{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Econ 101b Review: May 1, 2018\n", "\n", " | " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solow Growth Model" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Framework\n", "\n", "The Solow Growth Model (SGM) system of equations:\n", "\n", "         \n", "$ \\frac{d\\left(L_t\\right)}{dt} = nL_t $ :: labor force growth equation\n", "\n", "         \n", "$ \\frac{d\\left(E_t\\right)}{dt} = gE_t $ :: efficiency of labor growth equation\n", "\n", "         \n", "$ \\frac{d\\left(K_t\\right)}{dt} = sY_t - \\delta{K_t} $ :: capital stock growth equation\n", "\n", "         \n", "$ Y_t = \\left(K_t\\right)^{\\alpha}\\left(L_tE_t\\right)^{1-\\alpha} $ :: production function\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Balanced-Growth Path\n", "\n", "* $ \\lim\\limits_{t\\to\\infty}\\left(\\frac{K_t}{Y_t}\\right) = \\frac{s}{n+g+\\delta} $ :: steady-state balanced-growth path capital-output ratio\n", "\n", "* $ \\lim\\limits_{t\\to\\infty}\\left(\\frac{Y_t}{L_t}\\right) = \\left(\\frac{s}{n+g+\\delta}\\right)^{\\frac{\\alpha}{1-\\alpha}} \\left(E_0{e^{gt}}\\right) $ :: steady-state balanced-growth path output-per-worker ratio\n", "\n", "* $ \\lim\\limits_{t\\to\\infty}\\left(\\frac{K_t}{L_t}\\right) = \\left(\\frac{s}{n+g+\\delta}\\right)^{\\frac{1}{1-\\alpha}} \\left(E_0{e^{gt}}\\right) $ :: steady-state balanced-growth path capital-worker ratio\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Convergence\n", "\n", "* convergence rate $ = -(1-\\alpha)(n+g+\\delta) $\n", "\n", "* $ \\frac{K_t}{Y_t} = \\left(1- e^{-(1-\\alpha)(n+g+\\delta)t}\\right)\\left(\\frac{K}{Y}\\right)^* + \\left(e^{-(1-\\alpha)(n+g+\\delta)t}\\right)\\left(\\frac{K_o}{Y_o}\\right) $\n", ":: convergence to the steady-state balanced-growth capital-output ratio\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Malthusian Efficiency of Labor Growth\n", "\n", "H: ideas—non-rival: growth rate h\n", "\n", "E: efficiency of labor: growth rate g\n", "\n", "L: labor force: growth rate n\n", "\n", "N: natural resources—rival: growth rate 0\n", "\n", "* $ g = \\left(\\frac{\\gamma}{1+\\gamma}\\right)h - \\left(\\frac{1}{1+\\gamma}\\right)n $\n", "\n", "* $ n = {\\gamma}h $ :: steady-state balanced-growth path with g = 0\n", "\n", "* $ n = {\\phi}\\ln\\left(\\frac{Y/L}{y^s}\\right) $ :: Malthusian population growth\n", "\n", "* $ g = \\left(\\frac{\\gamma}{1+\\gamma}\\right)h - {\\phi}\\left(\\frac{1}{1+\\gamma}\\right)\\ln\\left(\\frac{Y/L}{y^{s}}\\right) $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### How Did We Escape?\n", "\n", "Two sets of theories for escape:\n", "\n", "* Eye of the needle\n", " * Cultural-scientific\n", " * Resource-technology\n", " * Plunder-exploitation\n", " * Variants: \"We almost got there many times\" and \"we never got close before\" variants\n", " * Variants: Commercial Revolution, Industrial Revolution, or Modern Economic Growth?\n", "\n", "Or:\n", "\n", "* Two heads are better than one...\n", " * $ h = \\left(h_1\\right)L^{\\lambda} $ :: idea generation\n", " \n", "Plus:\n", "\n", "* Demographic transition...