EXAMPLE 5.17 The Optimal Level of Advertising Economic theory tells us to undertake all those actions for which the marginal benefit is greater than the marginal cost. This optimizing principle applies to Big Andy's Burger Barn as it attempts to choose the optimal level of advertising expenditure. Recalling that SALES denotes sales revenue or total revenue, the marginal benefit in this case is the marginal revenue from more advertising. From (5.24), the required marginal revenue is given by the marginal effect of more advertising B3 + 24ADVERT. The marginal cost of $1 of advertising is $1 plus the cost of preparing the additional products sold due to effective advertising. If we ignore the latter costs, the marginal cost of $1 of advertising expenditure is $1. Thus, advertising should be increased to the point where B3 + 2B4ADVERT = 1 with ADVERTo denoting the optimal level of advertising. Using the least squares estimates for 3 and B4 in (5.25), a point estimate for ADVERT is ADVERTo = 1-63 = 1-12.1512 2b4 2x(-2.76796) = 2.014 implying that the optimal monthly advertising expenditure is $2014. To assess the reliability of this estimate, we need a standard error and an interval estimate for (1 - b3)/2b4. This is a tricky problem, and one that requires the use of calculus to solve. What makes it more difficult than what we have done so far is the fact that it involves a nonlinear function of b3 and b4. Variances of nonlinear functions are hard to derive. Recall that the variance of a linear function, say, C3b3 + C4b4, is given by var(c3b3 + c4b4) = c3var(b3) + cavar(b4) + 2c3c4cov(63, b4) (5.47) Finding the variance of (1 - b3)/2b4 is less straightforward. The best we can do is find an approximate expression that is valid in large samples. Suppose A = (1 - B3)/2B4 and A = (1 - b3) / 2b4; then, the approximate variance expression is var (2) = var (b3 ) + 264) var(b4) (5.48) +2 32 3 cov(b3, b4)5.30 In Section 5.7.4, we discovered that the optimal level of advertising for Big Andy's Burger Barn, ADVERT, satisfies the equation B3 + 234ADVERT = 1. Using a 5% significance level, test whether each of the following levels of advertising could be optimal: (a) ADVERT = 1.75, (b) ADVERT, = 1.9, and (c) ADVERT, = 2.3. What are the p- values for each of the tests