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# The role of probability in economics

It is beyond doubt that Keynes extensively used probability notions in builing his economic theory. Moreover, the methodological attitude revealed by his economic writings is essentially the one he displays in TP (see Carabelli, 1988, p. 8). The results of his investigations into the foundations of probability and statistics has proven to be of fundamental importance in his economic theory. His works on money, prosperity and the general theory of employment cannot be understood without reference to TP together with its preparatory studies. To mention only a few examples, taken from many identified by Carabelli (1988; 1992; 1995), the essential relativity of a probability statement that greatly affects the notion of expectation, to which we shall return; the noncomparability of some probability relations, which Keynes transposed in the incommensurability of some economic quantities; the impossibility of a general measurement of probability relations, which Keynes extended to price relations; the inadequacy of addition or multiplication rules to tackle certain probabilities taken as a paradigm for the impossibility of adding or comparing the value of currencies.

However, Keynes did not use the notion of probability in constructing his economics. Only on very few occasions does he mention probabilistic notions. We focus on two cases. In dealing with long-term expectations, Keynes refers to the weight of arguments he had previously considered as the case in which

we have a more substantial basis upon which to rest our conclusion. I express this by saying that an accession to new evidence increases the weight of an argument. New evidence will sometimes decrease the probability of an argument, but it always increase its "weight".

(TP, 77)

The second is the discussion of various types of risk (Keynes, 1973b , ch. 11, s. 4) which cannot be fully understood without reference to his definition of "risk"

If A is the amount of good which may result, p its probability (p + q = 1), and E the value of the "mathematical expectation", so that E = pA, then the "risk" is R where R = p(A-E) = p(1 -p)A = pqA = qE.

(TP, p. 348)

However, in many cases in which Keynes did not explicitly mention the notion of probability it is impossible to gain a full insight into his economic writings without having a clear understanding of the relative character of the notion of probability that Keynes first (and in a very perspicuous mode) suggested:

No proposition is in itself either probable or improbable, just as no place can be intrinsically distant; and the probability of the same statement varies with the evidence presented, which is, as it were, its origin of reference.

(TP, p. 7)

This is especially true for the keystone of his economics, that is, expectation. As a notion which can be introduced only after probability is taken into account, expectation, too, is essentially relative. As a consequence, in order to obtain a full insight into the peculiarities of the notion of expectation, one must have a clear understanding of the conditional- ization process at the basis of the updating of probabilities. That is to say, just as a probability changes not only if the hypothesis changes but also when the evidence changes, in the same way the expectation of the unknown value of a quantity may change when the evidence changes.

The explicit acknowledgment of the relative character of probability as well as that of all concepts based on it played a revolutionary role in the history of science. Unfortunately, this role is far from fully understood. This has serious consequences. People which refuse to acknowledge the crucial role of the evidence in probability assertions—as all schools of thought that, notwithstanding the various schisms, pursue in founding probabilities on frequencies—are unable to obtain a clear insight into the notion of expectation. For example, the fact that a long-term expectation changes over time is indissolubly tied to a probability that is changing with the evidence. The notion of relevance quotient, too, cannot be introduced without having at one's disposal a relative notion of probability. And the relevance quotient is the powerful intellectual tool that Keynes envisaged to account for stochastic dependence.

On the other hand, it is well known that Keynes expressed sharp criticisms against the manner in which the character and causes of cyclical fluctuations were submitted to statistical test (see Carabelli, 1988, ch. 10). Keynes was against econometric studies as the mathematical way of formalizing economics (which at the time was done by means of the differential calculus). This could superficially appear to be, as it were, in manifest contrast to the attitude we outlined with examples in section 8.6. But this contrast disappears when we endeavor to reflect upon the role Keynes assigned to probability in economics: a reflection that cannot be carried out without clearly distinguishing between epis- temic and ontic probabilities.

