The Dawn of Mathematical Economics: From Jevons and Walras to Arrow and Debreu

The field of economics has long been intertwined with mathematical formalisms, with a significant evolution marked by two major shifts in the early 20th century. This article delves into the origins of these shifts, focusing on the paradigm shifts that initiated the era of mathematical economics, and the subsequent advancements that have reshaped economic discourse.

From Petty to Jevons: The Early Foundations

Although mathematical formalisms have been part of economics since the 17th century, with the pioneering work of William Petty in his Political Arithmetic, the true dawn of mathematical economics emerged with the Marginal Revolution in the 1880s. This period, initiated by scholars such as William Stanley Jevons and Léon Walras, marked a significant paradigm shift in economic discourse. The Marginal Revolution was characterized by the introduction of marginal analysis, which focuses on the effect of changes at the margin of economic quantities.

Central to this marginalism was the natural translation to the structure of differential calculus. Intermediate microeconomics, as it is now taught, is a direct result of this shift. The concept of marginal utility, as developed by Jevons, Walras, and others, was foundational to the development of microeconomic theory. Marginal analysis provided a more precise and predictive framework for understanding economic behavior.

Arrow and Debreu: The General Equilibrium Paradigm

The second major shift in mathematical economics came around 1950 with the pioneering work of Kenneth Arrow and Gerard Debreu. Their proof of the existence of general equilibrium represented a significant advancement in economic theory. This proof, drawing from fixed-point theorems in topology, was more mathematically sophisticated than previous economic exercises. It not only closed a major gap in previous theory but also demonstrated the application of powerful mathematical tools in economic analysis.

The proof of general equilibrium by Arrow and Debreu had far-reaching implications. It inspired a multitude of contemporary applications, including game theory, where proofs can also be quite involved. Game theory, with its mathematical underpinnings, has become a crucial tool in economic analysis, game theory, and other fields. These applications have further solidified the importance of mathematical economics in modern economic research and policy-making.

Beyond General Equilibrium: The Lucas Critique and Recursive Methods

The Lucas Critique, introduced in the 1970s, brought a new level of complexity to economic modeling. This critique highlighted the importance of recursive methods in economic analysis. The Lucas Critique challenged the assumptions of stable policy effects, suggesting that policy changes lead to adaptive behavior by economic agents. This critique, along with the developments in recursive methods, has significantly influenced contemporary economic modeling and policy formulation.

The Evolution of Economic Thought: From Marx to Neoclassical and Beyond

The field of economics has seen a significant evolution in thought, with various schools of economic thought emerging and evolving over time. Michael Hudson, an economist and historian, provides important insights into this evolution. Hudson notes that economics used to be a more diverse field, with Marx studied alongside Smith and Ricardo, with Marx representing a fringe perspective that had evolved off classical liberalism and other socialisms.

The term "neoclassical" is, however, seen by many as a propaganda term, used to legitimize modern economics as a legitimate evolution from classical principles of the 1700s, but excluding any ideas and principles that could contain a whiff of socialism. New Keynesians are generally considered a bastardization of Keynes adapted for neoliberal views and outcomes. Steve Keen also criticizes the neoclassical adherence to equilibrium models, suggesting that these models are less suitable for the macroeconomic world, which is more suited to chaos theory.

The neoclassical view of mankind as utility-maximizing automations is another point of critique. This view has been criticized for lending itself to a kind of totalitarian form of economic thinking. These criticisms highlight the ongoing debates and evolving perspectives within the field of economics.

Conclusion

From the Marginal Revolution of the 1880s to the proofs of general equilibrium in the 1950s, and the Lucas Critique and recursive methods in the 1970s, the field of mathematical economics has undergone a significant transformation. These shifts have not only reshaped economic theory but have also influenced economic policy-making.

The evolution of economic thought, from classical liberals and socialists to the current neoclassical and Keynesian perspectives, further underscores the dynamic and evolving nature of economic theory. As we continue to grapple with complex economic challenges, the tools of mathematical economics remain crucial for accurately modeling and predicting economic behavior. The future of economic thought is likely to be shaped by ongoing debates and new developments in mathematical economics.

For a deeper understanding of the evolution of economic thought, we recommend consulting the works of Michael Hudson and other economists and historians. Their insights provide valuable context and nuanced perspectives on the development of economic theory over time.