A structual approach is used to test the following hypothesis: Franchisors have high levels of control over franchisees in metropolitan cities?
I empirically estimate the level of control of franchisors over franchisees by using the demand and supply estimation.
The indirect utility function that consumers who purchase product \(j\) in market \(t\) is
\[\begin{array} {lll} u_{ijt} & = & \alpha p_{jt} + X_{jt}\beta + \xi_{j} + \zeta_{ig} + (1-\sigma)\epsilon_{ijt} \\ & = & \delta_{jt} + \zeta_{ig} + (1-\sigma)\epsilon_{ijt} \end{array} \] - \(g=0, 1,\cdots, G\) represents a group in which a product is associated. - \(F_g\) is a set of products in group \(g\). - \(\zeta\) is common for all products in group \(g\) - \(\epsilon\) is IID with the extreme value distribution.
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 \[ D_g = \sum_{j\in F_g} \exp(\frac{\delta_j}{1-\sigma}) \]
\(s_g\) 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). For detail of analysis, see http://rpubs.com/jhkoh17/488032.
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.
Studies of mergers and product differentiation assumes the
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.
The nested logit demand estimation
# Nested Logit Model ------------------------------------------------------
# y.fld
load(file="df_hou4.Rdata")
load(file="blp_mmc.Rdata")
y.fld <- "y2"
# # prc.fld
end.fld <- "adr"
# x.fld
x.fld <- c( "n_todo", "n_room_amenity", "n_service", "cbd", "air", "log(sjg2)")
# iv.fld
iv.fld <- colnames(blp)
iv.fld <- iv.fld[c(3,4,7,8,11,12, 13,14, 15, 16)]
# nest.var.fld
nest.var.fld <- c("log(sjg2)")
ols.eq <- paste0(y.fld, " ~ ", end.fld, " + ", paste(x.fld, collapse = " + "))
ols.eq## [1] "y2 ~ adr + n_todo + n_room_amenity + n_service + cbd + air + log(sjg2)"iv.eq <- paste0(ols.eq, " | ", paste(x.fld, collapse = " + "), " + ", paste(iv.fld, collapse=" + "))
iv.eq## [1] "y2 ~ adr + n_todo + n_room_amenity + n_service + cbd + air + log(sjg2) | n_todo + n_room_amenity + n_service + cbd + air + log(sjg2) + sum_same_n_todo + sum_other_n_todo + sum_same_n_service + sum_other_n_service + sum_same_n_room_amenity + sum_other_n_room_amenity + sum_same_cbd + sum_other_cbd + sum_same_air + sum_other_air"nl.est <- ivreg(iv.eq, data=df_hou4)
summary(nl.est)##
## Call:
## ivreg(formula = iv.eq, data = df_hou4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -4.08924 -0.47677 0.04444 0.56038 4.51659
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 0.649766 0.136789 4.750 2.23e-06 ***
## adr -0.015212 0.001501 -10.134 < 2e-16 ***
## n_todo 0.114472 0.023562 4.858 1.31e-06 ***
## n_room_amenity -0.032589 0.020432 -1.595 0.110920
## n_service 0.060663 0.017722 3.423 0.000636 ***
## cbd 1.194139 0.182347 6.549 7.93e-11 ***
## air -0.087959 0.140836 -0.625 0.532360
## log(sjg2) 0.919510 0.024590 37.394 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.062 on 1513 degrees of freedom
## Multiple R-Squared: 0.3366, Adjusted R-squared: 0.3335
## Wald test: 228.5 on 7 and 1513 DF, p-value: < 2.2e-16chain.dummy.fld <- names(df_hou4)[grep("chain", names(df_hou4))][7:30]
w.fld <- c("const", "room", "s_rating", "n_todo", "n_room_amenity", chain.dummy.fld)
