The most peculiar features of Walras's auction market are that, rather than selling each commodity one at a time, the 'auctioneer' attempts to sell all goods at once; and rather than treating each commodity independently, this auctioneer refuses to accept any price for a commodity until supply equals demand for all commodities. In Walras's words: First, let us imagine a market in which only consumer goods and services are bought and sold [...] Once the prices or the ratios of exchange of all these goods and services have been cried at random in terms of one of them selected as numeraire, each party to the exchange will offer at these prices those goods or services of which he thinks he has relatively too much, and he will demand those articles of which he thinks he has relatively too little for his consumption during a certain period of time. The quant.i.ties of each thing effectively demanded and offered having been determined in this way, the prices of those things for which the demand exceeds the offer will rise, and the prices of those things of which the offer exceeds the demand will fall. New prices now having been cried, each party to the exchange will offer and demand new quant.i.ties. And again prices will rise or fall until the demand and the offer of each good and each service are equal. Then the prices will be current equilibrium prices and exchange will effectively take place. (Walras 1954 [1874]) This is clearly not the way markets work in the real world.3 Nonetheless, this mythical construct became the way in which economics attempted to model the behavior of real-world markets.

Walras's auctioneer starts the market process by taking an initial stab at prices. These arbitrarily chosen prices are almost certainly not going to equate demand and supply for each and every commodity instead, for some commodities, demand will exceed supply, while for others supply will exceed demand. The auctioneer then refuses to allow any sale to take place, and instead adjusts prices increasing the price of those commodities where demand exceeded supply, and decreasing the price where demand was less than supply. This then results in a second set of prices, which are also highly unlikely to balance demand and supply for all commodities; so another round of price adjustments will take place, and another, and another.

Walras called this iterative process of trying to find a set of prices which equates supply to demand for all commodities 'tatonnement' which literally translates as 'groping.' He believed that this process would eventually converge to an equilibrium set of prices, where supply and demand are balanced in all markets (so long as trade at disequilibrium prices can be prevented).

This was not necessarily the case, since adjusting one price so that supply and demand are balanced for one commodity could well push demand and supply farther apart for all other commodities. However, Walras thought that convergence would win out because the direct effects on demand of increasing the price of a commodity where demand exceeds supply, which directly reduces demand would outweigh the indirect effects of changes in demand for other commodities. In his words: This will appear probable if we remember that the change from p'b to p''b, which reduced the above inequality to an equality, exerted a direct influence that was invariably in the direction of equality at least so far as the demand for (B) was concerned; while the [consequent] changes from p'c to p''c, p'd to p''d, which moved the foregoing inequality farther away from equality, exerted indirect influences, some in the direction of equality and some in the opposite direction, at least so far as the demand for (B) was concerned, so that up to a certain point they cancelled each other out. Hence, the new system of prices (p''b, p''c, p''d) is closer to equilibrium than the old system of prices (p'b, p'c, p'd); and it is only necessary to continue this process along the same lines for the system to move closer and closer to equilibrium. (Ibid.) 'Generalizing' Walras Walras's ruse, of an auctioneer who stopped any trades taking place until such time as demand equaled supply in all markets, was clearly artificial. However, it enabled economists to make use of the well-known and relatively simple techniques for solving simultaneous linear equations.

The alternative was to describe the dynamics of a multi-commodity economy, in which trades could occur at non-equilibrium prices in anywhere from a minimum of two to potentially all markets. At a technical level, modeling non-equilibrium phenomena would have involved nonlinear difference or differential equations. In the nineteenth century, the methodology for them was much less developed than it is now, and they are inherently more difficult to work with than simultaneous linear equations.

Walras's auctioneer was therefore arguably a justifiable abstraction at a time when, as Jevons put it, it would have been 'absurd to attempt the more difficult question when the more easy one is yet so imperfectly within our power' (Jevons 1888: ch. 4, para. 25).

But it suggests an obvious, dynamic, research agenda: why not see what happens when the artifact of no non-equilibrium trades is dispensed with? Why not generalize Walras's general equilibrium by removing the reliance upon the concept of equilibrium itself? Why not generalize Walras by dropping the fiction that everything happens at equilibrium?

This potential path was, for economics, the path not chosen.

Instead, the neocla.s.sical 'Holy Grail' became to formalize Walras's concept of equilibrium: to prove that general equilibrium existed, and that it was the optimum position for society.

Unfortunately, reality had to be seriously distorted to 'prove' that general equilibrium could be attained. But, for the reasons given in Chapter 8, economists would rather sacrifice generality than sacrifice the concept of equilibrium.

