notebooks/tut4_solow.ipynb

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   "source": [
    "# <span style=\"color:#2c061f\"> Macro 318: Lecture 13 - 16 / Tutorial 4 </span>  \n",
    "\n",
    "<br>\n",
    "\n",
    "## <span style=\"color:#374045\"> Solow-Swan model </span>\n",
    "\n",
    "\n",
    "#### <span style=\"color:#374045\"> Lecturer: </span> <span style=\"color:#d89216\"> <br> Dawie van Lill (dvanlill@sun.ac.za) </span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These are basic notes for the Solow model in Julia. In this notebook we will also be going over the basics of difference equations and how they relate the Solow model. \n",
    "\n",
    "You will notice that the Solow model is different from our models that we previously covered in that it contains a time component. This is why we need to consider **dynamics**. Evolution of variables over time. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Below are the packages that we will be using for this tutorial. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import Pkg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
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     "text": [
      "\u001b[32m\u001b[1m    Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n",
      "\u001b[32m\u001b[1m   Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Project.toml`\n",
      "\u001b[32m\u001b[1m  No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n"
     ]
    }
   ],
   "source": [
    "Pkg.add(\"LaTeXStrings\")\n",
    "Pkg.add(\"Plots\")\n",
    "Pkg.add(\"Symbolics\")\n",
    "Pkg.add(\"Zygote\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LaTeXStrings\n",
    "using Plots\n",
    "using Symbolics\n",
    "using Zygote"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Learning outcomes"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Table of contents\n",
    "\n",
    "1. Consumers\n",
    "2. Firms\n",
    "3. Competitive equilibrium\n",
    "4. Steady state\n",
    "5. Shocks\n",
    "6. Difference equations (optional)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Consumers "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In class we discussed the Solow model in detail. We started with some of the following assumption, \n",
    "\n",
    "$$\n",
    "N_{t+1} = (1 + n) N_{t}\n",
    "$$\n",
    "\n",
    "This means that the population grows at a constant rate $n$. \n",
    "\n",
    "**Notation**: The notation has changed a bit here. We are now working with time subscripts. This is more traditional in the literature and also will help with our explanations down the line. \n",
    "\n",
    "An important consideration for this model is the behaviour of consumers with respect to saving and investment. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1.1 Saving and investment"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consumption and savings decisions are not microfounded in the Solow model. Decisions are made in a stylised way. \n",
    "\n",
    "The savings rate $s \\in (0, 1)$ is **exogenous**!\n",
    "\n",
    "Saving is a portion of income $\\rightarrow S_t = s Y_{t}$\n",
    "\n",
    "Agents consume the rest of their income $ \\rightarrow C_{t} = (1 - s)Y_{t}$\n",
    "\n",
    "Saving is a key component since it will finance **investment activities** in the model.\n",
    "\n",
    "Investment equals saving in this economy $\\rightarrow I_t = sY_t = S_t$\n",
    "\n",
    "In a closed economy with no government $\\rightarrow C_t = Y_t - I_t$\n",
    "\n",
    "Next we look at the **firm side** of the problem. We start with the production function. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Firms"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the most important considerations in the Solow model is the production function. The production function is given by \n",
    "\n",
    "$$\n",
    "Y_{t}\\left(K_{t}, N_{t}\\right) = z \\cdot F\\left(K_{t}, N_{t}\\right)\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.1. Cobb-Douglas production function "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this example we can assume a Cobb-Douglas production function in discrete time $(t)$ with Hicks-neutral techonology $(z)$, and with constant returns to scale where $\\alpha \\in (0, 1)$ is the output elasticity of capital.\n",
    "\n",
    "$$\n",
    "\\begin{equation*}\n",
    "    Y_{t}\\left(K_{t}, N_{t}\\right) = z \\cdot F\\left(K_{t}, N_{t}\\right) =  z \\cdot \\left(K_{t} ^{\\alpha} N_{t}^{1-\\alpha}\\right)\n",
    "\\end{equation*}\n",
