{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Part II: Economic Growth\n", "\n", "The theory of long-run economic growth is not very closely connected to the study of business cycles—of recessions, unemployment, inflation, and stabilization policy—that makes up the bulk of this book.\n", "\n", "Why, then, include it? The principal reason is that long-run economic growth is such an important topic. It is quite likely the most important topic in macroeconomics. And so it is the subject of this Part II. \n", "\n", "Our group of humans spread out from the Horn of Africa more than 50000 years ago. Humans developed agriculture and herding more than 10000 years ago. We see cities, complex and sophisticated divisions of labor, and substantial short- and long-distance trade starting 5000 years ago. We now live on all seven and have for millennia lived on six of the seven continents of the globe. And while 10000 years ago there were perhaps 5 million people on the globe with a total annual income at today's prices of perhaps 5 billion dollars a year, today there are 7.5 billion people on the globe with a total annual income at today's prices of perhaps 10 trillion dollars a year.\n", "\n", "What has taken us from the state we were in 10000 years ago to the state we are in today is the process of economic growth. The most important thing to know about economic growth is its magnitude and its pace—and how uneven the pace of economic growth has been across time.\n", "\n", "The second most important thing to know about economic growth is how uneven the process has been across space. Some of us are lucky enough to have chosen the right parents, and so we live in the Global North—North America, Western Europe, Australia and New Zealand, and Japan and South Korea—and today have a material standard of living at least fifteen times what our predecessors had 250 years ago. This level of material prosperity—remarkable in comparative and astonishing in historical perspective—is the result of a similar upward leap in economic productivity. Almost all citizens of the Global North today live better, and we hope happier, lives than even the narrowest and the richest of the elites of past centuries.\n", "\n", "By contrast, 1.5 billion of us—one-fifth of us—still live lives that are not that much different from the lives of our material ancestors (with the biggest difference being life expectancy: nearly all babies born today can expect to live to 60 or more, while nearly all born into pre-industrial societies had a life expectancy of perhaps 25). And the rest of us are spread out between those two extremes, with the average person in China today perhaps five times as well-off as our typical pre-industrial ancestors and perhaps one-third as well off-off as those of us in the Global North. \n", "\n", "The study of long-run economic growth aims at understanding its sources and causes. It aims at determining what government policies will promote or retard long-run economic growth. It aims to build and then apply theories to understand the worldwide pattern of economic growth—both across time and across space—that we have seen.\n", "\n", "One final thing: Note that the lack of close connection between the economics of long-run growth and the economics of business cycles means that, by and large, the models developed and this the conclusions reached in Part II will by and large not be central to the subsequent parts of the book that turn to the analysis of business cycles.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 4: Long-Run Economic Growth: Theory\n", "\n", "**QUESTIONS**:\n", "\n", "1. What are the causes of long-run economic growth—that is, of sustained and significant growth in an economy's level of output per worker?\n", "\n", "2. What is the \"efficiency of labor\"?\n", "\n", "3. What is an economy's \"capital intensity\"?\n", "\n", "4. What is an economy's \"balanced-growth path\"?\n", "\n", "5. How important is faster labor-force growth as a drag on economic growth?\n", "\n", "6. How important is a high saving rate as a cause of economic growth?\n", "\n", "7. How important is technological and organizational progress for economic growth?\n", "\n", " \n", "\n", "**GLOSSARY**\n", "\n", "**economic growth**: The process by which productivity, living standards, and output increase.\n", "\n", "**capital intensity**: The ratio of the capital stock to total potential output, K/Y: how valuable the capital stock is as a multiple of annual production.\n", "\n", "**efficiency of labor**: Determined skills and education of the labor force, the ability of the labor force to handle modern technologies, and the efficiency with which the economy's businesses and markets function. Technology, organization, skills, and education as an amplifier of human productivity and the principle source of economic growth and of high levels of productivity and standards of living\n", "\n", "**labor force**: The numer of those who are employed plus the number of those who are actively looking for work.\n", "\n", "**capital stock**: The economy's total accumulated stock of buildings, roads, other infrastructure, machines, and inventories.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4.1 Sources of Long-Run Economic Growth\n", "\n", "### 4.1.1 The Eagle's Eye View\n", "\n", "#### 4.1.1.1 The Shape of Global Growth\n", "\n", "Step back and take a broad, sweeping view of the economy.\n", "\n", "Look at it, but do not focus on the economists' “short-run” of calendar-year quarters or even of a year or two in which shifts in investment spending and other shocks push the flow of production and the unemployment rate up or down—that’s what we will do in Part IV. \n", "\n", "Look at it, but do not focus on the “long-run” period of 3 to 10 years or so, in which prices have time to adjust to return the economy to a full-employment equilibrium, but in which the economy’s productive resources do not change much—that’s what we will also look at in Part III. \n", "\n", "What do we do here? We take a step back and focus on the very long-run. We focus on periods of generations or more—periods over which everything else dwindles into insignificance except the sustained and significant increases in standards of living that we call long-run economic growth.\n", "\n", "When we take this broad, sweeping view, it is clear that what we are calling long-run economic growth is _the_ only truly important factor determining a market economy's potential for generating prosperity. Even the biggest business-cycle depressions have only pushed production per worker and real incomes down by a quarter. Yet the process of economic growth has, over the past two centuries, boosted production per worker and incomes at least fifteen-fold \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.1.1.2 Economic Prosperity Gaps Across Space\n", "\n", "Material standards of living and levels of economic productivity today in the United States are more than three times what they are in, say, Mexico (and more than nine times those of Nigeria, and more than 25 times those of Afghanistan). These differences matter for more than just how comfortable people are: richer societies have much longer life expectancies in addition to higher levels of consumption and greater control over resources by typical citizens. Even a large economic depression would not reduce U.S. levels of prosperity to those of Mexico, let alone Nigeria or Afghanistan.\n", "\n", " \n", "\n", "\"Gapminder\n", "\n", ">**U.S. Output per Capita in International Perspective** Source: _Gapminder_ \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.1.1.3 Economic Prosperity Gaps Over Time\n", "\n", "Moreover U.S. average living standards and productivity levels have not been near their current high levels for long. Standard estimates show that U.S. GDP per capita has grown roughly fifteen-fold since 1870. And life expectancy at birth in the U.S. has doubled over the past century and a half. Even a large economic depression would not reduce U.S. levels of productivity to those of the U.S. a generation, let alone a century ago.\n", "\n", " \n", "\n", "\"4\n", "\n", ">**U.S. Output per Capita in Historical Perspective**: Source: _Gapminder_ \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.1.1.4 Productivity—Labor Efficiency—Gaps as Key\n", "\n", "These differences and changes are huge.\n", "\n", "At most a trivial part of them are due to whether unemployment in a country is currently above or below its average level, or whether various bad macroeconomic policies are currently disrupting the functioning of the price system. \n", "\n", "The overwhelming bulk of these differences are the result of differences in economies’ productive potentials. The most important differences spring from differences in the level of technology and of organization currently used in production, which are themelves the result of differences in the skills of workers, the availability and competence of engineers and engineering knowledge, and the creation and maintenance of organizations with internal structures that impel them toward profductive efficiency. \n", "\n", "Secondary differences spring from differences in the presence or absence of key natural resources, and in the value of the capital stock—the structures and buildings, machines and programs, and inventories and work flows that have themselves been produced by humans earlier and are necessary for a productive economy.\n", "\n", "The enormous gaps between the productive potentials of different nations spring from favorable initial conditions and successful growth-promoting economic policies in the United States—and from less favorable initial conditions and less successful subsequent policies in Mexico and downright unsuccessful policies in Nigeria and Afghanistan.\n", "\n", "Moreover, the bulk of today’s gap between living standards and productivity levels in the United States and Mexico (and Nigeria, and Afghanistan) opened up in the past century and a half; the bulk of success (or failure) at boosting an economy’s productive potential is thus—to a historian at least—of relatively recent origin. A century and a half ago U.S. GDP per capita was four times Afghan, not twenty times. While U.S. productive potential has amplified 13-fold since 1870, Afghan has only amplified two-fold and Nigerian five-fold. (Do, however, note that life expectancies at birth have more than doubled everywhere: humanity has done better at distributing the technologies that reduce infant mortality and enable longer life expectancy across the globe than it has done at distributing the technologies that boost production.\n", "\n", " \n", "\n", "\"4\n", "\n", "**Growth since 1870: Afghan, Nigerian, Mexican, and American**: Source: _Gapminder_ \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.1.1.5 Reversals of Fortune\n", "\n", "Successful economic growth means that nearly all citizens of the United States today live better—along almost every dimension of material life—than did even the rich elites of preindustrial times. If good policies and good circumstances accelerate economic growth, bad policies and bad circumstances cripple long-run economic growth. Argentineans were richer than Swedes before World War I, but Swedes today have four times the standard of living and the productivity level of Argentineans.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.1.1.5.1 Sweden and Argentina: Some Details\n", "\n", "\"2017\n", "\n", ">**Long-Run Measured Economic Growth: Sweden and Argentina, 1890-2017**: Source: _Gapminder_ \n", "\n", " \n", "\n", "At the start of the twentieth century, Argentina was richer—and seen as having a brighter future—than Sweden. Europeans in large numbers were then migrating to Argentina, not Sweden—and Swedes were leaving their country in large numbers for greener pastures elsewhere, largely in the Americas. But economic policies that were mostly bad for long-run growth left Argentina far behind Sweden. The average Swede today looks to be more than sixteen times as well off in terms of material income as the average Swede of 1890. The average Argentinian today looks to be only between three and a third times as well off as their predecessor back in 1890.\n", " \n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.1.1.5.2 Gapminder: An Information Source\n", "\n", "On the World Wide Web at: is _Gapminder_. It is, its mission statement says:\n", "\n", ">an independent Swedish foundation... a fact tank, not a think tank.... fight[ing] devastating misconceptions about global development. Gapminder produces free teaching resources making the world understandable based on reliable statistics. Gapminder promotes a fact-based worldview everyone can understand...