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Review of Industrial Organization18: 175–182, 2001.© 2001Kluwer Academic Publishers. Printed in the Netherlands.

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Regulation and Cost Inefficiency

VAN KOLPINUniversity of Oregon, Department of Economics, College of Arts and Sciences, Eugene OR 974031285, U.S.A.

Abstract. The well known Averch Johnson effect states that rate-of-return regulation will inducecost inefficient production. This paper examines regulation induced inefficiency in broad set ofenvironments including arbitrary regulatory mechanisms, multiple outputs/inputs, uncertainty, timedynamics, price discrimination, and more. I show that the Averch Johnson effect applies throughouta wide variety of settings. Despite the generality of framework, my analysis is truly elementary anddoes not rely on Kuhn–Tucker analysis or three dimensional graphics. I also provide results anddiscussion which clarifies the limits to Averch and Johnson-like insights in practical applications.

Key words: Inefficiency, regulation.

I. Introduction

One of the best known results in regulatory economics was first presented byAverch and Johnson (1962). Their simple model of a regulated monopolist demon-strated that rate-of-return regulation induces cost inefficient production, a resultcommonly known as the Averch Johnson (AJ) effect. (See Baumol and Klevorick,1970; Sherman, 1985; and Train, 1991 for surveys of the literature.) This paperreexamines this result in the context of a much broader set of environments thanthat conceived in the traditional literature – including nonlinear regulatory mechan-isms, uncertainty, time dynamics, multiple outputs/inputs, price discrimination, andmore. Despite this added generality, my proofs are truly elementary and highlightthe essence of the AJ effect without the use of Kuhn–Tucker analysis or geometricgymnastics.

My work offers fresh insights in several dimensions. The first lesson is a ped-agogical one. The AJ effect is traditionally established either by solving a formalconstrained optimization problem with Kuhn–Tucker methods (as in Averch andJohnson’s seminal paper), or through the use of three dimensional geometric argu-ments (as in Zajac, 1970; and Bailey, 1973). I argue that the driving force of theresult is both more transparent and more general if one instead clearly identifiesthe two fundamental assumptions that are imposed. In particular, capital must bereplaceable with labor local to the regulated optimum and, secondly, the regulatedoptimum is not itself an unconstrained local optimum. By bringing these twopivotal assumptions to the forefront, proof of the AJ effect becomes transparent

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even to first time students of the material. (This is particularly noteworthy giventhat the AJ effect is standard material in popular undergraduate IO texts, such asCarlton and Perloff, 1999; and Viscusi et al., 1996.)

My results break new ground by dramatically expanding the realm of envir-onments in which the AJ effect is known to apply. Of equal importance, I alsoexamine the opposite perspective and provide a detailed analysis of when the AJeffect doesnot apply. I construct a general theorem characterizing all regulat-ory mechanisms “immune” to the effect as well as present a series of exampleshighlighting limits to the AJ effect in practical applications.

Another contribution of my analysis is less direct and relates to the “new reg-ulatory economics” promoted in such works as Laffont and Tirole (1993). Thisapproach stresses the importance of modelling the regulator’s objectives as well asthose of the regulated firm. A principal agent model emerges in which the regu-latory agent sets policy fully cognizant of the fact that the regulated firm will seekto optimize its own objective, subject to the regulatory constraints imposed. A dy-namic model such as this raises the possibility that a clever regulatory agent, armedwith a powerful arsenal of regulatory controls, might be capable of side-steppingthe AJ effect while still imposing limits on the return a monopoly can receive oncapital investment. To the contrary, I find that every less than comprehensive returnregulation, regardless of complexity, will induce cost inefficient production. If suchcost inefficiency is incompatible with the regulatory agent’s objectives, one canonly conclude that the agent must either micro-manage all inputs or abandon returnregulatory mechanisms altogether. This paper proceeds as follows. In Section II, Iexamine the classical Averch and Johnson framework. This analysis serves both tocontrast my approach with traditional presentations as well as motivate subsequentgeneralization. Section III presents my general model, assumptions, and results. Ishow that the AJ effect persists throughout all return regulatory environments. Ialso characterize all general regulatory mechanisms that do not suffer from the AJeffect. Section IV presents a series of examples showcasing limits to the Averchand Johnson result. Brief concluding remarks are offered in Section V.

