multi_agent_models/matsuyama.ipynb

{
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   "source": [
    "\n",
    "<a id='matsuyama'></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>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Globalization and Cycles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Contents\n",
    "\n",
    "- [Globalization and Cycles](#Globalization-and-Cycles)  \n",
    "  - [Overview](#Overview)  \n",
    "  - [Key Ideas](#Key-Ideas)  \n",
    "  - [Model](#Model)  \n",
    "  - [Simulation](#Simulation)  \n",
    "  - [Exercises](#Exercises)  \n",
    "  - [Solutions](#Solutions)  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Co-authored with Chase Coleman"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Overview\n",
    "\n",
    "In this lecture, we review the paper [Globalization and Synchronization of Innovation Cycles](http://www.centreformacroeconomics.ac.uk/Discussion-Papers/2015/CFMDP2015-27-Paper.pdf) by [Kiminori Matsuyama](http://faculty.wcas.northwestern.edu/~kmatsu/), [Laura Gardini](http://www.mdef.it/index.php?id=32) and [Iryna Sushko](http://irynasushko.altervista.org/)\n",
    "\n",
    "This model helps us understand several interesting stylized facts about the world economy\n",
    "\n",
    "One of these is synchronized business cycles across different countries\n",
    "\n",
    "Most existing models that generate synchronized business cycles do so by assumption, since they tie output in each country to a common shock\n",
    "\n",
    "They also fail to explain certain features of the data, such as the fact that the degree of synchronization tends to increase with trade ties\n",
    "\n",
    "By contrast, in the model we consider in this lecture, synchronization is both endogenous and increasing with the extent of trade integration\n",
    "\n",
    "In particular, as trade costs fall and international competition increases, innovation incentives become aligned and countries synchronize their innovation cycles"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Background\n",
    "\n",
    "The model builds on work by Judd [[Jud85]](zreferences.ipynb#judd1985), Deneckner and Judd [[DJ92]](zreferences.ipynb#deneckere1992) and Helpman and Krugman [[HK85]](zreferences.ipynb#helpman1985) by developing a two country model with trade and innovation\n",
    "\n",
    "On the technical side, the paper introduces the concept of [coupled oscillators](https://en.wikipedia.org/wiki/Oscillation#Coupled_oscillations) to economic modeling\n",
    "\n",
    "As we will see, coupled oscillators arise endogenously within the model\n",
    "\n",
    "Below we review the model and replicate some of the results on synchronization of innovation across countries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Key Ideas\n",
    "\n",
    "It is helpful to begin with an overview of the mechanism"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Innovation Cycles\n",
    "\n",
    "As discussed above, two countries produce and trade with each other\n",
    "\n",
    "In each country, firms innovate, producing new varieties of goods and, in doing so, receiving temporary monopoly power\n",
    "\n",
    "Imitators follow and, after one period of monopoly, what had previously been new varieties now enter competitive production\n",
    "\n",
    "Firms have incentives to innovate and produce new goods when the mass of varieties of goods currently in production is relatively low\n",
    "\n",
    "In addition, there are strategic complementarities in the timing of innovation\n",
    "\n",