The pinnacle of this warping of reality came with the publication in 1959 of Gerard Debreu's Theory of Value, which the respected historian of economic thought Mark Blaug has described as 'probably the most arid and pointless book in the entire literature of economics' (Blaug 1998). Yet this 'arid and pointless' tome set the mold for economics for the next forty years and won for its author the n.o.bel Prize for economics.

'The formal ident.i.ty of uncertainty with certainty'

Walras's vision of the market, though highly abstract, had some concept of process to it. Buyers and sellers would haggle, under the guidance of the auctioneer, until an equilibrium set of prices was devised. Exchange would then take place, and those prices would also determine production plans for the next period. There is at least some primitive notion of time in this series of sequential equilibria.

No such claim can be made for Debreu's vision of general equilibrium. In this model, there is only one market if indeed there is a market at all at which all commodities are exchanged, for all times from now to eternity. Everyone in this 'market' makes all their sales and purchases for all of time in one instant. Initially everything from now till eternity is known with certainty, and when uncertainty is introduced, it is swiftly made formally equivalent to certainty. A few choice extracts give a clearer picture of Debreu's total divorce from reality: For any economic agent a complete action plan (made now for the whole future), or more briefly an action, is a specification for each commodity of the quant.i.ty that he will make available or that will be made available to him, i.e., a complete listing of the quant.i.ties of his inputs and of his outputs.

For a producer, say the jth one, a production plan (made now for the whole future) is a specification of the quant.i.ties of all his inputs and all his outputs. The certainty a.s.sumption implies that he knows now what input-output combinations will be possible in the future (although he may not know the details of technical processes which will make them possible).

As in the case of a producer, the role of a consumer is to choose a complete consumption plan. His role is to choose (and carry out) a consumption plan made now for the whole future, i.e., a specification of the quant.i.ties of all his inputs and all his outputs.

The a.n.a.lysis is extended in this chapter to the case where uncertain events determine the consumption sets, the production sets, and the resources of the economy. A contract for the transfer of a commodity now specifies, in addition to its physical properties, its location and its date, an event on the occurrence of which the transfer is conditional. This new definition of a commodity allows one to obtain a theory of uncertainty free from any probability concept and formally identical with the theory of certainty developed in the preceding chapters. (Debreu 1959; emphases added) I can provide no better judgment of the impact this brazenly irrelevant theory had on economics than that given by Blaug: Unfortunately this paper soon became a model of what economists ought to aim for as modern scientists. In the process, few readers realized that Arrow and Debreu had in fact abandoned the vision that had originally motivated Walras. For Walras, general equilibrium theory was an abstract but nevertheless realistic description of the functioning of a capitalist economy. He was therefore more concerned to show that markets will clear automatically via price adjustments in response to positive or negative excess demand a property that he labeled 'tatonnement' than to prove that a unique set of prices and quant.i.ties is capable of clearing all markets simultaneously.

By the time we got to Arrow and Debreu, however, general equilibrium theory had ceased to make any descriptive claim about actual economic systems and had become a purely formal apparatus about a quasi economy. It had become a perfect example of what Ronald Coase has called 'blackboard economics,' a model that can be written down on blackboards using economic terms like 'prices,' 'quant.i.ties,' 'factors of production,' and so on, but that nevertheless is clearly and even scandalously unrepresentative of any recognizable economic system. (Blaug 1998) A hobbled general It is almost superfluous to describe the core a.s.sumptions of Debreu's model as unrealistic: a single point in time at which all production and exchange for all time is determined; a set of commodities including those which will be invented and produced in the distant future which is known to all consumers; producers who know all the inputs that will ever be needed to produce their commodities; even a vision of 'uncertainty' in which the possible states of the future are already known, so that certainty and uncertainty are formally identical. Yet even with these breathtaking dismissals of essential elements of the real world, Debreu's model was rapidly shown to need additional restrictive a.s.sumptions the Sonnenschein-Mantel-Debreu conditions discussed in Chapter 3. Rather than consumers being able to have any utility function consistent with what economists decreed as rational, additional restrictions had to be imposed which, as one economist observed, came 'very close to simply a.s.suming that the consumers in the aggregate have identical tastes and income' (Diewert 1977: 361).

This was not the end of the restrictions. As Blaug observes above, Walras hoped to show that the process of tatonnement would lead, eventually, to equilibrium being achieved, and that the same outcome would follow even if disequilibrium trading occurred. In mathematical terms, he hoped to show that general equilibrium was stable: that if the system diverged from equilibrium, it would return to it, and that if the process of tatonnement began with disequilibrium prices, it would eventually converge on the equilibrium prices. Debreu abandoned this aspect of Walras's endeavor, and focused solely on proving the existence of general equilibrium, rather than its stability. But stability cannot be ignored, and mathematicians have shown that, under fairly general conditions, general equilibrium is unstable.

