Rectangular Hyperbola

Definition

A rectangular hyperbola is a specific type of hyperbola where the product of the coordinates of any point on the curve is a constant. This unique property makes the rectangular hyperbola an important concept in the study of elasticity, particularly in the context of polar cases of elasticity and constant elasticity.

5 Must Know Facts For Your Next Test

1. The equation of a rectangular hyperbola is $xy = k$, where $k$ is a constant.
2. The rectangular hyperbola has the unique property that the product of the coordinates of any point on the curve is a constant.
3. In the context of elasticity, the rectangular hyperbola represents the case of constant elasticity, where the elasticity is the same at all points along the demand or supply curve.
4. The rectangular hyperbola is a special case of the general hyperbola, where the transverse and conjugate axes are perpendicular to each other.
5. The rectangular hyperbola is often used to model situations where the responsiveness of quantity demanded or supplied to changes in price or income is constant, such as in the case of perfectly elastic or perfectly inelastic demand or supply.

Review Questions

• Explain how the rectangular hyperbola is related to the concept of constant elasticity.
  • The rectangular hyperbola is closely related to the concept of constant elasticity because it represents a situation where the elasticity of demand or supply remains the same regardless of the level of the determinant, such as price or income. The unique property of the rectangular hyperbola, where the product of the coordinates of any point on the curve is a constant, allows for the representation of a demand or supply curve with a constant elasticity. This makes the rectangular hyperbola a useful tool in the analysis of elasticity, particularly in the context of polar cases of elasticity.
• Describe the key features of the rectangular hyperbola and how they relate to the study of elasticity.
  • The key features of the rectangular hyperbola that are relevant to the study of elasticity are its equation, $xy = k$, where $k$ is a constant, and the fact that the product of the coordinates of any point on the curve is a constant. These features allow the rectangular hyperbola to represent a demand or supply curve with a constant elasticity, where the responsiveness of quantity demanded or supplied to changes in price or income remains the same regardless of the level of the determinant. The perpendicular nature of the transverse and conjugate axes of the rectangular hyperbola also contributes to its usefulness in modeling situations of constant elasticity, as it reflects the inverse relationship between price and quantity that is characteristic of these cases.
• Analyze how the rectangular hyperbola can be used to illustrate the polar cases of elasticity, such as perfectly elastic and perfectly inelastic demand or supply.
  • The rectangular hyperbola can be used to illustrate the polar cases of elasticity, such as perfectly elastic and perfectly inelastic demand or supply. In the case of perfectly elastic demand or supply, the rectangular hyperbola would be a vertical or horizontal line, respectively, indicating that the quantity demanded or supplied is infinitely responsive to changes in price or income. Conversely, in the case of perfectly inelastic demand or supply, the rectangular hyperbola would be a horizontal or vertical line, respectively, indicating that the quantity demanded or supplied does not change at all in response to changes in price or income. By understanding the properties of the rectangular hyperbola and how it relates to the concept of constant elasticity, one can better analyze and interpret the polar cases of elasticity and their implications for economic decision-making.

Related terms

Hyperbola: A hyperbola is a type of conic section that is the set of all points in a plane where the absolute difference of the distances from two fixed points, called the foci, is a constant.

Elasticity:

Elasticity is a measure of the responsiveness of quantity demanded or supplied to changes in one of its determinants, such as price or income.

Constant Elasticity:

Constant elasticity refers to a situation where the elasticity of demand or supply remains the same regardless of the level of the determinant, such as price or income.