A comprehensive visual “wheel of duality” in consumer theory

Duality is the heart of advanced microeconomics. It exists everywhere throughout advanced microeconomics, from the beginning of consumer theory to the end of production theory. The complex, circular relationships among various theoretical microeconomic concepts involved in the setting of duality theory have caused it to be called the “wheel of pain” by many graduate economics students. Simply put, the main aim of this paper is to provide an instructional tool in order to turn this “wheel of pain” to a “wheel of joy”. To be more specific, the primary purpose of this paper is to graphically decode the logical, complex relationships among the quartet of dual functions as well as four demand-related concepts in a visual manner. The quartet of dual functions comprises Direct Utility Function, Indirect Utility Function, Expenditure Function, and Distance Function. The four demand-related concepts are as follows: Marshallian Demand Function, Inverse Demand Function, Hicksian Demand Function, and Antonelli Inverse Demand Function. Along with the eight concepts mentioned above, the visual “big picture” presented in this paper introduces so many other crucial microeconomic concepts, and explains in what ways these concepts are related to one another, aiming at building up a strong, working mental framework of the interrelatedly connected concepts in the duality theory in advanced microeconomics. Some of these concepts are as the following: Hotelling-Wold identity, Roy’s identity, Shephard’s lemma, Slutsky equation, Antonelli equation, budget constraint, among others. In total, the visual, comprehensive wheel of duality presented in this paper logically connects fourteen interrelatedly linked microeconomic concepts, and outlines how one can make sixteen microeconomically logical transitions among the mentioned dual and demand functions, which are commonly discussed and applied in the context of duality in consumer theory. In sum, as mentioned earlier, this paper is an attempt to show how one can turn a “wheel of pain” to a “wheel of joy” through a thoughtfully designed visual aid.  

Key Words: Theory of Duality, Consumer Theory, Teaching of Economics, Pedagogy, Graduate Teaching, Advanced Microeconomics, Wheel of Duality.