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#' @export
#' @title Overlapping Generations Financial Sequential Models with One Firm
#' @aliases gemOLGF_OneFirm
#' @description Some examples of overlapping generations financial sequential models with one firm.
#'
#' When there is a population growth, we will take the security-split assumption (see \code{\link{gemOLGF_PureExchange}}).
#' @param ... arguments to be passed to the function sdm2.
#' @references Samuelson, P. A. (1958) An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money. Journal of Political Economy, vol. 66(6): 467-482.
#' @references de la Croix, David and Philippe Michel (2002, ISBN: 9780521001151) A Theory of Economic Growth: Dynamics and Policy in Overlapping Generations. Cambridge University Press.
#' @seealso {
#' \code{\link{gemOLG_PureExchange}}
#' \code{\link{gemOLG_TimeCircle}}
#' }
#' @examples
#' \donttest{
#' #### an OLGF economy with a firm and two-period-lived consumers
#' beta.firm <- c(1 / 3, 2 / 3)
#' # the population growth rate
#' GRExg <- 0.03
#' saving.rate <- 0.5
#' ratio.saving.consumption <- saving.rate / (1 - saving.rate)
#'
#' dst.firm <- node_new(
#' "prod",
#' type = "CD", alpha = 5,
#' beta = beta.firm,
#' "lab", "prod"
#' )
#'
#' dst.age1 <- node_new(
#' "util",
#' type = "FIN",
#' rate = c(1, ratio.saving.consumption),
#' "prod", "secy" # security, the financial instrument
#' )
#'
#' dst.age2 <- node_new(
#' "util",
#' type = "Leontief", a = 1,
#' "prod"
#' )
#'
#' ge <- sdm2(
#' A = list(
#' dst.firm, dst.age1, dst.age2
#' ),
#' B = matrix(c(
#' 1, 0, 0,
#' 0, 0, 0,
#' 0, 0, 0
#' ), 3, 3, TRUE),
#' S0Exg = matrix(c(
#' NA, NA, NA,
#' NA, 1, NA,
#' NA, NA, 1
#' ), 3, 3, TRUE),
#' names.commodity = c("prod", "lab", "secy"),
#' names.agent = c("firm", "age1", "age2"),
#' numeraire = "lab",
#' GRExg = GRExg,
#' maxIteration = 1,
#' ts = TRUE
#' )
#'
#' ge$p
#' matplot(ge$ts.p, type = "l")
#' matplot(growth_rate(ge$ts.z), type = "l") # GRExg
#' addmargins(ge$D, 2) # the demand matrix of the current period
#' addmargins(ge$S, 2) # the supply matrix of the current period
#' addmargins(ge$S * (1 + GRExg), 2) # the supply matrix of the next period
#' addmargins(ge$DV)
#' addmargins(ge$SV)
#'
#' ## Suppose consumers consume product and labor (i.e. service) and
#' ## age1 and age2 may have different instantaneous utility functions.
#' dst.age1 <- node_new(
#' "util",
#' type = "FIN",
#' rate = c(1, ratio.saving.consumption),
#' "cc1", "secy" # security, the financial instrument
#' )
#' node_set(dst.age1, "cc1",
#' type = "Leontief",
#' a = c(0.5, 0.5),
#' "prod", "lab"
#' )
#' node_plot(dst.age1)
#'
#' dst.age2 <- node_new("util",
#' type = "Leontief",
#' a = c(0.2, 0.8),
#' "prod", "lab"
#' )
#'
#' ge <- sdm2(
#' A = list(
#' dst.firm, dst.age1, dst.age2
#' ),
#' B = matrix(c(
#' 1, 0, 0,
#' 0, 0, 0,
#' 0, 0, 0
#' ), 3, 3, TRUE),
#' S0Exg = matrix(c(
#' NA, NA, NA,
#' NA, 1, NA,
#' NA, NA, 1
#' ), 3, 3, TRUE),
#' names.commodity = c("prod", "lab", "secy"),
#' names.agent = c("firm", "age1", "age2"),
#' numeraire = "lab",
#' GRExg = GRExg,
#' priceAdjustmentVelocity = 0.05
#' )
#'
#' ge$p
#' addmargins(ge$D, 2)
#' addmargins(ge$S, 2)
#' addmargins(ge$DV)
#' addmargins(ge$SV)
#'
#' ## Aggregate the above consumers into one infinite-lived consumer,
#' ## who always spends the same amount on cc1 and cc2.
