REPRO-Bench

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Environment
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Import libraries
	library(tidyverse)
	library(magrittr)
	library(janitor)
	library(rio)
	library(rlist)
	library(boot)
	library(broom)
	library(purrr)
	library(parallel)
	library(ebal)
	library(rootSolve)
	library(stargazer)
	library(haven)
	library(latex2exp)
	library(estimatr)

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Constants
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Define basic constants for identification
	num_days <- 20
	hourly_rate <- 50
	max_hours <- 3

	param_initial <-
	list(
	delta = 0.997,
	p_B = (hourly_rate / num_days) / 60
	)

	# Define constants for balance
	us_adults <- 255200373
	avg_income <- 43.01
	avg_college <- 0.3009
	avg_male <- 0.4867
	avg_white <- 0.73581
	avg_age <- 47.6
	avg_usage <- 186

	# Define common random seed for bootstrap replicability
	size <- 2000
	seed <- 5
	RNGkind("L'Ecuyer-CMRG")
	set.seed(seed)

	# Define colors for plots
	maroon <- '#94343c'
	grey <- '#848484'
	skyblue <- '#87CEEB'
	black <- '#000000'
	deepskyblue <- '#B0C4DE'

	# Define base changes for parameters to be ported to tex
	# Note : tex names should not have symbols (_ or - or ,) and should not have numbers
	constants <- c('delta', 'p_B')
	name_changes <- list("omega" = "omegause",
	"omega_est" = "omegahat",
	"tau_L" = "tauL",
	"avg_use" = "avguse",
	"rho" = "rhohat",
	"lambda" = "lambdahat",
	"rho_res" = "rhoreshat",
	"lambda_res" = "lambdareshat",
	"delta" = "deltahat",
	"p_B" = "pB",
	"F_B" = "FB",
	"tau_B_1" = "tauBone",
	"tau_B_2" = "tauBtwo",
	"tau_B_3" = "tauBthree",
	"tau_B_4" = "tauBfour",
	"tau_B_5" = "tauBfive",
	"tau_tilde_B_3_2" = "tautildeBthreetwo",
	"tau_tilde_B_2_2" = "tautildeBtwotwo",
	"tau_tilde_L_3_2" = "tautildeLthreetwo",
	"tau_tilde_L_2_2" = "tautildeLtwotwo",
	"tau_L_2" = "tauLtwo",
	"tau_L_y" = "tauLy",
	"tau_B_y" = "tauBy",
	"tau_B_2_full" = "tauBtwofull",
	"tau_L_3" = "tauLthree",
	"mispredict" = "mispredict",
	"x_tilde" = "xtilde",
	"x_ss" = "xss",
	"limit_effect_last_week" = "limiteffectlastweek",
	"d_L" = "dL",
	"d_CL" = "dCL",
	"tau_tilde_B" = "tautildeB",
	"tau_tilde_L" = "tautildeL",
	"x_tilde_2_B" = "xtildetwoB",
	"v_B" = "vB",
	"v_L" = "vL",
	"attritionrate" = "attritionrate",
	"true_tau_tilde_L" = "truetautildeL",
	"MPL_S2" = "MPLStwo",
	"number_people_opted_out" = "numberpeopleoptedout",
	"percent_opted_out" = "percentoptedout",
	"gamma_spec_omega_hour" = "gammaspecomegahour",
	"alpha" = "alphahat",
	"alpha_res" = "alphares",
	"eta" = "etahat",
	"zeta" = "zetahat",
	"eta_hour" = "etahour",
	"zeta_hour" = "zetahour",
	"eta_res" = "etares",
	"zeta_res" = "zetares",
	"eta_res_hour" = "etareshour",
	"zeta_res_hour" = "zetareshour",
	"tau_B_2_desired" = "tauBtwodesired",
	"lambdarho" = "lambdarho",
	"lambdarhosquared" = "lambdarhosquared",
	"tauBthreehour" = "tauBthreehour",
	"pcttaubtwo" = "pcttaubtwo",
	"naivete" = "naivete",
	"gamma_L_effect" = "gammaLeffect",
	"gamma_tilde_L_effect" = "gammatildeLeffect",
	"gamma_L" = "gammaL",
	"gamma_tilde_L" = "gammatildeL",
	"gamma_B" = "gammaB",
	"gamma_tilde_B" = "gammatildeB",
	"gamma" = "gammacap",
	"gamma_tilde" = "gammatilde",
	"naivete_hour" = "naivetehour",
	"gamma_hour" = "gammahour",
	"gamma_tilde_B_hour" = "gammatildeBhour",
	"gamma_tilde_L_hour" = "gammatildeLhour",
	"gamma_tilde_hour" = "gammatildehour",
	"gamma_L_hour" = "gammaLhour",
	"gamma_B_hour" = "gammaBhour",
	"gamma_L_effect_omega_hour" = "gammaLeffectomegahour",
	"gamma_L_omega_hour" = "gammaLomegahour",
	"underestimatetemp" = "underestimatetemp",
	"attritionrate" = "attritionrate",
	"gamma_tilde_L_effect_omega" = "gammatildeLeffectomega",
	"gamma_L_effect_omega" = "gammaLeffectomega",
	"gamma_tilde_L_omega" = "gammatildeLomega",
	"gamma_L_omega" = "gammaLomega",
	"gamma_tilde_L_multiple" = "gammatildeLmultiple",
	"gamma_L_multiple" = "gammaLmultiple",
	"gamma_tilde_B_multiple" = "gammatildeBmultiple",
	"gamma_B_multiple" = "gammaBmultiple",
	"gamma_tilde_L_multiple_hour" = "gammatildeLmultiplehour",
	"gamma_L_multiple_hour" = "gammaLmultiplehour",
	"gamma_tilde_B_multiple_hour" = "gammatildeBmultiplehour",
	"gamma_B_multiple_hour" = "gammaBmultiplehour",
	"naivete_res" = "naiveteres",
	"underestimatetemp_res" = "underestimatetempres",
	"gamma_L_effect_res" = "gammaLeffectres",
	"gamma_tilde_L_effect_res" = "gammatildeLeffectres",
	"gamma_L_res" = "gammaLres",
	"gamma_B_res" = "gammaBres",
	"naivete_res_hour" = "naivetereshour",
	"gamma_tilde_L_effect_res_hour" = "gammatildeLeffectreshour",
	"gamma_L_res_hour" = "gammaLreshour",
	"gamma_B_res_hour" = "gammaBreshour",
	"gamma_L_effect_omega_res_hour" = "gammaLeffectomegareshour",
	"gamma_L_omega_res_hour" = "gammaLomegareshour",
	"gamma_tilde_L_effect_omega_res" = "gammatildeLeffectomegares",
	"gamma_L_effect_omega_res" = "gammaLeffectomegares",
	"gamma_L_omega_res" = "gammaLomegares",
	"gamma_L_multiple_res" = "gammaLmultipleres",
	"gamma_B_multiple_res" = "gammaBmultipleres",
	"gamma_L_effect_multiple_res" = "gammaLeffectmultipleres",
	"gamma_tilde_L_effect_multiple_res" = "gammatildeLeffectmultipleres",
	"gamma_L_multiple_res_hour" = "gammaLmultiplereshour",
	"gamma_B_multiple_res_hour" = "gammaBmultiplereshour",
	"gamma_L_effect_multiple_res_hour" = "gammaLeffectmultiplereshour",
	"gamma_tilde_L_effect_multiple_res_hour" = "gammatildeLeffectmultiplereshour",
	"x_ss_L_effect" = "xssLeffect",
	"x_ss_L_effect_eta_high" = "xssLeffectetahigh",
	"x_ss_L_effect_eta_low" = "xssLeffectetalow",
	"x_ss_B" = "xssB",
	"delta_x_cap" = "deltaxcap",
	"delta_x_cap_us" = "deltaxcapus",
	"delta_x_tilde_L" = "deltaxtildeL",
	"delta_x_tilde_L_us" = "deltaxtildeLus",
	"delta_x_tilde_B" = "deltaxtildeB",
	"delta_x_tilde_B_us" = "deltaxtildeBus",
	"x_ss_L" = "xssL",
	"mispredicthour" = "mispredicthour",
	"delta_x_L_multiple" = "deltaxLmultiple",
	"delta_B_multiple" = "deltaBmultiple",
	"delta_x_cap_omega" = "deltaxcapomega",
	"delta_x_L_omega" = "deltaxLomega",
	"x_ss_L_effect_res" = "xssLeffectres",
	"x_ss_L_effect_eta_high_res" = "xssLeffectetahighres",
	"x_ss_L_effect_eta_low_res" = "xssLeffectetalowres",
	"x_ss_B_res" = "xssBres",
	"delta_x_cap_res" = "deltaxcapres",
	"delta_x_cap_us_res" = "deltaxcapusres",
	"delta_x_tilde_L_res" = "deltaxtildeLres",
	"delta_x_tilde_L_us_res" = "deltaxtildeLusres",
	"delta_x_tilde_B_res" = "deltaxtildeBres",
	"delta_x_tilde_B_us_res" = "deltaxtildeBusres",
	"x_ss_L_res" = "xssLres",
	"delta_x_cap_multiple_res" = "deltaxcapmultipleres",
	"delta_x_L_multiple_res" = "deltaxLmultipleres",
	"delta_B_multiple_res" = "deltaBmultipleres",
	"delta_x_cap_omega_res" = "deltaxcapomegares",
	"delta_x_L_omega_res" = "deltaxLomegares",
	"delta_x_temptation" = "deltaxtemptation",
	"delta_x_naivete" = "deltaxnaivete",
	"delta_x_habit" = "deltaxhabit",
	"x_ss_temptation" = "xsstemptation",
	"x_ss_temptation_eta_high" = "xsstemptationetahigh",
	"x_ss_temptation_eta_low" = "xsstemptationetalow",
	"x_ss_naivete" = "xssnaivete",
	"x_ss_naivete_eta_high" = "xssnaiveteetahigh",
	"x_ss_naivete_eta_low" = "xssnaiveteetalow",
	"x_ss_habit" = "xsshabit",
	"x_ss_habit_eta_high" = "xsshabitetahigh",
	"x_ss_habit_eta_low" = "xsshabitetalow",
	"x_ss_habit_est" = "xsshabitest",
	"x_ss_habit_temptation" = "xsshabittemptation",
	"x_ss_habit_temptation_eta_high" = "xsshabittemptationetahigh",
	"x_ss_habit_temptation_eta_low" = "xsshabittemptationetalow",
	"x_ss_zero_temp" = "xsszerotemp",
	"delta_x_temptation_res" = "deltaxtemptationres",
	"delta_x_naivete_res" = "deltaxnaiveteres",
	"delta_x_habit_res" = "deltaxhabitres",
	"x_ss_temptation_res" = "xsstemptationres",
	"x_ss_temptation_eta_high_res" = "xsstemptationetahighres",
	"x_ss_temptation_eta_low_res" = "xsstemptationetalowres",
	"x_ss_naivete_res" = "xssnaiveteres",
	"x_ss_naivete_eta_high_res" = "xssnaiveteetahighres",
	"x_ss_naivete_eta_low_res" = "xssnaiveteetalowres",
	"x_ss_habit_res" = "xsshabitres",
	"x_ss_habit_eta_high_res" = "xsshabitetahighres",
	"x_ss_habit_eta_low_res" = "xsshabitetalowres",
	"x_ss_habit_est_res" = "xsshabitestres",
	"x_ss_habit_temptation_res" = "xsshabittemptationres",
	"x_ss_habit_temptation_eta_high_res" = "xsshabittemptationetahighres",
	"x_ss_habit_temptation_eta_low_res" = "xsshabittemptationetalowres",
	"x_ss_i_data" = "xssidata",
	"x_ss_spec" = "xssSpec",
	"tau_L_2_signed" = "tauLtwosignednotnice",
	"intercept_spec" = "interceptspec",
	"intercept_spec_hour" = "interceptspechour",
	"gamma_tilde_spec" = "gammatildespec",
	"gamma_spec" = "gammaspec",
	"delta_x_spec" = "deltaxspec",
	"gamma_tilde_spec_hour" = "gammatildespechour",
	"gamma_spec_hour" = "gammaspechour",
	"intercept_het_L_effect" = "intercepthetLeffect",
	"intercept_het_L_effect_eta_high" = "intercepthetLeffectetahigh",
	"intercept_het_L_effect_eta_low" = "intercepthetLeffectetalow",
	"intercept_het_B" = "intercepthetB",
	"intercept_het_L" = "intercepthetL",
	"intercept_het_B_hour" = "intercepthetBhour",
	"intercept_het_L_hour" = "intercepthetLhour",
	"intercept_het_L_effect_hour" = "intercepthetLeffecthour",
	"gamma_spec_omega_hour" = "gammaspecomegahour",
	"intercept_spec_omega_hour" = "interceptspecomegahour",
	"intercept_het_L_effect_omega_hour" = "intercepthetLeffectomegahour",
	"intercept_het_L_omega_hour" = "intercepthetLomegahour",
	"intercept_het_B_multiple" = "intercepthetBmultiple",
	"intercept_het_L_multiple" = "intercepthetLmultiple",
	"intercept_het_L_effect_omega" = "intercepthetLeffectomega",
	"intercept_het_L_omega" = "intercepthetLomega",
	"x_ss_spec_res" = "xssSpecres",
	"intercept_spec_res" = "interceptspecres",
	"intercept_spec_res_hour" = "interceptspecreshour",
	"gamma_tilde_spec_res" = "gammatildespecres",
	"gamma_spec_res" = "gammaspecwres",
	"delta_x_spec_res" = "deltaxspecres",
	"gamma_tilde_spec_res_hour" = "gammatildespecreshour",
	"gamma_spec_res_hour" = "gammaspechourres",
	"intercept_het_L_effect_res" = "intercepthetLeffectres",
	"intercept_het_L_effect_eta_high_res" = "intercepthetLeffectetahighres",
	"intercept_het_L_effect_eta_low_res" = "intercepthetLeffectetalowres",
	"intercept_het_B_res" = "intercepthetBres",
	"intercept_het_L_res" = "intercepthetLres",
	"intercept_het_B_res_hour" = "intercepthetBreshour",
	"intercept_het_L_res_hour" = "intercepthetLreshour",
	"intercept_het_L_effect_res_hour" = "intercepthetLeffectreshour",
	"gamma_spec_omega_res_hour" = "gammaspecomegareshour",
	"intercept_spec_omega_res_hour" = "interceptspecomegareshour",
	"intercept_het_L_effect_omega_res_hour" = "intercepthetLeffectomegareshour",
	"intercept_het_L_omega_res_hour" = "intercepthetLomegareshour",
	"intercept_het_L_effect_multiple_res" = "intercepthetLeffectmultipleres",
	"intercept_het_B_multiple_res" = "intercepthetBmultipleres",
	"intercept_het_L_multiple_res" = "intercepthetLmultipleres",
	"intercept_het_L_effect_omega_res" = "intercepthetLeffectomegares",
	"intercept_het_L_omega_res" = "intercepthetLomegares",
	"gamma_L_effect_res_hour" = "gammaLeffectreshour",
	"delta_x_temptation_wo_habit" = "deltaxtemptationwohabit",
	"delta_x_temptation_wo_habit_res" = "deltaxtemptationwohabitres",
	"DWLStatic" = "DWLStatic",
	"DWLStaticThreeWeeks" = "DWLStaticThreeWeeks",
	"YearlyWelfare" = "YearlyWelfare"
	)

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Utility functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	find_bottom <- function(x){
	return(as.numeric(quantile(x, 0.025, na.rm=TRUE)))
	}
	find_top <- function(x){
	return(as.numeric(quantile(x, 0.975, na.rm=TRUE)))
	}

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Data functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	import_data <- function(){
	# Import data
	df <- import('input/final_data_sample.dta')

