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Assume that the demand curve D(p) given below is the market demand for widgets:

Q=D(p)=2631−23p𝑄=𝐷(𝑝)=2631-23𝑝, p > 0




Let the market supply of widgets be given by:

Q=S(p)=−4+8p𝑄=𝑆(𝑝)=-4+8𝑝, p > 0




where p is the price and Q is the quantity. The functions D(p) and S(p) give the number of widgets demanded and supplied at a given price. What is the price elasticity of supply?

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.

Explore how to calculate the price elasticity of supply (PES) for widgets using the given demand and supply functions. Learn the formula and step-by-step calculation with specific examples. Suitable for advanced undergraduate students and learners of economics.

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