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applicability and a much more rigorous underpinning than had previously .... in terms of the theory of national income a

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Scand.J of Economics103(1), 1-23, 2001

A Contributionto the Theoryof Welfare Accounting Martin L. Weitzman* HarvardUniversity,Cambridge,MA 02138, USA [email protected]

Abstract A kindof "unifiedtheory"is proposedas a dynamicgeneralizationof the standardconsumersurplusmethodologyfor evaluatingwelfare changes.The "unifiedtheory"allows rigorous dynamicwelfarecomparisonsto be inferredbetweenany two economicsituations-fromjust knowingcurrentincomesandobservinga short-runmarketdemandschedule.Essentially,the changein presentdiscountedfutureutility is exactly capturedby the formula:differencein currentincomeplus consumersurplus.This well-knownformulais therebyshownto covera farwiderclass of welfarecomparisonsthanis customarilytreatedin the textbookstaticcase. Keywords:Nationalincomeaccounting;dynamicwelfare JELclassification:C43;D6; D9

I. Introduction The contributionin the paper'stitle refersto a kind of "dynamicwelfarecomparisonprinciple,"which extendsinto the dynamicrealmthe standard static methodology of using currentlyobservable incomes and market demandschedulesto infer welfare differencesamong economic situations. The theory might be considered"unified"not just in the sense that the standardstatic methodologybecomes a special case of the more general dynamictheory,but also because some basic unifying connectionscan be made with consumer-surplus theoryand index-numbertheory.The connection with consumer-surplustheory is especially striking, because this familiarlyuseful methodologyis shown to possess a far wider domainof applicabilityand a much more rigorousunderpinningthan had previously been suspected.There are severalpossible motivationsfor the paper,then, but I think that the best introductionis by way of seeing the contribution literature. placedinitiallyin the contextof the national-income-accounting *I am indebtedto Geir B. Asheim for helpfuldiscussionsand useful detailedcriticisms.The researchwas supportedby a grantfrom the National Science Foundation.I would like to dedicatethis paperto the memoryof TjallingC. Koopmans.The originalprogenitorpaperfor this line of researchwas writtensome 30 years ago when I was at the Cowles Foundation. Tjalling,then the director,recruitedme there from graduateschool, served as intellectual mentor,andencouragedmy fledglingresearch-includingthatearlypaper. ? The editorsof the ScandinavianJournalof Economics2001. Publishedby BlackwellPublishers,108 Cowley Road, OxfordOX4 1JF,UK and 350 Main Street,Malden,MA 02148, USA.

2 M. L. Weitzman Recent times have witnessed a greatly heightened awareness of the interactions between economic, social and environmental issues. People throughout the world have become much more sensitive to the important possible links between their own human societies and the natural environmental surroundings within which these societies may thrive or fail. Terms like "green accounting" and "sustainability" have found their way into the lexicon of popular jargon. There has appeared a widespread interest in the idea of extending the concepts and measurement of national income to include importantnon-market activities in related areas that bear on welfare and productivity-in particularenvironmental goods and services (including natural resources), but also human capital formation, unpaid home production (possibly including leisure-time activities), the services of consumerowned durables, near-marketresearch and development, and so on.1 Many questions have been raised about augmented national income accounting, ranging from broad concerns, posed at a high level of abstraction, about its welfare foundations, through basic issues touching on the design of green national income accounts, down to narrow advice on which particular activities to include and how to include them. In response, as if wanting to be able to answer such questions, has arisen a branch or application of economic analysis that might be called the "pure theory of comprehensive national income accounting." Through the core of this theory runs a common strand attempting to connect a currently observable index of comprehensive net national income or product with some appropriate but not-currentlyobservable welfare measure of future power to consume, which typically has a "sustainability-like" flavor or undertone. We seem presently to have created at least a partially successful body of theory.2 However, some big pieces of the conceptual puzzle do not yet fit snugly into a fully coherent overall picture. One piece is obvious because the existing theory is typically built around a fictitious entity of "aggregate consumption," while what we really want is a general theory that incorporates heterogeneous consumption as seamlessly as it weaves in heterogeneous capital.3 Another piece of the puzzle is a seeming disconnection between the idea that, to be observable in the first place, net national product 'For morebackgroundand details,set in a practicalcontext,see NationalAcademyPanelon andEconomicAccounts(1999). IntegratedEnvironmental 2Fora good overview,with an extensivebibliography,see Aronsson,Johanssonand L6fgren (1997);see also Asheim(2000). 3Thefailureto treatseriouslyheterogeneousconsumptionis actuallya muchmoresignificant shortcomingthanis commonlyrealized,becausemanyof the key conceptsused in the greenmeasures,arenot accountingliterature,suchas the various"welfare-equivalent-consumption" even well defined for the general case. The essential problemhere is that with multiple consumptiongoods, the distinctionbetween"aggregateconsumption"and "utility"becomes untenableat themostfundamental conceptuallevel. ? The editors of the Scandinavian Journal of Economics 2001.

