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Cobb-Douglas Production Function
This document discusses the Cobb-Douglas production function and its application to estimating production at a lumber company. It begins by defining the Cobb-Douglas production function and its typical form. It then presents the specific production function estimated for Washington-Pacific Lumber, which models lumber output (Q) as a function of labor hours (L), machine hours (K), and energy input (BTUs) (E). It provides the estimated coefficients and standard errors from regressing data. The rest of the document works through examples calculating the effect on output from changes in inputs and determining the returns to scale for this production system based on summing the exponent coefficients.
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Cobb-Douglas Production Function
- 1. Cobb-Douglas Production Function Pasakorn S. 5520212001 NabduanD. 5520212002 Ata K. 552022009
- 2. What is Cobb-DouglasProduction Function? During 1900–1947, Charles Cobb and Paul Douglas formulated and tested the Cobb– Douglas production function through various statistical evidence. Q = b0 X Y b1 b2 The Cobb–Douglas functional form of production functions is widely used to represent the relationship of an output and two inputs.
- 3. Question 7.2 Production FunctionEstimation. WashingtonPacific, Inc., manufactures and sells lumber, plywood, veneer, particle board, medium-density fiber board, and laminated beams. The company has estimated the following multiplicative production function for basic lumber products in the Pacific Northwest market: Q = b0 L K E b1 b2 b3 Q = output, L = labor input in worker hours, K = capital input in machine hours and E = energy input in BTUs (British Thermal Unit)
- 4. Each of theparameters of this model was estimated by regression analysis using monthly data over a 3-years period. Coefficient estimation results were as follows: ˆ ˆ ˆ ˆ b0 = 0.9; b1 = 0.4; b2 = 0.4; b3 = 0.2 The standard error estimates for each coefficient are σ b 0 = 0.6; σ b1 = 0.1; σ b 2 = 0.2; σ b 3 = 0.1
- 5. Question 1. Estimatethe effect on output of a 1% decline in worker hours (holding K and E constant) Given, Q = b0 L K E b1 b2 b3 Take the first derivation with respect to worker hours (L) Q = 0L K b b1 b2 E b3
- 6. ∂ Q 1 =b0b1 Lb1 −K b2 E b3 ∂ L ∂ Q 1 =b1b0 Lb1 K b2 E b3 L− ∂ L ∂ Q 1 =b1QL− ∂ L ∂ Q Q ∂ Q ∂ L =b1 =b1 * Q L ∂ L L ∂ Q ∂ Q L = 0.4( − .01) 0 * =b1 Q ∂ L Q ∂ Q = − .004 = − .4% 0 0 ∂ Q ∂ L Q b1 = ÷ Q L
- 7. Question 2 .Estimate the effect on output of a 5% reduction in machine hours availability accompanied by a 5% decline in energy input (holding L constant) Solution: From part A it is clear that, ∂Q = b2 (∆K / K ) + b3 (∆E / E ) Q ∂Q = 0.4(−0.05) + 0.2(−0.05) Q ∂Q = −0.03 = −3% Q ˆ b0 = 0.9 ˆ b = 0 .4 1 ˆ b2 = 0.4 ˆ b3 = 0.2
- 8. Question 3. Estimatethe returns to scale for this production system. Solution: In case of Cobb Douglas production function, the returns to scale are determined by summing up exponents because: Q =b0 L K b1 b2 E b3 hQ =b0 ( kL) ( kK ) b1 b2 hQ =k b1 + 2 + 3 b b b0 L K hQ =k b1 + 2 + 3 b b Q b1 ( kE ) b2 E b3 b3
- 9. Thus, summing upthe value of the exponents, we get, b1 + b2 + b3 = 0.4 + 0.4 + 0.2 = 1 hQ = k Q 1 h=k 1 This indicates constant returns to scale estimation.
- 10. Graph hQ = kn.f(X.Y.Z) Constant n=1h=k Increasing n>1 h>k decreasing n<1 h<k returns-to-scale estimation
- 11. Conclusion Returns to Scale isthe quantitative change in output of a firm or industry resulting from a proportionate increase in all inputs. Adding the value of the exponents, we can determine the returns to scale of a production function.
- 12.