Theory Of Cost PDF Study Material II

Lesson : 56,57 & 58 THEORY OF COST Structure 6.0 Introduction 6.1 Unit Objectives 6.2 Cost Concepts 6.2.1 Some Accounting Cost Concepts; 6.2.2 Some Analytical Cost Concepts 6.3 The Theory of Cost: The Cost-Output Relations 6.3.1 Short-run Cos...

Lesson : 56,57 & 58 THEORY OF COST Structure 6.0 Introduction 6.1 Unit Objectives 6.2 Cost Concepts 6.2.1 Some Accounting Cost Concepts; 6.2.2 Some Analytical Cost Concepts 6.3 The Theory of Cost: The Cost-Output Relations 6.3.1 Short-run Cost-Output Relations; 6.3.2 Short-run Cost Functions and Cost Curves; 6.3.3 Cost Curves and the Law of Diminishing Returns; 6.3.4 Some Important Cost Relationships; 6.3.5 Output Optimization in the Short-Run; 6.3.6 Long-run Cost-Output Relations 6.4 Economies and Diseconomies of Scale 6.4.1 Economies of Scales; 6.4.2 Diseconomies of Scales 6.5 Some Empirical Cost Functions 6.0 INTRODUCTION The cost of production is the second most important aspect in almost all business analysis and decisions. Cost data and data analysis play a very important role in most business decisions, specially those pertaining to (a) locating the weak points in production management; (b) minimizing the cost; (c) finding the optimum level of output; (d) determination of price and dealers margin; and (e) estimating or projecting the cost of business operations. Also, cost analysis assumes a great significance in all major business decisions because the term ‘cost’ has different meaning under different settings and is subject to varying interpretations. It is, therefore, essential that only the relevant concept of costs is used in business decisions. Inputs multiplied by their respective prices and added together give the money value of the inputs, i.e., the cost of production. This unit is divided into three sections. Section 6.2 discusses various cost concepts used in business decisions and section 6.3 analyses cost-output relations. Break-even analysis is discussed in section 6.4. 6.1 UNIT OBJECTIVES z To introduce cost concepts used in business analysis z To discuss theory of cost—the nature of relationship between cost and output, in the short-run and in the long-run  z To discuss economies and diseconomies of scale of production and their impact on cost behaviour z To introduce the concept of break-even and show its application in business decision-making 6.2 COST CONCEPTS The cost concepts which are relevant to business operations and decisions can be grouped, on the basi s of their nature and purpose, under two overlapping categories: (i) concepts used for accounting purposes, and (ii) analytical cost concepts used in economic analysis of business activities. We will discuss here some important concepts of the two categories. It is important to note here that this classification of cost concepts is only a matter of analytical convenience. 6.2.1 Some Accounting Cost Concepts 1. Opportunity Cost and Actual Cost. Resources available to any perosn, firm or society are scarce but have alternative uses with different returns. Income maximizing resource owners put their scarce resources to their most productive use and thus, they forego the income expected from the second best use of the resources. Thus, the opportunity cost may be defined as the expected returns form the second best use of the resources which are foregone due to the scarcity of resources. The opportunity cost is also called alternative cost. Had the resource available to a person, a firm or a society been unlimited there would be no opportunity cost. For example, suppose that a firm has a sum of Rs. 100,000 for which it has only two alternative uses. It can buy either a printing machine or alternatively a lathe machine both having productive life of 10 years. From the printing machine, the firm expects an annual income of Rs. 20,000 and from the lathe, Rs. 15,000. A profit maximizing firm would invest its money in the printing machine and forego the expected income from the lathe. The opportunity cost of the income from printing machine is the expected income from the lathe, i.e., Rs. 15,000. In assessing the alternative cost, both explicit and implicit costs are taken into account. Associated with the concept of opportunity cost is the concept of economic rent or economic profit. In our example of expected earnings firm printing machine and eco- nomic rent of the printing machine is the excess of its earning over the income expected from the lathe. That is, economic rent equals Rs. 20,000 – Rs. 15,000 = Rs. 5,000. The implication of this concept for a business man is that investing in the printing machine is preferable so long as its economic rent is greater than zero. Also, if firms know the economic rent of the various alternative uses of their resources, it will be helpful in the choice of the best investment avenue. In contrast to the concept of opportunity cost, actual costs are those which are actually incurred by the firm in payment for labour, material, plant, building, machinery, equipment, travelling and transport, advertisement, etc. The total money expenses, recorded in the books of accounts are for all practical purposes, the actual costs. In our example, the cost of printing machine, i.e., Rs. 100,000 is the actual cost. Actual cost comes under the accounting cost concept. 