In this section, we explain the mechanism that links costs, price and margin. Note that a strong assumption is made: we consider that any number of available items can be sold.
10 1 Introduction to Pricing
1.5.1 Notations
Price affects margin, but may also affect cost. The higher the price, the greater the margin per item, but increasing the price may result in a fall-off in sales. On the contrary, when the price decreases, the number of items sold may increase, and thus the production costs may decrease due to the scale effect.
Let us introduce the following variables and assumptions:
• c(s) is the variable cost per item when the total number of items sold during a given period (a month for instance) is s. The function c(s) is a non-increasing function of s.
• f(s) is the fixed cost, that is to say the cost that applies whatever the number of items sold during the same period as the one chosen for the variable cost.
Nevertheless, f(s) may increase with regard to s if some additional facilities are necessary to pass a given production threshold. The function f(s) is a non- decreasing function of s.
In fact, c(s) (respectively, f(s)) is either constant, or piecewise-constant.
• R is the margin for the period considered.
• p is the price of one item.
1.5.2 Basic Relation
We assume that all the items produced are sold, which implies that the market ex- ists and the price remains attractive to customers. Under this hypothesis, which is strong, Relation 1.2 is straightforward:
) ( ) (s f s c
s p s
R= − − (1.2)
As we will see below, it is often necessary to compute the number of items to be sold in order to reach a given margin, knowing the price, the variable cost per item and the fixed cost. Let us consider two cases.
1.5.2.1 Both the Variable Cost per Item and the Fixed Cost are Constant In this case, Equation 1.2 becomes:
f c s p s
R= − −
which leads to Relation 1.3:
⎥⎥
⎢ ⎤
⎢
⎡
−
= + c p
f
s R (1.3)
Remember that ⎡ ⎤a is the smallest integer greater than or equal to a.
Indeed, the price is always greater than the variable cost per item.
The margin being fixed, it is easy to see that the number of items to be sold is a decreasing function of the price and an increasing function of the variable cost per item.
1.5.2.2 At Least One of the Two Costs is Piecewise-constant
In this case, the series of positive integer numbers is divided into consecutive in- tervals for each cost, and the value of this cost is constant on each one of these in- tervals. To find the number of items to be sold, we consider all the pairs of inter- vals, a pair being made with an interval associated with the variable cost per item and an interval associated with the fixed cost. We use the costs corresponding to these intervals to apply Relation 1.3. If the resulting number of items to be sold belongs to both intervals, this number is a solution to the problem; otherwise an- other pair of intervals is tested. This approach is illustrated by the following ex- ample.
Example
We assume that the price of one item is 120 € and that the required value of the margin is 100 000 €. The fixed cost is 8000 € if the number of items sold is less than 2000 and 11 000 € if this number is greater than or equal to 2000. Similarly, the variable cost per item is 50 € if the number of items sold is less than 1000 and 40 € otherwise.
1. We first assume that the fixed cost is 8000 € and the variable cost per item is 50 €. Applying Relation 1.3 leads to:
50 1543 120
8000 000
100 ⎥⎥=
⎢ ⎤
⎢
⎡
−
= + s
To be allowed to select 50 € as variable cost per item, the number of items sold must be less than 1000, which is not the case. Thus, this solution is rejected.
2. We now assume that the fixed cost is 8000 € and the variable cost per item is 40 €. In this case, we obtain:
12 1 Introduction to Pricing
40 1350 120
8000 000
100 ⎥⎥=
⎢ ⎤
⎢
⎡
−
= + s
This value is less than 2000, which fits with the fixed cost used, and greater than 1000, which corresponds to a variable cost per item of 40 €. This is an op- timal solution.
3. If we consider that the fixed cost is 11 000 € and the variable cost per item is 50 €, then we obtain s = 1586 items. This fits neither with the variable cost per item nor with the fixed cost. Therefore, this solution is rejected.
