<p>The following assumes that the agent starts with no inventory.</p>

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

<p>For the third scheme, both <code>y</code> and <code>min[x,z]</code> would be <code>&gt;0</code> and hence <code>u</code> would also be <code>&gt;0</code>.</p>

CC BY-SA 3.0
2017-01-18T14:20:41.270Z
2017-01-18T14:20:41.270Z
Klas Lindbäck
<p>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:</p>

<p>A) Get $1$ unit of $z$ with $1$ unit of $x$.</p>

<p>B) Get $1$ unit of $z$ with $1$ unit of $y$.</p>

<p>C) Get $1/2$ units of $x$ and $1/2$ units of $z$ with $1$ unit of $y$.</p>

<p>Correct Answer is C).</p>

<p>I know that in the optimal bundle $x^*$=$z^*$. I can intuitively tell that scheme <em>C)</em> is the utility maximizing scheme, but am unable to prove it algebraically. </p>

<p>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!</p>

<p><em>(Source: Delhi School of Economics, Entrance Exam 2016.)</em></p>

CC BY-SA 3.0
2017-01-18T13:27:11.867Z
Show what you have tried. Unless, the users in this community does not like to give answers to this kind of questions.
optimal control
<microeconomics><consumer-theory>
Utility Maximization
Andy
# Utility Maximization

(Asked by: username_0 on 2017-01-18)

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.)*

**Tags:** microeconomics, consumer-theory
> **Comments:**
> **username_1**: Show what you have tried. Unless, the users in this community does not like to give answers to this kind of questions.  

---

## 1 answers

### Answer by username_2 on 2017-01-18. Score: 1

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`.


---

