Question

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) = c₀L^c₁C^c₂

where c₀, c₁, and c₂ are constants. The variable L represents the units of input of labor, and the variable C represents the units of input of capital. In this example, assume c₀ = 5, c₁ = 0.25, and c₂ = 0.75. Assume each unit of labor costs $20 and each unit of capital costs $70. With $75,000 available in the budget, develop an optimization model to determine how the budgeted amount should be allocated between capital and labor in order to maximize output.

f(L,C) =  × L × C

Find the optimal solution to the model you formulated. (Hint: When using Excel Solver, use bounds 0 ≤ L ≤ 3,000 and 0 ≤ C ≤ 1,000.) Round your answers to the nearest whole number, if necessary.

Input	Amount
Labor		units
Capital		units

	Amount
Output		units

GreenLawns provides a lawn fertilizing and weed control service. The company is adding a special aeration treatment as a low-cost extra service option, which it hopes will help attract new customers. Management is planning to promote this new service in two media: radio and direct-mail advertising. A media budget of $3,000 is available for this promotional campaign. Based on past experience in promoting its other services, GreenLawns has obtained the following estimate of the relationship between sales and the amount spent on promotion in these two media:

S = -2R ² - 10M ² - 8RM + 18R + 34M,

Where

   S = total sales in thousands of dollars

   R = thousands of dollars spent on radio advertising

   M = thousands of dollars spent on direct-mail advertising

GreenLawns would like to develop a promotional strategy that will lead to maximum sales subject to the restriction provided by the media budget.

(a)	What is the value of sales if $2,000 is spent on radio advertising and $1,000 is spent on direct-mail advertising? Enter amounts in thousands i. e. $1,012, should be written as $1,012.
	$
(b)	Formulate an optimization problem that can be solved to maximize sales subject to the media budget of spending no more than $3,000 on total advertising. If the constant is "1" it must be entered in the box. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300). If your answer is zero enter “0”.
	Max R 2 + M 2 + RM +  R +  M s.t R + M - Select your answer -≤≥<>=Item 9 R, M - Select your answer -≤≥<>=Item 11
(c)	Determine the optimal amount to spend on radio and direct-mail advertising. How much in sales will be generated? Enter amounts in thousands i. e. $1,012, should be written as $1,012.
	Amount spent on radio advertising = $
	Amount spent on direct-mail advertising = $
	Total Sales = $

Answer 1

2.