Are franchisees controlled or infleuenced by franchisors in Pricing at the franchised units?
Why do I try to answer this question?
In order to make collusion fesiable in the hotel industry through multimarket contacts, franchisors should be able to control their franchised units so the nesseary condition is satisfied: firms meet their competitiors at multiple markets. If not, franchisees would not compete with the same franchisees in different markets.
In order to make collusion fesible (any types of collusion), the number of firms in the markets should be small. If franchisors control franchisees’ pricing policies, I would be able to consider franchisors as individal firms, making the number of firms in the market smaller.
I use two different approaches:
Through reviewing prior literature and franchise information document (FID), a document in which franchiosrs publically explain their franchising contracts to potential franchisees, I have found some evidence that franchisors have some ways of controlling franchisees’ pricing policies through franchisors’ central reservation systems, revenuce management consulting services, or national/regional sales promotions (See my recent draft).
To completement this argument, I have conducted empirical exercises to show to what extent franchisors have influenced franchisees’ pricing(Focuse of this project).
Staring from the demand side model, I construct a strcutual model to estimate a conduct parameter capturing the control of franchisors over their franchisees, \(\lambda \in [0,1]\):
My estimation of \(\lambda\) is \(0.3\) at the \(1\%\) significant level, indicating that franchisors have some levels of influences on franchisees. Thus, my current work of the effects of the multimarket contact on collusion are somewhat free from the critistism on my assumption that franchisors have influence on franchisees in pricing.
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,
\[\Lambda^1 = \begin{bmatrix} 1 & \lambda^1 & 0\\ \lambda^1 & 1 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \]
This matrix can be expended if collusion among firms occurs as follows:
\[\Lambda^{12} = \begin{bmatrix} 1 & \lambda^1 & \lambda^2\\ \lambda^1 & 1 & \lambda^2 \\ \lambda^2 & \lambda^2 & 1 \\ \end{bmatrix} \] I assume \(\lambda^1\) is the same for all firms under the same franchisors, while \(\lambda^2\) is the same for all competitiors.
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.
| Dependent variable: | |
| ln(sj) -ln(s0) | |
| Price | -0.019*** |
| (0.002) | |
| No. of Activities | 0.157*** |
| (0.027) | |
| No. of Room Amenity | -0.031 |
| (0.023) | |
| No. of Services | 0.057*** |
| (0.020) | |
| Downdown | 1.008*** |
| (0.167) | |
| Airport | -0.015 |
| (0.132) | |
| ln(sjg) | 0.910*** |
| (0.028) | |
| Constant | 0.848*** |
| (0.148) | |
| Observations | 1,521 |
| R2 | 0.164 |
| Adjusted R2 | 0.160 |
| Residual Std. Error | 1.192 (df = 1513) |
| Note: | p<0.1; p<0.05; p<0.01 |
| Marginal Costs(Estimated) | |
| mc1 | |
| No. of Rooms | 0.087*** |
| (0.015) | |
| No. of Room Amenity | 1.299 |
| (1.284) | |
| No. of Room Types | -1.745 |
| (2.223) | |
| No. of Services | -0.642 |
| (0.956) | |
| Constant | 79.516*** |
| (3.366) | |
| Observations | 1,521 |
| R2 | 0.048 |
| Adjusted R2 | 0.030 |
| Residual Std. Error | 56.248 (df = 1492) |
| F Statistic | 2.669*** (df = 28; 1492) |
| Note: | p<0.1; p<0.05; p<0.01 |
| Coeff. | Std.Err. | p-value | |
|---|---|---|---|
| Lambda | 0.3000 | 0.0561 | 0.0000 |
| No. of Rooms | 0.1048 | 0.0187 | 0.0000 |
| No. of Room Amenity | 2.7579 | 0.7252 | 0.0001 |
| No. of Room Type | -0.1075 | 3.0764 | 0.4861 |
| No. of Services | -0.7974 | 1.0493 | 0.2237 |
| Constant | 32.0608 | 3.3567 | 0.0000 |
| 1 GMM Objective value = 0.0025 | |||
| 2 Fixed Effects Included: Hotel Chain Effects |
Here is the quick summary of the GMM estimation in this project for the supply side.
Demand Side Estiatmation: Used the nested logit model for the consumer utility function in this paper (A single nest design is used: standard hotel ratings).
Given the demand estimates and an assumption of the conduct parameter (\(\lambda\) = 0.5), the initial marginal costs are recovered.
With the recovered marginal costs, conduct the GMM estimation of the supply side model (Assume that the marginal cost is a function of the conduct parameter).
\[ g(\theta) = E(Z\omega) = 1/n \sum Z_i(mc_i(\lambda) - w_i \gamma) = Z'(mc(\lambda) - W\gamma)/n \]
where \(\theta = {\lambda, \gamma}\), \(Z\) is a vector of instruments, including \(X\).
\[ mc(\lambda) = p - \Lambda * \partial s/ \partial p \cdot s \]
where \(\Lambda\) is an ownership matrix: ones on the diagnoal in the martix, \(\lambda\) is one off-diagonal if firms are controlled by one franchisor.
\[ V(\hat{\theta}) = \frac{1}{N}(G'AG)^{-1} G'A\Omega A G (G'AG)^{-1} ) \] where \(G(\theta) = \partial g(\theta) / \partial \theta, \Omega = \frac{1}{n}\sum g_i()g_i()'\).
