definition and calculation of total utility and marginal utility

7.1 Utility – definition, calculation, diminishing MU, equi‑marginal principle

1. What is utility?

• Utility is a theoretical measure of the satisfaction or pleasure a consumer derives from consuming goods and services.
• In modern micro‑economics utility is ordinal: it allows us to rank bundles of goods but does not assign an absolute numerical value.
• Utility provides the behavioural foundation for the demand theory in Cambridge IGCSE (0470) and A‑Level (9708) economics.

2. Total Utility (TU)

Definition: the cumulative satisfaction obtained from consuming a given quantity of a good.

2.1 Discrete (unit‑by‑unit) form

When consumption is measured in whole units, TU is the sum of the marginal utilities of each unit:

\[ TU(q)=\sum_{i=1}^{q} MU_i \]

2.2 Continuous form

If the quantity can vary smoothly, TU is the area under the MU curve:

\[ TU(q)=\int_{0}^{q} MU(x)\,dx \]

2.3 How to calculate TU (discrete example)

Slices of pizza (q)	Total Utility (TU)	Marginal Utility (MU)
0	0	—
1	10	10
2	18	8
3	24	6
4	28	4
5	30	2

Step‑by‑step:

1. Identify the MU of each additional slice (difference in TU between successive rows).
2. Add the MU values cumulatively to obtain TU for each quantity.

3. Marginal Utility (MU)

Definition: the additional satisfaction gained from consuming one more unit of a good.

3.1 Discrete form

\[ MU_n = TU_n - TU_{n-1} \]

3.2 Continuous form

\[ MU(q)=\frac{dTU}{dq} \]

3.3 Illustration (coffee vs. tea – marginal utilities)

Quantity of coffee (C)	MUC	Quantity of tea (T)	MUT
1	12	1	10
2	9	2	8
3	6	3	6
4	4	4	4

4. Law of Diminishing Marginal Utility

• Holding all else constant, MU falls as more units of a good are consumed.
• It explains why the MU curve slopes downwards and why the demand curve is downward‑sloping.

5. Equi‑marginal (Utility‑maximisation) Principle

The consumer maximises total utility when the last unit of each good provides the same amount of utility per unit of money spent:

\[ \frac{MU_X}{P_X}= \frac{MU_Y}{P_Y}= \dots =\frac{MU_Z}{P_Z} \]

At this point any re‑allocation of expenditure would lower total utility.

5.1 Step‑by‑step application (two‑good case)

1. Calculate MU for each possible unit of each good.
2. Divide each MU by the good’s price to obtain the MU/P ratio.
3. Spend the next unit of income on the good with the highest remaining MU/P.
4. Repeat until the budget is exhausted or the MU/P ratios are equalised.

6. Deriving the Individual Demand Curve from Utility Maximisation

1. Specify the consumer’s income (I) and the price of the good (P) together with the prices of all other goods.
2. Draw the budget line: P_X X + P_Y Y = I.
3. Identify the point where the highest attainable indifference curve is tangent to the budget line – this gives the optimal quantity Q* at that price.
4. Change the price of the good (keeping income and other prices constant); the budget line pivots.
5. Locate the new tangency point → new optimal quantity Q*'.
6. Plot each (P, Q*) pair on a price‑quantity graph. Connecting the points yields the downward‑sloping individual demand curve.

7. Numerical Examples

7.1 Single‑good – Pizza (TU & MU)

Slices (q)	Total Utility (TU)	Marginal Utility (MU)
0	0	—
1	10	10
2	18	8
3	24	6
4	28	4
5	30	2

The MU values fall from 10 to 2, illustrating the law of diminishing marginal utility.

7.2 Two‑good – Coffee vs. Tea (Equi‑marginal allocation)

Assumptions:

• Income I = £20
• Price of coffee P_C = £2 per cup
• Price of tea P_T = £1 per cup

Using the MU table above, calculate MU/P for each possible unit:

Good	Unit	MU	Price (P)	MU/P
Coffee	1	12	2	6.0
Tea	1	10	1	10.0
Tea	2	8	1	8.0
Coffee	2	9	2	4.5
Tea	3	6	1	6.0
Coffee	3	6	2	3.0
Tea	4	4	1	4.0
Coffee	4	4	2	2.0

Allocation process (budget £20):

1. Buy the highest MU/P first → 1 cup of tea (cost £1, remaining £19).
2. Next highest MU/P → 1 cup of coffee (cost £2, remaining £17).
3. Continue selecting the highest remaining MU/P until the budget is spent.

Resulting optimal bundle (illustrative): 2 cups of tea and 3 cups of coffee, spending £8 on these two goods and leaving £12 for other items. At this bundle:

\[ \frac{MU_C}{P_C}= \frac{6}{2}=3 \quad\text{and}\quad \frac{MU_T}{P_T}= \frac{6}{1}=6 \]

The consumer would shift one unit of coffee to tea until the ratios converge (≈ 4), achieving the equi‑marginal condition.

8. Consumer Equilibrium

• Graphical condition: the highest attainable indifference curve is tangent to the budget line.
• Mathematical condition (equimarginal rule): \[ \frac{MU_X}{P_X}= \frac{MU_Y}{P_Y} \]
• When the condition holds, any re‑allocation of expenditure would lower total utility, so the consumer is at equilibrium.

9. Limitations of Utility Theory

• Utility is ordinal; the cardinal numbers used in calculations are a simplifying convention.
• Assumes rational behaviour and full information – not always realistic.
• Ignores psychological influences such as habits, expectations, and social norms.
• Utility cannot be measured directly; it is an abstract construct.
• Early textbook models treat utility as cardinal, whereas modern theory recognises only ranking.

10. Summary of Key Points (AO3)

• Utility ranks consumer preferences; total utility aggregates satisfaction, marginal utility measures the extra satisfaction from one more unit.
• The law of diminishing marginal utility makes the MU curve fall, providing a micro‑foundations for the downward‑sloping demand curve.
• The equi‑marginal principle (MU/P equalisation) links utility maximisation to consumer equilibrium and to the individual demand curve.
• Indifference curves and budget lines give a graphical method for locating the optimum bundle.
• Despite its abstractions and assumptions, utility theory remains a core analytical tool in Cambridge IGCSE and A‑Level economics.