# Chapter 5 Merger Analysis

## 5.1 Motivations

• Measuring the market power of firms and predicting the possible consequence of horizontal merger cases is one of the primary goal of empirical industrial organization.
• This is important for antitrust authority to review merger cases.
• To do so, we integrate the product/cost function estimation and demand function estimation techniques.
• We introduce the last piece of parameters that characterize the market competition, conduct parameter, and discuss its identification.
• Then, we conduct the first kind of counterfactual analysis, the merger simularion.
• In this exercise, we predict the market response when the ownership structure of product is changed due to a hypothetical merger.
• Every market institution needs its own model for merger simulation:
• :
• In the U.S. hospitals and managed care organizations (MGO) negotiate the hospital prices and the coinsurance rates.
• What if hospitals are merged? How much do the hospital prices and the coinsurance rates increase?
• :
• Sometimes the same service is sold through multiple stores such as in the supermarket industry.
• How does this multi-store nature affect the merger effects?
• reviews the cases in the European Commission.

## 5.2 Identification of Conduct

### 5.2.1 Identification of Conduct

• So far we have been concerned with the two types of parameters:
• Production and/or cost function.
• Demand function.
• To identify the marginal cost by the revealed preference approach, we have assumed that firms are engaging in a price competition.
• The mode of competition is another parameter of interest.
• Can we infer the mode of competition instead of assuming it?

### 5.2.2 Marginal Revenue Function

• To be specific, consider firms producing homogeneous product.
• Under what conditions can we distinguish across Bertrand competition, Cournot competition, and collusion? .
• Consider the following marginal revenue function: $\begin{equation} MR(Q) \equiv \lambda Q P'(Q) + P(Q), \end{equation}$ where $$Q$$ is the aggregate quantity, $$P(Q)$$ is the inverse demand function.
• This formula nests Bertrand, Cournot, and collusion:
• Bertrand:
• In Bertrand, a firm cannot change the market price.
• If a firm increases the production by one unit, whose revenue increases by $$P(Q)$$.
• Therefore, $$\lambda = 0$$.
• Cournot:
• In Cournot, the marginal revenue of firm $$f$$ is: $q_f P'(Q) + P(Q).$
• Therefore, $$\lambda = s_f$$, the quantity share of the firm $$f$$.
• Collusion:
• Under collusion, firms behave like a single monopoly.
• Then, the marginal revenue is: $QP'(Q) + P(Q).$
• Therefore, $$\lambda = 1$$.
• The identification of the mode of competition in this context is equivalent to the identification of $$\lambda$$, the conduct parameter.

### 5.2.3 First-Order Condition

• Let $$MC(q_f)$$ be the marginal cost of firm $$f$$.
• Given the previous general marginal revenue function, the first-order condition for profit maximization for firm $$f$$ is written as: $\begin{equation} \lambda Q P'(Q) + P(Q) = MC(q_f). \end{equation}$
• The system of equations for $$f = 1, \cdots, F$$ determine the market equilibrium.

### 5.2.4 Linear Model

• To be simple, consider a linear inverse demand function: $\begin{equation} P^D(Q) = \frac{\alpha_0}{\alpha_1} + \frac{1}{\alpha_1}Q + \frac{\alpha_2}{\alpha_1} X + \frac{1}{\alpha_1}u^D, \end{equation}$ where $$X_t$$ is a vector of observed demand sifters.
• Consider a linear marginal cost function: $\begin{equation} MC(q_{f}) = \beta_0 + \beta_1 q_{f} + \beta_2 W + u^S, \end{equation}$ where $$W$$ is a vector of observed cost sifters.

### 5.2.5 Pricing Equation

• Inserting the inverse demand function and marginal cost function to the optimality condition: $\begin{equation} \frac{\lambda}{\alpha_1} Q + P^S(Q) = \beta_0 + \beta_1 q_f + \beta_2 W + u^S \end{equation}$
• Summing them up and dividing by the number of firms $$N$$: $\begin{equation} \frac{\lambda}{\alpha_1} Q + P^S(Q) = \beta_0 + \frac{\beta_1}{N} Q + \beta_2 W_t + u^S, \end{equation}$
• This determines the aggregate pricing equation: $\begin{equation} \begin{split} P^S(Q) &= \beta_0 + (\frac{\beta_1}{N} - \frac{\lambda}{\alpha_1})Q + \beta_2 W + u^S\\ &= \beta_0 + \gamma Q + \beta_2 W + u^S. \end{split} \end{equation}$
• The key parameter is: $\begin{equation} \gamma \equiv \frac{\beta_1}{N} - \frac{\lambda}{\alpha_1}. \end{equation}$

