The generalizations of equilibrium theory are not true universal statements. Preferences are not always complete or transitive. Firms do not always aim to maximize profits. Individuals are sometimes satiated. Yet the generalizations that constitute equilibrium theory are informative, and mainstream economists have constructed useful models that incorporate them. How is one to understand the content and value of such “inexact” (i.e., false) claims?

9.1 Mill on Tendencies

In “On the Definition of Political Economy and the Method of Investigation Proper to It,” John Stuart Mill (Reference Mill1836a) argues that political economy is a science of “tendencies”: that its claims are “true in the abstract” and would be true in the concrete were it not for disturbing causes. What can he mean?

When Mill returns to these issues in A System of Logic (1843), his language is a little different and clearer. He maintains that in an inexact science:

Mill cites the science of tides as an example. Scientists know the laws of the greater causes – that is, the gravitational attraction of the sun and the moon – but they are ignorant of the laws of minor causes, and they do not know the precise initial conditions, such as the configuration of the shore and ocean bottom. One might suggest that there are no exact sciences, although in some cases for some purposes the inexactness of a science might be negligible. Mill disagrees. He believes that astronomy is an exact science, “because its phenomena have been brought under laws comprehending the whole of the causes by which the phenomena are influenced … and assigning to each of those causes the share of the effect which really belongs to it” (1843, 6.3.1).

Mill regards motives as analogous to forces. When he speaks of “compounding the effects” of causes, he has in mind the vector addition of forces in mechanics. Compounding of causes need not be additive. Perhaps it can be understood more generally as deducing a prediction from some principle of combination and a group of lawlike generalizations, the effects of which when operating singly are known.

When Mill talks about an “inexact science,” he is not concerned mainly with imprecision in the predictions of a science. Even if knowledge of relevant causal factors were complete, economists might still be unable to make accurate predictions because of difficulties in learning the initial conditions or because of computational or measurement limitations. Mill is concerned with inexactness within theories – within the set of lawlike statements that constitutes a theory (see §6.3).

Mill is also not mainly concerned with rough empirical generalizations such as “birds fly” or “trees shed their leaves in winter.” In his view, these are not explanatory. Instead, they express patterns in the data for which one seeks explanations.Footnote ¹ In Mill’s view, the “empirical laws” of the social sciences are typically just rough generalizations, not laws at all (compare Rescher Reference Rescher1970, pp. 164–7):

Although rough generalizations such as the Phillips curve lack explanatory power,Footnote ² they are the raw material for theorizing and may play an important role in models. In Mill’s view, the explanatory or causal laws of inexact sciences are not rough generalizations, which are mere correlations among features of human action as these are ordinarily classified. The “science of Human Nature” counts as a science, insofar as its rough empirical laws can be connected deductively to genuine laws of human nature.

Mill writes:

The generalizations concerning market demand and supply discussed in Chapters 2 and 3 are somewhere inbetween empirical laws and universal laws of human nature. These generalizations are causal claims, not merely statements of correlations. However, they are shallow and their explanatory power is limited. The “laws” of equilibrium theory from which the generalizations concerning supply and demand can be derived are, as stated, false and hence hardly “universal laws of human nature,” though they seem to identify genuine causes and function in economics as if they were laws. Tendencies are the causal powers underlying the regularities that inexact laws express.Footnote ³

In Mill’s view, knowing only the laws of the “greater causes” of the phenomena, economists are unable reliably to infer from them what will occur. Economics is in this way an inexact science. This inability is a consequence of inexactness within the theory, not merely of faulty data or mathematical limitations. Economics employs inexact laws and thus inexact theories. Although the fundamental generalizations of equilibrium theory are not true, it seems that there is a good deal of truth to them. But what does it mean to say that a claim has “a good deal of truth to it” other than putting a happy face of the admission that it is false? What exactly is inexactness? How should one analyze this inexactness and make precise the idea that economists possess true causal laws that nevertheless capture only the behavior of the most important causes of economic phenomena?