Considering first epistemic probabilities, we note that, as is well known, probabilities are extensively used to perform statistical (inductive) inferences. In this respect it should be emphasized that epistemological probabilities may be objective as well as subjective. Statistical inferences are essentially of two types: the control of hypotheses and the estimation of unknown values of quantities. To the control of hypotheses belong tests of significance, that is, the attempt to falsify hypotheses. This is done by assuming the validity of a statistical hypothesis to be used in order to determine hypothetical probabilities. If the observed result has a low probability under the assumed hypothesis, then the hypothesis is rejected. From this point of view statistical estimation is distinct from a test of significance. Such an estimation aims at evaluating the unknown value of a quantity by proceeding from experimental observations, which can be carried out in different ways. Frequentists use methods suitable for identifying the value of that quantity; this is, for example, the goal of Fisher's maximum likelihood estimation. But estimation can also be conducted in probability terms, that is, by specifying a probability distribution on the possible values of the quantity; this is what Bayesians, whether classical or modern, do. A variant of the probabilistic approach to estimation is predictive inferences by means of the relevance quotient and the condition of invariance. Thus, the probabilities used in statistical inferences are merely tools by which it is possible to quantify one's ignorance about a world imagined, in the great majority of cases, as being governed by deterministic laws. The work done by econometricians is of this type. The notion of probability used by econometricians is epistemic. Keynes was opposed to this way of using probability notions in economics. We believe that this is the way in which Keynes's criticism of Tinbergen's view should be interpreted (Keynes, 1939).

Nobody can deny that the use of epistemic probabilities is widespread. We maintain that probability has a true ontic aspect besides its old epistemic aspect. This statement is far from being new. With quantum mechanics it has become popular to acknowledge the existence of ontic probabilities. But we deny that the only ontic use of probability is linked to quantum mechanics. Ontic probabilities appear because of the noncausal character of a theory. This was clearly shown by Paul and Tatiana Ehrenfest, interpreteting and clarifying the work of Ludwig Boltzmann, especially his theorem on the probable increase in entropy over time. In genetics the ontic role of probability has been recognized since the time of Mendel. By contrast, this is not the case in economics. In general, economists do not recognize the fundamental lack of causal relations in the external world and the ensuing unpredictability of the future. This was not Keynes's way of thinking. In his economic writings he used neither tests of significance nor methods of estimation, but he referred to probability. We believe that Keynes took the indeterminateness of the economy for granted and that in order to account for it he referred to ontic probabilities. We are well aware that it is not easy to show the ontic aspect of probability in the economic writings of Keynes. This is for various reasons. First, at the beginning of the twentieth century only very few examples of the ontic use of probability were known. The most important one is the vindication of Boltzmann's Я-theorem accomplished by means of the stochastic process— the dogs and fleas model—envisaged by the Ehrenfests (Ehrenfest P. and

Ehrenfest T., 1907). Stochastic processes play a very important role in the ontic use of probability. In the bibliography of the Treatise, Keynes (1973a ) included about ten works of Markov, yet he was not acquainted with the revolutionary research on stochastic processes. As a consequence he could not use those formal results in his economic theory, and was content with an intuitive treatment of stochastic dependence, which he referred to as organicism (we shall return to this issue). Second, Keynes's economics is not a formal theory and neither stochastic processes nor transition probabilities can be dealt with in it. Third, and most important, Keynes based his probability theory upon a propositional calculus, and could not account for individual random variables. Since that time a lot has changed. As we have shown in section 8.6, a stochastic representation of economic processes is available. Referring to these examples, we shall attempt to clarify the way in which ontic probabilities may enter into a "non-causal" economic theory.

The distinctive mark of the examples we have considered consists in abandoning the attempt to follow the causal (deterministic) changes in state that would take place in a particular economic agent—as is done by the mathematical method usually introduced to formalize economics—and in studying from a probabilistic perspective the economic behavior of single economic agents. The transition probabilities (29), (33), and (35) refer to single agents. To obtain a full insight into the meaning of our examples, one must bear clearly in mind that the probability we have worked with accounts for single agents. We have inquired into the change which takes place in the strategy of an agent over time. The change is governed by probability conditions. These conditions produce the statistical behavior of the economic system expressed by a probability distribution, hence its average behavior too. This approach is clearly adopted in the two examples in section 8.6, especially the example concening customers and shops. We proceeded from a detailed probabilistic description of the behavior of agents with respect to strategies, considering the change of strategy as a mechanism leading actual economic systems to change over time. This is worth noting. The question we have answered is: what is the probability of choosing a new strategy when an agent gives up the old one? The macroscopic properties of the economic system result from the statistical behavior of agents, expressed by means of transition probabilities. Much more important, the analysis we have undertaken leads to a probability distribution that, as it were, characterizes the statistical equilibrium.

The stochastic process and its equilibrium probability distribution follow from the laws governing the probabilistic behavior of economic agents. These are: the basic rules of probability, exchangeability, invariance, the values of the initial probabilities and those of the relevance quotient. As a consequence it is quite natural to suppose that these agents obey the probability conditions from which follow the stochastic processes we have described. Because these conditions probabilistically describe properties of economic agents, they are not outside economics but inside it. Being inside the theory, the probability with which these conditions are expressed is ontic in character. This, too, holds true for the probability notions used by Keynes in economics. For Keynes expectations were not merely tools by which ignorance could be overcome but a way to understand and explain the behavior of economic agents. That is to say, the probability notions, directly or indirectly, used by Keynes are ontic because they supply a way of explaining and understanding economic reality.

