A. Set up a table or spreadsheet for CSI, that illustrates the relationships among quantity (Q), price (P), the optimal level of advertising (A), the advertising-sales ratio (A/S), and sales revenue (S). In this spreadsheet, use formula functions to set

Q = 5,000  10P + 40A + PA 0.8A2  0.5P2

A = $25 +$0.625P

A/S = (100×A)/S

S = P×Q

Establish a range for P from 0 to $125 in increments of $5 (i.e., $0, $5, $10, ..., $125). To test the sensitivity of all other variables to extreme bounds for the price variable, also set price equal to $1,000, $2,500, $10,000.

B. Based on the CSI table or spreadsheet, determine the price-advertising combination that will maximize the number of units sold.

C. Give an analytical explanation of the negative quantity and sales revenue levels observed at very high price-advertising combinations. Do these negative values have an economic interpretation as well?

The concept of multivariate optimization is important in managerial economics because many demand and supply relations involve more than two variables. In demand analysis, it is typical to consider the quantity sold as a function of the price of the product itself, the price of other goods, advertising, income, and other factors. In cost analysis, cost is determined by output, input prices, the nature of technology, and so on..

To explore the concepts of multivariate optimization and the optimal level of advertising, consider a hypothetical multivariate product demand function for CSI, Inc., where the demand for product Q is determined by the price charged, P, and the level of advertising, A:

Q = 5,000  10P + 40A + PA  0.8A2 - 0.5P2

When analyzing multivariate relations such as these, one is interested in the marginal effect of each independent variable on the quantity sold, the dependent variable. Optimization requires an analysis of how a change in each independent variable affects the dependent variable, holding constant the effect of all other independent variables. The partial derivative concept is used in this type of marginal analysis.

In light of the fact that the CSI demand function includes two independent variables, the price of the product itself and advertising, it is possible to examine two partial derivatives: the partial of Q with respect to price, or ΔQ/ΔP, and the partial of Q with respect to advertising expenditures, or ΔQ/ΔA.

In determining partial derivatives, all variables except the one with respect to which the derivative is being taken remain unchanged. In this instance, A is treated as a constant when the partial derivative of Q with respect to P is analyzed; P is treated as a constant when the partial derivative of Q with respect to A is evaluated. Therefore, the partial derivative of Q with respect to P is:

ΔQ/ΔP = 0 - 10 + 0 + A - 0 - P

= -10 + A - P

The partial with respect to A is:

ΔQ/ΔA
=0 - 0 + 40 + P - 1.6A – 0

= 40 + P - 1.6A

Solving these two equations simultaneously yields the optimal price-output-advertising combination. Because -10 + A - P = 0, P = A -10. Substituting this value for P into 40 + P  1.6A = 0, gives 40 + (A - 10) - 1.6A = 0, which implies that 0.6A = 30 and A = 50(00) or $5,000. Given this value, P = A  10 = 50 - 10 = $40.Inserting these numbers for P and A into the CSI demand function results in a value for Q of 5,800. Therefore, the maximum value of Q is 5,800 reflects an optimal price of $40 and optimal advertising of $5,000.

One attractive use of computer spreadsheets is to create simple numerical examples that can be used to conclusively show the change in sales, profits and other variables that occur as one approaches and moves beyond the profit-maximizing activity level.