II. The Classical Model

The classical Averch and Johnson model considers a monopoly employing labor(L) and capital (K) to produce a single homogeneous output (q) via a differentiableproduction technologyf , i.e.,L,K ≥ 0 and 0≤ q ≤ f (L,K). Labor and capitalare purchased at strictly positive unit pricesw andr respectively. A strictly positiveoutput price is determined by a differentiable inverse demand functionP(q) andprofit is characterized byπ(L,K, q) = P(q)q − (wL + rK). I shall say that aproduction plan(L,K, q) is fully employedif f (L,K) = q and let5(L,K) =π(L,K, f (L,K)) denote the full employment profit function.

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LEMMA 1. A production plan is cost efficient if and only if it is fully employedand there is no other equally productive and fully employed production plan thatis more profitable.

This result is virtually a tautology. Cost efficiency must require inputs to be fullyemployed, else the original output would be feasible with a sufficiently small re-duction in input purchases. The fact that revenue depends only on output quantityin turn implies that the cheapest way to produce a given output level is also themost profitable and vice versa.

The AJ effect considers the implications of rate-of-return regulation. Such regu-lation constrains the profit the firm is allowed to earn from its capital. Lettingδ > 0represent a given regulated rate of return, regulated profit effectively becomesπδ(L,K, q) = min{π(L,K, q), δK}.LEMMA 2. If (L∗,K∗, q∗) maximizes rate-of-return constrained profits, then giventhe capital level K∗, the labor employment level L∗ also maximizes unconstrainedprofits.

This result also follows immediately. Indeed, if labor employment could be unilat-erally modified to increase unconstrained profit then a sufficiently small increasein capital would at most partially offset this gain while simultaneously relaxingthe profit constraint. As regulated profit would increase, the starting point couldnot have been an optimal response to regulation as assumed. (The fact that rate-of-return regulation will not dissuade the monopolist from employing labor in anunconstrained profit maximizing fashion is highlighted in Sherman, 1972.)

I now piece these two elementary lemmas together. Suppose a rate-of-returnregulation locally binds at the regulated optimum (in the sense that first orderconditions of an unconstrained optimum fail) and that positive levels of both inputsare employed. Lemma 2 implies that at the regulated optimum, the full employmentisoprofit curve has slope zero (as d5/dL = 0 and d5/dK 6= 0). But marginalproduct of labor must be nonzero (else d5/dLwould be strictly negative) implyingthe isoquant at the regulated optimum must have nonzero slope. It follows thatequally productive but more profitable production plans must exist and thus Lemma1 confirms cost inefficiency. As capital increases relax profit constraints, one mayalso conclude that too much (rather than too little) capital is employed.

PROPOSITION 1 (Averch and Johnson, 1962).Optimal responses to rate-of-return regulation use more capital than is cost effective whenever regulation islocally binding and both inputs are employed.

Recapping my proof in plain English: the traditional assumption that the regulatoryconstraint is locally binding and both inputs are employed directly implies that thecorresponding isoprofit has zero slope at the regulated optimum. As the corres-

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ponding isoquant cannot be zero sloped, there must be more profitable methods ofproducing the same output and cost efficiency must fail. (Note that I do notassumea positive marginal product for capital. See Sherman, 1985 for a discussion of thisissue’s significance, as well as a survey of relevant literature.)

