# How to go from demand elasticities to a demand function? (merger simulation)

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?