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country’s lawmakers. Fisher equation \| A formula for converting from nominal interest rates to real interest rates: the real interest rate equals the nominal interest rate minus the inflation rate. Fisher equation \| A formula for converting from nominal interest rates to real interest rates: the real interest rate equals the nominal interest rate minus the inflation rate. Fisher equation \| A formula for converting from nominal interest rates to real interest rates: the real interest rate equals the nominal interest rate minus the inflation rate. Fisher equation \| A formula for converting from nominal interest rates to real interest rates: the real interest rate equals the nominal interest rate minus the inflation rate. fixed exchange rate \| A regime in which a central bank uses its tools to target the value of the domestic currency in terms of a foreign currency. fixed exchange rate \| A regime in which a central bank uses its tools to target the value of the domestic currency in terms of a foreign currency. flexible exchange rates \| A system in which market forces determine exchange rates. flexible exchange rates \| A system in which market forces determine exchange rates. Flexible prices \| Prices that adjust immediately to shifts in supply and demand curves so that markets are always in equilibrium. Flexible prices \| Prices that adjust immediately to shifts in supply and demand curves so that markets are always in equilibrium. 2 https://socialsci.libretexts.org/@go/page/71917 foreign exchange markets \| The place where suppliers and demanders of currencies meet and trade. growth rate \| The change in a variable over time divided by its value in the beginning period. foreign exchange markets \| The place where suppliers and demanders of currencies meet and trade. growth rate \| The change in a variable over time divided by its value in the beginning period. frictional unemployment \| The unemployment that occurs when workers are moving between jobs. Human capital \| The skills and knowledge that are embodied within workers. frictional unemployment \| The unemployment that occurs when workers are moving between jobs. Human capital \| The skills and knowledge that are embodied within workers. fully funded Social Security system \| A system in which the government taxes income and invests it on behalf of the household, paying back the saving with interest during retirement years. fully funded Social Security system \| A system in which the government taxes income and invests it on behalf of the household, paying back the saving with interest during retirement years. globalization \| The increasing ability of goods, capital, to flow among countries. information labor, and globalization \| The increasing ability of goods, capital, to flow among countries. information labor,
and government budget constraint \| A limit stating issuing that government debt. the deficit must be financed by government budget constraint \| A limit stating that issuing government debt. the deficit must be financed by government debt obligations of a government at a point in time. \| The total outstanding government debt obligations of a government at a point in time. \| The total outstanding government deficit \| The difference between government outlays and revenues. government deficit \| The difference between government outlays and revenues. Government purchases government on goods and services. \| Spending by Government purchases government on goods and services. \| Spending by the the government revenues \| Money that flows into the government sector from households and firms, largely through taxation. government revenues \| Money that flows into the government sector from households and firms, largely through taxation. government surplus \| Total tax revenues collected by the governments less its purchases of goods and services and transfers to households. government surplus \| Total tax revenues collected by the governments less its purchases of goods and services and transfers to households. growth accounting \| An accounting method that shows how changes in real GDP in an economy are due to changes in available inputs. growth accounting \| An accounting method that shows how changes in real GDP in an economy are due to changes in available inputs. growth rate \| The change in a variable over time divided by its value in the beginning period. growth rate \| The change in a variable over time divided by its value in the beginning period. growth rate \| The change in a variable over time divided by its value in the beginning period. growth rate \| The change in a variable over time divided by its value in the beginning period. hyperinflation \| A period of very high and often escalating inflation. hyperinflation \| A period of very high and often escalating inflation. illiquid \| Not capable of being easily and quickly exchanged for cash. illiquid \| Not capable of being easily and quickly exchanged for cash. income effect \| As income increases, households choose to consume more of everything, including leisure. income effect \| As income increases, households choose to consume more of everything, including leisure. individual labor supply curve \| A curve that indicates how many hours of labor an individual supplies at different values of the real wage. individual labor supply curve \| A curve that indicates how many hours of labor an individual supplies at different values of the real wage. inflation rate \| The growth rate of the price index from one year to the next. inflation rate \| The growth rate of the price index from one year to the next. inflation rate \| The growth rate of the price
index from one year to the next. inflation rate \| The growth rate of the price index from one year to the next. inflation rate \| The growth rate of the price index from one year to the next. inflation rate \| The growth rate of the price index from one year to the next. inflation targeting \| A regime in which the central bank uses its tools to set the inflation rate as close as possible to a target. inflation targeting \| A regime in which the central bank uses its tools to set the inflation rate as close as possible to a target. inflation tax \| A tax occurring when the government prints money to finance its deficit. inflation tax \| A tax occurring when the government prints money to finance its deficit. inflation tax \| A tax occurring when the government prints money to finance its deficit. inflation tax \| A tax occurring when the government prints money to finance its deficit. intermediate goods and services \| Products that are used—and completely used up—in the production of other goods and services. intermediate goods and services \| Products that are used—and completely used up—in the production of other goods and services. intertemporal budget constraint \| A limit stating that the discounted present value of taxes minus the discounted present value of outlays (excluding interest on the debt) must equal the current stock of debt outstanding. intertemporal budget constraint \| A limit stating that the discounted present value of taxes minus the discounted present value of outlays (excluding interest on the debt) must equal the current stock of debt outstanding. Inventory investment \| The change in inventories of final goods. Inventory investment \| The change in inventories of final goods. investment \| The purchase of new goods that increase capital stock, allowing an economy to produce more output in the future. investment \| The purchase of new goods that to increase capital stock, allowing an economy produce more output in the future. Investment \| The purchase of new goods that increase capital stock, allowing an economy to produce more output in the future. Investment \| The purchase of new goods that increase capital stock, allowing an economy to produce more output in the future. investment rate \| The total investment as a fraction of GDP. investment rate \| The total investment as a fraction of GDP. Knowledge production process. \| The blueprints Knowledge production process. \| The blueprints that describe a that describe a labor demand \| The amount of labor that firms want to hire at a given real wage. labor demand \| The amount of labor that firms want to hire at a given real wage. labor force \| All (civilian) individuals who
are either working or actively looking for work. labor force \| All (civilian) individuals who are either working or actively looking for work. Labor hours \| The total hours worked in an economy, measured as the number of people employed times the average hours they work. Labor hours \| The total hours worked in an economy, measured as the number of people employed times the average hours they work. labor market \| The market that brings together households who supply labor services and firms who demand labor as an input into the production process. labor market \| The market that brings together households who supply labor services and firms who demand labor as an input into the production process. labor market \| The market that brings together households who supply labor services and firms who demand labor as an input into the production process. labor market \| The market that brings together households who supply labor services and firms who demand labor as an input into the production process. law of demand \| People consume more of a good when its price decreases and less of a good when its price increases. law of demand \| People consume more of a good when its price decreases and less of a good when its price increases. law of one price \| Different prices for the same good or service will not persist because arbitrage eliminates such differences. law of one price \| Different prices for the same good or service will not persist because arbitrage eliminates such differences. 3 https://socialsci.libretexts.org/@go/page/71917 life-cycle model \| A model studying how an individual chooses a lifetime pattern of saving and consumption given a lifetime budget constraint. life-cycle model \| A model studying how an individual chooses a lifetime pattern of saving and consumption given a lifetime budget constraint. life-cycle model of consumption \| A model studying how an individual chooses a lifetime pattern of saving and consumption given a lifetime budget constraint. life-cycle model of consumption \| A model studying how an individual chooses a lifetime pattern of saving and consumption given a lifetime budget constraint. loan-deposit ratio \| The total amount of loans made by banks divided by the total amount of deposits in banks. loan-deposit ratio \| The total amount of loans made by banks divided by the total amount of deposits in banks. long-run neutrality of money \| The fact that changes in the money supply have no long-run effect on real variables. long-run neutrality of money \| The fact that changes in the money supply have no long-run effect on real variables. Macroeconomics \| The study of the economy as a
whole. Macroeconomics \| The study of the economy as a whole. marginal product of capital \| The extra output obtained from one more unit of capital. marginal product of capital \| The extra output obtained from one more unit of capital. marginal product of labor \| The amount of extra output produced from one extra hour of labor input. marginal product of labor \| The amount of extra output produced from one extra hour of labor input. marginal propensity to consume \| The amount by which consumption increases when disposable income increases by a dollar. marginal propensity to consume \| The amount by which consumption increases when disposable income increases by a dollar. marginal propensity to consume \| The amount consumption income increases by a dollar. increases when disposable marginal propensity to consume \| The amount income consumption increases by a dollar. increases when disposable marginal propensity to save \| The amount by which saving increases when disposable income increases by a dollar. marginal propensity to save \| The amount by which saving increases when disposable income increases by a dollar. marginal tax rates \| The tax rate paid on additional income. marginal tax rates \| The tax rate paid on additional income. market demand curve \| The number of units of a good or a service demanded at each price. market demand curve \| The number of units of a good or a service demanded at each price. market supply curve \| The number of units of a good or a service supplied at each price. market supply curve \| The number of units of a good or a service supplied at each price. maturities \| The term in which an asset comes due. maturities \| The term in which an asset comes due. maturity \| The time when the final payment on a loan is due. maturity \| The time when the final payment on a loan is due. national income identity \| The condition that production is the sum of consumption, investment, government purchases, and net exports. national income identity \| Production equals the sum of consumption plus investment plus government purchases plus net exports. national income identity \| Production equals the sum of consumption plus investment plus government purchases plus net exports. medium of exchange \| Anything that will be widely accepted in exchange for goods and services. national savings \| The sum of private and government saving. medium of exchange \| Anything that will be widely accepted in exchange for goods and services. national savings \| The sum of private and government saving. microeconomics \| The study of the choices made by individuals and firms, as well as how individuals and firms interact with each other through markets and other mechanisms. microeconomics \| The study of the choices made by individuals and
firms, as well as how individuals and firms interact with each other through markets and other mechanisms. monetary policy \| Changes in interest rates and other tools that are under the control of the monetary authority of a country (the central bank). monetary policy \| Changes in interest rates and other tools that are under the control of the monetary authority of a country (the central bank). monetary transmission mechanism \| A mechanism explaining how the actions of a central bank affect aggregate economic variables, in particular real GDP. monetary transmission mechanism \| A mechanism explaining how the actions of a central bank affect aggregate economic variables, in particular real GDP. monetary transmission mechanism \| A mechanism explaining how the actions of a central bank affect aggregate economic variables, in particular real GDP. monetary transmission mechanism \| A mechanism explaining how the actions of a central bank affect aggregate economic variables, in particular real GDP. moral hazard \| An incentive problem that arises when the provision of insurance leads individuals to make riskier choices. moral hazard \| An incentive problem that arises when the provision of insurance leads individuals to make riskier choices. multiplier \| The amount by which a change in autonomous spending must be multiplied to give the change in output, equal to 1 divided by (1 – the marginal propensity to spend). multiplier \| The amount by which a change in autonomous spending must be multiplied to give the change in output, equal to 1 divided by (1 – the marginal propensity to spend). multiplier \| The amount by which a change in autonomous spending must be multiplied to give the change in output, equal to 1 divided by (1 – the marginal propensity to spend). multiplier \| The amount by which a change in autonomous spending must be multiplied to give the change in output, equal to 1 divided by (1 – the marginal propensity to spend). national income identity \| The condition that production is the sum of consumption, investment, government purchases, and net exports. national savings \| The sum of private and government saving. national savings \| The sum of private and government saving. National savings \| The sum of private and government saving. National savings \| The sum of private and government saving. natural rate of unemployment \| The amount of unemployment we expect in an economy that is operating at full employment. natural rate of unemployment \| The amount of unemployment we expect in an economy that is operating at full employment. natural resources \| Oil, coal, and other mineral deposits; agricultural and forest lands; and other resources used in the production process. natural resources \| Oil, coal, and other mineral deposits; agricultural and forest lands; and other resources used in
the production process. Natural resources \| Oil, coal, and other mineral deposits; agricultural and forest lands; and other resources used in the production process. Natural resources \| Oil, coal, and other mineral deposits; agricultural and forest lands; and other resources used in the production process. net exports \| Exports minus imports. net exports \| Exports minus imports. net exports \| Exports minus imports. net exports \| Exports minus imports. Net exports \| Exports minus imports. Net exports \| Exports minus imports. Net exports \| Exports minus imports. Net exports \| Exports minus imports. nominal exchange rate \| The price of one currency in terms of another. nominal exchange rate \| The price of one currency in terms of another. nominal gross domestic product (nominal GDP) \| The market value of the final goods and services produced by an economy in a given period of time. nominal gross domestic product (nominal GDP) \| The market value of the final goods and services produced by an economy in a given period of time. nominal interest factor \| A factor, equal to 1 + the nominal interest rate, used to convert dollars today into dollars next year. nominal interest factor \| A factor, equal to 1 + the nominal interest rate, used to convert dollars today into dollars next year. 4 https://socialsci.libretexts.org/@go/page/71917 nominal interest rate \| The number of additional dollars that must be repaid for every dollar that is borrowed. nominal interest rate \| The number of additional dollars that must be repaid for every dollar that is borrowed. nominal interest rate \| The number of additional dollars that must be repaid for every dollar that is borrowed. nominal interest rate \| The number of additional dollars that must be repaid for every dollar that is borrowed. Nominal variables \| A variable defined and measured in terms of money. Nominal variables \| A variable defined and measured in terms of money. nominal wages \| The wage in dollars paid to workers per unit of time. nominal wages \| The wage in dollars paid to workers per unit of time. nonexcludable good \| A good (or resource) for which it is impossible to selectively deny access. nonexcludable good \| A good (or resource) for which it is impossible to selectively deny access. nonrenewable (exhaustible) resource \| A resource that does not regenerate over time. nonrenewable (exhaustible) resource \| A resource that does not regenerate over time. \| A good nonrival for which one person
’s consumption of that good does not prevent others from also consuming it. \| A good nonrival for which one person’s consumption of that good does not prevent others from also consuming it. nonrival good \| A good where one person’s consumption of that good does not prevent others from also consuming it. nonrival good \| A good where one person’s consumption of that good does not prevent others from also consuming it. open economy \| An economy that trades with other countries. open economy \| An economy that trades with other countries. open-market operations \| Purchases and sales of government debt by a central bank. Physical capital \| The total amount of machines and production facilities used in production. planned spending economy except for unplanned inventory investment. \| All expenditures in an planned spending economy except for unplanned inventory investment. \| All expenditures in an potential output \| The amount of real GDP the economy produces when the labor market is in equilibrium and capital goods are not lying idle. potential output \| The amount of real GDP the economy produces when the labor market is in equilibrium and capital goods are not lying idle. potential output \| The amount of real GDP the economy produces when the labor market is in equilibrium and capital goods are not lying idle. potential output \| The amount of real GDP the economy produces when the labor market is in equilibrium and capital goods are not lying idle. potential output \| The amount of real GDP the economy produces when the labor market is in equilibrium and capital goods are not lying idle. potential output \| The amount of real GDP the economy produces when the labor market is in equilibrium and capital goods are not lying idle. price level \| A measure of average prices in the economy. price level \| A measure of average prices in the economy. price level \| A measure of average prices in the economy. price level \| A measure of average prices in the economy. price level \| A measure of average prices in the economy. price level \| A measure of average prices in the economy. price level \| A measure of average prices in the economy. price level \| A measure of average prices in the economy. price-adjustment equation \| An equation that describes how prices adjust in response to the output gap, given autonomous inflation. price-adjustment equation \| An equation that describes how prices adjust in response to the output gap, given autonomous inflation. open-market operations \| Purchases and sales of government debt by a central bank. primary surplus \| The inverse of the primary deficit. opportunity cost \| What you must give up to
carry out an action. primary surplus \| The inverse of the primary deficit. opportunity cost \| What you must give up to carry out an action. output gap \| The difference between potential output and actual output. output gap \| The difference between potential output and actual output. overvalued \| The price of a currency is too high compared to the ratio of price levels in the two countries. overvalued \| The price of a currency is too high compared to the ratio of price levels in the two countries. prisoners’ dilemma \| There is a cooperative outcome that both players would prefer to the Nash equilibrium of the game. prisoners’ dilemma \| There is a cooperative outcome that both players would prefer to the Nash equilibrium of the game. private savings \| The net amount saved by households in the economy. private savings \| The net amount saved by households in the economy. private savings \| The net amount saved by households in the economy. Physical capital \| The total amount of machines and production facilities used in production. private savings \| The net amount saved by households in the economy. Process development \| Finding improvements in a firm’s operations and methods of manufacture to reduce the costs of production. Process development \| Finding improvements in a firm’s operations and methods of manufacture to reduce the costs of production. procyclical \| An economic variable that typically moves in the same direction as real GDP, increasing when GDP increases and decreasing when GDP decreases. procyclical \| An economic variable that typically moves in the same direction as real GDP, increasing when GDP increases and decreasing when GDP decreases. productivity \| The effectiveness of an economy for producing output. productivity \| The effectiveness of an economy for producing output. property rights \| An individual’s (or institution’s) legal right to make all decisions regarding the use of a particular resource. property rights \| An individual’s (or institution’s) legal right to make all decisions regarding the use of a particular resource. property rights \| An individual’s (or institution’s) legal right to make all decisions regarding the use of a particular resource. property rights \| An individual’s (or institution’s) legal right to make all decisions regarding the use of a particular resource. quantity equation \| An equation stating that the supply of money times the velocity of money equals nominal GDP. quantity equation \| An equation stating that the supply of money times the velocity of money equals nominal GDP. quantity theory of money \| A relationship among money, output, and prices that is used to study
