My new paper The mathematics of profit maximisation is incorrect has been published in real-world economics review.

Abstract: Profit maximisation is one of the two main optimising principles of neoclassical economics, the other being utility maximisation. In this paper we draw on Chapter 6 of John Maynard Keynes's General Theory to show that the mathematics of profit maximization is incorrect. We show, moreover, that marginal cost, a variable fundamental to neoclassical economics, cannot be calculated. We explore the implications for sticky prices, increasing returns, the shape of the supply curve, and market clearing. Finally, we argue that an important reason for the failure of neoclassical economics is that while it pays a great deal of attention to the influence of future expectations on present decisions, it completely ignores the past.

The stagflation of the 1970s is widely believed to have heralded the eclipse of Keynesian economics and the rise of monetarism. In a previous post (What caused the stagflation of the 1970s? Answer: Monetarism) I had argued that this was not the case and that the stagflation was directly caused by the monetarist policies of the time.

But there is an even more important question that does not seem to have been answered, or even seriously raised: How could the US have avoided stagflation? Considering our experience in handling the Great Recession and the Covid pandemic and in the light of theoretical advances, especially Modern Monetary Theory, this is a surprising omission.

Through 1974 the price of crude rose by an average of 170% year-on-year, because OPEC raised oil prices. If an oil importer in the US had been importing 1 million barrels of oil at a price of $5 and the price went up to $10 then it would need twice the amount of money it did earlier if it wished to import the same quantity of crude as it had previously. Down the line, those to whom it sold refined oil products would need a greater amount of money to buy the same amount of gasoline, etc that they used to. If money in the economy was not expanded to the required extent, then less crude would be imported from abroad and less gasoline consumed down the line. The result would be a contraction in overall output, or what we would call a recession. This is indeed what happened when the Fed tried to constrict money supply growth.

In hindsight, we can see that the policy was completely wrong. The inflation of the mid-1970s and after was caused by a huge rise in oil prices. The inflation should not have been fought; it needed to be accommodated by an increase in money supply. The question we should really ask is: How should the money supply have been increased? There were many options.

The Fed could have bought financial assets and increased money supply. But as we saw after the Great Recession, if money supply is increased in this manner it tends to stay in the market for financial assets instead of moving into the market for real goods and services. The usual result is an inflation in the price of financial assets, and little more.

The Fed could have printed more money equal to the difference in crude import costs, given it to the importers of crude to pay foreign crude exporters, and in return asked that the domestic price of oil be kept unchanged. Developing countries could not have exercised this option because the willingness of crude exporters to accept newly printed currency depended on the existence of goods and services (or investments) that could be bought with the newly minted currency. Also, it would have amounted to a blanket endorsement of any price increase by OPEC.

Another problem was that domestic oil producers too increased oil prices, though to a lesser extent because of price controls imposed by the Carter administration. They too could have been compensated in the same manner as crude importers but then there would have been no incentive for crude producers, domestic and foreign, to control prices.

But after the experience of the Covid pandemic and the spread of MMT, we know that there was another option. The government could have directly put money into the accounts of ordinary individuals and left it to them to decide whether the money should be entirely spent on higher priced oil and other goods or saved. There would still have been inflation, and given the international price of oil this was inevitable. But the pain would have been mitigated to a great extent and the problems associated with price control could have been avoided.

In a recent paper that went viral Jeremy B. Rudd, a Fed economist, wrote: "Mainstream economics is replete with ideas that 'everyone knows' to be true, but that are actually arrant nonsense."

Here I deal with the second "nonsense" idea mentioned by Rudd: "Over a sufficiently long span--specifically, one that allows necessary price adjustments to be made--the economy will return to a state of full market clearing."

Expositions of General Equilibrium Theory and neoclassical economics usually begin by assuming market clearing. In reality we know that markets do not always clear, a fact conceded even by Kenneth Arrow when he observed in his Nobel Prize lecture that "the history of the capitalist system has been marked by recurring periods in which the supply of available labour and of productive equipment available for the production of goods has been in excess of their utilisation, sometimes, as in the 1930s, by very considerable magnitudes."

So, when do markets clear and when do they not? That is what this post will examine.

Fig 1 shows a demand curve and a supply curve for a good that intersect at a point of equilibrium P. If the price is set at a level above the equilibrium price, then, at that point, the quantity that consumers are willing to buy is less than the quantity that suppliers are willing to sell; the oversupply is given by the length of QR. Suppliers therefore reduce the price until demand and supply are once again equal. P is a point of stable equilibrium because deviations from that point set into motion forces that return the equilibrium to P. The price at P is also a market-clearing price because the quantity that consumers are willing to buy at that price is equal to the quantity that suppliers are willing to sell at that price.

