From ce8f95a4ee88dee7ddf44d28c4b78b4ff373cb42 Mon Sep 17 00:00:00 2001 From: mmcky Date: Wed, 7 Feb 2024 12:51:20 +1100 Subject: [PATCH 01/18] MAINT: integrate book-dp1-public-companion into repo --- .github/workflows/ci.yml | 86 ++ .github/workflows/publish.yml | 94 ++ .gitignore | 7 +- LICENSE | 29 + README.md | 36 +- build_jb.py | 30 + code-book/_config.yml | 70 + code-book/_static/includes/header.raw | 7 + .../_static/includes/lecture_howto_py.raw | 7 + code-book/_static/lectures-favicon.ico | Bin 0 -> 1561 bytes code-book/_static/qe-logo-large.png | Bin 0 -> 1351 bytes code-book/_static/qe-logo.png | Bin 0 -> 2189 bytes code-book/_toc.yml | 31 + code-book/intro.md | 35 + code-book/jl/Manifest.toml | 905 ++++++++++++ code-book/jl/Project.toml | 12 + code-book/jl/figures/american_option_1.pdf | Bin 0 -> 200549 bytes code-book/jl/figures/american_option_2.pdf | Bin 0 -> 2640 bytes code-book/jl/figures/ez_utility_1.pdf | Bin 0 -> 136916 bytes code-book/jl/figures/ez_utility_2.pdf | 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=> website/slides}/ch_3.pdf (100%) rename {slides => website/slides}/ch_4.pdf (100%) rename {slides => website/slides}/ch_5.pdf (100%) rename {slides => website/slides}/ch_6.pdf (100%) rename {slides => website/slides}/ch_7.pdf (100%) rename {slides => website/slides}/ch_8.pdf (100%) rename {slides => website/slides}/ch_9.pdf (100%) diff --git a/.github/workflows/ci.yml b/.github/workflows/ci.yml new file mode 100644 index 0000000..ab874a9 --- /dev/null +++ b/.github/workflows/ci.yml @@ -0,0 +1,86 @@ +name: Build Project [using jupyter-book] +on: [push] +jobs: + tests: + runs-on: ubuntu-latest + steps: + - name: Checkout + uses: actions/checkout@v4 + - name: Setup Anaconda + uses: conda-incubator/setup-miniconda@v3 + with: + auto-update-conda: true + auto-activate-base: true + miniconda-version: 'latest' + python-version: "3.10" + environment-file: environment.yml + activate-environment: dp1 + - name: Install latex dependencies + run: | + sudo apt-get -qq update + sudo apt-get install -y \ + texlive-latex-recommended \ + texlive-latex-extra \ + texlive-fonts-recommended \ + texlive-fonts-extra \ + texlive-xetex \ + latexmk \ + xindy \ + dvipng \ + cm-super \ + msttcorefonts + - name: Set up Julia + uses: julia-actions/setup-julia@v1 + with: + version: 1.9.3 + - name: Install IJulia and Setup Project + shell: bash + run: | + julia -e 'using Pkg; ENV["PYTHON"]="/usr/share/miniconda3/python"; Pkg.add(["PyCall", "Conda", "PyPlot", "IJulia"]); using Conda; Conda.add("matplotlib"); using PyPlot;' + - name: Display Conda Environment Versions + shell: bash -l {0} + run: conda list + - name: Display Pip Versions + shell: bash -l {0} + run: pip list + + # - name: Download "build" folder (cache) + # uses: dawidd6/action-download-artifact@v2 + # with: + # workflow: cache.yml + # branch: main + # name: build-cache + # path: _build + + # Build Code Book (HTML) + - name: Build Code Book (HTML) + shell: bash -l {0} + run: | + python create_code_book.py + - name: Upload Execution Reports + uses: actions/upload-artifact@v2 + if: failure() + with: + name: execution-reports + path: code-book/_build/html/reports + # Assemble Website + - name: Build Code Book (HTML) + shell: bash -l {0} + run: | + mkdir -p website/code-book + cp -r code-book/_build/html/ website/code-book/ + - name: Save Build as Artifact + uses: actions/upload-artifact@v1 + with: + name: website + path: website + - name: Preview Deploy to Netlify + uses: nwtgck/actions-netlify@v2 + with: + publish-dir: website + production-branch: main + github-token: ${{ secrets.GITHUB_TOKEN }} + deploy-message: "Preview Deploy from GitHub Actions" + env: + NETLIFY_AUTH_TOKEN: ${{ secrets.NETLIFY_AUTH_TOKEN }} + NETLIFY_SITE_ID: ${{ secrets.NETLIFY_SITE_ID }} diff --git a/.github/workflows/publish.yml b/.github/workflows/publish.yml new file mode 100644 index 0000000..0303222 --- /dev/null +++ b/.github/workflows/publish.yml @@ -0,0 +1,94 @@ +# TODO: UPDATE +name: Build & Publish to GH-PAGES +on: + push: + tags: + - 'publish*' + branch: + - main +jobs: + publish: + if: github.event_name == 'push' && startsWith(github.event.ref, 'refs/tags') + runs-on: ubuntu-latest + steps: + - name: Checkout + uses: actions/checkout@v2 + - name: Setup Anaconda + uses: conda-incubator/setup-miniconda@v2 + with: + auto-update-conda: true + auto-activate-base: true + miniconda-version: 'latest' + python-version: "3.10" + environment-file: environment.yml + activate-environment: dse2023 + - name: Install latex dependencies + run: | + sudo apt-get -qq update + sudo apt-get install -y \ + texlive-latex-recommended \ + texlive-latex-extra \ + texlive-fonts-recommended \ + texlive-fonts-extra \ + texlive-xetex \ + latexmk \ + xindy \ + dvipng \ + cm-super \ + msttcorefonts + - name: Set up Julia + uses: julia-actions/setup-julia@v1 + with: + version: 1.9.3 + - name: Install IJulia and Setup Project + shell: bash + run: | + julia -e 'using Pkg; ENV["PYTHON"]="/usr/share/miniconda3/python"; Pkg.add(["PyCall", "Conda", "PyPlot", "IJulia"]); using Conda; Conda.add("matplotlib"); using PyPlot;' + - name: Display Conda Environment Versions + shell: bash -l {0} + run: conda list + - name: Display Pip Versions + shell: bash -l {0} + run: pip list + # - name: Download "build" folder (cache) + # uses: dawidd6/action-download-artifact@v2 + # with: + # workflow: cache.yml + # branch: main + # name: build-cache + # path: _build + # Build Assets (Download Notebooks and PDF via LaTeX) + # - name: Build PDF from LaTeX + # shell: bash -l {0} + # run: | + # jb build lectures --builder pdflatex --path-output ./ -n --keep-going + # - name: Copy LaTeX PDF for GH-PAGES + # shell: bash -l {0} + # run: | + # mkdir -p _build/html/_pdf + # cp -u _build/latex/*.pdf _build/html/_pdf + # - name: Build Download Notebooks (sphinx-tojupyter) + # shell: bash -l {0} + # run: | + # jb build lectures --path-output ./ --builder=custom --custom-builder=jupyter + # zip -r download-notebooks.zip _build/jupyter + # - uses: actions/upload-artifact@v2 + # with: + # name: download-notebooks + # path: download-notebooks.zip + # - name: Copy Download Notebooks for GH-PAGES + # shell: bash -l {0} + # run: | + # mkdir -p _build/html/_notebooks + # cp -u _build/jupyter/*.ipynb _build/html/_notebooks + # Build HTML (Website) + - name: Build HTML + shell: bash -l {0} + run: | + python create_code_book.py + - name: Deploy website to gh-pages + uses: peaceiris/actions-gh-pages@v3 + with: + github_token: ${{ secrets.GITHUB_TOKEN }} + publish_dir: code-book/_build/html/ + # cname: networks.quantecon.org diff --git a/.gitignore b/.gitignore index fd253a1..4d7105c 100644 --- a/.gitignore +++ b/.gitignore @@ -1,2 +1,7 @@ .DS_Store -*./DS_Store \ No newline at end of file +*./DS_Store +website/_build +website/code-book/ +code-book/_build +.ipynb_checkpoints +__pycache__ \ No newline at end of file diff --git a/LICENSE b/LICENSE new file mode 100644 index 0000000..acddd64 --- /dev/null +++ b/LICENSE @@ -0,0 +1,29 @@ +BSD 3-Clause License + +Copyright (c) 2022, QuantEcon +All rights reserved. + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are met: + +1. Redistributions of source code must retain the above copyright notice, this + list of conditions and the following disclaimer. + +2. Redistributions in binary form must reproduce the above copyright notice, + this list of conditions and the following disclaimer in the documentation + and/or other materials provided with the distribution. + +3. Neither the name of the copyright holder nor the names of its + contributors may be used to endorse or promote products derived from + this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" +AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE +IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE +DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE +FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL +DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR +SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER +CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, +OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE +OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. diff --git a/README.md b/README.md index 0f6fc9e..2723f50 100644 --- a/README.md +++ b/README.md @@ -4,6 +4,38 @@ Public repo for the textbook **Dynamic Programming Volume 1** by [Thomas J. Sargent](http://www.tomsargent.com/) and [John Stachurski](https://johnstachurski.net/). -## Website +| Folder | Description | +|--------|-------------| +| website | HTML files that build the website | +| code-book | jupyter-book powered code-book as a companion to the book | +| pdf | PDF of the book | -The website is currently being built with `jekyll` +## PDF + +The PDF is available in `pdf/dp.pdf` + +Comments and feedback are very welcome. +The easiest way to provide feedback is to open an issue above. + +## Codebooks + +> The repo also contains all of the Julia and Python code found in the code snippets within +> the textbook, as well as Julia and Python code to reproduce all figures. + +The repo contains a Jupyter Book with all code and figures used in the book. To build it locally, + +1. Clone the repo +2. Create an environment using `environment.yml` +3. You will have to install Julia 1.9 if you haven't already (Note: There is no Julia version on Conda for Windows systems). +4. Run `python3 create_code_book.py` from the root directory of the repo. This should take a few minutes to finish running. + +Figures generated by the Julia code can be found in the `./code-book/figures/` folder. + +To contribute to the codebook, + +1. Edit the scripts in `source_code_jl/` and `source_code_py/`. +2. Edit the chapter_meta dictionaries in `create_code_book.py` to +- include or remove scripts +- change the order of the scripts +- change function calls +3. Run `create_code_book.py` to generate the codebook. \ No newline at end of file diff --git a/build_jb.py b/build_jb.py new file mode 100644 index 0000000..aa07f67 --- /dev/null +++ b/build_jb.py @@ -0,0 +1,30 @@ +import os +import shutil + +julia_src_dir = './source_code_jl/' +python_src_dir = './source_code_py/' +julia_dst_dir = './code-book/jl/' +python_dst_dir = './code-book/py/' + +def remove_files(root_src_dir, root_dst_dir): + for file in os.listdir(root_src_dir): + if os.path.isfile(os.path.join(root_src_dir, file)): + os.remove(os.path.join(root_dst_dir, file)) + else: + shutil.rmtree(os.path.join(root_dst_dir, file)) + +shutil.copytree(julia_src_dir, julia_dst_dir, dirs_exist_ok=True) +shutil.copytree(python_src_dir, python_dst_dir, dirs_exist_ok=True) + +try: + os.system("jupyter-book build ./code-book/") + remove_files(python_src_dir, python_dst_dir) + remove_files(julia_src_dir, julia_dst_dir) + print("Success!") + exit(0) +except Exception as e: + print("Error in building Jupyter Book: " + str(e)) + remove_files(python_src_dir, python_dst_dir) + remove_files(julia_src_dir, julia_dst_dir) + print("Failed! - Directory cleaned") + exit(1) diff --git a/code-book/_config.yml b/code-book/_config.yml new file mode 100644 index 0000000..4f17acc --- /dev/null +++ b/code-book/_config.yml @@ -0,0 +1,70 @@ +title: Dynamic Programming +subtitle: Julia and Python Code +author: Thomas J. Sargent & John Stachurski +logo: _static/qe-logo.png +#description: This website presents a set of lectures on dynamic programming for economics, designed and written by Thomas J. Sargent and John Stachurski. + +execute: + execute_notebooks: "cache" + timeout: 600 # 10 minutes + allow_errors: true + +only_build_toc_files: true + + + +#html: + #baseurl: https://python.quantecon.org/ + +#latex: + #latex_documents: + #targetname: quantecon-dynamic-programming.tex + +sphinx: + extra_extensions: [sphinx_multitoc_numbering, sphinxext.rediraffe, sphinx_tojupyter] + config: + html_favicon: _static/lectures-favicon.ico + html_theme: quantecon_book_theme + html_static_path: ['_static'] + html_theme_options: + header_organisation_url: https://quantecon.org + header_organisation: QuantEcon + repository_url: https://github.com/orectique/book-dp1 + #nb_repository_url: https://github.com/QuantEcon/lecture-python-programming.notebooks + twitter: quantecon + twitter_logo_url: https://assets.quantecon.org/img/qe-twitter-logo.png + og_logo_url: https://assets.quantecon.org/img/qe-og-logo.png + #description: This website presents a set of lectures on python programming for economics, designed and written by Thomas J. Sargent and John Stachurski. + #keywords: Python, QuantEcon, Quantitative Economics, Economics, Sloan, Alfred P. Sloan Foundation, Tom J. Sargent, John Stachurski + #analytics: + #google_analytics_id: G-X7DH1M2DPY + launch_buttons: + notebook_interface : classic # The interface interactive links will activate ["classic", "jupyterlab"] + binderhub_url : https://mybinder.org # The URL of the BinderHub (e.g., https://mybinder.org) + colab_url : https://colab.research.google.com + thebe : false # Add a thebe button to pages (requires the repository to run on Binder) + mathjax3_config: + tex: + macros: + "argmax" : "arg\\,max" + "argmin" : "arg\\,min" + mathjax_path: https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js + rediraffe_redirects: + index_toc.md: intro.md + tojupyter_static_file_path: ["source/_static", "_static"] + tojupyter_target_html: true + #tojupyter_urlpath: "https://python-programming.quantecon.org/" + #tojupyter_image_urlpath: "https://python-programming.quantecon.org/_static/" + tojupyter_lang_synonyms: ["julia-1.9", "julia"] + tojupyter_kernels: + julia: + kernelspec: + display_name: Julia + language: julia + name: julia-1.9 + python: + kernelspec: + display_name: Python + language: python + name: python3 + tojupyter_images_markdown: true diff --git a/code-book/_static/includes/header.raw b/code-book/_static/includes/header.raw new file mode 100644 index 0000000..3bbb09b --- /dev/null +++ b/code-book/_static/includes/header.raw @@ -0,0 +1,7 @@ +.. raw:: html + +
+ + QuantEcon + +
diff --git a/code-book/_static/includes/lecture_howto_py.raw b/code-book/_static/includes/lecture_howto_py.raw new file mode 100644 index 0000000..36e5eca --- /dev/null +++ b/code-book/_static/includes/lecture_howto_py.raw @@ -0,0 +1,7 @@ +.. raw:: html + +
+ + QuantEcon + +
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Sargent](http://www.tomsargent.com/) + +This website provides Julia and Python code for the textbook [Dynamic Programming](https://dp.quantecon.org/). + +```{tableofcontents} +``` + +```{code-cell} +:tags: [remove-cell] + +using Pkg; +Pkg.activate("./jl/"); +Pkg.instantiate(); +``` + +```{code-cell} + +``` diff --git a/code-book/jl/Manifest.toml b/code-book/jl/Manifest.toml new file mode 100644 index 0000000..9d6cca9 --- /dev/null +++ b/code-book/jl/Manifest.toml @@ -0,0 +1,905 @@ +# This file is machine-generated - editing it directly is not advised + +julia_version = "1.9.1" +manifest_format = "2.0" +project_hash = "fa783f9c41ac5737ddf83fd90054493a12ce6251" + +[[deps.AbstractFFTs]] +deps = ["LinearAlgebra"] +git-tree-sha1 = "d92ad398961a3ed262d8bf04a1a2b8340f915fef" +uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" +version = "1.5.0" +weakdeps = ["ChainRulesCore", "Test"] + + [deps.AbstractFFTs.extensions] + AbstractFFTsChainRulesCoreExt = "ChainRulesCore" + AbstractFFTsTestExt = "Test" 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z-I8Y<{Q&eZfG;+7cc*@Izpf%}H#>JQbrOGHMkF$xNd7hE2n+^+lxKoMH4Jb}{{bB* B=obJ0 literal 0 HcmV?d00001 diff --git a/code-book/jl/introductions.md b/code-book/jl/introductions.md new file mode 100644 index 0000000..00f1284 --- /dev/null +++ b/code-book/jl/introductions.md @@ -0,0 +1,384 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 1: Introduction)= +```{raw} jupyter +

    +``` +# Chapter 1: Introduction + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## two_period_job_search.jl +```{code-cell} julia +""" +Two period job search in the IID case. +""" + +using Distributions + +"Creates an instance of the job search model, stored as a NamedTuple." +function create_job_search_model(; + n=50, # wage grid size + w_min=10.0, # lowest wage + w_max=60.0, # highest wage + a=200, # wage distribution parameter + b=100, # wage distribution parameter + β=0.96, # discount factor + c=10.0 # unemployment compensation + ) + w_vals = collect(LinRange(w_min, w_max, n+1)) + ϕ = pdf(BetaBinomial(n, a, b)) + return (; n, w_vals, ϕ, β, c) +end + +" Computes lifetime value at t=1 given current wage w_1 = w. " +function v_1(w, model) + (; n, w_vals, ϕ, β, c) = model + h_1 = c + β * max.(c, w_vals)'ϕ + return max(w + β * w, h_1) +end + +" Computes reservation wage at t=1. " +function res_wage(model) + (; n, w_vals, ϕ, β, c) = model + h_1 = c + β * max.(c, w_vals)'ϕ + return h_1 / (1 + β) +end + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering + +default_model = create_job_search_model() + +" Plot the distribution of wages. " +function fig_dist(model=default_model, fs=14) + fig, ax = plt.subplots() + ax.plot(model.w_vals, model.ϕ, "-o", alpha=0.5, label="wage distribution") + ax.legend(loc="upper left", fontsize=fs) +end + + +" Plot two-period value function and res wage. " +function fig_v1(model=default_model; savefig=false, + figname="./figures/iid_job_search_0.pdf", fs=18) + + (; n, w_vals, ϕ, β, c) = model + + v = [v_1(w, model) for w in model.w_vals] + w_star = res_wage(model) + continuation_val = c + β * max.(c, w_vals)'ϕ + min_w, max_w = minimum(w_vals), maximum(w_vals) + + fig, ax = plt.subplots(figsize=(9, 5.5)) + ax.set_ylim(0, 120) + ax.set_xlim(min_w, max_w) + ax.vlines((w_star,), (0,), (continuation_val,), lw=0.5) + ax.set_yticks((0, 50, 100)) + ax.set_yticklabels((0, 50, 100), fontsize=12) + ax.set_xticks((min_w, w_star, max_w)) + ax.set_xticklabels((min_w, L"$w^*_1$", max_w), fontsize=12) + ax.plot(w_vals, w_vals + β * w_vals, "-", alpha=0.8, lw=3, + label=L"$w_1 + \beta w_1$") + ax.plot(w_vals, fill(continuation_val, n+1), lw=3, alpha=0.8, + label=L"$c + \beta \sum_{w'} \max\{c, w'\} \varphi(w')$" ) + ax.plot(w_vals, v, "k--", ms=2, alpha=1.0, lw=2, label=L"$v_1(w_1)$") + ax.legend(frameon=false, fontsize=fs, loc="upper left") + if savefig + fig.savefig(figname) + end +end + + + +``` + +```{code-cell} julia +fig_v1(savefig=true) +``` +## compute_spec_rad.jl +```{code-cell} julia +using LinearAlgebra +ρ(A) = maximum(abs(λ) for λ in eigvals(A)) # Spectral radius +A = [0.4 0.1; # Test with arbitrary A + 0.7 0.2] +print(ρ(A)) + +``` +## power_series.jl +```{code-cell} julia +using LinearAlgebra + +# Primitives +A = [0.4 0.1; + 0.7 0.2] + +# Method one: direct inverse +B_inverse = inv(I - A) + +# Method two: power series +function power_series(A) + B_sum = zeros((2, 2)) + A_power = I + for k in 1:50 + B_sum += A_power + A_power = A_power * A + end + return B_sum +end + +# Print maximal error +print(maximum(abs.(B_inverse - power_series(A)))) + +``` +## s_approx.jl +```{code-cell} julia +""" +Computes an approximate fixed point of a given operator T +via successive approximation. + +""" +function successive_approx(T, # operator (callable) + u_0; # initial condition + tolerance=1e-6, # error tolerance + max_iter=10_000, # max iteration bound + print_step=25) # print at multiples + u = u_0 + error = Inf + k = 1 + + while (error > tolerance) & (k <= max_iter) + + u_new = T(u) + error = maximum(abs.(u_new - u)) + + if k % print_step == 0 + println("Completed iteration $k with error $error.") + end + + u = u_new + k += 1 + end + + if error <= tolerance + println("Terminated successfully in $k iterations.") + else + println("Warning: hit iteration bound.") + end + + return u +end + +``` +## linear_iter.jl +```{code-cell} julia +include("s_approx.jl") +using LinearAlgebra + +# Compute the fixed point of Tu = Au + b via linear algebra +A, b = [0.4 0.1; 0.7 0.2], [1.0; 2.0] +u_star = (I - A) \ b # compute (I - A)^{-1} * b + +# Compute the fixed point via successive approximation +T(u) = A * u + b +u_0 = [1.0; 1.0] +u_star_approx = successive_approx(T, u_0) + +# Test for approximate equality (prints "true") +print(isapprox(u_star, u_star_approx, rtol=1e-5)) + + +``` +## linear_iter_fig.jl +```{code-cell} julia +include("linear_iter.jl") +using PyPlot + +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering + +fig, ax = plt.subplots() + +e = 0.02 + +marker_size = 60 +fs=14 + +colors = ("red", "blue", "orange", "green") +u_0_vecs = ([2.0; 3.0], [3.0; 5.2], [2.4; 3.6], [2.6, 5.6]) +iter_range = 8 + +for (u_0, color) in zip(u_0_vecs, colors) + u = u_0 + s, t = u + ax.text(s+e, t-4e, L"u_0", fontsize=fs) + + for i in 1:iter_range + s, t = u + ax.scatter((s,), (t,), c=color, alpha=0.3, s=marker_size) + u_new = T(u) + s_new, t_new = u_new + ax.plot((s, s_new), (t, t_new), lw=0.5, alpha=0.5, c=color) + u = u_new + end +end + +s_star, t_star = u_star +ax.scatter((s_star,), (t_star,), c="k", s=marker_size * 1.2) +ax.text(s_star-4e, t_star+4e, L"u^*", fontsize=fs) + +ax.set_xticks((2.0, 2.5, 3.0)) +ax.set_yticks((3.0, 4.0, 5.0, 6.0)) +ax.set_xlim(1.8, 3.2) +ax.set_ylim(2.8, 6.1) + +fig.savefig("./figures/linear_iter_fig_1.pdf") + + +``` +## iid_job_search.jl +```{code-cell} julia +""" +VFI approach to job search in the infinite-horizon IID case. + +""" + +include("two_period_job_search.jl") +include("s_approx.jl") + +" The Bellman operator. " +function T(v, model) + (; n, w_vals, ϕ, β, c) = model + return [max(w / (1 - β), c + β * v'ϕ) for w in w_vals] +end + +" Get a v-greedy policy. " +function get_greedy(v, model) + (; n, w_vals, ϕ, β, c) = model + σ = w_vals ./ (1 - β) .>= c .