Law of Equimarginal Utility: Two-Good Consumer Equilibrium

You have Rs 60 to spend at a school fair. One more serving of your favourite snack would be lovely, but so would a drink. How do you decide where the next rupee should go?

Simply choosing the item you like more is not enough. Its price matters too. Something that gives twice as much extra satisfaction but costs four times as much may be the poorer use of a limited budget.

The law of equimarginal utility explains this choice. In the standard two-good model, a consumer maximises satisfaction by allocating the available budget so that marginal utility per rupee is equal across the two goods, with the budget fully spent. This equality describes an interior choice, where the consumer buys some of both goods and quantities can adjust suitably.

Let’s turn that sentence into a method you can actually use.

Start with the next unit, not all the units

Total utility (TU) is satisfaction from the entire quantity consumed. Marginal utility (MU) is the extra satisfaction from an additional unit.

For a one-unit increase:

MU of the nth unit = TU from n units - TU from n - 1 units

Suppose two servings give 54 utils and three servings give 74 utils. The third serving adds 20 utils. For a decision about that serving, use 20, not 74.

The numbers called utils are a classroom assumption for representing satisfaction. They are not readings from a real measuring instrument. The distinction between total and additional satisfaction is explained in NCERT’s Theory of Consumer Behaviour.

If schedules still feel confusing, first work through our guide to total utility, marginal utility and the saturation point.

The formula and the budget condition

For two commodities, X and Y:

MUx / Px = MUy / Py

Here MUx and MUy are the marginal utilities of X and Y, while Px and Py are their prices per unit. Each ratio measures additional satisfaction per rupee.

You also need:

Px × Qx + Py × Qy = M

Qx and Qy are the quantities bought, and M is the budget available for these goods. The equality assumes desirable additional consumption is possible and the model allocates this entire budget to X and Y.

With diminishing marginal utility and the usual independent-utility assumptions, moving spending away from the best allocation cannot increase total satisfaction. The equality is therefore used alongside the budget and these assumptions, not in isolation.

The price-adjusted comparison is the central rule in OpenStax’s lesson on consumption choices.

Why equal marginal utilities can be wrong

Imagine MUx = 30 and MUy = 20. Is X the better choice?

You cannot tell until you know the prices.

If Px = Rs 10 and Py = Rs 5:

• X provides 30 / 10 = 3 utils per rupee.
• Y provides 20 / 5 = 4 utils per rupee.

Y gives less extra satisfaction per unit, but more per rupee. In the standard divisible-goods model, shifting a small amount of expenditure towards Y improves satisfaction at this starting point.

Only when prices are equal does comparing MU directly give the same ranking as comparing MU divided by price.

The assumptions behind the law

The traditional cardinal explanation assumes that the consumer seeks maximum satisfaction, can represent utility numerically, faces given prices and income, and has stable tastes during the comparison. Marginal utility diminishes with additional consumption of each good. In the simple additive version, satisfaction from one good depends on its own quantity, so utilities from the two goods can be added.

The traditional treatment also assumes constant marginal utility of money. Textbooks may write the condition as:

MUx / Px = MUy / Py = MUm

MUm means the marginal utility of money. If it is supplied in a question, use it as the given benchmark. If the question instead supplies a fixed budget and utility schedules, find an affordable allocation rather than inventing a numerical MUm.

These assumptions belong to a simplified explanation of choice. Real purchases can involve habits, uncertainty and goods that work together. The INFLIBNET lesson on cardinal utility sets out the traditional assumptions and equimarginal rule.

A complete numerical: spend Rs 60 on two goods

Suppose X costs Rs 10 per unit and Y costs Rs 20 per unit. The consumer has Rs 60. The following numbers are an imaginary utility schedule for one person.

Unit consumed	MU of X	MUx / 10	MU of Y	MUy / 20
1st	30	3.0	60	3.0
2nd	24	2.4	48	2.4
3rd	20	2.0	36	1.8
4th	16	1.6	28	1.4
5th	12	1.2	20	1.0
6th	8	0.8	12	0.6

Step 1: Identify a possible equality

At two units of each good:

MUx / Px = 24 / 10 = 2.4

MUy / Py = 48 / 20 = 2.4

Both last purchased units provide the same marginal utility per rupee.

Step 2: Check whether the bundle fits the budget

Expenditure = (2 × 10) + (2 × 20) = Rs 60.

The bundle is affordable and uses the whole budget.

Notice that one unit of each good also gives equal ratios, both 3. But it costs only Rs 30. Since further affordable units give positive utility, stopping there would leave a better choice available.

This is why circling the first equality in a table is not a complete solution.

Step 3: Calculate total satisfaction

TU from two X = 30 + 24 = 54 utils.

TU from two Y = 60 + 48 = 108 utils.

Combined TU = 54 + 108 = 162 utils.

Do not add only the final marginal utilities, 24 and 48. That would ignore the satisfaction from the first units.

