A well-known charity is interested in conducting a television campaign to solicit contributions. The campaign will be conducted in two metropolitan areas. Past experience indicates that, in each city, the total contributions are a function of the amount of money expended for TV advertisements. Specifically, the charity has estimated response functions that indicate the percentage of the population making a donation as a function of the dollars spent on TV advertising. The form of this function is y = 1 exp (αx), where x represents the advertising expenditure (in thousands of dollars), y represents the percentage donating, and α is a parameter that differs from city to city. The charity has earmarked a fund of $400,000 for advertising between the two cities and wants to determine the best allocation of these funds.

a. Suppose the charity decides to maximize total donations, given its budget limit on advertising. How should advertising funds be spent, and what amount will be raised in donations as a result?
b. Is the budget binding in the solution of (a)? If so, how much additional budget would the charity like to obtain? If not, how much of the budget should actually be spent?
c. Suppose the charity maximizes total donations net of the cost of advertising. (This would make sense if surplus advertising funds can be put toward the charity's other services.) From the perspective of this objective, how should advertising funds be spent, and what amount will be raised in donations as a result?

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City 2 City 1 Response parameter (a) Population Average donation per donor 0.006 0.004 750,000 600,000 $1.50 $2.00