(a) Explanation Given Information: A company has the production function, z = 300 x 2 / 3 y 1 / 3 Where, x is the number of units of labor, y is the number of units of capital, and z is the units of production. The labor costs $ 50 per unit, capital costs are $ 50 per unit, and the total cost are limited to $ 75 , 000 . Formula used: According to the Lagrange multipliers method to obtain maxima or minima for a function z = f ( x , y ) subject to the constraint g ( x , y ) = 0 , Step 1: Find the critical values of f ( x , y ) using the new variable λ to form the objective function F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . Step 2: The critical points of f ( x , y ) are the critical values of F ( x , y , λ ) which satisfies g ( x , y ) = 0 . Step 3: The critical points of F ( x , y , λ ) are the points that satisfy ∂ F ∂ x = 0 , ∂ F ∂ y = 0 , and ∂ F ∂ λ = 0 , that is, the points which make all the partial derivatives of zero. For a function f ( x , y ) , the partial derivative of f with respect to y is calculated by taking the derivative of f ( x , y ) with respect to y and keeping the other variable x constant. The partial derivative of f with respect to y is denoted by f y . Power of x rule for a real number n is such that, if f ( x ) = x n then f ′ ( x ) = n x n − 1 . Constant function rule for a constant c is such that, if f ( x ) = c then f ′ ( x ) = 0 . Coefficient rule for a constant c is such that, if f ( x ) = c ⋅ u ( x ) , where u ( x ) is a differentiable function of x , then f ′ ( x ) = c ⋅ u ′ ( x ) . Calculation: Consider the function, z = 300 x 2 / 3 y 1 / 3 Also, if number of units of labor is x , which costs $ 50 per unit, and if number of units of capital is y , which costs $ 50 per unit, then the total cost is 50 x + 50 y . But the total cost of labor and capital is limited to $ 75 , 000 . Thus, 50 x + 50 y = 75 , 000 . Thus, the constraint is 50 x + 50 y = 75 , 000 . According to the Lagrange multipliers method, The objective function is F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) . Thus, f ( x , y ) = 300 x 2 / 3 y 1 / 3 and g ( x , y ) = 50 x + 50 y − 75 , 000 . Substitute 300 x 2 / 3 y 1 / 3 for f ( x , y ) and 50 x + 50 y − 75 , 000 for g ( x , y ) in F ( x , y , λ ) = f ( x , y ) + λ g ( x , y ) , F ( x , y , λ ) = 300 x 2 / 3 y 1 / 3 + λ ( 50 x + 50 y − 75 , 000 ) Since, the critical points of F ( x , y , λ ) are the points that satisfy ∂ F ∂ x = 0 , ∂ F ∂ y = 0 , and ∂ F ∂ λ = 0 (b) To determine To calculate: The marginal productivity of money and interpret the answer if the production function is z = 300 x 2 / 3 y 1 / 3 . (c) To determine To graph: The constraint with the production function when z = 180 , 000 , z = 300 , 000 , and when the z-value is optimal if the production function is z = 300 x 2 / 3 y 1 / 3 .

Mathematical Applications for the Management, Life, and Social Sciences

12th Edition

ISBN: 9781337625340

Author: Ronald J. Harshbarger, James J. Reynolds

Publisher: Cengage Learning

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Question

Chapter 14, Problem 34RE

(a)

To determine

To calculate: The number of units of labor and the number of units of capital that maximize production if the production function is z=300x2/3y1/3.

(b)

To determine

To calculate: The marginal productivity of money and interpret the answer if the production function is z=300x2/3y1/3.

(c)

To determine

To graph: The constraint with the production function when z=180,000, z=300,000, and when the z-value is optimal if the production function is z=300x2/3y1/3.

