<p>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 <strong>non-negative real numbers</strong> or <strong>strictly positive real numbers</strong>. 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.</p>

CC BY-SA 4.0
2018-11-17T08:43:26.090Z
2018-11-17T08:43:26.090Z
Deepanshu Yadav
<p>What's the difference <strong>between Strongly and strictly increasing utility functions</strong>?</p>

<p>What I know is that if <span class="math-container">$x'&gt;&gt;x $</span> where <span class="math-container">$x'$</span> has all elements strictly greater than <span class="math-container">$x$</span> then <span class="math-container">$U(x')&gt;U(x)$</span>, I think this is the definition of <em>Strictly increasing</em> utility function. And if <span class="math-container">$x'&gt;&gt;x$</span> , then <span class="math-container">$U(x')\geq U(x)$</span>, this is the definition of <em>Increasing function(monotone)</em> 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) </p>

<p>Reference is from GEOFFREY A. JEHLE PHILIP J. RENY, Advanced Microeconomic Theory. </p>

<microeconomics><utility><consumer-theory>
Strongly and strictly increasing utility functions
Henam
# Strongly and strictly increasing utility functions

(Asked by: username_0 on 2018-11-12)

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.

**Tags:** microeconomics, utility, consumer-theory

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## 1 answers

### Answer by username_1 on 2018-11-17. Score: 2

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.


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