growth-vfi.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Stochastic growth model\n",
    "\n",
    "I present below a simple version of the growth model solved mapped to a planner's problem. The planner has to choose a sequence of consumption, capital stock, investment, leisure and labor $(c_t,k_t,x_t,l_t,h_t)_{t=0}^\\infty$ to maximize the discounted utility of the representative consumer: $\\sum_{t=0}^\\infty N_t\\beta^t [ log(c_t) +\\varphi log(l_t)]$ \n",
    "\n",
    "The production function of the representative firm is $f(k,h)=k_t^\\theta(Z_t h_t)^{1-\\theta}$. Population grows at a constant rate such that $N_t = (1+\\gamma_n)N_{t-1}$. Labor augmenting productivity has a deterministic exogenous trend and a stochastic AR(1) component such that: $Z_t = z_t(1+\\gamma_z)^t$ where $z_t = \\rho_z z_{t-1}+\\varepsilon_t,\\; \\varepsilon_t \\sim N(0,\\sigma^2)$. Capital depreciates at rate $\\delta$.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The Planner's problem is:\n",
    "\n",
    " $$\\max_{\\{c_t,x_t,l_t\\}} E\\left[\\sum_{t=0}^\\infty \\beta^t N_t ( log(c_t) +\\varphi log(l_t))\\right]$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "st\n",
    "$$c_t + x_t = k_t^\\theta (Z_t h_t)^{1-\\theta} $$\n",
    "$$ N_{t+1} k_{t+1} = \\left((1-\\delta) k_t + x_t\\right) N_t  $$\n",
    "$$\\log z_t = \\rho \\log z_{t-1}+\\epsilon_t,\\; \\epsilon \\sim N(0,\\sigma^2)$$\n",
    "$$ h_t+l_t=1 $$\n",
    "$$c_t,x_t\\geq 0 $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "With some change in variables, the problem can be mapped to the following detrended equivalent functional problem:\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " $$V(k_t,z_t) = \\max_{\\{c_t,x_t,l_t\\}}     log(c_t) +\\varphi log(l_t) + \\beta(1+\\gamma_n) E[V(k_{t+1},z_{t+1})|z_t] $$\n",
    " st\n",
    "$$c_t + x_t = k_t^\\theta (z_t h_t)^{1-\\theta} $$\n",
    "$$ (1+\\gamma_z)(1+\\gamma_n) k_{t+1} = \\left((1-\\delta) k_t + x_t\\right)   $$\n",
    "$$\\log z_t = \\rho \\log z_{t-1}+\\epsilon_t,\\; \\epsilon \\sim N(0,\\sigma^2)$$\n",
    "$$ h_t+l_t=1 $$\n",
    "$$c_t,x_t\\geq 0 $$\n",
    "\n",
    "This is the problem we will be solving for now on."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Defining Parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Defining Prameters\n",
    "δ = 1.0    #depreciation rate\n",
    "θ = 1/3  #capital share of output\n",
    "β = 0.9  #Discouting\n",
    "ρ = 0.5  #AR coefficient\n",
    "σ = 0.5  #AR shock SD\n",
    "μ = 0.0    #AR(1) constant term\n",
    "ϕ = 0.0    #Labor parameter\n",
    "γn= 0.0    #Population growth rate\n",
    "γz= 0.0    #Productivitu growth rate\n",
    "\n",
    "Xnneg = true; #Non negative investment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving the basic growth model via Value Function Iteration\n",
    "\n",
    "VFI is a traditional method for solving dynamic economic models. It is a straightforward application of the contraction mapping theorem. The idea is simple, for an initia guess for the value function, $V_0$ solve the maximization problem above and replace the guess with the new value function implied by this solution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "VFI (generic function with 10 methods)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Optim, Distributions, LinearAlgebra\n",
    "#This code defines the VFI function as well its auxiliary functions, notably Tauchen method for discretizing an AR(1) process.\n",
    "\n",
    "#Consumption function implied by a decision of k, z and h with non-negative investment option\n",
    "function consumption!(c,k,z,h,γn=0,γz=0,ϕ=0,δ = 1,θ=1/3, InvestmentNonNegative = false)\n",
    "    K=length(k)\n",
    "    Z=length(z)\n",
    "    H=length(h)\n",
    "    #Preallocating memory. Note loops are fast in Julia.\n",
