In the following case, what is the optimum number of bottles a company should include in their super saver pack? How much can the pack be priced for?
A company has decided to create a super saver pack of a set of bottles it sells. Extensive research done by them has determined that the quantity demanded by the customers is Q = 100 – 8P. The cost of production is given by R = 4x + .5.
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We will use the concepts of marginal cost of production and marginal revenue to solve this problem.
First, we see that the demand function is given by Q = 100 – 8P, where P is the price of one bottle. We can write this as
8P = 100 – Q
=> P = 100/8 – Q/8.
The cost of the pack if it has Q bottles is P*Q = Q*(100/8 – Q/8)
=> 12.5Q - 0.125Q^2
This gives us the revenue function. The marginal revenue is given by its derivative. That is equal to 12.5 - .25Q
Now the cost function is 4Q + .5, the marginal cost is given by its derivative as 4.
Equating the marginal cost and marginal revenue we get 4 = 12.5 - .25Q
=> .25Q = 12.5 – 4
=> Q = 8.5 / .25 = 34
Therefore to maximise profits, the company should include 34 bottles in one pack and they can price the pack at $280.5.
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