{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "
\n", "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# *Numeric Methods for Economists:* Introduction to Dynamic Programming\n", "\n", "Dynamic programming is a fundamental tool of economic analysis. The objective of this lab is to show you how to solve, via numerical dynamic programming, a textbook dynamic stochastic general equilibrium (DSGE) model of optimal consumption/savings. For those interested in learning more about numerical dynamic programming, I suggest starting with John Stachurski and Thomas Sargent's excellent set of [Python-based online lecture notes](http://quant-econ.net/). The lecture notes are also available as a [PDF](http://quant-econ.net/_static/pdfs/quant-econ.pdf). " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# A model of optimal consumption/savings\n", "\n", "In this section I fully specify the decision problems faced by firms and households in a stochastic optimal savings with both inelastic labor supply. I begin by deriving the equilibrium conditions and the dynamic programming formulation of a model where the only choice variable for the household is consumption. \n", "\n", "## Firm behavior\n", "There are a large number of firms, each with access to an identical constant returns to scale, CES production technology.\n", "\n", "$$ y_t = f(k_t, l_t, z_t) = e^{z_t}\\bigg(\\alpha k_t^{\\rho} + (1 - \\alpha)l_t^{\\rho}\\bigg)^{\\frac{1}{\\rho}} \\tag{1.1}$$\n", "\n", "where $0 < \\alpha < 1$ and $\\rho = \\frac{\\sigma - 1}{\\sigma}$. Recall that $f_k > 0$, $f_{kk} < 0$, and that $f$ satisfies the usual Inada conditions. Total factor productivity shocks, $z_t$, follow a standard AR(1) process.\n", "\n", "$$ z_t = \\rho z_{t-1} + \\epsilon_t,\\ \\epsilon \\sim N(0, \\sigma) \\tag{1.2}$$\n", "\n", "All markets (i.e., for both factors of production and the final output good) are competitive and therefore capital and labor earn their respective marginal products, and firms earn zero profits. \n", "\\begin{align}\n", "r_t =& \\alpha e^{z_t} k_t^{\\rho - 1}\\bigg(\\alpha k_t^{\\rho} + (1 - \\alpha)l_t^{\\rho}\\bigg)^{\\frac{1}{\\rho}-1} = \\left(\\frac{\\alpha k_t^{\\rho-1}}{\\alpha k_t^{\\rho} + (1 - \\alpha)l_t^{\\rho}}\\right)f(k_t,l_t,z_t) \\tag{1.3}\\\\\n", "w_t =& (1 - \\alpha) e^{z_t} l_t^{\\rho-1}\\bigg(\\alpha k_t^{\\rho} + (1 - \\alpha)l_t^{\\rho}\\bigg)^{\\frac{1}{\\rho}-1} = \\left(\\frac{(1-\\alpha) l_t^{\\rho-1}}{\\alpha k_t^{\\rho} + (1 - \\alpha)l_t^{\\rho}}\\right)f(k_t,l_t,z_t) \\tag{1.4}\\\\\n", "y_t =& r_tk_t + w_tl_t \\tag{1.5}\n", "\\end{align} \n", "\n", "## Household behavior\n", "There are a large number of identical households with constant relative risk aversion (CRRA) preferences: \n", "\n", "$$ u(c_t) = \\frac{c_t^{1-\\theta} - 1}{1 - \\theta} \\tag{1.6}$$\n", "\n", "where the parameter $0<\\theta$ is the coefficient of relative risk aversion. The representative household's objective is to choose an infinite consumption sequence, $\\{c_t\\}_{t=0}^{\\infty}$, in order to \n", "\n", "$$ \\max \\ E\\left\\{\\sum_{t=0}^{\\infty} \\beta^t u(c_t)\\right\\} \\tag{1.7} $$\n", "\n", "subject to the following constraints:\n", "\n", "\\begin{align}\n", "c_t + i_t =& w_tl_t + r_t k_t \\tag{1.8}\\\\\n", "k_{t+1} =& (1-\\delta)k_t + i_t \\tag{1.9} \n", "\\end{align}\n", "\n", "where $0 < \\beta < 1$ is the discount factor.\n", "\n", "The representative household is endowed with a single unit of labor which we assume it supplies inelastically to firms (i.e., we assume that household chooses $l_t=1$ regardless of the real wage). The first order necessary condition for the household's decision problem is the consumption Euler equation\n", "\n", "$$ c_t^{-\\theta} = \\beta E\\left\\{(1 + r_{t+1} - \\delta)c_{t+1}^{-\\theta} | z_t\\right\\} \\tag{1.8}$$\n", "\n", "which, together with the budget constraint\n", "\n", "$$ c_t + k_{t+1} = w_t + (1 + r_t - \\delta)k_t \\tag{1.9} $$\n", "\n", "completely specifies the the behavior of the representative household.