\n", "\n", "and Gapminder exists because:\n", "\n", ">We humans are born with a craving for... drama. We pay attention to dramatic stories and we get bored if nothing happens. Journalists and lobbyists tell dramatic stories... about extraordinary events and unusual people. The piles of dramatic stories pile up in people’s minds into an overdramatic worldview and strong negative stress feelings: “The world is getting worse!”, “It’s we vs. them!” , “Other people are strange!”, “The population just keeps growing!” and “Nobody cares!” For the first time in human history reliable statistics exist. There’s data for almost every aspect of global development. The data shows a very different picture: a world where most things improve.... [where] decisions [are] based on universal human needs... easy to understand....\n", "\n", ">Fast population growth will soon be over. The total number of children in the world has stopped growing.... We live in a globalized world, not only in terms of trade and migration. More people than ever care about global development! The world has never been less bad. Which doesn’t mean it’s perfect. The world is far from perfect.\n", "\n", ">The dramatic worldview has to be dismantled, because it is stressful... wrong.... leads to bad focus and bad decisions. We know this because we have measured the global ignorance... [of] top decision makers... journalists, activists, teachers and the general public. This has nothing to do with intelligence. It’s a problem of factual knowledge. Facts don’t come naturally. Drama and opinions do. Factual knowledge has to be learned. We need to teach global facts in schools and in corporate training. This is an exciting problem to work on and we invite all our users to join the Gapminder movement for global factfulness. The problem can be solved, because the data exists...\n", "\n", "Do not be globally ignorant! Explore—and use—Gapminder! And watch Ola and Hans Rosling's \"How Not to Be Ignorant about the World\" talk at: \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.1.2 Factors Determining the Shape of Global Growth\n", "\n", "The overwhelming bulk of differences in prosperity levels both between countries and across large spans of time—the very long-runs that are our focus of analysis here in this Part II—are the result of differences in economies’ productive potentials. We classify the most important factors that generate differences in economies’ productive potentials into two broad groups:\n", "\n", "* First, differences in the economy’s efficiency of labor—how technology is deployed and how organization is used to increase the amount of output a worker can produce with the same amount of capital and other resources. As noted above, these differences are primarily the result of differences in three factors:\n", " * The community of engineering practice\n", " * The skills of workers\n", " * The incentives and efficiency of organizations\n", "\n", "* Second, differences in how large a multiple of current production has been set aside in the form of useful machines, buildings, and infrastructure to boost the productivity of workers, even with the same technology or organization:\n", " * With differences in natural resources—which are sometimes thought of as separate from and sometimes as a form of capital.\n", "\n", "We will call the first group \"the efficiency of labor\" and the second \"capital intensity\". It has been an important finding by economists over the past three generations that capital and investment have played a substantial role: it is not _all_ increases in knowledge, for increases in thrift and resources have played a substantial role. But it has been an even more important finding that the But an even more important finding is that the lion’s share of economic growth both acroos long spans of time and between countries has come from the factors that affect the efficiency of labor. They are: \n", "\n", "1. the advance of knowledge that enables the creation of communities of engineering practice, \n", "2. the diffusion of knowledge and engineering practice, \n", "3. the education and training of workers, and\n", "4. the creation and maintenance of efficient productive organizations.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.1.2.1 The Efficiency of Labor\n", "\n", "The biggest reason that Americans and other Global North in habitanta today are vastly richer and more productive than their predecessors of a couple of centuries ago is that they have enjoyed an extraordinary ampli fication of the efficiency of labor. Efficiency has risen for two reasons: advances in technology and in organization. We now know how to make electric motors, dope semiconductors, transmit signals over fiber optics, fly jet airplanes, machine internal combustion engines, build tall and durable structures out of concrete and steel, record entertainment programs on DVDs, make hybrid seeds, fertilize crops with nutrients, organize assembly lines, and do a host of other things our predecessors did not know how to do. \n", "\n", "These technological advances allow American workers to easily perform value-generating tasks that were unimaginable, or at least extremely difficult to accomplish, a century ago. Moreover, the American economy is equipped to make use of all these technological capabilities. First, we have engineers who understand and can apply modern technologies. There is a very large gap between theoretical knowledge of how the world works and what kinds of things might be possible on the one hand, and actually getting an energy, materials, and machine-based production process working smoothly on the other. An economy that has a large and productive community of engineers with their knowledge and practices will be effective at applying scientific knowledge of how things work and what production processes are conceivable. An economy that does not have such a community of engineering practice does not.\n", "\n", "But all the engineering advice and plans turned into blueprints for factory design and manuals for task performance will do no good if the people actually doing the work—the workers—lack the skills and experience to understand and implement them. Education and training are essential components of an economy with a high efficiency of labor.\n", "\n", "And engineers and workers produce little if they are unwilling to work, or if their work is not coordinated. An honest government and a society that rewards productive work—rather than acquisition by fraud or force—is essential. As Adam Smith wrote back in 1776, The United States today has the forms of business organization, it has the stability and honesty of government, and it has the other socioeconomic institutions needed to successfully utilize modern technology. So it is both better technology and advances in organization that have led to vast increases in the efficiency of labor and thus to American standards of living.\n", "\n", "We have to admit that we economists know less than we should about the processes by which this better technology and these advances in organization come to be. Economists are much better at analyzing the consequences of advances in technology and improvements in organization and other factors that make for a high efficiency of labor, but they have less to say about their sources.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.1.2.2 Capital Intensity\n", "\n", "A secondary but still large part of America’s very long run economic growth—and a secondary but still large component of differences in material standards of living across countries today—has been generated by the second source of growth: capital intensity. Does the economy have a low ratio of capital per unit of output, that is, relatively little in the way of machines, buildings, roads, bridges, and so on? Then it is likely to be poor.\n", "\n", "The higher is the economy’s capital intensity, the more prosperous the economy will be: A more capital-intensive economy will be a richer and a more productive economy.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.1.3 Fitting It All Together\n", "\n", "Our task here is to get our heads straight on what the important factors are and how they work. This chapter thus presents economists’ basic theory of economic growth. It presents the concepts and models economists use to organize their thinking. \n", "\n", "Thus we build economists’ standard model of long-run economic growth, a model into which we can fit these broad groups of factors. This standard model is called the Solow growth model, after Nobel Prize-winning MIT economist Robert Solow, who wrote two of the four foundation-cornerstone articles of this branch of economics back in the mid-1950s: [A Contribution to the Theory of Economic Growth](http://piketty.pse.ens.fr/files/Solow1956.pdf) and [Technical Change and the Aggregate Production Function](https://faculty.georgetown.edu/mh5/class/econ489/Solow-Growth-Accounting.pdf). (The third and fourth articles are by Moses Abramovitz: [Resource and Output Trends in the United States Since 1870](https://www.nber.org/chapters/c5650.pdf) and [Catching Up, Forging Ahead, and Falling Behind](http://sites-final.uclouvain.be/econ/DW/DOCTORALWS2004/bruno/adoption/abramovitz.pdf)).\n", "\n", "The Solow growth model is a dynamic model of the economy: It describes how the economy changes and grows over time as saving and investment, labor-force growth, and progress in advancing technol ogy and improving social organization raise the economy’s level of output per worker and thus its material standard of living. Saving and investment are the driv ers leading to increases in capital intensity. Progress in technology and organization are the drivers leading to increases in the efficiency of labor.\n", "\n", "This chapter is thus, primarily, a tour of how economists think about growth. Chapter 5 will apply that thinking to udnerstand the world.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4.1.4 RECAP: Sources of Long-Run Growth\n", "\n", "In the very long run, economic growth is the most important aspect of economic performance. Two major factors determine economic growth: growth in the efficiency of labor—a product of advances in technology, the implementation of those advances through communities of engineering practice, the education and training of workers, and improvements in the economic and social organization of both businesses and the government—and the economy’s capital intensity. Policies that accelerate innovation and diffusion, educate and train workers, improve institutions and so boost the efficiency of labor do the bulk of accelerating economic growth and create prosperity. Policies that boost investment and raise the economy’s capital intensity to a higher level assist.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.1.4.1 In-Class Assessments\n", "\n", "##### When We Speak of the \"Long-Run\"...\n", "\n", "...in the context of our models of economic growth, we are talking, roughly about a time span of:\n", "\n", "A. Between today and 4 years from now. \n", "B. Between 3 and 10 years from now. \n", "C. Between 7 and 20 years from now. \n", "D. More than 20 years from now. \n", "E. No answer.\n", "\n", " \n", "\n", "##### About How Rich Is the U.S. Today...\n", "\n", "...compared to Nigeria?:\n", "\n", "A. Abourt 3 times as rich. \n", "B. About 10 times as rich. \n", "C. About 25 times as rich. \n", "D. About 100 times as rich. \n", "E. No answer.\n", "\n", " \n", "\n", "...compared to Mexico?:\n", "\n", "A. Abourt 3 times as rich. \n", "B. About 10 times as rich. \n", "C. About 25 times as rich. \n", "D. About 100 times as rich. \n", "E. No answer.\n", "\n", " \n", "\n", "...compared to Afghanistan?:\n", "\n", "A. Abourt 3 times as rich. \n", "B. About 10 times as rich. \n", "C. About 25 times as rich. \n", "D. About 100 times as rich. \n", "E. No answer.\n", "\n", " \n", "\n", "...compared to itself 250 years ago?:\n", "\n", "A. Abourt 3 times as rich. \n", "B. About 6 times as rich. \n", "C. About 13 times as rich. \n", "D. About 40 times as rich. \n", "E. No answer.\n", "\n", " \n", "\n", "...compared to itself 200 years ago?:\n", "\n", "A. Abourt 3 times as rich. \n", "B. About 6 times as rich. \n", "C. About 13 times as rich. \n", "D. About 40 times as rich. \n", "E. No answer.\n", "\n", " \n", "\n", "##### What Is the Pace of U.S. Growth?\n", "\n", "\"4\n", "\n", "* 60000 today\n", "* 4000 150 years ago\n", "* What’s the growth rate?\n", "\n", "A. 2.8%/year \n", "B. 1.8%/year \n", "C. 0.8%/year \n", "D. 5.6%/year \n", "E. None of the above\n", "\n", " \n", "\n", "##### Why Do You Suppose...\n", "\n", "...increases in life expectancy are much closer together than increases in income per worker?:\n", "\n", "Email your less-than-200-word answer to: INSTRUCTOR@UNIVERSITY.EDU\n", "\n", " \n", "\n", "##### Consider Argentina and Sweden...\n", "\n", "A. Sweden was richer in 1870, and is richer today. \n", "B. Sweden was richer in 1870, and is poorer today. \n", "C. Sweden was poorer in 1870, and is richer today. \n", "D. Sweden was poorer in 1870, and is poorer today. \n", "E. No answer\n", "\n", " \n", "\n", "##### Which Is More Important in Driving Differences...\n", "\n", "...in income per worker over time?\n", "\n", "A. Efficiency-of-labor \n", "B. Capital intensity \n", "C. Efficiency-of-labor and capital intensity are equally important \n", "D. No answer\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "## 4.1 REFLECT: Catch Our Breath\n", "\n", "\n", "\n", "* Ask me two questions…\n", "* Make two comments…\n", "* Further reading…\n", " * **Robert Solow** (1956): \"[A Contribution to the Theory of Economic Growth](http://piketty.pse.ens.fr/files/Solow1956.pdf)\", _Quarterly Journal of Economics_ 70:1 (February), pp. 65-94.\n", " * **Robert Solow** (1957): \"[Technical Change and the Aggregate Production Function](https://faculty.georgetown.edu/mh5/class/econ489/Solow-Growth-Accounting.pdf)\", _Review of Economics and Statistics_ 39:3 (August), pp. 312-320.\n", " * **Robert Solow** (1970): _Growth Theory: An Exposition_ (Oxford: Oxford University Press: 0195109031) .\n", " * **Moses Abramovitz** (1956): \"[Resource and Output Trends in the United States Since 1870](https://www.nber.org/chapters/c5650.pdf)\" (New York: NBER: 0870143662).\n", " * **Moses Abramovitz** (1986): \"[Catching Up, Forging Ahead, and Falling Behind](http://sites-final.uclouvain.be/econ/DW/DOCTORALWS2004/bruno/adoption/abramovitz.pdf)\", _Journal of Economic History_ 46:2 (June), pp. 385-406.\n", "\n", "
