(Economyst Only Please):

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 $1200 plus $2 for each DVD; 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 it order? From which supplier?

B. Suppose instead that the station seeks to maximize its profit from sales of DVDS. What price should be charged? How many DVD should it order from which supplier? 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.

I cannot get this set up to work the problem can you help me?

Thanks

Sure! I can help you set up and solve this problem.

A. Suppose the station plans to give away the videos for free. In this case, the station doesn't need to consider the price equation. The goal is to minimize the cost. To determine which supplier to choose, we need to compare the costs of each supplier for the desired quantity of DVDs.

For Supplier A, the cost can be calculated as follows:
Cost_A = Set-up fee + (Cost per DVD) * (Quantity of DVDs)
Cost_A = $1200 + $2 * (Quantity of DVDs)

For Supplier B, the cost can be calculated as follows:
Cost_B = (Cost per DVD) * (Quantity of DVDs)
Cost_B = $4 * (Quantity of DVDs)

To find the quantity of DVDs that minimizes the cost, we can equate the costs of the two suppliers:
Set-up fee + $2 * (Quantity of DVDs) = $4 * (Quantity of DVDs)

Simplifying the equation, we have:
1200 + 2Q = 4Q
2Q = 1200
Q = 600

So, the station should order 600 DVDs to minimize the cost. In this case, Supplier B is preferred because it has a lower cost.

B. Now let's consider that the station wants to maximize its profit from selling the DVDs. Profit is calculated as revenue minus cost. Revenue is equal to the price per DVD multiplied by the quantity of DVDs sold. The cost can be determined using the same equations as in part A.

To determine the price that maximizes profit, we need to find the quantity that maximizes revenue and then calculate the corresponding price. We can do this by applying the MR = MC rule. MR (Marginal Revenue) is the derivative of the revenue function with respect to quantity, and MC (Marginal Cost) is the derivative of the cost function with respect to quantity.

For Supplier A:
Revenue_A = (Price per DVD) * (Quantity of DVDs)
Revenue_A = (8 - Q/200) * (Quantity of DVDs)

For Supplier B:
Revenue_B = (Price per DVD) * (Quantity of DVDs)
Revenue_B = (8 - Q/200) * (Quantity of DVDs)

To find the quantity that maximizes revenue, we need to find the derivative of the revenue functions with respect to quantity and set them equal to zero:

For Supplier A:
Revenue_A' = (8 - Q/200) + (Quantity of DVDs) * (-1/200)
Setting Revenue_A' = 0, we have:
8 - Q/200 - (Quantity of DVDs)/200 = 0

For Supplier B:
Revenue_B' = (8 - Q/200) + (Quantity of DVDs) * (-1/200)
Setting Revenue_B' = 0, we have:
8 - Q/200 - (Quantity of DVDs)/200 = 0

Simplifying the equations, we get:
Supplier A: Q = 800
Supplier B: Q = 400

Now, we can calculate the corresponding prices using the price equation P = 8 - Q/200:

Supplier A: P = 8 - 800/200 = 8 - 4 = 4
Supplier B: P = 8 - 400/200 = 8 - 2 = 6

So, for Supplier A, the station should order 800 DVDs and charge a price of $4 to maximize profit. For Supplier B, the station should order 400 DVDs and charge a price of $6 to maximize profit. Finally, compare the profits by taking the revenue and subtracting the cost for each supplier to determine which supplier yields higher profit.