Mathematical economics uses mathematical optimization, game theory and analysis to study allocation, choice and market coordination.
Utility represents how a decision maker ranks outcomes. A utility function assigns numbers to alternatives consistently with those preference relations; ordinal comparisons depend on the ordering rather than the particular numerical scale.
General equilibrium theory studies simultaneous market clearing through a common price vector, allowing demand in one market to depend on prices and resources in other markets.
At any price vector, the total value of aggregate excess demand is zero when consumers exhaust their budgets. After normalization to a price simplex, choosing a price that maximizes the value of a candidate excess-demand vector is a useful Kakutani fixed-point theorem construction.
An exchange economy specifies consumers, their preference relations, their consumption sets and their initial endowments. A consumption allocation is feasible when aggregate consumption equals aggregate endowment; free-disposal models instead allow a coordinatewise inequality. These conventions must be distinguished when specifying supporting prices.
An exchange economy is regular when the derivative of its reduced aggregate excess demand has full rank at every equilibrium. For two goods and normalized prices , this means at every zero of first-good excess demand. Regular equilibria are locally isolated and persist smoothly under small parameter changes.
If zero is a regular value of the joint smooth aggregate excess demand , almost every parameter gives a regular economy. The regular level set theorem makes the zero set a manifold. Its projection to parameter space fails to have full rank exactly where the derivative with respect to prices is singular; Sard theorem makes these exceptional parameter values a null set.
Aggregate excess demand subtracts total endowment from the sum of consumers' demand correspondences. A market-clearing price satisfies . Under Walras law, one equation is redundant after price normalization.
A price quasi-equilibrium with transfers is an exactly feasible consumption allocation and a nonzero price vector for which, almost everywhere, every strictly preferred bundle has cost at least . The transfer-adjusted wealth is ; transfers sum to zero. Equal-cost strict improvements may remain, which distinguishes a quasi-equilibrium from a utility-maximizing equilibrium. Nonnegative prices need extra assumptions if goods may be undesirable.
For an atomless measure of consumers with measurable, complete, transitive and continuous preference relations, every exactly feasible Pareto efficient consumption allocation admits nonzero quasi-equilibrium supporting prices. Countably many rational strict-improvement directions remove all exceptional null sets simultaneously. If their convex hull contained zero, sufficiently small disjoint cohorts could make those improvements in exactly balancing proportions, violating Pareto efficiency. A supporting hyperplane theorem then supplies the prices. Individual convexity of preferences is unnecessary.
A demand correspondence assigns a nonempty set of chosen consumption bundles to each price and wealth pair. Every chosen bundle satisfies the budget constraint. A utility function rationalizes the demand correspondence when the chosen bundles are exactly its maximizers over the budget set.
Revealed preference infers rankings from choices: a chosen bundle is directly revealed at least as desirable as any bundle affordable in the same budget. Indirect comparisons use chains of such observations.
The weak indirect revealed preference is the reflexive closure of the transitive closure of a relation of direct revealed comparisons. Equivalently, it consists of finite chains, including chains of length zero. The diagonal convention is essential if the relation is to be an intersection of complete weak preference relations on all alternatives.
The congruence axiom says that an affordable alternative indirectly revealed at least as desirable as a chosen alternative must also be chosen. For a chosen bundle and an affordable unchosen bundle , the axiom rules out ; because , every strict-part-preserving completion ranks strictly above .
In an economy with a measure space of consumers, a consumption allocation is a measurable, integrable function assigning each consumer a feasible bundle. Exact aggregate feasibility means .
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Mathematical economics is a field that applies mathematical methods and techniques to represent economic theories, analyze economic problems, and derive economic relationships. It utilizes mathematical concepts such as calculus, linear algebra, and optimization to model economic behaviors and interactions quantitatively. The primary objectives of mathematical economics include: 1. **Modeling Economic Behavior**: Creating models that describe how individuals, firms, and markets behave under various conditions. This includes utility functions, production functions, and demand and supply models.