If The Cost Function Is Given By C = Q 2 4 − 40 C = \frac{Q^2}{4} - 40 C = 4 Q 2 ​ − 40 And The Revenue Function Is Given By R = 8 Q + 5 R = 8Q + 5 R = 8 Q + 5 , Find The Marginal Cost Function, Marginal Revenue, And Equilibrium Price And Quantity.(Ans. M C = Q 2 MC = \frac{Q}{2} MC = 2 Q ​ ,

Introduction

In business, understanding the cost and revenue functions is crucial for making informed decisions about production levels, pricing strategies, and overall profitability. The cost function represents the total cost of producing a certain quantity of goods or services, while the revenue function represents the total revenue generated from selling those goods or services. In this article, we will explore the marginal cost function, marginal revenue, and equilibrium price and quantity using the given cost and revenue functions.

The Cost Function

The cost function is given by the equation $C = \frac{Q^2}{4} - 40$, where $C$ represents the total cost and $Q$ represents the quantity produced. This equation indicates that the cost of production increases quadratically with the quantity produced.

The Revenue Function

The revenue function is given by the equation $R = 8Q + 5$, where $R$ represents the total revenue and $Q$ represents the quantity sold. This equation indicates that the revenue increases linearly with the quantity sold.

Marginal Cost Function

The marginal cost function represents the rate of change of the total cost with respect to the quantity produced. To find the marginal cost function, we need to take the derivative of the cost function with respect to $Q$. Using the power rule of differentiation, we get:

$$MC = \frac{dC}{dQ} = \frac{d}{dQ} \left( \frac{Q^2}{4} - 40 \right) = \frac{Q}{2}$$

Therefore, the marginal cost function is given by $MC = \frac{Q}{2}$.

Marginal Revenue

The marginal revenue function represents the rate of change of the total revenue with respect to the quantity sold. To find the marginal revenue function, we need to take the derivative of the revenue function with respect to $Q$. Using the power rule of differentiation, we get:

$$MR = \frac{dR}{dQ} = \frac{d}{dQ} (8Q + 5) = 8$$

Therefore, the marginal revenue function is given by $MR = 8$.

Equilibrium Price and Quantity

The equilibrium price and quantity occur when the marginal revenue equals the marginal cost. To find the equilibrium price and quantity, we need to set the marginal revenue equal to the marginal cost and solve for $Q$.

$$MR = MC$$

$$8 = \frac{Q}{2}$$

$$Q = 16$$

Now that we have found the equilibrium quantity, we can find the equilibrium price by substituting the equilibrium quantity into the revenue function.

$$R = 8Q + 5$$

$$R = 8(16) + 5$$

$$R = 128 + 5$$

$$R = 133$$

Therefore, the equilibrium price is $133$ and the equilibrium quantity is $16$.

Conclusion

In conclusion, we have found the marginal cost function, marginal revenue, and equilibrium price and quantity using the given cost and revenue functions. The marginal cost function is given by $MC = \frac{Q}{2}$, the marginal revenue function is given by $MR = 8$, and the equilibrium price and quantity are $133$ and $16$, respectively. These results provide valuable insights for businesses to make informed decisions about production levels, pricing strategies, and overall profitability.

References

• [1] Cost and Revenue Functions. (n.d.). Retrieved from https://www.investopedia.com/terms/c/costfunction.asp
• [2] Marginal Cost and Revenue. (n.d.). Retrieved from https://www.investopedia.com/terms/m/marginalcost.asp

Discussion

• What are the implications of the marginal cost function on business decisions?
• How does the marginal revenue function impact pricing strategies?
• What are the benefits of finding the equilibrium price and quantity in business?

Related Topics

• Cost-Benefit Analysis: A method of evaluating the costs and benefits of a project or decision.
• Break-Even Analysis: A method of determining the point at which the total revenue equals the total cost.
• Pricing Strategies: Methods of setting prices for goods or services to maximize revenue and profitability.
  Frequently Asked Questions (FAQs) on Cost and Revenue Functions ====================================================================

Q: What is the cost function, and how is it used in business?

A: The cost function is a mathematical equation that represents the total cost of producing a certain quantity of goods or services. It is used in business to determine the optimal production level, pricing strategy, and overall profitability.

Q: What is the revenue function, and how is it related to the cost function?

A: The revenue function is a mathematical equation that represents the total revenue generated from selling a certain quantity of goods or services. It is related to the cost function in that the revenue function represents the income generated from sales, while the cost function represents the expenses incurred in producing those goods or services.

Q: What is the marginal cost function, and how is it used in business?

A: The marginal cost function is a mathematical equation that represents the rate of change of the total cost with respect to the quantity produced. It is used in business to determine the optimal production level and pricing strategy.

Q: What is the marginal revenue function, and how is it used in business?

A: The marginal revenue function is a mathematical equation that represents the rate of change of the total revenue with respect to the quantity sold. It is used in business to determine the optimal pricing strategy and revenue maximization.

Q: What is the equilibrium price and quantity, and how are they determined?

A: The equilibrium price and quantity are the price and quantity at which the marginal revenue equals the marginal cost. They are determined by setting the marginal revenue equal to the marginal cost and solving for the quantity.

Q: How do cost and revenue functions impact business decisions?

A: Cost and revenue functions have a significant impact on business decisions, including production levels, pricing strategies, and overall profitability. By analyzing the cost and revenue functions, businesses can make informed decisions about investments, pricing, and resource allocation.

Q: What are the benefits of using cost and revenue functions in business?

A: The benefits of using cost and revenue functions in business include:

• Improved decision-making: Cost and revenue functions provide a framework for making informed decisions about investments, pricing, and resource allocation.
• Increased profitability: By analyzing the cost and revenue functions, businesses can identify opportunities to increase profitability and reduce costs.
• Enhanced competitiveness: Cost and revenue functions enable businesses to stay competitive by identifying areas for cost reduction and revenue maximization.

Q: What are some common applications of cost and revenue functions in business?

A: Some common applications of cost and revenue functions in business include:

• Pricing strategies: Cost and revenue functions are used to determine the optimal price for goods or services.
• Production planning: Cost and revenue functions are used to determine the optimal production level and resource allocation.
• Investment analysis: Cost and revenue functions are used to evaluate the feasibility of investments and determine the optimal investment strategy.

Q: What are some common challenges in using cost and revenue functions in business?

A: Some common challenges in using cost and revenue functions in business include:

• Data quality: Cost and revenue functions require accurate and reliable data, which can be challenging to obtain.
• Complexity: Cost and revenue functions can be complex and difficult to analyze, requiring specialized knowledge and expertise.
• Interpretation: Cost and revenue functions require careful interpretation and analysis to ensure accurate decision-making.

Q: What are some best practices for using cost and revenue functions in business?

A: Some best practices for using cost and revenue functions in business include:

• Regular analysis: Regular analysis of cost and revenue functions is essential to ensure accurate decision-making.
• Data quality: Ensuring data quality is critical to obtaining accurate results from cost and revenue functions.
• Expertise: Seeking the expertise of professionals with knowledge of cost and revenue functions is essential to ensure accurate analysis and interpretation.

Conclusion

In conclusion, cost and revenue functions are essential tools for businesses to make informed decisions about investments, pricing, and resource allocation. By understanding the cost and revenue functions, businesses can identify opportunities to increase profitability and reduce costs, ultimately enhancing their competitiveness and success.