Grigorios Kostakos wrote:

To be the function \[C(x,y)=\begin{cases}

100+1.5\,qx+1.5\,qy\,, & qx>0,qy>0\\

0\,, & qx=qy=0

\end{cases}\] subadditive it should be: \begin{align*}

C\big((qx_1,qy_1)+(qx_2,qy_2)\big)\leqslant C(qx_1,qy_1)+C(qx_2,qy_2)

\end{align*} for all $(qx_1,qy_1),(qx_2,qy_2)\in X\times Y$. But to proceed, we should know first if

\[C\big((qx_1,qy_1)+(qx_2,qy_2)\big)=C(qx_1+qx_2,\,qy_1+qy_2)\] holds. Does it?

Thanks for the help!

I am not quite sure how to calculate the function of only producing one good for the equation $$C((qx_1,qy_1)+(qx_2,qy_2)\big)=C(qx_1+qx_2,\,qy_1+qy_2)$$

As the first function is about producing both goods jointly, I am not sure hoe to figure out the costs for producing the goods separately.

Is it simply $100 + qx^{0.5} + qx$ and $100 + qy^{0.5} + qy$ ?