<p>The Marshallian demand correspondence is given by:
<span class="math-container">$$
x(p_x, p_y,m) = \begin{cases} \{m/p_x\} &amp;\text{ if } p_x &lt; p_y\\ \{0, m/p_x\} &amp;\text{ if } p_x = p_y\\ \{0\} &amp;\text{ if } p_x &gt; p_y \end{cases}.
$$</span></p>
<p>The indirect utility function is given by:
<span class="math-container">$$
v(p_x, p_y, m) = \left(\frac{m}{\min\{p_x, p_y\}}\right)^2
$$</span></p>
<p>Now consider the cost minimization problem:
<span class="math-container">$$
e(p_x, p_y, \bar u) = \min p_x x + p_y y \text{ s.t. } x^2 + y^2 \ge \bar u.
$$</span>
If <span class="math-container">$p_x &lt; p_y$</span> it will be best to set <span class="math-container">$x = \sqrt{\bar u}$</span> and if <span class="math-container">$p_y &lt; p_x$</span> it will be best to set <span class="math-container">$y = \sqrt{\bar u}$</span>. So we get that the Hicksian demand correspondence is given by:
<span class="math-container">$$
h_x(p_x, p_y, \bar u) = \begin{cases}\{\sqrt{\bar u}\} &amp; \text{ if } p_x &lt; p_y,\\ \{0, \sqrt{\bar u}\} \text{ if } p_x = p_y,\\ \{0\} &amp;\text{ if } p_x &gt; p_y\end{cases}.
$$</span>
The expenditure function is given by:
<span class="math-container">$$
e(p_x, p_y, \bar u) = \min\{p_x, p_y\}\sqrt{\bar u}.
$$</span></p>
<p>Now, if <span class="math-container">$p_x &lt; p_y$</span> then:
<span class="math-container">$$
h_x(p_x, p_y, v(p_x, p_y, m)) = \left\{\sqrt{v(p_x, p_y, m)}\right\} = \left\{\frac{m}{\min\{p_x, p_y\}}\right\} = \left\{\frac{m}{p_x}\right\} = x(p_x, p_y, m).
$$</span>
If <span class="math-container">$p_x &gt; p_y$</span> then:
<span class="math-container">$$
h_x(p_x, p_y, v(p_x, p_y, m)) = \{0\} = x(p_x, p_y, m)
$$</span>
If <span class="math-container">$p_x = p_y$</span> we have:
<span class="math-container">$$
h_x(p_x, p_y, v(p_x,p_y,m)) = \left\{0, \sqrt{v(p_x, p_y, m)}\right\} = \left\{0, \frac{m}{p_x}\right\} = x(p_x, p_y, m).
$$</span>
This shows that strong duality does hold in this case.</p>
<p>The conceptual difficulty for this example is that if <span class="math-container">$p_x = p_y$</span>, then demand is not single-valued. As such, we have to work with demand correspondences rather than demand functions.</p>

CC BY-SA 4.0
2024-04-27T12:58:33.410Z
Wow thank you for this thorough answer! I completely missed that method for deriving Hicksian demand!
Martin
CC BY-SA 4.0
2024-04-27T09:23:04.390Z
2024-04-27T09:23:04.390Z
tdm
<p>I'm posting here, because I don't have enough reputation points to just comment. So, if I understand this correctly, monotonic means that preferences, do not change, right? Like, I'm familiar with the concept of monotonic transformation, but what do you mean by duality? Also, is there a scenario, where preferences are not monotonic?</p>
<p>I don't mean to be rude. I genuinely want to know.
Also, pls don't feel like you need to write long paragraphs to explain it (although that would be great), if you could recommend a video or an article I would really appreciate it too.</p>

CC BY-SA 4.0
2024-05-01T20:59:19.647Z
2024-05-01T20:59:19.647Z
KEll
<p>Could you please help me evaluate this logic?</p>
<p>I've been told that &quot;if preferences are strongly monotonic, duality holds.&quot;</p>
<p>In the case of utility u(x,y) = x^2 + y^2, we will get a corner solution depending on the relative price of x and y. Suppose p_x &gt; p_y =&gt; x* = 0.</p>
<p>Then MUx = 0 at x*, so preferences are not strongly monotonic around the optimum. Hence, duality may not hold (Hicksian demand ≠ Marshallian demand). After plugging in the indirect utility function into Hicksian demand, I found that indeed duality does not hold here.</p>
<p>Does duality hold? If yes, please show why mathematically.</p>

CC BY-SA 4.0
2024-04-26T15:44:43.087Z
This utility function is strongly monotonic.
Michael Greinecker
CC BY-SA 4.0
2024-04-26T15:54:42.257Z
Not globally though? Marginal utility is equal to zero for x or y equal to zero, which is exactly the interesting case? Although this may not be the formal definition of strong monotonicity, I feel like it interferes with duality here. Do you think duality holds?
Martin
CC BY-SA 4.0
2024-04-26T16:26:55.107Z
You could actually provide your calculations. Your utility function is strictly increasing everywhere. A strictly positive derivative everywhere is a sufficient but not necessary condition for a function to be strictly increasing.
Michael Greinecker
<microeconomics><consumer-theory><optimization><theory><academic-graduate>
Does duality hold for u(x, y) = x^2 + y^2? (Corner solution)
Martin
# Does duality hold for u(x, y) = x^2 + y^2? (Corner solution)

(Asked by: username_0 on 2024-04-26)

Could you please help me evaluate this logic?