inflation. quantity theory of money \| A relationship among money, output, and prices that is used to study inflation. rate of inflation \| The growth rate of the price index from one year to the next. rate of inflation \| The growth rate of the price index from one year to the next. real exchange rate \| A measure of the price of goods and services in one country relative to another when prices are expressed in a common currency. real exchange rate \| A measure of the price of goods and services in one country relative to another when prices are expressed in a common currency. real gross domestic product (real GDP) \| A measure of production that has been corrected for any changes in overall prices. real gross domestic product (real GDP) \| A measure of production that has been corrected for any changes in overall prices. Real gross domestic product (real GDP) \| A measure of production that has been corrected for any changes in overall prices. Real gross domestic product (real GDP) \| A measure of production that has been corrected for any changes in overall prices. real gross domestic product [real GDP] \| A measure of production that has been corrected for any changes in overall prices. 5 https://socialsci.libretexts.org/@go/page/71917 real gross domestic product [real GDP] \| A measure of production that has been corrected for any changes in overall prices. real interest rate \| The rate of return specified in terms of goods, not money. real interest rate \| The rate of return specified in terms of goods, not money. real interest rate \| The rate of return specified in terms of goods, not money. real interest rate \| The rate of return specified in terms of goods, not money. real interest rate \| The rate of return specified in terms of goods, not money. real interest rate \| The rate of return specified in terms of goods, not money. real interest rate \| The rate of return specified in terms of goods, not money. real interest rate \| The rate of return specified in terms of goods, not money. real variables \| A variable defined and measured in terms other than money, often in terms of real GDP. real variables \| A variable defined and measured in terms other than money, often in terms of real GDP. real wage \| The nominal wage (the wage in dollars) divided by the price level. real wage \| The nominal wage (the wage in dollars) divided by the price level. real wage \| The nominal wage (the wage in dollars) divided by the
price level. real wage \| The nominal wage (the wage in dollars) divided by the price level. real wage \| The nominal wage (the wage in dollars) divided by the price level. real wage \| The nominal wage (the wage in dollars) divided by the price level. real wage \| The nominal wage (the wage in dollars) divided by the price level. real wage \| The nominal wage (the wage in dollars) divided by the price level. real wage \| The nominal wage corrected for inflation. real wage \| The nominal wage corrected for inflation. reserve requirement \| The level of reserves that the monetary authority requires a bank to hold. reserve requirement \| The level of reserves that the monetary authority requires a bank to hold. reserves \| Deposits received by a bank that it must set aside rather than loan to firms and households. reserves \| Deposits received by a bank that it must set aside rather than loan to firms and households. Ricardian equivalence \| A balance that occurs when a decrease in taxes leads to an equal increase in private saving and thus no change in either the real interest rate or investment. Ricardian equivalence \| A balance that occurs when a decrease in taxes leads to an equal increase in private saving and thus no change in either the real interest rate or investment. risk premium \| A part of the interest rate needed to compensate the lender for the risk of default. risk premium \| A part of the interest rate needed to compensate the lender for the risk of default. savings rate \| The ratio of household savings to disposable income. savings rate \| The ratio of household savings to disposable income. Search theory \| A framework for understanding the flows of workers across periods of employment and unemployment along with job vacancies by firms. the creation of Search theory \| A framework for understanding the flows of workers across periods of employment and unemployment along with job vacancies by firms. the creation of Social infrastructure \| The general business climate, including any relevant features of the culture. Social infrastructure \| The general business climate, including any relevant features of the culture. Stabilization policy \| The use of monetary and fiscal policies to prevent large fluctuations in real GDP. Stabilization policy \| The use of monetary and fiscal policies to prevent large fluctuations in real GDP. standardized deficit (or structural deficit) \| A deficit that occurs when a government budget is in deficit because of expansionary fiscal policy. standardized deficit (or structural deficit) \| A deficit that occurs when a government budget is in deficit because of expansionary fiscal policy. Sticky prices \| Prices that
do not adjust immediately to shifts in supply and demand curves so that markets are not always in equilibrium. Sticky prices \| Prices that do not adjust immediately to shifts in supply and demand curves so that markets are not always in equilibrium. stock \| Any variable that can be measured in principle at an instant of time. stock \| Any variable that can be measured in principle at an instant of time. store of value \| Anything that can reliably be expected to maintain its worth over time. store of value \| Anything that can reliably be expected to maintain its worth over time. substitution effect \| As the real wage increases, households substitute away from toward consumption of goods and service and thus supply more labor. leisure substitution effect \| As the real wage increases, households substitute away from toward consumption of goods and service and thus supply more labor. leisure Supply and demand \| A framework that explains and predicts the equilibrium price and equilibrium quantity of a good. Supply and demand \| A framework that explains and predicts the equilibrium price and equilibrium quantity of a good. tax smoothing \| To reduce the distortionary effects of taxes, a government will finance some current spending by issuing debt to spread the tax burden over time. tax smoothing \| To reduce the distortionary effects of taxes, a government will finance some current spending by issuing debt to spread the tax burden over time. Taylor rule \| A rule for monetary policy in which the target real interest rate increases when inflation is too high and decreases when output is too low. Taylor rule \| A rule for monetary policy in which the target real interest rate increases when inflation is too high and decreases when output is too low. Taylor rule \| A rule for monetary policy in which the target real interest rate increases when inflation is too high and decreases when output is too low. Taylor rule \| A rule for monetary policy in which the target real interest rate increases when inflation is too high and decreases when output is too low. technology \| A catchall term comprising knowledge, social infrastructure, natural resources, and any other inputs to aggregate production that have not been included elsewhere. technology \| A catchall term comprising knowledge, social infrastructure, natural resources, and any other inputs to aggregate production that have not been included elsewhere. technology frontier \| Where the most advanced production technologies are available. technology frontier \| Where the most advanced production technologies are available. transfer technology \| The movement of knowledge and advanced production techniques across national borders. transfer technology \| The movement of knowledge and advanced production techniques across national borders. term structure of \| The relationship between the actual and expected returns on assets, which are
their maturities. identical except for interest rates term structure of \| The relationship between the actual and expected returns on assets, which are their maturities. identical except for interest rates time budget constraint \| The restriction that the sum of the time you spend on all your different activities must be exactly 24 hours each day. time budget constraint \| The restriction that the sum of the time you spend on all your different activities must be exactly 24 hours each day. tradable goods \| Goods that can be easily and cheaply transported from one place to another, which means they can be easily arbitraged. tradable goods \| Goods that can be easily and cheaply transported from one place to another, which means they can be easily arbitraged. transfers \| A cash payment from the government to individuals and firms. transfers \| A cash payment from the government to individuals and firms. transfers \| A cash payment from the government to individuals and firms. transfers \| A cash payment from the government to individuals and firms. uncovered interest parity \| A relationship between interest rates in two countries and the expected change in the exchange rate that holds when the return earned by investing in assets denominated in different currencies are equal. uncovered interest parity \| A relationship between interest rates in two countries and the expected change in the exchange rate that holds when the return earned by investing in assets denominated in different currencies are equal. unemployment duration \| The amount of time a typical worker spends searching for a new job. 6 https://socialsci.libretexts.org/@go/page/71917 unemployment duration \| The amount of time a typical worker spends searching for a new job. unemployment insurance \| A payment made by the government to those who are unemployed. unemployment insurance \| A payment made by the government to those who are unemployed. unemployment rate \| The number of unemployed individuals divided by the sum of the number employed and the number unemployed. unemployment rate \| The number of unemployed individuals divided by the sum of the number employed and the number unemployed. unemployment rate \| The percentage of people who are not currently employed but are actively seeking a job. unemployment rate \| The percentage of people who are not currently employed but are actively seeking a job. unemployment rate \| The number of unemployed individuals divided by the sum of the number employed and the number unemployed. unemployment rate \| The number of unemployed individuals divided by the sum of the number employed and the number unemployed. unit of account \| A standardized measure for economic transactions. unit of account \| A standardized measure for economic transactions. Unplanned inventory investment \| An increase in
inventories that comes about because firms have sold less than they anticipated. Unplanned inventory investment \| An increase in inventories that comes about because firms have sold less than they anticipated. velocity of money \| Nominal GDP divided by the money supply. velocity of money \| Nominal GDP divided by the money supply. wealth effect \| The effect on consumption of a change in wealth. wealth effect \| The effect on consumption of a change in wealth. yield curve \| The current annual return for assets of different maturities. yield curve \| The current annual return for assets of different maturities. zero lower bound \| The fact that the Fed cannot push nominal interest rates below zero. zero lower bound \| The fact that the Fed cannot push nominal interest rates below zero. 7 https://socialsci.libretexts.org/@go/page/71917 Detailed Licensing Overview Title: Principles of Macroeconomics (LibreTexts) Webpages: 117 Applicable Restrictions: Noncommercial All licenses found: CC BY-NC-SA 3.0: 94.9% (111 pages) Undeclared: 5.1% (6 pages) By Page Principles of Macroeconomics (LibreTexts) - CC BY-NC-SA 3.0 Front Matter - CC BY-NC-SA 3.0 TitlePage - CC BY-NC-SA 3.0 InfoPage - CC BY-NC-SA 3.0 Table of Contents - Undeclared Licensing - Undeclared 1: Economics: The Study of Choice - CC BY-NC-SA 3.0 1.1: Defining Economics - CC BY-NC-SA 3.0 1.2: The Field of Economics - CC BY-NC-SA 3.0 1.3: The Economists’ Tool Kit - CC BY-NC-SA 3.0 1.4: Review and Practice - CC BY-NC-SA 3.0 2: Confronting Scarcity: Choices in Production - CC BYNC-SA 3.0 2.1: Factors of Production - CC BY-NC-SA 3.0 2.2: The Production Possibilities Curve - CC BY-NCSA 3.0 2.3: Applications of the Production Possibilities Model - CC BY-NC-SA 3.0 2.4: Review and Practice - CC BY-NC-SA 3.0 3: Demand and Supply - CC BY-NC-SA 3
.0 3.1: Demand - CC BY-NC-SA 3.0 3.2: Supply - CC BY-NC-SA 3.0 3.3: Demand, Supply, and Equilibrium - CC BY-NCSA 3.0 3.4: Review and Practice - CC BY-NC-SA 3.0 4: Applications of Demand and Supply - CC BY-NC-SA 3.0 4.1: Putting Demand and Supply to Work - CC BYNC-SA 3.0 4.2: Government Intervention in Market Prices - Price Floors and Price Ceilings - CC BY-NC-SA 3.0 4.3: The Market for Health-Care Services - CC BYNC-SA 3.0 4.4: Review and Practice - CC BY-NC-SA 3.0 5: Macroeconomics: The Big Picture - CC BY-NC-SA 3.0 5.1: Growth of Real GDP and Business Cycles - CC BY-NC-SA 3.0 5.2: Price-Level Changes - CC BY-NC-SA 3.0 5.3: Unemployment - CC BY-NC-SA 3.0 5.4: Review and Practice - CC BY-NC-SA 3.0 6: Measuring Total Output and Income - CC BY-NC-SA 3.0 6.1: Measuring Total Output - CC BY-NC-SA 3.0 6.2: Measuring Total Income - CC BY-NC-SA 3.0 6.3: GDP and Economic Well-Being - CC BY-NC-SA 3.0 6.4: Review and Practice - CC BY-NC-SA 3.0 7: Aggregate Demand and Aggregate Supply - CC BYNC-SA 3.0 7.1: Aggregate Demand - CC BY-NC-SA 3.0 7.2: Aggregate Demand and Aggregate Supply: The Long Run and the Short Run - CC BY-NC-SA 3.0 7.3: Recessionary and Inflationary Gaps and LongRun Macroeconomic Equilibrium - CC BY-NC-SA 3.0 7.4: Review and Practice - CC BY-NC-SA 3.0 8: Economic Growth - CC BY-NC-SA 3.0 8.1: The Significance of Economic Growth - CC BYNC-SA 3.0 8.2: Growth and the Long-Run Aggregate Supply Curve - CC
BY-NC-SA 3.0 8.3: Determinants of Economic Growth - CC BY-NCSA 3.0 8.4: Review and Practice - CC BY-NC-SA 3.0 9: The Nature and Creation of Money - CC BY-NC-SA 3.0 9.1: What Is Money? - Undeclared 9.2: The Banking System and Money Creation - CC BY-NC-SA 3.0 9.3: The Federal Reserve System - CC BY-NC-SA 3.0 9.4: Review and Practice - CC BY-NC-SA 3.0 10: Financial Markets and the Economy - CC BY-NC-SA 3.0 1 https://socialsci.libretexts.org/@go/page/168979 10.1: The Bond and Foreign Exchange Markets - CC BY-NC-SA 3.0 10.2: Demand, Supply, and Equilibrium in the Money Market - CC BY-NC-SA 3.0 10.3: Review and Practice - CC BY-NC-SA 3.0 11: Monetary Policy and the Fed - CC BY-NC-SA 3.0 11.1: Monetary Policy in the United States - CC BYNC-SA 3.0 11.2: Problems and Controversies of Monetary Policy - CC BY-NC-SA 3.0 11.3: Monetary Policy and the Equation of Exchange - CC BY-NC-SA 3.0 11.4: Review and Practice - CC BY-NC-SA 3.0 12: Government and Fiscal Policy - CC BY-NC-SA 3.0 12.1: Government and the Economy - CC BY-NC-SA 3.0 12.2: The Use of Fiscal Policy to Stabilize the Economy - CC BY-NC-SA 3.0 12.3: Issues in Fiscal Policy - CC BY-NC-SA 3.0 12.4: Review and Practice - CC BY-NC-SA 3.0 13: Consumption and the Aggregate Expenditures Model - CC BY-NC-SA 3.0 13.1: Determining the Level of Consumption - CC BY-NC-SA 3.0 13.2: The Aggregate Expenditures Model - CC BYNC-SA 3.0 13.3: Aggregate Expenditures and Aggregate Demand - CC BY-NC-SA 3.0 13.4: Review and Practice
- CC BY-NC-SA 3.0 14: Investment and Economic Activity - CC BY-NC-SA 3.0 14.1: The Role and Nature of Investment - CC BYNC-SA 3.0 14.2: Determinants of Investment - CC BY-NC-SA 3.0 14.3: Investment and the Economy - CC BY-NC-SA 3.0 14.4: Review and Practice - CC BY-NC-SA 3.0 15: Net Exports and International Finance - CC BY-NCSA 3.0 15.1: The International Sector: An Introduction - CC BY-NC-SA 3.0 15.2: International Finance - CC BY-NC-SA 3.0 15.3: Exchange Rate Systems - CC BY-NC-SA 3.0 15.4: Review and Practice - CC BY-NC-SA 3.0 16: Inflation and Unemployment - CC BY-NC-SA 3.0 16.1: Relating Inflation and Unemployment - CC BYNC-SA 3.0 16.2: Explaining Inflation–Unemployment Relationships - CC BY-NC-SA 3.0 16.3: Inflation and Unemployment in the Long Run CC BY-NC-SA 3.0 16.4: Review and Practice - CC BY-NC-SA 3.0 17: A Brief History of Macroeconomic Thought and Policy - CC BY-NC-SA 3.0 17.1: The Great Depression and Keynesian Economics - CC BY-NC-SA 3.0 17.2: Keynesian Economics in the 1960s and 1970s CC BY-NC-SA 3.0 17.3: 32.3:. An Emerging Consensus: Macroeconomics for the Twenty-First Century - CC BY-NC-SA 3.0 17.4: Review and Practice - CC BY-NC-SA 3.0 18: Inequality, Poverty, and Discrimination - CC BY-NCSA 3.0 18.1: Income Inequality - CC BY-NC-SA 3.0 18.2: The Economics of Poverty - CC BY-NC-SA 3.0 18.3: The Economics of Discrimination - CC BY-NCSA 3.0 18.4: Review and Practice - CC BY-NC-SA 3.0 19: Economic Development - CC BY-NC-SA 3.0 19.1: The Nature and Challenge of Economic Development - CC
BY-NC-SA 3.0 19.2: Population Growth and Economic Development - CC BY-NC-SA 3.0 19.3: Keys to Economic Development - CC BY-NCSA 3.0 19.4: Review and Practice - CC BY-NC-SA 3.0 20: Socialist Economies in Transition - CC BY-NC-SA 3.0 20.1: The Theory and Practice of Socialism - CC BYNC-SA 3.0 20.2: Socialist Systems in Action - CC BY-NC-SA 3.0 20.3: Economies in Transition: China and Russia CC BY-NC-SA 3.0 20.4: Review and Practice - CC BY-NC-SA 3.0 Appendix A: Graphs in Economics - CC BY-NC-SA 3.0 21.1: Nonlinear Relationships and Graphs without Numbers - CC BY-NC-SA 3.0 21.2: Using Graphs and Charts to Show Values of Variables - CC BY-NC-SA 3.0 21.3: How to Construct and Interpret Graphs - CC BY-NC-SA 3.0 21.4: Extensions of the Aggregate Expenditures Model - CC BY-NC-SA 3.0 Appendix B: Extensions of the Aggregate Expenditures Model - CC BY-NC-SA 3.0 22.1: The Algebra of Equilibrium - CC BY-NC-SA 3.0 22.2: The Aggregate Expenditures Model and Fiscal Policy - CC BY-NC-SA 3.0 2 https://socialsci.libretexts.org/@go/page/168979 Back Matter - CC BY-NC-SA 3.0 Index - Undeclared Glossary - Undeclared Detailed Licensing - Undeclared 3 https://socialsci.libretexts.org/@go/page/168979................. 423 19 Eﬀects of Shocks in the Neoclassical Model 426 19.1 Equilibrium........................................ 426 14 19.2 The Eﬀects of Changes in Exogenous Variables on the Endogenous Variables.
428 19.2.1 Productivity Shock: Increase in At:..................... 429 19.2.2 Expected Future Productivity Shock: Increase in At+1.......... 431............... 434 19.2.3 Government Spending Shock: Increase in Gt: 19.2.4 An Increase in the Money Supply: Increase in Mt............. 441 19.2.5 Expected Future Inﬂation: Increase in πe t+1................. 442 19.2.6 Summary of Qualitative Eﬀects........................ 443 19.3 Summary......................................... 443 20 Taking the Neoclassical Model to the Data 448 20.1 Measuring the Business Cycle............................. 448 20.2 Can the Neoclassical Model Match Business Cycle Facts?............ 451 20.3 Is there Evidence that At Moves Around in the Data?.............. 454 20.4 Summary......................................... 457 21 Money, Inﬂation, and Interest Rates 460 21.1 Measuring the Quantity of Money........................... 460 21.1.1 How is the Money Supply Set?........................ 462 21.2 Money, the Price Level, and Inﬂation.
........................ 466 21.3 Inﬂation and Nominal Interest Rates......................... 472 21.4 The Money Supply and Real Variables........................ 474 21.5 Summary......................................... 477 22 Policy Implications and Criticisms of the Neoclassical Model 481 22.1 Criticisms......................................... 481 22.1.1 Measurement of TFP.............................. 482 22.1.2 What are these Productivity Shocks?.................... 484 22.1.3 Other Quantitative Considerations...................... 484 22.1.4 An Idealized Description of the Labor Market............... 484 22.1.5 Monetary Neutrality.............................. 485 22.1.6 The Role of Other Demand Shocks...................... 485 22.1.7 Perfect Financial Markets........................... 486 22.1.8 An Absence of Heterogeneity......................... 487 22.2
A Defense of the Neoclassical Model......................... 487 22.3 Summary......................................... 488 15 23 Open Economy Version of the Neoclassical Model 490 23.1 Exports, Imports, and Exchange Rates........................ 490 23.2 Graphically Characterizing the Equilibrium..................... 495................... 503 23.3 Eﬀects of Shocks in the Open Economy Model 23.3.1 Positive IS Shock................................ 503.................................. 507 23.3.2 Increase in At.................................. 510 23.3.3 Increase in Qt.................................. 513 23.3.4 Increase in Mt 23.3.5 Increase in P F.................................. 513 t 23.3.6 Summary of Qualitative Eﬀects........................ 513 23.4 Summary......................................... 514 V The
Short Run 517 24 The New Keynesian Demand Side: IS-LM-AD 521 24.1 The LM Curve...................................... 522 24.2 The IS Curve....................................... 525 24.3 The AD Curve...................................... 527 24.4 Summary......................................... 531 25 The New Keynesian Supply Side 534 25.1 The Neoclassical Model................................. 534 25.2 New Keynesian Model.................................. 539 25.2.1 Simple Sticky Price Model........................... 539 25.2.2 Partial Sticky Price Model........................... 543 25.3 Summary......................................... 552 26 Eﬀect of Shocks in the New Keynesian Model 554 26.1 The Neoclassical Model................................. 554.........................
...... 561 26.2 Simple Sticky Price Model............................... 571 26.3 Partial Sticky Price Model........ 583 26.4 Comparing the New Keynesian Model to the Neoclassical Model 26.5 Summary......................................... 585 27 Dynamics in the New Keynesian Model: Transition from Short Run to Medium Run 27.1 Simple Sticky Price Model 588............................... 589 16 27.2 Partial Sticky Price Model 27.1.1 A Non-Optimal Short Run Equilibrium................... 589 27.1.2 Dynamic Responses to Shocks......................... 592............................... 600 27.2.1 A Non-Optimal Short Run Equilibrium................... 600 27.2.2 Dynamic Responses to Shocks......................... 603 27.3 The Phillips Curve.................................... 612 27.3.1 Implications of the Phillips Curve for Monetary Policy.......... 616 27.3.2 The Possibility of Costless Disinﬂation................... 619 27.4 Summary.......................
.................. 623 28 Monetary Policy in the New Keynesian Model 626 28.1 Policy in the Partial Sticky Price Model....................... 627 28.2 The Case for Price Stability.............................. 634 28.3 The Natural Rate of Interest and Monetary Policy................. 639 28.4 The Taylor Rule..................................... 644 29 The Zero Lower Bound 647 29.1 The IS-LM-AD Curves with the ZLB......................... 649 29.2 Equilibrium Eﬀects of Shocks with a Binding ZLB................. 655 29.3 Why is the ZLB Costly?................................ 658 29.4 Fiscal Policy at the ZLB................................ 663 29.5 How to Escape the ZLB................................. 665 29.6 How to Avoid the ZLB................................. 667 29.7 Summary......................................... 668 30 Open Economy Version of the New Keynesian Model 671 30.1 Deriving the AD Curve
in the Open Economy................... 672 30.2 Equilibrium in the Open Economy Model...................... 674 30.3 Comparing the Open and Closed Economy Variants of the Model........ 675 30.3.1 Comparison in the Small Open Economy Version of the Model..... 681.... 685.................................. 685.................................. 687 30.5 Fixed Exchange Rates.................................. 689 30.6 Summary......................................... 696 30.4 Eﬀects of Foreign Shocks in the Open Economy New Keynesian Model 30.4.1 Increase in rF t 30.4.2 Increase in Qt 17 VI Money, Credit, Banking, and Finance 698 31 The Basics of Banking 701 31.1 Asymmetric Information: Adverse Selection and Moral Hazard......... 703 31.2 The Bank Balance Sheet................................ 708 31.3 Managing the Balance Sheet.............................. 711 31.3.1 Credit Risk.................................... 712 31.3.2 Liquidity Risk........................
