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7 Cost estimation Outline Objectives 7.1 7.2 254 Introduction Importance of cost estimation for decision-making Types of cost scenario Methodology page 255 255 256 256 Short-run cost estimation Types of empirical study Problems in short-run cost estimation Different forms of cost function, interpretation and selection Implications of empirical studies 259 260 260 7.3 Long-run cost estimation Types of empirical study Problems in long-run cost estimation Different forms of cost function Implications of empirical studies Case study 7.1: Banking 265 266 266 268 268 270 7.4 The learning curve Types of specification Case study 7.2: Airlines Case study 7.3: Electricity generation Application of the learning curve Example of a solved problem Implications of empirical studies 271 271 272 273 275 275 276 7.5 A problem-solving approach Examples of solved problems 277 278 263 265 Cost estimation Summary Review questions Problems Notes 280 280 281 282 Objectives 1 To explain the importance of cost estimation for managerial decision-making. 2 To explain the different methods of cost estimation and their relative advantages and disadvantages. 3 To describe the different types of empirical study which are used in cost estimation. 4 To explain the types of problem which are encountered in statistical cost estimation. 5 To explain how these problems apply in different ways to short-run and long-run situations. 6 To explain how these problems can be overcome. 7 To describe and interpret the different types of cost function in mathematical terms. 8 To explain the specification and estimation of the learning curve. 9 To examine and interpret the findings of various empirical studies. 7.1 Introduction After discussing the theory of demand in Chapter 3 we proceeded to examine the estimation of demand functions in the following chapter. Now that we have discussed the theoretical aspects of production and cost we need to examine the estimation of cost functions. 7.1.1 Importance of cost estimation for decision-making We have already seen in the previous chapter why managers need to understand the nature and application of cost functions in various aspects of decisionmaking. Since these functions are not self-evident, they have to be estimated using an appropriate method. Managers cannot apply their knowledge of the firm’s cost functions unless these have been estimated in the first place. Therefore, cost estimation is important for the same reasons as an understanding of cost theory, that is: 1 To determine the pricing of the firm’s products. 2 To determine the other components of the marketing mix. 255 256 PRODUCTION AND COST ANALYSIS 3 To determine the optimal output of the firm in the short run and the relevant input mix. 4 To determine the optimal scale of operation of the firm in the long run. 5 To determine whether to accept or refuse an order from a potential customer at a particular price. 6 To determine the impact of potential mergers and acquisitions on unit costs. 7.1.2 Types of cost scenario The word scenario is used here to indicate a distinction between a cost function’s time period and a cost function’s mathematical form. Scenario in this context refers to time period, although we shall see that there is some relationship between time period and mathematical form. It is vital for an investigator to determine which type of cost scenario is desired before collecting the data which are necessary to estimate the relevant cost function. This is because different scenarios require different types of data in order to estimate them. Essentially there are three possible main scenarios involved in cost estimation: 1 The short run – discussed in section 7.2. 2 The long run – discussed in section 7.3. 3 The learning curve – discussed in section 7.4. The differences between these scenarios have already been explained in the previous two chapters, but the implications in terms of estimation have still to be discussed. 7.1.3 Methodology As with demand estimation, there are different methods of estimation that can be used. However, once again we find that statistical analysis tends to be the preferred method, with its advantages being essentially the same as for demand analysis. Most of the remainder of this chapter will examine various aspects of statistical cost estimation, but before we move on to this it is helpful to consider two alternative methods of estimation: engineering analysis and the survivor method. a. Engineering analysis This is mainly concerned with estimating physical relationships between inputs and outputs in order to estimate a production function. This in turn can be used to obtain the cost function of the firm, once we know the prices of the inputs, and assuming that the firm produces with economic efficiency. In the short run the production process will often determine how the inputs are combined. Technical factors frequently determine, for example, how many workers can use a particular machine, how many hours a machine can operate in a day, Cost estimation how much raw material and what components are required to manufacture a finished product, how much fuel and power is required, and so on. Often relationships here may be linear, so that, for example, producing twice as much output requires twice as much material; sometimes, however, as with fuel requirements, the relationship may be non-linear; a machine operating twice as fast may require more than twice as much fuel. Having established production relationships, the next step is to price these inputs. This may involve some of the problems mentioned in the previous chapter, in that accounting data may be historical rather than current. Also the data may be aggregated in such a way as to make it difficult to estimate prices for particular types of labour or materials. The final step with this approach is to derive the cost function on the basis that optimal production methods are used. This may also present problems, since there may be some flexibility in terms of input substitution; for example, taking the variable factors in the short run, one material might be substituted for another, such as steel for aluminium in car production. Thus the firm has to determine the least-cost cost function, in other words the cost function that involves producing each output at minimum cost. This is the familiar mathematical problem that we have seen in Chapter 5, where the objective is to minimize total cost subject to a series of output constraints (shown as shifting isoquants). The only difference is that in this case both inputs are variable in the short run. There are three main shortcomings of engineering analysis. 