Equilibrium Systems and Maximization

The methodological claims made in the first chapter would be hollow if Samuelson could not justify them by showing how the goal of deriving operational theorems could be achieved.

c. Note that he was implicitly rejecting the representative agent model, which came to dominate macroeconomics in the 1970s, long before it had been enunciated. “Equilibrium Systems and Comparative Statics.” His starting point was the need to abstract from reality. Theories dealt not with the whole of reality but with carefully selected aspects of reality. The variables in which we are inter­ested (prices, quantities, etc.) are the unknowns, and their values are assumed to be determined by equations or functional relationships that describe the situation under consideration. For example, the price and quantity of a good are assumed to be determined by supply and demand—two functional rela­tionships between price and quantity.^d We need enough equations to deter­mine all the unknowns.

This was all well known. Where Samuelson went beyond this was in mak­ing the observation, considered trivial today but not so when he was writing, that we need something more. We need to introduce parameters (variables not determined by the system), changes in which will produce changes in the variables of interest. Suppose that we wish to ascertain the effect of a sales tax on the price and quantity of a good being sold. The functional relationships are the demand and supply curves for the product. If the sales tax is assumed to constitute a cost for producers (defining the market price as inclusive of the tax), then a change in the tax rate will shift the supply curve upward, chang­ing the equilibrium price and quantity—the price and quantity at which supply equals demand. The task of the economic theorist, then, is to work out how changes in parameters such as the tax rate will change the values of the unknowns.

Samuelson made three important points about this use of equilibrium. The first was that equilibrium, as he was using the term, had no normative connotations: there was no reason to believe that equilibrium was desirable or undesirable. Equilibrium meant “the value of the variables determined by a set of conditions.”¹⁵ Any system can be represented as an equilibrium system. The second was that it abstracted from issues of time, an assumption that he

d. It is generally assumed that as the price rises, the amount demanded will fall and the amount supplied will rise, and that the two curves will intersect at some point. This point where supply and demand are equal is the equilibrium price. If either curve shifts—for example, a rise in income might cause consumers to demand more of a good—the point at which the two curves intersect will change, implying that the equilibrium price and quantity have changed.

proposed to tackle separately. The third was that there was no firm rule about which variables should be taken as data (parameters) and which should be explained by the theory. Traditionally, economists had taken as data those factors that they felt unable to explain—“tastes, technology, the govern­mental and institutional framework”¹⁶—but there was nothing fundamental about this. Systems could be as broad or as narrow as the theorist wanted. His illustration of this was that although government policy might be a parameter for many economic problems, understanding the business cycle might require a theory that explained government policy.

In the remainder of the chapter, Samuelson translated these arguments into mathematics—his subheading was simply “Symbolic Representation.” His argument was presented in terms of variables, parameters, and func­tional relationships, without specifying any economic content. By keeping the analysis at a highly abstract level he made the point that he was present­ing a method that could be applied to any economic problem, an important part of his argument being that it was “precisely because theoretical economics does not confine itself to specific narrow types of functions that it is able to achieve wide generality in its initial formulation.”¹⁸ However, the equations he wrote down were, he claimed, not completely without content, for he started with a set of equations describing equilibrium. He then manipulated these equations to derive equations that showed each variable as a function of the parameters.^e

Given that his two sets of equations were equivalent, why not omit the first step? Why not omit talk of equilibrium and simply start with relationships between variables and parameters? For example, Cassel, whose general equilibrium system Director had introduced to him as an undergraduate, had argued that it was pointless starting with the assump­tion that consumers maximized utility (an equilibrium problem) and that instead the economist should start by postulating consumer demand

e. He started with n equations of the form f^t(x₁,x₂,...x_n;d_λ,..d_m) = 0, where i = 1,...n the %’s are the variables, and the ds are the parameters.

More important, the fact that relationships between variables and param­eters were derived from an equilibrium system might imply something about them. To return to the example, the fact that consumer demand functions were derived from utility maximization might provide some testable restriction on their form, giving them meaning. Simply to state that demand for a commodity depended on prices of all goods had little content unless something could be said about functional forms, and the hypothesis of equilibrium might be able to do this. Samuelson then showed how testable predictions could be derived in this highly abstract framework, before working through two simple examples—a tax example where there was just a single variable of interest, and a market example where there were two.

This short chapter, covering highly abstract material, has been dis­cussed in detail because it encapsulates the most important arguments in Samuelson’s thesis:

(ι) Many economic problems had the same structure, something that could be shown only by abstracting from the details of specific problems.

(2) When this was done, mathematical techniques (partial differential equations and matrix algebra) could be used to derive propositions that might not be obvious to those who confined themselves to verbal reasoning or simpler mathematics.

(3) Postulating that a set of equations was an equilibrium system might in itself be sufficient to provide information about relationships between variables and parameters.

