Albert Breton

Pierre Salmon

Are Discriminatory Procurement Policies Motivated by Protectionism?

Keywords: protectionnisme, achats publics, contrats incomplets. JEL: FI 3, H57 protectionism, government procurement, incomplete contracts. JEL: F13, H57

Dans leurs achats de biens et services, les Etats discriminent souvent en faveur des producteurs nationaux. On attribue généralement ce comportement à une motivation protectionniste. Cette interprétation est parfois valide, mais elle est aussi problématique.

Après avoir exposé certaines des interrogations qu'elle suscite, le texte propose une autre explication de la discrimination dans les achats publics, fondée sur l'hypothèse que les acheteurs publics veulent se procurer des biens et services au coût minimum mais, en raison de la présence de services non-vérifiables, doivent le faire dans un contexte de contrats incomplets. L'article essaie de montrer que la discrimination peut garantir une fourniture efficace de ces services non-vérifiables.

Are Discriminatory Procurement Policies Motivated by Protectionism?

Albert Breton and Pierre Salmon*

I. PUZZLES CREATED BY THE DOMINANT INTERPRETATION OF PREFERENTIAL PURCHASING

When purchasing goods and services, governments often discriminate against foreign suppliers. The available estimates of the size of the bias are precarious, but the existence of a bias is well-documented for both central and sub-central (i.e., regional and local) governments1. In this paper, we take the discrimination as fact. There is, in addition, a general agreement that discriminatory practices should be eliminated. That prescription has been largely endorsed, as a matter of principle, in the agreements that concluded GATT's Tokyo and Uruguay Rounds, in a series of Directives from the European Union (formerly European Community), in the decisions of the European Court of Justice, and in the comments of almost all economics and law scholarship concerned with the issue. We also note that governments, even though collectively the source of the prescription, have been severally reluctant to implement it.

Our main concern is not normative. The prescription is interesting to us because it reflects the quasi-unanimous belief that preferential procurement policies are exclusively motivated by protectionism. We therefore ask: Are discriminatory procurement policies always motivated by protectionism? Although it seems to be perfectly obvious, a positive answer to this question is puzzling in several ways. The main puzzle stems from a point made several years ago by [START_REF] Baldwin | Government Purchasing Policies, Other NTB's, and the International Monetary Crisis[END_REF] which has not, however, received sufficient attention. Baldwin and Richardson showed that when imported goods and domestically produced goods are perfect substitutes, and when government purchases of these goods are only a fraction of domestic supply, discriminatory government procurement is completely ineffective as a protectionist device. Because it plays an important role in the development of our own argument, we reproduce the essential of their basic proposition.

[Figure 1 to be inserted about here] Figure 7 is a simplified version of Baldwin's Figure 6.1 (1984, p. 602) and is similar to the figure used by [START_REF] Baldwin | Government Purchasing Policies, Other NTB's, and the International Monetary Crisis[END_REF]. For a good whose characteristics are as stated above, let Dq be the demand curve of the public sector, Dy the total demand curve in the country (public plus private sector demand), Sd the supply curve of domestic producers, and S j the total supply curve in the country (domestic supply plus imports). In the absence of discrimination, the price of the good is equal to OP, total consumption equal to OC, domestic supply to OD, government purchases to OG and imports to DC. Even if the government decides to buy exclusively from domestic suppliers, the values taken by these variables do not change. The upward pressure on price resulting from the government's exclusive patronage of domestic suppliers is offset by an increase in the purchase of imports by the private sector. In Baldwin's opinion, this result is not an artefact of modelling but is highly relevant in the real world. He writes: "Since government purchases of most non-military items represent only a fraction of the domestic supply of these items, the preceding analysis suggests that government preferential purchasing policies are generally ineffective in raising the price received by domestic suppliers. Nor will they affect domestic employment, production, profits, or growth" (p. 603).

Baldwin and Richardson's analysis was limited to competitive industries but it has been extended to oligopolistic settings by [START_REF] Miyagiwa | Oligopoly and Discriminatory Government Procurement Policy[END_REF]. Although the results are ambiguous in some cases, the proposition that discriminatory procurement policies are ineffective is generally confirmed -and indeed reinforced (imports increase) -under various assumptions about substitutability, the precise form given to discrimination, and returns to scale. When the conditions for Baldwin and Richardson's and Miyagiwa's results hold, even exogenous tariff or pressure groups explanations cannot be used to support the interpretation of discrimination as a manifestation of protectionism. The only possibility that remains is that governments and/or voters are ignorant -an assumption that we should use sparingly at least for persistent and widespread behaviours. When government demand is not a fraction of domestic supply, but instead exceeds that supply, discrimination raises domestic prices and production and reduces imports. But the price of imports to private consumers falls and the private sector imports more2. As a consequence, the aggregate effect of discrimination on imports is likely to be small (see especially Richardson 1972). This means that if discrimination is to be used as an instrument of protectionism, it is likely to be expensive in terms of procurement costs and yield little in terms of protection -it is likely to be cost-ineffective. If we now assume that central and sub-central governments, not unlike business enterprise, are competitive, we must then suppose that these governments supply goods and services to their citizens at the lowest possible unit cost. A well-known mechanism that drives governments to minimise costs is based on the mobility of people and capital [START_REF] Tiebout | A Pure Theory of Local Expenditures[END_REF][START_REF] Stigler | Economic Competition and Political Competition[END_REF]. Another less wellknown mechanism is based on the assumption that voters assess the performance of their government -in respect of the volume and quality of the goods and services which it makes available to them at specified prices -by comparing that performance with the performance of governments in corresponding jurisdictions elsewhere, in respect of the same variables (see [START_REF] Salmon | Decentralisation as an Incentive Scheme[END_REF][START_REF] Breton | Competitive Governments: An Economic Theory o f Politics and Public Finance[END_REF]. With electoral competition in each jurisdiction and uncertainty on the part of office-holders about the assessments of voters, the comparison of performances provides these office-holders with an incentive to perform as best as they can on all dimensions and, therefore, to minimise costs. Recent empirical work [START_REF] Besley | Incumbent Behavior Vote-Seeking, Tax-Setting, and Yardstick Competition[END_REF] as well as casual observation of budgetary discussion and of budget-making at all levels of government suggest that governments are indeed constrained both in the raising of funds and in the allocation of these funds between competing requirements. On the basis of this analysis, we consider a policy likely to be cost-ineffective as also puzzling3.

A third puzzle is the following. In every country there are many public sector buyers rather than a single unitary actor called "the government". Each one of them is submitted to budgetary constraints and to competition. Why, insofar as they remain free -or even exhorted -not to discriminate against foreign suppliers, would these public sector buyers discriminate more against foreign suppliers than against out-of-jurisdiction suppliers in general? In the case of a state or local government, preferential treatment of own-jurisdiction suppliers can be understood within the dominant paradigm, but favourable treatment of domestic albeit out-of jurisdiction suppliers is more difficult to explain. Still, that is what we observe in a number of countries. For example, a number of states in the United States impose upon themselves "buy-American" constraints, in addition to the constraints that are imposed on them by the federal government (see, e.g., Peterson 1980, p. 336).