Step 4: Verify the whole-unit alternatives

Because the example uses whole units, we can check every combination that spends exactly Rs 60.

Units of X	Units of Y	Expenditure	Combined total utility
6	0	Rs 60	110
4	1	Rs 60	90 + 60 = 150
2	2	Rs 60	54 + 108 = 162
0	3	Rs 60	144

The largest total is 162 utils, so the best bundle is two X and two Y. Any affordable bundle that leaves money unused can be improved here by adding X, because costs are multiples of Rs 10 and the relevant marginal utilities are positive.

Step 5: Explain the result in words

A strong final sentence is:

“The consumer purchases two units of X and two units of Y, spending Rs 60. Marginal utility per rupee is 2.4 for each good, and this bundle gives the highest total utility of 162 utils among the affordable whole-unit combinations.”

That sentence connects the quantities, budget, condition and objective.

Why moving money restores equilibrium

Suppose, at a starting bundle:

MUx / Px > MUy / Py.

An additional rupee devoted to X offers more satisfaction than the rupee sacrificed from Y. The consumer therefore increases spending on X and reduces spending on Y.

As X consumption rises, its marginal utility falls. As Y consumption falls, its marginal utility rises. These adjustments narrow the gap until no beneficial small transfer remains, under the model’s assumptions.

If MUx / Px < MUy / Py, reverse the direction: spend more on Y and less on X.

Here is a concrete whole-unit check using our schedule. Start at four X and one Y, which costs Rs 60 and gives 150 utils. Give up the fourth and third X to release Rs 20:

Utility lost = 16 + 20 = 36 utils.

Use the Rs 20 to buy the second Y:

Utility gained = 48 utils.

Net gain = 48 - 36 = 12 utils.

The new bundle, two X and two Y, therefore gives 150 + 12 = 162 utils.

Trying another exchange in the same direction would give up the first two X, losing 54 utils, to obtain the third Y, gaining only 36. Satisfaction would fall by 18 utils.

Missing-value questions made simple

Find marginal utility

At an interior equilibrium, Px = Rs 8, Py = Rs 12 and MUy = 36 utils. Find MUx.

MUx / 8 = 36 / 12 = 3

Therefore MUx = 24 utils.

This finds the marginal utility required by the equality. Without quantities and income, it does not independently prove that a complete bundle exhausts the budget.

Find an unknown price

MUx = 40, MUy = 25 and Py = Rs 5. Find Px at equilibrium.

40 / Px = 25 / 5 = 5

Therefore Px = Rs 8.

A quick check: 40 / 8 and 25 / 5 both equal 5.

Use the marginal utility of money

Suppose MUm = 4 utils per rupee and Px = Rs 6. At the stated equilibrium:

MUx / 6 = 4

Therefore MUx = 24 utils.

The units help: utils per rupee multiplied by rupees per unit gives utils per unit.

When exact equality is impossible

A cinema ticket cannot usually be divided into a tenth of a ticket. Whole-unit choices can prevent the consumer from reaching exact equality between MU-to-price ratios.

Consider a separate example with a budget of Rs 10:

• X costs Rs 6; its first unit gives 15 utils.
• Y costs Rs 5; its first unit gives 12 utils and its second gives 11 utils.

The first X has the higher ratio: 15 / 6 = 2.5, compared with 12 / 5 = 2.4 for the first Y.

But buying X leaves Rs 4, which cannot buy either good. Total satisfaction is then 15 utils. Buying two Y uses Rs 10 and gives 23 utils.

The available bundles are no goods, one X, one Y, and two Y. Two Y is the best of them, even though a mechanical “take the highest ratio first” rule would start with X.

For indivisible goods, comparing all feasible bundles is the reliable final check. Exact ratio equality is not required at every discrete optimum. A choice containing only one good is called a corner solution; the usual two-good interior equality need not hold there either.

What happens if the price of X falls?

Suppose both ratios initially equal 4 utils per rupee. At the original bundle, MUx is 40 utils and Px is Rs 10.

If Px falls to Rs 8, the ratio at that unchanged quantity becomes:

40 / 8 = 5 utils per rupee.

X now offers more marginal satisfaction per rupee than Y at the old bundle. This creates an incentive to adjust consumption towards X. In the elementary independent-utility model, consuming more X lowers its MU and helps restore balance.

Be careful about what this calculation proves. It identifies the initial incentive. It does not tell you the exact new quantities without the utility information and the new budget calculation. Nor should you automatically claim that spending on X rises: a lower price and a larger quantity can have different effects on total expenditure.

Diminishing utility and equimarginal utility are connected

Question	Diminishing marginal utility	Equimarginal utility
What are we examining?	Successive units of one good	Allocation of a budget across goods
What is the key idea?	Additional satisfaction falls as quantity rises	Compare additional satisfaction per rupee
How do they connect?	Explains the changing return from further consumption	Uses those changing returns to explain spending adjustments

Think of watering two gardens from a limited reservoir. Another cup of water may help the drier garden much more than the already well-watered one. The sensible allocation changes as each garden receives more. You are looking for the best contribution from the remaining water, not necessarily an equal split between the gardens.