    "$$\n",
    "\n",
    "From our notes we have that there are decreasing and positive returns to scale in each factor individually. We can observe this from the partial derivatives. The first and second derivatives with respect to capital and labor are:\n",
    "$$\n",
    "\\begin{align*}\n",
    "    \\frac{\\partial   Y_{t}}{\\partial K}   = \\alpha \\cdot z\\left(\\frac{N_{t}}{K_{t}}\\right)^{1-\\alpha} \n",
    "  = \\alpha \\cdot \\frac{Y_{t}}{K_{t}} > 0 \\; \\text{and} \\;\n",
    "    \\frac{\\partial^2 Y_{t}}{\\partial K^2} = -\\alpha (1-\\alpha) \\cdot z\\left(\\frac{N_{t}}{K_{t}}\\right)^{1-\\alpha} < 0 \\\\\n",
    "    \\frac{\\partial   Y_{t}}{\\partial N}   = (1-\\alpha) \\cdot z\\left(\\frac{K_{t}}{N_{t}}\\right)^{\\alpha} \n",
    "  = (1-\\alpha) \\cdot \\frac{Y_{t}}{N_{t}} > 0 \\; \\text{and} \\;\n",
    "    \\frac{\\partial^2 Y_{t}}{\\partial N^2} = -\\alpha (1-\\alpha) \\cdot z\\left(\\frac{K_{t}}{N_{t}}\\right)^{1-\\alpha} < 0\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We know from our experience with Julia so far that we can calculate these partial derivatives using the `Symbolics` package. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "@variables K N z α\n",
    "\n",
    "Y = z * (K ^ α) * (N ^ (1 - α)) \n",
    "\n",
    "∂Y_∂K = Differential(K)\n",
    "∂Y_∂N = Differential(N)\n",
    "∂2Y_∂K2 = Differential(K) ∘ Differential(K)\n",
    "∂2Y_∂N2 = Differential(N) ∘ Differential(N);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "N^{1 - \\alpha} K^{-1 + \\alpha} z \\alpha\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "z*α*(K^(α - 1))*(N^(1 - α))"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_derivatives(∂Y_∂K(Y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "N^{ - \\alpha} K^{\\alpha} z \\left( 1 - \\alpha \\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "z*(K^α)*(N^(-α))*(1 - α)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_derivatives(∂Y_∂N(Y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       "N^{1 - \\alpha} K^{-2 + \\alpha} z \\alpha \\left( -1 + \\alpha \\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "z*α*(K^(α - 2))*(N^(1 - α))*(α - 1)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_derivatives(∂2Y_∂K2(Y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation}\n",
       " - N^{-1 - \\alpha} K^{\\alpha} z \\alpha \\left( 1 - \\alpha \\right)\n",
       "\\end{equation}\n"
      ],
      "text/plain": [
       "-z*α*(K^α)*(N^(-1 - α))*(1 - α)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand_derivatives(∂2Y_∂N2(Y))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.2 Production per capita"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we write our production function in its intensive form. This means that we write the production function in per capita terms, \n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "    Y_{t} &= z \\cdot \\left(K_{t}^{\\alpha}N_{t}^{1-\\alpha} \\right) \\\\\n",
    "    y_{t} &= \\frac{Y_{t}}{N_{t}} = z \\cdot \\left[ \\left(\\frac{K_{t}}{N_{t}} \\right)^{\\alpha} \\left(\\frac{N_{t}}{N_{t}} \\right)^{1-\\alpha} \\right] \\\\\n",
    "    y_{t} &= z \\cdot f\\left(k_{t}\\right) = z \\cdot k_{t}^{\\alpha}\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next we are going to plot the intensive form of the production function, as in the textbook."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 2.3 Production function plot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by plotting the two dimensional version of the production function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "K = 100\n",
    "α = 0.5\n",
    "k = 0:0.5:K\n",
    "z = 1\n",
    "\n",
    "function production_per_capita(k, z)\n",
    "    return y = z .* k .^ α\n",
    "end\n",
    "\n",
    "plot(production_per_capita(k, z), label = \"z = 1\", legend = :topleft, lw = 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we will generate the same plot, but this time we will include a tangent line that provides the value for the marginal product of capital at a point. This is similar to Figure 7.12 in Chapter 7 of the textbook. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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   "source": [
    "g(k; z = 1) = z .* k .^ α\n",
    "\n",
    "a = round(g'(20), digits = 2)\n",
    "b = round(g'(65), digits = 2)\n",
    "\n",
    "gradient_line(f, x_0) = (x -> f(x_0) + f'(x_0) * (x - x_0))\n",
    "\n",
    "plot(g, 0:1:100, legend = :bottomright, label = L\"y = zf(k)\", xlabel = L\"$k = $capital per worker\", ylabel = \"\\$y = \\$ output per worker\", lw = 2.5)\n",
