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How can we find a utility function that represents buying a car that receive as an input $x,y, z$ when:

  • $x$ - The color of the car - The most important thing when red is better than blue that better which is better than green.
  • $y$ - Engine - The second most important thing - An electric vehicle is better than gasoline which is better than diesel.
  • $z$ - Car manufacturer - The third most important thing - When German manufacturer is better than French which is better than British.
Pedro Gómez
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Given your requirement there is no unique way of setting the utility as far as I can see.

One alternative could be having the following utility:

$$U = x^\alpha y^\beta z^\gamma$$

Where $\alpha> \beta > \gamma$, next let $x=3$ if color is red, 2 if green, 1 if blue. Apply the logic to $y$ and $z$.

1muflon1
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  • The OP seems to describe a lexicographic utility function: "The most important thing [...] The second most important thing [...]" This is not that. (Though given the limited input space the parameters could be set in such a way that it does become lexicographic.) – Giskard Jul 18 '22 at 15:13
  • @Giskard that would maybe work even better but I think the above does job as well, I was just giving an simple example – 1muflon1 Jul 18 '22 at 15:27
  • @1muflon1 If the input space isn't limited, what should be the utility function? – Pedro Gómez Jul 18 '22 at 16:49
  • @PedroGómez you could use the same function with i =1,2,3 ... n to order the unlimited set of possible inputs or you can follow the giskards suggestion – 1muflon1 Jul 18 '22 at 22:00
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    @1muflon1 My suggestion is not constructive; I wrote that you did not answer the question, which seems to have been about lexicographic preferences, not Cobb-Douglas. It was indeed a mistake to write "lexicographic utility function" instead of "lexicographic preferences". – Giskard Jul 19 '22 at 06:24