ipynb/julien_pascal-aiyagaribkm_quantecon.ipynb

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            "cell_type": "markdown",
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            "source": [
                "# Aiyagari Model with Aggregate Uncertainty"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Author: Julien Pascal\n",
                "\n",
                "https://julienpascal.github.io/\n",
                "\n",
                "---"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Introduction\n",
                "\n",
                "The Bewley-Huggett-Aiyagari-Imohoroğlu economies are the workhorse of modern macroeconomics. In these economies, markets are \"incomplete\". Agents cannot fully insure against risk and decide to self-insure by holding a safe asset to smooth their consumption (see [Ljungqvist and Sargent (2018)](https://mitpress.mit.edu/books/recursive-macroeconomic-theory-fourth-edition) for a textbook treatment of this topic).\n",
                "\n",
                "In this post, I consider the model of [Aiyagari (1994)](https://www.jstor.org/stable/2118417?seq=1#metadata_info_tab_contents). While the original model abstracts from aggregate fluctuations, Economists have since developed several techniques to simulate out-of-steady-state dynamics for this class of models.\n",
                "\n",
                "Here, I use a methodology that is quite general. It is a **3-step procedure**, which proceeds as follows:\n",
                "\n",
                "1. Solve for the non-stochastic steady-state\n",
                "2. Perturbe the model around its non-stochastic steady-state\n",
                "3. Use the perturbation to calculate out-of-steady-state dynamics\n",
                "\n",
                "I use the [BKM and GenBKM](https://notes.quantecon.org/submission/5ea288cb833c72001a988e4d) algorithms for step 2 and 3, which means that the only theoretical tool needed is **backward induction** (i.e. knowing the value tomorrow, what is the value today?).\n",
                "\n",
                "What are the cons of the methodology presented here? First, the methodology assumes a **small aggregate shock**. An implicit assumption is that aggregate shocks do not modify the value of the non-stochastic steady-state. Said differently, the cycle does not change the trend, [which seems to be violated in some instances](https://www.youtube.com/watch?v=K2X1GiTaxHw). If the shock is large, the value of steady-state may be altered and the methodology presented here is not adequate. An example of a large shock could be the disruption caused by COVID-19.\n",
                "\n",
                "This methodology also fails when the non-stochastic steady-state is not relevant for the dynamic economy. This can problematic in portfolio choice problems, in which portfolios are indeterminate when aggregate uncertainty vanishes (see [Coeurdacier et al (2011)](https://hal-sciencespo.archives-ouvertes.fr/hal-00972801/document)).\n",
                "\n",
                "On the pro side, the methodology presented here is fast (orders of magnitude faster than [Krusell-Smith (1998)](https://notes.quantecon.org/submission/5bb58d1e11611400157fdc8d)) and transparent. The code can be [downloaded here](https://github.com/JulienPascal/AiyagariAggregateUncertainty)."
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## I. The Model\n",
                "\n"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "In the model of Aiyagari (1994), there is continuum of agents each maximizing an infinite flow of discounted utility:\n",
                "\n",
                "$$ E_{0} \\sum_{t=0}^{\\infty} \\beta^t U(c_t) $$\n",
                "\n",
                "subject to the constraint that consumption plus savings in period $t$ (the left hand side of the next equation) is equal to labor earnings plus savings inherited from the last period (the right hand side of the next equation):\n",
                "\n",
                "$$ c_t + a_{t+1} = w_t l_t + (1 + r_t) a_t $$ \n",
                "\n",
                "The variable $l_t$ captures idiosyncratic risk in labor earnings and could be interpreted as unemployment risk. We also make the assumptions that consumption cannot be negative and that agents cannot borrow more than a certain amount $B$:\n",
                "\n",
                "$$ c_t \\geq 0 $$\n",
                "\n",
                "$$ a_t \\geq -B $$ \n",
                "\n",
                "The behavior of firms can be summarized by a representative firm hiring workers and capital:\n",
                "\n",
                "$$ Y_t = z_t K_t^{\\alpha} L_t^{1-\\alpha} $$ \n",
                "\n",
                "where $Y_t$ is total output, $K_t$ is the aggregate capital level and $L_t$ is the aggregate labor supply. The variables $w_t$ and $r_t$ are pinned down each period by the first order conditions (FOCs) of the representative firm. Note that at the non-stochastic steady-state $z_t = z_{SS} = 1$ (by definition) and both $w_t$ and $r_t$ are constant. Importantly, because agents have to take into consideration the value of $w_t$ and $r_t$ when making decisions, the cross-sectional distribution of agents across capital and idiosyncratic states matters (through the FOCs of the representative firm)."
            ]
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            "cell_type": "markdown",
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            "source": [
                "## II. Methodology\n",
                "\n",
                "To solve for individual policy functions, I use the endogenous grid method (EGM) of Carroll (2006). The main idea of this method is to start from the end-of-period level of capital. Using the Euler equation, one may recover the beginning-of-period consumption and level of capital without using a root-finding algorithm. \n",
                "\n",
                "To determine the non-stochastic equilibrium, I solve for the fixed-point problem over the aggregate capital level f(K*) = 0 using [Brent's method](https://en.wikipedia.org/wiki/Brent%27s_method).\n",
                "\n",
                "To calculate the response of the economy to one-period unforeseen aggregate shock (an \"MIT shock\"), I use a standard backward-forward [\"shooting\" method](https://en.wikipedia.org/wiki/Shooting_method): \n",
                "1. holding the path of aggregate capital $\\{K_t\\}_{t=1}^{T}$, calculate the policy functions\n",
                "2. holding constant the policy functions, calculate the aggregate capital $\\{K_t\\}_{t=1}^{T}$\n",
                "3. repeat until convergence of the path for aggregate capital $\\{K_t\\}_{t=1}^{T}$\n",
                "\n",
                "To simulate out-of-steady-state dynamics, I use the [BKM algorithm](https://ideas.repec.org/a/eee/dyncon/v89y2018icp68-92.html), which relies on the assumption that the response of the economy to an aggregate shock $d_t$ is **linear** with respect to the **aggregate state** $z_t$:\n",
                "\n",
                "$$ d_t = z_t d(1, 0, 0, ...) + z_{t-1}d(0, 1, 0, ...) + z_{t-2}d(0, 0, 1, ...) + ... $$ \n",
                "\n",
                "or more compactly:\n",
                "\n",
                "$$ d_t = \\sum_{k=0}^{+\\infty} z_{t-k} d_{k} $$ \n",
                "\n",
                "where \n",
                "\n",
                "$$ d_{1} = d(1,0,0,...)$$\n",
                "$$ d_{2} = d(0,1,0,...)$$\n",
                "$$ d_{3} = d(0,0,1,...)$$\n",
                "\n",
                "That is, the evolution of equilibrium variables is a moving average of past shocks. However, when calculating several trajectories after an MIT shock, one sees that the linearity assumption is slightly violated. Hence, I use the [GenBKM algorithm](https://irihs.ihs.ac.at/id/eprint/4500/), which is a refinement of the BKM algorithm taking into consideration these slight violations of linearity."
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## III. Implementation\n",
                "\n",
                "These ideas are implemented using [Julia](https://julialang.org/)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 1,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Julia Version 1.3.0\n",
                        "Commit 46ce4d7933 (2019-11-26 06:09 UTC)\n",
                        "Platform Info:\n",
                        "  OS: Linux (x86_64-pc-linux-gnu)\n",
                        "  CPU: Intel(R) Core(TM) i7-8850H CPU @ 2.60GHz\n",
                        "  WORD_SIZE: 64\n",
                        "  LIBM: libopenlibm\n",
                        "  LLVM: libLLVM-6.0.1 (ORCJIT, skylake)\n"
                    ]
                }
            ],
            "source": [
                "versioninfo()"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### II. A Dependencies"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 2,
            "metadata": {},
            "outputs": [],
            "source": [
                "using Distributions\n",
                "using Plots\n",
                "using DataFrames\n",
                "using Random\n",
                "using ForwardDiff\n",
                "using LinearAlgebra\n",
                "using Interpolations\n",
                "using DataFrames\n",
                "using Optim\n",
                "#using IterativeSolvers\n",
                "#plotlyjs()\n",
                "Random.seed!(1234);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 3,
            "metadata": {},
            "outputs": [],
            "source": [
                "using NBInclude #To load stuct and functions from other notebooks"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 4,
            "metadata": {
                "scrolled": true
            },
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Steady state: [1.0, 1.0204081632653061, 45.63711005516377, 4.610279485164506, 3.697537284061231]Epsilon state: [0.0]Convergence flag = 1"
                    ]
                },
                {
                    "data": {
                        "text/plain": [
                            "GenBKM_path! (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 4,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "@nbinclude(\"utils.ipynb\") #mutable structs and primitive functions for the model\n",
                "@nbinclude(\"RBC.ipynb\")   #Aiyagari model, without the borrowing constraint (standard RBC)\n",
                "@nbinclude(\"EGM.ipynb\")   #implementation of the EGM method\n",
                "@nbinclude(\"SteadyState.ipynb\") #to calculate the non-stochastic steady-state\n",
                "@nbinclude(\"GenBKM.ipynb\") #to simulate the stochastic model using the GenBKM algorithm"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### II.B Steady-state"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Finding the steady-state value of capital\n",
                "\n",
                "Finding the non-stochastic equilibrium is a fixed-point problem over the aggregate capital level f(K*) = 0, which can be solved using [Brent's method](https://en.wikipedia.org/wiki/Brent%27s_method):"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 5,
            "metadata": {
                "scrolled": true
            },
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "  7.499330 seconds (8.18 M allocations: 1.695 GiB, 5.92% gc time)\n",
                        "Steady-state value of capital K* = 46.09393601898816\n"
                    ]
                }
            ],
            "source": [
                "p = Params() #struct with model parameters\n",
                "z_ss = 1.0 #aggregate productivity at the non-stochastic steady-state\n",
                "@time oo = optimize(K -> eq_K(K,p), 10, 100, Brent()) #solve for the steady-state value of capital using Brent method\n",
                "K_star = oo.minimizer;\n",
                "println(\"Steady-state value of capital K* = $(K_star)\")\n",
                "# Store the optimal policy function at the steady-state\n",
                "g_star, c_star, g_low_star, g_high_star, success_flag= solve_EGM(x->log(x), x->log(x), R(K_star, z_ss, p), W(K_star, z_ss, p), p); #solve for policy functions\n",
                "# Store the stationary distribution at the steady-state\n",
                "t_star = make_trans_mat(g_star, p)    #generate transition matrix\n",
                "d_star = get_stationary_dist(t_star); #stationary distribution"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Visualizing transition probabilities"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 6,
            "metadata": {
                "scrolled": true
            },
            "outputs": [
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                    },
                    "execution_count": 6,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "xs = [string(\"x\", i) for i = 1:size(t_star,1)]\n",
                "ys = [string(\"y\", i) for i = 1:size(t_star,2)]\n",
                "heatmap(t_star, aspect_ratio = 1, color=:plasma, clim=(0., 0.25))"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: This graph shows the transition probabilities across capital and idiosyncratic probability states.*"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Visualizing convergence toward the steady-state\n",
                "\n",
                "One may visually check the equilibrium exists and is unique:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 7,
            "metadata": {},
            "outputs": [
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                    "execution_count": 7,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "# Solve for the demand and supply of capital for different values of the interest rate\n",
                "K_grid = collect(range(oo.minimizer-0.5, stop=oo.minimizer+0.5, length=20))\n",
                "K_implied_grid = similar(K_grid)\n",
                "R_grid = similar(K_grid)\n",
                "\n",
                "for (K_index, K_value) in enumerate(K_grid)\n",
                "    R_s, W_s = R(K_value, z_ss, p), W(K_value, z_ss, p) #calculate interest rate R and wage W\n",
                "    gg, c_star, g_low, g_high, success_flag= solve_EGM(x -> log(x), x -> 2*log(x), R_s, W_s, p) #solve for policy functions\n",
                "    tt = make_trans_mat(gg, p)   #generate transition matrix\n",
                "    dd = get_stationary_dist(tt) #stationary distribution\n",
                "    K_implied = aggregate_K(dd, p) #implied level of capital\n",
                "    R_grid[K_index] = R_s #store interest rate\n",
                "    K_implied_grid[K_index] = K_implied #store demand of capital\n",
                "    K_grid[K_index] = K_value #store supply of capital\n",
                "end\n",
                "\n",
                "# Plot demand and supply of capital\n",
                "plot(K_grid, R_grid, label = \"Demand\", ylabel=\"Interest rate\", xlabel=\"Capital\")\n",
                "plot!(K_implied_grid, R_grid, label = \"Supply\")"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: This graph shows the demand and the supply of capital as a function of the interest rate.*"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### II.C MIT Shock"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Backward and forward updates"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "The next block of code implements the backward-forward shooting method:\n",
                "1. holding the path of aggregate capital $\\{K_t\\}_{t=1}^{T}$, calculate the policy functions\n",
                "2. holding constant the policy functions, calculate the aggregate capital $\\{K_t\\}_{t=1}^{T}$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 8,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "forward_update (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 8,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "function backward_update(g_low_ss::Function, g_high_ss::Function, K_path_guess::Array{Float64,1}, z_path::Array{Float64,1}, p::Params)\n",
                "    \"\"\"\n",
                "    Update policy functions backward, holding {K_t,z_t} constant\n",
                "    \"\"\"\n",
                "    nT = length(z_path)\n",
                "    g_low_path = Array{Function}(undef,nT) #initialize two lists of functions\n",
                "    g_high_path =  Array{Function}(undef,nT)\n",