    "plot!(gradient_line(g, 20), 5:0.1:40, label = L\"MP_K = %$a\", lw = 3, alpha = 0.6, color = :black)\n",
    "scatter!([20], [g(20)], label = \"\", markersize = 6, color = :black)\n",
    "\n",
    "plot!(gradient_line(g, 65), 45:0.1:85, label = L\"MP_K = %$b\", lw = 3, alpha = 0.6, color = :red)\n",
    "scatter!([65], [g(65)], label = \"\", markersize = 6, color = :red)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From above one can observe that the marginal product of capital (the slope of the graph) is decreasing with an increase in capital per worker. This illustrates the concavity of the function with respect to capital. \n",
    "\n",
    "### 2.4 Changes in capital over time\n",
    "\n",
    "\n",
    "We make another assumption regarding capital. It wears out over time. This is referred to as depreciation $d$. Capital stock then changes over time according to the following equation, \n",
    "\n",
    "$$\n",
    "K_{t+1} = (1 - d)K_t + I_{t}\n",
    "$$\n",
    "\n",
    "Now that we have the behaviour of firms and consumers we can put everything together to retrieve the competitive equilibrium result. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Competitive equilibrium"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the equilibrium conditions from the consumer side of the economy is that, \n",
    "\n",
    "$$\n",
    "Y_t = C_t + I_t\n",
    "$$\n",
    "\n",
    "This means that current output is equal to aggregate consumption plus aggregate investment. From our equation for the change in capital over time we can substitute the investment component as follows, \n",
    "\n",
    "$$\n",
    "K_{t+1} = (1 - d)K_{t} + Y_t - C_t\n",
    "$$\n",
    "\n",
    "We can then rearrange this equation to write in terms of $Y_t$, \n",
    "\n",
    "$$\n",
    "Y_{t} = C_{t} + K_{t+1} - (1 - d)K_{t}\n",
    "$$\n",
    "\n",
    "Next, we use $C_{t} = (1-s)Y_{t}$ from the consumer's saving and investment decisions, \n",
    "\n",
    "$$\n",
    "Y_{t} = (1-s)Y_{t} + K_{t+1} - (1 - d)K_{t}\n",
    "$$\n",
    "\n",
    "Finally, we rearrange and simplify, \n",
    "\n",
    "$$ \n",
    "\\begin{align*}\n",
    "Y_{t} - (1 - s)Y_{t} &= K_{t+1} - (1 - d)K_{t} \\\\\n",
    "Y_{t}(1-1+s) & = K_{t+1} - (1 - d)K_{t} \\\\\n",
    "K_{t+1} &=  sY_{t} + (1 - d)K_{t} \n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "This means that capital stock in the future period is the quantity of aggregate savings in current period plus the capital stock left over from the current period that has not depreciated. \n",
    "\n",
    "We can now substitute $Y$ using our definition from of the production function, \n",
    "\n",
    "$$\n",
    "K_{t+1} =  sz \\cdot F\\left(K_{t}, N_{t}\\right)  + (1 - d)K_{t} \n",
    "$$\n",
    "\n",
    "We would like to express this equation of motion in per capita terms. We can do this by dividing through by the number of workers, \n",
    "\n",
    "$$\n",
    "\\frac{K_{t+1}}{N_t} =  sz \\cdot \\frac{F\\left(K_{t}, N_{t}\\right)}{N_t}  + (1 - d)\\frac{K_{t}}{N_t} \n",
    "$$\n",
    "\n",
    "Next, we multiply the left hand side by $1 = \\frac{N_{t+1}}{N_{t+1}}$, which then gives us, \n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "\\frac{K_{t+1}}{N_t}\\frac{N_{t+1}}{N_{t+1}} & =  sz \\cdot \\frac{F\\left(K_{t}, N_{t}\\right)}{N_t}  + (1 - d)\\frac{K_{t}}{N_t} \\\\\n",
    "\\frac{K_{t+1}}{N_{t+1}}\\frac{N_{t+1}}{N_{t}} & =  sz \\cdot F\\left(\\frac{K_{t}}{N_t}, \\frac{N_{t}}{N_t}\\right)  + (1 - d)\\frac{K_{t}}{N_t}  \\\\\n",
    "k_{t+1}(1 + n) & =  sz \\cdot f\\left(k_t, 1\\right)  + (1 - d)k_t \\\\ \n",
    "k_{t+1} & =  \\frac{sz \\cdot f\\left(k_t\\right)}{1+n}  + \\frac{(1 - d)k_t}{1+n}\n",
    "\\end{align*}\n",
    "$$\n",
    "\n",
    "The key equation that we want to use in this tutorial is then, \n",
    "\n",
    "$$\n",
    "k_{t+1} =  \\frac{sz \\cdot f\\left(k_t\\right)}{1+n}  + \\frac{(1 - d)k_t}{1+n}\n",
    "$$\n",
    "\n",
    "This equation summarises what we need to know about competitive equilibrium in the Solow growth model. \n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 4. Steady state"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In order to find the steady state of the Solow model, we can utilise the equation of motion for capital. The steady state in this model is going to be the point where there is no change in capital over time. In other words, it is the value that maintains a constant capital stock per capita over time. We will first determine this steady state algebraically, but then in our section on **difference equations** we will come back to a more accurate depiction of the notion of steady state. \n",
