The Use of Models and Mathematics in Economics

Economics has come to be distinguished from the other social sciences mainly by its use of mathematics rather than by its subject matter. There is a difference between mathematics and numbers. It is true that economics studies quantifiable aspects of human behaviour; yet students of other social sciences, together with the intelligent lay reader who is quite capable of reading a chart, find themselves excluded by the mathematical language of economics.

Economists use mathematics to build models of the real world. Any model, whether physical or mathematical, is necessarily an abstraction from reality. For example, a map is a model of terrain; a map at a scale of 1:1 is not useful. Niels Bohr's model of the atom as a tiny solar system with electrons orbiting a nucleus is not a complete representation of reality, yet it allows us to understand the properties of atoms well enough to do advanced chemistry.

Undergraduate, let alone graduate, degree courses in economics involve learning a series of mathematical models of such complexity that most students have no time or energy left to question their limitations as an explanation of the world around us. Leading employers of economists such as the UK Government Economics Service and the Bank of England have complained that an economics degree no longer equips a student to be an economist outside a self-perpetuating academia. Many economics students are fed up and, remarkably for such short-lived creatures, have organised themselves into a permanent global association, Rethinking Economics, which demands what its name declares: a rethinking of economics.

Yet Keynes himself was a modeller. He wrote to Roy Harrod (1900–78):

It seems to me that economics is a branch of logic, a way of thinking; and that you do not repel sufficiently firmly attempts … to turn it into a pseudo-natural science … Progress in economics consists almost entirely in a progressive improvement in the choice of models. The grave fault of the later classical school, exemplified by Pigou, has been to overwork a too simple or out-of-date model, and in not seeing that progress lay in improving the model …

Economics is a science of thinking in terms of models joined to the art of choosing models which are relevant to the contemporary world. It is compelled to be this, because, unlike the typical natural science, the material to which it is applied is, in too many respects, not homogeneous through time. The object of a model is to segregate the semi-permanent or relatively constant factors from those which are transitory or fluctuating so as to develop a logical way of thinking about the latter, and of understanding the time sequences to which they give rise in particular cases. Good economists are scarce because the gift for using vigilant observation to choose good models, although it does not require a highly specialised intellectual technique, appears to be a very rare one. (CW 14, 296–7)

However, even if economics is a branch of logic, a science of thinking in terms of models, it does not follow that models can be expressed only in mathematics. Keynes writes:

The object of our analysis is, not to provide a machine, or method of blind manipulation, which will furnish an infallible answer, but to provide ourselves with an organised and orderly method of thinking out particular problems … Too large a proportion of recent ‘mathematical’ economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols. (CW 7, 297–8)

Furthermore Keynes understood that economics is also an essay in persuasion:

When we write economic theory, we write in a quasi-formal style; and there can be no doubt, in spite of the disadvantages, that this is our best available means of conveying our thoughts to one another. … [Yet in] economics you cannot convict your opponent of error; you can only convince him of it. And, even if you are right, you cannot convince him, if there is a defect in your own powers of persuasion and exposition or if his head is already so filled with contrary notions that he cannot catch the clues to your thought which you are trying to throw to him … Time rather than controversy … will sort out the true from the false. (CW 13, 469–471)

The Models in This Book

As I have noted, modern economics is full of models, most of which are written in mathematical squiggles incomprehensible to the untrained reader. The models in this book are simple. There are a few tables, charts and the odd equation but rather less than is demanded of today's 16-year-old. Economics has to use models to say anything coherent, because everything connects and changes together. Without a model, however simple, we lose our way in a shifting maze.

The use of a simple model leads to the accusation of setting up a straw man in order to demolish it. This is not the intention. The corn model used to explain classical theory in chapter 2 is a simplification of more advanced mathematical models, but not a distortion. The purpose is to show the essence of classical analysis and to demonstrate the power of its central ideas, notably the concepts of competitive equilibrium and marginal productivity.