	# Clean data
	df %<>%
	mutate(L = ifelse(S2_LimitType != 0, 1, 0)) %>%
	mutate(B = ifelse(S3_Bonus == 1, 1, 0)) %>%
	mutate(S = as.character(Stratifier)) %>%
	mutate(w = 1)

	return(df)
	}

	balance_data <- function(df, magnitude=2){
	cols_used <- c('balance_income', 'balance_college', 'balance_male',
	'balance_white', 'balance_age')

	balance_df <- df[, c('UserID', cols_used)]
	balance_df$treat <- 0

	avg_df <- data.frame(0,
	avg_income,
	avg_college,
	avg_male,
	avg_white,
	avg_age,
	1)
	names(avg_df) <- c('UserID', cols_used, 'treat')

	balance_df <- rbind(balance_df, avg_df[rep(1, nrow(df)),])

	ebal.out <- ebalance(Treatment = balance_df$treat,
	X = balance_df[,cols_used],
	print.level = 0)

	df$w <- ebal.out$w
	df$w <- sapply(df$w, function(x){ return(min(magnitude, x))})
	df$w <- sapply(df$w, function(x){ return(max(1/magnitude, x))})
	return(df)
	}

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Solve functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	#solves for rho and lambda
	solve_sys_eq_1 <- function(param) {
	# Define
	tau_B_2 <- param[['tau_B_2']]
	tau_B_3 <- param[['tau_B_3']]
	tau_B_4 <- param[['tau_B_4']]
	tau_B_5 <- param[['tau_B_5']]

	# Solve
	a <- tau_B_2*tau_B_4
	b <- tau_B_3tau_B_4 - tau_B_2tau_B_5
	c <- tau_B_4^2 - tau_B_3*tau_B_5



	#solve for rho using the quadratic formula and choose the larger value
	rho <- ifelse((tau_B_2==0), tau_B_5/tau_B_4 - tau_B_4/tau_B_3, (-b + sqrt(b^2 - 4ac)) / (2*a))
	lambda <- tau_B_4 / (rhotau_B_3 + rho^2tau_B_2)

	rho <- ifelse(rho<0, 0, rho)
	lambda <- ifelse(lambda<0, 0, lambda)


	# Solve for tau_b_2=0, i.e. restricted alpha=1 model
	rho_res <- tau_B_5/tau_B_4 - tau_B_4/tau_B_3
	rho_res <- ifelse(rho_res<0, 0, rho_res)
	lambda_res <- tau_B_4 / (rho_res*tau_B_3)
	lambda_res <- ifelse(lambda_res<0, 0, lambda_res)


	# Return
	solution <- list(
	rho = rho,
	lambda = lambda,
	rho_res = rho_res,
	lambda_res = lambda_res
	)

	return(solution)
	}

	# Solves for alpha, eta, zeta
	solve_sys_eq_2 <- function(param, display_warning=FALSE) {
	rho = param[['rho']]
	lambda = param[['lambda']]
	rho_res = param[['rho_res']]
	lambda_res = param[['lambda_res']]
	delta = param[['delta']]
	tau_B_2 = param[['tau_B_2']]
	tau_tilde_B = param[['tau_tilde_B']]
	tau_B_3 = param[['tau_B_3']]
	tau_B_4 = param[['tau_B_4']]
	tau_tilde_B_3_2 = param[['tau_tilde_B_3_2']]
	p_B = param[['p_B']]

	find_y <- function(alpha, s, rho=rho, lambda=lambda){
	num <- -tau_B_4 + (1-alpha) * delta * (rho^2) * lambda * s
	denom <- rho * tau_B_3 + (rho^2) * tau_B_2 - (1-alpha) * delta * (rho^2) * (1 - lambda) * s

	return(num / denom)
	}

	find_x <- function(alpha, s, y, rho=rho, lambda=lambda){
	x1 <- tau_B_3 - (1-alpha)delta(rho^2)lambda(rho*tau_B_2 + tau_B_3)
	x2 <- rho * tau_B_2 - (1-alpha)delta(rho^2)(1 - lambda)(rho*tau_B_2 + tau_B_3)

	return(x1 + y * x2)
	}

	solve_alpha <- function(alpha){
	s <- (rho^2) * tau_B_2 + rho * tau_B_3 + tau_B_4
	y <- find_y(alpha, s, rho=rho, lambda=lambda)
	x <- find_x(alpha, s, y, rho=rho, lambda=lambda)

	eta <- p_B / x
	zeta <- y * eta

	num_z <- eta*tau_B_2
	den_z <- deltarho(-p_B + (eta - zeta)tau_tilde_B + zeta rho * tau_B_2)

	z <- (num_z / den_z) - (1-alpha)
	return(z)
	}

	#Unrestricted model
	print(tau_B_2)
	if (tau_B_2 >= 0){
	alpha <- 1
	} else {
	alpha <- uniroot(solve_alpha, interval=c(0,1.2))$root
	}






	alpha <- ifelse(alpha > 1, 1, alpha)
	alpha <- ifelse(alpha<0, 0, alpha)

	s <- (rho^2) * tau_B_2 + rho * tau_B_3 + tau_B_4
	y <- find_y(alpha, s, rho=rho, lambda=lambda)
	x <- find_x(alpha, s, y, rho=rho, lambda=lambda)
	eta <- p_B / x
	zeta <- y * eta

	#Winsorise parameters at 0
	eta <- ifelse(eta>0, 0, eta)
	zeta <- ifelse(zeta<0, 0, zeta)

	denom <- -eta - (1-alpha)deltarho(zeta-eta)-zeta((rho-(1-alpha)deltarho^2)/(1-rho))

	#eta <- ifelse(denom <0, NA, eta)
	#zeta <- ifelse(denom <0, NA, zeta)

	#Restricted model
	# note: alpha=1 in restricted model

	alpha_res <- 1
	eta_res <- p_B / tau_B_3
	zeta_res <- (-eta_restau_B_4) / (rho_restau_B_3)


	#Winsorise parameters at 0
	eta_res <- ifelse(eta_res>0, 0, eta_res)
	zeta_res <- ifelse(zeta_res<0, 0, zeta_res)


	# Display error for concavity assumptions
	if (display_warning){
	if(rho*(1+lambda)>1){
	stop("Concavity condition for rho(1+lambda) not satisfied")
	}
	}


	#Get hourly estimates
	eta_hour <- eta*3600
	zeta_hour <- zeta*3600

	eta_res_hour <- eta_res*3600
	zeta_res_hour <- zeta_res*3600



	#Export desired variables for in-text references
	lambdarho <- signif(lambda*rho, digits=2)
	lambdarhosquared <- lambda*rho^2
	lambdarhosquared <- signif(lambdarhosquared, digits=1)


	tauBthreehour <- signif(tau_B_3/60, digits=3)



	find_y_tau <- function(tau_B_2){
	s <- (rho^2) * tau_B_2 + rho * tau_B_3 + tau_B_4
	num <- -tau_B_4 + delta * (rho^2) * lambda * s
	denom <- rho * tau_B_3 + (rho^2) * tau_B_2 - delta * (rho^2) * (1 - lambda) * s

	return(num / denom)
	}

	find_x_tau <- function(tau_B_2, y){
	s <- (rho^2) * tau_B_2 + rho * tau_B_3 + tau_B_4
	x1 <- tau_B_3 - delta(rho^2)lambda(rhotau_B_2 + tau_B_3)
	x2 <- rho * tau_B_2 - delta(rho^2)(1 - lambda)(rhotau_B_2 + tau_B_3)

	return(x1 + y * x2)
	}

	solve_tau <- function(tau_B_2){
	y <- find_y_tau(tau_B_2)
	x <- find_x_tau(tau_B_2, y)

	left <- delta * rho * (-x + (1-y)*tau_tilde_B)

	right <- tau_B_2 * (1 - y * delta * rho^2)

	return(left - right)
	}

	tau_B_2_desired <- NA
	try(tau_B_2_desired <- uniroot(solve_tau, interval=c(-60,5))$root, silent = T)

	pcttaubtwo <- 100*(tau_B_2)/tau_B_2_desired

	oneminusalphapct <- (1-alpha)*100

	# Return
	solution <- list(
	alpha = alpha,
	eta = eta,
	zeta = zeta,
	eta_hour = eta_hour,
	zeta_hour = zeta_hour,
	alpha_res = alpha_res,
	eta_res = eta_res,
	zeta_res = zeta_res,
	eta_res_hour = eta_res_hour,
	zeta_res_hour = zeta_res_hour,
	tau_B_2_desired = tau_B_2_desired,
	lambdarho = lambdarho,
	lambdarhosquared = lambdarhosquared,
	tauBthreehour = tauBthreehour,
	pcttaubtwo = pcttaubtwo,
	oneminusalphapct = oneminusalphapct,
	denom = denom
	)

	# Return
	return(solution)
	}


	solve_sys_eq_3 <- function(param) {
	# Define
	rho <- param[['rho']]
	lambda <- param[['lambda']]
	rho_res <- param[['rho_res']]
	lambda_res <- param[['lambda_res']]
	alpha <- param[['alpha']]
	alpha_res <- param[['alpha_res']]

	delta <- param[['delta']]
	eta <- param[['eta']]
	zeta <- param[['zeta']]
	eta_res <- param[['eta_res']]
	zeta_res <- param[['zeta_res']]

	p_B <- param[['p_B']]
	F_B <- param[['F_B']]
	tau_B_2 <- param[['tau_B_2']]
	tau_B_3 <- param[['tau_B_3']]
	tau_tilde_B <- param[['tau_tilde_B']]
	tau_tilde_L <- param[['tau_tilde_L']]
	tau_L_2 <- param[['tau_L_2']]
	tau_L_y <- param[['tau_L_y']]
	tau_B_y <- param[['tau_B_y']]
	mispredict <- param[['mispredict']]
	d_L <- param[['d_L']]
	d_CL <- param[['d_CL']]
	v_L <- param[['v_L']]
	v_B <- param[['v_B']]
	x_ss <- param[['x_ss']]
	x_tilde_2_B <- param[['x_tilde_2_B']]


	# Compute estimated Omega
	omega_est <- (d_L - d_CL) / -d_CL
	omega <- 1
	p <- p_B


	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Unrestricted alpha
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Naivete
	term <- -eta + (1-alpha)delta(rho^2)(etalambda + zeta*(1 - lambda))
	naivete <- mispredict*term

	# Gamma-tilde-L
	gamma_tilde_L <- (2v_L) / (-tau_tilde_L(2-omega))
	gamma_L <- naivete + gamma_tilde_L

	# Gamma-tilde-L omega
	gamma_tilde_L_omega <- (2v_L) / (-tau_tilde_L(2-omega_est))
	gamma_L_omega <- naivete + gamma_tilde_L_omega


	# Gamma-tilde-L multiple good
	gamma_tilde_L_multiple <- v_L/-(tau_tilde_L*(2-omega)/2+tau_L_y)
	gamma_L_multiple <- naivete + gamma_tilde_L_multiple


	# Gamma-tilde-B
	term <- v_B - F_B + p*x_tilde_2_B
	gamma_tilde_B <- term / (-tau_tilde_B)
	gamma_B <- naivete + gamma_tilde_B

	# Gamma-tilde-B multiple good
	gamma_tilde_B_multiple <- (v_B - F_B + p*x_tilde_2_B)/-(tau_tilde_B+tau_B_y)
	gamma_B_multiple <- naivete + gamma_tilde_B_multiple



	# Gamma-L-effect
	num <- etatau_L_2/omega - (1-alpha)deltarho(((eta-zeta)tau_tilde_L/omega+zetarhotau_L_2/omega) + (1+lambda)mispredict(-eta+(1-alpha)deltarho^2((eta-zeta)*lambda+zeta)))
	denom <- 1 - (1-alpha)deltarho*(1+lambda)


	gamma_L_effect <- num/denom
	gamma_tilde_L_effect <- gamma_L_effect - naivete


	# Gamma-L-effect omega
	num_omega <- etatau_L_2/omega_est - (1-alpha)deltarho(((eta-zeta)tau_tilde_L/omega_est+zetarhotau_L_2/omega_est) + (1+lambda)mispredict(-eta+(1-alpha)deltarho^2((eta-zeta)*lambda+zeta)))


	gamma_L_effect_omega <- num_omega/denom
	gamma_tilde_L_effect_omega <- gamma_L_effect_omega - naivete


	# Select Gammas
	gamma_tilde <- gamma_tilde_L_effect
	gamma <- gamma_L_effect


	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Restricted alpha=1
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Naivete
	naivete_res <- mispredict*(-eta_res)

	# Gamma-tilde-L
	gamma_tilde_L <- (2v_L) / (-tau_tilde_L(2-omega))
	gamma_L_res <- naivete_res + gamma_tilde_L

	# Gamma-tilde-L omega
	gamma_tilde_L_omega <- (2v_L) / (-tau_tilde_L(2-omega_est))
	gamma_L_omega_res <- naivete_res + gamma_tilde_L_omega


	# Gamma-tilde-L multiple goods
	gamma_tilde_L_multiple <- v_L/-(tau_tilde_L*(2-omega)/2+tau_L_y)
	gamma_L_multiple_res <- naivete_res + gamma_tilde_L_multiple



	# Gamma-tilde-B
	gamma_B_res <- naivete_res + gamma_tilde_B

	# Gamma tilde-B multiple goods
	gamma_B_multiple_res <- naivete_res + gamma_tilde_B_multiple


	# Gamma-L-effect
	gamma_L_effect_res <- eta_res*tau_L_2
	gamma_tilde_L_effect_res <- gamma_L_effect_res - naivete_res

	# Gamma-L-effect omega
	gamma_L_effect_omega_res <- eta_res*tau_L_2/omega_est
	gamma_tilde_L_effect_omega_res <- gamma_L_effect_omega_res - naivete_res


	# Gamma-L-effect multiple goods (note: only for restricted case)
	gamma_L_effect_multiple_res <- (tau_L_2*p/tau_B_3)/(1+tau_L_y/tau_L_2)
	gamma_tilde_L_effect_multiple_res <- gamma_L_effect_multiple_res - naivete_res



	#Hourly variables for the tables
	naivete_hour <- naivete*60
	gamma_hour <- gamma*60
	gamma_L_hour <- gamma_L*60
	gamma_B_hour <- gamma_B*60

	gamma_tilde_hour <- gamma_tilde*60

	gamma_tilde_B_hour <- gamma_tilde_B*60
	gamma_tilde_L_hour <- gamma_tilde_L*60
	gamma_L_effect_omega_hour <- gamma_L_effect_omega*60

	gamma_L_omega_hour <- gamma_L_omega*60
	underestimatetemp <- naivete_hour/gamma_hour*100

	gamma_tilde_L_multiple_hour <- gamma_tilde_L_multiple*60
	gamma_L_multiple_hour <- gamma_L_multiple*60
	gamma_tilde_B_multiple_hour <- gamma_tilde_B_multiple*60
	gamma_B_multiple_hour <- gamma_B_multiple*60


	naivete_res_hour <- naivete_res*60
	gamma_L_effect_res_hour <- gamma_L_effect_res*60
	gamma_L_res_hour <- gamma_L_res*60
	gamma_B_res_hour <- gamma_B_res*60
	underestimatetemp_res <- naivete_res_hour/gamma_L_effect_res_hour*100
	gamma_tilde_L_effect_res_hour <- gamma_tilde_L_effect_res*60
	gamma_L_effect_omega_res_hour <- gamma_L_effect_omega_res*60
	gamma_L_omega_res_hour <- gamma_L_omega_res*60
	gamma_L_multiple_res_hour <- gamma_L_multiple_res*60
	gamma_B_multiple_res_hour <- gamma_B_multiple_res*60
	gamma_L_effect_multiple_res_hour <- gamma_L_effect_multiple_res*60
	gamma_tilde_L_effect_multiple_res_hour <- gamma_tilde_L_effect_multiple_res*60




	attritionrate <- (1-1933/2048)

	DWLStatic <- -(tau_L_2/60*gamma_L_effect_res_hour/2)
	DWLStaticThreeWeeks <- DWLStatic*21
	YearlyWelfare <- DWLStaticThreeWeeks52/30.24

