**Cobb-Douglas Production Function and Returns to Scale**The Cobb-Douglas production function can be shown to be a special case of a larger class of linear homogeneous production functions having the following mathematical form:\[ Q = \gamma [\delta K^{-\rho} + (1 - \delta)L^{-\rho}]^{-\nu / \rho} \]where $ \gamma $ is an efficiency parameter that shows the output resulting from given quantities of inputs; $ \delta $ is a distribution parameter $ (0 \leq \delta \leq 1) $ that indicates the division of factor income between capital and labor; $ \rho $ is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and $ \nu $ is a scale parameter $ (\nu > 0) $ that indicates the type of returns to scale (increasing, constant, or decreasing).**Objective:**Complete the following derivation to show that when $ \nu = 1 $, this function exhibits constant returns to scale.**Derivation:**First of all, if $ \nu = 1 $:\[Q = \gamma [\delta K^{-\rho} + (1 - \delta)L^{-\rho}]^{-1 / \rho}\]\[= \gamma [\delta K^{-\rho (1/\rho)} + (1 - \delta)L^{-\rho (1/ \rho)}]\]\[= \gamma [\delta K^{-1} + (1 - \delta)L^{-1}]\]Then, increase the capital $ K $ and labor $ L $ each by a factor of $ \lambda $, or $ K^* = (\lambda)K $ and $ L^* = (\lambda)L $. If the function exhibits constant returns to scale, then $ Q^* = (\lambda)Q $.\[Q^* = \gamma [\delta (\lambda)K^{-1} + (1 - \delta)(\lambda)L^{-1}]\]\[= \gamma [\delta(\lambda^{-1/\rho}) + (1-\delta)\lambda^{-1/\rho}]\]\[= \gamma (\lambda^{-1/\rho}) [\delta K^{-1/\rho} + (1-\delta)L^{-

Where Y is an Efficiency Parameter ∂ is Distribution Parameter P is the Substitution Parameter

lowing mathematical form: Q=y[SK+ (1-8)L-P]-V/P here y is an efficiency parameter that shows the output resulting from given quantities of inputs; ō is a distribution parameter (0 ≤ 6 ≤ 1) that dicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor vice versa) in the production process; and v is a scale parameter (v> 0) that indicates the type of returns to scale (increasing, constant, or creasing). mplete the following derivation to show that when v= 1, this function exhibits constant returns to scale. st of all, if v 1: = Y/SK P + (1-6)L-P]-1/P = Y/SK-P(-1/P) + (1-8)L-P(-1/P)] en, increase the capital K and labor L each by a factor of A, or K* = (A)K and L = (A)L. If the function exhibits constant returns to scale, then Q* = Q. * = y[5(A)K¯² + (1 - 5)(X)L¯P]-¹/P = y[SAK-P(-1/P) + (1 - 8)AL¯P(-1/P)] = AQ

lowing mathematical form: Q=y[SK+ (1-8)L-P]-V/P here y is an efficiency parameter that shows the output resulting from given quantities of inputs; ō is a distribution parameter (0 ≤ 6 ≤ 1) that dicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor vice versa) in the production process; and v is a scale parameter (v> 0) that indicates the type of returns to scale (increasing, constant, or creasing). mplete the following derivation to show that when v= 1, this function exhibits constant returns to scale. st of all, if v 1: = Y/SK P + (1-6)L-P]-1/P = Y/SK-P(-1/P) + (1-8)L-P(-1/P)] en, increase the capital K and labor L each by a factor of A, or K* = (A)K and L = (A)L. If the function exhibits constant returns to scale, then Q* = Q. * = y[5(A)K¯² + (1 - 5)(X)L¯P]-¹/P = y[SAK-P(-1/P) + (1 - 8)AL¯P(-1/P)] = AQ

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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answer question d Q5. Jason is running a cleaning business, there are available labours (L) and machines (M) for Jason to use as inputs to produce cleaning service for his clients. a) Suppose Jason must use both labours and machines without any specific ratio to complete the service for his clients, please write down the general production function formula for Jason. hint: you could use any letter if you want] b) Under the function form of a), assume Jason's Marginal Rate of Technical Substitution (MRTSL, M) equals to 2, how do you interpret it? c) Assume Jason now can use only single input, either labour or machine to complete the service, please write down the general production function formula for Jason. hint: you could use any letter if you want] d) Under the function form of c), assume Jason's Marginal Rate of Technical Substitution (MRTSL, M) is always larger than the market price ratio between labour and machine (w/r), what should Jason do? why?
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