Using the structural approach, I estimate the effect on MMC on firm behavior in an industry in where firms are in the vertical relationship (i.e., franchisors who control their franchisees act as somewhat single firms and could reach to collusion with other franchisors).
To deal with this vertical structure in the industry, I estimate the parameter measuring vertical control (\(\lambda_v\)) as well as the one measuiring the effect of MMC (\(\lambda_m\)) with the following models:
From the results of Model 2, I find that franchsiors have some level of control on franchisees. Through this vertical control, the franchiosors could reach to collusion with other franchisors via MMC.
Model 3 is designed to test the effect of MMC given an assumption that franchisors have a complete control over franchisors (i.e., \(\lambda_v = 1\)). This would highlight the importance of empirical estimation of vertical control.
In short, I have found that the siginificant results of the vertical control and MMC (from Model 2). Without the empirical estimation of \(\lambda_v\), the effect of MMC would be upward biased (\(\lambda_m\) in Model 2 < \(\lambda_m\) in Model 3).
The indirect utility function that consumers who purchase product \(j\) in market \(t\) is
\[\begin{array} {lll} u_{ijt} & = & \alpha p_{jt} + X_{jt}\beta + \xi_{j} + \zeta_{ig} + (1-\sigma)\epsilon_{ijt} \\ & = & \delta_{jt} + \zeta_{ig} + (1-\sigma)\epsilon_{ijt} \end{array} \] where
With a dummy variable \(d_{jg}\) (it is one if $j F_g $), the utility function can be rewritten:
\[ u_{ijt} = \delta_{jt} + \sum_g[d_{jg} \zeta_{ig}] + (1-\sigma)\epsilon_{ijt} \]
The market share of product \(j\) (the probability that consumers choose product \(j\)) is
\[ s_{jt} = s_{jgt} \cdot s_{g} \] where \(s_{jg}\) represents the probability that product \(j\) is chosen given all products in \(g\), while \(s_g\) is the probability that group \(g\) is chosen.
\(s_jg\) is defined as follows:
\[ s_{jg} = \frac{\exp(\frac{\delta_j}{1-\sigma})}{\sum_{j\in F_g}\exp(\frac{\delta_j}{1-\sigma})} = \exp(\frac{\delta_j}{1-\sigma}) / D_g \] where $ D_g = _{jF_g} () $
\(s_g\) is the following:
\[ s_g = \frac{D_g^{(1-\sigma)}}{\sum_{g'}D_{g'}^{(1-\sigma)}} \]
Market share of product \(j\) is
\[ s_{j} = s_{jg} \cdot s_g = \frac{\exp(\frac{\delta_j}{1-\sigma})}{D_g^{\sigma}[\sum_{g'}D_{g'}^{(1-\sigma)}]} \]
Outside option is defined as follows: \[ s_0 = \frac{1}{\sum_{g'}D_{g'}^{(1-\sigma)}} \]
Take the log of market share of product \(j\) and within group market share of product \(j\),
\[ \begin{align} \ln(s_j) - \ln(s_0) & = \delta_j / (1-\sigma) - \sigma \ln(D_g) && Eq. 1 \\ \ln(s_{jg}) & = \delta_j / (1-\sigma) - \ln(D_g) && Eq. 2 \end{align} \]
Substracting \(\sigma \cdot Eq. 2\) from \(Eq. 1\), the equation is rewritten as follows:
\[
\ln(s_{jt}) - \ln(s_{0t}) = \alpha p_{jt} + x_{jt} \beta + \sigma\ln(s_{jg}) + \epsilon_{jt}
\]
There are three different formulat for price elasticities under the nested logit model:
\[\begin{eqnarray} \eta_{jk} & = & \frac{\partial s_{j}}{\partial p_{k}} \frac{p_{k}}{s{j}} \\ & = & [s_j \cdot \frac{\alpha}{1-\sigma}(1- \sigma s_{jg} -(1-\sigma)s_j )] \frac{p_j}{s_j} && (j = k) \\ & = &[- s_j \cdot\frac{\alpha}{1-\sigma} (\sigma s_{kg} + (1-\sigma)s_k )] \frac{p_k}{s_j} && (j \neq k; j,k \in g)] \\ & = & [-s_j \cdot \alpha \cdot s_k ] \frac{p_k}{s_j} && (j\neq k; j\in g, k\notin g ) \end{eqnarray}\]
When calculating \(\partial s / \partial p\), the estimated market shares (\(s_j, s_{jg}\)) are used, and $<0 $.