2. Business Costs and Full Costs. Business cost include all the expenses which are incurred to carry out a business. The concept of business costs is similar to the actual or real costs. Business costs “include all the payments and contractual obligations made by the firm together with the book cost of depreciation on plant and equipment.”1 These cost concepts are used for calculating business profits and losses and for filing returns for income-tax and also for other legal purposes. The concept of full cost, includes business costs, opportunity cost and normal profit. The opportunity cost includes the expected earning from the second best use of the resources, or the market rate of interest on the total money capital and also the value of an entrepreneur’s own services which are not charged for in the current business. Normal profit is a necessary minimum earning in addition to the opportunity cost, which a firm must receive to remain in its present occupation. 3. Explicit and Implicit or Imputed Costs Explicit costs. are those which fall under actual or business costs entered in the books of accounts. The payments for wages and salaries, materials, license fee, insurance premium, depreciation charges are the examples of explicit costs. These costs involve cash payment and are recorded in normal accounting practices. In contrast to explicit costs, there are certain other costs which do not take the form of cash outlays, nor do they appear in the accounting system. Such costs are known as implicit or imputed costs. Opportunity cost is an important example of implicit cost. For example, suppose an entrepreneur does not utilize his services in his own business and works as a manager in some other firm on a salary basis. If he sets up his own business, he foregoes his salary as manager. This loss of salary is the opportunity cost of income from his own business. This is an implicit cost of his own business. Thus, implicit wages, rent, and implicit interest are the wages, rents and interest which an owner’s labour, building and capital, respectively, can earn from their second best use. Implicit costs are not taken into account while calculating the loss or gains of the business, but they form an important consideration in whether or not a factor would remain its present occupation. The explicit and implicit costs together make the economic cost. 4. Out-of-Pocket and Book Costs. The items of expenditure which involve cash payments or cash transfers, both recurring and non-recurring, are known as out-of-pocket costs. All the explicit costs (e.g., wages, rent, interest, cost of materials and maintenance, transport expenditure, etc.) fall in this category. On the contrary, there are certain actual business costs which do not involve cash payments, but a provision is therefore made in the books of account and they are taken into account while finalising the profit and loss accounts. Such expenses are known as book costs. In a way, these are payments made by a firm to itself. Depreciation allowances and unpaid interest on the owner’s own fund are the example of book costs. 6.2.2 Some Analytical Cost Concepts 1. Fixed and Variable Costs. Fixed costs are those which are fixed in volume for a certain given output. Fixed cost does not vary with variation in the output between zero and a certain given level of output. In other words, costs that do not vary for a certain level of output are known as fixed costs. The fixed costs include (i) costs of managerial and administrative staff, (ii) depreciation of machinery, building and other fixed assets, (iii) maintenance of land, etc. The concept of fixed cost is associated with the short- run. Variable costs are those which vary with the variation in the total output. Variable costs include cost of raw material, running cost of fixed capital, such as fuel, repairs, routine maintenance expenditure, direct labour charges associated with the level of output, and the costs of all other inputs that vary with output. 