Equilibrium in a Competitive Market

Market Equilibrium in a Competitive Market

A market consists of buyers (consumers) and sellers (producers). The demand curve (downward-sloping) shows quantity demanded at each price, ceteris paribus. The supply curve (upward-sloping) shows quantity supplied at each price, ceteris paribus. When plotted together, they intersect at a single point – the market equilibrium. The price at this point is the equilibrium price ($P^*$) and the quantity is the equilibrium quantity ($Q^*$).

At $P^*$, the quantity demanded exactly equals the quantity supplied: the market clears. No inherent force pushes price away from $P^*$; instead, self-interested behaviour of buyers and sellers drives any deviation back toward equilibrium – like a ball in a bowl returning to the bottom.

Forces Driving Toward Equilibrium: Excess Supply & Excess Demand

1. Price Above Equilibrium ($P > P^*$)

• Quantity supplied ($Q_S$) exceeds quantity demanded ($Q_D$) → excess supply (surplus). $\text{Excess supply} = Q_S - Q_D > 0$.
• Symptoms: inventories pile up, customers are infrequent.
• Seller response: offer discounts to clear stock → downward pressure on price.
• Process persists until price falls back to $P^*$.

2. Price Below Equilibrium ($P < P^*$)

• $Q_D$ exceeds $Q_S$ → excess demand (shortage). $\text{Excess demand} = Q_D - Q_S > 0$.
• Symptoms: inventories vanish quickly, long queues, waiting times.
• Seller response: some buyers offer to pay more; sellers can raise prices without losing customers → upward pressure on price.
• Process continues until price rises to $P^*$.

3. Price at Equilibrium ($P = P^*$)

• $Q_D = Q_S$. No excess demand or supply. No pressure for price to change.

Exam tip: The only price where quantity demanded equals quantity supplied is the equilibrium price. Any other price creates a surplus or shortage, which pushes the price back to $P^*$.

Worked Example: Finding Equilibrium with Linear Curves

Given:

• Inverse demand: $P_X = 25 - 0.005 Q + 0.15 P_Y$
• Inverse supply: $P_X = 5 + 0.004 Q$
• $P_Y$ is the price of a related good Y.

Part A: Equilibrium when $P_Y = 10$

1. Substitute $P_Y = 10$ into the demand equation: $P_X = 25 - 0.005 Q + 0.15 \times 10 = 26.5 - 0.005 Q$

2. At equilibrium, quantity demanded = quantity supplied. Set the two inverse curves equal: $26.5 - 0.005 Q^* = 5 + 0.004 Q^*$ $21.5 = 0.009 Q^* \quad \Rightarrow \quad Q^* = \frac{21.5}{0.009} = \frac{21500}{9} \approx 2388.89$

3. Find $P^*$ by substituting $Q^*$ into either equation (e.g., supply side): $P^* = 5 + 0.004 \times \frac{21500}{9} = 5 + \frac{86}{9} \approx 14.56$

   Thus, $P^* \approx 14.56$, $Q^* \approx 2388.89$ (pounds/week).

Part B: Are X and Y substitutes or complements?

Rewrite the demand function to see how $Q$ responds to $P_Y$: $0.005 Q = 25 - P_X + 0.15 P_Y \quad\Rightarrow\quad Q = \frac{25 - P_X + 0.15 P_Y}{0.005}$

As $P_Y$ increases, the right-hand side increases, so $Q$ increases. This means a rise in the price of Y raises demand for X – goods X and Y are substitutes (used in place of each other).

Part C & D: New equilibrium when $P_Y = 20$

1. Substitute $P_Y = 20$ into demand: $P_X = 25 - 0.005 Q + 0.15 \times 20 = 28 - 0.005 Q$

2. Equate with supply: $28 - 0.005 Q^*_N = 5 + 0.004 Q^*_N \quad\Rightarrow\quad 23 = 0.009 Q^*_N \quad\Rightarrow\quad Q^*_N = \frac{23000}{9} \approx 2555.56$

3. Find new price $P^*_N$ using supply: $P^*_N = 5 + 0.004 \times \frac{23000}{9} = 5 + \frac{92}{9} \approx 15.22$

4. Compare old ($P^* \approx 14.56$) and new ($P^*_N \approx 15.22$): the new equilibrium price is higher. At the old price $P^*$, after the shift in demand (due to higher $P_Y$), there will be excess demand because buyers now want more at that price than sellers are willing to supply. (Exact calculation left as exercise.)

The Invisible Hand of Prices

Even without a central planner, competitive markets achieve coordination – a feature called the invisible hand of prices. Prices simultaneously:

• Ration scarce resources – e.g., housing in Mumbai: price adjusts until only those willing and able to pay get housing.
• Allocate production – higher expected fish prices attract more fishermen; price signals determine who becomes a seller and how much to produce.
• Coordinate buyers and sellers – if price is too low, shortage pushes it up; if too high, surplus pushes it down – all through self-interested actions.

