The Cost Function

1. The Cost Function

2. Cost Function • The econometrical model which is used to analyze costs is a model in which explanatory variable represents total costs and endogenous variables represent factors that influence their level. Production quantity is the most important factor which determines the level of total costs.

3. The Costs Model The Costs model can be: • Dynamic (in long or short intervals) • Static In case of short-term dynamic model it is assumed that technical conditions, organizational conditions, the structure of products etc. are fixed. In long-term dynamic models we will try to figure out the influence of changes in technology and work organization on the production level.

4. Function Form • Linear • Power But in most cases it is polynomial of 3rd or lower degree.

5. Function Form total fixed costs total variable costs where: K – total production cost, Q – production quantity

6. Total Costs Total costs consist of two parts: • total fixed costs, which appear independently of the production quantity (when production level is zero) • total variable costs, which are dependent only on the production quantity

7. Total Cost • It is an expected (theoretical) value of endogenous variable that corresponds with given production quantity (explanatory variable) which consequents the total costs model equation.

8. Total Fixed Cost • It is an expected (theoretical) value of endogenous variable which does not depend on production quantity and consequents of the costs model (it is a constant value)

9. Total Variable Cost • It is an expected (theoretical) value of endogenous variable by the given production level (explanatory variable)

10. Average (Unitary) Cost • describes the average value of the cost for single production unit

11. Marginal Cost • It is the average raise of the total cost, caused by incrementation of the production by one unit. • When the cost function is a continuous function the marginal cost is its derivative:

12. Marginal Unit Cost • It is the average raise of the unitary cost, caused by incrementation of the production by one unit. • When the unitary cost function in a continuous function the marginal unitary is its derivative:

13. Total Profit • It is the profit from selling the production with the unit price of cj

14. Marginal Profit • It is the profit corresponding with one unit of the production • in other way it is the profit of selling one unit of the production

15. Profitability Interval • It is an interval where the incomes are higher than total costs. • At this production level the company gains positive profit so it is the interval of rational activity. • Profitability threshold can be found by solving the inequality below:

16. Optimal Production Quantity • It is the production quantity where the marginal profit has the highest value or the marginal cost is the lowest.

17. Example • There is a total cost function [in thousands of zł] in dependence of the production quantity [in thousands of units]: total fixed costs total variable costs

18. Example cont. Assumptions: • production: 5000 units • selling price of one production unit: 190zł

19. Example cont. • Total Cost • Total Fixed Cost 144,438 thou. zl • Total Variable Cost thou. zl thou. zl

20. Example cont. • Interpretation: • By the production level of 5000 units it can be expected that total costs will be 709 411 zł, which consists of 144 438 zł of fixed costs and 564 973,5 zł of variable costs.

21. Example cont. • Average Cost (unitary) zł per unit or zł per unit The manufacturing cost of one thousand units equals 141,88 thou. zl (or the manufacturing cost of one unit equals 141,88 zl).

22. Example cont. • Marginal Cost zł per unit Raising the production by 1000 units will cause the raise of total cost by 161,3662 thou. zł. (by raising the production by 1 unit total cost will increase by 161,366 zł per unit).

23. Example cont. • Marginal Unitary Cost • Total Profit – profit from selling the production with the unit price of cj. zł per unit Raising the production by 1000 units will cause the raise of unitary cost by 3,897 thou. zł (by raising the production by 1 unit unitary cost will increase by 3,897 zł per unit. thou. zł By selling 5000 units for 190zł each, profit will be equal 240,589 thou. zł.

24. Example cont. • Marginal Profit - profit corresponding with one unit of production, in other words, profit from selling one unit of the production. zł per unit By selling 1000 units the profit achieved will be equal 48,12 thou. zł ( the profit from selling one unit will be 48,12 zł).

25. Profitability interval

26. Example cont. • Interpretation: The interval described by inequality 1,278 < Q < 11,6817 is theprofitability interval that we were looking for. The boundaries of this interval are the thresholds of profitability. It means that production is profitable when it is higher than 1,278 and lower than 11,6817 thou. units.

27. Example cont. • Optimal Production Amount • By the criteria of the minimum cost thou. units zł per unit By producing 3,864 thou. units unitary cost will be the smallest and will be equal 139,38 thou. zł per thou. unit (which is 139,38 zł per unit)

28. Example cont. • Optimal Production Amount • by the criteria of the maximum profit thou. zł By producing 3,864 thou. units it is possible to reach the maximum profit which equals 195,58 thou. zł (that is 195,58 zł per unit)