Shell has over $13,000$ gas stations in the United States. In addition to purchasing gasoline, customers can also purchase snacks and beverages at gas stations. Most gas stations sell fountain soda which can be purchased in various sizes.

Suppose you manage a gas station and must determine how much you will charge for fountain soda. The sizes you will make available are $8$oz $($S$),16$oz $($M$),$ and $24$oz $($L$).$ For simplicity, we will assume customer demand $($Q$)$ is fountain soda in $8$oz increments; for example, a customers who demands $2$ is demanding a $16$oz soda. Demand for soda is Q $=3,000-1,500$P$.$ Assume zero fixed costs. Marginal cost per $8$oz of soda is $$\mathrm{.}25.$ The demand function and the cost value are gathered below.

$$ Size Number of $8$oz increments $8$oz $($S$)116$oz $($M$)224$oz $($L$)3$ Q $=3,000-1,500$ P $$ Marginal cost per $8$oz of soda $$0\mathrm{.}25$ a$)$ Set up formulas to calculate quantity demanded, total revenue, total cost, and profit. Using solver, find the profit maximizing price and quantity. What are the profit maximizing price, quantity, and profit? How much would medium and large sodas cost under this pricing structure?

$$ Price $($S$)$ Quantity Demanded$$ Total Revenue $$ Total Cost $$ Profit $$ Price $($M$)$ Price $($L$)$ b$)$ Suppose you wanted to engage in second degree price discrimination. You decide to charge $$1\mathrm{.}50$ for a S $($the first $8$oz$).$ How many $8$ oz portions do you sell and what is your profit?

$$ Price $($S$)$ $$1\mathrm{.}50$ Quantity Demanded$$ Total Revenue $$ Total Cost $$ Profit $$ c$)$ Suppose you charge $$\mathrm{.}75$ extra to increase the size from S to M$.$ What is the price for a Medium soda? How many additional $8$oz portions do you sell $($subtract the demand found in part b from the new demand$)?$ How much additional profit is made $($find TR and TC for the increased sales and subtract them$)?$ Charge $$ $$0\mathrm{.}75$ Price $($M$)$ New Quantity Demanded $$ Additional Quantity Demanded $$ Additional Total Revenue $$ Additional Total Cost $$ Additional Profit $$ d$)$ Suppose you charge $$\mathrm{.}50$ extra to increase the size from M to L$.$ How many additional $8$oz portions do you sell $($subtract the demand found in parts b and c from the new demand$)?$ How much additional profit is made $($find TR and TC for the increased sales and subtract them$)?$ Charge $$ $$0\mathrm{.}50$ Price $($L$)$ New Quantity Demanded $$ Additional Quantity Demanded $$ Additional Total Revenue $$ Additional Total Cost $$ Additional Profit $$ e$)$ What is total profit using second degree price discrimination?$$ Total Profit $$ Would one decide to price discriminate? Why or why not?$$ One$$decide to discriminate because the total profit $$