This spontaneous order is a striking property of perfectly competitive markets. Prices contain all necessary information for individuals to act, achieving equilibrium without central control.

Exam tip: The invisible hand is not a physical force; it describes how the price mechanism guides self-interested decisions toward a socially coordinated outcome (market clearing). Be prepared to explain with real-world examples like fish markets, housing, or agricultural produce.

Key Takeaways

• Market equilibrium: $Q_D = Q_S$ at $P^*$; the only point with no inherent pressure to change.
• Prices above $P^*$ create surplus → downward pressure; prices below $P^*$ create shortage → upward pressure.
• The adjustment process is automatic, driven by seller responses to inventory/build-up and buyer competition.
• Equilibrium is found by setting quantity demanded equal to quantity supplied (or inverse curves equal).
• Related goods: if demand for X increases when $P_Y$ rises, X and Y are substitutes.
• The invisible hand of prices coordinates the actions of many independent buyers and sellers, allocating resources without central authority.

Market Equilibrium: Real World Examples

The market equilibrium model (supply meeting demand) isn't just theory — it predicts real price changes. A four‑step checklist turns any news event into a prediction:

Four‑step checklist

1. Market – which good(s) are we analysing?
2. Curve shift – does the event affect supply (production cost, technology, number of sellers) or demand (tastes, income, prices of related goods)?
3. Direction – left (decrease) or right (increase)?
4. New equilibrium – combine shifts in a supply‑and‑demand diagram to get new price and quantity.

Exam tip: Step 2 is the most error‑prone. Always ask: does the event change producers’ costs/ability to sell, or consumers’ willingness to buy? Never guess — use the standard shift determinants.

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Example 1: Fire at an auto‑chip plant → new car prices (India, March 2021)

The news: A fire destroys a major semiconductor chip plant. Step 1 – Markets: Semiconductor chips → new cars. Step 2 – Supply or demand?

• Chips: production capacity destroyed → supply shifts left. Demand unchanged.
• New cars: chips are an input. Input price rises → supply of new cars shifts left. Demand unchanged (no direct effect on consumer preferences).

Step 3 – Direction: Both supply curves shift left (decrease). Step 4 – Result: Equilibrium price of new cars increases; equilibrium quantity decreases.

This analysis tells the Honda manager: expect higher car prices in coming months.

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Example 2: New car price rise → used car market (India)

The news: As above, new car prices will rise. Step 1 – Market: Used cars. Step 2 – Curve shift?

• Supply of used cars – unaffected (semiconductor fire doesn’t change stock of used cars).
• Demand for used cars – new cars and used cars are substitutes. Higher price of new cars → consumers switch → demand for used cars shifts right.

Step 3 – Direction: Demand right, supply unchanged. Step 4 – Result: Used car price increases; quantity increases.

Exam tip: Substitutes link markets. When the price of good A rises, demand for substitute B shifts right — always check for cross‑market effects.

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Example 3: Russia‑Ukraine war (Feb 2022) → crude oil & cotton prices

The event: Russia is a large crude‑oil supplier; war reduces oil supply.

Step 1 – Markets involved

• Crude oil (primary market)
• Synthetic fibres (nylon, polyester) – an intermediate market
• Cotton (final market, substitute for synthetic fibres)

Step 2 & 3 – Shifts and directions

Step 4 – New equilibrium prices

The shock propagates: $\text{Crude oil supply decreases} \rightarrow \text{Oil price increases} \rightarrow \text{Synthetic fibre supply decreases} \rightarrow \text{Synthetic fibre price increases} \rightarrow \text{Cotton demand increases} \rightarrow \text{Cotton price increases}$

This explains the WSJ observation: crude oil and cotton prices move together — because crude oil indirectly affects cotton via the synthetic‑fibre substitute channel.

Exam tip: A shock in one market can “travel” through related markets. Always trace the chain: input markets → producer goods → substitutes/complements. A left shift in supply upstream can create a right shift in demand downstream.

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Key takeaways

• Use the four‑step checklist: Market → Supply or demand? → Direction → New equilibrium.
• Input price increases shift supply left; substitute price increases shift demand right.
• One event can affect multiple markets sequentially (e.g., chip fire → new cars → used cars; war → oil → synthetics → cotton).
• Predictions are directional (price up/down), not numerical. The model gives qualitative trends.

Consumer Surplus

Consumer surplus measures the benefit buyers receive from participating in a market. Intuitively: a buyer is willing to pay up to a certain amount for a good (the value they place on it), but often pays less. The gap between what they’d pay and what they actually pay is their surplus – a direct measure of gain.

Formal definition

Consumer surplus = Willingness to pay – Amount actually paid

• Willingness to pay (WTP) – the maximum price a buyer would pay for a good; reflects the monetary value of the utility they expect from consuming it.
• Amount paid – the market price (or auction price) they actually hand over.

If a buyer does not purchase, their consumer surplus is zero (no gain, no loss).