2. Total, Average and Marginal. Costs Total cost (TC) is the total expenditure incured on the production of goods and service. It refers to the total outlays of money expenditure, both explicit and implicit, on the resources used to produce a given level of output. It includes both fixed and variable costs. The total cost for a given output is given by the cost function. Average cost (AC) is of statistical nature—it is not actual cost. It is obtained by dividing the total cost (TC) by the total output (Q), i.e., TC AC = Q Marginal cost (MC) is the addition to the total cost on account of producing one additional unit of the product. Or, marginal cost is the cost of the marginal unit produced. Marginal cost is calculated as TCn – TCn – 1 where n is the number of units produced. Alternatively, given the cost function, MC can be defined as ∂TC MC = ∂Q These cost concepts are discussed in further detail in the following section. Total, average and marginal cost concepts are used in the economic analysis of firm’s production activities. 3. Short-Run and Long-Run Costs. Short-run and long-run cost concepts are related to variable and fixed costs, respectively, and often figure in economic analysis inter- changeably. Short-run costs are the costs which vary with the variation in output, the size of the firm remaining the same. In other words, short-run costs are the same as variable costs. Long-run costs, on the other hand, are the costs which are incurred on the fixed assets like plant, building, machinery, etc. It is important to note that the running cost and depreciation of the capital assets are included in the short-run or variable costs. Long-run costs are by implication the same as fixed costs. In the long-run, however, even the fixed costs become variable costs as the size of the firm or scale of production increases. Broadly speaking, ‘the short-run costs are those associated with variables in the utilization of fixed plant or other facilities whereas long-run costs are associated with the changes in the size and kind of plant.’2 4. Incremental Costs and Sunk Costs. Conceptually, incremental costs are closely related to the concept of marginal cost but with a relatively wider connotation. While marginal cost refers to the cost of the marginal unit of output, incremental cost refers to the total additional cost associated with the decisions to expand the output or to add a new variety of product, etc. The concept of incremental cost is based on the fact that in the real world, it is not practicable (for lack of perfect divisibility of inputs) to employ factors for each unit of output separately. Besides, in the long run, when firms expand their production, they hire more of men, materials, machinery and equipments. The expenditures of this nature are incremental costs and not the marginal cost (as defined earlier). Incremental costs arise also owing to the change in product lines, addition or introduction of a new product, replacement of worn out plant and machinery, replace- ment of old technique of production with a new one, etc. The sunk costs are those which cannot be altered, increased or decreased, by varying the rate of output. For example, once it is decided to make incremental investment expenditure and the funds are allocated and spent, all the preceding costs are considered to be the sunk costs since they accord to the prior commitment and cannot be revised or reversed or recovered when there is a change in market conditions or change in business decisions. 5. Historical and Replacement Costs. Historical cost refers to the cost of an asset acquired in the past whereas replacement cost refers to the outlay which has to be made for replacing an old asset. These concepts owe their significance to the unstable nature of price behaviour. Stable prices over time, other things given, keep historical and replacement costs on par with each other. Instability in asset prices makes the two costs differ from each other. Historical cost of assets is used for accounting purposes, in the assessment of the net worth of the firm. The replacement cost figures in business decisions regarding the renovation of the firm. 6. Private and Social Costs. We have so far discussed the cost concepts that are related to the working of the firm and that are used in the cost-benefit analysis of business decisions. There are, however, certain other costs which arise due to the functioning of the firm but do not normally figure in the business decisions nor are such costs explicitly borne by the firms. The costs on this category are borne by the society. Thus, the total cost generated by a firm’s working may be divided into two categories: (i) those paid out or provided for by the firms, and (ii) those not paid or borne by the firms including the use of resources freely available plus the disutility created in the process of production. The costs of the former category are known as private costs and of the latter category are known as external or social costs. To mention a few examples of social cost: Mathura Oil Refinery discharging its wastage in the Yamuna river causes water pollution. Mills and factories located in a city cause air pollution, and so on. Such costs are termed as external