The publication of this book represents the culmination of a very lengthy process. The underpinning research began as part of my ESRC Professorial Fellowship project, Rethinking the Market (grant number ES/K010697/1, 2013–19). I remain extremely grateful to the ESRC for providing the funding to allow me to think my way into a brand new project and for giving me the time to see this book through to its natural conclusion. This has enabled me to educate myself in a number of fields that previously I knew next to nothing about. In particular, I have spent many fascinating hours discovering new leads in the history of mathematics, the history of metamathematics, the history of science and the history of hypothetical scientific modelling, before then trying to work out how they should influence my understanding of economics imperialism. As a consequence, the scope of my argument has extended far beyond where I would have been able to locate it at the start of the process. The result, I hope, is a study that self-consciously operates in the interstitial spaces between different disciplines, different traditions of thought and different bodies of work. There would have been little point in labouring hard for a decade only to have ended up where the existing literature was already situated. The synthesis of various intellectual registers is a deliberate attempt to chart new ground.

It is not entirely straightforward to enter the debate about economics imperialism while avoiding the overtly partisan tone of the existing literature. Not all of it falls into the same trap, but typically analysis from first principles is forced to give way to judgemental assertions from those who never doubt that their position has been the right one all along. Respectful invitations to discuss issues that might always have the ability to polarize therefore often get drowned out by raised voices. My readers will ultimately decide whether I have managed to stay away from the tendencies I regret in the writing of others. Even having selected the title of False Prophets of Economics Imperialism might be seen by some as a provocation too far.

INTRODUCTION

Most proponents of economics imperialism show scant regard for trying to make their case historically. Declarative statements of the “it has long been known” variety serve their purpose much more effectively. Thus, if it has long been known that the most tractable models of human behaviour are based on maximizing principles, if it has long been known that the economic content of maximization revolves around individuals following their own rational self-interest, if it has long been known that mathematical expression brings extra precision to the economic content, and if it has long been known that good social science requires such precision, then this is often enough to negate the case against the imperialists. In a single step, it seems, the debate shifts from the nature of explanation in general to the assertion of explanatory unification through the use of mathematical market models. No concern is given for what type of unification is entailed, nor yet for what particular market models assume about the external reality that is actually unknown there. A rush to judgement ensures that the market world is always a credible world in Robert Sugden's (2000: 24, 2009a: 17) sense, thus licencing an inferential capacity that far exceeds what subject specialists can claim of their own work.

It appears to make no difference that these four “it has long been known” claims came to economics at different times, were hotly contested in methodological terms in their own day and have remained significant points of contestation. What counts is that they are widely considered to be true, beliefs that are embedded when learning how to think like an economist. Economic models are often constructed to test theoretical claims that are already assumed to hold (Ireland 2003: 1624; Winther 2006: 709; Knuuttila 2009: 76; see Chapter 3). This leaves unexplained, though, how such assumptions formed, especially in the face of historiographical evidence that the authors with whom they are most usually associated all seemed to harbour reservations about the advances with which they are now credited.

INTRODUCTION

Judging by their actions, economists generally seemed happy to be taken to where Paul Samuelson wanted them to go. In the years between him starting his Harvard PhD in 1935 and receiving his Nobel Prize in 1970, the content of economics journals increasingly came to prioritize reasoning through mathematical objects (Backhouse 1998b: 86). However, Samuelson cannot claim sole credit for such a shift, despite his own history of mathematical economics repeatedly stressing the blank canvas he initially faced (Samuelson 1986: 797). This might have been true of his immediate environs in Cambridge, Massachusetts, but by no means elsewhere. The Cowles Commission was simultaneously bringing together a collection of mathematically minded scholars in Chicago, but outside the University's Economics Department where Samuelson had enjoyed a less than wholly fulfilling time as an undergraduate. All were influenced either through direct participation or inherited intellectual objectives by Karl Menger's Mathematical Colloquium in pre-war Vienna (Leonard 2010: 154).

While Samuelson was always both a willing and an effective advocate for economists with an interest in mathematizing their field, a significant proportion of the Cowlesmen were mathematicians first and foremost, who had merely happened upon economics as a convenient focus for their writings (Mirowski 2012a: 141). What looked like a significant advance in mathematization from Samuelson's perspective was often met with a shrug of indifference by the more mathematically sophisticated Cowlesmen. Samuelson might have been able to help them add more convincing economic narratives as counterparts to their mathematical reasoning, but that was all. From their perspective, he had produced a very basic mathematical framework for facilitating his economic arguments, whereas they were interested in determining the full logical implications of mathematical propositions, the economic relevance of which was at best of only secondary importance. The achievements that gave the Cowlesmen most satisfaction were consequently often fundamentally unreadable from a Samuelsonian perspective, so unforgiving were their mathematical demands on the reader. Samuelson, though, was reluctant to take his usurpation lying down. Looking back at the clash of styles in postwar mathematical economics, he used the 1983 Introduction to the enlarged version of Foundations to say, “More can be less.

