Module 1 — Choice, Duality, and Revealed Preference

Consumer and producer choice, value functions, envelope results, demand properties, and revealed-preference tests.

Module 1 — Choice, Duality, and Revealed Preference

Core question

What can we infer about preferences or technology from observed choices, and what does an optimisation solution still leave unidentified?

Learning outcomes

You will be able to:

state the assumptions needed for utility representation and well-behaved demand;
derive Marshallian and Hicksian demand and use Roy's identity and Shephard's lemma;
recognise corner solutions and verify demand properties;
connect consumer and producer duality;
test a small dataset for revealed-preference violations.

1. Preferences are rankings, not measured happiness

A preference relation ≽ is usually assumed:

Assumption	Meaning	What it permits
completeness	any two bundles can be ranked	a choice is defined
transitivity	rankings contain no cycle	internally consistent ordering
continuity	nearby bundles do not cause arbitrary jumps	continuous utility representation
local nonsatiation	every neighbourhood contains something better	budget exhaustion at positive prices
convexity	mixtures are weakly preferred to extremes	convex demand problem; diversification

A utility function represents the ordering:

x\succeq y\iff u(x)\ge u(y).

Any strictly increasing transformation f(u) represents the same ordinal preferences. Marginal utility levels therefore depend on representation; the marginal rate of substitution does not:

MRS_{12}=\frac{\partial u/\partial x_1}{\partial u/\partial x_2}.

Convex preferences mean the upper contour sets are convex; they do not require the chosen utility representation to be concave. A monotone transformation can change second derivatives without changing choice.

2. Utility maximisation

At prices p ≫ 0 and income m, Marshallian demand solves:

x(p,m)\in\arg\max_{x\ge0}\{u(x):p\cdot x\le m\}.

For a regular interior solution,

\nabla u(x^*)=\lambda p.

The condition says marginal utility per pound is equalised. It is not enough on its own: verify feasibility, curvature, and whether a non-negativity constraint binds.

Worked example: Cobb–Douglas demand

Let

u(x,y)=x^\alpha y^{1-\alpha},\qquad 0<\alpha<1.

The tangency condition and budget exhaustion give:

x(p_x,p_y,m)=\frac{\alpha m}{p_x},\qquad y(p_x,p_y,m)=\frac{(1-\alpha)m}{p_y}.

With α = 0.4, m = 100, p_x = 2, and p_y = 5, the optimum is (20,12). Check:

2(20)+5(12)=100.

The indirect utility function is:

v(p_x,p_y,m)= \left(\frac{\alpha m}{p_x}\right)^\alpha \left(\frac{(1-\alpha)m}{p_y}\right)^{1-\alpha}.

When tangency fails

For u(x,y)=ax+by, compare a/p_x with b/p_y; usually buy one good only.
For u(x,y)=min{x,y}, choose the kink x=y.
For u(x,y)=v(x)+y, an interior x satisfies v'(x)=p_x/p_y; x, not the numeraire y, has zero income effect once income is high enough.

KKT conditions handle these cases without pretending all optima are interior.

3. Expenditure minimisation and duality

Hicksian demand holds utility fixed:

h(p,\bar u)\in\arg\min_{x\ge0}\{p\cdot x:u(x)\ge\bar u\}.

The expenditure function is the minimum value:

e(p,\bar u)=p\cdot h(p,\bar u).

For Cobb–Douglas utility,

e(p_x,p_y,\bar u)= \frac{\bar u\,p_x^\alpha p_y^{1-\alpha}} {\alpha^\alpha(1-\alpha)^{1-\alpha}},

and expenditure shares remain α and 1−α.

The four links

Result	Formula	Interpretation
Shephard's lemma	`∂e/∂p_i = h_i`	price derivative of required spending is compensated demand
Roy's identity	`x_i = −(∂v/∂p_i)/(∂v/∂m)`	recover ordinary demand from indirect utility
inverse relation	`e(p,v(p,m))=m`	spending to recover achieved utility equals income
inverse relation	`v(p,e(p,u))=u`	minimum spending attains target utility

These are envelope results: the derivative of the optimised value ignores the first-order effect of re-optimising the choice.