The view that discrimination is obviously motivated by protectionism rests on an implicit model of what government procurement would be in the absence of protectionist motivations. This model is itself based on a conception of exchange in which transactions between buyers and sellers are embodied in complete contracts that are legally enforceable at zero cost. Because of this assumption , any difference between the long-run market-clearing price and marginal cost is prima facie evidence of imperfections in competition of which discrimination and protectionism are examples. Under the influence of Armen Alchian, Benjamin Klein, Keith Leffler and others, this conception has been modified to permit a reinterpretation of practices such as price discrimination, resale price maintenance, exclusive territories, franchising and other manifestations of vertical restraint which is not only more realistic but which is consistent with resource-economising behaviour on the part of economic agents.

Incomplete contracts are an essential ingredient in this conceptual transformation. They embody transactions in which the parties can be symmetrically informed, can monitor each other (at the limit at zero cost), but which, because of the high cost of ascertaining all relevant circumstances, cannot be monitored by a third party4. Performance is then said to be nonverifiable, which implies that incomplete contracts cannot be legally enforced. Still, as long as net gains to trade are assumed to exist, it must be presumed that rational actors will want to exploit these gains and that they will, as a consequence, strive to invent an effective enforcement mechanism. On the assumption of repeat business, such a mechanism will exist if a buyer (say) is capable of creating, for the benefit of the seller, a stream of quasi-rents whose present worth is at least as large as the value to the seller of defaulting on the contract.

In the theory of efficiency wages for example, employees are offered the prospect of a flow of quasi-rents -a premium over the market clearing wage -if they behave according to the wishes of the employer and are dismissed if they behave otherwise, which implies losing the quasi-rents. This logic is basic to our analysis of governmental procurements. Indeed, we attempt to show that preferential purchasing, by making the creation of a flow of quasi-rents feasible, can be an effective way of securing seller performance5.

The objective of the paper is a relatively simple one. It is to construct a model capable of explaining why a practice such as preferential purchasing by governments continues to be observed in the face of a nearly universal consensus, endorsed by these governments themselves, that these preferences are to be condemned. In other words, our purpose is to provide a rationale derived from, and consistent with, recent developments in the theory of contracts to explain the long-term persistence of preferential procurement policies by governments.

II. BUYING AS A QUASI-HIERARCHICAL RELATIONSHIP

In their well-known paper on the organisational forms of production structures, [START_REF] Alchian | Production, Information Costs, and Economic Organization[END_REF] argued that there is no more authority in firms than there is in markets, in which, they claimed also, there is none. This view that authority is not an essential characteristic of firms has been widely criticised. One objection is that most contracts within firms are incomplete. But Alchian and Demsetz's proposition provides us with an important insight, namely that relations in markets are not in all dimensions different from those in enterprises. In markets as in firms, quasi-rents are created by one party to the benefit of others as inducements to deliver what has been promised. One application of this enforcement mechanism is to vertical relationships between firms, namely to the relation of industrial buyers with their suppliers and to the relation of firms with their franchisees or retailers [START_REF] Klein | Vertical Restraints as Contract Enforcement Mechanisms[END_REF]. In this context, the enforcement mechanism which is often the most powerful and the most robust to changing circumstances is also the simplest : it relies exclusively on the unilateral creation of a prospective stream of quasi-rents in exchange for performance. To borrow a phrase coined by Arthur [START_REF] Stinchcombe | Contracts as Hierarchical Documents[END_REF], the buyer-seller relationship, then, "simulates the operations of hierarchies".

Notwithstanding national accounting conventions, governments are seldom (final) consumers of the goods and services they buy. For the great bulk of their activities they are, indeed, better conceived as producers and suppliers of goods and services and, therefore, as buyers of inputs. Furthermore, they are not single monolithic providers, but generally operate in competitive environments. As suggested earlier, there is competition, based on relative performance, between the multiplicity of governments that make up governmental systems. In addition, within each government, there is competition between different branches, departments and agencies. It is, therefore, reasonable to assume that governmental agencies will require and expect from their own suppliers, at least in part, the same kind of services that business firms require from their suppliers. It is also reasonable to suppose, consequently, that the analyses which have been developed to understand buyer-seller relationships in regard to business enterprises have some relevance for the analysis of similar relationships in the context of governmental agencies.

When we turn our attention to the literature that addresses these concerns -especially to the work on marketing -we note that it begins by stressing the importance of the services which accompany the delivery of goods to industrial firms. The following list of the more salient of these services is adapted from that literature (see for example Noordewier, John and Nevin 1990, and the references therein). It includes:

Information

The provision, prior to signing the contract, of adequate information about the goods and services to be supplied; 2. Client's needs A recognition of the particular needs of the buyer and a willingness to adjust to these needs; 3. Maintenance The capacity and willingness to provide after-sale maintenance services, to deal with follow-up problems, to deliver replacement parts, etc.;

Purchasing

The ease with which the buyer can deal with invoicing, accounting, stock keeping and other problems related to the acquisition or purchase itself; 5. Assistance

The capacity and willingness to install the purchased input, to train personnel in using it, to smooth out the psychological costs of using the input, etc.; 6 . Exclusivity and secrecy

Buyers may want that one or more inputs be of exclusive design and not made available to other suppliers, at least for a period of time, and they may also require that the exact nature of the inputs be kept secret. The willingness and ability of the seller to oblige is, then, a valuable complementary service to buyers; 7. Monitoring

The ease with which the behaviour of sellers can be monitored not only in respect of the inputs supplied, but also in respect of other actions which could be a source of public embarrassment to buyers; 8 . Flexibility

The willingness and capacity of sellers to forgive payment delays, alterations in the conditions of supply while delivery is in progress, trespass on agreed rules or conditions, cancellations, short notice acceleration of delivery, etc.6

Most purchases will not require all of these services. For some inputs, it is easy to verify quantity, quality and other characteristics of services; for others, buyers can rely on the reputation of sellers. But, for some of the services just listed -and, no doubt, for othersneither third-party enforcement nor the reputation of sellers will do. For these, the prospect of repeat business and of quasi-rents is essential to insure performance.

To proceed, we assume without loss of generality that a particular government agency, Gx, buys the same bundle of goods every period, say every year. For that bundle of goods, Gx is one among M buyers. Over a single period, together with the goods themselves, Gx buys from supplier Si (one among N potential suppliers) the provision of a number of complementary services. We make no assumption about the acquisition procedure used, which can range from public bidding to free negotiation, except that we suppose that Gx selects the proposal that it finds the most advantageous. The delivery of the goods and their quality, as well as the provision of some associated services, are verifiable; the other services are not. To simplify, we further assume that all transactions take place at the beginning of the period (thus eliminating discounting problems) and according to the following timing:

. Gx selects Si and signs a contract with that supplier;

. Sj delivers the bundle of goods and the associated verifiable services;

. Gx controls their quality;

. Gx pays the agreed price (for the whole package);

. Sj provides the unverifiable services;

. Gx observes the unverifiable services.

