We present a new model to test the convergence hypothesis based on the ideas of concordance and then employ the model to test empirically for GDP per capita convergence across 97 countries. Our results suggest the presence of switching, while there is more ‘strong divergence’ than ‘strong convergence’.Convergence; Concordance; Income per capita.

Don J Webber

Paul White

English

Research Papers in Economics

  1
Concordant Convergence Empirics 
 
 
Abstract 
 
We present a new model to test the convergence hypothesis based on the ideas 
of concordance and then employ the model to test empirically for GDP per 
capita convergence across 97 countries. Our results suggest the presence of 
switching, while there is more ‘strong divergence’ than ‘strong convergence’. 
 
 
JEL Classification: C14; F19 
 
Keywords: Convergence; Concordance; Income per capita. 
 
1.  Introduction 
 
Neo-classical growth theory suggests an inverse relation between the capital-
labour ratio and the productivity of capital. Generated by movements in the 
wage rate, the Solow-Swan model of economic growth suggests that income per 
capita will converge due to differences in the rates of return to capital and capital 
will move from economies with lower rates of return to those with higher rates 
of return (see Solow, 1956; Swan, 1956; or Barro and Sala-i-Martin, 1995, for a 
review of the literature). This leads to the proposition that poor economies 
should grow faster than rich economies and that output per capita should 
converge across economies. 
Regression models and time series analysis of sample data can be used to assess 
whether economies display convergent or divergent characteristics. These 
approaches are not without criticism (Thomas, 1997) and the problem of 
assessing distributional convergence or divergence over time is particularly 
awkward when data is available only for a small number of points in time or for 
only a small number of units, while some other methods can only analyse a 
small sample.  This paper presents a new method for identifying distributional 
dynamic properties of economies using the ideas of concordance and 
discordance.  To perform the empirical investigation we employ real GDP per 
capita figures for 97 countries between 1960 and 2000.  The analysis suggests 
that there is more ‘strong-divergence’ than ‘strong-convergence’ between 
countries although the distribution exhibits both convergent and divergent 
characteristics. This analysis lends support to the proposition that we should be 
questioning the theoretical underpinnings of the traditional Solow-Swan 
neoclassical growth literature. 
 
2.  Data and Model 
 
Convergence can be understood to mean several different things. Let  t i s ,  be a 
metric for output of country i at time t.  For instance, convergence could be 
inferred to be the gradual reduction in the magnitude of the difference between a 
richer economy (i) and a poorer economy (j) between two periods of time  t (  and 
) k t + .  ) ( ) ( , , , , k t j k t i t j t i s s s s + + − > − . Convergence using this reduction in ‘the   2
difference’ necessarily requires a slower rate of growth for i than for j.  Another 
definition may be based on the ‘absolute difference’  | | , , t j t i s s −  whereas other 
concepts of convergence could be based on ratios such as  t j t i s s , , / o r  
{ } { } k t j k t i t j t i s s s s + + _ , , , , , min / , max .   Accordingly, the data in empirical 
investigations of convergence could be either in ‘ratios’ (based on the original 
data) or in ‘differences’ possible with the original data being transformed using 
a new baseline value to reduce the bias due to natural growth. This allows us to 
identify a ‘strong’ form of convergence, which we define as occurring when the 
observations have converged in ‘ratios’ and in ‘differences’, and a ‘weak’ form 
of convergence, where convergence has occurred in either ratios or in 
differences (but not in both). A further theoretical possibility referring to the 
distributional dynamic properties of a sample is that i and j have 
converged/diverged (in either or both ratio and difference values) but their 
relative position has changed; this would be the case if they switched position. 
To capture this strong and weak form of convergence/divergence, we introduce a 
method that permits the identification of convergence and divergence, with and 
without switching, which is based on the ideas of concordance and discordance. 
For development purposes consider two economies indexed by i and j and 
without loss of generality assume  t j t i s s , , >  and define  it X  to be the solution of: 
 
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If: 
1 , > j i X   then countries i and  j exhibit divergence in ratio without switching  
1 0 , < < j i X  then countries i and  j exhibit convergence in ratio without 
switching  
0 1 , < < − j i X  then countries i and j  exhibit convergence in ratio with switching  
1 , − < j i X  then countries i  and j exhibit divergence in ratio with switching  
 
Similarly let  t i r ,  denote a normalising transformation of  t i s ,  (e.g. 
t t t i t i s s s r / ) ( , , − =  where  t s  is the sample mean at time t) and define 
t j t i
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j i r r
r r
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, ,
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, −
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+ + . If: 
 
1 , > j i Y  then countries i and j exhibit divergence in difference without switching  
1 0 , < < j i Y  then countries i  and j exhibit convergence in difference without 
switching   3
0 1 , < < − j i Y  then countries i and j exhibit convergence in difference with 
switching  
1 , − < j i Y  then countries i and j exhibit divergence in difference with switching  
The calculation of all possible pairwise combinations of ( j i j i Y X , , , ) may then be 
summarised by counting instances of convergence and divergence with and 
without switching in a matrix as shown in Table 1.   Note that the matrix in 
Table 1 contains 8 shaded cells that denote infeasible combinations in that they 
involve the ratio but not the difference switching, or vice versa. 
 
