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simulations.R
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1230 lines (899 loc) · 37.7 KB
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##################################################
## Filename: simulations.R
## Project: Vertical Sequential Model
## Author: mtp
## Date: 8.21.2023
## Task: Model Simulations
##################################################
## This script runs in about 17 minutes on a laptop with an AMD Ryzen 5 7640U
## processor and 16 GB of RAM.
rm(list=ls()) # clear workspace
library(BB)
library(rootSolve)
library(Matrix)
library(ggplot2)
library(beepr)
library(numDeriv) # for jacobian() function
library(kableExtra)
library(dplyr) # this is needed in conjunction with kable
options(knitr.kable.NA = '-')
##################################################
#### Load necessary functions
##################################################
source("my-functions.R")
##################################################
#### Define market primitives
##################################################
# structural parameters
alpha <- 1.2
R <- 2
W <- 2
delta <- c(3.5, 3.5, 3.5, 3.5)
c_R_vec <- matrix(.20, nrow = (R*W), ncol = 1)
c_W_vec <- matrix(.20, nrow = (R*W), ncol = 1)
lambda <- 0.5
J <- length(c_R_vec)
# ownership
own_down_pre <- paste0("R",rep(c(1,2),each=2))
own_up_pre <- paste0("W",rep(c(1,2),2))
own_down_pre
own_up_pre
own_up_post <- own_up_pre
own_down_post <- own_down_pre
own_down_post[own_down_post == "R1"] <- "W1"
##################################################
#### 1. Baseline Model Figure
##################################################
x_vals <- c(0,0.2,0.4,0.6,0.8,1.0)
numsims <- length(x_vals)
CS_pre <- as.vector(rep(0,numsims))
CS_post <- as.vector(rep(0,numsims))
p_R_pre <- matrix(data = 0, nrow = J, ncol = numsims)
p_R_post <- matrix(data = 0, nrow = J, ncol = numsims)
shares_pre <- matrix(data = 0, nrow = J, ncol = numsims)
shares_post <- matrix(data = 0, nrow = J, ncol = numsims)
convflag_pre <- as.vector(rep(0,numsims))
convflag_post <- as.vector(rep(0,numsims))
profits_pre <- as.vector(rep(0,numsims))
profits_post <- as.vector(rep(0,numsims))
p_W_pre <- matrix(data = 0, nrow = J, ncol = numsims)
p_W_post <- matrix(data = 0, nrow = J, ncol = numsims)
useBBoptim <- FALSE
for (num in 1:numsims) {
print("Scenario")
print(num)
sigma_val <- x_vals[num]
## Pre-merger
error <- rep(1,J)
tol <- 0.01
p_W0 <- c_W_vec*2
p_R0 <- as.numeric((c_R_vec+c_W_vec)*3)
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x]
if (useBBoptim == FALSE) {
outtest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
maxJointProfits = FALSE,
lower = 0, upper = 2.0)
pj_test <- outtest$minimum
}
if (useBBoptim == TRUE) {
outtest <- BBoptim(par = w_start, fn = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE)
pj_test <- outtest$par
}
error[x] <- abs(pj_test - p_W0[x])
print(error)
p_W0[x] <- pj_test
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha,
delta = delta, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
p_W1 <- p_W0
p_R1 <- p_R0
CS_pre[num] <- (1/alpha) * log(1 + sum(exp(delta - alpha * p_R1)) )
p_R_pre[,num] <- p_R1
p_W_pre[,num] <- p_W1
