«Contents 1. The Post-Keynesians and the Problem 2. The Logical, Subjective, and Intersubjective Interpretations of Probability 3. Probability in ...»
PROBABILITY AND UNCERTAINTY IN KEYNES’S GENERAL THEORY
by Donald Gillies, University College London.
1. The Post-Keynesians and the Problem
2. The Logical, Subjective, and Intersubjective Interpretations of Probability
3. Probability in Keynes’s Theory of Long-Term Expectation
4. Some Concluding Remarks in favour of the Post-Keynesians
1. The Post-Keynesians and the Problem
In the last two decades, a great deal of attention has been devoted to the question of probability and uncertainty in Keynes’s General Theory by a group often referred to as the ‘Post-Keynesians’. As I will be making a good deal of use of the researches of this group in the present paper, I will begin by saying a little in general terms about the group and its ideas.
After the second world war, Keynesian economics became dominant in the British academic community, and British governments to a large extent followed the advice of Keynesian economists. Keynesian economics had a similarly important (even if not always quite so dominant) rôle in other advanced capitalist countries in the same period. During the 1970’s, however, Keynesian economics came under increasing criticism from the monetarist school, and Keynesian economists began to lose both academic and political influence. In Britain the election of Margaret Thatcher in 1979 signalled the end of the government’s use of Keynesian policies, and the adoption instead of free market policies based on monetarist economic theory. Many academic economists went over to the new (or rather revived) free market ideas. However, some remained convinced of the value of Keynesian ideas in economics.
The remaining followers of Keynes were at this point faced with the unhappy situation that the academic and political influence of their ideas was declining, and that these ideas were being increasingly criticized as inadequate. The Post-Keynesians reacted to this crisis in a way which has parallels in other intellectual schools at a time of difficulty. They argued that the Keynesian economics which had prevailed in the period 1945-75, and which was now increasingly being rejected, was not in fact the economics which Keynes himself had proposed in his General Theory, but rather a simplified and unsatisfactory version of what Keynes had said. They suggested that Keynes’s approach could be revived by a return to Keynes’s original ideas.
The object of the Post-Keynesian attack was the standard text-book account of Keynesian economics based on Hicks’s IS-LM diagram. Skidelsky explains the origin of this kind of Keynesianism with characteristic clarity and historical erudition.
He writes (1992, 538):
‘The IS-LM diagram, first drawn by John Hicks in 1936, is the General Theory as it has been taught to economics students ever since: 384 pages of argument whittled down to four equationsand two curves. Hicks, Harrod, Meade and Hansen in America, the leading constructors of ‘IS-LM’ Keynesianism, had a clear motive: to reconcile Keynesians and nonKeynesians, so that the ground for policy could be quickly cleared. These early theoretical models incorporated features which were not at all evident in the magnum opus, but which conformed more closely to orthodox theory. The constructors of these models also thought they were improving the original building.’ A little later in a section significantly entitled: ‘Vision into Algebra’, Skidelsky writes (1992, 611):
‘The mathematisation of the General Theory started immediately it was published but it was left to Hicks to map the mathematics on to a two-curve diagram which became the accepted form of the General Theory. His famous paper ‘Mr. Keynes and the Classics: A Suggested Reinterpretation’ was published in Econometrica in April 1937. What Hicks does is to turn Keynes’s logical chain of reasoning designed to expose the causes which drive the economy towards a low employment trap into a generalised system of simultaneous equations, devoid of causal significance, with the behavioural characteristics of the propensities to be filled in according to assumption. The ‘generalised’ system has room for Keynes’s ‘special theory’, but also, for example, for the Treasury view, which Keynes wrote the General Theory to refute.’ IS-LM Keynesianism does not include any reference to probability and uncertainty.