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Worked example: single‑ticket auction

Three friends – Sanju, Rahul, Shreyas – bid for one IPL final ticket.

Auction starts at ₹3,000; bids rise. Sanju wins with a bid just above ₹8,000 (Rahul and Shreyas drop out at ₹8,000). Sanju pays ₹8,000 for a ticket he values at ₹10,000.

$\text{Sanju's consumer surplus} = 10,\!000 - 8,\!000 = 2,\!000 \text{ rupees}$

Rahul and Shreyas get zero surplus.

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Worked example: two tickets (uniform price)

Two tickets auctioned at the same price. Bidding stops when Sanju and Rahul bid just above ₹5,000; Shreyas drops out.

• Sanju gets a ticket at ₹5,000 (WTP ₹10,000) → surplus = ₹5,000.
• Rahul gets a ticket at ₹5,000 (WTP ₹8,000) → surplus = ₹3,000.
• Shreyas gets nothing → surplus = ₹0.

$\text{Total consumer surplus} = 5,\!000 + 3,\!000 = 8,\!000 \text{ rupees}$

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Consumer surplus on the demand curve

The demand curve for three buyers (step function) shows the willingness to pay of the marginal buyer at each quantity.

The height of the demand curve at a given quantity equals the WTP of the marginal buyer (the one who would exit the market at a slightly higher price).

Measuring consumer surplus graphically

At any price $P$, consumer surplus is the area below the demand curve and above the price, up to the quantity traded.

• Single ticket at ₹8,000: area = rectangle 2,000 × 1 = ₹2,000 (Sanju’s surplus).
• Two tickets at ₹5,000: area = two rectangles – Sanju’s (5,000 × 1) and Rahul’s (3,000 × 1) – total ₹8,000.

For a smooth downward‑sloping demand curve:

• At price $P_1$, quantity $Q_1$ → consumer surplus = triangle $ABC$.
• At a lower price $P_2$, quantity $Q_2$ → consumer surplus = triangle $ADF$.

Effect of a price decrease

Lowering price from $P_1$ to $P_2$ increases consumer surplus through two channels:

1. Existing buyers who already purchased at $P_1$ now pay less → gain rectangle $(P_1 - P_2) \times Q_1$.
2. New buyers (who valued the good between $P_2$ and $P_1$) now enter the market → gain triangle with base $(Q_2 - Q_1)$ and height $(P_1 - P_2)$.

Exam tip: Consumer surplus ≠ profit. It measures buyer welfare, not seller revenue. On a graph, always shade the area below demand and above the market price – that's the surplus.

Key takeaways

• Consumer surplus = willingness to pay – price paid.
• It captures the benefit buyers get from trading at a price lower than their maximum.
• Graphically: area below the demand curve and above the price.
• A price drop increases surplus both for existing buyers (rectangular gain) and new buyers (triangular gain).
• The demand curve's height at each quantity shows the marginal buyer's willingness to pay – the last buyer who would still buy at that price.

Producer Surplus

Producer surplus (PS) measures the economic benefit a seller receives from selling a good. Intuitively: a seller has a minimum acceptable price (their cost). If they actually get a higher price, the difference is pure gain — their surplus. The concept mirrors consumer surplus but on the supply side.

Definition and Example

$\text{Producer Surplus} = \text{Price received} - \text{Seller's cost}$

A seller’s cost includes all out‑of‑pocket expenses plus the opportunity cost of their time. It is the lowest price they would accept — their willingness to sell.

Three painters, one room

An auction for painting one room starts high; bidders drop out as price falls. The price settles at just below ₹4,000 — below Rahul’s cost but above Shreyas’s cost. Only Shreyas accepts.

• Shreyas receives ≈ ₹4,000, his cost is ₹3,000. $\text{PS} = 4{,}000 - 3{,}000 = \text{Rs. }1{,}000$.
• Rahul and Sanju do not work → no payment, no cost → PS = ₹0 each.
• Total PS = ₹1,000.

Two rooms, same painters

Now two rooms must be painted; each painter can do at most one room, and both rooms are paid the same price. The auction stops when the price reaches just below ₹5,000. At that price:

• Shreyas (cost ₹3,000) and Rahul (cost ₹4,000) are willing.

• Sanju (cost ₹5,000) drops out.

• Shreyas: $\text{PS} = 5{,}000 - 3{,}000 = \text{Rs. }2{,}000$.

• Rahul: $\text{PS} = 5{,}000 - 4{,}000 = \text{Rs. }1{,}000$.

• Sanju: PS = ₹0.

• Total PS = ₹3,000.

Exam tip: In the multi‑seller case, total producer surplus is the sum of individual surpluses. Only sellers who actually transact earn surplus.

Supply Schedule and Supply Curve

The supply schedule shows quantity supplied at each price. From the three painters:

Plotting this gives a step‑function supply curve. The height of each step equals the cost of the marginal seller — the one who would exit if price fell further:

• At $Q=1$, height = ₹3,000 (Shreyas’s cost).
• At $Q=2$, height = ₹4,000 (Rahul’s cost).
• At $Q=3$, height = ₹5,000 (Sanju’s cost).

