are interested in mathematically formalized Neoclassical models, should be logical conclusions derived from assumptions. And those assumptions should be on the primitives of the model. For example, one might have some assumptions about the tastes, production functions, or patterns of initial endowments.
Here Edwin Burmeister states that no such vulgar neoclassical theory is known to exist, albeit in the context of the analysis of an aggregate market for capital:
"Imposing some set of conditions on the technology ... should be sufficient to assure that real Wicksell effects are always negative. Such conditions would be of interest - especially if they could be empirically tested - since they would validate the qualitative conclusions derived from the one-good models often used in macroeconomics without any theoretical justification for ignoring aggregation problems. Moreover, Burmeister ... has proved that a negative real Wicksell effect is a necessary and sufficient condition for the existence of an index of capital ..., and a neoclassical aggregate production function defined across steady state equilibria such that (i) [consumption per head is a function of the index of capital per head], (ii) [the equilibrium interest rate is equal to the marginal product of the index of capital], and (iii) [This index of capital exhibits declining marginal returns]. Unfortunately, no set of such sufficient conditions is known, but the literature on capital aggregation suggests that they would impose severe restrictions on the technology." -- Edwin Burmeister (1987).
Economists who hold fast to vulgar neoclassical economics may present formal models. But often their mathematics is imprecise, disguising muddle and confusion. It is not a matter of having unrealistic assumpions; it is a matter of having assumptions that do not imply one's conclusions.
5.0 ConclusionAs far as I can tell, most economists do not learn the literature of their subject. Not only do most economists stay ignorant of the view of the English side of the CCC. They also remain ignorant of the most sophisticated neoclassical response. Thus, they ask to see irrelevant empirical evidence5 for the existence of reswitching, and do not acknowledge their applied stories6 are without foundation in rigorous neoclassical price theory.
FootnotesJohn Weeks' book, Economics of the 1%: How Mainstream Economics Serves the Rich, Obscures Reality and Distorts Policy (Verso, 2014) is a popular, polemical work.
"...By what logic do econfakers conclude that wage increases reduce employment and why is it contradicted by reality? The logic, if one might call it such, is from the same full-employment fantasy world as 'supply and demand,' dissected in Chapter 4. As in that discussion, I have to begin with clear specification of the fakeconomics trade-off hypothesis. It does not assert that a wage increase in a specific company will reduce employment. The precise hypothesis is: 'From an initial position of full employment for an economy that produces only one commodity under conditions of perfect competition, an increase in the real wage will reduce employment.'
A rational person might ask: why on earth state a simple proposition (wage up, employment down) in such an absurdly complex manner? They do so because the proposition is not simple. It is valid only under extremely restricted conditions. The hypothesis begins with the economy as a whole, not individual companies or industries. This reason for this will soon be clear. The full-employment caveat is necessary in order to exclude the effect of the most important determinant of the level of employment and unemployment: the total expenditure, public and private, in the economy as a whole (aggregate demand).
As should be obvious, if the analysis begins in conditions of unemployment, an increase in real wages should contribute to an increase in employment by increasing consumer demand. The econfakers exclude this possibility by starting from full employment (maximum output), so any increase in demand could only cause inflation.
But starting at full employment means that the analysis cannot apply at the level of individual companies except as part of the economy as a whole. This implies that the trade-off hypothesis has no relevance to real-world decisions made in companies about employment levels.
Moving on to the next absurdity, allowing the economy only one output is an unavoidable technical requirement. With only one product, either there is no input or the input is the output itself (which is quite strange when you think about it). The following example reveals why the econfakers enter into such contorted illogic. In an economy with an output that has an input different from itself (e.g., wheat and fertilizer), the result of a wage increase in both industries cannot be predicted. A possible logical (and practical) sequence might be as follows: the higher wage prompts farmers to use more fertilizer to raise yields and make labor more productive. Employment in the production of fertilizer increases, with little change in labor used on farms, so total employment expands. In a real economy with thousands of products, the result of increases and decreases in wages can only be known after the event.
This is no abstract, arcane issue. The unpredictable outcome of a general wage increase can be easily demonstrated using what are call 'input-output' tables. These tables are available via the Internet for most countries of the world. They show the flow of inputs through the productive system, which eventually results in what are called 'final products' - those bought by households and governments, and businesses for investment.
Finally, the trade-off hypothesis requires a competitive economy in which no collusion exists among employers or employees. This condition ensures that the demand for labor varies independent of the supply..., which rules out a feedback from higher wages to employment.
