Cost behaviour and estimation

To plan, a manager must predict what costs will be at the activity level expected next period. To control, she must know what costs should have been at the activity level actually achieved. Both require an understanding of cost behaviour: how a cost changes in total as an activity, such as units produced, labour hours or kilometres driven, changes. Lesson 2 introduced variable, fixed and mixed costs. This lesson looks more closely at the patterns costs follow, the relevant range within which those patterns hold, and three practical ways to estimate a cost's fixed and variable components from data: account analysis, the high–low method and least-squares regression. It ends with the contribution margin income statement, the report format that makes cost behaviour visible and that lesson 4 relies on.

Cost drivers and the relevant range

A cost driver is the activity that causes a cost to change. For hardwood, the driver is chairs produced. For machine maintenance, it may be machine hours. For the cost of processing customer orders, it may be the number of orders. Choosing the right driver matters: a cost that looks unpredictable against units may be closely related to batches or set-ups. Managers usually start with volume-based drivers such as units or labour hours, and lesson 7 adds activity-based drivers.

Cost behaviour patterns hold only within a relevant range of activity, the band within which the company expects to operate and within which the assumed relationships are reasonable. Cedar Ridge's chair line can make between about 5,000 and 8,500 chairs a year with its present building, equipment and supervisors. Within that range, fixed manufacturing overhead is $216,000. Beyond it, the company would need another shift supervisor, more equipment or a larger building, and the fixed cost would jump. Outside the range, the economics are different and the usual formulas should not be trusted.

Common cost behaviour patterns

Committed fixed costs arise from long-term decisions about capacity, such as buildings, equipment and key managers; they cannot be cut quickly without harming the business. Discretionary fixed costs result from annual management decisions, such as advertising, training and research; they can be adjusted more readily, though cutting them may hurt the future. Distinguishing the two helps managers know where savings are possible in a downturn.

Economists draw total cost curves that bend, because variable cost per unit falls at first as workers specialize and later rises as the plant becomes congested. Accountants usually assume straight lines. The two views are reconciled by the relevant range: over the band where a company actually operates, the economist's curve is close to a straight line, so the linear approximation is good enough for planning.

Mixed costs and the cost formula

A mixed cost can be written as a cost formula: total cost = fixed cost + (variable cost per unit of driver × activity), or Y = a + bX. Here Y is total cost, a is the fixed component (the intercept), b is the variable cost per unit of the driver (the slope) and X is the activity level. Once a and b are estimated, the formula predicts the cost at any activity within the relevant range.

Cedar Ridge's factory maintenance and utilities cost is a typical mixed cost. Part of it is fixed: basic electricity for lighting and heat, a maintenance contract, the base charge on the gas bill. Part varies with machine use: power for the saws, router and dust collectors, and wear parts. Managers want to know the split to budget the cost for next year and to judge whether a month's bill was reasonable.

Method 1: account analysis

The simplest method is account analysis: an experienced accountant reviews each account in the ledger and classifies it as variable, fixed or mixed, using judgement and knowledge of the operation. Hardwood is variable; the factory lease is fixed; the electricity bill is mixed and might be split using the utility's tariff, which shows the fixed service charge and the energy charge per kilowatt-hour. Account analysis is quick and uses knowledge that statistical methods lack, but it is subjective and may miss changes in behaviour.

The engineering approach is a more rigorous variation: engineers study the process and estimate how much of each input should be used per unit, for example by timing operations or measuring the board feet a chair design requires. It is the basis for standard costs (lesson 11) and works even when there is no past data, such as for a new product.

Method 2: the high–low method

The high–low method estimates the cost formula from just two observations: the periods with the highest and lowest activity. The change in cost between them is assumed to be caused entirely by the change in activity, so the variable rate is the change in cost divided by the change in activity. The fixed component is then the total cost at either point minus the variable part.

The high–low method is easy, but it rests on only two points, and those two are by definition the most extreme. If either month was unusual, such as a month with a large one-off repair, the estimate will be distorted. Always pick the high and low points by the activity measure, never by cost, and check a scatter plot first to make sure neither point is an outlier.

Method 3: least-squares regression

Least-squares regression uses all the observations. It fits the straight line that minimizes the sum of the squared vertical distances between the observed costs and the line. Spreadsheets and statistical software do the arithmetic, but the formulas are short enough to code directly. The slope b equals the covariance of activity and cost divided by the variance of activity, and the intercept a equals mean cost minus b times mean activity. The coefficient of determination, R², measures the proportion of the variation in cost explained by the activity; values close to 1 mean the driver explains most of the variation.

