Expressing Demand as a Function of Price

Oct 13, 2025 by recoveryshake.com 25 views

The price-demand equation is a fundamental concept in economics that describes the relationship between the price of a product and the quantity demanded by consumers. In this article, we will explore the price-demand equation and answer the following questions: Express the demand $x$ as a function of the price $p$ .

What is the Price-Demand Equation?

The price-demand equation is a mathematical representation of the relationship between the price of a product and the quantity demanded by consumers. It is typically represented by the equation:

p + 0.001x = 40

where $p$ is the price of the product, $x$ is the quantity demanded, and $0.001$ is a constant that represents the slope of the demand curve.

Expressing Demand as a Function of Price

To express the demand $x$ as a function of the price $p$ , we need to isolate $x$ on one side of the equation. We can do this by subtracting $0.001x$ from both sides of the equation:

p + 0.001x - 0.001x = 40 - 0.001x

This simplifies to:

p = 40 - 0.001x

Now, we can solve for $x$ by subtracting $40$ from both sides of the equation and then dividing both sides by $-0.001$ :

p - 40 = -0.001x

-0.001x = p - 40

x = \frac{40 - p}{0.001}

x = 40,000 - \frac{p}{0.001}

x = 40,000 - 1,000p

Therefore, the demand $x$ can be expressed as a function of the price $p$ as follows:

x = 40,000 - 1,000p

Interpreting the Demand Function

The demand function $x = 40,000 - 1,000p$ represents the relationship between the price of a product and the quantity demanded by consumers. The function indicates that as the price of the product increases, the quantity demanded decreases, and vice versa.

The constant term $40,000$ represents the maximum quantity that consumers are willing to buy at a price of $0$ . This is known as the y-intercept of the demand curve.

The slope of the demand curve is represented by the coefficient $-1,000$ , which indicates that for every $1$ unit increase in price, the quantity demanded decreases by $1,000$ units.

Graphing the Demand Curve

The demand curve can be graphed by plotting the demand function $x = 40,000 - 1,000p$ on a coordinate plane. The graph will be a straight line with a negative slope, indicating that as the price increases, the quantity demanded decreases.

Elasticity of Demand

The elasticity of demand is a measure of how responsive the quantity demanded is to changes in the price of a product. It can be calculated using the following formula:

\text{Elasticity of Demand} = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}

Using the demand function $x = 40,000 - 1,000p$ , we can calculate the elasticity of demand as follows:

\text{Elasticity of Demand} = \frac{\frac{\Delta x}{x}}{\frac{\Delta p}{p}}

\text{Elasticity of Demand} = \frac{\frac{\Delta (40,000 - 1,000p)}{40,000 - 1,000p}}{\frac{\Delta p}{p}}

\text{Elasticity of Demand} = \frac{\frac{-1,000\Delta p}{40,000 - 1,000p}}{\frac{\Delta p}{p}}

\text{Elasticity of Demand} = \frac{-1,000}{40,000 - 1,000p} \cdot \frac{p}{\Delta p}

\text{Elasticity of Demand} = \frac{-1,000p}{40,000 - 1,000p}

Therefore, the elasticity of demand is given by the formula:

\text{Elasticity of Demand} = \frac{-1,000p}{40,000 - 1,000p}

Conclusion

In conclusion, the price-demand equation is a fundamental concept in economics that describes the relationship between the price of a product and the quantity demanded by consumers. We have expressed the demand $x$ as a function of the price $p$ and interpreted the demand function. We have also calculated the elasticity of demand and graphed the demand curve. The price-demand equation is a powerful tool for understanding the behavior of consumers and making informed decisions in business and economics.

References

[1] Mankiw, G. N. (2017). Principles of Economics. Cengage Learning.
[2] Varian, H. R. (2014). Microeconomic Analysis. W.W. Norton & Company.
[3] Krugman, P. R. (2012). Macroeconomics. Worth Publishers.
Price-Demand Equation Q&A =============================

Frequently Asked Questions

Q: What is the price-demand equation?

A: The price-demand equation is a mathematical representation of the relationship between the price of a product and the quantity demanded by consumers.

Q: How is the price-demand equation typically represented?

A: The price-demand equation is typically represented by the equation:

p + 0.001x = 40

where $p$ is the price of the product, $x$ is the quantity demanded, and $0.001$ is a constant that represents the slope of the demand curve.

Q: How do I express the demand $x$ as a function of the price $p$ ?

