It. J. Cotemp. Math. Scieces, Vol. 7, 202, o. 2, 59-598 A Cobb - Douglas Fuctio Based Idex for Huma Developmet i Egypt E. Khater Istitute of Statistical Studies ad Research Dept. of Biostatistics ad Demography Cairo Uiversity, Egypt dr_said_khater@yahoo.com Abstract I this paper, a ew idex for huma developmet is established for the purpose of measurig ad assessig developmet amog differet overrates i Egypt. The model uses a idea based o Cobb-Douglas productio fuctio attemptig several combiatios of elasticities of the explaatory variables ivolved (coupled with data take from Huma Developmet Report, Egypt 2003). The model has come up with the optimal combiatios of elasticities. It has also bee used to predict the 200 idices ad put them i comparisos with other available idices. Our idices outperformed the available idices. Keywords: Idex for Huma Developmet, productio fuctio, margial rate of techical substitutio, Margial Product, the Average Product, Elasticit.. Itroductio A productio fuctio describes the techical relatio that trasforms iputs (resources) ito outputs (commodities). The geeral form of the productio fuctio [see for istace Bridge, J. L. (97)] is
592 E. Khater G [x, x 2,..., x, y, y 2,..., y m, q L, q L,..., q L, q k, q k,, q k ] (.) Where x: is output of good i, y: is iput of good j; q L : is iput of labor services of type r; q : is iput of capital services of type s: Lad is excluded as a factor as this is usually cosidered costat. Equatio (.) is still too geeral ad ivolves too may variables for ay possibility of estimatio. The stadard simplificatio ivolves the aggregatio of the argumets of the fuctio. A atural way to combie x i ad y j is to use their prices to give Z = F [q L, q L,.., q L, q k, q k,..., q k ] (.2) Where Z = P io X i m j= P jo X j, ad p io, p jo are fixed base period prices. Notice that z represets real icome origiatig i the firm. The productio fuctio still cotais too may factors for us to hadle so further groupig is ecessary. The type of productio fuctio estimated by ecoometricias is Z = f(q k, q L ) (.3) where q k = K [q k, q k,..., q k ] (.4) ad q L = L [q L, q L,..., q L ] (.5) It ca be show [see, e.g., Gree, 964, p.2] that this aggregatio is possible, i.e., f is a sigle valued fuctio; if ad oly if the margial rate of techical substitutio (MRTS) betwee ay two types of labor (capital) is idepedet of ay type of capital (labor). Furthermore (Gree, 964, p. 25). If the aggregatio fuctios K (q k, q k,..., q k ) & L (q L, q L,..., q L ) are homogeeous of degree oe, the aggregates K ad L ca be dealt with as though they are actual idividual iputs. Properties of Productio Fuctio Ecoometric theory provides some properties o the geeral shape of f i the very simple case where oly oe output (good) is produced by the services of oe piece of capital ad oe type of labor. (i) The Margial Product (MP) refers to the chage i output associated with a icremetal chage i the use of a iput. The MP of each iput factor oe would expect to be positive are decreasig, with
Cobb - Douglas fuctio based idex 593 MRTS = MP ( L) MP ( K ) (ii) The Average Product (AP) is defied as the ratio of output to a iput z AP(L) =, q L AP(K) = z q k (.6) The Elasticity of Productio (Output) is defied as the percetage chage i output divided by the percetage chage i a iput. (iii) Isoquats arid the Margial Rate of Techical Substitutio For the productio fuctio Z = f(q L, q k ), the level curve q K = h(q L ) defied by the set {( ql, qk ) : Z = f (ql, qk)} => qk = h(ql) represets the iput combiatios which will geerate the same level of output Z =. The graph of this curve is called "isoquat" (where "iso" meas equal) because all iput combiatios that satisfy the coditio above lead to the same quatity of output beig produced. 2. The Cobb - Douglas Model The Cobb-Douglas productio fuctio was first published i the joural "America Ecoomic Review" i 928. The fuctio proposed i the 928 article was α β Z = f((q L, q K ) = AqK ql A > 0,0 < α, β <, (2.) where q K = iput of capital, q L = iput of labor, ad A, α, β = techological parameters. The margial product fuctios are z MP( L) = ql z MP( K ) = q K = βaq α k = αaq q α k β L q β L z = β ql z = α q k (2.2)
594 E. Khater The Cobb-Douglas fuctio restricts the substitutio betwee factors such that the elasticity of substitutio is always uity. 3. Huma Developmet through the Cobb- Douglas Fuctio: I this sectio we make use of the Cobb-Douglas fuctio as a basis to establish a ew idex for huma developmet i the goveress of Egypt. The basic idea is to write Y = a X b X b2 2 X b, (3.) where Y is the desired huma developmet idex to be built. X,..., X are idepedet measures of developmet used i buildig Y b,..., b are the respective elasticities of X,..., X We ote that the impact of X,..., X o Y is traslated through the values of b,..., b, which have to be estimated from the available data. A logarithmic form of (3.) is: L Y = L a + b L x + b 2 L x 2 +... + b L x (3.2) This form is a flexible o liear form i X, X 2,..., X but liear i L Y, L x,, L x. We ote that the sum of elasticities b + b 2 + + b explais the extet to which Y respods to chages i X,..., X. We distiguish three cases (i) =. I this case, if the all primal factors x,..., x are icreased by a certai percetage, Y is icreased by the same percetage. (ii) <. I this case, if all the primal factors are icrease by Δ the correspodig icrease i Y is less tha Δ. (iii) >. I this case, if all the factors X,..., X are icreased by Δ, the correspodig icrease i Y is greater tha Δ.
