Consumer theory · Lesson 5 of 17

Utility maximisation: the interior optimum

Now both pieces are on the table: tastes (indifference curves) and limits (the budget line). The consumer picks the best bundle she can afford. This is the most-examined calculation in the subject, and there is one recipe: find where the indifference curve just touches the budget line.

← Consumer theory moduleLesson 5 of 17 · about 45 minutes

A question to hold on to

For the utility function u(x, y) = xy and the budget constraint M = p_xx + p_yy, what is the optimum bundle of consumption of x and y? (M/p_x, M/p_y); (M/2p_x, M/2p_y); (M/p_y, M/p_x); or (M/2p_y, M/2p_x).

CUET-PG 2023, shift 1Solved at the end ↓

The problem

The best affordable bundle

She chooses the bundle she likes best among those she can afford. With more-is-better, the best bundle uses all her income, so it lies on the budget line (any bundle below it has a better one nearby). So the problem is: reach the highest indifference curve that still touches the budget line.

Written out: maximise u(x, y) subject to p_xx + p_yy = m. The answer, as a function of prices and income, is her demand (called Marshallian or uncompensated demand): x*(p_x, p_y, m) and y*(p_x, p_y, m).

The condition

The tangency condition

Suppose she is on the budget line and an indifference curve cuts the line there. Then the curve’s slope differs from the line’s, and there are bundles on the line that lie above the curve: better bundles she can still afford. So she can only be at her best where the curve touches the line and does not cross it: the slopes are equal.

The tangency condition

M R S = \frac{M U _{x}}{M U _{y}} = \frac{p _{x}}{p _{y}}

The rate at which she is willing to swap equals the rate at which the market lets her swap. Holds at an interior optimum with smooth, convex tastes.

Read it as a rule of trade. If her MRS is 3, she would give up 3 units of y for one more x. If the market asks only 2 units of y for an x (p_x/p_y = 2), she gains by buying x. She keeps buying x, which lowers her MRS, until the two are equal. Solve this together with the budget equation to get both quantities.

Interactive

Try it: slide along the line

Choose one of the exam utility functions. Slide the bundle along the budget line. The blue curve through the dot cuts the line almost everywhere; the dashed curve through the hollow dot is the highest one she can reach. Watch the three equalities at the end: MRS against the price ratio, marginal utility per rupee, and the multiplier λ.

u = x^0.5 y^0.5. Slide the bundle along the budget line. Its curve (blue) cuts the line until she is at the optimum, where the curve (dashed) only touches it.

Where on the line(25, 75/2)Weight on x, a0.5Income, Rs100Price of x1Price of y2

Curve through ●○ the optimum (curve dashed)

Utility and slopes

Utility 30.619 (the best affordable is 35.355). Her MRS is 3/2; the price ratio p_x/p_y is 1/2.

Not the optimum

Her MRS is above the price ratio: she would give up more y for an extra x than the market asks. Move right and buy more x.

Marginal utility per rupee

On x: 0.6124. On y: 0.2041. Unequal, so a rupee moved from the lower to the higher would raise utility.

Shares of income

She spends 0.25 of income on x and 0.75 on y. At the optimum of a Cobb–Douglas consumer with weight a, the share on x is a = 0.5 whatever the prices.

The method

The Lagrangian, and what λ means

The same condition comes out of the Lagrangian, which handles any number of goods:

The Lagrangian

L = u (x, y) + λ (m - p_{x} x - p_{y} y)

λ multiplies the amount of income left unspent. The first-order conditions are three equations in x, y and λ.

\frac{\partial L}{\partial x} = M U_{x} - λ p_{x} = 0, \frac{\partial L}{\partial y} = M U_{y} - λ p_{y} = 0, m = p_{x} x + p_{y} y

Dividing the first two gives the tangency condition. Each also says λ = MU_x/p_x = MU_y/p_y, the equal marginal utility per rupee of Lesson 2. So λ is the marginal utility of income: the extra utility from one more rupee. For u = x₁x₂ maximised on x₁ + x₂ = 6, the best bundle is (3, 3), and λ = MU_x1/p₁ = x₂/1 = 3 (Hyderabad 2020). A unit more of income would raise the maximum of u by about 3.

The workhorse

Cobb–Douglas demand

For u = x^ay^1−a, the tangency condition and the budget give a result worth memorising:

Cobb–Douglas demand

x^{*} = \frac{a m}{p _{x}}, y^{*} = \frac{( 1 - a ) m}{p _{y}}

She spends the fixed fraction a of her income on x and 1 − a on y, whatever the prices.

