demo_notebooks/growth_recursive_demo.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "dae963a0",
   "metadata": {},
   "source": [
    "**Note**: This notebook should take around **a minute** to run.\n",
    "\n",
    "For a more comprehensive and detailed treatment of this problem, please refer to the [GitHub repository](https://github.com/HighDimensionalEconLab/transversality)."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ad24e909",
   "metadata": {},
   "source": [
    "# The Economic Problem:\n",
    "\n",
    "Consider the following Bellman equation:\n",
    "\n",
    "$\\begin{align}\n",
    "    v(k) = & \\max_{c, k'}\\left\\{u(c) + \\beta v(k')\\right\\} \\\\\n",
    "    \\text{s.t.}~\\quad & k' = f(k) + (1 - \\delta)k - c \\\\\n",
    "    & k' \\geq 0,\n",
    "\\end{align}\n",
    "$\n",
    "\n",
    "where $v(\\cdot)$ is the value function, $u(\\cdot)$ is the utility function, $f(\\cdot)$ is the production function, $\\beta$ is the discount factor, and $\\delta$ is the depreciation rate.\n",
    "\n",
    "\n",
    "#### First Order Condition\n",
    "The Euler equation can be written as:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad & u'(c) = \\beta u'(c')\\big[f'(k')+(1-\\delta)\\big]\n",
    "\\end{align}$\n",
    "\n",
    "To pin down the optimal solution a transversality condition is required, described as:\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad & \\lim_{T\\rightarrow \\infty} \\beta^T u'(c^T(k))k^{'T}(k) = 0\n",
    "\\end{align}$\n",
    "\n",
    "for any initial condition of capital $k_0$.\n",
    "\n",
    "#### The System of Equations Describing the Optimal Solution:\n",
    "\n",
    "\n",
    "$\\begin{align}\n",
    "\\quad & \\beta u'\\big(c(k)\\big)\\bigg[f'\\big(k'(k)\\big)+(1-\\delta)\\bigg] - u'\\big(c(k')\\big) = 0 \\\\\n",
    "\\quad & c(k) = f\\big(k\\big) + (1-\\delta) k - k'(k)  \\\\\n",
    "\\end{align}$\n",
    "\n",
    "#### Note on the Transversality Condition:\n",
    "\n",
    "One of the main contributions of this paper is that one can ignore the transversality condition and completely rely on the implicit bias in deep learning to obtain the optimal solution.\n",
    "\n",
    "\n",
    "\n",
    "In this example we focus on the case of:\n",
    " 1. $u(c) =log(c)$, and $f(k) = k^\\alpha$,\n",
    "\n",
    " 2.  $\\beta = 0.9$, $\\alpha = \\frac{1}{3}$, and $\\delta = 0.1$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ec32d74",
   "metadata": {},
   "source": [
    "#### Importing Packages"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "0be8994c",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import torch\n",
    "import torch.nn as nn\n",
    "from torch.utils.data import DataLoader\n",
    "import matplotlib.pyplot as plt\n",
    "from matplotlib import cm"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "f559ba55",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<torch._C.Generator at 0x23a938d90b0>"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "torch.manual_seed(123) # fixing the seed"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8c326069",
   "metadata": {},
   "source": [
    "#### Setting Up the Economic Parameters \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "196b6caf",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "alpha = 1.0/3.0\n",
    "beta = 0.9\n",
    "delta = 0.1\n",
    "k_0 = 1.0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8dc5db0e",
   "metadata": {},
   "source": [
    "#### Setting Up the Grid ($\\mathcal{X}_{\\text{train}}$) and the DataLoader\n",
    "\n",
    "We are looking for the policy function for capital $k'(k): \\mathcal{X} \\rightarrow \\mathbb{R}_+ $, where $\\mathcal{X} \\subset \\mathbb{R}_+$. \n",
    "\n",
    "We pick a grid of the following form:\n",
    "\n",
    "$\\begin{align*}\n",
    "\t\\mathcal{X}_{\\text{train}} \\equiv \\{k_{\\text{min}},\\ldots,k_{\\text{max}}\\}.\n",
    "\\end{align*}$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "3605c936",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "k_min = 0.4\n",
    "k_max =4.0\n",
    "grid_size = 16\n",
    "capital_grid = torch.linspace(k_min, k_max, grid_size)\n",
    "capital_grid = capital_grid.unsqueeze(dim =1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "31126853",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "data_loader = DataLoader(capital_grid, batch_size=len(capital_grid), shuffle = False)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "58bffd2a",
   "metadata": {},
   "source": [
    "#### Setting up the Neural Network\n",
    "\n",
    "This is an approximation for capital $\\hat{k}': \\mathcal{X} \\rightarrow \\mathbb{R}_+$.\n",
    "\n",
    "$\\hat{k}'(\\cdot)$ takes capital as an input. The positivity of capital is going to be implemented by using a `Softplus` output layer."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "9b14524b",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "class K_prime_NN(nn.Module):\n",
    "    def __init__(self,\n",
    "                 dim_hidden = 128,):\n",
    "        super().__init__()\n",
    "        \n",