shares_pre[,num] <- (exp(delta - alpha*p_R1))/(1+sum(exp(delta - alpha*p_R1)))
convflag_pre[num] <- outtest_r$convergence
profits_pre[num] <- (p_R1[1] - p_W1[1] - c_R_vec[1,]) * shares_pre[,num][1] +
(p_R1[2] - p_W1[2] - c_R_vec[2,]) * shares_pre[,num][1] +
(p_W1[1] - c_W_vec[1]) * shares_pre[,num][1] +
(p_W1[3] - c_W_vec[3]) * shares_pre[,num][3]
## Post-merger
error <- rep(1,J)
tol <- 0.01
p_W0 <- p_W1
p_R0 <- p_R1
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x]
if (useBBoptim == FALSE) {
outtest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
maxJointProfits = FALSE,
lower = 0, upper = 2.0)
pj_test <- outtest$minimum
}
if (useBBoptim == TRUE) {
outtest <- BBoptim(par = w_start, fn = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE)
pj_test <- outtest$par
}
error[x] <- abs(pj_test - p_W0[x])
print(error)
p_W0[x] <- pj_test
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha,
delta = delta, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
## save values from the merger sim
p_W1_post <- p_W0
p_R1_post <- p_R0
CS_post[num] <- (1/alpha) * log(1 + sum(exp(delta - alpha * p_R1_post)) )
p_R_post[,num] <- p_R1_post
p_W_post[,num] <- p_W1_post
shares_post[,num] <- (exp(delta - alpha*p_R1_post))/(1+sum(exp(delta - alpha*p_R1_post)))
convflag_post[num] <- outtest_r$convergence
profits_post[num] <- (p_R1_post[1] - p_W1_post[1] - c_R_vec[1,]) * shares_post[,num][1] +
(p_R1_post[2] - p_W1_post[2] - c_R_vec[2,]) * shares_post[,num][1] +
(p_W1_post[1] - c_W_vec[1]) * shares_post[,num][1] +
(p_W1_post[3] - c_W_vec[3]) * shares_post[,num][3]
}
## Create the figure
df_fig1 <- data.frame(sigma = x_vals )
df_fig1$cs_pre <- CS_pre
df_fig1$cs_post <- CS_post
df_fig1$pct_change_CS <- (df_fig1$cs_post-df_fig1$cs_pre)/df_fig1$cs_pre*100
shares_pre; shares_post; df_fig1
plot1 <- ggplot(data=df_fig1, aes(x=sigma, y=pct_change_CS)) +
geom_point() +
xlab("Sigma") + ylab("Percent CS change")
plot1
# check convergence flags
convflag_pre
convflag_post
# check if mergers are profitable
profits_pre
profits_post
is_profitable <- (profits_post > profits_pre)
is_profitable
#### Output table
df_tab1 <- data.frame(sigma = rep(x_vals, each = J),
good = rep((1:J), times = numsims) )
df_tab1$p_R_pre <- as.vector( round(p_R_pre, 1) )
df_tab1$p_R_post <- as.vector( round(p_R_post, 1) )
df_tab1$p_W_pre <- as.vector( round(p_W_pre, 1) )
df_tab1$p_W_post <- as.vector( round(p_W_post, 1) )
## VI good in output table:
df_tab1$p_W_post[c(1,5,9,13,17,21)] <- NA
df_tab1$shares_pre <- as.vector( round(shares_pre*100,1) )
df_tab1$shares_post <- as.vector( round(shares_post*100,1) )
df_tab1$pct_change_CS <- rep(round(df_fig1$pct_change_CS, 1), each = J)
df_tab1$pct_change_p_R <- as.vector( round((p_R_post-p_R_pre)/p_R_pre*100, 1) )
##################################################
#### 2. Proportion integrated figure
##################################################
x_vals <- c(c(1.5,2.5,3.5,4.5,5.5),
c(1.5,2.5,3.5,4.5,5.5))
numsims <- length(x_vals)
sigma_values <- rep( c(0,1), each = length(x_vals)/2)
CS_pre <- as.vector(rep(0,numsims))
CS_post <- as.vector(rep(0,numsims))
p_R_pre <- matrix(data = 0, nrow = J, ncol = numsims)
p_R_post <- matrix(data = 0, nrow = J, ncol = numsims)
p_W_pre <- matrix(data = 0, nrow = J, ncol = numsims)