.......... 714 31.4 Modern Banking and Shadow Banking........................ 717 31.5 Summary......................................... 723 32 The Money Creation Process 725 32.1 Some Deﬁnitions and Algebra............................. 725 32.2 Open Market Operations and the Simple Deposit Multiplier with T-Accounts 729 32.3 The Money Multiplier with Cash and Excess Reserve Holdings......... 736 32.4 Two Monetary Episodes: The Great Depression and Great Recession..... 749 32.4.1 Great Depression................................ 750 32.4.2 Great Recession................................. 753 32.4.3 Fractional Reserve Banking.......................... 757 32.5 Summary......................................... 758 33 A Model of Liquidity Transformation and Bank Runs 759 33.1 Model Assumptions................................... 759 33.2 Enter a Bank....................................... 761.............................
........... 764 33.3 Bank Runs 33.4 Policies to Deal with Bank Runs........................... 768 33.5 Summary......................................... 771 34 Bond Pricing and the Risk and Term Structures of Interest Rates 773 34.1 Bond Cash Flow Repayment Plans.......................... 774 34.1.1 Yield to Maturity................................ 775 34.2 Bond Pricing with No Uncertainty: A General Equilibrium Approach..... 779 34.3 Default Risk and the Risk Structure of Interest Rates............... 784 34.3.1 No Income Risk................................. 789 34.3.2 No Default Risk................................. 792 34.3.3 Income Risk and Default Risk......................... 794 34.4 Time to Maturity and the Term Structure of Interest Rates........... 797 18 34.4.1 No Uncertainty: The Expectations Hypothesis............... 800 34.4.2 Uncertainty and the Term Premium..................... 805 34.5 Conventional versus Unconventional Monetary Policy............... 818 34.5.1 A Model with Short and Long Term Riskless
Debt and Long Term Risky Debt.................................... 819 34.5.2 Conventional Monetary Policy........................ 826 34.5.3 Unconventional Policy............................. 828 34.6 Summary......................................... 836 35 The Stock Market and Bubbles 35.3.1 The Gordon Growth Model 838 35.1 Equity Pricing in a Two Period General Equilibrium Model........... 840 35.2 Comparing Diﬀerent Kinds of Stocks......................... 847 35.3 Moving Beyond Two Periods.............................. 852.......................... 862 35.4 Bubbles and the Role of the Terminal Condition.................. 864 35.4.1 A Numerical Example with Bubbles..................... 867 35.4.2 Should Monetary Policy Attempt to Prick Bubbles?........... 874 35.5 Equilibrium Stock Prices with Endogenous Production: the Neoclassical Model876 35.6 Summary......................................... 880 36 Financial Factors in a Macro Model 883 36.1 Incorporating an Exogenous Credit Spread........
............. 884 36.2 Detailed Foundations.................................. 887 36.3 Equilibrium Eﬀects of an Increase in the Credit Spread.............. 890 36.4 The Financial Accelerator............................... 892 36.5 Summary......................................... 900 37 Financial Crises and The Great Recession 902 37.1 Financial Crises: The Great Depression and Great Recession.......... 902 37.2 The Great Recession: Some More Speciﬁcs on the Run.............. 909 37.3 Thinking About the Great Recession in the AD-AS Model............ 916 37.4 Unconventional Policy Actions............................. 923 37.4.1 Federal Reserve Lending............................ 924 37.4.2 Fiscal Stimulus.................................. 927 37.4.3 Unconventional Monetary Policy....................... 930 37.5 Lingering Questions................................... 937 37.6 Summary................................
......... 938 19 VII Appendices 962 A Mathematical Appendix 963 A.1 Variables and Parameters................................ 963 A.2 Exponents and Logs................................... 964 A.3 Summations and Discounted Summations...................... 965 A.4 Growth Rates....................................... 967 A.5 Systems of Equations.................................. 969 A.6 Calculus.......................................... 970 A.7 Optimization....................................... 976 B Probability and Statistics Appendix 986 B.1 Measures of Central Tendency: Mean, Median, Mode............... 986 B.2 Expected Value...................................... 987 B.3 Measures of Dispersion: Variance and Standard Deviation............ 991 B.4 Measures of Association: Covariance and Correlation............... 994 C The Neoclassical Model with an Upward-Sloping Y s Curve 998 C.1 The Neoclassical Model with an Intertemporal Dimension to Labor Supply..
999 C.2 Eﬀects of Shocks with Upward-Sloping Y s...................... 1003 C.3 Sources of Output Fluctuations with an Upward-Sloping Y s Curve....... 1007 D The New Keynesian Model with Sticky Wages D.1 Equilibrium Eﬀects of Shocks in the Sticky Wage Model 1009............. 1014 D.1.1 Comparing the Sticky Wage Model to the Neoclassical Model...... 1024......................... 1028 D.2.1 A Non-Optimal Short Run Equilibrium................... 1028 D.2.2 Dynamic Responses to Shocks......................... 1030 D.2 Dynamics in the Sticky Wage Model E Replacing the LM Curve with the MP Curve 1038 E.1 The AD Curve when the MP Curve Replaces the LM Curve........... 1038............................... 1044 E.2 The Modiﬁed Supply Side............................... 1046 E.3 The IS-MP-AD-AS Model............ 1048 E.3.1 The Eﬀects of Shocks in the IS-MP-AD-AS Model 20 Part I Introduction 21 Part I serves as an introduction to the book and a review of materials from a principles course. Chapter 1 reviews some basics concerning national income and product accounts (NIPA), discusses the distinction between real and nominal variables and how to construct an aggregate price index, and discusses diﬀerent measures of labor market variables. Chapter 2 explains what an economic model is and why models are useful when thinking about
the economy, particularly at a high level of aggregation. Chapter 3 includes a brief discussion of the history of macroeconomics. In so doing, it provides some context for how modern macroeconomics as it is now practiced came to be. 22 Chapter 1 Macroeconomic Data In this chapter we deﬁne some basic macroeconomic variables and statistics and go over their construction as well as some of their properties. For those of you who took principles of macroeconomics, this should be a refresher. We start by describing what is perhaps the single most important economic indicator, GDP. 1.1 Calculating GDP Gross domestic product (GDP) is the current dollar value of all ﬁnal goods and services that are produced within a country within a given period of time. “Goods” are physical things that we consume (like a shirt) while “services” are things that we consume but which are not necessarily tangible (like education). “Final” means that intermediate goods are excluded from the calculation. For example, rubber is used to produce tires, which are used to produce new cars. We do not count the rubber or the tires in used to construct a new car in GDP, as these are not ﬁnal goods – people do not use the tires independently of the new car. The value of the tires is subsumed in the value of the newly produced car – counting both the value of the tires and the value of the car would “double count” the tires, so we only look at “ﬁnal” goods.1 “Current” means that the goods are valued at their current period market prices (more on this below in the discussion of the distinction between “real” and “nominal”). GDP is frequently used as a measure of the standard of living in an economy. There are many obvious problems with using GDP as a measure of well-being – as deﬁned, it does not take into account movements in prices versus quantities (see below); the true value to society of some goods or services may diﬀer from their market prices; GDP does not measure non-market activities, like meals cooked at home as opposed to meals served in a restaurant (or things that are illegal); it does not say anything about the distribution of resources among society; etc. Nevertheless, other measures of well-being have issues as well, so we will focus 1There are many
nuances in the NIPA accounts, and this example is no exception. Tires included in the production of a new car are not counted in GDP because these are not ﬁnal goods, but replacement tires sold at an auto shop for an already owned car are. More generally, depending on circumstances sometimes a good is an intermediate good and other times it is a ﬁnal good. 23 on GDP. Let there be n total ﬁnal goods and services in the economy – for example, cell phones (a good), haircuts (a service), etc. Denote the quantities of each good (indexed by i) produced in year t by yi,t for i = 1, 2,..., n and prices by pi,t. GDP in year t is the sum of prices times quantities: GDPt = p1,ty1,t + p2,ty2,t + ⋅ ⋅ ⋅ + pn,tyn,t = pi,tyi,t n ∑ i=1 As deﬁned, GDP is a measure of total production in a given period (say a year). It must also be equal to total income in a given period. The intuition for this is that the sale price of a good must be distributed as income to the diﬀerent factors of production that went into producing that good – i.e. wages to labor, proﬁts to entrepreneurship, interest to capital (capital is some factor of production, or input, that itself has to be produced and is not used up in the production process), etc. For example, suppose that an entrepreneur has a company that uses workers and chain-saws to produce ﬁrewood. Suppose that the company produces 1000 logs at $1 per log; pays its workers $10 per hour and the workers work 50 hours; and pays $100 to the bank, from which it got a loan to purchase the chain-saw. Total payments to labor are $500, interest is $100, and the entrepreneur keeps the remaining $400 as proﬁt. The logs contribute $1000 to GDP, $500 to wages, $100 to interest payments, and $400 to proﬁts, with $500 + $100 + $400 = $1,000. The so-called “expenditure” approach to GDP measures GDP as the sum of consumption, C; investment
, I; government expenditure, G; and net exports, N X. Net exports is equal to exports, X, minus imports, IM, where exports are deﬁned as goods and services produced domestically and sold abroad and imports are deﬁned as goods and services produced abroad and purchased domestically. Formally: GDPt = Ct + It + Gt + (Xt − IMt) (1.1) Loosely speaking, there are four broad actors in an aggregate economy: households, ﬁrms, government (federal, state, and local), and the rest of the world. We measure aggregate expenditure by adding up the spending on ﬁnal goods and services by each of these actors. Things that households purchase – food, gas, cars, etc. – count as consumption. Firms produce stuﬀ. Their expenditures on new capital, which is what is used to produce new goods (e.g. a bulldozer to help build roads), is what we call investment. Government expenditures includes everything the government spends either on buying goods (like courthouses, machine guns, etc.) or on services (including, in particular, the services provided by government employees). The latter half – basically counting government payments to workers as expenditure – is making use of the fact that income = expenditure from above, as there is no other feasible 24 way to “value” some government activities (like providing defense). This number does not include transfer payments (social security, Medicaid, etc.) and interest payments on debt from the government (which together amount to a lot). The reason transfer payments do not count in government expenditure is that these transfers do not, in and of themselves, constitute expenditure on new goods and services. However, when a retiree takes her Social Security payment and purchases groceries, or when a Medicaid recipient visits a doctor, those expenditures get counted in GDP. Finally, we add in net exports (more on this in a minute). In summary, what this identity says is that the value of everything produced, GDPt, must be equal to the sum of the expenditure by the diﬀerent actors in the economy. In other words, the total value of production must equal the total value of expenditure. So we shall use the words production, income, and expenditure interchangeably. If we want to sum up expenditure to get the total value of production, why do we subtract imports (IM in the notation above)? After all, GDP is a
measure of production in a country in a given period of time, while imports measure production from other countries. The reason is because our notion of GDP is the value of goods and services produced within a country; the expenditure categories of consumption, investment, and government spending do not distinguish between goods and services that are produced domestically or abroad. So, for example, suppose you purchase an imported Mercedes for $50,000. This causes C to go up, but should not aﬀect GDP. Since this was produced somewhere else, IM goes up by exactly $50,000, leaving GDP unaﬀected. Similarly, you could imagine a ﬁrm purchasing a Canadian made bulldozer – I and IM would both go up in equal amounts, leaving GDP unaﬀected. You could also imagine the government purchasing foreign-produced warplanes which would move G and IM in oﬀsetting and equal directions. As for exports, a Boeing plane produced in Seattle but sold to Qatar would not show up in consumption, investment, or government spending, but it will appear in net exports, as it should since it is a component of domestic production. There are a couple of other caveats that one needs to mention, both of which involve how investment is calculated. In addition to business purchases of new capital (again, capital is stuﬀ used to produce stuﬀ), investment also includes new residential construction and inventory accumulation. New residential construction is new houses. Even though households are purchasing the houses, we count this as investment. Why? At a fundamental level investment is expenditure on stuﬀ that helps you produce output in the future. A house is just like that – you purchase a house today (a “stock”), and it provides a “ﬂow” of beneﬁts for many years going forward into the future. There are many other goods that have a similar feature – we call these “durable” goods – things like cars, televisions, appliances, etc. At some level we ought to classify these as investment too, but for the purposes of national income 25 accounting, they count as consumption. From an economic perspective they are really more like investment; it is the distinction between “ﬁrm” and “household” that leads us to put new durable goods expenditures into consumption. However, even though residential homes are purchased by households, new home construction is counted as a component
of investment. Inventory “investment” is the second slightly odd category. Inventory investment is the accumulation (or dis-accumulation) of unsold, newly produced goods. For example, suppose that a company produced a car in 1999 but did not sell it in that year. It needs to count in 1999 GDP since it was produced in 1999, but cannot count in 1999 consumption because it has not been bought yet. Hence, we count it as investment in 1999, or more speciﬁcally inventory investment. When the car is sold (say in 2000), consumption goes up, but GDP should not go up. Here inventory investment would go down in exactly the same amount of the increase in consumption, leaving GDP unaﬀected. We now turn to looking at the data, over time, of GDP and its expenditure components. Figure 1.1 plots the natural log of GDP across time. These data are quarterly and begin in 1947.2 The data are also seasonally adjusted – unless otherwise noted, we want to look at seasonally adjusted data when making comparisons across time. The reason for this is that there are predictable, seasonal components to expenditure that would make comparisons between quarters diﬃcult (and would introduce some systematic “choppiness” into the plots – download the data and see for yourself). For example, there are predictable spikes in consumer spending around the holidays, or increases in residential investment in the warm summer months. When looking at aggregate series it is common to plot series in the natural log. This is nice because, as you can see in Appendix A, it means that we can interpret diﬀerences in the log across time as (approximately) percentage diﬀerences – reading oﬀ the vertical diﬀerence between two points in time is approximately the percentage diﬀerence of the variable over that period. For example, the natural log of real GDP increases from about 6.0 in 1955 to about 6.5 in 1965; this diﬀerence of 0.5 in the natural logs means that GDP increased by approximately 50 percent over this period. For reasons we will discuss more in detail below, plotting GDP without making a “correction” for inﬂation makes the series look smoother than the “real” series actually is. To the eye, one observes that GDP appeared to grow at a faster rate in the 1970s than it did later in the 1980s and
1990s. This is at least partially driven by higher inﬂation in the 1970s (again, more on this below). 2You can download the data for yourselves from the Bureau of Economic Analysis. 26 Figure 1.1: Logarithm of Nominal GDP Figure 1.2 plots the components of GDP, expressed as shares of total GDP. We see that consumption expenditures account for somewhere between 60-70 percent of total GDP, making consumption by far the biggest component of aggregate spending. This series has trended up a little bit over time; this upward trend is largely mirrored by a downward trend in net exports. At the beginning of the post-war sample we exported more than we imported, so that net exports were positive (but nevertheless still a small fraction of overall GDP). As we’ve moved forward into the future net exports have trended down, so that we now import more than we export. Investment is about 15 percent of total GDP. Even though this is a small component, visually you can see that it appears quite volatile relative to the other components. This is an important point to which we shall return later. Finally, government spending has been fairly stable at around 20 percent of total GDP. The large increase very early in the sample has to do with the Korean War and the start of the Cold War. 27 56789105055606570758085909500051015 Figure 1.2: GDP Components as a Share of Total GDP 1.2 Real versus Nominal Measured GDP could change either because prices or quantities change. Because we are interested in the behavior of quantities (which is ultimately what matters for well-being), we would like a measure of production (equivalent to income and expenditure) that removes the inﬂuence of price changes over time. This is what we call real GDP. Subject to the caveat of GDP calculation below, in principle real prices are denominated in units of goods, whereas nominal prices are denominated in units of money. Money is anything which serves as a unit of account. As we’ll see later in the book, money solves a bartering problem and hence makes exchange much more eﬃcient. To make things clear, let’s take a very simple example. Suppose you only have one good, call it y. People trade this good using money, call it M. We are going to set money to be the numeraire: it serves as the “unit of account,”
i.e. the units by which value is measured. Let p be the price of goods relative to money – p tells you how many units of M you need to buy one unit of y. So, if p = 1.50, it says that it takes 1.50 units of money (say dollars) to buy a good. Suppose an economy produces 10 units of y, e.g. y = 10, and the price of goods in terms of money is p = 1.50. This means that nominal output is 15 units of money (e.g. 1.50 × 10, or p ⋅ y). It is nominal because it is denominated in units of M – it says how many units of M the quantity of y is worth. The real value is of course just y – that is the quantity 28.58.60.62.64.66.68.705055606570758085909500051015Consumption/GDP.12.14.16.18.20.225055606570758085909500051015Investment/GDP.14.16.18.20.22.24.265055606570758085909500051015Government/GDP-.06-.04-.02.00.02.04.065055606570758085909500051015Net Exports/GDP of goods, denominated in units of goods. To get the real from the nominal we just divide by the price level: Real = Nominal Price = py p = y. Ultimately, we are concerned with real variables, not nominal variables. What we get utility from is how many apples we eat, not whether we denominate one apple as one dollar, 100 Uruguayan pesos, or 1.5 euros. Going from nominal to real becomes a little more diﬃcult when we go to a multi-good world. You can immediately see why – if there are multiple goods, and real variables are denominated in units of goods, which good should we use as the numeraire? Suppose you have two goods, y1 and y2. Suppose that the price measured in units of money of the ﬁrst good is p1 and the price of good 2 is p2. The nominal quantity of goods is: Nominal = p1y1 + p2y2. Now, the real relative price between y1 and y2 is
just the ratio of nominal prices, p1/p2. p1 is “dollars per unit of good 1” and p2 is “dollars per unit of good 2”, so the ratio of the prices is “units of good 2 per units of good 1.” Formally: = p1 p2 $ good 1 $ good 2 = good 2 good 1 (1.2) In other words, the price ratio tells you how many units of good 2 you can get with one unit of good 1. For example, suppose the price of apples is $5 and the price of oranges is $1. The relative price is 5 – you can get ﬁve oranges by giving up one apple. You can, of course, deﬁne the relative price the other way as 1/5 – you can buy 1/5 of an apple with one orange. We could deﬁne real output (or GDP) in one of two ways: in units of good 1 or units of good 2: y2 Real1 = y1 + p2 p1 Real2 = p1 y1 + y2 p2 (Units are good 1) (Units are good 2). As you can imagine, this might become a little unwieldy, particularly if there are many goods. It would be like walking around saying that real GDP is 14 units of Diet Coke, or 6 29 cheeseburgers, if Diet Coke or cheeseburgers were used as the numeraire. As such, we have adopted the convention that we use money as the numeraire and report GDP in nominal terms as dollars of output (or euros or lira or whatever). But that raises the issue of how to track changes in GDP across time. In the example above, what if both p1 and p2 doubled between two periods, but y1 and y2 stayed the same? Then nominal GDP would double as well, but we’d still have the same quantity of stuﬀ. Hence, we want a measure of GDP that can account for this, but which is still measured in dollars (as opposed to units of one particular good). What we typically call “real” GDP in the National Income and Products Accounts is what would more accurately be called “constant dollar GDP.” Basically, one arbitrarily picks a year as a baseline. Then in subsequent years one multiplies quantities by base year prices. If