	# Return
	solution <- list(
	naivete = naivete,
	omega_est = omega_est,
	omega = omega,
	gamma_L_effect = gamma_L_effect,
	gamma_tilde_L_effect = gamma_tilde_L_effect,
	gamma_L = gamma_L,
	gamma_tilde_L = gamma_tilde_L,
	gamma_B = gamma_B,
	gamma_tilde_B = gamma_tilde_B,
	gamma = gamma,
	gamma_tilde = gamma_tilde,
	naivete_hour = naivete_hour,
	gamma_hour = gamma_hour,
	gamma_tilde_B_hour = gamma_tilde_B_hour,
	gamma_tilde_L_hour = gamma_tilde_L_hour,
	gamma_tilde_hour = gamma_tilde_hour,
	gamma_L_hour = gamma_L_hour,
	gamma_B_hour = gamma_B_hour,
	gamma_L_effect_omega_hour = gamma_L_effect_omega_hour,
	gamma_L_omega_hour = gamma_L_omega_hour,
	underestimatetemp = underestimatetemp,
	attritionrate = attritionrate,
	gamma_tilde_L_effect_omega = gamma_tilde_L_effect_omega,
	gamma_L_effect_omega = gamma_L_effect_omega,
	gamma_tilde_L_omega = gamma_tilde_L_omega,
	gamma_L_omega = gamma_L_omega,
	gamma_tilde_L_multiple = gamma_tilde_L_multiple,
	gamma_L_multiple = gamma_L_multiple,
	gamma_tilde_B_multiple = gamma_tilde_B_multiple,
	gamma_B_multiple = gamma_B_multiple,
	gamma_tilde_L_multiple_hour = gamma_tilde_L_multiple_hour,
	gamma_L_multiple_hour = gamma_L_multiple_hour,
	gamma_tilde_B_multiple_hour = gamma_tilde_B_multiple_hour,
	gamma_B_multiple_hour = gamma_B_multiple_hour,
	naivete_res = naivete_res,
	underestimatetemp_res = underestimatetemp_res,
	gamma_L_effect_res = gamma_L_effect_res,
	gamma_tilde_L_effect_res = gamma_tilde_L_effect_res,
	gamma_L_res = gamma_L_res,
	gamma_B_res = gamma_B_res,
	naivete_res_hour = naivete_res_hour,
	gamma_tilde_L_effect_res_hour = gamma_tilde_L_effect_res_hour,
	gamma_L_res_hour = gamma_L_res_hour,
	gamma_B_res_hour = gamma_B_res_hour,
	gamma_L_effect_omega_res_hour = gamma_L_effect_omega_res_hour,
	gamma_L_omega_res_hour = gamma_L_omega_res_hour,
	gamma_tilde_L_effect_omega_res = gamma_tilde_L_effect_omega_res,
	gamma_L_effect_omega_res = gamma_L_effect_omega_res,
	gamma_L_omega_res = gamma_L_omega_res,
	gamma_L_multiple_res = gamma_L_multiple_res,
	gamma_B_multiple_res = gamma_B_multiple_res,
	gamma_tilde_L_effect_multiple_res = gamma_tilde_L_effect_multiple_res,
	gamma_L_effect_multiple_res = gamma_L_effect_multiple_res,

	gamma_L_multiple_res_hour = gamma_L_multiple_hour,
	gamma_B_multiple_res_hour = gamma_B_multiple_res_hour,
	gamma_tilde_L_effect_multiple_res_hour = gamma_tilde_L_effect_multiple_res_hour,
	gamma_L_effect_multiple_res_hour = gamma_L_effect_multiple_res_hour,
	gamma_L_effect_res_hour = gamma_L_effect_res_hour,
	DWLStatic = DWLStatic,
	DWLStaticThreeWeeks = DWLStaticThreeWeeks,
	YearlyWelfare = YearlyWelfare
	)

	return(solution)
	}


	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate effects from different identification strategies
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


	solve_effects <- function(param, df){
	# Define
	eta <- param[['eta']]
	zeta <- param[['zeta']]
	eta_res <- param[['eta_res']]
	zeta_res <- param[['zeta_res']]
	alpha <- param[['alpha']]
	alpha_res <- param[['alpha_res']]
	rho <- param[['rho']]
	rho_res <- param[['rho_res']]
	lambda <- param[['lambda']]
	lambda_res <- param[['lambda_res']]
	delta <- param[['delta']]
	p_B <- param[['p_B']]

	naivete <- param[['naivete']]
	naivete_res <- param[['naivete_res']]
	mispredict <- param[['mispredict']]
	x_ss <- param[['x_ss']]



	intercept_het_L_effect <- param[['intercept_het_L_effect']]
	intercept_het_L_effect_eta_high <- param[['intercept_het_L_effect_eta_high']]
	intercept_het_L_effect_eta_low <- param[['intercept_het_L_effect_eta_low']]
	intercept_het_B <- param[['intercept_het_B']]
	intercept_het_L <- param[['intercept_het_L']]
	intercept_het_B_multiple <- param[['intercept_het_B_multiple']]
	intercept_het_L_multiple <- param[['intercept_het_L_multiple']]
	intercept_het_L_effect_omega <- param[['intercept_het_L_effect_omega']]
	intercept_het_L_omega <- param[['intercept_het_L_omega']]

	intercept_het_L_effect_res <- param[['intercept_het_L_effect_res']]
	intercept_het_L_effect_eta_high_res <- param[['intercept_het_L_effect_eta_high_res']]
	intercept_het_L_effect_eta_low_res <- param[['intercept_het_L_effect_eta_low_res']]
	intercept_het_B_res <- param[['intercept_het_B_res']]
	intercept_het_L_res <- param[['intercept_het_L_res']]
	intercept_het_L_effect_multiple_res <- param[['intercept_het_L_effect_multiple_res']]
	intercept_het_B_multiple_res <- param[['intercept_het_B_multiple_res']]
	intercept_het_L_multiple_res <- param[['intercept_het_L_multiple_res']]
	intercept_het_L_effect_omega_res <- param[['intercept_het_L_effect_omega_res']]
	intercept_het_L_omega_res <- param[['intercept_het_L_omega_res']]



	gamma_L_effect <- param[['gamma_L_effect']]
	gamma_tilde_L_effect <- param[['gamma_tilde_L_effect']]
	gamma_L_effect_res <- param[['gamma_L_effect_res']]
	gamma_tilde_L_effect_res <- param[['gamma_tilde_L_effect_res']]

	gamma_L_res <- param[['gamma_L_res']]
	gamma_L <- param[['gamma_L']]
	gamma_tilde_L <- param[['gamma_tilde_L']]

	gamma_B <- param[['gamma_B']]
	gamma_B_res <- param[['gamma_B_res']]
	gamma_tilde_B <- param[['gamma_tilde_B']]

	gamma_tilde_L_multiple <- param[['gamma_tilde_L_multiple']]
	gamma_L_multiple <- param[['gamma_L_multiple']]
	gamma_L_multiple_res <- param[['gamma_L_multiple_res']]

	gamma_tilde_B_multiple <- param[['gamma_tilde_B_multiple']]
	gamma_B_multiple <- param[['gamma_B_multiple']]
	gamma_B_multiple_res <- param[['gamma_B_multiple_res']]

	gamma_L_effect_multiple_res <- param[['gamma_L_effect_multiple_res']]
	gamma_tilde_L_effect_multiple_res <- param[['gamma_tilde_L_effect_multiple_res']]


	gamma_tilde_L_effect_omega <- param[['gamma_tilde_L_effect_omega']]
	gamma_L_effect_omega <- param[['gamma_L_effect_omega']]
	gamma_tilde_L_effect_omega_res <- param[['gamma_tilde_L_effect_omega_res']]
	gamma_L_effect_omega_res <- param[['gamma_L_effect_omega_res']]

	gamma_tilde_L_omega <- param[['gamma_tilde_L_omega']]
	gamma_L_omega <- param[['gamma_L_omega']]
	gamma_L_omega_res <- param[['gamma_L_omega_res']]


	# set price to zero
	p <- 0




	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Unrestricted alpha model
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Estimating using gamma from limit effect
	x_ss_L_effect <- max(0, calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect))
	delta_x_cap <- x_ss_L_effect - max(0, calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect))
	delta_x_cap_us <- delta_x_cap * 365 * us_adults

	# Estimating using gamma from limit effect eta high (eta1.1) and low (eta0.9)
	x_ss_L_effect_eta_high <- max(0, calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect_eta_high, eta_scale=1.1))
	x_ss_L_effect_eta_low <- max(0, calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect_eta_low, eta_scale=0.9))



	# Estimating using gamma_tilde from limit valuation
	x_ss_L <- max(0, calculate_steady_state(param, gamma_tilde_L, gamma_L, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L))
	delta_x_tilde_L <- x_ss_L - max(0, calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L))
	delta_x_tilde_L_us <- delta_x_tilde_L * 365 * us_adults





	# Estimating using gamma_tilde from bonus valuation
	x_ss_B <- max(0, calculate_steady_state(param, gamma_tilde_B, gamma_B, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_B))
	delta_x_tilde_B <- x_ss_B - max(0, calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_B))
	delta_x_tilde_B_us <- delta_x_tilde_B * 365 * us_adults


	mispredicthour <- mispredict/60


	# Estimating time effects using multiple goods moments

	x_ss_L_multiple <- max(0, calculate_steady_state(param, gamma_tilde_L_multiple, gamma_L_multiple, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L_multiple))
	delta_x_L_multiple <- max(0,x_ss_L_multiple - max(0, calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_multiple)))

	x_ss_B_multiple <- max(0, calculate_steady_state(param, gamma_tilde_B_multiple, gamma_B_multiple, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_B_multiple))
	delta_B_multiple <- max(0,x_ss_B_multiple - max(0, calculate_steady_state(param, 0, 0, alpha, rho,lambda, 0, eta, zeta, intercept=intercept_het_B_multiple)))


	# Estimating time effects using omega hat moments

	# limit effect; omega_hat
	x_ss_L_effect_omega <- max(0, calculate_steady_state(param, gamma_tilde_L_effect_omega, gamma_L_effect_omega, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect_omega))
	delta_x_cap_omega <- max(0,x_ss_L_effect_omega - max(0, calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect_omega)))

	# limit valuation: omega_hat
	x_ss_L_omega <- max(0, calculate_steady_state(param, gamma_tilde_L_omega, gamma_L_omega, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L_omega))
	delta_x_L_omega <- max(0,x_ss_L_omega - max(0, calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_omega)))




	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Restricted alpha=1 model
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Estimating using gamma from limit effect
	x_ss_L_effect_res <- max(0, calculate_steady_state(param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_res))
	delta_x_cap_res <- x_ss_L_effect_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_res))
	delta_x_cap_us_res <- delta_x_cap_res * 365 * us_adults

	# Estimating using gamma from limit effect eta high (eta1.1) and low (eta0.9)
	x_ss_L_effect_eta_high_res <- max(0, calculate_steady_state(param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_high_res, eta_scale=1.1))
	x_ss_L_effect_eta_low_res <- max(0, calculate_steady_state(param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_low_res, eta_scale=0.9))



	# Estimating using gamma_tilde from limit valuation (note: gamma_tilde_L is same for restricted and unrestricted models)
	x_ss_L_res <- max(0, calculate_steady_state(param, gamma_tilde_L, gamma_L_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_res))
	delta_x_tilde_L_res <- x_ss_L_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_res))
	delta_x_tilde_L_us_res <- delta_x_tilde_L_res * 365 * us_adults



	# Estimating using gamma_tilde from bonus valuation (note: gamma_tilde_B is same for restricted and unrestricted models)
	x_ss_B_res <- max(0, calculate_steady_state(param, gamma_tilde_B, gamma_B_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_B_res))
	delta_x_tilde_B_res <- x_ss_B_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_B_res))
	delta_x_tilde_B_us_res <- delta_x_tilde_B_res * 365 * us_adults


	# Estimating time effects using multiple goods moments

	x_ss_L_effect_multiple_res <- max(0, calculate_steady_state(param, gamma_tilde_L_effect_multiple_res, gamma_L_effect_multiple_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_multiple_res))
	delta_x_cap_multiple_res <- max(0,x_ss_L_effect_multiple_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_multiple_res)))

	x_ss_L_multiple_res <- max(0, calculate_steady_state(param, gamma_tilde_L_multiple, gamma_L_multiple_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_multiple_res))
	delta_x_L_multiple_res <- max(0,x_ss_L_multiple_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_multiple_res)))

	x_ss_B_multiple_res <- max(0, calculate_steady_state(param, gamma_tilde_B_multiple, gamma_B_multiple_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_B_multiple_res))
	delta_B_multiple_res <- max(0,x_ss_B_multiple_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_B_multiple_res)))


	# Estimating time effects using omega hat moments (limit effects)

	x_ss_L_effect_omega_res <- max(0, calculate_steady_state(param, gamma_tilde_L_effect_omega_res, gamma_L_effect_omega_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_omega_res))
	delta_x_cap_omega_res <- max(0,x_ss_L_effect_omega_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_omega_res)))

	x_ss_L_omega_res <- max(0, calculate_steady_state(param, gamma_tilde_L_omega, gamma_L_omega_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_omega_res))
	delta_x_L_omega_res <- max(0,x_ss_L_omega_res - max(0, calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_omega_res)))



	#xero temp x_ss
	x_ss_zero_temp <- max(0, calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect))


	# Return
	solution <- list(
	x_ss_L_effect = x_ss_L_effect,
	x_ss_L_effect_eta_high = x_ss_L_effect_eta_high,
	x_ss_L_effect_eta_low = x_ss_L_effect_eta_low,
	x_ss_B = x_ss_B,
	delta_x_cap = delta_x_cap,
	delta_x_cap_us = delta_x_cap_us,
	delta_x_tilde_L = delta_x_tilde_L,
	delta_x_tilde_L_us = delta_x_tilde_L_us,
	delta_x_tilde_B = delta_x_tilde_B,
	delta_x_tilde_B_us = delta_x_tilde_B_us,
	x_ss_L = x_ss_L,
	mispredicthour = mispredicthour,
	delta_x_L_multiple = delta_x_L_multiple,
	delta_B_multiple = delta_B_multiple,
	delta_x_cap_omega = delta_x_cap_omega,
	delta_x_L_omega = delta_x_L_omega,
	x_ss_L_effect_res = x_ss_L_effect_res,
	x_ss_L_effect_eta_high_res = x_ss_L_effect_eta_high_res,
	x_ss_L_effect_eta_low_res = x_ss_L_effect_eta_low_res,
	x_ss_B_res = x_ss_B_res,
	delta_x_cap_res = delta_x_cap_res,
	delta_x_cap_us_res = delta_x_cap_us_res,
	delta_x_tilde_L_res = delta_x_tilde_L_res,
	delta_x_tilde_L_us_res = delta_x_tilde_L_us_res,
	delta_x_tilde_B_res = delta_x_tilde_B_res,
	delta_x_tilde_B_us_res = delta_x_tilde_B_us_res,
	x_ss_L_res = x_ss_L_res,
	delta_x_cap_multiple_res = delta_x_cap_multiple_res,
	delta_x_L_multiple_res = delta_x_L_multiple_res,
	delta_B_multiple_res = delta_B_multiple_res,
	delta_x_cap_omega_res = delta_x_cap_omega_res,
	delta_x_L_omega_res = delta_x_L_omega_res,
	x_ss_zero_temp = x_ss_zero_temp
	)


	return(solution)
	}

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate effects from different counterfactuals (heterogeneous)
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	solve_counterfactuals <- function(param, df){

	# Define

	alpha <- param[['alpha']]
	alpha_res <- param[['alpha_res']]
	rho <- param[['rho']]
	rho_res <- param[['rho_res']]
	eta <- param[['eta']]
	eta_res <- param[['eta_res']]
	zeta <- param[['zeta']]
	zeta_res <- param[['zeta_res']]
	lambda <- param[['lambda']]
	lambda_res <- param[['lambda_res']]
	delta <- param[['delta']]

	mispredict <- param[['mispredict']]
	x_ss <- param[['x_ss']]

	gamma_L_effect <- param[['gamma_L_effect']]
	gamma_tilde_L_effect <- param[['gamma_tilde_L_effect']]
	gamma_tilde_L_effect_res <- param[['gamma_tilde_L_effect_res']]
	intercept_het_L_effect <- param[['intercept_het_L_effect']]
	intercept_het_L_effect_eta_high <- param[['intercept_het_L_effect_eta_high']]
	intercept_het_L_effect_eta_low <- param[['intercept_het_L_effect_eta_low']]
	gamma_L_effect_res <- param[['gamma_L_effect_res']]
	intercept_het_L_effect_res <- param[['intercept_het_L_effect_res']]
	intercept_het_L_effect_eta_high_res <- param[['intercept_het_L_effect_eta_high_res']]
	intercept_het_L_effect_eta_low_res <- param[['intercept_het_L_effect_eta_low_res']]



