3-Milestones Page 77 Wednesday, February 4, 2004 12:47 PM
Milestones in Financial Modeling and Investment Management 77Convinced that economics should become a mathematical science,
Walras set himself the task of writing the first mathematical general
equilibrium system. The British economist Stanley Jevons and the Aus-
trian economist Carl Menger had already formulated the idea of eco-
nomic equilibrium as a situation where supply and demand match in
interrelated markets. Walras’s objective—to prove that equilibrium was
indeed possible—required the explicit formulation of the equations of
supply-and-demand equilibrium.
Walras introduced the idea of tatonemment (French for groping) as a
process of exploration by which a central auctioneer determines equilib-
rium prices. A century before, in 1776, in his book An Inquiry into the
Nature and Causes of the Wealth of Nations, Adam Smith had introduced
the notion of the “invisible hand” that coordinates the activity of inde-
pendent competitive agents to achieve desirable global goals.^2 Walras was
to make the hand “visible” by defining the process of price discovery.
Pareto followed Walras in the Chair of Economics at the University of
Lausanne. Pareto’s focus was the process of economic decision-making. He
replaced the idea of supply-and-demand equilibrium with a more general
idea of the ordering of preferences through utility functions.^3 Equilibrium
is reached where marginal utilities are zero. The Pareto system hypothe-
sized that agents are able to order their preferences and take into account
constraints in such a way that a numerical index—“utility” in today’s ter-
minology—can be associated to each choice.^4 Economic decision-making
is therefore based on the maximization of utility. As Pareto assumed utility
to be a differentiable function, global equilibrium is reached where mar-
ginal utilities (i.e., the partial derivatives of utility) vanish.
Pareto was especially interested in the problem of the global opti-
mum of utility. The Pareto optimum is a state in which nobody can be
better off without making others worse off. A Pareto optimum does not
imply the equal division of resources; quite the contrary, a Pareto opti-
mum might be a maximally unequal distribution of wealth.(^2) In the modern parlance of complex systems, the “invisible hand” would be called
an “emerging property” of competitive markets. Much recent work on complex sys-
tems and artificial life has focused on understanding how the local interaction of in-
dividuals might result in complex and purposeful global behavior.
(^3) Pareto used the word “ophelimity” to designate what we would now call utility.
The concept of ophelimity is slightly different from the concept of utility insofar as
ophelimity includes constraints on people’s preferences.
(^4) It was not until 1944 that utility theory was formalized in a set of necessary and
sufficient axioms by von Neumann and Morgenstern and applied to decision-making
under risk and uncertainty. See John von Neumann and Oskar Morgenstern, Theory
of Games and Economic Behavior (Princeton, NJ: Princeton University Press,
1944).