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Drivers of Growth: Technology, Policy, and Institutions

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In the last chapter, we used the Solow model to examine the role of capital accumulation in determining why some countries have experienced high economic growth, while others have not grown at all, leaving their citizens mired in abject poverty. A deeper question remains: Why do capital and productivity grow at rapid rates in some countries but not in others?

To answer this question, we start this chapter by discussing how technology differs from the conventional production inputs of capital and labor, which we discussed in the previous chapter. We will then explore policies to promote productivity growth, such as building physical infrastructure (roads and ports), increasing the knowledge and skills of workers (human capital), and providing incentives to stimulate research and development. We will also explore why the right set of basic institutions—including property rights and an effective legal system—is key to attaining high growth in capital and productivity, and hence economic growth. Rich countries like Japan, the United States, and those in Western Europe feature strong institutions that promote economic growth, while poor nations in parts of Africa, the Middle East, and elsewhere have weak institutional frameworks, leaving large fractions of their populations unnecessarily stuck at low levels of income.

Finally, to understand more deeply why policies to promote institutional development and productivity growth are so crucial to improving living standards (economic growth), we develop a theory of economic growth that analyzes productivity changes endogenously through technological advancement.

Technology as a Production Input

An important conclusion from both growth accounting and the Solow growth model is that increases in total factor productivity, $A$, are a key driver of economic growth. Up until now, we have treated capital and labor as the only inputs of production and the productivity term, $A$, as exogenous, or given. We now put aside that assumption and look closely at $A$, which we can also think of as technology. Because technology is what increases the efficiency of capital and labor, we can think of it as an input of production, similar to capital and labor.

Technology Versus Conventional Production Inputs

To explain what causes $A$ to grow over time, we first need to understand that technology differs from the conventional production inputs, capital and labor, in an important way. Capital and labor are physical things, or objects, while technology is a set of instructions, designs, or ideas. Objects are rival in their use, because when they are used in one activity, they cannot be used in another. Ideas, on the other hand, can be used by multiple people in more than one activity simultaneously, so we call them nonrival. For example, you and your roommate cannot both use the same computer to write a term paper at the same time, nor can you simultaneously write a term paper and wait tables in a restaurant. Your capital and labor are objects. A suggestion for a term paper topic posted on the Web might be used by many students across the country: it is an idea. Technology is a set of ideas, making it a special form of capital that is nonphysical, nonrivalrous, and endlessly reusable.

Technology and Excludability

A big idea—say, using bed nets to repel mosquitoes and thus prevent malaria—can sweep the world and improve standards of living everywhere. The flip side of the nonrivalry of ideas is their low excludability. That is, a technology cannot be padlocked like a factory and kept from unauthorized use. More formally, excludability is the ability of the owner of a piece of property to deny its use to others unless they pay for it. Owners of capital earn a rate of return by renting out excludable items, so investments in them can have high payoffs. On the other hand, it is hard to keep unauthorized users from using a technology, reducing the potential benefits you can reap from producing it. This nonexcludability may discourage investment in technology, which has important implications for policy, as we shall see next.

Policies to Promote Productivity

Our analysis of the Solow model indicates that policies that promote saving and productivity can lead to higher standards of living. We have already discussed policies to stimulate saving in Chapter 4; we now focus on government policies that stimulate the accumulation of capital and productivity.

Building Infrastructure

The responsibilities of laying roads and building ports and bridges often fall to governments, since such costly projects wouldn’t be economical for any individual firm. Investments in public infrastructure have generated high economic returns and have raised living standards.

But just how much should governments spend on physical infrastructure? In 2009, the Obama administration cited the sad state of many U.S. roads and bridges as a rationale for its 2009 stimulus package. It argued that government spending on infrastructure had fallen too low, leading to some devastating bridge collapses over recent years. Critics, however, dispute evidence that infrastructure investments boost productivity and income. It may be instead that governments spend more on infrastructure in periods of rapid economic growth. Also, the government may overspend on infrastructure projects. It is tempting for businesses to contribute to political campaigns, or even pay bribes, to win contracts for infrastructure building, creating a recipe for wasteful spending.

The U.S. Interstate Highway System is one successful example of government spending on infrastructure. The Interstate Highway System, formally named the Dwight D. Eisenhower National System of Interstate and Defense Highways, is the largest highway system in the world, with nearly 50,000 miles of roads. It is also the largest public works program in history, costing several hundred billion in today’s dollars. The Eisenhower administration sought legislation to create this system, which was started in 1956 and was largely completed by 1990. The Interstate Highway System has promoted tourism (almost all of us have taken family vacations using the system) and has substantially reduced the cost of shipping goods from one part of the country to another. Without the Interstate Highway System, U.S. productivity would likely be much lower and the U.S. way of life would be substantially different.

