Superstores like Costco and Sam’s Club serves as wholesalers to businesses but also target consumers who are willing to pay a fixed fee in order to get access to the lower wholesale prices offered in these stores. For purposes of this exercise, suppose that you can denote goods sold at Superstores as x1 and “dollars of other consumption” as x2.
A. Suppose all consumers have the same homothetic tastes over x1 and x2 but they differ in their income. Every consumer is offered the same option of either shopping at stores with somewhat higher prices for x1 or paying the fixed fee c to shop at a Superstore at somewhat lower prices for x1.
(a) On a graph with x1 on the horizontal axis and x2 on the vertical, illustrate the regular budget (without a Superstore membership) and the Superstore budget for a consumer whose income is such that these two budgets cross on the 45 degree line. Indicate on your graph a vertical distance that is equal to the Superstore membership fee c.
(b) Now consider a consumer with twice that much income. Where will this consumer’s two budgets intersect relative to the 45 degree line?
(c) Suppose consumer 1 (from part (a)) is just indifferent between buying and not buying the Superstore membership. How will her behavior differ depending on whether or not she buys the membership?
(d) If consumer 1 was indifferent between buying and not buying the Superstore membership, can you tell whether consumer 2 (from part (b)) is also indifferent? (Hint: Given that tastes are homothetic and identical across consumers, what would have to be true about the intersection of the two budgets for the higher income consumer in order for the consumer to also be indifferent between them?)
(e) True or False: Assuming consumers have the same homothetic tastes, there exists a “marginal” consumer with income  such that all consumers with income greater than  will buy the Superstore membership and no consumer with income below  will buy that membership.
(f) True or False: By raising c and/or p1, the Superstore will lose relatively lower income customers and keep high income customers.
(g) Suppose you are a Superstore manager and you think your store is overcrowded. You’d like to reduce the number of customers while at the same time increasing the amount each customer purchases. How would you do this?
B. Suppose you manage a Superstore and you are currently charging an annual membership fee of $50. Since x2 is denominated in dollar units, p2 = 1. Suppose that p1 = 1 for those shopping outside the Superstore but your store sells x1 at 0.95. Your statisticians have estimated that your consumers have tastes that can be summarized by the utility function
u(x1,x2) = x10.15 x20.85 .
(a) What is the annual discretionary income (that could be allocated to purchasing x1 and x2) of your “marginal” consumer?
(b) Can you show that consumers with more income than the marginal consumer will definitely purchase the membership while consumers with less income will not?
(c) If the membership fee is increased from$50 to $100, how much could the Superstore lower p1 without increasing membership beyond what it was when the fee was $50 and p1 was 0.95?