    "    if InvestmentNonNegative == true #Non Negative investment\n",
    "        for k0 in 1:K, k1 in 1:K\n",
    "            x = (1+γz)*(1+γn)*k[k1]-(1-δ)*k[k0] #define the investment variable\n",
    "            if x>=0     #If it is positive, we assign a normal value for consumption\n",
    "                for z0 in 1:Z , h0 in 1:H\n",
    "                    c[k0,z0,k1,h0]=k[k0]^θ *(z[z0]*h[h0])^(1-θ) + (1-δ)k[k0] - (1+γz)*(1+γn)k[k1]\n",
    "                end\n",
    "            else #If not, consumption is negative and will\n",
    "                                      #be ruled out by Utility function\n",
    "                c[k0,:,k1,:] = -ones(1,Z,1,H)\n",
    "            end\n",
    "        end\n",
    "    else #Standard case: investment can be negative\n",
    "        for k0 in 1:K, z0 in 1:Z, k1 in 1:K, h0 in 1:H\n",
    "            c[k0,z0,k1,h0]=k[k0]^θ *(z[z0]*h[h0])^(1-θ) + (1-δ)k[k0] - (1+γz)*(1+γn)k[k1]\n",
    "        end\n",
    "    end\n",
    "    return c\n",
    "end\n",
    "\n",
    "function consumption(k::Real,k1::Real,z::Real,h::Real,γn=0,γz=0,ϕ=0,δ = 1,θ=1/3, InvestmentNonNegative = false)\n",
    "\n",
    "    x = (1+γz)*(1+γn)*k1-(1-δ)*k #define the investment variable\n",
    "    \n",
    "    if x<0   && InvestmentNonNegative                                       #c is negative and will be ruled out by Utility function\n",
    "        return -1.0\n",
    "    else\n",
    "        return k^θ *(z*h)^(1-θ) - x\n",
    "    end\n",
    "end\n",
    "\n",
    "function log_utility(c,h,ϕ)\n",
    "    if c<=0\n",
    "        u = -1e10 #any value of negative consumtpion in the grid will be avoided\n",
    "    else\n",
    "        u = log(c) + (ϕ > 0 ? ϕ * log(1-h) : 0.0)\n",
    "    end\n",
    "    return u\n",
    "end\n",
    "\n",
    "\n",
    "function Tauchen(ρ,σ,Y,μ = 0.0,m = 3.0)\n",
    "    #This function is to discretize an AR(1) process following Tauchen(1986) method\n",
    "    # y_{t+1} = μ + ρy_t + ϵ\n",
    "    #ϵ~N(0,σ^2)\n",
    "    #Y is the number of y states\n",
    "if Y>1\n",
    "    ybar = μ/(1-ρ)\n",
    "    ymax= ybar + m*(σ^2/(1-ρ^2))^(1\\2) #maximum y\n",
    "    ymin= ybar - m*(σ^2/(1-ρ^2))^(1\\2) #minimum y\n",
    "\n",
    "    Δ = (ymax-ymin)/(Y-1)# #distance between each y\n",
    "    y=ymin:Δ:ymax #vector of possible states of p\n",
    "\n",
    "    d=Normal()\n",
    "\n",
    "    pdfY=ones(Y,Y) #preallocate memory and create the transition matrix in the following loop\n",
    "    for i in 1:Y\n",
    "        pdfY[i,1]=cdf(d,(y[1] + Δ/2 -ρ*y[i]) / σ^0.5);\n",
    "        pdfY[i,Y]=1-cdf(d,(y[Y] -Δ/2 - ρ*y[i]) / σ^0.5);\n",
    "        for j in 2:Y-1\n",
    "            pdfY[i,j]=cdf(d,(y[j] + Δ/2 - ρ*y[i])/σ^0.5) - cdf(d,(y[j] - Δ/2 - ρ*y[i]) / σ^0.5);\n",
    "        end\n",
    "    end\n",
    "else\n",
    "    y=μ:μ\n",
    "end\n",
    "\n",
    "    return pdfY, y\n",
    "end\n",
    "\n",
    "function hstar(k,k1,z,γn=0,γz=0,ϕ=0,δ = 1,θ=1/3, InvestmentNonNegative = false)\n",
    "    hs = 1.0\n",
    "    ch(h) = consumption(k,k1,z,h,γn,γz,ϕ,δ,θ, InvestmentNonNegative )\n",
    "    uh(h) = -log_utility(ch(h),h,ϕ)\n",
    "    if ϕ > 0\n",
    "        hs = optimize( uh, 0.0,1.0).minimizer\n",
    "    end\n",
    "    \n",
    "\n",
    "    return hs \n",
    "end\n",
    "\n",
    "\n",
    "function VFI(δ,θ,β,Π, z,γn=0,γz=0,ϕ=0,K=500,kmax=10,kmin=0.01,Z=10,Xnneg=false,tol=1e-5;verbose=false)\n",
    "    #parameters:\n",
    "    #δ - depreciation rate\n",
    "    #θ - capital share of output\n",
    "    #β - Discouting\n",
    "    #ϕ - Labor parameter\n",
    "    #K - number of gridpoints for capital AND LABOR\n",
    "    #Xnneg - Non negative investment (true or false)\n",
    "    #Π - Productivity probability\n",
    "    #z -productivity vector\n",
