ipynb/min-chang_julia_notebook.ipynb

{
    "cells": [
        {
            "cell_type": "markdown",
            "metadata": {
                "slideshow": {
                    "slide_type": "slide"
                }
            },
            "source": "# Chang (1998)"
        },
        {
            "cell_type": "markdown",
            "metadata": {
                "slideshow": {
                    "slide_type": "slide"
                }
            },
            "source": "by Min Kim (min.kim@rutgers.edu), Jan 14, 2020"
        },
        {
            "cell_type": "markdown",
            "metadata": {
                "slideshow": {
                    "slide_type": "slide"
                }
            },
            "source": "Reference:\n - [Chang (1998)](https://www.sciencedirect.com/science/article/pii/S002205319892395X)\n - Ljungqvist and Sargent (4th edition, Figure 25.6.1 and 25.6.2)\n - [Quantecon Python code](https://python-advanced.quantecon.org/chang_credible.html) by Sebastian Graves\n - [Yeltekin's Matlab code](http://sevinyeltekin.com/code/) for [Judd et al. (2003)](https://onlinelibrary.wiley.com/doi/abs/10.1111/1468-0262.t01-1-00445)"
        },
        {
            "cell_type": "code",
            "execution_count": 27,
            "metadata": {
                "slideshow": {
                    "slide_type": "slide"
                }
            },
            "outputs": [],
            "source": [
                "using Parameters\n",
                "using ForwardDiff, LinearAlgebra, QuantEcon\n",
                "using Plots, LaTeXStrings\n",
                "using Gurobi, JuMP # for lienar programming\n",
                "using SharedArrays, Distributed # for parallel loop\n",
                "using LazySets # for convexification"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 28,
            "metadata": {
                "scrolled": true,
                "slideshow": {
                    "slide_type": "subslide"
                }
            },
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Academic license - for non-commercial use only\r\n"
                    ]
                },
                {
                    "name": "stderr",
                    "output_type": "stream",
                    "text": [
                        "WARNING: redefining constant GUROBI_ENV\n"
                    ]
                },
                {
                    "data": {
                        "text/plain": [
                            "Gurobi.Env(Ptr{Nothing} @0x0000000090bf7010)"
                        ]
                    },
                    "execution_count": 28,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "const GUROBI_ENV = Gurobi.Env() # to suppress Grubi license output"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 29,
            "metadata": {
                "slideshow": {
                    "slide_type": "slide"
                }
            },
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "##NamedTuple_kw#395 (generic function with 2 methods)"
                        ]
                    },
                    "execution_count": 29,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "parameters = @with_kw (β = 0.8, m_min = 1e-9, mbar = 30, h_min = 0.9, h_max = 1/β)\n",
                "functions = @with_kw (uc = c -> log(c), v = m -> 1/2000 * (mbar .* m .- 0.5 .* m.^2).^0.5,\n",
                "   f = (h,m) -> 180 .- (0.4 .* m.*(h .- 1)).^2)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 30,
            "metadata": {
                "slideshow": {
                    "slide_type": "slide"
                }
            },
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "(uc = var\"#49#55\"(), v = var\"#50#56\"(), f = var\"#51#57\"())"
                        ]
                    },
                    "execution_count": 30,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "# set 1: LS figure 25.6.2\n",
                "@unpack β, mbar, m_min, h_min, h_max = parameters()\n",
                "@unpack uc, v, f = functions() # utility and production functions\n",
                "\n",
                "# set 2: LS figure 25.6.1\n",
                "# @unpack β, mbar, h_min, h_max = parameters(β = 0.3, h_max = 2)\n",
                "# @unpack uc, v, f = functions() # utility and production functions\n",
                "\n",
                "# try\n",
                "# @unpack β, mbar, h_min, h_max = parameters(β = 0.4, h_max = 2, m_min = 10, mbar = 30)\n",
                "\n",
                "# set 3: Chang(1998)\n",