costs from the firm’s point of view and social costs from the society’s point of view. The relevance of the social costs lies in the social cost-benefit analysis of the overall impact of a firm’s operation on the society as a whole and in working out the social cost of private gains. A further distinction between private cost and social cost is, therefore, in order. Private costs are those which are actually incurred or provided for by an individual or a firm on the purchase of goods and services from the market. For a firm, all the actual costs both explicit and implicit are private costs. Private costs are internalized costs that are incorporated in the firm’s total cost of production. Social costs, on the other hand, refer to the total cost borne by the society due to production of a commodity. Social cost includes both private cost and the external cost. Social cost includes (a) the cost of resources for which the firm is not compelled to pay a price, i.e., atmosphere, rivers, lakes, and also for the use of public utility services 3 like roadways, drainage system, etc., and (b) the cost in the form of ‘disutility’ created through air, water and noise pollution, etc. The costs of category (b) are generally assumed to equal the total private and public expenditure incurred to safeguard the individual and public interest against the various kinds of health hazards created by the production system. The private and public expenditure, however, serve only as an indicator of ‘public disutility’—they do not give the exact measure of the public disutility or the social costs. 6.3 THE THEORY OF COST: THE COST-OUTPUT RELATIONS The theory of cost deals with the behaviour of cost in relation to a change in output. In other words, the cost theory deals with cost-output relations. The basic principle of the cost behaviour is that the total cost increases with increase in output. This simple statement of an observed fact is of little theoretical and practical importance. What is of importance from a theoretical and managerial point of view is not the absolute increase in the total cost but the direction of change in the average cost (AC) and the marginal cost (MC). The direction of change in AC and MC—whether AC and MC decrease or increase or remain constant—depends on the nature of the cost function. A cost function is a symbolic statement of the technological relationship between the cost and output. The general form of the cost function is written as TC = f (Q), D T C/D Q > 0 …(6.1) The specific form of the cost function depends on whether the time framework chosen for cost analysis is short-run or long-run. It is important to recall here that some costs remain constant in the short-run while all costs are variable in the long-run. Thus, depending on whether cost analysis pertains to short-run or to long run, there are two kinds of cost functions: (i) short-run cost functions, and (ii) long-run cost functions, Accordingly, the cost output relations are analyzed in short-run and long-run framework. In this section, we will analyse the cost-output relations in the short-run. The long-run cost output relations are discussed in the following section. 6.3.1 Short-Run Cost-Output Relations Before we discuss the cost-output relations, let us first look at the cost concepts and the components used to analyse the short-run cost-output relations. The basic analytical cost concepts used in the analysis of cost behaviour are Total, Average and Marginal costs. The total cost (TC) is defined as the actual cost that must be incurred to produce a given quantity of output. The short-run TC is composed of two major elements: (i) total fixed cost (TFC), and (ii) total variable cost (TVC). That is, in the short-run, TC = TFC + TVC...(6.2) As mentioned earlier, TFC (i.e., the cost of plant, building, etc.) remains fixed in the short-run, whereas T VC varies with the variation in the output. For a given quantity of output (Q), the average total cost, (AC), average fixed cost (AFC) and average variable cost (AVC) can be defined as follows. TC T F C + TV C AC = Q = Q T FC AFC = Q TV C AVC = Q and AC = AFC + AVC...(6.3) Marginal cost (MC) is defined as the change in the total cost divided by the change in the total output, i.e., ΔTC MC = ΔQ...(6.4) ∂TC or as the first derivative of cost function, i.e., ∂Q It may be added here that since ΔTC = ΔTFC + ΔTVC and, in the short-run, ΔTFC = 0, therefore, ΔTC = ΔTVC. Furthermore, under the marginality concept, where ΔQ = 1, MC = ΔTVC. Now we turn to cost function and derivation of cost curves. 6.3.2 Short-Run Cost Functions and Cost Curves The cost-output relations are determined by the cost function and are exhibited through cost curves. The shape of the cost curves depends on the nature of the cost function. Cost functions are derived from actual cost data of the firms. Given the cost data, cost functions may take a variety of forms, yielding different kinds of cost curves. The cost curves produced by linear, quadratic and cubic cost functions are illustrated below. 