Growth and convergence
Contents
1. The Solow model ............................................................................................................................. 2
1.1 The model .................................................................................................................................. 2
1.2 The steady state .......................................................................................................................... 3
2. Empirical evidence........................................................................................................................... 5
2.1 « Stylized facts » ........................................................................................................................ 5
2.2 Econometric evidence ................................................................................................................ 6
3. Conditional convergence ................................................................................................................. 7
3.1 The message ............................................................................................................................... 7
3.2 Empirical evidence..................................................................................................................... 8
4. Growth and poverty: long run tendencies ...................................................................................... 10
4.1 Growth per region .................................................................................................................... 10
4.2 Growth and global poverty ...................................................................................................... 14
References .......................................................................................................................................... 18
List of tables
Table 1: Regression results of Mankiw, Romer and Weill (1992) ...................................................... 8
Table 2: Absolute and conditional convergence .................................................................................. 9
List of figures
Figure 1: Graphical interpretation ........................................................................................................ 4
Figure 2: Evolution of the log of GDP per capita of a sample of countries from 1820 onwards ........ 5
Figure 3: Absolute convergence : growth as a function of initial GDP ............................................... 5
Figure 4: Initial GDP and growth, (a) 1980-90.................................................................................... 6
Figure 5: Distribution of national endowments in capital per worker, 1997 and 2007 ....................... 9
Figure 6: Divergence and convergence, 1990-2010 .......................................................................... 10
Figure 7: Growth across countries and time ...................................................................................... 11
Figure 8: Growth and poverty ............................................................................................................ 11
Figure 9: GDP per capita, 1960-95 .................................................................................................... 11
Figure 10: Real growth and price effects: Exports of East African countries, 2000-2010 ................ 12
Figure 11: Average growth per country and income growth of lowest quintile ................................ 14
Figure 12: Distribution of global household incomes, in dollars PPA .............................................. 15
Figure 13: Estimation of the absolute number of the poor, 1820-2020 ............................................. 17
Figure 14: GDP per capita and poverty in the DRC .......................................................................... 17
1
1. The Solow model
Basic question: Do poor countries catch up to rich countries? « Should » they catch up (according
to the models)?
1.1 The model
Growth model with growth being « exogenous » (not determined in the model), as opposed to
models with “endogenous growth”.
Notation
Y
Aggregate output
K
Aggregate stock of capital
L
Labor force (# number of workers)
k
K/L : capital per worker
y
Y/L : output per worker
I
Investment
s
S/Y : Saving rate
δ
Rate of capital depreciation (7%)
n
Rate of growth in population.
*
k
Capital per worker in steady state
Production function
Y  K  L1
(1)
Y

 K  L   K / L 
L
y  k
(2)
K  I K
(3)
L̂  n
(4)
Laws of motion
Mathematical Digression: The derivative of the log of a variable with respect to time is
approximately equal to its growth rate. In continuous time, the two are just equal.
Take a process in time such as
x  f  t   100t
Where t denotes the time, which we take as continuous variable. The derivative of x with respect to
time is given as
2
x  f ' t  
dx
 100
dt
And its growth rate is
xˆ 
1 dx 100 100 1



x dt
x
100t t
But if we express this in logs
ln  x   ln 100t   ln 100   ln  t 
The derivative of this log is
d ln  x  1

dt
t
Which is equal to the growth rate.
Assumption about investment
I  sY
(5)
Investment equals savings, which in turn is a constant fraction of output. Strong simplification ! We
assume a closed economy.
1.2 The steady state
We can show that under certain not too unrealistic assumptions, there exists a steady state (true for
the Cobb-Douglas function used in this chapter)
Def. We define the steady state by k  0 (capital per worker K / L is constant) or k̂  n .
K
KL  KL KL K L
 k
 2 
L
L2
L
L L
K
  kLˆ
L
k
(6)
sY
I K
 kLˆ
L
sY   K

 kn
L
 sy   k  kn  sy    n  k
k
k  sk     n  k
3
(7)
By definition, k  0 in the steady state
  n  k  sy
sy
sk 

