Concept and application of cd and ces production function in resource management.

Natural Resource Economics:M.Sc.1st year
Institute of Forestry, Pokhara Campus, Pokhara, Nepal

1. Concept and Application of Cobb-Douglas and Constant Elasticity of Substitution Production Function in Resource M anagement Team members: Nar Bahadur Chhetri, Roll no:17 Rajesh Pokhrel , Roll no:34 Prabin Poudel , Roll no:51 Welcome and Namaste!!!

2. Definition of Production function Production, is transformation of physical inputs into physical outputs Indicates the highest output that a firm can produce for every specified combination of inputs given the state of technology The functional relationship between physical inputs and physical output of a firm is know as production function A production function of a wheat farm : W=F(L,A,M,F,T,R) Where, That is, ( W)= production of wheat in tons L=labor measured in days , A=land in ha, M=machinery in NRs, F= fertilizer in tons , T=mean summer temperature in degrees, and R= rainfall in inches., u nspecific relationship is denoted by letter of F. Types of Production function: a) CD production function b) CES production function c) Linear Homogeneous Production Function d) Variable Elasticity Substitution Production Function

3. What is Cobb-Douglas Production Function?? Most widely used empirical production function in industrial field and a griculture sector ( not to the production process of an individual firm) It was proposed by Knut Wicksell(1851-1926) , and tested against statically evidence by Charles Cobb and Paul Douglas in 1928 There are two factor of inputs, labor and capital Mathematically, Q(L,K) = AL α K β , Where, Q= Total production(the monetary value of all goods production in a year), L= Labor input(the total no of person worked in a year), K = Capital input(the monetary worth of all machinery, equipment, and buildings) α , β =Output elasticities of capital and labor respectively , A= Efficiency parameter( Always positive )

4. Cobb-Douglas Cont … Cobb- Duglas found that about 75% production increases due to labor and remaining 25% was due to capital input. CD production function is multicative production function because both input are required to get an output ( If amount of one factor is zero, no output can be produced ). CD is the one that is most heavily used in economic analyses. The reason that that , under competition, this is the only production function with the property that factor income shares are independent of relative factor prices. This property of the Cobb-Douglas production function is consistent with the data.

5. What is Constant of Substitution(CES) Production Function?? H ave any constant value of elasticity of factor substitution has been developed jointly by Arrow, Chenery , Minhas and Solow Elasticity of factor substitution are zero , infinity and one but elasticity equal to zero, and infinity are rarely found in practical phenomena CD production function with elasticity of factor substitution being equal to one has been found to be approximately valid for several industries and agricultural crops CES production function is quite a general production function wherein elasticity of factor substitution can take any positive constant value. Mathematical form; A = technological determines productivity and A ϵ [0, ᾳ ], α=Optimum distribution of inputs , α ϵ [0,1] ρ = elasticity of substitution where , ρ belong[-1, 0)U(0, ᾳ )

6. Properties of Cobb-Douglas Production Function: 1. Homogeneous of degree (  +  ). If  +  > 1, increasing returns to scale (IRS ) If  +  = 1, constant returns to scale (CRS ) If  +  < 1, decreasing return to scale (DRS ) In special case i.e. when  +  =1, it is linearly homogenous of degree one. 2. APP and MPP of the factors are the function of the ratio of factors

7. Properties of CD Function……… 3 . Negative Slope and Strict Convexity 4. Diminishing returns to each input for all positive level of inputs. (MP is decreasing and second derivative of MP L ( d 2 Q/d L 2 ) and MP K is positive) 5. The expansion path is linear, homogeneous and passes through origin 6. Elasticity of Substitution is constant and unity (Percentage change in factor quantity ratio / percentage change in inverse of their price ratio) 7. Proves Euler's Theorem: Q=MP of Capital× K+ MP of labour × L