\n", "\n", "----\n", "\n", "Lecture Slides: Long-Run Economic Growth Theory—Sources: \n", "Lecture Support and Scratch: \n", "\n", " \n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4.2 Presenting the Solow Growth Model\n", "\n", "### 4.2.1 Model Overview\n", "\n", "#### 4.2.1.1 What Are Economic Models?\n", "\n", "As is the case for all economic models, the Solow growth model consists of:\n", "\n", "* _Variables_: economic quantities of interest that we can calculate and measure, denoted by letters (or maybe two letters) like L for the number of workers in the labor force or Y for the amount of useful goods and services produced in a year—real GDP.\n", "\n", "* _Behavioral relationships_: relationships that (1) describe how humans, making economic decisions given their opportunities and opportunity costs, decide what to do, and (2) that thus determine the values of the economic variables, represented by equations that have an economic variable on the left hand side and, on the right, a set of factors that determine the value of the variable and a rule of thumb for what that value is currently.\n", "\n", " * _Parameters_: determine which out of a broad family of potential behavioral relations describes the behavior of the particular economic scenario at hand; the ability to work algebraically with parameters allows one to perform an enormous number of potential \"what-if?\" calculations very quickly and in a very small space\n", "\n", "* _Equilibrium conditions_: conditions that tell us what is the case the economy is in a position of balance, when some subset of the variables are \"stable\"—that is, are either constant are changing in simple and predictable ways, usually represented by solutions of some system of the equations that are behavioral relationships.\n", "\n", "* _Accounting identities_: statements about the relationships between variables that are automatically and necessarily true because of the way the variables are defined, represented by equations. \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.1.2 Two Keys to the Solow Long-Run Growth Model\n", "\n", "**The Key Variable**: Almost every economic model has a single key principal economic variable at its heart: the variable that those using the model are typically the most interested in, and the one that the model is organized around.\n", "\n", "In the case of the Solow growth model, the key variable is labor productivity: income and output per worker, how much the average worker in the economy is able to produce in the way of making the things people find useful—necessary, convenient, or luxurious. \n", "\n", "We calculate output per worker by simply taking the economy’s level of national income and output Y, and dividing it by the economy’s labor force L. National income and output is the estimated annual value of goods and services produced for final sale: that somebody buys them (or, at least, somebody thinks they will be bought) tells us that they are _useful_. The labor force is the number of workers with or actively looking for jobs. \n", "\n", "Output per worker, Y/L, is a different concept from output per capita, Y/Pop. Y/Pop is the average income that can be spent to benefit the average person people, received by them or somebody else. Output per worker Y/L is how much in the way of useful goods and services the average worker can produce annually. It is a proxy, an imperfect proxy, but a useful simple proxy for how prosperous an economy is.\n", "\n", "**The Key Equilibrium Condition**: In every economic model economists proceed with their analysis by looking for an equilibrium: a point of balance, a condition of rest, a state of the system toward which the model will converge over time. Economists look for equilibrium for a simple reason: either an economy is at its (or one of its) equilibrium position(s), or it is moving—and probably (or hopefully?) moving rapidly—to an equilibrium position. \n", "\n", "The Solow growth model is no exception.\n", "\n", "Once economists have found the equilibrium position toward which the economy tends to move, they then understand how the model will behave. And, if they have built the right model, it will tell you in broad strokes how the economy will behave.\n", "\n", "In economic growth, the equilibrium economists look for is an equilibrium in which the economy’s capital stock per worker, its level of real income and output per worker, and the value of a variable called the _efficiency-of-labor_ are all three growing at exactly the same proportional rate—a rate that remains constant over time. \n", "\n", "We will call this common rate of growth g. Why \"g\"? To remind us that it is about the rate of proportional economic growth.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2.2 The Production Function\n", "\n", "Let’s start with the most important behavioral relationship in the Solow growth model: the production function. \n", "\n", " \n", "\n", "### 4.2.2.1 Determinants of Output per Worker\n", "\n", "Consider an average worker, a worker whose productivity is simply Y/L, the economywide average for income and output divided by the labor force. \n", "\n", "This average worker uses the economy’s current level of technology and organization. These are captured by the current value of the variable we will call the efficiency-of-labor, and write E. Why \"E\"? To remind us that it is about how efficient workers are in this economy. \n", "\n", "This average worker also uses an average share of the economy’s capital stock, which variable we will denote by \"K\". (Why not \"C\"? Because we are reserving \"C\" for spending on consumption goods and services, as we will see later on.). The average worker thus has K/L worth of capital to amplify his or her productivity. \n", "\n", "We want to describe how these two—efficiency of labor E and the capital-to-labor ratio K/L—affect the average worker’s productivity Y/L.\n", "\n", "To do this, we write down a behavioral relationship that tells us how the average worker’s productivity Y/L is related to the efficiency-of-labor E and the amount of capital K/L at the average worker’s disposal. We give this behavioral relationship a name: the production function. Tell the production function what resources the economy’s average worker has available, and it will tell you how much output the typical worker can produce. \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2.2.2 Two Rules of Thumb We Want Our Production Function to Have\n", "\n", "In the Solow Growth Model, we seek a production function that will satisfy two rules of thumb:\n", "\n", "1. A one percent increase in the capital at the disposal of the average worker—a one percent increase in K/L—will carry with it the same proportional increase in the level of labor productivity—of output per worker Y/L—with a factor of proportionality we will call α, a number between 1 and 0 that serves as a gauge of how quickly opportunities to amplify productivity via using more structures and machines are exhausted, and how rapidly diminishing returns set in. (Note that, when we apply the Solow growth model, different economies will have different values of α; the same economy may well have a different value of α at different times; and asking \"what would happen if the value of the factor of proportionality α were different?\" is one of the principal what-if questions and exercises we will want to undertake and analyze.)\n", "\n", "2. A one percent increase in how well technology and organization are used to boost the productivity of the average worker—a one percent increase in the _efficiency of labor_, which we will label E—will carry with it a 1 - α percent increase in the level of labor productivity—of output per worker Y/L. This is best thought of as a definition of what we mean by the _efficiency of labor_ E. (And why the same α here as in the first rule of thumb? Because it makes the ultimate equations we will work with in the model simpler: it is harder to make mistakes when things are simple).\n", "\n", "Why do we require that our production function satisfy these two rules of thumb? Because they are the simplest requirements we can think of that have the potential for generating useful conclusions we can then apply to study real-world economies.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.2.3 The Resulting Production Function\n", "\n", "There is one and only one algebraic functional form that satisfies these two rule-of-thumb conditions. It is the so-called Cobb-Douglas function:\n", "\n", ">$ \\frac{Y}{L} = \\left({\\frac{K}{L}}\\right)^α \\left(E\\right)^{(1-α)} $\n", " \n", "This gives us our quantitative rule for calculating what the economy’s level of annual output per worker Y/L will be. It will be equal to the capital stock per worker K/L raised to the power of a parameter α, and then multiplied by the current efficiency of labor E itself raised to the power (1—α).\n", "\n", " \n", "\n", "\n", "\n", ">**Graphing the Cobb-Douglas Production Function**: The Cobb-Douglas production function for α = 1/2, E = 1, for capital per worker K/L from 0 to 1000.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.2.4 Alternative Forms of the Production Function\n", "\n", "The production function has other, equivalent forms. When we are interested in not the level of output per worker but in the total level of output Y in the economy, we simply multiply both the left- and right-hand sides of the equation above by the labor force L to get the total output form of this Cobb-Douglas production function:\n", "\n", ">$ Y = K^α(EL)^{1-α} $\n", "\n", "When we are interested in output per worker as a function not of capital per worker K/L but of the so-called _capital intensity_ of the economy K/Y, the ratio of the capital stock to output, we use a little bit of simple algebra to get from:\n", "\n", ">$ \\frac{Y}{L} = \\left({\\frac{K}{L}}\\right)^α \\left(E\\right)^{(1-α)} $\n", "\n", ">$ \\frac{Y}{L} = \\left({\\frac{Y}{L}\\frac{K}{Y}}\\right)^α \\left(E\\right)^{(1-α)} $\n", "\n", ">$ \\frac{Y}{L} = \\left({\\frac{Y}{L}}\\right)^α \\left({\\frac{K}{Y}}\\right)^α \\left(E\\right)^{(1-α)} $\n", "\n", ">$ \\left(\\frac{Y}{L}\\right)^{(1-α)} = \\left({\\frac{K}{Y}}\\right)^α \\left(E\\right)^{(1-α)} $\n", "\n", "to:\n", "\n", ">$ \\left(\\frac{Y}{L}\\right) = \\left({\\frac{K}{Y}}\\right)^\\left(\\frac{α}{1-α}\\right) E $\n", "\n", "We could—and economists do—work with some other functional form for the production function, or simply leave it abstract as that output per worker Y/L is some function or other of the capital-labor ratio Y/L and the efficiency of labor E:\n", "\n", ">$ \\frac{Y}{L} = F\\left({\\frac{K}{L}}, E\\right) $\n", "\n", "This says just that there is a systematic relationship between output per worker Y/L—real GDP divided by the number of workers—and the economy’s available resources: the capital stock per worker K/L and the efficiency of labor E. The pattern of this relationship is prescribed by the form of the function F( ).\n", "\n", "Why then assume the Cobb-Douglas form? Because it incorporates our two conditions above. These conditions, both of them, (a) are very reasonable rules of thumb, and (b) make our calculations much simpler and easier to understand. We can then give quantitative answers to any questions we are asked about the effects of changes in economic policy and the economic environment on economic growth, and those answers will depend solely on (a) how policy and the environment shift the values of K/L and E, and (b) the value of the economy's factor of proportionality α.