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?
a: if price is zero, then q=1600
how many to order is q.
then, cost to order from A is 1200+3200=4400
cost to order from b=4*1600=5200
order from A.
B: R=PQ-cost
dR/dq=dP/dq q + P- dCost/dq
now, for supplier A
dcost/dq=2
dR/dq=0=-1/200 q + 800-q/200 -2
q(1/100)=798
q= 79800
For supplier B
dR/dq=0=-q/200 +800-q/200 -4
q=79600
check that, it is late at night.
Thanks, bobpursely
Yes the B: R=PQ-cost is what I was missing at first, but after plugging it in and checking, it's correct. Thanks
You do great work when it's late!
Sure, let's break down the problem step by step.
A. When the station plans to give away the videos, it means the price (P) will be zero. We can substitute P=0 into the demand equation Q = 1,600 - 200P to find the quantity demanded.
Q = 1,600 - 200(0)
Q = 1,600 DVDs
Now we need to determine which supplier to choose. We'll calculate the cost for each supplier and choose the one with the lowest cost.
1. Supplier A:
Set-up fee = $1200
Cost per DVD = $2
Total cost = set-up fee + (cost per DVD × quantity)
Total cost = 1200 + (2 × 1600) = $4000
2. Supplier B:
Set-up fee = $0
Cost per DVD = $4
Total cost = set-up fee + (cost per DVD × quantity)
Total cost = 0 + (4 × 1600) = $6400
Comparing the total costs, it is more cost-effective for the station to order DVDs from Supplier A when giving them away.
B. To maximize profit, the station needs to find the price and quantity combination that maximizes the profit. We'll use the MR = MC rule.
1. Supplier A:
Profit = Revenue - Cost
Revenue = Price × Quantity
Revenue = (8 - Q/200) × Q
Revenue = 8Q - (Q^2)/200
Cost = Set-up fee + (Cost per DVD × Quantity)
Cost = 1200 + (2 × Q)
Profit = Revenue - Cost
Profit = 8Q - (Q^2)/200 - (1200 + 2Q)
Profit = 6Q - (Q^2)/200 - 1200
To find the quantity that maximizes profit, differentiate the profit equation with respect to Q and set it equal to zero.
d(Profit)/dQ = 6 - Q/100 = 0
Q/100 = 6
Q = 600
We can substitute this value of Q back into the price equation to find the price.
P = 8 - Q/200
P = 8 - 600/200
P = 8 - 3
P = $5
So, with Supplier A, the station should charge $5 per DVD and order 600 DVDs to maximize profit.
2. Supplier B:
Profit = Revenue - Cost
Revenue = Price × Quantity
Revenue = (8 - Q/200) × Q
Revenue = 8Q - (Q^2)/200
Cost = Set-up fee + (Cost per DVD × Quantity)
Cost = 0 + (4 × Q)
Cost = 4Q
Profit = Revenue - Cost
Profit = 8Q - (Q^2)/200 - 4Q
Profit = 4Q - (Q^2)/200
Differentiating the profit equation with respect to Q and setting it equal to zero, we can find the quantity that maximizes profit.
d(Profit)/dQ = 4 - Q/100 = 0
Q/100 = 4
Q = 400
Substituting this value of Q back into the price equation to find the price.
P = 8 - Q/200
P = 8 - 400/200
P = 8 - 2
P = $6
So, with Supplier B, the station should charge $6 per DVD and order 400 DVDs to maximize profit.
To compare the profits:
1. Supplier A:
Profit = 6 × 600 - (600^2)/200 - 1200
Profit = $1800
2. Supplier B:
Profit = 4 × 400 - (400^2)/200 - 0
Profit = $800
Therefore, when maximizing profit, Supplier A offers higher profit for the station.
Sure, I can help you with this problem.
Let's start with part A: Suppose the station plans to give away the DVDs. In this case, the station is not concerned about making a profit, so the DVD cost is the main consideration. We need to find the supplier that offers the lowest cost per DVD.
For Supplier A, the cost per DVD is $2, so the total cost for ordering Q DVDs would be given by:
Cost_A = $1200 + $2 * Q
For Supplier B, there is no set-up fee, but the cost per DVD is $4. The total cost for ordering Q DVDs from Supplier B would be:
Cost_B = $4 * Q
To determine which supplier to choose, we need to find the quantity that minimizes the cost for each supplier. We can do this by finding the derivative of the respective cost functions and setting it to zero.
For Supplier A:
dCost_A/dQ = 2
For Supplier B:
dCost_B/dQ = 4
Since the derivatives do not depend on Q, this means that the cost functions are linear, and the cost per DVD is constant regardless of the quantity ordered. Therefore, the station should order from the supplier with the lowest cost per DVD, which is Supplier A with a cost of $2 per DVD.
To find the number of DVDs the station should order, we need to plug the quantity into the demand equation Q = 1,600 - 200P. Since the DVDs are being given away, the price P is zero. Plugging this into the demand equation, we get:
Q = 1,600 - 200(0)
Q = 1,600
So, the station should order 1,600 DVDs from Supplier A since they offer the lowest cost per DVD.
Now let's move on to part B: Suppose the station seeks to maximize its profit from DVD sales. In this case, we need to consider both the cost of ordering the DVDs and the revenue from selling them. We can determine the price and quantity that maximizes profit by applying the MR = MC rule.
To find the quantity and price for Supplier A, we need to determine the marginal revenue and marginal cost functions. Marginal revenue (MR) is given by the derivative of the revenue function, which is the quantity multiplied by the price:
MR = d(R)/dQ = P + Q * d(P)/dQ
From the price equation P = 8 - Q/200, we can find d(P)/dQ:
d(P)/dQ = -1/200
Substituting this into the marginal revenue equation, we get:
MR_A = P + Q * (-1/200) = 8 - Q/200 + Q * (-1/200) = 8 - 2Q/200
Now, the marginal cost (MC) is equal to the cost per DVD from Supplier A, which is $2. So, MR_A = MC when:
8 - 2Q/200 = 2
Simplifying this equation, we find:
6 = 2Q/200
Q = 600
Plugging this quantity into the price equation P = 8 - Q/200, we get:
P = 8 - 600/200 = 5
Therefore, the station should order 600 DVDs from Supplier A and charge a price of $5 per DVD to maximize its profit.
To find the quantity and price for Supplier B, we can follow the same steps. For Supplier B, the marginal cost (MC) is $4, and the marginal revenue (MR_B) is:
MR_B = P + Q * d(P)/dQ = 8 - Q/200
Setting MR_B = MC, we have:
8 - Q/200 = 4
Q = 800
Plugging this quantity into the price equation P = 8 - Q/200, we get:
P = 8 - 800/200 = 4
Therefore, the station should order 800 DVDs from Supplier B and charge a price of $4 per DVD to maximize its profit.
To compare profits, we need to calculate the total revenue and total cost for each supplier. The total revenue is given by:
Revenue = Price * Quantity
For Supplier A:
Revenue_A = $5 * 600 = $3000
For Supplier B:
Revenue_B = $4 * 800 = $3200
The total cost for both suppliers will be the same, as it is determined by the quantity ordered:
Cost_A = $2 * 600 = $1200
Cost_B = $4 * 800 = $3200
Therefore, the profit for Supplier A is Revenue_A - Cost_A:
Profit_A = $3000 - $1200 = $1800
And the profit for Supplier B is Revenue_B - Cost_B:
Profit_B = $3200 - $3200 = $0
Hence, the station would have a higher profit of $1800 by ordering 600 DVDs from Supplier A and charging a price of $5 per DVD.