What seemed so simple and obvious - lower wages, cheaper labor, more employment - proves impossible to establish as a general rule. At the level of the company, lower wages may allow for lower prices, and the lower wage company takes business away from its rivals. The 'higher wages cause unemployment' accusation is quite different. It alleges a fakeconomics faux law that a general increase in wages for the economy as a whole will reduce employment (and vice versa). This allegation cannot be established in theory, nor is it supported by empirical evidence. It is an ideological construction intended to justify lower wages and higher profits, and to blame unemployment on workers themselves.
In practice the econfakers and those they have indoctrinated trumpet this argument as a law of nature, and use it against all attempts to improve the conditions and hours of work. For example, laws that regulate working hours and require additional pay for overtime allegedly reduce employment because they increase labor costs. The same ideological illogic applies to workplace protection, health and safety legislation, and protection of vulnerable workers. They all raise the cost of employing people. Therefore, they must contribute to unemployment. All attempts to improve the conditions of labor, either through the collective action of workers or legislation are self-defeating. These arguments are wrong, technically, empirically, and morally. In civilized societies all people are paid decently and work in healthy conditions to the extent that the level of economic development allows." -- John F. Weeks (pages 36-38)
Can you see that the middle part of the above quotation is about the application of the Cambridge Capital Controversy to so-called labor markets?
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Here are some ways of classifying capital. This post does not talk much about profit on alienation (buying low, selling high). Nor does it talk about analogies (for example, "human capital", "social capital") extending beyond production and, maybe, even economics. The definitions are my attempt to give an off-hand elaboration of the meaning of terms. I have no objection to those offering more authoritative definitions.
First division:Update (10 June 2014): I want to note this passage - more succinct than my writing - from Josh Mason:
"We shouldn't ask what capital 'really' is. It really is a quantity of money in a process of self-expansion, and it really is a mass of means of production, and it really is authority over the production process. But the particular historical questions Piketty is interested in may be better suited to thinking of capital as a claim on the social surplus than as a physical quantity of means of production. Seth Ackerman has some very interesting thoughts along these lines in his contribution to the Jacobin symposium on the book. "
Some economists sometimes say that neoclassical economics will be abandoned when a better theory is available. And they use this as an excuse for not pursuing, say, heterodox economics.
I am tempted to respond that a better theory already exists, for example, some combination of Post Keynesian and institutional economics. But two properties of most varieties of heterodox economics make it difficult for this answer to register for mainstream economists. Before I get to to those properties, let me caveat my answer.
Economists have a number of applied fields. Is heterodox economics developed to such a state that one can say economists would (or should) know how it should impact their work in every applied field? I cannot say how my concerns would apply to, say, transport economics, although I know that work exists there that draws on behavioral economics. Anyways, I doubt that many applied economists are aware of or concerned with how there work fits better with some approaches in heterodox economics. Much applied work seems to me agnostic between paradigms. For example, would you say that most work applying linear programming is Sraffian, even though the historical origins of linear programming are entwined with John Von Neumann's Sraffa-like, classically inspired growth model? How about work with National Income and Product Accounts (NIPA), which, to me, fit comfortably with the classical focus on the generation, distribution, and use of the surplus? Furthermore, I know of some applied fields to which heterodox economists have contributed. I doubt that I can count Walter Isard as a heterodox economist, but I think of economic geography and regional analysis as compatible with Sraffian economics. Wassily Leontief, as far as I am concerned, was a heterodox economist, although, as far as I know, he never commented on the theoretical controversies in which I am interested. I am fairly sure that Bertram Schefold has work in energy economics inspired by Sraffian theory. Is not Amartya Sen's work on the capabilities approach a heterodox contribution to welfare economics? Given Paolo Sylos Labini's importance to the field, I suspect that many models exist in Industrial Organization (IO) that are compatible with a broad definition of Post Keynesianism. To conclude, the connection between applied fields and debates about heterodox economics are not clear to me.
Anyways, heterodox economists tend to emphasize a need for open systems models in economics. This emphasis is the first property of heterodox economics that I want to mention that makes it difficult for mainstream economists to accept the superiority of much heterodox work. Tony Lawson has provided lots of elaboration on the need for an open systems approach to studying human society. But I do not need to go into such philosophy. My favorite approach to price theory takes the distribution of income as given for the most abstract theory of the prices of commodities. Likewise, the composition of final uses is exogenous in Sraffa's theory. The structure of the theory is open to elaborations at lower levels of abstractions that include inputs from other social sciences than economics. Consumption might be explained by a substantive theory, in contrast to the empty formal theory of neoclassical utility-maximization.
