3.2 Total cost and marginal cost
Here’s another example of a situation where it’s useful to think of the gradient of a graph (and hence the derivative of a function) as a rate of change. This example shows you one way in which the idea of differentiation can be used in economics.
Suppose that a small confectionery company has found that the weekly cost of making its milk chocolate consists of a fixed cost of £3000 (to pay for the rent and maintenance of its premises and equipment, for example), plus £8 per kilogram of chocolate made (to pay for the ingredients and staff time, for example). In other words, the weekly cost, (in £), is modelled by the equation
where is the amount of chocolate made in the week, in kilograms. The graph of this equation is shown in Figure 35.
Suppose that the company is currently making a particular quantity of milk chocolate per week, but is thinking of increasing its production. Then the additional weekly cost of making the extra chocolate will, of course, be £8 per extra kilogram made. This is known as the marginal cost per kilogram of making the extra chocolate. It’s different from the real cost per kilogram of making the chocolate (usually called the unit cost or average cost), which is equal to the total cost, (in £), divided by the quantity in kilograms of chocolate made, .
In fact, the marginal cost of making the extra chocolate is the gradient of the graph in Figure 35. This is because the gradient of the graph is the rate at which the quantity on the vertical axis (the total cost, in £) increases as the quantity on the horizontal axis (the quantity of chocolate made, in kg) increases. The fact that the gradient is 8 pounds per kilogram, which is normally written as £8 per kilogram, tells you that the total cost is increasing at a rate of £8 for every kilogram by which the quantity of chocolate increases.
Often an equation that models the total cost of making a quantity of a product doesn’t have a straight-line graph. For example, suppose that the confectionery company has found that a better model for the total weekly cost (in £) of making a quantity (in kg) of milk chocolate is
The graph of this equation is shown in Figure 36.
As before, the gradient of the graph tells you the rate at which the total weekly cost increases as the amount of chocolate made increases. In other words, the gradient of the graph is the marginal cost per kilogram of making extra chocolate. You can see that the gradient of the graph – the marginal cost per kilogram – increases as the quantity of chocolate made increases. This tells you that the more chocolate the company makes, the larger is the cost of making an extra kilogram of chocolate.
There are various reasons why this might be the case. For example, to increase its production of chocolate the company might have to pay its staff at higher rates, for overtime or non-standard hours.
For any particular quantity (in kg) of chocolate being made, the marginal cost per kilogram of making any extra chocolate is the gradient of the graph at that value of , which as you know is given by . Of course, this is an ‘instantaneous’ value, valid only for that value of .
Activity 12 Working with marginal cost
Suppose that the confectionery company discussed above has decided that the second equation above, namely
is an appropriate model for the total weekly cost of making its milk chocolate. Here is the quantity of chocolate made (in kg), and is the total cost (in £). Let the marginal cost per kilogram of making any extra chocolate be (in £).
a.Use differentiation to find an equation for the marginal cost in terms of the quantity .
b.What is the marginal cost per kilogram of making extra chocolate when the amount of chocolate already being made is kilograms?
c.The company sells all the chocolate that it makes at a price of £ per kilogram. It decides to keep increasing its weekly production of chocolate until the marginal cost is equal to the price at which it sells the chocolate. (This is because if it increases its production any further, then it will cost more money to make the extra chocolate than the company will obtain by selling it.) By writing down and solving a suitable equation, find the weekly quantity of chocolate that the company should make.
Answer
a.The equation for in terms of is
Hence
That is, an equation for in terms of is
b.Substituting into the equation found in part (a) gives
So the marginal cost per kilogram of making extra chocolate when the amount of chocolate already being made is kilograms is £.
c.The marginal cost is equal to the selling price when
By part (a), the value of for which this happens is given by
Solving this equation gives
that is,
So the company should make 750 kg of chocolate per week. (As a check on parts (b) and (c), you can see that the gradients of the graph in Figure 36 at the points with -coordinates and kilograms are about £ per kilogram and about £ per kilogram, respectively. The tangents at these points are shown below.)
The fact that marginal cost is the derivative of total cost is used in many economic models.