Rivista di diritto finanziario e scienza delle finanze. 1997, Anno 56, giugno, n.2

Spedizione in a.p. - 45% - art. 2 comma 20/b legge 662/96 - Filiale di Varese

GIUGNO 1997 Pubblicazione trimestrale Anno LVI - N. 2

RIVISTA DI DIRITTO FINANZIARIO

E S C I E N Z A D E L L E F I N A N Z E

Fondata da BENVENUTO GRIZIOTTI

(e R IV IS T A IT A L IA N A DI D IR IT T O F IN A N Z IA R IO )

D I R E Z I O N E

EMILIO GERELLI - GIULIO TREMONTI

C O M I T A T O S C I E N T I F I C O

E N R IC O DE M IT A - A N D R E A FED ELE - FRANCESCO FO RTE AMEDEO FO SSA TI - FRA N C O G ALLO - SA LVATO RE LA ROSA IGNAZIO MANZONI - GIANNINO PARRAVICINI - ANTONIO PEDONE

SERGIO STEVE

C O M I T A T O D I R E T T I V O

ROBERTO ARTONI - FILIPPO CAVAZZUTI - AUGUSTO FANTOZZI G. FRANCO GAFFURI - DINO PIERO GIARDA - EZIO LANCELLOTT1 ITALO MAGNANI - GILBERTO MURARO - LEONARDO PERRONE E N RICO P O T IT O - P ASQ U ALE RUSSO - GIULIANO TABF.T

FRANCESCO TESAURO - ROLANDO VALIANI

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Pubblicità inferiore al 45%

INDICE-SOMMARIO

P A R T E P R I M A

Mauro Mark - Carmelo Salleo- Antitrust Enforcement and Rent Seeking: A Strategie Approach ... ...

Anna Cristina D ’Addio - Efficiency Analysis of Public Transport In Italian Cities: A Non-Parametric Approach ...

Franco Fichera - L ’armonizzazione delle accise ...

Alvaro Rodriguez Berelio- Le sanzioni fiscali in Spagna ...

A P P U N T I E RASSEGNE

Piergiorgio Valente- Riorganizzazioni d’impresa: rassegna delle operazioni ri­ comprese nell’art. 10 L. n. 408 del 1990 ...

NUOVI L IB R I .... ... ... ...

RASSEGNA D I PU B BLICA ZIO N I RECENTI ...

P A R T E S E C O N D A

Giulio Tremo nti - In materia di inammissibilità di referendum sulle leggi tribu­ tarie ... Francesco Tesauro - In tema di difesa tecnica nel processo dinanzi alle commis­ sioni tributarie ...

SENTENZE A N N O TATE

D iritto tributario costituzionale - Referendum - Art. 75, comma 2, Cost. - R ite ­ nute alla fonte - A rtt. 23 e 25, comma 1, D .P .R . n. 600 del 1973 - A broga­ zione - Ablazione e versamento di somme trattenute dal datore di lavoro - Apprestamento di mezzi necessari al fabbisogno dello Stato - Legge tribu­ taria sostanziale - Sussistenza - Richiesta referendaria - Inammissibilità (Corte Cost., 10 febbraio 1997) (con nota di G. Trem o n ti) ... D iritto processuale tributario - Assistenza e rappresentanza del contribuente -

Mandato conferito a procuratori e avvocati - Lim ite territoriale - Obbligo di elezione di domicilio - Sussistenza (Comm. Trib. Prov. di Trieste, Sez. IV , 7 gennaio 1997).

sintesi dell’esercizio 1996

Ü Gruppo Generati si consolida sul mercato intemazionale premi per 35 mOa miliardi utili di 1.438miliardi

il Gruppo Generali in cifre 101 compagnie di assicurazioni attive

in 50 Paesi

61 società finanziarie, immobiliari e agricole consolidate

126 società controllate diverse non consolidate

35.000 miliardi di lire di prem i (+10,7% sul 1995)

112.336 miliardi di accantonamenti tecnici

121.700 miliardi di investimenti

1.43S miliardi di utile consolidato

40000 professionisti dell'assicurazione al servizio della clientela la crescita dell’utile in m iliardi d i lire 3 ® 1994 1995 1996 Il risultato del bilancio consolidato beneficia del­ l'importante plusvalenza realizzata con la cessione delle azioni AXA; senza considerare tale effetto l’u­ tile sarebbe dell'ordine di 800 miliardi di lire, con un

aumento del 14,3%. ripartizione

dei premi consolidati

vita danni

Ita lia 6.399

L’attività del Gruppo

Nel 1996 la Compagnia si è mossa coerentemente con le linee guida degli ultimi anni. In Italia, con l’acquisto del gruppo Prime, dotato di un’articolata rete di prom otori spe­ cializzati nella distribuzione di prodotti finanziari e previdenziali, sono aumentate le potenzialità nel mercato del risparmio. Un ulteriore passo per una gestione integrata delle esigenze finanziarie e previdenziali della clientela sarà rappresentato dalla costi­ tuzione di una banca telematica, che opererà a supporto delle reti di vendita in tutto il territorio nazionale. Nuovi accordi di collaborazione sono stati sottoscritti con Isti­ tuti di credito che, aggiungendosi alle intese già in atto con primarie banche, hanno ampliato la capacità distributiva della Compagnia e del Gruppo.

In Francia è stata ceduta, da una compagnia interamente controllata, la quota di par­ tecipazione in Axa, che non rivestiva più interesse strategico, realizzando una forte plusvalenza ed una elevata liquidità che verrà impiegata nei piani di sviluppo del G ruppo a livello intemazionale. Nel mercato transalpino è proseguita la razionalizza zion e delle strutture operative delle diverse Com pagnie e d è stata deliberata la fusio­ ne per incorporazione di France IARD nella Concorde.

