Catching Up and Falling Behind

Nancy L. Stokey

University of Chicago

September 1, 2010

Preliminary Draft

Abstract

This paper studies the interaction between technology, which ows in

from abroad, and human capital, which is accumulated domestically, as the twin

engines of growth in a developing economy. The model displays two types of

long run behavior, depending on policies and initial conditions. One is sustained

growth, where the economy keeps pace with the technology frontier. The other

is stagnation, where the economy converges to a minimal technology level that

is independent of the world frontier. Transitions to the balanced growth path

display features seen in modern growth miracles: a high savings rate and rapid

investment in education.

I am grateful to Emilio Espino, Peter Klenow, Robert Lucas, Juan Pablo Nicolini,

Robert Tamura, and Alwyn Young for helpful comments.

1

This paper develops a model of growth that can accommodate the enormous

di�erences in observed outcomes across countries and over time: periods of rapid

growth as less developed countries catch up to the income levels of those at the

frontier, long periods of sustained growth in developed countries, and substantial

periods of decline in countries that at one time seemed to be catching up. The

sources of growth are technology, which ows in from abroad, and human capital,

which is accumulated domestically.

The framework shares several features with the one in Parente and Prescott

(1994), including a world technology frontier that grows at a constant rate and \bar-

riers" that impede the in ow of new technologies into particular countries. Technology

in ows here are modeled as a pure external e�ect, with the rate of in ow governed

by three factors: (i) the domestic technology gap, relative to a world frontier, (ii)

the domestic human capital stock, also relative to the world frontier technology, and

(iii) the domestic \barrier." The growth rate of the local technology is an increasing

function of the technology gap, re ecting the fact that a larger pool of untapped ideas

o�ers more opportunities for the adopting country. It is also increasing in the local

human capital stock, re ecting the role of education in enhancing the ability to ab-

sorb new ideas. Finally, the \barrier" re ects tari�s, internal taxes, capital controls,

currency controls, or any other policy measures that retard the in ow of ideas and

technologies.

A key feature of the model is the interaction between local technology and local

human capital. There are many channels through which human capital can facilitate

the in ow of ideas. Better educated entrepreneurs and managers are better able to

identify new products and processes that are suitable for the local market. In addi-

tion, a better educated workforce makes a wider range of new products and processes

viable for local production, an important consideration for both domestic entrepre-

neurs interested in producing locally and foreign multinationals seeking attractive

2

destinations for direct investment.

Human capital is modeled here as a private input into production, accumulated

with a technology that uses the agent's own time (current human capital) and the local

technology as inputs. Human capital also has an external e�ect, however, through its

impact on the rate of technology in ow. Because of this externality, public subsidies

to education a�ect the long run behavior of the economy.

For �xed values of the technology and preference parameters, the model displays

two types of long run behavior, depending on the policies in place and the initial

conditions. If the technology barrier is low, the subsidy to education is high, and

the initial levels for local technology and local human capital are not too far below

the frontier, the economy displays sustained growth in the long run. In this region of

policy space, and for suitable initial conditions, higher barriers and lower subsidies to

education imply slower convergence to the economy's balanced growth path (BGP)

and wider steady state gaps between the local technology and the frontier. But inside

this region of policy space, changes in the technology barrier or the education subsidy

do not a�ect the long run growth rate. Policies that widen the steady state technology

gap also produce lower levels for capital stocks, output and consumption along the

BGP, but the long run growth rate is equal to growth rate of the frontier.

Thus, the model predicts that high and middle income countries can, over long

periods, grow at the same rate as the world frontier. In these countries the gap

between the local technology and the world frontier is constant in the long run, and

not too large.

Alternatively, if the technology barrier is su�ciently high, the subsidy to edu-

cation is su�ciently low, or some combination, balanced growth is not possible: the

economy stagnates in the long run. An economy with policies in this region converges

to a minimal technology level that is independent of the world frontier, and a human

capital level that depends on the local technology and local education subsidy. In

3

addition, even for parameters that permit balanced growth, for su�ciently low ini-

tial levels of technology and human capital, the economy converges to the stagnation

steady state instead of the balanced growth path.

Thus, low income countries|those with large technology gaps|cannot display

modest, sustained growth, as middle and high income countries can. They can adopt

policies that trigger a transition to a BGP, or they can stagnate, falling ever farther

behind the frontier. Moreover, economies that enjoyed technology in ows in the past

can experience technological regress if they raise their barriers: local TFP and per

capita income can actually decline during the transition to a stagnation steady state.

Two policy parameters are included in the model, the barrier to technology

in ows and a subsidy to human capital accumulation. Although both can be used to

speed up transitional growth, the simulations here suggest that stimulating technology

in ows is more the potent tool. The intuition for this is twofold. First, human capital

accumulation takes resources away from production, reducing consumption in the

short run. In addition, human capital accumulation is necessarily slow. Thus, while

it eventually leads to higher technology in ows, the process is prolonged. Faster

technology in ows increase output immediately, and in addition they increase the

returns to human (and physical) capital, thus stimulating further investment and

growth.

The rest of the paper is organized as follows. Section 1 discusses evidence on the

importance of technology in ows as a potential source of growth. It also documents

the fact that many countries are not enjoying these in ows. Indeed, they are falling

ever farther behind the world frontier. The model is described in section 2. Section

3 looks at economies that converge to balanced growth paths, and section 4 looks

at economies that stagnate. Section 5 describes the implications of the model for

growth and development accounting, and also discusses methods for estimating a key

technology parameter. In section 6 the model is calibrated, and in section 7 transition

4

paths are simulated for economies that reduce their barriers and/or increase their

subsidies to education. Section 8 concludes.

1. EVIDENCE ON THE SOURCES OF GROWTH

Five types of evidence point to the conclusion that di�erences in technology are

critical for explaining di�erences in income levels over time and across economies, that

there is a common (growing) `frontier' technology that developed economies share,

and that poor economies can grow rapidly by tapping into that world technology.1

First, growth accounting exercises for individual developed countries, starting

with those in Solow (1957) and Denison (1974), invariably attribute a large share of

the increase in output per worker to an increase in total factor productivity (TFP).

Although measured TFP in these exercises|the Solow residual|surely includes the

in uence of other (omitted) factors, the search for the missing factors has been exten-

sive, covering a multitude of potential explanatory variables, many countries, many

time periods, and many years of e�ort. It is di�cult to avoid the conclusion that

technical change is a major ingredient.

Second, development accounting exercises using cross-country data arrive at a

similar conclusion, �nding that di�erences in physical and human capital explain only

a modest portion of the di�erences in income levels across countries. For example,

Hall and Jones (1999) �nd that of the 35-fold di�erence in GDP per worker between

the richest and poorest countries, inputs|physical and human capital per worker|

account for 4.5-fold, while di�erences in TFP|the residual|accounts for 7.7-fold.

Klenow and Rodriguez-Clare (1997b) arrive at a similar conclusion.

To be sure, the cross-country studies have a number of limitations. Data on hours

are not available for many countries, so output is measured per worker rather than

1See Prescott (1997) and Klenow and Rodriguez-Clare (2005) for further evidence supporting

this conclusion.

5

per manhour. No adjustment is made for potential di�erences between education

attainment in the workforce and the population as a whole, which might be much

larger in countries with lower average attainment. Human capital is measured very

imprecisely, consisting of average years of education in the population with at best

a rough adjustment for educational quality.2 Nor is any adjustment made for other

aspects of human capital, such as health.

In development accounting exercises, as in growth accounting, the �gure for TFP

is a residual, so it is surely biased upward. Nevertheless, it is large enough to absorb

a substantial amount of downward revision and survive as a key determinant of cross-

country di�erences.

A third piece of evidence is Baumol's (1986) study of the OECD countries. Al-

though criticized on methodological grounds (see DeLong, 1988, and Baumol and

Wol�, 1988), the data nevertheless convey an important message: the OECD coun-

tries (and a few more) seem to share common technologies. It is hard to explain in any

other way the harmony|over many decades|in both their income levels and growth

rates. Moreover, as Prescott (2002, 2004) and Ragan (2006) have shown, much of the

persistent di�erences in income levels can be explained by di�erences in �scal policy

that a�ect work incentives.