p_W_post <- matrix(data = 0, nrow = J, ncol = numsims)
shares_pre <- matrix(data = 0, nrow = J, ncol = numsims)
shares_post <- matrix(data = 0, nrow = J, ncol = numsims)
profits_pre <- as.vector(rep(0,numsims))
profits_post <- as.vector(rep(0,numsims))
for (num in 1:numsims) {
print("Scenario")
print(num)
delta_val <- delta
delta1 <- x_vals[num]
delta_val[1] <- delta1
sigma_val <- sigma_values[num]
## Pre-merger
error <- rep(1,J)
tol <- 0.01
p_W0 <- c_W_vec*4
p_R0 <- as.numeric((c_R_vec+c_W_vec)*6)
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x]
outtest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha, delta = delta_val,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
lower = 0, upper = 2.0)
pj_test <- outtest$minimum
error[x] <- abs(pj_test - p_W0[x])
print(error)
p_W0[x] <- pj_test
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha,
delta = delta_val, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
p_W1 <- p_W0
p_R1 <- p_R0
CS_pre[num] <- (1/alpha) * log(1 + sum(exp(delta_val - alpha * p_R1)) )
p_R_pre[,num] <- p_R1
p_W_pre[,num] <- p_W1
shares_pre[,num] <- (exp(delta_val - alpha*p_R1))/(1+sum(exp(delta_val - alpha*p_R1)))
profits_pre[num] <- (p_R1[1] - p_W1[1] - c_R_vec[1,]) * shares_pre[,num][1] +
(p_R1[2] - p_W1[2] - c_R_vec[2,]) * shares_pre[,num][1] +
(p_W1[1] - c_W_vec[1]) * shares_pre[,num][1] +
(p_W1[3] - c_W_vec[3]) * shares_pre[,num][3]
## Post-merger
error <- rep(1,J)
tol <- 0.01
p_W0 <- p_W1
p_R0 <- p_R1
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x]
outtest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha, delta = delta_val,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
lower = 0, upper = 2.0)
pj_test <- outtest$minimum
error[x] <- abs(pj_test - p_W0[x])
print(error)
p_W0[x] <- pj_test
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha,
delta = delta_val, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
## save values from the merger sim
p_W1_post <- p_W0
p_R1_post <- p_R0
CS_post[num] <- (1/alpha) * log(1 + sum(exp(delta_val - alpha * p_R1_post)) )
p_R_post[,num] <- p_R1_post
p_W_post[,num] <- p_W1_post
shares_post[,num] <- (exp(delta_val - alpha*p_R1_post))/(1+sum(exp(delta_val - alpha*p_R1_post)))
profits_post[num] <- (p_R1_post[1] - p_W1_post[1] - c_R_vec[1,]) * shares_post[,num][1] +
(p_R1_post[2] - p_W1_post[2] - c_R_vec[2,]) * shares_post[,num][1] +
(p_W1_post[1] - c_W_vec[1]) * shares_post[,num][1] +
(p_W1_post[3] - c_W_vec[3]) * shares_post[,num][3]
}
#### Create the figure
df_fig2 <- data.frame(int_pre_share = (shares_pre[1,]*100) )
df_fig2$cs_pre <- CS_pre
df_fig2$cs_post <- CS_post
df_fig2$sigma <- rep(c("Linear pricing","Two-part tariff"), each = numsims/2)
df_fig2$pct_change_CS <- (df_fig2$cs_post-df_fig2$cs_pre)/df_fig2$cs_pre*100
plot2 <- ggplot(data=df_fig2,
aes(x=int_pre_share, y=pct_change_CS, group=sigma, color=sigma)) +
geom_point() +
xlab("Pre-Merger Integrated Good Share") + ylab("Percent CS change") +
scale_color_brewer(palette="Dark2") +
theme(legend.position="bottom")
plot2
# check if mergers are profitable
profits_pre
profits_post
is_profitable <- (profits_post > profits_pre)
is_profitable
#### Output table
df_tab2 <- data.frame(sigma = rep(sigma_values, each = J),