Increasing Human Capital

So far we have focused on two factors of production in discussing economic growth: physical capital and labor. However, another kind of capital, human capital, the knowledge and skills that workers have built up through education and training programs, has a very large effect on productivity. For example, scientists require many years of education and experience in the lab before they can produce new technology. Research finds that human capital is at least as important as physical capital in explaining differences in per-capita GDP across countries.

With higher education and training, workers are much more productive and therefore earn much higher wages. You will be glad to know that your college education leads to a college premium, a higher wage over their lifetime for college graduates relative to high-school graduates. The return to education has risen substantially since 1980, with the college premium rising from $50\%$ in 1980 to $80\%$ today—so it is well worth your while to study hard and stay in school. (We will discuss why this college premium has increased in Chapter 20.)

Policy and Practice: Government Measures to Increase Human Capital

For most of the twentieth century, the United States set the global gold standard for educating its populace. Building on its global lead in providing free elementary education, the United States in the early part of the century invested heavily in universal high school education. By mid-century, its colleges and universities had become the world’s best. The rest of the world lagged badly behind for much of the century. Fewer than $40\%$ of older European teenagers attended secondary schools in the 1950s, compared with over $70\%$ of older U.S. teenagers.

As our productivity model would predict, investments in human capital through education spending made a sizable impact on economic growth. Between 1915 and 2005, educational improvements in the U.S. labor force accounted for about $15\%$ of real per-capita GDP growth, or about 0.34 percentage point of growth a year. Data strongly suggest that without an educated population, countries will struggle to become productive members of the global economy. In a study of 114 nations, economists Claudia Goldin and Lawrence Katz of Harvard University identified a positive correlation between enrollment levels in secondary school and real GDP. They noted, however, that educational improvements alone were not guaranteed to make a poor country wealthy.

The rest of the world has learned its lesson from U.S. successes. Today, even poor nations invest large sums in education, and wealthy regions like Europe have largely caught up to U.S. educational achievements. In a 2006 study of twenty-six wealthy nations, eighteen nations graduated a greater proportion of citizens from high school than did the United States. U.S. educational achievements began to stagnate in the 1970s, sparking a debate about whether the country had hit some natural limit on educational achievement, or whether a new educational strategy—perhaps focusing more on quality, and less on quantity, of schooling—was required.

Nevertheless, the United States continues to allocate large sums to finance education. With a budget of around $7$ billion, the federally-funded Head Start Program supports local schools and other organizations that provide comprehensive child development services for poor pre-school children, giving them a leg up that will help them succeed when they go to elementary school. State and local governments spend on the order of \500$ billion annually to provide fifty million children free public school education up through twelfth grade. Also, the government funds many colleges and universities and subsidizes loans to students who cannot afford college.

While not a direct investment in human capital, government spending on public health has the effect of keeping workers healthier and more productive. For example, in 2009, the U.S. government spent \1$billiononanewswinefluvaccinetocontaintheoutbreakoftheswineflupandemic.Governmentspendingonhealthcarehasbecomeoneofthelargestitemsinthefederalbudgetandisnowcloseto$$1$ trillion per year.

Encouraging Research and Development

The importance of technology to economic growth suggests that countries can improve their standards of living by devoting increased resources to research and development (R&D). Private companies recognize the value of investing in research and spend a great deal on R&D. For instance, in 2009, Google invested \2.8$billion,ormorethan$12%$ of its revenue, in R&D. But because technology has the downside that it may be nonexcludable—that is, others can make use of it without paying for it—many economists believe that the private sector naturally underspends on R&D. Government encouragement of R&D is therefore less controversial among economists than is government spending on infrastructure.

There are three basic ways that governments can encourage R&D: government spending on R&D, government tax incentives, and patents.

Government Spending on R&D

Governments can directly increase R&D by engaging in research and development at government facilities. For example, many technological innovations have come from government labs, including nuclear power, jet planes, and the electronic computer. Governments also provide grants to universities and private researchers for basic research through agencies such as the National Science Foundation and the National Institutes of Health. Governments recognize that research universities can be an important source of economic growth for particular regions. For example, Boston has gained tremendously from having top research universities in its local area, such as Harvard University, MIT, Tufts, Boston University, and Brandeis. Similarly, Silicon Valley grew up around Stanford University. And India’s own high-tech hub, Bangalore, flourished around the highly prestigious Indian Institute of Science. State and local governments, as well as the U.S. federal government, give direct subsidies to research universities. In recent years, Europeans have increased their support for research universities, recognizing the benefits that have accrued in the United States.

