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Knowing that a utility function's primary purpose it to be used as a tool to rank consumption bundles, I was wondering:

"do we really need to be concerned with accurate identification utility functions or would simple structural assumptions do?"

The main argument against this would be that by using misspecified utility functions we would end up with inaccuracy in estimation of demand.

But in terms of using it as an tool to index consumer welfare, does it really matter?

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    $\begingroup$ Perhaps you could provide tangible examples of an "accurately identified utility function" and of a "simple structural assumption". These are very vague verbal descriptions, and I don't see how one could explore and present their pros and cons as they are. $\endgroup$ – Alecos Papadopoulos Jun 3 '18 at 19:58
  • $\begingroup$ @AlecosPapadopoulos For example if I said simply that the types of goods this particualar consumer consumes is in an arbitrary cobb douglas manner (with all output elasticities summing to 1) as opposed to running the appropriate econometric tests for existence of such a pattern. $\endgroup$ – EconJohn Jun 4 '18 at 21:06
  • $\begingroup$ To clarify, are you only interested in the importance of properly specified utility functions for use in consumer welfare calculations, or in economics more generally? Ultimately, I'd imagine the answer to your question is "it depends on what you're using the utility function for." $\endgroup$ – AndrewC Jun 5 '18 at 11:36
  • $\begingroup$ @AndrewC i'm talking about consumer welfare calculations. $\endgroup$ – EconJohn Jun 5 '18 at 16:02
  • $\begingroup$ Could you perhaps also link to an example of someone trying to accurately identify a utility function, rather than simply assume a functional form? $\endgroup$ – Giskard Jun 6 '18 at 11:02
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To the extent you are constructing marginal decisions from preferences, the general answer is no. Any utility function that preserves preference orderings maps to the same partial or total order ranking. As a consequence, $U(w)=\log(w),w>0$ and $U(w)=\sqrt{w},w>0$ preserve preference orderings with respect to $w$. Issues can exist in the multivariate case as all orderings and partial orderings must be preserved.

In the other direction, though, it does matter. If you are beginning with utility and mapping to preferences, then any transformation results in a different preference structure unless it results in the same indifference curve. As an example, $U(w)=\log(w)$ and $U(w)=\log(w)+5$ have the same indifference curve and so could be treated as the same preference structure.

So the answer is, unfortunately, dependent on the exact question you are asking. If only the ordering matters, then no, it does not matter. If the structure matters, then it can matter.

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