Positive prices and negative stability Walras's a.s.sumption that the direct effects of the price change would outweigh the indirect effects so that the process of tatonnement would converge on the set of equilibrium prices was reasonable, given the state of mathematics at the time. However, mathematical theorems worked out in the twentieth century established that, in general, this a.s.sumption is wrong.

These theorems established that the conditions which ensure that an economy can experience stable growth simultaneously guarantee that Walras's tatonnement process is unstable (Blatt 1983). Therefore if the auctioneer's first stab at prices is only a tiny bit different from the set of prices which would put all markets in equilibrium, his next stab derived by increasing prices for goods where demand exceeded supply, and vice versa will be farther away from the equilibrium set of prices. The process of tatonnement will never converge to the equilibrium set of prices, so if equilibrium is a prerequisite for trade, trade will never take place.

These theorems4 are too complex to be conveyed accurately by either words or figures, but in keeping with the objectives of this book, I'll attempt an explanation. If you don't want to twist your mind around the mathematical concepts involved, then please skip to the following heading ('A transitional methodology?').

The 'general equilibrium problem' is to find a set of prices which result in the amount consumers demand of each and every product equaling the amount supplied. Prices obviously have to be positive, as do the quant.i.ties demanded and the quant.i.ties produced.5 Before commodities can be demanded, they must be produced, and the means of production are simply other commodities. If the economy is going to last indefinitely, the system of production must be able to generate growth.

This can be described by a set of equations in which the prices are the variables, and the quant.i.ties required to produce each commodity are the coefficients. A single equation adds up the cost of inputs needed to produce a given commodity at a given price. There will be as many equations as there are commodities to produce.

It is then possible to separate the prices into a column of numbers called a vector, and the quant.i.ties into a square of numbers called a matrix where, as noted earlier, every element is either a positive number or zero. The properties of this matrix can then be a.n.a.lyzed mathematically, and its mathematical properties can be used to answer economic questions.

This matrix is known as a Leontief input-output matrix, after the Russian economist who first developed this method of a.n.a.lysis. The first row of such a matrix effectively says that 'a units of commodity a combined with b units of commodity b and z units of commodity z will produce 1 unit of commodity a.' It is the simplest method of describing a system of production, in that it implies that there is one and only one best way to make each commodity: no subst.i.tution of one technology for another is allowed.

While this is a much simpler model of production than economists like to work with, it turns out that the properties of this very simple system determine whether the equilibrium of any more general model is stable. If this simple system can't guarantee stability, then no more complex system is going to either (this is a general property of dynamic models: the stability of the system very close to its equilibrium is determined by its 'linear' parts, and Leontief's matrix is the linear component of any more complex model of production).

There are two stability conditions in the simple Leontief system: the quant.i.ties produced each year have to enable the system to reproduce itself (this won't happen if, for example, the required inputs of iron for year 10 exceed the output of iron in year 9); and the prices must be feasible (the iron-producing sector can't depend on the price of some required input to producing iron being negative, for example).

It turns out that the first stability condition is governed by a characteristic of the input-output matrix, whereas the second stability condition is governed by the same characteristic of the inverse of that matrix. As with simple constants, a matrix and its inverse have, to some extent, opposite properties. Thus if you have a constant a which is less than 1, then a squared will be much less than 1, a cubed even more so, and higher powers of a will eventually converge to zero. However, the inverse of a, 1/a, will be greater than 1, and powers of 1/a will blow out to infinity. If the stability of some system depends upon both a and the inverse of a being less than 1, then no number can fulfill both requirements, and the system is going to be unstable.

Since economic models are supposed to concern themselves with real economies, which can and do change in size, the general conclusion is that a real economy will never be in a state of general equilibrium. If economics is to have any relevance to the real world if economics is even to be internally consistent then it must be formulated in a way which does not a.s.sume equilibrium. Time, and dynamic a.n.a.lysis, must finally make an appearance in economic a.n.a.lysis.

A transitional methodology?

The founding fathers of economics had no problem accepting such a conclusion. In fact, to them, static a.n.a.lysis was merely a stop-gap measure, a transitional methodology which would be superseded by dynamic a.n.a.lysis as economics reached maturity. Jevons, for example, argued that 'If we wished to have a complete solution we should have to treat it as a problem of dynamics.' But he instead pioneered static a.n.a.lysis because 'it would surely be absurd to attempt the more difficult question when the more easy one is yet so imperfectly within our power' (Jevons 1888).

Similarly, and at more length, Marshall noted that.