A contributionto the theoryof welfare accounting 3

index of the or incomemustessentiallytake the formof a production-based moneyvalue of currentaggregateoutput,whereasthe conceptsthatshowup naturallyin optimalgrowththeory,such as the Hamiltonian,are essentially utility-basedmeasures. To my mind, the biggest and most critical piece of the puzzle not yet fitting neatly into the existing body of theory concerns an answerto the following question.At least in principle,how are we actuallysupposedto use nationalincome statisticsand othercurrentlyobservablemarketinformationto make rigorouswelfare comparisonsamong differenteconomies, or the same economy over time? Taken seriously,such inferenceswould appear to require the computationof inherentlydynamic, wealth-like, magnitudes.4Is there a way to circumventthese present-discounted-utility or dauntingcalculations, at least to relate such wealth-likewelfare-stock magnitudesto some simpler,andmorereadilyobservable,staticincome-like surrogateslocated within the national income statistician's"production boundary"? Posedthis way,the paperaddressesthe dynamicversionof a fundamental question of welfare economics. When we have two inherentlydynamic situationswhose welfare we wish to compare,then in theory we should directly evaluate the two conceptually correct wealth-like present-discounted-utilitymagnitudes.But such welfare-stockmeasures seem very remote from anythingout there that is actuallyobserved,or that is even observablein principle.Meanwhile,within the "productionboundary"of observablestatisticswe have some flow informationaboutcurrentpricesand quantities(includingprice-quantitypairs observedalong the currentconsumer-demand curves).The fundamentalquestionis this. Whatrelationship connectsthe currentlyobservableincomes(andconsumerdemands)with the not-directlyobservable difference in dynamic welfare between the two situationswe wish to compare? From a national-incomeperspectivethe paper is aimed primarilyat showing how to make rigorousdynamicwelfare-stockcomparisonsbased only on directlyobservablemarketinformationaboutcurrentincomeflows. The theorytreatsfully disaggregatedconsumptionas a naturalformulation, and also shows implicitlyhow a money-valuedproduction-basednational productcan be reconciledwith utility-basedwelfare.Lastbutnot least,when static consumer-welfaretheory is correctlyplaced in its properdynamic setting,the analysisactuallybecomesmuchsimpler-and considerablymore revealing.By embeddingshort-runconsumerbehaviorwithin the unified theoryof an optimalgrowthframework,the papercasts new light on some old but importantcontroversiesin consumer-surplustheory and indexnumbertheory. 4This point was first made forcefully by Samuelson (1961). ? The editors of the Scandinavian Journal of Economics 2001.