Fig 2 shows what happens when the demand curve falls from DD to D1D1. The point of equilibrium moves from P to P1. At this new point of equilibrium both the price and the quantity are lower than at P. What is more important is that the shift to P1 does not set into motion forces that restore the equilibrium to P. That happens only when the demand curve moves back to its earlier position. However, P1 can still be regarded as a market-clearing point because suppliers do not wish to supply a larger quantity at that price.

Thus, when the demand curve for a good shifts, we are confronted not with a single stable equilibrium but with multiple equilibria.

Curiously, Fig 2, which depicts the microeconomics of a single good, is consistent with Keynesian macroeconomics: when aggregate demand falls there is no automatic mechanism that returns the economy to its original position. The lower point of intersection of the aggregate demand curve with the aggregate supply curve is also a point of equilibrium. A common charge against Keynesian macroeconomics is that it has no microeconomics. And yet, as Fig 2 suggests, the microeconomics of demand and supply yield the same conclusion as Keynesian macroeconomics.

But how would market non-clearing be depicted in a diagram? In other words, if suppliers were willing to sell more than they do at the current market price what would the supply curve look like? Fig 3 shows the situation.

When the demand curve shifts from DD to DD1 the point of equilibrium moves from P to P1. Between P1 and P the supply curve is horizontal. Suppliers are willing to sell a greater quantity than at P1 at the same price as at P1 but cannot because demand is not high enough.

When the good being considered is labour Fig 3 looks plausible. During a severe recession, the unemployed are willing to work at the market wage but cannot find work. That wages do not rise for a long time after a recession even though employment increases lends credibility to Fig 3. The General Theory went a step further and considered the labour supply curve as sloping downward during a recession: "Men are involuntarily unemployed if, in the event of a small rise in the price of wage-goods relatively to the money-wage, both the aggregate supply of labour willing to work for the current money-wage and the aggregate demand for it at that wage would be greater than the existing volume of unemployment." Keynes was of course thinking of real wages, not nominal wages. Workers are willing to work for a wage even lower than the market wage because whether they obtain employment or not, they have to incur the expense of keeping themselves and their families alive.

The assumption of market clearing amounts to saying that the situation of Fig 3 is not possible. It thus depends on two other assumptions: 1. That the supply curve is nowhere horizontal, and 2. That the demand curve does not shift. We shall consider each of these assumptions in turn.

The idea that the supply curve is nowhere horizontal is refuted by the ideas in Chapter 6 of The General Theory, probably one of the most important chapters in the book and the least understood. I think of it as an anticipatory demolition of the Samuelson kind of neoclassical economics. The chapter can be summarised in two sentences: 1. Marginal cost is not equal to marginal factor cost, and 2. Fixed investment does not become zero at the margin of production. In my book Economics Redefined I have showed, for example, that the ideas in Chapter 6 disprove the mathematics of profit maximisation. But here we are concerned with its implications for the shape of the supply curve.

Rather than explain the idea using Keynes's terminology I use an example. Imagine that you are a manufacturer of shoes and have invested $1 million in a new production line. Let the variable cost of a pair of shoes (leather, labour, power etc) be $50. You set the price of a pair of shoes at $100. When you sell one pair of shoes you pay off $50 from the cost of your fixed investment. Imagine that you are at point P1 of Fig 3 where you sell 2,000 pairs of shoes a month and can pay off $100,000 of fixed investment. If the demand curve moves to DD where you can sell 4,000 pairs of shoes a month you would still be happy to sell at the same price because at that point you can pay off an additional $200,000 of your fixed investment each month. (Remember that it is only after you have paid off your entire fixed investment that you begin to make a profit.) Even if the variable cost goes up by, say, $5, it would still make economic sense to sell at the same price as before or even lower.

But this means that the supply curve can be horizontal or even slope downwards. And if supply curves are horizontal or slope downwards in an interval then it is obvious that the market does not clear over that interval.

That brings us to the second assumption in market clearing: that the demand curve does not shift, i.e., we are dealing with Fig 1 and not Fig 2. A common objection to using the three figures above is that they depict situations of partial equilibrium and, in particular, that they assume all other prices are constant. In a paper published in 2017 in real-world economics review (A diagrammatic derivation of involuntary unemployment from Keynesian microfoundations), and more elegantly in my book, I showed that this is incorrect. A downward sloping demand curve of the kind in Figs 1 to 3 cannot simultaneously fulfil the conditions that income is constant and all other prices remain constant. The proof requires nothing more sophisticated than school mathematics. There is only one exception to this rule and that is the demand curve in the form of the rectangular hyperbola.

