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4.1 Linear Functions & The High-Low Method
ACCA Applied Knowledge — Management Accounting MA
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1 Chapter 1: Nature, Purpose & Source of Management Information
1.1 Role of Management Accounting, Planning Levels & Information Quality 1.2 Sources of Data, Big Data & Data Capture Costs 1.3 Cost Classification, Cost Behaviour & Coding Systems
2 Chapter 2: Cost Centres, Cost Units & Presenting Information
2.1 Cost Objects, Responsibility Centres & Managerial Needs 2.2 Presenting Management Information & Visualisation
3 Chapter 3: Sampling Methods, Statistical Summaries & Spreadsheets
3.1 Sampling Methods & Practical Selection 3.2 Data Summarisation, Dispersion & Probability in Decisions 3.3 Normal Distribution & Spreadsheet Applications in Accounting
4 Chapter 4: Forecasting Techniques, Time Series & Index Numbers
4.1 Linear Functions & The High-Low Method 4.2 Correlation, Linear Regression & Product Life Cycle 4.3 Time Series Analysis & Index Numbers
5 Chapter 5: Accounting for Materials & Inventory Control
5.1 Material Procedures, Physical Control & Inventory Valuation 5.2 Economic Order Quantity & Bulk Purchase Discounts 5.3 Economic Batch Quantity & Inventory Reorder Levels
6 Chapter 6: Accounting for Labour & Overhead Costs
6.1 Accounting for Labour, Remuneration Systems & Labour Ratios 6.2 Overhead Allocation, Apportionment & Reciprocal Reapportionment 6.3 Overhead Absorption Rates & Under/Over Absorption
7 Chapter 7: Marginal Costing & Absorption Costing
7.1 Marginal Costing, Contribution & Cost-Volume-Profit Analysis 7.2 Profit Comparison & Reconciliation Between Absorption and Marginal Costing
8 Chapter 8: Costing Methods & Alternative Principles
8.1 Job Costing, Batch Costing & Service/Operation Costing 8.2 Process Costing, Losses, Gains & Joint/By-Products 8.3 Alternative Cost Accounting: ABC, Target Costing & Life-Cycle Costing
9 Chapter 9: Budgetary Planning, Preparation & Behaviour
9.1 Budgetary Framework, Administration & The Principal Budget Factor 9.2 Functional Budgets, Cash Budgets & Master Budgets 9.3 What-If Analysis, Motivation & Behavioural Aspects of Budgeting
10 Chapter 10: Flexible Budgets & Capital Investment Appraisal
10.1 Flexible Budgets & Budgetary Control Reporting 10.2 Capital Expenditure, Compounding & Discounting Fundamentals 10.3 Discounted Cash Flow: NPV, IRR & Payback Methods
11 Chapter 11: Standard Costing & Basic Cost Variances
11.1 Standard Costing Purpose, Principles & Standard Cost Cards 11.2 Sales Price, Sales Volume & Direct Material Variances 11.3 Direct Labour & Overhead Variances
12 Chapter 12: Variance Investigation & Operating Statements
12.1 Variance Investigation, Interrelationships & Reverse Calculations 12.2 Operating Statements & Profit Reconciliations
13 Chapter 13: Performance Measurement Framework & Financial Analysis
13.1 Mission Statements, Objectives & Contextual Influences 13.2 Financial Performance: Profitability & Operating Efficiency Ratios 13.3 Financial Performance: Liquidity & Financial Gearing Ratios
14 Chapter 14: Divisional Performance, Multidimensional Models & Quality
14.1 Divisional Performance: Return on Investment & Residual Income 14.2 The Balanced Scorecard, Critical Success Factors & KPIs 14.3 Value for Money (3Es), Resource Utilisation & Non-Profit Performance 14.4 Quality Management, Benchmarking, Cost Control & Cost Reduction
4.1 Linear Functions & The High-Low Method
Key Takeaways Explain This Topic AI
The linear cost equation y = a + bx models semi-variable costs within a relevant range, where y is total cost, a is total fixed cost, b is variable cost per unit, and x is the activity level.
The high-low method isolates variable cost per unit by calculating the ratio of cost difference to activity difference between the highest and lowest activity levels—never based on highest and lowest costs.
Stepped fixed costs must be normalized before applying high-low by adjusting the cost at either the high or low activity level to ensure both observations reside on an identical fixed-cost structure.