w.fld <- c("const", "room", "class2", "n_todo", "n_room_amenity", "factor(brand)")
w.fld <- c("const", "room", "n_todo", "n_room_type", "n_room_amenity", "n_service", "factor(brand)")
w.fld <- c("const", "room", "n_todo", "n_room_type", "n_room_amenity", "n_service", "factor(chain)")
w.fld <- c("const", "room", "n_todo", "n_room_type", "n_room_amenity", "n_service", "factor(chain)")
w.fld <- c("const", "n_todo", "n_room_amenity", "n_service", "cbd", "air", "n.same.brand", "n.same.chain")
w.fld <- c("const", "room", "n_todo", "n_room_amenity", "n_service", "cbd", "air", "factor(chain)")
w.fld <- c("const", "s_rating", "n_todo", "n_room_amenity", "n_service", "cbd", "air", "factor(brand)")
w.fld <- c("const", "room", "s_rating", "n_todo", "n_room_amenity", "n_service", "cbd", "air", "factor(brand)")
# The below specification is the best specification
w.fld <- c("const", "room", "n_room_amenity", "n_room_type", "n_service", chain.dummy.fld)
ols.supply <- paste0("mc1 ~ ", paste(w.fld[-1], collapse = " + "))
ols.supply## [1] "mc1 ~ room + n_room_amenity + n_room_type + n_service + chain1 + chain2 + chain3 + chain4 + chain5 + chain7 + chain11 + chain12 + chain13 + chain14 + chain15 + chain16 + chain17 + chain18 + chain19 + chain22 + chain23 + chain24 + chain26 + chain27 + chain28 + chain29 + chain30 + chain40"est.supply <- lm(ols.supply, data =df_hou4)
summary(est.supply)##
## Call:
## lm(formula = ols.supply, data = df_hou4)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2203.84 -8.48 13.71 37.07 348.68
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -0.24398 7.46599 -0.033 0.97394
## room 0.19881 0.03278 6.066 1.66e-09 ***
## n_room_amenity 1.53320 2.84700 0.539 0.59029
## n_room_type -0.67819 4.93005 -0.138 0.89061
## n_service -1.79414 2.12062 -0.846 0.39766
## chain1 102.36169 12.02989 8.509 < 2e-16 ***
## chain2 106.26326 16.05974 6.617 5.10e-11 ***
## chain3 83.82054 13.47421 6.221 6.41e-10 ***
## chain4 26.97229 14.94558 1.805 0.07132 .
## chain5 60.05010 13.46375 4.460 8.80e-06 ***
## chain7 10.51704 30.25752 0.348 0.72820
## chain11 15.95243 22.51450 0.709 0.47872
## chain12 68.67512 22.95114 2.992 0.00281 **
## chain13 6.25523 44.78437 0.140 0.88894
## chain14 35.75558 45.26771 0.790 0.42973
## chain15 59.62543 37.14829 1.605 0.10869
## chain16 101.93693 22.11358 4.610 4.38e-06 ***
## chain17 40.16555 18.82108 2.134 0.03300 *
## chain18 192.20576 63.67495 3.019 0.00258 **
## chain19 380.33937 125.12916 3.040 0.00241 **
## chain22 5.40600 23.39219 0.231 0.81727
## chain23 46.70469 17.84744 2.617 0.00896 **
## chain24 15.87384 21.85492 0.726 0.46775
## chain26 93.62352 45.81261 2.044 0.04117 *
## chain27 15.74198 29.35814 0.536 0.59190
## chain28 53.35809 65.11541 0.819 0.41267
## chain29 -5.00702 44.74111 -0.112 0.91091
## chain30 12.11475 37.32690 0.325 0.74556
## chain40 -7.12241 15.17018 -0.470 0.63878
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 124.7 on 1492 degrees of freedom
