8. A company produces and sells a consumer product, and thus far has been able t

Question

8. A company produces and sells a consumer product, and thus far has been able to control the volume of the product by varying the selling price. The company is seeking to maximize its net profit. It has been concluded that the relationship between price and demand, per month, is approximately D=500-5p, where p is the price per unit in dollars, and the variable cost is $20 per unit. Obtain the answer mathematically to the following questions: a. What is the optimal number of units that should be produced and sold per month? b. What is the maximum profit per month? c. What are the breakeven sales quantities (range of profitable demand volume)?

Explanation / Answer

Revenues from the product = D*P= (500-5P)*P
To maximize revenues, derivative of revenues =0
=>d(500P-5p^2)/dp=0
=>500-10p=0
=>P=50
Therefore, the price at which revenues will be maximized is 50
Demand = 500-5*50=250 Units
Thus, 250 Units should be produced per month

Maximum profit = Maximum revenues - Cost
= D*P - 20*D - Fixed Cost
=250*(50-20) - Fixed Cost
=7500-Fixed Cost

At Break even, Revenues= Cost
=>D*P=20*D + Fixed Cost
=>D= Fixed Cost/(P-20)
=>D= Fixed Cost/30

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