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Note: These explanations are in the process of being adapted from my textbook.
I'm trying to make them each a "standalone" treatment of a concept, but there may still
be references to the narrative flow of the book that I have yet to remove.
This work is under development and has not yet been professionally edited.
If you catch a typo or error, or just have a suggestion, please submit a note here. Thanks!

Defining “Optimal Choice”

The consumer’s choice problem is an example of constrained optimization. In its most general sense, a constrained optimization problem has three elements:

With this general framework, we can say that $X^\star$ is an optimal choice if:

In the context of consumer choice, this has a good intuitive interpretation: a $X^\star$ is optimal if:

A bit more informally, we can say that if $X^\star$ is an optimal choice, then any other choice which is better isn’t affordable, and any other choice which is affordable isn’t better.

To get a feel for this, try playing with the following four diagrams. Each diagram shows a budget set, and as you drag the bundle $X$ around the choice space, it shows the set of bundles preferred to $X$. As you drag $X$ around, the graph will indicate if the point is affordable, and whether there is any overlap between the purple “preferred” region and the green budget set. Try to find the four optimal points for these situations by dragging the bundles until all the green check marks light up!

Some things to notice:

There are a wide variety of preferences and utility functions (and constraints) that may be analyzed using this framework.

Next: The “Gravitational Pull” Toward Optimality
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