Technical Paper Series

Congressional Budget Office

Washington, DC

MODELING LONG-RUN ECONOMIC GROWTH

Robert W. Arnold

Congressional Budget Office

Washington, D.C. 20515

arnold@cbo.gov

June 2003

2003-4

Technical papers in this series are preliminary and are circulated to stimulate

discussion and critical comment. These papers are not subject to CBO’s formal

review and editing processes. The analysis and conclusions expressed in them

are those of the author and should not be interpreted as those of the

CongressionalBudget Office. References in publications should be cleared with

the author. Papers in this series can be obtained from www.cbo.gov.
Abstract

This paper reviews the recent empirical literature on long-run growth to determine

what factors influence growth in total factor productivity (TFP) and whether there are

any channels of influence that should be added to standard models of long-run

growth. Factors affecting productivity fall into three general categories: physical

capital, human capital, and innovation (including other factors that might influence

TFP growth). Recent empirical evidence provides little support for the idea that there

are extra-normal returns to physical capital accumulation, nor is there solid

justification for adding a separate channel of influence from capital to TFP growth.

The paper finds evidence that human capital—as distinct from labor hours

worked—is an important factor for growth but also that there is not yet a consensus

about exactly how it should enter the model. Some argue that human capital should

enter as a factor of production, while others argue that it merely spurs innovation.

The forces governing TFP growth are not well understood, but there is evidence that

R&D spending is a significant contributor and that its benefit to society may exceed

its benefit to the company doing the spending—that is, it is a source of spillovers.

The paper concludes with some examples of how standard models of growth could

be modified to reflect some of the channels of influence identified in the empirical

review.

The author gratefully acknowledges helpful comments from Doug Hamilton, Bob

Dennis, Ben Page, and John Peterson. Any errors or omissions are the responsibility

of the author.
1 Total factor productivity is average real output per unit of combined labor and capital input. The

growth of TFP is defined as the growth of real output that is not explained by the growth of labor and

capital.

1

INTRODUCTION

Long-run economic growth is the key determinant of living standards over time and

among countries. Despite its importance, economists do not have complete answers

to some fundamental questions about long-run growth. Why were living standards

essentially stagnant for thousands of years, before accelerating rapidly in certain

European countries during the Industrial Revolution? Why are some countries

twenty or thirty times as wealthy as others? Long-run growth is of interest to the

Congressional Budget Office (CBO) as it analyzes a number of long-run issues,

including possible future burdens on the economy of the growing number of retirees

and their medical costs. 

In 1994, CBO published an evaluation of models in the endogenous growth

literature, which were put forward as an alternative to the more traditional

neoclassical model. That study concluded that the balance of evidence did not

support discarding the neoclassical model in favor of a newer one. In fact, the best

of the new theories, namely those that model the factors that explain technological

change, can be interpreted as extensions of the neoclassical model rather than as

replacements for it.

Since that paper was released, there has been a steady stream of empirical

investigations of the sources of long-run growth. That body of empirical work

provides CBO with an opportunity to update its models of economic growth with

new insights. In addition, CBO is developing a new model that will be used to

analyze Social Security and other issues related to the budget outlook over a 75-year

horizon. In so doing, CBO will have to make a host of choices about how to model

long-run economicgrowth, including the specification of the production function, the

values of key parameters, and the channels of influence for changes in government

policy. The latter issue is particularly important since the major purpose of the model

will be to analyze the effects of fiscal policy on the economy. Although improving

CBO’s models for making baseline projections is an important task in and of itself,

a major challenge facing CBO is to develop a better understanding of the economic

effects of changes of policy. With those considerations in mind, have any new

channels of influence been uncovered that should be included in the new model? 

CBO’s current projections for real gross domestic product (GDP) come from

a neoclassical (Solow) growth model in which the saving rate (including net saving

from abroad) determines the rate of investment which, in turn, determines the growth

in the amount of capital per worker in the economy. (For a given saving rate, growth

of per-capita income is driven by growth of total factor productivity—TFP—in the

long run).1 The key advantage of this model is that it is a simple, robust model that
2

See CBO (1994) for a more complete discussion of the neoclassical model and several examples of

endogenous growth models.

3 Convergence is a process by which economies with low levels of per-capita output grow faster than

those with higher levels. The Solow model predicts conditional convergence: economies that are

identical in terms of parameters such as tastes and technology will converge to the same steady-state

level of output per capita.

2

automatically captures the effects of changes in the rate of saving (including changes

in government saving). It also produces results that are intuitive and relatively

transparent. The primary disadvantage of the model is that it is silent on the

determinants of TFP growth, which is projected using a deterministic time trend.

In contrast, the endogenous growth literature offers the prospect of perpetual

growth in output per capita that does not rely on exogenous growth in TFP. Although

all endogenous growth models share the same basic idea—they lack the Solow

model’s restrictive property that growth in per-capita output can only be increased

temporarily when an economy shifts to a higher rate of saving and investment—they

rely on different mechanisms to drive long-run growth. Some focus on explaining

technological change, while others modify the structure of the model so that

investment in physical or human capital sustains growth. These models are of

interest to policymakers because they imply that some policy changes can have much

larger effects on growth in the long run than the Solow model would predict.2

This paper reviews the recent empirical literature on long-run growth and

productivity to determine what factors, if any, influence productivity growth. The

intent is not to revisit the debate on endogenous versus neoclassical growth but

instead to identify channels of influence that drive the growth in labor productivity

and TFP. Are there any lessons from the recent literature that can inform our

modeling effort? Are there any promising areas of research where CBO should

concentrate its efforts?