The Mecca of the economist lies in economic biology rather than in economic dynamics. But biological conceptions are more complex than those of mechanics; a volume on Foundations must therefore give a relatively large place to mechanical a.n.a.logies; and frequent use is made of the term 'equilibrium,' which suggests something of statical a.n.a.logy. This fact, combined with the predominant attention paid in the present volume to the normal conditions of life in the modern age, has suggested the notion that its central idea is 'statical,' rather than 'dynamical.' But in fact it is concerned throughout with the forces that cause movement: and its key-note is that of dynamics, rather than statics. (Marshall 1920 [1890]: Preface, para. 19) At the end of the nineteenth century, J. B. Clark, the economist who developed the marginal productivity theory of income distribution (critiqued in Chapter 5), looked forward to the twentieth century as the period during which economic dynamics would supplant economic statics: A point on which opinions differ is the capacity of the pure theory of Political Economy for progress. There seems to be a growing impression that, as a mere statement of principles, this science will fairly soon be complete. It is with this view that I take issue. The great coming development of economic theory is to take place, I venture to a.s.sert, through the statement and solution of dynamic problems. (Clark 1898) In this paper, Clark gave many good reasons why economics should be a.n.a.lyzed using dynamics rather than statics. Foremost among these was that 'A static state is imaginary. All actual societies are dynamic; and those that we have princ.i.p.ally to study are highly so. Heroically theoretical is the study that creates, in the imagination, a static society' (ibid.).

One century later, economic dynamics has indeed been developed but not by the school to which J. B. Clark belonged. Instead, neocla.s.sical economics still by and large ignores the issue of time. Students are often told that dynamics is important, but they are taught nothing but statics. A typical undergraduate macroeconomics textbook, for example, states that 'the examination of the process of moving from one equilibrium to another is important and is known as dynamic a.n.a.lysis.' However, it then continues that 'Throughout this book we will a.s.sume that the economic system is stable and most of the a.n.a.lysis will be conducted in the comparative static mode' (Taslim and Chowdhury 1995).

The leading textbook used today to teach graduate students makes a similar claim that while other disciplines use dynamics, economists model processes as if they occur in equilibrium because economists are good at identifying equilibrium! Two-thirds through his voluminous 1,000-page tome, Mas-Colell, the current doyen of neocla.s.sical instruction, writes: We have, so far, carried out an extensive a.n.a.lysis of equilibrium equations. A characteristic feature that distinguishes economics from other scientific fields is that, for us, the equations of equilibrium const.i.tute the center of our discipline. Other sciences, such as physics or even ecology, put comparatively more emphasis on the determination of dynamic laws of change. In contrast, up to now, we have hardly mentioned dynamics.

The reason, informally speaking, is that economists are good (or so we hope) at recognizing a state of equilibrium but poor at predicting how an economy in disequilibrium will evolve.

Certainly there are intuitive dynamic principles: if demand is larger than supply, then the price will increase, if price is larger than marginal cost then production will expand, if industry profits are positive and there are no barriers to entry, then new firms will enter and so on. The difficulty is in translating these informal principles into precise dynamic laws. (Mas-Colell et al. 1995: 620) This is nonsense, and to give Mas-Colell his due I think he realizes it here. Economists model in equilibrium, not because they are 'good (or so we hope) at recognizing a state of equilibrium,' but simply because they can't get the results they want in dynamic a.n.a.lysis and have therefore not made the leap from static to dynamic modeling that has occurred in all other disciplines.

Mas-Colell admits this when he discusses the attempts to generalize Walras's tatonnement process to a disequilibrium one. While he argues that a two-commodity exchange economy is stable,6 he admits that this result does not generalize to three or more commodities: 'Unfortunately, as soon as [there are more than two goods] neither the local conclusions nor the global conclusions of the two-commodity case generalize' (ibid.: 622).

This may be unfortunate, but the correct reaction to it is to abandon static a.n.a.lysis and work in disequilibrium. This, clearly, is not what neocla.s.sical economists have done and unfortunately, economists of many other persuasions also use static a.n.a.lysis because they believe that equilibrium is the enduring state of the economy, while dynamics merely captures the transient moments between different equilibria. For example, a Sraffian economist defended static methodology in economics by arguing that '"static" a.n.a.lysis does not "ignore" time. To the contrary, that a.n.a.lysis allows enough time for changes in prime costs, markups, etc., to have their full effects' (Steedman 1992).

As this chapter shows, this confidence that 'the end point of a dynamic process is the state of static equilibrium' is false. Equally false was the belief of the founding fathers of economics, that dynamic a.n.a.lysis 'does not invalidate the conclusions of a static theory' (Clark 1898). But even if they were right, even if dynamic forces did lead, eventually, to static outcomes, it would still be invalid to model the economy using static techniques. Keynes put the case best in 1923, when he made his oft-quoted but rarely appreciated observation that 'in the long run we are all dead.' The full statement gives a rather better picture of his intent: 'But this long run is a misleading guide to current affairs. In the long run we are all dead. Economists set themselves too easy, too useless a task if in tempestuous seasons they can only tell us that when the storm is long past the ocean is flat again' (Keynes 1971 [1923]).

