# Graduated quantity discounts

Incremental quantity tiers, threshold conventions and the difference between marginal rates, blended averages and total goods charges.

Reviewed: 2026-10-30

Tags: commercial, pricing, reference

## Meaning and related terms

This article uses **graduated quantity discount** for an incremental schedule: a lower rate applies only to quantity beyond a breakpoint, while earlier quantity keeps its earlier rate. MIT's October 2006 lecture distinguishes this from an all-units discount, which reprices the whole eligible quantity. Its incremental diagram joins successive purchase-charge segments. [Caplice, slides 10 and 13][mit]

“Graduated,” “tiered” and “declining-block” describe the structure here; the cited sources call it **incremental**. A label alone does not establish whether a particular schedule is incremental or retroactive. See the separate [all-units quantity discounts reference](all-units-quantity-discounts.md) for the latter structure.

## Hypothetical arithmetic

Under these invented rates, **250 pieces cost 2,300 CU**. Charge 10 CU per piece for the first 100, 9 CU for the next 100 and 8 CU for the final 50. The average is **9.2 CU per piece**. CU means hypothetical currency units, and the example covers goods charges alone.

| Quantity | Goods charge | Average goods charge |
| --- | --- | --- |
| 100 pieces | `100 × 10 = 1,000 CU` | 10 CU/piece |
| 101 pieces | `1,000 + 1 × 9 = 1,009 CU` | approximately 9.990 CU/piece |
| 200 pieces | `1,000 + 100 × 9 = 1,900 CU` | 9.5 CU/piece |
| 250 pieces | `1,000 + 900 + 50 × 8 = 2,300 CU` | 9.2 CU/piece |

At 250 pieces, the marginal rate is 8 CU. An [all-units schedule](all-units-quantity-discounts.md) granting 8 CU for every piece would charge **2,000 CU**, or 300 CU less.

## Quantity, rate and threshold

The model below concerns one purchase of one priced quantity. Quantity `q` and breakpoints use the same unit, such as pieces or kilograms; rates use one currency per that unit. A threshold in pieces and a rate per kilogram require a defined conversion before they describe a complete charge.

**Model convention:** the first `b` units receive rate `p0`; quantity beyond `b` receives `p1`, with `b > 0` and `0 ≤ p1 < p0`. At exactly `q = b`, none of the quantity receives `p1`. For whole pieces and a positive integer `b`, the first lower-priced piece is piece `b + 1`. For continuously measured quantity, any positive amount beyond `b` receives the lower rate; there is no smallest such amount without a measurement increment.

These boundaries make “first 100 pieces, then subsequent pieces” explicit. A rule saying “the 100th piece receives the lower rate” instead leaves only 99 pieces in the first block. The distinction follows from counting; generic breakpoint terminology does not resolve an actual offer's inclusivity.

## Goods-charge model

**Explanatory synthesis:** let `G(q)` be the goods charge, excluding setup, freight, tax, minimum charges and rounding. For `q ≥ 0`:

| Quantity | Graduated goods charge `G(q)` |
| --- | --- |
| `0 ≤ q ≤ b` | `p0 × q` |
| `q > b` | `p0 × b + p1 × (q − b)` |

This restates the joined incremental segments in MIT's diagram. Wang's original supplier-allocation model similarly adds the accumulated charge for completed intervals to the current interval's charge; its setup cost is separate. [Caplice, slide 13][mit]; [Wang, table 2 and equation 13][wang]

For multiple blocks, define finite boundaries `0 = b0 < b1 < … < bm`, rates `r1, …, rm` for those blocks, and a final rate `r(m+1)` beyond `bm`. The quantity charged in block `i` is `max(0, min(q, bi) − b(i−1))`; the last block contains `max(0, q − bm)`. Multiplying each block's quantity by its rate and summing gives the goods charge. This is a mathematical generalization of the two-block model, with every boundary and `q` expressed in the same unit.

**Conditional deductions:** the two-block charge is continuous at `b`: both expressions give `p0 × b`. Its slopes are `p0` below and `p1` above the break, so it has a corner rather than a downward jump. With nonnegative rates, adding quantity cannot reduce this goods charge. For whole pieces and integer `b`, `G(b) − G(b − 1) = p0`, while `G(b + 1) − G(b) = p1`. These finite changes differ from a derivative at the corner.

For the two-block model above `b`, the blended goods average is `G(q)/q = p1 + (p0 − p1) × b/q`. It exceeds `p1` at finite `q` and approaches `p1` as `q` increases without bound. This is an algebraic deduction under the stated assumptions; a finite final tier or another price change limits its applicability. The average is undefined at `q = 0`.

## Additional charges and applicability

Wang's equation 36 divides accumulated interval charges, current-interval charges and setup cost by quantity. Thus its average includes more than the marginal tier rate. [Wang, equation 36][wang] **Conditional extension:** if a fixed charge `F` applies once to a positive purchase quantity, total charge is `F + G(q)` and average charge is `F/q + G(q)/q`. The accumulated charge from earlier blocks is not an additional setup fee.

The examples establish no aggregation across purchase orders, deliveries, products or rebate periods. Currency, eligible quantity, validity, conversions, fixed charges and rounding remain properties of a particular schedule. No actual customer terms, buying quantity, inventory policy, contract enforceability or financial-reporting treatment is established here.

## Sources and coverage

Reviewed 30 October 2026 PDT. Both sources describe mathematical structures; neither supplies current manufacturing market prices or a binding commercial offer.

| Primary source | Version and inspected evidence |
| --- | --- |
| [Chris Caplice, MIT, *Inventory Management: Extensions to EOQ*][mit] | ESD.260/15.770/1.260 Logistics Systems, October 2006; cover and slides 10, 13–15 visually inspected. Used for discount structures; inventory optimization and order recommendations are outside this article. |
| [Qing Wang, *Flexible supplier selection and order allocation in the big data era with various quantity discounts*][wang] | PLOS ONE 18(3), e0283585, published 27 March 2023; figures 3–4, tables 2–3 and equations 13 and 36 visually inspected in the open-access PDF. A supplier-allocation research model, not evidence of any supplier's accepted terms. |

[mit]: https://ocw.mit.edu/courses/esd-260j-logistics-systems-fall-2006/4851a27cb396c065d7b7c0fafa50d039_lect8.pdf
[wang]: https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0283585