Question

This problem concerns the relationship between demand and marginal revenue curves for a few functional forms. Show that: a. for a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. b. for any linear demand curve, the vertical distance between the demand and marginal revenue curves is \(-V b \cdot q,\) where \(b(<0)\) is the slope of the demand curve. c. for a constant elasticity demand curve of the form \(q=a P^{\prime},\) 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. 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) through (d) of this problem.

Short answer

#Answer: (a) For a linear demand curve, the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price. The marginal revenue curve is a straight line with a steeper slope than the demand curve. (b) The vertical distance between a linear demand and marginal revenue curve is given by (-Vb * q), where b is the slope of the demand curve, and q is the quantity demanded. The vertical distance depends on the slope and quantity demanded, increasing as both of these factors increase. (c) For a constant elasticity demand curve, the vertical distance between the demand and marginal revenue curve is given by a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand. (d) For a downward-sloping demand curve, the vertical distance between the demand and marginal revenue curve can be found using a linear approximation and applying the procedure from part (b). This means the vertical distance depends on the slope of the demand curve and the quantity demanded, with a steeper slope resulting in a larger vertical distance and increasing distance as quantity demanded increases. (e) Graphing the results for each part visually demonstrates these relationships between the demand and marginal revenue curves for each case.

Step-by-step solution

Write the linear demand curve equation

The linear demand curve is given by the equation: \(Q = a - bP\), where \(Q\) is the quantity demanded, \(a\) is a constant, \(P\) is the price, and \(b\) is the slope of the demand curve (which is negative).

Calculate total revenue

Total revenue (\(TR\)) is calculated as the product of the price and the quantity demanded, \(TR = PQ\). From step 1, we have the equation \(Q = a - bP\). Substitute this into the total revenue equation to get: \(TR = P(a - bP)\).

Calculate marginal revenue

Marginal revenue (\(MR\)) is the derivative of total revenue with respect to quantity, \(MR = \frac{dTR}{dQ}\). Differentiate the total revenue equation from step 2 with respect to \(Q\) to get: \(MR = a - 2bP\).

Show that the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price

To show that the \(MR\) curve bisects the distance between the vertical axis and the demand curve for any price, we need to find the halfway point between the demand curve equation and the marginal revenue equation when \(P\) is constant. We have: Demand curve: \(Q = a - bP\) Marginal revenue curve: \(MR = a - 2bP\) Let's calculate the average between these two, or the "midpoint": Midpoint: \(\frac{1}{2}(Q + MR) = \frac{1}{2}(a - bP + a - 2bP) = a - \frac{3}{2}bP\) Here, we can observe that the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price, as the equation represents a straight line with a slope steeper than the demand curve and crosses halfway between the demand curve and vertical axis at any price. #b. Vertical distance between the demand and marginal revenue curves#

Calculate the vertical distance

Now we will calculate the vertical distance between the demand and marginal revenue curves, given the slope \(b\) and quantity \(q\). From part (a), we have the demand curve equation \(Q = a - bP\) and the marginal revenue equation \(MR = a - 2bP\). The vertical distance between these two curves (\(VD\)) is the difference in quantity demanded at the same price level: \(VD = Q - MR\) Substitute the equations: \(VD = (a - bP) - (a - 2bP) = -Vb \cdot q\), since \(q = -V \cdot b\).

Explain the result

The vertical distance between the demand and marginal revenue curves is \(-Vb \cdot q\). This means that the distance between the two curves depends on the slope of the demand curve and the quantity demanded. A steeper slope (\(b\)) will result in a larger vertical distance, and as quantity demanded (\(q\)) increases, the vertical distance between the curves also increases. #c. Constant elasticity demand curve and the vertical distance#

Write the constant elasticity demand curve equation

In this case, the constant elasticity demand curve is given by the equation: \(q = aP^{e}\), where \(a\) is a constant and \(e\) is the price elasticity of demand.

Calculate total revenue

Total revenue (\(TR\)) is calculated as the product of the price and the quantity demanded, \(TR = PQ\). From step 1, we have the equation \(q = aP^{e}\). Substitute this into the total revenue equation to get: \(TR = P(aP^{e})\).

Calculate marginal revenue

Marginal revenue (\(MR\)) is the derivative of total revenue with respect to quantity, \(MR = \frac{dTR}{dQ}\). Differentiate the total revenue equation from step 2 with respect to \(Q\) to get: \(MR = aP^{e-1}\).