{Insert Table 1 about here} 
 
3.  Empirical Results 
 
Countries were included in the sample if and only if data for their GDP per 
capita was available for ten-year intervals between 1960 and 2000. The countries 
included in the data set maximise both the time period (1960-2000) and the 
number of countries (97) subject to the constraint that there are observations for 
each country at each point in time. Observations correspond to the level of real 
GDP per capita (in 1995 constant $US) and were drawn from the World Bank 
(2003). 
The variables  ij X  were estimated using GDP per capita ( t i s , ) and the variables 
t i Y ,  were estimated using  t t t i t i s s s r / ) ( , , − = .  Enumeration of  ij X  and  t i Y ,  for all 
pairs of countries is summarised in matrix form given in the appendix.  Tables 
A1 – A4 present the results relative to 1960, while Tables A5 – A7, A8 – A9, 
and A10 present the results relative to 1970, 1980 and 1990 respectively. The 
numbers representing switching are in the bottom left quadrant of the tables. 
Several patterns can be identified from the tables. First, the longer the period of 
time under examination then the greater the number of pairs of economies that 
have switched their position. This suggests that the growth path of a national 
economy is neither uniform nor smooth across the sample: some countries grow 
faster than others and an economy can overtake another. 
Second, there is more evidence of divergence than there is of convergence. The 
number of pairs of economies that have strongly diverged (irrespective of 
whether or not they switched) are summarised in Table 2, while the number of 
pairs of economies that have strongly converged (irrespective of whether or not 
they have switched) are summarised in Table 3. 
 
{Insert Table 2 about here} 
{Insert Table 3 about here} 
 
As the total possible number of converging pairs for a sample of 97 economies 
is equal to 4656, the evidence does not support the proposition that there was 
strong convergence throughout the period. When the observations are converted 
into percentages, it is easily identifiable that the maximum number of ‘strongly 
converging’ pairs is only 34.9% of the entire sample whereas the minimum 
number of ‘strongly diverging’ pairs is 40.1%. 
The above analysis has been repeated replacing country i and country j with the 
i-th and j-th percentile i.e. working directly with the properties of the empirical 
distribution rather than the countries which make up the empirical distribution.    4
The analysis based on percentiles leads to the same broad conclusions (e.g. 
when comparing 1960 with 2000 there are 2783 pairs of percentiles for which 
1 ) ( ), ( > j i X   and   1 ) ( ), ( > j i Y ,  1300 pairs of percentile for which  1 ) ( ), ( > j i X  and 
1 0 ) ( ), ( < < j i Y , 543 pairs of percentiles for which  1 0 ) ( ), ( < < j i X  and  1 0 ) ( ), ( + < < j i Y , 
and 30 pairs of percentiles for which  1 0 ) ( ), ( < < j i X  and  1 ) ( ), ( > j i Y . The evidence 
provided in these analyses does not support the proposition that the sample is 
strongly-converging over time and questions the realism of the traditional 
Solow-Swan neo-classical growth model. 
 
 
4.  Conclusion 
 
This paper has presented a new method for identifying the convergence and 
divergence empirics of a sample that permits the researcher to identify switching 
simultaneously. When these methods are employed to identify some convergent 
properties of 97 countries’ real GDP per capita between 1960 and 2000, the 
results suggest that divergence is the dominant property, but there is evidence of 
convergence and switching in the sample also. The results question empirically 
the traditional theoretical approach to growth, which suggest that economies 
should converge to the long-run steady-state. Moreover, it questions empirically 
the role of capital mobility in equalising income levels and reinforces the need to 
find other explanations to understand cross-economies patterns of growth. 
 
Acknowledgements 
 
The authors would like to thank David Allen and Sam Perlo-Freeman for their 
helpful comments. 
 