good = rep((1:J), times = numsims) )
df_tab2$p_R_pre <- as.vector( round(p_R_pre, 1) )
df_tab2$p_R_post <- as.vector( round(p_R_post, 1) )
df_tab2$p_W_pre <- as.vector( round(p_W_pre, 1) )
df_tab2$p_W_post <- as.vector( round(p_W_post, 1) )
## VI good in output table:
df_tab2$p_W_post[((1:10) * 4 - 3)] <- NA
df_tab2$shares_pre <- as.vector( round(shares_pre*100,1) )
df_tab2$shares_post <- as.vector( round(shares_post*100,1) )
df_tab2$pct_change_CS <- rep(round(df_fig2$pct_change_CS, 1), each = J)
df_tab2$pct_change_p_R <- as.vector( round((p_R_post-p_R_pre)/p_R_pre*100, 1) )
##################################################
#### 4. Size of outside option
##################################################
# x_vals is all delta's, to scale down oo
x_vals <- c(c(2.5,3.5,4.5,5.5,6.0)*0.9,
c(1.5,2.5,3.5,4.5,5.5) )
numsims <- length(x_vals)
sigma_values <- rep( c(0,1), each = length(x_vals)/2)
#### Simulations
CS_pre <- as.vector(rep(0,numsims))
CS_post <- as.vector(rep(0,numsims))
p_R_pre <- matrix(data = 0, nrow = J, ncol = numsims)
p_R_post <- matrix(data = 0, nrow = J, ncol = numsims)
p_W_pre <- matrix(data = 0, nrow = J, ncol = numsims)
p_W_post <- matrix(data = 0, nrow = J, ncol = numsims)
shares_pre <- matrix(data = 0, nrow = J, ncol = numsims)
shares_post <- matrix(data = 0, nrow = J, ncol = numsims)
profits_pre <- as.vector(rep(0,numsims))
profits_post <- as.vector(rep(0,numsims))
for (num in 1:numsims) {
print("Scenario")
print(num)
delta_val <- rep(x_vals[num], J)
sigma_val <- sigma_values[num]
## Pre-merger
error <- rep(1,J)
tol <- 0.01
p_W0 <- c_W_vec*2
p_R0 <- as.numeric((c_R_vec+c_W_vec)*3)
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x]
outtest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha, delta = delta_val,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
lower = 0, upper = 2.0)
pj_test <- outtest$minimum
error[x] <- abs(pj_test - p_W0[x])
print(error)
p_W0[x] <- pj_test
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha,
delta = delta_val, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
p_W1 <- p_W0
p_R1 <- p_R0
CS_pre[num] <- (1/alpha) * log(1 + sum(exp(delta_val - alpha * p_R1)) )
p_R_pre[,num] <- p_R1
p_W_pre[,num] <- p_W1
shares_pre[,num] <- (exp(delta_val - alpha*p_R1))/(1+sum(exp(delta_val - alpha*p_R1)))
profits_pre[num] <- (p_R1[1] - p_W1[1] - c_R_vec[1,]) * shares_pre[,num][1] +
(p_R1[2] - p_W1[2] - c_R_vec[2,]) * shares_pre[,num][1] +
(p_W1[1] - c_W_vec[1]) * shares_pre[,num][1] +
(p_W1[3] - c_W_vec[3]) * shares_pre[,num][3]
## Post-merger
error <- rep(1,J)
tol <- 0.01
p_W0 <- p_W1
p_R0 <- p_R1
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x]
outtest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha, delta = delta_val,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
lower = 0, upper = 2.0)
pj_test <- outtest$minimum
error[x] <- abs(pj_test - p_W0[x])
print(error)
p_W0[x] <- pj_test
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha,
delta = delta_val, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
## save values from the merger sim
p_W1_post <- p_W0
p_R1_post <- p_R0
CS_post[num] <- (1/alpha) * log(1 + sum(exp(delta_val - alpha * p_R1_post)) )
p_R_post[,num] <- p_R1_post
p_W_post[,num] <- p_W1_post
