dolo_framework.ipynb

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     "data": {
      "text/html": [
       "<p> edited manually </p>"
      ]
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   "source": [
    "using Dolang, Dolo, PlotlyJS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "# The Dolo Modeling Framework\n",
    "\n",
    "**Spencer Lyon**\n",
    "\n",
    "JuliaCon 2017"
   ]
  },
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   "source": [
    "## Outline\n",
    "\n",
    "- Bio\n",
    "- Three packages:\n",
    "    - Dolang.jl: manipulation/compilation of symbolic mathematical expresssions\n",
    "    - Dolo.jl: rational expectations modeling\n",
    "    - Dyno.jl: prototype Dynare in Julia\n",
    "- Will talk more about tools than economics -- catch me after if you want to talk econ!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
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   "source": [
    "## Bio\n",
    "\n",
    "- 4+ years into PhD in Economics at NYU Stern\n",
    "- Live with wife and 3 kids in NYC (UWS)\n",
    "- Lead developer/core member of [QuantEcon](https://quantecon.org/) (wrote [Julia libs](https://github.com/QuantEcon?language=julia)/bulit [lecture site](https://lectures.quantecon.org/jl/index.html)\n",
    "- Varying activity in Julia community over the last 3 years\n",
    "    - Authored packages: QuantEcon, PlotlyJS, BasisMatrices, IterationManagers, ...\n",
    "    - Contributions: Plots, Blink, DSGE, Interpolations, NLsolve, atom-language-julia, ...\n",
    "    - Help Stefan and Jared Lander run the NYC Julia meetup group"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "# Dolang.jl\n",
    "\n",
    "- What is Dolang? Toolbox for working with mathematical objects in symbolic form\n",
    "- Symbolic manipulation\n",
    "    - Substitutions\n",
    "    - Shifting time periods\n",
    "    - Symbol extraction\n",
    "- Compiler\n",
    "    - Flexible: multiple methods, allocating/mutating, derivatives, ...\n",
    "    - Smart: `@generated` symbolic differentiation\n",
    "- Used as a building block for DSLs and similar libraries"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "subslide"
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   },
   "source": [
    "## Basics: Symbolic manipulation\n",
    "\n",
    "- These are primitives used to build compiler..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
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     "text": [
      "Dolang.normalize(:x) = :_x_\n",
      "Dolang.normalize($(Expr(:quote, :(x + y)))) = :(_x_ + _y_)\n",
      "Dolang.normalize($(Expr(:quote, :(x(1))))) = :_x__1_\n",
      "Dolang.normalize($(Expr(:quote, :(x(0))))) = :_x__0_\n",
      "Dolang.normalize($(Expr(:quote, :(x(-1))))) = :_x_m1_\n"
     ]
    }
   ],
   "source": [
    "using Dolang\n",
    "@show Dolang.normalize(:x)        # symbols\n",
    "@show Dolang.normalize(:(x + y))  # expressions with symbols\n",
    "@show Dolang.normalize(:(x(1)))   # notion of timing\n",
    "@show Dolang.normalize(:(x(0)))   # current period \n",
    "@show Dolang.normalize(:(x(-1))); # previous period"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
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     "slide_type": "fragment"
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    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Dolang.time_shift($(Expr(:quote, :(x + y + z))), 1) = :(x + y + z)\n",
      "Dolang.time_shift($(Expr(:quote, :(x(0) + y(1) + z))), 0) = :(x(0) + y(1) + z)\n",
      "Dolang.time_shift($(Expr(:quote, :(x(0) + y(1) + z))), 1) = :(x(1) + y(2) + z)\n",
      "Dolang.time_shift($(Expr(:quote, :(x(0) + y(1) + z))), -1) = :(x(-1) + y(0) + z)\n"
     ]
    }
   ],
   "source": [
    "# time_shift: shift all _timed variables_ by an integer\n",
    "@show Dolang.time_shift(:(x + y + z), 1)\n",
    "@show Dolang.time_shift(:(x(0) + y(1) + z), 0)\n",
    "@show Dolang.time_shift(:(x(0) + y(1) + z), 1)\n",
    "@show Dolang.time_shift(:(x(0) + y(1) + z), -1);"
   ]
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   "outputs": [
    {
     "data": {
      "text/plain": [
       ":(a + (2a) / a)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# csubs: _recursively_ substitue subexpressions\n",
    "d = Dict(:b => :(c/a), :c => :(2a))\n",
