<p>Let <span class="math-container">$q_x(p_x, p_y, m)$</span> and <span class="math-container">$q_y(p_x, p_y, m)$</span> be the Marshallian demands for good <span class="math-container">$x$</span> and <span class="math-container">$y$</span> respectively. Let <span class="math-container">$h_x(p_x, p_y, u)$</span> and <span class="math-container">$h_y(p_x, p_y,u)$</span> be the Hicksian demands for the two goods. Here <span class="math-container">$p_x, p_y$</span> are the prices for <span class="math-container">$x$</span> and <span class="math-container">$y$</span>, <span class="math-container">$m$</span> is the income and <span class="math-container">$u$</span> is the utility level.</p>
<p>Goods <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are <strong>gross subsitutes</strong> if:
<span class="math-container">$$
\frac{\partial q_x}{\partial p_y} &gt;0 \text{ and } \frac{\partial q_y}{\partial p_x} &gt; 0.
$$</span>
<span class="math-container">$x$</span> and <span class="math-container">$y$</span> are <strong>net substitutes</strong> if:
<span class="math-container">$$
\frac{\partial h_x}{\partial p_y} &gt;0 \text{ and } \frac{\partial h_y}{\partial p_x} &gt; 0.
$$</span></p>
<p>As your utility function is Cobb-Douglass, we have that:
<span class="math-container">$$
q_x = \frac{2}{3} \frac{m}{p_x} \text{ and } q_y = \frac{1}{3} \frac{m}{p_y}.
$$</span>
This should allow you to determine whether <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are gross subsitutes (or not).</p>
<p>Next, to get the answer for the net subsitutes, you can use the Slutsky equation:
<span class="math-container">$$
\frac{\partial h_x}{\partial p_y} = \frac{\partial q_x}{\partial p_y} + \frac{\partial q_x}{\partial m} q_y,\\
\frac{\partial h_y}{\partial p_x} = \frac{\partial q_y}{\partial p_x} + \frac{\partial q_y}{\partial m}q_x.
$$</span>
(Note that <span class="math-container">$\dfrac{\partial h_y}{\partial p_x} = \dfrac{\partial h_x}{\partial p_y}$</span> by symmetry of the Slutsky-matrix, so you only need to compute one of them.)</p>

CC BY-SA 4.0
2023-08-25T06:33:34.090Z
2023-08-25T06:33:34.090Z
tdm
<p>Question: For the utility function U(x,y)=x^(2/3)y^(1/3), are x and y net substitutes, gross substitutes, net complements, or gross complements?</p>
<p>I'm having trouble understanding IF I need to derive the function or if there's an intuitive way of reaching the answer and HOW to derive the function if necessary. I've scoured the internet and read my textbook, but I can't seem to piece it together. Can anyone provide any insight please? Thank you.</p>

<microeconomics>
Determining if two goods are net/gross substitutes/complements from a utility function
bruined7
# Determining if two goods are net/gross substitutes/complements from a utility function

(Asked by: username_0 on 2023-08-25)

Question: For the utility function U(x,y)=x^(2/3)y^(1/3), are x and y net substitutes, gross substitutes, net complements, or gross complements?

I'm having trouble understanding IF I need to derive the function or if there's an intuitive way of reaching the answer and HOW to derive the function if necessary. I've scoured the internet and read my textbook, but I can't seem to piece it together. Can anyone provide any insight please? Thank you.

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2023-08-25. Score: 1

Let $q\_x(p\_x, p\_y, m)$ and $q\_y(p\_x, p\_y, m)$ be the Marshallian demands for good $x$ and $y$ respectively. Let $h\_x(p\_x, p\_y, u)$ and $h\_y(p\_x, p\_y,u)$ be the Hicksian demands for the two goods. Here $p\_x, p\_y$ are the prices for $x$ and $y$, $m$ is the income and $u$ is the utility level.

Goods $x$ and $y$ are **gross subsitutes** if:
$$
\frac{\partial q\_x}{\partial p\_y} >0 \text{ and } \frac{\partial q\_y}{\partial p\_x} > 0.
$$
$x$ and $y$ are **net substitutes** if:
$$
\frac{\partial h\_x}{\partial p\_y} >0 \text{ and } \frac{\partial h\_y}{\partial p\_x} > 0.
$$

As your utility function is Cobb-Douglass, we have that:
$$
q\_x = \frac{2}{3} \frac{m}{p\_x} \text{ and } q\_y = \frac{1}{3} \frac{m}{p\_y}.
$$
This should allow you to determine whether $x$ and $y$ are gross subsitutes (or not).

Next, to get the answer for the net subsitutes, you can use the Slutsky equation:
$$
\frac{\partial h\_x}{\partial p\_y} = \frac{\partial q\_x}{\partial p\_y} + \frac{\partial q\_x}{\partial m} q\_y,\\
\frac{\partial h\_y}{\partial p\_x} = \frac{\partial q\_y}{\partial p\_x} + \frac{\partial q\_y}{\partial m}q\_x.
$$
(Note that $\dfrac{\partial h\_y}{\partial p\_x} = \dfrac{\partial h\_x}{\partial p\_y}$ by symmetry of the Slutsky-matrix, so you only need to compute one of them.)


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