In the extended Solow growth model of Mankiw, Romer and Weil (1992) human capital has only permanent level and no growth effects. In the endogenous growth models human capital is a growth improving variable. Human capital may have both a permanent level and a permanent growth effect. We show, with data from India, that both the level and growth effects of human capital can be estimated with an extension to the Solow model.Solow model, Level and growth effects of human capital and India

Rao, B. Bhaskara

Vadlamannati, Krishna Chaitanya

English

Research Papers in Economics

 
Estimates of the Level and Growth Effects  
of Human Capital in India 
 
 
B. Bhaskara Rao 
School of Economics and Finance 
University of Western Sydney, Sydney, Australia 
raob123@bigpond.com 
 
and 
 
Krishna Chaitanya Vadlamannati 
Development Economics & International Economics  
University of Goettingen, Goettingen, Germany 
kcv_dcm@yahoo.co.in 
 
 
 
ABSTRACT 
 
In the extended Solow growth model of Mankiw, Romer and Weil (1992) human capital has 
only permanent level and no growth effects. In the endogenous growth models human capital 
is a growth improving variable. Human capital may have both a permanent level and a 
permanent growth effect. We show, with data from India, that both the level and growth 
effects of human capital can be estimated with an extension to the Solow model. 
 
Keywords: Solow model, Level and growth effects of human capital and India  
 
JEL:  O4, O53 1.  Introduction 
 
In the empirical literature on growth the role of human capital (H) is interesting. In the well 
known extension to the exogenous growth model of Solow (1956) Mankiw, Romer and Weil 
(1992, MRW hereafter) have treated H as an additional factor of production. Therefore, H 
has only permanent level effects on per worker output and no permanent growth effects. 
With this modification MRW have argued that the Solow model can explain observed facts 
as well as the endogenous growth models. On other hand H is treated as a growth improving 
policy variable in the endogenous models. Lucas (1988 and 1990) and Benhabib and Spiegel 
(1994) discuss the channels through which H can improve the growth rate. However, the 
literature is silent on if H has both a permanent level and a permanent growth effect because 
it is not known how in practice such effects can be estimated. This paper shows that both the 
level and growth effects of H can be estimated with a further extension to the Solow model. 
For illustration we shall use data from India from 1970 to 2007. 
 
2.  Specification 
 
Let the Cobb-Douglas production function, with constant returns, be as follows.
1  
 
   
(1 ) ( )                                                             (1) t t t t t Y AK H L
a a - = ´  
 
where Y = output, A = stock of knowledge, K = stock of capital, H = an index of human 
capital formation through education and L = employment. The intensive form of (1) is:  
 
                                                                                   (2) t t t y Ak
a =  
 
where ( / ) Y H L y ´ = and   ( / ). K H L k ´ =  In  (2) the variables are in per worker terms adjusted 
for skill improvement. To estimate (1) or (2) it is necessary to check the time series 
properties of the variables  , , , Y K LH y and  . k  We have conducted the ADF, KPSS and DF-
GLS tests to find that these are  (1) I in levels and  (0) I in their first differences. To conserve 
space these results are not reported but obtained from the authors. 
 
The steady state properties of the Solow model are well known where the steady state level 
of output (y
*) is: 
                                                 
1 This is slightly different from the one used by MRW where labour (L) and HK are separated but helps to 
increase the degrees of freedom in estimation. 
 1
*
*
 y                                                (3)
and lny ln                                                      (4)
s
A
d g n
A
a
a -  
=  + +  
D = D
 
  
where a =share of profits, s= investment rate, d = depreciation rate, g= growth rate and n = 
rate of growth of population. Since  , , , s g n d and a remain constant in the steady state the 
steady state rate of growth of output equals total factor productivity (TFP). Thus the steady 
state growth rate in MRW’s extended Solow model is the same as in Solow’s (1956) original 
model and H does not have any permanent growth effects. The level effects of H on per 
worker income is as follows: 
 
1
                                     (5)
Y s
A H
L d g n
a
a -     = ´     + +    
 
 
However, the Solow model can be extended to estimate both the level and growth effects as 
follows. We can assume that the stock of knowledge  t A evolves over time (t) as follows. 
 