The exponents, normalised to sum to one, are the budget shares. For u = xy they are ½ and ½: x* = m/2p_x and y* = m/2p_y (CUET 2023, JAM 2025, Hyderabad 2012, 2013, 2020). For u = x^0.25y^0.75 with prices 2 and 5 and income 100, she spends 25 on x and 75 on y: x* = 12.5 and y* = 15 (Hyderabad 2018). For u = xy² the shares are ⅓ and ⅔, so with prices 2 and 3 and income 9 she buys x = 1.5 and y = 2 (JNU 2011, 2015). The demand for y when U = a√(XY), M = 656 and P_x = 8 is 656/2 ÷ P_y = 328/P_y (Hyderabad 2010).

The converse is how examiners test it: someone who spends fixed percentages of income on each good whatever the prices, such as 30% on housing and 25% on food, behaves as a Cobb–Douglas consumer (Hyderabad 2015). Only Cobb–Douglas has budget shares that stay constant at every price and income.

Pause and predict

u = x^0.4 y^0.6, income 100, p_x = 2, p_y = 5. What does she spend on y?

Beyond Cobb–Douglas

Other smooth tastes, step by step

The same two steps work for any smooth, convex utility: set MRS = p_x/p_y, then put the answer in the budget.

Shifted Cobb–Douglas, u = (x + 2)(y + 1), prices 4 and 6, budget 130. MRS = (y + 1)/(x + 2) = 4/6, so y + 1 = (2/3)(x + 2), that is y = (2x + 1)/3. The budget 4x + 6y = 130 becomes 4x + 2(2x + 1) = 130, so 8x = 128, x = 16, y = 11 (UGC NET 2025).
Sum of roots, u = √x + √y, prices 5 and 1, income 60. MRS = √y/√x = 5, so y = 25x. The budget 5x + y = 60 gives 30x = 60, x = 2 and y = 50 (JNU 2013).
Equal shares in a different dress, u = √(X₁X₂), prices 10 and 4, income 200. The exponents are equal, so half of income, Rs 100, goes on each good: 10 samosas and 25 jalebis. The question asks only for the proportions: 50% and 50% (JNU 2020).

The statements

What is true at the optimum

UGC NET 2025 asked which of five statements describe the outcome of a consumer’s optimisation. Four are true and one is false:

Statement	True?
The marginal utility per rupee is the same on every good	Yes
The MRS equals the price ratio	Yes
She reaches the highest indifference curve she can	Yes
The indifference curve is tangent to the budget line	Yes
She is indifferent between any two points on the budget line	No: only the optimum is on the highest curve; the others are on lower ones

When the recipe fails

Tangency is necessary only for a smooth, convex, interior optimum. With kinks (perfect complements), straight lines (perfect substitutes) or curves that bow away from the origin, the best bundle can sit at a corner of the budget line. That is Lesson 6.

Watch for

Traps

Watch out for

The mistakes examiners are fishing for

Putting the exponent on the wrong good. The share on x is the exponent of x. For xy² it is ⅓, not ⅔.
Mixing up spending and quantity. a·m is how many rupees go on x; divide by p_x to get units.
Tangency without checking the corner. Solve, then confirm both quantities are non-negative; with shifted functions like (x+2)(y+1) a high price of x can push x below zero.
Calling the multiplier a price. λ is the marginal utility of income, in utils per rupee. It changes if you change the utility function’s labelling, though the optimum does not.
Forgetting the budget. The tangency gives the ratio of y to x; the budget fixes the scale.

Worked questions

The opening question, and four more

CUET 2023 shift 1 · standard

u = xy and M = p_xx + p_yy. Find the optimum bundle.

Marginal utilities.
MU_x = y and MU_y = x, so MRS = y/x.
Tangency.
y/x = p_x/p_y, so p_yy = p_xx: she spends equal amounts on the two goods.
Budget.
Each takes half: p_xx = M/2, so x = M/2p_x and y = M/2p_y.

Answer

(M/2p_x, M/2p_y).

UGC NET 2025 · advanced

U = (x + 2)(y + 1), P_x = 4, P_y = 6 and budget 130. Find the optimum.

Tangency.
(y + 1)/(x + 2) = 4/6, so 6(y + 1) = 4(x + 2), giving y = (2x + 1)/3.
Budget.
4x + 6(2x + 1)/3 = 130 gives 4x + 4x + 2 = 130, so x = 16.
Back-substitute.
y = (32 + 1)/3 = 11.

Answer

x* = 16, y* = 11.

Hyderabad 2017 · standard

U = x₁^0.5x₂^0.5, p₁ = 2, p₂ = 5 and income 100. Find the demands.

