Difficulty: Intermediate | Prerequisites: Understanding of contribution margin and basic cost behaviour.
This topic covers how a firm decides which products to make and in what quantities when resources are limited. It builds on the cost-behaviour foundations (fixed vs variable) and applies them to short-run decision-making. The core skill is ranking products by their contribution margin per unit of the binding constraint, then filling demand in that order. This is one of the most calculation-heavy areas on the exam and frequently appears as a multi-part problem.
When there is no constraint, produce everything up to demand. When a single resource is scarce, rank products by contribution margin per unit of that resource and produce in that order until the resource runs out. The shadow price of the constraint is the contribution margin per unit of resource of the last product produced (the marginal product). Special-order pricing must cover direct costs plus the opportunity cost of displaced production.
Contribution margin (CM)
Sales price minus all variable costs (direct materials + direct labour, and any other variable costs). This is the amount each unit contributes toward covering fixed costs and generating profit.
In simple terms: what is left from each sale after paying the costs that vary with production.
Contribution margin per unit of constraint
Contribution margin divided by the amount of the scarce resource the product requires. This is the metric used to rank products when a resource is limited.
Think of it as: how much profit you squeeze out of each minute (or hour, or kilogram) of the bottleneck.
Binding constraint (bottleneck)
The resource whose limited supply prevents the firm from producing everything the market demands. Only the binding constraint matters for product-mix decisions; non-binding constraints have excess capacity.
In simple terms: the thing you run out of first.
Shadow price
The maximum amount the firm would be willing to pay for one additional unit of the scarce resource. It equals the contribution margin per unit of constraint of the marginal product (the last product being produced, or the one being partially produced, in the optimal mix).
Think of it as: the value of loosening the bottleneck by one unit.
Optimal product mix
The combination of product quantities that maximises total contribution margin, given the constraint. Found by ranking products by CM per unit of constraint and filling demand from highest-ranked to lowest until the resource is exhausted.
Special order
A one-off order from an outside customer, typically at a non-standard price. The decision to accept depends on whether the price covers the incremental cost plus the opportunity cost of any displaced regular production.
Opportunity cost
The contribution margin lost on the regular product that must be cut back to free up capacity for the special order. If there is slack capacity, opportunity cost is zero.
In simple terms: what you give up by saying yes.
When the only limit is market demand, produce every unit you can sell, provided each product has a positive contribution margin.
Texasonic example:
DIP: CM = $45 - $11 - $17 = $17 per unit. Total = 30,000 x $17 = $510,000.
SOT: CM = $76 - $18 - $10 = $48 per unit. Total = 20,000 x $48 = $960,000.
PLCC: CM = $108 - $42 - $39 = $27 per unit. Total = 10,000 x $27 = $270,000.
Total monthly profit if all demand is met: $510,000 + $960,000 + $270,000 = $1,740,000.
When a resource is scarce, rank products by CM per unit of that resource, then allocate the resource to the highest-ranked product first, up to its demand, then the next, and so on.
Texasonic with 25,000 machine minutes available:
DIP: CM per minute = $17 / 0.25 = $68 per minute. Rank: 1st.
SOT: CM per minute = $48 / 0.75 = $64 per minute. Rank: 2nd.
PLCC: CM per minute = $27 / 0.50 = $54 per minute. Rank: 3rd.
Allocate minutes:
DIP: 30,000 units x 0.25 min = 7,500 minutes used. Remaining: 17,500 minutes.
SOT: 20,000 units x 0.75 min = 15,000 minutes used. Remaining: 2,500 minutes.
PLCC: 2,500 / 0.50 = 5,000 units produced (out of 10,000 demanded). Remaining: 0 minutes.
Optimal mix: 30,000 DIP, 20,000 SOT, 5,000 PLCC.
Maximum profit: (30,000 x $17) + (20,000 x $48) + (5,000 x $27) = $510,000 + $960,000 + $135,000 = $1,605,000.
The shadow price equals the CM per minute of the marginal product, which is the product being partially produced. Here, PLCC is the marginal product.