One possible channel of influence runs through the stock of physical capital.

Some economists noticed that the observed empirical correlation between the

investment share of GDP and long-run growth in real GDP (or TFP) was larger than

would be predicted by the Solow growth model. The elevated correlation could be

the result of externalities, or spillovers, from the installation and use of physical

capital. If true, such spillovers would suggest a long-run model with a coefficient on

capital that is higher than the Solow model’s or a model with a direct link from

capital to TFP growth. However, estimates of the correlation between investment

and growth have been criticized on econometric grounds; researchers have found

that such estimates are sensitive to the specification of the regression and they are not

statistically significant after controlling for simultaneity bias. Stronger evidence

against this type of model comes from its inability to account for the conditional

convergence of income per (effective) worker observed among regions and certain

countries.3
4 Growth in TFP is often interpreted as technological change, which it is if all of the assumptions of the

neoclassical model are satisfied and if all concepts are measured correctly. However, in practice, TFP

growth reflects technological change and anything else that causes GDP growth to exceed the growth

attributed to labor and capital.

3

Human capital will undoubtedly play an important role in any convincing

theory of long-run growth. The key question is how it should enter the model. Some

would argue that it should be a separate factor of production while others would

argue that it spurs innovation. There is not yet a consensus about how human capital

should be modeled, and further, any approach would have to rely on imperfect

measures of human capital. However, some researchers have included human capital

in empirical growth models, and CBO could follow a similar approach.

It is generally agreed that innovation is the fundamental source of true (as

opposed to measured) technological change.

4

 However, data are scarce, making it

difficult to empirically evaluate models of endogenous innovation. Researchers rely

on public and private R&D expenditures—a subset of innovative effort—because

that is where data exist. Empirical evidence indicates that the social rate of return to

R&D is higher than the private rate of return, suggesting the possibility that R&D

spending by one company benefits other companies in the economy. Although

measuring the degree of spillovers is quite challenging, the consensus seems to be

that they do exist and could be quite large. Some researchers have developed models

that link R&D spending to growth, and CBO could do the same. 

Channels of Influence

The resurgence of interest in long-run growth, spurred in part by Paul Romer’s 1986

paper suggesting the possibility of endogenous growth, has brought to light many

possible channels of influence that could be used to model productivity growth. The

channels of influence fall into three general categories, physical capital, human

capital and total factor productivity. One recurring question is whether a particular

variable has a spillover or externality associated with it. If so, then the Solow model

underestimates the coefficient on that variable and, therefore, its contribution to

growth.

Physical Capital. Accumulation of physical capital is at the heart of the growth

mechanism in the neoclassical model and the core of the production sector in CBO’s

models of medium-term and long-term growth. After the neoclassical model was

developed, economists devised a method to estimate the contributions of the factor

inputs (labor and capital) to the growth of output. Early efforts found that a large

fraction of growth, over 40 percent by some estimates, went unexplained by the

inputs and had to be attributed to TFP and, by implication, technological progress.

This finding called into question the usefulness of the neoclassical model since the

model does not address the causes of technological progress. Subsequent refinements, however, have reduced the proportion of unexplained growth and point to a
5

See Denison (1967) for an example of an early growth accounting estimate. Dale Jorgenson has been

responsible for many of the subsequent refinements. See, for example, Jorgenson (1990), Jorgenson,

Gollop, and Fraumeni (1987), and Christensen and Jorgenson (1970).

6

See, for example, Paul Romer (1987), Kormendi & Meguire (1985), and Levine and Renelt (1992).

DeLong and Summers (1991) argue that equipment investment is the category that has extra-normal

returns.

7 Note that the Solow model predicts a positive relationship between real GDP growth and the saving

rate. Since data on the saving rate is unavailable for a wide sample of countries, most researchers use

the investment rate as a proxy for the saving rate.

8 Romer (1987) suggested the possibility of such spillovers, but later returned his focus to spillovers

associated with knowledge. See also Romer (1994). If the spillover is large enough, then endogenous

growth is possible. CBO (1994) provides a more complete description of so-called “AK” models of

endogenous growth.

4

larger contribution from the inputs, and from capital in particular.5

CBO’s projections for real GDP growth, because they rely on models built

on the neoclassical framework, incorporate the effects of changes in saving and

investment. For example, changes in government policy that affect investment—by

changing national saving or through other channels—will affect output by changing

the size of the capital stock. Similarly, changes in the outlook for business investment that are not the result of policy changes will also influence the long-run rate of

growth through their impact on capital accumulation.

The explosion of empirical research spurred by the endogenous growth

revolution hinted that the influence of saving and investment was more powerful than

was assumed by the neoclassical model. During the 1980s, researchers found a

strong correlation between the ratio of investment to GDP and economic growth in

cross-section regressions across countries. That is, countries with high rates of

investment during the post-war period tended to grow faster than those that did not.6

That correlation is not inconsistent with the Solow model, which predicts a positive

relationship between real GDP growth and the investment rate, at least during the

transition from one steady state to another.7 Once the economy completes the

transition to a new steady state, the effect of capital accumulation tapers off and only

TFP growth raises per-capita GDP, according to the neoclassical model.