Second, heterodox economists tend to be skeptical of the possibility of economic laws applying to all societies across all of human existence. By contrast, Lionel Robbins' definition of economics as the study of the allocation of scarce means among alternative uses is often claimed to be universal. There is nothing in the mainstream about capitalism or how social norms might differ among societies. Contrast, for example, with Luigi Pasinetti's work on structural dynamics, in which the theoretical structure builds in a place for variations in institutions. Or look at suggestions in Sraffa's book that taking one of wages or the rate of profits as the independent variable might be more appropriate at different times or places. Or look at the emphasis on conventions in Keynes' analysis of investment. I could go on.
So, by adopting theories appropriate for the problem domain, heterodox economists have developed theories that are superior to many orthodox theories. But, because of this very appropriateness, mainstream economists are socialized to fail to perceive this superiority.
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| Profits by Selected Industry as Percent of Total Profits in the United States |
The data for the graph are taken from Use tables for the United States. I am thinking of trying my hand again at some empirical exploration of input-output tables. When graphs for such start looking like those I know how to generate from Java code, I will have begun to start to make some progress. I have found a tool, Apache POI, for Java programs to read Excel spreadsheets.
Some brief observations:
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| Table 1: Components, As Percentage Of GDP |
I thought I'd expand on a recent graph. I was curious to see how state and federal government spending break down in the United States. To draw the graphs in this post I performed some aggregation from the data:
I suppose I could find a price index, and plot absolute amounts, rather than percentages of GDP. Then you could see, for example, that GDP in 2009 is actually lower than the 2008 value, as a result of the global crash. Does the breakdown of government spending into consumption and gross investment components reflect the influence of Robert Eisner?
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| Table 2: Selected Components, As Percentage Of GDP |
Consider:
These quotes are from Robert P. Murphy. (See here also.) I have written about Murphy before, including his take on Sraffa.
Some publishers have made it easy to get an overview of currently existing paradigms in economics, including non-neoclassical paradigms:
The books in the Edward Elgar series are probably too expensive for most individual purchases. Perhaps you can request some of interest from an academic library. Some of these books are more than a decade old, with articles in the collections going back even more decades. But, by looking at authors and journals, you can can get a hint of where to look for more contemporary work in a school of interest.
Compare and contrast Paul Krugman on this topic.
I have not even started to read Thomas Piketty's Capital in the Twenty-First Century, but I have heard of Piketty as compared to Marx.
Whatever else Marx was, he was a very learned man. He read the works of virtually all political economists who came before him. And in his lengthy tomes, he would comment on them, not always fairly. He did not confine himself to ones that were politically influential among the elite. For example, consider Marx on the Ricardian socialists.
So if Piketty is like Marx, can I expect to find comments on the Cambridge equation, r = g/sc? Can I expect to find something about the models of growth and distribution put forward by Richard Kahn, Nicholas Kaldor, Luigi Pasinetti, and Joan Robinson? (Joshua Gans has also noticed a parallelism between the work of Piketty and the Post Keynesian theory of distribution.) Or maybe the analogy is not complete.
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I thought that Krugman had a post about Paul Ryan stating, incorrectly, that Obama had increased the size of the government. And he wondered why conservatives make factual statements that can be easily shown to be wrong. But I cannot find such a post. I can find this one on Rand Paul making a different incorrect factual claim. I am fairly sure I am thinking of something more recent than this post about Rand Paul being confused in 2012.
(By the way, Paul Krugman is wrong about what heterodox economists believe about marginal productivity theory. If he reads this, though, I would rather read his comments about the empirical correlation between increased government size and increased equality.)
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| Figure 1: Distribution for Mixture Distribution |
What is the probability that the Dow Jones Industrial Average (DJIA) will rise by at least 5% tomorrow? By 10%? Very few samples can be found in the data for a large enough rise, and, eventually, you will be asking about a rise beyond all historical experience. Some have argued that Extreme Value Theory can be applied to financial data to extrapolate these sorts of tail probabilities. In this post, I attempt to explain this theory. For purposes of exposition, I here disregard the possibility of such rises as being associated with states that might be impossible to foresee from the past history of the data-generating process.
