utility-function, consumer-theoryMy question is “Does consumer theory depend on the utility function?”
That is, can you draw indifference curves or something similar without the utility function.
Consumer Theory does not need a utility function to function. Most results can be obtained if preferences are (1) complete; (2) transitive; (3) continuous; (4) locally non-satiated [no matter how small a neighbourhood is chosen around a bundle x, there is a bundle y in the neighbourhood which is preferred to x]. In particular, the above axioms will ensure that preference maximisation with prices and income given would be well defined, and the consumption bundle would lie on the budget line (not in the interior of the budget set).
Utility functions are merely representations of preferences; they are not fundamental. They are useful in order to find closed-form solutions to consumer theory problems, and that’s why they become important in practice. But to show that they are not necessary even to find closed-form solutions, consider lexicographical preferences (google it) - which satisfy axioms (1), (2) and (4). These preferences cannot be represented by a utility function, and yet solving the consumer’s problem completely is a piece of cake. Also note that the indifference curve for lexicographic preferences is a point!
One final comment: it is true that the indifference relation can be derived from preferences even if there is no utility function, but it is not fundamental. Consumer theory is usually derived starting with weak preference or, if you follow David Kreps, strict preference.
There are at least three fundamental ways of representing "how people think" about consumption, with utility functions simply being the mathematically easiest to write into models. The three approaches are "preferences," "choice," and "utility," and are often (hopefully) taught in the first section of a graduate-level core sequence in microeconomics.
These are in fact entirely separate theoretical models of how people think, but with some (arguably) minor assumptions, they can be shown to represent each other. Thus, if you find "choice theory" or "preferences" more intuitively appealing, if you accept some assumptions, they are entirely representable with a utility function. Of course, some of the interesting cases are when those concepts aren't equivalent.
A very nice overview of these three approaches can be found in the first three chapters of Ariel Rubinstein's absolutely free grad-level microeconomics textbook, posted on his website here. I highly recommend it if you ever want to delve deeply into these topics. Mas-Colell, of course, also covers these topics nicely, but tends to be more expensive than free.
Consumer Theory does depend on a consumer who can rank alternatives and make consistent choices. If you look at Debreu’s proof of existence of utility functions and various simplifications, it becomes clear how little is required of consumer preferences (in terms of axioms) to guarantee the existence of a utility function.
Now, this does not imply that everything you see in consumer and demand theory is as general as the axioms required for the existence of a utility function corresponding for every preference relation. It just means that, maybe, we could discover/construct such function for particular consumers.
For the most part, utility functions are convenient mathematical entities for working with preferences.
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