- Email: [email protected]
Echoes of Edgeworth
European
Economic
Echoes
Review
37 (1993) 491499.
North-Holland
of Edgeworth
The problem of indeterminacy John Sutton London
School of Economics,
London.
‘Where all are monopolists occupatton ___ There would congemal to their mentahty.’
UK
the abstract economists survrve only the emprctal
. . would be deprived of theu school, flourishing in a chaos Edgeworth (1925, p. 139)
1. A very old problem The game-theoretic 1.0. literature of the 1980s has rediscovered one of the oldest issues in economics; it has also provided us at last with some machinery with which to attack it. The issue is easily stated: outside the confines of the simple perfect competition and monopoly models, economic mechanisms typically fail to pin down a unique market outcome. Indeterminacy is the norm. The question is, how do we learn to live with this? This problem was well understood by a number of economists from the late 1860s onwards. [See especially the Jevons-Jenkins correspondence in Black (1977).] Edgeworth’s contribution to the question is well known. Alone among his contemporaries, he set out an analytical response, which was to be the only fundamental contribution to the question for generations. To Edgeworth, the problem of indeterminacy was central; the question was, how was it to be handled analytically? Already in 1881, he stated The problem to which attention is specially directed in this introductory summary is: How fur contract is indeterminate - an inquiry of more than theoretical importance, if it is shown not only that indeterminateness tends to [be present]’ widely, but also in what direction an escape from its evils is to be sought. There are two possible responses to the problem of indeterminacy. The first is to add a mixture of extra assumptions and extra institutional structure to the problem in order to pin down a unique outcome. This response was implicit in the mainstream literature for a century; from 1880 Correspondence
‘As amended
to: John Sutton, London School of Economrcs, London WCZA 2AE, UK. by hand in Edgeworth’s own copy (LSE Library), the original reads ‘prevent’.
00142921/93/$06.00
EER-
K
ICI 1993-Elsevier
Science Publishers
B.V. All rights reserved
492
J. Sutton, The problem of indeterminacy
to 1980 it reigned supreme. I will argue that this response has now unravelled. The second response is to accept the fact that economic theory simply does not constrain outcomes as tightly as that: and that the only sound way forward is to come to terms with this; this was where Edgeworth’s instincts led.
2. The exchange economy: Blocking arguments Edgeworth’s analysis of the Exchange Economy is reviewed by Werner Hildenbrand in the present volume. It begins from the observation that two agents may agree on any outcome which lies on that segment of the contract curve along which each receives at least his ‘no trade’ level of utility. Now let two A’s bargain with two B’s. Suppose a trade was now proposed at which the utility of the A’s is reduced to its ‘no-trade’ minimum. Such an outcome can be broken by a coalition of one B and two A’s, for trades within this group can make all three better off than they were. If we exclude allocations which can be ‘blocked’ by some coalition of traders, the segment of the contract curve along which trades can occur (the ‘core’) will shrink. This observation led Edgeworth to state that as the number of traders is replicated indefinitely, the core shrinks to the competitive equilibrium [Edgeworth (1881, pp. 35-38)]. What is interesting about this argument in the present context is Edgeworth’s appeal to a simple mechanism, which appears to be independent of many institutional details of bargaining, which allows him to place a bound on the set of outcomes which can be supported as equilbria. It is to this point that I shall return in section 5. In Appendix F of his Principles of Economics (1890), Marshall argued that indeterminacy was not materially affected by increasing the number of agents [Marshall (1890, Vol. 1, p. 792)]. Rather, he identified the source of indeterminacy in the fact that agents were assumed to have non-linear utility functions. He suggested a model in which agents’ utility functions were linear in one of the goods (money). In such a setting, the contract curve becomes a straight line, parallel to one of the axes of the box. Marshall argued that, in this setting, there was no indeterminacy. This, however, is false. In Marshall’s model, nothing substantial has changed.2 In an article published in the Giornale degli Economiste in 1891, ‘The contract curve delines a utihty possibility frontier which constitutes the ‘cake’ over which agents may bargam As usual. we can ask the question, let some pomt on this frontier represent the outcome of bargaining. At what rate of exchange ~111 agents be Indifferent to small transfers m the neighbourhood of this point? In general, this rate of exchange WIII differ from point to point on the frontier. In Marshall’s example It IS the same at all points. This special feature of Marshall’s example IS a curiosum; it doesn’t bear on the point at issue at all.
J. Sutton, The problemof indeterminuc)
493
Edgeworth took issue with Marshall, and this led to an exchange of letters. Nothing captures so pungently the flavour of the relationship between Marshall and his diffident critic as Marshall’s de hut en bas letter of April 4, 1891.3 In the end, Edgeworth appeared to yield, not to logic, but to rhetoric. In his introduction to the translation of his 1891 article published in his Collected Papers (1925) however, Edgeworth restates his position quite explicitly: indeterminacy is a central issue. These concerns, which Edgeworth first expressed in 1881, were to resurface throughout his later investigations in the theory of oligopoly.
