IO Demand estimation: Exercise 4

Tuomas Markkula

2. Merger simulation

The profit function of the merged firm, assuming no efficiencies, is

\begin{align*} \pi_{1 \cup 3} = s_1(p)(p_1 - mc_1) + s_3(p)(p_3 - mc_3). \end{align*}

And profit functions for the single profit firms are

\begin{align*} \pi_{i} = s_i(p)(p_i - mc_i) \; \text{where} \; i \in \{2,4\} \end{align*}

Profit maximizing first order conditions are

\begin{align*} \frac{ \partial \pi_{1 \cup 3}}{\partial p_1} &= (p_1 - mc_1) + \left( \frac{\partial s_1(p)}{\partial p_1} \right)^{-1} s_1(p) + \left( \frac{\partial s_1(p)}{\partial p_1} \right)^{-1} \frac{\partial s_3}{\partial p_1} (p_3 - mc_3) = 0 \\ \frac{ \partial \pi_{1 \cup 3}}{\partial p_3} &= (p_3 - mc_3) + \left( \frac{\partial s_3(p)}{\partial p_3} \right)^{-1} s_3(p) + \left( \frac{\partial s_3(p)}{\partial p_3} \right)^{-1} \frac{\partial s_1}{\partial p_3} (p_1 - mc_1) = 0 \\ \frac{ \partial \pi_{2}}{\partial p_2} &= (p_2 - mc_2) + \left( \frac{\partial s_2(p)}{\partial p_2} \right)^{-1} s_2(p) = 0\\ \frac{ \partial \pi_{4}}{\partial p_4} &= (p_4 - mc_4) + \left( \frac{\partial s_4(p)}{\partial p_4} \right)^{-1} s_4(p) = 0 \end{align*}

And

\begin{align*} \frac{\partial s_j(p)}{\partial p_j} &= \int \alpha(1 - s_{jt}(v))s_{jt}(v)\: f(v) dv = \frac{1}{6000} \sum_{i = 1}^{6000} \alpha(1 - s_{jt}(v_{i}))s_{jt}(v_{i}) \\ \frac{\partial s_k(p)}{\partial p_j} &= \int -\alpha s_{kt}(v)s_{jt}(v)\: f(v) dv = \frac{1}{6000} \sum_{i = 1}^{6000} -\alpha s_{kt}(v_{i})s_{jt}(v_{i}) \end{align*}

where $v \sim N(4, 4)$ and $v_i$ is one draw from that distribution. $S_{jt}$ is as in exercise 2.

Interstingly I had problems with negative prices, likely due to the non-convexity of the problem. However, changing initial values for the prices fro zeroes to positives helped. I set initial values to 3 and all prices are positive.

Average price difference before and after the merger (%)

\begin{array}{|c|c|c|} \hline \text{Firm} & \text{Price difference}\\ 1.0 & 1.32\\ 2.0 & 0.37\\ 3.0 & 5.96\\ 4.0 & 0.49\\ \hline \end{array}

3. Empirical merger simulation

Supply side estimation equation withouth the merger is

\begin{align*} p_{ijt} + \left( \frac{\partial s_{ijt}(p)}{\partial p_i} \right)^{-1} s_{ijt}(p) = \gamma mc_{ijt} + \omega_{ijt} \; \text{where} \; i \in \{1,2,3,4\} \end{align*}

Here the LHS is observable or a function of the demand parameters. Supply side instruments need to be correlated with the LHS of the equation and the marginal costs and independent of the $\omega_{ijt}$. Specifically, we need instruments for the marginal costs, that depend on unobserved quality. The unobserved quality is correlated with the price and demand on the LHS. Following Berry et al. (1995) cost side instruments are product characteristics that shift marginal costs but do not shift prices or demand. So some variable affecting costs that the consumers do not care about. For us this is firm's own cost shifter.

3.1 Demand estimation

3.2 Average margins and average diversion ratios between firm 1 and 3

3.2 Upward pricing pressure

The formula for the UPP is the last term of the merged firm's FOC wrt. price of good $j$, where producer of good $j$ has bought producer of good $k$. The last term is

\begin{align*} D_{jk}(p_k - mc_k) = \left( \frac{\partial s_j(p)}{\partial p_j} \right)^{-1} \frac{\partial s_k}{\partial p_j} (p_k - mc_k) \end{align*}

where $D_{jk}$ is diversion ratio from good $j$ to good $k$. This expression tells how the merger allows producer of good $j$ to increase prices as some consumers change to good $k$, which is now owned by the producer of good $j$.

3.3 Merger simulation using PyBLP

Average price difference before and after the merger with simulation, PyBLP and UPP

\begin{array}{|c|c|c|c|c|c|} \hline \text{Firm} & \text{Price diff. \% My sim.} & \text{Price diff. \% PyBLP} & \text{Price diff. \% Nevo} & \text{UPP (levels)} \\ \hline 1.0 & 1.32 & 1.32 & 2.89 & 0.08\\ 2.0 & 0.38 & 0.43 & 0.00 & 0.00\\ 3.0 & 5.96 & 5.97 & 8.96 & 0.17\\ 4.0 & 0.50 & 0.35 & 0.00 & 0.00\\ \hline \end{array}

To compare exercise section 1 to section 3, the most relevant columns to compare are the first and the second ones. It seems that the results are really close for the price changes, as price change for all goods are quite similar with some differencies. The differences likely stem from the fact that here we have empirically estimated the demand side, which resulted in slightly different estimates for the demand parameters and for the marginal costs. The third column presents the mergers price changes assuming that market share and price derivatives stay constant. Hence, it is no wonder that the values are much larger and unplausible. Increasing prices should decrease market share and for example the own price derivative should become larger in absolute value as own price increases.

UPP tell us how large efficiency gains (decreases in MC) we need to negatiate the mergers price effects.

UPP is quite close to the Proper PyBLP merger simulation. The values for UPP are around 0.01 larger than the price changes from the full merger simulation. UPP's larger estimate likely results from it ignoring equilibrium results. The price changes affect the own and cross-price elasticity of the buyer's product as these derivatives are nonlinear in price, and hence the diversion from 1 to 3 and from 3 to 1 also changes. In addition, other firms also change prices in response to the merger. Because of these considerations the results of UPP and merger simulation differ. UPP is also really similar to section 1 merger simulation.