<p>I guess by <span class="math-container">$med$</span> you mean the median value. For example: <span class="math-container">$med(2,7,19) = 7$</span>.</p>
<p>The preferences are not strictly monotonic because increasing only one of the goods, doesn’t necessarily increase the utility value. (Only increasing the maximum or, only increasing the minimum value by a bit (not enough to surpass the former median value), keeps the same utility value).</p>
<p>The sufficient condition you meant is <span class="math-container">$UMG_i &gt; 0$</span>.</p>
<p>Also, here we have <span class="math-container">$UMG_i \geq 0$</span> wherever it is defined, but there are points where <span class="math-container">$UMG_i$</span> isn’t defined as the function is not differentiable where at least two of the <span class="math-container">$x_i$</span>’s are equal.</p>

CC BY-SA 4.0
2023-03-17T23:59:11.987Z
2023-03-17T23:59:11.987Z
Nicolas Torres
<p>I have the following utility function: <span class="math-container">$$ u(x_1, x_2, x_3) = med(x_1, x_2, x_3) $$</span> Given that <span class="math-container">$UMG_{i}$</span> ≥ 0, the utility function represents a strictly monotonic preference. Does this assertion make sense?&quot;</p>

<utility><consumer-theory>
Prove strict monotonicity of utility function
carlos marena
# Prove strict monotonicity of utility function

(Asked by: username_0 on 2023-03-17)

I have the following utility function: $$ u(x\_1, x\_2, x\_3) = med(x\_1, x\_2, x\_3) $$ Given that $UMG\_{i}$ ≥ 0, the utility function represents a strictly monotonic preference. Does this assertion make sense?"

**Tags:** utility, consumer-theory


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## 1 answers

### Answer by username_1 on 2023-03-17. Score: 1 | Accepted answer

I guess by $med$ you mean the median value. For example: $med(2,7,19) = 7$.

The preferences are not strictly monotonic because increasing only one of the goods, doesn’t necessarily increase the utility value. (Only increasing the maximum or, only increasing the minimum value by a bit (not enough to surpass the former median value), keeps the same utility value).

The sufficient condition you meant is $UMG\_i > 0$.

Also, here we have $UMG\_i \geq 0$ wherever it is defined, but there are points where $UMG\_i$ isn’t defined as the function is not differentiable where at least two of the $x\_i$’s are equal.


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