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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.)

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

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not meet the standards for homework questions as spelled out in the relevant meta posts. For more information, see our policy on homework question and the general FAQ." – optimal control, Giskard, ml0105, emeryville, BKay

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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
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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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