# Net Surplus and first party products calculation on digital platform

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:   $$\begin{equation} n_s = r_s + \beta_s n_b - A_s \tag{1} \end{equation}$$

$$\begin{equation} n_b = r_b + \beta_b n_s - A_b \tag{2} \end{equation}$$

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.

• Thank you very much for all the help :) Jun 25 at 17:38