Module 2 — Comparative Statics and Welfare Measurement

Core question

When prices, income, or policy rules change, how much behaviour changes—and how much money represents the resulting welfare gain or loss?

Learning outcomes

You will be able to:

• derive a local comparative static from equilibrium or first-order conditions;
• decompose a price response into substitution and income effects;
• calculate compensating variation, equivalent variation, and consumer-surplus change;
• state when these measures coincide and when distribution matters;
• connect a theoretical welfare statistic to evidence and government cost.

1. Comparative statics is a conditional derivative

If an optimum or equilibrium is characterised by

and F_z is nonsingular, the implicit function theorem gives:

This is a local result. Its sign is meaningful only after stating:

• what θ changes;
• what remains fixed;
• which optimum/equilibrium is followed;
• whether a corner, discontinuity, or equilibrium switch is possible.

2. Slutsky decomposition

Marshallian demand can be written as Hicksian demand evaluated at attained utility:

Differentiating with respect to price p_j gives:

For an own-price change, the compensated effect is non-positive. Ordinary demand can nevertheless slope upward only if the good is sufficiently inferior for the income effect to dominate—a Giffen case.

For many goods, the Hicksian substitution matrix is symmetric, negative semidefinite, and singular under standard regularity. These are testable restrictions, not descriptive slogans.

3. Worked finite change

Let

and let p_x rise from 1 to 4.

Marshallian demand is:

Hicksian demand for x is:

At new prices but old utility, h_x=25. The Hicks decomposition of the finite change is therefore:

• substitution: 50 → 25, or −25;
• purchasing-power effect: 25 → 12.5, or −12.5;
• total: −37.5.

The differential Slutsky equation is exact for infinitesimal changes. For a large change, Hicks and Slutsky finite compensation use different reference rules and need not produce identical intermediate bundles.

4. Put a money value on the welfare change

For a price increase from p⁰ to p¹, define welfare losses as positive numbers:

the compensation required after the change, and

the amount the consumer would pay beforehand to avoid the change.

For u=√(xy), the expenditure function is:

Thus:

The Marshallian consumer-surplus loss is:

For this normal good and price rise:

Earlier versions of these notes reversed the CV–EV ordering; the reference utility makes the correct ordering clear. With quasi-linear utility and no income effect on the priced good, CV = EV = ΔCS.

5. From one consumer to policy appraisal

Adding willingness to pay is defensible under quasi-linear utility or an explicit social-welfare approximation. Outside that benchmark, one pound to a low-income household and one pound to a high-income household need not carry the same social weight.

A policy appraisal should separate:

1. behavioural response;
2. willingness to pay or accept;
3. producer and government surplus;
4. external effects;
5. distribution and social weights;
6. implementation and fiscal spillovers.

Marginal value of public funds

One modern statistic is:

The denominator includes behavioural fiscal effects, not only the programme's sticker cost. A 2025 guide explains its use for marginal tax-policy comparisons while emphasising that the question and spending margin must be defined carefully (Bastani, 2025). It is not a complete social welfare function and does not eliminate distributional judgment.

6. Worked policy: a transit fare increase

A city raises a fare from £2 to £3. A commuter's monthly trip demand is q(p)=60−10p over this range.

• trips fall from 40 to 30;
• consumer-surplus loss is the trapezoid:

• fare revenue changes from 2×40=£80 to 3×30=£90.

The £10 revenue gain is not the commuter's full welfare loss. Nor does the partial-equilibrium calculation include congestion, operator cost, crowding, service quality, low-income incidence, or substitution to cars.

7. Evidence discipline

Hicksian demand is rarely observed directly. Researchers combine demand estimation, experiments or quasi-experiments, price variation, and assumptions about utility or market structure. Welfare results can be sensitive to:

• which consumers are represented;
• intensive versus extensive margins;
• equilibrium price responses;
• quality and product entry;
• extrapolation beyond observed variation.

A precise estimate under a restrictive model is not automatically a complete welfare judgment.

Practice

1. Derive the Slutsky equation from x(p,m)=h(p,v(p,m)).
2. Repeat the worked example for p_x: 1 → 2 and compare the three losses.
3. Show why quasi-linearity removes the income effect on the non-numeraire good, subject to an interior solution.
4. Evaluate a £1 per-trip subsidy using consumer surplus, fiscal cost, congestion, and distribution.
5. Give one case where a local derivative gives the wrong intuition for a large policy change.

Quick check

• Comparative statics is conditional on an equilibrium and “other things equal.”
• Substitution holds utility fixed; ordinary demand holds income fixed.
• CV and EV answer different counterfactual compensation questions.
• Consumer surplus is exact only under particular preference conditions.
• Welfare appraisal requires incidence, external effects, and a normative criterion.

Next: value risky prospects and design insurance.