UK Overtime in Job Costing


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We learn that a cost is a variable cost if the total cost (y) varies with the level of output (x) like this: y = kx, where k is a constant


However, I am not so sure if this makes sense in job costing:


For example, for job A, we need 1 worker to work for the basic number of hours (40 hours/week) for 1 week, plus overtime in that week.

Basic rate is £8 an hour. Overtime is time and a half (£12 an hour)

Then:

Let x be the number of labour hours worked, where x is greater than or equal to 40

Am I correct to say that the graph for the total labour cost (y) is:

y = 8 x 40 + 12 (x - 40) = 12x - 160?

Then, the graph does not have the form y = kx, so the total labour cost here is no longer a variable cost, which seems strange?
 
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Your function's restricted domain confuses the picture. If overtime is time-and-a-half, your total labor cost y is given by:

y = 8t + 4 * max(0, t - 40)

The cost is a variable cost because the cost depends with the amount of time worked. If you worked zero hours, your labor cost would be zero. The function doesn't look like y = kt, but that isn't required for the cost to be variable.

Contrast that with monthly rent or straight-line annual depreciation, where the same cost is charged each period.
 
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Thanks. Yes, I am okay with having the domain expanded.

However, as far as my study is concerned (I am doing AAT Level 3 at the moment -- sorry if I should have posted this in the Exams and Studying area, which I discovered only afterwards), both AAT Level 2 and AAT Level 3 taught that the definition of a variable cost is one that changes proportionately with output level: its graph always cuts through the origin (which really is the graph of the form y = kx) and the variable cost per unit stays the same (which is k).

This is especially relevant when I needed to calculate contribution.

Suppose the income for this job is given. By definition, contribution = revenue - variable costs. Do you exclude all labour costs (including overtime) when calculating the contribution?
 
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Most instructors teach that variable costs mean y = kx because they're trying not to turn an accounting lesson into a math lesson. They're oversimplifying the situation in order to make it easy to connect the math to the accounting.

Most costs are mixed costs, partly fixed and partly variable. And in general, costs aren't linear functions of their drivers. But a lot of costs are close to linear over some range of reasonable values of the drivers, and linear functions are easy to use, understand and explain. So in most real-world situations, you'll do some kind of linear regression to estimate costs over a range of driver values, effectively forcing your cost to be a linear function y = mx + n.

The "mx" part is the variable part of the cost, and the "n" is the fixed part of the cost. If your fixed costs are zero, your cost looks like y = mx.
When you have mixed costs, contribution margin = revenue - variable costs, and net income = contribution margin - fixed costs. In this case, if your mixed cost is y = mx + n, then contribution margin = revenue - mx, and net income = contribution margin - n.
 
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Thank you so much. I completely get what you meant: not turning an accounting lesson into a maths lesson, as you then lose focus. I was just very confused actually, not being pedantic. Thanks for your help!
 

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