COST REVENUE AND PROFIT FUNCTIONS
Many business situations allow us to model how cost, revenue and vary
with respect to different parameters, and how they combine to yield a
functional expression for profit.
In most cases, it is then possible to attempt to maximise and minimise
these functions using calculus techniques.
In any commercial environment, the gross profit can be considered as a
simple functions of the difference between the revenue obtainable from
the sale of a number of products and the costs involved in producing those
products, i.e.
Profit(P) = Revenue(R) − Costs(C).
eg, if a wholesaler can buy items at $6.50
each and sells them at $7.20, the profit per item is $7.20 - $6.50 =$0.70. The total profit realised from the handling of x items can be
expressed in a similar form:
Profit from x items
= Revenue from x items − Cost of x items
= (7.20)x − (6.50)x
= (0.7)x
The Profit Function
Normally, both cost and revenue functions of x (the number of items
demanded or supplied) and so an expression for profit, also as a function
of x, can be derived as the difference between the revenue and cost
functions:
Definition: The Profit Function
P(x) = R(x) −C(x)
where
x
is the quantity of items demanded
(supplied or produced)
P(x) is the profit function in terms of x
R(x) is the revenue function in terms of x
C(x) is the cost function in terms of x
It is usual to assume that supply and demand of items (or products) is
identical unless specifically stated otherwise.
A maximum profit point for some process can be defined through normal
calculus methods.
Definition: Maximum Profit Point
The Maximum Profit Point for some process can be
solved by solving the equation
P′ (x) = 0
for x, where
x is the quantity of items demanded (supplied or produced)
P is the profit function in terms of x
In general however, the precise form of cost and revenue functions
are not known and it is more likely that it will be necessary to derive
them from supplied information.
Cost Functions
There are normally a number of components to the cost of produced:
Fixed (or setup) costs This type of cost is normally associated with the
purchase, rent or lease of equipment and fixed overheads. It can
sometimes include the transportation and manpower movement costs
also.
In general, it can be considered as all those costs that need to be
borne before production can physically begin (and thus is
independant of the number of items to be produced).
Variable costs These are costs normally associated with the supply of
the raw materials and overheads necessary to manufacture each
product. Thus, they will depend on the number of items produced.
For example, if x is the number of items produced in a day and each
item costs $2.50, then the daily variable cost is (2.5)x.
Special costs This option cost factor is sometimes included in a total cost
function and might cover costs relating to storage, maintenance or
deterioration. The total costs involved are normally expressed in the
form cx2 , where c is a relatively small value and x is the number of
items under consideration.
The effects of this type of cost would only be significant for large
production rungs or other large processes. Hence, a total cost
function takes a general form as follows:
Definition: Form of a Cost Function
C(x) = a + bx + cx2
where
x
is the quantity of items demanded
(supplied or produced)
a is the fixed cost associated with the product
b is the variable cost per item
c
is the (optional) special cost factor
Revenue Functions
We can write the revenue function from the output of some process as
follows:
Definition: Form of a Revenue Function
R(x) = x pr (x)
where
x
is the quantity of items demanded
pr (x) is the fixed cost associated with the product.
The Demand Function
It is quite usual in a business environment for item price to depend on the
number of items in demand. That is, the more items that are in demand,
the less the price per unit is. Hence, the reason that price is expressed in
terms of x.
It is usually known as the demand function, where it should be
remembered that (unless otherwise stated) it is assumed that supply and
demand are equal.
The following two sections outline its form and the technique that
sometimes need to be used to identify it.
Demand functions are used to vary the price of an item according to how
many items are being considered. As mentioned in the previous section,
the more the number of items, the less the price per item and vice versa.
This is simply the standard business principle of ‘economies of scale,’
where it is generally more efficient to operate on as large a scale as can be
coped with.
Demand functions are normally linear.
Definition: The Demand Function
pr (x) = a + bx
where
x
a, b
is the quantity of items demanded
∈R
pr (x) is the price function
Thus, if we have pr (x) = 25 − 2x then R(x) = x(25 − 2x) = 25x − x2 .
Constructing the Demand Function
In many cases the demand function will not be stated specifically, but
rather given in terms of the price of the unit at two different demand
levels.
In these cases, we assume that the demand function is linear, and
construct the demand function accordingly.
As an example, suppose the price of a product is $0.85 per item when
100 products are demanded, but is $0.68 when 500 items.
So, we assume the demand function is linear and so has the form
pr (x) = a + bx
Substituting our two sets of values gives us:
(0.85) = a + b(100)
(0.68) = a + b(500)
Subtracting
=⇒ (0.17) = −400b
and so we have b = −0.000425 and so a = 0.85 + 0.0425 = 0.8925
Thus, our demand function is
pr (x) = 0.8925 − (0.000425)x
Figure below represents a graph of (a) the hypothetical total revenue, total cost, and profit functions, and (b) the marginal revenue, marginal cost, and marginal profit functions.
