# Strongly and strictly increasing utility functions

What's the difference between Strongly and strictly increasing utility functions?

What I know is that if $$x'>>x$$ where $$x'$$ has all elements strictly greater than $$x$$ then $$U(x')>U(x)$$, I think this is the definition of Strictly increasing utility function. And if $$x'>>x$$ , then $$U(x')\geq U(x)$$, this is the definition of Increasing function(monotone) function. I've no idea about Strongly Increasing function. Can anyone show a graphical example if this strongly increasing assumption is violated, how will the graph look like? (Utility function's graph)

Reference is from GEOFFREY A. JEHLE PHILIP J. RENY, Advanced Microeconomic Theory.

The term strongly increasing function is non-standard in economics (and I believe also in math) and should have been clearly defined in the main text.

The following definitions are given only in their Appendix 1, p. 529 (2011, 3rd edition):

As the following text states:

a strictly increasing function need not be strongly increasing, but every strongly increasing function is strictly increasing

• Hey, thanks a lot. Can you show and example where this strongly increasing assumption is violated ? – Henam Nov 12 '18 at 10:08
• @Henam: Any constant function. – Kenny LJ Nov 12 '18 at 10:11
• So, what does this statement $0x_0\geq x_1$ and whenever they are distinct, since in a constant function, $f(x_0)=f(x_1)$ but $x_0>x_1$. And a constant function is also increasing function. So, does that imply, Strongly increasing $\Rightarrow$ Increrasing function. – Henam Nov 12 '18 at 10:30
• $\mathbf{x}$ is supposed to be an $n$-dimensional vector. With this definition, a function which gave the sum of the elements of $\mathbf{x}$ would be strongly increasing (and strictly increasing), while a function which gave the maximum element of $\mathbf{x}$ would be strictly increasing but not strongly increasing. Consider what they do to $(1,1)$, $(1,2)$ and $(2,2)$ and how these fit the definitions – Henry Nov 12 '18 at 23:42

The difference between strongly and strictly increasing functions depends on the set on which functions are defined. In reference to the book mentioned, you are asking the difference between strongly and strictly increasing utility functions. The domains of such functions are non-negative real numbers or strictly positive real numbers. Now take an example of Cobb-Douglas utility functions and non-negative real numbers as the domain. Now compare two bundles (0,1),(0,2). You will find that Cobb-Douglas utility functions are not strongly increasing. Now consider CES utility function with non-negative real numbers. Now compare the same bundle (0,1),(0,2). You will find that CES utility functions are strongly increasing on non-negative real numbers. If you compare Cobb-Douglas and CES utility function defined on strictly positive real numbers then both are strongly increasing.