Producer Surplus as Area Below Price and Above Supply

For any price $P$, producer surplus = area below $P$ and above the supply curve, up to the quantity traded.

Step‑function case

• Single room, $P = \text{Rs. }4{,}000$, $Q=1$ Area = rectangle of height $(4{,}000 - 3{,}000) = 1{,}000$, width 1 → ₹1,000 (Shreyas’s surplus).

• Two rooms, $P = \text{Rs. }5{,}000$, $Q=2$ Area splits into two rectangles:

  • Blue: $(5{,}000 - 3{,}000) \times 1 = \text{Rs. }2{,}000$ (Shreyas).
  • Red: $(5{,}000 - 4{,}000) \times 1 = \text{Rs. }1{,}000$ (Rahul). Total = ₹3,000.

Smooth upward‑sloping supply curve

In a market with many sellers, the supply curve becomes smooth. At price $P_1$ and quantity $Q_1$, producer surplus is triangle ABC (area below $P_1$, above supply).

If price rises to $P_2 > P_1$, new equilibrium quantity $Q_2$. The increase in producer surplus decomposes into three parts:

• Triangle ABC – original PS at $P_1$.
• Rectangle BCED – extra surplus for the $Q_1$ existing sellers (each gets $P_2 - P_1$ more).
• Triangle CEF – surplus of $(Q_2 - Q_1)$ new sellers now willing to produce at the higher price.

Thus, higher price increases producer surplus through both intensive (higher per‑unit gain) and extensive (new sellers) margins.

Key takeaways

• $\text{PS} = \text{Price} - \text{Seller's cost}$ (cost includes opportunity cost).
• Only sellers who actually sell earn surplus; total PS = sum over all sellers.
• Graphically: PS = area below price and above the supply curve.
• For a step supply curve, each step’s height = marginal seller’s cost; area decomposes into rectangles per seller.
• A price rise boosts PS via existing sellers (rectangle) and new entrants (triangle).

Intuition and Definition

Total surplus measures society’s overall economic well‑being from a market. It combines the welfare of buyers and sellers:

$$\text{Total Surplus} = \text{Consumer Surplus} + \text{Producer Surplus}$$

Recall:

• Consumer surplus (CS) = value to buyers – price paid.
• Producer surplus (PS) = price received – cost to sellers.

Since the price paid by buyers equals the price received by sellers, these terms cancel when adding CS and PS. Thus total surplus simplifies to:

$$\boxed{\text{Total Surplus} = \text{Value to Buyers} - \text{Cost to Sellers}}$$

Intuition: Society gains when a good is produced and consumed only if its value to the buyer exceeds the seller’s cost of producing it. Total surplus is the sum of those net gains across all units traded.

Graphical Representation

In a competitive market equilibrium (price $P^*$, quantity $Q^*$):

• CS = area below the demand curve and above $P^*$ (triangle $ABC$ in the figure).
• PS = area above the supply curve and below $P^*$ (triangle $DBC$).
• Total surplus = area between the demand and supply curves, up to $Q^*$ (triangle $ADC$).

Demand curve (downward sloping)
Supply curve (upward sloping)
Intersection at (Q*, P*)

Consumer surplus: triangle above P* under demand
Producer surplus: triangle below P* above supply
Total surplus: triangle from demand to supply, left of Q*

Efficiency

An allocation of resources is efficient if it maximizes total surplus. If not, it is inefficient.

Three symptoms of inefficiency:

1. Unrealised gains from trade – Some mutually beneficial transactions do not occur. Example: Keeping an IPL ticket instead of selling it; total surplus is zero instead of positive.

2. Goods not allocated to the highest‑value buyers – A buyer with a lower willingness to pay receives the good, while a higher‑value buyer does not. Example: Selling the IPL ticket to Shreyas (WTP ₹5,000) instead of Sanju (WTP ₹10,000) reduces total surplus from ₹10,000 to ₹5,000.

3. Goods not produced by the lowest‑cost sellers – A seller with higher cost produces, while a lower‑cost seller does not. Example: Giving the painting job to Sanju (cost ₹5,000) instead of Shreyas (cost ₹3,000) reduces total surplus from ₹4,000 to ₹2,000.

Why Market Equilibrium is Efficient

The competitive equilibrium (where demand = supply) automatically avoids all three inefficiencies.

1. All gains from trade are realised

• At any quantity $Q < Q^*$, the value to buyers (height of demand) exceeds cost to sellers (height of supply). Producing these units adds positive surplus.
• At any $Q > Q^*$, cost exceeds value – producing those units would reduce total surplus.
• The equilibrium quantity $Q^*$ is exactly where value = cost on the marginal unit, so no gains are left unrealised and no wasteful units are produced.