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.2.5 The Orientation-of-Growth Toward Investment Parameter in the Production Function\n", "\n", "The value of α in the Cobb-Douglas production function tells us how rapidly the economic usefulness of additional investment in buildings, machines, and inventories declines as the economy accumulates more and more of them. That is, α measures how fast diminishing returns to investment set in in the economy. A value of α near zero means that the extra amount of output made possible by an additional unit of capital declines very quickly as the capital stock rises. A value of α near 1 means that each additional unit of capital makes possible almost as large an increase in output as the last additional unit. As α varies from a high number near 1 to a low number near 0, the force of diminishing returns to investment gets stronger.\n", "\n", "One way of illustrating this point in algebra is to use a little calculus to calculate the marginal product of capital, the MPK: how much total output increases as a result of a one-unit increase in the capital stock, dY/dK. For the Cobb-Douglas production function:\n", "\n", ">$ MPK = \\frac{dY}{dK} = αK^{α-1}(EL)^{1-α} = α\\left(\\frac{Y}{K}\\right) $\n", "\n", "The higher the current capital-to-output ratio, the lower is the marginal product of capital. And the lower the parameter α, the lower is the marginal product of capital.\n", "\n", "Examine what the production function looks like for five different values of α—from a low value of 0.15, in which case diminishing returns set in very quickly, to a high value of 0.75, in which case additional capital is very useful in production over a wide domain of values of capital per worker K/L. All five of the alternative production functions are calibrated so that the level of output per worker Y/L is 40 when the capital-labor ratio K/L is 200. Take away from this graph the flexibility of the Cobb-Douglas production function. It can be used to analyze economies in which the profits from investment and the boost to labor productivity from increasing the capital stock is high. It can be used to analyze economies in which the profits from investment and the boost to labor productivity from increasing the capital stock is low. It can be used to analyze economies in which the profits from investment and the boost to labor productivity from increasing the capital stock is intermediate.\n", "\n", " \n", "\n", "\"4\n", "\n", ">**Cobb-Douglas Production Function for Various Parameter Values**\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.2.6 The Efficiency-of-Labor Parameter in the Production Function\n", "\n", "The value of E in the Cobb-Douglas production function tells us how good the levels of technology invented and diffused, of worker skills acquired through education and training and experience, and of organizational efficiency both public and private are. That is, E measures the scale of the economy's underlying efficiency and powre. A value of E near zero means that the economy is not very productive at all. A high value of E means that the economy is very productive indeed. Figure 4.2.3 shows what the production function looks like for five different values E, in all cases keeping the diminishing-returns parameter α equal to 1/2.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.2.7 The Flexibility of the Cobb-Douglas Production Function\n", "\n", "Thus the Cobb-Douglas production function is a flexible one, and, indeed, its flexibility is what makes it useful. It can be “tuned” to fit any of a wide variety of different economic situations. \n", "\n", "Are we studying an economy in which productivity is high? Use the Cobb-Douglas production function with a high value of the efficiency of labor E. The value of the efficiency of labor E tells us how high the production function rises: A higher level of E means that more output per worker is produced for each possible value of the capital stock per worker.\n", "\n", "Does the economy rapidly hit the wall as investment proceeds, with little increase in the level of production? Use the Cobb-Douglas function with a low value—near zero—of α. Is the speed with which diminishing returns to investment set in moderate? Pick a middle value of α. The Cobb-Douglas function will once again fit. The value of α determines how quickly the Cobb-Douglas production function flattens out when output per worker is plotted on the vertical axis and capital per worker is plotted on the horizontal axis.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.2.7.1: Calculating Using the Cobb-Douglas Production Function: An Example\n", "\n", "Given values for the diminishing-returns-to-investment parameter α, the efficiency of labor E, the economy’s capital stock K, and the labor force L, we can calculate the level of output per worker Y/L in the economy.\n", "\n", "Suppose we know that the current value of the efficiency of labor E is 10,000 a year and that the diminishing-returns-to-investment parameter α is 0.3. Then determining how the level of output per worker Y/L depends on the capital stock per worker K/E is straightforward.\n", "\n", "Let’s start with the case in which the capital stock per worker is 125,000. The Cobb-Douglas production function is:\n", "\n", "          \n", "$ \\frac{Y}{L} = \\left(\\frac{K}{L}\\right)^α\\left(E\\right)^{1-α} $\n", "\n", "Substitute in the known values of K/L, E, and α to get:\n", "\n", "          \n", "$ \\frac{Y}{L} = \\left({125000}\\right)^{0.3}\\left(10000\\right)^{1-0.3} $\n", "\n", "Use your calculator—who do I think I am fooling? Nobody has a calculator anymore—smartphone or larger computer to evaluate the effect of raising these numbers to these exponents to get:\n", "\n", "          \n", "$ \\frac{Y}{L} = \\left(33.81\\right)\\left(630.96\\right) $\n", "\n", "And then multiply to get:\n", "\n", "          \n", "$ \\frac{Y}{L} = 21333.92 $\n", "\n", "If we are interested in a capital stock per worker level of $250,000, the calculations are:\n", "\n", "          \n", "$ \\frac{Y}{L} = \\left({250000}\\right)^{0.3}\\left(10000\\right)^{1-0.3} $\n", "\n", "          \n", "$ \\frac{Y}{L} = \\left(41.63\\right)\\left(630.96\\right) $\n", "\n", "          \n", "$ \\frac{Y}{L} = 26266.86 $\n", "\n", "Note that the first 125,000 of capital per worker boosted output per worker from 0 to 21,334, while the second $125,000 of capital per worker boosted output per worker only from 21,334 to 26,265—by less than one-quarter as much. These substantial diminishing returns to investment should come as no surprise: The value of α is quite low at 0.3, and low values of a produce rapidly diminishing returns to capital accumulation.\n", "\n", "This example offers another important lesson: Keep your calc... your personal information technology device handy! Nobody expects anyone to raise 250,000 to the 0.3 power in her or his head and come up with 41.63. The Cobb-Douglas form of the production function, with its fractional exponents, carried the drawback that students (or professors) can do problems in their heads or with just pencil and paper only if the problems have been carefully rigged beforehand. Nevertheless, we used the Cobb-Douglas production function because of its convenience: by varying just two parameters we can fit the model to an enormous variety of potential economic situations.\n", "\n", "And in the infotech age that inconvenience is no longer a significant inconvenience.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2.3 Saving, Investment, and Capital Accumulation\n", "\n", "#### 4.2.3.1 A Glance at Some Not-Yet-Important Details\n", "\n", "In Part III and Part IV we will talk in detail about the circular-flow relationship you learned in your principles of cconomics class: Total production—the amount of output an economy produces—equals aggregate demand—total spending—equals total income. So we use the variable for Y for all three (although we will write \"AD\" on those rare occasions when we want to distinguish total spending from national income and product).\n", "\n", "The most commonly-used measure of this circular flow of economic activity is Gross Domestic Product, abbreviated \"GDP\"—but do not get hung up on either the \"gross\", the \"domestic\", or the \"product\" part of this particular most easily-measured definition. Think of it as the circular flow of purchasing power, production, and useful goods through the economy.\n", "\n", "Y-national income and product, aggregate demand—is divided into four parts: \n", "\n", "* consumption spending C, \n", "* investment spending I, \n", "* government purchases G, and \n", "* net exports NX, which equal gross exports minus imports, NX = GX — IM. \n", "\n", "Society can spend its national income on consumption goods and government purchases, or income can be saved. Total savings directed at the domestic economy has three components: private—business and household—saving $ S^p = Y - C - T $ where T are total net taxes collected by the government, government saving $ S^g = T - G$, and saving from foreigners directed at the domestic economy $ S^f = - NX $. Note well: savings that actually occur _ex post_ and are translated into investment that increases the capital stock may well not be equal to what people had planned to save _ex ante_. This will matter at lot in Parts III and IV.\n", "\n", "But none of that is of the essence here. \n", "\n", "The details of saving and the circular flow are not our concern—yet.\n", "\n", "Simply flag the idea that additional complexites will be introduced later.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.3.2 Ruthless Simplification: The Savings-Investment Rate s\n", "\n", "Here we will follow economists' custom of ruthless simplification, assume that the workings of the circular flow are unproblematic, and assume that total domestic saving S is equal to a constant parameter s times real GDP Y. And we will assume that total savings is unproblematically translated into investment I that adds to the capital stock:\n", "\n", ">$ I = S = sY $\n", "\n", "In this chapter we will almost always assume that s is constant. We may think about the consequences of its taking an upward or downward jump or two at some particular moment of time. The background assumption, however—made because it makes formulas much simpler—will be that s will remain at its current value as far as we look into the future.\n", "\n", "We call s the economy’s saving rate or, more completely, its saving-investment rate (to remind us that s is measuring both the flow of saving into the economy’s financial markets and also the share of total production that is invested and used to build up and increase the economy’s capital stock).\n", "\n", " \n", "\n", "#### 4.2.3.3 The Growth of the Capital Stock\n", "\n", "While the saving-investment rate s is constant, the economy's capital stock K is not. It changes from year to year. \n", "\n", "Adopt the convention of using a subscript when we need to identify the year to which we are referring. K0 denotes the capital stock at some initial year, usually the year at which we begin the analysis; K3 denotes the capital stock three years after the year at which we begin the analysis; K2003 means the capital stock in 2003; Kt denotes the capital stock when some year t is the current year; Kt+1 denotes the capital stock in the year after the current year; and Kt-1 means the capital stock in the year before the current year.\n", "\n", "Over time, investment makes the capital stock tend to grow. But investment is not the only factor that changes the capital stock. There is depreciation. Over time depreciation makes the capital stock tend to shrink—old capital becomes obsolete, or breaks, or simply wears out.\n", "\n", "Sometimes you will hear people refer to the change in the capital stock as net investment—investment net of depreciation—and refer to the amount of plant and equipment purchased and installed as gross investment. In this book “investment” will mean gross investment, and we will use “investment minus depreciation” in place of net investment.\n", "\n", " \n", "\n", "\"Investment\"\n", "\n", ">**Gross Investment Increases and Depreciation Decreases the Capital Stock**\n", "\n", " \n", "\n", "We make a simple assumption for depreciation: The amount of capital that wears out, breaks, and becomes obsolete in any year is simply a constant parameter δ (lowercase Greek delta, the depreciation rate) times the current capital stock. Thus we can write that next year’s capital stock will be:\n", "\n", ">$ K_{t+1} = K_{t} + I - δK_{t} = (1-δ)K_{t} + sY_{t} $\n", "\n", "From these definitions, we can see that the capital stock is constant—that this year’s capital stock Kt is equal to next year’s Kt+1—when\n", "\n", ">$ sY_t = δK_t $\n", "\n", ">$ \\frac{K_t}{Y_t} = \\frac{s}{δ} $\n", "\n", "We will—often—want to talk about the proportional growth rate of the capital stock—not the _amount_ by which the capital stock is growing in a year, but the _proportional rate_ relative to its current value at which it is growing. We will call this proportional growth rate $ g_K $: \"g\" for growth, and subscript \"K\", for capital stock. From these definitions, we can see that this proportional growth rate can be written:\n", "\n", ">$ g_K = \\frac{K_{t+1} - K_t}{K_t} = \\frac{I_t}{K_t} - δ $\n", "\n", "If you are comfortable with calculus, you will find that you make your life simpler by working with an almost equivalent expression for the growth rate of the capital stock: that the instantaneous rate of change is equal to s multiplied by the current level of output minus δ multiplied by the urrent level of the capital stock:\n", "\n", ">$ \\frac{dK_{t}}{dt} = sY_{t} - δK_{t} $\n", "\n", "which then tells us:\n", "\n", ">$ g_K = \\frac{1}{K_t}\\frac{dK_{t}}{dt} = s\\left(\\frac{Y_{t}}{K_{t}}\\right) - δ $\n", "\n", "or the equivalent logarithmic form:\n", "\n", ">$ g_K = \\frac{dln(K_{t})}{dt} = s\\left(\\frac{Y_{t}}{K_{t}}\\right) - δ $\n", "\n", "These are not all _exactly_ identical. But differences in calculations typically show up only in the second decimal point or so. We will not sweat the differences for those of you for whom math is more of an intellectual language and force multiplier and less of a ritualistic obstacle.