    "Firms have incentives to innovate in the same period, so as to avoid competing with substitutes that are competitively produced\n",
    "\n",
    "This leads to temporal clustering in innovations in each country\n",
    "\n",
    "After a burst of innovation, the mass of goods currently in production increases\n",
    "\n",
    "However, goods also become obsolete, so that not all survive from period to period\n",
    "\n",
    "This mechanism generates a cycle, where the mass of varieties increases through simultaneous innovation and then falls through obsolescence"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Synchronization\n",
    "\n",
    "In the absence of trade, the timing of innovation cycles in each country is decoupled\n",
    "\n",
    "This will be the case when trade costs are prohibitively high\n",
    "\n",
    "If trade costs fall, then goods produced in each country penetrate each other’s markets\n",
    "\n",
    "As illustrated below, this leads to synchonization of business cycles across the two countries"
   ]
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   "source": [
    "## Model\n",
    "\n",
    "Let’s write down the model more formally\n",
    "\n",
    "(The treatment is relatively terse since full details can be found in [the original paper](http://www.centreformacroeconomics.ac.uk/Discussion-Papers/2015/CFMDP2015-27-Paper.pdf))\n",
    "\n",
    "Time is discrete with $ t = 0, 1, \\dots $\n",
    "\n",
    "There are two countries indexed by $ j $ or $ k $\n",
    "\n",
    "In each country, a representative household inelastically supplies $ L_j $ units of labor at wage rate $ w_{j, t} $\n",
    "\n",
    "Without loss of generality, it is assumed that $ L_{1} \\geq L_{2} $\n",
    "\n",
    "Households consume a single nontradeable final good which is produced competitively\n",
    "\n",
    "Its production involves combining two types of tradeable intermediate inputs\n",
    "via\n",
    "\n",
    "$$\n",
    "Y_{k, t} = C_{k, t} = \\left( \\frac{X^o_{k, t}}{1 - \\alpha} \\right)^{1-\\alpha} \\left( \\frac{X_{k, t}}{\\alpha} \\right)^{\\alpha}\n",
    "$$\n",
    "\n",
    "Here $ X^o_{k, t} $ is a homogeneous input which can be produced from labor using a linear, one-for-one technology\n",
    "\n",
    "It is freely tradeable, competitively supplied, and homogeneous across countries\n",
    "\n",
    "By choosing the price of this good as numeraire and assuming both countries find it optimal to always produce the homogeneous good, we can set $ w_{1, t} = w_{2, t} = 1 $\n",
    "\n",
    "The good $ X_{k, t} $ is a composite, built from many differentiated goods via\n",
    "\n",
    "$$\n",
    "X_{k, t}^{1 - \\frac{1}{\\sigma}} = \\int_{\\Omega_t} \\left[ x_{k, t}(\\nu) \\right]^{1 - \\frac{1}{\\sigma}} d \\nu\n",
    "$$\n",
    "\n",
    "Here $ x_{k, t}(\\nu) $ is the total amount of a differentiated good $ \\nu \\in \\Omega_t $ that is produced\n",
    "\n",
    "The parameter $ \\sigma > 1 $ is the direct partial elasticity of substitution between a pair of varieties and $ \\Omega_t $ is the set of varieties available in period\n",
    "$ t $\n",
    "\n",
    "We can split the varieties into those which are supplied competitively and those supplied monopolistically; that is, $ \\Omega_t = \\Omega_t^c + \\Omega_t^m $"
   ]
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   "source": [
    "### Prices\n",
    "\n",
    "Demand for differentiated inputs is\n",
    "\n",
    "$$\n",
    "x_{k, t}(\\nu) = \\left( \\frac{p_{k, t}(\\nu)}{P_{k, t}} \\right)^{-\\sigma} \\frac{\\alpha  L_k}{P_{k, t}}\n",
    "$$\n",
    "\n",
    "Here\n",
    "\n",
