ACCA PM December 2026: Shadow Prices in Linear Programming (Why They Are NOT the Contribution)

Shadow price = extra contribution from ONE more unit of the scarce resource. Not the contribution per unit.

That one distinction is where PM students lose the marks on linear programming. If your "shadow price" is the same number as a product's contribution, it is almost certainly wrong.

What the examiner actually wants

In PM limiting-factor questions, the shadow (dual) price is tested as a number and as a sentence. The number is the increase in total contribution if you get one extra unit of a binding constraint. The sentence is what it means for the business: the maximum premium above the normal cost you would pay for that extra unit and still be no worse off.

Three rules come up again and again. A constraint that is not binding (there is slack) has a shadow price of zero. The shadow price is only valid within a range, because once another constraint starts to bite, the value changes. And you must always state it per unit of the resource, such as per machine hour or per labour hour.

Worked example

X earns £10 contribution and uses 2 machine hours and 1 labour hour. Y earns £12 and uses 3 machine hours and 1 labour hour. Machine hours are limited to 600 and labour hours to 250. Both constraints bind at the optimum: X = 150, Y = 100, total contribution £2,700.

Wrong answer: "The shadow price of a machine hour is £10 / £12 / £4 (contribution divided by hours)." That is contribution per hour, and it ignores the other constraint completely.

Correct answer: add one machine hour (601) and re-solve the two constraints simultaneously. Y rises to 101, X falls to 149, and contribution becomes £2,702. Shadow price of a machine hour = £2. Do the same with 251 labour hours: X = 153, Y = 98, contribution £2,706, so the shadow price of a labour hour = £6.

Quick check: solve 2a + b = 10 and 3a + b = 12 (one equation per product, using each resource's usage). You get a = £2 and b = £6. The dual prices times the resource limits should give the original contribution: (2 x 600) + (6 x 250) = £2,700. If it does not, you have an error.

What to do

1. Use the "add one unit and re-solve" method. It is slower than the dual equations but it is almost impossible to get wrong under pressure, and it earns method marks even if your arithmetic slips.

2. Always finish with the interpretation. Write one sentence: "The business should pay up to £2 per hour above the current cost for extra machine time." That sentence is routinely worth its own mark.

3. Check for slack before you calculate. If a resource is not fully used at the optimal point, say its shadow price is nil and move on. Do not waste five minutes solving for it.

The bottom line

Linear programming is a predictable source of marks in PM Section C because the method is fixed. Learn the three rules above and you turn a feared topic into a checklist.

Shadow price: one more unit in, extra contribution out. Say it in a sentence and bank the mark.