**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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Consider the production function: F(L, K) = L0:2K0.7. %3D The wage rate (price per unit of labour) is w = 2 and the capital rental rate (price per unit of capital) is r = 7. (a) Does this production function exhibit increasing, decreasing or constant returns to scale? Explain. What is the marginal productivity of labour and the marginal productivity of capital for (L, K) = (1,1)? Would a firm (which minimises costs) use this combination of labour and capital? Explain. If your answer is yes, then what would be the quantity of production for which the company would use this combination? r (b) Compute the quantity of labour and capital that this firm would use to produce y = 2 at the minimum cost. How much would this cost be? What is the average cost and the marginal cost for that production level? Hint: for this part, you can use directly (without providing the derivation) any results derived in the lecture or tutorials. (c) Derive the equation of the isoquant for y = 2 (with K in the…
***PLEASE NOTE - An answer is NOT needed for parts A, B and C; these are included to assist with answering part D. Only an answer for part D is required, but it is derived from the previous answers*** Given: A farmer raises peaches using land (K) and labor (L), and has an output of ?(?,?)= ?0.5?0.5 bushels of apples. a. Find several input combinations that give the farmer 6 bushels of apples. Sketch the associated isoquant on a graph, with L on the x-axis and K on the y-axis. b. In the short run, the farmer only has 4 units of land. What is his short-run production function? Graph it for values of L from 0 to 16, with L on the x-axis and output on the y-axis. What is the name of the slope of this curve? c. Assuming the farmer still only has 4 units of land, how much extra output does he get from adding 1 extra unit of labor if he is already using only 1 unit of labor? How much extra output does he get from adding 1 extra unit of labor if he is already using 4 units of labor?…
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?
The Cobb-Douglas production function is a classic model from economics used to model output as a function of capital and labor. It has the form f(L, C)=²1C²2 where co. ₁, and care constants. The variable L represents the units of input of labor and the variable C represents the units of input of capital. (a) In this example, assume co5, c, 0.25, and c₂-0.75. Assume sach unit of labor costs $25 and each unit of capital costs $75. With $70,000 available in the budget, devalop an optimization model for determining how the budgeted amount should be allocated between capital and labor in order to maximize output. Max s.t. L, CZO € 70,000 (b) Find the optimal solution to the model you formulated in part (a). What is the optimal solution value (in units)? (Hint: When using Excel Solver, use the bounds 0S LS 3,000 and 0 s Cs 1,000. Round your answers to the nearest integer when necessary.) units at (L. C)=(
In economics and econometrics, the Cobb-Douglas production function is a particular functional form of ne production function, widely used to represent the technological relationship between the amounts of two r more inputs (particularly physical capital and labor) and the amount of output that can be produced by nose inputs. The function they used to model production is defined by, P(L, K) = 6LªK!-a where P is the total production (the monetary value of all goods produced in a year), L is the amount f labor (the total number of person-hours worked in a year), and K is the amount of capital invested (the onetary worth of all machinery, equipment, and buildings). Its domain is {(L, k)|L > 0, K > 0} because L nd K represent labor and capital and are therefore never negative. Show that the Cobb-Douglas production function can be written as P P(L, K) = 6LªK1-a → In K L In b+ a ln K
For each of the following production functions Q = f(K,L), find the marginal productivity with respect to K and L a) Q = 5KL – 2K² – 2L² at K = 1 and L = 1 %3D 1 b) Q = 0.03K³ – 0.4KL + 0.5L7 at K = 8 and L = 4 -
Suppose a firm’s production function is ? = ?2?. a) Determine the labor and capital demand functions. b) Are capital and labor normal or inferior inputs in this production process?