Un’importante acquisizione è stata completata nei primi mesi di que­ st’anno in Israele o v e è entrata a far parte del Gruppo Genera­ li la compagnia Migdal, società leader del mercato, la quale a sua volta controlla altre quattro com pagnie di assicu razione. Le Generali hanno così assunto una posizio­ ne preminente in un’area che presenta interessanti prospettive di sviluppo e redditività. In Austria, venuta meno la possibilità di partecipare alla privatizzazione della Creditanstalt, è stato sti­ pulato quest’anno con le tre maggiori banche regionali un accordo che prevede la sottoscri­ zione di una quota del loro capitale e la com ­ mercializzazione dei prodotti assicurativi e dei servizi finanziari del Gruppo. Un’altra iniziativa intrapresa nel settore della bancas- sicurazione è quella avviata in Brasile, assie­ me al Banco Sudameris, con la costituzione di una società che opererà nel settore vita e pensioni attraverso gli sportelli del Banco. È proseguita la graduale espansione della Com­ pagnia e del Gruppo nei territori che presenta­ n o buone prospettive di sviluppo assicurativo: alle presenze già consolidate nei Paesi europei cen­ tro-orientali e in Estremo Oriente si sono aggiunte due nuove società, una nel corso del 1996 in Slovenia e l’altra nei primi mesi del corrente anno nella Repubblica Slovacca, mentre a Pechino è stato aperto un ufficio di rap­ presentanza, passo preliminare per ottenere l’autorizzazione all’e­ sercizio d ell’attività assicurativa in Cina.

I risultati della Capogruppo

L’assemblea degli azionisti delle Assicurazioni Generali S.p.A., capofila del Gruppo Generali, riunita a Trieste il 28 giu gno scorso, ha approvato il bilancio 1996, chiusosi con un utile netto di 519,7 miliardi (482,7 miliardi nel 1995), e la distribuzione di un dividendo unitario di lire 375 per azione (+10%, tenuto conto d ell’aumento di capita­ le effettuato nel 1996); il dividendo risulta, includendo il credito d ’imposta, di lire 585,9 e d è in pagamento dal 21 luglio 1997.

Proseguendo nella tradizionale politica di rafforzamento patrimoniale, i soci della Compagnia hanno deciso di accantonare alla riserva straordinaria 161 miliardi di lire prelevati dall’utile.

II Consiglio di Amministrazione post-assembleare ha confermato Presidente Antoine Bemheim, Vicepresidente e Amministratore Delegato Gianfranco Gutty e Vicepresi­ dente Francesco Cingano.

Oltre che in Italia il Gruppo Generali opera in Argentina, Australia, Austria, Belgio, Brasile, Canada, Colombia, Danimarca, Ecuador, Fgit- to, Emirati Arabi Uniti, Francia, Germania, Giappone, Gibilterra, Gran Bretagna, Grecia, Guatemala, Guernsey, Hong Kong, Irlanda, Isole Vergini, Israele, Jersey, Libano, Liechtenstein, Lussemburgo, Malta, Marocco, Messico, Nigeria, Olanda, Panama, Perù, Polonia, Portogal­ lo, Repubblica Ceca, Repubblica San Marino, Repubblica Slovacca. Romania, Singapore, Slovenia, Spagna, Stati Uniti, Sud Africa, Svizzera. Tunisia, Turchia, Ungheria.

(¡HI ITO

GENERALI

Com pagnie del G ru p p o Generali in Italia: Adria Vita, Agricoltura,

Rivista di diritto finanziario e scienza delle finanze, L V I , 2, I, 165-185 (1997)

A N T IT R U S T E N F O R C E M E N T A N D R E N T S E E K IN G : A S T R A T E G IC A P P R O A C H

of Ma u r o Ma r è and Ca r m e lo Sa l le o ( * )

University of Rome and Bank o f Italy and Harvard University

Summary: 1. Introduction. — 2. R ent seeking and game theory. — 8. The antitrust game with no rent seeking. - 8.1. Set up. - 3.2. The p ay off o f the firm. - 3.3. The p ay off o f the agency. - 3.4. Expected payoffs. 4. The game be­ tween agency and firm with rent seeking. - 4.1. The expected payoffs. 5. Stackelberg antitrust game with no rent seeking. - 5.1. The firm as a leader. - 5.2. The agency as a leader. — 6. Stackelberg antitrust game with rent seek­ ing. - 6.1. The firm as a leader. - 6.2. The agency as a leader. — 7. Policy im ­ plications. — 8. Conclusions. — References.

1. Introduction.

I t is a well-known result of the rent-seeking approach that the social cost of monopoly exceeds the deadweight loss (the classical Harberger triangle) and could be as big as the entire profit o f the mo­ nopolist. The standard model of antitrust enforcement (see for exam­ ple: Lee (1980a, 1980b), Besanko-Spulber (1989a, 1989b) and Spulber (1989)) considers only the classical deadweight loss (due to the markup) to derive the optimal policy for the public agency and the firms. In fact, an enforcement effort by the antitrust agency could re­ sult in an opposite rent-seeking effort by the collusive firms. These costs should be considered by a welfare-maximizing social planner in designing an optimal policy.

The aim of this paper is to use the rent-seeking theory in the context of the antitrust problem to explore how it affects the choices of an optimal policy by the agency in the game that takes place be­ tween the agency and the firm. In the second section, we briefly re­

(* ) A first version of this paper was written while the Authors were at the Depart­ ment of Economics at Harvard University. The authors are grateful to G. Ellison and A. Redone for helpful comments. Mare wrote sections 3, 5 and 6; Salleo wrote sections 2, 4 and 7.

call the main outcomes of the game theoretical approach to the rent seeking theory. In the third section, we develop a simultaneous Nash equilibrium game of antitrust enforcement and we study the equilib­ rium of the game with pure and mixed strategies. In the fourth sec­ tion, with the same simultaneous Nash framework we introduce ex­ plicitly a rent-seeking behaviour and we study how it affects the ef­ fort of the monopolistic firm and of the antitrust agency and the equilibria o f the game. In the fifth section, we analyse a Stackelberg antitrust game, where the agency can choose whether to move first, while in the sixth section, we study the same Stackelberg game with the presence of the rent seeking cost. W e show that the elasticity of demand and the expenditure of the agency are the crucial variables in deciding whether to behave as a leader or as a follower and whether to introduce an antitrust agency. In the seventh section, we outline the policy implications and we compare the Nash and Stack­ elberg equilibria with and without rent seeking. In the last section, after recalling some possible extensions we conclude.

2. Rent seeking and game theory.

As it is well known, Gordon Tullock in his seminal paper of 1967 (Tullock, 1967) suggested a rent seeking approach to the study of the welfare effects of monopoly, pioneering a field of research that since then has grown enormously. Previous work on the monopoly effect on welfare dates back to the classical article of 1954 by Harberger (Harberger, 1954), in which he estimated the social cost (welfare loss) o f monopoly for the United States. Harberger’s very conservative es­ timates o f this loss (less than 1 per cent o f US G N P ) raised a wide controversy on how the social cost of monopoly should be measured and on the methods to be used (1).