    "\n",
    "We are going to replicate Figure 7.13 from the textbook and indicate the steady state point on the graph. We are looking for the point where $k_{t} = k_{t+1} = k^{*}$. The value of $k^{*}$ is known as the steady state value of capital. This value will occur where the transition equation for capital intersects with the $45^{\\degree}$ line (since this line reflects all the points where $k_{t+1} = k_{t}$).\n",
    "\n",
    "We would like to know what the steady state value is going to be, so we first calculate it algebraically. Solving for $k^{*}$ from the equilibrium condition entails the following, \n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "k^{*} &=  \\frac{sz \\cdot f\\left(k^{*}\\right)}{1+n}  + \\frac{(1 - d)k^{*}}{1+n} \\\\\n",
    "k^{*} &=  \\frac{sz \\cdot (k^{*})^{\\alpha}}{1+n}  + \\frac{(1 - d)k^{*}}{1+n} \\\\\n",
    "(1 + n)k^{*} &=  sz \\cdot (k^{*})^{\\alpha}  + (1 - d)k^{*} \\\\\n",
    "(d + n)k^{*} &= sz \\cdot (k^{*})^{\\alpha}         \\\\\n",
    " {k^*}^{1-\\alpha} &= z \\cdot \\frac{s}{d + n} \\\\\n",
    "  k^* &= \\left[ z \\cdot \\frac{s}{d + n} \\right]^{\\frac{1}{1-\\alpha}}\n",
    "\\end{align*}\n",
    "$$\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
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       "  404.712,1382.82 407.233,1352.75 409.755,1342.47 412.277,1334.73 414.799,1328.25 417.32,1322.54 419.842,1317.37 422.364,1312.6 424.886,1308.15 427.407,1303.94 \n",
       "  429.929,1299.95 432.451,1296.13 434.973,1292.46 437.494,1288.92 440.016,1285.5 442.538,1282.18 445.06,1278.96 447.581,1275.81 450.103,1272.74 452.625,1269.74 \n",
       "  455.147,1266.81 457.668,1263.93 460.19,1261.1 462.712,1258.32 465.234,1255.59 467.755,1252.9 470.277,1250.26 472.799,1247.65 475.32,1245.08 477.842,1242.54 \n",
       "  480.364,1240.03 482.886,1237.55 485.407,1235.11 487.929,1232.69 490.451,1230.29 492.973,1227.92 495.494,1225.57 498.016,1223.25 500.538,1220.95 503.06,1218.67 \n",
       "  505.581,1216.41 508.103,1214.16 510.625,1211.94 513.147,1209.73 515.668,1207.54 518.19,1205.37 520.712,1203.21 523.234,1201.07 525.755,1198.94 528.277,1196.83 \n",
       "  530.799,1194.73 533.32,1192.64 535.842,1190.57 538.364,1188.51 540.886,1186.46 543.407,1184.43 545.929,1182.4 548.451,1180.39 550.973,1178.38 553.494,1176.39 \n",
       "  556.016,1174.41 558.538,1172.44 561.06,1170.48 563.581,1168.52 566.103,1166.58 568.625,1164.64 571.147,1162.72 573.668,1160.8 576.19,1158.89 578.712,1156.99 \n",
       "  581.234,1155.1 583.755,1153.21 586.277,1151.33 588.799,1149.46 591.321,1147.6 593.842,1145.75 596.364,1143.9 598.886,1142.05 601.407,1140.22 603.929,1138.39 \n",
       "  606.451,1136.57 608.973,1134.75 611.494,1132.94 614.016,1131.14 616.538,1129.34 619.06,1127.55 621.581,1125.76 624.103,1123.98 626.625,1122.21 629.147,1120.44 \n",
       "  631.668,1118.67 634.19,1116.91 636.712,1115.16 639.234,1113.41 641.755,1111.67 644.277,1109.93 646.799,1108.19 649.321,1106.46 651.842,1104.74 654.364,1103.02 \n",
       "  656.886,1101.3 659.407,1099.59 661.929,1097.88 664.451,1096.18 666.973,1094.48 669.494,1092.79 672.016,1091.1 674.538,1089.41 677.06,1087.73 679.581,1086.05 \n",
       "  682.103,1084.37 684.625,1082.7 687.147,1081.03 689.668,1079.37 692.19,1077.71 694.712,1076.05 697.234,1074.4 699.755,1072.75 702.277,1071.1 704.799,1069.46 \n",
       "  707.321,1067.82 709.842,1066.19 712.364,1064.55 714.886,1062.92 717.408,1061.3 719.929,1059.67 722.451,1058.05 724.973,1056.44 727.494,1054.82 730.016,1053.21 \n",
       "  732.538,1051.6 735.06,1050 737.581,1048.39 740.103,1046.79 742.625,1045.2 745.147,1043.6 747.668,1042.01 750.19,1040.42 752.712,1038.84 755.234,1037.25 \n",
       "  757.755,1035.67 760.277,1034.09 762.799,1032.52 765.321,1030.94 767.842,1029.37 770.364,1027.8 772.886,1026.24 775.408,1024.67 777.929,1023.11 780.451,1021.55 \n",
       "  782.973,1020 785.494,1018.44 788.016,1016.89 790.538,1015.34 793.06,1013.8 795.581,1012.25 798.103,1010.71 800.625,1009.17 803.147,1007.63 805.668,1006.09 \n",
       "  808.19,1004.56 810.712,1003.02 813.234,1001.49 815.755,999.965 818.277,998.439 820.799,996.914 823.321,995.392 825.842,993.872 828.364,992.353 830.886,990.837 \n",
       "  833.408,989.322 835.929,987.809 838.451,986.298 840.973,984.789 843.495,983.282 846.016,981.777 848.538,980.273 851.06,978.771 853.581,977.271 856.103,975.773 \n",
