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(11) Last month the Financial Times reported that Japanese firm Akagi Nyuguyo was increasing the price of its popular children’s ice cream bar Garigari-kun for the first time in 25 years, from ¥60 to ¥70. Suppose that a 7-11 store typically sold 500 ice cream bars a day, but now expects to sell only 400 bars a day. Assuming a linear relationship between price (p) and quantity demanded (x), write down the linear price-demand function p(x) (hint: y−y1 = m(x−x1)).
(12) Based on your insight from question (11), Akagi Nyuguyo has hired you as their analyst. They need to know the following information:
(a) What sales per 7-11 store will maximize revenue, R(x)?
(b) What is the maximum revenue per store?
(13) Pleased with your work, the managers at Akagi Nyuguyo ask for a more complete analysis of their business. Using linear regression models, you estimate a total cost function C(x) per 7-11 retailer with a fixed cost of ¥5000 (for freezers, electricity, and marketing) and variable costs of ¥5 (for sticks, packaging, etc.) per bar.
(a) Derive the ice cream bar profit function P(x) per store.
(b) Graph P(x).
(14) Lastly, your Japanese employers need to know the following:
(a) What sales per 7-11 store will maximize profit, P(x)?
(b) What is the marginal average profit function, per store? (hint: P ′(x) = d P (x)) dx