However, some researchers noticed that the correlation between growth and

investment was larger than would be predicted by the neoclassical model. That

finding led them to question one of the model’s fundamental assumptions, that the

coefficient on capital in the model’s production function is equal to capital’s share

in income. Instead, they speculated that there were externalities or spillovers

associated with investment and that the coefficient on capital was larger than what

was assumed by the neoclassical model.8 That observation led some to propose the

so-called “AK” endogenous growth model, in which output is modeled as a function

of a single input, capital. Others interpreted the evidence to mean that capital has
9

See Dowrick & Nguyen (1989), Wolff (1991) and Bernanke & Gurkaynak (2001).

10 See Lucas (1990), D. Romer (1996), Temple (1999). 

11 Romer (1986) used the stock of physical capital as a proxy for the stock of knowledge.

5

two influences, a direct one through the factor input and an indirect one via TFP.

Empirical studies have documented a positive correlation between investment and

TFP growth, which is consistent with the idea that there is a second channel of

influence.9

Such a finding would be significant because it would overcome one of the

primary shortcomings of the neoclassical model, namely the inability to explain large

differences in per-capita income across countries. According to the neoclassical

model, differences in the level of per-capita output across countries must stem from

one of two forces: increases in either the amount of capital per worker or increases

in TFP. Researchers have found that differences in per-capita output across countries

are much larger than what can be explained by differences in capital per worker.10

Hence those relying on the neoclassical model must attribute the rest of the

differences to TFP, which is exogenous in the neoclassical model. New growth

models do not have to rely on unexplained TFP growth to generate large differences

in per-capita output. In 1994, CBO concluded that, although such work was in its

infancy, empirical research did not support the idea of spillovers to physical capital

(and, by implication, the AK model). 

The intervening years have not provided compelling empirical evidence to

revise that view. Although it is hard to reject almost any hypothesis convincingly

using macro data, empirical evidence released since 1994 has not favored the idea of

extra-normal returns to capital. The evidence has not overturned the observed

correlation between capital and growth—it is one of the more robust correlations to

be found in cross-sectional regression. However, it probably does not arise because

of spillovers.

There are several reasons to be skeptical of the idea that the high correlation

between capital and growth implies the existence of spillovers to physical capital.

First, microeconomic evidence of such spillovers is lacking. It is not clear why a new

machine installed in one factory should benefit the productivity of a machine

installed in another. (Knowledge embodied in a new machine would have

widespread benefits, but that is a separate issue). One possible justification, known

as learning-by-doing, is that knowledge is created as a by-product of the production

of new capital goods, which implies that productivity increases along with

investment.11

Second, it is possible that the strong correlation may arise in part because

some technological change is embodied in new capital goods. If the price indexes
12 For more on the link between technology and the output elasticity of investment, see Benhabib and

Jovanovic (1991) or Benhabib and Spiegel (1994).

13 See Romer (1986) and DeLong and Summers (1991).

14 See Barro and Sala-i-Martin (1995), p. 434.

15 See Auerbach et. al. (1994).

6

used to deflate investment spending are not adjusted for changes in quality, then any

technological change that is embodied in capital goods will not be captured in the

capital stock and will show up in TFP. The U.S. adjusts the prices of some

investment goods for quality change in its national accounts, but most countries do

not adjust any. If foreign countries followed the U.S. in their measurement of

investment prices, then some of their economic growth that is currently attributed to

the residual would be moved into the capital input. Since the component that gets

moved would be correlated with investment, this shift would probably lower the

correlation between TFP and the investment ratio in cross-sectional regressions.

Third, the correlation between the investment ratio and growth could arise

because of the influence of an unobserved third variable. Technological progress

would be an obvious example. A country experiencing rapid technical change would

not only have faster TFP growth, but would have many profitable investment

opportunities. That combination could cause the country to have a high investment

share and rapid real growth over prolonged periods, perhaps as long as several

decades. In addition, the rapid technological change would likely reduce the relative

price of capital goods, encouraging further investment.12

Another reason to be suspicious of the correlation between physical capital

and growth is that changes in the method used to estimate this correlation have

reduced its magnitude. Early estimates found a very strong correlation between

physical investment—especially equipment investment—and growth in output,

which was interpreted by some as evidence of spillovers.13 It appears that this

conclusion was premature. Subsequent research showed that, although it is one of

the more robust correlations in the growth literature, it is not as strong as early

estimates indicated. Moreover, it does not necessarily imply an inconsistency with

the neoclassical model. For example, changing the specification of the basic growth

regression equation by adding explanatory variables—meant to control for

differences in the fundamental characteristics of different economies—or estimating

the equation using instrumental variables reduces the statistical significance of the

investment ratio in cross-sectional regressions.14 In addition, the idea that equipment

investment conveys extra-normal returns has lost favor in light of evidence that the

relationship is not robust to changes in sample. For example, the relationship holds

for a sample that include both high- and low-growth countries, but not for a sample

that includes only OECD nations, or for a sample of LDCs if it excludes Botswana.15
16 This concern has been used repeatedly in surveys of the empirical growth literature. See, for example,

McGratten and Schmitz (1999) or Temple (1999).

17 See Barro & Sala-i-Martin (1995). For related work, see , Carroll et. al. (2000), Attanasio et. al. (2000),

Blomstrom, Lipsey, and Zejan (1996), and Carroll and Weil (1994). Note that this is an empirical

regularity. Some theoretical models of consumption behavior predict the opposite result—faster

income growth will cause people to save less.

