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) = (60+xJ )xC −2x2C, and the total profits per day of the jewelry store are ΠJ(xC, xJ ) = (105 + xC)xJ − 2x2J. (In each case, these are profits net of all costs, including advertising.)
(a) If each store believes that the other store’s amount of advertising is independent of its own advertising expenditure, then we can find the equilibrium amount of advertising for each store by solving two equations in two unknowns. One of these equations says that the derivative of the clothing store’s profits with respect to its own advertising is zero. The other equation requires that the derivative of the jeweler’s profits with respect to its own advertising is zero. These two equations are written as _______________ The equilibrium amounts of advertising are ______________ Profits of the clothing store are __________ and profits of the jeweler are _____________
(b) The extra profit that the jeweller would get from an extra dollar’s worth of advertising by the clothing store is approximately equal to the derivative of the jeweller’s profits with respect to the clothing store’s advertising expenditure. When the two stores are doing the equilibrium amount of advertising that you calculated above, a dollar’s worth of advertising by the clothing store would give the jeweller an extra profit of about __________ and an extra dollar’s worth of advertising by the jeweler would give the clothing store an extra profit of about ___________
(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 jeweller 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. When I do this, I find that xJ = ______________ If I substitute this value of xJ into my profit function and then choose xC to maximize my own profits, I will choose xC = _______________ and he will choose xJ ____________ In this case my profits will be $1,062.72 and his profits will be ___________
d) Suppose that the clothing store and the jewelry store have the same profit functions as before but are owned by a single firm that chooses the amounts of advertising so as to maximize the sum of the two stores’ profits. The single firm would choose xC = __________ and xJ = _________ Without calculating actual profits, can you determine whether total profits will be higher, lower, or the same as total profits would be when they made their decisions independently? ______________ How much would the total profits be? __________