CHAPTER 6

Economic Growth

One of the most important challenges facing policy makers is the question of how to increase economic growth. Developing countries stuck with low living standards strive to overcome centuries of poverty. Developed countries facing an extended economic slump search for an answer as to how to return to “normal” growth.

One of the enduring myths, exposed in vivid detail by William Easterly in his book, The Elusive Quest for Economic Growth, is that the path to more rapid growth lies in bringing more investment to bear on the problem (Easterly 2001, p. 4). This is an appealing solution, based as it is in what amounts to a half-truth about the power of investment to propel an economy forward. It is also a misleading solution, as it is based on wishful thinking about the ability of government to manage economic growth through top-down policies.

So what do we want to grow when we try to increase economic growth? The answer must be expressed in terms of measurable aggregates. There is no systematic way to address the problem of national “happiness” under this heading (see Chapter 2). The measurable aggregates that are ordinarily used and that give us a handle on the problem are output, or more specifically, GDP per person and consumption per person.

Let’s continue with our practice of using the letter C to stand for aggregate consumption, and let’s now let POP stand for population. We can then say that the two most important measures of economic growth are the growth of Y/POP and C/POP. Output per person is important because it provides the broadest measure of economic performance. But economic welfare depends on how much of the output goes to consumption. As Adam Smith said, “consumption is the sole end and purpose of all production.” A benevolent dictator might set about doing what he could to maximize the growth of C/POP for his country.

Let’s approach the problem of growth by using the model developed by Robert M. Solow (Solow 1956). As we work through this chapter it will be convenient to use population as a proxy for the total number of people employed. It is in fact a pretty good proxy. In April 2000, 64.7% of the U.S. civilian population that was 16 and older had jobs. In December 2017, the number stood at 60.1%. Part of the recent decline is due to the lingering effects of the recession of 2007–2009 and part is the retirement of baby boomers, now in full sway. In discussing growth theory, however, we cannot readily allow for such swings in employment. Here we will talk interchangeably about “population” and “labor,” as if the two were the same.

Another convenient, probably necessary, procedure is to treat the quantity of labor L as the number of persons working and to assume that everyone who wants to work is working. There is no recognition of the sensitivity of labor supply to current wages, future wages, and the state of the economy in this assumption. In effect, we assume that the country has reached an equilibrium in which everyone, young and old alike, is working. Fluctuations in the labor-force participation rate are the subject of the next volume, but not this one.

Finally, we assume that the saving rate s is fixed (never mind the extensive discussion in Chapter 4 about how individuals adjust their saving rate according to the parameters r, ρ, and IES). Whatever decisions or “propensities” determine the aggregate saving rate, they are “exogenous” or outside the model we will be using.

Proceeding in this spirit, two ratios become important: the size of the capital stock per person and output per person . We will begin by focusing on

In the preceding chapter, we established the links between saving and capital formation and between capital formation and production. Let’s take the Cobb-Douglas production function, introduced there,

and divide both sides by L to get

Reconfigured in terms of economic growth, this equation becomes

To grasp the relevance of this equation, imagine that Z doesn’t change. Then, if capital and labor are rising at the same rate and if Z is constant, the left-hand side of equation (6.3) is zero.

And now we can see a dark and foreboding truth about economic growth: There is a “steady state” of the economy in which K and L are rising at the same rate and in which Y and L are also therefore rising at the same rate. Growth of the kind that everyone wants requires output to rise faster than population. So if the economy reaches a state in which both output and population are rising at the same rate, we have to see the economy as expanding only fast enough to keep living standards at what might well be the same, deplorably low, level.

To see this let’s think of what the steady state looks like in an economy in which population is not changing. There is no growth of L.

