(10158) (4247) the cost function

8/14/2019 (10158) (4247) the Cost Function

1/28

The CostThe Cost

FunctionFunction


2/28

Cost FunctionCost Function

The econometrical model which isThe econometrical model which is

used to analyze costs is a model inused to analyze costs is a model in

which explanatory variablewhich explanatory variable

represents total costs andrepresents total costs and

endogenous variables representendogenous variables represent

factors that influence their level.factors that influence their level.

Production quantity is the mostProduction quantity is the mostimportant factor which determinesimportant factor which determines

the level of total costs.the level of total costs.


3/28

The Costs ModelThe Costs Model

The Costs model can be:The Costs model can be: Dynamic (in long or short intervals)Dynamic (in long or short intervals) StaticStatic

In case of short-term dynamic model it isIn case of short-term dynamic model it isassumed that technical conditions,assumed that technical conditions,organizational conditions, the structure oforganizational conditions, the structure of

products etc. are fixed. In long-termproducts etc. are fixed. In long-termdynamic models we will try to figure out thedynamic models we will try to figure out theinfluence of changes in technology and workinfluence of changes in technology and workorganization on the production level.organization on the production level.


4/28

Function FormFunction Form

LinearLinear

PowerPower

But in mostBut in most cases it is polynomial of 3cases it is polynomial of 3 rdrd

or lower degree.or lower degree.


5/28

Function FormFunction Form

total fixed coststotal fixed costs total variabletotal variable

costscosts

where:where:K total production cost,K total production cost,

Q production quantityQ production quantity

3

3

2

210 QQQK +++=3

3

2

210 QQQK +++=

3

3

2

210 QQQK +++=


6/28

Total CostsTotal Costs

Total costs consist of two parts:Total costs consist of two parts:

total fixed coststotal fixed costs, which appear, which appearindependently of the productionindependently of the production

quantity (when production level is zero)quantity (when production level is zero)

total variable coststotal variable costs, which are, which aredependent only on the productiondependent only on the production

quantityquantity


7/28

Total CostTotal Cost

ItIt is an expected (theoretical) valueis an expected (theoretical) value

of endogenous variable thatof endogenous variable that

corresponds with given productioncorresponds with given productionquantity (explanatory variable) whichquantity (explanatory variable) which

consequents the total costs modelconsequents the total costs model

equation.equation.


8/28

Total Fixed CostTotal Fixed Cost

ItIt is an expected (theoretical) valueis an expected (theoretical) value

of endogenous variable which doesof endogenous variable which does

not depend on production quantitynot depend on production quantityand consequents of the costs modeland consequents of the costs model

(it is a constant value)(it is a constant value)


9/28

Total Variable CostTotal Variable Cost

It isIt is an expected (theoretical) valuean expected (theoretical) value

of endogenous variable by the givenof endogenous variable by the given

production level (explanatoryproduction level (explanatoryvariable)variable)


10/28

Average (Unitary) CostAverage (Unitary) Cost

describes the average value of thedescribes the average value of the

cost for single production unitcost for single production unit

Q

KKj =


11/28

Marginal CostMarginal Cost

ItIt is the average raise of the totalis the average raise of the total

cost, caused by incrementation ofcost, caused by incrementation of

the production by one unit.the production by one unit.

When the cost function is aWhen the cost function is a

continuous function the marginalcontinuous function the marginal

cost is its derivativecost is its derivative::

Q

KKK

=

Q

KKK

=

Q

KKK

=

Q

KKK

=


12/28

Marginal Unit CostMarginal Unit Cost

ItIt is the average raise of the unitaryis the average raise of the unitary

cost, caused by incrementation ofcost, caused by incrementation of

the production by one unit.the production by one unit.

When the unitary cost function in aWhen the unitary cost function in a

continuous function the marginalcontinuous function the marginal

unitary is its derivative:unitary is its derivative:

Q

KKK

j

j

=


13/28

Total ProfitTotal Profit

ItIt is the profit from selling theis the profit from selling the

production with the unit price ofproduction with the unit price ofccjj

KcQZ j=


14/28

Marginal ProfitMarginal Profit

ItIt is the profit corresponding with oneis the profit corresponding with one

unit of the productionunit of the production

in other way it is the profit of sellingin other way it is the profit of selling

one unit of the productionone unit of the production

jjj KcQ

ZZ ==


15/28

Profitability IntervalProfitability Interval

ItIt is ais ann interval where the incomes areinterval where the incomes are

higher than total costs.higher than total costs.

At this production level the company gainsAt this production level the company gains

positive profit so it is the interval ofpositive profit so it is the interval ofrational activity.rational activity.