### 5.2.6 Identification of Inverse Demand Function and Pricing Equation

• We have two systems of reduced-form equations: $\begin{equation} \begin{split} &P^D(Q) = \frac{\alpha_0}{\alpha_1} + \frac{1}{\alpha_1}Q + \frac{\alpha_2}{\alpha_1} X + \frac{1}{\alpha_1}u^D,\\ &P^S(Q) = \beta_0 + \gamma Q + \beta_2 W + u^S. \end{split} \end{equation}$
• If we observe a demand shifter $$X$$, then it can be used as an instrument for $$Q$$ in the pricing equation to identify the parameters in the pricing equation.
• If we observe a cost shifter $$W_t$$, then it can be used as an instrument for $$Q$$ in the inverse demand function to identify the parameters in the pricing equation.
• Thus, we can identify the reduced-form parameters $$(\alpha_0, \alpha_1, \alpha_2)$$ and $$(\beta_0, \gamma, \beta_2)$$ if we observe a demand shifter $$X$$ and a cost shifter $$W$$.
• However, this is not enough to separately identify the structural-form parameters $$\beta_1$$ and $$\lambda$$ in $$\gamma$$.

### 5.2.7 The Conduct Parameter is Unidentified

• Even if the demand function and pricing equation (supply function) are identified, we still cannot identify the conduct parameter $$\lambda$$.
• The price at a quantity may be high either because of the high marginal cost or because of the high markup.
• Remember that the identification of $$\gamma$$ and $$\alpha_1$$ do not determine the value of $$\lambda$$ and $$\beta_1$$ in: $\begin{equation} \gamma = \frac{\beta_1}{N} - \frac{\lambda}{\alpha_1}. \end{equation}$
• $$\beta_1$$ is the derivative of the marginal cost. Figure 5.1: Figure 6.2 of Davis (2006)

### 5.2.8 Identification of the Conduct Parameter: When Cost Data is Available

• If there is reliable cost data, we can directly identify the marginal cost function: $\begin{equation} MC(q_f) = \beta_0 + \beta_1 q_f + \beta_2 W + u^S, \end{equation}$ and so $$\beta_1$$.
• Then, in a combination with the identification of inverse demand function and pricing equation, $$\lambda$$ is identified as: $\begin{equation} \lambda = \alpha_1 \left(\frac{\beta_1}{N} - \gamma\right). \end{equation}$

### 5.2.9 Identification of the Conduct Parameter: When Cost Data is Not Available

• Remember the first-order condition: $\begin{equation} \lambda Q P^{D\prime}(Q) + P^S(Q) = MC(q_f), \end{equation}$ where we can identify $$P^D(Q)$$ and $$P^S(Q)$$ if we have demand and cost shifters.
• It is clear from this expression that we need a variation in $$P^{D\prime}(Q)$$ with a fixed $$Q$$ to identify $$\lambda$$, i.e., something that rotates the inverse demand function.
• Intuition: If demand becomes more elastic, prices will decrease and quantity will increase in a market with a high degree of market power.

### 5.2.10 Identification of the Conduct Parameter: Demand Rotater is Available

• Let’s formalize the idea.

• To identify the conduct parameter, we needed a demand rotater: $\begin{equation} P^D(Q) = \frac{\alpha_0}{\alpha_1} + \frac{1}{\alpha_1}Q + \frac{\alpha_2}{\alpha_1} X + \frac{\alpha_3}{\alpha_1} Q \underbrace{Z}_{\text{demand rotater}} + \frac{1}{\alpha_1}u^D. \end{equation}$

• Inserting this into the first-order condition yields: $\begin{equation} \frac{\lambda}{\alpha} Q_t (1 + \alpha Z_t)+ P^S(Q) = \beta_0 + \frac{\beta_1}{N} Q + \beta_2 W + u^s, \end{equation}$

• This determines the pricing equation: $\begin{equation} \begin{split} P^S(Q) &= \beta_0 - \frac{\lambda}{\alpha_1} Q(1 + \alpha_3 Z_t) + \frac{\beta_1}{N} Q_t + \beta_2 W + u^S\\ &\equiv \beta_0 + \gamma_1 Q + \gamma_2 Z Q + \beta_2 W + u^S, \end{split} \end{equation}$ where: $\begin{equation} \gamma_1 \equiv \frac{\beta_1}{N} - \frac{\lambda}{\alpha_1}, \gamma_2 \equiv - \frac{\lambda \alpha_3}{\alpha_1}. \end{equation}$ ### Identification of the Conduct Parameters: Demand Rotater is Available

• If we have cost shifters $$W$$, it can be used as instruments for $$Q$$ in the inverse demand function to identify the demand parameters $$(\alpha_0, \alpha_1, \alpha_2, \alpha_3)$$.

