Catching Up and Falling Behind

Nancy L. Stokey

University of Chicago

October 22, 2009

Preliminary Draft

Abstract

This paper studies the interaction between technology, which flows in

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

engines of growth in a developing economy. Technology inflows are modeled as

a pure external effect, with the rate of inflow governed by a policy parameter,

a barrier, and by the average level of local human capital. Human capital, a

private input, is accumulated using time and the local technology as inputs.

The model produces two distinct 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 condition is not too far from the

frontier, the economy displays sustained growth, keeping pace with the tech-

nology frontier. Otherwise the economy stagnates in the long run, converging

to a minimal technology level that is independent of the world frontier. Transi-

tion paths are computed for economies that reduce their barriers to technology

adoption and/or change their subsidies to education.

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

and Alwyn Young for helpful comments.

1

This paper develops a framework that is consistent with the available evidence on

the underlying sources of growth in developing countries, and that can accommodate

the enormous observed differences across countries and over time in the income levels

and growth rates of particular economies. The sources of growth are technology,

which flows in from abroad, and human capital, which is accumulated domestically,

and the resulting model can generate both catching up and falling behind.

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 inflow of new technologies into particular countries. Technology

inflows here are modeled as a pure external effect, with the rate of inflow 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, reflecting the fact that a larger pool of untapped ideas

offers more opportunities for the adopting country. It is also increasing in the local

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

sorb new ideas. Finally, the “barrier” reflects tariffs, internal taxes, capital controls,

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

technologies.

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

human capital. There are many reasons why human capital might be important in

facilitating the inflow of ideas. Better educated domestic entrepreneurs are better

able to identify new products and processes that are suitable for the local market.

In addition, a better educated workforce makes a wider range of new products and

processes viable for local production, an important consideration for both domes-

tic entrepreneurs interested in producing locally and foreign multinationals seeking

attractive destinations for direct investment.

2

In the model here human capital is a private input into production, accumulated

with a technology that uses time (current human capital) and the local technology as

inputs. In addition, human capital has an external effect through its impact on the

rate of technology inflow. Because of this externality, public subsidies to education

are an important determinant of the long run behavior of the economy.

The model produces two distinct 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 affect 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 sufficiently high or the subsidy to ed-

ucation is sufficiently low (or some combination), 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.

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

3

the same pattern of sustained growth as middle income countries. They can change

their policies and start the transition to a BGP. In this case the dynamics of the

transition depend on the new policy parameters and on the level of local human

capital when the barrier is reduced. Low post-reform barriers and high initial levels

for local human capital produce fast transitions: growth miracles.

Low income countries that do not lower their barriers eventually stagnate, falling

ever farther behind the frontier. Moreover, economies that enjoyed technology inflows

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 the stagnation

steady state.

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

on the importance of technology inflows as a potential source of growth. It also

documents the fact that many countries are not enjoying these inflows. 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. In section 5 the model is calibrated, and in section 6

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

their subsidies to education.

1. EVIDENCE ON THE SOURCES OF GROWTH

Five types of evidence point to conclusion that differences in technology are

critical for explaining differences 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 countries almost invariably pro-

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

this conclusion.

4

duce large increases in total factor productivity (TFP). And if the accounting is done

in a way that incorporates the effect of TFP growth in stimulating the accumulation

of human and physical capital, its role becomes even larger. Although the measured

Solow residuals may include the influence of other (omitted) factors as well, the search

for the missing factors has been extensive, covering a multitude of potential explana-

tory variables, many countries, many time periods, and many years of effort. It is

difficult to avoid the conclusion that technical change is a major ingredient.

Second, development accounting exercises have produced similar conclusions,

finding that differences in physical and human capital explain only a modest portion

of the differences in income levels across countries. For example, Hall and Jones

(1999) find that of the 35-fold difference in GDP per worker between the richest

and poorest countries, inputs–physical and human capital per worker–account for

4.5-fold, while differences 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

per manhour. No adjustment is made for potential differences 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 figure for TFP

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

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

reports that many studies find that immigrants’ earnings are within 25% of earnings of native-born

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

5

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

country differences.

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

Wolff (1988)), the data nevertheless make an important message: the OECD coun-

tries (and a few more) seem very clearly to enjoy a common technology. It is hard

to explain in any other way the harmony–over many decades–in both their in-

come levels and growth rates. Moreover, as Prescott (2002, 2004) and Ragan (2006)

have shown, much of the persistent differences in income levels can be explained by

differences in fiscal policy that affect work incentives.

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

‘late bloomers.’ As first 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, updated to include data up to 2006, is

displayed in Figure 1. Each point in this figure represents one of the 55 countries

that have reached a per capita GDP of $4000 by 2006. On the horizontal axis is the

year that the country first reached $2000, and on the vertical axis is the number of

years required to first reach $4000.

As the figure shows, there is a strong downward trend. Countries that arrived at

the $2000 figure later have doubled their incomes more quickly. An obvious interpre-

tation is that advances in world technology provide an ever-growing stock of ideas.

Later developers have enjoyed the advantage of fishing from a richer pool of ideas.

The figure does have a built-in bias, which should be noted. Among countries

that have more recently reached the $2000 figure, many have not yet reached $4000. In

particular, the slow growers have not yet reached that goal. The dotted line indicates

6

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.

A fifth and final 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 implies that an especially bad decade in terms of growth rates is likely

to be followed by a good one. But mean reversion (recovery from a disaster) is not a

miracle.