INTRODUCTION

The 1970 Nobel laureate in Economics, Paul Samuelson, was a strong devotee of the so-called SMMS narrative of disciplinary development, whereby he sits at the apex of a lineage that passes uninterrupted through Adam Smith, John Stuart Mill and Alfred Marshall (Feiwel 1982: 3). A series of advances are posited through which the best bits of the preceding theory were preserved in successive conciliatory syntheses, shorn of their troublesome features. “The giant steps that have been made in this narrative are steps of codifications of the body of knowledge”, writes Till Duppe (2011: 41) of Samuelson's approach to the history of economic thought. “The disagreements economists had with their predecessors did not have to be argued, but simply vanished in their recodification.” Samuelson was only too happy to assume the role of recodifier-in-chief, because the only justification for studying what economists once did, he said, is to understand how they were now doing the same thing better. He described his own approach through the “good, if ugly title [of] … ‘Presentistic history’” (Samuelson 1991: 6), as well as through the potentially better if definitely even less appealing “Cliowhiggism” (Samuelson 1987: 56). “Inside every classical writer”, he argued, “there is a modern economist trying to get born” (Samuelson 1992: 5). From this perspective, the person responsible for the last recodification must necessarily be best placed to speak on behalf of the whole subject field, both now and for everyone from the past who has helped it get to its current position.

The text that allowed Samuelson (1983a: xxvi) to claim the status of disciplinary standard-setter was his 1947 Foundations of Economic Analysis. Others have hardly been less stinting in their praise for it. Foundations has been described by luminaries within the profession as “a remarkable performance” (Fischer 1987: 236), “the most important book in economics since the war” (Kenneth Boulding, cited in Lucas 2001: 7), the text that “really formed [the successor] generation of economists” (Robert Solow, cited in Breit & Hirsch 2009: 156).

We continue discussion of row operations to solve linear systems. In particular, we see how to characterise when a system has no solutions (is inconsistent) and, if consistent, we show how the method can be used to find all (possibly infinitely many) solutions, and to express these in vector notation. Here, the notion of the rank of the system, which determines the number of free parameters in the general solutions, is shown to be important. Continuing the earlier discussion of portfolios, we explain how the existence of an arbitrage portfolio is determined by the existence or otherwise of state prices.

We start the chapter with a mathematical model of how consumers might anticipate market trends and what effect this will have on the evolution of prices. This leads us to second-order differential equations. We then embark on describing how to solve linear constant-coefficient second-order differential equations. The general solution is the sum of the solution of a corresponding homogeneous equation and a particular solution. In an analogous way to the way in which second-order recurrence equations are solved, there is a general method for determining the solution of the homogeneous equation, involving the solution of a corresponding quadratic equation known as the auxiliary equation. We explain how to find particular solutions and how to use initial conditions. We also discuss the behaviour of the solutions obtained.

The derivative is introduced as an instantaneous rate of change and it is shown how this can be determined from first principles. Techniques (sum, product, quotient and composite function rules) are then explained and the connection with small changes is illustrated. Economic interpretations via marginals are given.

Elasticity of demand is introduced and it is shown how this characterises how revenue will change upon a price increase (the two distinct possibilities being represented by elastic and inelastic demand). Profit maximisation is considered in general, and it is shown that, when maximising profit, marginal revenue and marginal cost must be equal. Two very different cases are then studied: that in which the firm is a monopoly and that of perfect competition.

Optimisation of two-variable functions is motivated via the example of a firm producing two goods, where the concepts of complementary and substitute goods are discussed. The general idea of a critical point is discussed and it is explained that there can be different types of critical point: maxima, minima and saddle points. It is explained how the second partial derivatives may be used to determine what type of critical point one has. This is shown explicitly for the case in which the two-variable function is quadratic, and then stated in general. Examples involving profit-maximisation are given.

The determinant of a 2 × 2 and a 3 × 3 matrix are defined explicitly, and a more general way of (defining and) calculating determinants of larger matrices is described, involving the use of row operations to transform a matrix to upper-triangular form. It is then explained that a non-zero determinant is equivalent to invertibility. Cramer's rule is presented and a general method (based on the co-factor matrix) is given for inverting 3 × 3 matrices, an alternative to the row operations procedure described in .

Continuing from the previous chapter, this chapter explores the powerful applications of diagonalisation. We demonstrate first how it can be used to determine the powers of a matrix, which can then be applied to solve a coupled system of recurrence equations. An alternative approach to solving such systems also uses diagonalisation, but uses it to effect a change of variable so that the corresponding system in the new variables is much simpler to solve (and can then be used to revert to the solution in the original variables). We show how an analogous approach can be used to solve coupled systems of differential equations. This closing chapter provides an interesting link between the calculus and linear algebra aspects of the course.

This chapter involves the input--output model. Here, there are several goods under production, and some of each is needed to meet the production of the others, and there is also an external demand for each good. The model involves a matrix known as the technology matrix and a related matrix know as the Leontief matrix. It is shown how to solve such problems and it is explained that, in general (under very reasonable conditions), there always will be a solution. It is also shown how to approximate the solution using powers of the technology matrix.

This chapter studies the case of a small efficient firm in a perfectly competitive market. Breakeven and startup points are defined. Relationships between marginal cost, average cost and average variable cost at breakeven and startup points are investigated, and it is shown how to derive the supply set of such firms.

The chapter starts by discussing how we can determine the long-term qualitative behaviour of the solutions to second-order recurrence equations. In particular, in some cases, it can be seen that the solution is oscillatory. In the context of the multiplier-accelerator model, this corresponds to what are known as business cycles. The chapter concludes with an analysis of a dynamic macroeconomic model that is more realistic than the multiplier-accelerator one.

The chapter starts by formulating the standard problem in the theory of the firm: namely, to minimise combined capital and labour costs while producing a certain output. A general formulation of a constrained optimisation problem (with one constraint) is given and it is explained how to solve such problems by the method of Lagrange multipliers. This leads to a method to calculate the cost function of firms given their production functions and capital and labour unit costs. This enables us to derive the supply sets for efficient small firms with Cobb--Douglas production functions.