                "# @unpack β, mbar, m_min, h_min, h_max = parameters(β = 0.9, mbar = 40, h_min = .25, h_max = 1.25)\n",
                "# @unpack uc, v, f = functions(uc = c -> 10000*log(c),\n",
                "    # v = m -> mbar .* m .- 0.5 .* m.^2, f = (h,m) -> 64 .- (0.2 .* m.*(h .- 1)).^2)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 31,
            "metadata": {
                "scrolled": true,
                "slideshow": {
                    "slide_type": "slide"
                }
            },
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "1×2 Array{Float64,2}:\n",
                            " 0.0  1.09649"
                        ]
                    },
                    "execution_count": 31,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "m_max = mbar\n",
                "# n_h = 5 # number of actions for h\n",
                "n_h = 8 # number of actions for h\n",
                "n_m = 35 # number of actions for m\n",
                "# n_m = 5 # number of actions for m\n",
                "n_a = n_h*n_m # number of combinations of actions\n",
                "# n_g = 24 # number of subgradients\n",
                "n_g = 48 # number of subgradients\n",
                "\n",
                "x = (h,m) -> m.*(h .- 1) # tax from government budget constraint\n",
                "u = (h,m) -> uc.(f.(h, m)) .+ v(m) #total utility after market clearing\n",
                "D(f) = x -> ForwardDiff.derivative(f, x) # returns derivative of f\n",
                "uc_p = D(uc)\n",
                "v_p = D(v)\n",
                "Euler = (h,m) -> m.*uc_p.(f.(h, m)).-v_p.(m) # left hand side of the Euler equation\n",
                "Θ = (h,m) -> uc_p.(f.(h, m)) .* (m .*h) # promised utility\n",
                "\n",
                "# grids\n",
                "A_h = range(h_min,h_max, length = n_h)\n",
                "A_m = range(m_min,m_max, length = n_m)\n",
                "A = gridmake(A_h,A_m)\n",
                "\n",
                "# Precompute utility and output vectors\n",
                "Euler_vec = Euler(A[:,1],A[:,2])\n",
                "f_vec = f(A[:,1],A[:,2])\n",
                "u_vec = u(A[:,1],A[:,2])\n",
                "Θ_vec = Θ(A[:,1],A[:,2])\n",
                "\n",
                "# initial space as in Chang (1998)\n",
                "# w_space = [maximum([u(h_min,0)/(1-β), u(h_max,0)/(1-β)]), u(1,m_max)/(1-β)]\n",
                "# θ_space = [0, Θ(h_max,m_max)]\n",
                "\n",
                "# initial space as in LS\n",
                "p_vec = Θ_vec\n",
                "w_vec = u_vec./(1-β)\n",
                "w_space = [minimum(w_vec) maximum(w_vec)]\n",
                "θ_space = [0 maximum((Θ_vec)./(1-β))]\n"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 32,
            "metadata": {
                "scrolled": true
            },
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "initialize (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 32,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "function initialize()\n",
                "\n",
                "    # center and radius\n",
                "    cen = [(w_space[2]+w_space[1])/2 (θ_space[2]+θ_space[1])/2]\n",
                "    r1 = maximum([(w_space[2]- cen[1])^2, (cen[1] - w_space[1])^2])\n",
                "    r2 = maximum([(θ_space[2]- cen[2])^2, (cen[2] - θ_space[1])^2])\n",
                "    rad = sqrt(r1 + r2)\n",
                "    degree = range(0, 2π, length=n_g)\n",
                "    H = [cos.(degree) sin.(degree)]\n",
                "    Z = cen .+ rad .* H\n",
                "    C = Z .* H * [1,1] # hyperplane level\n",
                "    return (C, H, Z)\n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 33,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "([26.452626668543168, 26.29470053708557, 25.677592360886464, 24.612314449969674, 23.117876713892418, 21.220947429458878, 18.955377344688866, 16.361595611417446, 13.485888326122687, 10.37957255343743  …  9.356348199225152, 12.524320422387305, 15.47884336018107, 18.167193480787613, 20.541397112616497, 22.5590865356118, 24.184256034220223, 25.387904420238122, 26.14855255970204, 26.452626668543168], [1.0 0.0; 0.9910774881547801 0.13328695537377883; … ; 0.99107748815478 -0.13328695537377913; 1.0 -2.4492935982947064e-16], [26.452626668543168 0.5482456140350878; 26.447602913905268 0.6232918534624999; … ; 26.447602913905268 0.47319937460767536; 26.452626668543168 0.5482456140350876])"
                        ]
                    },
                    "execution_count": 33,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "# Initial hyperplane levels, subgradients, and verticies\n",