(4) To derive testable relationships it was necessary not to describe equilibrium but to analyze how it changed in response to changes in parameters—to perform comparative static analysis.

Samuelson was providing an argument about how to do economics, arguing that the use of mathematical methods that were new to economists made it possible to turn long-held theories about economics operational. Given that he also described operational theories as meaningful, that was not just an argument that economists could use mathematical methods: it was an argument that they should do so.

Samuelson believed that, even if the behavior of individuals could be ana­lyzed as the solution to a maximization problem, the behavior of groups of individuals could not be studied in the same way.^f Given this, it was impor­tant to him that equilibrium systems might be of two types: equilibrium might be the result of maximizing behavior or they might be the point of rest in a dynamic system in which nothing was being maximized. Postponing the latter to the last two chapters of the thesis, he started by analyzing maxi­mization, beginning with a long chapter on “The Theory of Maximizing Behavior.” This was obviously relevant for situations where economic agents were consciously maximizing something, as when companies chose to pro­duce the output at which their profit was maximized. The case of consum­ers was similar in that choosing their most preferred bundle of goods could be represented as maximizing an ordinal utility function. However, there might also be cases where no conscious optimization was involved, but which behavior could be represented as the solution to an optimization problem. He drew an analogy with physics:

In some cases as we shall see later, it is possible to formulate our condi­tions of equilibrium as those of an extremum problem, even though it is admittedly not a case of any individual’s behaving in a maximiz­ing manner, just as it is possible in classical dynamics to express the path of a particle as one which maximizes (minimizes) some quantity despite the fact that the particle is obviously not acting consciously or purposively.²⁰

He assumed that his readers would be familiar with the relevant physics, and did not provide any example.^g He argued that even when economists defended their theories on other grounds—for example, as resting on plausi­ble laws such as diminishing marginal productivity—they frequently rested on some implicit underlying optimization problem.

f. This is related to what economists commonly call aggregation problems: the behavior of an aggregate need not be the same as the behavior of the individuals who make up the group. It can be proved rigorously that group behavior will be a scaled-up version of individual behavior only under conditions that are so special they can rarely be satisfied in any real-world context. For example, even if every individual’s demand for a commodity falls when the price of the commodity rises, it is possible that, unless people are identical, market demand may not fall.

g. One of the simplest examples is the catenary—the shape of a cable suspended between two points—which can be calculated as the path that minimizes the potential energy of the cable. The example of the consumer did not fall into this category because, though consumers might not consciously maximize anything, their behavior was purposive. theory of maximizing behavior could unify much, though not all, economic theory.

Samuelson’s account of maximizing behavior, provided in chapter 3, emphasized three points. The first of these was Samuelson’s discussion of what he called “the Generalized Le Chatelier Principle.”²¹ The principle is named after the French chemist Henri Le Chatelier, who in 1884 observed that, starting with a chemical system that is in equilibrium, if one of the variables is changed, the equilibrium will be changed so as to counteract the effect of the change.¹¹ Samuelson had learned from Wilson that this principle was not something specific to chemistry but was a general mathematical rela­tion—a property of any maximum or minimum system—and that it might, therefore, be applicable to economics.²² In Samuelson’s hands, it became the theorem that if a system is at an maximum or minimum, the effect of relax­ing a constraint is reduced by the presence of additional constraints. It was a “generalized” Le Chatelier Principle because it was shorn of any reference to chemical equilibrium.^{[31] [32] To see its implications for economics, consider the example of a company employing labor. If the wage rate rises, the com­pany may choose to employ less labor, possibly because it will choose to use mechanized production methods. However, if the company is prevented from employing the optimal amount of machinery, this will reduce the effect of the wage increase on the demand for labor. The implication that a company’s demand for labor would be more elastic (more responsive to changes in the wage rate) in the long run, when the stock of machinery and other factors of production could be adjusted, than in the short run, when other factors could not be adjusted, was well known. Samuelson’s point was that this had noth­ing to do specifically with economic arguments—it was simply the result of assuming that the company was in an equilibrium in which it was minimiz­ing or maximizing some objective function.}

The second point, one on which Wilson had insisted, was that Samuelson emphasized the importance of finite changes. It was very useful, and math­ematically convenient, to use differential calculus and hence to analyze infinitesimal changes. However, real-world problems always concerned finite changes, and considering infinitesimal changes was useful only insofar as it provided information about finite changes. Finite changes were fundamental. This was connected to the point that the most general statements of equilib­rium conditions typically involved inequalities, rather than equalities.

Finally, Samuelson argued that many economic problems, though they might not appear to involve maximization or minimization problems, could be restated as such problems. This was a variation of the well-known “inte­grability problem” in demand theory, which concerned whether, given a set of demand functions, it was possible to represent those as the outcome of maximizing some utility function. Once again, Samuelson was taking a spe­cific problem and generalizing it, arguing that it was important to focus, not on the details of the specific economic example, but on the properties of the general mathematical problem.