       "  858.625,974.276 861.147,972.782 863.668,971.289 866.19,969.797 868.712,968.307 871.234,966.819 873.755,965.333 876.277,963.848 878.799,962.365 881.321,960.883 \n",
       "  883.842,959.403 886.364,957.925 888.886,956.448 891.408,954.973 893.929,953.499 896.451,952.027 898.973,950.556 901.495,949.087 904.016,947.619 906.538,946.153 \n",
       "  909.06,944.688 911.581,943.224 914.103,941.762 916.625,940.302 919.147,938.843 921.668,937.385 924.19,935.929 926.712,934.474 929.234,933.021 931.755,931.568 \n",
       "  934.277,930.118 936.799,928.668 939.321,927.22 941.842,925.773 944.364,924.328 946.886,922.884 949.408,921.441 951.929,919.999 954.451,918.559 956.973,917.12 \n",
       "  959.495,915.682 962.016,914.246 964.538,912.81 967.06,911.376 969.582,909.943 972.103,908.512 974.625,907.081 977.147,905.652 979.668,904.224 982.19,902.797 \n",
       "  984.712,901.372 987.234,899.947 989.755,898.524 992.277,897.101 994.799,895.68 997.321,894.26 999.842,892.841 1002.36,891.424 1004.89,890.007 1007.41,888.592 \n",
       "  1009.93,887.177 1012.45,885.764 1014.97,884.352 1017.49,882.94 1020.02,881.53 1022.54,880.121 1025.06,878.713 1027.58,877.306 1030.1,875.9 1032.63,874.495 \n",
       "  1035.15,873.091 1037.67,871.688 1040.19,870.287 1042.71,868.886 1045.23,867.486 1047.76,866.087 1050.28,864.689 1052.8,863.292 1055.32,861.896 1057.84,860.501 \n",
       "  1060.36,859.107 1062.89,857.714 1065.41,856.322 1067.93,854.931 1070.45,853.541 1072.97,852.151 1075.49,850.763 1078.02,849.376 1080.54,847.989 1083.06,846.604 \n",
       "  1085.58,845.219 1088.1,843.835 1090.63,842.452 1093.15,841.07 1095.67,839.689 1098.19,838.309 1100.71,836.929 1103.23,835.551 1105.76,834.173 1108.28,832.796 \n",
       "  1110.8,831.42 1113.32,830.045 1115.84,828.671 1118.36,827.297 1120.89,825.925 1123.41,824.553 1125.93,823.182 1128.45,821.812 1130.97,820.443 1133.49,819.074 \n",
       "  1136.02,817.707 1138.54,816.34 1141.06,814.974 1143.58,813.608 1146.1,812.244 1148.63,810.88 1151.15,809.517 1153.67,808.155 1156.19,806.794 1158.71,805.433 \n",
       "  1161.23,804.073 1163.76,802.714 1166.28,801.356 1168.8,799.998 1171.32,798.641 1173.84,797.285 1176.36,795.93 1178.89,794.575 1181.41,793.221 1183.93,791.868 \n",
       "  1186.45,790.516 1188.97,789.164 1191.49,787.813 1194.02,786.463 1196.54,785.113 1199.06,783.764 1201.58,782.416 1204.1,781.068 1206.63,779.722 1209.15,778.376 \n",
       "  1211.67,777.03 1214.19,775.686 1216.71,774.341 1219.23,772.998 1221.76,771.655 1224.28,770.313 1226.8,768.972 1229.32,767.631 1231.84,766.291 1234.36,764.952 \n",
       "  1236.89,763.613 1239.41,762.275 1241.93,760.938 1244.45,759.601 1246.97,758.265 1249.49,756.93 1252.02,755.595 1254.54,754.261 1257.06,752.927 1259.58,751.594 \n",
       "  1262.1,750.262 1264.63,748.93 1267.15,747.599 1269.67,746.269 1272.19,744.939 1274.71,743.609 1277.23,742.281 1279.76,740.953 1282.28,739.625 1284.8,738.299 \n",
       "  1287.32,736.972 1289.84,735.647 1292.36,734.322 1294.89,732.997 1297.41,731.673 1299.93,730.35 1302.45,729.027 1304.97,727.705 1307.49,726.384 1310.02,725.063 \n",
       "  1312.54,723.742 1315.06,722.422 1317.58,721.103 1320.1,719.784 1322.63,718.466 1325.15,717.149 1327.67,715.832 1330.19,714.515 1332.71,713.199 1335.23,711.884 \n",
       "  1337.76,710.569 1340.28,709.255 1342.8,707.941 1345.32,706.628 1347.84,705.315 1350.36,704.003 1352.89,702.691 1355.41,701.38 1357.93,700.069 1360.45,698.759 \n",
       "  1362.97,697.45 1365.49,696.141 1368.02,694.832 1370.54,693.524 1373.06,692.217 1375.58,690.91 1378.1,689.604 1380.63,688.298 1383.15,686.992 1385.67,685.687 \n",
       "  1388.19,684.383 1390.71,683.079 1393.23,681.776 1395.76,680.473 1398.28,679.17 1400.8,677.868 1403.32,676.567 1405.84,675.266 1408.36,673.965 1410.89,672.665 \n",
       "  1413.41,671.366 1415.93,670.067 1418.45,668.768 1420.97,667.47 1423.49,666.173 1426.02,664.876 1428.54,663.579 1431.06,662.283 1433.58,660.987 1436.1,659.692 \n",
       "  1438.63,658.397 1441.15,657.102 1443.67,655.809 1446.19,654.515 1448.71,653.222 1451.23,651.93 1453.76,650.638 1456.28,649.346 1458.8,648.055 1461.32,646.764 \n",
       "  1463.84,645.474 1466.36,644.184 1468.89,642.895 1471.41,641.606 1473.93,640.317 1476.45,639.029 1478.97,637.741 1481.49,636.454 1484.02,635.167 1486.54,633.881 \n",
       "  1489.06,632.595 1491.58,631.31 1494.1,630.025 1496.63,628.74 1499.15,627.456 1501.67,626.172 1504.19,624.889 1506.71,623.606 1509.23,622.323 1511.76,621.041 \n",
       "  1514.28,619.759 1516.8,618.478 1519.32,617.197 1521.84,615.917 1524.36,614.636 1526.89,613.357 1529.41,612.078 1531.93,610.799 1534.45,609.52 1536.97,608.242 \n",