3. Oligopoly: Irregular vibrations Edgeworth’s views on oligopoly are set out in his well-known paper, ‘The Pure Theory of Monopoly’, and his earlier note ‘On the Determinateness of Economic Equilibrium’, first published in Italian in 1897 and 1891 respectively, and later reprinted in translation [Edgeworth (1925)]. He begins with a rather cavalier dismissal of Cournot’s duopoly model: ‘Cournot’s conclusion has been shown to be erroneous by Bertrand for the case in which there is no cost of production; by Professor Marshall for the case in which cost follows the law of increasing returns; and by the present writer for the case in which the cost follows the law of diminishing returns.’ This three-way dismissal of Cournot is less than helpful. The arguments of Bertrand and Marshall are each quite different in kind to that of Edgeworth, and the final form of Edgeworth’s argument can be extended beyond the confines of the diminishing returns case. Edgeworth’s argument involves two lines of attack. To the passage just quoted, he attaches a long footnote spelling out his own refutation for the diminishing returns case, by reference to a figure in his earlier Mathematical Psychics ( 1881). That figure relates to two agents trading in an exchange economy, and is a precursor of the ‘Edgeworth Box’ diagram. What Edgeworth asks us to do, is to think of the duopoly mode1 as being analogous to a situation where two type B traders meet with a very large number of type A’s He then invites the reader to see that a ‘monopoly solution’, here defined as one in which all trading surplus accrues to the B’s,
‘Partial quotations cannot do It Justice: 1 can only refer readers to Gudlebaud’s editlon for the original [Marshall (1890, Vol. 2, pp. 79%798)]. 4Bertrand’s argument was the familiar ‘undercutting’ one, whereas Marshall argued that the only candidate equdibrlum under Increasing returns is a monopoly outcome [Bertrand (1883); Marshall (1890)].
494
J. Sutton, The problem of indeterminuc)
is not sustainable, ‘for it will always be in the interests of one or more of the A’s to re-contract with one or both of the B’s ‘. Now while this analogy is an imperfect one, what is of interest is the notion that the appropriate way to look at the oligopoly problem is to think of a candidate equilibrium as being untenable because of the availability of blocking coalitions. In the light of this argument, Edgeworth says that his original conclusion, that some stable outcome would emerge [presented on p. 116 of Mathematical Psychics (lSSl)], is erroneous. He now, in 1897, realises that ‘there will be an indeterminate tract through which the index of value will oscillate, or rather will vibrate irregularly for an indefinite length of time’ [Edgeworth (1925, p. 118)]. Edgeworth now turns to his second line of attack. He sets out a simple duopoly model in which the firms face the market demand schedule x = n( 1 - p), n being the number of consumers. A zero cost monopolist would set price p = i and sell ($rr units; while the market is supplied by two firms, each with zero cost up to a fixed capacity level of (2)~ This is Edgeworth’s model. His analysis now proceeds very much in spirit of Bertrand; instead of a direct appeal to some blocking argument, he imagines each lit-m to set a price at a point in time. He sets up what is in effect a reaction function (his ‘watershed of the utility surface’) and then proceeds to argue that no prices exist at which, in modern language, each firm is using an optimal reply to the other. What he does not do is to compute cycles; instead, his emphasis is on the existence of a zone of indeterminacy; and his next topic lies in discussing how the size of that zone might be narrowed if the goods were complements, rather that (perfect) substitutes. Given Edgeworth’s first line of attack - the analogy with the 2-person exchange economy - what we might expect to find here is some discussion of whether the zone of indeterminacy might not be narrowed as the number of agents increases. But this is not the direction Edgeworth takes. Where would it lead to? This is an interesting question, which we postpone to section 5 below. Edgeworth’s main concern lay elsewhere: What he wanted to emphasise was that indeterminacy is the norm. It is this view of Edgeworth’s which was to fall out of favour in the decades that followed. 4. A century of (in)determinacy One way of dealing with the problem of indeterminacy is to simply add in more assumptions and more ‘institutional’ structure to the problem, until a unique outcome is pinned down. It was this approach, in a series of different guises, which was to become the most popular approach in the decades that followed. The story of how this trend developed is a complex and fascinating one; I have space here to touch only on a few high points in the story.
