Question

Explain the law of equi-marginal utility with the help of a numerical example.

Answer 1

The utility maximization The final unit of money (rupee) spent on each commodity must provide the consumer with the same (equal) marginal value, thus he must divide his income among these commodities.

To keep things simple, we'll imagine that a customer wants to spend ₹ 40 on two items, X and Y, which have respective pricing of ₹ 5 and no. The marginal utility of these two goods, X and Y, are displayed in the table:

Utility Schedule for X and Y:

Units	MUX (utils)	MUY (utils)	`(MU_X)/P_X` (utils)	`MU_y/P_Y` (utils)
(1)	(2)	(3)	(4)	(5)
1	50	80	`50/5=10`	`80/10=8`
2	45	70	`45/5=9`	`70/10=7`
3	40	60	`40/5=8`	`60/10=6`
4	35	50	`35/5=7`	`50/10=5`
5	30	40	`30/5=6`	`40/10=4`
6	25	30	`25/5=5`	`30/10=3`

The marginal utility of X and Y is shown in columns (2) and (3). The marginal utility of a rupee spent on the purchase of two commodities, or the ratios of marginal utility to price of the two commodities, are shown in columns (4) and (5). Because the marginal utility of each commodity decreases as we consume more of it, you can see in the table that the marginal utility per rupee spent on each good decreases.

The consumer can solve the challenge of choosing the quantities of each commodity that maximize his utility with the help of the marginal utility per rupee. The proportionality rule, which reads `(MU_X)/P_X = (MU_Y)/(P_Y)`, can be satisfied at a variety of points, including 3 units of X and 1 unit of Y, 4 units of X and 2 units of Y, 5 units of X and 3 units of Y, and 6 units of X and 4 units of Y, as can be seen from the table.

As shown in Table Column (2), the consumer will need to spend varying sums of money in order to buy these various combinations of X and Y.

Keep in mind that the customer just needs to spend ₹40 to buy X and Y. He will only be able to spend ₹25 if he buys combo (i). He would have to pay ₹55 and ₹70, respectively, if he bought combinations (iii) and (iv), which is beyond his means. His only opportunity to spend ₹ 40 will be if he buys combo (ii). Therefore, when the customer purchases 4 units of X and 2 units of Y, spending a total of ₹40, he will be in balance. The customer will be in equilibrium when `(MU_x)/(P_x) = (MU_y)/P_y`, and they must also spend their full income on buying the two items. This is a generalization of the equilibrium condition.

Expenditure on Purchase of X and Y:

Combinations (1)	Total Expenditure (2)
(i) 3 units of X + 1 unit of Y	₹ 25 (3 × 5 + 1 × 10 = 15 + 10 = 25)
(ii) 4 units of X + 2 units of Y	₹ 40 (4 × 5 + 2 × 10 = 20 + 20 = 40
(iii) 5 units of X + 3 units of Y	₹ 55 (5 × 5 + 3 × 10 = 25 + 30 = 55)
(iv) 6 units of X + 4 units of Y	₹ 70 (6 × 5 + 4 × 10 = 30 + 40 = 70)

The sum of the marginal utilities from the purchases of each product can be used to get the total utility. The marginal utilities for the first four units of X are 50 + 45 + 40 + 35 = 170 utils, as may be seen by consulting the marginal utilities table. For the first two units of Y, the marginal utilities are 80 + 70 = 150 utils. The customer will receive 320 (170 + 150 = 320) utils of utility when they buy 4 units of X and 2 units of Y. When income is ₹40, no other combination of X and Y can provide as much benefit.