9.2 Four Kinds of Inexactness

There are at least four ways, which are not mutually exclusive, in which to analyze inexact laws:

1. 1. Inexact laws are probabilistic or statistical. Instead of stating how human beings always behave, economic laws state how they usually behave.

2. 2. Inexact laws are approximate. They are true within some margin of error.

3. 3. Inexact laws are qualified with ceteris paribus clauses.

4. 4. Inexact laws state tendencies that causal factors exert both singly and in combination.

As I argue in this chapter, the first two construals do not capture the important ways in which economic generalizations are inexact, even though it may sometimes be useful to identify the approximations and probabilistic aspects of economics. In contrast, the third and the fourth interpretations go to the heart of the matter. J. N. Keynes (Reference Keynes1917) appears to endorse the third interpretation, as I did in the first edition of this book. However, the view that the generalizations of economics express tendencies seems most faithful to Mill and to the thinking of most economists. Despite apparent metaphysical commitments in invoking tendencies, the fourth construal is more natural then the third. I argue that the differences between the third and fourth interpretations are subtle and may matter little to the practice of economics.

9.3 The Meaning or Truth Conditions of Inexact (Causal) Generalizations

Economic laws are qualified with ceteris paribus clauses in two different ways. In partial equilibrium theories and practical work, it is common practice to consider separately the effects of different known causal factors. As discussed in Section 2.1, for example, demand for some commodity or service depends on its price, the prices of substitutes and complements, income, and tastes. Yet economists may want to consider demand for coffee as a function (ceteris paribus) of the price of coffee only. In the language of tendencies, they may want to consider how a change in the price of coffee tends to affect the quantity of coffee demanded when acting by itself. Here the constituents of the ceteris paribus clause are those factors that economic theory itself identifies as other causal determinants of demand for coffee. Such ceteris paribus qualifications are of philosophical interest in the analysis of the causal structure of partial equilibrium explanations, but the meaning and justification of “laws” with only such qualifications is unproblematic. If one takes for granted fundamental economic theory, the term “ceteris paribus” in generalizations such as the law of demand can be replaced with a list of specific causal factors, the effects of which are considered separately. Moreover, in principle, changes in the price of coffee, the prices of complements and substitutes, and incomes all increase or decrease the quantity of coffee demanded, and there should be an algorithm predicting the net effect of the separate tendencies. Exactly how to add in the effect of changes in tastes is murkier. Although the ceteris paribus clauses attached to derivative laws introduce no additional vagueness, they inherit the vague qualifications attached to the fundamental “laws” of equilibrium theory.

The ceteris paribus laws or statements of tendencies I am concerned with in this section and the next are more problematic. Fundamental economic theory considers only some of the causes of economic phenomena. The remaining causes are not enumerated and are often unknown. The basic claims of economics are true only under various conditions that are not fully specified. Without specifying the disturbing causes, can one still make substantive claims concerning the “greater” economic causes? What precisely is a vague ceteris paribus clause? (Or, alternatively, what makes a generalization a statement of a tendency?) What does it mean to say that people’s preferences tend to be transitive or that, ceteris paribus, people’s preferences are transitive? What must the world be like if such claims are true?

The same sentence can be used to say different things in different contexts. Following Stalnaker (Reference Stalnaker, Davidson and Harman1972, pp. 380–97), let us distinguish the meaning of a sentence (the context-invariant interpretation of the sentence) from its content (the proposition expressed by the sentence), which may vary in different contexts. “I’m confused by this book” has a single meaning, but its content depends on who utters it and when and where it is uttered. Stalnaker suggests that one should regard the meaning of a sentence as a function from contexts to contents or propositions. The meaning of a sentence determines a content in a given context.

Adapting this terminology, one might suggest that ceteris paribus clauses, both explicit and implicit, have one meaning – “other things being equal” – which in different contexts picks out different propositions or properties.Footnote ¹² The context – especially the economist’s background understanding – determines what the “other things” are and what it is for them to be “equal.” So, for example, in the simpler case of the precise ceteris paribus clauses of partial equilibrium analyses, the term “ceteris paribus” might pick out the proposition “other prices, tastes, and incomes do not change.”

