Chapter 2 gave us the supply-and-demand model: curves, equilibrium, shifts, and interventions. But that model tells us only the direction of price and quantity changes, not their magnitudes. When demand increases, how much does price rise? When the government imposes a tax, who actually bears the burden — buyers or sellers? To answer these questions, we need a measure of responsiveness: elasticity.
This chapter also introduces the welfare framework — consumer surplus, producer surplus, and deadweight loss — that allows us to evaluate whether a market outcome is efficient and measure the cost of interventions. Together, elasticity and surplus analysis give us the tools to make quantitative judgments about markets and policies, not just qualitative ones.
Saying "quantity demanded falls when price rises" is qualitative. A business owner needs to know: by how much? If I raise my price by 10%, will I lose 5% of my customers or 50%? The answer determines whether the price increase is profitable or ruinous. Elasticity provides the answer.
By the law of demand, $\varepsilon_d$ is typically negative (quantity moves opposite to price). Convention varies — some texts take the absolute value. We keep the negative sign and use $|\varepsilon_d|$ when comparing magnitudes.
Why use percentages? Because they make elasticity unit-free and comparable across goods. A \$1 price increase means very different things for a \$1 cup of coffee and a \$10,000 car. But a 10% price increase is a meaningful comparison regardless of the unit.
| $|\varepsilon_d|$ | Term | Meaning | Example |
|---|---|---|---|
| $> 1$ | Elastic | Quantity responds more than proportionally | Restaurant meals, vacation travel |
| $= 1$ | Unit elastic | Quantity responds proportionally | The revenue-maximizing point |
| $< 1$ | Inelastic | Quantity responds less than proportionally | Gasoline (short run), insulin |
| $= 0$ | Perfectly inelastic | Quantity does not respond (vertical curve) | Life-saving medication with no substitute |
| $= \infty$ | Perfectly elastic | Any price increase kills demand (horizontal curve) | Wheat from one farmer in a competitive market |
For a continuous demand function $Q_d = a - bP$, the derivative $dQ_d/dP = -b$, so:
Notice something important: even though the slope $-b$ is constant along a linear demand curve, the elasticity is not constant. It depends on the ratio $P/Q$, which changes as you move along the curve. At high prices (where $P$ is large and $Q$ is small), $P/Q$ is large, making $|\varepsilon_d|$ large — demand is elastic. At low prices (where $P$ is small and $Q$ is large), $P/Q$ is small, making $|\varepsilon_d|$ small — demand is inelastic. At the midpoint of the demand curve, $|\varepsilon_d| = 1$.
This is a subtlety that trips up many students: a steep demand curve is not the same as an inelastic one, and a flat curve is not the same as an elastic one. Slope and elasticity are different concepts. Slope ($\Delta Q/\Delta P$) uses absolute changes; elasticity uses percentage changes.
Figure 3.1. Elasticity varies along a linear demand curve even though the slope is constant. The upper portion is elastic ($|\varepsilon_d| > 1$), the midpoint is unit elastic ($|\varepsilon_d| = 1$), and the lower portion is inelastic ($|\varepsilon_d| < 1$). Hover over any point on the curve to see the exact elasticity.
When we don't have a continuous function but only two discrete data points $(P_1, Q_1)$ and $(P_2, Q_2)$, computing elasticity faces an asymmetry problem: using $(P_1, Q_1)$ as the base gives a different answer than using $(P_2, Q_2)$. The midpoint (arc) method resolves this by using the average of the two points as the base:
The arc elasticity gives the same answer regardless of which direction you compute the change — from point 1 to point 2 or from point 2 to point 1.
Using $Q_d = 100 - 20P$:
Point elasticity at $P = 3$, $Q = 40$:
$\varepsilon_d = -20 \cdot \frac{3}{40} = -1.5$ — elastic. A 1% price increase would reduce quantity demanded by 1.5%.
Point elasticity at $P = 1$, $Q = 80$:
$\varepsilon_d = -20 \cdot \frac{1}{80} = -0.25$ — inelastic. A 1% price increase would reduce quantity by only 0.25%.
