ACCA PM Syllabus C. Decision-making Techniques - Shadow Prices (Dual Prices) - Notes ^{4 / 5}

Any scarce resource that is fully utilised in the optimal solution will have a shadow price.

It would be worth paying more than the ‘normal’ price to obtain more of the scarce resource because of the contribution foregone by not being able to satisfy the sales demand.

Therefore, if more critical (scarce) resource becomes available, then the feasible region would tend to expand and this means that the optimal point would tend to move outward away from the origin, thus earning more contribution.

Hence the shadow price of a binding constraint is the amount by which the total contribution would increase if one more unit of the scarce resource became available.

Management can use the shadow price as a measure of how much they would be willing to pay to gain more of a scarce resource over and above the normal price subject to any non-financial issues that may be present.

If the availability of a non-critical scarce resource increased then the feasible region would not tend to expand and therefore no more contribution could be earned. In this case, extra non-critical scarce resource has no value and a nil shadow price.

Calculating shadow prices

1. add one unit to the constraint concerned while leaving the other critical constraint unchanged

2. solve the revised simultaneous equations to derive a new optimal solution

3. calculate the revised optimal contribution and compare to the old contribution. 

   The increase in contribution is the shadow price for the constraint under consideration.

Illustration

Following from the previous illustration, find the shadow price of Material X.

20A + 30B = 3001  (add 1 kg to Material X)
15A + 50B = 2700

20A + 30B = 3001 ….. x5
15A + 50B = 2700 ….. x3

100A + 150B = 15005
minus  45A + 150B =   8100

Totals   55A  /    =   6905
      
A   =   125.55

100 A       + 150B = 15005
100(125.55) + 150B = 15005

B = 16.33

Therefore New Contribution = 
20 (125.55) + 45 (16.33)

= 2511+ 734.85
= 3245.85
                  
Old Contribution = 3245.20
                       
Shadow price =  0.65

Extra contribution / kg of Material X