The debate was strongly reinvigorated by Tullock’s fundamental contribution which aimed to demonstrate that the welfare loss from the existence of a monopoly would largely exceed the classical Har­ berger triangle. In Figure 1, we have reported the demand function D for a product sold in a monopolistic market. In the case of perfect competition, we have, as usual, a competitive price P c and a quantity Or I f instead the producer can set a monopolistic price P m, we have a

— 166 —

167

welfare loss given by the triangle ABC, which indicates the reduction of the consumers’ surplus following the monopolized product. Before Tullock’s paper, the welfare loss due to the presence of a monopoly was identified with area WL, while the shaded area P rP mAC was in­ terpreted as an income transfer from consumers to the monopolistic producer, which therefore did not affect social welfare.

Tullock showed that a consistent share, if not all of the monopo­ list profits - the transfer from consumers to the monopolist, ie, the rectangular shaded area labelled with R in Figure 1 - would be lost in the process to gain the right to be the monopolist or afterwards in the effort to defend this right. The incumbent monopolist and the poten­ tial (or actual) entrants would consume these resources in the effort to defend their prospective rents. The chance to gain the right to be a monopolist and to be one for a long period of time is an activity that for a firm can well be worth a remarkable investment o f resources and expenditures - lawyers, lobbyists, etc. - which can completely absorb the expected value of the rents. In the same way, the govern­ ment can also be asked to invest resources in monitoring the potential monopolists and in obtaining benefits from this process - for example by collecting compensation fees for violations of the competition law. So the true social cost o f monopoly is given by the sum of the Har- berger triangle W L and of the Tullock rectangle R.

Since Tullock’s work, a vast amount of theoretical and empirical research has been produced (2). One major direction of research has tried to assess whether monopoly rents are fully wasted by the efforts of groups, firms and government (3). Another interesting field of re­ search has focussed on the market conditions which govern the rent seeking process: the existence of a competition for monopoly and the effects of this competition on the size of the rent and on its distribu­ tion among the different interest groups.

Finally, in the last decade, moving from the Tullock’s paper of 1980 several scholars have applied a game theoretical approach to the rent seeking process. In a first contribution, Lee (1980a) stressed the importance of the cost o f antitrust enforcement in assessing the opti­ mal antitrust policy. When setting guidelines for the antitrust agen­ cy, a cost-benefit approach should be applied to the specific cases:

(2) See among others, Posner(1975), Krueger(1974), Buchanan(1980) Tu l- i.ook (1980) and the two anthologies Buciianan-Toi.lison-Tui.lock (1980) and R o- w i.e y-To ixiso n-Tui.lock (1988).

— 168 —

moreover, the value of costs and benefits depends on the type o f the game played: a simultaneous Nash game or a sequential Stackelberg game.

In another interesting paper Block, Nold and Sidak (Block-Nold- Sidak, 1981) checked empirically how the collusive pricing practice of firms is affected by the antitrust enforcement policy: they found that the relevant variable is the credibility of the threat by the agency. Besanko and Spulber (Besanko-Spulber, 1989a) and Spulber (1989) developed some strategic models of antitrust enforcement under asymmetric information in a Stackelberg setting. They found th a t« it is optimal to induce a low-cost industry to engage in price-fixing » and that this is indipendent from the magnitude o f enforcement costs. They also extended to a dynamic framework the usual Nash and Stackelberg game and showed that over time the antitrust au­ thority may improve its policy by learning about the costs of firms and by building up a credible reputation.

More recently, Glazer and McMillan (1992), Linster (1993), Perez- Castrillo and Verdier (1992) and Leidy (1994) extended in some other directions the basic rent seeking games: in particular, to the analysis of the effects of technology on the outcomes o f various rent seeking games, and to how the value o f the prize for the different players af­ fects the choice to behave as a leader or as a follower in a Stackelberg game.

3. The antitrust game with no rent seeking. 8.1. Set up.

L e t’s suppose a market with a monopolistic firm and an antitrust agency. As usual, we define the social welfare function of the agency as follows:

W = r P(x,y)dx-Q q + a ( q , K ) ( A - K ) - c [ P ( q ) - d ] q [1] Jo

— 169 —

not considered as a waste given that it can be used in a productive way (eg. investment, compensation of victims).

In this first simple game, we make the case of a simultaneous Nash equilibrium: the agency and the firm choose their actions tak­ ing the strategy of the other party as given, so as to obtain a Nash equilibrium. In other words, the agency will choose K so as to maxi­ mize W (q,K ), the social welfare function, while the firm will choose the quantity q to maximize Yl(q,K ), ie., the profit function. There­ fore, the problem for the agency is to find K * (q * ) s.t. W (q *,K *) > W (q*,K ), while for the firm it is to find q * (K * ) s.t. Y l(q*,K *) > Yl(q,K *). W e assume that the usual individual rationality constraints apply: A > K and n > 0.

The antitrust agency maximizes social welfare by setting K and taking the firm’s optimal strategy as given.

The firm maximizes its net profit taking K as given:

E [Y l(q ,K )] = P (q )q - Qq - a(q,KJA - o [P (q )q - 9q]. [2] Rearranging we get:

E [T l(q ,K )] = P (q )q [1 - a ] - 6q f l - c ] - a(q ,K )A . [3] In Figure 2 we show the strategic form o f the game. The firm F can choose to behave as a monopolist (with probability 1 — P) or as a competitive firm (with probability P).

The profit functions of the firm in these two cases are respective­ ly P (q )q M - CqM for a monopolist and Pq - Cq for a competitive firm. Given this choice of the firm, the antitrust agency can choose whether to enforce or not (with a probability a and 1 — a respective­

ly)-Horizontally, we have the firm choice between competition ( C) and monopoly (M ), while vertically we have the agency choice be­ tween to enforce (E ) and not to enforce (N E ). As usual, the first hori­ zontal figure in the cells of the matrix denotes the payoff to the firm, while the other the payoff to the agency. For simplicity, let define the social welfare in the case that the firm chooses C (a competitive firm) as follows: SiVc r qc / P(x)dx - Qqc Jo [4]

and the social welfare in the case of a monopolistic firm as follows:

rqM

SWM = / P(x)dx-QqM. Jo

170 —

W e assume the usual individual rationality constraint for the agency: 0 < K < A. Moreover we define with nc and km the two levels o f the firm profits in the case of a competitive and monopolistic be­ havior. The individual rationality constraint for the firm is the fol­ lowing:

P (q )q M -

0<7v/ >

9c >

e'/.u -

[6]

In Figure 3 we have simplified the payoffs in the cells: with nc we denote the payoff when the firm behaves as a competitive firm (this payoff is indipendent from the agency’s choice to enforce); with nAI the payoff when the firm is a monopoly; if the monopolistic firm is detected the payoff becomes Km — A; SWr — K is the social welfare when the agency enforces controls on a competitive firm, while SWc and SWM are the payoffs when the firm is not detected; finally SWM + A - K\s the payoff of the agency when a monopolistic firm is de­ tected, net of the cost o f detection K and gross of the fine A. For completeness, we also show in Figure 4 the usual extensive form of the game.

