Assume the utility function is $u(x,y,z)=y*min[x,z]$. The prices of all three goods are equal. The agent has an amount $M$ to spend on the goods. He has to choose one of the following schemes:
A) Get $1$ unit of $z$ with $1$ unit of $x$.
B) Get $1$ unit of $z$ with $1$ unit of $y$.
C) Get $1/2$ units of $x$ and $1/2$ units of $z$ with $1$ unit of $y$.
Correct Answer is C).
I know that in the optimal bundle $x^*$=$z^*$. I can intuitively tell that scheme C) is the utility maximizing scheme, but am unable to prove it algebraically.
I tried writing out the budget equations, which come out to be the same for scheme C) and B). Any tips on how to solve this will be appreciated!
(Source: Delhi School of Economics, Entrance Exam 2016.)