Production functions

Firm, production, optimal input combinationCoverageThe concept of production functionProperties of the production functionsConditions for achieving peak production efficiency and for optimizing the mix of resource inputs in short and long runTechnical and economic efficiency**

Production?**

ProductionAny activity which creates value is production.e.g. transporting sand, collecting tax, operating a jeweler store, drilling for oil, recruiting new employees, driving a garbage truck, designing a system to measure air pollution.**

Production processAn integrated system of activities by which inputs are transformed into the production of goods and services over some period of time.Production processes composed of sequentially organized phases e.g.;Adding laborsaving equipment > material specifications and modifying skill levels of labor > procurement pattern and hiring and training programmes

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**Production FunctionQ= f(x1, x2, x3, x4, . . . . . . . Xn)Where x1 = labourx2 = landx3 = capitalx4 = organizationtools, machines, infrastructure, transport, electricity, fuel, time, advertisement, supervision, planning, control and coordination, leadership, managerial talent, R&D, government policies - licenses, tax, rules, and regulations etc.

**Factors of production

Factor definitionsLand- all natural resources, including the sea and outer space. Fixed supply and a factor in its unimproved stateLabour- number of people and physical and intellectual skills and effortsCapital- capable of generating incomes, holding stored value means it represents deferred consumption or useOrganizer or entrepreneur- who perceives market opportunities in uncertainty. Involved in risk taking, invention **

Inputs fixed and variableTime periods- market, short and longRational producerProductivity- evaluates the effectiveness of production processMeasurements of productivity total production (TP), average production (AP), marginal production (MP)

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Short run production function The law of diminishing returns or The law of variable proportionsQ = f (L, K)Where L = labourK = capitalAs the use of one input increases keeping other inputs fixed, a point will eventually be reached at which the resulting additions to output decrease.**

The law of variable proportions**

KLQAPL = Q/LMPL = Q/L1000--101101010102301520103602030104802020105951915106108181310711216410811214010910812-4101010010-8

**Example - law of diminishing returnsYield of wheat under different fertilizer rates, 1989-90, q/haAnnual Report, Agricultural Research Station, Arjia, Bhilwara, Rajasthan, 1990N- Nitrogen, P2O5- Phosphorus pentoxide

FERTILIZER LEVEL Kg/ha

EXPERIMENTAL FARM (Y1)

DEMONSTRATION FARM (Y2)

N

P2O5

((N+P2O5)

Y1

(Y1

Y2

(Y2

F1

15

7.5

22.5

19.76

19.76

29.85

29.85

F2

30

15.0

22.5

32.30

12.54

38.99

9.14

F3

45

22.5

22.5

35.82

3.52

37.16

-1.83

F4

60

30.0

22.5

36.10

0.28

29.50

-7.66

Thomas Malthus (1766-1834)1798 -1826 six editions An Essay on the Principle of PopulationThe law of diminishing returns and the food crisisIndex of world food consumption (per capita)**

YearIndex1948-19521001955109196011519651161970123197812819871331991142

Determining optimal input proportions long run analysis

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**

maximize output for a given costminimize cost for a given outputproduce output that max. profit

Production function for long run - three options

**Production table

Rate of Capital Input (K)

8

283

400

490

565

632

693

748

800

7

265

374

458

529

592

648

700

748

6

245

346

424

490

548

600

648

693

5

224

316

387

447

500

548

592

632

4

200

283

346

400

447

490

529

565

3

173

245

300

346

387

424

458

490

2

141

200

245

283

316

346

374

400

1

100

141

173

200

224

245

265

283

1

2

3

4

5

6

7

8

Rate of Labor Input (L)

Isoquants or Isoproduct curveA line showing all the alternative combinations of two factors that can produce a given level of output.