J. Sutton. The problem of indeterminacy
495
It was not until the 1930s that widespread attention was once again focused on the middle ground between perfect competition and monopoly. Both the bargaining problem and the ‘imperfect competition’ model became major concerns in the literature. In both areas, however, the issue of indeterminacy as such seemed to fade into the background. On the bargaining side, Hicks opened up a new agenda through his concerns with the modelling of wage bargaining between the firm and a union. Hicks set out explicitly, and at length, his response to the views of Marshall and Edgeworth [Hicks (1930)]. He emphasised the importance of the indeterminacy issue to Edgeworth’s generation, saying it at least appeared to be the ‘centre of the great controversy’ between Edgeworth and Marshall. Having spelt out Edgeworth’s argument however, he then proceeded to play down its importance, claiming that the range of indeterminacy is not likely to be wide in practice. He does this in two stages. First, he points to its disappearance as the number of agents on each side rises, and them simply asserts that ‘in fact, of course, not many competitors will be required to make the indeterminateness vanish!’ Second, he describes, quite informally, a specific setting in which competition between rival firms and employers takes such a form that the equilibrium outcome is unique up to an integer constraint. He then states that ‘the range of indeterminacy involved is only that between the marginal product of n men and n + 1. This difference is usually regarded as the atom of economics, into which we need not pry’. What has happened here, albeit implicitly, is that a particular bargaining model has been adduced within which a unique outcome indeed appears; the ‘integer problem’ is neither here nor there. As Hicks was to do in his late modelling of wage bargaining [Hicks (1932)], his is postulating a reasonable but arbitrary bargaining framework within which uniqueness is assured. This line of approach is quite in line with the kind of response which was to be followed a little later in the analysis of product markets (‘imperfect competition’, broadly defined). Here, too, the question of indeterminacy seemed to have faded away. Instead, the central question becomes the one prompted by Chapter 4 of Joan Robinson’s Economics of Imperfect Competition (1933). This issue concerned the notion that the theory of imperfect competition might be ‘empirically empty’. Initially, concerns arose because standard comparative static results seemed ambiguous within Joan Robinson’s own (monopoly) model; but later these concerns widened to the point where the whole enterprise was called into question. Matters came to a head in the ‘Chamberlin versus Chicago’ debate in the Review of Economic Studies in 1961 [a summary of which will be found in Sutton (1989)]. It was becoming clear that different ways of designing the ‘imperfect competition’ model seemed equally plausible a priori, and that different variants of the model
496
J. Sutton. The
prohlrm of rndeternwuc~
might lead to different results. At first glance, this appears to be a quite different issue to Edgeworth’s indeterminacy - but in fact, it is merely the obverse side of that same problem. One of the nicest features of the gametheoretic literature of the 1980s was that it made this relationship transparent. In the game-theoretic literature, a remarkably wide range of problems came to be attacked using a simple recipe: first specify the structure of the model by writing down an extensive form game, and then look for a (subgame perfect or other) Nash Equilibrium of that game. Oligopoly and the Bargaining problems, were both reformulated within this framework. In each case, the challenge of explicitly specifying a game tree made it transparently obvious that a huge range of extensive forms might be postulated as reasonable a priori. It also became clear that (a) within any particular game (extensive form) a range of (subgame perfect) equilibria might be present: (b) the (set of) equilibrium outcome(s) might be very sensitive to the choice of extensive form; (c) it was always possible to narrow the range set of equilibrium outcomes by adding restrictions to the model, or by way of modifying the extensive form(s) considered. Now this makes clear at once that the problem of indeterminacy and the problem of ‘empirical emptiness’ are merely two sides of the same coin [problems (a) and (b) above, respectively]. Edgeworth, placing minimal restrictions on the bargaining process, faced a problem of indeterminacy. Choosing a particular extensive form in the manner of the post-Rubinstein literature can always pin down an answer, but the answer may be a fragile one [see Sutton (1986)]. This seems to be the right way of looking at Hicks’s dismissal of the indeterminacy problem in bargaining; the elimination of indetermmacy merely introduces another problem. Where does all this lead to? I have argued elsewhere that the best way forward lies in first exploring the degree to which empirically observed outcomes are constrained (bounded) by reference to predictions which are ‘robust’. Explicitly, these predictions are those which hold good across some minimal relevant class of models between which we cannot choose on a priori grounds, and between which we cannot distinguish by reference to observable market characteristics [Sutton (1990, 1991)]. By accepting that, within the set of bounds thus defined, in general ‘anything can happen’, we are then led to explore. step by step, special subsets of this class of models which may be justified a priori for some subset of markets. In this way we sacrifice the breadth of application of our (shrinking) set of models, for increasing precision in our predictions (set of equilibrium outcomes). In this way that we can steer a course between claiming too much for the
J. Sutton, The problem of zndeterminacy
497
predictive content of theory, and abandoning the field. Edgeworth was all too conscious of the dangers ahead, as the opening quote indicates. The emphasis, then, needs to be focused on how certain basic mechanisms may partially constrain the set of outcomes; and this brings us back to where we started from. 5. Edgeworth’s Blocking Argument revisited At first sight, Edgeworth’s ‘Blocking Argument’ seems to be exactly the kind of simple robust mechanism to which we should first appeal in placing (first order) restrictions on the data. But is it? The answer turns out to be rather complicated; like all such questions, the answer depends on the context. There are three contexts of interest. The first context is that of the bargaining problem; here, the answer is a qualified ‘yes’. The qualification derives from the fact that we have learned, within the non-cooperative bargaining literature, that the incorporation of mechanisms allowing any agent(s) to exit the negotiation and take up an ‘outside option’ must be built into the extensive form in a particular way, if ‘intuitively plausible’ results are to be obtained. [The most elementary example of this is the ‘Outside Option Principle’ of Shaked and Sutton (1984); see also Sutton (1986); Shaked (1987)J Edgeworth’s ‘Blocking Argument’ rests on the assumption that any subet of bargainers may opt out at any time into a subgame in which they alone, now irrevocably isolated, may turn to a division of some ‘smaller’ cake of their own. To capture this within a non-cooperative bargaining game would require this particular feature to be built into the extensive form; and it would then become a matter of judgement as to whether that restriction was empirically valid in any particular context of application. So the ‘Blocking Argument’ is a good one here, albeit in this qualified sense. In the second context, that of oligopoly, it fares less well. In an insightful paper published in 1970, Farrell took up the following question: Suppose we carry through the logic of Edgeworth’s appeal to the ‘Blocking Argument’ in the context of oligopoly. What restrictions does it place on the set of outcomes? Farrell was able to show formally how the argument could indeed restrict the space of outcomes - but, as he rightly pointed out, the ‘Blocking Argument’ is hard to justify here. The difficulty is that, one-shot game representations aside, the empirically relevant pricing model is the ‘dynamic’ game in which each firm sets, and varies, its prices over time, and the firms’ sales fluctuate passively as relative prices vary. Now in this context, a long literature on the ‘Folk Theorem’ assures us that it is possible to write down certain models in which the full range of prices between the monopoly price and the perfectly competitive price can be supported as Nash equilibrium outcomes. The literature also
