§8 Preparation for economics November 2003 Yuichiro Hayashi      
Please refer to  'Outline of This Website'. The income statement  Table 1 in the website,  is changed to Table 1 below, where:  

η (ε) = AX(-) (ε) - AX(+)(ε)

(1)
                              Table 1 Income statement of a company

Table 1 is transformed to Table 2, where the notation (ε) has been omitted, and:
V( variable cost) = D ; which mainly consists of the amount paid to sub-constructors, variable labor costs, and material costs. 
F = C + η +G ; Fixed cost.
                                     Table 2 Income statement-2 

In the income statement-2 in Table 2,  we will examine the break-even chart when sales X naturally increases or decreases to be X + Δ X, keeping F constant. The expected income statement is obtained as shown in Table 3. 
                                Table 3 Income statement when F = constant

From Table 2 and Table 3 we have: 
Δ X = Δ V + Δ P (2)
When we denote a = V / X ( Variable cost ratio), Eq. (2) is expressed as follows:
Δ X = a Δ X + Δ P (3)
Then we get:
Δ P =  (1 - a ) Δ X (4-1)
 Δ X = Δ P / (1 - a )  (4-2)
Table 3 is transformed into Fig. 1. In Fig. 1, Δ X takes any value, or we can give any value to Δ X. Ratio Δ P / Δ X = 1 - a . Eq. (2)  holds whether Δ X moves in the plus or minus direction. Eq. (2) shows that the incremental sales consist of the incremental profit and the incremental variable cost.  Eq. (4-2) indicates that Δ X can be expressed by using Δ P and ' a '.  This equation does not mean that Δ P is naturally multiplied by 1 / (1 - a ) to be Δ P / (1 - a ). Both Δ P and  Δ V are variable energy inputs as explained later. The expression Δ P / (1 - a ) is another expression of Δ X, in the same meaning, the expression Δ V / a =Δ X is. 

Fig. 1. Break-even chart when X → X + Δ X and F = constant
There is another case that  the current, fixed cost F needs to decrease in such a manner that the current profit P is kept constant when the next year's sales  are expected to decrease as much as Δ X because of recession. In this case, profit P is set to be constant, that is P = P0, in Table 2.  The next year's intended income statement is shown in Table 4.
                                 Table 4 Next year's intended income statement 

From Table 2 and Table 4 we have: 
Δ X = Δ V + Δ F (5)
Δ X = Δ F / (1 - a )  (6)
This case is shown in Fig. 2. In this case, the decrement Δ X consists of a proportional part of Δ X and an independent part of  Δ X. The former is Δ V = a Δ X , and the latter is Δ F = P0. The figure shows that the total costs decrease as much as Δ F + Δ V, and the sales decrease as much as, Δ X keeping P constant.

              Fig. 2 Break-even chart when X → X - Δ X , keeping P constant
Consider the case where Δ F is a given amount in F → F + Δ F keeping P( = P0) constant, that is Δ P = 0. In a similar manner as the previous case, we have Table 5 and a break-even chart Fig. 3 for Table 5. In this case,  Δ F = P0 holds.
                 Table 5 Income statement when F → F + Δ F, keeping P constant 

                            Fig. 3 Break-even chart when X → X + Δ X, keeping P constant
In Fig. 3, we should not interpret as if F → F + Δ F（ = P0 ) and that Δ X, therefore, naturally approaches  P0 / (1 - a ). Fig.3 should be interpreted as if F → F + Δ F（ = P0 ). The next year's sales X need to increase to be X + P 0 / (1 - a ) in order to keep P constant . This result will not be necessarily assured because there is no relationship between incremental fixed costs and incremental sales in the firms, when  P cannot be kept constant. 
The relationship between Δ X (sales= goods sold, output), Δ N (inventory, goods not sold, output ), Δ P (profit, input), Δ V'( injected variable costs, input), and  Δ F'( injected fixed costs,  input) will be studied a little more deeply. By the principle of conservation of energy, that is, Input energy ( Injected  energy into goods )  = Output (produced goods with the energy ), we have: 
Δ X + Δ N =  Δ P + Δ V' + Δ F' (7)
where the left side shows output and the right side, input energy.
If we define: 
Δ N = Δ V' + Δ F' -   (Δ V + Δ F ) (8)
we obtain :
Δ X  =  Δ P + Δ V + Δ F (9)
where
Δ V  =  a Δ X  (10)
a  = V /  X  (11)
In firms' income statements, Δ N  is capitalized to be assets. Δ X is sales. The injected energy Δ P means that Δ P is incremental energy evaluated or evaluated value by buyers. Δ V is a variable cost including labor costs,  amount paid to sub-contractors and the other variable costs. The term 'fixed' in 'fixed cost' means that the cost does not relate to the increase and decrease of sales Δ X. Therefore, the fixed cost can vary independently of sales or unaffectedly by sales fluctuation. 
The relationship between the terms in Eq.(9) is obtained by Table 6.

Table 6 Relationship between incremental inputs and output in firms
Table 6 shows firstly that Δ X( output)  is determined by demand after negotiations with buyers. Secondly, variable input energy Δ V =a Δ X is automatically needed to produce Δ X.  Thus, Δ P and Δ F have a competing relationship for sharing Δ X - Δ V. Thirdly, entrepreneurs determine the amount of Δ F not affected by the amount of Δ X, by the definition of the fixed cost. The residual value( evaluated cost by the buyers) is Δ P. 
Through these tables, we have the following:
Case-1 When F is constant, that is to say Δ F = 0, we have Table 7.

Table 7 Relationship between incremental inputs and output when Δ F = 0

In Table 7 the following holds:
 Δ P  =  (1 - a ) Δ X (12-1)
Δ X = Δ P / (1 - a )   (12-2)
The expression Δ P / (1 - a ) is another expression of Δ X. This shows that Δ P (input)  is directly proportional to Δ X (output). These relations are expressed in Fig. 1. 
Case-2 When P is constant, that is to say Δ P = 0, we have Table 8.

Table 8 Relationship between incremental inputs and output when Δ P = 0
In table 8 the following holds:
Δ F =  (1 - a )  Δ X (13-1)
Δ X = Δ F / (1 - a )  (13-2)
Eq. (13-1) shows that if Δ X (output) is given as a determined amount, Δ F( input energy) will be determined according to Δ X. So, Δ F is not independent of the amount of Δ X under the condition of keeping P constant. If F varies a little from Eq. (13-1), Δ P will inevitably be generated a little. Therefore, the condition of both Eq. (13-1) and Δ P = 0 is one pair in Table 8.
Case-3 In the case of Table 6 , we have:
 Δ P =  (1 - a ) Δ X -  Δ F (14)
When Δ X (output) is a given amount. Δ P and Δ F make a linear relation which will be explained later.
There is another case where the ratio a (= V / X) itself varies. However, we will not  pursue this further. Through the cases, we know that the income statement expresses the relationship between input energy ( costs and profit) and output (sales).
In conclusion, cost-volume-profit analysis under standard costing (absorption costing or full costing) is possible.