Quantitative Techniques for Competition and Antitrust Analysis_1 pptx

128 3. Estimation of Cost Functions Investment Fixed proportion depreciation Life Value of capital Life Depreciation Accelerated depreciation (a) (b) Figure 3.1. Depreciation schedules. Source: Tom Stoker, MIT Sloan. With a constant depreciation rule, the accounting cost of capital will be Cost of capital t D ıK t ; where K t is the original capital investment. An economist, on the other hand, would ideally define the user cost of capital (UCC) as the opportunity cost of the capital employed (whether financed by debt or equity) plus economic depreciation: UCC t D Opportunity cost t C Economic depreciation t : The opportunity cost will be an appropriate interest rate times the amount of capital employed so that Opportunity cost t D rV t ; where V t is the value of the capital good at time t . An “appropriate” interest rate r will control for risk since not all investments are equally risky. The question then arises about what we mean by an “appropriate” interest rate. One popular answer to that question is to use the weighted average cost of capital (WACC) for the firm. 5 5 At its most basic the WACC takes the various sources of funds (usually debt and equity, but there can be different types of debt and equity: senior and subordinated debt or ordinary and preference shares) and takes the weighted average return required for each source of funds where the weights allocated to each required return are the proportions of debt and equity. An important complication emerges in that tax treatments can differ by source of funds, in particular, in many jurisdictions corporate taxes are paid on profits after interest is deducted as an expense, which means interest is accounted for as an expense while taxes are due on the returns to equity. While we may determine the weights in a WACC calculation by the amount of various sources of funds, the underlying return on each source of funds must, of course, 3.1. Accounting and Economic Revenue, Costs, and Profits 129 The second component of the user cost of capital is economic depreciation, which can be calculated as the (expected) change in the value of the asset over the period of use: Economic depreciation t D V t  V tC1 : The difference between economic depreciation and accounting depreciation can be a source of substantial differences between economic and accounting profits. In fact, there are many accounting methods used to “write off” capital. In general, the firm deducts each year a share of the value of the investment either at a constant or decreasing rate (see figure 3.1). The choice of the method used to write off the capital can potentially have an enormous impact on the annual cost figures and hence result in substantial reallocations of profits across periods. Accounting depreciation is rarely negative, but economic depreciation certainly can be when assets appreciate in value. To illustrate the differences, consider a firm which buys a new car. An accounting treatment might calculate the depreciation charge by writing off the value of the investment using a straight-line depreciation charge over ten years, so that depreci- ation would be one tenth of the purchase price each year. The economist, however, when looking at the economic depreciation might go to a price book for second- hand cars and compare today’s price differential between a new car and the identical model of car which is one year old. Doing so would give you an estimate of the economic depreciation—the decline in value of the asset—that results from holding it for one year. Using a 2007 Belgian car magazine, we see, for instance, that a new Volkswagen Passat Variant Comfortline costs €28,050 while a similar one-year-old model can be found for €21,000. 6 We can calculate the economic cost or the UCC as UCC t D Opportunity cost t C Economic depreciation t D rV t C .V t  V tC1 / D r 28;050 C .28;050  21;000/; where V is the market value of the capital good and r is often measured using the WACC. 7 With r D 10% the user cost of capital is €9,855. be calculated. Debt costs can often be obtained from accounting statements while obtaining those for equity leads us toward methods such as those associated with the CAPM (see, for example, White et al. 2001). 6 Le Moniteur Automobile, September 2007. 7 Doing so involves calculating a weighted average of the cost of equity and the cost of debt according to their respective participation in the value of the firm. WACC D .D=V /.1  t/r d C .E=V /r e , where D=V and E=V are, respectively, the ratio of debt and equity to the value of the firm, r d is the cost of debt, r e is the cost of equity, and t is the marginal corporate tax rate (assuming that tax is not paid on debt). Authors often use the book-value of debt for D and the market value of equity (number of shares outstanding times share price) for E, while by definition V D E C D. The cost of debt r d can be obtained as the ratio of interest expenses to debt for a given firm, whereas the cost of equity is often obtained from an asset pricing model, although within the context of a case it may also come from company documents. 