BRAC Production Function

8/8/2019 BRAC Production Function

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Production Function

Chapter 6


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Production Function

Inputs Process Output

Land

Labor

Capital

Product orservice

generated

value added


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Basic Concepts of Production Theory

y Production functiony Maximum amount of output that can be produced

from any specified set of inputs, given existingtechnology

y Technical efficiencyyAchieved when maximum amount of output is

produced with a given combination of inputsy Economic efficiency

yAchieved when firm is producing a given output atthe lowest possible total cost


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y Inputsy variable or

y Fixed

y Variable inputy An input for which the level of usage may be changed quite

readily

y Fixed inputy level of usage cannot readily be changed

y

must be paid even if no output is producedy Quasi-fixed input

y An input employed in a fixed amount for any positive level ofoutput that need not be paid if output is zero


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y Short run

y At least one input is fixed

y All changes in output achieved by changing usage ofvariable inputs

y Long run

y All inputs are variable

y Output changed by varying usage of all inputs


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Short Run Productiony In the short run, capital is fixed

y Only changes in the variable labor input can change the

level of outputy Short run production function

Q f ( L,K ) f ( L )! !


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Production with one variable inputTPL = f(K , L )

Total product of labor (TPL) is defined as maximum rate of out putforthcoming from combining varying rates of labor input with a fixed capitalinput.Similarly,TPk = f(K,L )Marginal Product of LaborMPL = Q/ LMarginal Product of CapitalMPk = Q/ KUsing Cobb-Douglas:

Q =AKLMPL = dQ/dL = AKL-1MPk = dQ/dK= AK-1LAverage Product of LaborAPL =TPL/LAverage Product of Labor

APK =TPK/K


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Average & Marginal Products

yAverage product of labor

yAP = Q/L

yMarginal product of labor

y MP = (Q/(L

y WhenAPis rising,MPis greater thanAP

y WhenAPis falling,MPis less thanAP

yWhenAPreaches it maximum,AP = MP

y Law of diminishing marginal producty As usage of a variable input increases, a point is reached

beyond which its marginal product decreases


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Total,Average, & Marginal Products of Labor, K = 2

Number of

workers (L)

Total product (Q) Average product

(AP=Q/L)

Marginal product

(MP=(Q/(L)

0 0

1 52

2 112

3 170

4 220

5 258

6 2867 304

8 314

9 318

10 314

--

55

51.6

52

56

56.7

47.743.4

39.3

35.3

31.4

--

50

38

52

60

58

2818

10

4

-4


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T tal, Av rage argi al r cts,


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Total,Average & Marginal Product Curves

Panel A

Panel B

Total

product

Averageproduct

Marginal product

Q1

L1

L1

L2

Q2

L2

L0

Q0

L0


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Explaining TP,AP and MP Marginal product is the change in output associated with a one-unit

change in the variable input (i.e., MPL= Q/L) or the first derivativeof the production function with respect to the variable input (MPL=dQ/dL),

Average product is the rate of output produced per unit of the variableinput employed (APL= Q/L),

The law of diminishing marginal returns states that when increasingrates of a variable input are combined with a fixed rate of anotherinput, a point will be reached where marginal product will decline.

Marginal revenue product (MRP) is found by multiplying the

marginal product function by marginal revenue ( i.e., MRP = MR.MP), The marginal revenue product function for a productive factor is the

demand curve for that factor, Additional units of productive factor should be hired until the value ofthe marginal product of the input is equal to the prices of that input.


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Cobb Douglas Production Function

There are various types of production functions, one is Cobb-Douglasproduction function:

Q =AKLor, log Q = log A + log K + log L

Combination K L

A 6 1

B 3 2

C 2 3

D 1 6


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Three important relationships can be found

1. Substitutability between Factors: There are a variety of waysto produce a particular rate of output (example: to produce a fixedunits, any combination can be used). Therefore, the question oflabor or capital-intensive production arises.

