I have read in many books that since utility cannot be measured - so ordinal concept or comparison concept is used. If that is so, how can one define a mathematical function for utility which gives a cardinal value for inputs. The two things mentioned above are inconsistent with each other.

Can somebody explain what I am missing here.

  • $\begingroup$ What is a "cardinal value?" $\endgroup$ Oct 4 '18 at 19:44

Suppose Alex, Bryan, and Chris run in a race. Alex is the fastest and Chris is the slowest. So far I have only given you ordinal information about where they finished.

I guess you'd be okay with me taking that ordinal information and saying "Alex finished first, Bryan finished second, and Chris finished third". But then I just defined a function that assigns a number to each of the ordinal places: $$f(x)=\begin{cases}1\text{ if }x=\text{Alex}\\2\text{ if }x=\text{Bryan}\\3\text{ if }x=\text{Chris}.\end{cases}$$ Note that the numbers 1, 2, 3 don't tell us how much faster Alex was than Bryan, only that he was somewhat faster.

Utility functions are just like this: they give bigger numbers to the best alternatives. But the size of those numbers don't tell us how much better the best alternative is, only that it is better. That is the sense in which the function is ordinal.

  • $\begingroup$ Suppose I have a utility function $U(x,y)=xy$ Bundle (x=5,y=2) gives twice as much utility as the Bundle(x=5,y=1). Can't I say that the first bundle is twice better than the second bundle? $\endgroup$ Oct 5 '18 at 4:29
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    $\begingroup$ @fuckir No, you cannot! When people say that utility is ordinal, exactly what they mean is that you can't make comparisons like the one you describe. All you can say is that $(5,2)$ is better than $(5,1)$. One of the remarkable things about consumer choice theory is that, even with only this much information, we have all we need to go ahead and solve the consumer's choice problem. $\endgroup$
    – Ubiquitous
    Oct 5 '18 at 6:32
  • $\begingroup$ Thank you, Sir. I wonder why don't they use such an easy real life language to explain such concepts in conventional microeconomics books. Thanks and respects from India. $\endgroup$ Oct 5 '18 at 15:56
  • $\begingroup$ Along with this one more question, the concepts of consumer theory are the same as of producer theory with different variables of course. But in producer theory the output is measurable in cardinal numbers. Can I say in that case $(Capital=5, Labour=2)$ produces twice as much output as $(Capital=5, Labour=1)$ in the production function $f(k,l) = kl$. $\endgroup$ Oct 5 '18 at 16:03

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