 n  n
s
k 1 
 n
k
1
 s 1
k 

 n 
*
 s 
y*  

 n 

1
(8)
The idea that the value of the capital stock per worker will at one day stop growing may sound a bit
weird, but is ok in this context.
Figure 1: Graphical interpretation
Prop. 1. (steady state) If all countries have (more or less) the same rates of population growth, if
the technology (denoted by α) is the same, then the differences in savings rates (which is to say also
the rates of investment as investment and savings are equal in this model) completely explain the
differences in output per capita if all countries are in their steady state.
Prop. 2 (transition) For countries that are not yet in their steady state, the rate of growth of the
capital stock and of output per worker slow down the more they approach the steady state.
So if the poor countries are far away from their steady state, they should have a growth rate that is
higher. One should therefore expect to observe convergence. Note that if we were to relax the
4
assumption that investment equals savings (no capital mobility), capital would migrate from rich
countries (lots of capital, hence low return) to poor ones, which would reinforce convergence.
2. Empirical evidence
2.1 « Stylized facts »
At a first glance, income levels seem to diverge instead of displaying convergence:
Figure 2: Evolution of the log of GDP per capita of a sample of countries from 1820 onwards
And at a second glance ? Baumol (1980) shows, using historical data of Maddison, that the OECD
countries have experienced an astonishing convergence. The basic graph that we will use
throughout the course is a scatterplot (one point per country) in a diagram
o The initial GDP on the horizontal axis
o The growth rate on the vertical axis.
If the regression line has a positive slope, we abserve an unconditional divergence (which means
divergence independent of other characteristics of the countries); if the slop is negative, there is
convergence.
Figure 3: Absolute convergence : growth as a function of initial GDP
5
Source : De Long (1988)
But De Long (1988) has shown that the observation of Baumol (1980) that was limited on OECD
countries suffered from a severe selection bias and that the data ofBen David (1991) does not
suggest any convergence. The graph is very easy to reproduce with data of the World Development
Indicators :
-1
-.5
0
.5
1
Figure 4: Initial GDP and growth, (a) 1980-90
4
6
8
lggdppc1980
GDP per cap real growth 1980-1990
10
12
Fitted values
Source : WDI
2.2 Econometric evidence
Take as an example the classic paper of Mankiw, Romer et Weill (1992).
sy   n      k
6
(9)
where π is an adjustment factor of the differences in the productivity of labor. Slightly
rewriting,
1
1
 y 
y