2.0 A Random Sample from a Mixture DistributionThis exposition includes an example. I need a probability distribution in which tails differ from the portion of the distribution clustered around the center, in some sense. Consider a random variable X which can take on any real number. The probability distribution for this random variable is defined by the Cumulative Distribution Function (CDF). The CDF specifies the probability that a realization of the random variable is less than or equal to a given value:
F(x) = Pr(X ≤ x)
where:
(Conventionally, uppercase letters toward the end of the alphabet denote random variables. The corresponding lowercase letter denotes a realization of that random variable resulting from the outcome of conducting the underlying experiment.)
To obtain a distribution with heavy tails, I consider a mixture distribution. (Mixture distributions are often used in the theory for robust statistics. I would appreciate a reference arguing that robust statistics and Extreme Value Theory are complementary, in some sense.) Suppose F1 and F2 are CDFs for Gaussian (also known as normal or bell shaped) distributions with possibly different means and standard deviations. And let p be a real number between zero and one. F is the CDF for a mixture distribution if it is defined as follows:
F(x) = p F1(x) + (1 - p) F2(x)
For definiteness, let the parameters for this distribution be as in Table 1. The two Gaussian distributions have equal means. The distribution with the 90% weight also has the smaller standard deviation. In other words, the distribution that is selected less frequently will have realizations that tend towards the tails of the overall mixture distribution.
| Parameter | Value |
| Probability Variate from First Distribution | 90% |
| First Gaussian Distribution | |
| Mean | 0.0 |
| Standard Deviation | 1.0 |
| Second Gaussian Distribution | |
| Mean | 0.0 |
| Standard Deviation | 3.0 |
2.1 A Random Sample
Suppose X1, X2, ..., Xn are mutually stochastically independent random variables, each of which has the probability distribution with CDF F. Under these conditions, these random variables comprise a random sample. I wrote a computer program to generate a realization of such a random sample of size n. Table 2 shows some statistics for this realization. I use this realization of a random sample to illustrate the application of various statistical techniques below.
| Value | |
| Sample Size | 500 |
| Realizations from 1st Distribution | 443 |
| Realizations from 2nd Distribution | 57 |
| Sample Mean | -0.0086 |
| Standard Deviation | 1.3358 |
| Minimum | -6.0125 |
| Median | -0.0034 |
| Maximum | 9.932 |
2.2 Goodness of Fit
It is difficult to determine that a realization of the random sample is not from the distribution F1. In other words, the existence of sample values, often in the tails, from F2 is not readily apparent from a straightforward statistical test for the goodness-of-fit. Consider the order statistics found by sorting the random sample:
X(1) ≤ X(2) ≤ ... ≤ X(n)
(By convention, a subscript for a random variable without parentheses denotes a random variable from a random sample. Parentheses denotes an order statistic.)
An empirical CDF can be constructed from the order statistics. The probability that a random variable from the distribution generating the random sample is less than or equal to x(i) is estimated as i/n, the proportion of the sample less than or equal to the given order statistic. Figure 1, above, shows the empirical CDF for my realization of the random sample, as well as the CDFs for the two Gaussian distributions in the mixture distribution. Both Gaussian CDFs have a value of 1/2 for an argument of zero, since that is their mean. The Gaussian distribution with the smaller standard deviation has a CDF with a steeper slope around the mean, since more of its probability is clustered around zero. The empirical CDF, estimated from the data, is a step function, with equal size steps occurring at each realization of a random variable in the sample. One needs to sort the data to calculate the empirical CDF.
The maximum vertical distance between a theoretical distribution and an empirical CDF is known as the Kolmogorov-Smirnov statistic. Under the null hypothesis that the random sample is drawn from the theoretical distribution, the Kolmogorov-Smirnov statistic will be a small positive number. Table 3 shows the Kolmogorov-Smirnov statistics for the data. This statistic is not statistically significant for the first Gaussian distribution. The probability that one would observe such a large value for the Kolmogorov-Smirnov statistic for the second Gaussian distribution is less than 1%. Thus, one could conclude that this data was not generated from the second distribution, but (incorrectly) conclude that it was generated from the first.
| 1st Gaussian Distribution | 2nd Gaussian Distribution | |
| Kolmogorov-Smirnov Statistic | 0.0354 | 0.228 |
| p Value | 54.59% | 0.00% |
3.0 Distribution for the Tail
With the description of the data out of the way, tail probabilities can now be defined. I concentrate on the upper tail.
