Understanding Cost Curves and the Rectangular Hyperbola

In economics, cost curves visually represent the relationship between the cost of production and the quantity of output produced. Different cost curves have distinct shapes based on how costs behave as output changes.

What is a Rectangular Hyperbola?

A rectangular hyperbola is a specific type of curve where, for any point on the curve, the product of its x and y coordinates is a constant. In the context of cost curves, if a curve is a rectangular hyperbola, it means that the product of the quantity produced (x-axis) and the cost per unit (y-axis) remains constant.

Analyzing the Given Cost Curves

Let's look at the different cost curves provided in the options:

• Total Fixed Cost (TFC) Curve: Total fixed costs do not change with the level of output. The TFC curve is a horizontal line parallel to the x-axis. This is not a rectangular hyperbola.
• Total Variable Cost (TVC) Curve: Total variable costs change with the level of output. The TVC curve typically starts from the origin and increases as output increases, often initially at a decreasing rate and then at an increasing rate, forming an inverse S-shape. This is not a rectangular hyperbola.
• Average Variable Cost (AVC) Curve: Average variable cost is calculated as Total Variable Cost divided by the quantity of output ($\text{AVC} = \frac{\text{TVC}}{\text{Quantity}}$). The AVC curve is typically U-shaped. This is not a rectangular hyperbola.
• Average Fixed Cost (AFC) Curve: Average fixed cost is calculated as Total Fixed Cost divided by the quantity of output ($\text{AFC} = \frac{\text{TFC}}{\text{Quantity}}$). Total Fixed Cost (TFC) is constant.

Why the AFC Curve is a Rectangular Hyperbola

The formula for Average Fixed Cost (AFC) is:

$$\text{AFC} = \frac{\text{TFC}}{\text{Quantity}}$$

Let Q represent the Quantity of output and AFC represent the Average Fixed Cost. So, the formula is:

$$\text{AFC} = \frac{\text{TFC}}{\text{Q}}$$

Rearranging this equation, we get:

$$\text{AFC} \times \text{Q} = \text{TFC}$$

Since TFC is a constant value (it doesn't change with output), the product of AFC (on the y-axis) and Quantity (on the x-axis) is always equal to this constant TFC value.

This relationship ($\text{AFC} \times \text{Q} = \text{Constant}$) is the defining characteristic of a rectangular hyperbola. As quantity (Q) increases, AFC must decrease proportionally so that their product remains equal to the constant TFC. The curve gets closer and closer to both the x-axis and the y-axis but never actually touches them.

Therefore, the cost curve which is a rectangular hyperbola is the AFC curve.