economics
letters
EI~gEVIER
Economics Letters 50 (1996) 285-290
The other side of conditional convergence
Dongchul Cho*, Stephen Graham
Department of Economics, Texas A&M University, College Station, TX 77843-4228, USA
Received 3 January 1995; accepted 22 August 1995
Abstract
The empirics of conditional convergence of Mankiw et al. (Quarterly Journal of Economics, 1992, 407-437)
imply a peculiar result: on average, countries with a lower income per adult are above their steady-state positions,
while countries with a higher income are below their steady-state positions.
Keywords: Conditional convergence; Neoclassical growth model; Steady state
JEL classification: O11; 047
The empirical evidence reveals conditional convergence in the sense that economies grow faster per capita if they
start further below their steady-state positions (Barro et al., 1993, Abstract).
1. Introduction
The most p r o m i n e n t purpose of the conditional convergence literature is to protect the
traditional neoclassical growth model (see, for example, Solow, 1956) from the t u m u l t u o u s
attack of the e n d o g e n o u s growth camp. Since R o m e r (1986) and Lucas (1988), the recent
e n d o g e n o u s growth literature has claimed postwar cross-country growth experiences as the
main evidence for the failure of the neoclassical model. In particular, rich countries have
grown, on average, faster (or no slower, at least) than poor countries, which has been
interpreted as being in sharp conflict with the neoclassical model that predicts convergence of
standards of living to a steady-state level.
H o w e v e r , more scrupulous empirical assessments have found that steady-state levels m a y be
substantially different across countries. Thus, a proper way to test the neoclassical convergence is to examine the correlation between the growth rate and the initial level of income per
* Corresponding author. Present address: Korea Development Institute, P.O. Box 113, Cheong Ryang, Seoul
130-012, South Korea.
0165-1765/96/$12.00 © 1996 Elsevier Science S.A. All rights reserved
S S D I 0165-1765(95)00745-8
286
D. Cho, S. Graham / Economics Letters 50 (1996) 285-290
capita conditional on each economy's steady state. Here the empirical results appear to
support conditional convergence (Barro, 1991; Mankiw et al., 1992; Barro and Sala-i-Martin,
1992; and many others).
This paper presents a direct implication of this success that is contrary to what we believe is
a c o m m o n conjecture rooted in the neoclassical growth model. The c o m m o n conjecture is that
lower income countries converge to their steady states f r o m farther below, as is revealed not
only in the quotation given at the beginning of the paper but also in m a n y articles, such as
R o m e r (1987):
Countries starting from a low level of the capital labor ratio are assumed to accumulate capital more rapidly and
catch up with the countries that start from a higher initial position. For example, countries like Germany and
Japan that suffered large losses during World War II, or developing countries like Korea, are thought to have
grown faster than the United States in the 1950s and 1960s because of more rapid capital accumulation as they
approached the capital-labor ratio in the United States.
This paper follows the analysis of Mankiw et al. (1992) ( M R W hereafter), which seems the
most articulated model for conditional convergence. Using their data set, we found that,
unlike the above c o m m o n conjecture, the conditional convergence regression estimates imply
that poor countries are converging f r o m farther above to their steady-state income levels.
Specifically, these countries' steady-state levels are only half of their income levels in 1960.
With this result, one may find hardly any grounds for policies such as the World Bank's
investment in these poorer countries with the intent of pushing them towards their steady
states.
2. Initial positions, steady state positions, and growth rates
A standard manipulation of the human capital a u g m e n t e d Solow model in M R W yields the
steady-state income per capita at time t, y*(t),
ln(y*(t)) = ( l n A ( 0 ) + g t ) - a 1 ln(n + g + 6) + a 2 ln(s~) + a 3 ln(sh),
(1)
w h e r e A(0) is the initial level of technology, g is the exogenous progress rate of A(t), 6 is the
c o m m o n rate of depreciation of physical and h u m a n capital, n is the rate of labor growth, s~
and s h are the fractions of income invested in physical and h u m a n capital respectively. Taking
an approximation of the model around the steady-state position y* yields
d ln(y(t))/dt = g + A[ln(y*(t)) - ln(y*(t))],
(2)
w h e r e A is the speed of convergence. Eqs. (1) and (2) lead to the regression equation:
ln(y(t)) - In(y(0)) = gt - yl[In(y*(0)) - I n ( y ( 0 ) ) ]
(3a)
=% +ylln(y(O))+yzln(n+g+6)+Y31n(sk)+%ln(sh),
where %=gt-%(lnA(O)),
--a3")/1 > O.