shares_post[,num] <- (exp(delta_val - alpha*p_R1_post))/(1+sum(exp(delta_val - alpha*p_R1_post)))
profits_post[num] <- (p_R1_post[1] - p_W1_post[1] - c_R_vec[1,]) * shares_post[,num][1] +
(p_R1_post[2] - p_W1_post[2] - c_R_vec[2,]) * shares_post[,num][1] +
(p_W1_post[1] - c_W_vec[1]) * shares_post[,num][1] +
(p_W1_post[3] - c_W_vec[3]) * shares_post[,num][3]
}
#### Create the figure
df_fig4 <- data.frame(oo_share = ((1-colSums(shares_pre))*100) )
df_fig4$cs_pre <- CS_pre
df_fig4$cs_post <- CS_post
df_fig4$sigma <- rep(c("Linear pricing","Two-part tariff"), each = numsims/2)
df_fig4$pct_change_CS <- (df_fig4$cs_post-df_fig4$cs_pre)/df_fig4$cs_pre*100
plot4 <- ggplot(data=df_fig4,
aes(x=oo_share, y=pct_change_CS, group=sigma, color=sigma)) +
geom_point() +
xlab("Pre-Merger Outside Option Share") + ylab("Percent CS change") +
scale_color_brewer(palette="Dark2") +
theme(legend.position="bottom")
plot4
# check if mergers are profitable
profits_pre
profits_post
is_profitable <- (profits_post > profits_pre)
is_profitable
#### Output table
df_tab4 <- data.frame(sigma = rep(sigma_values, each = J),
good = rep((1:J), times = numsims) )
df_tab4$p_R_pre <- as.vector( round(p_R_pre, 1) )
df_tab4$p_R_post <- as.vector( round(p_R_post, 1) )
df_tab4$p_W_pre <- as.vector( round(p_W_pre, 1) )
df_tab4$p_W_post <- as.vector( round(p_W_post, 1) )
## VI good in output table:
df_tab4$p_W_post[((1:10) * 4 - 3)] <- NA
df_tab4$shares_pre <- as.vector( round(shares_pre*100,1) )
df_tab4$shares_post <- as.vector( round(shares_post*100,1) )
df_tab4$pct_change_CS <- rep(round(df_fig4$pct_change_CS, 1), each = J)
df_tab4$pct_change_p_R <- as.vector( round((p_R_post-p_R_pre)/p_R_pre*100, 1) )
##################################################
#### 5. Comparison to simultaneous equilibrium (S&T)
##################################################
c_R_vec <- as.numeric(c_R_vec)
c_W_vec <- as.numeric(c_W_vec)
## 5.1 and 5.2.
## Here, loop through x_vals (lambda barg weights) twice times Once for sigma=0, then
## for sigma=1. Then separately for simultaneous code.
x_vals1 <- rep(c(0.30,.40,0.50,0.60,.70), 2)
x_vals2 <- c(0.30,.40,0.50,0.60,.70)
numsims1 <- length(x_vals1)
numsims2 <- length(x_vals2)
sigma_values <- rep( c(0,1), each = length(x_vals1)/2)
CS_pre <- as.vector(rep(0,numsims1))
CS_post <- as.vector(rep(0,numsims1))
p_R_pre <- matrix(data = 0, nrow = J, ncol = numsims1)
p_R_post <- matrix(data = 0, nrow = J, ncol = numsims1)
p_W_pre <- matrix(data = 0, nrow = J, ncol = numsims1)
p_W_post <- matrix(data = 0, nrow = J, ncol = numsims1)
shares_pre <- matrix(data = 0, nrow = J, ncol = numsims1)
shares_post <- matrix(data = 0, nrow = J, ncol = numsims1)
convflag_up_pre <- matrix(data = 0, nrow = J, ncol = numsims1)
convflag_up_post <- matrix(data = 0, nrow = J, ncol = numsims1)
convflag_down_pre <- matrix(data = 0, nrow = J, ncol = numsims1)
convflag_down_post <- matrix(data = 0, nrow = J, ncol = numsims1)
profits_pre <- as.vector(rep(0,numsims1))
profits_post <- as.vector(rep(0,numsims1))
for (num in 1:numsims1) {
print("Scenario")
print(num)
print("Pre")
lambda <- x_vals1[num]
sigma_val <- sigma_values[num]
## Pre-merger
error <- rep(1,J)
tol <- 0.003
eps <- rep(0, J)
p_W0 <- c_W_vec*5.0