W e assume the usual participation constraints:

•

K\i

>

K

(■

>

K

—

A\

• SWm < SWr ; • 0 < K < A.

The first constraint implies that it is convenient to behave as a monopolist only when there is no enforcement by the agency; the sec­ ond constraint says that the social welfare of competition is greater than that of monopoly; the third constraint ensures that the agency obtains a positive net benefit from enforcement. By inspection, it is easy to see that there are no Nash equilibria in pure strategies. There­ fore we look at the equilibria in mixed strategies.

3.2. The payoff of the firm.

The expected profit of the firm (in the case of a competitive and a monopolistic behavior) is the following:

n Ff = P/a7Cr + (1 - a)K c] = P«c [7]

and

n iV„

= (1 - $) [a (n M - A ) + (1 - a )nMJ

= (1 -

p;

[n M - a A ].

[8]

Therefore the total expected profit of the firm is:

— 171

I f we maximize the expected profit with respect to p we get the prob­ ability a:

^ = n c - K M + a A = 0

°P

[

]

from which we get:

a = Km — TCc

[

]

which we assume to be smaller than 1. Given that the profit of the monopolist is inversely correlated with the elasticity of demand, we notice that the probability o f enforcement a will increase as the elas­ ticity of demand decreases, ie, when there is more scope for anticom­ petitive behavior.

3.3. The payoff of the agency.

Similarly, the expected payoff for the agency (A T ) in the case of enforcement is

A Te = o [$ (S W C - K ) + (1 - p; (SW M + A - K )J [12] and

A Tne = ( 1 - 0 . ) fflSWc + (1 - P)S W M] [13] when the agency does not enforce. Therefore, the total expected pay­ off for the agency is:

E [ A T ] = pSWc + (1 - p )S W M - a K + a (1 - p )A . [14] B y maximizing the expected payoff with respect to a we obtain the probability P:

f) = ± ^ < l . [15]

— 172 —

decision-making process and therefore it chooses its strategy by com­ paring net and gross fine.

3.4. Expected payoffs.

Replacing the equilibrium probabilities a and P in the equations [9] and [14] we get the equilibrium expected payoffs:

E[Yl] =

+

-

K

-

K

~

A ~ nc

t10]

for the firm, and

E [A T ] = ^ 5Wc + 1LSWm _ X"-a *c_k + x m- xjl JL A [17]

= p S W c + ( I - P ) S Wm

for the antitrust agency. In a simple Nash game between a firm and an antitrust agency, the firm makes ex ante only the profit of compe­ tition nc. Apparently, the introduction of the antitrust agency has a positive effect on the firm behavior; however, in our game 7tc is not the lowest profit attainable, given that if the firm behaves as a mo­ nopolist and is caught, its payoff is lower (nM - A , see the participa­ tion constraints). The expected payoff of the agency is the sum of so­ cial welfare in competition and monopoly weighted by the respective probabilities.

Note that in this first set up, we have assumed that A and K are exogenous. W e could also consider the case where the agency chooses A or K, ie the case where the control power over the choice variables is potentially in the hands of the agency.

4. The game between agency and firm with rent seeking.

— 173 —

4.1. The expected payoffs.

The total expected profit of the firm is the following:

E[TlrJ = pjic + ( l - p; U (n M( l - a ) - A ) a ] + ( 1 - a) n M(1 - oJ}.[18] By deriving the expected profit with respect to p we obtain the prob­ ability of enforcement a:

km ( l - o ) - n c [19]

LX>rs = 1 ■■ < J'

A

The effect of rent seeking is to reduce the probability o f enforce­ ment by the agency (oc„. < a ) and to reduce its sensitivity to the elas­ ticity of demand.

The expected payoff of the agency is:

E [A T ] = a/p(SW C - K ) + (1 - p j (S W M - onM - K + A ) ] + +(1 - a ) [$SW C + (1 - PJ (S W „ - onM) ]

= P»SWr + (1 - PJ SWM - o ( l - $)nM + a ( l - fi)A - aK. [20] By deriving the expected payoff with respect to a we get the proba­ bility of competitive behavior p:

P " = A j r - t21i

Note that the rent seeking parameter a enters only in the deci­ sion of the agency; furthermore P„s = P ie, the firm does not consider the rent seeking cost in deciding its behavior.

By replacing the equilibrium probabilities in the respective payoffs we obtain the equilibrium expected values:

£[n„] =

- j I*« 0 - °) -

(i - c> “

[

]

for the firm, and:

T—r , -r. A - K K _ K _ nM ( \ - a ) - n c K

E[ATrs] = — ^— SWc + SWm — <7— km + --- —--- ~A A +

tcm (1 — o ) — Jtc K

PSWc + ( I - P ) S Wm ~ ( 1 ~ P ) o nM [23]

repli-— 174 repli-—

cates the classieal Tullock result where the monopoly rent is com­ pletely dissipated by the firm’s rent seeking effort.

5. Stackelberg antitrust game with no rent seeking.

W e have modelled in the previous section the case where no player knows the story o f the game and the opponents move as in a Nash game with a simultaneous equilibrium in mixed strategies. But it is realistic to reckon that one of the two players may have an infor­ mational advantage or simply be asked to move first. The simplest way to model such a situation is to analyze a Stackelberg game where one of the two actors is asked to move first (leader). W e begin with­ out considering expliticly the rent seeking effort. In this framework, we will always assume that the information is perfect and complete: ie, that one actor knows always the move of the other in the previous stage of the game (4).

5.1. The firm as a leader.

In Figure 6 we show the extensive form of the Stackelberg game where the firm moves first and the antitrust agency is the follower. As before, the firm can choose to behave competitively where it gains a payoff Itc - or as a monopolistic firm - where it gains 1tM if is not de­ tected and nM - A if detected - while the agency has to decide whether to enforce or not. W e solve the game by backward induction.