                "    g_low_path[nT] = g_low_ss\n",
                "    g_high_path[nT] = g_high_ss\n",
                "    a_path = zeros(p.nI, p.grid_size, nT) #to store policy functions on savings grid\n",
                "    R_path = zeros(nT) #to store the interest rate on path\n",
                "    W_path = zeros(nT) #to store the wage on path\n",
                "    \n",
                "    #Start from the steady-state and iterate backward\n",
                "    #holding constant the path for {K_t,z_t}\n",
                "    #---------------------------------------------------\n",
                "    for t=nT:-1:2 #iterate backward\n",
                "        # Next period\n",
                "        R_path[t], W_path[t] = R(K_path_guess[t], z_path[t], p), W(K_path_guess[t], z_path[t], p)\n",
                "        # Current period\n",
                "        R_path[t-1], W_path[t-1] = R(K_path_guess[t-1], z_path[t-1], p), W(K_path_guess[t-1], z_path[t-1], p)\n",
                "        # Current period's policy, given next period\n",
                "        a_path[:,:,t-1], c_new, g_low_path[t-1], g_high_path[t-1] = euler_back(g_low_path[t], g_high_path[t], R_path[t-1], W_path[t-1], R_path[t], W_path[t], p)\n",
                "    end\n",
                "    \n",
                "    return a_path, g_low_path, g_high_path\n",
                "end\n",
                "\n",
                "function forward_update(K_star::Float64, a_path::Array{Float64,3}, d_ss::Array{Float64,1}, p::Params)\n",
                "    \"\"\"\n",
                "    Update forward the distribution of agents + aggregate capital\n",
                "    \"\"\"\n",
                "    nT = length(z_path)\n",
                "    K_path_forward = zeros(nT)\n",
                "    K_path_forward[1] = K_star\n",
                "    dd_path_forward = zeros(size(d_ss,1), nT)\n",
                "    dd_path_forward[:,1] = d_ss\n",
                "    #2. Iterate forward {K_t,z_t}, using the policy\n",
                "    #functions from step 1\n",
                "    #-----------------------------------------------\n",
                "    for t=2:nT\n",
                "        tt = make_trans_mat(a_path[:,:,t-1], p) #generate transition matrix\n",
                "        dd_path_forward[:,t] = tt*dd_path_forward[:,t-1]\n",
                "        K_path_forward[t] = aggregate_K(dd_path_forward[:,t], p)\n",
                "    end\n",
                "    \n",
                "    return dd_path_forward, K_path_forward\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Finding the transition path"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "One problem with backward-forward shooting method is that updating the path for $\\{K_t\\}_{t=1}^{T}$ \"too quickly\" may result in the overall procedure to diverge. An easy fix is to take a convex combination of the \n",
                "previous guess and the newly calculated path, with $\\lambda$ small:\n",
                "\n",
                "$\\{K^{NEW}_t\\}_{t=1}^{T} = \\lambda \\{K_t\\}_{t=1}^{T} + (1-\\lambda)\\{K^{OLD}_t\\}_{t=1}^{T}$\n",
                "\n",
                "The next function implement this idea, with the extra feature that $\\lambda$ increases when distance between two iterations is getting small (too speed up convergence) and decreases when the distance is getting bigger (to prevent divergence) (see [this excellent notebook](https://notes.quantecon.org/submission/5b3faf1fb9eab00015b89f9a))."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 9,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "solve_mit! (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 9,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "function solve_mit!(K_path, g_low_ss::Function, g_high_ss::Function, d_ss::Array{Float64,1}, \n",
                "                    K_ss::Float64, z_path::Array{Float64,1}, p::Params; convex_combination::Float64 = 0.2,\n",
                "                    shrink_factor::Float64 = 0.5, expand_factor::Float64 = 1.05,\n",
                "                    max_iter::Int64 = 1000, tol::Float64=1e-6, verbose::Bool=true, display_iter::Int64 = 20)\n",
                "    \"\"\"\n",
                "    Finds the path for aggregate capital K_path\n",
                "    \"\"\"\n",
                "    diff = Inf #initialization\n",
                "    diff_old = Inf #initialization\n",
                "    convergence_flag = 0 #initialization\n",
                "    damp = convex_combination #initial dampening parameter\n",
                "    \n",
                "    for i_mit=1:max_iter\n",
                "    \n",
                "        # Step 1. Solve backward the policy functions {g_t(a,e_low), g_t(a,e_high)}, keeping {K_t,z_t} constant:\n",
                "        a_path, g_low_path, g_high_path = backward_update(g_low_ss, g_high_ss, K_path[i_mit], z_path, p);\n",
                "        \n",
                "        #2. Solve forward {K_t,z_t}, keeping policy functions {g_t(a,e_low), g_t(a,e_high)} constant:\n",
                "        dd_path_forward, K_path_forward = forward_update(K_ss, a_path, d_star, p);\n",
                "        \n",
                "        # Distance between guess for {K_t} and implied values:\n",
                "        diff = maximum(abs.(K_path_forward - K_path[i_mit]))\n",
                "        \n",
                "        # Display every display_iter iterations\n",
                "        if verbose==true\n",
                "            if mod(i_mit,display_iter) == 0\n",
                "                println(\"Iteration $(i_mit). diff = $(diff)\")\n",
                "            end\n",
                "        end\n",
                "\n",
                "        if diff < tol\n",
                "            if verbose==true\n",
                "                println(\"Convergence reached after $(i_mit) iterations.\")\n",
                "            end\n",
                "            convergence_flag = 1\n",
                "            break\n",
                "        else\n",
                "            # Update the guess for the path {K_t}\n",
                "            # Decrease the dampening factor\n",
                "            if diff > diff_old\n",
                "                damp = max(min(damp * shrink_factor, 1.0-eps()), eps())\n",
                "            # Increase the dampening factor\n",
                "            else\n",
                "                damp = max(min(damp * expand_factor, 1.0-eps()), eps())\n",
                "            end\n",
                "            if mod(i_mit, 10) == 0\n",
                "                if verbose==true\n",
                "                    println(\"damp = $(damp); diff = $(diff)\")\n",
                "                end\n",
                "            end\n",
                "            # Store the updated path for {K_t}\n",
                "            push!(K_path, damp.*K_path_forward .+ (1.0 - damp).*K_path[i_mit])\n",
                "            diff_old = diff\n",
                "                \n",
                "        end\n",
                "    end\n",
                "    \n",
                "    return K_path, convergence_flag\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Find the path for {K_t} for a 1 std. dev positive productivity shock"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 10,
            "metadata": {
                "scrolled": false
            },
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "damp = 0.19391602699731453; diff = 0.5214624279931286\n",
                        "Iteration 20. diff = 0.019740426540607814\n",
                        "damp = 0.31586877442195505; diff = 0.019740426540607814\n",
                        "damp = 0.5145169494226985; diff = 7.184339025911868e-5\n",
                        "Iteration 40. diff = 9.339544817521528e-7\n",
                        "Convergence reached after 40 iterations.\n",
                        "  7.603950 seconds (7.16 M allocations: 23.457 GiB, 12.45% gc time)\n"
                    ]
                }
            ],
            "source": [
                "max_t = 300 #Let us assume that the economy is back to the steady state after max_t periods\n",
                "z_ss = 1.0  #Value of aggregate productivity at the non-stochastic steady-state\n",
                "z_shock = 2.0 #Value of the inital shock\n",
                "\n",
                "# Let's generate a path for the aggregate shock\n",
                "z_path = ones(max_t)\n",
                "z_path[1] = z_ss*z_shock #initial shock\n",
                "\n",
                "# Evolution of aggregate productivity in level:\n",
                "for t_index=2:max_t \n",
                "    z_path[t_index] = z_path[t_index-1]^p.rho \n",
                "end\n",
                "\n",
                "# Heroic guess for the initial path of {K_t}: K_t = K* for all t\n",
                "K_path = []\n",
                "push!(K_path, repeat([K_star], max_t))\n",
                "\n",
                "# Find the path for {K_t}:\n",
                "@time K_path, convergence_flag = solve_mit!(K_path, g_low_star, g_high_star, d_star, K_star, z_path, p, convex_combination=0.25);\n",
                "\n",
                "# Find the path for other aggregates:\n",
                "R_path = zeros(length(z_path)) #to store the interest rate on path\n",
                "W_path = zeros(length(z_path)) #to store the wage on path\n",
                "for t=length(z_path):-1:1 #iterate backward\n",
                "    # Next period\n",
                "    R_path[t], W_path[t] = R(K_path[end][t], z_path[t], p), W(K_path[end][t], z_path[t], p)\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Visualize convergence of the transition path\n",
                "\n",
                "The first guess for $\\{K_t\\}_{t=1}^{T}$ is that it equal to $K*$ for each period. Very quickly, the path for $\\{K_t\\}_{t=1}^{T}$ converges to the perfect foresight transition path:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 11,
            "metadata": {
                "scrolled": true
            },
            "outputs": [
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                            "  1885.31,1376.49 1892.29,1376.83 1899.27,1377.17 1906.25,1377.5 1913.23,1377.82 1920.21,1378.13 1927.19,1378.44 1934.17,1378.74 1941.15,1379.03 1948.13,1379.32 \n",
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                            "  1885.31,1376.56 1892.29,1376.9 1899.27,1377.23 1906.25,1377.56 1913.23,1377.88 1920.21,1378.19 1927.19,1378.5 1934.17,1378.79 1941.15,1379.09 1948.13,1379.37 \n",
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                            "  1885.31,1376.56 1892.29,1376.9 1899.27,1377.23 1906.25,1377.56 1913.23,1377.88 1920.21,1378.19 1927.19,1378.49 1934.17,1378.79 1941.15,1379.08 1948.13,1379.37 \n",
                            "  1955.11,1379.65 1962.09,1379.92 1969.07,1380.18 1976.05,1380.44 1983.03,1380.7 1990.01,1380.95 1996.99,1381.19 2003.97,1381.43 2010.95,1381.66 2017.93,1381.89 \n",
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                            "  1745.71,1367.83 1752.69,1368.37 1759.67,1368.9 1766.65,1369.41 1773.63,1369.91 1780.61,1370.4 1787.59,1370.88 1794.57,1371.35 1801.55,1371.81 1808.53,1372.25 \n",
                            "  1815.51,1372.69 1822.49,1373.12 1829.47,1373.54 1836.45,1373.95 1843.43,1374.34 1850.41,1374.74 1857.39,1375.12 1864.37,1375.49 1871.35,1375.85 1878.33,1376.21 \n",
                            "  1885.31,1376.56 1892.29,1376.9 1899.27,1377.23 1906.25,1377.56 1913.23,1377.88 1920.21,1378.19 1927.19,1378.49 1934.17,1378.79 1941.15,1379.08 1948.13,1379.37 \n",
                            "  1955.11,1379.64 1962.09,1379.92 1969.07,1380.18 1976.05,1380.44 1983.03,1380.7 1990.01,1380.95 1996.99,1381.19 2003.97,1381.43 2010.95,1381.66 2017.93,1381.89 \n",
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                            "  1466.51,1332.42 1473.49,1333.75 1480.47,1335.05 1487.45,1336.32 1494.43,1337.57 1501.41,1338.79 1508.39,1339.97 1515.37,1341.14 1522.35,1342.27 1529.33,1343.38 \n",
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                            "  1745.71,1367.83 1752.69,1368.37 1759.67,1368.9 1766.65,1369.41 1773.63,1369.91 1780.61,1370.4 1787.59,1370.88 1794.57,1371.35 1801.55,1371.81 1808.53,1372.25 \n",
                            "  1815.51,1372.69 1822.49,1373.12 1829.47,1373.54 1836.45,1373.95 1843.43,1374.34 1850.41,1374.73 1857.39,1375.12 1864.37,1375.49 1871.35,1375.85 1878.33,1376.21 \n",
                            "  1885.31,1376.56 1892.29,1376.9 1899.27,1377.23 1906.25,1377.56 1913.23,1377.88 1920.21,1378.19 1927.19,1378.49 1934.17,1378.79 1941.15,1379.08 1948.13,1379.37 \n",
                            "  1955.11,1379.64 1962.09,1379.92 1969.07,1380.18 1976.05,1380.44 1983.03,1380.7 1990.01,1380.95 1996.99,1381.19 2003.97,1381.43 2010.95,1381.66 2017.93,1381.89 \n",
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                    },
                    "execution_count": 11,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "p0 = plot(1:max_t, K_path[1], label= \"Iteration 0\", title=\"Convergence of K(t)\")\n",
                "plot!(p0, 2:max_t, K_path[2][2:end], label = \"Iteration 1\")\n",
                "show_every = 5 #display {K_t} for each multiple of show_every\n",
                "for k in 2:length(K_path)\n",
                "    if mod(k,show_every) == 0\n",
                "        plot!(p0, 2:max_t, K_path[k][2:end], xlabel=\"t\", label = \"Iteration $(k)\", title=\"Convergence of K(t)\", legend=:best)\n",
                "    end\n",
                "end\n",
                "\n",
                "p0"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: This graph shows the path for capital {K_t}^(i) for different iterations of the backward-forward algorithm*"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Removing the borrowing constraint"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "The next graph compares shows what happens to an Aiyagari model without borrowing constraint. With no borrowing constraint, the aggregate level of capital reacts less to an aggregate shock in productivity:"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 12,
            "metadata": {},
            "outputs": [
                {
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                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1973.27, 283.175)\" x=\"1973.27\" y=\"283.175\">K_t Aiyagari</text>\n",
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                        ]
                    },
                    "execution_count": 12,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "p1 = plot(1:max_t, K_path[end], label= \"K_t Aiyagari\", title=\"IRF Aiyagari versus RBC\")\n",
                "plot!(p1, xx[RBCp.iK,2:end] .+ K_star, label = \"K_t RBC\", color = \"black\", xlabel=\"t\")"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: This graph shows the impulse response of K_t of for the Aiyagari model and a RBC model.*"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 13,
            "metadata": {},
            "outputs": [
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                            "  2264.61,1407.85 2267.76,1407.91 2270.91,1407.98 2274.06,1408.04 2277.21,1408.1 2280.36,1408.16 2283.52,1408.22 2286.67,1408.28 2289.82,1408.34 2292.97,1408.4 \n",
                            "  2296.12,1408.45 2299.27,1408.51 2302.43,1408.56 2305.58,1408.61 2308.73,1408.67 2311.88,1408.72 2315.03,1408.77 2318.18,1408.82 2321.33,1408.87 2324.49,1408.92 \n",
                            "  \n",
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                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2155.21, 1008.74)\" x=\"2155.21\" y=\"1008.74\">K(t)</text>\n",
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                        ]
                    },
                    "execution_count": 13,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "p1 = plot(1:max_t, z_path./z_path[end] .-1, label = \"z(t)\", xlabel= \"t\")\n",
                "p2 = plot(1:max_t, R_path./R_path[end] .-1, label= \"R(t)\", xlabel= \"t\")\n",
                "p3 = plot(1:max_t, W_path./W_path[end].-1 , label= \"W(t)\", xlabel= \"t\")\n",