A fourth piece of evidence for the importance of technology comes from data on

`late bloomers.' As �rst noted by Gerschenkron (1962), economies that develop later

have an advantage over the early starters exactly because they can adopt technologies,

methods of organization, and so on developed by the leaders. Followers can learn from

the successes of their predecessors and avoid their mistakes. Parente and Prescott's

(1994, 2000) evidence on doubling times makes this point systematically. Figure 1

2Di�erences in educational quality probably have a modest impact, however. Hendricks (2002)

reports that many studies �nd that immigrants' earnings are within 25% of earnings of native-born

workers with the same age, sex, and educational attainment.

6

reproduces their scatter plot, updated to include data through 2006. Each point in

this �gure represents one of the 55 countries that had reached a per capita GDP of

$4000 by 2006. On the horizontal axis is the year that the country �rst reached $2000,

and on the vertical axis is the number of years required to �rst reach $4000.

As Figure 1 shows, there is a strong downward trend: countries that arrived at

the $2000 �gure later, doubled their incomes more quickly. The later developers seem

to have enjoyed the advantage of �shing from a richer pool of ideas, ideas provided

by advances in an ever-improving world technology.3

A �fth and �nal piece of evidence supporting the importance of technology is

the occurrence, infrequently, of `growth miracles.' The term is far from precise, and a

stringent criterion should be used in classifying countries as such, since growth rates

show little persistence from one decade to the next. Indeed, mean reversion in income

levels following a �nancial crisis or similar event implies that an especially bad decade

in terms of growth rates is likely to be followed by a good one. But recovery from a

disaster is not a miracle.

Nevertheless, over the period 1950-2006 twelve countries (i) enjoyed at least one

20-year episode where average per capita GDP growth exceeded 5%, and (ii) in 2006

had GDP per capita that was at least 45% of the U.S. This group has �ve members

in Europe (Germany, Italy, Greece, Portugal, and Spain), �ve in east Asia (Japan,

Taiwan, Hong Kong, Singapore, and Korea), and two others (Israel and Puerto Rico).

3

Figure 1 has a built-in bias, which should be noted. Among countries that have recently reached

the $2000 �gure, many have not yet reached $4000. In particular, the slow growers have not yet

reached that goal. The dotted line indicates a region where, by construction, there cannot yet be

any observations. Ignoring the pool of countries that in the future will occupy this space biases the

impression in favor of the `advantage of backwardness' hypothesis. An easy way to mitigate the bias

is to truncate the last twenty years of data, which eliminates many of the observations for which the

$4000 goal lies in the future. The strong downward trend seems to survive this truncation.

7

The jury is still out on several others candidates: China, Thailand, Malaysia, and

Botswana have met criterion (i) but not (yet) accomplished (ii).4

Although each of these �ve sources of evidence has individual weaknesses, taken

together they make a strong case for the importance of international technology

spillovers in keeping income levels loosely tied together in the developed countries,

and occasionally allowing a less developed country to enjoy a growth spurt during

which it catches up to the more developed group.

Not all countries succeed in tapping into the global technology pool, however.

Figure 2 shows the world pattern of catching up and falling behind, with the U.S.

taken as the benchmark for growth. It plots per capita GDP relative to the U.S. in

2000, against per capita GDP relative to the U.S. in 1960 for 104 countries. Countries

that are above the 45o line have gained ground over that 40-year period, and those

below it have lost ground. It is striking how few have gained. The geographic pattern

of gains and losses is striking as well. The countries that are catching up are almost

exclusively European (plus Israel) and East Asian. With only a few exceptions,

countries in Latin America, Africa, and South Asia have fallen behind.

Figure 3 shows 69 of the poorest countries, those in Asia and Africa, in more

detail. The six Asian miracles (Israel, Japan, Taiwan, Hong Kong, Singapore, and

Korea) are omitted, since they signi�cantly alter the scale.5 Here the plot shows per

capita GDP in 2000 against per capita GDP in 1960, both in levels. The rays from

the origin correspond to various average growth rates, and keeping pace with the U.S.

over this period requires a growth rate of 2%. The number of countries that have

4It is sobering to see how many countries enjoyed 20-year miracles, yet gave up all their (relative)

gains or even lost ground over the longer period. Among countries in this groups are Bulgaria,

Yugoslavia, Jamaica, North Korea, Iran, Gabon, Libya, and Swaziland.5Also excluded are the oil producers Bahrain, Oman, Saudia Arabia, and countries with popula-

tion under 1.2 million in 2000, Bahrain, Cape Verde, Comoro Islands, Djibouti, Equatorial Guinea,

Mauritius, Reunion, Seychelles Islands.

8

gained ground relative to the U.S. (points above the 2% growth line) is modest, while

the number that have fallen behind (points below the line) is much larger. A shocking

number, mostly in Africa, have su�ered negative growth over the whole period.

The model developed below focuses on technology in ows as the only source of

sustained growth. Other factors are neglected, although some are clearly important.

For example, it is well documented that in most developing economies, TFP in

agriculture is substantially lower than it is in the non-agricultural sector.6 Thus, an

important component of growth in almost every fast-growing economy has been the

shift of labor out of agriculture and into other occupations. The e�ect of this shift

on aggregate TFP, which is signi�cant, cannot be captured in a one-sector model.

More recently, detailed data for manufacturing has allowed similar estimates for

the gains from re-allocation across �rms within that single sector. Such misallocation

can result from �nancial market frictions, from frictions that impede labor mobility,

or from taxes (or other policies) that distort factor prices.

For example, Hsieh and Klenow (2009) �nd that in China, improvements in

allocative e�ciency contributed about 1/3 of the 6.2% TFP growth in manufacturing

over the period 1993-2004. Although this gain is substantial, it is a modest part of

overall TFP growth in China|in all sectors|over that period.

In addition, evidence from other countries gives reallocation a smaller role. In-

deed, Hsieh and Klenow �nd that in India allocative e�ciency declined over the same

period, although per capita income grew. In addition, Bartelsman, Haltiwanger, and

Scarpetta (2008) �nd that for the two countries (Slovenia and Hungary) for which

they have time series, most of the (substantial) TFP gains both countries enjoyed dur-

ing the 1990s came from other sources: improvements in allocative e�ciency played a

minor role. And most importantly, TFP gains from resource reallocation are one-time

6See Caselli (2005) for recent evidence that TFP di�erences across countries are much greater in

agriculture than they are in the non-agricultural sector.

9

gains, not a recipe for sustained growth.7

The model here is silent about the ultimate source of advances in the technology

frontier, which are taken as exogenous. Thus, it is complementary to the many models

that address the sources of technical change more directly, looking at the incentives

to invest in R&D, the role of learning by doing, and other factors that a�ect the pace

of innovation.

Nor does the model her have anything to say about the societal factors that lead

countries to develop institutions or adopt policies that stimulate or hinder growth, by

stimulating technology in ows, encouraging factor accumulation, or any other means.

In this sense is complementary to the work of Acemoglu, Johnson, and Simon (2001,

2002, 2005) and others, that looks at the country characteristics associated with

economic success, without specifying the more proximate mechanism.

2. THE MODEL

The model is a variant of the technology di�usion model �rst put forward in Nel-

son and Phelps (1966) and subsequently developed elsewhere in many speci�c forms.8

7In addition, it is not clear what the standard for allocative e�ciency should be in a fast-growing

economy. Restuccia and Rogerson (2008) develop a model with entry, exit, and �xed costs that

produces a non-degenerate distribution of productivity across �rms, even in steady state. Their

model has the property that the stationary distribution across �rms is sensitive to the �xed cost

of staying in business and the distribution of productivity draws for potential entrants. There is

no direct evidence for either of these important components, although they can be calibrated to

any observed distribution. Thus, it is not clear if di�erences across countries re ect distortions that

a�ect the allocation of factors, or if they represent di�erences in fundamentals, especially in the

`pool' of technologies that new entrants are drawing from. In particular, one might suppose that the

distribution of productivities for new entrants would be quite di�erent in a fast-growing economy

(like China) and a mature, slow-growing economy (like the U.S.).8See Benhabib and Spiegel (2005) for an excellent discussion of the long-run dynamics of various

versions.