The term “ceteris paribus” need not determine a property or proposition in every context. Sometimes in uttering a sentence containing such a clause, one fails to express any proposition. For example, I suggest that the ceteris paribus clause in the sentence “ceteris paribus all dogs have three heads” has no content. Moreover, the properties ceteris paribus clauses pick out in different uses may vary greatly in clarity and precision. At one extreme are examples such as those in supply and demand explanations or in some laws of physics such as Coulomb’s law.Footnote ¹³ Consider, in contrast, clauses such as “holding technology and other inputs constant,” which one finds in the law of diminishing returns. Such clauses do not have a precise extension, but they are not completely vague either. Although there are formal difficulties with vague predicates, such predicates abound in science and ordinary language, and we cannot do without them.

What proposition does a vaguely qualified law, such as “ceteris paribus people’s preferences are transitive,” express? Suppose that the logical form of an inexact law were “ceteris paribus everything that is an $F$ is a $G$,” where $F$ and $G$ are predicates with definite extensions.Footnote ¹⁴ Consider first the unqualified generalization, “everything that is an $F$ is a $G$.” Logicians interpret sentences with this form to mean that there is nothing in the extension of the predicate $F$ that is not in the extension of the predicate $G$. (Recall that the extension of a predicate is the set of all things of which the predicate is true.)

In the case of qualified generalizations such as “ceteris paribus everything that is an $F$ is a $G$,” some things that belong to the extension of $F$ do not belong to the extension of $G$ – otherwise the qualification would be unnecessary. One view, which I endorsed in the first edition, is to regard “ceteris paribus everything that is an $F$ is a $G$” as a true universal statement if and only if, in the given context, the ceteris paribus clause picks out a property – call it $C$ – and everything that is both $C$ and $F$ is $G$. The extension of the vague predicate $C$ must contain only properties that economists consider to be nomologically relevant to $G$. Otherwise one might take $C$ to be $G$ itself or some property whose extension includes the extension of $G$ and trivializes the analysis (Earman and Roberts Reference Earman and Roberts1999, p. 475). If one considers only the interior of region $C$ in Figure 9.1, one sees that all of region $F$ that is contained there (i.e., the intersection of regions $F$ and $C$) lies within region $G$. In offering a qualified generalization, one is only asserting that, once the qualifications are met, all of region $F$ lies within region $G$. The predicate $C$ belongs in the antecedent of the law, although it may be awkward to state the law in this form. I have drawn $C$ without a solid boundary only to suggest that economists do not know precisely what the extension of the ceteris paribus predicate is and not to suggest that it does not have a definite extension – which it must have if the qualified claim is truly to be a law. In committing oneself to a law qualified with a ceteris paribus clause, one envisions that the imprecision in the extension of the predicate one is picking out will diminish without limit as one’s scientific knowledge increases.

Figure 9.1 Ceteris paribus clauses.

To believe that, ceteris paribus, everybody’s preferences are transitive is to believe that anything that satisfies the ceteris paribus condition and is a human being has transitive preferences. One need not be disturbed by intransitive preferences caused by, for example, changes in tastes, because such counterexamples to the unqualified generalization lie outside region $C$. In my analysis, sentences qualified with ceteris paribus clauses may be laws. A sentence with the form “ceteris paribus everything that is an $F$ is a $G$” is a law just in case the ceteris paribus clause determines a property $C$ in the given context, and it is a law that everything that is $C$ and $F$ is also $G$.