• If we have demand shifters $$X$$, it can be used as instruments for $$Q$$ in the pricing equation to identify the supply parameters $$(\beta_0, \gamma_1, \gamma_2, \beta_2)$$.

• Now we can identify the reduced-form parameters $$\alpha_1, \alpha_3$$ and $$\gamma_2$$.

• Then, we can not identify the conduct parameter: $\begin{equation} \lambda = - \frac{\gamma_2 \alpha_1}{\alpha_3}. \end{equation}$

### 5.2.11 Identification of Conduct in Differentiated Product Market

• Consider the identification of conduct when there are two differentiated substitutable products and companies compete in price .

• Are the prices determined independently or jointly?

• The general first-order condition is: $\begin{equation} \begin{split} &(p_1 - c_1) \frac{\partial Q_1(p)}{\partial p_1} + Q_1^S(p) + \Delta_{12}(p_2 - c_2) \frac{\partial Q_2 (p)}{\partial p_1} = 0\\ &\Delta_{21}(p_1 - c_1)\frac{\partial Q_1(p)}{\partial p_2} + Q_2^S(p) + (p_2 - c_2) \frac{\partial Q_2 (p)}{\partial p_2} = 0, \end{split} \end{equation}$ where

• $$\Delta_{12} = \Delta_{21} = 0$$ if prices are determined independently.

• $$\Delta_{12} = \Delta_{21} = 1$$ if price are determined jointly.

• Under what conditions can we identify $$\Delta_{12}$$ and $$\Delta_{21}$$?

• We will have to rotate $$\frac{\partial Q_1(p)}{\partial p_2}$$ and $$\frac{\partial Q_2 (p)}{\partial p_1}$$ while keeping the other variables at the same values.

• infers $$\Delta$$ after the MillerCoors, a joint venture of SABMiller PLC and Molson Coors Brewing, is formed.

• The unobserved year-specific and region-specific cost shocks are identified from the outsiders and the unobserved product-specific cost shocks are assumed to be the same before and after the merger.

## 5.3 Merger Simulation

### 5.3.1 Unilateral and Coordinated Effects of a Horizontal Merger

• There are two effects associated with a merge episode:
1. Unilateral effect:
• The new merged firm usually have a unilateral incentive to raise prices above their pre-merger level.
• This unilateral effect may lead to the other firms to have an unilateral incentive to raise price, and the reaction continues to reach the new equilibrium.
• The latter chain reaction is sometimes called the multi-lateral effect.
• In economic theory, it is the price change when the same mode of competition (say, the Bertrand-Nash equilibrium) is played.
• $$\Delta$$ is either 0 or 1: firms internalize the profits from owned product but do not internalize the other products.
2. Coordinated effect:
• After the merger, the mode of competition may change.
• For example, the tacit collusion becomes easier and it can happen.
• This effect, caused by the change in the mode of competition, is called the coordinated effect of a merger.
• In this case, the conduct parameters $$\Delta$$ may take positive values for products owned by the rival firms.

### 5.3.2 Merger Simulation

• In merger simulation, we compute the equilibrium under different ownership structure.
• This amounts to run a counterfactual simulation hypothetically changing the conduct parameter $$\Delta$$.
• The idea stems back to , , .
• Before running the simulation, you have to be very careful about the model assumptions:
• In music industry, firms compete not in price but in advertisement.
• If technological diffusion is important, firms may set dynamic pricing.
• This is the first counterfactual analysis we study in this lecture.
• The results are valid only if the modeling assumptions are correct.

### 5.3.3 Quantifying the Unilateral Effect

• See Nevo (2000) and Nevo (2001).
• There are $$J$$ products, $$\mathcal{J} = \{1, \cdots, J\}$$.
• We can have multiple markets but we suppress the market index.
• Firm $$f$$ produces a set of products which we denote $$\mathcal{J}_f \subset \mathcal{J}$$.
• Let $$mc_j$$ be the constant marginal cost of producing good $$j$$.
• Thus assuming a separable cost function.
• We can relax this assumption for the estimation.
• However, once you admit that the costs can be non-separable, you will start to wonder what happens to the cost function if two firms merged and started to produce the two products that were previously produced by separate firms.
• Let $$D_j(p)$$ be the demand for product $$j$$ when the price vector is $$p$$.
• The problem for firm $$f$$ given the price of other firms $$p_{-f}$$ is: $\begin{equation} \max_{p_f} \sum_{j \in \mathcal{J}_f} \Pi_j(p_f, p_{-f}) = \sum_{j \in \mathcal{J}_f} (p_j - mc_j) D_j(p_f, p_{-f}), \end{equation}$ where $$p_f = \{p_j\}_{j \in \mathcal{J}_f}$$, $$p_{-f} = \{p_j\}_{j \in \mathcal{J} \setminus \mathcal{J}_f}$$.