                "c0, H, z0 = initialize()\n",
                "# gr() # GR backend\n",
                "# scatter(z0[:,1], z0[:,2], legend = false)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 34,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "([25.7613507911023 26.01781726304005], [0.0 1.0964912280701755])"
                        ]
                    },
                    "execution_count": 34,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "c0_c, z0_c = c0_s, z0_s = copy(c0), copy(z0)\n",
                "w_space_c, θ_space_c = w_space_s, θ_space_s = copy(w_space), copy(θ_space)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 35,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "outer_competitive (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 35,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "## Outer apporximaiton\n",
                "# for competitive equilibria\n",
                "function outer_competitive(c0_c,z0_c,w_space_c,θ_space_c)\n",
                "\n",
                "   # c1_c, z1_c = SharedArray{Float64}(n_g), SharedArray{Float64}(n_g,2) # to be shared by workers\n",
                "    c1_c, z1_c = copy(c0_c), copy(z0_c)\n",
                "\n",
                " @sync @distributed for i in 1:n_g\n",
                "        c_action = -Inf*ones(n_a,1) # levels\n",
                "        wp_action = zeros(n_a,2) # values\n",
                "        for j in 1:n_a\n",
                "            if !(f_vec[j]>0)\n",
                "                break # nonpositive consumption\n",
                "            end\n",
                "#             m = Model(with_optimizer(Gurobi.Optimizer, GUROBI_ENV, Presolve=0, OutputFlag=0))\n",
                "            m = Model(optimizer_with_attributes(() -> Gurobi.Optimizer(GUROBI_ENV), \"Presolve\" => 0, \"OutputFlag\" => 0))            \n",
                "            @variable(m, w_space_c[1] <= w <=w_space_c[2])\n",
                "            @variable(m, θ_space_c[1] <= θ <= θ_space_c[2])\n",
                "            @objective(m, Max, dot(H[i,:],[w, θ]))\n",
                "            # @objective(m, Max, dot(H[i,:],[u_vec[j,1]+β*w, Θ_vec[j,1]]))\n",
                "            @constraint(m, constraint1, H*[w, θ] .<= c0_c)\n",
                "            if A[j,2] == mbar\n",
                "                @constraint(m, constraint2, Euler_vec[j] <= β*θ )\n",
                "            else # m < mbar, then adding one equality constraint\n",
                "                @constraint(m, constraint2, Euler_vec[j] == β*θ )\n",
                "            end\n",
                "            optimize!(m)\n",
                "            if termination_status(m) == MOI.OPTIMAL\n",
                "                c_action[j,1] = dot(H[i,:],[u_vec[j,1]+β*value(w), Θ_vec[j,1]]) # actual hyperplane level\n",
                "                #c_action[j,1] = objective_value(m)\n",
                "                wp_action[j,:] = [value(w) value(θ)]\n",
                "            end\n",
                "        end\n",
                "        c1_c[i], idx = findmax(c_action) # the maximum\n",
                "        z1_c[i,:] = [u_vec[idx]+β*wp_action[idx] Θ_vec[idx]] #taking into account other constraints\n",
                "    end\n",
                "    return c1_c, z1_c\n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 36,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "worst_spe (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 36,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "function worst_spe(w_space_s,θ_space_s)\n",
                "\n",
                "    P_vec = SharedArray{Float64}(n_a)\n",
                "#     P_vec = fill(missing,n_a)\n",
                "\n",
                " @sync @distributed for j in 1:n_a\n",
                "    if !(f_vec[j]>0) \n",
                "        break # nonpositive consumption\n",
                "    end\n",
                "#     m = Model(with_optimizer(Gurobi.Optimizer, GUROBI_ENV, Presolve=0, OutputFlag=0))\n",
                "    m = Model(optimizer_with_attributes(() -> Gurobi.Optimizer(GUROBI_ENV), \"Presolve\" => 0, \"OutputFlag\" => 0))        \n",
                "    @variable(m, w_space_s[1] <= w <=w_space_s[2])\n",
                "    @variable(m, θ_space_s[1] <= θ <= θ_space_s[2])\n",
                "    @objective(m, Min, u_vec[j,1] + β * w)\n",
                "    @constraint(m, constraint1, H*[w, θ] .<= c0_s)\n",