4. Finally, if the fixed cost is 11 000 € and the variable cost per item is 40 €, then s = 1388 items: this does not fit with the fixed cost. This solution is rejected. To conclude, we have to sell 1350 items to reach the margin of 100 000 €.
1.5.3 Equilibrium Point
The equilibrium point is the minimum number of items to sell in order not to lose money. We simply have to assign the value 0 to R and apply Relation 1.3. We still assume that all the items that are produced are sold. We will examine how the equilibrium point evolves with regard to the price in the following example.
Example
We consider the case when c(s)=c=120€, f (s)= f =100000€ and we as- sume that the price evolves from 200 € to 320 €. The data obtained by applying Relation 1.3 are collected in Table 1.1.
Table 1.1 Equilibrium point with regard to price
Price 200 210 220 230 240 250 260 270 280 290 300 310 320 Equil.
point 1250 1112 1000 910 834 770 715 667 625 589 556 527 500
It is not surprising (see Equation 1.3) that the equilibrium point is a decreasing function of the price, and that the slope of the curve decreases as the price in- creases.
The equilibrium point as a function of price is represented in Figure 1.2.
450 550 650 750 850 950 1050 1150 1250 1350
200 210 220 230 240 250 260 270 280 290 300 310 320 Price
Equilibrium point
Figure 1.2 Equilibrium point function of price
1.5.4 Items Sold with Regard to Price (Margin Being Constant)
We still assume that both the variable cost per item and the fixed cost are constant.
In this case, Relation 1.3 holds. If we relax the integrity constraint, this relation becomes:
c p
f s R
−
= + (1.4)
Assume that the price increases by ε, which results in a variation of η in the number of items sold. Relation 1.4 becomes:
c p
f s R
− +
= +
+η ε (1.5)
Subtracting (1.4) from (1.5) we obtain:
) (
) )(
(R f p c p c
− + + −
−
= ε
η ε
Taking into account Relation 1.4, this equality can be rewritten as:
c p
s
−
− +
= ε
η ε
14 1 Introduction to Pricing In terms of ratio, we obtain:
c p
p p s =− + −
ε ε
η (1.6)
Assume that R, f and c are given. Let p0 be a price and s0 the corresponding number of items to be sold in order to reach the margin R.
According to Relation 1.6:
c p
p
−
− +
= ε ε
η
0
% 0
%
where %ε is the percentage the price increases with regard to p0 and %η the percentage of items sold decreases with regard to s0, to reach a given margin R.
Consider the function:
c p f p
−
− +
= ε
ε
0
) 0
(
This function is increasing and tends to 0 when ε tends to infinity. Further- more, f (ε) is equal to –1 for ε=c, less than –1 when ε<c, and greater than –1 when ε>c. As a consequence, the decrease in the percentage of items sold is faster than the augmentation in the percentage of price increase as long as the in- crease in price remains less than the variable cost per item. When the increase of price becomes greater than the variable cost per item, the percentage of price in- crease is greater than the percentage of increase of items sold. Note that these re- marks hold for a given margin and a given initial price. Remember also that we assume that the market exists or, in other words, that the market can absorb all items produced. Function f (ε) is represented in Figure 1.3.
ε
) (ε f
c
–1
– p0 / ( p0 – c )
Figure 1.3 Function f(ε)
Example
In this example, the variable cost per item is equal to 80 €, the initial price p0 is equal to 200 € and the fixed cost is equal to 100 000 €. Both the variable cost per item and the fixed cost are constant. Table 1.2 provides the decrease in the per- centage of the number of items sold according to the percentage increase in price.
Table 1.2 Effect of the price on the number of items sold
% of increase in price 0 10 20 30 40 50 60 70 80 90 100
% of decrease in the number of items sold
0 14.3 25 33.3 40 45.4 50 53.8 57.1 60 62.5
The increase in price of 40% corresponds to an increase of 80 €, which is the variable cost per item. As mentioned before, we observe that the evolution of the difference between the percentage of increase in price and the percentage of de- crease in the number of items sold is reversed starting from this point. In other words, the sign of this difference changes from this point.