<p>The utility function looks like this:</p>
<p><span class="math-container">$(big)^2 - (small)^2$</span></p>
<p>Since <span class="math-container">$small$</span> is something that takes away utility, you want <span class="math-container">$small = 0$</span>. Otherwise, you’re spending some money in getting unhappier, definitely not an optimal bundle.</p>
<p>This implies you wouldn’t consume anything at all from one good.</p>
<p>Therefore, you either spend everything on <span class="math-container">$x$</span> or everything on <span class="math-container">$y$</span>.</p>
<ul>
<li>If you spend everything on <span class="math-container">$x$</span>:</li>
</ul>
<p><span class="math-container">$U(x,y) = (\max\{x,y\})^2 - (\min\{x,y\})^2 = x^2 - 0^2 = x^2 = (\frac{I}{p_x})^2$</span></p>
<ul>
<li>If you spend everything on <span class="math-container">$y$</span>:</li>
</ul>
<p><span class="math-container">$U(x,y) = (\max\{x,y\})^2 - (\min\{x,y\})^2 = y^2 - 0^2 = y^2 = (\frac{I}{p_y})^2$</span></p>
<p>Now you have to compare the prices of both goods.</p>
<ul>
<li><p><span class="math-container">$p_x &lt; p_y \implies (\frac{I}{p_x})^2 &gt; (\frac{I}{p_y})^2 \implies$</span> spend everything on <span class="math-container">$x \implies x^{m} = \frac{I}{p_x}, y^{m} = 0$</span></p>
</li>
<li><p><span class="math-container">$p_x &gt; p_y \implies (\frac{I}{p_y})^2 &gt; (\frac{I}{p_x})^2 \implies$</span> spend everything on <span class="math-container">$y \implies x^{m} = 0, y^{m} = \frac{I}{p_y}$</span></p>
</li>
<li><p><span class="math-container">$p:= p_x = p_y \implies (\frac{I}{p_x})^2 = (\frac{I}{p_y})^2 \implies$</span> indifferent between spending everything on x and spending everything on y <span class="math-container">$\implies (x^{m},y^{m}) = (0,\frac{I}{p})$</span> or <span class="math-container">$(\frac{I}{p},0)$</span>.</p>
</li>
</ul>

CC BY-SA 4.0
2022-11-20T12:51:47.870Z
That is a great explanation and thank you very much. And can I request any website or book to work on those? I appreciate any kind of source and have a nice day.
Tatanik501
CC BY-SA 4.0
2022-11-20T14:57:32.647Z
@Tatanik501 well, the book that is in my Intermediate Microeconomics course’s syllabus is Nicholson Microeconomic Theory 11th edition but I haven’t seen the fancier utility functions (other than the ones that admit the Lagrangian method and the linear/standard Leontief). For these fancier ones, I think for a little bit and use some logic.
Nicolas Torres
CC BY-SA 4.0
2022-11-20T17:04:59.963Z
Thanks for the recommendation.
Tatanik501
CC BY-SA 4.0
2022-11-19T18:31:11.807Z
2022-11-19T18:31:11.807Z
Nicolas Torres
<p><strong>For only max or min problems, I understand we should proceed they are complements but for that
type of function, how do we really get demand functions? should we graph but can this be done without a computer?</strong></p>
<p><span class="math-container">$$ 
u\left( x,y\right) = \left( \max \left\{ x,y\right\} \right) ^{2}-\left( \min \left\{ x,y\right\} \right) ^{2}
$$</span></p>

CC BY-SA 4.0
2022-11-19T13:04:51.247Z
1. These are not really Leontief-preferences, the utility just contains the $\min$ operator.
Giskard
CC BY-SA 4.0
2022-11-19T13:05:28.107Z
2. Have you considered solving the utility maximization problem by looking at the cases $x \geq y$ and $x < y$ separately?
Giskard
CC BY-SA 4.0
2022-11-19T16:58:38.803Z
to be honest I am not good at some parts of micro and I can't understand the nature of some assumptions. so basically I really don't know how to continue. If you don't mind, could you explain what do you mean by solving separately by imposing these restrictions? Can I even form a Lagrange here?
Tatanik501
<microeconomics><optimization><leontief><lagrangian>
how to derive marshallian demand functions from leontief preferences?
Tatanik501
# how to derive marshallian demand functions from leontief preferences?

(Asked by: username_0 on 2022-11-19)

**For only max or min problems, I understand we should proceed they are complements but for that
type of function, how do we really get demand functions? should we graph but can this be done without a computer?**

$$
u\left( x,y\right) = \left( \max \left\{ x,y\right\} \right) ^{2}-\left( \min \left\{ x,y\right\} \right) ^{2}
$$

**Tags:** microeconomics, optimization, leontief, lagrangian

> **Comments:**
> **username_1**: 1. These are not really Leontief-preferences, the utility just contains the $\min$ operator.  
> **username_1**: 2. Have you considered solving the utility maximization problem by looking at the cases $x \geq y$ and $x < y$ separately?  
> **username_0**: to be honest I am not good at some parts of micro and I can't understand the nature of some assumptions. so basically I really don't know how to continue. If you don't mind, could you explain what do you mean by solving separately by imposing these restrictions? Can I even form a Lagrange here?  

---

## 1 answers

### Answer by username_2 on 2022-11-19. Score: 2 | Accepted answer

The utility function looks like this:

$(big)^2 - (small)^2$

Since $small$ is something that takes away utility, you want $small = 0$. Otherwise, you’re spending some money in getting unhappier, definitely not an optimal bundle.

This implies you wouldn’t consume anything at all from one good.

Therefore, you either spend everything on $x$ or everything on $y$.

* If you spend everything on $x$:

$U(x,y) = (\max\{x,y\})^2 - (\min\{x,y\})^2 = x^2 - 0^2 = x^2 = (\frac{I}{p\_x})^2$

* If you spend everything on $y$:

$U(x,y) = (\max\{x,y\})^2 - (\min\{x,y\})^2 = y^2 - 0^2 = y^2 = (\frac{I}{p\_y})^2$

Now you have to compare the prices of both goods.

* $p\_x < p\_y \implies (\frac{I}{p\_x})^2 > (\frac{I}{p\_y})^2 \implies$ spend everything on $x \implies x^{m} = \frac{I}{p\_x}, y^{m} = 0$
* $p\_x > p\_y \implies (\frac{I}{p\_y})^2 > (\frac{I}{p\_x})^2 \implies$ spend everything on $y \implies x^{m} = 0, y^{m} = \frac{I}{p\_y}$
* $p:= p\_x = p\_y \implies (\frac{I}{p\_x})^2 = (\frac{I}{p\_y})^2 \implies$ indifferent between spending everything on x and spending everything on y $\implies (x^{m},y^{m}) = (0,\frac{I}{p})$ or $(\frac{I}{p},0)$.

> **Comments:**
> **username_0**: That is a great explanation and thank you very much. And can I request any website or book to work on those? I appreciate any kind of source and have a nice day.  
> **username_2**: @username_0 well, the book that is in my Intermediate Microeconomics course’s syllabus is Nicholson Microeconomic Theory 11th edition but I haven’t seen the fancier utility functions (other than the ones that admit the Lagrangian method and the linear/standard Leontief). For these fancier ones, I think for a little bit and use some logic.  
> **username_0**: Thanks for the recommendation.  

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