Question

If a 12 percent rise in the price of orange juice decreases the quantity of orange juice demanded by 22 percent and increases the quantity of apple juice demanded by 14 percent, calculate the a. Price elasticity of demand for orange juice. b. Cross elasticity of demand for apple juice with respect to the price of orange juice.

Short answer

The price elasticity of demand for orange juice is \(-1.83\). The cross elasticity of demand for apple juice with respect to the price of orange juice is \(1.17\).

Step-by-step solution

Understand the Price Elasticity of Demand Formula

The price elasticity of demand (PED) measures the responsiveness of quantity demanded to a change in price. It is given by the formula: \[PED = \frac{\% \text{change in quantity demanded}}{\% \text{change in price}}\]

Calculate the Price Elasticity of Demand for Orange Juice

Given a 12% rise in price causing a 22% decrease in demand, use the formula: \[PED_{OJ} = \frac{-22\%}{12\%} = -1.83\]The negative sign indicates that the demand moves inversely to the price.

Understand the Cross Elasticity of Demand Formula

The cross elasticity of demand (CED) measures the responsiveness of quantity demanded for one good in response to a change in price of another good. It is given by the formula: \[CED = \frac{\% \text{change in quantity demanded of good B}}{\% \text{change in price of good A}}\]

Calculate the Cross Elasticity of Demand for Apple Juice with Respect to the Price of Orange Juice

Given a 12% rise in orange juice price causing a 14% increase in apple juice demand, use the formula: \[CED_{Apple/OJ} = \frac{14\%}{12\%} = 1.17\]

Key concepts

Cross Elasticity of Demand

Cross elasticity of demand measures how the quantity demanded of one good (say, apple juice) responds to changes in the price of another good (like orange juice). This concept helps us understand the relationship between two goods and whether they are substitutes or complements. When two goods are substitutes, an increase in the price of one leads to an increase in the demand for the other. Conversely, if they are complements, an increase in the price of one results in a decrease in the demand for the other. In our example, the cross elasticity of demand between apple juice and orange juice is 1.17. This positive value indicates that apple juice and orange juice are substitutes. When the price of orange juice rises by 12%, the quantity demanded for apple juice increases by 14%.

Responsiveness of Quantity Demanded

The responsiveness of quantity demanded shows how much the quantity demanded of a good changes in response to changes in various factors such as price, income, or the price of related goods. It's a key concept in economics because it tells us how sensitive consumers are to these changes. A highly responsive (or elastic) demand means that consumers will buy a lot less of a product if the price goes up slightly. This is evident in the case of orange juice's price elasticity of demand (PED), which is -1.83. This negative and greater-than-one (in absolute value) number indicates that a 12% increase in the price leads to a 22% decrease in quantity demanded. Thus, the demand for orange juice is highly elastic.

Inverse Relationship in Economics

An inverse relationship in economics occurs when two variables move in opposite directions. One common example is the relationship between price and quantity demanded, illustrated by the negative sign in the PED formula. When the price of a good increases, the quantity demanded generally decreases, and vice versa. This fundamental concept explains consumer behavior: higher prices tend to discourage purchases, while lower prices attract more buyers. In our orange juice example, the increase in price (12%) results in a significant decrease in demand (22%). This is a clear example of an inverse relationship at work.