# Estimate the maintenance cost formula two ways (illustrative data)
import statistics as st

hours = [1800, 2400, 2100, 3000, 2700, 2200, 1950, 2600, 2850, 2300, 2050, 2550]
cost  = [21400, 24800, 22900, 28000, 26500, 23700, 22300, 25400, 27300, 24100, 22200, 25300]

# High-low: pick extremes by activity
hi, lo = hours.index(max(hours)), hours.index(min(hours))
b_hl = (cost[hi] - cost[lo]) / (hours[hi] - hours[lo])
a_hl = cost[hi] - b_hl * hours[hi]

# Least squares
mx, my = st.mean(hours), st.mean(cost)
b_ls = sum((x - mx) * (y - my) for x, y in zip(hours, cost)) / sum((x - mx) ** 2 for x in hours)
a_ls = my - b_ls * mx
r2 = st.correlation(hours, cost) ** 2

for name, a, b in (("High-low", a_hl, b_hl), ("Regression", a_ls, b_ls)):
    print(f"{name:<11} fixed ${a:,.0f}/month, variable ${b:.2f}/hour, "
          f"cost at 2,500 h ${a + b * 2500:,.0f}")
print(f"R-squared {r2:.3f}")

High-low    fixed $11,500/month, variable $5.50/hour, cost at 2,500 h $25,250
Regression  fixed $11,175/month, variable $5.61/hour, cost at 2,500 h $25,193
R-squared 0.989

Regression puts the fixed cost at about $11,175 a month and the variable rate at about $5.61 per machine hour, and R² of about 0.99 shows that machine hours explain almost all the month-to-month variation. The two methods give predictions within about $60 of each other at 2,500 hours, because these invented data happen to be well behaved. Real data are noisier, and regression is then clearly better, because it is not hostage to two extreme months. Python's statistics module also provides a linear_regression function that returns the same slope and intercept.

Three cautions apply to any estimate. First, the intercept is an estimate of fixed cost within the relevant range; it is not the cost the company would incur at zero activity, which was never observed. Second, check that the driver makes economic sense: a high R² between two unrelated series is a coincidence, not a cost formula. Third, adjust past data for known changes, such as a new utility rate, before relying on it. Regression with several drivers, such as machine hours and number of set-ups, is possible and is covered in ACCT-341.

The contribution margin income statement

Once costs are separated into variable and fixed, they can be reported in contribution format. The contribution margin income statement subtracts all variable costs, both manufacturing and selling, from sales to give the contribution margin, the amount available to cover fixed costs and then provide profit. It then subtracts all fixed costs. The traditional (functional) income statement used in external reporting, by contrast, subtracts cost of goods sold, which mixes variable and fixed manufacturing costs, to give gross margin, and then subtracts selling and administrative expenses, which also mix variable and fixed costs.

# Classic chair line: traditional versus contribution format (illustrative)
units, price = 7_000, 180
v_mfg, v_sell = 102, 6                 # variable manufacturing and selling per chair
f_mfg, f_sa = 216_000, 144_000         # fixed manufacturing and fixed S&A
sales = units * price

# Contribution format
vc = units * (v_mfg + v_sell)
cm = sales - vc
oi_cm = cm - f_mfg - f_sa

# Traditional format (assumes production equals sales)
cogs = units * v_mfg + f_mfg
gross = sales - cogs
sa = units * v_sell + f_sa
oi_trad = gross - sa

print(f"Contribution: sales {sales:,} - variable {vc:,} = CM {cm:,}; - fixed {f_mfg + f_sa:,} = {oi_cm:,}")
print(f"Traditional:  sales {sales:,} - COGS {cogs:,} = gross {gross:,}; - S&A {sa:,} = {oi_trad:,}")
print(f"CM ratio {cm / sales:.0%}; operating income identical: {oi_cm == oi_trad}")

Contribution: sales 1,260,000 - variable 756,000 = CM 504,000; - fixed 360,000 = 144,000
Traditional:  sales 1,260,000 - COGS 930,000 = gross 330,000; - S&A 186,000 = 144,000
CM ratio 40%; operating income identical: True

Both statements show the same operating income of $144,000 when production equals sales; lesson 8 shows what happens when it does not. What differs is the information they reveal. The contribution format shows at a glance that each additional $1 of sales adds 40 cents of contribution margin, which is exactly what a manager needs to predict the effect of changes in volume. The traditional format's gross margin of $330,000 says nothing about how costs will move if volume changes, because it buries $216,000 of fixed overhead inside cost of goods sold.