+ β * v'ϕ # Boolean policy vector + return σ +end + +" Solve the infinite-horizon IID job search model by VFI. " +function vfi(model=default_model) + (; n, w_vals, ϕ, β, c) = model + v_init = zero(model.w_vals) + v_star = successive_approx(v -> T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star +end + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering + +# A model with default parameters +default_model = create_job_search_model() + + +" Plot a sequence of approximations. " +function fig_vseq(model=default_model; + k=3, + savefig=false, + figname="./figures/iid_job_search_1.pdf", + fs=16) + + v = zero(model.w_vals) + fig, ax = plt.subplots(figsize=(9, 5.5)) + for i in 1:k + ax.plot(model.w_vals, v, lw=3, alpha=0.6, label="iterate $i") + v = T(v, model) + end + + for i in 1:1000 + v = T(v, model) + end + ax.plot(model.w_vals, v, "k-", lw=3.0, label="iterate 1000", alpha=0.7) + + #ax.set_ylim((0, 140)) + ax.set_xlabel("wage offer", fontsize=fs) + ax.set_ylabel("lifetime value", fontsize=fs) + + ax.legend(fontsize=fs, frameon=false) + + if savefig + fig.savefig(figname) + end +end + + +" Plot the fixed point. " +function fig_vstar(model=default_model; + savefig=false, fs=18, + figname="./figures/iid_job_search_3.pdf") + + (; n, w_vals, ϕ, β, c) = model + v_star, σ_star = vfi(model) + + fig, ax = plt.subplots(figsize=(9, 5.5)) + ax.plot(w_vals, v_star, "k-", lw=1.5, label="value function") + cont_val = c + β * v_star'ϕ + ax.plot(w_vals, fill(cont_val, n+1), + "--", + lw=5, + alpha=0.5, + label="continuation value") + + ax.plot(w_vals, + w_vals / (1 - β), + "--", + lw=5, + alpha=0.5, + label=L"w/(1 - \beta)") + + #ax.set_ylim(0, v_star.max()) + ax.legend(frameon=false, fontsize=fs, loc="lower right") + + if savefig + fig.savefig(figname) + end +end + + + + +``` + +```{code-cell} julia +fig_vseq(savefig=true) +``` + +```{code-cell} julia +fig_vstar(savefig=true) +``` diff --git a/code-book/jl/markov_decision_processes.md b/code-book/jl/markov_decision_processes.md new file mode 100644 index 0000000..72422f1 --- /dev/null +++ b/code-book/jl/markov_decision_processes.md @@ -0,0 +1,1209 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 5: Markov Decision Processes)= +```{raw} jupyter + +``` +# Chapter 5: Markov Decision Processes + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## inventory_dp.jl +```{code-cell} julia +include("s_approx.jl") +using Distributions +m(x) = max(x, 0) # Convenience function + +function create_inventory_model(; β=0.98, # discount factor + K=40, # maximum inventory + c=0.2, κ=2, # cost paramters + p=0.6) # demand parameter + ϕ(d) = (1 - p)^d * p # demand pdf + x_vals = collect(0:K) # set of inventory levels + return (; β, K, c, κ, p, ϕ, x_vals) +end + +"The function B(x, a, v) = r(x, a) + β Σ_x′ v(x′) P(x, a, x′)." +function B(x, a, v, model; d_max=100) + (; β, K, c, κ, p, ϕ, x_vals) = model + revenue = sum(min(x, d) * ϕ(d) for d in 0:d_max) + current_profit = revenue - c * a - κ * (a > 0) + next_value = sum(v[m(x - d) + a + 1] * ϕ(d) for d in 0:d_max) + return current_profit + β * next_value +end + +"The Bellman operator." +function T(v, model) + (; β, K, c, κ, p, ϕ, x_vals) = model + new_v = similar(v) + for (x_idx, x) in enumerate(x_vals) + Γx = 0:(K - x) + new_v[x_idx], _ = findmax(B(x, a, v, model) for a in Γx) + end + return new_v +end + +"Get a v-greedy policy. Returns a zero-based array." +function get_greedy(v, model) + (; β, K, c, κ, p, ϕ, x_vals) = model + σ_star = zero(x_vals) + for (x_idx, x) in enumerate(x_vals) + Γx = 0:(K - x) + _, a_idx = findmax(B(x, a, v, model) for a in Γx) + σ_star[x_idx] = Γx[a_idx] + end + return σ_star +end + +"Use successive_approx to get v_star and then compute greedy." +function solve_inventory_model(v_init, model) + (; β, K, c, κ, p, ϕ, x_vals) = model + v_star = successive_approx(v -> T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star +end + +# == Plots == # + +using PyPlot +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering + +# Create an instance of the model and solve it +model = create_inventory_model() +(; β, K, c, κ, p, ϕ, x_vals) = model +v_init = zeros(length(x_vals)) +v_star, σ_star = solve_inventory_model(v_init, model) + +"Simulate given the optimal policy." +function sim_inventories(ts_length=400, X_init=0) + G = Geometric(p) + X = zeros(Int32, ts_length) + X[1] = X_init + for t in 1:(ts_length-1) + D = rand(G) + X[t+1] = m(X[t] - D) + σ_star[X[t] + 1] + end + return X +end + + +function plot_vstar_and_opt_policy(; fontsize=16, + figname="./figures/inventory_dp_vs.pdf", + savefig=false) + fig, axes = plt.subplots(2, 1, figsize=(8, 6.5)) + + ax = axes[1] + ax.plot(0:K, v_star, label=L"v^*") + ax.set_ylabel("value", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=false) + + ax = axes[2] + ax.plot(0:K, σ_star, label=L"\sigma^*") + ax.set_xlabel("inventory", fontsize=fontsize) + ax.set_ylabel("optimal choice", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=false) + if savefig == true + fig.savefig(figname) + end +end + +function plot_ts(; fontsize=16, + figname="./figures/inventory_dp_ts.pdf", + savefig=false) + X = sim_inventories() + fig, ax = plt.subplots(figsize=(9, 5.5)) + ax.plot(X, label=L"X_t", alpha=0.7) + ax.set_xlabel(L"t", fontsize=fontsize) + ax.set_ylabel("inventory", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=false) + ax.set_ylim(0, maximum(X)+4) + if savefig == true + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +plot_vstar_and_opt_policy(savefig=true) +``` + +```{code-cell} julia +plot_ts(savefig=true) +``` +## finite_opt_saving_0.jl +```{code-cell} julia +using QuantEcon, LinearAlgebra, IterTools + +function create_savings_model(; R=1.01, β=0.98, γ=2.5, + w_min=0.01, w_max=20.0, w_size=200, + ρ=0.9, ν=0.1, y_size=5) + w_grid = LinRange(w_min, w_max, w_size) + mc = tauchen(y_size, ρ, ν) + y_grid, Q = exp.(mc.state_values), mc.p + return (; β, R, γ, w_grid, y_grid, Q) +end + +"B(w, y, w′, v) = u(R*w + y - w′) + β Σ_y′ v(w′, y′) Q(y, y′)." +function B(i, j, k, v, model) + (; β, R, γ, w_grid, y_grid, Q) = model + w, y, w′ = w_grid[i], y_grid[j], w_grid[k] + u(c) = c^(1-γ) / (1-γ) + c = w + y - (w′ / R) + @views value = c > 0 ? u(c) + β * dot(v[k, :], Q[j, :]) : -Inf + return value +end + +"The Bellman operator." +function T(v, model) + w_idx, y_idx = (eachindex(g) for g in (model.w_grid, model.y_grid)) + v_new = similar(v) + for (i, j) in product(w_idx, y_idx) + v_new[i, j] = maximum(B(i, j, k, v, model) for k in w_idx) + end + return v_new +end + +"The policy operator." +function T_σ(v, σ, model) + w_idx, y_idx = (eachindex(g) for g in (model.w_grid, model.y_grid)) + v_new = similar(v) + for (i, j) in product(w_idx, y_idx) + v_new[i, j] = B(i, j, σ[i, j], v, model) + end + return v_new +end + + + + +``` +## finite_opt_saving_1.jl +```{code-cell} julia +include("finite_opt_saving_0.jl") + +"Compute a v-greedy policy." +function get_greedy(v, model) + w_idx, y_idx = (eachindex(g) for g in (model.w_grid, model.y_grid)) + σ = Matrix{Int32}(undef, length(w_idx), length(y_idx)) + for (i, j) in product(w_idx, y_idx) + _, σ[i, j] = findmax(B(i, j, k, v, model) for k in w_idx) + end + return σ +end + +"Get the value v_σ of policy σ." +function get_value(σ, model) + # Unpack and set up + (; β, R, γ, w_grid, y_grid, Q) = model + w_idx, y_idx = (eachindex(g) for g in (w_grid, y_grid)) + wn, yn = length(w_idx), length(y_idx) + n = wn * yn + u(c) = c^(1-γ) / (1-γ) + # Build P_σ and r_σ as multi-index arrays + P_σ = zeros(wn, yn, wn, yn) + r_σ = zeros(wn, yn) + for (i, j) in product(w_idx, y_idx) + w, y, w′ = w_grid[i], y_grid[j], w_grid[σ[i, j]] + r_σ[i, j] = u(w + y - w′/R) + for (i′, j′) in product(w_idx, y_idx) + if i′ == σ[i, j] + P_σ[i, j, i′, j′] = Q[j, j′] + end + end + end + # Reshape for matrix algebra + P_σ = reshape(P_σ, n, n) + r_σ = reshape(r_σ, n) + # Apply matrix operations --- solve for the value of σ + v_σ = (I - β * P_σ) \ r_σ + # Return as multi-index array + return reshape(v_σ, wn, yn) +end + + + + +``` +## finite_opt_saving_2.jl +```{code-cell} julia +include("s_approx.jl") +include("finite_opt_saving_1.jl") + +"Value function iteration routine." +function value_iteration(model, tol=1e-5) + vz = zeros(length(model.w_grid), length(model.y_grid)) + v_star = successive_approx(v -> T(v, model), vz, tolerance=tol) + return get_greedy(v_star, model) +end + +"Howard policy iteration routine." +function policy_iteration(model) + wn, yn = length(model.w_grid), length(model.y_grid) + σ = ones(Int32, wn, yn) + i, error = 0, 1.0 + while error > 0 + v_σ = get_value(σ, model) + σ_new = get_greedy(v_σ, model) + error = maximum(abs.(σ_new - σ)) + σ = σ_new + i = i + 1 + println("Concluded loop $i with error $error.") + end + return σ +end + +"Optimistic policy iteration routine." +function optimistic_policy_iteration(model; tolerance=1e-5, m=100) + v = zeros(length(model.w_grid), length(model.y_grid)) + error = tolerance + 1 + while error > tolerance + last_v = v + σ = get_greedy(v, model) + for i in 1:m + v = T_σ(v, σ, model) + end + error = maximum(abs.(v - last_v)) + end + return get_greedy(v, model) +end + +# == Simulations and inequality measures == # + +function simulate_wealth(m) + + model = create_savings_model() + σ_star = optimistic_policy_iteration(model) + (; β, R, γ, w_grid, y_grid, Q) = model + + # Simulate labor income (indices rather than grid values) + mc = MarkovChain(Q) + y_idx_series = simulate(mc, m) + + # Compute corresponding wealth time series + w_idx_series = similar(y_idx_series) + w_idx_series[1] = 1 # initial condition + for t in 1:(m-1) + i, j = w_idx_series[t], y_idx_series[t] + w_idx_series[t+1] = σ_star[i, j] + end + w_series = w_grid[w_idx_series] + + return w_series +end + +function lorenz(v) # assumed sorted vector + S = cumsum(v) # cumulative sums: [v[1], v[1] + v[2], ... ] + F = (1:length(v)) / length(v) + L = S ./ S[end] + return (; F, L) # returns named tuple +end + +gini(v) = (2 * sum(i * y for (i,y) in enumerate(v))/sum(v) + - (length(v) + 1))/length(v) + + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + +function plot_timing(; m_vals=collect(range(1, 600, step=10)), + savefig=false) + model = create_savings_model(y_size=5) + println("Running Howard policy iteration.") + pi_time = @elapsed σ_pi = policy_iteration(model) + println("PI completed in $pi_time seconds.") + println("Running value function iteration.") + vfi_time = @elapsed σ_vfi = value_iteration(model) + println("VFI completed in $vfi_time seconds.") + @assert σ_vfi == σ_pi "Warning: policies deviated." + opi_times = [] + for m in m_vals + println("Running optimistic policy iteration with m=$m.") + opi_time = @elapsed σ_opi = optimistic_policy_iteration(model, m=m) + @assert σ_opi == σ_pi "Warning: policies deviated." + println("OPI with m=$m completed in $opi_time seconds.") + push!(opi_times, opi_time) + end + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(m_vals, fill(vfi_time, length(m_vals)), + lw=2, label="value function iteration") + ax.plot(m_vals, fill(pi_time, length(m_vals)), + lw=2, label="Howard policy iteration") + ax.plot(m_vals, opi_times, lw=2, label="optimistic policy iteration") + ax.legend(fontsize=fontsize, frameon=false) + ax.set_xlabel(L"m", fontsize=fontsize) + ax.set_ylabel("time", fontsize=fontsize) + if savefig + fig.savefig("./figures/finite_opt_saving_2_1.pdf") + end + return (pi_time, vfi_time, opi_times) +end + +function plot_policy(; method="pi") + model = create_savings_model() + (; β, R, γ, w_grid, y_grid, Q) = model + if method == "vfi" + σ_star = value_iteration(model) + elseif method == "pi" + σ_star = policy_iteration(model) + else + σ_star = optimistic_policy_iteration(model) + end + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_grid, w_grid, "k--", label=L"45") + ax.plot(w_grid, w_grid[σ_star[:, 1]], label=L"\sigma^*(\cdot, y_1)") + ax.plot(w_grid, w_grid[σ_star[:, end]], label=L"\sigma^*(\cdot, y_N)") + ax.legend(fontsize=fontsize) +end + + +function plot_time_series(; m=2_000, + savefig=false, + figname="./figures/finite_opt_saving_ts.pdf") + + w_series = simulate_wealth(m) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_series, label=L"w_t") + ax.set_xlabel("time", fontsize=fontsize) + ax.legend(fontsize=fontsize) + if savefig + fig.savefig(figname) + end +end + +function plot_histogram(; m=1_000_000, + savefig=false, + figname="./figures/finite_opt_saving_hist.pdf") + + w_series = simulate_wealth(m) + g = round(gini(sort(w_series)), digits=2) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.hist(w_series, bins=40, density=true) + ax.set_xlabel("wealth", fontsize=fontsize) + ax.text(15, 0.4, "Gini = $g", fontsize=fontsize) + + if savefig + fig.savefig(figname) + end +end + +function plot_lorenz(; m=1_000_000, + savefig=false, + figname="./figures/finite_opt_saving_lorenz.pdf") + + w_series = simulate_wealth(m) + (; F, L) = lorenz(sort(w_series)) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(F, F, label="Lorenz curve, equality") + ax.plot(F, L, label="Lorenz curve, wealth distribution") + ax.legend() + + if savefig + fig.savefig(figname) + end +end + +``` + +```{code-cell} julia +plot_timing(savefig=true) +``` + +```{code-cell} julia +plot_policy() +``` + +```{code-cell} julia +plot_time_series(savefig=true) +``` + +```{code-cell} julia +plot_histogram(savefig=true) +``` + +```{code-cell} julia +plot_lorenz(savefig=true) +``` +## finite_lq.jl +```{code-cell} julia +using QuantEcon, LinearAlgebra, IterTools +include("s_approx.jl") + +function create_investment_model(; + r=0.04, # Interest rate + a_0=10.0, a_1=1.0, # Demand parameters + γ=25.0, c=1.0, # Adjustment and unit cost + y_min=0.0, y_max=20.0, y_size=100, # Grid for output + ρ=0.9, ν=1.0, # AR(1) parameters + z_size=25) # Grid size for shock + β = 1/(1+r) + y_grid = LinRange(y_min, y_max, y_size) + mc = tauchen(y_size, ρ, ν) + z_grid, Q = mc.state_values, mc.p + return (; β, a_0, a_1, γ, c, y_grid, z_grid, Q) +end + +""" +The aggregator B is given by + + B(y, z, y′) = r(y, z, y′) + β Σ_z′ v(y′, z′) Q(z, z′)." + +where + + r(y, z, y′) := (a_0 - a_1 * y + z - c) y - γ * (y′ - y)^2 + +""" +function B(i, j, k, v, model) + (; β, a_0, a_1, γ, c, y_grid, z_grid, Q) = model + y, z, y′ = y_grid[i], z_grid[j], y_grid[k] + r = (a_0 - a_1 * y + z - c) * y - γ * (y′ - y)^2 + return @views r + β * dot(v[k, :], Q[j, :]) +end + +"The policy operator." +function T_σ(v, σ, model) + y_idx, z_idx = (eachindex(g) for g in (model.y_grid, model.z_grid)) + v_new = similar(v) + for (i, j) in product(y_idx, z_idx) + v_new[i, j] = B(i, j, σ[i, j], v, model) + end + return v_new +end + +"The Bellman operator." +function T(v, model) + y_idx, z_idx = (eachindex(g) for g in (model.y_grid, model.z_grid)) + v_new = similar(v) + for (i, j) in product(y_idx, z_idx) + v_new[i, j] = maximum(B(i, j, k, v, model) for k in y_idx) + end + return v_new +end + +"Compute a v-greedy policy." +function get_greedy(v, model) + y_idx, z_idx = (eachindex(g) for g in (model.y_grid, model.z_grid)) + σ = Matrix{Int32}(undef, length(y_idx), length(z_idx)) + for (i, j) in product(y_idx, z_idx) + _, σ[i, j] = findmax(B(i, j, k, v, model) for k in y_idx) + end + return σ +end + +"Value function iteration routine." +function value_iteration(model; tol=1e-5) + vz = zeros(length(model.y_grid), length(model.z_grid)) + v_star = successive_approx(v -> T(v, model), vz, tolerance=tol) + return get_greedy(v_star, model) +end + + + +"Get the value v_σ of policy σ." +function get_value(σ, model) + # Unpack and set up + (; β, a_0, a_1, γ, c, y_grid, z_grid, Q) = model + yn, zn = length(y_grid), length(z_grid) + n = yn * zn + # Function to extract (i, j) from m = i + (j-1)*yn" + single_to_multi(m) = (m-1)%yn + 1, div(m-1, yn) + 1 + # Allocate and create single index versions of P_σ and r_σ + P_σ = zeros(n, n) + r_σ = zeros(n) + for m in 1:n + i, j = single_to_multi(m) + y, z, y′ = y_grid[i], z_grid[j], y_grid[σ[i, j]] + r_σ[m] = (a_0 - a_1 * y + z - c) * y - γ * (y′ - y)^2 + for m′ in 1:n + i′, j′ = single_to_multi(m′) + if i′ == σ[i, j] + P_σ[m, m′] = Q[j, j′] + end + end + end + # Solve for the value of σ + v_σ = (I - β * P_σ) \ r_σ + # Return as multi-index array + return reshape(v_σ, yn, zn) +end + + +"Howard policy iteration routine." +function policy_iteration(model) + yn, zn = length(model.y_grid), length(model.z_grid) + σ = ones(Int32, yn, zn) + i, error = 0, 1.0 + while error > 0 + v_σ = get_value(σ, model) + σ_new = get_greedy(v_σ, model) + error = maximum(abs.(σ_new - σ)) + σ = σ_new + i = i + 1 + println("Concluded loop $i with error $error.") + end + return σ +end + +"Optimistic policy iteration routine." +function optimistic_policy_iteration(model; tol=1e-5, m=100) + v = zeros(length(model.y_grid), length(model.z_grid)) + error = tol + 1 + while error > tol + last_v = v + σ = get_greedy(v, model) + for i in 1:m + v = T_σ(v, σ, model) + end + error = maximum(abs.(v - last_v)) + end + return get_greedy(v, model) +end + + +# Plots + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=12 + +function plot_policy() + model = create_investment_model() + (; β, a_0, a_1, γ, c, y_grid, z_grid, Q) = model + σ_star = optimistic_policy_iteration(model) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(y_grid, y_grid, "k--", label=L"45") + ax.plot(y_grid, y_grid[σ_star[:, 1]], label=L"\sigma^*(\cdot, z_1)") + ax.plot(y_grid, y_grid[σ_star[:, end]], label=L"\sigma^*(\cdot, z_N)") + ax.legend(fontsize=fontsize) +end + +function plot_sim(; savefig=false, figname="./figures/finite_lq_1.pdf") + ts_length = 200 + + fig, axes = plt.subplots(4, 1, figsize=(9, 11.2)) + + for (ax, γ) in zip(axes, (1, 10, 20, 30)) + model = create_investment_model(γ=γ) + (; β, a_0, a_1, γ, c, y_grid, z_grid, Q) = model + σ_star = optimistic_policy_iteration(model) + mc = MarkovChain(Q, z_grid) + + z_sim_idx = simulate_indices(mc, ts_length) + z_sim = z_grid[z_sim_idx] + y_sim_idx = Vector{Int32}(undef, ts_length) + y_1 = (a_0 - c + z_sim[1]) / (2 * a_1) + y_sim_idx[1] = searchsortedfirst(y_grid, y_1) + for t in 1:(ts_length-1) + y_sim_idx[t+1] = σ_star[y_sim_idx[t], z_sim_idx[t]] + end + y_sim = y_grid[y_sim_idx] + y_bar_sim = (a_0 .- c .+ z_sim) ./ (2 * a_1) + + ax.plot(1:ts_length, y_sim, label=L"Y_t") + ax.plot(1:ts_length, y_bar_sim, label=L"\bar Y_t") + ax.legend(fontsize=fontsize, frameon=false, loc="upper right") + ax.set_ylabel("output", fontsize=fontsize) + ax.set_ylim(1, 9) + ax.set_title(L"\gamma = " * "$γ", fontsize=fontsize) + end + + fig.tight_layout() + if savefig + fig.savefig(figname) + end +end + + +function plot_timing(; m_vals=collect(range(1, 600, step=10)), + savefig=false, + figname="./figures/finite_lq_time.pdf" + ) + model = create_investment_model() + #println("Running Howard policy iteration.") + #pi_time = @elapsed σ_pi = policy_iteration(model) + #println("PI completed in $pi_time seconds.") + println("Running value function iteration.") + vfi_time = @elapsed σ_vfi = value_iteration(model, tol=1e-5) + println("VFI completed in $vfi_time seconds.") + #@assert σ_vfi == σ_pi "Warning: policies deviated." + opi_times = [] + for m in m_vals + println("Running optimistic policy iteration with m=$m.") + opi_time = @elapsed σ_opi = + optimistic_policy_iteration(model, m=m, tol=1e-5) + println("OPI with m=$m completed in $opi_time seconds.") + #@assert σ_opi == σ_pi "Warning: policies deviated." + push!(opi_times, opi_time) + end + fig, ax = plt.subplots(figsize=(9, 5.2)) + #ax.plot(m_vals, fill(pi_time, length(m_vals)), + # lw=2, label="Howard policy iteration") + ax.plot(m_vals, fill(vfi_time, length(m_vals)), + lw=2, label="value function iteration") + ax.plot(m_vals, opi_times, lw=2, label="optimistic policy iteration") + ax.legend(fontsize=fontsize, frameon=false) + ax.set_xlabel(L"m", fontsize=fontsize) + ax.set_ylabel("time", fontsize=fontsize) + if savefig + fig.savefig(figname) + end + return (vfi_time, opi_times) + #return (pi_time, vfi_time, opi_times) +end + +``` + +```{code-cell} julia +plot_policy() +``` + +```{code-cell} julia +plot_sim(savefig=true) +``` + +```{code-cell} julia +plot_timing(savefig=true) +``` +## firm_hiring.jl +```{code-cell} julia +using QuantEcon, LinearAlgebra, IterTools + +function create_hiring_model(; + r=0.04, # Interest rate + κ=1.0, # Adjustment cost + α=0.4, # Production parameter + p=1.0, w=1.0, # Price and wage + l_min=0.0, l_max=30.0, l_size=100, # Grid for labor + ρ=0.9, ν=0.4, b=1.0, # AR(1) parameters + z_size=100) # Grid size for shock + β = 1/(1+r) + l_grid = LinRange(l_min, l_max, l_size) + mc = tauchen(z_size, ρ, ν, b, 6) + z_grid, Q = mc.state_values, mc.p + return (; β, κ, α, p, w, l_grid, z_grid, Q) +end + +""" +The aggregator B is given by + + B(l, z, l′) = r(l, z, l′) + β Σ_z′ v(l′, z′) Q(z, z′)." + +where + + r(l, z, l′) := p * z * f(l) - w * l - κ 1{l != l′} + +""" +function B(i, j, k, v, model) + (; β, κ, α, p, w, l_grid, z_grid, Q) = model + l, z, l′ = l_grid[i], z_grid[j], l_grid[k] + r = p * z * l^α - w * l - κ * (l != l′) + return @views r + β * dot(v[k, :], Q[j, :]) +end + + +"The policy operator." +function T_σ(v, σ, model) + l_idx, z_idx = (eachindex(g) for g in (model.l_grid, model.z_grid)) + v_new = similar(v) + for (i, j) in product(l_idx, z_idx) + v_new[i, j] = B(i, j, σ[i, j], v, model) + end + return v_new +end + +"Compute a v-greedy policy." +function get_greedy(v, model) + (; β, κ, α, p, w, l_grid, z_grid, Q) = model + l_idx, z_idx = (eachindex(g) for g in (model.l_grid, model.z_grid)) + σ = Matrix{Int32}(undef, length(l_idx), length(z_idx)) + for (i, j) in product(l_idx, z_idx) + _, σ[i, j] = findmax(B(i, j, k, v, model) for k in l_idx) + end + return σ +end + +"Optimistic policy iteration routine." +function optimistic_policy_iteration(model; tolerance=1e-5, m=100) + v = zeros(length(model.l_grid), length(model.z_grid)) + error = tolerance + 1 + while error > tolerance + last_v = v + σ = get_greedy(v, model) + for i in 1:m + v = T_σ(v, σ, model) + end + error = maximum(abs.(v - last_v)) + end + return get_greedy(v, model) +end + + +# Plots + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=14 + +function plot_policy(; savefig=false, + figname="./figures/firm_hiring_pol.pdf") + model = create_hiring_model() + (; β, κ, α, p, w, l_grid, z_grid, Q) = model + σ_star = optimistic_policy_iteration(model) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(l_grid, l_grid, "k--", label=L"45") + ax.plot(l_grid, l_grid[σ_star[:, 1]], label=L"\sigma^*(\cdot, z_1)") + ax.plot(l_grid, l_grid[σ_star[:, end]], label=L"\sigma^*(\cdot, z_N)") + ax.legend(fontsize=fontsize) +end + + +function sim_dynamics(model, ts_length) + + (; β, κ, α, p, w, l_grid, z_grid, Q) = model + σ_star = optimistic_policy_iteration(model) + mc = MarkovChain(Q, z_grid) + z_sim_idx = simulate_indices(mc, ts_length) + z_sim = z_grid[z_sim_idx] + l_sim_idx = Vector{Int32}(undef, ts_length) + l_sim_idx[1] = 32 + for t in 1:(ts_length-1) + l_sim_idx[t+1] = σ_star[l_sim_idx[t], z_sim_idx[t]] + end + l_sim = l_grid[l_sim_idx] + + y_sim = similar(l_sim) + for (i, l) in enumerate(l_sim) + y_sim[i] = p * z_sim[i] * l_sim[i]^α + end + + t = ts_length - 1 + l_g, y_g, z_g = zeros(t), zeros(t), zeros(t) + + for i in 1:t + l_g[i] = (l_sim[i+1] - l_sim[i]) / l_sim[i] + y_g[i] = (y_sim[i+1] - y_sim[i]) / y_sim[i] + z_g[i] = (z_sim[i+1] - z_sim[i]) / z_sim[i] + end + + return l_sim, y_sim, z_sim, l_g, y_g, z_g + +end + + + +function plot_sim(; savefig=false, + figname="./figures/firm_hiring_ts.pdf", + ts_length = 250) + + model = create_hiring_model() + (; β, κ, α, p, w, l_grid, z_grid, Q) = model + l_sim, y_sim, z_sim, l_g, y_g, z_g = sim_dynamics(model, ts_length) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(1:ts_length, l_sim, label=L"\ell_t") + ax.plot(1:ts_length, z_sim, alpha=0.6, label=L"Z_t") + ax.legend(fontsize=fontsize, frameon=false) + ax.set_ylabel("employment", fontsize=fontsize) + ax.set_xlabel("time", fontsize=fontsize) + + if savefig + fig.savefig(figname) + end +end + + +function plot_growth(; savefig=false, + figname="./figures/firm_hiring_g.pdf", + ts_length = 10_000_000) + + model = create_hiring_model() + (; β, κ, α, p, w, l_grid, z_grid, Q) = model + l_sim, y_sim, z_sim, l_g, y_g, z_g = sim_dynamics(model, ts_length) + + fig, ax = plt.subplots() + ax.hist(l_g, alpha=0.6, bins=100) + ax.set_xlabel("growth", fontsize=fontsize) + + #fig, axes = plt.subplots(2, 1) + #series = y_g, z_g + #for (ax, g) in zip(axes, series) + # ax.hist(g, alpha=0.6, bins=100) + # ax.set_xlabel("growth", fontsize=fontsize) + #end + + plt.tight_layout() + if savefig + fig.savefig(figname) + end +end + + + +``` + +```{code-cell} julia +plot_policy() +``` + +```{code-cell} julia +plot_sim(savefig=true) +``` + +```{code-cell} julia +plot_growth(savefig=true) +``` +## modified_opt_savings.jl +```{code-cell} julia +using QuantEcon, LinearAlgebra, IterTools + +function create_savings_model(; β=0.98, γ=2.5, + w_min=0.01, w_max=20.0, w_size=100, + ρ=0.9, ν=0.1, y_size=20, + η_min=0.75, η_max=1.25, η_size=2) + η_grid = LinRange(η_min, η_max, η_size) + ϕ = ones(η_size) * (1 / η_size) # Uniform distributoin + w_grid = LinRange(w_min, w_max, w_size) + mc = tauchen(y_size, ρ, ν) + y_grid, Q = exp.(mc.state_values), mc.p + return (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) +end + + + +## == Functions for regular OPI == ## + +""" +B(w, y, η, w′) = u(w + y - w′/η)) + β Σ v(w′, y′, η′) Q(y, y′) ϕ(η′) +""" +function B(i, j, k, l, v, model) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + w, y, η, w′ = w_grid[i], y_grid[j], η_grid[k], w_grid[l] + u(c) = c^(1-γ)/(1-γ) + c = w + y - (w′/ η) + exp_value = 0.0 + for m in eachindex(y_grid) + for n in eachindex(η_grid) + exp_value += v[l, m, n] * Q[j, m] * ϕ[n] + end + end + return c > 0 ? u(c) + β * exp_value : -Inf +end + +"The policy operator." +function T_σ(v, σ, model) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + grids = w_grid, y_grid, η_grid + w_idx, y_idx, η_idx = (eachindex(g) for g in grids) + v_new = similar(v) + for (i, j, k) in product(w_idx, y_idx, η_idx) + v_new[i, j, k] = B(i, j, k, σ[i, j, k], v, model) + end + return v_new +end + +"Compute a v-greedy policy." +function get_greedy(v, model) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + w_idx, y_idx, η_idx = (eachindex(g) for g in (w_grid, y_grid, η_grid)) + σ = Array{Int32}(undef, length(w_idx), length(y_idx), length(η_idx)) + for (i, j, k) in product(w_idx, y_idx, η_idx) + _, σ[i, j, k] = findmax(B(i, j, k, l, v, model) for l in w_idx) + end + return σ +end + + +"Optimistic policy iteration routine." +function optimistic_policy_iteration(model; tolerance=1e-5, m=100) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + v = zeros(length(w_grid), length(y_grid), length(η_grid)) + error = tolerance + 1 + while error > tolerance + last_v = v + σ = get_greedy(v, model) + for i in 1:m + v = T_σ(v, σ, model) + end + error = maximum(abs.