<p>As previously stated by VARulle you simply compute your Marshallian demand for your goods, fix your income and just vary your prices.</p>
<p>To illustrate for cobb douglas preferences <span class="math-container">$U(x_1,x_2)=x_1^\alpha x_2^{1-\alpha}$</span> the marshallian demands for <span class="math-container">$x_1$</span> and <span class="math-container">$x_2$</span> are:
<span class="math-container">$$x_1(p_1,m)=\frac{\alpha m}{p_1}, x_2(p_1,m)=\frac{(1-\alpha)m}{p_2}$$</span></p>
<p>all you need to do is hold <span class="math-container">$m$</span> constant to see the shape of these &quot;single price changing&quot; expansion paths.</p>

CC BY-SA 4.0
2023-03-23T16:17:29.020Z
2023-03-23T16:17:29.020Z
EconJohn
<p>You compute it by setting <span class="math-container">$MRS$</span> equals relative prices:</p>
<p><span class="math-container">$\frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = \frac{p_x}{p_y}$</span>,</p>
<p>solving for one variable in terms of the other.</p>

CC BY-SA 4.0
2023-03-23T08:40:54.797Z
Thank you very much! However, this is not correct answer. I will edit the question such that it is more clear what I want. The answer you proposed gives me just a line between optimum and the zero/zero point. It is not the expansion path for prices. The expansion path for prices should not change with the change in price.
Athaeneus
CC BY-SA 4.0
2023-03-23T02:33:26.677Z
2023-03-23T02:43:35.930Z
2023-03-23T02:43:35.930Z
Nicolas Torres
<p>The answer depends on what you mean by &quot;<em>computing the expansion path</em>&quot;.</p>
<p><strong>Variant (i):</strong> For fixed <span class="math-container">$p_2$</span> and <span class="math-container">$M$</span>, you want to compute the optimal points <span class="math-container">$\boldsymbol{x}=(x_1,x_2)$</span> as a function of <span class="math-container">$p_1$</span>.</p>
<p>That's easy: <span class="math-container">$\boldsymbol{x}=\boldsymbol{D}(p_1,p_2,M)$</span>.</p>
<p><strong>Variant (ii):</strong> For fixed <span class="math-container">$p_2$</span> and <span class="math-container">$M$</span>, you want to compute the <span class="math-container">$x_2$</span>-component of the optimal points <span class="math-container">$\boldsymbol{x}=(x_1,x_2)$</span> as a function of <span class="math-container">$x_1$</span>.</p>
<p>This needs some work: You are given the functions <span class="math-container">$x_1=D_1(p_1,p_2,M)$</span> and <span class="math-container">$x_2=D_2(p_1,p_2,M)$</span>. From the equation for <span class="math-container">$x_1$</span> express (if possible) <span class="math-container">$p_1$</span> as a function <span class="math-container">$p_1=P_1(x_1,p_2,M)$</span> and substitute for <span class="math-container">$p_1$</span> in <span class="math-container">$D_2$</span> to get <span class="math-container">$x_2=D_2(P_1(x_1,p_2,M),p_2,M)$</span>. For fixed <span class="math-container">$p_2$</span> and <span class="math-container">$M$</span>, this gives <span class="math-container">$x_2$</span> as a function of <span class="math-container">$x_1$</span>, the graph of which is the red curve in <span class="math-container">$x_1$</span>-<span class="math-container">$x_2$</span>-space.</p>

CC BY-SA 4.0
2023-03-23T16:17:31.567Z
Thank you very much! Yes, I was searching for variant (ii).
Athaeneus
CC BY-SA 4.0
2023-03-23T14:53:29.950Z
2023-03-23T14:53:29.950Z
VARulle
<h2>Is there a way how to precisely compute the expansion path?</h2>
<p>I know a consumer's utility function <span class="math-container">$U(\boldsymbol{x})$</span>, I know the budget constraint <span class="math-container">$\sum P_i x_i \leq M$</span>, I am able to compute the marshallian demands <span class="math-container">$\boldsymbol{D}(\boldsymbol{P}, M)$</span>, therefore I am able to gain optimal point for any given prices and income.</p>
<p>If I plot this, using sliders, I can observe how the optimal point moves with changes in one single price. This movement, however, seems to be on a curve, <strong>the expansion path</strong>. Is there some way how to compute it analythically? Would the differentiation of demands be reasonable approach or is this too naive:</p>
<p><span class="math-container">$\qquad \qquad x_1^{EP} = \frac{\partial D_1(\boldsymbol{P}, M)}{\partial P_1} \qquad \qquad x_2^{EP} = \frac{\partial D_2(\boldsymbol{P}, M)}{\partial P_1}$</span></p>
<p>What would be the next step? This would be two variables but I would like to plug it into single curve...</p>
<p>Or is there another approach?</p>
<p><strong>EDIT:</strong> The expansion path I search for is also called <strong>price consumption curve</strong> in literature. I could find it by fitting few points (see illustration below), however, <strong>how can I find this analythically</strong>?</p>
<p><a href="https://i.sstatic.net/grVgT.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/grVgT.png" alt="expansion path" /></a></p>

