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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?

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    $\begingroup$ What exactly do you mean by the monotonicity property? Can you provide a definition? (Doing so may help you answer the question yourself.) $\endgroup$ – Giskard Sep 21 '19 at 16:29
  • $\begingroup$ 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 $\endgroup$ – neto333 Sep 21 '19 at 17:06
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    $\begingroup$ 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 $\endgroup$ – Brennan Sep 21 '19 at 23:34
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Yes. But you should review the definition of monotonicity (and perhaps of marginal utility) so that you understand why.

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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 $U(x,y)<U(x+a,y)$ and $U(x,y)<U(x,y+b)$. 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,

$$U(x,y)<U(x+a,y)<U(x+a,y+b)$$

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