## Multiple R-squared: 0.1691, Adjusted R-squared: 0.1535
## F-statistic: 10.84 on 28 and 1492 DF, p-value: < 2.2e-16est.data <- nl.est
prc.fld <- "adr"
nest.var.fld <- "log(sjg2)"
nest.fld <- "class2"
mkt.id.fld <- "mkt.id"
brand.id.fld <- "id.brand2"
lambda <- 1
dat <- df_hou4
moments <- function(lambda, est.data, prc.fld, nest.var.fld, nest.fld, mkt.id.fld,
brand.id.fld, w.fld, iv.fld, mc0,
dat, ...){
# GMM momment conditiosn
# Parameters to be estimated: lambda and gamma
# data prepration
est <- est.data$coefficient # coefficients from demeand side
mkt.id <- as.matrix(dat[ , mkt.id.fld]) #
brand.id <- as.matrix(dat[, brand.id.fld]) # brand.id
w <- as.matrix(ones=rep(1, nrow(dat)), dat[w.fld]) # RHS variable in MC: MC = w*gamma + omega
iv <- as.matrix(dat[iv.fld]) # instruments
z <- cbind(w, iv) # Z = {w, iv}
#
brand.id[brand.id==1] <- (max(brand.id)+1):(max(brand.id)+sum(brand.id==1))
price <- as.matrix(dat[, prc.fld])
nest <- as.matrix(dat[, nest.fld])
beta <- as.matrix(est[!names(est) %in% nest.var.fld]) # coefficients of price and product attributes
alpha <- beta[prc.fld,] # price coefficients
sigma <- est[nest.var.fld] # nest coefficient
# dsdp
dsdp <- dshare.dp(est.data = est.data, prc.fld=prc.fld,
nest.var.fld=nest.var.fld,
nest.fld=nest.fld, mkt.id.fld=mkt.id.fld,
brand.id.fld=brand.id.fld,
lambda = lambda, dat = dat)
sj.dat <- sj.fn(est = est.data, prc.fld=prc.fld, nest.var.fld=nest.var.fld,
nest.fld=nest.fld, mkt.id.fld=mkt.id.fld, dat=dat)
print("share calculated")
sj <- as.matrix(sj.dat[,1])
sjg <- as.matrix(sj.dat[,2])
# initial value of gamma
gamma0 <- solve(t(w) %*% w) %*% t(w) %*% mc0
theta0 <- cbind(lambda, t(gamma0))
# updated weighting matrix
omega.hat <- mc0 - w%*%gamma0
z.hat <- z * matrix(rep(omega.hat, ncol(z)), ncol = ncol(z))
gmm_obj <- function(theta){
lambda <- theta[1]
gamma <- theta[-1]
mc <- mc(est.data = est.data, prc.fld = prc.fld, nest.var.fld = nest.var.fld,
nest.fld = nest.fld, mkt.id.fld = mkt.id.fld, brand.id.fld = brand.id.fld,
lambda = lambda, dat = dat)
# Projection matrix with solve(t(Z)'Z) (2sls)
PZ <- z %*% solve(t(z) %*% z) %*% t(z)
# updating weighting matrix
omega.hat <- mc - w %*% gamma
z.hat <- z * matrix(rep(omega.hat, ncol(z)), ncol = ncol(z))
a.inv <- try(solve(t(z.hat) %*% z.hat), silent =FALSE)
if("matrix" == class(a.inv)){
print("GMM step 2 updated")
PZ <<- z %*% a.inv %*% t(z);
PX.inv <- solve(t(w) %*% PZ %*% w)
# theta1 <<- PX.inv %*% t(X) %*% PZ %*% delta
# xi.hat <<- delta - X %*% theta1
# X.hat <- (PZ %*% X) * matrix(rep(xi.hat, K), ncol = K)
# tsls.se <- sqrt(diag(PX.inv %*% t(X.hat) %*% X.hat %*% PX.inv))
# print("GMM step 2 updated theta1 estimate:")
# print(beta.est <<- list ( linear = data.frame(beta.est = theta1, beta.se = tsls.se),nonlinear = theta2))
}