Even if a more stringent criterion is used, two decades instead of one, growth

‘miracles’ are likely to be reversed. Table 1 lists 26 countries that had at least one

episode of growth in per capita GDP exceeding an average of 5% per year over a

period of 20 years. The first column shows the maximum value for the difference in

log income over twenty-year periods, less the 20× 005 = 100 criterion for inclusionin the list. Thus, it measures the extent to which the country cleared the hurdle. The

second column shows the terminal dates for which the hurdle was surpassed. Many

dates indicate the hurdle was cleared for many 20-year stretches.

The third and fourth column show per capita GDP relative to the U.S. in 1950

and 2006. It 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 that gave back most or all of their gains are Bulgaria, Yugoslavia, Jamaica,

North Korea, Iran, Gabon, Libya, and Swaziland.

The last column categorizes countries as ‘sustained miracles’ only if they managed

7

to gain substantial ground, relative to the U.S., over the 55-year period. Twelve

countries are clearly in this group, the five Western European examples, five of the

East Asian examples (Japan and the four tigers–Taiwan, Hong Kong, Singapore, and

Korea), Puerto Rico, and Israel. The jury is still out on China, Thailand, Malaysia,

and Botswana.

Although each piece of evidence has individual weaknesses, taken together they

make a strong case for the importance of international technology spillovers in keeping

technology 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 and East Asian (plus Israel). 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. Six Asian success stories (Taiwan, Hong Kong Singapore, ....) are omitted,

since they significantly alter the scale. Here the plot shows per capita GDP in 2000

against per capita GDP in 1960, both in levels. The rays from the origin are loci

corresponding to various average growth rates. Keeping pace with the U.S. over

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

ground relative to the U.S., those above the 2% growth line, is modest. The number

that have fallen behind is much larger. A shocking number, mostly in Africa, have

8

suffered negative growth over the whole period. Since many of these countries became

independent in the 1960’s, the figures clearly dispel the idea that self government is

necessarily beneficial to economic performance.

One factor, neglected here, deserves special mention: the contribution of the

misallocation of resources across sectors or firms to TFP differences across countries.

Such misallocation can result from financial market frictions, from frictions that im-

pede labor mobility, or from taxes (or other policies) that distort factor prices.

It is well documented that in most developing economies, TFP is substantially

lower in agriculture than it is in the non-agricultural sector.3 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 effect of this shift on aggregate

TFP, which is significant, 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 firms within that sector. For example, Bartelsman,

Haltiwanger, and Scarpetta (2008) find that manufacturing TFP in the western Euro-

pean countries suffers due to resource misallocation (relative the U.S.), and that the

effect is even stronger for central and eastern Europe. However, in the two countries

(Slovenia and Hungary) for which they report time series, most of the (substantial)

TFP gains both countries enjoyed during the 1990s came from other sources: im-

provements in allocative efficiency played a minor role.

Hsieh and Klenow (2009) find bigger effects in China and India. For China they

find that improvements in allocative efficiency contributed about 1/3 of the 6.2%

TFP growth in manufacturing over the period 1993-2004. In India they find that

allocative efficiency declined over the same period, at about 1.8% per year, reducing

TFP growth in manufacturing to only 0.3% per year. Although the gain in China is

3See Caselli (2005) for recent evidence that TFP differences across countries are much greater in

agriculture than they are in the non-agricultural sector.

9

substantial, it is a modest boost relative to total TFP growth in China over the last

25 years. More importantly, it is a one-time gain, not a recipe for sustained growth.4

2. THE MODEL

The model is a variant of the technology diffusion model first put forward in

Nelson and Phelps (1966) and subsequently developed elsewhere in many specific

forms.5 It has several components. There is a frontier (world) technology () that

grows at a constant (exogenously given) rate,

() = () (1)

where 0 In addition each country has a local technology () which affects

returns to physical and human capital. Growth in is exogenous to the individual

household in country but it depends on the average level of local human capital.

Thus, there is a feedback from households decisions that govern the accumulation of

human capital to the rate of technology inflow.

Specifically, the technology () in country grows at a rate that depends on

4In addition, it is not clear what the standard for allocative efficiency should be in a fast-growing

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

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

model has the property that the stationary distribution across firms is sensitive to the fixed 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 differences across countries reflect distortions that

affect the allocation of factors, or if they represent differences 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 different in a fast-growing economy

(like China) and a mature, slow-growing economy (like the U.S.).5See Benhabib and Spiegel (2005) for an excellent discussion of the long-run dynamics of various

versions.

10

three factors: the ratio of its current level to the level of the frontier technology ,

the ratio of the average human capital in country to the frontier technology

and a ‘barrier’ that represents policies in country that impede technology inflows.

Technology depreciates at the rate 0 but there is a floor on its level. The

floor allows the economy to have a ‘stagnation’ steady state in which the technology

is constant. In particular,

= 0 if = and

0

µ1−

¶

(2)

=0

µ1−

¶

− otherwise.

Above the floor technology growth has a logistic form: its growth rate is pro-

portional to the relative gap 1− between its current level and the frontier. A

larger gap–a lower value for –means there is a larger pool of technologies that

have not yet been adopted, which speeds up the adoption rate. A higher level of

human capital relative to the frontier technology, , also speeds up adoption.

Thus a higher value for the frontier technology has two effects. It increases

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

capacity, which tends to retard growth. The first effect dominates if the ratio

is high and the second if is low. Benhabib and Spiegel (2005, Table 2) find

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.

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

anything that impedes access to or adoption of new ideas, or reduces the profitability

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

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

or civil conflict that impedes the flow of people and ideas across border. Notice that

11

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

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

Next consider a typical household in country The household is endowed with

one unit of time (a flow), which it allocates between human capital accumulation

and goods production. The technology for human capital accumulation uses the

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

In particular,

() = 0()()

()1− − () (3)

where is time devoted to human capital accumulation, 0 is the depreciation

rate for human capital, and 0 1. This technology has CRS jointly in

the stocks () and DRS in the time input. The former fact permits sustained

growth at a constant rate, and the assumption that 0 rules out the possibility of

endogenous growth in the absence of technology diffusion (without increases in )

The assumption 1 insures that time allocated to human capital accumulation is

always strictly positive.

