Abstract
There are two distinct approaches to modeling individual behaviors. The first approach starts with preferences and proceeds to rationalizing the preferences and then showing the existence of a utility representation. The second approach is based on observations of individual choices and proceeds to reveal individual preferences with the aid of some consistency assumptions on individual preferences.
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Notes
- 1.
The collective preferences of a group of rational consumers may not have such preferences at all.
- 2.
For any two sets A and B, and any function f, we have
\( f^{ - 1} \left( {A^{c} } \right) = \left[ {f^{ - 1} \left( A \right)} \right]^{c} ,\quad f^{ - 1} (A \cup B) = f^{ - 1} (A) \cup f^{ - 1} (B). \)
- 3.
For example, if we believe that ordinary demand has the form \( x^{*} = Ap^{{a_{1} }} I^{{a_{2} }} \) with positive constants A, \( \alpha_{1} \) and \( \alpha_{2} , \) we can try to collect data on \( \left( {x^{*} ,p,I} \right) \) and use regression model \( \ln x^{*} = \alpha_{0} + \alpha_{1} \ln p + \alpha_{2} \ln I + \varepsilon \) to estimate the parameters \( \alpha_{0} , \alpha_{1} \) and \( \alpha_{2} . \)
- 4.
The general duality is the equivalence between
\( \begin{array}{*{20}c} {U = { \hbox{max} }\,u(x)} \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,s.t.\,v(x) \le 0} \\ \end{array} \,\,\,\,\,\,\,\,\,\,\,{\text{and}}\,\,\,\,\,\,\,\,\,\begin{array}{*{20}c} {V = { \hbox{min} }\,v(x)} \\ {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,s.t.\,u(x) \ge 0,} \\ \end{array} \)
for some arbitrary functions u and v. Since for our case we have a linear function \( v(x) = p \cdot x, \) we have stronger properties.
- 5.
Essentially, the only requirement is that the compensated own price effect is negative.
- 6.
To rule out trivial cases, we have to require the utility function to be at least locally nonsatiated.
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Wang, S. (2018). Consumer Theory. In: Microeconomic Theory. Springer Texts in Business and Economics. Springer, Singapore. https://doi.org/10.1007/978-981-13-0041-7_2
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DOI: https://doi.org/10.1007/978-981-13-0041-7_2
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