In the left tree o f the game, the antitrust agency chooses in the second stage whether to enforce or not: it compares the payoff S W f - K when it enforces with the payoff NW(- when it does not enforce. Given that, by assumption, iSWr > SWc — K, it will choose not to en­ force, NE. Similarly, on the right tree o f the game, the agency com­ pares whether to enforce or not according to the following payoffs: N W w + A - K, if it enforces, and SWM, if it does not enforce. Given that SWM + A - K > SWM, it will choose to enforce, E.

The firm anticipates the behavior of the agency: it knows what the move o f the agency will be in the two cases and therefore it com­ pares nM - A with itc. Given that nc > nM - A it will choose to behave as a competitive firm and get the payoff nc.

Note that the equilibrium outcome is determined by the fact

— 175 —

that the firm anticipates that it will be always be fined if it behaves anticompetitively. Thus, the key assumption is that the agency is perfectly able to discriminate between competitive and monopolistic behavior. In this case, the outcome is a first best.

5.2. The agency as a leader.

In the second stage o f the game, the firm has to choose between nc and nM - A and nr and (see Figure 7). Given the usual assump­ tions, it will choose, on the left side, nc (nc > ft.v - A ) and nM (nM > 7tr ) on the right side. The agency anticipates this choice and therefore it chooses between SWC - K and SWM.

The decision to enforce regulation requires a slightly different in­ terpretation: it should be viewed as a committment to act in case of violations; it can be assimilated to an investment. It is not evident which of the two payoffs is greater; The agency will enforce only if SWC - K > SWM.

Enforcement is more probable when the demand is elastic. In fact, when r| (the elasticity of demand) is high, although the agency has to pay a cost K o f enforcement, the difference between the social welfare of monopoly SWM and the social welfare of competition STT, is likely to be greater than K. This result confirms, on the other hand, that it is not always good to have an antitrust policy: in presence of an inelastic demand, it could be better not to have an antitrust agen­ cy at all (5).

Here, the key assumption is that the agency’s commitment to enforce is a credible threat; that is, the firm must be convinced that the agency will go after any non-competitive behavior. In this case, the optimal policy for the agency and the maximization o f social wel­ fare depend both on the sector of activity that is considered. In sec­ tors with high elasticity of demand, there will probably be firms that behave competitively, and an antitrust agency credibly committed to enforcing regulation.

6. Stackelberg antitrust game with rent seeking.

In this section we study how the rent seeking effort in a Stackel­ berg framework affects the equilibrium of the game. As before, we

— 176 —

consider two particular situations: in the first, the firm is the leader and the agency is the follower; in the second, the roles o f the players are reversed.

6.1. The firm as a leader.

In Figure 8 we describe the extensive form o f the Stackelberg game when the firm plays as a leader and the agency as a follower. In the second stage, the agency has to choose whether to enforce (SWC -

K) or not to enforce (SW(-), on the left tree, and whether to enforce

(SWM - anM - K + A ) or not to enforce (SWM - anM), on the right

tree.

Given the previous assumptions, we have that SW(: > SWC - K and SWy - any - K + A > SW y - any: therefore, the agency chooses N E on the left side and E on the right side. The firm anticipates this

result and chooses to behave as a competitive firm, given that nc > nM — any — A.

6.2. The agency as a leader.

In Figure 9, we show the extensive form of the Stackelberg game when the agency plays as a leader and the firm as a follower. In the second stage of the game, the firm chooses to behave competitively C (7tr > nM - an - A ) on the left tree. On the right tree, the decision de­ pends on the magnitude of any, ie, on the fraction o f profits that goes

wasted in rent seeking efforts. The probability of choosing a monopo­ listic behavior increases as the elasticity o f demand becomes smaller: it is plausible to assume in this case that ny - an y > nc and therefore

that the firm chooses M.

The agency anticipates the choice o f the firm; if the firm chooses to behave competitively, C (in case of high elasticity o f demand), the agency does not enforce (N E) because <SWr > iSWf — K. I f the firm

chooses to behave as a monopolist, the decision of the agency is less straightforward. The agency will choose to enforce if

SWC - K - SWy + any > 0. [24] Notice that NITC - SW y = W L (welfare loss from monopoly: see Fig­ ure 1), therefore equation [24] can be rewritten as:

W L + any > K. [25]

— 177 —

I f it is high, the welfare loss W L is also very high, while the prof­ it of the monopolist nM is low; if the elasticitiy is low, W L is low and nM is high. In both extreme cases, the agency is likely to enforce (see Figure 10). For intermediate values of T|, the agency could decide not to enforce if K is high enough; that is, the cost of enforcement could be so high as to discourage a policy of enforcement.

7. Policy implications.

In a Nash setting without rent seeking, the firm’s expected pay­ off is equal to the profit of competition; the cost of the option to act as a monopolist is reflected in the social welfare, which is a weighted average of the social welfare for the monopolist and competitive out­ comes.

The introduction of rent seeking does not change the payoff of the firm; however, social welfare is reduced and can even be lower than the social welfare in the case o f monopoly. This suggests that for sectors with low elasticity of demand regulation through an antitrust agency can be suboptimal and should be carefully sized.

In a more realistic Stackelberg setting, the presence o f rent seek­ ing becomes relevant when setting guidelines for the agency. I f there is no rent seeking and the agency can choose, it will always want to play follower and thus realize the first best outcome. But if the agen­ cy cannot choose (for example, if it is not credible that it can always ascertain the firm behavior) and is forced to act as a leader, then its policy must depend on the sector characteristics. It may be counter­ productive to invest resources in industries with low elasticity o f de­ mand; the increase in social welfare might be more than offset by the cost of the enforcement. In some cases a monopolistic behavior by the firm is the lesser of two evils.

The introduction of rent seeking by the firm does not only alter the payoffs; it changes the interpretation of the game. Now, the an­ titrust agency interiorizes the cost of rent seeking, which in turn is proportional to the profit of the monopolist. In a broader sense, de­ mand isn’t anymore the sole strategic determinant; supply enters the picture too.

— 178 —

In this case, the only way to avoid excessive intervention is to delegate the antitrust policy to various agencies, each specialized in a particular industrial sector. For the sectors with low elasticity of de­ mand, in which the presence o f an agency would lead to a suboptimal outcome, a different kind of intervention should be devised (eg, taxa­ tion of the profit of the monopolist).

8. Conclusions.

In this paper, we have applied a strategic approach to antitrust enforcement with rent seeking. First, we have modelled the case of imperfect information, where no player knows the story of the game and the opponents move as in a Nash game with a simultaneous equi­ librium in mixed strategies. Second, we have analyzed the case where one of the two players is asked to behave as a leader - the Stackelberg game.