2. Return to Scale: If input rates are doubled, the output ratealso doubles. [example: 200 = 1K + 4L, if 2K + 8L the Q would be =400]. The relationship between output change and proportionatechanges in both inputs is referred to Return to Scale.

3. Returns to Factor: When output changes because one inputchangeswhile the other remains constant, the changes in theoutput rates are referred to as Return to Factor. [example: 200 = 1K +4L 1K + 8L = 250


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Costsy Short run Diminishing marginal returns results

from adding successive quantities of variable factors to

a fixed factory Long run Increases in capacity can lead to

increasing, decreasing or constant returns to scale


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Short Run Production Costsy Total variable cost (TVC)

y Total amount paid for variable inputs

y Increases as output increasesy Total fixed cost (TFC)

y Total amount paid for fixed inputs

y Does not vary with output

y Total cost (TC)

y TC= TVC+ TFC


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Short-Run Total Cost Schedules

Output (Q) Total fixed cost

(TFC) taka

Total variable cost

(TVC) Taka

Total Cost Taka

TC=TFC+TVC)

0 6,000

100 6,000

200 6,000

300 6,000

400 6,000

500 6,000600 6,000

0

14,000

22,000

4,000

6,000

9,000

34,000

6,000

20,000

28,000

10,000

12,000

15,000

40,000


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Total Cost Curves


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Average Costs

!

TVCAVC

Q

!TFC

AFC

Q

! ! TC

ATC AVC AFC Q

( AFC )Average fixed cost

( ATC )Average total cost

( AVC )Average variable cost


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Short Run Marginal Costy Short run marginal cost (SMC) measures rate of

change in total cost (TC) as output varies

( (

( (

TC TVCSMC

Q Q


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Average & Marginal Cost Schedules

Output

(Q)

Average

fixed cost

(AFC=TFC/Q)

Average

variable cost

(AVC=TVC/Q)

Average total

cost

(ATC=TC/Q=

AFC+AVC)

Short-run

marginal cost

(SMC=(TC/(Q)

0

100

200

300

400

500

600

--

15

12

60

30

20

10

--

35

44

40

30

30

56.7

--

50

56

100

60

50

66.7

--

50

80

40

20

30

120


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Average & Marginal Cost Curves


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Short Average argi al ost rves


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Short Run Cost Curve Relations

yAFCdecreases continuously as output increases

y Equal to vertical distance betweenATC&AVC

yAVCis U-shaped

y EqualsSMCatAVCs minimum

yATCis U-shaped

y EqualsSMCatATCsminimum


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Short Run Cost Curve Relations

y SMCis U-shaped

y IntersectsAVC&ATCat their minimum points

y Lies belowAVC&ATCwhenAVC&ATCarefalling

y Lies aboveAVC&ATCwhenAVC&ATCarerising


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y In the case of a single variable input, short-run costsare related to the production function by tworelations

RelationsBetween Short-Run Costs &

Production

! !

w w

AVC SMC MP MP

and

wWhere is the price of the variable input

A


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Short- roduction ost elations


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RelationsBetween Short-Run Costs &

Production

yWhen marginal product (average product)is increasing, marginal cost (average cost) isdecreasing

yWhen marginal product (average product)is decreasing, marginal cost (average

variable cost) is increasing

yWhen marginal product = average product

at maximumAP, marginal cost = averagevariable cost at minimumAVC


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OPTIMAL EMPLOYMENT OF A

FACTOR OF PRODUCTIONThe marginal revenue product (MRP) of the last unt employed isequal to the cost of input


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Marginal Revenue Product is the

labor demand function for the firm


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Revenuey Total revenue the total amount receivedfrom selling a given output

y TR= P x Qy

Average Revenue the average amountreceived from selling each unityAR= TR/ Q

y Marginal revenue the amount received fromselling one extra unitof output

y MR= TRn TRn-1 units


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Profity Profit = TR TC

y The reward for enterprise

y Profits help in the process of directing resources toalternative uses in free markets

y Relating price to costs helps a firm to assessprofitability in production


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Profity Normal Profit the minimum amount

required to keep a firm in its current line ofproduction

yAbnormal or Supernormal profit profitmade over and above normal profity Abnormal profit may exist in situations where firms

have market powery Abnormal profits may indicate the existence of

welfare lossesy Could be taxed away without altering resource

allocation


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Profity Sub-normal Profit profit below normal profit

y Firms may not exit the market even if sub-normal profits

made if they are able to cover variable costsy Cost of exit may be high

y Sub-normal profit may be temporary (or perceived assuch!)