1. Linear Cost Function. A linear cost function takes the following form. TC = a + bQ …(6.5) (where TC = total cost, Q = quantity produced, a = TFC, and bQ = TVC). Given the cost function (Eq. 6.5), AC and MC can be obtained as follows. TC a + bQ a AC = = = + b Q Q Q ∂TC and MC = =b ∂Q Note that since ‘b’ is a constant, MC remains constant throughout in case of a linear cost function. Assuming an actual cost function given as TC = 60 + 10Q …(6.6) the cost curves (TC, TVC and TFC) are graphed in Fig. 6.1. Fig. 6.1: Linear Cost Functions Given the cost function (6.6), 60 AC = + 10 Q and MC = 10 Figure 6.1 shows the behaviour of TC, TVC and TFC. The straight horizontal line shows TFC and the line marked TVC = 10Q shows the movement in TVC. The total cost function is shown by TC = 60 + 10Q. Fig. 6.2: AC and MC Curves Derived from Linear Cost Function More important is to notice the behaviour of AC and MC curves in Fig. 6.2. Note that, in case of a linear cost function, MC = AVC and it remains constant, while AC continues to decline with the increase in output. This is so simply because of the logic of the linear cost function. 2. Quadratic Cost Function. A quadratic cost function is of the form TC = a + bQ + Q2....(6.7) where a and b are constants and TC and Q are total cost and total output, respectively. Given the cost function (6.7), AC and MC can be obtained as follows. TC a + bQ + Q 2 AC = =...(6.8) Q Q a = + b + Q Q ∂TC MC = = b + 2Q...(6.9) ∂Q Let the actual (or estimated) cost function be given as TC = 50 + 5Q + Q2 …(6.10) Given the cost function (6.10), 50 AC = Q + 5 Q ∂C and = 5 + 2Q MC = ∂Q The cost curves that emerge from the cost function (6.10) are graphed in Fig. 6.3 (a) and (b). As shown in panel (a), while fixed cost remains constant at 50, TVC is increasing at an increasing rate. The rising TVC sets the trend in the total cost (TC). Panel (b) shows the behaviour of AC, MC and AVC in a quadratic cost function. Note that MC and AVC are rising at a constant rate whereas AC first declines and then increases. 200 (a) (b) 180 60 Average and Marginal costs 160 2 Q 50 140 + 5Q Total Cost 120 + 40 50 2 100 = Q TC + 30 80 5Q = 60 T VC 20 MC 50 FC (= 50) 40 AC 10 AVC 20 1 2 3 4 5 6 7 8 9 10 11 0 1 2 3 4 5 6 7 8 9 10 11 Output (Q) Output (Q) Fig. 6.3: Cost Curves Derived from a Quadratic Cost Function 3. Cubic Cost Function. A cubic cost function is of the form TC = a + bQ – cQ2 + Q3 …(6.11) where a, b and c are the parametric constants. From the cost function (6.11), AC and MC can be derived as follows. TC a + bQ − cQ 2 + Q3 AC = = Q Q  a = + b – cQ + Q2 Q ∂TC and MC = = b – 2 cQ + 3Q2 ∂Q Let us suppose that the cost function is empirically estimated as TC = 10 + 6Q – 0.9Q2 + 0.05Q3 …(6.12) and TVC = 6Q – 0.9Q2 + 0.05Q3 …(6.13) Fig. 6.4: TC, TFC and TVC Curves The TC and TVC, based on Eqs. (6.12) and (6.13), respectively, have been calculated for Q = 1 to 16 and presented in Table 6.1. The TFC, TVC and TC have been graphically presented in Fig. 6.4. As the figure shows, TFC remains fixed for the whole range of output, and hence, takes the form of a horizontal line—TFC. The TVC curve shows that the total variable cost first increases at a decreasing rate and then at an increasing rate with the increase in the output. The rate of increase can be obtained from the slope of TVC curve. The pattern of change in the TVC stems directly from the law of increasing and diminishing returns to the variable inputs. As output increases, larger quantities of variable inputs are required to produce the same quantity of output due to diminishing returns. This causes a subsequent increase in the variable cost for producing the same output. Table 6.1: Cost-Output Relations Q FC TVC TC AFC AVC AC MC (1) (2) (3) (4) (5) (6) (7) (8) 0 10 0.0 10.00 - - - - 1 10 5.15 15.15 10.00 5.15 15.15 5.15 2 10 8.80 18.80 5.00 4.40 9.40 3.65 3 10 11.25 21.25 3.33 3.75 7.08 2.45 4 10 12.80 22.80 2.50 3.20 5.70 1.55 5 10 13.75 23.75 2.00 2.75 4.75 0.95 6 10 14.40 24.40 1.67 2.40 4.07 0.65 7 10 15.05 25.05 1.43 2.15 3.58 0.65 8 10 16.00 26.00 1.25 2.00 3.25 0.95 9 10 17.55 27.55 1.11 1.95 3.06 1.55 (Contd...)  10 10 20.00 30.00 1.00 2.00 3.00 2.45 11 10 23.65 33.65 0.90 2.15 3.05 3.65 12 10 28.80 38.80 0.83 2.40 3.23 5.15 13 10 35.75 45.75 0.77 2.75 3.52 6.95 14 10 44.80 54.80 0.71 3.20 3.91 9.05 15 10 56.25 66.25 0.67 3.75 4.42 11.45 16 10 70.40 80.40 0.62 4.40 5.02 14.15 From Equation. (6.12) and (6.13), we may derive the behavioural equations for AFC, AVC and AC. Let us first consider AFC. Average Fixed Cost (AFC). As already mentioned, the costs that remain fixed for a certain level of output make the total fixed cost in the short-run. The fixed cost is represented by the constant term ‘a’ in Eq. (6.11) and a = 10. We know that TFC AFC =....