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Linear Cost Model from Costs or Two Points

Calculate total and average cost from fixed and per-unit costs, or infer a linear model from two quantity-and-cost observations.

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Calculator inputs

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How to use this calculator

  1. Enter the known values in the units shown. Results update as you type.
  2. Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
  3. Use the worked example to check the method. Reset restores the starting fields.

Use the result with context

Results are estimates for planning and education. Confirm rates, taxes, fees, and legal requirements with the relevant institution or a qualified professional before making a financial decision.

Formula and method

Fixed cost of $100 and variable cost of $5 per unit give C(x) = 100 + 5x. At 20 units, total cost is $200 and average cost is $10 per unit.

C(x) = fixed cost + variable cost per unit × x; fitted slope = change in total cost / change in quantity

Worked example

Enter these known values and leave the other values blank.

Fixed-cost intercept
100 USD
Variable cost per quantity unit
5 USD
Quantity to evaluate
20
Total cost
200 USD
Average cost per unit
10 USD
Total variable cost
100 USD

Assumptions and limitations

  • The model is linear: the fixed-cost intercept stays constant and each additional quantity unit adds the same variable cost. Capacity steps, discounts, nonlinear production effects and changing prices are excluded.
  • Use consistent quantity units and currency bases for both observations. The two observed quantities must be positive and distinct. Their ordering may be reversed as long as each cost stays paired with its quantity.
  • This cost model requires nonnegative observed costs, fixed-cost intercept and variable-cost slope. A two-point line that implies a negative coefficient is rejected under these assumptions.
  • The evaluated quantity must be positive because average costs divide by quantity. Zero fixed or variable cost is supported, including an all-zero cost model at a positive quantity.
  • Two observations determine a line algebraically; they do not validate its fit over other quantities. Currency conversions are fixed captured factors rather than current quotes.

Common questions

How are fixed and variable costs inferred from two observations?

The change in cost divided by the change in quantity gives the variable cost per unit. Subtract that rate times either observed quantity from its total cost to obtain the fixed-cost intercept.

Why is the variable cost multiplied by x in the summary?

It is a cost per quantity unit. Multiplying by x produces the variable cost at that quantity; adding a per-unit cost alone would not give the total cost function.

References

Bookify allows zero fixed and variable costs, requires nonnegative cost observations and fitted cost coefficients, and distinguishes the two observed quantities exactly.

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