Consumer theory · Lesson 2 of 17

Utility, the MRS and marginal utility

A ranking is hard to calculate with. A utility function turns it into a number, and a number can be differentiated. This lesson says exactly what that number does and does not mean, and builds the one slope the whole subject runs on: the marginal rate of substitution.

← Consumer theory moduleLesson 2 of 17 · about 40 minutes

A question to hold on to

Match each utility function U(x₁, x₂) with its marginal rate of substitution. Utility functions: (A) C x₁^a x₂^b; (B) (x₁ + 2)(x₂ + 1); (C) a x₁ + b√x₂; (D) (x₁^a + x₂^a)^b. MRS: (I) −(x₂ + 1)/(x₁ + 2); (II) −(x₁/x₂)^a−1; (III) −a x₂/(b x₁); (IV) −2a√x₂/b.

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

The idea

From a ranking to a number

A utility function u(x, y) represents her preferences if it gives a bigger number to the bundle she likes better: u(A) > u(B) exactly when A ≻ B, and u(A) = u(B) exactly when A ∼ B.

It is a scoring scheme and nothing more. If two bundles score 12 and 6, the only fact is that the first is better. It does not mean she likes it twice as much. Every ranking from Lesson 1 that is complete, transitive and continuous can be scored this way. (Lexicographic preferences cannot: Lesson 3.)

Ordinal, not cardinal

Which changes keep the same preferences?

Many different functions score the same ranking. If u represents her preferences and f is any increasing function, then f(u) ranks the bundles in the same order, so it represents the same preferences. Doubling, squaring (for positive u), taking a log (for positive u) and adding a constant are all examples. Such a change is called a monotonic transformation.

Modern theory is therefore ordinal: only the order of the scores means anything. An older cardinal view treats utility as measurable, so that differences and ratios of utility have meaning. The ordinal view asks less, and it is enough for everything in demand theory.

The test for whether a function is a valid replacement is whether it preserves the order of every pair of bundles. Not every function of x and y does:

If u = xy represents her, then	Same preferences?	Why
x²y², or xy + (xy)³, or √(xy)	Yes	An increasing function of xy (JAM 2023: u + u³ and u^(1/n))
−xy	No	Decreasing: it reverses every ranking
xy + x + y	No	Adds a term that depends on the bundle, not just on u, so two bundles with equal xy can now differ (JAM 2023)
√(x + y) against 2x + 3y	Yes, both perfect substitutes	√(x+y) is an increasing function of x + y, and 2x + 3y has the same straight-line curves (JNU 2019)
min{x, y} + max{x, y}	Equals x + y	The smaller plus the larger is the sum (JNU 2012)

Pause and predict

She has u = xy. Which of these represents the same preferences?

What survives a transformation

The indifference curves, the MRS, the ranking of every pair, and the best affordable bundle all stay the same. The utility numbers, the marginal utilities and the size of utility gaps all change. So any claim that depends on those cannot be a claim about her tastes.

Interactive

Try it: what survives a transformation

Her tastes are u = xy. Choose a transformation and move bundle A. The curves do not move at all. The right-hand column of the table is the label she would read off the new function; the last column says whether it matches the old one. Watch the marginal utility of x: it can rise, fall or stay flat as you change the labelling, while the MRS stays put.

Her tastes are u = xy. Pick a transformation and move bundle A. The curves are drawn once, because they do not depend on how you label them; watch which rows of the table change.

Bundle A: x2Bundle A: y4

Curve through ACurve through B = (4, 2.5)

What you read off	u = xy	u²	Same?
Utility of A	8	64	no
Utility of B	10	100	no
Which is preferred	B	B	yes
MRS at A	2	2	yes
Marginal utility of x at A	4	64	no
Utility gap, A minus B	−2	−36	no

Does marginal utility fall as x rises?

With u = xy, the marginal utility of x stays the same when x goes up by one. With u² it rises. Diminishing marginal utility is a property of the labelling, not of her tastes.

The key slope

The marginal rate of substitution

Move a little along an indifference curve, changing x by dx and y by dy. Utility does not change, so the gain from x and the loss from y cancel. The marginal utility of a good is the extra utility from one more unit of it, written MU_x = ∂u/∂x, so

M U_{x} d x + M U_{y} d y = 0 ⟹ - \frac{d y}{d x}_{u fixed} = \frac{M U _{x}}{M U _{y}}

Marginal rate of substitution

M R S_{x y} = \frac{M U _{x}}{M U _{y}} = - \frac{d y}{d x}_{u fixed}

How many units of y she will give up for one more unit of x and stay on the same curve. It is minus the slope of the indifference curve.

