Identifying utility function

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]

• Hi welcome to SE, what exactly is your question here? Are you just looking for special cases of ‘weird’ looking utility functional forms? – Brennan Oct 5 '19 at 18:11
• Sharing the "confusing" function may be helpful. – Giskard Oct 5 '19 at 19:02
• 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 Oct 5 '19 at 19:03
• @Brennan Yes exactly and any tools with which I can find out the accurate kind of utility function. – user708015 Oct 5 '19 at 21:03
• @Giskard I have edited the question now. – user708015 Oct 5 '19 at 21:05

1 Answer

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: $$u(x_1,x_2) = x_1^ax_2^b \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

Then you have a perfect-substitute preferences which can be represented by utility function of the form: $$u(x_1,x_2) = ax_1 + bx_2 \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

You can also have a perfect-complement preferences which can be represented by utility function of the form: $$u(x_1,x_2) = \min\{ax_1,bx_2\} \:\:\: \textrm{for} \:\:\:a>0 \:\: \textrm{and} \:\: b>0$$

Finally, you can have a quasi-linear preferences which can be represented by utility function of the form: $$u(x_1,x_2) = f(x_1) +x_2 \:\:\: \textrm{where f is increasing and concave}$$

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):

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 $$a=b=1$$.

• Just added a small note about monotonic transformations which should be the last "case" of confusing examples. – bajun65537 Oct 9 '19 at 18:50