Yonsei University
Chae-Yeon Xon
Yonsei University, School of Economics
Home/Resources/Structural Models and Economic Computation/IV-02
Demand, Supply, Bundles, and Welfare · Chapter IV-02

Residual-Income Demand and Approximation Domains

When utility depends on income remaining after purchase, price enters through a nonlinear residual-income term. Replacing that term by a linear price coefficient is a local approximation whose error depends on price-income ratios, affordability, and heterogeneity. The approximation domain must therefore travel with any derivative, share, or welfare calculation.

Conceptual map

  1. IV-02.01Exact residual-income utility
  2. IV-02.02Linear price branches
  3. IV-02.03Taylor error
  4. IV-02.04Affordability
  5. IV-02.05Income heterogeneity

1. Affordability is part of the utility domain

Residual-income utility is defined only when income after purchase is positive. Its local price slope and curvature vary with income and the expansion point.

For income \(y>0\) and price \(p<y\), consider systematic utility difference \(v(p;y)=q+\log(y-p)-\log y\) relative to an outside option. Its derivatives are \(v^{\prime}=-1/(y-p)\) and \(v^{\prime\prime}=-1/(y-p)^{2}\).

Proposition 1 · Controlled local price approximation

Fix \(p_{0}<y\). If every price between \(p_{0}\) and \(p\) leaves residual income at least \(m>0\), then

\[|v(p;y)-v(p_{0};y)+(p-p_{0})/(y-p_{0})|\le (p-p_{0})^{2}/(2m^{2}).\](1)

Proof. Taylor's theorem gives a remainder \(v^{\prime\prime}(\xi)(p-p_{0})^{2}/2\) for an intermediate \(\xi\). The residual-income bound controls \(|v^{\prime\prime}(\xi)|\) by 1/\(m^{2}\). ∎

2. A common linear coefficient fits different income groups poorly

Take quality \(q=1\), incomes 10 and 20 with equal population weights, base price \(p_{0}=2\), and binary logit choice. At price four, exact purchase probabilities are approximately 0.61991 and 0.68500. Linearizing each group's utility at price two gives probabilities approximately 0.62875 and 0.68644.

The exact own-price derivative is \(-P(1-P)/(y-p)\). The linear approximation uses the fixed denominator \(y-p_{0}\) and its own approximated probability. Both the share error and derivative error grow as the evaluation price approaches an income boundary.

3. Aggregate demand integrates affordability and choice

For income distribution \(F\), aggregate share is \(\int 1\{p<y\}\,\Lambda[v(p;y)]\,dF(y)\), where Λ is the logistic cdf. A mass of consumers can cross the affordability threshold when price changes. Differentiating under the integral then requires treatment of the moving support boundary in addition to the smooth within-support derivative.

A linear price coefficient estimated near one price and income distribution represents a local average slope. Reusing it under a different price-income support alters both curvature and the set of available purchases.

4. The log boundary cannot be crossed by Taylor expansion

Failure case · Linear utility evaluated beyond affordability

The linear approximation remains finite at \(p\ge y\), while \(\operatorname{log}(y-p)\) is undefined. The approximation has crossed the domain of the structural utility.

At \(p=y\), curvature diverges and no positive uniform residual-income bound \(m\) exists. At zero or negative prices, the formula remains mathematically defined when residual income is positive, while the economic environment must determine whether such prices are admissible.

5. Implementation, exercises, and sources

Check affordability before evaluating logarithms, retain the income distribution and weights, and report the smallest residual income on every counterfactual path. Compare exact and linear utilities, probabilities, and derivatives at the same draws and choice sets.

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Exercises

  1. Reproduce the two-income probabilities in Section 2.
  2. Compute exact and linearized derivatives at prices two and four.
  3. Add an income mass at four and examine one-sided aggregate demand as price crosses four.
Partial solutions

1. Apply Λ to 1+\(\operatorname{log}(1-p/y)\) and to its first-order expansion at two. 2. Use \(-P(1-P)/(y-p)\) for the exact model. 3. The mass becomes unaffordable at the boundary, creating a discrete share change under strict affordability.

  1. Jerry A. Hausman and David A. Wise (1978), “A Conditional Probit Model for Qualitative Choice: Discrete Decisions Recognizing Interdependence and Heterogeneous Preferences,” Econometrica 46, 403–426.Observed heterogeneity in discrete choice.
  2. Kenneth Train, Discrete Choice Methods with Simulation.Aggregation over observed heterogeneity.
  3. Walter Rudin, Principles of Mathematical Analysis, chapters on Taylor's theorem.Finite-radius remainder control.

6. Audit checkpoint

State income and price supports, strict affordability rule, base price, smallest residual income, exact utility, linear coefficient, remainder bound, heterogeneity weights, moving support boundary, and exact versus approximated derivatives.

7. Scope boundary

The chapter treats logarithmic residual-income utility and a local price approximation. Flexible demand estimation and general expenditure systems require separate models.

Prerequisites