How competitive markets operate: formula sheet

Key formulas

Law of demand

Price ↑ ⇒ Quantity demanded ↓ (ceteris paribus)

Gives a downward-sloping demand curve. Valid only if other factors stay constant.

Linear demand function

Qd = a − bP

a is the quantity demanded at zero price, b > 0 is the slope in quantity terms. A change in P moves you along the line. A change in a shifts the line.

General demand function

Qd = f(P, Y, Pr, T, E, N)

P own price, Y income, Pr related goods' prices, T tastes, E expectations, N population. Only P causes movement along the curve.

Market demand

Q market = Σ Q individual at each price

Add quantities horizontally at each price, not prices.

Direction of shifts

Substitute price ↑ ⇒ demand ↑; Complement price ↑ ⇒ demand ↓; Normal good income ↑ ⇒ demand ↑; Inferior good income ↑ ⇒ demand ↓

Use these four rules to decide the direction of any shift.

Law of supply

Price ↑ → quantity supplied ↑ (ceteris paribus)

Holds for the usual upward-sloping supply curve. It is a movement along the curve, not a shift.

Linear supply function

Qs = a + bP, with b > 0

A change in a (the intercept) shifts the curve. A change in P moves you along it. Check which variable has changed.

Inverse supply function

P = (Qs − a) ÷ b

Use this form to add a per-unit tax or subsidy.

Per-unit tax shift

New supply price = old supply price + t

The curve shifts vertically up by t, which is a leftward shift. The price paid by buyers includes the tax.

Per-unit subsidy shift

New supply price = old supply price − s

The curve shifts vertically down by s, which is a rightward shift.

Market supply

Market Qs = sum of each firm's Qs at the same price

Add quantities horizontally at each price. Do not add prices.

Equilibrium condition

Qd = Qs

Set quantity demanded equal to quantity supplied and solve for the price P*.

Linear demand and supply

Qd = a − bP and Qs = c + dP, so P* = (a − c) ÷ (b + d)

Valid for b + d ≠ 0. Substitute P* into either equation to get Q*.

Excess demand

Excess demand = Qd − Qs at a given price

Positive when price is below equilibrium. Price tends to rise.

Excess supply

Excess supply = Qs − Qd at a given price

Positive when price is above equilibrium. Price tends to fall.

Single-shift rules

Demand up: P↑, Q↑. Demand down: P↓, Q↓. Supply up: P↓, Q↑. Supply down: P↑, Q↓

Assumes normal downward-sloping demand and upward-sloping supply, with the other curve unchanged.

Price elasticity of demand

PED = (% change in Qd) ÷ (% change in P)

Usually negative. Compare the absolute value with 1.

Percentage change

% change = (new − old) ÷ old × 100

Using the old value as base gives point-style elasticity for a given change. Midpoint method uses the average of old and new.

Midpoint (arc) elasticity

E = [(Q2 − Q1) ÷ ((Q1 + Q2) ÷ 2)] ÷ [(P2 − P1) ÷ ((P1 + P2) ÷ 2)]

Gives the same value whichever direction the change goes. Use it when the question asks for arc elasticity.

Point elasticity

PED = (dQ/dP) × (P ÷ Q)

Use when a demand function Q = f(P) is given.

Income elasticity of demand

YED = (% change in Qd) ÷ (% change in income)

Positive: normal good. Negative: inferior good. Above 1: luxury.

Cross elasticity of demand

XED = (% change in Qd of A) ÷ (% change in price of B)

Positive: substitutes. Negative: complements. Near zero: unrelated.

Price elasticity of supply

PES = (% change in Qs) ÷ (% change in P)

Positive for an upward-sloping supply curve.

Revenue rule

|PED| > 1: price up, revenue down. |PED| < 1: price up, revenue up. |PED| = 1: revenue unchanged.

Reverse the direction for a price cut.

Total revenue

TR = P × Q

Use it to check any revenue conclusion numerically.

Shortage under a binding ceiling

Shortage = Qd(Pc) − Qs(Pc), where Pc < P*

Quantity traded equals Qs(Pc), the smaller quantity.

Surplus under a binding floor

Surplus = Qs(Pf) − Qd(Pf), where Pf > P*

Quantity traded equals Qd(Pf), the smaller quantity.

Tax wedge

Pb − Ps = t (per-unit tax)

Pb is the price buyers pay, Ps the price sellers keep after tax.

Tax incidence split

Buyer share : Seller share = 1/Ed : 1/Es (using magnitudes)

The less elastic side bears the larger share. This is a rule for small taxes and approximately linear curves.

Tax revenue

Revenue = t × Q after tax

Use the new, lower quantity, not the old equilibrium quantity.

