A company will soon be introducing a new product into a very competitive market and is currently planning its marketing strategy. The decision has been made to introduce the product in three phases. Phase 1 will feature making a special introductory offer of the product to the public at a greatly reduced price to attract first-time buyers. Phase 2 will involve an intensive advertising campaign to persuade these first-time buyers to continue purchasing the product at a regular price. It is known that another company will be introducing a new competitive product at about the time that phase 2 will end. Therefore, phase 3 will involve a follow-up advertising and promotion campaign to try to keep the regular purchasers from switching to the competitive product.
A total of $4 million has been budgeted for this marketing campaign. The problem now is to determine how to allocate this money most effectively to the three phases. Let m denote the initial share of the market (expressed as a percentage) attained in phase 1, f2 the fraction of this market share that is retained in phase 2, and f3 the fraction of the remaining market share that is retained in phase 3. Use dynamic programming to determine how to allocate the $4 million to maximize the final share of the market for the new product, i.e., to maximize mf2 f3.
(a) Assume that the money must be spent in integer multiples of $1 million in each phase, where the minimum permissible multiple is 1 for phase 1 and 0 for phases 2 and 3. The following table gives the estimated effect of expenditures in each phase:

(b) Now assume that any amount within the total budget can be spent in each phase, where the estimated effect of spending an amount xi (in units of millions of dollars) in phase i (i = 1, 2, 3) is
m = 10x1 €“ x21
f2 = 0.40 + 0.10x2
f3 = 0.60 + 0.07x3.

Effect on Market Share Millions of Dollars Expended In 2 0 20 30 40 50 0.2 0.4 0.5 0.6 0.3 0.5 0.6 0.7 4