    "csubs(:(a + b), d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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    {
     "data": {
      "text/plain": [
       ":(a + (c(0) + d(1)) + (c(1) + d(2)))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# interaction with timing\n",
    "d2 = Dict(:b => :(c(0) + d(1)))\n",
    "csubs(:(a + b(0) + b(1)), d2)"
   ]
  },
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   "cell_type": "markdown",
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   "source": [
    "### Other functions:\n",
    "\n",
    "- `steady_state`: shift all timed symbols to time 0\n",
    "- `list_symbols`: list all timed and untimed symbols\n",
    "- `list_variables`: list all timed symbols\n",
    "- `list_parameters`: list all untimed symbols\n",
    "- `subs`: direct expression substitution (non-recursive)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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   "source": [
    "## Compiler\n",
    "\n",
    "- Compiles `equations`, `variables` and `parameters` into Julia functions\n",
    "- Various methods for each function\n",
    "    - Allocating\n",
    "    - Mutating\n",
    "    - Partially or fully vectorized (similar to broadcasting)\n",
    "- Produces \"smart\" `@generated` functions that compile analytical derivatives at runtime"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "### Basic Examples\n",
    "\n",
    "- Compiler implemented as methods on `FunctionFactory` type"
   ]
  },
  {
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       "FunctionFactory\n",
       "            name: myfun\n",
       "  # of equations: 2\n",
       "       # of args: 6\n",
       "     # of params: 1\n",
       "    Has targets?: true\n"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqs = [\n",
    "    :(foo(0) = log(a(0))+b(0)/x(-1)),\n",
    "    :(ping(0) = c(1)+u*d(1))\n",
    "]\n",
    "defs = Dict(:x=>:(a(0)/(1-c(1))))\n",
    "args = [(:a, -1), (:a, 0), (:b, 0), (:c, 0), (:c, 1), (:d, 1)]\n",
    "params = [:u]\n",
    "targets = [(:foo, 0), (:ping, 0)]\n",
    "\n",
    "ff = FunctionFactory(\n",
    "    eqs, args, params, targets=targets, defs=defs, funname=:myfun\n",
    ")"
   ]
  },
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   "cell_type": "code",
   "execution_count": 7,
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     "data": {
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       "5 methods for generic function <b>myfun</b>:<ul><li> myfun(::<b>Type{Dolang.Der{0}}</b>, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun<i>{D}</i>(::<b>Type{Dolang.Der{D}}</b>, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun(V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun(::<b>Type{Dolang.Der{0}}</b>, V::<b>AbstractArray</b>, p)</li> <li> myfun(V::<b>AbstractArray</b>, p)</li> </ul>"
      ],
      "text/plain": [
       "# 5 methods for generic function \"myfun\":\n",
       "myfun(::Type{Dolang.Der{0}}, V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun(::Type{Dolang.Der{D}}, V::AbstractArray{T,1} where T, p) where D in Main\n",
       "myfun(V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun(::Type{Dolang.Der{0}}, V::AbstractArray, p) in Main\n",
       "myfun(V::AbstractArray, p) in Main"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# `make_function` produces code (a Julia Expr) for \n",
    "# evaluating our function\n",
    "code = make_function(ff) \n",
    "\n",
    "# need to evaluate it to make function callable\n",
    "# could also save to a file and `include` it elsewhere\n",
    "eval(code)  \n",
    "methods(myfun)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
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       "5 methods for generic function <b>myfun!</b>:<ul><li> myfun!(::<b>Type{Dolang.Der{0}}</b>, out, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun!<i>{D}</i>(::<b>Type{Dolang.Der{D}}</b>, out, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun!(out, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun!(::<b>Type{Dolang.Der{0}}</b>, out, V::<b>AbstractArray</b>, p)</li> <li> myfun!(out, V::<b>AbstractArray</b>, p)</li> </ul>"
      ],
      "text/plain": [
       "# 5 methods for generic function \"myfun!\":\n",
       "myfun!(::Type{Dolang.Der{0}}, out, V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun!(::Type{Dolang.Der{D}}, out, V::AbstractArray{T,1} where T, p) where D in Main\n",