0                                                                  (6)
gt
t A A e =  
 
where  0 A is the initial stock of knowledge and g is its growth rate. If H has some permanent 
growth effects, (6) can be extended by assuming that  ( ) g f H = and a linear specification is 
as follows. 
 
0 1 ( )
0                                                         (7)
t g g H t
t A A e
+ =  
 
where  0 g captures the growth effects of trended but ignored variables and  1 g is an estimate of 
the growth effects of H. With these modifications the production function (2) will be:
2 
 
0 1 ( )
0
0 0 1
                                                   
ln ln (  +g ) ln                            (8)
t g g H t
t t
t t t
y A e k
y A g H t k
a
a
+ =
\ = + +
 
 
Equation (8) can now be estimated with a suitable nonlinear method. 
                                                 
2 This specification was originally developed by Rao and used in his several empirical works on the growth 
models; see the next footnote for references.  
3. Empirical Results 
 
We shall use the London School of Economics approach, known as the general to specific 
method (GETS ), for estimating (8). Professor David Hendry is its most ardent exponent and 
supporter.
3 The general GETS specification for (8) is as follows. 
 
( )
1 1 1
1 2 3
0 0 1
1 0 0 ln
+ ln + ln
ln [ ln ( (
                    
) ]
                     (9)
t t
n n n
i t i i t i i t i
i i i
t t g k
k H y
y y a g H t a
g t h
l
- -
- - -
= = =
- +
D D D
D = - - + +
+ ∑ ∑ ∑
 
 
A parsimonious version of (9) can be derived by deleting the insignificant lagged changes in 
the variables and this is a well known procedure in the estimation of the short run dynamic 
equations from the cointegrating equations. Parsimonious estimates of alternative 
specifications of (9) are given in Table 1 for India for the period 1973 to 2007. These 
estimates are made with the non-linear two stage least squares method with the internal 
instrumental variables option (NL2SLSIV). Definitions of the variables and sources of data 
are in the Appendix. 
 
Prior to  estimation it is necessary to note some difficulties in estimating a production 
function for India since to the best of our knowledge they do not exist. Recently, in an 
influential growth accounting exercise for India, Bosworth and Collins (2008) have assumed 
that  0.4 a = instead of estimating this parameter with a production function. The main 
problem seems to be due to large negative shocks caused by monsoon failures, wars with 
Pakistan, bad economic policies due to regulation and bureaucracy, the license Raj,  and 
some political instability due to the emergency rule during 1978-1979 and the uncertain 
outcome of the elections of 2004. We have added a few dummy variables for these shocks 
but found that in most cases three dummy variables viz., DUM79, DUM91 and DUM04 are 
significant. DUM79 is to capture the adverse effects of the emergency rule and DUM91 is for 
the economic crisis of 1991 after which India has devalued its currency and implemented 
liberalisation policies under the pressure of the World Bank and IMF. DUM04 captures a 
somewhat smaller negative shock caused by the uncertain 2004 election outcome and the 
change of government. It was not significant in some regressions. 
                                                 
3 GETS has been extensively used in the empirical works of Rao and Singh (2005) for the demand for money, 
Rao and Rao (2009a), Rao, Gounder and Loeining (2009) and Rao, Tamazian and Vadlamannati (2009) for 
growth models and Rao and Rao (2009b) to estimate the demand for gasoline. Rao, Sing and Kumar (2009) 
defend GETS approach over time series methods.  
Estimates without the growth effects for H but with only its level effects are given in column 
(1) of Table 1. This equation is estimated with a correction for first order serial correlation, 
which is -0.5 and significant. The other summary statistics for misspecification ( ) ff c and 
non-normality of residuals( ) nn c are significant only at about 70% and the adjusted R-Bar 
square is high at 0.812. The Sargan test indicates that the selected instruments are valid.  The  
2 dummy variables for negative shocks viz., DUM79 and DUM91 are significant but DUM04 
was insignificant (not shown). However, the estimate of profit share a  at more than 75% 
seems to be high and significant only at 10% and the coefficient of autonomous TFP is  
insignificant. The high estimate for a may be partly due to the neglect of the growth effects 
of H.  
 
