Quantitative Techniques for Competition and Antitrust Analysis_1 pptx

An economist, on the other hand, would ideally define the user cost of capitalUCC as the opportunity cost of the capital employed whether financed by debt orequity plus economic deprecia

Figure 3.1. Depreciation schedules Source: Tom Stoker, MIT Sloan.

With a constant depreciation rule, the accounting cost of capital will be

Cost of capitaltD ıKt;where Ktis 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 orequity) plus economic depreciation:

UCCtD Opportunity costtC Economic depreciationt:

The opportunity cost will be an appropriate interest rate times the amount of capitalemployed so that

Opportunity costt D rVt;where Vtis the value of the capital good at time t An “appropriate” interest rate rwill control for risk since not all investments are equally risky The question thenarises about what we mean by an “appropriate” interest rate One popular answer tothat 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,

The second component of the user cost of capital is economic depreciation, whichcan be calculated as the (expected) change in the value of the asset over the period

of use:

Economic depreciationt D Vt
Vt C1:The difference between economic depreciation and accounting depreciation can be

a source of substantial differences between economic and accounting profits Infact, there are many accounting methods used to “write off” capital In general, thefirm 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 thecapital can potentially have an enormous impact on the annual cost figures and henceresult 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 accountingtreatment might calculate the depreciation charge by writing off the value of theinvestment 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 identicalmodel of car which is one year old Doing so would give you an estimate of theeconomic 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 newVolkswagen Passat Variant Comfortline costs€28,050 while a similar one-year-oldmodel can be found for€21,000.6We can calculate the economic cost or the UCC as

UCCtD Opportunity costtC Economic depreciationt

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 rdcan 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.

One can also perform the equivalent calculation:

UCCtD r C Depreciation rate/Vt;where

Depreciation rate D Vt
Vt C1

Vt

D 28;050
21;00028;050

3.1.2 Comparing Costs and Revenues: Discounted Cash Flows

Sometimes, we want to compare the cost of an investment with the value of itsexpected return A common method to calculate the flow of revenues is to calcu-late the discounted cash flow generated by the investment This means calculatingthe present value of the future revenue stream generated by the current capitalexpenditure

The discounted cash flow is calculated as follows:

DCF D

TX

t D1

Rt.1 C r/t C FVT;where Rtis 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 FVT is the final valuation ofthe investment at the end period of the project T

Discounted cash flows can be used to compare the value of revenue streams withthe 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 whenevaluating 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 Theidea 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 sacrificetest is that such a sacrifice would only be rational if it were followed by sufficientlyhigher profits in the future

3.2 Estimation of Production and Cost Functions

The traditional estimation of cost and production functions can be a complex taskthat 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 mainempirical issues in cost estimation and proceed to discuss some seminal illustrativeexamples 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 estimatingthem is a significant body of literature Chapter 1 in this volume covers the basictheoretical framework underlying the empirical estimation of cost functions Wereview here the basic conclusions of that discussion and then proceed to presentsome practical examples of cost function estimation since that is undoubtedly thebest 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 oftencomplicated 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 toproduce output To see why, let us consider a case when costs do just add up becausethere are no substitution possibilities, namely the case of producing according to afixed 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 ingredientstimes the quantities they are required in A fixed-proportions production functionhas the form

˛1; : : : ; ˛ndescribe 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 ˛2D 6 and the ratio 16I2tells

us the number of cakes that we have enough eggs to make However, now supposethat 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 therelative prices of capital and labor our cake producer may choose to use them indifferent proportions The “fixed-proportions” production function may therefore

suffice as a model for the materials piece of the production function, but we willrequire a production function that embeds that piece into a full production functionwhich allows for substitution possibilities in capital and labor.