The first thing that Gx does at the beginning of the period therefore is to select Si. That supplier is the successful bidder because (if "most advantageous" is equated with "least costly") the total cost of the bundle of goods plus the verifiable and unverifiable services it offers is lower than that of the other firms in the pool of potential suppliers. If we let g be the cost of the bundle of goods plus that of the verifiable services, u the cost of the unverifiable services, q the quasi-rent and m the monopoly-oligopoly rent (see below), the total cost of procurement is g + u + c, where c = q + m. To simplify, and again without loss of generality, we assume throughout that u is the same for all suppliers; we also assume that once the size of the pool of potential suppliers (N) has been determined (Sections III and IV), the excess costs c are also the same for all firms. The selection of the successful bidder is, therefore, made on the basis of total cost or of g alone because, once N is known, both lead to the same selection.

In the remainder of this section and in the two that follow, we propose a model of the determination of the pool of successful suppliers and of the level of c (= q + m) and, therefore, a model of total cost that Gx must pay to purchase a bundle of goods (plus accompanying services). We begin by recognising that S; may not provide the unverifiable services expected by Gx. To avoid this outcome, Gx includes in the price a quasi-rent (a premium) in addition to the total cost of the bundle of goods plus the verifiable and unverifiable services. Associated with the prospect of repeat business, this may prove effective in inducing Sj to perform. Sj will compare Vp, the remuneration expected in the case of performance and Vnp, the remuneration expected in the case of non-performance. It will decide to perform if Vp > Vnp.

Let us consider Vp first. It is assumed that in future periods Gx will continue to buy the same unverifiable services (along with the same goods and verifiable services) from some supplier. But even if satisfied with the unverifiable services provided by Si, Gx by no means commits itself to again select Si in these future periods. Since costs vary from firm to firm -and, for each firm, from period to period -in any future period another supplier may make a more advantageous offer to Gx than Si could profitably make. Gx would then turn to this new supplier (this does not imply that Gx would not, at a still later date, contract anew with Si)7.

What then is the probability that Si will be selected by Gx in each future period when Sj provides the unverifiable services it has promised to deliver? That question, it should be stressed, will be central to Si's decision as to whether it should perform or shirk. In what follows, we assume that this probability is related to the size of the pool of potential suppliers of the goods and services under consideration. To be more specific, assume that Si is riskneutral and that, in period 1, it forms the following expectations: first, that the number of firms (N) able to supply the relevant inputs is constant over time; second, that the probability for any of these firms (including itself) to make the most favourable offer in each future period is equal to 1/N; and, third, that both the cost of the unverifiable services and the size of the quasi-rent remain constant over time. If r is the rate of interest and q the quasi-rent received in each period, then the present (expected) value of the future stream of quasi-rents (i.e., Vp), assuming an infinite time horizon, is equal to q/rN.

Suppose that Sj does not provide the unverifiable services. The simplest and, at the same time, the limiting case is when Sj assumes that this will lead to a permanent discontinuance of every kind of relation with Gx. Then, future quasi-rents are completely lost8, and the gain from reneging on the promise to deliver unverifiable services is simply the cost of these services, that is, u. In that simple case, Vnp = u and the condition that Vp £ Vnp reduces to:

q/rN > u or, assuming strict equality:

q = urN (1)
This expression tells us that as N increases continuously, the size of the quasi-rent that the buyer must create to prevent shirking by the seller also rises. For example, with 3 potential suppliers and an interest rate equal to 5 percent, the quasi-rent must be equal to 15 percent of the cost of the unverifiable services to discourage shirking; with 100 suppliers and the same rate of interest, the quasi-rent must be 5 times the cost of the unverifiable services to achieve the same end.

It should be clear however that the buyer will not pay any sum whatsoever for the unverifiable services and, consequently, that the linear relationship between N and q [equation ( 1 (height equal to OJ), the point at which the value of the unverifiable services to the buyer, or the cost of an alternative arrangement, is equal to the cost of the quasi-rent.

[Figure 2 to be inserted about here]

Following a long tradition which to our knowledge begins with Edgeworth, it is reasonable to assume that, as the number of enterprises in an industry increases from one to a very large number -at the theoretical limit, infinity -the monopoly-oligopoly rents accruing to the firms and, therefore, the price in excess of the competitive price which the buyer must pay, decrease giving rise to a curve such as MK in Figure 2. Two features of that curve should be noted. One, the height of point M is determined by the equilibrium monopoly price -the monopoly rent when there is only one firm in the industry. Two, the reduction in the monopoly-oligopoly rent brought about by the addition of one firm will be smaller when there are many firms than when there are a few. This is depicted in Figure 2 by the shape of the MK curve: falling rapidly at first and more slowly afterwards. Two rationales can be given for that shape of MK. First, under, certain types of interactions -for example, under

Bertrand competition -the competitive result is obtained after the entry of only one additional firm. Under other types of interaction, the competitive outcome is obtained or approached less rapidly. Being uncertain about the exact weight that should be given to the various types of interaction, it must be assumed that on the average MK will fall more rapidly when there are only a few firms than when there are many. Second, collusion and the probability of its incidence will be larger the smaller the number of firms -a result that can be ascribed to the more than proportional increase in coordination and monitoring costs as the number of firms increases (Scherer and Ross 1990, chapter 8 ).

Taking the vertical sum of the two curves (ORQ and MK), and neglecting possible comer solutions, we obtain a curve such as ZT which has an interior minimum at E corresponding to an optimal pool of suppliers N* -at which the expected excess costs (c = q + m) paid by the public purchaser to the successful bidder is at a minimum.

III. DISCRIMINATION AGAINST OUT-OF-JURISDICTION SUPPLIERS

So far we have derived the magnitudes of the quasi-rents and of the monopoly oligopoly rents, as they are related to the size of the pool of potential suppliers (N), that Gx must expect to pay if it wishes to be provided with non-verifiable services by Si. Assuming that Gx's decisions are made on the basis of ZT, Gx's only decision, if N is given, is whether to purchase or not to purchase the unverifiable services, that is, for any N, whether the associated expected cost of the quasi-rent is lower or higher than OJ.

In the remainder of the paper, we examine alternative ways that Gx may use to alter the size of the pool of potential suppliers so as to minimise the expected excess costs of nonverifiable services. A basic consideration that we will have to keep in mind is that Gx is a governmental or public decision-maker and subjected to constraints that are not binding for private purchasers that are in other respect similarly positioned.

