Microeconomics: Consumer Preferences, Utility, and Demand Analysis

Axiom: Completeness

Consumer can compare any two bundles A and B: A ≽ B, B ≽ A, or both.

Axiom: Transitivity

If A ≽ B and B ≽ C then A ≽ C.

Axiom: Continuity

Small changes in a bundle do not create jumps in preferences; indifference curves have no gaps.

Monotonicity

More of any good is weakly better; indifference curves slope downward.

Convexity

Consumers prefer averages to extremes; indifference curves are bowed inward.

Strict Convexity

Consumers strictly prefer averages; guarantees unique interior optimum.

Homothetic Preferences

Preferences where MRS depends only on x2/x1; income expansion paths are straight.

Utility Function

Numerical representation of preferences; only ordinal information.

Monotonic Transformation

Any strictly increasing transformation of utility; preserves preferences.

Indifference Curve

Set of bundles with equal utility; slope downward and do not cross.

Marginal Rate of Substitution (MRS)

MRS = MU1/MU2; equals slope of indifference curve (negative).

Cobb-Douglas Utility

U = x1^a x2^b; MRS = (a/b)(x2/x1); Marshallian: x1 = (a/(a+b))(I/p1), x2 = (b/(a+b))(I/p2).

Perfect Substitutes

U = ax + by; ICs are straight lines; corner solutions likely.

Perfect Complements

U = min(ax, by); L-shaped ICs; goods consumed in fixed proportions.

Budget Constraint

p1x1 + p2x2 = I.

Utility Maximization Condition

Choose x1,x2 such that MRS = p1/p2.

Lagrangian (Consumer)

L = U(x1,x2) + λ(I − p1x1 − p2x2).

Interpretation of λ (Consumer)

λ = marginal utility of income.

Marshallian Demand

Utility-maximizing demand x(p1,p2,I).

Corner Solution

Optimal bundle on boundary when interior FOC fails.

Indirect Utility Function

v(p1,p2,I); utility achieved at optimal choices; homogeneous degree 0 in (p,I).

Roy's Identity

xi = −(∂v/∂pi)/(∂v/∂I).

Lump-Sum Principle

Lump-sum taxes cause less distortion than per-unit taxes.

Expenditure Minimization

Choose x1,x2 to minimize cost given target utility ū.

Hicksian Demand

Cost-minimizing demand h(p1,p2,ū).

Lagrangian (Expenditure)

L = p1x1 + p2x2 + λ(ū − U(x1,x2)).

Interpretation of λ (Expenditure)

Marginal cost of raising utility by 1 unit.

Expenditure Function

e(p1,p2,ū); minimum cost to reach ū; homogeneous degree 1 in prices; concave in prices.

Shephard's Lemma

h_i(p,ū) = ∂e/∂p_i.

Magic Square

Relationships: Marshallian ↔ Indirect Utility ↔ Hicksian ↔ Expenditure (via Roy + Shephard).

Price Elasticity of Demand

ε = (%ΔQ)/(%ΔP).

Income Elasticity

η = (%ΔQ)/(%ΔI).

Cross-Price Elasticity

ε12 = (%ΔQ1)/(%Δp2).

Normal Good

Demand increases with income.

Inferior Good

Demand decreases with income.

Luxury Good

Income elasticity greater than 1.

Engel Aggregation

Σ (budget share × income elasticity) = 1.

Giffen Good

Price ↑ causes Q ↑ because income effect > substitution effect.

Hicksian Substitution Effect

Effect of price change holding utility constant.

Income Effect

Effect of change in purchasing power from price change.

Slutsky Equation

Total effect = substitution effect + income effect.

Gross Substitutes

∂xi/∂pj > 0.

Gross Complements

∂xi/∂pj < 0.

Net Substitutes

∂hi/∂pj > 0.

Net Complements

∂hi/∂pj < 0.

Compensating Variation

Money needed after price change to restore original utility.

Equivalent Variation

Money required before price change to reach new utility.

Consumer Surplus

Area under demand curve above price.

Returns to Scale

Description of output change when all inputs scaled proportionally.

Marginal Product of Labor (MPL)

∂F/∂L.

Average Product of Labor (APL)

F(K,L)/L.

Diminishing MPL

MPL decreases as L increases.

Isoquant

Set of input bundles producing same output.

Isocost Line

All input bundles with equal cost; slope = −w/r.

MRTS

MRTS = MPL/MPK = slope of isoquant.

Cobb-Douglas Production

F(K,L) = K^a L^b; MRTS = (a/b)(L/K).

Perfect Substitutes Production

F = aK + bL.

Perfect Complements Production

F = min(aK, bL).

Cost Minimization Condition

MRTS = w/r.

Conditional Factor Demands

Cost-minimizing K(w,r,q) and L(w,r,q).

Cost Function

C(w,r,q); homogeneous degree 1 in prices; concave in prices.

Average Cost

AC = C(q)/q.

Marginal Cost

MC = dC/dq.

Short-Run Costs

Some inputs fixed; includes AFC, AVC, MC.

Long-Run Costs

All inputs variable; LR cost is envelope of SR costs.

Profit Maximization

Choose q where MR = MC.

Perfect Competition

Price taker; MR = P.

Short-Run Supply

Firm supplies where P ≥ AVC and P = MC.

Shut-Down Condition

Produce if P ≥ AVC.

Unconditional Factor Demands

Profit-maximizing K(p,w,r) and L(p,w,r).

Profit Function

π(p,w,r); homogeneous degree 1 in prices; convex in output price; decreasing in w,r.

Envelope Theorem (Firm)

Derivative of profit wrt price = supply.

Producer Welfare Change

Change in profit from output price shifts.

Monopoly MR

MR = P(1 + 1/ε).

Monopoly Markup Rule

(P − MC)/P = −1/ε.

Price Discrimination

Firms charge different prices to different groups or units.

Market Demand

Sum of individual demands horizontally.

Market Supply

Sum of firm supply curves horizontally.

Consumer Surplus

Area between WTP and price.

Producer Surplus

Area above supply curve and below price.

Very Short Run

Quantity fixed; no adjustment.

Short Run

Some inputs fixed.

Long Run

All inputs variable.

Long-Run Comp. Equilibrium

P = MC = min AC; zero economic profit.

Economic Efficiency

Maximizes total surplus; no DWL.

Quantity Cap

Quantity limit; creates DWL unless at efficient level.

Price Ceiling

Max legal price; causes shortage; DWL.

Price Floor

Min legal price; causes surplus; DWL.

Per-Unit Tax

Shifts supply up by tax amount; DWL.

Subsidy

Opposite of tax; raises Q; costs government.

Tariff

Tax on imports; raises domestic price.

Quota

Limit on imports.

Minimum Wage

Price floor in labor market.

Monopsony

Single buyer sets wage; hires where MRP = ME.