In general, we describe a production function as

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 byintroducing an “input” variable over which, in the simplest (static) version of thesetheories, the firm is typically assumed to have no choice.8This unobserved “input”will become our econometric error term and is sometimes described as measuringfirms’(total factor) “productivity.” We shall denote a firm’s (total factor) productivity

be less than that feasible according to the production function

Describing the costs of producing output in this way makes it rather clear thatcosts 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 theresearcher interested in eliciting information about the cost structure of firms in anindustry, 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 functiondirectly or instead learn about the production function and estimate costs indi-rectly Finally, readers may recall that there is a relationship between cost functionsand 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).

program can be described as

Ij D @C.Q; p1; : : : ; pnI ˛; u/

@pj

: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 fromestimates of input demand equations

An extremely important fact for the investigator is that each of these threeapproaches to understanding costs requires somewhat different variables to be inour data sets For example, to empirically estimate a production function such as

deriva-Before discussing some empirical applications, we first discuss four substantiveissues that must be squarely faced by the investigator when attempting to learn aboutcosts or technology using econometrics Each is introduced here and subsequentlyfurther 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 estimationexercises: endogeneity, functional form, technological change, and multiproductfirms

First we note that in each of the three estimation approaches described above, wemay 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 areattempting to estimate the production function Q D f I1; : : : ; Im; uI ˛/

For OLS estimation, even if the true model is assumed linear in parameters and theunobserved (productivity) term is assumed additively separable, productivity mustnot be correlated with the independent variables in the regression, i.e., the choseninputs We will face an endogeneity problem if, for example, the high-productivityfirms, those with high unobserved productivity u, also demand a lot of inputs On

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 competitiveadvantage As a result, efficient firms will tend to be both high productivity and usehigh levels of inputs These observations suggest that the key condition required forOLS to provide a consistent estimator will not be satisfied Namely, OLS requires

ui and Ij 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 ourendogenous input will be biased upward.9To solve this problem by instrumentalvariable regression we would need to find an identifying variable that can explainthe 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 themethods described in Olley and Pakes (1996), who suggest using investment as aproxy for productivity and use it to control for endogeneity.10Levinsohn and Petrin(2003) suggest an alternative approach, but in an important paper Ackerberg et al.(2006) critique the identification arguments in those papers, particularly Levinsohnand Petrin (2003), and suggest alternative methodologies

A second consideration is that we must carefully specify the functional form totake into account the technological realities of the production process In particular,the functional form needs to reflect the plausible input substitution possibilities andthe 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 toallow the data to determine the existence of scale effects It is not uncommon toimpose restrictions such as requiring the production function have the same returns

to scale over the whole range of output and such potentially restrictive assumptionsshould only be made when deemed reasonable over the data range of the analysis.Overly flexible specifications, on the other hand, may produce estimated cost orproduction functions that do not behave in a way that is plausible, for example, byproducing negative marginal costs The reason is that data sets are often limited andunable to identify parameters in overly flexible specifications Clearly, we want touse any actual knowledge of the production process we have before we move toestimation, but ideally not impose more than we know on the data

Third, particularly when the data for the cost or production function estimationcome from time series data, we will need to take into account technological changegoing 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.

and input prices associated with the corresponding output cannot therefore 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

immedi-is generally included in specifications using time series data Clearly, with a crosssection of firms there is less likely to be a direct problem with technological progressbut, equally, if firms are using different technologies or the same technologies withdifferent level of aptitude, then it would be important to attempt to account for suchdifferences

When the firms involved produce more than one product or service, costs andinputs can be hard to allocate to the different outputs and constructing the dataseries for the different products may turn into a challenge Estimating multiproductcost or production functions will also further complicate the exercise by increasingthe number of parameters to estimate Of course, such efforts may nonetheless bewell worthwhile

In the next sections, we use well-known estimation examples to discuss these andother 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 estimatethe technological characteristics of the production process and provide informationabout the nature of technology in an industry The estimation of cost functions issometimes preferred to other approaches since, at its best, it subsumes all of therelevant information about production into a single function which is very familiarfrom our theoretical models Doing so can of course be done only in cases wherefirms behave in the manner assumed by the model: they must minimize costs and theymust 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 examplespresented are not meant to be comprehensive or to reflect the state of the art in theliterature, but rather to introduce the rationale of the methodology and point to theeconometric issues that are likely to arise We also hope that they provide a solidbasis 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 byNerlove (1963) He calculated a baseline regression model derived from the commonCobb–Douglas production function, Q D ˛0L˛LK˛KF˛Fu, where Q, K, L, and

