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Production function [ management ]


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  • 1. Chapter 3: PRODUCTION FUNCTION Chapter 3: Contents:  Market Structures  Price – Output Decision in case of Perfect and Monopoly  Production Function  Isoquants and Isocosts  MRTS Least Cost Combination of Inputs  Law Of Return , Economies of Scale 1. INTRODUCTION TO PRODUCTION FUNCTION : A production function shows the relationship between inputs of capital and labour and other factors and the outputs of goods and services.Production of goods requires resources or inputs. These inputs are called factors of production named as land, labour, capital and organization. What Is Production Function in Economics with one or two variables input? A rational producer is always interested that he should get the maximum output from the set of resources or inputs available to him. He would like to combine these inputs in a technical efficient manner so that he obtains maximum desired output of goods. The production function represents that how much output can be produced by using the various combinations of the various factors of productions. The theory of production revolves around the production function. “A production function can be an equation, table or graph presenting the maximum amount of a commodity that a firm can produce from a given set of inputs during a period of time.” The production function considers the combination of various inputs in the company related to the production factor considering the four factors of production i.e. land, labour, capital and entrepreneur. The production function considers the various forms of technology to be used for the production area which helps to reduce the cost of the production and increases the output of the product. The equation of the production function can be expressed in terms of equation is as follows: Q = ƒ (N,L,K,E,T) In the equation the factors which are included decides the Q= Quantity supplied with the factors of production which includes Land, Labour, Capital, and entrepreneur so by which it can be concluded that the combinations of factors of production the level of output is decided. In general, economic output is not a (mathematical) function of input, because any given set of inputs can be used to produce a range of outputs. To satisfy the mathematical definition of a function, a production function is customarily assumed to specify the maximum output Page 1
  • 2. Chapter 3: PRODUCTION FUNCTION obtainable from a given set of inputs. The production function, therefore, describes a boundary or frontier representing the limit of output obtainable from each feasible combination of input. (Alternatively, a production function can be defined as the specification of the minimum input requirements needed to produce designated quantities of output, given available technology.) By assuming that the maximum output, which is technologically feasible, from a given set of inputs, is obtained, economists are abstracting away from technological, engineering and managerial problems associated with realizing such a technical maximum, to focus exclusively on the problem of allocative efficiency, associated with the economic choice of how much of a factor input to use, or the degree to which one factor may be substituted for another. In the production function itself, the relationship of output to inputs is non-monetary; that is, a production function relates physical inputs to physical outputs, and prices and costs are not reflected in the function. In the decision frame of a firm making economic choices regarding production—how much of each factor input to use to produce how much output—and facing market prices for output and inputs, the production function represents the possibilities afforded by an exogenous technology. Under certain assumptions, the production function can be used to derive a marginal product for each factor. The profit-maximizing firm in perfect competition (taking output and input prices as given) will choose to add input right up to the point where the marginal cost of additional input matches the marginal product in additional output. This implies an ideal division of the income generated from output into an income due to each input factor of production, equal to the marginal product of each input. The inputs to the production function are commonly termed factors of production and may represent primary factors, which are stocks. Classically, the primary factors of production were Land, Labour and Capital. Primary factors do not become part of the output product, nor are the primary factors, themselves, transformed in the production process. The production function, as a theoretical construct, may be abstracting away from the secondary factors and intermediate products consumed in a production process. The production function is not a full model of the production process: it deliberately abstracts from inherent aspects of physical production processes that some would argue are essential, including error, entropy or waste, and the consumption of energy or the co-production of pollution. Moreover, production functions do not ordinarily model