Law of EquiMarginal Utility

A rational consumer will maximize his utility subject to their budget constraint or income.
A consumer will be at equilibrium when he allocates his given income in the purchase of different goods in such a way that he maximizes his utility.
Law of Equi-Marginal Utility solves the above problem of consumers.

 The law says that-
“The Consumer maximizing his total utility will allocate his income among various commodities in such a way that the marginal utility of the last unit of money (rupee) spent on each commodity is equal.”

Now, to understand what is - marginal utility of the last unit of money is (rupee) spent on each commodity

Marginal utility of a rupee we spend on a good say ‘X’ is calculated by
( MUx / Px )

Where,  MUx is marginal utility of the good ‘X’ and Px is the price

Similarly, Marginal utility of a rupee we spend on a good say ‘Y’ is
calculated by ( MUy / Py )

Now we will understand why a consumer maximizes his total utility when the marginal utility of the last rupee spent on each good is equal.

If a consumer is getting more utility form a rupee spend on commodity X than with Y, he will switch one rupee from Y to get commodity X. Thus total utility of his will rise.

The utility maximizing consumer will continue to switch his expenditure from Y to X as long as one rupee spent on X brings him more utility than Y.

This will leads to more quantity of X and less of  Y. But remember the law of diminishing marginal utility (with increase in each successive units, the utility derived from the additional unit goes on decreasing), marginal utility derived from X will start falling.

This process of reallocation of expenditure will ultimately lead to equalization of marginal utility of last rupee spent on each of the two commodities.
This is the satiation point for the consumer, he will not gain by further reallocation of expenditure from commodity Y to commodity X.

The consumer will spend his money income on different goods in such a way that marginal utility of each good is proportional to its price.

(MUx / Px)  = (MUx / Py)  = MU per unit of money

Let us illustrate the law of equi-marginal utility by taking a numerical example:

Calculate which combination of goods A and B provide maximum utility to the consumer?

Units	MUx	MUy	MUx/Px	MUy/Py
1	10	30	5	7.5
2	8	24	4	6
3	6	20	3	5
4	4	16	2	4
5	2	14	1	3.5
6	1	8	0.5	2

Price of X= Rs.2

Price of Y=Rs.4

Income= Rs.20

In the above numerical we are given with MUx and MUy for various units, using the formula MUx/Px and MUy/Py (given the prices of both) , we calculated the marginal utility of the last unit of money spent on both goods ‘X’ and ‘Y’.

Now after calculating the Marginal utilities for various units, 
it is clear that the proportionality rule
i.e. MUx/Px = MUy/Py , can be fulfilled at 3 points, as shown below

Combinations	Total Expenditure
1) 1 unit of X and 3   units of Y 2)  2 unit of X and 4 units of Y 3) 4unit of X and 6 units of Y	1x2+3x4= Rs. 14 2x2+4x4= Rs. 20 4x2+6x4= Rs. 32

The question is which is a utility maximizing combination.

Remember the proportionality rule i.e. MUx/Px = MUy/Py should also equal be to MU per unit of money.

Here the consumer income is Rs. 20, so he will not choose combination (1), as there expenditure is Rs. 14, he will be left with money, more ever not fully satisfied.
He can’t choose combination (2) as that is beyond his budget.

When 2 unit of X and 4 units of Y are purchased, the consumer will be able to incur an expenditure of Rs. 20.This is his equilibrium combination.

No other combination of X and Y can give as much utility when income is Rs. 20.

This explains the Law of Equi-Marginal Utility.