2. Goods go to the highest‑value buyers

• In equilibrium, the good is bought only by consumers whose willingness to pay is at least $P^*$ (points on the demand curve at and above $A$–$C$). Buyers with lower willingness to pay choose not to purchase.

3. Goods are produced by the lowest‑cost sellers

• Only sellers whose cost is at most $P^*$ supply the good (points on the supply curve at and below $B$–$C$). Higher‑cost sellers choose not to produce.

Because the equilibrium achieves all three properties, it maximises total surplus. A benevolent central planner would have no reason to alter the market outcome.

Exam tip: Efficiency of a competitive market is one of the most important results in microeconomics. Be ready to explain (i) why $Q^*$ maximises total surplus, (ii) how the price mechanism allocates goods to those who value them most, and (iii) why production is done by the cheapest sellers. The three inefficiency symptoms often appear in welfare analysis questions.

Key Takeaways

• Total surplus = value to buyers – cost to sellers = CS + PS.
• An efficient allocation maximises total surplus.
• Inefficiency arises from unrealised trades, misallocation to low‑value buyers, or production by high‑cost sellers.
• In a competitive equilibrium, all three problems are automatically solved: $Q^*$ maximises surplus, buyers with highest WTP purchase, and sellers with lowest cost produce.
• Hence the market is efficient without any central planner intervention.

Price Elasticity of Demand

Price elasticity of demand (PED) measures how responsive the quantity demanded of a good is to a change in its own price. The law of demand tells only the direction of change (price up → quantity down), but not the magnitude. Managers and policymakers need a quantitative measure:

• A pricing manager deciding whether to raise refrigerator prices when costs rise must know how much sales will drop.
• A government trying to cut cigarette consumption by 20% via a tax needs to set the tax such that the resulting price hike reduces quantity by exactly 20%.

PED supplies that measure.

Formal definition

Holding all other determinants of demand constant,

$$\text{PED} = \frac{\%\,\Delta Q_d}{\%\,\Delta P}$$

where

$$\%\,\Delta P = \frac{P_2 - P_1}{P_1} \times 100,\qquad \%\,\Delta Q_d = \frac{Q_2 - Q_1}{Q_1} \times 100$$

Because price and quantity move in opposite directions (law of demand), the signs of the two percentage changes are always opposite → PED is always negative. To simplify interpretation, economists work with the absolute value:

$$\lvert\text{PED}\rvert = \frac{\lvert\%\,\Delta Q_d\rvert}{\lvert\%\,\Delta P\rvert}$$

Interpreting the number

Worked example

Consider two markets. In both, price rises by 10%.

• Market 1: Quantity falls by 20%. $\lvert\text{PED}\rvert = 20/10 = 2$ → elastic.
• Market 2: Quantity falls by 40%. $\lvert\text{PED}\rvert = 40/10 = 4$ → even more elastic.

The same price change causes a larger quantity response in Market 2; higher elasticity means greater responsiveness.

Relationship with the slope of the inverse demand curve

The inverse demand curve expresses price as a function of quantity, e.g. $P = A - BQ$. Slopes of these curves relate to elasticity:

Rule of thumb: The steeper the inverse demand curve through a point, the lower the elasticity; the flatter it is, the higher the elasticity.

Exam tip: Do not confuse elasticity with slope. Slope measures absolute changes ($\Delta P / \Delta Q$), while elasticity measures percentage changes. A vertical curve has infinite slope but zero elasticity; a horizontal curve has zero slope but infinite elasticity.

Key takeaways

• PED $= \frac{\%\Delta Q}{\%\Delta P}$; use absolute values because the sign is always negative.
• Elastic if $>1$, inelastic if $<1$, unit elastic if $=1$.
• Perfectly inelastic ($\text{PED}=0$) → vertical demand; perfectly elastic ($\text{PED}=\infty$) → horizontal demand.
• Steeper demand → lower elasticity; flatter demand → higher elasticity.
• PED quantifies how much quantity responds to price, solving problems where only direction (law of demand) is not enough.

Elasticity and Linear Demand Curve

Price elasticity of demand (PED) measures how responsive quantity demanded is to a price change. For a linear inverse demand curve, elasticity varies at every point—it is not constant—and we can partition the curve into distinct zones.

Deriving Elasticity for a Linear Inverse Demand Curve

The inverse demand curve is: $P = A - BQ, \quad A,B > 0$

Let the initial point be $(P_1, Q_1)$ with $Q_1 = (A - P_1)/B$. After a price increase to $P_2$, quantity falls to $Q_2 = (A - P_2)/B$. Compute the absolute percentage changes:

• $\% \Delta P = \frac{P_2 - P_1}{P_1}$
• $\% \Delta Q = \frac{Q_1 - Q_2}{Q_1} = \frac{(A-P_1)/B - (A-P_2)/B}{Q_1} = \frac{P_2 - P_1}{B Q_1}$

Then elasticity is: $E_d = \frac{\% \Delta Q}{\% \Delta P} = \frac{(P_2 - P_1)/(B Q_1)}{(P_2 - P_1)/P_1} = \frac{P_1}{B Q_1}$

Generalising to any point $(P,Q)$ on the linear curve: $E_d = \frac{1}{B} \cdot \frac{P}{Q}$

Since $P = A - BQ$, substitute to get an expression in $Q$ only: $E_d = \frac{A}{BQ} - 1$

How Elasticity Varies Along the Curve

Because $E_d = A/(BQ) - 1$, it decreases as $Q$ increases. The following table summarises the five zones:

The corresponding price at the midpoint ($Q = A/(2B)$) is $P = A/2$; at the intercepts, $P = A$ when $Q=0$, and $P=0$ when $Q = A/B$.