Observe that the capital stock in any given period equals the capital stock in the previous period, adjusted for depreciation and investment in that period:

Because dK_t = depreciation in period t, D_t,

We know that

which is to say that gross saving equals gross investment. In the previous chapter, in which we ignored population and technological change, we found that profit-maximizing firms will reach an equilibrium in which ΔK is zero. When the firm has achieved this equilibrium,

In order to prevent the capital stock from eroding, saving must equal the rate of economic depreciation times the capital stock. To put it differently, once the firm has reached equilibrium, the level of saving is only high enough to prevent the capital stock from eroding. Indeed, it is possible for governments to increase the saving rate s through policies aimed at reducing the cost of capital. And it is true that an increase in the saving rate would then bring about an increase in the demand for capital and a new equilibrium, at a higher capital stock, where the equilibrium (6.7) is satisfied. But note that this would be a once-and-for-all adjustment in K, which would not repeat itself without a further decrease in the cost of capital, and so on.

Now let’s keep Z fixed but allow that L is growing at the rate Then

where is the percentage change in the labor force from one year to the next and Ŷ is the percentage change in output from one year to the next.

Let’s put some hypothetical numbers on this now. Let:

We note that the capital-to-labor ratio in period t is

and the output-to-labor ratio is

Whether the capital-to-labor ratio actually falls, rises, or remains the same depends on what’s going on with saving. Suppose, for now, that s = 0, which is to say that there is no saving in period t. Then the capital-to-labor ratio will fall by

from $10,000 to $9,500. If the numerator of falls by 2% because of depreciation and if the denominator rises by 3% because of growth in the labor force, the ratio will fall by 5%.

Now let s = 20% and consider two scenarios. In the first scenario, saving per person is

Thus,

and there is more than enough saving per person to maintain the current capital-to-labor ratio. Hence the capital-to-labor ratio will rise.

Now consider scenario 2, in which the capital-to-labor ratio has tripled so that it equals $30,000. Under the law of diminishing returns the output-to-labor ratio will rise by less than the capital-to labor ratio. Suppose then that the output-to-labor ratio has risen by 10%, from $5,000 to $5,500.

Now

Because saving per person is less than the amount needed to sustain the capital-to-labor ratio, the capital-to-labor ratio must fall.

The economy thus reaches a steady-state equilibrium when

that is, when the capital-to-labor ratio is just high enough that the actual saving per person equals the saving per person needed to keep the capital-to-labor ratio from either rising or falling. In this example, this could be where the capital-to-labor ratio is $21,000 and the output-per-labor ratio is $5,250:

From this example, we can see that as long as s, and d are fixed, the output-per-labor ratio must remain fixed at $5,250 and the capital-to-labor ratio must remain fixed at $21,000. But if is fixed at 3%, then the growth of output and of capital, and , must also remain fixed at 3%. The economy is said to have reached a “steady state” when the growth of labor, capital, and output, are all equal.

Now assume that L, K, and Y are all growing at the same rate. If we go back to equation (6.2), we see that output per worker remains constant if Z remains constant (which we currently assume) and if K and L are rising at the same rate.

This is to be expected, given a Cobb-Douglas production function, which has built into it the assumption that a given percentage change in both labor and capital will yield the same percentage change in output.

From equation (6.1), we know that

Letting

Given that

We can show graphically how this comes about. See Figure 6.1.

The graph shows different levels of output per person, , and saving per person, , for different values of the capital-to-labor ratio, . The line cuts the , which tells us that the amount of saving per person needed to sustain the corresponding capital-to-labor ratio is just matched by the amount of saving per person that is forthcoming. For points to the left of would exceed and the capital-to-labor ratio would rise. For points to the right, would exceed and the capital-to-labor ratio would fall. Only where , does and only there is the capital-to-labor ratio constant.

Figure 6.1 The Solow growth model

The capital-labor ratio, , is called the steady state ratio because it is just high enough to keep constant. Equation (6.20) holds true when the system is in the steady state. But then, as equation (6.3) implies, must be zero, meaning that output per capita cannot rise unless Z is rising. More generally

The only way to bring about growth of per-capita output is to bring about growth of Z.