[Figure 3 to be inserted about here]

In particular, Gx is bound by what is in effect a basic principle of democracy which asserts the equality of citizens before government agencies or, as it applies to procurements, the equal opportunity for all 'domestic' supply sources to bid for government contracts. At any moment, there will be derogations of various sorts from that principle but we neglect these in what immediately follows. It is this principle which will lead Gx, if it wants to reduce the pool of potential suppliers, to discriminate against 'foreign' or, more precisely, out-of-jurisdiction suppliers. To understand how discrimination under the principle of democracy operates consider Figure 3. Assume first that the total (world) pool of suppliers is equal to N'w of which Nd are 'domestic' enterprises. Under these circumstances, the expected costs of unverifiable services to Gx are less when no restrictions are placed on the size of N; that is, because c'w < Cd -the expected excess costs associated with N'w and Nd respectivelythere is no need to discriminate. N'w is the size of the pool of suppliers that make expected excess costs as small as possible. If, however, the world pool of suppliers is Nw of which the domestic component is still Nd, the expected excess costs are cw and Cd, and Cd < cw. Under these circumstances, Gx will want to reduce N and, because of the principle of democracy, will do so by discriminating against foreign or out-of-jurisdiction enterprises9.

[Figure 4 to be inserted about here]

The reader will recall our discussion, identified above as the 'third puzzle', of the situation which is sometimes encountered of local jurisdictions which discriminate more against foreign suppliers than against other out-of-jurisdiction enterprises, as happens when American state, Canadian provincial, or French regional authorities decide to abide by a 'buy American', 'buy Canadian' or a 'buy French' policy. That such a policy may be efficient can be seen by reference to Figure 4 in which Nw is again the worldwide pool of suppliers, while Nd and N tare respectively its 'domestic' (American, Canadian, or French) and 'local' (state, provincial or regional) components. As drawn, Figure 4 makes clear that 'local'jurisdictions can reduce the expected excess costs of non-verifiable services by adopting 'buy-domestic' policies. We note that the discrimination enacted by the 'local' authorities when they adopt such policies is based on a differentiation between 'foreign' (genuinely foreign plus domestic out-of jurisdiction) sources of supply but not between their 'own' suppliers. Thus, it does not violate the democratic principle discussed earlier.

IV. FINE-TUNING THE NUMBER OF SUPPLIERS

In the discussion of the last section, we were concerned with the analysis of discrimination under the equalitarian principle of democracy. Gx, the public decision-maker, was therefore limited to actions that would restrict or increase the pool of potential suppliers through discrimination that was respectful of that principle. It is time to recognise that, except by accident, the pool of suppliers obtained through that instrumentality will not be N*, the size of the pool that would minimise the expected excess costs of the unverifiable services.

We should therefore expect that government purchasers will seek to reduce -or, possibly, increase -the pool of suppliers by making use of different instruments. This section is concerned with the analysis of the operation of a subset of these instruments and begins by focusing on instruments used to reduce N.

Consider the situation depicted in Figure 5 when the w,orld pool of suppliers is of which Nd are domestic firms. Gx will first discriminate against foreign suppliers, but will in addition wish to reduce the pool of 'domestic' firms to attain N*. How can this objective be achieved? We suggest that this is done by implementing rules of transparency (such as open bidding and public disclosure of sealed bids) and regulatory controls (like bookkeeping requirements, proof of internal monitoring, and certification of compliance) which in the words of Kovacik (1992, p. 34) "discourage otherwise eligible firms from taking risks and bearing costs uniquely associated with regulatory compliance in public contracting."

[Figure 5 to be inserted about here] Kovacik, though he decries the regulatory measures as inefficient, argues that they act as "barriers to entry in government procurement". We agree, but contend that, as long as Nd > N* in Figure 5, the barriers promote efficiency by allowing government purchasers to reduce the excess costs of buying non-verifiable services. Kovacik's general line of reasoning raises an interesting problem. He points out that the Competition in Contracting Act of 1984 "required each executive agency to establish a 'competition advocate' who would promote ...'full and open competition' and 'challenge barriers to such competition' "(p. 39). The pursuit of these objectives, Kovacik argues, was frustrated by the expansion of regulatory controls which, in effect, led to the opposite result (p. 30).

On the general precept, formulated earlier, that it is inadvisable to presume persistent and widespread ignorance and irrationality, we would argue, on the contrary, that the Act of 1984 was always intended by Congress to generate barriers that would induce purchasers to economise on scarce resources. In addition, we insist that the Act had to be formulated in a language consistent with the equalitarian principle of democracy as well as acceptable to the era's Zeitgeist.

We must now consider briefly the situation depicted in Figure 5 when the world pool of suppliers is Nw, but the domestic pool is N'd< N*. To reduce the excess cost of buying unverifiable services, the public purchaser will want to increase the number of domestic suppliers. This situation is not likely to be encountered for the bulk of the goods purchased by governments, but can still be important, measured by dollars of expenditures, for certain goods such as military hardware, nuclear installations and transportation equipment such as trains, aeroplanes, buses and subway cars when public purchasers prefer to have three or, at least, two potential suppliers instead of one. Under these circumstances, we expect governments to subsidise either directly or indirectly the development of one or two domestic enterprises.

Figure 5 is not very useful to analyse this case because the public subsidies will displace the ZT curve upward and to the right -the subsidy will increase the unit cost of non-verifiable services and the increase in N will move Nw to the right, because N'd plus the additional firms are part of the world total -but the general idea that the excess cost of nonverifiable serv ices can in some cases be lowered by the public subsidisation of new ventures is, however, easily intuitioned with the help of the figure which makes clear that an increase in the number of domestic suppliers will lead to a downward movement on the new curve, signalling a reduction in the excess costs of unverifiable services which can, in some cases, exceed the cost of the subsidy.

V. IMPLICATIONS

Some of the implications of the foregoing analysis have already been noted. In this section, after a brief discussion of how the approach we are proposing can in principle be tested, we examine two additional implications of the model, both of which pertain to the relationship between the size of jurisdictions and the incidence of non-protectionist procurement policies.

It would be a simple matter to test the hypothesis if information on the following variables (measured across categories or classes of goods and services) was readily available:

-the level of discrimination and/or the speed (or lack thereof) at which discrimination is reduced;

-the degree to which the conditions for the Baldwin-Richardson-Miyagiwa ineffectiveness proposition are satisfied;

-the relative importance of unverifiable services;

-the effect of discrimination on the excess cost of procurement (as measured vertically in Figures 3, 4 and 5 ) or, more precisely, on the two components of that cost, namely the monopoly-oligopoly rent and the quasi-rent.

Then, for any class of goods, if discrimination in government procurement is significant and has resisted liberalisation efforts, if the Baldwin-Richardson-Miyagiwa conditions are satisfied, if unverifiable services are important, and if the reduction in the pool of suppliers caused by the exclusion of out-of-jurisdiction supply has a negative effect on the quasi-rent which is larger than the reduction's positive effect on the monopoly-oligopoly rent, we would have to conclude that, with respect to these goods, our analysis is capable of accounting for the facts and that, as a consequence public procurement is not motivated by protectionism.

