Problem 16
Question
Table 2.14 shows the cost, \(C(q),\) and revenue, \(R(q),\) in terms of quantity \(q .\) Estimate the marginal cost. \(M C(q)\) and marginal revenue, \(M R(q),\) for \(q\) between 0 and 6, $$\begin{array}{r|r|r|r|r|r|r}\hline q & 1 & 2 & 3 & 4 & 5 & 6 \\\\\hline C(q) & 20 & 60 & 120 & 200 & 300 & 420 \\\\\hline R(q) & 100 & 220 & 330 & 410 & 450 & 480 \\\\\hline\end{array}$$
Step-by-Step Solution
Verified Answer
MC and MR are calculated for each q by the change in C and R respectively between 1 and 6.
1Step 1: Understand the Problem
We need to estimate the marginal cost, \(MC(q)\), and marginal revenue, \(MR(q)\), for quantity \(q\) between 0 and 6 based on the given cost and revenue data. Marginal cost and revenue indicate the change in cost and revenue for each additional unit produced.
2Step 2: Calculate Marginal Cost (MC)
The marginal cost \(MC(q)\) is estimated by calculating the change in cost divided by the change in quantity. We use the formula \( MC(q) = \frac{C(q+1) - C(q)}{q+1 - q}\) for each \(q\). For example:- For \(q = 1\), \( MC(1) = \frac{C(2) - C(1)}{2 - 1} = \frac{60 - 20}{1} = 40 \)Repeat this process for each \(q\) from 1 to 5.
3Step 3: Calculate Marginal Revenue (MR)
The marginal revenue \(MR(q)\) is calculated similarly. Use the formula \( MR(q) = \frac{R(q+1) - R(q)}{q+1 - q}\) to find the marginal revenue for each \(q\). For example:- For \(q = 1\), \( MR(1) = \frac{R(2) - R(1)}{2 - 1} = \frac{220 - 100}{1} = 120 \)Repeat this for \(q\) from 1 to 5.
4Step 4: Calculate Values for Each q
Calculate the values for both \(MC(q)\) and \(MR(q)\) for each \(q\):- For \(q=1\): - \(MC(1) = 40\), \(MR(1) = 120\)- For \(q=2\): - \(MC(2) = 60\), \(MR(2) = 110\)- For \(q=3\): - \(MC(3) = 80\), \(MR(3) = 80\)- For \(q=4\): - \(MC(4) = 100\), \(MR(4) = 40\)- For \(q=5\): - \(MC(5) = 120\), \(MR(5) = 30\)
5Step 5: Interpret the Results
Based on the calculated values, we can compare the marginal cost and revenue at each \(q\). These results show how the cost and revenue change with each increase in quantity, which is crucial for understanding profitability at different levels of production.
Key Concepts
Marginal CostMarginal RevenueCost and Revenue Analysis
Marginal Cost
Understanding marginal cost is essential for making informed production decisions. Marginal cost (MC) refers to the cost of producing one additional unit of a good. It informs businesses on the financial impact of increasing production by just one unit.
To calculate marginal cost, we analyze the difference in total cost when quantity increases by one unit. This can be described with the formula:
\[ MC(q) = \frac{C(q+1) - C(q)}{q+1 - q} \]
This calculation gives insight into the incremental cost associated with producing additional units.
The exercise simplifies this idea by using specific data points for cost associated with different quantities. By going through each step of marginal cost calculation as shown in the solution, a student can see how costs evolve as production ramps up.
Understanding MC helps companies determine optimal production levels. Producing beyond the point where marginal cost equals marginal revenue could lead to losses, as costs outweigh additional revenue.
To calculate marginal cost, we analyze the difference in total cost when quantity increases by one unit. This can be described with the formula:
\[ MC(q) = \frac{C(q+1) - C(q)}{q+1 - q} \]
This calculation gives insight into the incremental cost associated with producing additional units.
The exercise simplifies this idea by using specific data points for cost associated with different quantities. By going through each step of marginal cost calculation as shown in the solution, a student can see how costs evolve as production ramps up.
Understanding MC helps companies determine optimal production levels. Producing beyond the point where marginal cost equals marginal revenue could lead to losses, as costs outweigh additional revenue.
Marginal Revenue
Marginal revenue (MR) is the additional income from selling one more unit of a good or service. It is crucial for pricing strategies and maximizing profit. Essentially, while MR indicates how revenue changes with each additional unit sold, understanding it helps firms in setting a profit-maximizing price.
The formula for marginal revenue is:
\[ MR(q) = \frac{R(q+1) - R(q)}{q+1 - q} \]
This formula calculates the change in revenue corresponding to an increase by a single unit.
A table of calculated MR helps visualize how additional sales contribute to total revenue, allowing businesses to adjust production and pricing strategies efficiently.
Comparing MR to MC is a vital part of the decision-making process. When the marginal cost is equal to marginal revenue, a firm achieves maximum profit. Therefore, a deep understanding of MR, alongside MC, is integral to economic success.
Pushing production when MR falls below MC can result in decreased profitability.
The formula for marginal revenue is:
\[ MR(q) = \frac{R(q+1) - R(q)}{q+1 - q} \]
This formula calculates the change in revenue corresponding to an increase by a single unit.
A table of calculated MR helps visualize how additional sales contribute to total revenue, allowing businesses to adjust production and pricing strategies efficiently.
Comparing MR to MC is a vital part of the decision-making process. When the marginal cost is equal to marginal revenue, a firm achieves maximum profit. Therefore, a deep understanding of MR, alongside MC, is integral to economic success.
Pushing production when MR falls below MC can result in decreased profitability.
Cost and Revenue Analysis
Cost and revenue analysis merges insights from both marginal cost and marginal revenue to help businesses make data-driven decisions. This analytical approach is not only about calculating numbers but interpreting them to guide strategic planning.
The original exercise provided a simplified view through data tables, showing how these principles apply directly to theoretical quantities. This method helps conceptualize what specific changes in costs and revenues imply on a larger scale.
Ultimately, a thorough cost and revenue analysis is vital for stakeholders at all levels, providing the tools necessary to align production strategies with financial goals.
- Analytical Insights: Observing trends in both MC and MR enables a business to understand where cost increases are outweighed by revenue increases, indicating potential gains or losses.
- Profit Strategy: Ideally, production should occur where MR equals MC. This scenario is where profit maximization occurs, ensuring that every produced unit contributes positively to overall gain.
The original exercise provided a simplified view through data tables, showing how these principles apply directly to theoretical quantities. This method helps conceptualize what specific changes in costs and revenues imply on a larger scale.
Ultimately, a thorough cost and revenue analysis is vital for stakeholders at all levels, providing the tools necessary to align production strategies with financial goals.
Other exercises in this chapter
Problem 15
The following table gives the percent of the US population living in urban areas as a function of year. $$\begin{array}{c|c|c|c|c|c}\hline \text { Year } & 1800
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