Shadow price = $54 per minute.
This means Texasonic would pay up to $54 for one additional machine minute, because that minute would produce 2 additional PLCC units (each taking 0.5 min), each contributing $27. So one minute yields $54.
Note from the Merton Truck case: if a department has excess capacity, the shadow price of an additional hour in that department is $0. There is no value in adding more of a resource you are not fully using.
A special order should be accepted if the price covers incremental costs plus the opportunity cost of displaced production.
Texasonic receives an order for 5,000 SDIP units:
Direct material: $12/unit
Direct labour: $20/unit
Machine time: 0.4 min/unit
Total machine time needed: 5,000 x 0.4 = 2,000 minutes
Which product is displaced? The lowest-ranked product currently in the mix: PLCC (the marginal product, producing only 5,000 of 10,000 demanded). Cut 2,000 / 0.50 = 4,000 PLCC units.
Opportunity cost per SDIP unit: displaced PLCC's CM per minute x SDIP minutes per unit = $54 x 0.4 = $21.60.
Indifference price = Direct costs + Opportunity cost = $12 + $20 + $21.60 = $53.60 per unit.
At $53.60, Texasonic earns exactly the same profit whether it accepts or rejects the order.
If the special order requires more capacity than the marginal product alone can free up, you must displace units from the next-highest-ranked product as well.
Texasonic, 10,000 SDIP (instead of 5,000):
Machine time needed: 10,000 x 0.4 = 4,000 minutes.
PLCC can release: 5,000 units x 0.5 min = 2,500 minutes. This fully eliminates PLCC from the mix.
Remaining minutes needed: 4,000 - 2,500 = 1,500 minutes. These come from SOT.
SOT displaced: 1,500 / 0.75 = 2,000 units.
Opportunity cost has two layers:
From PLCC: 2,500 min x $54/min = $135,000.
From SOT: 1,500 min x $64/min = $96,000.
Total opportunity cost: $231,000.
Direct costs: 10,000 x ($12 + $20) = $320,000.
Indifference price per unit: ($320,000 + $231,000) / 10,000 = $55.10 per unit.
Note: this is higher than the 5,000-unit case ($53.60) because the firm must now displace some SOT production, which has a higher CM per minute than PLCC.
\text{Contribution margin per unit} = \text{Price} - \text{Variable costs per unit}\text{CM per unit of constraint} = \frac{\text{CM per unit}}{\text{Constraint usage per unit}}\text{Shadow price} = \text{CM per unit of constraint of the marginal product}\text{Indifference price (special order)} = \text{Direct costs per unit} + \text{Opportunity cost per unit}\text{Opportunity cost per unit} = \text{CM/constraint of displaced product} \times \text{Constraint per unit of special order}When a large special order displaces more than one product, compute the opportunity cost in layers: first exhaust the lowest-ranked product, then move up to the next.
Product-mix optimisation under constraints is the logic behind every factory scheduler that decides which orders to run first when the shop floor is full. Airlines use the same principle when deciding which routes to serve with a limited fleet: contribution per seat-mile is the ranking metric, just as CM per machine-minute is here.
Special-order analysis is how contract manufacturers decide whether to take a one-off job: they quote the customer a price that covers direct costs plus the profit they forfeit by bumping a regular customer's order.
Students often rank products by contribution margin per unit rather than per unit of the constraint. The highest CM per unit is not necessarily the best use of a scarce resource. In the Texasonic example, SOT has the highest CM per unit ($48) but DIP ranks first when machine minutes are the constraint ($68/min vs $64/min).
Students often think the shadow price is the total profit from one more unit of the product. It is the CM earned from one additional unit of the resource, not one additional unit of a product.
Students forget that the shadow price changes when the constraint shifts. If you buy enough additional capacity that the binding constraint moves to a different resource, the shadow price from the original calculation no longer applies.
Students sometimes use allocated fixed costs in the special-order decision. Only incremental (variable) costs and opportunity costs matter. Fixed costs do not change with the special order.