Raw labor–time allocated to production–is augmented by both technology and

human capital to produce effective labor input. The augmentation function, which

is Cobb-Douglas, is the same across economies and over time. Specifically, the labor

offered by the representative household in country is augmented by the country

technology , which it takes as exogenous, and its own human capital Thus its

effective labor is

() = [1− ()] ()

1− () (4)

where (1− ) denotes time allocated to goods production.

The technology for goods production is also the same across economies and over

time, and it is also Cobb-Douglas, with physical capital and effective labor as

inputs,

=

1−

12

Hence factor returns are

() =

µ

¶−1 (5)

() ≡ (1− )

µ

¶

where is the return to capital and is the return to a unit of effective labor. The

wage for a worker with human capital is therefore

( ) =

1− (6)

Many different ownership structures are equivalent. Here we will assume that

households accumulate capital, firms rent capital and labor services from households,

and households use income (goods) for consumption and investment.

In addition, we will be interested in studying the role of subsidies to education.

To this end, suppose that time spent investing in human capital is subsidized at the

rate ( ), where is the average human capital level in the economy and

∈ [0 1) is the subsidy rate. Thus, the subsidy is a fraction of the current averagewage. The subsidy is financed with a lump sum tax and the government’s budget

is balanced at all dates. Hence the budget constraint for the household is

() = (1− )() + ( − ) − + ()− (7)

where 0 is the depreciation rate for physical capital.

Household’s problem

The representative household has constant elasticity preferences, with parameters

0 Hence the household’s problem is

max{()()}∞=0

Z ∞

0

−1−()1−

s.t. (3) and (7),

13

given 00 and the paths©() () ( )() ()

ª where to simplify the

notation the subscript ’s have been dropped.

The Hamiltonian for the household’s problem is

H =1−

1− + Λ

£0

1− − ¤

+Λ

£(1− )() + (− ) − + ()−

¤

Taking the first order conditions for a maximum, using the fact that in equilibrium

= () and = substituting for () and its derivative, and simplifying

gives

Λ0−1 = Λ (1− )

µ

¶− (8)

− = Λ

Λ

Λ

= + − 0

µ

¶ ∙(1− ) +

1−

1− −1 (1− )

¸

Λ

Λ

= + −

= 0

µ

¶

−

=

µ

¶−1−

−

In equilibrium is as in (2), with = The transversality conditions are

lim→∞

−Λ()() = 0 and lim→∞

−Λ()() = 0 (9)

The system of equations in (2) and (8) completely characterizes the competitive

equilibrium for the economy, given initial values for the state variables and

, and given the policy parameters and Two types of behavior are possible in

the long run.

If the barrier is low enough and/or the subsidy to education is high enough,

balanced growth is possible. Specifically, for some initial conditions the economy

14

converges asymptotically to a BGP, along which and all grow at the rate

Economies of this type are studied in section 3.

But the set of initial conditions for which balanced growth occurs varies with

the policy parameters and For any fixed the trapping region for balanced

growth shrinks with increases in , at some point disappearing altogether. Thus,

if is sufficiently large and/or is sufficiently low, or if the initial values for

and are sufficiently small relative to the economy stagnates in the long run.

In these economies the technology level converges to its stagnation level the

stocks of human and physical capital adjust accordingly to and consumption

converges to a constant level . Economies of this sort are studied in section 4.

3. CATCHING UP: ECONOMIES THAT GROW

Consider BGPs where the time allocation 0 is constant and the state variables

grow at constant rates. By assumption grows at the constant rate The

law of motion for requires and to be constant along a BGP, so and

must also grow at the rate Consequently the law of motion for human capital

must have exponents that sum to one, as in (3). If the function producing effective

labor also has exponents that sum to one, as in (4), then effective labor also grows

at the rate Since the production function for goods has constant returns to scale,

physical capital also grows at the rate as do output and consumption, and the

marginal utility of consumption grows at the rate − The FOC for consumptionthen implies that Λ grows at the rate − The factor returns and are constantalong a BGP, and the FOC for time allocation implies that Λ grows at the same

rate as Λ

To study economies that grow, it is convenient to define the ratios

() ≡ ()

() () ≡ ()

()

15

() ≡ ()

() () ≡ ()

()

() ≡ Λ()

−() () ≡ Λ()

−()

Along a BGP these normalized variables are constant, so the transversality conditions

in (9) hold if and only if

(1− ) (10)

This (standard) condition insures that the discounted value of lifetime utility is finite.

From (2) and (8), the conditions for an equilibrium with growth are

0−1 = (1− )

³

´− (11)

− =

= + + − 0

³

´∙(1− ) +

1−

1−

µ1

− 1¶¸

= + + −

= 0

³

´− −

= −1 −

− −

=

0

(1− )− −

where

≡ = (1− ) 1− (12)

= −1 = (1− )

a. The steady state of the normalized economy

Along a BGP the normalized variables and so on are constant, as is the

time allocation The interest rate is

= +

16

so (10) implies The interest rate determines the input ratio ,

= ¡¢−1

= +

which then determines the return to effective labor

To determine the time allocation along the BGP, use the third and fifth equations

in (11) to find that

=

∙1 +

1

1−

1−

µ +

−

+

¶¸−1 (13)

Since the second term in brackets is positive and ∈ (0 1) Notice thattime allocated to human capital accumulation is increasing in the subsidy with

→ 1 as → 1 Note that is also increasing in 1 − the share for human

capital in producing effective labor from and and in the exponent on in

the production function for human capital. A higher value for 1 − increases the

sensitivity of the wage rate () to (private) human capital, increasing the incentives

to invest. A higher value for reduces the force of diminishing returns. Finally, is

increasing and reflecting the fact that human capital grows at the rate along

the BGP, and that the time allocation must offset depreciation.

