{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Intermediate Macroeconomics Review\n", "\n", "\n", "\n", "\n", "\n", " \n", "\n", "## The Structure of Economic Models\n", "\n", "* **Variables**: Quantities of interest that we can measure, calculate, or at least hypothesize about...\n", "* **Identities**: Relationships between variables that are true \"by the metaphysical necessity of the case\"; they do not tell us what happens or how it happens, but they constrain the possibilities that might happen.\n", "* **Behavioral Relationships**: Descriptions of (1) what humans, given their opportunities and opportunity costs presented to them by the economic environment and economic policies, decide what to do; and (2) thus determine the values of economic variables.\n", "* **Equilibrium Conditions**: Conditions that, if satisfied, place the economy or at least some subset of economic variables at stable values, or in some sort of balance.\n", " * **Convergence Mechanisms**: A dynamic process in the economy that drives the economy toward an equilibrium position of some sort. \n", " \n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solow Growth Model\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Framework\n", "\n", "The Solow Growth Model (SGM) system of equations:\n", "\n", ">$ g_L = \\frac{d\\left(L_t\\right)}{dt} = nL_t $ :: labor-force growth equation\n", "\n", ">$ g_E = \\frac{d\\left(E_t\\right)}{dt} = gE_t $ :: efficiency-of-labor growth equation\n", "\n", ">$ g_K = \\frac{d\\left(K_t\\right)}{dt} = sY_t - \\delta{K_t} $ :: capital stock growth equation\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": [ "#### Some Facts: Looking Across Countries\n", "\n", "\"Banners\n", "\n", "* U.S. today:\n", " * 3x richer than Mexico\n", " * 9x richer than Nigeria\n", " * 20x richer than Afghanistan\n", "\n", "* Afghans back at where U.S. was 200 years ago\n", " * Differences between Afghanistan and U.S. then much smaller than now\n", " * In spite of improved communications, transport...\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Some Facts: Looking Over Time\n", "\n", "\"Macroeconomics\n", "\n", "* 15x in productivity since 1870\n", " * And this might be an underestimate\n", " * Doubling of life expectancy since 1870\n", "* Where do you think the 1919 “Spanish Influenza” epidemic is?\n", "* Divide $4000 by 13 and you are at a dollar a day\n", " * Anything less, and you are starving to death\n", "* ln(60000/4000)/150 = what? <https://www.wolframalpha.com/input/?i=ln(60000%2F4000)%2F150>\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Balanced-Growth Path\n", "\n", "* $ \\left(\\frac{K}{Y}\\right)^* = \\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", "* $ \\left(\\frac{Y_t}{L_tE_t}\\right)^* = \\lim\\limits_{t\\to\\infty}\\left(\\frac{Y_t}{L_tE_t}\\right) = \\left(\\frac{s}{n+g+\\delta}\\right)^{\\frac{\\alpha}{1-\\alpha}} $ :: steady-state balanced-growth path output-per-worker ratio\n", "\n", "* $ \\left(\\frac{K_t}{L_tE_t}\\right)^* = \\lim\\limits_{t\\to\\infty}\\left(\\frac{K_t}{L_tE_t}\\right) = \\left(\\frac{s}{n+g+\\delta}\\right)^{\\frac{1}{1-\\alpha}} $ :: steady-state balanced-growth path capital-worker ratio\n", "\n", " * $ E_t = \\left(E_o{e^{gt}}\\right) $ \n", " * $ E_t = (1+g)^tE_o $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Convergence\n", "\n", "convergence rate $ = -(1-\\alpha)(n+g+\\delta) $\n", "\n", "convergence to the steady-state balanced-growth capital-output ratio:\n", "\n", ">$ \\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", "\n", "> $ \\frac{K_t}{Y_t} = \\left(1 - \\frac{1}{[(1+(1-\\alpha)(n+g+\\delta)]^t}\\right)\\left(\\frac{K}{Y}\\right)^* + \\left(\\frac{1}{[(1+(1-\\alpha)(n+g+\\delta)]^t}\\right)\\left(\\frac{K_o}{Y_o}\\right) $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Exercises\n", "\n", "Start with an economy on its balanced-growth path, with:\n", "\n", "* $ \\alpha $\n", "* $ \\delta $\n", "* $ n_{ini}, s_{ini}, g_{ini} $\n", "* $ L_o, E_o, K_o $\n", "\n", "And let's change something: $ s_{alt} = s_{ini} + {\\Delta}s $...