<p>The mathematical steps to arrive from 7.12 to 7.13 are the following:</p>
<p><span class="math-container">\begin{aligned}
&amp; 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}</span>
Considering that
<span class="math-container">\begin{aligned}
&amp; \int_{t_k}^v r d w=r\left(v-t_k\right)
\end{aligned}</span>
We have:
<span class="math-container">\begin{aligned}
&amp; \pi(k) \int_{t_k}^{t_{k+1}} e^{-r\left(v-t_k\right)} d v =\\
&amp;= \pi(k)\left[\frac{e^{-r\left(v-t_k\right)}}{-r}\right]_{v=t_k}^{t_k+1}= \\
&amp; =\pi(k)\left(\frac{-e^{-r\left(t_{k+1}-t_k\right)}+1}{r}\right)= \\
&amp; =\pi(k) \frac{1-e^{-r T_k}}{r}. \\
&amp;
\end{aligned}</span></p>

CC BY-SA 4.0
2023-07-07T14:25:38.397Z
2023-07-07T14:25:38.397Z
Don
<p>I'm studying Schumpeterian Models of Quality Ladders from the Barro's book (Ch.7).</p>
<p>I have problems in getting the derivation of equation <span class="math-container">$(7.13)$</span> as follows
<a href="https://i.sstatic.net/ZpXGo.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZpXGo.png" alt="enter image description here" /></a></p>
<p>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 <span class="math-container">$j$</span> industry at time <span class="math-container">$t_{k_j}$</span> the monopolist will enjoy a stream of profits equal to <span class="math-container">$\pi(k_j)$</span> until a new (different innovator) will invent a better quality of the product variety <span class="math-container">$j$</span> at time <span class="math-container">$t_{k_{j+1}}$</span>. Once this will occur, the profits of the previous monopolist will go to zero. So, <span class="math-container">$T(k_j)$</span> is the time span when the monopolist making an innovation at <span class="math-container">$t_{k_j}$</span> makes positive profits.</p>

<macroeconomics><microeconomics><mathematical-economics><economic-growth><endogenous-growth>
Present Value of flow of profits with constant interest rate
Maximilian
# Present Value of flow of profits with constant interest rate

(Asked by: username_0 on 2023-07-07)

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
[![enter image description here](https://i.sstatic.net/ZpXGo.png)](https://i.sstatic.net/ZpXGo.png)

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.

**Tags:** macroeconomics, microeconomics, mathematical-economics, economic-growth, endogenous-growth


---

## 1 answers

### Answer by username_1 on 2023-07-07. Score: 4 | Accepted answer

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}


---