Keynes was right: it is not valid to ignore the transient state of the economy. As Fisher later observed in very similar terms, equilibrium conditions in the absence of disturbances are irrelevant, because disturbances will always occur. Whether equilibrium is stable or not, disequilibrium will be the state in which we live: We may tentatively a.s.sume that, ordinarily and within wide limits, all, or almost all, economic variables tend, in a general way, toward a stable equilibrium [...]

It follows that, unless some outside force intervenes, any 'free' oscillations about equilibrium must tend progressively to grow smaller and smaller, just as a rocking chair set in motion tends to stop.

But the exact equilibrium thus sought is seldom reached and never long maintained. New disturbances are, humanly speaking, sure to occur, so that, in actual fact, any variable is almost always above or below the ideal equilibrium [...]

Theoretically there may be in fact, at most times there must be over- or under-production, over- or under-consumption, over- or under-spending, over- or under-saving, over- or under-investment, and over or under everything else. It is as absurd to a.s.sume that, for any long period of time, the variables in the economic organization, or any part of them, will 'stay put,' in perfect equilibrium, as to a.s.sume that the Atlantic Ocean can ever be without a wave. (Fisher 1933: 339) We also live in a changing and normally growing economy. Surely we should be concerned, not with absolute levels of variables, but with their rates of change? Should not demand and supply a.n.a.lysis, for instance, be in terms of the rate of change of demand, and the rate of change of supply? Should not the outcome of supply and demand a.n.a.lysis be the rate of change of price and quant.i.ty over time, rather than static levels? Should not macroeconomics concern itself with the rate of change of output and employment, rather than their absolute levels?

Of course they should. As Keynes also once remarked, 'equilibrium is blither.' So why, fifty years after Keynes, are economists still blithering? Why do economists persist in modeling the economy with static tools when dynamic ones exist; why do they treat as stationary ent.i.ties which are forever changing?

There are many reasons, but the main one, as outlined in the previous chapter, is the extent to which the core ideological beliefs of neocla.s.sical economics are bound up in the concept of equilibrium. As a by-product of this, economists are driven to maintain the concept of equilibrium in all manner of topics where dynamic, non-equilibrium a.n.a.lysis would not only be more relevant, but frankly would even be easier. This obsession with equilibrium has imposed enormous costs on economics.

First, unreal a.s.sumptions are needed to maintain conditions under which there will be a unique, 'optimal' equilibrium. These a.s.sumptions are often justified by an appeal to Friedman's methodological 'a.s.sumptions don't matter' argument, but as Chapter 8 pointed out, this notion is easily debunked. However, most economists take it as an article of faith, with insidious results. If you believe you can use unreality to model reality, then eventually your grip on reality itself can become tenuous as Debreu's bizarre model of general equilibrium indicates.

Secondly, as shown in this chapter, even the unreal a.s.sumptions of general equilibrium theory are insufficient to save it from irrelevance, since even the model of general equilibrium has been shown to be unstable, so that no modeled or real economy could ever be in a state of equilibrium. Many of those who pioneered general equilibrium a.n.a.lysis are grudgingly conceding that these results require economics to radically alter direction. But they are also quite aware that lesser economists are, as Alan Kirman put it, 'not even concerned over the sea-worthiness of the vessel in which they are sailing' (Kirman 1989).

Thirdly, the emphasis on modeling everything as an equilibrium phenomenon has isolated economics from most if not all other sciences, where dynamic a.n.a.lysis and in particular evolutionary a.n.a.lysis is now dominant. Economists are now virtually the only 'scientists' who attempt to model a real-world system using static, equilibrium tools. As a result of this isolation, economists have been shielded from developments in mathematics and other sciences which have revolutionized how scientists perceive the world.

This isolation is to some extent fortuitous, because if economists really knew what is common knowledge in other sciences, then they would finally have to abandon their obsession with equilibrium, and economics as outlined in this book would cease to exist. Most modern-day economists believe, as did the founding fathers of economics, that dynamic a.n.a.lysis would simply 'fill in the dots' between the static snapshots, thus replacing a series of still photographs with a moving picture. In fact, modern research in mathematics, physics, biology and many other disciplines has shown that dynamic a.n.a.lysis normally leads to results which contradict those of static a.n.a.lysis.

In the long run, we are all in the short run.