Changes in variable cost per unit (such as bulk purchase discounts or overtime wage rates) require expressing variable costs across tiers before solving for the base variable rate.
The primary limitation of the high-low method is its total reliance on two extreme data points, rendering it vulnerable to distortion by unrepresentative outliers and non-linear cost dynamics.
Last updated: September 2026
4.1 Linear Functions & The High-Low Method
In management accounting, planning, budgeting, and control require accurate predictions of future expenditure across varying operational volumes. Many organizational costs are neither purely fixed nor purely variable; instead, they exhibit semi-variable (mixed) cost behaviour , containing both a constant base charge and a usage-dependent component. Before an organization can construct flexible budgets, calculate break-even points, or evaluate variance reports, management accountants must separate mixed costs into their underlying fixed and variable elements.
1. Structure of Linear Functions and Cost Equations
Linear functions provide the foundational mathematical framework for cost modeling in management accounting. Within a defined operational window, mixed costs are represented by the standard straight-line equation: y = a + b x y = a + bx y = a + b x
Where:
y y y = Total Cost: The dependent variable plotted on the vertical axis ( y y y -axis). Its magnitude depends directly on the level of activity undertaken.
a a a = Total Fixed Cost: The vertical intercept on the y y y -axis. It represents the cost that would theoretically be incurred at zero activity level within the relevant range. Fixed costs include facility rent, basic equipment leasing, and executive salaries.
b b b = Variable Cost per Unit: The slope (gradient) of the line. It reflects the marginal rate of expenditure per unit of output or activity. Variable costs include direct raw materials, standard direct labor hours, and production electricity consumption.
x x x = Activity Level: The independent variable plotted on the horizontal axis ( x x x -axis). It serves as the primary cost driver (e.g., units manufactured, machine hours run, direct labor hours worked, or vehicle miles driven). Total Cost ($)
- ^
- | / Total Cost Line (y = a + bx)
- | /
- | / <-- Slope = b (Variable Cost per Unit)
- | /
- | /
- a |--------------------/ <------- Vertical Intercept = a (Fixed Cost)
- | |
- | Fixed Cost (a) |
- | |
- 0 +--------------------+----------------------------> Activity Level (x)
- 0 Relevant Range
- The Concept of the Relevant Range
A fundamental assumption underpinning linear cost equations is the relevant range . The relevant range represents the specific band of activity over which budgeted sales and production volumes are expected to operate, and within which established cost behaviours remain valid:
Total fixed costs remain constant in absolute terms.
Variable cost per unit remains constant regardless of output fluctuations.
Operational efficiency, technology, and factor prices remain unchanged.
If activity expands beyond the relevant range, the enterprise may need to acquire additional factory floor space (causing fixed costs to step up) or hire weekend workers at premium wage rates (causing variable cost per unit to rise).
2. Scatter Diagrams and Visual Inspection
A scatter diagram (or scattergraph) is a graphical representation of historical bivariate cost data. Historical observations are plotted as paired coordinates ( x i , y i ) (x_i, y_i) ( x i , y i ) , where each point reflects the total cost incurred at a specific recorded activity level. Cost ($)
- ^
- | * * (Outlier: Equipment Breakdown)
- | *
- | * *
- | * *
- | * * <------- Line of Best Fit (Drawn by Eye)
- | *
- | *
- a |------------------------------------
- |
- 0 +-----------------------------------> Activity Level (x)
- Mechanics and Purpose
Visual Inspection: Management accountants plot 12 to 24 months of historical cost observations. By inspecting the resulting cluster of points, analysts evaluate whether a genuine linear relationship exists between activity and expenditure.
Identifying Outliers: Scatter diagrams reveal unrepresentative historical periods—such as a month where costs spiked due to an unprecedented flood or machine breakdown, or dropped due to a prolonged labor strike. Outliers can be flagged and excluded before formal mathematical modeling.
Line of Best Fit by Eye: An analyst uses a straight edge to draw a line through the center of the plotted points, balancing points above and below the line. The point where this line intersects the vertical axis provides a rough estimate of total fixed costs ( a a a ), and the slope provides an approximation of the variable rate ( b b b ).