    "        self.dim_hidden= dim_hidden\n",
    "        \n",
    "        self.k_prime = nn.Sequential(\n",
    "            nn.Linear(1, dim_hidden, bias=True),\n",
    "            nn.Tanh(),\n",
    "            nn.Linear(dim_hidden, dim_hidden, bias=True),\n",
    "            nn.Tanh(),\n",
    "            nn.Linear(dim_hidden, dim_hidden, bias=True),\n",
    "            nn.Tanh(),\n",
    "            nn.Linear(dim_hidden, dim_hidden, bias=True),\n",
    "            nn.Tanh(),\n",
    "            nn.Linear(dim_hidden, 1),\n",
    "            nn.Softplus(beta=1.0) #To make sure capital stays positive\n",
    "        )\n",
    "\n",
    "    def forward(self, x):\n",
    "        return self.k_prime(x)\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5c7bfac7",
   "metadata": {},
   "source": [
    "#### Initializing the Neural Network and Defining the Optimizer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "9f802a33",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "k_prime= K_prime_NN()\n",
    "learning_rate = 1e-3\n",
    "optimizer = torch.optim.Adam(k_prime.parameters(), lr=learning_rate) #Adam Optimizer \n",
    "scheduler = torch.optim.lr_scheduler.StepLR(optimizer, step_size=200, gamma=0.95) #dampening the learning rate"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "66c43887",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "num_epochs = 2001 # step of optimization\n",
    "print_epoch_frequency = 200 # how often printing the results"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fca37882",
   "metadata": {},
   "source": [
    "### Implementing the Economics\n",
    "\n",
    "Here, we implement the economic conditions in the form of residuals to be minimized. Specifically:\n",
    "\n",
    "1. The Euler residuals: the difference between the left-hand and right-hand sides of the Euler equation (i.e., Equation 6).\n",
    "\n",
    "Note: The feasibility condition (i.e., Equation 7) is used to construct the consumption function. Therefore, by construction, Equation 7 always holds."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8e70792c",
   "metadata": {},
   "source": [
    "#### Defining the Marginal Utility, Production, and Marginal Product of Capital Functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "3bd02ad6",
   "metadata": {},
   "outputs": [],
   "source": [
    "def f(k): # Production function\n",
    "    return k**alpha\n",
    "\n",
    "def df_k(k): # Marinal product of capital\n",
    "    return  alpha*(k**(alpha -1))\n",
    "\n",
    "def du_c(c): # Mariginal utility\n",
    "    return c**(-1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84cb8ced",
   "metadata": {},
   "source": [
    "#### Defining the Residuals Function to be Used in the Optimization\n",
    "This funtion defines the residuals of the Euler equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "eaf07748",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "def residuals(batch):\n",
    "    k_t = batch\n",
    "\n",
    "# iterate forwards\n",
    "    k_tp1 = k_prime(k_t)\n",
    "    k_tp2 = k_prime(k_tp1)\n",
    "\n",
    "# compute consumptions level\n",
    "    c_t = f(k_t) + (1 - delta) * k_t - k_tp1\n",
    "    c_tp1 = f(k_tp1) + (1 - delta) * k_tp1 - k_tp2\n",
    "\n",
    "# Euler residual\n",
    "    residual = (du_c(c_t) / du_c(c_tp1)) - beta * (1 - delta + df_k(k_tp1))\n",
    "    return residual\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "350f4ed0",
   "metadata": {},
   "source": [
    "#### Training Loop (Optimization Process)\n",
    "\n",
    "Here, we minimize the mean squared of price residuals over the grid points:\n",
    "\n",
    "$\n",
    "\\begin{align*}\n",
    "\tmin_{\\theta\\in \\Theta} \\frac{1}{|\\mathcal{X}_{\\text{train}}|} \\sum_{k \\in \\mathcal{X}_{\\text{train}}} \\Bigg( &\t\\frac{u'\\big(c\\big(k,k'(\\cdot;\\theta)\\big)\\big)}{u'\\big(c\\big(k'(k;\\theta);k'(\\cdot;\\theta)\\big)\\big)} - \\beta \\big[f'\\big(k'(k;\\theta)\\big) + (1-\\delta)\\big]\\Bigg)^2.\n",
    "\\end{align*}\n",
    "$\n",
    "\n",
    "where $\\theta$ is the coefficients of the neural network, and the optimization id done with gradient-based optimizers such as `Gradient Descent (GD)` and `Adam`.  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "7f4f2c52",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "epoch = 0, loss = 6.71e-01\n",
      "epoch = 200, loss = 1.45e-04\n",
      "epoch = 400, loss = 1.02e-04\n",
      "epoch = 600, loss = 7.38e-05\n",
      "epoch = 800, loss = 5.36e-05\n",
      "epoch = 1000, loss = 3.79e-05\n",
      "epoch = 1200, loss = 2.55e-05\n",
      "epoch = 1400, loss = 1.63e-05\n",
      "epoch = 1600, loss = 9.71e-06\n",
      "epoch = 1800, loss = 5.49e-06\n",
      "epoch = 2000, loss = 3.37e-06\n"
     ]
    }
   ],
   "source": [
    "for epoch in range(num_epochs):\n",
    "    for batch in data_loader:\n",
    "        \n",
    "        optimizer.zero_grad()\n",
    "        \n",
    "        res = residuals(batch)\n",
    "        \n",
    "        loss = res.pow(2).mean()\n",
    "        \n",
