Assignment Background: A Television Station Is Considering
Assignment background: A television station is considering the sale of
A television station is contemplating the sale of promotional DVDs and is evaluating options for sourcing these DVDs from two different suppliers. The decision-making process involves analyzing costs, revenue, and profit maximization strategies under specific demand and pricing equations. The station wants to determine the optimal quantity and supplier choices under two scenarios: giving away the DVDs for free and selling them for profit. Both cases require applying economic principles such as the marginal revenue equals marginal cost rule to identify the most profitable quantities and prices. This analysis aims to provide a comprehensive understanding of how supplier costs influence optimal decisions and overall profitability, using the given cost structures and demand functions.
Paper For Above instruction
The analysis of the television station’s decision to source promotional DVDs involves a detailed examination of cost structures, demand functions, and profit maximization strategies. In particular, the station faces two suppliers: Supplier A with a fixed setup fee plus a variable per-unit cost, and Supplier B with no setup fee but a higher per-unit charge. The core challenge is to determine optimal order quantities and pricing strategies under two scenarios—free distribution and profit maximization—using marginal revenue and marginal cost principles.
Demand and Price Relationship
The demand for DVDs is given by Q = 1 + P, where Q represents the quantity demanded and P is the price in dollars. Rearranged into the price function, P = 8 - Q/200. This inverse relationship indicates that as quantity increases, the price decreases, reflecting typical demand behavior. When analyzing profit maximization, the key is to determine the quantity where marginal revenue (MR) equals marginal cost (MC). The MR is derived from the demand function, while MC depends on supplier costs.
Part A: DVDs Given Away for Free
In this scenario, the station is not seeking revenue from the DVDs but wants to determine the optimal quantity to distribute without considering profit. Since the DVDs are free, the station's primary concern is maximizing the reach or fulfilling promotional objectives. Hence, the quantity ordered should be where the demand function indicates maximum possible distribution, bounded by supplier capacity constraints or cost considerations.
Given the demand function P = 8 - Q/200, when the DVDs are free (P=0), solving for Q gives: P=0, so 0 = 8 - Q/200. Solving for Q, we have Q = 8 * 200 = 1600 DVDs. Thus, if the station distributes DVDs for free, the maximum demand—assuming full willingness to accept free DVDs—is 1600 units.
Choosing between suppliers under these conditions depends largely on cost efficiency. Supplier A incurs a setup fee plus a variable component, while Supplier B has no setup fee but higher per-unit cost. Since the station is distributing at no cost, the cost analysis centers on total expense relative to maximum demand. The variable costs for supplying 1600 DVDs are:
- Supplier A: Total cost = $1200 + ($2 * 1600) = $1200 + $3200 = $4400
- Supplier B: Total cost = $4 * 1600 = $6400
Given the large quantity, Supplier A is more cost-effective for full distribution if the station intends to distribute all 1600 DVDs free of charge. Therefore, the station should order 1600 DVDs from Supplier A under free distribution objectives.
Part B: Profit Maximization Scenario
When the station aims to maximize profit from DVD sales, it must determine the optimal quantity and price where marginal revenue equals marginal cost for each supplier separately.
Step 1: Derive Total Revenue and Marginal Revenue
Revenue (R) is the product of price (P) and quantity (Q): R = P * Q. Substituting P from the demand function, P = 8 - Q/200, yields:
R = (8 - Q/200) * Q = 8Q - Q^2 / 200
The Marginal Revenue (MR) is the derivative of R with respect to Q:
MR = dR/dQ = 8 - Q/100
Step 2: Determine Marginal Costs (MC) for Each Supplier
- Supplier A: Total cost (TC) = $1200 + $2Q
- Marginal cost (MC) = dTC/dQ = $2
- Supplier B: Total cost (TC) = $4Q
- Marginal cost (MC) = dTC/dQ = $4
Step 3: Calculate Optimal Quantity (Q*) using MR = MC for each supplier
- Supplier A: Set MR = MC, hence 8 - Q/100 = 2. Solving for Q:
8 - Q/100 = 2
Q/100 = 8 - 2 = 6
Q = 600 units
- Supplier B: Set MR = MC, hence 8 - Q/100 = 4. Solving for Q:
8 - Q/100 = 4
Q/100 = 8 - 4 = 4
Q = 400 units
Step 4: Calculate Corresponding Optimal Prices and Total Revenues
Using P = 8 - Q/200:
- Supplier A: Q = 600, P = 8 - 600/200 = 8 - 3 = $5
- Supplier B: Q = 400, P = 8 - 400/200 = 8 - 2 = $6
Step 5: Calculate Profits for Each Supplier
Profit (π) = Total Revenue - Total Cost
- Supplier A:
- TR = P Q = $5 600 = $3000
- TC = $1200 + $2 * 600 = $1200 + $1200 = $2400
- Profit = $3000 - $2400 = $600
- Supplier B:
- TR = $6 * 400 = $2400
- TC = $4 * 400 = $1600
- Profit = $2400 - $1600 = $800
Comparison and Recommendations
When given away for free, the station should order 1600 DVDs from Supplier A because it is more cost-efficient at high demand volumes given the fixed setup fee. For profit maximization, ordering from Supplier B yields higher profits ($800 vs. $600), primarily because the higher per-unit selling price offsets the higher marginal cost and fixed setup fee associated with Supplier A.
These analyses illustrate the importance of understanding cost structures and demand elasticity in decision-making. By applying the MR=MC rule, the station can optimize its profit margins and choose suppliers based on expected demand and pricing strategies. The exercise underscores the relevance of economic principles in practical business decisions such as sourcing and pricing.
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