Exam tip: For a linear demand curve, the midpoint is always the point of unit elasticity. Above the midpoint demand is elastic; below it is inelastic. This is a classic test item.

Worked Example: Recovering the Linear Demand Curve from a Point and Elasticity

Problem: Tata Motors expects to sell 2 lakh cars at a price of 20 lakhs per car, with a price elasticity of 2. Find the linear inverse demand curve (assumed linear).

Solution: Let inverse demand be $P = A - BQ$. At $(P,Q) = (20, 2)$: $E_d = \frac{1}{B} \cdot \frac{P}{Q} \quad \Rightarrow \quad 2 = \frac{1}{B} \cdot \frac{20}{2} = \frac{10}{B} \quad \Rightarrow \quad B = 5$

Now use $P = A - 5Q$: $20 = A - 5 \cdot 2 \quad \Rightarrow \quad A = 30$

The inverse demand curve is: $P = 30 - 5Q$

Key insight: A single point $(P,Q)$ plus the local elasticity is enough to determine the entire linear demand curve.

Factors Affecting Price Elasticity of Demand

Five determinants are:

Exam tip: Memorise these five factors. The most common exam errors are confusing necessity/luxury and confusing short-run/long-run effects.

Key Takeaways

• For a linear inverse demand curve $P = A - BQ$, elasticity at any point is $E_d = \frac{1}{B}\frac{P}{Q} = \frac{A}{BQ} - 1$.
• Elasticity falls as quantity increases along the curve.
• Zones: $Q=0$ (perfectly elastic), $0<Q<A/(2B)$ (elastic), $Q=A/(2B)$ (unitary), $A/(2B)<Q<A/B$ (inelastic), $Q=A/B$ (perfectly inelastic).
• Given one $(P,Q)$ and $E_d$, the full linear demand curve can be found if linearity is assumed.
• Five factors determine elasticity: substitutes, necessity/luxury, narrowness, budget share, time horizon.

Elasticity and Revenue

Price elasticity of demand directly answers a manager’s core question: Will a price hike raise or lower my revenue? This section bridges the elasticity concept to real pricing decisions, using a graphical intuition that becomes a crisp decision rule.

Revenue as a rectangle

Total revenue $R = P \times Q$. On an inverse demand curve, revenue at price $P_1$ is the area of the rectangle with height $P_1$ and width $Q_1$ (area $R$ in the basic diagram). When price changes from $P_1$ to $P_2$, quantity moves from $Q_1$ to $Q_2$ (by the law of demand, $Q_2 < Q_1$). The new revenue $P_2 Q_2$ is a different rectangle.

Decomposing the change: gains and losses

When price rises, two forces act on revenue:

Let:

• Area A = $(P_2 - P_1) \times Q_2$ — the extra revenue from selling $Q_2$ units at the higher price (gain).
• Area B = $(Q_1 - Q_2) \times P_1$ — the revenue lost because $Q_1 - Q_2$ fewer units are sold (loss).
• Old revenue $R_1 = P_1 Q_1 = B + C$
• New revenue $R_2 = P_2 Q_2 = A + C$

The change in revenue is $R_2 - R_1 = A - B$. The common area $C$ cancels.

Key insight: The decision to raise price reduces to comparing the gain rectangle $A$ and the loss rectangle $B$.

The decision rule: elasticity tells you which rectangle is bigger

Whether $A > B$, $A < B$, or $A = B$ depends on the price elasticity of demand at the original operating point $(P_1, Q_1)$.

Graphically:

• Inelastic demand → steep demand curve; $A$ visibly larger than $B$.
• Elastic demand → flat demand curve; $B$ dominates.

Exam tip: The single most-tested result: Raising price increases total revenue only when demand is inelastic at the current point. The converse: cutting price increases revenue only when demand is elastic.

Worked example: Gurugram Metro

Demand: $Q = 30 - 3P$ (thousands of km travelled) Inverse demand: $P = 10 - \frac{Q}{3}$ (so $A=10$, $B=1/3$ in $P = A - BQ$ format). Operating cost rising → COO considers a price increase.