Limitations of Scatter Diagrams
Subjectivity: Visual inspection relies heavily on individual judgment. Two management accountants viewing the same scatter diagram will invariably draw slightly different lines, yielding divergent fixed and variable cost parameters.
Inability to Minimize Errors: Visual inspection does not mathematically minimize estimation errors across all observed data points.
3. High-Low Method Mechanics
The high-low method is an algebraic technique used to separate mixed costs into fixed and variable components by comparing total costs at two distinct historical activity levels.
The Cardinal Rule of Selection
CRITICAL ACCA EXAM RULE: The high and low points must be selected based strictly on the HIGHEST AND LOWEST ACTIVITY LEVELS ( x x x ) , never on the highest and lowest dollar costs ( y y y ).
Selecting periods based on extreme costs rather than extreme activity introduces severe distortion whenever uncharacteristic operational events or non-volume spending occur during a mid-volume period.
Four-Step Mathematical Procedure
Step 1: Identify High and Low Activity Levels: Scan the historical dataset to locate the observation with the maximum activity volume ( x high , y high ) (x_{\text{high}}, y_{\text{high}}) ( x high , y high ) and the observation with the minimum activity volume ( x low , y low ) (x_{\text{low}}, y_{\text{low}}) ( x low , y low ) .
Step 2: Calculate Variable Cost per Unit ( b b b ): Compute the difference in total cost between the two points and divide by the difference in activity volume: b = Δ Cost Δ Activity = Cost at High Activity − Cost at Low Activity High Activity Level − Low Activity Level = y high − y low x high − x low b = \frac{\Delta \text{Cost}}{\Delta \text{Activity}} = \frac{\text{Cost at High Activity} - \text{Cost at Low Activity}}{\text{High Activity Level} - \text{Low Activity Level}} = \frac{y_{\text{high}} - y_{\text{low}}}{x_{\text{high}} - x_{\text{low}}} b = Δ Activity Δ Cost = High Activity Level − Low Activity Level Cost at High Activity − Cost at Low Activity = x high − x low y high − y low
Step 3: Calculate Total Fixed Cost ( a a a ): Substitute the calculated unit variable rate ( b b b ) into either the high activity equation or the low activity equation and solve for a a a : a = y high − ( b × x high ) or a = y low − ( b × x low ) a = y_{\text{high}} - (b \times x_{\text{high}}) \quad \text{or} \quad a = y_{\text{low}} - (b \times x_{\text{low}}) a = y high − ( b × x high ) or a = y low − ( b × x low ) (Both substitutions yield the exact same fixed cost figure, providing an immediate arithmetic check).
Step 4: Formulate the Forecasting Equation: Construct the finalized cost model y = a + b x y = a + bx y = a + b x and predict total expenditure for the budgeted activity volume ( x target x_{\text{target}} x target ): y target = a + ( b × x target ) y_{\text{target}} = a + (b \times x_{\text{target}}) y target = a + ( b × x target )
- 4. Fully Worked Example: Standard High-Low Application
- Operational Data
Vanguard Precision Engineering Ltd records total maintenance overhead expenditure across six consecutive operating months:
- Month Machine Hours ( x x x ) Total Maintenance Cost ( y y y )
- January 4,000 $44,000
- February 6,500 $59,000
- March 3,200 $39,200
- April 5,000 $50,000
- May 7,000 $62,000
- June 5,800 $54,800
Management wishes to forecast total maintenance overheads for July, when budgeted production requires 6,000 machine hours .