Tax Incentives for R&D

Because private businesses are likely to be more efficient than the government at producing practical R&D that can be utilized immediately in developing new products and technology, governments also encourage R&D by giving private businesses tax breaks for research. U.S. tax credits for R&D were first implemented with the Economic Recovery Act of 1981. Congress has renewed these credits several times over the intervening years. Companies that qualify for the credit can deduct $20\%$ of their research expenses above a base amount that is determined by historical expenses over a base period.

Almost all advanced economies offer some form of tax incentive for R&D. These incentives go beyond the form of tax credits by allowing firms to deduct $100\%$ and sometimes more of their research expenses from their income in calculating their taxes. Other incentives allow firms to depreciate machinery and equipment that are involved in R&D at a higher rate than normal, thereby reducing their tax bills.

Patents

Another approach to managing the nonexcludability of technology is to grant intellectual property rights to inventors through a system of patents that give inventors the sole right to use, make, or sell licensing rights to others for a set period of time, typically around twenty years. For example, a drug company that earns a patent on a cholesterol-lowering medicine can sue anyone who tries to produce that drug without permission for decades. These property rights help businesses earn higher profits and recoup the investments they make in research and development, and encourage others to invest in R&D.

Patents are not a new idea. Indeed, at the very founding of the United States, the Constitution authorized Congress to enact laws to “promote the progress of science and useful arts, by securing for limited times to authors and inventors the exclusive right to their respective writings and discoveries.” For example, firms in the drug industry have spent billions on research to develop drugs such as Lipitor or Tamiflu because their sole right to produce them for a number of years has provided them with enormous profits.

Effective design of a patent system is crucial. If patents are given too freely or for too long a period, technological growth may slow, since patent owners might refuse to share their knowledge or charge prohibitively high prices to license its use. Particularly worrisome in recent years has been the rise of colorfully named “patent trolls,” who buy up patents and then try to extract large payments from companies that try to develop or use similar technologies. On the other hand, if the patent right is difficult to enforce, or if patents are not given for certain types of inventions, then there might not be enough investment in R&D. Alternatively, the inventor may be unwilling to share ideas with others in publicly available patent applications, thereby hindering technological advance. For example, the Coca-Cola Company has chosen to keep its soda formula a secret, rather than disclose it in a patent application, because it does not believe its patent rights would be enough to keep competitors from copying the recipe. Patents that do not give sufficient rights to the inventor may therefore leave technology nonexcludable, and so weaken the incentives for R&D and technological progress.

Institutions and Property Rights

We now go on to analyze the role of institutions in economic growth. Institutions are sets of rules, organizations, and customs that govern the behavior of individuals and firms. The most basic and fundamental set of institutions that affect economic growth are property rights, the protection of property from expropriation by the government or other parties. Property rights are necessary if people are to have incentives to make investments. Otherwise, the fruits of their investments (the profits) could be easily taken away. A farmer who does not have clear-cut ownership of his land, or who worries that armed marauders may take away his crops or farm equipment, is unlikely to buy enough seeds, fertilizer, and tractors to boost his farm’s production. Weak property rights thus lead to low investment and low accumulation of capital.

In this section, we outline the key characteristics of legal systems that enforce property rights and examine the key obstacles faced by these legal systems.

The Legal System and Property Rights

Residents of advanced nations like the United States often take property rights for granted. The founding fathers and their British forbearers understood the importance of property rights to the success of an economy, and a vast body of U.S. law protects private property. The government cannot take your property when it wants to; people who steal property are jailed; and people who make use of our property (either physical or intellectual, that is, property based on ideas) without our permission can be sued in courts of law.

Property rights are established by the writing and enforcement of legal contracts. Strong property rights, needed to provide incentives to invest and accumulate capital, therefore require a legal system that operates quickly and at low cost. Douglass North, recipient of a Nobel Prize for his work on the role of institutions in economic development, has emphasized the importance of the legal system to economic growth: “The inability of societies to develop effective, low-cost enforcement of contracts is the most important source of both historical stagnation and contemporary underdevelopment in the Third World.”

There are several elements that make up an effective legal system: the ability to enforce contracts, adequate resources, and lawyers.

Ability to Enforce Contracts

Enacting a good system of laws is the first step in creating strong property rights. Not all legal systems are alike in this regard. A legal system based on common law, in which the law is continually reinterpreted by judges, originated in England. The common-law legal system is in use in the United Kingdom and its former colonies, including the United States, Canada, Australia, and New Zealand, as well as in India and many countries in Africa. Alternative legal systems based on the Napoleonic Code, which was first developed in France, determine the law primarily by statutes. The Napoleonic legal system spread to much of continental Europe during the Napoleonic Wars at the beginning of the nineteenth century. The German and Swedish legal systems have elements of both the Napoleonic and the English common law systems, with laws determined by statutes that are subject to substantial modification by judges.