(6.14) Q Substituting 10 for TFC in Eq. 6.14, we get 10 AFC =....(6.15) Q Eq. (6.15) expresses the behaviour of AFC in relation to change in Q. The behaviour of AFC for Q from 1 to 16 is given in Table 6.1 (col. 5) and presented graphically by the AFC curve in Fig. 6.5. The AFC curve is a rectangular hyperbola. TVC Average Variable Cost (AVC). As defined above, AVC = Q Given the TVC function (Eq. 6.13), we may express AVC as follows. 6Q − 0.9Q 2 + 0.05Q3 AVC = = 6 – 0.9Q + 0.05Q2...(6.16) Q Having derived the AVC function in Eq. (6.16), we can easily obtain the behaviour of AVC in response to change in Q. The behaviour of AVC for Q = 1 to 16 is given in Table 6.1 (col. 6), and graphically presented in Fig. 6.5 by the AVC curve. Fig. 6.5: Short-run Curves Critical Value of AVC. From Eq. (6.10), we may compute the critical value of Q in respect of AVC. The critical value of Q (in respect of AVC) is one that minimizes AVC. The AVC will be minimum when its (decreasing) rate of change equals zero. This can be accomplished by differentiating Eq. (6.16) and setting it equal to zero. Thus, critical value of Q can be obtained as ∂AVC Critical value of Q= = – 0.9 + 0.10 Q = 0 ∂Q 0.10 Q = 0.9 Q= 9 In our example, the critical value of Q = 9. This can be verified from Table 6.1. The AVC is minimum (1.95) at output 9. TC. Average Cost (AC). The average cost (AC) is defined as AC = Q Substituting cost function given in Eq. (6.12) for TC in the above equation, we get 10 + 6Q − 0.9Q 2 + 0.05Q3 AC = Q 10 = + 6 – 0.9Q + 0.05Q2...(6.17) Q The Eq. (6.17) gives the behaviour of AC in response to change in Q. The behaviour of AC for Q = 1 to 16 is given in Table 6.1 and graphically presented in Fig. 6.5 by the AC curve. Note that AC curve is U-shaped. Minimization of AC. One objective of business firms is to minimize AC of their product or, which is the same as, to optimize the output. The level of output that minimizes AC can be obtained by differentiating Eq. 6.17 and setting it equal to zero. Thus, the optimum value of Q can be obtained as follows. ∂AC 10 = − 0.9 + 0.1Q = 0 ∂Q Q 2 When simplified this equation takes the form of a quadratic equation as – 10 – 0.9Q2 + 0.1Q3 = 0 or Q3 – 9Q2 – 100 = 0...(6.18) By solving4 equation (6.18) we get Q = 10. Thus, the critical value of output in respect of AC is 10. That is AC reaches its minimum at Q = 10. This can be verified from Table 6.1. Marginal Cost (MC). The concept of marginal cost (MC) is particularly useful in economic analysis. MC is technically the first derivative of the TC function. Given the TC function in Eq. (6.12), the MC function can be obtained as ∂TC MC = ∂Q = 6 – 1.8Q + 0.15Q2...(6.19) Equation (6.19) represents the behaviour of MC. The behaviour of MC for Q = 1 to 16 computed as MC = TCn– TCn– 1 is given in Table 6.1 (col. 8) and graphically presented by the MC curve in Fig. 6.5. The critical value of Q with respect to MC is 6 or 7. This can be seen from Table 6.1. 6.3.3 Cost Curves and the Law of Diminishing Returns We now return to the law of variable proportions and explain it through the cost curves. Figures 6.4 and 6.5 present the short-term law of production, i.e., the law of diminishing returns. Let us recall the law: it states that when more and more units of a variable input are applied, other inputs held constant, the returns from the marginal units of the variable input may initially increase but it decreases eventually. The same law can also be interpreted in terms of decreasing and increasing costs. The law can then be stated as, if more and more units of a variable input are applied to a given amount of a fixed input, the marginal cost initially decreases, but eventually increases. Both interpretations of the law yield the same information—one in terms of marginal productivity of the variable input, and the other in terms of the marginal cost. The former is expressed through a production function and the latter through a cost function. Figure 6.5 presents the short-run laws of return in terms of cost of production. As the figure shows, in the initial stage of production, both AFC and AVC are declining because of some internal economies. Since AC = AFC + AVC, AC is also declining. This shows the operation of the law of increasing returns. But beyond a certain level of output (i.e., 9 units in our example), while AFC continues to fall, AVC starts increasing because of a faster increase in the TVC. Consequently, the rate of fall in AC decreases. The AC reaches its minimum when output increases to 10 units. Beyond this level of output, AC starts increasing which shows that the law of diminishing returns comes into operation. The MC curve represents the change in both the TVC and TC curves due to change in output. A downward trend in the MC shows increasing marginal productivity of the variable input due mainly to internal economy resulting from increase in production. Similarly, an upward trend in the MC shows increase in TVC, on the one hand, and decreasing marginal productivity of the variable input, on the other. 