Question

A soda company is selling cans of soda to two types of consumers, but it is unable to identify which type is which. Each high-type consumer has a demand for cans of Q = 40 - P. The low type consumer has a demand of Q = 30 - P. The marginal cost of a can of soda is 4. The soda company decides to adopt a menu pricing strategy, meaning it packages a specific number of sodas together and sells the entire package at some price. (Hint: This problem is very similar to one described in the lecture notes on menu pricing.)

1. What is the package size the soda company hopes to sell to low-demand consumers? How much should it charge for this package?
2. What is the surplus of low-demand consumers who purchase this package?
3. If high-demand customers were to purchase the package you described, what would their surplus be?
4. Given this, how many sodas are in the package the firm designs to sell to high-demand consumers, and how much does it charge for this package?

Answer 1

To sell to low-demand consumers, the soda company should package 26 sodas together and charge $4. The surplus of low-demand consumers who purchase this package is 52. If high-demand customers were to purchase this package, their surplus would be zero. The soda company designs a package of 36 sodas to sell to high-demand consumers, charging $4 for this package.

Step-by-step explanation:

To determine the package size and price for low-demand consumers, we need to find the price at which the quantity demanded equals the package size. For low-demand consumers, the demand function is Q = 30 - P. Setting Q equal to the package size, we have 30 - P = package size. Rearranging the equation, we get P = 30 - package size. To find the package size, we substitute the marginal cost (4) into the equation and set it equal to the price. 4 = 30 - package size. Solving for the package size, we have package size = 26. For low-demand consumers, the package size should be 26 and the price should be $4.

The surplus of low-demand consumers who purchase this package can be calculated by finding the area of the consumer surplus triangle. The consumer surplus triangle is formed by the demand curve and the price line for the package. The height of the triangle is (30 - P), which in this case is the package size (26). The base of the triangle is the price of the package (4). Using the formula for the area of a triangle (0.5 * base * height), the consumer surplus is 0.5 * 4 * 26 = 52.

If high-demand customers were to purchase the package designed for low-demand consumers, their surplus would be zero since their demand function is Q = 40 - P. If the package size is 26, the price they would pay is 30 - 26 = $4, which is the same as the low price for the low-demand package. Therefore, their consumer surplus would be zero since they would be paying the same price as in the low-demand package.

For high-demand consumers, we need to determine the package size and price that maximize their surplus. We can use a similar process as before, setting the quantity demanded equal to the package size and finding the price that maximizes their surplus. For high-demand consumers, the demand function is Q = 40 - P. Setting Q equal to the package size, we have 40 - P = package size. Rearranging the equation, we get P = 40 - package size. Plugging in the marginal cost (4), we can solve for the package size: 4 = 40 - package size. The package size for high-demand consumers is 36. To find the price, we substitute the package size back into the demand function to get P = 40 - 36 = $4. Therefore, the package size for high-demand consumers is 36 and the price is $4.