I've been told that "if preferences are strongly monotonic, duality holds."

In the case of utility u(x,y) = x^2 + y^2, we will get a corner solution depending on the relative price of x and y. Suppose p\_x > p\_y => x\* = 0.

Then MUx = 0 at x\*, so preferences are not strongly monotonic around the optimum. Hence, duality may not hold (Hicksian demand ≠ Marshallian demand). After plugging in the indirect utility function into Hicksian demand, I found that indeed duality does not hold here.

Does duality hold? If yes, please show why mathematically.

**Tags:** microeconomics, consumer-theory, optimization, theory, academic-graduate
> **Comments:**
> **username_1**: This utility function is strongly monotonic.  
> **username_0**: Not globally though? Marginal utility is equal to zero for x or y equal to zero, which is exactly the interesting case? Although this may not be the formal definition of strong monotonicity, I feel like it interferes with duality here. Do you think duality holds?  
> **username_1**: You could actually provide your calculations. Your utility function is strictly increasing everywhere. A strictly positive derivative everywhere is a sufficient but not necessary condition for a function to be strictly increasing.  

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## 2 answers

### Answer by username_2 on 2024-04-27. Score: 1 | Accepted answer

The Marshallian demand correspondence is given by:
$$
x(p\_x, p\_y,m) = \begin{cases} \{m/p\_x\} &\text{ if } p\_x < p\_y\\ \{0, m/p\_x\} &\text{ if } p\_x = p\_y\\ \{0\} &\text{ if } p\_x > p\_y \end{cases}.
$$

The indirect utility function is given by:
$$
v(p\_x, p\_y, m) = \left(\frac{m}{\min\{p\_x, p\_y\}}\right)^2
$$

Now consider the cost minimization problem:
$$
e(p\_x, p\_y, \bar u) = \min p\_x x + p\_y y \text{ s.t. } x^2 + y^2 \ge \bar u.
$$
If $p\_x < p\_y$ it will be best to set $x = \sqrt{\bar u}$ and if $p\_y < p\_x$ it will be best to set $y = \sqrt{\bar u}$. So we get that the Hicksian demand correspondence is given by:
$$
h\_x(p\_x, p\_y, \bar u) = \begin{cases}\{\sqrt{\bar u}\} & \text{ if } p\_x < p\_y,\\ \{0, \sqrt{\bar u}\} \text{ if } p\_x = p\_y,\\ \{0\} &\text{ if } p\_x > p\_y\end{cases}.
$$
The expenditure function is given by:
$$
e(p\_x, p\_y, \bar u) = \min\{p\_x, p\_y\}\sqrt{\bar u}.
$$

Now, if $p\_x < p\_y$ then:
$$
h\_x(p\_x, p\_y, v(p\_x, p\_y, m)) = \left\{\sqrt{v(p\_x, p\_y, m)}\right\} = \left\{\frac{m}{\min\{p\_x, p\_y\}}\right\} = \left\{\frac{m}{p\_x}\right\} = x(p\_x, p\_y, m).
$$
If $p\_x > p\_y$ then:
$$
h\_x(p\_x, p\_y, v(p\_x, p\_y, m)) = \{0\} = x(p\_x, p\_y, m)
$$
If $p\_x = p\_y$ we have:
$$
h\_x(p\_x, p\_y, v(p\_x,p\_y,m)) = \left\{0, \sqrt{v(p\_x, p\_y, m)}\right\} = \left\{0, \frac{m}{p\_x}\right\} = x(p\_x, p\_y, m).
$$
This shows that strong duality does hold in this case.

The conceptual difficulty for this example is that if $p\_x = p\_y$, then demand is not single-valued. As such, we have to work with demand correspondences rather than demand functions.

> **Comments:**
> **username_0**: Wow thank you for this thorough answer! I completely missed that method for deriving Hicksian demand!  

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### Answer by username_3 on 2024-05-01. Score: 0

I'm posting here, because I don't have enough reputation points to just comment. So, if I understand this correctly, monotonic means that preferences, do not change, right? Like, I'm familiar with the concept of monotonic transformation, but what do you mean by duality? Also, is there a scenario, where preferences are not monotonic?

I don't mean to be rude. I genuinely want to know.
Also, pls don't feel like you need to write long paragraphs to explain it (although that would be great), if you could recommend a video or an article I would really appreciate it too.


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