#' Identify the dynamic optimum using backward iteration (dynamic programming)
#' @param SDP_Mat the stochastic transition matrix at each h value
#' @param x_grid the discrete values allowed for the population size, x
#' @param h_grid the discrete values of harvest levels to optimize over
#' @param OptTime the stopping time
#' @param xT the boundary condition population size at OptTime
#' @param delta the discounting rate (1-delta)
#' @param epsilon value iteration tolerance
#' @param profit the profit function
#' (i.e. enforces the boundary condition)
#' @return list containing the matrices D and V. D is an x_grid by OptTime
#' matrix with the indices of h_grid giving the optimal h at each value x
#' as the columns, with a column for each time.
#' V is a matrix of x_grid by x_grid, which is used to store the value
#' function at each point along the grid at each point in time.
#' The returned V gives the value matrix at the first (last) time.
#' @export
value_iteration <- function(SDP_Mat, x_grid, h_grid,
OptTime=100, xT, profit,
delta, epsilon = 1e-4){
## Initialize space for the matrices
gridsize <- length(x_grid)
HL <- length(h_grid)
D <- matrix(NA, nrow=gridsize, ncol=1)
V <- rep(0,gridsize) # initialize BC,
Vp <- rep(10, gridsize) # initialize previous value (not equal to current value)
# loop through time
#for(time in 1:OptTime){
time <- 1
while(max(abs(Vp - V)) >= (delta * epsilon / (2 - 2 * delta)) && time < OptTime){
# try all potential havest rates
V1 <- sapply(1:HL, function(i){
# Transition matrix times V gives dist in next time
SDP_Mat[[i]] %*% V +
# then (add) harvested amount times discount
profit(x_grid, h_grid[i]) * (1 - delta)
})
# find havest, h that gives the maximum value
out <- sapply(1:gridsize, function(j){
value <- max(V1[j,], na.rm = T) # each col is a diff h, max over these
index <- which.max(V1[j,]) # store index so we can recover h's
c(value, index) # returns both profit value & index of optimal h.
})
# Sets V[t+1] = max_h V[t] at each possible state value, x
Vp <- V
V <- out[1,] # The new value-to-go
D <- out[2,] # The index positions
time <- time+1
}
# check for convergence in V
# Reed derives a const escapement policy saying to fish the pop down to
# the largest population for which you shouldn't harvest:
ReedThreshold <- x_grid[max(which(D == 1))]
# Format the output
list(D=D, V=V, S=ReedThreshold)
}
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