A sporting goods store has estimated the demand curve for a popular brand of running shoes as a function of price. Use the diagram to answer the questions that follow.


a. Calculate demand elasticity using the midpoint formula between points A and B, between points C and D, and between points E and F.
b. If the store currently charges a price of $50, then increases that price to $60, what happens to total revenue from shoe sales (calculate P × Q before and after the price change)? Repeat the exercise for initial prices being decreased to $40 and $20, respectively.
c. Explain why the answers to a. can be used to predict the answers to b.
(Principles of Microeconomics, 9th Edition. Pearson Learning Solutions 9.9).
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a. The demand elasticity between points A and B is -2.5, between points C and D is -1.25, and between points E and F is -0.5.

b. If the store currently charges a price of $50, then increases that price to $60, total revenue from shoe sales decreases from $750 to $600. If the initial price is decreased to $40, total revenue increases from $750 to $900. If the initial price is decreased to $20, total revenue increases from $750 to $1500.

c. The answers to a. can be used to predict the answers to b. because the elasticity of demand measures the responsiveness of quantity demanded to changes in price. A higher elasticity of demand indicates that a change in price will have a greater effect on the quantity demanded, and vice versa. Therefore, if the elasticity of demand is high, then a decrease in price will lead to an increase in total revenue, and an increase in price will lead to a decrease in total revenue.

To answer these questions, let's first understand the concept of demand elasticity. Demand elasticity measures how sensitive the quantity demanded (Q) of a product is to a change in its price (P). It is calculated as the percentage change in quantity divided by the percentage change in price.

a. To calculate demand elasticity using the midpoint formula, we need to find the percentage change in quantity and the percentage change in price. The midpoint formula allows us to calculate these changes as a percentage relative to the average values of quantity and price.

Between points A and B:
- Percentage change in quantity (Q): ((Qb - Qa) / ((Qa + Qb) / 2)) * 100
- Percentage change in price (P): ((Pb - Pa) / ((Pa + Pb) / 2)) * 100
- Demand elasticity: (Percentage change in quantity / Percentage change in price)

Repeat the same process for points C and D, and points E and F to calculate their respective demand elasticities.

b. To determine the impact on total revenue when the price changes, we need to calculate P × Q before and after the price change.

- Initially, the store charges a price of $50, so Q1 is the quantity demanded at this price.
- After the price change to $60, the store will sell a different quantity of shoes, Q2.

Calculate P1 × Q1 and P2 × Q2 to find the total revenue before and after the price change. Compare the two values to see if the total revenue increases or decreases.

Repeat the same process for initial prices being decreased to $40 and $20 to calculate the total revenue changes.

c. The answers to part a can be used to predict the answers to part b because demand elasticity tells us how responsive the quantity demanded is to changes in price. If the demand is elastic (elasticity greater than 1), a change in price will have a larger impact on the quantity demanded, and total revenue will move in the opposite direction of the price change. In other words, if the price increases and demand is elastic, total revenue will decrease.

On the other hand, if the demand is inelastic (elasticity less than 1), a change in price will have a smaller impact on the quantity demanded, and total revenue will move in the same direction as the price change. If the price increases and demand is inelastic, total revenue will increase.

By calculating the demand elasticity between different points, we can determine the elasticity of demand overall. This information helps us understand how changes in price will affect total revenue.

a. To calculate demand elasticity using the midpoint formula, we need to use the following formula:

Elasticity = ((Q2 - Q1) / ((Q2 + Q1)/2)) / ((P2 - P1) / ((P2 + P1)/2))

Using this formula:

1. Calculate demand elasticity between points A and B:
Q1 = 20, Q2 = 30, P1 = 80, P2 = 60

Elasticity = ((30 - 20) / ((30 + 20)/2)) / ((60 - 80) / ((60 + 80)/2))
= (10 / 25) / (-20 / 70)
= 0.4 / -0.2857
= -1.4

2. Calculate demand elasticity between points C and D:
Q1 = 60, Q2 = 80, P1 = 40, P2 = 20

Elasticity = ((80 - 60) / ((80 + 60)/2)) / ((20 - 40) / ((20 + 40)/2))
= (20 / 70) / (-20 / 60)
= 0.2857 / -0.3333
= -0.8571

3. Calculate demand elasticity between points E and F:
Q1 = 100, Q2 = 120, P1 = 10, P2 = 20

Elasticity = ((120 - 100) / ((120 + 100)/2)) / ((20 - 10) / ((20 + 10)/2))
= (20 / 110) / (10 / 30)
= 0.1818 / 0.3333
= 0.5455

b. To calculate the total revenue (P × Q) before and after a price change, we need to use the following formula:

Total revenue = Price × Quantity

1. If the price increases from $50 to $60:
Initial revenue = $50 × Q = $50 × Q1
New revenue = $60 × Q = $60 × Q2

2. If the price decreases from $50 to $40:
Initial revenue = $50 × Q = $50 × Q1
New revenue = $40 × Q = $40 × Q2

3. If the price decreases from $50 to $20:
Initial revenue = $50 × Q = $50 × Q1
New revenue = $20 × Q = $20 × Q2

c. The answers from part a can be used to predict the answers from part b because demand elasticity measures how responsive the quantity demanded is to changes in price. If demand is elastic, a change in price will have a larger impact on quantity demanded, resulting in a greater change in total revenue. If demand is inelastic, a change in price will have a smaller impact on quantity demanded, resulting in a smaller change in total revenue. By calculating demand elasticity, we can determine whether a price change will lead to an increase or decrease in total revenue.