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1 Theme No. 5: Costs classification, functions (models), analysis Department of Economics and Management UCT PRAGUE 1

2 Department of Economics and Management 2

3 As an introduction we will repeat definitions from the area of revenues, expenses and profit. Revenue is the amount of money received by an enterprise during a specific time period (significant part comes from the sale of goods and services to customers). Expense is the money spent or cost incurred in an enterprise's effort to generate revenue. Earning is the difference between revenues and expenses. It can be profit positive earning or loss negative earning. The main kinds of earning according the level of grossness: earnings before taxes (EBT) and earnings after taxes (EAT). Income statement is a financial statement covering the certain time period and containing revenues, expenses and earnings. 3

4 Costs classification Cost classification: By their type: material and energy consumption, depreciation and amortization, personal costs (wages and other personal costs as insurance), financial costs, service costs and others. By their function: according to outputs or subparts of the enterprise. By their traceability: direct or indirect costs, respectively unit costs and overhead costs; direct or unit costs are closely connected with one kind of product or service provided by the enterprise, on the other hand indirect costs or overhead costs are connected with more or even with all kinds of products and services provided by the enterprise. By their level of activity: fixed costs (FC) and variable costs (VC). Total costs are equal to fixed costs plus variable costs (TC = FC + VC). The fixed costs do not change and the variable costs change detail further. And others. Costs classification 4

5 Further we need the distinguish of costs (but also of revenues or profits) according to size absolute, relative Total costs TC, or only C (per specific production volume) Average costs AC (per a unit of production) Marginal costs MC (costs of additional unit of production) Costs classification 5

6 Example 5-1: The enterprise A has the fixed costs equal to per a month for all production volumes and variable costs 28 per one produced unit. This enterprise has a homogenous production (only one kind of product) and the linear dependency between production volume and costs. The linear dependency can be displayed by the linear cost function, in our case TC = * q, where q is the enterprise production volume per a month. Compute the following table: q (number of units/ month) FC VC TC ( ) ( ) ( ) AFC ( /unit) AVC ( /unit) AC ( /unit) MC ( /unit) Costs classification 6

7 Solution of example 5-1: q (number of units/ month) FC VC TC ( ) ( ) ( ) AFC ( /unit) AVC ( /unit) AC ( /unit) MC ( /unit) , , , , , , , , , ,56 28 Costs classification 7

8 Remarks: 1) Take attention that AC are lower when the production volume is higher. It is caused by the decreasing AFC. This positive thing is called degression of fixed costs and dit comes when the enterprise is able to achieve higher (better) usage of production capacities. Higher amount of the production has a sense only in the case when the enterprise is able to sell it on the market. The total production capacity of this enterprise is units per month. Higher production than units is possible only with the capacity increasing which causes an increase of fixed costs (jump) because the enterprise need to buy additional long term assets other buildings, machinery etc. 2) Take attention that AVC and MC do not change with the increasing of production volume and they are equal. It is only a case of enterprises with linear cost function. Further detail. Costs classification 8

9 Relations among productivity, production function and cost function Enterprise productivity (total factor productivity TFP) is a ratio outputs/inputs. It shows the efficiency of transformation inputs (production factors) into outputs t (goods and services provided dby the enterprise). Production function shows the relationship between inputs and outputs. It is used as a model which is a mathematical function displaying the relationship between quantity of inputs (production factors) as independent variable (horizontal axis in the following figure) and quantity of outputs as dependent variable (vertical axis in the following figure). The production function is specific for the particular enterprise or particular kind of production. The same is valid for revenueorcost function. The character of production function and its values are concentrated equation of production. For example the shape of production function is affected by the situation if the enterprise productivity according to the production volume changes or does not change. If the productivity does not change it leads to linear production function. If the productivity decreases the production function is nonlinear degressivelly increasing (its growths are decreasing) and even after particular production volume per the time period progressively decreasing. If the productivity increases then the production function is progressively increasing (its growths are increasing). The character of production function influences the character of cost function (and then as well the shape of profit function). In the cost function the volume of production is the independent variable (on the horizontal axis) and costs are dependent variable (on vertical axis). Linear cost function is implicated (almost, if the price changes are not overweighting) by linear production function, progressively increasing cost function is implicated by degressivelly increasing production function, degressivelly increasing cost function is implicated by progressivelly increasing production function and finally the cost function in the shape of inverted S by production S. It is logic that production and cost function have opposite character/shape. They react to productivity changes oppositely. These relationships are displayed on following figures. Note: This lecture is based on the short term cost and production function for example period of months (the inputs are divided id d into fixed and variable, in the long term all inputs are/can be variable time period of years or decades). The explanatory power of short term function is especially important for enterprise decision making process. Therefore total costs (TC) are marked as short term total costs (STC) in the following figures. Costs classification 9