More specifically,
\[ G(\theta) = [ \partial g(\theta) / \partial \lambda, \partial g(\theta) / \partial \gamma] \]
\[ \begin{aligned} \partial g() / \partial \lambda = Z' \cdot \partial mc(\lambda) / \partial \lambda \end{aligned} \]
\[\begin{aligned} \partial mc (\lambda) / \partial \lambda & = \frac{1}{n} \Omega(\lambda)^{-2} \frac{\partial \Omega(\lambda)}{\partial \lambda} \cdot s \\ & = 1/n \left[\begin{array}{ccc} 0 & \frac{1}{\lambda^2 \partial s_1 / \partial p_2} & 0 \\ \frac{1}{\lambda^2 (\partial s_2 / \partial p_1)} & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] \cdot \left[ \begin{array}{r} s_1\\ s_2\\ s_3 \end{array} \right] \\ & = 1/n \left[ \begin{array}{c} \frac{s_2}{\lambda^2 \partial s_1 / \partial p_2} \\ \frac{s_1}{\lambda^2 \partial s_2 / \partial p_1} \\ \end{array} \right] \\ \end{aligned}\]where
\[\begin{align} \Omega(\lambda)^{-2} & = \left( \left[\begin{array}{rrr} 1 & \lambda & 0 \\ \lambda & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] * \left[\begin{array}{rrr} \partial s_1 / \partial p_1 & \partial s_1 / \partial p_2 & \partial s_1 / \partial p_3 \\ \partial s_2 / \partial p_1 & \partial s_2 / \partial p_2 & \partial s_2 / \partial p_3 \\ \partial s_3 / \partial p_1 & \partial s_3 / \partial p_2 & \partial s_3 / \partial p_3 \end{array}\right] \right) ^{-2} = 1/ \left[\begin{array}{rrr} (\partial s_1 / \partial p_1)^2 & \lambda^2 (\partial s_1 / \partial p_2)^2 & 0 \\ \lambda^2 (\partial s_2 / \partial p_1)^2 & (\partial s_2 / \partial p_2)^2 & 0 \\ 0 & 0 & (\partial s_3 / \partial p_3)^2 \end{array}\right]\\ \frac{\partial \Omega()}{\partial \lambda} & = \left[\begin{array}{rrr} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] * \left[\begin{array}{rrr} \partial s_1 / \partial p_1 & \partial s_1 / \partial p_2 & \partial s_1 / \partial p_3 \\ \partial s_2 / \partial p_1 & \partial s_2 / \partial p_2 & \partial s_2 / \partial p_3 \\ \partial s_3 / \partial p_1 & \partial s_3 / \partial p_2 & \partial s_3 / \partial p_3 \end{array}\right] = \left[ \begin{array}{rrr} 0 & \partial s_1 / \partial p_2 & 0 \\ \partial s_2 / \partial p_1 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right] \end{align}\]
where \(*\) represents the element by element multiplication.
Rewrite
\[\begin{align} \frac{\partial g(\theta)}{\partial \lambda} = & 1/n \left[ \begin{array}{cc} z_{11} & z_{21} \\ z_{12} & z_{22} \\ z_{13} & z_{23} \end{array} \right]' \cdot \left[ \begin{array}{c} \frac{s_2}{\lambda^2 \partial s_1 / \partial p_2} \\ \frac{s_1}{\lambda^2 \partial s_2 / \partial p_1} \\ 0 \end{array} \right] \end{align}\]
where \(Z\) is a \(n \times l\) matrix (Here, I assume that only two instruments included for nontational convenience).
The second term of \(G(\theta)\) is
\[\begin{align} \frac{\partial g()}{\partial \gamma} = & - \frac{1}{n} Z'W = - \frac{1}{n} \left[\begin{array}{cc} z_{11} & z_{21} \\ z_{12} & z_{22} \\ z_{13} & z_{23} \end{array} \right]' \cdot \left[ \begin{array}{r} w_{11} \\ w_{12} \\ w_{13} \end{array} \right] \end{align}\] where \(W\) is a \(n \times k\) matrix (Here, I assume that only one variable included for nontational convenience).
Finially, the derivative of \(g(\theta)\) with respect to \(\theta\) is a $ l (k+1)$ matrix: \[\begin{align} G(\theta) & = \left[ \begin{array}{cc} \partial g() / \partial \lambda & \partial g() / \partial\gamma \end{array} \right] \end{align}\]
Thus, the variance and covariance matrix is a \((k+1) \times (k+1)\) matrix:
\[ V(\hat{\theta}) = \frac{1}{N}(G'AG)^{-1} G'A\Omega A G (G'AG)^{-1} ) \] where \(\Omega = \frac{1}{n}\sum g_i()g_i()'\).