Assume that the demand function for a certain commodity has...

Assume that the demand function for a certain commodity has the form $$ p=\sqrt{-a x^{2}+b} \quad(a \geq 0, b \geq 0) $$ where $x$ is the quantity demanded, measured in units of a thousand and $p$ is the unit price in dollars. Suppose the quantity demanded is $6000(x=6)$ when the unit price is $\$ 8$ and $8000(x=8)$ when the unit price is $\$ 6 .$ Determine the demand equation. What is the quantity demanded when the unit price is set at $\$ 7.50 ?$

Assume that the demand function for a certain commodity has the form
$$ p=\sqrt{-a x^{2}+b} \quad(a \geq 0, b \geq 0) $$
where $x$ is the quantity demanded, measured in units of a thousand and $p$ is the unit price in dollars. Suppose the quantity demanded is $6000(x=6)$ when the unit price is $\$ 8$ and $8000(x=8)$ when the unit price is $\$ 6 .$ Determine the demand equation. What is the quantity demanded when the unit price is set at $\$ 7.50 ?$

Key Concepts

Parameter Estimation

Parameter estimation involves using known data points to determine unknown constants in a mathematical model. By substituting observed values for price and quantity into the demand function, a system of equations is constructed and solved to find the specific parameters that best describe the relationship.

Algebraic Manipulation and Equation Solving

Algebraic manipulation is key to rearranging and simplifying equations to isolate and solve for unknown variables. Techniques such as squaring both sides to eliminate a square root or rewriting equations ensure that the model can be solved either directly or through a system of equations.

Demand Function

A demand function in economics represents the relationship between the quantity of a good that consumers are willing to buy and its price. It serves as a fundamental tool to analyze consumer behavior, market equilibrium, and the effects of price changes on overall consumption.

Functional Form and Square Root Relationships

Using a specific functional form, such as incorporating a square root, allows the model to capture non-linear relationships between price and quantity demanded. The square root indicates that changes in quantity have a diminishing or moderated effect on price, which can reflect realistic market behaviors.

Inverse Function Analysis

Inverse function analysis is used when the goal is to determine the quantity demanded corresponding to a particular price. This involves inverting the demand function—solving for quantity when given a price value—thus enabling predictions and decision-making based on market conditions.

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