Calculate the vertical distance

Now we will calculate the vertical distance between the demand and marginal revenue curves for the constant elasticity demand curve. The vertical distance between these two curves is the difference in quantity demanded at the same price level: \(VD = q - MR\) Substitute the equations: \(VD = aP^{e} - aP^{e-1} = aP^{e-1}(P - 1)\) Since, the ratio \(\frac{VD}{q} = \frac{aP^{e-1}(P-1)}{aP^{e}} = \frac{P - 1}{P}\), 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. Vertical distance for a downward-sloping demand curve#

Approximate the demand curve using a linear approximation

For any downward-sloping demand curve, we can use a linear approximation to the demand curve at that point. This means that we can represent the demand curve using a linear equation, similar to part (a).

Apply the procedure described in part (b)

Based on our analysis in part (b), we have found the vertical distance (\(VD\)) between the demand curve and the marginal revenue curve for any linear demand curve as \(-Vb \cdot q\). Since we have approximated the demand curve using a linear equation, we can apply the same formula to find the vertical distance for any downward-sloping demand curve.

Explain the result

The vertical distance between the demand and marginal revenue curves for any downward-sloping demand curve can be found using a linear approximation and applying the procedure from part (b). This means that the vertical distance depends on the slope of the demand curve and the quantity demanded. A steeper slope will result in a larger vertical distance, and as quantity demanded increases, the vertical distance between the curves also increases. #e. Graph the results#

Graph the results for part (a)

Plot a linear demand curve (\(Q = a - bP\)) and a corresponding marginal revenue curve (\(MR = a - 2bP\)). Show that the marginal revenue curve bisects the distance between the vertical axis and the demand curve for any price.

Graph the results for part (b)

Show that the vertical distance between the linear demand curve and the marginal revenue curve is \(-Vb \cdot q\).

Graph the results for part (c)

Plot a constant elasticity demand curve (\(q = aP^{e}\)) and a corresponding marginal revenue curve (\(MR = aP^{e-1}\)). Show that the vertical distance between the curves is a constant ratio of the height of the demand curve, with this constant depending on the price elasticity of demand.

Graph the results for part (d)

Plot a downward-sloping demand curve with a linear approximation. Show that the vertical distance between the demand and marginal revenue curves can be found using the procedure from part (b). By following these steps, you should have a visual representation of the results from parts (a) through (d) for the relationship between demand and marginal revenue curves for various functional forms.

Key concepts

Demand Curve

In economics, a demand curve is a graphical representation that shows the relationship between the price of a good or service and the quantity demanded by consumers over a given period. Generally, a demand curve slopes downwards from left to right, indicating that as the price decreases, the quantity demanded increases. This is due to the law of demand, which states that all else being equal, consumers will purchase more of a good as its price falls. This curve is crucial for understanding how changes in price influence consumer purchasing decisions.

Key points about the demand curve include:

• It illustrates consumers' willingness to purchase at varying price points.
• The slope can indicate sensitivity to price changes, known as price elasticity.
• It often assumes that all other factors, such as consumer income and preferences, remain constant.

Economists use demand curves to predict consumer behavior, analyze market dynamics, and make informed decisions about pricing strategies.

Downward-Sloping Demand

A downward-sloping demand curve is a fundamental concept in economics that depicts a negative relationship between price and quantity demanded. The downward slope implies that lower prices lead to higher quantities demanded, and vice versa. This phenomenon can be attributed to several reasons, such as the substitution effect and income effect.

The substitution effect occurs when consumers opt for cheaper alternatives as the price of a good rises. Simultaneously, the income effect reflects a consumer's increased purchasing power as prices decline. Together, these effects account for the downward slope of the demand curve, underscoring that consumers will likely buy more of a product when it is cheaper.

Important points to remember about downward-sloping demand are:

• This slope is observed in normal goods where demand increases with a decrease in price.
• It demonstrates consumer behavior in terms of price sensitivity.
• Not all demand curves are strictly downward-sloping; exceptions include Giffen and Veblen goods.

Understanding the nuances of a downward-sloping demand curve helps businesses tailor pricing strategies to optimize sales and revenues.

Price Elasticity of Demand

Price elasticity of demand is a measure used by economists to explain how the quantity demanded of a good responds to a change in its price. It quantifies the sensitivity or responsiveness of consumers to price changes, expressed as a percentage change in quantity demanded divided by a percentage change in price.

If the demand for a product is elastic, consumers are very responsive to price changes; a small decrease in price will lead to a significant increase in quantity demanded. Conversely, with inelastic demand, consumers are less responsive, meaning that changes in price will result in a smaller change in the quantity demanded.

Here are some core aspects of price elasticity of demand:

• Elastic demand has an elasticity greater than 1, indicating high sensitivity to price changes.
• Inelastic demand has an elasticity less than 1, indicating low sensitivity to price changes.
• Unitary elasticity, where elasticity is exactly 1, means the percentage change in quantity demanded is equal to the percentage change in price.

Recognizing the price elasticity of demand is crucial for businesses in setting prices, estimating consumer reactions, and determining optimal pricing points to balance revenue and sales.