                "    if A[j,2] == mbar\n",
                "    # m = mbar, then adding one more inequality constraint\n",
                "        @constraint(m, constraint2, Euler_vec[j] <= β*θ)\n",
                "    else # m < mbar, then adding one equality constraint\n",
                "       @constraint(m, constraint2, Euler_vec[j] == β*θ)\n",
                "    end\n",
                "    optimize!(m)\n",
                "    if termination_status(m) == MOI.OPTIMAL\n",
                "       P_vec[j] = objective_value(m)\n",
                "    end\n",
                "    end\n",
                "# Max over h and min over other variables\n",
                "BR = maximum(minimum(reshape(P_vec,n_h, n_m), dims=1))\n",
                "    return BR\n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 37,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "outer_sustainable (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 37,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "# for sustainable plans\n",
                "function outer_sustainable(BR,c0_s,z0_s,w_space_s,θ_space_s)\n",
                "\n",
                "    # c1_s, z1_s = SharedArray{Float64}(n_g), SharedArray{Float64}(n_g,2) # to be shared by workers\n",
                "    c1_s, z1_s = copy(c0_s), copy(z0_s)\n",
                "\n",
                " @sync @distributed for i in 1:n_g\n",
                "        c_action = -Inf*ones(n_a,1) # levels\n",
                "        wp_action = zeros(n_a,2) # values\n",
                "        for j in 1:n_a\n",
                "            if !(f_vec[j]>0)\n",
                "                break # nonpositive consumption\n",
                "            end\n",
                "#             m = Model(with_optimizer(Gurobi.Optimizer, GUROBI_ENV, Presolve=0, OutputFlag=0))\n",
                "            m = Model(optimizer_with_attributes(() -> Gurobi.Optimizer(GUROBI_ENV), \"Presolve\" => 0, \"OutputFlag\" => 0))\n",
                "            @variable(m, w_space_s[1] <= w <=w_space_s[2])\n",
                "            @variable(m, θ_space_s[1] <= θ <= θ_space_s[2])\n",
                "            #@objective(m, Min, u_vec[j,1] + β * w)\n",
                "            @objective(m, Max, dot(H[i,:],[w, θ]))\n",
                "            @constraint(m, constraint1, H*[w, θ] .<= c0_s)\n",
                "            @constraint(m, constraint3, BR <= u_vec[j]+β*w)\n",
                "            if A[j,2] == mbar\n",
                "                @constraint(m, constraint2, Euler_vec[j] <= β*θ )\n",
                "            else # m < mbar, then adding one equality constraint\n",
                "                @constraint(m, constraint2, Euler_vec[j] == β*θ )\n",
                "            end\n",
                "            optimize!(m)\n",
                "            if termination_status(m) == MOI.OPTIMAL\n",
                "                c_action[j,1] = dot(H[i,:],[u_vec[j,1]+β*value(w), Θ_vec[j,1]]) # actual hyperplane level\n",
                "                wp_action[j,:] = [value(w) value(θ)]\n",
                "            end\n",
                "        end\n",
                "        c1_s[i], idx = findmax(c_action) # the maximum\n",
                "        z1_s[i,:] = [u_vec[idx]+β*wp_action[idx] Θ_vec[idx]] #taking into account other constraints\n",
                "    end\n",
                "    return c1_s, z1_s\n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 38,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "ChangModel (generic function with 1 method)"
                        ]
                    },
                    "execution_count": 38,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "function ChangModel(c0_c,z0_c,c0_s,z0_s,w_space_c,θ_space_c,w_space_s,θ_space_s;\n",
                "    # tol = 1e-3,\n",
                "    tol = 1e-5,\n",
                "    # tol = 1e-9,\n",
                "    max_iter = 250,\n",
                "    diff = tol + 1,\n",
                "    diff_z = tol + 1)\n",
                "\n",
                "     c1_c, z1_c = copy(c0_c), copy(z0_c)\n",
                "\n",
                "     c1_s, z1_s = copy(c0_s), copy(z0_s)\n",
                " for iters in 1:max_iter\n",
                "        c1_c, z1_c = outer_competitive(c0_c,z0_c,w_space_c,θ_space_c)\n",
                "\n",
                "        BR = worst_spe(w_space_s,θ_space_s)\n",
                "        c1_s, z1_s =  outer_sustainable(BR,c0_s,z0_s,w_space_s,θ_space_s)\n",
                "\n",
