Economics

Microeconomics: Elasticity

Price elasticity of demand, revenue, and incidence — worked from the definitions.

Price elasticity of demand measures how responsive quantity is to price. Using the midpoint (arc) formula, $E_{d} = \frac{%Δ Q}{%Δ P} = \frac{( Q _{2} - Q _{1} ) / [( Q _{1} + Q _{2} ) /2 ]}{( P _{2} - P _{1} ) / [( P _{1} + P _{2} ) /2 ]}$ . Demand is elastic when $∣ E_{d} ∣ > 1$ , inelastic when $∣ E_{d} ∣ < 1$ .

1. Arc elasticity
easy
When the price of a good rises from $\$ 4 $t o$ $6 $, q u an t i t y d e man d e df a l l s f r o m$ 120 $t o$ 80$ units. Compute the price elasticity of demand using the midpoint formula and classify demand.
Show solution
$%Δ Q = \frac{80 - 120}{( 80 + 120 ) /2} = \frac{- 40}{100} = - 40%$ .

$%Δ P = \frac{6 - 4}{( 6 + 4 ) /2} = \frac{2}{5} = 40%$ .

$E_{d} = \frac{- 40%}{40%} = - 1$ . In absolute value $∣ E_{d} ∣ = 1$ : demand is unit elastic over this range.
2. Elasticity and revenue
medium
A cinema currently sells $1000$ tickets at $\$ 10 $. I t es t ima t es$ |E_d| = 0.5 $o v er t h er e l e v an t r an g e . I f i t r ai ses t h e p r i ce b y$ 10%$, what happens to quantity and to total revenue?
Show solution
$%Δ Q = E_{d} \times %Δ P = - 0.5 \times 10% = - 5%$ . Quantity falls to $1000 \times 0.95 = 950$ tickets.

Old revenue: $1000 \times \$ 10 = $10{,}000 $. N e w r e v e n u e :$ 950 \times $11 = $10{,}450$.

Revenue rises by $\$ 450 $. B ec a u se d e man d i s in e l a s t i c ($ |E_d|<1$), a price increase raises total revenue.
3. Tax incidence
hard
A market has supply $Q_{s} = 2 P$ and demand $Q_{d} = 120 - P$ . The government imposes a per-unit tax of $t = \$ 12$ paid by sellers. Find the new price paid by buyers, the price received by sellers, and the share of the tax borne by buyers.
Show solution
Without tax: $2 P = 120 - P \Rightarrow 3 P = 120 \Rightarrow P^{*} = 40$ , $Q^{*} = 80$ .

With a seller tax $t$ , sellers supply based on the net price $P_{b} - t$ : $Q_{s} = 2 (P_{b} - 12)$ . Set equal to demand:

$2 (P_{b} - 12) = 120 - P_{b} \Rightarrow 2 P_{b} - 24 = 120 - P_{b} \Rightarrow 3 P_{b} = 144 \Rightarrow P_{b} = 48$ .

Sellers receive $P_{s} = P_{b} - 12 = 36$ .

Buyers' price rose from $40$ to $48$ (up $\$ 8 $); se l l er s^{'} f e l l f r o m$ 40 $t o$ 36 $(d o w n$ $4 $) . B u y er s b e a r$ \dfrac{8}{12} = 66.7%$ of the tax — the side with less elastic behaviour (steeper curve) bears more.

Microeconomics: Elasticity

1. Arc elasticity

2. Elasticity and revenue

3. Tax incidence

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