Expert Solution & Answer

Chapter 14 Solutions

Mathematical Applications for the Management, Life, and Social Sciences

Ch. 14.1 - CHECKPOINT 1. Find the domain of the function Ch. 14.1 - CHECKPOINT 2. (a) If . (b) If . Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E Ch. 14.1 - Prob. 6E Ch. 14.1 - Prob. 7E Ch. 14.1 - Prob. 8E

Ch. 14.1 - Prob. 9E Ch. 14.1 - Prob. 10E Ch. 14.1 - Prob. 11E Ch. 14.1 - Prob. 12E Ch. 14.1 - Prob. 13E Ch. 14.1 - Prob. 14E Ch. 14.1 - Prob. 15E Ch. 14.1 - Prob. 16E Ch. 14.1 - In Problems 15-22, evaluate each function as...Ch. 14.1 - Prob. 18E Ch. 14.1 - Prob. 19E Ch. 14.1 - Prob. 20E Ch. 14.1 - Prob. 21E Ch. 14.1 - In Problems 15-22, evaluate each function as...Ch. 14.1 - Prob. 23E Ch. 14.1 - Prob. 24E Ch. 14.1 - 25. Curve speeds One method traffic planners use...Ch. 14.1 - Prob. 26E Ch. 14.1 - Prob. 27E Ch. 14.1 - Prob. 28E Ch. 14.1 - 29. Mortgage The following tables show that a...Ch. 14.1 - 30. Wind chill Wind and cold temperatures combine...Ch. 14.1 - Prob. 31E Ch. 14.1 - Prob. 32E Ch. 14.1 - Prob. 33E Ch. 14.1 - Prob. 34E Ch. 14.1 - Prob. 35E Ch. 14.1 - 36. Profit The Kirk Kelly Kandy Company makes two...Ch. 14.1 - Prob. 37E Ch. 14.1 - Prob. 38E Ch. 14.2 - Prob. 1CP Ch. 14.2 - Prob. 2CP Ch. 14.2 - Prob. 3CP Ch. 14.2 - Prob. 4CP Ch. 14.2 - Prob. 5CP Ch. 14.2 - Prob. 1E Ch. 14.2 - Prob. 2E Ch. 14.2 - Prob. 3E Ch. 14.2 - Prob. 4E Ch. 14.2 - Prob. 5E Ch. 14.2 - Prob. 6E Ch. 14.2 - Prob. 7E Ch. 14.2 - Prob. 8E Ch. 14.2 - Prob. 9E Ch. 14.2 - Prob. 10E Ch. 14.2 - Prob. 11E Ch. 14.2 - Prob. 12E Ch. 14.2 - Prob. 13E Ch. 14.2 - Prob. 14E Ch. 14.2 - Prob. 15E Ch. 14.2 - Prob. 16E Ch. 14.2 - Prob. 17E Ch. 14.2 - Prob. 18E Ch. 14.2 - 19. Find the slope of the tangent in the...Ch. 14.2 - 20. Find the slope of the tangent in the...Ch. 14.2 - Prob. 21E Ch. 14.2 - Prob. 22E Ch. 14.2 - Prob. 23E Ch. 14.2 - Prob. 24E Ch. 14.2 - Prob. 25E Ch. 14.2 - Prob. 26E Ch. 14.2 - Prob. 27E Ch. 14.2 - Prob. 28E Ch. 14.2 - Prob. 29E Ch. 14.2 - Prob. 30E Ch. 14.2 - Prob. 31E Ch. 14.2 - 32. If , find the following. (a) (b) (c) (d) Ch. 14.2 - 33. If , find the following. Ch. 14.2 - 34. If , find the following. Ch. 14.2 - Prob. 35E Ch. 14.2 - Ch. 14.2 - Ch. 14.2 - Prob. 38E Ch. 14.2 - Prob. 39E Ch. 14.2 - Prob. 40E Ch. 14.2 - 41. . Ch. 14.2 - . Ch. 14.2 - Prob. 43E Ch. 14.2 - Ch. 14.2 - 45. Mortgage When a homeowner has a 25-year...Ch. 14.2 - 46. Mass transportation ridership Suppose that in...Ch. 14.2 - 47. Wilson's lot size