Lecture 3 · Week 4 · Besanko & Braeutigam ch. 3 · about 45 min

Consumer Preferences and the Concept of Utility

How economists represent what consumers want: the axioms behind rational preferences, indifference curves and their properties, the marginal rate of substitution, and utility functions as a bookkeeping device for preference rankings.

By the end you can

State the three basic assumptions about consumer preferences
Read an indifference curve map and explain why curves cannot cross
Compute the marginal rate of substitution from a utility function
Recognise perfect substitutes, perfect complements, Cobb-Douglas and quasilinear preferences from their indifference curves
Find an indifference curve from a utility function and compute marginal utility

Key idea · Exam recipe: preferences, utility and MRS (memorise this)

Three preference axioms: Completeness (can rank any two bundles), Transitivity ( $A \succ B \text{ and } B \succ C \implies A \succ C$ ), and More is better (non-satiation: $MU \gt 0$ , indifference curves cannot cross).
Indifference curves: Level curves where utility is constant: $U(x, y) = \bar{U}$ . Solve for $y$ to get curve equation $y = f(x)$ . Slope is $-MRS_{xy}$ .
Marginal utility: Partial derivatives $MU_x = \partial U/\partial x$ and $MU_y = \partial U/\partial y$ . Positive under more-is-better; diminishing if $\partial^2 U/\partial x^2 \lt 0$ .
Marginal rate of substitution: $MRS_{xy} = -dy/dx\big|_{U = \bar{U}} = MU_x / MU_y$ . It measures the units of $y$ the consumer is willing to sacrifice for 1 extra unit of $x$ .
Canonical preference shapes:
- Cobb-Douglas ( $U = x^a y^b$ ): $MRS = \frac{a}{b}\frac{y}{x}$ , bowed-in convex curves, diminishing MRS.
- Perfect substitutes ( $U = ax + by$ ): $MRS = a/b$ , constant slope, downward straight lines.
- Perfect complements ( $U = \min(ax, by)$ ): right-angle L-shaped curves, kink ray $ax = by$ .
- Quasilinear ( $U = v(x) + y$ ): $MRS = v'(x)$ depends on $x$ only; curves are vertical parallel shifts.

How the exam asks it: three core exam templates

Exam questions on consumer preferences test your ability to rank consumption bundles using utility functions, compute partial derivatives to find $MRS$ , and identify indifference curve geometry.

Exam template 1: Ranking bundles and level curves of utility

Question: A consumer has the utility function $U(x, y) = \sqrt{xy}$ .
a) Calculate the utility level for bundles $A = (4, 9)$ , $B = (6, 6)$ , and $C = (2, 18)$ , and state the preference relation between them.
b) Derive the algebraic equation of the indifference curve passing through these three bundles.
c) If the consumer consumes 1 unit of good $x$ , how many units of good $y$ are required to maintain this same utility level?

Exam template 2: Cobb-Douglas marginal utilities and diminishing MRS

Question: A consumer has Cobb-Douglas preferences represented by $U(X, Y) = X^{0.5} Y^{0.5}$ .
a) Find the marginal utility functions $MU_X$ and $MU_Y$ .
b) Derive the marginal rate of substitution $MRS_{XY}$ .
c) Evaluate the $MRS_{XY}$ at bundle $(4, 16)$ and at bundle $(10, 5)$ , and explain the economic significance of the difference.

Exam template 3: Quasilinear utility and vertical parallelism

Question: A consumer has quasilinear utility given by $U(X, Y) = 2\sqrt{X} + Y$ .
a) Derive $MU_X$ , $MU_Y$ , and the $MRS_{XY}$ function.
b) Calculate $MRS_{XY}$ at bundle $(4, 10)$ and at bundle $(4, 30)$ .
c) Interpret why $MRS_{XY}$ does not change between these two bundles and describe what this implies for the indifference map.