This is not the only analysis of a ceteris paribus law in economics. It is plausible to believe that events and tendencies in separate sciences such as economics or psychology supervene on a variety of physical states and physical laws. Suppose that microstates $f_1,f_2,\dots f_n$ are realizers of some economic state $F$, micro states such that necessarily anything that is $f_i$ is $F$. In other words, each $f_i$ is one way of being $F$. Suppose then that for almost all of the realizers of $F$, there are completers $c_1$, $c_2$, $c_m$, such that it is a true exceptionless law that if $f_i$ and $c_j$, then $G$. In such a case, because $F$ supervenes on the microstates, there would clearly be a nomological connection between $F$ and $G$, but there would be no exceptionless law in the vocabulary of economics connecting them. This possibility describes a different way in which the generalizations of economics can be inexact laws or tendencies. We can call such generalizations nomological but irreparably inexact. If any of the many different physical realizations of any preference ordering do not guarantee that preferences are transitive, then there will be no condition $C$ that is stateable in the language of economics that will make, “if $C$, then preferences are transitive” true (Schiffer Reference Schiffer1991; Fodor Reference Fodor1991). But this statement can be an (irreparably) inexact law nevertheless.

9.4 Qualification or Independent Specification

James Woodward criticizes the strategy of taking the antecedents of inexact laws to contain a ceteris paribus qualification, which he calls “exception incorporating” (2000, pp. 228–35; 2003, pp. 273–9). His name for strategies such as the one I have so far defended is tendentious, because the qualifications that I envision as implicit in the inexact generalizations of economics are not ad hoc exclusions of apparent falsifications, but instead characterize significant causal factors that enhance or impede the action of the explicitly specified causes.

Instead of regarding generalizations such as profit maximization as universal truths once they are properly qualified, Woodward suggests that one might regard them as possessing a limited scope that is specified independently. With a different interpretation, Figure 9.1 might represent the independent specification interpretation just as well as it represents the exception-incorporating view. The difference is that $C$ is now an independent specification of the scope of the generalization, not an antecedent in the law.

Woodward has an additional and much more radical critique (2002; see also Lange Reference Lange2002) of regarding ceteris paribus conditions as antecdents in laws. In his view, causal laws are not exceptionless regularities. Instead, they are statements of relations among variables that are “invariant to interventions.” What this means, roughly, is that for some interventions that change the values of variables in the antecedents of causal laws, the generalization will correctly state the values of the variables in the consequent. $Y=F(X,Z)$ explains why $Y=y*$, if $y*=F(x*,z*)$ and for some interventions that set $X=x\text{'}$, for some value $x\text{'}$ of $X$, $F$ is invariant – that is $Y=F(x\text{'},z*)$. An intervention on a variable, $X$, that changes the value of $X$ is a cause of $X$ that has no causal connection to any other variables except in virtue of changing the value of $X$. Although not endorsing Woodward’s view of causality, Nancy Cartwright also argues that laws have a much less significant role in modeling and explanation. In her view, laws result from stable arrangements of tendencies, which, rather than laws, are the fundamental building blocks of scientific theorizing (1999, chapter 6).

Woodward’s and Cartwright’s views are tempting but controversial, and I do not want to stake my account of tendencies or inexact laws on them. The solutions that their views on explanatory generalizations offer to the puzzle of how the laws of equilibrium theory can be explanatory despite not being true universal generalizations would be straightforward, but I am unwilling to stake my analysis on an idiosyncratic minimalist take on what causal explanation requires.

Setting aside Woodward’s and Cartwright’s accounts of causal explanation, one can nevertheless appreciate how the independent specification approach avoids complicating the generalizations of economics with qualifications, which are now off-loaded to a codicil specifying that this law is limited to a domain in which condition $C$ holds. One is only concerned with states of affairs in which $C$ is satisfied, and within $C$, on an old-fashioned view of laws, all $Fs$ are $Gs$. In both the exception-incorporating and independent specification views, all $Fs$ that are also $C$ are $Gs$. But the independent specification approach allows the generalizations of economics to remain simple and unqualified, if limited in scope. The task of specifying the disturbing causes will be handed off to something like a commentary detailing when one can and cannot make use of economic laws either singly or in combination.

