Week 1 · Intermediate Microeconomics
Mathematical Review, Utility, Preferences & Indifference Curves
Prof. Naveen Sunder
Learning Objectives
What You Will Be Able to Do
- LO1Apply the three core differentiation rules (constant, power, sum) and compute slopes of linear and non-linear functions.
- LO2Define opportunity cost and explain why sunk costs should be ignored in forward-looking decisions.
- LO3Explain what utility represents and state the three assumptions on rational preferences: completeness, transitivity, and monotonicity.
- LO4Draw and interpret indifference curves; explain why they slope downward and bow inward toward the origin.
- LO5Define marginal utility and explain the law of diminishing marginal utility with examples.
- LO6Calculate the MRS from a graph and from a utility function; explain its economic meaning as the consumer’s internal trade-off rate.
Chapter Sections
Topics
Every economic trade-off is a rate of change, and slope is how we measure it. This section covers three tools used throughout the course: the slope of linear functions, differential calculus (the three rules), and solving systems of linear equations.
For a linear function \(y = mx + b\), the slope is \(m = \Delta y / \Delta x\). For non-linear functions, slope varies at each point and is found using the derivative. The constant, power, and sum rules cover all differentiation needed in this course.
Opportunity cost is the value of the best foregone alternative. Nothing is ever truly free: every choice uses a scarce resource, even if that resource is only time.
Sunk costs are costs already incurred and unrecoverable. They are irrelevant to forward-looking decisions. If you paid $15 for a bad film and are 20 minutes in, the $15 is gone regardless; only the opportunity cost of the remaining time should affect your choice.
Utility is an ordinal index of satisfaction: it ranks bundles from most to least preferred, but the numbers carry no cardinal meaning. \(U = 10\) is not twice as good as \(U = 5\); only the ordering matters. Any order-preserving transformation of a utility function represents the same preferences.
Three assumptions underpin rational preferences: completeness (any two bundles can be ranked), transitivity (if A ≻ B and B ≻ C then A ≻ C), and monotonicity (more of both goods is always better). Together they guarantee a well-behaved utility function.
An indifference curve (IC) is the set of all bundles \((X,Y)\) giving the consumer equal utility. The full set of ICs (one per utility level) is the indifference map.
Four key properties follow from the preference assumptions: ICs slope downward (monotonicity), are convex and bow inward (diminishing MRS), represent higher utility when further from the origin, and never cross (transitivity).
Marginal utility is the additional utility from consuming one more unit of a good: \(MU_X = \partial U / \partial X\). It is the partial derivative of the utility function with respect to X.
The law of diminishing marginal utility states that \(MU_X\) decreases as X increases: the first cup of coffee is the most satisfying, the fifth less so. Mathematically, \(\partial^2 U / \partial X^2 < 0\). Diminishing MU is what makes indifference curves convex.
The MRS is the number of units of Y the consumer is willing to give up per unit of X gained, staying on the same IC. It equals the negative slope of the IC: \(\text{MRS} = -MU_X / MU_Y\).
Diminishing MRS: As the consumer moves right along an IC (more X, less Y), |MRS| falls. They become less willing to sacrifice Y for additional X because X grows relatively abundant, producing the characteristic inward bow.
Why it matters: The MRS is the consumer’s internal trade-off rate. In Week 2 we compare it to the market’s trade-off rate (the price ratio \(P_X/P_Y\)) to find the optimal bundle: the core result of consumer theory.
Quick Reference
The Three Differentiation Rules
| Rule | Formula | Course Example |
|---|---|---|
| Constant Rule | \(d(c)/dx = 0\) | In \(U = X^{1/2}Y^{2/3}\), when taking \(\partial U/\partial X\), the term \(Y^{2/3}\) is a constant (derivative is 0); it stays as a multiplier. |
| Power Rule | \(d(x^n)/dx = nx^{n-1}\) | \(U = X^{1/2} \Rightarrow MU_X = \tfrac{1}{2}X^{-1/2}\). Positive but decreasing; captures diminishing marginal utility. |
| Sum Rule | \(d(f+g)/dx = f' + g'\) | \(U = X^2 + Y^{2/3}\): taking \(\partial U/\partial X\) gives \(2X\); the \(Y^{2/3}\) term is constant (Rule 1), so \(MU_X = 2X\). |
Quick Reference
Three Assumptions on Rational Preferences
Quick Reference
Properties of Indifference Curves
- Downward sloping: from monotonicity. Taking away Y must be compensated by gaining X to stay on the same curve.
- Convex (bowed inward): from diminishing MRS. As X increases, the consumer gives up less and less Y per unit of X.
- Higher = better: an IC further from the origin represents higher utility (from monotonicity).
- Never cross: a crossing implies the same bundle simultaneously yields two utility levels, violating transitivity.
Equations
Key Equations
Exam Prep
Five Common Mistakes
Watch out for these
- Including sunk costs in forward-looking decisions. Once a cost is sunk, it is irrelevant. Ask only: what are the future costs and benefits from this point forward?