498
J. Sutton.
The problem
oj indetermmacq
shows, especially in the context of models of cartel stability under uncertainty, that for some classes of models, the range of prices that can be supported as equilibria shrinks to a narrow band close to the competitive price as the number of competitors increases [see for example, Slade (1990)]. Meanwhile, on the empirical front, a substantial body of literature strongly underwrites the assertion that, on average, market prices tend to fall as concentration declines [Weiss (1989)]. Forging a rigorous link between this empirical regularity and the modern theoretical literature is a major challenge in the 1.0. field. In trying to forge such links, an appeal to a ‘blocking argument’ just doesn’t help: once we have a context in which the relevant space of outcomes is a sequence of price vectors which evolve over time, then any ‘blocking coalition’ which drives us away from some price vector today, may itself be blocked by a new coalition tomorrow - and once we allow this, we are back in the now-familiar world of Folk Theorems. This idea that the blocking coalition may itself be subject to disruption, either from within or from without, leads us to the third context of application: the general case of an exchange economy, and so, to the foundation of cooperative game theory. The way we motivate the ‘core’ idea is by way of a negative argument: allocations which are not in the core will not be observed because any such allocation, were it proposed, would be blocked. Now this appeal to a negative argument is fine, insofar as it goes. After all. this is just how, within non-cooperative game theory, we define Nash Equilibrium: we point out that any outcome which is not supported as a Nash Equilibrium will break down because some agent will have available a profitable deviation which breaks that outcome. To provide any generul positive motivation, by showing how the set of agents can ‘find’ or ‘move to’ a Nash Equilibrium within an_r non-cooperative game is notoriously difficult. With the core, the analogous problem is to provide an explicit mechanism by way of which agents would move to an allocation in the core. The context being a quite abstract one, we have little to hold on to by way of ‘institutional constraints’ which might delimit the models (extensive forms) considered ‘reasonable’ here. Of all the current research questions bequeathed us by Edgeworth, this is one of the most elusive. References Bertrand, J., 1983, Review of ‘Theorte mathemattque de la rtchesse sociale’ (Leon Walras). and Recherches sur les prmciples mathematiques de le theories des richesse (Augustin Cournot). Journal de Savants, 4999508. Black, RD. Collison, 1977, Paper and correspondence of Wtlltam Stanley Jevons. Vol. III: Correspondence 186331872 (Macmtllan, London). Edgeworth, F.Y., 1881, Mathemattcal psychtcs: An essay on the applicatton of mathematics to the moral sciences (C Keegan Paul & Co, London). Edgeworth, F.Y , 1925. Papers relating to pohttcal economy, Vols. I-III (Macmtllan, London).
J. Sutton,
The problem
of indeterminacy
499
Farrell, M.J., 1970. Edgeworth bounds for oligopoly prtces. Economica 37. 341-361 Hicks, J.R.. 1930, Edgeworth. Marshall and the Indeterminateness of wages. Economic Journal 11, 215-231. Htcks, J.R.. 1932, The theory of wages (Macmtllan, London). Marshall, Alfred. 1890, Principles of economics. 9th (variorum) edn. (1961). C.W. Guilleband, ed., 2 vols (Macmillan, London). Robmson, Joan, 1953, The economics of imperfect competttion (Macmillan, London). Shaked, Avner, 1987. Optmg out: Bazaars versus ‘high-tech’ markets. STICERD working paper no 87/159 (London School of Economtcs. London). Shaked, A. and J. Sutton, 1984. The semt-Walrasian economy, STICERD dtscusston paper no. 98 (London School of Economtcs, London) Slade, Margaret E.. 1990, Strategic pricing models and mterpretatton of prtce-war data. European Economtc Revtew 34, 524-537 Sutton. John, 1986. Non-cooperative bargammg theory: An mtroduction. Review of Economtc Studtes 53, 709-724. Sutton. John, 1989. Is imperfect competttion empntcally empty?, m G. Fetwel. ed., The economtcs of imperfect competttton and employment: Joan Robinson and beyond (Macmillan. London). Sutton, John. 1990, Explammg everything, explaimng nothmg: Game theoretic models m mdustrial orgamsation. European Economtc Revtew 34. 5055512 Sutton. John, 1991. Sunk costs and market structure (MIT Press, Cambridge, MA). Weiss, Leonard W., 1989, Concentration and price (MIT Press. Cambridge. MA).