    "    #kmax - maximum value for k (set low if no productivity shocks!)\n",
    "    #kmin - minimum value for k\n",
    "    #tol - convergence parameter\n",
    "    #grid for k\n",
    "        k = range(kmin,stop = kmax, length=K)\n",
    "   \n",
    "\n",
    "    #Initial value function guess:\n",
    "        V  = zeros(K,Z)\n",
    "        Vf = zeros(K,Z)\n",
    "        EVf = copy(V)\n",
    "        \n",
    "        hs = ones(K,K,Z)\n",
    "        if ϕ >0.0\n",
    "        for (zi,z0) in enumerate(z),(ki,k0) in enumerate(k),(ki1,k1) in enumerate(k)\n",
    "            hs[ki1,zi,ki] = hstar(k0,k1,z0,γn,γz,ϕ,δ,θ, Xnneg)\n",
    "        end\n",
    "        end\n",
    "\n",
    "        #preallocation\n",
    "        policy_k = ones(Int64,K,Z)\n",
    "        policy_h=  ones(K,Z)\n",
    "\n",
    "        #loop stuff\n",
    "        distance = tol+1.0\n",
    "\n",
    "    #iterating on Value functions:\n",
    "    while distance >= tol\n",
    "        #distance, Vf, policy_k\n",
    "        Vf .= V #save V to compare later\n",
    "\n",
    "        #find the expected Vf: E[V(k,z')|z] = π(z1|z)*V(k,z1) +π(z2|z)*V(k,z2)\n",
    "        EVf .= (Π*Vf')'  #The transposing fix the dimensions so it works!\n",
    "        for  (zi,z0) in enumerate(z), (ki,k0) in enumerate(k)\n",
    "            v = -Inf\n",
    "            for (ki1,k1) in enumerate(k) \n",
    "                hh = hs[ki1,ki,zi]\n",
    "                cc = consumption(k0,k1,z0,hh,γn,γz,ϕ,δ ,θ, Xnneg)\n",
    "                v1 = log_utility(cc ,hh,ϕ) + β *(1+γn) * EVf[ki1,zi]\n",
    "                if v1 > v\n",
    "                    V[ki,zi] = v1\n",
    "                    policy_k[ki,zi] = ki1\n",
    "                    v = v1 \n",
    "                elseif policy_k[ki,zi] < ki1 \n",
    "                    break\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    distance = norm(V .- Vf)\n",
    "\n",
    "    end\n",
    "    if ϕ > 0.0\n",
    "    for (zi,z0) in enumerate(z),(ki,k0) in enumerate(k) \n",
    "        policy_h[ki,zi] = hs[policy_k[ki,zi],ki,zi]\n",
    "    end\n",
    "    end\n",
    "        return V, policy_k, k,policy_h\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "#Defining grids and running code\n",
    "\n",
    "#grid for k\n",
    "\n",
    "K = 200 #number of gridpoints for capital\n",
    "kmax = 1/5 # maximum value for k (set low if no productivity shocks!)\n",
    "kmin = 0.01 #minimum value for k\n",
    "#grid for z\n",
    "Z = 40 #number of productivity states\n",
    "Π, z = Tauchen(ρ,σ,Z,μ) #Transition matrix and values for log(z)\n",
    "z=exp.(z) #recover z\n",
    "\n",
    "\n",
    "V, policy_k, k,policy_h = VFI(δ,θ,β,Π,z,γn,γz,ϕ,K,kmax,kmin,Z,Xnneg);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots,Printf\n",
    "\n",
    "#plot the Value function griven 2 level of z\n",
    "labels = [@sprintf(\"z = %.2f\",z[2]) @sprintf(\"z = %.2f\",z[9]) ]\n",
    "data = [V[:,2],V[:,9]]\n",
    "plot(k, data, label = labels, legend = :bottomright )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#plot the capital policy function given 2 level of z\n",
    "labels = [@sprintf(\"z = %.2f\",z[2]) @sprintf(\"z = %.2f\",z[9]) \"45\"]\n",
    "data = [k[policy_k[:,2]],k[policy_k[:,9]],k]\n",
    "plot(k, data, label = labels,legend = :bottomright )\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
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     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#plot the Labor policy function given 2 level of z. Note, labor choice is constant!\n",
    "labels = [@sprintf(\"z = %.2f\",z[2]) @sprintf(\"z = %.2f\",z[9]) ]\n",
    "data = [policy_h[:,3],policy_h[:,9]]\n",
    "plot(k, data, label = labels,ylim = [0,1.01],legend = :bottomright )"
   ]
  }
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