The English common law system is particularly effective at enforcing contracts because it is able to evolve with changing economic circumstances. Countries with legal systems derived from English common law outperform those with systems based on the Napoleonic Code in terms of financial development and economic growth, with the performance of the German and Swedish systems somewhere in between. (See Table 7.1 for a list of countries and their style of legal system.) It also matters how the legal system was imposed on a country to begin with, a subject explored in the box, “Geography, the Legal System, and Economic Growth.”

Adequate Resources

A well-functioning legal system requires sufficient funding for courts and qualified judges. India’s legal system, for example, is based on the English common-law system, which suggests it should strongly protect property rights. However, India’s legal system is starved for resources and is incredibly overburdened.

Table 7.1: Legal System Origins

English Common Law	Napoleonic Law	German/Swedish
Australia	Argentina	Austria
Botswana	Brazil	Switzerland
Canada	Chile	Germany
U.K.	Côte d’Ivoire	Denmark
Hong Kong	Egypt	Japan
India	Spain	Korea
Ireland	France	Finland
Jamaica	Greece	Iceland
Singapore	Haiti	Taiwan
United States	Italy	Norway
South Africa	Turkey	Sweden

Source: Thorsten Beck, Asli Demirguc-Kunt, and Ross Levine. 2003. Law, endowments, and finance. Journal of Financial Economics 70, no. 2 (November): 137-181.

Access to Lawyers

Love them or hate them, lawyers are needed to promote property rights. When someone encroaches on your land or makes use of your property without your permission, you turn to a lawyer to stop them. Lawyers are necessary to protect your investments, and knowing you can hire one encourages you to invest in property in the first place. Without lawyers, there would be little to no investment, and thus little economic growth.

Obstacles to Effective Property Rights

Countries with strong property rights laws on the books still must enforce them to ensure a flow of investment in the economy. We will now look at obstacles to property rights systems, including corruption, costly legal processes, and greedy government officials.

Corruption

Corruption, which is endemic in many less-developed countries, is a key obstacle to a well-functioning property rights system. If judges can be bribed, then property rights can be expropriated by the highest bidder for the judge’s “services.” An entrepreneur with a good idea or investment cannot protect it from the rich and powerful, those who can use the courts as a weapon to take it away and reap any rewards for themselves. If a government official has to be bribed to allow you to conduct your business, then he or she has in effect appropriated part of the value of your property. Corruption is like a cancer in the body of an economy: it weakens and sickens the economy by reducing the incentives for entrepreneurs to make investments and work hard to make profits. Corruption creates much uncertainty for entrepreneurs, who are left wondering if the bribe money is sufficient and if there will be future demands. Research finds that both lower investment relative to GDP and lower economic growth are associated with increases in corruption.

The High Cost of Establishing Legal Businesses

The high cost of setting up a legal business is another barrier to establishing clear property rights in many developing countries. In countries like the United States, opening a legal business is a simple procedure that requires filling out a form and paying a nominal licensing fee. But, as Peruvian economist Hernando De Soto documented in his fascinating book The Mystery of Capital, setting up a business in a less-developed country can be a nightmare. To legally register a small garment workshop with one worker in Peru required 289 days, working six hours a day, at a total cost of \1,231$—a figure thirty-one times the monthly minimum wage. In contrast, setting up a similar U.S. business takes only a few days at a small fraction of the cost. Peru may be an extreme example, but this problem is endemic in less-developed countries. These barriers to legally registering businesses allow only the rich to operate legal businesses and are a serious impediment to economic growth.

The Grabbing Hand

Simply enacting property laws on the books does not guarantee their enforcement. Government officials in many less-developed countries can arbitrarily expropriate an idea, a business, or an investment for personal gain. Robert Mugabe, the president of Zimbabwe, has impoverished his country by expropriating land and giving it to his cronies. When land can be taken away from a landowner without adequate compensation, landowners will stop investing in their land, and farm production will fall. Zimbabwe, once a highly successful producer of farm products that exported over 700,000 tons of corn in 1990, now exports less than one-twentieth of that amount. Andrei Schleifer of Harvard University and Robert Vishny of the University of Chicago have coined the term “the grabbing hand” to describe the often rapacious behavior of such governments, which have been given the name kleptocracies because they steal from their citizens in a variety of ways.

Kleptocracies have been particularly common in Africa since former European colonies gained independence in the 1960s. Kleptocratic behavior by African governments explains, in large part, why Africa has failed to keep up with other regions in its economic growth, and has even seen a decline in income per person and an increase in poverty in many countries. Haiti, the poorest country in the Americas, has suffered a similar fate. The Duvaliers, Papa Doc and Baby Doc, dictators from 1957 to 1986, were notorious for stealing on a grand scale. Years of strife and misrule in Haiti prevented investment in strong building materials. As a result, the poor infrastructure became a death sentence for the hundreds of thousands of Haitians who died in the 2010 earthquake. Meanwhile, a much more powerful earthquake in Chile a few months later killed only a tiny fraction of the number killed in Haiti, in no small part because Chile had invested in quake-resistant buildings.

