There are many definitions relating to cost, including the following 7 terms: marginal cost, total cost, fixed cost, total variable cost, average total cost, average fixed cost and average variable cost.

When asked to compute these 7 figures on an assignment or on a test, the data you need is likely to come in one of three forms:

- In a table which provides data on total cost and quantity produced.
- A linear equation relating total cost (TC) and quantity produced (Q).

- A non-linear equation relating total cost (TC) and quantity produced (Q).

Let's first define each of the 7 terms of cost, and then see how the 3 situations should be dealt with.

### Defining Terms of Cost

Marginal cost** **is the cost a company incurs when producing one more good. Suppose we're producing two goods, and we would like to know how much costs would increase if we increase production to 3 goods. This difference is the marginal cost of going from 2 to 3. It can be calculated by:

Marginal Cost(2 to 3) = Total Cost of Producing 3 – Total Cost of Producing 2.

For example, let's say it costs 600 to produce 3 goods and 390 to produce 2 goods. The difference between the two figures is 210, so that is our marginal cost.

Total cost is simply all the costs incurred in producing a certain number of goods.

Fixed costs are the costs that are independent of the number of goods produced, or more simply, the costs incurred when no goods are produced.

Total variable cost is the opposite of fixed costs. These are the costs that do change when more is produced. For instance, the total variable cost of producing 4 units is calculated by:

Total Variable Cost of Producing 4 units = Total Cost of Producing 4 Units – Total Cost of Producing 0 units.

In this case, let’s say it costs 840 to produce 4 units and 130 to produce 0.

Then total variable costs when 4 units are produced is 710 since 810-130=710.

Average total cost** **is the fixed cost over the number of units produced. So if we produce 5 units our formula is:

Average Total Cost of Producing 5 = Total Cost of Producing 5 units / Number of Units

If the total cost of producing 5 units is 1200, average total cost is 1200/5 = 240.

Average fixed cost** **is fixed costs over the number of units produced, given by the formula:

Average Fixed Cost = Fixed Costs / Number of Units

As you might have guessed, the formula for average variable costs is:

Average Variable Cost = Total Variable Costs / Number of Units

### Table of Given Data

Sometimes a table or chart will give you the marginal cost, and you'll need to figure out the total cost. You can figure out the total cost of producing 2 goods by using the equation:

*Total Cost of Producing 2 = Total Cost of Producing 1 + Marginal Cost(1 to 2)*

A chart will typically provide information regarding the cost of producing one good, the marginal cost and fixed costs. Let's say the cost of producing one good is 250, and the marginal cost of producing another good is 140. In this case, the total cost would be 250 + 140 = 390. So the total cost of producing 2 goods is 390.

### Linear Equations

This section will look at how to calculate marginal cost, total cost, fixed cost, total variable cost, average total cost, average fixed cost and average variable cost when given a linear equation regarding total cost and quantity. Linear equations are equations without logs. As an example, let’s use the equation TC = 50 + 6Q.

Given the equation TC = 50 + 6Q, that means the total cost goes up by 6 whenever an additional good is added, as shown by the coefficient in front of the Q. This means there is a constant marginal cost of 6 per unit produced.

Total cost is represented by TC. Thus, if we want to calculate the total cost for a specific quantity, all we need to do is substitute the quantity in for Q. So the total cost of producing 10 units is 50 + 6*10 = 110.

Remember that fixed cost is the costs we incur when no units are produced.

So to find the fixed cost, substitute in Q = 0 to the equation. The result is 50 + 6*0 = 50. So our fixed cost is 50.

Recall that total variable costs are the non-fixed costs incurred when Q units are produced. So total variable costs can be calculated with the equation:

Total Variable Costs = Total Costs – Fixed Costs

Total cost is 50 + 6Q and, as just explained, fixed cost is 50 in this example. Therefore, total variable cost is (50 +6Q) – 50, or 6Q. Now we can calculate total variable cost at a given point by substituting for Q.

Now on to average total costs. To find the average total cost (AC), you need to average total costs over the number of units we produce. Take the total cost formula of TC = 50 + 6Q, and divide the right hand side to get average total costs. This looks like AC = (50 + 6Q)/Q = 50/Q + 6. To get average total cost at a specific point, substitute for the Q. For example, average total cost of producing 5 units is 50/5 + 6 = 10 + 6 = 16.

Similarly, just divide fixed costs by the number of units produced to find average fixed costs. Since our fixed costs are 50, our average fixed costs are 50/Q.

As you may have guessed, to calculate average variable costs you divide variable costs by Q. Since variable costs are 6Q, average variable costs are 6. Notice that average variable cost does not depend on quantity produced and is the same as marginal cost. This is one of the special features of the linear model, but will not hold with a non-linear formulation.

### Non-Linear Equations

In this final section, we will consider non-linear total cost equations. These are total cost equations that tend to be more complicated than the linear case, particularly in the case of marginal cost where calculus is used in the analysis. For this exercise, let’s consider the following 2 equations:

TC = 34Q3 – 24Q + 9

TC = Q + log(Q+2)

The most accurate way of calculating the marginal cost is with calculus. Marginal cost is essentially the rate of change of total cost, so it is the first derivative of total cost.

So using the 2 given equations for total cost, take the first derivate of total cost to find the expressions for marginal cost:

TC = 34Q3 – 24Q + 9

TC’ = MC = 102Q2 – 24TC = Q + log(Q+2)

TC’ = MC = 1 + 1/(Q+2)

So when total cost is 34Q3 – 24Q + 9, marginal cost is 102Q2 – 24, and when total cost is Q + log(Q+2), marginal cost is 1 + 1/(Q+2). To find the marginal cost for a given quantity, just substitute the value for Q into each expression for marginal cost.

For total cost, the formulas are given.

Fixed cost is found when Q = 0 to the equations. When total costs are = 34Q3 – 24Q + 9, fixed costs are 34*0 – 24*0 + 9 = 9. This is the same answer we get if we eliminate all the Q terms, but this will not always be the case. When total costs are Q + log(Q+2), fixed costs are 0 + log(0 + 2) = log(2) = 0.30. So although all the terms in our equation have a Q in them, our fixed cost is 0.30, not 0.

Remember that total variable costs is found by:

Total Variable Costs = Total Costs – Fixed Costs

Using the first equation, total costs are 34Q3 – 24Q + 9 and fixed costs is 9, so total variable costs are 34Q3 – 24Q. Using the second total cost equation, total costs are Q + log(Q+2) and fixed cost is log(2), so total variable costs are Q + log(Q+2) – 2.

To get the average total cost, take the total cost equations and divide them by Q. So for the first equation with a total cost of 34Q3 – 24Q + 9, the average total cost is 34Q2 – 24 + (9/Q). When total costs are Q + log(Q+2), average total costs are 1 + log(Q+2)/Q.

Similarly, divide fixed costs by the number of units produced to get average fixed costs. So when fixed costs are 9, average fixed costs are 9/Q. And when fixed costs are log(2), average fixed costs are log(2)/9.

To calculate average variable costs, divide variable costs by Q. In the first given equation, total variable cost is 34Q3 – 24Q, so average variable cost is 34Q2 – 24. In the second equation, total variable cost is Q + log(Q+2) – 2, so average variable cost is 1 + log(Q+2)/Q – 2/Q.