10 If the productivity (the ratio outputs/inputs) does not change in the enterprise when the amount of used inputs increases, the production function is linear (the figure above, where L are inputs, respectively variable inputs, and Q is output, respectively the production volume). This usually leads to linear cost function (the figure bellow) (if the price changes do not work significantly). Costs classification 10

11 If the productivity (the ratio outputs/inputs) decreases in the enterprise when the amount of used inputs increases, the production function is degressivelly increasing (the figure above, where L are inputs, respectively variable inputs, and Q is output, respectively the production volume). This usually leads to progressively increasing cost function (the figure bellow) (if the price changes do not overweight). Costs classification 11

12 If the productivity (the ratio outputs/inputs) increases in the enterprise when the amount of used inputs increases, the production function is progressively increasing (the figure above, where L are inputs, respectively variable inputs, and Q is output, respectively the production volume). This usually leads to degressively increasing cost function (the figure bellow) (if the price changes do not overweight). Costs classification 12

13 If the productivity (the ratio outputs/inputs) increases in the case of low production volume, by higher production volume the productivity does not change and by even higher production volume the productivity decreases, the production function has S shape. It means that this function is first progressively increasing, then linear and finally degressively increasing (the figure above, where L are inputs, respectively variable inputs, and Q is output, respectively the production volume). This usually leads to the cost function (the figure bellow) which has the shape of invert S. It means that the cost function is first degressively increasing, then linear and finally progressively increasing (if the price changes do not overweight). Costs classification 13

14 Cost function (models, curves) How to find out (estimate) fixed and variable costs for a particular enterprise, and therefore its cost functions? There are several methods, for example the method of classification analysis (the method of logic distinguishing of costs) the least squares method (using regression analysis) the method of two time periods. Cost function (models, curves) 14

15 The following example is focused on the principle of the method of classification analysis (the method of logic distinguishing of costs). Example 5-2 (method of classification analysis): The enterprise B produces only one kind of output. (The cases when enterprise produces more kinds of product will be solved later, discussing global cost functions.) The enterprise B produced units of the product per the particular month. Enterprise costs are displayed in the table further (in ). The production capacity is units per month. The productivity analysis has shown that the enterprise productivity does not change with the increase of inputs and therefore the enterprise production function is linear and the cost function as well. It means TC = a + b * q, TC are total month cost in a are FC in per month b are AVC in per one unit of production, and therefore b * q are VC. cost consumption wages of production workers salaries of administrative employees depreciation and amortization rent consumption of (technological) energy consumption of energy for lightening transport promotion Total costs Cost function (models, curves) 15

16 Estimate for this enterprise (a) Fixed costs per month in, FC (we use the symbol a) and Variable costs per a unit in, AVC (we use the symbol b). Use the method of classification analysis firstly distinguish the cost items as fixed or variable costs, or almost fixed and variable costs. Question: Will be such sorting always smooth and clear? According to the solved part write the enterprise cost function. (b) After estimate enterprise costs when the enterprise produces units per month or when the enterprise does not produce anything per month. Cost function (models, curves) 16

17 Solution of example 5-2: a) V cost consumption V wages of production workers F salaries of administrative employees F depreciation and amortization F rent V consumption of (technological) energy F consumption of energy for lightening V transport F promotion Total costs FC: = ( ) VC: = ( ) ) a = b = / = 920 TC = a + b * q TC = * q Cost function (models, curves) 17

18 Continued solution of example 5-2: b) TC (q = 2 000) = * = ( ) TC (q = 0) = * 0 = ( ) = FC Cost function (models, curves) 18

19 In the following example (example 5-3) we will show other method leading to the determination (calculation) of the cost model (function). This approach does not need information about cost structure and it uses only information about total costs and total production volumes in the particular enterprise, but in several time periods (at least two). Cost function (models, curves) 19

20 Example 5-3 (the method of two time periods): The enterprise C also produces only one kind of product. We have information about total production volume (q) and about total costs (TC) which are displayed bellow. The production capacity is units per month. Write the appropriate enterprise cost function. Use the method of two time periods, choose two very typical costs periods and calculate the linear model, the line, which is passing through these two points displaying periods. Note: Using the least squares method we would compute that more precisely because of usage regression (and correlation) analysis which works not only with these two time periods but with all relevant points (time periods). Cost function (models, curves) 20