#' dst.consumer <- node_new("util",
#' type = "CD", alpha = 1,
#' beta = c(0.5, 0.5),
#' "cc1", "cc2"
#' )
#' node_set(dst.consumer, "cc1",
#' type = "Leontief",
#' a = c(0.5, 0.5),
#' "prod", "lab"
#' )
#' node_set(dst.consumer, "cc2",
#' type = "Leontief",
#' a = c(0.2, 0.8),
#' "prod", "lab"
#' )
#'
#' ge <- sdm2(
#' A = list(
#' dst.firm, dst.consumer
#' ),
#' B = matrix(c(
#' 1, 0,
#' 0, 0
#' ), 2, 2, TRUE),
#' S0Exg = matrix(c(
#' NA, NA,
#' NA, 1
#' ), 2, 2, TRUE),
#' names.commodity = c("prod", "lab"),
#' names.agent = c("firm", "consumer"),
#' numeraire = "lab",
#' GRExg = GRExg,
#' priceAdjustmentVelocity = 0.05
#' )
#'
#' ge$p
#' addmargins(ge$D, 2)
#' addmargins(ge$S, 2)
#' addmargins(ge$DV)
#' addmargins(ge$SV)
#'
#' #### an OLGF economy with a firm and two-period-lived consumers
#' ## Suppose each consumer has a Leontief-type utility function min(c1, c2/a).
#' beta.firm <- c(1 / 3, 2 / 3)
#' # the population growth rate, the equilibrium interest rate and profit rate
#' GRExg <- 0.03
#' rho <- 1 / (1 + GRExg)
#' a <- 0.9
#'
#' dst.firm <- node_new(
#' "prod",
#' type = "CD", alpha = 5,
#' beta = beta.firm,
#' "lab", "prod"
#' )
#'
#' dst.age1 <- node_new(
#' "util",
#' type = "FIN",
#' rate = c(1, ratio.saving.consumption = a * rho),
#' "prod", "secy" # security, the financial instrument
#' )
#'
#' dst.age2 <- node_new(
#' "util",
#' type = "Leontief", a = 1,
#' "prod"
#' )
#'
#' ge <- sdm2(
#' A = list(
#' dst.firm, dst.age1, dst.age2
#' ),
#' B = matrix(c(
#' 1, 0, 0,
#' 0, 0, 0,
#' 0, 0, 0
#' ), 3, 3, TRUE),
#' S0Exg = matrix(c(
#' NA, NA, NA,
#' NA, 1, NA,
#' NA, NA, 1
#' ), 3, 3, TRUE),
#' names.commodity = c("prod", "lab", "secy"),
#' names.agent = c("firm", "age1", "age2"),
#' numeraire = "lab",
#' GRExg = GRExg,
#' maxIteration = 1,
#' ts = TRUE
#' )
#'
#' ge$p
#' matplot(ge$ts.p, type = "l")
#' matplot(growth_rate(ge$ts.z), type = "l") # GRExg
#' addmargins(ge$D, 2)
#' addmargins(ge$S, 2)
#' addmargins(ge$DV)
#' addmargins(ge$SV)
#'
#' ## the corresponding time-cycle model
#' n <- 5 # the number of periods, consumers and firms.
#' S <- matrix(NA, 2 * n, 2 * n)
#'
#' S.lab.consumer <- diag((1 + GRExg)^(0:(n - 1)), n)
#' S[(n + 1):(2 * n), (n + 1):(2 * n)] <- S.lab.consumer
#'
#' B <- matrix(0, 2 * n, 2 * n)
#' B[1:n, 1:n] <- diag(n)[, c(2:n, 1)]
#' B[1, n] <- rho^n
#'
#' dstl.firm <- list()
#' for (k in 1:n) {
#' dstl.firm[[k]] <- node_new(
#' "prod",
#' type = "CD", alpha = 5,
#' beta = beta.firm,
#' paste0("lab", k), paste0("prod", k)
#' )
#' }
#'
#' dstl.consumer <- list()
#' for (k in 1:(n - 1)) {
#' dstl.consumer[[k]] <- node_new(
#' "util",
#' type = "FIN",
#' rate = c(1, ratio.saving.consumption = a * rho),
#' paste0("prod", k), paste0("prod", k + 1)
#' )
#' }
#'
#' dstl.consumer[[n]] <- node_new(
#' "util",
#' type = "FIN",
#' rate = c(1, ratio.saving.consumption = a * rho),
#' paste0("prod", n), "cc1"
#' )
#' node_set(dstl.consumer[[n]], "cc1",
#' type = "Leontief", a = rho^n, # a discounting factor
#' "prod1"
#' )
#'
#' ge2 <- sdm2(
#' A = c(dstl.firm, dstl.consumer),
#' B = B,
#' S0Exg = S,
#' names.commodity = c(paste0("prod", 1:n), paste0("lab", 1:n)),
#' names.agent = c(paste0("firm", 1:n), paste0("consumer", 1:n)),
#' numeraire = "lab1",
#' policy = makePolicyMeanValue(30),
#' maxIteration = 1,
#' numberOfPeriods = 600,
#' ts = TRUE
#' )
#'
#' ge2$p
#' growth_rate(ge2$p[1:n]) + 1 # rho
#' growth_rate(ge2$p[(n + 1):(2 * n)]) + 1 # rho
#' ge2$D
#' }
#'
gemOLGF_OneFirm <- function(...) sdm2(...)
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