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Characteristics of an Isoquant

all rational combinations of inputs lie on negatively sloped and convex to the origin portion nonintersectinghigher isoquant gives higher outputinputs are imperfectly substitutable

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MRTSThe rate at which firm is able to substitute labor for capitalLoss in Q due to decline in K= K.MPKGain in Q due to increase in L= L.MPLLoss = Gain- K.MPK = L.MPLK/ L = MPL/ MPKImperfect substitution, perfect substitution and perfectly complimentary

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Isocost curve or cost constraint

TC = PL.L + PK.KSlope of isocost = TC/PK/TC/PL= - PL/PK the rate at which firm is economically able to substitute labor for capital

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**The optimum mix of resource inputsLeast-cost input combination - Optimization where isocost and isoquent are tangentialMRTSLK = - PL/PKMPL/MPK = PL/PKMaximum-output input combinationMaximum-profit input combination - the expansion path

**Profit maximizationmaximize = TR - TC = PQ PL.L - PK.Kwhere = profit P = product priceQ = 549.92 +12.98L + 26.72K + 0.196 LK - 0.104L2 - 0.319K2

maximize = (0.1585) (549.92 + 12.98L + 26.72K + 0.196LK- 0.104L2- 0.319K2) - 0.75L - 0.50K L,K 0

**Unconstrained max. problem for which two first - order partial derivatives be zerod /dL = (0.1585) (12.98 + 0.196K 0.208L) 0.75 = 0d /dK = (0.1585) (26.72 + 0.196L 0.638K) 0.50 = 0Simplifying these we get(0.1585) (12.98 + 0.196K 0.208L) = 10.75And (0.1585) (26.72 + 0.196L 0.638K) = 10.50Yields profit max. input combinationL =103K = 68

Q = 549.92 + 12.98 (103) + 26.72 (68) + 0.196 (103)(68) 0.104 (103)2 0.319 (68)2

= 2498 profit max. output

corresponding max. profit = (0.1585)(2498) 0.75(103) 0.50(68)= 395.9 111.2= 284.7**

Ridge linesThe impact of change in resource price

Cost effect = substitution effect + output effect

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**Energy consumption (in thousand Btu) per dollar of value added in selected industriesSource: U.S. Department Of Commerce, Bureau Of The Census, Statistical Abstract Of The United States: 1981 (Washington, D.C.:U.S. Government Printing Office, 1981).SectorsYear AllManufacturingPaper OrganicChemicalsPetroleumRefiningSteel Aluminum 1971 52.5 316.2 277.9 631.4 314.7 418.51977 42.3 308.7 193.9 573.4 282.7 379.9Percent -19.4 -2.4 -30.2 -9.2 -10.2 -9.2ChangePrices rose 1971-80Crude oil 240%, natural gas 347%, coal 113%

Returns to scaleRefers to the character of changes in output when all resource inputs are changed in equal proportions.increasing returns to scale-short range(b>a)constant returns to scale- lengthy range(b=a)decreasing returns to scale- (b

Exercise - Returns to scale**

LKQI115II2231III3359IV4472V66107

Factors responsible for returns to scaleEconomiesDiseconomies

InternalInternal

External External

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Factors for increasing returns to scalelabor economies, division of labor, specializationindivisibility of factors of production - technical economiesdimensional economieseconomies of mass production low cost, less spare partsmanagerial economiesmarketing economiesfinancial economieseconomies of risk spreadingexternal economies

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Factors for constant returns to scaleproducing in the most efficient plant size

Factors for decreasing returns to scalediseconomies related to management, labor, transport

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All input elasticity of outputResponsiveness of output to the change in all inputsEQ,I = %Q/% in all inputs=DQ/Di.I/Q=Q/I.I/QEQ,I > 1 Increasing RTSEQ,I = 1Constant RTSEQ,I < 1Decreasing RTS