What Is Needed to Spur Growth

What is needed for per-capita output to grow? There are two portals through which government can exert its influence. Government can directly affect s by instituting policies that either encourage or discourage saving. The other portal, its influence on Z, is harder to pin down. (Because the growth of the labor force is considered fixed here, we ignore the influence government can have over the supply and demand for labor.)

Suppose that the economic planners, or government officials responsible for tax policy as it affects s, decide to cut taxes on saving in order to increase s. The line rotates upward in a counterclockwise direction, pushing the economy up along the line, to a new steady-state capital-to-labor ratio, .

What has this action accomplished? Well, it has brought about a once-and-for all increase in output per capita (a higher ratio of Y to L). But once the capital-to-labor ratio reaches , the economy settles back into a steady state, in which, K, L, and Y are all rising at the same rate as they were before the increase in s. Output per person is higher than it was at , but the growth of output per person experienced during the transition to this higher capital-to-labor ratio comes to a halt.

This might seem to imply that the government could keep pushing the economy further up the curve just by taking steps to increase s. That, however, is neither practical nor desirable. A country that combines political with economic freedom will not respond, beyond a certain point, to government policies aimed at increasing the saving rate. Taxes on the return to saving are a deterrent to saving but if all such taxes were removed there would be little left in the government policy arsenal to encourage further saving. Even under a command economy, political resistance to ever-higher “forced saving” would put practical limits on this policy. After all, people will not tolerate a state of affairs in which the government compels them to save 100% of their income.

Insofar as it is government policy to maximize consumption per person, there does exist a theoretically ideal saving rate. To see this, observe that the marginal product of capital measures the increase in output that results from adding another dollar to the stock of capital, while holding labor constant. Suppose that the capital-to-labor ratio is at in Figure 6.1, so that the condition is satisfied. But suppose also that MP_K, given by the slope of , is 6% but that equals 5%. Whereas saving per person is just high enough to preserve the existing ratio of capital to labor, the government could notch up consumption per person by pushing up the saving rate.

Let’s see why. Under our assumptions, it is necessary to add only 5 cents to output per person to sustain the capital-to-labor ratio if that ratio rises by one dollar. But if MP_K = 0.06, the rise in the ratio will add 6 cents to output. Where would the other 1 cent go? The answer is to consumption. (Note that in this world C + S = C + I = Y, so every dollar of Y that does not go to S or therefore I must go to C.) Thus a policy aimed at increasing the saving rate (and therefore moving further up the curve) would, for a while at least, increase consumption per person. In the steady state in Figure 6.1, the consumption-to-labor ratio is . An increase in the saving rate under these circumstances would increase to a higher level.

Now consider another steady state, this one reached at . Here, we assume that MP_K is only 4%, the law of diminishing returns having operated in its ineluctable way. Now think about the last dollar added to the capital stock (per unit of labor). That dollar increased output per person by only 4%, whereas it is necessary for 5% of output to go to saving and investment in order to sustain the capital-to-labor ratio at this level. The missing 1% must come out of consumption. So now to increase consumption-per person, the s should fall to bring down into line with some intermediate capital-to-labor ratio, , at which

Given the Cobb-Douglas function, this would mean that

Equation (6.22) is called the golden rule for economic growth, in that, by satisfying it, the economy maximizes current consumption per person. The saving rate is brought just high enough so that any other steady-state capital-to-labor ratio would yield a lower level of consumption per capita. This designation might not quite fit the meaning behind the Biblical golden rule, insofar as it might be godlier for the current generation to save for the purpose of lifting the next generation’s standard of living. Yet, the idea is clear. The thus-stated golden rule has a degree of moral authority in that it puts individual welfare ahead of some central authority that might wish to distinguish itself by forcing growth on the current generation to create a kind of shrine to its command-and-control efforts. See Stalinist Russia.

This line of thought reminds us, though, of the limited scope of efforts to influence the saving rate for the purpose of increasing economic growth. An increase in the saving rate will increase output per person but, once the new steady state is attained, it will not increase the growth of output per person.