Price elasticity formula for a linear demand: $\lvert \text{PED} \rvert = \frac{1}{B} \cdot \frac{P}{Q}$

Scenario 1: current price ₹8 per km

• $Q = 30 - 3\times 8 = 6$
• $\lvert \text{PED} \rvert = \frac{1}{1/3} \cdot \frac{8}{6} = 3 \times \frac{4}{3} = 4$
• Since $4 > 1$, demand is elastic.
• Advice: Do NOT raise price – revenue would fall.

Scenario 2: current price ₹4 per km

• $Q = 30 - 3\times 4 = 18$
• $\lvert \text{PED} \rvert = 3 \cdot \frac{4}{18} = 3 \times \frac{2}{9} = \frac{2}{3} \approx 0.67$
• Since $0.67 < 1$, demand is inelastic.
• Advice: Raise price – revenue will increase.

Key takeaways

• Revenue change from a price change = gain rectangle $(P_2-P_1)Q_2$ minus loss rectangle $(Q_1-Q_2)P_1$.
• The comparison of these rectangles is governed by price elasticity at the initial point.
• Inelastic → price and revenue move together; elastic → price and revenue move opposite.
• Unitary elasticity is a theoretical knife-edge; real decisions involve elastic or inelastic ranges.
• Always compute $\lvert \text{PED} \rvert$ at the current operating point before recommending a price change for revenue maximisation.

Income Elasticity of Demand

Income elasticity of demand ($E_I$) measures how responsive quantity demanded is to a change in consumer income, holding all other determinants constant.

$E_I = \frac{\%\,\Delta Q}{\%\,\Delta I} = \frac{(Q_2 - Q_1)/Q_1}{(I_2 - I_1)/I_1}$

• Normal goods: $E_I > 0$ — income increase raises quantity demanded.
• Inferior goods: $E_I < 0$ — income increase reduces quantity demanded.

Exam tip: The sign of income elasticity directly classifies a good as normal or inferior. No extra steps.

Cross Price Elasticity of Demand

Cross price elasticity of demand ($E_{XY}$) measures how responsive the quantity demanded of good $X$ is to a change in the price of good $Y$, holding all other determinants of demand for $X$ constant.

$E_{XY} = \frac{\%\,\Delta Q_X}{\%\,\Delta P_Y} = \frac{(Q_{X2} - Q_{X1})/Q_{X1}}{(P_{Y2} - P_{Y1})/P_{Y1}}$

• Substitutes: $E_{XY} > 0$ (e.g., tea and coffee). A rise in $P_Y$ increases $Q_X$.
• Complements: $E_{XY} < 0$ (e.g., petrol and petrol cars). A rise in $P_Y$ decreases $Q_X$.

Exam tip: Substitutes → positive cross elasticity; complements → negative cross elasticity.

Worked example

You run a grocery store. The price of recreation is expected to increase by 15%. The table below gives cross-price elasticities between selected categories.

To find the effect on food sales:

$\frac{\%\,\Delta Q_F}{\%\,\Delta P_R} = 0.15 \quad\Rightarrow\quad \%\,\Delta Q_F = 0.15 \times 15\% = 2.25\%$

Food demand rises by 2.25%, confirming that food and recreation are substitutes.

Price Elasticity of Supply

Price elasticity of supply ($E_S$) measures how responsive quantity supplied is to a change in the selling price of the good, holding all other determinants of supply constant.

$E_S = \frac{\%\,\Delta Q_S}{\%\,\Delta P}$

• Larger $E_S$ → more sensitive supply.
• Examples: a 2% price rise with a 16% quantity increase gives $E_S = 8$; with a 1% quantity increase gives $E_S = 0.5$.

Factors affecting supply elasticity

Supply curve shapes and elasticity

Exam tip: Elasticity of supply changes along a typical upward-sloping supply curve — it is larger at lower quantities and smaller at higher quantities (due to capacity constraints).

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Key takeaways

• Income elasticity: positive for normal goods, negative for inferior goods.
• Cross price elasticity: positive for substitutes, negative for complements.
• Price elasticity of supply: driven by production flexibility and time horizon.
• Supply curve shapes: vertical (perfectly inelastic), steep (inelastic), 45° line (unit elastic), flat (elastic), horizontal (perfectly elastic).
• Elasticity of supply varies along the curve, generally decreasing at higher output levels.

Tax Incidence

When a government imposes a specific tax (a fixed amount per unit), it drives a wedge between the price buyers pay ($P_b$) and the price sellers receive ($P_s$). The key insight: the side of the market that is more inelastic bears a larger share of the tax, regardless of whether the tax is legally levied on sellers or buyers. The tax also reduces the quantity traded, creates government revenue, and generates a deadweight loss.

Specific Tax on Sellers: The Wedge

A specific tax of $t$ rupees per unit is imposed on sellers. In equilibrium:

$P_b - P_s = t$

The supply curve in terms of $P_b$ shifts vertically upward by exactly $t$. The demand curve (already in $P_b$) stays unchanged.