1. Some production processes are highly complex, involving a large number of inputs. This makes it difficult to determine the mathematical model of the relationships between these inputs and output. The problem is even more difficult if there is more than one output and there are interdependencies between these outputs, for example if a plant is producing different models in a product line. 2. No consideration is given to indirect costs. Thus management, administrative and distribution inputs and costs are ignored in the analysis. Since these are also related to output in practice, they can have a considerable impact on the firm’s cost function. 3. There is often no consideration of optimization of the input-output relationships. Existing combinations of inputs and outputs are often used in the analysis, with no regard as to whether these are optimal. In order to determine the optimal combinations the aspect of flexibility of input substitution discussed above must be examined; this can make the analysis considerably more difficult, and may require some costly, and risky, experimentation by the firm in terms of varying its production processes. b. Survivor method This method was originally developed by Stigler,1 and only applies to long-run cost estimation. The method does not rely on the use of accounting data, and therefore avoids the problems arising from the unreliability of such data. The 257 258 PRODUCTION AND COST ANALYSIS method simply involves categorizing the firms in an industry according to size, observing the industry over a relatively long time period, and recording the growth or decline of the different size categories. Some conclusion is then formed regarding the most efficient size of firms in the industry according to which size categories are growing fastest. Stigler applied this technique to the steel industry in the United States, using data from 1930 to 1951, and observed that the medium-size firms (between 2.5 per cent and 25 per cent of industry capacity) had gained most in terms of total market share. He concluded that these firms must be the most efficient and that therefore the industry featured a U-shaped LAC curve. Apart from its avoidance of unreliable accounting data, the other advantage of the survivor method is its simplicity. It does however suffer from three serious problems. 1. It does not estimate costs or unit costs at any level of output, and therefore is not very useful for most aspects of managerial decision-making. One exception may be the consideration of mergers and acquisitions. 2. It assumes that the industry is highly competitive, so that firms that are performing below maximum efficiency will not survive. It thus ignores market power and barriers to entry, discussed in the next chapter. 3. It ignores changing technology and its impact on optimal size. In the steel industry for example it has become easier in recent decades for smaller firms to compete by using smaller plant size with new technology, resulting in unit costs comparable to larger firms. The survivor method may not reveal such a reduction in minimum efficient scale if it has only occurred in the more recent years of the long period under consideration. c. Statistical analysis In view of the problems described above, statistical methods are often used to estimate cost functions. These overcome some of the problems, but are not without problems of their own, as we shall see. Much of the pioneering work in this area was carried out by Dean,2 who also wrote the first textbook on managerial economics. He conducted some of the empirical studies that are discussed later in the chapter. Johnston reviewed and extended this work,3 and much of the material in this chapter is based on their work. The procedure for statistical cost estimation is essentially the same as for demand estimation. The same seven steps are involved: 1 Statement of a hypothesis. An example here might be that a firm’s short-run total cost function is linear. 2 Model specification. This means determining what variables should be included in the cost function and what mathematical form or forms such a relationship should take. These issues are again determined on the basis of economic theory and prior empirical studies; obviously output is the most important explanatory variable. Various alternative models may be specified at this stage, since economic theory is often not robust enough to be definitive regarding the details of the form of model. Cost estimation 3 Data collection. This stage can only be performed after the cost function has been specified, otherwise it is not known for which variables we have to collect data. 4 Estimation of parameters. This means computing the values of the coefficients of the variables in the model. These effects can be measured in different ways, for example in terms of the marginal effects and elasticities already discussed. Clearly we must use some technique to estimate these values and the method of OLS regression will again be used in this context. 