the business processes, either, ignoring the role of strategic and operational business management. (For a primer on the fundamental elements of microeconomic production theory, see production theory basics). The production function is central to the marginalist focus of neoclassical economics, its definition of efficiency as allocative efficiency, its analysis of how market prices can govern the achievement of allocative efficiency in a decentralized economy, and an analysis of the distribution of income, which attributes factor income to the marginal product of factor input. The firm is assumed to be making allocative choices concerning how much of each input factor to use and how much output to produce, given the cost (purchase price) of each factor, the selling price of the output, and the technological determinants represented by the production function. Page 2
  • 3. Chapter 3: PRODUCTION FUNCTION 2. LONG RUN – SHORT RUN PROCESS: The production function decides the process of the production and according to the process cycle. The production function helps the manager to take various decisions related to the business related to the production so that the equilibrium position of the company can be maintained. Thus the production technique is divided into two parts : 1. LONG RUN FUNCTION 2. SHORT RUN FUNCTION Thus the process of the function decides the profitability of the company. According to the function the factors of the production are been decided and according to that the production cycle of the firms product is decided. In the short run function the plant size is fixed and the variable factor labour keeps on changing. Thus the factor is remains fixed then the other factors if kept on increasing it results into negative return for the firms. Thus in the short run as the firm earns a maximum production level and after that if the units of production are kept on increasing then the production function starts giving the negative return as the plant size is fixed so, the labours becomes more and so the machines can be used more or the labours remains ideal in the firm. Thus the company earns expenses on the production cycle but the production is not earned as per the expectation which results in loss area in the production function resulting into negative returns. Thus in the short run function the firms focuses on production function up to a certain level of profitability return. In a long run production function the plant size as well as the labour size as variable. Thus the company’s analysis the production function cycle in the market for the long term and according the market situation the production of the company is decided. In long term if it is considered that the land is a variable factor then that means that the plant size is variable in the planning process and the increment in the land sector results in more productivity which results in increment in profit. Thus in the long run the planning of the production function which results in to the profit area according to the situation. Thus both the long run and the short run production function is to be considered which ultimately decides the market structure. Thus the production function considers the various types of the cost which includes average revenue, marginal revenue and the total productivity. Page 3
  • 4. Chapter 3: PRODUCTION FUNCTION There are three major ways to measure the productivity of labour which helps to measure the productivity of the labour along with which the production function moves in the market. 1. TOTAL PRODUCTIVITY: Total productivity (also known as total physical product) is defined as the total quantity of output produced by a firm for a given quantity of input necessities. Total product identifies the specific outputs which are possible using variable levels of input. An understanding of total product is essential to the short-run analysis of a firm's production. Changes in total product are taken into account closely when there are changes in variable costs (labour) of production. Thus the total productivity shows the relationship between number of workers and the total number of outputs been produced which is termed as Q, holding the concept that the other factors remains constant.  Thus for e.g. For a coffee shop, output would be measured in ―number of coffee cups a day‖  For a steel mill, output would be measured in ―tons of steel produced a day‖ Thus the total productivity shows the three following conditions: The first situation shows that as the labour increases the number of output also increases in an increasing, constant or in a decreasing form. But as the three cases are seen the total output increases thus the total productivity is profitable for the firm. But as the marginal revenue starts decreasing it give low total productivity which results in losses. 2. Marginal Productivity : Productivity is, at its most basic, the output gained from a unit of input. For example, a clothing company’s productivity could be the number of jeans sewn per worker or per hour. To increase productivity, you have to increase the input. To use the clothing factory example, you would have to buy more machines, hire more workers or find some means of increasing efficiency to increase productivity. The term ―marginal productivity‖ refers to the extra output gained by adding one unit of labor; all other inputs are held constant. So, the technology and efficiency of the factory stays Page 4
