<p>The integral of interest is $\int_{w^*}^Bw'dF(w')$, which we get after bringing $\frac{1}{1-\beta}$ out from the last integral on the second line. I assume that $F(B)=1$. First, we have:</p>

<p>$$
\int_{w^*}^Bw'dF(w')=w'F(w')\vert_{w^*}^B - \int^B_{w^*}
F(w')dw'
$$
from integration by parts with $u=w'$ and $dv=dF(w')$. We can evaluate the first term on the right-hand side and both add and subtract $B-w^*=\int^B_{w^*}dw'$ to get:</p>

<p>$$
\int_{w^*}^Bw'dF(w')= [B - w^*F(w^*)]-[B-w^*]+\int^B_{w^*}(1-F(w')dw'.
$$</p>

<p>Note that I used my initial assumption here, and that I brought $B-w^*=\int^B_{w^*}dw'$ under the pre-existing integral.It should be clear after a bit of algebra that:</p>

<p>$$
\int_{w^*}^Bw'dF(w')= w^*(1 - F(w^*))+\int^B_{w^*}(1-F(w')dw'.
$$</p>

<p>The rest should be easy after multipling both sides by the $\frac{\beta}{1-\beta}$ factor.</p>

CC BY-SA 3.0
2016-11-01T21:17:30.227Z
Thanks so much! I wonder why textbooks so often neglect to mention they are using algebraic 'tricks' like adding and subtracting B-w* in this case?
MHall
CC BY-SA 3.0
2016-11-01T02:39:32.913Z
2016-11-01T02:39:32.913Z
MJRS
<p><a href="https://i.sstatic.net/szTOI.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/szTOI.png" alt="integrating value function by parts"></a></p>

<p>This is a derivation from "Exercises in Recursive Macroeconomic Theory
preliminary and incomplete by 
Stijn Van Nieuwerburgh
Pierre-Olivier Weill
Lars Ljungqvist
Thomas J. Sargent"</p>

<p>which uses integration by parts. Can someone explain how this derivation works? I have tried to replicate the result but to no avail. The text says that they use integration by parts on the second integral on the second line [the integral from w to B of w'dF(w') ]</p>

<macroeconomics>
integrating CDF functions by parts
MHall
# integrating CDF functions by parts

(Asked by: username_0 on 2016-10-31)

[![integrating value function by parts](https://i.sstatic.net/szTOI.png)](https://i.sstatic.net/szTOI.png)

This is a derivation from "Exercises in Recursive Macroeconomic Theory
preliminary and incomplete by
Stijn Van Nieuwerburgh
Pierre-Olivier Weill
Lars Ljungqvist
Thomas J. Sargent"

which uses integration by parts. Can someone explain how this derivation works? I have tried to replicate the result but to no avail. The text says that they use integration by parts on the second integral on the second line [the integral from w to B of w'dF(w') ]

**Tags:** macroeconomics

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## 1 answers

### Answer by username_1 on 2016-11-01. Score: 3 | Accepted answer

The integral of interest is $\int\_{w^\*}^Bw'dF(w')$, which we get after bringing $\frac{1}{1-\beta}$ out from the last integral on the second line. I assume that $F(B)=1$. First, we have:

$$
\int\_{w^\*}^Bw'dF(w')=w'F(w')\vert\_{w^\*}^B - \int^B\_{w^\*}
F(w')dw'
$$
from integration by parts with $u=w'$ and $dv=dF(w')$. We can evaluate the first term on the right-hand side and both add and subtract $B-w^\*=\int^B\_{w^\*}dw'$ to get:

$$
\int\_{w^\*}^Bw'dF(w')= [B - w^\*F(w^\*)]-[B-w^\*]+\int^B\_{w^\*}(1-F(w')dw'.
$$

Note that I used my initial assumption here, and that I brought $B-w^\*=\int^B\_{w^\*}dw'$ under the pre-existing integral.It should be clear after a bit of algebra that:

$$
\int\_{w^\*}^Bw'dF(w')= w^\*(1 - F(w^\*))+\int^B\_{w^\*}(1-F(w')dw'.
$$

The rest should be easy after multipling both sides by the $\frac{\beta}{1-\beta}$ factor.

> **Comments:**
> **username_0**: Thanks so much! I wonder why textbooks so often neglect to mention they are using algebraic 'tricks' like adding and subtracting B-w* in this case?  

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