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2.4 Adding in Labor-Force and Labor-Efficiency Growth\n", "\n", "#### 4.2.4.1 Labor Force Growth\n", "\n", "If the labor force L were constant and technological and organizational progress plus education that add to the efficiency of labor E were non-existent, we could immediately move on. But the economy’s labor force grows as more people turn 18 or so and join the labor force than retire, and as immigrants continue to arrive. And the efficiency of labor rises as science and technology progress, people keep thinking of new and more efficient forms of business organization, and people go to school and learn on the job\n", "\n", "We assume—once again making a simplifying leap—that the economy’s labor force L is growing at a constant proportional rate n every year. Note that n is not the same across countries. Note that it can and does shift over time in any one country. But we want to tackle simple cases first. A constant labor force growth rate n is simple. Thus our background assumption will be that n is constant as far as we can see into the future. Then between this year and the next the labor force grows according to the formula:\n", "\n", ">$ L_{t+1} = (1 + n)L_{t} $\n", "\n", "We will—often—want to talk about the proportional growth rate of the labor force—not the _amount_ by which the labor force is growing in a year, but the _proportional rate_ relative to its current value at which it is growing. We will call this proportional growth rate $ g_L $: \"g\" for growth, and subscript \"L\", for labor force:\n", "\n", ">$ g_L = \\frac{L_{t+1}}{L_t} - 1 $\n", "\n", "Next year’s labor force will thus be n percent higher than this year’s labor force:\n", "\n", ">$ g_L = n $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.4.1.1 Labor Force Growth: An Example\n", "\n", "If this year’s labor force is 140 million and its growth rate n is 1.3 percent per year, then next year’s labor force will be:\n", "\n", ">$ L_{t+1} = (1 + n)L_{t} $\n", "\n", ">$ L_{t+1} = (1 + 1.3{\\%})140000000 $\n", "\n", ">$ L_{t+1} = (1.013)140000000 $\n", "\n", ">$ L_{t+1} = 141820000 $\n", "\n", "Again, we assume that the rate of growth of the labor force is constant not because we believe that labor-force growth is unchanging, but because the assumption allows us to start our analysis of the model with a simple case. \n", "\n", "The trade-off between realism in the model’s description of the world and simplicity as a way to make the model easier to analyze'is one that economists always face, and economists have a strong bias toward resolving this trade-off in favor of simplicity.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.4.1.2 The Continuous-Calculus Version\n", "\n", "If you are comfortable with calculus, you will find that you make your life simpler by working with an almost equivalent expression for the growth rate of the labor force: that the instantaneous rate of change of the labor force is equal to n multiplied by its current level:\n", "\n", ">$ \\frac{dL_{t}}{dt} = nL_{t} $\n", "\n", "which tells us that the proportional growth rate $ g_L $ of the labor force is:\n", "\n", ">$ g_L = \\frac{1}{L_t}\\frac{dL_{t}}{dt} = n $\n", "\n", "or in the logarithmic form:\n", "\n", ">$ g_L = \\frac{dln(L_{t})}{dt} = n $\n", "\n", "Once again, these formulas are not _exactly_ equivalent. But differences in calculations typically show up only in the second decimal point or so. \n", "\n", "Once again, we will not sweat the difference for those of you for whom math is more of an intellectual language and force multiplier and less of a ritualistic obstacle to thought and analysis.\n", "\n", " \n", "\n", "\"Labor\n", "\n", ">**Labor Force Growth at Constant Rate n**\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### 4.2.4.2 Efficiency-of-Labor Growth\n", "\n", "We also assume—once again making a simplifying leap—that the economy’s efficiency of labor E is growing at a constant proportional rate g every year. Note that g is not the same across countries. Note that it can and does shift over time in any one country. But we want to tackle simple cases first. A constant efficiency-of-labor growth rate g is simple. Thus our background assumption will be that g is constant as far as we can see into the future. Then between this year and the next the efficiency of labor grows according to the formula:\n", "\n", ">$ E_{t+1} = (1 + g)E_{t} $\n", "\n", "We will—often—want to talk about the proportional growth rate of the efficiency-of-labor—not the _amount_ by which the efficiency-of-labor is growing in a year, but the _proportional rate_ relative to its current value at which it is growing. We will call this proportional growth rate $ g_K $: \"g\" for growth, and subscript \"E\", for efficiency-of-labor. In the year-over-year non-calculus formulation:\n", "\n", ">$ g_E = \\frac{E_{t+1}}{E_t} - 1 = g $\n", "\n", "Next year’s efficiency will thus be g percent higher than this year’s labor force. \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.4.1.2 Efficiency-of-Labor Growth: An Example\n", "\n", "If this year’s efficiency is 1.000 and its growth rate n is 1.8 percent per year, then next year’s effiency will be:\n", "\n", ">$ E_{t+1} = (1 + g)E_{t} $\n", "\n", ">$ E_{t+1} = (1 + 1.8{\\%})1.000 $\n", "\n", ">$ E_{t+1} = 1.018 $\n", "\n", "Again, we assume that the rate of growth of labor efficiency is constant not because we believe it actuallt is unchanging, but because the assumption allows us to start our analysis of the model with a simple case. \n", "\n", "The trade-off between realism in the model’s description of the world and simplicity as a way to make the model easier to ana lyze'is one that economists always face, and economists have a strong bias toward resolving this trade-off in favor of simplicity.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.4.2.2 The Continuous-Calculus Version\n", "\n", "If you are comfortable with calculus, you will find that you make your life simpler by working with an almost equivalent expression for the growth rate of labor efficiency: that the instantaneous rate of change is equal to g multiplied by its current level:\n", "\n", ">$ \\frac{dE_{t}}{dt} = gE_{t} $\n", "\n", "which gives us for the proportional growth rate of the efficiency-of-labor $ g_E $:\n", "\n", ">$ g_E = \\frac{1}{E_t}\\frac{dE_{t}}{dt} = g $\n", "\n", "and in its logarithmic form:\n", "\n", ">$ g_E = \\frac{dln(E_{t})}{dt} = g $\n", "\n", "These, again, are not _exactly_ identical. But differences in calculations typically show up only in the second decimal point or so, and so we will not sweat the difference for those of you for whom math is more of an intellectual language and force multiplier and less of a ritualistic obstacle.\n", "\n", " \n", "\n", "\"2018\n", "\n", "**Efficiency of Labor Growth at Constant Rate g**\n", "\n", " \n", "\n", "At this point you may be puzzled: why make a song-and-dance about how $ g_E $ is the proportional rate of growth of the labor force when we already defined bare $ g $ as the same thing? \n", "\n", "The reason is that when we work through our model, we will discover that bare $ g $ is, in some but in very important situations, not just the proportional growth rate of the efficiency-of-labor but the growth rate of other things as well—like the amount of capital-per-worker, and the level of output-per-worker as well.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2.5 How the Model Behaves: An Initial Take\n", "\n", "That is it: that is the entire model. It has a production function, and equations describing how the capital stock, the labor force, and the efficiency of labor grow. What can we learn from this model? The easiest way to start is to simply set the model up and see how it behaves. \n", "\n", " \n", "\n", "#### 4.2.5.1 The Model without Labor Force Growth or Efficiency Growth\n", "\n", "##### 4.2.5.1.1 This Is the Simplest Case\n", "\n", "Let us start with a simple case:\n", "\n", "* No labor force growth: n = 0\n", "* No labor efficiency growth: g = 0\n", "* Positive rates s of savings of 15&percent; and δ of depreciation of 3&percent; per year\n", "* A diminishing-returns-to-investment parameter α of 0.5\n", "\n", "And let us start with a \"simulated\" economy—an econoy that satisfies the model equations—in a more-or-less arbitary condition. Let's see what would happen as time were to pass for an economy that behaved according to the model. We thus examine graphs showing the levels and the change over time in the labor force, the efficiency of labor, the capital stock, output, ouput per worker, and the capital-output ratio.\n", "\n", " \n", "\n", "\"Constant\n", "\n", ">**Solow Growth Model Simulation Run: 200 Years**: No labor force growth and no efficiency-of-labor growth case.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.5.1.2 What Does the Simulation Tell Us?\n", "\n", "The efficiency of labor and the labor force do not grow. The are constant. This is no surprise: in this simulation of the economy we excluded the possibility of growth in either. We set both n and g equal to zero.\n", "\n", "the capital stock, which stared out at 5000 with output at about 2300, initially grows rapidly. This also is no surprise. 15% of the starting level of output as gross investment is 345. 3% of the starting level of the capital stock is 150. With gross investment more than twice depreciation, the capital stock grows rapidly. And output grows too: with the diminishing returns parameter α = 1/2, additional capital from investment is quite productive.\n", "\n", "But as we follow this simulation down the hypothetical years growth slows down. It looks as though the capital stock is headed for a value of 25000, with output at 5000 and the capital-output ratio at 5.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.5.1.3 Is This Convergence an Accident?\n", "\n", "Why is this simulated economy headed for that particular configuration? Is this some kind of accident? Let's use our model as a \"what-if\" machine, and ask what would have happened had we started it in a different initial configuration. Let's keep everything the same, but start with an initial capital stock not of 5000 but of 40000. What would happen then?\n", "\n", " \n", "\n", "\"K\n", "\n", ">**Solow Growth Model: 200 Year Simulation Run**: No labor force growth, no efficiency-of-labor growth, high initial capital stock K = 40000 case.**\n", "\n", " \n", "\n", "In this simulation run the economy started out much richer and more prosperous: an initial level of output of not 2300 but rather 6300. The extra initial capital endowment was valuable and productive. But in this run depreciation starts out larger than gross investment: the capital stock falls. And by the end of the simulation run the capital stock is again approaching 25000, and output again approaching 5000, only this time both approaching from above rather than from below.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.5.1.4 Understanding What Is Going on\n", "\n", "Why is it approaching the same configuration?\n", "\n", "Insight comes from noting that since neither the labor force nor the efficiency of labor are changing, the only productive resource that is changign is the capital stock. Recall the equation for the growth rate of the capital stock:\n", "\n", ">$ K_{t+1} = K_{t} + sY_{t} - δK_{t} $\n", "\n", "The capital stock will grow as long as $ sY_{t} > δK_{t} $. It will shrink whenever $ sY_{t} < δK_{t} $. It will stay constant whenever $ sY_{t} = δK_{t} $.\n", "\n", "Thus the capital stock will be growing—and output, and output per worker will be growing—as long as:\n", "\n", ">$ \\frac{K_t}{Y{t}} < \\frac{s}{δ} $\n", "\n", "And the capital stock will be constant—and output, and output per worker—will be constant whenever:\n", "\n", ">$ \\frac{K_t}{Y{t}} = \\frac{s}{δ} $\n", "\n", "This is thus the equilibrium condition for this model of the economy _in this particular case and only in this particular case in which we have assumed no labor-force and no efficiency-of-labor growth:_ n = g = 0. \n", "\n", "When this equilibrium condition equation is satisfied, the condition of the economy will not change. When this equation is not satisfied, the economy will be changing over time to make the gap between K/Y and s/δ smaller. What else can we say about this equilibrium? We could use a little algebra to calculate it. Or we could use a more friendly, graphical method to calculate it.