Choosing Cost Drivers & Nonlinear Costs

Overview

• What you’ll learn: Multiple regression analysis with two or more cost drivers, nonlinear cost functions, learning curves and their impact on cost estimation, criteria for selecting cost drivers, and the limitations of quantitative cost analysis.
• Prerequisites: Lesson 7 — Cost Estimation Methods
• Estimated reading time: 20 minutes

Introduction

The Grand Historian records: In the preceding seven lessons, we have journeyed from the management accountant’s mandate through cost objects, cost behavior, CVP analysis, and cost estimation. We have wielded the high-low method and regression analysis like swords in the cost accountant’s dojo. But all our models thus far have shared a simplifying assumption: that one cost driver explains the behavior of one cost, and that the relationship is linear.

Reality, that perennial spoiler of elegant models, is rarely so accommodating. Many costs are driven by multiple factors simultaneously — maintenance costs may depend on both machine hours and machine age. Some costs follow curves rather than straight lines — the hundredth unit takes less time to produce than the first, thanks to the learning curve. And choosing the right cost driver from a multitude of candidates requires not just statistical skill but business judgment of the highest order.

This final lesson of Module 4 ventures into advanced territory. Here, cost estimation becomes both an art and a science — the art of choosing the right model, and the science of fitting it to data. Those who master this lesson will possess the complete toolkit of cost behavior analysis.

Multiple Regression Analysis

Multiple regression extends simple regression by including two or more independent variables (cost drivers) in a single equation:

Y = a + b₁X₁ + b₂X₂ + … + bₙXₙ

Where Y = total cost, a = fixed cost, and each b represents the variable cost rate for its corresponding driver X.

When to Use Multiple Regression

Use multiple regression when you suspect that more than one factor drives a cost. Common scenarios:

• Manufacturing overhead may depend on both machine hours (X₁) and number of setups (X₂)
• Shipping costs may depend on both weight shipped (X₁) and number of shipments (X₂)
• Customer service costs may depend on both number of customers (X₁) and number of complaints (X₂)

Example: Maintenance Cost with Two Drivers

Suppose maintenance costs are driven by both machine hours (X₁) and number of maintenance calls (X₂). A multiple regression might produce:

Y = $3,200 + $2.80X₁ + $45.00X₂

Interpretation: Fixed maintenance costs are $3,200 per month. Each machine hour adds $2.80 to maintenance costs. Each maintenance call adds $45.00. If in a given month X₁ = 2,000 hours and X₂ = 50 calls:

Predicted cost = $3,200 + $2.80(2,000) + $45(50) = $3,200 + $5,600 + $2,250 = $11,050

Evaluating Multiple Regression

Multiple regression uses adjusted R-squared rather than plain R² to evaluate fit. Adjusted R² penalizes the addition of variables that don’t genuinely improve the model:

• Plain R² always increases (or stays the same) when you add another variable — even a random one
• Adjusted R² increases only if the new variable actually improves the model’s explanatory power
• If adjusted R² drops when you add a variable, that variable is not useful — remove it

Additional evaluation criteria for multiple regression:

• Each coefficient should be statistically significant (p-value < 0.05)
• Signs should be economically plausible: A positive coefficient means the driver increases cost (expected for most costs)
• No multicollinearity: The independent variables should not be highly correlated with each other. If machine hours and direct labor hours are 95% correlated, including both creates instability in the estimates

The Danger of Multicollinearity

Multicollinearity occurs when two or more independent variables are highly correlated. Symptoms include:

• Individual coefficients have unexpected signs (negative when positive is expected)
• Individual coefficients are not statistically significant, but the overall model has high R²
• Coefficients change dramatically when one variable is added or removed

Solution: Choose one driver from each group of correlated variables, or combine correlated variables into a single composite measure.

Nonlinear Cost Functions

Not all costs follow straight lines. Several types of nonlinearity are common in practice:

Economies and Diseconomies of Scale

At low volumes, per-unit costs may decrease as volume increases (economies of scale — bulk purchasing, spreading fixed costs). But beyond a certain point, per-unit costs may increase (diseconomies of scale — overtime, congestion, coordination complexity). The total cost function is an S-curve rather than a straight line.

Step Functions

As discussed in Lesson 3, some costs are constant within a range and then jump. Supervisory costs, warehouse capacity, and licensing fees often follow step patterns. These are handled by:

• Treating as fixed within the relevant range
• Using dummy variables in regression (1 = above threshold, 0 = below)
• Segmenting the data and estimating separate functions for each step

Exponential and Power Functions

Some costs follow exponential growth (e.g., environmental remediation costs that accelerate as contamination worsens) or power functions (e.g., construction costs that increase at less than a proportional rate with building size). These can be estimated using:

• Logarithmic transformations to linearize the data
• Nonlinear regression software
• Polynomial regression (adding X² or X³ terms)

The Learning Curve

The learning curve (also called the experience curve) describes the systematic reduction in the time (and cost) required to perform a task as cumulative experience increases. The more you do something, the faster and cheaper it becomes.