The fifth equation in (11) then determines the ratio of technology to human

capital,

≡

=

µ +

0 ()

¶1 (14)

Hence this ratio is decreasing in the subsidy rate Note that and do not

depend on the barrier

The last equation in (11) determines the BGP levels , if any exist, as solutions

to the quadratic

¡1−

¢= ( + )

0 (15)

The solutions are as follows.

17

Proposition 1: If

( + )

01

4 (16)

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

lower solutions and

with

0

1

2

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

Figure 4 displays the solutions as functions of for two values of and fixed

values of the other parameters. As shown there, with fixed a small increase in

moves both solutions toward the value 1/2 For fixed a small decrease in which

increases shifts the curve to the left, also moving the solutions toward the value

1/2. Hence a smaller subsidy shrinks the range for where a solution exists.

Notice that the higher solution has the expected comparative statics, increas-

ing as the barrier falls or the subsidy rises, while the lower solution has the

opposite pattern. As we will see later, for reasonable parameter values is stable

and is not. Therefore, since

∈ [12 1] this model produce BGP productivity

(and income) ratios of no more than two across growing economies.6 Poor economies

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

If a BGP exists, changes in do not affect the growth rate , the interest rate

the time allocation or the ratio 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 differently, an economy with a higher barrier looks like its neighbor

with a lower one, but with a time lag.

6Other 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.)

18

b. Transitional dynamics

The normalized system has three state variables, and and two costates,

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 and Then = 1 = 0

and all output is consumed, so

= (1− ) 1−

and − takes the place of The first equation in (11) then implies that the time

allocation satisfies

−1 (1− )=1−

0∆−(∆+)−1 (17)

where

∆ ≡ (1− )−

The transitional dynamics are then described by

=

0

(1− )− ( + ) (18)

= 0

³

´− ( + )

=

¡ +

¢− 0

³

´∙(1− ) +

1−

1−

µ1

− 1¶¸

with given by (17). The stability properties of this system determine the long run

behavior of the economy. The Appendix describes conditions under which at the

roots are real and two are negative, and at there is one negative real root, and two

that have positive real parts. That is the pattern in all of the simulations reported

here. Under these conditions the higher steady is locally stable: for any pair of

initial conditions 0 0 in the neighborhood of³

´, there exists a unique initial

19

condition 0 for the costate with the property that the laws of motion in (18) produce

asymptotic convergence to the steady state. Transitional dynamics near this steady

state can be computed by perturbing the (3-dimensional) system in (18) away from

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

manifold of initial values for which the system converges to³

´ This manifold,

which defines the boundary of the trapping region for the stable (high) steady state,

is found by perturbing with the eigenvector associated with the single negative root.

The dynamics will be described in more detail in section 7.

4. FALLING BEHIND: ECONOMIES THAT STAGNATE

In economies where the barrier is sufficiently high and/or the educational

subsidy is sufficiently 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 sufficiently 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

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

In the stagnation steady state the technology level = is constant. Con-

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

= The stocks of physical and human capital adjust accordingly, and the time

20

allocation is constant. The stagnation levels ( Λ) satisfy

Λ0

−1 =¡¢−

(1− ) (1− )

µ

¶µ

¶−

+ = 0

µ

¶ ∙(1− )

¡¢+1−

1− ¡¢−1 ¡

1− ¢¸

+ =

µ

¶−1

= 0¡¢ µ

¶

=¡1−

¢ ¡¢ ¡

¢1−

=¡

¢ ¡¢1− −

The steady state time and input ratio, the analogs of (13) and (14), are

=

∙1 +

1

1−

1−

µ +

¶¸−1 (19)

≡

=

µ

0 ()

¶1 (20)

Thus, the time allocation and input ratio in a stagnation steady state are like those

along the BGP in a growing economy, except that the interest rate is and there is

no growth.

Comparing (13) and (19), we find that for economies with the same educational

subsidy , more time is allocated to human capital accumulation in the growing

economy, if and only if

( − 1)

If is sufficiently large, the low willingness to substitute intertemporally discourages

investment in the growing economy. For = and = 2 both economies devote

the same fraction of time to human capital accumulation, =

21

Comparing (14) and (20), we see that in economies with the same time allocation,

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

=

µ1 +

¶1 1

b. Transitions to stagnation

To study dynamics near the stagnation steady state, set = 0 in (8) to eliminate

physical capital, as before. This gives = 1 and

= (1− )1−

Then using − for Λ we find that the time allocation satisfies the analog of (17),

−1 (1− )=1−

0∆−(∆+)Λ−1 (21)

where as before ∆ ≡ (1− )− Then from (2) and (8), along the transition path,

=

0

µ1−

¶

− or = 0 (22)

= 0

µ

¶

−

Λ

Λ

= + − 0

µ

¶

∙1− +

1−

1−

1−

¸

Equations (21) and (22) are similar to the analogous system (17) and (18) for

the BGP, except that the state variables and are not normalized, the growth

rate is zero, and the interest rate is =

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.

22

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

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

where ≥ is of interest, but the second branch of the law of motion nevertheless

comes into play if reaches before reaches . That is, approaches to the

stagnation steady state ( ) are likely to occur with = constant and

falling. In addition, since the world technology appears, the system in (22) is not

autonomous.