Shares.
Half of income on each: Rs 50 on x₁, Rs 50 on x₂.
Quantities.
x₁ = 50 ÷ 2 = 25 and x₂ = 50 ÷ 5 = 10.

Answer

x₂ = 10 and x₁ = 25.

UGC NET 2023 · standard

U = q₁q₂, p₁ = 1, p₂ = 2 and she spends k. What are her demands?

Equal shares.
Half of k on each good: k/2 on q₁, k/2 on q₂.
Divide by prices.
q₁ = (k/2)/1 = k/2 and q₂ = (k/2)/2 = k/4.

Answer

(q₁, q₂) = (k/2, k/4).

Hyderabad 2020 · standard

U = x₁x₂ is maximised subject to x₁ + x₂ = 6 using the Lagrangian. What is λ at the optimum?

Optimum.
Equal prices and equal exponents: x₁ = x₂ = 3.
First-order condition.
MU_x1 = x₂ = λ × 1, so λ = 3.

Answer

λ = 3.

Revise

The lesson on one screen

The problem: max u(x, y) subject to p_x x + p_y y = m. With more-is-better she is on the line.
Tangency: MRS = MU_x/MU_y = p_x/p_y: the curve touches the line.
Lagrangian: MU_x = λp_x, MU_y = λp_y, budget. λ = marginal utility of income.
Cobb–Douglas: u = x^a y^(1−a): x* = am/p_x, y* = (1−a)m/p_y. Constant budget shares.
u = xy: x* = m/2p_x, y* = m/2p_y.
Shifted: (x + 2)(y + 1): MRS = (y+1)/(x+2). Solve tangency then the budget; check the answer is non-negative.
Statements: At the optimum MU/p is equal, MRS = price ratio, highest curve, tangent. Not: indifferent along the line.
Recipe fails: For kinks, straight lines and non-convex curves: corners (Lesson 6).

Practice

Check yourself

0 of 10 correct so far

Quick questions

1At an interior optimum with smooth convex tastes,

2A consumer has u = xy. If her income doubles and prices do not change, her demand for x

3For u = x^a y^b, the share of income spent on x is

4In the Lagrangian approach, λ at the optimum is the

5Which statement is NOT true at the optimum of a consumer with convex, monotone tastes?

Numerical problems

Cobb–Douglas demand

A consumer has u = x^0.4y^0.6, income Rs 120, p_x = Rs 4 and p_y = Rs 6. How many units of x does she buy?

Units of x

Shifted Cobb–Douglas

A consumer has u = (x + 2)(y + 1), income Rs 19, p_x = Rs 2 and p_y = Rs 1. Find the optimum.

x*y*

The marginal utility of income

A consumer has u = xy, income Rs 80, p_x = Rs 2 and p_y = Rs 4. What is the marginal utility of income, λ, at her optimum?

A demand quantity

A consumer has u = x^0.25y^0.75, income Rs 200 and p_y = Rs 5. How many units of y does she buy?

Units of y

Real questions

Past papers, by level

Foundation

Definitions and one-step calculations: the shape of a curve, a budget line, an optimum for Cobb–Douglas tastes. Most CUET, UGC NET, Hyderabad and early JNU questions sit here.

6 questionsPractise →

Standard

Statements to judge, matching lists, and demands to compute with unusual tastes, up to JAM, GATE and JNU level.

12 questionsPractise →

Delhi School level

This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 17: Min, max and kinked utilities, and deals with three goods.

10 questionsLesson 17 →

All 18 past-paper questions for this lesson, easiest first: practise the full set →

These are questions from the TestStop bank. A free account answers 10 a day, each with its worked solution; All-Access opens every set with no daily limit.

← Lesson 4The budget set and the budget line Lesson 6 →Corner solutions and demand for special utilities

Utility maximisation: the interior optimum

The best affordable bundle

The tangency condition

Try it: slide along the line

The Lagrangian, and what λ means

Cobb–Douglas demand

Other smooth tastes, step by step

What is true at the optimum

Traps

The mistakes examiners are fishing for

The opening question, and four more

The lesson on one screen

Check yourself

Quick questions

Numerical problems

Past papers, by level

Foundation

Standard

Delhi School level

Read next

Utility maximisation: the interior optimum

The best affordable bundle

The tangency condition

Try it: slide along the line

The Lagrangian, and what λ means

Cobb–Douglas demand

Other smooth tastes, step by step

What is true at the optimum

Traps

The mistakes examiners are fishing for

The opening question, and four more

The lesson on one screen

Check yourself

Quick questions

Numerical problems

Past papers, by level

Foundation

Standard

Delhi School level

Read next