<p>Given your requirement there is no unique way of setting the utility as far as I can see.</p>
<p>One alternative could be having the following utility:</p>
<p><span class="math-container">$$U = x^\alpha y^\beta z^\gamma$$</span></p>
<p>Where <span class="math-container">$\alpha&gt; \beta &gt; \gamma$</span>, next let <span class="math-container">$x=3$</span> if color is red, 2 if green, 1 if blue. Apply the logic to <span class="math-container">$y$</span> and <span class="math-container">$z$</span>.</p>

CC BY-SA 4.0
2022-07-18T15:13:31.710Z
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
CC BY-SA 4.0
2022-07-18T15:27:34.213Z
@Giskard that would maybe work even better but I think the above does job as well, I was just giving an simple example
1muflon1
CC BY-SA 4.0
2022-07-18T16:49:56.907Z
@1muflon1 If the input space isn't limited, what should be the utility function?
Pedro Gómez
CC BY-SA 4.0
2022-07-18T22:00:23.093Z
@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
CC BY-SA 4.0
2022-07-19T06:24:44.800Z
@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
CC BY-SA 4.0
2022-07-18T11:48:30.783Z
2022-07-18T11:48:30.783Z
1muflon1
<p>How can we find a <strong>utility function</strong> that represents buying a car that receive as an input <span class="math-container">$x,y, z$</span> when:</p>
<ul>
<li><span class="math-container">$x$</span> - The color of the car - The most important thing when red is better than blue that better which is better than green.</li>
<li><span class="math-container">$y$</span> - Engine - The second most important thing - An electric vehicle is better than gasoline which is better than diesel.</li>
<li><span class="math-container">$z$</span> - Car manufacturer - The third most important thing - When German manufacturer is better than French which is better than British.</li>
</ul>

<utility>
Find a utility function for a specific problem
Pedro Gómez
# Find a utility function for a specific problem

(Asked by: username_0 on 2022-07-18)

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.

**Tags:** utility

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## 1 answers

### Answer by username_1 on 2022-07-18. Score: 1 | Accepted answer

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 $z$.

> **Comments:**
> **username_2**: 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.)  
> **username_1**: @Giskard that would maybe work even better but I think the above does job as well, I was just giving an simple example  
> **username_0**: @1muflon1 If the input space isn't limited, what should be the utility function?  
> **username_1**: @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  
> **username_2**: @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".  

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