7

Aside from questions about whether the correlation implies the presence of

spillovers, there are reasons to question whether the correlation is valid in the first

place. First and foremost is the problem of simultaneity bias, which arises because

these regressions probably violate an assumption of ordinary least squares

(OLS)—that the explanatory variables and the equation’s error term are independent.

For example, the rate of investment is a function of, among other things, output

growth. If so, any unobserved shock to productivity that raises output could

indirectly raise investment, inducing a correlation between an explanatory variable

(the investment rate) and the error term in the output equation. Under these

circumstances, output and capital are simultaneously determined, and OLS estimates

of the coefficient on investment will be biased. Simultaneity bias is a concern for

many explanatory variables in cross-sectional growth regressions, but is particularly

acute for investment.16 This problem is most pronounced in time-series regression

equations, but it also affects cross-sectional regressions using data that has been

averaged over long periods, such as the ones described above.

The presence of simultaneity bias means that one cannot be certain about the

direction of causality between the investment ratio and the rate of growth. One

would expect a positive correlation based on the Solow model, at least for economies

on the transition path, because an increase in the saving rate would raise investment

and the capital stock, which in turn raises output. Hence, the original interpretation

was that the positive correlation implied that causality runs from the investment ratio

to growth. However, investment is clearly endogenous, and using a cross-sectional

regression with data averaged over long periods may not totally avoid the statistical

problems caused by using an endogenous variable on the right-hand-side of a

regression equation. Some analysts have concluded from the empirical evidence that

causation runs in the other direction: faster growth causes countries to save more,

which in turn raises the rate of investment.17 This question is by no means settled,

but it raises a red flag about the correlation between investment and growth.

One of the primary motivations of the literature examining cross-sectional

regressions was to determine whether economies converged, as predicted by the

Solow model. Convergence, or catch-up, is a process by which poor countries (those

with low levels of per-capita GDP) grow faster than rich counties. Evidence of

convergence would constitute another strike against the AK model because it predicts

no convergence of any sort. A vast literature arose during the 1990s in which

researchers used cross-sectional regressions of growth across countries to examine

the convergence question. A consensus in favor of convergence emerged from that
18 The empirical literature on convergence is enormous. For surveys, see CBO (1994), Pack (1994),

Barro and Sala-i-Martin (1995) and Temple (1999).

19 See Barro & Sala-i-Martin (1995).

20 Arnold Harberger quipped that you can’t make too much of “regression lines that draw much of their

slope from the differences between Sudan and Switzerland.” See Harberger (1998), p.21.

8

literature, at least from the studies that included variables in the regressions to control

for differences in steady states across countries.18 Convergence of this sort is known

as “conditional convergence” and is entirely consistent with the neoclassical model

of growth. It implies that countries with wider gaps between per-capita output and

its steady-state level will grow faster than those with smaller gaps.

Even stronger evidence comes from studies of conditional convergence that

use data from regional data sets (e.g., states, provinces or prefectures within a single

country or region). These studies are important because the data sets better

correspond to the assumptions that underlie the Solow model, meaning that the

regions are more similar in terms of their preferences, endowments, and, most

importantly, access to technology. Essentially, regressions that use regional data

implicitly control for ‘other’ factors without the need to include regressors that

measure those other factors. Cross-section regressions that use regional data sets find

solid evidence that poorer regions tend to catch up to richer regions, meaning that

areas with lower levels of capital per worker grow faster than areas with higher

levels, on average.19 This is stronger evidence of convergence than cross-sectional

regressions that use country-level data.20

Corroborating evidence supporting the existence of convergence comes from

other studies that employ time-series methods instead of cross-section estimates.

Most of the evidence relating to the question of convergence and the returns to

physical capital comes from cross-sectional regressions, which use data averaged

over long periods for individual countries or regions within countries. Their results

stem from the differences across countries in certain variables, generally finding, for

example, that countries with higher average rates of investment have grown more

quickly during the post-war period than countries with lower rates of investment. In

contrast, time-series studies rely on differences in certain variables through time,

either in a single country or for a collection of countries. 

For example, regressions using panel data from a variety of countries exploit

variation in the long-run growth rates of individual countries while allowing

researchers to relax the assumption (implicit in most cross-section regressions) that

aggregate production functions are identical across countries. Instead of assuming

a common production technology, panel data studies allow for country-specific

effects, meaning differences in tastes or technology across countries. Ignoring such

effects will bias the estimated coefficients in an OLS regression if the effects are
21 See Islam (1995), Evans (1998), Evans and Karras (1996), Caselli, et. al. (1996), Lee et. al. (1997),

Knight et. al. (1993), and Duffy and Papageorgiou (2000). Among these studies, only the paper by

Caselli et. al. attempts to correct for simultaneity bias as well as country-specific effects.

22 See Jones (1995). The evidence for the OECD countries is less conclusive, but Jones notes that those

economies were much more heavily affected by the two World Wars. McGrattan (1998) argues that

there is a positive correlation if one uses a longer historical sample and a broader definition of physical

investment.