       "myfun!(out, V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun!(::Type{Dolang.Der{0}}, out, V::AbstractArray, p) in Main\n",
       "myfun!(out, V::AbstractArray, p) in Main"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# methods for mutating version\n",
    "methods(myfun!)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " 0.249624\n",
       " 3.55    "
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# evaluation\n",
    "V = [1.5, 2.5, 1.0, 2.0, 3.0, 1.1]\n",
    "p = [0.5]\n",
    "myfun(V, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Float64,2}:\n",
       " 0.249624  3.55\n",
       " 0.249624  3.55"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# vectorized\n",
    "myfun([V V]', p)  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "@allocated(myfun!(out, V, p)) = 0\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       " 0.249624\n",
       " 3.55    "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# mutating\n",
    "out = zeros(2)\n",
    "myfun!(out, V, p)  # eval once to jit\n",
    "@show @allocated myfun!(out, V, p)\n",
    "out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
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    {
     "data": {
      "text/plain": [
       "2×6 Array{Float64,2}:\n",
       " 0.444444  0.4  -0.666667  -0.666667  0.0  0.0\n",
       " 0.0       0.0   0.0        0.0       1.0  0.5"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Jacobian (first derivative matrix)\n",
    "myfun(Der{1}, V, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
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     "slide_type": "fragment"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.344998 seconds (142.66 k allocations: 7.587 MiB)\n",
      "  0.000012 seconds (17 allocations: 2.109 KiB)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2×36 SparseMatrixCSC{Float64,Int64} with 8 stored entries:\n",
       "  [1 ,  1]  =  -0.592593\n",
       "  [1 ,  3]  =  0.444444\n",
       "  [1 ,  4]  =  0.444444\n",
       "  [1 ,  8]  =  -0.16\n",
       "  [1 , 13]  =  0.444444\n",
       "  [1 , 16]  =  -0.666667\n",
       "  [1 , 19]  =  0.444444\n",
       "  [1 , 21]  =  -0.666667"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# second derivative.\n",
    "@time myfun(Der{2}, V, p)\n",
    "@time myfun(Der{2}, V, p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Notice time for first call is much slower\n",
    "- 2 reasons:\n",
    "    1. Actually doing symbolic differentiation upon first call\n",
    "    2. Typical jit compilation cost for first evaluation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
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     "slide_type": "fragment"
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.473450 seconds (205.77 k allocations: 10.211 MiB, 1.74% gc time)\n",
      "  0.000011 seconds (13 allocations: 3.688 KiB)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "2-element Array{Dict{NTuple{10,Int64},Float64},1}:\n",
       " Dict((2, 2, 2, 2, 2, 2, 2, 2, 2, 2)=>-38.0507,(1, 1, 1, 1, 1, 1, 1, 1, 1, 3)=>6292.89,(1, 1, 1, 1, 1, 1, 1, 1, 1, 4)=>6292.89,(1, 1, 1, 1, 1, 1, 1, 1, 1, 1)=>-41952.6,(1, 1, 1, 1, 1, 1, 1, 1, 3, 4)=>-1048.82)\n",
       " Dict{NTuple{10,Int64},Float64}()                                                                                                                                                                                "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# 10th derivative \n",
    "@time myfun(Der{10}, V, p)\n",
    "@time myfun(Der{10}, V, p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example with dispatch arg\n",
    "\n",
    "- Can optionally specify a type to drive dispatch\n",
    "- Allows you to define multiple methods for same function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
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   "outputs": [
    {
     "data": {
      "text/html": [