                "p4 = plot(1:max_t, K_path[end]./K_path[end][end] .-1, label= \"K(t)\", xlabel= \"t\" )\n",
                "\n",
                "p5 = plot(p1, p2, p3, p4)"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: This graph shows the percentage deviation from steady-state values of (i) aggregate productivity (ii) the interest rate (iii) wages (iv) capital.*"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "#### Linearity checks"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "To simulate the stochastic economy, the BKM algorithm makes the assumption that an MIT shock is linear with \n",
                "respect to the aggregate shock. That is, doubling the initial shock will simply double the value of aggregates\n",
                "along the transition path without changing the shape of the path. The next block\n",
                "calculates several transition paths for different initial aggregate shocks. "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 14,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "  9.390779 seconds (6.21 M allocations: 23.989 GiB, 16.50% gc time)\n",
                        "  8.103079 seconds (6.19 M allocations: 23.988 GiB, 15.77% gc time)\n",
                        " 10.796896 seconds (6.80 M allocations: 26.328 GiB, 14.34% gc time)\n",
                        "  9.913545 seconds (5.89 M allocations: 22.817 GiB, 14.93% gc time)\n",
                        " 10.290372 seconds (6.34 M allocations: 24.573 GiB, 14.44% gc time)\n",
                        "  6.917209 seconds (6.49 M allocations: 25.158 GiB, 12.66% gc time)\n"
                    ]
                }
            ],
            "source": [
                "max_t = 300 #Let us assume that the economy is back to the steady state after max_t periods\n",
                "z_ss = 1.0  #Value of aggregate productivity at the non-stochastic steady-state\n",
                "# Different initial shocks\n",
                "array_sigma = collect(range(-0.75, stop=0.75, step=0.25))\n",
                "# Let's exclude sigma = 0\n",
                "array_sigma = array_sigma[array_sigma .!= 0.]\n",
                "# To store the different scaled IRF:\n",
                "x_mit_scaled_sigma = zeros(max_t, length(array_sigma))\n",
                "# To store path of aggregate productivity:\n",
                "z_path_sigma = zeros(max_t, length(array_sigma))\n",
                "# To store the path for the %deviation of aggregate productivity from its steady-state value\n",
                "z_path_sigma_dev = zeros(max_t, length(array_sigma))\n",
                "\n",
                "for (index_sigma, sigma) in enumerate(array_sigma)\n",
                "    \n",
                "    # Let's generate a path for the aggregate shock\n",
                "    z_path = ones(max_t)\n",
                "    z_path[1] = z_ss + z_ss*sigma\n",
                "\n",
                "    # Evolution of aggregate productivity in level:\n",
                "    for t_index=2:max_t \n",
                "        z_path[t_index] = z_path[t_index-1]^p.rho\n",
                "    end\n",
                "\n",
                "    # Heroic guess for the initial path of {K_t}: K_t = K* for all t\n",
                "    K_path = []\n",
                "    push!(K_path, repeat([K_star], max_t))\n",
                "\n",
                "    # Find the path for {K_t}:\n",
                "    @time K_path, convergence_flag = solve_mit!(K_path, g_low_star, g_high_star, d_star, K_star, z_path, p, convex_combination=0.2, verbose=false);\n",
                "\n",
                "    # Check for convergence\n",
                "    if convergence_flag!=1\n",
                "        error(\"No convergence for z(1) = $(z_path[1]).\")\n",
                "    end\n",
                "    \n",
                "    # store the path for z:\n",
                "    z_path_sigma[:, index_sigma] = z_path\n",
                "    \n",
                "    # store for the %deviation of aggregate productivity from its steady-state value\n",
                "    z_path_sigma_dev[:, index_sigma] = z_path./z_ss .- 1.0\n",
                "    \n",
                "    # Scaled IRF: how a percentage deviation in z_t from its steady-state results in a % deviation of k_t\n",
                "    x_mit_scaled_sigma[:, index_sigma] = (K_path[end]./K_star .- 1.0)./z_path_sigma_dev[1, index_sigma]\n",
                "    \n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 15,
            "metadata": {},
            "outputs": [],
            "source": [
                "p0 = plot()\n",
                "p1 = plot()\n",
                "p2 = plot()\n",
                "p3 = plot()\n",
                "\n",
                "for (index_sigma, sigma) in enumerate(array_sigma)\n",
                "    if index_sigma == 1\n",
                "        p0 = plot(100 .*z_path_sigma_dev[:, index_sigma], label=\"z(t) with z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\")\n",
                "        p1 = plot(z_path_sigma[:, index_sigma], label=\"z(t) z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\")\n",
                "        p2 = plot(x_mit_scaled_sigma[:, index_sigma], label=\"z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\")\n",
                "        p3 = plot(sign(z_path_sigma[1, index_sigma] - z_ss)*x_mit_scaled_sigma[:, index_sigma], label=\"z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\")\n",
                "    else\n",
                "        plot!(p0, 100 .*z_path_sigma_dev[:, index_sigma], label=\"z(t) with z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\", title = \"% deviation aggregate productivity\", xlabel=\"t\")\n",
                "        plot!(p1, z_path_sigma[:, index_sigma], label=\"z(t) with z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\", title = \"Aggregate productivity\", xlabel=\"t\")\n",
                "        plot!(p2, x_mit_scaled_sigma[:, index_sigma], label=\"z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\", title = \"Impact of shock's size on scaled IRF: x\", xlabel=\"t\")\n",
                "        plot!(p3, sign(z_path_sigma[1, index_sigma] - z_ss)*x_mit_scaled_sigma[:, index_sigma], label=\"z(1) = $(round(100 .*z_path_sigma_dev[1, index_sigma], digits=2))%\", title = \"Impact of shock's sign on scaled IRF: sign(x - x_ss)*x\", xlabel=\"t\")\n",
                "    end\n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 16,
            "metadata": {},
            "outputs": [
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                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 144.627, 1127.76)\" x=\"144.627\" y=\"1127.76\">1.2</text>\n",
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                            "  1470.88,1173.69 1477.77,1173.69 1484.66,1173.69 1491.55,1173.68 1498.44,1173.68 1505.33,1173.68 1512.22,1173.68 1519.12,1173.68 1526.01,1173.68 1532.9,1173.67 \n",
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                            "  1953.27,1173.65 1960.16,1173.65 1967.05,1173.65 1973.94,1173.65 1980.83,1173.65 1987.72,1173.65 1994.61,1173.65 2001.51,1173.65 2008.4,1173.65 2015.29,1173.65 \n",
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                            "  2228.92,1173.65 2235.81,1173.65 2242.7,1173.65 2249.59,1173.65 2256.48,1173.65 2263.38,1173.65 2270.27,1173.65 2277.16,1173.65 2284.05,1173.65 2290.94,1173.65 \n",
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                            "</svg>\n"
                        ]
                    },
                    "execution_count": 16,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "plot(p0,p1, fg_legend = :transparent, legend=:best, layout=(2,1))"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: This graph shows the percentage deviation of aggregate productivity from its steady-state value (top panel) and aggregate productivity in level (bottom panel) for different initial shocks.*"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 17,
            "metadata": {},
            "outputs": [
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                            "  585.898,184.916 592.747,189.259 599.597,193.709 606.446,198.258 613.295,202.895 620.144,207.613 626.993,212.403 633.842,217.257 640.691,222.167 647.54,227.127 \n",
                            "  654.389,232.129 661.238,237.168 668.087,242.236 674.936,247.329 681.786,252.44 688.635,257.564 695.484,262.696 702.333,267.831 709.182,272.965 716.031,278.094 \n",
                            "  722.88,283.212 729.729,288.318 736.578,293.406 743.427,298.473 750.276,303.517 757.125,308.533 763.975,313.52 770.824,318.474 777.673,323.393 784.522,328.275 \n",
                            "  791.371,333.117 798.22,337.917 805.069,342.674 811.918,347.385 818.767,352.049 825.616,356.665 832.465,361.231 839.314,365.746 846.163,370.209 853.013,374.618 \n",
                            "  859.862,378.974 866.711,383.274 873.56,387.519 880.409,391.708 887.258,395.839 894.107,399.914 900.956,403.932 907.805,407.892 914.654,411.794 921.503,415.638 \n",
                            "  928.352,419.424 935.202,423.152 942.051,426.822 948.9,430.435 955.749,433.99 962.598,437.488 969.447,440.928 976.296,444.312 983.145,447.64 989.994,450.912 \n",
                            "  996.843,454.128 1003.69,457.289 1010.54,460.395 1017.39,463.448 1024.24,466.446 1031.09,469.392 1037.94,472.285 1044.79,475.126 1051.64,477.916 1058.49,480.655 \n",
                            "  1065.33,483.344 1072.18,485.984 1079.03,488.574 1085.88,491.117 1092.73,493.612 1099.58,496.06 1106.43,498.461 1113.28,500.818 1120.13,503.129 1126.98,505.396 \n",
                            "  1133.82,507.619 1140.67,509.8 1147.52,511.938 1154.37,514.034 1161.22,516.09 1168.07,518.105 1174.92,520.081 1181.77,522.018 1188.62,523.917 1195.47,525.777 \n",
                            "  1202.32,527.601 1209.16,529.389 1216.01,531.14 1222.86,532.857 1229.71,534.539 1236.56,536.187 1243.41,537.802 1250.26,539.384 1257.11,540.933 1263.96,542.452 \n",
                            "  1270.81,543.939 1277.66,545.396 1284.5,546.823 1291.35,548.221 1298.2,549.591 1305.05,550.932 1311.9,552.245 1318.75,553.531 1325.6,554.791 1332.45,556.024 \n",
                            "  1339.3,557.232 1346.15,558.415 1353,559.573 1359.84,560.707 1366.69,561.818 1373.54,562.905 1380.39,563.969 1387.24,565.011 1394.09,566.032 1400.94,567.03 \n",
                            "  1407.79,568.008 1414.64,568.966 1421.49,569.903 1428.34,570.82 1435.18,571.718 1442.03,572.597 1448.88,573.457 1455.73,574.299 1462.58,575.124 1469.43,575.931 \n",
                            "  1476.28,576.72 1483.13,577.493 1489.98,578.25 1496.83,578.99 1503.68,579.715 1510.52,580.424 1517.37,581.118 1524.22,581.797 1531.07,582.462 1537.92,583.112 \n",
                            "  1544.77,583.749 1551.62,584.372 1558.47,584.982 1565.32,585.578 1572.17,586.162 1579.02,586.733 1585.86,587.292 1592.71,587.839 1599.56,588.374 1606.41,588.898 \n",
                            "  1613.26,589.411 1620.11,589.912 1626.96,590.403 1633.81,590.883 1640.66,591.352 1647.51,591.812 1654.36,592.262 1661.2,592.702 1668.05,593.132 1674.9,593.553 \n",
                            "  1681.75,593.965 1688.6,594.368 1695.45,594.763 1702.3,595.148 1709.15,595.526 1716,595.895 1722.85,596.257 1729.69,596.61 1736.54,596.956 1743.39,597.294 \n",
                            "  1750.24,597.625 1757.09,597.949 1763.94,598.265 1770.79,598.575 1777.64,598.878 1784.49,599.175 1791.34,599.465 1798.19,599.748 1805.03,600.026 1811.88,600.297 \n",
                            "  1818.73,600.563 1825.58,600.822 1832.43,601.077 1839.28,601.325 1846.13,601.568 1852.98,601.806 1859.83,602.038 1866.68,602.265 1873.53,602.488 1880.37,602.705 \n",
                            "  1887.22,602.918 1894.07,603.126 1900.92,603.329 1907.77,603.528 1914.62,603.723 1921.47,603.913 1928.32,604.099 1935.17,604.281 1942.02,604.459 1948.87,604.633 \n",
                            "  1955.71,604.803 1962.56,604.969 1969.41,605.132 1976.26,605.291 1983.11,605.446 1989.96,605.598 1996.81,605.747 2003.66,605.892 2010.51,606.034 2017.36,606.172 \n",
                            "  2024.21,606.308 2031.05,606.44 2037.9,606.57 2044.75,606.696 2051.6,606.82 2058.45,606.94 2065.3,607.058 2072.15,607.173 2079,607.286 2085.85,607.396 \n",
                            "  2092.7,607.503 2099.55,607.608 2106.39,607.71 2113.24,607.81 2120.09,607.908 2126.94,608.003 2133.79,608.096 2140.64,608.187 2147.49,608.275 2154.34,608.361 \n",
                            "  2161.19,608.446 2168.04,608.528 2174.89,608.607 2181.73,608.685 2188.58,608.761 2195.43,608.835 2202.28,608.907 2209.13,608.977 2215.98,609.046 2222.83,609.112 \n",
                            "  2229.68,609.177 2236.53,609.239 2243.38,609.3 2250.23,609.359 2257.07,609.417 2263.92,609.473 2270.77,609.527 2277.62,609.579 2284.47,609.63 2291.32,609.679 \n",
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                            "  311.935,305.346 318.784,286.32 325.633,268.925 332.482,253.064 339.331,238.643 346.181,225.574 353.03,213.775 359.879,203.168 366.728,193.681 373.577,185.243 \n",
                            "  380.426,177.79 387.275,171.261 394.124,165.598 400.973,160.748 407.822,156.659 414.671,153.283 421.52,150.577 428.37,148.496 435.219,147.001 442.068,146.055 \n",
                            "  448.917,145.622 455.766,145.668 462.615,146.162 469.464,147.073 476.313,148.375 483.162,150.04 490.011,152.044 496.86,154.362 503.709,156.972 510.558,159.853 \n",
                            "  517.408,162.985 524.257,166.348 531.106,169.926 537.955,173.702 544.804,177.658 551.653,181.78 558.502,186.054 565.351,190.465 572.2,195.002 579.049,199.652 \n",
                            "  585.898,204.403 592.747,209.245 599.597,214.168 606.446,219.161 613.295,224.215 620.144,229.322 626.993,234.474 633.842,239.663 640.691,244.882 647.54,250.123 \n",
                            "  654.389,255.381 661.238,260.65 668.087,265.922 674.936,271.194 681.786,276.46 688.635,281.716 695.484,286.957 702.333,292.178 709.182,297.377 716.031,302.549 \n",
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                            "  791.371,356.875 798.22,361.525 805.069,366.12 811.918,370.66 818.767,375.143 825.616,379.568 832.465,383.935 839.314,388.244 846.163,392.493 853.013,396.682 \n",
                            "  859.862,400.812 866.711,404.881 873.56,408.89 880.409,412.838 887.258,416.726 894.107,420.554 900.956,424.322 907.805,428.029 914.654,431.677 921.503,435.265 \n",
                            "  928.352,438.793 935.202,442.263 942.051,445.674 948.9,449.027 955.749,452.322 962.598,455.56 969.447,458.742 976.296,461.867 983.145,464.937 989.994,467.952 \n",
                            "  996.843,470.912 1003.69,473.818 1010.54,476.671 1017.39,479.472 1024.24,482.221 1031.09,484.919 1037.94,487.566 1044.79,490.163 1051.64,492.711 1058.49,495.21 \n",
                            "  1065.33,497.662 1072.18,500.067 1079.03,502.425 1085.88,504.737 1092.73,507.005 1099.58,509.228 1106.43,511.407 1113.28,513.544 1120.13,515.638 1126.98,517.691 \n",
                            "  1133.82,519.703 1140.67,521.675 1147.52,523.607 1154.37,525.5 1161.22,527.356 1168.07,529.173 1174.92,530.954 1181.77,532.699 1188.62,534.408 1195.47,536.083 \n",
                            "  1202.32,537.723 1209.16,539.329 1216.01,540.902 1222.86,542.443 1229.71,543.952 1236.56,545.43 1243.41,546.878 1250.26,548.295 1257.11,549.683 1263.96,551.041 \n",
                            "  1270.81,552.372 1277.66,553.674 1284.5,554.95 1291.35,556.198 1298.2,557.42 1305.05,558.617 1311.9,559.788 1318.75,560.935 1325.6,562.057 1332.45,563.156 \n",
                            "  1339.3,564.231 1346.15,565.284 1353,566.314 1359.84,567.322 1366.69,568.309 1373.54,569.274 1380.39,570.219 1387.24,571.144 1394.09,572.049 1400.94,572.935 \n",
                            "  1407.79,573.802 1414.64,574.65 1421.49,575.48 1428.34,576.292 1435.18,577.087 1442.03,577.864 1448.88,578.625 1455.73,579.37 1462.58,580.098 1469.43,580.811 \n",
                            "  1476.28,581.508 1483.13,582.19 1489.98,582.857 1496.83,583.51 1503.68,584.149 1510.52,584.774 1517.37,585.385 1524.22,585.984 1531.07,586.569 1537.92,587.141 \n",
                            "  1544.77,587.701 1551.62,588.249 1558.47,588.785 1565.32,589.309 1572.17,589.822 1579.02,590.323 1585.86,590.814 1592.71,591.294 1599.56,591.763 1606.41,592.222 \n",