%=-(1-e
z')<0,
y2=a1%<0,
"y3=--Of2")/l>0,
(3b)
and
y4 =
D. Cho, S. Graham / Economics Letters 50 (1996) 285-290
287
T o save space, we confine ourselves to the discussion of the results using the data for the 98
l
'non-oil' countries provided by MRW. A p a r t from the residual terms, the results are
ln(y(t)) - In(y(0)) =- - 0 . 2 7 + 0.094 In(y(0))
= 3.04 - 0.289 ln(y(0)) - 0.505 ln(n + g + 6) + 0.524 ln(Sk)
+ 0.233 ln(Sh) ,
w h e r e time 0 and t denote the years 1960 and 1985 respectively. That is to say, the coefficient
on In(y(0)) is positive unconditionally, 2 but negative conditionally. We state this finding as:
Finding 1. E c o n o m i e s converge conditionally, although they diverge unconditionally.
Noting that the growth rate is determined by the (log) distance from the steady state (Eq.
(2)), the higher (unconditional) growth rates of rich countries, on average, imply the following
alternative statement:
Implication 1. E c o n o m i e s with higher income per capital are, in general, further b e l o w their
steady-state positions, provided all the economies are below their steady states.
The steady state of each country can be c o m p u t e d from Eq. (1) if the relevant p a r a m e t e r s
are identified. Unfortunately, 3'0 in Eq. (3b) does not identify A ( 0 ) from g. To identify A ( 0 ) ,
we t o o k a widely used a priori value of g, 2% per year (see, for example, Barro et al., 1993). 3
Using the estimates of regression (3b) and noting that the sample period is 25 years, we can
estimate the steady state of each country by using the following:
ln(y*(O))={yo--(O.O2)(25)+Y21n(n+g+6)+Y31n(s~)+Y41n(sh)}/(-%).
(3b')
Fig. 1 plots these estimates of y* against current income y in 1960. E a c h point represents an
individual country's steady-state level of income, y*, the d o t t e d line is obtained by regressing
l n ( y * ) on In(y), 4 and the solid line (45 ° line) represents y. Consistent with the first part of
Implication 1, the d o t t e d line appears steeper than the solid line. A m o n g 98 countries in the
1These results can be found in tables III and Vof MRW. See MRW for the details of the data. Briefly, GDP per
adult is used for y, the rate of growth of the working-age population for n, the share of investment for sk, and the
percentage of the working-age population that is in secondary school for s h.
2 This is a general finding for a wide set of countries (see, for example, Barro, 1991; Barro and Sala-i-Martin,
1992; and others), even though it is often reported that this coefficient estimate is not statistically different from
zero for other sets of countries, excluding the poorest countries (e.g. MRW's 'intermediate countries').
3 An anonymous referee suggested an alternative normalization to anchor the steady-state positions, which
assumes the average steady-state (log) income across countries to be equal to the average actual income. This
experiment yields almost identical results (the implied value of g is 1.8%).
4 The result is In(y*(0)) = -2.65 + 1.32 In(y(0)), where the standard error of the slope is 0.12. Thus, the slope is
significantly greater than 1.
