tax_smoothing_1.ipynb

{
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   "metadata": {},
   "source": [
    "\n",
    "<a id='tax-smoothing-1'></a>\n",
    "<div id=\"qe-notebook-header\" style=\"text-align:right;\">\n",
    "        <a href=\"https://quantecon.org/\" title=\"quantecon.org\">\n",
    "                <img style=\"width:250px;display:inline;\" src=\"https://assets.quantecon.org/img/qe-menubar-logo.svg\" alt=\"QuantEcon\">\n",
    "        </a>\n",
    "</div>\n",
    "\n",
    "\n",
    "<a id='index-0'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# How to Pay for a War: Part 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contents\n",
    "\n",
    "- [How to Pay for a War: Part 1](#How-to-Pay-for-a-War:-Part-1)  \n",
    "  - [Reader’s Guide](#Reader’s-Guide)  \n",
    "  - [Public Finance Questions](#Public-Finance-Questions)  \n",
    "  - [Barro (1979) Model](#Barro-%281979%29-Model)  \n",
    "  - [Python Class to Solve Markov Jump Linear Quadratic Control Problems](#Python-Class-to-Solve-Markov-Jump-Linear-Quadratic-Control-Problems)  \n",
    "  - [Barro Model with a Time-varying Interest Rate](#Barro-Model-with-a-Time-varying-Interest-Rate)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Co-author:** [Sebastian Graves](https://github.com/sebgraves)\n",
    "\n",
    "In addition to what’s in Anaconda, this lecture will need the following libraries:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": true
   },
   "outputs": [],
   "source": [
    "!pip install --upgrade quantecon"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Reader’s Guide\n",
    "\n",
    "Let’s start with some standard imports:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "import quantecon as qe\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This lecture uses the method of   **Markov jump linear quadratic dynamic programming** that is described in lecture\n",
    "[Markov Jump LQ dynamic programming](https://lectures.quantecon.org/markov_jump_lq.html)\n",
    "to extend the [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979) model of optimal tax-smoothing and government debt in a\n",
    "particular direction.\n",
    "\n",
    "This lecture has two sequels that offer further extensions of the Barro model\n",
    "\n",
    "> 1. [How to Pay for a War: Part 2](https://lectures.quantecon.org/tax_smoothing_2.html)  \n",
    "1. [How to Pay for a War: Part 3](https://lectures.quantecon.org/tax_smoothing_3.html)  \n",
    "\n",
    "\n",
    "\n",
    "The extensions are modified versions of\n",
    "his 1979 model later suggested by Barro (1999 [[Bar99]](https://lectures.quantecon.org/zreferences.html#barro1999determinants), 2003 [[BM03]](https://lectures.quantecon.org/zreferences.html#barro2003religion)).\n",
    "\n",
    "Barro’s original 1979 [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979) model is about a government that borrows and lends\n",
    "in order to minimize an intertemporal measure of distortions\n",
    "caused by taxes.\n",
    "\n",
    "Technical tractability induced Barro [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979) to assume that\n",
    "\n",
    "- the government trades only one-period risk-free debt, and  \n",
    "- the one-period risk-free interest rate is constant  \n",
    "\n",
    "\n",
    "By using  *Markov jump linear quadratic dynamic\n",
    "programming* we  can allow interest rates to move over time in\n",
    "empirically interesting ways.\n",
    "\n",
    "Also, by expanding the dimension of the\n",
    "state, we can add a maturity composition decision to the government’s\n",
    "problem.\n",
    "\n",
    "It is by doing these two things that we extend Barro’s 1979 [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979)\n",
    "model along lines he suggested in Barro (1999 [[Bar99]](https://lectures.quantecon.org/zreferences.html#barro1999determinants), 2003 [[BM03]](https://lectures.quantecon.org/zreferences.html#barro2003religion)).\n",
    "\n",
    "Barro (1979) [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979) assumed\n",
    "\n",
    "- that a government faces an **exogenous sequence** of expenditures\n",
    "  that it must finance by a tax collection sequence whose expected\n",
    "  present value equals the initial debt it owes plus the expected\n",
    "  present value of those expenditures.  \n",
    "- that the government wants to minimize the following measure of tax\n",
    "  distortions: $ E_0 \\sum_{t=0}^{\\infty} \\beta^t T_t^2 $, where $ T_t $ are total tax collections and $ E_0 $\n",
    "  is a mathematical expectation conditioned on time $ 0 $\n",
    "  information.  \n",