But the Post-Keynesians argue that probability and uncertainty were central to the real Keynes who wrote a Treatise on Probability in 1921, and in his General Theory of 1936 made implicit use of probability in his theory of long-term expectation. The Post-Keynesians have accordingly carried out a great deal of valuable historical research on the evolution of Keynes’s ideas on probability, and his use of probability in the General Theory.
Post-Keynesianism began in the 1980’s as a reaction to the decline in academic and political influence of post-war IS-LM Keynesianism. Perhaps the first significant PostKeynesian book was the first volume of Skidelsky’s masterly life of Keynes which appeared in
1983. This covers Keynes’s life up to 1920, and discusses Keynes’s early philosophical work on probability and induction - a topic which had been ignored for many years. Other PostKeynesian books to appear in the 1980’s include Carabelli (1988), Fitzgibbons (1988), and O’Donnell (1989). In 1985 a collection of papers edited by Lawson and Pesaran appeared.
This contains articles by Victoria Chick, Alexander and Sheila Dow, Tony Lawson, and John Pheby. Somewhat younger Post-Keynesians include Bateman (1987, 1988, and 1996), Davis (1994), and Runde (1994, 1996). In what follows I will make use of this Post-Keynesian work on the reconstruction of Keynes’s ideas.1 Let us now turn to Keynes General Theory of 1936, which I will take in conjunction with his 1937 article: ‘The General Theory of Employment’, written to summarise and defend his book. In these works Keynes argues that the amount of investment is the key factor in determining the performance of the economy as a whole. As we shall see he regards it as the ‘causa causans’ of ‘the level of output and employment as a whole’ (1937, 121). Let us start therefore with Keynes’s analysis of investment. We shall consider two of the concepts which Keynes introduces in this connection, namely: prospective yield and demand price of the
investment. Keynes defines these as follows (1936, 135 & 137):
‘When a man buys an investment or capital-asset, he purchases the right to the series of prospective returns, which he expects to obtain from selling its output, after deducting the running expenses of obtaining that output, during the life of the asset. This series of annuities Q1, Q2,... Qn it is convenient to call the prospective yield of the investment....
If Qr is the prospective yield from an asset at time r, and dr is the present value of £1 deferred r years at the current rate of interest, Qrdr is the demand price of the investment; and investment will be carried to the point where Qrdr becomes equal to the supply price of the investment as defined above. If, on the other hand, Qrdr falls short of the supply price, there will be no current investment in the asset in question.’ So any decision to invest depends crucially on the quantity Qrdr (the demand price of the investment) which is the sum of the prospective annual yields discounted at the current rate of interest. But now the crucial problem arises, because the prospective yield Q1, Q2,... Qn of an investment is not known, and and consequently Qrdr cannot be calculated.
As Keynes puts it (1936, 149-50):
‘The outstanding fact is the extreme precariousness of the basis of knowledge on which our estimates of prospective yield have to be made. Our knowledge of the factors which will govern the yield of an investment some years hence is usually very slight and often negligible.
If we speak frankly, we have to admit that our basis of knowledge for estimating the yield ten years hence of a railway, a copper mine, a textile factory, the goodwill of a patent medicine, an Atlantic liner, a building in the City of London amounts to little and sometimes to nothing; or even five years hence.’ Since the actual future yields are unknown, they must be replaced in calculating Qrdr to make an investment decision by expected yields. A decision to invest consequently depends on what Keynes calls the state of long-term expectation (the title of the famous chapter 12 of the General Theory). Now the notions of expectation and of probability are interdefinable. If we take expectation as the starting point, we can define probabilities in terms of expectations, and vice versa.2 If then Keynes is using the notion of expectation in its standard sense, he is implicitly operating with a concept of probability, and it is natural to ask what should be the interpretation of the probabilities involved. This then brings us to the fundamental question with which this paper is concerned, namely: ‘what is the most appropriate interpretation of probability in Keynes’s General Theory?’ The Post-Keynesians have devoted a great deal of attention to this problem, but, before we can consider their arguments in detail, it will be necessary to give a brief explanation of the various interpretations of probability.3
2. The Logical, Subjective, and Intersubjective Interpretations of Probability
Different versions of the logical interpretation of probability have been developed by different authors, but here, naturally, we will be concerned with Keynes’s version as expounded in his 1921 Treatise on Probability. In the case of deductive logic a conclusion is entailed by the premises, and is certain given those premises. Thus, if our premises are that all ravens are black, and George is a raven, it follows with certainty that George is black. But now let us consider an inductive, rather than deductive, case. Suppose our premises are the evidence (e say) that several thousand ravens have been observed, and that they were all black.