CC BY-SA 4.0
2023-03-23T14:06:16.007Z
So you want $x_i$ as a function of $P_i$ is that correct? At first glance I suggest maximizing the Lagrangian.
Daniel
<microeconomics><consumer-theory><optimization><demand>
The formula for expansion path
Athaeneus
# The formula for expansion path

(Asked by: username_0 on 2023-03-22)

## Is there a way how to precisely compute the expansion path?

I know a consumer's utility function $U(\boldsymbol{x})$, I know the budget constraint $\sum P\_i x\_i \leq M$, I am able to compute the marshallian demands $\boldsymbol{D}(\boldsymbol{P}, M)$, therefore I am able to gain optimal point for any given prices and income.

If I plot this, using sliders, I can observe how the optimal point moves with changes in one single price. This movement, however, seems to be on a curve, **the expansion path**. Is there some way how to compute it analythically? Would the differentiation of demands be reasonable approach or is this too naive:

$\qquad \qquad x\_1^{EP} = \frac{\partial D\_1(\boldsymbol{P}, M)}{\partial P\_1} \qquad \qquad x\_2^{EP} = \frac{\partial D\_2(\boldsymbol{P}, M)}{\partial P\_1}$

What would be the next step? This would be two variables but I would like to plug it into single curve...

Or is there another approach?

**EDIT:** The expansion path I search for is also called **price consumption curve** in literature. I could find it by fitting few points (see illustration below), however, **how can I find this analythically**?

[![expansion path](https://i.sstatic.net/grVgT.png)](https://i.sstatic.net/grVgT.png)

**Tags:** microeconomics, consumer-theory, optimization, demand

> **Comments:**
> **username_1**: So you want $x_i$ as a function of $P_i$ is that correct? At first glance I suggest maximizing the Lagrangian.  

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## 3 answers

### Answer by username_4 on 2023-03-23. Score: 4 | Accepted answer

The answer depends on what you mean by "*computing the expansion path*".

**Variant (i):** For fixed $p\_2$ and $M$, you want to compute the optimal points $\boldsymbol{x}=(x\_1,x\_2)$ as a function of $p\_1$.

That's easy: $\boldsymbol{x}=\boldsymbol{D}(p\_1,p\_2,M)$.

**Variant (ii):** For fixed $p\_2$ and $M$, you want to compute the $x\_2$-component of the optimal points $\boldsymbol{x}=(x\_1,x\_2)$ as a function of $x\_1$.

This needs some work: You are given the functions $x\_1=D\_1(p\_1,p\_2,M)$ and $x\_2=D\_2(p\_1,p\_2,M)$. From the equation for $x\_1$ express (if possible) $p\_1$ as a function $p\_1=P\_1(x\_1,p\_2,M)$ and substitute for $p\_1$ in $D\_2$ to get $x\_2=D\_2(P\_1(x\_1,p\_2,M),p\_2,M)$. For fixed $p\_2$ and $M$, this gives $x\_2$ as a function of $x\_1$, the graph of which is the red curve in $x\_1$-$x\_2$-space.

> **Comments:**
> **username_0**: Thank you very much! Yes, I was searching for variant (ii).  

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### Answer by username_2 on 2023-03-23. Score: 1

As previously stated by username_4 you simply compute your Marshallian demand for your goods, fix your income and just vary your prices.

To illustrate for cobb douglas preferences $U(x\_1,x\_2)=x\_1^\alpha x\_2^{1-\alpha}$ the marshallian demands for $x\_1$ and $x\_2$ are:
$$x\_1(p\_1,m)=\frac{\alpha m}{p\_1}, x\_2(p\_1,m)=\frac{(1-\alpha)m}{p\_2}$$

all you need to do is hold $m$ constant to see the shape of these "single price changing" expansion paths.


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### Answer by username_3 on 2023-03-23. Score: 1

You compute it by setting $MRS$ equals relative prices:

$\frac{\frac{\partial U}{\partial x}}{\frac{\partial U}{\partial y}} = \frac{p\_x}{p\_y}$,

solving for one variable in terms of the other.

> **Comments:**
> **username_0**: Thank you very much! However, this is not correct answer. I will edit the question such that it is more clear what I want. The answer you proposed gives me just a line between optimum and the zero/zero point. It is not the expansion path for prices. The expansion path for prices should not change with the change in price.  

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