    "- that the government trades only one asset, a risk-free one-period\n",
    "  bond.  \n",
    "- that the gross interest rate on the one-period bond is constant and\n",
    "  equal to $ \\beta^{-1} $, the reciprocal of the factor\n",
    "  $ \\beta $ at which the government discounts future tax distortions.  \n",
    "\n",
    "\n",
    "Barro’s model can be mapped into a discounted linear quadratic dynamic\n",
    "programming problem.\n",
    "\n",
    "Partly inspired by Barro\n",
    "(1999) [[Bar99]](https://lectures.quantecon.org/zreferences.html#barro1999determinants) and Barro (2003) [[BM03]](https://lectures.quantecon.org/zreferences.html#barro2003religion),\n",
    "our generalizations of Barro’s (1979) [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979) model assume\n",
    "\n",
    "- that the government borrows or saves in the form of risk-free bonds\n",
    "  of maturities $ 1, 2, \\ldots , H $.  \n",
    "- that interest rates on those bonds are time-varying and in particular,\n",
    "  governed by a jointly stationary stochastic process.  \n",
    "\n",
    "\n",
    "Our generalizations are designed to fit within a generalization of an\n",
    "ordinary linear quadratic dynamic programming problem in which matrices\n",
    "that define the quadratic objective function and the state transition\n",
    "function are **time-varying** and **stochastic**.\n",
    "\n",
    "This generalization, known as a **Markov jump linear quadratic dynamic\n",
    "program**, combines\n",
    "\n",
    "- the computational simplicity of **linear quadratic dynamic\n",
    "  programming**, and  \n",
    "- the ability of **finite state Markov chains** to represent\n",
    "  interesting patterns of random variation.  \n",
    "\n",
    "\n",
    "We want the stochastic time variation in the matrices defining the\n",
    "dynamic programming problem to represent variation over time in\n",
    "\n",
    "- interest rates  \n",
    "- default rates  \n",
    "- roll over risks  \n",
    "\n",
    "\n",
    "As described in [Markov Jump LQ dynamic programming](https://lectures.quantecon.org/markov_jump_lq.html),\n",
    "the idea underlying **Markov jump linear quadratic dynamic programming**\n",
    "is to replace the constant matrices defining a **linear quadratic\n",
    "dynamic programming problem** with matrices that are fixed functions of\n",
    "an $ N $ state Markov chain.\n",
    "\n",
    "For infinite horizon problems, this leads to $ N $ interrelated\n",
    "matrix Riccati equations that pin down $ N $ value functions and\n",
    "$ N $ linear decision rules, applying to the $ N $ Markov\n",
    "states."
   ]
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   "source": [
    "## Public Finance Questions\n",
    "\n",
    "Barro’s 1979 [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979) model is designed to answer questions such as\n",
    "\n",
    "- Should a government finance an exogenous surge in government\n",
    "  expenditures by raising taxes or borrowing?  \n",
    "- How does the answer to that first question depend on the exogenous\n",
    "  stochastic process for government expenditures, for example, on\n",
    "  whether the surge in government expenditures can be expected to be\n",
    "  temporary or permanent?  \n",
    "\n",
    "\n",
    "Barro’s 1999 [[Bar99]](https://lectures.quantecon.org/zreferences.html#barro1999determinants) and 2003 [[BM03]](https://lectures.quantecon.org/zreferences.html#barro2003religion)\n",
    "models are designed to answer more fine-grained\n",
    "questions such as\n",
    "\n",
    "- What determines whether a government wants to issue short-term or\n",
    "  long-term debt?  \n",
    "- How do roll-over risks affect that decision?  \n",
    "- How does the government’s long-short *portfolio management* decision\n",
    "  depend on features of the exogenous stochastic process for government\n",
    "  expenditures?  \n",
    "\n",
    "\n",
    "Thus, both the simple and the more fine-grained versions of Barro’s\n",
    "models are ways of precisely formulating the classic issue of *How to\n",
    "pay for a war*.\n",
    "\n",
    "This lecture describes:\n",
    "\n",
    "- An application of Markov jump LQ dynamic programming to a model in\n",
    "  which a government faces exogenous time-varying interest rates for\n",
    "  issuing one-period risk-free debt.  \n",
    "\n",
    "\n",
    "A [sequel to this\n",
    "lecture](https://lectures.quantecon.org/tax_smoothing_2.html)\n",
    "describes applies Markov LQ control to settings in which a government\n",
    "issues risk-free debt of different maturities."