       "  1539.49,606.964 1542.02,605.687 1544.54,604.41 1547.06,603.134 1549.58,601.858 1552.1,600.582 1554.63,599.307 1557.15,598.032 1559.67,596.757 1562.19,595.483 \n",
       "  1564.71,594.209 1567.23,592.936 1569.76,591.663 1572.28,590.39 1574.8,589.118 1577.32,587.846 1579.84,586.575 1582.36,585.304 1584.89,584.033 1587.41,582.763 \n",
       "  1589.93,581.493 1592.45,580.223 1594.97,578.954 1597.49,577.685 1600.02,576.416 1602.54,575.148 1605.06,573.88 1607.58,572.613 1610.1,571.346 1612.63,570.079 \n",
       "  1615.15,568.813 1617.67,567.547 1620.19,566.281 1622.71,565.016 1625.23,563.751 1627.76,562.487 1630.28,561.222 1632.8,559.959 1635.32,558.695 1637.84,557.432 \n",
       "  1640.36,556.169 1642.89,554.907 1645.41,553.644 1647.93,552.383 1650.45,551.121 1652.97,549.86 1655.49,548.599 1658.02,547.339 1660.54,546.079 1663.06,544.819 \n",
       "  1665.58,543.56 1668.1,542.301 1670.63,541.042 1673.15,539.784 1675.67,538.526 1678.19,537.268 1680.71,536.01 1683.23,534.753 1685.76,533.497 1688.28,532.24 \n",
       "  1690.8,530.984 1693.32,529.728 1695.84,528.473 1698.36,527.218 1700.89,525.963 1703.41,524.709 1705.93,523.454 1708.45,522.201 1710.97,520.947 1713.49,519.694 \n",
       "  1716.02,518.441 1718.54,517.188 1721.06,515.936 1723.58,514.684 1726.1,513.433 1728.63,512.181 1731.15,510.93 1733.67,509.68 1736.19,508.429 1738.71,507.179 \n",
       "  1741.23,505.929 1743.76,504.68 1746.28,503.431 1748.8,502.182 1751.32,500.933 1753.84,499.685 1756.36,498.437 1758.89,497.189 1761.41,495.942 1763.93,494.695 \n",
       "  1766.45,493.448 1768.97,492.202 1771.49,490.956 1774.02,489.71 1776.54,488.464 1779.06,487.219 1781.58,485.974 1784.1,484.729 1786.63,483.485 1789.15,482.241 \n",
       "  1791.67,480.997 1794.19,479.753 1796.71,478.51 1799.23,477.267 1801.76,476.024 1804.28,474.782 1806.8,473.54 1809.32,472.298 1811.84,471.056 1814.36,469.815 \n",
       "  1816.89,468.574 1819.41,467.333 1821.93,466.093 1824.45,464.853 1826.97,463.613 1829.49,462.373 1832.02,461.134 1834.54,459.895 1837.06,458.656 1839.58,457.418 \n",
       "  1842.1,456.179 1844.63,454.941 1847.15,453.704 1849.67,452.466 1852.19,451.229 1854.71,449.992 1857.23,448.756 1859.76,447.519 1862.28,446.283 1864.8,445.047 \n",
       "  1867.32,443.812 1869.84,442.577 1872.36,441.342 1874.89,440.107 1877.41,438.872 1879.93,437.638 1882.45,436.404 1884.97,435.17 1887.49,433.937 1890.02,432.704 \n",
       "  1892.54,431.471 1895.06,430.238 1897.58,429.006 1900.1,427.774 1902.63,426.542 1905.15,425.31 1907.67,424.079 1910.19,422.847 1912.71,421.617 1915.23,420.386 \n",
       "  1917.76,419.156 1920.28,417.925 1922.8,416.696 1925.32,415.466 1927.84,414.237 1930.36,413.007 1932.89,411.778 1935.41,410.55 1937.93,409.321 1940.45,408.093 \n",
       "  1942.97,406.865 1945.49,405.638 1948.02,404.41 1950.54,403.183 1953.06,401.956 1955.58,400.729 1958.1,399.503 1960.63,398.277 1963.15,397.051 1965.67,395.825 \n",
       "  1968.19,394.6 1970.71,393.374 1973.23,392.149 1975.76,390.924 1978.28,389.7 1980.8,388.476 1983.32,387.251 1985.84,386.028 1988.36,384.804 1990.89,383.581 \n",
       "  1993.41,382.357 1995.93,381.135 1998.45,379.912 2000.97,378.689 2003.49,377.467 2006.02,376.245 2008.54,375.023 2011.06,373.802 2013.58,372.581 2016.1,371.359 \n",
       "  2018.63,370.139 2021.15,368.918 2023.67,367.698 2026.19,366.477 2028.71,365.257 2031.23,364.038 2033.76,362.818 2036.28,361.599 2038.8,360.38 2041.32,359.161 \n",
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     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# parameter values (selected arbitrarily, can you think of values that make sense economically?)\n",
    "s = 0.4             # savings rate at 90%! This seems quite high.\n",
    "z = 1               # total factor productivity\n",
    "d = 0.15             # depreciation rate\n",
    "n = 0.5            # population growth rate   \n",
    "α = 0.4             # output elasticity of capital\n",
    "\n",
    "# calculate k_star from the formula above\n",
    "k_star = round((z .* s ./ (d .+ n)) .^ (1 / (1 - α)), digits = 2)\n",
    "\n",
    "# equation of motion for capital\n",
    "h(k; s = s, z = z, d = d, n = n) = (s .* z .* g(k)) ./ (1 .+ n) .+ ((1 .- d) .* k) ./ (1 .+ n)  \n",
    "\n",
    "# code for plotting the function -- try to read one line at a time. \n",
    "capital_plot = plot(h, 0:0.001:0.75, legend = :topleft, label = L\"k_{t+1} = \\frac{sz \\cdot f\\left(k_t\\right)}{1+n}  + \\frac{(1 - d)k_t}{1+n}\", xlabel = L\"k_t\", ylabel = L\"k_{t+1}\", lw = 2.5, color = :blue, xticks = ([k_star], [L\"k^{*} = %$k_star\"]), y_ticks = ([k_star], [L\"k^{*} = %$k_star\"]))\n",