The independent specification view has some drawbacks. It can encourage a lazy instrumentalism (which is definitely not true of Woodward’s own views). On the independent specification view, a mistaken prediction need not call for the revision of an economic generalization such as acquisitiveness. It may only call for an adjustment in the scope specified for the generalization, which itself appears not to be subject to empirical scrutiny. Of course, if one has to keep cutting down the scope of a generalization such as acquisitiveness, there may come a time when it will make sense to abandon it. But according to the independent specification view, unfavorable evidence seems to have very little direct or immediate bearing on the credibility of purported laws themselves.

A second problem is that sometimes adding a small qualification to a generalization can vastly increase its explanatory and predictive power. Adding the qualification to the generalization might in addition point the way toward a deeper theoretical grounding for the generalization. In such a case, independent specification would impede scientific progress.

Nevertheless, the independent specification view fits the practice of economics much better than the exception-incorporating view. Economic essays are not clogged with myriad qualifications. Moreover, the independent specification view simplifies the task (see Chapter 10) of clarifying when it is reasonable or unreasonable to regard a generalization that is qualified with a ceteris paribus clause as an inexact law. The test is instead whether the independently specified ceteris paribus condition is met in the domain that economists are studying.

The term “ceteris paribus” may be used in other ways. In offering a rough generalization, such as “birds fly,” one need not believe that there is any set of conditions in which being a bird is sufficient for flying.Footnote ¹⁵ One might simply believe that almost all birds fly. Indeed, scientists may sometimes believe that the true law will not involve the current predicates used in the generalization at all. One may regard a rough generalization, such as “$Fs$ are generally $Gs$,” as having some predictive force, even though one expects it to be superseded in the course of further inquiry. My analysis is not intended to deny these truths, nor am I taking any position concerning whether there are probabilistic or statistical laws. All I am claiming is that when one takes an inexact generalization to be an explanatory law, one supposes that the ceteris paribus clause picks out conditions in which the purported law no longer faces counterexamples.

9.5 Mechanical Phenomena and the Composition of Economic Causes

Suppose one has two qualified laws: (1) ceteris paribus for every $1 increase in $p_x$, $q^D{}_x$ drops by 1,000 units; and (2) for every $1 increase in the price $p_z$, of a substitute, $z$, $q^D{}_x$ increases by 200 units. The ceteris paribus clause attached to (1) (whether it be incorporated as a qualification or independently specified) maintains that there are no effects from changes in income, from changes in prices of any complements or substitutes, from changes in tastes, or from a miscellany of other possible interferences whether they be earthquakes or alien invasions. The ceteris paribus clause attached to (2) has the same content except that it precludes changes in $p_x$, and it does not preclude changes in $p_z$.

From these two laws and the claim that the total effect is the sum of the two separate effects, one can apparently deduce (3) ceteris paribus if $p_x$ increases by $5 and $p_z$ increases by $10, then $q^D{}_x$ decreases by 5,000–2,000 or 3,000 units. Notice that the drop in demand will be the simple sum of the two effects only if the demand is the sum of separable functions of $p_x$ and $p_z$ (i.e., $q^D{}_x=f(p_x)+g(p_z)$. If the proportional change in demand is a linear function of the proportional change in price, then one composes the effects of the causes by multiplying their separate effects rather than adding them.Footnote ¹⁶ The “composition of causes” is not always addition.

Furthermore, it may be complicated to keep track of the contents of ceteris paribus clauses when there are multiple causes acting. The qualification in (3) clearly cannot include all the qualifications in (1) and (2), because the ceteris paribus qualification that specifies the domain of application for each law rules out the change considered in the other law. The ceteris paribus condition for the combination is the intersection of the conditions ruled out for each of the separate laws.