It is unchanged by any monotonic transformation, because f′(u) multiplies both MU_x and MU_y and cancels. That is why the MRS, unlike marginal utility, is a fact about tastes. Some standard values:

Utility function	MRS = MU_x/MU_y
$x^{a} y^{b}$	$\frac{a y}{b x}$
$(x + 2) (y + 1)$	$\frac{y + 1}{x + 2}$
$a x + b y$ (perfect substitutes; CUET 2024 asks 2x + 3y)	$\frac{a}{b}$ , the same everywhere
$a x + b y$	$\frac{2 a y}{b}$
$x + y$ (quasi-linear)	$\frac{1}{2 x}$ , depends on x only
$(x^{a} + y^{a})^{b}$ (CES)	$(\frac{x}{y})^{a - 1}$

Read the wording carefully. “An MRS of 2 eggs for 1 sandwich” (Hyderabad 2010) says she will give up 2 eggs for 1 sandwich. So sandwiches are worth twice as much at the margin: MU_sandwich/MU_egg = 2, and the ratio the question wants, MU_egg/MU_sandwich, is ½. At any point on an indifference curve, the absolute value of its slope is the MRS (Hyderabad 2018).

Curvature

Diminishing MRS and convexity

Ordinary tastes show a diminishing MRS: the more x she holds relative to y, the less y she will give up for another unit of x. As she moves down a curve, the MRS falls, so the curve gets flatter. A curve that gets flatter as x rises is convex to the origin (bowed in towards it). Convexity is the formal name for a taste for variety: she prefers a balanced mixture of two equally good bundles to either extreme (Hyderabad 2012: she gives up successively less of one good for each extra unit of the other).

Convexity can fail. “I would rather spend all my time in the country or all in the town than divide myself between the two” prefers the extremes, which is the opposite of convex (Hyderabad 2010). A utility function such as √(x² + y²) has curves that are quarter-circles around the origin: the midpoint of two equally good bundles is worse, and the curves are concave to the origin (Hyderabad 2021).

Pause and predict

Along a typical indifference curve, as x rises, the MRS

Diminishing MRS is not diminishing marginal utility

The two are different. The JNU 2019 table lists bundles worth 1, 4 and 16 utils. They fit u = x²y², so the marginal utility of x, 2xy², rises with x. Yet the MRS, y/x, falls as x rises along a curve, and the curves are convex. Convexity needs a diminishing MRS only.

The cardinal rule

Marginal utility per rupee

The older, cardinal approach asks the same question with money. The extra utility from a rupee spent on x is MU_x/P_x. If it is larger than MU_y/P_y, a rupee moved from y to x raises her utility. She stops moving rupees when the two are equal. That is the equi-marginal principle:

Equi-marginal principle

\frac{M U _{x}}{P _{x}} = \frac{M U _{y}}{P _{y}} = M U_{money}

At her best bundle every good gives the same extra utility per rupee. The common value is the marginal utility of money.

Rearranged, it says MU_x/MU_y = P_x/P_y: the MRS equals the price ratio. So the cardinal rule and the indifference-curve rule are the same condition (Lesson 5 draws it). Questions use it as a ratio: if MU_x = 50, MU_y = 100 and P_y = Rs 20, then 100/20 = 5 per rupee, so P_x = 50/5 = Rs 10 (CUET 2019).

Watch for

Traps

Watch out for

The mistakes examiners are fishing for

Treating utility numbers as amounts. “She gets twice the utility” has no meaning. Only the order of bundles does.
Calling any function of x and y a valid replacement. It must be an increasing function of u. Adding a term in x or y changes the ranking.
Putting the goods the wrong way round in the MRS. MRS_xy is MU_x/MU_y: the y given up per x gained. Read “2 eggs for 1 sandwich” from the first-named side.
Confusing diminishing MRS with diminishing marginal utility. They are different, and an exam table can have one without the other (JNU 2019).
Forgetting that a curve can be concave. If utility is a sum of squares, √(x²+y²), curves bow away from the origin and the “optimum” at a tangency is a worst point.

Worked questions

The opening question, and four more

CUET 2023 shift 1 · advanced

Match (A) Cx₁^ax₂^b; (B) (x₁+2)(x₂+1); (C) ax₁ + b√x₂; (D) (x₁^a + x₂^a)^b with their MRS.

(A).
MU₁ = a·Cx₁^(a−1)x₂^b and MU₂ = b·Cx₁^a x₂^(b−1). Divide: a x₂ / (b x₁). That is (III).
(B).
MU₁ = x₂ + 1 and MU₂ = x₁ + 2. Divide: (x₂ + 1)/(x₁ + 2). That is (I).
(C).
MU₁ = a and MU₂ = b/(2√x₂). Divide: 2a√x₂/b. That is (IV).
(D).
Using the chain rule, MU₁ = b(…)^(b−1) a x₁^(a−1) and MU₂ = b(…)^(b−1) a x₂^(a−1). The common factor cancels, leaving (x₁/x₂)^(a−1). That is (II).

Answer

(A)–(III), (B)–(I), (C)–(IV), (D)–(II) (option C). The signs in the list are the minus sign of the slope.