These considerations lead us to the core of the problem. We have shown that our approach is able to deduce the equilibrium distributions of an economic system. As a consequence, the following question arises in a very natural way: do these distributions reflect the actual behavior of macroeconomic systems? An answer can only be provided by using a statistical inference. This is a typical task of epistemic probabilities. For example, with reference to stock price dynamics, only a statistical inference can reveal whether the behavior of bulls and bears is governed by negative binomial distributions. This means that the ultimate validity of the considered conditions is to be regarded as resting on the correspondence between deduced results and empirical findings. However, if such a control gives a positive answer, then the probabilistic conditions we have considered describe and explain the behavior of the agents trading in a single asset. In other words, this ought to mean that we have worked with probabilities able to understand and explain some aspects of economic reality.

At this point, another question arises: what about the probability of A Treatise on Probability? Or, is the probability studied in TP epistemic or ontic? Our answer is very clear: neither or both. The reason is simple. In TP Keynes dealt with probability in a formal way. In stating the fundamental theorems of probability he clearly asserts that he is dealing with the formal logic of probable knowledge (TP p. 125). Following Leibniz, he treat probability calculus as a branch of logic. For Keynes the probability relation is a primitive notion. As such it needs no explicit definition. All Keynes does is to introduce some axioms, like addition and multiplication rules, to be used in transforming probabilities, that is, in calculating probabilities of events given those of other events.

As a notion only implicitly defined through the axioms of an abstract theory, probability can be seen as both epistemic (objective or subjective) and ontic. As a matter of fact, Keynes used probability in both epistemic and ontic modes. In Part V of IP, which is devoted to the foundations of statistical inference, probability is clearly epistemic. This is not the case with the notion of probability used in The General Theory of Employment, Interest and Money (Keynes, 1973b ). This probability is ontic.

Unfortunately, the introduction of probability in economics took the epistemic route, and this route is far removed from that suggested by Keynes. The notion of probability that in the first half of the twentieth century began to be explicitly used in economics was epistemological in character. Having in mind a frequentistic notion of probability, econometricians narrowed the use of probability in economics to hypothetical probability, that is, to likelihood, the notion, strongly biased by frequentistic philosophy, which Ronald A. Fisher opposed to probability (see Fisher, 1956). As a matter of fact, tests of significance are based on likelihood. This is also the case with estimation, mainly aimed at specify values of deterministic quantities, which is performed in most cases with the method of maximum likelihood. An econometric model, such as a linear equation characterized by a set of parameters previously estimated, has to be tested against empirical observations. The result in the case of a small hypothetical probability (a fixed significance level) is the rejection (falsification) of the model, or else its (provisional) acceptance. In both testing and estimating econometricians aim at discovering the "true" values of the parameters, that is, the "true" description of a deterministic world. For econometricians probability is no more than a tool to be used under circumstances in which the knowledge of the conditions of the system is less than the maximal knowledge which would be theoretically possible: in other words, a tool able to ascertain the validity of a causal model. This attitude is in plain contrast with the Keynesian philosophy of probability.

We are convinced that a probabilistic perspective is at the core of Keynes's economic theory. But how is it possible to reconcile this perspective with the organicist approach to economics that on many occasions Keynes opposed to the atomistic approach? As Carabelli (1995) notes, this attitude is also present in Keynes's approach to the foundations of probability. The answer has to be found in the Laplacean theory of probability, that is, in the theory of probability Keynes was acquainted with. Moreover, one must not forget that the Laplacean theory was formulated with a view to explaining the outcomes of gambling systems which, from a probabilistic point of view, are independent systems. That theory was well suited to dealing with atomic event. Keynes's criticism is directed against the probability theory of the nineteenth century. Criticism of the principle of indifference is correct but completely out of date. The reason is simply that nobody defends that principle any more. Nowadays the theory of probability is very different from the classical theory criticized by Keynes.

We may now consider Keynes's organicistic attitude. According to Carabelli, this attitude is typical of his approach both to economics and probability. As Carabelli writes:

The final evidence of Keynes's non-atomistic attitude can be found through the analysis of the economic works he wrote between 1908 and 1921 while revising the Treatise. An examination of them clearly reveals that Keynes does not radically change his organicist attitude.