year t is the base year, then what we call real GDP in year t + s is equal to the sum of quantities of stuﬀ produced in year t + s weighted by the prices from year t. This diﬀers from nominal GDP in that base year prices are used instead of current year prices. Let Yt+s denote real GDP in year t + s, s = 0, 1, 2,.... Let there be n distinct goods produced. For quantities of goods y1,t+s, y2,t+s,..., yn,t+s, we have: Yt = p1,ty1,t + p2,ty2,t + ⋅ ⋅ ⋅ + pn,tyn,t Yt+1 = p1,ty1,t+1 + p2,ty2,t+1 + ⋅ ⋅ ⋅ + pn,tyn,t+1 Yt+2 = p1,ty1,t+2 + p2,ty2,t+2 + ⋅ ⋅ ⋅ + pn,tyn,t+2. Or, more generally, using the summation notation covered in Appendix A: Yt+h = n ∑ i=1 pi,tyi,t+h for h = 0, 1, 2. From this we can implicitly deﬁne a price index (an implicit price index) as the ratio of nominal to real GDP in a given year: = 1 Pt = p1,ty1,t + p2,ty2,t + ⋅ ⋅ ⋅ + pn,tyn,t p1,ty1,t + p2,ty2,t + ⋅ ⋅ ⋅ + pn,tyn,t Pt+1 = p1,t+1y1,t+1 + p2,t+1y2,t+1 + ⋅ ⋅ ⋅ + pn,t+1yn,t+1 Pt+2 = p1,t+2y1,t+2 + p2,t+2y2,t+2 + ⋅ ⋅ ⋅ + pn,t+2yn,t+2 p1,ty1,
t+1 + p2,ty2,t+1 + ⋅ ⋅ ⋅ + pn,tyn,t+1 p1,ty1,t+2 + p2,ty2,t+2 + ⋅ ⋅ ⋅ + pn,tyn,t+2. Or, more succinctly, Pt+h = ∑n i=1 pi,t+hyi,t+h ∑n i=1 pi,tyi,t+h for h = 0, 1, 2. 30 A couple of things are evident here. First, we have normalized real and nominal GDP to be the same in the base year (which we are taking as year t). This also means that we are normalizing the price level to be one in the base year (what you usually see presented in national accounts is the price level multiplied by 100). Second, there is an identity here that nominal GDP divided by the price level equals real GDP. If prices on average are rising, then nominal GDP will go up faster than real GDP, so that the price level will rise. A problem with this approach is that the choice of the base year is arbitrary. This matters to the extent that the relative prices of goods vary over time. To see why this might be a problem, let us consider a simple example. Suppose that an economy produces two goods: haircuts and computers. In year t, let the price of haircuts be $5 and computers by $500, and there be 100 hair cuts and 10 computers produced. In year t + 1, suppose the price of haircuts is $10, but the price of computers is now $300. Suppose that there are still 100 haircuts produced but now 20 computers. Nominal GDP in year t is $5,500, and in year t + 1 it is $7,000. If one uses year t as the base year, then real GDP equals nominal in year t, and real GDP in t + 1 is $10,500. Using year t as the base year, one would conclude that real GDP grew by about 91 percent from t to t + 1. What happens if we instead use year t + 1 as the base year? Then real GDP in year t + 1 would be $7,000, and in year t real GDP would be $4,000. One would conclude that real GDP grew between t and t + 1 by 75 percent, which
is substantially diﬀerent than the 91 percent one obtains when using t as the base year. To deal with this issue, statisticians have come up with a solution that they call chainweighting. Essentially they calculate real GDP in any two consecutive years (say, 1989 and 1990) two diﬀerent ways: once using 1989 as the base year, once using 1990 as the base year. Then they calculate the growth rate of real GDP between the two years using both base years and take the geometric average of the two growth rates. Chain-weighting is a technical detail that we need not concern ourselves with much, but it does matter in practice, as relative prices of goods have changed a lot over time. For example, computers are far cheaper in relative terms now than they were 10 or 20 years ago. Throughout the book we will be mainly dealing with models in which there is only one good – we’ll often refer to it as fruit, but it could be anything. Fruit is a particularly convenient example for reasons which will become evident later in the book. This is obviously an abstraction, but it’s a useful one. With just one good, real GDP is just the amount of that good produced. Hence, as a practical matter we won’t be returning to these issues of how to measure real GDP in a multi-good world. Figure 1.3 below plots the log of real GDP across time in the left panel. Though considerably less smooth than the plot of log nominal GDP in Figure 1.1, the feature that sticks out most from this ﬁgure is the trend growth – you can approximate log real GDP 31 pretty well across time with a straight line, which, since we are looking at the natural log, means roughly constant trend growth across time. We refer to this straight line as a “trend.” This is meant to capture the long term behavior of the series. The average growth rate (log ﬁrst diﬀerence) of quarterly nominal GDP from 1947-2016 was 0.016, or 1.6 percent. This translates into an annualized rate (what is most often reported) of about 6 percent (approximately 1.6 × 4). The average growth rate of real GDP, in contrast, is signiﬁcantly lower at about 0.008, or 0.8 percent per quarter, translating into about 3.2 percent at an annualized rate. From the identities above, we
know that nominal GDP is equal to the price level times real GDP. As the growth rate of a product is approximately equal to the sum of the growth rates, growth in nominal GDP should approximately equal growth in prices (inﬂation) plus growth in real GDP. Figure 1.3: Real GDP Figure 1.4 plots the log GDP deﬂator and inﬂation (the growth rate or log ﬁrst diﬀerence of the GDP deﬂator) in the right panel. On average inﬂation has been about 0.008, or 0.8 percent per quarter, which itself translates to about 3 percent per year. Note that 0.008 + 0.008 = 0.016, so the identity appears to work. Put diﬀerently, about half of the growth in nominal GDP is coming from prices, and half is coming from increases in real output. It is worth pointing out that there has been substantial heterogeneity across time in the behavior of inﬂation – inﬂation was quite high and volatile in the 1970s but has been fairly low and stable since then. 32 7.58.08.59.09.510.05055606570758085909500051015Real GDPTrendLog real GDP and its trend-.20-.15-.10-.05.00.05.105055606570758085909500051015Detrended real GDP Figure 1.4: GDP Deﬂator Turning our focus back to the real GDP graph, note that the blips are very minor in comparison to the trend growth. The right panel plots “detrended” real GDP, which is deﬁned as actual log real GDP minus its trend. In other words, detrended GDP is what is left over after we subtract the trend from the actual real GDP series. The vertical gray shaded areas are “recessions” as deﬁned by the National Bureau of Economic Research. There is no formal deﬁnition of a recession, but loosely speaking they deﬁne a recession as two or more quarters of a sustained slowdown in overall economic activity. For most of the recession periods, in the left plot we can see GDP declining if we look hard enough. But even in the most recent recession (oﬃcial dates 2007Q4–2009Q2), the
decline is fairly small in relation to the impressive trend growth. You can see the “blips” much more clearly in the right plot. During most of the observed recessions, real GDP falls by about 5 percentage points (i.e. 0.05 log points) relative to trend. The most recent recession really stands out in this respect, where we see GDP falling by more than 10 percent relative to trend. A ﬁnal thing to mention before moving on is that at least part of the increase in real GDP over time is due to population growth. With more people working, it is natural that we will produce more products and services. The question from a welfare perspective is whether there are more goods and services per person. For this reason, it is also quite common to look at “per capita” measures, which are series divided by the total population. Population growth has been pretty smooth over time. Since the end of WW2 it has averaged about 0.003 per quarter, or 0.3 percent, which translates to about 1.2 percent per year. Because population growth is so smooth, plotting real GDP per capita will produce a similar looking ﬁgure to that shown in Figure 1.3, but it won’t grow as fast. Across time, the average growth rate of real GDP per capita has been 0.0045, 0.45 percent, or close to 2 percent per year. 33 0204060801001205055606570758085909500051015GDP Deflator-.01.00.01.02.03.045055606570758085909500051015Inflation - GDP Deflator Doing a quick decomposition, we can approximate the growth rate of nominal GDP as the sum of the growth rates of prices, population, and real GDP per capita. This works out to 0.008 + 0.003 + 0.0045 = 0.0155 ≈ 0.016 per quarter, so again the approximation works out well. At an annualized rate, we’ve had population growth of about 1.2 percent per year, price growth of about 3.2 percent per year, and real GDP per capita growth of about 2 percent per year. Hence, if you look at the amount of stuﬀ we produce per person, this has grown by about 2 percent per year since 1947. 1.3 The Consumer Price Index The consumer price index (CPI) is
another popular macro variable that gets mentioned a lot in the news. When news commentators talk about “inﬂation” they are usually referencing the CPI. The CPI is trying to measure the same thing as the GDP deﬂator (the average level of prices), but does so in a conceptually diﬀerent way. The building block of the CPI is a “consumption basket of goods.” The Bureau of Labor Statistics (BLS) studies buying habits and comes up with a “basket” of goods that the average household consumes each month. The basket includes both diﬀerent kinds of goods and diﬀerent quantities. The basket may include 12 gallons of milk, 40 gallons of gasoline, 4 pounds of coﬀee, etc. Suppose that there are N total goods in the basket, and let xi denote the amount of good i (i = 1,..., N ) that the average household consumes. The total price of the basket in any year t is just the sum of the prices in that year times the quantities. Note that the quantities are held ﬁxed and hence do not get time subscripts – the idea is to have the basket not change over time: Costt = p1,tx1 + p2,tx2 + ⋅ ⋅ ⋅ + pN,txN. The CPI in year t, call it P cpi t, is the ratio of the cost of the basket in that year relative to the cost of the basket in some arbitrary base year, b: P cpi t = Costt Costb = p1,tx1 + p2,tx2 + ⋅ ⋅ ⋅ + pN,txN p1,bx1 + p2,bx2 + ⋅ ⋅ ⋅ + pN,bxN ∑N ∑N i=1 pi,txi i=1 pi,bxi =. As in the case of the GDP deﬂator, the choice of the base year is arbitrary, and the price level will be normalized to 1 in that year (in practice they multiply the number by 100 when 34 presenting the number). The key thing here is that the basket – both the goods in the basket and the quantities – are held ﬁxed across time (of course in practice the basket is periodically rede�
�ned). The idea is to see how the total cost of consuming a ﬁxed set of goods changes over time. If prices are rising on average, the CPI will be greater than 1 in years after the base year and less than 1 prior to the base year (as with the implicit price deﬂator it is common to see the CPI multiplied by 100). Figure 1.5 plots the natural log of the CPI across time. It broadly looks similar to the GDP deﬂator – trending up over time, with an acceleration in the trend in the 1970s and something of a ﬂattening in the early 1980s. There are some diﬀerences, though. For example, at the end of 2008 inﬂation as measured by the CPI went quite negative, whereas it only dropped to about zero for the GDP deﬂator. On average, the CPI gives a higher measure of inﬂation relative to the deﬂator and it is more volatile. For the entire sample, the average inﬂation by the GDP deﬂator is 0.8 percent per quarter (about 3.2 percent annualized); for the CPI it is 0.9 percent per quarter (about 3.6 percent annualized). The standard deviation (a measure of volatility) of deﬂator inﬂation is 0.6 percent, while it is 0.8 percent for the CPI. Figure 1.5: CPI The reason for these diﬀerences gets to the basics of how the two indices are constructed and what they are intended to measure. A simple way to remember the main diﬀerence is that the CPI ﬁxes base year quantities and uses updated prices, whereas the deﬂator is based on the construction of constant dollar GDP, which ﬁxes base year prices and uses updated quantities. The ﬁxing of quantities is one of the principal reasons why the CPI gives a higher measure of inﬂation. From principles of microeconomics we know that when relative prices change, people will tend to substitute away from relatively more expensive goods and into relatively cheaper goods – the so-called substitution eﬀect. By ﬁxing quantities, the CPI 35 0501001502002505055606570758085909500051015Consumer Price Index-.020-.015-.010-.005.000.
005.010.015.0205055606570758085909500051015Inflation - CPI does not allow for this substitution away from relatively expensive goods. To the extent that relative prices vary across time, the CPI will tend to overstate changes in the price of the basket. It is this substitution bias that accounts for much of the diﬀerence between inﬂation as measured by the CPI and the deﬂator. There are other obvious diﬀerences – the CPI does not include all goods produced in a country, and the CPI can include goods produced in other countries. Because the deﬂator is based on what the country actually produces, whereas the CPI is based on what the country consumes (which are diﬀerent constructs due to investment, exports, and imports), it follows that if a country produces much more of a particular product than it consumes, then this product will have a bigger impact on the implicit price deﬂator than on the CPI. For getting a sense of overall price inﬂation in US produced goods, the GDP deﬂator is thus preferred. For getting a sense of nominal changes in the cost of living for the average household, the CPI is a good measure. Chain weighting can also be applied to the CPI. As described above in the context of the GDP deﬂator, chain-weighting attempts to limit the inﬂuence of the base year. This is an attempt to deal with substitution biases in a sense because relative price changes will result in the basket of goods that the typical household consumes changing. Whether to chain-weight or not, and what kind of price index to use to index government transfer payments like Social Security, is a potentially important political issue. If inﬂation is really 2 percent per year, but the price index used to update Social Security payments measures inﬂation (incorrectly) at 3 percent per year, then Social Security payments will grow in real terms by 1 percent. While this may not seem like much in any one year, over time this can make a big diﬀerence for the real burden of Social Security transfers for a government. 1.4 Measuring the Labor Market One of the key areas on which the press is focused is the labor market. This usually takes the form of talking about the unemployment rate, but there are other ways to measure the “strength” or “
health” of the aggregate labor market. The unemployment rate is nevertheless a fairly good indicator of the overall strength of the economy – it tends to be elevated in “bad” times and low in “good” times. An economy’s total labor input is a key determinant of how much GDP it can produce. What is relevant for how much an economy produces is the size of the total labor input. There are two dimensions along which we can measure labor input – the extensive margin (bodies) and the intensive margin (amount of time spent working per person). Deﬁne L as the total population, E as the number of people working (note that E ≤ L), and h as the average number of hours each working person works (we’ll measure the unit of time as an 36 hour, but could do this diﬀerently, of course). Total hours worked, N, in an economy are then given by: N = h × E. Total hours worked is the most comprehensive measure of labor input in an economy. Because of diﬀerences and time trends in population, we typically divide this by L to express this as total hours worked per capita (implicitly per unit of time, i.e. a year or a quarter). This measure represents movements in two margins – average hours per worker and number of workers per population. Denote hours per capita as n = N /L: n = h × E L. As you may have noticed, the most popular metric of the labor market in the press is the unemployment rate. To deﬁne the unemployment rate we need to introduce some new concepts. Deﬁne the labor force, LF, as everyone who is either (i) working or (ii) actively seeking work. Deﬁne U as the number of people who are in the second category – looking for work but not currently working. Then: LF = E + U. Note that LF ≤ L. We deﬁne the labor force participation rate, lf p, as the labor force divided by the total working age population: lf p = LF L. Deﬁne the unemployment rate as the ratio of people who are unemployed divided by the labor force: u = U LF = U U + E. Figure 1.6 plot these diﬀerent measures of the labor market: (i) the unemployment rate; (ii) the employment to population
ratio, E L ; (iii) the natural log of average hours worked per person; (iv) the labor force participation rate; and (iv) log hours worked per capita, n.3 To get an idea for how these series vary with output movements, we included NBER “recession 3Note that there is no natural interpretation of the units of the graphs of average hours per worker and total hours per capita. The underlying series are available in index form (i.e. unitless, normalized to be 100 in some base year) and are then transformed via the natural log. 37 dates” as indicated by the shaded gray bars. Figure 1.6: Labor Market Variables A couple of observations are in order. First, hours worked per capita ﬂuctuates around a roughly constant mean – in other words, there is no obvious trend up or down. This would indicate that individuals are working about as much today as they did ﬁfty years ago. But the measure of hours worked per capita masks two trends evident in its components. The labor force participation rate (and the employment-population ratio) have both trended up since 1950. This is largely driven by women entering the labor force. In contrast, average hours per worker has declined over time – this means that, conditional on working, most people work a shorter work week now than 50 years ago (the units in the ﬁgure are log points of an index, but the average workweek itself has gone from something like 40 hours per week to 36). So the lack of a trend in total hours worked occurs because the extra bodies in the labor force have made up for the fact that those working are working less on average. In terms of movements over the business cycle, these series display some of the properties you might expect. Hours worked per capita tends to decline during a recession. For example, from the end of 2007 (when the most recent recession began) to the end of 2009, hours worked per capita fell by about 10 percent. The unemployment rate tends to increase during recessions – in the most recent one, it increased by about 5-6 percentage points, from around 5 percent to a maximum of 10 percent. Average hours worked tends to also decline during recessions, but this movement is small and does not stand out relative to the trend. The 38 2345678910115055606570758085909500051015Unemployment Rate5456586062646650556065707580859
09500051015Employement-Population Ratio5860626466685055606570758085909500051015Labor Force Participation Rate5.705.755.805.855.905.955055606570758085909500051015Hours Per Capita4.604.624.644.664.684.704.724.744.764.785055606570758085909500051015Average Hours employment to population ratio falls during recessions, much more markedly than average hours. In the last several recessions, the labor force participation rate tends to fall (which is sometimes called the “discouraged worker” phenomenon, to which we will return below), with this eﬀect being particularly pronounced (and highly persistent) around the most recent recession. In spite of its popularity, the unemployment rate is a highly imperfect measure of labor input. The unemployment rate can move because (i) the number of unemployed changes or (ii) the number of employed changes, where (i) does not necessarily imply (ii). For example, the number of unemployed could fall if some who were oﬃcially unemployed quit looking for work, and are therefore counted as leaving the labor force, without any change in employment and hours. We typically call such workers “discouraged workers” – this outcome is not considered a “good” thing, but it leads to the unemployment rate falling. Another problem is that the unemployment rate does not say anything about intensity of work or part time work. For example, if all of the employed persons in an economy are switched to part time, there would be no change in the unemployment rate, but most people would not view this change as a “good thing” either. In either of these hypothetical scenarios, hours worked per capita is probably a better measure of what is going on in the aggregate labor market. In the case of a worker becoming “discouraged,” the unemployment rate dropping would be illusory, whereas hours worked per capita would be unchanged. In the case of a movement from full time to part time, the unemployment rate would not move, but hours per capita would reﬂect the downward movement in labor input. For these reasons the unemployment rate is a diﬃcult statistic to interpret. As a measure of total labor input, hours per capita is a preferred measure. For these reasons, many economists often focus on
hours worked per capita as a measure of the strength of the labor market. For most of the chapters in this book, we are going to abstract from unemployment, instead focusing on how total labor input is determined in equilibrium (without really diﬀerentiating between the intensive and extensive margins). It is not trivial to think about the existence of unemployment in frictionless markets – there must be some friction which prevents individuals looking for work from meeting up with ﬁrms who are looking for workers. However, later, in Chapter 17 we study a model that can be used to understand why an economy can simultaneously have ﬁrms looking for workers and unemployed workers looking for ﬁrms. Frictions in this setting can result in potential matches not occurring, resulting in unemployment. 39 1.5 Summary • Gross Domestic Product (GDP) equals the dollar value of all goods and services produced in an economy over a speciﬁc unit of time. The revenue from production must be distributed to employees, investors, payments to banks, proﬁts, or to the government (as taxes). Every dollar a business or person spends on a produced good or service is divided into consumption, investment, or government spending. • GDP is an identity in that the dollar value of production must equal the dollar value of all expenditure which in turn must equal the dollar value of all income. For this identity to hold, net exports must be added to expenditure since the other expenditure categories do not discriminate over where a consumed good was produced. • GDP may change over time because prices change or output changes. Changes in output are what we care about for welfare. To address this, real GDP uses constant prices over time to measure changes in output. • Changes in prices indexes and deﬂators are a way to measure inﬂation and deﬂation. A problem with commonly uses price indexes like the consumer price index is that they overstate inﬂation on average. • The most comprehensive measure of the labor input is total hours. Total hours can change because the number of workers are changing or because the average hours per worker changes. Other commonly used metrics of the labor market include hours per capita, the unemployment rate, and the labor force participation rate. Key Terms • Nominal GDP • Real GDP • GDP price deﬂator • Numeraire • Chain weighting • Consumer Price Index • Substitution bias • Unemployment rate • Labor force participation rate 40 Questions for Review 1.