A: To express the demand $x$ as a function of the price $p$ , you need to isolate $x$ on one side of the equation. You can do this by subtracting $0.001x$ from both sides of the equation:

p + 0.001x - 0.001x = 40 - 0.001x

This simplifies to:

p = 40 - 0.001x

Now, you can solve for $x$ by subtracting $40$ from both sides of the equation and then dividing both sides by $-0.001$ :

p - 40 = -0.001x

-0.001x = p - 40

x = \frac{40 - p}{0.001}

x = 40,000 - \frac{p}{0.001}

x = 40,000 - 1,000p

Therefore, the demand $x$ can be expressed as a function of the price $p$ as follows:

x = 40,000 - 1,000p

Q: What does the demand function $x = 40,000 - 1,000p$ represent?

A: The demand function $x = 40,000 - 1,000p$ represents the relationship between the price of a product and the quantity demanded by consumers. The function indicates that as the price of the product increases, the quantity demanded decreases, and vice versa.

Q: What is the y-intercept of the demand curve?

A: The y-intercept of the demand curve is the constant term $40,000$ , which represents the maximum quantity that consumers are willing to buy at a price of $0$ .

Q: What is the slope of the demand curve?

A: The slope of the demand curve is represented by the coefficient $-1,000$ , which indicates that for every $1$ unit increase in price, the quantity demanded decreases by $1,000$ units.

Q: How do I graph the demand curve?

A: To graph the demand curve, you can plot the demand function $x = 40,000 - 1,000p$ on a coordinate plane. The graph will be a straight line with a negative slope, indicating that as the price increases, the quantity demanded decreases.

Q: What is the elasticity of demand?

A: The elasticity of demand is a measure of how responsive the quantity demanded is to changes in the price of a product. It can be calculated using the following formula:

\text{Elasticity of Demand} = \frac{\text{Percentage Change in Quantity Demanded}}{\text{Percentage Change in Price}}

Using the demand function $x = 40,000 - 1,000p$ , you can calculate the elasticity of demand as follows:

\text{Elasticity of Demand} = \frac{\frac{\Delta x}{x}}{\frac{\Delta p}{p}}

\text{Elasticity of Demand} = \frac{\frac{\Delta (40,000 - 1,000p)}{40,000 - 1,000p}}{\frac{\Delta p}{p}}

\text{Elasticity of Demand} = \frac{\frac{-1,000\Delta p}{40,000 - 1,000p}}{\frac{\Delta p}{p}}

\text{Elasticity of Demand} = \frac{-1,000}{40,000 - 1,000p} \cdot \frac{p}{\Delta p}

\text{Elasticity of Demand} = \frac{-1,000p}{40,000 - 1,000p}

Therefore, the elasticity of demand is given by the formula:

\text{Elasticity of Demand} = \frac{-1,000p}{40,000 - 1,000p}

Q: What is the significance of the elasticity of demand?

A: The elasticity of demand is a crucial concept in economics that helps businesses and policymakers understand how changes in price affect the quantity demanded of a product. It can be used to make informed decisions about pricing, production, and investment.

Q: Can you provide more examples of the price-demand equation?

A: Yes, here are a few more examples of the price-demand equation:

Example 1: Suppose the price-demand equation is given by $p + 0.002x = 50$ . How do you express the demand $x$ as a function of the price $p$ ?

Answer: To express the demand $x$ as a function of the price $p$ , you need to isolate $x$ on one side of the equation. You can do this by subtracting $0.002x$ from both sides of the equation:

p + 0.002x - 0.002x = 50 - 0.002x

This simplifies to:

p = 50 - 0.002x

Now, you can solve for $x$ by subtracting $50$ from both sides of the equation and then dividing both sides by $-0.002$ :

p - 50 = -0.002x

-0.002x = p - 50

x = \frac{50 - p}{0.002}

x = 25,000 - \frac{p}{0.002}

x = 25,000 - 500p

Therefore, the demand $x$ can be expressed as a function of the price $p$ as follows:

x = 25,000 - 500p

Example 2: Suppose the price-demand equation is given by $p + 0.003x = 60$ . How do you express the demand $x$ as a function of the price $p$ ?

Answer: To express the demand $x$ as a function of the price $p$ , you need to isolate $x$ on one side of the equation. You can do this by subtracting $0.003x$ from both sides of the equation:

p + 0.003x - 0.003x = 60 - 0.003x

This simplifies to:

p = 60 - 0.003x

Now, you can solve for $x$ by subtracting $60$ from both sides of the equation and then dividing both sides by $-0.003$ :

p - 60 = -0.003x

-0.003x = p - 60

x = \frac{60 - p}{0.003}

x = 20,000 - \frac{p}{0.003}

x = 20,000 - 1,000p

Therefore, the demand $x$ can be expressed as a function of the price $p$ as follows:

x = 20,000 - 1,000p

These examples demonstrate how to express the demand $x$ as a function of the price $p$ using the price-demand equation.