130 3. Estimation of Cost Functions One can also perform the equivalent calculation: UCC t D .r CDepreciation rate/V t ; where Depreciation rate D V t  V tC1 V t D 28;050  21;000 28;050 D 0:25133; UCC t D .0:10 C 0:25133/  28;050 D 9;855: If the new car is expected to last five years, the accounting cost of the firm for the first year might be 28;050=5 D €5;610. The economic costs will nevertheless be €9,855 given the rapid decline in the market value of the car during its first year of usage, a percentage depreciation rate of 25.1%. 3.1.2 Comparing Costs and Revenues: Discounted Cash Flows Sometimes, we want to compare the cost of an investment with the value of its expected return. A common method to calculate the flow of revenues is to calcu- late the discounted cash flow generated by the investment. This means calculating the present value of the future revenue stream generated by the current capital expenditure. The discounted cash flow is calculated as follows: DCF D T X tD1 R t .1 C r/ t C FV T ; where R t is the revenue generated by the investment at time t , r is the discount rate, which is normally the cost of capital of the firm, and FV T is the final valuation of the investment at the end period of the project T . Discounted cash flows can be used to compare the value of revenue streams with the value of the cost streams when the time paths are different. In competition inves- tigations such calculations are commonplace. For example, they will be useful when evaluating prices that are cost reflective in investigations of industries where produc- tion involves investment in costly durable capital goods (e.g., telecommunications, where firms invest in networks). Another example involves the investigation of pre- dation cases, where many jurisdictions use a “sacrifice” or “recoupment” test. The idea is that a dominant firm charging a low price today to drive out a rival is fol- lowing a strategy that involves a sacrifice of current profits. The idea of the sacrifice test is that such a sacrifice would only be rational if it were followed by sufficiently higher profits in the future. 3.2. Estimation of Production and Cost Functions 131 3.2 Estimation of Production and Cost Functions The traditional estimation of cost and production functions can be a complex task that raises a number of difficult issues. In tandem with obtaining appropriate data, one must combine a sound theoretical framework that generates one or more esti- mating equation(s) with appropriate econometric techniques. We introduce the main empirical issues in cost estimation and proceed to discuss some seminal illustrative examples which help illustrate both the problems and usefulness of the approach. 3.2.1 Principles of Production and Cost Function Estimation The theory of production and cost functions and the empirical literature estimating them is a significant body of literature. Chapter 1 in this volume covers the basic theoretical framework underlying the empirical estimation of cost functions. We review here the basic conclusions of that discussion and then proceed to present some practical examples of cost function estimation since that is undoubtedly the best way to see how such exercises can be done. 3.2.1.1 Theoretical Frameworks and Data Implications Intuitively, costs simply add up. However, as we will see, that simple picture is often complicated because firms have input substitution possibilities—they can some- times, for example, substitute capital for labor. Substitution possibilities mean that we have to think harder than simply adding up a firm’s costs for the inputs required to produce output. To see why, let us consider a case when costs do just add up because there are no substitution possibilities, namely the case of producing according to a fixed recipe. To fix ideas, let us consider an example. To produce a cake, suppose we need 1 kg of flour, six eggs, a fixed quantity of milk, and so on. Ignoring divisibility issues, the cost of producing a cake may simply be the sum of the prices of the ingredients times the quantities they are required in. A fixed-proportions production function has the form Q D min  I 1 ˛ 1 ; I 2 ˛ 2 ;:::; I n ˛ n  ; where I 1 ;I 2 ;:::;I n are inputs such as flour, eggs, and milk, while the parameters ˛ 1 ;:::;˛ n describe the amount of each input required to produce a single cake. So if we require 1 kg of flour and six eggs ˛ 1 D 1 and ˛ 2 D 6 and the ratio 1 6 I 2 tells us the number of cakes that we have enough eggs to make. However, now suppose that some capital and labor are required to produce the cake. We could either have a small amount of labor and a cake-mixer or a large amount of labor and a spoon. In this case, we have capital–labor substitution possibilities and depending on the relative prices of capital and labor our cake producer may choose to use them in different proportions. The “fixed-proportions” production function may therefore 132 3. Estimation of Cost Functions suffice as a model for the materials piece of the production function, but we will require a production function that embeds that piece into a full production function which allows for substitution possibilities in capital and labor. In general, we describe a production function as Q D f.I 1 ;:::;I m I˛ 1 ;:::;˛ m /; where