Mixed costs and cost estimation

Learning goals

• Express a cost as Y = a + bX.
• Use high-low to estimate fixed and variable components.
• Choose high and low observations by activity, not total cost.
• Test a model against actual results and explain residuals.
• Recognize when a scatterplot, regression, engineering study, or new relevant range is needed.

Prerequisite check

1. How does total variable cost behave as activity changes?
2. What is the danger of using last month's fixed cost per unit in a new volume forecast?

Vocabulary

• Mixed cost: contains fixed and variable components.
• Cost equation: Y = a + bX, where Y is total cost, a fixed cost, b variable rate, and X activity.
• Cost driver: activity with a plausible cause-and-effect relationship to cost.
• High-low method: two-point estimate using highest and lowest activity observations.
• Residual: actual cost minus model-predicted cost.
• Scatterplot: graph of cost against activity used to inspect pattern, outliers, and range.
• Regression: statistical method that estimates a line using multiple observations and diagnostics.

Core idea

A mixed-cost model separates a capacity/base component from an activity-driven component:

Total cost = fixed cost + variable rate × activity

High-low is quick and transparent, but it uses only two points. It is a planning approximation, not proof of causation. A good model has a logical driver, clean data, stable process, defined time period, and stated valid range.

Why this treatment makes sense

Managers need a cost equation to budget at different activity levels and create flexible performance comparisons. Dividing total mixed cost by activity hides the fixed component and produces a unit rate that drifts with volume.

A repeatable method

Use PLOT–SELECT–SLOPE–FIXED–TEST:

1. Plot or scan data for errors, seasonality, step changes, and outliers.
2. Select highest and lowest activity within one relevant range.
3. Compute b = change in cost ÷ change in activity.
4. Compute a = total cost − bX using either selected point; verify with the other.
5. Test predictions against unused observations and investigate residuals.

Worked example

North Shore Clinic's maintenance data include a low point of 1,200 machine-hours at $7,600 and high point of 3,600 hours at $14,800.

Variable rate:

b = ($14,800 − $7,600) ÷ (3,600 − 1,200) = $3 per machine-hour.

Fixed cost:

a = $14,800 − $3(3,600) = $4,000.

Equation: Y = $4,000 + $3X.

At 2,800 hours, predicted maintenance is $4,000 + $3(2,800) = $12,400. If actual cost is $13,000, residual is +$600 unfavourable to prediction. That is a question, not automatically poor performance: an emergency repair, price change, timing difference, or omitted driver may explain it.

Journal, ledger, and statement connection

The equation creates a budget, not a journal entry. Actual maintenance invoices, accruals, and payments still enter the ledger. Managers compare the actual ledger total with a budget adjusted to actual activity. External statements report costs using applicable classification; the internal split supports planning.

Common mistakes

• Choosing the highest and lowest cost months instead of activity months.
• Calculating fixed cost by averaging total costs.
• Changing units mid-equation, such as hours in the slope but hundreds of hours in X.
• Forecasting below shutdown level or beyond capacity.
• Using an outlier without investigation because high-low requires it mechanically.
• Calling every residual inefficiency.
• Treating correlation from a regression as evidence the driver causes the cost.

Guided practice

Delivery cost is $5,600 at 800 stops and $9,200 at 2,000 stops. Estimate the cost equation and predict cost at 1,500 stops. If actual is $7,900, calculate and interpret the residual.

Independent practice

Campus Print's monthly utility data are:

Use high-low to estimate an equation and predict at 2,700 hours. Then explain why you would inspect June before relying on the model and show the residual for February.

Self-check and solutions

Guided: b = ($9,200 − $5,600)/(2,000 − 800) = $3 per stop. Fixed = $9,200 − $6,000 = $3,200. At 1,500 stops, predicted $7,700. Actual $7,900 gives +$200 residual; investigate price, routing, overtime, weather, or timing.

Independent: High activity June 3,600/$15,900; low April 1,400/$7,700. b = $8,200/2,200 = $3.7273 per hour. a ≈ $7,700 − $3.7273(1,400) = $2,481.82. At 2,700, predicted ≈ $12,545.45. February predicted $2,481.82 + $3.7273(2,500) = $11,800, so residual is −$1,400.

June appears high relative to nearby activity—May has 2,900/$11,600 and March 3,200/$12,300—so an outage, rate change, billing catch-up, or data error may distort both slope and forecast. A scatterplot and regression after evidence review would use more observations.

Retrieval practice

1. Define each symbol in Y = a + bX.
2. Recite the five high-low steps.
3. Why select points by activity?
4. What does a positive residual mean—and not mean?

Exam-style application

High activity is 10,000 tests at $58,000; low is 6,000 tests at $42,000. A manager divides $58,000 by 10,000 and calls $5.80 fully variable. Find the mixed equation and predict 8,000 tests.

Target: b = $16,000/4,000 = $4/test; a = $18,000. At 8,000, cost = $50,000. The $5.80 average at one volume embeds $1.80 of allocated fixed cost.

Lesson summary

Estimate mixed cost by separating base capacity from activity response, then test the model. High-low is useful for a first pass; evidence, range, and diagnostics decide whether it is credible.