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

procedure. Begin by conjecturing a path for () Then use (21) and the second

and third equations in (22) to compute () and Λ() These can be obtained by

linearizing in the usual fashion, using the eigenvector associated with the (single)

negative root to perturb away from the steady state, and integrating backward. Use

the result for () revise the solution for () and continue iterating.

5. DIGRESSION ON GROWTH ACCOUNTING

Consider a world with many economies, = 1 2 each with its own policies

capital taxes, and so on. In each economy output per capita at any date is

() = ()©()

()1− [1− ()]

ª1−

The income shares for capital and labor, and 1− could be obtained from NIPA

data for any country. Suppose in addition that can be estimated. A method for

doing so is described below.

Assume that all differences in time allocation between production and human

capital accumulation, all differences in take the form of differences in labor force

participation. Then each individual is engaged in only one activity, and hours per

worker are the same across countries. Let

() ≡ ()

1− ()and () ≡ ()

1− ()

23

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

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

() = ()£()

1−()¤1−

(23)

() =

µ()

()

¶(1−)()

1−()

() =

µ()

()

¶(1−)µ()

()

¶(1−)(1−)() = 1

where the second and third lines are obtained from the first. The first equation is a

standard basis for growth or development accounting, the second is the version used

in Hall and Jones (1999), Hendricks (2002), and elsewhere, and the third is a variation

suitable for the model here. Suppose that and are observable, as well as and

Then these three equations have the following interpretations.

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

line in (23) in general attributes growth in output per worker to growth in all three

inputs, and and along the BGP the shares are [(1− ) (1− )] and

(1− )

An accounting exercise based on the second line attributes some growth in output

per worker to physical capital only if the ratio is growing. The rationale for using

the ratio is that growth in or induces growth in , by raising its return. Using

the ratio means that the accounting exercise attributes to growth in capital

only increases in excess of those prompted by growth in effective labor. Output

growth that arises from the increases in physical capital required to keep the ratio

–and hence the return to capital–constant are attributed to the factor(s) that

induced the increase in effective labor. Along the BGP is constant, and the

exercise attributes all growth to and with shares (1− ) and

The third line applies the same logic to human capital as well. Thus, along the

BGP the growth accounting exercise attributes all growth in output per worker to

24

the residual

Next consider a world with many countries, with different values for capital

taxes, and so on. Suppose that each country is on a BGP or at its stagnation steady

state. Policy differences across countries lead to differences in the ratios and

as well as differences in so a development accounting exercise using any of

the three versions attributes some differences in labor productivity to differences in

physical and human capital, as well as technology differences. But the second line

attributes to physical capital, and the third to both types of capital, only differences

in excess of those induced by differences in

Estimating .–

An estimate of can be obtained using data on the wages of migrants. Continue

to suppose that all countries are on their BGPs, and in addition suppose there is a

little migration across countries. From (6), the wage of a migrant from country to

country depends on the return to effective labor and the technology in the

country where he is employed, and the human capital that he acquired in his

country of origin. Suppose that his human capital is the average value for his

country of origin. Then his wage in country is

( ) =

1−

= Γ()³

´ ³

´1−= Γ

³

´ Ã

!1−

where the second line uses Γ() ≡ () and the third uses the fact that =

If and have the same education subsidy then they also have a common

input ratio In this case the wage of an immigrant from relative to the wage of

25

a native-born worker in is

=

Ã

!1−

Recall that the ratio of per capita GDP in relative to is

=

Hence 1 − is the coefficient in a regression of ln on ln in a cross section of

immigrants in

ln = constant + (1− ) ln + error. (24)

Note that this is true even in the presence of physical capital, even if 0 If

varies across countries, then (24) should be modified to control for the educational

policy in the worker’s country of origin.

An alternative empirical approach starts with the observation that the wage of

an immigrant from working in relative to his pre-emigration wage in is

=

Ã

!

=

µ

¶

Thus is the coefficient in a regression of his log wage change from emigrating,

∆ = ln− ln on the log difference in per capita GDP between the destination

and source countries, ∆ = ln − ln

6. CALIBRATION

The model parameters are the long run growth rate ; the preference parameters

( ); the share for technology in the effective labor function; and the technology

parameters (0 ) for human capital accumulation and (0 ) for technology

diffusion. These parameters are fixed throughout the simulations. The policy para-

meters () vary as indicated.

26

The growth rate is = 0019 the rate of growth of per capita GDP in the U.S.

over the period 1870-2003, and the standard values = 003 and = 2 are used

for the preference parameters. The depreciation rates for both human capital and

technology are set at 3%. Notice that for these parameter values the time allocation

is the same along the BGP and in the stagnation steady state, =

The share parameter is calibrated using (24). Borjas (1987) reports regression

results for data on immigrants in the U.S. from 41 source countries. The estimated

coefficient there is about 0.12, which implies = 088 Hence the technology (coun-

try) component of the wage, is quite important relative to the human capital

(individual) component 1− An individual in a poor country is wise to focus on

emigrating rather than acquiring an education. The simulations here use = 075

Returns to education

With an estimate of (1− ) in hand, cross-sectional variation in educational

attainment across individuals can be used to estimate from a Mincer regression.

From (6), the wage of a native-born worker in with human capital possibly

different from the average level is

( )

( )=

Ã

!1−=

Ã

!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 instead

of then from (14) we find that along the BGP their human capital, relative to

the average, is

=

Ã

!