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.5.1.5 Calculating Equilibrium Output per Worker: An Example\n", "\n", "In this simulation run the labor force L is constant at a level of 1000, the efficiency of labor is constant at a level of 1, the savings rate s = 15%, the depreciation rate δ = 3%, and the production-function diminishing-returns parameter α = 1/2. We know that the equilibrium capital-output ratio is:\n", "\n", ">$ \\frac{K}{Y} = \\frac{s}{δ} $\n", "\n", "And the production function is:\n", "\n", ">$ Y = K^α(EL)^{1-α} $\n", "\n", "Substituting:\n", "\n", ">$ \\frac{K}{K^α(EL)^{1-α}} = \\frac{s}{δ} $\n", "\n", ">$ \\left(\\frac{K}{EL}\\right)^{1-α} = \\frac{s}{δ} $\n", "\n", ">$ \\frac{K}{EL} = \\left(\\frac{s}{δ}\\right)^{\\frac{1}{1-α}} $\n", "\n", ">$ K = \\left(\\frac{s}{δ}\\right)^{\\frac{1}{1-α}}\\left(EL\\right) $\n", "\n", "And:\n", "\n", ">$ Y = K^α(EL)^{1-α} = \\left(\\left(\\frac{s}{δ}\\right)^{\\frac{1}{1-α}}\\left(EL\\right)\\right)^α (EL)^{1-α} = \\left(\\frac{s}{δ}\\right)^{\\frac{α}{1-α}}\\left(EL\\right) $\n", "\n", "To determine what the long-run equilibrium capital stock will be:\n", "\n", "* Take the equilibrium capital-output ratio: s/δ\n", "* Raise it to the power of: 1/(1-α)\n", "* Multiply it by the product of the labor force and the efficiency of labor ratio EL\n", "\n", "To determine what the long-run equilibrium level of output per worker will be:\n", "\n", "* Take the equilibrium capital-output ratio: s/δ\n", "* Raise it to the power of: α /(1-α)\n", "* Multiply it by the product of the labor force and the efficiency of labor ratio EL\n", "\n", "In this particular case the savings rate s = 15%, the depreciation rate δ = 3%, and the production-function diminishing-returns parameter α = 1/2, the efficiency of labor E is 1 and the labor force L is 1000. And so:\n", "\n", ">$ K = \\left(\\frac{s}{δ}\\right)^{\\frac{1}{1-α}}\\left(EL\\right) = (5^2)(1000) = 25000 $\n", "\n", "And:\n", "\n", ">$ Y = \\left(\\frac{s}{δ}\\right)^{\\frac{α}{1-α}}\\left(EL\\right) = (5^1)(1000) = 5000 $\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.5.1.6 The Graphical Approach\n", "\n", "We can reach the same conclusion by a different, graphical, approach—an approach that is more friendly to those who prefer lines and curves on graphs to Greek variables and exponents in equations. Recall that we have already drawn a graph of the production function, plotting output per worker Y/L on the vertical axis and capita per worker K/L on the horizontal axis. That production function is our behaviorial relationship. Equilibrium is when and where that behavioral relationship satisfies our equilibrium condition. What is our equilibrium condition? It is that the capital-output ratio is at its equilibrium value:\n", "\n", ">$ \\frac{K}{Y} = \\frac{s}{δ} $\n", "\n", "What is that plotted on the same axes as our production function? It is the straight line starting at (0, 0) at which the capital stock is equal to s/δ times the level of output. So plot the production function. Draw the right straight line for the equilibrium condition. And the economy's equilibrium is where both are satisfied: where the curves cross.\n", "\n", " \n", "\n", "\"Equilibrium\n", "\n", ">**Equilibrium Output per Worker: Graphically**\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "##### 4.2.5.1.7 Using the Model to Analyze the Economy\n", "\n", "We can use either the algebra or the graphical method to think about the long-run consequences of, say, changes in a government’s fiscal policy. At the end of 2017 the Trump administration and the Republican majorities in Congress narrowly passed bills reducing federal tax revenues by about 150 billion dollars a year—0.75 percent of GDP—and increasing government purchases by about the same amount. \n", "\n", "That amount of money will have to be borrowed by the government in the future, and thus will be a subtraction from the governmernt savings component national saving—a \"small\" reduction in the national savings rate. Thus the value of the equilibrium condition s/δ is likely to be slightly lower as a result of this policy change. On the graph, the green equilibrium condition line will rotate slightly counterclockwise. The economy's equilibrium will have a lower capital intensity, and so the economy will be (slightly) poorer.\n", "\n", "This depressing effect of rising government deficits on the standard of living is one important reason that international agencies like the International Monetary Fund (IMF) and the World Bank and almost all economists advise governments at or near full employment to avoid large and prolonged government deficits.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2.6 RECAP: Presenting the Solow Growth Model\n", "\n", "The Solow growth model consists of:\n", "\n", "* **Variables**: economic quantities of interest that we can calculate and measure: L for the number of workers in the labor force, Y for annual production—real GDP, the amount of useful goods and services produced in a year—K for the economy's capital stock, S for savings, and E for the effiency of labor.\n", "\n", "* **Behavioral relationships**: relationships that (1) describe how humans, making economic decisions given their opportunities and opportunity costs, decide what to do, and (2) that thus determine the values of the economic variables, represented by equations: a production function describing how labor, capital, and efficiency enable the level of output; and equations for the savings rate, and for the growth of the labor force, of labor efficiency, and of the capital stock.\n", "\n", " * **Parameters**: determine which out of a broad family of potential behavioral relations describes the behavior of the particular economic scenario at hand; the ability to work algebraically with parameters allows one to perform an enormous number of potential \"what-if?\" calculations very quickly and in a very small space: the parameters of the Solow model are the labor force growth rate n, the labor efficiency growth rate g, the depreciation rate δ, the savings rate s, and the production function parameter α.\n", " \n", "* **Equilibrium conditions**: indicators that the economy is in a position of balance with some subset of its important variables stable, and others predictable: the important equilibrium condition sets the capital-output ratio Y/L equal to the quotient of the savings rate s and of investment requirements n+g+δ.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.2.6.1 In-Class Assessments\n", "\n", "##### A Rule-of-Thumb That We Require in Our Production Function Is:\n", "\n", ">A. \n", "B. \n", "C. \n", "D. \n", "E. None of the above\n", "\n", "\n", " \n", "\n", "### 4.2.6.2 Calculations and Exercises\n", "\n", "* \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## Catch Our Breath\n", "\n", "* Ask me two questions…\n", "* Make two comments…\n", "* Further reading…\n", "\n", "
\n", "\n", "----\n", "\n", "Long-Run Economic Growth Theory—Presenting the Model: <> \n", "Lecture Support: <>\n", "\n", " \n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.3 Understanding the Solow Growth Model\n", "\n", "#### 4.3.1 Getting an Exploratory Sense of How the Model Behaves\n", "\n", "The two simulation runs presented in the section above both made the hugely unrealistic assumptions of no labor force growth and no growth in the efficiency of labor. \n", "\n", "How does the model behave if we relax those unrealistic assumptions, and incorporate labor force and efficiency of labor growth? We can see the same way we saw above: assume various parameter values and initial conditions, and see what happens. Back when Robert Solow originally developed his model, computer time was so expensive that the only way to gain understanding was to solve the algebra and then think. Now, however, (as long as our programs run without bugs) we can directly see. Now look at the behavior of the model for the same parameter values as above, only with labor force growth at 1% per year and labor efficiency growth at 2% per year.\n", "\n", "This time the economy does not stagnate: there is no tendency for the economy's level of output or of output per worker or of the capital stock to head for some asymptote where they then stick. And there should not be. This model was built to understand the ongoing and—so far—fairly constant proportional growth that modern industrial economies have experienced since the Industrial Revolution. If our model predicted some eventual cessation of growth and subsequent stagnation, either we would have been doing it wrong, or it would be huge news. Indeed, there is no reason for output per worker to stagnate: labor is becoming more efficient and productive, and the other productive resource in the economy is capital, which is something labor can and does create. And with both efficiency and with both labor efficiency and the labor force increasing, there is no reason for output growth to approach any end. And with output increasing and savings a constant share of output, we would expect the capital stock to grow too.\n", "\n", " \n", "\n", "\"Full\n", "\n", ">**Solow Model Simulation: 2% Annual Efficiency of Labor Growth, 1% Annual Labor Force Growth**: α = 1/2, δ = 3%, s = 15%\n", "\n", " \n", "\n", "Only one economic variable plotted seems to be approaching any sort of limit or asymptote: the capital-output ratio. Is that a special feature of the particular initial conditions we chose? Let's choose a different set of initial conditions and see. \n", "\n", "Now let us plot a simulation run starting with a very high initial capital stock. In its first few periods the simulated economy is behaving substantially differently. The capital stock is not rising but falling, as depreciation overwhelms gross investment. Output per worker is falling as well, as increased efficiency of labor cannot keep up with the reduction in the buildings, machines, and inventories the average worker has at his or her disposal. The capital-output ratio is falling very rapidly at the start.\n", "\n", " \n", "\n", "\"High\n", "\n", ">**Solow Simulation Run with High Initial Capital Stock**\n", "\n", " \n", "\n", "But by the end of the simulation run, the economy is at the same place in both of these simulations. Whatever influence the initial conditions had on the state of the economy has been erased. That the labor force and the efficiency of labor are at the same levels and have the same growth rates is no surprise: that is compelled by the basic setup. But the capital stock, the level of output, the level of output per worker, and the capital-output ratio—all were very different in the first, initial years. And the capital-output ratio is approaching the same steady-state asymptote. \n", "\n", " \n", "\n", "### 4.3.2 Analyzing the Solow Model\n", "\n", "#### 4.3.2.1 Balanced Growth\n", "\n", "Earlier, when we had assumed that labor force and efficiency were both constant, and thus that n and g were both equal to 0, our equilibrium condition was K/Y = s/δ. Since the saving-investment rate s and the depreciation rate δ were both constant, our equilibrium condition required that the capital-output ratio K/Y be constant as well. There was where the economy was in equilibrium, in balance, with a stable level of not only the capital-output ratio but of the capital stock and output as well.\n", "\n", "Now we have added more realism back into our model by allowing the labor force and labor efficiency to both increase over time, at their constant rates n and g. This should, one would think, change how the economy behaves—and it would change the equilibrium condition that we look at to understand in which directions the economic variables of interest are changing.\n", "\n", "What effect does this added realism—allowing n and g to take on values other than 0—have on our equilibrium condition that the capital-output ratio be stable?\n", "\n", "In an important sense: None! Our equilibrium condition is, in a sense, the same: that the capital-output ratio be constant. But this time, when the capital-output ratio is constant the economy's equilibrium is not one of stable and stagnant balance but rather one of _balanced growth_. Output per worker is then growing at the same rate as the capital stock per worker. Output is growing and the same rate as the capital stock. The former two per worker variables are in balance, and they are both growing at the same rate as the efficiency of labor. The latter two economy-wide aggregate variables are in balance, and they are growing at the same rate as the sum of labor efficiency growth and labor force growth.