<p><span class="math-container">\begin{equation}
n_s = r_s + \beta_s n_b - A_s  \tag{1}
\end{equation}</span></p>
<p><span class="math-container">\begin{equation}
n_b = r_b + \beta_b n_s - A_b  \tag{2}
\end{equation}</span></p>
<p>Substituting (1) into (2) we get</p>
<p><span class="math-container">$$
n_b = r_b + \beta_b 
\left(
r_s + \beta_s n_b - A_s
\right) - A_b.  
$$</span></p>
<p>Rearranging all <span class="math-container">$n_b$</span>'s to the left:</p>
<p><span class="math-container">$$
n_b - \beta_b \beta_s n_b = r_b + \beta_b 
\left(
r_s - A_s
\right) - A_b
$$</span></p>
<p>Dividing by <span class="math-container">$1 - \beta_b \beta_s$</span> yields the desired result for <span class="math-container">$n_b$</span>. Similar calculations will yield the result for <span class="math-container">$n_s$</span> as well.</p>

CC BY-SA 4.0
2022-06-25T17:38:08.053Z
Thank you very much for all the help :)
Faizan Ahmed
CC BY-SA 4.0
2022-06-25T08:55:13.307Z
2022-06-25T08:55:13.307Z
Giskard
<p>I was reading the book &quot;The Economics of Platforms: Concepts and Strategy&quot; by Paul Belleflamme and I was confused about the solution of the seller and buyer surplus equation on page 132. The equations were derived by solving them simultaneously but I am unable to come up with the same solution as in the book. Futhermore the explaination on how we solved the equation to come up with 4.1 solution is uncomprehendable. Kindly help. Please see the following excerpts:
<a href="https://i.sstatic.net/cz1H9.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/cz1H9.png" alt="enter image description here" /></a>
<a href="https://i.sstatic.net/7bv7C.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/7bv7C.png" alt="enter image description here" /></a>
<a href="https://i.sstatic.net/42p4t.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/42p4t.png" alt="enter image description here" /></a></p>

<supply-and-demand><markets>
Net Surplus and first party products calculation on digital platform
Faizan Ahmed
# Net Surplus and first party products calculation on digital platform

(Asked by: username_0 on 2022-06-24)

I was reading the book "The Economics of Platforms: Concepts and Strategy" by Paul Belleflamme and I was confused about the solution of the seller and buyer surplus equation on page 132. The equations were derived by solving them simultaneously but I am unable to come up with the same solution as in the book. Futhermore the explaination on how we solved the equation to come up with 4.1 solution is uncomprehendable. Kindly help. Please see the following excerpts:
[![enter image description here](https://i.sstatic.net/cz1H9.png)](https://i.sstatic.net/cz1H9.png)
[![enter image description here](https://i.sstatic.net/7bv7C.png)](https://i.sstatic.net/7bv7C.png)
[![enter image description here](https://i.sstatic.net/42p4t.png)](https://i.sstatic.net/42p4t.png)

**Tags:** supply-and-demand, markets


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## 1 answers

### Answer by username_1 on 2022-06-25. Score: 1

\begin{equation}
n\_s = r\_s + \beta\_s n\_b - A\_s \tag{1}
\end{equation}

\begin{equation}
n\_b = r\_b + \beta\_b n\_s - A\_b \tag{2}
\end{equation}

Substituting (1) into (2) we get

$$
n\_b = r\_b + \beta\_b
\left(
r\_s + \beta\_s n\_b - A\_s
\right) - A\_b.
$$

Rearranging all $n\_b$'s to the left:

$$
n\_b - \beta\_b \beta\_s n\_b = r\_b + \beta\_b
\left(
r\_s - A\_s
\right) - A\_b
$$

Dividing by $1 - \beta\_b \beta\_s$ yields the desired result for $n\_b$. Similar calculations will yield the result for $n\_s$ as well.

> **Comments:**
> **username_0**: Thank you very much for all the help :)  

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