In a Nash setting, the firm's expected payoff is equal to the prof­ it of competition both with and without rent seeking; the cost of the monopoly is internalized by the social welfare function, which is a weighted average of the social welfare for the monopolist and compet­ itive outcome. However, with rent seeking the social welfare is re­ duced and can even be lower than the social welfare in the case of monopoly. In sectors with low elasticity of demand regulation through an antitrust agency should be therefore carefully sized.

In a Stackelberg setting without rent seeking, we show that, with the assumption that the agency is perfectly able to discriminate between competitive and monopolistic behavior of the firm, the out­ come is a first best when the agency is a follower.

If, instead the agency acts as a leader, the decision to enforce regulation should be viewed as a committment to act. The key as­ sumption is that the agency’s committment to enforce is a credible threat; that is, if the firm is convinced that the agency will go after any non-competitive behavior. This result confirms that it is not al­ ways good to have an antitrust policy: in presence of an inelastic de­ mand, it could be better not to have an antitrust agency at all.

In the Stackelberg game with rent seeking, we show that the cru­ cial variables for the agency are the elasticity o f demand and the cost of enforcement. The general conclusions are that the presence o f rent seeking becomes relevant when setting guidelines for the agency.

— 179 —

interestingly developed. The first one is that of extending our game to n periods: for example, by considering repeated Nash and Stackel- berg games, where the timing dimension can affect the choice of strategies by the two actors.

Another simple extension could be that of introducing incom­ plete information in the structure of the game, where players do not know the payoffs of the opponents. One last extension could endoge- nize the relevant parameters: in particular, by setting that cost of rent seeking a as a function of the enforcement cost of the agency K.

References

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Rent-Seeking Society, Texas A and M University Press, College Station.

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Lin s t e r, B ., (1993), « Stackelberg R en t Seeking », in Public Choice, vol. 77, n. 2,

— 180 —

Ma r iS, M. and C., Sa l l e o, (1994), « R e n t Seeking and O ptim al A n titru st P o lic y »,

mimeo, H a rva rd U n iversity , June.

Mu e l l e r, D., (1989) Public Choice II, Cam bridge, C am bridge U n ive rs ity Press. Pe r e z-Ca s t r il l o, J. and T ., Ve r d i e r, (1992), « A General An alysis o f R e n t

Seeking Games », in Public Choice, vol. 73, no. 3, April.

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Sp u l b e r, D ., (1989) Regulation and Markets, Mi t Press, C am bridge, Mass. Tu l l o c k, G., (1967), « T h e W e lfa re Cost o f T a riffs, M onopolies and T h e ft » , in

Western Economic Journal, vol. 5, June.

— 181 —

Fi g. 1 P

On

Qc

Q

Fi g. 2. - The strategic game with no rent seeking. Agency

Enforce Non enforce

— 182 —

Fig. 3. - The strategic game with no rent seeking.

C

P

Firm

M

1 - P

— 183 —

Fig. 5. - The Nash game with no rent seeking.

P

Firm

M

1 - P

Fi g. 6. - The Stackelberg game with no rent seeking.

— 184 —

Agency Leader

A

Fig. 7. - The Stackelberg game with no rent seeking.

Fi g. 8. - The Stackelberg game with rent seeking.

Firm Leader

— 185 —

Agency Leader

F

Fig. 9. - The Stackelberg game with rent seeking.

SWc-k

sWM- c n M+A-k

I f T] low —^ Tljyj — 07Cm TCc I f T| high —> TZq > 7C]yj — CT7C^j

Rivista di diritto finanziario e scienza delle finanze, L V I , 2, I, 186-215 (1997)

E F F IC IE N C Y A N A L Y S IS OF P U B L IC T R A N S P O R T IN I T A L IA N C ITIE S : A N O N -P A R A M E T R IC A P P R O A C H

di Anna Cr istin a D ’Au dio (*) Institut de recherches économiques et sociales - 1rese Core Université Catholique de Louvain

Su m m ary: 1. The problem. — 2. Non-parametric analysis o f efficiency. - 2.1. Data envelopment analysis and free disposal hull methodologies. - 2.2. Mathematical programming formulations o f efficiency measurement. 3. The empirical anal­ ysis. - 8.1. Description o f the data. - 3.2. Description o f the variables. - 3.3. Non-parametric estimation. - 3.3.1. Residts o f FD H m ethodology. - 3.3.2. Results o f D EA m ethodology. — 4. Conclusions. — References. — A p ­ pendix.

1. The problem.

The sector of public transport in Italy must be studied with par­ ticular attention. Public intervention in it is justified with the fact that a monopolistic regime would be non optimal: moreover to make transport easier is one o f the factors that improve the economic devel­ opment o f a society.

Public transport utilities may be seen as the way by which ev­ eryone can satisfy his needs of mobility regardless to his revenues and these needs are to be considered as essential nowadays.

In our society public transport has been considered a merit good w'hose consumption has been promoted through low user charges and conspicuous grants to the public utilities producing the service. This policy has led, through the use of ex-post subsidies, to a constant waste of resources and in particular to the big deficits and the finan­ cial crisis public transport utilities have to face. Moreover the growth of traffic congestion in urban areas, the duplication o f services in

— 187

towns where there is a low density of population, the need to allocate more rationally the resources, show that the way to improve the situ­ ation is to assess policies in which efficiency will play a central role. The need to rationalise the intervention in the sector of public transport has led to the emanation of the law n. 154/81, without ef­ fects, that stated the necessity to get the balance between costs and revenues in the following five years. The subsidy from the Central Government was determined ex-ante through an yearly allocation to the Fondo nazionale Trasporti which was divided among the urban transit companies. The grant from the Central Government represent­ ed the compensation to companies to provide the service at a price lower than its cost. A ll the other costs should have been covered with the revenues coming from the sale of the transport service. Other laws had, afterwards, the same target: in particular we may remember the art. 2 of the law n. 4031990 that obliged public utilities to improve their management before the 31st December 1996.

Local transport providers have to improve their efficiency with­ out delay. The observation o f the sharp increase in costs o f produc­ tion along with a level of output almost unchanged does not seem to have any rational economic explanations. This justify the need to study this problem more deeply.

In this paper I shall evaluate the efficiency of public transport units a non-parametric approach, a purpose being to compare the re­ sults obtained by each of them. The methodologies used are the fol­ lowing:

— Data Envelopment Analysis; — Free Disposal Hull.

