huggett_poisson_eqm.ipynb

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    "# Solving continuous time economic models: Poisson income process\n",
    "\n",
    "This Jupyter notebook presents Julia code that solves a continuous time version of a heterogeneous agents economy. Specifically, it solves a version of the Huggett model in which agents solve a consumption-savings problem subject to a borrowing constraint and idiosyncratic shocks. This version of the model assumes that the income process for agents follows a Markov process with a Poisson probability of switching between two possible income states.\n",
    "\n",
    "The model and it's solution method is described in detail in the accompanying PDF document. Note, too, that an additional Jupyter notebook solves the same model but with a diffusion process governing the income state variable. \n",
    "\n",
    "#### Defining a composite type for the model object\n",
    "\n",
    "The first thing we do before anything else is declare a composite type, which contains (most of) the fields that will be used in the solution to the model. Each model parameter, variable, and data storage matrix is given a particular type, which helps increase the speed with which the code can be run. "
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    "type Huggett_model\n",
    "  sigma::Float64            # risk aversion coefficient\n",
    "  rho::Float64              # discount rate.\n",
    "  z1::Float64               # Income in state 1\n",
    "  z2::Float64               # Income in state 2\n",
    "  z::Array{Float64}         # Vector for the two incomes\n",
    "  la1::Float64              # State 1 switching intensity\n",
    "  la2::Float64              # State 2 switching intensity\n",
    "  la::Array{Float64}        # Vector for the two intensities\n",
    "  I::Int64                  # Number of grids for assets\n",
    "  Ir::Int64                 # Number of interest rate iterations\n",
    "  amin::Float64             # Min assets = borrowing constraint\n",
    "  amax::Float64             # Max assets = upper bound of state space\n",
    "  rmin::Float64             # Min interest rate\n",
    "  rmax::Float64             # Max interest rate\n",
    "  agrid::Vector{Float64}    # Grid space for assets\n",
    "  aa::Array{Float64}        # Matrix for asset grid space\n",
    "  zz::Array{Float64}        # Matrix for income grid space\n",
    "  da::Float64               # This is the implied grid step size\n",
    "  maxit::Int64              # Maximum iterations for HJB solution algorithm\n",
    "  crit::Float64             # Tolerance for HJB equation solution   \n",
    "  crit_S::Float64           # Tolerance for asset mkt clearing\n",
    "  Delta::Float64            # Time step-size for use in approximating derivatives\n",
    "  Aswitch::SparseMatrixCSC{Float64,Int64}    # Transition matrix\n",
    "  v0::Array{Float64}        # Initial guess for value function\n",
    "  r0::Float64               # Intial guess for interest rate\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "Next we create the model object itself. We do this by defining a function, which takes the deep parameters of the model as default arguments, and produces all of the other model objects that can be pre-computed. This includes the grid spaces for assets and income, the initial guess for the HJB value function, and the transition matrix used in computation of the HJB equation.\n",
    "\n",
    "The final command in this cell produces an instance of the model object (`Huggett_model`), which we will use when calling any of the paramters, variables, data storage matrices, or pre-computed model objects. "
   ]
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  {
   "cell_type": "code",
   "execution_count": 12,
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   "source": [
    "function Huggett_model(;sigma::Float64=1.5,\n",
    "  rho::Float64=0.05,\n",
    "  I::Int64=1000,\n",
    "  Ir::Int64=40,\n",
    "  amin::Float64=-0.15,\n",
    "  amax::Float64=5.0,\n",
    "  rmin::Float64=0.01,\n",
    "  rmax::Float64=0.04,\n",
    "  z1::Float64=0.1,\n",
    "  z2::Float64=0.2,\n",
    "  la1::Float64=1.2,\n",
    "  la2::Float64=1.5,\n",
    "  maxit::Int64=100,\n",
    "  crit::Float64=1e-6,\n",
    "  crit_S::Float64=1e-5,\n",
    "  Delta::Float64=1000.0,\n",
    "  r0::Float64=0.03)\n",
    "\n",
    "  z       = hcat(z1,z2)\n",
    "  la      = hcat(la1,la2)\n",
    "  agrid   = collect(linspace(amin,amax,I))\n",
    "  aa      = hcat(agrid, agrid)\n",
    "  zz      = ones(I,1)*z\n",
    "  da      = (amax-amin)/(I-1)\n",
    "  v0 = hcat( (z[1] + r0.*agrid).^(1-sigma)/(1-sigma)/rho, (z[2] + r0.*agrid).^(1-sigma)/(1-sigma)/rho )\n",
