Problem 12 - Applied Calculus

Question

The demand for a product is given by \(p=90-10 q\). Find the elasticity of demand when \(p=50\). If this price rises by \(2 \%\), calculate the corresponding percentage change in demand.

Step-by-Step Solution

Answer

Elasticity of demand is -1.25; demand decreases by 2.5% for a 2% price rise.

Step 1: Write the formula for elasticity of demand

Elasticity of demand (Ed) is given by the formula: \[ E_d = \frac{dq}{dp} \times \frac{p}{q} \] where \(dq/dp\) is the derivative of the demand function with respect to price, and \(p\) and \(q\) are the price and quantity demanded, respectively.

Step 2: Find the derivative of the demand function

The demand function is given by the equation \(p = 90 - 10q\). To find the derivative in terms of \(q\): \[ q = \frac{90 - p}{10} \] \(\frac{dq}{dp} = -\frac{1}{10}\)

Step 3: Calculate the quantity demanded at \(p=50\)

Substitute \(p = 50\) into the demand equation to find \(q\): \[ 50 = 90 - 10q \] \[ 10q = 40 \] \[ q = 4 \] When the price is 50, the quantity demanded is 4.

Step 4: Calculate elasticity of demand at \(p=50\)

Substitute \(p = 50\), \(q = 4\), and \(\frac{dq}{dp} = -\frac{1}{10}\) into the elasticity formula: \[ E_d = -\frac{1}{10} \times \frac{50}{4} = -1.25 \] The elasticity of demand is -1.25.

Step 5: Interpret elasticity value

An elasticity of -1.25 means demand is slightly elastic; a 1% change in price results in a 1.25% change in quantity demanded in the opposite direction.

Step 6: Calculate percentage change in demand for 2% price increase

Using the elasticity value calculated (\(-1.25\)), the percentage change in demand due to a 2% price increase is: \[ \text{Percentage change in demand} = E_d \times \text{Percentage change in price} = -1.25 \times 2\% = -2.5\% \]

Key Concepts

Percentage Change in Demand

The percentage change in demand helps us understand how sensitive consumers are to price changes. This measure tells us by what percentage the quantity demanded changes in response to a percentage change in price. It's important when we want to predict how changes in price may impact consumer buying behavior.
In simple terms, if you know the elasticity of demand and the percentage by which the price changes, you can easily predict how much the demand will increase or decrease.

For instance, if the elasticity of demand is -1.25, and the price of a product increases by 2%, the demand will fall by 2.5%. This is because elasticity gives us the percentage change in quantity demanded for a 1% change in price. Here:

• Elasticity = -1.25
• Price Change = 2%

So, the calculation goes as follows:
\( \text{Percentage change in demand} = -1.25 \times 2\% = -2.5\% \)
Thus, a 2% increase in price results in a 2.5% decrease in demand.

Derivative of Demand Function

The derivative of the demand function tells us the rate at which quantity demanded changes with respect to price. It's a vital concept we use when calculating elasticity of demand. The derivative, often written as \( \frac{dq}{dp} \), helps to identify the relationship between price changes and demand shifts.
In our scenario, the demand function is given by the equation:
\( p = 90 - 10q \)
To find the derivative, we first need to express \( q \) in terms of \( p \) by rearranging the equation:
\( q = \frac{90 - p}{10} \)
Then, taking the derivative of \( q \) with respect to \( p \), we get:
\( \frac{dq}{dp} = -\frac{1}{10} \)
This value signifies that, as the price changes, the quantity demanded decreases at a constant rate of \(-\frac{1}{10}\).

Demand Equation

The demand equation provides a mathematical representation connecting price with quantity demanded. In economic terms, it's an equation that predicts how much of a product consumers will buy at a given price.
For our exercise, the equation used was:
\( p = 90 - 10q \)
This linear equation lets us predict the demand at various price points. By substituting a particular price into this equation, we can solve for the corresponding quantity.

• To find the quantity demanded when \( p = 50 \):

\( 50 = 90 - 10q \)
Subtract 90 from both sides:
\( 10q = 40 \)
Then, divide by 10:
\( q = 4 \)
This tells us that when the price is 50, 4 units of the product will be demanded. Using the demand equation, you can easily calculate the demand for different price levels, helping businesses make pricing decisions and forecast sales.