10

There is a frontier (world) technology W (t) that grows at a constant (exogenously

given) rate,

_W (t) = gW (t); (1)

where g > 0; and in addition each country i has a local technology Ai(t): Growth in

Ai(t) has the form

_AiAi

= 0; if Ai = Asti and 0Bi

H i

W

�1� Asti

W

�< �A; (2)

_AiAi

= 0Bi

H i

W

�1� Ai

W

�� �A; otherwise,

where Asti is a lower bound on the local technology level, Bi is a policy parameter,

H i is average human capital, and �A > 0 is the depreciation rate for technology.

The technology oor Asti allows the economy to have a `stagnation' steady state

with a constant technology. Above that oor technology growth is proportional to

the ratio H i=W of local human capital to the frontier technology and to the relative

gap 1�Ai=W between the current technology and the frontier. The former measures

the capacity of the economy to absorb technologies near the frontier, while the latter

measures the pool of technologies that have not yet been adopted.

A higher value for the frontier technology W thus has two e�ects. It widens

the technology gap, which tends to speed up growth, but also reduces the absorption

capacity, which tends to retard growth. The �rst e�ect dominates if the ratio Ai=W

is high and the second if Hi=W is low, resulting in a logistic form.9

As in Parente and Prescott (1994), the barrier Bi � 1 can be interpreted as any

policies that impede access to or adoption of new ideas, or reduce the pro�tability of

adoption. For example, it might represent impediments to international trade that

reduce contact with new technologies, taxes on capital equipment that is needed to

9Benhabib and Spiegel (2005, Table 2) �nd that cross-country evidence on the rate of TFP

growth supports the logistic form: countries with very low TFP also have slower TFP growth. Their

evidence also seems to support the inclusion of a depreciation term.

11

implement new technologies, poor infrastructure for electric power or transportation,

or civil con ict that impedes the ow of people and ideas across border. Notice that

because of depreciation, technological regress is possible. Thus, an increase in the

barrier can lead to the abandonment of once-used technologies.

Growth in Ai is exogenous to the individual household in country i; although it

depends on collective household decisions through H i: Next consider the decisions of

a typical household.

Households accumulate capital, which they rent to �rms, and in addition each

household is endowed with one unit of time (a ow), which it allocates between human

capital accumulation and goods production. Investment in human capital uses the

household's own time and human capital as inputs, as well as the technology level.

In particular,

_Hi(t) = �0vi(t)�Ai(t)

�Hi(t)1�� � �HHi(t); (3)

where vi is time devoted to human capital accumulation, �H > 0 is the depreciation

rate for human capital, and 0 < �; � < 1. This technology has CRS jointly in the

stocks (A;H) and DRS in the time input. The former fact permits sustained growth,

and the assumption that � > 0 rules out the possibility of endogenous growth in

the absence of technology di�usion (without increases in Ai): The assumption � < 1

insures that time allocated to human capital accumulation is always strictly positive.

Time allocated to production is also augmented by technology and human capital

to produce e�ective labor input. Speci�cally, the labor o�ered by a household in

country i is augmented by the country technology Ai, which it takes as exogenous,

and its own human capital Hi: Thus its e�ective labor is

Li(t) = [1� vi(t)]A�i (t)H

1��i (t); (4)

where (1� vi) is time allocated to goods production.

The technology for goods production is Cobb-Douglas, with physical capital Ki

12

and e�ective labor Li as inputs,

Yi = K�i L

1��i ;

so factor returns are

Ri(t) = �

�Ki

Li

���1; (5)

wi(t) � (1� �)

�Ki

Li

��;

where w is the return to a unit of e�ective labor. Hence the wage for a worker with

human capital H is

wi(H; t) = wiA�iH

1��: (6)

To allow balanced growth, subsidies to education can be incorporated as follows.

Time spent investing in human capital is subsidized at the rate �iwi(H i; t), where H i

is the average human capital level in the economy and �i 2 [0; 1) is a policy variable.

Thus, the subsidy is a fraction of the current average wage. The subsidy is �nanced

with a lump sum tax �i; and the government's budget is balanced at all dates,

�i(t) = v�iwi(H i); t � 0: (7)

Thus, the budget constraint for the household is

_Ki(t) = (1� vi)wi(Hi) + v�iwi(H i) + (Ri � �K)Ki � Ci � �i; (8)

where �K > 0 is the depreciation rate for physical capital. Households have constant

elasticity preferences, with parameters �; � > 0: Hence the household's problem is

maxfvi(t);Ci(t)g1t=0

Z 1

0

e��tC1��i (t)

1� �s.t. (3) and (8), (9)

given �i; Hi0; Ki0 and�Ai(t); H(t); Ri(t); wi(�; t); �i(t); t � 0

:

13

Definition: Given the policy parameters Bi and �i and initial values for the state

variablesW0; Ai0; Hi0; andKi0, a competitive equilibrium consists of�Ai(t); Hi(t); H i(t);

Ki(t); vi(t); Ci(t); Ri(t); wi(�; t); �i(t); t � 0g with the property that:

i. fvi; Ci; Hi; Kig solves (9), given �i; Hi0; Ki0; and�Ai; H i; Ri; wi; �i

;

ii. fRi; wig satisfy (5) and (6);

iii. fAig satis�es (2) and�H i(t) = Hi(t); t � 0

;

iv. f�ig satis�es (7).

The system of equations describing the equilibrium is developed in the Appen-

dix. Two types of behavior are possible in the long run. If the barrier Bi is low

enough and/or the subsidy �i to education is high enough, balanced growth is possi-

ble. Speci�cally, for some initial conditions the economy converges asymptotically to

a BGP, along which Ai; Hi; Ki and Ci all grow at the rate g: Economies of this type

are studied in section 3.

Balanced growth is not the only possibility, however. The set of initial conditions

that lead to balanced growth shrinks with increases in Bi and reductions in �i, at

some point disappearing altogether. Thus, if Bi is su�ciently large and/or �i is

su�ciently small, or if the initial values for Ai and Hi are su�ciently small relative

to W; the economy stagnates in the long run. In these economies the technology

level converges to its stagnation level Asti ; the stocks of human and physical capital

adjust accordingly to Hsti ; K

sti ; and consumption converges to a constant level C

sti .

Economies of this sort are studied in section 4.

3. CATCHING UP: ECONOMIES THAT GROW

For convenience drop the country subscript. In this section BGPs are charac-

terized, where the time allocation v > 0 is constant and the state variables A;H;K

grow at constant rates. It is easy to show that A;H;K;L, and C grow at the same

14

rate g as the frontier technology, the factor returns R and w are constant, the wage

w(H) grows at the rate g; and the costate variables �H and �K for the household's

problem grow at the rate ��g:

To study growing economies it is convenient to de�ne the normalized variables

a(t) � A(t)

W (t); h(t) � H(t)

W (t);

k(t) � K(t)

W (t); c(t) � C(t)

W (t);

�k(t) � �K(t)

W��(t); �h(t) �

�H(t)

W��(t):

The equilibrium conditions can then be written as

�h��0v��1 = (1� �)�kw

�ah

����; (10)

c�� = �k;_�h�h

= �+ �g + �H � �0

�ah

��v��(1� �) +

1� �

1� ��

�1

v� 1��

;

_�k�k

= �+ �g + �K �R;

_h

h= �0v

��ah

��� �H � g;

_k

k= ���1 � c

k� �K � g;

_a

a=

0Bih (1� a)� �A � g;

where

� � K=L = k= (1� v) a�h1��; (11)

R = ����1; w = (1� �)��;

and the transversality conditions hold if and only if

� > (1� �) g; (12)

which insures that the discounted value of lifetime utility is �nite.