Economic
Echoes
Review
37 (1993) 491499.
North-Holland
of Edgeworth
The problem of indeterminacy John Sutton London
School of Economics,
London.
‘Where all are monopolists occupatton ___ There would congemal to their mentahty.’
UK
the abstract economists survrve only the emprctal
. . would be deprived of theu school, flourishing in a chaos Edgeworth (1925, p. 139)
1. A very old problem The game-theoretic 1.0. literature of the 1980s has rediscovered one of the oldest issues in economics; it has also provided us at last with some machinery with which to attack it. The issue is easily stated: outside the confines of the simple perfect competition and monopoly models, economic mechanisms typically fail to pin down a unique market outcome. Indeterminacy is the norm. The question is, how do we learn to live with this? This problem was well understood by a number of economists from the late 1860s onwards. [See especially the Jevons-Jenkins correspondence in Black (1977).] Edgeworth’s contribution to the question is well known. Alone among his contemporaries, he set out an analytical response, which was to be the only fundamental contribution to the question for generations. To Edgeworth, the problem of indeterminacy was central; the question was, how was it to be handled analytically? Already in 1881, he stated The problem to which attention is specially directed in this introductory summary is: How fur contract is indeterminate - an inquiry of more than theoretical importance, if it is shown not only that indeterminateness tends to [be present]’ widely, but also in what direction an escape from its evils is to be sought. There are two possible responses to the problem of indeterminacy. The first is to add a mixture of extra assumptions and extra institutional structure to the problem in order to pin down a unique outcome. This response was implicit in the mainstream literature for a century; from 1880 Correspondence
‘As amended
to: John Sutton, London School of Economrcs, London WCZA 2AE, UK. by hand in Edgeworth’s own copy (LSE Library), the original reads ‘prevent’.
00142921/93/$06.00
EER-
K
ICI 1993-Elsevier
Science Publishers
B.V. All rights reserved
492
J. Sutton, The problem of indeterminacy
to 1980 it reigned supreme. I will argue that this response has now unravelled. The second response is to accept the fact that economic theory simply does not constrain outcomes as tightly as that: and that the only sound way forward is to come to terms with this; this was where Edgeworth’s instincts led.
2. The exchange economy: Blocking arguments Edgeworth’s analysis of the Exchange Economy is reviewed by Werner Hildenbrand in the present volume. It begins from the observation that two agents may agree on any outcome which lies on that segment of the contract curve along which each receives at least his ‘no trade’ level of utility. Now let two A’s bargain with two B’s. Suppose a trade was now proposed at which the utility of the A’s is reduced to its ‘no-trade’ minimum. Such an outcome can be broken by a coalition of one B and two A’s, for trades within this group can make all three better off than they were. If we exclude allocations which can be ‘blocked’ by some coalition of traders, the segment of the contract curve along which trades can occur (the ‘core’) will shrink. This observation led Edgeworth to state that as the number of traders is replicated indefinitely, the core shrinks to the competitive equilibrium [Edgeworth (1881, pp. 35-38)]. What is interesting about this argument in the present context is Edgeworth’s appeal to a simple mechanism, which appears to be independent of many institutional details of bargaining, which allows him to place a bound on the set of outcomes which can be supported as equilbria. It is to this point that I shall return in section 5. In Appendix F of his Principles of Economics (1890), Marshall argued that indeterminacy was not materially affected by increasing the number of agents [Marshall (1890, Vol. 1, p. 792)]. Rather, he identified the source of indeterminacy in the fact that agents were assumed to have non-linear utility functions. He suggested a model in which agents’ utility functions were linear in one of the goods (money). In such a setting, the contract curve becomes a straight line, parallel to one of the axes of the box. Marshall argued that, in this setting, there was no indeterminacy. This, however, is false. In Marshall’s model, nothing substantial has changed.2 In an article published in the Giornale degli Economiste in 1891, ‘The contract curve delines a utihty possibility frontier which constitutes the ‘cake’ over which agents may bargam As usual. we can ask the question, let some pomt on this frontier represent the outcome of bargaining. At what rate of exchange ~111 agents be Indifferent to small transfers m the neighbourhood of this point? In general, this rate of exchange WIII differ from point to point on the frontier. In Marshall’s example It IS the same at all points. This special feature of Marshall’s example IS a curiosum; it doesn’t bear on the point at issue at all.
J. Sutton, The problemof indeterminuc)
493
Edgeworth took issue with Marshall, and this led to an exchange of letters. Nothing captures so pungently the flavour of the relationship between Marshall and his diffident critic as Marshall’s de hut en bas letter of April 4, 1891.3 In the end, Edgeworth appeared to yield, not to logic, but to rhetoric. In his introduction to the translation of his 1891 article published in his Collected Papers (1925) however, Edgeworth restates his position quite explicitly: indeterminacy is a central issue. These concerns, which Edgeworth first expressed in 1881, were to resurface throughout his later investigations in the theory of oligopoly.
