Question

This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. a. Show that, for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. Show that, for any linear demand curve, the vertical distance between the demand and marginal revenue curves is \(-1 / b \cdot q,\) where \(b(<0)\) is the slope of the demand curve. c. Show that, for a constant elasticity demand curve of the form \(q=a P^{b},\) the vertical distance between the demand and marginal revenue curves is a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand. d. Show that, for any downward-sloping demand curve, the vertical distance between the demand and marginal revenue curves at any point can be found by using a linear approximation to the demand curve at that point and applying the procedure described in part (b). e. Graph the results of parts (a)-(d) of this problem.

Short answer

#Question# Explain the relationship between demand and marginal revenue curves for different functional forms of demand, focusing on both linear and constant elasticity demand curves. #Answer# For linear demand curves, the relationship between the demand curve and the marginal revenue curve is such that the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. The vertical distance between the demand and marginal revenue curves at any price level can be represented by the formula: -1/b × q. In the case of constant elasticity demand curves, the marginal revenue curve is a downward-sloping curved line that is parallel to the demand curve, with a constant ratio between their heights. This ratio is equal to -b, where b is the elasticity of demand. For any downward-sloping demand curve, a linear approximation can be used to find the vertical distance between the demand and marginal revenue curves by calculating the tangent line to the demand curve at a specific point and applying the procedure for linear demand curves.

Step-by-step solution

Linear demand curve equation

Let the linear demand curve be given by the equation: \(q = a - b \cdot P\), where \(a\) and \(b\) are positive constants.

Finding marginal revenue

We know that the total revenue is equal to the product of price and quantity, or \(R(P) = P \cdot q = P \cdot (a - b \cdot P)\). To find the marginal revenue, we take the first derivative of the revenue function with respect to price: $$MR(P) = \frac{dR(P)}{dP} = a - 2 \cdot b \cdot P$$

Show that MR bisects the distance between the vertical axis and the demand curve

At any given price level, both demand and marginal revenue can be expressed as vertical distances from the horizontal axis. The vertical distance from the demand curve is given by \((a - b \cdot P) - 0 = a - b\cdot P\), and the vertical distance from the MR curve is given by \((a - 2 \cdot b \cdot P) - 0 = a - 2 \cdot b\cdot P\). Thus, the distance between demand and MR is \((\)a - b\cdot P) - (a - 2\cdot b\cdot P) = b\cdot P$. Since the distance between demand and MR at any given price is half the distance from the vertical axis \((2 \cdot b\cdot P),\) we can conclude that the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. Vertical distance between demand and marginal revenue curves

Formula for vertical distance

As derived in part (a), the vertical distance between the demand and marginal revenue curve at any price level is given by: \(-1/b \cdot q\). c. Constant elasticity demand curve

Constant elasticity demand curve equation

The constant elasticity demand curve is given by the equation: \(q = a \cdot P^{b}\).

Finding marginal revenue

The total revenue function is given by \(R(P) = P \cdot q = P \cdot (a \cdot P^{b})\). Taking the first derivative with respect to price, we get: $$MR(P) = a \cdot P^{b} + b \cdot a \cdot P^{b-1}$$

Proportional distance between demand and MR curves

The vertical distance between the demand and MR curves is given by \((a \cdot P^{b}) - (a \cdot P^{b} + b\cdot a\cdot P^{b-1}) = -b \cdot a \cdot P^{b-1}\). By dividing both sides by the demand function \((a \cdot P^{b})\), we find the constant ratio between the heights of the demand and MR curves: $$\frac{-b \cdot a \cdot P^{b-1}}{a \cdot P^{b}} = -b$$ d. Downward-sloping demand curve

Linear approximation of demand curve

To use the procedure from part (b) for any downward-sloping demand curve, we need to find a linear approximation of the demand curve at a given point. This can be done by calculating the tangent line to the demand curve at that point. Using the results from part (b), we can find the vertical distance between the demand and marginal revenue curves. e. Graphical representation

Graphing linear demand and MR curves

For a linear demand curve, draw a downward-sloping straight line representing the demand curve. The marginal revenue curve will be another downward-sloping straight line that is parallel to the demand curve and bisects the distance between the vertical axis and the demand curve for any price.

Graphing constant elasticity demand and MR curves

For a constant elasticity demand curve, draw a downward-sloping curved line representing the demand curve. The marginal revenue curve will be another downward-sloping curved line parallel to the demand curve and maintaining a constant ratio between their heights.

Graphing downward-sloping demand curve and linear approximation

For any downward-sloping demand curve, draw a downward-sloping curved line representing the demand curve. Use the tangent line to the demand curve at a particular point as a linear approximation of the demand curve. Then, apply the procedure from part (b) and draw the marginal revenue curve as a downward-sloping straight line parallel to the linear approximation.

Key concepts

Demand Curve

The demand curve is a fundamental concept in economics representing the relationship between the price of a good and the quantity of that good consumers are willing to purchase at various prices. It graphically shows the inverse relationship between price and quantity demanded, typically sloping downward from left to right. This downward slope indicates that, as the price decreases, the quantity demanded increases, and conversely, as the price increases, the quantity demanded decreases.

Understanding the demand curve is crucial because it provides insight into consumer preferences and the potential revenue a company can generate at different prices. It also serves as a basis for calculating other important economic measures, such as marginal revenue, which is vital for determining optimal pricing strategies for maximising profit.

Linear Demand Curve

A linear demand curve is a specific type of demand curve that shows a straight-line relationship between price and quantity demanded. It can be represented by the simple equation, \(q = a - b \times P\), where \(q\) is the quantity demanded, \(P\) is the price, \(a\) is the y-intercept indicating the maximum quantity demanded at a price of zero, and \(b\) is the slope, which shows the rate at which quantity demanded changes with price. The slope is normally negative, reflecting the inverse relationship between price and quantity.

The simplicity of the linear demand curve makes it a popular choice for basic economic analysis and helps in illustrating economic principles with clarity. Moreover, when analyzing marginal revenue in the context of a linear demand curve, marginal revenue will always be less than the price, as the curve for marginal revenue falls at twice the rate of the demand curve due to increased competition or decreased consumer willingness to pay as quantity increases.

Constant Elasticity Demand Curve

The constant elasticity demand curve represents a situation where the price elasticity of demand is the same at all points along the curve. Price elasticity measures how sensitive the quantity demanded is to a change in price. A constant elasticity demand curve is depicted by the equation \(q = a \times P^b\), where \(a\) and \(b\) are constants, and the elasticity is represented by the exponent \(b\).

This type of curve is nonlinear and reflects different market behaviors compared to linear demand curves. One key feature is that percentage changes in price lead to a constant percentage change in quantity demanded, regardless of the initial price level. When it comes to marginal revenue, for a product with constant price elasticity, the marginal revenue curve is a straight line with a constant slope reflecting the consistent elasticity.

Downward-Sloping Demand Curve

The downward-sloping demand curve, a universal concept in economics, illustrates that as the price of a product decreases, the quantity demanded increases, and as the price increases, demand falls. This behavior is due to the law of demand, which is based on the substitutability of goods, income effect, and diminishing marginal utility.

Real-world demand curves can take on many shapes but will generally slope downwards. They may not always be linear or exhibit constant elasticity. When estimating marginal revenue for such curves, economists often use a linear approximation at a particular point on the curve, just as an engineer might use a straight-line to approximate the slope of a curve at a particular point. This approximation simplifies the complex relationship between price and quantity demanded, highlighting the importance of understanding how various types of demand curves affect the analysis of marginal revenue and market dynamics.