<p>Not sure about utility function being <span class="math-container">$\min$</span> and it being a case of perfect substitutes... </p>

<p>In general, I would say there are a few different types of utility functions but if you know more or less the theory and can imagine various functions' graphs, then (as mentioned in the comment) you will be fine.</p>

<p>So I think we should start with a <strong>Cobb-Douglas preferences</strong> which can be represented by utility function of the form: <span class="math-container">$$u(x_1,x_2) = x_1^ax_2^b \:\:\: \textrm{for} \:\:\:a&gt;0 \:\: \textrm{and} \:\: b&gt;0$$</span></p>

<p><a href="https://i.sstatic.net/JWzGa.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/JWzGa.gif" alt="enter image description here"></a></p>

<p>Then you have a <strong>perfect-substitute preferences</strong> which can be represented by utility function of the form: <span class="math-container">$$u(x_1,x_2) = ax_1 + bx_2 \:\:\: \textrm{for} \:\:\:a&gt;0 \:\: \textrm{and} \:\: b&gt;0$$</span></p>

<p><a href="https://i.sstatic.net/C4bv3.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/C4bv3.gif" alt="enter image description here"></a></p>

<p>You can also have a <strong>perfect-complement preferences</strong> which can be represented by utility function of the form: <span class="math-container">$$u(x_1,x_2) = \min\{ax_1,bx_2\} \:\:\: \textrm{for} \:\:\:a&gt;0 \:\: \textrm{and} \:\: b&gt;0$$</span></p>

<p><a href="https://i.sstatic.net/qMJAg.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qMJAg.gif" alt="enter image description here"></a></p>

<p>Finally, you can have a <strong>quasi-linear preferences</strong> which can be represented by utility function of the form: <span class="math-container">$$u(x_1,x_2) = f(x_1) +x_2 \:\:\: \textrm{where f is increasing and concave}$$</span></p>

<p><a href="https://i.sstatic.net/NcUWW.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/NcUWW.gif" alt="enter image description here"></a></p>

<p>In all of the above graphs I assumed <span class="math-container">$a=b=1$</span> for simplicity. In the last case the concave function was a square root.
Your example is (more or less):</p>

<p><a href="https://i.sstatic.net/n8q9f.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/n8q9f.gif" alt="enter image description here"></a></p>

<p>You can now try and argue what kind of preferences these are.</p>

<p>If you are interested in consumer theory, you should find some proper course book or other materials - there's plenty of it on the Internet - just google it.</p>

<p>The last piece of advice I can give you - try to draw as many graphs as you can think of and then try to find a story that the graph is explaining. Or the other way round. I remember my professors trying to "play" with us by giving us different kinds of utility functions - when you drew them, they looked "strange" but when you actually thought about them - they did have some story to tell. </p>

<p><strong>EDIT.</strong>
There is also a special case in a way - <strong>monotonic transformation</strong> is a real-valued function <span class="math-container">$f: \mathbb{R} \to \mathbb{R}$</span> which basically transforms numbers into numbers in a way that preserves the order of the numbers, that is, for every two numbers <span class="math-container">$p$</span> and <span class="math-container">$k$</span> it hods that <span class="math-container">$p&gt;k \implies f(p)&gt;f(k) $</span>. This means that if a function <span class="math-container">$f$</span> is a monotonic transformation of utility <span class="math-container">$u( \cdot )$</span>, then <span class="math-container">$f(u(\cdot))$</span> also represents the consumer’s preferences. <span class="math-container">$$(x_1,x_2) \succeq (y_1,y_2) \iff f(u(x_1,x_2)) &gt; f (u(y_1,y_2))$$</span></p>

<p>All this means is that you need to be careful for a case of utility function like: <span class="math-container">$\ln{x_1} + \ln{x_2}$</span> which is just a natural logarithm of Cobb-Douglas preferences <span class="math-container">$x_1x_2$</span> with <span class="math-container">$a=b=1$</span>.</p>

CC BY-SA 4.0
2019-10-09T18:50:30.197Z
Just added a small note about monotonic transformations which should be the last "case" of confusing examples.
bajun65537
CC BY-SA 4.0
2019-10-08T17:57:36.210Z
2019-10-09T18:48:33.787Z
2019-10-09T18:48:33.787Z
bajun65537
<p>I recently came across a utility function with min written at the start. I assumed that it was a case of a leontief utility function, and only after going ahead with the problem I found out that it is a case of perfect substitutes. Until then, I didn't even know that perfect substitutes could take this form. How do I identify various utility functions accurately? Are there other such confusing cases which I need to be aware of? </p>

<p>The example I have mentioned above looks something like this: 
      U = Min [2x + y, x + 2y]</p>

CC BY-SA 4.0
2019-10-05T18:11:22.193Z
Hi welcome to SE, what exactly is your question here? Are you just looking for special cases of ‘weird’ looking utility functional forms?
Brennan
CC BY-SA 4.0
2019-10-05T19:02:02.293Z
Sharing the "confusing" function may be helpful.
Giskard
CC BY-SA 4.0
2019-10-05T19:03:30.063Z
The final answer will probably be "you need to be aware of what your teacher thinks you need to aware of". Being able to map functions to indifference maps in general shows that you understand the material.
Giskard
CC BY-SA 4.0
2019-10-05T21:03:37.580Z
@Brennan Yes exactly and any tools with which I can find out the accurate kind of utility function.
user708015
CC BY-SA 4.0
2019-10-05T21:05:06.057Z
@Giskard I have edited the question now.
user708015
CC BY-SA 4.0
2019-10-05T23:49:40.070Z
I mean, using the standard tools for deriving indifference curves and understanding functions should serve you well enough. Here its clearly that its a combination of two “perfect substitutes” cases with a slight twist. All you would really be doing here would just solving both of the arguments in $min$ separately and compare. What do you mean “identify” them? Like come up with a label to describe them like perfect complements or something?
Brennan
CC BY-SA 4.0
2019-10-15T03:13:36.097Z
@Brennan when you draw the indifference curve for a certain utility level, why are there two segments?
user708015
CC BY-SA 4.0
2019-10-15T04:48:14.633Z
In your example there is two segments because when the first or second argument in the min function is smaller than the other it takes that slope and functional form. But above a certain threshold for say Y, the other would take form.
Brennan
<microeconomics><consumer-theory>
Identifying utility function
user708015
# Identifying utility function