4 M. L. Weitzman

It shouldbe understoodthatwhile the motivationhas thusfarbeen framed in termsof the theoryof nationalincomeaccounting,the essentialcontribution of the paper is to provide a proper dynamic generalizationof the standardstaticformulafor the welfareevaluationof economicchanges.As a contributionto the theoryof consumersurplusandcost-benefit analysis,the paper therefore also has potential applicationsin many other areas of economics.

II. The Setting of the Model It is importantto state clearlyat the outset that the resultsobtainedin the paperdo not dependon any trickyor unorthodoxassumptions.The assumptions are the usual familiar ingredientsof the conventionalmulti-sector optimalgrowthmodel. It is possible, of course,to criticize these assumptions,or this model.Forbetteror for worse,however,on no substantivepoint does the formulationhere deviate from the standardrepresentative-agent "consensus"versionof intertemporaloptimizationwith multipleconsumption andinvestmentgoods. Let the vector C representan m-dimensionalfully disaggregatedconsumptionbundle. (More specifically,componenti of C(t) measuresthe instantaneous flow of consumption services from consuming at the rate of

Ci(t) units of commodity i per unit time at time instant t, for i

=

1, 2, ...,

m.) The consumption vector C is conceptualized as a complete

list containing everything that influences current well-being, including environmentalamenitiesand other externalities.Consumptionhere would ideallyincludeall componentsthatinfluencethe true "standardof living"notjust the goodswe buy in storesandthe governmentservices"purchased" with our taxes,but also non-marketcommodities,such as those producedat home, andenvironmentalservices,such as those renderedby naturalcapital like forestsand clean air.Forthe sake of developingthe core theory,initial consumptionC(O) is presumed to be fully observable,along with its associatedm-vectorof competitiveor efficiencyprices.We also presumeto knowor be ableto observethe relevantshort-runmarketdemandfunctionin the domainoverwhichconsumptioncomparisonsareto be made. For any consumption-flowtime series {C(t)}, it is supposedthat it is meaningfulto measure overall intertemporalwell-being by the familiar expression: W({C(t)})

e-PtU(C(t))dt,

(1)

where U(C) is some given concave, non-decreasing,instantaneousutility functionwith continuoussecond derivativesdefinedover all non-negative ? The editors of the Scandinavian Journal of Economics 2001.

A contributionto the theoryof welfare accounting 5

consumptionflows C, while p is some given rateof puretime preference.As practicallyeveryeconomistwill attest,for betteror for worse,formula(1) is the standardworkhorseobjectivefunctionused widely in economics as a maximandin intertemporal optimizationproblems.Also for whatit is worth, a linear functionaltaking the form of W({C(t)}) in (1) can be given an axiomaticjustificationas representingthe appropriatedynamicpreference orderingwheneverindependence,stationarity,continuityand a few other seeminglyreasonable(to me) conditionsarepostulated.5 The notion of "capital"used in the model is intendedto be quite a bit more general than the traditionallyproducedmeans of productionlike equipmentand structures.Most immediately,subsoil mineralresourcesare unquestionablyconsideredto be forms of capital.Formsof humancapital, such as education,shouldin principlebe included,as well as the knowledge capital accumulatedfrom R&D-like activities. Generallyspeaking,every possible type of capitalought to be included-to the extent that we know how to measureand evaluatethe associatednet investmentflows. Undera broad interpretation,renewableresourcesin particularand environmental assets more generally should be treated as forms of capital. From this perspective,environmental qualitywouldbe viewedas a stockof capitalthat is depreciatedby pollutionand investedin by abatement.6The underlying ideal is for the list of capital goods to be as comprehensiveas possible, subjectto the practicallimitationthat meaningfulcompetitive-market-like efficiency prices are availablefor evaluatingthe correspondingnet investments. Suppose that altogetherthere are n capital goods, including stocks of naturalresourcesand other non-orthodoxforms. The stock of capital of type i (1 < i < n) in existence at time t is denoted Ki(t), and its corresponding net investment flow is Ii(t) = Ki(t). The n-vector K = {Ki}

denotes all capital stocks, while I = {Ii} standsfor the correspondingnvector of net investments.Note that the net investmentflow of a natural capital asset like a timber reserve would be negative if the overall extraction rate exceeds the replacementrate. Generally speaking, net investmentin environmentalcapitalshouldbe regardedas negativewhenever the underlyingasset is being depletedor run down more rapidlythan it is replacedor builtup. Again in the spirit of focusing sharplyfor the sake of developingthe core theory,we assume the "attainablepossibilities" of the production-