   ]
  },
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   "metadata": {},
   "source": [
    "## Barro (1979) Model\n",
    "\n",
    "We begin by solving a version of the Barro (1979) [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979) model by mapping it\n",
    "into the original LQ framework.\n",
    "\n",
    "As mentioned [in this lecture](https://lectures.quantecon.org/perm_income_cons.html), the\n",
    "Barro model is mathematically isomorphic with the LQ permanent income\n",
    "model.\n",
    "\n",
    "Let $ T_t $ denote tax collections, $ \\beta $ a discount factor,\n",
    "$ b_{t,t+1} $ time $ t+1 $ goods that the government promises to\n",
    "pay at $ t $, $ G_t $ government purchases, $ p_{t,t+1} $\n",
    "the number of time $ t $ goods received per time $ t+1 $ goods\n",
    "promised.\n",
    "\n",
    "Evidently, $ p_{t, t+1} $ is inversely related to\n",
    "appropriate corresponding gross interest rates on government debt.\n",
    "\n",
    "In the spirit of Barro (1979) [[Bar79]](https://lectures.quantecon.org/zreferences.html#barro1979), the stochastic process of government\n",
    "expenditures is exogenous.\n",
    "\n",
    "The government’s problem is to choose a plan\n",
    "for taxation and borrowing $ \\{b_{t+1}, T_t\\}_{t=0}^\\infty $ to\n",
    "minimize\n",
    "\n",
    "$$\n",
    "E_0 \\sum_{t=0}^\\infty \\beta^t T_t^2\n",
    "$$\n",
    "\n",
    "subject to the constraints\n",
    "\n",
    "$$\n",
    "T_t + p_{t,t+1} b_{t,t+1} = G_t + b_{t-1,t}\n",
    "$$\n",
    "\n",
    "$$\n",
    "G_t = U_{g,t} z_t\n",
    "$$\n",
    "\n",
    "$$\n",
    "z_{t+1} = A_{22,t} z_t + C_{2,t} w_{t+1}\n",
    "$$\n",
    "\n",
    "where $ w_{t+1} \\sim {\\cal N}(0,I) $\n",
    "\n",
    "The variables\n",
    "$ T_t, b_{t, t+1} $ are *control* variables chosen at $ t $,\n",
    "while $ b_{t-1,t} $ is an endogenous state variable inherited from\n",
    "the past at time $ t $ and $ p_{t,t+1} $ is an exogenous state\n",
    "variable at time $ t $.\n",
    "\n",
    "To begin, we  assume that\n",
    "$ p_{t,t+1} $ is constant (and equal to $ \\beta $)\n",
    "\n",
    "> - later  we will  extend the model to allow $ p_{t,t+1} $ to vary over time  \n",
    "\n",
    "\n",
    "\n",
    "To map into the LQ framework, we use\n",
    "$ x_t = \\begin{bmatrix} b_{t-1,t} \\\\ z_t \\end{bmatrix} $ as the\n",
    "state vector, and $ u_t = b_{t,t+1} $ as the control variable.\n",
    "\n",
    "Therefore, the $ (A, B, C) $ matrices are defined by the state-transition law:\n",
    "\n",
    "$$\n",
    "x_{t+1} = \\begin{bmatrix} 0 & 0 \\\\ 0 & A_{22} \\end{bmatrix} x_t + \\begin{bmatrix} 1 \\\\ 0 \\end{bmatrix} u_t + \\begin{bmatrix} 0 \\\\ C_2 \\end{bmatrix} w_{t+1}\n",
    "$$\n",
    "\n",
    "To find the appropriate $ (R, Q, W) $ matrices, we note that $ G_t $ and\n",
    "$ b_{t-1,t} $ can be written as appropriately defined functions of\n",
    "the current state:\n",
    "\n",
    "$$\n",
    "G_t = S_G x_t \\hspace{2mm}, \\hspace{2mm} b_{t-1,t} = S_1 x_t\n",
    "$$\n",
    "\n",
    "If we define $ M_t = - p_{t,t+1} $, and let $ S = S_G + S_1 $, then\n",
    "we can write taxation as a function of the states and control using the\n",
    "government’s budget constraint:\n",
    "\n",
    "$$\n",
    "T_t = S x_t + M_t u_t\n",
    "$$\n",
    "\n",
    "It follows that the $ (R, Q, W) $ matrices are implicitly defined by:\n",
    "\n",
    "$$\n",
    "T_t^2 = x_t'S'Sx_t + u_t'M_t'M_tu_t + 2 u_t'M_t'S x_t\n",
    "$$\n",
    "\n",
    "If we assume that $ p_{t,t+1} = \\beta $, then $ M_t \\equiv M = -\\beta $.\n",
    "\n",
    "In this case, none of\n",
    "the LQ matrices are time varying, and we can use the original LQ\n",
    "framework.\n",
    "\n",
    "We will implement this constant interest-rate version first, assuming\n",
    "that $ G_t $ follows an AR(1) process:\n",
    "\n",
    "$$\n",