This leaves us with the hope to influence Z. Returning to equation (6.3), assume that the economy is in a steady state, so that is zero. Now suppose that, through either public or private initiatives, Z rises by 2% annually. Then will also rise by 2% annually. This observation points to a truth that is both powerful and daunting, that is, that living standards will steadily improve with technical progress (as well as other kinds of progress associated with Z) but will only temporarily improve without such progress.

This should be seen as a cautionary tale on the matter of economic growth. This book argues that government can expand work and saving, and therefore output, by reducing taxes on work and saving. But a necessary condition for growth in living standards, as measured by output per capita, is steady growth in total factor productivity, Z, combined with the institutional and legal environment that is needed to encourage innovation and entrepreneurship.

Gordon on the Decline of Growth

There is currently a lot of pessimism on the prospects for U.S. economic growth. Figure 6.2 shows that the growth of total factor productivity has fallen sharply since 2010, after generally rising over the period 1996 to 2004. In 2016, the macroeconomist Robert J. Gordon published a massive volume titled The Rise and Fall of American Growth, in which he concluded that various “headwinds” are causing growth to fall. He predicted that “future growth in the disposable income of the bottom 99 percent of the income distribution... is likely to be barely positive and substantially lower than the growth of the labor productivity [output per man hour] of the total economy” (Gordon 2016, pp. 530–31).

Figure 6.2 Growth of U.S. total factor productivity 1988–2016

Source: U.S. Bureau of Labor Statistics.

Gordon predicts that the annual growth in labor productivity will be 1.2 percent over the next 35 years (compared to 2.71 percent from 1948 to 1970 and 1.38 percent from 1970 to 2015). Output per person will grow by 0.8 percent annually (compared to 2.41 percent from 1920 to 1970 and 1.77 percent from 1970 to 2014) (Gordon 2016, pp. 635, 637).

Gordon identifies three industrial revolutions through which the United States has passed:

He questions why the last of these has failed “to maintain productivity growth at a faster pace,” as shown in the declining productivity in recent years (Gordon 2012, p. 12). One answer, he believes, lies in the trend toward products such as smartphones and iPods, which have served mainly to enhance the lives of consumers, but away from labor-saving innovations (Gordon 2012, p. 13).

Despite the potential for productivity gains offered by artificial intelligence and by methods of extracting information from “big data,” future economic growth will be buffeted by six headwinds: “demography, education, inequality, globalization, energy/environment, and the overhang of consumer and government debt” (Gordon 2012).

The demographic trend affecting growth is the reduction in work hours resulting from the retirement of the baby boomers. As for education, “the U.S. is steadily slipping down the international league tables in the percentage of its population of a given age which has completed higher education” (Gordon 2012, p. 16).

Gordon reports that the top one percent of households experienced 52 percent of the growth in real income over the period 1993 to 2008. This trend toward inequality is, in his view, “the most important quantitatively in holding down the growth of our future income” (Gordon 2012, p. 17). The remaining factors are the decline in U.S. wages owing to outsourcing and imports from low-wage countries, the threat of a carbon tax aimed at global warming, and the growing volume of household and government debt (Gordon 2012, pp. 17, 18).

Gordon’s pessimism is countered by Michael Mandel and Bret Swanson in a paper that stresses the as-yet unexploited potential of artificial intelligence. “The pessimism about growth,” say Mandel and Swanson, “ignores the fact that information has revolutionized only 30% of the private-sector economy. Applying the power of information to the remaining 70% will replicate the gains of digital industries, but on a much larger scale” (Mandel and Swanson 2017, p. 30).

Indeed, everything, considered, Gordon is unpersuasive in his attempt to tie declining productivity to inequality and other factors that have nothing to do with the act of bringing ideas to the marketplace and realizing the potential for increased profits through the introduction of new products and production methods. Inequality is a growing concern among economists, but it has nothing to do with the ability to innovate in a society that respects property rights.

