<p>The Marshallian demands are functions <span class="math-container">$x_i(p_1,p_2,m)$</span>.</p>
<p>You treat <span class="math-container">$m$</span> as an independent variable rather than <span class="math-container">$p_1 x_1 + p_2 x_2$</span>.</p>
<p>Since the demands don’t depend on the cross prices (as they don’t appear in the formulas), i.e.,</p>
<p><span class="math-container">$\frac{\partial x_1}{\partial p_2} = 0$</span>,</p>
<p><span class="math-container">$\frac{\partial x_2}{\partial p_1} = 0$</span>,</p>
<p>they are not complements nor substitute goods, they’re unrelated goods.</p>
<p>Intuitively, if you were to spend all the budget on one good, you would buy</p>
<p><span class="math-container">$x_i = \frac{m}{p_i}$</span> units of that good.</p>
<p>The demands for both goods <span class="math-container">$x_1, x_2$</span> look like that multiplied by <span class="math-container">$\frac{1}{2}$</span>.</p>
<p>This means you’re spending half the budget on each good independently of their relative prices.</p>
<p>For example, let’s say I’m spending half my budget on pizzas and the other half on hamburgers.</p>
<p>If the price of pizzas suddenly increased, I’d still spend that <span class="math-container">$\frac{m}{2}$</span> on pizzas, being able to buy less pizzas with it of course.</p>
<p>But I’d still leave the other spending of <span class="math-container">$\frac{m}{2}$</span> on hamburgers intact. Since I’m not changing my spending on hamburgers and their price didn’t change, I’d buy the same amount of hamburgers independently of the price change on pizzas.</p>
<p>Note: A beautiful property of the Cobb-Douglas demand function is that when the exponents add up to <span class="math-container">$1$</span>, each exponent corresponds to the ratio of the budget you’d spend on that specific good.</p>
<p>If the exponents don’t add up to 1, just divide each exponent by the sum of the exponents to get the ratio.</p>
<p>The marshallian demands for a general Cobb-Douglas <span class="math-container">$u(x_1,x_2) = C x_{1}^{\alpha} x_{2}^{\beta}$</span> are</p>
<p><span class="math-container">$x_1 = \frac{\alpha}{\alpha + \beta} \frac{m}{p_1}$</span>,</p>
<p><span class="math-container">$x_2 = \frac{\beta}{\alpha + \beta} \frac{m}{p_2}$</span>.</p>
<p>Therefore, for every Cobb-Douglas utility function, the goods are unrelated (not complements nor substitutes).</p>

CC BY-SA 4.0
2023-01-26T03:05:28.317Z
2023-01-26T03:37:54.303Z
2023-01-26T03:37:54.303Z
Nicolas Torres
<p><strong>Substitutes and complements are determined by elasticity of substitution</strong>, which is a <em>characteristics of utility function</em>. You can compute it in the following way:</p>
<p><span class="math-container">$$\sigma_{ij} = \frac{\frac{\partial (x_j/x_i)}{x_j/x_i}}{\frac{\partial MRS_{ij}}{MRS_{ij}}}$$</span></p>
<p>See that this elasticity of substitution characterized per each pair of goods (any <span class="math-container">$x_i$</span> and <span class="math-container">$x_j$</span>) in utility function.</p>
<p>Why is this important? Because you know one important thing:</p>
<ul>
<li><strong>If elasticity of substitution is high</strong> (<span class="math-container">$\sigma_{i,j} &gt; 1$</span>), then it is (relatively) easy to substitute your pair of goods among each, therefore <strong>you work with substitutes</strong>.</li>
<li><strong>If elasticity of substitution is low</strong> (<span class="math-container">$\sigma_{i,j} &lt; 1$</span>), then it is (relatively) hard to substitute your pair of goods among each, therefore <strong>you work with complements</strong>.</li>
</ul>
<p>Why do we need to know? Because it can be shown that for</p>
<ul>
<li><span class="math-container">$\sigma_{i,j}&gt;1$</span> the cross-price elasticity in case of linear prices will be always positive!</li>
<li><span class="math-container">$\sigma_{i,j}&gt;1$</span> the cross-price elasticity in case of linear prices will be always negative!</li>
</ul>
<p>Cross price elasticity can be characterized as:</p>
<p><span class="math-container">$$e_{P_j}^{x_i} = \frac{\frac{\partial x_i}{x_i}}{\frac{\partial P_j}{P_j}}$$</span></p>
<p>These are percentage changes which can be rewritten in terms of logarithms:</p>
<p><span class="math-container">$$e_{P_j}^{x_i} = \frac{ln(x_i)}{ln(P_j)}$$</span></p>
<p>Therefore:</p>
<ul>
<li><strong>Positive cross-price elasticity</strong> means that increase of price of <em>other good by 1 %</em> implies <strong>an increase in demand for our good</strong>. Therefore, we have <strong>substitutes</strong>!</li>
<li><strong>Negative cross-price elasticity</strong> means that increase of price of <em>other good by 1 %</em> implies <strong>a decrease in demand for our good</strong>. Therefore, we have <strong>complements</strong>!</li>
</ul>
<p>So what you have to do is take your Marshallian demand and compute the cross-price elasticity in the following way:</p>
<p><span class="math-container">$$e_{P_j}^{x_i} = \frac{\partial x_i}{\partial P_j} * \frac{P_j}{x_i}$$</span></p>
<p>Where <span class="math-container">$x_i$</span> is the demand for good <span class="math-container">$i$</span>.</p>

