The analysis of production function is always at the heart of the economic analysis, it is used as a model of production of the output according to the inputs used. The progressive refinement during the recent year in the mesearument of the volume of physical production in manufacturing suggests the possibility of attempting to measure the change in the amount of labor and capital which has been used to turn out this volume goods and to determine what relationships existed between the three factors capital, labor and product (cobb douglass 1928). We analyzing for the new Saudi company the impact of the capital and labor variables on production. The analysis will be based on the cobb douglass production function.
I) Research objective:
The main objective of this report is to analyze the impact of capital and labor factors in the sale of the company. It will thus be a specific way to see on the one hand the role of the variables capital and work in terms of explanatory power and on the other hand the particular impact of the most important variable. As a result, an empirical model has been adopted.
II) Model and methodology:
The analysis is done from the cobb douglas production function Y= C.KαLβ
Y = sale K: capital L: labor
We can apply the logarithm: LogY = c + αlog (K) + βlog (L)
We will introduce the error term фi:
The econometric model estimated becomes: LogY = c + αlog (K) + βlog (L) + фi
The estimation methodology thus used is the ordinary least square method(OLS). the OLS method consists of minimizing the square of the residuals. It allows us, from a multiple regression to estimate the coefficients associated with capital and labor.
we have in our econometric model an endogenous variable (sales) and two explanatory variables (capital and labor)
Sales: it is the whole of the production sold during a period, in our econometric model sales constitute the endogenous variable.
Capital: it is defined as the whole of the means of sustainable production allowing the company to produce goods and services. It is introduced into the model as an explanatory variable.
Labor: It is the paid activity that allows the company to produce goods and services. It is essentially a factor of production. It is thus used as an explanatory variable in the model.
III) Analyse de résultat (Estimation with stata)
1)Function in log linear form :
The structure of the model is: LogY = c + αlog (K) + βlog (L) + фi
By replacing the coefficient α, β and c by their value, we obtain:
Logsale = 0.621+ 0.199logcapital + 0.771loglabor + фi
2) Test the coefficients of labor and capital:
To test the significant coefficients we will pose two hypotheses:
H0: the coefficient is not significant
H1: the coefficient is significant
If the probability associated with the test is less than 5%, we reject hypothesis H0 and conclude that the coefficient is statistically significant.
· For the coefficient associated with the capital
H0: the capital coefficient is not significant (α = 0)
H1: the capital coefficient is significant (α ≠ 0)
Estimation show that the probability associated with α is equal to 0.016 which is less than 5% and H0 is rejected. It is concluded that the coefficient associated with capital is statistically significant in explaining sales.
· For the coefficient associated with the labor
We pose: H0: the labor coefficient is not significant (β = 0)
H1: the capital coefficient is significant (β ≠ 0)
Estimation show that the probability associated with β is equal to (1.27E-05=0.000) which is less than 5% and H0 is rejected. We conclude that the coefficient associated with work is statistically significant in explaining sales.
3) Test of independant
Variable (overall significance of the model)
H0: the model is not globally significant (α = β = 0)
H1: the model is globally significant (α = β ≠ 0)
The probability of fisher test is equal to (5.5E-09 = 0.000) and less than 5% .we reject H0. Thus the model is globally significant so the variables have a significant impact on the sales explanation.
4) Explanatory of the model:
To Analysis the explanatory power of the model we interpret the value of R-squared (R2).
The value of R2 = 0.9580 which means that 95.80% of logarithm sale fluctuations are explained by variation of logarithm capital and labor. This shows essentially a very high explanatory power of the variables capital and labor on sales.
5) type of return to scale:
the production function is cobb douglas type:
The returns to scale are the way in which production varies if all the factors of production are increased in the same proportion.
Y = F (K, L) = C.Kα Lβ
λY = F (λK, λL) = C (λK) α (λL) β ==> F (λK, λL) = C.λ α K α λβ L β==> λY =F (λK, λL) = C. λ α λ β K α L β λY = F (λK, λL) = C. λ α+ β K α L β ==>
If α + β> 1 we conclude that the returns of scale are increasing.
If α + β <1 we conclude that the returns of scale are decreasing.
If α + β = 1 we conclude that the returns of scale are constant.
In our production function α + β = 0.771 + 0.199 = 0.97 <1 .Thus, we conclude that the returns of scales are decreasing. If we increase the factors in the same proportions, the production decreases less than proportionally.
6) The coefficient meaning:
The coefficients measure the elasticity, the increase in sales following an increase in a factor of sale (cetirus paribus)
α = estimate of capital elasticity in relation to sales. The capital cost factor equals 0.199 which means that if the capital factor increases by 10%, sales increase by 1.99%
β = estimate of labor elasticity in relation to sales. The work-related coefficient is equal to 0.777 which means that if the capital factor increases by 10%, sales increase by 7.77%
The econometric estimation of sales reveals a positive and significant impact of the capital and labor variables. However both variables have a strong explanatory power of the sales of the company. The econometric result also shows that the production function have return of sale decreasing , the new company does not have an interest in increasing the two factors in the same proportion. The work factor has a very significant impact, so the company must invest the most on this factor to increase its production in the best way.