I'm trying to learn how to do merger simulation with logit demand. I think I get the concept of it and most of the math, but I'm struggling to put everything together and see how you would actually empirically simulate the merger. I have a few questions, if someone could help me out here. I will present my current understanding below. (If you see any flaws, please point them out.)

Here is my current understanding of how you would do it:

In a logit model, demand for firm $i$'s good is given by:

$q_i=\frac{Le^{\beta x_i+\alpha p_i+\epsilon_i}}{1+\sum_k e^{\beta x_k+\alpha p_k+\epsilon_k}}$, where $L$ is market size, $x_i$ is the product characteristics of firm $i$'s good, and $p_i$ is the price of the good.

$L$ should be taken from exogenous data. (If we're looking at the national market for cars, we might use the number of households in the country.)

$\alpha$ and $\beta$ must be estimated. In particular, we estimate the equation: $ln(s_i/s_0)=\beta x_i+\alpha p_i+\epsilon_i$, where $s_0$ is the market share of the outside good, often using instrumental variables. To do this, we need data on pre-merger prices and pre-merger market shares (including pre-merger market shares for the outside good).

If we have data on pre-merger prices, as well as other necessary parameters, we can recover marginal costs by solving each firm's profit maximization problem, whereby each firm chooses prices to maximize $(p_i-c_i)q_i$.

We see that $c_i=p_i+\frac{1}{\alpha(1-s_i)}$, where $s_i$ is firm $i$'s market share.

With these marginal costs, we can solve the post-merger profit maximization problem of the merged firm to find post-merger prices. Suppose firms 1 and 2 merge. Then the merged firm will choose prices $p_1$ and $p_2$ to maximize $(p_1-c_1)q_1+(p_2-c_2)q_2$. Actually solving this problem is not trivial given the non-linear form of the demand function; it requires use of Newton's method.

Questions I have:

1) What do you do with the $\epsilon_i$ in the demand equation? Like what number do I put there when I'm numerically calculating demand. 0? The average error term from my parameter estimation?

2) When you calculate $c_i$, are you using the observed price and observed market share? Or an estimated price and estimated market share? Is $s_i$ the market share with the outside good in the denominator or without?

  • $\begingroup$ 3) What does "model calibration" mean? I keep seeing it everywhere as the step to do after simulating demand and before solving the model, but I don't understand what it is. $\endgroup$ – leecarvallo Feb 21 at 20:57

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