21 The pair of points has to be chosen carefully which leads to the line location which has the most appropriate p location to all relevant observations (respectively points, respectively time periods). Cost function (models, curves) 21

22 It is often (but not always) suitable to choose points with the lowest and highest q. It is done by visibility (on the other hand the least squares method instead of visibility uses more reliable and accurate mathematical-statistical apparatus regression and correlation analysis). The points with the lowest and highest production volume look as a good choice in the case of this enterprise. Question: Which months are displayed by these points? Cost function (models, curves) 22

23 Answer: It is February and April. We can compute now the parameters a and b from the set of two equations (corresponding to February and April) and two unknown variables. See the table with example. a =? b =? TC =? (TC = a + b * q) Cost function (models, curves) 23

24 Solution: = a + b * = a + b * = 480 * b b = : 480 = 47.1 a = * = TC = * q Cost function (models, curves) 24

25 Break-even point analysis and further analyses Which volume of production (per particular time period, for example month) does the enterprise have to produce for not being in red numbers (not being in loss)? And which volume of production (per particular time period, for example month) does the enterprise have to produce for creating specific amount of profit? How high/low should the product price be? Which volume of production (per particular time period, per month) does the enterprise have to produce and sell to achieve the highest profit? These questions and many others can be answered due to break-even point analysis and further connected analyses. Note: Partly simplified form of these analyses is called model (respectively analysis) CVP (Cost, Volume, Profit). Now we will calculate production volume, values of profit, costs, prices etc. in connected relationships. It is necessary to know actual enterprise cost model in practice. Break-even point analysis and further analyses 25

26 Continuing of example 5-3 (focus on break-even point analysis and further analyses): The enterprise C has the cost function TC = * q. This function has been already estimated during example solution. We know additional information: The product is sold for the price (p) 70 per a unit. This price is accepted by customers without t the respect how many units of product are supplied by this enterprise at the market. This leads to the linear shape of revenue curve (TR = p * q): TR = 70 * q. If the enterprise has to change the price for example for achieving higher sales, the revenue function is not linear. Break-even point analysis and further analyses 26

27 Continuing of example 5-3 Specify for this enterprise by computation and displaying in the figure: (a) Month break-even point, the minimal month production volume for not being in loss (b) The month production and sale volume which brings the profit equal to (operating profit defined as π = TR TC) (c) Product price (p) when the enterprise produces units and achieves the month profit (d) Maximal possible profit respecting cost and price condition and as well as production and sale possibilities Break-even point analysis and further analyses 27

28 Break-even point analysis and further analyses 28

29 Continuing solution of example 5-3: (a) Compute month break-even point, the minimal month production volume for not being in loss. Universally (for enterprises with linear revenues and linear costs): For q BEP is valid that π = 0 And because π =TR TC Therefore for q BEP is applied TR = TC It means p * q = a + b * q p * q b * q = a q * (p b) = a And therefore q BEP = a / (p b) q BEP = FC / (p AVC) For the enterprise C q BEP = / ( ) = 516 (units per month) Attention! This number has to be compared with constraints on the supply side (enterprise C its production capacity is units per month) and on the demand side (market, customers for enterprise C this constraint is above its production capacity from example text). Therefore the value 516 units per month is achievable by the enterprise C. Break-even point analysis and further analyses 29

30 Continuing solution of example 5-3: (b) Calculate the month production and sale volume which brings the profit equal to Universally (for enterprises with linear revenues and linear costs): For q π is valid that TR = TC + π Therefore p * q=a+b* q+π π p * q b * q = a + π q * (p b) = a + π And therefore q π = (a + π) / (p b) q π =(FC+π) /(p AVC) For the enterprise C q π = ( ) / ( ) = (units per month) Attention! Also this number has to be compared with constraints on the supply side (enterprise C its production capacity is units per month) and on the demand side (market, customers for enterprise C this constraint is above its production capacity from example text). Therefore the value units per month is achievable by the enterprise C. Break-even point analysis and further analyses 30