23 See Temple (1999), p.138.

9

correlated with other explanatory variables. Panel data studies generally provide

strong support for the idea of conditional convergence, although their results differ

with respect to the speed at which convergence occurs.21

Other types of time-series evidence can be used to examine the predictions

of endogenous growth models. For example, some endogenous growth models

predict the existence of “scale” effects, meaning that permanent increases in the

levels of certain variables can lead to a permanent increase in an economy’s growth

rate, rather than a permanent increase in the level of real GDP. In the AK model, for

example, the presence of constant returns to physical capital ensures that an increase

in the investment ratio will permanently raise the growth rate. If true, that property

suggests that policy actions can have very powerful effects on living standards in the

long run. However, the prediction of scale effects appears to be at odds with the

time-series evidence in both the U.S. and in the larger group of OECD economies.

Specifically, there has been a steady upward movement through time in the rate of

gross investment in the United States and other OECD countries without any concomitant increase in the rate of economic growth.22

To summarize, the best evidence suggests that the neoclassical model, with

its assumption of decreasing returns to physical capital, is appropriate for analyzing

and projecting long-run economic growth. This is Jonathan Temple’s conclusion in

his review of the “New Growth” evidence in a 1999 Journal of Economic Literature

survey. He states that the “strongest result in the investment-growth literature is that

the returns to physical capital are almost certainly diminishing, in agreement with the

Solow-Swan growth model and most theoretical work since.”23 Although it is

difficult to reject any hypothesis decisively in the area of long-run growth, the

empirical evidence favors models that assume decreasing returns to physical capital

rather than models that assume constant returns or externalities. 

Human Capital. Economists have long recognized that human capital plays a major

role in long-run economic growth. Workers who are better educated and trained are

better able to perform their tasks, to solve problems, and to embrace the latest

production techniques. Indeed, human capital can be viewed as the fundamental

source of technological progress since it is the means by which the stock of

knowledge is embodied and transmitted. Human capital allows less-developed
24 There is a large literature examining this question. Surveys can be found in Kreuger & Lindahl (2001)

and Card (1999). The arguments in this paragraph come from Kreuger & Lindahl (2001). 

25 There are models of endogenous growth that rely on spillovers to human capital, but they are not well

supported by the empirical evidence. See the discussion in CBO (1994).

26 Kreuger and Lindahl (2001) argue that the question is not fully settled, but that these estimates are not

merely capturing differences in unobserved ability.

27 This argument can be found in Pritchett (2001).

10

countries to imitate, and thereby to catch up to, countries at the technological frontier.

Undoubtedly, human capital will play a key role in any convincing theory of long-run

growth, but it is not yet clear how it should be modeled. Does human capital operate

through the factor inputs? Or is it a separate factor of production? Or does it

influence TFP growth, by affecting a country’s ability to innovate? Or is there

perhaps a spillover associated with human capital, in which an increase in the

economy-wide stock of human capital raises the productivity of all workers, even

those who didn’t receive any extra training or education.

The microeconomic evidence is fairly clear: workers with more education are

better paid than those with less education. Estimates of the return to education vary

slightly through time and across countries and are bit higher for women than for men,

but standard regression equations show that each additional year of schooling raises

earnings by roughly 10 percent.24 These estimates come from generally-accepted

micro-based equations that relate earnings to education (among other variables) and

generally fit the data well. 

A natural implication of these results is that a country could raise its long-run

growth rate if it were able to raise the average educational attainment of it population.

Unfortunately, that implication doesn’t necessarily follow. The micro equations

identify the private return to education, which is the increase in earnings reaped by

a worker with more education. What is relevant for long-run growth is the social

return to education, which is the increase in productivity that could result from an

increase in educational attainment. It is possible for the social return to education to

be larger than the private return if there are spillovers associated with education.25

It is also possible for the social return to be smaller than the private return. This

would occur if the increase in private earnings arose as a result of “signaling.” If

workers with a lot of innate ability tend to acquire more education than workers with

less ability, then more education will appear to “cause” an increase in wages, when

in fact it is merely signaling the presence of more ability.26 The social return to

education could be lower than the private return for another reason. If education does

in fact raise ability, but that increased ability is put to use in profitable, but

unproductive activities, such as corruption, then the increase in earnings would not

imply an increase in aggregate income.27 This phenomenon would be more likely in

less-developed countries rather than the United States. 
28 See Mankiw, Romer, and Weil (1992). Similar results are reported by Barro (1991) and Barro and

Sala-i-Martin (1995). For further discussion of Mankiw et. al., see CBO (1994). 

29 See Benhabib & Spiegel (1994). Other studies that report similar results include Klenow & RodriguezClare (1997), Islam (1995), and Pritchett (2001). Kreuger and Lindahl (2001) challenge these results,

arguing that the measures of human capital are tainted by measurement error.

30 See Benhabib & Spiegel (1994), p. 155. The authors cite the work of Nelson and Phelps (1966), who

argue that the pace of innovation is a function of the stock of capital rather than the flow of human

capital accumulation. See also Aghion and Howitt (1998), pp. 327-328.

11

Macroeconomic studies attempt to estimate the social return to education and,

unfortunately, the evidence from those studies is more equivocal than is the microeconomic evidence on the private return. Many macroeconomic studies, especially

cross-sectional empirical studies, have found a significant role for human capital in

long-run growth. A canonical example is the 1992 study by Mankiw, Romer and

Weil (MRW), in which the authors use a cross section of 98 countries to show that

growth during the 1960-1985 period is positively related to the level of human capital

during that period, measured by the average percentage of the working-age

population enrolled in secondary school from 1960 through 1985.28 Essentially,

those results indicate that countries with higher school enrollment rates tended to

grow faster during this period. Results like that of MRW seemed to confirm the

intuition that human capital is an important factor in long-run growth and were

consistent with the microeconomic evidence of a private return to education.