8. Uses of CD production function E mpirical studies of manufacturing industries and inter industries comparison D etermine the relative shares of labor and capital in to total outputs U sed to prove Euler's theorem E conomists have extended CD function to more than two functions F orecasting GDP and income shares

9. Properties of CES H omogeneities: function is homogeneous of degree r Marginal Products are Positive Capital and labor are complementary inputs ( Marginal product of capital increases as labor input increases and vice-versa) Negative Slope and Strict Convexity Proves Euler's Theorem Diminishing returns to each input for all positive levels of input Elasticity of Substitution is constant but not unitary

10. CD is a special case of CES function Elasticity of Substitution of CES function is  = W hen  = 0, it leads to unitary elasticity of substitution which is characteristic of the Cobb-Douglas production function Thus, it implies that the linearly homogeneous Cobb-Douglas production function is a special case of the linearly homogeneous CES production function.

11. CD data Log transformation for regression

12. Regression analysis of CD Here , α=0.206925 & Adding α & β: i.e.  + β 0.206925+0.952008 =1.158>1 Industry exhibits increasing returns to scale Results of normality test : As p(Q,K,L) 0.05 so series is Normally Distributed Identify the best model by estimating the different parameters .

13. Discussion According to Adinya et.al . (2011) Cobb-Douglas production function had the best fit in explaining the relationship between output of snail and inputs used, Log Y=Log a + b 1 LogX 1 + b 2 LogX 2 + b 3 LogX 3 + b 4 LogX 4 + b 5 LogX 5 +e Y = Value of total output of snail X 1 = Area of plots (farm size) devoted to snail production in (hectare ), X 2 = Total labor in (man-days/hectare, X 3 =Farm management practices, X 4 = Operating costs (Naira/hectare ) X 5 =Farming experience (years ), e = Error The regression analysis , revealed that farm size, operating costs and farming experience has positive influence on output of snail production Lameed (2006) reported similar results for operational cost and farming experience, he further stated that many snail farmers (46.7%) had been involved in snail production for the past two years and farming experience has positive influence on output of snail production

14. The Cobb-Douglas production function was linearized through transformation  into a double log  (Equation 1). Ordinary least square (OLS) was used for estimating finger millet production function (Doll and Orazem , 1984). LnY i =C+b 1 LnLB+b 2 LnK+b 3 LnLD+b 4 LnSEEDRT+b 5 LnFaExp+b 6 Weedpre+ b 7 rotation +e   ....... equation 1 Where, Yi=Finger millet yield (kg ha -1 )          C   = Ln b 1 (efficiency parameter ), Input facors = Labour , Capita, Seed distribution , farm size, farmer experience, rotation of finger millet, e=error The study found that farm size, farmer experience were effected on the production of the finger millet

15. The stochastic production frontier for wheat farmers is assumed to be of the Cobb Douglas form L n(Y)=C+b 1 Ln( LandSize )+b 2 Ln(FAMLAB )+ b 3 Ln(HIBLAB )+ b 4 Ln(FERT )+ b 5 Ln(MACHIN )+ b 6 Ln (MULE)+….+e Where , Y =amount of wheat harvested in kg; Land size, family labour , Hire labour , Fertilize, Machine …,& Mule =Input factors, e= error arise This research was conducted in Egypt Modified OLS regression result fount that; 10 percent increase in land-size would increase output by 5.8 percent, a 10 percent increase in the amount of fertilizer applied would translate into a 1.8 percent output rise . Increased support to improving irrigation techniques would raise output by another 14 percent. Providing extension services to farmers who reported zero or only 1 visit would raise their output by 177 kg/ feddan . (Andres Croppenstedt , 2005)