3. Oligopoly: Irregular vibrations Edgeworth’s views on oligopoly are set out in his well-known paper, ‘The Pure Theory of Monopoly’, and his earlier note ‘On the Determinateness of Economic Equilibrium’, first published in Italian in 1897 and 1891 respectively, and later reprinted in translation [Edgeworth (1925)]. He begins with a rather cavalier dismissal of Cournot’s duopoly model: ‘Cournot’s conclusion has been shown to be erroneous by Bertrand for the case in which there is no cost of production; by Professor Marshall for the case in which cost follows the law of increasing returns; and by the present writer for the case in which the cost follows the law of diminishing returns.’ This three-way dismissal of Cournot is less than helpful. The arguments of Bertrand and Marshall are each quite different in kind to that of Edgeworth, and the final form of Edgeworth’s argument can be extended beyond the confines of the diminishing returns case. Edgeworth’s argument involves two lines of attack. To the passage just quoted, he attaches a long footnote spelling out his own refutation for the diminishing returns case, by reference to a figure in his earlier Mathematical Psychics ( 1881). That figure relates to two agents trading in an exchange economy, and is a precursor of the ‘Edgeworth Box’ diagram. What Edgeworth asks us to do, is to think of the duopoly mode1 as being analogous to a situation where two type B traders meet with a very large number of type A’s He then invites the reader to see that a ‘monopoly solution’, here defined as one in which all trading surplus accrues to the B’s,
‘Partial quotations cannot do It Justice: 1 can only refer readers to Gudlebaud’s editlon for the original [Marshall (1890, Vol. 2, pp. 79%798)]. 4Bertrand’s argument was the familiar ‘undercutting’ one, whereas Marshall argued that the only candidate equdibrlum under Increasing returns is a monopoly outcome [Bertrand (1883); Marshall (1890)].
494
J. Sutton, The problem of indeterminuc)
is not sustainable, ‘for it will always be in the interests of one or more of the A’s to re-contract with one or both of the B’s ‘. Now while this analogy is an imperfect one, what is of interest is the notion that the appropriate way to look at the oligopoly problem is to think of a candidate equilibrium as being untenable because of the availability of blocking coalitions. In the light of this argument, Edgeworth says that his original conclusion, that some stable outcome would emerge [presented on p. 116 of Mathematical Psychics (lSSl)], is erroneous. He now, in 1897, realises that ‘there will be an indeterminate tract through which the index of value will oscillate, or rather will vibrate irregularly for an indefinite length of time’ [Edgeworth (1925, p. 118)]. Edgeworth now turns to his second line of attack. He sets out a simple duopoly model in which the firms face the market demand schedule x = n( 1 - p), n being the number of consumers. A zero cost monopolist would set price p = i and sell ($rr units; while the market is supplied by two firms, each with zero cost up to a fixed capacity level of (2)~ This is Edgeworth’s model. His analysis now proceeds very much in spirit of Bertrand; instead of a direct appeal to some blocking argument, he imagines each lit-m to set a price at a point in time. He sets up what is in effect a reaction function (his ‘watershed of the utility surface’) and then proceeds to argue that no prices exist at which, in modern language, each firm is using an optimal reply to the other. What he does not do is to compute cycles; instead, his emphasis is on the existence of a zone of indeterminacy; and his next topic lies in discussing how the size of that zone might be narrowed if the goods were complements, rather that (perfect) substitutes. Given Edgeworth’s first line of attack - the analogy with the 2-person exchange economy - what we might expect to find here is some discussion of whether the zone of indeterminacy might not be narrowed as the number of agents increases. But this is not the direction Edgeworth takes. Where would it lead to? This is an interesting question, which we postpone to section 5 below. Edgeworth’s main concern lay elsewhere: What he wanted to emphasise was that indeterminacy is the norm. It is this view of Edgeworth’s which was to fall out of favour in the decades that followed. 4. A century of (in)determinacy One way of dealing with the problem of indeterminacy is to simply add in more assumptions and more ‘institutional’ structure to the problem, until a unique outcome is pinned down. It was this approach, in a series of different guises, which was to become the most popular approach in the decades that followed. The story of how this trend developed is a complex and fascinating one; I have space here to touch only on a few high points in the story.