F denote output, capital, labor, and fuel, respectively:

ln C D ˇ C ˇ ln Q C ˇ ln p C ˇ ln p C ˇ ln p C V:

It can be shown that a Cobb–Douglas production function implies a cost function

of degree 1 in input prices, before estimating the equation.11That is, he imposes

ˇLC ˇKC ˇF D 1; which is equivalent to ˇK D 1
ˇF
ˇL:With modern computers we could just estimate the restricted model by telling ourregression package to impose the restriction directly Nerlove, on the other hand, atthe time estimated an unrestricted formulation of the restricted model:

ln C
ln pK D ˇ0C ˇQln Q C ˇF.ln pF
ln pK/ C ˇL.ln pL
ln pK/ C V:The restriction results in one parameter less to be estimated, namely ˇK, which can

be inferred from the other parameters In practice, intuitively, this may be helpful ifsuch variables as the capital price data are noisy, making estimation of an unrestricted

ˇK difficult On the other hand, the parameter restriction has not actually removedthe price of capital from the equation since that price is now used to normalizethe other input prices and cost Thus such an argument, while intuitive, does relyrather on the idea that there remains enough information in the relative prices (logdifferences) to infer ˇLand ˇF 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 costfunctions under strong assumptions regarding the unobserved efficiencies of thefirm, 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.

Table 3.1. Nerlove’s cost function estimation results.

Variable Parameter jt j-Statistic

.ln pL
ln pK/ 0.59 (2.90).ln pF
ln pK/ 0.41 (4.19)

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 ˇK 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:

As S > 1, we conclude that the production function exhibits economies of scale

To see why, consider that

@ ln C

@ ln Q D

QC

A log-linear cost function’s diseconomies or economies of scale are a globalproperty of the cost function and, as such, do not depend on the exact level of outputbeing 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 notplotted as a simple straight line

A basic specification check of any estimated regression equation involves plottingthe residuals of the estimated regression The residual is the difference between theactual 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 ofoutput which violates the requirement for OLS to generate consistent estimates Atboth low and high levels of output, the residuals are positive so that true cost is

Figure 3.2. Nerlove’s first model data and fitted values.

Source: Authors’ calculations from data provided in Nerlove (1963).

Figure 3.3. Residual plot calculated using Nerlove’s data

systematically higher than the estimated value On the other hand, for intermediatevalues of output the true value of costs is lower than the estimated value A plot ofthe residuals reveals a clear U-shaped pattern

Figure 3.4. Estimated and true cost function approximation.

Source: Authors’ rendition of figure 3 in Nerlove (1963).

This diagnosis suggests that the assumed shape of the cost function is incorrectand that the true shape is more likely to have the form as in figure 3.4

In fact, the data indicate that there are increasing returns to scale that are exhausted

at a certain level of output after which there are decreasing returns to scale Nerlovesuggests that the specification can be corrected by introducing a second-order term

in the natural log of output as an additional explanatory variable This generates amore flexible cost function that will allow the cost to vary with level of output in

a way that can generate economies of scale followed by diseconomies of scale asoutput rises Now we have

Note that now ˇK D 1
0:48
0:44 D 0:08 Figure 3.5 repeats Nerlove’s nostic check—a graph of residuals against the explanatory variable In contrast toour earlier findings, the graph shows that the expected value of the residuals fromthis regression is indeed independent of the level of output and seems to stay around

diag-Table 3.2. Nerlove’s cost function estimation results with flexible specification.

Variable Parameter jt j-Statistic

.ln pL
ln pK/ 0.48 (2.98).ln pF
ln pK/ 0.44 (5.73)

Estimated using data from 145 firms during 1955.

Figure 3.5. Diagnostic residual plot for Nerlove’s more flexible functional form

0 as we look across the graph, as required for consistent estimates in OLS On theother hand, the variance of the residuals does seem to be related to output, whichsuggests a heteroskedasticity problem Heteroskedasticity is less of a problem thanfunctional form misspecification, because it does not imply that our estimates areinconsistent However, the presence of heteroskedasticity does mean that we willhave to be careful when calculating standard errors, the measures of uncertaintyassociated with our parameter estimates Specifically, a conventional formula willassume homoskedasticity and will generate inconsistent estimates of the standarderrors even though we have consistent estimates of the parameters themselves For-tunately, it is usually possible to construct heteroskedasticity consistent standarderrors (HCSEs), i.e., estimates of standard errors that are robust to the presence ofheteroskedasticity We refer to chapter 2 for a discussion of heteroskedasticity

Size distribution of firms

Figure 3.6. The evolution of cost functions Source: Christensen and Greene (1976).