       "10 methods for generic function <b>myfun</b>:<ul><li> myfun(::<b>Type{Dolang.Der{0}}</b>, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun<i>{D}</i>(::<b>Type{Dolang.Der{D}}</b>, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun(V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun(::<b>Type{Dolang.Der{0}}</b>, V::<b>AbstractArray</b>, p)</li> <li> myfun(V::<b>AbstractArray</b>, p)</li> <li> myfun(::<b>Type{Dolang.Der{0}}</b>, dispatch::<b>Int64</b>, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun<i>{D}</i>(::<b>Type{Dolang.Der{D}}</b>, dispatch::<b>Int64</b>, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun(dispatch::<b>Int64</b>, V::<b>AbstractArray{T,1} where T</b>, p)</li> <li> myfun(::<b>Type{Dolang.Der{0}}</b>, dispatch::<b>Int64</b>, V::<b>AbstractArray</b>, p)</li> <li> myfun(dispatch::<b>Int64</b>, V::<b>AbstractArray</b>, p)</li> </ul>"
      ],
      "text/plain": [
       "# 10 methods for generic function \"myfun\":\n",
       "myfun(::Type{Dolang.Der{0}}, V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun(::Type{Dolang.Der{D}}, V::AbstractArray{T,1} where T, p) where D in Main\n",
       "myfun(V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun(::Type{Dolang.Der{0}}, V::AbstractArray, p) in Main\n",
       "myfun(V::AbstractArray, p) in Main\n",
       "myfun(::Type{Dolang.Der{0}}, dispatch::Int64, V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun(::Type{Dolang.Der{D}}, dispatch::Int64, V::AbstractArray{T,1} where T, p) where D in Main\n",
       "myfun(dispatch::Int64, V::AbstractArray{T,1} where T, p) in Main\n",
       "myfun(::Type{Dolang.Der{0}}, dispatch::Int64, V::AbstractArray, p) in Main\n",
       "myfun(dispatch::Int64, V::AbstractArray, p) in Main"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqs = [\n",
    "    :(sin(x(0)) + exp(2*x(1))),\n",
    "    :(y(0) / (2 * (1 - β))),\n",
    "]\n",
    "# NOTE: Arguments can be pre-normalized and contain unicode\n",
    "args = [(:x, 0), (:y, 0), (:x, 1)]\n",
    "params = [:β]\n",
    "code = make_function(FunctionFactory(Int, eqs, args, params, funname=:myfun))\n",
    "eval(code)\n",
    "\n",
    "# notice new methods that require passing an Int\n",
    "methods(myfun)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       "   4.95303\n",
       " 125.0    "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = [π, 5.0, 0.8]\n",
    "p = [0.98] \n",
    "myfun(1, V, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
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      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# actual integer doesn't matter\n",
    "myfun(42, V, p) == myfun(rand(Int), V, p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example with grouped arguments\n",
    "\n",
    "- We can also construct functions that take multiple vector arguments"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "5 methods for generic function <b>grouped_args</b>:<ul><li> grouped_args(::<b>Type{Dolang.Der{0}}</b>, a::<b>AbstractArray{T,1} where T</b>, b::<b>AbstractArray{T,1} where T</b>, p)</li> <li> grouped_args<i>{D}</i>(::<b>Type{Dolang.Der{D}}</b>, a::<b>AbstractArray{T,1} where T</b>, b::<b>AbstractArray{T,1} where T</b>, p)</li> <li> grouped_args(a::<b>AbstractArray{T,1} where T</b>, b::<b>AbstractArray{T,1} where T</b>, p)</li> <li> grouped_args(::<b>Type{Dolang.Der{0}}</b>, a::<b>AbstractArray</b>, b::<b>AbstractArray</b>, p)</li> <li> grouped_args(a::<b>AbstractArray</b>, b::<b>AbstractArray</b>, p)</li> </ul>"
      ],
      "text/plain": [
       "# 5 methods for generic function \"grouped_args\":\n",
       "grouped_args(::Type{Dolang.Der{0}}, a::AbstractArray{T,1} where T, b::AbstractArray{T,1} where T, p) in Main\n",
       "grouped_args(::Type{Dolang.Der{D}}, a::AbstractArray{T,1} where T, b::AbstractArray{T,1} where T, p) where D in Main\n",
       "grouped_args(a::AbstractArray{T,1} where T, b::AbstractArray{T,1} where T, p) in Main\n",
       "grouped_args(::Type{Dolang.Der{0}}, a::AbstractArray, b::AbstractArray, p) in Main\n",
       "grouped_args(a::AbstractArray, b::AbstractArray, p) in Main"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eqs = [\n",
    "    :(sin(x(0)) + exp(2*x(1))),\n",
    "    :(y(0) / (2 * (1 - β))),\n",
    "]\n",
    "args = Dict(\n",
    "    :a => [(:x, 0), (:y, 0)],\n",
    "    :b => [(:x, 1)]\n",
    ")\n",
    "params = [:β]\n",
    "ff2 = FunctionFactory(eqs, args, params, funname=:grouped_args)\n",
    "eval(make_function(ff2))\n",
    "\n",
    "# notice we have args [a, b, p] -- not just [V, p]\n",
    "methods(grouped_args) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{Float64,1}:\n",
       "   4.95303\n",
       " 125.0    "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = [π, 5.0]\n",