27

so their relative wage is

( )

( )=

Ã

!(1−)

Hence in a cross section where³

´vary across individuals,

ln( ) = constant + (1− )

ln (25)

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

counterpart to (25). They use information from wage regressions in Psacharopoulos

(1994) to construct

ln() = Φ()

where is educational attainment, and where Φ() is concave and piecewise linear,

with breaks at = 4 and = 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

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

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

function in (25) with (1− ) = 055. For = 075 this implies that = 22

There remains the problem of separating and The simulations here use

= 075 which gives modest DRS in the human capital production technology, and

implies = 07522 = 03409

Technology diffusion

For calibrating the remaining constants it is useful to have in mind a “frontier”

economy, which I will think of as the U.S. This economy has no barrier, = 1 and

an educational subsidy of In the simulations here = 035

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

normalization.7 It is convenient to choose 0 so that = = 1 in the frontier

7To see this fact, note that a change in 0 can be offset by changing = and 0

28

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

0 = ( + )³

´−= ( + )

∙1 +

1

1−

1−

µ +

−

+

¶¸ (26)

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

omy that has just reduced its barrier significantly. This choice is constrained only

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

= = 1 the requirement is

0 4 ( + ) = 01960 (27)

Here the value is 0 = 070 which produces rapid growth after a barrier is reduced.

To summarize, the benchmark parameters are

= 0019 = 003 = 2 = 075

= 003 = 003 = 075 = 03409

= 035 0 = 01795 0 = 070

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

Figure 6 displays the normalized variables³

´at the stable steady state,

relative to the levels in the frontier economy, as the education subsidy varies over the

range ∈ [0 035] and the technology barrier varies over the range ∈ [1 36] Bothvalues are decreasing in and increasing in but

is more sensitive to and

is more sensitive to The threshold in −space beyond which there is no BGPcan also be seen on these figures.

Figure 7 shows (the absolute value of) the negative roots at the stable steady

state. Call these | | || 0 for fast and slow. A higher barrier slows down

29

transitions: both roots fall in absolute value as increases. As the parameters

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

→ 0

Figure 8 shows the high and low steady states =³

´ = for

three barriers, = 1 2 3 with the education subsidy held at the level for the

frontier economy, = 035 Since 1 = 1 economy = 1 is the frontier economy.

Increasing the barrier has several effects. First, it moves the stable steady state

downward, so the BGP lags farther behind the frontier. It also moves the lower

steady state upward. The curve through each of the lower steady states is the one-

dimensional stable manifold for that point. Thus, an economy with policy parameters

= 035 and = 24 that has initial conditions on the solid curve through the point

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

to the point 2 and for initial conditions below that curve it converges to its

stagnation steady state. Thus, a higher barrier reduces the set of initial conditions

that lead to balanced growth in the long run.

Figure 9 shows the phase diagram for policy parameters of the frontier economy.

Notice that for initial conditions with low value for both state variables, the adjust-

ment paths are very flat, indicating that the technology level adjusts more rapidly

than human capital

Figures 10 - 12 displays the adjustment of an economy that eliminates a barrier.

Specifically, this experiment assumes that the economy has the same education sub-

sidy as the frontier economy, = 035 but started with a higher technology barrier

≈ 33 instead of = 10 Even under the old policy, the economy was on a BGP.

Thus, the transition path moves it from its old BGP to a new one with higher levels

for ( ) at every date. Since its education subsidy was already at the level of the

frontier economy, it steady state ratio = is unchanged. Both stocks start at

about 68% of the benchmark levels, ≈ 068×

30

Both the exact path (solid) and the linear approximation (dashed) are displayed.

The two are very close during the entire transition, but the linearized system sub-

stantially overstates the speed of adjustment. Over the first decade of the transition,

the normalized technology grows rapidly while the normalized human capital stock

remains approximately constant. Thus, over the first decade human capital grows

at about the same rate, 1.9%, as it would have under the old policy. Over several sub-

sequent decades the technology continues to grow, but more slowly, and the human

capital stock grows more quickly.

Figures 11 and 12 display time plots for the period after the policy change. In all

of these plots the solid line is the exact transition path and the broken line is linear

approximation.

Figure 11a shows the time allocated to human capital accumulation. Interest-

ingly, it falls rather sharply immediately after the policy change. Presumably this

is for two reasons. Since the technology is growing rapidly, and technology is an im-

portant input into human capital accumulation, there is an incentive to delay the

investment of time until after the complementary input has increased. In addition,

consumption smoothing provides a direct incentive to shift the time allocation toward

goods production.

Figure 11b shows the growth rate of consumption, which jumps from 1.9% to

about 6.6% immediately after the policy change. It then falls gradually back to the

steady state level.

Figures 12 shows the technology level , human capital level and consump-

tion (output) relative to the frontier economy. As the phase diagram showed, the

technology grows rapidly and human capital grows more slowly. The shift in the time

allocation just after the policy change allows consumption to adjust rapidly, jumping

immediately from 68% to about 73% of the value in the frontier economy.

Figures 13 - 15 show similar paths for an economy starting from an initial condi-

31

tion farther below the frontier. In particular, the initial conditions for this economy

lie in the region where they cannot be interpreted as a BGP for any () Instead,

this economy should be thought of as initially being in a stagnation steady state. At

the time of the policy change the technology level is about 15% of the level in the

frontier economy, and the human capital level about 19%. This transition will be

called a ‘miracle.’

Figure 13 shows the (exact) transition path. Qualitatively, the growth miracle

is similar to the previous transition. There is a long period–here several decades–

during which the technology grows rapidly and human capital grows very little. This

initial phase is followed by one where human capital grows more rapidly.

Figure 14 shows time allocated to human capital accumulation and the growth

rate of consumption. As before, time allocated to human capital accumulation

declines just after the policy change. It later overshoots the steady state level and

eventually declines back toward it asymptotically (not shown). The growth rate of

consumption jumps at the date of the policy change, a result of the shift of time

towards goods production.

Figure 15 shows the adjustment paths for the technology, human capital, and

consumption. They are qualitatively similar to the previous experiment.