Bibliography 
 
Barro, R. J. and Sala-i-Martin, X. (1995) Economic Growth, McGraw Hill, 
London 
Solow R.M. (1956) “A Contribution to the Theory of Economic Growth”, 
Quarterly Journal of Economics, 70, 65-94. 
Swan, T.W. (1956) “Economic Growth and Capital Accumulation”, Economic 
Record, 32, 334-61 
Thomas, R. L. (1997) Modern Econometrics: An Introduction (Addison-Wesley, 
UK). 
World Bank (2003) World Development Report - 2003, Oxford University Press, 
New York 
 
Appendix: 
The countries in the sample are: Algeria, Argentina, Australia, Austria, The 
Bahamas, Bangladesh, Barbados, Belgium, Belize, Benin, Bolivia, Botswana, 
Brazil, Burkina Faso, Burundi, Cameroon, Central African Republic, Chad, 
Chile, China, Colombia, Rep. of Congo, Costa Rica, Cote d'Ivoire, Denmark, 
Dominican Republic, Ecuador, Arab Rep. of Egypt, El Salvador, Fiji, Finland, 
France, Gabon, Ghana, Greece, Guatemala, Guyana, Haiti, Honduras, Hong 
Kong, China, Hungary, Iceland, India, Indonesia, Ireland, Israel, Italy, Jamaica, 
Japan, Kenya, Rep. of Korea, Lesotho, Luxembourg, Madagascar, Malawi,   5
Malaysia, Malta, Mauritania, Mauritius, Mexico, Morocco, Nepal, Netherlands, 
New Zealand, Nicaragua, Niger, Nigeria, Norway, Pakistan, Panama, Papua 
New Guinea, Paraguay, Peru, Philippines, Portugal, Rwanda, Saudi Arabia, 
Senegal, Seychelles, Sierra Leone, Singapore, South Africa, Spain, Sri Lanka, 
Sudan, Sweden, Switzerland, Syrian Arab Republic, Thailand, Togo, Trinidad 
and Tobago, United Kingdom, United States, Uruguay, Venezuela, Zambia and 
Zimbabwe. 
   6
 Table 1: All combinations of outcomes. 
  
Converged, Ratio, 
No Switching 
 
Diverged, Differences, 
No Switching 
Diverged
No Swi
Diverged, D
No Swi +1 
  
Converged, Ratio, 
No Switching 
 
Converged, Differences, 
No Switching 
Diverged
No Swi
Converged, D
No Swi
Diverged, Ratio, 
Switching 
 
Converged, Differences, 
Switching 
Converged, Ratio, 
Switching 
 
Converged, Differences, 
Switching 
 
D
i
f
f
e
r
e
n
c
e
s
 
A
x
i
s
 
-1 
Diverged, Ratio, 
Switching 
 
Diverged, Differences, 
Switching 
Converged, Ratio, 
Switching 
 
Diverged, Differences, 
Switching 
 
   -1  +1 
   Ratio  Axis 
 
 
 
 
Table 2: Number of Strongly Diverging Pairs of Economies 
  1970 1980 1990 2000 
1960  2104 (45.2%)  2220 (47.7%)  2125 (45.6%)  2152 (46.2%) 
1970  -  2155 (46.3%)  2063 (44.3%)  2006 (43.1%) 
1980  -  -  2002 (43.0%)  1869 (40.1%) 
1990  - - -  1932  (41.5%) 
 
 
 
 
Table 3: Number of Strong Converging Pairs of Economies 
  1970 1980 1990 2000 
1960  1446 (31.1%)  1462 (31.4%)  1411 (30.3%)  1355 (29.1%) 
1970  -  1623 (34.9%)  1570 (33.7%)  1465 (31.5%) 
1980  -  -  1625 (34.9%)  1518 (32.6%) 
1990  - - -  1619  (34.8%) 
   7
Table A1: 1960 to 1970 
********* *********  91  2025 
********* *********  1363  1006 
4 85  ******  ****** 
79 3  ******  ****** 
 
Table A2: 1960 to 1980 
********* *********  167  2069 
********* *********  1315  788 
14 147  ******  ****** 
151 5  ******  ****** 
 
Table A3: 1960 to 1990 
********* *********  169  1918 
********* *********  1212  912 
30 199  ******  ****** 
207 9  ******  ****** 
 
Table A4: 1960 to 2000 
********* *********  132  1914 
********* *********  1143  958 
43 212  ******  ****** 
238 16  ******  ****** 
 
Table A5: 1970 to 1980 
********* *********  299  2048 
********* *********  1514  569 
7 109  ******  ****** 
107 3  ******  ****** 
 
Table A6: 1970 to 1990 
********* *********  190  1892 
********* *********  1404  804 
17 166  ******  ****** 
171 12  ******  ****** 
 
Table A7: 1970 to 2000 
********* *********  158  1789 
********* *********  1268  993 
20 197  ******  ****** 
217 14  ******  ****** 
 
Table A8: 1980 to 1990 
********* *********  199  1901 
********* *********  1537  821 
5 88  ******  ****** 
101 4  ******  ****** 
 
Table A9: 1980 to 2000 
********* *********  139  1720 
********* *********  1389  1102 
20 129  ******  ****** 
149 8  ******  ****** 
 
Table A10: 1990 to 2000 
********* *********  199  1840 
********* *********  1549  900 
6 70  ******  ****** 
92 0  ******  ****** 
 

Concordant Convergence Empirics

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