Markets are defined based on the cluster analysis. I use the density based clustering analysis with noise (DBSAN), a unsupervised machiine learning. This technique creates observations that are not sorted into any clusters, or markets, (called noises). To deal with these unsorted observations, I use the For detail of analysis, see http://rpubs.com/jhkoh17/488032.
I use the instrumental variable approach by treating the price variable as a endogenous one. The BLP styple instruments are used in the context of the nested logit model: sums of product characteristcs in the same nest, and sumes of product characteristics in other nests.
The profit function of a firm is as follows:
\[\begin{eqnarray} \Pi_{jt} & = & \underbrace{(p_{jt} - mc_{jt})s_{jt}M_{t}}_{\text{own profits}(\pi_{jt})} + \underbrace{\sum_{k\neq j}\lambda(p_{kt}-mc_{kt})s_{kt}M_t}_{\text{considering others' profits}} \end{eqnarray}\]
where \(\lambda\) is a \((J-1) \times (J-1)\) matrix measuring the level of controls. This would vary depending on assumptions of market comeptition and/or vertical relationship between franchisors and franchisees.
If firm \(j\) only considers, \(\lambda\) would be a zero matrix and \(\Pi_{jt} = \pi_{jt}\).
However, firm \(j\) considers and is influenced by other firms under the same franchisors, \(\lambda\) is non-zero matrix. This equation can be rewritten in a matrix form:
\[\begin{eqnarray} \Pi = \Lambda(P - MC)SM_t \end{eqnarray}\]
where \(\Lambda\) is a \(J \times J\) matrix with ones in its diagonal elements.
Empirical studies analyzing M&A, or multi-brand firms have assumed that the ownership matrix \(\Lambda\) consists of values of ones and zeros, allowing the joint profit maximazation. In this paper, I assume that there would be some level of joint profitmaximzation from the viewpoint of franchisors(i.e., franchisors have strong control over franchisees, and choose their prices of all their franchisees to maximize their profits). However, this would be a strong assumption. Instead, I empirically estimate the franchisors’ influence on franchisees’ pricing by estimating this matrix \(\Lambda\).
The following two cases shows two possible cases in which I can estimates the influence of franchisors on franchisees’ pricing.
For example, there are three firms in the markets (Firm 1, 2, and 3). Firms 1 and 2 are under the same franchisor, while Firm 3 is associated with another franchisor. Thus,
$$=
\begin{bmatrix} 1 & f_v(I_{ij}; _v) & f_m(mmc;_m)\
f_v(I_{ij}; _v) & 1 & f_m(mmc;_m) \
f_m(mmc;_m) & f_m(mmc;_m) & 1 \ \end{bmatrix} $$
\[ f_v(I_{ij};\lambda_v) = \lambda_v I_{ij} \\ \] where \(I_{ij} = 1\) if firms \(i\) and \(j\) are under the same franchisor. Otherwise, it is zero.
\[ f_m(mmc_{ij}; \lambda_m) = \lambda_m mmc_{ij} \] where \(mmc_{ij}^t = \frac{\sum_t I_i^{t'} \cdot I_j^{t'}}{\sum_t I_i^{t'}}\) and \(I_i^t = 1\) if firm \(i\) presents at market \(t\).
$$=
\begin{bmatrix} 1 & 1 & f_m(mmc;_m)\
1 & 1 & f_m(mmc;_m) \
f_m(mmc;_m) & f_m(mmc;_m) & 1 \ \end{bmatrix} $$
The model for marginal costs is used to create the moment conditions for GMM for the supply side, like BLP (1995). The model is as follows:
\[ mc_{jt} = w_{jt}\gamma + \omega_{jt} \] where \(mc\) is derived from the demand side model, \(w\) is a matrix of factors affecting costs.
