Business Economics 06 Theory of Production

Production functions

Firm, production, optimal input combination

Coverage• The concept of production function• Properties of the production functions• Conditions for achieving peak production

efficiency and for optimizing the mix of resource inputs in short and long run

• Technical and economic efficiency

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Production?

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Production

Any 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.

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Production process

• An 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 labor–saving 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 = labour

x2 = land

x3 = capital

x4 = organization

tools, 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.

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Factors of production

Factor definitions

• Land- all natural resources, including the sea and outer space. Fixed supply and a factor in its unimproved state

• Labour- number of people and physical and intellectual skills and efforts

• Capital- capable of generating incomes, holding stored value means it represents deferred consumption or use

• Organizer or entrepreneur- who perceives market opportunities in uncertainty. Involved in risk taking, invention

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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.

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The law of variable proportionsK L Q APL = Q/L MPL =

ΔQ/ΔL

10 0 0 - -

10 1 10 10 10

10 2 30 15 20

10 3 60 20 30

10 4 80 20 20

10 5 95 19 15

10 6 108 18 13

10 7 112 16 4

10 8 112 14 0

10 9 108 12 -4

10 10 100 10 -8 1104/08/23

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Example - law of diminishing returns

FERTILIZERLEVEL Kg/ha

EXPERIMENTAL FARM (Y1)

DEMONSTRATION FARM (Y2)

N P2O5(N+P2O5) Y1

Y1 Y2Y2

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

Yield of wheat under different fertilizer rates, 1989-90, q/ha

Annual Report, Agricultural Research Station, Arjia, Bhilwara, Rajasthan, 1990

N- Nitrogen, P2O5- Phosphorus pentoxide

Thomas Malthus (1766-1834)1798 -1826 six editions An Essay on the Principle of Population

The law of diminishing returns and the food crisisIndex of world food consumption (per capita)

Year Index

1948-1952 100

1955 109

1960 115

1965 116

1970 123

1978 128

1987 133

1991 142 1304/08/23

Determining optimal input proportions – long run analysis

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maximize output for a given cost

minimize cost for a given output

produce output that max. profit

Production function for long run - three options

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Production table

Rate of CapitalInput (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 curve

A 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 • nonintersecting• higher isoquant gives higher output• inputs 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= ΔK.MPK

Gain in Q due to increase in L= ΔL.MPL

Loss = Gain- ΔK.MPK = ΔL.MPL

ΔK/ ΔL = MPL/ MPK

Imperfect 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 inputs Least-cost input combination - Optimization

where isocost and isoquent are tangential MRTSLK = - PL/PK

MPL/MPK = PL/PK

Maximum-output input combination

Maximum-profit input combination - the

expansion path

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Profit maximization

maximize = 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

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Unconstrained max. problem for which two first - order partial derivatives be zero

d /dL = (0.1585) (12.98 + 0.196K – 0.208L) – 0.75 = 0

d /dK = (0.1585) (26.72 + 0.196L – 0.638K) – 0.50 = 0

Simplifying these we get

(0.1585) (12.98 + 0.196K – 0.208L) = 1

0.75

And (0.1585) (26.72 + 0.196L – 0.638K) = 1

0.50

Yields profit max. input combination

L =103 K = 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

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Ridge lines The 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 industries

Source: 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).

Sectors

Year AllManufacturing

Paper OrganicChemicals

PetroleumRefining

Steel Aluminum

1971 52.5 316.2 277.9 631.4 314.7 418.5

1977 42.3 308.7 193.9 573.4 282.7 379.9

Percent -19.4 -2.4 -30.2 -9.2 -10.2 -9.2Change

Prices rose 1971-80

Crude 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<a)L + K = QaL + aK = bQRTS is reflection on TFP

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Exercise - Returns to scale

L K Q

I 1 1 5

II 2 2 31

III 3 3 59

IV 4 4 72

V 6 6 107

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Factors responsible for returns to scale

Economies Diseconomies

Internal Internal

External External

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Factors for increasing returns to scale

• labor economies, division of labor, specialization• indivisibility of factors of production - technical economies• dimensional economies• economies of mass production – low cost, less spare parts• managerial economies• marketing economies• financial economies• economies of risk spreading• external economies04/08/23 30

Factors for constant returns to scale• producing in the most efficient plant size

Factors for decreasing returns to scale• diseconomies related to management, labor,

transport

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All input elasticity of output

Responsiveness of output to the change in all inputsEQ,I = %ΔQ/%Δ in all inputs

=DQ/Di.I/Q=ΔQ/ΔI.I/Q

EQ,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

inputs

Q = AKL

Q = 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 one decreasing

equal to one constant

greater than one increasing

Q = 10K0.5L0.6

+ = 0.5+0.6 = 1.1 > 1

increasing return to scale

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Case on C-D function

Agricultural 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 pounds

G = 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

Industry Country 1

2

3

1+

2+3

Gas France .83 .10 - 0.93Railroads United States .89 .12 .28 1.29Goal United Kingdom .79 .29 - 1.08Food United States .72 .35 - 1.07Metals and machinery United States .71 .26 - 0.97Communications Russia .80 .38 - 1.18Cotton India .92 .12 - 1.04Jute India .84 .14 - 0.98Sugar India .59 .33 - 0.92Coal India .71 .44 - 1.15Paper India .64 .45 - 1.09Chemicals India .80 .37 - 1.17Electricity India .20 .67 - 0.87Food2 United States .63 .44 - 1.07Paper2 United States .62 .37 - 098Telephone Canada 1972 .70 .41 - 1.11Chemicalsb United States .54 .38 .11 1.03Aircraftb United States .79 .18 .04 1.01

Q = AL1K 2M 3 L - laborK - capitalM - raw material

Economies of Scope

• Savings when two or more products are produced jointly

• S = 50,000+30,000 – 70,000 ------------------------------ = 0.14 70,000

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Factor productivity

Single 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/L

• 200/10=20, 240/11=21.8• Whether labour productivity has increased by 9%?• No consideration for capital used

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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 process

TFP 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)04/08/23 41

TFP = Q/r.K + w.LCase 1 Q=500 K1=8 L1=20 r1=4 w1=2

K2=11 L2=10 r2=2 w2=4

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94.6)20(2)8(4

500 1

16.13% increase in TFP in terms of output per rupee of inputs

2 = 500/ 2(11) + 4(10) = 8.06

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0 10 20 L

8

11

K

B

A

500

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

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50.2)40(4)20(2

500 1

78.2)43(4)22(2

600 2

11.2% increase in TFP

Relative contribution of factors in TFP in India 1960-80

Industry type TFP K (%) L(%) O(%)

All selected 1.1385 15.7 47.8 36.5

Basic goods 1.1070 5.5 63.7 30.8

Intermediate goods

1.1591 25.5 45.9 28.6

Consumer goods 1.1357 39.5 36.5 24.0

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Trivedi, 2002, Managerial Economics,Tata McGraw, p 288

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The impact of technological advance upon production function

Several forms of technical efficiency

a 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.

.

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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