5 Selecting the best model. Once several alternative models have been estimated, it is necessary to examine how well each model fits the data and which model fits best. The situation is somewhat different here from the demand situation. In the latter case, as we have seen, economic theory is not usually robust enough for us to specify a mathematical form. However, in cost situations, theoretical considerations may be dominant in selecting suitable forms, as will be seen in the remainder of this chapter. If the fit is not good it may be necessary to return to step 2 and respecify the model before moving on to the next stage. 6 Testing a hypothesis. In the example above this is determined by comparing the goodness of fit, as described in the previous stage. 7 Forecasting. Once the appropriate cost function has been estimated cost forecasts for different outputs can be computed. 7.2 Short-run cost estimation Short-run cost functions assume, as we have seen, that at least one factor is fixed. Thus changes in cost are caused mainly by changes in the level(s) of the variable factor inputs. This has important implications regarding the type of empirical study used. Different measures of cost are sometimes used as the dependent variable, depending on the availability of data; thus the cost function may be specified as: VC ¼ f ðQ Þ (7:1) TC ¼ a þ f ðQ Þ (7:2) AVC ¼ f ðQ Þ=Q (7:3) ATC ¼ ½a þ f ðQ Þ
=Q (7:4) MC ¼ f 0 ðQ Þ (7:5) In each case it is assumed that there is only one explanatory variable, output. Once one particular form of cost function has been estimated, other forms can be obtained mathematically from that form. 259 260 PRODUCTION AND COST ANALYSIS 7.2.1 Types of empirical study Various approaches are possible here. In some cases, experiments can be specifically designed for the purposes of cost estimation, or panel data may be used, as discussed in the chapter on demand estimation. However, as with demand studies, there are two main types of study that can be performed: time-series and cross-section. Each involves different constraints and problems. These are explained briefly here and then expanded in the next subsection. a. Time-series studies In this case the researcher needs data relating to a specific plant or firm over a period of time. However, it is important that the time period involved be relatively short, since in order to qualify as a short-run analysis neither the size of the plant, the scale of the firm nor the technology in use should change. There must be sufficient time periods, and there must also be some variation in production levels from one period to another, for the analysis to be reliable. Otherwise the estimated coefficients of the variables in the equation will have large standard errors. To obtain sufficient observations in the short time period such observations may have to be made at monthly intervals, or even more frequently. This may pose problems, which are discussed later. b. Cross-section studies A researcher can estimate a short-run cost function for an industry by examining different firms of the same scale. However, a firm can only use a crosssection study to estimate its particular cost function if it has several plants. Since managers are usually concerned with the analysis of firm-specific data, this places a constraint on the use of cross-section studies. If the firm does operate several plants, these plants must be of the same size, and use the same technology, in order for the study to qualify as a short-run analysis. Furthermore, these plants must all be producing the same product, and involve the same stage of the production process, in other words the firm must be horizontally integrated. Again, for the results to be reliable, there must be some variation between the outputs of the different plants. Given the constraints described above, timeseries studies are more commonly used for short-run cost estimation. 7.2.2 Problems in short-run cost estimation In Chapter 4 on demand estimation it was seen that there were a number of problems related to the estimation process. There are again problems with cost estimation, but they tend to be of a different nature. The main problematic area is in data collection and measurement, as explained below. a. Dynamic environment If time-series analysis is used it must be recognized that the cost function is really a dynamic relationship that is changing over time. Thus there is an Cost estimation identification problem similar to the situation discussed under demand estimation. This means that each observation could relate to a different cost function if care is not taken to keep other factors equal; if other factors cannot be kept equal then certain adjustments must be made to the values of the variables recorded. One of the most important factors that needs to be kept equal is the product quality; this is more difficult to achieve with a time-series study, as quality improvements may be being made unknown to the researcher. This problem of a dynamic environment is explained in more detail in conjunction with the next problem. b. Use of accounting data Researchers are inevitably constrained to using data collected and recorded by the firm’s accountants. As already seen in the last chapter, this information has been collected and recorded for different purposes from those of the researcher: to conform to legal requirements and externally imposed accounting standards, and for other external uses like providing shareholder information. Managers who want to use the data for internal purposes, therefore, are faced with a number of problems. The principles involved are the same as those discussed in the previous chapter: the relevant costs for decision-making purposes are economic costs rather than accounting costs, current costs rather than historical costs, and incremental rather than sunk costs. The application of these principles to cost estimation involves the following specific aspects. 1. Adjustment for changes in prices. Prices of labour, materials and other inputs must be adjusted so that current prices are used. The best way of doing this is to use a specific cost index for the relevant input. Say, for example, that labour costs were recorded as being £10,000 for a particular time period, and that wage rates had increased by 12 per cent since then; these labour costs should be recorded at the estimated current costs of £11,200. If a specific cost index is not available, a general price index like the wholesale price index can be used (this is preferable to the retail price index, as being more relevant to input prices). 