equi-marginal principle

Utility – Equi‑marginal Principle (Cambridge A‑Level Economics 9708)

1. Utility

• Total Utility (TU): the overall satisfaction a consumer derives from a given quantity of a good (or a bundle of goods).
• Marginal Utility (MU): the additional satisfaction obtained from consuming one more unit of a good. $$MU = \frac{\Delta TU}{\Delta Q}$$
• Law of Diminishing Marginal Utility: ceteris paribus, the MU of a good falls as the quantity consumed rises.

Relationship: total utility is the sum of the marginal utilities of each unit. $$TU = \sum_{i=1}^{Q} MU_i$$

2. Indifference‑curve analysis (Syllabus 7.2)

• Indifference curve (IC): a curve joining all bundles that give the consumer the same level of total utility.
• Key properties
  • Downward sloping – more of one good requires less of the other to keep utility constant.
  • Convex to the origin – reflects a diminishing marginal rate of substitution (MRS).
  • Higher ICs represent higher utility.
• Marginal Rate of Substitution (MRS): the rate at which a consumer is willing to give up good Y for an additional unit of good X while remaining on the same IC. $$MRS_{XY}= \frac{MU_X}{MU_Y}= -\frac{dY}{dX}\Big|_{IC}$$
• Preference assumptions required for the standard model
  • Completeness – the consumer can rank any two bundles.
  • Transitivity – if A ≽ B and B ≽ C, then A ≽ C.
  • Convexity – mixtures of two bundles are at least as preferred as the original bundles (gives a convex IC).
  • No satiation within the relevant range – more of a good never reduces utility.
• Non‑convex preferences (e.g. perfect complements) are part of the syllabus; in such cases the tangency condition may not give the optimum, and the optimum can occur at a corner of the budget line.

3. Budget constraint (Syllabus 7.2.1)

For two goods X and Y with prices \(P_X\) and \(P_Y\) and a consumer’s income (budget) \(B\):

$$P_X Q_X + P_Y Q_Y = B$$

• Intercepts: $$Q_X^{\max}= \frac{B}{P_X},\qquad Q_Y^{\max}= \frac{B}{P_Y}$$
• Slope of the budget line: \(-\dfrac{P_X}{P_Y}\) – the opportunity cost of one unit of X in terms of Y.
• Effect of changes
  • Income change – shifts the line outward (higher B) or inward (lower B) without altering its slope.
  • Relative‑price change – rotates the line about the intercept of the good whose price is unchanged.

Numeric illustration: if income rises from £20 to £30 (with \(P_X=£2\), \(P_Y=£1\)), the intercepts become \(Q_X^{\max}=15\) and \(Q_Y^{\max}=30\); the whole line moves outward, allowing a higher attainable indifference curve.

4. Consumer equilibrium – the Equi‑marginal principle (Syllabus 7.1.4)

Maximum total utility is achieved when the last unit of money spent on each good yields the same extra satisfaction.

4.1 Algebraic (equimarginal) rule

$$\frac{MU_X}{P_X}= \frac{MU_Y}{P_Y}= \dots = \frac{MU_n}{P_n}$$

Each “MU / price” term is the marginal utility obtained per £ (or per unit of currency) spent on that good.

4.2 Graphical (tangency) condition

$$\frac{MU_X}{MU_Y}= \frac{P_X}{P_Y}\qquad\Longleftrightarrow\qquad MRS_{XY}= \frac{P_X}{P_Y}$$

When the two conditions hold simultaneously, the budget line is tangent to the highest attainable indifference curve (IC*).

4.3 Diagram (placeholder)

5. From the equi‑marginal rule to the individual demand curve (Syllabus 7.1.5)

1. Choose a price for good X, keep the price of Y and income constant.
2. Apply the equi‑marginal rule to determine the optimal bundle \((Q_X, Q_Y)\).
3. Record the quantity \(Q_X\) demanded at that price.
4. Repeat for other prices of X.
5. Plot the pairs \((P_X, Q_X)\); the resulting curve is the consumer’s individual demand curve for X.