15

a. The balanced growth path

Let abg; hbg; and so on denote the constant values for the normalized variables

along the BGP, determined by setting _�h=�h = _h=h = ::: = 0 in (10). The interest

rate is

rbg � Rbg � �K = �+ �g;

so (12) implies rbg > g: The input ratio �bg then satis�es

���bg���1

= Rbg = rbg + �K ;

which in turn determines the return to e�ective labor wbg:

Then use the third and �fth equations in (10) to �nd that time allocated to

human capital accumulation is

vbg =

�1 +

1

�

1� �

1� �

�� +

rbg � g

g + �H

���1: (13)

Since rbg > g; the second term in brackets is positive and vbg 2 (0; 1) : Notice that

vbg is increasing in the subsidy �; with vbg ! 1 as � ! 1: It is also increasing in

1 � �; the share for human capital, and in �; the exponent on v; in the production

function for human capital. A higher value for 1 � � increases the sensitivity of the

wage rate w(H) to private human capital, increasing the incentives to invest, while a

higher value for � reduces the force of diminishing returns. Finally, vbg is increasing

�H and g; re ecting the fact the time allocated to investment in human capital must

o�set depreciation and allow the stock H to keep pace with W:

The �fth equation in (10) then determines the ratio of technology to human

capital,

zbg � abg

hbg=

�g + �H�0 (vbg)

�

�1=�: (14)

Hence this ratio is decreasing in the subsidy rate �: Note that vbg and zbg do not

depend on the barrier B:

16

The last equation in (10) then implies that a BGP level for the relative technology

abg, if any exists, satis�es the quadratic

abg�1� abg

�= (g + �A)

B

0zbg: (15)

The solutions are as follows.

Proposition 1: If

(g + �A)Bzbg

0<1

4; (16)

then (15) has two solutions, and they are symmetric around the value 1/2. Call the

higher and lower solutions abgH and abgL ; with

0 < abgL <1

2< abgH :

If the inequality in (16) is reversed, then no BGP exists.

Figure 4 displays the solutions as functions of B; for two values of � and �xed

values for the other parameters. With � �xed, a small increase in B moves both

solutions toward the value 1/2: For �xed B; a small decrease in �; which increases

zbg; also moves the solutions toward 1/2. For B su�ciently large or � su�ciently

small, the inequality in (16) fails, and no BGP exists.

Notice that the higher solution abgH has the expected comparative statics|it

increases as the barrier B falls or the subsidy � rises|while the lower solution abgL

has the opposite pattern. As we will see later, for reasonable parameter values abgH

is stable and abgL is not. Therefore, since abgH 2 [1=2; 1] ; this model produce BGP

productivity (and income) ratios of no more than two across growing economies.10

Poor economies cannot grow along BGPs, in parallel with richer ones.

10Other factors, like taxes on labor income (which reduce labor supply) can increase the spread

in incomes across growing economies. See Prescott 2002, 2004, and Ragan, 2006, for models of this

type.

17

If a BGP exists, changes in B do not a�ect the growth rate g, the interest rate

rbg; the time allocation vbg; or the ratio zbg of technology to human capital. Thus,

looking across economies with similar education policies, along BGPs those with

higher barriers lag farther behind the frontier, but in all other respects are similar.

Stated a little di�erently, an economy with a higher barrier looks like its neighbor

with a lower one, but with a time lag.

b. Transitional dynamics

The normalized system has three state variables, a; h; and k; and two costates,

�h and �k; making the transitional dynamics complicated. The more interesting

interactions involve technology and human capital, with physical capital playing a

less important role. Thus, for simplicity we will drop physical capital.

To this end, set � = 0 and drop the equations for _k and _�k: Then w = 1; R = 0;

and all output is consumed, so

c = (1� v) a�h1��;

and c�� takes the place of �k: The �rst equation in (10) then implies that the time

allocation satis�es

v��1 (1� v)� =1� �

��0a�h�(�+�)��1h ; (17)

where

� � � (1� �)� �:

The transitional dynamics are then described by the remaining three equations in

(10), for _�h=�h; _h=h; and _a=a:

This system of three equations can be linearized around each of the two steady

states, and the characteristic roots determine the long run behavior of the economy.

As shown in the Appendix, a su�cient condition for all of the roots to be real is

18

� � 1. Further conditions that insure exactly two negative roots at abgH and exactly

one negative root at abgL are also discussed. All of the simulations reported here have

this pattern for the roots..

Under these conditions the higher steady is locally stable: for any pair of ini-

tial conditions a0; h0 in the neighborhood of�abgH ; h

bgH

�, there exists a unique initial

condition �h0 for the costate with the property that the system converges asymptoti-

cally. The lower steady state is not locally stable. Instead, there is a one-dimensional

manifold of initial conditions a0; h0 in the neighborhood of�abgL ; h

bgL

�for which the

system converges. This manifold de�nes the boundary of the trapping region for the

high (stable) steady state. The dynamics are described in more detail in section 7.

4. FALLING BEHIND: ECONOMIES THAT STAGNATE

In economies where the barrier B is su�ciently high and/or the educational

subsidy � is su�ciently low, condition (16) in Proposition 1 fails and there is no

BGP. These economies stagnate in the long run. In addition, even if the policies

permit a BGP, for su�ciently low initial conditions the economy converges to the

stagnation steady state rather than the BGP. These economies may grow|slowly|

during a transition period. But their rate of technology adoption is always less than

g; and it declines over time, eventually converging to zero. In this section we will

describe the stagnation steady state and transitions to it.

a. The stagnation steady state

At the stagnation steady state the technology level, capital stocks, factor returns,

consumption, and time allocation are constant, as are the costates for the household's

problem. Let Ast; Hst; Kst; and so on denote these levels. Since consumption is

19

constant, the interest rate is equal to the rate of time preference,

rst = Rst � �K = �;

the input ratio �st satis�es

���st���1

= Rst = �+ �K ;

and the return to e�ective labor wst depends only on �st: The steady state time

allocation and input ratio, the analogs of (13) and (14), are

vst =

�1 +

1

�

1� �

1� �

�� +

�

�H

���1; (18)

zst � Ast

Hst=

��H

�0 (vst)�

�1=�; (19)

so they are like those on a BGP except that the interest rate is r = � and there is

no growth.

Notice that for economies with the same educational subsidy �, more time is

allocated to human capital accumulation in the growing economy, vbg > vst; if and

only if

� > (� � 1) �H :

If � is su�ciently large, the low willingness to substitute intertemporally discourages

investment in the growing economy.

In growing and stagnant economies with the same time allocation, the one that

is growing has a higher ratio of technology to human capital,

zbg

zst=

�1 +

g

�H

�1=�> 1:

For � = �H and � = 2; as will be assumed below, the steady state time allocations

are the same, vst = vbg:

20

b. Transitional dynamics

To study dynamics near the stagnation steady state we can proceed as before,

setting � = 0 to eliminate physical capital, so w = 1 and all output is consumed. The

time allocation during the transition then satis�es the analog of (17),

v��1 (1� v)� =1� �

��0A�H�(�+�)��1H ; (20)

where as before � � � (1� �)� �: The laws of motion for A;H and �H can then be

used to study the transition.