DEA and F D H (1) are procedures that allow to evaluate whether inputs are used to produce output efficiently without taking as given a specific parametric functional form. As already said both of them are non-parametric ones but the assumptions used are quite dif­ ferent and lead to the obtention of different results that are worth to be examined. The paper will be organised in the following way: in the second section I will present the two methods, to point out the differ­ ences and the advantages of both of them; in section three the results of the analysis will be presented and the last section contains the con­ clusions.

— 188 —

2. Non-parametric analysis of efficiency.

In the formulation and estimation of frontiers, research has re­ cently focused on three problems. First, since it is the production (cost) frontier rather than some fitted « average » function that corre­ sponds to the theoretical notion of a production (cost) function the interest has been concentrated on the determination of the shape and location of the frontier. Second the research has been concentrated on the nature of the relationship between the frontier and the fitted « av­ erage » function (2). Finally, and most importantly frontiers have im­ plications for the computation of efficiency of production. Although under certain conditions a fitted « average » function allows a rank­ ing of observations by technical efficiency, it provides no quantita­ tive information on technical inefficiency in the sample. I he estima­ tion of the frontier carries to the determination of the actual magni­ tude o f technical inefficiency.

One o f the assumptions of the neo-classical theory of the firm is that decisions taken by the managers are based on the maximisation of the profit. This target is reached by using inputs in the most effi­ cient way (technical efficiency) and by producing the quantity of out­ puts that allows the maximisation of the difference between revenues and costs (allocative efficiency). I f these conditions are satisfied the firm is said to operate at the boundary of its production and cost frontier.

The way in which econometricians look at the treatment of fron­ tiers has undergone substantial modifications in recent years.

As said before the building of frontiers is very useful for the com­ putation o f efficiency. The earliest work on frontier assumed what we would call a deterministic frontier. The first empirical treatment of it was developed by Farrell (1957). So from that moment on efficiency studies have been become synonymous with Farell efficiency mea­ surement. The method of Farrell consisted of the measurement of the technical efficiency o f the observed firms in relation to a frontier of possibilities that is not a theoretical curve but a statistical one, it is built starting from the observations, meaning that only the best of them are drawn from the sample.

189 —

In parallel to the growing literature using parametric estima­ tion (3) another part o f efficiency analysis is rapidly developing and corresponds to Data Envelopment Analysis { DEA) and Free Disposal Hull (F D H ) methodologies.

Both of them are non-parametric ones which do not require a priori the form of technology used. This implies that normally the re­ vealed technology will be a closer estimate of the true one underlying the data; moreover in employing a series of optimisation, one for each individual unit ( DMU) , non-parametric procedures provide a better fit to each observation and a better approximation to the scale prop­ erties of individual D M Us.

2.1. Data envelopment analysis and free disposal hull methodolo­ gies.

In the following paragraph I will describe the postulates on which each methodology rests. In particular, as shown afterwards, for both of them it is possible to compute the degree of efficiency of a firm according to the assumption either of constant or of variable re­ turns to scale (4). Most of the researches that have been carried over in the sector of public transport agree on the assumption of constant returns to scale in the short-term and of variable returns when con­ sidering the long-term. I will use F D H and DEA methodologies to carry over the empirical analysis in both cases; the results obtained will be useful to have an idea of the kind o f returns to scale existing in the sector o f public transport in Italy.

In all empirical studies of productive efficiency a central rôle is played by the choice of the « reference technology » that is the pro­ duction possibilities set whose frontier is used to evaluate the ob­ served productive activities (5). The results obtained are indeed quite sensitive to the alternative specifications of this set.

The purpose of this paper will be to consider the economic pro­ duction-theoretical assumptions on which the F D H methodology rests and the question of how efficiency computations of F D H com­ pare with the linear programming techniques used in the DEA liter­ ature.

(3) I want to refer to methodologies that are based to the formulation o f a spe­ cific functional form.

(4) I t is necessary to point out at the fact that the methodology concerning the computation o f FD H efficiency degree in the case o f constant returns to scale has been applied here for the first time.

— 190 —

The production set that economic theory associates with any productive activity is usually unknown; this means that the person who is interested in efficiency analysis must construct the set he needs. I f he does this from statistical data on outputs produced and inputs used his method is necessarily based on a relationship between the statistical observations and the elements of the set that has been built. W hat I will point out here is that differences in the assump­ tions on which the relationship is based, cause differences in the dif­ ference sets used and hence in the results obtained through efficiency measurements.

In F D H methodology, performance is evaluated in terms of dominance and o f efficiency. Dominance is measured by simple counts of the number o f public transport utilities providing no less of each service with no more of each resource, and of the number of public transport utilities providing no more of each service with no less of each resource. Being undominated is associated with best prac­ tice operation, while being dominated implies the existence of at least one better performing company that can serve as a role model. In each case the number of dominating or dominated urban transit com- paniescredit provides an important component of the performance evaluation.

The efficiency o f a public transport company is measured by its proximity to best practice operation, as defined by the resource usage and the service provision of the most dominant of the public trans­ port utility that dominates it. Efficiency quantifies the dominance re­ lationship by showing how much improvement is feasible in each re­ source and service dimension.

Given a set Y0- [(a t , if ) I k=l,...,n] of n observed production plans of a given organisation, where x* is an /-dimensional nonnega­ tive vector of the quantities of I inputs used and f is a J-dimensional nonnegative vector of the quantities of J outputs achieved, this de­ notes a set of n actually observed production plans; using the follow­ ing two postulates (See Tulkens (1993)):

(i) : (determinist postulate) every observed production plan belongs to the constructed production set

— 191 —

we obtain a « free disposal hull ( FDH) exhibit reference production s e t»(6 ) (because these postulates make this set to be the « free dispos­ al h u ll» of the observed plans), YFDH, built from Y0 (defined before as being the set of observed production plans) that can be written as fol­ lows

where e{ denote an /-dimensional zero vector with the ith component equal to 1 and similarly ef denotes a /-dimensional zero vector with the jth component equal to 1.

The first postulate makes the ensuing methodology o f efficiency measurement a deterministic one; the second is the one of free dispos­ al of inputs and of outputs, from which the name of the set is de­ rived. It may be expressed in terms of « dominance »; that is to say that in the previous set, observation (cch,yh) weakly dominates ( x,y)) in inputs if 3 i:\Lf>0, and in output if 3 j:\p-0. The values taken by the variables |i, (denoting disposal of the inputs) and Vy (disposal of the outputs) thus express the free disposal assumption.

