Some Contributions of Rolf R. Mantel to Economists' Formation

Omar O. Chisari

Introducing James Tobin at the XXI Annual Meeting of the Argentine Association of Political Economy, Rolf Mantel remembered a qualification applied formerly to Tobin: "An Economist to economists" (Tobin (1986), page 17).

This definition of the man is very apt. Because Rolf Mantel channeled most of his own work and efforts in two directions: trying to make more consistent and logical the theoretical construction of economics, and giving us, as economists, better tools to take or recommend policy decisions.

Rolf Mantel taught us several things, both to people working at the theoretical frontier of economics as to practicing professionals, in part because economics is a science where there is a very thin line between theory and professional practice.

Let us remember some examples of the contributions of Rolf Mantel in these two spheres, some of them drawn up by himself (Mantel (1985)). I have selected a few of them, only three, under the influence of my own interests and preferences.

Rolf Mantel (1974) replied with an elegant proof to the Sonnenschein Conjecture (1973), "…the last step lacking to Uzawa's theorem, to permit concluding that the Kakutani mathematical problem and the Walras economic one are equivalent from the purely logical point of view" (see Mantel (1985)).

Given that aggregate excess demand functions are continuous, homogeneous and conform to Walras' law, the problem was to demonstrate the reciprocal proposition, that is, arbitrary functions, with the properties of being continuous, homogeneous and verifying Walras' law could be deemed excess demand functions of some economy (in which these functions could be derived from consumers and producers behaviour as usually postulated).

Mantel's proof was superior to the Sonnenschein one because he didn't constrain himself to polinomial functions in prices, using instead specifications that differed "slightly" from convex functions (Debreu (1974) generalized thereafter the proposition). In fact, in his synthesis of microeconomics, Mas-Colell, Whinston and Green (1995) name the corresponding section of their book (17.E) as "Anything Goes: The Sonnenschein-Mantel-Debreu Theorem".

The consequences of those papers are more than relevant for our science. The consistency of the Walrasian model, so elegant and fruitful, is a necessary corollary of the logical equivalence between the general equilibrium model of Walras and fixed point theorems; the Walrasian theory of general equilibrium, when it is founded on basic data such as preferences, endowments and technologies, is mathematical by its own nature.

Moreover, every method of calculation of a solution must have a fixed point theorem incorporated. Mantel himself noted this property (Mantel (1985)) when summarizing his research on computing general equilibrium.

There, in the discovery of such algorithms of calculation, we have a second instance of the contributions of Professor Mantel to our discipline. His algorithm minimized the distance between actual and equilibrium, unknown prices, and discarded fixed points theorems.

Scarf corrected Mantel's method, searched for prices at which quantities demanded and supplied were approximately equal, and his algorithm - which neither computes an equilibrium but only approximates to it (see Ellickson (1993)) - was improved upon by Merril. Mantel (1978) offered thereafter a proposal with a superior procedure; without his research and his initial impetus, the field of general computable equilibrium would have arrived, no doubt, later to its current state.

Models of general computable equilibrium have been and are useful tools for decision-making, as well as forecasting and policy design. Mantel (1985) summarized a great deal of applications; also, Shoven and Whalley (1992) present an interesting discussion.

Now let us see the last example. In recent years Professor Mantel insisted on a line of research that he had initiated in the seventies, on the utilisation of a variable rate of discount in models of optimal control, under the influence of developments by I. Fisher, Uzawa and Koopmans.

I had the opportunity of commenting on a paper presented by him on this subject, at a conference in Tafí del Valle, organized by the National University of Tucumán and the University of Chile. He offered an improved presentation on the same subject at a Miguel Sidrauski Lecture, at the XV Latin-American Meeting of the Econometric Society, organized at Santiago de Chile in August 1998.

Professor Mantel had several important insights when he insisted on the need to have consistent models of optimal control with a variable rate of temporal preference.

As he used to say (Mantel (1993b)), in models of discret time there is the possibility of relaxing the hypothesis of time-independent preferences, and obtaining then a greater wealth of results. This is impossible with continuous time analysis.

A very important consequence is derived from this generalisation: the dependence of asymptotic trajectories on initial conditions and, in this sense, the proposition that the final position of an economy depends crucially on the stock of resources it owns, that is, our aspirations can depend on our endowments - see Mantel (1993a) and also Mantel (1998) for a synthesis of the main results.

The main insight was that differences in growth between countries "could be explained in terms of interacting preferences and actual resources, more than in terms of technological factors" (see Mantel (1998)).

As a consequence, it is convenient to emphasize two aspects of this line of research. Even when data used by economists are represented in discrete time, we cannot dismiss that economic agents take their decisions in continuous time; Mantel's last papers close the gap between the process of perceiving information and the building of theoretical and planning-oriented models of growth, from an axiomatic point of departure.

Besides, he was providing useful material to the new literature on endogenous growth. In place of emphasizing factor externalities or the public-good characteristic of some inputs, which impose a lower bound to growth impending the occurrence of a stationary state, he pointed to the dependence of a preference parameter on control variables, for example the consumption level. We can add that some of the recent literature analyzes also the processes of habit formation in determining asset prices, abandoning the restrictive hypothesis of inter-temporal separability (e.g. Chapman (1998)).

In the cases we have revised, as in his other areas of research, Rolf Mantel always preferred using mathematical tools and building from an axiomatic base, clear and consistent. Notwithstanding, I have obtained from this revision of his publications, studies and contributions an interesting corollary: for him, the mathematical method always was useful to solve an economic problem. Does this imply that some theorems in Walrasian economics seem mathematical theorems? Obviously this is as it should be in his field of research. After all, Uzawa proved that fixed point theorems and the existence of an equilibrium solution are equivalent (see again Mantel (1985)).

Of course, his field of research was much larger than the mentioned subjects. There are in his papers research on international trade, growth, optimal taxes, equity and income distribution, even the history of economic thought; several contributions were co-authored with his wife.

It is not difficult to assess his high quality not only as an economist, but as a human being. To be tolerant towards young economists - several steps down the value scale! - who are, on occasions, both imprudent and irreverent, is one of those relevant proxy variables of personal value. When I first met Rolf Mantel he was already a distinguished personality and economist. But this didn't impede his being kind to me and to other colleagues, still novel concerning theoretical economics.

One of the distinct characteristics of human beings is that we are able to conceive states of nature in which we no longer exist. It is a paradox that this is easier than to antipate a scenario in which other people, necessary to us or whose work provides sense to the world, are lost. Rolf Mantel's passing catches us by surprise, especially those of us who are economists, and we feel that our logical edifice is suddenly more fragile than before.

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