                            "  1613.26,592.671 1620.11,593.111 1626.96,593.541 1633.81,593.961 1640.66,594.372 1647.51,594.774 1654.36,595.167 1661.2,595.552 1668.05,595.928 1674.9,596.296 \n",
                            "  1681.75,596.656 1688.6,597.008 1695.45,597.353 1702.3,597.689 1709.15,598.019 1716,598.341 1722.85,598.656 1729.69,598.964 1736.54,599.265 1743.39,599.56 \n",
                            "  1750.24,599.848 1757.09,600.13 1763.94,600.405 1770.79,600.675 1777.64,600.939 1784.49,601.196 1791.34,601.448 1798.19,601.695 1805.03,601.936 1811.88,602.172 \n",
                            "  1818.73,602.402 1825.58,602.628 1832.43,602.848 1839.28,603.064 1846.13,603.275 1852.98,603.481 1859.83,603.683 1866.68,603.88 1873.53,604.073 1880.37,604.261 \n",
                            "  1887.22,604.446 1894.07,604.626 1900.92,604.802 1907.77,604.975 1914.62,605.143 1921.47,605.308 1928.32,605.469 1935.17,605.627 1942.02,605.781 1948.87,605.932 \n",
                            "  1955.71,606.079 1962.56,606.223 1969.41,606.364 1976.26,606.502 1983.11,606.637 1989.96,606.768 1996.81,606.897 2003.66,607.023 2010.51,607.146 2017.36,607.267 \n",
                            "  2024.21,607.385 2031.05,607.5 2037.9,607.612 2044.75,607.722 2051.6,607.83 2058.45,607.935 2065.3,608.038 2072.15,608.139 2079,608.237 2085.85,608.334 \n",
                            "  2092.7,608.428 2099.55,608.52 2106.39,608.61 2113.24,608.698 2120.09,608.784 2126.94,608.868 2133.79,608.951 2140.64,609.031 2147.49,609.11 2154.34,609.187 \n",
                            "  2161.19,609.263 2168.04,609.337 2174.89,609.409 2181.73,609.479 2188.58,609.548 2195.43,609.616 2202.28,609.682 2209.13,609.747 2215.98,609.81 2222.83,609.872 \n",
                            "  2229.68,609.933 2236.53,609.992 2243.38,610.05 2250.23,610.107 2257.07,610.163 2263.92,610.217 2270.77,610.271 2277.62,610.323 2284.47,610.375 2291.32,610.425 \n",
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                            "  859.862,433.537 866.711,437.289 873.56,440.971 880.409,444.585 887.258,448.13 894.107,451.609 900.956,455.022 907.805,458.369 914.654,461.651 921.503,464.87 \n",
                            "  928.352,468.025 935.202,471.119 942.051,474.152 948.9,477.124 955.749,480.038 962.598,482.893 969.447,485.69 976.296,488.431 983.145,491.117 989.994,493.747 \n",
                            "  996.843,496.324 1003.69,498.849 1010.54,501.321 1017.39,503.742 1024.24,506.113 1031.09,508.435 1037.94,510.709 1044.79,512.935 1051.64,515.114 1058.49,517.248 \n",
                            "  1065.33,519.337 1072.18,521.382 1079.03,523.384 1085.88,525.343 1092.73,527.261 1099.58,529.138 1106.43,530.975 1113.28,532.772 1120.13,534.532 1126.98,536.253 \n",
                            "  1133.82,537.938 1140.67,539.587 1147.52,541.2 1154.37,542.778 1161.22,544.322 1168.07,545.833 1174.92,547.312 1181.77,548.758 1188.62,550.173 1195.47,551.557 \n",
                            "  1202.32,552.911 1209.16,554.235 1216.01,555.531 1222.86,556.799 1229.71,558.039 1236.56,559.251 1243.41,560.437 1250.26,561.598 1257.11,562.733 1263.96,563.843 \n",
                            "  1270.81,564.928 1277.66,565.99 1284.5,567.028 1291.35,568.044 1298.2,569.037 1305.05,570.009 1311.9,570.959 1318.75,571.888 1325.6,572.796 1332.45,573.685 \n",
                            "  1339.3,574.554 1346.15,575.403 1353,576.234 1359.84,577.047 1366.69,577.841 1373.54,578.618 1380.39,579.378 1387.24,580.12 1394.09,580.847 1400.94,581.557 \n",
                            "  1407.79,582.251 1414.64,582.931 1421.49,583.594 1428.34,584.244 1435.18,584.878 1442.03,585.499 1448.88,586.106 1455.73,586.699 1462.58,587.279 1469.43,587.846 \n",
                            "  1476.28,588.401 1483.13,588.943 1489.98,589.473 1496.83,589.991 1503.68,590.498 1510.52,590.993 1517.37,591.477 1524.22,591.951 1531.07,592.414 1537.92,592.866 \n",
                            "  1544.77,593.309 1551.62,593.741 1558.47,594.164 1565.32,594.577 1572.17,594.981 1579.02,595.377 1585.86,595.763 1592.71,596.14 1599.56,596.509 1606.41,596.87 \n",
                            "  1613.26,597.223 1620.11,597.568 1626.96,597.905 1633.81,598.234 1640.66,598.556 1647.51,598.871 1654.36,599.179 1661.2,599.48 1668.05,599.774 1674.9,600.062 \n",
                            "  1681.75,600.343 1688.6,600.617 1695.45,600.886 1702.3,601.149 1709.15,601.405 1716,601.656 1722.85,601.901 1729.69,602.141 1736.54,602.375 1743.39,602.604 \n",
                            "  1750.24,602.828 1757.09,603.047 1763.94,603.261 1770.79,603.47 1777.64,603.674 1784.49,603.874 1791.34,604.069 1798.19,604.26 1805.03,604.447 1811.88,604.629 \n",
                            "  1818.73,604.808 1825.58,604.982 1832.43,605.152 1839.28,605.319 1846.13,605.481 1852.98,605.641 1859.83,605.796 1866.68,605.948 1873.53,606.097 1880.37,606.242 \n",
                            "  1887.22,606.384 1894.07,606.523 1900.92,606.659 1907.77,606.791 1914.62,606.921 1921.47,607.048 1928.32,607.172 1935.17,607.293 1942.02,607.411 1948.87,607.527 \n",
                            "  1955.71,607.641 1962.56,607.751 1969.41,607.86 1976.26,607.966 1983.11,608.069 1989.96,608.17 1996.81,608.269 2003.66,608.366 2010.51,608.461 2017.36,608.554 \n",
                            "  2024.21,608.644 2031.05,608.733 2037.9,608.82 2044.75,608.905 2051.6,608.988 2058.45,609.069 2065.3,609.148 2072.15,609.226 2079,609.302 2085.85,609.377 \n",
                            "  2092.7,609.45 2099.55,609.522 2106.39,609.592 2113.24,609.66 2120.09,609.727 2126.94,609.793 2133.79,609.858 2140.64,609.921 2147.49,609.983 2154.34,610.044 \n",
                            "  2161.19,610.104 2168.04,610.162 2174.89,610.22 2181.73,610.276 2188.58,610.332 2195.43,610.386 2202.28,610.439 2209.13,610.492 2215.98,610.544 2222.83,610.595 \n",
                            "  2229.68,610.645 2236.53,610.694 2243.38,610.743 2250.23,610.791 2257.07,610.838 2263.92,610.885 2270.77,610.931 2277.62,610.976 2284.47,611.021 2291.32,611.066 \n",
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                            "  1762.43,265.675 1906.43,265.675 \n",
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                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1930.43, 283.175)\" x=\"1930.43\" y=\"283.175\">z(1) = -75.0%</text>\n",
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                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1930.43, 343.655)\" x=\"1930.43\" y=\"343.655\">z(1) = -50.0%</text>\n",
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                            "  1762.43,386.635 1906.43,386.635 \n",
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                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1930.43, 404.135)\" x=\"1930.43\" y=\"404.135\">z(1) = -25.0%</text>\n",
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                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 1930.43, 464.615)\" x=\"1930.43\" y=\"464.615\">z(1) = 25.0%</text>\n",
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                            "  1762.43,507.595 1906.43,507.595 \n",
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                            "  1887.22,1178.76 1894.07,1178.65 1900.92,1178.55 1907.77,1178.45 1914.62,1178.35 1921.47,1178.26 1928.32,1178.17 1935.17,1178.07 1942.02,1177.98 1948.87,1177.9 \n",
                            "  1955.71,1177.81 1962.56,1177.73 1969.41,1177.65 1976.26,1177.57 1983.11,1177.49 1989.96,1177.41 1996.81,1177.34 2003.66,1177.27 2010.51,1177.19 2017.36,1177.13 \n",
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                            "  1681.75,1181.9 1688.6,1181.72 1695.45,1181.55 1702.3,1181.38 1709.15,1181.22 1716,1181.06 1722.85,1180.9 1729.69,1180.74 1736.54,1180.59 1743.39,1180.44 \n",
                            "  1750.24,1180.3 1757.09,1180.16 1763.94,1180.02 1770.79,1179.88 1777.64,1179.75 1784.49,1179.62 1791.34,1179.5 1798.19,1179.37 1805.03,1179.25 1811.88,1179.13 \n",
                            "  1818.73,1179.02 1825.58,1178.9 1832.43,1178.79 1839.28,1178.68 1846.13,1178.58 1852.98,1178.48 1859.83,1178.37 1866.68,1178.28 1873.53,1178.18 1880.37,1178.08 \n",
                            "  1887.22,1177.99 1894.07,1177.9 1900.92,1177.81 1907.77,1177.73 1914.62,1177.64 1921.47,1177.56 1928.32,1177.48 1935.17,1177.4 1942.02,1177.32 1948.87,1177.25 \n",
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                            "  1613.26,1183.06 1620.11,1182.86 1626.96,1182.66 1633.81,1182.47 1640.66,1182.28 1647.51,1182.09 1654.36,1181.91 1661.2,1181.73 1668.05,1181.55 1674.9,1181.38 \n",
                            "  1681.75,1181.22 1688.6,1181.05 1695.45,1180.9 1702.3,1180.74 1709.15,1180.59 1716,1180.44 1722.85,1180.29 1729.69,1180.15 1736.54,1180.01 1743.39,1179.87 \n",
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                    },
                    "execution_count": 17,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "plot(p2,p3, fg_legend = :transparent, legend=:best, layout=(2,1))"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: The top panel shows the scaled impulse response function of capital for different aggregate shocks. The bottom panel shows the scaled impulse response function of capital for different aggregate shocks, multiplied by the sign of the aggregate shock.*"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### II.D. Out-of-steady-state dynamics"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 18,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "  0.231372 seconds (1.06 M allocations: 50.189 MiB, 5.34% gc time)\n"
                    ]
                }
            ],
            "source": [
                "max_t = 2000\n",
                "shocks_t = rand(Normal(0,0.005), max_t) # Series of aggregate shocks\n",
                "# Let's generate a path for the aggregate shock\n",
                "z_path = ones(max_t)\n",
                "z_path[1] = z_ss\n",
                "\n",
                "# Evolution of aggregate productivity in level:\n",
                "for t_index=2:max_t \n",
                "    z_path[t_index] = z_path[t_index-1]^p.rho + shocks_t[t_index]\n",
                "end\n",
                "\n",
                "# Calculation of GenBKM path:\n",
                "XT_GenBKM = zeros(max_t);# Initialization\n",
                "@time GenBKM_path!(XT_GenBKM, max_t, x_mit_scaled_sigma, z_path./z_ss .- 1.0, array_sigma)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 19,
            "metadata": {
                "scrolled": false
            },
            "outputs": [
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                            "  281.98,544.798 283.011,550.914 284.042,561.26 285.072,571.932 286.103,570.034 287.134,541.92 288.165,455.71 289.195,458.857 290.226,413.315 291.257,410.483 \n",
                            "  292.288,439.189 293.319,464.369 294.349,411.155 295.38,423.204 296.411,467.716 297.442,456.532 298.472,436.037 299.503,442.906 300.534,425.597 301.565,401.174 \n",
                            "  302.595,391.676 303.626,323.874 304.657,368.123 305.688,361.765 306.718,383.374 307.749,375.125 308.78,422.146 309.811,479.03 310.842,469.025 311.872,436.921 \n",
                            "  312.903,454.306 313.934,439.785 314.965,466.855 315.995,453.596 317.026,472.432 318.057,458.957 319.088,488.421 320.118,454.764 321.149,443.606 322.18,398.751 \n",
                            "  323.211,425.735 324.241,430.616 325.272,417.068 326.303,379.194 327.334,333.212 328.365,330.297 329.395,356.065 330.426,368.359 331.457,418.162 332.488,429.818 \n",
                            "  333.518,418.863 334.549,454.079 335.58,415.457 336.611,439.419 337.641,410.162 338.672,446.382 339.703,450.016 340.734,402.989 341.764,353.834 342.795,328.075 \n",
                            "  343.826,307.52 344.857,305.128 345.888,282.938 346.918,312.743 347.949,260.105 348.98,277.752 350.011,236.976 351.041,241.039 352.072,261.578 353.103,278.149 \n",
                            "  354.134,262.375 355.164,258.068 356.195,187.409 357.226,197.491 358.257,205.808 359.287,198.398 360.318,234.534 361.349,213.649 362.38,191.579 363.411,192.706 \n",
                            "  364.441,190.367 365.472,238.692 366.503,122.842 367.534,106.924 368.564,128.164 369.595,148.508 370.626,89.7582 371.657,123.972 372.687,164.52 373.718,192.608 \n",
                            "  374.749,212.073 375.78,283.639 376.81,272.202 377.841,282.847 378.872,260.046 379.903,260.08 380.934,321.834 381.964,324.49 382.995,347.773 384.026,352.603 \n",
                            "  385.057,350.041 386.087,364.096 387.118,324.307 388.149,338.642 389.18,343.295 390.21,315.511 391.241,362.709 392.272,441.663 393.303,407.021 394.333,356.486 \n",
                            "  395.364,401.356 396.395,423.326 397.426,416.904 398.457,387.551 399.487,411.595 400.518,369.274 401.549,399.308 402.58,432.269 403.61,452.257 404.641,391.458 \n",
                            "  405.672,427.315 406.703,424.2 407.733,389.122 408.764,350.51 409.795,339.719 410.826,341.718 411.856,326.511 412.887,321.74 413.918,311.504 414.949,337.808 \n",
                            "  415.98,278.309 417.01,293.44 418.041,241.722 419.072,171.493 420.103,139.286 421.133,164.527 422.164,131.005 423.195,163.259 424.226,225.152 425.256,240.964 \n",
                            "  426.287,274.371 427.318,251.171 428.349,260.438 429.379,283.8 430.41,328.959 431.441,335.184 432.472,372.896 433.503,339.859 434.533,325.756 435.564,341.092 \n",
                            "  436.595,293.883 437.626,256.7 438.656,226.954 439.687,226.382 440.718,209.025 441.749,217 442.779,220.772 443.81,167.898 444.841,149.095 445.872,168.386 \n",
                            "  446.902,169.24 447.933,183.306 448.964,136.786 449.995,142.193 451.026,131.407 452.056,193.452 453.087,121.046 454.118,179.467 455.149,185.623 456.179,203.592 \n",
                            "  457.21,208.696 458.241,256.525 459.272,312.65 460.302,299.87 461.333,322.264 462.364,314.994 463.395,270.248 464.425,285.628 465.456,256.912 466.487,232.693 \n",
                            "  467.518,209.824 468.549,232.767 469.579,251.302 470.61,285.088 471.641,254.864 472.672,287.976 473.702,313.05 474.733,288.607 475.764,334.05 476.795,377.466 \n",
                            "  477.825,371.88 478.856,388.169 479.887,358.087 480.918,352.439 481.949,346.591 482.979,336.282 484.01,346.28 485.041,322.636 486.072,365.864 487.102,394.574 \n",
                            "  488.133,345.597 489.164,341.909 490.195,362.44 491.225,339.071 492.256,341.295 493.287,332.681 494.318,330.431 495.348,376.719 496.379,432.259 497.41,407.546 \n",
                            "  498.441,445.558 499.472,427.703 500.502,459.414 501.533,429.662 502.564,407.412 503.595,413.394 504.625,396.829 505.656,331.587 506.687,368.06 507.718,354.124 \n",
                            "  508.748,343.614 509.779,328.948 510.81,323.533 511.841,342.601 512.871,390.943 513.902,428.815 514.933,455.043 515.964,452.345 516.995,438.219 518.025,405.71 \n",
                            "  519.056,424.005 520.087,407.81 521.118,438.82 522.148,431.874 523.179,442.085 524.21,388.044 525.241,364.475 526.271,375.503 527.302,385.61 528.333,359.4 \n",
                            "  529.364,359.759 530.394,352.295 531.425,320.458 532.456,326.456 533.487,301.013 534.518,274.535 535.548,263.044 536.579,234.565 537.61,255.328 538.641,282.16 \n",
                            "  539.671,231.455 540.702,224.566 541.733,227.634 542.764,250.558 543.794,258.499 544.825,262.178 545.856,237.012 546.887,281.288 547.917,274.794 548.948,352.835 \n",
                            "  549.979,355.643 551.01,351.633 552.041,346.105 553.071,367.049 554.102,374.251 555.133,416.299 556.164,440.565 557.194,420.642 558.225,414.821 559.256,356.657 \n",
                            "  560.287,376.294 561.317,350.543 562.348,375.585 563.379,370.989 564.41,368.552 565.44,353.619 566.471,357.849 567.502,357.403 568.533,355.154 569.564,304.097 \n",
                            "  570.594,234.848 571.625,259.224 572.656,269.41 573.687,303.925 574.717,257.404 575.748,268.842 576.779,232.849 577.81,255.465 578.84,291.273 579.871,286.83 \n",
                            "  580.902,362.132 581.933,349.067 582.963,332.593 583.994,327.248 585.025,351.602 586.056,351.607 587.087,383.968 588.117,373.296 589.148,357.739 590.179,436.679 \n",