To reduce the size of the level effects of H, we reestimated this equation by assuming first 
that  0.4 a = as by Bosworth and Collins and second  0.33, a = which is its stylised value in 
many growth accounting exercises. These estimates are in columns (2) and (3) respectively. 
Their summary statistics are as good as those for the equation in column (1) but the R Bar 
squares are reduced. The serial correlation test indicates that it is absent at the 5% level in 
both equations. DUM04 and autonomous TFP have now become significant and the latter  
indicates that the long run growth rate of the Indian economy is about 2%. Both equations 
have similar statistical properties by we prefer the one in column (3) because the assumed 
value for a is widely used in the growth accounting exercises. 
 
To estimate both the level and growth effects of H, we estimated  our modified specification 
in (8) and (9) first with the assumption that a equals 0.4 and then 0.33 as in the two earlier 
estimates with only level effects. Both gave very similar results and to conserve space only 
the latter is reported in column (4). The summary statistics of this equation are similar to the 
one in column (3) except that (a) serial correlation in its residuals is significant at the 5% but 
not at the 1% level; (b) the coefficient of autonomous TFP  0 ( ) g is negative and insignificant;  
(c) the coefficient of 
2
1 ln ( ) t k g D is insignificant and most importantly (d) the growth effect of 
H 1 ( ) g is significant and estimated to be 1.6%. When this equation is reestimated with first 
order serial correlation transformation the first order serial correlation coefficient  1 ( ) r was 
insignificant even at the 10% level and this is not reported to conserve space.  
 
Since the coefficient of autonomous TFP is insignificant, this equation is reestimated with 
the constraint that  0 0. g = Furthermore, we have removed the constraint that  0.33 a = and   
Table – 1 
Level and Growth Effects of Human Capital for India  
Dependent variable:  ln y D  
NL2SLS IV Estimates, 1973-2007 
Models  1  2  3  4  5 
 
Intercept  0 ( ) a   -1.841 
(-1.145)             
-3.036  
(-93.497) ** 
-3.322  
(-101.063) ** 
-3.287  
(-124.524) ** 
-3.241  
(-5.578) ** 
1 ln ( ) t y l -   -0.112            
(4.974)** 
0.133  
(3.947) ** 
0.135  
(3.867) ** 
0.173  
(3.920) ** 
0.173  
(3.887) ** 
0 ( ) t g   0.007            
(0.577) 
0.019  
(7.061) ** 
0.021  
(8.173) ** 
-0.001 
(-0.1698) 
0.015  
(4.245) ** 
1 1) ( t H t g - ´         0.016  
(3.027) ** 
 
1 ln ( ) t k a -   0.755   
(1.891)*           
0.4 (c)  0.33 (c)  0.33 (c)  0.343  
(2.351) ** 
2
1 ln ( ) t k g D   0.866  
(15.254)**           
0.137  
(2.270) ** 
0.128  
(2.126) ** 
0.007 
(0.059) 
0.017 
(0.148) 
1 2 ln ( ) t k g - D   0.935  
(20.022)**           
       
DUM71  -0.098 
(-31.496)**           
-0.101  
(-34.645) ** 
-0.101  
(-34.434) ** 
-0.103  
(-35.830) ** 
-0.103  
(-36.549) ** 
DUM91  -.0479 
(-12.780)**           
-0.049  
(-16.203) ** 
-0.049  
(-16.294) ** 
-0.049  
(-16.792) ** 
-0.049  
(-18.624) ** 
DUM04    -0.086  
(-11.345) ** 
-0.087  
(-11.909) ** 
-0.094  
(-13.934) ** 
-0.093  
(-12.515) ** 
 