In contrast, other macroeconomic studies that approach the question from a

different angle find little impact on long-run growth from human capital. These

studies examine the relationship between growth in per-capita output and growth in

human capital (rather than the stock of human capital) have not generally found a

significant effect for human capital. For example, Jess Benhabib and Mark Spiegel

regressed growth in per-capita income over the 1965-1985 period on the growth in

physical capital, human capital, and labor force for a sample of 121 countries. They

found that growth in physical capital and labor force entered the regression

significantly, with the correct sign and expected magnitude, but that two different

measures of growth in human capital were insignificant.29 Benhabib and Spiegel

interpret these results as support for the idea that human capital is important because

it facilitates innovation and the adoption of new technologies, rather than because it

directly contributes to the productive process as a factor of production.30

What are we to make of these disparate strands of evidence? Why has the

striking increase in educational enrollments in the U.S. and other countries not been

matched by a corresponding increase in economic growth? It’s hard to believe that

human capital has no effect on growth, but there are several reasons why it is difficult

to tease out the effects of human capital on economic growth:
31 See Bils and Klenow (2000).

32 Although growth in total factor productivity is often interpreted as technical change, the difficulties

inherent in measuring TFP mean that it captures technical change and any other factor that contributes

to GDP growth apart from labor and capital, including measurement error and cyclical variation. 

33 R&D spending is not the only activity that expands the knowledge stock. However, it is the activity

that is most amenable to measurement and statistical estimation. Hence, it has been the focus of most

empirical work.

12

 o Measurement error. Many studies measure educational attainment using

school enrollment rates, which are a poor proxy for total years of schooling.

Even those studies that measure schooling more accurately still do not

measure the quality of education and they miss other forms of human capital

accumulation, such as training programs and the informal training that occurs

on the job. 

 o Timing issues. Economic theory is unable to predict how long it takes for

increases in educational attainment to affect growth. The lags could be as

short as one year, or for as long as twenty years. Moreover, the lags could

vary through time. This means that empirical equations may be misspecified.

 o Endogeneity. Similar to the problem estimating the effect of physical capital,

it is hard to tell if increased education causes an increase in per-capita GDP

or if rich countries merely demand more education because they can afford

it.31 Estimation using instrumental variables can correct for this problem, but

it requires variables that are correlated with education but are uncorrelated

with the error term in the growth regression. Such variables are hard to find.

In conclusion, the results of these empirical studies do not help us answer the

basic question that needs to be answered for our models. If educational attainment

in the U.S. were to increase, would productivity rise? It is hard to predict that

outcome with confidence based on the research to date. Moreover, it’s quite likely

that the social return to education would shrink if such a policy were implemented

because the marginal students (i.e., those who would increase their educational

attainment under a policy change) would tend to be less able than the average

student. 

Total Factor Productivity. Technological progress is the fundamental source of

long-run growth in living standards. The neoclassical model is silent on the causes

of TFP growth, largely because economists have yet to provide a complete

description of the sources of growth in technology.32 However, few economists

believe that TFP growth is truly exogenous. Clearly, companies and governments try

to expand the technological frontier with explicit expenditures on R&D and other

activities.33 Research on the link between R&D and growth has focused on three
34 See Griliches (1994) and Griliches (1988).

35 See Jones (2002).

36 See CBO (1994) for a more complete discussion. Also, see Aghion & Howitt (1998) or Grossman &

Helpman (1991) for more discussion of endogenous innovation models.

13

questions: what drives R&D spending, how big R&D’s contribution to growth is, and

whether there are externalities to R&D spending.

It is possible to analyze R&D spending squarely in the neoclassical tradition

by treating it as a factor of production. Under this approach, R&D spending

accumulates into a stock of R&D “capital,” which earns a rate of return that is

captured by the innovating firm.34 If there are any external benefits, they are purely

accidental. Studies of this type typically find that private R&D spending has a

positive and significant impact on productivity growth across firms and industries,

with social rates of return that range from 20 to 50 percent. (The link between

government R&D spending and productivity is generally estimated to be smaller and

is less robust statistically). A reasonable approach would be to calculate a stock of

private R&D capital and add it to the production function in CBO’s long-term

models. Another approach would be to follow the lead of some researchers who have

developed growth accounting models based on the growth of capital, raw labor, R&D

labor, and human capital.35

Treating R&D as a separate factor input formalizes the process of innovation

and reduces the contribution to growth made by the residual, but leaves the residual

as an exogenous variable. An important contribution made by Paul Romer, as well

as other analysts who developed endogenous growth models, was to stress two key

differences between technical knowledge and other types of capital goods. First,

technical knowledge is nonrival in consumption, which means that it can be used by

an unlimited number of people at the same time, and second, it is at least partially

nonexcludable, which means that the owner of an idea cannot completely prevent

others from using it.36

An important implication of these properties is the existence of increasing

returns to scale, meaning that a given percentage increase in the factors of production

will lead to a larger percentage increase in output. To see this concept, consider a

firm that produces $10 million using one factory, 10 workers, and a certain amount

of technical knowledge. This firm could produce $20 million by building a second

plant, hiring 10 more workers, and using the same amount of knowledge. Since

output has doubled with a less-than-doubling of inputs, output would more than

double if all inputs were doubled. Thus, production is characterized by increasing

returns to scale.
37 National Science Board, Science and Engineering Indicators 2002, (Arlington, VA: National Science

Foundation, 2002).

