The Algebra of John Maynard Keynes

Essay by 


March 2010  
Economics imitating Physics
John Maynard Keynes's most influential book, The General Theory of Employment, Interest, and Money, clearly follows the example of the earlier generations of mathematical economists from Jevons to Marshall in presenting itself as a kind of social physics. The title evokes Albert Einstein's general relativity, and in doing so suggests that just as there is a narrower theory of special relativity, so there is a narrower "special economics" on which Keynes's ideas are an advance. Keynes even identifies the restrictive assumption of that special or classical economics, as embodied in Arthur Cecil Pigou's theory of employment: Say's Law, or the principle that there cannot be a general shortage of demand for all goods. And Keynes's own theories are expressed in a whole series of mathematical formulae, though often the expression is difficult to follow, and each chapter's mathematical exposition seems to be largely independent of any other's. But arguably, Keynes's central mathematical argument is his theory of the multiplier. This is the basis for all the practical applications and misapplications of his policy (for it must be noted, to do Keynesians justice, that they advocate giving the economy fiscal "stimulus" when it's slow, but not when it's speeded up, as in the early years of George W. Bush's presidency). Keynes's argument starts out with the marginal propensity to consume (MPC). (I will simplify his notation slightly, but preserve the same mathematical relationships.) He notes that people divide their income between consumption and investment, which he expresses as Y = C + I. Considering a marginal increase in income, he argues that it will be divided between consumption and investment: dY = dC + dI. In his view, a given community tends to spend a certain fraction of its income, and of any changes in its income, on consumption, which can be written as dC/dY. This is what he calls the marginal propensity to consume. The increase in investment is whatever is left over after consumption; it's a residual amount. If we write MPC for the marginal propensity to consume, then we can say that C = MPC x Y, and that I = Y  C = Y  (MPC x Y) = (1  MPC) x Y. That is, income and marginal propensity to consume together determine what a community will have left over for new investments. But now comes the trick! Having an expression for I in terms of Y, Keynes points out that the equation can be solved for Y. The result is that Y = I / (1  MPC). For simplicity of notation, he defines k = 1 / (1  MPC). This lets him write Y = k x I. Because I is multiplied by k, he calls k the multiplier. Suppose, for example, that people in Ekonomistan habitually spend 80% of their income on food, fuel, clothing, rent, and other consumer goods. MPC is then 0.80. So k = 1 / (1  0.80) = 1 / 0.20 = 5; that is, income is five times investment. Now, says Keynes, suppose we add to investment by an amount dI. Then we can apply the multiplier to get an increase in income by dY = k x dI. In Ekonomistan, if a billion dollars are invested, national income will go up by five billion dollars. But in doing this, he's no longer treating income as an independent variable, that is, as a cause that determines other features of the economy. Rather, his independent variables are investment and the multiplier; income is a dependent variable or an effect. The multiplier is just another way of expressing the marginal propensity to consume, so it's still being treated as a cause in both cases; but in the first case, income is the cause and investment the effect, and in the second, it's the other way around. Certainly, in engineering, there are cases where this kind of interchange of cause and effect occurs; they're called feedback systems. But in feedback systems, it's almost never the case that the two causal relations are mirror images of each other in the way Keynes supposes. Rather, the old value of the cause we start with determines some value of the effect; and then the effect turns around and determines a new value of the cause, which then determines a new value of the effect. The system evolves dynamically through a series of different states, like a homing missile tracking its target. Keynes's treating the multiplier as having a fixed mathematical relation to the MPC doesn't work; the causal effect of income on investment and the causal effect of investment on income can't be expected to be simple backward and forward versions of the same magnitude. The people of Ekonomistan might save 20% of their income, and invest it, but the investment might produce an increment of income not of five times its own size, but twenty times, or twice ... or, if it's managed really badly, no increment at all. Keynes's simplification of his analysis for ease of presentation has simplified away all the complexities of how economies actually work. There's one case where it's equally legitimate to treat either quantity as the cause and the other as the effect ... or to treat the two causal relations as mirror images of each other. That case is static equilibrium, where the whole system has come to rest and is no longer changing. (In terms of the homing missile, this is rather like the missile having finally hit the target!) If the whole economy is in stasis, it makes no difference whether we think income is the cause and investment the effect, or the other way round. And, in fact, Keynes's disciples do spend a lot of attention on statics; their analysis of the economy as a whole, or macroeconomics, seems often to assume that it has stopped changing and has settled down to an equilibrium. Keynes tends to assume that the economy naturally reaches a point of stability and rests there, until it's jolted out by some external force . . . a natural catastrophe, or a "fiscal stimulus" from government. The way to get the slumbering economy to wake up and grow is for the government to prescribe carefully measured infusions of new "investment", which will then be multiplied and produce a substantial boost in income ... just sufficient, in fact, so that the part of the income that people consume leaves a residue exactly big enough to maintain the investment, at least if the economy straightforwardly follows Keynes's multiplier relation. In fact, if we get rid of the reversibility of the two relations, that's far from the most likely outcome. Suppose that the total income of Ekonomistan is $10 billion, of which $8 billion is consumed and $2 billion invested. Now the government steps in and adds an extra $2 billion. We can't just multiply investment by k to get the new income; we don't know what k is. But we have consumption of $8 billion and investment of $2 billion plus $2 billion, or $4 billion in total; they add up to a total income of $12 billion. With an MPC of 80%, we can expect that the Ekonomistan's will now consume $9.6 billion and invest $2.4 billion; that is, they will treat the government "investment" as a windfall and shift their own income out of investment to consumption to compensate. And reversibility is extremely unlikely, precisely because capitalist economies are not static. They have not only longterm continuing growth, but a constant emergence of new industries and failure of old ... the "creative destruction" that Joseph Schumpeter wrote about. Anyone who follows any market is aware that it's never safe to assume it's gone to sleep and won't wake up. Keynes's basic model applies only to economies of a kind that has not existed at least since the start of the Industrial Revolution. The actual mathematical content of Keynes's formulae reveals assumptions that don't work in the real world. Certainly, some later Keynesians have attempted to generalize Keynes's general theory, developing versions of it suited to a dynamic rather than a static economy. But this seems like a doubtful enterprise. Wouldn't it be better, not to work out a complete theory based on a static economy, and then try to adjust it here and there to fit a dynamic one, but to start out by assuming a dynamic, selforganizing system in the first place?


© 2010 William H. Stoddard 


Coining at Troynovant 



