# What is the concept of ordinal utility?

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.

• What is a "cardinal value?" – Michael Greinecker Oct 4 '18 at 19:44

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.
• 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? – DrStrangeLove Oct 5 '18 at 4:29
• @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. – Ubiquitous Oct 5 '18 at 6:32
• 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 $f(k,l) = kl$. – DrStrangeLove Oct 5 '18 at 16:03