Explain why the three methods of calculating GDP are always equal to each other. 2. Why are intermediate goods subtracted when calculating GDP under the production method? 3. Why are imports subtracted when calculating GDP under the expenditure method? 4. Discuss the expenditure shares of GDP over time. Which ones have gotten bigger and which ones have gotten smaller? 5. Explain the diﬀerence between real and nominal GDP. 6. Discuss the diﬀerences between the CPI and the GDP deﬂator. 7. Discuss some problems with using the unemployment rate as a barometer for the health of the labor market. 8. Hours worked per worker has declined over the last 50 years yet hours per capita have remained roughly constant. How is this possible? Exercises 1. An economy produces three goods: houses, guns, and apples. The price of each is $1. For the purposes of this problem, assume that all exchange involving houses involves newly constructed houses. (a) Households buy 10 houses and 90 apples, eating them. The government buys 10 guns. There is no other economic activity. What are the values of the diﬀerent components of GDP (consumption, investment, government spending, exports/imports)? (b) The next year, households buy 10 houses and 90 apples. The government buys 10 guns. Farmers take the seeds from 10 more apples and plant them. Households then sell 10 apples to France for $1 each and buy 10 bananas from Canada for $2 each, eating them too. What are the values of the components of GDP? (c) Return to the economy in part 1a. The government notices that the two richest households consume 40 apples each, while the ten poorest consume one each. It levies a tax of 30 apples on each of the rich households, and gives 6 apples each to the 10 poorest households. All 41 other purchases by households and the government are the same as in (a). Calculate the components of GDP. 2. Suppose the unemployment rate is 6%, the total working-age population is 120 million, and the number of unemployed is 3.5 million. Determine: (a) The participation rate. (b) The size of the labor force. (c) The number of employed workers. (d) The Employment-Population rate. 3. Suppose an economy produces steel, wheat, and oil. The steel industry produces $100,000 in revenue, spends $4,000 on oil, $10,
000 on wheat, pays workers $80,000. The wheat industry produces $150,000 in revenue, spends $20,000 on oil, $10,000 on steel, and pays workers $90,000. The oil industry produces $200,000 in revenue, spends $40,000 on wheat, $30,000 on steel, and pays workers $100,000. There is no government. There are neither exports nor imports, and none of the industries accumulate or deaccumulate inventories. Calculate GDP using the production and income methods. 4. This question demonstrates why the CPI may be a misleading measure of inﬂation. Go back to Micro Theory. A consumer chooses two goods x and y to minimize expenditure subject to achieving some target level of utility, ¯u. Formally, the consumer’s problem is min x,y E = pxx + pyy s.t. ¯u = xαyβ Total expenditure equals the price of good x times the number of units of x purchased plus the price of good y times the number of units of y purchased. α and β are parameters between 0 and 1. px and py are the dollar prices of the two goods. All the math required for this problem is contained in Appendix A. (a) Using the constraint, solve for x as a function of y and ¯u. Substitute your solution into the objective function. Now you are choosing only one variable, y, to minimize expenditure. (b) Take the ﬁrst order necessary condition for y. (c) Show that the second order condition is satisﬁed. Note, this is a one 42 variable problem. (d) Use your answer from part b to solve for the optimal quantity of y, y∗. y∗ should be a function of the parameters α and β and the exogenous variables, px, py and ¯u. Next, use this answer for y∗ and your answer from part a to solve for the optimal level of x, x∗. Note, the solutions of endogenous variables, x∗ and y∗ in this case, only depend on parameters and exogenous variables, not endogenous variables. (e) Assume α = β = 0.5 and ¯u = 5. In the year 2000, px = py = $10. Calculate 2000 and total expenditure, E2000. We will use these quantities as 2000, y∗ x�
� our “consumption basket” and the year 2000 as our base year. (f) In 2001, suppose py increases to $20. Using the consumption basket from part e, calculate the cost of the consumption basket in 2001. What is the inﬂation rate? (g) Now use your results from part d to calculate the 2001 optimal quantities 2001 and total expenditures, E2001. Calculate the percent x∗ 2001 and y∗ change between expenditures in 2000 and 2001. (h) Why is the percent change in expenditures less than the percent change in the CPI? Use this to explain why the CPI may be a misleading measure of the cost of living. 5. [Excel Problem] Download quarterly, seasonal adjusted data on US real GDP, personal consumption expenditures, and gross private domestic investment for the period 1960Q1-2016Q2. You can ﬁnd these data in the BEA NIPA Table 1.1.6, “Real Gross Domestic Product, Chained Dollars”. (a) Take the natural logarithm of each series (“=ln(series)”) and plot each against time. Which series appears to move around the most? Which series appears to move the least? (b) The growth rate of a random variable x, between dates t − 1 and t is deﬁned as gx t = xt − xt−1 xt−1. Calculate the growth rate of each of the three series (using the raw series, not the logged series) and write down the average growth rate of each series over the entire sample period. Are the average growth rates of each series approximately the same? (c) In Appendix A we show that that the ﬁrst diﬀerence of the log is 43 approximately equal to the growth rate: gx t ≈ ln xt − ln xt−1. Compute the approximate growth rate of each series this way. Comment on the quality of the approximation. (d) The standard deviation of a series of random variables is a measure of how much the variable jumps around about its mean (“=stdev(series)”). Take the time series standard deviations of the growth rates of the three series mentioned above and rank them in terms of magnitude. (e) The National Bureau of Economic Research (NBER) declares business cycle peaks and troughs (i.e. recessions and
expansions) through a subjective assessment of overall economic conditions. A popular deﬁnition of a recession (not the one used by the NBER) is a period of time in which real GDP declines for at least two consecutive quarters. Use this consecutive quarter decline deﬁnition to come up with your own recession dates for the entire post-war period. Compare the dates to those given by the NBER. (f) The most recent recession is dated by the NBER to have begun in the fourth quarter of 2007, and oﬃcially ended after the second quarter of 2009, though the recovery in the last three years has been weak. Compute the average growth rate of real GDP for the period 2003Q1–2007Q3. Compute a counterfactual time path of the level of real GDP if it had grown at that rate over the period 2007Q4-2010Q2. Visually compare that counterfactual time path of GDP, and comment (intelligently) on the cost of the recent recession. 44 Chapter 2 What is a Model? Jorge Luis Borges “On Exactitude in Science”: In that Empire, the Art of Cartography attained such perfection that the map of a single province occupied the entirety of a city, and the map of the Empire, the entirety of a province. In time, those unconscionable maps no longer satisﬁed, and the cartographers guilds struck a map of the Empire whose size was that of the Empire, and which coincided point for point with it. The following generations, who were not so fond of the study of cartography as their forebears had been, saw that that vast map was useless, and not without some pitilessness was it, that they delivered it up to the inclemencies of sun and winters. In the Deserts of the West, still today, there are tattered ruins of that map, inhabited by animals and beggars; in all the land there is no other relic of the disciplines of geography. Su´arez Miranda Viajes de varones prudentes, Libro IV, Cap. XLV, L´erida, 1658 2.1 Models and Why Economists Use Them To an economist, a model is a simpliﬁed representation of the economy; it is essentially a representation of the economy in which only the main ingredients are being accounted for. Since we are interested in analyzing the direction of relationships (
e.g. does investment go up or down when interest rates increase?) and the quantitative impact of those (e.g. how much does investment change after a one percentage point increase in interest rates?), in economics, a model is composed of a set of mathematical relationships. Through these mathematical relationships, the economist determines how variables (like an interest rate) aﬀect each other (e.g. investment). Models are not the only way to study human behavior. Indeed, in the natural sciences, scientists typically follow a diﬀerent approach. Imagine a chemist wants to examine the eﬀectiveness of a certain new medicine in addressing a speciﬁc illness. After testing the eﬀects of drugs on guinea pigs the chemist decides to perform experiments on humans. How would she go about it? Well, she will select a group of people willing to participate – providing the right incentives as, we know from 45 principles, incentives can aﬀect behavior – and, among these, she randomly divides members in two groups: a control and a treatment group. The control group will be given a placebo (something that resembles the medicine to be given but has no physical eﬀect in the person who takes it). The treatment group is composed by the individuals that were selected to take the real medicine. As you may suspect, the eﬀects of the medicine on humans will be based on the diﬀerences between the treatment and control group. As these individuals were randomly selected, any diﬀerence to which the illness is aﬀecting them can be attributed to the medicine. In other words, the experiment provides a way of measuring the extent to which that particular drug is eﬀective in diminishing the eﬀects of the disease. Ideally, we would like to perform the same type of experiments with respect to economic policies. Variations of lab experiments have proven to be a useful approach in some areas of economics that focus on very speciﬁc markets or group of agents. In macroeconomics, however, things are diﬀerent. Suppose we are interested in studying the eﬀects of training programs in improving the chances unemployed workers ﬁnd jobs. Clearly the best way to do this would be to split the pool of unemployed workers in two groups, whose members are randomly selected. Here is where the problem with experiments of this sort
becomes clear. Given the cost associated with unemployment, would it be morally acceptable to prevent some workers from joining a program that could potentially reduce the time without a job? What if we are trying to understand the eﬀects a sudden reduction in income has on consumption for groups with diﬀerent levels of savings? Would it be morally acceptable to suddenly conﬁscate income from a group? Most would agree not. As such, economists develop models, and in these models we run experiments. A model provides us a ﬁctitious economy in which these issues can be analyzed and the economic mechanisms can be understood. All models are not created equal and some models are better to answer one particular question but not another one. Given this, you may wonder how to judge when a model is appropriate. This is a diﬃcult question. The soft consensus, however, is that a model should be able to capture features of the data that it was not artiﬁcially constructed to capture. Any simpliﬁed representation of reality will not have the ability to explain every aspect of that reality. In the same way, a simpliﬁed version of the economy will not be able to account for all the data that an economy generates. A model that can be useful to study how unemployed workers and ﬁrms ﬁnd each other will not necessarily be able to account for the behavior of important economic variables such as the interest rate. What is expected is that the model matches relevant features of the process through which workers and ﬁrms meet. While this exercise is useful to understand and highlights relevant economic mechanisms, it is not suﬃcient for providing policy prescriptions. For the latter, we would expect the 46 model to be able to generate predictions that are consistent with empirical facts for which the model was not designed to account. That gives us conﬁdence that the framework is a good one, in the sense of describing the economy. Returning to our example, if the model designed to study the encounter of workers and ﬁrms is also able to describe the behavior of, for instance, wages, it will give us conﬁdence that we have the mechanism that is generating these predictions in the model. The more a model explains, the more conﬁdence economists have in using that model to predict the eﬀects of various policies. In
addition to providing us with a “laboratory” in which experiments can be performed, models allow us to disentangle speciﬁc relationships by focusing on the most fundamental components of an economy. Reality is extremely complex. People not only can own a house or a car but they can also own a pet. Is it important to account for the latter when studying the eﬀects of monetary policy? If the answer is no, then the model can abstract from that. Deciding what “main ingredients” should be included in the model depends on the question at hand. For many questions, assuming individuals do not have children is a ﬁne assumption, as long as that is not relevant to answering the question at hand. If we are trying to understand how saving rates in China are aﬀected by the one-child policy, however, then we would need to depart from the simplifying assumption of no children and incorporate a richer family structure into our framework. In other words, what parts of the reality should be simpliﬁed would depend on the question at hand. While some assumptions may seem odd, the reality is that abstraction is part of every model in any scientiﬁc discipline. Meteorologists, physicists, biologists, and engineers, among others, rely on these parsimonious representations of the real world to analyze their problems. New York is signiﬁcantly larger than the screen of your smartphone. However, for the purpose of navigating Manhattan or ﬁnding your way to Buﬀalo, it is essential that the map does not provide the level of detail you see while driving or walking around. As the initial paragraph in this section suggests, a map the size of the place that is being represented is useless. By the same token, a model that accounts for all or most aspects of reality would be incomprehensible. Criticizing a model purely for its simplicity, while easy to do, misunderstands why we use models in the ﬁrst place. Always remember the words of George Box: “All models are wrong, but some are useful.” 2.2 Summary • A model is a simpliﬁed representation of a complex reality. • We use models to conduct experiments which we cannot run in the real world and use the results from these experiments to inform policy-making 47 • If a model is designed to explain phenomenon x, a test for the usefulness of
a model is whether it can explain phenomenon y which the model was not designed to explain. However, a model not being able to explain all features of reality is not a knock against the model. • All models are wrong, but some are useful. Questions for Review 1. Suppose that you want to write down a model to explain the observed relationship between interest rates and aggregate economic spending. Suppose that you want to test other predictions of your model. You consider two such predictions. First, your model predicts that there is no relationship between interest rates and temperature, but in the data there is a mild negative relationship. Second, your model predicts that consumption and income are negatively correlated, whereas they are positively correlated in the data. Which of these failures is problematic for your model and which is not? Why? 2. During recessions, central banks tend to cut interest rates. You are interested in understanding the question of how interest rates aﬀect GDP. You look in the data and see that interest rates tend to be low when GDP is low (i.e. the interest rate is procylical). Why do you think this simple correlation might give a misleading sense of the eﬀect of changes in the interest rate on GDP? How might a model help you answer this question? 3. Suppose that you are interested in answering the question of how consumption reacts to tax cuts. In recent years, recessions have been countered with tax rebates, wherein households are sent a check for several hundred dollars. This check amounts to a “rebate” of past taxes paid. If you could design an ideal experiment to answer this question, how would you do so? Do you think it would be practical to use this experiment on a large scale? 48 Chapter 3 Brief History of Macroeconomic Thought Macroeconomics as a distinct ﬁeld did not exist until the 1930s with the publication of John Maynard Keynes’ General Theory of Employment, Interest, and Money. That is not to say economists did not think about aggregate outcomes until then. Adam Smith, for instance, discussed economic growth in The Wealth of Nations which was published over 150 years before Keynes wrote his book. Likewise, in the later part of the 18th century, John Baptiste Say and Thomas Malthus debated the self stabilizing properties of the economy in the short run.1 However, macroeconomics as a ﬁeld is a child of the Great Depression and it is where we start the discussion. 3.1 The
Early Period: 1936-1968 Keynes published his seminal book in the throes of the Great Depression of 1936. Voluminous pages have been ﬁlled posing answers to the question, “What did Keynes really mean?” Unfortunately, since he died in 1946, he did not have much time to explain himself.2 The year following the General Theory’s debut, John Hicks oﬀered a graphical interpretation of Keynes’ work and it quickly became a go-to model for macroeconomic policy (Hicks 1937). As time progressed, computational power continually improved. This allowed researchers to build statistical models containing the key economic aggregates (e.g. output, consumption, and investment) and estimate the relationships implied by Keynes’ model. In the 1950s Lawrence Klein and his colleagues developed sophisticated econometric models to forecast the path of the economy. The most complicated of these models, Klein and Goldberger (1955), contained dozens of equations (each of which were inspired by some variant of the Keynesian theory) that were solved simultaneously. The motivation was that, after estimating these models, economists and policy makers could predict the dynamic path of the economic variables after a shock. For instance, if oil prices unexpectedly go up, one could take the estimated model and trace out the eﬀects on output, consumption, inﬂation, and any other variable of interest. In the face of such shocks policy makers could choose the appropriate 1Econlib provides a short and nice summary on Malthus. 2For more on Keynes see Econlib. 49 ﬁscal and monetary policies to combat the eﬀects of an adverse shock. In contrast to the rich structure of the Klein model, it was a single equation which perhaps carried the most weight in policy circles: the “Phillips Curve.” Phillips (1958) showed a robust downward-sloping relationship between the inﬂation and unemployment rates. Economists reasoned that policy makers could conduct monetary and ﬁscal policy in such a way as to achieve a target rate of inﬂation and unemployment. The tradeoﬀ was clear: if unemployment increased during a recession, the central bank could increase the money supply thereby increasing inﬂation but lowering unemployment. Consequently, decreasing unemployment was not costless, but the tradeoﬀ between unemployment and inﬂation was clear, predictable and exploitable by policy
makers. Until it wasn’t. In a now famous 1968 presidential address to the American Economic Association, Milton Friedman explained why a permanent tradeoﬀ between unemployment and inﬂation is theoretically dubious.3 The reason is that to achieve a lower unemployment rate the central bank would need to cut interest rates thereby increasing money supply and inﬂation. This increase in inﬂation in the medium to long run would increase nominal interest rates. To keep unemployment at this low level, there would need to be an even bigger expansion of the money supply and more inﬂation. This process would devolve into an inﬂationary spiral where more and more inﬂation would be needed to achieve the same level of unemployment. Friedman’s limits of monetary policy was a valid critique of the crudest versions of Keynesianism, but it was only the beginning of what was to come. 3.2 Blowing Everything Up: 1968-1981 In microeconomics you learn that supply and demand curves come from some underlying maximizing behavior of households and ﬁrms. Comparing various tax and subsidy policies necessitates going back to the maximization problem and ﬁguring out what exactly changes. There was no such microeconomic behavior at the foundation of the ﬁrst-generation Keynesian models; instead, decision rules for investment, consumption, labor supply, etc. were assumed rather than derived. For example, consumption was assumed to be a function of current disposable income. This was not the solution to a household’s optimization problem, but rather just seemed “natural.” Later generations of economists recognized this shortcoming and attempted to rectify it by providing microeconomic foundations for consumption-saving decisions (Ando and Modigliani 1963), portfolio choice (Tobin 1958), and investment (Robert E. Lucas 1971). While each of these theories improved the theoretical underpinnings of the latest vintages of the Keynesian model, they were typically analyzed in isolation. They 3Friedman (1968) and also see Phelps (1967) for a formal derivation. 50 were also analyzed in partial equilibrium, so, for instance, a consumer’s optimal consumption and savings schedule was derived taking the interest rate as given. Moreover, econometric forecasting continued to be conducted in an ad hoc framework. In “Econometric Policy Evaluation: A Critique,” Robert Lucas launched a devastating