5Seee.g. Koopmans(1960). This is not the place to get embroiledin controversy, butI believe a majorityof economistswould agree that the critics of the utilitarianform (1) have yet to delivera workablealternativeobjectivefunction. 6Maler(1991) containsa good discussionof some of the relevantissueshere. ? The editors of the Scandinavian Journal of Economics 2001.

6 M. L. Weitzman distribution system are time autonomous.7 For theoretical purposes, we thus imagine an idealized world where the coverage of capital goods is so comprehensive, and the national accounting system is so complete, that there remain no unaccounted-for residual "atmospheric" growth factors. In the paper, all sources of future growth have been attributed as proper investments, which are fully "accounted for" by being valued at their proper efficiency prices and included in the national product. Unfortunately, we do not now live in a world where national income accounting is complete, even though our theoretical models typically assume this feature. Completeness is perhaps best envisioned as a limiting case, which some real-world accounting systems approach in coverage but few attain. In our actual world we cannot measure all investments accurately, many externalities are not internalized, it is often difficult to impute marketlike prices for non-market goods, there are various "atmospheric"sources of positive or negative growth, which we cannot or do not include in net national product, etc. (The omitted "atmospheric" contributions are identified primarily as a residual, which is obtained by subtractingoff from actual growth the effects of all known, properly attributed,sources of growth.) The justification traditionally given for studying the pure theory of complete accounting in a real world of incomplete accounting is that the pure theory can serve as a beacon guiding the way toward greater completeness-by suggesting what activities to include, and how best to include them, to "green up" national income into a more comprehensive aggregate reflecting more accurately what the future portends relative to the present. The motivation here has a slightly different nuance. For this paper, the pure theory of complete accounting is important because it indicates how to use currentincome-like data to make rigorous dynamic welfare comparisons-at least in principle. In a setting where the comparisons take the form of a hypothetical costbenefit evaluation of the welfare difference between two economic situations (i.e., attainablepossibilities "with" and "without" the proposed project), the assumption of accounting completeness may not be so much of a practical constraint. Loosely speaking, in such a context it matters only that the accounting be complete for the relevant subset of goods that are changed between the two situations. Mathematically, the national-income accounting system is complete or comprehensive if the attainable-possibilities set at any time t can be 7Forsome treatmentsof the time-dependent case, see Nordhaus(1995), Weitzman(1997), or WeitzmanandL6fgren(1997), andthe referencescitedtherein.Timedependenceintroducesa host of messy complications,but a modified(and much less pretty)version of the result presentedhere can usuallybe found,contingenton some simplifyingassumptionsaboutthe particularformof time dependency. ? The editors of the Scandinavian Journal of Economics 2001.

A contribution to thetheoryof welfareaccounting 7 describedin reducedform as a function only of the capital stocks K(t) existingat thattime. Therefore,by makingthis assumption,we are allowed set here as S(K). to denotethe (m + n)-dimensionalattainable-possibilities Thenthe consumption-investment pair(C(t), I(t)) is attainableat time t if andonly if (C(t), I(t)) E S(K(t)).

(2)

As usual,the set of attainablepossibilitiesS(K) is presumedto be convex. This completesthe backgrounddescriptionrequiredto formulatethe basic problemof the paper.

