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I am reading this paper: https://econpapers.repec.org/article/oupjleorg/v_3a10_3ay_3a1994_3ai_3a2_3ap_3a407-26.htm (Werden and Froeb 1994) about merger simulation.

The paper is above my level, so I'm struggling through it a bit. The part I'm having the most difficulty with is how the authors go from demand elasticities to a demand function.

My general question is, if I have own-price elasticity of demand for every good $j$ $e_j$ and cross-price elasticity of demand between all goods $j$ and $k$ $e_{jk}$, how can I derive a demand function $q(p_j, p_{-j})$ where $p_j$ is the price of good $j$ and $p_{-j}$ is a vector of prices for all non-$j$ goods?

It seems to me that I would need some observed price-quantity pair and an assumption that the demand function is linear. Is this all I would need? Is this what the authors are doing?

To confirm my understanding of the overall procedure for merger simulation used in this paper, my understanding is that I would then solve the merging firms' profit maximization problems pre- and post-transaction. This problem is

$max_{p_j} (p_j-c_j)q_j$, where $q_j$ is a function of $e_j$ and $e_{jk}$, which in turn depend on market shares, which will be the varying factor between the pre- and post-transaction calculations.

Is this what the authors are doing in this paper?

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