A manufacturer of personal computers currently sells 10,000 units per month of a basic model. The cost of manufacture is $700/unit, and the wholesale price is $950. During the last quarter the manufacturer lowered the price $100 in a few test markets, and the result was a 50% increase in sales. The company has been advertising its product nationwide at a cost of $50,000 per month. The advertising agency claims that increasing the advertising budget by $10,000/month would result in a sales increase of 200 units/month. Management has agreed to consider an increase in the advertising budget to no more than $100,000/month.
(a) Determine the price and the advertising budget that will maximize profit. Use the five-step method. Model as a constrained optimization problem, and solve using the method of Lagrange multipliers.
(b) Determine the sensitivity of the decision variables (price and advertising) to price elasticity (the 50% number).
(c) Determine the sensitivity of the decision variables to the advertising agency's estimate of 200 new sales each time the advertising budget is increased by $10,000 per month.
(d) What is the value of the multiplier found in part (a)? What is the real-world significance of the multiplier? How could you use this information to convince top management to lift the ceiling on advertising expenditures?