Question: a clothing store and a jewelry store are located side...
A clothing store and a jewelry store are located side by side in a small shopping mall. The number of customers who come to the shopping mall intending to shop at either store depends on the amount of money that the store spends on advertising per day. Each store also attracts some customers who came to shop at the neighboring store. If the clothing store spends $xC per day on advertising, and the jeweler spends $xJ on advertising per day, then the total profits per day of the clothing store are 𝜋𝐶(𝑥𝐶, 𝑥𝐽) = 60𝑥𝐶 + 𝑥𝐶𝑥𝐽 − 2𝑥𝐶 2 , and the total profits per day of the jeweler are 𝜋𝐽(𝑥𝐶, 𝑥𝐽) = 105𝑥𝐶 + 𝑥𝐶𝑥𝐽 − 2𝑥𝐽 2 . (In each case, these are profits net of all costs, including advertising.)
a) Find the equilibrium amount of advertising for each store if each of them believes that the other store’s amount of advertising is independent of its own advertising expenditure. (Hint: calculate a derivative of each store’s profits with respect to its own advertising and set it equal to zero.) (PREVIOUSLY POSTED)
b) The extra profit that each store would get from an extra dollar’s worth of advertising by the otherstore is approximately equal to the derivative of that store’s profits with respect to the other store’s advertising expenditure. Estimate the jeweler’s extra profit from a dollar’s worth of advertising by the clothing store and the clothing store’s extra profit from an extra dollar’s worth of advertising by the jeweler. (PREVIOUSLY POSTED)
c) Suppose that the owner of the clothing store knows the profit functions of both stores. She reasons to herself as follows, “Suppose that I can decide how much advertising I will do before the jeweler decides what he is going to do. When I tell him what I am doing, he will have to adjust his behavior accordingly. I can calculate his reaction function to my choice of xC, by setting the derivative of his profits with respect to his own advertising equal to zero and solving for his amount of advertising as a function of my own advertising. Then, if I substitute the jeweler’s reaction function xJ(xC) into my profit function, I will be able to find xC to maximize my own profits.” Following this line of reasoning calculate advertising expenditures and profits of each store.