p_R0 <- as.numeric((c_R_vec+c_W_vec)*5.5)
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x] - .001 # add epsilon
checktest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
lower = 0, upper = 1.05)
pj_test <- checktest$minimum
error[x] <- abs(pj_test - p_W0[x])
print(max(error))
p_W0[x] <- pj_test
if (convflag_up_pre[x] > 0){
eps[x] <- .0001
}
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha,
delta = delta, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
p_W1 <- p_W0
p_R1 <- p_R0
CS_pre[num] <- (1/alpha) * log(1 + sum(exp(delta - alpha * p_R1)) )
p_R_pre[,num] <- p_R1
p_W_pre[,num] <- p_W1
shares_pre[,num] <- (exp(delta - alpha*p_R1))/(1+sum(exp(delta - alpha*p_R1)))
profits_pre[num] <- (p_R1[1] - p_W1[1] - c_R_vec[1]) * shares_pre[,num][1] +
(p_R1[2] - p_W1[2] - c_R_vec[2]) * shares_pre[,num][1] +
(p_W1[1] - c_W_vec[1]) * shares_pre[,num][1] +
(p_W1[3] - c_W_vec[3]) * shares_pre[,num][3]
## Post-merger
print("Post")
error <- rep(1,J)
tol <- 0.005
eps <- rep(0, J)
p_W0 <- c_W_vec*5
p_R0 <- as.numeric((c_R_vec+c_W_vec)*6.5)
iter <- 1
while (max(error) > tol) {
for (x in 1:J) {
w_start <- p_W0[x] - .01
checktest <- optimize(f = Barg_NP_seq3_vert,
product_max = x, p_W = p_W0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R0 = p_R0, sigma = sigma_val, showAll = FALSE,
lower = 0, upper = 1.05)
pj_test <- checktest$minimum
error[x] <- abs(pj_test - p_W0[x])
print(error)
p_W0[x] <- pj_test
if (convflag_up_post[x] > 0){
eps[x] <- -.01
}
# recover p_R at these p_W and update r_R0
outtest_r <- BBoptim(f = Bert_foc_vert, par = p_R0,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha,
delta = delta, c_R = c_R_vec,
p_W = p_W0, c_W = c_W_vec, sumFOC = TRUE,
control = list(trace=FALSE),
quiet = TRUE)
p_R0 <- outtest_r$par
iter <- iter + 1
}
}
## save values from the merger sim
p_W1_post <- p_W0
p_R1_post <- p_R0
CS_post[num] <- (1/alpha) * log(1 + sum(exp(delta - alpha * p_R1_post)) )
p_R_post[,num] <- p_R1_post
p_W_post[,num] <- p_W1_post
shares_post[,num] <- (exp(delta - alpha*p_R1_post))/(1+sum(exp(delta - alpha*p_R1_post)))
profits_post[num] <- (p_R1_post[1] - p_W1_post[1] - c_R_vec[1]) * shares_post[,num][1] +
(p_R1_post[2] - p_W1_post[2] - c_R_vec[2]) * shares_post[,num][1] +
(p_W1_post[1] - c_W_vec[1]) * shares_post[,num][1] +
(p_W1_post[3] - c_W_vec[3]) * shares_post[,num][3]
}
#### 5.3 Simultaneous Model (code copied from simulations-vertical-mergers.Rmd)
CS_pre_C <- as.vector(rep(0,numsims2))
CS_post_C <- as.vector(rep(0,numsims2))
profits_pre_C <- as.vector(rep(0,numsims2))
profits_post_C <- as.vector(rep(0,numsims2))
p_R_pre_C <- matrix(data = 0, nrow = J, ncol = numsims2)
p_R_post_C <- matrix(data = 0, nrow = J, ncol = numsims2)
p_W_pre_C <- matrix(data = 0, nrow = J, ncol = numsims2)
p_W_post_C <- matrix(data = 0, nrow = J, ncol = numsims2)
shares_pre_C <- matrix(data = 0, nrow = J, ncol = numsims2)
shares_post_C <- matrix(data = 0, nrow = J, ncol = numsims2)
for (num in 1:numsims2) {
lambda <- x_vals2[num]
tol <- .0001
error <- 1
p_W0 <- matrix(.25, nrow = (R*W), ncol = 1) # dummy wholesale prices
p_r_start <- as.numeric((c_R_vec+c_W_vec)*6)
p_R0 <- p_r_start
while (error > tol) {
out1 <- BBoptim(f = Bert_foc ,par = p_R0,
own_down = own_down_pre, alpha= alpha,
