In the competitive world of manufacturing and retail, determining the right price for a product can be the deciding factor between maximizing profit and potentially losing sales. When it comes to products as ubiquitous as printer paper, even small adjustments in pricing can lead to significant changes in demand and, consequently, the profit margins. Let’s dive into a scenario involving a manufacturer and their decision-making process regarding printer paper pricing.

**UNDERSTANDING THE INITIAL SCENARIO**

A manufacturer produces printer paper with a production cost of $2 per ream. At its current selling price of $6 per ream, consumers have displayed an appetite for purchasing, with sales reaching 4000 reams each month. These figures provide a starting point for the manufacturer to consider potential changes in pricing and the subsequent impact on demand.

**PREDICTING CONSUMER BEHAVIOR**

The manufacturer’s market research suggests a direct relationship between price hikes and decreasing demand. Specifically, for every $1 increase in the price of the printer paper, there’s an estimated decrease in sales by 400 reams each month. This inverse relationship between price and demand is a critical factor to consider in the quest for maximum profit.

**CALCULATING THE OPTIMAL PRICE FOR MAXIMUM PROFIT**

Profit is calculated as the difference between the selling price and the cost of production, multiplied by the number of units sold. In this scenario, the challenge is to determine the selling price that will maximize this profit, considering the changing demand with each price increment.

Using the provided information and setting up an equation to represent the profit based on varying price increments, we can derive the optimal price point.

**Answer:**

After performing the necessary calculations, we find that the price per ream that corresponds to the maximum profit is **(dollars)**.

Pricing strategy is an intricate balancing act. By understanding consumer behavior and utilizing mathematical models, manufacturers can make informed decisions that lead to optimal profits. As demonstrated in this scenario, even commonplace products like printer paper require careful analysis to ensure financial success.

Profit PP is given by the formula: P=(Selling Price−Cost)×QuantityP=(Selling Price−Cost)×Quantity

Given:

- Cost per ream = $2
- Current selling price = $6
- Current quantity sold = 4000 reams

For each $1 increase in price, the quantity sold decreases by 400 reams.

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Let’s represent the increase in price by xx.

New price = $6 + xx New quantity sold = 4000 – 400xx

P(x) = ($6 + x – $2) \times (4000 – 400x) P(x)=(4+x)(4000−400x)P(x)=(4+x)(4000−400x) P(x)=16000+4000x−1600x−400x2P(x)=16000+4000x−1600x−400×2 P(x)=−400×2+2400x+16000P(x)=−400×2+2400x+16000

To find the value of xx for which the profit is maximum, we’ll differentiate PP with respect to xx and set it to zero.

P′(x)=−800x+2400P′(x)=−800x+2400 Set P′(x)=0P′(x)=0 for maximum/minimum: −800x+2400=0−800x+2400=0 −800x=−2400−800x=−2400 x=3x=3

So, the manufacturer should increase the price by $3 to maximize the profit.

New selling price = Current selling price + Increase = $6 + $3 = **$9**

Thus, the price per ream that corresponds to the maximum profit is **$9**.

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