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I'am in the midst of a paper on mutual fund product differentiation by Li and Qiu. Here, the authors model the utility an investor derives from investing in a mutual fund using a Discrete Choice Model approach. At the heart of their paper they argue that the expected price elasticity is lower if a fund differentiates herself from another fund in the market in terms of her factor loading on a random variable $f_{M2}$.

What I do not understand is the proof of Proposition 2 and I hope someone can help me here.

In particular, I cannot see why the second equation equals the third: \begin{align} |\eta_j^e|=|-b_0\frac{P_j}{S^e_j}\int s_{j2}(1-s_{j2})dF(f_{M2})|=|b_0 P_j(1-{S^e_j}-Var(S_{j2})/S_j^e)| \end{align}

where $|\eta_j^e|=|\frac{\partial S_j^e/S_j^e}{\partial P_j/P_j}|$ is the expected price elasticity of demand; $s_{j2}=\frac{exp(A_j-\frac{1}{2} \phi_2(\beta_{jM}-\beta^*_{M2})^2\sigma^2_M)}{\sum_{l=1}^{2}exp(A_l-\frac{1}{2} \phi_2(\beta_{lM}-\beta^*_{M2})^2\sigma^2_M}$ with $A_j=\phi_0\alpha_j-b_0P_j-\frac{1}{2}\phi_2\sigma_{\epsilon_j}^2+\mathbf{X'_jB}+\xi_j$ and $\beta_{M2}^*=\frac{\phi_1 f_{M2}}{\phi_2 \sigma^2_M}$ is the fund's market share at time 2 and $S_j^e$ is the expected market share with respect to time 1 information.

In other words, I am looking for a nice explanation on how to solve for the integral stated in the equation. A literature reference may also do the job.

Thanks for your time and help.

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