    "plot!(capital_plot, x -> x, lw = 2, color =:black, lab = L\"k_{t+1} = k_{t}\") # 45 degree line plot added\n",
    "scatter!([k_star], [k_star], label = \"\", markersize = 6, color = :black) # point of intersection"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see from the graph above that the two lines intersect at the point where $k^{*} = 0.45$. The value will change if we select a different parameterisation of the model. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can move on to plotting Figure 7.14 from the textbook. From this graph we can easily determine where the steady state value is. This graph basically has two components. First, there is an investment curve $szf(k^{*})$. Second, we have a curve that captures the reasons why capital might decrease. There are two reasons in this model why capital might decrease, depreciation and dilution. Depreciation accurs via the depreciation rate and dilution through the population growth rate. Capital is diluted among the population if the population grows faster than the capital growth rate. In the literature, the line $(n + d)k^{*}$ is known as the break-even line. It shows the level of investment needed to compensate for depreciation and population growth. \n",
    "\n",
    "From our steady state equations above we had that, \n",
    "\n",
    "$$\n",
    "(d + n)k^{*} = sz \\cdot f(k^{*})\n",
    "$$\n",
    "\n",
    "This solves for the steady state capital stock per worker. Next we replicate Figure 7.14 using this equation. Afterwards we will determine what happens to the steady state value of capital if shocks are imposed on the savings rate $s$, population growth rate $n$ and total factor productivity $z$. \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "y_star = z .* (k_star .^ α)\n",
    "i_star = s .* y_star\n",
    "c_star = y_star - i_star\n",
    "\n",
    "break_even(k; n = n, d = d) = (n .+ d) .* k\n",
    "inv_curve(k; s = s, z = z) =  s .* z .* g(k) \n",
    "\n",
    "inv = plot(inv_curve, 0:0.01:1.2, leg = :topleft, lw = 2, lab = L\"szf(k^{*})\", color = :blue, alpha = 0.7)\n",
    "break_crv = plot!(inv, break_even, lw = 2, lab = L\"(n + d)k^{*}\", color = :black, alpha = 0.7)\n",
    "plot!(break_crv, g, lw = 2, lab = L\"zf(k^{*})\", color = :red, alpha = 0.5)\n",
    "vline!([k_star], lab = \"\", color = :black, alpha = 0.5, ls = :dash, xticks = ([k_star], [L\"k^{*}\"]))\n",
    "hline!([y_star], lab = \"\", color = :black, alpha = 0.5, ls = :dash, yticks = ([round(y_star, digits = 2), round(i_star, digits = 2)], [L\"y^{*}\", L\"i^{*}\"]))\n",
    "hline!([i_star], lab = \"\", color = :black, alpha = 0.5, ls = :dash)\n",
    "scatter!([k_star], [i_star], label = \"\", markersize = 6, color = :black) "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Shocks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we start from the steady state, can we determine what the impact of different shocks are going to be? If we have an increase in the savings rate what impact will this have on the steady state level of capital? What about changes in the population growth rate, or for that matter productivity? \n",
    "\n",
    "The reason why we ask these questions is because we want to know what the long term drivers of growth are going to be. Which of the model component are going to contribute most to growth? \n",
    "\n",
    "Each of the shocks in this model are going to occur in the tenth time period, $t=10$. We will then trace out how these changes in variables impact the dynamics of capital, investment and output. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.1 Savings rate"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.2 Population growth"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.3 Technology"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One of the things we can easily see is the effect that an increase in productivity will have on the production function. We use a `for loop` to iterate over the different values of $z$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
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       "  1232.47,416.742 1243.04,412.638 1253.6,408.559 1264.17,404.503 1274.74,400.471 1285.31,396.463 1295.88,392.477 1306.45,388.514 1317.02,384.573 1327.59,380.653 \n",
       "  1338.15,376.756 1348.72,372.879 1359.29,369.023 1369.86,365.188 1380.43,361.373 1391,357.579 1401.57,353.804 1412.14,350.048 1422.7,346.312 1433.27,342.595 \n",
