A television station is considering the sale of promotional DVDs. It can have the DVDs produced by one of two suppliers. Supplier A will charge the station a set-up fee of $1,200 plus $2 for each DVDs; supplier B has no set-up fee and will charge $4 per DVD. The station estimates its demand for the DVDs to be given by Q=1,600 – 200P, where P is the price in dollars and Q is the number of DVDs. (The price equation is P=8 – Q/200)

a. Suppose the station plans to give away the videos. How many DVDs should be order? From which supplier?

b. Suppose instead that the station seeks to maximize its profit from sales of the DVDs. What price should it charge? How many DVDs should it order from which supplier? (Hint: Solve two separate problems, one with supplier A and one with supplier B, and then compare profits. In each case, apply the MR=MC rule)

a. To determine how many DVDs the station should order and from which supplier when giving them away, we need to compare the costs of supplier A and supplier B.

For supplier A:
Cost per DVD = $2
Set-up fee = $1,200

For supplier B:
Cost per DVD = $4
No set-up fee

To determine how many DVDs should be ordered, we need to compare the costs of the two suppliers at different quantities of DVDs.

Let's start by looking at the cost of supplier A:
Cost(A) = Set-up fee + (Cost per DVD * Quantity)
Cost(A) = $1,200 + ($2 * Quantity)
Cost(A) = $1,200 + $2Q

Now let's look at the cost of supplier B:
Cost(B) = Cost per DVD * Quantity
Cost(B) = $4Q

To determine the number of DVDs to order, we need to find the quantity at which the costs of the two suppliers are equal. So we set the two cost equations equal to each other:

$1,200 + $2Q = $4Q

Simplifying the equation, we get:
$1,200 = $2Q

Dividing both sides by $2:
600 = Q

Therefore, if the station plans to give away the DVDs, it should order 600 DVDs from supplier A.

b. To determine the price and quantity that will maximize profit from the sale of DVDs, we need to consider the demand equation Q = 1,600 - 200P and apply the MR (Marginal Revenue) = MC (Marginal Cost) rule.

Let's start with supplier A:

Profit(A) = Revenue(A) - Cost(A)
Revenue(A) = Price * Quantity
Revenue(A) = P * Q
Since P = 8 - Q/200, we can substitute it in the revenue equation:
Revenue(A) = (8 - Q/200) * Q

The marginal revenue is the derivative of the revenue function:
MR(A) = d(Revenue(A))/dQ
MR(A) = d((8 - Q/200) * Q)/dQ
MR(A) = (8 - Q/100) - Q/200

To find the quantity that maximizes profit, we set MR(A) equal to MC(A):

MR(A) = MC(A)
(8 - Q/100) - Q/200 = 2

Simplifying the equation, we get:
16 - Q/100 - Q/200 = 2
12 - Q/100 = Q/200

Simplifying further, we get:
24 - Q/100 = Q/100
24 = Q/100 + Q/100
24 = 2Q/100
2400 = 2Q
Q = 1200

Therefore, if the station wants to maximize profit with supplier A, it should order 1200 DVDs.

Now let's calculate the profit for supplier B:

Profit(B) = Revenue(B) - Cost(B)
Revenue(B) = Price * Quantity
Revenue(B) = P * Q
Since P = 8 - Q/200, we can substitute it in the revenue equation:
Revenue(B) = (8 - Q/200) * Q

The marginal revenue is the derivative of the revenue function:
MR(B) = d(Revenue(B))/dQ
MR(B) = d((8 - Q/200) * Q)/dQ
MR(B) = (8 - Q/100) - Q/200

To find the quantity that maximizes profit, we set MR(B) equal to MC(B):

MR(B) = MC(B)
(8 - Q/100) - Q/200 = 4

Simplifying the equation, we get:
16 - Q/100 - Q/200 = 4
12 - Q/100 = Q/200

Simplifying further, we get:
24 - Q/100 = Q/100
24 = Q/100 + Q/100
24 = 2Q/100
2400 = 2Q
Q = 1200

Therefore, if the station wants to maximize profit with supplier B, it should order 1200 DVDs.

To determine the price, we can substitute the value of Q (1200) into the demand equation:
P = 8 - Q/200
P = 8 - 1200/200
P = 2

Therefore, the station should charge a price of $2 and order 1200 DVDs from either supplier A or supplier B to maximize profit.