Quarto Research Blog - A Simple Life Cycle Model

Model

This note examines the model of Parallel Computing by Rcpp.

For $t = 1, \dots, T$, households solve

\[ \begin{aligned} V(t, e, x) =& \max_{c, x'} \frac{c^{1 - \sigma}}{1 - \sigma} + \beta \mathbb{E}V(t + 1, e', x') \\ &\text{s.t. } \\ & c + x' \le (1 + r)x + ew \\ & \text{Pr}(e' | e) = \Gamma(e) \\ & x' \ge 0 \end{aligned} \]

Results

Simulations

viewof t = Inputs.range(
  [1, 10], 
  {value: 1, step: 1, label: "t"}
)
viewof ssigma = Inputs.range(
  [2, 4], 
  {value: 2.0, step: 0.1, label: "σ"}
)
viewof bbeta = Inputs.range(
  [0.92, 0.99], 
  {value: 0.97, step: 0.01, label: "β"}
)

filtered = transpose(data).filter(function(sim) {
    return sim.ssigma == ssigma && sim.bbeta == bbeta && sim.t == t
})

Plot.plot({
    marginLeft: 50,
    height: 400,
    color: {domain: ["e₁", "e₈", "e₁₅"], legend: true},
    x: {domain: [0.0, 4.0]},
    y: {domain: [-2.5, 0.0]},
    marks: [
        Plot.lineY(filtered, {x: "x", y: "v", stroke: "lbl"}),
    ]
})

--- title: "A Simple Life Cycle Model" author: Kazuharu Yanagimoto date: 2024-03-21 categories: [model] --- ```{r} #| include: false library(here) library(tidyverse) library(Rtauchen) source(here("code", "modules", "graphics.R")) library(Rcpp) library(RcppThread) sourceCpp(here("posts/240321_lifecycle/model.cpp")) ``` ## Model This note examines the model of [Parallel Computing](https://github.com/davidzarruk/Parallel_Computing/) by Rcpp. For $t = 1, \dots, T$, households solve $$ \begin{aligned} V(t, e, x) =& \max_{c, x'} \frac{c^{1 - \sigma}}{1 - \sigma} + \beta \mathbb{E}V(t + 1, e', x') \\ &\text{s.t. } \\ & c + x' \le (1 + r)x + ew \\ & \text{Pr}(e' | e) = \Gamma(e) \\ & x' \ge 0 \end{aligned} $$ ```{r} # Grid for x nx = 100 xmin = 0.1 xmax = 4.0 x_grid = seq(from = xmin, to = xmax, by = (xmax - xmin) / (nx - 1)) # Grid for e: parameters for Tauchen ne = 15 ssigma_eps = 0.02058 llambda_eps = 0.99 m = 1.5 P = Rtauchen(ne, ssigma_eps, llambda_eps, m) e_grid = exp(Tgrid(ne, ssigma_eps, llambda_eps, m)) # Utility function ssigma = 2 bbeta = 0.97 T = 10 # Prices r = 0.07 w = 5 ``` ```{r} # Take care of the array shape V <- solve(nx, x_grid, ne, e_grid, P, ssigma, bbeta, T, r, w) ``` ## Results ```{r} df <- tibble(e = rep(1:ne, T * nx), v = V) |> mutate(t = (row_number() - 1) %/% (nx * ne) + 1) |> arrange(e, t) |> mutate(x = rep(x_grid, T * ne), e = factor(e)) ``` ```{r} #| classes: preview-image df |> filter(t %in% c(1, 4, 7, 10), e %in% c(1, 8, 15)) |> mutate(lbl_facet = factor(paste0("t = ", t), levels = c("t = 1", "t = 4", "t = 7", "t = 10"))) |> ggplot(aes(x = x, y = v, color = e)) + geom_line() + scale_color_discrete(labels = ~latex2exp::TeX(sprintf("$e_{%s}$", .x))) + facet_wrap(~ lbl_facet) + labs(x = "x", y = NULL, title = "Value Function") + theme_notebook() + theme(legend.position = c(0.9, 0.1), panel.grid.major.x = element_blank()) ``` ### Simulations ```{r} simulate <- function(ssigma, bbeta) { V <- solve(nx, x_grid, ne, e_grid, P, ssigma, bbeta, T, r, w) tibble(ssigma = ssigma, bbeta = bbeta, e = rep(1:ne, T * nx), v = V) |> mutate(t = (row_number() - 1) %/% (nx * ne) + 1) |> arrange(e, t) |> mutate(x = rep(x_grid, T * ne)) } ssigmas <- seq(2, 4, by = 0.1) bbetas <- seq(0.92, 0.99, by = 0.01) exgrid <- expand.grid(ssigmas, bbetas) df_sim <- map2(exgrid$Var1, exgrid$Var2, simulate) |> list_rbind() |> filter(e %in% c(1, 8, 15)) |> mutate(lbl = recode_factor(e, `1` = "e₁", `8` = "e₈", `15` = "e₁₅")) ``` ```{r} ojs_define(data = df_sim) ``` ```{ojs} //| panel: input viewof t = Inputs.range( [1, 10], {value: 1, step: 1, label: "t"} ) viewof ssigma = Inputs.range( [2, 4], {value: 2.0, step: 0.1, label: "σ"} ) viewof bbeta = Inputs.range( [0.92, 0.99], {value: 0.97, step: 0.01, label: "β"} ) ``` ```{ojs} filtered = transpose(data).filter(function(sim) { return sim.ssigma == ssigma && sim.bbeta == bbeta && sim.t == t }) ``` ```{ojs} Plot.plot({ marginLeft: 50, height: 400, color: {domain: ["e₁", "e₈", "e₁₅"], legend: true}, x: {domain: [0.0, 4.0]}, y: {domain: [-2.5, 0.0]}, marks: [ Plot.lineY(filtered, {x: "x", y: "v", stroke: "lbl"}), ] }) ```