Question

Suppose that the following table gives data on the price of rye and the number of bushels of rye sold in 2017 and 2018 : $$ \begin{array}{c|c|c} \hline \text { Year } & \begin{array}{c} \text { Price (dollars per } \\ \text { bushel) } \end{array} & \text { Quantity (bushels) } \\ \hline 2017 & \$ 3 & 8 \text { million } \\ \hline 2018 & 2 & 12 \text { million } \\ \hline \end{array} $$ a. Calculate the change in the quantity of rye demanded divided by the change in the price of rye. Measure the quantity of rye in bushels. b. Calculate the change in the quantity of rye demanded divided by the change in the price of rye, but this time measure the quantity of rye in millions of bushels. Compare your answer to the one you computed in (a). c. Assuming that the demand curve for rye did not shift between 2017 and \(2018,\) use the information in the table to calculate the price elasticity of demand for rye. Use the midpoint formula in your calculation. Compare the value for the price elasticity of demand to the values you calculated in (a) and (b).

Short answer

Solution to part a is -4 million bushels per dollar, part b is -4 bushels per dollar, and part c is -1.6, illustrating that demand for rye is price elastic.

Step-by-step solution

Calculate the Change in Quantity and Price

To find the change in the quantity of rye demanded divided by the change in the price of rye for part (a), you need to subtract the quantity demanded in 2017 from the quantity demanded in 2018 (12 million bushels - 8 million bushels = 4 million bushels). Then, subtract the price in 2017 from the price in 2018 (2 dollars per bushel - 3 dollars per bushel = -1 dollar per bushel). Dividing the change in quantity by the change in price gives \(-4\) million bushels per dollar.

Convert Quantity into Millions of Bushels

For part (b), the change in quantity and price is the same except you measure the quantity in millions of bushels. Therefore, you divide the 4 million bushel change in quantity by 1 million to get a change of 4 bushels divided by 1 dollar. So, the change in quantity demanded divided by the change in price is \(-4\) bushels per dollar.

Calculate Price Elasticity of Demand

The price elasticity of demand, for part (c), is calculated using the midpoint formula: \((\Delta Q/ \frac{Q1+Q2}{2}) / (\Delta P/ \frac{P1+P2}{2}).\) \(\Delta Q\) is the change in quantity, which is 4 million bushels, \((Q1+Q2)/2\) is the average quantity, which is 10 million bushels. Similarly, \(\Delta P\) is the change in price, which is -1 dollar, and \((P1+P2)/2\) is the average price, which is 2.5 dollars. Plugging these values in, the price elasticity of demand is \(-1.6\).

Key concepts

Demand Curve

The demand curve is a graphical representation that shows how much of a good consumers are willing to purchase at different price levels. It generally slopes downward from left to right, indicating that as the price decreases, the quantity demanded increases, and vice versa. This is known as the law of demand.
In the context of the rye example, the demand curve illustrates the relationship between the price of rye and the quantity sold in the years 2017 and 2018.
These points on the curve, 8 million bushels at $3 and 12 million bushels at $2, reflect consumer behavior under different pricing scenarios. When prices drop from $3 to $2, the demand increases, exemplifying the inverse relationship inherent in the demand curve.

• The slope of the demand curve is crucial as it shows how sensitive quantity demanded is to a change in price.
• Steeper curves reveal less sensitivity to price changes, while flatter curves indicate higher sensitivity.

Midpoint Formula

The midpoint formula is a method used to calculate the price elasticity of demand more accurately. It uses the average of starting and ending values to find elasticity between two points on the demand curve.
This approach is considered more precise because it accounts for the relative change instead of the absolute change.
The midpoint formula is given by:
\[\left(\frac{\Delta Q}{\frac{Q1+Q2}{2}}\right) / \left(\frac{\Delta P}{\frac{P1+P2}{2}}\right)\]
To use it, we identify \(\Delta Q\) as the change in quantity demanded and \(\Delta P\) as the change in price. \(Q1\) and \(Q2\) are the initial and final quantities, while \(P1\) and \(P2\) are the initial and final prices, respectively.

• This formula provides a measure of elasticity that does not depend on the direction of movement along the curve.
• In the rye example, it is used to calculate the price elasticity of demand, resulting in a value of -1.6, showing that demand is elastic.

Change in Quantity Demanded

Change in quantity demanded refers to the difference in the quantity of a good consumers are willing to buy as a result of a change in its price. It is depicted as a movement along the demand curve.
To find this change, we simply subtract the initial quantity from the final quantity. In our rye example, the change is calculated as:
\[12\text{ million bushels} - 8\text{ million bushels} = 4\text{ million bushels}\]
The quantity demanded increased as the price declined. This highlights the fundamental principle of demand that consumers will typically purchase more of a good when its price is lower.

• This concept is crucial for understanding consumer behavior and making pricing decisions.
• Knowing how much quantity will change with price helps businesses set competitive prices and predict sales volume.

Change in Price

The change in price describes the difference between the initial and final price levels of a good or service. It acts as the driving force behind movements along the demand curve, affecting how much quantity is demanded.
To calculate this, one subtracts the original price from the new price. In the rye example, this change is calculated as follows:
\[2\text{ dollars per bushel} - 3\text{ dollars per bushel} = -1\text{ dollar per bushel}\]
This negative change indicates a price drop, which typically leads to an increase in quantity demanded, as seen with the increase from 8 million to 12 million bushels.

• The change in price is a key determinant of demand elasticity, showing how sensitive demand is to price variations.
• Understanding price change effects helps businesses and economists forecast market behavior.