    "- $ p_{k, t}(\\nu) $ is the price of the variety $ \\nu $ and  \n",
    "- $ P_{k, t} $ is the price index for differentiated inputs in $ k $,\n",
    "  defined by  \n",
    "\n",
    "\n",
    "$$\n",
    "\\left[ P_{k, t} \\right]^{1 - \\sigma} = \\int_{\\Omega_t} [p_{k, t}(\\nu) ]^{1-\\sigma} d\\nu\n",
    "$$\n",
    "\n",
    "The price of a variety also depends on the origin, $ j $, and destination, $ k $, of the goods because shipping\n",
    "varieties between countries incurs an iceberg trade cost\n",
    "$ \\tau_{j,k} $\n",
    "\n",
    "Thus the effective price in country $ k $ of a variety $ \\nu $ produced in country $ j $ becomes $ p_{k, t}(\\nu) = \\tau_{j,k} \\, p_{j, t}(\\nu) $\n",
    "\n",
    "Using these expressions, we can derive the total demand for each variety,\n",
    "which is\n",
    "\n",
    "$$\n",
    "D_{j, t}(\\nu) = \\sum_k \\tau_{j, k} x_{k, t}(\\nu) = \\alpha A_{j, t}(p_{j, t}(\\nu))^{-\\sigma}\n",
    "$$\n",
    "\n",
    "where\n",
    "\n",
    "$$\n",
    "A_{j, t} := \\sum_k \\frac{\\rho_{j, k}  L_{k}}{(P_{k, t})^{1 - \\sigma}}\n",
    "\\quad \\text{and} \\quad\n",
    "\\rho_{j, k} = (\\tau_{j, k})^{1 - \\sigma} \\leq 1\n",
    "$$\n",
    "\n",
    "It is assumed that $ \\tau_{1,1} = \\tau_{2,2} = 1 $ and $ \\tau_{1,2} = \\tau_{2,1} = \\tau $ for some $ \\tau > 1 $, so that\n",
    "\n",
    "$$\n",
    "\\rho_{1,2} = \\rho_{2,1} = \\rho := \\tau^{1 - \\sigma} < 1\n",
    "$$\n",
    "\n",
    "The value $ \\rho \\in [0, 1) $ is a proxy for the degree of globalization\n",
    "\n",
    "Producing one unit of each differentiated variety requires $ \\psi $ units of labor, so the marginal cost is equal to $ \\psi $ for $ \\nu \\in \\Omega_{j, t} $\n",
    "\n",
    "Additionally, all competitive varieties will have the same price (because of equal marginal cost), which means that, for all $ \\nu \\in \\Omega^c $,\n",
    "\n",
    "$$\n",
    "p_{j, t}(\\nu) = p_{j, t}^c := \\psi\n",
    "\\quad \\text{and} \\quad\n",
    "D_{j, t} = y_{j, t}^c := \\alpha A_{j, t} (p_{j, t}^c)^{-\\sigma}\n",
    "$$\n",
    "\n",
    "Monopolists will have the same marked-up price, so, for all $ \\nu \\in \\Omega^m $ ,\n",
    "\n",
    "$$\n",
    "p_{j, t}(\\nu) = p_{j, t}^m := \\frac{\\psi }{1 - \\frac{1}{\\sigma}}\n",
    "\\quad \\text{and} \\quad\n",
    "D_{j, t} = y_{j, t}^m  := \\alpha A_{j, t} (p_{j, t}^m)^{-\\sigma}\n",
    "$$\n",
    "\n",
    "Define\n",
    "\n",
    "$$\n",
    "\\theta\n",
    ":= \\frac{p_{j, t}^c}{p_{j, t}^m} \\frac{y_{j, t}^c}{y_{j, t}^m}\n",
    "= \\left(1 - \\frac{1}{\\sigma} \\right)^{1-\\sigma}\n",
    "$$\n",
    "\n",
    "Using the preceding definitions and some algebra, the price indices can now be rewritten as\n",
    "\n",
    "$$\n",
    "\\left(\\frac{P_{k,t}}{\\psi}\\right)^{1-\\sigma} = M_{k,t}  + \\rho M_{j,t}\n",
    "\\quad \\text{where} \\quad\n",
    "M_{j,t} := N_{j,t}^c + \\frac{N_{j,t}^m}{ \\theta}\n",
    "$$\n",
    "\n",
    "The symbols $ N_{j, t}^c $ and $ N_{j, t}^m $ will denote the measures of $ \\Omega^c $ and $ \\Omega^m $ respectively"
   ]
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   "source": [
    "### New Varieties\n",
    "\n",
    "To introduce a new variety, a firm must hire $ f $ units of labor per variety in each country\n",
    "\n",
    "Monopolist profits must be less than or equal to zero in expectation, so\n",
    "\n",
    "$$\n",
    "N_{j,t}^m \\geq 0, \\quad\n",
    "\\pi_{j, t}^m := (p_{j, t}^m - \\psi) y_{j, t}^m - f \\leq 0\n",
    "\\quad \\text{and} \\quad\n",
    "\\pi_{j, t}^m N_{j,t}^m = 0\n",
    "$$\n",
    "\n",
    "With further manipulations, this becomes\n",