For linear inverse demand and supply:

• Demand: $P_b = a - b Q$ (buyers’ price)
• Supply (before tax): $P_s = c + d Q$ (sellers’ price)
• After tax, supply in terms of $P_b$: $P_b = c + t + d Q$

New Equilibrium

Solve $P_b = a - b Q = c + t + d Q$.

$Q_{\text{new}} = \frac{a - c - t}{b + d} = Q^* - \frac{t}{b + d}$

$P_b = \frac{ad + bc + b t}{b + d} = P^* + \frac{b}{b+d}\,t$

$P_s = P_b - t = \frac{ad + bc - d t}{b + d} = P^* - \frac{d}{b+d}\,t$

Where $P^*$ and $Q^*$ are the no‑tax equilibrium.

Exam tip: The formulas for $P_b$, $P_s$, and $Q_{\text{new}}$ are derived by equating quantity demanded and supplied after shifting the supply curve up by $t$. The split between $P_b$ and $P_s$ depends only on the slopes $b$ (absolute slope of demand) and $d$ (slope of supply).

Tax Incidence Depends on Elasticities

The consumer burden (share of $t$ paid by buyers) is:

$\text{Consumer burden} = \frac{P_b - P^*}{t} = \frac{b}{b+d}$

The producer burden is $\frac{d}{b+d}$.

Multiply numerator and denominator by $\frac{Q^*}{P^*}$ to convert slopes into elasticities:

$\frac{b}{b+d} \; \Rightarrow \; \frac{ \frac{1}{\epsilon_d} }{ \frac{1}{\epsilon_d} + \frac{1}{\epsilon_s} } = \frac{\epsilon_s}{\epsilon_s + \epsilon_d}$

where $\epsilon_d$ is price elasticity of demand and $\epsilon_s$ is price elasticity of supply (both evaluated at the no‑tax equilibrium).

Key rule: The more inelastic side bears the larger burden.

Example: Luxury Tax on Yachts

Yachts are luxury goods with highly elastic demand (many substitutes). Supply (shipbuilding) is relatively inelastic. Therefore, when a specific tax is imposed on yacht sales, producers bear most of the burden – the price buyers pay rises only slightly, while the price sellers receive falls substantially.

Exam tip: Do not assume that taxing sellers always hurts sellers more. The legal incidence (who writes the cheque) is irrelevant; the economic incidence depends solely on relative elasticities.

Deadweight Loss (DWL)

The tax reduces quantity from $Q^*$ to $Q_{\text{new}}$. The lost surplus (consumer + producer) that is not transferred to government revenue is the deadweight loss – a pure efficiency loss.

With linear curves:

$\text{DWL} = \frac{1}{2}\,t\,(Q^* - Q_{\text{new}}) = \frac{1}{2}\,t \cdot \frac{t}{b+d} = \frac{t^2}{2(b+d)}$

In elasticity terms:

$\text{DWL} = \frac{1}{2}\,t^2 \cdot \frac{Q^*}{P^*} \cdot \frac{1}{\frac{1}{\epsilon_d} + \frac{1}{\epsilon_s}}$

As either $\epsilon_d$ or $\epsilon_s$ increases, DWL increases. More elastic curves mean a larger quantity reduction for a given tax, hence a larger deadweight loss.

Surplus Flows Diagram

• Areas $R$ and $L$ are transferred from consumers and producers to the government.
• Areas $A$ and $B$ are lost – the deadweight loss.

Tax on Buyers (Equivalent)

Imposing the same specific tax on buyers (e.g., a sales tax) shifts the demand curve downward by $t$ instead of shifting supply upward. The final equilibrium $P_b$, $P_s$, $Q_{\text{new}}$, and the incidence are identical. The legal designation does not matter.

Price Caps and Price Floors (brief)

Governments sometimes intervene with price controls, which also create surpluses or shortages and deadweight loss.

Price Ceiling (Cap)

• Legally maximum price $P_c$.
• Effective only if $P_c < P^*$ → excess demand (shortage).
• Transfers surplus from producers to consumers, but causes rationing and DWL.

Price Floor (Minimum)

• Legally minimum price $P_f$.
• Effective only if $P_f > P^*$ → excess supply (surplus).
• Protects sellers (e.g., agricultural price supports) but creates wasted output and DWL.

Exam tip: An ineffective price control (above equilibrium for ceiling, below for floor) has no effect on the market.

Key takeaways

• A specific tax creates a wedge $P_b - P_s = t$; the supply curve shifts up by $t$ in terms of $P_b$.
• The side with more inelastic demand or supply bears a larger share of the tax.
• Linear formulas: $P_b = P^* + \frac{b}{b+d}t$, $P_s = P^* - \frac{d}{b+d}t$, $Q = Q^* - \frac{t}{b+d}$.
• Deadweight loss = $\frac{t^2}{2(b+d)}$; increases with elasticities.
• Tax incidence is independent of who legally pays the tax.
• Price controls (caps and floors) cause shortages/surpluses and DWL when effective.