Property rights cannot exist if the rule of the gun supersedes the rule of law. Continual wars and rebellions diminish property rights because the threat of force allows aggressors to take property. We usually think of the cost of war in terms of the number of dead and wounded, but the economic cost is horrendous as well. Because the threat of war or rebellion makes it hard to retain the profits from productive investment, investment will not occur. A continual state of warfare and years of kleptocratic rule go a long way toward explaining the awful growth experience of many of the countries of sub-Saharan Africa since they gained independence in the 1960s.

Endogenous Growth Theory

To more deeply understand what promotes economic growth, we need to study a theory of what causes technological advancement, that is, what causes growth in $A$. This theory, which we outline here, is referred to as endogenous growth theory, because it explains why advances in technology endogenously (from within the system) fuel sustained economic growth. Because Paul Romer was so instrumental in developing this theory, endogenous growth models are often referred to as Romer models.

Here we outline a simple version of a Romer model that highlights many of the important implications of endogenous growth theory. We extend the Solow model of the last chapter, using many of the same building blocks, but add a key feature—the production of ideas.

Allocation of Labor

The Solow model assumes that all labor, $L$, is allocated to the production of goods and services. The Romer model changes that assumption and states that some labor, $L_P$, is devoted to producing goods and services, while the rest, $L_A$, is devoted to producing new technology, which we will refer to as research and development (R&D). The total amount of labor is fixed at the total amount of the population, $\overline{N}$.

We can then write total labor, which we assume is the same as the population, $\overline{N}$, as follows:

$$\overline{N}=L_P+L_A(1)$$

We assume that a constant fraction $\alpha$ of the population is engaged in R&D:

$$L_A=\alpha \overline{N}(2)$$

which from Equation (1) implies that the amount of labor devoted to producing goods and services is:

To find $L_P$, we rearrange Equation (1) and substitute Equation (2):

$$L_P=\overline{N}-L_A$$ $$L_P=\overline{N}-\alpha \overline{N}$$ $$L_P=(1-\alpha )\overline{N}(3)$$

Production Function

The production of goods and services in the Romer model is described with largely the same Cobb-Douglas production function as in the previous chapter. Also, we add time subscripts, $t$, to reflect variables that can change over time.

$$Y_t=A_t{K}_t^{0.3}{L}_t^{0.7}(4)$$

Because labor in Equation (4) is only that subset of labor devoted to producing goods and services, we replace $L_t$ with $L_P$ to yield the following:

$$Y_t=A_t{K}_t^{0.3}{L}_P^{0.7}(5)$$

Next, we rewrite Equation (5) in terms of output per worker, just as we did in the previous chapter, but recognize that the workers are only the ones engaged in the production of goods and services:

To express output per worker, $y_{Pt}$, we divide total output $Y_t$ by the labor engaged in production $L_P$:

$$y_{Pt}=\frac{Y_t}{L_P}=\frac{A_t{K}_t^{0.3}{L}_P^{0.7}}{L_P}$$

This simplifies to:

$$y_{Pt}=A_t{\left(\frac{K_t}{L_P}\right)}^{0.3}$$

Defining capital per worker as $k_{Pt}=\frac{K_t}{L_P}$, we get:

$$y_{Pt}=A_t{k}_{Pt}^{0.3}(6)$$

Notice that output and capital per worker now have two subscripts, one to indicate that they can change over time, and the other to indicate that they are now defined in terms of workers that are devoted to the production of goods and services:

$$y_{Pt}=\frac{Y_t}{L_P},k_{Pt}=\frac{K_t}{L_P}$$

The production function in Equation (6) has the same form as that used in the Solow model, so that all the analysis we applied in the previous chapter applies equally here.

Production of Technology

We now assume a very simple form of a production function for technology. Technology is produced by building on old ideas to create new ones, along the lines suggested by the famous quote from Sir Isaac Newton: “If I have seen farther than others, it is because I have stood on the shoulders of giants.” In addition, we assume the increase in technology is proportional to the number of workers that are engaged in research and development, $L_A$. The production function for technology is thus as follows:

$$A_{t+1}-A_t=\Delta A_t=\chi A_t{L}_A(7)$$

where the $\chi$ term indicates how productive labor is in producing ideas. When $\chi$ rises, a unit of labor engaged in research and development produces more knowledge than it did before.