dat[, "omega.hat"] <<- omega.hat
f <- t(omega.hat) %*% PZ %*% omega.hat
print("Updated GMM objective:")
print(f <- as.numeric(f))
return(f)
}
# # 2sls for gamma
# str.ivreg.y <- "mc ~ "
# str.ivreg.x <- paste(w.fld, collapse = " + ")
# str.ivreg.iv <- paste(iv.fld, collapse = " + ")
# print("2SLS specification:")
# print(fm.ivreg <- paste0(str.ivreg.y, str.ivreg.x , " | ", str.ivreg.x, " + ", str.ivreg.iv))
# rm(str.ivreg.y, str.ivreg.x, str.ivreg.prc, str.ivreg.iv)
# mo.ivreg <- ivreg(fm.ivreg, data = dat, x = TRUE))
# stop condition
tol <- 1e-8
max.it <- 1000
# lower and upper bounds
lb <- c(0, rep(-Inf, ncol(z)))
up <- c(1, rep(Inf, ncol(z)))
# optimization
theta.est <- neldermead(x0 = theta0, fn = gmm_obj, lower=NULL, upper=NULL,
control = list(xtol_rel = tol, maxeval = max.it))
theta <- theta.est$par
gmm.obj.value <- gmm_obj(theta)/nrow(dat)
return(list(theta=theta, gmm.obj.value = gmm.obj.value))
}std.err <- function(lambda, est.data, prc.fld, nest.var.fld, nest.fld, mkt.id.fld,
brand.id.fld, w.fld, iv.fld, mc0,
dat, ...){
# Data Prep
est <- est.data$coefficient # coefficients from demeand side
mkt.id <- as.matrix(dat[ , mkt.id.fld]) #
brand.id <- as.matrix(dat[, brand.id.fld]) # brand.id
w <- as.matrix(ones=rep(1, nrow(dat)), dat[w.fld]) # RHS variable in MC: MC = w*gamma + omega
iv <- as.matrix(dat[iv.fld]) # instruments
z <- cbind(w, iv) # Z = {w, iv}
#
brand.id[brand.id==1] <- (max(brand.id)+1):(max(brand.id)+sum(brand.id==1))
price <- as.matrix(dat[, prc.fld])
nest <- as.matrix(dat[, nest.fld])
beta <- as.matrix(est[!names(est) %in% nest.var.fld]) # coefficients of price and product attributes
alpha <- beta[prc.fld,] # price coefficients
sigma <- est[nest.var.fld] # nest coefficient
# dsdp
# dshare.dp with lambda == 1
dsdp <- dshare.dp(est.data = est.data, prc.fld=prc.fld,
nest.var.fld=nest.var.fld,
nest.fld=nest.fld, mkt.id.fld=mkt.id.fld,
brand.id.fld=brand.id.fld,
lambda = 1, dat = dat)
}## var coefficient
## 1 Lambda 0.305436988717096
## 2 Constant 82.6197969408459
## 3 Room -0.00132904231676757
## 4 n_room_amenity -1.55471989587518
## 5 n_room_type 0.249417985125057
## 6 n_service 1.35556165353253
## 7 chain1 -7.11104458685857
## 8 chain2 -14.0892094958736
## 9 chain3 3.9706692686823
## 10 chain4 -0.952853103676081
## 11 chain5 -4.38379640225017
## 12 chain7 -7.42942718257215
## 13 chain11 -19.2502799115492
## 14 chain12 12.5379067508233
## 15 chain13 -13.9757292841679
## 16 chain14 1.025200487434
## 17 chain15 7.54807320683419
## 18 chain16 -2.36948737590514
## 19 chain17 -1.4170547342752
## 20 chain18 -13.4261110148514
## 21 chain19 -76.260601221126
## 22 chain22 9.53312450790367
## 23 chain23 0.397425969270333
## 24 chain24 -15.6698956189974
## 25 chain26 66.4201490102843
## 26 chain27 -1.24061882743597
## 27 chain28 18.5406147809286
## 28 chain29 7.3896709141192
## 29 chain30 -17.7907512630858
## 30 chain40 8.67948024079653