General equilibrium theory studies whether all markets can function together, not just one at a time. Debreu reformulated Walrasian equilibrium with strict mathematics, deriving demand and supply functions from the assumption that consumers maximize utility and firms maximize profit. Using properties of convex sets and fixed-point theorems, he identified conditions—such as compactness and continuity—under which a price vector exists that clears every market. This provided a unified framework for evaluating taxes, trade, and many other policies. His approach helped transform economics from a mostly descriptive discipline into a mathematical theoretical science.

In his 1959 book “Theory of Value,” Debreu fully formalized the Arrow–Debreu model and proved both an existence theorem and the two welfare theorems. Treating each agent’s preferences as convex, closed sets embedded in a real vector space, he established continuity and homogeneity of the aggregate excess-demand function. By applying the Kakutani fixed-point theorem, he showed that a fixed point on the price simplex corresponds to an equilibrium price vector. This framework became the starting point for research on informational asymmetry, incomplete markets, modern macro-dynamics, and financial economics. Owing to its mathematical sophistication, the work influenced not only economists but also applied mathematicians interested in convex analysis and topology.

Debreu (1959, 1962) developed a choice-theoretic framework in Hausdorff topological vector spaces, proving that the Walrasian excess-demand correspondence is upper hemicontinuous and convex-valued. Consequently, a Kakutani fixed point on the price simplex yields an equilibrium even with nondifferentiable or non-transitive preferences. Treating technological production sets as closed convex cones, he guaranteed the existence of strictly positive shadow prices. His “Differentiability of Utility” paper invoked the Lyapunov convexity theorem to derive convexity of randomized consumption plans, opening equilibrium analysis under probabilistic uncertainty. Debreu also presented conditions for market completeness via payoff matrices and provided the first core–Walras equivalence theorem without divisibility assumptions. These contributions inspired the Sonnenschein–Mantel–Debreu results and continue to underpin research on equilibrium stability and market dynamics.