    "G_{t+1} = \\bar G + \\rho G_t + \\sigma w_{t+1}\n",
    "$$\n",
    "\n",
    "To do this, we set\n",
    "$ z_t = \\begin{bmatrix} 1 \\\\ G_t \\end{bmatrix} $, and consequently:\n",
    "\n",
    "$$\n",
    "A_{22} = \\begin{bmatrix} 1 & 0 \\\\ \\bar G & \\rho \\end{bmatrix} \\hspace{2mm} ,\n",
    "\\hspace{2mm} C_2 = \\begin{bmatrix} 0 \\\\ \\sigma \\end{bmatrix}\n",
    "$$"
   ]
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   "source": [
    "# Model parameters\n",
    "β, Gbar, ρ, σ = 0.95, 5, 0.8, 1\n",
    "\n",
    "# Basic model matrices\n",
    "A22 = np.array([[1,    0],\n",
    "                [Gbar, ρ],])\n",
    "\n",
    "C2 = np.array([[0],\n",
    "               [σ]])\n",
    "\n",
    "Ug = np.array([[0, 1]])\n",
    "\n",
    "# LQ framework matrices\n",
    "A_t = np.zeros((1, 3))\n",
    "A_b = np.hstack((np.zeros((2, 1)), A22))\n",
    "A = np.vstack((A_t, A_b))\n",
    "\n",
    "B = np.zeros((3, 1))\n",
    "B[0, 0] = 1\n",
    "\n",
    "C = np.vstack((np.zeros((1, 1)), C2))\n",
    "\n",
    "Sg = np.hstack((np.zeros((1, 1)), Ug))\n",
    "S1 = np.zeros((1, 3))\n",
    "S1[0, 0] = 1\n",
    "S = S1 + Sg\n",
    "\n",
    "M = np.array([[-β]])\n",
    "\n",
    "R = S.T @ S\n",
    "Q = M.T @ M\n",
    "W = M.T @ S\n",
    "\n",
    "# Small penalty on the debt required to implement the no-Ponzi scheme\n",
    "R[0, 0] = R[0, 0] + 1e-9"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can now create an instance of `LQ`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "LQBarro = qe.LQ(Q, R, A, B, C=C, N=W, beta=β)\n",
    "P, F, d = LQBarro.stationary_values()\n",
    "x0 = np.array([[100, 1, 25]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see the isomorphism by noting that consumption is a martingale in\n",
    "the permanent income model and that taxation is a martingale in Barro’s\n",
    "model.\n",
    "\n",
    "We can check this using the $ F $ matrix of the LQ model.\n",
    "\n",
    "Because $ u_t = -F x_t $, we have\n",
    "\n",
    "$$\n",
    "T_t = S x_t + M u_t = (S - MF) x_t\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "T_{t+1} = (S-MF)x_{t+1} = (S-MF)(Ax_t + B u_t + C w_{t+1}) = (S-MF)((A-BF)x_t + C w_{t+1})\n",
    "$$\n",
    "\n",
    "Therefore, the conditional expectation of $ T_{t+1} $ at time\n",
    "$ t $ is\n",
    "\n",
    "$$\n",
    "E_t T_{t+1} = (S-MF)(A-BF)x_t\n",
    "$$\n",
    "\n",
    "Consequently, taxation is a martingale ($ E_t T_{t+1} = T_t $) if\n",
    "\n",
    "$$\n",
    "(S-MF)(A-BF) = (S-MF) ,\n",
    "$$\n",
    "\n",
    "which holds in this case:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "S - M @ F, (S - M @ F) @ (A - B @ F)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This explains the gradual fanning out of taxation if we simulate the\n",
    "Barro model a large number of times:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "T = 500\n",
    "for i in range(250):\n",
    "    x, u, w = LQBarro.compute_sequence(x0, ts_length=T)\n",
    "    plt.plot(list(range(T+1)), ((S - M @ F) @ x)[0, :])\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Taxation')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see a similar, but a smoother pattern, if we plot government debt\n",
    "over time.\n",
    "\n",
    "Debt is smoother due to the persistence of the government\n",
    "spending process."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "T = 500\n",
    "for i in range(250):\n",
    "    x, u, w = LQBarro.compute_sequence(x0, ts_length=T)\n",
    "    plt.plot(list(range(T+1)), x[0, :])\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Taxation')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Python Class to Solve Markov Jump Linear Quadratic Control Problems\n",
    "\n",
    "To implement the extension to the Barro model in which $ p_{t,t+1} $\n",
    "varies over time, we must allow the M matrix to be time-varying.\n",
    "\n",
