Consumer demand and behaviour: formula sheet

Key formulas

Linear demand

Qd = a − bP

b > 0 gives a downward-sloping curve. a is the quantity demanded at price zero.

Linear supply

Qs = c + dP

d > 0 gives an upward-sloping curve. c may be negative, meaning no supply below a minimum price.

Equilibrium condition

Qd = Qs

Solve for P* first, then substitute into either equation for Q*.

Equilibrium price (linear case)

P* = (a − c) ÷ (b + d)

Valid for Qd = a − bP and Qs = c + dP.

Surplus or shortage

Surplus = Qs − Qd (if positive); Shortage = Qd − Qs (if positive)

Evaluate both at the given non-equilibrium price.

Direction of shifts

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

Applies to a single shift with normal sloping curves.

Price elasticity of demand (point or simple form)

PED = (%ΔQ) ÷ (%ΔP) = (ΔQ ÷ Q) ÷ (ΔP ÷ P)

Normally negative. For a linear demand curve, the value changes along the curve.

Arc (midpoint) PED

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

Gives the same answer for a rise or a fall between two points. Use it when the question says midpoint or arc.

Point PED from a demand function

PED = (dQ/dP) × (P ÷ Q)

For linear Q = a − bP, dQ/dP = −b.

Income elasticity of demand

YED = (%ΔQ) ÷ (%ΔY)

YED > 1 luxury; 0 < YED < 1 normal necessity; YED < 0 inferior.

Cross elasticity of demand

XED = (%ΔQ of A) ÷ (%ΔP of B)

Positive: substitutes. Negative: complements. Sign depends on the pair, not on one good alone.

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.

Marginal utility

MU = ΔTU ÷ ΔQ

For one extra unit, MU = TU(n) − TU(n−1). In calculus form, MU = dTU/dQ.

Total utility from marginal utilities

TU(n) = MU(1) + MU(2) + ... + MU(n)

TU is the running sum of MU. TU is maximum where MU = 0.

Law of diminishing marginal utility

MU falls as Q rises (other things constant)

Holds for a given period, with unchanged tastes and income and identical units.

Equimarginal principle (consumer equilibrium)

MUx ÷ Px = MUy ÷ Py = ... = MU of money

Must also satisfy the budget: Px·x + Py·y + ... = income.

Single good equilibrium

MU of good = Price × MU of money

If MU of a rupee is 1 util, buy until MU = P.

Budget line

Px·X + Py·Y = M

M is income. Px and Py are the prices of X and Y. All income is spent.

Budget line slope and intercepts

Slope = -Px ÷ Py; X-intercept = M ÷ Px; Y-intercept = M ÷ Py

Intercepts show the most of one good you can buy if you buy none of the other.

Marginal rate of substitution

MRS = -dY/dX along an indifference curve = MUx ÷ MUy

MRS is quoted as a positive number. It is the slope of the curve in absolute value.

Consumer optimum (interior)

MRS = Px ÷ Py, equivalently MUx ÷ Px = MUy ÷ Py

Needs convex curves and a point where both goods are bought. Also check the budget is fully spent.

Budget line shifts

Income change: parallel shift. Change in Px only: X-intercept moves, Y-intercept fixed. Change in Py only: Y-intercept moves, X-intercept fixed.

Equal proportional change in both prices and income leaves the line unchanged.

Decomposition

Total effect = Substitution effect + Income effect

Each effect is the change in quantity demanded of the good. Keep the signs when you add them.

Slutsky compensated income

m' = m + (p1' − p1) × x0

x0 is the original quantity of the good, p1 is the old price and p1' is the new price. After a price fall, p1' − p1 is negative, so income is reduced. After a price rise, income is increased.

Slutsky substitution effect

SE = x(p1', m') − x(p1, m)

Demand at the new price with compensated income m', minus original demand.

Income effect

IE = x(p1', m) − x(p1', m')

Moves from the compensated income back to the actual income, at the new price.

Hicks substitution effect

Utility held at the original level: SE = x(p1', u0) − x(p1, u0)

Uses compensated demand on the original indifference curve.

Sign rules

SE always opposite to price change. Normal good: IE opposite to price change. Inferior good: IE same direction as price change. Giffen: |IE| > |SE| and good is inferior.