                            "  591.21,400.836 592.24,379.143 593.271,374.947 594.302,349.285 595.333,354.762 596.363,358.744 597.394,354.175 598.425,344.316 599.456,272.167 600.486,223.259 \n",
                            "  601.517,239.135 602.548,217.379 603.579,220.733 604.61,259.934 605.64,294.508 606.671,309.005 607.702,339.106 608.733,369.12 609.763,373.716 610.794,372.951 \n",
                            "  611.825,444.107 612.856,429.212 613.886,397.29 614.917,350.013 615.948,338.949 616.979,331.606 618.009,282.312 619.04,241.498 620.071,274.649 621.102,278.551 \n",
                            "  622.133,285.503 623.163,278.076 624.194,291.265 625.225,285.664 626.256,332.064 627.286,341.417 628.317,350.046 629.348,356.954 630.379,386.743 631.409,416.652 \n",
                            "  632.44,455.075 633.471,474.674 634.502,464.685 635.532,403.346 636.563,403.769 637.594,397.55 638.625,380.206 639.656,394.308 640.686,425.772 641.717,431.075 \n",
                            "  642.748,418.866 643.779,404.274 644.809,391.678 645.84,424.311 646.871,393.217 647.902,373.444 648.932,392.231 649.963,396.636 650.994,420.055 652.025,384.566 \n",
                            "  653.055,346.383 654.086,384.225 655.117,404.062 656.148,425.877 657.179,430.31 658.209,437.099 659.24,349.412 660.271,311.682 661.302,295.524 662.332,300.466 \n",
                            "  663.363,279.185 664.394,267.501 665.425,251.593 666.455,267.185 667.486,281.782 668.517,333.7 669.548,319.63 670.578,297.035 671.609,245.233 672.64,298.05 \n",
                            "  673.671,320.723 674.702,330.732 675.732,297.92 676.763,279.365 677.794,258.583 678.825,276.54 679.855,280.537 680.886,276.906 681.917,273.252 682.948,251.74 \n",
                            "  683.978,236.954 685.009,226.529 686.04,218.334 687.071,223.164 688.101,190.608 689.132,218.272 690.163,260.104 691.194,183.169 692.225,147.44 693.255,156.188 \n",
                            "  694.286,194.536 695.317,229.551 696.348,232.516 697.378,245.094 698.409,258.945 699.44,289.793 700.471,284.894 701.501,284.051 702.532,294.559 703.563,342.175 \n",
                            "  704.594,368.988 705.624,392.132 706.655,407.341 707.686,366.42 708.717,342.788 709.748,374.328 710.778,402.051 711.809,363.734 712.84,335.376 713.871,270.265 \n",
                            "  714.901,264.309 715.932,222.103 716.963,246.709 717.994,210.901 719.024,195.715 720.055,219.364 721.086,249.281 722.117,192.81 723.147,137.67 724.178,120.44 \n",
                            "  725.209,148.039 726.24,186.723 727.271,183.638 728.301,182.937 729.332,220.054 730.363,241.609 731.394,275.376 732.424,278.596 733.455,255.92 734.486,291.723 \n",
                            "  735.517,284.521 736.547,316.922 737.578,286.729 738.609,282.271 739.64,280.376 740.67,311.451 741.701,311.199 742.732,286.065 743.763,338.806 744.794,349.682 \n",
                            "  745.824,303.71 746.855,356.249 747.886,386.388 748.917,413.58 749.947,394.349 750.978,439.267 752.009,421.435 753.04,449.387 754.07,492.936 755.101,499.421 \n",
                            "  756.132,474.468 757.163,472.936 758.193,488.494 759.224,440.651 760.255,494.258 761.286,455.741 762.317,454.02 763.347,395.427 764.378,335.903 765.409,353.705 \n",
                            "  766.44,340.245 767.47,316.63 768.501,333.791 769.532,315.789 770.563,308.495 771.593,291.087 772.624,226.075 773.655,245.184 774.686,308.715 775.717,314.149 \n",
                            "  776.747,324.075 777.778,340.207 778.809,374.853 779.84,314.054 780.87,320.869 781.901,314.313 782.932,264.126 783.963,260.269 784.993,310.134 786.024,399.606 \n",
                            "  787.055,385.137 788.086,392.34 789.116,385.446 790.147,388.036 791.178,401.124 792.209,386.225 793.24,394.053 794.27,402.174 795.301,387.272 796.332,415.672 \n",
                            "  797.363,475.376 798.393,487.922 799.424,467.526 800.455,449.66 801.486,476.173 802.516,490.741 803.547,489.433 804.578,468.182 805.609,471.265 806.639,513.729 \n",
                            "  807.67,496.474 808.701,490.136 809.732,489.287 810.763,405.179 811.793,415.504 812.824,404.883 813.855,365.738 814.886,337.709 815.916,332.577 816.947,359.122 \n",
                            "  817.978,420.363 819.009,391.334 820.039,451.101 821.07,467.814 822.101,506.638 823.132,517.37 824.162,533.534 825.193,493.198 826.224,454.785 827.255,433.457 \n",
                            "  828.286,421.498 829.316,409.541 830.347,361.2 831.378,338.319 832.409,395.241 833.439,372.848 834.47,375.358 835.501,379.504 836.532,350.221 837.562,370.87 \n",
                            "  838.593,347.083 839.624,363.047 840.655,350.936 841.685,348.895 842.716,357.885 843.747,312.249 844.778,284.21 845.809,277.207 846.839,278.108 847.87,306.247 \n",
                            "  848.901,323.875 849.932,369.893 850.962,334.694 851.993,382.366 853.024,366.378 854.055,351.411 855.085,393.617 856.116,332.363 857.147,291.953 858.178,246.602 \n",
                            "  859.208,292.618 860.239,311.697 861.27,354.504 862.301,311.893 863.332,338.306 864.362,291.448 865.393,310.573 866.424,309.355 867.455,282.812 868.485,335.691 \n",
                            "  869.516,387.723 870.547,424.501 871.578,421.093 872.608,365.784 873.639,405.235 874.67,418.398 875.701,429.804 876.731,462.164 877.762,484.488 878.793,499.986 \n",
                            "  879.824,452.357 880.855,431.081 881.885,450.642 882.916,447.473 883.947,391.748 884.978,398.379 886.008,395.356 887.039,376.55 888.07,359.574 889.101,307.458 \n",
                            "  890.131,318.701 891.162,335.363 892.193,324.025 893.224,367.722 894.254,332.823 895.285,297.427 896.316,358.533 897.347,387.36 898.378,382.082 899.408,374.16 \n",
                            "  900.439,394.758 901.47,323.465 902.501,283.891 903.531,271.862 904.562,259.837 905.593,292.82 906.624,293.285 907.654,264.164 908.685,284.733 909.716,271.658 \n",
                            "  910.747,253.165 911.777,256.439 912.808,250.669 913.839,235.031 914.87,238.322 915.901,267.705 916.931,291.945 917.962,328.282 918.993,297.861 920.024,277.503 \n",
                            "  921.054,281.55 922.085,244.388 923.116,189.751 924.147,221.455 925.177,168.946 926.208,189.925 927.239,189.649 928.27,212.896 929.3,216.191 930.331,201.814 \n",
                            "  931.362,203.482 932.393,237.79 933.424,249.767 934.454,221.881 935.485,208.569 936.516,223.224 937.547,200.659 938.577,196.467 939.608,177.663 940.639,219.323 \n",
                            "  941.67,254.558 942.7,307.518 943.731,243.28 944.762,216.023 945.793,226.812 946.823,241.698 947.854,199.371 948.885,281.01 949.916,306.263 950.947,313.719 \n",
                            "  951.977,315.998 953.008,292.186 954.039,330.689 955.07,384.302 956.1,354.791 957.131,335.635 958.162,277.523 959.193,321.522 960.223,263.424 961.254,273.807 \n",
                            "  962.285,251.909 963.316,261.952 964.346,318.384 965.377,303.197 966.408,287.99 967.439,345.829 968.47,349.933 969.5,402.158 970.531,393.69 971.562,411.102 \n",
                            "  972.593,350.442 973.623,338.933 974.654,359.322 975.685,396.641 976.716,328.264 977.746,256.34 978.777,242.065 979.808,250.9 980.839,270.36 981.869,305.146 \n",
                            "  982.9,283.746 983.931,349.465 984.962,387.614 985.993,386.807 987.023,426.965 988.054,405.027 989.085,410.974 990.116,424.711 991.146,364.755 992.177,343.213 \n",
                            "  993.208,382.764 994.239,443.062 995.269,473.514 996.3,471.437 997.331,521.573 998.362,545.103 999.392,555.12 1000.42,592.57 1001.45,552.476 1002.48,590.24 \n",
                            "  1003.52,589.552 1004.55,581.814 1005.58,554.937 1006.61,546.241 1007.64,535.425 1008.67,518.285 1009.7,525.02 1010.73,521.371 1011.76,502.394 1012.79,520.623 \n",
                            "  1013.82,548.446 1014.85,571.778 1015.88,531.642 1016.92,479.575 1017.95,522.148 1018.98,514.31 1020.01,501.171 1021.04,527.175 1022.07,529.279 1023.1,513.188 \n",
                            "  1024.13,447.513 1025.16,430.803 1026.19,451.902 1027.22,417.419 1028.25,468.513 1029.28,403.127 1030.32,414.48 1031.35,399.116 1032.38,406.75 1033.41,366.982 \n",
                            "  1034.44,329.822 1035.47,334.111 1036.5,300.45 1037.53,317.731 1038.56,331.914 1039.59,255.407 1040.62,274.355 1041.65,278.053 1042.68,278.993 1043.72,285.857 \n",
                            "  1044.75,295.374 1045.78,235.505 1046.81,229.893 1047.84,274.114 1048.87,297.722 1049.9,349.715 1050.93,372.285 1051.96,339.874 1052.99,314.004 1054.02,246.213 \n",
                            "  1055.05,243.318 1056.08,280.375 1057.12,227.877 1058.15,182.893 1059.18,196.494 1060.21,269.692 1061.24,267.993 1062.27,274.794 1063.3,293.446 1064.33,265.634 \n",
                            "  1065.36,299.034 1066.39,299.067 1067.42,333.02 1068.45,316.44 1069.48,299.301 1070.52,269.94 1071.55,235.7 1072.58,286.001 1073.61,325.001 1074.64,281.898 \n",
                            "  1075.67,261.745 1076.7,244.419 1077.73,216.343 1078.76,219.795 1079.79,270.718 1080.82,289.169 1081.85,281.547 1082.88,230.134 1083.92,231.034 1084.95,282.453 \n",
                            "  1085.98,260.286 1087.01,300.485 1088.04,303.783 1089.07,314.762 1090.1,381.934 1091.13,371.447 1092.16,371.435 1093.19,370.086 1094.22,364.046 1095.25,320.46 \n",
                            "  1096.28,323.614 1097.32,396.049 1098.35,344.765 1099.38,313.757 1100.41,346.284 1101.44,292.518 1102.47,331.057 1103.5,317.483 1104.53,285.087 1105.56,257.264 \n",
                            "  1106.59,222.433 1107.62,219.28 1108.65,246.683 1109.68,265.054 1110.72,244.79 1111.75,266.025 1112.78,291.425 1113.81,269.067 1114.84,285.195 1115.87,329.124 \n",
                            "  1116.9,329.497 1117.93,285.459 1118.96,317.022 1119.99,346.442 1121.02,328.484 1122.05,342.76 1123.08,337.747 1124.12,336.556 1125.15,338.649 1126.18,367.503 \n",
                            "  1127.21,370.54 1128.24,389.014 1129.27,387.415 1130.3,354.743 1131.33,399.37 1132.36,406.029 1133.39,387.603 1134.42,417.129 1135.45,387.536 1136.48,406.675 \n",
                            "  1137.51,404.074 1138.55,434.431 1139.58,441.402 1140.61,435.574 1141.64,370.307 1142.67,349.795 1143.7,424.699 1144.73,444.278 1145.76,441.903 1146.79,425.92 \n",
                            "  1147.82,479.458 1148.85,455.757 1149.88,460.126 1150.91,470.638 1151.95,526.833 1152.98,539.974 1154.01,554.521 1155.04,482.817 1156.07,506.659 1157.1,548.311 \n",
                            "  1158.13,609.25 1159.16,593.584 1160.19,570.788 1161.22,588.549 1162.25,568.382 1163.28,577.155 1164.31,560.929 1165.35,518.918 1166.38,518.065 1167.41,502.618 \n",
                            "  1168.44,501.416 1169.47,485.112 1170.5,426.179 1171.53,394.113 1172.56,411.771 1173.59,444.222 1174.62,471.435 1175.65,498.023 1176.68,443.24 1177.71,539.205 \n",
                            "  1178.75,541.103 1179.78,502.742 1180.81,470.134 1181.84,402.06 1182.87,396.346 1183.9,340.823 1184.93,291.491 1185.96,302.842 1186.99,283.223 1188.02,295.042 \n",
                            "  1189.05,312.241 1190.08,280.721 1191.11,256.273 1192.15,312.41 1193.18,339.399 1194.21,354.247 1195.24,355.261 1196.27,320.712 1197.3,285.399 1198.33,298.443 \n",
                            "  1199.36,320.716 1200.39,342.358 1201.42,369.041 1202.45,371.837 1203.48,399.29 1204.51,429.477 1205.55,442.449 1206.58,433.119 1207.61,451.75 1208.64,429.941 \n",
                            "  1209.67,406.96 1210.7,381.655 1211.73,314.644 1212.76,289.847 1213.79,287.933 1214.82,241.781 1215.85,163.111 1216.88,198.672 1217.91,165.515 1218.95,183.943 \n",
                            "  1219.98,197.42 1221.01,184.66 1222.04,185.622 1223.07,236.168 1224.1,238.25 1225.13,171.142 1226.16,154.196 1227.19,176.023 1228.22,201.903 1229.25,204.113 \n",
                            "  1230.28,241.429 1231.31,314.584 1232.35,335.59 1233.38,305.319 1234.41,273.985 1235.44,209.904 1236.47,252.856 1237.5,227.061 1238.53,202.228 1239.56,210.459 \n",
                            "  1240.59,230.926 1241.62,213.161 1242.65,226.441 1243.68,255.22 1244.71,241.426 1245.75,293.117 1246.78,297.977 1247.81,275.688 1248.84,321.364 1249.87,331.433 \n",
                            "  1250.9,299.898 1251.93,287.671 1252.96,325.491 1253.99,334.142 1255.02,355.642 1256.05,340.256 1257.08,381.854 1258.11,357.592 1259.15,333.555 1260.18,337.093 \n",
                            "  1261.21,368.016 1262.24,380.718 1263.27,401.686 1264.3,415.514 1265.33,405.418 1266.36,434.135 1267.39,402.364 1268.42,418.144 1269.45,468.589 1270.48,419.905 \n",
                            "  1271.51,413.681 1272.55,336.436 1273.58,324.403 1274.61,349.254 1275.64,372.624 1276.67,351.985 1277.7,369.897 1278.73,377.787 1279.76,397.119 1280.79,408.774 \n",
                            "  1281.82,389.279 1282.85,353.882 1283.88,328.381 1284.91,354.932 1285.95,331.797 1286.98,329.6 1288.01,265.279 1289.04,279.882 1290.07,290.842 1291.1,258.349 \n",
                            "  1292.13,284.153 1293.16,226.504 1294.19,220.885 1295.22,231.674 1296.25,269.653 1297.28,275.596 1298.31,244.218 1299.35,226.454 1300.38,285.385 1301.41,264.998 \n",
                            "  1302.44,298.432 1303.47,347.761 1304.5,345.827 1305.53,329.672 1306.56,360.235 1307.59,364.691 1308.62,384.361 1309.65,416.045 1310.68,356.992 1311.71,306.99 \n",
                            "  1312.75,281.865 1313.78,284.411 1314.81,299.652 1315.84,292.118 1316.87,246.643 1317.9,290.531 1318.93,248.826 1319.96,272.157 1320.99,252.53 1322.02,342.736 \n",
                            "  1323.05,289.335 1324.08,320.008 1325.11,295.145 1326.14,289.845 1327.18,344.897 1328.21,323.542 1329.24,301.678 1330.27,296.03 1331.3,373.942 1332.33,394.23 \n",
                            "  1333.36,440.866 1334.39,429.539 1335.42,439.386 1336.45,447.672 1337.48,473.901 1338.51,434.797 1339.54,441.673 1340.58,446.739 1341.61,448.937 1342.64,464.924 \n",
                            "  1343.67,486.749 1344.7,428.708 1345.73,401.599 1346.76,420.948 1347.79,416.914 1348.82,377.643 1349.85,363.794 1350.88,390.182 1351.91,359.558 1352.94,324.121 \n",
                            "  1353.98,321.437 1355.01,421.06 1356.04,418.128 1357.07,368.582 1358.1,397.293 1359.13,396.212 1360.16,394.29 1361.19,360.945 1362.22,368.626 1363.25,344.768 \n",
                            "  1364.28,325.494 1365.31,359.059 1366.34,337.075 1367.38,295.385 1368.41,320.262 1369.44,324.427 1370.47,334.212 1371.5,329.575 1372.53,333.371 1373.56,317.619 \n",
                            "  1374.59,279.608 1375.62,264.367 1376.65,243.131 1377.68,237.528 1378.71,237.849 1379.74,192.09 1380.78,163.163 1381.81,166.663 1382.84,203.635 1383.87,193.233 \n",
                            "  1384.9,172.839 1385.93,206.649 1386.96,182.294 1387.99,193.558 1389.02,164.122 1390.05,206.651 1391.08,260.547 1392.11,286.339 1393.14,308.96 1394.18,316.75 \n",
                            "  1395.21,322.233 1396.24,259.336 1397.27,298.877 1398.3,321.974 1399.33,293.051 1400.36,345.929 1401.39,352.467 1402.42,338.31 1403.45,318.32 1404.48,301.163 \n",
                            "  1405.51,362.178 1406.54,366.778 1407.58,410.883 1408.61,391.798 1409.64,366.65 1410.67,397.106 1411.7,366.858 1412.73,332.873 1413.76,394.262 1414.79,413.961 \n",
                            "  1415.82,394.918 1416.85,355.171 1417.88,380.385 1418.91,397.566 1419.94,402.374 1420.98,383.409 1422.01,364.265 1423.04,323.768 1424.07,308.749 1425.1,289.292 \n",
                            "  1426.13,260.635 1427.16,256.307 1428.19,205.998 1429.22,195.849 1430.25,207.421 1431.28,203.98 1432.31,212.4 1433.34,228.213 1434.38,158.358 1435.41,139.632 \n",
                            "  1436.44,166.25 1437.47,207.039 1438.5,188.929 1439.53,179.936 1440.56,150.534 1441.59,98.3197 1442.62,89.8889 1443.65,66.161 1444.68,71.1299 1445.71,98.0673 \n",
                            "  1446.74,132.139 1447.78,151.213 1448.81,162.792 1449.84,162.592 1450.87,174.898 1451.9,222.358 1452.93,200.166 1453.96,226.26 1454.99,243.147 1456.02,223.726 \n",
                            "  1457.05,171.993 1458.08,210.546 1459.11,236.438 1460.14,262.334 1461.18,310.882 1462.21,312.281 1463.24,300.112 1464.27,332.899 1465.3,339.331 1466.33,331.605 \n",