	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate effects for unrestricted model
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Remember the order of the arguments for calculate steady state is
	# calculate_steady_state(param, gamma_tilde, gamma, alpha, rho, mispredict, eta, zeta, intercept=NA, eta_scale=1){

	x_ss_L_effect <- max(calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect) ,0)

	x_ss_temptation <- max(calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect) ,0)
	x_ss_temptation_eta_high <- max(calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect_eta_high, eta_scale=1.1) ,0)
	x_ss_temptation_eta_low <- max(calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect_eta_low, eta_scale=0.9) ,0)

	x_ss_habit <- max(calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, 0, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect) ,0)
	x_ss_habit_eta_high <- max(calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, 0, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect_eta_high, eta_scale=1.1) ,0)
	x_ss_habit_eta_low <- max(calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, 0, lambda, mispredict, eta, zeta, intercept=intercept_het_L_effect_eta_low, eta_scale=0.9) ,0)

	x_ss_naivete <- max(calculate_steady_state(param, gamma_L_effect, gamma_L_effect, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect) ,0)
	x_ss_naivete_eta_high <- max(calculate_steady_state(param, gamma_L_effect, gamma_L_effect, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect_eta_high, eta_scale=1.1) ,0)
	x_ss_naivete_eta_low <- max(calculate_steady_state(param, gamma_L_effect, gamma_L_effect, alpha, rho, lambda, 0, eta, zeta, intercept=intercept_het_L_effect_eta_low, eta_scale=0.9) ,0)

	x_ss_habit_temptation <- max(calculate_steady_state(param, 0, 0, alpha, 0, lambda, 0, eta, zeta, intercept=intercept_het_L_effect) ,0)
	x_ss_habit_temptation_eta_high <- max(calculate_steady_state(param, 0, 0, alpha, 0, lambda, 0, eta, zeta, intercept=intercept_het_L_effect_eta_high, eta_scale=1.1) ,0)
	x_ss_habit_temptation_eta_low <- max(calculate_steady_state(param, 0, 0, alpha, 0, lambda, 0, eta, zeta, intercept=intercept_het_L_effect_eta_low, eta_scale=0.9) ,0)


	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate effects for restricted alpha=1 model
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Remember the order of the arguments for calculate steady state is
	# calculate_steady_state(param, gamma_tilde, gamma, alpha, rho, mispredict, eta, zeta, intercept=NA, eta_scale=1){


	x_ss_L_effect_res <- max(calculate_steady_state(param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_res) ,0)

	x_ss_temptation_res <- max(calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_res) ,0)
	x_ss_temptation_eta_high_res <- max(calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_high_res, eta_scale=1.1) ,0)
	x_ss_temptation_eta_low_res <- max(calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_low_res, eta_scale=0.9) ,0)

	x_ss_habit_res <- max(calculate_steady_state(param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, 0, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_res) ,0)
	x_ss_habit_eta_high_res <- max(calculate_steady_state(param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, 0, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_high_res, eta_scale=1.1) ,0)
	x_ss_habit_eta_low_res <- max(calculate_steady_state(param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, 0, lambda_res, mispredict, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_low_res, eta_scale=0.9) ,0)

	x_ss_naivete_res <- max(calculate_steady_state(param, gamma_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_res) ,0)
	x_ss_naivete_eta_high_res <- max(calculate_steady_state(param, gamma_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_high_res, eta_scale=1.1) ,0)
	x_ss_naivete_eta_low_res <- max(calculate_steady_state(param, gamma_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_low_res, eta_scale=0.9) ,0)

	x_ss_habit_temptation_res <- max(calculate_steady_state(param, 0, 0, alpha_res, 0, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_res) ,0)
	x_ss_habit_temptation_eta_high_res <- max(calculate_steady_state(param, 0, 0, alpha_res, 0, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_high_res, eta_scale=1.1) ,0)
	x_ss_habit_temptation_eta_low_res <- max(calculate_steady_state(param, 0, 0, alpha_res, 0, lambda_res, 0, eta_res, zeta_res, intercept=intercept_het_L_effect_eta_low_res, eta_scale=0.9) ,0)



	# Return
	solution <- list(
	delta_x_temptation = x_ss_L_effect - x_ss_temptation,
	delta_x_naivete = x_ss_L_effect - x_ss_naivete,
	delta_x_habit = x_ss_L_effect - x_ss_habit,
	delta_x_temptation_wo_habit = x_ss_habit - x_ss_habit_temptation,
	delta_x_temptation_wo_habit_res = x_ss_habit_res - x_ss_habit_temptation_res,
	x_ss_temptation = x_ss_temptation,
	x_ss_temptation_eta_high = x_ss_temptation_eta_high,
	x_ss_temptation_eta_low = x_ss_temptation_eta_low,
	x_ss_naivete = x_ss_naivete,
	x_ss_naivete_eta_high = x_ss_naivete_eta_high,
	x_ss_naivete_eta_low = x_ss_naivete_eta_low,
	x_ss_habit = x_ss_habit,
	x_ss_habit_eta_high = x_ss_habit_eta_high,
	x_ss_habit_eta_low = x_ss_habit_eta_low,
	x_ss_habit_temptation = x_ss_habit_temptation,
	x_ss_habit_temptation_eta_high = x_ss_habit_temptation_eta_high,
	x_ss_habit_temptation_eta_low = x_ss_habit_temptation_eta_low,
	delta_x_temptation_res = x_ss_L_effect_res - x_ss_temptation_res,
	delta_x_naivete_res = x_ss_L_effect_res - x_ss_naivete_res,
	delta_x_habit_res = x_ss_L_effect_res - x_ss_habit_res,
	x_ss_temptation_res = x_ss_temptation_res,
	x_ss_temptation_eta_high_res = x_ss_temptation_eta_high_res,
	x_ss_temptation_eta_low_res = x_ss_temptation_eta_low_res,
	x_ss_naivete_res = x_ss_naivete_res,
	x_ss_naivete_eta_high_res = x_ss_naivete_eta_high_res,
	x_ss_naivete_eta_low_res = x_ss_naivete_eta_low_res,
	x_ss_habit_res = x_ss_habit_res,
	x_ss_habit_eta_high_res = x_ss_habit_eta_high_res,
	x_ss_habit_eta_low_res = x_ss_habit_eta_low_res,
	x_ss_habit_temptation_res = x_ss_habit_temptation_res,
	x_ss_habit_temptation_eta_high_res = x_ss_habit_temptation_eta_high_res,
	x_ss_habit_temptation_eta_low_res = x_ss_habit_temptation_eta_low_res)

	return(solution)

	}




	solve_effects_individual <- function(x_ss_i_data, param, tau_tilde_L, tau_L_2, w){
	rho <- param[['rho']]
	lambda <- param[['lambda']]
	rho_res <- param[['rho_res']]
	lambda_res <- param[['lambda_res']]
	delta <- param[['delta']]
	alpha <- param[['alpha']]
	alpha_res <- param[['alpha_res']]
	omega <- param[['omega']]
	omega_est <- param[['omega_est']]
	mispredict <- param[['mispredict']]
	d_L <- param[['d_L']]
	d_CL <- param[['d_CL']]

	eta <- param[['eta']]
	zeta <- param[['zeta']]

	naivete <- param[['naivete']]

	gamma_L_effect <- param[['gamma_L_effect']]
	gamma_tilde_L_effect <- param[['gamma_tilde_L_effect']]
	gamma_tilde_L_effect_omega <- param[['gamma_tilde_L_effect_omega']]
	gamma_L_effect_omega <- param[['gamma_L_effect_omega']]

	gamma_L <- param[['gamma_L']]
	gamma_tilde_L <- param[['gamma_tilde_L']]
	gamma_tilde_L_omega <- param[['gamma_tilde_L_omega']]
	gamma_L_omega <- param[['gamma_L_omega']]
	gamma_tilde_L_multiple <- param[['gamma_tilde_L_multiple']]
	gamma_L_multiple <- param[['gamma_L_multiple']]


	gamma_B <- param[['gamma_B']]
	gamma_tilde_B <- param[['gamma_tilde_B']]
	gamma_tilde_B_multiple <- param[['gamma_tilde_B_multiple']]
	gamma_B_multiple <- param[['gamma_B_multiple']]

	eta_res <- param[['eta_res']]
	zeta_res <- param[['zeta_res']]

	naivete_res <- param[['naivete_res']]

	gamma_L_effect_res <- param[['gamma_L_effect_res']]
	gamma_tilde_L_effect_res <- param[['gamma_tilde_L_effect_res']]
	gamma_tilde_L_effect_omega_res <- param[['gamma_tilde_L_effect_omega_res']]
	gamma_L_effect_omega_res <- param[['gamma_L_effect_omega_res']]
	gamma_tilde_L_effect_multiple_res <- param[['gamma_tilde_L_effect_multiple_res']]
	gamma_L_effect_multiple_res <- param[['gamma_L_effect_multiple_res']]


	gamma_L_res <- param[['gamma_L_res']]
	gamma_L_omega_res <- param[['gamma_L_omega_res']]
	gamma_L_multiple_res <- param[['gamma_L_multiple_res']]


	gamma_B_res <- param[['gamma_B_res']]
	gamma_B_multiple_res <- param[['gamma_B_multiple_res']]


	tau_L_2_signed <- param[['tau_L_2']]*-1


	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate individual intercepts and steady states under different strategies - Unrestricted alpha
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Gamma-spec
	num <- etatau_L_2/omega - (1-alpha)deltarho(((eta-zeta)tau_tilde_L/omega+zetarhotau_L_2/omega) + (1+lambda)mispredict(-eta+(1-alpha)deltarho^2((eta-zeta)*lambda+zeta)))
	denom <- 1 - (1-alpha)deltarho*(1+lambda)

	num_omega <- etatau_L_2/omega_est - (1-alpha)deltarho(((eta-zeta)tau_tilde_L/omega_est+zetarhotau_L_2/omega) + (1+lambda)mispredict(-eta+(1-alpha)deltarho^2((eta-zeta)*lambda+zeta)))

	gamma_spec <- num/denom
	gamma_spec_omega <- num_omega/denom
	gamma_tilde_spec <- gamma_spec - naivete
	gamma_tilde_spec_omega <- gamma_spec_omega - naivete



	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate individual intercepts and steady states under different strategies
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	intercept_spec <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_spec, gamma_spec, alpha, rho, lambda, mispredict, eta, zeta)
	intercept_het_L_effect <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, lambda, mispredict, eta, zeta)
	intercept_het_B <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_B, gamma_B, alpha, rho, lambda, mispredict, eta, zeta)
	intercept_het_L <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L, gamma_L, alpha, rho, lambda, mispredict, eta, zeta)

	intercept_spec_omega <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_spec_omega, gamma_spec_omega, alpha, rho, lambda, mispredict, eta, zeta)
	intercept_het_L_effect_omega <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect_omega, gamma_L_effect_omega, alpha, rho, lambda, mispredict, eta, zeta)
	intercept_het_L_omega <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_omega, gamma_L_omega, alpha, rho, lambda, mispredict, eta, zeta)

	intercept_het_B_multiple <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_B_multiple, gamma_B_multiple, alpha, rho, lambda, mispredict, eta, zeta)
	intercept_het_L_multiple <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_multiple, gamma_L_multiple, alpha, rho, lambda, mispredict, eta, zeta)


	intercept_het_L_effect_eta_high <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, lambda, mispredict, eta, zeta, eta_scale=1.1)
	intercept_het_L_effect_eta_low <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, lambda, mispredict, eta, zeta, eta_scale=0.9)

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate individual counterfactuals
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	x_ss_spec <- calculate_steady_state(param, gamma_tilde_spec, gamma_spec, alpha, rho, lambda, mispredict, eta, zeta, intercept_spec)
	x_ss_zero_un <- calculate_steady_state(param, 0, 0, alpha, rho, lambda, 0, eta, zeta, intercept_spec)
	x_ss_zero <- ifelse(x_ss_zero_un<0, 0, x_ss_zero_un)
	delta_x <- x_ss_spec - x_ss_zero



	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate individual intercepts and steady states under different strategies - Restricted alpha
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	# Gamma-spec
	gamma_spec_res <- eta_res*tau_L_2
	gamma_spec_omega_res <- eta_res*tau_L_2/omega_est
	gamma_tilde_spec_res <- gamma_spec_res - naivete_res
	gamma_tilde_spec_omega_res <- gamma_spec_omega_res - naivete_res


	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate individual intercepts and steady states under different strategies
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	intercept_spec_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_spec_res, gamma_spec_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)
	intercept_het_L_effect_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)
	intercept_het_B_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_B, gamma_B_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)
	intercept_het_L_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L, gamma_L_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)

	intercept_spec_omega_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_spec_omega_res, gamma_spec_omega_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)
	intercept_het_L_effect_omega_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect_omega_res, gamma_L_effect_omega_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)
	intercept_het_L_omega_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_omega, gamma_L_omega_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)

	intercept_het_L_effect_multiple_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect_multiple_res, gamma_tilde_L_effect_multiple_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)
	intercept_het_B_multiple_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_B_multiple, gamma_B_multiple_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)
	intercept_het_L_multiple_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_multiple, gamma_L_multiple_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res)


	intercept_het_L_effect_eta_high_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res, eta_scale=1.1)
	intercept_het_L_effect_eta_low_res <- calculate_intercept_spec(x_ss_i_data, param, gamma_tilde_L_effect_res, gamma_L_effect_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res, eta_scale=0.9)

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Calculate individual counterfactuals
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	x_ss_spec_res <- calculate_steady_state(param, gamma_tilde_spec_res, gamma_spec_res, alpha_res, rho_res, lambda_res, mispredict, eta = eta_res, zeta = zeta_res, intercept_spec_res)
	x_ss_zero_un_res <- calculate_steady_state(param, 0, 0, alpha_res, rho_res, lambda_res, 0, eta = eta_res, zeta = zeta_res, intercept_spec_res)
	x_ss_zero_res <- ifelse(x_ss_zero_un_res<0, 0, x_ss_zero_un_res)
	delta_x_res <- x_ss_spec_res - x_ss_zero_res