288
D. Cho, S. Graham / Economics Letters 50 (1996) 285-290
-
/
/
%
/
o
//
tO
O~
.E
-~
o
~-
/
/
0
0
~:~
o
0
D
e~
%
~-
0
~
o
~ac~
o
b
-0
0
YO~
c,n
0
/
0
O[]
/
[]
D
/
OD
O
£3O
/
,t
~
i
t
I
I
102
I
I III
103
t
I
I
I
I
f I1~
104
i
I
I
I
I
I I
105
GDP Per A d u lt in 1960
Fig. 1. Current vs. steady-state positions, 98 countries. Data from Mankiw et al. (1992).
sample, however, a half of countries (49 countries) appear to be above their respective
steady-state positions. Thus, Fig. 1 leads to the following statement:
Finding 2. On average, relatively poor countries converge to their steady-state positions from
above, while rich countries converge from below.
To make this statement clear, we divided the 98 countries evenly into four groups in terms of
the 1960 GDP per adult, and computed the (geometric) average GDP per adult for each
group. Table 1 shows the results. Probably the most surprising result in this table is that the
average GDP per adult in the 'low income' countries is approximately twice as high as the
Table 1
Number of countries
G D P per adult in 1960
Steady-state GDP per adult
Low
income
Lower
middle
income
Upper
middle
income
High
income
24
$683
$333
25
$1282
$938
25
$2553
$3247
24
$7126
$8389
D. Cho, S. Graham / Economics Letters 50 (1996) 285-290
289
corresponding value in steady states. In fact, Finding 2 appears more surprising, noting that it
implies the following:
I m p l i c a t i o n 2. In poor countries rather than rich countries, the current capital to (effective)
labor ratio tends to be higher than the steady-state level, and thus this ratio decreases over
time to converge to the steady state.
The usual name for poor countries, 'less developed countries', seems a misnomer in this sense,
because poor countries, in general, are already 'over-developed' compared with their potential
or steady-state positions.
While Implication is simply a mirror statement of Finding 1, which is the main finding of the
conditional convergence, Finding 2 (and hence Implication 2) relies on the particular
specification and data of MRW. s In particular, the number of countries that are positioned
above the steady-state levels depends heavily upon our assumption about g: a higher value of
g implies more countries positioned above their steady-state levels.6
3. Concluding remarks
This paper derives a peculiar implication of conditional convergence. By combining the
backbone idea of the neoclassical convergence (Eq. (2)) with the main finding of M R W
(economies converge conditionally but diverge unconditionally), we are led to state that lower
income countries are far above their steady-state positions, while higher income countries are
below their steady states. These results do not logically contradict the neoclassical growth
model per se: they are the unexpected, but inevitable, implications to preserve the conventional neoclassical model.
Acknowledgements
We are grateful for the helpful comments of an anonymous referee. All remaining errors are
OURS.
5 We have also used Summers and Heston's data (1988) and found consistent results.
6 Note that apart from random errors Eq. (3a) implies,
{ln(y(t)) - In(y(0))}/t ~ g ~ In(y*(0)) ~ In(y(0)).
That is, countries with an annual growth rate less than g are the countries whose positions were above their
steady-state positions.
290
D. Cho, S. Graham / Economics Letters 50 (1996) 285-290
References
Barro, R.J., 1991, Economic growth in a cross section of countries, Quarterly Journal of Economics, 407-443.
Barro R.J. and X. Sala-i-Martin, 1992, Convergence, Journal of Political Economy, 223-251.
Barro, R.J., N.G. Mankiw and X. Sala-i-Martin, 1993, Capital mobility in neoclassical growth models, NBER
Working Paper No, 4206.
Lucas, R.E., 1988, On the mechanics of economic development, Journal of Monetary Economics, 3-42.
Mankiw, N.G., D. Romer and D.N. Weil, 1992, A contribution to the empirics of economic growth, Quarterly
Journal of Economics, 407-437.
Romer, P.M., 1986, Increasing returns and long-run growth, Journal of Political Economy, 1-002-1037.
Romer, P.M., 1987, Crazy explanations for the productivity slowdown, NBER Macroeconomics Annual, 167-201.
Solow, R.M., 1956, A contribution to the theory of economic growth, Quarterly Journal of Economics, 65-94.
Summers, R. and A. Heston, 1988, A new set of international comparisons of real product and price levels:
Estimates for 130 countries, The Review of Income and Wealth, 1-25.