Price Elasticity of Demand (PED) Along a Straight-Line Demand Curve

Author

A Level Economics Notes

Definition of Price Elasticity of Demand (PED)

Price elasticity of demand (PED) measures the responsiveness of the quantity demanded of a good to a change in its price.

Formally:

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

or, using calculus for a demand function \(Q = f(P)\):

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

Key points:

  • PED is usually negative for normal goods (because of the law of demand), but we often refer to its absolute value \(|\text{PED}|\).
  • Interpretation:
    • \(|\text{PED}| > 1\): Elastic demand (quantity responds strongly to price changes).
    • \(|\text{PED}| = 1\): Unitary elastic demand.
    • \(|\text{PED}| < 1\): Inelastic demand (quantity responds weakly to price changes).

How PED Changes Along a Straight-Line Demand Curve

On a typical downward-sloping straight-line (linear) demand curve, PED is not constant:

  • High prices / low quantities (upper part of the curve): demand is more elastic.
  • Midpoint of the curve: demand is unitary elastic (\(|\text{PED}| = 1\)).
  • Low prices / high quantities (lower part of the curve): demand is more inelastic.

Why PED changes even though the slope is constant

For a linear demand curve, the slope \(\frac{dP}{dQ}\) (or \(\frac{dQ}{dP}\)) is constant, but PED depends on both the slope and the ratio \(\frac{P}{Q}\):

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

  • \(\frac{dQ}{dP}\) is constant for a straight line.
  • But \(\frac{P}{Q}\) changes as you move along the curve.

So even with a constant slope, the percentage responsiveness changes because the base levels of \(P\) and \(Q\) change.

Intuitively:

  • At the top-left (high \(P\), low \(Q\)): a given absolute change in price is a small percentage change, but the same absolute change in quantity is a large percentage change → high elasticity.
  • At the bottom-right (low \(P\), high \(Q\)): the same absolute price change is a large percentage change, but the quantity change is a small percentage change → low elasticity.

Numerical Example

Consider a simple linear demand function:

\[ P = 100 - 2Q \quad \text{or} \quad Q = 50 - 0.5P \]

Here, \(\frac{dQ}{dP} = -0.5\) (constant).

PED at any point:

\[ \text{PED} = \frac{dQ}{dP} \times \frac{P}{Q} = -0.5 \times \frac{P}{Q} \]

Point A: High price, low quantity (elastic region)

Let \(Q = 10\):

  • \(P = 100 - 2(10) = 80\)
  • \(\text{PED} = -0.5 \times \frac{80}{10} = -0.5 \times 8 = -4\)

\(|\text{PED}| = 4 > 1\) → elastic demand.

Point B: Midpoint (unitary elasticity)

Midpoint in quantity terms: \(Q = 25\)

  • \(P = 100 - 2(25) = 50\)
  • \(\text{PED} = -0.5 \times \frac{50}{25} = -0.5 \times 2 = -1\)

\(|\text{PED}| = 1\) → unitary elastic.

Point C: Low price, high quantity (inelastic region)

Let \(Q = 40\):

  • \(P = 100 - 2(40) = 20\)
  • \(\text{PED} = -0.5 \times \frac{20}{40} = -0.5 \times 0.5 = -0.25\)

\(|\text{PED}| = 0.25 < 1\) → inelastic demand.

So along the same straight line:

  • Top segment: \(|\text{PED}| > 1\) (elastic)
  • Midpoint: \(|\text{PED}| = 1\) (unitary)
  • Bottom segment: \(|\text{PED}| < 1\) (inelastic)

Using the Percentage-Change (Midpoint) Formula Between Two Points

You can also see this by computing PED between pairs of points using the midpoint formula:

\[ \text{PED} = \frac{\%\ \Delta Q}{\%\ \Delta P} = \frac{\frac{Q_2 - Q_1}{(Q_1 + Q_2)/2}}{\frac{P_2 - P_1}{(P_1 + P_2)/2}} \]

Using the same curve \(P = 100 - 2Q\):

Between upper points (elastic region)

Between \((P_1, Q_1) = (80, 10)\) and \((P_2, Q_2) = (60, 20)\):

\[ \%\ \Delta Q = \frac{20 - 10}{(10 + 20)/2} = \frac{10}{15} \approx 0.667 \] \[ \%\ \Delta P = \frac{60 - 80}{(80 + 60)/2} = \frac{-20}{70} \approx -0.286 \] \[ \text{PED} \approx \frac{0.667}{-0.286} \approx -2.33 \quad (\text{elastic}) \]

Between lower points (inelastic region)

Between \((P_1, Q_1) = (40, 30)\) and \((P_2, Q_2) = (20, 40)\):

\[ \%\ \Delta Q = \frac{40 - 30}{(30 + 40)/2} = \frac{10}{35} \approx 0.286 \] \[ \%\ \Delta P = \frac{20 - 40}{(40 + 20)/2} = \frac{-20}{30} \approx -0.667 \] \[ \text{PED} \approx \frac{0.286}{-0.667} \approx -0.43 \quad (\text{inelastic}) \]

Same straight line, but elasticity falls as you move down and to the right.

Key Pattern to Remember (for Exams)

On a typical downward-sloping straight-line demand curve:

  • Upper half: \(|\text{PED}| > 1\) → elastic
  • Midpoint: \(|\text{PED}| = 1\) → unitary
  • Lower half: \(|\text{PED}| < 1\) → inelastic
  • At the price-axis intercept (where \(Q = 0\)): \(|\text{PED}| \to \infty\) (perfectly elastic in the limit)
  • At the quantity-axis intercept (where \(P = 0\)): \(\text{PED} = 0\) (perfectly inelastic in the limit)

This is why total revenue behaves differently along the curve:

  • In the elastic region, a price decrease raises total revenue.
  • In the inelastic region, a price increase raises total revenue.
  • Maximum total revenue occurs at the unitary-elastic midpoint.