If those terms make you groan, please ask yourself whether you really want to understand Keynes. His thought cannot be understood, especially in the light of the classical restoration, without understanding the essence of the classical model. You cannot understand his revolution without a grasp of what he was revolting against, let alone of what he actually wrote. Many criticisms of both classical economics and Keynes's are misinformed. As he himself wrote,

The worst of economics is that it really is a technical and complicated subject. One can make approximate statements in a common-sense sort of way which may appear superficially satisfactory. But if someone begins to ask one intelligent and penetrating questions it is only possible to deal with them by means of something much more complicated. (CW 20, 469)

Nevertheless this book tries to make the learning process as painless as possible. Readers who already know what is meant by competitive equilibrium and marginal productivity will find chapter 2 an easy read. The intention is that students beginning economics, students of other social sciences, and interested lay readers should find that chapter 2 equips them with the basic knowledge required to understand the real issues at stake.

The Nature of This Book

This book is a biography of sorts, but not of Keynes the man. There are several hundred books about Keynes. Many better qualified authors have written about his life and his loves, the extraordinary, sparkling combination of statesman, journalist, philosopher, scientist, financial wizard, patron of the arts and bon vivant that he was. Recommendations of some such biographies can be found in the Further Reading at the end.

Rather, this book charts the development of Keynes's economic thought over the course of his life and considers its legacy. The method is analytical. That means tracing the progression of economic models from the ‘classical’ theory – as he called it – that he inherited, through to the final form of his own models in The General Theory and, beyond that, to the model implicit in his contribution to the Bretton Woods conference at the end of his life. Most of the biographical element is contained in chapters 5 and 6; the story is not told in chronological order.

The principal purpose of this book is to restate Keynes's critique of classical economics in terms accessible to today's students of both economics and other social sciences. The arguments dividing Keynes from the classicals have become clearer now than they were eighty years ago. So, too, are the differences between Keynes and the Keynesians of the Keynesian era – and even between Keynes and the post-Keynesians, the ‘keepers of the flame’ today.

However, the book is not just about theoretical models and what may have been going on inside Keynes's head. Chapter 7 looks at the historical record in some detail and considers the extent of Keynes's influence on subsequent events. Chapter 8 considers the relevance of his thinking and policy ideas to today's problems.

The Book in a Nutshell

What is the central difference between the conclusions of Keynes and those of the classicals? On the one hand, classical theory holds that in a free society everyone willing to work at the going rate can, in the long run, find employment. Unemployment is the result of a poorly functioning labour market, for which the remedy is greater flexibility, that is, more competition. On the other hand, Keynes holds that much unemployment and consequent loss of national income are the result of a failure of the system as a whole, not of the hapless workers alone.

Yet the answer, for Keynes, is not the revolution of Karl Marx. The remedy for unemployment and the precondition for sustained economic growth lie in the financial and monetary system, both at national and at international level. In his view, the free enterprise system can be made to work for everyone if we are clever enough to create the right institutions – if only the world is wise enough to adopt them.

Why do Keynes and the classicals reach such different policy conclusions? To explain this is the burden of the next five chapters. Chapter 2 sets out the classical theory of employment, interest and money using a simple model. Chapter 3 sets out Keynes's alternative general theory of employment, interest and money, a theory that furnishes the title of his magnum opus. The interpretation offered here is markedly different from the standard treatment of Keynesian economics in economics textbooks. Chapter 4 addresses the great confusion between saving and money that plays a major role in allowing modern classical economists to continue resisting Keynes's technical critique. Chapter 5 charts the development of Keynes's thinking about money and inflation over the course of his life, during his long struggle to escape from classical theory. Chapter 6 addresses the international economy, the real world of sovereign states and multiple currencies, and explains the thinking that led Keynes from walking out of the Versailles Peace Conference in 1919 to supporting the compromise at the Bretton Woods Conference in 1944.

