Residual-Income Demand across Implementations

Conceptual map

1. V-10.01Exact and approximate operators
2. V-10.02A finite demand witness
3. V-10.03Transparent reconstruction
4. V-10.04Downstream propagation
5. V-10.05Reproducibility boundaries

1. Exact and linear demand implementations have different domains

A first-order price term can match residual-income utility at a base price while changing affordability, derivatives, shares, markups, and welfare away from that point.

For income \(y>0\), quality \(q\), and price \(p<y\), define exact utility relative to the outside option as \(u(p)=q+\log[(y-p)/y]\). Around base price \(p_{0}<y\), the linear implementation is

2. The remainder is controlled by residual income

A uniform bound follows if \(y-\xi\ge r>0\) throughout the domain: the absolute utility error is at most \((p-p_{0})^{2}/(2r^{2})\). Without a positive residual-income margin, no finite uniform bound of this form exists.

3. A two-income witness propagates downstream

Set \(q=1\), \(p_{0}=2\), and \(p=4\). With a zero-utility outside option and standard logistic shock difference, purchase probability is Λ(\(u\)). At income 10, exact and linear shares are approximately 0.619912 and 0.628750. At income 20, they are 0.685002 and 0.686440.

The exact price derivative is \(-1/(y-p)\), while (1) uses constant derivative \(-1/(y-p_{0})\). At income 10 and price four these are \(-1/6\) and \(-1/8\). Multiplying by \(\Lambda(u)[1-\Lambda(u)]\) gives different demand slopes. Those slopes enter markup inversion; share differences alter purchaser composition; utility differences alter compensating variation. Agreement at \(p_{0}\) does not reconcile these downstream objects.

4. Affordability is an economic support restriction

As \(p\) approaches \(y\) from below, exact utility tends to \(-\infty\) and purchase probability tends to zero. At \(p=y\), the logarithm is undefined; a model can define the alternative as unavailable by a separate boundary rule. Income support must be checked household by household. Zero purchase shares make conditional purchaser moments undefined. A singular multi-product demand Jacobian blocks unconstrained markup inversion. Numerical log evaluation should use stable log1p transformations when price-income ratios are small.

5. Implementation, exercises, and sources

Locate every price-utility function, record units and affordability checks, and test exact and approximate values at the base, interior, and near-boundary points. Propagate both implementations through shares, demographic moments, derivatives, supply first-order conditions, and welfare. Report the maximum domain-specific discrepancy and identify which results rely on the approximation.

Exercises

1. Reproduce all four shares in Section 3.
2. Compute the exact and linear share derivatives at income 10.
3. Find a price at which the linear model gives positive demand while exact residual-income utility is unavailable.

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.Choice probabilities and aggregation.
3. Kenneth A. Small and Harvey S. Rosen (1981), “Applied Welfare Economics with Discrete Choice Models,” Econometrica 49, 105–130.Welfare propagation from utility.