                            "  1467.36,355.729 1468.39,374.617 1469.42,358.35 1470.45,339.531 1471.48,328.639 1472.51,303.328 1473.54,232.113 1474.58,264.745 1475.61,269.914 1476.64,223.609 \n",
                            "  1477.67,261.922 1478.7,286.629 1479.73,292.06 1480.76,262.519 1481.79,255.986 1482.82,236.219 1483.85,241.672 1484.88,263.362 1485.91,263.403 1486.94,272.979 \n",
                            "  1487.98,285.155 1489.01,260.309 1490.04,274.718 1491.07,283.544 1492.1,316.358 1493.13,295.642 1494.16,316.561 1495.19,286.485 1496.22,232.478 1497.25,215.077 \n",
                            "  1498.28,229.221 1499.31,224.899 1500.34,221.359 1501.38,274.957 1502.41,218.422 1503.44,240.979 1504.47,195.493 1505.5,164.631 1506.53,130.715 1507.56,143.031 \n",
                            "  1508.59,237.492 1509.62,306.922 1510.65,362.379 1511.68,363.145 1512.71,354.199 1513.74,344.179 1514.77,331.803 1515.81,333.433 1516.84,300.699 1517.87,303.97 \n",
                            "  1518.9,320.011 1519.93,293.662 1520.96,347.154 1521.99,345.195 1523.02,315.684 1524.05,309.198 1525.08,286.21 1526.11,245.444 1527.14,278.776 1528.17,246.945 \n",
                            "  1529.21,256.697 1530.24,271.012 1531.27,329.928 1532.3,337.51 1533.33,298.654 1534.36,304.896 1535.39,319.745 1536.42,378.374 1537.45,357.613 1538.48,353.231 \n",
                            "  1539.51,353.098 1540.54,332.298 1541.57,370.471 1542.61,394.878 1543.64,376.84 1544.67,365.436 1545.7,347.873 1546.73,349.158 1547.76,354.489 1548.79,356.624 \n",
                            "  1549.82,325.396 1550.85,338.827 1551.88,341.404 1552.91,307.028 1553.94,274.084 1554.97,280.026 1556.01,235.378 1557.04,193.207 1558.07,190.029 1559.1,263.291 \n",
                            "  1560.13,237.144 1561.16,210.275 1562.19,218.539 1563.22,252.468 1564.25,276.992 1565.28,266.482 1566.31,247.14 1567.34,245.899 1568.37,243.108 1569.41,252.492 \n",
                            "  1570.44,230.578 1571.47,207.162 1572.5,175.768 1573.53,242.845 1574.56,217.89 1575.59,236.807 1576.62,263.263 1577.65,300.549 1578.68,259.407 1579.71,279.065 \n",
                            "  1580.74,343.914 1581.77,320.078 1582.81,312.634 1583.84,307.58 1584.87,341.005 1585.9,343.041 1586.93,337.559 1587.96,343.415 1588.99,418.902 1590.02,405.076 \n",
                            "  1591.05,380.214 1592.08,383.348 1593.11,391.409 1594.14,420.436 1595.17,461.962 1596.21,474.486 1597.24,468.554 1598.27,420.821 1599.3,389.259 1600.33,342.086 \n",
                            "  1601.36,390.28 1602.39,397.883 1603.42,399.607 1604.45,438.068 1605.48,399.422 1606.51,376.229 1607.54,395.65 1608.57,385.783 1609.61,428.945 1610.64,410.365 \n",
                            "  1611.67,415.452 1612.7,360.17 1613.73,334.704 1614.76,340.342 1615.79,365.242 1616.82,305.687 1617.85,277.501 1618.88,269.029 1619.91,243.248 1620.94,224.088 \n",
                            "  1621.97,168.892 1623.01,167.138 1624.04,174.407 1625.07,172.308 1626.1,163.173 1627.13,144.792 1628.16,207.406 1629.19,263.656 1630.22,280.405 1631.25,281.363 \n",
                            "  1632.28,272.927 1633.31,369.142 1634.34,327.465 1635.37,337.028 1636.41,347.478 1637.44,327.323 1638.47,335.681 1639.5,263.677 1640.53,304.558 1641.56,317.521 \n",
                            "  1642.59,350.444 1643.62,331.805 1644.65,282.337 1645.68,296.663 1646.71,234.925 1647.74,212.744 1648.77,202.958 1649.81,222.557 1650.84,281.832 1651.87,333.184 \n",
                            "  1652.9,351.161 1653.93,356.356 1654.96,355 1655.99,370.29 1657.02,368.618 1658.05,260.823 1659.08,340.414 1660.11,343.35 1661.14,337.543 1662.17,328.536 \n",
                            "  1663.21,312.627 1664.24,303.641 1665.27,318.194 1666.3,326.362 1667.33,279.078 1668.36,307.149 1669.39,318.667 1670.42,393.347 1671.45,420.698 1672.48,389.66 \n",
                            "  1673.51,390.066 1674.54,421.692 1675.57,432.225 1676.61,395.186 1677.64,424.376 1678.67,388.048 1679.7,362.072 1680.73,366.769 1681.76,396.259 1682.79,374.992 \n",
                            "  1683.82,384.129 1684.85,347.597 1685.88,298.249 1686.91,296.621 1687.94,285.015 1688.97,279.25 1690,217.73 1691.04,223.317 1692.07,201.648 1693.1,177.725 \n",
                            "  1694.13,164.203 1695.16,231.93 1696.19,247.792 1697.22,232.557 1698.25,226.559 1699.28,243.979 1700.31,260.493 1701.34,250.329 1702.37,209.062 1703.4,226.314 \n",
                            "  1704.44,212.344 1705.47,295.049 1706.5,332.852 1707.53,294.011 1708.56,356.387 1709.59,368.295 1710.62,410.065 1711.65,337.22 1712.68,339.644 1713.71,311.988 \n",
                            "  1714.74,256.757 1715.77,249.134 1716.8,276.784 1717.84,293.33 1718.87,301.373 1719.9,287.989 1720.93,314.128 1721.96,291.816 1722.99,289.77 1724.02,249.636 \n",
                            "  1725.05,258.217 1726.08,267.683 1727.11,337.416 1728.14,358.849 1729.17,324.059 1730.2,362.39 1731.24,329.046 1732.27,342.739 1733.3,359.255 1734.33,378.954 \n",
                            "  1735.36,365.651 1736.39,388.468 1737.42,379.987 1738.45,425.353 1739.48,451.399 1740.51,420.057 1741.54,438.796 1742.57,410.792 1743.6,370.9 1744.64,347.112 \n",
                            "  1745.67,349.813 1746.7,328.434 1747.73,352.82 1748.76,387.734 1749.79,373.922 1750.82,398.558 1751.85,399.187 1752.88,381.279 1753.91,358.682 1754.94,358.429 \n",
                            "  1755.97,388.824 1757,428.513 1758.04,403.271 1759.07,396.561 1760.1,393.161 1761.13,378.815 1762.16,330.231 1763.19,279.516 1764.22,251.629 1765.25,261.182 \n",
                            "  1766.28,277.096 1767.31,226.842 1768.34,214.705 1769.37,246.402 1770.4,230.325 1771.44,211.916 1772.47,255.903 1773.5,244.11 1774.53,210.555 1775.56,213.154 \n",
                            "  1776.59,215.218 1777.62,195.347 1778.65,164.53 1779.68,205.1 1780.71,195.363 1781.74,193.299 1782.77,168.146 1783.8,174.234 1784.84,189.947 1785.87,201.915 \n",
                            "  1786.9,225.689 1787.93,271.848 1788.96,296.991 1789.99,350.314 1791.02,277.176 1792.05,282.595 1793.08,288.86 1794.11,329.832 1795.14,261.408 1796.17,280.353 \n",
                            "  1797.2,225.77 1798.24,180.212 1799.27,188.147 1800.3,198.4 1801.33,203.233 1802.36,182.942 1803.39,151.371 1804.42,192.846 1805.45,213.516 1806.48,183.68 \n",
                            "  1807.51,205.976 1808.54,202.771 1809.57,203.434 1810.6,232.95 1811.64,201.025 1812.67,189.808 1813.7,152.209 1814.73,182.501 1815.76,180.123 1816.79,243.427 \n",
                            "  1817.82,281.659 1818.85,280.022 1819.88,278.405 1820.91,209.833 1821.94,228.118 1822.97,211.912 1824,180.911 1825.04,185.06 1826.07,161.579 1827.1,118.908 \n",
                            "  1828.13,106.834 1829.16,131.905 1830.19,94.4921 1831.22,84.3379 1832.25,83.6033 1833.28,113.183 1834.31,102.971 1835.34,87.2568 1836.37,109.378 1837.4,144.532 \n",
                            "  1838.44,163.438 1839.47,155.465 1840.5,82.7405 1841.53,63.6132 1842.56,112.874 1843.59,119.499 1844.62,131.645 1845.65,142.352 1846.68,131.226 1847.71,175.738 \n",
                            "  1848.74,167.87 1849.77,181.252 1850.8,254.982 1851.84,271.613 1852.87,341.043 1853.9,355.296 1854.93,392.017 1855.96,415.187 1856.99,417.893 1858.02,354.762 \n",
                            "  1859.05,313.937 1860.08,360.063 1861.11,341.368 1862.14,311.16 1863.17,289.58 1864.2,243.68 1865.24,232.668 1866.27,220.599 1867.3,259.297 1868.33,230.315 \n",
                            "  1869.36,262.243 1870.39,308.806 1871.42,356.929 1872.45,349.894 1873.48,391.311 1874.51,297.939 1875.54,305.971 1876.57,296.033 1877.6,347.923 1878.63,413.156 \n",
                            "  1879.67,358.87 1880.7,328.912 1881.73,286.097 1882.76,302.404 1883.79,340.41 1884.82,326.156 1885.85,362.993 1886.88,416.051 1887.91,400.014 1888.94,369.613 \n",
                            "  1889.97,395.928 1891,410.121 1892.03,377.141 1893.07,348.495 1894.1,337.429 1895.13,306.375 1896.16,280.905 1897.19,369.605 1898.22,343.079 1899.25,366.569 \n",
                            "  1900.28,355.575 1901.31,324.657 1902.34,318.669 1903.37,344.092 1904.4,311.064 1905.43,241.325 1906.47,289.021 1907.5,250.764 1908.53,314.686 1909.56,286.881 \n",
                            "  1910.59,247.584 1911.62,271.502 1912.65,248.049 1913.68,227.419 1914.71,223.337 1915.74,252.325 1916.77,294.788 1917.8,325.279 1918.83,329.704 1919.87,248.76 \n",
                            "  1920.9,223.091 1921.93,227.48 1922.96,263.512 1923.99,211.967 1925.02,186.995 1926.05,256.809 1927.08,240.08 1928.11,238.787 1929.14,252.562 1930.17,276.082 \n",
                            "  1931.2,274.94 1932.23,256.694 1933.27,302.452 1934.3,290.729 1935.33,265.122 1936.36,246.352 1937.39,268.746 1938.42,287.076 1939.45,356.548 1940.48,394.913 \n",
                            "  1941.51,394.318 1942.54,452.169 1943.57,418.481 1944.6,401.435 1945.63,371.915 1946.67,345.429 1947.7,350.693 1948.73,339.171 1949.76,330.745 1950.79,323.807 \n",
                            "  1951.82,423.688 1952.85,399.741 1953.88,410.854 1954.91,418.915 1955.94,421.728 1956.97,434.055 1958,435.665 1959.03,488.876 1960.07,522.326 1961.1,477.6 \n",
                            "  1962.13,490.311 1963.16,435.442 1964.19,405.293 1965.22,443.57 1966.25,456.804 1967.28,488.592 1968.31,465.931 1969.34,478.879 1970.37,496.603 1971.4,403.367 \n",
                            "  1972.43,423.692 1973.47,440.506 1974.5,417.752 1975.53,418.615 1976.56,417.957 1977.59,412.44 1978.62,464.362 1979.65,397.182 1980.68,367.002 1981.71,352.878 \n",
                            "  1982.74,381.573 1983.77,397.648 1984.8,386.313 1985.83,378.422 1986.87,408.854 1987.9,400.007 1988.93,351.293 1989.96,361.469 1990.99,386.158 1992.02,381.738 \n",
                            "  1993.05,369.985 1994.08,378.217 1995.11,437.604 1996.14,480.142 1997.17,440.944 1998.2,440.025 1999.23,433.115 2000.27,442.553 2001.3,412.798 2002.33,446.518 \n",
                            "  2003.36,454.452 2004.39,474.562 2005.42,476.888 2006.45,493.289 2007.48,455.222 2008.51,495.08 2009.54,460.34 2010.57,399.006 2011.6,397.054 2012.63,351.9 \n",
                            "  2013.67,339.887 2014.7,346.172 2015.73,379.588 2016.76,377.935 2017.79,355.597 2018.82,389.216 2019.85,393.721 2020.88,438.943 2021.91,377.248 2022.94,387.609 \n",
                            "  2023.97,319.674 2025,283.811 2026.03,266.239 2027.07,290.795 2028.1,230.456 2029.13,219.707 2030.16,223.674 2031.19,185.802 2032.22,203.196 2033.25,235.091 \n",
                            "  2034.28,269.604 2035.31,264.095 2036.34,271.338 2037.37,287.448 2038.4,319.234 2039.43,307.286 2040.47,315.32 2041.5,279.712 2042.53,293.36 2043.56,302.298 \n",
                            "  2044.59,305.205 2045.62,281.815 2046.65,311.798 2047.68,372.625 2048.71,409.889 2049.74,453.398 2050.77,413.132 2051.8,403.375 2052.83,414.214 2053.87,419.621 \n",
                            "  2054.9,400.628 2055.93,422.689 2056.96,400.144 2057.99,378.35 2059.02,400.108 2060.05,341.306 2061.08,336.539 2062.11,326.424 2063.14,364.721 2064.17,370.498 \n",
                            "  2065.2,382.718 2066.23,400.911 2067.26,437.874 2068.3,429.425 2069.33,414.676 2070.36,372.129 2071.39,377.299 2072.42,353.032 2073.45,373.491 2074.48,367.952 \n",
                            "  2075.51,360.55 2076.54,329.684 2077.57,286.664 2078.6,305.291 2079.63,301.774 2080.66,284.284 2081.7,268.799 2082.73,258.496 2083.76,252.22 2084.79,249.983 \n",
                            "  2085.82,278.223 2086.85,300.332 2087.88,277.649 2088.91,305.131 2089.94,295.87 2090.97,304.454 2092,312.454 2093.03,335.603 2094.06,285.301 2095.1,296.234 \n",
                            "  2096.13,305.728 2097.16,326.237 2098.19,328.111 2099.22,337.434 2100.25,363.322 2101.28,378.928 2102.31,393.684 2103.34,389.592 2104.37,317.647 2105.4,286.187 \n",
                            "  2106.43,335.719 2107.46,315.327 2108.5,318.652 2109.53,295.218 2110.56,300.496 2111.59,326.429 2112.62,334.976 2113.65,322.012 2114.68,295.475 2115.71,322.787 \n",
                            "  2116.74,286.336 2117.77,315.42 2118.8,285.432 2119.83,249.338 2120.86,237.078 2121.9,268.469 2122.93,299.115 2123.96,269.493 2124.99,253.275 2126.02,324.963 \n",
                            "  2127.05,325.505 2128.08,359.948 2129.11,355.25 2130.14,381.419 2131.17,403.566 2132.2,403.544 2133.23,418.186 2134.26,377.011 2135.3,340.983 2136.33,347.181 \n",
                            "  2137.36,334.79 2138.39,355.854 2139.42,380.918 2140.45,405.567 2141.48,433.655 2142.51,386.532 2143.54,405.936 2144.57,376.433 2145.6,375.643 2146.63,350.912 \n",
                            "  2147.66,318.656 2148.7,324.313 2149.73,288.532 2150.76,256.303 2151.79,244.854 2152.82,217.868 2153.85,212.713 2154.88,224.666 2155.91,208.308 2156.94,199.724 \n",
                            "  2157.97,182.463 2159,160.016 2160.03,161.413 2161.06,109.21 2162.1,113.901 2163.13,117.351 2164.16,136.418 2165.19,73.6787 2166.22,152.588 2167.25,182.216 \n",
                            "  2168.28,191.304 2169.31,187.972 2170.34,241.774 2171.37,196.138 2172.4,239.529 2173.43,214.044 2174.46,190.472 2175.5,121.663 2176.53,171.179 2177.56,224.226 \n",
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                            "  601.517,1191.82 602.548,1191.61 603.579,1191.02 604.61,1190.09 605.64,1188.98 606.671,1187.81 607.702,1186.66 608.733,1185.61 609.763,1184.76 610.794,1184.13 \n",
                            "  611.825,1183.68 612.856,1183.64 613.886,1183.94 614.917,1184.44 615.948,1184.97 616.979,1185.49 618.009,1185.97 619.04,1186.25 620.071,1186.21 621.102,1185.98 \n",
                            "  622.133,1185.58 623.163,1185.06 624.194,1184.39 625.225,1183.64 626.256,1182.79 627.286,1182.01 628.317,1181.33 629.348,1180.77 630.379,1180.33 631.409,1180.13 \n",
                            "  632.44,1180.23 633.471,1180.74 634.502,1181.71 635.532,1183.05 636.563,1184.55 637.594,1186.18 638.625,1187.92 639.656,1189.69 640.686,1191.55 641.717,1193.58 \n",
                            "  642.748,1195.8 643.779,1198.13 644.809,1200.54 645.84,1202.95 646.871,1205.49 647.902,1208.04 648.932,1210.52 649.963,1213.01 650.994,1215.51 652.025,1218.1 \n",
                            "  653.055,1220.65 654.086,1223.03 655.117,1225.39 656.148,1227.77 657.179,1230.26 658.209,1232.86 659.24,1235.57 660.271,1238.1 661.302,1240.33 662.332,1242.22 \n",
                            "  663.363,1243.81 664.394,1245.05 665.425,1245.92 666.455,1246.39 667.486,1246.55 668.517,1246.46 669.548,1246.32 670.578,1246.09 671.609,1245.69 672.64,1244.97 \n",
                            "  673.671,1244.12 674.702,1243.23 675.732,1242.35 676.763,1241.35 677.794,1240.19 678.825,1238.81 679.855,1237.28 680.886,1235.63 681.917,1233.86 682.948,1231.96 \n",
                            "  683.978,1229.87 685.009,1227.57 686.04,1225.02 687.071,1222.22 688.101,1219.21 689.132,1215.9 690.163,1212.4 691.194,1208.88 692.225,1205.08 693.255,1200.89 \n",
                            "  694.286,1196.39 695.317,1191.73 696.348,1187.05 697.378,1182.36 698.409,1177.72 699.44,1173.18 700.471,1168.83 701.501,1164.66 702.532,1160.65 703.563,1156.84 \n",
                            "  704.594,1153.37 705.624,1150.31 706.655,1147.72 707.686,1145.61 708.717,1143.82 709.748,1142.25 710.778,1140.98 711.809,1140.09 712.84,1139.43 713.871,1138.89 \n",
                            "  714.901,1138.23 715.932,1137.45 716.963,1136.4 717.994,1135.21 719.024,1133.74 720.055,1131.97 721.086,1130.01 722.117,1127.97 723.147,1125.67 724.178,1122.94 \n",
                            "  725.209,1119.75 726.24,1116.24 727.271,1112.56 728.301,1108.73 729.332,1104.75 730.363,1100.77 731.394,1096.86 732.424,1093.15 733.455,1089.62 734.486,1086.21 \n",
                            "  735.517,1083.01 736.547,1080.01 737.578,1077.3 738.609,1074.75 739.64,1072.35 740.67,1070.08 741.701,1068.03 742.732,1066.21 743.763,1064.49 744.794,1063.07 \n",
                            "  745.824,1061.95 746.855,1060.97 747.886,1060.28 748.917,1059.97 749.947,1060.1 750.978,1060.58 752.009,1061.53 753.04,1062.85 754.07,1064.61 755.101,1066.92 \n",
                            "  756.132,1069.76 757.163,1073.01 758.193,1076.63 759.224,1080.64 760.255,1084.85 761.286,1089.41 762.317,1094.18 763.347,1099.12 764.378,1104.02 765.409,1108.68 \n",
                            "  766.44,1113.17 767.47,1117.46 768.501,1121.48 769.532,1125.29 770.563,1128.86 771.593,1132.16 772.624,1135.15 773.655,1137.63 774.686,1139.7 775.717,1141.6 \n",