delta = delta, cost = c_R_vec,
p_W = p_W0, sumFOC = TRUE)
p_R1 <- out1$par
out2 <- BBoptim(par = as.numeric(p_W0), fn = Barg_foc,
own_down = own_down_pre, own_up = own_up_pre,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R = p_R1)
p_W1 <- out2$par
error <- max(abs(c(p_W1-p_W0,p_R1-p_R0)))
print(error)
p_W0 <- p_W1
p_R0 <- p_R1
}
print(p_R1)
print(p_W1)
shares_pre1 <- (exp(delta - alpha*p_R1))/(1+sum(exp(delta - alpha*p_R1)))
as.numeric(shares_pre1)
sum(shares_pre1)
tol <- .0001
error <- 1
p_W0_post <- matrix(.25, nrow = (R*W), ncol = 1) # dummy wholesale prices
p_R0_post <- p_r_start
while (error > tol) {
out1 <- BBoptim(f = Bert_foc_vert, par = p_R0_post,
own_down = own_down_post, own_up = own_up_post,
alpha = alpha, delta = delta, c_R = c_R_vec,
p_W = p_W0_post, c_W = c_W_vec, sumFOC = TRUE)
p_R1_post <- out1$par
out2 <- BBoptim(par = as.numeric(p_W0_post), fn = Barg_foc_vert,
own_down = own_down_post, own_up = own_up_post,
alpha= alpha, delta = delta,
c_W = c_W_vec, c_R = c_R_vec, lambda = lambda,
p_R = p_R1_post)
p_W1_post <- out2$par
error <- max(abs(c(p_W1_post-p_W0_post,p_R1_post-p_R0_post)))
print(error)
p_W0_post <- p_W1_post
p_R0_post <- p_R1_post
}
p_R1
p_W1
p_R1_post
p_W1_post
# retail price change with vertical integration
(p_R1_post-p_R1)/p_R1
shares_post1 <- (exp(delta - alpha*p_R1_post))/(1+sum(exp(delta - alpha*p_R1_post)))
CS_pre_C[num] <- (1/alpha) * log(1 + sum(exp(delta - alpha * p_R1)) )
CS_post_C[num] <- (1/alpha) * log(1 + sum(exp(delta - alpha * p_R1_post)) )
shares_pre_C[,num] <- shares_pre1
shares_post_C[,num] <- shares_post1
profits_pre_C[num] <- (p_R1[1] - p_W1[1] - c_R_vec[1]) * shares_pre1[1] +
(p_R1[2] - p_W1[2] - c_R_vec[2]) * shares_pre1[1] +
(p_W1[1] - c_W_vec[1]) * shares_pre1[1] +
(p_W1[3] - c_W_vec[3]) * shares_pre1[3]
profits_post_C[num] <- (p_R1_post[1] - p_W1_post[1] - c_R_vec[1]) * shares_post1[1] +
(p_R1_post[2] - p_W1_post[2] - c_R_vec[2]) * shares_post1[1] +
(p_W1_post[1] - c_W_vec[1]) * shares_post1[1] +
(p_W1_post[3] - c_W_vec[3]) * shares_post1[3]
p_R_pre_C[,num] <- p_R1
p_R_post_C[,num] <- p_R1_post
p_W_pre_C[,num] <- p_W1
p_W_post_C[,num] <- p_W1_post
}
# check if mergers are profitable
is_profitable <- (profits_post > profits_pre)
is_profitable
is_profitable_C <- (profits_post_C > profits_pre_C)
is_profitable_C
#### Create the figure
df_fig5 <- data.frame(lambda = x_vals1)
df_fig5$cs_pre <- CS_pre
df_fig5$cs_post <- CS_post
df_fig5$model <- rep(c("sequential linear pricing","sequential two part tariff"), each = numsims1/2)
df_fig5_C <- data.frame(lambda = x_vals2)
df_fig5_C$cs_pre <- CS_pre_C
df_fig5_C$cs_post <- CS_post_C
df_fig5_C$model <- "simultaneous"
df_fig5 <- rbind.data.frame(df_fig5,df_fig5_C)
df_fig5$pct_change_CS <- (df_fig5$cs_post-df_fig5$cs_pre)/df_fig5$cs_pre*100
plot5 <- ggplot(data=df_fig5,
aes(x=lambda, y=pct_change_CS, group=model, color=model)) +
geom_point() +
xlab("Retailer bargaining power") + ylab("Percent CS change") +
scale_color_brewer(palette="Dark2") +
theme(legend.position="bottom")
plot5
#### Table showing underlying pre and post CS
df_fig5$cs_pre <- round(df_fig5$cs_pre, 2)
df_fig5$cs_post <- round(df_fig5$cs_post, 2)
df_fig5$pct_change_CS <- round(df_fig5$pct_change_CS, 1)