An Estimation of Cobb-Douglas production function in Egyptian Tourism Sector

Transcription

1 An Estimation of Cobb-Douglas production function in Egyptian Tourism Sector 1-1 Introduction (Dr: Khaled Abd El-Moaty Mohamed El-Shawadfy) lecturer of statistic in Institute of productivity zagazig university Research into the production function has a long history. Since the first work, in 1928, many studies have tended to support the hypothesis that production processes are well described by a linear homogeneous function with an elasticity of substitution of one between factors, P H Douglas, (March 1948) K Arrow, H Chenery, B Minhas and R Solow, (August, 1961). New results are presented here, using 40 years of observations on Egyptian Tourism Sector during the 1960s and 2000s in Monetary Model (Ministry of Planning in Egypt 2000), using 19 years of observations during 1982 and 2000 in both monetary formula and Real Model for comparison between 2 formulas. This article consists of the following parts: 1-2 the theoretical model 1-3 The Monetary Model 1-4 The Real Model 1-5 The comparison between two models In addition to conclusions and recommendations, reference and appendix 1-2The Theoretical Model Production functions can be applied to a single firm, an industry, or an entire nation. Note, however, that they are limited to producing a single output, so that joint production is disallowed, although multiple inputs are used. The simplest production function used frequently in economics is a Cobb-Douglas production function. This two-input production function takes on the form Where Q = A.L α.k β e u (1.1)

2 Q = Output : number of tourist nights L = Labor : number of laborers in tourism sector K = Capital : number of tourist rooms in all Hotels in Egypt A is a constant term u is the error term, with E E(e i ) = 0. E(e i e j )=ψ where the covariance matrix results from the first-order autoregressive process. Taking logs, the model becomes a linear regression model: lnq = lna + αlnl + βlnk + u, (1.2) where lna is the intercept. Now suppose that we want to estimate this model under the restriction of homogeneity of degree 1, i.e., if both K and L increase with say 10% then so wills Q. This condition is equivalent to: α + β = 1. Thus, replace α with 1 - β: LnQ = lna + (1 - β) LnL + βlnk + u. (1.4) This model can be reformulated and estimated as an unrestricted linear regression model, as follows: Y = lnq - lnl = lna + β(lnk - lnl) + u Then Where Y = lnq - lnl X = lnk - lnl β 0 = lna β 1 = β. Y = β 0 + β 1 X + u, (1.5) 2

3 1-3 The Monetary Model The idea of Monetary Model is summarize in substituting the real variables by monetary variables in equation (1.1) as follows: Q = Output : number of tourist nights will become the gross domestic production (GDP) in tourism sector per year ( ) in million L E, L = Labor : number of workers in tourism sector will become the wages of workers in tourism sector per year ( ) in million L E, K = Capital : number of tourist rooms in all Hotels in Egypt will become the investment in tourism sector per year ( ) in million L E, When apply the model (1.5) using data in appendix Table (I) the results appear in the following table (1-1) Table (1-1) Results for Monetary Model with constant Variables Prais- Winsten Β (P.W) Cochrane- Orcutt Β (C.O) Maximum likelihood Β (M.L) Ln (k) Ln(L) Ln(A) R D.W

4 From Table (1.1)first the three methods appear that all B`s take the expected positive signs which prove the direct relationship between gross domestic production GDP( LnQ )and each of investment(ln K) and wags (Ln L) In the same table(1.1) using the three methods notice that all Β`s are highly significant with significant t less than 0.05 without exception. But in the prais-winsten method the adjusted R 2 =.707 greater than its value in Cochrane-Orcutt method and D.W value =1.998 is more close to 2 value than the same value in Cochrane- Orcutt method.then the estimated monetary model using prais-winsten is better than the others in table(1.1). So the estimated monetary model is LnQ = Ln K Ln L (1.6) When estimate the Monetary Model without constant term, the results can be shown in the following table (1.2) Table (1-2) Results for Monetary Model without Constant Variables Β (P.W) Prais- Winsten Β (C.O) Cochrane- Orcutt Β (M.L) Maximum likelihood Ln (k) Ln(L) R D.W

