# Is the leontief utility function homogeneous of degree zero? And if that is true, how can that be prove? [closed]

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

• Do you know what the definition of being homogeneous of degree 0 is? Commented Oct 18, 2022 at 2:14
• 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]. Commented Oct 18, 2022 at 6:37
• @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. Commented Oct 18, 2022 at 7:55
• 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! Commented Oct 18, 2022 at 8:05
• @ 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. Commented Nov 13, 2022 at 22:35

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 $$1$$.