2.6.1. Elasticity along a Linear Demand Curve

Let us consider a linear demand curve q = a – bp. Note that at any point on the demand curve, the change in demand per unit change in the price = –b.

Elasticity measures how responsive the demand for a good is to changes in its price. When we have a linear demand curve, such as q = a - bp, 'q' represents the quantity demanded, 'a' is the y-intercept, and 'b' is the slope of the line. The slope of the demand curve (–b) indicates how much the quantity demanded will change when the price changes by one unit.

Imagine a lemonade stand. If the price of a cup of lemonade increases slightly, you might still pay for it because you really want it. But if the price increases significantly, you might decide to buy a different drink instead. This shows how demand can change based on price adjustments.

Substituting the value of in (2.16b), we obtain, e = – b / (q) = – (a – bp), thus e = -bp / (a – bp).

To calculate price elasticity of demand (e) along a linear demand curve, we start with the formula e = (∆q / ∆p) * (p / q). Here, the price elasticity (e) equals the absolute value of the slope of the demand curve multiplied by the ratio of the price (p) to the quantity demanded (q). This formula allows us to see how sensitivity changes at different points on the demand curve.

Returning to our lemonade example, if the price goes from $1 to$2 and the quantity sold drops significantly, we've observed high elasticity. Conversely, if the price changes slightly and customer numbers stay steady, this reflects low elasticity.

From (2.17), it is clear that the elasticity of demand is different at different points on a linear demand curve. At p = 0, the elasticity is 0, at q = 0, elasticity is ∞.

This means that elasticity levels vary along the demand curve. For instance, at the point where the price is zero, a tiny change in price will not affect demand at all (elasticity = 0). However, when the quantity reaches zero, even a minuscule price change will dramatically impact demand (elasticity = ∞). This concept allows economists to analyze consumer responsiveness based on current price and quantity levels.

Picture a concert where entry is free. Here, everyone would want to attend (elasticity = 0 because price has no role). Conversely, if the concert were sold out but the price remained high, even a slight dip in price might lead to a sudden increase in attendance, illustrating high elasticity.

The elasticity of a linear demand curve can easily be measured geometrically. The elasticity of demand at any point on a straight line demand curve is given by the ratio of the lower segment and the upper segment of the demand curve at that point.

Graphically, if you look at the linear demand curve, you can visualize elasticity as the ratio of the segments formed along the curve. This demonstrates the relationship between price changes and resulting quantity changes as you move along the demand curve—showing how elasticity factors in at various price levels.

Think of a seesaw. The distance from the fulcrum on each side corresponds to how much changes in price will affect demand for the product. A longer distance indicates higher sensitivity (greater elasticity), while a shorter distance indicates less sensitivity (lower elasticity).

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