III. A Generalized Model

The classical Averch and Johnson model has been criticized for its lack of dynamicstructure and narrow view of regulatory controls. I now introduce a general modelof monopoly regulation which allows for multiple inputs, outputs, and produc-tion periods, uncertainty, arbitrary regulation schemes, etc. This exercise providesseveral useful lessons. In particular, I show that the classical AJ-effect retains itscentral spirit in a very general framework and I characterize the class of all arbitraryregulatory controls that do not suffer from this effect.

The basic components of my general model includeX – the space of all inputprofiles,Q – the space of all output profiles,f : X→ Q – the production function,R: Q→R+ – the revenue function,C: X→R+ – the cost function (defined overinputs),π : X ×Q→R+ – the unregulated profit function defined by the differencebetween revenue and costπ(x, q) = R(q) − C(x), and lastly,πδ: X × Q→ R+– the regulated profit function whose form is determined by the underlying marketdemand/costs and a given set of regulatory rulesδ. A more detailed description ofthese components, along with expository examples follows below.

The input space of the classical model is assumed to be just two dimensional,while my general model imposes no fixed limits on the dimensionality ofX. Thisallows for multiple physical inputs as well as stochastic and dynamic features.To be more precise, uncertainty is modelled by introducing state contingent inputconsumption. As a simple example, imagine a world in which demand is determ-ined by a stochastic variable which may either be “high” or “low”. If the state ofdemand is observed in advance, input consumption can be made state-contingent,effectively doubling the number of input variables the firm must choose. Inherentlydynamic situations can likewise be modelled by introducing time contingent inputemployment. As the dimension ofX is arbitrary, such scenarios fit readily into thisgeneral framework.

The richness of the output spaceQ is analogous to that of the input spaceX as ittoo may be multi-dimensional and allows for the production of multiple outputs inmultiple states at multiple times. The monopoly’s production function is assumedto be a differentiable mappingf : X→ Q.

The only requirements of revenues and costs are that both are differentiable andcosts are strictly increasing inX. This generality allows for multiple inputs andoutputs as well as market power in their purchase and sale. Firm profits (uncon-strained by regulation) are denoted byπ(x, q) = R(q) − C(x) for each feasibleproduction plan(x, q).

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It is important to note that unlike the classical model, the cheapest way to pro-duce a particular output profile need not necessarily result in the full employmentof inputs. Indeed, if one commodity is produced as a byproduct of another, someoutput profiles may not be feasible unless excess quantities of the byproduct areproduced and then discarded. While this is surely a theoretical possibility, I shallassume that there is at least some nominal input usage affiliated with producingand bringing each product to market. (Thus I make a distinction between simplegeneration of a byproduct and the manufacture of this product into a saleable good.)I refer to this as thebyproduct assumptionand with it Lemma 1 can be extended tomy general model. (Details are presented in an earlier version of this paper whichis available upon request.)

The precise manner in which regulated profitπδ differs from the unconstrainedprofit function π depends on the nature of regulation imposed. Examples mayinclude profit constraints, price constraints, taxes, subsidies, or virtually any otherregulatory mechanism. I shall say thatδ is a return regulationif it represents anallowed return which is a strictly increasing continuous function of some subset Kof input variables (“capital”). The regulation constrained profit function is in thiscase defined byπδ(x, q) = min{π(x, q), δ(xK)}; where xK denotes the vector of“capital” goods purchased andδ(xK) the corresponding maximal allowed return.Note thatδ need not represent a rate of return as in the classical model, but isinstead an aggregate level of return which need not be linear. (This generality isparticularly germane given that “capital” may in practice often be a composite ofplant, equipment, etc.) The classical fixed rate-of-return scenario of course remainsand is characterized by the special cases for whichδ is linear.

