Optional calculus proof to show that MR has twice slope of demand | Khan Academy

TL;DR

The video provides an optional proof that the slope of the marginal revenue curve for a monopolist is twice the slope of the demand curve, assuming the demand curve is a line.

Transcript

For those of you who are curious and have a little bit of a background in calculus, I thought I would do a very optional and when I say it's optional, you don't have to understand this in order to progress with the economics playlist, but a very optional proof showing you that in general, the slope of the marginal revenue curve for a monopolist is ... Read More

Key Insights

• 🫥 The slope of the marginal revenue curve for a monopolist is twice the slope of the demand curve when the demand curve is a line.
• 😑 The price can be expressed as a function of quantity using the slope-intercept form.
• 📉 Total revenue as a function of quantity follows a downward sloping parabola.
• 🫡 The derivative of the total revenue function with respect to quantity gives the equation of the marginal revenue curve.
• ❓ The marginal revenue curve has twice the slope and is generally steeper than the demand curve for a monopolist.
• ❓ Calculus concepts, such as derivatives, are used to analyze economics concepts like marginal revenue.
• ❓ This optional proof is not necessary to understand economics topics but provides a deeper understanding for those with a background in calculus.

Questions & Answers

Q: What is the relationship between the slope of the marginal revenue curve and the demand curve for a monopolist?

The slope of the marginal revenue curve for a monopolist is twice the slope of the demand curve, assuming the demand curve is a line. This means that the marginal revenue curve is steeper than the demand curve.

Q: How can the price be expressed as a function of quantity using the slope-intercept form?

The price can be written as P = mQ + b, where P is the price, Q is the quantity, m is the slope, and b is the y-intercept. This expression allows us to find the slope of the demand curve.

Q: What is the formula for total revenue as a function of quantity?

Total revenue can be calculated by multiplying the price function (mQ + b) by the quantity (Q), resulting in total revenue = mQ^2 + bQ. This formula represents a downward sloping parabola.

Q: How is the marginal revenue curve derived from the total revenue function?

The marginal revenue curve is obtained by taking the derivative of the total revenue function with respect to quantity (TR/DQ). This derivative results in 2mQ + b, which represents the equation of the marginal revenue curve.

Summary & Key Takeaways

• The video offers a proof showing that the slope of the marginal revenue curve for a monopolist is twice the slope of the demand curve if the demand curve is a line.

• It explains how to write the price as a function of quantity using the slope-intercept form, and then shows how to calculate total revenue as a function of quantity.

• The video demonstrates how to find the derivative of the total revenue function to obtain the marginal revenue curve equation.