Cheat-sheet & the 4-step method

· Cheat-sheet: preference types, utility functions and MRS

Preference Type	Utility Function $U(X, Y)$	$MU_X, MU_Y$	$MRS_{XY}$	Indifference Curve Shape
Cobb-Douglas	$X^a Y^b$	$MU_X = aX^{a-1}Y^b$ , $MU_Y = bX^a Y^{b-1}$	$\dfrac{a}{b}\dfrac{Y}{X}$	Smooth, bowed-in (convex), asymptotic to axes
Perfect Substitutes	$aX + bY$	$MU_X = a$ , $MU_Y = b$	$\dfrac{a}{b}$ (constant)	Downward-sloping straight lines (slope $-a/b$ )
Perfect Complements	$\min(aX, bY)$	Undefined at corner	Not defined at corner ( $0$ on horizontal, $\infty$ on vertical)	L-shaped right angles; vertices lie on ray $aX = bY$
Quasilinear	$v(X) + Y$	$MU_X = v'(X)$ , $MU_Y = 1$	$v'(X)$ (independent of $Y$ )	Vertically parallel curves; same slope at any given $X$
Good and Bad	$X - Y$ or $X/Y$	$MU_X \gt 0$ , $MU_Y \lt 0$	Positive slope	Upward-sloping curves; higher utility toward good’s axis

Key idea · The 4-step method for preferences, utility and MRS questions

Step 1: Identify the utility function type.
- Is it Cobb-Douglas ( $X^a Y^b$ ), linear ( $aX+bY$ ), Leontief ( $\min(aX, bY)$ ), or quasilinear ( $v(X)+Y$ )?
Step 2: Calculate marginal utilities.
- Differentiate $U$ with respect to $X$ (holding $Y$ constant) for $MU_X$ .
- Differentiate $U$ with respect to $Y$ (holding $X$ constant) for $MU_Y$ .
Step 3: Compute $MRS_{XY}$ and evaluate at the given bundle.
- Write $MRS_{XY} = MU_X / MU_Y$ .
- Simplify powers cleanly: $\dfrac{X^{a-1}}{X^a} = \dfrac{1}{X}$ and $\dfrac{Y^b}{Y^{b-1}} = Y$ .
- Plug in numeric coordinates $(X, Y)$ to obtain the numeric rate.
Step 4: Check and interpret in bold.
- Does $MRS$ decrease as $X$ increases (diminishing $MRS$ )?
- State the trade-off in plain English: “The consumer is willing to give up $MRS$ units of $Y$ for 1 extra unit of $X$ .”

Assumptions about preferences

Before building any model of consumer choice, microeconomics needs a precise way to describe what a consumer wants. Three assumptions do the job.

Assumptions about preferencesAxiomsFormula sheet →

1.\ \text{Completeness} \qquad 2.\ \text{Transitivity} \qquad 3.\ \text{More is better (non-satiation)}

Use it when you need to justify why preferences can be represented by well-behaved indifference curves, a common short-answer question.

Completeness: a consumer can rank any two bundles — prefer A to B, prefer B to A, or be indifferent between them. There is no “I don’t know.”
Transitivity: if A is preferred to B, and B is preferred to C, then A is preferred to C. Without transitivity, a consumer’s choices could cycle endlessly.
More is better: for goods (not “bads” like pollution), a bundle with more of at least one good and no less of any other is always preferred.

Indifference curves

An indifference curve connects all bundles of two goods that give the consumer the same level of satisfaction — the consumer is indifferent among every point on it.

Properties of indifference curvesKey propertiesFormula sheet →

1.\ \text{Downward sloping} \qquad 2.\ \text{Curves further from the origin represent higher utility} \qquad 3.\ \text{Curves cannot cross} \qquad 4.\ \text{Typically convex to the origin}

Use it when you need to identify a badly drawn indifference curve diagram, or explain why a diagram is wrong, a frequent short-answer prompt.

Utility functions and marginal utility

A utility function $U(x, y)$ gives every bundle a number so that preferred bundles get higher numbers. It is a compact way to write down a preference ranking.

Marginal utility $MU_x$ is the extra utility from one more unit of $x$ , holding $y$ fixed: the slope of the total utility curve. More is better means $MU \gt 0$ . Many utility functions also show diminishing marginal utility: each extra unit adds less than the one before. Total utility still rises, just more slowly.

The marginal rate of substitution

The marginal rate of substitution (MRS) is the rate at which a consumer is willing to trade one good for another while staying on the same indifference curve — the magnitude of the curve’s slope at a point.

