A firm has decided through regression analysis that its sales (S) are a function of the amount of advertising (measured in units) in two different media,
television (x) and magazines (y): S(x, y) = 100 - x^2 + 30x - y^2 + 40y
(a) Find the level of TV and magazine advertising units that maximizes the firm's sales.
(b) Suppose that the advertising budget is restricted to 31 units. Determine the level of advertising (in units) that maximizes sales subject to this budget constraint.
(c) Give an economic interpretation for the value of the Lagrangian Multiplier obtained in part (b) above.
(d) The marketing department of the firm is lobbying to have the advertising budget increased to 40. Do you agree with the marketing department? If not, what advertising budget would you recommend and why?