\n", " * $ n = \\min\\left({\\phi}\\ln\\left(\\frac{Y/L}{y^s}\\right), \\frac{n_1}{Y/L}\\right) $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Escape: Industrial Revolution and Modern Economic Growth\n", "\n", "* Elasticity of Demand as a Key (not on final)\n", "* Productivity Trends in the North Atlantic\n", " * Britain the First Industrial Nation\n", " * Britain richer—but with low real wages\n", " * British growth acceleration\n", " * But America growing faster from 1800\n", " * And American growth acceleration—modern economic growth and the industrial research lab\n", " * Until the productivity growth slowdon of the 1970s\n", " * And then the speed up of the new-economy 1990s\n", " * And then the growth collapse of the Great Recession\n", " \n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Income and Wealth Inequality\n", "\n", "(not on exam)\n", "\n", "Kaldor facts:\n", "\n", "* Constant r (=αK/Y)\n", "*C onstant wL/Y (= 1-α)\n", "* Constant K/Y\n", "* Constant g\n", "* d(ln(w))/dt = g\n", "\n", "Piketty facts:\n", "\n", "* Increase in W/K\n", "* Increase in market-to-book ratio for K\n", "* Divergence between marginal product of capital and average return\n", "* Substantial decrease in real interest rates in financial markets\n", "\n", "Plutocracy and its fear of creative destruction\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Measuring Economic Growth Truly\n", "\n", "(not on exam)\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Global Patterns\n", "\n", "Divergence, 1800-1975\n", "\n", "* Britain and U.S. growing together\n", "* OECD convergence 1945-present\n", "* Behind Iron Curtain divergence\n", "* General divergence 1800-1975\n", "* From a fivefold to a fifty-fold divergence\n", "\n", "Convergence 1975-present?\n", "\n", "* East Asia\n", "* Japan\n", "* China\n", "\n", "\n", "How to understand?\n", "\n", "* $ \\alpha = 3/5 $\n", "* Schooling very important for the efficiency of labor\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Modeling Global Patterns\n", "\n", "We need a high capital share α:\n", "\n", "* To make “convergence” take a long time\n", "* To amplify the effects of differences in (K/Y)* on prosperity\n", "\n", "We need n to be inversely and s strongly correlated with E\n", "\n", "* Demographic transition\n", "* Favorable relative price structure\n", "\n", "And we need education to be a key link:\n", "\n", "* We need technology transfer to a poorly educated population to be nearly impossible…\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Business Cycles\n", "\n", "\"2018\n", "\n", "* Okun's Law\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Flexible-Price Models\n", "\n", "Full employment (because of flexible wages and prices and debt)\n", "\n", "* Unemployment rate equal to NAIRU\n", "* Production equal to potential output\n", "\n", "Shifts of production and spending across categories\n", "\n", "* In response to changes in the economic environment\n", "* And in response to changes in economic policy\n", "* As a result of shifts in the long-term real risky interest rate r\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Business Cycle NIPA Framework\n", "\n", "* $ Y = C + I + G + (GX - IM) $ :: national income and product\n", "* $ C = c_o + c_y(1-t)Y $ :: consumption function—consumer confidence; marginal propensity to consume; net taxes-less-transfers rate\n", "* $ I = I_o - I_r{r} $ :: investment spending; \"animal spirits\"\n", "* $ G $\n", "* $ IM = im_y{Y} $ :: imports\n", "* $ \\epsilon = \\epsilon_o + \\epsilon_r(r^f - r) $ :: exchange rate; foreign exchange speculators; \"gnomes of Zurich\"\n", "* $ GX = x_f{Y^f} + x_\\epsilon{\\epsilon} $ :: gross exports\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Flexible-Price Model IS Curve Equation\n", "\n", "$ Y^* = Y = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Sticky-Price Models\n", "\n", "### The Sticky-Price Model IS Curve Equation\n", "\n", "$ Y = E = \\mu\\left(c_o + I_o + G\\right) + \\mu\\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right)r $\n", "\n", "Causation from left to right:\n", "\n", "* Spending determines aggregate demand\n", "* Aggregage demand via the inventory adjustment channel determines national income and product\n", "\n", " \n", "\n", "Influences on spending from:\n", "\n", "* Policy variables: G, t, $ r = i - \\pi +\\rho $\n", "* Expectations: $c_o, I_o, \\epsilon_o $\n", "* Foreign economic conditions: $ Y^f, r^f $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Keynesian Multiplier\n", "\n", "$ Y = C + I + G + (GX - IM) $\n", "\n", "$ Y = (c_o + c_y(1-t)Y) + I + G + (GX - im_y{Y}) $\n", "\n", "$ (1 - c_y(1-t) + im_y)Y = c_o + I + G + GX $\n", "\n", "$ Y = \\frac{c_o + I + G + GX}{(1 - c_y(1-t) + im_y)} $\n", "\n", "$ Y = {\\mu}(c_o + I + G + GX) $\n", "\n", "$ \\mu = \\frac{1}{(1 - c_y(1-t) + im_y)} $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Monetary Policy and the Zero Lower Bound\n", "\n", "The interest rate in the IS Curve is the long-term risky real interest rate: r\n", "\n", "The interest rate the central bank controls is the short-term safe nominal interest rate: i\n", "\n", "* $ r = i - \\pi + \\rho $ subject to $ i ≥ 0 $\n", "* $ \\rho = \\rho^R + \\rho^T $\n", " * $ \\rho^R $ :: the risk premium for lending to privates rather than to the government\n", " * Moral hazard\n", " * Adverse selection\n", " * \"Skin in the game\" from borrowers\n", " * Financial crises\n", " * $ \\rho^T $ :: lack of confidence that the central bank will keep i where it currently is\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Phillips Curve\n", "\n", "$ {\\pi_t} = {\\pi_t}^e - \\beta\\left(u_t - u^*\\right) + SS_t$\n", "\n", "Expectations:\n", "\n", "* Static: $ {\\pi_t}^e = \\pi^{*} $\n", "* Adaptive: $ {\\pi_t}^e = \\pi_{t-1} $\n", "* Rational: $ {\\pi_t}^e = \\pi_{t} $\n", "* Hybrids: $ {\\pi_t}^e = \\lambda(\\pi_{t}) + (1-\\lambda)(\\pi_{t-1}) $ or $ {\\pi_t}^e = (1-\\lambda)(\\pi^*) + \\lambda(\\pi_{t-1}) $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Inflation Dynamics\n", "\n", "* Static: $ {\\pi_t} = \\pi^* - \\beta\\left(u_t - u^*\\right) + SS_t$\n", "* Adaptive: $ {\\pi_t} = {\\pi_{t-1}} - \\beta\\left(u_t - u^*\\right) + SS_t$\n", "* Rational: $ {\\pi_t} = {\\pi_t}^e $ and $ u_t = u^* - \\frac{SS_t}{\\beta} $\n", "* Hybrids:\n", " * $ {\\pi_t} = {\\pi_{t-1}} - \\frac{\\beta\\left(u_t - u^*\\right) + SS_t}{1-\\lambda} $\n", " * $ {\\pi_t} - \\pi^* = \\lambda({\\pi_{t-1}}-\\pi^*) - \\beta\\left(u_t - u^*\\right) + SS_t $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Monetary Policy Reaction Function\n", "\n", "$ r_t = r^{**} + r_{\\pi}(\\pi_t - \\pi^T) - r_u(u_t - u^{**}) $\n", "\n", "$ r_t = r^{**} + r_{\\pi}(\\pi_t - \\pi^T) $\n", "\n", "$ u_t - u^* = \\phi(\\pi_{t-1} - \\pi^T) + \\psi(r^{**} - r^*) + \\delta_t $\n", "\n", "Combine the MPRF with the \"inflation dynamics\" version of the Phillips Curve...\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Hysteresis and Budget Arithmetic in a Depression\n", "\n", "Boost government purchases by ΔG—if no Federal Reserve offset because at ZLB \n", "\n", "* Get boost to real GDP by μΔG\n", "* Get boost to taxes by tμΔG\n", "* Increase in debt of (1 - tμ)ΔG = ΔD\n", "* Financing cost of this debt: (r-g)ΔD = (r-g)(1 - tμ)ΔG\n", "\n", "“Hysteresis” parameter η\n", "\n", "* Gain tημΔG in tax revenue from heading off “hysteresis”\n", "* (r-g)(1 - tμ)ΔG greater or less than ηtμΔG?\n", " * t = 0.33\n", " * μ = 2\n", " * 0.33(r - g) greater or less than 0.66η?\n", "\n", "r - g greater or less than 2η?" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.1" } }, "nbformat": 4, "nbformat_minor": 2 }