16. This is case study of Wheat, in Thailand The basic Cobb-Douglas model specified is LnY = Lna + b 1 LnX 1 + b 2 LnX 2 + b 3 LnX 3 + b 4 LnX 4 + b 5 LnX 5 + b 6 LnX 6 + b 7 LnX 7 (bi)= production coefficients or exponents in the Cobb-Douglas form are the elasticities of production, X 1 = cost of feed (US $), X 2 = cost of fertilizer (US $), X 3 = cost of fingerling (US$) X 4 = cost of fuel (US $), X 5 = cost of labour (US $), X 6 = size of fingerling (cm) X 7 = survival rate (%) Ordinary least Squares(OLS) methods were used to estimate the regression mode Of seven explanatory variables, only four, feed cost ( X 1 ), fingerlings ( X 3 ), labour ( X 5 ), and survival rate ( X 7 ), were statistically significant. This model could explain 83 percent of the variation in yield. Cost of fingerlings was more powerful explanatory variable with the high partial output elasticity (0.4865), which indicates that 10 percent increase in cost of fingerlings (the stocking rate), holding other inputs constant, will increase yield by 4.86 percent . ( Pongpat et.al., 2007)

17. Discussion…… For the lumber and the veneer and plywood industries in Ontario the Cobb-Douglas functions have been found to be reasonable approximations Q = 0.1456K 0.3803 L 0.5015F F 0.1326 Where, Q is the output of lumber in a year(1,000 cubic meters), K is the value of capital services used in that year(1,000 dollars), L is the labor and F the quantity of round wood(1,000 cubic meters in a year) ( Conto , 2002)

18. Discussion……… CES is also widely used in manufacturing industry When ρ >1, increasing return to scale ρ <1, decreasing return to scale ρ =1, constant return to scale When ρ =0(i.e. σ =1) this function collapse into Cobb-Douglas function

19. Discussion…… For the province of Ontario the production process of its pulp and paper industry has been estimated as the CES function: Q= 2.353[0.18K -.9 +0.19L -.9+ 0.63F -0.9 ] -.83 Where, Q=Output of pulp and paper in thousand metric tones produced in a year In this industry the coefficient denoting the economies of scale is ρ =(-.83)*(-.9)=.75, means decreasing return to scale The constant elasticity of substitution between each pair of inputs is σ = =10, means that if the price ration of labor to capital changes by 1% then the ratio of capital to labor will change by 10%, a high degree of substitution indee d

20. Merits of CD and CES ( i ) It suits to the nature of all industries. (ii) It is convenient in international and inter-industry comparisons. (iii) It is the most commonly used function in the field of econometrics. (iv) It can be fitted to time series analysis and cross section analysis. (v) The function can be generalized in the case of ‘n’ factors of production. (vi) The unknown parameters a and p in the function can be easily computed. (vii) It becomes linear function in logarithm. (viii) It is more popular in empirical research. ( i ) CES production function is more general. (ii) CES covers all types of returns. (iii) CES function takes account of a number of parameters. (iv) CES function takes account of raw material among its inputs. (v) CES function is very easy for estimation. (vi) CES function is free from unrealistic assumptions

21. Limitations of CD and CES Considers only two inputs, labor and capital Constant returns to scale, limited scope Substitutability of factors but neglects the complementary of factors The assumption of stagnant technology is unrealistic Assumes all industries are homogenous Problems of aggregation Elasticity of substitution is equal to unity which is not possible everywhere Very difficult and complicated to use more than two inputs Capital intensity factor, α, is not dimensionless Value of A can not be made independent of Q or the units of Q, K and L Cannot be used to describe the aggregate production function of all the firms in the industry Assumes elasticity of substitution is constant but empirical study shows that the elasticity of substitution also changes due to changed factor production CD CES

22. Conclusion CD used in empirical studies of manufacturing industries and inter industries comparison D etermine the relative shares of labor and capital in to total outputs CES production function is more appropriate to reality as compared to CD function When empirical estimates are restricted to the CD form, it tends to be quite good For analysis of policies affecting factor returns, such as taxes on capital and labor income, CES is preferred However, the estimation of CES function parameters is more difficult, in this respect CD is preferred

23. Thanks everyone for your kind attention Any suggestion and queries??