### 5.3.4 Pre-Merger Equilibrium

• The first-order condition for firm $$f$$ is: $\begin{equation} D_k(p) + \sum_{j \in \mathcal{J}_f} (p_j - mc_j) \frac{\partial D_j(p)}{\partial p_k} = 0, \forall k \in \mathcal{J}_f. \end{equation}$
• Let $$\Delta_{jk}^{pre}$$ takes 1 if same firm produces $$j$$ and $$k$$ and 0 otherwise before the merger.
• The first-order condition can be written as: $\begin{equation} D_k(p) + \sum_{j \in \mathcal{J}} \Delta_{jk}^{pre}(p_j - mc_j) \frac{\partial D_j(p)}{\partial p_k} = 0, \forall k \in \mathcal{J}_f. \end{equation}$
• In terms of the product share: $\begin{equation} s_k(p) + \sum_{j \in \mathcal{J}} \Delta_{jk}^{pre}(p_j - mc_j) \frac{\partial s_j(p)}{\partial p_k} = 0, \forall k \in \mathcal{J}_f. \end{equation}$
• Let $$\Delta^{pre}$$ be a $$J \times J$$ matrix whose $$(j, k)$$-element is $$\Delta_{jk}$$.
• At the end of the day, performing a merger simulation is to recompute the equilibrium with different ownership structure encoded in $$\Delta$$.

### 5.3.5 Post-Merger Equilibrium

• Let $$\Omega^{pre}(p)$$ is a matrix whose $$(j, k)$$-element is:

$\begin{equation} - \frac{\partial s_{j}(p)}{\partial p_k} \Delta_{jk}^{pre}. \end{equation}$

• Then, by the first-order condition, the marginal cost should be: $\begin{equation} mc = p - \Omega^{pre}(p)^{-1} s(p). \end{equation}$

• If the ownership structure $$\Delta^{pre}$$ is changed to $$\Delta^{post}$$, and $$\Omega^{pre}$$ changed to $$\Omega^{post}$$, the post-merger price is determined by solving the non-linear equation: $\begin{equation} p^{post} = mc + \Omega^{post}(p^{post})^{-1}s(p^{post}). \end{equation}$

• The post-merger share is given by: $\begin{equation} s^{post} = s(p^{post}). \end{equation}$

### 5.3.6 Consumer Surplus

• Suppose that the demand function is based on the mixed-logit model such that the indirect utility is: $\begin{equation} u_{ijt} = x_{jt} \beta_i + \alpha_i p_{jt} + \xi_{j} + \xi_t + \Delta \xi_{jt} + \epsilon_{ijt}, \end{equation}$ with $$\epsilon_{ijt}$$ is drawn from i.i.d. Type-I extreme value distribution and the consumer-level heterogeneity:

$\begin{equation} \begin{pmatrix} \alpha_i \\ \beta_i \end{pmatrix} = \begin{pmatrix} \alpha\\ \beta \end{pmatrix} + \Pi z_i + \Sigma \nu_i, \nu_i \sim N(0, I_{K + 1}). \end{equation}$

• Then, the compensated variation due to the price change for consumer $$i$$ is: $\begin{equation} CV_{it} = \frac{\ln (\sum_{j = 0}^J \exp(V_{ijt}^{post}) ) - \ln (\sum_{j = 0}^J \exp(V_{ijt}^{pre})) }{\alpha_i}, \end{equation}$ where $$V_{ijt}^{post}$$ and $$V_{ijt}^{pre}$$ are indirect utility for consumer $$i$$ to purchase good $$j$$ at the prices after and before the merger.
• This formula holds only if the price enters linearly in the indirect utility (no income effect).
• For general case, see and .

### 5.3.7 Quantifying the Coordinated Effect

• The repeated-game theory suggests that a collusion is sustainable if and only if it it incentive compatible: the collusion profit is no less than the deviation profit for each member of the collusion.
• The theory provides a check list that affects the incentive compatibility such as the market share, cost asymmetry, and demand dynamics.
• But it is often hard to judge the coordinated effects from these qualitative information, because mergers simultaneously change many factors and the factors may encourage or hinder collusion.
• retrospectively studies the coordinated effect of a merger.
• Is prospective analysis of coordinated effects possible as well as the analysis of unilateral effects?
• If we can identify the demand and cost functions, we can calculate the collusion profits and deviation profits.
• If we specify the collusion strategy, we can write down the incentive compatibility.
• We can check how the incentive compatibility change when a hypothetical merger happens.
• The problem is the identification of conduct: to identify the cost function, we need to know the conduct.
• Thus, the stated strategy will work only if we have a data during which we are sure that there was no collusion, or there was a particular type of collusion.
• use the detailed information of vitamin C cartel case and apply this approach.