In their study of cigarette advertising, Roberts and...

In their study of cigarette advertising, Roberts and Samuelson (1988) found that the advertising of a particular brand affects overall market demand for cigarettes but does not affect the brand's share of market sales. Suppose the demand for brand $i$ is $q_{i}=a+b\left(A_{i}+A_{j}\right)^{0.5},$ where $A_{i}$ is brand $i$ 's advertising expenditure. Brand $i$ 's profit function is $\pi_{i}=p_{i}\left(a+b\left(A_{i}+A_{j}\right)^{0.5}\right)-A_{i}$ a. Does brand $\mathrm{B}$ 's advertising expenditure affect A's market share, $q_{A} /\left(q_{A}+q_{B}\right)$ ? b. In terms of $a$ and $b,$ what are the Nash equilibrium advertising expenditures? How does an increase in $b$ affect the equilibrium expenditures? $C$

In their study of cigarette advertising, Roberts and Samuelson (1988) found that the advertising of a particular brand affects overall market demand for cigarettes but does not affect the brand's share of market sales. Suppose the demand for brand
$i$ is $q_{i}=a+b\left(A_{i}+A_{j}\right)^{0.5},$ where $A_{i}$ is brand $i$ 's advertising expenditure. Brand $i$ 's profit function is $\pi_{i}=p_{i}\left(a+b\left(A_{i}+A_{j}\right)^{0.5}\right)-A_{i}$ a. Does brand $\mathrm{B}$ 's advertising expenditure affect A's market share, $q_{A} /\left(q_{A}+q_{B}\right)$ ?
b. In terms of $a$ and $b,$ what are the Nash equilibrium advertising expenditures? How does an increase in $b$ affect the equilibrium expenditures? $C$

Key Concepts

Comparative Statics in Advertising Models

Comparative statics involves analyzing how the equilibrium outcomes, such as advertising expenditures, respond to changes in key parameters of the model. For instance, a parameter that measures the effectiveness of advertising (often denoted by a variable) can shift the marginal returns on advertising. An increase in such a parameter typically affects the equilibrium expenditures by altering the tradeoff between the benefits from increased demand and the cost incurred from advertising.

Advertising and Overall Market Demand

This concept involves understanding that advertising expenditures are used to stimulate overall market demand. In many models, the combined advertising efforts of all firms affect the size of the market by increasing consumer awareness or preferences, rather than redistributing demand among competitors. Thus, while total demand may rise with more advertising, the relative shares of individual brands might remain unchanged if the effect is symmetric.

Market Share Determination

Market share in many advertising models is determined by the relative performance of a product or brand in attracting consumers from the total market. If advertising influences the market demand additively or symmetrically across brands without affecting the competitive differentials, then the advertising efforts of one brand might not change its share relative to rivals. This means that while overall sales volume may increase, the proportionate distribution of sales across brands may remain constant.

Profit Maximization in Advertising Models

In the context of advertising, firms typically set their advertising expenditures by maximizing a profit function that balances the incremental revenue from increased demand against the cost of advertising. The profit function often incorporates the price received per unit and the advertising cost, and the optimal spending decision is reached when the marginal benefit from advertising equals the marginal cost.

Nash Equilibrium in Strategic Advertising

This concept refers to a scenario in competitive settings where each firm’s advertising decision is optimal, given the other firms' advertising levels. In Nash equilibrium, no firm can unilaterally increase its profit by changing its advertising level, given that competitors are making their own prudent choices. This equilibrium analysis is essential in determining how advertising expenditures are strategically chosen in an oligopolistic market.

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