3.1 Definition of a TailThe upper tail is defined in terms of the lower bound u for the tail and the tail probability q. These parameters are related like so:
q = Pr(X > u) = 1 - F(u)
The upper tail is defined as those values of the random variable such that the probability of exceeding such a value is less than the given parameter:
{x | Pr(X > x) < q}
In other words, the tail consists of values of the random variable that lie above the lower bound on the tail. It is sometimes convenient to define a new random variable, Y, for outcomes that lie in the tail:
Y = X - u
This new random variable is the distance from the lower bound of the tail, given that a realization of X lies in the tail. One could give a symmetrical definition of the lower tail and a corresponding random variable. Table 4 shows how many samples in my realization of the random sample, defined above, happen to come from the Gaussian distribution with the larger standard deviation, where, the parameter q is taken to be 10%.
| Value | |
| Number in Lower Tail | 18 |
| Number in Center | 23 |
| Number in Upper Tail | 16 |
| Percentage of Lower Tail from 2nd Distribution | 36.7% |
| Percentage of Center from 2nd Distribution | 5.7% |
| Percentage of Upper Tail from 2nd Distribution | 32.7% |
For y > 0, the conditional probability that X exceeds any given value in the tail, given that X lies in the tail is:
Pr(X > y + u | X > u) = Pr[(X > y + u) and (X > u)]/Pr(X > u)
The above formula simply follows from the definition of conditional probability. The second clause in the "and" expression is redundant. So the above can be rewritten as:
Pr(X > y + u | X > u) = Pr(X > y + u)/Pr(X > u), y > 0
Let G(y) be the CDF for the distribution for the random variable Y. One can then rewrite the above formula as follows"
1 - G(y) = [1 - F(y + u)]/[1 - F(u)], y > 0
Substituting for the definition of the parameter q, one obtains:
F(y + u) = (1 - q) + q G(y), y > 0
Or:
F(x) = (1 - q) + q G(x - u), x > u
The above two expressions relate the CDFs for the distributions of the random variables X and Y.
3.2 Generalized Pareto DistributionA theorem states that if X is a continuous random variable, the distribution of the tail is from a Generalized Pareto Distribution with the following CDF:
G(y) = 1 - [1 + (c/a)y]-1/c
The parameter a is called the scale parameter, and it must be positive. The parameter c is the shape parameter. It can take on any real number. When the shape parameter is zero, the Generalized Pareto Distribution reduces, by a limit theorem, to the exponential distribution.
Below, I will need the following expression for the Probability Density Function (PDF) for the Generalized Pareto Distribution:
g(y, a, c) = (1/a)[1 + (c/a)y]-(1 + c)/c
The PDF is the derivative of the CDF. For any set A to which a probability can be assigned, the probability that Y lies in A is the integral, over A, of the PDF for Y.
3.3 Parameter EstimatesThe parameters defining the upper tail are easily estimated. Let r be an exogenously specified number of variates in the tail. The lower bound on the upper tail is estimated as:
uestimate = X(n - r)
The corresponding tail probability is estimated as:
qestimate = r/n
Several methods exist for estimating the scale and shape parameters for the Generalized Pareto Distribution. I chose to apply the method of maximum likelihood. Since the random variables in a random sample are stochastically independent, their joint PDF is merely the product of the their individual PDFs. The log-likelihood function is the natural logarithm of the joint PDF, considered as a function of the parameters of the PDF.
ln g(a, c) = ln g(y1, a, c) + ... ln g(yr, a, c)
Maximum likelihood estimates are the values of the parameters that maximize the log-likelihood function for the observed realization of the random sample. I found these estimates by applying the Nelder-Mead algorithm to the additive inverse of the log-likelihood function. Table 5 shows estimates for the example.
| Parameter | Estimate |
| Tail Probability (q) | 10% |
| Lower Bound on Tail (u) | 1.368 |
| Scale Parameter (a) | 0.7332 |
| Shape Parameter (c) | 0.2144 |
The above has described how to estimate parameters for a distribution characterizing a tail of any continuous distribution. Given these estimates, one can calculate the conditional probability that Y lies above any value in the tail. Figure 2 plots this probability for the example. Notice that this probability is noticeably higher, for much of the tail, for the mixture distribution, as compared to the probability found from the Gaussian distribution with the smaller standard deviation in the mixture. And the Kolmogorov-Smirnov goodness-of-fit would not have led one to reject estimates from the first Gaussian distribution. But the estimates from Extreme Value Theory are closer to the higher (and correct) probabilities from the true theoretical distribution.
|
| Figure 2: Tail Probabilities |
4.0 Conclusion
This post has illustrated:
References
I have been reading a book, edited by Gavin Kitching and Nigel Pleasants, comparing and contrasting Ludwig Wittgenstein and Karl Marx. This is the later Wittgenstein of the Philosophical Investigations, not of the Tractatus. The authors of the papers from the conference generating this work do not seem too concerned with arguments about the differences between the young Marx and the mature Marx, albeit many quote a passage from the German Ideology about language. (I think this post is more disorganized than many others here.)