                "        # difference in hyplerplane levels\n",
                "        diff = minimum([maximum(abs.(c0_c - c1_c)), maximum(abs.(c0_s - c1_s))]) # sooner\n",
                "        # diff = maximum([maximum(abs.(c0_c - c1_c)), maximum(abs.(c0_s - c1_s))]) # later\n",
                "        # difference in extreme points\n",
                "        diff_z = maximum([maximum(abs.(z1_c-z0_c)./(1 .+ abs.(z0_c))), maximum(abs.(z1_s-z0_s)./(1 .+ abs.(z0_s)))])\n",
                "        # diff_z = minimum([maximum(abs.(z1_c-z0_c)./(1 .+ abs.(z0_c))), maximum(abs.(z1_s-z0_s)./(1 .+ abs.(z0_s)))])\n",
                "\n",
                "        # print results\n",
                "        if mod(iters,1) == 0\n",
                "            println(\"iteration #$iters: difference is $diff, $diff_z\")\n",
                "        end\n",
                "\n",
                "        # break\n",
                "        if  diff < tol || diff_z < tol\n",
                "            break\n",
                "        end\n",
                "\n",
                "        # update\n",
                "        c0_c, z0_c = copy(c1_c), copy(z1_c)\n",
                "        c0_s, z0_s = copy(c1_s), copy(z1_s)\n",
                "        w_space_c = [minimum(z1_c[:,1]), maximum(z1_c[:,1])]\n",
                "        θ_space_c = [minimum(z1_c[:,2]), maximum(z1_c[:,2])]\n",
                "        w_space_s = [minimum(z1_s[:,1]), maximum(z1_s[:,1])]\n",
                "        θ_space_s = [minimum(z1_s[:,2]), maximum(z1_s[:,2])]\n",
                "end\n",
                "    return  c1_c, z1_c, c1_s, z1_s\n",
                "end\n"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 39,
            "metadata": {
                "scrolled": true
            },
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "iteration #1: difference is 0.8875214031201635, 0.42240011359321034\n",
                        "iteration #2: difference is 0.032800710280348255, 0.21394257951742784\n",
                        "iteration #3: difference is 0.02481438979947015, 0.025974025973133953\n",
                        "iteration #4: difference is 0.018337328847962908, 0.2086335368312195\n",
                        "iteration #5: difference is 0.012960231968328628, 0.0833712515121931\n",
                        "iteration #6: difference is 0.010083575992304361, 0.1156226779547478\n",
                        "iteration #7: difference is 0.007022416525877873, 0.04378387441913222\n",
                        "iteration #8: difference is 0.005324000273006391, 0.02947735788276953\n",
                        "iteration #9: difference is 0.004259200218410086, 0.012316543242811333\n",
                        "iteration #10: difference is 0.0034230415209854925, 0.013014129135273236\n",
                        "iteration #11: difference is 0.002848704186252604, 0.012858146226508072\n",
                        "iteration #12: difference is 0.002278963349002794, 0.006529787797708546\n",
                        "iteration #13: difference is 0.0018231706792057878, 0.005481485072661978\n",
                        "iteration #14: difference is 0.0014824344337327489, 0.005497953657707505\n",
                        "iteration #15: difference is 0.0012518283384181927, 0.005515484535854002\n",
                        "iteration #16: difference is 0.0010979367206864765, 0.005334949219475804\n",
                        "iteration #17: difference is 0.0008807608432555014, 0.005553794006251393\n",
                        "iteration #18: difference is 0.0007371913408036335, 0.004337082332939675\n",
                        "iteration #19: difference is 0.0005897530726457489, 0.0026534393301004093\n",
                        "iteration #20: difference is 0.000484818807262144, 0.005574605143283524\n",
                        "iteration #21: difference is 0.00039134684265107467, 0.005840327837549827\n",
                        "iteration #22: difference is 0.00031307747411801756, 0.003442791773777823\n",
                        "iteration #23: difference is 0.0002504619792951246, 0.004336362016986978\n",
                        "iteration #24: difference is 0.00020036958343538913, 1.8897333062824806e-5\n",
                        "iteration #25: difference is 0.00016029566674902185, 0.005860916064951887\n",
                        "iteration #26: difference is 0.0001282365334027702, 0.00476190476172487\n",
                        "iteration #27: difference is 0.00010258922671724235, 0.0035138108957611055\n",
                        "iteration #28: difference is 8.207138137450443e-5, 1.1961961775710869e-5\n",