(v - last_v)) + println("OPI current error = $error") + end + return get_greedy(v, model) +end + + +## == Functions for modified OPI == ## + +"D(w, y, η, w′, g) = u(w + y - w′/η) + β g(y, w′)." +@inline function D(i, j, k, l, g, model) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + w, y, η, w′ = w_grid[i], y_grid[j], η_grid[k], w_grid[l] + u(c) = c^(1-γ)/(1-γ) + c = w + y - (w′/η) + return c > 0 ? u(c) + β * g[j, l] : -Inf +end + + +"Compute a g-greedy policy." +function get_g_greedy(g, model) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + w_idx, y_idx, η_idx = (eachindex(g) for g in (w_grid, y_grid, η_grid)) + σ = Array{Int32}(undef, length(w_idx), length(y_idx), length(η_idx)) + for (i, j, k) in product(w_idx, y_idx, η_idx) + _, σ[i, j, k] = findmax(D(i, j, k, l, g, model) for l in w_idx) + end + return σ +end + + +"The modified policy operator." +function R_σ(g, σ, model) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + w_idx, y_idx, η_idx = (eachindex(g) for g in (w_grid, y_grid, η_grid)) + g_new = similar(g) + for (j, i′) in product(y_idx, w_idx) # j indexes y, i′ indexes w′ + out = 0.0 + for j′ in y_idx # j′ indexes y′ + for k′ in η_idx # k′ indexes η′ + out += D(i′, j′, k′, σ[i′, j′, k′], g, model) * + Q[j, j′] * ϕ[k′] + end + end + g_new[j, i′] = out + end + return g_new +end + + +"Modified optimistic policy iteration routine." +function mod_opi(model; tolerance=1e-5, m=100) + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + g = zeros(length(y_grid), length(w_grid)) + error = tolerance + 1 + while error > tolerance + last_g = g + σ = get_g_greedy(g, model) + for i in 1:m + g = R_σ(g, σ, model) + end + error = maximum(abs.(g - last_g)) + println("OPI current error = $error") + end + return get_g_greedy(g, model) +end + + +# == Simulations and inequality measures == # + +function simulate_wealth(m) + + model = create_savings_model() + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + σ_star = mod_opi(model) + + # Simulate labor income + mc = MarkovChain(Q) + y_idx_series = simulate(mc, m) + + # IID Markov chain with uniform draws + l = length(η_grid) + mc = MarkovChain(ones(l, l) * (1/l)) + η_idx_series = simulate(mc, m) + + w_idx_series = similar(y_idx_series) + w_idx_series[1] = 1 + for t in 1:(m-1) + i, j, k = w_idx_series[t], y_idx_series[t], η_idx_series[t] + w_idx_series[t+1] = σ_star[i, j, k] + end + + w_series = w_grid[w_idx_series] + return w_series +end + + +function lorenz(v) # assumed sorted vector + S = cumsum(v) # cumulative sums: [v[1], v[1] + v[2], ... ] + F = (1:length(v)) / length(v) + L = S ./ S[end] + return (; F, L) # returns named tuple +end + + +gini(v) = (2 * sum(i * y for (i,y) in enumerate(v))/sum(v) + - (length(v) + 1))/length(v) + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + + + +function plot_contours(; savefig=false, + figname="./figures/modified_opt_savings_1.pdf") + + model = create_savings_model() + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + σ_star = optimistic_policy_iteration(model) + + fig, axes = plt.subplots(2, 1, figsize=(10, 8)) + y_idx, η_idx = eachindex(y_grid), eachindex(η_grid) + H = zeros(length(y_grid), length(η_grid)) + + w_indices = (1, length(w_grid)) + titles = "low wealth", "high wealth" + for (ax, w_idx, title) in zip(axes, w_indices, titles) + + for (i_y, i_ϵ) in product(y_idx, η_idx) + w, y, η = w_grid[w_idx], y_grid[i_y], η_grid[i_ϵ] + H[i_y, i_ϵ] = w_grid[σ_star[w_idx, i_y, i_ϵ]] / (w+y) + end + + cs1 = ax.contourf(y_grid, η_grid, transpose(H), alpha=0.5) + plt.colorbar(cs1, ax=ax) #, format="%.6f") + + ax.set_title(title, fontsize=fontsize) + ax.set_xlabel(L"y", fontsize=fontsize) + ax.set_ylabel(L"\varepsilon", fontsize=fontsize) + end + + plt.tight_layout() + if savefig + fig.savefig(figname) + end +end + + +function plot_policies(; savefig=false, + figname="./figures/modified_opt_savings_2.pdf") + + model = create_savings_model() + (; β, γ, η_grid, ϕ, w_grid, y_grid, Q) = model + σ_star = mod_opi(model) + y_bar = floor(Int, length(y_grid) / 2) # Index of mid-point of y_grid + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_grid, w_grid, "k--", label=L"45") + + for (i, η) in enumerate(η_grid) + label = L"\sigma^*" * " at " * L"\eta = " * "$η" + ax.plot(w_grid, w_grid[σ_star[:, y_bar, i]], label=label) + end + ax.legend(fontsize=fontsize) + + plt.tight_layout() + if savefig + fig.savefig(figname) + end +end + + +function plot_time_series(; m=2_000, + savefig=false, + figname="./figures/modified_opt_savings_ts.pdf") + + w_series = simulate_wealth(m) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_series, label=L"w_t") + ax.legend(fontsize=fontsize) + ax.set_xlabel("time", fontsize=fontsize) + if savefig + fig.savefig(figname) + end + +end + +function plot_histogram(; m=1_000_000, + savefig=false, + figname="./figures/modified_opt_savings_hist.pdf") + + w_series = simulate_wealth(m) + g = round(gini(sort(w_series)), digits=2) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.hist(w_series, bins=40, density=true) + ax.set_xlabel("wealth", fontsize=fontsize) + ax.text(15, 0.7, "Gini = $g", fontsize=fontsize) + if savefig + fig.savefig(figname) + end + +end + + +function plot_lorenz(; m=1_000_000, + savefig=false, + figname="./figures/modified_opt_savings_lorenz.pdf") + + w_series = simulate_wealth(m) + (; F, L) = lorenz(sort(w_series)) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(F, F, label="Lorenz curve, equality") + ax.plot(F, L, label="Lorenz curve, wealth distribution") + ax.legend() + if savefig + fig.savefig(figname) + end + +end + +``` + +```{code-cell} julia +plot_contours(savefig=true) +``` + +```{code-cell} julia +plot_policies(savefig=true) +``` + +```{code-cell} julia +plot_time_series(savefig=true) +``` + +```{code-cell} julia +plot_histogram(savefig=true) +``` + +```{code-cell} julia +plot_lorenz(savefig=true) +``` diff --git a/code-book/jl/markov_dynamics.md b/code-book/jl/markov_dynamics.md new file mode 100644 index 0000000..39868b7 --- /dev/null +++ b/code-book/jl/markov_dynamics.md @@ -0,0 +1,445 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 3: Markov Dynamics)= +```{raw} jupyter + +``` +# Chapter 3: Markov Dynamics + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## inventory_sim.jl +```{code-cell} julia +using Distributions, IterTools, QuantEcon + +function create_inventory_model(; S=100, # Order size + s=10, # Order threshold + p=0.4) # Demand parameter + ϕ = Geometric(p) + h(x, d) = max(x - d, 0) + S*(x <= s) + return (; S, s, p, ϕ, h) +end + +"Simulate the inventory process." +function sim_inventories(model; ts_length=200) + (; S, s, p, ϕ, h) = model + X = Vector{Int32}(undef, ts_length) + X[1] = S # Initial condition + for t in 1:(ts_length-1) + X[t+1] = h(X[t], rand(ϕ)) + end + return X +end + +"Compute the transition probabilities and state." +function compute_mc(model; d_max=100) + (; S, s, p, ϕ, h) = model + n = S + s + 1 # Size of state space + state_vals = collect(0:(S + s)) + P = Matrix{Float64}(undef, n, n) + for (i, j) in product(1:n, 1:n) + P[i, j] = sum((h(i-1, d) == j-1)*pdf(ϕ, d) for d in 0:d_max) + end + return MarkovChain(P, state_vals) +end + +"Compute the stationary distribution of the model." +function compute_stationary_dist(model) + mc = compute_mc(model) + return mc.state_values, stationary_distributions(mc)[1] +end + + +# Plots + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering + + +function plot_ts(model; fontsize=16, + figname="./figures/inventory_sim_1.pdf", + savefig=false) + (; S, s, p, ϕ, h) = model + X = sim_inventories(model) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(X, label=L"X_t", lw=3, alpha=0.6) + ax.set_xlabel(L"t", fontsize=fontsize) + ax.set_ylabel("inventory", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=false) + ax.set_ylim(0, S + s + 20) + + if savefig == true + fig.savefig(figname) + end +end + + +function plot_hist(model; fontsize=16, + figname="./figures/inventory_sim_2.pdf", + savefig=false) + (; S, s, p, ϕ, h) = model + state_values, ψ_star = compute_stationary_dist(model) + X = sim_inventories(model; ts_length=1_000_000) + histogram = [mean(X .== i) for i in state_values] + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(state_values, ψ_star, "k-", lw=3, alpha=0.7, label=L"\psi^*") + ax.bar(state_values, histogram, alpha=0.7, label="frequency") + ax.set_xlabel("state", fontsize=fontsize) + + ax.legend(fontsize=fontsize, frameon=false) + ax.set_ylim(0, 0.015) + + if savefig == true + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +model = create_inventory_model() +``` + +```{code-cell} julia +plot_ts(model; savefig=true) +``` + +```{code-cell} julia +plot_hist(model; savefig=true) +``` +## is_irreducible.jl +```{code-cell} julia +using QuantEcon +P = [0.1 0.9; + 0.0 1.0] +mc = MarkovChain(P) +print(is_irreducible(mc)) + +``` +## laborer_sim.jl +```{code-cell} julia +function create_laborer_model(; α=0.3, β=0.2) + return (; α, β) +end + +function laborer_update(x, model) # update X from t to t+1 + (; α, β) = model + if x == 1 + x′ = rand() < α ? 2 : 1 + else + x′ = rand() < β ? 1 : 2 + end + return x′ +end + +function sim_chain(k, p, model) + X = Array{Int32}(undef, k) + X[1] = rand() < p ? 1 : 2 + for t in 1:(k-1) + X[t+1] = laborer_update(X[t], model) + end + return X +end + +function test_convergence(; k=10_000_000, p=0.5) + model = create_laborer_model() + (; α, β) = model + ψ_star = (1/(α + β)) * [β α] + + X = sim_chain(k, p, model) + ψ_e = (1/k) * [sum(X .== 1) sum(X .== 2)] + error = maximum(abs.(ψ_star - ψ_e)) + approx_equal = isapprox(ψ_star, ψ_e, rtol=0.01) + println("Sup norm deviation is $error") + println("Approximate equality is $approx_equal") + end + + +``` +## markov_js.jl +```{code-cell} julia +""" +Infinite-horizon job search with Markov wage draws. + +""" + +using QuantEcon, LinearAlgebra +include("s_approx.jl") + +"Creates an instance of the job search model with Markov wages." +function create_markov_js_model(; + n=200, # wage grid size + ρ=0.9, # wage persistence + ν=0.2, # wage volatility + β=0.98, # discount factor + c=1.0 # unemployment compensation + ) + mc = tauchen(n, ρ, ν) + w_vals, P = exp.(mc.state_values), mc.p + return (; n, w_vals, P, β, c) +end + +" The Bellman operator Tv = max{e, c + β P v} with e(w) = w / (1-β)." +function T(v, model) + (; n, w_vals, P, β, c) = model + h = c .+ β * P * v + e = w_vals ./ (1 - β) + return max.(e, h) +end + +" Get a v-greedy policy." +function get_greedy(v, model) + (; n, w_vals, P, β, c) = model + σ = w_vals / (1 - β) .>= c .+ β * P * v + return σ +end + +"Solve the infinite-horizon Markov job search model by VFI." +function vfi(model) + v_init = zero(model.w_vals) + v_star = successive_approx(v -> T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star +end + + + +# == Policy iteration == # + +"Get the value of policy σ." +function get_value(σ, model) + (; n, w_vals, P, β, c) = model + e = w_vals ./ (1 - β) + K_σ = β .* (1 .- σ) .* P + r_σ = σ .* e .+ (1 .- σ) .* c + return (I - K_σ) \ r_σ +end + + + +"Howard policy iteration routine." +function policy_iteration(model) + σ = Vector{Bool}(undef, model.n) + i, error = 0, 1.0 + while error > 0 + v_σ = get_value(σ, model) + σ_new = get_greedy(v_σ, model) + error = maximum(abs.(σ_new - σ)) + σ = σ_new + i = i + 1 + println("Concluded loop $i with error $error.") + end + return σ +end + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + +default_model = create_markov_js_model() + + +function plot_main(; model=default_model, + method="vfi", + savefig=false, + figname="./figures/markov_js_1.pdf") + (; n, w_vals, P, β, c) = model + + + if method == "vfi" + v_star, σ_star = vfi(model) + else + σ_star = policy_iteration(model) + v_star = get_value(σ_star, model) + end + + h_star = c .+ β * P * v_star + e = w_vals / (1 - β) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_vals, h_star, lw=4, ls="--", alpha=0.4, label=L"h^*(w)") + ax.plot(w_vals, e, lw=4, ls="--", alpha=0.4, label=L"w/(1-\beta)") + ax.plot(w_vals, max.(e, h_star), "k-", alpha=0.7, label=L"v^*(w)") + ax.legend(frameon=false, fontsize=fontsize) + ax.set_xlabel(L"w", fontsize=fontsize) + if savefig + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +plot_main(savefig=true) +``` +## markov_js_with_sep.jl +```{code-cell} julia +""" +Infinite-horizon job search with Markov wage draws and separation. + +""" + +include("s_approx.jl") +using QuantEcon, LinearAlgebra + +"Creates an instance of the job search model with separation." +function create_js_with_sep_model(; + n=200, # wage grid size + ρ=0.9, ν=0.2, # wage persistence and volatility + β=0.98, α=0.1, # discount factor and separation rate + c=1.0) # unemployment compensation + mc = tauchen(n, ρ, ν) + w_vals, P = exp.(mc.state_values), mc.p + return (; n, w_vals, P, β, c, α) +end + +" The Bellman operator for the value of being unemployed." +function T(v, model) + (; n, w_vals, P, β, c, α) = model + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * P * v) + reject = c .+ β * P * v + return max.(accept, reject) +end + +" Get a v-greedy policy." +function get_greedy(v, model) + (; n, w_vals, P, β, c, α) = model + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * P * v) + reject = c .+ β * P * v + σ = accept .>= reject + return σ +end + +"Solve by VFI." +function vfi(model) + v_init = zero(model.w_vals) + v_star = successive_approx(v -> T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star +end + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + +default_model = create_js_with_sep_model() + + +function plot_main(; model=default_model, + method="vfi", + savefig=false, + figname="./figures/markov_js_with_sep_1.pdf") + (; n, w_vals, P, β, c, α) = model + v_star, σ_star = vfi(model) + + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * P * v_star) + h_star = c .+ β * P * v_star + + w_star = Inf + for (i, w) in enumerate(w_vals) + if accept[i] ≥ h_star[i] + w_star = w + break + end + end + @assert w_star < Inf "Agent never accepts" + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_vals, h_star, lw=4, ls="--", alpha=0.4, label="continuation value") + ax.plot(w_vals, accept, lw=4, ls="--", alpha=0.4, label="stopping value") + ax.plot(w_vals, v_star, "k-", alpha=0.7, label=L"v_u^*(w)") + ax.legend(frameon=false, fontsize=fontsize) + ax.set_xlabel(L"w", fontsize=fontsize) + if savefig + fig.savefig(figname) + end +end + +function plot_w_stars(; α_vals=LinRange(0.0, 1.0, 10), + savefig=false, + figname="./figures/markov_js_with_sep_2.pdf") + + w_star_vec = similar(α_vals) + for (i_α, α) in enumerate(α_vals) + print(i_α, α) + model = create_js_with_sep_model(α=α) + (; n, w_vals, P, β, c, α) = model + v_star, σ_star = vfi(model) + + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * P * v_star) + h_star = c .+ β * P * v_star + + w_star = Inf + for (i_w, w) in enumerate(w_vals) + if accept[i_w] ≥ h_star[i_w] + w_star = w + break + end + end + @assert w_star < Inf "Agent never accepts" + w_star_vec[i_α] = w_star + end + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(α_vals, w_star_vec, lw=2, alpha=0.6, label="reservation wage") + ax.legend(frameon=false, fontsize=fontsize) + ax.set_xlabel(L"\alpha", fontsize=fontsize) + ax.set_xlabel(L"w", fontsize=fontsize) + if savefig + fig.savefig(figname) + end +end + + + +``` + +```{code-cell} julia +plot_main(savefig=true) +``` + +```{code-cell} julia +plot_w_stars(savefig=true) +``` diff --git a/code-book/jl/nonlinear_valuation.md b/code-book/jl/nonlinear_valuation.md new file mode 100644 index 0000000..86cdb6a --- /dev/null +++ b/code-book/jl/nonlinear_valuation.md @@ -0,0 +1,291 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 7: Nonlinear Valuation)= +```{raw} jupyter + +``` +# Chapter 7: Nonlinear Valuation + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## rs_utility.jl +```{code-cell} julia +include("s_approx.jl") +using LinearAlgebra, QuantEcon + +function create_rs_utility_model(; + n=180, # size of state space + β=0.95, # time discount factor + ρ=0.96, # correlation coef in AR(1) + σ=0.1, # volatility + θ=-1.0) # risk aversion + mc = tauchen(n, ρ, σ, 0, 10) # n_std = 10 + x_vals, P = mc.state_values, mc.p + r = x_vals # special case u(c(x)) = x + return (; β, θ, ρ, σ, r, x_vals, P) +end + +function K(v, model) + (; β, θ, ρ, σ, r, x_vals, P) = model + return r + (β/θ) * log.(P * (exp.(θ*v))) +end + +function compute_rs_utility(model) + (; β, θ, ρ, σ, r, x_vals, P) = model + v_init = zeros(length(x_vals)) + v_star = successive_approx(v -> K(v, model), + v_init, tolerance=1e-10) + return v_star +end + + +# Plots + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + +function plot_v(; savefig=false, + figname="./figures/rs_utility_1.pdf") + + fig, ax = plt.subplots(figsize=(10, 5.2)) + model = create_rs_utility_model() + (; β, θ, ρ, σ, r, x_vals, P) = model + + a = 1/(1 - (ρ*β)) + b = (β /(1 - β)) * (θ/2) * (a*σ)^2 + + v_star = compute_rs_utility(model) + v_star_a = a * x_vals .+ b + ax.plot(x_vals, v_star, lw=2, alpha=0.7, label="approximate fixed point") + ax.plot(x_vals, v_star_a, "k--", lw=2, alpha=0.7, label=L"v(x)=ax + b") + ax.set_xlabel(L"x", fontsize=fontsize) + + ax.legend(frameon=false, fontsize=fontsize, loc="upper left") + if savefig + fig.savefig(figname) + end +end + + + + +function plot_multiple_v(; savefig=false, + figname="./figures/rs_utility_2.pdf") + + fig, ax = plt.subplots(figsize=(10, 5.2)) + σ_vals = 0.05, 0.1 + + for σ in σ_vals + model = create_rs_utility_model(σ=σ) + (; β, θ, r, x_vals, P) = model + v_star = compute_rs_utility(model) + ax.plot(x_vals, v_star, lw=2, alpha=0.7, label=L"\sigma="*"$σ") + ax.set_xlabel(L"x", fontsize=fontsize) + ax.set_ylabel(L"v(x)", fontsize=fontsize) + end + + ax.legend(frameon=false, fontsize=fontsize, loc="upper left") + if savefig + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +plot_v(savefig=true) +``` + +```{code-cell} julia +plot_multiple_v(savefig=true) +``` +## ez_utility.jl +```{code-cell} julia +""" +Epstein--Zin utility: solving the recursion for a given consumption +path. + +""" + +include("s_approx.jl") +using LinearAlgebra, QuantEcon + +function create_ez_utility_model(; + n=200, # size of state space + ρ=0.96, # correlation coef in AR(1) + σ=0.1, # volatility + β=0.99, # time discount factor + α=0.75, # EIS parameter + γ=-2.0) # risk aversion parameter + + mc = tauchen(n, ρ, σ, 0, 5) + x_vals, P = mc.state_values, mc.p + c = exp.(x_vals) + + return (; β, ρ, σ, α, γ, c, x_vals, P) +end + +function K(v, model) + (; β, ρ, σ, α, γ, c, x_vals, P) = model + + R = (P * (v.^γ)).^(1/γ) + return ((1 - β) * c.^α + β * R.^α).^(1/α) +end + +function compute_ez_utility(model) + v_init = ones(length(model.x_vals)) + v_star = successive_approx(v -> K(v, model), + v_init, + tolerance=1e-10) + return v_star +end + + +# Plots + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + +function plot_convergence(; savefig=false, + num_iter=100, + figname="./figures/ez_utility_c.pdf") + + fig, ax = plt.subplots(figsize=(10, 5.2)) + model = create_ez_utility_model() + (; β, ρ, σ, α, γ, c, x_vals, P) = model + + + v_star = compute_ez_utility(model) + v = 0.1 * v_star + ax.plot(x_vals, v, lw=3, "k-", alpha=0.7, label=L"v_0") + + greys = [string(g) for g in LinRange(0.0, 0.4, num_iter)] + greys = reverse(greys) + + for (i, g) in enumerate(greys) + ax.plot(x_vals, v, "k-", color=g, lw=1, alpha=0.7) + for t in 1:20 + v = K(v, model) + end + end + + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, lw=3, alpha=0.7, label=L"v^*") + ax.set_xlabel(L"x", fontsize=fontsize) + + ax.legend(frameon=false, fontsize=fontsize, loc="upper left") + if savefig + fig.savefig(figname) + end +end + + +function plot_v(; savefig=false, + figname="./figures/ez_utility_1.pdf") + + fig, ax = plt.subplots(figsize=(10, 5.2)) + model = create_ez_utility_model() + (; β, ρ, σ, α, γ, c, x_vals, P) = model + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, lw=2, alpha=0.7, label=L"v^*") + ax.set_xlabel(L"x", fontsize=fontsize) + + ax.legend(frameon=false, fontsize=fontsize, loc="upper left") + if savefig + fig.savefig(figname) + end +end + + +function vary_gamma(; gamma_vals=[1.0, -8.0], + savefig=false, + figname="./figures/ez_utility_2.pdf") + + fig, ax = plt.subplots(figsize=(10, 5.2)) + + for γ in gamma_vals + model = create_ez_utility_model(γ=γ) + (; β, ρ, σ, α, γ, c, x_vals, P) = model + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, lw=2, alpha=0.7, label=L"\gamma="*"$γ") + ax.set_xlabel(L"x", fontsize=fontsize) + ax.set_ylabel(L"v(x)", fontsize=fontsize) + end + + ax.legend(frameon=false, fontsize=fontsize, loc="upper left") + if savefig + fig.savefig(figname) + end +end + + +function vary_alpha(; alpha_vals=[0.5, 0.6], + savefig=false, + figname="./figures/ez_utility_3.pdf") + + fig, ax = plt.subplots(figsize=(10, 5.2)) + + for α in alpha_vals + model = create_ez_utility_model(α=α) + (; β, ρ, σ, α, γ, c, x_vals, P) = model + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, lw=2, alpha=0.7, label=L"\alpha="*"$α") + ax.set_xlabel(L"x", fontsize=fontsize) + ax.set_ylabel(L"v(x)", fontsize=fontsize) + end + + ax.legend(frameon=false, fontsize=fontsize, loc="upper left") + if savefig + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +plot_convergence(savefig=true) +``` + +```{code-cell} julia +plot_v(savefig=true) +``` + +```{code-cell} julia +vary_gamma(savefig=true) +``` + +```{code-cell} julia +vary_alpha(savefig=true) +``` diff --git a/code-book/jl/operators_fixed_points.md b/code-book/jl/operators_fixed_points.md new file mode 100644 index 0000000..b46eae5 --- /dev/null +++ b/code-book/jl/operators_fixed_points.md @@ -0,0 +1,178 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 2: Operators and Fixed Points)= +```{raw} jupyter + +``` +# Chapter 2: Operators and Fixed Points + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## lake.jl +```{code-cell} julia +using LinearAlgebra + +α, λ, d, b = 0.01, 0.1, 0.02, 0.025 +g = b - d +A = [(1 - d) * (1 - λ) + b (1 - d) * α + b; + (1 - d) * λ (1 - d) * (1 - α)] + +ū = (1 + g - (1 - d) * (1 - α)) / + (1 + g - (1 - d) * (1 - α) + (1 - d) * λ) + +ē = 1 - ū +x̄ = [ū; ē] + +println(isapprox(A * x̄, (1 + g) * x̄)) # prints true + +# == Plots == # +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=14 + + + +function plot_paths(; figname="./figures/lake_1.pdf", + savefig=false) + path_length = 100 + x_path_1 = zeros(2, path_length) + x_path_2 = zeros(2, path_length) + x_0_1 = 5.0, 0.1 + x_0_2 = 0.1, 4.0 + x_path_1[1, 1] = x_0_1[1] + x_path_1[2, 1] = x_0_1[2] + x_path_2[1, 1] = x_0_2[1] + x_path_2[2, 1] = x_0_2[2] + + + for t in 1:(path_length-1) + x_path_1[:, t+1] = A * x_path_1[:, t] + x_path_2[:, t+1] = A * x_path_2[:, t] + end + + fig, ax = plt.subplots() + + # Set the axes through the origin + for spine in ["left", "bottom"] + ax.spines[spine].set_position("zero") + end + for spine in ["right", "top"] + ax.spines[spine].set_color("none") + end + + ax.set_xlim(0, 6) + ax.set_ylim(0, 6) + ax.set_xlabel("unemployed workforce", fontsize=fontsize) + ax.set_ylabel("employed workforce", fontsize=fontsize) + ax.set_xticks((0, 6)) + ax.set_yticks((0, 6)) + s = 10 + ax.plot([0, s * ū], [0, s * ē], "k--", lw=1) + ax.scatter(x_path_1[1, :], x_path_1[2, :], s=4, c="blue") + ax.scatter(x_path_2[1, :], x_path_2[2, :], s=4, c="green") + + ax.plot([ū], [ē], "ko", ms=4, alpha=0.6) + ax.annotate(L"\bar x", + xy=(ū, ē), + xycoords="data", + xytext=(20, -20), + textcoords="offset points", + fontsize=fontsize, + arrowprops=Dict("arrowstyle" => "->")) + + x, y = x_0_1[1], x_0_1[2] + lb = latexstring("\$x_0 = ($(x), $(y))\$") + ax.plot([x], [y], "ko", ms=2, alpha=0.6) + ax.annotate(lb, + xy=(x, y), + xycoords="data", + xytext=(0, 20), + textcoords="offset points", + fontsize=fontsize, + arrowprops=Dict("arrowstyle" => "->")) + + x, y = x_0_2[1], x_0_2[2] + lb = latexstring("\$x_0 = ($(x), $(y))\$") + ax.plot([x], [y], "ko", ms=2, alpha=0.6) + ax.annotate(lb, + xy=(x, y), + xycoords="data", + xytext=(0, 20), + textcoords="offset points", + fontsize=fontsize, + arrowprops=Dict("arrowstyle" => "->")) + + if savefig + fig.savefig(figname) + end +end + + +function plot_growth(; savefig=false, figname="./figures/lake_2.pdf") + + + path_length = 100 + x_0 = 2.1, 1.2 + x = zeros(2, path_length) + x[1, 1] = 0.6 + x[2, 1] = 1.2 + + for t in 1:(path_length-1) + x[:, t+1] = A * x[:, t] + end + + fig, axes = plt.subplots(3, 1) + u = x[1, :] + e = x[2, :] + n = x[1, :] .