    "b = [0.8]\n",
    "p = [0.98]\n",
    "grouped_args(a, b, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2×2 Array{Float64,2}:\n",
       " 4.95303  125.0\n",
       " 4.11156  150.0"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# partially vectorized (b is repeated for each row of a)\n",
    "# similar to broadcasting\n",
    "grouped_args([a a+1]', b, p)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "  0.486266 seconds (226.63 k allocations: 12.029 MiB, 1.85% gc time)\n",
      "  0.000010 seconds (7 allocations: 400 bytes)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "([-1.0 0.0; 0.0 25.0], [9.90606; 0.0])"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# only first derivatives -- note doing differentiation at runtime still\n",
    "@time grouped_args(Der{1}, a, b, p)\n",
    "@time grouped_args(Der{1}, a, b, p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Implementation detail\n",
    "\n",
    "- How to do symbolic derivatives at runtime?\n",
    "- Where to store symbolic data so it is available when derivatives are asked for?\n",
    "- We know `FunctionFactory` has all data/knows how to do derivatives..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "- Solution: _Embed_ instance of `FunctionFactory` in body of `@generated` function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       ":(@generated function anon{D}(::Dolang.TDer{D}, V::AbstractVector, p)\n",
       "            ff = Dolang.FunctionFactory{Array{Tuple{Symbol,Int64},1},Array{Symbol,1},Dict{Symbol,Any},DataType}(Expr[:(_a_ + _b__0_)], Tuple{Symbol,Int64}[(:b, 0)], Symbol[:a], Symbol[], Dict{Symbol,Any}(), :anon, Dolang.SkipArg, Dolang.IncidenceTable(Dict(1=>Dict(:b=>Set([0]))), Dict(:b=>Set([0])), Dict(0=>Set(Symbol[:b]))))\n",
       "            Dolang.func_body(ff, Der{D})\n",
       "        end)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ff_simple = FunctionFactory([:(a + b(0))], [(:b, 0)], [:a])\n",
    "make_function(ff_simple).args[1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Two lines: (1) embed `ff` and (2) call `Dolang.func_body` with ff\n",
    "- Tips:\n",
    "    - `@generated` functions see _types_ of arguments and return `Expr`s\n",
    "    - `Dolang.func_body` returns an `Expr` for evaluating the body of the function"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Dolo.jl\n",
    "\n",
    "- What is Dolo? A library for working with rational expectations models\n",
    "- Two main components\n",
    "    1. A _language_ for describing models via their key equations\n",
    "    2. A reference implementation of various global, non-linear solution algorithms\n",
    "        - Time iteration (basic, direct, improved)\n",
    "        - Value function iteration\n",
    "        - Parameterized expectations\n",
    "        - Perfect foresight\n",
    "        - Perturbation (linear)\n",
    "- So far no focus on performance, but plan to have algorithms be more efficient than most hand coded\n",
    "    - Already doing pretty well\n",
    "    - Optimistic about Julia letting us do this\n",
    "- Leverages Dolang as a compiler"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Why (or, how is it unique)?\n",
    "\n",
    "- 100% Julia with permissive license ==> easier to use in industry/government settings\n",
    "- Models usually written in mathematical notation in YAML files (see example)\n",
    "    - Match notation used in papers/on whiteboard\n",
    "    - Separate model _definition_ from _solution_ algorithms (DRY)\n",
    "- Automate defining Julia functions for evaluating model equations/derivatives\n",
    "- Implements _global_, _non-linear_  algorithms that handle _occasionally binding constraints_ (example later)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true,
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example 1: Sudden stop model\n",
    "\n",
    "- Simple model of \"sudden stops\" in access to financial markets\n",
    "- Shows importance of modeling constraints"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "# This file adapts the model described in\n",
      "# \"From Sudden Stops to Fisherian Deflation, Quantitative Theory and Policy\"\n",
      "# by Anton Korinek and Enrique G. Mendoza\n",
      "\n",
      "name: Sudden Stop (AR1)\n",
      "model_type: dtcc\n",
      "symbols:\n",
      "    exogenous: [ly]\n",
      "    states: [l]\n",