Figures 16 - 18 show transitions for two economies with different initial conditions

but equal initial productive capacity. Economy A (the solid line) is the one from

Figure 10, which had a higher barrier than the frontier economy and the same subsidy

to education. Economy B (the dashed line) has an even higher barrier, but also has

a higher education subsidy. Consequently Economy B has a higher initial stock

of human capital but a lower technology level. As shown in Figure 16, Economy

B enjoys a much more rapid transition to the steady state. As Figure 17 shows,

Economy B makes a more dramatic shift in its time allocation, away from human

capital accumulation and toward production. Hence it enjoys a larger immediate

32

jump in consumption growth. As shown in Figure 18, Economy B’s technology soon

overtakes Economy A’s, and its human capital and consumption exceed A’s over the

entire transition path.

7. CONCLUSIONS

1. If the local technology depreciates, countries with sufficiently high barriers have

no BGP.

2. A higher barrier shrinks the set of initial conditions for which the economy

converges to the BGP.

3. Among less developed countries with high barriers, those with better educational

attainment grow more rapidly when the barrier is reduced.

4. ‘Miracles’ typically feature a (short) period of very rapid technology inflow,

followed by a (longer) period of gradual human capital accumulation.

33

APPENDIX: LINEAR APPROXIMATIONS AND STABILITY

Define the log deviations

1 = ln¡

¢ 2 = ln

¡

¢ 3 = ln

³

´

Taking a first-order approximation to (17) gives

−

= Γ3 [∆1 − (∆+ )2 − 3]

where

∆ = (1− )−

Γ3 = −∙1− +

1−

¸−1 0

Then linearize (18) to find that

1 ≈ ( + )

"−

1−

1 + 2

#

2 ≈ ( + )

∙ (1 − 2) +

−

¸= ( + ) { (1 − 2) + Γ3 [∆ (1 − 2)− 2 − 3]}

3 ≈ − ¡ + ¢ ∙

(1 − 2) + Γ2 −

¸= − ¡ +

¢ { (1 − 2) + Γ2Γ3 [∆ (1 − 2)− 2 − 3]}

where

Γ2 ≡ −

∙1−

1− (1− ) +

µ1

− 1¶¸−1

Hence ⎛⎜⎜⎜⎝1

2

3

⎞⎟⎟⎟⎠ ≈⎛⎜⎜⎜⎝−1 2 0

3 −3 − 4 −4−6 6 + 5 5

⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝

1

2

3

⎞⎟⎟⎟⎠

34

where

1 = ( + ) ³1−

´

2 = +

3 = ( + ) ( + Γ3∆)

4 = ( + ) Γ3

5 =¡ +

¢Γ2Γ3

6 =¡ +

¢( + Γ2Γ3∆)

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

teristic equation,

0 = det

⎛⎜⎜⎜⎝−− 1 2 0

3 −− 3 − 4 −4−6 6 + 5 −+ 5

⎞⎟⎟⎟⎠

= det

⎛⎜⎜⎜⎝− (+ 1) −− (1 − 2) 0

3 − −4−6 −+ 5

⎞⎟⎟⎟⎠= − (+ 1)

£2 + (4 − 5)

¤+ [+ (1 − 2)] [−3+ 35 − 46]

= −3 − (4 − 5)2 − 1

2 − 1 (4 − 5)

−32 + (35 − 46)− 3 (1 − 2)+ (35 − 46) (1 − 2)

= −3 + (5 − 4 − 1 − 3)2 (28)

+ [(35 − 46)− 3 (1 − 2) + 1 (5 − 4)]

+(35 − 46) (1 − 2)

Write this equation as

0 = Ψ() ≡ 3 −12 −2−3 =

35

where

1 ≡ (5 − 4 − 3)− 1

2 ≡ (35 − 46) + 23 + (5 − 4 − 3) 1

3 ≡ (35 − 46) (1 − 2) =

As shown below, (35 − 46) 0 so

Ψ(0) = −3 0 Ψ(0) = −3 0

Hence Ψ has at least one positive real root, and Ψ has at least one negative real

root. The other roots of Ψ are real and both are negative if and only if

Ψ() 0 for some 0

This condition holds if and only if Ψ has (real) inflection points, the lower one occurs

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

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

Ψ() 0 for some 0

Consider inflection points of Ψ points satisfying

0 = Ψ0() = 3

2 − 21 −2 =

Then

=1

3

h1 ±

p

i

where

≡ 21 + 32

Hence Ψ has real inflection points if and only if 0 or

0 21 + 32

36

= (5 − 4 − 3 − 1)2+ 3 [35 − 46 + 23 + (5 − 4 − 3) 1 ]

= (5 − 4 − 3)2+ 21 + (5 − 4 − 3) 1 + 3 (35 − 46 + 23)

= (5 − 4 − 3 + 1)2 − (5 − 4 − 3) 1 + 3 (35 − 46 + 32)

If (5 − 4 − 3) 0 then the first two terms in the last line are positive. If

(5 − 4 − 3) 0 then the first three terms in the preceeding line are positive. Con-

sider the last term, which is the same in both lines. As shown below, 35− 46 0

Since Γ3 0 it follows that ∆ ≤ 0 implies 23 0 Hence

∆ ≤ 0 (29)

is a sufficient condition for 0, =

To locate the lower inflection point for = , note that (5 − 4 − 3) 0

implies 1 0 and (5 − 4 − 3) 0 implies 2 0 In either case

≡ 13

h1 −

p

i 0

To complete the argument we must show that Ψ() 0 The required condition is

0

∙³1 −

p

´2−2

¸³1 −

p

´−1

³1 −

p

´2−3

= −³1 −

p

´2p −2

³1 −

p

´−3

Note that ³1 −

p

´2= 221 + 32 − 21

p

Hence the required condition is

0 −³221 + 32 − 21

p

´p −2

³1 −

p

´−3

= − ¡221 + 22¢p + 21 −21 −3

= −2 ¡21 +2¢p

+ 21¡21 + 32

¢−21 −3

= −2 ¡21 +2¢p

+ 231 + 512 −3 (30)

37

Hence the (normalized) steady state is locally stable if 0 and in addition

(30) holds.