       "  1443.84,338.896 1454.41,335.216 1464.98,331.554 1475.55,327.91 1486.12,324.283 1496.69,320.674 1507.25,317.083 1517.82,313.508 1528.39,309.951 1538.96,306.41 \n",
       "  1549.53,302.885 1560.1,299.377 1570.67,295.885 1581.24,292.408 1591.8,288.947 1602.37,285.502 1612.94,282.072 1623.51,278.657 1634.08,275.257 1644.65,271.872 \n",
       "  1655.22,268.501 1665.79,265.145 1676.35,261.803 1686.92,258.475 1697.49,255.161 1708.06,251.861 1718.63,248.574 1729.2,245.301 1739.77,242.041 1750.34,238.794 \n",
       "  1760.9,235.561 1771.47,232.34 1782.04,229.132 1792.61,225.937 1803.18,222.754 1813.75,219.584 1824.32,216.426 1834.89,213.28 1845.46,210.146 1856.02,207.024 \n",
       "  1866.59,203.914 1877.16,200.815 1887.73,197.728 1898.3,194.652 1908.87,191.587 1919.44,188.534 1930.01,185.492 1940.57,182.461 1951.14,179.44 1961.71,176.431 \n",
       "  1972.28,173.432 1982.85,170.444 1993.42,167.466 2003.99,164.498 2014.56,161.541 2025.12,158.594 2035.69,155.657 2046.26,152.73 2056.83,149.813 2067.4,146.906 \n",
       "  2077.97,144.008 2088.54,141.12 2099.11,138.242 2109.67,135.373 2120.24,132.514 2130.81,129.663 2141.38,126.823 2151.95,123.991 2162.52,121.168 2173.09,118.354 \n",
       "  2183.66,115.55 2194.22,112.754 2204.79,109.967 2215.36,107.188 2225.93,104.418 2236.5,101.657 2247.07,98.9041 2257.64,96.1597 2268.21,93.4236 2278.77,90.6958 \n",
       "  2289.34,87.9763 \n",
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      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p1 = plot()\n",
    "\n",
    "for i in z\n",
    "    plot!(p1, production_per_capita(k, i), label = \"z = $i\", legend = :topleft, lw = 1.5)\n",
    "end\n",
    "\n",
    "plot(p1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that with an increase in $z$ the boundary of the production function shifts upward. This indicates that we are able to produce a higher level of output at each given level of capital. In other words, the productivity increase augments our ability to generate output from an existing capital stock. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. Difference equations (optional)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The key equation for the Solow model is a type of **difference equation**.  \n",
    "\n",
    "In the next section we are going to provide a brief introduction to difference equations and then link this back to how we can solve and analyse the behaviour of the components of the Solow model. These notes are an almost direct port of the Python ones at [QuantEcon](https://python.quantecon.org/scalar_dynam.html). Please look at that website for a more complete exploration of difference equations. \n",
    "\n",
    "This section is perhaps a bit more technical, so be sure to **slowly** go through the material. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.1 Foundations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A **time homogeneous first order difference equation** is of the form, \n",
    "\n",
    "$$\n",
    "x_{t+1} = g(x_t)\n",
    "$$\n",
    "\n",
    "where $g$ is a function from some subset of the real numbers to itself. In other words, \n",
    "\n",
    "$$\n",
    "g: S \\rightarrow S\n",
    "$$\n",
    "\n",
    "\n",
    "$S$ is called the **state space**, and $x$ is the **state variable**.\n",
    "\n",
    "Time homogeneity means that $g$ is the same at each time $t$\n",
    "\n",
    "First order means dependence on only one lag (only depends $x_t$, not on earlier points such as $x_{t-1}$)\n",
    "\n",
    "If $x_0 \\in S$, then our difference equation above **recursively** defines the following sequence, \n",
    "\n",
    "$$\n",
    "x_0, x_1 = g(x_0), x_2 = g(x_1) = g(g(x_0)), \\quad \\text{etc.}\n",
    "$$\n",
    "\n",
    "This sequence defines the trajectory of $x_0$ under the function $g$.\n",
    "\n",
    "In other words, if we started at $x_0$, this sequence maps out how the variable would move over time given the function $g$ being applied to the variable multiple times.\n",
    "\n",
    "If we think of $g^{n}$ to be $n$ compositions of $g$ with itself, then we can write the trajectory as\n",
    "\n",
    "$$\n",
    "x_t = g^{t}(x_{0})\n",
    "$$\n",
    "\n",
    "If $n = 2$, the composition would be $g^{2}(x_0) = g(g(x_0))$. In other words, applying the function $g$ twice. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.2 Example: Linear model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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