Likewise, consumer equilibrium does not require equal quantities, equal expenditure or equal total utilities.

A short answer you can build on

If asked to explain equilibrium with two commodities, organise your response in this order:

1. State that the consumer seeks maximum total satisfaction within a given budget.
2. Write MUx / Px = MUy / Py for an interior equilibrium.
3. Include the budget condition PxQx + PyQy = M.
4. Mention diminishing marginal utility and the relevant assumptions.
5. Explain the direction of expenditure adjustment if the ratios differ.

If a numerical accompanies the question, show the ratio columns, expenditure and chosen quantities. For a whole-unit schedule, use total utility to verify the feasible alternatives.

The most avoidable errors are dividing by the wrong price, using TU in place of MU, ignoring leftover income, and saying “buy more X because MUx is higher” without checking prices.

Practise before checking the answers

All figures below are hypothetical. Use the standard interior-equilibrium assumptions unless the question says otherwise.

1. MUx = 36, Px = Rs 6, MUy = 32 and Py = Rs 8. In which direction should expenditure shift?
2. MUx = 18, Px = Rs 3 and Py = Rs 5. Find MUy at equilibrium.
3. MUx = 42, MUy = 28 and Py = Rs 4. Find Px at equilibrium.
4. A consumer buys three X at Rs 10 each and two Y at Rs 15 each. The budget is Rs 90 and the two MU-to-price ratios are equal. Is this enough to establish equilibrium if additional consumption remains desirable and affordable?
5. In the Rs 60 schedule above, what is the gain from moving from six X to four X and one Y?
6. In the indivisible-goods example, why does buying the first X fail to maximise total utility?
7. If two goods have equal prices and equal marginal utilities, must their total utilities also be equal?
8. MUm is 3 utils per rupee and Y costs Rs 9. What MUy satisfies the textbook condition?

Answers and reasoning

1. Shift expenditure towards X and away from Y. The ratios are 6 and 4 utils per rupee respectively.
2. MUy / 5 = 18 / 3 = 6, so MUy = 30 utils.
3. 42 / Px = 28 / 4 = 7, so Px = Rs 6.
4. No. The bundle costs Rs 60, leaving Rs 30. Equality alone is insufficient when affordable additional consumption would increase satisfaction.
5. Giving up the fifth and sixth X loses 12 + 8 = 20 utils. Buying the first Y gains 60. The net gain is 40 utils, taking total utility from 110 to 150.
6. It leaves Rs 4 unusable in that exercise and gives 15 utils. Two Y give 23 utils, which is the larger affordable total.
7. No. Equal marginal utilities concern the final units. Earlier units may have contributed different amounts of satisfaction.
8. MUy / 9 = 3, so MUy = 27 utils.

Frequently asked questions

What is the law of equimarginal utility in simple words?

It explains how to distribute a limited budget so that no small transfer of spending between goods can improve satisfaction. At the usual interior equilibrium, the final rupee devoted to each good provides equal marginal utility.

Is equimarginal utility the same as equal marginal utility?

Not when prices differ. The relevant equality is MUx / Px = MUy / Py. Marginal utilities themselves are equal only if the prices are also equal at that interior equilibrium.

What are the two main conditions for consumer equilibrium?

The standard two-good answer combines equal marginal utility per rupee with full use of the given budget. It also relies on assumptions such as diminishing marginal utility and a suitable interior choice.

Does the consumer spend equal amounts on both goods?

No. In our main example, Rs 20 is spent on X and Rs 40 on Y. Equal marginal returns per rupee can occur with unequal expenditure.

Why is diminishing marginal utility important here?

It explains why continuing to direct expenditure towards the initially more attractive good reduces its extra benefit. This helps explain why a balanced allocation can emerge rather than spending everything on that good.

What if MUx divided by Px is greater than MUy divided by Py?

Under the standard assumptions, the consumer can improve satisfaction by shifting a small amount of expenditure from Y to X. For indivisible goods, compare affordable whole-unit exchanges before choosing the final bundle.

Can there be equilibrium without exact equality of the ratios?

Yes. Indivisible goods and corner solutions can prevent the interior equality from applying. The essential objective remains the highest total satisfaction among affordable choices.

Is consumer equilibrium the same as the saturation point?

No. Saturation concerns a point where additional consumption of a good adds no utility. A consumer with a limited budget can reach equilibrium while the marginal utility of each purchased good is still positive.

Can I compare one person’s utils with another person’s utils?

Not as an objective measure of who is happier. Utility numbers in these exercises represent one person’s assumed satisfaction. They are useful for comparing that person’s possible bundles.

What should I check last in a numerical?

Recalculate expenditure, check that you used marginal rather than total utility in each price ratio, and state the chosen quantities. If goods come in whole units, verify the total satisfaction of the feasible alternatives.