NotesAtHKU

Price elasticity of demand

PED Definition

The price elasticity of demand $\eta_d$ is a measure of responsiveness of quantity demanded to a change in price.

\eta_d = \frac{\% \Delta Q}{\% \Delta P}

$\eta < -1$ : elastic.
$\eta > -1$ : inelastic.
$\eta = -1$ : unit elastic.
$\eta = 0$ : perfectly inelastic.
$\eta = -\infty$ : perfectly elastic.

Formulas

Midpoint formula When two points $(Q_0, P_0)$ and $(Q_1, P_1)$ are given:

\eta = \frac{(Q_1-Q_0)(P_0+P_1)}{(Q_0+Q_1)(P_1-P_0)}

Point formula When the curve and a point (P, Q) are given, where $Q_x$ is the x-intercept and $P_y$ is the y-intercept:

\eta = \frac{P}{Q}\times\frac{1}{\text{Slope}} = -\frac{PQ_x}{QP_y}

These formulas are the same for the price elasticity of supply.

Linear demand elasticity observations

Consider the mid-point $M = (Q_m, P_m) = (\frac{Q_x}2, \frac{P_y}2)$ of a linear demand curve:

$P > P_m\to\eta_d < -1$ (elastic).
$P < P_m\to\eta_d > -1$ (inelastic).

Consider two curves, the flatter curve is more elastic when P is at a common point of the two curves.

Total revenue

Total revenue is the product of price and quantity:

PQ \text{ (total revenue)} = P \times Q

Therefore, it is also the area of the rectangle formed by the price and quantity to the axes.

Total revenue is maximized at the midpoint on a linear demand curve.

Elasticity, price change and revenue

The effects of price change to the revenue $PQ$ is affected by the elasticity of demand:

Demand elasticity	$P\uparrow$	$P\downarrow$
Elastic	$P\uparrow \times Q\Downarrow = PQ\downarrow$	$P\downarrow \times Q\Uparrow = PQ\uparrow$
Inelastic	$P\uparrow \times Q\Uparrow = PQ\uparrow$	$P\downarrow \times Q\Downarrow = PQ\downarrow$

Where the arrows indicate the direction and magnitude of change. The comparison of magnitude can be visualized by overlaps of the $PQ$ rectangles as well.

Unitary elasticity demand curve

Unitary elastic means that $PQ = \text{constant}$ , and hence the total revenue is constant at every point. The price elasticity equals -1 at every point.

If given an equation in the following form:

\ln Q = a + b \ln P

The price elasticity is always $b$ .

Example

If John would purchase $10 of a good regardless of the price, then $\eta_d = -1$ , and the revenue is constant at $10.

Determinants of the Price Elasticity of Demand

Determinants (increase)	Elasticity (more negative)	Explaination
Substitutes	+	Switching when price change is easier
Time horizon	+	More time to adjust and find substitutes
Specificity of classification	+	More specific goods are easier to find substitutes
Nature of good (luxurious)	+	Necessities have inelastic demand
Price	+	Higher price goods have more elastic demand

Other elasticities of demand for goods

Income elasticity of demand

Measures sensitivity of $\Delta Q_d$ to $\Delta I$ :

\eta = \frac{I}{Q} \times \frac{\Delta Q}{\Delta I}

The value of income elasticity of demand can deduce the type of good.

Cross-price elasticity of demand

Measures sensitivity of $\Delta Q_d$ of good A to $\Delta P$ of good B:

\eta = \frac{P_B}{Q_A} \times \frac{\Delta Q_A}{\Delta P_B}

The value of cross-price elasticity of demand can deduce the relationship between goods.

Questions might give you a demand relation for two goods, and asks for the above elasticities. They are in the form of:

Q_A = c - \frac{\Delta Q_A}{\Delta P_A}P_A + \frac{\Delta Q_A}{\Delta P_B}P_B + \frac{\Delta Q_A}{\Delta I}I

If the quantities are under logarithm, the elasticity is the coefficient of the price, income or cross-price term.

Price elasticity of supply

PES Definition

The price elasticity of supply $\eta_s$ is a measure of responsiveness of quantity supplied to a change in price. The equations are identical to the price elasticity of demand.

$\eta_s > 1$ : elastic.
$\eta_s < 1$ : inelastic.
$\eta_s = 1$ : unit elastic.
$\eta_s = 0$ : perfectly inelastic.
$\eta_s = \infty$ : perfectly elastic.

Linear supply curve observations

The y-intercept relates to the elasticity of the supply curve. Consider $P = mQ_s + c$ :

$c > 0\to\eta_s > 1$ (elastic at every point)
$c = 0\to\eta_s = 1$ (unit elastic at every point)
$c < 0\to\eta_s < 1$ (inelastic at every point)

Also notice that when $P$ is large enough, $\eta_{s} \approx 1$ , as the term $\frac{P}{Q} \to 1$ .

Example

If $MQ(q) = 2q$ , we can notice that it passes through the origin, and hence $\eta_s = 1$ at every point.

Determinants of the Price Elasticity of Supply

Determinants (larger)	Elasticity	Explaination
Change in per-unit costs with production increase	-	If it is easier to produce more, then supply curve is more elastic
Time horizon	+	If it takes longer to increase capacity, supply curve is more elastic
Availability of production inputs	+	If inputs are easily available, supply curve is more elastic
Geographic scope (wide)	-	If the scope is wide, their would be fewer places to supply the good

Elasticities and quick predictions

Quick predictions

We can use the following equations to predict the change in price or quantity given a change in the other:

\text{Increase }\%\Delta P = \frac{\%\Delta Q_d}{|\eta_d|+\eta_s} = -\frac{\%\Delta Q_s}{|\eta_d|+\eta_s}

When asked about the percentage change in quantity, use definition of price elasticity:

\text{Increase }\%\Delta Q = \eta_s \times \%\Delta P

Elasticity