# 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:
- Gowrisankaran, Nevo, & Town (2015):
- 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?

- Smith (2004):
- 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?

- Ivaldi & Verboven (2005) 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? (Timothy F. Bresnahan, 1982).
- 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.

### 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:
\[\begin{equation}
\gamma_1 \equiv \frac{\beta_1}{N} - \frac{\lambda}{\alpha_1}, \gamma_2 \equiv - \frac{\lambda \alpha_3}{\alpha_1}.
\end{equation}\]
### 5.2.10 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 (Nevo, 1998).
- 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.
- N. H. Miller & Weinberg (2017) 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:

**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.

**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 Farrell & Shapiro (1990), Werden & Frobe (1993), Hausman et al. (1994).
- 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:

- 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:

- 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 Small & Rosen (1981) and D. Mcfadden et al. (1995).

### 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.
- N. H. Miller & Weinberg (2017)
**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.
- Igami et al. (2018) use the detailed information of vitamin C cartel case and apply this approach.

### References

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Hausman, Leonard, & Zona. (1994). Competitive Analysis with Differenciated Products. *Annales d’Économie et de Statistique*, (34), 159–180.

Nevo, A. (2000). *Mergers with differentiated products: The case of the ready-to-eat cereal industry* (No. 3) (Vol. 31).

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Mcfadden, D., Geweke, J., Hajivassiliou, V., Johnson, R., Lancaster, T., Rossi, P., … Waters, S. (1995). *Computing Willingness-to-pay in Random Utility Models*.

Igami, M., Sugaya, T., Thank, W., Harrington, J., Asker, J., Porter, R., … Kaplow, L. (2018). *Measuring the Incentive to Collude: The Vitamin Cartels, 1990-1999*.