    "\n",
    "$$\n",
    "N_{j,t}^m = \\theta(M_{j,t} - N_{j,t}^c) \\geq 0,\n",
    "\\quad\n",
    "\\frac{1}{\\sigma}\n",
    "\\left[\n",
    "    \\frac{\\alpha L_j}{\\theta(M_{j,t} + \\rho M_{k,t})}\n",
    "    +\n",
    "    \\frac{\\alpha L_k}{\\theta(M_{j,t} + M_{k,t} / \\rho)}\n",
    "\\right]\n",
    "\\leq f\n",
    "$$"
   ]
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   "metadata": {},
   "source": [
    "### Law of Motion\n",
    "\n",
    "With $ \\delta $ as the exogenous probability of a variety becoming obsolete,\n",
    "the dynamic equation for the measure of firms becomes\n",
    "\n",
    "$$\n",
    "N_{j, t+1}^c = \\delta (N_{j, t}^c + N_{j, t}^m) = \\delta (N_{j, t}^c + \\theta(M_{j, t} - N_{j, t}^c))\n",
    "$$\n",
    "\n",
    "We will work with a normalized measure of varieties\n",
    "\n",
    "$$\n",
    "n_{j, t} := \\frac{\\theta \\sigma f N_{j, t}^c}{\\alpha (L_1 + L_2)},\n",
    "\\quad\n",
    "i_{j, t} := \\frac{\\theta \\sigma f N_{j, t}^m}{\\alpha (L_1 + L_2)},\n",
    "\\quad\n",
    "m_{j, t} := \\frac{\\theta \\sigma f M_{j, t}}{\\alpha (L_1 + L_2)} = n_{j, t} + \\frac{i_{j, t}}{\\theta}\n",
    "$$\n",
    "\n",
    "We also use $ s_j := \\frac{L_j}{L_1 + L_2} $ to be the share of labor employed in country $ j $\n",
    "\n",
    "We can use these definitions and the preceding expressions to obtain a law of\n",
    "motion for $ n_t := (n_{1, t}, n_{2, t}) $\n",
    "\n",
    "In particular, given an initial condition, $ n_0 = (n_{1, 0}, n_{2, 0}) \\in \\mathbb{R}_{+}^{2} $, the equilibrium trajectory, $ \\{ n_t \\}_{t=0}^{\\infty} = \\{ (n_{1, t}, n_{2, t}) \\}_{t=0}^{\\infty} $, is obtained by iterating on $ n_{t+1} = F(n_t) $ where $ F : \\mathbb{R}_{+}^{2} \\rightarrow \\mathbb{R}_{+}^{2} $ is given by\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "  F(n_t)\n",
    "  &=\n",
    "  \\begin{cases}\n",
    "      \\big( \\delta (\\theta s_1(\\rho) + (1-\\theta) n_{1, t}), \\delta (\\theta s_2(\\rho) + (1-\\theta) n_{2, t}) \\big) \\; & \\text{for } n_t \\in D_{LL} \\\\\n",
    "      \\big( \\delta n_{1, t}, \\delta n_{2, t} \\big) \\; &\\text{for } n_t \\in D_{HH}  \\\\\n",
    "      \\big( \\delta n_{1, t}, \\delta (\\theta h_2(n_{1, t}) + (1-\\theta) n_{2, t}) \\big) &\\text{for } n_t \\in D_{HL}  \\\\\n",
    "      \\big( \\delta (\\theta h_1(n_{2, t}) + (1-\\theta) n_{1, t}, \\delta n_{2, t}) \\big) &\\text{for } n_t \\in D_{LH}\n",
    "  \\end{cases}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "Here\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "      D_{LL} & := \\{ (n_1, n_2) \\in \\mathbb{R}_{+}^{2} | n_j \\leq s_j(\\rho) \\} \\\\\n",
    "      D_{HH} & := \\{ (n_1, n_2) \\in \\mathbb{R}_{+}^{2} | n_j \\geq h_j(\\rho) \\} \\\\\n",
    "      D_{HL} & :=  \\{ (n_1, n_2) \\in \\mathbb{R}_{+}^{2} | n_1 \\geq s_1(\\rho) \\text{ and } n_2 \\leq h_2(n_1) \\} \\\\\n",
    "      D_{LH} & :=  \\{ (n_1, n_2) \\in \\mathbb{R}_{+}^{2} | n_1 \\leq h_1(n_2) \\text{ and } n_2 \\geq s_2(\\rho) \\}\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "while\n",
    "\n",
    "$$\n",
    "s_1(\\rho) = 1 - s_2(\\rho)\n",
    "= \\min \\left\\{ \\frac{s_1 - \\rho s_2}{1 - \\rho}, 1 \\right\\}\n",
    "$$\n",
    "\n",
    "and $ h_j(n_k) $ is defined implicitly by the equation\n",
    "\n",
    "$$\n",
    "1 = \\frac{s_j}{h_j(n_k) + \\rho n_k} + \\frac{s_k}{h_j(n_k) + n_k / \\rho}\n",
    "$$\n",
    "\n",
    "Rewriting the equation above gives us a quadratic equation in terms of $ h_j(n_k) $\n",
    "\n",
    "Since we know $ h_j(n_k) > 0 $ then we can just solve the quadratic equation and return the positive root\n",