III. A Tale of Two Economies Supposewe are interestedin comparingthe dynamicwelfareachievableby two differenteconomies across space or two differenteconomic situations overtime. The formulationhere is intendedto be quitegeneral,in principle coveringactualreal-worldwelfarecomparisonsacrossspace and over time, as well as "with project"and "withoutproject"hypotheticalbenefit-cost evaluations.(Benefit-cost evaluationsaredoneprospectively"withproject" and "withoutproject"by comparingthe welfare attainablefrom a hypothetical "after-the-project-is-included" set with the attainable-possibilities welfare delivered by the existing "sans-project"status quo attainablepossibilities set.) In what follows, let the economy "type" or "role" be indexed by the superscriptindicatorvariablej. The index value j = 1 indicatesthe given base economy.The index value j = 2 indicatessome particularcomparisoneconomy.Both economiessharethe samepreferences, but they may have arbitrarilydifferentendowmentsand/orarbitrarilydifferent attainablepossibilities.Themaincontributionof this paperis to compare (1) across the two economies relyingonly on currentlyobservablemarket information. Both economiesor economicsituationsj = 1 andj = 2 are postulatedto exhibitdynamicbehavioras if they are solutions,respectively,to a pair of optimalgrowthproblemsof the form: maximize

U(CJ(t))e-Ptdt,

(3)

subjectto the constraints

(CJ(t),I(t)) e SJ(KJ(t)),

(4)

? The editors of the Scandinavian Journal of Economics 2001.

8 M. L. Weitzman and the differential equations Ki(t) = Ii(t),

(5)

and obeying the initial conditions KJ(0) = KJ,

(6)

where Ko is the initially given capital stocks-all of the above holding for j= 1 andj= 2. Concerning the above formulation (3)-(6), note that the "attainablepossibilities sets" (or "technologies") SJ(K) in (4) and the "endowments" Ko in (6) are allowed to differ arbitrarilybetween the base economy (j = 1) and the comparison economy (j = 2), while "preferences" are identical,8 as indicated by the shared objective (3). The goal of the paper is to infer the difference in the value of the optimized objective function (3) between the two economies from currently observable market information alone-without actually having to solve the pair of optimal growth problems (3)-(6). This might appear to be a formidable task since no additional structure is being imposed on the technologies or endowments of the two economies. In what follows, it is assumed, purely for ease of exposition, that the two optimal solutions of (3)-(6) corresponding to j = 1 and j = 2 not only exist, but are unique. Let {C*J(t), I*i(t), K*j(t)} represent the optimal trajectory for economy j. As is well known from duality theory, the solutions of (3)-(6) for both economies will generate corresponding dynamic competitive prices, denoted here by the m-vector time series {p*i(t)} for consumption-goods (money) prices, and by the n-vector time series {q*j(t)} for investment-goods (money) prices. Then (money) national income or product for economy j at time t is

Y*J(t) p*(t) C*J(t)+ q*J(t). I*(t).

(7)

Let )i(t) represent the non-observable (to an outsider) marginal utility of money income along an optimal trajectory in economy j (= 1, 2) at time t. The investment-goods price n-vector, expressed in real current-valueutility terms for economy j (= 1, 2) at time t is then AJ(t)q*J(t),

(8)

8Unless preferences are postulated to be comparable in some way across any two situations, it is impossible to make rigorous general welfare comparisons. The standard static framework yields a bona fide welfare-change indicator only by assuming that, in essence, the same consumer faces two different price-income situations. ? The editors of the Scandinavian Journal of Economics 2001.

A contributionto the theory of welfare accounting 9

while the correspondingconsumption-goodsprice m-vector,expressedin real current-value utilitytermsfor economyj (= 1, 2) at time t is 2J(t)p*J(t).

