**High/Low and Linear Regression Analysis Overview**

The total costs associated with a business are the SUM of fixed costs and variable costs i,e. the total cost is semi-variable in nature.

- Total costs can be divided into fixed costs /variable cost per unit of output.

Formula for total costs:

**Y=a+bx**

where:**y**= total cost in a period**a**= the fixed costs in the period**b**= the variable cost per unit of output or unit of activity**x**= the number of output or the volume of activity in the period.

## Constructing Total cost function

## High/Low and Linear Regression Analysis

**The total cost function can be used to estimate costs associated with different levels of activities. Its useful for forecasting and decision making.**

There are **two** methods for constructing the total cost function equation.

- High/Low Analysis.
- Linear Regression Analysis.

## High/Low and Linear Regression Analysis

High/Low analysis;

can be used to estimate fixed costs and variable costs per unit, whenever:

- figures available for total costs at two different level of output.
- it can be used that fixed costs are same and variable cost per unit is constant at both levels of activity.
- the different between the total costs at high level and low level of output is entirely variable cost.

**The following circumstances are to be considered:**

## When No Change

**Step 1**: Take activity level and cost for:

- Highest level
- Lowest level

**Step 2**: Calculate variable cost per unit (**b**) as:

difference in total cost (highest minus lowest) **divide** by difference in no. of units (highest minus lowest).

**Step 3**: Now for fixed cost (**a**) put the variable cost per unit into one of the cost expressions (mostly high level).

**Step 4**: Construct total cost function for any activity level:

Total cost=a+bx

*For Example:*

Step 1

Highest level 7,000 (units) costs $38,800

Lowest level 4,500 (units) costs $30,400

Step 2: Difference

Therefore: variable cost per unit= 8400/2500 = $ 3.36

Step 3: Cost expressions: Total cost of 7,000 units

Fixed cost + variable cost= 38,800

Fixed cost + 7,000 x 3.36= 38,800

Fixed cost + 23,520= 38,800

Fixed cost=38,800–23,520=15,280

Step 4: Construct total cost function

Total cost=a+bx= 15,280+ 3.36x

## A step change in fixed costs as money value And the amount is known

**Step 1**: Take activity level and cost for:

- Highest level
- Lowest level

**Step 2**: Make adjustment for the step in fixed cost:

- add the step in fixed cost in lower level; or
- deduct it from higher level

Now calculate variable cost per unit (**b**) as:

difference in total cost (highest minus lowest) divide by difference in no. of units (highest minus lowest).

**Step 3**: Now for fixed cost (**a**) put the variable cost per unit into one of the cost expressions (mostly high level).

**Step 4**: Construct total cost function for any activity level:

Total cost=a+bx

*For Example:*

Step 1

Highest level 7,000 (units) costs $ 38,800

Lowest level 4,500 (units) costs $ 30,400

Step 2: Make an adjustment for step in fixed cost. *For example fixed costs increase by $3,000 when the activity level exceeds or equals 10,000 units.*

Add the increase in cost in lowest activity level.

Therefore: variable cost per unit= 5400/2500 = $ 2.16

Step 3: Cost expressions: Total cost of 7,000 units

Fixed cost + variable cost= 38,800

Fixed cost + 7,000 x 2.16= 38,800

Fixed cost + 15,120= 38,800

Fixed cost=38,800–15,120=23,680

Step 4: Construct total cost function (un-adjusted levels):

- Above 10,000 units

Total cost=a+bx= 23,680+ 2.16x

- Below 10,000 units

Total cost=a+bx=(23,680-3,000)+2.16x

Total cost=a+bx= 20,680+2.16x

## A step change in fixed costs is given as a Percentage amount

When there is a percentage change after a particular level, this means there are **TWO** levels which share *same* fixed cost.

**Step 1**: Take activity level and cost for (3 levels):

- Highest level
- Middle level
- Lowest level

**Step 2**: Choose the pair which is on the same side as the step.

Now calculate variable cost per unit (**b**) as:

difference in total cost divide by difference in no. of units.

**Step 3**: Now for fixed cost (**a**) put the variable cost per unit into one of the cost expressions (mostly high level).

**Step 4**: Construct total cost function for any activity level:

Total cost=a+bx

*For Example:*

Step 1

- Highest level 7,000 (units) costs $ 38,800
- Middle level 5,500 (units) costs $ 35,000
- Lowest level 4,500 (units) costs $ 30,400

Step 2: Pair with same percentage change. (**assume a 10% increase in fixed costs when the activity level exceeds or equals 5,500 units**)

- Highest level 7,000 (units) costs $ 38,800
- Middle level 5,500 (units) costs $ 35,000

Therefore: variable cost per unit= 3800/1500 = $ 2.53

Step 3: Cost expressions: Total cost of 7,000 units

Fixed cost + variable cost= 38,800

Fixed cost + 7,000 x 2.53= 38,800

Fixed cost + 17,710= 38,800

Fixed cost=38,800–17,710=21,090

Step 4: Construct total cost function (un-adjusted levels):

- Above 5,500 units

Total cost=a+bx= 21,090+ 2.53x

- Below 5,500 units

Total cost=a+bx=(21,090 x 100/110)+2.53x

Total cost=a+bx= 19,173+2.53x

## A step change in variable costs (is given as a Percentage/money amount)

- When there is a step change in variable cost per unit as money value the same approach is needed as for a step change in fixed cost. (
**above**) - When the change in variable cost per unit is given as a percentage amount, then variable cost(s) per unit should be calculated for
**TWO**levels:

- variable cost per unit used above change ; and
- variable cost per unit used below change.

## High/Low and Linear Regression Analysis

In summary, **linear regression is a better technique then high/low analysis** because;

- it is more reliable ; and
- it’s reliability can be measured.

**Formula:**

Line of best fit (y=a+bx) can be constructed by calculating values for “**a**” and “**b**” using:

**a** = ∑y – b∑x

n n**b**= ∑xy – ∑x ∑y

n∑x2 – (∑x)2

where:

**x** = units**y** = costs

Enter the values into the line of best fit (y=a+bx) and solve for b and then a .

The following table should be used for calculating values of “x” and “y”

**Table**

X | Y | X^{2} | XY |

High/Low and Linear Regression Analysis