Anyways, I want to first consider a reading of Capital, consonant with the approach of Friedrich Engels and the Second International, but at variance with an analogy to Wittgenstein's later philosophy. One might think of the labor theory of value as a scientific approach revealing hidden forces and structures that are at a deeper level than observed empirical reality. Think about how, for example, physicists have an atomic theory that explains why tables are hard and water is wet. Even though a table may be seem solid, we know, if we accept science, that it is mostly empty space. Somewhere Bertrand Russell writes something like, "Naive realism leads to physics, and physics shows naive realism is wrong. Hence naive realism is false". Similarly, you may think purchases and sales on markets under capitalism are made between equals, freely contracting. But the science of Marxism reveals an underlying reality in which the source of profits is the exploitation of the workers.
Wittgenstein, in rejecting his early approach to language, rejects the idea of a decontextualized analysis of the sentences of our languages into an ultimate underlying uniform atomic structure which explains their meaning. Rather, in his later philosophy, he gathers togethers descriptions of the use of language, to dispel and dissolve the illusions characteristic of traditional philosophy. He is hostile to ideal of an ultimate essence for meaning, and points out the multifarious uses to which language is put. Some of his famous aphorisms include, "Nothing is hidden" and his explanation of the point of his philosophical investigations as "To show the fly the way out of the fly bottle". Some of his descriptions are not from actually existing societies, but from imagined primitive societies. Some of these imagined societies are described near the beginning of the Philosophical Investigations, much as in the first chapter of Piero Sraffa's Production of Commodities by Means of Commodities.
Can Marx be read in an analogous manner, as attempting to dispel illusions, while claiming that no hidden essence or foundation underlies capitalist economies? Such a reading, I think, will emphasize Marx's remarks on commodity fetishism and "real illusions" that come with non-reflective participation in a market economy. It also makes sense of Marx's literary style. Both Marx and Wittgenstein are attempting to encourage a fundamental change so that our form of life will not generate these illusions.
Perhaps such a reading is in tension with the view of Marx's account of exploitation as descriptive, not normative. What about Wittgenstein's saying that philosophy "leaves everything as it is"? How can one read Wittgenstein and Marx as pursuing complementary projects when Marx writes, "Philosophers have hitherto only interpreted the world in various ways; the point is to change it"? Various essays in this book address these issues. I guess what concerns me more is Marx's Hegelian style, quite different from Wittgenstein. (I rely on English translations.)
This book also alerted me to some issues in Wittgenstein interpretation. When Wittgenstein writes of a form of life, is he writing of human life in general (in contrast, say, to the form of life of a lion)? Or would different human cultures and societies have different forms of life? Does Wittgenstein encourage a political quietism since he does not provide an external standpoint outside of language to criticize rules? (I think the last objection draws lines more firm than is compatible with Wittgenstein's comments on family resemblances.)
I also have two new books to look up, Gellner (1959) and Winch (1963). Gellner sounds like an unscholarly polemic that yet was influential in turning philosophy away from the linguistic philosophy of the later Wittgenstein, J. L. Austen, and Gilbert Ryle. Winch seems to argue those studying society must use the terms that members of a culture use, and with the same understanding. So perhaps this is a Wittgensteinian argument that social science is not possible, or at least must lower its aims. But I have not read it yet.
ReferencesI finally finished Kartik Athreya's book, Big Ideas in Economics: A Nontechnical View. I have already offered two comments on it. I do not expect it to be successful. Do not look here for a discussion of the theory of the second best, the aggregation of production functions, the distinction between risk and uncertainty, or the problems with microeconomics (despite its point being that macroeconomics, as the author understands it, is applied microeconomics). Athreya does select and address some theoretical objections, such as the Sonnenschein-Debreu-Mantel theorem, related difficulties with using a representative agent, and the folk theorem in game theory. I was disappointed not to see an informed discussion of the relationship of steady state models, such as the Solow growth model, to very short run models such as the Arrow-Debreu model. On the other hand, you will find a lot of rationalization of assumptions on the ground that they are needed (useful?) to get definite conclusions, independent of any discussion of whether or not models with those models work empirically.
