Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)
14th Edition
ISBN: 9781305506381
Author: James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Publisher: Cengage Learning
expand_more
expand_more
format_list_bulleted
Question
Chapter 7, Problem 10E
To determine
Whether the given function shows constant returns to scale or not at v=1
Expert Solution & Answer
Trending nowThis is a popular solution!
Students have asked these similar questions
10. Extension of the Cobb-Douglas Production Function-The Cobb-Douglas produc-
tion function (Equation 7.16) can be shown to be a special case of a larger class of
linear homogeneous production functions having the following mathematical
form:"
Q = y[@K* + (1 – a)L°]W/P
where y is an efficiency parameter that shows the output resulting from given
quantities of inputs; a is a distribution parameter (0 sas 1) that indicates the
division of factor income between capital and labor; p is a substitution parameter
that is a measure of substitutability of capital for labor (or vice versa) in the pro-
duction process; and v is a scale parameter (v > 0) that indicates the type of re-
turns to scale (increasing, constant, or decreasing). Show that when v = 1, this
function exhibits constant returns to scale. [Hint: Increase capital K and labor L
each by a factor of A, or K* = (A)K and L* = (1)L, and show that output Q also
increases by a factor of A, or Q* = (A)(Q).]
Use second image for reference,
for part b here is referene;
The maximum profit is found at the tangency between the production function and the
isoprofit line.
In other words, the slope of the production function and the slope of the
isoprofit line must be the same. This is written as
MPL = w
where w is the slope of the isoprofit line. Then we get
sqrt1 / 2L = w
=>
1/2w = sqrtL
=>
L*D = 1/4w^2
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
-
Chapter 7 Solutions
Managerial Economics: Applications, Strategies and Tactics (MindTap Course List)
Knowledge Booster
Similar questions
- 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=γ[δK−ρ+(1 - δ)L−ρ]−ν/ρ�=�[δK−ρ+(1 - δ)�−ρ]−ν/ρ where γ is an efficiency parameter that shows the output resulting from given quantities of inputs; δ is a distribution parameter (0 ≤ δ ≤ 1) that indicates the division of factor income between capital and labor; ρ is a substitution parameter that is a measure of substitutability of capital for labor (or vice versa) in the production process; and ν is a scale parameter (ν > 0) that indicates the type of returns to scale (increasing, constant, or decreasing). Complete the following derivation to show that when ν = 1, this function exhibits constant returns to scale. First of all, if ν = 1: Q� = = γ[δK−ρ+(1 - δ)L−ρ]−1/ρ�[δK−ρ+(1 - δ)�−ρ]−1/ρ = = γ[δK−ρ(−1/ρ)+(1 - δ)L−ρ(−1/ρ)]�[δK−ρ(−1/ρ)+(1 - δ)�−ρ(−1/ρ)] = = Then, increase the…arrow_forwardThe 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=y[SKP + (1-8)L-P]-v/P where y is an efficiency parameter that shows the output resulting from given quantities of inputs; ō is a distribution parameter (0 ≤ 6 ≤ 1) that indicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor (or 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 decreasing). Complete the following derivation to show that when v = 1, this function exhibits constant returns to scale. First of all, if v = 1: Q = Y/SK P+ (1-6)L-P]-1/P = Y/SK-P(-1/P) + (1 - 6)L-P(-1/P)] Then, 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,…arrow_forwardConsider the following production function of DVDS: Q = K0.5L 0.5, where Q represents DVDS (boxes per hour), K denotes capital input (units per hour), and L denotes labour input (units of worker time per hour). The unit cost of capital and labour are $40 and $20, respectively. a) What is the optimal input ratio of labour and capital for the production? b) Determine the appropriate input mix to produce 800 boxes of DVDS. Compute the cost of production. c) Suppose the government decided to offer a subsidy that would make the cost of labour $15. What is the optimal input mix to produce the same level of output, and the new cost of production? Also, compute the substitution effect in the employment of labour.arrow_forward