formula In economics, the...Ch. 14.2 - 48. Cost Suppose that the total cost (in dollars)...Ch. 14.2 - 49. Pesticide Suppose that the number of thousands...Ch. 14.2 - 50. Profit Suppose that the profit (in dollars)...Ch. 14.2 - Prob. 51E Ch. 14.2 - Prob. 52E Ch. 14.2 - 53. Production Suppose that the output Q (in...Ch. 14.2 - Prob. 54E Ch. 14.2 - Prob. 55E Ch. 14.2 - Prob. 56E Ch. 14.3 - CHECKPOINT If the joint cost in dollars for two...Ch. 14.3 - Prob. 2CP Ch. 14.3 - Prob. 3CP Ch. 14.3 - Prob. 1E Ch. 14.3 - Prob. 2E Ch. 14.3 - 3. The total cost of producing 1 unit of a product...Ch. 14.3 - Prob. 4E Ch. 14.3 - Prob. 5E Ch. 14.3 - Prob. 6E Ch. 14.3 - Prob. 7E Ch. 14.3 - Prob. 8E Ch. 14.3 - 9. If the joint cost function for two products is ...Ch. 14.3 - 10. Suppose the joint cost function for x units of...Ch. 14.3 - 11. Suppose that the joint cost function for two...Ch. 14.3 - Prob. 12E Ch. 14.3 - Prob. 13E Ch. 14.3 - Prob. 14E Ch. 14.3 - Prob. 15E Ch. 14.3 - Prob. 16E Ch. 14.3 - Prob. 17E Ch. 14.3 - Prob. 18E Ch. 14.3 - Prob. 19E Ch. 14.3 - Prob. 20E Ch. 14.3 - 21. Suppose the Cobb-Douglas production function...Ch. 14.3 - Prob. 22E Ch. 14.3 - Prob. 23E Ch. 14.3 - Prob. 24E Ch. 14.3 - Prob. 25E Ch. 14.3 - Prob. 26E Ch. 14.3 - Prob. 27E Ch. 14.3 - Prob. 28E Ch. 14.3 - Prob. 29E Ch. 14.3 - Prob. 30E Ch. 14.4 - CHECKPOINT Suppose that Find Ch. 14.4 - Prob. 2CP Ch. 14.4 - Prob. 3CP Ch. 14.4 - Prob. 4CP Ch. 14.4 - Prob. 1E Ch. 14.4 - Prob. 2E Ch. 14.4 - Prob. 3E Ch. 14.4 - Prob. 4E Ch. 14.4 - Prob. 5E Ch. 14.4 - Prob. 6E Ch. 14.4 - Prob. 7E Ch. 14.4 - Prob. 8E Ch. 14.4 - Prob. 9E Ch. 14.4 - Prob. 10E Ch. 14.4 - Prob. 11E Ch. 14.4 - Prob. 12E Ch. 14.4 - Prob. 13E Ch. 14.4 - Prob. 14E Ch. 14.4 - Prob. 15E Ch. 14.4 - Prob. 16E Ch. 14.4 - Prob. 17E Ch. 14.4 - Prob. 18E Ch. 14.4 - 19. Profit Suppose that the quarterly profit from...Ch. 14.4 - Prob. 20E Ch. 14.4 - 21. Nutrition A new food is designed to add weight...Ch. 14.4 - Prob. 22E Ch. 14.4 - 23. Production Suppose that tons is the...Ch. 14.4 - 24. Production Suppose that x units of one input...Ch. 14.4 - 25. Profit Suppose that a manufacturer produces...Ch. 14.4 - Prob. 26E Ch. 14.4 - Prob. 27E Ch. 14.4 - Prob. 28E Ch. 14.4 - 29. Profit A company manufactures two products, A...Ch. 14.4 - Prob. 30E Ch. 14.4 - Prob. 31E Ch. 14.4 - The manager of the Sea Islands Chicken Shack is...Ch. 14.4 - Prob. 33E Ch. 14.4 - Prob. 34E Ch. 14.4 - Prob. 35E Ch. 14.4 - Prob. 36E Ch. 14.5 - Prob. 1CP Ch. 14.5 - Prob. 2CP Ch. 14.5 - Prob. 3CP Ch. 14.5 - Prob. 4CP Ch. 14.5 - Prob. 1E Ch. 14.5 - Prob. 2E Ch. 14.5 - Prob. 3E Ch. 14.5 - Prob. 4E Ch. 14.5 - Prob. 5E Ch. 14.5 - Prob. 6E Ch. 14.5 - Prob. 7E Ch. 14.5 - Prob. 8E Ch. 14.5 - Prob. 9E Ch. 14.5 - Prob. 10E Ch. 14.5 - Prob. 11E Ch. 14.5 - Prob. 12E Ch. 14.5 - Prob. 13E Ch. 14.5 - Prob. 14E Ch. 14.5 - Prob. 15E Ch. 14.5 - 16. Utility Suppose that the budget constraint in...Ch. 14.5 - 17. Utility Suppose that the utility function for...Ch. 14.5 - 18. Utility Suppose that the utility function for...Ch. 14.5 - Prob. 19E Ch. 14.5 - Prob. 20E Ch. 14.5 - 21. Cost A firm has two plants, X and Y. Suppose...Ch. 14.5 - Prob. 22E Ch. 14.5 - Prob. 23E Ch. 14.5 - Prob. 24E Ch. 14.5 - 25. Manufacturing Find the dimensions (in...Ch. 14.5 - Prob. 26E Ch. 14 - 1. What is the domain of ? Ch. 14 - Prob. 2RE Ch. 14 - Prob. 3RE Ch. 14 - 4. If . Ch. 14 - Prob. 5RE Ch. 14 - Prob. 6RE Ch. 14 - Prob. 7RE Ch. 14 - Prob. 8RE Ch. 14 - Prob. 9RE Ch. 14 - Prob. 10RE Ch. 14 - Prob. 11RE Ch. 14 - Prob. 12RE Ch. 14 - Prob. 13RE Ch. 14 - Prob. 14RE Ch. 14 - In Problems 15-18, find the second partials. Ch. 14 - Prob. 16RE Ch. 14 - Prob. 17RE Ch. 14 - Prob. 18RE Ch. 14 - Prob. 19RE Ch. 14 - Prob. 20RE Ch. 14 - Prob. 21RE Ch. 14 - Prob. 22RE Ch. 14 - Prob. 23RE Ch. 14 - Prob. 24RE Ch. 14 - Prob. 25RE Ch. 14 - Prob. 26RE Ch. 14 - Prob. 27RE Ch. 14 - Prob. 28RE Ch. 14 - Prob. 29RE Ch. 14 - Prob. 30RE Ch. 14 - Prob. 31RE Ch. 14 - Prob. 32RE Ch. 14 - Prob. 33RE Ch. 14 - Prob. 34RE Ch. 14 - 35. Modeling US. average wage The table gives the...Ch. 14 - Prob. 36RE Ch. 14 - Prob. 1T Ch. 14 - Prob. 2T Ch. 14 - Prob. 3T Ch. 14 - Prob. 4T Ch. 14 - Prob. 5T Ch. 14 - Prob. 6T Ch. 14 - 7. Suppose the demand functions for two products...Ch. 14 - Prob. 8T Ch. 14 - 9. Find x and y that maximize the utility function...Ch. 14 - Prob. 10T

Knowledge Booster

The number of units of labor and the number of units of capital that maximize production if the production function is z = 300 x 2 / 3 y 1 / 3 .

To calculate: The number of units of labor and the number of units of capital that maximize production if the production function is z=300x2/3y1/3.

To calculate: The marginal productivity of money and interpret the answer if the production function is z=300x2/3y1/3.

To graph: The constraint with the production function when z=180,000, z=300,000, and when the z-value is optimal if the production function is z=300x2/3y1/3.

Chapter 14 Solutions