Notice that technology, $A_t$, is used in producing more technology, $\Delta A_t$, in Equation (7), and also in producing goods and services in Equations (5) and (6). How is this possible? Technology is nonrivalrous because it is a set of ideas that can be used over and over again to produce both new technology and also goods and services. This nonrivalry of ideas and technology in the Romer model gives technology a one-two punch in terms of economic growth. With rivalry, as in the Solow model, diminishing returns to capital eventually lead output per capita to come to rest at a steady state without growth. Incorporating nonrivalry into our economic model allows us to explain sustained growth in per capita income.

Dividing both sides of Equation (7) by $A_t$ and replacing $L_A$ by $\alpha \overline{N}$ from Equation (2), we can write the production function in terms of the growth rate of technology, $g_A$, as follows:

First, divide Equation (7) by $A_t$ to find the growth rate of technology, $\frac{\Delta A_t}{A_t}$:

$$\frac{\Delta A_t}{A_t}=\frac{\chi A_t{L}_A}{A_t}=\chi L_A$$

Then, substitute $L_A=\alpha \overline{N}$ from Equation (2):

$$g_A=\frac{\Delta A_t}{A_t}=\chi \alpha \overline{N}(8)$$

This technology production function tells us that the growth rate of technology is positively correlated to the productivity of research and development ($\chi$), the fraction of the population that is devoted to R&D ($\alpha$), and the total population of the economy ($\overline{N}$).

Sustained Growth in the Romer Model

With these building blocks in place, we can see how sustained growth in per capita income is an implication of the Romer model. From the Solow model, the production function in Equation (6) has a steady state $k_P^{\ast }$:

$$y_{Pt}^{\ast }=A_t{k}_P^{\ast 0.3}(9)$$

Equation (9) tells us that although there is a steady state at $k_P^{\ast }$, $y_{Pt}^{\ast }$ will have a time subscript and changes over time because $A_t$ changes over time. We can write this production function in terms of output per worker, which is the same as output per person, $y_t=Y_t/\overline{N}$, which can be written as $y_t=y_{Pt}\cdot (L_P/\overline{N})$. Since $L_P/\overline{N}=(1-\alpha )$ from Equation (3), we can get an expression for $y_t$ by multiplying Equation (9) by $(1-\alpha )$:

We start with the definition of output per person, $y_t=Y_t/\overline{N}$. From $y_{Pt}=Y_t/L_P$, we know $Y_t=y_{Pt}{L}_P$. Substituting this into the definition of $y_t$:

$$y_t=\frac{y_{Pt}{L}_P}{\overline{N}}=y_{Pt}\left(\frac{L_P}{\overline{N}}\right)$$

From Equation (3), we derived $L_P=(1-\alpha )\overline{N}$, which means $\frac{L_P}{\overline{N}}=(1-\alpha )$. Substituting this and Equation (9) ($y_{Pt}^{\ast }=A_t{k}_P^{\ast 0.3}$) into the expression for $y_t$:

$$y_t^{\ast }=(1-\alpha )A_t{k}_P^{\ast 0.3}(10)$$

From the production function for technology, we see that the $A_t$ term is growing at a constant rate of $g_A$, and therefore output per person grows as well. Indeed, from our analysis of the Solow model in the previous chapter, this growth rate in $A_t$ is amplified by a direct effect from the production function and an additional positive effect from a growing capital-labor ratio. The size of this amplification effect, with the Cobb-Douglas production function used here, was derived to be 1.43 in an appendix in Chapter 6, so with a growth rate of technology of $g_A$, the per-worker growth rate of output is $1.43\times g_A$.

Figure 7.1: Balanced Growth Path in the Romer Model

(Figure shows output per capita on a ratio scale against time, as a straight line with slope $1.43\times g_A$)

In the Romer model, output per person keeps growing, but grows at a steady rate. Economists refer to growth at a constant rate as a balanced growth path. This constant growth rate contrasts with the Solow model, in which output per person eventually reaches a steady state from which it grows no further. We show the Romer model’s balanced growth path in Figure 7.1. The vertical axis is a ratio scale (also called a logarithmic scale) in which equal distances reflect the same percentage change. When we plot a variable in a ratio scale against time, if it is growing at a constant rate per year, then it will appear as a straight line, with the slope representing the rate of growth. We show output per capita, $y_t$, in the Romer model as a straight line in Figure 7.1, with a slope of $1.43\times g_A$, indicating that output per capita is growing at that rate.

Factors That Affect Endogenous Growth

Three factors in the Romer model can change the economy’s growth rate: 1) the fraction of the population that is engaged in R&D, $\alpha$; 2) the productiveness of R&D, $\chi$; and 3) the total population in the economy, $\overline{N}$. Let’s look at how growth responds to changes in each of these factors in turn.