5. DIGRESSION ON GROWTH ACCOUNTING

Consider a world with many economies, i = 1; 2; :::; I. In each economy i; output

per capita at any date t is

Yi(t) = Ki(t)��Ai(t)

�H i(t)1�� [1� vi(t)]

1��:

Assume that at each date, any individual is engaged in only one activity, and hours

per worker are the same over time and across countries. Then all di�erences in v|in

time allocation between production and human capital accumulation|take the form

of di�erences in labor force participation. Let

yi(t) �Yi(t)

1� vi(t)and ki(t) �

Ki(t)

1� vi(t)

denote output and capital per worker, and note that human capital H i is already so

measured. Then output per worker can be written three ways,

yi(t) = ki(t)��H i(t)

1��Ai(t)��1��

; (21)

yi(t) =

�ki(t)

yi(t)

��=(1��)H i(t)

1��Ai(t)�;

yi(t) =

�ki(t)

yi(t)

��=�(1��)�H i(t)

yi(t)

�(1��)=�(1��)Ai(t):

21

Suppose that � and � are known. The income shares for capital and labor, � and

1��; could be obtained from NIPA data for any country, and a method for estimating

� is described below. Suppose in addition that H i is observable, as well as yi and

ki: The �rst equation in (21) is the standard basis for a growth accounting exercise,

as in Solow (1957); the second is the version used for the development accounting

exercises in Hall and Jones (1999), Hendricks (2002), and elsewhere; and the third is

a variation suitable for the model here. In each case the technology level Ai is treated

as a residual.

First consider a single economy. A growth accounting exercise based on the

�rst line in (21) in general attributes growth in output per worker to growth in all

three inputs, ki; H i and Ai; and along a BGP the shares are �; (1� �) (1� �) ; and

� (1� �) :

An accounting exercise based on the second line attributes some growth to phys-

ical capital only if the ratio ki=yi is growing. The rationale for using the ratio is that

growth in H i or Ai induces growth in Ki, by raising its return. Here the accounting

exercise attributes to growth in capital only increases in excess of those prompted by

growth in e�ective labor. Along a BGP ki=yi is constant, and the exercise attributes

all growth to H i and Ai; with shares (1� �) and �:

The third line applies the same logic to human capital, since growth in Ai induces

growth in H i as well as Ki: Since H i=yi is also constant along a BGP, here the

accounting exercise attributes all growth on a BGP to the residual Ai:

Next consider a development accounting exercise involving many countries. Dif-

ferences across countries in capital taxes, public support to education, and other

policies lead to di�erences in the ratios ki=yi and H i=yi; so a development accounting

exercise using any of the three versions attributes some di�erences in labor produc-

tivity to di�erences in physical and human capital. But the second line attributes to

physical capital|and the third to both types of capital|only di�erences in excess of

22

those induced by changes in the supplies of the complementary factor(s), H i and Ai

in the second line and Ai in the third.

Estimating �.|

Suppose that there is a little migration across countries. Then an estimate of �

can be obtained by using data on the wages of migrants. From (6), the wage of a

migrant from country j to country i depends on the return to e�ective labor wi and

the technology Ai in the country i where he is employed, and the human capital Hj

that he acquired in his country of origin. Suppose that his human capital is Hj; the

average value for his country of origin. Then his wage in country i is

Wij(t) = wiA�iH

1��j :

Recall that the input ratio zbgj = abgj =hbgj and time allocation vbgj on the BGP

depend only on the education subsidy �j: Thus, if i and j are on their BGPs, and if

both have the same education subsidy �; then they also have a common input ratio

zbg: They also have a common time allocation vbg; so the ratio of output per worker

in the two countries at any date t is

yj(t)

yi(t)=Yj(t)

Yi(t)=

�Kj

Ki

�� "�Hj

Hi

�1�� �AjAi

��#1��=abgj

abgi;

where the last equality uses the fact that along a BGP both kinds of capital grow in

proportion to A: The wage of an immigrant from j relative to the wage of a native-

born worker in i can then be written as

Wij(t)

Wii(t)=

�Hj

H i

�1��=

abgj

abgi

!1��=

�yjyi

�1��:

Hence 1 � � is the coe�cient in a regression of lnWij on ln yj in a cross section of

immigrants in i from various source countries,

lnWij = constant + (1� �) ln yj + error. (22)

23

If �i varies across countries, then (22) must be modi�ed to control for the educational

policy in the worker's country of origin. Similarly, if capital taxes or other policies

that distort the returns to physical and human capital vary across countries, then

(22) should also take those policies into account.

An alternative empirical strategy involves looking at a cross section of emigrants

from j to various destination countries i: As before assume that all countries are on

their BGPs. Recall that in the absence of capital taxes and similar distortions, the

return wbg to a unit of e�ective labor is the same on all BGPs. Hence the wage of an

immigrant from j working in i; relative to his pre-emigration wage in j; is

Wij

Wjj

=

�AiAj

��=

�YiYj

��: (23)

Thus � is the coe�cient on the log wage gain from emigrating, �Wij = lnWij� lnWjj;

on the log di�erence in output per capita in the destination and source countries,

�Yij = lnYi � lnYj, in a cross section of emigrants from j. Note that this approach

does not require any assumption about educational policies. If some of the destination

countries are not on their BGPs, then (23) must be modi�ed to take into account

di�erences in w across destinations.

6. CALIBRATION

The model parameters are g for the long run growth rate, ( 0; �A) for technology

di�usion, (�0; �; �; �H) for human capital accumulation, � for e�ective labor, and (�; �)

for preferences. These parameters are �xed throughout the simulations. The policy

parameters (�;B) vary as indicated.

The growth rate used is g = 0:019; the rate of growth of per capita GDP in

the U.S. over the period 1870-2003, and the standard values � = 0:03 and � = 2 are

used for the preference parameters. The depreciation rates for technology and human

capital are set at �A = �H = 0:03: For these values the time allocation is the same

24

along the BGP and in the stagnation steady state, vbg = vst:

For the share parameter �; (22) suggest looking at data on wages of migrants.

Borjas (1987) reports regression results for data on immigrants in the U.S. from

41 source countries. The estimated coe�cient there is about 0.12, which implies

� = 0:88: Hence the technology (country) component of the wage, A�i ; is quite im-

portant relative to the human capital (individual) component H1��j ; suggesting that

an individual in a poor country is wise to focus on emigrating rather than acquiring

an education. The simulations here use a slightly more conservative �gure, � = 0:75:

Returns to education

With an estimate of 1�� in hand, cross-sectional variation in educational attain-

ment across individuals can be used to estimate �=� from a Mincer regression. From

(6), the wage of a native-born worker in i with human capital H; possibly di�erent

from the average level H i; is

wi(H; t) = wi(t)A�i H

1��:

Consider a family (dynasty) that invests more (or less) time than the economy-wide

average in accumulating human capital. If they allocate the share of time v instead

of vbgi ; then from (3) or (14) we �nd that along the BGP their human capital, relative

to the average, is

H

H i

=

v

vbgi

!�=�:

Hence in a cross section where�v; H

�varies across individuals,

lnwi(H; t) = constant + (1� �)�

�ln v: (24)

We can follow the procedure in Hall and Jones (1999) to obtain an empirical

counterpart to (24). They use information from wage regressions in Psacharopoulos

25

(1994) to construct

lnw(e) = �(e);

where e is educational attainment, and where �(e) is concave and piecewise linear,

with breaks at e = 4 and e = 8; and slopes 13.4%, 10.1%, and 6.8%. Suppose that

an individual's time in market activities (education plus work) is 60 years. Then

v = e=60 is the fraction of time devoted to education. In this case, as shown in

Figure 5, the function �(e) calculated by Hall and Jones is well approximated by the

function in (24) with (1� �) �=� = 0:55. For � = 0:75; this implies that �=� = 2:2:

There remains the problem of separating � and �: The simulations here use

� = 0:75; which gives modest DRS in the human capital production technology, and

implies � = 0:75=2:2 = 0:3409:

Technology di�usion

For calibrating the remaining parameters it is useful to have in mind a \frontier"

economy. This economy has no barrier, BF = 1; and an educational subsidy of �F :

In the simulations here �F = 0:35:

The constant �0 involves units for A and H, so we can choose any convenient

normalization.11 It is convenient to choose �0 so that zbg = abg=hbg = 1 in the frontier

economy. From (13) and (14), this requires

�0 = (g + �H)�vbgF

���= (g + �H)

�1 +

1

�

1� �F1� �

�� +

rbg � g

g + �H

���: (25)

The level parameter 0 can then be calibrated to �t the growth rate of an econ-

omy that has just reduced its barrier signi�cantly. This choice is constrained only

by the requirement that condition (16) should hold for the frontier economy. Since

11To see this fact, note that a change in �0 can be o�set by changing z = a=h; �h; and 0:

26

BF = zbgF = 1; the requirement is

0 > 4 (g + �A) = 0:1960: (26)

Here the value is 0 = 0:70; which produces rapid growth after a barrier is reduced.