Problem 3 on the practice exam (Texasonic) is a five-part product-mix problem. Expect a similar structure: compute CM, rank by CM per constraint unit, find the optimal mix, compute the shadow price, and analyse a special order.
The Giberson Glass Studio case tests the same concept: when prices are fixed, produce the product with the highest contribution margin per minute of the bottleneck resource first.
The Merton Truck case tests shadow prices: if a department has excess capacity, the price Merton would pay for an extra hour there is zero.
Special-order questions can span two products if the order is large enough. Be ready to compute a layered opportunity cost.
True or false: When choosing the optimal product mix under a single constraint, rank products by contribution margin per unit. (False, rank by CM per unit of the constraint.)
Fill in the blank: The shadow price of a constraint equals the CM per unit of constraint of the ______ product. (Marginal)
True or false: If a department has excess capacity, the shadow price of an additional hour in that department is zero. (True)
True or false: When evaluating a special order, fixed costs should be included in the analysis. (False)
Fill in the blank: A special order that displaces existing production has an opportunity cost equal to the ______ lost on displaced units. (Contribution margin)
Q: Texasonic has three products (DIP, SOT, PLCC) and 25,000 machine minutes available. DIP has CM $17 and uses 0.25 min; SOT has CM $48 and uses 0.75 min; PLCC has CM $27 and uses 0.5 min. What is the optimal product mix?
A: Rank by CM/min: DIP $68, SOT $64, PLCC $54. Produce 30,000 DIP (7,500 min), then 20,000 SOT (15,000 min), then 5,000 PLCC with the remaining 2,500 min. Maximum profit: $1,605,000.
Q: In the Giberson Glass Studio case, which product should Mr. Giberson produce first when prices are fixed?
A: The one with the highest contribution margin per minute. This is not necessarily the one with the highest CM per unit or the highest price.
Q: What is the maximum Texasonic would pay for one additional machine minute?
A: $54, the CM per minute of PLCC, which is the marginal product. An extra minute produces 2 more PLCC (at 0.5 min each), contributing $27 each, totalling $54.
Q: Texasonic receives a special order for 5,000 SDIP units requiring $12 DM, $20 DL, and 0.4 min machine time per unit. At what price is Texasonic indifferent?
A: Direct cost = $32. Machine time needed = 2,000 min. This displaces PLCC (the marginal product). Opportunity cost per unit = $54/min x 0.4 min = $21.60. Indifference price = $32 + $21.60 = $53.60.
Q: If the special order is for 10,000 SDIP instead, what is the indifference price?
A: Machine time needed = 4,000 min. PLCC can release only 2,500 min (5,000 units x 0.5 min). Remaining 1,500 min come from SOT. Opportunity cost = (2,500 x $54) + (1,500 x $64) = $135,000 + $96,000 = $231,000. Direct costs = 10,000 x $32 = $320,000. Indifference price = $551,000 / 10,000 = $55.10.
Product-mix decisions depend on having accurate contribution margins, which ties directly back to cost-system accuracy (ABC vs traditional). If your costing system misallocates overhead, your contribution margins will be wrong, and the product-mix ranking may be distorted.
Capacity management connects here as well: the shadow price tells you the value of additional capacity, which feeds into capital-budgeting decisions about whether to invest in more machines or hire more labour.
Joint-product decisions share the theme of opportunity cost. In joint costing, the sell-or-process-further decision uses incremental revenue vs incremental cost, ignoring sunk (joint) costs. In product-mix, the special-order decision uses incremental revenue vs incremental cost plus opportunity cost of displaced production.
Product mix, constrained optimisation, contribution margin per unit of constraint, bottleneck, binding constraint, shadow price, special order analysis, opportunity cost, indifference price, Texasonic, Giberson Glass Studio, Merton Truck, machine minutes, linear programming, scarce resources, capacity constraint, marginal product, ACC 359, ACC 387, managerial accounting, cost accounting, UT Austin, McCombs