(Asked by: username_0 on 2019-10-05)

I recently came across a utility function with min written at the start. I assumed that it was a case of a leontief utility function, and only after going ahead with the problem I found out that it is a case of perfect substitutes. Until then, I didn't even know that perfect substitutes could take this form. How do I identify various utility functions accurately? Are there other such confusing cases which I need to be aware of?

The example I have mentioned above looks something like this:
U = Min [2x + y, x + 2y]

**Tags:** microeconomics, consumer-theory
> **Comments:**
> **username_1**: Hi welcome to SE, what exactly is your question here? Are you just looking for special cases of ‘weird’ looking utility functional forms?  
> **username_2**: Sharing the "confusing" function may be helpful.  
> **username_2**: The final answer will probably be "you need to be aware of what your teacher thinks you need to aware of". Being able to map functions to indifference maps in general shows that you understand the material.  
> **username_0**: @Brennan Yes exactly and any tools with which I can find out the accurate kind of utility function.  
> **username_0**: @Giskard I have edited the question now.  
> **username_1**: I mean, using the standard tools for deriving indifference curves and understanding functions should serve you well enough. Here its clearly that its a combination of two “perfect substitutes” cases with a slight twist. All you would really be doing here would just solving both of the arguments in $min$ separately and compare. What do you mean “identify” them? Like come up with a label to describe them like perfect complements or something?  
> **username_0**: @Brennan when you draw the indifference curve for a certain utility level, why are there two segments?  
> **username_1**: In your example there is two segments because when the first or second argument in the min function is smaller than the other it takes that slope and functional form. But above a certain threshold for say Y, the other would take form.  

---

## 1 answers

### Answer by username_3 on 2019-10-08. Score: 2

Not sure about utility function being $\min$ and it being a case of perfect substitutes...

In general, I would say there are a few different types of utility functions but if you know more or less the theory and can imagine various functions' graphs, then (as mentioned in the comment) you will be fine.

So I think we should start with a **Cobb-Douglas preferences** which can be represented by utility function of the form: $$u(x\_1,x\_2) = x\_1^ax\_2^b \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

[![enter image description here](https://i.sstatic.net/JWzGa.gif)](https://i.sstatic.net/JWzGa.gif)

Then you have a **perfect-substitute preferences** which can be represented by utility function of the form: $$u(x\_1,x\_2) = ax\_1 + bx\_2 \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

[![enter image description here](https://i.sstatic.net/C4bv3.gif)](https://i.sstatic.net/C4bv3.gif)

You can also have a **perfect-complement preferences** which can be represented by utility function of the form: $$u(x\_1,x\_2) = \min\{ax\_1,bx\_2\} \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

[![enter image description here](https://i.sstatic.net/qMJAg.gif)](https://i.sstatic.net/qMJAg.gif)

Finally, you can have a **quasi-linear preferences** which can be represented by utility function of the form: $$u(x\_1,x\_2) = f(x\_1) +x\_2 \:\:\: \textrm{where f is increasing and concave}$$

[![enter image description here](https://i.sstatic.net/NcUWW.gif)](https://i.sstatic.net/NcUWW.gif)

In all of the above graphs I assumed $a=b=1$ for simplicity. In the last case the concave function was a square root.
Your example is (more or less):

[![enter image description here](https://i.sstatic.net/n8q9f.gif)](https://i.sstatic.net/n8q9f.gif)

You can now try and argue what kind of preferences these are.

If you are interested in consumer theory, you should find some proper course book or other materials - there's plenty of it on the Internet - just google it.

The last piece of advice I can give you - try to draw as many graphs as you can think of and then try to find a story that the graph is explaining. Or the other way round. I remember my professors trying to "play" with us by giving us different kinds of utility functions - when you drew them, they looked "strange" but when you actually thought about them - they did have some story to tell.

**EDIT.**
There is also a special case in a way - **monotonic transformation** is a real-valued function $f: \mathbb{R} \to \mathbb{R}$ which basically transforms numbers into numbers in a way that preserves the order of the numbers, that is, for every two numbers $p$ and $k$ it hods that $p>k \implies f(p)>f(k) $. This means that if a function $f$ is a monotonic transformation of utility $u( \cdot )$, then $f(u(\cdot))$ also represents the consumer’s preferences. $$(x\_1,x\_2) \succeq (y\_1,y\_2) \iff f(u(x\_1,x\_2)) > f (u(y\_1,y\_2))$$

All this means is that you need to be careful for a case of utility function like: $\ln{x\_1} + \ln{x\_2}$ which is just a natural logarithm of Cobb-Douglas preferences $x\_1x\_2$ with $a=b=1$.

> **Comments:**
> **username_3**: Just added a small note about monotonic transformations which should be the last "case" of confusing examples.  

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