To summarize, the benchmark parameters are

g = 0:019; � = 0:03 � = 2; � = 0:75;

�H = 0:03; �A = 0:03; � = 0:75; � = 0:3409;

�F = 0:35; �0 = 0:1795; 0 = 0:70:

In all of the simulations the roots are real, two are negative at the high steady state,

and two are positive at the low steady state.

7. SIMULATIONS

a. Transitions to BGPs

Transitional dynamics to the BGP can be computed by perturbing the system

away from the normalized steady state�abgH ; h

bgH

�and running the ODEs backward.

Any linear combinations of the eigenvectors associated with the two negative roots,

with small weights, can be used as an initial perturbation, giving a (2-dimensional)

set of allowable perturbations.

The boundary of the basin of attraction for the BGP is found by perturbing

around the normalized steady state�abgL ; h

bgL

�with the eigenvector associated with

the single negative root.

Figures 6a and 6b display the normalized variables�abgH ; h

bgH

�at the stable steady

state as the education subsidy varies over the range � 2 [0; 0:35] and the technology

barrier varies over the range B 2 [1; 3:6] : Both values are decreasing in B and in-

creasing in �; but hbgH is more sensitive to �: The threshold in (B; �)�space beyond

which there is no BGP can also be seen on these �gures.

27

Figures 6c and 6d show (the absolute value of) the negative roots at the stable

steady state. Call these jRf j > jRsj > 0; for fast and slow. A higher barrier slows

down transitions: both roots fall in absolute value as B increases. And as the para-

meters approach the threshold where a BGP ceases to exist, the slow root converges

to zero.

Figure 7a shows the high and low steady states sJi =�abgJi; h

bgJi

�; J = H;L; for

three barriers, B1 < B2 < B3; with the education subsidy held at the level for the

frontier economy, � = 0:35: Since B1 = 1; that is the frontier economy.

For each of the lower steady states, the one-dimensional stable manifold for that

point is also displayed. Thus, an economy with policy parameters � = 0:35 and

B = Bi; with an initial condition on the solid curve through the point ssLi; converges

to that point. For any initial condition above that curve it converges to the point

ssHi; and for initial conditions below that curve it converges to its stagnation steady

state.

Increasing the barrier B has several e�ects. First, it moves the stable steady

state downward, so the levels for the BGP lag farther behind the frontier. It also

moves the lower steady state upward, Thus, a higher barrier reduces the region of

initial conditions that produce balanced growth. For B > Bcrit � 3:57 the system

does not have a balanced growth path.

Figure 7b shows the phase diagram for policy parameters of the frontier economy,

with B = 1. Notice that for initial conditions with low value for both state variables,

the adjustment paths are very at, indicating that the technology level a adjusts more

rapidly than human capital h:

Figure 8 displays the adjustment for an economy that reduces its barrier. Specif-

ically, this economy has the same education subsidy � = 0:35 as the frontier economy,

but starts with a higher technology barrier. The old barrier is Bold = B3 = 3:5 from

Figure 7, and the new one is Bnew = B1 = 1:0: The initial condition is the (stable)

28

steady state for the old parameters, the point ssH3 in Figure 7. Since � is constant

throughout, the steady state ratio z = a=h is the same on both BGPs. Both stocks

start at about 68% of the benchmark levels, ssold � 0:68� ssnew: In each panel both

the exact path (solid) and the linear approximation (broken) are displayed. (The

exact path is calculated for the �rst 26 years of the transition, and a linear approxi-

mation is used for the continuation after that period.)

Figure 8a shows the transition path in a; h�space. Over the �rst decade of the

transition, the normalized technology a grows rapidly while the normalized human

capital stock h remains approximately constant. Thus, over the �rst decade human

capital grows at about the same rate, 1.9%, as it would have under the old policy.

Over several subsequent decades the technology continues to grow, but more slowly,

and the human capital stock grows more quickly. The linear approximation is close

to the exact path over the whole transition, although it substantially overstates the

speed of adjustment.

Figures 8b - 8d display time plots for the 100 years after the policy change. Figure

8b shows the time v allocated to human capital accumulation. Interestingly, it falls

rather sharply immediately after the policy change. Two factors are at work. First,

since technology is an important input into human capital accumulation, there is an

incentive to delay the investment of time in that activity until after the complementary

input has increased. In addition, consumption smoothing provides a direct incentive

to shift the time allocation toward goods production.

Figure 8c shows the growth rate of output (consumption), which jumps from

1.9% to about 7.2% immediately after the policy change, and then falls gradually

back to the steady state level.

Figure 8d shows the technology level a, human capital level h; and consumption

(output) c relative to the frontier economy. As the phase diagram showed, the tech-

nology adjusts rapidly toward the BGP, while human capital adjusts quite slowly.

29

The shift in the time allocation just after the policy change allows consumption to

adjust rather rapidly, jumping immediately from 68% to about 73% of the value in

the frontier economy.

Figure 9 shows the transitions for two economies with di�erent initial conditions.

Economy D (the solid line) is the one from Figure 8. Thus, its initial condition is the

(normalized) steady state for an economy with B = 3:5 and � = :035: Economy E (the

dashed line) has a higher initial stock of human capital but a lower technology level.

This point could be interpreted as the BGP for a regime with an even higher initial

barrier than economy D, but also has a higher education subsidy. The initial condition

for Economy E is chosen so that it has the same productive capacity as Economy D,

it all time is allocated to goods production. Both economies are assumed to adopt

the policies of the frontier economy.

As Figure 9a shows, Economy E enjoys a much more rapid transition. It makes

a more dramatic shift in its time allocation, away from human capital accumulation

and toward production, as shown in Figure 9b, and hence it enjoys a larger immediate

jump in consumption growth, as shown in Figure 9c. As shown in Figure 9d, Economy

E's technology soon overtakes Economy D's, and its human capital and consumption

exceed D's over the entire transition path.

The experiment in Figure 9 shows that poor but well educated countries grow

rapidly when technology in ows accelerate. It does not imply that policies to promote

education are the most valuable ones, however.

Figure 10 displays the transition paths for two di�erent policies, for an economy

with �xed initial conditions. Policy 1 (solid line) involves a low barrier to technology

adoption, B1 = 1:3, but a low subsidy to education, �1 = 0:087: Policy 2 (broken

line) involves a higher barrier and a higher subsidy, B2 = 3:0 and �2 = 0:35: These

policies are constructed so that consumption|and hence welfare|on the BGP is the

same under both.

30

As Figure 10a shows, the economy enjoys a much more rapid transition under

Policy 1: technology increases very quickly. Figure 10b shows the time allocations.

Under Policy 2 the time allocation is very close to its BGP level throughout the (very

long) transition. Under Policy 1, the incentive to smooth consumption leads to a

sharp reduction in time devoted to human capital accumulation, in order to increase

production. The result is a high growth rate for output (consumption) in the �rst

decade of the transition. Under policy 2 consumption growth adjusts very gradually

to the new steady state level.

Figures 10d { 10f show the transition paths for technology, human capital, and

consumption. The two policies lead to di�erent BGP levels for a and h; but the same

level for c: Under Policy 1, human capital is lower over the whole transition, but

technology and consumption are higher. Thus, welfare is higher under Policy 1.

b. Transitions to the stagnation steady state

Our interest here is in transitions to stagnation from above. Such transitions

would be observed in countries which had enjoyed growth above the stagnation level,

by reducing their barrier to technology and increasing subsidies to education, and

then reversed those policies. As shown in Figure 3, a number of African countries

have had negative growth in per capita GDP over the four decade period 1960-2000.