2 __
R  
 
0.812 
 
0.701 
 
0.702 
 
0.707 
 
0.705 
Sargan’s 
2 c   3.132 
[0.680] 
7.420 
[0.284] 
7.289 
[0.295] 
5.057 
[0.409] 
5.079 
[.406] 
SEE  0.015     0.018  0.018  0.0180  0.0180 
1 r   -0.501 
(-2.835)**             
-----  -----  -----  ----- 
) (
2 sc c   ----  3.302 
[0.069] 
3.387 
[0.066] 
4.899 
[0.027] 
4.841 
[0.028] 
) (
2 ff c   0.172 
[0.678]       
0.042 
[0.838] 
0.062 
[.803] 
0.615 
[0.433] 
0.105 
[0.746] 
) (
2 n c   0.606 
[0.738]       
3.973 
[0.137] 
3.840 
[0.147] 
1.204 
[0.548] 
1.359 
[0.507] 
Notes:  t-ratios  (White-adjusted)  are  in  the  parentheses  below  the  coefficients;  5%  and  10% 
significance are denoted with ** and * respectively; p-values are in the square brackets for the 
2 c  
tests; constrained estimates are denoted with (c). 
 
reestimated our specification of level and growth effects. This is shown in column (5) and its 
summary statistics are very similar to those in columns (2) to (4). The noteworthy feature of 
this estimate is that both the level and growth effects of H are significant. The latter is about 
1.5% per year and the level effect of H with an elasticity of 0.65 is consistent with the 
assumed values for the share of profits in many growth accounting exercises. When this 
equation was reestimated correcting for first order serial correlation  1 r  was insignificant. 
These estimates are not reported to conserve space. Although the summary statistics of the 
estimates of the equations in columns (2) to (5) are very similar, the estimate of our modified specification in column (5) is preferred because it can explain both the level and growth 
effects of H.  
 
4. Conclusion 
 
In this paper we have shown that the Solow (1956) growth model can be extended to estimate 
both the level and growth effects of human capital. This is an improvement because only one 
of these two effects is estimated in the existing empirical works such as Mankiw, Romer and 
Weil (1992). Our estimates for India showed that the elasticity of the level of output with 
respect to human capital is about 0.65 and that human capital formation permanently 
increases the rate of growth of output. The sample average value of H was 1.131, implying 
that the contribution of H to India’s growth rate was 1.7%. If this average is increased by 
20%, then the permanent growth rate in India will increase to 2%. there are some limitations 
in our study of which the most important is the insignificance of the effects of other 
neglected growth enhancing variables like trade openness, investment ratio and reforms etc. 
Hopefully other investigators will pay attention to these gaps. 
 Data Appendix 
 
All data from 1970-2003 are from the database of Bosworth and Collins (2008). From 2004 
to 2007 these variables are computed from the sources indicated in the parentheses. 
 
Y = GDP in National currency 2000 Constant prices (2004 to 
2007 from WDI, 2008) 
 
K = National currency 2000 Constant prices (2004 to 2007 
investment data are from WDI, 2008 and K is computed with 
the perpetual inventory method)   
 
L = Labour force (2004 to 2007 from WDI, 2008)     
 
Inflation = Rate of change in GDP Deflator (2004 to 2007 from 
WDI, 2008) 
 
H = Human Capital (2004 to 2007 proxied with the Secondary 
School Enrolment Ratio of the Ministry of HRD, Govt. of India) 
 
  
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Mankiw, N. G., D. Romer and D. N. Weil (1992) A Contribution to the Empirics of 
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A Contribution to the Empirics of Economic Growth,

A contribution to the theory of economic growth,

Cointegration and the Demand for Gasoline,

Demand for money in India: 1953-2003&quot;,

Exports and Determinants of Growth Rate: Some Evidence with Time Series Data from Fiji,

forthcoming) Do we need time series econometrics? Applied Economics Letters.

On the Mechanics of Economic Development,

The Role of Human Capital in Economic Development: Evidence from Aggregate Cross-Country Data,

Why Doesn't Capital Flow from Rich to Poor Countries?,

Estimates of the level and growth effects of human capital in India

Estimates of the level and growth effects of human capital in India

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