The roots at the lower steady state are more problematic. It is not hard to

produce examples with reasonable parameters where the remaining pair of roots is

complex, with a positive real part. For very high discount rates, one can also generate

examples in which all three roots are real and negative, or in which there is a pair

of complex roots with a negative real part. In these examples evidently some initial

conditions allow multiple equilibria.

38

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707-720.

41

Table 1: Miracles and misses

GDP per capitamax excess End dates for relative to US Sustained20 years Miracles 1950 2006 miracle?

Western Europe Germany 0.027 0.05 1970 40.6% 64.4% yesItaly 0.021 0.05 1970 36.6% 63.8% yesGreece 0.282 0.05 1970-80 20.0% 50.7% yesPortugal 0.123 0.05 1972-74, 1978-80 21.8% 45.8% yesSpain 0.109 0.05 1970-80 22.9% 60.8% yes Eastern Europe Bulgaria 0.062 0.05 1970 17.3% 25.0% noYugoslavia 0.078 0.05 1972, 1974, 1976 16.2% 19.0% no

Latin AmericaJamaica 0.065 0.05 1970 13.9% 12.1% noPuerto Rico 0.103 0.05 1970-74 22.4% 49.3% yes

AsiaChina 0.221 0.05 1993-2006 4.7% 19.5% ???Japan 0.621 0.05 1970-83 20.1% 72.4% yesSouth Korea 0.436 0.05 1976-2006 8.9% 58.2% yesThailand 0.211 0.05 1990-97 8.5% 26.5% ???Taiwan 0.416 0.05 1970-2000 9.7% 63.3% yesHong Kong 0.239 0.05 1972-74, 1976-95 23.2% 94.9% yesMalaysia 0.000 0.05 1995 16.3% 30.9% ???Singapore 0.460 0.05 1975-97 23.2% 84.3% yes North Korea 0.269 0.05 1971-74, 1976 8.9% 3.5% no Iran 0.432 0.05 1971-78 18.0% 19.2% noIsrael 0.198 0.05 1970-75 29.5% 59.2% yesOman 0.827 0.05 1970-87 6.5% 24.9% ???Saudi Arabia 0.443 0.05 1970-82 23.3% 26.6% no

AfricaBotswana 0.699 0.05 1973-93, 1995 3.6% 17.0% ???Gabon 0.200 0.05 1974, 1976 32.5% 11.8% noLibya 1.364 0.05 1970-81 9.0% 8.7% noSwaziland 0.133 0.05 1970, 1972-79, 7.5% 9.4% no

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 2 3 4 5 6 70

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 σ1 < σ

2

abgH

abgL

σ2

σ1

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 8: trapping regions for balanced growth

a

ho

o

ssH1

ssL1

o

o

ssH2

ssL2

o

o

ssH3

ssL3

o

sscrit

B3 = 3.5

B2 = 2.4

B1 = 1

σ = 0.35

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

1Figure 9: phase diagram

a

h

ossH

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Figure 10: catching up, the transition

o

o

ssold

ssnew

2

4

10

20

50

2 4

10

20

50

exact transition path

linear approximation

Bold = 3.337

Bnew = 1,

a

h

0 5 10 15 20 25 30 35 40 45 500.1

0.15

0.2

0.25

time

vFigure 11a: time allocation

0 5 10 15 20 25 30 35 40 45 500.01

0.02

0.03

0.04

0.05

0.06

0.07Figure 11b: consumption growth rate

time

exact transition path

linear approximation

steady state

0 5 10 15 20 25 30 35 40 45 500.6

0.7

0.8

0.9

1Figure 12a: technology relative to a "frontier" country

time

a/ass

exact transition path

linear approximation

0 5 10 15 20 25 30 35 40 45 500.6

0.7

0.8

0.9

1Figure 12b: human capital relative a "frontier" country

time

h/hss

0 5 10 15 20 25 30 35 40 45 500.6

0.7

0.8

0.9

1Figure 12c: consumption relative a "frontier" country

time

c/css

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 13: transition for a miracle

a

ho

o

ssH

ssL

0 2040

60

80

100

0 10 20 30 40 50 60 70 80 90 1000

0.05

0.1

0.15

0.2

0.25

time

vFigure 14a: time allocation for a miracle

0 10 20 30 40 50 60 70 80 90 100

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Figure 14b: consumption growth rate for a miracle

time

0 10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Figure 15: technology, human capital and consumption, relative to a frontier country

time

technology, a

consumption, c

human capital, h

0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Figure 16: Two transitions, equal initial productive capacity

a/abg

h/hbg

ossH,new

Economy B

Economy A2 5

10

20

40

50

2 5 10

20

0 5 10 15 20

0.08

0.1

0.12

0.14

0.16

0.18

0.2

time

vFigure 17a: time allocation

Economy A

Economy B

0 5 10 15 200

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Figure 17b: consumption growth rate

time

0 5 10 15 200.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Figure 18a: technology relative to benchmark

time

a/ass

Economy A

Economy B

0 5 10 15 200.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Figure 18b: human capital relative to benchmark

time

h/hss

0 5 10 15 200.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1Figure 18c: consumption relative to benchmark

time

c/css