<p>There is probably an error in your formula for <span class="math-container">$\sum_0^s v_i$</span>.
You can use a trick to directly compute <span class="math-container">$E_0 e^{(1-\gamma)v_i}$</span> using
<span class="math-container">\begin{align}
E_0 e^{(1-\gamma)v_i}&amp; = \Pi_0^s E_0e^{(1-\gamma)v_i}.
\end{align}</span>
Notice <span class="math-container">$e^{(1-\gamma)v_i}$</span> is a random variable equal to 1 with probability <span class="math-container">$1-p$</span> and <span class="math-container">$(1-b)^{1-\gamma}$</span> with probability <span class="math-container">$p$</span>. Therefore,
<span class="math-container">\begin{align}
E_0 e^{(1-\gamma)v_i}&amp; = \Pi_0^s (1-p+pE_0(1-b)^{1-\gamma}),\\
&amp; = (1-p+pE_0(1-b)^{1-\gamma})^s.
\end{align}</span></p>
<p>The term <span class="math-container">$E(1-b)^{1-\gamma}$</span> now appears your expression.</p>

CC BY-SA 4.0
2020-10-13T16:20:03.350Z
2020-10-13T16:20:03.350Z
GuiWil
<p>In Barro (2009) <a href="http://piketty.pse.ens.fr/files/Barro2009.pdf" rel="nofollow noreferrer">http://piketty.pse.ens.fr/files/Barro2009.pdf</a>
My question is reference to equation #5, whereby Barro is deriving the reciprocal of the market value 1/v, and I am trying to derive this equation, and having trouble. In particular, without further specifications of distribution of v.</p>
<p>So far, I have:
<a href="https://i.sstatic.net/lZwo8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/lZwo8.png" alt="enter image description here" /></a></p>
<p>But can't seem to arrive at the same result, without further information/assumptions on the distribution of the disasters.<br />
Barro's Result for equation (5)
<a href="https://i.sstatic.net/Pzu51.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Pzu51.png" alt="enter image description here" /></a></p>
<p>Any help would be greatly appreciated.</p>

<macroeconomics><risk>
In Barro's (2009) Rare Disaster Model in AER: How to derive equation (5)
cgguy
# In Barro's (2009) Rare Disaster Model in AER: How to derive equation (5)

(Asked by: username_0 on 2020-10-09)

In Barro (2009) <http://piketty.pse.ens.fr/files/Barro2009.pdf>
My question is reference to equation #5, whereby Barro is deriving the reciprocal of the market value 1/v, and I am trying to derive this equation, and having trouble. In particular, without further specifications of distribution of v.

So far, I have:
[![enter image description here](https://i.sstatic.net/lZwo8.png)](https://i.sstatic.net/lZwo8.png)

But can't seem to arrive at the same result, without further information/assumptions on the distribution of the disasters.  
Barro's Result for equation (5)
[![enter image description here](https://i.sstatic.net/Pzu51.png)](https://i.sstatic.net/Pzu51.png)

Any help would be greatly appreciated.

**Tags:** macroeconomics, risk


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## 1 answers

### Answer by username_1 on 2020-10-13. Score: 2

There is probably an error in your formula for $\sum\_0^s v\_i$.
You can use a trick to directly compute $E\_0 e^{(1-\gamma)v\_i}$ using
\begin{align}
E\_0 e^{(1-\gamma)v\_i}& = \Pi\_0^s E\_0e^{(1-\gamma)v\_i}.
\end{align}
Notice $e^{(1-\gamma)v\_i}$ is a random variable equal to 1 with probability $1-p$ and $(1-b)^{1-\gamma}$ with probability $p$. Therefore,
\begin{align}
E\_0 e^{(1-\gamma)v\_i}& = \Pi\_0^s (1-p+pE\_0(1-b)^{1-\gamma}),\\
& = (1-p+pE\_0(1-b)^{1-\gamma})^s.
\end{align}

The term $E(1-b)^{1-\gamma}$ now appears your expression.


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