Question

If a rise in the price of sushi from 98 t to \(\$ 1.02\) a piece decreases the quantity of soy sauce demanded from 101 units to 99 units an hour and decreases the quantity of sushi demanded by 1 percent an hour, calculate the: a. Price elasticity of demand for sushi. b. Cross elasticity of demand for soy sauce with respect to the price of sushi.

Short answer

a. PED for sushi is approximately -0.24. b. CED for soy sauce with respect to the price of sushi is approximately -0.49.

Step-by-step solution

- Define Price Elasticity of Demand (PED) for sushi

Price elasticity of demand (PED) measures how much the quantity demanded of a good responds to a change in the price of that good. The formula for PED is given by: \[ PED = \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}} \]

- Calculate Percentage Change in Quantity Demanded for Sushi

We are given the percentage change in quantity demanded for sushi as 1 percent. Therefore, \[ \text{Percentage change in quantity demanded} = -1\text{\textbackslash%} \]

- Calculate Percentage Change in Price for Sushi

The initial price of sushi is 98 t, which converts to 0.98 dollars. The new price is $\text 1.02. The percentage change in price is: \[ \text{Percentage change in price} = \frac{1.02 - 0.98}{0.98} \times 100 = \frac{0.04}{0.98} \times 100 \approx 4.08\text{\textbackslash%} \]

- Calculate PED for Sushi

Using the formula for PED: \[ PED = \frac{-1}{4.08} \approx -0.24 \]

- Define Cross Elasticity of Demand (CED) for Soy Sauce

Cross elasticity of demand (CED) measures how much the quantity demanded of one good responds to a change in the price of another good. The formula for CED is given by: \[ CED = \frac{\text{Percentage change in quantity demanded of soy sauce}}{\text{Percentage change in price of sushi}} \]

- Calculate Percentage Change in Quantity Demanded for Soy Sauce

The initial quantity of soy sauce demanded is 101 units and it decreases to 99 units. The percentage change in quantity demanded is: \[ \text{Percentage change in quantity demanded} = \frac{99 - 101}{101} \times 100 = \frac{-2}{101} \times 100 \approx -1.98\text{\textbackslash%} \]

- Calculate CED for Soy Sauce

Using the formula for CED: \[ CED = \frac{-1.98\text{\textbackslash%}}{4.08\text{\textbackslash%}} \approx -0.49 \]

Key concepts

Cross Elasticity of Demand

Cross Elasticity of Demand (CED) helps us understand how the demand for one product changes when the price of another product changes. For instance, if the price of sushi goes up, we might see a change in the demand for soy sauce. The formula for CED is: \[ CED = \frac{\text{Percentage change in quantity demanded of soy sauce}}{\text{Percentage change in price of sushi}} \] This formula tells us how sensitive the quantity demanded of one good is to price changes of another good. In our example, when the price of sushi increases, fewer people buy soy sauce, showing a negative cross elasticity. This indicates that sushi and soy sauce are complementary goods. If the relationship was positive, the goods would be considered substitutes.

Percentage Change in Quantity Demanded

The percentage change in quantity demanded is a key component in calculating elasticity. It measures how much the quantity of a good demanded changes in response to a change, whether in price or another factor. Here's how you calculate it: \[ \text{Percentage change in quantity demanded} = \frac{\text{New Quantity} - \text{Initial Quantity}}{\text{Initial Quantity}} \times 100 \] In the context of the exercise, if the quantity of soy sauce changes from 101 units to 99 units, we compute: \[ \text{Percentage change in quantity demanded} = \frac{99 - 101}{101} \times 100 \approx -1.98\% \] A negative result indicates a decrease in the quantity demanded in response to some factor, which in our case, is the price of sushi.

Percentage Change in Price

Percentage change in price is essential in calculations for both Price Elasticity of Demand (PED) and Cross Elasticity of Demand (CED). It shows the rate at which the price of a good has changed over a certain period. The formula is: \[ \text{Percentage change in price} = \frac{\text{New Price} - \text{Initial Price}}{\text{Initial Price}} \times 100 \] In our given problem, if the price of sushi rises from \(0.98 to \)1.02, the calculation is: \[ \text{Percentage change in price} = \frac{1.02 - 0.98}{0.98} \times 100 \approx 4.08\% \] This percentage tells us how much the price has increased, which is crucial for further calculations like PED and CED. Understanding and calculating this concept correctly lets us analyze how price variations impact the demand for goods.