- (2) The general Cobb-Douglas production function for two inputs is given by q = f(k,I) = Ak“1® where 0 0, f; > 0, fx 0 (b) Show that (the elasticity of output with respect to capital) e = a and (the elasticity of output with respect to labor) e = Barrow_forwardA company manufactures two products, X and Y, and the joint-cost function of these products is given by c=x+4y where c is the total cost of producing v units of X and y units of Y Determine the marginal cost with respect to y when x=36 and y=16 b. The production function for a company's product is O-20, where is the output that results from /units of labor and & units of capital. The unit cost of labor and capital are $225 and $150, respectively. The company wants the total cost of puts to be $270,000 Find the number of units of capital and labor that maximize productionarrow_forward7. A fast-food company in Hermanus has the following cost function: TC = 10 + 3K + 2L, where L is the number of labour hours and K is the number of machines they have to use. They have a production function of Q = 2KL, where Q represents the number of meals produced per day. The company has to make at least 75 meals every day. Using the Lagrangian method of production optimization, calculate the optimal number of labourers and machines needed to meet this quota. (6)arrow_forward
- Problem 5 The production function of COVID vaccines for firm O is given by v° = VKL. and requires at least one unit of labor L and one unit of capital K, i.e. L>1 und K > 1. a) After a year and some thorough research, the production function changes to VO = V4KL. (I) Compute the marginal products with respect to labor L and capital K for both production functions ! (II) Which company's the marginal products is/are larger ? What happened to the original level of production after a year ? (III) Do the production functions exhibit increasing, constant or decreasing returns to scale ? b) Suppose now, we compare production functions of company O that discovered the vaccine to a second company M. The production function of M is given by VM = K0.6 L0.4 (I) Compute the marginal products with respect to labor L and capital K for this pro- duction function ! (II) If both companies used the same equal amount of labor L and capital K, which of them will generate more output ? (III) If the…arrow_forwardQUESTION 12: Let the production function be as follows Q = K²/3L1/3 Find the cost-minimizing level of K and L that costs C=500 when w=1, r=2arrow_forwardSuppose a brewery uses a Cobb-Douglas production function for his production. He studies the production process and finds the following. An additional machine-hour of fermentation capacity would increase output by 600 bottles per day (i. e. MPK = 600). An additional man-hour of labor would increase output by 1200 bottles per day (i. e. MP₁ = 1200). The price of a man-hour of labor is $40 per hour. The price of a machine-hour of fermentation capacity is $8 per hour. 1. Is the brewery currently minimizing its cost of production? Check using the minimization condition. 2. It turns out, the brewery is not optimally choosing the factors of production. To lower its production cost, which factor of production should the brewery increase and which factor should he decrease? 3. Suppose that the price of a machine-hour of fermentation capacity rises to $20 per hour. How does this change the answer from part 1?arrow_forward
- Does the production function y=0.8x_1x_2 exhibit constant returns to scale, increasing returns to scale, decreasing returns to scale, or homothetic returns to scalearrow_forward1) The Cobb-Douglas production function is y = 12 with a > 0 and 3 > 0. Show that MP₁ = ay/2₁, MP₂= By/22. What is the MRT S21? How does it vary with: (a) y: (b) 2/1. 2) Show that the cost function for a firm with the constant returns Cobb-Douglas production function y = Actis given by C(p, y) = ypip-B, where B is a function of A and a only. Derive the conditional input demands.arrow_forwardThe 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 = y[SK¯ + (1 - 8)L¯P]¯V / P where y is an efficiency parameter that shows the output resulting from given quantities of inputs; d is a distribution parameter (0 ≤ d ≤ 1) that indicates the division of factor income between capital and labor; p is a substitution parameter that is a measure of substitutability of capital for labor (or 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 decreasing). Complete the following derivation to show that when v = 1, this function exhibits constant returns to scale. First of all, if v = 1: Q = Y[SK¯º + (1 - 8)L-P]-1/p Y[SK−P(−1/P) + (1 - 8)L−p(−1/p)] Then, increase the capital K and labor L each by a factor of λ, or K* = (A)K and L* = (A)L. If the function exhibits constant returns to…arrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Managerial Economics: Applications, Strategies an...EconomicsISBN:9781305506381Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. HarrisPublisher:Cengage Learning
Managerial Economics: Applications, Strategies an...
Economics
ISBN:9781305506381
Author:James R. McGuigan, R. Charles Moyer, Frederick H.deB. Harris
Publisher:Cengage Learning