# In Barro's (2009) Rare Disaster Model in AER: How to derive equation (5)

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:

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)

Any help would be greatly appreciated.

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.