Even more have had shorter episodes|one or two decades|of negative growth.

Linearizing around the stagnation steady state is problematic, for two reasons.

First, the law of motion for A has a kink at Ast: Only the region of the state space

where A � Ast is of interest, but kink nevertheless comes into play if A reaches Ast

before H reaches Hst. In addition, since the frontier technology W appears in (2),

the system is not autonomous.

A computational method that deals with both issues is the following iterative

procedure. Begin by conjecturing a path for A(t): Then use (3), (20) and the equation

31

for _�H to compute H(t) and �H(t); by linearizing in the usual fashion around the

steady state, using the eigenvector associated with the (single) negative root to per-

turb away from that point, and integrating backward. Use the resulting solution for

H(t) in (2) to revise the conjecture for A(t); and repeat until the solution converges.

[To be completed.]

8. CONCLUSIONS

The model developed here has a number of empirical implications. First, as

shown in Proposition 1, economies with su�ciently unfavorable policies|a combi-

nation of high barriers to technology in ows and low subsidies to human capital

accumulation|have no BGP. Thus, the model implies that only middle and upper

income economies should grow like the technology frontier over long periods. Low

income economies can grow faster (if they have reduced their barriers) or slower (if

they have raised them) or stagnate. The empirical evidence is at least consistent with

this prediction. Higher income countries grow at very similar rates over long periods,

while low income countries show more heterogeneity (in cross section) and variability

(over time).

Second, unfavorable policies reduce the level of income and consumption on the

BGP, and also shrink the set of initial conditions for which an economy converges to

that path. Thus, the model predicts that a middle-income countries that grow with

the frontier should have displayed slower growth during their transitional phases.

Third, as shown in Figure 8, the transition to a (higher income) BGP features

a modest period of very rapid TFP growth, accompanied by rapid growth in income

and consumption. This initial phase is followed by a (longer) period during which

human (and physical) capital are accumulated. Income growth declines toward the

rate of frontier growth as the income level approaches the frontier level. Thus, the

model has predictions for some features of the transition paths of growth `miracles.'

32

Fourth, as shown in Figure 9, low income countries with higher human capital

are better candidates to become growth miracles, since TFP and income grow more

rapidly in such economies.

Finally, as shown in Figure 10, the model suggests that policies stimulating

technology transfer are more e�ective in accelerating growth than policies stimulating

human capital accumulation. Investment in human|and physical|capital responds

to returns, and those returns are high when technology is growing rapidly. Thus, the

empirical association between high investment rates and rapid growth is not causal:

technology drives both.

33

APPENDIX

A. Equilibrium conditions

The Hamiltonian for the household's problem is

H =C1��

1� �+ �H

��0v

�A�H1�� � �HH�

+�K�(1� v)w(H) + (R� �K)K � C + v�w(H)� �

�;

where to simplify the notation the subscript i's have been dropped. Taking the �rst

order conditions for a maximum, using the fact that in equilibrium � = v�w(H) and

H = H; substituting for w(H) and its derivative, and simplifying gives

�H��0v��1 = �K (1� �) w

�A

H

����; (27)

C�� = �K ;_�H�H

= �+ �H � �0

�A

H

�� �(1� �) v� +

1� �

1� ��v��1 (1� v)

�;

_�K�K

= �+ �K �R;

_H

H= �0v

�

�A

H

��� �H ;

_K

K=

�K

L

���1� C

K� �K ;

with _A=A in (2). The transversality conditions are

limt!1

e��t�K(t)K(t) = 0 and limt!1

e��t�H(t)H(t) = 0: (28)

The system of equations in (1), (2), (27) and (28) completely characterizes the com-

petitive equilibrium, given initial values for the state variables W;A;H;K and the

policy parameters B; �:

The law of motion for A requires H=W and A=W to be constant along a BGP,

so A and H must also grow at the rate g: Since the production functions for e�ective

34

labor in (4) and for goods have constant returns to scale, the factor inputs L and K

also grows at the rate g; as do output and consumption, and the marginal utility of

consumption grows at the rate ��g: The FOC for consumption then implies that �Kgrows at the rate ��g: The factor returns R and w are constant along a BGP, and

the FOC for time allocation implies that �H grows at the same rate as �K :

The normalized conditions in (10) follow directly from (2) and (27).

B. Linear approximations and stability

De�ne the constants

� = � (1� �)� �; � � �1� �

1� �;

M � rbg � g

g + �H; M + 1 =

�H + rbg

�H + g;

vbg =

�1 +

1

�(M + �)

��1; �

�1

vbg� 1�=M + �;

�bg3 =

�� � 1� �

1=vbg � 1

��1�� � 1� ��

M + �

��1=

� +M

(� � 1) (� +M)� ��< 0;

1

�bg3= � � 1� ��

M + �:

and

�bg2 � � ���vbg��1

1� � � �+ � (vbg)�1

= � � 1� �+ � � 11� � + � (1=vbg � 1)

= � � 1� �+ � � 11 +M

:

35

the log deviations

x1 = ln�a=abg

�; x2 = ln

�h=hbg

�; x3 = ln

��h=�

bgh

�:

Take a �rst-order approximation to (17) to get

v � vbg

vbg= �3 [�x1 � (� + �)x2 � x3] ;

Then linearize (??) to �nd that

_x1 � (g + �A)

"� abgJ1� abgJ

x1 + x2

#;

_x2 � (g + �H)

�� (x1 � x2) + �

v � vbg

vbg

�= (g + �H)

n� (x1 � x2) + ��bg3 [�x1 � (� + �)x2 � x3]

o;

_x3 � ��rbg + �H

� �� (x1 � x2) + �

bg2

v � vbg

vbg

�= �

�rbg + �H

�n� (x1 � x2) + �

bg2 �

bg3 [�x1 � (� + �)x2 � x3]

o:

Hence 0BBB@_x1

_x2

_x3

1CCCA �

0BBB@�c1J c2 0

c3 �c3 � �c4 �c4�c6 c6 + �c5 c5

1CCCA0BBB@

x1

x2

x3

1CCCA ;

where

c1J = (g + �A) abgJ =�1� abgJ

�;

c2 = g + �A;

c3 = (g + �H) (� + ��3�) ;

c4 = (g + �H) ��3;

c5 =�rbg + �H

��2�3;

c6 =�rbg + �H

�(� + �2�3�)

36

The stability of each steady state depends on the roots of the associated charac-

teristic equation,

0 = det

0BBB@�R� c1J c2 0

c3 �R� (c3 + �c4) �c4�c6 c6 + �c5 �R + c5

1CCCA= �R3 + (c5 � c3 � �c4 � c1J)R

2

� [ciJ (c3 + �c4)� c1Jc5 � (c3 + �c4) c5]R

+c1J (c3 + �c4) c5 + c2c4c6

� (R + c1J) c4 (c6 + �c5) + c2c3 (R� c5)

= �R3 + (m1 � c1J)R2 + (m2 + c2c3 +m1c1J)R

+m2 (c1J � c2) :

where

m1 � c5 � c3 � �c4; m2 � c3c5 � c4c6:

Write this equation as

0 = J(R) � R3 � A1JR2 � A2JR� A3J ; J = H;L;

where

A1J � m1 � c1J ;

A2J � m2 + c2c3 +m1c1J ;

A3J � m2 (c1J � c2) ; J = H;L:

Since �3 < 0 and �2 � � < 0; it follows that

m2 = (�H + g)��H + rbg

��3� (�2 � �) > 0;

37

and since abgH > 1=2 > abgL ; it follows that c1H > c2 > c1L: Hence

H(0) = �A3H < 0; L(0) = �A3L > 0;

so H has at least one positive real root, and L has at least one negative real root.