31 Continuing solution of example 5-3: (c) Calculate product price (p) when the enterprise produces units and achieves the month profit Universally (for enterprises with linear revenues and linear costs): For p for achieving profit π when the production is q is valid that TR = TC + π Therefore p * q = a + b * q + π And therefore p = (a + π) / q + b p = (FC + π) / q + AVC For the enterprise C p = ( ) / = per one unit Attention! Also this number has to be compared with constraints on the supply side (enterprise C its production capacity is units per month) and on the demand side (market, customers for enterprise C this constraint is above its production capacity from example text). Customers are able to accept the price 73.9 per one unit. Break-even point analysis and further analyses 31

32 Continuing solution of example 5-3: Calculate maximal possible profit respecting cost and price condition and as well as production and sale possibilities Universally (for enterprises with linear revenues and linear costs): If TR and also TC with the growth of q increase linear and if the enterprise is able to achieve the profit, the highest profit is achieved at the highest possible production and sale volume see the figure displayed on the slide 28. (Attention! For enterprises with non linear costs and/or revenues this is not universally valid see following slide.) Because p * q = a + b * q + π So π = p * q a b * q Therefore π = q * (p b) a, where for achieving π MAX we put q MAX For the enterprise C By the highest production and sale volume (1 500 units per month given by the production capacity of the enterprise C) the enterprise C achieves the highest month profit: π = * ( ) = per month Break-even point analysis and further analyses 32

33 Task: The following figure displays an example of non-linear shape of costs and revenues. Mark the production and sale volumes which respond with the break-even point and with profit maximization. Break-even point analysis and further analyses 33

34 Solution of the task: Note: These computations are done with the finding q where the curves TC and TR intersect (breakeven points) and maximum of profit function is. For these solutions it is possible to use computation as well as analysis of marginal variables. These computations will not be done during this course. Question: What is the reason of linearity and non-linearity of costs? Which reasons exist for non-linearity of revenues? (It has been already explained on the previous slides repeat answers.) Break-even point analysis and further analyses 34

35 Question: How graphically looks the displaying of different cases (from the point of reasons) when the enterprise is not even able to achieve break-even even point/profit? Break-even point analysis and further analyses 35

36 Solution of the question: Break-even point analysis and further analyses 36

37 Question: How are computations and analyses realized in the case of non-homogeneous production and what are the global cost functions? Answer: In the case of different production (several kinds of products) it is not possible to express the total production volume in natural values (for example in pieces, tons) but it is possible to use monetary values and global cost functions. Global cost function of linear shape can be displayed TC = a + c * Q Where Q is the volume of different production in as the value of sales for this production (Q = p 1 * q 1 + p 2 * q p n * q n ), for i = 1, 2,, n kinds of products, q i are quantities and p i are prices per unit of product i) a are FC c are VC attributable for 1 of sales value. The production and sale volume attributable to Q BEP in the case of heterogeneous production and the use of global cost function is computable following For Q BEP valid TR = TC, Therefore Q = a + c * Q, Q c * Q = a, Q * (1 c) = a, And therefore Q BEP = a / (1 c) Break-even point analysis and further analyses 37

38 Other important kind of computations and analyses in the area of costs is the comparison of cost options. Example 5-4 (choice of cost option): The enterprise can choose among 3 options of machinery. Option A has fixed costs per year and variable cost 5.72 per unit. Option B has fixed costs per year and variable cost 2.86 per unit. Option C has fixed costs per year and variable cost 1.43 per unit. The production capacity is units per year in option A, units per year in option B, units per year in option C. Tasks and question: (a) Sketch the figure (b) Compute for which production volumes (range) per year are options from the point of view of costs optimal (of course respecting production capacity) (c) Would the situation change if the production capacity in option A is only units? (d) We have answered the question which option the enterprise would choose according the volume of production. How the volume of production is chosen? Break-even point analysis and further analyses 38

39 Break-even point analysis and further analyses 39

40 Solution of Example 5-4 From the previous slide it is clear that the option A would be optimal (because of the lowest cost) for annual production volume in range 0 to q 1. The q 1 is the intersection point of curves TC A and TC B. The option B would be optimal (because of the lowest costs) for q between q 1 and q 2. The q 2 is the intersection point of curves TC B and TC C. The option C would be optimal (because of the lowest costs) for the production volume higher than q. q 1 can be computed from relationship TC A = TC B * q = * q q 1 = units per year q 2 can be computed from relationship TC B = TC C * q = * q q 2 = units per year Interpretation of results: The option A would be optimal (because of the lowest TC from all options) for the production volume between 0 and units per year. The option B would be optimal (because of the lowest TC from all options) for the production volume between and units per year. The option C would be optimal (because of the lowest TC from all options) for the production volume exceeding units per year. Break-even point analysis and further analyses 40