The demand for tickets to a children's museum can be modeled...

The demand for tickets to a children's museum can be modeled as $$ D(p)=0.03 p^{2}-1.6 p+21 \text { thousand tickets } $$ where $p$ is the market price. a. What is the price elasticity of demand at a market price of $\$ 15$ per ticket? b. Is demand elastic or inelastic at $\$ 15$ per ticket? c. Explain in context what elasticity (or inelasticity) at $\$ 15$ per ticket means.

The demand for tickets to a children's museum can be modeled as
$$
D(p)=0.03 p^{2}-1.6 p+21 \text { thousand tickets }
$$
where $p$ is the market price.
a. What is the price elasticity of demand at a market price of $\$ 15$ per ticket?
b. Is demand elastic or inelastic at $\$ 15$ per ticket?
c. Explain in context what elasticity (or inelasticity) at $\$ 15$ per ticket means.

Key Concepts

Demand Function

A demand function mathematically represents the relationship between the price of a good and the quantity demanded by consumers. It provides a way to model consumer behavior and analyze how changes in market prices can influence the overall demand for a product or service.

Differentiation and Marginal Analysis

Differentiation is a calculus tool used to determine the rate at which a function changes with respect to its variable—in this case, understanding how minute changes in price affect quantity demanded. Marginal analysis uses the derivative of the demand function to assess the marginal change in quantity, which is crucial for calculating elasticity.

Price Elasticity of Demand

This concept measures the responsiveness of the quantity demanded of a good or service to a change in its price. It is defined as the percentage change in quantity demanded divided by the percentage change in price. Understanding this concept allows economists and businesses to predict how changes in pricing can affect sales and revenue.

Elastic versus Inelastic Demand

This concept categorizes demand based on the calculated elasticity value. If the absolute value of the elasticity is greater than one, the demand is considered elastic, meaning consumers are highly responsive to price changes. If it is less than one, the demand is inelastic, indicating that consumer response to price changes is relatively low. This distinction is vital for making decisions about pricing strategies and understanding revenue implications.

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