{
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"source": [
"# Foundations of Computational Economics #22\n",
"\n",
"by Fedor Iskhakov, ANU\n",
"\n",
""
]
},
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"cell_type": "markdown",
"metadata": {
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"slide_type": "fragment"
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"source": [
"## Successive approximations (fixed point iterations)\n",
"\n",
""
]
},
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"metadata": {
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"slide_type": "subslide"
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"source": [
"\n",
"\n",
"[https://youtu.be/AQt9Q9qc3io](https://youtu.be/AQt9Q9qc3io)\n",
"\n",
"Description: Scalar and multivariate solver. Equilibrium in market of platforms."
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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},
"source": [
"### What is successive approximations?\n",
"\n",
"- root finding method for the equations of the type \n",
"\n",
"\n",
"$$\n",
"x = F(x) \\quad \\Leftrightarrow \\quad F(x) - x = 0\n",
"$$\n",
"\n",
"- **fixed point equation** \n",
"- conversion is always possible (although does not guaranteed to work as we’ll see below) \n",
"\n",
"\n",
"$$\n",
"g(x) = 0 \\quad \\Leftrightarrow \\quad \\underbrace{g(x) + x}_{F(x)} = x\n",
"$$"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"#### Main algorithm\n",
"\n",
"$$\n",
"x = F(x) \\rightarrow x_{n+1} = F(x_n)\n",
"$$\n",
"\n",
"1. Initialize the iterations at $ x_0 $ \n",
"1. Iterate on $ x_{i+1} = F(x_i) $ until $ x_{i+1} $ is close enough to $ x_i $ "
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"#### Stationary distribution of Markov chains\n",
"\n",
"For aperiodic and irreducible Markov chain with transition probability matrix $ P $, the stationary distribution\n",
"is given by $ \\psi^\\star = \\psi^\\star P $\n",
"\n",
"Successive approximation applicable with $ F(\\psi) = \\psi P $\n",
"\n",
"Have seen this is last two videos"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
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"source": [
"#### Will it converge?\n",
"\n",
"*The central question for the successive approximations method*\n",
"\n",
"- it will from any starting point for aperiodic and irreducible Markov chains \n",
"- but what are the general conditions? \n",
"\n",
"\n",
"Will come back to this question in general terms when talking about the dynamic models and Bellman operator\n",
"(which is a *contraction mapping*)"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"#### Scalar example\n",
"\n",
"$$\n",
"\\frac{1}{2} - \\exp\\big(-(x-2)^2\\big) = 0\n",
"$$\n",
"\n",
"Using the trick above convert to $ x = F(x) $\n",
"\n",
"$$\n",
"F(x) = x - \\exp\\big(-(x-2)^2\\big) + \\frac{1}{2}\n",
"$$\n",
"\n",
"$$\n",
"x_{i+1} = x_i - \\exp\\big(-(x_i-2)^2\\big) + \\frac{1}{2}\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"hide-output": false,
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [],
"source": [
"import numpy as np\n",
"eq = lambda x: - np.exp(-(x-2)**2) + .5 # initial equation\n",
"F = lambda x: x - np.exp(-(x-2)**2) + .5 # fixed point equation\n",
"dF = lambda x: 1 + np.exp(-(x-2)**2)*2*(x-2) # derivative (for later)"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"hide-output": false,
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [],
"source": [
"def solve_sa(F,x0,tol=1e-6,maxiter=100,callback=None):\n",
" '''Computes the solution of fixed point equation x = F(x)\n",
" with given initial value x0 and algorithm parameters\n",
" Method: successive approximations\n",
" '''\n",
" for i in range(maxiter): # main loop\n",
" x1 = F(x0) # update approximation\n",
" err = np.amax(np.abs(x0-x1)) # allow for x to be array\n",
" if callback != None: callback(iter=i,err=err,x=x1,x0=x0)\n",
" if err2.8"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"hide-output": false,
"slideshow": {
"slide_type": "slide"
}
},
"outputs": [],
"source": [
"# plot to understand what is going on!\n",
"import numpy as np\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"plt.rcParams['figure.figsize'] = [12, 8]"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"hide-output": false,
"slideshow": {
"slide_type": "slide"
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},
"outputs": [
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\n",
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