the logo after 1 em space; see diagram B), or under the logo
the right of the logo (centered vertically along the height of
(the space between the logo and the words "THE WORLD
the logo after 1 em space; see diagram B), or under the logo
BANK" should be the same as the cap height of the words,
(the space between the logo and the words "THE WORLD
aligned flush left; see diagram C). When the logo is placed
46268
BANK" should be the same as the cap height of the words,
on aalignedorflush
dark black background, the logo should reverse to
left; see diagram C). When the logo is placed
whiteon(see diagram D).
a dark or black background, the logo should reverse to
PRMED Knowledge Brief
white (see diagram D).
2. The words "THE WORLD BANK" should be set in ALL
Growth andlogoIncome Convergence
IncomeCAPS, Convergence
2. The words "THE WORLD BANK" should be set in ALL
Univers Bold. The size of the type in relation to the
CAPS, Univers Bold. The size of the type in relation to the
should remain constant. Always use the art provided in
Jesús Crespo Cuaresma
an electronic file or in CRC.
logo should remain constant. Always use the art provided in
Jes´us Crespo Cuaresma
Department of Economics, University of Innsbruck, Austria
an electronic file or in CRC.
Department of Economics, University of Innsbruck, Austria
jesus.crespocuaresma@uibk.ac.at
jesus.crespocuaresma@uibk.ac.at
Placement of Logo
Placement of Logo
All World Bank books must display the World Bank logo on
All World Bank books must display the World Bank logo on
The Solow model of economic growth (Solow, 1956, Swan, 1956) concludes that poorer countries will
TheSolowmodelofeconomicgrowth(Solow,1956,Swan,1956)concludesthatpoorercountrieswill
the front and back covers, the spine, and the title page.
the front and back covers, the spine, and the title page.
tend to grow faster than richer onesprovidedcover countries share the same production function,
tend to grow faster than richer onesprovidedcover countries share the same production function,
that that
savings rate and population growth, and labouraugmenting technology grows at the same rate in
labouraugmenting technologybegrows atattheeither rate in
all countries. The existence of income convergence hascover, beenshould
savings rate and population growth,FrontFrontthe
allexogenousgrowthmodelversusendogenousgrowthmodelsthatdonotnecessarilyconcludeontheof
countries. The existence of income convergence has thus beenusually taken tosameatest
On the frontfront
andOn cover, the logologo
thusthe should be placed at either the
usuallyplaced
taken to be athe
be testof
lowerlowerupper left left corner and should be accompaniedby the
or or upper corner and should be accompanied by the
exogenous growth model versus endogenous growth modelsthat do not necessarily conclude on the
existenceofconvergenceinincomepercapitaamongeconomies. Herewedescribedierentconcepts
words "THE WORLD BANK." Placement of the World Bank
words "THE WORLD BANK." Placement of the World Bank
existence of convergence in income per capita among economies. Here we describe different concepts
ofconvergenceusedintheempiricalliteratureoneconomicgrowthandsummarizetheresultsofthis
logo logo blockthe front cover should bebe as follows:the outside
block on on the front cover should as follows: the outside
of convergence used in the empirical literature edgeeconomic growth and summarize the results of this
literature.
literature. edge onthe logo (the outer box) should be between 2.25 picas
of of the logo (the outer box) should be between 2.25 picas
(3/8 inch) and 3.75 picas (5/8 inch) from the trim. The logo
(3/8 inch) and 3.75 picas (5/8 inch) from the trim. The logo
should be placed equidistant from both trim and spine. See
should be placed equidistant from both trim and spine. See
samples on next page.
samples on next page.
The Solow model and incomeThe
The Solow model and income convergence
The World Bank logo is the only logo to appear on front covers
convergence
World Bank logo is the only logo to appear on front covers
and spines of publications published by EXTOP. Any exception
Total output (Yt) is assumed to depend on physicalCobbDouglas),productioncopublishers)
Total output (Yt) is assumed to depend on physicalneeds
and spines of publications published by EXTOP. Any exception
augmenting) technology (At) accordingAdditionala
augmenting) technology (At) accordingAdditional
to this guidelinelogos (totbe labouror by the publisher.
to this guideline needs to be approved by the publisher.
capital (Ktapproved input (Lt) and (labour
), labour
returnsreturns on all inputs,
to scale on all inputs, to atoCobbDouglas production function with costant
capital(forKcosponsors input (Lt) and (labour
logos (for cosponsors or copublishers) appear at the
function with costant
appear at the
bottom of the back cover, along with the World Bank logo.