(9)

In the model, {AJ(t)}may be chosen arbitrarilybecause it representsan extra degree of freedomthat merely parameterizesthe marginalutility of money income, which can be given a life of its own, relatedbehind the scenes of the real economy to the money supply and other background, purelymonetary,factorsthatdeterminethe price level. Whatmattersfor the allocationof resourcesin the real economy-through the classical-dichotomy veil of arbitrary{%J(t)},so to speak-are the real prices (8) and (9), which are denominatedin terms of the contemporaneousvalue of utility servingas numeraire,and are thereforeinvariantto {AJ(t)}. In otherwords, changing the exogenous specification of {AJ(t)} would merely induce inverselyproportionalchangesin {q*j(t)} and {p*j(t)} withoutaltering(8) or (9). (Typically,a paper on optimal growth theory specifies, without ceremony,all prices to be expressed in "real" utility-valuedunits. The reasonwe haveto deal carefullywith the issuesraisedby arbitrary{Ji(t)} in this paper is that the ultimategoal here is to translateobservablemarket values, denominatedin the arbitrarymonetaryunits of the two different economies, into a statementabouttheir real welfare difference,expressed, ultimately,in utiles.) As is well known,the dualityconditionscorrespondingto (3)-(6) can be as if describinga decentralizedperfectlycompetitive given an interpretation economyin dynamicequilibriumwith a single representativeagent having the preferenceordering(1). We emphasizethis decentralizedmarketinterpretationthroughoutthe paper,concentratingespeciallyon how the observable short-runmarketdemandfunctionof the representative consumer-agent can be usedto revealcriticalaspectsof the agent'sunderlyingpreferences. Hamiltonianexpression Definethe maximizedcurrent-value HJ(K; q, X)

max

q. I}. {U(C) + q

(C,I)ESj(K)

(10)

The first type of optimality condition requires that the Hamiltonian expression(10) should actually attain its maximumeverywherealong an optimal trajectory.In the representative-agentinterpretation,maximizing the Hamiltonianis equivalentto the combinationof a conditiondescribing the representativeconsumer'sdecentralizedbehaviorin choosing among consumptiongoods C andaggregatenet savingsor investmentZ: ? The editors of the Scandinavian Journal of Economics 2001.

10 M. L. Weitzman U(C*J(t)) + AJ(t)q*J(t) I*J(t) -

max

{ U(C) + )J(t)Z},

p*J(t)'C+Z=Y*J(t)

(11)

along with a conditiondescribingthe representativeproducer'sdecentralized static-equilibrium behavior: p*i(t). C*i(t) + q*J(t). I*i(t) =

max

{p*(t) * C + q*J(t) I}.

(C,)ESi(K*i(t))

(12) A second set of optimalityconditionscan be translatedas describinga perfectcapital/stockmarketin dynamiccompetitiveequilibrium: dt [i(t)q*j(t)] - p[LJ(t)q*J(t)]= dt a *K

t

(13)

where the notation ]*j(t)means evaluationalong the optimaltrajectoryof economyj at time t. Finally,the thirdoptimalityconditionhere is the transversalityrequirement lime-PtPYJ(t)q*i(t)

t--oo

K*J(t) = 0.

(14)

If conditions(13) or (14) did not hold,thenpurepositiveprofitscould be madeby intertemporal arbitrageoperations,whichwouldinducea changein (13), (14)-meaning these equations could not have been describinga dynamiccompetitiveequilibriumin the firstplace. Because of the underlyingconvexity of problem (3)-(6), the duality conditions (11)-(14) are both necessary and sufficient for an optimal solution.9Thus,wheneverwe postulateor observehere a dynamiccompetitive equilibriumof the form (11)-(14), it is as if we are postulatingor observingthe solutionto an optimalgrowthproblemof the form(3)-(6). IV. Current Directly Observable Market Information Fromthis pointon, the paperdealswith market-behavior observationsmade only at the presenttime t = 0. More precisely,we take on faith that the conditionsdescribingthe coupled system dynamicoptimality-equilibrium (3)-(6), (10)-(14) will hold overall futuretime. But, asidefromthis general 9This aspect, along with the representative-agent dynamic-competitive-equilibrium interpretation of duality, is discussed in several advanced theory treatises. For an exposition whose notation is very close to this paper, see Weitzman (1970) and/or Weitzman (1973). (

The editors of the Scandinavian Journal of Economics 2001.

