<p>I made a very elementary error in forgetting that <span class="math-container">$f_i(v_i=P_j-\gamma_iq_j,\gamma_i) = f_i(\gamma_i|v_i=P_j-\gamma_iq_j)\text{Pr}(v_i=P_j-\gamma_iq_j)$</span>. This is not the same as <span class="math-container">$f_i(\gamma_i|v_i=P_j-\gamma_iq_j)$</span>. So the correction amounts to:</p>
<p><span class="math-container">$$\frac{\partial D_i}{\partial P_i}=\text{Pr}(v_i=P_j-\gamma_iq_j)$$</span></p>

CC BY-SA 4.0
2022-05-04T19:54:27.257Z
2022-05-04T19:54:27.257Z
bcguy
<p>More of a comment than a fully fledged answer. But my sense is that you should consider the effect of <span class="math-container">$P_i$</span> on <span class="math-container">$q_j$</span>, as well (see Rochet and Tirole (2006, RAND)).</p>
<p>Minor notes:</p>
<ol>
<li><p>the condition is <span class="math-container">$v_i + \gamma_i q_j &gt; P_i$</span> (not <span class="math-container">$v_i + \gamma_i q_j &gt; P_j$</span>)</p>
</li>
<li><p>In the Jullien et al. paper <span class="math-container">$\theta_i = (v_i, \gamma_i)$</span> (not <span class="math-container">$(v_i,\theta_i)$</span>)</p>
</li>
</ol>

CC BY-SA 4.0
2022-05-05T18:13:30.243Z
2022-05-05T18:13:30.243Z
Evangelos Con
<p>I have a very basic question about the elasticity of demand/participation in the two-sided market framework, as created by <a href="https://www.aeaweb.org/articles?id=10.1257/aer.100.4.1642" rel="nofollow noreferrer">Weyl (2010)</a> and described by <a href="https://faculty.wcas.northwestern.edu/apa522/Two-Sided-Market-and-Network-Effects.pdf" rel="nofollow noreferrer">Jullien, Pavan, and Rysman (2021)</a>. Using the notation of Jullien et al., let <span class="math-container">$q_i$</span> be the participation rate on side <span class="math-container">$i$</span>. The intermediary/platform sets a price <span class="math-container">$P_i$</span> for a side <span class="math-container">$i$</span> agent to participate. An agent on side <span class="math-container">$i$</span> pulls a type <span class="math-container">$(v_i,\theta_i)$</span> from some bivariate, twice-continuously differentiable distribution with pdf <span class="math-container">$f_i(v_i, \theta_i)$</span>. Here <span class="math-container">$v_i$</span> represents the utility the agent gets from participating and <span class="math-container">$\gamma_i$</span> represents a sort of interaction benefit the agent gets from participation on the other side (which is denoted <span class="math-container">$q_j$</span>).</p>
<p>So, the utility for an agent on side <span class="math-container">$i$</span> is:</p>
<p><span class="math-container">$u_i(v_i, \gamma_i, q_j) := v_i + \gamma_iq_j$</span></p>
<p>And the participation on side <span class="math-container">$i$</span> is:</p>
<p><span class="math-container">$D_i(P_i,q_j) := q_i = \text{Pr}(v_i + \gamma_i q_j &gt; P_j)$</span></p>
<p>It appears to me that <span class="math-container">$\frac{\partial q_i}{\partial P_i} = -1$</span>, but this outcome feels so strong and isn't directly stated in the literature (as far as I know), so I'm left with doubts. The reasoning is as follows:</p>
<p>Participation on side <span class="math-container">$i$</span> is:</p>
<p><span class="math-container">$D_i(P_i, q_j) = \int_{-\infty}^\infty \int_{P_j-\gamma_iq_j}^\infty f_i(v_i,\gamma_i)d v_id\gamma_i$</span></p>
<p>So, using Leibniz rule and the FTOC:</p>
<p><span class="math-container">$\frac{\partial D_i}{\partial P_i} = \int_{-\infty}^{\infty} f_i(v_i = P_j-\gamma_iq_j,\gamma_i) d \gamma_i = -1$</span></p>
<p>What is wrong with this logic?</p>

<microeconomics><industrial-organisation><networks>
Two-sided market framework
bcguy
# Two-sided market framework

(Asked by: username_0 on 2022-05-02)

I have a very basic question about the elasticity of demand/participation in the two-sided market framework, as created by [Weyl (2010)](https://www.aeaweb.org/articles?id=10.1257/aer.100.4.1642) and described by [Jullien, Pavan, and Rysman (2021)](https://faculty.wcas.northwestern.edu/apa522/Two-Sided-Market-and-Network-Effects.pdf). Using the notation of Jullien et al., let $q\_i$ be the participation rate on side $i$. The intermediary/platform sets a price $P\_i$ for a side $i$ agent to participate. An agent on side $i$ pulls a type $(v\_i,\theta\_i)$ from some bivariate, twice-continuously differentiable distribution with pdf $f\_i(v\_i, \theta\_i)$. Here $v\_i$ represents the utility the agent gets from participating and $\gamma\_i$ represents a sort of interaction benefit the agent gets from participation on the other side (which is denoted $q\_j$).

So, the utility for an agent on side $i$ is:

$u\_i(v\_i, \gamma\_i, q\_j) := v\_i + \gamma\_iq\_j$

And the participation on side $i$ is:

$D\_i(P\_i,q\_j) := q\_i = \text{Pr}(v\_i + \gamma\_i q\_j > P\_j)$

It appears to me that $\frac{\partial q\_i}{\partial P\_i} = -1$, but this outcome feels so strong and isn't directly stated in the literature (as far as I know), so I'm left with doubts. The reasoning is as follows:

Participation on side $i$ is:

$D\_i(P\_i, q\_j) = \int\_{-\infty}^\infty \int\_{P\_j-\gamma\_iq\_j}^\infty f\_i(v\_i,\gamma\_i)d v\_id\gamma\_i$

So, using Leibniz rule and the FTOC:

$\frac{\partial D\_i}{\partial P\_i} = \int\_{-\infty}^{\infty} f\_i(v\_i = P\_j-\gamma\_iq\_j,\gamma\_i) d \gamma\_i = -1$

What is wrong with this logic?

**Tags:** microeconomics, industrial-organisation, networks

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## 2 answers

### Answer by username_0 on 2022-05-04. Score: 2

I made a very elementary error in forgetting that $f\_i(v\_i=P\_j-\gamma\_iq\_j,\gamma\_i) = f\_i(\gamma\_i|v\_i=P\_j-\gamma\_iq\_j)\text{Pr}(v\_i=P\_j-\gamma\_iq\_j)$. This is not the same as $f\_i(\gamma\_i|v\_i=P\_j-\gamma\_iq\_j)$. So the correction amounts to:

$$\frac{\partial D\_i}{\partial P\_i}=\text{Pr}(v\_i=P\_j-\gamma\_iq\_j)$$


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### Answer by username_1 on 2022-05-05. Score: 0

More of a comment than a fully fledged answer. But my sense is that you should consider the effect of $P\_i$ on $q\_j$, as well (see Rochet and Tirole (2006, RAND)).

Minor notes:

1. the condition is $v\_i + \gamma\_i q\_j > P\_i$ (not $v\_i + \gamma\_i q\_j > P\_j$)
2. In the Jullien et al. paper $\theta\_i = (v\_i, \gamma\_i)$ (not $(v\_i,\theta\_i)$)


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