Consequences such as (3) can only be drawn reliably when one has what Mill calls “mechanical phenomena” (1843, 3.6.1 and 3.6.2). Mill maintains that in mechanical phenomena the effect of two causal factors acting simultaneously is the sum of the effects of each acting separately. As we have seen, this definition is too narrow. Even though in note 16 the existence of a change in the price of a substitute changes the absolute amount that $q^D{}_x$ decreases in response to a change in $p_x$, it does not change the functional relationship. Mill needs instead something like the claim that the relationships between the effects of two causes $x$ and $z$ and the value of $y$ are mechanical if and only from the relations ceteris paribus $y=f(x)$ and ceteris paribus $y=g(z)$ it follows that ceteris paribus, $y$ is a mathematical function of both $f(x)$ and $g(z)$, such as the sum or product. Each factor continues to “operate” no matter what other causes are operating (1843, 3.10.5; Cartwright Reference Cartwright1983, pp. 44–73). When one has such “mechanical phenomena” the causal factor captured in the qualified law is responsible for a “tendency” in the phenomena that is present whenever the causal factor is.

When one is not dealing with causal factors that compose in this way or when one simply does not know how various causal factors will interact, one may still use laws qualified with ceteris paribus clauses. Qualified laws dealing with nonmechanical phenomena will, however, be more provisional and will have a more restricted scope. They may apply only when there are no appreciable interfering factors (Elster Reference Elster1989a, p. 216). Even if the basic generalizations of equilibrium theory are inexact laws, they will not help one to understand real economies with their inevitable disturbing causes, unless economic phenomena are mechanical phenomena.

Mill simply asserts that economic phenomena are mechanical: that the basic economic causal factors continue to act as component “forces” in the total complicated effect (1843, 6.7.1). Such a supposition is implicit in many applications of economic models. I see no justification for it other than the empirical confirmation of the implications of composing the effects of multiple causes. For an illustration of how Mill treats economic phenomena as mechanical, see his discussion of the combined effects on rents, profits, and wages of an increase in capital and labor and of technological change (1871, book IV, chapter 3).

Since scientists do not know exactly what property a ceteris paribus clause picks out, why regard it as picking out any property at all? Is there enough clarity in the independent specification of a ceteris paribus clause that rules out “other interferences” as in the example just discussed? Does such a specification identify a domain in which the basic “laws” of equilibrium theory are true? One can recognize that the generalizations of equilibrium theory may guide research and help economists to interpret data without regarding them as laws. If the interferences vaguely specified by the implicit ceteris paribus clause are absent, then economists can regard the generalization in that domain as a restricted law.Footnote ¹⁷ Without the limitation to a specific domain provided by the independent specification of the ceteris paribus condition, economists can regard the generalizations of equilibrium theory merely as assumptions in models. To regard inexact general “laws” as merely assumptions in models highlights the elusiveness of ceteris paribus clauses, which I have perhaps understated, and it emphasizes that economists regard inexact “laws” differently when they use them to give explanations than when they rely on them in doing speculative research.

Because theorists use basic economic “laws” to try to explain economic phenomena, they cannot regard them as mere assumptions, but must take them as expressing some truth, however rough (see §A.3). Otherwise their attempts to use them to explain economic phenomena would be incomprehensible (Reiss Reference Reiss2012; 2013, chapter 7). At some point, with respect to some domains, economists must construe the assumptions of the basic equilibrium models either as true qualified lawlike assertions or as true statements of tendencies.

Countenancing qualified laws forthrightly, one need not make invidious comparisons between the natural sciences and social sciences. One finds instead gradations of inexactness. Scientists strive for exactness, but possessing, as they typically do (whether in economics or chemistry), only qualified generalizations or generalizations with restricted scope, they nevertheless have learned something about their subject matter and can explain some of the phenomena in the domain.

9.6 Conclusions

Mill’s views of tendencies and inexactness are of value only if there is some way to tell whether generalizations express genuine tendencies. Can those generalizations that express tendencies be distinguished from rough generalizations that happen by accident occasionally to give the right answers? Since it is entirely consistent with the claim that $F$ tends to cause $G$ that we observe instances in which $F$ is not followed by $G$, how can claims about tendencies be tested? Chapter 10 attempts to provide the answer that economists from Mill in the first half of the nineteenth century to Lionel Robbins in the first half of the twentieth century have given.