  • 5. Chapter 3: PRODUCTION FUNCTION the same. Marginal productivity is the extra jeans sewn, that is output gained, by hiring an extra worker, for example. The additional output that can be produced by adding one more worker while holding plant size constant. MP = ΔQ/ΔL8 Is the slope of the Total Product Function. 3. AVERAGE PRODUCTIVITY: Average productivity is measured by taking the total output and dividing the quantity by the number of workers. For example, if the combined number of phone calls handled in a week is 1,300 and the company has 10 employees each working the same shift length, the average productivity per worker is 130. Businesses use average productivity figures to gain a perspective on the performance of its workforce: by pooling the labour of every individual, it focuses less on how to improve a problematic worker's output and more as an estimate of the output currently being given. If low productivity is the result of a systemic issue within the company and for reasons that affect all workers, measuring average productivity as opposed to per-worker productivity is the better plan. If average productivity is more than marginal productivity, average productivity will decrease. If the average productivity is less than marginal productivity, average productivity will increase. Thus the average productivity shows the average change in the production theory of the organisation. Thus if the average production decreases and it affects the marginal productivity which in turn affects the total productivity which frames the law of return. LAW OF DIMINISHING RETURN : Diminishing Returns occurs in the short run when one factor is fixed (e.g. Capital) If the variable factor of production is increased, there comes a point where it will become less productive and therefore there will eventually be a decreasing marginal and then average product This is because if capital is fixed extra workers will eventually get in each other’s way as they attempt to increase production. E.g. think about the effectiveness of extra workers in a small café. If more workers are employed production could increase but more and more slowly. This law only applies in the short run because in the long run all factors are variable. Assume the wage rate is £10, then an extra worker Costs £10. The Marginal Cost (MC) of a sandwich will be the Cost of the worker divided by the number of extra sandwiches that are produced Therefore as MP increases MC declines and vice versa A good example of Diminishing Returns includes the use of chemical fertilizers- a small quantity leads to a big increase in output. However, increasing its use further may lead to declining Marginal Product (MP) as the efficacy of the chemical declines. Page 5
  • 6. Chapter 3: PRODUCTION FUNCTION Thus the law of diminishing return states that ―As more of a variable input (labor) is added to a fixed input (plant), additions to output eventually slow down.” Thus the productivity is acceptable only upto the level where the total productivity and marginal productivity increases. The following example shows the marginal rate of return: EXAMPLE : As the law of diminishing considers the labour and productivity to change and other factors keeps on changing, here we consider the land factor to be fixed in the market. In the example we can see that the total productivity keeps on changing but then also when the returns are considered it starts decreasing the return and still the production is continued of the firm it may result into negative return. As the total productivity starts from 10 to 68 units the average and marginal units are been calculated as under: Average productivity = total productivity/ Labour units, thus 10/1=10 and so on. Marginal productivity = 22-10= 12 marginal difference of two total productivity. Page 6
  • 7. Chapter 3: PRODUCTION FUNCTION Thus as the marginal productivity is calculate the company when inputs 1 or 2 labour it earn increasing return, for 3 and 4 number of units the return becomes constant and finally for 5 and 6 unit the return has started diminishing. It proves that now suppose if the labour starts increasing then the firm will result in negative profits. Thus the law of return says that upto a certain level the firm are increasing and constant return the firm falls in part of diminishing return and still of the production is continued it may result in negative return which indicates the loss of the firm. MRTS : Prof. R.G.D. Allen and J.R. Hicks introduced the concept of MRS (marginal rate of substitution) in the theory of demand. The similar concept is used in the explanation of producers’ equilibrium and is named as marginal rate of technical substitution (MRTS). This theory finds the combinations of inputs in such a way that the total productivity is been maintained at a lower production cost in the organisation. This means that input factors are combined in such a way that the maximum production is achieved at the minimum cost. This is called