Optimal response to return regulation requires employment of “labor” inputs(those outsideK) in an unconstrained profit maximizing fashion. This result fol-lows analogously to the classical case and thus Lemma 2 also extends to mygeneralized setting. Fundamental to any investigation of “AJ-like” effects is a de-gree of replaceability amongst inputs. Given an input profilex∗, I shall say that“capital” inputs arelocally replaceableat x∗ if the marginal impact they have onproduction can be replicated by some marginal change in “labor” inputs. Note thatif multiple outputs are produced, incremental changes in one input may inducechanges in more than one output and replicating this effect may be nontrivial.While replaceability may be essential for a discussion on “AJ-like” effects, itis not a vacuous requirement. As in the classical setting, Lemmas 1 and 2 canbe used to confirm the presence of an AJ-effect in my general framework. If areturn regulation locally binds at the regulated optimum (in the sense that firstorder conditions of an unconstrained optimum fail) and “capital” inputs are locallyreplaceable, Lemma 2 can be used to show that the full employment isoprofit andthe isoquant corresponding to the regulated optimum cannot be tangent. Lemma 1then immediately implies that cost inefficiency must prevail. (Details are presentedin an earlier version of this paper and are available upon request.)

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PROPOSITION 2.Optimal responses to return regulation are cost inefficientwhenever regulation is locally binding and regulated inputs are locally replace-able.

The question naturally arises as to which regulatory mechanismsdo not sufferfrom an AJ effect. I shall say thatδ is demand basedif the regulatory constraintaffects only revenues, not costs, i.e., there exists a functionRδ: Q → R+ suchthatπδ(x, q) = Rδ(q) − C(x). I shall say thatδ is effectively demand basedif thebehavior induced is identical to that which would emerge from some demand basedregulation. It is worth emphasizing thatπδ completely characterizes the firm’s “postregulation preferences” over feasible production plans(x, q) as it incorporates allconsiderations for demand, market power, regulatory constraints, etc. For instance,a desire to avoid inelastic demand (a scenario highlighted in Westfield, 1965) willbe reflected inπδ.

I conclude this section by noting that the regulatory mechanisms which inducecost efficient behavior are in fact characterized by the class of all effectively de-mand based rules. In other words any regulation not suffering from an AJ effectmust be equivalent to a demand based regulation. A formal proof appeared in anearlier version of this paper and is available upon request. Informally, the resultfollows by first noting that demand based regulation clearly induces cost efficientbehavior (regulated revenues depend only on output, so the firm will choose toproduce its output of choice in the cheapest way possible). As effectively de-mand based regulation induces demand based behavior, it too must lead to costefficiency. Finally, the reverse implication is established by directly demonstratingthat whenever a regulatory mechanism induces cost efficient behavior, this samebehavior can be induced by a demand based rule.

PROPOSITION 3.Regulation induces cost efficiency if and only if it is effectivelydemand based.

IV. Clarifying Examples

This section supplements my theoretical results with several informal exampleswhich help illuminate the extent to which the AJ effect applies in real worldsettings. In particular, these examples respectively reveal the significance of dy-namics, differentiability, and continuity in the real world application of Averchand Johnson style insights. Let us first consider a dynamic model in which thefirm seeks to maximize the sum of discounted profits in a series of classicalAverch and Johnson stage games. Suppose further that regulatory policy imposesa rate-of-return regulation at each and every stage, but the allowed rate-of-returnis decreasing in the cost inefficiency observed in previous periods. In effect, inef-ficient behavior today triggers more restrictive regulatory constraints in the future.It is easy to see that if the future drops in allowed rate-of-return are sufficiently

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severe, they may forever deter cost inefficient behavior. (Readers familiar withthe static/single stage graduated return mechanism proposed in Klevorick, 1966should not confuse it with the dynamic/multi-stage example above.) It is importantto note that even though theobservedrate of return is constant every period, this“trigger” policy is not a return regulation. Indeed, the allowed returns in futureperiods depend on the labor decisions in past periods, contrary to the definitionof return regulation which requires allowed returns to depend only on “capital”employment. Such examples draw attention to the special care that must be takenwhen empirically testing for an AJ-effect. Even though a fixed rate-of-return regu-lation may be observed period by period, if the monopolist perceives there is evenan implicit policy in which past behavior may influence future allowed returns, theAJ effect does not apply and one need not expect cost inefficiency to be observed.

