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.)


closed as off-topic by optimal control, Giskard, ml0105, emeryville, BKay Jan 23 '17 at 15:00

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    $\begingroup$ Show what you have tried. Unless, the users in this community does not like to give answers to this kind of questions. $\endgroup$ – optimal control Jan 18 '17 at 13:27

The following assumes that the agent starts with no inventory.

If you can only choose a single scheme, then y would be 0 for the first scheme and min[x,z] would be 0 for the second scheme. And hence u would also be 0 for either of the first two schemes.

For the third scheme, both y and min[x,z] would be >0 and hence u would also be >0.


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