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Unification theorem for the two basic dualities of homothetic demand theory
1Department of Economics, Massachusetts Institute of Technology, Cambridge, Mass. 02139.
Summary
This study introduces a unifying theorem relating output and cost functions in economics. It demonstrates how these dual functions, under specific conditions, can be used to measure real output and minimize costs, offering new insights into economic theory.
Area of Science:
- Economics
- Mathematical Economics
- Consumer Theory
Background:
- Economic theory often utilizes utility functions to model consumer behavior and measure economic welfare.
- Understanding the relationship between utility, output, and cost is crucial for economic analysis.
Purpose of the Study:
- To establish a unifying theorem that connects production-dual functions (output and cost) with indirect-utility dual functions.
- To demonstrate how these dualities can serve as measures of real output and minimized cost per unit.
Main Methods:
- The study employs mathematical economics, focusing on utility functions with specific properties (homogeneous-first-degree, concave).
- It analyzes the dual relationships between output, cost, and utility functions under conditions of unity income elasticities of demand.
Main Results:
- A homogeneous-first-degree, concave utility function is identified as a measure of real output when income elasticities are unity.
- The minimized-cost-per-unit-of-output function is presented as the production dual to the utility function.
- A unifying theorem is proven, linking the logarithms of these production-dual functions to their respective indirect-utility duals.
Conclusions:
- The findings provide a cohesive framework for understanding dualities in economic theory, particularly between output, cost, and utility.
- The established relationships offer a novel perspective on measuring economic output and optimizing production costs.
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