Understanding Price Elasticity of Supply

Price elasticity of supply ($\text{E}_\text{s}$) measures how much the quantity supplied of a good changes in response to a change in its price. It is a crucial concept in economics as it helps us understand the responsiveness of producers to price signals.

The formula for price elasticity of supply is:

$\text{E}_\text{s} = \frac{\text{% change in quantity supplied}}{\text{% change in price}}$

This can also be written using calculus as:

$\text{E}_\text{s} = \frac{\text{dQ}}{\text{dP}} \times \frac{\text{P}}{\text{Q}}$

Where:

• $\text{Q}$ is the quantity supplied
• $\text{P}$ is the price
• $\frac{\text{dQ}}{\text{dP}}$ is the slope of the supply curve (specifically, the reciprocal of the slope of the price-quantity graph, $\frac{1}{\text{slope}}$)

Analyzing a Linear Supply Curve Through the Origin

A linear supply curve has a constant slope. When this linear supply curve also passes through the origin (0,0), its equation can be represented as:

$\text{P} = \text{mQ}$ or $\text{Q} = \frac{1}{\text{m}} \text{P}$

Where $\text{m}$ is the constant slope of the supply curve when price is on the y-axis and quantity on the x-axis. In our elasticity formula, we use $\frac{\text{dQ}}{\text{dP}}$, which is the slope of the quantity-price relationship, which is $\frac{1}{\text{m}}$. Let's denote $\text{k} = \frac{1}{\text{m}}$. So, the equation is $\text{Q} = \text{kP}$.

For this linear relationship $\text{Q} = \text{kP}$, the derivative of Q with respect to P is constant:

$\frac{\text{dQ}}{\text{dP}} = \text{k}$

Now, let's substitute this into the elasticity formula:

$\text{E}_\text{s} = \frac{\text{dQ}}{\text{dP}} \times \frac{\text{P}}{\text{Q}}$

$\text{E}_\text{s} = \text{k} \times \frac{\text{P}}{\text{Q}}$

Since the curve passes through the origin, for any point ($\text{Q}, \text{P}$) on the curve (other than the origin itself), the relationship $\text{Q} = \text{kP}$ holds. We can substitute $\text{Q} = \text{kP}$ into the elasticity formula:

$\text{E}_\text{s} = \text{k} \times \frac{\text{P}}{\text{kP}}$

Assuming $\text{P} \neq 0$ and $\text{k} \neq 0$ (a meaningful supply curve), we can cancel out k and P:

$\text{E}_\text{s} = \frac{\text{kP}}{\text{kP}}$

$\text{E}_\text{s} = 1$

Conclusion on Price Elasticity of Supply

For any linear supply curve that passes through the origin, the price elasticity of supply is always equal to 1. This means that the percentage change in quantity supplied is always equal to the percentage change in price, regardless of the specific price level. Such a supply curve is considered to have unitary elasticity.

Revision Table: Types of Price Elasticity of Supply

Additional Information on Supply Curves and Elasticity

• A linear supply curve that intersects the price (y) axis above the origin will have $\text{E}_\text{s} > 1$ at points above the intersection with the quantity axis and $\text{E}_\text{s} < 1$ at points below it.
• A linear supply curve that intersects the quantity (x) axis to the right of the origin will have $\text{E}_\text{s} < 1$ at points to the right of the intersection with the price axis and $\text{E}_\text{s} > 1$ at points to the left of it.
• The elasticity of supply is influenced by factors like the availability of inputs, time horizon (short run vs. long run), and ease of switching production.