    "        loss.backward()\n",
    "        optimizer.step()\n",
    "        \n",
    "    scheduler.step()\n",
    "    \n",
    "    if epoch % print_epoch_frequency == 0:\n",
    "        print(f\"epoch = {epoch}, loss = {loss.detach().numpy():.2e}\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "27604ac5",
   "metadata": {},
   "source": [
    "### Plotting the Results"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "30306ecf",
   "metadata": {},
   "source": [
    "#### Calculating the Steady-State for Capital and Consumption\n",
    "The steady-state for capital and consumption is used to test the convergence to the optimal solution "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "6bdac4b4",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "base = ((1.0/beta)-1.0+ delta)/alpha\n",
    "exponent = 1.0/(alpha-1)\n",
    "\n",
    "k_star = base**exponent\n",
    "c_star = f(k_star)-delta*k_star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "079a3256",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "# creating the test data: we test for the capital path that starts k_0=0.5\n",
    "test_T = 50\n",
    "k_0 =0.5\n",
    "k_hat_path  = torch.empty(test_T+1).unsqueeze(1)\n",
    "c_hat_path = torch.empty(test_T).unsqueeze(1)\n",
    "time_grid = torch.arange(test_T).unsqueeze(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "adb74f95-f34d-4358-b9e9-962614724773",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "k_hat_path[0] =k_0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6e725238",
   "metadata": {},
   "source": [
    "#### Calculating the Approximate Capital and Consumption Path\n",
    "\n",
    "We use approximated $\\hat{k}'(k)$ (by neural network) to calculate capital and consumption paths.  \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "7a23891b",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "for t in range(0, test_T):\n",
    "    k_hat_path[t+1] = k_prime(k_hat_path[t])\n",
    "    c_hat_path[t] = f(k_hat_path[t]) + (1-delta)*k_hat_path[t] - k_hat_path[t+1]\n",
    "    \n",
    "k_hat_path =k_hat_path.detach()\n",
    "c_hat_path =c_hat_path.detach()\n",
    "time_grid = time_grid.detach()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "45c88413",
   "metadata": {},
   "source": [
    "#### Calculating the Euler Residuals\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "ed5dc57c",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "res = residuals(k_hat_path[0:test_T])\n",
    "res = res.detach()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ecd7ee4c",
   "metadata": {},
   "source": [
    "#### Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3517f24a",
   "metadata": {},
   "source": [
    "We firstly set up the plots parmeters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "697287b6",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "fontsize= 14\n",
    "ticksize = 14\n",
    "figsize = (14, 4.5)\n",
    "params = {'font.family':'serif',\n",
    "    \"figure.figsize\":figsize,\n",
    "    'figure.dpi': 80,\n",
    "    'figure.edgecolor': 'k',\n",
    "    'font.size': fontsize,\n",
    "    'axes.labelsize': fontsize,\n",
    "    'axes.titlesize': fontsize,\n",
    "    'xtick.labelsize': ticksize,\n",
    "    'ytick.labelsize': ticksize\n",
    "}\n",
    "plt.rcParams.update(params)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "10b42a91",
   "metadata": {},
   "source": [
    "Next we plot the approximate capital and consumption plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "47476cba",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x23a9a1a3fd0>"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1120x360 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.subplot(1, 2 , 1)\n",
    "\n",
    "plt.plot(time_grid,k_hat_path[0:test_T], color='k',  label = r\"$\\hat{k}(t)$\")\n",
    "plt.axhline(y=k_star, linestyle='--',color='k', label=\"k Steady-State\")\n",
    "plt.xlabel(r\"Time(t)\")\n",
    "plt.title(r\"Approximate Capital Path\")\n",
    "plt.ylim([k_0-0.1,k_star*(1+0.35) ])\n",
    "plt.legend(loc='best')\n",
    "\n",
    "plt.subplot(1, 2, 2)\n",
    "\n",
    "plt.plot(time_grid,c_hat_path, color='b',  label = r\"$\\hat{c}(t)$\")\n",
    "plt.axhline(y=c_star, linestyle='--',color='b', label=\"c Steady-State\")\n",
    "plt.xlabel(r\"Time(t)\")\n",
    "plt.title(r\"Approximate Consumption Path\")\n",
    "plt.ylim([c_hat_path[0]-0.1,c_star*(1+0.25) ])\n",
    "plt.legend(loc='best')"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a6fac4b1",
   "metadata": {},
   "source": [
    "Lastly we plot residuals "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "61beb014",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1120x360 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(time_grid,res**2, color='k',  label = r\"Euler Residuals$^2$\")\n",
    "plt.xlabel(r\"Time(t)\")\n",
    "plt.title(r\"Euler Residuals Squared\")\n",
    "plt.legend(loc='lower right')\n",
    "plt.yscale('log')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "517c7525",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}