Numerical illustration (three price points)

Price of X (\(P_X\))	Optimal \(Q_X\)	Optimal \(Q_Y\)
£2	4	12
£3	3	14
£4	2	16

Connecting the three points yields a downward‑sloping demand curve for X, confirming the usual law of demand.

Demand‑curve sketch (placeholder)

6. Numerical example – applying the equi‑marginal rule step‑by‑step

Budget \(B = £20\); prices \(P_X = £2\) and \(P_Y = £1\). The marginal utilities for successive units are given below.

Unit of X	MUX	MUX/PX	Unit of Y	MUY	MUY/PY
1	10	5.0	1	8	8.0
2	8	4.0	2	7	7.0
3	6	3.0	3	6	6.0
4	4	2.0	4	5	5.0
5	2	1.0	5	4	4.0

Allocation algorithm (running total after each purchase)

Step	Spending (£)	Good bought	Qty X	Qty Y	Next MUX/PX	Next MUY/PY	Remaining budget
0	0	–	0	0	5.0	8.0	20
1	1	Y	0	1	5.0	7.0	19
2	2	Y	0	2	5.0	6.0	18
3	3	Y	0	3	5.0	5.0	17
4	4	Y	0	4	5.0	4.0	16
5	6	X	1	4	4.0	4.0	14
6	7	Y	1	5	4.0	3.0	13
7	9	X	2	5	3.0	3.0	11
8	10	Y	2	6	3.0	2.0	10
9	12	X	3	6	2.0	2.0	8
10	13	Y	3	7	2.0	1.0	7
11	15	X	4	7	1.0	1.0	5
12	16	Y	4	8	1.0	0.5	4
13	18	Y	4	9	1.0	0.4	2
14	20	Y	4	10	—	0.3	0

Optimal bundle

• Good X: 4 units (cost £8) → total MU from X = 10 + 8 + 6 + 4 = 28.
• Good Y: 12 units (cost £12) → total MU from Y ≈ 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 + 0.5 + 0.4 + 0.3 + 0.2 = 43.4.
• Total utility ≈ 71.4.
• MU per £ for the last unit of X = 1.0 / 2 = 0.5; for the last unit of Y = 0.3 / 1 = 0.3 – the two ratios are as close as possible given the discrete units, satisfying the equi‑marginal condition.

7. Limitations of the marginal‑utility approach (Syllabus 7.1.5)

• Assumes rational, fully informed consumers who can rank every possible bundle (complete & transitive preferences).
• Relies on the ceteris paribus condition; real‑world changes in income, tastes, or expectations can shift MU curves.
• Works best with divisible (continuous) goods; many goods are purchased in discrete units, making the “last‑unit” rule an approximation.
• Ignores behavioural factors such as habit formation, loss aversion, and reference‑point effects.
• Does not handle non‑convex preferences (e.g., perfect complements, satiation points) where the tangency condition may not give a maximum.

8. Connections to later A‑Level topics

• Price elasticity of demand: the steeper the MU‑derived demand curve, the lower the price elasticity; a flatter curve implies a higher elasticity.
• Consumer surplus: the area between the demand curve (derived from MU) and the market price measures the extra benefit to consumers.
• Welfare analysis: shifts in income or prices move the budget line; the resulting movement along or between indifference curves helps evaluate changes in consumer welfare.

9. Key take‑aways

• The equi‑marginal principle states that utility is maximised when the marginal utility per unit of expenditure is equal across all goods: \(\displaystyle \frac{MU_X}{P_X}= \frac{MU_Y}{P_Y}= \dots\).
• This rule is algebraically identical to the tangency condition \(MRS_{XY}=P_X/P_Y\) in indifference‑curve analysis.
• Applying the rule (either numerically or graphically) yields the consumer’s optimal bundle; varying the price of a good and re‑applying the rule generates the individual demand curve.
• While powerful, the marginal‑utility framework rests on strong assumptions (rationality, convex preferences, divisibility) and has recognised limitations when analysing real‑world consumer behaviour.