    "Our $ Q $ and $ W $ matrices must also vary over time.\n",
    "\n",
    "We can solve such a\n",
    "model using the `LQMarkov` class that solves Markov jump linear\n",
    "quandratic control problems as described above.\n",
    "\n",
    "The code for the class can be viewed\n",
    "[here](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lqcontrol.py#L334).\n",
    "\n",
    "The class takes lists of matrices that corresponds to $ N $ Markov states.\n",
    "\n",
    "The value and policy functions are then found by iterating on the system of\n",
    "algebraic matrix Riccati equations.\n",
    "\n",
    "The solutions for $ Ps,Fs,ds $ are stored as attributes.\n",
    "\n",
    "The class also contains a “method” for simulating the model."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Barro Model with a Time-varying Interest Rate\n",
    "\n",
    "We can use the above class to implement a version of the Barro model\n",
    "with a time-varying interest rate. The simplest way to extend the model\n",
    "is to allow the interest rate to take two possible values. We set:\n",
    "\n",
    "$$\n",
    "p^1_{t,t+1} = \\beta + 0.02 = 0.97\n",
    "$$\n",
    "\n",
    "$$\n",
    "p^2_{t,t+1} = \\beta - 0.017 = 0.933\n",
    "$$\n",
    "\n",
    "Thus, the first Markov state  has a low-interest rate, and the\n",
    "second Markov state has a high-interest rate.\n",
    "\n",
    "We also need to specify a transition matrix for the Markov state.\n",
    "\n",
    "We use:\n",
    "\n",
    "$$\n",
    "\\Pi = \\begin{bmatrix} 0.8 & 0.2 \\\\ 0.2 & 0.8 \\end{bmatrix}\n",
    "$$\n",
    "\n",
    "(so each Markov state is persistent, and there is an equal chance\n",
    "of moving from one state to the other)\n",
    "\n",
    "The choice of parameters means that the unconditional expectation of\n",
    "$ p_{t,t+1} $ is 0.9515, higher than $ \\beta (=0.95) $.\n",
    "\n",
    "If we\n",
    "were to set $ p_{t,t+1} = 0.9515 $ in the version of the model with\n",
    "a constant interest rate, government debt would explode."
   ]
  },
  {
   "cell_type": "code",
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   "source": [
    "# Create list of matrices that corresponds to each Markov state\n",
    "Π = np.array([[0.8, 0.2],\n",
    "              [0.2, 0.8]])\n",
    "\n",
    "As = [A, A]\n",
    "Bs = [B, B]\n",
    "Cs = [C, C]\n",
    "Rs = [R, R]\n",
    "\n",
    "M1 = np.array([[-β - 0.02]])\n",
    "M2 = np.array([[-β + 0.017]])\n",
    "\n",
    "Q1 = M1.T @ M1\n",
    "Q2 = M2.T @ M2\n",
    "Qs = [Q1, Q2]\n",
    "W1 = M1.T @ S\n",
    "W2 = M2.T @ S\n",
    "Ws = [W1, W2]\n",
    "\n",
    "# create Markov Jump LQ DP problem instance\n",
    "lqm = qe.LQMarkov(Π, Qs, Rs, As, Bs, Cs=Cs, Ns=Ws, beta=β)\n",
    "lqm.stationary_values();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The decision rules are now dependent on the Markov state:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "lqm.Fs[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "lqm.Fs[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Simulating a large number of such economies over time reveals\n",
    "interesting dynamics.\n",
    "\n",
    "Debt tends to stay low and stable but\n",
    "recurrently surges temporarily to higher levels."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "T = 2000\n",
    "x0 = np.array([[1000, 1, 25]])\n",
    "for i in range(250):\n",
    "    x, u, w, s = lqm.compute_sequence(x0, ts_length=T)\n",
    "    plt.plot(list(range(T+1)), x[0, :])\n",
    "plt.xlabel('Time')\n",
    "plt.ylabel('Taxation')\n",
    "plt.show()"
   ]
  }
 ],
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  "filename": "tax_smoothing_1.rst",
  "kernelspec": {
   "display_name": "Python",
   "language": "python3",
   "name": "python3"
  },
  "title": "How to Pay for a War: Part 1"
 },
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}