Q.State and explain the Law of Equi-Marginal Utility with a suitable numerical illustration showing the consumer's equilibrium.

This law explains how a consumer with a fixed income spread across two goods reaches the combination that maximises total satisfaction.

Statement of the Law. A consumer, with a fixed income to spend on two or more goods, obtains maximum total satisfaction when the marginal utility of the last rupee spent on each good is equal, and the whole of the income is spent — formally, $\dfrac{MU_x}{P_x} = \dfrac{MU_y}{P_y}$, where each ratio is the marginal utility per rupee spent on that good.

Numerical illustration. Suppose a consumer has a fixed income of Rs 20, to be spent on goods X (price $P_x$ = Rs 4) and Y (price $P_y$ = Rs 2), with the following marginal utility schedules:

Finding the equilibrium. The consumer compares $MU_x/P_x$ and $MU_y/P_y$ and, being rational, always spends the next rupee on whichever good currently offers the higher marginal utility per rupee, continuing until income is exhausted and the two ratios are equalised. At X = 3 units, Y = 4 units: $MU_x/P_x = 6$ and $MU_y/P_y = 6$ — equal, as the law requires. Checking the budget: $3 \times 4 + 4 \times 2 = 12 + 8 = 20$, exactly equal to the given income of Rs 20. Both conditions of the law are satisfied simultaneously, so X = 3, Y = 4 is the consumer's equilibrium combination.

Why no other combination is better. Consider instead X = 4, Y = 2, which also costs Rs 20 ($4 \times 4 + 2 \times 2 = 16 + 4 = 20$): here $MU_x/P_x = 4$ but $MU_y/P_y = 10$. Since the last rupee spent on Y yields far more utility than the last rupee spent on X, the consumer is clearly not maximising total utility — shifting one rupee's worth of spending from X toward Y would raise total satisfaction. Only at X = 3, Y = 4, where the two ratios are exactly equal, is there no such profitable rearrangement possible, confirming it as the true point of consumer's equilibrium. …

Students often find a combination where the ratios are equal but forget to also check that it exactly exhausts the given income — both conditions (equal ratio AND ful …