    "\n",
    "This gives us\n",
    "\n",
    "$$\n",
    "h_j(n_k)^2 + \\left( (\\rho + \\frac{1}{\\rho}) n_k - s_j - s_k \\right) h_j(n_k) + (n_k^2 - \\frac{s_j n_k}{\\rho} - s_k n_k \\rho) = 0\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulation\n",
    "\n",
    "Let’s try simulating some of these trajectories\n",
    "\n",
    "We will focus in particular on whether or not innovation cycles synchronize\n",
    "across the two countries\n",
    "\n",
    "As we will see, this depends on initial conditions\n",
    "\n",
    "For some parameterizations, synchronization will occur for “most” initial conditions, while for others synchronization will be rare\n",
    "\n",
    "Here’s the main body of code"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Setup"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": true
   },
   "outputs": [],
   "source": [
    "using InstantiateFromURL\n",
    "github_project(\"QuantEcon/quantecon-notebooks-julia\", version = \"0.2.0\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "using LinearAlgebra, Statistics, Compat\n",
    "using Plots, Parameters\n",
    "gr(fmt = :png);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "function h_j(j, nk, s1, s2, θ, δ, ρ)\n",
    "    # Find out who's h we are evaluating\n",
    "    if j == 1\n",
    "        sj = s1\n",
    "        sk = s2\n",
    "    else\n",
    "        sj = s2\n",
    "        sk = s1\n",
    "    end\n",
    "\n",
    "    # Coefficients on the quadratic a x^2 + b x + c = 0\n",
    "    a = 1.0\n",
    "    b = ((ρ + 1 / ρ) * nk - sj - sk)\n",
    "    c = (nk * nk - (sj * nk) / ρ - sk * ρ * nk)\n",
    "\n",
    "    # Positive solution of quadratic form\n",
    "    root = (-b + sqrt(b * b - 4 * a * c)) / (2 * a)\n",
    "\n",
    "    return root\n",
    "end\n",
    "\n",
    "DLL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =\n",
    "    (n1 ≤ s1_ρ) && (n2 ≤ s2_ρ)\n",
    "\n",
    "DHH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =\n",
    "    (n1 ≥ h_j(1, n2, s1, s2, θ, δ, ρ)) && (n2 ≥ h_j(2, n1, s1, s2, θ, δ, ρ))\n",
    "\n",
    "DHL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =\n",
    "    (n1 ≥ s1_ρ) && (n2 ≤ h_j(2, n1, s1, s2, θ, δ, ρ))\n",
    "\n",
    "DLH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =\n",
    "    (n1 ≤ h_j(1, n2, s1, s2, θ, δ, ρ)) && (n2 ≥ s2_ρ)\n",
    "\n",
    "function one_step(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "    # Depending on where we are, evaluate the right branch\n",
    "    if DLL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "        n1_tp1 = δ * (θ * s1_ρ + (1 - θ) * n1)\n",
    "        n2_tp1 = δ * (θ * s2_ρ + (1 - θ) * n2)\n",
    "    elseif DHH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "        n1_tp1 = δ * n1\n",
    "        n2_tp1 = δ * n2\n",
    "    elseif DHL(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "        n1_tp1 = δ * n1\n",
    "        n2_tp1 = δ * (θ * h_j(2, n1, s1, s2, θ, δ, ρ) + (1 - θ) * n2)\n",
    "    elseif DLH(n1, n2, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "        n1_tp1 = δ * (θ * h_j(1, n2, s1, s2, θ, δ, ρ) + (1 - θ) * n1)\n",
    "        n2_tp1 = δ * n2\n",
    "    end\n",
    "\n",
    "    return n1_tp1, n2_tp1\n",
    "end\n",
    "\n",
    "new_n1n2(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ) =\n",
    "    one_step(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "\n",
    "function pers_till_sync(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ,\n",
    "                        maxiter, npers)\n",
    "\n",