Profitability threshold can be found byProfitability threshold can be found by

solving the inequality below:solving the inequality below:

KD >

KcQ j >


16/28

Optimal ProductionOptimal Production

QuantityQuantity

ItIt is the production quantity whereis the production quantity wherethe marginal profit has the highestthe marginal profit has the highest

value or the marginal cost is thevalue or the marginal cost is the

lowest.lowest.


17/28

ExampleExample

There is a total cost function [inThere is a total cost function [in

thousands of z] in dependence ofthousands of z] in dependence of

the production quantity [inthe production quantity [in

thousands of units]:thousands of units]:

26743,96232,64438,144 QQK ++=

total fixed costs total variable

costs


18/28

Example cont.Example cont.

Assumptions:Assumptions: production: 5000 unitsproduction: 5000 units

selling price of one production unit:selling price of one production unit:

190z190z


19/28


Total CostTotal Cost

Total Fixed CostTotal Fixed Cost

144,438 thou.144,438 thou. zlzl

Total Variable CostTotal Variable Cost

411,70956743,956232,64438,1442=++=K

thou. zl

9735,56456743,956232,642=+ thou. zl


20/28


Interpretation:Interpretation:

By the production level of 5000 unitsBy the production level of 5000 unitsit can be expected that total costsit can be expected that total costs

will bewill be 709 411 z, which consists of709 411 z, which consists of

144 438 z of fixed costs and 564144 438 z of fixed costs and 564

973,5 z of variable costs.973,5 z of variable costs.


21/28


Average Cost (marginal)Average Cost (marginal)

88,14156743,96232,645

438,144

21

0

2

210

=++=

=++=++

== Q

QQ

QQ

Q

KK

j

z per unit

or 88,1415

411,709=== Q

KKj z per unit

The manufacturing cost of one thousand unitsequals 141,88 thou. zl (or the manufacturing costof one unit equals 141,88 zl).


22/28


Marginal CostMarginal Cost

3662,16156743,926232,642 21 =+=+= QKK

z per unit

Raising the production by 1000 units will cause the raise of totalcost by 161,3662 thou. z. (by raising the production by 1 unit totalcost will increase by 161,366 z per unit).


23/28


Marginal Unitary CostMarginal Unitary Cost

Total ProfitTotal Profit profit from selling the production with theprofit from selling the production with theunit price ofunit price ofcj.cj.

897,36743,9

25

438,14422

0=+=+=

=

QQ

KKK

j

j z per unit

Raising the production by 1000 units will cause the raise of unitarycost by 3,897 thou. z (by raising the production by 1 unit unitarycost will increase by 3,897 z per unit.

589,240411,7091905 === KcQZ j thou. z

By selling 5000 units for 190z each, profit will be equal 240,589 thou. z.


24/28


Marginal Profit -Marginal Profit - profit corresponding with oneprofit corresponding with oneunit of production, in other words, profit from selling oneunit of production, in other words, profit from selling one

unit of the production.unit of the production.

12,4888,141190 ==== jjj KcQ

ZZ z per unit

By selling 1000 units the profit achieved will be equal 48,12thou. z ( the profit from selling one unit will be 48,12 z).


25/28

Profitability intervalProfitability intervalKD >

KcQ j >26743,96232,64438,144190 QQQ ++>

06743,96232,64438,144190 2 > QQQ

0438,1443768,1256743,9 2 >+ QQ

1013034,558934,15719

)438,144)(6743,9(43768,125 2

==

==

6479,100=

6817,113486,19

0274,226

)6743,9(2

6479,1003768,1251 =

=

=Q

278,13486,19

7289,24

)6743,9(2

6479,1003768,1252 =

=

+=Q


26/28


Interpretation:Interpretation:

The interval described by inequalityThe interval described by inequality 1,278 < Q


27/28


Optimal Production AmountOptimal Production Amount ByBy the criteria of the minimum costthe criteria of the minimum cost

jKKK=

Q

Q

Q 6743,96232,64438,144

3486,196232,64 ++=+

QQ

QQ = /438,144

6743,93486,19

438,1446743,9 2 =Q

864,393,14

2==

QQ

38,13938,376232,6438,37

864,36743,96232,64864,3

438,144min

=++=

=++=jK

thou.

units

z per unitBy producing 3,864 thou. units unitary cost will bethe smallest and will be equal 139,38 thou. z perthou. unit (which is 139,38 z per unit)


28/28


Optimal Production AmountOptimal Production Amount by the criteria of the maximum profitby the criteria of the maximum profit

58,195442,1447,249438,14416,734

864,36743,9864,36232,64438,144190864,3

max

2

==

==

== KcQZ j

thou. z

By producing 3,864 thou. units it is possible toreach the maximum profit which equals 195,58thou. z (that is 195,58 z per unit)

(10158) (4247) the cost function

Documents