Express the price elasticity in terms of MR and AR

Price elasticity of demand (\(E_d\)) measures the responsiveness of quantity demanded to a change in price. It is defined as the percentage change in quantity demanded divided by the percentage change in price:

\[ E_d = \frac{\%\text{ change in quantity demanded}}{\%\text{ change in price}} \]

For a linear demand curve, price elasticity can be expressed in terms of marginal revenue (\(MR\)) and average revenue (\(AR\)). 

1. **Marginal Revenue (\(MR\)):**
   Marginal revenue is the additional revenue generated from selling one more unit of a product. It is calculated as the derivative of total revenue with respect to quantity (\(Q\)).

2. **Average Revenue (\(AR\)):**
   Average revenue, also known as price, is the total revenue divided by the quantity sold (\(Q\)).

The relationship between price elasticity (\(E_d\)), marginal revenue (\(MR\)), and average revenue (\(AR\)) is as follows:

\[ E_d = \frac{dQ}{dP} \times \frac{P}{Q} \]

Where:
- \(dQ/dP\) is the rate of change of quantity with respect to price, which is the inverse of the slope of the demand curve. This is equivalent to the reciprocal of the slope of the demand curve.
- \(P/Q\) is the ratio of price to quantity, which represents the average revenue (\(AR\)).

So, we can rewrite the price elasticity of demand (\(E_d\)) in terms of marginal revenue (\(MR\)) and average revenue (\(AR\)) as:

\[ E_d = \frac{dQ}{dP} \times \frac{P}{Q} = \frac{1}{\text{slope of demand curve}} \times \frac{P}{Q} = \frac{MR}{AR} \]

Therefore, the price elasticity of demand (\(E_d\)) can be expressed as the ratio of marginal revenue (\(MR\)) to average revenue (\(AR\)).