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ProfityAssumption that firms aim to maximise profit

y May not always hold true

there are other objectivesy Profit maximising output would be where MC = MR


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Profit Why?Cost/Revenue

Output

MR

MR the additionto total revenue asa result ofproducing onemore unit ofoutput the pricereceived fromselling that extraunit.

MC MC The cost ofproducing ONEextra unit ofproduction

100

Assume output is at100 units. The MC ofproducing the 100th

unit is 20.

The MR received fromselling that 100th unitis 150. The firm canadd the difference ofthe cost and therevenue received fromthat 100th unit toprofit (130)

20

150

Totaladded

toprofit

If the firm decides toproduce one more unit the 101st the additionto total cost is now 18,

the addition to totalrevenue is 140 the firmwill add 128 to profit. it is worth expandingoutput.

101

18

140

Added tototalprofit

30

120

Addedto totalprofit

The process continuesfor each successiveunit produced.Provided the MC isless than the MR it

will be worthexpanding output asthe differencebetween the two isADDED to total profit

102

40

145

104103

Reducestotalprofit bythisamount

If the firm were toproduce the 104th unit,this last unit would costmore to produce than itearns in revenue (-105)

this would reduce totalprofit and so would notbe worth producing.

The profit maximisingoutput is where MR =MC


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ExampleyA micro-entrepreneur produces caps and hats for

women. The output-cost data of the business isreproduced below:

Output TotalCost

50 870

100 920

150 990

200 1240

250 1440

300 1940

350 2330

a. Estimate the total cost function and then usethat equation to determine the average andmarginal cost functions. Assume a costfunction.

b. Determine the output rate that will minimize

average cost and the per-unit cost at that rate ofoutput.

c. The current market price of caps and hats perunit is Tk. 6.00 and is expected to remain atthat level for the foreseeable future. Should thefirm continue its production?


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Gettingan Ideaabout the form of

the equation

0

500

1000

1500

2000

2500

50 100 150 200 250 300 350

Output-Cost


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Estimate ofExampley First we assume the cost function asTC = c0+c1Q + c2Q

2 +c3Q3

y ResultsTC= 954.29 -2.46Q +0.02Q2 -.0002Q3

(5.9) (-0.75) (1.04) (-0.07)R2 = 0.99 F = 197.78

y Comments: t-statistics are not acceptable though R2 and F are good.y Second, we assume the cost function asTC = c0+c1Q + c2Q

2

Resultsy TC = 944.29 -2.24Q + 0.02Q2

t Stat (12.51) (-2.58) (8.45)R2 = 0.99 F = 394.86

y Comments: t-statistics are acceptable and R2 and F are good.


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Answer to Question (a)y a. The t-statistics, shown in the parenthesis of

the second estimation, indicate that the

coefficient of each of the independent variablesare significantly different from zero. The valueof the co-efficient of determination means that99 percent of the variation in total cost is

explained by changes in the rate of output.


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Answer (a) contd.


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Answer (b)y The output rate that results in minimum per-unit cost

is found by taking the first derivative of the averagecost function, setting it equal to zero, and solving forQ.


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Answer (b) contd.


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Answer (c)y Because the lowest possible cost is Tk. 6.45 per

unit, which is above the market price of Tk. 6.00, theproduction should not be continued.


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AssignmentOutput Total Cost

25 700

100 920150 990

200 1240

280 1440360 1940

460 2330

600 3500

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BRAC Production Function

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