An Outline of the Argument

A summary of the argument of the forthcoming chapters may be helpful, even if at this stage it is perhaps not fully comprehensible. Classical theory (chapter 2) is built upon the foundation created by the concepts of diminishing returns and marginal productivity. Under conditions of free competition, these concepts are sufficient to explain the levels of production, employment and investment, together with the distribution of income between wages and other incomes. Classical theory is a remarkable achievement: it has produced powerful and compelling results from a few key ideas. So long as wages are flexible, in line with productivity, everyone who wants to work will be fully employed. In a similar fashion, the rate of interest balances the forces of productivity and thrift; investment has to be attractive enough for savers to be persuaded to be patient and postpone jam today for more jam tomorrow.

Money does not play an essential role in the classical model. All the key results can be worked out without money. The theory of value and distribution is distinct from the theory of money and prices, which comes later. This separation is justified by what is known as Say's law, the proposition that there cannot be a general glut or shortfall of demand for what is produced as a whole, at least not in the long run. Keynes can be understood in part as refuting Say's law by showing how money does indeed play an essential role in the market economy.

At the heart of Keynes's critique, in The General Theory (chapter 3), is the concept of expected income. The level of production and employment depends upon the income expected by employers. In a monetary economy, this expected income may not be enough to make it profitable to employ everyone who is willing to work at the going rate. In the classical model, the level of production and employment is determined by the wage that workers are willing to accept. In Keynes's model, expected income also depends on the spending decisions of investors and consumers. Consumption, in turn, depends on income, so that expected income turns out to depend mainly on investment.

Why does investment not adjust so as to ensure full employment, in line with Say's law? This is the question addressed by chapter 4. In the classical model, the rate of interest governs the division of income between consumption and investment. Income is determined first, then divided between consumption and investment (see Figure 1.1). In Keynes's model, consumption follows investment according to a relation called the multiplier. Investment is determined first, then income and consumption follow (see Figure 1.2).

In both models, investment depends on the rate of interest. In the classical model, the rate of interest balances saving (i.e. the decision not to consume) and investment. In Keynes's model, the rate of interest is independent of saving; it balances instead the supply of and the demand for money, which is quite different from saving or savings. The rate of interest, then, determines investment – and hence income, consumption, employment and the real wage.

The rate of interest depends on confidence in expectations about the inherently uncertain long-term future. The future cannot be reduced to an extrapolation from the present and the past. When confidence is lacking in the face of such uncertainty, people want to hold on to their liquid money, and a higher rate of interest is required if they are to lend or spend money for investment. When money is plentiful or confidence is high, the rate of interest falls and investment increases.

In fact expectations about the long-term future have a direct influence on investment, and not only through the rate of interest. It is irreducible, unquantifiable uncertainty that requires the ‘animal spirits’ of enterprise and makes investment, especially in new products and processes, the engine of capitalism. Fluctuations, in both confidence and long-term expectation itself, lead to cycles in production and employment. There is no market mechanism to regulate the level of investment, certainly not the rate of interest, so as to ensure that investment is neither above nor below what is necessary to warrant full employment. Say's law does not hold.

It took Keynes most of his working life to develop his final understanding of the nature and role of money in the market economy (chapter 5). Money does not appear at all in Figure 1.1. He began as an advocate of the Cambridge version of the quantity theory of money, the idea – dating back at least to David Hume (1711–76) – that price inflation is the result of an excessive increase in the quantity of money. This theory seemed well supported by the experience of hyperinflation in the 1920s.

Over time Keynes probed the nature of money and the motives for holding it. He came to place more emphasis on changes in the demand for money than on changes in its supply. Money can affect prices only through spending, but then it is clear that spending can affect output and employment as well as prices. He developed and discarded a series of sophisticated models within the classical framework to explain the role of money in the economy before arriving at The General Theory. He remained as concerned about inflation as ever and his analysis helped prevent its recurrence in Britain during the Second World War.

