{
"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",
"\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",
"\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) $\n",
"\n",
"E. None of the above\n",
"\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### The Shape of World Economic History\n",
"\n",
"\n",
"\n",
"* The Solow growth model was built to account for the experience of the modern economic growth era\n",
" * 1900-today\n",
"* But there was a lot of history before then\n",
"* And things were different then: the past is another country\n",
"* Slower population growth, and much much slower growth in real standards of living\n",
"\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Malthus and Efficiency-of-Labor Growth\n",
"\n",
"H: ideas: non-rival factors: growth rate h\n",
"\n",
"E: efficiency of labor: growth rate g\n",
"\n",
"L: labor force: growth rate n\n",
"\n",
"N: natural resources: rival factors: growth rate 0 (in most applications)\n",
"\n",
">$ 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",
"\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",
"\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",
"----"
]
}
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