I 1 ;I 2 ;:::;I m are inputs such as labor, capital, and other materials, and the alphas are parameters. Probably the most famous production function that captures an ability to produce output using capital and labor in different proportions is the Cobb–Douglas production function (Cobb and Douglas 1928): Q D ˛ 0 L ˛ 1 K ˛ 2 : First note that a Cobb–Douglas production function requires that each firm must have at least some capital and also some labor if it is to produce any output. Second, we note that when writing down an econometric model, we will often suppose that at least one of the “inputs” is a variable which is not observed by the investigator. For clarity of exposition we distinguish the observed and unobserved inputs by introducing an “input” variable over which, in the simplest (static) version of these theories, the firm is typically assumed to have no choice. 8 This unobserved “input” will become our econometric error term and is sometimes described as measuring firms’(totalfactor) “productivity.” We shall denote a firm’s (total factor) productivity by u and ˛ D .˛ 1 ;:::;˛ m /. Given a production function, we may describe the minimum cost of producing a given level of output as the solution to C.QI˛; u/ D min I 1 ;:::;I m p 1 I 1 CCp m I m subject to Q 6 f.I 1 ;:::;I m ;uI˛/; where Q 6 f.I 1 ;:::;I m ;uI˛/ describes the fact that the amount of output must be less than that feasible according to the production function. Describing the costs of producing output in this way makes it rather clear that costs and technological possibilities—as encapsulated in the production function— are rather fundamentally related. This interrelation is discussed in appropriate depth in chapter 1. That fact has important implications for both the theorist and the researcher interested in eliciting information about the cost structure of firms in an industry, namely that such information can be obtained in several ways. If we want to learn about the way costs vary with output, we can either examine a cost function directly or instead learn about the production function and estimate costs indi- rectly. Finally, readers may recall that there is a relationship between cost functions and input demand equations, via Shephard’s lemma, which states that under some- times reasonable assumptions, the input demands which solve the cost-minimization 8 For a model in which the firms do make investments to boost their productivity, see Pakes and Maguire (1994). 3.2. Estimation of Production and Cost Functions 133 program can be described as I j D @C.Q; p 1 ;:::;p n I˛; u/ @p j : Thus input demand equations and the cost function are also intimately related and, as a result, much information about technology can also sometimes be inferred from estimates of input demand equations. An extremely important fact for the investigator is that each of these three approaches to understanding costs requires somewhat different variables to be in our data sets. For example, to empirically estimate a production function such as Q D f.I 1 ;:::;I m ;uI˛/; where I 1 ;I 2 ;:::;I m are inputs such as labor, capital, and other materials, we need data on input quantities for different levels of quantity produced Q. A cost function, on the other hand, will relate the minimum possible cost to the quantity produced and input prices so will take the form C D C.Q;p 1 ;:::;p m ;uI˛/: Input demand schedules relate the optimal demand for inputs to the quantity pro- duced and the input prices I j D D i .Q; p 1 ;:::;p m ;uI˛/. Shephard’s lemma makes it clear that the input demand approach contains information about the first deriva- tives of the cost function rather than the level of costs. For that reason, not all information about costs can be inferred from input demands. Before discussing some empirical applications, we first discuss four substantive issues that must be squarely faced by the investigator when attempting to learn about costs or technology using econometrics. Each is introduced here and subsequently further explored below. 3.2.1.2 Empirical Issues with Cost and Production Estimation There are four issues that are likely to arise in cost or production function estimation exercises: endogeneity, functional form, technological change, and multiproduct firms. First we note that in each of the three estimation approaches described above, we may face a problem with endogeneity. To see why, consider a data set consisting of a large number of firm-level observations on outputs and inputs and suppose we are attempting to estimate the production function Q D f.I 1 ;:::;I m ;uI˛/. For OLS estimation, even if the true model is assumed linear in parameters and the unobserved (productivity) term is assumed additively separable, productivity must not be correlated with the independent variables in the regression, i.e., the chosen inputs. We will face an endogeneity problem if, for example, the high-productivity firms, those with high unobserved productivity u, also demand a lot of inputs. On 134 3. Estimation of Cost Functions the one hand, according to the model, the efficient firms may require fewer inputs to produce any given level of output. On the other hand, and probably dominating, we will expect the efficient firms to be large—they are the ones with a competitive advantage. As a result, efficient firms will tend to be both high productivity and use high levels of inputs. These observations suggest that the key condition required for OLS to provide a consistent estimator will not be satisfied. Namely, OLS requires u i and I j to be uncorrelated but these arguments suggest they will not be. If we do not account for this endogeneity problem, our estimate of the coefficient on our endogenous input will be biased upward. 