	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Compute population averages
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	x_ss_spec_w <- weighted.mean(x_ss_spec, w, na.rm=T)
	gamma_tilde_spec_w <- weighted.mean(gamma_tilde_spec, w, na.rm=T)
	gamma_spec_w <- weighted.mean(gamma_spec, w, na.rm=T)
	gamma_spec_omega_w <- weighted.mean(gamma_spec_omega, w, na.rm=T)
	delta_x_spec <- weighted.mean(delta_x, w, na.rm=T)
	x_ss_i_data <- weighted.mean(x_ss_i_data, w, na.rm=T)



	intercept_spec <- weighted.mean(intercept_spec, w, na.rm=T)
	intercept_spec_omega <- weighted.mean(intercept_spec_omega, w, na.rm=T)
	intercept_het_L_effect_omega <- weighted.mean(intercept_het_L_effect_omega, w, na.rm=T)
	intercept_het_L_effect <- weighted.mean(intercept_het_L_effect, w, na.rm=T)
	intercept_het_L_effect_eta_high <- weighted.mean(intercept_het_L_effect_eta_high, w, na.rm=T)
	intercept_het_L_effect_eta_low <- weighted.mean(intercept_het_L_effect_eta_low, w, na.rm=T)
	intercept_het_L_omega <- weighted.mean(intercept_het_L_omega, w, na.rm=T)
	intercept_het_B <- weighted.mean(intercept_het_B, w, na.rm=T)
	intercept_het_L <- weighted.mean(intercept_het_L, w, na.rm=T)


	intercept_het_B_multiple <- weighted.mean(intercept_het_B_multiple, w, na.rm=T)
	intercept_het_L_multiple <- weighted.mean(intercept_het_L_multiple, w, na.rm=T)

	x_ss_spec_w_res <- weighted.mean(x_ss_spec_res, w, na.rm=T)
	gamma_tilde_spec_w_res <- weighted.mean(gamma_tilde_spec_res, w, na.rm=T)
	gamma_spec_w_res <- weighted.mean(gamma_spec_res, w, na.rm=T)
	gamma_spec_omega_w_res <- weighted.mean(gamma_spec_omega_res, w, na.rm=T)
	delta_x_spec_res <- weighted.mean(delta_x_res, w, na.rm=T)



	intercept_spec_res <- weighted.mean(intercept_spec_res, w, na.rm=T)
	intercept_spec_omega_res <- weighted.mean(intercept_spec_omega_res, w, na.rm=T)
	intercept_het_L_effect_omega_res <- weighted.mean(intercept_het_L_effect_omega_res, w, na.rm=T)
	intercept_het_L_effect_res <- weighted.mean(intercept_het_L_effect_res, w, na.rm=T)
	intercept_het_L_effect_eta_high_res <- weighted.mean(intercept_het_L_effect_eta_high_res, w, na.rm=T)
	intercept_het_L_effect_eta_low_res <- weighted.mean(intercept_het_L_effect_eta_low_res, w, na.rm=T)
	intercept_het_L_omega_res <- weighted.mean(intercept_het_L_omega_res, w, na.rm=T)
	intercept_het_B_res <- weighted.mean(intercept_het_B_res, w, na.rm=T)
	intercept_het_L_res <- weighted.mean(intercept_het_L_res, w, na.rm=T)


	intercept_het_L_effect_multiple_res <- weighted.mean(intercept_het_L_effect_multiple_res, w, na.rm=T)
	intercept_het_B_multiple_res <- weighted.mean(intercept_het_B_multiple_res, w, na.rm=T)
	intercept_het_L_multiple_res <- weighted.mean(intercept_het_L_multiple_res, w, na.rm=T)





	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Compute hourly variables
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	intercept_spec_hour <- intercept_spec60-1
	gamma_tilde_spec_hour <- gamma_tilde_spec_w*60
	gamma_spec_hour <- gamma_spec_w*60
	gamma_spec_omega_hour <- gamma_spec_omega_w*60
	intercept_het_B_hour <- intercept_het_B60-1
	intercept_het_L_hour <- intercept_het_L60-1
	intercept_het_L_effect_hour <- intercept_het_L_effect60-1
	intercept_spec_omega_hour <- intercept_spec_omega60-1
	intercept_het_L_effect_omega_hour <- intercept_het_L_effect_omega60-1
	intercept_het_L_omega_hour <- intercept_het_L_omega60-1

	intercept_spec_res_hour <- intercept_spec_res60-1
	gamma_tilde_spec_res_hour <- gamma_tilde_spec_w_res*60
	gamma_spec_res_hour <- gamma_spec_w_res*60
	gamma_spec_omega_res_hour <- gamma_spec_omega_w_res*60
	intercept_het_B_res_hour <- intercept_het_B_res60-1
	intercept_het_L_res_hour <- intercept_het_L_res60-1
	intercept_het_L_effect_res_hour <- intercept_het_L_effect_res60-1
	intercept_spec_omega_res_hour <- intercept_spec_omega_res60-1
	intercept_het_L_effect_omega_res_hour <- intercept_het_L_effect_omega_res60-1
	intercept_het_L_omega_res_hour <- intercept_het_L_omega_res60-1



	# Return
	solution <- list(
	x_ss_i_data = x_ss_i_data,
	x_ss_spec = x_ss_spec_w,
	tau_L_2_signed = tau_L_2_signed,
	intercept_spec = intercept_spec,
	intercept_spec_hour = intercept_spec_hour,
	gamma_tilde_spec = gamma_tilde_spec_w,
	gamma_spec = gamma_spec_w,
	delta_x_spec = delta_x_spec,
	gamma_tilde_spec_hour = gamma_tilde_spec_hour,
	gamma_spec_hour = gamma_spec_hour,
	intercept_het_L_effect = intercept_het_L_effect,
	intercept_het_L_effect_eta_high = intercept_het_L_effect_eta_high,
	intercept_het_L_effect_eta_low = intercept_het_L_effect_eta_low,
	intercept_het_B = intercept_het_B,
	intercept_het_L = intercept_het_L,
	intercept_het_B_hour = intercept_het_B_hour,
	intercept_het_L_hour = intercept_het_L_hour,
	intercept_het_L_effect_hour = intercept_het_L_effect_hour,
	gamma_spec_omega_hour = gamma_spec_omega_hour,
	intercept_spec_omega_hour = intercept_spec_omega_hour,
	intercept_het_L_effect_omega_hour = intercept_het_L_effect_omega_hour,
	intercept_het_L_omega_hour = intercept_het_L_omega_hour,
	intercept_het_B_multiple = intercept_het_B_multiple,
	intercept_het_L_multiple = intercept_het_L_multiple,
	intercept_het_L_effect_omega = intercept_het_L_effect_omega,
	intercept_het_L_omega = intercept_het_L_omega,
	x_ss_spec_res = x_ss_spec_w_res,
	intercept_spec_res = intercept_spec_res,
	intercept_spec_res_hour = intercept_spec_res_hour,
	gamma_tilde_spec_res = gamma_tilde_spec_w_res,
	gamma_spec_res = gamma_spec_w_res,
	delta_x_spec_res = delta_x_spec_res,
	gamma_tilde_spec_res_hour = gamma_tilde_spec_res_hour,
	gamma_spec_res_hour = gamma_spec_res_hour,
	intercept_het_L_effect_res = intercept_het_L_effect_res,
	intercept_het_L_effect_eta_high_res = intercept_het_L_effect_eta_high_res,
	intercept_het_L_effect_eta_low_res = intercept_het_L_effect_eta_low_res,
	intercept_het_B_res = intercept_het_B_res,
	intercept_het_L_res = intercept_het_L_res,
	intercept_het_B_res_hour = intercept_het_B_res_hour,
	intercept_het_L_res_hour = intercept_het_L_res_hour,
	intercept_het_L_effect_res_hour = intercept_het_L_effect_res_hour,
	gamma_spec_omega_res_hour = gamma_spec_omega_res_hour,
	intercept_spec_omega_res_hour = intercept_spec_omega_res_hour,
	intercept_het_L_effect_omega_res_hour = intercept_het_L_effect_omega_res_hour,
	intercept_het_L_omega_res_hour = intercept_het_L_omega_res_hour,
	intercept_het_L_effect_multiple_res = intercept_het_L_effect_multiple_res,
	intercept_het_B_multiple_res = intercept_het_B_multiple_res,
	intercept_het_L_multiple_res = intercept_het_L_multiple_res,
	intercept_het_L_effect_omega_res = intercept_het_L_effect_omega_res,
	intercept_het_L_omega_res = intercept_het_L_omega_res
	)
	return(solution)
	}



	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Check functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~



	check_sys_eq_1 <- function(param) {
	# Define
	rho <- param[['rho']]
	lambda <- param[['lambda']]
	tau_B_2 <- param[['tau_B_2']]
	tau_B_3 <- param[['tau_B_3']]
	tau_B_4 <- param[['tau_B_4']]
	tau_B_5 <- param[['tau_B_5']]
	diff <- rho + lambda

	# Rho
	diff <- diff - tau_B_5 / (tau_B_4*(1 + lambda))

	# Lambda
	diff <- diff - tau_B_4 / (rhotau_B_3 + rho^2tau_B_2)

	# Return
	return(diff)
	}


	calculate_steady_state <- function(param, gamma_tilde, gamma, alpha, rho, lambda, mispredict, eta, zeta, intercept=NA, eta_scale=1){

	eta <- eta * eta_scale
	delta <- param[['delta']]

	# Calculate
	term1 <- intercept
	term2 <- (1-alpha)deltarho
	term3 <- (zeta-eta) * mispredict - gamma_tilde*(1+lambda)
	num <- term1 + term2*term3 + gamma

	terma <- -eta -((1-alpha)deltarho*(zeta-eta))
	termb <- zeta((rho - (1-alpha)delta(rho*2)) / (1-rho))
	denom <- terma - termb

	x_ss_calc <- num /denom
	return(x_ss_calc)
	}



	calculate_x_ss_i_spec <- function(df){
	x_ss_i_df <- df %>%
	mutate(x_ss_i_data = PD_P1_UsageFITSBY)


	return(x_ss_i_df$x_ss_i_data)

	}

	get_avg_use <- function(df){
	estimate <- list(signif(weighted.mean(df$PD_P1_UsageFITSBY,df$w, na.rm=TRUE), digits=3))
	names(estimate) <- c('avg_use')

	# Return
	return(estimate)

	}

	get_fb <- function(df){
	df <- df %>%
	mutate(F_B_uncensored = 50*PD_P1_UsageFITSBY/20) %>%
	mutate(F_B_min = ifelse(F_B_uncensored<150, F_B_uncensored, 150)) %>%
	mutate(F_B = F_B_min/num_days)

	estimate <- list(signif(weighted.mean(df$F_B,df$w, na.rm=TRUE), digits=3))
	names(estimate) <- c('F_B')

	# Return
	return(estimate)
	}


	calculate_intercept_spec <- function(x_ss_i_data, param, gamma_tilde, gamma, alpha, rho, lambda, mispredict, eta, zeta, eta_scale=1){
	delta <- param[['delta']]
	eta <- eta*eta_scale

	terma <- (1-alpha)deltarho
	termb <- (rho - terma*rho) / (1-rho)

	term1 <- -eta -terma(zeta-eta)-zetatermb
	term2 <- terma((zeta-eta)mispredict - (1+lambda)*gamma_tilde)

	intercept <- x_ss_i_data*term1 - term2 - gamma

	return(intercept)
	}

	check_steady_state <- function(param){

	# Define
	x_ss <- param[['x_ss']]
	gamma_L_effect <- param[['gamma_L_effect']]
	gamma_tilde_L_effect <- param[['gamma_tilde_L_effect']]
	mispredict <- param[['mispredict']]
	rho <- param[['rho']]
	alpha <- param[['alpha']]
	intercept_het_L_effect <- param[['intercept_het_L_effect']]
	intercept_het_L_effect_eta_high <- param[['intercept_het_L_effect_eta_high']]
	intercept_het_L_effect_eta_low <- param[['intercept_het_L_effect_eta_low']]

	print('\|\|\|\|\|\|\|\|\|\|\|\|\| Initial calculation \|\|\|\|\|\|\|\|\|\|\|\|\|')
	print(x_ss)

	#print('\|\|\|\|\|\|\|\|\|\|\|\|\| Checked calculation rho_tilde=0 \|\|\|\|\|\|\|\|\|\|\|\|\|')
	#print(calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, mispredict, intercept = intercept_het_L_effect))
	# print(paste0('alpha 1.0eta:', intercept_het_L_effect))

	# print('\|\|\|\|\|\|\|\|\|\|\|\|\| Checked calculation rho_tilde=0, eta*1.1 \|\|\|\|\|\|\|\|\|\|\|\|\|')
	# print(calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, mispredict, intercept = intercept_het_L_effect_eta_high))
	# print(paste0('alpha 1.1eta:', intercept_het_L_effect_eta_high))

	# print('\|\|\|\|\|\|\|\|\|\|\|\|\| Checked calculation rho_tilde=0, eta*0.9 \|\|\|\|\|\|\|\|\|\|\|\|\|')
	# print(calculate_steady_state(param, gamma_tilde_L_effect, gamma_L_effect, alpha, rho, mispredict, intercept = intercept_het_L_effect_eta_low))
	# print(paste0('alpha 0.9eta:', intercept_het_L_effect_eta_low))

	}



	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Moment functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	get_bonus <- function(df, t) {
	# Specify regression
	dep_var <- sprintf('PD_P%s_UsageFITSBY', t)

	if (t == 1) {
	eq <- paste0(dep_var, '~ L + B + S')
	} else if (t > 1) {
	eq <- paste0(dep_var, '~ PD_P1_UsageFITSBY + L + B + S')
	}

	# Run regression
	fit <- lm(data = df,
	formula = eq,
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['B']]>0, 0,fit$coefficients[['B']]))
	names(estimate) <- c(sprintf('tau_B_%s', t))
	# Return
	return(estimate)
	}

	get_multiple_good_taus <- function(df) {
	# Run regression
	fit <- lm(data = df,
	formula = S4_Substitution_W ~ L+B+S,
	weights = w)

	estimate <-
	list(fit$coefficients[['L']],
	fit$coefficients[['B']])

	names(estimate) <- c('tau_L_y', 'tau_B_y')

	return(estimate)
	}

	get_full_bonus_2 <- function(df, t) {
	# Run regression
	fit <- lm(data = df,
	formula = 'PD_P2_UsageFITSBY ~ PD_P1_UsageFITSBY + L + B + S',
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['B']]>0, 0,fit$coefficients[['B']]))
	names(estimate) <- c('tau_B_2_full')
	# Return
	return(estimate)
	}



	get_bonus_2b <- function(df, full=FALSE) {
	if (full){
	fit <- lm(data = df,
	formula = 'PD_P2_UsageFITSBY ~ PD_P1_UsageFITSBY + L + B + S',
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['B']]>0, 0,fit$coefficients[['B']]))
	names(estimate) <- c('tau_B_2')
	} else{
	days <- 33:42

	fit <- lm(data = df %>%
	mutate(PD_P2b_Usage_FITSBY =
	rowSums(.[paste0('PD_DailyUsageFITSBY_',days)], na.rm=TRUE)/length(days)),
	formula = 'PD_P2b_Usage_FITSBY ~ PD_P1_UsageFITSBY + L + B + S',
	weights = w)

	coef <- fit$coefficients[['B']]
	coef <- ifelse(coef>0, 0, coef)
	estimate <- list(coef)
	names(estimate) <- c('tau_B_2')

	}
	# Return
	return(estimate)
	}

	get_bonus_tilde <- function(df) {
	# Specify regression
	dep_var <- sprintf('S3_PredictUseNext_1_W')
	eq <- paste0(dep_var, '~ PD_P1_UsageFITSBY + L + B + S')

	# Run regression
	fit <- lm(data = df,
	formula = eq,
	weights = w)
	estimate <- list(fit$coefficients[['B']])
	names(estimate) <- c(sprintf('tau_tilde_B_3_2', t))
	# Return
	return(estimate)
	}


	get_limit_tilde <- function(df) {
	# Specify regression
	eq <- 'S3_PredictUseNext_1_W ~ L + B + S'

	# Run regression
	fit <- lm(data = df,
	formula = eq,
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['L']]>0, 0,fit$coefficients[['L']]))
	names(estimate) <- c('tau_tilde_L')

	# Return
	return(estimate)
	}

	get_true_limit_tilde <- function(df) {
	# Specify regression
	eq <- 'S3_PredictUseNext_1_W ~ L + B + S'

	# Run regression
	fit <- lm(data = df,
	formula = eq,
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['L']]>0, 0,fit$coefficients[['L']]))
	names(estimate) <- c('true_tau_tilde_L')

	# Return
	return(estimate)
	}



	get_limit_tildes <- function(df, t) {
	# Specify regression
	dep_var <- sprintf('S2_PredictUseNext_%s_W', t-1)
	eq <- paste0(dep_var, '~ PD_P1_UsageFITSBY + L + B + S')

	# Run regression
	fit <- lm(data = df,
	formula = eq,
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['L']]>0, 0,fit$coefficients[['L']]))
	names(estimate) <- c(sprintf('tau_tilde_L_%s_2', t))
	# Return
	return(estimate)
	}

	get_limit_2b <- function(df) {
	# Run regression
	fit <- lm(data = df ,
	formula = 'PD_P2_UsageFITSBY ~ PD_P1_UsageFITSBY + L + B + S',
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['L']]>0, 0,fit$coefficients[['L']]))
	names(estimate) <- c('tau_L_2')
	# Return
	return(estimate)
	}

	get_limit <- function(df) {
	#Ryb regressuib
	fit <- lm(data = df ,
	formula = 'PD_P3_UsageFITSBY ~ PD_P1_UsageFITSBY + L + B + S',
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['L']]>0, 0,fit$coefficients[['L']]))
	names(estimate) <- c('tau_L_3')