    "\n",
    "  Aswitch =  vcat( hcat(-speye(I)*la[1], speye(I)*la[1]), hcat(speye(I)*la[2],-speye(I)*la[2]) )\n",
    "\n",
    "  Huggett_model(sigma, rho, z1, z2, z, la1, la2, la, I, Ir, amin, amax, rmin, rmax, agrid, aa, zz, da, maxit, crit, crit_S, Delta, Aswitch,  v0, r0)\n",
    "\n",
    "end\n",
    "\n",
    "# Creates an instance of the model\n",
    "mod = Huggett_model();"
   ]
  },
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    "Next I define several useful functions for use in solving the model. The first three are, the CRRA utility function, the derivative of the utility function, and the inverse of the derivative of the utility function. The final function solves the Kolmogorov Forward Equation for the model using the transition matrix found by solving the HJB equation. "
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    {
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      "text/plain": [
       "KFE (generic function with 1 method)"
      ]
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     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
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   "source": [
    "function util(x; sigma::Float64=mod.sigma)\n",
    "  u = x.^(1-sigma)/(1-sigma)\n",
    "  return u\n",
    "end\n",
    "\n",
    "function utilprime(x; sigma::Float64=mod.sigma)\n",
    "  uprime = x.^(-sigma)\n",
    "  return uprime\n",
    "end\n",
    "\n",
    "function utilprimeinv(x; sigma::Float64=mod.sigma)\n",
    "  uprimeinv = x.^(-1/sigma)\n",
    "  return uprimeinv\n",
    "end\n",
    "\n",
    "# Function to compute the KFE\n",
    "function KFE(A::SparseMatrixCSC{Float64,Int64}; I::Int64=mod.I, da::Float64=mod.da)\n",
    "  AT          = A'\n",
    "  b           = zeros(2*I,1)\n",
    "  i_fix       = 1    # fix one value so that matrix is not singular\n",
    "  b[i_fix]    = 0.1\n",
    "  row         = hcat(zeros(1,i_fix-1), 1, zeros(1,2*I-i_fix))\n",
    "  AT[i_fix,:] = row\n",
    "  #Solve linear system\n",
    "  gg          = AT\\b          # A'*gg = b --> gg = (A')^(-1)*b\n",
    "  g_sum       = gg'*ones(2*I,1)*da\n",
    "  gg          = gg./g_sum     # Normalize so sum(gg)*da) = 1\n",
    "  g           = hcat(gg[1:I], gg[I+1:2*I])\n",
    "  #check1      = g[:,1]'*ones(mod.I,1)*mod.da\n",
    "  #check2      = g[:,2]'*ones(mod.I,1)*mod.da\n",
    "  return g\n",
    "end"
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   "cell_type": "markdown",
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   "source": [
    "Next, we initialize several data storage arrays. This turns out to be important, as Julia does not store arrays/data that are initialized and produced inside loops. Thus, anything that we might want access to after convergence of the algorithm (e.g. the equilibrium interest rate, the value function, policy functions) needs to be initialized here. "
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    "dVf         = zeros(mod.I,2)          # Forward difference in each state\n",
    "dVb         = zeros(mod.I,2)          # Backward difference in each state\n",
    "c           = zeros(mod.I,2)          # Consumption in each state\n",
    "A           = sparse(zeros(2*mod.I,2*mod.I))   # Transition matrix\n",
    "r_r         = zeros(mod.Ir,1)         # Interest rate guesses\n",
    "rmin_r      = zeros(mod.Ir,1)         # Lower bound for bisection\n",
    "rmax_r      = zeros(mod.Ir,1)         # Upper bound for bisection\n",
    "g_r         = zeros(mod.I,2,mod.Ir)   # Stationary distribution\n",
    "adot        = zeros(mod.I,2,mod.Ir)   # Policy function\n",
    "V_r         = zeros(mod.I,2,mod.Ir)   # Value function\n",
    "V           = zeros(mod.I,2)          # Value function\n",
    "S           = zeros(mod.Ir,1)         # Aggregate asset supply\n",
    "r           = mod.r0                  # Current interest rate guess\n",
    "dist        = zeros(mod.maxit,1)      # Distance between value function guesses\n",
    "r_end       = 0                       # Last iteration value"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Now we get to the meat of the code. This code block runs the algorithm for solving the model and the associated equilibrium. \n",
    "\n",
    "#### Inner loop of the algorithm\n",
    "After the outer loop has updated (initalized) the interest rate guess, the inner loop proceeds as follows:\n",
    "* Construct the forward difference approximations of the derivatives of the value function\n",
    "* Use the \"upwind scheme\" to compute consumption, utility, and the savings policy rules\n",
    "* Use the savings policy rules to construct the transition matrix. This construction is carried out using sparse matrix methods to save memory (these methods are also fast). \n",