CC BY-SA 4.0
2023-02-25T06:37:47.880Z
2023-02-25T06:37:47.880Z
Athaeneus
<p>U should check this video, it helped me anyway
<a href="https://www.youtube.com/watch?v=0BJ4GUpvKHQ" rel="nofollow noreferrer">https://www.youtube.com/watch?v=0BJ4GUpvKHQ</a></p>

CC BY-SA 4.0
2023-01-27T09:30:57.790Z
Please make your answer more explicit!
emeryville
CC BY-SA 4.0
2023-01-23T23:53:00.127Z
2023-01-23T23:53:00.127Z
Zooey Wang
<p>So I have a consumer with a utility function of the Cobb-Douglas form <span class="math-container">$v(x_1,x_2)=x^{\frac{1}{2}}_1x^{\frac{1}{2}}_2$</span>.</p>

<p>From that I constructed the demand function for good 1 and good 2:<br>
<span class="math-container">$x_1=\frac{1}{2} \cdot \frac{m}{p_1}$</span></p>

<p><span class="math-container">$x_2=\frac{1}{2} \cdot \frac{m}{p_2}$</span></p>

<p>From here, I am to determine whether good 1 and good 2 are substitutes or complements. I was told by my teacher to take the derivative with respect to <span class="math-container">$m, p_1$</span> and <span class="math-container">$p_2$</span>, however I can't get a sensible result. As far as I've come to understand from my book, I need to find <span class="math-container">$\frac{dx_1}{dp_2}$</span>, the change in the demand for good 1 as the price of good 2 changes. Substituting m back to <span class="math-container">$p_1x_1+p_2x_2$</span> becomes messy as well. I understand the problem theoretically, but I simply can't figure out how to properly argue using math, rather than logic.</p>

<p>I hope someone is willing to help</p>

CC BY-SA 4.0
2019-12-11T04:29:14.067Z
Would be useful to tell us what you get for each step before it gets "messy."
Art
<microeconomics><utility><cobb-douglas><demand>
Determine if goods are substitutes or Complements based on demand function
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# Determine if goods are substitutes or Complements based on demand function

(Asked by: username_0 on 2019-12-10)

So I have a consumer with a utility function of the Cobb-Douglas form $v(x\_1,x\_2)=x^{\frac{1}{2}}\_1x^{\frac{1}{2}}\_2$.

From that I constructed the demand function for good 1 and good 2:  
$x\_1=\frac{1}{2} \cdot \frac{m}{p\_1}$

$x\_2=\frac{1}{2} \cdot \frac{m}{p\_2}$

From here, I am to determine whether good 1 and good 2 are substitutes or complements. I was told by my teacher to take the derivative with respect to $m, p\_1$ and $p\_2$, however I can't get a sensible result. As far as I've come to understand from my book, I need to find $\frac{dx\_1}{dp\_2}$, the change in the demand for good 1 as the price of good 2 changes. Substituting m back to $p\_1x\_1+p\_2x\_2$ becomes messy as well. I understand the problem theoretically, but I simply can't figure out how to properly argue using math, rather than logic.

I hope someone is willing to help

**Tags:** microeconomics, utility, cobb-douglas, demand
> **Comments:**
> **username_1**: Would be useful to tell us what you get for each step before it gets "messy."  

