## Maximization

## Maximization

# Maximization

In modern economics agents are assumed to maximize, so naturally the structure of the maximization assumption has had a significant impact on the structure of modern economics. On one hand, the widespread use of maximization has given economics an intellectual unity not seen in most of the social sciences; on the other hand, the reliance on maximization threatens to halt empirical and theoretical progress in economics.

Nobel Prize winner Paul Samuelson should be credited with making maximization the foundation of modern economics. In his influential *Foundations of Economic Analysis* (1947), Samuelson demonstrates that consumer and firm behavior could be usefully modeled as solutions to what are now known, in mathematics, as classical programming problems.

In general, the classical programming problem takes the following mathematical form:

Max _{x1, x2, …, xn } *F* (*x* _{1}, *x* _{2}, …, *x _{n}* ) subject to:

*g* _{1} (*x* _{1}, *x* _{2}, …, *x _{n}* ) =

*b*

_{1}

*g* _{2} (*x* _{1}, *x* _{2}, …, *x _{n}* ) =

*b*

_{2}

*g* _{m } (*x* _{1}, *x* _{2}, …, *x _{n}* ) =

*b*

_{m }.

The *n* variables *x* _{1}, *x* _{2}, …, x_{n} are the instruments. The function *F* (·) is the objective function, and the *m* functions *g* _{1}(·), *g* _{2}(·), …, g_{m}(·) are the constraint functions. The constants *b* _{1}, *b* _{2}, …, *b _{m}* are the constraint constants. Interpret the instruments as a representation of the choices available to the agent. Interpret the objective function as a representation of the agent’s desires with respect to the choices. Finally, interpret the constraint functions and constants as a representation of the limits the environment imposes upon the agent’s choices. Under these interpretations, the classical programming problem becomes a model of constrained maximization by an agent. If certain mathematical assumptions are met, a solution to the classical programming problem exists (Intriligator 1971). The method of LaGrange multipliers can be used to find a solution. Once a solution is deduced, compare its characteristics against the agent’s observable behavior. If there is a match, the agent’s behavior has been explained. If not, change the mathematical structure and repeat the process.

It is important to recognize that although the classical programming version of constrained maximization was new to most economists at the time of Samuelson’s *Foundations*, the idea of constrained maximization was familiar. Constrained maximization is merely a sophisticated version of what the philosopher Daniel Dennett (1987) calls the “intentional stance.” The intentional stance is a strategy for prediction and explanation. The first step in taking an intentional stance toward something—call it *Z* —is to attribute beliefs, desires, and rationality to *Z*. Since *Z* is assumed to be rational, the attributed beliefs and desires ought to make sense in the context of *Z* ’s circumstances. One can then predict (or explain) *Z* ’s behavior by determining what is rational given *Z* ’s attributed beliefs and desires. Since there is always leeway in the attribution of beliefs and desires, if a prediction turns out wrong, or an explanation does not impress, the intentional stance need not be questioned for we can always revise the beliefs and desire we attribute to *Z*. The intentional stance is practically irrefutable.

As Dennett notes, there is nothing particularly profound about the intentional stance. It is “folk psychology,” “familiar to us since childhood and used effortlessly by us all every day” (Dennett 1987, p. 7). Yet when Samuelson merged the intentional stance with classical programming, he elevated this folk psychology to science. Think of Thomas Kuhn’s (1962) definition of normal science as the relatively routine puzzle-solving activity of trained professionals. All can use the intentional stance effortlessly; few can do classical programming effortlessly, even with training.

Economic theorists at the research frontier quickly mastered the classical programming version of constrained maximization and began rapidly to transform the discipline. Mechanisms were created to select and reward mathematical skill. In one generation—the late 1940s to the late 1970s—the leading periodicals in economics, the leading economics departments in America, the minimal mathematical proficiency levels of American graduate students in economics, and the standards defining scholarly success in economics all changed. The mathematical goals of rigor, generality, and simplicity became widely shared imperatives in economics. The Nobel Prize winner Gerard Debreu uses the term: “the mathematization of economic theory” (Debreu 1991) to refer to this evolution.

To illustrate the nature of this evolution, consider the discovery of nonlinear programming by Harold Kuhn and Albert Tucker (1951). The nonlinear programming problem is similar to the classical programming problem described above; the only differences are that the constraint functions are inequalities (≤) and the instruments *x _{i}* (

*i*= 1, 2, …,

*n*) are nonnegative (≥ 0). However, surface similarities mask deep differences; mathematically, nonlinear programming is better than classical programming. Nonlinear programming imposes fewer restrictions than classical programming, and it is based on a less complex mathematical foundation (convexity) than classical programming (differentiability). Given the new emphasis on rigor, generality, and simplicity in economics, on the battlefield of economic theory, nonlinear programming wiped out classical programming. Yet empirically speaking there is no difference between the two types of mathematical programming. Not surprisingly, modeling with nonlinear programming did not lead to new empirical generalizations.