Turning our attention to the first of the two implications we wish to discuss, we note that, if for some bundle of goods, Cd < cw , but the number of potential suppliers within the jurisdiction is large such that Nd lies to the right of N* (as in Figures 4 or 5), we expect to observe, as noted earlier, not only discrimination against out-of-jurisdiction sources of supply but also the implementation of policies aimed at reducing Nd-In addition to what has been said about this in Section IV, the model also suggests that, ceteris paribus, the flow of regulatory measures whose indirect effect is to reduce Nd would increase as the size of markets increases and with it the number of potential suppliers. A ceteris paribus comparison between large and small jurisdictions for example, should reveal that the conditions for the Baldwin-Richardson-Miyagiwa ineffectiveness proposition will be satisfied more often and that Nd will be large more frequently in larger jurisdictions than in the smaller ones. As a consequence, we should expect both the incidence of non-protectionist public procurement and the incidence of measures designed to reduce Nd to be greater in larger than in smaller jurisdictions.

The second implication adds another dimension -the presence of economies of scale in production -to the size of the jurisdiction. The liberalisation of procurement policies -the elimination or reduction of discrimination and preferential treatment -currently implemented in Europe (see, e.g., [START_REF] Hartley | The Single European Market and Public Procurement Policy: The Case of the United Kingdom[END_REF] is mainly motivated by the gains expected from the exploitation of scale economies. The quantitative estimates presented in the Atkins Report (1988) indicate that these economies constitute the main part of the benefit expected from liberalisation. As a result of discrimination, and thus of fragmentation of markets, the size of many European firms in sectors in which government procurement is important or even dominant (e.g., defence or telecommunications) -or the scale of production for particular goods (e.g., a particular fighter aircraft) -is small relative to the optimum.

Liberalisation of procurement within Europe is expected to generate a reorganisation of European industries that will entail the survival, in each of these sectors, of only a few large firms. According to the Atkins Report, this will considerably reduce the cost of procurement for all European governments (for defence procurement, see some impressive estimates in [START_REF] Fontanel | A European Defence Union? Economic Policy[END_REF].

Our hypothesis can account for the willingness of member states to accept a liberalisation of government procurements which leads to diminished procurement costs for goods and verifiable services and which, at the same time, maintains a small number of suppliers and therefore a lower cost for unverifiable services. It can also account for the reluctance of the same member states to liberalise on a wider scale -that is, on one in which the number of potential suppliers would be much larger. Hence, their attachment to préférence communautaire and their timidity in implementing liberalisation in a GATT setting -a phenomenon which international relations specialists find puzzling (see [START_REF] Grieco | Cooperation Among Nations: Europe, America, and Non-Tariff Barriers to Trade[END_REF]).

V. CONCLUSION

Discriminatory governmental procurement policies are habitually taken to be prima facie evidence of protectionism. Without denying that discrimination may sometimes be motivated by protectionism, we have suggested a hypothesis based on the idea that discrimination may be an efficient method of dealing with the problems posed by the existence of incomplete contracts, namely contracts whose terms, while observable by the signing parties, cannot be verified by third parties and are, therefore, legally unenforceable.

Discrimination is, in that framework, a way of assisting in the creation of a stream of quasi rents to the benefit of suppliers, which, if they decide to shirk on their contractual commitments, they will lose. Discrimination is then an instrument that serves to make incomplete contracts enforceable. Granted that our hypothesis has some validity, it becomes important to design empirical work in such a way as to permit identification of those instances when procurement policies are protectionist and those when they are used to enhance overall efficiency. Though we have not considered normative issues, it is also clear that the hypothesis has far-reaching policy implications which future work will have to draw out. Bénard (Université de Paris I Panthéon-Sorbonne), as well as to an anonymous referee. We are of course solely responsible for the shortcomings. We are also grateful to the Lynde and Harry Bradley Foundation for its support.

*. For some estimates at the level of central governments see, e.g., [START_REF] Baldwin | Nontariff Distorsions o f International Trade[END_REF][START_REF] Lowinger | Discrimination in Government Procurement of Foreign Goods in the U.S. and Western Europe[END_REF]), Atkins (1988) and the references therein. At the level of sub-central governments see, e.g., [START_REF] Breton | Discriminatory Government Policies in Federal Countries[END_REF], [START_REF] Finsinger | Non-Competitive and Protectionist Government Purchasing Behavior[END_REF], [START_REF] Dalpé | Les marchés publics québécois et canadiens: la définition des politiques d'achat dans le contexte d'une économie ouverte[END_REF] and, again, the references therein. The bias is typically measured by contrasting public sector with overall purchases of foreign (or out-of-jurisdiction) goods. In European countries, for instance, according to the Atkins Report, imports represented in 1986-88, on average, 22 percent of GNP but only 2 percent of public purchasing. For some evidence that figures such as these underestimate public purchasing of foreign goods and services, see [START_REF] Hartley | The Single European Market and Public Procurement Policy: The Case of the United Kingdom[END_REF].

2. This formulation, borrowed from [START_REF] Baldwin | Trade Policies in Developed Countries[END_REF], implies that the government will pay for a homogeneous good a price which is higher than the price paid by private buyers, albeit, contrary to [START_REF] Herander | The Impact of Government Price Discrimination and its Equivalence with the Tariff[END_REF], not from the same sellers.

3. One could argue that the same kind of puzzle would be created by the assumption that office-holders maximise their rents and thus have no reason to buy above minimum cost even when not constrained by competition. But office-holders may have other objectives than maximising their (pecuniary) rents. They may, for instance, seek "a quiet life". The information provided to voters by competition of the relative performance variety impedes this kind of behaviour. For a similar argument in the context of agency costs within firms, see Holmstrom and Tirole (1989, pp. 95-96).

4. As a consequence, the problem of incomplete contracting can arise not only in the context of long-run agreements but also when the relationship is modelled as a series of short-run contracts. This point is stressed by [START_REF] Williamson | Economic Institutions: Spontaneous and Intentional Governance[END_REF] in his critical comments on the assumptions made in [START_REF] Fudenberg | Short-Term Contracts and Long-Term Agency Relationship[END_REF].

5. In the volume edited by Leitzel and Tirole (1993), several contributions analyse government procurement under the assumption that contracts are incomplete, and the importance of quasi-rents and repeat business as a way to ensure supplier performance is stressed in Julio Rothenberg's brief comment (pp. 72-73); discrimination against out-ofjurisdiction suppliers is not envisaged, though.

6 . We note that in the case of "flexibility", third-party enforcement is ruled out almost by definition: as it is conceived in the text, flexibility typically consists in giving-up third-party enforcement.