least cost combination. The analysis of production function has shown that alternative combinations of factors of production, which are technically efficient, can be used to produce a given level of output. Of these, the firm will have to choose that combination of factors which will cost it the least. In this way the firm can maximise its profits. The choice of any particular method from a set of technically efficient methods is an economic one and it is based on the prices of factors of production at a particular time. The firm can maximise its profits either by maximising the level of output for a given cost or by minimising the cost of producing a given output. In either case, the factors will have to be employed in optimal combination at which the cost of production will be minimum. Page 7
  • 8. Chapter 3: PRODUCTION FUNCTION Thus in this method the variables are been such arranged that the company combines the factors of production and gets the maximum level of output at the given budgeted rate. In this every input should be combined in such a way that marginal productivity of a factor and the marginal utility of the money spent on it are equal. Thus in MRTS the cost of production is minimum and is also known as the optimal combination of input for the production theory. The equation of MRTS is as follows: Where MP is the marginal productivity and P stands for price and is continued till the number of inputs been used by the firm. So a,b,c,d ,n are the factors of inputs of the variables. Assumption of the MRTS Theory : 1. MRTS works in a prefect competitive market as this theory works in lowering the prices at the maximum output which is the need of the perfect competitive world rather then the monopoly companies. 2. This theory considers that the factors of production is going to be mobile thus any factors can be changed as there is a change in the market situation. 3. The prices of each input variable is been decided and once decided it should not be changed and all the inputs should consider the similar value as a part of the price. 4. The production theory as works on marginal theory, the marginal rate of individual input unit is to be considered then only the optimal value is created. Example: Suppose a firm, uses A,B,C as inputs in the process of the production. The prices of A ,B and C are 6,4 and 2 respectively. On all these three Rs.122 has been spent. The Marginal productivity is as follow: So here the prices of A is 6 RS , for B is 4 RS and for the c the price is 2 respectively. So to find the optimal value first the MPA, MPB and MPC is to divided by the prices individually Page 8
  • 9. Chapter 3: PRODUCTION FUNCTION so that the optimal combination values are obtained. Thus where the MPA/PA= MPB/PB = MPC/PC is there that values are been selected. Thus the calculation of MRTS is as follows: Thus here for Product A optimum value of combination 3 is achieved at 9 level of units, whereas for product B the optimum value of combination 3 is achieved at 11 units and for the product C 12 units of production is required. Thus the optimal combination defines the maximum production and the minimum price. Thus the , Minimum Price : A= 9*6 = 54 B= 11*4 = 44 C= 12*2 = 24 Thus the optimal Production is as follows: A= 30+28+24+18 = 100 B= 24+22+18+12 = 76 C= 20+18+14+6 = 58 Thus according to the MRTS technique the minimum price for the production is 122 at which the optimal production available for the company is 234 units. Thus the MRTS works for calculating the expected units of production. Economies of scale: he cost advantage that arises with increased output of a product. Economies of scale arise because of the inverse relationship between the quantity produced and per-unit fixed costs; i.e. the greater the quantity of a good produced, the lower the per-unit fixed cost because Page 9
  • 10. Chapter 3: PRODUCTION FUNCTION these costs are shared over a larger number of goods. Economies of scale may also reduce variable costs per unit because of operational efficiencies and synergies. Economies of scale can be classified into two main types: Internal – arising from within the company; and External – arising from extraneous factors such as industry size. Economies of scale can arise in several areas within a large enterprise. While the benefits of this concept in areas such as production and purchasing are obvious, economies of scale can also impact areas like finance. For example, the largest companies often have a lower cost of capital than small firms because they can borrow at lower interest rates. As a result, economies of scale are often cited as a major rationale when two companies announce a merger or takeover. However, there is a finite upper limit to how large an organization can grow to achieve economies of scale. After reaching a certain size, it becomes increasingly expensive to manage a gigantic organization for a number of reasons, including its complexity, bureaucratic nature and operating inefficiencies. This undesirable phenomenon is referred to as "diseconomies of scale". Internal economies of Scale: The internal economies of scales are the production advantages which the companies earns from the internal working of the