# All-units quantity discounts

How all-units price breaks differ from incremental tiers, with explicit thresholds, units and purchase-charge arithmetic.

Reviewed: 2026-10-30

Tags: commercial, reference, quantity-pricing

## Meaning and related terms

An **all-units quantity discount** applies a qualifying lower rate to the entire eligible purchase quantity, including units below the threshold. An **incremental** discount applies the lower rate only to the quantity beyond a breakpoint; earlier units retain their earlier rates. Chris Caplice's MIT logistics lecture distinguishes these two structures. [MIT, October 2006, slides 10–14][mit]

A marginal purchase charge is the change in the total goods charge as the quantity changes. For the other constraints on an order, see [purchasing quantities and price breaks](../../sourcing/purchasing-units-minimums-and-price-breaks.md).

## Hypothetical totals and marginal change

In this invented example, the all-units charge drops **from 990 CU at 99 pieces to 900 CU at 100**. Under the incremental schedule, 100 pieces cost **1,000 CU**. Both schedules use 10 CU per piece before the break and 9 CU for the eligible quantity. The all-units rate starts at 100 pieces. The incremental rate starts with piece 101. CU means hypothetical currency units, and the comparison covers goods charges alone.

| Quantity | All-units goods charge | All-units average | Incremental goods charge | Incremental average |
| --- | --- | --- | --- | --- |
| 99 pieces | 990 CU | 10 CU/piece | 990 CU | 10 CU/piece |
| 100 pieces | 900 CU | 9 CU/piece | 1,000 CU | 10 CU/piece |
| 120 pieces | 1,080 CU | 9 CU/piece | 1,180 CU | approximately 9.833 CU/piece |

The 100th piece changes the charge by **−90 CU** under all-units pricing and **+10 CU** under incremental pricing. Piece 101 adds **9 CU** under incremental pricing.

## Quantity and threshold conventions

The following models use one eligible item, one quantity unit and one currency. Quantity `q` is the number of units in one purchase; `b` is a positive breakpoint in the same unit. Rates `p0` and `p1` are currency per unit, with `0 < p1 < p0`. Charges are currency per purchase. These are mathematical assumptions, not supplier terms.

**Boundary limitation:** MIT slide 10 describes exceeding a breakpoint, while slide 11 prints both price ranges with equality at that breakpoint. Slide 12 gives an example explicitly using `q ≥ 500`. Those lecture conventions do not resolve an offer's wording. The models here assign the all-units discount at `q ≥ b`, following the nonoverlapping all-units tariff in O'Brien's model. [MIT, slides 10–12][mit]; [O'Brien, section III, printed pp. 7–8][obrien]

For whole pieces, “100 or more” includes 100, whereas “more than 100” begins at 101. For continuously measured quantity, `q > 100` has no smallest qualifying value unless a measurement increment is also specified. These are deductions from the inequalities. A threshold in pieces and a rate per kilogram do not form a complete price function without a defined conversion.

## Two purchase-charge models

**Simplified restatement:** exclude setup, freight, tax, minimum charges and rounding. The two-price tariffs in O'Brien's section III become the following when a separate fixed fee is zero. [O'Brien, printed pp. 7–8][obrien]

| Quantity | All-units goods charge `G_A(q)` | Incremental goods charge `G_I(q)` |
| --- | --- | --- |
| `0 ≤ q < b` | `p0 × q` | `p0 × q` |
| `q ≥ b` | `p1 × q` | `p0 × b + p1 × (q − b)` |

At `q = b`, the incremental formula still charges the first `b` units at `p0`; only units beyond `b` receive `p1`. In the all-units column, every unit receives `p1` as soon as the threshold is met.