Christensen and Greene (1976) reestimate the same cost function adjusting the

1955 data and adding 1970 data They try various models, some of which areillustrated in figure 3.6

The lowest line on the graph is the cost curve estimated using 1970 data while thetop two lines are estimates using different model specifications each with data from

1955 First note that the 1955 models differ a great deal at high levels of output.Looking at the table underneath the figure, which reports the number of observations

at each scale of output in each data set, it is easy to see why At high levels of outputthere are simply very few data points and therefore little information about the shape

of the curves at those high output levels Where there is a substantial amount of data,

at lower output levels, the two 1955 regression results seem far more in agreement

A second nice feature of this graph is that it demonstrates very clearly the impact

of technological progress over time First, the technological progress seems to havechanged the minimum efficient size (MES) of operations An increase in MESwould be indicated by a movement to the right of the point at which the averagecost function reaches a minimum Secondly, and more dramatically evident in thegraph, it is clear that by 1970 technological progress has shifted the average costfunction downward At all levels of output, the average cost of producing a kilowatthour of electricity is lower in 1970 than it was in 1955

So far, in presenting Nerlove’s study we have carefully examined the econometricresults, but to ease exposition we have omitted a crucial step in a proper analysis,one that would ordinarily need to be undertaken before progressing this far down

the path in an empirical exercise Namely, we have not examined the validity thetheoretical model’s basic assumptions In this case we would want to know that aplausible view of the firm’s activities is that it (1) minimizes costs for a given level

of output at a given point of time, and (2) is a price-taker in the input markets.These kinds of basic modeling framework assumptions are usually best considered

by developing an understanding of the industry being studied In electricity tion, it is a fact that the electricity cannot generally be stored and has to be supplied

genera-on demand so market dynamics tend to be fairly straightforward.12 Firms do try

to supply the electricity at the lowest possible costs.13 With regards to the inputmarkets, however, the assumption of price-taking behavior may sometimes be moredifficult to motivate On the one hand, relatively small generators are unlikely to

be other than price-takers on the markets for capital and for fuel, although Nerlovenotes that fuel was purchased on long-term contracts Labor, on the other hand, washeavily unionized so that wages were also set via negotiated long-term contracts.Today, many labor economists would not recognize “price-taking” as the relevantassumption for negotiated labor market outcomes, except perhaps as an approxi-mation (see, for example, Manning 2005) On the other hand, if input prices areeffectively fixed when firms are deciding how much labor capital and fuel to use—even if they are fixed by long-term contracts rather than fixed to the firm in the

“price-taking” sense—then firms will choose the mix of inputs that minimize thecost of producing any given level of output treating input prices as fixed and ourassumptions may not be implausible even if they are not motivated in the way thetheorists initially envisioned

In addition to such immediate concerns, a myriad of other factors may also raiseissues that need careful consideration For example, in electricity there could beopportunities in the industry for a strategic withdrawal of capacity by suppliers

to exploit bottlenecks resulting in output varying independently of demand andcosts.14Also, for a study like Nerlove’s which uses variation across firms, it will

be very important that input prices vary sufficiently across firms to tell us the waycosts differ in response to changes in relative input prices If major inputs are, say,commodities, then we may be unlikely to see sufficient variation in input pricesacross firms

12 Current researchers may have a harder time since today there are some exceptions to this general rule available by using hydroelectric generators While electricity is hard to store, engineers realized that water can be both stored and also used to generate electricity The Dinorwig power station in Wales, for instance, has used its reversible pump/turbines since 1984 It uses cheap off-peak electricity to pump water up the mountain and then uses that water to move its turbines in order to generate electricity at peak times.