- Treating utility numbers as cardinal. \(U = 10\) is not “twice as good” as \(U = 5\). Only the ranking matters; any order-preserving transformation gives identical predictions.
- Confusing movement along a curve with a shift of the curve. A price change causes movement along demand or supply. A change in income or tastes shifts the entire curve.
- Dropping the negative sign in the MRS. \(\text{MRS} = -MU_X/MU_Y\) is negative. Always work with \(|\text{MRS}|\) when comparing magnitudes to the price ratio.
- Applying the power rule without rewriting negative exponents first. Rewrite \(1/x^2\) as \(x^{-2}\) before differentiating: the single most common algebraic error on calculus questions.
Practice
Practice Questions
(a) Slope \(= (1-7)/(4-1) = -2\).
(b) Rearrange: \(y = 2x+4\). Slope \(= 2\).
(c) As x increases by 1, y increases by 2; both move in the same direction.
(a) \(4x^3\). (b) \(\tfrac{2}{5}x^{-3/5}\). (c) Rewrite as \(x^{-1/2}\): \(-\tfrac{1}{2}x^{-3/2}\). (d) \(15x^2 + x^{-1/2}\).
(a) \(MU_X = \tfrac{1}{3}X^{-2/3}Y^{2/3}\). (b) \(MU_Y = \tfrac{2}{3}X^{1/3}Y^{-1/3}\).
(c) Both positive for \(X,Y>0\). Reflects monotonicity: more of either good always increases utility.
(a) Best foregone alternative is working at $15/hr for 3 hours = $45.
(b) No: the 3 hours are scarce and have an opportunity cost even though there is no tuition charge.
(c) Every choice uses a scarce resource, even time. Nothing is ever truly free.
A sunk cost is already incurred and unrecoverable: it is the same whether the restaurant stays open or closes. The correct question is only: do expected future revenues exceed expected future costs? The $50,000 never enters this forward-looking calculation.
(a) Transitivity. (b) Completeness. (c) Monotonicity. (d) Monotonicity: (4,6) has strictly more of both goods and must be preferred.
(a) Supply left → P rises, Q falls. (b) Demand right → P and Q both rise. (c) Supply right → P falls, Q rises.
An indifference curve connects all combinations of two goods that leave you equally happy. If someone takes away some of one good you become worse off: the only way to get back to the same happiness is to receive more of the other good as compensation.
So as you give up Y you must gain X to stay equally happy: when Y goes down, X goes up. That is a downward slope. If the curve sloped upward, giving you more of both goods would leave you equally happy, which contradicts the simple fact that more is better.
(a) \(y = 15-3x\); sub in: \(-11x=-44 \Rightarrow x=4, y=3\).
(b) \(y=4-x\); sub in: \(7x=16 \Rightarrow x=16/7, y=12/7\).
(a) \(U=20\), IC: \(XY=20\). (b) \(U=20\), same IC. (c) Yes: both give equal utility on \(XY=20\).
(d) Equal utility. Consistent with monotonicity: it only requires strictly more of both goods to be preferred. These bundles trade off X and Y, which is exactly what the IC captures.
(a) \(MU_X = 3\) (constant), \(MU_Y = \tfrac{1}{2}Y^{-1/2}\) (decreasing in Y).
(b) At (2,9): \(MU_X = 3\), \(MU_Y = 1/6\).
(c) \(MU_Y\) falls as Y rises: the law of diminishing marginal utility.
(a) \(|\text{MRS}| = 4/2 = 2\). (b) \(|\text{MRS}| = 2/3\).
(c) Yes: |MRS| fell from 2 to 2/3. Implies the IC is convex (bowed inward).
(d) Downward-sloping convex curve through P=(1,10), Q=(3,6), R=(6,4); steep near P, flatter near R.
(a) \(-3Y/X\). (b) \(-(Y/X)^{1/2}\). (c) \(-Y/(2X)\).
(d) \(MU_X=2, MU_Y=3\), MRS \(= -2/3\): constant. X and Y are perfect substitutes; ICs are straight lines of slope \(-2/3\).
(a) B ≻ Z (higher curve). Z ≻ A (lower curve). But Z lies on both, implying Z ~ A and Z ~ B simultaneously.
(b) Z ~ A and Z ~ B requires A ~ B by transitivity. But B is on U2 > U1, so B ≻ A: contradiction.
(c) Any crossing creates a logical inconsistency. ICs must be distinct and never intersect.
(a) \(100-2P=10+3P \Rightarrow P^*=18\), \(Q^*=64\).
(b) At P=15: \(Q_d=70, Q_s=55\), shortage of 15 units.
(c) At P=25: \(Q_d=50, Q_s=85\), surplus of 35 units.
(a) A:16, B:16, C:12, D:12, E:18. Ranking: E ≻ A=B ≻ C=D.
(b) A and B lie on \(XY=16\); C and D lie on \(XY=12\).
(a) MRS \(= -Y/X\). (b) At (2,8): |MRS|=4; at (4,4): |MRS|=1. Yes, diminishing.
(c) Consumer values X at 4 units of Y internally, but market only charges 2. X is underpriced: buy more X and less Y until |MRS|=2.