Step-by-Step Calculation
Step 1: Identify extreme activity points:
Highest Activity (May): x high = 7 , 000 x_{\text{high}} = 7,000 x high = 7 , 000 machine hours, y high = $ 62 , 000 y_{\text{high}} = \text{\textdollar}62,000 y high = $ 62 , 000
Lowest Activity (March): x low = 3 , 200 x_{\text{low}} = 3,200 x low = 3 , 200 machine hours, y low = $ 39 , 200 y_{\text{low}} = \text{\textdollar}39,200 y low = $ 39 , 200
Step 2: Calculate variable cost per machine hour ( b b b ): Δ Activity = 7 , 000 − 3 , 200 = 3 , 800 machine hours \Delta \text{Activity} = 7,000 - 3,200 = 3,800 \text{ machine hours} Δ Activity = 7 , 000 − 3 , 200 = 3 , 800 machine hours Δ Cost = $ 62 , 000 − $ 39 , 200 = $ 22 , 800 \Delta \text{Cost} = \$62,000 - \$39,200 = \$22,800 Δ Cost = $62 , 000 − $39 , 200 = $22 , 800 b = $ 22 , 800 3 , 800 hours = $ 6.00 per machine hour b = \frac{\$22,800}{3,800 \text{ hours}} = \$6.00 \text{ per machine hour} b = 3 , 800 hours $22 , 800 = $6.00 per machine hour
Step 3: Solve for total fixed costs ( a a a ):
Using May (High Point): a = $ 62 , 000 − ( 7 , 000 × $ 6.00 ) = $ 62 , 000 − $ 42 , 000 = $ 20 , 000 a = \$62,000 - (7,000 \times \$6.00) = \$62,000 - \$42,000 = \$20,000 a = $62 , 000 − ( 7 , 000 × $6.00 ) = $62 , 000 − $42 , 000 = $20 , 000
Verification using March (Low Point): a = $ 39 , 200 − ( 3 , 200 × $ 6.00 ) = $ 39 , 200 − $ 19 , 200 = $ 20 , 000 a = \$39,200 - (3,200 \times \$6.00) = \$39,200 - \$19,200 = \$20,000 a = $39 , 200 − ( 3 , 200 × $6.00 ) = $39 , 200 − $19 , 200 = $20 , 000
The maintenance cost equation is: y = $ 20 , 000 + $ 6.00 x y = \$20,000 + \$6.00x y = $20 , 000 + $6.00 x
Step 4: Forecast maintenance costs for July ( x = 6 , 000 x = 6,000 x = 6 , 000 machine hours): y July = $ 20 , 000 + ( 6 , 000 × $ 6.00 ) = $ 20 , 000 + $ 36 , 000 = $ 56 , 000 y_{\text{July}} = \$20,000 + (6,000 \times \$6.00) = \$20,000 + \$36,000 = \$56,000 y July = $20 , 000 + ( 6 , 000 × $6.00 ) = $20 , 000 + $36 , 000 = $56 , 000
5. Handling Stepped Fixed Costs in the High-Low Method
In many production environments, fixed overheads do not remain strictly flat across all activity tiers. Instead, they behave as stepped fixed costs , jumping by a lump sum when operational thresholds are breached (e.g., leasing a secondary warehouse or appointing an additional quality supervisor). Total Fixed Cost ($)
- ^
- | +------------------ Higher Fixed Cost ($43,000)
- | |
- | | <-- Step Increase = $15,000
- | |
- $28,000|------------------------------+
- | Base Fixed Cost ($28,000)
- | |
- 0 +------------------------------+-------------------> Activity Level (Units)
- 0 6,000 Units
- Analytical Adjustment Rule
To calculate the variable cost per unit ( b b b ) accurately, both data points must be adjusted to the same fixed cost baseline . If fixed costs step up at an intermediate activity level between the low and high observations:
Subtract the step increase from the high activity total cost (bringing it down to the low-volume fixed cost tier); OR
Add the step increase to the low activity total cost (bringing it up to the high-volume fixed cost tier).
- Worked Example: Stepped Fixed Costs
- Low Activity: 4,000 units, Total Cost = $52,000
- High Activity: 9,000 units, Total Cost = $97,000
Operating Condition: Total fixed costs increase by $15,000 for production volumes exceeding 6,000 units.
Objective: Forecast total costs for an output volume of 8,500 units .