	# Return
	return(estimate)
	}

	get_MPL_tilde <- function(df) {
	regression_df <- df %>%
	mutate(S2_reduction = PD_P3_UsageFITSBY - S2_PredictUseInitial_W) %>%
	select(S2_reduction, S2_PredictUseInitial_W, S2_PredictUseBonus, PD_P3_UsageFITSBY, B, S, L, PD_P1_UsageFITSBY,w)

	# Run regression
	fit <- lm(data = regression_df,
	formula = S2_reduction ~ B + S + L + PD_P1_UsageFITSBY,
	weights = w)
	estimate <- list(fit$coefficients[['B']])
	names(estimate) <- c(sprintf('MPL_S2'))
	# Return
	return(estimate)
	}




	get_limit_last_week <- function(df) {
	# Run regression
	eq <- 'PD_WeeklyUsageFITSBY_15 ~ PD_P1_UsageFITSBY + L + B + S'
	fit <- lm(data = df,
	formula = eq,
	weights = w)
	estimate <- list(fit$coefficients[['L']])
	names(estimate) <- c('limit_effect_last_week')

	# Return
	return(estimate)
	}

	get_limit_avg <- function(df) {
	# Reshape data
	df %<>%
	select(
	UserID,
	w,
	L,
	B,
	S,
	PD_P1_UsageFITSBY,
	PD_P2_UsageFITSBY,
	PD_P3_UsageFITSBY,
	PD_P4_UsageFITSBY,
	PD_P5_UsageFITSBY
	)

	df %<>%
	gather(
	key = 'period',
	value = 'usage',
	-UserID,
	-w,
	-L,
	-B,
	-S,
	-PD_P1_UsageFITSBY
	)

	# Run regression
	eq <- 'usage ~ PD_P1_UsageFITSBY + L + B + S'
	fit <- lm(data = df,
	formula = eq,
	weights = w)
	estimate <- list(ifelse(fit$coefficients[['L']]>0, 0,fit$coefficients[['L']]))
	names(estimate) <- c('tau_L')

	# Return
	return(estimate)
	}

	get_taus <- function(df, winsorize=F, full=full) {
	# Bonus
	bonus <-
	3:5 %>%
	map(~get_bonus(df, .)) %>%
	list.flatten

	# Bonus Tilde
	bonus_tilde <- get_bonus_tilde(df)

	# Bonus 2B
	bonus_2b <- get_bonus_2b(df, full=full)

	bonus_2_full <- get_full_bonus_2(df)

	#Get MPL
	MPL <- get_MPL_tilde(df)


	# Limit
	limit <- get_limit(df)
	limit_tilde <- get_limit_tilde(df)
	true_limit_tilde <- get_true_limit_tilde(df)

	limit_tilde[[1]] <- limit[[1]]

	# Limit Tilde
	limit_tildes <-
	2:3 %>%
	map(~get_limit_tildes(df, .)) %>%
	list.flatten

	limit_2 <- get_limit_2b(df)

	limit_avg <- get_limit_avg(df)

	multiple_good_taus <- get_multiple_good_taus(df)

	# Return
	tau <- list.merge(bonus, bonus_tilde, bonus_2b, bonus_2_full, limit, limit_tilde, limit_tildes,
	limit_2,limit_avg, true_limit_tilde, MPL, multiple_good_taus)

	if (winsorize){
	tau_tilde_limit <- -4
	ratio_limit <- 0.18

	tau$tau_tilde_L <- min(-2, tau$tau_tilde_L)
	tau$tau_B_5 <- min(tau$tau_B_5, tau$tau_B_4 - ratio_limit * tau$tau_B_3)
	}

	return(tau)
	}

	get_mispredict <- function(df) {
	# Filter to control
	df_unfiltered <- df



	df %<>% filter(B == 0, L == 0)

	# Reshape data
	names(df) <- gsub('PD_(.*)_UsageFITSBY', 'UsageActual_\\1', names(df))
	names(df) <- gsub('S(.*)_PredictUseNext_1_W', 'UsagePredicted_P\\1', names(df))

	df %<>%
	select(
	UserID,
	w,
	starts_with('UsagePredicted'),
	starts_with('UsageActual')
	) %>%
	select(
	UserID,
	w,
	ends_with('P2'),
	ends_with('P3'),
	ends_with('P4')
	)

	df %<>%
	gather(key = 'period', value = 'value', -UserID, -w) %>%
	separate(period, c('key', 'period'), sep = '_') %>%
	spread(key = 'key', value = 'value')

	# Get mean
	df %<>% mutate(mispredict = UsageActual - UsagePredicted)

	estimate <-
	list(
	weighted.mean(df$mispredict, df$w, na.rm = T),
	weighted.mean(df_unfiltered$PD_P1_UsageFITSBY, df_unfiltered$w, na.rm=T),
	weighted.mean(df$UsagePredicted, df$w, na.rm = T)
	)
	names(estimate) <- c('mispredict', 'x_ss', 'x_tilde')

	# Return
	return(estimate)
	}

	get_ideal <- function(df) {
	# Reshape data
	df %<>%
	select(
	UserID,
	w,
	L,
	B,
	S3_PhoneUseChange
	)

	df %<>% gather(key = 'key', value = 'value', -UserID, -w, -L, -B)

	# Get mean
	estimate <-
	list(
	weighted.mean(filter(df, L == 1)$value, filter(df, L == 1)$w, na.rm = T),
	weighted.mean(filter(df, L == 0)$value, filter(df, L == 0)$w, na.rm = T)
	)

	names(estimate) <- c('d_L', 'd_CL')

	# Return
	return(estimate)
	}

	get_predict <- function(df) {
	# Filter to control
	df %<>% filter(B == 1)

	# Reshape data
	df %<>%
	select(
	UserID,
	w,
	ends_with('PredictUseInitial'),
	ends_with('PredictUseBonus')
	)

	#reformats any winsorization
	names(df) <- gsub('(.*)_W', '\\1', names(df))

	df %<>%
	gather(key = 'period', value = 'value', -UserID, -w) %>%
	separate(period, c('period', 'key'), sep = '_') %>%
	spread(key = 'key', value = 'value')

	# Get mean
	df %<>%
	mutate(PredictUseBonus = PredictUseInitial * (1 - (PredictUseBonus / 100))) %>%
	mutate(bonus_effect = PredictUseBonus - PredictUseInitial) %>%
	mutate(average = (PredictUseBonus + PredictUseInitial) / 2)

	estimate <- list(
	weighted.mean(df$bonus_effect, df$w, na.rm = T),
	weighted.mean(filter(df, period == 'S2')$average, filter(df, period == 'S2')$w, na.rm = T)
	)

	names(estimate) <- c('tau_tilde_B', 'x_tilde_2_B')

	# Return
	return(estimate)
	}

	get_wtp <- function(df) {
	df %<>%
	select(
	UserID,
	w,
	ends_with('MPL'),
	ends_with('MPLLimit')
	)

	df %<>%
	gather(key = 'period', value = 'value', -UserID, -w) %>%
	separate(period, c('period', 'key'), sep = '_') %>%
	spread(key = 'key', value = 'value')

	# Get mean
	estimate <- list(
	weighted.mean(df$MPL, df$w, na.rm = T) / num_days,
	weighted.mean(df$MPLLimit, df$w, na.rm = T) / num_days
	)

	names(estimate) <- c('v_B', 'v_L')

	# Return
	return(estimate)
	}

	find_tau_spec <- function(df, save_reg=FALSE){
	tau_data <- df %>%
	select(
	UserID,
	w, L, B, S,
	PD_P1_UsageFITSBY,
	PD_P2_UsageFITSBY,
	PD_P3_UsageFITSBY,
	PD_P4_UsageFITSBY,
	PD_P5_UsageFITSBY,
	PD_P2_LimitTightFITSBY
	)

	tau_data %<>%
	gather(
	key = 'period',
	value = 'usage',
	-UserID,
	-w, -L, -B, -S,
	-PD_P1_UsageFITSBY,
	-PD_P2_LimitTightFITSBY
	)

	fit <- tau_data %>%
	mutate(tightness=ifelse(L,PD_P2_LimitTightFITSBY, 0)) %>%
	lm(formula = 'usage ~ PD_P1_UsageFITSBY + L + tightness + B + S',
	weights = w)

	if(save_reg){
	stargazer(fit,
	omit.stat = c("adj.rsq","f","ser"),
	covariate.labels = c("Period 1 Usage", "Limit Treatment",
	"Limit Treatment x Period 2 Limit Tightness",
	"Bonus Treatment"),
	dep.var.labels = "FITSBY Usage (mins/day)",
	title = "",
	omit = c("S1", "S2", "S3", "S4",
	"S5", "S6", "S7", "S8"),
	type = "latex",
	omit.table.layout = "n",
	float = FALSE,
	dep.var.caption = "",
	star.cutoffs = NA,
	out = "output/heterogeneity_reg.tex"
	)
	}

	const <- fit$coefficients['L']
	slope <- fit$coefficients['tightness']

	return(const + slope * df$PD_P2_LimitTightFITSBY)
	}


	find_tau_L2_spec <- function(df, save_reg=FALSE){
	tau_data <- df %>%
	select(
	UserID,
	w, L, B, S,
	PD_P1_UsageFITSBY,
	PD_P2_UsageFITSBY,
	PD_P3_UsageFITSBY,
	PD_P4_UsageFITSBY,
	PD_P5_UsageFITSBY,
	PD_P2_LimitTightFITSBY
	)

	fit <- tau_data %>%
	mutate(tightness=ifelse(L,PD_P2_LimitTightFITSBY, 0)) %>%
	lm(formula = 'PD_P2_UsageFITSBY ~ PD_P1_UsageFITSBY + L + tightness + B + S',
	weights = w)

	const <- fit$coefficients['L']
	slope <- fit$coefficients['tightness']

	return(const + slope * df$PD_P2_LimitTightFITSBY)
	}

	find_tau_L3_spec <- function(df, save_reg=FALSE){
	tau_data <- df %>%
	select(
	UserID,
	w, L, B, S,
	PD_P1_UsageFITSBY,
	PD_P2_UsageFITSBY,
	PD_P3_UsageFITSBY,
	PD_P4_UsageFITSBY,
	PD_P5_UsageFITSBY,
	PD_P2_LimitTightFITSBY
	)

	fit <- tau_data %>%
	mutate(tightness=ifelse(L,PD_P2_LimitTightFITSBY, 0)) %>%
	lm(formula = 'PD_P3_UsageFITSBY ~ PD_P1_UsageFITSBY + L + tightness + B + S',
	weights = w)

	const <- fit$coefficients['L']
	slope <- fit$coefficients['tightness']

	return(const + slope * df$PD_P3_LimitTightFITSBY)
	}



	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Model functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	estimate_model <- function(df, param, winsorize=F, full=F, display_warning=FALSE) {
	# Empirical moments
	param %<>%
	list.merge(
	#get_opt(df),
	get_taus(df, winsorize=winsorize, full=full),
	get_mispredict(df),
	get_ideal(df),
	get_predict(df),
	get_wtp(df),
	get_avg_use(df),
	get_fb(df),
	get_limit_last_week(df)
	)

	# Solve system of equation #1
	param %<>%
	solve_sys_eq_1 %>%
	as.list %>%
	list.merge(param)

	# Solve system of equations #2
	param %<>%
	solve_sys_eq_2(display_warning=display_warning) %>%
	as.list %>%
	list.merge(param)

	# Solve system of equations #3
	param %<>%
	solve_sys_eq_3 %>%
	as.list %>%
	list.merge(param)

	# Solve for individual effects
	tau_L_2_spec <- find_tau_L2_spec(df)
	tau_tilde_spec <- find_tau_L3_spec(df)
	x_ss_i_data <- calculate_x_ss_i_spec(df)

	param %<>%
	solve_effects_individual(x_ss_i_data= x_ss_i_data, tau_tilde_L=tau_tilde_spec, tau_L_2=tau_L_2_spec, w=df$w)%>%
	as.list %>%
	list.merge(param)

	# Solve for effects
	param %<>%
	solve_effects(df) %>%
	as.list %>%
	list.merge(param)

	# Solve for counterfactuals
	param %<>%
	solve_counterfactuals(df) %>%
	as.list %>%
	list.merge(param)

	# Return
	return(param)
	}

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Plotting functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	plot_time_effects <- function(median, bottom, top, filename=""){
	order_results_time <- function(df1){
	r <- c(df1[['delta_x_cap_res']],
	df1[['delta_x_tilde_B_res']],
	df1[['delta_x_tilde_L_res']],
	df1[['delta_x_spec_res']])
	return(r)
	}

	order_results_time_est <- function(df1){
	r <- c(df1[['delta_x_cap']],
	df1[['delta_x_tilde_B']],
	df1[['delta_x_tilde_L']],
	df1[['delta_x_spec']])
	return(r)
	}


	x <- c('Limit\n effect', 'Bonus \n valuation', 'Limit \n valuation', 'Heterogeneous\n limit effect')

	names <- rep(factor(x, levels=x),2)

	counterfactuals_zero <- order_results_time(param_full)
	lower_counterfactuals_zero <- order_results_time(bottom)
	upper_counterfactuals_zero <- order_results_time(top)
	counterfactuals_est <- order_results_time_est(param_full)
	lower_counterfactuals_est <- order_results_time_est(bottom)
	upper_counterfactuals_est <- order_results_time_est(top)


	df <- data.frame(names, counterfactuals_zero, lower_counterfactuals_zero, upper_counterfactuals_zero,
	counterfactuals_est, lower_counterfactuals_est, upper_counterfactuals_est)


	cols <- c("No projection bias"=maroon,
	"Perceived projection bias"=grey)


	a <- ggplot(df, aes(x=names, width=.2)) +
	geom_point(aes(y=counterfactuals_zero, colour="No projection bias"), stat="identity", position = position_nudge(x = -.1)) +
	geom_point(aes(y=counterfactuals_est, colour="Perceived projection bias"), stat="identity", position = position_nudge(x = .1))+
	geom_errorbar(aes(ymin=lower_counterfactuals_zero, ymax=upper_counterfactuals_zero), stat="identity", colour=maroon, position = position_nudge(x = -.1)) +
	geom_errorbar(aes(ymin=lower_counterfactuals_est, ymax=upper_counterfactuals_est), stat="identity", colour=grey, position=position_nudge(x = .1)) +
	scale_y_continuous(name="Time effect (minutes/day)") +
	theme_classic() +
	labs(x = "") +
	scale_colour_manual(name = "", values=cols,
	labels = unname(TeX(c(paste0("Zero projection bias", " (", "$\\alpha=1$", ")"),
	paste0("Estimated projection bias", " (" ,"$\\alpha=\\hat{\\alpha}$", ")")
	))))+
	theme(legend.text.align = 0,
	legend.key.height = unit(1, "cm"),
	legend.position="bottom") +
	theme(legend.margin=margin(0,0,0,0),
	legend.box.margin=margin(-10,-10,-10,-10)) +
	theme(axis.text.x = element_text(colour="black"))
	coord_cartesian(ylim = c(0, 180))


	ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")
	}

	plot_time_effects_bal <- function(param_full, param_balanced, bottom, top, bottom_bal, top_bal, filename=""){
	order_results_time <- function(df1){
	r <- c(df1[['delta_x_cap_res']],
	df1[['delta_x_tilde_B_res']],
	df1[['delta_x_tilde_L_res']],
	df1[['delta_x_spec_res']])
	return(r)
	}

	x <- c('Limit\n effect', 'Bonus \n valuation', 'Limit \n valuation', 'Heterogeneous\n limit effect')

	names <- rep(factor(x, levels=x),2)



	counterfactuals_zero <- order_results_time(param_full)
	lower_counterfactuals_zero <- order_results_time(bottom)
	upper_counterfactuals_zero <- order_results_time(top)
	counterfactuals_est <- order_results_time(param_balanced)
	lower_counterfactuals_est <- order_results_time(bottom_bal)
	upper_counterfactuals_est <- order_results_time(top_bal)


	df <- data.frame(names, counterfactuals_zero, lower_counterfactuals_zero, upper_counterfactuals_zero,
	counterfactuals_est, lower_counterfactuals_est, upper_counterfactuals_est)


	cols <- c("No perceived habit formation"=maroon ,
	"Perceived habit formation"=skyblue)


	a <- ggplot(df, aes(x=names, width=.2)) +
	geom_point(aes(y=counterfactuals_zero, colour="No perceived habit formation"), stat="identity", position = position_nudge(x = -.1)) +
	geom_point(aes(y=counterfactuals_est, colour="Perceived habit formation"), stat="identity", position = position_nudge(x = .1))+
	geom_errorbar(aes(ymin=lower_counterfactuals_zero, ymax=upper_counterfactuals_zero), stat="identity", colour=maroon, position = position_nudge(x = -.1)) +
	geom_errorbar(aes(ymin=lower_counterfactuals_est, ymax=upper_counterfactuals_est), stat="identity", colour=skyblue, position=position_nudge(x = .1)) +
	scale_y_continuous(name="Time effect (minutes/day)") +
	theme_classic() +
	labs(x = "") +
	scale_colour_manual(name = "", values=cols,
	labels = c("Unweighted sample", "Weighted toward U.S. adults"))+
	theme(legend.text.align = 0,
	legend.key.height = unit(1, "cm"),
	legend.position="bottom") +
	theme(legend.margin=margin(0,0,0,0),
	legend.box.margin=margin(-10,-10,-10,-10)) +
	theme(axis.text.x = element_text(colour="black")) +
	coord_cartesian(ylim = c(0, 165))

	ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")
	}


	plot_time_effects_both <- function(param_full, param_balanced, bottom, top, bottom_bal, top_bal, filename=""){
	order_results_time <- function(df1){
	r <- c(df1[['delta_x_cap_res']],
	df1[['delta_x_tilde_B_res']],
	df1[['delta_x_tilde_L_res']],
	df1[['delta_x_cap_multiple_res']],
	df1[['delta_B_multiple_res']],
	df1[['delta_x_L_multiple_res']],
	df1[['delta_x_cap_omega_res']],
	df1[['delta_x_L_omega_res']],
	df1[['delta_x_spec_res']])
	return(r)
	}

	order_results_bal <- function(df1){
	r <- c(df1[['delta_x_cap_res']])
	return(r)
	}


	x <- c('Limit effect',
	'Bonus valuation',
	'Limit valuation',
	'Limit effect \n multiple-good model',
	'Bonus valuation \n multiple-good model',
	'Limit valuation \n multiple-good model',
	'Limit effect \n w=w',
	'Limit valuation \n w=w',
	'Heterogeneous limit effect' )



	names <- factor(x, levels=x)

	x_bal <- c( 'Limit effect \n reweighted sample')
	names_bal <- factor(x_bal, levels=x_bal)

	counterfactuals <- order_results_time(param_full)
	lower_counterfactuals <- order_results_time(bottom)
	upper_counterfactuals <- order_results_time(top)
	counterfactuals_bal <- order_results_bal(param_balanced)
	lower_counterfactuals_bal <- order_results_bal(bottom_bal)
	upper_counterfactuals_bal<- order_results_bal(top_bal)


	df_normal <- data.frame(names, counterfactuals,lower_counterfactuals, upper_counterfactuals)

	df_bal <- data.frame(names_bal, counterfactuals_bal, lower_counterfactuals_bal, upper_counterfactuals_bal) %>%
	rename(names = names_bal,
	counterfactuals = counterfactuals_bal,
	lower_counterfactuals = lower_counterfactuals_bal,
	upper_counterfactuals = upper_counterfactuals_bal)

	df <- rbind(df_normal, df_bal)

	xaxislabels <- c('Limit effect',
	'Bonus valuation',
	'Limit valuation',
	'Limit effect, \n multiple-good model',
	'Bonus valuation, \n multiple-good model',
	'Limit valuation, \n multiple-good model',
	unname(TeX(paste0("Limit effect,", "$\\omega = \\hat{\\omega}$"))),
	unname(TeX(paste0("Limit valuation,", "$\\omega = \\hat{\\omega}$"))),
	'Heterogeneous limit effect',
	'Limit effect, \n weighted sample')


	a <- ggplot(df, aes(x=names, width=.2)) +
	geom_point(aes(y=counterfactuals), colour=maroon)+
	geom_errorbar(aes(ymin=lower_counterfactuals, ymax=upper_counterfactuals), colour=maroon) +
	scale_y_continuous(name="Effect of temptation on FITSBY use (minutes/day)") +
	theme_classic() +
	labs(x = "") +
	theme(axis.text.x = element_text(colour="black")) +
	coord_cartesian(ylim = c(0, 180)) +
	theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
	scale_x_discrete(labels= xaxislabels)+
	theme(axis.title.y = element_text(size = 10))


	ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")
	}


	# plot_time_effects_robust <- function(param_full, bottom, top, filename=""){
	# order_results_time <- function(df1){
	# r <- c(df1[['delta_x_cap']],
	# df1[['delta_x_tilde_B']],
	# df1[['delta_x_tilde_L']],
	# df1[['delta_x_cap_multiple']],
	# df1[['delta_x_L_multiple']],
	# df1[['delta_B_multiple']],
	# df1[['delta_x_cap_omega']],
	# df1[['delta_x_L_omega']],
	# df1[['delta_x_spec']])
	# return(r)
	# }



	# x <- c( 'Limit effect',
	# 'Bonus valuation',
	# 'Limit valuation',
	# 'Limit effect \n multiple-good model',
	# 'Limit valuation \n multiple-good model',
	# 'Bonus valuation \n multiple-good model',
	# 'Limit effect \n w=w',
	# 'Limit valuation \n w=w',
	# 'Heterogeneous limit effect' )



	# names <- factor(x, levels=x)


	# counterfactuals <- order_results_time(param_full)
	# lower_counterfactuals <- order_results_time(bottom)
	# upper_counterfactuals <- order_results_time(top)


	# df <- data.frame(names, counterfactuals)


	# xaxislabels <- c('Limit effect',
	# 'Bonus valuation',
	# 'Limit valuation',
	# 'Limit effect \n multiple-good model',
	# 'Limit valuation \n multiple-good model',
	# 'Bonus valuation \n multiple-good model',
	# unname(TeX(paste0("Limit effect,", "$\\omega = \\hat{\\omega}$"))),
	# unname(TeX(paste0("Limit valuation,", "$\\omega = \\hat{\\omega}$"))),
	# 'Heterogeneous limit effect')


	# a <- ggplot(df, aes(x=names, width=.2)) +
	# geom_point(aes(y=counterfactuals), colour=maroon)+
	# geom_errorbar(aes(ymin=lower_counterfactuals, ymax=upper_counterfactuals), colour=maroon) +
	# scale_y_continuous(name="Effect of temptation on FITSBY use (minutes/day)") +
	# theme_classic() +
	# labs(x = "") +
	# theme(axis.text.x = element_text(colour="black")) +
	# #coord_cartesian(ylim = c(-50, 200)) +
	# theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
	# scale_x_discrete(labels= xaxislabels)+
	# theme(axis.title.y = element_text(size = 10))


	# ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")
	# }


	plot_time_effects_both_est <- function(param_full, bottom, top, filename=""){
	order_results_time <- function(df1){
	r <- c(df1[['delta_x_cap']],
	df1[['delta_x_tilde_B']],
	df1[['delta_x_tilde_L']],
	df1[['delta_B_multiple']],
	df1[['delta_x_L_multiple']],
	df1[['delta_x_cap_omega']],
	df1[['delta_x_L_omega']],
	df1[['delta_x_spec']])
	return(r)
	}



	x <- c('Limit effect',
	'Bonus valuation',
	'Limit valuation',
	'Bonus valuation, \n multiple-good model',
	'Limit valuation, \n multiple-good model',
	'Limit effect \n w=w',
	'Limit valuation \n w=w',
	'Heterogeneous limit effect' )



	names <- factor(x, levels=x)


	counterfactuals <- order_results_time(param_full)
	lower_counterfactuals <- order_results_time(bottom)
	upper_counterfactuals <- order_results_time(top)


	df <- data.frame(names, counterfactuals)


	xaxislabels <- c('Limit effect',
	'Bonus valuation',
	'Limit valuation',
	'Bonus valuation, \n multiple-good model',
	'Limit valuation, \n multiple-good model',
	unname(TeX(paste0("Limit effect,", "$\\omega = \\hat{\\omega}$"))),
	unname(TeX(paste0("Limit valuation,", "$\\omega = \\hat{\\omega}$"))),
	'Heterogeneous limit effect')


	a <- ggplot(df, aes(x=names, width=.2)) +
	geom_point(aes(y=counterfactuals), colour=maroon)+
	geom_errorbar(aes(ymin=lower_counterfactuals, ymax=upper_counterfactuals), colour=maroon) +
	scale_y_continuous(name="Effect of temptation on FITSBY use (minutes/day)") +
	theme_classic() +
	labs(x = "") +
	theme(axis.text.x = element_text(colour="black")) +
	coord_cartesian(ylim = c(0, 200)) +
	theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
	scale_x_discrete(labels= xaxislabels)+
	theme(axis.title.y = element_text(size = 10))


	ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")
	}


	plot_decomposition_boot <- function(median, bottom, top, filename){
	order_results_zero <- function(df1){
	r <- c(df1[['x_ss_L_effect_res']],
	df1[['x_ss_naivete_res']],
	df1[['x_ss_temptation_res']],
	df1[['x_ss_habit_res']],
	df1[['x_ss_habit_temptation_res']])
	return(r)
	}

	order_results_est <- function(df1){
	r <- c(df1[['x_ss_L_effect']],
	df1[['x_ss_naivete']],
	df1[['x_ss_temptation']],
	df1[['x_ss_habit']],
	df1[['x_ss_habit_temptation']])
	return(r)
	}

	x <- c('Baseline','No naivete', 'No temptation', 'No habit\nformation',
	'No temptation\n or habit formation')

	names <- rep(factor(x, levels=x),2)

	counterfactuals_zero <- order_results_zero(param_full)
	lower_counterfactuals_zero <- order_results_zero(bottom)
	upper_counterfactuals_zero <- order_results_zero(top)
	counterfactuals_est <- order_results_est(param_full)
	lower_counterfactuals_est <- order_results_est(bottom)
	upper_counterfactuals_est <- order_results_est(top)


	df <- data.frame(names, counterfactuals_zero, lower_counterfactuals_zero, upper_counterfactuals_zero,
	counterfactuals_est, lower_counterfactuals_est, upper_counterfactuals_est)


	cols <- c("No projection bias"=maroon ,
	"Perceived projection bias"=grey)


	a <- ggplot(df, aes(x=names, width=.2)) +
	geom_point(aes(y=counterfactuals_zero, colour="No projection bias"), stat="identity", position = position_nudge(x = -.1)) +
	geom_point(aes(y=counterfactuals_est, colour="Perceived projection bias"), stat="identity", position = position_nudge(x = .1))+
	geom_errorbar(aes(ymin=lower_counterfactuals_zero, ymax=upper_counterfactuals_zero), stat="identity", colour=maroon, position = position_nudge(x = -.1)) +
	geom_errorbar(aes(ymin=lower_counterfactuals_est, ymax=upper_counterfactuals_est), stat="identity", colour=grey, position=position_nudge(x = .1)) +
	scale_y_continuous(name="FITSBY use (minutes/day)") +
	theme_classic() +
	#theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
	labs(x = "") +
	scale_colour_manual(name = "", values=cols,
	labels = unname(TeX(c(paste0("Restricted model", " (", "$\\alpha$", "$=$", "$1$", ")"),
	paste0("Unrestricted model", " (" ,"$\\alpha$", "$=$", "$\\hat{\\alpha}$", ")")
	))))+
	theme(legend.text.align = 0,
	legend.key.height = unit(1, "cm"),
	legend.position="bottom") +
	theme(legend.margin=margin(0,0,0,0),
	legend.box.margin=margin(-10,-10,-10,-10)) +
	theme(axis.text.x = element_text(colour="black")) +
	coord_cartesian(ylim = c(0, 200))





	ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")

	}

	plot_decomposition_boot_etas <- function(median, bottom, top, filename){

	order_results_zero <- function(df1){
	r <- c(df1[['x_ss_L_effect_res']],
	df1[['x_ss_naivete_res']],
	df1[['x_ss_temptation_res']],
	df1[['x_ss_habit_res']],
	df1[['x_ss_habit_temptation_res']])
	return(r)
	}

	order_results_zero_eta_high <- function(df1){
	r <- c(df1[['x_ss_L_effect_eta_high_res']],
	df1[['x_ss_naivete_eta_high_res']],
	df1[['x_ss_temptation_eta_high_res']],
	df1[['x_ss_habit_eta_high_res']],
	df1[['x_ss_habit_temptation_eta_high_res']])
	return(r)
	}

	order_results_zero_eta_low <- function(df1){
	r <- c(df1[['x_ss_L_effect_eta_low_res']],
	df1[['x_ss_naivete_eta_low_res']],
	df1[['x_ss_temptation_eta_low_res']],
	df1[['x_ss_habit_eta_low_res']],
	df1[['x_ss_habit_temptation_eta_low_res']])
	return(r)
	}

	x <- c('Baseline','No naivete', 'No temptation', 'No habit\nformation',
	'No temptation\n or habit formation')

	names <- rep(factor(x, levels=x),2)

	counterfactuals_zero <- order_results_zero(param_full)
	lower_counterfactuals_zero <- order_results_zero(bottom)
	upper_counterfactuals_zero <- order_results_zero(top)

	counterfactuals_zero_eta_high <- order_results_zero_eta_high(param_full)
	lower_counterfactuals_zero_eta_high <- order_results_zero_eta_high(bottom)
	upper_counterfactuals_zero_eta_high <- order_results_zero_eta_high(top)

	counterfactuals_zero_eta_low <- order_results_zero_eta_low(param_full)
	lower_counterfactuals_zero_eta_low <- order_results_zero_eta_low(bottom)
	upper_counterfactuals_zero_eta_low <- order_results_zero_eta_low(top)


	df <- data.frame(names, counterfactuals_zero, lower_counterfactuals_zero, upper_counterfactuals_zero,
	counterfactuals_zero_eta_high, lower_counterfactuals_zero_eta_high, upper_counterfactuals_zero_eta_high,
	counterfactuals_zero_eta_low, lower_counterfactuals_zero_eta_low, upper_counterfactuals_zero_eta_low)


	cols <- c("eta_0.9"=deepskyblue,
	"eta_1.0"=maroon,
	"eta_1.1"=grey)


	a <- ggplot(df, aes(x=names, width=.2)) +
	geom_point(aes(y=counterfactuals_zero_eta_low, colour="eta_0.9"), stat="identity", position = position_nudge(x = -.3))+
	geom_point(aes(y=counterfactuals_zero, colour="eta_1.0"), stat="identity", position = position_nudge(x = -.1)) +
	geom_point(aes(y=counterfactuals_zero_eta_high, colour="eta_1.1"), stat="identity", position = position_nudge(x = .1)) +
	geom_errorbar(aes(ymin=lower_counterfactuals_zero_eta_low, ymax=upper_counterfactuals_zero_eta_low), stat="identity", colour=deepskyblue, position=position_nudge(x = -.3)) +
	geom_errorbar(aes(ymin=lower_counterfactuals_zero, ymax=upper_counterfactuals_zero), stat="identity", colour=maroon, position = position_nudge(x = -.1)) +
	geom_errorbar(aes(ymin=lower_counterfactuals_zero_eta_high, ymax=upper_counterfactuals_zero_eta_high), stat="identity", colour=grey, position=position_nudge(x = .1)) +
	scale_y_continuous(name="FITSBY use (minutes/day)") +
	theme_classic() +
	#theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
	labs(x = "") +
	scale_colour_manual(name = "", values=cols,
	labels = unname(TeX(c(paste0("$\\eta$", "$=0.9\\hat{\\eta}$", " ,(", "$\\alpha$", "$=1$", ")"),
	paste0("$\\eta$", "$=1.0\\hat{\\eta}$", " ,(", "$\\alpha$", "$=1$", ")"),
	paste0("$\\eta$", "$=1.1\\hat{\\eta}$", " ,(", "$\\alpha$", "$=1$", ")")
	)))) +
	theme(legend.text.align = 0,
	legend.key.height = unit(1, "cm"),
	legend.position="bottom") +
	theme(legend.margin=margin(0,0,0,0),
	legend.box.margin=margin(-10,-10,-10,-10)) +
	theme(axis.text.x = element_text(colour="black")) +
	coord_cartesian(ylim = c(0, 200))



	ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")

	}

	plot_decomposition_boot_unique <- function(median, bottom, top, filename){
	order_results_zero <- function(df1){
	r <- c(df1[['x_ss_L_effect_res']],
	df1[['x_ss_temptation_res']],
	df1[['x_ss_habit_res']],
	df1[['x_ss_habit_temptation_res']])
	return(r)
	}

	x <- c('Baseline', 'No temptation', 'No habit\nformation',
	'No temptation\n or habit formation')

	names <- rep(factor(x, levels=x),2)

	counterfactuals_zero <- order_results_zero(param_full)
	lower_counterfactuals_zero <- order_results_zero(bottom)
	upper_counterfactuals_zero <- order_results_zero(top)



	df <- data.frame(names, counterfactuals_zero, lower_counterfactuals_zero, upper_counterfactuals_zero)


	cols <- c("No perceived habit formation"=maroon)


	a <- ggplot(df, aes(x=names, width=.2)) +
	geom_point(aes(y=counterfactuals_zero, colour="No perceived habit formation"), stat="identity", position = position_nudge(x = -.1)) +
	geom_errorbar(aes(ymin=lower_counterfactuals_zero, ymax=upper_counterfactuals_zero), stat="identity", colour=maroon, position = position_nudge(x = -.1)) +
	scale_y_continuous(name="FITSBY use (minutes/day)") +
	theme_classic() +
	#theme(axis.text.x = element_text(angle = 45, hjust = 1)) +
	labs(x = "") +
	scale_colour_manual(name = "", values=cols,
	labels = unname(TeX(c(paste0("Restricted model", " (", "$\\alpha$ ", "$=1$", ")")
	))))+
	theme(legend.text.align = 0,
	legend.key.height = unit(1, "cm"),
	legend.position="bottom") +
	theme(legend.margin=margin(0,0,0,0),
	legend.box.margin=margin(-10,-10,-10,-10)) +
	theme(axis.text.x = element_text(colour="black")) +
	coord_cartesian(ylim = c(0, 165))





	ggsave(sprintf('output/%s.pdf', filename), plot=a, width=6.5, height=4.5, units="in")

	}




	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Bootstraps functions
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	run_boot_procedure <- function(fun){
	results <- mclapply(1:(size*1.1), fun, mc.set.seed = TRUE)
	results %<>% bind_rows

	nrowsunfiltered <- nrow(results)

	winsortaubtwopre <- nrow(results %>% filter(tau_B_2 ==0))

	winsortaubthreepre <- nrow(results %>% filter(tau_B_3 ==0))

	percenttaubthreepre <- winsortaubthreepre/nrowsunfiltered*100

	percenttaubthreepre <- signif(percenttaubthreepre, digits=2)



	winsortaubfourpre <- nrow(results %>% filter(tau_B_4 ==0))
	percenttaubfourpre <- winsortaubfourpre/nrowsunfiltered*100

	percenttaubfourpre <- signif(percenttaubfourpre, digits=2)


	winsortaubfivepre <- nrow(results %>% filter(tau_B_5 ==0))

	percenttaubfivepre <- winsortaubfivepre/nrowsunfiltered*100

	percenttaubfivepre <- signif(percenttaubfivepre, digits=2)


	winsortaultwopre <- nrow(results %>% filter(tau_L_2 ==0))

	percenttaultwopre <- winsortaultwopre/nrowsunfiltered*100

	percenttaultwopre <- signif(percenttaultwopre, digits=2)




	winsortautildelpre <- nrow(results %>% filter(tau_tilde_L ==0))

	percenttautildelpre <- winsortautildelpre/nrowsunfiltered*100

	percenttautildelpre <- signif(percenttautildelpre, digits=2)




	winsortautildebthreewtwopre <- nrow(results %>% filter(tau_tilde_B_3_2 ==0))

	percenttautildebthreewtwopre <- winsortautildebthreewtwopre/nrowsunfiltered*100

	percenttautildebthreewtwopre <- signif(percenttautildebthreewtwopre, digits=2)


	winsoreta <- nrow(results %>% filter(eta ==0))

	percenteta <- winsoreta/nrowsunfiltered*100

	percenteta <- signif(percenteta, digits=2)

	winsorzeta <- nrow(results %>% filter(zeta ==0))

	percentzeta <- winsorzeta/nrowsunfiltered*100

	percentzeta <- signif(percentzeta, digits=2)



	winsorlambda <- nrow(results %>% filter(lambda ==0))

	percentlambda <- winsorlambda/nrowsunfiltered*100

	percentlambda <- signif(percentlambda, digits=2)

	winsorlambdares <- nrow(results %>% filter(lambda_res ==0))

	percentlambdares <- winsorlambdares/nrowsunfiltered*100

	percentlambdares <- signif(percentlambdares, digits=2)



	winsorrho <- nrow(results %>% filter(rho ==0))

	percentrho <- winsorrho/nrowsunfiltered*100

	percentrho <- signif(percentrho, digits=2)





	winsoretares <- nrow(results %>% filter(eta_res ==0))

	percentetares <- winsoretares/nrowsunfiltered*100

	percentetares <- signif(percentetares, digits=2)



	winsorzetares <- nrow(results %>% filter(zeta_res ==0))

	percentzetares <- winsorzetares/nrowsunfiltered*100

	percentzetares <- signif(percentzetares, digits=2)



	winsorrhores <- nrow(results %>% filter(rho_res ==0))

	percentrhores <- winsorrhores/nrowsunfiltered*100

	percentrhores <- signif(percentrhores, digits=2)







	#dropdrawsdenom <- nrow(results %>% filter(is.na(rho_tilde)))

	#percentdenom <- dropdrawsdenom/nrowsunfiltered*100

	# percentdenom <- signif(percentdenom, digits=2)

	# dropdrawsdenomstm <- nrow(results %>% filter(is.na(rho_tilde_658)))

	#percentdenomstm <- dropdrawsdenomstm/nrowsunfiltered*100

	#percentdenomstm <- signif(percentdenomstm, digits=2





	winsortauspre <- nrow(results %>% filter(tau_tilde_B_3_2==0 \| tau_tilde_L==0 \| tau_L_2==0 \|tau_B_3 ==0 \| tau_B_5 ==0 \| tau_B_4 ==0))

	paramwinsorisepre <- nrow(results %>% filter(eta==0 \| lambda ==0 \| zeta==0 \| rho==0 \| alpha==1 \| eta_res==0 \| zeta_res==0 \|rho_res==0))

	percenttaubtwopre <- winsortaubtwopre/nrowsunfiltered*100

	percenttaubtwopre <- signif(percenttaubtwopre, digits=2)


	alphapre <- nrow(results %>% filter(alpha == 0 \| alpha == 1))

	percentalphapre <- alphapre/nrowsunfiltered*100

	percentalphapre <- signif(percentalphapre, digits=2)




	dropdrawsdenom <- nrow(results %>% filter(is.na(eta)))

	NegSSDenom <- dropdrawsdenom/nrowsunfiltered*100

	NegSSDenom <- signif(NegSSDenom, digits=2)



	alphaundefined <- nrow(results %>% filter(is.na(alpha)))

	percentalphaundefined <- alphaundefined/nrowsunfiltered*100

	percentalphaundefined <- signif(percentalphaundefined, digits=2)





	percenttauspre <- winsortauspre/nrowsunfiltered*100

	percenttauspre <- signif(percenttauspre, digits=2)

	percentparampre <- paramwinsorisepre/nrowsunfiltered*100

	percentparampre <- signif(percentparampre, digits=2)

	xsszeropre <- nrow(results %>% filter(x_ss_zero_temp==0))

	xsszeropre <- xsszeropre/nrowsunfiltered*100

	xsszeropre <- signif(xsszeropre, digits=2)




	combined <- list(nrowsunfiltered,
	percenttaubtwopre, percenttauspre, percentparampre, xsszeropre, percenttaubthreepre, percenttaubfourpre,
	percenttaubfivepre, percenttaultwopre, percenttautildelpre, percenttautildebthreewtwopre,percenteta,
	percentzeta, percentlambda, percentrho, percentetares, percentzetares, percentlambdares, percentrhores,
	percentalphapre, alphapre, NegSSDenom, dropdrawsdenom,
	paramwinsorisepre, percentalphaundefined)


	names(combined) <- c("nrowsunfiltered",
	"percenttaubtwopre", "percenttauspre", "percentparampre", "xsszeropre", "percenttaubthreepre", "percenttaubfourpre",
	"percenttaubfivepre", "percenttaultwopre", "percenttautildelpre", "percenttautildebthreewtwopre","percenteta",
	"percentzeta", "percentlambda", "percentrho", "percentetares", "percentzetares", "percentlambdares", "percentrhores",
	"percentalphapre", "alphapre", "NegSSDenom", "dropdrawsdenom",
	"paramwinsorisepre", "percentalphaundefined")

	save_nrow(combined, filename ="bootsdrawsnumbers", suffix="")



	return(results)
	}



	run_boot_iter <- function(s) {
	if (s %% 50 == 0){ print(s) }
	sample <- sample_n(df, nrow(df), replace = T)

	sample %<>%
	mutate(UserID = 1:n())

	param <- param_initial

	#You can chose to print statements that display warning when concavity conditions are not met when estimating the model
	try(param <- estimate_model(sample, param, display_warning=F))
	return(param)
	}

	run_boot_iter_full <- function(s) {
	if (s %% 50 == 0){ print(s) }
	sample <- sample_n(df, nrow(df), replace = T)

	sample %<>%
	mutate(UserID = 1:n())

	param <- param_initial

	try(param <- estimate_model(sample, param, full=T))
	return(param)
	}

	run_boot_iter_bal2 <- function(s) {
	if (s %% 50 == 0){ print(s) }
	sample <- sample_n(df, nrow(df), replace = T)

	sample %<>%
	mutate(UserID = 1:n())
	sample %<>% balance_data(magnitude=2)

	param <- param_initial
	try(param <- estimate_model(sample, param, winsorize=T))
	return(param)
	}

	run_boot_iter_bal <- function(s) {
	if (s %% 50 == 0){ print(s) }
	sample <- sample_n(df, nrow(df), replace = T)

	sample %<>%
	mutate(UserID = 1:n())
	sample %<>% balance_data(magnitude=3)

	param <- param_initial
	try(param <- estimate_model(sample, param, winsorize=T))
	return(param)
	}


	run_boot_iter_bal4 <- function(s) {
	if (s %% 50 == 0){ print(s) }
	sample <- sample_n(df, nrow(df), replace = T)

	sample %<>%
	mutate(UserID = 1:n())
	sample %<>% balance_data(magnitude=4)

	param <- param_initial
	try(param <- estimate_model(sample, param, winsorize=T))
	return(param)
	}
	run_boot_iter_bal5 <- function(s) {
	if (s %% 50 == 0){ print(s) }
	sample <- sample_n(df, nrow(df), replace = T)

	sample %<>%
	mutate(UserID = 1:n())
	sample %<>% balance_data(magnitude=5)

	param <- param_initial
	try(param <- estimate_model(sample, param, winsorize=T))
	return(param)
	}

	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
	# Latex saving functions for different formating numbers
	# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

	signnumber <- function(x){
	significant <- ifelse(x==0, sprintf("0"),sprintf(paste0("%0.", pmax(floor(3 - log(abs(x)+0.001, 10)), 0), "f"), x))
	return(significant)
	}

	signnumber2 <- function(x){
	significant <- ifelse(x==0, sprintf("0"),sprintf(paste0("%0.", pmax(floor(2 - log(abs(x)+0.01, 10)), 0), "f"), x))
	return(significant)
	}


	save_tex <- function(param, filename, suffix=""){
	tex_string <- ""
	for(x in names(param)){
	name <- x
	if (x %in% names(name_changes)){
	name <- name_changes[[x]]
	}
	new_line <- (sprintf("\\newcommand{\\%s%s}{%s}", name, suffix, signnumber(param[[x]])))
	tex_string <- sprintf("%s\n%s", tex_string, new_line)
	}
	# saves it as a .tex file
	sink(file=sprintf('output/%s.tex', filename))
	cat(tex_string)
	sink(NULL)
	}


	save_tex_nice <- function(param, filename, suffix=""){
	tex_string <- ""
	for(x in names(param)){
	name <- x
	if (x %in% names(name_changes)){
	name <- name_changes[[x]]
	}
	new_line <- (sprintf("\\newcommand{\\%s%s}{%s}", name, suffix, signif(param[[x]], digits=2)))
	tex_string <- sprintf("%s\n%s", tex_string, new_line)
	}
	# saves it as a .tex file
	sink(file=sprintf('output/%s.tex', filename))
	cat(tex_string)
	sink(NULL)
	}

	save_tex_one <- function(param, filename, suffix=""){
	tex_string <- ""
	for(x in names(param)){
	name <- x
	if (x %in% names(name_changes)){
	name <- name_changes[[x]]
	}
	new_line <- (sprintf("\\newcommand{\\%s%s}{%s}", name, suffix, format(param[[x]], digits=1)))
	tex_string <- sprintf("%s\n%s", tex_string, new_line)
	}
	# saves it as a .tex file
	sink(file=sprintf('output/%s.tex', filename))
	cat(tex_string)
	sink(NULL)
	}



	save_tex2 <- function(param, filename, suffix=""){
	tex_string <- ""
	for(x in names(param)){
	name <- x
	if (x %in% names(name_changes)){
	name <- name_changes[[x]]
	}
	new_line <- (sprintf("\\newcommand{\\%s%s}{%s}", name, suffix, signnumber2(param[[x]])))
	tex_string <- sprintf("%s\n%s", tex_string, new_line)
	}
	# saves it as a .tex file
	sink(file=sprintf('output/%s.tex', filename))
	cat(tex_string)
	sink(NULL)
	}

	save_boot_tex <- function(median, se, filename, suffix=""){
	tex_string <- ""
	for(x in names(median)){
	name <- x
	if (x %in% names(name_changes)){
	name <- name_changes[[x]]
	}
	if (x %in% constants){
	new_line <- (sprintf("\\newcommand{\\%s%s}{%s}", name, suffix, signnumber(median[[x]])))
	} else {
	new_line <- (sprintf("\\newcommand{\\%s%s}{%s\\:(%s)}", name, suffix, signnumber(median[[x]]), signnumber(se[[x]])))
	}
	tex_string <- sprintf("%s\n%s",tex_string, new_line)
	}
	sink(file=sprintf('output/%s.tex', filename))
	cat(tex_string)
	sink(NULL)
	}

	save_boot_tex_percentile <- function(bottom, top, filename, suffix=""){
	tex_string <- ""
	for(x in names(bottom)){
	name <- x
	if (x %in% names(name_changes)){
	name <- name_changes[[x]]
	}
	new_line <- (sprintf("\\newcommand{\\%s%s}{[%s, %s]}", name, suffix, signnumber(bottom[[x]]), signnumber(top[[x]])))
	tex_string <- sprintf("%s\n%s",tex_string, new_line)
	}
	sink(file=sprintf('output/%s.tex', filename))
	cat(tex_string)
	sink(NULL)
	}

	save_nrow <- function(nrows_list, filename, suffix=""){
	tex_string <- ""
	for(x in names(nrows_list)){
	name <- x
	new_line <- (sprintf("\\newcommand{\\%s%s}{%s}", name, suffix, nrows_list[[x]]))
	tex_string <- sprintf("%s\n%s",tex_string, new_line)
	}
	sink(file=sprintf('output/%s.tex', filename))
	cat(tex_string)
	sink(NULL)
	}