2. Measurement of depreciation. Accountants often measure depreciation on an essentially arbitrary basis from an economist’s viewpoint, for example to perform easier calculations (straight-line method), or to reduce tax liability. This means that depreciation is often calculated on a time-related basis, whereas economists are interested in the usage-related aspect of depreciation. Furthermore, accountants calculate depreciation based on historical cost, whereas economic depreciation should be based on replacement value. c. Multiproduct firms We have assumed up to this point that the firm is producing a single product. If a firm produces different products from different plants cost estimation may 261 262 PRODUCTION AND COST ANALYSIS not be too difficult; different cost functions can be estimated for each product from each plant. There is still a problem of overhead allocation, discussed below, but the situation is still more simple than if more than one product is produced from a single plant, as is often the case with a product line. There are two possible approaches to this problem. 1. Combination of products into a single output variable. This is most easily explained by means of an example. Say that a firm produces two models of car, model A and model B. In a particular time period a plant may produce 2,000 of A and 3,000 of B. Since the models may have different prices according to the different production costs involved, it may be preferable to take these prices into account in measuring total output, rather than just adding the crude output figures together. Thus if A is priced at £8,000 and B is priced at £12,000, the total value of output is £52 million for the period. This can then be divided by an average price of £10,000 to obtain a proxy or combined measure of output of 5,200 units. This figure can then be compared with outputs from other periods, provided that once again allowance is made for any changes in prices, this time of the outputs. 2. Estimating separate cost functions for different products. The main problem here is the allocation of fixed overhead costs. Accountants again tend to allocate these on an arbitrary basis, sometimes on the basis of the net realizable value of the different products. Net realizable value is usually defined as revenues minus direct costs. This basis for allocating overheads causes problems for the estimation of cost functions because it treats a cost category as being variable when it is really fixed. The result is that this method can result in nonsensical cost functions with negative average variable costs, as shown by Dobbs.4 The problem of estimating cost functions becomes even more serious when there are cost interdependencies among the different products. d. Timing of costs It has already been stated that in a time-series approach to short-run cost estimation the total time period for the study must be of relatively short duration, and this can necessitate the use of short time intervals, such as a month or less, for observations in order to obtain sufficient observations. The lower time limit for these intervals is set by the accounting periods for the firm. Costs, or at least some costs, may not be recorded on a weekly basis. However, there is another problem related to short observation intervals, and this concerns spillover effects. There is a danger of recording costs in the wrong time period; for example, billing dates may not match production and usage. Some costs may appear before or after the period to which they relate. In addition to this, some costs can be scheduled according to convenience rather than production; this applies particularly to maintenance, which is often scheduled for quiet periods when production is low. In reality this maintenance cost is related to previous periods when production levels were higher. Only Cost estimation when researchers are very knowledgeable regarding production activity and the accounting practices of the firm are they likely to be able to avoid these pitfalls. In addition to the problems described above, it should also be noted that the function being estimated is based on actual costs that have been incurred by the firm(s) in the study; if the firm or firms have not been operating efficiently the cost function estimated will not be an optimal cost function. It will tend to overstate the ‘true’ values of the cost parameters. 7.2.3 Different forms of cost function, interpretation and selection a. Forms We saw in the previous chapter that cost functions can take a variety of forms in the short run. The polynomial form is the most common form, consisting of linear, quadratic and cubic functions. In each case the relationships between total cost, average cost and marginal cost were examined. Therefore, as stated earlier in this chapter, it does not matter which of these is specified in the cost model, since the other measures can be calculated mathematically from the measure specified. This is briefly reviewed below. The marginal cost and average variable cost relationships have been seen to have a U shape in many situations, because of increasing and then diminishing returns. This translates mathematically into a quadratic unit cost function of the general form: MC ¼ b þ cQ þ dQ 2 (7:6) We can then obtain the TC function by integration, producing the following cubic: TC ¼ a þ bQ þ ðc=2ÞQ 2 þ ðd=3ÞQ 3 In this case (7:7) FC ¼ a and VC ¼ bQ þ ðc=2ÞQ 2 þ ðd=3ÞQ 3 Thus AFC ¼ a=Q and AVC ¼ b þ ðc=2ÞQ þ ðd=3ÞQ 2 (7:8) and ATC ¼ a=Q þ b þ ðc=2ÞQ þ ðd=3ÞQ 2 (7:9) Alternatively, a particular form of TC function may be specified; in this case the MC function can be derived by differentiation. It can be easily seen that if the TC function is the quadratic: TC ¼ a þ bQ þ cQ 2 (7:10) 263