Step 1: Normalize costs to a single baseline: Deduct the $15,000 step from the high activity cost to align both points with the baseline fixed cost tier: Adjusted High Cost = $ 97 , 000 − $ 15 , 000 = $ 82 , 000 \text{Adjusted High Cost} = \$97,000 - \$15,000 = \$82,000 Adjusted High Cost = $97 , 000 − $15 , 000 = $82 , 000
Step 2: Calculate variable cost per unit ( b b b ): Δ Activity = 9 , 000 − 4 , 000 = 5 , 000 units \Delta \text{Activity} = 9,000 - 4,000 = 5,000 \text{ units} Δ Activity = 9 , 000 − 4 , 000 = 5 , 000 units Δ Cost = $ 82 , 000 − $ 52 , 000 = $ 30 , 000 \Delta \text{Cost} = \$82,000 - \$52,000 = \$30,000 Δ Cost = $82 , 000 − $52 , 000 = $30 , 000 b = $ 30 , 000 5 , 000 units = $ 6.00 per unit b = \frac{\$30,000}{5,000 \text{ units}} = \$6.00 \text{ per unit} b = 5 , 000 units $30 , 000 = $6.00 per unit
Step 3: Determine base and stepped fixed costs:
Baseline fixed cost (for activity ≤ 6 , 000 \le 6,000 ≤ 6 , 000 units): a base = $ 52 , 000 − ( 4 , 000 × $ 6.00 ) = $ 52 , 000 − $ 24 , 000 = $ 28 , 000 a_{\text{base}} = \$52,000 - (4,000 \times \$6.00) = \$52,000 - \$24,000 = \$28,000 a base = $52 , 000 − ( 4 , 000 × $6.00 ) = $52 , 000 − $24 , 000 = $28 , 000
Stepped fixed cost (for activity > 6 , 000 > 6,000 > 6 , 000 units): a stepped = $ 28 , 000 + $ 15 , 000 = $ 43 , 000 a_{\text{stepped}} = \$28,000 + \$15,000 = \$43,000 a stepped = $28 , 000 + $15 , 000 = $43 , 000
Reconciliation at High Point: $ 43 , 000 + ( 9 , 000 × $ 6.00 ) = $ 43 , 000 + $ 54 , 000 = $ 97 , 000 \text{\textdollar}43,000 + (9,000 \times \text{\textdollar}6.00) = \text{\textdollar}43,000 + \text{\textdollar}54,000 = \text{\textdollar}97,000 $ 43 , 000 + ( 9 , 000 × $ 6.00 ) = $ 43 , 000 + $ 54 , 000 = $ 97 , 000 (Confirmed).
Step 4: Forecast total cost for 8,500 units: Because 8,500 units exceeds the 6,000-unit threshold, the higher fixed cost applies: y = a stepped + b x = $ 43 , 000 + ( 8 , 500 × $ 6.00 ) = $ 43 , 000 + $ 51 , 000 = $ 94 , 000 y = a_{\text{stepped}} + bx = \$43,000 + (8,500 \times \$6.00) = \$43,000 + \$51,000 = \$94,000 y = a stepped + b x = $43 , 000 + ( 8 , 500 × $6.00 ) = $43 , 000 + $51 , 000 = $94 , 000
6. Handling Changes in Variable Cost per Unit
Variable costs per unit can shift across operational volumes due to structural supply factors, such as bulk quantity discounts on raw materials (which lower b b b ) or overtime premiums on direct labor (which increase b b b ).
- Worked Example: Quantity Discount Threshold
- Low Activity: 3,000 units, Total Cost = $46,000
- High Activity: 8,000 units, Total Cost = $80,000
Operating Condition: Direct material quantity discounts reduce the variable cost per unit by 25% for all units produced in excess of 5,000 units. Total fixed costs remain completely unchanged.
Objective: Calculate the base variable cost per unit, total fixed costs, and forecast expenditure for 7,000 units .
Step 1: Set up the algebraic cost equations: Let b b b represent the standard variable cost per unit for output up to 5,000 units.