38 The issues are discussed at length in Griliches (1979) and in the Appendix to Chapter 12 in Aghion

& Howitt (1998). David Romer (1996) notes that the spillovers to R&D could be positive or negative.

39 See Griliches (1992), p. S43.

40 See Coe & Helpman (1995).

14

Models of endogenous innovation have a lot of intuitive appeal. R&D

spending in the U.S. is not trivial. In 2000, for example, private companies spent

$181 billion and the federal government spent $70 billion on research and

development.37 And that is only the spending that is caught in the government’s

statistical net. Entrepreneurs devote resources to invent, innovate, and nurture

companies that sell products derived from new ideas. It seems unlikely that

innovation would have continued for so long if these companies and individuals were

not getting some sort of return to their efforts. To be successful, innovators must be

able to prevent others to some extent from using their discovery, through patents and

other means. Many models of endogenous innovation assume that inventors earn

monopoly rents, at least temporarily, to compensate them for their investments. 

Does innovation have significant spillovers? It is one thing to say that

spillovers to R&D exist. It is another matter to say that they are large enough to drive

TFP growth at the economy-wide level. Measuring the degree of spillovers at the

firm or industry level is quite challenging, largely because the requisite data do not

exist.38 Nevertheless, after surveying the micro evidence, Zvi Griliches concluded

that: 

there has been a significant number of reasonably well done studies all

pointing in the same direction: R&D spillovers are present, their magnitude

may be quite large, and social rates of return remain significantly above

private rates.39

The existence of international spillovers has been hinted at in an oft-cited paper that

found positive and significant effects for domestic and foreign R&D stocks on

domestic TFP growth.40 Foreign R&D was found to be particularly important for

smaller countries.

The macro evidence supporting the stronger versions of the endogenous

innovation models is less persuasive. This evidence would include the class of

models that predict so-called “scale” effects and permanent effects on TFP growth

resulting from policy changes. Scale effects refers to a prediction of some models

that changes in the level of resources dedicated to R&D can affect the economy’s

growth rate. For example, there has been a secular increase in the share of labor

devoted to R&D in the U.S. and other industrialized nations during the postwar
41 See Jones (1995b). Jones (1995a) presents a model of endogenous innovation in which long-run

growth is a positive function of variables, most notably population growth, that are normally treated

as exogenous in growth models. Kremer (1983) has found evidence to support such “semiendogenous” growth models. His results indicate that technological progress increaseswithpopulation

size, at least over very long periods of time.

42 See Griliches (1988).

43 See Phelps (1995).

44 See Barro (1991), Barro & Sala-i-Martin (1995), Hall & Jones (1999), and Easterly (1993).

45 See Prescott (1998), Harberger (1998), Olson (1996). Schmitz (2001) documents a doubling of out

per worker in the iron-ore mining industry due solely to a relaxation of restrictive work rules.

15

period. According to the predictions of some models of endogenous innovation, such

an increase should cause an increase in the growth of TFP in those countries, but that

hasn’t been observed.41 In addition, the data do not seem to support the idea that

R&D spending playing a significant role in the post-1973 productivity slowdown.42

Other factors that influence TFP growth. There is a large and growing literature

that argues that there is no single ‘silver bullet’ to explain TFP growth. Instead, the

sources of productivity growth are diffuse. Aggregate productivity growth arises as

different firms in different industries aggressively cut costs and try to produce as

efficiently as possible. Governments need to get the fundamentals right by first

establishing a system of property rights and a functioning legal system and then by

avoiding policies that would restrain productivity growth. As an example, consider

the socialist economies in eastern Europe during the postwar period. Those countries

failed to grow as quickly as other countries with comparable levels of physical and

human capital, in part because their political system did not provide the incentive for

entrepreneurs to innovate or for workers to increase efficiency.43

Empirical studies that employ cross-sectional regressions have found that

measures of a country’s policies and institutions help to explain differences in output

per capita across countries.

44 Variables such as the degree of political instability, and

the black-market premium on foreign exchange have been found to have a negative

relationship with growth. Other researchers emphasize factors that allow resistance

to the adoption of international best practice techniques. Examples include anything

from zoning laws that prohibit large-scale retailers to compete with smaller

operations, to laws that make the labor market less flexible, all the way up to outright

corruption on the part of government officials.45 These findings augur well for the

U.S. economy because it ranks high with respect to the various measures that have

been found to be associated with faster growth. However, none of the factors

described in this section can be incorporated easily into a long-term model of the

U.S. economy.
46 Jorgenson’s method is first described in Jorgenson and Griliches (1967). More recent examples

include Jorgenson, Ho, and Stiroh (2002), and Jorgenson and Stiroh (2000). Bureau of Labor Statistics

(1993) describes a similar approach used to calculate multifactor productivity.

16

Possible Approaches for Extending CBO’s Long-Term Growth Model

Which approaches to modeling long-run growth are suggested by the latest empirical

research? With regard to physical capital, the conclusion of CBO’s 1994 paper has

been reaffirmed—the core of the model should follow the standard neoclassical

model in assuming that the returns to physical capital are decreasing. The latest

research does not support the existence of spillovers to physical capital.