The Basic Learning Curve Model

The most common formulation is the cumulative average-time learning model:

When cumulative output doubles, the cumulative average time per unit falls to a fixed percentage of the previous average.

For example, with an 80% learning curve:

The learning curve has profound implications for cost estimation, pricing, and bidding:

• Early production is expensive: The first units cost significantly more per unit than later ones. Pricing based on early production costs will overstate the long-run cost.
• Cost predictions must account for learning: If you use early production data to estimate a cost function, you will overestimate costs for future production.
• Competitive bidding: Companies with more cumulative experience have lower costs and can bid more aggressively.

Where Learning Curves Apply

• Labor-intensive manufacturing (aircraft, electronics assembly)
• Software development (subsequent projects using similar technology)
• Professional services (consultants become more efficient with repeated engagements)
• Surgical procedures (outcome quality improves with surgeon experience)

Limitations of Learning Curves

• Learning eventually plateaus — the curve flattens as workers approach maximum proficiency
• Worker turnover resets the curve — new employees start at the beginning
• Process changes disrupt the curve — new technology requires relearning
• The learning rate varies by industry and task complexity

Criteria for Choosing Cost Drivers

Selecting the right cost driver is perhaps the most consequential decision in cost estimation. Horngren provides five criteria, listed in order of importance:

1. Economic Plausibility

The cost driver must have a logical, cause-and-effect relationship with the cost. Machine hours should drive machine-related costs. Number of orders should drive order-processing costs. If you cannot explain why the driver causes the cost, the statistical relationship may be spurious — a coincidence that will not hold in the future.

2. Goodness of Fit (R²)

The driver should explain a high proportion of cost variation. Among economically plausible candidates, prefer the one with the highest R².

3. Significance of the Relationship

The coefficient on the driver should be statistically significant (p-value < 0.05). A driver that appears to fit well on average but has a high p-value may be unreliable — the relationship could be due to chance.

4. Specification Analysis

Examine the residuals (the differences between predicted and actual costs):

• Residual plot: Plot residuals against the predicted values. Ideally, residuals are randomly scattered around zero. Patterns (fan shapes, curves, clusters) indicate problems with the model.
• Linearity check: If residuals show a curved pattern, the true relationship may be nonlinear.
• Heteroscedasticity: If residuals fan out as the predicted value increases, the variance is not constant — this violates a regression assumption.

5. Data Collection Feasibility

The driver must be measurable and the data must be available. A theoretically perfect driver is useless if the company cannot collect the data needed to use it. Sometimes a slightly inferior driver with readily available data is preferable to a superior driver that requires an expensive new measurement system.

Integrating Cost Estimation with Decision-Making

The ultimate purpose of cost estimation is not to produce elegant equations — it is to support better decisions. A few principles to keep in mind:

• Use multiple methods: Triangulate by comparing results from account analysis, high-low, and regression. If all three point in the same direction, confidence is high.
• Involve operations managers: Accountants crunch numbers; operations managers understand the physical processes that generate costs. The best cost functions emerge from collaboration.
• Update regularly: Cost functions are perishable. As technology changes, as prices shift, as processes evolve, the old cost function becomes unreliable. Re-estimate at least annually.
• Communicate uncertainty: Present a range of estimates, not a single point. “We estimate maintenance costs will be between $10,000 and $12,000” is more honest and more useful than “$11,000 exactly.”

Key Takeaways

• Multiple regression uses two or more cost drivers to explain cost behavior — evaluated using adjusted R-squared and checking for multicollinearity.
• Nonlinear cost functions (economies of scale, step functions, exponential curves) require specialized estimation techniques or data transformations.
• Learning curves describe the systematic reduction in per-unit cost as cumulative production increases — critical for pricing, bidding, and cost forecasting.
• Cost driver selection follows five criteria: economic plausibility, goodness of fit, statistical significance, specification analysis (residual patterns), and data collection feasibility.
• Economic plausibility is the most important criterion — a statistically beautiful model without logical justification is a trap.
• Cost estimation is a means to better decisions, not an end in itself. Use multiple methods, involve operations experts, update regularly, and communicate uncertainty.

What’s Next

Congratulations — you have completed Module 4: Cost Terminology and Behavior. You now command the full vocabulary and analytical toolkit of cost accounting: cost objects, cost drivers, variable and fixed costs, CVP analysis, and the quantitative methods for estimating cost functions. Module 5 awaits, where we will apply these tools to product costing systems — job-order costing and process costing. The theory meets practice; the equations meet the factory floor.