How is it that if the slope of the demand curve is constant, elasticity can change along the demand curve?

The linear demand curve can be represented by the equation: \(Q_d = a - bP\), where \(Q_d\) is the quantity demanded, \(a\) and \(b\) are constants, and \(P\) is the price. The slope of the demand curve is represented by the constant \(b\).

Price elasticity of demand (E_d) is the percentage change in quantity demanded divided by the percentage change in price. Mathematically, it can be represented as: \(E_d = \frac{\% \Delta Q_d}{\% \Delta P}\). Using calculus, we can express this as: \(E_d = \frac{dQ_d}{dP} \cdot \frac{P}{Q_d}\).

First, we'll find the derivative of the demand curve function with respect to price, \(dQ_d/dP\), which will tell us the rate at which quantity demanded changes with respect to the price. Since our demand curve equation is \(Q_d = a - bP\), its derivative with respect to price is simply: \(\frac{dQ_d}{dP} = -b\). Now, we can substitute this result into the elasticity equation: \(E_d = (-b) \cdot \frac{P}{a - bP}\).

From the equation we derived in step 3, \(E_d = (-b) \cdot \frac{P}{a - bP}\), we can see that while the slope (\(-b\)) remains constant, the elasticity (\(E_d\)) changes as the values of price (\(P\)) and quantity demanded (\(Q_d\)) change. In other words, elasticity depends on the ratio of price to quantity demanded, not just on the slope of the demand curve. At higher prices (closer to the vertical intercept on the price axis), the quantity demanded is lower, and the elasticity value is higher (more elastic, as buyers are very responsive to price changes). As we move down the demand curve, the price decreases while the quantity demanded increases. Lower prices correspond to lower values of elasticity (less elastic, as buyers become less responsive to price changes).

The slope of the linear demand curve remains constant, but the price elasticity of demand changes along the curve because elasticity is determined by the ratio of price to quantity demanded, which varies along the curve. At higher prices and lower quantities demanded, the demand curve is more elastic, while at lower prices and higher quantities demanded, the demand curve is less elastic.