Once maximization opened the gate to mathematization, constrained maximization became even more entrenched as the preferred method of doing economics. If you like rigor, generality, and simplicity then you like mathematical programming. So after years of selecting and rewarding mathematical skill, by the late 1970s the field of economics had as much consensus in the National Science Foundation’s peer review system as the fields of chemical dynamics and solid-state physics (Cole, Cole, and Simon 1981).

Of course the consensus is incomplete. One of the earliest and most profound critics of constrained maximization is the Nobel Prize winner Herbert Simon. In 1955 Simon proposed satisficing as an alternative explanation of economic behavior, and throughout his career he urged economists to take a more empirical approach to their science. For more than four decades Simon carefully observed agents making choices, created theories, and devised experiments and computer simulations to test his theories. His work helped create two brand-new disciplines: artificial intelligence and cognitive science. Further, the primarily empirical research produced by two cognitive scientists inspired by Simon—Daniel Kahneman and Amos Tversky—received one of the 2002 Nobel Prizes in economics.

The work of Simon, Kahneman, and Tversky is very different from the economics produced after mathematization. To explain the difference, let us return to Dennett. In developing his notion of the intentional stance, Dennett has often referred to a contrasting strategy for prediction and explanation; he calls this strategy the

Price theory | Applications of price theory | Behavioral economics | |
---|---|---|---|

Empirical | 40 | 27 | 6 |

Theoretical | 10 | 20 | 10 |

design stance (Dennett 1987, pp. 16–17). Simon, Kahneman, and Tversky can be interpreted as applying the design stance toward human beings, and the first step in taking this sort of design stance is to figure out “how the machinery which Nature has provided us works” (Dennett 1987, p. 33). Figuring out the human machinery and implementing the intentional stance on a system of mathematical objects are two tasks divergent in empirical content. In which direction will economics go? Will it remain on the deeply mathematical track of constrained maximization, or will it move toward Simon and become more empirical?

Pierre-Andre Chiappori and Steven Levitt (2003) categorized every microeconomics paper in three prestigious economics journals between the years of 1999 and 2001. Consider three of their categories: (1) *price theory,* which “refers to basic economic principles and techniques used by economists in the 1950s and before” (p. 152); (2) *applications of price theory,* which “refer to the testing of simple economic ideas … in domains outside the traditional purview of the field” (p. 152); and (3) *behavioral economics,* which gives us a rough indicator of research in the spirit of Simon, Kahneman, and Tversky. The table includes the percentages of the paper types that Chiappori and Levitt placed in these three categories.

The pattern displayed in the table invites the question: If a high proportion of the most talented current economists are exploring economic ideas that were well known a half century ago, and few seem inspired by the current revolutions in cognitive science and artificial intelligence, will economics continue to achieve empirical and theoretical progress?

**SEE ALSO** *Debreu, Gerard; Economics, Neoclassical; Kuhn, Thomas; Maximin Principle; Minimization; Optimizing Behavior; Programming, Linear and Nonlinear; Samuelson, Paul A.; Satisficing Behavior; Science*

## BIBLIOGRAPHY

Chiappori, Pierre-Andre, and Steven D. Levitt. 2003. An Examination of the Influence of Theory and Individual Theorists on Empirical Research in Microeconomics. *American Economic Review* 93 (2): 151–155.

Cole, Stephen, Jonathan R. Cole, and Gary A. Simon. 1981. Chance and Consensus in Peer Review. *Science* 214 (November): 881–886.

Debreu, Gerard. 1991. Economic Theory in the Mathematical Mode. *American Economic Review* 74 (3): 267–278.

Dennett, Daniel. 1987. *The Intentional Stance.* Cambridge, MA: MIT Press.

Intriligator, Michael. 1971. *Mathematical Optimization and Economic Theory.* Englewood Cliffs, NJ: Prentice Hall.

Kuhn, H. W., and A. W. Tucker. 1951. Nonlinear Programming. In *Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability,* ed. J. Neyman. Berkeley: University of California Press.

Kuhn, Thomas. 1962. *The Structure of Scientific Revolutions*. Chicago: University of Chicago Press.

Samuelson, Paul. 1947. *Foundations of Economic Analysis.* Cambridge, MA: Harvard University Press.

Simon, Herbert. 1955. A Behavioral Model of Rational Choice. *Quarterly Journal of Economics* 69 (1): 99–118.

*Gregory A. Lilly*