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## 3 answers

### Answer by username_2 on 2023-01-26. Score: 1

The Marshallian demands are functions $x\_i(p\_1,p\_2,m)$.

You treat $m$ as an independent variable rather than $p\_1 x\_1 + p\_2 x\_2$.

Since the demands don’t depend on the cross prices (as they don’t appear in the formulas), i.e.,

$\frac{\partial x\_1}{\partial p\_2} = 0$,

$\frac{\partial x\_2}{\partial p\_1} = 0$,

they are not complements nor substitute goods, they’re unrelated goods.

Intuitively, if you were to spend all the budget on one good, you would buy

$x\_i = \frac{m}{p\_i}$ units of that good.

The demands for both goods $x\_1, x\_2$ look like that multiplied by $\frac{1}{2}$.

This means you’re spending half the budget on each good independently of their relative prices.

For example, let’s say I’m spending half my budget on pizzas and the other half on hamburgers.

If the price of pizzas suddenly increased, I’d still spend that $\frac{m}{2}$ on pizzas, being able to buy less pizzas with it of course.

But I’d still leave the other spending of $\frac{m}{2}$ on hamburgers intact. Since I’m not changing my spending on hamburgers and their price didn’t change, I’d buy the same amount of hamburgers independently of the price change on pizzas.

Note: A beautiful property of the Cobb-Douglas demand function is that when the exponents add up to $1$, each exponent corresponds to the ratio of the budget you’d spend on that specific good.

If the exponents don’t add up to 1, just divide each exponent by the sum of the exponents to get the ratio.

The marshallian demands for a general Cobb-Douglas $u(x\_1,x\_2) = C x\_{1}^{\alpha} x\_{2}^{\beta}$ are

$x\_1 = \frac{\alpha}{\alpha + \beta} \frac{m}{p\_1}$,

$x\_2 = \frac{\beta}{\alpha + \beta} \frac{m}{p\_2}$.

Therefore, for every Cobb-Douglas utility function, the goods are unrelated (not complements nor substitutes).


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### Answer by username_3 on 2023-02-25. Score: 0

**Substitutes and complements are determined by elasticity of substitution**, which is a *characteristics of utility function*. You can compute it in the following way:

$$\sigma\_{ij} = \frac{\frac{\partial (x\_j/x\_i)}{x\_j/x\_i}}{\frac{\partial MRS\_{ij}}{MRS\_{ij}}}$$

See that this elasticity of substitution characterized per each pair of goods (any $x\_i$ and $x\_j$) in utility function.

Why is this important? Because you know one important thing:

* **If elasticity of substitution is high** ($\sigma\_{i,j} > 1$), then it is (relatively) easy to substitute your pair of goods among each, therefore **you work with substitutes**.
* **If elasticity of substitution is low** ($\sigma\_{i,j} < 1$), then it is (relatively) hard to substitute your pair of goods among each, therefore **you work with complements**.

Why do we need to know? Because it can be shown that for

* $\sigma\_{i,j}>1$ the cross-price elasticity in case of linear prices will be always positive!
* $\sigma\_{i,j}>1$ the cross-price elasticity in case of linear prices will be always negative!

Cross price elasticity can be characterized as:

$$e\_{P\_j}^{x\_i} = \frac{\frac{\partial x\_i}{x\_i}}{\frac{\partial P\_j}{P\_j}}$$

These are percentage changes which can be rewritten in terms of logarithms:

$$e\_{P\_j}^{x\_i} = \frac{ln(x\_i)}{ln(P\_j)}$$

Therefore:

* **Positive cross-price elasticity** means that increase of price of *other good by 1 %* implies **an increase in demand for our good**. Therefore, we have **substitutes**!
* **Negative cross-price elasticity** means that increase of price of *other good by 1 %* implies **a decrease in demand for our good**. Therefore, we have **complements**!

So what you have to do is take your Marshallian demand and compute the cross-price elasticity in the following way:

$$e\_{P\_j}^{x\_i} = \frac{\partial x\_i}{\partial P\_j} \* \frac{P\_j}{x\_i}$$

Where $x\_i$ is the demand for good $i$.


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### Answer by username_4 on 2023-01-23. Score: -1

U should check this video, it helped me anyway
<https://www.youtube.com/watch?v=0BJ4GUpvKHQ>

> **Comments:**
> **username_5**: Please make your answer more explicit!  

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