Arc elasticity between $(P_1 = 2, Q_1 = 60)$ and $(P_2 = 3, Q_2 = 40)$:
$\varepsilon_d^{arc} = \frac{40 - 60}{3 - 2} \cdot \frac{2 + 3}{60 + 40} = \frac{-20}{1} \cdot \frac{5}{100} = -1.0$ — unit elastic over this range.
What makes demand for some goods elastic and others inelastic? Five factors matter:
1. Availability of close substitutes. This is the most important determinant. If many alternatives exist, consumers easily switch when the price rises — demand is elastic. If few or no substitutes exist, consumers are stuck — demand is inelastic.
The key insight: elasticity depends on how narrowly you define the market. Demand for "beverages" is very inelastic. Demand for "coffee" is somewhat inelastic. Demand for "Starbucks coffee" is quite elastic. Demand for "a tall latte at the Starbucks on 5th and Main" is extremely elastic.
2. Necessities vs. luxuries. Necessities — insulin for diabetics, basic food staples, heating fuel in winter — have inelastic demand. Luxuries — vacation travel, fine dining, designer clothing — have elastic demand.
3. Time horizon. Demand is more elastic in the long run than the short run. Short-run gasoline demand is very inelastic ($|\varepsilon_d| \approx 0.2$); long-run demand is more elastic ($|\varepsilon_d| \approx 0.7$).
4. Share of budget. Goods that account for a large share of the consumer's budget have more elastic demand.
5. How broadly or narrowly the market is defined. Narrower markets have more elastic demand. "Food" is inelastic. "Organic heirloom tomatoes from the farmers' market" is very elastic.
The elasticity concept extends beyond own-price demand.
| $\varepsilon_I$ | Classification | Examples |
|---|---|---|
| $> 1$ | Luxury (income-elastic normal good) | Organic food, international travel, private education |
| \$1 < \varepsilon_I < 1$ | Necessity (income-inelastic normal good) | Basic groceries, utilities, clothing staples |
| $< 0$ | Inferior good | Instant noodles, bus tickets, generic store brands |
As income rises, the budget share of necessities falls (Engel's law) and the share of luxuries rises.
$\varepsilon_{xy} > 0$: goods are substitutes. $\varepsilon_{xy} < 0$: goods are complements. $\varepsilon_{xy} = 0$: goods are unrelated.
Cross-price elasticities matter enormously in antitrust economics. Regulators use them to define markets: if two products have high cross-price elasticity (strong substitutes), they are in the same market.
Supply elasticity is typically positive. It depends on spare capacity, input availability, and the time horizon.
Total revenue is $TR = P \times Q$. When price changes, two forces work in opposite directions: a higher price means more revenue per unit (price effect), but fewer units sold (quantity effect). Which force wins depends on elasticity.
Taking the derivative:
Since $\varepsilon_d < 0$, the sign of $dTR/dP$ depends on whether $|\varepsilon_d|$ is greater or less than 1:
| If demand is... | $|\varepsilon_d|$ | Price rise → TR... | Price fall → TR... |
|---|---|---|---|
| Elastic | $> 1$ | Falls (quantity effect dominates) | Rises |
| Unit elastic | $= 1$ | Unchanged | Unchanged |
| Inelastic | $< 1$ | Rises (price effect dominates) | Falls |
Using $Q_d = 100 - 20P$: $TR = P(100 - 20P) = 100P - 20P^2$.
To find the maximum: $dTR/dP = 100 - 40P = 0 \implies P = 2.50$.
At $P = 2.50$: $Q = 50$, $TR_{max} = 125$. Elasticity: $\varepsilon_d = -20 \times (2.50/50) = -1.0$. Unit elastic — revenue is maximized where $|\varepsilon_d| = 1$.
Figure 3.2. Move the price slider. Left: the demand curve with the current price highlighted. Right: the total revenue curve — an inverted parabola peaking at $P = 2.50$ where demand is unit elastic.