9 To solve this problem by instrumental variable regression we would need to find an identifying variable that can explain the firm’s demand for the input but that is not linked to the productivity of a firm. Recent advances in the production function estimation literature have included the methods described in Olley and Pakes (1996), who suggest using investment as a proxy for productivity and use it to control for endogeneity. 10 Levinsohn and Petrin (2003) suggest an alternative approach, but in an important paper Ackerberg et al. (2006) critique the identification arguments in those papers, particularly Levinsohn and Petrin (2003), and suggest alternative methodologies. A second consideration is that we must carefully specify the functional form to take into account the technological realities of the production process. In particular, the functional form needs to reflect the plausible input substitution possibilities and the plausible nature of returns of scale. If we are unsure about the nature of the returns to scale in an industry, we should adopt a specification that is flexible enough to allow the data to determine the existence of scale effects. It is not uncommon to impose restrictions such as requiring the production function have the same returns to scale over the whole range of output and such potentially restrictive assumptions should only be made when deemed reasonable over the data range of the analysis. Overly flexible specifications, on the other hand, may produce estimated cost or production functions that do not behave in a way that is plausible, for example, by producing negative marginal costs. The reason is that data sets are often limited and unable to identify parameters in overly flexible specifications. Clearly, we want to use any actual knowledge of the production process we have before we move to estimation, but ideally not impose more than we know on the data. Third, particularly when the data for the cost or production function estimation come from time series data, we will need to take into account technological change going on in the industry—and therefore driving a part of the variation in our data. Technological progress will result in new production and cost functions and the cost 9 In fact, this intuition, discussed in chapter 2, is really only valid for the case of a single endogenous input. If we have multiple endogeneity problems, establishing the sign of the OLS bias is unfortunately substantially harder. 10 While capital stock is already in a production function, investment—the change in capital stock—is not, at least provided that the resulting capital stock increases only in the next period. 3.2. Estimation of Production and Cost Functions 135 and input prices associated with the corresponding output cannot therefore immedi- ately be compared over time without controlling for such changes. For this reason, one or more variables attempting to account for the effect of technological progress is generally included in specifications using time series data. Clearly, with a cross section of firms there is less likely to be a direct problem with technological progress but, equally, if firms are using different technologies or the same technologies with different level of aptitude, then it would be important to attempt to account for such differences. When the firms involved produce more than one product or service, costs and inputs can be hard to allocate to the different outputs and constructing the data series for the different products may turn into a challenge. Estimating multiproduct cost or production functions will also further complicate the exercise by increasing the number of parameters to estimate. Of course, such efforts may nonetheless be well worthwhile. In the next sections, we use well-known estimation examples to discuss these and other issues as they are commonly encountered in actual cost estimation exercises. 3.2.2 Practical Examples of Cost Function Estimation Numerous empirical exercises have shown that cost functions can be used to estimate the technological characteristics of the production process and provide information about the nature of technology in an industry. The estimation of cost functions is sometimes preferred to other approaches since, at its best, it subsumes all of the relevant information about production into a single function which is very familiar from our theoretical models. Doing so can of course be done only in cases where firms behave in the manner assumed by the model: they must minimize costs and they must typically be price-takers in the input markets (see the discussion in chapter 1). In what follows we provide a discussion of two empirical exercises. The examples presented are not meant to be comprehensive or to reflect the state of the art in the literature, but rather to introduce the rationale of the methodology and point to the econometric issues that are likely to arise. We also hope that they provide a solid basis for going on to explore more advanced techniques. 