Anyways, I read the book on my Kindle. I found it difficult to keep the thread. So I have prepared the following analytical table of contents for my own use, if I should reread sections. I think Athreya could have gone through a couple more edits, reconsidering this structure. For example, maybe the book would have been more understandable with shorter and more chapters.
I want to highlight the similarity in conclusions in Mirowski's recent book and Reed's controversial essay (see references below). Their understanding of the current conjuncture is fairly dispiriting. The right is winning in mass consciousness, despite their ideas being incoherent and vicious from an intellectual perspective. And their ideas extend over the entirety of the political spectrum, at least if one restricts oneself to what is seen to be practical. Arguments over how to make existing markets work better or to address current problems by constructing new markets, for example, accept the inevitability of capitalism.
Both Mirowski and Reed have something to say about what must be done by the left now. What is needed is a collective development of a leftist alternative. Those developing such an alternative need to be part of a group, like the Mont Pelerin Society was for the development of neoliberalism. And those developing this alternative, at least in their role in such a group, should not be overly concerned with the vagaries of this or that election in this or that country. This is a long term project, which, if successful, will spawn other groups over decades more concerned with implementation in specific times and places.
Are these authors correct in arguing the left does not currently have an inspiring vision to put before the public? You can talk about social democracy, but is that a way forward now? Are there powerful institutionalized groups working to improve our societies based on an architectonic view of what is possible? It seems to me more of a rearguard movement in advanced industrialized countries. And what about further left? I am aware of various statements of ideals - for example, Davidson and Davidson (1996), Rorty (1999)- but, without being built upon by a movement, these seem kind of idiosyncratic and quixotic to me.
An aside: If Mirowski is going to read literature produced by well-known writers who taught at Syracuse University, I wish he would mix some Raymond Carver in with the David Foster Wallace he has been reading.
ReferencesI continue to read Kartik Athreya's supposedly popular account of contemporary macroeconomics. Today I focus on the misleading presentation of the theory of economic growth.
Athreya presents the Solow-Swan Neoclassical Growth Model (NGM) as a contrast to Malthus' model of economic growth. He briefly alludes to Real Business Cycle (RBC) theory as the result of appending random shocks to the Solow-Swan model. He then goes on to discuss what he calls the Ramsey-Cass-Koopmans model. There are two problems here. (I bracket off the grouping of the Ramsey model of a central planning authority determining an optimal savings rate with models of household savings decisions.)
First, Solow developed his model in the context of many other economists also developing growth models. This setting is totally missing from Athreya's book. Neither "Harrod" nor "Domar" appear anywhere in the book. Yet Solow's work was a neoclassical response to the Harrod-Domar model. The Post Keynesian approach to steady-state growth, associated with such economists as Richard Kahn, Nicholas Kaldor, and Joan Robinson provided an alternative at the time. (I might also mention Michal Kalecki and Frank Hahn's doctoral thesis, if I recall correctly.) Maybe this approach is missing because Athreya is not aware of its existence.
Second, Athreya does not even get classical growth theory correct, as presented by Malthus or others. According to Athreya, Malthus' theory abstracts from the existence of capital. I guess income is supposedly distributed only in the form of wages and rents. Athreya then claims to consider the effects of a technological innovation, namely, the introduction of a vaccine in Malthus' theory. Supposedly, the effect is to lower the death rate, while leaving birth rates unchanged. That is, population increases. Since the quantity of land is fixed, the theory exhibts diminishing marginal returns to labor. So Athreya misrepresents Malthus as claiming that improved technology, while increasing total output, ultimately leads to lower average income per worker.
In the classical theory of value, the natural wage is given by habit and custom. Malthus, building on his predecessors, argued that transitory wages higher than the natural wage might lead to changes in habits, through what we now might call hysteresis. This effect would be to increase the natural rate of wages. At any rate, population was expected to increase when wages exceeded the natural wage. But, maybe, the classical economists emphasized more reactions to opportunities for jobs than reactions to wages. They accepted that unemployment could be persistent and expected lower and higher periods of unemployment to encourage increases and decreases of the rate of growth of population. Anyways, Athreya is right, at least, about the response to increased productivity being an initial increase in the population of workers.