Another class of examples emerges when production and/or profits fail differ-entiability. For instance, it is easy to construct examples in which a firm endowedwith Leontief production technology will continue to employ labor and capital inefficient proportions when exposed to rate-of-return regulation. More generally,any scenario in which the marginal productivity of “capital” varies discontinuouslywith the availability of those inputs necessary for their operation is subject to thefailure of the AJ effect.

In practical application, capital inputs may often be composed of “stock” com-ponents that are readily available in a few sizes but prohibitively expensive inothers. As a consequence, capital choice in such examples may be effectively dis-crete. The replacement of “capital” with “labor” may thus cause discrete jumpsin profits which may enable return regulation and cost efficient behavior to becompatible. While a continuous version of a given model may necessarily inducecost inefficiency, a discrete version of the same model may leave a cost efficientchoice as the best available alternative, thereby disarming Averch and Johnson styleconclusions.

V. Conclusion

Averch and Johnson (1962) had an enormous impact on the subsequent literaturein regulatory economics. Even though contemporary work by Laffont, Tirole, andothers suggest its ongoing influence has sharply curtailed, its historical signific-ance surely remains. For this reason alone it seems worthwhile to demonstratethat Kuhn–Tucker analysis, 3-D graphics, and the like can be pushed aside asunnecessary clutter when examining the AJ-effect. Indeed, I have shown that trulyelementary logic is sufficient for the pedagogy of this problem. My work also helpsto resolve any debate regarding whether the AJ effect is simply historical trivia orif it is a factor of ongoing concern in regulation design. Both sides of this issue areconsidered by first demonstrating that the AJ effect extends to an extraordinarilygeneral setting and then clearly delineating where the effect falters. The lessons

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learned from this theoretical exercise should be useful both in terms of policyformulation and interpretation of empirical analysis.

Acknowledgements

The helpful comments of an anonymous referee, William G. Shepherd, and BruceBlonigen are gratefully acknowledged.

References

Averch, Harvey, and Leland L. Johnson (1962) ‘Behavior of the Firm under Regulatory Constraint’,American Economic Review, 52, 1052–1069.

Bailey, Elizabeth E. (1973)Economic Theory of Regulatory Constraint. Lexington, MA: LexingtonBooks.

Baumol, William J., and Alvin K. Klevorick (1970) ‘Input Choices and Rate-of-Return Regulation:An Overview of the Discussion’,Bell Journal of Economics and Management Science, 1, 162–190.

Carlton, Dennis W., and Jeffrey M. Perloff (1999)Modern Industrial Organization, 3rd edn.Addison-Wesley-Longman.

Klevorick, Alvin K. (1966) ‘The Graduated Fair Return: A Regulatory Proposal’,AmericanEconomic Review, 56, 477–484.

Laffont, Jean-Jacques, and Jean Tirole (1993)A Theory of Incentives in Procurement and Regulation.Cambridge, MA: MIT Press.

Sherman, Roger (1972) ‘The Rate-of-Return Regulated Public Utility Firm is Schizophrenic’,Applied Economics, 4, 23–31.

Sherman, Roger (1985) ‘The Averch and Johnson Analysis of Public Utility Regulation Twenty YearsLater’, Review of Industrial Organization, 2, 178–193.

Train, Kenneth E. (1991)Optimal Regulation: The Economic Theory of Natural Monopoly.Cambridge, MA: MIT Press.

Viscusi, W. Kip, John M. Vernon, and Joseph E. Harrington Jr. (1996 )Economics of Regulation andAntitrust, 2nd edn. Cambridge, MA: MIT Press.

Westfield, Fred M. (1965) ‘Regulation and Conspiracy’,American Economic Review, 55, 424–443.Zajac, E. E. (1970) ‘A Geometric Treatment of Averch–Johnson’s Behavior of the Firm Model’,

American Economic Review, 60, 117–125.


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