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.


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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