s  s  1/   sy    n  
k
y 
or
y
1
1


  n 
(10)
s


s
 n      1 
1
y
 

s


 n   
(11)
Finally we we take logs to get a linear expression in order to use OLS
  
  
ln yi  
 ln si  
 ln   ni   i   ui
1 
1 
1
(12)
2
We want to test the predictions of the model. Those are 1  0 ,  2  0 ,  2   1 . In addition, it can
be shown (I spare you the proof) that α is the share of capital in GDP. We know from national
accounts in most countries that this share is about one third. So we should have approximately
1  1 / 3 / 1  1 / 3   1 / 2 . All we have to do now is to take a cross-section of countries and to
test Proposition 1.
Results : ˆ1  1.42 , standard deviation  2  0.14. So t = 1.42/0.14 = 9, significant at 1%.
Qualitatively (sign and significance of the coefficient), that’s alright. Quantitatively, however, the
size of the coefficient is way too large (1.42 instead of about 0.5). So we have a problem.
3. Conditional convergence
3.1 The message
We’re still using Mankiw, Romer and Weill. The idea is to go back to the model and argue that the
steady state should depend on factors other than just the savings rate. For instance, suppose that the
production function is
Y  K  L1  H 
(13)
where H is « human capital ». The steady state now depends on human capital: it is different for
each country, and a poor country may have low growth if it is near its own, low steady state.
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3.2 Empirical evidence
We see in Table 1 that when including the initial level of education (ln(SCHOOL)) we get a
coefficient on ln(I/GDP) that is closer to the theoretical value (0.69 for non-exporters of petrol, 0.7
for middle-income countries and 0.28 for OECD countries).
The second part of the table shows the result of a regression in which the constraint on the equality
of coefficients on the savings rate and on the term ln(n + g + δ) is imposed; this constraint is
implied by theory, see (12). The test of this restriction has a p-value of 0.41, which means that we
cannot recjet the null hypothesis of the equality of the coefficients. But ist that because they are
really equal or is it that the sample is too small to reject the null?
Finally, observe that if we regress growth on initial GDP and nothing else (equivalent to the
scatterplot), convergence is not verified: The coefficient on initial GDP is positive in the first
column of Table III (non-oil developing countries), insignificant in the second (intermediateincome countries, and negative and significant only for OECD countries—back to the “Baumol
problem”. However, convergence is back once we include other factors as implied by theory.
Table 1: Regression results of Mankiw, Romer and Weill (1992)
8
Table 2: Absolute and conditional convergence
Error! Reference source not found. (continued)
Having said this, the capital accumulation seems to follow more agglomeration forces than yield
spreads, because the data on capital endowment indicates divergence.
Figure 5: Distribution of national endowments in capital per worker, 1997 and 2007
9
.01
.008
.006
.004
.002
distribution
World
1997
World
2007
Uganda 2007
0
Uganda 1997
0
50
100
150
200
Physical capital per worker, thousand dollars
250
Data source: UNCTAD
4. Growth and poverty: long run trends
4.1 Growth at the region level
Despite this, we observe global and unconditional convergence in the last ten years (the first time
since the data is collected) in Figure 6.
Figure 6: Divergence and convergence, 1990-2010
(b) 2000-2010
-1
-1
0
0
1
1
2
2
3
3
(a) 1990-2000
4
6
8
lggdppc1990
GDP per cap real growth 1990-2000
10
12
4
Fitted values
6
8
lggdppc2000
GDP per cap real growth 2000-2010
Source : WDI
… which explains the renewed growth in the last decade (Figure 7)
10
10
Fitted values
12
Figure 7: Growth across countries and time
(a) all countries
(b) Developed countries vs. Developing countries
Note : Each point is an average for all countries of a category in one year. We could therefore connect all the points (or
all the triangles, or all the squares) to get one curve. Instead of doing this, the plotted curves are smoothed over time to
eliminate visible annual fluctuations in the dispersion of the points
Source: Rodrik (2011)
And the gain in growth is particularly spectacular for Africa :
Figure 8: Growth and poverty
(a) Growth of GDP per capita, 1950-2010, per (b) GDP per capita and absolute poverty rate in Subregion
Saharan Africa
Note : In the first panel (a), growth is displayed in algebraical value, so 0.02 = 2% etc.
Source: Rodrik (2011) for panel (a) and Sala-i-Martin and Pinkowskiy (2010) for panel (b).
…which begins to partly revers the « great divergence between Asia and Africa after the 1960s
(Figure 9—compare what is measured on the vertical axes in Figure 8 and Figure 9)
Figure 9: GDP per capita, 1960-95
(a) Base 100 = 1960
(b) Real
11
600
3'000
500
East Asia & Pacific
(developing only)
2'500
East Asia & Pacific
(developing only)
400
Sub-Saharan Africa
(developing only)
2'000
Sub-Saharan Africa
(developing only)
300
1'500
200
1'000
100
500
-
-
1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010
Note : GDP per capita (constant 2005 US$)
Source World Bank, World Development Indicators
Is this just another “commodities boom”? Not only. During the previous commodities booms, the
price effects dominated the quantity effects. This is not the case this time. Example: Growth of
exports of a sample of East African countries, 2001-2010
Figure 10: Real growth and price effects: Exports of East African countries, 2000-2010
16.00
14.00
Volume
12.00
Price
10.00
8.00
6.00
4.00
2.00
0.00
-2.00
Uganda
Tanzania
Kenya
Rwanda
Mauritius
-4.00
Source : World Bank (2012)
However, all is not rosy for Africa. Rodrik (2014) argues that convergence takes place only in
manufacturing, as shown in Figure 11:
Figure 11: Absolute convergence, economywide and manufacturing sector only
(a) Economy-wide
(b) Manufacturing sector only
12
Croissance a/
Croissance suséquente a/
Log PIB par travailleur initial
Log productivité par travailleur initiale
Source: Rodrik (2014), Figure I and III
This implies that only manufacturing is a growth engine sufficiently powerful to ensure
convergence. The problem is that sub-Saharan Africa is not industrializing. In fact, it is deindustrializing. In general, the share of manufacturing value-added in GDP follows an inverse-U
shape with the (log) level of income, as shown in Figure 12. In the case of African countries, the is
a big difference between the resource-rich ones (red) and the resource-poor ones (blue ones), with
the resource-rich attaining their peak very early. In the case of resource-poor ones, there doesn’t
seem to be a peak but it’s all driven by two special cases, Mauritius (a middle-income country) and
Swaziland (a South-African “colony”).
.3
SWZ
.2
VA manuf./PIB
.4
.5
Figure 12: Manufacturing value-added and development
0
.1
MUS
4
6
8
Log PIB/hab
Autres pays
ASS PRR
Tendance, ASS PPR
10
12
ASS PPR
Tendance, autres pays
Tendance, ASS PRR
If one looks at individual trajectories, the de-industrialization is clear, even for countries that are
considered, by and large, to be doing okay like Ghana (Figure 13):
Figure 13: Ghana’s premature de-industrialization
13
15.0
1975
1976
.
1969
1970
1973 1971
1972
1974
1977
1986
1965
1993 1966
2006
.
. . .. 2002
. .
. 2003
1989
. 2000
.
2007
. 1978 2004 2005
1988
1991
1985
10.0
VA manuf., % du PIB
1967
1980
2008
1984
2011
2009 2010
2012
2013
5.0
1981
1982
300
400
500
600
700
800
PIB par hab.
VA manuf., % du PIB
VA manuf., % du PIB
From this data, Rodrik concludes that Africa is not “out of the wood” yet. That’s probably true, but
the idea that productivity in services never converges is probably a bit exaggerated, as shown by the
example of the transportation sector in Figure 14.
Figure 14: Productivity convergence in transportation
Note: The horizontal axis is the log of labor productivity in the initial year of the database and the vertical one measures
the growth in labor productivity over the subsequent 10 years or so.
Source: Compiled from firm-level data from the Orbis database by Dany Bahar.
4.2 Growth and global poverty
Does the growth imply a reduction in poverty? A cross sectional regression of countries (Dollar et
Kray, 2001) suggests that the average income growth per country transmits more or less one to one
to growth in income of poorest 20% of the population of each country – in other words, there is no
divergence within countries (a bit sloppy as demonstration, but ok)
Figure 15: Average growth per country and income growth of lowest quintile
14
Source : Dollar et Kray (2005)
Sala-i-Martin (2003) reconstructs the distribution of global income from surveys of existing
households in countries where they are available, repeating the exercise for 1970, 80, 90 et 98. The
results were controversial, but nonetheless interesting:
Figure 16: Distribution of global household incomes, in dollars PPA
a) 1970
b) 1980
15
c) 1990
d) 1998
Source : Sala i Martin (2002)
16
For certain countries like China, there has been both an increase in average income and rising
inequality (the distribution becomes “flatter”). In total, the share of the global population that lives
in absolute poverty (less than one dollar PPA per day adjusted for inflation) has been substantially
reduced over the last 30 years, especially in China and India.
We find similar indications in Bourguignon (2004) who calculated the absolute number of the poor
in the world since 1870. Whatever the uncertainties in the calculations for the 19th century, the
surprising fact is the rapid decline in the number of poor people during the last 20 years. The global
financial crisis has only partly distorted this trend.
Figure 17: Estimation of the absolute number of the poor, 1820-2020
1800
1600
1400
1200
1000
800
Nombre absolu de pauvres
sur la planète, en millions
600
400
200
1820
1830
1840
1850
1860
1870
1880
1890
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
2010
2020
0
Sources : Data combined from the World Bank (2002), Globalization, Growth and Poverty: Building an Inclusive
World Economy (until 2000) and Chandy, Laurence, and G. Gertz 2011, "Poverty in numbers: the changing state of
global poverty from 2005 to 2015; Brookings, 2011, for 2000-2013 (estimations)
The result is that the extreme poverty is more and more concentrated in «fragile states », where the
most tragic is the Democratic Republic of Congo (Figure 18).
Figure 18: GDP per capita and poverty in the DRC
Source: Sala-i-Martin and Pinkowskiy (2010)
17
References
Chandy, Laurence, and G. Gertz 2011, "Poverty in numbers: the changing state of global poverty
from 2005 to 2015; mimeo; Washington: Brookings Institution.
Dollar, David, and A. Kraay (2004), “Trade, Growth, and Poverty”; Economic Journal 114, F22F49.
Rodrik, Dani (2011), « The Future of convergence »; Kennedy School Discussion paper RWP 11033; Cambridge, MA: Harvard University.
— (2013), “Unconditional Convergence in Manufacturing”, Quarterly Journal of Economics Vol.
128(1), pages 165-204.
— (2014) “An Africa Growth Miracle”, CEPR DP #10005.
Sala-i-Martin, Xavier (2002), “The World Distribution of Income (estimated from Individual
Country Distributions)”; NBER Working Papers 8933, Boston, MA: National Bureau of Economic
Research.
Sala-i-Martin, Xavier, and Maxim Pinkovskiy (2010), “African Poverty Is Falling...Much Faster
Than You Think!” NBER Working Paper 15775; Boston, MA: National Bureau of Economic
Research.
World Bank (2002), Globalization, Growth and Poverty: Building an Inclusive World Economy;
Washington, DC: The World Bank.
18