**Conditional deductions:** within either all-units range, the average goods charge `G_A(q)/q` equals that range's rate for `q > 0`. Above the incremental breakpoint, the average is `p1 + (p0 − p1) × b/q`, which exceeds `p1` for finite positive `q`. Thus a marginal tier rate and a blended average answer different questions. Wang's original supplier-allocation paper likewise distinguishes all-units and incremental costs, and defines average cost using the costs included in its model. [Wang, 2023, equations 6, 13 and 35–36][wang]

For multiple incremental tiers, the same arithmetic sums each completed tier's quantity times its rate, then adds the current tier's quantity times its rate. Multiple all-units breaks instead select the applicable rate for the entire eligible quantity. This is a synthesis of the two structures, also illustrated by MIT's multiple-break incremental diagram. [MIT, slides 13–14][mit]

For a continuous quantity model, the all-units charge has a downward jump of `b × (p0 − p1)` at the breakpoint. Its derivative within a range equals that range's rate, but no ordinary derivative exists at the jump. For whole units with a positive integer breakpoint `b`, the crossing change from `b − 1` to `b` units is `b × p1 − (b − 1) × p0`; it is negative only when `b × (p0 − p1) > p0`. These are algebraic deductions from the table, distinguishing a continuous discontinuity from a finite one-unit change. O'Brien discusses the threshold discontinuity; Wang plots the contrasting cost structures. [O'Brien, introduction][obrien]; [Wang, figures 3–4][wang]

## Charge scope and documentary limits

Daniel O'Brien describes both discounts applied at purchase and periodic rebates paid when quantity thresholds are met; he calls the latter form **retroactive rebates**. The timing of payment and the quantity receiving the discount are separate characteristics. [O'Brien, revised working paper, introduction][obrien]

Wang's model includes a separate setup cost, and its all-units average-cost formula is the selected unit rate plus setup cost divided by quantity. That illustrates why an average including other charges differs from a goods rate. [Wang, table 2 and equation 35][wang] **Conditional extension:** if a fixed charge `F` applies once per purchase, the total is `F + G(q)` and its average is `F/q + G(q)/q` for `q > 0`. Additional charges or rounding would require their own stated rules; the examples above do not incorporate them.

A per-purchase model establishes no aggregation across orders, deliveries, revisions or a rebate period. Nor does its arithmetic establish price validity, currency conversion, stock availability, contract formation or enforceability. Those facts are outside these sources' price functions. No supplier schedule or actual customer terms are represented here, and no order quantity is recommended.

## Sources and coverage

Reviewed 30 October 2026 PDT. These sources establish terminology and mathematical structures, not current law or a binding offer in any jurisdiction.

| Source | Version and evidence limits |
| --- | --- |
| [Chris Caplice, MIT, *Inventory Management: Extensions to EOQ*][mit] | ESD.260/15.770/1.260, October 2006; slides 10–15 inspected. US university teaching material. Its inventory-optimization procedures and numerical recommendations are outside this article's scope. |
| [Daniel P. O'Brien, *All-units Discounts and Double Moral Hazard*][obrien] | FTC Bureau of Economics working paper 316; cover revised July 2014, internal title dated June 2014. Author's preliminary research, expressly not necessarily FTC views. Used for definitions and price functions, not its historical legal discussion. The [FTC HTML summary](https://www.ftc.gov/reports/all-units-discounts-double-moral-hazard) differs from the PDF on entry-related conclusions; neither conclusion is adopted. |
| [Qing Wang, *Flexible supplier selection and order allocation in the big data era with various quantity discounts*][wang] | PLOS ONE 18(3), e0283585, 27 March 2023; open-access original research. Figures 3–4, tables 2–3 and relevant equations inspected. A mathematical supplier-allocation model, not evidence of real supplier prices or acceptance conditions. |

The cited working paper is the inspected 2014 version, not a claim to have reviewed the full 2017 journal publication. Direct FTC PDF download returned HTTP 403; the retrieved browser text and the [GovInfo preserved PDF](https://www.govinfo.gov/content/pkg/GOVPUB-FT-PURL-gpo65721/pdf/GOVPUB-FT-PURL-gpo65721.pdf), whose relevant pages were visually inspected, provide the version evidence. No present-day legal classification or financial-reporting treatment is inferred.

[mit]: https://ocw.mit.edu/courses/esd-260j-logistics-systems-fall-2006/4851a27cb396c065d7b7c0fafa50d039_lect8.pdf
[obrien]: https://www.ftc.gov/system/files/documents/reports/all-units-discounts-double-moral-hazard/wp316.pdf
[wang]: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0283585