PRODUCTION FUNCTION(Manufacture) “ Production is the process of combining the factors of production in

the suitable manner for the creation of utility of a product or service” Production function is a function that mathematically quantify the

output of a firm, an industry, or an entire economy for all combinations of inputs. This function is an assumed technological relationship, based on the current state of technical knowledge knowledge;.

Factors of productionThe factors of production are all physical non-physical factors

involved in the process of production. The physical factors are land, building, Plant, machinery, tools,

materials and the power inputs. Depending on the nature of operations these factors considerably change. The behavior of these factors varies according to the nature inherent and attributed values

The non-physical factors are essential input factors in the production process which is non-physical nature. The copy and patents, goodwill of various forms, visual and none—visual aids, financial services and consultancy services etc form non-physical inputs. In modern business these are very important factors for production

A production function can be expressed in a functional form as

Q = f(X1,X2,X3,...,Xn)where: Q = quantity of outputX1,X2,X3,...,Xn = quantities of factor inputs

(such as capital, labour, land or raw materials).

Production function

FACTORS IF PRODUCTIONPhysical Inputs1.Labour2.capital3.Materials4.Plant,Machinery etc5.lLand & BuildingNon-Physical1.Goodwill2.Inellectual Property(,Patents, copy right etc)3.Market & marketing factors(design, packing etc)

Inter-relationships analysisIf Q is not a direct (i.e. a scalar, a vector, or even a diagonal matrix), then this form does not encompass joint production, which is a production process that has multiple co-products. On the other hand, if f maps from Rn to Rk then it is a joint production function expressing the determination of k different types of output based on the joint usage of the specified quantities of the n inputs.One formulation, unlikely to be relevant in practice, is as a linear function:Q = a + bX1 + cX2 + dX3 + ...where a,b,c, and d are parameters that are determined from past standards

Cobb-Douglas Production Function

Cobb-Douglas production function which can be estimated in three factor combination is:

q = A * (L^alpha) * (K^beta) * (M^gamma) = f(L,K,M).

where L = labour, K = capital, M = materials and supplies, and q = product. The symbol "^" means "raise to the power," i.e. L^alpha means "raise the value of L to the power of the value of alpha."Production functions need to have certain properties, to ensure to solve the least-cost problem:. If for given values of L,K, and M, the Hessian of the production function f is negative definite, then its isoquants at that point are concave to the origin.

f Labour Material Capital Machinery Q0.2 80 30 10 20 96000

Illustration.Ajanta cement Company is employ 800 workers. The monthly consumption

of rawmaterials include lime stone 50 tons.The capital invested is Rs 10 crores on average rate. The machine capacity used is 2000 hours.

The production function is calculated in terms of the number of bags.The functional index of constant f=.2. The labour index L is taken as 10% of labour number. The material is as 0.6 of weight in tons. The capital index is 1one for every Rs in crores The machinery capacity taken as 1%

Calculate the quantity of cement producedAnswerQ = f(X1,X2,X3,...,Xn)where: Q = quantity of outputX1,X2,X3,...,Xn = quantities of factor inputs (such as capital, labour, land or raw

materials).Q=0.2 (800 X10%) (50X.6) (10) (2000x.1)Q=0.2 x80 x30 x10 x20 =96000 bagsThe production is 96000 number bags of cement

ILLUSTRATIONSangeevini Hospital, Kalyan is one of the pioneers in field of general Medicine

in the local area. The capital investment of the hospital is Rs 50 crores. There are 200 persons working including 4 specialists, 4 general

practiseioniers, 8 Junior Doctor and other medical paramedical, accounting, housekeeping and security staff. The average pay of the personal is Rs 8000/ per month

The rolling stock of medicine and other items on an average is Rs 150 crores

The hospital works on medical/treatment hour’s basis for the purpose of operation planning .The credit values are 0.6 for labour on the number of persons, 0.2 on capital invested and 0.2 for materials on the investments/expenditure in crores . The constants used for the hospital in general is 8

The cost of hospital are calculated on the basis of actual payment to all persons, 24% per year on capital invested and 12 %per year on average materials stocks

The rate chargeable/applicable to hospital is Rs 100 per treatment/medical hours

You are required to find out the medical/treatment hours of the hospital and cost operation. Find out the operative profit per month of the hospital?

Calculate profit per operating treatment hour?How much profit the hospital can make at this level of operation?

Factors of production

Const Labour credit Capital Credit Materials Credit Q medical hours

Q 8 200 0.6 120 50 0.2 10 150 0.2 30 288000

AnswerCalculation of medical/treatment hoursA production function can be expressed in a functional form as Q = f(1X,2X,3X,...,nX)where: Q = quantity of output, f is the constant,X1,X2,X3,...,Xn = quantities of factor

inputs (such as capital, labour, land or raw material, 1,2,3 etc are production index of each factors

Function of labolurThere are 200 persons working including 4 specialists, 4 general practiseioniers, 8 Junior

Doctor and other medical paramedical, accounting, housekeeping and security staff. The credit index is .6

The function of Labour=200 X 0.6 = 120 Function of CapitalThe capital investment of the hospital is Rs 50 crores. The production function index is 0.2The production index ol capital =50 X 0 .2 =10Function ofMedicenesThe rolling stock of medicine and other items on an average is Rs 150 crores. The production

credit is0.2 on investment in croresProduction function of Madiceines-= 150 X0.2 =30Total production in treatment hoursQ=8 ( 120 X10 X30) =288000The production function is 288000 medical treatment hours per month in the hospital

cost of productionLabour pay Total Capital Interest Materials Chrge Total Total Cost

200 8000 1600000 500000000 2% 10000000 150000000 10% 15000000 26600000

Calculation of cost of operation

Cost of labolur

There are 200 persons working including 4 specialists, 4 general practiseioniers, 8 Junior Doctor and other medical paramedical, accounting, housekeeping and security staff. The average pay of the personal is Rs 8000/ per month.