\n", "\n", " \n", "\n", "What is $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* $ going to be?\n", "\n", "A. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\left(\\frac{{\\Delta}s}{n+g+\\delta}\\right)^{\\alpha} $\n", "\n", "B. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\left(\\frac{{\\Delta}s}{n+g+\\delta}\\right)^\\left({\\frac{\\alpha}{1-\\alpha}}\\right) $\n", "\n", "C. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\frac{{\\Delta}s}{n+g+\\delta} $\n", "\n", "D. $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* = \\frac{(1-\\alpha){\\Delta}s}{n+g+\\delta} $\n", "\n", "E. None of the above\n", "\n", " \n", "\n", "If $ \\alpha = 2/3, \\delta = 0.025, n_{ini} = 0.005, $ \n", "$ s_{ini} = 0.16, g_{ini} = 0.01,$ and $ {\\Delta}s = 0.04, $ \n", "then $ {\\Delta}\\left(\\frac{K}{Y}\\right)^* $ is going to be?\n", "\n", "A. 9\n", "\n", "B. 0.404\n", "\n", "C. 1\n", "\n", "D. 1/3\n", "\n", "E. None of the above\n", "\n", " \n", "\n", "3 years after a jump $ {\\Delta}s $ in the savings rate, an economy that started out on its initial balanced-growth path would have seen its capital-output ratio reduce approximately what fraction of the gap between the initial and the alternative balanced-growth path values?\n", "\n", "A. $ (1-\\alpha)(n+g+\\delta) $\n", "\n", "B. $ e^{(1-\\alpha)(n+g+\\delta)} $\n", "\n", "C. $ e^{3(1-\\alpha)(n+g+\\delta)} $\n", "\n", "D. $ 3(1-\\alpha)(n+g+\\delta) $\n", "\n", "E. None of the above\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Shape of World Economic History\n", "\n", "\"Macroeconomics\n", "\n", "* The Solow growth model was built to account for the experience of the modern economic growth era\n", " * 1900-today\n", "* But there was a lot of history before then\n", "* And things were different then: the past is another country\n", "* Slower population growth, and much much slower growth in real standards of living\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Malthus and Efficiency-of-Labor Growth\n", "\n", "H: ideas: non-rival factors: growth rate h\n", "\n", "E: efficiency of labor: growth rate g\n", "\n", "L: labor force: growth rate n\n", "\n", "N: natural resources: rival factors: growth rate 0 (in most applications)\n", "\n", ">$ g = \\left(\\frac{\\gamma}{1+\\gamma}\\right)h - \\left(\\frac{1}{1+\\gamma}\\right)n $\n", "\n", ">$ 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": [ "#### Exercises\n", " \n", "In a Malthusian equilibrium, what is \n", "the relationship between the rate of \n", "growth $ h $ of the useful ideas stock \n", "and the rate of population growth $ n $?\n", "\n", "A. $ n = h $\n", "\n", "B. $ n = {\\gamma}h $\n", "\n", "C. $ n = \\frac{h}{\\gamma} $\n", "\n", "D. None of the above\n", "\n", " \n", "\n", "Between the year 1 and the year 1800, \n", "$ \\gamma = 3, n = 0.075% $ per year, and \n", "the economy was in Malthusian equilibrium. \n", "What, roughly, was the rate of growth of h?