+ x[2, :] + paths = u, e, n + labels = L"u_t", L"e_t", L"n_t" + for (ax, path, label) in zip(axes, paths, labels) + ax.plot(path, label=label) + ax.legend(frameon=false, fontsize=14) + ax.set_xlabel(L"t") + end + + plt.tight_layout() + if savefig + fig.savefig(figname) + end + +end + +``` + +```{code-cell} julia +plot_paths(savefig=true) +``` + +```{code-cell} julia +plot_growth(savefig=true) +``` diff --git a/code-book/jl/optimal_stopping.md b/code-book/jl/optimal_stopping.md new file mode 100644 index 0000000..e448d50 --- /dev/null +++ b/code-book/jl/optimal_stopping.md @@ -0,0 +1,315 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 4: Optimal Stopping)= +```{raw} jupyter + +``` +# Chapter 4: Optimal Stopping + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## firm_exit.jl +```{code-cell} julia +""" +Firm valuation with exit option. + +""" + +using QuantEcon, LinearAlgebra +include("s_approx.jl") + +"Creates an instance of the firm exit model." +function create_exit_model(; + n=200, # productivity grid size + ρ=0.95, μ=0.1, ν=0.1, # persistence, mean and volatility + β=0.98, s=100.0 # discount factor and scrap value + ) + mc = tauchen(n, ρ, ν, μ) + z_vals, Q = mc.state_values, mc.p + return (; n, z_vals, Q, β, s) +end + + +"Compute value of firm without exit option." +function no_exit_value(model) + (; n, z_vals, Q, β, s) = model + return (I - β * Q) \ z_vals +end + +" The Bellman operator Tv = max{s, π + β Q v}." +function T(v, model) + (; n, z_vals, Q, β, s) = model + h = z_vals .+ β * Q * v + return max.(s, h) +end + +" Get a v-greedy policy." +function get_greedy(v, model) + (; n, z_vals, Q, β, s) = model + σ = s .>= z_vals .+ β * Q * v + return σ +end + +"Solve by VFI." +function vfi(model) + v_init = no_exit_value(model) + v_star = successive_approx(v -> T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star +end + + +# Plots + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + + +function plot_val(; savefig=false, + figname="./figures/firm_exit_1.pdf") + + fig, ax = plt.subplots(figsize=(9, 5.2)) + + model = create_exit_model() + (; n, z_vals, Q, β, s) = model + + v_star, σ_star = vfi(model) + h = z_vals + β * Q * v_star + + ax.plot(z_vals, h, "-", lw=3, alpha=0.6, label=L"h^*") + ax.plot(z_vals, s * ones(n), "-", lw=3, alpha=0.6, label=L"s") + ax.plot(z_vals, v_star, "k--", lw=1.5, alpha=0.8, label=L"v^*") + + ax.legend(frameon=false, fontsize=fontsize) + ax.set_xlabel(L"z", fontsize=fontsize) + + if savefig + fig.savefig(figname) + end +end + + +function plot_comparison(; savefig=false, + figname="./figures/firm_exit_2.pdf") + + fig, ax = plt.subplots(figsize=(9, 5.2)) + + model = create_exit_model() + (; n, z_vals, Q, β, s) = model + + v_star, σ_star = vfi(model) + w = no_exit_value(model) + + ax.plot(z_vals, v_star, "k-", lw=2, alpha=0.6, label=L"v^*") + ax.plot(z_vals, w, lw=2, alpha=0.6, label="no-exit value") + + ax.legend(frameon=false, fontsize=fontsize) + ax.set_xlabel(L"z", fontsize=fontsize) + + if savefig + fig.savefig(figname) + end +end + + + +``` + +```{code-cell} julia +plot_val(savefig=true) +``` + +```{code-cell} julia +plot_comparison(savefig=true) +``` +## american_option.jl +```{code-cell} julia +""" +Valuation for finite-horizon American call options in discrete time. + +""" + +include("s_approx.jl") +using QuantEcon, LinearAlgebra, IterTools + +"Creates an instance of the option model with log S_t = Z_t + W_t." +function create_american_option_model(; + n=100, μ=10.0, # Markov state grid size and mean value + ρ=0.98, ν=0.2, # persistence and volatility for Markov state + s=0.3, # volatility parameter for W_t + r=0.01, # interest rate + K=10.0, T=200) # strike price and expiration date + t_vals = collect(1:T+1) + mc = tauchen(n, ρ, ν) + z_vals, Q = mc.state_values .+ μ, mc.p + w_vals, φ, β = [-s, s], [0.5, 0.5], 1 / (1 + r) + e(t, i_w, i_z) = (t ≤ T) * (z_vals[i_z] + w_vals[i_w] - K) + return (; t_vals, z_vals, w_vals, Q, φ, T, β, K, e) +end + +"The continuation value operator." +function C(h, model) + (; t_vals, z_vals, w_vals, Q, φ, T, β, K, e) = model + Ch = similar(h) + z_idx, w_idx = eachindex(z_vals), eachindex(w_vals) + for (t, i_z) in product(t_vals, z_idx) + out = 0.0 + for (i_w′, i_z′) in product(w_idx, z_idx) + t′ = min(t + 1, T + 1) + out += max(e(t′, i_w′, i_z′), h[t′, i_z′]) * + Q[i_z, i_z′] * φ[i_w′] + end + Ch[t, i_z] = β * out + end + return Ch +end + +"Compute the continuation value function by successive approx." +function compute_cvf(model) + h_init = zeros(length(model.t_vals), length(model.z_vals)) + h_star = successive_approx(h -> C(h, model), h_init) + return h_star +end + + +# Plots + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + + +function plot_contours(; savefig=false, + figname="./figures/american_option_1.pdf") + + model = create_american_option_model() + (; t_vals, z_vals, w_vals, Q, φ, T, β, K, e) = model + h_star = compute_cvf(model) + + fig, axes = plt.subplots(3, 1, figsize=(7, 11)) + z_idx, w_idx = eachindex(z_vals), eachindex(w_vals) + H = zeros(length(w_vals), length(z_vals)) + + for (ax_index, t) in zip(1:3, (1, 195, 199)) + + ax = axes[ax_index, 1] + + for (i_w, i_z) in product(w_idx, z_idx) + H[i_w, i_z] = e(t, i_w, i_z) - h_star[t, i_z] + end + + cs1 = ax.contourf(w_vals, z_vals, transpose(H), alpha=0.5) + ctr1 = ax.contour(w_vals, z_vals, transpose(H), levels=[0.0]) + plt.clabel(ctr1, inline=1, fontsize=13) + plt.colorbar(cs1, ax=ax) #, format="%.6f") + + ax.set_title(L"t=" * "$t", fontsize=fontsize) + ax.set_xlabel(L"w", fontsize=fontsize) + ax.set_ylabel(L"z", fontsize=fontsize) + + end + + fig.tight_layout() + if savefig + fig.savefig(figname) + end +end + + +function plot_strike(; savefig=false, + fontsize=12, + figname="./figures/american_option_2.pdf") + + model = create_american_option_model() + (; t_vals, z_vals, w_vals, Q, φ, T, β, K, e) = model + h_star = compute_cvf(model) + + # Built Markov chains for simulation + z_mc = MarkovChain(Q, z_vals) + P_φ = zeros(length(w_vals), length(w_vals)) + for i in eachindex(w_vals) # Build IID chain + P_φ[i, :] = φ + end + w_mc = MarkovChain(P_φ, w_vals) + + + y_min = minimum(z_vals) + minimum(w_vals) + y_max = maximum(z_vals) + maximum(w_vals) + fig, axes = plt.subplots(3, 1, figsize=(7, 12)) + + for ax in axes + + # Generate price series + z_draws = simulate_indices(z_mc, T, init=Int(length(z_vals) / 2 - 10)) + w_draws = simulate_indices(w_mc, T) + s_vals = z_vals[z_draws] + w_vals[w_draws] + + # Find the exercise date, if any. + exercise_date = T + 1 + for t in 1:T + if e(t, w_draws[t], z_draws[t]) ≥ h_star[w_draws[t], z_draws[t]] + exercise_date = t + end + end + + @assert exercise_date ≤ T "Option not exercised." + + # Plot + ax.set_ylim(y_min, y_max) + ax.set_xlim(1, T) + ax.fill_between(1:T, ones(T) * K, ones(T) * y_max, alpha=0.2) + ax.plot(1:T, s_vals, label=L"S_t") + ax.plot((exercise_date,), (s_vals[exercise_date]), "ko") + ax.vlines((exercise_date,), 0, (s_vals[exercise_date]), ls="--", colors="black") + ax.legend(loc="upper left", fontsize=fontsize) + ax.text(-10, 11, "in the money", fontsize=fontsize, rotation=90) + ax.text(-10, 7.2, "out of the money", fontsize=fontsize, rotation=90) + ax.text(exercise_date-20, 6, #s_vals[exercise_date]+0.8, + "exercise date", fontsize=fontsize) + ax.set_xticks((1, T)) + ax.set_yticks((y_min, y_max)) + end + + if savefig + fig.savefig(figname) + end +end + +``` + +```{code-cell} julia +plot_contours(savefig=true) +``` + +```{code-cell} julia +plot_strike(savefig=true) +``` diff --git a/code-book/jl/recursive_decision_processes.md b/code-book/jl/recursive_decision_processes.md new file mode 100644 index 0000000..b34c490 --- /dev/null +++ b/code-book/jl/recursive_decision_processes.md @@ -0,0 +1,182 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 8: Recursive Decision Processes)= +```{raw} jupyter + +``` +# Chapter 8: Recursive Decision Processes + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## quantile_function.jl +```{code-cell} julia +import Distributions.quantile, Distributions.DiscreteNonParametric + +"Compute the τ-th quantile of v(X) when X ∼ ϕ and v = sort(v)." +function quantile(τ, v, ϕ) + for (i, v_value) in enumerate(v) + p = sum(ϕ[1:i]) # sum all ϕ[j] s.t. v[j] ≤ v_value + if p ≥ τ # exit and return v_value if prob ≥ τ + return v_value + end + end + +end + +"For each i, compute the τ-th quantile of v(Y) when Y ∼ P(i, ⋅)" +function R(τ, v, P) + return [quantile(τ, v, P[i, :]) for i in eachindex(v)] +end + + +function quantile_test(τ) + ϕ = [0.1, 0.2, 0.7] + v = [10, 20, 30] + + d = DiscreteNonParametric(v, ϕ) + return quantile(τ, v, ϕ), quantile(d, τ) +end + + + + +``` +## quantile_js.jl +```{code-cell} julia +""" +Job search with Markov wage draws and quantile preferences. + +""" + +using QuantEcon +include("quantile_function.jl") + +"Creates an instance of the job search model." +function create_markov_js_model(; + n=100, # wage grid size + ρ=0.9, # wage persistence + ν=0.2, # wage volatility + β=0.98, # discount factor + c=1.0, # unemployment compensation + τ=0.5 # quantile parameter + ) + mc = tauchen(n, ρ, ν) + w_vals, P = exp.(mc.state_values), mc.p + return (; n, w_vals, P, β, c, τ) +end + +""" +The policy operator + + (T_σ v)(w) = σ(w) (w / (1-β)) + (1 - σ(w))(c + β (R_τ v)(w)) + +""" +function T_σ(v, σ, model) + (; n, w_vals, P, β, c, τ) = model + h = c .+ β * R(τ, v, P) + e = w_vals ./ (1 - β) + return σ .* e + (1 .- σ) .* h +end + +" Get a v-greedy policy." +function get_greedy(v, model) + (; n, w_vals, P, β, c, τ) = model + σ = w_vals / (1 - β) .≥ c .+ β * R(τ, v, P) + return σ +end + + +"Optimistic policy iteration routine." +function optimistic_policy_iteration(model; tolerance=1e-5, m=100) + (; n, w_vals, P, β, c, τ) = model + v = ones(n) + error = tolerance + 1 + while error > tolerance + last_v = v + σ = get_greedy(v, model) + for i in 1:m + v = T_σ(v, σ, model) + end + error = maximum(abs.(v - last_v)) + println("OPI current error = $error") + end + return v, get_greedy(v, model) +end + + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + + +function plot_main(; tau_vals=(0.1, 0.25, 0.5, 0.6, 0.7, 0.8), + savefig=false, + figname="./figures/quantile_js.pdf") + + w_star_vals = zeros(length(tau_vals)) + + for (τ_idx, τ) in enumerate(tau_vals) + model = create_markov_js_model(τ=τ) + (; n, w_vals, P, β, c, τ) = model + v_star, σ_star = optimistic_policy_iteration(model) + for i in 1:n + if σ_star[i] > 0 + w_star_vals[τ_idx] = w_vals[i] + break + end + end + end + + model = create_markov_js_model() + (; n, w_vals, P, β, c, τ) = model + mc = MarkovChain(model.P) + s = stationary_distributions(mc)[1] + + fig, ax = plt.subplots() + ax.plot(tau_vals, w_star_vals, "k--", lw=2, alpha=0.7, label="reservation wage") + ax.barh(w_vals, 32 * s, alpha=0.05, align="center") + ax.legend(frameon=false, fontsize=fontsize, loc="upper center") + ax.set_xlabel("quantile", fontsize=fontsize) + ax.set_ylabel("wages", fontsize=fontsize) + + if savefig + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +plot_main(savefig=true) +``` diff --git a/code-book/jl/stochastic_discounting.md b/code-book/jl/stochastic_discounting.md new file mode 100644 index 0000000..d1cb749 --- /dev/null +++ b/code-book/jl/stochastic_discounting.md @@ -0,0 +1,287 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(Chapter 6: Stochastic Discounting)= +```{raw} jupyter + +``` +# Chapter 6: Stochastic Discounting + + +```{contents} Contents +:depth: 2 +``` + + +```{code-cell} julia +:tags: ["remove-cell"] +using Pkg; +Pkg.activate("./"); + +using PyCall; +pygui(:tk); +``` + +## plot_interest_rates.jl +```{code-cell} julia +# Nominal interest rate from https://fred.stlouisfed.org/series/GS1 +# Real interest rate from https://fred.stlouisfed.org/series/WFII10 +# +# Download as CSV files +# + +using DataFrames, CSV, PyPlot + +df_nominal = DataFrame(CSV.File("data/GS1.csv")) +df_real = DataFrame(CSV.File("data/WFII10.csv")) + +function plot_rates(df; fontsize=16, savefig=true) + r_type = df == df_nominal ? "nominal" : "real" + fig, ax = plt.subplots(figsize=(9, 5)) + ax.plot(df[!, 1], df[!, 2], label=r_type*" interest rate") + ax.plot(df[!, 1], zero(df[!, 2]), c="k", ls="--") + ax.set_xlim(df[1, 1], df[end, 1]) + ax.legend(fontsize=fontsize, frameon=false) + if savefig + fig.savefig("./figures/plot_interest_rates_"*r_type*".pdf") + end +end + +``` + +```{code-cell} julia +plot_rates(df_nominal, savefig=true) +``` + +```{code-cell} julia +plot_rates(df_real, savefig=true) +``` +## pd_ratio.jl +```{code-cell} julia +""" +Price-dividend ratio in a model with dividend and consumption growth. + +""" + +using QuantEcon, LinearAlgebra + +"Creates an instance of the asset pricing model with Markov state." +function create_asset_pricing_model(; + n=200, # state grid size + ρ=0.9, ν=0.2, # state persistence and volatility + β=0.99, γ=2.5, # discount and preference parameter + μ_c=0.01, σ_c=0.02, # consumption growth mean and volatility + μ_d=0.02, σ_d=0.1) # dividend growth mean and volatility + mc = tauchen(n, ρ, ν) + x_vals, P = exp.(mc.state_values), mc.p + return (; x_vals, P, β, γ, μ_c, σ_c, μ_d, σ_d) +end + +" Build the discount matrix A. " +function build_discount_matrix(model) + (; x_vals, P, β, γ, μ_c, σ_c, μ_d, σ_d) = model + e = exp.(μ_d - γ*μ_c + (γ^2*σ_c^2 + σ_d^2)/2 .+ (1-γ)*x_vals) + return β * e .* P +end + +"Compute the price-dividend ratio associated with the model." +function pd_ratio(model) + (; x_vals, P, β, γ, μ_c, σ_c, μ_d, σ_d) = model + A = build_discount_matrix(model) + @assert maximum(abs.(eigvals(A))) < 1 "Requires r(A) < 1." + n = length(x_vals) + return (I - A) \ (A * ones(n)) +end + + +# == Plots == # + +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering +fontsize=16 + +default_model = create_asset_pricing_model() + + +function plot_main(; μ_d_vals = (0.02, 0.08), + savefig=false, + figname="./figures/pd_ratio_1.pdf") + fig, ax = plt.subplots(figsize=(9, 5.2)) + + for μ_d in μ_d_vals + model = create_asset_pricing_model(μ_d=μ_d) + (; x_vals, P, β, γ, μ_c, σ_c, μ_d, σ_d) = model + v_star = pd_ratio(model) + ax.plot(x_vals, v_star, lw=2, alpha=0.6, label=L"\mu_d="*"$μ_d") + end + + ax.legend(frameon=false, fontsize=fontsize) + ax.set_xlabel(L"x", fontsize=fontsize) + if savefig + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +plot_main(savefig=true) +``` +## inventory_sdd.jl +```{code-cell} julia +""" +Inventory management model with state-dependent discounting. +The discount factor takes the form β_t = Z_t, where (Z_t) is +a discretization of the Gaussian AR(1) process + + X_t = ρ X_{t-1} + b + ν W_t. + +""" + +include("s_approx.jl") +using LinearAlgebra, Distributions, QuantEcon + +function create_sdd_inventory_model(; + ρ=0.98, ν=0.002, n_z=20, b=0.97, # Z state parameters + K=40, c=0.2, κ=0.8, p=0.6) # firm and demand parameters + ϕ(d) = (1 - p)^d * p # demand pdf + y_vals = collect(0:K) # inventory levels + mc = tauchen(n_z, ρ, ν) + z_vals, Q = mc.state_values .+ b, mc.p + ρL = maximum(abs.(eigvals(z_vals .* Q))) + @assert ρL < 1 "Error: ρ(L) ≥ 1." # check ρ(L) < 1 + return (; K, c, κ, p, ϕ, y_vals, z_vals, Q) +end + +m(y) = max(y, 0) # Convenience function + +"The function B(x, a, v) = r(x, a) + β(x) Σ_x′ v(x′) P(x, a, x′)." +function B(x, i_z, a, v, model; d_max=100) + (; K, c, κ, p, ϕ, y_vals, z_vals, Q) = model + z = z_vals[i_z] + revenue = sum(min(x, d)*ϕ(d) for d in 0:d_max) + current_profit = revenue - c * a - κ * (a > 0) + cv = 0.0 + for i_z′ in eachindex(z_vals) + for d in 0:d_max + cv += v[m(x - d) + a + 1, i_z′] * ϕ(d) * Q[i_z, i_z′] + end + end + return current_profit + z * cv +end + +"The Bellman operator." +function T(v, model) + (; K, c, κ, p, ϕ, y_vals, z_vals, Q) = model + new_v = similar(v) + for (i_z, z) in enumerate(z_vals) + for (i_y, y) in enumerate(y_vals) + Γy = 0:(K - y) + new_v[i_y, i_z], _ = findmax(B(y, i_z, a, v, model) for a in Γy) + end + end + return new_v +end + +"Get a v-greedy policy. Returns a zero-based array." +function get_greedy(v, model) + (; K, c, κ, p, ϕ, y_vals, z_vals, Q) = model + n_z = length(z_vals) + σ_star = zeros(Int32, K+1, n_z) + for (i_z, z) in enumerate(z_vals) + for (i_y, y) in enumerate(y_vals) + Γy = 0:(K - y) + _, i_a = findmax(B(y, i_z, a, v, model) for a in Γy) + σ_star[i_y, i_z] = Γy[i_a] + end + end + return σ_star +end + + +"Use successive_approx to get v_star and then compute greedy." +function solve_inventory_model(v_init, model) + (; K, c, κ, p, ϕ, y_vals, z_vals, Q) = model + v_star = successive_approx(v -> T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star +end + + +# == Plots == # + +using PyPlot +using PyPlot +using LaTeXStrings +PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering + +# Create an instance of the model and solve it +model = create_sdd_inventory_model() +(; K, c, κ, p, ϕ, y_vals, z_vals, Q) = model +n_z = length(z_vals) +v_init = zeros(Float64, K+1, n_z) +println("Solving model.") +v_star, σ_star = solve_inventory_model(v_init, model) +z_mc = MarkovChain(Q, z_vals) + +"Simulate given the optimal policy." +function sim_inventories(ts_length; X_init=0) + i_z = simulate_indices(z_mc, ts_length, init=1) + G = Geometric(p) + X = zeros(Int32, ts_length) + X[1] = X_init + for t in 1:(ts_length-1) + D′ = rand(G) + X[t+1] = m(X[t] - D′) + σ_star[X[t] + 1, i_z[t]] + end + return X, z_vals[i_z] +end + +function plot_ts(; ts_length=400, + fontsize=16, + figname="./figures/inventory_sdd_ts.pdf", + savefig=false) + X, Z = sim_inventories(ts_length) + fig, axes = plt.subplots(2, 1, figsize=(9, 5.5)) + + ax = axes[1] + ax.plot(X, label="inventory", alpha=0.7) + ax.set_xlabel(L"t", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=false) + ax.set_ylim(0, maximum(X)+3) + + # calculate interest rate from discount factors + r = (1 ./ Z) .- 1 + + ax = axes[2] + ax.plot(r, label=L"r_t", alpha=0.7) + ax.set_xlabel(L"t", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=false) + #ax.set_ylim(0, maximum(X)+8) + + plt.tight_layout() + if savefig == true + fig.savefig(figname) + end +end + + +``` + +```{code-cell} julia +plot_ts(savefig=true) +``` diff --git a/code-book/py/introductions.md b/code-book/py/introductions.md new file mode 100644 index 0000000..07032da --- /dev/null +++ b/code-book/py/introductions.md @@ -0,0 +1,415 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 1: Introduction)= +```{raw} jupyter + +``` +# Chapter 1: Introduction + + +```{contents} Contents +:depth: 2 +``` + + + +## two_period_job_search.py +```{code-cell} python3 +from quantecon.distributions import BetaBinomial + +import numpy as np +from numba import njit +from collections import namedtuple + + +# NamedTuple Model +Model = namedtuple("Model", ("n", "w_vals", "φ", "β", "c")) + + +def create_job_search_model( + n=50, # wage grid size + w_min=10.0, # lowest wage + w_max=60.0, # highest wage + a=200, # wage distribution parameter + b=100, # wage distribution parameter + β=0.96, # discount factor + c=10.0 # unemployment compensation + ): + """ + Creates the parameters for job search model and returns the + instance of namedtuple Model + """ + w_vals = np.linspace(w_min, w_max, n+1) + φ = BetaBinomial(n, a, b).pdf() + return Model(n=n, w_vals=w_vals, φ=φ, β=β, c=c) + + +@njit +def v_1(w, model): + """ + Computes lifetime value at t=1 given current wage w_1 = w + """ + β, c = model.β, model.c + s = np.maximum(c, model.w_vals) + h_1 = c + β * np.sum(s * model.φ) + return np.maximum(w + β * w, h_1) + + +@njit +def res_wage(model): + """ + Computes reservation wage at t=1 + """ + β, c = model.β, model.c + s = np.maximum(c, model.w_vals) + h_1 = c + β * np.sum(s * model.φ) + return h_1 / (1 + β) + + + +##### Plots ##### + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +default_model = create_job_search_model() + + +def fig_dist(model=default_model, fs=10): + """ + Plot the distribution of wages + """ + fig, ax = plt.subplots() + ax.plot(model.w_vals, model.φ, "-o", alpha=0.5, label="wage distribution") + ax.legend(loc="upper left", fontsize=fs) + + +def fig_v1(model=default_model, savefig=False, + figname="./figures/iid_job_search_0_py.pdf", fs=18): + """ + Plot two-period value function and res wage + """ + n, w_vals, φ, β, c = model + + v = [v_1(w, model) for w in w_vals] + w_star = res_wage(model) + s = np.maximum(c, w_vals) + continuation_val = c + β * np.sum(s * φ) + min_w, max_w = np.min(w_vals), np.max(w_vals) + + fontdict = {'fontsize': 10} + fig, ax = plt.subplots(figsize=(9, 5.5)) + ax.set_ylim(0, 120) + ax.set_xlim(min_w, max_w) + ax.vlines((w_star,), (0,), (continuation_val,), lw=0.5) + ax.set_yticks((0, 50, 100)) + ax.set_yticklabels((0, 50, 100), fontdict=fontdict) + ax.set_xticks((min_w, w_star, max_w)) + ax.set_xticklabels((min_w, r"$w^*_1$", max_w), fontdict=fontdict) + ax.plot(w_vals, w_vals + β * w_vals, alpha=0.8, linewidth=3, + label=r"$w_1 + \beta w_1$") + ax.plot(w_vals, [continuation_val]*(n+1), linewidth=3, alpha=0.8, + label=r"$c + \beta \sum_{w'} \max\{c, w'\} \varphi(w')$" ) + ax.plot(w_vals, v, "k--", markersize=2, alpha=1.0, linewidth=2, + label=r"$v_1(w_1)$") + ax.legend(frameon=False, fontsize=fs, loc="upper left") + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +fig_v1() +``` +## compute_spec_rad.py +```{code-cell} python3 +import numpy as np + +# Spectral radius +ρ = lambda A: np.max(np.abs(np.linalg.eigvals(A))) + +# Test with arbitrary A +A = np.array([ + [0.4, 0.1], + [0.7, 0.2] +]) +print(ρ(A)) + +``` +## power_series.py +```{code-cell} python3 +import numpy as np + +# Primitives +A = np.array([ + [0.4, 0.1], + [0.7, 0.2] +]) + + +# Method one: direct inverse +I = np.identity(2) +B_inverse = np.linalg.inv(I - A) + + +# Method two: power series +def power_series(A): + B_sum = np.zeros((2, 2)) + A_power = np.identity(2) + for k in range(50): + B_sum += A_power + A_power = np.dot(A_power, A) + return B_sum + + +# Print maximal error +print(np.max(np.abs(B_inverse - power_series(A)))) + +``` +## s_approx.py +```{code-cell} python3 +""" +Computes the approximate fixed point of T via successive approximation. +""" + +import numpy as np + +def successive_approx(T, # Operator (callable) + x_0, # Initial condition + tolerance=1e-6, # Error tolerance + max_iter=10_000, # Max iteration bound + print_step=25): # Print at multiples + x = x_0 + error = tolerance + 1 + k = 1 + while (error > tolerance) and (k <= max_iter): + x_new = T(x) + error = np.max(np.abs(x_new - x)) + if k % print_step == 0: + print(f"Completed iteration {k} with error {error}.") + x = x_new + k += 1 + if error <= tolerance: + print(f"Terminated successfully in {k} iterations.") + else: + print("Warning: hit iteration bound.") + return x + +``` +## linear_iter.py +```{code-cell} python3 +from s_approx import successive_approx +import numpy as np + +# Compute the fixed point of Tx = Ax + b via linear algebra +A = np.array([ + [0.4, 0.1], + [0.7, 0.2] +]) + +b = np.array([ + [1.0], + [2.0] +]) + +I = np.identity(2) +x_star = np.linalg.solve(I - A, b) # compute (I - A)^{-1} * b + + +# Compute the fixed point via successive approximation +T = lambda x: np.dot(A, x) + b +x_0 = np.array([ + [1.0], + [1.0] +]) +x_star_approx = successive_approx(T, x_0) + +# Test for approximate equality (prints "True") +print(np.allclose(x_star, x_star_approx, rtol=1e-5)) + +``` +## linear_iter_fig.py +```{code-cell} python3 +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) +import numpy as np + +from linear_iter import x_star, T + +def plot_main(savefig=False, figname="./figures/linear_iter_fig_1.pdf"): + + fig, ax = plt.subplots() + + e = 0.02 + + marker_size = 60 + fs = 10 + + colors = ("red", "blue", "orange", "green") + u_0_vecs = ([[2.0], [3.0]], [[3.0], [5.2]], [[2.4], [3.6]], [[2.6], [5.6]]) + u_0_vecs = list(map(np.array, u_0_vecs)) + iter_range = 8 + + for (x_0, color) in zip(u_0_vecs, colors): + x = x_0 + s, t = x + ax.text(s+e, t-4*e, r"$u_0$", fontsize=fs) + + for