7. To simplify, we assume that only one supplier is awarded a contract in each period.

8 . Because Si cannot really form the expectation that Gx will refuse to do business with it (Si) under all circumstances, we should merely assume that, when Gx is dissatisfied with Si's performance, it places a lower value on Si's promise to perform in the future (see Laffont and Tirole 1993, p. 227, for a related discussion). Suppliers will remain uncertain about the exact effects of their actions but, with expected quasi-rents that are sufficiently attractive, they will nonetheless be induced to please Gx as much as they can. [This is exactly what happens in a hierarchy, as already argued by Albert [START_REF] Breton | The Logic of Bureaucratic Conduct[END_REF], when subordinates provide informal services to their superiors in the expectation of an undisclosed future reward]. Under these circumstances, the precise calculation of the quasi-rent provided in the text is no longer valid. Indeed, because S; is now excluded from repeat business over a finite number of periods rather than forever and because it remains necessary that the expected stream of quasi-rents be superior to the given gain from shirking (u), the value of the quasi-rent per period (q) must be higher. The main qualitative result of the analysis holds however: reducing the number of potential suppliers reduces the quasi-rent that must be paid by Gx and thus makes possible the obtaining, on "reasonable terms", of the desired unverifiable services. 9. Within the theory of auctions, some researchers have formally demonstrated that, in a context of competitive bidding, discrimination against foreign suppliers may significantly lower procurement costs. For two different analyses along that line, see [START_REF] Mcafee | Government Procurement and International Trade[END_REF] and [START_REF] Mougeot | Marchés publics et intégration européenne: le rôle des coûts d'entrée[END_REF]. Both assume complete contracting. expected

  )] is upwardly bounded by a constraint defined either by the monetary equivalent of the utility derived from the services or by the money cost of the best alternative available, the smaller of the two being shown as OJ in Figure 2. In other words, that Figure 2 s ORQ curve, passing through the origin and having slope ur [equation (1)] has a discontinuity at R

  Figure1

  University of Toronto, Department of Economics, 150 St. George Street, Toronto, Ontario M5S 1A1, Canada; Université de Bourgogne, Faculté de Science économique, 4 boulevard Gabriel, 21000 Dijon, France. For helpful comments and suggestions, we are grateful to Jean Magnan de Bornier, Michel Mougeot, Martin Paldam, participants at the European Public Choice Society Meetings, European Economic Association Congress and Séminaire Jean

NOTES

*


Louis De Mesnard

THE MICROECONOMIC MAXIMISATION OF A RATIO OF PROFIT: A RESTATEMENT

Keywords: . Maximisation, Profit, Rate of Profit, Microeconomics

On the traditional microeconomic theory, firms are supposed to maximise the aggregate pure profit. We study other objective functions which are ratios profit over capital. We explore various combinations, proposing a typology of the ratios of profit, particularly the rate of profit (accounting profit over capital), the rate of gross profit (gross profit over capital), the rate of accounting profit (accounting profit over capital). The cases of monopoly with variable coefficient of capital, monopoly with fix coefficient of capital, competition with fix coefficient of capital, are studied. The solutions given by the maximisation of the aggregate pure profit and the maximisation of the rate of profit are compared. We conclude to a usually lower volume of production with the maximisation of the rate of profit. Thus, as firms are smaller, competition may be greater; however the clearing of the market must be studied; in competition at the very long term, with the rate of profit maximisation, the pure profit remain positive.

INTRODUCTION

I) Maximisation of the aggregate profit is one of the foundations of microeconomic theory (and of the theory of industrial organisation). However, it is well accepted, its basis are not very much studied. In many handbook of microeconomic theory, the principle of the maximisation of the aggregate profit is asserted without justifications: "it goes without saying". In the other side, economic calculus (actualisation, comparison of projects of investment, etc.) required to think in terms of maximisation of profitability ratios. There is a gap between microeconomic theory and the practice. In fact, microeconomic theory has some special characteristics. For example, it considers a very special type of firm confused with the capitalist-shareholder (apart from the fact that this firm have no intern structure: interactions between individuals are not taken into account).

In one hand, we can say that in the pre-industrial age and from the beginning of the industrial revolution, capitalists (or landowners) often had no alternative possibility of investment than their own business (or property). The modern legal form of firms, like company, was not widespread and the financial market was non-existent. That is why the Classics authors, A. Smith et D. Ricardo principally, have adopted naturally the principle the more in accordance with the observed reality of their era, the maximisation of the aggregate profit: why maximise a ratio of profitability when your funds cannot be employed elsewhere (this profitability have no signification is this case)? In this point of view, capitalists (or landowners) maximise the aggregate profit, because they must invest all their funds in their own business (or property): as long as they had funds, they invested if the profit remains non negative; the not invested funds yield nothing.

In the other hand, we must consider the cases where there are alternative possibilities of investment with the concept of opportunity cost of the capital. The cost of the accounting capital 1 is an opportunity cost, because these free funds may be invested in another place at a normal interest rate: the accounting capital looks to come from an operation of renting similar to a loan, and its opportunity cost looks like an interest of which the rate is the average rate of the market (calculated with a standard alternative investment) 2 . This leads to calculate the pure 1 In the following, the words accounting capital refers to the funds advanced by the shareholders to the firm: there is nor obligatory pay back, nor obligatory payment of a remuneration, but it is the usual counterpart of a property right. This remuneration is free ex ante as ex post, and varies between zero and the amount of accounting profit (less the funds putted in reserve and taxes), depending on the policy of dividend of the firm. Accounting capital remains in the balance sheet of the firm and remains the property of the shareholder (to the contrary of a debt, which disappear of the balance sheet of the firm when paid back).

In the opposite, loans are paid back obligatorily at the settlement date (except if bankruptcy or at the dissolution of the firm) with obligatory payment of an interest to the moneylender, without giving to him a property right (the question of the control of the shareholders over the firm differs): interest is a cost.

Self-financing correspond to a withdraw over a reserve (a past profit or over an actual profit): it is comparable to a withdraw over the shareholder and then similar to an increase of accounting capital (self-financing plus accounting capital forms the own funds of the firm). Also, the reselling of shares is only a transfer between shareholders and does not concern the firm directly (it determine only the future success of the operations of increasing the accounting capital).

The words accounting profit denotes the accounting profit able to remunerate the shareholders, that is to say before payment of dividends. It is not used to pay the eventual interest (yet paid). If accounting profit is positive, a dividend is optionally paid, equally to every shareholders, in order to remunerate this accounting capital. Dividend is not a cost. Then, accounting capital is free. In this paper, the costs corresponding to the accounting profits are the accounting costs.

Note that capital gains over reselling of shares may constitute a remuneration of shareholders. We do not take this question into account.

2

In [de MESNARD 1992] the following arguments are given. The opportunity cost is due to an abstinence, as in Senior, or whether it is due to an expectation as in Marshall, or whether a preference for the present as in Bohm-Bawerk or an impatience as in Fisher. It is not easy to calculate and maximise the aggregate pure profit. Indeed, the alternative investment of the funds must be well defined, unique and known, in order to calculate the opportunity cost. The marshallian conception sees in the pure profit the salary of the manager. Another point of view (EH. Knight, M. Allais) sees in the pure profit a conjectural yield, created by some errors of forecast; the remuneration of the capital is integrated in interests and then in the costs, including the insurable risks; the accidental phenomenon, the non insurable risks, create a temporary profit. More recently, Shih-Yen Wu argues as it follows [START_REF] Wu | Production, Entrepreneurship and Profits[END_REF] ]: the remuneration of the capital included interests is fixed ex ante, and dividends fixed ex post, is a cost. This position seems to confuse between the cost to obtain the capital and the its remuneration; it supposes a perfect financial market, with a unique interest rate; it supposes that the funds obtained by increasing the accounting capital must be remunerated, what is contrary to the intrinsically risked (and linked to a property right) nature of the capital, the accounting capital may be remunerated. profit (or economical profit): it is the profit obtained by the firm including every costs and opportunity costs, that is to say, including the normal remuneration of the accounting capital.