to scale bottom of the back cover, along with the World Bank logo.
where (0,1). Labour input and technology are assumed to grow at constant rates n and g,
Yt =YKt (AtLt)
t = Kt (At1Lt)1
,

,
where (0,1). Labour input and technology are assumed to grow at constant rates n and g,
respectively. Physical capital is accumulated through savings (with a constant savings rate s)
respectively. Physical capital is accumulated through savings (with a constant savings rate s)
and depreciates at a constant rate ,
and depreciates at a constant rate , dKt
dKt dt (1)
(1)
We can write (1) in terms of eective labour as
dt = K t = sYt  Kt.
= K t = sYt  Kt.
We can write (1) in terms of effective labour as kt = syt  (n + + g)kt, (2)
where kt = Kt/(AtLt) and y =tYt/(AttLt) = kt . The tsteady state level of capital per unit of
eective labour (k) can be found by setting kt = 0, which leads to
k = sy  (n + + g)k , (2)
where kt = Kt/(AtLt) and y = Yt/(AtLt) = kt . The steady state level of capital per unit of
effective labour (k) can be found by setting kt = 0, which leads to
s(k) = (n + + g)k. (3)
Graphically, the equilibrium level ofk is given by theintersection point of the investment per
s(k) = (n + + g)k . (3)
unit of eective labour curve, s(k), with the breakeven investment line, [(n++g)k], as shown
Graphically, the equilibrium level of k is given by the intersection point of the investment per
in Figure 1.1 Countries with levels of capital per unit of eective labour below k (see k1 in
unit of Figure labour curve, s(k), with the breakeven investment line, [(n++g)k], as shown
effective1) present positive growth in the stock of capital per unit of eective labour (see (2)),
in Figure 1. Countries with levels of capital per unit of effective labour below k (see k1 in Fig
while1countries to the right of k will tend to decrease their stock of capital per unit of eective
ure 1) present positive growth in the stock of capital per unit of effective labour (see (2)), while
labour.
countriesLoglinearizingofaround tend to decrease their stock of capital per unit of effective labour.
to the right k willthe steady state level of income per unit of eective labour,
Loglinearizing around the steady statedlevel=ofincome per unit of effective labour,
dln(yt)
t [ln(y)  ln(yt)], (4)
1Thebreakeven investment line represents the investment needed to avoid the capital stock from falling.
dln(yt)
dt = [ln(y)  ln(yt)], (4)
1
1The breakeven investment line represents the investment needed to avoid the capital stock from falling.
Figure 1: The steady state in the Solow model
y,sy (n+ +g)k y=f(k)
y*
dk/dt sf(k)
dk/dt
k 1 k* k 2 k
which implies that the growth rate of income per unit of eective labour (and thus income
per capita) is related to the distance to the steady state level of income. This means that the
Solow model concludes that (after controlling for those factors that determine dierences in the
steady state level of income per capita) poorer countries should grow at higher rates that richer
countries.
Unconditional and conditional convergence
A natural empirical test of income convergence from (4) is based on regressing the growth rate
of income per capita on initial income levels for a crosssection of countries or regions (see
Barro and SalaiMartin, 1992). A negative (positive) correlation between these two variables
indicates the existence of socalled unconditional convergence (divergence). Figure 2 presents
the corresponding scatterplot (growth rates of GDP per capita versus initial GDP per capita in
the period 19702000) for all countries in the world for which Penn World Table data are available
and the same scatterplot for Spanish provinces in the period 19802005 (source: Cambridge
Econometrics). As can be seen from the scatterplots, considering more homogeneous groups of
economic units (which are more likely to be succesfully modelled through the theoretical setting
put forward above), the empirical relevance of unconditional convergence appears more evident.