Deadweight loss of a tax (linear curves)

DWL = ½ × t × (Q* − Q_tax)

Triangle between the curves, with height t and base the fall in quantity.

Subsidy wedge

Ps − Pb = s (per-unit subsidy)

Sellers receive Ps, buyers pay Pb. Cost to government = s × Q after subsidy.

Consumer surplus (CS)

CS = area under demand curve above price, up to quantity Q

For a straight-line demand curve, CS = ½ × (maximum price − market price) × Q.

Producer surplus (PS)

PS = area above supply curve below price, up to quantity Q

For a straight-line supply curve starting at the minimum price, PS = ½ × (market price − minimum supply price) × Q. If supply starts from the origin, the minimum price is 0.

Total welfare

Total welfare = CS + PS

Maximised at the competitive equilibrium, where demand equals supply.

Efficiency condition

Marginal benefit (demand) = Marginal cost (supply)

This is allocative efficiency. In perfect competition, P = MC.

Deadweight loss (DWL)

DWL = ½ × (price gap at restricted quantity) × (Q* − Q1)

Q* is equilibrium quantity, Q1 is the restricted quantity. The price gap is the demand price minus the supply price at Q1. This triangle formula holds for linear curves.

Quick revision

• Law of demand: other things equal, a higher price lowers quantity demanded.
• Law of supply: other things equal, a higher price raises quantity supplied.
• A price change moves you along a curve; a change in any other factor shifts the curve.
• Equilibrium is where quantity demanded equals quantity supplied, so there is no shortage or surplus.
• Price above equilibrium creates excess supply; price below equilibrium creates excess demand.
• Price elasticity of demand = % change in quantity demanded ÷ % change in price.
• The more inelastic side of the market bears more of a tax.
• A binding price ceiling is set below equilibrium and causes a shortage; a binding price floor is set above equilibrium and causes a surplus.
• A subsidy shifts the supply curve right and lowers the price paid by buyers.
• Consumer surplus is the gap between willingness to pay and price; producer surplus is the gap between price and the minimum acceptable price.
• A competitive equilibrium maximises total surplus, and a tax or binding control creates deadweight loss.

Common mistakes

• Saying a fall in price 'increases demand'. Fix: A price fall increases quantity demanded. Demand itself, meaning the curve, does not change.
• Shifting the curve when the own price changes. Fix: Keep the curve fixed and move along it. Only draw a new curve for non-price factors.
• Calling a change in price a 'change in supply'. Fix: Use 'change in quantity supplied' for price-driven movements along the curve. Reserve 'change in supply' for shifts caused by other factors.
• Shifting the curve when the good's own price changes. Fix: The own price never shifts its own supply curve. It only moves you along it.
• Confusing a movement along a curve with a shift of the curve. Fix: A change in the good's own price moves you along the curve. Only other factors shift the curve. Price is the result, not the cause, of the shift.
• Saying a rise in demand raises price and lowers quantity. Fix: After demand shifts right, price rises and supply responds, so quantity supplied also rises. Both rise.
• Treating PED as the slope of the demand curve. Fix: Slope uses absolute changes. Elasticity uses percentage changes. A straight line has constant slope but varying elasticity.
• Dropping the sign or misreading a negative value as inelastic. Fix: Take the absolute value for PED before comparing with 1. Keep the sign for income and cross elasticity, where it carries meaning.
• Saying a price ceiling always causes a shortage. Fix: Check the ceiling is below the equilibrium price. If it is above, nothing changes.
• Using the quantity demanded as the traded quantity under a ceiling. Fix: Traded quantity is the smaller of Qd and Qs. Under a ceiling that is Qs; under a floor that is Qd.

Exam tips

• Use the exact terms. Write 'change in quantity demanded' for own-price changes and 'change in demand' for shifts. Examiners award marks for this wording.
• Always label your diagram: axes P and Q, curves D1 and D2, and an arrow for the direction. An unlabelled sketch earns little.
• In MCQs, first check if the good's own price changed. Many wrong options tempt you with the word 'demand'.
• In written answers, state ceteris paribus and name the specific factor. Do not just write 'other factors'.
• Link demand to equilibrium when asked about price outcomes. A rightward demand shift with fixed supply raises both equilibrium price and quantity.
• Examiners often reward correct vocabulary. Write 'change in quantity supplied' or 'change in supply' exactly, and never swap them.
• In graph answers, label both axes, the old and new curves (S1, S2) and the old and new equilibrium points. Unlabelled diagrams lose marks.
• For tax and subsidy questions, state whether the amount is per unit. A per-unit amount gives a parallel vertical shift in the curve.