Secular Stagnation

In his March 1939 presidential address before the American Economic Association, Alvin Hansen raised his concerns over “secular stagnation,” which he defined as “sick recoveries which die in their infancy and depressions that feed on themselves and leave a hard and seemingly immovable core of unemployment” (Hansen 1939). Hansen spoke when the U.S. economy was recovering from a two-year plunge in real GDP, after having recovered somewhat from the worst of the Great Depression. Prior to the Great Depression, less than full employment was a passing feature of the business cycle.

Sounding much like Gordon, Hansen compared the contemporary (1939) U.S. economy unfavorably with the 19th century. The relatively fast economic growth experienced in the earlier period was due in large part to population growth:

The problem, as Hansen saw it, was the declining birth rate seen in the twentieth century. This, in light of the experience of the previous century, could be expected to suppress private investment, with consequent negative effects on economic growth. The problem for his own generation, said Hansen, “is, above all, the problem of inadequate private investment outlets. What we need is not a slowing down in the progress of science and technology, but rather an acceleration of that rate” (Hansen 1939, p. 10).

What is most interesting about Hansen’s article is that, as a leading exponent of Keynesian economics, Hansen expressed his concern that the expansion of government would arise as an obstacle to the investment and innovation. “Can a rising public debt,” he asked,

For Hansen, the danger lay in “vastly enlarged government activities” (Hansen 1939, p. 15). It is useful to observe, in this context, that current government spending in the United States was 16% of GDP in 1939 and is now 33% of GDP.

Now, almost 80 years later, the same threat looms over the U.S. economy. In a recent speech, Larry Summers opined as follows: “Six years ago, macroeconomics was primarily about the use of monetary policy to reduce the already small amplitude of fluctuations about a given trend, while maintaining price stability. That was the preoccupation” (Summers 2014, p. 65). “Today,” Summers continued, “we wish for the problem of minimizing fluctuations around a satisfactory trend.” Continuing, he said, “It is increasingly clear that the trend in growth can be adversely affected over the longer term by what happens in the business cycle” (Summers 2014, pp. 65, 66).

The problem, as Summers described it, is “secular stagnation, the idea that the economy re-equilibrates; hysteresis, the shadow cast forward on economic activity by adverse cyclical developments; and the significance of the zero lower bound for the relative efficacy of monetary and fiscal policy” (Summers 2014, p. 66).

Summers observed that, five years into the recovery from the Great Contraction of 2007 to 2009, actual real GDP was still well below potential real GDP. In particular, the employment-population ratio was still well below the historical norm. The culprit, said Summers, was not total factor productivity (Z in our model) but a lack of investment. For Summers, the principal problem is the declining real interest rate (the rate that equilibrates the supply and demand for capital). The source of the problem is declining population growth and investment, combined with rising saving. With actual nominal interest rates already near zero, there is little to be gained from further reductions in real interest rates. The answer, argued Summers, is for government to undertake a policy of increasing aggregate demand through increased deficit spending. We will have more to say about this idea in Volume II.

Piketty’s Laws of Capitalism

Thomas Piketty became world famous in 2014 with the publication of his book, Capital in the Twenty-First Century. The book provides a lengthy and fact-filled argument that capitalist countries are characterized by growing income inequality. Among his suggested cures for this unwanted trend are a “global tax on capital” and the imposition of a marginal tax rate of 80 percent on U.S. incomes of $500,000 to $1,000,000 a year (Piketty 2014, pp. 513, 515–39). Because Piketty’s book has received widespread attention and approval and because the argument here is that taxes, particularly taxes on capital, suppress economic activity, it is necessary to give his book consideration.