The set of feasible production activities consists of all possible production activities having resources no smaller than, and services no larger than, those o f each public transport company in the sample. Yfdh( Yo) is in fact the smallest possible set of feasible production ac­ tivities that contains all observed data and satisfies the property of free disposability of resources and services.

The previous definition refers to the methodology known as FDH -V, where the V implies the fact that we are making the as­ sumption of variable returns to scale. In the case of constant returns

192 —

to scale (FDH-C), the relative expression is written in the following way:

Yfdhc(Y<>)

e / r

■B' +I> ^ -B

In this expression, the value k that may take any value in the inter­ val (0,°°); when k=l returns to scale are variable (7).

In the two-dimensional case (one output and one input) the re­ sults of FD H -C will be equivalent to D EA-C ones. As we will see lat­ er, DEA methodology rests on, beyond the others, the postulate of convexity that is excluded in F D H: this makes results different when the analysis is carried on in the multi-dimensional case (more than one input and more than one output) (8).

In the DEA literature, efficiency is measured by referring to pro­ duction sets, similarly constructed from the data, and using a third postulate in addition to the two mentioned above. The third postu­ late differs, however, depending upon which DEA postulate is used: the one o f Deprins and Simar (1983) or the original of Farrell (1957) and Charnes, Cooper and Rhodes (1978) (9).

(iii-1): (convexity postulate) Every non observed production plan that is a convex combination of production plans induced by postulate ( i ) , ex­ cluding the origin, and by postulate ( i i ) , also belongs to the constructed production set (10).

The set induced by this postulate is to be called the « convex free disposal hull » of the data. Because the origin is excluded from the da­ ta set, the constructed set does not contain the origin either. The

(7) W hen k > l we have FD H -N D RS (N on Decreasing Returns to Scale); when

k < l we have FD H -N IRS (N on Increasing Returns to Scale). (8) This can be shown graphically (see the appendix). (9) W e examine them in chronologically reverse order.

— 193

piecewise linear frontier of this set is interpreted as exhibiting « vari­ able » returns to scale ( VRS) (11).

(iii-2): (convexity and partial proportionality postulate) Every non ob­ served production plan that is a convex combination of production plans induced by postulates ( i ) and ( i i ) also belongs to the constructed produc­ tion set (12).

In this case the set contains the origin and it means that the piecewise linear frontier can only exhibit constant returns up to some point and then decreasing ones. This specification will be identified by the term « CD ».

( iii- 3 ): Every non observed production plan that is a linear combination of production plans induced by postulates ( i ) and ( i i ) also belongs to the constructed production set (13).

Because o f the proportionality allowed by the weights of the lin­ ear combination, this set is a cone, to be called « free disposal cone » constructed from the data. - The linearity of its frontier implies that it exhibits constant returns to scale, hence the subscript « C ».

I t can be seen that for a given data set Y0: Y F D H - V ^ Yd e a . j < Z Y FDtf.cC FD E A C l f — Y Ü E A C

-The postulates have, thus, been stated in order of increasing strength. Formally the relative strength of the postulates is expressed by the alternative restrictions put on the weights yh and on the con­ vention to include or not the origin in the observation set.

As regards the convexity assumption, the F D H methodology shows that it may be excluded, without loosing the economically es­ sential element of efficiency analysis, that is to say a well defined ref­ erence production set. Moreover it is worth to underline that while in DEA methodology the efficiency degree is computed by the relative distance between the actual observed production and the nearest benchmark production, in F D H one, efficiency appears to be a prop­ erty of an observation when compared to the other observations.

2.2. Mathematical programming formulations of efficiency mea­ surement.

The data used in the empirical analysis are in cross-section form, therefore we have a number of observations at time t.

(11) This is only true in the special sense that returns are icnreasing (I) only in the lower range o f the inputs up to some point, and are decreasing (D) beyond. This set will be identify by the letter V.

(12) Deprins and Simar, (1983).

— 194 —

Let us recall that radial efficiency measures with respect to the D E A -F reference technology are obtained as follows. The efficiency degrees in inputs of observation k is the value 0‘ * of the optimal solu­ tion of the linear programming problem:

M in 9* subject to

(0* Y,h=l,..n}

&x';

-YyV ¿o ;=i... /

n A-1

Y f y) ¿yf i=

km

<9‘ 7‘ > 0 h = l , . . . . , n

and the efficiency degree in outputs of this observation is obtained as the value of ( 1\’kK* ), where XK* is the optimal solution of the pro­ gramming problem: Max XK h = l,. : , n } < x ‘ A - l

1=1

1...

I t is well known that adding to either one of these constraints

iy<-> [ i ]

*»i

amounts to take YDEA.cn as the reference set; changing further this con­ straint into

ir-1

[

]

— 195 —

and

T r - i

ye {0 ,1}, h=l...n

radial efficiency measures are obtained with respect to the reference set Yfdh.

The interpretation for D EA is quite simple. The kth branch is relatively efficient if and only if the efficiency ratio 0k equals unity and the slacks are equal to zero. DEA also allows to evaluate how the inefficient units could be made more efficient. This can be done using the concept o f the peer group. Best practice units could in fact be used as peers for the improvement o f inefficient production. The peer group for an inefficient unit can be defined as the set of best-practice establishments who are likely to be producing at a broadly similar scale than the inefficient unit. The target for this one is then a weighted average of the performances of the best-practice.

The problems concerning DEA can be computed using the appa­ ratus of the standard linear programming algorithms. In the case of F D lith e problems are seen to be ones of mixed integer programming: the standard algorithms do not apply. Solutions are obtained by means of a simple vector comparison procedure.

Given the observation to be evaluated (x?,yk) it is necessary to associate with it the set U ( k ) containing the indices of the observa­ tion (x k,yk) itself and of the subset of observations that weakly domi­ nate it in inputs, that is, the subset of vectors ( xf',yk) e Y0 such that x k<xik, v=l,..I with strict inequality for at least one i and y/^yf, j= l,..,J . The procedure goes on with the selection within D '(k ) of the vector (x *(k),y*(k)) for which the component x is the smallest, and with the computation the ratio x*,k,\x!‘. From the definition of D (x k,yk) this number is equal to or smaller to one according to whether (x * (k,,y*(k)) is the same point as (x k,yk), or a different, i.e. dominating, one. When there are more than two elements in D (k ), the identified vector (x * lkl,y*,kl) (14) is in addition the « b es t» example, in the sense of be­ ing the most input-saving one.

The solution 0*’ of the linear programming problem [1] stated above augmented with constraints [3] and [4] reads then: