Gerhard Tintner’s memorial survey evaluates Wald’s contributions to economic equilibrium, cost-of-living indexes, statistical decision theory, seasonal adjustment, identification, and stochastic difference equations. Its opening establishes the significance of the loss:
The untimely death of Professor Wald has deprived econometrics of a man who has made many important contributions to mathematical economics and statistics.
Tintner combines appreciation of mathematical rigor with attention to economic applicability. Wald’s importance lies in establishing when formal systems admit economically meaningful solutions and when statistical methods justify inference, while specifying the assumptions on which those achievements depend.
The first section examines production systems, Walrasian exchange, and Cournot duopoly. Rather than treating the formulation of equations as sufficient, Wald investigates existence, uniqueness, positivity, and stability under specified conditions. Production theory accommodates surplus commodities as free goods, while exchange theory imposes restrictions on initial endowments and utility. Tintner explains the economic rationale for one restriction:
It is derived from a proposition in value theory: the marginal utility of each commodity is much more influenced by changes in the quantity of this commodity than of any other commodity.
The connection between an economic proposition and a mathematical condition illustrates the survey’s central concern: proofs must establish economically admissible outcomes, not merely formal solutions. In duopoly, Wald considers firms that take their competitor’s supply as given and maximize profit, neglecting costs. Demand restrictions yield a unique, continuous, monotonically decreasing reaction function and support analysis of equilibrium and stability. Tintner characterizes the technical achievement succinctly:
The mathematical methods used in the proofs are essentially topological and related to the theory of sets.
He nevertheless calls for analogous results covering other economic systems and market organizations, and connects this foundational work with activity analysis.
The discussion of cost-of-living indexes shifts from equilibrium to the limits of welfare measurement. Given observed prices and consumption in two situations and an unchanged indifference system, Wald derives bounds that cannot be tightened without further information. Additional assumptions thus have an explicit informational role. A locally quadratic utility function permits an approximate index and implies linear Engel curves whose coefficients can be estimated from family budgets; under special conditions, Wald’s formula yields Fisher’s ideal index. Related investigations reconstruct approximate indifference systems from Engel curves subject to integrability and stability conditions. Tintner emphasizes the gap between these theoretical possibilities and their limited empirical exploitation despite available budget data.
Statistical decision theory receives a more qualified appraisal. Wald generalizes hypothesis testing through losses attached to errors and risks associated with decisions. Tintner notes the tentative character of Wald’s advocacy of minimax and his interest in a less conservative principle. Industrial quality control offers a persuasive application, but social policy poses difficulties in measuring and comparing gains and losses. Invoking Arrow’s social-choice findings, Tintner questions whether the requisite evaluations can generally be supplied. He also challenges the representation of Nature as an adversary: guarding against its worst possible action encourages extreme conservatism, although Nature is indifferent to the statistician. These objections do not diminish Wald’s contribution to multiple-choice problems, such as selecting a polynomial’s degree or determining the number of systematic linear relationships.
The shorter treatment of seasonal adjustment presents Wald’s procedure as valuable because its assumptions are explicit and economically plausible. It separates trend and cycle, seasonal variation, and random variation, using twelve-month moving averages and least-squares approximations to estimate slowly changing seasonality. Rapid seasonal change requires more elaborate treatment.
Identification connects statistical estimation with economic policy by asking whether behavioral equations can be uniquely determined within a stochastic system. Tintner expects Wald’s general results on identification and incomplete systems to retain their importance despite limited empirical application. The concluding discussion addresses consistent and asymptotically normal estimation in stationary linear stochastic difference equations and particular simultaneous systems. More general uniqueness depends on prior restrictions on coefficients or disturbance covariances. Tintner’s call for difficult but practically valuable small-sample results closes a survey in which mathematical achievement continually prompts further empirical and methodological work.
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