Demand curves in the shape of the rectangular hyperbola have several interesting properties. For one, they can be aggregated arithmetically, so the aggregate demand curve can be derived from the demand curves of heterogeneous agents, thus dispensing with a representative agent; in fact, all demand curves have the same shape. Two, they do not assume that tastes are constant or that other prices are constant or that aggregate demand is constant. They are thus the most general demand curves. Three, the rectangular hyperbola which is a curve of constant demand is also a curve of constant money; this resolves the dispute between monetarists and Keynesians, the one contending that recessions are caused by a contraction of money and the other that recessions are caused by a contraction of aggregate demand.

But here we are concerned with another interesting property. In conventional Marshallian economics, when demand changes, we cannot be sure whether we must stay on the same demand curve or move to a different demand curve. When the demand curve is a rectangular hyperbola there is no such ambiguity. When demand (measured in money) changes, the demand curve has of necessity to shift; in other words, we have to deal with a different rectangular hyperbola. Where the curve in Fig 1 is a static curve, the rectangular hyperbola is a dynamic curve; it moves with every change in demand.

For small changes in demand, however, we can approximate the shift in demand with a single demand curve as in Fig 4 and thus revert to the static version. The figure shows two demand curves xy = a and xy = b. If demand shifts by small amounts then the two curves can be replaced by a single curve and the analysis thus reduces from the general Keynesian case to the particular Marshallian case. This is of course consistent with Keynes's claim that his book dealt with the general case in which aggregate demand changes whereas his predecessors dealt with the particular case in which aggregate demand does not change.

So, to return to the title of this post: when do markets fail to clear?

There are two conditions to be met. One, the demand curve should shift. But, as we have seen, when the demand curve is the rectangular hyperbola, it shifts with every change in demand. So, this condition is always fulfilled. However, for small changes in demand, the two rectangular hyperbolas can be replaced by a single curve. We can thus say that as long as demand stays within narrow bounds of full-employment equilibrium the market clears.

The second condition is that when the supply curve is horizontal over any interval, the market does not clear over that interval. This happens when a firm has made a substantial investment expecting rising demand and, instead, demand falls. In that case, the first priority of the firm is to recover its fixed investment as quickly as possible, and to do that it is willing to maintain its price or even lower it if demand rises.

In a recession both these conditions are fulfilled and it can be said definitely that there is then no market clearing in many or most markets.

In the 18th and 19th centuries, economists spent a lot of time thinking about what determined the division of income between land, labour and capital. As the importance of agriculture diminished, the question transformed into one of determining how income was distributed between capital and labour. The problem was difficult and an answer elusive.

Then neoclassical economics resolved the problem in a nifty way. It drew up a production function showing how output was mathematically determined by physical inputs of capital and labour. When a partial derivative of the production function with respect to labour was taken one arrived at the marginal product of labour. A partial derivative of the production function with respect to capital gave the marginal return on capital. Clearly, the use of mathematics could resolve puzzles that verbal arguments could not.

On closer examination, however, the solution turns out to be not so nifty. A partial derivative of the production function with respect to labour means holding capital constant and determining the change in output due to a marginal change in labour alone. A partial derivative of the production function with respect to capital means holding labour constant and determining the change in output due to a marginal change in capital alone.

The problem of course is that a capitalist cannot produce an output without the aid of a worker. And a worker cannot produce an output unless he makes use of capital supplied by a capitalist. Else, the capitalist would not be a capitalist and the worker would not be a worker. It is not possible to produce an output by holding labour constant and varying only capital or by holding capital constant and varying only labour So, the partial derivative of the production function with respect to labour and the partial derivative with respect to capital do not exist. A production function like the Cobb-Douglas production function must therefore be ruled out because it has both a partial derivative with respect to labour and a partial derivative with respect to capital. Other production functions in common use in neoclassical economics must be rejected for the same reason.

Tens of well-known economics papers are therefore invalid. One thinks of such classics as Technical Change and the Aggregate Production Function by Robert Solow (1957) or Capital-Labor Substitution and Economic Efficiency by Kenneth Arrow, H.B. Chenery, B.S. Minhas and Robert Solow (1961). Real Business Cycle theory which makes use of similar production functions must also be disqualified for the same reason.