JNU 2019 · advanced

A table shows bundles (X, Y) giving 1 util: (1, 1), (2, ½), (4, ¼); 4 utils: (1, 2), (2, 1), (4, ½); 16 utils: (1, 4), (2, 2), (4, 1). Which statement is false: (A) the marginal utility of X diminishes as X rises, holding Y; (B) the MRS of X for Y diminishes as X rises; (C) the curves are convex; (D) more X raises the marginal utility of Y?

Find the formula.
On the 1-util curve XY = 1. On the 4-util curve XY = 2, on the 16-util curve XY = 4. So utils = (XY)² = X²Y².
(A).
MU_X = 2XY², which rises with X for fixed Y. So (A) is false.
(B), (C), (D).
MRS = MU_X/MU_Y = Y/X falls as X rises, so the curves are convex; and MU_Y = 2X²Y rises with X. All three are true.

Answer

(A) is false.

JAM 2023 · standard

u represents a complete, transitive, continuous preference on ℝⁿ₊. Which also represent it: f = u + u³; g = u + Σxᵢ; h = u^(1/n); m = u + (Σxᵢ²)^0.5?

Test for an increasing function of u.
f is u + u³, increasing for u ≥ 0. h is u^(1/n), increasing too. Both are fine.
Test the others.
g and m add a term that depends on the bundle itself. Take u = x₁x₂ and the equally good bundles (1, 4) and (2, 2): g gives 4 + 5 = 9 and 4 + 4 = 8, so they are no longer tied. m differs too.

Answer

f and h only (options A and C).

Hyderabad 2014 · foundation

The prices of ice cream and chocolate are Rs 40 and Rs 30, and the marginal utility of chocolate is 150. At equilibrium, what is the marginal utility of ice cream?

Equal per rupee.
MU_c/P_c = 150/30 = 5 per rupee.
Scale to ice cream.
MU_i = 5 × 40 = 200.

Answer

200.

JNU 2019 · standard

Sally has 5 hours for two projects and plans 3 hours on the first and 2 on the second. The last minute on the first adds 3 points; the last minute on the second adds 20. What should she do?

Compare marginal returns.
A minute on project 2 is worth 20 points, a minute on project 1 only 3. The marginal utility per minute is unequal.
Move time to the better use.
Take a minute from project 1 (lose 3) and give it to project 2 (gain 20): net +17. Repeat until the two marginal values are equal.

Answer

Decrease the time on project 1: her plan does not allocate time efficiently.

Revise

The lesson on one screen

Utility function: A score that ranks bundles in her order. Only the order is meaningful (ordinal).
Monotonic transformation: Any increasing f(u) represents the same preferences. Decreasing ones reverse them; adding a term in x or y changes them.
What survives: Indifference curves, the MRS, every ranking and the optimum. Utility levels, marginal utility and utility gaps do not.
MRS: MU_x/MU_y = −dy/dx along the curve: the y she gives up for one more x.
Common MRS: x^a y^b: ay/bx. ax + by: a/b. Quasi-linear √x + y: 1/(2√x). (x+2)(y+1): (y+1)/(x+2).
Diminishing MRS: Curves convex to the origin: a taste for variety. It does not need diminishing marginal utility.
Equi-marginal rule: MU_x/P_x = MU_y/P_y = MU of money at her best bundle. The same as MRS = P_x/P_y.

Practice

Check yourself

0 of 8 correct so far

Quick questions

1If u(x, y) represents a consumer's preferences, which of these does NOT represent the same preferences?

2The absolute value of the slope of an indifference curve equals

3Her MRS falls as she moves down an indifference curve. Her curves are

4At a consumer's best bundle, MU_x/P_x = MU_y/P_y. This common value is the

5Diminishing marginal utility of x holding y fixed is

Numerical problems

MRS of a Cobb–Douglas function

A consumer has u = x²y³. What is her MRS (MU_x/MU_y) at the bundle (x, y) = (2, 6)?

MRS at (2, 6)

The equi-marginal rule

MU_x = 60, MU_y = 90 and P_y = Rs 15. For the consumer to be in equilibrium, what must P_x be (in rupees)?

Price of x

A quasi-linear MRS

A consumer has u = √x + y. What is her MRS at x = 4?

MRS at x = 4

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.

7 questionsPractise →

Standard

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

15 questionsPractise →

Delhi School level

This topic’s hardest questions, from the DSE entrance, are worked through in Lesson 17: Orderings, representation and revealed choice.

13 questionsLesson 17 →

All 22 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 1Preferences and indifference curves Lesson 3 →Substitutes, complements, lexicographic, homothetic and quasi-linear preferences

Utility, the MRS and marginal utility

From a ranking to a number

Which changes keep the same preferences?

Try it: what survives a transformation

The marginal rate of substitution

Diminishing MRS and convexity

Marginal utility per rupee

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, the MRS and marginal utility

From a ranking to a number

Which changes keep the same preferences?

Try it: what survives a transformation

The marginal rate of substitution

Diminishing MRS and convexity

Marginal utility per rupee

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