But he is mistaken about the ultimate effect. Suppose the market wage falls below the natural wage, in a period in which the accumulation of capital has declined. Then the classical economists, such as Malthus, expected the rate of increase in population to fall. Emigration would increase, birth rates would fall, and workers would form families later in their lives. (It is unclear to me how the classical economists envisioned such mechanisms to kick in fast enough for their theories. At any rate, I can quote Ricardo suggesting that the stationary state was far away.) The ultimate effect of declining population would be for workers to obtain their natural wage, with the level of employment and distribution between wages, profits, and rent being consistent with technological possibilities after a change. That is, the ultimate effect, in Malthus' theory, of an improvement is not lower real wages. (I am here bracketing out any consideration of whether Malthus presented a stylized theory consistent with the empirical experience in the centuries prior to his time or overlooked the effects of the ongoing industrial revolution.)I cannot recommend Athreya's book, either for the general reader curious about macroeconomics or for the advanced undergraduate or beginning graduate student. It is too misleading. The above is only one of many examples. I suppose some professional economists might find it of interest to catalog the misconceptions, mistakes, inconsistencies, tendentious statements, and occasional insights.
Update: I want to recall the comments of David Glasner, John Quiggin, Noah Smith, and Stephen Williamson.
ReferencesThe following is reproduced from "An Essay on Post-Keynesian Theory: A New Paradigm in Economics", Al Eichner and Jan Kregel's 1975 Journal of Economic Literature article. Of course, the table being a summary, all entries are highly stylized.
| Aspect | Post Keynesian Theory | Neoclassical Theory |
| Dynamic properties | Assumes pronounced cyclical pattern on top of a clearly discernible growth path | Either no growth, or steady-state expansion with market mechanisms assumed to preclude any but a temporary deviation from that growth path |
| Explanation of how income is distributed | Institutional factors determine a historical division of income between residual and non-residual shareholders, with changes in that distribution depending on changes in the growth rate | The distribution of income explained solely by variable factor inputs and the marginal productivity of those variable factor inputs |
| Amount of information assumed to be available | Only the past is known, the future is uncertain | Complete foresight exists as to all possible events |
| Conditions that must be met before the analysis is considered complete | Discretionary income must be equal to discretionary expenditures | All markets cleared with supply equal to demand in each of those markets |
| Microeconomic base | Imperfect markets with significant monopolistic elements | Perfect markets with all micro units operating as price takers |
| Purpose of the theory | To explain the real world as observed empirically | To demonstrate the social optimality if the real world were to resemble the model |
In this passage, Roxana is preparing to move from Paris to Amsterdam. She liquidates her possessions, and uses jewelry and bills of exchange as money to carry with her.
"I could not but approve all his measures, seeing they were so well contrived, and in so friendly a manner, for my benefit; and as he seemed to be so very sincere, I resolved to put my life in his hands. Immediately I went to my lodgings, and sent away Amy with such bundles as I had prepared for my travelling. I also sent several parcels of my fine[Pg 181] furniture to the merchant's house to be laid up for me, and bringing the key of the lodgings with me, I came back to his house. Here we finished our matters of money, and I delivered into his hands seven thousand eight hundred pistoles in bills and money, a copy of an assignment on the townhouse of Paris for four thousand pistoles, at three per cent. interest, attested, and a procuration for receiving the interest half-yearly; but the original I kept myself.
I could have trusted all I had with him, for he was perfectly honest, and had not the least view of doing me any wrong. Indeed, after it was so apparent that he had, as it were, saved my life, or at least saved me from being exposed and ruined—I say, after this, how could I doubt him in anything?
When I came to him, he had everything ready as I wanted, and as he had proposed. As to my money, he gave me first of all an accepted bill, payable at Rotterdam, for four thousand pistoles, and drawn from Genoa upon a merchant at Rotterdam, payable to a merchant at Paris, and endorsed by him to my merchant; this, he assured me, would be punctually paid; and so it was, to a day. The rest I had in other bills of exchange, drawn by himself upon other merchants in Holland. Having secured my jewels too, as well as I could, he sent me away the same[Pg 182] evening in a friend's coach, which he had procured for me, to St. Germain, and the next morning to Rouen. He also sent a servant of his own on horseback with me, who provided everything for me, and who carried his orders to the captain of the ship, which lay about three miles below Rouen, in the river, and by his directions I went immediately on board. The third day after I was on board the ship went away, and we were out at sea the next day after that; and thus I took my leave of France, and got clear of an ugly business, which, had it gone on, might have ruined me, and sent me back as naked to England as I was a little before I left it." -- Daniel Defoe, Roxana: The Fortunate Mistress (1724).
Defoe's novel, Robinson Crusoe, is more well-known among economists. For example, one can read Stephen Hymer's "Robinson Crusoe and the secret of primitive accumulation" (Monthly Review, 1971).
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