i in range(iter_range): + s, t = x + ax.scatter((s,), (t,), c=color, alpha=0.2, s=marker_size) + x_new = T(x) + s_new, t_new = x_new + ax.plot((s, s_new), (t, t_new), marker='.',linewidth=0.5, alpha=0.5, color=color) + x = x_new + + s_star, t_star = x_star + ax.scatter((s_star,), (t_star,), c="k", s=marker_size * 1.2) + ax.text(s_star-4*e, t_star+4*e, r"$u^*$", fontsize=fs) + + ax.set_xticks((2.0, 2.5, 3.0)) + ax.set_yticks((3.0, 4.0, 5.0, 6.0)) + ax.set_xlim(1.8, 3.2) + ax.set_ylim(2.8, 6.1) + + if savefig: + fig.savefig(figname) + +``` +## iid_job_search.py +```{code-cell} python3 +""" +VFI approach to job search in the infinite-horizon IID case. + +""" + +from quantecon import compute_fixed_point + +from two_period_job_search import create_job_search_model + +from numba import njit +import numpy as np + + +# A model with default parameters +default_model = create_job_search_model() + + +@njit +def T(v, model): + """ The Bellman operator. """ + n, w_vals, φ, β, c = model + return np.array([np.maximum(w / (1 - β), + c + β * np.sum(v * φ)) for w in w_vals]) + + +@njit +def get_greedy(v, model): + """ Get a v-greedy policy. """ + n, w_vals, φ, β, c = model + σ = w_vals / (1 - β) >= c + β * np.sum(v * φ) # Boolean policy vector + return σ + + +def vfi(model=default_model): + """ Solve the infinite-horizon IID job search model by VFI. """ + v_init = np.zeros_like(model.w_vals) + v_star = compute_fixed_point(lambda v: T(v, model), v_init, + error_tol=1e-5, max_iter=1000, print_skip=25) + σ_star = get_greedy(v_star, model) + return v_star, σ_star + + + +# == Plots == # + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def fig_vseq(model=default_model, + k=3, + savefig=False, + figname="./figures/iid_job_search_1.pdf", + fs=10): + + v = np.zeros_like(model.w_vals) + fig, ax = plt.subplots(figsize=(9, 5.5)) + for i in range(k): + ax.plot(model.w_vals, v, linewidth=3, alpha=0.6, + label=f"iterate {i}") + v = T(v, model) + + for i in range(1000): + v = T(v, model) + + ax.plot(model.w_vals, v, "k-", linewidth=3.0, + label="iterate 1000", alpha=0.7) + + fontdict = {'fontsize': fs} + ax.set_xlabel("wage offer", fontdict=fontdict) + ax.set_ylabel("lifetime value", fontdict=fontdict) + + ax.legend(fontsize=fs, frameon=False) + + if savefig: + fig.savefig(figname) + + +def fig_vstar(model=default_model, + savefig=False, fs=10, + figname="./figures/iid_job_search_3.pdf"): + """ Plot the fixed point. """ + n, w_vals, φ, β, c = model + v_star, σ_star = vfi(model) + + fig, ax = plt.subplots(figsize=(9, 5.5)) + ax.plot(w_vals, v_star, "k-", linewidth=1.5, label="value function") + cont_val = c + β * np.sum(v_star * φ) + ax.plot(w_vals, [cont_val]*(n+1), + "--", + linewidth=5, + alpha=0.5, + label="continuation value") + + ax.plot(w_vals, + w_vals / (1 - β), + "--", + linewidth=5, + alpha=0.5, + label=r"$w/(1 - \beta)$") + + ax.legend(frameon=False, fontsize=fs, loc="lower right") + + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +fig_vseq() +``` + +```{code-cell} python +fig_vstar() +``` diff --git a/code-book/py/markov_decision_processes.md b/code-book/py/markov_decision_processes.md new file mode 100644 index 0000000..3ab6f40 --- /dev/null +++ b/code-book/py/markov_decision_processes.md @@ -0,0 +1,1246 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 5: Markov Decision Processes)= +```{raw} jupyter + +``` +# Chapter 5: Markov Decision Processes + + +```{contents} Contents +:depth: 2 +``` + + + +## inventory_dp.py +```{code-cell} python3 +from quantecon import compute_fixed_point + +import numpy as np +from collections import namedtuple +from numba import njit + + +# NamedTuple Model +Model = namedtuple("Model", ("β", "K", "c", "κ", "p")) + + +def create_inventory_model(β=0.98, # discount factor + K=40, # maximum inventory + c=0.2, κ=2, # cost parameters + p=0.6): # demand parameter + return Model(β=β, K=K, c=c, κ=κ, p=p) + + +@njit +def demand_pdf(d, p): + return (1 - p)**d * p + + +@njit +def B(x, a, v, model, d_max=101): + """ + The function B(x, a, v) = r(x, a) + β Σ_x′ v(x′) P(x, a, x′). + """ + β, K, c, κ, p = model + x1 = np.array([np.minimum(x, d)*demand_pdf(d, p) for d in np.arange(d_max)]) + reward = np.sum(x1) - c * a - κ * (a > 0) + x2 = np.array([v[np.maximum(0, x - d) + a] * demand_pdf(d, p) + for d in np.arange(d_max)]) + continuation_value = β * np.sum(x2) + return reward + continuation_value + + +@njit +def T(v, model): + """The Bellman operator.""" + β, K, c, κ, p = model + new_v = np.empty_like(v) + for x in range(0, K+1): + x1 = np.array([B(x, a, v, model) for a in np.arange(K-x+1)]) + new_v[x] = np.max(x1) + return new_v + + +@njit +def get_greedy(v, model): + """ + Get a v-greedy policy. Returns a zero-based array. + """ + β, K, c, κ, p = model + σ_star = np.zeros(K+1, dtype=np.int32) + for x in range(0, K+1): + x1 = np.array([B(x, a, v, model) for a in np.arange(K-x+1)]) + σ_star[x] = np.argmax(x1) + return σ_star + + +def solve_inventory_model(v_init, model): + """Use successive_approx to get v_star and then compute greedy.""" + β, K, c, κ, p = model + v_star = compute_fixed_point(lambda v: T(v, model), v_init, + error_tol=1e-5, max_iter=1000, print_skip=25) + σ_star = get_greedy(v_star, model) + return v_star, σ_star + + +# == Plots == # + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +# Create an instance of the model and solve it +model = create_inventory_model() +β, K, c, κ, p = model +v_init = np.zeros(K+1) +v_star, σ_star = solve_inventory_model(v_init, model) + + +def sim_inventories(ts_length=400, X_init=0): + """Simulate given the optimal policy.""" + global p, σ_star + X = np.zeros(ts_length, dtype=np.int32) + X[0] = X_init + # Subtracts 1 because numpy generates only positive integers + rand = np.random.default_rng().geometric(p=p, size=ts_length-1) - 1 + for t in range(0, ts_length-1): + X[t+1] = np.maximum(X[t] - rand[t], 0) + σ_star[X[t] + 1] + return X + + +def plot_vstar_and_opt_policy(fontsize=10, + figname="./figures/inventory_dp_vs.pdf", + savefig=False): + fig, axes = plt.subplots(2, 1, figsize=(8, 6.5)) + + ax = axes[0] + ax.plot(np.arange(K+1), v_star, label=r"$v^*$") + ax.set_ylabel("value", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=False) + + ax = axes[1] + ax.plot(np.arange(K+1), σ_star, label=r"$\sigma^*$") + ax.set_xlabel("inventory", fontsize=fontsize) + ax.set_ylabel("optimal choice", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=False) + if savefig: + fig.savefig(figname) + + +def plot_ts(fontsize=10, + figname="./figures/inventory_dp_ts.pdf", + savefig=False): + X = sim_inventories() + fig, ax = plt.subplots(figsize=(9, 5.5)) + ax.plot(X, label="$X_t$", alpha=0.7) + ax.set_xlabel("$t$", fontsize=fontsize) + ax.set_ylabel("inventory", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=False) + ax.set_ylim(0, np.max(X)+4) + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_vstar_and_opt_policy() +``` + +```{code-cell} python +plot_ts() +``` +## finite_opt_saving_0.py +```{code-cell} python3 +from quantecon.markov import tauchen +import numpy as np +from collections import namedtuple +from numba import njit, prange + + +# NamedTuple Model +Model = namedtuple("Model", ("β", "R", "γ", "w_grid", "y_grid", "Q")) + + +def create_savings_model(R=1.01, β=0.98, γ=2.5, + w_min=0.01, w_max=20.0, w_size=200, + ρ=0.9, ν=0.1, y_size=5): + w_grid = np.linspace(w_min, w_max, w_size) + mc = tauchen(y_size, ρ, ν) + y_grid, Q = np.exp(mc.state_values), mc.P + return Model(β=β, R=R, γ=γ, w_grid=w_grid, y_grid=y_grid, Q=Q) + + +@njit +def U(c, γ): + return c**(1-γ)/(1-γ) + + +@njit +def B(i, j, k, v, model): + """ + B(w, y, w′, v) = u(R*w + y - w′) + β Σ_y′ v(w′, y′) Q(y, y′). + """ + β, R, γ, w_grid, y_grid, Q = model + w, y, w_1 = w_grid[i], y_grid[j], w_grid[k] + c = w + y - (w_1 / R) + value = -np.inf + if c > 0: + value = U(c, γ) + β * np.dot(v[k, :], Q[j, :]) + return value + + +@njit(parallel=True) +def T(v, model): + """The Bellman operator.""" + β, R, γ, w_grid, y_grid, Q = model + v_new = np.empty_like(v) + for i in prange(w_grid.shape[0]): + for j in prange(y_grid.shape[0]): + x_tmp = np.array([B(i, j, k, v, model) for k + in np.arange(w_grid.shape[0])]) + v_new[i, j] = np.max(x_tmp) + return v_new + + +@njit(parallel=True) +def T_σ(v, σ, model): + """The policy operator.""" + β, R, γ, w_grid, y_grid, Q = model + v_new = np.empty_like(v) + for i in prange(w_grid.shape[0]): + for j in prange(y_grid.shape[0]): + v_new[i, j] = B(i, j, σ[i, j], v, model) + return v_new + +``` +## finite_opt_saving_1.py +```{code-cell} python3 +import numpy as np +from finite_opt_saving_0 import U, B +from numba import njit, prange + +@njit(parallel=True) +def get_greedy(v, model): + """Compute a v-greedy policy.""" + β, R, γ, w_grid, y_grid, Q = model + σ = np.empty((w_grid.shape[0], y_grid.shape[0]), dtype=np.int32) + for i in prange(w_grid.shape[0]): + for j in range(y_grid.shape[0]): + x_tmp = np.array([B(i, j, k, v, model) for k in + np.arange(w_grid.shape[0])]) + σ[i, j] = np.argmax(x_tmp) + return σ + + +@njit +def single_to_multi(m, yn): + # Function to extract (i, j) from m = i + (j-1)*yn + return (m//yn, m%yn) + +@njit(parallel=True) +def get_value(σ, model): + """Get the value v_σ of policy σ.""" + # Unpack and set up + β, R, γ, w_grid, y_grid, Q = model + wn, yn = len(w_grid), len(y_grid) + n = wn * yn + # Build P_σ and r_σ as multi-index arrays + P_σ = np.zeros((wn, yn, wn, yn)) + r_σ = np.zeros((wn, yn)) + for i in range(wn): + for j in range(yn): + w, y, w_1 = w_grid[i], y_grid[j], w_grid[σ[i, j]] + r_σ[i, j] = U(w + y - w_1/R, γ) + for i_1 in range(wn): + for j_1 in range(yn): + if i_1 == σ[i, j]: + P_σ[i, j, i_1, j_1] = Q[j, j_1] + + # Solve for the value of σ + P_σ = P_σ.reshape(n, n) + r_σ = r_σ.reshape(n) + + I = np.identity(n) + v_σ = np.linalg.solve((I - β * P_σ), r_σ) + # Return as multi-index array + v_σ = v_σ.reshape(wn, yn) + return v_σ + + + +``` +## finite_opt_saving_2.py +```{code-cell} python3 +from quantecon import compute_fixed_point + +import numpy as np +from numba import njit +import time +from finite_opt_saving_1 import get_greedy, get_value +from finite_opt_saving_0 import create_savings_model, T, T_σ +from quantecon import MarkovChain + + +def value_iteration(model, tol=1e-5): + """Value function iteration routine.""" + vz = np.zeros((len(model.w_grid), len(model.y_grid))) + v_star = compute_fixed_point(lambda v: T(v, model), vz, + error_tol=tol, max_iter=1000, print_skip=25) + return get_greedy(v_star, model) + + +@njit(cache=True, fastmath=True) +def policy_iteration(model): + """Howard policy iteration routine.""" + wn, yn = len(model.w_grid), len(model.y_grid) + σ = np.ones((wn, yn), dtype=np.int32) + i, error = 0, 1.0 + while error > 0: + v_σ = get_value(σ, model) + σ_new = get_greedy(v_σ, model) + error = np.max(np.abs(σ_new - σ)) + σ = σ_new + i = i + 1 + print(f"Concluded loop {i} with error: {error}.") + return σ + + +@njit +def optimistic_policy_iteration(model, tolerance=1e-5, m=100): + """Optimistic policy iteration routine.""" + v = np.zeros((len(model.w_grid), len(model.y_grid))) + error = tolerance + 1 + while error > tolerance: + last_v = v + σ = get_greedy(v, model) + for i in range(0, m): + v = T_σ(v, σ, model) + error = np.max(np.abs(v - last_v)) + return get_greedy(v, model) + +# Simulations and inequality measures + +def simulate_wealth(m): + + model = create_savings_model() + σ_star = optimistic_policy_iteration(model) + β, R, γ, w_grid, y_grid, Q = model + + # Simulate labor income (indices rather than grid values) + mc = MarkovChain(Q) + y_idx_series = mc.simulate(ts_length=m) + + # Compute corresponding wealth time series + w_idx_series = np.empty_like(y_idx_series) + w_idx_series[0] = 1 # initial condition + for t in range(m-1): + i, j = w_idx_series[t], y_idx_series[t] + w_idx_series[t+1] = σ_star[i, j] + w_series = w_grid[w_idx_series] + + return w_series + +def lorenz(v): # assumed sorted vector + S = np.cumsum(v) # cumulative sums: [v[1], v[1] + v[2], ... ] + F = np.arange(1, len(v) + 1) / len(v) + L = S / S[-1] + return (F, L) # returns named tuple + +gini = lambda v: (2 * sum(i * y for (i, y) in enumerate(v))/sum(v) - (len(v) + 1))/len(v) + +# Plots + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_timing(m_vals=np.arange(1, 601, 10), + savefig=False): + model = create_savings_model(y_size=5) + print("Running Howard policy iteration.") + t1 = time.time() + σ_pi = policy_iteration(model) + pi_time = time.time() - t1 + print(f"PI completed in {pi_time} seconds.") + print("Running value function iteration.") + t1 = time.time() + σ_vfi = value_iteration(model) + vfi_time = time.time() - t1 + print(f"VFI completed in {vfi_time} seconds.") + + assert np.allclose(σ_vfi, σ_pi), "Warning: policies deviated." + + opi_times = [] + for m in m_vals: + print(f"Running optimistic policy iteration with m={m}.") + t1 = time.time() + σ_opi = optimistic_policy_iteration(model, m=m) + t2 = time.time() + assert np.allclose(σ_opi, σ_pi), "Warning: policies deviated." + print(f"OPI with m={m} completed in {t2-t1} seconds.") + opi_times.append(t2-t1) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(m_vals, [vfi_time]*len(m_vals), + linewidth=2, label="value function iteration") + ax.plot(m_vals, [pi_time]*len(m_vals), + linewidth=2, label="Howard policy iteration") + ax.plot(m_vals, opi_times, linewidth=2, + label="optimistic policy iteration") + ax.legend(frameon=False) + ax.set_xlabel(r"$m$") + ax.set_ylabel("time") + if savefig: + fig.savefig("./figures/finite_opt_saving_2_1.png") + return (pi_time, vfi_time, opi_times) + + +def plot_policy(method="pi", savefig=False): + model = create_savings_model() + β, R, γ, w_grid, y_grid, Q = model + if method == "vfi": + σ_star = value_iteration(model) + elif method == "pi": + σ_star = policy_iteration(model) + else: + method = "OPT" + σ_star = optimistic_policy_iteration(model) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_grid, w_grid, "k--", label=r"$45$") + ax.plot(w_grid, w_grid[σ_star[:, 0]], label=r"$\sigma^*(\cdot, y_1)$") + ax.plot(w_grid, w_grid[σ_star[:, -1]], label=r"$\sigma^*(\cdot, y_N)$") + ax.legend() + plt.title(f"Method: {method}") + if savefig: + fig.savefig(f"./figures/finite_opt_saving_2_2_{method}.png") + +def plot_time_series(m=2_000, savefig=False): + + w_series = simulate_wealth(m) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_series, label="w_t") + ax.set_xlabel("time") + ax.legend() + if savefig: + fig.savefig("./figures/finite_opt_saving_ts.pdf") + +def plot_histogram(m=1_000_000, savefig=False): + + w_series = simulate_wealth(m) + w_series.sort() + g = round(gini(w_series), ndigits=2) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.hist(w_series, bins=40, density=True) + ax.set_xlabel("wealth") + ax.text(15, 0.4, f"Gini = {g}") + + if savefig: + fig.savefig("./figures/finite_opt_saving_hist.pdf") + +def plot_lorenz(m=1_000_000, savefig=False): + + w_series = simulate_wealth(m) + w_series.sort() + (F, L) = lorenz(w_series) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(F, F, label="Lorenz curve, equality") + ax.plot(F, L, label="Lorenz curve, wealth distribution") + ax.legend() + + if savefig: + fig.savefig("./figures/finite_opt_saving_lorenz.pdf") + +``` + +```{code-cell} python +plot_timing() +``` + +```{code-cell} python +plot_policy() +``` + +```{code-cell} python +plot_time_series() +``` + +```{code-cell} python +plot_histogram() +``` + +```{code-cell} python +plot_lorenz() +``` +## finite_lq.py +```{code-cell} python3 +from quantecon import compute_fixed_point +from quantecon.markov import tauchen, MarkovChain + +import numpy as np +from collections import namedtuple +from numba import njit, prange +import time + + +# NamedTuple Model +Model = namedtuple("Model", ("β", "a_0", "a_1", "γ", "c", + "y_grid", "z_grid", "Q")) + + +def create_investment_model( + r=0.04, # Interest rate + a_0=10.0, a_1=1.0, # Demand parameters + γ=25.0, c=1.0, # Adjustment and unit cost + y_min=0.0, y_max=20.0, y_size=100, # Grid for output + ρ=0.9, ν=1.0, # AR(1) parameters + z_size=25): # Grid size for shock + β = 1/(1+r) + y_grid = np.linspace(y_min, y_max, y_size) + mc = tauchen(y_size, ρ, ν) + z_grid, Q = mc.state_values, mc.P + return Model(β=β, a_0=a_0, a_1=a_1, γ=γ, c=c, + y_grid=y_grid, z_grid=z_grid, Q=Q) + + +@njit +def B(i, j, k, v, model): + """ + The aggregator B is given by + + B(y, z, y′) = r(y, z, y′) + β Σ_z′ v(y′, z′) Q(z, z′)." + + where + + r(y, z, y′) := (a_0 - a_1 * y + z - c) y - γ * (y′ - y)^2 + + """ + β, a_0, a_1, γ, c, y_grid, z_grid, Q = model + y, z, y_1 = y_grid[i], z_grid[j], y_grid[k] + r = (a_0 - a_1 * y + z - c) * y - γ * (y_1 - y)**2 + return r + β * np.dot(v[k, :], Q[j, :]) + + +@njit(parallel=True) +def T_σ(v, σ, model): + """The policy operator.""" + v_new = np.empty_like(v) + for i in prange(len(model.y_grid)): + for j in prange(len(model.z_grid)): + v_new[i, j] = B(i, j, σ[i, j], v, model) + return v_new + + +@njit(parallel=True) +def T(v, model): + """The Bellman operator.""" + v_new = np.empty_like(v) + for i in prange(len(model.y_grid)): + for j in prange(len(model.z_grid)): + tmp = np.array([B(i, j, k, v, model) for k + in np.arange(len(model.y_grid))]) + v_new[i, j] = np.max(tmp) + return v_new + + +@njit(parallel=True) +def get_greedy(v, model): + """Compute a v-greedy policy.""" + n, m = len(model.y_grid), len(model.z_grid) + σ = np.empty((n, m), dtype=np.int32) + for i in prange(n): + for j in prange(m): + tmp = np.array([B(i, j, k, v, model) for k + in np.arange(n)]) + σ[i, j] = np.argmax(tmp) + return σ + + + +def value_iteration(model, tol=1e-5): + """Value function iteration routine.""" + vz = np.zeros((len(model.y_grid), len(model.z_grid))) + v_star = compute_fixed_point(lambda v: T(v, model), vz, + error_tol=tol, max_iter=1000, print_skip=25) + return get_greedy(v_star, model) + + +@njit +def single_to_multi(m, zn): + # Function to extract (i, j) from m = i + (j-1)*zn + return (m//zn, m%zn) + + +@njit(parallel=True) +def get_value(σ, model): + """Get the value v_σ of policy σ.""" + # Unpack and set up + β, a_0, a_1, γ, c, y_grid, z_grid, Q = model + yn, zn = len(y_grid), len(z_grid) + n = yn * zn + # Allocate and create single index versions of P_σ and r_σ + P_σ = np.zeros((n, n)) + r_σ = np.zeros(n) + for m in prange(n): + i, j = single_to_multi(m, zn) + y, z, y_1 = y_grid[i], z_grid[j], y_grid[σ[i, j]] + r_σ[m] = (a_0 - a_1 * y + z - c) * y - γ * (y_1 - y)**2 + for m_1 in prange(n): + i_1, j_1 = single_to_multi(m_1, zn) + if i_1 == σ[i, j]: + P_σ[m, m_1] = Q[j, j_1] + + I = np.identity(n) + # Solve for the value of σ + v_σ = np.linalg.solve((I - β * P_σ), r_σ) + # Return as multi-index array + v_σ = v_σ.reshape(yn, zn) + return v_σ + + +@njit +def policy_iteration(model): + """Howard policy iteration routine.""" + yn, zn = len(model.y_grid), len(model.z_grid) + σ = np.ones((yn, zn), dtype=np.int32) + i, error = 0, 1.0 + while error > 0: + v_σ = get_value(σ, model) + σ_new = get_greedy(v_σ, model) + error = np.max(np.abs(σ_new - σ)) + σ = σ_new + i = i + 1 + print(f"Concluded loop {i} with error: {error}.") + return σ + + +@njit +def optimistic_policy_iteration(model, tol=1e-5, m=100): + """Optimistic policy iteration routine.""" + v = np.zeros((len(model.y_grid), len(model.z_grid))) + error = tol + 1 + while error > tol: + last_v = v + σ = get_greedy(v, model) + for i in range(m): + v = T_σ(v, σ, model) + error = np.max(np.abs(v - last_v)) + return get_greedy(v, model) + + +# Plots + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_policy(savefig=False, figname="./figures/finite_lq_0.pdf"): + model = create_investment_model() + β, a_0, a_1, γ, c, y_grid, z_grid, Q = model + σ_star = optimistic_policy_iteration(model) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(y_grid, y_grid, "k--", label=r"$45$") + ax.plot(y_grid, y_grid[σ_star[:, 0]], label=r"$\sigma^*(\cdot, z_1)$") + ax.plot(y_grid, y_grid[σ_star[:, -1]], label="$\sigma^*(\cdot, z_N)$") + ax.legend() + if savefig: + fig.savefig(figname) + + +def plot_sim(savefig=False, figname="./figures/finite_lq_1.pdf"): + ts_length = 200 + + fig, axes = plt.subplots(4, 1, figsize=(9, 11.2)) + + for (ax, γ) in zip(axes, (1, 10, 20, 30)): + model = create_investment_model(γ=γ) + β, a_0, a_1, γ, c, y_grid, z_grid, Q = model + σ_star = optimistic_policy_iteration(model) + mc = MarkovChain(Q, z_grid) + + z_sim_idx = mc.simulate_indices(ts_length) + z_sim = z_grid[z_sim_idx] + + y_sim_idx = np.empty(ts_length, dtype=np.int32) + y_1 = (a_0 - c + z_sim[1]) / (2 * a_1) + + y_sim_idx[0] = np.searchsorted(y_grid, y_1) + for t in range(ts_length-1): + y_sim_idx[t+1] = σ_star[y_sim_idx[t], z_sim_idx[t]] + y_sim = y_grid[y_sim_idx] + y_bar_sim = (a_0 - c + z_sim) / (2 * a_1) + + ax.plot(np.arange(1, ts_length+1), y_sim, label=r"$Y_t$") + ax.plot(np.arange(1, ts_length+1), y_bar_sim, label=r"$\bar Y_t$") + ax.legend(frameon=False, loc="upper right") + ax.set_ylabel("output") + ax.set_ylim(1, 9) + ax.set_title(r"$\gamma = $" + f"{γ}") + + fig.tight_layout() + if savefig: + fig.savefig(figname) + + +def plot_timing(m_vals=np.arange(1, 601, 10), + savefig=False, + figname="./figures/finite_lq_time.pdf" + ): + # NOTE: Uncomment the following lines in this function to + # include Policy iteration plot + model = create_investment_model() + # print("Running Howard policy iteration.") + # t1 = time.time() + # σ_pi = policy_iteration(model) + # pi_time = time.time() - t1 + # print(f"PI completed in {pi_time} seconds.") + print("Running value function iteration.") + t1 = time.time() + σ_vfi = value_iteration(model) + vfi_time = time.time() - t1 + print(f"VFI completed in {vfi_time} seconds.") + opi_times = [] + for m in m_vals: + print(f"Running optimistic policy iteration with m={m}.") + t1 = time.time() + σ_opi = optimistic_policy_iteration(model, m=m, tol=1e-5) + t2 = time.time() + print(f"OPI with m={m} completed in {t2-t1} seconds.") + opi_times.append(t2-t1) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(m_vals, [vfi_time]*len(m_vals), + linewidth=2, label="value function iteration") + # ax.plot(m_vals, [pi_time]*len(m_vals), + # linewidth=2, label="Howard policy iteration") + ax.plot(m_vals, opi_times, linewidth=2, label="optimistic policy iteration") + ax.legend(frameon=False) + ax.set_xlabel(r"$m$") + ax.set_ylabel("time") + if savefig: + fig.savefig(figname) + return (vfi_time, opi_times) + +``` + +```{code-cell} python +plot_policy() +``` + +```{code-cell} python +plot_sim() +``` + +```{code-cell} python +plot_timing() +``` +## firm_hiring.py +```{code-cell} python3 +import numpy as np +from quantecon.markov import tauchen, MarkovChain + +from collections import namedtuple +from numba import njit, prange + + +# NamedTuple Model +Model = namedtuple("Model", ("β", "κ", "α", "p", "w", "l_grid", + "z_grid", "Q")) + + +def create_hiring_model( + r=0.04, # Interest rate + κ=1.0, # Adjustment cost + α=0.4, # Production parameter + p=1.0, w=1.0, # Price and wage + l_min=0.0, l_max=30.0, l_size=100, # Grid for labor + ρ=0.9, ν=0.4, b=1.0, # AR(1) parameters + z_size=100): # Grid size for shock + β = 1/(1+r) + l_grid = np.linspace(l_min, l_max, l_size) + mc = tauchen(z_size, ρ, ν, b, 6) + z_grid, Q = mc.state_values, mc.P + return Model(β=β, κ=κ, α=α, p=p, w=w, + l_grid=l_grid, z_grid=z_grid, Q=Q) + + +@njit +def B(i, j, k, v, model): + """ + The aggregator B is given by + + B(l, z, l′) = r(l, z, l′) + β Σ_z′ v(l′, z′) Q(z, z′)." + + where + + r(l, z, l′) := p * z * f(l) - w * l - κ 1{l != l′} + + """ + β, κ, α, p, w, l_grid, z_grid, Q = model + l, z, l_1 = l_grid[i], z_grid[j], l_grid[k] + r = p * z * l**α - w * l - κ * (l != l_1) + return r + β * np.dot(v[k, :], Q[j, :]) + + +@njit(parallel=True) +def T_σ(v, σ, model): + """The policy operator.""" + v_new = np.empty_like(v) + for i in prange(len(model.l_grid)): + for j in prange(len(model.z_grid)): + v_new[i, j] = B(i, j, σ[i, j], v, model) + return v_new + + +@njit(parallel=True) +def get_greedy(v, model): + """Compute a v-greedy policy.""" + β, κ, α, p, w, l_grid, z_grid, Q = model + n, m = len(l_grid), len(z_grid) + σ = np.empty((n, m), dtype=np.int32) + for i in prange(n): + for j in prange(m): + tmp = np.array([B(i, j, k, v, model) for