13 On the other hand, this might not be the case in heavily regulated sectors.

14 See, for example, the 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.

3.2.2.2 Estimating Scale and Scale Effects in a Multiproduct Firm

In the case of multiproduct firms, efficiencies can arise not only from economies

of scale but also from economies of scope, the efficiency gain from using a uniqueproduction entity for several goods or services In an interesting paper, Evans andHeckman (1984a,b) undertake an empirical estimation of the cost function of theU.S telecommunications giant AT&T in order to determine whether economies ofscale and scope in the production of local and long-distance services justified theexistence of a single national carrier.15

In 1982, the U.S government accused AT&T of foreclosing the long-distancetoll market by leveraging its monopoly on local exchanges and decided to break upthe company into different providers for local exchange and for the long-distanceservices This led eventually to the proposal to create the “baby Bells,” providers oflocal toll services which were barred from entering the long-distance market, and ofAT&T as a long-distance carrier.AT&T argued that there were significant efficienciesfrom managing all telecommunications services within one company and that thebreakup of the company by region and activity would cause the irremediable loss

of those efficiencies

Evans and Heckman (1984a,b), hereafter EH, try to empirically examine theseclaims by testing for the “subadditivity” of the cost function, a property implying thatthe cost of production is lower when production is carried out by one firm comparedwith when it is carried out by several smaller firms The property of subadditivity,which we define formally below, suffices to ensure that productive efficiency will

be achieved by a single firm rather than multiple firms and therefore could provide

a rationale for allowing a single large firm to provide both local and long-distancetelecommunications services rather than two more specialized firms

To proceed, define the following two-product cost function:

C D C.qL; qT; r; m; w; t /;

where qLis the output level of local calls L, qT is the output level of toll calls T

As usual, cost functions depend on input prices so that r is the rate of return ofcapital, w is the wage rate, and m is the price of materials In addition, EH use timeseries data so they must correct for changes in the cost function over time For thatreason, t is a variable capturing the current trend in technology EH obtained theiroutput data by dividing the revenues generated with the two different services bythe average prices for local and toll services respectively

The cost function defined above is a two-product cost function More generally, wecan define a J input and M output cost function For example, EH used a two-product

15 See Evans and Heckman (1984a,b, 1986) The latter corrects some important errors that crept into the reporting of the authors’ results in their initial article.

and three-input variant of the general multiproduct Translog cost function:

ln C D ˛0C

JX

j D1

˛jln pj C

MXmD1

ˇmln qmC1

2

JX

j D1

JXkD1

j kln pjln pk

C 1

2

MXmD1

MX

i D1

ımiln qmln qiC 1

2

JX

j D1

MXmD1

mln qmln RnD C  ln RnD C .ln RnD/2

:

Evidently, this cost function is much more general than the Cobb–Douglas costfunction used by Nerlove It shows a greater flexibility and it can be shown tolocally approximate any cost function For EH’s application, we will set J D 3

as the number of inputs and M D 2 as the number of outputs In addition, wewill follow EH and capture technological progress by using lagged research anddevelopment expenditure of Bell laboratories, which we shall denote RnD.The Translog cost function as presented is an unrestricted formulation In estima-tion we may wish to impose the restrictions on cost functions that are suggested bytheory For example, in estimation, EH impose the homogeneity restriction in inputprices in a strategy analogous to Nerlove’s approach discussed above In addition,they impose symmetry restrictions with respect to input prices

In fact, EH estimate a system of equations including the Translog cost functionand the three-input cost-share equations:

sj D ˛j C

JXkD1

j kln pkC

MXmD1

j mln qmC jln RnD:

Let us motivate the equations Evans and Heckman actually estimate To do so, recallShephard’s lemma, which states that one can obtain the input demand functions bytaking the derivative of the cost function with respect to input prices

Define Ij as the input demand function, which by Shephard’s lemma is

@C

@pj

D @ ln C

@ ln pj:

as the number of inputs and M D as the number of outputs... inputs and M D as the number of outputs In addition, wewill follow EH and capture technological progress by using lagged research anddevelopment expenditure of Bell laboratories, which we shall denote... function as presented is an unrestricted formulation In estima-tion we may wish to impose the restrictions on cost functions that are suggested bytheory For example, in estimation, EH impose the