critique on using these econometric models for policy evaluation (Lucas 1976). Lucas showed that while the ad hoc models might ﬁt the data well, one cannot validly analyze the eﬀects of policy within them. The reason is that the relationships between macroeconomic aggregates (e.g. output, wages, consumption) are the consequence of optimizing behavior. For example, if people consume about ninety percent of their income it might seem that an appropriate prediction of a $1,000 tax cut is that people would consume $900 of it. However, if this tax cut was ﬁnanced by running a deﬁcit, then the person receiving the tax cut might anticipate that the deﬁcit will have to be repaid and will therefore save more than ten percent of the tax cut. Naively looking at the historical correlation between consumption and income would lead to an incorrect prediction of the eﬀects of a tax cut on consumption.4 The magnitude of the consumption increase in this example is a function of the household’s expectations about the future. If the household is myopic and does not realize the government will eventually raise taxes, then consumption will go up by more than if the household anticipates that its future tax burden will be higher. In summary, the relationship between macroeconomic variables cannot be assumed to be invariant to policy as in the Klein model, but instead actually depends on policy. Lucas and his followers contended that individuals maximizing their utility or ﬁrms maximizing their proﬁts would also optimize their expectations. What does it mean to “optimize” expectations? In Lucas’ framework it means that households use all available information to them when making their forecasts. This has come to be known as “rational expectations” and is now ubiquitous in macroeconomics. The implications of the rational expectations hypothesis were sweeping. First, it implied that predictable changes in monetary policy would not stimulate aggregate demand (Sargent and Wallace 1975). If everyone knows that the central bank is going to raise the money supply by ten percent, then all prices and wages will increase by ten percent simultaneously. Since there is no change in relative prices, expansionary monetary policy will not stimulate output. In terms of tax policy, governments have an incentive to promise to keep the tax rate on capital gains low to encourage investment in capital goods. Once the capital goods are completed, however, the government has an incentive to renege on its promise
and tax the capital gains. Since a tax on a perfectly inelastic good like capital causes no deadweight loss, even a perfectly benevolent government would have an incentive to renege on its promise. 4A similar point was made by Barro (1974). 51 Rational individuals would anticipate the government’s incentive structure and not ever invest in capital goods.5 Hence, what is optimal in a static sense is not optimal in a dynamic sense. This time inconsistency problem pervades many areas of policy and regulation and implies that any policy designed to trick people (even if it is for their own good) is doomed to fail. These critiques led economists to be skeptical of the monetary and ﬁscal ﬁne tuning policies of the 1960s and 70s, but the adverse economic conditions in the 1970s put the ﬁnal nail in the Keynesian coﬃn. A mix of rising oil prices and slower productivity growth led to simultaneously high unemployment and inﬂation. The Phillips curve had shifted. If the relationship between unemployment and inﬂation was unstable then it could not necessarily be exploited by policy makers. Of course, this was Lucas’ point: any policy designed to exploit a historical relationship between aggregate variables without understanding the microeconomic behavior that generated the relationship is misguided. By the early 1980s it was clear that the Keynesian orthodoxy was fading. In a 1979 paper, Bob Lucas and Tom Sargent put it best in discussing how to remedy Keynesian models: In so doing, our intent will be to establish that the diﬃculties are fatal: that modern macroeconomic models are of no value in guiding policy, and that this condition will not be remedied by modiﬁcations along any line which is currently being pursued. Lucas and Sargent (1979). The demise of Keynesian models was not in question. The relevant question was what would come to replace them. 3.3 Modern Macroeconomics: 1982-2016 In addition to the lack of microfoundations (i.e. the absence of ﬁrms and individuals maximizing objective functions), macroeconomic models suﬀered because models designed to address the short run question of business cycles were incompatible with models designed to address the long run questions of economic growth. While of course models are abstractions that will not capture every ﬁne detail in the data, the inconsistency between short and long run models was especially severe. In 1982 Finn K
ydland and Ed Prescott developed the “Real Business Cycle theory” which addressed this concern (Kydland and Prescott 1982). Kydland and Prescott’s model extended the basic Neoclassical growth model to include a labor-leisure decision and random ﬂuctuations in technology.6 The model consists of utility maximizing households and proﬁt maximizing ﬁrms. Everyone has rational expectations and there are no 5See Kydland and Prescott (1977) for a discussion and more examples. 6The Neoclassical growth model was developed independently by Cass (1965) and Koopmans (1963). The version with randomly ﬂuctuating technology was developed in Brock and Mirman (1972). 52 market failures. Kydland and Prescott showed that a large fraction of economic ﬂuctuations could be accounted for by random ﬂuctuations in total factor productivity alone. Total factor productivity (TFP) is simply the component of output that cannot be account for by observable inputs (e.g. capital, labor, intermediate goods). The idea that period-to-period ﬂuctuations in output and other aggregates could be driven by changing productivity ﬂew in the face of conventional wisdom which saw recessions and expansions as a product of changes in consumer sentiments or the mismanagement of ﬁscal and monetary policy. Their model also had the implication that pursuing activist monetary or ﬁscal policy to smooth out economic ﬂuctuations is counterproductive. While Kydland and Prescott’s approach was certainly innovative, there were caveats to their stark conclusions. First, by construction, changes in TFP were the only source of business cycle movement in their model. How TFP is measured, however, depends on which inputs are included in the production function. A production function which includes capital, labor, and energy will give a diﬀerent measure of TFP than a production function that includes only capital and labor. Second, because all market failures were assumed away, there was no role for an activist government by construction. Despite these potential shortcomings, Kydland and Prescott’s model served as a useful benchmark and was the starting point for essentially all business cycle models up to the present day. Over the following decades, researchers developed the Real Business Cycle model to include diﬀerent sources of ﬂuctuations (McGrattan, Rogerson, and Wright 1997 and Greenwood, Hercow
itz, and Krusell 2000), productive government spending (Baxter and King 1993), and market failures such as coordination problems, sticky prices and wages, and imperfect information. Models built from the neoclassical core but which feature imperfectly ﬂexible prices are often called New Keynesian models.7 Since the Great Recession, economists have worked hard to incorporate ﬁnancial frictions and realistic ﬁnancial intermediation into their models. This broad class of models have come to be grouped by an acronym: DSGE. All of them are Dynamic in that households and ﬁrms make decisions over time. They are Stochastic which is just a fancy word for random. That is, the driving force of business cycles are random ﬂuctuations in exogenous variables. Also, all of them are consistent with General Equilibrium. We discuss general equilibrium later in the book, but for now think of it as a means of accounting. Markets have to clear, one person’s savings is another’s borrowing, etc. This accounting procedure guards against the possibility of misidentifying something as a free lunch and is pervasive through all of economics, not just macroeconomics. DSGE models have become the common tools of the trade for academic researchers and central bankers all over the world. They incorporate many of the frictions discussed by 7For an early discussion see Mankiw (1990) and for a more up-to-date description see Woodford (2003). 53 Keynes and his followers but are consistent with rational expectations, long-run growth, and optimizing behavior. While some of the details are beyond the scope of what follows, all of our discussion is similar in spirit to these macroeconomic models. At this point you may be wondering how macroeconomics is distinct from microeconomics. Following the advances in “microfoundations” of macro that followed Kydland and Prescott, it is now fairly accurate to say that all economics is microeconomics. When you want to know how labor supply responds to an increase in the income tax rate, you analyze the question with indiﬀerence curves and budget constraints rather than developing some sort of alternative economic theory. Similarly, macroeconomics is simply microeconomics at an aggregate level. The tools of analysis are exactly the same. There are preferences, constraints, and equilibrium just as in microeconomics, but the motivating questions are diﬀerent. In the next chapter, we start to look
at these motivating questions, in particular those related to long-run. 3.4 Summary • As a distinct ﬁeld of inquiry, macroeconomics began with Keynes’ The General Theory in 1936 • By the late 1960s, a consensus had emerged in macroeconomics, theoretically based on Hicks (1937)’s graphical interpretation of Keynes’ book and the Phillips Curve, and empirically implemented in the so-called large scale macroeconometric models. • The 1970s witnessed an upheaval in macroeconomics. The end of the macroeconomic consensus of the 1960s came about because of an empirical failure (the breakdown of the Phillips Curve relationship) and theoretical inadequacies in the Keynesian models of the day. These models were not micro-founded and did not take dynamics seriously. • A new consensus emerged in the 1980s. Loosely speaking, modern macroeconomic models can be divided into two camps – neoclassical / real business cycle models and New Keynesian models. Both of these are DSGE models in the sense that they are dynamic, feature an element of randomness (i.e. are stochastic), and study general equilibrium. • Modern macroeconomics is microeconomics, but at a high level of aggregation. 54 Part II The Long Run 55 Nobel Prize winning economist Robert Lucas once famously said that “Once you start to think about growth, it is diﬃcult to think about anything else.” The logic behind Lucas’s statement is evident from a time series plot of real GDP, for example that shown in Figure 1.3. Visually it is diﬃcult to even see the business cycle – what stands out most from the picture is trend growth. In a typical recession, real GDP declines by a couple of percentage points. This pales in comparison to what happens over longer time horizons. Since World War II, real GDP in the US has increased by a factor of 8. This means that real GDP has doubled roughly three times in the last 70 or so years. Given the power of compounding, the potential welfare gains from increasing the economy’s longer run rate of growth dwarf the potential gains from eliminating short run ﬂuctuations. Understanding what drives growth is also key for understanding poverty in the developing world and how to lift the poorest of countries out of this poverty. We begin the core of the book here in Part II by studying long run economic growth. We think of the long run as
describing frequencies of time measured in decades. When economists talk about “growth,” we are typically referencing the rate of growth of GDP over these long stretches of time. This should not be confused with the usage of the word “growth” in much of the media, which typically references quarter-over-quarter percentage changes in real GDP. Chapter 4 presents some basic facts about economic growth. The presentation is centered around the “Kaldor stylized facts” based on Kaldor (1957). Here we also present some facts concerning cross-country comparisons of standards of living. Chapter 5 presents the classical Solow model of economic growth, based on Solow (1956). Chapter 6 augments the basic Solow model with exogenous population and productivity growth. The main take-away from the Solow model is that sustained growth must primarily come from productivity growth, not from the accumulation of physical capital. This conclusion has important implications for policy. In Chapter 7, we use the basic Solow model from Chapter 5 to study the large diﬀerences in standards of living across countries. The principal conclusion of this analysis echoes the conclusion about the sources of long run trend growth – a key determinant in diﬀerences in GDP per capita across countries is productivity, with factor accumulation playing a more limited role. This too has important policy implications, particularly for those interested in lifting the developing world out of dire poverty. Chapter 8 studies an overlapping generations economy in which at any point in time there are two generations – young and old. Diﬀerently than the Solow model, young households choose saving to maximizes lifetime utility, and thus the saving decision is endogenized. At least in some cases, the OLG economy is nevertheless quite similar to the Solow economy, although we are able to address some interesting questions related to intergenerational transfers and the role of government. Because Chapter 8 considers a micro-founded optimization model, it provides a 56 nice bridge to Part III. 57 Chapter 4 Facts About Economic Growth In this chapter we set the table for the growth model to come. Before jumping into the economic model, we start by describing some basic facts of economic growth. First, we look at the time series growth in the United States which is more or less representative of the average high-income country. Next, we look at economic growth over the world. 4.1 Economic Growth over Time: The Kaldor Facts In an inﬂuential 1957 article Nicholas
Kaldor listed a set of stylized facts characterizing the then relatively recent economic growth across countries (Kaldor (1957)).1 “Stylized” means that these facts are roughly true over suﬃciently large periods of time – they do not exactly hold, especially over short time frequencies. The “Kaldor Facts” continue to provide a reasonably accurate description of economic growth across developed countries including the United States. 1. Output per worker grows at a sustained, roughly constant, rate over long periods of time. How rich are Americans today relative to several generations ago? To make such a comparison requires a standard unit of account. As we discussed in Chapter 1, it is common to use price indexes to distinguish changes in prices from changes in quantities. Hence, we focus on real, rather than nominal, GDP. A natural measure of the productive capacity of an economy is real GDP per worker. GDP can go up either because there are more people working in an economy or because the people working in an economy are producing more. For thinking about an economy’s productive capacity, and for making comparisons across time, we want a measure of GDP that controls for the number of people working in an economy. The log of real GDP per worker in the US is shown in Figure 4.1. Why do we plot this relationship in logs? GDP grows exponentially over time which implies the slope gets 1Also see the Wikipedia entry on this. 58 steeper as time goes by. The log of an exponential function is a linear function which is much easier to interpret. The slope of a plot in the log is approximately just the growth rate of the series. Figure 4.1: Real GDP per Worker in the US 1950-2011 The ﬁgure also plots a linear time trend, which is depicted with the dotted straight line. While the actual series is occasionally below or above the trend line, it is clear that GDP per worker grows at a sustained and reasonably constant rate. The average growth rate over this period is about 1.7 percent annually. How does a 1.7 percent annual growth rate translate into absolute diﬀerences in income over time? A helpful rule of thumb is called the “Rule of 70.” The Rule of 70 (or sometimes rule of 72) is a way to calculate the approximate number of years it takes a variable to double. To calculate this, divide 70 by the average growth rate of this series. This gives you the approximate number of
years it takes the variable to double. To see why it is called the “Rule of 70” consider the following example. Let Y0 be a country’s initial level of income per person and suppose the annual rate of growth is g percent per year. We can ﬁnd 59 19501960197019801990200020102020Year10.210.410.610.81111.211.411.6log(RGDP per Worker)Actual SeriesTrend Series how long it takes income per person to double by solving the following equation for t: 2Y0 = (1 + g)tY0 ⇔ 2 = (1 + g)t ⇔ ln 2 = t ln(1 + g). Provided g is suﬃcient small, ln(1 + g) ≈ g. ln 2 is approximately equal to 0.7. Hence, 70 divided by the annual percent rate of growth equals the required time for a country’s income per person to double. In the U.S. case, t = 70/1.7 ≈ 41. This means that, at this rate, GDP per worker in the US will double twice every 80 years or so. Measured in current dollar terms, US GDP per capita in 1948 was about $32,000. In 2016, it is $93,000. In other words, over this 60 year period GDP per work in the US has doubled about 1.5 times. Small diﬀerences in average growth rates can amount to large diﬀerences in standards of living over long periods of time. Suppose that US real GDP per capita were to continue to grow at 1.7 percent for the next one hundred years. Using the rule of 70, this means that real GDP per capita would double approximately 2.5 times over the next century. Suppose instead that the growth rate were to increase by a full percentage point to 2.7 percent per year. This would imply that real GDP per capita would double approximately 4 times over the next century, which is a substantial diﬀerence relative to double 2.5 times. 2. Capital per worker grows at a sustained, approximately constant, rate over long periods of time. Figure 4.2 shows the time series of the log of capital per worker over the period 1950-2011 along with a linear time trend. Capital constitutes the plant, machinery, and equipment that is used to produce output. Similar
to output per worker, the upward trend is unmistakable. Over the period under consideration, on average capital per worker grew about 1.5 percent per year, which is only slightly lower than the growth rate in output per capita. 60 Figure 4.2: Capital per worker in the US 1950-2011. The fact that capital and output grow at similar rates leads to the third of Kaldor’s facts. 3. The capital to output ratio is roughly constant over long periods of time. If capital and output grew at identical rates from 1950 onwards, the capital to output ratio would be a constant. However, year to year the exact growth rates diﬀer and, on average, capital grew a little slower than did output. Figure 4.3 plots the capital to output ratio over time. The capital to output ratio ﬂuctuated around a roughly constant mean from 1950 to 1990, but then declined substantially during the 1990s. The capital output ratio picked up during the 2000s. Nevertheless, it is not a bad ﬁrst approximation to conclude that the capital to output ratio is roughly constant over long stretches of time. 61 19501960197019801990200020102020Year11.411.611.81212.212.412.6log(Capital per Worker)Actual SeriesTrend Series Figure 4.3: Capital to Output Ratio in the U.S. 1950–2011 Over the entire time period, the ratio moved from 3.2 to 3.1 with a minimum value a little less than 3 and a maximum level of about 3.5. However, the ratio moves enough to say that the ratio is only approximately constant over fairly long periods of time. 4. Labor’s share of income is roughly constant over long periods of time.. Who (or what) earns income? This answer to this question depends on how broadly (or narrowly) we deﬁne the factors of production. For instance, should we make a distinction between those who collect rent from leasing apartments to those who earn dividends from owning a share of Facebook’s stock? Throughout most of this book, we take the broadest possible classiﬁcation and group income into “labor income” and “capital income.” Clearly, when people earn wages from their jobs, that goes into labor income and when a tractor owner rents his tractor to a farmer, that goes into capital income. Classifying every type of
income beyond these two stark cases is sometimes not as straightforward. For instance, a tech entrepreneur may own his computers (capital), but also supply his labor to make some type of software. The revenue earned by the entrepreneur might reasonably be called capital income or labor income. In practice, countries have developed ways to deal with the assigning income problem in their National Income and Product Accounts (also known by the acronym, NIPA). For now, assume that everything is neatly categorized as wage income or capital income. 62 19501960197019801990200020102020Year2.933.13.23.33.43.5K/Y Labor’s share of income at time t equals total wage income divided by output, or: LABSHt = wtNt Yt. Here wt is the (real) wage, Nt total labor input, and Yt output. Output must equal income, and since everything is classiﬁed as wage or capital income, capital’s share is CAP SHt = 1 − LABSHt. Clearly, these shares are bounded below by 0 and above by 1. Figure 4.4 shows the evolution of the labor share over time. Figure 4.4: Labor Share in the US 1950-2011. The labor share is always between a 0.62 and 0.7 with an average of 0.65. Despite a downward trend over the last decade, labor’s share has been relatively stable. This also implies capital’s share has been stable with a mean of about 0.35. The recent trends in factor shares have attracted attention from economists and we return to this later in the book, but for now take note that the labor share is relatively stable over long periods of time. 5. The rate of return on capital is relatively constant. 63 19501960197019801990200020102020Year0.620.630.640.650.660.670.68Labors Share The return to capital is simply the value the owner gets from “renting” capital to someone else. For example, if I own tractors and lease them period-by-period to a farmer for $10 per tractor, capital income is simply $10 times the number of tractors. If a producer owns his or her own capital this “rent” is implicit. We will lose Rt to denote the rental rate on capital and Kt the total stock of capital.
