Construction of idealized models

By ‘model’ economists usually understand a set of sentences describing a puta­tive object, but I shall use the term here with a definite philosophical meaning. By ‘model’ I understand here, precisely, a set-theoretical structure seen as the realization of a set of sentences.

Although [set-theoretic] structures do play an important role in scientific modeling,... model systems cannot be identified with structures. What is missing in the structuralist conception is an analysis of the physical character of model systems. The view of model systems that I advocate regards them as imagined physical systems, i.e. as hypothetical entities that, as a matter of fact, do not exist spatio-temporally but are nevertheless not purely mathemat­ical or structural in that they would be physical things if they were real. If the Newtonian model system of sun and earth were real, it would consist of two spherical bodies with mass and other concrete properties such as hardness and colour, properties that structures do not have; likewise, the populations in the Lotka-Volterra model would consist of flesh-and-blood animals if they were real, and the agents in Edgeworth’s economic model would be rational human beings.

(Frigg 2010: 253)

Friggs attributes to Patrick Suppes the claim that model systems are to be iden­tified with set-theoretical structures, but this attribution is unfair. Actually, Suppes acknowledges that many scientists think of their models as imagined concrete objects, different from set-theoretical structures. For instance, “many physicists want to think of a model of the orbital theory of the atom as being more than a set-theoretical entity.

Nevertheless, Frigg considers that there are two reasons to prefer ‘his view’ to the structuralist. “The first is that scientists often talk about model systems as if they were physical things, which is a natural thing to do if models are imagined physical entities” (Frigg 2010: 253).

The second reason has to do with how model systems relate to the world. A structure is not about anything in the world, let alone about a particular target system. Those who take model systems to be structures suggest connecting structures to target systems by setting up a morphism between them (the most common morphism is isomorphism; other suggestions include partial isomorphism, homomorphism, and embedding). But a morphism holds between two structures and not between a structure and a part of the world per se. In order to make sense of the notion that there is a morphism between a model system and its target we have to assume that the target exemplifies a particular structure, and this cannot be had without bringing non-structural features into play.

(Ibid.: 254)

Frigg observes that “in order for it to be true that a target system possesses a particular structure, a more concrete description must be true of the system as well”, because “structural claims are abstract in the sense that they cannot be true unless some more concrete claims are true as well” and accepts that “this by itself would not have to worry the structuralist. The problem, and this is the second step, arises when we realise that the descriptions we choose to ground structural claims are almost never true descriptions of the target system” (ibid.). And he adds:

Taken literally, descriptions that ground structural claims (almost always) fail to be descriptions of the intended target system.

(Ibid.: 254-5)

In his interesting pretension theory Frigg distinguishes p-representations, which are representations of merely fictitious objects (like Don Quijote de la Mancha) from t-representations, which also intend to represent a certain target system:

These two senses of ‘representation’ need to be clearly distinguished, and for this reason I call the former ‘p-representation’ (‘p’ for ‘prop’) and the latter ‘t-representation’ (‘t’ for target). Using this idiom, the two acts mentioned in the introduction can be described as, first, introducing a p-representation specifying a hypothetical object and, second, claiming that this imagined object t-represents the relevant target system.

(Ibid.: 264)

Frigg believes that “the apparent comparison with a nonexistent object eventu­ally comes down to the unproblematic comparison of properties, and the state­ment making this comparison is true iff the statement comparing the properties with each other is true” (ibid.: 263-264). I would like to discuss, with a certain accuracy, by means of an example in economics, how this comparison is usually made, or should be made, and how both the target and the model system relate to set-theoretical structures used to make claims about both.

Accepting the distinction between model systems and set-theoretical structures, and acknowledging that model systems are an important part of the apparatus of empirical sciences, the first thing to be noticed is that model systems are ideali­zations par excellence.

As I explained above, it is usual in economics to produce a Gedankenkonkre- tum, which is a non-idealized concept, stated in broad and general terms, of a class of target systems. A relatively simple and common example of such a concept is that of the consumer. Granted, Marx criticized the idea of the “individ­ual and isolated” producer, a criticism that we may generalize to the notion of the individual and isolated economic agent. But, seen within the frame of a concrete capitalist economy (or of any other mode of production), it is possible to theorize over the specific determinations of the consumer. To attribute to the consumer a preference relation, or the capacity to make rational choices, does not mean that the consumer is a Robinson Crusoe, let alone a naturally independent, autono­mous individual detached of all natural bonds (cf. Marx 1973: 83). For there is nothing in the theory of consumer choice that prevents accepting that the prefer­ence relation of the individual is the result of imbued traditions and tastes within a given community (like the family) that can be transformed by social forces (like advertisement). It is indeed a gross confusion to believe that Marx’s view of the economic system as an organic whole conflicts with methodological individualism. Marx never denied that only individuals had the possibility of making choices. Rather, one of his complaints was that the conditions of the pro­letariat were such that the workers were forced to consume a restricted consump­tion basket, without many options. His claim that some social processes go on “behind the backs of the producers” (cf.