For output above 5,000 units, variable cost per unit is: b discounted = ( 1.00 − 0.25 ) b = 0.75 b b_{\text{discounted}} = (1.00 - 0.25)b = 0.75b b discounted = ( 1.00 − 0.25 ) b = 0.75 b
At Low Activity (3,000 units ≤ 5 , 000 \le 5,000 ≤ 5 , 000 ): Total Cost = a + 3 , 000 b = $ 46 , 000 — (Equation 1) \text{Total Cost} = a + 3,000b = \$46,000 \quad \text{--- (Equation 1)} Total Cost = a + 3 , 000 b = $46 , 000 — (Equation 1)
At High Activity (8,000 units):
Output consists of 5,000 units at standard rate b b b , plus 3,000 incremental units at discounted rate 0.75 b 0.75b 0.75 b : Total Variable Cost = 5 , 000 b + 3 , 000 ( 0.75 b ) = 5 , 000 b + 2 , 250 b = 7 , 250 b \text{Total Variable Cost} = 5,000b + 3,000(0.75b) = 5,000b + 2,250b = 7,250b Total Variable Cost = 5 , 000 b + 3 , 000 ( 0.75 b ) = 5 , 000 b + 2 , 250 b = 7 , 250 b Total Cost = a + 7 , 250 b = $ 80 , 000 — (Equation 2) \text{Total Cost} = a + 7,250b = \$80,000 \quad \text{--- (Equation 2)} Total Cost = a + 7 , 250 b = $80 , 000 — (Equation 2)
Step 2: Solve simultaneously for b b b and a a a : Subtract Equation 1 from Equation 2: ( a + 7 , 250 b ) − ( a + 3 , 000 b ) = $ 80 , 000 − $ 46 , 000 (a + 7,250b) - (a + 3,000b) = \$80,000 - \$46,000 ( a + 7 , 250 b ) − ( a + 3 , 000 b ) = $80 , 000 − $46 , 000 4 , 250 b = $ 34 , 000 4,250b = \$34,000 4 , 250 b = $34 , 000 b = $ 34 , 000 4 , 250 = $ 8.00 per unit b = \frac{\$34,000}{4,250} = \$8.00 \text{ per unit} b = 4 , 250 $34 , 000 = $8.00 per unit
Discounted variable rate (units above 5,000): 0.75 × $ 8.00 = $ 6.00 per unit 0.75 \times \$8.00 = \$6.00 \text{ per unit} 0.75 × $8.00 = $6.00 per unit
Total fixed cost ( a a a ): a = $ 46 , 000 − ( 3 , 000 × $ 8.00 ) = $ 46 , 000 − $ 24 , 000 = $ 22 , 000 a = \$46,000 - (3,000 \times \$8.00) = \$46,000 - \$24,000 = \$22,000 a = $46 , 000 − ( 3 , 000 × $8.00 ) = $46 , 000 − $24 , 000 = $22 , 000
Step 3: Forecast total cost for 7,000 units: Production of 7,000 units incorporates 5,000 units at $8.00 and 2,000 units at $6.00: y = $ 22 , 000 + ( 5 , 000 × $ 8.00 ) + ( 2 , 000 × $ 6.00 ) = $ 22 , 000 + $ 40 , 000 + $ 12 , 000 = $ 74 , 000 y = \$22,000 + (5,000 \times \$8.00) + (2,000 \times \$6.00) = \$22,000 + \$40,000 + \$12,000 = \$74,000 y = $22 , 000 + ( 5 , 000 × $8.00 ) + ( 2 , 000 × $6.00 ) = $22 , 000 + $40 , 000 + $12 , 000 = $74 , 000
7. Advantages and Limitations of the High-Low Method
Dimension Advantages Limitations
Computational Ease Extremely fast and straightforward; requires no complex statistical software, matrix algebra, or statistical training. Severely oversimplifies complex cost structures by attempting to define an entire operating curve from just two data points.
Data Requirements Requires only two historical data points (highest and lowest activity levels). Ignores all intermediate historical observations, discarding valuable operational data.
Sensitivity to Outliers Easy to explain to operational managers and non-financial executives. Highly vulnerable to outlier distortion. If either the highest or lowest activity month was distorted by abnormal events (equipment failure, strikes), the resulting equation is severely biased.
Linearity Assumption Provides a reasonable preliminary approximation within tightly bounded relevant ranges. Assumes cost behaviour is strictly linear, failing to capture learning curve effects, productivity gains, or curved cost curves.
Inflation Impact Effective for short-term forecasts in non-inflationary environments. Distorted by historical inflation: if the high activity period occurred recently and the low period occurred years ago, general price rises artificially inflate b b b .
8. ACCA Exam Traps & Pitfalls
Exam Trap 1: Selecting by Cost Rather Than Activity. ACCA exam questions frequently feature a month with moderate activity but inflated costs (e.g., due to emergency repair work). Candidates who pick the highest dollar cost rather than the highest volume score zero on the calculation.
Exam Trap 2: Omitting Stepped Overheads in Future Projections. When calculating forecasts above a step threshold, candidates often determine the base fixed cost correctly from the low point, but forget to add the step increment back when generating the final budgeted figure.
Exam Trap 3: Extrapolating Beyond the Relevant Range. Applying high-low parameters to production volumes far above the observed high point assumes an infinite relevant range. In reality, capacity limits will trigger stepped costs or overtime penalties.