We can infer the rate of return on capital from the information we have already seen. Start with the formula for capital’s share of income: CAP SHt = 1 − LABSHt = RtKt Yt ⇒ Rt = (1 − LABSHt) Yt Kt Rt is the rate of return on capital. Since we already have information on labor’s share and the capital to output ratio, we can easily solve for Rt. It’s also straightforward to see why, given previously documented facts, Rt ought to be approximately constant – LABSHt and Yt are both roughly constant, so the product of the two series ought Kt also to be close to constant. The implied time series for Rt is displayed in Figure 4.5. 64 Figure 4.5: Return on capital in the US 1950-2011. The rate of return on capital varies between 0.095 and 0.125 with an upwards trend since the mid 1980s. Therefore, the upward trend in capital’s share since 2000 can be attributed more to the rise in the real return of capital rather than an increase in the capital to output ratio. The rate of return on capital is closely related to the real interest rate, as we will see in Part III. In particular, in a standard competitive framework the return on capital equals the real interest rate plus the depreciation rate on capital. If the depreciation rate on capital is roughly 0.1 per year, these numbers suggest that real interest rates are quite low on average. An interesting fact not necessarily relevant for 65 19501960197019801990200020102020Year0.0950.10.1050.110.1150.120.125Return on Capital growth is that the return on capital seems to be high very recently in spite of extremely low real interest rates. 6. Real wages grow at a sustained, approximately constant, rate. Finally, we turn our attention to the time series evolution of wages. By now, you should be able to guess what such a time path looks like. If the labor share is relatively stable and output per worker rises at a sustained rate, then wages must also be rising at a sustained rate. To see this, go back to the equation for the labor share. If Yt/Nt is going up then Nt/Yt must be going down. But the only way for the left hand side to be approximately constant is for the wage to increase
over time. Figure 4.6 plots the log of wages against time and shows exactly that. Figure 4.6: Wages in the US 1950-2011. Annual wage growth averaged approximately 1.8 percent over the entire time period. Resembling output and capital per worker, there is a clear sustained increase. Moreover, wages grow at approximately the same rate as output per worker and capital per worker. The Kaldor facts can be summarized as follows. Wages, output per worker, and capital per worker grow at approximately the same sustained rate and the return on capital is approximately constant. All the other facts are corollaries to these. A perhaps surprising implication of these facts is that economic growth seems to beneﬁt labor 66 19501960197019801990200020102020Year2.22.42.62.833.23.43.6log(Wages)Actual SeriesTrend Series (real wages rise over time) and not capital (the return on capital is roughly constant). Over the next two chapters we show that our benchmark model of economic growth is potentially consistent with all these facts. 4.2 Cross Country Facts Kaldor’s facts pertain to the economic progress of rich countries over long periods of time. However, there is immense variation in income across countries at any given point in time. Some of these countries have failed to grow at all, essentially remaining as poor today as they were forty years ago. On the other hand, some countries have become spectacularly wealthy over the last several decades. In this section we discuss the variation in economic performance across a subset of countries. We measure economic performance in terms of output per person. Because not everyone works, this is more indicative of average welfare across people than output per worker. This does not mean output per capita is necessarily an ideal way to measure economic well-being. Output per capita does not capture the value of leisure, nor does it capture many things which might impact both the quality and length of life. For example, output per capita does not necessarily capture the adverse consequences of crime or pollution. In spite of these diﬃculties, we will use output per capita as our chief measure of an economy’s overall standard of living. When comparing GDP across countries, a natural complication is that diﬀerent countries have diﬀerent currencies, and hence diﬀerent units of GDP. In the analysis which follows, we measure GDP across country in terms of US
dollars using a world-wide price index that accounts for cross-country diﬀerences in the purchasing power of diﬀerent currencies. 1. There are enormous variations in income across countries. Table 4.1 shows the diﬀerences in the level of output per person in 2011 for a selected subset of countries. 67 Table 4.1: GDP Per Capita for Selected Countries High income countries GDP per Person Middle income countries Low income countries Canada Germany Japan Singapore United Kingdom United States China Dominican Republic Mexico South Africa Thailand Uruguay Cambodia Chad India Kenya Mali Nepal $35,180 $34,383 $30,232 $59,149 $32,116 $42,426 $8,640 $8,694 $12,648 $10,831 $9,567 $13,388 $2,607 $2,350 $3,719 $1636 $1,157 $1,281 Notes: This data comes from the Penn World Tables, version 8.1. The real GDP is in terms of chain-weighted PPPs. In purchasing power parity terms, the average person in the United States was 36.67 times ($42,426/$1,157) richer than the average person in Mali. This is an enormous diﬀerence. In 2011, 29 countries had an income per capita of ﬁve percent or less of that in the U.S. Even among relatively rich countries, there are still important diﬀerences in output per capita. For example, in the US real GDP per capita is about 30 percent larger than it is in Great Britain and about 25 percent larger than in Germany. 2. There are growth miracles and growth disasters. Over the last four decades, some countries have become spectacularly wealthy. The people of Botswana, for instance, subsisted on less than two dollars a day in 1970, but their income increased nearly 20 fold over the last forty years. 68 Table 4.2: Growth Miracles and Growth Disasters Growth Miracles 2011 Income % change South Korea Taiwan China Botswana 1970 Income $1918 $4,484 $1,107 $721 $27,870 $33,187 $8,851 $14,787 1353 640 700 1951 Madagascar Niger Burundi Central African Republic Growth Disasters 1970 Income $1,321 $1,304 $712 $1,148 $937 $651 $612 $762 -29 -50 -14 -34
2011 Income % change Notes: This data comes from the Penn World Tables, version 8.1. The real GDP is in terms of chain-weighted PPPs. As Table 4.2 shows, the countries of East Asia are well represented in the accounting of growth miracles. On the other hand, much of continental Africa has remained mired in poverty. Some countries, like those shown under the growth disasters column actually saw GDP per person decline over the last forty years. Needless to say, a profound task facing leaders in developing countries is ﬁguring out how to get to the left side of this table and not fall on the right side. 3. There is a strong, positive correlation between income per capita and human capital. Human capital refers to the stock of knowledge, social attributes, and habits possessed by individuals or groups of individuals. It is capital in the sense that human capital must itself by accumulated over time (i.e. it is not something with which an individual is simply endowed), is useful in producing other goods, and does not get completely used up in the process of producing other goods. Unlike physical capital, it is intangible and possessed by individuals or groups of individuals. As such, measuring what economists call “human capital” is rather diﬃcult. A natural proxy for human capital is years of education. On one hand, calculating the average number of years citizens spend in school seems like a reasonable proxy, but what if teachers do not show up to school? What if there are no books or computers? Clearly, the quality of education matters as much as the quantity of education. While imperfect, economists have devised measures to deal with cross country heterogeneity in quality. With that caveat in mind, Figure 4.7 shows how the level of human capital varies with income per person. 69 Figure 4.7: Relationship Between Human Capital and Income per Person As the level of human capital per person increases, income per person also increases. This does not mean that more education causes an increase in income. Indeed, the arrow of causation could run the other direction. If education is a normal good, then people in richer countries will demand more education. However, it is reasonable to think that people who know how to read, write, and operate a computer are more productive than those who do not. Understanding the direction of causation is diﬃcult, but carries very important policy implications. 4.3 Summary • In this chapter we covered a number of cross country and within country growth facts. •
The main time series facts are that output, capital, and wages grow at a sustained rate and that the capital to output ratio and real interest rate do not have sustained growth. • From a cross country perspective there are enormous variations in living standards. • Rich countries tend to have more educated populations. Questions for Review 70 67891011log(RGDP per Person)11.522.533.5Index of Human Capital 1. Write down and brieﬂy discuss the six Kaldor stylized facts about economic growth in the time series dimension. 2. Write down and discuss the three stylized facts about economic growth in the cross-sectional dimension. 3. It has been widely reported that income inequality within the US and other industrialized countries is growing. Yet one of the stylized facts is that capital does not seem to beneﬁt from economic growth (as evidenced by the approximate constancy of the return to capital across time). If this is the case, what do you think must be driving income inequality? Exercises 1. [Excel Problem] Download quarterly data on output per worker in the nonfarm business sector for the US for the period 1947 through 2016. You can do so here. (a) Take natural logs of the data (which appears as an index) and then compute ﬁrst diﬀerences of the natural logs (i.e. compute the diﬀerence between the natural log of the index in 1947q2 with the value in 1947q1, and so on). What is the average value of the ﬁrst diﬀerence over the entire sample? To put this into annualized percentage units, you may multiply by 400. (b) Compute the average growth rate by decade for the 1950s, 1960s, 1970s, 1980s, 1990s, 2000s, and 2010s (even though this decade isn’t complete). Does the average growth rate look to be constant by decade here? What pattern do you observe? 2. [Excel Problem] In this problem we investigate the relationship between the size of government and growth in GDP per person between 1960-2010 for the following countries: Australia, Canada, Germany, Japan, Spain, and the United States. (a) Go to this Saint Louis FRED page, and download the “Share of Government Consumption at Current Purchasing Power Parities” for the relevant subset of countries. Plot the trends of government’s
share over time for the countries. Comment on the trends. Do they seem to be moving in the same direction for all the countries? (b) Next, we have to construct real GDP per capita. First, go to this, page, and download ”‘Expenditure-Side Real GDP at Chained Purchasing 71 Power Parities” for the subset of countries. Next go to this page, and download “Population” for each country. Real GDP per capita is Real GDP divided by population. Calculate real GDP per person at each point in time for each country. Plot the log level of real GDP per capita over time for each country. Do the countries appear to be getting closer together or fanning out? (c) For every country, calculate the average share of government expenditures and the average rate of growth in output per worker over ten years. For example, calculate the average share of government expenditures in Canada from 1960-1969 and the average growth rate in GDP per capita between 1961-1970. You will have ﬁve decade pairs for each country. Once these are constructed, create a scatter plot of real GDP growth on the vertical axis and government’s share of expenditure on the horizontal axis. What is the correlation between these variables? 72 Chapter 5 The Basic Solow Model The Solow Model is the principal model for understand long run growth and cross-country income diﬀerences. It was developed by Robert Solow in Solow (1956), work for which he would later win a Nobel Prize. This chapter develops the simplest version of the Solow model. The theoretical framework is rather simple but makes powerful predictions that line up well with the data. We do not explicitly model the microeconomic underpinnings of the model. The key equations of the model are an aggregate production function, a consumption/saving function, and an accumulation equation for physical capital. In the sections below, we present the equations summarizing the model and graphically work through some implications of the theory. 5.1 Production, Consumption, and Investment The Solow model presumes that there exists an aggregate production function which maps capital and labor into output. Labor is denominated in units of time. Capital refers to something which (i) must itself be produced, (ii) helps you produce output, and (iii) does not get fully used up in the production process. Capital and labor are said to both be factors of production. Capital and labor share the similarity that both help you produce
output. They diﬀer in that capital is a stock whereas labor is a ﬂow concept. They also diﬀer in that labor/time is an endowment – there is nothing one can do to increase the number of hours available to work in a day, for example. In contrast, capital can be accumulated. As an example, suppose that your output is lawns mowed, your capital is your lawn mower, and your labor is time spent mowing. Each period, there is a ﬁxed amount of hours in the day in which you can spend mowing – this is the endowment feature of labor input. Furthermore, the amount of hours available tomorrow is independent of how many hours you spend mowing today – in other words, how many hours you worked in the past doesn’t inﬂuence how many hours you can work in the future. This is the ﬂow feature of labor. Capital is diﬀerent in that how much capital you had in the past inﬂuences how much capital you’ll have in the future. If you had two mowers yesterday, you’ll probably still have two mowers tomorrow (or nearly the equivalent of two mowers tomorrow should the mowers experience some depreciation). 73 This is the stock feature of capital – how much you had in the past inﬂuences how much capital you have in the present and future. Furthermore, you can accumulate capital – you can go to Home Depot and buy another mower if you want to increase your future productive capacity.1 Let us now turn to a formal mathematical description of the aggregate production function. Denote Kt as the stock of capital and Nt as the total time spent working in period t. Let Yt denote output produced in period t. Suppose that there is a single, representative ﬁrm which leases labor and capital from a single, representative household each period to produce output. The production function is given by: Yt = AtF (Kt, Nt). (5.1) Here At is an exogenous variable which measures productivity. It is exogenous and can in principle vary across time. However, we shall assume that if it does change, it does so permanently, meaning that future values of At+j, for j > 1, equal the current value, At. Therefore, to simplify notation we will drop the t subscript and simply denote this exogenous
variable with A. F (⋅) is a function which relates capital and labor into output. The bigger is A, the more Yt you get for given amounts of Kt and Nt – i.e. you are more eﬃcient at turning inputs into output. The function F (⋅) is assumed to have the following properties: FK > 0 and FN > 0 (i.e. the marginal products, or ﬁrst partial derivatives with respect to each argument, are always positive, so more of either input means more output); FKK < 0, FN N < 0 (i.e. there are diminishing marginal products in both factors, so more of one factor means more output, but the more of the factor you have, the less an additional unit of that factor adds to output); FKN > 0 (i.e. if you have more capital, the marginal product of labor is higher); and F (γKt, γNt) = γF (Kt, Nt), which means that the production function if you double both inputs, γ = 2, you double output). has constant returns to scale (i.e. Finally, we assume that both capital and labor are necessary to produce, which means that F (0, Nt) = F (Kt, 0) = 0. An example functional form for F (⋅) which we will use throughout the course is the Cobb-Douglas production function: F (Kt, Nt) = K α t N 1−α t, with 0 < α < 1. (5.2) 1Although we are here focusing on physical capital (e.g. mowers), the logic also applies to human capital, which was discussed brieﬂy in Chapter 4. For example, suppose that you are in the business of air conditioner repair. You can go to school to learn how to repair air conditioners. This stock of knowledge helps you produce output (repaired air conditioners). Using your stock of knowledge on Tuesday doesn’t prevent you from also using your stock of knowledge on Wednesday. Furthermore, you can go back to school to learn more about air conditioners so as to increase your productive capacity. This knowledge you accumulate is not tangible like physical capital (e.g. lawn mowers), but it is like physical capital in that it is a stock and in that it can be accumulated.
74 Example Suppose that the production function is Cobb-Douglas, as in (5.2). Let’s verify that this production function satisﬁes the properties laid out above. The ﬁrst partial derivatives are: FK(Kt, Nt) = αK α−1 t N 1−α t FN (Kt, Nt) = (1 − α)K α t N −α t. Since 0 < α < 1, and Kt and Nt cannot be negative, the marginal products of capital and labor are both positive. Now, let’s look at the second derivatives. Diﬀerentiating the ﬁrst derivatives, we get: FKK(Kt, Nt) = α(α − 1)K α−2 FN N (Kt, Nt) = −α(1 − α)K α FKN (Kt, Nt) = (1 − α)αK α−1 t N 1−α t t N −α−1 t t N −α. t Again, since 0 < α < 1, FKK and FN N are both negative, while FKN > 0. Now, let’s verify the constant returns to scale assumption. F (γKt, γNt) = (γKt)α(γNt)1−α = γαK α t γ1−αN 1−α = γK α t N 1−α. t t Eﬀectively, since the exponents on Kt and Nt sum to one, scaling them by a factor γ simply scales the production function by the same factor. If the exponents summed to less than 1, we would say that the production function has decreasing returns to scale. If the exponents summed to greater than 1, we would say that the production function had increasing returns to scale. Finally, let’s verify that both inputs are necessary for any output to be produced: F (0, Nt) = 0αN 1−α t F (Kt, 0) = K α t 01−α. Since 0 raised to any power other than 0 is 0, as long as α ≠ 1 or α ≠ 0, both inputs are necessary for production. The optimization problem of the ﬁrm is to choose capital and labor so as to maximize 75 the di�
�erence between revenue and total costs or, more simply, maximize proﬁt (denoted by Πt). Stated in math, the problem is: max Kt,Nt Πt = AF (Kt, Nt) − wtNt − RtKt. (5.3) where wt denotes the real wage paid to labor and Rt denotes the real return to capital. Note that revenue equals output rather than output times a price. The output good, let’s say fruit, is the unit in which every other price is denominated. For example, if wt = 3, workers receive three units of fruit per unit of time spent working.2 The ﬁrst order conditions for the representative ﬁrm are: wt = AFN (Kt, Nt) Rt = AFK(Kt, Nt). (5.4) (5.5) These conditions say that the ﬁrm ought to hire capital and labor up until the point at which the marginal beneﬁt of doing so (the marginal product of capital or labor) equals the marginal cost of doing so (the factor price). As you will see in a question at the end of the chapter, the assumption of constant returns to scale implies that proﬁt is equal to zero and the number of ﬁrms is indeterminate. Consequently, nothing is lost by assuming one representative ﬁrm. There exists a representative household in the economy. This household is endowed with time, Nt, and an initial stock of capital, Kt. It earns income from supplying capital and labor to the ﬁrm, wtNt + RtKt. It can consume its income, Ct, or invest some of it in additional capital, It. Formally, its budget constraint is: Ct + It ≤ wtNt + RtKt + Πt. (5.6) Although separate decision-making units, the household owns the ﬁrm through common stock, and Πt is a dividend payment equal to any proﬁt earned by the ﬁrm. As discussed above, however, the ﬁrm earns no proﬁt under the assumption of constant returns to scale, and wtNt + RtKt = Yt. This simply says that total income equals total output. (5.
6) holding with equality (i.e. replacing the ≤ with a = sign) means that total expenditure, Ct + It, equals total income which equals total output. In other words: Yt = Ct + It. (5.7) 2When we introduce money later in this book, we will denominate all goods in terms of money, i.e. in in nominal, rather than real, terms. 76 Although the initial level of capital, Kt, is given, future levels of capital can be inﬂuenced through investment. In particular, investment in period t yields new capital in period t + 1. Furthermore, some existing capital depreciates (or becomes obsolete) during production. Formally, capital accumulates according to: Kt+1 = It + (1 − δ)Kt. (5.8) We refer to (5.8) as the capital accumulation equation, or sometimes as a “law of motion” for capital. This equation says that your capital stock in t + 1 equals your investment in period t plus the non-depreciated stock of capital with which you started, where 0 < δ < 1 is the depreciation rate (the fraction of the capital stock that goes bad or becomes obsolete each period). In writing (5.8), we have implicitly assumed that one unit of investment yields one unit of future capital. We could have this transformation be greater than or less than one without fundamentally changing anything. We also assume that there is a one period delay between when investment is undertaken and when the new capital becomes productive. Because capital must itself be produced, there must be some delay between when investment is undertaken and when the capital becomes usable. For example, if a ﬁrm decides to build a new manufacturing plant, it cannot use the new plant to produce output in the period in which it decided to build the new plant because it takes some time for the new plant to be built. We could assume a longer than one period delay without fundamentally changing any of the subsequent analysis. Let’s return to the lawn mower example given above. Suppose your capital stock is ten lawn mowers, Kt = 10. Suppose that the depreciation rate is δ = 0.1. Suppose you produce 3 units of output in period t, Yt = 3. If you choose to consume all of your output in period t, Ct = 3, you will have It = 0, so you will have only Kt+
1 = 9 lawn mowers in the next period. If instead you consume two units of output in period t, Ct = 2, you will have It = 1 and hence Kt+1 = 10, the same as it was in period t. If you consume only one unit of output, Ct = 1, then you’ll have It = 2 and hence Kt+1 = 11. The beneﬁt of not consuming all your output in period t is that it leaves you more capital in t + 1, which means you can produce more output in the future (since FK > 0), which aﬀords you the opportunity to consume more in the future. Hence, the decision of how much to invest (equivalently, how much to not consume, i.e. how much to save) is fundamentally an intertemporal decision about trading oﬀ current for future consumption. The Solow model assumes that investment is a constant fraction of output. In particular, 77 let 0 < s < 1 denote the saving rate (equivalently, the investment rate): Combining (5.9) with (5.7) implies: It = sYt. Ct = (1 − s)Yt. (5.9) (5.10) The Solow model is therefore assuming that the economy as a whole consumes a constant fraction of its output each period, investing the other fraction. The assumption of a constant saving rate is not, in general, going to be optimal from a microeconomic perspective in the short run. But over long periods of time, it seems consistent with the data, as documented in Chapter 4. Finally, we assume that the household supplies labor inelastically. This means that the amount of time the household spends working is independent of the factor price to supplying labor, wt. Hence, we can take the overall quantity of Nt as exogenous and ﬁxed across time. This is also not consistent with optimizing microeconomic behavior in the short run, but is again consistent with long run trends, where total labor hours per capita is roughly trendless. All together, the Solow model is characterized by the following equations all simultaneously holding: Yt = AF (Kt, Nt) Yt = Ct + It Kt+1 = It + (1 − δ)Kt It = sYt wt = AFN (Kt, Nt) Rt = AFK(Kt