Consequently, in microeconornic theory, the firm invests as long as the pure profit is not negative, even it is very very small. A caricatural example is the construction of a plant for one billion dollars, because the pure profit will be one cent! Nobody do that for many reasons: the risk of investment, the cost of managing the investment. However, these two reasons may be taken into account in the pure profit: the risk have a cost, the management have a cost. There is a third reason: the sunk costs or the immobilisation of the funds over a certain period. It has also a cost in the actualisation theory, but we think that every man prefers to conserve the liquidity of the capital putting it in a remunerated account if the pure profit is nil (that is to say if the net remuneration of its funds including all the opportunity cost is equal to the remuneration in the financial market for liquid funds) However, the evolution of firms leads toward modern companies, principally limited companies and other forms where the shareholders are distinct from the managers, and toward the financial market and the Stock Exchange. Capitalist become shareholders, with alternative possibilities of investment: the funds not invested in one business yield something in another business and the comparison between the profitability of businesses is based on ratios. Thus shareholders maximise the profitability of their funds, that is to say the rate of profit \ In the view point of the agency theory, the objective function of the firm depends on the objective function of the shareholders. If shareholders think in terms of profitability, the firm must think in terms of profitability. The theory of the choice of investment, the theory of demand of capital (Keynes, among others), the portfolio theory, etc., thinks into terms of profitability.

In addition, the calculating the pure profit is difficult because the opportunity cost are hard to know.

Note that the same behaviour, take into account the available means to reach the objective, is observed in any human activities, when a problem of choice exists. In some cases, human people do not respect this principle, for example when the objective is vital (high level sports competitions, military fight, etc.).

2) So, the question is: why the subtractive form of the function of profit (profit minus opportunity cost of the capital) prevailed over the "divisive" form of the function of profit (profit over capital)?

To decide if a firm produces or not, there are two possibilities. The firm produces if: In the first case, we compare the pure profit to zero. In the second case, we compare the rate of profit to the opportunity costs per unit of capital. These two possibilities are obviously equivalent. However, this indicates nothing about the optimal amount of production: we shall show that it is not the same thing to maximise the pure profit and to maximise the rate of profit. The two theories differs from this point and this shows that it is necessary to explore the alternative theory of the maximisation of the rate of profit. So, following [ de MESNARD 1992 ], we want to link up microeconomic theory and practice, suggesting an alternative principle to the maximisation of the profit, the maximisation of the rate of profit (that is to say the ratio of the accounting profit over the whole capital engaged in the production), and making a distinction between shareholders and producers 4 .

The maximisation of the rate of profit has not been studied, even if it is an old idea in the business world 5 . The maximisation of a rate of profit margin have been studied even it is less realistic than the maximisation of the rate of profit 6 . The model of financial theory introduce the rate of profit. However, there is not maximisation of the rate of profit. We obtain the present actualised value of the firm for each level of the capital:

V = f(AT). 4
Remark that asserting a distinction between shareholders and producers leads to a point of view near those of O Lange [START_REF] Lange | The Place of Interest in the Theory of Production[END_REF] ]or those of J. Tobin (and its "q") [ TOBIN 1969 ]or those of F. [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF], but also those of J M. Keynes, for whom investment comes from the comparison between the anticipated profitability and the interest rate. These two categories of agents, the firms and the capitalists-shareholders have their own logic.

In fact, nobody invest one billion to obtain one billion and one franc finally: then it is the rate and not the aggregate profit the capitalist optimises.

6

In the case of a firm installed in an economic space, R. Lantner and J.-F. Thisse have envisaged the maximisation of a rate of profit margin calculated as the ration of the profit over the costs. They show that the marginal cost of production (which is, with the transportation cost, a component of the marginal cost of production), may be decreasing and that the optimum may be to the left of the point of the minimum cost of production, at the condition that the marginal transportation cost would be increasing [START_REF] Lantner | Une révision des conditions d'équilibre de la firme: l'intégration du facteur spatial[END_REF].

This approach introduces the rate of profit. However, V, the present actualised value of the firm, is maximised. Thus this model (and the financial models which maximise V) remains classical: basically, the present actualised value is homogeneous to a profit (with a nil rate of actualisation, Vis a part of a sum of profits). Then, the maximisation of V is homogeneous to the maximisation of the profit N .

3) In [ de MESNARD 1992 ], the accounting capital was putted at the denominator of the ratio of profit, that is to say did not comprise loans in the capital8 : here we call this ratio\ the rate of accounting profit. It leaded to a non constant denominator if the financial structure varies, which make the economic calculus difficult: for a project, it is not easy to know the part of its funds financed by increasing accounting capital and the part financed by loans because, often, operations of increasing accounting capital are decided independently from operations of investment. Moreover, as the capital loaned is not taken into account, it is supposed that the accounting capital may increase independently from the loaned capital: for example, the first derivative of the function of accounting capital were supposed positive which is not sure, while it is more acceptable to suppose that the first derivative of the function of the whole capital is positive.

These arguments push the theory of the maximisation of the rate of accounting profit back to mieroeconomic theory and economic calculus.

To avoid this pitfall, after proposing a typology of the possible ratios of profit, we retain two other types in addition to the maximisation of the rate of accounting profit: a new ratio that we call the rate ofprofit, relying on the accounting profit at the numerator, and putting the whole capital at the denominator (and not only to the accounting capital), and another ratio the rate of gross profit, with the gross profit (accounting profit without debts) at the numerator and the whole capital at the denominator. These ratios are more convenient regarding to the practice and these new versions of the theory of the maximisation of the rate of profit are more compatible with the mieroeconomic customs. The maximisation of the rate of profit introduces a financial structure, including both the loans and the increase of the accounting capital as a way to finance the investment; particularly, we study the case of a variable financial structure in a first time, and the more simple case of a constant financial structure in a second time.

In [ de MESNARD 1992 ], the problem of the fixity of the capital is treated by a distinction between long term (capital is variable) and short term (capital is fixed). We propose an alternative but not exclusive point of view: the calculus is made ex ante, where K(Q) is freely fixed, while ex post, K(Q) is fixed. However, we shall see that a condition of triviality is not only the fixity of the capital but also the fixity of the financial structure.

As in [de MESNARD 1992], we shall not explore all the improvements brought to mieroeconomic theory by the authors: uncertainty and stochastic models, dynamic models, incomplete information, theory of the bounded rationality, theory of the agency, theory of imperfect markets and fixed prices equilibrium, financial theory, managerial and behaviourist theory of the firm, etc. These improvements are useful, and we want not call them into question.