organisation. Internal economies of scale are the advantages of large scale production. They are enjoyed by the firm when it increases its scale of production. They accrue to the firm from their own actions. They affect the shape of the longrun average cost curve. They are responsible for increasing returns to scale. According to many economists, internal economies arise due to indivisibility of some factors. As the output increases the large indivisible factors can be used more efficiently and, therefore, the firm experiences increasing returns to scale.Thus the internal economies of scale includes the following points: 1. Technical Economies: The important technical economies result from the use of specialised capital equipment, which comes into effect only when the output is produced on a large scale. Technical economies also arise from the indivisibilities, which are the characteristics of the modern techniques of production. In other words, as the scale of production increases the firm reaps the advantages of mechanisation of using mass production methods. This will reduce the unit cost of production. 2. Managerial Economies : Large scale production makes possible the division of managerial functions. Thus, there exists a production manager, a sales manager, a finance manager, a personnel manager and so on in a large firm. However, all or most of the managerial decisions are taken by a single manager in a small firm. This division of managerial functions increases their efficiency. The decentralisation of managerial decision making also increases the efficiency of management. Large firms are also in a position to introduce mechanisation of managerial functions through the use of telex machines, computers and so on. Hence, as output increases the managerial costs per unit of output continue to decline. Page 10
  • 11. Chapter 3: PRODUCTION FUNCTION 3. Marketing Economies : They are allied with the selling of the product of the firm. They arise from advertising economies. Since, advertising expenses increase less thanproportionately with the increase in output, the advertising costs per unit of output fallsas the output increases. Similarly, other sales promotion expenditures like samples, salesmen force etc. also increase less than proportionately with the output. Further, a large firm can have special arrangements with exclusive dealers to maintain a good service department for the product of the firm. Hence, the average selling costs fall with the increase in the size of the firm. 4. Financial Economies : The role of finance is to aid the firm in meeting random changes in the input and the output sides of the operations of the firm. The purpose of inventories is to smooth out the supply of inputs and the supply of outputs. Inventories on spare parts, raw materials and finished products increase with the scale of production, but they do not increase proportionately with the increase in the size of output. Therefore, as the size of output amplifies the firm can hold smaller percentage of inventories to meet random changes. External Economies of Scale: The external economies arise outside the firm as a result of improvement in the industrial environment in which the firm operates. They are external to the firm, but internal to the industry to which the firms belong. They may be realised from the actions of other firms in the same industry or in another industry. Their effect is to cause a change in the prices of factors employed by the firm. They cause a shift in the short-run and long-run cost curves of the firm. The important external economies are the following: 1. Concentration Economies: Expansion of an industry increases the demand for various kinds of materials and capital equipment’s. This will lead to large scale production of materials and equipment’s. Large scale production will reduce their cost of production and therefore, their prices. Hence, the firms using them will get them at lower prices.Expansion of an industry may lead to the discovery of new technical know-how. As a result of this the firms may be able to use improved and better machinery which will increase the productivity of the firms and therefore, reduce the cost of production. Thus the place where the productional units are been established and their nearby areas gives the firm the benefit of concentration of economies. 2. Information Economies : External economies also arise from the interchange of technical information between firms. With the expansion of an industry the firms may give the information about the technical knowledge through the publication of trade and technical journals. The firms may also set up jointly research institutes to develop new improved techniques. Thus the information which is received is very helpful for the companies development. Thus this advantage is been got by the large scale companies as compared to the small scale units. For eg. Ahmedabad Textile Page 11