There is no denying that substitution between labour and capital can take place. A capitalist who employs 50 workers to manufacture shoes manually can set up an automated assembly line with five workers. But that substitution does not take place at the margin. Since it is not possible to disentangle the contributions of capital and labour in the final product, neoclassical economics is far from having found a solution to the question that exercised the minds of economists in the 18th and 19th centuries.

Both the capitalist and the worker can lay claim to the marginal product, which is a joint result of capital and labour. But the decision about how the rewards flowing from the marginal product are divided is made by the capitalist. He does not make it in a vacuum, though. If labour unions are strong or there is demand for the worker's labour from other capitalists the worker will get a greater share than otherwise. Similarly, if he has sufficient accumulated saving, he can hold out for a greater share whereas if he is asset-destitute he has to settle for whatever the capitalist gives him. The division depends on the balance of power between the capitalist and the worker. There are clear parallels between the marginal product of labour and marginal utility. In both cases, neoclassical economics invents a variable which has no basis in reality. Then it justifies this by constructing a function which it assumes to be differentiable, when, even by internal logic, there is no evidence of differentiability.

Looking back, it is astonishing how much of neoclassical economics is based on mathematical errors.

This may seem like the daydream of a fond old man. But I am confident that my new ebook Economics Redefined deals a death blow to neoclassical economics. I expect few economists to read it and fewer still to understand it. Even so, I expect that eventually it will become the conventional wisdom. Those who have been intimidated by the mathematics of mainstream economics -- the Lagrangians and the bordered Hessians -- will read this book and rejoice because it shows that the math is hogwash.

PREFACE TO THE BOOK

In September 2014, Jonathan Barzilai, a mathematician at Dalhousie University in Canada, wrote an open letter to William Nordhaus, who was the president of the American Economic Association at the time. Among other things Barzilai pointed out: "In economic preference theory, the claim that ordinal functions can be differentiated is based on elementary errors: Theorems of differential calculus are applied where the assumptions of these theorems are not satisfied and mathematical operations are applied where they are not applicable. The notion that ordinal functions are differentiable is an error. It has no parallel in mathematics and science."

Since a large part of economics rests on the idea of utility maximisation (which is untenable if Barzilai is right) his letter was expectedly not met with enthusiasm. To describe the response as a "deafening silence" might be cliched but quite appropriate and hardly surprising. His claim was after all akin to someone telling physicists that the law of conservation of momentum is incorrect. Nordhaus did not respond. If other economists took note of Barzilai's letter they probably regarded him as crazy.

Unfortunately for mainstream economics Barzilai is right. And it does not take much thought to arrive at that conclusion. If utility is ordinal, then three values of utility a, b and c are of the kind that we can say a > b > c and nothing more. The three numbers 3, 2 and 1 satisfy that ordering but so do 25, 3 and 1. Obviously the operations of differential calculus cannot be applied to them. That is all there is to it. It took me quite some time to realise this because, like most economists, I was thinking of baskets of goods as in indifference curves. Not being a mathematician but an engineer by training, I needed to think in pictures, and it was a couple of years before I convinced myself that Barzilai's argument applies to baskets of goods as well.

In June 2017, Real-World Economics Review, a heterodox economics journal, published a paper of mine where I showed that the demand curve must necessarily take the shape of a rectangular hyperbola, and that when this is the case, macroeconomics can be built from the microeconomics of heterogeneous agents, and that involuntary unemployment must follow as a matter of course. It was a month or so after this, while looking through old issues of Real-World Economics Review, that I first chanced upon two papers by Barzilai as well as a putative refutation of one of his papers, which was actually nothing of the kind.

Four years later, in a moment of transcendental joy, the realisation struck me that Barzilai's idea and mine, though seemingly oceans apart, actually amounted to the same thing. It is because utility is not differentiable that demand curves must be rectangular hyperbolas.

I hope the reader of this book too will be able to savour the joy of that realisation.

A summary of the book is as below:

Chapter 1 shows that the mathematics of profit maximisation is incorrect.

Chapter 2 shows that the mathematics of utility maximisation is incorrect.

Chapter 3 uses the ideas of the previous two chapters to arrive at a non-equilibrium macroeconomics derived from the microeconomics of heterogeneous agents and having involuntary unemployment as a key feature.

Chapter 4 explains how our conception of money is different from that of Keynes and Friedman.

Chapter 5 shows that the idea of the money multiplier is incorrect.

Chapter 6 sets out a way of measuring money.

Chapter 7 shows that the velocity of money has nothing to do with the speed at which money travels from hand to hand and, indeed, has nothing to do with time at all.

Chapter 8 shows why recessions occur.