k + in np.arange(n)]) + σ[i, j] = np.argmax(tmp) + return σ + + +@njit +def optimistic_policy_iteration(model, tolerance=1e-5, m=100): + """Optimistic policy iteration routine.""" + v = np.zeros((len(model.l_grid), len(model.z_grid))) + error = tolerance + 1 + while error > tolerance: + last_v = v + σ = get_greedy(v, model) + for i in range(m): + v = T_σ(v, σ, model) + error = np.max(np.abs(v - last_v)) + return get_greedy(v, model) + + +# Plots + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_policy(savefig=False, + figname="./figures/firm_hiring_pol.pdf"): + model = create_hiring_model() + β, κ, α, p, w, l_grid, z_grid, Q = model + σ_star = optimistic_policy_iteration(model) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(l_grid, l_grid, "k--", label=r"$45$") + ax.plot(l_grid, l_grid[σ_star[:, 0]], label=r"$\sigma^*(\cdot, z_1)$") + ax.plot(l_grid, l_grid[σ_star[:, -1]], label=r"$\sigma^*(\cdot, z_N)$") + ax.legend() + if savefig: + fig.savefig(figname) + + +def sim_dynamics(model, ts_length): + β, κ, α, p, w, l_grid, z_grid, Q = model + σ_star = optimistic_policy_iteration(model) + mc = MarkovChain(Q, z_grid) + z_sim_idx = mc.simulate_indices(ts_length) + z_sim = z_grid[z_sim_idx] + l_sim_idx = np.empty(ts_length, dtype=np.int32) + l_sim_idx[0] = 32 + for t in range(ts_length-1): + l_sim_idx[t+1] = σ_star[l_sim_idx[t], z_sim_idx[t]] + l_sim = l_grid[l_sim_idx] + + y_sim = np.empty_like(l_sim) + for (i, l) in enumerate(l_sim): + y_sim[i] = p * z_sim[i] * l_sim[i]**α + + t = ts_length - 1 + l_g, y_g, z_g = np.zeros(t), np.zeros(t), np.zeros(t) + + for i in range(t): + l_g[i] = (l_sim[i+1] - l_sim[i]) / l_sim[i] + y_g[i] = (y_sim[i+1] - y_sim[i]) / y_sim[i] + z_g[i] = (z_sim[i+1] - z_sim[i]) / z_sim[i] + + return l_sim, y_sim, z_sim, l_g, y_g, z_g + + +def plot_sim(savefig=False, + figname="./figures/firm_hiring_ts.pdf", + ts_length = 250): + model = create_hiring_model() + β, κ, α, p, w, l_grid, z_grid, Q = model + l_sim, y_sim, z_sim, l_g, y_g, z_g = sim_dynamics(model, ts_length) + fig, ax = plt.subplots(figsize=(9, 5.2)) + x_grid = np.arange(ts_length) + ax.plot(x_grid, l_sim, label=r"$\ell_t$") + ax.plot(x_grid, z_sim, alpha=0.6, label=r"$Z_t$") + ax.legend(frameon=False) + ax.set_ylabel("employment") + ax.set_xlabel("time") + + if savefig: + fig.savefig(figname) + + +def plot_growth(savefig=False, + figname="./figures/firm_hiring_g.pdf", + ts_length = 10_000_000): + + model = create_hiring_model() + β, κ, α, p, w, l_grid, z_grid, Q = model + l_sim, y_sim, z_sim, l_g, y_g, z_g = sim_dynamics(model, ts_length) + + fig, ax = plt.subplots() + ax.hist(l_g, alpha=0.6, bins=100) + ax.set_xlabel("growth") + + #fig, axes = plt.subplots(2, 1) + #series = y_g, z_g + #for (ax, g) in zip(axes, series): + # ax.hist(g, alpha=0.6, bins=100) + # ax.set_xlabel("growth") + + plt.tight_layout() + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_policy() +``` + +```{code-cell} python +plot_sim() +``` + +```{code-cell} python +plot_growth() +``` +## modified_opt_savings.py +```{code-cell} python3 +from quantecon import tauchen, MarkovChain + +import numpy as np +from collections import namedtuple +from numba import njit, prange +from math import floor + + +# NamedTuple Model +Model = namedtuple("Model", ("β", "γ", "η_grid", "φ", + "w_grid", "y_grid", "Q")) + + +def create_savings_model(β=0.98, γ=2.5, + w_min=0.01, w_max=20.0, w_size=100, + ρ=0.9, ν=0.1, y_size=20, + η_min=0.75, η_max=1.25, η_size=2): + η_grid = np.linspace(η_min, η_max, η_size) + φ = np.ones(η_size) * (1 / η_size) # Uniform distributoin + w_grid = np.linspace(w_min, w_max, w_size) + mc = tauchen(y_size, ρ, ν) + y_grid, Q = np.exp(mc.state_values), mc.P + return Model(β=β, γ=γ, η_grid=η_grid, φ=φ, w_grid=w_grid, + y_grid=y_grid, Q=Q) + +## == Functions for regular OPI == ## + +@njit +def U(c, γ): + return c**(1-γ)/(1-γ) + +@njit +def B(i, j, k, l, v, model): + """ + The function + + B(w, y, η, w′) = u(w + y - w′/η)) + β Σ v(w′, y′, η′) Q(y, y′) ϕ(η′) + + """ + β, γ, η_grid, φ, w_grid, y_grid, Q = model + w, y, η, w_1 = w_grid[i], y_grid[j], η_grid[k], w_grid[l] + c = w + y - (w_1 / η) + exp_value = 0.0 + for m in prange(len(y_grid)): + for n in prange(len(η_grid)): + exp_value += v[l, m, n] * Q[j, m] * φ[n] + return U(c, γ) + β * exp_value if c > 0 else -np.inf + + +@njit(parallel=True) +def T_σ(v, σ, model): + """The policy operator.""" + β, γ, η_grid, φ, w_grid, y_grid, Q = model + v_new = np.empty_like(v) + for i in prange(len(w_grid)): + for j in prange(len(y_grid)): + for k in prange(len(η_grid)): + v_new[i, j, k] = B(i, j, k, σ[i, j, k], v, model) + return v_new + + +@njit(parallel=True) +def get_greedy(v, model): + """Compute a v-greedy policy.""" + β, γ, η_grid, φ, w_grid, y_grid, Q = model + w_n, y_n, η_n = len(w_grid), len(y_grid), len(η_grid) + σ = np.empty((w_n, y_n, η_n), dtype=np.int32) + for i in prange(w_n): + for j in prange(y_n): + for k in prange(η_n): + _tmp = np.array([B(i, j, k, l, v, model) for l + in range(w_n)]) + σ[i, j, k] = np.argmax(_tmp) + return σ + + +def optimistic_policy_iteration(model, tolerance=1e-5, m=100): + """Optimistic policy iteration routine.""" + β, γ, η_grid, φ, w_grid, y_grid, Q = model + w_n, y_n, η_n = len(w_grid), len(y_grid), len(η_grid) + v = np.zeros((w_n, y_n, η_n)) + error = tolerance + 1 + while error > tolerance: + last_v = v + σ = get_greedy(v, model) + for i in range(m): + v = T_σ(v, σ, model) + error = np.max(np.abs(v - last_v)) + print(f"OPI current error = {error}") + return get_greedy(v, model) + + +## == Functions for modified OPI == ## + + +@njit +def D(i, j, k, l, g, model): + """D(w, y, η, w′, g) = u(w + y - w′/η) + β g(y, w′).""" + β, γ, η_grid, φ, w_grid, y_grid, Q = model + w, y, η, w_1 = w_grid[i], y_grid[j], η_grid[k], w_grid[l] + c = w + y - (w_1 / η) + return U(c, γ) + β * g[j, l] if c > 0 else -np.inf + + +@njit(parallel=True) +def get_g_greedy(g, model): + """Compute a g-greedy policy.""" + β, γ, η_grid, φ, w_grid, y_grid, Q = model + w_n, y_n, η_n = len(w_grid), len(y_grid), len(η_grid) + σ = np.empty((w_n, y_n, η_n), dtype=np.int32) + for i in prange(w_n): + for j in prange(y_n): + for k in prange(η_n): + _tmp = np.array([D(i, j, k, l, g, model) for l + in range(w_n)]) + σ[i, j, k] = np.argmax(_tmp) + return σ + + +@njit(parallel=True) +def R_σ(g, σ, model): + """The modified policy operator.""" + β, γ, η_grid, φ, w_grid, y_grid, Q = model + w_n, y_n, η_n = len(w_grid), len(y_grid), len(η_grid) + g_new = np.empty_like(g) + for j in prange(y_n): + for i_1 in prange(w_n): + out = 0.0 + for j_1 in prange(y_n): + for k_1 in prange(η_n): + out += D(i_1, j_1, k_1, σ[i_1, j_1, k_1], g, + model) * Q[j, j_1] * φ[k_1] + g_new[j, i_1] = out + return g_new + + +def mod_opi(model, tolerance=1e-5, m=100): + """Modified optimistic policy iteration routine.""" + β, γ, η_grid, φ, w_grid, y_grid, Q = model + g = np.zeros((len(y_grid), len(w_grid))) + error = tolerance + 1 + while error > tolerance: + last_g = g + σ = get_g_greedy(g, model) + for i in range(m): + g = R_σ(g, σ, model) + error = np.max(np.abs(g - last_g)) + print(f"OPI current error = {error}") + return get_g_greedy(g, model) + + +def simulate_wealth(m): + + model = create_savings_model() + σ_star = mod_opi(model) + β, γ, η_grid, φ, w_grid, y_grid, Q = model + + # Simulate labor income + mc = MarkovChain(Q) + y_idx_series = mc.simulate(ts_length=m) + + # IID Markov chain with uniform draws + l = len(η_grid) + mc = MarkovChain(np.ones((l, l)) / l) + η_idx_series = mc.simulate(ts_length=m) + + w_idx_series = np.empty_like(y_idx_series) + w_idx_series[0] = 1 # initial condition + for t in range(m-1): + i, j, k = w_idx_series[t], y_idx_series[t], η_idx_series[t] + w_idx_series[t+1] = σ_star[i, j, k] + w_series = w_grid[w_idx_series] + + return w_series + +def lorenz(v): # assumed sorted vector + S = np.cumsum(v) # cumulative sums: [v[1], v[1] + v[2], ... ] + F = np.arange(1, len(v) + 1) / len(v) + L = S / S[-1] + return (F, L) # returns named tuple + +gini = lambda v: (2 * sum(i * y for (i, y) in enumerate(v))/sum(v) - (len(v) + 1))/len(v) + +# Plots + + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_contours(savefig=False, + figname="./figures/modified_opt_savings_1.pdf"): + + model = create_savings_model() + β, γ, η_grid, φ, w_grid, y_grid, Q = model + σ_star = optimistic_policy_iteration(model) + + fig, axes = plt.subplots(2, 1, figsize=(10, 8)) + y_n, η_n = len(y_grid), len(η_grid) + y_idx, η_idx = np.arange(y_n), np.arange(η_n) + H = np.zeros((y_n, η_n)) + + w_indices = (0, len(w_grid)-1) + titles = "low wealth", "high wealth" + for (ax, w_idx, title) in zip(axes, w_indices, titles): + + for i_y in y_idx: + for i_η in η_idx: + w, y, η = w_grid[w_idx], y_grid[i_y], η_grid[i_η] + H[i_y, i_η] = w_grid[σ_star[w_idx, i_y, i_η]] / (w + y) + + cs1 = ax.contourf(y_grid, η_grid, np.transpose(H), alpha=0.5) + + plt.colorbar(cs1, ax=ax) #, format="%.6f") + + ax.set_title(title) + ax.set_xlabel(r"$y$") + ax.set_ylabel(r"$\varepsilon$") + + plt.tight_layout() + if savefig: + fig.savefig(figname) + +def plot_policies(savefig=False): + model = create_savings_model() + β, γ, η_grid, φ, w_grid, y_grid, Q = model + σ_star = mod_opi(model) + y_bar = floor(len(y_grid) / 2) # index of mid-point of y_grid + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_grid, w_grid, "k--", label=r"$45$") + + for (i, η) in enumerate(η_grid): + label = r"$\sigma^*$" + " at " + r"$\eta = $" + f"{η.round(2)}" + ax.plot(w_grid, w_grid[σ_star[:, y_bar, i]], label=label) + + ax.legend() + if savefig: + fig.savefig(f"./figures/modified_opt_saving_2.pdf") + +def plot_time_series(m=2_000, savefig=False): + + w_series = simulate_wealth(m) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_series, label=r"$w_t$") + ax.set_xlabel("time") + ax.legend() + if savefig: + fig.savefig("./figures/modified_opt_saving_ts.pdf") + +def plot_histogram(m=1_000_000, savefig=False): + + w_series = simulate_wealth(m) + w_series.sort() + g = round(gini(w_series), ndigits=2) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.hist(w_series, bins=40, density=True) + ax.set_xlabel("wealth") + ax.text(15, 0.4, f"Gini = {g}") + + if savefig: + fig.savefig("./figures/modified_opt_saving_hist.pdf") + +def plot_lorenz(m=1_000_000, savefig=False): + + w_series = simulate_wealth(m) + w_series.sort() + (F, L) = lorenz(w_series) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(F, F, label="Lorenz curve, equality") + ax.plot(F, L, label="Lorenz curve, wealth distribution") + ax.legend() + + if savefig: + fig.savefig("./figures/modified_opt_saving_lorenz.pdf") +``` + +```{code-cell} python +plot_contours() +``` + +```{code-cell} python +plot_policies() +``` + +```{code-cell} python +plot_time_series() +``` + +```{code-cell} python +plot_histogram() +``` + +```{code-cell} python +plot_lorenz() +``` diff --git a/code-book/py/markov_dynamics.md b/code-book/py/markov_dynamics.md new file mode 100644 index 0000000..1ea08ae --- /dev/null +++ b/code-book/py/markov_dynamics.md @@ -0,0 +1,459 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 3: Markov Dynamics)= +```{raw} jupyter + +``` +# Chapter 3: Markov Dynamics + + +```{contents} Contents +:depth: 2 +``` + + + +## inventory_sim.py +```{code-cell} python3 +import numpy as np +from scipy.stats import geom +from itertools import product +from quantecon import MarkovChain +from collections import namedtuple + +# NamedTuple Model +Model = namedtuple("Model", ("S", "s", "p", "φ", "h")) + +def create_inventory_model(S=100, # Order size + s=10, # Order threshold + p=0.4): # Demand parameter + φ = geom(p, loc=-1) # loc sets support to {0,1,...} + h = lambda x, d: max(x - d, 0) + S*(x <= s) + return Model(S=S, s=s, p=p, φ=φ, h=h) + + +def sim_inventories(model, ts_length=200): + """Simulate the inventory process.""" + S, s, p, φ, h = model + X = np.empty(ts_length) + X[0] = S # Initial condition + for t in range(0, ts_length - 1): + X[t+1] = h(X[t], φ.rvs()) + return X + + +def compute_mc(model, d_max=100): + """Compute the transition probabilities and state.""" + S, s, p, φ, h = model + n = S + s + 1 # Size of state space + state_vals = np.arange(n) + P = np.empty((n, n)) + for (i, j) in product(range(0, n), range(0, n)): + P[i, j] = sum((h(i, d) == j)*φ.pmf(d) for d in range(d_max+1)) + return MarkovChain(P, state_vals) + + +def compute_stationary_dist(model): + """Compute the stationary distribution of the model.""" + mc = compute_mc(model) + return mc.state_values, mc.stationary_distributions[0] + + + +# Plots + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +default_model = create_inventory_model() + + +def plot_ts(model, fontsize=16, + figname="./figures/inventory_sim_1.pdf", + savefig=False): + S, s, p, φ, h = model + X = sim_inventories(model) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(X, label=r"$X_t$", linewidth=3, alpha=0.6) + fontdict = {'fontsize': fontsize} + ax.set_xlabel(r"$t$", fontdict=fontdict) + ax.set_ylabel("inventory", fontdict=fontdict) + ax.legend(fontsize=fontsize, frameon=False) + ax.set_ylim(0, S + s + 20) + + if savefig: + fig.savefig(figname) + + +def plot_hist(model, fontsize=16, + figname="./figures/inventory_sim_2.pdf", + savefig=False): + S, s, p, φ, h = model + state_values, ψ_star = compute_stationary_dist(model) + X = sim_inventories(model, 1_000_000) + histogram = [np.mean(X == i) for i in state_values] + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(state_values, ψ_star, "k-", linewidth=3, alpha=0.7, + label=r"$\psi^*$") + ax.bar(state_values, histogram, alpha=0.7, label="frequency") + fontdict = {'fontsize': fontsize} + ax.set_xlabel("state", fontdict=fontdict) + + ax.legend(fontsize=fontsize, frameon=False) + ax.set_ylim(0, 0.015) + + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +model = create_inventory_model() +``` + +```{code-cell} python +plot_ts(model) +``` + +```{code-cell} python +plot_hist(model) +``` +## is_irreducible.py +```{code-cell} python3 +from quantecon import MarkovChain +import numpy as np + +P = np.array([ + [0.1, 0.9], + [0.0, 1.0] +]) + +mc = MarkovChain(P) +print(mc.is_irreducible) + +``` +## laborer_sim.py +```{code-cell} python3 +import numpy as np +from collections import namedtuple + + +# NamedTuple Model +Model = namedtuple("Model", ("α", "β")) + + +def create_laborer_model(α=0.3, β=0.2): + return Model(α=α, β=β) + + +def laborer_update(x, model): # update X from t to t+1 + if x == 1: + x_ = 2 if np.random.rand() < model.α else 1 + else: + x_ = 1 if np.random.rand() < model.β else 2 + return x_ + + +def sim_chain(k, p, model): + X = np.empty(k) + X[0] = 1 if np.random.rand() < p else 2 + for t in range(0, k-1): + X[t+1] = laborer_update(X[t], model) + return X + + +def test_convergence(k=10_000_000, p=0.5): + model = create_laborer_model() + α, β = model + ψ_star = (1/(α + β)) * np.array([β, α]) + X = sim_chain(k, p, model) + ψ_e = (1/k) * np.array([sum(X == 1), sum(X == 2)]) + error = np.max(np.abs(ψ_star - ψ_e)) + approx_equal = np.allclose(ψ_star, ψ_e, rtol=0.01) + print(f"Sup norm deviation is {error}") + print(f"Approximate equality is {approx_equal}") + +``` +## markov_js.py +```{code-cell} python3 +""" +Infinite-horizon job search with Markov wage draws. + +""" + +from quantecon.markov import tauchen +import numpy as np +from collections import namedtuple +from s_approx import successive_approx + + +# NamedTuple Model +Model = namedtuple("Model", ("n", "w_vals", "P", "β", "c")) + +def create_markov_js_model( + n=200, # wage grid size + ρ=0.9, # wage persistence + ν=0.2, # wage volatility + β=0.98, # discount factor + c=1.0 # unemployment compensation + ): + """ + Creates an instance of the job search model with Markov wages. + """ + mc = tauchen(n, ρ, ν) + w_vals, P = np.exp(mc.state_values), mc.P + return Model(n=n, w_vals=w_vals, P=P, β=β, c=c) + + +def T(v, model): + """ + The Bellman operator Tv = max{e, c + β P v} with e(w) = w / (1-β). + """ + n, w_vals, P, β, c = model + h = c + β * np.dot(P, v) + e = w_vals / (1 - β) + return np.maximum(e, h) + + +def get_greedy(v, model): + """Get a v-greedy policy.""" + n, w_vals, P, β, c = model + σ = w_vals / (1 - β) >= c + β * np.dot(P, v) + return σ + + + +def vfi(model): + """Solve the infinite-horizon Markov job search model by VFI.""" + v_init = np.zeros(model.w_vals.shape) + v_star = successive_approx(lambda v: T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star + + + +# == Policy iteration == # + + +def get_value(σ, model): + """Get the value of policy σ.""" + n, w_vals, P, β, c = model + e = w_vals / (1 - β) + K_σ = β * ((1 - σ) * P.T).T + r_σ = σ * e + (1 - σ) * c + I = np.identity(K_σ.shape[0]) + return np.linalg.solve((I - K_σ), r_σ) + + +def policy_iteration(model): + """ + Howard policy iteration routine. + """ + σ = np.zeros(model.n, dtype=bool) + i, error = 0, True + while error: + v_σ = get_value(σ, model) + σ_new = get_greedy(v_σ, model) + error = np.any(σ_new ^ σ) + σ = σ_new + i = i + 1 + print(f"Concluded loop {i} with error: {error}.") + return σ + + +# == Plots == # + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +default_model = create_markov_js_model() + + +def plot_main(model=default_model, + method="vfi", + savefig=False, + figname="./figures/markov_js_vfix.png"): + n, w_vals, P, β, c = model + + if method == "vfi": + v_star, σ_star = vfi(model) + else: + σ_star = policy_iteration(model) + v_star = get_value(σ_star, model) + + h_star = c + β * np.dot(P, v_star) + e = w_vals / (1 - β) + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_vals, h_star, linewidth=4, ls="--", alpha=0.4, label=r"$h^*(w)$") + ax.plot(w_vals, e, linewidth=4, ls="--", alpha=0.4, label=r"$w/(1-\beta)$") + ax.plot(w_vals, np.maximum(e, h_star), "k-", alpha=0.7, label=r"$v^*(w)$") + ax.legend(frameon=False) + ax.set_xlabel(r"$w$") + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_main() +``` +## markov_js_with_sep.py +```{code-cell} python3 +""" +Infinite-horizon job search with Markov wage draws and separation. + +""" + +from quantecon.markov import tauchen +import numpy as np +from collections import namedtuple +from s_approx import successive_approx + + +# NamedTuple Model +Model = namedtuple("Model", ("n", "w_vals", "P", "β", "c", "α")) + + +def create_js_with_sep_model( + n=200, # wage grid size + ρ=0.9, ν=0.2, # wage persistence and volatility + β=0.98, α=0.1, # discount factor and separation rate + c=1.0): # unemployment compensation + """Creates an instance of the job search model with separation.""" + mc = tauchen(n, ρ, ν) + w_vals, P = np.exp(mc.state_values), mc.P + return Model(n=n, w_vals=w_vals, P=P, β=β, c=c, α=α) + + +def T(v, model): + """The Bellman operator for the value of being unemployed.""" + n, w_vals, P, β, c, α = model + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * np.dot(P, v)) + reject = c + β * np.dot(P, v) + return np.maximum(accept, reject) + + +def get_greedy(v, model): + """ Get a v-greedy policy.""" + n, w_vals, P, β, c, α = model + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * np.dot(P, v)) + reject = c + β * np.dot(P, v) + σ = accept >= reject + return σ + + +def vfi(model): + """Solve by VFI.""" + v_init = np.zeros(model.w_vals.shape) + v_star = successive_approx(lambda v: T(v, model), v_init) + σ_star = get_greedy(v_star, model) + return v_star, σ_star + + + +# == Plots == # + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +default_model = create_js_with_sep_model() + + +def plot_main(model=default_model, + savefig=False, + figname="./figures/markov_js_with_sep_1.pdf"): + n, w_vals, P, β, c, α = model + v_star, σ_star = vfi(model) + + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * np.dot(P, v_star)) + h_star = c + β * np.dot(P, v_star) + + w_star = np.inf + for (i, w) in enumerate(w_vals): + if accept[i] >= h_star[i]: + w_star = w + break + + assert w_star != np.inf, "Agent never accepts" + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(w_vals, h_star, linewidth=4, ls="--", alpha=0.4, + label="continuation value") + ax.plot(w_vals, accept, linewidth=4, ls="--", alpha=0.4, + label="stopping value") + ax.plot(w_vals, v_star, "k-", alpha=0.7, label=r"$v_u^*(w)$") + ax.legend(frameon=False) + ax.set_xlabel(r"$w$") + if savefig: + fig.savefig(figname) + + +def plot_w_stars(α_vals=np.linspace(0.0, 1.0, 10), + savefig=False, + figname="./figures/markov_js_with_sep_2.pdf"): + + w_star_vec = np.empty_like(α_vals) + for (i_α, α) in enumerate(α_vals): + print(i_α, α) + model = create_js_with_sep_model(α=α) + n, w_vals, P, β, c, α = model + v_star, σ_star = vfi(model) + + d = 1 / (1 - β * (1 - α)) + accept = d * (w_vals + α * β * np.dot(P, v_star)) + h_star = c + β * np.dot(P, v_star) + + w_star = np.inf + for (i_w, w) in enumerate(w_vals): + if accept[i_w] >= h_star[i_w]: + w_star = w + break + + assert w_star != np.inf, "Agent never accepts" + w_star_vec[i_α] = w_star + + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(α_vals, w_star_vec, linewidth=2, alpha=0.6, + label="reservation wage") + ax.legend(frameon=False) + ax.set_xlabel(r"$\alpha$") + ax.set_xlabel(r"$w$") + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_main() +``` + +```{code-cell} python +plot_w_stars() +``` diff --git a/code-book/py/nonlinear_valuation.md b/code-book/py/nonlinear_valuation.md new file mode 100644 index 0000000..b82f423 --- /dev/null +++ b/code-book/py/nonlinear_valuation.md @@ -0,0 +1,284 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 7: Nonlinear Valuation)= +```{raw} jupyter + +``` +# Chapter 7: Nonlinear Valuation + + +```{contents} Contents +:depth: 2 +``` + + + +## rs_utility.py +```{code-cell} python3 +from quantecon import compute_fixed_point +from quantecon.markov import tauchen + +import numpy as np +from numba import njit +from collections import namedtuple + + +# NamedTuple Model +Model = namedtuple("Model", ("β", "θ", "ρ", "σ", "r", "x_vals", "P")) + + +def create_rs_utility_model( + n=180, # size of state space + β=0.95, # time discount factor + ρ=0.96, # correlation coef in AR(1) + σ=0.1, # volatility + θ=-1.0): # risk aversion + mc = tauchen(n, ρ, σ, 0, 10) # n_std = 10 + x_vals, P = mc.state_values, mc.P + r = x_vals # special case u(c(x)) = x + return Model(β=β, θ=θ, ρ=ρ, σ=σ, r=r, x_vals=x_vals, P=P) + + +@njit +def K(v, model): + β, θ, ρ, σ, r, x_vals, P = model + return r + (β/θ) * np.log(np.dot(P, (np.exp(θ*v)))) + + +def compute_rs_utility(model): + β, θ, ρ, σ, r, x_vals, P = model + v_init = np.zeros(len(x_vals)) + v_star = compute_fixed_point(lambda v: K(v, model), v_init, + error_tol=1e-10, max_iter=1000, print_skip=25) + return v_star + + +# Plots + + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_v(savefig=False, + figname="./figures/rs_utility_1.pdf"): + + fig, ax = plt.subplots(figsize=(10, 5.2)) + model = create_rs_utility_model() + β, θ, ρ, σ, r, x_vals, P = model + + a = 1/(1 - (ρ*β)) + b = (β /(1 - β)) * (θ/2) * (a*σ)**2 + + v_star = compute_rs_utility(model) + v_star_a = a * x_vals + b + ax.plot(x_vals, v_star, linewidth=2, alpha=0.7, + label="approximate fixed point") + ax.plot(x_vals, v_star_a, "k--", linewidth=2, alpha=0.7, + label=r"$v(x)=ax + b$") + ax.set_xlabel(r"$x$") + + ax.legend(frameon=False, loc="upper left") + if savefig: + fig.savefig(figname) + + +def plot_multiple_v(savefig=False, + figname="./figures/rs_utility_2.pdf"): + + fig, ax = plt.subplots(figsize=(10, 5.2)) + σ_vals = 0.05, 0.1 + + for σ in σ_vals: + model = create_rs_utility_model(σ=σ) + β, θ, ρ, σ, r, x_vals, P = model + v_star = compute_rs_utility(model) + ax.plot(x_vals, v_star, linewidth=2, alpha=0.7, + label=r"$\sigma=$" + f"{σ}") + ax.set_xlabel(r"$x$") + ax.set_ylabel(r"$v(x)$") + + ax.legend(frameon=False, loc="upper left") + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_v() +``` + +```{code-cell} python +plot_multiple_v() +``` +## ez_utility.py +```{code-cell} python3 +""" +Epstein--Zin utility: solving the recursion for a given consumption +path. + +""" + +from quantecon import compute_fixed_point +from quantecon.markov import tauchen + +import numpy as np +from numba import njit +from collections import namedtuple + + +# NamedTuple Model +Model = namedtuple("Model", ("β", "ρ", "σ", "α", "γ", "c", "x_vals", "P")) + + +def