                        "iteration #29: difference is 6.565710510031408e-5, 9.56968389270897e-6\n",
                        "645.944166 seconds (2.71 G allocations: 163.045 GiB, 5.43% gc time)\n"
                    ]
                },
                {
                    "data": {
                        "text/plain": [
                            "([25.99322399897939, 25.765641368263367, 25.07940813404647, 23.947190907987306, 22.389149479160512, 20.43395924547124, 18.11708834284791, 15.481067577196495, 12.576622807535685, 9.463617219477053  …  9.332233840087929, 12.481859747195758, 15.408746571414113, 18.060663948025617, 20.390288348619535, 22.356145402825472, 23.923650804164403, 25.064272524359517, 25.758198687565557, 25.99322399897939], [25.99322399897939 0.02941176471045752; 25.992759414282887 0.036032466424797176; … ; 25.992469184968932 0.01764901304391933; 25.99322399897939 0.02941176471045752], [25.99267644953289, 25.764635751165663, 25.0778740533394, 23.944617047247643, 22.384627523873668, 20.425338766025902, 18.10155935401314, 15.454757186494119, 12.532164510852109, 9.385935062621453  …  9.332196389713674, 12.481809670464113, 15.40868476194554, 18.060591508811548, 20.390206572339462, 22.355985877475025, 23.92348010829476, 25.064059269824206, 25.757840271395445, 25.99267644953289], [25.99267644953289 0.02450980392630719; 25.99223850751157 0.032361003438417774; … ; 25.992107542035463 0.01764901304391933; 25.99267644953289 0.02450980392630719])"
                        ]
                    },
                    "execution_count": 39,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "# Results\n",
                "@time c1_c, z1_c, c1_s, z1_s = ChangModel(c0_c,z0_c,c0_s,z0_s,w_space_c,θ_space_c,w_space_s,θ_space_s)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 40,
            "metadata": {
                "scrolled": true
            },
            "outputs": [
                {
                    "data": {
                        "text/plain": [
                            "10-element Array{Array{Float64,1},1}:\n",
                            " [25.98672764970181, 0.008823773673181524]\n",
                            " [25.9883933240489, 0.008823773673181524]\n",
                            " [25.99103925748768, 0.013236118517131395]\n",
                            " [25.992107542035463, 0.01764901304391933]\n",
                            " [25.99267644953289, 0.02450980392630719]\n",
                            " [25.99223850751157, 0.032361003438417774]\n",
                            " [25.990919547828792, 0.03949091493037129]\n",
                            " [25.988395294156188, 0.04714234214760601]\n",
                            " [25.98737323452967, 0.04915567893202606]\n",
                            " [25.98672764970181, 0.04915567893202606]"
                        ]
                    },
                    "execution_count": 40,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "v_c, v_s = fill(Float64[], n_g), fill(Float64[], n_g)\n",
                "for i in 1:n_g\n",
                "    v_c[i] = z1_c[i,:]\n",
                "    v_s[i] = z1_s[i,:]\n",
                "end\n",
                "hull_c = convex_hull(v_c)\n",
                "hull_s = convex_hull(v_s)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 41,
            "metadata": {
                "scrolled": false
            },
            "outputs": [
                {
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                        ]
                    },
                    "execution_count": 41,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "p = plot(VPolygon(hull_c), alpha=0.2, fillcolor = :red, xlabel = \"w\", ylabel = \"θ\", label = \"commetitive equilibria\")\n",
                "plot!(p, VPolygon(hull_s), alpha=0.2, fillcolor = :blue, label = \"sustainable plans\")\n",
                "scatter!([v_c[1][1]],[v_c[1][2]], label = \"ramsey\")\n",
                "# savefig(\"b$β\"*\"g\"*\"$n_g\"*\".png\")\n",
                "# savefig(\"b$β\"*\"g\"*\"$n_g\"*\"_2\"*\".png\")"
            ]
        }
    ],
    "metadata": {
        "@webio": {
            "lastCommId": null,
            "lastKernelId": null
        },
        "kernelspec": {
            "display_name": "Julia 1.4.2",
            "language": "julia",
            "name": "julia-1.4"
        },
        "language_info": {
            "file_extension": ".jl",
            "mimetype": "application/julia",
            "name": "julia",
            "version": "1.4.2"
        }
    },
    "nbformat": 4,
    "nbformat_minor": 2
}