With regard to human capital, the empirical results are mixed, with crosssectional studies generally finding a significant link and panel-data studies not.

Hence, there is not yet a consensus among researchers about precisely how human

capital should enter empirical models. Some would argue that it should enter the

production function as a separate factor, while others would contend that it only

affects TFP. This is not to say that there are not operational models that incorporate

human capital, several examples exist.

One example of a model that reflects the impact of human capital

accumulation derives from the growth accounting literature. This approach, used

most notably by Dale Jorgenson and several co-authors, accounts for labor

heterogeneity, or differences in the relative productivity of workers with different

levels of education or experience.46 Economic theory predicts that an hour worked

by a person with more skills will yield a larger flow of “labor services” and more

output than a lesser-skilled worker. Models that use total hours worked as the labor

input, implicitly weight each hour worked identically, no matter whether it is worked

by a veteran pilot or a newly-hired clerk. In contrast, Jorgenson’s index of labor

input reflects changes in both hours worked and labor quality. It does so by

weighting the hours growth of different demographic groups—workers are divided

into groups based on age, sex, and educational attainment—using data on relative

earnings of each group. Earnings are used to weight the hours of each demographic

group because worker productivity is difficult to observe directly. Economic theory

predicts that, under certain conditions, a worker’s marginal product will be equated

to his or her earnings.

In principle, Jorgenson’s method would provide a more accurate picture of

the sources of economic growth over history because it would reclassify a portion of

GDP growth as arising from growth in the labor input instead of attributing it to the

unexplained residual, TFP. Using Jorgenson’s approach might improve the long-run

projection for GDP as well, to the extent that the demographic factors that go into the

estimate can be projected. However, this method relies on the assumption that

differences in relative earnings reflect differences in productivity across categories

of workers, as implied by the assumption competitive labor markets. If markets fail
47 As noted in the discussion of human capital earlier in this paper, microeconomic wage equations

indicate that earnings increase with education and experience, which suggests that earnings are

correlated with productivity, as required by Jorgenson’s model. However, there are alternate

interpretations of the microeconomic evidence. Under these alternatives, workers with more education

are also paid more on average, but an increase in education does directly increase an individual

worker’s marginal product. An excellent discussion of these alternate explanations can be found in

BLS (1993), especially pages 41-43.

48 For details, see Pritchett (2001) or Jorgenson and Fraumeni (1989).

17

to assign earnings in this fashion, due possibly to market failure or because the

competitive model does not accurately describe the workings of labor markets, then

this method will give a faulty estimate of TFP during history and a misleading

projection of GDP for the future.47 

The preceding discussion also highlights a more general problem. Adopting

Jorgenson’s procedure for adjusting the labor input for quality would require

projections of a greater number of demographic variables than are currently

necessary. Doing so would complicate the model and potentially introduce more

sources of forecast error. Although this problem is pronounced with Jorgenson’s

model—due to the large number of additional variables that would be required—it

would arise in any alternate model with more variables.

Another approach would be to incorporate the stock of human capital

explicitly into long-run models as a factor of production separate from hours worked.

While following the spirit of Mankiw, Romer, and Weil, this approach would not

have to rely on the rather crude estimate of the human capital stock used by MRW.

Instead, it could use the research of Dale Jorgenson and Barbara Fraumeni or by Lant

Pritchett, who each calculate stocks of human capital by combining estimates of the

educational attainment of the labor force with micro earnings functions that relate

earnings to years of schooling.48 Alternatively, one could model TFP as a function

of the human capital. This approach would follow from the results of Benhabib and

Spiegel, who argue that differences in growth rates across countries arise due to

differences in stocks of human capital rather than differences in rates of

accumulation.

There are problems with all of these approaches. First and foremost, there is

not yet a consensus among economists about how human capital should enter the

production function, largely because empirical evidence on this question is

ambiguous. If human capital truly is a factor of production, then it should enter

significantly in regressions,whether those equations are estimated in levels or growth

rates. Another problem is that these approaches would dramatically complicate

CBO’s models, by forcing CBO to model the process by which the stock of human

capital grows. Examples of such models of human capital accumulation exist, but

it’s not clear that using them would improve the projection of GDP, which is the
49 See for example, Ho and Jorgenson (2002).

50 See Jones (2002).

18

variable that ultimately matters.49

Another approach would be to include both human capital and R&D spending

in the model along the lines suggested by Charles Jones in a recent article in the

American Economic Review.

50 In Jones’ model, which follows in the spirit of Paul

Romer’s models of endogenous innovation, output is produced using physical capital,

human capital, and the stock of ideas. Growth in the stock of ideas, in turn, arises

from research effort (measured by the share of human capital devoted to research and

development) as well as from the size of the stock of ideas itself. In steady state,

growth arises solely from increases in the stock of ideas, which are a function of

population growth. Jones used his model to carry out a growth accounting exercise.

He found that the vast majority of U.S. growth during the 1950-1993 period can be

attributed to transitional dynamics (associated with increases in research effort and

educational attainment) rather than steady-state growth.

A major challenge of using any of these approaches for 75-year projections

is that they would require making additional assumptions about variables that affect

labor quality and technological progress, including educational attainment, returns

to education, and R&D spending. (This, of course, leaves aside the important

question about whether changes in government policy can affect educational

attainment or R&D effort). Expanding CBO’s model to include any of these

channels of influence would require guesses about parameters on which little reliable

evidence exists.
19

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