El propósito de este estudio es analizar el cambio en magnitud, composición y distribución del gasto de los hogares en los años 1997 y 2003, usando las encuestas de calidad de vida. Este análisis encuentra una gran disminución del gasto per cápita (23.9%), especialmente en zonas urbanas y regiones más adelantadas, reflejo de la crisis económica que golpea al país desde 1998. Este decrecimiento desigual generó una mejor distribución en el 2003 y ocasionó un cambio en la estructura de gastos, llevando a un importante crecimiento de la proporción de los gastos en alimentos y una disminución de los gastos de educación y salud. En el documento se utilizó estadística descriptiva, discriminando por quintiles de cada región y clase (urbano rural). Posteriormente se realizaron estimaciones paramétricas y no paramétricas de las curvas de Engel, para cada rubro, con el fin de identificar su naturaleza.*************************************************************************************************************The purpose of this study is to analyze the change in magnitude, composition and distribution of the household's expenditure in 1997 and 2003, using quality of life surveys. The analysis shows a decrease in the per capita expenditure (23,9%), specially in rural zones and richer regions, which reflects the economic recession that the country lives since 1998. This unequal decrease leads to a better distribution and a change in the expenditure structure where the portion of foodexpenses increase while the education and health ones reduces. In the paper we use descriptive statistics taking into acount the quantiles by region and zones (rural and urban). Then, we make parametric and non-parametric estimations of the Engel curve for each group of commodities in order to identify their nature.Gasto de los Hogares, magnitud, composición, distribución, curvas de Engel

Andrés Zambrano

Manuel Muñoz

Manuel Ramirez

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
SERIE DOCUMENTOS DE TRABAJO 
No. 67 
Agosto 2009 
IF FACTOR SHARES ARE NOT CONSTANT THEN WE HAVE A 
MEASURMENT PROBLEM. CAN WE SOLVE IT? 
 
 
 
Hernando Zuleta   1
 
IF FACTOR SHARES ARE NOT CONSTANT THEN WE HAVE A 
MEASURMENT PROBLEM. CAN WE SOLVE IT?
 1 
 
 
Hernando Zuleta 
Universidad del Rosario 
American University in Bulgaria 
hernando.zuleta@gmail.com 
 
 
Abstract 
Recent evidence show that factor shares, if properly measured, are far from constant. Moreover, the 
shares of natural resources and raw labor seem to be negatively correlated with income per capita 
while the share of human and physical capital is positively correlated with income per capita.  Now, if 
factor shares are not constant then (i) growth accounting exercises rely on a false assumption and (ii) 
there is a measurement problem. The effect that changes in factor shares have on output depend on the 
relative abundance of factors and, for this reason, it is necessary to have correct measures. We propose 
an empirical methodology to solve the measurement issue and estimate TFP growth.  
 
 
 
JEL Codes: O11, O30, O41  
 
Keyword: Factor Shares, Production Function, Measurement. 
                                                 
1 The findings, recommendations, interpretations and conclusions expressed in this paper are those of the 
authors and not necessarily reflect the view of the Department of Economics of the Universidad del 
Rosario.   2
 
1.  Introduction 
 
The idea that labor income share is roughly constant, namely, the Cobb-Douglas-Kaldor 
paradigm
2, produced important consequences in the area of economic growth. Almost all of the 
literature on growth accounting assumes that the elasticity of output with respect to capital is 
constant and concludes that the major part of economic growth is explained by growth in TFP
1 
(see Easterly and Levine, 2002; Solow, 1957 or Young, 1994 among others). 
 
Recent empirical work shows that the Cobb-Douglas-Kaldor paradigm is not really 
supported by the data. Kahn and Lim (1998) show that the shares of equipment, 
production workers and non production workers have clear trends. Blanchard (1997) 
observes the share of labor decreases in continental Europe after the 80s and suggests 
that the reason of such decline may be technological bias. Other authors calculate the 
income share of reproducible factors (human and physical capital) and not reproducible factors 
and it turns out that the later is positively correlated with the income level (see Krueger, 1999; 
Caselli and Feyrer, 2007; Zuleta, 2008a; Sturgill, 2009 and Zuleta, Parada and Campo, 2008)
3.  
 
Additionally, the variability of factor shares may have different effects on output depending on 
the factor abundance of the economy. If the income share of abundant factors is growing the 
effect of these changes on the income level is positive but if income share of abundant factors is 
decreasing the effect of these changes on the income level is negative. 
 
In this paper we propose an empirical methodology to estimate the correct measures of factor 
shares. In the next section we explain why measurement is an issue when factor shares are not 
constant. In the third section we present the methodology. In the last section we present the 
conclusions. 
 
 
 
                                                 
2 See Cobb and Douglas (1928) and Kaldor (1961). 
3 In the last decades some authors revisited the theory of biased innovations and challenged the Cobb-
Douglas-Kaldor paradigm. Zeira (1998) provides a model of technological innovations that reduce labor 
requirements and find that innovations are only adopted in countries with high wages. Seater (2005), 
Zeira (2006), Peretto y Seater (2007) and Zuleta (2008b) among others, present models where the scarcity 
of not reproducible factors generates incentives to adopt technologies that reduce the need of the these 
factors. In a market economy, this type of technological change affects factor income shares in such a 
way that the share or reproducible factors is positively correlated to the income level of the economy. 
   3
2.  Factor Shares and Measurement 
 
To illustrate the importance of correct measures for the factors consider the simplest Cobb-
Douglas technology with two factors: K and L. Output per worker, 
L
Y
y  , can be expressed 
as a function of capital per worker 
L
K
k  :  
 Ak y   
Now, suppose that there is an increase in the share of capital. The effect of the change in 
factor shares on income per worker depends on the relative abundance of capital, 
k Ak
y
ln





 
Therefore, if  1  k  the effect is positive and if  1  k  the effect is negative.  Figure 1 
illustrates this fact. 
 
