<p>Yes. But you should review the definition of monotonicity (and perhaps of marginal utility) so that you understand why.</p>

CC BY-SA 4.0
2020-07-09T08:04:00.460Z
For strict monotonicity, the marginal utility of both goods should be strictly positive everywhere. However, marginal utility of x will be 0 at y=0 (and similarly for marginal utility of y). Is the function still considered strictly monotone?
PGupta
CC BY-SA 4.0
2019-09-23T15:59:52.167Z
2019-09-23T15:59:52.167Z
heh
<p>Consider <span class="math-container">$f:\mathbb{R}\to\mathbb{R}$</span> which has a positive derivative. Then by the fundamental theorem of calc we have</p>

<p><span class="math-container">$$f(a)=\int_{0}^a\frac{df}{dx}dx-f(0)$$</span></p>

<p>Therefore, <span class="math-container">$f$</span> is monotonic (you are integrating a positive function over a larger area when you increase <span class="math-container">$x$</span>). For a function <span class="math-container">$U$</span> of two variables with positive partial derivatives we have similar results: If <span class="math-container">$a,b&gt;0$</span> then <span class="math-container">$U(x,y)&lt;U(x+a,y)$</span> and <span class="math-container">$U(x,y)&lt;U(x,y+b)$</span>. This follows because we can always fix <span class="math-container">$y_0$</span> and view <span class="math-container">$U(x,y_0)$</span> as a function of one variable call it <span class="math-container">$f_{y_0}(x)$</span>. Then <span class="math-container">$\frac{\partial U}{\partial x}(x,y_0)=\frac{df_{y_0}}{dx}(x)$</span> and apply the above results.</p>

<p>Hence we have,</p>

<p><span class="math-container">$$U(x,y)&lt;U(x+a,y)&lt;U(x+a,y+b)$$</span></p>

CC BY-SA 4.0
2019-09-23T12:21:09.947Z
2019-09-23T12:21:09.947Z
guest12382
<p>For example I have a Cobb-Douglas utility function <span class="math-container">$U(x,y)$</span> and I want to check the monotonicity property. 
Can I use the marginal utility functions to see that they are always positive to conclude that <span class="math-container">$U(x,y)$</span> has the monotonicity property?</p>

CC BY-SA 4.0
2019-09-21T16:29:04.017Z
What exactly do you mean by the monotonicity property? Can you provide a definition? (Doing so may help you answer the question yourself.)
Giskard
CC BY-SA 4.0
2019-09-21T17:06:40.173Z
Hi Giskard, Monotone is that for any $x,y\in X$, if $y\gg x$, then $y\succ x$. That means that  the agent prefers all consumption bundles that have more of all goods
neto333
CC BY-SA 4.0
2019-09-21T23:34:35.570Z
I suppose you could, but why not just look at that function and determine whether it is increasing with each argument? Or look at the marginal utility of $y$ and determine if it had a larger effect on the utility function than $x$ does such that it satisfies the definition of monotone you provided in the above comment
Brennan
<microeconomics>
Can i check monotonicity with the marginal utility?
neto333
# Can i check monotonicity with the marginal utility?

(Asked by: username_0 on 2019-09-21)

For example I have a Cobb-Douglas utility function $U(x,y)$ and I want to check the monotonicity property.
Can I use the marginal utility functions to see that they are always positive to conclude that $U(x,y)$ has the monotonicity property?

**Tags:** microeconomics

> **Comments:**
> **username_1**: What exactly do you mean by the monotonicity property? Can you provide a definition? (Doing so may help you answer the question yourself.)  
> **username_0**: Hi username_1, Monotone is that for any $x,y\in X$, if $y\gg x$, then $y\succ x$. That means that  the agent prefers all consumption bundles that have more of all goods  
> **username_2**: I suppose you could, but why not just look at that function and determine whether it is increasing with each argument? Or look at the marginal utility of $y$ and determine if it had a larger effect on the utility function than $x$ does such that it satisfies the definition of monotone you provided in the above comment  

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## 2 answers

### Answer by username_3 on 2019-09-23. Score: 1 | Accepted answer

Yes. But you should review the definition of monotonicity (and perhaps of marginal utility) so that you understand why.

> **Comments:**
> **username_4**: For strict monotonicity, the marginal utility of both goods should be strictly positive everywhere. However, marginal utility of x will be 0 at y=0 (and similarly for marginal utility of y). Is the function still considered strictly monotone?  

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### Answer by username_5 on 2019-09-23. Score: 0

Consider $f:\mathbb{R}\to\mathbb{R}$ which has a positive derivative. Then by the fundamental theorem of calc we have

$$f(a)=\int\_{0}^a\frac{df}{dx}dx-f(0)$$

Therefore, $f$ is monotonic (you are integrating a positive function over a larger area when you increase $x$). For a function $U$ of two variables with positive partial derivatives we have similar results: If $a,b>0$ then $U(x,y) and $U(x,y). This follows because we can always fix $y\_0$ and view $U(x,y\_0)$ as a function of one variable call it $f\_{y\_0}(x)$. Then $\frac{\partial U}{\partial x}(x,y\_0)=\frac{df\_{y\_0}}{dx}(x)$ and apply the above results.

Hence we have,

$$U(x,y)


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