Marginal rate of substitutionKey formulaFormula sheet →

MRS_{XY} = -\frac{dY}{dX}\bigg|_{U = \bar{U}} = \frac{MU_X}{MU_Y}

$MU_X, MU_Y$: marginal utility of X and Y, the extra utility from one more unit of each

Use it when you need the slope of an indifference curve at a point, either from the curve itself or from a utility function's partial derivatives.

Diminishing MRS is what gives indifference curves their typical convex (bowed-in) shape: as a consumer gives up more and more of good Y for more X, each additional unit of X is worth less and less Y to them, because X becomes relatively abundant and Y relatively scarce.

Switch between the three preference types and slide along the middle curve. Watch the MRS: it falls as you move right on a Cobb-Douglas curve, never changes on a straight line, and jumps from infinite to zero at the corner of an L.

Utility: ordinal, not cardinal

Marginal utilityFormula sheet →

MU_X = \frac{\partial U}{\partial X}

$MU_X$: the extra utility from one more unit of X, holding Y fixed

Use it when you need the individual building block before forming a marginal rate of substitution, or to discuss diminishing marginal utility of a single good.

Special preference shapes

Besanko and Braeutigam use four standard utility functions. Know each one’s formula, MRS and indifference-curve shape, and be able to draw them.

Cobb-DouglasSpecial caseFormula sheet →

U(X,Y) = AX^{a}Y^{b} \quad\Rightarrow\quad MRS_{XY} = \frac{aY}{bX}

$A, a, b$: positive constants

Use it when the utility function is a product of powers. Indifference curves are smooth, bowed in toward the origin, and never touch the axes; MRS diminishes along each curve.

Two extreme cases:

Perfect substitutesSpecial caseFormula sheet →

U(X,Y) = aX + bY \quad\Rightarrow\quad MRS_{XY} = \frac{a}{b} \text{ (constant)}

$a, b$: fixed per-unit utility weights

Use it when two goods are essentially interchangeable to the consumer at a fixed rate, such as 500ml and 1L bottles of the same water brand. Indifference curves are straight lines.

Perfect complementsSpecial caseFormula sheet →

U(X,Y) = \min(aX,\, bY)

$a, b$: the fixed proportion in which the goods must be consumed together

Use it when two goods are always used together in a fixed ratio, such as left shoes and right shoes. Indifference curves are L-shaped, with the kink at the fixed ratio, and MRS is undefined at the kink.

QuasilinearSpecial caseFormula sheet →

U(X,Y) = v(X) + Y,\ \text{e.g.}\ U = 2\sqrt{X} + Y \quad\Rightarrow\quad MRS_{XY} = \frac{1}{\sqrt{X}}

$v(X)$: a concave function of X only
$Y$: enters linearly, so MU_Y = 1

Use it when utility is linear in one good. The MRS depends on X only, so every indifference curve is a vertical shift of every other: at a given X they all have the same slope.

Bads and neutral goods

Not everything is a good. A bad is something the consumer would rather have less of (pollution, noise, homework): $MU \lt 0$ . With one good on one axis and a bad on the other, indifference curves slope upward: to accept more of the bad, the consumer needs more of the good. A neutral good has $MU = 0$ : the consumer does not care about it, so indifference curves are vertical or horizontal lines.

Exam practice

Summary and review

Preferences are assumed complete, transitive, and satisfy more-is-better.
A utility function ranks bundles; an indifference curve is the set of bundles with the same $U$ (set $U$ = constant and solve for $y$ ).
Marginal utility is the slope of total utility; more is better means $MU \gt 0$ ; diminishing MU means each unit adds less.
Four standard forms: Cobb-Douglas ( $MRS = aY/bX$ ), perfect substitutes (constant MRS), perfect complements (L-shapes), quasilinear (MRS depends on X only).
A bad has $MU \lt 0$ and gives upward-sloping indifference curves.
Indifference curves: downward sloping, farther out means higher utility, never cross, typically convex.
$MRS_{XY} = MU_X/MU_Y$ : the rate of trade-off that keeps utility constant.
Diminishing MRS produces the typical convex (bowed-in) indifference curve shape.
Utility is ordinal: only the ranking of bundles matters, not the size of the numbers.
Perfect substitutes: straight-line indifference curves, constant MRS. Perfect complements: L-shaped curves, fixed-ratio consumption.