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Calculating the Price Elasticity of Supply (PES) involves determining how the quantity supplied of a good or service responds to changes in its price. There are several methods used to calculate PES, each applicable in different contexts and depending on the availability of data. Here are the primary methods:

###### 1. **Percentage Method (Point Elasticity)**

This is the most common method and involves calculating the percentage change in quantity supplied in response to a percentage change in price. The formula is:

\[ PES = \frac{\% \text{ change in quantity supplied}}{\% \text{ change in price}} \]

This method gives the elasticity of supply at a specific point on the supply curve.

###### 2. **Arc Elasticity Method**

Arc elasticity is useful when calculating the elasticity over a range of the supply curve, rather than at a single point. This method averages the two end points of the price and quantity changes, providing a measure of elasticity over a segment of the supply curve. The formula is:

\[ PES = \frac{\left( \frac{Q_2 – Q_1}{Q_2 + Q_1} \right)}{\left( \frac{P_2 – P_1}{P_2 + P_1} \right)} \]

Where \(Q_1\) and \(Q_2\) are the initial and final quantities, and \(P_1\) and \(P_2\) are the initial and final prices, respectively.

###### 3. **Midpoint Method**

A specific case of the arc elasticity method, the midpoint method is used to calculate the elasticity between two points on the supply curve. It mitigates the issue of which point to use as a reference by using the midpoint between the two points as the base for percentage changes. The formula is:

\[ PES = \frac{\left( \frac{Q_2 – Q_1}{(Q_1 + Q_2)/2} \right)}{\left( \frac{P_2 – P_1}{(P_1 + P_2)/2} \right)} \]

###### 4. **Linear Supply Curve Method**

If the supply curve is linear, PES can vary along the curve. The general form of a linear supply curve is \(Q = a + bP\), where \(a\) and \(b\) are constants, \(Q\) is quantity, and \(P\) is price. The elasticity can be calculated at any point using the slope of the curve (\(b\)) and the price and quantity at the point of interest.

###### 5. **Logarithmic Method**

When the relationship between price and quantity supplied is expected to be log-linear, economists might use logarithmic regression to estimate the elasticity. This involves estimating a regression equation of the form:

\[ \ln(Q) = a + b \ln(P) \]

Here, \(b\) represents the elasticity of supply, as it indicates the percentage change in quantity supplied for a 1% change in price.

Each of these methods has its applications and is chosen based on the nature of the supply curve, the range of data available, and the specific requirements of the analysis.

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