Effects of an Increase in the Fraction of the Population Engaged in R&D, $\alpha$

Suppose that at time 0, a country decides to devote more resources to developing technology. The fraction of the population engaged in R&D rises from ${\alpha }_1$ to ${\alpha }_2$ in Figure 7.2. From Equation (8), we see that the growth rate of technology rises from $g_{A1}$ to $g_{A2}$. As in Figure 7.2, this increasing growth rate eventually will lead to balanced growth, where the growth rate of output per capita will rise from $1.43g_{A1}$ to $1.43g_{A2}$. However, there are two additional effects on the production function that must be considered.

1. As more labor is devoted to R&D production, less labor is devoted to the production of goods and services. Equation (10) indicates that there is an immediate decline in output per capita, as shown in Figure 7.2.
2. The decline in $L_P$ pushes up the capital-labor ratio $k_{Pt}$. We know from the Solow analysis that, over time, $k_{Pt}$ will decline toward the steady state value of $k_P^{\ast }$, so that the growth rate will for a time be below the long-run growth rate of $1.43g_{A2}$. As the capital-labor ratio approaches the steady state over time, the growth rate of the economy will rise toward the long-run growth rate of $1.43g_{A2}$, and the economy will be on a new, higher balanced growth path.

We show the resulting path of per capita income in Figure 7.2. The reasoning behind the path is a bit complicated. However, the basic conclusion from the analysis is fairly straightforward: when more resources are devoted to research and development, the level of per-capita output at first falls, but the growth rate of per-capita output will rise permanently. The analysis in the Romer model therefore suggests a trade-off to increasing the resources devoted to research and development: short-run living standards decline, while long-run living standards improve as per-capita output grows at a higher, sustained rate.

Endogenous growth theory provides an important rationale for why governments should spend money on R&D. With increased government expenditures on R&D, the resources going to production of technology increase, that is, $\alpha$ rises. As we see from the preceding analysis, this leads to a permanently higher growth rate of per-capita output.

Figure 7.2: Response to an Increase in the Fraction of Population Engaged in R&D

(Figure shows output per capita initially falling, then rising along a steeper balanced growth path)

Effect of Changes in the Productiveness of R&D, $\chi$

Inventors have become particularly productive at certain times in history, causing productivity of R&D to suddenly rise. Let’s see how the resulting increase in $\chi$, the productivity of R&D, affects growth in income per capita using the Romer model. Suppose that at time 0, marked in Figure 7.3, $\chi$ rises from ${\chi }_1$ to ${\chi }_2$. The production function for technology in Equation (8) tells us that the growth rate of technology rises from $g_{A1}$ to $g_{A2}$. Since there are no changes in the other elements of the production function in Equation (10), the steady state $k_P^{\ast }$ is unchanged and only the $A_t$ term changes over time. The growth rate of output per capita rises from $1.43g_{A1}$ to $1.43g_{A2}$. The line showing how output per capita is growing over time immediately achieves a higher slope, as shown in Figure 7.3, indicating that the economy has immediately moved to a new, higher balanced growth path. The conclusion: when R&D becomes more productive, output per capita grows at a more rapid rate.

Government policies aimed at making R&D more productive receive strong support from the analysis in endogenous growth theory. Spending on education, which results in an increase in human capital, can make workers in the R&D sector more productive, thus raising $\chi$. As we see in Figure 7.3, this will lead to a higher growth rate of output per capita, making the nation’s citizens richer over time.

Because the private sector is often more efficient than the government in developing new technology, government tax incentives to encourage private spending on R&D, rather than direct government spending on R&D, can help improve the efficiency of technology production, thereby increasing $\chi$ and economic growth. In addition, tax incentives to spur private spending on R&D can lead to increased resources going into R&D, which raises $\alpha$ and also promotes higher growth in per-capita output.

Figure 7.3: Response to an Increase in the Productiveness of R&D, $\chi$

(Figure shows output per capita immediately shifting to a steeper balanced growth path)

Response to an Increase in the Total Population, $\overline{N}$

In the Solow model in the previous chapter, we saw that an increase in population lowers the standard of living because it leads to capital dilution. The Romer model, to the contrary, finds that there can be improvements in the standard of living from population increases.

To illustrate, suppose that at time 0 there is a surge in immigration that leads to an increase in population from $\overline{N_1}$ to $\overline{N_2}$ in Figure 7.4. As in the previous examples, we again see from Equation (8) that the growth rate of technology rises from $g_{A1}$ to $g_{A2}$. More labor is engaged in producing new technology and, as before, there eventually will be balanced growth, in which the growth rate of output per capita will rise from $1.43g_{A1}$ to $1.43g_{A2}$. However, the rise in population will lead to an immediate decline in the capital-labor ratio $k_{P1}$, so the production function in Equation (10) tells us that output per capita will at first fall. Because $k_{P1}$ has fallen below the steady state value of $k_{P1}^{\ast }$, it will begin to rise, so that the growth rate will for a time be above the long-run growth rate of $1.43g_{A2}$. Over time, as the capital-labor ratio approaches the steady state, the growth rate of the economy will fall toward the long-run growth rate of $1.43g_{A2}$.