Equilibrium can be the long-run destination of the economy only if it is stable if any divergence sets up forces which will return the economy to equilibrium. Even after the proofs of the instability of general equilibrium, most economists believe that this is a non sequitur: surely, the equilibrium of any real-world system must be stable, since if it were unstable, wouldn't it break down? John Hicks articulated this view when he criticized one of the earliest dynamic models developed by an economist. He commented that Harrod (1939) welcomes the instability of his system, because he believes it to be an explanation of the tendency to fluctuation which exists in the real world. I think, as I shall proceed to show, that something of this sort may well have much to do with the tendency to fluctuation. But mathematical instability does not in itself elucidate fluctuation. A mathematically unstable system does not fluctuate; it just breaks down. The unstable position is one in which it will not tend to remain. (Hicks 1949) The modern discipline known colloquially as chaos theory has established that this belief, though still widespread among economists today, is quite simply wrong. The equilibrium of a real-world system can be unstable without the system itself breaking down.

The first and best ill.u.s.tration of this occurred, not in economics, but in meteorology. I'll give a brief exposition of this model, because it ill.u.s.trates several ways in which the conventional economic understanding of dynamics is profoundly wrong. But first, we need a brief technical interlude to explain the difference between the mathematical methods used in static a.n.a.lysis and those used in dynamics (you can skip to 'The weather and the b.u.t.terfly' if you'd like to avoid mathspeak).

Straight lines and curved paths What static a.n.a.lysis means in technical terms is that the equations most neocla.s.sical economists (and many non-orthodox economists) use in their mathematical models are 'algebraic' rather than 'differential.'

Algebraic equations are simply larger and more complicated versions of the equations we all did at school in geometry, when we were asked to work out the intersection of two lines. Given two equations for Y in terms of X, with different slopes and Y intercepts, we worked out the only X point where the two formulas gave the same Y point. Continuing with the geometry a.n.a.logy, most of the equations used by economists use only straight lines, rather than more complicated shapes like parabolas, etc. Algebraic techniques with these equations scale indefinitely you can have equations with hundreds of 'straight lines' and still get unique solutions.

9.2 The time path of one variable in the Lorenz model.

Differential equations, on the other hand, are more complicated descendants of the technique of differentiation, which you might have learnt if you did calculus at school or college. Rather than being expressed in terms of X and Y, these equations are expressed in terms of the rate of change of X and the rate of change of Y. While school calculus dealt only with 'the rate of change of Y with respect to X,' differential equations typically are in terms of 'the rate of change of Y with respect to Y itself, other variables, and time.'

Most differential equation models also involve curved relationships between variables, rather than straight lines. A straight line is in fact the simplest type of relationship which can exist between two variables (other than that of no relationship at all). Straight-line relationships in differential equation models with unstable equilibria lead to ultimately absurd outcomes, such as negative prices, or cycles which approach infinite amplitude as time goes on. Nonlinear relationships, however, result in bounded behavior: the forces which repel the system when it is very close to equilibrium are eventually overwhelmed by attractive forces when the system is substantially distant from the equilibrium.

Unlike linear algebraic equations, nonlinear differential equations don't scale well. Only a very few simple nonlinear differential equations can be solved the vast majority can't be solved at all. Once there are more than two variables in a system of nonlinear differential equations, there is in fact no a.n.a.lytic solution. Such systems must be simulated to see what is actually going on.

The weather and the b.u.t.terfly In 1963, the meteorologist E. N. Lorenz devised a simple mathematical model of turbulent flow in a weather cell, using a simplified version of a well-known mathematical model of turbulent flow. His model had just three equations, with three variables and three constants. The first (x) equation described the intensity of convective motion, the second (y) the temperature difference between ascending and descending columns of air, and the third (z) described the divergence from linearity of the temperature gradient across the weather cell.7 9.3 Structure behind the chaos.

It would be hard to think of a simpler set of three equations, and yet the behavior they generated was unbelievably complex. Figure 9.2 shows the time path of the eastwest fluid displacement.

The y and z patterns were equally complex. Even more mysteriously, a tiny difference in the initial x, y or z values led, very quickly, to a totally different time path. It had been thought in the past that a tiny difference in any initial measurement would mean only a tiny error in predicting the future behavior of a variable. However, in this model, a tiny difference initially has no apparent effect, but then abruptly leads to a totally different outcome.

9.4 Sensitive dependence on initial conditions.

Finally, though the pattern for any one variable appeared erratic, behind this apparent randomness lay a beautiful structure which is visible when the three variables are plotted on the one graph. Figure 9.3 shows the 'b.u.t.terfly' behind the superficial chaos.