Hence, even though the typical human consumer lives within an organic social whole, it can be described as a normal human being, a psychophysical acting unit endowed with memory, able to store and process information, capable of per­forming arithmetic calculations, having preferences regarding the consumption of goods, able to make choices, and so on. Hence, the Gedankenkonkretum ‘con­sumer’ can be characterized by means of the conjunction of the following predicates:

(10) x possesses information.

(20) x has memory.

(30) x is able to perform calculations.

(40) x has preferences.

(50) x is able to make choices.

The consumer - the story goes - has information about and remembers the available consumption menus within a determined set of commodity bundles X, there are several menus for him to choose from (X is nonempty and has more than one element), the amount of wealth w he has available to buy a menu is limited, this wealth can be applied to obtain any of the menus x 2 X whose monetary value px (under a ruling system of prices p) is not greater than w, and not all menus are equally indifferent to him. (This last condition is supposed to mean that the consumer has a non-trivial transitive preference rela­tion ≥ defined over X.) Among the menus within B = {x 2 X ∣px ≤ wg, the con­sumer always chooses one that is most preferred among those in B. This last condition can be expressed by means of the predicate

(6⁰) x is rational.

This concept of consumer can be applied to typical human persons under a situation in which they have to choose consumption baskets. It is not at all ide­alized. The concept seems roughly correct and applicable in general terms. Ide­alization starts when the economists demand more specification and the performance of measurements: it is then that the scientists become interested in constructing a quantitative theory to match the intuitive ideas of the original theory.

the mathematical method is not an experimental method; it is a rational method,....the pure science of economics should then abstract and define ideal-type concepts in terms of which it carries its reasoning. The return to reality should not take place until the science is completed and then only with a view to practical applications.

Let us call ‘idealized concepts’ (or, briefly, idealizations) the concepts obtained by deformation out of concepts which are truly predicable of real concrete beings if, in spite of this deformation, the same are meaningful - even if they are false - of these real objects, and keep an intension akin to those.

Idealizations obtained by deforming predicates (10)—(60) are the following

(1) x possesses perfect information.

(2) x has perfect recall.

(3) x has unlimited computational powers.

(4) x has regular preferences.

(5) x is able to make choices.

(6) x always chooses a menu that is optimal within

{x 2 X ∣pχ ≤ wg for any (p, w).

The conjunction of predicates (1)-(6) defines a type or ideal object — what we have called a model system - nonexistent in reality but required to channel math­ematical reasoning. How is this model system related to set-theoretical structures?

First of all, perfect information and memory are required in order to postulate ‘large’ consumption sets, in fact as large as the nonnegative orthant of R^l (L a positive integer). Moreover, the consumer is supposed to produce or possess a quasiconcave, locally nonsatiated, continuously differentiable utility function rep­resenting his preference relation (this implies that the relation is connected and transitive). Given this function, the consumer is supposed to be able to maximize the utility function for any level (p, w) of prices and wealth. And he is indeed able, and sufficiently ‘rational’, to choose for consumption precisely the optimal menu.

Thus, on top of (1)-(6), the theory assumes the following axioms:

(7) x has as option set X the nonnegative orthant Ω of R^l.

(8) x has kitschy tastes (since his preference relation is strictly convex, he strictly prefers combinations of variegated styles).

(9) x is nonsatiated; he always prefers to consume more to less.

(10) the rate of change of x’s tastes, as he moves away from a consumption menu, is almost constant; i.e. the rate of change of his tastes is constant within an infinitesimal neighborhood of any of the con­sumption menus in the interior of Ω.

Axioms (1)-(10) delineate a very important idealization in economic theory, a model system depicting a very regular type of consumer. They also define a par­ticular type of set-theoretical structure. By itself, they do not involve any empir­ical claim, but can be used to make empirical claims about real-concrete consumers (the theory requires that the utility functions be invariant with respect to monotonic increasing transformations; i.e. that they be ordinal).