Cost of labour per month is Rs8000 X 200nos =1600000/ Cost ofCapital

The capital investment of the hospital is Rs 50 crores.The interest rate is 24% per year and 2% per month.

The cost of capital is Rs I crore per month (Rs50 crores X .2)

Cost ofMedicenes

The rolling stock of medicine and other items on an average is Rs 150 crores. The cost is taken10% per month on average stock.The cost is Rs15 crores per month

The total cost isRs26600000/ for a month

Calculation of operative profit

The Income chargeable for 288000 medical hours is at the rate of Rs per Hour Rs 100/ for hospitalIncome =288000 hours X Rs100 = Rs28800000 for the month

Cost is Rs26600000 for the monthProfit = 28800000-26600000 = 2200000/for the month

Rs22 lakhs is the profit for the month

It is Rs6.6 per hours per medical hours

Rs 63400 profit for an operating day

Stages of productionProduction function, it is commonly to divide range into 3 stages. In Stage 1 (from the origin the variable input is being used with increasing output

per unit, the latter reaching a maximum (since the average physical product is at its maximum at that point). Because the output per unit of the variable input is improving throughout stage 1, a price-taking firm will always operate beyond this stage.In Stage 2, output increases at a decreasing rate, and the average and marginal

physical product are declining. However, the average product of fixed inputs (not shown) is still rising, because output is rising while fixed input usage is constant. In this stage, the employment of additional variable inputs increases the output per unit of fixed input but decreases the output per unit of the variable input. The optimum input/output combination for the price-taking firm will be in stage 2, although a firm facing a downward-sloped demand curve might find it most profitable to operate in Stage 1. In Stage 3, too much variable input is being used relative to the available fixed

inputs: variable inputs are over-utilized in the sense that their presence on the margin obstructs the production process rather than enhancing it. The output per unit of both the fixed and the variable input declines throughout this stage. At the boundary between stage 2 and stage 3, the highest possible output is being obtained from the fixed input.

http://en.wikipedia.org/wiki/File:Stages_of_production_small.png

Cobb-Douglas Production Function DataIncreasing Returns to Scale

q L K M cost ave.cost marg. cost5 5.57 3.43 5.57 117.01 23.4 22.2910 10.78 6.64 10.78 226.43 22.64 21.5615 15.86 9.76 15.86 333.14 22.21 21.1520 20.86 12.84 20.86 438.15 21.91 20.8625 25.8 15.88 25.8 541.9 21.68 20.6430 30.7 18.89 30.7 644.65 21.49 20.4735 35.55 21.88 35.55 746.6 21.33 20.3240 40.37 24.85 40.37 847.84 21.2 20.1945 45.17 27.79 45.17 948.49 21.08 20.0750 49.93 30.73 49.93 1048.6 20.97 19.97

Production with increasing return

(The cost come down and return increases)

Cobb-Douglas Production Function DataDecreasing Returns to Scale

q L K M cost ave.cost marg. cost5 7.16 5.07 4.9 145.46 29.09 31.6210 15.21 10.77 10.41 308.99 30.9 33.5915 23.63 16.74 16.18 480.12 32.01 34.7920 32.31 22.89 22.12 656.38 32.82 35.6725 41.18 29.17 28.19 836.55 33.46 36.3730 50.2 35.56 34.37 1019.91 34 36.9535 59.36 42.05 40.64 1205.95 34.46 37.4540 68.63 48.61 46.98 1394.32 34.86 37.8945 78.01 55.25 53.4 1584.76 35.22 38.2850 87.47 61.96 59.88 1777.05 35.54 38.63

Production with decreasing return

(The cost go up and return decrease)

Shifting a production functionBy definition, in the long run the firm can change its scale of operations by adjusting the level of inputs that are fixed in the short run, thereby shifting the production function upward as plotted against the variable input. If fixed inputs are lumpy, adjustments to the scale of operations may be more significant than what is required to merely balance production capacity with demand. For example, you may only need to increase production by a million units per year to keep up with demand, but the production equipment upgrades that are available may involve increasing productive capacity by 2 million units per year.

Shifting a Production FunctionIf a firm is operating at a profit-maximizing level in stage one, it might, in the long run, choose to reduce its scale of operations (by selling capital equipment). By reducing the amount of fixed capital inputs, the production function will shift down. The beginning of stage 2 shifts from B1 to B2.

Levels of production Function. Almost all economic theories presuppose a production function, either

on the firm level, industry level, national level and international level. At each level the production function differ greatly. In this sense, the production function is one of the key concepts of operation Management.

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