# A comprehensive visual "Wheel of Duality" in consumer theory.

JEL Classification A22 * A23 * D10 * D11

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 led it to be called the "wheel of pain" by many graduate economics students. Put simply, the aim of the present paper is to turn this "wheel of pain" into a "wheel of joy." This paper graphically decodes the logical, complex relationships among a quartet of dual functions which present preferences as well as a quartet of demand-related functions in a visual manner. The quartet of dual functions comprises the Direct Utility Function (DUF), Indirect Utility Function (IUF), Expenditure Function (EF), and Distance Function (DF). The quartet of demand-related functions consists of the Marshallian Demand Function (MDF), Hotelling-style Inverse Demand Function (HIDF), Hicksian Demand Function (HDF), and Antonelli-style Inverse Demand Function (AIDF).

The visualization of duality introduced herein is more comprehensive and innovative than those put forth before. It decodes at least ten more connections visuallly, and discovers several more symmetric loops. The basic idea behind this paper lies in the fact that providing one-by-one relationships may not present the whole picture of the wheel of duality at once. However, providing a comprehensive, visual "wheel of duality" can clear up any possible confusion.

Along with the eight concepts mentioned above, the visual "big picture" introduces many other essential microeconomic concepts, and explains how these concepts are related to one another. Some of these concepts are the following: Hotelling-Wold identity, Roy's identity, Shephard's lemma, Antonelli equation, Slutsky equation, budget constraint, expenditure amount function, among others. In total, the visual, comprehensive wheel of duality presented herein logically connects 14 interconnected microeconomic concepts, and outlines how one can make 16 microeconomically logical transitions among the aforementioned dual and demand functions, which are commonly discussed and applied in the context of duality in consumer theory. For example, as shown in the figure, Roy's identity enables us to make a transition from IUF to MDF. More specifically, this transition shows us how to generate a full system of MDFs out of an IUF. As another example, the transition in front of Roy's identity demonstrates how one can substitute a full system of MDFs into DUF in order to end up with IUF.

According to Comes (2008), "dual arguments have, in recent years, become standard tools for analysis of problems involving optimization by consumers and producers" (Comes, Richard, Cambridge University Press, 2008, p. 1). He then points out that in recent decades the dual techniques have been widely used among economists. In his opinion, familiarity with basic duality theory is now becoming a necessity for graduate economics students, whether they are interested in economic theory or in empirical applications of economic theory. Comes (2008) also adds that "the paucity of simple introductory exposition of these techniques is both surprising and disappointing" (Comes, Richard, Cambridge University Press, 2008, p. xi). This note attempts to fill the aforementioned gap, helping graduate economics students easily understand and readily remember theoretical concepts in advanced microeconomics.

According to Zeytoon Nejad (Journal of Economics and Political Economy, 2016), the plurality and variety of concepts and models involved in economics can be a source of confusion for many economics students. Such a confusion could be effectively cleared up by furnishing students with a visual "big picture" that illustrates the ways through which those numerous, diverse concepts are connected. The visual "big picture" that is introduced in this paper can be used as a resource in advanced microeconomics courses. The focus is on the ways through which the main elements of duality theory in advanced microeconomics are connected to each other. This holistic visual wheel of duality graphically demonstrates these connections, and also shows how to make transitions among these connections using the essential mathematical formulas provided in the figure.

DOI 10.1007/s11294-016-9586-8

Seyyed Ali Zeytoon Nejad Moosavian [1]

Published online: 6 June 2016

[mail] Seyyed Ali Zeytoon Nejad Moosavian

szeytoo@ncsu.edu

[1] North Carolina State University, Raleigh, North Carolina, USA