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A demand function represents the relationship between the quantity of a good demanded and its price. It shows how the quantity consumers are willing to purchase responds to changes in price. In the case of the breakfast cereal Grape Nuts, the demand function is given as:\[ x = A p^{-2.0711} \]where:

• \( x \) is the quantity demanded,
• \( A \) is a constant, and
• \( p \) is the price.

This specific function indicates that the quantity sold is inversely related to the price raised to a certain power. The exponent \( -2.0711 \) tells us the degree of responsiveness the quantity has to changes in price.

The price and quantity relationship is a fundamental concept in economics. It describes how the quantity demanded of a good varies with its price. In the given demand function:\[ x = A p^{-2.0711} \]we see that the quantity \( x \) is negatively related to the price \( p \), with an exponent of \(-2.0711\).This means that as the price increases, the quantity demanded decreases, and vice versa, at a specific rate determined by the exponent. Essentially, if the price of Grape Nuts goes up, people buy less of it, and if the price goes down, they buy more. This inverse relationship is typical for most goods and highlights the sensitivity of consumers to price changes.

Differentiation is a mathematical process used to find the derivative of a function, which describes how a change in one variable affects another variable. In the context of elasticity, we need to differentiate the demand function with respect to price.For the demand function \( x = A p^{-2.0711} \), the derivative with respect to price \( p \) is calculated as follows:\[ \frac{dx}{dp} = -2.0711 A p^{-3.0711} \]This derivative tells us the rate at which quantity changes when there is a small change in price. Negative differentiation, as seen here, signifies that as the price increases, the quantity demanded decreases, which aligns with the law of demand.

Elastic demand refers to a situation where the quantity demanded of a good is highly responsive to changes in its price. To quantify this responsiveness, we calculate the elasticity of demand:\[ E_d = \frac{dQ}{dP} \times \frac{P}{Q} \]By substituting the differentiated values from the demand equation:\[ E_d = \frac{(-2.0711 A p^{-3.0711}) \cdot p}{A p^{-2.0711}} \]After simplifying the algebraic expression, we get \( E_d = -2.0711 \). With \( E_d = -2.0711 \), this indicates a very responsive or elastic demand, as values less than \(-1\) signify elasticity. This means that for every 1% increase in the price of Grape Nuts, the quantity demanded decreases by 2.0711%. This level of elasticity suggests consumers of Grape Nuts are sensitive to any price changes.