Suppose further that we are considering the hypothesis (h say) that all ravens are black, or the prediction (d say) that the next observed raven will be black. Hume argued, and this is in agreement with modern logic, that neither h nor d follow logically from e. Yet even though e does not entail either h or d, could we not say that e partially entails h and d, since e surely gives some support for these conclusions? This line of thought suggests that there might be a logical theory of partial entailment which generalises the ordinary theory of full entailment which is found in deductive logic. This is the starting point of Keynes’s approach to
probability. He writes (1921, 52):
‘Inasmuch as it is always assumed that we can sometimes judge directly that a conclusion follows from a premiss, it is no great extension of this assumption to suppose that we can sometimes recognise that a conclusion partially follows from, or stands in a relation of probability to a premiss.’ So a probability is the degree of a partial entailment. Keynes further makes the assumption that if e partially entails h to degree p, then, given e, it is rational to believe h to degree p. For Keynes probability is degree of rational belief not simply degree of belief. As
he says (1921, 4):
‘... in the sense important to logic, probability is not subjective. It is not, that is to say, subject to human caprice. A proposition is not probable because we think it so. When once the facts are given which determine our knowledge, what is probable or improbable in these circumstances has been fixed objectively, and is independent of our opinion. The Theory of Probability is logical, therefore, because it is concerned with the degree of belief which it is rational to entertain in given conditions, and not merely with the actual beliefs of particular individuals, which may or may not be rational.’ Here Keynes speaks of probabilities as being fixed objectively, but he is not using objective to refer to things in the material world. He means objective in the Platonic sense, referring to something in a supposed Platonic world of
The next question which might be asked regarding Keynes’s approach is the following:
‘how do we obtain knowledge about this logical relation of probability?’ Keynes’s answer is that we get to know at least some probability relations by direct acquaintance or immediate logical intuition. As Keynes says (1921, 13): ‘We pass from a knowledge of the proposition a to a knowledge about the proposition b by perceiving a logical relation between them. With this logical relation we have direct acquaintance.’ A problem which arises on this account is how we can ever assign numerical values to probabilities. Keynes indeed thinks that this is possible only in some cases, and writes on this point (1921, 41): ‘In order that numerical measurement may be possible, we must be given a number of equally probable alternatives.’ So in order to get numerical probabilities we have to be able to judge that a number of cases are equally probable and to enable us to make this judgement we need an a priori principle. This a priori principle is called by Keynes the
Principle of Indifference, and he gives the following statement of it (1921, 42):
‘The Principle of Indifference asserts that if there is no known reason for predicating of our subject one rather than another of several alternatives, then relatively to such knowledge the assertions of each of these alternatives have an equal probability.’ Unfortunately the Principle of Indifference leads to a number of paradoxes. Keynes gives a full account of these in chapter IV of his Treatise, and makes an attempt to solve them. Yet is has to be said that his solution is far from satisfactory. This concludes my brief account of Keynes’s version of the logical theory of probability. Let us now turn to the subjective interpretation.
The subjective theory of probability was discovered independently and at about the same time by Frank Ramsey in England, and Bruno de Finetti in Italy. Their two versions of the theory are broadly similar, though there are important differences which are well described in Galavotti (1991). In what follows I will concentrate mainly on Ramsey since his work is directly connected with that of Keynes.