Elasticity tells us how much quantities respond to prices. Surplus analysis tells us how much benefit buyers and sellers receive from market transactions — and how much is lost when markets are distorted.
A fundamental result: total surplus is maximized at the competitive equilibrium quantity. Any deviation from $Q^*$ — whether from taxes, price controls, monopoly, or quotas — reduces total surplus. The lost surplus is called deadweight loss.
Using $Q_d = 100 - 20P$ and $Q_s = 20P - 10$. Equilibrium: $P^* = 2.75$, $Q^* = 45$.
$CS = \frac{1}{2}(5.00 - 2.75)(45) = 50.63$
$PS = \frac{1}{2}(2.75 - 0.50)(45) = 50.63$
$TS = 50.63 + 50.63 = 101.25$
Figure 3.3. Drag the price away from equilibrium (\$1.75) to see how CS and PS change. A deadweight loss triangle appears whenever the price deviates from the equilibrium — these are mutually beneficial trades that no longer occur.
A question that surprises most people: when the government imposes a tax on sellers, do sellers actually bear the burden? The answer: not necessarily. Tax incidence — who truly pays — depends on the relative elasticities of supply and demand, not on who legally remits the tax.
A per-unit tax of $t$ imposed on sellers drives a wedge between the price buyers pay ($P_B$) and the price sellers receive ($P_S$): $P_B = P_S + t$.
The rule: the more inelastic side bears more of the tax. The party with fewer alternatives cannot easily escape the tax by adjusting behavior. They are "stuck" — and the tax burden falls on them.
A $t = 0.50$ per-cup tax on lemonade sellers (with $Q_d = 100 - 20P$, $Q_s = 20P - 10$):
$P_B = 2.75 + 0.5(0.50) = 3.00$ | $P_S = 2.75 - 0.5(0.50) = 2.50$
$Q_{new} = 100 - 20(3.00) = 40$
Buyers bear \$1.25 of the \$1.50 tax (50%). Sellers bear the other \$1.25 (50%). The even split occurs because $b = d = 20$ — equal absolute slopes.
Figure 3.4. A fixed \$1.00 tax. Change the demand slope to see the burden shift: steeper (more inelastic) demand means buyers bear more of the tax because they cannot easily reduce consumption. Flatter (more elastic) demand means sellers bear more.
DWL is not a transfer from one group to another. Tax revenue is a transfer (from private parties to the government). But DWL is a net loss — it goes to nobody. It is the cost of inefficiency.
where $\Delta Q = Q^*_{no\,tax} - Q^*_{tax}$ is the reduction in quantity caused by the tax.
From Example 3.4: $t = 0.50$, $\Delta Q = 45 - 40 = 5$.
$DWL = \frac{1}{2}(0.50)(5) = 1.25$
Verification: $TS_{original} = 101.25$. With tax: $CS = 40.00$, $PS = 40.00$, Revenue $= 20.00$, so $TS = 100.00$. The \$1.25 difference is the deadweight loss.
For linear supply and demand, $\Delta Q$ is proportional to $t$. Since $DWL = \frac{1}{2} t \cdot \Delta Q$ and $\Delta Q \propto t$:
Doubling the tax rate quadruples the deadweight loss. This has a profound implication: it is more efficient to spread taxes across many goods at low rates than to concentrate them on a few goods at high rates.
Figure 3.5. Drag the tax slider from \$1 to \$1. Watch the DWL triangle (yellow) grow with the square of the tax rate. At $t = 1$, DWL = \$1.00. At $t = 2$, DWL = \$10.00 — four times as much. The purple rectangle is tax revenue, which eventually shrinks as high taxes destroy too many transactions.
DWL is larger when supply and demand are more elastic. Elastic markets are responsive — the tax eliminates many transactions. Inelastic markets are unresponsive — the tax barely changes behavior, so few transactions are lost.