\n", "\n", "A. 0.025% per year\n", "\n", "B. 0.075% per year\n", "\n", "C. 0.225% per year\n", "\n", "D. None of the above \n", "\n", " \n", "\n", "In Malthusian equilibrium, what is the \n", "Malthusian balanced-growth path level \n", "of output per worker (Y/L)*?\n", "\n", "A. $ \\left(Y/L\\right)^* = e^{\\left(\\frac{{\\gamma}h}{\\phi}\\right)}y^s $\n", "\n", "B. $ \\left(Y/L\\right)^* = {\\left(\\frac{{\\gamma}h}{\\phi}\\right)}y^s $\n", "\n", "C. $ \\left(Y/L\\right)^* = {\\phi}{\\gamma}hy^s $\n", "\n", "D. None of the above \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 $\n", "\n", "Alternatively:\n", "\n", "$ 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)(i - \\pi + \\rho) $\n", "\n", " \n", "\n", "#### Exercises\n", "\n", "$ Y^* = Y = AD = \\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", "* flexprice causation runs from left to right...\n", "* real wage W/P adjusts to get full employment and Y = Y*\n", "* price level P adjusts to get AD = Y\n", "* interest rate r adjusts to get the components of AD to match up...\n", "\n", "Start from one equilibrium \"ini\" and consider a shift to another \"alt\"...\n", "\n", "What things change?\n", "\n", "* The initial shock to economic policy or the economic environment...\n", "* The interest rate: $ {\\Delta}r ≠ 0 $\n", "* Anything determined, directly or indirectly, by r:\n", " * $ I $ :: investment spending\n", " * $ \\epsilon $ :: the exchange rate\n", " * $ GX $ :: exports\n", "\n", " \n", "\n", "#### A Rise in Interest Rates Abroad\n", "\n", "$ Y^* = Y = AD = \\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 $\n", "\n", "$ 0 = \\mu\\left(x_{\\epsilon}{\\epsilon}_r{{\\Delta}r^f}\\right) - \\mu\\left(I_r + x_{\\epsilon}{\\epsilon}_r\\right){\\Delta}r $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "What is $ {\\Delta}r $?\n", "\n", "A. $ {\\Delta}r = {\\Delta}r^f $\n", "\n", "B. $ {\\Delta}r = \\frac{x_{\\epsilon}{\\epsilon}_r}{I_r + x_{\\epsilon}{\\epsilon}_r}{\\Delta}r^f $\n", "\n", "C. $ {\\Delta}r = \\frac{I_r}{I_r + x_{\\epsilon}{\\epsilon}_r}{\\Delta}r^f $\n", "\n", "D. None of the above\n", "\n", " \n", "\n", "If $ I_r = 20 , \\mu = 2, x_{\\epsilon} = 1, $ and \n", "$ {\\epsilon_r = 10 } $, then $ {\\Delta}r^f = + 0.03 $ will \n", "have what effect on domestic investment I?\n", "\n", "A. $ {\\Delta}I = + 0.6 $\n", "\n", "B. $ {\\Delta}I = - 0.6 $\n", "\n", "C. $ {\\Delta}I = - 0.2 $\n", "\n", "D. None of the above\n", "\n", " \n", "\n", "If $ I_r = 20 , \\mu = 2, x_{\\epsilon} = 1, $ and \n", "$ {\\epsilon_r = 10 } $, then $ {\\Delta}G = + 0.3 $ will have \n", "what effect on domestic investment I?\n", "\n", "A. $ {\\Delta}I = \\left(x_{\\epsilon}{\\epsilon_r}{I_r + x_{\\epsilon}{\\epsilon}_r}\\right){\\Delta}G $\n", "\n", "B. $ {\\Delta}I = \\left(\\frac{I_r}{I_r + x_{\\epsilon}{\\epsilon}_r}\\right){\\Delta}G $\n", "\n", "C. $ {\\Delta}I = \\left(\\frac{1}{I_r + x_{\\epsilon}{\\epsilon}_r}\\right){\\Delta}G $\n", "\n", "D. None of the above" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Sticky-Price Models\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Sticky-Price Model IS Curve Equation\n", "\n", "$ Y = AD = \\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 $\n", "\n", "Alternatively:\n", "\n", "$ 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)(i - \\pi + \\rho) $\n", "\n", "Causation from right to left:\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:\n", " * Fiscal: G, t\n", " * Monetary: $ r = i - \\pi +\\rho $\n", " * risk premium $ \\rho $ sum of term $ \\rho^T $ and counterparty risk $ \\rho^r $\n", "* Expectations: confidence/animal