A contribution to thetheoryof welfareaccounting 11 knowledge,everythingwe are permittedto knowor inferat the presenttime t = 0 must be based solely on what is, at least in principle,the current directly observable market behavior of the representativeconsumer.In keeping with this restrictionon knowableinformation,the symbol X*i(O) (for all pertinentvariablesX) is hereafterreferencedsimplyby the symbol

XJ. Consistentwith long-standingeconomicusage, all consumptionenumerations, includingthe currentconsumptionvector Ci, are conceptualizedas flows of services.At least in theory,this meansthe "shortrun"is envisioned as a period of arbitrarilyshort duration,with the correspondingprices of consumerdurables,like owner-occupiedhouses (or automobiles,or refrigrentalflow rates.10Pracerators),imputedas competitive-market-equivalent tically,for most commoditiesand for most applications,it probablysuffices to thinkin termsof short-runconsumptionas occurringovera periodof, say, a year, or, for the most extreme cases, maybe a month. Theoretically, however,we are eliminatingtime aggregationaltogetherby going to the limit in distinguishingamong commoditiesconsumed at each instant of time. The currentshort-runconsumerdemandfunction in economy j is the representativeconsumer-agent'sresponseto the following counterfactual question.At what ratewould you choose to consume(throughouta vanishingly shorttime intervalstartingnow) if the instantaneousrentalprices of consumptionservice flows (duringthis interval)were p (but the rest of the pricepathdoes not change)?The traditionalway of formalizingthis question is to representconsumptionchoices over time by an intertemporalbudget constraintof the form e-Ptj (t)p * Cdt + Jo

J e-Pt(t)p*J(t)

* C(t)dt

a

=

J

e-Ptjl(t)p*J(t). C*J(t)dt (15)

for some "vanishinglysmall" 6. The short-runconsumerdemandfunction in economy j is the limiting optimized value of C (= C(O)), expressed parametricallyas a functionof p (= p(O)),which maximizesthe intertemporalutilityfunctionsubjectto budgetconstraint(15), as - 0+. An equivalent(but much neater)descriptionof the short-runconsumer demandfunctioncomes straightout of the maximumprincipleof optimal l0See e.g. Boskin, Dulberger,Gordon,Griliches,and Jorgenson(1998) for a discussionof imputedrentsaimedat applications. ? The editors of the Scandinavian Journal of Economics 2001.

12 M. L. Weitzman

controltheory.The act of "maximizingthe Hamiltonian"translatesbehaviorally from (11) into havingthe representativeconsumer-agentin situation j (= 1, 2) solve a decentralizedproblemof the reducedform: maximize

U(C) + jZ,

(16)

subjectto the budgetconstraint

p. C + Z = Y,

(17)

fixed short-runmoney where p standsfor the counterfactual parametrically national-income YJ as-if-fixed the given represents consumptionprices, as-if-fixed to an Ai is the observable marginal outsider) given (not budget, to or net and Z of investment, income, savings symbolizesaggregate utility consumerin j. be chosenalongwith C > 0 by the representative The short-rundemandfunctionis simply the optimizedvalue of C in (16), (17) expressed parametricallyas a function of p. The important implicationhere for consumerdemandtheoryis that the Hamiltonianis a quasi-linearutilityfunction.Intuitively,this quasi-linearHamiltonianobjective form (16) is inherentin a continuous-timeformulationbecause the consumercan fully offset, via changes in savings behavior,any and all possible income effects of a short-runprice change-merely by shiftingthe tiniest bit of investmentincome acrosstime. It is for this reasonthat (16), (17) with Ai constantdescribesthe same short-runinstantaneousdemand functionof p as would a rigorouslimiting argumentwhen 6 -* 0+ in the budgetconstraint(15). 11 intertemporal We write the directly observable short-run consumer-demandfunction in

economyj (= 1, 2) as DJ(p). ThevectorfunctionDJ(p) is the implicitnonnegativesolutionof the aboveproblem(16), (17), which thereforesatisfies, for all parametricallygiven hypotheticalvalues of p > 0, the standard dualityconditions U'(DJ(p)) 0. An immediateconsequenceof comparing(Al) with the definitions(21), (22) is that 0(C)=0 o