Demetris Portides (2013) has pointed out that there are three kinds of idealiza­tion: isolation, stabilization, and decomposition. The first two kinds have been described by Uskali Maki (1992, 1994), who distinguished among two types of idealizing assumptions: nullifying idealizations and stabilizing idealizations. Nullifying assumptions express that a certain factor, which is usually present affecting the powers of the isolated universal, is missing. If p(x) is the degree in which factor p is present at x, a nullifying assumption can be expressed in the form p(x) = 0. A stabilizing assumption expresses that the rate of change of factor p at x is nullified, and is expressed as p(x). Additionally, decomposition, according to Portides, “is the conceptual act of setting apart factors, clusters of factors, processes, or mechanisms in our model descriptions; a rough way to put it, that decomposition is the conceptual act of abstracting from interconnec­tion and interaction” (Portides 2013: 260). But we are in a position to see that these kinds of idealization are only particular cases of idealization. Another type of idealization, as important as the former, is deformation: the assumptions in this case do not mean that some factor is supposed nil or disconnected from others, but rather that the target system (or some of its parts) satisfies certain deformed predicates. Clearly, none of these kinds of idealization can be carried out if the target system - the Subjekt - has not been previously fixed by means of a Gedankenkonkretum.

A model system is an ideal system built by means of conjunctions of predicates expressing idealizations. The model system t-represents its respective target systems in the first place because the predicates defining the former are intelligi­ble deformations of predicates which are literally true of the second. But this rep­resentation is conventional up to a point and it is deemed as such a representation just because it was built with the intention of representing that target system in the first place, the scientific community sees it that way, and that’s the way it is pre­sented in the textbooks of the discipline. If model system S is taken to represent target system σ, we write S = s, following a notation introduced by Bunge (1972). How well σ is represented with S is an empirical matter that has to be settled in terms of how well the set-theoretical structure A associated to S matches the empirical data generated by σ.

It is not unusual in the economic sciences to find that, far from being formulated in complete generality, scientific theories start with the concoction of a very special model system. This is the case, for instance, of the stylized capitalist economy Marx describes in Capital, where (among other things) labor is sup­posed to be homogeneous. It is also the history of game theory, where zero­sum, two-person games were the first to be formulated.

What this means, in structuralist terms, is that, historically, economic theories begin with some specialization and it is only later that their basic theory-element is determined (if at all). Whereas in physics certain magnitudes (like weights) are closely related to the senses and common practices, and in some cases simple operations can be performed to measure them, the opposite situation is true in economics. Economic value, in terms of public prices, can be observed without the development of much abstract theory. Nevertheless, the central magnitudes of important theories like the labor theory of value, game theory, or consumer choice are not that close to the experience of the common man. For instance, in order to prove the existence of labor-values, very stringent assumptions must be made on the structure of the productive system. Something analogous happens in game theory, where the calculation, or just the proof of the existence of expected utilities, can be very complex.

Concretization is in some ‘virtual’ sense the opposite of specialization. According to the structuralist view, specializations are seen as specializations of a previously given basic theory-element, defined by an explicit fundamental law. Specialization consists of the introduction of special laws or theoretical sys­tematizations that describe with more detail putative specific applications of the theory. Historically, however, at least in economics, a very special theory­element appears first and then the problem is to find out that basic theory­element of which the special theory-element is a specialization. In other words, sometimes the special laws defining the typical theory-element - like Delphi’s oracle - neither reveal nor conceal the general fundamental law, but give signs. Moulines is absolutely right when he says:

From my own point of view, I consider approximation at least as fundamen­tal for empirical science as idealization, and moreover I think that the two notions, though interrelated, have to be conceptually distinguished; they are in need of different explications - both from a logical and a methodolog­ical point of view. Very roughly speaking, idealization is rather connected with model construction, whereas approximation is a relationship between already constructed models.

(Moulines 2007: 258-9)

As Walras has taught us, idealization has to do with the use of the mathematical method; it cares about nothing but the construction of model systems in which to carry out its reasoning. Once this problem has been solved, then it is possible to deal with the problem of better approximations, as part of the problem of the “return to reality”.

But it is usual to apply even theory-elements that are very idealized. How good these theory-elements are is - as I said above - an empirical matter. Wade Hands pointed out that some intended applications of some economic theories are not ‘concrete’, in the sense in which “our solar system” is a concrete application of Newtonian mechanics or “the U.S. economy in 1960” is a concrete application of macroeconomic theory:

While it may not be that general equilibrium theory is utterly devoid of such concrete intended applications, it is certainly safe to say “most” applications of the theory are not of this type. It seems that the sense in which general equilibrium is “empirical”, that it has concrete applications, is much more complex than can be captured in the standard structuralist definition of the set of intended applications.

(Hands 1985: 329)

Indeed, I have long stressed⁴ that the cause of this problem is a confusion between the intended applications in the structuralist sense (which are set- theoretical structures) and the target systems the theory presumptively applies to. I have distinguished above between the Gedankenkonkreten, the intended applications (which are set-theoretical structures), the empirical structures (models of data; also set-theoretical structures), the model system precisely described by the theoretical systematizations of the theory-element (an idealized, imagined system), the target or real-concrete systems the model systems intend to represent, and how all these elements are combined to produce an empirical claim. I think this explains, in a detailed and thorough way, the complex sense in which eco­nomic theories (or any empirical theory, for that matter) has concrete applications. If there are theories which will never apply to any real-concrete system, not even in a remotely approximate way, then these theories are not empirical at all; they are not scientific theories. But the structuralist view is not to be blamed for their failure.

The structure of scientific theories, and the ways they relate to their putative real-concrete Subjekte is far from being simple. The typical narrative of normal scientific practice runs like this. Having in sight a real-concrete economic system or phenomenon σ described by means of a general concept g - a Gedan- kenkonkretum - the scientist wants to use theory T in order to explain certain empirical data related to σ recorded in an empirical structure D. Two ways are open here. The first one, let me call it ‘parametrical’, consists of trying to ‘fit’ (imbed) the data in D into a partial potential model p of T in order to try to apply T (usually in an approximate way) to p. The second consists of using the data in D in order to directly determine the theoretical terms of T, by means of the laws or theoretical systematizations of T, build a model A of T, and then to generate a reduct r(A) into which the data are to be imbedded. The first way, the parametric one, was historically followed by Kepler, who intended to find curves (first circles and later ellipses) to fit the astronomical data left by Tycho Brahe. The second one has been proposed by Varian and other authors in order to determine the utility function of a consumer. The first method intends to obtain model A by means of integration; the second method by means of a certain algorithm.⁵

Hands also believes that the structuralist view cannot account for the inverse process to ‘virtual’ specialization:

Often theoretical progress occurs just in the reverse manner; the theory is made not more specific, but more general. Much of the history of general equilibrium theory can be characterized as a search for increasingly more general conditions which preserve the basic properties of the theory. This type of “generalizing” theoretical progress is outside the standard structural­ist view of theoretical progress and thus represents one more way in which the fit seems less than perfect.

(Hands 1985: 330)

Actually, the structuralist view is particularly suited to explain this process. It con­sists of postulating a theory-element T₀ of which the given, more idealized theory­element T₁ would be a specialization (in the usual structuralist sense). I claim, by the way, that this is the most important sense of concretization. Nowak’s view can be seen as a case of concretization in which the special conditions defining T₁ are isolations. But sometimes concretization is not merely de-isolation: it must also figure out the form of the fundamental law defining the theory (and hence also T₀). All my effort in (Garcia de la Sienra 1992) was devoted precisely to a task of this type,⁶ namely to find a more general form of the law of value in order to generalize the (then) standard model systems of the labor theory of value, taking into account very general productive structures with heterogeneous labor.⁷

De Donato (2011: 83) proposes understanding idealization “basically as a rela­tion between theory-elements just as any other intertheoretical relation”. I think he is right, but his explication only accounts for the case in which idealized theory-elements are obtained by means of nullifying assumptions. Thus, it would seem that concretization consists simply of dropping the nullifying assumptions in order to obtain a more general theory-element. But, as I have been trying to stress, finding more general versions of the fundamental law implicitly involved in the definition of the idealized theory-element can be harder than what such a description suggests, as it may involve unsuspected con­ceptual transformations of the required notions.

Notes

1 Cf. “Die Methode der politischen Okonomie” in Marx (1983); I follow closely the Penguin edition (Marx 1973).

2 I myself believed this when I wrote that a philosophia de ente was necessary to that end (cf. Garcia de la Sienra 1990). Stigum (2003: 37ff) also seems to believe it when he introduces his “world of facts”.

3 For a survey of this last project, cf. Crangle (2015).

4 For instance, Garcia de la Sienra (1988: 77).

5 These methods will be discussed in Chapter 5 when I deal with the topic of measurement.

6 Cf. Garcia de la Sienra (2007) for a survey of the approaches to the problem.

7 A similar case in biology has been reported by Lorenzano (2014).

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