6.3.4 Some Important Cost Relationships Some important relationships between costs used in analysing the short-run cost- behaviour may now be summed up as follows: (a) Over the range of output AFC and AVC fall, AC also falls because AC = AFC + AVC. (b) When AFC falls but AVC increases, change in AC depends on the rate of change in AFC and AVC. (i) if decrease in AFC > increase in AVC, then AC falls, (ii) if decrease in AFC = increase in AVC, AC remains constant, and (iii) if decrease in AFC < increase in AVC, then AC increase. (c) The relationship between AC and MC is of a varied nature. It may be described as follows: (i) When MC falls, AC follows, over a certain range of initial output. When MC is falling, the rate of fall in MC is greater than that of AC, because in the case of MC the decreasing marginal cost is attributed to a single marginal unit while, in case of AC, the decreasing marginal cost is distributed over the entire output. Therefore, AC decreases at a lower rate than MC. (ii) Similarly, when MC increases, AC also increases but at a lower rate for the reason given in (i) There is, however, a range of output over which the relationship does not exist. Compare the behaviour of MC and AC over the range of output from 6 units to 10 units (see Fig. 6.5). Over this range of output, MC begins to increase while AC continues to decrease. The reason for this can be seen in Table 6.1: when MC starts increasing, it increases at a relatively lower rate which is sufficient only to reduce the rate of decrease in AC—not sufficient to push the AC up. That is why AC continues to fall over some range of output even if MC increases. (iii) MC intersects AC at its minimum point. This is simply a mathematical relationship between MC and AC curves when both of them are obtained from the same TC function. In simple words, when AC is at its minimum,  it is neither increasing nor decreasing: it is constant. When AC is constant, AC = MC. 6.3.5 Output Optimization in the Short-Run The technique of output optimization has already been discussed in unit 5. Optimization of output in the short-run has been illustrated graphically in Fig. 6.5 at the point of interaction of AC and MC. Optimization technique is shown here algebraically by using a TC–function. Let us suppose that a short run cost function is given as TC = 200 + 5Q + 2Q2 …(6.20) We have noted above that an optimum level of output is one that equalizes AC and MC. In other words, at optimum level of output, AC = MC. Given the cost function in Eq. (6.20), 200 + 5Q + 2Q 2 AC = Q 200 = + 5 + 2Q... (6.21) Q ∂TC and MC = = 5 + 4Q... (6.22) ∂Q By equating AC and MC equations, i.e., Eqs. (6.21) and (6.22), respectively, and solving them for Q, we get the optimum level of output. Thus, 200 + 5 + 2Q = 5 + 4Q Q 200 = 2Q Q 2Q2 = 200 Q = 10 Thus, given the cost function (6.20), the optimum output is 10. 6.3.6 Long-Run Cost-Output Relations By definition, long-run is a period in which all the inputs become variable. The variability of inputs is based on the assumption that in the long-run supply of all the inputs, including those held constant in the short-run, becomes elastic. The firms are, therefore, in a position to expand the scale of their production by hiring a larger quantity of all the inputs. The long-run-cost-output relations, therefore, imply the relationship between the changing scale of the firm and the total output, whereas in the short-run this relationship is essentially one between the total output and the variable cost (labour). To understand the long-run-cost-output relations and to derive long-run cost curves it will be helpful to imagine that a long-run is composed of a series of short-run production decisions. As a corollary of this, long-run cost curve is composed of a series of short- run cost curves. We may now derive the long-run cost curves and study their relationship with output. Long-run Total Cost Curve (LTC). In order to draw the long-run total cost curve, let us begin with a short-run situation. Suppose that a firm having only one plant has its short-run total cost curve as given by STC1, in panel (a) of Fig. 6.6. Let us now suppose that the firm decides to add two more plants to its size over time, one after the other. As a result, two more short-run total cost curves are added to STC1, in the manner shown by STC2 and STC3 in Fig. 6.6(a). The LTC can now be drawn through the minimum points of STC1, STC2 and STC3 as shown by the LTC curve corresponding to each STC. Fig. 6.6: Long-run Total and Average Cost Curves Long-run