\n", "\n", "## General Equilibrium\n", "Equations 1.3, 1.4, 1.5, 1.8, and 1.9 reduce to a system of two non-linear equations in two unknowns, $k$ and $c$ that describe the equilibrium of the model.\n", "\n", "\\begin{align}\n", "c_t^{-\\theta} =& \\beta E\\left\\{\\left(1 + \\alpha e^{z_{t+1}}k_{t+1}^{\\rho-1}\\bigg(\\alpha k_{t+1}^{\\rho} + (1 - \\alpha)\\bigg)^{\\frac{1}{\\rho}-1} - \\delta\\right)c_{t+1}^{-\\theta} \\bigg| z_t\\right\\} \\tag{1.10}\\\\\n", "k_{t+1} =& (1 - \\delta)k_t + e^{z_{t+1}}\\bigg(\\alpha k_{t+1}^{\\rho} + (1 - \\alpha)\\bigg)^{\\frac{1}{\\rho}} - c_t \\tag{1.11}\n", "\\end{align} \n", " \n", "\n", "## Dynamic programming specification\n", "The stochastic optimal savings model with inelastic labor supply can be formulated as a simple dynamic programming problem with capital, $k$, and productivity $z$ as the only state variables. The most natural control variable is, perhaps, $c$. \n", "\n", "\\begin{align} \n", "V(k, z) =& \\max_{c\\in \\Gamma(k, z)}\\ \\frac{c^{1-\\theta}-1}{1-\\theta} + \\beta E\\bigg\\{V(k', z') \\bigg| z\\bigg\\} \\tag{1.13}\\\\\n", "\\Gamma(k, z) =& (1 - \\delta)k + f(k, z) \\tag{1.14} \\\\\n", "k' =& (1 - \\delta)k + f(k, z) - c \\tag{1.15}\\\\\n", "z' =& \\rho_z z + \\epsilon',\\ \\epsilon \\sim N(0, \\sigma_z) \\tag{1.16}\\\\\n", "\\end{align}\n", "\n", "Equation 1.13 is the familiar Bellman equation. Equation 1.14 defines the set of feasible values of the control (in this case consumption) as a function of the current states. **Note that this definition of $\\Gamma(k,z)$ assumes that capital can be \"eaten\".** Equations 1.15 and 1.16 are the equations of motion for capital and the productivity shock, respectively.\n", "\n", "We are going to replace 1.13 with \n", "\n", "$$W(k, z) = \\max_{c \\in \\Gamma(k, z)}\\ (1-\\beta)\\frac{c^{1-\\theta}-1}{1-\\theta} + \\beta E\\bigg\\{W(k', z') \\bigg| z\\bigg\\}. \\tag{1.13b} $$\n", "\n", "Why? Note that the current value, $W(k,z)$, is now a convex combination of the flow payoff, $u(c)$, and the expected future value (conditional on the current productivity shock!), $E\\bigg\\{W(k', z') \\bigg| z\\bigg\\}$. Transforming the value function in this manner makes the computation of the value function more numerically stable and efficient. **This transformation, which is really just a rescaling of utility by a positive constant, will not impact the optimal policy function in any way**.\n", "\n", "The optimal policy function $c(k, z)$ obeys the following first order necessary condition for optimality. \n", "\n", "$$ 0 = (1 - \\beta)c(k, z)^{-\\theta} - \\beta E\\left\\{W'(k', z') \\bigg| z\\right\\} \\tag{1.17}$$\n", "\n", "The envelope theorem applied to equation 1.14b yields \n", "\n", "$$ W'(k, z) = \\beta\\left(1 + \\alpha e^{z}k^{\\rho-1}\\bigg(\\alpha k^{\\rho} + (1 - \\alpha)\\bigg)^{\\frac{1}{\\rho}-1} - \\delta\\right)E\\left\\{W'(k', z') \\bigg| z\\right\\} \\tag{1.18} $$\n", "\n", "which together with equation 1.17 implies that\n", "\n", "$$ (1-\\beta)c(k, z)^{-\\theta}\\left(1 + \\alpha e^{z}k^{\\rho-1}\\bigg(\\alpha k^{\\rho} + (1 - \\alpha)\\bigg)^{\\frac{1}{\\rho}-1} - \\delta\\right) = W'(k, z). \\tag{1.19} $$\n", "\n", "Thus the solution to the dynamic programming problem specified in equations 1.13b through 1.16 is a policy function, $c(k, z)$, and a value function, $W(k, z)$, that jointly satisfy equations 1.19 and \n", "\n", "$$ W(k, z) = (1-\\beta)\\frac{c(k, z)^{1-\\theta}-1}{1-\\theta} + \\beta E\\left\\{W\\bigg((1-\\delta)k + e^{z}\\bigg(\\alpha