Law of Equi-Marginal Utility

A consumer rarely spends income on only one commodity; income has to be allocated across many goods. The Law of Equi-Marginal Utility, also called Gossen's Second Law or the Law of Substitution, explains how a rational consumer with a fixed income and given prices distributes that income among different goods so as to obtain the maximum possible total satisfaction.

The law states that a consumer maximises total utility when the last rupee spent on every commodity yields the same marginal utility — equivalently, when the marginal utility of each good divided by its price (the marginal utility per rupee) is equalised across all goods purchased:

$MU_x/P_x = MU_y/P_y = ... = MU_m$

where $MU_x$, $MU_y$ are the marginal utilities of goods $X$, $Y$, $P_x$, $P_y$ are their prices, and $MU_m$ is the marginal utility of money (assumed constant).

How equilibrium is reached: suppose the marginal utility per rupee spent on good $X$ is currently higher than on good $Y$. A rational consumer gains more extra satisfaction by shifting a rupee of spending away from $Y$ towards $X$. As more of $X$ is bought, its marginal utility falls (by the Law of Diminishing Marginal Utility); as less of $Y$ is bought, its marginal utility rises. The consumer keeps substituting spending from the good with the lower marginal utility per rupee to the one with the higher marginal utility per rupee until the two ratios become equal — at which point no further reallocation can raise total utility, and the consumer is in equilibrium. …