We show the resulting path of per capita income in Figure 7.4—and once again the reasoning is a little complicated. However, the basic conclusion holds: a rise in population at first leads to a decline in per-capita output, but the growth rate of per-capita output will rise permanently. The Romer model therefore suggests that an increase in population improves living standards in the long run. This result is a very different result from that given by the Solow model, although there is a decline in living standards in the short run.

Figure 7.4: Response to an Increase in the Total Population

(Figure shows output per capita initially falling, then rising along a steeper balanced growth path)

Application: Does Population Growth Improve Living Standards?

When a country’s population grows, does capital dilution lower living standards, as the Solow model predicts, or does the higher rate of technological progress increase living standards, as the Romer model predicts? As strange as it sounds, the conclusions of both models may be correct.

The Romer model’s optimistic view of population growth may not apply well to individual countries. Suppose India, a country with a fast-growing population, develops a new process for growing wheat that helps farmers produce more wheat in less time. The process will improve living standards in India. However, the nonrivalry of ideas will lead the technology to spread to India’s neighbors and eventually around the world. This phenomenon of technological spillover might explain why countries with high rates of population growth don’t have higher per-capita income (see Figure 6.11 in the last chapter). If the productivity frontier $A_t$ is similar across countries, then the Solow result still holds and rising population will be associated with lower per-capita income.

However, there is evidence that the Romer model’s approach to population growth applies better to the world as a whole than the Solow model’s approach. As Figure 7.5 shows, the world population has steadily grown from a little over 200 million people at the beginning of the Christian era to over seven billion today, while per-capita GDP has risen dramatically, from under \500$toover$$7,000$ currently. This evidence suggests that higher populations are positively associated with higher living standards, as endogenous growth theory predicts.

Figure 7.5: World Population and Per-Capita GDP over the Very Long Run

(Figure shows world population and per-capita GDP both rising over time)

Indeed, Michael Kremer of Harvard University has produced some tantalizing evidence that suggests that higher populations stimulate technological progress and raise living standards. The melting of the polar ice caps around 10,000 BC separated regions of the world and isolated their populations for thousands of years. Kremer found that the region that began with the highest population, the Eurasian-African land mass, developed more advanced technology and higher per capita income than the next most populated, the Americas. In turn, the Americas had a higher initial population than Australia, and developed better technology and a higher standard of living. A small island between Australia and Tasmania named Flinders Island had the lowest population at the time of separation and ended up with the fewest technological advances. Indeed, Flinders’ population died off by 3,000 BC. Since the separation was exogenous—that is, caused by events outside of human control—the results indicate that population growth causes higher living standards, as endogenous growth theory predicts.

The Romer Model and Saving

A central feature of the Solow model is that an increase in saving leads to a higher level of per capita income and a temporary increase in economic growth, but does not lead to a sustained rise in economic growth. The Romer model comes to the same conclusion that saving does not drive endogenous growth—as Figure 7.6 illustrates.

When the saving rate rises at time 0, the growth rate of technology remains unchanged at $g_A$ since none of the variables in the technology production function in Equation (8) change. Over time, though, greater saving generates investment in capital, lifting the capital-labor ratio $k_{Pt}$. The production function in Equation (10) tells us that output per capita will rise, so the growth rate will for a time be above the long-run growth rate of $1.43g_A$. As the capital-labor ratio comes to rest at a higher steady-state value, the economy will return to a balanced growth path in which per-capita output is growing at the same rate of $1.43g_A$. We show the path of per-capita output resulting from the rise in the saving rate at time 0 in Figure 7.6. This path shows that a higher saving rate results in a higher level of output per capita, but not a higher sustained growth rate.

More sophisticated models of technology production come to a more optimistic conclusion on the effects of saving. In these models, higher saving lifts the steady state capital-labor ratio and also generates investment in productivity-enhancing capital like computers and communications equipment. (In the simple model we have built, technology is a function of labor, not capital.) As saving increases and the capital stock grows, more technology is produced, leading to a more rapid rate of growth for both $A_t$ and per capita income.

Figure 7.6: Response to an Increase in the Saving Rate

(Figure shows output per capita rising to a new level but then continuing to grow at the same rate as before)

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I hope this structured presentation enhances your understanding and makes the concepts even more engaging. We’ve taken raw information and, by applying precision and clarity, transformed it into a more navigable landscape of knowledge. Now, with this solid map, what specific aspects are you most curious to explore further? What questions are bubbling up in your mind about the interplay of technology, policy, and institutions? Don’t hesitate to ask! Let’s delve deeper together.