J. Sutton. The problem of indeterminacy
495
It was not until the 1930s that widespread attention was once again focused on the middle ground between perfect competition and monopoly. Both the bargaining problem and the ‘imperfect competition’ model became major concerns in the literature. In both areas, however, the issue of indeterminacy as such seemed to fade into the background. On the bargaining side, Hicks opened up a new agenda through his concerns with the modelling of wage bargaining between the firm and a union. Hicks set out explicitly, and at length, his response to the views of Marshall and Edgeworth [Hicks (1930)]. He emphasised the importance of the indeterminacy issue to Edgeworth’s generation, saying it at least appeared to be the ‘centre of the great controversy’ between Edgeworth and Marshall. Having spelt out Edgeworth’s argument however, he then proceeded to play down its importance, claiming that the range of indeterminacy is not likely to be wide in practice. He does this in two stages. First, he points to its disappearance as the number of agents on each side rises, and them simply asserts that ‘in fact, of course, not many competitors will be required to make the indeterminateness vanish!’ Second, he describes, quite informally, a specific setting in which competition between rival firms and employers takes such a form that the equilibrium outcome is unique up to an integer constraint. He then states that ‘the range of indeterminacy involved is only that between the marginal product of n men and n + 1. This difference is usually regarded as the atom of economics, into which we need not pry’. What has happened here, albeit implicitly, is that a particular bargaining model has been adduced within which a unique outcome indeed appears; the ‘integer problem’ is neither here nor there. As Hicks was to do in his late modelling of wage bargaining [Hicks (1932)], his is postulating a reasonable but arbitrary bargaining framework within which uniqueness is assured. This line of approach is quite in line with the kind of response which was to be followed a little later in the analysis of product markets (‘imperfect competition’, broadly defined). Here, too, the question of indeterminacy seemed to have faded away. Instead, the central question becomes the one prompted by Chapter 4 of Joan Robinson’s Economics of Imperfect Competition (1933). This issue concerned the notion that the theory of imperfect competition might be ‘empirically empty’. Initially, concerns arose because standard comparative static results seemed ambiguous within Joan Robinson’s own (monopoly) model; but later these concerns widened to the point where the whole enterprise was called into question. Matters came to a head in the ‘Chamberlin versus Chicago’ debate in the Review of Economic Studies in 1961 [a summary of which will be found in Sutton (1989)]. It was becoming clear that different ways of designing the ‘imperfect competition’ model seemed equally plausible a priori, and that different variants of the model
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might lead to different results. At first glance, this appears to be a quite different issue to Edgeworth’s indeterminacy - but in fact, it is merely the obverse side of that same problem. One of the nicest features of the gametheoretic literature of the 1980s was that it made this relationship transparent. In the game-theoretic literature, a remarkably wide range of problems came to be attacked using a simple recipe: first specify the structure of the model by writing down an extensive form game, and then look for a (subgame perfect or other) Nash Equilibrium of that game. Oligopoly and the Bargaining problems, were both reformulated within this framework. In each case, the challenge of explicitly specifying a game tree made it transparently obvious that a huge range of extensive forms might be postulated as reasonable a priori. It also became clear that (a) within any particular game (extensive form) a range of (subgame perfect) equilibria might be present: (b) the (set of) equilibrium outcome(s) might be very sensitive to the choice of extensive form; (c) it was always possible to narrow the range set of equilibrium outcomes by adding restrictions to the model, or by way of modifying the extensive form(s) considered. Now this makes clear at once that the problem of indeterminacy and the problem of ‘empirical emptiness’ are merely two sides of the same coin [problems (a) and (b) above, respectively]. Edgeworth, placing minimal restrictions on the bargaining process, faced a problem of indeterminacy. Choosing a particular extensive form in the manner of the post-Rubinstein literature can always pin down an answer, but the answer may be a fragile one [see Sutton (1986)]. This seems to be the right way of looking at Hicks’s dismissal of the indeterminacy problem in bargaining; the elimination of indetermmacy merely introduces another problem. Where does all this lead to? I have argued elsewhere that the best way forward lies in first exploring the degree to which empirically observed outcomes are constrained (bounded) by reference to predictions which are ‘robust’. Explicitly, these predictions are those which hold good across some minimal relevant class of models between which we cannot choose on a priori grounds, and between which we cannot distinguish by reference to observable market characteristics [Sutton (1990, 1991)]. By accepting that, within the set of bounds thus defined, in general ‘anything can happen’, we are then led to explore. step by step, special subsets of this class of models which may be justified a priori for some subset of markets. In this way we sacrifice the breadth of application of our (shrinking) set of models, for increasing precision in our predictions (set of equilibrium outcomes). In this way that we can steer a course between claiming too much for the