(Carabelli, 1995, p. 153)

We agree with this assessment. Keynes envisages a theory of probability that is not well suited to dealing with stochastic dependence. Rightly, Keynes does not limit his attention to atomic events and consistently rejects this possibility when working in economics. In order to understand this, one has to consider the meaning of stochastic dependence. Without a clear idea of the very meaning of this notion, it is impossible to fully understand the organic interdependence that according to Keynes characterizes a monetary economy. The only way to understand this interdependence is to grasp the difference between the initial (absolute) probability of any given hypothesis and its final (conditional) probability.

At the very beginning of the nineteenth century, stochastic dependence was considered by Albert Einstein in order to explain Brownian motion. Keynes was not aware of this revolutionary turning point, which made it possible to obtain an insight into entangled phenomena. Now, by using appropriate stochastic processes, we are able to deal with stochastic dependence in a very effective way. In section 6 we have given two examples of this. Keynes warns against using correlation coefficients in economics. This was and is an important warning insofar as this coefficient is concerned with descriptive or inductive statistics. But, as we have seen, correlation can be used within the theory, and Keynes is not aware of this possibility. Once again the reason is that Keynes takes neither the works of Einstein nor those of Markov into account.

In the spirit of modern approaches to stochastic processes, we have used the notion of relevance quotient. We have shown that this quotient can be widely used in ascertaining the equilibrium distributions of systems ruled by exchangeability and invariance; that the notion of equilibrium is essentially probabilistic; and, what is more important, how it is possible to describe stochastic macroeconomic patterns by referring to the stochastic microeconomic behaviors of individual agents. In this respect, the notion of relevance quotient plays a crucial role. This quotient on the one hand makes it evident that economic behaviors are essentially stochastic and on the other hand measures the degree of stochastic dependence one will introduce in the theory.

The ontic use of Keynes's relevance quotient is crucial in describing and interpreting the stochastic behavior of economic agents. We are convinced that the only feasible route open to those who wish to follow the approach pioneered by Keynes is to introduce this approach into economics. Accepting this point of view has the notable consequence of getting rid of the paradoxical behavior of econometricians, who are used to maintaining, at least theoretically, a deterministic attitude towards both what happened and what happens but are compelled to refer to probabilities in order to foresee what actually will happen.

A probabilistic theory supplies a form of knowledge that leads to the understanding of both the microscopic and the macroscopic sides of our probabilistic world. We are acquainted with the fact that most writers maintain that probabilistic explanations of economic phenomena are not good enough. We have shown that a good explanation of some economic phenomena can be provided in terms of conditional probabilities. In this respect, our example points with great clarity to the role of conditional probabilities in economics. This role is clearly seen by Keynes, the first author who recognized that probability is a relation. Failure to appreciate the ontic status of probabilities is the true reason why probabilistic explanations have not been considered to be good explanations.

Finally, we would like to conclude our comment on probability and economics by recalling the words with which Keynes ends A Treatise on Probability:

The physicists of the nineteenth century have reduced matter to the collisions and arrangements of particles, between which the ultimate qualitative differences are very few; and the Mendelian biologists are deriving the various qualities of men from the collisions and arrangements of chromosomes. In both cases the analogy with a perfect game of chance is really present; and the validity of some current modes of inferences may depend on the assumption that it is to material of this kind that we are applying them. Here, though I have complained sometimes at their want of logic, I am in fundamental sympathy with the deep underlying conceptions of the statistical theory of the day. If the contemporary doctrines of biology and physics remain tenable, we may have a remarkable, if undeserved, justification of some of the methods of the traditional calculus of probabilities. [...] and it may turn—reversing Quetelet's expression—that "La nature que nous interrogeons c'est une urne."

{TP, p. 468)

Biology and statistical physics are still tenable, and much more so than at the beginning of the nineteenth century. The probability conditions we have supposed to hold in our examples have reduced to transition probabilities the study of agents who are in a condition which is changing over time. This probability mechanism is the same one that governs, as Keynes says above, the "collisions and arrangements of particles" {see for example, Costantini and Garibaldi, 2004). From this perspective, what Keynes expresses when concluding his Treatise can be interpreted as being "in fundamental sympathy" with the stochastic approach to economics we have outlined. However, a difference must be emphasized. The particles Keynes refers to are molecules, that is, classical particles whose behavior is governed by stochastic independence. The agents we have considered are comparable not to classical particles but to quantum particles. They are not atomistic agents but organicistic agents, that is, agents whose probabilistic behavior is governed by stochastic dependence.

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