The other roots of H are real and both are negative if and only if

H(I) > 0; for some I < 0:

This condition holds if and only if H has (real) in ection points, the lower one occurs

at a negative value IH , and H takes a positive value at this point. Similarly, the

other roots of L are real and both are positive if and only if

L(IL) < 0; for some IL > 0:

The in ection points of J satisfy the quadratic equitation

0 = 0J(I) = 3I2 � 2A1JI � A2J ; J = H;L;

so

IJ =1

3

hA1J �

pDJ

i; where DJ � A21J + 3A2J :

Hence J has real in ection points if and only if DJ > 0; or

0 < A21J + 3A2J

= (m1 � c1J)2 + 3 (m2 + c2c3 +m1c1J)

= (m1 + c1J)2 +m1c1J + 3 (m2 + c2c3) :

As shown above, m2 > 0: In addition, since �3 < 0; it follows that c2c3 > 0 if � � 0;

or

� (1� �) < �: (29)

If � � 1 this condition always holds. If � < 1 it puts a lower bound on �=�:

38

Since c1J > 0; for the �nal term it su�ces to show that

0 < m1 = c5 � c3 � �c4

= (�H + g) �bg3

��H + rbg

�H + g�bg2 �

�

�bg3� � (� + �)

�;

or

0 >m1

(�H + g) �3= (M + 1) (� � 1)� (�+ � � 1)

���� � 1� ��

M + �

�� � [� (1� �)� � + �]

= M (� � 1) + � ��

M + �� � (� � 1) (1� �)� �

= M (� � 1)� �

�� ��

M + �+ (1� �) (� � 1) + 1

�= M (� � 1)� �

��

M

M + �+ � (1� �)

�:

Since � < 1; the �rst term on the RHS is negative. The second is also negative if

� (1� �) < �M

M + �: (30)

If � � 1; this condition holds for any �: If � < 1; it puts an upper bound on �: Hence

DJ > 0, J = H;L; if (29) and (30) hold, and � � 1 insures both.

To complete the argument that there are two negative (real) roots at abgH ; we

must show that H(IH) > 0 for the in ection point

IH �1

3

�A1H �

pDH

�< 0:

The required condition is

0 < I3H � A1HI2H � A2HIH � A3H

= (IH � A1H) I2H � A2HIH � A3H :

39

The �rst term is positive if

0 < 3 (IH � A1H) = �2A1H �pDH ;

which requires A1H < 0 and

DH = A21H + 3A2H < 4A21H ;

or

A2H < A21H ;

The sum of the last two terms positive if

3A3H < �3A2HIH = �A2H�A1H �

pDH

�:

These conditions hold for the benchmark calibration.

To complete the argument that there are two positive (real) roots at abgL ; we must

show that L(IL) < 0 for

IL �1

3

�A1L +

pDL

�> 0:

The required condition is

0 > I3L � A1LI2L � A2LIL � A3L

= (IL � A1L) I2L � A2LIL � A3L

=1

3

�pDL � 2A1L

�I2L � A2LIL � A3L:

For the benchmark calibration the term in parenthesis is positive. Since A3J < 0; the

last term is also positive. Nevertheless, the whole expression is (slightly) negative.

[Question: are there examples where the inequality fails?]

Check sum of roots for � = 0; x1 � 0.|

40

For the case where x1 = _x1 � 0 the system is0@ _x2

_x3

1A �

0@ �c3 � �c4 �c4c6 + �c5 c5

1A0@ x2

x3

1A ;

so the roots satisfy

0 = [�R� (c3 + �c4)] [�R + c5] + c4 (c6 + �c5)

= R2 �m1R�m2:

Since m2 > 0; the roots are real and of opposite sign.

For � = 0; the household's optimization problem is undistorted, so the sum of

the roots should equal the discount rate,

R1 +R2 = m1 = �� g (1� �) = rbg � g;

orm1

(� + g) �bg3=rbg � g

� + g

1

�bg3=M

�bg3:

Use the expression above for m1 to �nd that this condition holds, since

M (� � 1)� ��M

M + �=M

�bg3:

41

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45

1820 1840 1860 1880 1900 1920 1940 1960 1980 20000

10

20

30

40

50

60

70

80

90Figure 1: doubling times for per capita GDP, 55 countries

Year reached $2000

Yea

rs to

rea

ch $

4000

Western Europe & OffshootsEastern Europe Latin America Asia Africa

theoretical limit

for 2006

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

Figure 2: catching up and falling behind, 1960-2000

1960 per capita GDP relative to U.S.

2000

per

cap

ita G

DP

rel

ativ

e to

U.S

.

W. Europe & Offshoots E. Europe Latin America Asia Africa

0 500 1000 1500 2000 2500 3000 3500 4000 45000

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000Figure 3: Asia and Africa, excluding 6 successes

1960 per capita GDP

2000

per

cap

ita G

DP

E. & S. Asia

W. Asia

Africa

Malaysia

Thailand Turkey

Botswana

China Indo

Sri Lanka

W. Bank & Gaza

Tunsia

Iraq

4% growth 3% growth

2% growth

1% growth

0% growth

1 1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

B

Figure 4: technology ratios abg, for σ2 < σ

1

abgH

abgL

σ1

σ2

0.05 0.1 0.15 0.2 0.25

0.2

0.4

0.6

0.8

1

1.2

1.4

Figure 5: approximation of Mincer returns

v = e/60

retu

rn to

edu

catio

n

y = b0 + .55 ln(v)

y = Φ(e)

e = 4 e = 8 e = 12

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Figure 7a: basins of attraction for BGPs

a

h

σ = 0.35B

3 = 3.5

B2 = 2.4

B1 = 1

o

o

oo

o

o

o

ssH1

ssH2

ssH3

sscrit

ssL3

ssL2

ssL1

0.65 0.7 0.75 0.8 0.85 0.90.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 7b: phase diagram for B1

a

h

oss

H1

0.5 0.6 0.7 0.8 0.9 10.5

0.6

0.7

0.8

0.9

1Figure 8a: catching up, the transition

o

o

ssold

ssnew

2 414

26

2 4

14

26

Bold = 3.52

Bnew = 1

a

h

0 20 40 60 80 1000.08

0.1

0.12

0.14

0.16

0.18

0.2

time

v

Figure 8b: time allocation

exact transition path

linear approximation

Balance Growth Path

0 20 40 60 80 1000.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08Figure 8c: growth rate (output)

time0 20 40 60 80 100

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Figure 8d: a, h, and c, relative to BGP

time

a/aBGP

c/cBGP

h/hBGP

0.5 0.6 0.7 0.8 0.9 10.5

0.6

0.7

0.8

0.9

1Figure 9a: Two transitions

a

ho

o

o

ss

E

D2 5

20

502 5

20

50

0 10 20 30 40 500.05

0.1

0.15

0.2

time

v

Figure 9b: time allocation

Economy D

Economy E

0 10 20 30 40 500

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18Figure 9c: growth rate (output)

time0 10 20 30 40 50

0.5

0.6

0.7

0.8

0.9

1Figure 9d: a, h, c relative to BGP

time

a/ass

c/css

h/hss

0.5 0.55 0.6 0.65 0.7 0.75 0.8

0.3

0.4

0.5

0.6

0.7

Figure 10a: Transitions for two policies

a

h

o

o

ss2

ss1

5 12 3060 100

5 1230

60

100

0 10 20 30 40 50 60 70 80 90 100

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time

v

Figure 10b: time allocations

B1 = 1.3097

B2 = 3

σ1 = 0.086723

σ2 = 0.35

0 10 20 30 40 50 60 70 80 90 1000.01

0.02

0.03

0.04

0.05Figure 10c: growth rates (output)

time

policy (B1, σ

1)

policy (B2, σ

2)

steady state

0 10 20 30 40 50 60 70 80 90 1000.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8Figure 10d: technology levels, a

time

a

0 10 20 30 40 50 60 70 80 90 1000.2

0.3

0.4

0.5

0.6

0.7

0.8Figure 10e: human capital, h

time

h

0 10 20 30 40 50 60 70 80 90 1000.3

0.35

0.4

0.45

0.5

0.55

Figure 10f: consumption, c

time

c

policy (B1, σ

1)

policy (B2, σ

2)