Elasticity of Demand for Exercise Bicycles The demand...

Elasticity of Demand for Exercise Bicycles The demand function for a certain make of exercise bicycle sold exclusively through cable television is $$ p=\sqrt{9-0.02 x} \quad(0 \leq x \leq 450) $$ where $p$ is the unit price in hundreds of dollars and $x$ is the quantity demanded per week. Compute the elasticity of demand and determine the range of prices corresponding to inelastic, unitary, and elastic demand. Hint: Solve the equation $E(p)=1.$

Elasticity of Demand for Exercise Bicycles The demand function for a certain make of exercise bicycle sold exclusively through cable television is
$$
p=\sqrt{9-0.02 x} \quad(0 \leq x \leq 450)
$$
where $p$ is the unit price in hundreds of dollars and $x$ is the quantity demanded per week. Compute the elasticity of demand and determine the range of prices corresponding to inelastic, unitary, and elastic demand.
Hint: Solve the equation $E(p)=1.$

Key Concepts

Elastic Demand

Elastic demand is characterized by an absolute elasticity value greater than one. Here, the percentage change in quantity demanded is greater than the percentage change in price, showing that consumers are highly responsive to variations in price.

Differentiation in Economics

Differentiation, a core concept in calculus, is applied in economics to compute the rate of change of a function, such as a demand or supply function. In elasticity calculations, taking the derivative of the demand function is crucial to determining how quantity responds to small changes in price, thus enabling the analysis of elasticity.

Unitary Elasticity

Unitary elasticity arises when the absolute value of elasticity is equal to one. In this case, the percentage change in quantity demanded exactly matches the percentage change in price, meaning that the total revenue remains constant as price varies.

Price Elasticity Formula

The price elasticity formula, typically expressed as E = (dQ/dP) × (P/Q) or alternatively using the specific functions for price and quantity, is used to calculate the responsiveness of demand. This formula is essential for analyzing how small changes in price can result in larger proportional changes in quantity demanded.

Elasticity of Demand

Elasticity of demand measures how the quantity demanded of a good responds to a change in its price. It is quantified as the percentage change in quantity divided by the percentage change in price, serving as a key indicator of consumer sensitivity to price variations.

Inelastic Demand

Inelastic demand occurs when the absolute value of elasticity is less than one. This means that the percentage change in quantity demanded is less than the percentage change in price, indicating that consumers are relatively insensitive to price changes in this range.

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