3.2.2.1 Estimating Economies of Scale A wonderful empirical example of an attempt to estimate economies of scale using a cost function is the classic study on the U.S. electric power generation industry by Nerlove (1963). He calculated a baseline regression model derived from the common Cobb–Douglas production function, Q D ˛ 0 L ˛ L K ˛ K F ˛ F u, where Q, K, L, and F denote output, capital, labor, and fuel, respectively: ln C D ˇ 0 C ˇ Q ln Q Cˇ L ln p L C ˇ K ln p K C ˇ F ln p F C V: 136 3. Estimation of Cost Functions It can be shown that a Cobb–Douglas production function implies a cost function of the form C D kQ 1=r p ˛ L =r L p ˛ K =r K p ˛ F =r F v; where v D u 1=r , r D ˛ L C ˛ K C ˛ F , and k D r.˛ 0 ˛ ˛ L L ˛ ˛ K K ˛ ˛ F F / 1=r . The parameter r can be interpreted as the degree of economies of scale (see the discussion below). The model restricts the economies of scale to be constant for all quantities. Taking a natural log transformation of this cost function, Nerlove obtained an equation which is linear in the parameters and can be easily estimated using standard regression packages: ln C D ˇ 0 C ˇ Q ln Q Cˇ L ln p L C ˇ K ln p K C ˇ F ln p F C V; where ˇ 0 D ln k, ˇ Q D 1=r, ˇ L D ˛ L =r, ˇ K D ˛ K =r, ˇ F D ˛ F =r, and V D ln v. The cost equation above is an unrestricted model, i.e., there are no restrictions imposed on the parameters of the cost function. On the other hand, cost functions in theory are expected to satisfy some conditions. For example, Nerlove imposes the theoretical “homogeneity” restriction, that cost functions should be homogeneous of degree 1 in input prices, before estimating the equation. 11 That is, he imposes ˇ L C ˇ K C ˇ F D 1; which is equivalent to ˇ K D 1  ˇ F  ˇ L : With modern computers we could just estimate the restricted model by telling our regression package to impose the restriction directly. Nerlove, on the other hand, at the time estimated an unrestricted formulation of the restricted model: ln C  ln p K D ˇ 0 C ˇ Q ln Q Cˇ F .ln p F  ln p K / C ˇ L .ln p L  ln p K / C V: The restriction results in one parameter less to be estimated, namely ˇ K , which can be inferred from the other parameters. In practice, intuitively, this may be helpful if such variables as the capital price data are noisy, making estimation of an unrestricted ˇ K difficult. On the other hand, the parameter restriction has not actually removed the price of capital from the equation since that price is now used to normalize the other input prices and cost. Thus such an argument, while intuitive, does rely rather on the idea that there remains enough information in the relative prices (log differences) to infer ˇ L and ˇ F even though we have introduced measurement error in each of the relative price variables remaining in the equation. Nerlove estimates the model using the OLS using cost and input price data from 145 firms in 1955. His results are presented in table 3.1. As we have described, OLS is only an appropriate estimation technique for cost functions under strong assumptions regarding the unobserved efficiencies of the firm, particularly that they be conditional mean uncorrelated with choice of quantity 11 If, for example, we double the price of all inputs, the total cost of producing the same level of output will also double. 3.2. Estimation of Production and Cost Functions 137 Table 3.1. Nerlove’s cost function estimation results. Variable Parameter jt j-Statistic ln Q 0.72 (41.33) .ln p L  ln p K / 0.59 (2.90) .ln p F  ln p K / 0.41 (4.19) Constant 4.69 (5.30) R 2 0.927 — Source: Estimation results from the model presented in Nerlove (1963). The dependent variable is ln C  ln P K . Estimated from data from 145 firms during 1955. The full data set is made available in the original paper. produced. Note that Nerlove’s initial estimates suggest rather surprisingly that ˇ K D 1  0:59  0:41 D 0, a matter to which we shall return. We can retrieve a measure of economies of scale S as follows: S D  @ ln C @ ln Q à 1 D .0:72/ 1 D 1:39 > 1: As S>1, we conclude that the production function exhibits economies of scale. To see why, consider that @ ln C @ ln Q D Q C @C @Q D MC AC ; so that S D  @ ln C @ ln Q à 1 D AC MC : Thus, the estimated cost function implies S>1so that AC > MC, i.e., AC is declining so that there are economies of scale. A log-linear cost function’s diseconomies or economies of scale are a global property of the cost function and, as such, do not depend on the exact level of output being considered. We will see below that with more general cost functions, the value of S will depend on the level of output. Figure 3.2 shows Nerlove’s data (in natural logs) and also the estimated costs as a function of output. Note that the model involves prices so the fitted values are not plotted as a simple straight line. A basic specification check of any estimated regression equation involves plotting the residuals of the estimated regression. The residual is the difference between the actual and the estimated “explained” variable. For consistent estimation using OLS, the residuals need to have an expected value conditional on explanatory variables of zero. In figure 3.3, it is apparent that residuals are dependent on the level of output which violates the requirement for OLS to generate consistent estimates. At both low and high levels of output, the residuals are positive so that true cost is [...]