Piketty’s book and its title suggest that he reckons himself as a modern successor to Karl Marx, whose Capital provided the intellectual foundation for communism. Like Marx, Piketty sees capitalism as governed by certain fundamental laws that presage growing concentration of wealth and power on the part of capitalists, with “terrifying” consequences for democracy and social justice. Piketty’s book is very readable, and he makes his point with a few simple formulas. Because his formulas have become widely quoted, let’s write them down using his notation. He postulates two “fundamental laws of capitalism” and then a “central contradiction” of capitalism. These are as follows:

The first and second fundamental laws are not laws at all and are not limited in their applicability to capitalist economies. They are, rather, truisms that apply to any economy and which fall out of the growth model presented previously. As for the variables in his system:

In the forgoing sections of this book, the share of income going to capital is

consistent with Piketty’s first law.

We can rewrite his second law as

or

Then, as in the Solow model,

in the steady state.

One might think that Piketty’s central contradiction of capital would follow from his fundamental laws, but this is not the case. To see why, let’s revert to his notation and insert his second law into his first to get

It turns out that his central contradiction (r > g) holds only if

Suppose, for example that a = 0.3 and s = 0.2. Then, if r = 0.05 (which Piketty takes as a reasonable assumption), g = 0.033. Thus r > g. But this example does not, in and of itself, imply increasing inequality. Rather, we can say only (1) that the economy will settle into its steady state and (2) that the saving rate and therefore the capital-to-output ratio would have to rise in order for the economy to reach the golden-rule capital-to-labor ratio.

To understand point (1), return to Piketty’s second law, using our notation:

Substituting

Dividing by L,

or

which is the condition for achieving the steady-state growth of labor, capital, and income as derived in the Solow model.

To understand the second point, as noted previously, recall the golden rule condition for the optimal capital-to-labor ratio:

or

From equation (5.25) in Chapter 5, the firm maximizes profit by setting

If

the golden rule is not satisfied. Then also

and, because

as in Piketty’s “central contradiction.”

But equation (6.40) means only that the saving rate is too low to achieve the golden-rule capital-to-labor ratio. Society needs to increase the saving rate and therefore the capital-to-labor ratio in order to maximize consumption per capita.

Piketty devotes some space to a consideration of this reality (Piketty 2014, pp. 562–65), concluding that any suggestion to achieve the golden-rule capital-to-labor ratio is “not very useful in practice” (Piketty 2014, p. 563). Piketty is certainly correct in saying that we cannot usefully set our sights on maximizing consumption per capita. All we can do, in practice, is aim for ways to increase output per capita since we don’t have the policy tools to align the marginal product of capital with the sum of population growth and depreciation. The question, again, is what any of this has to do with income inequality.

The answer is, “Nothing at all.” In the Solow model presented previously,

in the steady state. Given the return to capital r and the wage rate w, both capital income (rK) and labor income wL will rise at the same rate as total income, (rK + wL), no matter what the value of r and Ŷ. If, as Piketty assumes, the share of income going to capital (a) is a given, then so also is the share going to labor (1 – a), and that share is independent of the relationship between r and Ŷ.

Piketty believes, and provides strong support for his argument, that income inequality is on the rise. He speculates that the trend toward increased inequality is strong and permanent. The question is what his “central contradiction of capitalism” has to do with this, given that it is neither a contradiction nor an inherent feature of capitalism. Indeed, he could have skipped the entire theoretical edifice on which his book hangs and simply focused on the growing literature on labor’s declining share of income (Lee and Jayadev 2005; Karabarbounis and Neiman 2013; Rodriguez and Jayadev 2010). One can make an argument that labor is losing out to capital because of capital-augmenting innovation, without resorting to purposeless theorizing.

It is interesting that Piketty devotes some space in his book to squaring Marx’s central contradiction with his own (Piketty 2014, pp. 227–30). For Marx, capitalism was doomed because r would steadily fall as the capital-to-labor ratio rose, driving capitalists to ever-more frantic efforts to raise their profit rates—these including wars for new territory and efforts to push down wages—thus ineluctably driving capitalism to collapse. This contradicts Piketty’s central contradiction that r is high and rising. One can only wonder why, considering Marx’s take on the death spiral affecting capitalism, Piketty sees a connection between his warnings about capitalism and Marx’s.