The Law of Equi-Marginal Utility (Consumer's Equilibrium)

While the Law of Diminishing Marginal Utility explains a consumer's behaviour with a single good, real consumers must divide a limited income across many goods. The Law of Equi-Marginal Utility (also called the Law of Substitution, or Gossen's Second Law) explains exactly how a rational consumer allocates a fixed income among different goods so as to obtain maximum total satisfaction.

Statement of the Law. "A consumer, with a fixed income to spend on two or more goods, will obtain maximum total satisfaction when the marginal utility of the last rupee spent on each good is equal, and the whole of income is spent." Formally, for two goods X and Y with prices $P_x$ and $P_y$:

where $\dfrac{MU_x}{P_x}$ and $\dfrac{MU_y}{P_y}$ are each read as the marginal utility per rupee spent on that good. If more goods (Z, W, …) are involved, the condition simply extends: $\dfrac{MU_x}{P_x} = \dfrac{MU_y}{P_y} = \dfrac{MU_z}{P_z} = \dots = MU$ of money (the marginal utility of the last rupee itself, held in reserve or spent on any good).

Assumptions of the Law.

1. The consumer's income is fixed/given and must be spent entirely on the two (or more) goods.
2. Utility is cardinally measurable in utils, exactly as in the Law of DMU.
3. The Law of Diminishing Marginal Utility operates for each good separately as more units of it are bought.
4. The consumer is rational and aims to maximise total satisfaction.
5. Prices of the goods are given and constant throughout the period.
6. The consumer's tastes, preferences and habits remain unchanged.

Numerical illustration (worked example). A consumer has a fixed income of Rs 20 to spend on two goods, X (price $P_x$ = Rs 4 per unit) and Y (price $P_y$ = Rs 2 per unit). The marginal utility schedules for successive units of each good, and the marginal utility per rupee ($MU/P$) derived from them, are:

To find the consumer's equilibrium combination, compare $MU_x/P_x$ and $MU_y/P_y$ at every possible combination and find where the two ratios are equal while the entire income of Rs 20 is exactly exhausted. At 3 units of X and 4 units of Y, $MU_x/P_x = 6$ and $MU_y/P_y = 6$ — the two ratios are equal, satisfying the equi-marginal condition. Checking the budget: expenditure on X = 3 × Rs 4 = Rs 12; expenditure on Y = 4 × Rs 2 = Rs 8; total expenditure = Rs 12 + Rs 8 = Rs 20, exactly equal to income. Both conditions of the law — equal marginal utility per rupee, and full utilisation of income — are satisfied, so X = 3 units, Y = 4 units is the consumer's equilibrium (utility-maximising) combination.

Why this maximises total utility. At any other combination that still costs Rs 20, the consumer could rearrange spending to gain more utility. For example, at X = 4, Y = 2 (cost = 16 + 4 = Rs 20), $MU_x/P_x = 4$ while $MU_y/P_y = 10$ — the last rupee spent on Y is buying far more utility per rupee than the last rupee spent on X, so the consumer would be better off shifting a rupee away from X toward Y. Only when the two ratios are brought into equality, as at X = 3, Y = 4, is there no such profitable rearrangement left — this is precisely why that point, and only that point (among those exhausting the full income), maximises total utility. …