\n", "\n", " \n", "\n", "#### 4.3.2.2 Mathematical Tools for Analyzing Growth\n", "\n", "This is a good place to introduce four mathematical rules of thumb to make life easier. They are all only approximations. But they are good enough for our purposes. They are:\n", "\n", "1. **The growth-of-a-product rule**: The growth rate of a product is equal to the sum of the growth rates of its components. Since total output Y is equal to output per worker Y/L times the number of workers L, the growth rate of total output Y will be equal to the growth rate of Y/L plus the growth rate of L: $ g_Y = g_{(Y/L)} + g_L $\n", "\n", "2. **The growth-of-a-quotient rule**: The proportional change of a quotient is equal to the difference between the proportional changes of its components. Since capital per worker K/L is equal to the quotient of capital K and the number of workers L, its growth rate will be the difference between their growth rates: $ g_{(K/L)} = g_K - g_L $\n", "\n", "3. **The growth-of-a-power rule**: The proportional change of a quantity raised to a power is equal to the proportional change in the quantity times the power to which it is raised. For example, suppose that we have a situation in which output per worker Y/L is equal to the capital stock per worker K/L raised to raised to the power $\\alpha$ times some constant E: $ Y/L = (K/L)^{\\alpha}E $. Then because the growth rate of E is zero and becasue of the growth-of-a-power rule the growth rate of Y will be equal to $ \\alpha $ times the growth rate of K: $ g_{(Y/L)} = {\\alpha}g_{(K/L)} $\n", "\n", "4. **The rule of 72**: A quantity growing at k percent per year doubles in 72/k years. A quantity shrinking at k percent per year halves itself in 72/k years.\n", "\n", "You may hear people say that a background in calculus is needed to understand intermediate macroeconomics. That is not true. These four mathematical rules of thumb contain 95 percent of what calculus is used for in intermediate macro economics. (Of course, you do need calculus if you want to do more than just take them on faith, and instead have a deep understanding of just why these rules of thumb work.)\n", "\n", " \n", "\n", "### 4.3.2.3 The Balanced Growth Capital-Output Ratio\n", "\n", "But at what value will the economy’s capital-output ratio be constant? Here is where allowing n and g to take on values other than 0 matters. The capital-output ratio will be constant—and therefore we’ll be in balanced-growth equilibrium — when K/Y = s/(n + g + δ). Add up the economy’s labor-force growth rate, efficiency-of-labor growth rate, and depreciation rate; divide the saving-investment rate by that sum; and that is your balanced-growth equilibrium capital-output ratio.\n", "\n", "Why is s/(n + g + δ) the capital-output ratio in equilibrium? Think of it this way: Suppose the economy is in balanced growth. How much is it investing? There must be investment equal to δK to replace depreciated capital. There must be investment equal to nK to provide the extra workers in the labor force, which is expanding at rate n, with the extra capital they will need. And, since the efficiency of labor is growing at rate g, there must be investment equal to gK in order for the capital stock to keep up with increasing efficiency of labor.\n", "\n", "Adding these three parts of investment requirements together and setting the sum equal to the gross investment sY actually going on gets us to:\n", "\n", ">$ (n + g + δ)K = sY $\n", "\n", "This is a condition for capital and output to be in balance—for savings at that capital-output ratio to be such so to equal the investment requirements for capital and output to grow at the same rate. Thus the economy’s investment requirements for balanced growth equal the actual flow of investment when the capital-ouput ratio is:\n", "\n", ">$ \\left(\\frac{K}{Y}\\right)_* = \\frac{s}{n+g+δ} $\n", "\n", "Where we add the star (\\*) to denote that this is an _equilibrium condition_: this is not the current capital-output ratio, but rather the equilibrium capital-output ratio: the one toward which the economy is heading and at which it will rest.\n", "\n", "This is thus the balanced-growth equilibrium condition. \n", "\n", "When it is attained, what the capital-output ratio K/Y will be is constant because s, n, g, and δ are all constant. So when there is balanced growth—when output per worker Y/L and capital per worker K/L are growing at the same rate—the capital-output ratio K/Y will be constant. If the capital-output ratio K/Y is lower than s/(n + g + δ), then depreciation (δK) plus the amount (n + g)K that capital needs to grow to keep up with growing output will be less than investment (sY), so the capital-output ratio will grow. It will keep growing until K/Y reaches s/(n + g + δ). If the capital-output ratio K/Y is greater than s/(n + g + δ), then depreciation (δK) plus the amount (n + g)K that capital needs to grow to keep up with growing output will be greater than investment (sY), so the capital-output ratio will shrink. It will keep shrinking until K/Y falls to s/(n + g + δ). \n", "\n", " \n", "\n", "### 4.3.2.4 Some Algebra\n", "\n", "To see more formally that K/Y = s/(n + g + δ) is the balanced-growth equilibrium condition requires a short march through algebra—simple algebra, we promise. Start with the production function in its per worker form:\n", "\n", ">$ \\frac{Y}{L} = \\left(\\frac{K}{L}\\right)^α \\left(E\\right)^{1-α} $\n", "\n", "Break the capital-labor ratio down into the capital-output ratio times output per worker:\n", "\n", ">$ \\frac{Y}{L} = \\left(\\frac{K}{Y}\\frac{Y}{L}\\right)^α \\left(E\\right)^{1-α} $\n", "\n", "Regroup:\n", "\n", ">$ \\frac{Y}{L} = \\left(\\frac{Y}{L}\\right)^α\\left(\\frac{K}{Y}\\right)^α \\left(E\\right)^{1-α} $\n", "\n", "Collect terms:\n", "\n", ">$ \\left(\\frac{Y}{L}\\right)^{1-α} = \\left(\\frac{K}{Y}\\right)^α \\left(E\\right)^{1-α} $\n", "\n", "And clean up:\n", "\n", ">$ \\left(\\frac{Y}{L}\\right) = \\left(\\frac{K}{Y}\\right)^{\\frac{α}{1-α}} \\left(E\\right) $\n", "\n", "This tells us that _if the capital-output ratio K/L is constant, then the proportional growth rate of output per worker is the same as the proportional growth rate of E_. And the proportional growth rate of labor efficiency E is the constant g. \n", "\n", "Recall that the labor force is growing at a constant proportional rate n. With output per worker growing at rate g and the number of workers growing at rate n, total output is growing at the constant rate n + g. Thus for the capital-output ratio K/Y to be constant, the capital stock also has to be growing at rate n + g. \n", "\n", "This means that the annual change in the capital stock must be: (n + g)K. Add in investment necessary to compensate for depreciation, and that is why we have:\n", "\n", ">$ \\frac{K}{Y} = \\frac{s}{n+g+δ} $\n", "\n", "as our balanced-growth equilibrium condition.\n", "\n", " \n", "\n", "### 4.3.2.5 From Algebra Back to Economics\n", "\n", "In economic terms, the balanced-growth equilibrium capital-output ratio is equal to the share of production that is saved and invested for the future—the economy’s saving-investment rate s—divided by the sum of three things:\n", "\n", "* The growth rate of the labor force n.\n", "* The growth rate of the efficiency of labor g.\n", "* The depreciation rate δ at which capital breaks down and wears out.\n", "\n", "We’ll sometimes call s/(n + g + δ) the “equilibrium” capital-output ratio, and we’ll sometimes call it the “balanced-growth” capital-output ratio. To be always saying “balanced-growth equilibrium” capital-output ratio is too much of a mouthful.\n", "\n", "How do we know K/Y = s/(n + g + δ) gives us balanced growth, where capital per worker K/L and output per worker Y/L grow at the same rate? \n", "\n", "Suppose the current capital-output ratio is lower than s/(n + g + δ). Then (n + g + δ)K would be less than the economy’s total investment which is equal to sY, the saving rate s times the level of output Y. Thus saving and investment will more than provide new workers with the capital they need to be fully productive, more than cover the increase in output due to the increase in labor efficiency, and more than com pensate for the wearing out of capital through depreciation. The capital stock would grow faster than n + g. Since n + g is the rate at which output grows, the capital-output ratio would rise.\n", "\n", "Suppose instead the current capital-output ratio were above s/(n + g + δ). Then sY will be less then the economy’s total investment sY will be insufficient to keep the capital stock growing at rate n + g. And since n + g is the rate at which output grows, the capital-output ratio will fall.\n", "\n", "Thus a capital-output ratio greater than s/(n + g + δ) makes the capital-output ratio fall. And a capital-output ratio less than s/(n + g + δ) makes the capital- output ratio rise. So a capital-output ratio equal to s/(n + g + δ) is indeed the balanced-growth equilibrium condition.\n", "\n", "We have now solved our Solow growth model.\n", "\n", "It delivers one equilibrium condition: telling us that the stable capital-output ratio will be s/(n + g + δ), and that the economy will evolve over time to reduce the gap between the current capital-output ratio K/Y and its equilibrium balanced-growth value.\n", "\n", "It consists of five behavioral relationships:\n", "\n", "* A production function.\n", "* A rate of labor-force growth: n.\n", "* A rate of growth in the efficiency of labor: g.\n", "* A rate of depreciation of capital: δ.\n", "* The saving-investment rate: s.\n", "\n", "“Bob Solow got the Nobel Prize for that?!” you may ask. Ah, but what he got the Nobel Prize for was taking a complicated subject and making a useful model of it that was very simple indeed. The model is simple to write down. But it is powerful. And now we get to use it to generate insights.\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 4.3.3 Understanding the Solow Model\n", "\n", "#### 4.3.3.1 Deriving the Balanced Growth Path for Output per Worker\n", "\n", "Along what path for output per worker will the balanced-growth equilibrium condition be satisfied? Y/L is, after all, our best simple proxy for the economy’s overall level of prosperity: for material standards of living and for the possession by the economy of the resources needed to diminish poverty. Let’s calculate the level of output per worker Y/L along the balanced-growth path.\n", "\n", "Begin with the capital-output ratio version of the production function that we just calculated above:\n", "\n", ">$ \\left(\\frac{Y}{L}\\right) = \\left(\\frac{K}{Y}\\right)^{\\frac{α}{1-α}} \\left(E\\right) $\n", "\n", "Since the economy is on its balanced-growth path, it satisfies the equilibrium con dition K/Y = s/(n + g + δ). Substitute that in:\n", "\n", ">$ \\left(\\frac{Y}{L}\\right) = \\left(\\frac{s}{n+g+δ}\\right)^{\\frac{α}{1-α}} \\left(E\\right) $\n", "\n", "s, n, g, δ, and α are all constants, and so [s/(n + g + δ)](α/(1-α));is a constant as well. This tells us that along the balanced-growth path, output per worker is simply a constant multiple of the efficiency of labor, with the multiple equal to:\n", "\n", ">$ \\left(\\frac{s}{n+g+δ}\\right)^{\\frac{α}{1-α}} $\n", "\n", "Over time, the efficiency of labor grows. Each year it is g percent higher than the last year. Since along the balanced-growth path output per worker Y/L is just a constant multiple of the efficiency of labor, it too must be growing at the same proportional rate g.\n", "\n", "Now it is time to introduce time subscripts, for we want to pay attention to where the economy is now, where it was whence, and where it will be when. So rewrite as:\n", "\n", ">$ \\left(\\frac{Y_t}{L_t}\\right) = \\left(\\frac{s}{n+g+δ}\\right)^{\\frac{α}{1-α}} \\left(E_t\\right) $\n", "\n", "Paying attention to the equations for how labor efficiency and the labor force grow over time, $ E_t = E_0(1 + g)^t $ and $ L_t = L_0(1 + n)^t $, we can plug in and solve for what Y/L and Y will be at any time t—as long as the economy is on its balanced-growth path.\n", "\n", " \n", "\n", "##### 4.3.3.1.1 Along the Balanced-Growth Path, Output per Worker Is a Constant Multiple of the Efficiency of Labor**\n", "\n", "Along its balanced-growth path, the level of output per worker is a constant multiple of the efficiency of labor. What that multiple is depends on all the parameters of the growth model: the saving rate s, the labor-force growth rate n, the efficiency-of-labor growth rate g, the depreciation rate δ, and the diminishing-returns-to-investment parameter α. The equation is:\n", "\n", ">$ \\frac{Y_t}{L_t} = \\left(\\frac{s}{n+g+\\delta}\\right)^\\left(\\frac{\\alpha}{1-\\alpha}\\right)E_t $\n", "\n", "with:\n", "\n", ">$ E_t = E_0{(1 + g)^t} $\n", "\n", " \n", "\n", "\"Output\n", "\n", ">**Output per Worker and the Efficiency of Labor on the Balanced Growth Path**\n", "\n", " \n", "\n", "#### 4.3.3.2 Interpreting the Balanced Growth Path for Output per Worker\n", "\n", "We now see how capital intensity and technological and organizational progress drive economic growth. Capital intensity—the economy’s capital-output ratio—determines what is the multiple of the current efficiency of labor E that balanced-growth path output per worker Y/L is. Things that increase capital intensity—raise the capital-output ratio—make balanced-growth output per worker a higher multiple of the efficiency of labor. Thus they make the economy richer. Things that reduce capital intensity make balanced-growth output per worker a lower multiple of the efficiency of labor, and so make the economy poorer.