The price elasticity of supply (PES) measures the responsiveness of the quantity supplied to a change in price. It is calculated using the following formula:

$\text{PES} = \frac{dQ_s}{dp} \times \frac{p}{Q_s}$

Given the supply function:

$Q_s = S(p) = -4 + 8p$

1. Find the derivative of the supply function with respect to price $p$:

$\frac{dQ_s}{dp} = 8$

2. Substitute $\frac{dQ_s}{dp} = 8$, $p$, and $Q_s = -4 + 8p$ into the PES formula:

$\text{PES} = 8 \times \frac{p}{-4 + 8p}$

Thus, the price elasticity of supply is:

$\text{PES} = \frac{8p}{-4 + 8p}$

This equation shows that the elasticity depends on the price $p$. For specific values of $p$, you can substitute to find the exact PES value.

Would you like to evaluate the elasticity at a particular price or need further clarification?

Follow-up Questions:

1. How does the price elasticity of supply change as the price increases?
2. What is the elasticity of supply when the price $p = 1$?
3. At what price does the supply elasticity become unitary (PES = 1)?
4. How would the supply elasticity differ if the supply function had a different coefficient for $p$?
5. What does a high price elasticity of supply indicate about the market for widgets?

Tip:

Price elasticity of supply is generally higher in markets where producers can quickly increase production when prices rise.