  • 12. Chapter 3: PRODUCTION FUNCTION Industry Research Association (ATIRA), is a unit working for the collection of the vital information which is helpful for the overall development of the firms. 3. Disintegration Economies: As the industry grows the training facilities for labourwill increase. This helps the development of skilled labour, which will increase the productivity of workers in the firms.Expansion of an industry may facilitate the growth of subsidiary and ancillary industries to produce tools, equipment’s, machines etc. and to provide them to the main industry at the lower prices. Likewise, firms may also come up to transform the waste of the industry into some useful products. This tends to reduce the cost of production.The expansion of an industry may expedite the development of transportation and marketing facilities which will reduce the cost of transportation. Thus the disintegration facilities help for the overall advantage for the production areas of the firm. ISOQUANTS AND ISOCOSTS: An isoquant shows all those combinations of factors which produce the same level of output. An isoquant is also known as equal product curve or iso-product curve. Thus the isoquant curve shows the combinations of the two factors of production for which the total outlay of the product remains the same. Thus the cost earned for the prodcutional area is called iscosts. Certain Assumptions of this theory includes :  Uses capital and labour combination for production  All other factors remains fixed.  Production method is given, and no change can be made. In economics, an isocost line represents all combinations of inputs which cost the same total amount. Although, similar to the budget constraint in consumer theory, the use of the isocost line pertains to cost-minimisation in production, as opposed to utilitymaximisation. For the two production inputs, labour and capital, with fixed unit costs of the inputs, the equation of the Isocosts line is Where w represents the wage rate of labour, r represents the interest rate of capital, K is the amount or units of capital used, L is the amount of labour used and C is the total costof acquiring these inputs. PROPERTIES OF ISOQUANTS: (CHSRECTERISTICS): 1. Isoquants slope downwards to the right: It means that, in order to keep the output constant; when the amount of one factor is increased the quantity of other factor mustbe reduced. Page 12
  • 13. Chapter 3: PRODUCTION FUNCTION An upward sloping isoquant demonstrates that a given product can be produced with less of both the factors of production. An entrepreneur, who is maximising profits, would not use any combinations of factors shown on an upward sloping portion of an isoquant. Therefore, the points on the upward sloping portion of an isoquant cannot represent an equilibrium position. Similarly, a horizontal or vertical range of an isoquant cannot also represent a possible position of equilibrium. In this case, the same output could be obtained at a reduced cost by reducing the amount of one of the factors. Thus, isoquants slope downwards to the right as in fig 3.15. 2. Isoquants are convex to the origin: The slope, at any point of an isoquant, is negative. Its numerical value measures the marginal rate of technical substitution between labour and capital. It equals the ratio of the marginal product of labour to the marginal product of capital. Thus, the slope of an isoquant is Where ΔK is the change in capital, ΔL is the change in labour, MRTSLK is the marginal rate of technical substitution of labour for capital, MPL is the marginal product of labour and MPK is the marginal product of capital. The convexity of isoquant means that as we move down the curve less and less of capital is given up for an additional unit of labour so as to keep constant the level of output. This can be observed from the Fig. 3.16. Page 13
  • 14. Chapter 3: PRODUCTION FUNCTION It can be seen from the figure above that as we increase labour at a constant rate the amount of capital given up (ΔK) for an additional unit of labour goes on falling. Thus, the convexity of the isoquant shows that the marginal rate of technical substitution of labour for capital is diminishing. 3. Isoquants do not intersect: By definition isoquants, like indifference curves, can never cut each other. If they cut each other it would be a logical contradiction. 4. The higher the isoquant curve the more output it indicates. 5. All points on isoquant curve shows equal production. ISOQUANT GRAPH: The firm uses an input as a substitute of another input in such a manner that at each combinations of inputs output remains same. The list of equal output at different combinations is prepared so that its forms a convex shapes. Those convex curves are considered as isoquant curves and the cost related to that combination for production is called the isocost. Thus here the various combinations of the labour and capital is considered and based on that the marginal rate ration is been calculated and it can be observed that all the possible combinations the variable output remains the same. Page 14
  • 15. Chapter 3: PRODUCTION FUNCTION Thus in the above graph: - On OX axis the capital units are considered and on OY axis the units of labour is mentioned. R,P,N and M shows the various combinations of the isoquant curves which is a combination of capital and labour. Thus the convex shapes been formed are considered as isoquant curves which shows equal production units i.e. 100 at each combines combination. Page 15