      "    controls: [b, lam]\n",
      "    parameters: [beta, R, sigma, mu, lam_inf]\n",
      "\n",
      "\n",
      "definitions:\n",
      "    y: exp(ly)\n",
      "    c: 1 + y + l*R - b\n",
      "\n",
      "equations:\n",
      "    transition:\n",
      "        - l = b(-1)\n",
      "\n",
      "    arbitrage:\n",
      "        - lam = b/c\n",
      "        - 1 - beta*(c(1)/c)^(-sigma)*R    |  lam_inf <= lam <= inf\n",
      "\n",
      "calibration:\n",
      "    beta: 0.95\n",
      "    R: 1/beta\n",
      "    sigma: 2.0\n",
      "    mu: 0.8\n",
      "    lam_inf: -0.2\n",
      "    ly: 0.0\n",
      "    l: 0.0\n",
      "    b: 0.0\n",
      "    lam: 0.0\n",
      "    y: exp(ly)\n",
      "    c: 1.0 + y\n",
      "\n",
      "exogenous: !VAR1\n",
      "    rho: 0.01\n",
      "    Sigma: 0.065^2\n",
      "    N: 2\n",
      "\n",
      "domain:\n",
      "    l: [-1.0, 1.0]\n",
      "\n",
      "options:\n",
      "    grid: !Cartesian\n",
      "        orders: [100]\n"
     ]
    }
   ],
   "source": [
    "# Let's view the yaml file defining the model\n",
    "using Dolo, PlotlyJS\n",
    "print(readstring(joinpath(Dolo.pkg_path, \"examples\", \"models\", \"sudden_stop_ar1.yaml\")))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "     <table>\n",
       "         <td><b>Model</b></td>\n",
       "     <tr>\n",
       "        <td>name</td>\n",
       "        <td>Sudden Stop (AR1)</td>\n",
       "      </tr>\n",
       "      <tr>\n",
       "        <td>filename</td>\n",
       "        <td>sudden_stop_ar1.yaml</td>\n",
       "      </tr>\n",
       "    </table>\n",
       "<table width=\"100%\"><tr><td><b>Type</b></td><td><b>Equation</b></td></tr>\n",
       "<tr><td>transition</td><td>\\[l_{t} = b_{t-1}\\]</td></tr><tr><td>arbitrage</td><td>\\[lam_{t} = \\frac{b_{t}}{c_{t}}\\]</td></tr><tr><td></td><td>\\[1-\\beta \\left(\\frac{c_{t+1}}{c_{t}}\\right)^{-\\sigma} R\\]</td></tr><table>"
      ],
      "text/plain": []
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model\n"
     ]
    }
   ],
   "source": [
    "# Load model\n",
    "model_ss = yaml_import(joinpath(Dolo.pkg_path, \"examples\", \"models\", \"sudden_stop_ar1.yaml\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Solving the model\n",
    "\n",
    "- Two methods:\n",
    "    - Linear perturbation around \"steady state\"\n",
    "    - Time iteration"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Perturbation Results\n",
       " * Decision Rule type: Dolo.PerturbationResult\n",
       " * Blanchard-Kahn: true\n",
       "   * stable < true\n",
       "   * determined < true\n",
       "   * unique < true\n"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res_linear_ss = Dolo.perturbate(model_ss)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Results of Time Iteration Algorithm\n",
       " * Complementarities: true\n",
       " * Discretized Process type: Dolo.DiscretizedProcess{Dolo.CartesianGrid}\n",
       " * Decision Rule type: Dolo.DecisionRule{Dolo.CartesianGrid,Dolo.CartesianGrid}\n",
       " * Number of iterations: 156\n",
       " * Convergence: true\n",
       "   * |x - x'| < 1.0e-07: true\n"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res_nonlin_ss = Dolo.time_iteration(model_ss, verbose=false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Visualizing constraints"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "plot_policies (generic function with 1 method)"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function plot_policies(mod, state, variable, res_lin, res_nonlin, bounds)\n",
    "    s0, m0 = mod.calibration[:states, :exogenous]\n",
    "    linear_data = Dolo.tabulate(mod, res_lin.solution, state, bounds, s0, m0)\n",
    "    nonlin_data = Dolo.tabulate(mod, res_nonlin.dr, state, bounds, s0, m0);\n",
    "    t1 = scatter(x=linear_data[state], y=linear_data[variable], name=\"Linear policy\")\n",
    "    t2 = scatter(\n",
    "        x=nonlin_data[state], y=nonlin_data[variable], name=\"Non-linear policy\",\n",
    "        fill=\"tonextx\", fillcolor=\"RGBA(255, 0, 0, 0.2)\"\n",
    "    )\n",
    "    k_ss = mod.calibration.flat[state]\n",
    "    v_ss = mod.calibration.flat[variable]\n",
    "    layout = Layout(\n",
    "        annotations=[attr(x=k_ss, y=v_ss, text=\"Perturbation point\", ay=-50)],\n",
    "        xaxis_title=\"Capital\",\n",
    "    )\n",
