8. Calculating Samson loves to have lunch at the Italian restaurant "At Pablo's". During his next visit to the restaurant, Samson was offered to purchase a loyalty card for a period of 1 year at a price of 30000 rubles, which gives the client a $30 \%$ discount on the bill amount.

(a) Suppose that during the week Samson visits the restaurant three times (regardless of whether he has a loyalty card or not) with an average bill amount of 900 rubles without the discount. Is it profitable for him to buy the loyalty card?

(b) According to data from the information portal, the average bill amount per person in this restaurant is 600 rubles. On average, how many visits per year does a customer need to make to be indifferent about buying the restaurant's card or not?

(c) Recently, many stores and organizations operating in the service sector have been issuing so-called customer cards. Some of these cards are cumulative and provide discounts only after accumulating a certain amount on the card's account, while others immediately provide the customer with a discount that can be used with any purchase. Provide at least two strong reasons and substantiate why it is profitable for trading and service enterprises to issue such cards and use them in their work with customers. (10 points)
# Solution:

(a) After purchasing the card, one lunch will cost the client $900 * 0.3 = 270$ rubles less. Since there are 52 full weeks in a year $(365 / 7 = 52.14)$, Samson will save $270 * 3 * 52 = 42120$ rubles on discounts over the year, which is more than the cost of the card. Therefore, it is beneficial for him to buy the card.

(b) On average, a lunch with the card will be $600 * 0.3 = 180$ rubles cheaper. Then, from the condition of equal expenses, we determine that one needs to visit $30000 / 180 = 166$,(6), i.e., it will be beneficial to purchase the card if the restaurant is visited 167 times a year or more, and it will be unprofitable to purchase the card if the restaurant is visited 166 times or less.

(c) The use of cards with a cumulative discount system allows firms to "retain" regular customers, thereby preventing them from "switching" to the services of other firms.

The use of cards that provide discounts without a cumulative system also helps to attract new and "retain" old customers with discounts.

Often, providing such cards to customers requires their authorization in the company's databases. In this case, companies can inform their customers about planned promotions and also provide customer databases to their partners, albeit not without a cost. Additionally, issuing such cards requires additional expenses from the company owners. Therefore, it can signal to customers about the stability of the company, its concern for its customers, and its willingness to provide them with discounts.

## Grading Criteria:

(a) Correct calculations allowing for a comparison of expenses - 3 points, Correct conclusion about the profitability of purchasing the card

(b) Correctly found number of visits - 3 points, (c) 2 points each (total 4 points) for the reason and corresponding explanation.