The Demand Function { Q $}$ And Total Cost Function { T(q) $}$ Of A Commodity Are Given By The Equations:${ P = 100 - 4q } A N D And An D { TC = 320 + 4q \} Where { P $}$ And { Q $}$ Are The Price

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The Demand Function and Total Cost Function of a Commodity: Understanding the Relationship Between Price and Quantity

In the world of economics, understanding the relationship between price and quantity is crucial for businesses to make informed decisions about production and pricing strategies. The demand function and total cost function are two fundamental concepts that help businesses analyze the behavior of consumers and the costs associated with producing a commodity. In this article, we will delve into the demand function { Q $}$ and total cost function { T(q) $}$ of a commodity, and explore how they are related.

The demand function, denoted by { Q $}$, represents the quantity of a commodity that consumers are willing to buy at a given price. It is typically represented by a downward-sloping curve, indicating that as the price of the commodity increases, the quantity demanded decreases. The demand function is often expressed as a mathematical equation, which can be used to predict the quantity demanded at different price levels.

In the case of the commodity in question, the demand function is given by the equation:

P=100−4q{ P = 100 - 4q }

where { P $}$ is the price of the commodity and { q $}$ is the quantity demanded.

Interpreting the Demand Function

To understand the demand function, let's break down the equation:

P=100−4q{ P = 100 - 4q }

The equation states that the price of the commodity is equal to 100 minus 4 times the quantity demanded. This means that as the quantity demanded increases, the price of the commodity decreases, and vice versa.

For example, if the quantity demanded is 10 units, the price of the commodity would be:

P=100−4(10)=100−40=60{ P = 100 - 4(10) = 100 - 40 = 60 }

This means that if 10 units of the commodity are demanded, the price of the commodity would be $60.

The total cost function, denoted by { T(q) $}$, represents the total cost of producing a commodity as a function of the quantity produced. It is typically represented by a linear or quadratic equation, which can be used to predict the total cost of production at different quantity levels.

In the case of the commodity in question, the total cost function is given by the equation:

TC=320+4q{ TC = 320 + 4q }

where { TC $}$ is the total cost of production and { q $}$ is the quantity produced.

Interpreting the Total Cost Function

To understand the total cost function, let's break down the equation:

TC=320+4q{ TC = 320 + 4q }

The equation states that the total cost of production is equal to 320 plus 4 times the quantity produced. This means that as the quantity produced increases, the total cost of production also increases.

For example, if the quantity produced is 10 units, the total cost of production would be:

TC=320+4(10)=320+40=360{ TC = 320 + 4(10) = 320 + 40 = 360 }

This means that if 10 units of the commodity are produced, the total cost of production would be $360.

The demand function and total cost function are related in that they both depend on the quantity produced. The demand function determines the quantity demanded at a given price, while the total cost function determines the total cost of production at a given quantity.

To understand the relationship between the two functions, let's consider the following scenario:

Suppose the price of the commodity is $60, and the quantity demanded is 10 units. The total cost of production would be:

TC=320+4(10)=360{ TC = 320 + 4(10) = 360 }

However, if the quantity produced is 10 units, the price of the commodity would be:

P=100−4(10)=60{ P = 100 - 4(10) = 60 }

This means that if the quantity produced is 10 units, the price of the commodity would be $60, and the total cost of production would be $360.

To optimize production, businesses need to find the quantity that maximizes profit. Profit is equal to revenue minus total cost. Revenue is equal to price times quantity, while total cost is equal to the total cost function.

Let's consider the following scenario:

Suppose the price of the commodity is $60, and the quantity demanded is 10 units. The revenue would be:

R=Pq=60(10)=600{ R = Pq = 60(10) = 600 }

The total cost of production would be:

TC=320+4(10)=360{ TC = 320 + 4(10) = 360 }

The profit would be:

π=R−TC=600−360=240{ \pi = R - TC = 600 - 360 = 240 }

This means that if the quantity produced is 10 units, the profit would be $240.

In conclusion, the demand function and total cost function are two fundamental concepts that help businesses analyze the behavior of consumers and the costs associated with producing a commodity. The demand function determines the quantity demanded at a given price, while the total cost function determines the total cost of production at a given quantity. By understanding the relationship between the two functions, businesses can optimize production and maximize profit.

  • Microeconomics: Theory and Applications by R. Glenn Hubbard and Anthony P. O'Brien
  • Managerial Economics: Theory and Applications by William J. McEachern
  • Economics: Principles, Problems, and Policies by Gregory Mankiw
  • The Law of Demand by Investopedia
  • The Law of Supply by Investopedia
  • The Total Cost Function by Khan Academy
    Frequently Asked Questions: Demand Function and Total Cost Function

In our previous article, we explored the demand function and total cost function of a commodity, and how they are related. In this article, we will answer some of the most frequently asked questions about these concepts.

Q: What is the demand function?

A: The demand function is a mathematical equation that represents the quantity of a commodity that consumers are willing to buy at a given price. It is typically represented by a downward-sloping curve, indicating that as the price of the commodity increases, the quantity demanded decreases.

Q: What is the total cost function?

A: The total cost function is a mathematical equation that represents the total cost of producing a commodity as a function of the quantity produced. It is typically represented by a linear or quadratic equation, which can be used to predict the total cost of production at different quantity levels.

Q: How are the demand function and total cost function related?

A: The demand function and total cost function are related in that they both depend on the quantity produced. The demand function determines the quantity demanded at a given price, while the total cost function determines the total cost of production at a given quantity.

Q: What is the relationship between price and quantity?

A: The relationship between price and quantity is determined by the demand function. As the price of the commodity increases, the quantity demanded decreases, and vice versa.

Q: How do I calculate the total cost of production?

A: To calculate the total cost of production, you need to use the total cost function, which is typically represented by a linear or quadratic equation. The equation will depend on the specific commodity and production process.

Q: What is the optimal quantity to produce?

A: The optimal quantity to produce is the quantity that maximizes profit. Profit is equal to revenue minus total cost. Revenue is equal to price times quantity, while total cost is equal to the total cost function.

Q: How do I determine the optimal price?

A: To determine the optimal price, you need to consider the demand function and the total cost function. The optimal price is the price that maximizes revenue, while minimizing total cost.

Q: What is the difference between the demand function and the supply function?

A: The demand function represents the quantity of a commodity that consumers are willing to buy at a given price, while the supply function represents the quantity of a commodity that producers are willing to sell at a given price.

Q: How do I use the demand function and total cost function in business decision-making?

A: The demand function and total cost function can be used to inform business decisions about pricing, production, and investment. By understanding the relationship between price and quantity, businesses can optimize production and maximize profit.

In conclusion, the demand function and total cost function are two fundamental concepts that help businesses analyze the behavior of consumers and the costs associated with producing a commodity. By understanding the relationship between these two functions, businesses can make informed decisions about pricing, production, and investment.

  • Microeconomics: Theory and Applications by R. Glenn Hubbard and Anthony P. O'Brien
  • Managerial Economics: Theory and Applications by William J. McEachern
  • Economics: Principles, Problems, and Policies by Gregory Mankiw
  • The Law of Demand by Investopedia
  • The Law of Supply by Investopedia
  • The Total Cost Function by Khan Academy