What changes?
All changes are income-label offers minus one pooled offer. Income and valuation are separate: either group can contain buyers who value the product highly.
Poorer consumers' surplus
Richer consumers' surplus
Seller's profit
Net total surplus
Calculating the comparison…
Who keeps more surplus?
Surplus is the buyer's value from the purchase minus the payment. Bars show the change per person of each type, averaged over label errors.
The full accounting
| Measure | One offer | Income labels | Change |
|---|
Net total surplus adds consumer surplus and profit, then subtracts administration cost. The weighted criterion below applies your chosen weights; it does not change the seller's offers.
What does the seller offer?
| Menu | High-value share | Low-value buyer quality / payment | High-value buyer quality / payment |
|---|
Where does the conclusion change?
Vary one assumption while holding all the others at your current settings. Each line is a change per person in the full population; the dotted vertical line marks your current setting. Curves connect 51 sampled settings; switches in who is served can create jumps.
Read a sample of the plotted numbers
| Assumption | Consumer surplus Δ | Net total Δ | Weighted Δ |
|---|
How the calculation works
The dashboard makes the overlapping-income-groups mechanism in the OpportunityCard technical appendix runnable. The price-and-quality view uses a new, explicit quadratic-cost specification. It illustrates the mechanism; it is not a numerical reproduction of every archival model.
1. People, information and costs
A share s of the population is poorer. Within each income group, a chosen share has high valuation. Low valuation is θL; high valuation is θH = θL + gap. These are preferences for the product, not income levels or budget limits.
A buyer's utility is θq − t: valuation times quality, minus payment. Producing quality q costs kq²/2. The firm is a monopolist, knows the distribution of preferences, and cannot observe an individual's valuation. Buyers can decline to buy. Resale and sharing of offers across labels are ruled out.
With income labels, the probability of a correct label is the selected accuracy. Errors depend on income, not valuation within an income group. The firm recomputes the high-valuation share in each label and optimizes its menu. At 50% accuracy the labels carry no information about income; at 100% they reveal the income group exactly.
2. Price and quality menus
For a segment whose high-valuation share is h, the optimal menu is:
qL = max(0, [θL − h/(1−h) × (θH − θL)] / k)
tL = θL × qL
tH = θH × qH − (θH − θL) × qL
At h = 1, the unused low-type offer is set to zero. The high type's quality is efficient in this specification. Low-type quality is reduced to limit the rent that high types keep. If the expression for low-type quality is negative, low types are excluded.
Low types keep zero surplus in this model. High types keep (θH − θL)qL, just enough to stop them choosing the basic offer. This is why better basic quality can raise high types' surplus while low types pay for their quality improvement. Ties are resolved in favor of the designated menu entry.
3. One fixed product
The alternative view fixes quality at one, so unit cost is k/2. For each segment, the seller compares selling to everyone at θL, selling only to high types at θH, and making no sales. It chooses the most profitable option. A positive-profit tie goes to the lower price; zero profit goes to no trade. There is no choice of product quality in this view.
4. Welfare and the comparison
The one-offer baseline is optimized for the pooled population. The income-label case optimizes each label's menu. The firm can reproduce its pooled offer, so information cannot reduce its gross profit here. Administration cost is a separate social resource cost, charged per person in the full population only in the income-label case; it is not a fee paid by buyers or the firm.
Weighted criterion = wP × poorer consumers' total surplus
+ richer consumers' total surplus + λ × profit − administration cost
The weighted criterion is a judgment you can change. Poorer and richer consumer surpluses are aggregated using their population shares before applying the weights. Weight one on all surpluses reproduces net total surplus. These weights do not model marginal utility of income or a tax-and-transfer system.
What this does not solve
Everyone receives an observed income label. Optional disclosure, endogenous take-up, stigma and privacy costs are not solved. Nor are competition, a discount-only constraint with a changing standard price, budgets, income reporting, labor supply, multiple goods or a government's optimal credential rule. A more accurate label is not itself proof of a better policy.
Defaults are illustrative, not estimated from data. Welfare changes can reverse across assumptions and across the two model views. Formal equations and automated numerical checks make the assumptions inspectable; they do not establish empirical relevance.
Further reading: proposed model extensions · literature map · original 2015 discussion paper · dashboard source and calculation notes.
Questions to carry into research
Explore here
- Who gains within each income group? Separate valuation from income, and inspect both the type-level changes and the group averages.
- How informative must the label be? Vary accuracy and compare extra gross surplus with the administration cost.
- Does quality choice change the story? Compare screening with the fixed-product view.
- Whose surplus counts? Change the poorer-consumer and profit weights while keeping the seller's incentives fixed.
Require a further model or evidence
- Who would disclose a credential voluntarily? The firm learns from both disclosure and non-disclosure.
- Can a discount-only rule protect buyers? Firms can change the standard offer and product range.
- How do firms compete and people respond? Entry, switching, stigma, reporting and privacy may alter the result.
- Which settings can identify these effects? See the proposed empirical designs, with their data requirements and identification threats.