, Nt). (5.11) (5.12) (5.13) (5.14) (5.15) (5.16) This is six equations and six endogenous variables – Yt, Ct, It, Kt+1, wt, and Rt.3 Nt, Kt, and A are exogenous variables (taken as given) and s and δ are parameters. It is useful to think about an example economy. Suppose that output, Yt, is units of fruit. Capital, Kt, is trees. Labor, Nt, is hours spent picking fruit from the trees. Trees have to be planted from unconsumed fruit, and we assume that one unit of unconsumed fruit yields one tree in the next period. Labor and capital are paid in terms of units of fruit – so the units of 3The reason that (5.10) is not listed here is because it is redundant given that both (5.12) and (5.14) must hold. 78 wt and Rt are units of fruit. So, the household “wakes up” in period t with a stock of capital (say 10 trees) and an endowment of time (say 24 hours). It leases its trees to a ﬁrm for Rt fruits per tree, and follows a rule of thumb where it supplies a ﬁxed amount of labor (say Nt = 8 hours) for wt fruits per unit of time. The ﬁrm transforms the trees and time into fruit. The household’s total income equals total fruit production, Yt. The household follows a rule of thumb in which it consumes a constant fraction of its income (say 80 percent, so s = 0.2), and plants the remainder in the ground, which yields additional trees (capital) in the future. Equations (5.11), (5.13), and (5.14) can be combined into one central equation describing the evolution of capital over time. In particular, we have It = sAF (Kt, Nt) from combining (5.11) with (5.14). Plugging it into (5.13), we are left with: Kt+1 = sAF (Kt, Nt) + (1 − δ)Kt. (5.17) This equation describes the evolution of Kt. Given an exogenous current value of Kt
, it tells you how much Kt+1 an economy will have, given exogenous values for A and Nt, and values for the parameters s and δ. It is helpful to write this in terms of capital per work. Divide both sides of (5.18) by Nt: Kt+1 Nt = sAF (Kt, Nt) Nt + (1 − δ)Kt Nt. (5.18) Let’s deﬁne kt ≡ Kt/Nt. We will call this variable capital per worker, or sometimes capital per capita (since the model has inelastically supplied labor, capital per worker and capital per capita will be the same up to a scale factor reﬂecting the labor force participation rate, which we are not modeling here).4 Using the properties of the production function, in particular the assumption that it is constant returns to scale, we can write: F (Kt, Nt) Nt = F (Kt Nt, Nt Nt ) = F (kt, 1). So as to economize on notation, we will deﬁne f (kt) ≡ F (kt, 1) as the per worker production function. We can therefore write (5.18) as: Kt+1 Nt = sAf (kt) + (1 − δ)kt. Multiply and divide the left hand side by Nt+1, re-arranging terms so as to write it in terms 4This is admittedly somewhat poor terminology because taken literally there is only one worker in the economy (the representative household) which supplies an exogenous amount of time in the form of labor to the ﬁrm. Therefore, it would be more appropriate to refer to kt as capital per labor input, but we will henceforth engage in an abuse of terminology and call it capital per worker. 79 of capital per worker: Kt+1 Nt+1 Nt+1 Nt = sAf (kt) + (1 − δ)kt. Since we are assuming that labor is constant across time, this means that Nt+1/Nt = 1. So we can write: kt+1 = sAf (kt) + (1 − δ)kt. (5.19) Equation (5.19) is the central equation of the Solow model. It describes how capital per worker
evolves over time, given an initial value of the capital stock, an exogenous value of A, and parameter values s and δ. Once we know the dynamics of kt, we can back out the dynamics of all other variables. We can deﬁne yt, ct, and it as output, consumption, and investment per worker. In terms of kt, these can be written: yt = Af (kt) ct = (1 − s)Af (kt) it = sAf (kt). (5.20) (5.21) (5.22) To get expressions for wt and Rt in terms of kt, we need to use something called Euler’s theorem, explained below in the Mathematical Diversion. The rental rate and wage can be written: Rt = Af ′(kt) wt = Af (kt) − ktAf ′(kt). (5.23) (5.24) Mathematical Diversion Referring back to the assumed mathematical properties of the production function, we assumed that the production function has constant returns to scale. In words, this means that doubling both inputs results in a doubling of output. A fancier term for constant returns to scale is to say that the function is homogeneous of degree 1. More generally, a function is homogeneous of degree ρ if: F (γKt, γNt) = γρF (Kt, Nt). where γ = 1 corresponds to the case of constant returns to scale. γ < 1 is what is called decreasing returns to scale (meaning that doubling both inputs results in a less than doubling of output), while γ > 1 is increasing returns to 80 scale (doubling both inputs results in a more than doubling of output). Euler’s theorem for homogeneous functions states (see Mathworld (2016)) if a function is homogeneous of degree ρ, then: ρF (Kt, Nt) = FK(Kt, Nt)Kt + FN (Kt, Nt)Nt. (5.25) If ρ = 1 (as we have assumed), this says that the function can be written as the sum of partial derivatives times the factor being diﬀerentiated with respect to. To see this in action for the Cobb-Douglas production function, note: K α t N 1−
α t = αK α−1 = αK α = K α t N 1−α t N 1−α t t N 1−α t t Kt + (1 − α)K α + (1 − α)K α t N 1−α t (α + 1 − α) = K α t N 1−α t. t N −α t Nt Euler’s theorem also states that, if a function is homogeneous of degree ρ, then its ﬁrst partial derivatives are homogeneous of degree ρ − 1. This has the implication, for example, that: FK(γKt, γNt) = γρ−1FK(Kt, Nt). Since we are working with a constant returns to scale function, meaning ρ = 1, this means that you can scale both inputs by a factor and not change the partial derivative. Concretely, this means that: FK(Kt, Nt) = FK ( Kt Nt, Nt Nt ) = f ′(kt). (5.26) In other words, (5.26) means that the partial derivative of F (⋅) with respect to Kt is the same thing as the partial derivative of f (⋅) with respect to kt. This yields (5.23) above. To get (5.24), use this result plus (5.25) to get: F (Kt, Nt) = f ′(kt)Kt + FN (Kt, Nt)Nt f (kt) = f ′(kt)kt + FN (Kt, Nt) FN (Kt, Nt) = f (kt) − f ′(kt)kt. (5.27) The second line in (5.27) follows by dividing both sides of the ﬁrst line by Nt. The last line is just re-arrangement. Since wt = AFN (Kt, Nt), using the last line of (5.27) we get the expression in (5.24). Example Suppose that we have the Cobb-Douglas production function. The 81 central equation of the Solow model can be written: kt+1 = sAkα t + (1 − δ)kt. (5.28) The other variables are determined
as a function of kt. These can be written: yt = Akα t ct = (1 − s)Akα t it = sAkα t Rt = αAkα−1 wt = (1 − α)Akα t. t (5.29) (5.30) (5.31) (5.32) (5.33) 5.2 Graphical Analysis of the Solow Model We will use both graphs and math to analyze the Solow model. We will start with graphical analysis. Consider the central equation of the Solow model, (5.19). Let’s graph kt+1 as a function of kt (which is predetermined in period t and therefore exogenous). If kt = 0, then kt+1 = 0 given that we assume capital is necessary for production. This means that in a graph with with kt on the horizontal axis and kt+1 on the vertical axis, the graph starts in the origin. How will kt+1 vary as kt changes? To see this, let’s take the derivative of kt+1 with respect to kt: dkt+1 dkt = sAf ′(kt) + (1 − δ). (5.34) Equation (5.34) is an expression for the slope of the graph of kt+1 against kt. The magnitude of this slope depends on the value of kt. Since f ′(kt) is positive and δ < 1, the slope is positive – so kt+1 is increasing in kt. Since f ′′(kt) < 0, the term sAf ′(kt) gets smaller as kt gets bigger. This means that kt+1 is an increasing function of kt, but at a decreasing rate. Let’s assume two additional conditions, which are sometimes called “Inada conditions.” In particular, assume that: f ′(kt) = ∞ f ′(kt) = 0. lim kt→0 lim kt→∞ (5.35) (5.36) In words, (5.35) says that the marginal product of capital is inﬁnite when there is no capital, while (5.36) says that the marginal product of capital goes to zero as the capital stock per worker gets inﬁnitely large
. These conditions together imply that dkt+1 starts out dkt 82 at the origin at positive inﬁnity but eventually settles down to 1 − δ, which is positive but less than one. Suppose that the production function is Cobb-Douglas, so that the Example central equation of the Solow model is given by (5.28). The expression for the slope of the central equation is: dkt+1 dkt = αsAkα−1 t + (1 − δ). This can equivalently be written: dkt+1 dkt = αsA ( 1 kt ) 1−α + (1 − δ). (5.37) (5.38) If kt = 0, then 1 → ∞. Since 1 − α > 0, and inﬁnity raised to any positive number kt → 0. 0 raised to is inﬁnity, the slope is inﬁnity. Likewise, if kt → ∞, then 1 kt any positive power is 0. Hence, the Inada conditions hold for the Cobb-Douglas production function. Figure 5.1 plots kt+1 as a function of kt. The plot starts in the origin, starts out steep, and ﬂattens out as kt gets bigger, eventually having a slope equal to 1 − δ. We add to this plot what is called a 45 degree line – this is a line which plots all points where the horizontal and vertical axes variables are the same, i.e. kt+1 = kt. It therefore has a slope of 1. Since it splits the plane in half, it is often called a 45 degree line. The 45 degree line and the plot of kt+1 both start at the origin. The kt+1 plot starts out with a slope greater than 1, and hence initially lies above the 45 degree line. Eventually, the plot of kt+1 has a slope less than 1, and therefore lies below the 45 degree line. Since it is a continuous curve, this means that the plot of kt+1 cross the 45 degree line exactly once away from the origin. We indicate this point with k∗ – this is the value of kt for which kt+1 will be the same as kt, i.e. kt+1 = kt = k∗. We will refer to this point, k
∗, as the “steady state.” 83 Figure 5.1: Plot of Central Equation of Solow Model It is useful to include the 45 degree line in the plot of kt+1 against kt because this makes it straightforward to use the graph to analyze the dynamics of the capital stock per worker. The 45 degree line allows one to “reﬂect” the horizontal axis onto the vertical axis. Suppose that the economy begins with a period t capital stock below the steady state, i.e. kt < k∗. One can read the current capital stock oﬀ of the vertical axis by reﬂecting it with the 45 degree line. This is labeled as “initial point in period t” in Figure 5.2. The next period capital stock, kt+1, is determined at the initial kt from the curve. Since the curve lies above the 45 degree line in this region, we see that kt+1 > kt. To then think about how the capital stock will evolve in future periods, we can functionally iterate the graph forward another period. Use the 45 degree to reﬂect the new value of kt+1 down onto the horizontal axis. This becomes the initial capital stock in period t + 1. We can determine the capital stock per worker in period t + 2 by reading that point oﬀ of the curve at this new kt+1 (labeled as “initial point in period t + 1” in the graph). We can continue iterating with this procedure as we move “forward” in time. We observe that if kt starts below k∗, then the capital stock will be expected to grow. 84 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑠𝑠𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡)+(1−𝛿𝛿)𝑘𝑘𝑡𝑡 𝑘𝑘∗ 𝑘𝑘∗
Figure 5.2: Convergence to Steady State from kt < k∗ Figure 5.3 repeats the analysis but assumes the initial capital stock per worker lies about the steady state, kt > k∗. The process plays out similar, but in reverse. Since in this region the line lies above the curve, the capital stock per worker will get smaller over time, eventually approaching the steady state point. 85 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑠𝑠𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡)+(1−𝛿𝛿)𝑘𝑘𝑡𝑡 𝑘𝑘∗ 𝑘𝑘∗ 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+2 𝑘𝑘𝑡𝑡+3 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+2 𝑘𝑘𝑡𝑡+3 Initial point in period 𝑡𝑡 Initial point in period 𝑡𝑡+1 Initial point in period 𝑡𝑡+2 Initial point in period 𝑡𝑡+3 Figure 5.3: Convergence to Steady State from kt > k∗ The analysis displayed in Figures 5.2 and 5.3 reveals a crucial point. For any non-zero starting value of kt, the capital stock per worker ought to move toward k∗ over time. In other words, the steady state capital stock per work is, in a sense, a point of attraction – starting from any initial point, the dynamics embedded in the model will continuously move the economy toward that point. Once the economy reaches kt = k∗,
it will stay there (since kt+1 = kt at that point), hence the term “steady.” Furthermore, the capital stock will change quite a bit across time far from the steady state (i.e. at these points the vertical gap between the curve and the line is large) and will change very little when the initial capital stock is close to the steady state (the curve is close to the line in this region). Figure 5.4 plots hypothetical time paths of the capital stock, where in one case kt > k∗ and in the other kt < k∗. In the former case, kt declines over time, approaching k∗. In the latter, kt increases over time, also approaching k∗. 86 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑠𝑠𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡)+(1−𝛿𝛿)𝑘𝑘𝑡𝑡 𝑘𝑘∗ 𝑘𝑘∗ 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+2 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+2 Initial point in period 𝑡𝑡 Initial point in period 𝑡𝑡+1 Initial point in period 𝑡𝑡+2 Figure 5.4: Convergence to Steady State The steady state is a natural point of interest. This is not because the economy is always at the steady state, but rather because, no matter where the economy starts (provided it does not start with kt = 0), it will naturally gravitate towards this point. An alternative way to graphically analyze the Solow model, one that is commonly presented in textbooks,
is to transform the central equation of the Solow model, (5.19), into ﬁrst diﬀerences. In particular, deﬁne ∆kt+1 = kt+1 − kt. Subtracting kt from both sides of (5.19), one gets: ∆kt+1 = sAf (kt) − δkt. (5.39) In (5.39), the ﬁrst term on the right hand side, sAf (kt), is total investment. The second term, δkt, is total depreciation. This equation says that the change in the capital stock is equal to the diﬀerence between investment and depreciation. Sometimes the term δkt is called “break-even investment,” because this is the amount of investment the economy must do so as to keep the capital stock from falling. 87 𝑗 0 𝑘∗ 𝑘𝑡 𝑘𝑡+1 𝑘𝑡+2 𝑘𝑡+1 𝑘𝑡 𝑘𝑡+2 𝑘𝑡+𝑗 1 2 Figure 5.5: Alternative Plot of Central Equation of Solow Model Figure 5.5 plots the two diﬀerent terms on the right hand side of (5.39) against the initial capital stock per worker, kt. The ﬁrst term, sAf (kt), starts at the origin, is increasing (since f ′(kt) > 0), but has diminishing slope (since f ′′(kt) < 0). Eventually, as kt gets big enough, the slope of this term goes to zero. The second term is just a line with slope δ, which is positive but less than one. The curve must cross the line at some value of kt, call it k∗. This single crossing is guaranteed if the Inada conditions hold, which we assume they do. This is the same steady state capital stock derived using the alternative graphical depiction. For values of kt < k∗, we have the curve lying above the line, which means that investment, sAf (kt), exceeds depreciation, δkt, so that the capital stock will be expected to grow over time. Alternatively, if kt > k�
�, then depreciation exceeds investment, and the capital stock will decline over time. We prefer the graphical depiction shown in Figure 5.1 because we think it is easier to use the graph to think about the dynamics of capital per worker across time. That said, either graphical depiction is correct, and both can be used to analyze the eﬀects of changes in exogenous variables or parameters. 88 𝑠𝑠𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡), 𝛿𝛿𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡 𝛿𝛿𝑘𝑘𝑡𝑡 𝑠𝑠𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡) 𝑘𝑘∗ ∆𝑘𝑘𝑡𝑡+1= 𝑠𝑠𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡)−𝛿𝛿𝑘𝑘𝑡𝑡 5.3 The Algebra of the Steady State with Cobb-Douglas Produc- tion Suppose that the production function is Cobb-Douglas, so that the central equation of the model is given by (5.28) and the other variables are determined by (5.29). To algebraically solve for the steady state capital stock, take (5.28) and set kt+1 = kt = k∗: k∗ = sAk∗α + (1 − δ)k∗. This is one equation in one unknown. k∗ is: k∗ = (sA δ ) 1 1−α. (5.40) We observe that k∗ is increasing in s and A and decreasing in δ. All the other variables in the model can be written as functions of kt and parameters. Hence, there will exist a steady state in these other variables as well. Plugging (5.40) in wherever kt shows up, we get: y∗ = Ak∗α c∗ = (1 − s)Ak∗α i�
� = sAk∗α R∗ = αAk∗α−1 w∗ = (1 − α)Ak∗α. (5.41) (5.42) (5.43) (5.44) (5.45) 5.4 Experiments: Changes in s and A We want to examine how the variables in the Solow model react dynamically to changes in parameters and exogenous variables. Consider ﬁrst an increase in s. This parameter is exogenous to the model. In the real world, increases in the saving rate could be driven by policy changes (e.g. changes to tax rates which encourage saving), demographics (e.g. a larger fraction of the population is in its prime saving years), or simply just preferences (e.g. households are more keen on saving for the future). Suppose that the economy initial sits in a steady state, where the saving rate is s0. Then, in period t, the saving rate increases to s1 > s0 and is forever expected to remain at s1. In terms of the graph, an increase in s has the eﬀect of shifting the curve plotting kt+1 against kt up. It is a bit more nuanced than simply a shift up, as an increase in s also has the eﬀect of making the curve steeper at every value of kt. This eﬀect can be seen in Figure 89 5.6 below with the blue curve. The 45 degree line is unaﬀected. This means that the curve intersects the 45 degree line at a larger value, k∗ 0. In other words, a higher value of the 1 saving rate results in a larger steady state capital stock. This can be seen mathematically in (5.40) for the case of a Cobb-Douglas production function. > k∗ Figure 5.6: Exogenous Increase in s, s1 > s0 Now, let’s use the graph to think about the process by which kt transitions to the new, higher steady state. The period t capital stock cannot jump – it is predetermined and hence exogenous within period. We can determine the t + 1 value by reading oﬀ the new, blue curve at the initial kt. We see that kt+1 > kt, so the capital stock per worker will grow after an increase in the saving rate. From that point
on, we continue to follow the dynamics we discussed above in reference to Figure 5.2. In other words, when s increases, the economy is suddenly below its steady state. Hence, the capital stock will grow over time, eventually approaching the new, higher steady state. 90 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑠𝑠0𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡)+(1−𝛿𝛿)𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡=𝑘𝑘0∗ 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+1=𝑠𝑠1𝐴𝐴𝑓𝑓(𝑘𝑘𝑡𝑡)+(1−𝛿𝛿)𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+2 𝑘𝑘𝑡𝑡+1 𝑘𝑘1∗ 𝑘𝑘1∗ Figure 5.7: Dynamic Responses to Increase in s We can trace out the dynamic path of the capital stock per worker to an increase in s, which is shown in the upper left panel of Figure 5.7. Prior to period t, assume that the economy sits in a steady state associated with the saving rate s0. In period t (the period in which s increases), nothing happens to the capital stock per worker. It starts getting bigger in period t + 1 and continues to get bigger, though at a slower rate as time passes. Eventually, 1. it will approach the new steady state associated with the higher saving rate, k∗ Once we have the dynamic path of kt, we can back out the dynamic paths of all other 91 𝑘𝑡 𝑦𝑡 �
��𝑡 𝑖𝑡 𝑤𝑡 𝑅𝑡 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡 𝑡 𝑡 𝑡 𝑡 𝑡 𝑘0∗ 𝑦0∗ 𝑐0∗ 𝑖0∗ 𝑤0∗ 𝑅0∗ 𝑘1∗ 𝑦1∗ 𝑐1∗ 𝑖1∗ 𝑤1∗ 𝑅1∗ variables in the model. Since yt = Af (kt), output will follow a similar looking path to kt – it will not change in period t, and then will grow for a while, approaching a new, higher steady state value. Note that the response graphs in Figure 5.7 are meant to be qualitative and are not drawn to scale, so do not interpret anything about the magnitudes of the responses of kt and other variables. Since ct = (1 − s)yt, consumption per worker must initially decline in the period in which the saving rate increases. Eﬀectively, the “size of the pie,” yt, doesn’t initially change, but a smaller part of the pie is being consumed. After the initial decrease, consumption will begin to increase, tracking the paths of kt and yt. Whether consumption ends up in a higher or lower steady state than where it began is unclear, though we have drawn the ﬁgure where consumption eventually ends up being higher. Investment is it = syt. Hence, investment per worker must jump up in the period in which the saving rate increases. It will thereafter continue to increase as capital accumulates and transitions to the new steady state. wt will not react in period t, but will follow a similar dynamic path as the other variables thereafter. This happens because of our underlying assumption that FN K > 0 – so having more capital raises the
marginal product of labor, and hence the wage. The rental rate on capital, Rt, will not react in the period s increases but then will decrease thereafter. This is driven by the assumption that FKK < 0. As capital accumulates following the increase in the saving rate, the marginal product of capital falls. It will continue to fall and eventually ends up in a lower steady state. What happens to the growth rate of output after an increase in s? Using the approximation that the growth rate is approximately the log ﬁrst diﬀerence of a variable, deﬁne gy = t ln yt − ln yt−1 as the growth rate of output. Since output per worker converges to a steady state, in steady state output growth is 0. In the period of the increase in s, nothing happens to output, so nothing happens to output growth. Since output begins to increase starting in period t + 1, output growth will jump up to some positive value in period t + 1. It will then immediately begin to decrease (though remain positive), as we transition to the new steady state, in which output growth is again zero. This is displayed graphically in Figure 5.8. 92 Figure 5.8: Dynamic Path of Output Growth The analysis portrayed graphically in Figure 5.8 has an important and powerful implication – output will not forever grow faster if an economy increases the saving rate. There will be an initial burst of higher than normal growth immediately after the increase in s, but this will dissipate and the economy will eventually return to a steady state no growth. Next, consider the experiment of an exogenous increase in A. In particular, suppose that, prior to period t, the economy sits in a steady state associated with A0. Then, suppose that A increases to A1. This change is permanent, so all future values of A will equal A1. How will this impact the economy? In terms of the main graph plotting kt+1 against kt, this has very similar eﬀects to an increase in s. For every value of kt, kt+1 will be higher when A is higher. In other words, the curve shifts up (and has a steeper slope at every value of kt). This is shown in Figure 5.9 below. We can use the ﬁgure to think about the dynamic eﬀects on kt. Since the curve is shifted up relative
to where it was with A0,t, we know that the curve will intersect the 45 degree line at a higher value, meaning that the steady state capital stock will be higher, k∗ 0. In 1 period t, nothing happens to kt. But since the curve is now shifted up, we will have kt+1 > kt. Capital will continue to grow as it transitions toward the new, higher steady state. > k∗ 93 𝑔𝑔𝑡𝑡𝑦𝑦 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 𝑡𝑡 𝑡𝑡+1 0 Figure 5.9: Exogenous Increase in A, A1 > A0 Given a permanent higher value of A, once we know the dynamic path of kt we can determine the dynamic paths of the other variables just as we did in the case with an increase in s. These are shown in Figure 5.10. The capital stock per worker does not jump in period t, but grows steadily thereafter, eventually approaching a new higher steady state. Next, consider what happens to yt. Since yt = Af (kt), yt jumps up initially in period t (unlike the case of an increase in s). This increase in period t is, if you like, the “direct eﬀect” of the increase in A on yt. But yt continues to grow thereafter, due to the accumulation of more capital. It eventually levels oﬀ to a new higher steady state. ct and it follow similar paths as yt, since they are just ﬁxed fractions of output. The wage also follows a similar path – it jumps up initially, and then continues to grow as capital accumulates. The rental rate on capital, Rt, initially jumps up. This is because higher A makes the marginal product of capital higher. But as capital accumulates, the marginal product of capital starts to decline. Given the assumptions we have made on the production function, one can show that Rt eventually settles back to where it began – there is no eﬀect of A on the steady state value of Rt. 94 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡 𝑘𝑘�
��𝑡+1=𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+1=𝑠𝑠𝐴𝐴0𝑓𝑓(𝑘𝑘𝑡𝑡)+(1−𝛿𝛿)𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡=𝑘𝑘0∗ 𝑘𝑘𝑡𝑡+1 𝑘𝑘𝑡𝑡+1=𝑠𝑠𝐴𝐴1𝑓𝑓(𝑘𝑘𝑡𝑡)+(1−𝛿𝛿)𝑘𝑘𝑡𝑡 𝑘𝑘𝑡𝑡+2 𝑘𝑘𝑡𝑡+1 𝑘𝑘1∗ 𝑘𝑘1∗ Figure 5.10: Dynamic Responses to Increase in A Mathematical Diversion How does one know that there is no long run eﬀect of A on Rt? Suppose that the 95 𝑘𝑡 𝑦𝑡 𝑐𝑡 𝑖𝑡 𝑤𝑡 𝑅𝑡 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑡𝑖𝑚𝑒 𝑘0∗ 𝑦0∗ 𝑐0∗ 𝑖0∗ 𝑤0∗ 𝑅0∗ 𝑘1∗ 𝑦1∗ 𝑐1∗ 𝑖1∗ 𝑤1∗ 𝑡 𝑡 𝑡 𝑡 𝑡 �

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