k^{\\rho} + (1 - \\alpha)\\bigg)^{\\frac{1}{\\rho}} - c(k, z), \\rho_z z + \\epsilon')\\right\\}. \\tag{1.20}$$" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [], "source": [ "import numpy as np\n", "import matplotlib.pyplot as plt\n", "from mpl_toolkits.mplot3d import Axes3D\n", "from scipy import interpolate, optimize, stats" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exercise:\n", "\n", "Using `alpha`, `sigma`, `delta`, `theta`, and `rho_z` as parameters, fill in the blanks in the code below in order to define Python functions for equations 1.1, 1.2, 1.3, 1.6, 1.11, and 1.14. Confirm that your functions are correct before moving forward!" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# equation 1.1\n", "def ces_output(k, z=0.0):\n", " \"\"\"\n", " Output is generated by a CES production function. Note that output is a \n", " function of state variables (i.e., capital and possibly a productivity shock).\n", " \n", " Arguments:\n", " \n", " k: (array) Current value of capital. Inherited from previous period!\n", " z: (array) Current value of the productivity shock.\n", " \n", " Returns:\n", " \n", " y: (array) Output produced from k and z\n", " \n", " \"\"\"\n", " rho = (sigma - 1) / sigma\n", " \n", " # nest Cobb-Douglas output as special case\n", " if rho == 0:\n", " y = np.exp(z) * k**alpha\n", " else:\n", " y = np.exp(z) * (alpha * k**rho + (1 - alpha))**(1 / rho)\n", " \n", " return y\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.4999995181380563" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# confirm that ces_output is functioning correctly...\n", "alpha = 0.5\n", "sigma = 1e6\n", "ces_output(4, 0)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# equation 1.2\n", "def productivity_motion(z, eps):\n", " \"\"\"\n", " Equation of motion for total factor productivity.\n", " \n", " Arguments:\n", " \n", " z: (array) Current value of the total factor productivity.\n", " eps: (array) Productivity shock.\n", " \n", " Returns:\n", " \n", " zplus: (array) Next period's value of capital.\n", " \n", " \"\"\"\n", " zplus = rho_z * z + eps\n", " return zplus" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.95" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# confirm that productivity_motion is functioning correctly...\n", "rho_z = 0.95\n", "productivity_motion(1, 0)" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# equation 1.3\n", "def ces_mpk(k, z=0.0):\n", " \"\"\"\n", " Marginal product of capital for the CES production function.\n", " \n", " Arguments:\n", " \n", " k: (array) Current value of capital. Inherited from previous period!\n", " z: (array) Current value of the productivity shock.\n", " \n", " Returns:\n", " \n", " mpk: (array) Output produced from k.\n", " \n", " \"\"\"\n", " rho = (sigma - 1) / sigma\n", " \n", " # nest Cobb-Douglas output as special case\n", " if rho == 0:\n", " mpk = alpha * (ces_output(k, z) / k)\n", " else:\n", " mpk = (alpha * k**(rho - 1) / (alpha * k**rho + (1 - alpha))) * ces_output(k, z)\n", " \n", " return mpk" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "0.49999976499814425" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# confirm that ces_mpk is functioning correctly...\n", "alpha = 0.5\n", "sigma = 1e6\n", "ces_mpk(4, 0)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# equation 1.6\n", "def crra_utility(c):\n", " \"\"\"\n", " Agent has CRRA preferences over consumption. Function makes use of np.log \n", " and np.exp functions which are optimized for array arguments.\n", "\n", " Arguments:\n", " \n", " c: (array) Current value of consumption.\n", " \n", " Returns: (array-like) \n", " \n", " utility: (array) Utility from consumption.