... Cost information can be highly relevant for a diverse range of competition policy and regulatory investigations For example, cost information can shed light on margins and efficiencies Firm’s accounting and financial data can provide useful information about both costs and profits However, the data may need to be carefully adjusted in order to correctly represent economic rather than accounting costs and/ or... chapter, we first explain the main concepts used in market definition and then go on to explore quantitative methods that are used to define the relevant market(s) for a competition investigation We will review different methods in order of complexity, starting with the use of price correlations, survey techniques, shock analysis, and formal and semiformal tests such as diversion ratio analysis, critical loss... engineers familiar with the planning and design of plants and produces direct and detailed industry specific data (see also Stennek and Verboven 2001) As the name suggests, the objective is to determine the shape of the cost function or the nature of the production function by collecting specific and detailed information first hand from people knowledgeable of the cost and scale implications of their businesses... discussion in Joskow and Kahn (2001), who note that during the summer of 2000 wholesale electricity prices in California were almost 500% higher than they were in the same months in 1998 or 1999 See also Borenstein et al (2002) If supply and demand are inelastic and supply is less than demand, then prices will skyrocket 3.2 Estimation of Production and Cost Functions 143 3.2.2.2 Estimating Scale and Scale Effects... proponents of SFA and other model-based frontier methods) For example, the proponents of SFA initially observed that DEA methods did not appear to have a statistical foundation, while more recent authors have shown that is incorrect and it is now standard to calculate standard errors and confidence intervals for DEA models using a class of methods called the “bootstrap” (see Simar and Wilson 1998).24... concentration) industries (see Sutton 1991) If, for example, price competition is very intensive, we will tend to find high concentration For intuition, think about the incentives for a second firm to enter a market if, when she enters, the two firms will play a pure Bertrand equilibrium Knowing she will face very intense price competition, a potential entrant who must pay some form of entry cost would never actually... analysis for firm k, Âk , could be constructed by solving the minimization problem: Â; min 1 ;:::; J ˇ n X ˇ qk ¡ 6 ˇ  i D1 i qi I Ij k > n X i Ij i ; j D 1; : : : ; J I i D1  > 0I i > 0; i D 1; : : : ; n : To understand this minimization problem first note that the observed data are the inputs and output levels for each firm and that the nonnegative weighted sums Pn Pn iD1 i qi and iD1 i Ij i for j D... / for j D 1; : : : ; J inputs for firm k, the superscript “obs” denotes that the variable is observed data and we retain our earlier notation for outputs and inputs, an efficient cost frontier 20 If the reader would like a data set to try such an exercise on, one is provided in table 1 of Thanassoulis (1993) That data set relates to fifteen hypothetical hospitals and was also used in Sherman (1984) and. .. its price above that of close substitutes and competition between firms will ensure that its price is driven down close to its cost Thus market definition for competition policy purposes is directly related to the concept of market power Indeed, a common description of a competition policy market is one which is “worth monopolizing.” Before presenting the tools for the empirical investigation of market... when calculating standard errors, the measures of uncertainty associated with our parameter estimates Specifically, a conventional formula will assume homoskedasticity and will generate inconsistent estimates of the standard errors even though we have consistent estimates of the parameters themselves Fortunately, it is usually possible to construct heteroskedasticity consistent standard errors (HCSEs), . Production and Cost Functions 14 7 C(s 21 q 21 , s 22 q 22 ) C(s 11 q 11 , s 12 q 12 ) C( q ) Costs q 2 q 1 (s 11 q 11 , s 12 q 12 ) + (s 21 q 21 , s 22 q 22 ) = (q 1 , q 2 ) (s 11 q 11 , s 12 q 12 ). (billion kWh) 19 55-I 19 55-II 19 70 88 26 11 7 3 4 2 2 1 1 7 615 883722 2 1 2622 710 916 43 214 111 2 212 111 213 Size distribution of firms Figure 3.6. The evolution of cost functions. Source: Christensen and Greene. Estimation of Production and Cost Functions 14 1 4.725 4.883 5.040 5 .19 8 5.355 5. 513 5.670 5.828 5.985 6 .14 3 6.300 6.458 6. 615 036 912 1 518 212 427303336 19 55-I 19 55-II 19 70 Average cost ($ /10 00 kWh) Output