    "    # Initialize the status of synchronization\n",
    "    synchronized = false\n",
    "    pers_2_sync = maxiter\n",
    "    iters = 0\n",
    "\n",
    "    nsync = 0\n",
    "\n",
    "    while (~synchronized) && (iters < maxiter)\n",
    "        # Increment the number of iterations and get next values\n",
    "        iters += 1\n",
    "\n",
    "        n1_t, n2_t = new_n1n2(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "\n",
    "        # Check whether same in this period\n",
    "        if abs(n1_t - n2_t) < 1e-8\n",
    "            nsync += 1\n",
    "        # If not, then reset the nsync counter\n",
    "        else\n",
    "            nsync = 0\n",
    "        end\n",
    "\n",
    "        # If we have been in sync for npers then stop and countries\n",
    "        # became synchronized nsync periods ago\n",
    "        if nsync > npers\n",
    "            synchronized = true\n",
    "            pers_2_sync = iters - nsync\n",
    "        end\n",
    "        n1_0, n2_0 = n1_t, n2_t\n",
    "    end\n",
    "    return synchronized, pers_2_sync\n",
    "end\n",
    "\n",
    "function create_attraction_basis(s1_ρ, s2_ρ, s1, s2, θ, δ, ρ,\n",
    "                                 maxiter, npers, npts)\n",
    "    # Create unit range with npts\n",
    "    synchronized, pers_2_sync = false, 0\n",
    "    unit_range = range(0.0,  1.0, length = npts)\n",
    "\n",
    "    # Allocate space to store time to sync\n",
    "    time_2_sync = zeros(npts, npts)\n",
    "    # Iterate over initial conditions\n",
    "    for (i, n1_0) in enumerate(unit_range)\n",
    "        for (j, n2_0) in enumerate(unit_range)\n",
    "            synchronized, pers_2_sync = pers_till_sync(n1_0, n2_0, s1_ρ, s2_ρ,\n",
    "                                                       s1, s2, θ, δ, ρ,\n",
    "                                                       maxiter, npers)\n",
    "            time_2_sync[i, j] = pers_2_sync\n",
    "        end\n",
    "    end\n",
    "    return time_2_sync\n",
    "end\n",
    "\n",
    "# model\n",
    "function MSGSync(s1 = 0.5, θ = 2.5, δ = 0.7, ρ = 0.2)\n",
    "    # Store other cutoffs and parameters we use\n",
    "    s2 = 1 - s1\n",
    "    s1_ρ = min((s1 - ρ * s2) / (1 - ρ), 1)\n",
    "    s2_ρ = 1 - s1_ρ\n",
    "\n",
    "    return (s1 = s1, s2 = s2, s1_ρ = s1_ρ, s2_ρ = s2_ρ, θ = θ, δ = δ, ρ = ρ)\n",
    "end\n",
    "\n",
    "function simulate_n(model, n1_0, n2_0, T)\n",
    "    # Unpack parameters\n",
    "    @unpack s1, s2, θ, δ, ρ, s1_ρ, s2_ρ = model\n",
    "\n",
    "    # Allocate space\n",
    "    n1 = zeros(T)\n",
    "    n2 = zeros(T)\n",
    "\n",
    "    # Simulate for T periods\n",
    "    for t in 1:T\n",
    "        # Get next values\n",
    "        n1[t], n2[t] = n1_0, n2_0\n",
    "        n1_0, n2_0 = new_n1n2(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2, θ, δ, ρ)\n",
    "    end\n",
    "\n",
    "    return n1, n2\n",
    "end\n",
    "\n",
    "function pers_till_sync(model, n1_0, n2_0,\n",
    "                        maxiter = 500, npers = 3)\n",
    "    # Unpack parameters\n",
    "    @unpack s1, s2, θ, δ, ρ, s1_ρ, s2_ρ = model\n",
    "\n",
    "    return pers_till_sync(n1_0, n2_0, s1_ρ, s2_ρ, s1, s2,\n",
    "                        θ, δ, ρ, maxiter, npers)\n",
    "end\n",
    "\n",
    "function create_attraction_basis(model;\n",
    "                                 maxiter = 250,\n",
    "                                 npers = 3,\n",
    "                                 npts = 50)\n",
    "    # Unpack parameters\n",
    "    @unpack s1, s2, θ, δ, ρ, s1_ρ, s2_ρ = model\n",