\n", " \n", " \"\"\"\n", " # nest log utility as a special case\n", " if theta == 1:\n", " utility = np.log(c) \n", " else:\n", " utility = (c**(1 - theta) - 1) / (1 - theta)\n", " \n", " return utility" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.9999974548238537" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# confirm that crra_utility is working!\n", "theta = 1e-6\n", "crra_utility(4.0)" ] }, { "cell_type": "code", "execution_count": 13, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# equation 1.11\n", "def capital_motion(k, z, c):\n", " \"\"\"\n", " Equation of motion for capital.\n", " \n", " Arguments:\n", " \n", " k: (array) Current value of capital. Inherited from previous period!\n", " z: (array) Current value of the productivity shock.\n", " c: (array) Current value of consumption.\n", " \n", " Returns:\n", " \n", " kplus: (array) Next period's value of capital.\n", " \n", " \"\"\"\n", " kplus = ces_output(k, z) + (1 - delta) * k - c\n", " return kplus" ] }, { "cell_type": "code", "execution_count": 14, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "1.0" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# confirm that capital_motion is functioning correctly...\n", "alpha = 0.5\n", "sigma = 1e0\n", "delta = 1.0\n", "capital_motion(4, 0, 1)" ] }, { "cell_type": "code", "execution_count": 15, "metadata": { "collapsed": false }, "outputs": [], "source": [ "# equation 1.14\n", "def Gamma(k, z=0.0):\n", " \"\"\"\n", " The correspondence of feasible controls given current state.\n", "\n", " Arguments:\n", " \n", " k: (array) Current value of capital. Inherited from previous period!\n", " z: (array) Current value of the productivity shock.\n", " \n", " Returns: \n", " \n", " c_upper: (scalar) The upper bound on the correspondence of feasible \n", " controls given current state.\n", " \n", " \"\"\"\n", " # note that we are allowing capital to be eaten!\n", " c_upper = ces_output(k, z) + (1 - delta) * k\n", " return c_upper" ] }, { "cell_type": "code", "execution_count": 16, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "2.4999995181380563" ] }, "execution_count": 16, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# confirm that Gamma is working\n", "alpha = 0.5\n", "sigma = 1e6\n", "delta = 1.0\n", "Gamma(4, 0)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Solving the model:\n", "\n", "From above, our objective is to find functions $c(k,z)$ and $W(k, z)$ that jointly solve the following system of functional equations.\n", "\n", "\\begin{align}\n", "W'(k, z) =& (1-\\beta)c(k, z)^{-\\theta}\\left(1 + \\alpha e^{z}k^{\\rho-1}\\bigg(\\alpha k^{\\rho} + (1 - \\alpha)\\bigg)^{\\frac{1}{\\rho}-1} - \\delta\\right) \\\\\n", "W(k, z) =& (1-\\beta)\\frac{c(k, z)^{1-\\theta}-1}{1-\\theta} + \\beta E\\left\\{W\\bigg((1-\\delta)k + e^{z}\\bigg(\\alpha k^{\\rho} + (1 - \\alpha)\\bigg)^{\\frac{1}{\\rho}} - c(k, z), \\rho_z z + \\epsilon')\\right\\}\n", "\\end{align}\n", "\n", "The *key* theorem in infinite horizon dynamic programming is the contraction mapping theorem applied to the Bellman equation. First, a couple of definitions:\n", "\n", "* A **map** $T:Y \\rightarrow Z\\ $ on ordered spaces $Y$ and $Z$ is **monotone** if and only if $y_1 \\ge y_2 \\implies Ty_1 \\ge Ty_2$.\n", "* A **map** $T:Y \\rightarrow Y\\ $ on a metric space $Y$ is a **contraction mapping** with modulus $\\beta \\lt 1$ if and only if $||Ty_1 - Ty_2|| \\le \\beta||y_1 - y_2||.$\n", "\n", "**Theorem:** If $X$ is compact (i.e., closed and bounded), $\\beta \\lt 1$, and $u$ is bounded, then the map\n", "\n", "$$ TW = \\max_{c \\in \\Gamma(k, z)} (1-\\beta)u(c) + \\beta E\\left\\{W(k', z')| z\\right\\} $$\n", "\n", "is monotone in $W$, and is a contraction mapping with modulus $\\beta$ in the space of bounded functions, and has a *unique* fixed point.