create_ez_utility_model( + n=200, # size of state space + ρ=0.96, # correlation coef in AR(1) + σ=0.1, # volatility + β=0.99, # time discount factor + α=0.75, # EIS parameter + γ=-2.0): # risk aversion parameter + mc = tauchen(n, ρ, σ, 0, 5) + x_vals, P = mc.state_values, mc.P + c = np.exp(x_vals) + return Model(β=β, ρ=ρ, σ=σ, α=α, γ=γ, c=c, x_vals=x_vals, P=P) + + +@njit +def K(v, model): + β, ρ, σ, α, γ, c, x_vals, P = model + R = np.dot(P, v**γ)**(1/γ) + return ((1 - β) * c**α + β * R**α)**(1/α) + + +def compute_ez_utility(model): + v_init = np.ones(len(model.x_vals)) + v_star = compute_fixed_point(lambda v: K(v, model), v_init, + error_tol=1e-6, max_iter=1000, print_skip=100) + return v_star + + + +# Plots + + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_convergence(savefig=False, + num_iter=100, + figname="./figures/ez_utility_c.pdf"): + + fig, ax = plt.subplots(figsize=(10, 5.2)) + model = create_ez_utility_model() + β, ρ, σ, α, γ, c, x_vals, P = model + + + v_star = compute_ez_utility(model) + v = 0.1 * v_star + ax.plot(x_vals, v, "k-", linewidth=3, alpha=0.7, label=r"$v_0$") + + greys = [str(g) for g in np.linspace(0.0, 0.4, num_iter)] + greys.reverse() + + for g in greys: + ax.plot(x_vals, v, "k-", linewidth=1, alpha=0.7) + for t in range(20): + v = K(v, model) + + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, linewidth=3, alpha=0.7, label=r"$v^*$") + ax.set_xlabel(r"$x$") + + ax.legend(frameon=False, loc="upper left") + if savefig: + fig.savefig(figname) + + +def plot_v(savefig=False, + figname="./figures/ez_utility_1.pdf"): + + fig, ax = plt.subplots(figsize=(10, 5.2)) + model = create_ez_utility_model() + β, ρ, σ, α, γ, c, x_vals, P = model + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, linewidth=2, alpha=0.7, label=r"$v^*$") + ax.set_xlabel(r"$x$") + + ax.legend(frameon=False, loc="upper left") + if savefig: + fig.savefig(figname) + + +def vary_gamma(gamma_vals=[1.0, -8.0], + savefig=False, + figname="./figures/ez_utility_2.pdf"): + + fig, ax = plt.subplots(figsize=(10, 5.2)) + + for γ in gamma_vals: + model = create_ez_utility_model(γ=γ) + β, ρ, σ, α, γ, c, x_vals, P = model + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, linewidth=2, alpha=0.7, label=r"$\gamma=$" + f"{γ}") + ax.set_xlabel(r"$x$") + ax.set_ylabel(r"$v(x)$") + + ax.legend(frameon=False, loc="upper left") + if savefig: + fig.savefig(figname) + + +def vary_alpha(alpha_vals=[0.5, 0.6], + savefig=False, + figname="./figures/ez_utility_3.pdf"): + + fig, ax = plt.subplots(figsize=(10, 5.2)) + + for α in alpha_vals: + model = create_ez_utility_model(α=α) + β, ρ, σ, α, γ, c, x_vals, P = model + v_star = compute_ez_utility(model) + ax.plot(x_vals, v_star, linewidth=2, alpha=0.7, label=r"$\alpha=$"+f"{α}") + ax.set_xlabel(r"$x$") + ax.set_ylabel(r"$v(x)$") + + ax.legend(frameon=False, loc="upper left") + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_convergence() +``` + +```{code-cell} python +plot_v() +``` + +```{code-cell} python +vary_gamma() +``` + +```{code-cell} python +vary_alpha() +``` diff --git a/code-book/py/operators_fixed_points.md b/code-book/py/operators_fixed_points.md new file mode 100644 index 0000000..5e1ed58 --- /dev/null +++ b/code-book/py/operators_fixed_points.md @@ -0,0 +1,157 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 2: Operators and Fixed Points)= +```{raw} jupyter + +``` +# Chapter 2: Operators and Fixed Points + + +```{contents} Contents +:depth: 2 +``` + + + +## lake.py +```{code-cell} python3 +import numpy as np + +α, λ, d, b = 0.01, 0.1, 0.02, 0.025 +g = b - d +A = np.mat(np.array([[(1 - d) * (1 - λ) + b, (1 - d) * α + b], + [(1 - d) * λ, (1 - d) * (1 - α)]])) + +ū = (1 + g - (1 - d) * (1 - α)) / (1 + g - (1 - d) * (1 - α) + (1 - d) * λ) + +ē = 1 - ū +x̄ = np.array([[ū], [ē]]) + +print(np.allclose(A * x̄, (1 + g) * x̄)) # prints true + +# == Plots == # + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_paths(figname="./figures/lake_1.pdf", savefig=False): + + path_length = 100 + x_path_1 = np.zeros((2, path_length)) + x_path_2 = np.zeros((2, path_length)) + x_0_1 = 5.0, 0.1 + x_0_2 = 0.1, 4.0 + x_path_1[0, 0] = x_0_1[0] + x_path_1[1, 0] = x_0_1[1] + x_path_2[0, 0] = x_0_2[0] + x_path_2[1, 0] = x_0_2[1] + + + for t in range(path_length-1): + x_path_1[:, t+1] = (A * x_path_1[:, t][np.newaxis].T).flatten() + x_path_2[:, t+1] = (A * x_path_2[:, t][np.newaxis].T).flatten() + + + fig, ax = plt.subplots() + + # Set the axes through the origin + for spine in ["left", "bottom"]: + ax.spines[spine].set_position("zero") + for spine in ["right", "top"]: + ax.spines[spine].set_color("none") + + ax.set_xlim(0, 6) + ax.set_ylim(0, 6) + ax.set_xlabel("unemployed workforce") + ax.set_ylabel("employed workforce") + ax.set_xticks((0, 6)) + ax.set_yticks((0, 6)) + s = 10 + ax.plot([0, s * ū], [0, s * ē], "k--", lw=1) + ax.scatter(x_path_1[0, :], x_path_1[1, :], s=4, c="blue") + ax.scatter(x_path_2[0, :], x_path_2[1, :], s=4, c="green") + + ax.plot([ū], [ē], "ko", ms=4, alpha=0.6) + ax.annotate(r"$\bar{x}$", + xy=(ū, ē), + xycoords="data", + xytext=(20, -20), + textcoords="offset points", + arrowprops={"arrowstyle" : "->"}) + + x, y = x_0_1[0], x_0_1[1] + #lb = r"\$x_0 = ($(x), $(y))\$" + ax.plot([x], [y], "ko", ms=2, alpha=0.6) + ax.annotate(rf"$x_0 = ({x}, {y})$", + xy=(x, y), + xycoords="data", + xytext=(0, 20), + textcoords="offset points", + arrowprops={"arrowstyle" : "->"}) + + x, y = x_0_2[0], x_0_2[1] + #lb = r"\$x_0 = ($(x), $(y))\$" + ax.plot([x], [y], "ko", ms=2, alpha=0.6) + ax.annotate(rf"$x_0 = ({x}, {y})$", + xy=(x, y), + xycoords="data", + xytext=(0, 20), + textcoords="offset points", + arrowprops={"arrowstyle" : "->"}) + + if savefig: + fig.savefig(figname) + + + +def plot_growth(savefig=False, figname="./figures/lake_2.pdf"): + + path_length = 100 + x_0 = 2.1, 1.2 + x = np.zeros((2, path_length)) + x[0, 0] = 0.6 + x[1, 0] = 1.2 + + for t in range(path_length-1): + x[:, t+1] = (A * x[:, t][np.newaxis].T).flatten() + + fig, axes = plt.subplots(3, 1) + u = x[0, :] + e = x[1, :] + n = x[0, :] + x[1, :] + paths = u, e, n + labels = r"$u_t$", r"$e_t$", r"$n_t$" + for (ax, path, label) in zip(axes, paths, labels): + ax.plot(path, label=label) + ax.legend(frameon=False, fontsize=14) + ax.set_xlabel(r"t") + + plt.tight_layout() + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_paths() +``` + +```{code-cell} python +plot_growth() +``` diff --git a/code-book/py/optimal_stopping.md b/code-book/py/optimal_stopping.md new file mode 100644 index 0000000..ac3dde3 --- /dev/null +++ b/code-book/py/optimal_stopping.md @@ -0,0 +1,329 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 4: Optimal Stopping)= +```{raw} jupyter + +``` +# Chapter 4: Optimal Stopping + + +```{contents} Contents +:depth: 2 +``` + + + +## firm_exit.py +```{code-cell} python3 +""" +Firm valuation with exit option. + +""" + +from quantecon.markov import tauchen +from quantecon import compute_fixed_point + +import numpy as np +from collections import namedtuple +from numba import njit + + +# NamedTuple Model +Model = namedtuple("Model", ("n", "z_vals", "Q", "β", "s")) + + +def create_exit_model( + n=200, # productivity grid size + ρ=0.95, μ=0.1, ν=0.1, # persistence, mean and volatility + β=0.98, s=100.0 # discount factor and scrap value + ): + """ + Creates an instance of the firm exit model. + """ + mc = tauchen(n, ρ, ν, mu=μ) + z_vals, Q = mc.state_values, mc.P + return Model(n=n, z_vals=z_vals, Q=Q, β=β, s=s) + + +@njit +def no_exit_value(model): + """Compute value of firm without exit option.""" + n, z_vals, Q, β, s = model + I = np.identity(n) + return np.linalg.solve((I - β * Q), z_vals) + + +@njit +def T(v, model): + """The Bellman operator Tv = max{s, π + β Q v}.""" + n, z_vals, Q, β, s = model + h = z_vals + β * np.dot(Q, v) + return np.maximum(s, h) + + +@njit +def get_greedy(v, model): + """Get a v-greedy policy.""" + n, z_vals, Q, β, s = model + σ = s >= z_vals + β * np.dot(Q, v) + return σ + + +def vfi(model): + """Solve by VFI.""" + v_init = no_exit_value(model) + v_star = compute_fixed_point(lambda v: T(v, model), v_init, error_tol=1e-6, + max_iter=1000, print_skip=25) + σ_star = get_greedy(v_star, model) + return v_star, σ_star + + +# Plots + + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_val(savefig=False, + figname="./figures/firm_exit_1.pdf"): + + fig, ax = plt.subplots(figsize=(9, 5.2)) + + model = create_exit_model() + n, z_vals, Q, β, s = model + + v_star, σ_star = vfi(model) + h = z_vals + β * np.dot(Q, v_star) + + ax.plot(z_vals, h, "-", linewidth=3, alpha=0.6, label=r"$h^*$") + ax.plot(z_vals, s * np.ones(n), "-", linewidth=3, alpha=0.6, label=r"$s$") + ax.plot(z_vals, v_star, "k--", linewidth=1.5, alpha=0.8, label=r"$v^*$") + + ax.legend(frameon=False) + ax.set_xlabel(r"$z$") + + if savefig: + fig.savefig(figname) + + +def plot_comparison(savefig=False, + figname="./figures/firm_exit_2.pdf"): + + fig, ax = plt.subplots(figsize=(9, 5.2)) + + model = create_exit_model() + n, z_vals, Q, β, s = model + + v_star, σ_star = vfi(model) + w = no_exit_value(model) + + ax.plot(z_vals, v_star, "k-", linewidth=2, alpha=0.6, label="$v^*$") + ax.plot(z_vals, w, linewidth=2, alpha=0.6, label="no-exit value") + + ax.legend(frameon=False) + ax.set_xlabel(r"$z$") + + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_val() +``` + +```{code-cell} python +plot_comparison() +``` +## american_option.py +```{code-cell} python3 +""" +Valuation for finite-horizon American call options in discrete time. + +""" + +from quantecon.markov import tauchen, MarkovChain +from quantecon import compute_fixed_point + +import numpy as np +from collections import namedtuple +from numba import njit, prange + + +# NamedTuple Model +Model = namedtuple("Model", ("t_vals", "z_vals","w_vals", "Q", + "φ", "T", "β", "K")) + + +@njit +def exit_reward(t, i_w, i_z, T, K): + """Payoff to exercising the option at time t.""" + return (t < T) * (i_z + i_w - K) + + +def create_american_option_model( + n=100, μ=10.0, # Markov state grid size and mean value + ρ=0.98, ν=0.2, # persistence and volatility for Markov state + s=0.3, # volatility parameter for W_t + r=0.01, # interest rate + K=10.0, T=200): # strike price and expiration date + """ + Creates an instance of the option model with log S_t = Z_t + W_t. + """ + t_vals = np.arange(T+1) + mc = tauchen(n, ρ, ν) + z_vals, Q = mc.state_values + μ, mc.P + w_vals, φ, β = np.array([-s, s]), np.array([0.5, 0.5]), 1 / (1 + r) + return Model(t_vals=t_vals, z_vals=z_vals, w_vals=w_vals, Q=Q, + φ=φ, T=T, β=β, K=K) + + +@njit(parallel=True) +def C(h, model): + """The continuation value operator.""" + t_vals, z_vals, w_vals, Q, φ, T, β, K = model + Ch = np.empty_like(h) + z_idx, w_idx = np.arange(len(z_vals)), np.arange(len(w_vals)) + for i in prange(len(t_vals)): + t = t_vals[i] + for i_z in prange(len(z_vals)): + out = 0.0 + for i_w_1 in prange(len(w_vals)): + for i_z_1 in prange(len(z_vals)): + t_1 = min(t + 1, T) + out += max(exit_reward(t_1, w_vals[i_w_1], z_vals[i_z_1], T, K), + h[t_1, i_z_1]) * Q[i_z, i_z_1] * φ[i_w_1] + Ch[t, i_z] = β * out + + return Ch + + +def compute_cvf(model): + """ + Compute the continuation value function by successive approx. + """ + h_init = np.zeros((len(model.t_vals), len(model.z_vals))) + h_star = compute_fixed_point(lambda h: C(h, model), h_init, + error_tol=1e-6, max_iter=1000, + print_skip=25) + return h_star + + +# Plots + + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_contours(savefig=False, + figname="./figures/american_option_1.pdf"): + + model = create_american_option_model() + t_vals, z_vals, w_vals, Q, φ, T, β, K = model + h_star = compute_cvf(model) + fig, axes = plt.subplots(3, 1, figsize=(7, 11)) + z_idx, w_idx = np.arange(len(z_vals)), np.arange(len(w_vals)) + H = np.zeros((len(w_vals), len(z_vals))) + for (ax_index, t) in zip(range(3), (1, 195, 198)): + + ax = axes[ax_index] + for i_w in prange(len(w_vals)): + for i_z in prange(len(z_vals)): + H[i_w, i_z] = exit_reward(t, w_vals[i_w], z_vals[i_z], T, + K) - h_star[t, i_z] + + cs1 = ax.contourf(w_vals, z_vals, np.transpose(H), alpha=0.5) + ctr1 = ax.contour(w_vals, z_vals, np.transpose(H), levels=[0.0]) + plt.clabel(ctr1, inline=1, fontsize=13) + plt.colorbar(cs1, ax=ax) + + ax.set_title(f"$t={t}$") + ax.set_xlabel(r"$w$") + ax.set_ylabel(r"$z$") + + fig.tight_layout() + if savefig: + fig.savefig(figname) + + +def plot_strike(savefig=False, + fontsize=9, + figname="./figures/american_option_2.pdf"): + model = create_american_option_model() + t_vals, z_vals, w_vals, Q, φ, T, β, K = model + h_star = compute_cvf(model) + + # Built Markov chains for simulation + z_mc = MarkovChain(Q, z_vals) + P_φ = np.zeros((len(w_vals), len(w_vals))) + for i in range(len(w_vals)): # Build IID chain + P_φ[i, :] = φ + w_mc = MarkovChain(P_φ, w_vals) + y_min = np.min(z_vals) + np.min(w_vals) + y_max = np.max(z_vals) + np.max(w_vals) + fig, axes = plt.subplots(3, 1, figsize=(7, 12)) + + for ax in axes: + + # Generate price series + z_draws = z_mc.simulate_indices(T, init=int(len(z_vals) / 2 - 10)) + w_draws = w_mc.simulate_indices(T) + s_vals = z_vals[z_draws] + w_vals[w_draws] + + # Find the exercise date, if any. + exercise_date = T + 1 + for t in range(T): + k = exit_reward(t, w_vals[w_draws[t]], z_vals[z_draws[t]], T, K) - \ + h_star[w_draws[t], z_draws[t]] + if k >= 0: + exercise_date = t + + if exercise_date >= T: + print("Warning: Option not exercised.") + else: + # Plot + ax.set_ylim(y_min, y_max) + ax.set_xlim(0, T+1) + ax.fill_between(range(T), np.ones(T) * K, np.ones(T) * y_max, alpha=0.2) + ax.plot(range(T), s_vals, label=r"$S_t$") + ax.plot((exercise_date,), (s_vals[exercise_date]), "ko") + ax.vlines((exercise_date,), 0, (s_vals[exercise_date]), ls="--", colors="black") + ax.legend(loc="upper left", fontsize=fontsize) + ax.text(-10, 11, "in the money", fontsize=fontsize, rotation=90) + ax.text(-10, 7.2, "out of the money", fontsize=fontsize, rotation=90) + ax.text(exercise_date-20, 6, + "exercise date", fontsize=fontsize) + ax.set_xticks((1, T)) + ax.set_yticks((y_min, y_max)) + + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_contours() +``` + +```{code-cell} python +plot_strike() +``` diff --git a/code-book/py/recursive_decision_processes.md b/code-book/py/recursive_decision_processes.md new file mode 100644 index 0000000..b509c13 --- /dev/null +++ b/code-book/py/recursive_decision_processes.md @@ -0,0 +1,162 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 8: Recursive Decision Processes)= +```{raw} jupyter + +``` +# Chapter 8: Recursive Decision Processes + + +```{contents} Contents +:depth: 2 +``` + + + +## quantile_function.py +```{code-cell} python3 +from scipy.stats import rv_discrete +import numpy as np + +from numba import njit + +"Compute the τ-th quantile of v(X) when X ∼ ϕ and v = sort(v)." +@njit +def quantile(τ, v, ϕ): + for (i, v_value) in enumerate(v): + p = sum(ϕ[:i+1]) # sum all ϕ[j] s.t. v[j] ≤ v_value + if p >= τ: # exit and return v_value if prob ≥ τ + return v_value + +"For each i, compute the τ-th quantile of v(Y) when Y ∼ P(i, ⋅)" +def R(τ, v, P): + return np.array([quantile(τ, v, P[i, :]) for i in range(len(v))]) + +def quantile_test(τ): + ϕ = [0.1, 0.2, 0.7] + v = [10, 20, 30] + + #d = DiscreteNonParametric(v, ϕ) + d = rv_discrete(values=(v, ϕ)) + return quantile(τ, v, ϕ), d.ppf(τ) + + + + +``` +## quantile_js.py +```{code-cell} python3 +""" +Job search with Markov wage draws and quantile preferences. + +""" +from quantecon import tauchen, MarkovChain +import numpy as np +from quantile_function import * + +"Creates an instance of the job search model." +def create_markov_js_model( + n=100, # wage grid size + ρ=0.9, # wage persistence + ν=0.2, # wage volatility + β=0.98, # discount factor + c=1.0, # unemployment compensation + τ=0.5 # quantile parameter + ): + mc = tauchen(n, ρ, ν) + w_vals, P = np.exp(mc.state_values), mc.P + return (n, w_vals, P, β, c, τ) + +""" +The policy operator + + (T_σ v)(w) = σ(w) (w / (1-β)) + (1 - σ(w))(c + β (R_τ v)(w)) + +""" +def T_σ(v, σ, model): + n, w_vals, P, β, c, τ = model + h = [x + c for x in β * R(τ, v, P)] + e = w_vals / (1 - β) + return σ * e + (1 - σ) * h + +" Get a v-greedy policy." +def get_greedy(v, model): + n, w_vals, P, β, c, τ = model + σ = w_vals / (1 - β) >= c + β * R(τ, v, P) + return σ + + +"Optimistic policy iteration routine." +def optimistic_policy_iteration(model, tolerance=1e-5, m=100): + n, w_vals, P, β, c, τ = model + v = np.ones(n) + error = tolerance + 1 + while error > tolerance: + last_v = v + σ = get_greedy(v, model) + for i in range(m): + v = T_σ(v, σ, model) + + error = max(np.abs(v - last_v)) + print(f"OPI current error = {error}") + + return v, get_greedy(v, model) + + +# == Plots == # + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +def plot_main(tau_vals=(0.1, 0.25, 0.5, 0.6, 0.7, 0.8), + savefig=False, + figname="./figures/quantile_js.pdf"): + + w_star_vals = np.zeros(len(tau_vals)) + + for (τ_idx, τ) in enumerate(tau_vals): + model = create_markov_js_model(τ=τ) + n, w_vals, P, β, c, τ = model + v_star, σ_star = optimistic_policy_iteration(model) + for i in range(n): + if σ_star[i] > 0: + w_star_vals[τ_idx] = w_vals[i] + break + + model = create_markov_js_model() + n, w_vals, P, β, c, τ = model + mc = MarkovChain(P) + s = mc.stationary_distributions[0] + + fig, ax = plt.subplots() + ax.plot(tau_vals, w_star_vals, "k--", lw=2, alpha=0.7, label="reservation wage") + ax.barh(w_vals, 32 * s, alpha=0.05, align="center") + ax.legend(frameon=False, loc="upper center") + ax.set_xlabel("quantile") + ax.set_ylabel("wages") + + if savefig: + fig.savefig(figname) + + +``` + +```{code-cell} python +plot_main() +``` diff --git a/code-book/py/stochastic_discounting.md b/code-book/py/stochastic_discounting.md new file mode 100644 index 0000000..3784b3c --- /dev/null +++ b/code-book/py/stochastic_discounting.md @@ -0,0 +1,383 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(Chapter 6: Stochastic Discounting)= +```{raw} jupyter + +``` +# Chapter 6: Stochastic Discounting + + +```{contents} Contents +:depth: 2 +``` + + + +## plot_interest_rates.py +```{code-cell} python3 +# Nominal interest rate from https://fred.stlouisfed.org/series/GS1 +# Real interest rate from https://fred.stlouisfed.org/series/WFII10 +# +# Download as CSV files +# + +import pandas as pd +import numpy as np +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + +df_nominal = pd.read_csv("./data/GS1.csv") +df_real = pd.read_csv("./data/WFII10.csv") + +def plot_rates(df, fontsize=16, savefig=False): + r_type = 'nominal' if df.equals(df_nominal) else 'real' + fig, ax = plt.subplots(figsize=(9, 5)) + ax.plot(df.iloc[:, 0], df.iloc[:, 1], label=f'{r_type} interest rate') + ax.plot(df.iloc[:, 0], np.zeros(df.iloc[:, 1].size), c='k', ls='--') + ax.set_xlim(df.iloc[0, 0], df.iloc[-1, 0]) + ax.legend(fontsize=fontsize, frameon=False) + if savefig: + fig.savefig(f'./figures/plot_interest_rates_{r_type}.pdf') +``` + +```{code-cell} python +plot_rates(df_nominal) +``` + +```{code-cell} python +plot_rates(df_real) +``` +## pd_ratio.py +```{code-cell} python3 +""" +Price-dividend ratio in a model with dividend and consumption growth. + +""" + +from quantecon.markov import tauchen +import numpy as np +from collections import namedtuple + + +# NamedTuple Model +Model = namedtuple("Model", ("x_vals", "P", "β", "γ", + "μ_c", "σ_c", "μ_d", "σ_d")) + + +def create_asset_pricing_model( + n=200, # state grid size + ρ=0.9, ν=0.2, # state persistence and volatility + β=0.99, γ=2.5, # discount and preference parameter + μ_c=0.01, σ_c=0.02, # consumption growth mean and volatility + μ_d=0.02, σ_d=0.1): # dividend growth mean and volatility + """ + Creates an instance of the asset pricing model with Markov state. + """ + mc = tauchen(n, ρ, ν) + x_vals, P = np.exp(mc.state_values), mc.P + return Model(x_vals=x_vals, P=P, β=β, γ=γ, + μ_c=μ_c, σ_c=σ_c, μ_d=μ_d, σ_d=σ_d) + + +def build_discount_matrix(model): + """Build the discount matrix A.""" + x_vals, P, β, γ, μ_c, σ_c, μ_d, σ_d = model + e = np.exp(μ_d - γ*μ_c + (γ**2 * σ_c**2 + σ_d**2)/2 + (1-γ)*x_vals) + return β * (e * P.T).T + + + +def pd_ratio(model): + """ + Compute the price-dividend ratio associated with the model. + """ + x_vals, P, β, γ, μ_c, σ_c, μ_d, σ_d = model + A = build_discount_matrix(model) + assert np.max(np.abs(np.linalg.eigvals(A))) < 1, "Requires r(A) < 1." + n = len(x_vals) + I = np.identity(n) + return np.linalg.solve((I - A), np.dot(A, np.ones(n))) + + +# == Plots == # + + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + + +default_model = create_asset_pricing_model() + + +def plot_main(μ_d_vals=(0.02, 0.08), + savefig=False, + figname="./figures/pd_ratio_1.pdf"): + fig, ax = plt.subplots(figsize=(9, 5.2)) + + for μ_d in μ_d_vals: + model = create_asset_pricing_model(μ_d=μ_d) + x_vals, P, β, γ, μ_c, σ_c, μ_d, σ_d = model + v_star = pd_ratio(model) + ax.plot(x_vals, v_star, linewidth=2, alpha=0.6, + label=r"$\mu_d$=" + f"{μ_d}") + + ax.legend(frameon=False) + ax.set_xlabel(r"$x$") + if savefig: + fig.savefig(figname) + +``` + +```{code-cell} python +plot_main() +``` +## inventory_sdd.py +```{code-cell} python3 +""" + +Inventory management model with state-dependent discounting. The discount +factor takes the form β_t = Z_t, where (Z_t) is a discretization of a +Gaussian AR(1) process + + X_t = ρ X_{t-1} + b + ν W_t. + +""" + +from quantecon import compute_fixed_point +from quantecon.markov import tauchen, MarkovChain + +import numpy as np +from time import time +from numba import njit, prange +from collections import namedtuple + +# NamedTuple Model +Model = namedtuple("Model", ("K", "c", "κ", "p", "r", + "R", "y_vals", "z_vals", "Q")) + + +@njit +def ϕ(p, d): + return (1 - p)**d * p + +@njit +def f(y, a, d): + return np.maximum(y - d, 0) + a # Inventory update + +def create_sdd_inventory_model( + ρ=0.98, ν=0.002, n_z=20, b=0.97, # Z state parameters + K=40, c=0.2, κ=0.8, p=0.6, # firm and demand parameters + d_max=100): # truncation of demand shock + d_vals = np.arange(d_max+1) + ϕ_vals = ϕ(p, d_vals) + y_vals = np.arange(K+1) + n_y = len(y_vals) + mc = tauchen(n_z, ρ, ν) + z_vals, Q = mc.state_values + b, mc.P + ρL = np.max(np.abs(np.linalg.eigvals(z_vals * Q))) + assert ρL < 1, "Error: ρ(L) >= 1." # check r(L) < 1 + + R = np.zeros((n_y, n_y, n_y)) + for i_y, y in enumerate(y_vals): + for i_y_1, y_1 in enumerate(y_vals): + for i_a, a in enumerate(range(K - y + 1)): + hits = [f(y, a, d) == y_1 for d in d_vals] + R[i_y, i_a, i_y_1] = np.dot(hits, ϕ_vals) + + + r = np.empty((n_y, n_y)) + for i_y, y in enumerate(y_vals): + for i_a, a in enumerate(range(K - y + 1)): + cost = c * a + κ * (a > 0) + r[i_y, i_a] = np.dot(np.minimum(y, d_vals), ϕ_vals) - cost + + + return Model(K=K, c=c, κ=κ, p=p, r=r, R=R, + y_vals=y_vals, z_vals=z_vals, Q=Q) + +@njit +def B(i_y, i_z, i_a, v, model): + """ + The function B(x, z, a, v) = r(x, a) + β(z) Σ_x′ v(x′) P(x, a, x′). + """ + K, c, κ, p, r, R, y_vals, z_vals, Q = model + β = z_vals[i_z] + cv = 0.0 + + for i_z_1 in prange(len(z_vals)): + for i_y_1 in prange(len(y_vals)): + cv += v[i_y_1, i_z_1] * R[i_y, i_a, i_y_1] * Q[i_z, i_z_1] + return r[i_y, i_a] + β * cv + +@njit(parallel=True) +def T(v, model): + """The Bellman operator.""" + K, c, κ, p, r, R, y_vals, z_vals, Q = model + new_v = np.empty_like(v) + for i_z in prange(len(z_vals)): + for (i_y, y) in enumerate(y_vals): + Γy = np.arange(K - y + 1) + new_v[i_y, i_z] = np.max(np.array([B(i_y, i_z, i_a, v, model) + for i_a in Γy])) + return new_v + +@njit +def T_σ(v, σ, model): + """The policy operator.""" + K, c, κ, p, r, R, y_vals, z_vals, Q = model + new_v = np.empty_like(v) + for (i_z, z) in enumerate(z_vals): + for (i_y, y) in enumerate(y_vals): + new_v[i_y, i_z] = B(i_y, i_z, σ[i_y, i_z], v, model) + return new_v + +@njit(parallel=True) +def get_greedy(v, model): + """Get a v-greedy policy. Returns a zero-based array.""" + K, c, κ, p, r, R, y_vals, z_vals, Q = model + n_z = len(z_vals) + σ_star = np.zeros((K+1, n_z), dtype=np.int32) + for (i_z, z) in enumerate(z_vals): + for (i_y, y) in enumerate(y_vals): + Γy = np.arange(K - y + 1) + i_a = np.argmax(np.array([B(i_y, i_z, i_a, v, model) + for i_a in