Average Cost Curve (LAC). The long-run average cost curve (LAC) is derived by combining the short-run average cost curves (SACs). Note that there is one SAC associated with each STC. Given the STC1, STC2, STC3 curves in panel (a) of Fig. 6.6 there are three corresponding SAC curves as given by SAC1, SAC2, and SAC3 curves in panel (b) of Fig. 6.6. Thus, the firm has a series of SAC curves, each having a bottom point showing the minimum SAC. For instance, C1Q1 is minimum AC when the firm has only one plant. The AC decreases to C2Q2 when the second plant is added and then rises to C3Q3 after the addition of the third plant. The LAC curve can be drawn through the SAC1, SAC2 and SAC3 as shown in Fig. 6.6 (b) The LAC curve is also known as the ‘Envelope Curve’ or ‘Planning Curve’ as it serves as a guide to the entrepreneur in his plans to expand production. The SAC curves can be derived from the data given in the STC schedule, from STC function or straightaway from the LTC curve5. Similarly, LAC and can be derived from LTC-schedule, LTC function or form LTC-curve. The relationship between LTC and output, and between LAC and output can now be easily derived. It is obvious from the LTC that the long-run cost-output relationship is similar to the short-run cost-output relation. With the subsequent increases in the output, LTC first increases at a decreasing rate, and then at an increasing rate. As a result, LAC initially decreases until the optimum utilization of the second plant and then it begins to increase. These cost-output relations follow the ‘laws of returns to scale’. When the scale of the firm expands, unit cost of production initially decreases, but ultimately increases as shown in Fig. 6.6(b). The decrease in unit cost is attributed to the internal and external economies and the eventual increase in cost, to the internal and external diseconomies. The economies and diseconomies of scale are discussed in the following section. Long-run Marginal Cost Curve (LMC). The long-run marginal cost curve (LMC) is derived from the short-run marginal cost curves (SMCs). The derivation of LMC is illustrated in Fig. 6.7 in which SACs and LAC are the same as in Fig. 6.6(b). To derive the LMC, consider the points of tangency between SACs and the LAC, i.e., points A, B and C. In the long-run production planning, these points determine the output levels at the different levels of production. For example, if we draw perpendiculars from points A, B and C to the X-axis, the corresponding output levels will be OQ1, OQ2 and OQ3. The perpendicular AQ1 intersects the SMC1 at point M. It means that at output OQ1, LMC is MQ1. If output increases to OQ2, LMC rises to BQ2. Similarly, CQ3 measures the LMC at output OQ3. A curve drawn through points M, B and N, as shown by the LMC, represents the behaviour of the marginal cost in the long-run. This curve is known as the long-run marginal cost curve, LMC. It shows the trends in the marginal cost in response to the changes in the scale of production. Some important inferences may be drawn from Fig. 6.7. The LMC must be equal to SMC for the output at which the corresponding SAC is tangent to the LAC. At the point of tangency, LAC = SAC. Another important point to notice is that LMC intersects LAC when the latter is at its minimum, i.e., point B. There is one and only one short-run plant size whose minimum SAC coincides with the minimum LAC. This point is B where SAC2 = SMC2 = LAC = LMC Optimum Plant Size and Long-Run Cost Curves. The short-run cost curves are helpful in showing how a firm can decide on the optimum utilization of the plant—the fixed factor, or how it can determine the least-cost-output level. Long-run cost curves, on the other hand, can be used to show how a firm can decide on the optimum size of the firm. Fig. 6.7 Derivation of LMC Conceptually, the optimum size of a firm is one which ensures the most efficient utilization of resources. Practically, the optimum size of the firm is one which minimises the LAC. Given the state of technology over time, there is technically a unique size of the firm and level of output associated with the least-cost concept. In Fig. 6.7, the optimum size consists of two plants which produce OQ2 units of a product at minimum long-run average cost (LAC) of BQ2. The downtrend in the LAC indicates that until output reaches the level of OQ2, the firm is of less than optimal size. Similarly, expansion of the firm beyond production capacity OQ2, causes a rise in SMC and, therefore, in LAC. It follows that given the technology, a firm aiming to minimize its average cost over time must choose a plant which gives minimum LAC where SAC = SMC = LAC = LMC. This size of plant assures the most efficient utilization of the resource. Any change in output level, increase or decrease, will make the firm enter the area of in optimality.