\[\begin{eqnarray} \frac{\partial \Pi_jt}{\partial p_jt} & = & 0 \end{eqnarray}\] \[ s_{jt} + (p_{jt}-mc_{jt})\frac{\partial s_jt}{\partial p_{jt}} + [\sum_{k} \lambda_{jk} (p_{kt} - mc_{kt})\frac{\partial s_{kt}}{ \partial p_{jt}}] = 0 \] \[ s_{jt} + (p_{jt}-mc_{jt})\frac{\partial s_{jt}}{\partial p_{jt}} + [\sum_{k} \lambda^1 (p_{kt} - mc_{kt})\frac{\partial s_{kt}}{ \partial p_{jt}}] = 0 \] where \(j\) and \(k\) are under the same franchisor.
Define \(\Omega = -\Lambda^1 \cdot \partial s / \partial p\)( \(\partial s / \partial p\) is a matrix form of \(\frac{\partial s_{jt}}{\partial p_{kt}}\)) and rewrite the above equation is the marginal cost fucntion
\[ mc(\lambda) = p - \Omega^{-1}(\lambda)\cdot s() \] With the function of \(mc\), the shock for the marginal cost function:
\[ \omega = p- w\gamma - \Omega^{-1}()\cdot s() \]
With the instruments for the supply side, \(Z_s\), the moment condits are defined as \[ g(\theta)=E(Z_s\omega) = 0 \] where \(\theta = [\lambda, \gamma]\).
Using the moment conditions, I estimate two set of parameters, \(\gamma\) and \(\lambda\). \(\lambda \in(0,1)\) is similar to conduct parameters used in the literature in multi-market contact (See Ciliberto and Williams(2014))
The estimated marginal costs from the supply side is the following: \[ \hat{mc} = W \cdot \hat{\gamma} \]
Rewrite the above the equation: \[\begin{align} \omega & = mc - \hat{mc} \\ & = p - \Omega^{-1}() s() - \hat{mc} \\ \end{align}\]
The sample momoent conditions are as follows: \[\begin{align} \hat{g(\theta)} & = \sum_i^n Z_i (mc_i - W_i\hat{\gamma})/n \\ & = Z' (mc - W\gamma) / n \end{align}\] Finally, the GMM objective function is the following: \[ \arg \min_\theta \hat{g}(\theta)' A \hat{g}(\theta) \] where \(A\) is a weighting matrix.
I use the estimated asympotic variance approach to obtain the variance-covarinace matrix.
| Coeff. | Std. Err. | p | Coeff. | Std. Err. | p | Coeff. | Std. Err. | p | |
|---|---|---|---|---|---|---|---|---|---|
| Lambda_v | NA | NA | NA | 0.387 | 0.080 | 0.000 | NA | NA | NA |
| Lambda_m | NA | NA | NA | 0.187 | 0.007 | 0.000 | 0.431 | 0.000 | 0.000 |
| Constant | 30.592 | 3.055 | 0.000 | 30.585 | 13.154 | 0.020 | 30.596 | 15.844 | 0.054 |
| No. of Rooms | 0.236 | 0.027 | 0.000 | 0.199 | 0.031 | 0.000 | 0.214 | 0.032 | 0.000 |
| No.of Room Amenities | 5.303 | 0.774 | 0.000 | 5.247 | 2.104 | 0.013 | 5.325 | 2.326 | 0.022 |
| No. of Room Types | 0.074 | 1.758 | 0.966 | 0.050 | 4.633 | 0.991 | 0.084 | 3.680 | 0.982 |
| No. of Services | 1.491 | 0.834 | 0.074 | 1.419 | 1.368 | 0.300 | 1.509 | 1.313 | 0.250 |
| GMM Objective Values | 121.133 | NA | NA | 120.687 | NA | NA | 79.255 | NA | NA |
| Note: | |||||||||
| Quarter Fixed Effect Included for all three model | |||||||||
| 1 Model 1: Neither MMC nor Vertical Control Considered | |||||||||
| 2 Model 2: Vertical control and MMC are estimated | |||||||||
| 3 Model 3: Vertical Control is given as one, and MMC is estimated |