Cobb - Douglas fuctio based idex 595 We may also ote that the amout by which is less tha is a idicator of the shortage ad iability of the populatio to geerate a mootoic icrease i huma developmet. O the other had, the more the amout by which exceeds, the more ability of the populatio to geerate potetials of icreased huma developmet. 4. Experimetatio with the Proposed Model: I this sectio we accommodate the available data to the model. We use values for: X : Life expectacy of birth. X 2, X 3 : Educatio (based o rate of adult literacy(x 2 ) ad rate ad erollmet rates i differet educatioal levels(x 3 ). X 4 : Icome (based o the pre-capital icome i US. dollars estimated by purchase power, the GDP idex). Y: Huma developmet idex. The proposed model, as show i (5.) below, is used i the followig way. First, 2003 data are used to estimate the parameters of the model. This was followed by usig the full estimated model to predict the 200 set of idices for the 27 goverates. Fially these set of idices are compared to the aalogous available 200 idices. The estimated results from the model are preseted i Table. 5. Coclusios: Whe the Cobb-Douglas Model was fitted to the data, it resulted i L Y = -3.856 + 0.338 l x + 0.262 l x 2 + 0.096 l x 3 + 0.063 l x 4 (5.) The elasticties b, b 2, b3, ad b4 are highly sigificat at the 0.0 level of; with R 2 = 0.99. It is clear that our evaluated idices outperform the correspodig available idices i the sese that: (i) The estimated parameters are highly sigificat( < 0.0)
596 E. Khater (ii) The explaied variatio ( R 2 = 0.99). Table : The estimated idices for year 200 versus the idices of the Istitute of Natioal Plaig Goverorates Huma developmet idex 200 based o Cobb- Douglas Model (5.) Value Level of Huma Developmet Order Huma developmet idex 200 Istitute of Natioal Plaig Value Level of Huma Developmet Order Cairo 0.735 Middle 4 0.743 Middle 7 Alex 0.753 Middle 6 0.765 Middle 6 Port-said 0.770 Middle 2 0.783 Middle 2 Suez 0.765 Middle 4 0.776 Middle 4 Damietta 0.750 Middle 7 0.764 Middle 7 Dakahlia 0.736 Middle 2 0.75 Middle 3 Sharkia 0.722 Middle 7 0.733 Middle 9 Kalyoubia 0.732 Middle 6 0.746 Middle 5 Kafr El- Middle 0.76 Middle 9 0.73 Sheikh 2 Gharbia 0.74 Middle 0 0.754 Middle 0 Meoufia 0.739 Middle 0.753 Middle Behera 0.74 Middle 20 0.733 Middle 20 Ismailia 0.747 Middle 8 0.758 Middle 8 Giza 0.746 Middle 9 0.752 Middle 2 Bei-Suef 0.696 Middle 22 0.77 Middle 22 Fayoum 0.683 Middle 27 0.699 Middle 27 Meia 0.685 Middle 26 0.702 Middle 26 Assyout 0.693 Middle 25 0.70 Middle 25 Suhag 0.695 Middle 23 0.7 Middle 23 Quea 0.699 Middle 2 0.7 Middle 24
Cobb - Douglas fuctio based idex 597 Table : The estimated idices for year 200 versus the idices of the Istitute of Natioal Plaig (Cotiued) Luxor 0.736 Middle 3 0.748 Middle 4 Aswa 0.735 Middle 5 0.745 Middle 6 Red Sea 0.764 Middle 5 0.773 Middle 5 New Valley 0.78 Middle 0.794 Middle Matrouh 0.77 Middle 8 0.734 Middle 8 North Siai 0.694 Middle 24 0.757 Middle 9 South Siai 0.767 Middle 3 0.778 Middle 3 Notes: Scale level of Huma Developmet. 0 0.5 0.8.00 Low Middle High Refereces []. C. W. Cobb., ad P. H.Douglas, A theory of Productio". America Ecoomic Review. 8 (928), 39-5. [2] H. A. J. Gree, Aggregatio i Ecoomic Aalysis, A Itroductory Survey". New-Jersey, Priceto Uiversity Press, 964. [3] J. L. Bridge, Applied Ecoometrics". Amsterdam, North-Hollad Publishig Compay, 97. [4] M. Ramada "Some Ecoometric Models of Multiproduct Firm". Thesis of Master i Statistic. Istitute of Statistical Studies ad Research, Cairo Uiversity, 2003. [5] The Egyptia Huma Developmet. Report, The Istitute of Natioal Plaig, Cairo, 2003.
598 E. Khater [6] The Egyptia Huma Developmet. Report, The Istitute of Natioal Plaig, Cairo, 200. Received: September, 20