Detailed a.n.a.lysis of this system reveals that it has not one equilibrium, but three. More importantly, all three equilibria are unstable. A slight divergence from any equilibrium causes the system moving to move away from it very rapidly. A tiny divergence from one equilibrium point leads to the system instantly being propelled from that equilibrium. It then approaches another, only to be flung off to a third. It orbits that equilibrium, only to be eventually repelled from it. Finally, it approaches and then is repelled from the second equilibrium back towards the first.

9.5 Unstable equilibria.

There are at least four lessons for economics in this model.

First, a system with unstable equilibria doesn't have to 'break down.' Instead, such a system can display complex cyclical behavior rather like that we see in real-world weather and, more to the point, in real-world economies.

Secondly, if the equilibria of a model are unstable, then neither the initial nor the final position of the model will be equilibrium positions. The economic belief that dynamic a.n.a.lysis simply plots the movement between one equilibrium and another is therefore wrong. Instead, even simple dynamic models will display 'far from equilibrium' behavior. As a result, rather than equilibrium being where the action is, equilibrium tells you where the model will never be.

Thirdly, extrapolating from models to the real world, actual economic variables are likely to always be in disequilibrium even in the absence of external shocks (or 'exogenous' shocks, as economists prefer to call them), which is the usual economic explanation for cycles and the conditions which economists have 'proved' apply at equilibrium will therefore be irrelevant in actual economies. In this sense, equilibrium truly is, as Keynes put it, 'blither.' Static economic a.n.a.lysis therefore can't be used as a simplified proxy for dynamic a.n.a.lysis: the two types of a.n.a.lysis will lead to completely different interpretations of reality. In all such cases, the static approach will be completely wrong and the dynamic approach will be at least partially right.

Finally, even as simple a system as Lorenz's, with just three variables and three constants, can display incredibly complex dynamics because the interactions between variables are nonlinear (if you check the equations in note 7, you will see terms like 'x times y'). As noted earlier, nonlinear relationships in differential equation models can lead to complex but bounded behavior.

From meteorology to economics There are many models in economics which have properties akin to those of Lorenz's weather model very few of which have been developed by neocla.s.sical economists. Most were instead developed by economists who belong to alternative schools, in particular complexity theorists and evolutionary economists. One of the best-known such models, Goodwin's model of cyclical growth, put in mathematical form a model first suggested by Marx.

Marx argued that in a highly simplified economy consisting of just capitalists and workers there would be cycles in employment and income shares. In Marx's words: A rise in the price of labor, as a consequence of acc.u.mulation of capital [... means that] acc.u.mulation slackens in consequence of the rise in the price of labor, because the stimulus of gain is blunted. The rate of acc.u.mulation lessens; but with its lessening, the primary cause of that lessening vanishes, i.e., the disproportion between capital and exploitable labor-power.

The mechanism of the process of capitalist production removes the very obstacles that it temporarily creates. The price of labor falls again to a level corresponding with the needs of the self-expansion of capital, whether the level be below, the same as, or above the one which was normal before the rise of wages took place [...]

To put it mathematically: the rate of acc.u.mulation is the independent, not the dependent, variable; the rate of wages, the dependent, not the independent, variable. (Marx 1867: ch. 25, section 1)8 In point form, the model is as follows: A high rate of growth of output led to a high level of employment.

The high level of employment encouraged workers to demand large wage rises, which reduced profits.

The reduced level of profits caused investment to decline, and growth to slow.

The slower rate of growth led to increasing unemployment, which in turn led to workers accepting lower wages.

Eventually the fall in workers' share of output restored profit to levels at which investment would resume, leading to a higher rate of growth and higher employment levels.

This in time led to high wage demands once more, thus completing the cycle.

This cycle can also be stated in terms of causal relationships between key economic variables the amount of capital, the level of output, and so on which shows that the process Marx describes was based on an accurate view of the overall structure of the economy, and also an accurate deduction that this would lead to cycles in income distribution and employment, rather than either equilibrium or breakdown: 1 The amount of physical capital determines the amount of output.

2 Output determines employment.

3 The rate of employment determines the rate of change of wages (the 'Phillips Curve' relationship I discuss in the addendum to this chapter).

4 Wages times employment determines the wage bill, and when this is subtracted from output, profit is determined.

5 Profit determines the level of investment.

6 Investment determines the rate of change of capital and this closes the causal loop of the model.

In mathematical form, this model reduces to two equations which are easily stated verbally: The rate of change of workers' share of output equals workers' wage demands minus the rate of productivity growth.

The rate of change of employment equals the rate of growth of output, minus population growth and technological change.9 This mathematical model generates the cycle envisaged by Marx. Rather than converging to equilibrium values, workers' share of output and the rate of employment both cycle indefinitely.