\n", "\n", "Although the assumption of boundedness and compactness of the state space required for the theorem to hold seem restrictive, they are not! We care only about **numerically feasible** dynamic programming problems which in general can examine only compact state spaces. The contraction mapping theorem tells us how to constuct an operator, $T$, that transforms a value function into another value function and that repeated application of this operator generates a sequence of functions that eventually converges to a unique fixed point (i.e., the \"optimal\" value function)!\n", "\n", "Before I explain the basic algorithm that we will use to solve for the value function and its implied optimal policy, we need to define a couple of maps. The map $T$, which is called the Bellman operator, takes next period's value function, $V^+$, and maps it into the current value function, $W$ and is defined by\n", "\n", "$$ W = \\max_{c\\in\\Gamma(k, z)} (1-\\beta)u(c) + \\beta E\\left\\{W^+(k', z')| z\\right\\} \\equiv TW^+.$$ \n", "\n", "We also need to define the map $\\mathcal{U}$, called the \"greedy\" operator, which maps next period's value function into the current policy function, $U$:\n", "\n", "$$ U = \\arg \\max_{c\\in\\Gamma(k, z)} (1-\\beta)u(c) + \\beta E\\left\\{W^+(k', z')| z\\right\\} \\equiv \\mathcal{U}W^+.$$ \n", "\n", "## Value function iteration\n", "\n", "The most basic numerical procedure for computing $W^*$ is called *value function iteration* is directly motivated by the contraction mapping properties of the Bellman equation. Specifically, value function iteration computes a sequence of functions\n", "\n", "$$ W^{l+1} = TW^l,\\ l=0, 1, 2, \\dots .$$\n", "\n", "The contraction mapping theorem tells us that the sequence $W^l$ will converge to the infinite-horizon value function for *any* initial guess $W^0$. Additionally, the contraction mapping theorem implies that the sequence of policy functions, $U^l$, as defined by \n", "\n", "$$U^{l+1} = \\mathcal{U}U^l\\ l=0, 1, 2, \\dots$$ \n", "\n", "will also converge to the optimal policy function. In theory, convergence is achieved only asymptotically. In practice, we will just iterate the operator $T$ until successive value functions change very little (and/or successive policy functions do not change). Below is some pseudo-code for computing the function, $W$, using value function iteration.\n", "\n", " while True:\n", " # apply the Bellman operator, T\n", " next_w = T(current_w, **kwargs)\n", " \n", " # compare successive iterates using some distance metric\n", " change = np.max(np.abs(next_w(pts) - current_w(pts)))\n", " n_iter += 1\n", " \n", " # check for convergence\n", " if change < tol:\n", " if mesg == True:\n", " print \"After\", n_iter, \"iterations, the final change is\", change\n", " final_w = next_w\n", " break\n", " \n", " current_w = next_w\n", " \n", "Once the code above terminates, we can compute the optimal policy by applying the map $\\mathcal{U}$ to `final_w`. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Approximation of value and policy functions\n", "\n", "In order to implement value function iteration, we need to first choose a method for representing the functions used to approximate successive value and policy function iterates on the computer. Two widely used function approximation schemes are linear interpolation and cubic spline interpolation. " ] }, { "cell_type": "code", "execution_count": 17, "metadata": { "collapsed": false }, "outputs": [ { "data": { "image/png": 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p0yb+/vtv5syZw6BBg/LtswuRr+Li4MwZ9MFBRB47SGLgP9ievYhNeDRXPawI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\n6+IuOtfpnOWlEREQEAD16jkwn4iIiJ0KTid87BgkJxP1+Bk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