    "    plot([t1, t2], layout)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
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       "               {\"xaxis\":{\"title\":\"Capital\"},\"annotations\":[{\"y\":0.0,\"ay\":-50,\"text\":\"Perturbation point\",\"x\":0.0},{\"y\":-0.2,\"ax\":20,\"ay\":40,\"text\":\"Constraint\",\"x\":-0.415}],\"margin\":{\"l\":50,\"b\":50,\"r\":50,\"t\":60},\"title\":\"Price of capital\"}, {showLink: false});\n",
       "\n",
       "    });\n",
       " </script>\n"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bounds_ss = [model_ss.grid.min[1]; model_ss.grid.max[1]];\n",
    "p_b = plot_policies(model_ss, :l, :lam, res_linear_ss, res_nonlin_ss, bounds_ss)\n",
    "restyle!(p_b, 2, fill=\"tonextx\", fillcolor=\"RGBA(255, 0, 0, .2)\")\n",
    "constraint = model_ss.calibration.flat[:lam_inf]\n",
    "relayout(\n",
    "    p_b, \n",
    "    title=\"Price of capital\", \n",
    "    annotations=[\n",
    "        p_b.plot.layout[:annotations];\n",
    "        attr(text=\"Constraint\", x=-0.415, y=constraint, ay=40, ax=20)\n",
    "    ]\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Example 2: RBC Model\n",
    "\n",
    "- Classic model of business cycles (recessions and booms) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "     <table>\n",
       "         <td><b>Model</b></td>\n",
       "     <tr>\n",
       "        <td>name</td>\n",
       "        <td>Real Business Cycle</td>\n",
       "      </tr>\n",
       "      <tr>\n",
       "        <td>filename</td>\n",
       "        <td>rbc_dtcc_iid.yaml</td>\n",
       "      </tr>\n",
       "    </table>\n",
       "<table width=\"100%\"><tr><td><b>Type</b></td><td><b>Equation</b></td></tr>\n",
       "<tr><td>expectation</td><td>\\[m_{t} = \\frac{\\beta}{\\left(c_{t+1}\\right)^{\\sigma}} 1-\\delta+rk_{t+1}\\]</td></tr><tr><td>value</td><td>\\[V_{t} = \\frac{\\left(c_{t}\\right)^{1-\\sigma}}{1-\\sigma}-\\frac{\\chi \\left(n_{t}\\right)^{1+\\eta}}{1+\\eta}+\\beta V_{t+1}\\]</td></tr><tr><td>transition</td><td>\\[z_{t} = \\rho z_{t-1}+e_{z,t}\\]</td></tr><tr><td></td><td>\\[k_{t} = 1-\\delta k_{t-1}+i_{t-1}\\]</td></tr><tr><td>direct_response</td><td>\\[n_{t} = \\left(\\frac{1-\\alpha \\text{exp}\\left(z_{t}\\right) \\left(k_{t}\\right)^{\\alpha} m_{t}}{\\chi}\\right)^{\\frac{1}{\\eta+\\alpha}}\\]</td></tr><tr><td></td><td>\\[i_{t} = \\text{exp}\\left(z_{t}\\right) \\left(k_{t}\\right)^{\\alpha} \\left(n_{t}\\right)^{1-\\alpha}-\\left(m_{t}\\right)^{\\frac{-1}{\\sigma}}\\]</td></tr><tr><td>felicity</td><td>\\[u_{t} = \\frac{\\left(c_{t}\\right)^{1-\\sigma}}{1-\\sigma}-\\frac{\\chi \\left(n_{t}\\right)^{1+\\eta}}{1+\\eta}\\]</td></tr><tr><td>arbitrage</td><td>\\[\\chi \\left(n_{t}\\right)^{\\eta} \\left(c_{t}\\right)^{\\sigma}-w_{t}\\]</td></tr><tr><td></td><td>\\[1-\\beta \\left(\\frac{c_{t}}{c_{t+1}}\\right)^{\\sigma} 1-\\delta+rk_{t+1}\\]</td></tr><table>"
      ],
      "text/plain": []
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Model\n"
     ]
    }
   ],
   "source": [
    "# import model\n",
    "model = yaml_import(joinpath(Dolo.pkg_path, \"examples\", \"models\", \"rbc_dtcc_iid.yaml\"))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Solving the RBC model\n",
    "\n",
    "- Use same two algorithms for this model as above\n",
    "- Uses the _same_ implementation of the algorithm (separate model description from solution)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Results of Time Iteration Algorithm\n",
       " * Complementarities: true\n",
       " * Discretized Process type: Dolo.DiscretizedIIDProcess\n",
       " * Decision Rule type: Dolo.DecisionRule{Dolo.EmptyGrid,Dolo.CartesianGrid}\n",
       " * Number of iterations: 134\n",
       " * Convergence: true\n",
       "   * |x - x'| < 1.0e-07: true\n"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res_linear = Dolo.perturbate(model)\n",
    "res_nonlin = time_iteration(model, verbose=false)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Visualizing nonlinearities\n",
    "\n",
    "- We will visualize the optimal choices under the linear and non-linear policy rules"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
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       "               {\"xaxis\":{\"title\":\"Capital\"},\"annotations\":[{\"y\":0.33,\"ay\":-50,\"text\":\"Perturbation point\",\"x\":9.354978290145986}],\"margin\":{\"l\":50,\"b\":50,\"r\":50,\"t\":60},\"title\":\"Labor supply choice\"}, {showLink: false});\n",