4. Properties that catch algebra mistakes

Marshallian demand satisfies, under local nonsatiation:

Walras' law: p·x(p,m)=m;
degree-zero homogeneity: x(tp,tm)=x(p,m);
adding-up and aggregation restrictions on demand derivatives.

The expenditure function is increasing in utility, homogeneous of degree one and concave in prices. Hicksian demand obeys the compensated law of demand; its substitution matrix is symmetric and negative semidefinite under regularity.

After deriving a demand, check units, budget exhaustion, homogeneity, signs, limiting cases, and corners. A formula that fails one check is not an economic solution even if the differentiation was correct.

5. Revealed preference: learn from budgets, not utility surveys

If bundle xᵗ is chosen when xˢ was affordable, then xᵗ is directly revealed at least as good as xˢ:

p^t\cdot x^t\ge p^t\cdot x^s.

A two-observation violation

Observation	Prices	Chosen bundle	Cost of other bundle
1	`(2,1)`	`A=(2,1)`, cost 5	`B=(1,2)` costs 4
2	`(1,2)`	`B=(1,2)`, cost 5	`A=(2,1)` costs 4

At observation 1, A is chosen although B is cheaper; at observation 2, B is chosen although A is cheaper. Both strict comparisons cannot come from one locally nonsatiated, stable preference ordering. This violates WARP.

For many observations, GARP rules out an indirect revealed-preference cycle with at least one strict step. Afriat's theorem says that a finite dataset satisfies GARP if and only if it can be rationalised by a continuous, monotone, concave utility function.

Recent extension

Chambers and Echenique show how finite choice data can support bounds on counterfactual welfare and whether an allocation could be Pareto efficient for some preferences consistent with the data (2025). The key boundary is incompleteness: observed budgets rarely rank every counterfactual bundle.

6. Producer choice uses the same dual logic

For technology f(z), output price q, and input prices w, profit maximisation is:

\pi(q,w)=\max_{z\ge0}\{qf(z)-w\cdot z\}.

Cost minimisation for target output \bar y is:

c(w,\bar y)=\min_{z\ge0}\{w\cdot z:f(z)\ge\bar y\}.

Hotelling's lemma recovers net supply from derivatives of profit.
Shephard's lemma recovers conditional input demand: ∂c/∂w_i=z_i^c.
Profit is convex and homogeneous of degree one in the price vector; cost is concave and homogeneous of degree one in input prices.

Worked cost problem

If f(L,K)=√(LK), w=r=1, and required output is 10, minimise L+K subject to LK≥100. The cost-minimising bundle is L=K=10, so c=20. If labour becomes relatively expensive, conditional demand substitutes toward capital; the output target remains fixed.

Practice

Derive Marshallian demand and v(p,m) for u=x^{1/3}y^{2/3}.
Derive its expenditure function and verify Shephard's lemma.
Solve u=2x+y at three price ratios and identify all corners/ties.
Construct three observations that pass WARP pairwise but fail GARP through a longer cycle.
For f(L,K)=L^{1/3}K^{1/3}, derive the conditional input ratio.

Quick check

Utility represents rankings; its numerical scale is usually arbitrary.
Tangency is a candidate, not a universal solution rule.
Marshallian demand holds income fixed; Hicksian demand holds utility fixed.
Dual value functions recover behaviour through envelope theorems.
Revealed preference disciplines—but does not fully identify—unobserved welfare.

Next: decompose responses and measure welfare changes.

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Advanced Microeconomics — Course Guide

A rigorous course from individual choice and duality to games, information, mechanisms, externalities, and market design.

Module 2 — Comparative Statics and Welfare Measurement

Implicit derivatives, Slutsky decomposition, compensating and equivalent variation, surplus, incidence, and empirical welfare.