{Use of Tech} Demand functions and elasticity Economists use demand functions to describe how much of a commodity can be sold at varying prices. For example, the demand function D(p) = 500 - 10p says that at a price of p = 10, a quantity of D(10) = 400 units of the commodity can be sold. The elasticity E = dD/dp p/D of the demand gives the approximate percent change in the demand for every 1% change in the price. (See Section 3.6 or the Guided Project Elasticity in Economics for more on demand functions and elasticity.)


b. If the price is $12 and increases by 4.5%, what is the approximate percent change in the demand? 

Verified step by step guidance

1. First, identify the demand function given in the problem: D(p) = 500 - 10p.
2. Next, calculate the derivative of the demand function with respect to price p, which is dD/dp. For D(p) = 500 - 10p, the derivative dD/dp = -10.
3. Now, use the formula for elasticity E = (dD/dp) * (p/D). Substitute dD/dp = -10, p = 12, and D(12) = 500 - 10*12 = 380 into the formula.
4. Calculate the elasticity E at p = 12 using the values obtained: E = (-10) * (12/380).
5. Finally, use the elasticity to find the approximate percent change in demand for a 4.5% increase in price. Multiply the elasticity E by the percent change in price (4.5%) to get the percent change in demand.

Key Concepts

Demand Function

A demand function expresses the relationship between the price of a commodity and the quantity demanded by consumers. It is typically represented as D(p), where D is the quantity demanded and p is the price. For instance, in the function D(p) = 500 - 10p, the quantity demanded decreases as the price increases, illustrating the law of demand.

Elasticity of Demand

Elasticity of demand measures how sensitive the quantity demanded is to a change in price. It is calculated using the formula E = (dD/dp) * (p/D), where dD/dp is the derivative of the demand function with respect to price. A higher elasticity value indicates that demand is more responsive to price changes, which is crucial for understanding consumer behavior.

Percent Change Calculation

Percent change is a way to express the change in a quantity relative to its original value, calculated as (new value - old value) / old value * 100%. In the context of demand elasticity, it helps determine how much the demand will change in response to a percentage change in price, providing insights into market dynamics.