       "\n",
       "    });\n",
       " </script>\n"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# let's look at  labor supply as capital varies\n",
    "bounds = [model.grid.min[2]; model.grid.max[2]];\n",
    "p_n = plot_policies(model, :k, :n, res_linear, res_nonlin, bounds)\n",
    "relayout!(p_n, title=\"Labor supply choice\")\n",
    "p_n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
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       "               {\"xaxis\":{\"title\":\"Capital\"},\"annotations\":[{\"y\":0.23387445725364966,\"ay\":-50,\"text\":\"Perturbation point\",\"x\":9.354978290145986}],\"margin\":{\"l\":50,\"b\":50,\"r\":50,\"t\":60},\"title\":\"Investment choice\"}, {showLink: false});\n",
       "\n",
       "    });\n",
       " </script>\n"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Now look at investment -- looks fairly linear\n",
    "p_i = plot_policies(model, :k, :i, res_linear, res_nonlin, bounds)\n",
    "relayout(p_i, title=\"Investment choice\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "#### Experiment\n",
    "\n",
    "- Does the \"error\" from linear policy change as we vary how much agents dislike working\n",
    "- Will change the parameter $\\eta$ that governs \"pain\" from working\n",
    "- Shows off some flexibility of Dolo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {},
     "metadata": {},
     "output_type": "display_data"
    },
    {
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       "               {\"xaxis\":{\"title\":\"Capital\"},\"annotations\":[{\"y\":0.33,\"ay\":-50,\"text\":\"Perturbation point\",\"x\":9.354978290145986}],\"margin\":{\"l\":50,\"b\":50,\"r\":50,\"t\":60},\"title\":\"Labor supply choice (η = 2)\"}, {showLink: false});\n",
       "\n",
       "    });\n",
       " </script>\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# one line to change a paramter value and have it propogate through the model\n",
    "Dolo.set_calibration(model, :eta, 2.0)\n",
    "\n",
    "# resolve the model with new parameters\n",
    "res_linear2 = Dolo.perturbate(model)\n",
    "res_nonlin2 = time_iteration(model, verbose=false)\n",
    "\n",
    "# produce a new plot for the labor choice\n",
    "p_n2 = plot_policies(model, :k, :n, res_linear2, res_nonlin2, bounds)\n",
    "relayout!(p_n2, title=\"Labor supply choice (η = 2)\")\n",
    "\n",
    "# repeat previous plot for comparison\n",
    "display(p_n)\n",
    "display(p_n2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
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       "               {\"xaxis\":{\"title\":\"Capital\"},\"annotations\":[{\"y\":0.23387445725364966,\"ay\":-50,\"text\":\"Perturbation point\",\"x\":9.354978290145986}],\"margin\":{\"l\":50,\"b\":50,\"r\":50,\"t\":60},\"title\":\"Investment choice (η = 2)\"}, {showLink: false});\n",
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   ],
   "source": [
    "# make new plot for investment\n",
    "# (intuition -- when we work less, capital is less valuable, so investment lowers...\n",
    "#  raising eta made things more non-linear, so linear model does worse)\n",
    "p_i2 = plot_policies(model, :k, :i, res_linear2, res_nonlin2, bounds)\n",
    "relayout!(p_i2, title=\"Investment choice (η = 2)\")\n",
    "display(p_i)\n",
    "display(p_i2)"
   ]
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  {
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   "metadata": {
    "slideshow": {
     "slide_type": "slide"
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   },
   "source": [
    "### Dyno.jl\n",
    "\n",
    "- Prototype of Julia version of Dynare\n",
    "- Leverages Dolang as compiler"
   ]
  },
  {
   "cell_type": "raw",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Thanks\n",
    "\n",
    "- Thanks to Pablo Winant -- started dolo.py for his thesis and continued at IMF and Bank of England\n",
    "- Dolo.jl: Started at Bank of England with Pablo, Anastasia Zhutova, James Graham, and me\n",
    "- Dolang.jl: Written by Pablo and me\n",
    "- Financial support from QuantEcon\n",
    "- Thanks JuliaCon committee for accepting the talk"
   ]
  },
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   "outputs": [],
   "source": []
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