<p>We have data for 3 goods (equilibrium prices and quantities) and we want to know the cross-price elasticities among them.</p>
<p>I think, this could be estimated as the following system of regressions:</p>
<p><span class="math-container">$$ \ln(x_1) = \beta_{1,1} \ln(P_1) + \beta_{1,2} \ln(P_2) + \beta_{1,3} \ln(P_3) + \epsilon_1$$</span></p>
<p><span class="math-container">$$ \ln(x_2) = \beta_{2,1} \ln(P_1) + \beta_{2,2} \ln(P_2) + \beta_{2,3} \ln(P_3) + \epsilon_2$$</span></p>
<p><span class="math-container">$$ \ln(x_3) = \beta_{3,1} \ln(P_1) + \beta_{3,2} \ln(P_2) + \beta_{3,3} \ln(P_3) + \epsilon_3$$</span></p>
<p>Where <span class="math-container">$\beta_{i,j}$</span> denotes the cross-price elasticity between <span class="math-container">$i$</span> and <span class="math-container">$j$</span>. While information about all prices is known, each product has its own dimension, meaning <span class="math-container">$dim(x_1) \neq dim(x_2)$</span>. The problem is, that after estimation, we have two estimates of this cross-price elasticity, that is <span class="math-container">$\beta_{i,j}$</span> and <span class="math-container">$\beta_{j,i}$</span>. <strong>Now, when I would like to say, the cross-price elasticity between <span class="math-container">$i$</span> and <span class="math-container">$j$</span> is this and that...</strong></p>
<p><strong>...Is the correct approach to average the both estimates?</strong></p>
<p><span class="math-container">$$\beta_{i,j}^* = \frac{\beta_{i,j} + \beta_{j,i}}{2}$$</span></p>
<p><strong>...Or can I report just one-way effects? Or do something else entirely?</strong></p>
<p>I know, that both estimates will differ as I am trying to estimate the cross-price elasticity on <strong>Marshallian</strong> demand, but they differ significantly. I guess, this may be caused by different dimensionality of products. Additionally, is there a good reference to this?</p>

CC BY-SA 4.0
2024-03-05T08:56:59.920Z
You could estimate them simultaneously while imposing the symmetry conditions $\beta_{i,j} = \beta_{j,i}$ using (for example) restricted least squares.
tdm
CC BY-SA 4.0
2024-03-05T13:03:17.373Z
Is this possible even when the amount of observations between $x_1$ and $x_2$ differ and LASSO is a necessary approach? Is there a way how to legitimately argue that averaging is not necessary?
Athaeneus
<microeconomics><reference-request><cross-price-elasticity>
Averaging estimates of cross-price elasticity
Athaeneus
# Averaging estimates of cross-price elasticity

(Asked by: username_0 on 2024-03-04)

We have data for 3 goods (equilibrium prices and quantities) and we want to know the cross-price elasticities among them.

I think, this could be estimated as the following system of regressions:

$$ \ln(x\_1) = \beta\_{1,1} \ln(P\_1) + \beta\_{1,2} \ln(P\_2) + \beta\_{1,3} \ln(P\_3) + \epsilon\_1$$

$$ \ln(x\_2) = \beta\_{2,1} \ln(P\_1) + \beta\_{2,2} \ln(P\_2) + \beta\_{2,3} \ln(P\_3) + \epsilon\_2$$

$$ \ln(x\_3) = \beta\_{3,1} \ln(P\_1) + \beta\_{3,2} \ln(P\_2) + \beta\_{3,3} \ln(P\_3) + \epsilon\_3$$

Where $\beta\_{i,j}$ denotes the cross-price elasticity between $i$ and $j$. While information about all prices is known, each product has its own dimension, meaning $dim(x\_1) \neq dim(x\_2)$. The problem is, that after estimation, we have two estimates of this cross-price elasticity, that is $\beta\_{i,j}$ and $\beta\_{j,i}$. **Now, when I would like to say, the cross-price elasticity between $i$ and $j$ is this and that...**

**...Is the correct approach to average the both estimates?**

$$\beta\_{i,j}^\* = \frac{\beta\_{i,j} + \beta\_{j,i}}{2}$$

**...Or can I report just one-way effects? Or do something else entirely?**

I know, that both estimates will differ as I am trying to estimate the cross-price elasticity on **Marshallian** demand, but they differ significantly. I guess, this may be caused by different dimensionality of products. Additionally, is there a good reference to this?

**Tags:** microeconomics, reference-request, cross-price-elasticity
> **Comments:**
> **username_1**: You could estimate them simultaneously while imposing the symmetry conditions $\beta_{i,j} = \beta_{j,i}$ using (for example) restricted least squares.  
> **username_0**: Is this possible even when the amount of observations between $x_1$ and $x_2$ differ and LASSO is a necessary approach? Is there a way how to legitimately argue that averaging is not necessary?  

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## 0 answers