This creates a tension: the most efficient taxes (smallest DWL) fall on goods with inelastic demand — but these are also the taxes where buyers bear the largest burden. Efficiency and equity can conflict.
Figure 3.6. The same tax applied to an elastic market (left, $b = 40$) and an inelastic market (right, $b = 5$). The elastic market loses far more transactions and has much larger DWL. Drag the tax slider to compare.
The city council, looking for revenue, imposes a \$1.50 per-cup tax on lemonade vendors.
Recall from Chapter 2: $Q_d = 100 - 20P$, $Q_s = 20P - 10$, equilibrium at $P^* = 2.75$, $Q^* = 45$.
Before tax: Revenue = \$1.75 \times 45 = \\$123.75$/day. CS = \$10.63, PS = \$10.63, TS = \$101.25.
After tax ($t = 0.50$): Buyers pay \$1.00; Maya receives \$1.50; she sells 40 cups.
Maya's revenue: \$1.50 \times 40 = \\$100.00$/day (down from \$123.75).
CS = \$10.00 (fell by \$10.63). PS = \$10.00 (fell by \$10.63). Tax revenue = \$10.00. DWL = \$1.25.
Maya's daily revenue of \$100.00 is now below her opportunity cost of \$120/day from the bookstore job (Chapter 1). The tax pushed her from barely viable to clearly unprofitable. The five cups that go unsold each day represent transactions that would have created value for both buyer and seller. The \$1.25 of deadweight loss is the total value those five transactions would have created.
| Label | Equation | Description |
|---|---|---|
| Eq. 3.1 | $\varepsilon_d = (\Delta Q_d / \Delta P)(P/Q)$ | Price elasticity of demand |
| Eq. 3.2 | $\varepsilon_d = -b \cdot P/Q$ | Point elasticity for linear demand |
| Eq. 3.3 | $\varepsilon_d^{arc} = \frac{Q_2-Q_1}{P_2-P_1} \cdot \frac{P_1+P_2}{Q_1+Q_2}$ | Arc (midpoint) elasticity |
| Eq. 3.4 | $\varepsilon_I = (\Delta Q_d / \Delta I)(I/Q_d)$ | Income elasticity of demand |
| Eq. 3.5 | $\varepsilon_{xy} = (\Delta Q_x / \Delta P_y)(P_y/Q_x)$ | Cross-price elasticity |
| Eq. 3.6 | $\varepsilon_s = (\Delta Q_s / \Delta P)(P/Q_s)$ | Price elasticity of supply |
| Eq. 3.7 | $TR = P \times Q$ | Total revenue |
| Eq. 3.8 | $dTR/dP = Q(1 + \varepsilon_d)$ | TR response to price change |
| Eq. 3.9 | $CS = \int_0^{Q^*} D(Q)\,dQ - P^* Q^*$ | Consumer surplus (general) |
| Eq. 3.10 | $CS = \frac{1}{2}(P_{max} - P^*)Q^*$ | Consumer surplus (linear demand) |
| Eq. 3.11 | $PS = P^* Q^* - \int_0^{Q^*} S(Q)\,dQ$ | Producer surplus (general) |
| Eq. 3.12 | $PS = \frac{1}{2}(P^* - P_{min})Q^*$ | Producer surplus (linear supply) |
| Eq. 3.13 | $TS = CS + PS$ | Total surplus |
| Eq. 3.14 | $Q_d(P_B) = Q_s(P_B - t)$ | Tax equilibrium condition |
| Eq. 3.15 | Buyer's share $= \varepsilon_s / (\varepsilon_s + |\varepsilon_d|)$ | Tax incidence — buyers |
| Eq. 3.16 | Seller's share $= |\varepsilon_d| / (\varepsilon_s + |\varepsilon_d|)$ | Tax incidence — sellers |
| Eq. 3.17 | $DWL = \frac{1}{2} t \cdot \Delta Q$ | Deadweight loss from per-unit tax |
| Eq. 3.18 | $DWL \propto t^2$ | DWL grows with square of tax rate |