;t2

(A2)

for all C > 0. U form of the conclusionto be Equation(A2) representsa stronger-than-required provedin the statementof the lemma,because0 here is not just "a constant,"but actuallyequalsA2/i1. The strongform(A2) is neededin the followingproofof the theorem. ? The editors of the Scandinavian Journal of Economics 2001.

A contributionto the theoryof welfare accounting 21 Proofof Theorem: A basic resultfromWeitzman(1970, p. 15, eq. (16)), transposedto the notationof this paper,is p

U(C*i(t))e-Ptdt = U(C) + AJqJ Ij.

(A3)

Takingthe differenceof (A3) betweencomparisonandbase economiesgives P[f

U(C*2(t))e-Pt

dt -J

U(C*1(t))e-Pt

dt

= U(C2) -+- q2 * 2 _ U(C1) - il *q I.

(A4)

Nowjust usingbasicmathematicalconsiderationsarisingfromsmoothdifferentiabilityof the functionU(C), we have C2

U'(C). dC = U(C2) - U(C1),

(A5)

Jcl

wherethe LHSintegralof (A5) is pathindependentbecausethe secondmixedpartial derivativesof U(C) areequalby the assumptionof continuoussecondderivatives.'5 Now (Al) impliesdirectlythat C2

Jcl

c2

U'(C)* dC =

A1' P'(C) dC.

(A6)

Jcl

Because {P1(.)} and D1(.)} from (18), (19), (20) are inversefunctionsto each other,integrationby partsalonganycontinuousconnectingpathyields the equation PC2

P'(C)

'dC

(C2)

= Pl(C2)

C2 - Pl(C')

Cl -

C1

D(p)

dp.

(A7)

JpI(cl)

Selecting C = C2 in (Al) for j = 1 and for j = 2, and then comparingthe resultingexpressionwith (A2) implies pl(C2) = 0p2(C2).

(A8)

15Actually,becausethe functionU(C) is concave,the assumptionof differentiabilityis not even requiredhere, since the singularpointswherethe secondderivativesfail to exist or are not continuoushave measure zero in the relevant domain. However,the slight gain in generalityof recastingthe paperwithoutany differentiabilityassumptionsis not worththe notationthatis therebyrequired.But it couldbe done! messy andexcessivelymathematical ? The editors of the Scandinavian Journal of Economics 2001.

22

M. L. Weitzman Now, by the definition (20), Pi(Ci) = p'.

(A9)

Making use of (A8) and (A9), expression (A7) can be transformed into the equivalent form op2

C2

P1(C).-dC = Op2. C2 pl .C1

_

(A10)

Dl(p)? dp

Next, substitute (A10) into (A6) into (A5) to yield the equation

U(C2)-

U(C1) = A1 p2 C2 -p

p2

C1 -

D1(p) d

.

(All)

Jpl

Finally, substitute (Al 1) into the RHS of equation (A4) and use (A2) to obtain the expression p

U(C*2(t))e-Pt

dt-

dt

U(C*l(t))e-pt

= j1 Op2 c2 + Oq2 12 _ pl . C1 - ql

1 -

Jp2

I

DZ(p) dp

(A12)

Using (7) to abbreviate (A12) and rearranging terms, we have, at last, equation (24), which is the result desired to be proved. I

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? The editorsof the ScandinavianJournalof Economics2001.

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