\n", "\n", "Suppose that α is 1/2, so that α/(l—α) is 1, and that s is equal to three times n + g + δ, so that the balanced-growth capital-output ratio is 3. Then balanced-growth output per worker is simply equal to three times the efficiency of labor. If we consider another economy with twice the saving rate s, its balanced-growth capital-output ratio is 6, and its balanced-growth level of output per worker is twice as great a multiple of the level of the efficiency of labor.\n", "\n", "The higher is the parameter α—that is, the slower diminishing returns to investment set in—the stronger is the effect of changes in the economy’s balanced-growth capital intensity on the level of output per worker, and the more important are thrift and investment incentives and other factors that influence s relative to those that influence the efficiency of labor.\n", "\n", "* Suppose that the balanced-growth capital-output ratio is 4. Then if α is 1/3, α/(1—α) is 1/2, and the balanced-growth path level of output per worker is twice the level of the efficiency of labor, as you then multiply the efficiency of labor by the square root of the capital-output ratio. Most economists think that 1/3 is a reasonable parameter value for the United States today.\n", "\n", "* By contrast, if α is 1/2, α/(l—α) is equal to 1, and again with a balanced-growth capital-output ratio of 4, the level of output per worker is fully four times the level of the efficiency of labor, as you then multiply the efficiency of labor by the capital-output ratio.. Most economists think that 1/2 is a reasonable parameter value for the United States a century ago or for relatively poor countries today.\n", "\n", "* But some economists—for example, Paul Romer, currently of the World Bank—believe that the evidence points at still higher values of α. If α = 2/3, and again with a balanced-growth capital-output ratio of 4, the level of output per worker is fully sixteen times the level of the efficiency of labor, as you then multiply the efficiency of labor by the square of the capital-output ratio.\n", "\n", "Note—this is important-that changes in the economy’s capital intensity shift the balanced-growth path up or down to a different multiple of the efficiency of labor, but the growth rate of Y/L along the balanced-growth path is simply the rate of growth g of the efficiency of labor E. The material standard of living grows at the same rate as labor efficiency. \n", "\n", "To change the very long run growth rate of the economy you need to change how fast the efficiency of labor grows. Changes in the economy that merely alter the capital-output ratio will not do it. They can have a large effect on the level. \n", "\n", "This is what tells us that technology, organization, worker skills, education—all those things that increase the efficiency of labor and keep on increasing it—are likely to be ultimately more important to growth in output per worker than saving and investment. The U.S. economy experienced a large increase in its capital-output ratio in the late nineteenth century. It may be experiencing a similar increase now, as we invest more and more in computers. But the Gilded Age industrialization came to an end, and the information technology revolution will run its course. Aside from these episodes, it is growth in the efficiency of labor E that sustains and accounts for the lion’s share of long-run economic growth.\n", "\n", " \n", "\n", "#### 4.3.3.2.1 Calculating Balanced Growth Path Output per Worker: An Example\n", "\n", "To see how to use the expression for output per worker when the economy is on its balanced-growth path, let’s work through an example. \n", "\n", "Suppose that the economy’s labor-force growth rate n is 1 percent per year, the efficiency-of-labor growth rate g is 2 percent per year, and the depreciation rate δ is 3 percent per year. Suppose further that the diminishing-returns-to-investment parameter α is 1/2, and the economy’s saving-investment rate s is 18 percent.\n", "\n", "Then the balanced-growth equilibrium capital-output ratio s/(n + g + δ) equals 3, and α/(l—α) equals 1. Substituting these values into equation (4.3.8) above:\n", "\n", ">$ \\frac{Y_t}{L_t} = \\left(\\frac{0.18}{0.06}\\right)^\\left(\\frac{0.5}{1-0.5}\\right)E_t $\n", "\n", ">$ \\frac{Y_t}{L_t} = \\left(3\\right)^\\left(1\\right)E_t $\n", "\n", ">$ \\frac{Y_t}{L_t} = 3(E_t) $\n", "\n", "For these parameter values, balanced-growth output per worker is simply three times the efficiency of labor, whatever the value of the efficiency of labor is. When the efficiency of labor is 10,000 per year, balanced-growth output per worker is 30,000 per year. When the efficiency of labor rises to 20,000 per year, balanced- growth output per worker rises to 60,000 per year.\n", "\n", "Over time, because balanced-growth output per worker is a constant multiple of the efficiency of labor, its growth rate is the same as g, the growth rate of the efficiency of labor: 2 percent per year.\n", "\n", " \n", "\n", "##### 4.3.3.2.2 A Graphical Approach\n", "\n", "The implications of the balanced-growth capital-output ratio for the balanced-growth level of output per worker, and how that level changes over time, can be seen in an alternative, diagrammatic way. Take a look at Figure 4.3.4. As before, draw the production-function curve that shows output per worker Y/L as a function of capital per worker K/L for the current level of the efficiency of labor Et. In addition, as before, draw the line that shows where the capital-output ratio is equal to its balanced-growth equilibrium value, K/Y = s/(n + g + δ). This line starts at the bottom left origin point (0, 0) and climbs toward the upper right. Because K/L is on the horizontal axis and Y/L is on the vertical axis, the slope of the line is not K/Y but instead Y/K or (n + g δ)/s\n", "\n", "Look once again at where the curves cross. That point shows the current level of output per worker along the balanced-growth path. Output per worker is given by the production function for the current levels of capital per worker and the efficiency of labor. And the capital-output ratio is at its balanced-growth path level. Anything that increases the balanced-growth capital-output ratio will lower Y/K. Thus rotating the equilibrium line clockwise raising the balanced-growth path level of output per worker. Anything that decreases the balanced-growth capital output ratio rotates the equilibrium line counterclockwise. It thus lowers the level of output per worker for a given value of the efficiency of labor E.\n", "\n", " \n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ " \n", "\n", "----\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 4.1 Slides: Sources of Long-Run Economic Growth\n", "\n", "### Solow Growth Model: Preliminaries\n", "\n", "* The Eagle's Eye View\n", "* Divergences\n", " * Differences in the efficiency of labor\n", " * Differences in capital intensity\n", "* Aim to understand why some countries…\n", " * …are growing rapidly and others are not\n", " * …are rich and others are not\n", " * …under conditions of industrial civilization\n", " \n", " \n", "\n", "### Some Facts: Looking Across Countries\n", "\n", "\"Gapminder\n", "\n", "* U.S. today:\n", " * 3x richer than Mexico\n", " * 9x richer than Nigeria\n", " * 20x richer than Afghanistan\n", " * PPP or RER?\n", "* Afghans back at where U.S. was 200 years ago\n", " * Differences between Afghanistan and U.S. then much smaller than now\n", " * In spite of improved communications, transport, etc…\n", " \n", " \n", "\n", "### Some Facts: Looking Over Time\n", "\n", "\"4\n", "\n", "* 15x in productivity since 1870\n", " * And this might be an underestimate\n", "* Doubling of life expectancy since 1870\n", " * Where do you think the 1919 “Spanish Influenza” epidemic is?\n", "* Divide $4000 by 13 and you are at a dollar a day\n", " * Anything less, and you are starving to death\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### What Is the Pace of U.S. Growth?\n", "\n", "\"4\n", "\n", "* 60000 today\n", "* 4000 150 years ago\n", "* What’s the growth rate?\n", "\n", " \n", "\n", "1. 2.8%/year\n", "2. 1.8%/year\n", "3. 0.8%/year\n", "4. 5.6%/year\n", "5. None of the above\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Reversals of Fortune\n", "\n", "\"2017\n", "\n", "* Sweden was about like Norway back in 1870\n", " * A “s—-hole country”\n", " * Swedes (and Norwegians) left in droves for the New World\n", "* By contrast, Argentina back in 1870 looked like it had it made\n", "* Reversals of fortune\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Structure of an Economic Model\n", "\n", "* **Variables**: economic quantities of interest that we can calculate and measure.\n", "* **Behavioral relationships**: relationships that (1) describe how humans, making economic decisions given their opportunities and opportunity costs, decide what to do, and (2) that thus determine the values of the economic variables.\n", "* **Equilibrium conditions**: conditions that tell us when the economy is in a position of balance, when some subset of the variables are \"stable\"—that is, are either constant are changing in simple and predictable ways.\n", "\n", " \n", "\n", "### The Structure of the Solow Growth Model\n", "\n", "* Production function:\n", " * A 1% increase in the efficiency of labor E increases production Y by 1-α%\n", " * A 1% increase in the capital-labor ratio K/L increases production per worker Y/L by α%\n", " * Why these rules of thumb?\n", " * They force us to: $ \\frac{Y}{L} = \\left(\\frac{K}{L}\\right)^α(E)^{(1-α)} $\n", "* Capital accumulation equation: dK/dt = sY - δK\n", "* Labor force growth: dL/dt = nL\n", "* Efficiency of labor (technology and organization) growth: dE/dt = gE\n", "\n", " " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Production Function II\n", "\n", "Alternative functional forms:\n", "\n", "> $ \\frac{Y}{L} = \\left(\\frac{K}{L}\\right)^α(E)^{(1-α)} $\n", "\n", "> $ \\ln\\left(\\frac{Y}{L}\\right) = α\\ln\\left(\\frac{K}{L}\\right) + (1-α)\\ln(E) $\n", "\n", "> $ \\frac{Y}{L} = \\left(\\frac{K}{Y}\\right)^\\left(\\frac{α}{1-α}\\right)(E) $\n", "\n", "> $ \\ln\\left(\\frac{Y}{L}\\right) = \\left(\\frac{α}{1-α}\\right)\\ln\\left(\\frac{K}{L}\\right) + \\ln(E) $\n", "\n", "> $ \\frac{d\\left(\\ln\\left(\\frac{Y}{L}\\right)\\right)}{dt} = α \\frac{d\\left(\\ln\\left(\\frac{K}{L}\\right)\\right)}{dt} + (1-α)\\frac{d\\left(\\ln\\left(E\\right)\\right)}{dt} $\n", "\n", "> $ \\frac{d\\left(\\ln\\left(\\frac{Y}{L}\\right)\\right)}{dt} = α \\frac{d\\left(\\ln\\left(K\\right)\\right)}{dt} - α \\frac{d\\left(\\ln\\left(L\\right)\\right)}{dt} + (1-α)\\frac{d\\left(\\ln\\left(E\\right)\\right)}{dt} $\n", "\n", "> $ \\frac{d\\left(\\ln\\left(\\frac{Y}{L}\\right)\\right)}{dt} = \\left(\\frac{α}{1-α}\\right) \\frac{d\\left(\\ln\\left(\\frac{K}{Y}\\right)\\right)}{dt} + \\frac{d\\left(\\ln\\left(E\\right)\\right)}{dt} $\n", "\n", "We will use whatever is most convenient for the task at hand at the moment...\n", "\n", " \n", "\n", "### Simplest Possible Accumulation Equations\n", "\n", "> $ \\frac{dK}{dt} = sY - {\\delta}K $ :: capital accumulation\n", "\n", "> $ \\frac{dL}{dt} = nL $ :: labor force growth\n", "\n", "> $ \\frac{dE}{dt} = gE $ :: efficiency-of-labor growth\n", "\n", " \n", "\n", "> $ \\frac{d}{dt}{\\ln}(K) = s\\left(\\frac{Y}{K}\\right) - {\\delta} $ :: capital accumulation\n", "\n", "> $ \\frac{d}{dt}{\\ln}(L) = n $ :: labor force growth\n", "\n", "> $ \\frac{d}{dt}{\\ln}(E) = g $ :: efficiency-of-labor growth\n", "\n", " \n", "\n", "### How Does a System Like This Behave Over Time?\n", "\n", "* Let’s start it up and see…\n", "* " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "## Catch Our Breath\n", "\n", "* Ask me two questions…\n", "* Make two comments…\n", "* Further reading…\n", "\n", "
\n", "\n", "----\n", "\n", "Long-Run Economic Growth Theory—Sources: \n", "Lecture Support: \n", "\n", " \n", "\n", "----" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Slides: Presenting the Solow Growth Model\n", "\n", "\n", "\n", "## Catch Our Breath\n", "\n", "* Ask me two questions…\n", "* Make two comments…\n", "* Further reading…\n", "\n", "
\n", "\n", "----\n", "\n", "Long-Run Economic Growth Theory—Presenting the Model: <> \n", "Lecture Support: <>\n", "\n", " \n", "\n", "----" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.6" } }, "nbformat": 4, "nbformat_minor": 2 }