    "* Construct and solve the linear system of equations that constitute the HJB equation(s) across asset and income states.\n",
    "\n",
    "#### Outer loop of the algorithm\n",
    "After the inner loop has solved the HJB equation, the transition matrix computed on the final iteration is used to solve the KFE for the stationary distribution. The outer loop then uses this stationary distribution to compute the aggregate asset supply. We then update the interest rate via a bisection method until assets are in zero net supply. \n",
    "\n"
   ]
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   "cell_type": "code",
   "execution_count": 5,
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     "text": [
      "-----------------------\n",
      "Interest rate = 0.030000\n",
      "Value Function Converged, Iteration = 8\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.035000\n",
      "Value Function Converged, Iteration = 5\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.037500\n",
      "Value Function Converged, Iteration = 5\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.038750\n",
      "Value Function Converged, Iteration = 5\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039375\n",
      "Value Function Converged, Iteration = 5\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039688\n",
      "Value Function Converged, Iteration = 4\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039844\n",
      "Value Function Converged, Iteration = 4\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039922\n",
      "Value Function Converged, Iteration = 4\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039961\n",
      "Value Function Converged, Iteration = 4\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039980\n",
      "Value Function Converged, Iteration = 4\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039990\n",
      "Value Function Converged, Iteration = 4\n",
      "Excess Demand\n",
      "-----------------------\n",
      "Interest rate = 0.039995\n",
      "Value Function Converged, Iteration = 3\n",
      "Excess Supply\n",
      "-----------------------\n",
      "Interest rate = 0.039993\n",
      "Value Function Converged, Iteration = 3\n",
      "Equilibrium Found, Interest rate = 0.039993/n"
     ]
    }
   ],
   "source": [
    "for ir in 1:mod.Ir\n",
    "  r_r[ir]     = r\n",
    "  rmin_r[ir]  = mod.rmin\n",
    "  rmax_r[ir]  = mod.rmax\n",
    "\n",
    "  if ir>1\n",
    "    mod.v0 = V_r[:,:,ir-1]\n",
    "  end\n",
    "\n",
    "  v = mod.v0\n",
    "\n",
    "  for n = 1:mod.maxit\n",
    "    V               = v\n",
    "    # forward difference\n",
    "    dVf[1:mod.I-1,:]    = (V[2:mod.I,:]-V[1:mod.I-1,:])/mod.da\n",
    "    dVf[mod.I,:]        = utilprime(mod.z + r.*mod.amax)\n",
    "    # backward difference\n",
    "    dVb[1,:]            = utilprime(mod.z + r.*mod.amin)  #state constraint boundary condition\n",
    "    dVb[2:mod.I,:]      = (V[2:mod.I,:]-V[1:mod.I-1,:])/mod.da\n",
    "\n",
    "    #consumption and savings with forward difference\n",
    "    cf              = utilprimeinv(dVf)\n",
    "    ssf             = mod.zz + r.*mod.aa - cf\n",
    "    #consumption and savings with backward difference\n",
    "    cb              = utilprimeinv(dVb)\n",
    "    ssb             = mod.zz + r.*mod.aa - cb\n",
    "    #consumption and derivative of value function at steady state\n",
    "    c0              = mod.zz + r.*mod.aa\n",
    "    dV0             =utilprime(c0)\n",
    "\n",
    "    # dV_upwind makes a choice of forward or backward differences based on the sign of the drift\n",
    "    If              = ssf .> 0    # positive drift --> forward difference\n",
    "    Ib              = ssb .< 0    # negative drift --> backward difference\n",
    "    I0              = (1-If-Ib)   # at steady state\n",
    "    dV_Upwind       = dVf.*If + dVb.*Ib + dV0.*I0    #important to include third term\n",
    "    c               = utilprimeinv(dV_Upwind)\n",
    "    u               = util(c)\n",
    "\n",
    "    # Construct transition matrix \n",
    "    X               = - min(ssb,0)/mod.da\n",
    "    Y               = - max(ssf,0)/mod.da + min(ssb,0)/mod.da\n",
    "    Z               = max(ssf,0)/mod.da\n",
    "        \n",
    "    A1              = spdiagm(Y[:,1],0,mod.I,mod.I) + spdiagm(X[2:mod.I,1],-1,mod.I,mod.I) + spdiagm(Z[1:mod.I-1,1],1,mod.I,mod.I)\n",
    "    A2              = spdiagm(Y[:,2],0,mod.I,mod.I) + spdiagm(X[2:mod.I,2],-1,mod.I,mod.I) + spdiagm(Z[1:mod.I-1,2],1,mod.I,mod.I)\n",
    "    A               = vcat(hcat(A1, sparse(zeros(mod.I,mod.I))), hcat(sparse(zeros(mod.I,mod.I)), A2)) + mod.Aswitch\n",
