<p>Let <span class="math-container">$x$</span> and <span class="math-container">$y$</span> be the quantities of the two goods and <span class="math-container">$t&gt;0$</span> and <span class="math-container">$U$</span> a standard Leontief utility function.</p>
<p><span class="math-container">$U(tx,ty) = min\{tx,ty\}$</span></p>
<p>Since <span class="math-container">$t&gt;0$</span>, it can be factored out of the min</p>
<p><span class="math-container">$U(tx,ty) = t \cdot min\{x,y\} = t \cdot U(x,y) = t^1 \cdot U(x,y)$</span></p>
<p>Therefore, we conclude that the standard Leontief is actually homogeneous of degree <span class="math-container">$1$</span>.</p>

CC BY-SA 4.0
2022-11-12T23:35:55.897Z
2022-11-12T23:35:55.897Z
Nicolas Torres
<p>I have not been able to find a mathematical prove is such statement.</p>

CC BY-SA 4.0
2022-10-18T02:14:45.577Z
Do you know what the definition of being homogeneous of degree 0 is?
Herr K.
CC BY-SA 4.0
2022-10-18T06:37:30.393Z
The minimum is either one or the other. If you augment both goods by the same factor, you get the same result, multiplied by the factor. Hence you have homogeneity of degree.... [Complete the sentence].
Matteo Bulgarelli
CC BY-SA 4.0
2022-10-18T07:55:45.410Z
@MatteoBulgarelli Hi! The functions $U(x,y) = \min(x,2y)$ and $U(x,y) = \min(x,2y)^3$ have different degrees of homogeneity, Also, here is a function with a ratios that is not homogeneous of degree zero: $\frac{\ln x}{y}$. So your hints don't seem to be very useful to a novice.
Giskard
CC BY-SA 4.0
2022-10-18T08:05:42.220Z
1) I said you need ratios to have homogeneity of degree zero, not that every function that features a ratio is homogenous of degree zero. BTW, I deleted.

2) $U(x,y)  = [\min\{x, y\}]^3$ is not a standard Leontief! I think that my first comment is straightforward enough for a novice to understand the answer by herself!
Matteo Bulgarelli
CC BY-SA 4.0
2022-11-13T22:35:08.283Z
@ Aaba I think you need to formulate better your question, also  specifying what is the Leontief Utility Function.  It is not good practice to leave undefined the concepts one is referring to, even if they are well known concepts.
BakerStreet
<consumer-theory><preferences><leontief>
Is the leontief utility function homogeneous of degree zero? And if that is true, how can that be prove?
Aaba
# Is the leontief utility function homogeneous of degree zero? And if that is true, how can that be prove?

(Asked by: username_0 on 2022-10-18)

I have not been able to find a mathematical prove is such statement.

**Tags:** consumer-theory, preferences, leontief
> **Comments:**
> **username_1**: Do you know what the definition of being homogeneous of degree 0 is?  
> **username_2**: The minimum is either one or the other. If you augment both goods by the same factor, you get the same result, multiplied by the factor. Hence you have homogeneity of degree.... [Complete the sentence].  
> **username_3**: @MatteoBulgarelli Hi! The functions $U(x,y) = \min(x,2y)$ and $U(x,y) = \min(x,2y)^3$ have different degrees of homogeneity, Also, here is a function with a ratios that is not homogeneous of degree zero: $\frac{\ln x}{y}$. So your hints don't seem to be very useful to a novice.  
> **username_2**: 1) I said you need ratios to have homogeneity of degree zero, not that every function that features a ratio is homogenous of degree zero. BTW, I deleted.  2) $U(x,y)  = [\min\{x, y\}]^3$ is not a standard Leontief! I think that my first comment is straightforward enough for a novice to understand the answer by herself!  
> **username_4**: @ Aaba I think you need to formulate better your question, also  specifying what is the Leontief Utility Function.  It is not good practice to leave undefined the concepts one is referring to, even if they are well known concepts.  

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## 1 answers

### Answer by username_5 on 2022-11-12. Score: 3

Let $x$ and $y$ be the quantities of the two goods and $t>0$ and $U$ a standard Leontief utility function.

$U(tx,ty) = min\{tx,ty\}$

Since $t>0$, it can be factored out of the min

$U(tx,ty) = t \cdot min\{x,y\} = t \cdot U(x,y) = t^1 \cdot U(x,y)$

Therefore, we conclude that the standard Leontief is actually homogeneous of degree $1$.


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