                            "  776.747,1143.36 777.778,1145.03 778.809,1146.65 779.84,1148.36 780.87,1149.93 781.901,1151.39 782.932,1152.74 783.963,1153.8 784.993,1154.59 786.024,1155.28 \n",
                            "  787.055,1156.18 788.086,1157.24 789.116,1158.46 790.147,1159.81 791.178,1161.28 792.209,1162.91 793.24,1164.64 794.27,1166.48 795.301,1168.44 796.332,1170.47 \n",
                            "  797.363,1172.66 798.393,1175.19 799.424,1178.07 800.455,1181.22 801.486,1184.53 802.516,1188.1 803.547,1191.95 804.578,1196.04 805.609,1200.28 806.639,1204.67 \n",
                            "  807.67,1209.33 808.701,1214.19 809.732,1219.21 810.763,1224.36 811.793,1229.35 812.824,1234.22 813.855,1238.93 814.886,1243.38 815.916,1247.47 816.947,1251.21 \n",
                            "  817.978,1254.72 819.009,1258.19 820.039,1261.55 821.07,1264.98 822.101,1268.54 823.132,1272.34 824.162,1276.4 825.193,1280.75 826.224,1285.22 827.255,1289.69 \n",
                            "  828.286,1294.06 829.316,1298.31 830.347,1302.4 831.378,1306.17 832.409,1309.57 833.439,1312.81 834.47,1315.83 835.501,1318.64 836.532,1321.26 837.562,1323.63 \n",
                            "  838.593,1325.8 839.624,1327.73 840.655,1329.48 841.685,1331.01 842.716,1332.34 843.747,1333.5 844.778,1334.36 845.809,1334.84 846.839,1334.94 847.87,1334.69 \n",
                            "  848.901,1334.21 849.932,1333.58 850.962,1332.96 851.993,1332.24 853.024,1331.58 854.055,1330.93 855.085,1330.23 856.116,1329.63 857.147,1328.92 858.178,1327.97 \n",
                            "  859.208,1326.63 860.239,1325.11 861.27,1323.48 862.301,1321.89 863.332,1320.19 864.362,1318.49 865.393,1316.63 866.424,1314.69 867.455,1312.67 868.485,1310.49 \n",
                            "  869.516,1308.34 870.547,1306.4 871.578,1304.77 872.608,1303.43 873.639,1302.17 874.67,1301.11 875.701,1300.29 876.731,1299.73 877.762,1299.52 878.793,1299.71 \n",
                            "  879.824,1300.32 880.855,1301.16 881.885,1302.14 882.916,1303.33 883.947,1304.68 884.978,1306.01 886.008,1307.34 887.039,1308.65 888.07,1309.88 889.101,1310.98 \n",
                            "  890.131,1311.78 891.162,1312.34 892.193,1312.74 893.224,1312.95 894.254,1313.13 895.285,1313.15 896.316,1312.93 897.347,1312.66 898.378,1312.47 899.408,1312.32 \n",
                            "  900.439,1312.18 901.47,1312.13 902.501,1311.91 903.531,1311.41 904.562,1310.61 905.593,1309.48 906.624,1308.16 907.654,1306.68 908.685,1304.94 909.716,1303.03 \n",
                            "  910.747,1300.93 911.777,1298.58 912.808,1296.01 913.839,1293.24 914.87,1290.2 915.901,1286.96 916.931,1283.61 917.962,1280.25 918.993,1277.03 920.024,1273.81 \n",
                            "  921.054,1270.55 922.085,1267.25 923.116,1263.8 924.147,1260.03 925.177,1256.07 926.208,1251.76 927.239,1247.2 928.27,1242.41 929.3,1237.49 930.331,1232.47 \n",
                            "  931.362,1227.31 932.393,1222.04 933.424,1216.78 934.454,1211.59 935.485,1206.36 936.516,1201.07 937.547,1195.77 938.577,1190.38 939.608,1184.92 940.639,1179.32 \n",
                            "  941.67,1173.74 942.7,1168.32 943.731,1163.22 944.762,1158.22 945.793,1153.21 946.823,1148.25 947.854,1143.38 948.885,1138.45 949.916,1133.76 950.947,1129.38 \n",
                            "  951.977,1125.32 953.008,1121.58 954.039,1118.04 955.07,1114.84 956.1,1112.12 957.131,1109.77 958.162,1107.69 959.193,1105.66 960.223,1103.84 961.254,1102.01 \n",
                            "  962.285,1100.22 963.316,1098.38 964.346,1096.53 965.377,1094.87 966.408,1093.34 967.439,1091.88 968.47,1090.68 969.5,1089.74 970.531,1089.22 971.562,1089.05 \n",
                            "  972.593,1089.27 973.623,1089.65 974.654,1090.14 975.685,1090.79 976.716,1091.73 977.746,1092.69 978.777,1093.44 979.808,1093.93 980.839,1094.21 981.869,1094.37 \n",
                            "  982.9,1094.53 983.931,1094.63 984.962,1094.88 985.993,1095.4 987.023,1096.18 988.054,1097.32 989.085,1098.72 990.116,1100.39 991.146,1102.35 992.177,1104.37 \n",
                            "  993.208,1106.39 994.239,1108.52 995.269,1110.96 996.3,1113.78 997.331,1116.96 998.362,1120.61 999.392,1124.79 1000.42,1129.49 1001.45,1134.79 1002.48,1140.5 \n",
                            "  1003.52,1146.72 1004.55,1153.4 1005.58,1160.48 1006.61,1167.83 1007.64,1175.39 1008.67,1183.1 1009.7,1190.9 1010.73,1198.79 1011.76,1206.73 1012.79,1214.66 \n",
                            "  1013.82,1222.63 1014.85,1230.72 1015.88,1238.99 1016.92,1247.29 1017.95,1255.43 1018.98,1263.56 1020.01,1271.63 1021.04,1279.62 1022.07,1287.59 1023.1,1295.56 \n",
                            "  1024.13,1303.45 1025.16,1311.04 1026.19,1318.3 1027.22,1325.31 1028.25,1331.96 1029.28,1338.44 1030.32,1344.54 1031.35,1350.33 1032.38,1355.75 1033.41,1360.87 \n",
                            "  1034.44,1365.55 1035.47,1369.71 1036.5,1373.39 1037.53,1376.5 1038.56,1379.15 1039.59,1381.4 1040.62,1383.03 1041.65,1384.14 1042.68,1384.77 1043.72,1384.97 \n",
                            "  1044.75,1384.79 1045.78,1384.28 1046.81,1383.27 1047.84,1381.77 1048.87,1379.98 1049.9,1377.98 1050.93,1375.99 1051.96,1374.07 1052.99,1372.11 1054.02,1370.03 \n",
                            "  1055.05,1367.6 1056.08,1364.85 1057.12,1361.93 1058.15,1358.67 1059.18,1354.94 1060.21,1350.83 1061.24,1346.62 1062.27,1342.32 1063.3,1337.96 1064.33,1333.61 \n",
                            "  1065.36,1329.19 1066.39,1324.82 1067.42,1320.5 1068.45,1316.35 1069.48,1312.3 1070.52,1308.29 1071.55,1304.23 1072.58,1300 1073.61,1295.8 1074.64,1291.75 \n",
                            "  1075.67,1287.71 1076.7,1283.61 1077.73,1279.39 1078.76,1274.98 1079.79,1270.4 1080.82,1265.85 1081.85,1261.39 1082.88,1257 1083.92,1252.5 1084.95,1247.89 \n",
                            "  1085.98,1243.39 1087.01,1238.89 1088.04,1234.55 1089.07,1230.38 1090.1,1226.4 1091.13,1222.83 1092.16,1219.61 1093.19,1216.72 1094.22,1214.13 1095.25,1211.8 \n",
                            "  1096.28,1209.57 1097.32,1207.45 1098.35,1205.67 1099.38,1204.04 1100.41,1202.46 1101.44,1201.01 1102.47,1199.52 1103.5,1198.13 1104.53,1196.77 1105.56,1195.34 \n",
                            "  1106.59,1193.74 1107.62,1191.88 1108.65,1189.75 1109.68,1187.49 1110.72,1185.15 1111.75,1182.69 1112.78,1180.18 1113.81,1177.72 1114.84,1175.23 1115.87,1172.78 \n",
                            "  1116.9,1170.51 1117.93,1168.41 1118.96,1166.32 1119.99,1164.37 1121.02,1162.63 1122.05,1161.03 1123.08,1159.61 1124.12,1158.35 1125.15,1157.22 1126.18,1156.24 \n",
                            "  1127.21,1155.47 1128.24,1154.93 1129.27,1154.65 1130.3,1154.61 1131.33,1154.68 1132.36,1155.01 1133.39,1155.6 1134.42,1156.37 1135.45,1157.4 1136.48,1158.58 \n",
                            "  1137.51,1159.96 1138.55,1161.51 1139.58,1163.33 1140.61,1165.4 1141.64,1167.7 1142.67,1169.98 1143.7,1172.18 1144.73,1174.55 1145.76,1177.15 1146.79,1179.93 \n",
                            "  1147.82,1182.84 1148.85,1186.04 1149.88,1189.43 1150.91,1193 1151.95,1196.77 1152.98,1200.92 1154.01,1205.45 1155.04,1210.38 1156.07,1215.43 1157.1,1220.68 \n",
                            "  1158.13,1226.24 1159.16,1232.29 1160.19,1238.72 1161.22,1245.44 1162.25,1252.48 1163.28,1259.73 1164.31,1267.21 1165.35,1274.83 1166.38,1282.44 1167.41,1290.03 \n",
                            "  1168.44,1297.54 1169.47,1304.97 1170.5,1312.25 1171.53,1319.2 1172.56,1325.72 1173.59,1331.9 1174.62,1337.87 1175.65,1343.71 1176.68,1349.53 1177.71,1355.14 \n",
                            "  1178.75,1360.85 1179.78,1366.67 1180.81,1372.46 1181.84,1378.1 1182.87,1383.37 1183.9,1388.27 1184.93,1392.64 1185.96,1396.34 1186.99,1399.46 1188.02,1401.96 \n",
                            "  1189.05,1403.94 1190.08,1405.48 1191.11,1406.5 1192.15,1406.96 1193.18,1407.08 1194.21,1406.99 1195.24,1406.75 1196.27,1406.37 1197.3,1405.74 1198.33,1404.77 \n",
                            "  1199.36,1403.52 1200.39,1402.09 1201.42,1400.56 1202.45,1399.04 1203.48,1397.53 1204.51,1396.12 1205.55,1394.91 1206.58,1393.93 1207.61,1393.13 1208.64,1392.57 \n",
                            "  1209.67,1392.14 1210.7,1391.77 1211.73,1391.38 1212.76,1390.73 1213.79,1389.76 1214.82,1388.49 1215.85,1386.78 1216.88,1384.39 1217.91,1381.5 1218.95,1378.03 \n",
                            "  1219.98,1374.09 1221.01,1369.76 1222.04,1365.04 1223.07,1359.95 1224.1,1354.72 1225.13,1349.35 1226.16,1343.63 1227.19,1337.54 1228.22,1331.18 1229.25,1324.68 \n",
                            "  1230.28,1318.05 1231.31,1311.45 1232.35,1305.13 1233.38,1299.14 1234.41,1293.37 1235.44,1287.71 1236.47,1281.92 1237.5,1276.17 1238.53,1270.38 1239.56,1264.47 \n",
                            "  1240.59,1258.48 1241.62,1252.5 1242.65,1246.47 1243.68,1240.45 1244.71,1234.54 1245.75,1228.69 1246.78,1223.09 1247.81,1217.73 1248.84,1212.54 1249.87,1207.66 \n",
                            "  1250.9,1203.1 1251.93,1198.75 1252.96,1194.55 1253.99,1190.63 1255.02,1187 1256.05,1183.71 1257.08,1180.69 1258.11,1178.06 1259.15,1175.71 1260.18,1173.56 \n",
                            "  1261.21,1171.6 1262.24,1169.91 1263.27,1168.53 1264.3,1167.51 1265.33,1166.86 1266.36,1166.52 1267.39,1166.58 1268.42,1166.89 1269.45,1167.49 1270.48,1168.53 \n",
                            "  1271.51,1169.81 1272.55,1171.29 1273.58,1172.7 1274.61,1174 1275.64,1175.28 1276.67,1176.63 1277.7,1177.95 1278.73,1179.32 1279.76,1180.76 1280.79,1182.31 \n",
                            "  1281.82,1184.03 1282.85,1185.82 1283.88,1187.55 1284.91,1189.15 1285.95,1190.72 1286.98,1192.18 1288.01,1193.52 1289.04,1194.53 1290.07,1195.28 1291.1,1195.83 \n",
                            "  1292.13,1196.09 1293.16,1196.15 1294.19,1195.83 1295.22,1195.15 1296.25,1194.16 1297.28,1193.02 1298.31,1191.77 1299.35,1190.3 1300.38,1188.58 1301.41,1186.82 \n",
                            "  1302.44,1184.95 1303.47,1183.11 1304.5,1181.45 1305.53,1179.97 1306.56,1178.6 1307.59,1177.43 1308.62,1176.46 1309.65,1175.75 1310.68,1175.39 1311.71,1175.14 \n",
                            "  1312.75,1174.85 1313.78,1174.42 1314.81,1173.88 1315.84,1173.28 1316.87,1172.61 1317.9,1171.71 1318.93,1170.75 1319.96,1169.6 1320.99,1168.34 1322.02,1166.93 \n",
                            "  1323.05,1165.67 1324.08,1164.38 1325.11,1163.18 1326.14,1161.96 1327.18,1160.71 1328.21,1159.63 1329.24,1158.63 1330.27,1157.64 1331.3,1156.62 1332.33,1155.85 \n",
                            "  1333.36,1155.39 1334.39,1155.36 1335.42,1155.69 1336.45,1156.39 1337.48,1157.46 1338.51,1158.96 1339.54,1160.73 1340.58,1162.77 1341.61,1165.08 1342.64,1167.64 \n",
                            "  1343.67,1170.48 1344.7,1173.66 1345.73,1176.95 1346.76,1180.25 1347.79,1183.62 1348.82,1187.04 1349.85,1190.37 1350.88,1193.58 1351.91,1196.74 1352.94,1199.76 \n",
                            "  1353.98,1202.53 1355.01,1205.06 1356.04,1207.69 1357.07,1210.4 1358.1,1213.01 1359.13,1215.63 1360.16,1218.25 1361.19,1220.86 1362.22,1223.35 1363.25,1225.74 \n",
                            "  1364.28,1227.98 1365.31,1229.98 1366.34,1231.9 1367.38,1233.65 1368.41,1235.1 1369.44,1236.36 1370.47,1237.45 1371.5,1238.42 1372.53,1239.26 1373.56,1239.99 \n",
                            "  1374.59,1240.56 1375.62,1240.86 1376.65,1240.85 1377.68,1240.47 1378.71,1239.74 1379.74,1238.69 1380.78,1237.17 1381.81,1235.13 1382.84,1232.61 1383.87,1229.78 \n",
                            "  1384.9,1226.63 1385.93,1223.12 1386.96,1219.39 1387.99,1215.38 1389.02,1211.15 1390.05,1206.62 1391.08,1201.96 1392.11,1197.38 1393.14,1192.96 1394.18,1188.78 \n",
                            "  1395.21,1184.84 1396.24,1181.16 1397.27,1177.51 1398.3,1174.02 1399.33,1170.76 1400.36,1167.62 1401.39,1164.78 1402.42,1162.24 1403.45,1159.94 1404.48,1157.79 \n",
                            "  1405.51,1155.73 1406.54,1153.95 1407.58,1152.45 1408.61,1151.37 1409.64,1150.6 1410.67,1150.06 1411.7,1149.81 1412.73,1149.74 1413.76,1149.72 1414.79,1149.96 \n",
                            "  1415.82,1150.49 1416.85,1151.25 1417.88,1152.06 1418.91,1153.02 1419.94,1154.17 1420.98,1155.52 1422.01,1156.97 1423.04,1158.47 1424.07,1159.86 1425.1,1161.11 \n",
                            "  1426.13,1162.15 1427.16,1162.91 1428.19,1163.38 1429.22,1163.41 1430.25,1163 1431.28,1162.22 1432.31,1161.08 1433.34,1159.65 1434.38,1158 1435.41,1155.9 \n",
                            "  1436.44,1153.32 1437.47,1150.4 1438.5,1147.3 1439.53,1143.99 1440.56,1140.43 1441.59,1136.56 1442.62,1132.22 1443.65,1127.43 1444.68,1122.13 1445.71,1116.39 \n",
                            "  1446.74,1110.34 1447.78,1104.13 1448.81,1097.84 1449.84,1091.51 1450.87,1085.18 1451.9,1078.88 1452.93,1072.78 1453.96,1066.8 1454.99,1061.02 1456.02,1055.51 \n",
                            "  1457.05,1050.17 1458.08,1044.82 1459.11,1039.61 1460.14,1034.61 1461.18,1029.92 1462.21,1025.67 1463.24,1021.85 1464.27,1018.39 1465.3,1015.38 1466.33,1012.8 \n",
                            "  1467.36,1010.62 1468.39,1008.87 1469.42,1007.61 1470.45,1006.73 1471.48,1006.15 1472.51,1005.81 1473.54,1005.61 1474.58,1005.29 1475.61,1004.98 1476.64,1004.69 \n",
                            "  1477.67,1004.26 1478.7,1003.83 1479.73,1003.49 1480.76,1003.25 1481.79,1003.01 1482.82,1002.73 1483.85,1002.35 1484.88,1001.9 1485.91,1001.46 1486.94,1001.03 \n",
                            "  1487.98,1000.64 1489.01,1000.34 1490.04,1000.02 1491.07,999.756 1492.1,999.556 1493.13,999.536 1494.16,999.61 1495.19,999.841 1496.22,1000.12 1497.25,1000.24 \n",
                            "  1498.28,1000.17 1499.31,999.971 1500.34,999.63 1501.38,999.145 1502.41,998.713 1503.44,998.137 1504.47,997.505 1505.5,996.668 1506.53,995.533 1507.56,994.006 \n",
                            "  1508.59,992.157 1509.62,990.338 1510.65,988.788 1511.68,987.676 1512.71,986.976 1513.74,986.628 1514.77,986.576 1515.81,986.759 1516.84,987.163 1517.87,987.663 \n",
                            "  1518.9,988.262 1519.93,989.008 1520.96,989.8 1521.99,990.813 1523.02,992.024 1524.05,993.318 1525.08,994.666 1526.11,995.981 1527.14,997.125 1528.17,998.221 \n",
                            "  1529.21,999.162 1530.24,999.992 1531.27,1000.77 1532.3,1001.69 1533.33,1002.78 1534.36,1003.89 1535.39,1005.04 1536.42,1006.27 1537.45,1007.77 1538.48,1009.46 \n",
                            "  1539.51,1011.29 1540.54,1013.27 1541.57,1015.3 1542.61,1017.51 1543.64,1019.97 1544.67,1022.59 1545.7,1025.32 1546.73,1028.1 1547.76,1030.91 1548.79,1033.78 \n",
                            "  1549.82,1036.7 1550.85,1039.55 1551.88,1042.4 1552.91,1045.24 1553.94,1047.95 1554.97,1050.42 1556.01,1052.7 1557.04,1054.63 1558.07,1056.1 1559.1,1057.13 \n",
                            "  1560.13,1057.99 1561.16,1058.61 1562.19,1058.92 1563.22,1058.95 1564.25,1058.86 1565.28,1058.73 1566.31,1058.54 1567.34,1058.21 1568.37,1057.76 1569.41,1057.18 \n",
                            "  1570.44,1056.52 1571.47,1055.71 1572.5,1054.67 1573.53,1053.33 1574.56,1051.93 1575.59,1050.39 1576.62,1048.79 1577.65,1047.23 1578.68,1045.83 1579.71,1044.45 \n",
                            "  1580.74,1043.14 1581.77,1042.15 1582.81,1041.35 1583.84,1040.71 1584.87,1040.2 1585.9,1039.93 1586.93,1039.88 1587.96,1040.02 1588.99,1040.37 1590.02,1041.14 \n",
                            "  1591.05,1042.27 1592.08,1043.65 1593.11,1045.27 1594.14,1047.13 1595.17,1049.31 1596.21,1051.93 1597.24,1055 1598.27,1058.45 1599.3,1062.1 1600.33,1065.82 \n",
                            "  1601.36,1069.45 1602.39,1073.15 1603.42,1076.94 1604.45,1080.81 1605.48,1084.88 1606.51,1089 1607.54,1093.1 1608.57,1097.22 1609.61,1101.33 1610.64,1105.57 \n",
                            "  1611.67,1109.87 1612.7,1114.22 1613.73,1118.45 1614.76,1122.47 1615.79,1126.3 1616.82,1130.05 1617.85,1133.51 1618.88,1136.59 1619.91,1139.31 1620.94,1141.59 \n",
                            "  1621.97,1143.39 1623.01,1144.56 1624.04,1145.13 1625.07,1145.17 1626.1,1144.71 1627.13,1143.76 1628.16,1142.28 1629.19,1140.54 1630.22,1138.74 1631.25,1136.96 \n",
                            "  1632.28,1135.19 1633.31,1133.41 1634.34,1131.95 1635.37,1130.65 1636.41,1129.53 1637.44,1128.61 1638.47,1127.81 1639.5,1127.16 1640.53,1126.39 1641.56,1125.65 \n",
                            "  1642.59,1125 1643.62,1124.52 1644.65,1124.16 1645.68,1123.73 1646.71,1123.28 1647.74,1122.61 1648.77,1121.65 1649.81,1120.4 1650.84,1118.94 1651.87,1117.5 \n",