Figure 1 
1  k 
y 
 
3.  How to solve the problem? 
 
For simplicity consider a production function with only two factors: 
 
(1)     
t t
t L t K t t L K A Y
   
 
1  
   4
Where Y is total income, K is physical capital L is raw labor. Note that factor shares have the 
subindex t, namely they are variables not parameters. Now, given that factor shares vary, the 
relative abundance of factors becomes very important and, for this reason, it is necessary to 
have correct measures of capital and labor. The parameters  K  and L  play this role. If 
1 
t L
t K
L
K


 then the economy is capital abundant. 
 
a.  The Methodology assuming away factor augmenting technological change. 
 
Differentiating equation 1,  
 
(2)  






 


 


 

 





t L
t K
t
t
t
t
t
t
t
t
t
t
t
L
K
L
L
K
K
A
A
Y
Y


   ln ) 1 (  
 
Now define  
(3)      


 

 
 




t
t
t
t
t
t
t
t
t L
L
K
K
Y
Y
S ) 1 (    
The variable S in equation 3 is the Solow residual. Now, from equations 2 and 3 it follows 
that the Solow residual is not only TFP growth. The Solow residual includes biased 
technological change: 
(4)      


 


 


t L
t K
t
t
t
t L
K
A
A
S


 ln  
Define   


 


  
t
t
t t t L
K
S S ln
~
  
So equation 4 can be rewritten  
(5)      


 


 


L
K
t
t
t
t A
A
S


 ln
~
 
Therefore, the reduced form to be estimated is the following, 
(6)     t t t C C S       1 0
~
 
Where  t
t
t C
A
A
  

0 ,    


 



L
K C


ln 1 .    5
Therefore, this methodology allows us to identify the correct measures of factors per worker 
L
K


. 
Finally, combining the previous results with equation 1 and taking logs we get 
(7)     ) log( log log log t L
t L
t K
t t t L
L
K
A Y 


   


 


   
We also know that  ) 1 log( log log 0 0 t t C A A      . Therefore, we can define 
t t
t L
t K
t t t L C
L
K
Y Y log ) 1 log( log log
~
0     







 


 


  


  so that equation 7 can be written 
as  ) log(
~
0 L t A Y   . Therefore, we can also identify the initial level of TFP multiplied by the 
correct measure of labor.  
 
b.  What if there is labor augmenting or capital augmenting technological 
change? 
We assume that there is factor augmenting technological change and the rate of technological 
change is constant. Equation 2 becomes 
 
(2A)  






















 

 



 





t t L
t t K
t
t
t
t
t
t
t
t L
t L
t
t K
t K
t
t
t
t
t
L
K
L
L
K
K
A
A
A
A
A
A
Y
Y
,
, ln ) 1 (
) 1 (


  
 
 
Where 
t K
t K
A
A 
 and 
t L
t L
A
A 
 are, respectively, capital and labor augmenting technological 
change. 
 
Equations 4 and 5 becomes 
(4A)   
 



 











 

 




t t L
t t K
t
t L
t L
t
t K
t K
t
t
t
t L
K
A
A
A
A
A
A
S
,
, ln ) 1 (


    
 
 (5A)  
 



 



 


 

 






 







 


 

 


  
t L
t L
t K
t K
t
t L
t L
t
t
t L
t L
t K
t K
L
K
t t A
A
A
A
A
A
A
A
A
A
A
A
S 


 1 ln
~
0 ,
0 ,  
 
   6
Therefore, the reduced form to be estimated is the following, 
 
(6)   t t t t C C C S         2 1 0
~
 
Where  








 


 

 


 
t L
t L
t K
t K
L
K
A
A
A
A
C 1 ln 1 

,     


 

 



t L
t L
t K
t K
A
A
A
A
C2  and 
t L
t L
t
t
t A
A
A
A
C



   0 . 
Therefore, we can identify the correct measures of factors per worker 
L
K


,  the difference 
between capital augmenting and labor augmenting technological change 
t L
t L
t K
t K
A
A
A
A 


 
and the sum of neutral plus labor augmenting technological change
t L
t L
t
t
A
A
A
A 


. 
 
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A Theory of Production&quot;,

An Empirical Note on Factor Shares,”

Augmentation or Elimination?&quot;, working paper 2006, http://ideas.repec.org/p/deg/conpap/c011_060.html  Sturgill,

Capital Natural, Capital Humano y Participación de los Factores. Una Revisión de los Métodos de Medición del Crecimiento Económico,&quot;

Directed Technical Change&quot;

Factor Saving Innovation&quot;

Factor Saving Innovations and Factor Income Shares,”

Getting Income Shares Right”,

Induced Bias in Innovation and the Theory of Distribution&quot;,

Machines as Engines of Growth,&quot; CEPR Discussion Papers 5429. 7  Zuleta,

Measuring Labor's Share&quot;

The Effect of Foreign Trade on the Distribution of Income”, Ekonomisk Tidskrift

The Tyranny of Numbers: Confronting the Statistical Realities of the East Asian Growth Experience.

Comparación del Gasto de los hogares colombianos entre 1997 y 2003, según resultados de encuestas de calidad de vida: magnitud, composición y distribu

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Comparación del Gasto de los hogares colombianos entre 1997 y 2003, según resultados de encuestas de calidad de vida: magnitud, composición y distribu

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