9.6 Cycles in employment and income shares When wages share and employment are plotted against each other, the result is a closed loop. This is a far less complex structure than Lorenz's model, but it has one thing in common with it: the model does not converge to its equilibrium (which lies in the center of the loop), but orbits around it indefinitely.

9.7 A closed loop in employment and wages share of output It is also easily extended to capture more aspects of the real world, and when this is done, dynamic patterns as rich as those in Lorenz's model appear as I detail in Chapters 13 and 14.

Real-world phenomena therefore simply cannot be modeled using 'comparative statics' or equilibrium unless we are willing to believe that cyclones are caused by something 'exogenous' to the weather, and stock market bubbles are caused by something outside the economy. Complexity theory has established that such phenomena can be modeled dynamically, so that abandoning static equilibrium a.n.a.lysis does not mean abandoning the ability to say meaningful things about the economy.

Instead, what has to be abandoned is the economic obsession with achieving some socially optimal outcome. As noted in this and the previous chapter, economists have conflated the concept of equilibrium with the vision of an 'economic utopia' in which no one could be made better off without making someone else worse off. But a free market economy could never remain in an optimal position, because economic equilibria are unstable. The real question is whether we can control such an unstable system whether we can constrain its instability within acceptable bounds.

This question was once at the heart of what is known as macroeconomics the study of the entire economy and the attempt to control it using government fiscal and monetary policy. Unfortunately, as we shall see in the next chapter, neocla.s.sical economists have emasculated this once virile area of a.n.a.lysis. As they did so, they ignored possibly the most important lesson to flow from the advances in dynamic a.n.a.lysis since Lorenz: the realization that complex systems have what are known as 'emergent behaviors' which mean that they cannot be understood by studying their const.i.tuent parts alone. This reality invalidates a key aspect of modern neocla.s.sical macroeconomics: the attempt to derive models of the macroeconomy from microeconomic models of the behavior of individuals. A discussion of emergent behavior properly belongs in this chapter, but its neglect by neocla.s.sical economists and the practice of its opposite philosophy, 'reductionism' has been so essential to the neocla.s.sical ruination of macroeconomics that I have delayed a discussion of it until the next chapter.

Before I move on, there is one other topic that also belongs in this chapter, rather than the next on macroeconomics, where it would normally be discussed in a conventional textbook: the 'Phillips Curve.' This is an alleged relationship between the level of unemployment and the rate of inflation that, though it is hotly disputed within economics, nonetheless plays a role in virtually every theory of macroeconomics, from Marx's at one extreme to neocla.s.sical economics at the other.

It belongs in this chapter on dynamics, because the real objective of the person after whom it was named the New Zealand-born engineer-turned-economist A. W. ('Bill') Phillips was to persuade economists to abandon their static methods and embrace dynamic a.n.a.lysis. This is precisely what I am attempting to do now, so Phillips's work including the 'Phillips Curve' deserves to be discussed here as a valiant but unsuccessful previous attempt to shake economists out of their static straitjackets.

Addendum: Misunderstanding Bill Phillips, wages and 'the Phillips Curve'

Bill Phillips the man was undoubtedly one of the most dynamic human beings of all time. Compared to that of Phillips, the lives of most economists even non-neocla.s.sical ones are as pale as the theories that neocla.s.sical economists have concocted about the world. He left school at fifteen, worked as a crocodile hunter and gold miner in Australia, learnt engineering by correspondence, was awarded an MBE for his role in the defence of Singapore in 1942, and, as a prisoner of war, made a miniaturized radio from components he stole from the camp commander's radiogram. Despite the effects of malnutrition and abuse in the camp, within five years of the war finishing and while still an undergraduate student of economics he had his first paper published in a leading journal (Phillips 1950). The paper described an a.n.a.log computer dynamic simulation model of the economy (MONIAC) that he constructed at a cost of 400, just three years after the first digital computer (ENIAC) had been constructed at a cost of US$500,000 (Leeson 1994, 2000).

MONIAC put into mechanical-hydraulic form the principles of dynamics that Phillips had learnt as an engineer, and it was this approach which he tried to communicate to economists, on the sound basis that their preferred methodology of comparative statics was inappropriate for economic modeling: 9.8 Phillips's functional flow block diagram model of the economy RECOMMENDATIONS for stabilizing aggregate production and employment have usually been derived from the a.n.a.lysis of multiplier models, using the method of comparative statics. This type of a.n.a.lysis does not provide a very firm basis for policy recommendations, for two reasons.

First, the time path of income, production and employment during the process of adjustment is not revealed. It is quite possible that certain types of policy may give rise to undesired fluctuations, or even cause a previously stable system to become unstable, although the final equilibrium position as shown by a static a.n.a.lysis appears to be quite satisfactory.