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predictive content of theory, and abandoning the field. Edgeworth was all too conscious of the dangers ahead, as the opening quote indicates. The emphasis, then, needs to be focused on how certain basic mechanisms may partially constrain the set of outcomes; and this brings us back to where we started from. 5. Edgeworth’s Blocking Argument revisited At first sight, Edgeworth’s ‘Blocking Argument’ seems to be exactly the kind of simple robust mechanism to which we should first appeal in placing (first order) restrictions on the data. But is it? The answer turns out to be rather complicated; like all such questions, the answer depends on the context. There are three contexts of interest. The first context is that of the bargaining problem; here, the answer is a qualified ‘yes’. The qualification derives from the fact that we have learned, within the non-cooperative bargaining literature, that the incorporation of mechanisms allowing any agent(s) to exit the negotiation and take up an ‘outside option’ must be built into the extensive form in a particular way, if ‘intuitively plausible’ results are to be obtained. [The most elementary example of this is the ‘Outside Option Principle’ of Shaked and Sutton (1984); see also Sutton (1986); Shaked (1987)J Edgeworth’s ‘Blocking Argument’ rests on the assumption that any subet of bargainers may opt out at any time into a subgame in which they alone, now irrevocably isolated, may turn to a division of some ‘smaller’ cake of their own. To capture this within a non-cooperative bargaining game would require this particular feature to be built into the extensive form; and it would then become a matter of judgement as to whether that restriction was empirically valid in any particular context of application. So the ‘Blocking Argument’ is a good one here, albeit in this qualified sense. In the second context, that of oligopoly, it fares less well. In an insightful paper published in 1970, Farrell took up the following question: Suppose we carry through the logic of Edgeworth’s appeal to the ‘Blocking Argument’ in the context of oligopoly. What restrictions does it place on the set of outcomes? Farrell was able to show formally how the argument could indeed restrict the space of outcomes - but, as he rightly pointed out, the ‘Blocking Argument’ is hard to justify here. The difficulty is that, one-shot game representations aside, the empirically relevant pricing model is the ‘dynamic’ game in which each firm sets, and varies, its prices over time, and the firms’ sales fluctuate passively as relative prices vary. Now in this context, a long literature on the ‘Folk Theorem’ assures us that it is possible to write down certain models in which the full range of prices between the monopoly price and the perfectly competitive price can be supported as Nash equilibrium outcomes. The literature also
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shows, especially in the context of models of cartel stability under uncertainty, that for some classes of models, the range of prices that can be supported as equilibria shrinks to a narrow band close to the competitive price as the number of competitors increases [see for example, Slade (1990)]. Meanwhile, on the empirical front, a substantial body of literature strongly underwrites the assertion that, on average, market prices tend to fall as concentration declines [Weiss (1989)]. Forging a rigorous link between this empirical regularity and the modern theoretical literature is a major challenge in the 1.0. field. In trying to forge such links, an appeal to a ‘blocking argument’ just doesn’t help: once we have a context in which the relevant space of outcomes is a sequence of price vectors which evolve over time, then any ‘blocking coalition’ which drives us away from some price vector today, may itself be blocked by a new coalition tomorrow - and once we allow this, we are back in the now-familiar world of Folk Theorems. This idea that the blocking coalition may itself be subject to disruption, either from within or from without, leads us to the third context of application: the general case of an exchange economy, and so, to the foundation of cooperative game theory. The way we motivate the ‘core’ idea is by way of a negative argument: allocations which are not in the core will not be observed because any such allocation, were it proposed, would be blocked. Now this appeal to a negative argument is fine, insofar as it goes. After all. this is just how, within non-cooperative game theory, we define Nash Equilibrium: we point out that any outcome which is not supported as a Nash Equilibrium will break down because some agent will have available a profitable deviation which breaks that outcome. To provide any generul positive motivation, by showing how the set of agents can ‘find’ or ‘move to’ a Nash Equilibrium within an_r non-cooperative game is notoriously difficult. With the core, the analogous problem is to provide an explicit mechanism by way of which agents would move to an allocation in the core. The context being a quite abstract one, we have little to hold on to by way of ‘institutional constraints’ which might delimit the models (extensive forms) considered ‘reasonable’ here. Of all the current research questions bequeathed us by Edgeworth, this is one of the most elusive. References Bertrand, J., 1983, Review of ‘Theorte mathemattque de la rtchesse sociale’ (Leon Walras). and Recherches sur les prmciples mathematiques de le theories des richesse (Augustin Cournot). Journal de Savants, 4999508. Black, RD. Collison, 1977, Paper and correspondence of Wtlltam Stanley Jevons. Vol. III: Correspondence 186331872 (Macmtllan, London). Edgeworth, F.Y., 1881, Mathemattcal psychtcs: An essay on the applicatton of mathematics to the moral sciences (C Keegan Paul & Co, London). Edgeworth, F.Y , 1925. Papers relating to pohttcal economy, Vols. I-III (Macmtllan, London).
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Farrell, M.J., 1970. Edgeworth bounds for oligopoly prtces. Economica 37. 341-361 Hicks, J.R.. 1930, Edgeworth. Marshall and the Indeterminateness of wages. Economic Journal 11, 215-231. Htcks, J.R.. 1932, The theory of wages (Macmtllan, London). Marshall, Alfred. 1890, Principles of economics. 9th (variorum) edn. (1961). C.W. Guilleband, ed., 2 vols (Macmillan, London). Robmson, Joan, 1953, The economics of imperfect competttion (Macmillan, London). Shaked, Avner, 1987. Optmg out: Bazaars versus ‘high-tech’ markets. STICERD working paper no 87/159 (London School of Economtcs. London). Shaked, A. and J. Sutton, 1984. The semt-Walrasian economy, STICERD dtscusston paper no. 98 (London School of Economtcs, London) Slade, Margaret E.. 1990, Strategic pricing models and mterpretatton of prtce-war data. European Economtc Revtew 34, 524-537 Sutton. John, 1986. Non-cooperative bargammg theory: An mtroduction. Review of Economtc Studtes 53, 709-724. Sutton. John, 1989. Is imperfect competttion empntcally empty?, m G. Fetwel. ed., The economtcs of imperfect competttton and employment: Joan Robinson and beyond (Macmillan. London). Sutton, John. 1990, Explammg everything, explaimng nothmg: Game theoretic models m mdustrial orgamsation. European Economtc Revtew 34. 5055512 Sutton. John, 1991. Sunk costs and market structure (MIT Press, Cambridge, MA). Weiss, Leonard W., 1989, Concentration and price (MIT Press. Cambridge. MA).