Consumer Welfare and Compensating Variation

Conceptual map

IV-05.01The valuation object
IV-05.02Log-sum welfare
IV-05.03Nonlinear income effects
IV-05.04Ex ante and ex post welfare
IV-05.05Coherent counterfactuals

1. Log-sum welfare belongs to a quasilinear utility model

A welfare formula inherits the utility, shock scale, income coefficient, normalization, and choice set that generated demand.

Let $U_{ij}=\delta _{j}-\alpha p_{j}+\varepsilon _{ij}$ , $\alpha>0$ , with iid standard type-I extreme-value shocks and a fixed finite choice set. Up to a policy-invariant constant, expected maximum utility is log $\sum _{j}\operatorname{exp}(\delta _{j}-\alpha p_{j})$ . The compensating monetary amount for a harmful price change from 0 to 1 is

Proposition 1 · Quasilinear log-sum compensating variation

\[CV=[\operatorname{log} \sum e^{\delta _{j}-\alpha p_{j}^{0}}-\operatorname{log} \sum e^{\delta _{j}-\alpha p_{j}^{1}}]/\alpha .\](1)

Proof. Adding income compensation $$c$$ raises every post-policy utility by $\alpha c$ . Equate pre-policy and compensated post-policy expected maxima and solve for $$c$$ . ∎

2. A one-product price change

Take an outside option with utility zero, one product with $\delta=2$ , $\alpha=1$ , and price rising from one to 1.5. The inclusive values are $\operatorname{log}(1+e)\approx1.31326$ and $\operatorname{log}(1+e^{0.5})\approx0.97408$ . Compensating variation is approximately 0.33918 in the price currency.

Changing the shock scale or marginal utility of income changes the monetary conversion. Product entry changes the set inside each logarithm and must be reflected in both demand and welfare.

3. Residual-income utility requires a root

For consumer income $$y$$ and realized shocks, let utility be $q_{j}+\operatorname{log}(y-p_{j})+\varepsilon _{j}$ for affordable products, with a consistently specified outside option. Consumer-specific compensation $$c$$ solves equality between the pre-policy maximum and the post-policy maximum evaluated at income $$y+c$$ .

When indirect utility is continuous and strictly increasing in compensated income on a bracket containing the solution, bisection gives a unique root. Aggregate welfare averages consumer or draw-specific compensation with declared population weights. A constant $\alpha$ log-sum conversion from another utility branch does not apply.

4. Affordability and changing choice sets delimit the root

Failure case · Demand and welfare evaluated under different utilities

Shares generated by residual-income utility and welfare computed from a constant- $\alpha$ log sum combine two structural models. The resulting number lacks a common indirect-utility interpretation.

A root bracket must respect positive residual income for every alternative treated as available. A product crossing affordability can change the choice set discontinuously. At a tie in realized maximum utility, the value remains well defined while the chosen alternative may be set valued. If indirect utility has flat segments or nonmonotone income effects, uniqueness requires a separate argument.

5. Implementation, exercises, and sources

Call the same utility function and shocks used by demand, carry the same alternative availability, and solve each nonlinear compensation equation within an admissible income bracket. Report root residuals, failed brackets, population weights, and distributional summaries alongside the mean.

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Exercises

Reproduce the log-sum calculation in Section 2.
Derive (1) with shock scale $\sigma$ and systematic utility divided by $\sigma$ .
Construct a residual-income example where a product becomes affordable only after compensation.

Partial solutions

1. Subtract the post-policy inclusive value from the pre-policy value because $\alpha=1$ . 2. Expected maximum utility carries factor $\sigma$ ; monetary conversion divides by $\alpha$ , giving $\sigma/\alpha$ times the difference in log sums formed with utilities divided by $\sigma$ . 3. Choose a post-policy price above initial income and a compensation bracket that crosses that price.

Daniel McFadden (1978), “Modelling the Choice of Residential Location,” in Spatial Interaction Theory and Planning Models.Random-utility inclusive value.
Kenneth Small and Harvey Rosen (1981), “Applied Welfare Economics with Discrete Choice Models,” Econometrica 49, 105–130.Log-sum welfare and compensating variation.
Kenneth Train, Discrete Choice Methods with Simulation, chapters on welfare.Simulation of individual welfare changes.

6. Audit checkpoint

State utility, shock scale and draws, marginal utility of income, price currency, pre- and post-choice sets, affordability, normalization, compensation sign, root bracket, monotonicity, residual tolerance, population weights, and aggregation target.

7. Scope boundary

The chapter covers consumer compensating variation in quasilinear logit and nonlinear residual-income utility. Producer surplus and social welfare weights require additional definitions.