Microeconomics Min Operator Derivative?

Question

A perfection complements model has a utility of $U\left(x,y\right)=min\left\{3x,y\right\}$.

I'm struggling to calculate the marginal utility of x and y (which is the partial derivative of x or y) when y>3x or when y<3x.

I do know that when:

y>3x

$MUx=\frac{\partial u}{\partial x}=3$

$MUy=\frac{\partial u}{\partial y}=0$

y<3x

$MUx=\frac{\partial u}{\partial x}=0$

$MUy=\frac{\partial u}{\partial y}=1$

I'm not sure how to mathematically generate these results. Any thoughts?

Thanks in advance!

Answer 1

I write down all $4$ cases:

1. For $y>3x$ we have $U=\min\{3x,y\}=3x$. That means for the two partial derivatives:

\begin{eqnarray*} & a. \qquad \qquad \frac{\partial U}{\partial x }\left.\right\vert_{y>3x}=3 \\ \\ & b.\qquad \qquad \frac{\partial U}{\partial y }\left.\right\vert_{y>3x}=0 \end{eqnarray*}

2. For $y<3x$ we have $U=\min\{3x,y\}=y$. That means for the two partial derivatives:

\begin{eqnarray*} & a. \qquad \qquad \frac{\partial U}{\partial x }\left.\right\vert_{y<3x}=0 \\ \\ & b.\qquad \qquad \frac{\partial U}{\partial y }\left.\right\vert_{y<3x}=1 \end{eqnarray*}

If you comprehend the resulting function of two case decisions then it is straightforward to get the partial derivatives.