Keynes's entire working life was concerned with the problems of the international economy (chapter 6). The problems of an economy such as Britain's cannot be understood except within the international context. This in turn requires an understanding of the relationships and exchange rates between different currencies. Keynes's views on the gold standard and on free trade reflected, and changed with, his views on the impact of the international monetary system on national economies as circumstances changed. In his policy work, from Versailles to Bretton Woods, he was painfully aware of the importance of the current account of the balance of payments and of the absolute necessity of foreign currency reserves for national sovereignty and even for survival.

Although Keynes was too ill or too busy to write another academic treatise after The General Theory, his thinking on the international economy was embodied in plans for a postwar international economic order. He recognised that a persistent tendency towards surplus on the current account had the same depressing effect on global economic activity as a tendency towards excessive saving within a national economy. His understanding of the nature of money led him to propose a radical solution for removing the balance of payments as an obstacle to full employment. His solution was too radical for the Americans at the time. Nevertheless the international agreements that formed the International Monetary Fund, the World Bank, and ultimately the World Trade Organisation owe a great deal to Keynes.

The Legacy of Keynes

Chapter 7 considers the impact of Keynes's ideas on the world during the thirty-year period after his death – the Keynesian era. A number of charts present the economic data for the 140 years from 1874–2013 for 17 countries and from 1874–2015 for the United Kingdom. The period is divided into five subperiods that correspond to major changes in economic policy: gold standard, interwar, Keynesian, neoliberal and austerity.

The bare fact is that the Keynesian era delivered on average the high income and low unemployment sought by Keynes without either excessive inflation or additional public borrowing. Keynes's primary prescription was to use monetary policy to reduce the long-term real rate of interest and stimulate investment. This policy was adopted during both the Keynesian and the austerity eras. The full employment and the high rate of investment of the Keynesian era were also underpinned by public consumption and by export growth, encouraged by the free trade regime established at Bretton Woods.

Although historical events never have one single cause, the outstanding reason for the end of the Keynesian era was the United States’ decision to unpeg the dollar from gold in 1971, itself the outcome of underlying political and economic changes. The years 1973–9 saw inflation and unemployment rise together to unprecedented levels, a phenomenon known as ‘stagflation’. In 1976 the prime minister ended the United Kingdom's political commitment to full employment. Although President Carter persisted for another few years in the United States, the writing was on the wall.

From about 1980 the influence of Keynes on economic theory became divided into two main currents, which represent alternative responses to the internal contradictions of the attempt to integrate Keynesian and classical economics during the 1950s. One current returns to a modified version of the classical theory of the equilibrium of supply and demand, which abandons Keynes's key insights and reduces them to a special case – the very opposite of his intent. The other current has largely abandoned supply and demand theory and, while accepting Keynes's insights and method of progression through the development of new and relevant models, has now moved substantially beyond and away from his Marshallian framework.

Finally, chapter 8 considers the relevance of Keynes's thought to us today. Here I have tried to distinguish the voice of Keynes himself from those of his followers of all shades. Keynes was a market-friendly economist who sought to reform classical theory, not to abandon it. He was pragmatic and politically astute in his policy recommendations and politically liberal in the British sense. He was an elite member of the establishment, with no particular loyalty to working people, but he despised the waste and stupidity of high unemployment. Today's economic policy is much closer to Keynes than is generally admitted, but academic theory remains stubbornly resistant to his insights.

In the current geopolitical environment, I suggest that his recommendations for a reform of the international monetary system would be limited to what he regarded as feasible rather than extending to what he considered ideal. In particular, he would regard the reform of the euro as an urgent priority, not only for the European Union but for the prosperity of the world as a whole. Drawing on this book's analysis, I venture to suggest what he might have proposed in today's circumstances.

By the time we reach the end of the book, you should be equipped to form your own conclusions about what Keynes means for us today. This book is a journey upwards, through Keynes's economic thought, and we start in chapter 2 where he did: from the classical theory that he inherited. The ascent is steep in places, but the view from the summit is worth it.