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 $8$dollar and $8000(x=8)$ when the unit price is $6$dollar. Determine the demand equation. What is the quantity demanded when the unit price is set at $7.50$dollar?

Key Concepts

Demand Function

A demand function describes the relationship between the price of a commodity and the quantity demanded by consumers. It is an essential tool in economics for understanding how various factors, notably price changes, influence consumer purchasing behavior and overall market demand.

Parameter Estimation in Demand Functions

Parameter estimation involves using real-world data points to determine the unknown constants in a theoretical demand function. This process is fundamental in ensuring the model accurately represents the market and can be used for predictions and economic analysis.

Solving Simultaneous Equations

When multiple data points are provided, simultaneous equations are used to solve for the unknown parameters in the function. This technique involves setting up and solving a system of equations that stem from the demand function evaluated at different observed values.

Square Root Transformation

The presence of a square root in the demand function means that the underlying quadratic expression must be non-negative to yield real prices. Handling a square root transformation in a function often requires careful consideration of the domain and the impact of the transformation on the shape and behavior of the demand curve.

Application for Prediction

After determining the complete demand function through parameter estimation, it can be used to make predictions about market behavior, such as calculating the quantity demanded at a specific price. This application is a key aspect of using demand functions for economic forecasting and decision-making.

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