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Got a question about the famous BLP paper (http://people.stern.nyu.edu/wgreene/Econometrics/BLP.pdf). When there is no interaction between product characteristic and consumer characteristic, the utility of consuming product $j$ is given by equation (2.3) in the paper: $x_j\beta-\alpha p_j+\xi_j+\epsilon_{ij}\equiv\delta_j+\epsilon_{ij}$. Suppose there are 4 options: the outside option, product 1, product 2 and product 3. Then it is clear that the market shares depend only on the mean utilities: $s_j=S_j(\delta_0,\delta_1,\delta_2,\delta_3)\equiv \int\prod _{p\neq j}Pr(\delta_j-\delta_p\geq\epsilon_{ip}-\epsilon_{ij}|\epsilon_{ij} )f(\epsilon_{ij})d\epsilon_{ij}\equiv \int\prod _{p\neq j}F_{\epsilon_{ip}-\epsilon_{ij}|\epsilon_{ij}}(\delta_j-\delta_p)f(\epsilon_{ij})d\epsilon_{ij}$

and this implies that cross-price derivatives $\frac{\partial S_1(\delta_0,\delta_1,\delta_2,\delta_3)}{\partial p_3}=\frac{\partial S_2(\delta_0,\delta_1,\delta_2,\delta_3)}{\partial p_3}$ as long as $s_1=s_2$ (paragraph 3 on page 847). My question is mathematically, why introducing interaction between consumer characteristic $v_i$ with product characteristic $x_j$ will allow for more reasonable substitution pattern such that if $x_1$ is closer to $x_3$ than $x_2$, then $|\frac{\partial S_1(\delta_0,\delta_1,\delta_2,\delta_3)}{\partial p_3}|$ will be larger than $|\frac{\partial S_2(\delta_0,\delta_1,\delta_2,\delta_3)}{\partial p_3}|$ (in plain words: if product 3 is more similar to product 1, then as product 3 becomes more expensive, increase in demand for product 1 should be larger). I'm asking this question as intuitively, it seems that we need to interact price with the product characteristic instead of interact consumer characteristic with product characteristic to create the pattern we want.

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