(a) Diminishing MRS means each additional unit of X is worth less and less in terms of Y as X increases. The IC flattens moving right, bowing inward. Any bundle between two extremes on the IC lies on a higher IC: the mix is preferred because diminishing MRS makes balanced consumption inherently more valuable.
(b) The MRS is the consumer’s internal trade-off rate; the price ratio \(P_X/P_Y\) is the market’s trade-off rate. When these differ, the consumer can improve welfare by trading along the budget line. Optimum requires them to be equal (developed fully in Week 2).
(c) Not at optimum. |MRS|=3 > 2=P_X/P_Y. She values X at 3 units of Y, but the market only charges 2. Buy more X and less Y until |MRS|=2.
(a) \(MP_L = \tfrac{1}{2}L^{-1/2}K^{1/2}\), \(MP_K = \tfrac{1}{2}L^{1/2}K^{-1/2}\).
(b) \(\partial MP_L/\partial L = -\tfrac{1}{4}L^{-3/2}K^{1/2} < 0\): diminishing returns to labour.
(c) MRTS \(= -K/L\).
(d) At (4,9): MRTS=−2.25; at (9,4): MRTS≈−0.44. At (4,9), labour is scarce and highly productive: the firm gives up more K per unit of L. This mirrors diminishing MRS exactly: the scarcer the input, the more valuable it is at the margin.
(a) \(MU_X = Y+2\), \(MU_Y = X+1\). (b) MRS \(= -(Y+2)/(X+1)\).
(c) \(Y = 30/(X+1) - 2\). (d) X=0: Y=28; X=2: Y=8; X=4: Y=4.
(e) |MRS|: 30 at (0,28), 10/3 at (2,8), 6/5 at (4,4). Diminishing ✓
(f) Convex curve through (0,28), (2,8), (4,4); steep near Y-axis, flattening toward X-axis.
(a) Economic profit = $90,000 − $50,000 − $120,000 = −$80,000.
(b) No: she is $80,000 worse off economically than staying employed.
(c) Any two of: long-run growth potential; non-monetary value of autonomy; risk preferences; option value of equity; personal passion for the project.
(a) False. Utility is ordinal; only rankings matter.
(b) False. ICs for an economic bad slope upward; the statement is too strong.
(c) False. Diminishing MRS requires \(MU_X/MU_Y\) to fall. Both MUs can change; diminishing \(MU_X\) alone is not sufficient.
(d) True. A crossing implies the same bundle yields two utility levels, violating transitivity.
(e) True. If an IC sloped upward, moving along it would increase both goods and reach higher utility, contradicting the constant-utility definition of an IC.
(a) \(MU_X = \alpha X^{\alpha-1}Y^{1-\alpha}\), \(MU_Y = (1-\alpha)X^\alpha Y^{-\alpha}\).
(b) MRS \(= -\dfrac{\alpha}{1-\alpha}\cdot\dfrac{Y}{X}\).
(c) \(\partial|\text{MRS}|/\partial X = -\dfrac{\alpha}{1-\alpha}\cdot\dfrac{Y}{X^2} < 0\). Diminishing MRS confirmed.
(d) As \(\alpha \to 1\), the coefficient \(\alpha/(1-\alpha) \to \infty\). The consumer values X enormously relative to Y; in the limit, only X matters.
(a) For any bundles A, B: \(U(A) > U(B) \Leftrightarrow U(A)^2 > U(B)^2\) (since \(x^2\) is strictly increasing for \(x>0\)). Rankings preserved.
(b) Let \(U = X^\alpha Y^\beta\), \(V = X^{2\alpha}Y^{2\beta}\). Then \(MRS_V = -(2\alpha/2\beta)(Y/X) = -(\alpha/\beta)(Y/X) = MRS_U\). ✓
(c) Any two utility functions related by a strictly increasing transformation represent the same preferences: same ICs, same rankings, same MRS. Only the ordinal structure matters.
(a) \(\ln X + \ln Y = 2 \Rightarrow XY = e^2\).
(b) The IC for \(V=XY\) at level \(e^2\) is \(XY=e^2\), the same curve.
(c) For \(U\): MRS \(= -(1/X)/(1/Y) = -Y/X\). For \(V\): MRS \(= -Y/X\). Equal ✓
(d) Since \(V = e^U\) is strictly increasing, both represent the same preferences with identical ICs and MRS.
Imagine deciding how many sandwiches and coffees to have for lunch. The MRS simply asks: how many coffees would you willingly give up to get one more sandwich, staying equally satisfied? If you currently have four coffees and one sandwich, you are swimming in caffeine and starving, so you would gladly give up three coffees for one extra sandwich. But if you already have four sandwiches and one coffee, you are full and under-caffeinated, so now you would barely give up any coffee for yet another sandwich.
That is diminishing MRS in everyday language. The more of something you already have, the less you are willing to sacrifice to get even more of it. This is why people naturally end up with a mix: going to extremes is irrational because the 10th coffee of the day is worth almost nothing while you are still hungry.
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