    "\n",
    "    ab = create_attraction_basis(s1_ρ, s2_ρ, s1, s2, θ, δ,\n",
    "                                 ρ, maxiter, npers, npts)\n",
    "    return ab\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Time Series of Firm Measures\n",
    "\n",
    "We write a short function below that exploits the preceding code and plots two time series\n",
    "\n",
    "Each time series gives the dynamics for the two countries\n",
    "\n",
    "The time series share parameters but differ in their initial condition\n",
    "\n",
    "Here’s the function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "function plot_timeseries(n1_0, n2_0, s1 = 0.5, θ = 2.5, δ = 0.7, ρ = 0.2)\n",
    "    model = MSGSync(s1, θ, δ, ρ)\n",
    "    n1, n2 = simulate_n(model, n1_0, n2_0, 25)\n",
    "    return [n1 n2]\n",
    "end\n",
    "\n",
    "# Create figures\n",
    "data_ns = plot_timeseries(0.15, 0.35)\n",
    "data_s = plot_timeseries(0.4, 0.3)\n",
    "\n",
    "plot(data_ns, title = \"Not Synchronized\", legend = false)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "plot(data_s, title = \"Synchronized\", legend = false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the first case, innovation in the two countries does not synchronize\n",
    "\n",
    "In the second case different initial conditions are chosen, and the cycles\n",
    "become synchronized"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basin of Attraction\n",
    "\n",
    "Next let’s study the initial conditions that lead to synchronized cycles more\n",
    "systematically\n",
    "\n",
    "We generate time series from a large collection of different initial\n",
    "conditions and mark those conditions with different colors according to\n",
    "whether synchronization occurs or not\n",
    "\n",
    "The next display shows exactly this for four different parameterizations (one\n",
    "for each subfigure)\n",
    "\n",
    "Dark colors indicate synchronization, while light colors indicate failure to synchronize\n",
    "\n",
    "\n",
    "<a id='matsrep'></a>\n",
    "![_static/figures/matsuyama_14.png](_static/figures/matsuyama_14.png)  \n",
    "As you can see, larger values of $ \\rho $ translate to more synchronization\n",
    "\n",
    "You are asked to replicate this figure in the exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercises"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1\n",
    "\n",
    "Replicate the figure [shown above](#matsrep) by coloring initial conditions according to whether or not synchronization occurs from those conditions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solutions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Exercise 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "function plot_attraction_basis(s1 = 0.5, θ = 2.5, δ = 0.7, ρ = 0.2; npts = 250)\n",
    "    # Create attraction basis\n",
    "    unitrange = range(0,  1, length = npts)\n",
    "    model = MSGSync(s1, θ, δ, ρ)\n",
    "    ab = create_attraction_basis(model,npts=npts)\n",
    "    plt = Plots.heatmap(ab, legend = false)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "hide-output": false
   },
   "outputs": [],
   "source": [
    "params = [[0.5, 2.5, 0.7, 0.2],\n",
    "          [0.5, 2.5, 0.7, 0.4],\n",
    "          [0.5, 2.5, 0.7, 0.6],\n",
    "          [0.5, 2.5, 0.7, 0.8]]\n",
    "\n",
    "plots = (plot_attraction_basis(p...) for p in params)\n",
    "plot(plots..., size = (1000, 1000))"
   ]
  }
 ],
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  "title": "Globalization and Cycles"
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