Present Value of flow of profits with constant interest rate

Question

I'm studying Schumpeterian Models of Quality Ladders from the Barro's book (Ch.7).

I have problems in getting the derivation of equation $(7.13)$ as follows

The basic idea of those models is: when an innovation arrives, in the form of quality improvements of the existing product varieties, (randomly) in the $j$ industry at time $t_{k_j}$ the monopolist will enjoy a stream of profits equal to $\pi(k_j)$ until a new (different innovator) will invent a better quality of the product variety $j$ at time $t_{k_{j+1}}$. Once this will occur, the profits of the previous monopolist will go to zero. So, $T(k_j)$ is the time span when the monopolist making an innovation at $t_{k_j}$ makes positive profits.

Answer 1

The mathematical steps to arrive from 7.12 to 7.13 are the following:

\begin{aligned} & V\left(k_j\right)=\int_{t_k}^{t_{k+1}} \pi(k) e^{-\frac{v-t_{k}}{v-t_{k}} \cdot \int_{t_k}^v r(\omega) d w} \end{aligned} Considering that \begin{aligned} & \int_{t_k}^v r d w=r\left(v-t_k\right) \end{aligned} We have: \begin{aligned} & \pi(k) \int_{t_k}^{t_{k+1}} e^{-r\left(v-t_k\right)} d v =\\ &= \pi(k)\left[\frac{e^{-r\left(v-t_k\right)}}{-r}\right]_{v=t_k}^{t_k+1}= \\ & =\pi(k)\left(\frac{-e^{-r\left(t_{k+1}-t_k\right)}+1}{r}\right)= \\ & =\pi(k) \frac{1-e^{-r T_k}}{r}. \\ & \end{aligned}