Figure 2: Income growth versus initial incomeWhole world and Spanish provinces
.08 Spanish provinces
.04
.06
19702000
.03
.04 19802005
GDPpc,
of .02 .02
income,
rate .00 rate .01
growth .02 growth .00
Annual .04 Annual .01
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
log(GDPpc), 1970 Income level, 1980
The Solow model predicts income convergence across countries which share the same production
function, investment rate, population growth, depreciation rate and common growth rate of
2
technology. In order to account for such dierences, we can control for these (and potentially
other) variables in the convergence regression. If a negative partial correlation between initial
income and subsequent income growth appears after controlling for other covariates, conditional
convergence is said to exist.
convergence and divergence
Intuitively, convergence takes place if the dispersion of income across countries is reduced over
time. This concept of convergence is known as convergence. Figure 3 presents the evolution
of the standard deviation of income per capita across world countries and European regions
(source: Cambridge Econometrics). The results in Figure 3 give evidence of divergence across
Figure 3: Dispersion of income per capitaWhole world and European regions
Standard deviation, log(GDPpc) Standard deviation, log(GDPpc), European regions
1.16 .37
1.14 .36
1.12 .35
.34
1.10
.33
1.08
.32
1.06 .31
1.04 .30
1.02 .29
1970 1975 1980 1985 1990 1995 2000 1980 1985 1990 1995 2000
countries at the world level but convergence within more homogeneous economic areas (in this
case, Europe). The statistical signicance of changes in the dispersion of income can be evaluated
using the test put forward by Carree and Klomp (1997).
It can be easily shown that convergence implies convergence, but the opposite does not
necessarily apply (see for example Furceri, 2005).
The dynamics of the world distribution of income
The convergence analyses presented above (and most of the studies existing on the dynamics of
income at the world level) take countries as the natural unit of analysis. The results concerning
convergence/divergence across countries do not necessarily imply convergence/divergence across
individuals at the world level. A rst hint at the dierences appearing from both approaches
can be obtained from weighted crosscountry convergence regressions using population as a
weight. The unweighted parameter estimate corresponding to the data presented in Figure 2
for the whole world is 0.002 (standard deviation = 0.002), while the weighted estimate is 0.012
(standard deviation = 0.001), which indicates convergence once that we take into account the
size of each country in terms of population.
SalaiMartin (2006) reconstructs the dynamics of world income across individuals by matching
macroeconomic estimates of income per capita (at purchasing power parity) with estimates of
income dispersion across individuals within countries. Notwithstanding the degree of uncertainty
implied by the fact that withincountry dispersion estimates are not available for all countries and
need to be projected from neighbouring countries, SalaiMartin (2006) nds evidence of income
convergence for individuals in the period 19702000. SalaiMartin's (2006) approach is not
without criticism. Milanovic (2003) critizises the approach heavily and pinpoints several reasons
3
why SalaiMartin's (2006) study has some drawbacks which tend to exagerate the decrease in
global income inequality. In particular, among other issues,
a) several nations where withincountry income inequality has risen in the period under study
(and where data are available!) are omitted of the analysis,
b) the data on the distribution of income across households is treated as if they reected
income distribution across individuals.
Nonlinearities and club convergence
Azariadis and Drazen (1990) present a theoretical model where heterogeneity in the marginal
productivity of capital across levels of the capital stock leads to multiple equilibria corresponding
to steady states at dierent income levels. This result implies that depending on the initial level
of income, countries may converge to dierent equilibria and thus may get stuck at relatively low
levels of GDP per capita, corresponding to a socalled poverty trap.
Empirically, the existence of club convergence can be tested by estimating piecewiselinear
models where the initial level of GDP per capita determines the parameters corresponding to
the other covariates in the regression equation. Durlauf and Johnson (1996) present empirical
evidence of this type of nonlinearities in crosscountry growth regressions.
References
[1] Azariadis, C. and A. Drazen (1990), Threshold externalities in economic development. The
Quarterly Journal of Economics, 105, 501526.
[2] Barro, R. J. and X. SalaiMartin (1992), Convergence. Journal of Political Economy, 100,
223251.
[3] Carree, M. A. and L. Klomp (1997), Testing the convergence hypothesis: A comment.
Review of Economics and Statistics, 79, 683686.
[4] Durlauf, S. and P. Johnson (1995), Multiple regimes and cross country growth behaviour.
Journal of Applied Econometrics, 10, 36584.
[5] Furceri, D. (2005), and convergence: A mathematical relation of causality. Economics
Letters,89, 212215.
[6] Milanovic, M. (2003), The Ricardian Vice: Why SalaiMartins calculations of world in
come inequality are wrong. HEW 0305003, EconWPA.
[7] SalaiMartin, X. (2006), The world distribution of income: Falling poverty and ... conver
gence, period. The Quarterly Journal of Economics, 121, 351397.
[8] Solow, R.M. (1956), A contribution to the theory of economic growth. TheQuarterlyJournal
of Economics, 70, 6594.
[9] Swan, T. (1956), Economic growth and capital accumulation, EconomicRecord, 32, 334361.
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