Γy])) + σ_star[i_y, i_z] = Γy[i_a] + return σ_star + +@njit +def get_value(v_init, σ, m, model): + """Approximate lifetime value of policy σ.""" + v = v_init + for _ in range(m): + v = T_σ(v, σ, model) + return v + +def solve_inventory_model(v_init, model): + """Use successive_approx to get v_star and then compute greedy.""" + v_star = compute_fixed_point(lambda v: T(v, model), v_init, + error_tol=1e-5, max_iter=1000, print_skip=25) + σ_star = get_greedy(v_star, model) + return v_star, σ_star + +def optimistic_policy_iteration(v_init, + model, + tolerance=1e-6, + max_iter=1_000, + print_step=10, + m=60): + v = v_init + error = tolerance + 1 + k = 1 + while (error > tolerance) and (k < max_iter): + last_v = v + σ = get_greedy(v, model) + v = get_value(v, σ, m, model) + error = np.max(np.abs(v - last_v)) + if k % print_step == 0: + print(f"Completed iteration {k} with error {error}.") + k += 1 + return v, get_greedy(v, model) + + +# == Plots == # + +import matplotlib.pyplot as plt +import matplotlib.pyplot as plt + +plt.rcParams.update({"text.usetex": True, "font.size": 14}) + +# Create an instance of the model and solve it +model = create_sdd_inventory_model() +K, c, κ, p, r, R, y_vals, z_vals, Q = model +n_z = len(z_vals) +v_init = np.zeros((K+1, n_z), dtype=float) +print("Solving model.") +v_star, σ_star = optimistic_policy_iteration(v_init, model) +z_mc = MarkovChain(Q, z_vals) + +def sim_inventories(ts_length, X_init=0): + """Simulate given the optimal policy.""" + global p, z_mc + i_z = z_mc.simulate_indices(ts_length, init=1) + X = np.zeros(ts_length, dtype=np.int32) + X[0] = X_init + rand = np.random.default_rng().geometric(p=p, size=ts_length-1) - 1 + for t in range(ts_length-1): + X[t+1] = f(X[t], σ_star[X[t], i_z[t]], rand[t]) + return X, z_vals[i_z] + +def plot_ts(ts_length=400, + fontsize=10, + figname="./figures/inventory_sdd_ts.pdf", + savefig=False): + + X, Z = sim_inventories(ts_length) + fig, axes = plt.subplots(2, 1, figsize=(9, 5.5)) + + ax = axes[0] + ax.plot(X, label="inventory", alpha=0.7) + ax.set_xlabel(r"$t$", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=False) + ax.set_ylim(0, np.max(X)+3) + + # calculate interest rate from discount factors + r = (1 / Z) - 1 + + ax = axes[1] + ax.plot(r, label=r"$r_t$", alpha=0.7) + ax.set_xlabel(r"$t$", fontsize=fontsize) + ax.legend(fontsize=fontsize, frameon=False) + + plt.tight_layout() + if savefig: + fig.savefig(figname) + +def plot_timing(m_vals=np.arange(1, 400, 10), + fontsize=16, + savefig=False): + print("Running value function iteration.") + t_start = time() + solve_inventory_model(v_init, model) + vfi_time = time() - t_start + print(f"VFI completed in {vfi_time} seconds.") + opi_times = [] + for m in m_vals: + print(f"Running optimistic policy iteration with m = {m}.") + t_start = time() + optimistic_policy_iteration(v_init, model, m=m) + opi_time = time() - t_start + print(f"OPI with m = {m} completed in {opi_time} seconds.") + opi_times.append(opi_time) + fig, ax = plt.subplots(figsize=(9, 5.2)) + ax.plot(m_vals, np.full(len(m_vals), vfi_time), + lw=2, label="value function iteration") + ax.plot(m_vals, opi_times, lw=2, label="optimistic policy iteration") + ax.legend(fontsize=fontsize, frameon=False) + ax.set_xlabel(r"$m$", fontsize=fontsize) + ax.set_ylabel("time", fontsize=fontsize) + if savefig: + fig.savefig("./figures/inventory_sdd_timing.pdf") + return (opi_time, vfi_time, opi_times) +``` + +```{code-cell} python +plot_ts() +``` diff --git a/code-book/status.md b/code-book/status.md new file mode 100644 index 0000000..312dfb6 --- /dev/null +++ b/code-book/status.md @@ -0,0 +1,21 @@ +--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +# Execution Statistics + +This table contains the latest execution statistics. + +```{nb-exec-table} +``` + +(status:machine-details)= + +These lectures are built on `linux` instances through `github actions`. \ No newline at end of file diff --git a/create_code_book.py b/create_code_book.py new file mode 100644 index 0000000..4ee8776 --- /dev/null +++ b/create_code_book.py @@ -0,0 +1,338 @@ +import os +import shutil + +julia_src_dir = './source_code_jl/' +python_src_dir = './source_code_py/' +julia_dst_dir = './code-book/jl/' +python_dst_dir = './code-book/py/' + +def remove_files(root_src_dir, root_dst_dir): + for file in os.listdir(root_src_dir): + if os.path.isfile(os.path.join(root_src_dir, file)): + os.remove(os.path.join(root_dst_dir, file)) + else: + shutil.rmtree(os.path.join(root_dst_dir, file)) + +#################################### Julia Version ################################################# + +shutil.copytree(julia_src_dir, julia_dst_dir, dirs_exist_ok=True) + +header = """--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(placeholder_text)= +```{raw} jupyter + +``` +""" + +contents = """ +```{contents} Contents +:depth: 2 +``` +""" +chapter_meta = { + "introductions": { + "name": "Chapter 1: Introduction", + "subs": { # order determines order in chapter + "two_period_job_search.jl": ["fig_v1(savefig=true)"], + "compute_spec_rad.jl": [], + "power_series.jl": [], + "s_approx.jl": [], + "linear_iter.jl": [], + "linear_iter_fig.jl": [], + "iid_job_search.jl": ["fig_vseq(savefig=true)", "fig_vstar(savefig=true)"] # plotting functions are added here to keep source code clean + } + }, + "operators_fixed_points": { + "name": "Chapter 2: Operators and Fixed Points", + "subs": { + "lake.jl": ["plot_paths(savefig=true)", "plot_growth(savefig=true)"], + # "lake_1.jl" + } + }, + "markov_dynamics": { + "name": "Chapter 3: Markov Dynamics", + "subs": { + "inventory_sim.jl": ["model = create_inventory_model()", "plot_ts(model; savefig=true)", "plot_hist(model; savefig=true)"], + "is_irreducible.jl": [], + # "inventory_sim.jl", + "laborer_sim.jl": [], + "markov_js.jl": ["plot_main(savefig=true)"], + "markov_js_with_sep.jl": ["plot_main(savefig=true)", "plot_w_stars(savefig=true)"] + } + }, + "optimal_stopping": { + "name": "Chapter 4: Optimal Stopping", + "subs": { + "firm_exit.jl": ["plot_val(savefig=true)", "plot_comparison(savefig=true)"], + "american_option.jl": ["plot_contours(savefig=true)", "plot_strike(savefig=true)"] + } + }, + "markov_decision_processes": { + "name": "Chapter 5: Markov Decision Processes", + "subs": { + "inventory_dp.jl": ["plot_vstar_and_opt_policy(savefig=true)", "plot_ts(savefig=true)"], + "finite_opt_saving_0.jl": [], + "finite_opt_saving_1.jl": [], + "finite_opt_saving_2.jl": ["plot_timing(savefig=true)", "plot_policy()", "plot_time_series(savefig=true)", "plot_histogram(savefig=true)", "plot_lorenz(savefig=true)"], + "finite_lq.jl": ["plot_policy()", "plot_sim(savefig=true)", "plot_timing(savefig=true)"], + "firm_hiring.jl": ["plot_policy()", "plot_sim(savefig=true)", "plot_growth(savefig=true)"], + "modified_opt_savings.jl": ["plot_contours(savefig=true)", "plot_policies(savefig=true)", "plot_time_series(savefig=true)", "plot_histogram(savefig=true)", "plot_lorenz(savefig=true)"] + } + }, + "stochastic_discounting": { + "name": "Chapter 6: Stochastic Discounting", + "subs": { + "plot_interest_rates.jl": ["plot_rates(df_nominal, savefig=true)", "plot_rates(df_real, savefig=true)"], + # "plot_interest_rates_real.jl", + "pd_ratio.jl": ["plot_main(savefig=true)"], + "inventory_sdd.jl": ["plot_ts(savefig=true)"] + } + }, + "nonlinear_valuation": { + "name": "Chapter 7: Nonlinear Valuation", + "subs": { + "rs_utility.jl": ["plot_v(savefig=true)", "plot_multiple_v(savefig=true)"], + "ez_utility.jl": ["plot_convergence(savefig=true)", "plot_v(savefig=true)", "vary_gamma(savefig=true)", "vary_alpha(savefig=true)"] + } + }, + "recursive_decision_processes": { + "name": "Chapter 8: Recursive Decision Processes", + "subs": { + "quantile_function.jl": [], + "quantile_js.jl": ["plot_main(savefig=true)"] + } + } +} + + +pkg = """```{code-cell} julia +:tags: [\"remove-cell\"] +using Pkg; +Pkg.activate(\"./\"); + +using PyCall; +pygui(:tk); +``` +""" + +try: + for chapter in chapter_meta.keys(): + chapter_name = chapter_meta[chapter]["name"] + chapter_subs = chapter_meta[chapter]["subs"] + + with open(f"./code-book/jl/{chapter}.md", "w", encoding='utf-8') as b: + new_header = header.replace("placeholder_text", chapter_name) + b.write(new_header) + b.write(f"# {chapter_name}\n\n") + b.write(contents) + b.write("\n\n") + + b.write(pkg) + + b.write("\n") + + for sub in chapter_subs.keys(): + with open(f"./code-book/jl/{sub}", "r", encoding='utf-8') as g: + + b.write(f"## {sub}\n") + b.write(f"```{{code-cell}} julia\n") + #b.write(":tags: [\"hide-input\"]\n") + + text = g.readlines() + + for line in text: + if "plt.show()" not in line: + b.write(line) + + b.write(f"\n```\n") + + if len(chapter_subs[sub]) > 0: + for func_call in chapter_subs[sub]: + code = f"\n```{{code-cell}} julia\n{func_call}\n```\n" + + b.write(code) +except Exception as e: + print("Error in Julia markdown generation: " + str(e)) + remove_files(julia_src_dir, julia_dst_dir) + print("Failed! - Directory cleaned") + exit(1) + +##################################################################################################### + +#################################### Python Version ################################################# + +shutil.copytree(python_src_dir, python_dst_dir, dirs_exist_ok=True) + +header = """--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(placeholder_text)= +```{raw} jupyter + +``` +""" + +contents = """ +```{contents} Contents +:depth: 2 +``` +""" +chapter_meta = { + "introductions": { + "name": "Chapter 1: Introduction", + "subs": { # order determines order in chapter + "two_period_job_search.py": ["fig_v1()"], + "compute_spec_rad.py": [], + "power_series.py": [], + "s_approx.py": [], + "linear_iter.py": [], + "linear_iter_fig.py": [], + "iid_job_search.py": ["fig_vseq()", "fig_vstar()"] # plotting functions are added here to keep source code clean + } + }, + "operators_fixed_points": { + "name": "Chapter 2: Operators and Fixed Points", + "subs": { + "lake.py": ["plot_paths()", "plot_growth()"], + # "lake_1.py" + } + }, + "markov_dynamics": { + "name": "Chapter 3: Markov Dynamics", + "subs": { + "inventory_sim.py": ["model = create_inventory_model()", "plot_ts(model)", "plot_hist(model)"], + "is_irreducible.py": [], + # "inventory_sim.py", + "laborer_sim.py": [], + "markov_js.py": ["plot_main()"], + "markov_js_with_sep.py": ["plot_main()", "plot_w_stars()"] + } + }, + "optimal_stopping": { + "name": "Chapter 4: Optimal Stopping", + "subs": { + "firm_exit.py": ["plot_val()", "plot_comparison()"], + "american_option.py": ["plot_contours()", "plot_strike()"] + } + }, + "markov_decision_processes": { + "name": "Chapter 5: Markov Decision Processes", + "subs": { + "inventory_dp.py": ["plot_vstar_and_opt_policy()", "plot_ts()"], + "finite_opt_saving_0.py": [], + "finite_opt_saving_1.py": [], + "finite_opt_saving_2.py": ["plot_timing()", "plot_policy()", "plot_time_series()", "plot_histogram()", "plot_lorenz()"], + "finite_lq.py": ["plot_policy()", "plot_sim()", "plot_timing()"], + "firm_hiring.py": ["plot_policy()", "plot_sim()", "plot_growth()"], + "modified_opt_savings.py": ["plot_contours()", "plot_policies()", "plot_time_series()", "plot_histogram()", "plot_lorenz()"] + } + }, + "stochastic_discounting": { + "name": "Chapter 6: Stochastic Discounting", + "subs": { + "plot_interest_rates.py": ["plot_rates(df_nominal)", "plot_rates(df_real)"], + # "plot_interest_rates_real.py", + "pd_ratio.py": ["plot_main()"], + "inventory_sdd.py": ["plot_ts()"] + } + }, + "nonlinear_valuation": { + "name": "Chapter 7: Nonlinear Valuation", + "subs": { + "rs_utility.py": ["plot_v()", "plot_multiple_v()"], + "ez_utility.py": ["plot_convergence()", "plot_v()", "vary_gamma()", "vary_alpha()"] + } + }, + "recursive_decision_processes": { + "name": "Chapter 8: Recursive Decision Processes", + "subs": { + "quantile_function.py": [], + "quantile_js.py": ["plot_main()"] + } + } +} + +try: + for chapter in chapter_meta.keys(): + chapter_name = chapter_meta[chapter]["name"] + chapter_subs = chapter_meta[chapter]["subs"] + + with open(f"./code-book/py/{chapter}.md", "w", encoding='utf-8') as b: + new_header = header.replace("placeholder_text", chapter_name) + b.write(new_header) + b.write(f"# {chapter_name}\n\n") + b.write(contents) + b.write("\n\n") + + b.write("\n") + + for sub in chapter_subs.keys(): + with open(f"./code-book/py/{sub}", "r", encoding='utf-8') as g: + + b.write(f"## {sub}\n") + b.write(f"```{{code-cell}} python3\n") + #b.write(":tags: [\"hide-input\"]\n") + + text = g.readlines() + + for line in text: + if "plt.show()" not in line: + b.write(line) + + if "import matplotlib.pyplot as plt" in line: + b.write(line + "\nplt.rcParams.update({\"text.usetex\": True, \"font.size\": 14})\n") + + b.write(f"\n```\n") + + if len(chapter_subs[sub]) > 0: + for func_call in chapter_subs[sub]: + code = f"\n```{{code-cell}} python\n{func_call}\n```\n" + + b.write(code) +except Exception as e: + print("Error in Python markdown generation: " + str(e)) + remove_files(julia_src_dir, julia_dst_dir) + remove_files(python_src_dir, python_dst_dir) + print("Failed! - Directory cleaned") + exit(1) + +##################################################################################################### + +try: + os.system("jupyter-book build ./code-book/") + remove_files(python_src_dir, python_dst_dir) + remove_files(julia_src_dir, julia_dst_dir) + print("Success!") + exit(0) +except Exception as e: + print("Error in building Jupyter Book: " + str(e)) + remove_files(python_src_dir, python_dst_dir) + remove_files(julia_src_dir, julia_dst_dir) + print("Failed! - Directory cleaned") + exit(1) diff --git a/create_jb.py b/create_jb.py new file mode 100644 index 0000000..cfc1fcb --- /dev/null +++ b/create_jb.py @@ -0,0 +1,328 @@ +import os +import shutil + +julia_src_dir = './source_code_jl/' +python_src_dir = './source_code_py/' +julia_dst_dir = './code-book/jl/' +python_dst_dir = './code-book/py/' + +def remove_files(root_src_dir, root_dst_dir): + for file in os.listdir(root_src_dir): + if os.path.isfile(os.path.join(root_src_dir, file)): + os.remove(os.path.join(root_dst_dir, file)) + else: + shutil.rmtree(os.path.join(root_dst_dir, file)) + +#################################### Julia Version ################################################# + +shutil.copytree(julia_src_dir, julia_dst_dir, dirs_exist_ok=True) + +header = """--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Julia + language: julia + name: julia-1.9 +--- + +(placeholder_text)= +```{raw} jupyter + +``` +""" + +contents = """ +```{contents} Contents +:depth: 2 +``` +""" +chapter_meta = { + "introductions": { + "name": "Chapter 1: Introductions", + "subs": { # order determines order in chapter + "two_period_job_search.jl": ["fig_v1(savefig=true)"], + "compute_spec_rad.jl": [], + "power_series.jl": [], + "s_approx.jl": [], + "linear_iter.jl": [], + "linear_iter_fig.jl": [], + "iid_job_search.jl": ["fig_vseq(savefig=true)", "fig_vstar(savefig=true)"] # plotting functions are added here to keep source code clean + } + }, + "operators_fixed_points": { + "name": "Chapter 2: Operators and Fixed Points", + "subs": { + "lake.jl": ["plot_paths(savefig=true)", "plot_growth(savefig=true)"], + # "lake_1.jl" + } + }, + "markov_dynamics": { + "name": "Chapter 3: Markov Dynamics", + "subs": { + "inventory_sim.jl": ["model = create_inventory_model()", "plot_ts(model; savefig=true)", "plot_hist(model; savefig=true)"], + "is_irreducible.jl": [], + # "inventory_sim.jl", + "laborer_sim.jl": [], + "markov_js.jl": ["plot_main(savefig=true)"], + "markov_js_with_sep.jl": ["plot_main(savefig=true)", "plot_w_stars(savefig=true)"] + } + }, + "optimal_stopping": { + "name": "Chapter 4: Optimal Stopping", + "subs": { + "firm_exit.jl": ["plot_val(savefig=true)", "plot_comparison(savefig=true)"], + "american_option.jl": ["plot_contours(savefig=true)", "plot_strike(savefig=true)"] + } + }, + "markov_decision_processes": { + "name": "Chapter 5: Markov Decision Processes", + "subs": { + "inventory_dp.jl": ["plot_vstar_and_opt_policy(savefig=true)", "plot_ts(savefig=true)"], + "finite_opt_saving_0.jl": [], + "finite_opt_saving_1.jl": [], + "finite_opt_saving_2.jl": ["plot_timing(savefig=true)", "plot_policy()", "plot_time_series(savefig=true)", "plot_histogram(savefig=true)", "plot_lorenz(savefig=true)"], + "finite_lq.jl": ["plot_policy()", "plot_sim(savefig=true)", "plot_timing(savefig=true)"], + "firm_hiring.jl": ["plot_policy()", "plot_sim(savefig=true)", "plot_growth(savefig=true)"], + "modified_opt_savings.jl": ["plot_contours(savefig=true)", "plot_policies(savefig=true)", "plot_time_series(savefig=true)", "plot_histogram(savefig=true)", "plot_lorenz(savefig=true)"] + } + }, + "stochastic_discounting": { + "name": "Chapter 6: Stochastic Discounting", + "subs": { + "plot_interest_rates.jl": ["plot_rates(df_nominal, savefig=true)", "plot_rates(df_real, savefig=true)"], + # "plot_interest_rates_real.jl", + "pd_ratio.jl": ["plot_main(savefig=true)"], + "inventory_sdd.jl": ["plot_ts(savefig=true)"] + } + }, + "nonlinear_valuation": { + "name": "Chapter 7: Nonlinear Valuation", + "subs": { + "rs_utility.jl": ["plot_v(savefig=true)", "plot_multiple_v(savefig=true)"], + "ez_utility.jl": ["plot_convergence(savefig=true)", "plot_v(savefig=true)", "vary_gamma(savefig=true)", "vary_alpha(savefig=true)"] + } + }, + "recursive_decision_processes": { + "name": "Chapter 8: Recursive Decision Processes", + "subs": { + "quantile_function.jl": [], + "quantile_js.jl": ["plot_main(savefig=true)"] + } + } +} + + +pkg = """```{code-cell} julia +:tags: [\"remove-cell\"] +using Pkg; +Pkg.activate(\"./\"); + +using PyCall; +pygui(:tk); +``` +""" + +try: + for chapter in chapter_meta.keys(): + chapter_name = chapter_meta[chapter]["name"] + chapter_subs = chapter_meta[chapter]["subs"] + + with open(f"./code-book/jl/{chapter}.md", "w", encoding='utf-8') as b: + new_header = header.replace("placeholder_text", chapter_name) + b.write(new_header) + b.write(f"# {chapter_name}\n\n") + b.write(contents) + b.write("\n\n") + + b.write(pkg) + + b.write("\n") + + for sub in chapter_subs.keys(): + with open(f"./code-book/jl/{sub}", "r", encoding='utf-8') as g: + + b.write(f"## {sub}\n") + b.write(f"```{{code-cell}} julia\n") + #b.write(":tags: [\"hide-input\"]\n") + + text = g.readlines() + + for line in text: + if "plt.show()" not in line: + b.write(line) + + b.write(f"\n```\n") + + if len(chapter_subs[sub]) > 0: + for func_call in chapter_subs[sub]: + code = f"\n```{{code-cell}} julia\n{func_call}\n```\n" + + b.write(code) +except Exception as e: + print("Error in Julia markdown generation: " + str(e)) + remove_files(julia_src_dir, julia_dst_dir) + print("Failed! - Directory cleaned") + exit(1) + +##################################################################################################### + +#################################### Python Version ################################################# + +shutil.copytree(python_src_dir, python_dst_dir, dirs_exist_ok=True) + +header = """--- +jupytext: + text_representation: + extension: .md + format_name: myst +kernelspec: + display_name: Python 3 + language: python + name: python3 +--- + +(placeholder_text)= +```{raw} jupyter + +``` +""" + +contents = """ +```{contents} Contents +:depth: 2 +``` +""" +chapter_meta = { + "introductions": { + "name": "Chapter 1: Introductions", + "subs": { # order determines order in chapter + "two_period_job_search.py": ["fig_v1()"], + "compute_spec_rad.py": [], + "power_series.py": [], + "s_approx.py": [], + "linear_iter.py": [], + "linear_iter_fig.py": [], + "iid_job_search.py": ["fig_vseq()", "fig_vstar()"] # plotting functions are added here to keep source code clean + } + }, + "operators_fixed_points": { + "name": "Chapter 2: Operators and Fixed Points", + "subs": { + "lake.py": ["plot_paths()", "plot_growth()"], + # "lake_1.py" + } + }, + "markov_dynamics": { + "name": "Chapter 3: Markov Dynamics", + "subs": { + "inventory_sim.py": ["model = create_inventory_model()", "plot_ts(model)", "plot_hist(model)"], + "is_irreducible.py": [], + # "inventory_sim.py", + "laborer_sim.py": [], + "markov_js.py": ["plot_main()"], + "markov_js_with_sep.py": ["plot_main()", "plot_w_stars()"] + } + }, + "optimal_stopping": { + "name": "Chapter 4: Optimal Stopping", + "subs": { + "firm_exit.py": ["plot_val()", "plot_comparison()"], + "american_option.py": ["plot_contours()", "plot_strike()"] + } + }, + "markov_decision_processes": { + "name": "Chapter 5: Markov Decision Processes", + "subs": { + "inventory_dp.py": ["plot_vstar_and_opt_policy()", "plot_ts()"], + "finite_opt_saving_0.py": [], + "finite_opt_saving_1.py": [], + "finite_opt_saving_2.py": ["plot_timing()", "plot_policy()", "plot_time_series()", "plot_histogram()", "plot_lorenz()"], + "finite_lq.py": ["plot_policy()", "plot_sim()", "plot_timing()"], + "firm_hiring.py": ["plot_policy()", "plot_sim()", "plot_growth()"], + "modified_opt_savings.py": ["plot_contours()", "plot_policies()", "plot_time_series()", "plot_histogram()", "plot_lorenz()"] + } + }, + "stochastic_discounting": { + "name": "Chapter 6: Stochastic Discounting", + "subs": { + "plot_interest_rates.py": ["plot_rates(df_nominal)", "plot_rates(df_real)"], + # "plot_interest_rates_real.py", + "pd_ratio.py": ["plot_main()"], + "inventory_sdd.py": ["plot_ts()"] + } + }, + "nonlinear_valuation": { + "name": "Chapter 7: Nonlinear Valuation", + "subs": { + "rs_utility.py": ["plot_v()", "plot_multiple_v()"], + "ez_utility.py": ["plot_convergence()", "plot_v()", "vary_gamma()", "vary_alpha()"] + } + }, + "recursive_decision_processes": { + "name": "Chapter 8: Recursive Decision Processes", + "subs": { + "quantile_function.py": [], + "quantile_js.py": ["plot_main()"] + } + } +} + +try: + for chapter in chapter_meta.keys(): + chapter_name = chapter_meta[chapter]["name"] + chapter_subs = chapter_meta[chapter]["subs"] + + with open(f"./code-book/py/{chapter}.md", "w", encoding='utf-8') as b: + new_header = header.replace("placeholder_text", chapter_name) + b.write(new_header) + b.write(f"# {chapter_name}\n\n") + b.write(contents) + b.write("\n\n") + + b.write("\n") + + for sub in chapter_subs.keys(): + with open(f"./code-book/py/{sub}", "r", encoding='utf-8') as g: + + b.write(f"## {sub}\n") + b.write(f"```{{code-cell}} python3\n") + #b.write(":tags: [\"hide-input\"]\n") + + text = g.readlines() + + for line in text: + if "plt.show()" not in line: + b.write(line) + + if "import matplotlib.pyplot as plt" in line: + b.write(line + "\nplt.rcParams.update({\"text.usetex\": True, \"font.size\": 14})\n") + + b.write(f"\n```\n") + + if len(chapter_subs[sub]) > 0: + for func_call in chapter_subs[sub]: + code = f"\n```{{code-cell}} python\n{func_call}\n```\n" + + b.write(code) +except Exception as e: + print("Error in Python markdown generation: " + str(e)) + remove_files(julia_src_dir, julia_dst_dir) + remove_files(python_src_dir, python_dst_dir) + print("Failed! - Directory cleaned") + exit(1) + +##################################################################################################### + +remove_files(python_src_dir, python_dst_dir) +remove_files(julia_src_dir, julia_dst_dir) diff --git a/environment.yml b/environment.yml new file mode 100644 index 0000000..379f481 --- /dev/null +++ b/environment.yml @@ -0,0 +1,25 @@ +name: dp1 +channels: + - default +dependencies: + - python=3.10 +# - julia=1.9 +# - anaconda=2023.03 + - pip + - pip: + - jupyter-book==0.15.1 + - quantecon-book-theme==0.4.1 + - sphinx-tojupyter==0.3.0 + - sphinxext-rediraffe==0.2.7 + - sphinx-exercise==0.4.1 + - ghp-import==2.1.0 + - sphinxcontrib-youtube==1.2.0 + - sphinx-togglebutton==0.3.2 + - arviz==0.13.0 + - kaleido + - matplotlib==3.7.1 + - numba==0.57.1 + - quantecon==0.7.1 + - pandas==2.0.2 + - pandas-datareader==0.10.0 + - yfinance==0.2.22 \ No newline at end of file diff --git a/pdf/dp.pdf b/pdf/dp.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e3129c1f497d77ca1717be14d2ef871c9123dad1 GIT binary patch literal 10174487 zcma&NQ?Mveuc$k1+qP}nwr$(CZQC~2v~7E?X`5&5TlZn_+UMT--^O_8(Uqi=uak_B zDu{^DGSaa^ksjY&zd^ARFc8=qSwZpe(2H5xxR^T8i`f{un2MMh+nbou%b41kyI2q~ 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