Question

In the United States, 2013 was a bad year for growing wheat. And as wheat supply decreased, the price of wheat rose dramatically, leading to a lower quantity demanded (a movement along the demand curve). The accompanying table describes what happened to prices and the quantity of wheat demanded. $$ \begin{array}{l|c|c} & 2012 & 2013 \\ \text { Quantity demanded (bushels) } & 2.2 \text { billion } & 2.0 \text { billion } \\ \text { Average price (per bushel) } & \$ 3.42 & \$ 4.26 \end{array} $$ a. Using the midpoint method, calculate the price elasticity of demand for winter wheat. b. What is the total revenue for U.S. wheat farmers in 2012 and \(2013 ?\) c. Did the bad harvest increase or decrease the total revenue of U.S. wheat farmers? How could you have predicted this from your answer to part a?

Short answer

The price elasticity of demand for winter wheat is 0.32 (inelastic), total revenue for 2012 is \$7.524 billion, total revenue for 2013 is \$8.52 billion. The bad harvest increased the total revenue of U.S. wheat farmers; this could be predicted from the inelastic price elasticity of demand.

Step-by-step solution

- Calculate percentage change in Quantity Demanded

Using the midpoint formula for elasticity, calculate the percentage change in quantity demanded. The formula is: \[\text{Percentage Change in Quantity} = \frac{Q_2 - Q_1}{(Q_2 + Q_1) / 2}\], where \(Q_1\) is the initial quantity and \(Q_2\) is the new quantity.

- Calculate percentage change in Price

Calculate the percentage change in price using the midpoint formula: \[\text{Percentage Change in Price} = \frac{P_2 - P_1}{(P_2 + P_1) / 2}\], where \(P_1\) is the initial price and \(P_2\) is the new price.

- Calculate the Price Elasticity of Demand

The price elasticity of demand is calculated as the absolute value of the percentage change in quantity demanded divided by the percentage change in price: \[\text{Price Elasticity of Demand} = \left|\frac{\text{Percentage Change in Quantity}}{\text{Percentage Change in Price}}\right|\].

- Calculate Total Revenue for 2012 and 2013

Total Revenue (TR) is calculated as the price (P) times the quantity demanded (Q): \[TR = P \times Q\]. Calculate this for both 2012 and 2013 to answer part b.

- Compare Total Revenues to Determine Impact on Revenue

Compare total revenue from 2012 to 2013 to determine if the revenue increased or decreased. If the price elasticity of demand computed in Step 3 is less than 1, it means demand is inelastic and total revenue would move in the direction of the price change.

Key concepts

Midpoint Method

The midpoint method is a technique used to calculate the percentage change in economic variables, such as quantity demanded or price, in a way that is symmetrical regardless of the direction of the change. It's particularly useful in price elasticity of demand calculations which provide insight into how much the quantity demanded of a good responds to a change in its price.

The formula for the midpoint method is:
\[\text{Percentage Change} = \frac{\text{New Value} - \text{Old Value}}{(\text{New Value} + \text{Old Value}) / 2}\]
By applying this method, we average the starting and ending values, which allows us to calculate percentage changes from a neutral middle ground, avoiding the problem of using either the new or the old value as a base for our percentage calculation. This is crucial for elasticity calculations because it doesn't overemphasize the change from either the beginning or the end perspective.

Quantity Demanded

Quantity demanded refers to the total amount of a good or service that consumers are willing and able to purchase at a particular price over a given period of time. It is represented graphically by the demand curve, which typically slopes downwards from left to right, indicating that higher prices will, all things being equal, lead to a lower quantity demanded.

It's important to distinguish between a 'change in quantity demanded', which is a movement along the demand curve due to a price change, and a 'change in demand', which implies a shift of the entire demand curve usually caused by factors other than the price of the good itself. The exercise given illustrates the former, where a poor wheat harvest led to an increased wheat price and consequently a movement along the demand curve to a lower quantity demanded of wheat.

Total Revenue

Total revenue (TR) is a fundamental concept in economics that measures the total amount of money a firm receives from the sale of its goods or services. It is calculated by multiplying the price of the good (P) by the quantity of the good that is sold (Q):
\[TR = P \times Q\]
In the context of the exercise, it involves calculating the product of the price per bushel of wheat and the quantity of bushels demanded for the years 2012 and 2013. Changes in total revenue can inform businesses about consumer behavior and the elasticity of their product, which implicates how sensitive consumers are to price changes. An inelastic demand, indicated by a price elasticity less than one, would suggest that an increase in price will lead to an increase in total revenue, holding all else constant.

Price Change

Price change refers to the variation in the cost of a good or service over time. It is a significant factor affecting both the quantity demanded and the seller's total revenue. In economics, the response of the quantity demanded to a price change is measured by the price elasticity of demand, which indicates if the quantity demanded is sensitive (elastic) or insensitive (inelastic) to price changes.

A price change can be due to factors such as changes in production costs, shifts in supply and demand conditions, or market competition levels. In the case of the wheat market described in the exercise, an external event (the bad harvest) caused a decrease in supply, leading to higher prices. It's critical to analyze whether such price changes lead to higher or lower total revenue, which depends on the elasticity of the product's demand.