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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

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

Any help would be greatly appreciated.

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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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