                            "  1652.9,1116.24 1653.93,1115.22 1654.96,1114.44 1655.99,1113.88 1657.02,1113.58 1658.05,1113.51 1659.08,1113.28 1660.11,1113.19 1661.14,1113.23 1662.17,1113.37 \n",
                            "  1663.21,1113.58 1664.24,1113.79 1665.27,1113.98 1666.3,1114.19 1667.33,1114.46 1668.36,1114.61 1669.39,1114.75 1670.42,1114.92 1671.45,1115.36 1672.48,1116.16 \n",
                            "  1673.51,1117.17 1674.54,1118.39 1675.57,1119.89 1676.61,1121.7 1677.64,1123.67 1678.67,1125.88 1679.7,1128.19 1680.73,1130.5 1681.76,1132.83 1682.79,1135.28 \n",
                            "  1683.82,1137.75 1684.85,1140.28 1685.88,1142.74 1686.91,1144.96 1687.94,1146.95 1688.97,1148.69 1690,1150.17 1691.04,1151.2 1692.07,1151.82 1693.1,1152 \n",
                            "  1694.13,1151.67 1695.16,1150.84 1696.19,1149.76 1697.22,1148.52 1698.25,1147.07 1699.28,1145.41 1700.31,1143.61 1701.34,1141.76 1702.37,1139.81 1703.4,1137.63 \n",
                            "  1704.44,1135.31 1705.47,1132.8 1706.5,1130.42 1707.53,1128.27 1708.56,1126.22 1709.59,1124.47 1710.62,1123.05 1711.65,1122.06 1712.68,1121.24 1713.71,1120.58 \n",
                            "  1714.74,1119.98 1715.77,1119.25 1716.8,1118.36 1717.84,1117.43 1718.87,1116.52 1719.9,1115.65 1720.93,1114.78 1721.96,1114 1722.99,1113.22 1724.02,1112.44 \n",
                            "  1725.05,1111.52 1726.08,1110.51 1727.11,1109.44 1728.14,1108.55 1729.17,1107.9 1730.2,1107.37 1731.24,1107.08 1732.27,1106.89 1733.3,1106.84 1734.33,1106.99 \n",
                            "  1735.36,1107.37 1736.39,1107.93 1737.42,1108.74 1738.45,1109.73 1739.48,1111.06 1740.51,1112.78 1741.54,1114.75 1742.57,1117.02 1743.6,1119.48 1744.64,1121.97 \n",
                            "  1745.67,1124.4 1746.7,1126.8 1747.73,1129.08 1748.76,1131.34 1749.79,1133.69 1750.82,1136.08 1751.85,1138.58 1752.88,1141.18 1753.91,1143.81 1754.94,1146.4 \n",
                            "  1755.97,1148.94 1757,1151.53 1758.04,1154.3 1759.07,1157.15 1760.1,1160.05 1761.13,1162.98 1762.16,1165.89 1763.19,1168.61 1764.22,1170.98 1765.25,1172.92 \n",
                            "  1766.28,1174.51 1767.31,1175.81 1768.34,1176.67 1769.37,1177.09 1770.4,1177.19 1771.44,1176.95 1772.47,1176.33 1773.5,1175.51 1774.53,1174.47 1775.56,1173.1 \n",
                            "  1776.59,1171.44 1777.62,1169.51 1778.65,1167.28 1779.68,1164.66 1780.71,1161.82 1781.74,1158.75 1782.77,1155.46 1783.8,1151.88 1784.84,1148.06 1785.87,1144.07 \n",
                            "  1786.9,1139.98 1787.93,1135.87 1788.96,1131.91 1789.99,1128.18 1791.02,1124.85 1792.05,1121.65 1793.08,1118.59 1794.11,1115.68 1795.14,1113.07 1796.17,1110.48 \n",
                            "  1797.2,1108 1798.24,1105.42 1799.27,1102.6 1800.3,1099.6 1801.33,1096.44 1802.36,1093.18 1803.39,1089.75 1804.42,1086.06 1805.45,1082.28 1806.48,1078.48 \n",
                            "  1807.51,1074.57 1808.54,1070.64 1809.57,1066.69 1810.6,1062.71 1811.64,1058.83 1812.67,1054.92 1813.7,1050.96 1814.73,1046.82 1815.76,1042.63 1816.79,1038.39 \n",
                            "  1817.82,1034.32 1818.85,1030.54 1819.88,1027.03 1820.91,1023.78 1821.94,1020.54 1822.97,1017.36 1824,1014.2 1825.04,1010.94 1826.07,1007.62 1827.1,1004.16 \n",
                            "  1828.13,1000.42 1829.16,996.397 1830.19,992.191 1831.22,987.694 1832.25,982.899 1833.28,977.828 1834.31,972.611 1835.34,967.229 1836.37,961.643 1837.4,955.954 \n",
                            "  1838.44,950.295 1839.47,944.735 1840.5,939.247 1841.53,933.581 1842.56,927.693 1843.59,921.774 1844.62,915.856 1845.65,909.985 1846.68,904.205 1847.71,898.477 \n",
                            "  1848.74,892.959 1849.77,887.614 1850.8,882.481 1851.84,877.804 1852.87,873.617 1853.9,870.126 1854.93,867.333 1855.96,865.312 1856.99,864.088 1858.02,863.613 \n",
                            "  1859.05,863.627 1860.08,863.955 1861.11,864.729 1862.14,865.856 1863.17,867.205 1864.2,868.685 1865.24,870.126 1866.27,871.492 1867.3,872.743 1868.33,874.016 \n",
                            "  1869.36,875.212 1870.39,876.444 1871.42,877.867 1872.45,879.628 1873.48,881.677 1874.51,884.129 1875.54,886.64 1876.57,889.229 1877.6,891.854 1878.63,894.684 \n",
                            "  1879.67,897.918 1880.7,901.344 1881.73,904.845 1882.76,908.261 1883.79,911.652 1884.82,915.146 1885.85,918.681 1886.88,922.374 1887.91,926.387 1888.94,930.638 \n",
                            "  1889.97,935.001 1891,939.555 1892.03,944.328 1893.07,949.187 1894.1,954.026 1895.13,958.804 1896.16,963.415 1897.19,967.777 1898.22,972.201 1899.25,976.592 \n",
                            "  1900.28,981.027 1901.31,985.459 1902.34,989.781 1903.37,993.974 1904.4,998.127 1905.43,1002.13 1906.47,1005.75 1907.5,1009.17 1908.53,1012.27 1909.56,1015.29 \n",
                            "  1910.59,1018.14 1911.62,1020.69 1912.65,1023.04 1913.68,1025.12 1914.71,1026.88 1915.74,1028.32 1916.77,1029.57 1917.8,1030.78 1918.83,1032.06 1919.87,1033.42 \n",
                            "  1920.9,1034.58 1921.93,1035.45 1922.96,1036.08 1923.99,1036.6 1925.02,1036.84 1926.05,1036.75 1927.08,1036.57 1928.11,1036.26 1929.14,1035.83 1930.17,1035.33 \n",
                            "  1931.2,1034.86 1932.23,1034.39 1933.27,1033.88 1934.3,1033.47 1935.33,1033.14 1936.36,1032.77 1937.39,1032.31 1938.42,1031.85 1939.45,1031.44 1940.48,1031.32 \n",
                            "  1941.51,1031.6 1942.54,1032.25 1943.57,1033.42 1944.6,1034.98 1945.63,1036.83 1946.67,1038.86 1947.7,1040.95 1948.73,1043.13 1949.76,1045.35 1950.79,1047.56 \n",
                            "  1951.82,1049.75 1952.85,1052.25 1953.88,1054.95 1954.91,1057.88 1955.94,1061.03 1956.97,1064.41 1958,1068.02 1959.03,1071.86 1960.07,1076.08 1961.1,1080.76 \n",
                            "  1962.13,1085.71 1963.16,1090.95 1964.19,1096.27 1965.22,1101.57 1966.25,1106.96 1967.28,1112.48 1968.31,1118.21 1969.34,1124.06 1970.37,1130.07 1971.4,1136.26 \n",
                            "  1972.43,1142.31 1973.47,1148.29 1974.5,1154.26 1975.53,1160.13 1976.56,1165.91 1977.59,1171.59 1978.62,1177.17 1979.65,1182.8 1980.68,1188.26 1981.71,1193.45 \n",
                            "  1982.74,1198.35 1983.77,1203.06 1984.8,1207.64 1985.83,1212.06 1986.87,1216.31 1987.9,1220.48 1988.93,1224.56 1989.96,1228.37 1990.99,1231.98 1992.02,1235.47 \n",
                            "  1993.05,1238.82 1994.08,1242.02 1995.11,1245.1 1996.14,1248.25 1997.17,1251.6 1998.2,1255.02 1999.23,1258.48 2000.27,1261.97 2001.3,1265.49 2002.33,1268.96 \n",
                            "  2003.36,1272.47 2004.39,1276.06 2005.42,1279.76 2006.45,1283.59 2007.48,1287.59 2008.51,1291.6 2009.54,1295.76 2010.57,1299.94 2011.6,1303.92 2012.63,1307.71 \n",
                            "  2013.67,1311.18 2014.7,1314.29 2015.73,1317.09 2016.76,1319.71 2017.79,1322.16 2018.82,1324.37 2019.85,1326.47 2020.88,1328.47 2021.91,1330.55 2022.94,1332.47 \n",
                            "  2023.97,1334.29 2025,1335.78 2026.03,1336.84 2027.07,1337.44 2028.1,1337.69 2029.13,1337.42 2030.16,1336.62 2031.19,1335.34 2032.22,1333.5 2033.25,1331.19 \n",
                            "  2034.28,1328.55 2035.31,1325.74 2036.34,1322.75 2037.37,1319.61 2038.4,1316.41 2039.43,1313.26 2040.47,1310.11 2041.5,1307 2042.53,1303.8 2043.56,1300.58 \n",
                            "  2044.59,1297.36 2045.62,1294.17 2046.65,1290.92 2047.68,1287.72 2048.71,1284.77 2049.74,1282.19 2050.77,1280.1 2051.8,1278.33 2052.83,1276.83 2053.87,1275.61 \n",
                            "  2054.9,1274.68 2055.93,1273.95 2056.96,1273.48 2057.99,1273.17 2059.02,1272.95 2060.05,1272.89 2061.08,1272.76 2062.11,1272.57 2063.14,1272.29 2064.17,1272.04 \n",
                            "  2065.2,1271.85 2066.23,1271.75 2067.26,1271.79 2068.3,1272.1 2069.33,1272.61 2070.36,1273.28 2071.39,1273.94 2072.42,1274.61 2073.45,1275.21 2074.48,1275.82 \n",
                            "  2075.51,1276.4 2076.54,1276.95 2077.57,1277.35 2078.6,1277.47 2079.63,1277.4 2080.66,1277.13 2081.7,1276.62 2082.73,1275.84 2083.76,1274.76 2084.79,1273.39 \n",
                            "  2085.82,1271.75 2086.85,1269.95 2087.88,1268.08 2088.91,1266.07 2089.94,1264.03 2090.97,1261.93 2092,1259.8 2093.03,1257.69 2094.06,1255.66 2095.1,1253.54 \n",
                            "  2096.13,1251.38 2097.16,1249.21 2098.19,1247.12 2099.22,1245.09 2100.25,1243.17 2101.28,1241.43 2102.31,1239.91 2103.34,1238.64 2104.37,1237.61 2105.4,1236.55 \n",
                            "  2106.43,1235.35 2107.46,1234.2 2108.5,1233.03 2109.53,1231.84 2110.56,1230.57 2111.59,1229.23 2112.62,1227.92 2113.65,1226.67 2114.68,1225.44 2115.71,1224.13 \n",
                            "  2116.74,1222.84 2117.77,1221.44 2118.8,1220.06 2119.83,1218.57 2120.86,1216.87 2121.9,1214.94 2122.93,1212.9 2123.96,1210.86 2124.99,1208.73 2126.02,1206.45 \n",
                            "  2127.05,1204.29 2128.08,1202.24 2129.11,1200.42 2130.14,1198.8 2131.17,1197.44 2132.2,1196.4 2133.23,1195.67 2134.26,1195.27 2135.3,1195.04 2136.33,1194.85 \n",
                            "  2137.36,1194.71 2138.39,1194.59 2139.42,1194.55 2140.45,1194.66 2141.48,1195.01 2142.51,1195.66 2143.54,1196.43 2144.57,1197.39 2145.6,1198.42 2146.63,1199.5 \n",
                            "  2147.66,1200.56 2148.7,1201.48 2149.73,1202.3 2150.76,1202.89 2151.79,1203.17 2152.82,1203.11 2153.85,1202.64 2154.88,1201.79 2155.91,1200.6 2156.94,1199.07 \n",
                            "  2157.97,1197.17 2159,1194.89 2160.03,1192.17 2161.06,1189.05 2162.1,1185.38 2163.13,1181.23 2164.16,1176.65 2165.19,1171.73 2166.22,1166.29 2167.25,1160.66 \n",
                            "  2168.28,1154.95 2169.31,1149.2 2170.34,1143.41 2171.37,1137.79 2172.4,1132.16 2173.43,1126.68 2174.46,1121.27 2175.5,1115.83 2176.53,1110.15 2177.56,1104.41 \n",
                            "  2178.59,1098.81 2179.62,1093.32 2180.65,1088.14 2181.68,1083.18 2182.71,1078.33 2183.74,1073.64 2184.77,1069.06 2185.8,1064.67 2186.83,1060.43 2187.86,1056.45 \n",
                            "  2188.9,1052.68 2189.93,1049.27 2190.96,1046.24 2191.99,1043.32 2193.02,1040.39 2194.05,1037.44 2195.08,1034.52 2196.11,1031.9 2197.14,1029.56 2198.17,1027.71 \n",
                            "  2199.2,1026.46 2200.23,1025.8 2201.26,1025.62 2202.3,1025.8 2203.33,1026.52 2204.36,1027.61 2205.39,1029.21 2206.42,1031.12 2207.45,1033.18 2208.48,1035.41 \n",
                            "  2209.51,1037.69 2210.54,1040.03 2211.57,1042.15 2212.6,1044.08 2213.63,1045.82 2214.66,1047.37 2215.7,1048.66 2216.73,1049.78 2217.76,1050.75 2218.79,1051.46 \n",
                            "  2219.82,1051.91 2220.85,1052.11 2221.88,1052.12 2222.91,1051.9 2223.94,1051.49 2224.97,1050.96 2226,1050.32 2227.03,1049.72 2228.06,1049.22 2229.1,1048.88 \n",
                            "  2230.13,1048.62 2231.16,1048.29 2232.19,1047.78 2233.22,1047.3 2234.25,1046.94 2235.28,1046.69 2236.31,1046.32 2237.34,1045.74 2238.37,1044.94 2239.4,1043.93 \n",
                            "  2240.43,1042.84 2241.46,1041.58 2242.49,1040.44 2243.53,1039.66 2244.56,1039.05 2245.59,1038.63 2246.62,1038.22 2247.65,1037.76 2248.68,1037.18 2249.71,1036.53 \n",
                            "  2250.74,1035.83 2251.77,1035.16 2252.8,1034.5 2253.83,1033.91 2254.86,1033.52 2255.89,1033.41 2256.93,1033.72 2257.96,1034.3 2258.99,1034.93 2260.02,1035.66 \n",
                            "  2261.05,1036.59 2262.08,1037.87 2263.11,1039.53 2264.14,1041.45 2265.17,1043.59 2266.2,1045.89 2267.23,1048.33 2268.26,1050.83 2269.29,1053.46 2270.33,1056.26 \n",
                            "  2271.36,1059.08 2272.39,1061.92 2273.42,1064.81 2274.45,1067.84 2275.48,1071.02 2276.51,1074.22 2277.54,1077.48 2278.57,1080.74 2279.6,1083.97 2280.63,1087.16 \n",
                            "  2281.66,1090.25 2282.69,1093.03 2283.73,1095.43 2284.76,1097.59 2285.79,1099.53 2286.82,1101.24 2287.85,1102.85 2288.88,1104.44 2289.91,1106.02 2290.94,1107.49 \n",
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                            "  1968.51,1051.72 2280.76,1051.72 2280.76,930.764 1968.51,930.764 1968.51,1051.72 \n",
                            "  \"/>\n",
                            "<polyline clip-path=\"url(#clip4700)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
                            "  1992.51,991.244 2136.51,991.244 \n",
                            "  \"/>\n",
                            "<g clip-path=\"url(#clip4700)\">\n",
                            "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:start;\" transform=\"rotate(0, 2160.51, 1008.74)\" x=\"2160.51\" y=\"1008.74\">K_t</text>\n",
                            "</g>\n",
                            "</svg>\n"
                        ]
                    },
                    "execution_count": 19,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "p1 = plot(100 .*(z_path./z_ss .- 1.0), label=\"z_t\", xlabel=\"t\")\n",
                "p2 = plot(100 .*XT_GenBKM, label = \"K_t\", xlabel=\"t\")\n",
                "plot(p1,p2, fg_legend = :transparent, legend=:best, layout=(2,1))"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "*Notes: The top panel shows the percentage deviation of aggregate productivity from its steady-state value. The bottom panel shows the percentage deviation of capital from its steady-state value.*"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Conclusion\n",
                "\n",
                "This post presents the model of Aiyagari (1994) and a general three-step procedure to simulate out-of-steady-state dynamics for models of this class for \"small\" shocks. Solving incomplete market models for large shocks seems to be [much more complicated](https://www.springer.com/gp/book/9783319564357) and is still an active area of research.\n"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## Links"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "* An excellent course on heterogeneous agent models with code in Matlab, Python and Julia: https://alisdairmckay.com/Notes/HetAgents/index.html\n",
                "* More on the EGM method: https://julia.quantecon.org/dynamic_programming/egm_policy_iter.html\n",
                "* More on the Aiyagari model: https://python.quantecon.org/aiyagari.html\n",
                "* Aiyagari model in continuous time: https://nbviewer.jupyter.org/github/QuantEcon/QuantEcon.notebooks/blob/master/aiyagari_continuous_time.ipynb\n",
                "* Similar model solved very efficiently with a Python package: https://github.com/shade-econ/sequence-jacobian/blob/master/krusell_smith.ipynb"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "## References"
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "* Aiyagari, S. Rao. \"Uninsured idiosyncratic risk and aggregate saving.\" The Quarterly Journal of Economics 109.3 (1994): 659-684.\n",
                "* Bewley, Truman. \"A difficulty with the optimum quantity of money.\" Econometrica: Journal of the Econometric Society (1983): 1485-1504.\n",
                "* Boppart, Timo, Per Krusell, and Kurt Mitman. “Exploiting MIT shocks in heterogeneous-agent economies: the impulse response as a numerical derivative.” Journal of Economic Dynamics and Control 89 (2018): 68-92.\n",
                "* Christopher D Carroll. The method of endogenous gridpoints for solving dynamic stochastic optimization problems. Economics Letters, 91(3):312–320, 2006.\n",
                "* Coeurdacier, Nicolas, Helene Rey, and Pablo Winant. \"The risky steady state.\" American Economic Review 101.3 (2011): 398-401.\n",
                "* Huggett, Mark. \"The risk-free rate in heterogeneous-agent incomplete-insurance economies.\" Journal of economic Dynamics and Control 17.5-6 (1993): 953-969.\n",
                "* İmrohoroğlu, Ayşe. \"The welfare cost of inflation under imperfect insurance.\" Journal of Economic Dynamics and Control 16.1 (1992): 79-91.\n",
                "* Ljungqvist, Lars, and Thomas J. Sargent. Recursive macroeconomic theory. MIT press, 2018.\n",
                "* Reiter, Michael. “Comments on” Exploiting MIT Shocks in Heterogeneous-Agent Economies: The Impulse Response as a Numerical Derivative” by T. Boppart, P. Krusell and K. Mitman.” Journal of Economic Dynamics and Control 89 (2018): 93-99."
            ]
        },
        {
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "---\n",
                "\n",
                "Originally published at https://julienpascal.github.io/post/aiyagariaggregateuncertainty/ on 19 May, 2020."
            ]
        }
    ],
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