Expected Price Elasticity in a Discrete Choice Fund Model

Summary

The document concerns an expected price elasticity result in a discrete choice model of mutual fund demand. The underlying paper argues that a fund's expected demand is less price-sensitive when it differentiates itself through its loading on a random market factor. The question focuses on turning an integral of the choice share times one minus that share into an expression involving expected market share and the variance of the share.

The supplied material states the elasticity formula and defines the choice probability, fund characteristics, and factor-related quantities, but it does not include an answer deriving the integral identity. It therefore presents a modeling question rather than a resolved method or empirical finding. The discussion is useful for understanding what quantities enter the proposed relationship, but it does not establish the algebraic step or provide evidence that the comparative-static claim holds beyond the cited paper's proposition.

Key ideas

• The model represents investor fund choice with probabilities that depend on fund attributes, price, and a random factor.
• The question asks how an expected share response can be rewritten using expected share and share variance.
• The stated proposition links factor-loading differentiation with lower expected price elasticity.
• The document supplies no derivation of the integral identity, so the key mathematical step remains unresolved.

Tags

# Understanding Price Elasticities in Discrete Choice Models (Derivative)


# Understanding Price Elasticities in Discrete Choice Models (Derivative)












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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