Question

Below is demand and cost information for Warmfuzzy Press, which holds the copyright on the new best-seller, Burping Your Inner Child. $$\begin{array}{ccc} \begin{array}{c} Q \\ \text { (No. of Copies) } \end{array} & \begin{array}{c} P \\ \text { (per Book) } \end{array} & \begin{array}{c} A T C \\ \text { (per Book) } \end{array} \\ \hline 100,000 & \$ 100 & \$ 20 \\ 200,000 & \$ 80 & \$ 15 \\ 300,000 & \$ 60 & \$ 16^{2 / 3} \\ 400,000 & \$ 40 & \$ 221 / 2 \\ 500,000 & \$ 20 & \$ 31 \end{array}$$ a. Determine what quantity of the book Warmfuzzy should print, and what price it should charge in order to maximize profit. b. What is Warmfuzzy's maximum profit? c. Prior to publication, the book's author rencgotiates his contract with Warmfuzzy. He will receive a great big hug from the CEO, along with a onetime bonus of \(\$ 1,000,000,\) payable when the book is published. This payment was not part of Warmfuzzy's original cost calculations. How many copies should Warmfuzzy publish now? Explain your reasoning.

Short answer

a. The quantity that Warmfuzzy should print to maximize profit is the one that corresponds to the maximum profit in our calculation from Step 1 and 2. Correspondingly, the price to charge would be the one that aligns with this quantity. b. Warmfuzzy's maximum profit would be the highest value calculated in Step 1 and 2. c. The quantity that Warmfuzzy should now publish after considering the author's bonus contract will be determined by the new profit calculations in Step 3 and 4. If the additional bonus significantly impacts the cost per book, the quantity of books to print will most likely change.

Step-by-step solution

Identify the profit at each quantity level

Profit equals total revenue minus total cost. Since price is equal to average revenue and average total cost is given, profit per book can be determined by subtracting average total costs from the price. Multiply each profit per book by the quantity to get total profit.

Determine the quantity and price for maximum profit

Scan through the total profits computed in Step 1, and find the maximum value. The quantity and price corresponding to this maximum profit are the optimal production quantity and price.

Calculate Profits including Author Bonus

The next step is to recalculate total profits, considering the one-time author bonus of $1,000,000 as an additional cost. If we distribute this additional cost evenly over the books produced, this would increase the cost per book. We repeat the profit calculation in step 1 and 2 to get new quantity and price for max profit.

Explain reasoning

In the last step, clarify how the additional author bonus affects the quantity to be published. If the cost increase exceeds the revenue generated from selling more books, the optimal quantity to publish will be different from what it was without the bonus considered.

Key concepts

Demand and Cost Analysis

When analyzing a business scenario like that of Warmfuzzy Press, understanding the demand and cost structure is crucial. The demand curve indicates how much consumers are willing to pay for different quantities of the product, in this case, books. This is reflected in the price column associated with various quantities.
Cost analysis involves assessing the average total cost (ATC) at each level of production. ATC shows the cost per unit, inclusive of both fixed and variable costs, as production scales up. In our exercise, we see how the ATC changes as the number of books produced increases from 100,000 to 500,000 copies. This helps forecast how cost efficiency may vary with production scale.

• Demand Curve: Shows consumer willingness to pay at different quantity levels.
• Average Total Cost (ATC): Reflects the cost per book as production volume changes, needing to consider both fixed and variable costs.

Linking these analyses helps determine the potential profitability at each production level by comparing prices (demand side) to costs (supply side).
Properly aligning demand and cost insights is essential to formulating a production strategy that ensures maximum profitability.

Optimal Production Quantity

The optimal production quantity is the level of output at which a firm maximizes its profit. This requires a careful balance between generating revenue and managing costs.
To find this optimal point for Warmfuzzy Press, one would first calculate the profit per book, which is the difference between the price per book and the average total cost per book. This step shows the profitability of producing each additional unit.
Next, compute the total profit by multiplying the profit per book by the number of books sold at each price level. The optimal quantity corresponds to the point where this total profit is highest, as it signifies the greatest financial gain for the firm.

• Calculate Profit per Book: Subtract ATC from Price.
• Total Profit: Profit per Book × Number of Books.
• Identify Maximum Profit: Find the point where total profit is at its peak.

This systematic approach ensures that the business isn't just covering its costs, but is optimizing its financial returns by aligning production with market demand.

Pricing Strategy

The pricing strategy involves setting a price that not only covers costs but also maximizes profits. For Warmfuzzy Press, this means setting a book price that balances consumer willingness to pay against production costs.
Using the demand and cost data, once the optimal quantity is known, the corresponding price can be pinpointed as the optimal pricing strategy. This price covers the average total cost and allows for the highest possible profit. However, external factors like renegotiated contracts (e.g., author bonuses) can significantly impact pricing decisions.
In scenarios where additional costs arise, such as a $1,000,000 author bonus, this amount must be spread across the entire production run to understand its impact on per-unit cost and profitability. The new adjusted price should then ensure profitability despite the increased costs.

• Set Initial Price: Based on demand and cost data to maximize profit.
• Adjust for External Costs: Include one-time costs, such as bonuses, in pricing considerations.
• Ensure Profitability: Even with added costs, maintain a price that supports profit margins.

Effective pricing strategies are dynamic, adjusting to both internal cost structures and external market conditions to ensure ongoing profitability.