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**Introduced to relate output in manufacturing industries from 1899-1922 to labor and capital inputsQ = AKLQ = A(2K)(2L) Q = 2 2(AKL)but Q= AKL. Hence the factor = 2+ and will be less than 2, equal to 2, or greater than 2, depending on + Cobb-Douglas production function

sum of exponents returns (+ ) to scale

less than onedecreasing

equal to oneconstant

greater than oneincreasing

Q = 10K0.5L0.6

+ = 0.5+0.6 = 1.1 > 1

increasing return to scale

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**Case on C-D functionAgricultural economist E O Heady conducted an experiment on 302 pigs weighing between 34-250 pound

G = 1.60P.30C.53 for the weight interval 34-75 poundsG = 0.71P.14C.77 for 75-150 pounds

G = 0.46P.09C.86 for 150-250 pounds

where G is weight gain, P is input of soybean oil meal (protein), and C is input of corn (carbohydrate) all measured in pounds per pig.

Source: E O Heady, An econometric investigation of the technology of agricultural production functions.

Econometrica, V25 (April, 1957)

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**Estimates of 1, 2, 3 for selected industries 1963-80 Q = AL1K 2M 3 L - laborK - capitalM - raw material

Industry

Country

(1

(2

(3

(1+(2+(3

Gas

France

.83

.10

-

0.93

Railroads

United States

.89

.12

.28

1.29

Goal

United Kingdom

.79

.29

-

1.08

Food

United States

.72

.35

-

1.07

Metals and machinery

United States

.71

.26

-

0.97

Communications

Russia

.80

.38

-

1.18

Cotton

India

.92

.12

-

1.04

Jute

India

.84

.14

-

0.98

Sugar

India

.59

.33

-

0.92

Coal

India

.71

.44

-

1.15

Paper

India

.64

.45

-

1.09

Chemicals

India

.80

.37

-

1.17

Electricity

India

.20

.67

-

0.87

Food2

United States

.63

.44

-

1.07

Paper2

United States

.62

.37

-

098

Telephone

Canada 1972

.70

.41

-

1.11

Chemicalsb

United States

.54

.38

.11

1.03

Aircraftb

United States

.79

.18

.04

1.01

Economies of ScopeSavings when two or more products are produced jointly

S = 50,000+30,000 70,000 ------------------------------ = 0.14 70,000**

**Factor productivitySingle factor productivity (SFP) - ratio of volume of output to the quantity of the factor of production for which productivity is to be estimated.

APL = Q/L200/10=20, 240/11=21.8Whether labour productivity has increased by 9%?No consideration for capital used**

Multi factor (or total factor) productivity (MFP or TFP)Ratio of volume of output to a weighted sum of the inputs used in the production processTFP tries to circumvent the problem encountered in interpretation of SFP estimates due to changing factor intensities Broadest measure of productivity and efficiency in resource use Decomposes changes in Q due to changes in quantity of inputs used and changes in all the residual factors Also called as index of ignorance (ABROMOVITZ, 1986)

**

TFP = Q/r.K + w.LCase 1 Q=500K1=8 L1=20 r1=4 w1=2 K2=11 L2=10 r2=2 w2=4

**16.13% increase in TFP in terms of output per rupee of inputs 2 = 500/ 2(11) + 4(10) = 8.06

**

Case 2 Q1=500K1=20 L1=40r1=2 W1=4 Q2=600K2=22 L2=43

**11.2% increase in TFP

Relative contribution of factors in TFP in India 1960-80** Trivedi, 2002, Managerial Economics,Tata McGraw, p 288

Industry typeTFPK (%)L(%)O(%)All selected1.138515.747.836.5Basic goods1.10705.563.730.8Intermediate goods1.159125.545.928.6Consumer goods1.135739.536.524.0

**The impact of technological advance upon production functionSeveral forms of technical efficiencya new production process permits same amount of resources combined differently to yield more.a new process uses same type of inputs but less of one or several inputs and no more of others to produce same type of output.a new process may require inputs or yield output, that are of a kind not used until now or available at all.

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Economic efficiency

Any change that harms no one and improves the lot of some people, if all such changes are carried out and thus no opportunity to make such changes remains - situation is termed as economic efficient**

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