    "    B               = (mod.rho + 1/mod.Delta)*speye(2*mod.I) - A\n",
    "\n",
    "    u_stacked       = vcat(u[:,1], u[:,2])\n",
    "    V_stacked       = vcat(V[:,1], V[:,2])\n",
    "\n",
    "    # Set up and solve linear system of eqns    \n",
    "    b               = u_stacked + V_stacked/mod.Delta\n",
    "    V_stacked       = B\\b             \n",
    "\n",
    "    V               = hcat(V_stacked[1:mod.I], V_stacked[mod.I+1:2*mod.I])\n",
    "\n",
    "    Vchange         = V - v\n",
    "    v               = V\n",
    "\n",
    "    dist[n] = maximum(maximum(abs(Vchange)))\n",
    "    if dist[n]<mod.crit\n",
    "      @printf(\"-----------------------\\n\")\n",
    "      @printf(\"Interest rate = %f\\n\", r)\n",
    "      @printf(\"Value Function Converged, Iteration = %i\\n\", n)\n",
    "      break\n",
    "    end\n",
    "\n",
    "  end\n",
    "\n",
    "  ## Solve the Kolmogorov Forward equations\n",
    "  g = KFE(A)\n",
    "\n",
    "  # Pack up output\n",
    "  g_r[:,:,ir] = g\n",
    "  adot[:,:,ir] = mod.zz + r.*mod.aa - c\n",
    "  V_r[:,:,ir] = V\n",
    "  S[ir] = (g[:,1]'*mod.agrid*mod.da + g[:,2]'*mod.agrid*mod.da)[1,1]\n",
    "\n",
    "  # Check for asset market clearing, update the interest rate\n",
    "  r_end = ir\n",
    "\n",
    "  if S[ir]>mod.crit_S\n",
    "    @printf(\"Excess Supply\\n\")\n",
    "    mod.rmax = r\n",
    "    r = 0.5*(r+mod.rmin)\n",
    "  elseif S[ir]<-mod.crit_S\n",
    "    @printf(\"Excess Demand\\n\")\n",
    "    mod.rmin = r\n",
    "    r = 0.5*(r+mod.rmax)\n",
    "  elseif abs(S[ir])<mod.crit_S\n",
    "    @printf(\"Equilibrium Found, Interest rate = %f/n\", r)\n",
    "    break\n",
    "  end\n",
    "\n",
    "end\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Graphs\n",
    "\n",
    "Finally, I use the Gadfly package to plot the savings policy function across the two income states, and to plot the distribution of assets across the two income states. Note that Gadfly does not seem to have an easily accessible legend function, and so let it be known that red lines are associated with the high income state, and blue lines are associated with the low income state. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "INFO: Precompiling module Gadfly...\n"
     ]
    }
   ],
   "source": [
    "using Gadfly\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Savings policy function\n",
    "\n",
    "Note that the savings policy function differs across income states. In the low income state (blue), agents are always dissaving, regardless of their current wealth/assets. Notice, too, that at the borrowing constraint, the derivative of the low income savings function is negative infinity: any additional wealth would be entirely dissaved away, taking the agent back to the borrowing constraint again. \n",
    "\n",
    "In the high income state (red), agents are saving over most of the asset state space. These agents will dissave when assets are very high. \n",
    "\n",
    "We can see that high income agents have a precautionary saving motive, and are self-insuring against the possibility of hitting the borrowing constraint if/when they enter the low income state. Low income agents always dissave in order to smooth consumption across income states. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
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   "outputs": [
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    "#### Stationary distributions\n",
    "\n",
    "In the figure below we see the stationary distribution of agents across the two income states. Note that in the low income state (blue), there is a Dirac mass point on the borrowing constraint. That is, a large proportion of the low income agents are at the borrowing constraint in the stationary distribution. \n",
    "\n",
    "We can see that the high income state distribution is shifted right relative to the low income state distribution. This reflects the fact that high income state agents are saving while in that state."
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   "source": [
    "draw(SVG(20cm, 15cm), \n",
    "plot(layer(x=mod.agrid,y=g_r[:,1,r_end],Geom.line,Theme(line_width=1.5mm)), layer(x=mod.agrid,y=g_r[:,2,r_end],Geom.line, Theme(line_width=1.5mm,default_color=colorant\"red\")),\n",
    "xintercept=[mod.amin], Geom.vline(color=colorant\"black\", size=0.5mm), yintercept=[0], Geom.hline(color=colorant\"black\", size=0.5mm),\n",
    "Guide.title(\"Stationary distributions\"),\n",
    "Guide.xlabel(\"Wealth, <i>a\"),\n",
    "Guide.ylabel(\"Density, <i>g<sub>i</sub>(a)\"),\n",
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    "minor_label_font_size=4mm)) )"
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