spirits/greed vs. fear: $c_o, I_o, \\epsilon_o $\n", "* Foreign economic conditions: $ Y^f, r^f $\n", "* The multiplier: $ \\mu = \\frac{1}{1 - c_y(1-t) + im_y} $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Keynesian Multiplier\n", "\n", "$ 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": [ "### Playing with the Multiplier\n", "\n", "**The Simple Multiplier**: $ \\frac{{\\Delta}Y}{{\\Delta}G} = \\mu $ :: holding r, t fixed\n", "\n", " \n", "\n", "**The Balanced Budget Multiplier**:\n", "\n", "> Allowing t to vary with G, so that: $ {\\Delta}t = \\frac{{\\Delta}G}{Y} $ so that $ {\\Delta}(G - T) = 0 $... \n", "\n", ">Then: $ {\\Delta}Y = \\frac{{\\Delta}G - c_y(1-t){\\Delta}G}{1-c_y(1-t)+im_y} = \\frac{1-c_y(1-t)}{1-c_y(1-t)+im_y}{\\Delta}G $\n", "\n", " \n", "\n", "**The Monetary-Offset Multiplier**:\n", "\n", "> Allowing r to vary with G, so that: $ {\\Delta}r = \\frac{{\\Delta}G}{I_r + x_{\\epsilon}{\\epsilon}_r} $... \n", "\n", ">Then: $ {\\Delta}Y = 0{\\Delta}G $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Exercises\n", "\n", "$ \\mu = \\frac{1}{(1 - c_y(1-t) + im_y)} $\n", "\n", " \n", "\n", "Suppose $ c_y = 5/6, t = 1/4, im_y = 1/8 $, \n", "then if $ {\\Delta}G = 0.4 $ and r and t are fixed:\n", "\n", "A. $ {\\Delta}Y = 0.4 $\n", "\n", "B. $ {\\Delta}Y = 0.5 $\n", "\n", "C. $ {\\Delta}Y = 0.8 $\n", "\n", "D. $ {\\Delta}Y = 0.6 $\n", "\n", "E. None of the above\n", "\n", " \n", "\n", "Suppose $ c_y = 2/3, t = 1/4, im_y = 1/4 $, \n", "then if $ {\\Delta}G = 0.3 $ and r and t are fixed:\n", "\n", "A. $ {\\Delta}Y = 0.4 $\n", "\n", "B. $ {\\Delta}Y = 0.5 $\n", "\n", "C. $ {\\Delta}Y = 0.8 $\n", "\n", "D. $ {\\Delta}Y = 0.6 $\n", "\n", "E. None of the above\n", "\n", " \n", "\n", "Suppose $ c_y = 1/3, t = 1/4, im_y = 1/4 $, \n", "then if $ {\\Delta}G = 0.6 $ and r and t are fixed:\n", "\n", "A. $ {\\Delta}Y = 0.4 $\n", "\n", "B. $ {\\Delta}Y = 0.5 $\n", "\n", "C. $ {\\Delta}Y = 0.8 $\n", "\n", "D. $ {\\Delta}Y = 0.6 $\n", "\n", "E. None of the above\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Solving IS-Curve Problems\n", "\n", "$ Y = AD = \\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 right to left...\n", "\n", "* Government policy determines G, t via fiscal policy and (influences) r via monetary policy that determines i in $ r = i + \\rho - \\pi $\n", "\n", "* The economic environment:\n", " * Expectations: $ c_o, I_o, {\\epsilon}_o $\n", " * International influences: $ Y^f, r^f $\n", " * Parameters in behavioral relationships: $ I_r, c_y $ (influences $ \\mu $ ), $ x_f, x_{\\epsilon}, {\\epsilon}_r $\n", " \n", "* Add these up (and apply the multiplier $ \\mu $) to calculate aggregate demand AD\n", "* Aggregate demand AD determines production Y through the inventory-adjustment process\n", " * And consequent hiring and firing alters income so it equals Y—and at that point $ I = S^p + S^g + S^f $ no matter what the interest rate r is.\n", " \n", "* And potential output $ Y^* $ is nowheresville here...\n", " \n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Exercises\n", "\n", "$ Y = AD = \\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", "Suppose $ \\mu = 2 $ and $ {\\Delta}G = 0.3, $ then:\n", "\n", "A. $ {\\Delta}Y = 0.2 $\n", "\n", "B. $ {\\Delta}Y = 0.3 $\n", "\n", "C. $ {\\Delta}Y = 0.5 $\n", "\n", "D. $ {\\Delta}Y = 0.6 $\n", "\n", "E. None of the above\n", "\n", " \n", "\n", "Suppose $ \\mu = 2, I_r = 20, x_{\\epsilon} = 4, $ and $ {\\epsilon}_r = 5 $. \n", "Then if $ {\\Delta}r = -0.015 $ :\n", "\n", "A. $ {\\Delta}Y = 0.2 $\n", "\n", "B. $ {\\Delta}Y = 0.3 $\n", "\n", "C. $ {\\Delta}Y = 0.5 $\n", "\n", "D. $ {\\Delta}Y = 0.6 $\n", "\n", "E. None of the above\n", "\n", " \n", "\n", "Suppose $ \\mu = 2, {\\Delta}G = 0.3, I_r = 20, x_{\\epsilon} = 4, $ and $ {\\epsilon}_r = 5 $. \n", "Then if $ {\\Delta}r = -0.015 $ :\n", "\n", "A. $ {\\Delta}Y = 0.2 $\n", "\n", "B. $ {\\Delta}Y = 0.3 $\n", "\n", "C. $ {\\Delta}Y = 0.5 $\n", "\n", "D. $ {\\Delta}Y = 0.6 $\n", "\n", "E. None of the above\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Phillips Curve\n", "\n", "$ {\\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", "* Hybrid: Adaptive and Rational:\n", " * $ {\\pi_t} = \\lambda{\\pi_t} + (1-\\lambda)\\pi_{t-1} - \\beta\\left(u_t - u^*\\right) + SS_t$\n", " * $ {\\pi_t} = {\\pi_{t-1}} - \\frac{\\beta\\left(u_t - u^*\\right) + SS_t}{1-\\lambda} $\n", " \n", "* Hybrid: Adaptive and Static:\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^{n} + r_{\\pi}(\\pi_t - \\pi^T) $\n", "\n", "$ u_t - u^* = {u_r}({r_t}^n - r^*) + \\delta_t $\n", "\n", "$ \\pi_t = {\\pi_t}^e - \\beta(u_t - u^*) + SS_t $\n", "\n", "Combine the MPRF with the \"inflation dynamics\" version of the Phillips Curve...\n", "\n", " \n", "\n", "$ u_t - u^* = {u_r}r_{\\pi}(\\pi_t - \\pi^T) $\n", "\n", "$ \\pi_t = {\\pi_t}^e - \\beta{u_r}r_{\\pi}(\\pi_t - \\pi^T) + SS_t $\n", "\n", "$ \\pi_t = \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}{\\pi_t}^e + \\frac{\\beta{u_r}r_{\\pi}}{1 + \\beta{u_r}r_{\\pi}}\\pi^T\\right) + \n", "\\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t $\n", "\n", " \n", "\n", "**Expectations: Static**:\n", "\n", "* $ \\pi_t = \\pi^T + \\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\left(\\pi^* - \\pi^T\\right) + \n", "\\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t $\n", "* $ u_t = u^* + {u_r}r_{\\pi}(\\pi_t - \\pi^T) $\n", "\n", " \n", "\n", "**Expectations: Adaptive**: \n", "\n", "* $ \\pi_t = \\pi^T + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}({\\pi_{t-1}}- \\pi^T) + \n", "\\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t\\right) $\n", "* $ u_t = u^* + {u_r}r_{\\pi}(\\pi_t - \\pi^T) $\n", "* $ \\pi_t = \\pi^T + \\frac{1}{1 + \\beta{u_r}r_{\\pi}}SS_t + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\right)^2SS_{t-1} + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\right)^3SS_{t-2} + \\left(\\frac{1}{1 + \\beta{u_r}r_{\\pi}}\\right)^4SS_{t-3}... $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The \"Neutral\" Interest Rate\n", "\n", "$ Y^* = Y = AD = \\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 $ :: Flexprice IS\n", "\n", "$ r^* = \\frac{\\left[\\frac{Y^*}{\\mu} - \\left(c_o + I_o + G\\right) + \\left(x_f{Y^f} + x_{\\epsilon}{\\epsilon}_o + x_{\\epsilon}{\\epsilon}_r{r^f}\\right)\\right]}{I_r + x_\\epsilon\\epsilon_r} $ :: r-star\n", "\n", "* Drive the long-term risky real interest rate to this $ r^* $ value...\n", " * Do so as $ r^* $ is pushed around by shocks to the economic environment and to fiscal policy...\n", "\n", "\"Consumer\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": [ "### 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?\n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Catch Our Breath\n", "\n", "\n", "\n", "* Ask me two questions…\n", "* Make two comments…\n", "\n", " \n", "\n", "* Further reading…\n", "\n", "
\n", "\n", "----\n", "\n", "Keynote: \n", "\n", "Lecture Support: \n", "\n", "\n", " \n", "\n", "----" ] } ], "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": "3.6.7" } }, "nbformat": 4, "nbformat_minor": 2 }