5 From table (1.2) notice that the adjusted R square became.25 in comparison to.707 in the monetary model with constant term. 1-3 The Real Model The problem with Real Model is the limitation of data, where the available observations not more than 19 points through , by Ministry of Tourism in Egypt (2000). Where Q (Output) is the number of tourist nights, L (Labor) is the number of workers in tourism sector and K (Capital) is the number of tourist rooms in all Hotels in Egypt When apply the model (1.5) using data in appendix Table (2) the results appear in the following table (1-3) Table (1-3) Results for Real Model with Constant Term Variables Β (P.W) Prais- Winsten Β (C.O) Cochrane- Orcutt Β (M.L) Maximum likelihood Ln (k) Ln(L) Ln(A) R D.W

6 From Table (1.3) using the three methods notice that all Β`s are highly significant with significant t less than 0.05 without exception, the results is closed to results of monetary model in table (1.1) except the value of adjusted R square and D.W statistic, this happened may be because of small sample size in real model. In appendix, Diagram (1) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method When estimating the real model without constant term using the same data in appendix table (2), the results can be shown in the following table (1.4) Table (1-4) Results for Real Model without Constant Term Variables Prais-Winsten Β (P.W) Β (C.O) Cochrane- Orcutt Β (M.L) Maximum likelihood Ln (k) Ln(L) R D.W From Table (1.4) using the three methods notice that all Β`s of Ln (k) not take the expected positive signs which prove the direct relationship between gross domestic production GDP (LnQ) and investment (Ln 6

7 K). In appendix Diagram (2) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method 1-4 The comparison between two models In order to make the comparison between two models,it should be estimate the monetary model using 19 observation only as the case of real model during Table (1-5) Results for Monetary Model with constant Variables Β (P.W) Prais- Winsten Β (C.O) Cochrane- Orcutt Β (M.L) Maximum likelihood Ln (k) Ln(L) Ln(A) F 2 R D.W From Table (1.5) using the three methods notice that all Β`s are not significant except intercept, in addition to bad adjusted R squre In appendix Diagram (3) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method 7

8 When estimate the Monetary Model without constant term, the results can be shown in the following table (1.6) Table (1.6) Results for Monetary Model without constant Variables Β (P.W) Prais- Winsten Β (C.O) Cochrane- Orcutt Β (M.L) Maximum likelihood Ln (k) Ln(L) F 2 R D.W From Table (1.5) using the three methods notice that all Β`s are not significant, in addition to bad adjusted R square In appendix Diagram (4) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method 8

9 1-5 Conclusions and recommendations 1- Although Cobb-Douglas production function shows physical output as the Douglas labor and capital inputs, this article prove that Monetary Model is good alternative to Real Model (Physical Model) at least in Egyptian tourism sector. 2- Cobb-Douglas production function in Monetary Model may be good formula in services sectors. 3- Cobb-Douglas production function with intercept has good results than Cobb-Douglas production function without intercept. 4- The Egyptian tourism sector is more sensitive for regional and national security, but it is disappear in Cobb-Douglas production function 9

10 Reference 1- P H Douglas, (March, 1948), 'Are There Laws of Production?', American Economic Review, vol. XXXVIII, K Arrow, H Chenery, B Minhas and R Solow, (August, 1961)'Capital- Labor Substitution and Economic Efficiency', Review of Economics and Statistics, 43, 3, /EasyReg International\OLS.HTM

11 APPENDX, Table (1) Monetary Data for Tourism Sector Year Wages in Tourism GDP in Tourism Investment in Tourism , , , ,

12 Table (2) Real Data For Tourism Sector Year Tourist Nights Tourist Rooms Labor in Tourism

13 Real Model with constant Diagram (1) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method Y MOD_1 MOD_2 MOD_3 year 13

14 Real Model without constant Diagram (2) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method Y 4 MOD_5 2 MOD_ MOD_7 year 14

15 Monetary Model with constant Diagram (2) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method Y MOD_1 MOD_2 MOD_3 year 15

16 Monetary Model without constant Diagram (2) demonstrates the true values of GDP in Tourism and its predicted values of Cochran- Orcutt method, Prais- Winsten and Maximum likelihood method Y MOD_5 MOD_6 MOD_7 year 16