<p>You correctly derived the Marshallian demand function for Cobb-Douglas utility, you notice that the optimal level of consumption of $x$ or $y$ is a function only of the individual's income and the price of said good.  This is an interesting feature of CD-utility, that when the price of good $y$ changes the demand for $x$ doesn't change.  This means that $x$ and $y$ are neither substitutes nor complements to one another. <a href="http://www.econport.org/content/handbook/consumerdemand/Demand.html" rel="nofollow noreferrer">Econport</a> has a nice figure of this:</p>

<p>$\hskip2in$ <a href="https://i.sstatic.net/cbRha.gif" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cbRha.gif" alt="enter image description here"></a></p>

<p>The first figure shows that as the price of good $x$ changes the quantity of $x$ changes but the quantity of $y$ does not change.  The second figure shows that as price increases the amount demanded decreases and vice-versa when price decreases.</p>

<p>As a side note, CD-utility further implies strict quasi-concave utility so that the indifference set is strictly convex, hence $\vec{x}^* (\vec{p},I)$ is single-valued.  With constant prices for $y$ and prices $p_a, p_b, p_c$ for $x,$ we choose the optimal consumption level of each commodity at the points $a = (x_a, y_a) ,b = (x_b, y_b) ,c = (x_c, y_c)$, such that the slope of the budget hyperplane and the tangent slope of the indifference curve are equal.</p>

CC BY-SA 3.0
2016-10-16T02:20:34.967Z
2016-10-16T02:20:34.967Z
Sunhwa
<p>To briefly add to Sunhwa's answer, the intuitive reasoning for why the price of $y$ does not affect consumption of good $x$ is because of the unique properties of the Cobb-Douglas utility function. Changing the price of a good changes the price ratio, which is part of what determines the ratio of goods consumed. So even if the price of $y$ were to go up and cause you to buy less $y$, that would also change the ratio of goods you are buying, even if you don't change $x$. In this case, the optimization condition ends up being so that you don't have to change $x$ at all (for all other things equal).</p>

<p>$$\frac{p_y}{p_x} \cdot \frac{\alpha}{\beta} = \frac{x}{y}$$</p>

<p>Peruse this condition as you see fit.</p>

CC BY-SA 3.0
2016-11-15T03:43:28.810Z
2016-11-15T03:43:28.810Z
Kitsune Cavalry
<p>Suppose we have a Cobb-Douglas utility function: $$U(x,y)=x^\alpha y^\beta$$ and a budget constraint: $$p_{x}x+p_{y}y=I$$ where $\alpha+\beta=1$.</p>

<p>It can be shown that the Marshallian demand for $x$ and $y$ is $x^*=\frac{\alpha I}{p_{x}}$ and $y^*=\frac{\beta I}{p_{y}}$ respectively.</p>

<p>Does a change in $p_{y}$ affect the quantity of $x$ demanded? By looking at $x$'s Marshallian Demand, it seems as though such a change will not affect the quantity of x demanded but $p_{x}$ and $p_{y}$ are related by the budget constraint which makes me not quite sure.</p>

CC BY-SA 3.0
2016-10-15T15:53:19.113Z
$p_x$ and $p_y$ are not related by the budget constraint, they are not variables but parameters therein.
Giskard
CC BY-SA 3.0
2016-10-15T16:04:44.583Z
@denesp, thank you! So a change in $p_[y}$ does not affect quantity demanded of x is the answer, right?
Omrane
<microeconomics><utility><optimization><cobb-douglas>
Indirect changes in Marshallian Demand
Omrane
# Indirect changes in Marshallian Demand

(Asked by: username_0 on 2016-10-15)

Suppose we have a Cobb-Douglas utility function: $$U(x,y)=x^\alpha y^\beta$$ and a budget constraint: $$p\_{x}x+p\_{y}y=I$$ where $\alpha+\beta=1$.

It can be shown that the Marshallian demand for $x$ and $y$ is $x^\*=\frac{\alpha I}{p\_{x}}$ and $y^\*=\frac{\beta I}{p\_{y}}$ respectively.

Does a change in $p\_{y}$ affect the quantity of $x$ demanded? By looking at $x$'s Marshallian Demand, it seems as though such a change will not affect the quantity of x demanded but $p\_{x}$ and $p\_{y}$ are related by the budget constraint which makes me not quite sure.

**Tags:** microeconomics, utility, optimization, cobb-douglas
> **Comments:**
> **username_1**: $p_x$ and $p_y$ are not related by the budget constraint, they are not variables but parameters therein.  
> **username_0**: @denesp, thank you! So a change in $p_[y}$ does not affect quantity demanded of x is the answer, right?  

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## 2 answers

### Answer by username_2 on 2016-10-16. Score: 3

You correctly derived the Marshallian demand function for Cobb-Douglas utility, you notice that the optimal level of consumption of $x$ or $y$ is a function only of the individual's income and the price of said good. This is an interesting feature of CD-utility, that when the price of good $y$ changes the demand for $x$ doesn't change. This means that $x$ and $y$ are neither substitutes nor complements to one another. [Econport](http://www.econport.org/content/handbook/consumerdemand/Demand.html) has a nice figure of this:

$\hskip2in$ [![enter image description here](https://i.sstatic.net/cbRha.gif)](https://i.sstatic.net/cbRha.gif)

The first figure shows that as the price of good $x$ changes the quantity of $x$ changes but the quantity of $y$ does not change. The second figure shows that as price increases the amount demanded decreases and vice-versa when price decreases.

As a side note, CD-utility further implies strict quasi-concave utility so that the indifference set is strictly convex, hence $\vec{x}^\* (\vec{p},I)$ is single-valued. With constant prices for $y$ and prices $p\_a, p\_b, p\_c$ for $x,$ we choose the optimal consumption level of each commodity at the points $a = (x\_a, y\_a) ,b = (x\_b, y\_b) ,c = (x\_c, y\_c)$, such that the slope of the budget hyperplane and the tangent slope of the indifference curve are equal.


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### Answer by username_3 on 2016-11-15. Score: 1

To briefly add to Sunhwa's answer, the intuitive reasoning for why the price of $y$ does not affect consumption of good $x$ is because of the unique properties of the Cobb-Douglas utility function. Changing the price of a good changes the price ratio, which is part of what determines the ratio of goods consumed. So even if the price of $y$ were to go up and cause you to buy less $y$, that would also change the ratio of goods you are buying, even if you don't change $x$. In this case, the optimization condition ends up being so that you don't have to change $x$ at all (for all other things equal).

$$\frac{p\_y}{p\_x} \cdot \frac{\alpha}{\beta} = \frac{x}{y}$$

Peruse this condition as you see fit.


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