Use these to classify any good from the signs.

Consumer surplus (per unit)

Consumer surplus = willingness to pay − price paid

Add across all units bought to get the total.

Consumer surplus, linear demand

CS = ½ × Q × (P_max − P)

P_max is the price-axis intercept of demand, P is the market price, Q is quantity demanded at P. Valid only for a straight-line demand curve.

Change in consumer surplus

ΔCS = area between the old and new price lines, under the demand curve

For a price fall it is a gain. Use a trapezium: ½ × (Q1 + Q2) × (P1 − P2) for linear demand.

Expected utility

E[U(W)] = Σ p_i × U(W_i)

p_i are probabilities of outcomes and W_i the wealth in each outcome.

Risk aversion

U(E[W]) > E[U(W)] when U is concave (U'' < 0)

This is Jensen's inequality. The person prefers the certain expected wealth to the gamble.

Quick revision

• A change in the good's own price moves along the demand curve; a change in income, tastes or related prices shifts the curve.
• At equilibrium, quantity demanded equals quantity supplied; a price above it gives a surplus and below it a shortage.
• Price elasticity of demand = % change in quantity demanded ÷ % change in price; it is usually negative, and is often quoted in absolute value.
• Demand is elastic if |elasticity| > 1, inelastic if < 1 and unit elastic if = 1.
• Income elasticity is positive for normal goods and negative for inferior goods.
• Cross elasticity is positive for substitutes and negative for complements.
• Marginal utility is the extra utility from one more unit and is assumed to diminish.
• A consumer maximises utility where the marginal rate of substitution equals the price ratio, for well-behaved preferences with an interior solution.
• The budget line shifts in parallel when income changes and pivots when one price changes.
• Indifference curves slope downwards, do not cross and are convex to the origin under standard assumptions.
• A price fall has a substitution effect and an income effect; for a normal good both raise quantity demanded.
• Consumer surplus is the gap between what buyers are willing to pay and what they pay; behavioural economics adds biases such as framing and loss aversion.

Common mistakes

• Calling a price change a shift in demand. Fix: Ask what caused it. If the good's own price changed, it is a movement along the curve. Only other factors shift the curve.
• Showing a rise in price as the cause of a fall in demand. Fix: Use 'quantity demanded' for movements and 'demand' for shifts. Write the correct term every time.
• Using the absolute change instead of the percentage change. Fix: Always divide each change by the base value before taking the ratio.
• Calling a good elastic because PED = −0.4 is 'smaller than −1'. Fix: Compare the absolute value with 1. |−0.4| = 0.4 < 1, so demand is inelastic.
• Saying diminishing marginal utility means total utility falls. Fix: TU keeps rising while MU is positive, even if MU is falling. TU falls only when MU is negative.
• Equating MU of goods instead of MU per rupee. Fix: Always compare MU ÷ P. Equal MU is correct only when prices are equal.
• Taking the budget line slope as Py ÷ Px. Fix: With X on the horizontal axis, slope = -Px ÷ Py. Check with intercepts: rise M ÷ Py over run M ÷ Px.
• Writing MRS = MUy ÷ MUx. Fix: MRS of X for Y = MUx ÷ MUy. It is the Y given up per extra X, which uses X's marginal utility on top.
• Saying the substitution effect can be positive for a price fall. Fix: The substitution effect always moves quantity against the price change. Only the income effect can work in the opposite way for an inferior good.
• Calling every inferior good a Giffen good. Fix: A good is Giffen only if the income effect is larger than the substitution effect, so the total effect has the wrong sign.

Exam tips

• Use the exact terms 'change in quantity demanded' and 'change in demand'. Examiners reward this precision.
• In written answers, draw a clear diagram with labelled axes, curves, and old and new equilibrium points, and then explain it in words.
• For numerical questions, show the equating step, the value of P*, and the check on Q*.
• Work through the cause and effect chain: event, which curve, which direction, new price and quantity.
• When asked about exceptions to the law of demand, name Giffen and Veblen goods, give the reason for each, and say they are rare.
• Write the formula and the signed result. Then interpret in words, because marks are often given for the conclusion.
• In multiple-choice questions, check whether the options use signed or absolute values before you choose.
• For revenue questions, a quick P × Q before and after catches sign errors.