Determining How Costs Behave

Overview

• What you’ll learn: The general approach to estimating cost functions, the industrial engineering method, account analysis, scatter plots and visual fitting, criteria for a good cost function, and practical data issues.
• Prerequisites: Lesson 5 — CVP Analysis: Sensitivity & What-If
• Estimated reading time: 18 minutes

Introduction

The Grand Historian records: In Lessons 3 through 5, we assumed that we knew which costs were variable, which were fixed, and which were mixed. We blithely wrote equations like Y = a + bX and plugged in tidy numbers. But the seasoned practitioner asks an uncomfortable question: “Where did those numbers come from?”

In reality, costs do not arrive with labels reading “I am variable at $5 per unit” or “I am fixed at $12,000 per year.” Costs are raw, unclassified data — a sprawl of invoices, paychecks, and utility bills. The management accountant’s task is to analyze this historical data and extract the underlying cost function: the mathematical relationship between cost and activity.

This lesson introduces the foundational methods for determining how costs behave. We begin with qualitative approaches (the industrial engineering method and account analysis) and then move to the scatter plot — the visual tool that reveals cost patterns hiding in the data. In Lesson 7, we will add the quantitative rigor of the high-low method and regression analysis.

The General Approach to Cost Estimation

Estimating a cost function follows a systematic process:

1. Choose the dependent variable (Y): The cost you want to explain or predict (e.g., total maintenance cost).
2. Identify potential cost drivers (X): The activity variables that might cause the cost to change (e.g., machine hours, units produced, number of setups).
3. Collect data: Gather historical observations of both the dependent variable and the independent variable(s).
4. Plot the data: Create a scatter plot to visualize the relationship.
5. Estimate the cost function: Use one of several methods (account analysis, high-low, regression) to determine the equation Y = a + bX.
6. Evaluate the cost function: Assess whether it is a good fit — does it accurately describe the cost behavior?

This six-step process is iterative. If the first cost driver you try doesn’t produce a good fit, try another. If the data is noisy, investigate outliers. If the relationship is clearly nonlinear, consider more advanced methods (Lesson 8).

The Industrial Engineering Method

The industrial engineering method (also called the work-measurement method) estimates cost functions by analyzing the physical relationship between inputs and outputs. Rather than looking at historical costs, it asks: “What should this cost, based on engineering analysis?”

• Time-and-motion studies determine how much labor each unit requires
• Material specifications determine how much raw material goes into each unit
• Engineering analysis determines machine time, energy consumption, and other resource requirements

Advantages and Disadvantages

• Advantages: Does not require historical data (useful for new products); based on physical relationships, not accounting artifacts; can identify inefficiencies by comparing actual to engineered standards.
• Disadvantages: Time-consuming and expensive (requires engineers); focuses on physical inputs, may miss overhead; standards may not reflect actual operating conditions.

Account Analysis

The account analysis method asks an experienced accountant or manager to review each cost account and classify it as variable, fixed, or mixed based on professional judgment:

1. List all cost accounts in the general ledger
2. For each account, determine whether it is primarily variable, primarily fixed, or mixed
3. For mixed accounts, estimate the fixed and variable components based on experience
4. Sum all variable costs to get the total variable cost rate; sum all fixed costs to get total fixed costs

Example

If total activity was 10,000 units: Variable rate = $149,000 / 10,000 = $14.90/unit. Cost function: Y = $66,000 + $14.90X

Advantages: Quick, inexpensive, uses insider knowledge. Limitations: Subjective, relies on single-period data, depends on analyst’s expertise and potential biases.

Scatter Plots: The Conference of Dots

A scatter plot graphs historical data points with the cost driver (X) on the horizontal axis and the cost (Y) on the vertical axis. Each dot represents one observation — one month, one quarter, or one batch.

What Scatter Plots Reveal

• Linearity: Do the dots approximate a straight line? If yes, a linear cost function is appropriate.
• Direction: Does the line slope upward (positive relationship) or neither (no relationship)?
• Tightness: Are dots clustered closely around the line (good fit) or scattered widely (poor fit)?
• Outliers: Dots far from the pattern may represent unusual events that should be investigated.
• Nonlinearity: If dots curve rather than following a straight line, a nonlinear cost function may be needed.

Visual Fit Method

After plotting, draw a line that “best fits” the data by eye. Read the y-intercept (fixed cost, a) and calculate the slope (variable rate, b) from two points on the line. This is quick and intuitive, but subjective — two analysts may draw different lines. Quantitative methods (high-low and regression) are preferred when precision matters.

Criteria for a Good Cost Function

Horngren identifies three criteria for evaluating cost functions:

1. Economic plausibility: The relationship must make logical sense. Machine maintenance should be driven by machine hours, not by the number of office workers.
2. Goodness of fit: Estimated values should be close to actual observed values. A large gap between predicted and actual suggests a poor model.
3. Significance of the cost driver: Changes in the cost driver should result in significant changes in the cost. If the cost barely moves when the driver changes, you have the wrong driver.

Practical Data Issues

Real-world data is never as clean as textbook examples. Common problems that the management accountant must navigate:

• Time-period mismatch: Costs recorded in one period may relate to activity in another (e.g., a repair invoice received in March for work done in February).
• Outliers: Unusual observations that distort the analysis. Investigate before excluding — they may signal real changes in the cost structure.
• Inflation: Costs may increase over time due to inflation, not activity. Adjust for price-level changes when using multi-year data.
• Missing data: Gaps in the data set reduce reliability. Collect more observations if possible.
• Insufficient variation: If activity levels barely changed during the observation period, the data cannot reveal cost behavior. You need periods of both high and low activity.
• Technology or method changes: A cost function estimated from old data may not apply after a process change or technology upgrade. Use only data from the current operating environment.

Key Takeaways

• Estimating cost functions follows a six-step process: choose the cost, identify drivers, collect data, plot, estimate, and evaluate.
• The industrial engineering method estimates costs from physical input-output relationships — best for new products but expensive.
• Account analysis classifies each cost account as variable, fixed, or mixed using professional judgment — quick but subjective.
• Scatter plots visualize the relationship between cost and activity, revealing linearity, outliers, and tightness of fit.
• A good cost function must be economically plausible, show goodness of fit, and use a significant cost driver.
• Practical data issues (outliers, inflation, time-period mismatches) must be addressed before estimation.

What’s Next

In Lesson 7, we add quantitative precision with the high-low method and regression analysis — the two workhorses of cost estimation that replace visual guessing with mathematical rigor.