The theory of the maximisation of the rate of profit is upstream from them. Moreover, the notion of profit itself will not be called into question: we remain into the logic of capitalism 9 .

After presenting a typology of the ratios of profit and recalling some simple results in a few classical situations (monopoly, competition), we shall study two ratios of profit: the rate of profit itself (accounting profit over capital) and the rate of gross profit (gross profit over capital). We shall focus on the problem of the aggregate pure profit: as the opportunity costs of the capital are taken into account in the pure profit, is it necessary to maximise a ratio of profit over capital? The answer will lead to a comparison of the optimal production when the aggregate pure profit is maximised and when the rate of profit is maximised.

VARIABLES AND NOTATIONS

We call argument of the pure profit the following argument of triviality of the maximisation of the rate of profit: the pure profit yet includes the remuneration of the accounting capital. To dismiss this argument, in the following, we shall introduce explicitly the pure profit, in order to show that the question is not to know if we must or not work with pure profit, but to know if we must work in term of profit maximisation or in terms of rate of profit maximisation.

The calculus of the accounting profit do not includes the remuneration of the accounting capital, while the calculus of the pure profit leads to remunerate this capital at the interest / (as if it was loaned).

The different variables are.

• P(Q) i s the inverse of the curve of demand. R(Q) = P(Q)Q is the receipt.

• Cits) the cost of production, excluding the interest of loans and the remuneration of the accounting capital. Introducing the pure profit, the cost C(Q) is no more undetermined: it includes the whole of the costs of production, excluding the interest of eventual loans and the remuneration of the accounting capital subscribed to shareholders. The opportunity cost associated to this remuneration of the accounting capital will be introduced explicitly.

• K(Q) the positive (financial) capital used in the production. By simplification, the capital is entirely dedicated to the actual production.

• KA (Q) the positive accounting capital used in the production (we confuse the accounting capital with the totality of the own funds excluding the long term debts).

• KL(Q) the loaned capital used in the production. We have by hypothesis *{Q)=K A {Q) + K L {Q).

9

Note that, even it treats of the rate of profit, the theory must not be confused with some aspects of the marxian theory (like the Marx's law of the decreasing rate of profit).

• What happens when the financial structure varies? The more simple way is to suppose that the financial structure depends on Q. A functional parameter a{Q) , depending on Q, is the part of the capital K{Q) financed by the shareholders, that is to say it is the ratio

a (Q) =
-it is the part of the accounting capital in the total of the capital, varying between 0 and 1; we suppose that a(Q) is a derivable function, (l -a(Q)) is the part of the capital K(Q) financed by loans.

• / the constant and positive interest rate in the financial market, valid for the borrower and the moneylender: the same rate is used for the loans of the firm and for the opportunity cost of the funds. The accounting capital is remunerated at the fix rate of interests, giving an opportunity cost of:

C K (Q) = t a(Q) K(Q)
• The total cost when calculating the accounting profit is

C A (Q) = C(Q) + / [l -a(Q)] K{Q) .
• The total cost when calculating the pure profit includes the opportunity cost of the accounting

capital C P {Q) = C A (Q) + C K ((?) = C(Q) + / K{Q) . • Ttegrossprofitisn G {Q)=R{Q)-C(Q).
• The accounting profit is:

n A (Q) = R(Q)-C A (Q) = R{Q)-C(Q)-/ (l-O{Q))K(Q) = n G ((?)-t(\-O(Q))K{Q).
• The pure profit is independent from a{Q) :

n{Q) = R(Q)-C L P {Q) = R{Q)-C,(Q)-t K{Q) = Yl G {Q)-t K{Q)
and

I1(Q) = R{Q)-C T P (Q) = R(Q)-C T A {Q)-C K (Q) = (Q)-1 a{Q)K{Q) .

TYPOLOGY OF THE RATIOS OF PROFIT

The different types of ratios

Considering the gross profit

U. G {Q)= R(Q)-C(Q), the accounting profit U A {Q) = n G {Q)-t(\-a{Q))K(Q) and the pure profit Yl{Q) = U G {Q)-1 K{Q),
considering the capital K{Q) and the accounting capital K A (Q), there are six great types of rate of profit possible:

2 the rate of accounting profit

^A(Q) U A {Q) 4Q) x G (Q)-t(\-ciQ)) *A (Q) K A (Q) cc{Q)K{Q) a{Q) a(Q) 3. the rate of gross profit KQ (Q) = -^Q) +t (l -a{Q)): it is independent of a{Q)
4. the rate of "accounting gross profit"

* (IAQ) 'YJQ)'^Q)= Ms) -* Aeh '-4or
5. the rate of pure profit

n P (Q) = ^||| = ^^J)^ = *G (Q) ~t = *(Q) -^6) • * is independent of 6. the rate of "accounting pure profit" n PA {Q) = OV ^ ^ J-/ = ^(g)-f .
The optimum of the last two ratios are confused with the optimum of other ratios. The optimum of the rate of pure profit is identical to the optimum of the rate of gross profit; the optimum of the rate of "accounting pure profit" is identical to the optimum of the rate of accounting profit. Note that the rate of pure profit is the lower bound of the rate of profit and the rate of gross profit is the upper bound.

Then, regarding to the object of this paper, these two ratios may be pushed back. It remains the four first ratios. The rate of "accounting gross profit" is an hybrid ratio: is indicates how the accounting capital creates the gross profit, disregarding the role of the loaned capital Thus, it may be dismissed.

It remains three ratios.

• The rate of profit n{Q) takes into account the financial structure and seems to be operational.

• The rate of accounting profit TC A (Q) studied in [de MESNARD 1992], but with a fix financial structure.

• The rate of gross profit 7TQ{Q) indicates how the whole capital creates the gross profit: it does not depends on the financial structure.

As the maximisation of the rate of gross profit is a particular case of the maximisation of the rate of profit (a{Q) = 1 for every Q), we shall study only the more general case: the maximisation of the rate of profit.

Example

We keep the same example as in [ de MESNARD 1992 ], adapted: • Not surprisingly, the optimum is identical to the point of long term equilibrium of the firm;.

C(Q) = 5Q 3 +\0Q 2 -5Q + 20, p{Q) = -5Q + №, R{Q) = PQ, K{Q) = 20QQ
However, the profit is here the accounting profit and not the pure profit; and, not said in the quoted paper, the profit is not nil at the optimum.

...under some simple hypothesis, it is easy to show that the maximisation of the rate of profit leads to invest and produce less the maximisation of the aggregate profit does. • The optimum does not depend on the price, the production does not vary if the price vary, what corresponds to an observed behaviour. As the price is no more an indicator for the volume of production of the firm, the optimum does not automatically clears the market as supposed by the neo-classical theory.

And there is an equilibrium with rationing because the marginal cost is not equal to the price.

Comparison of the optimal levels of production

It is not sufficient to say that the optimum point of the maximisation of a ratio of profit is to the left of the optimum of the maximisation of the corresponding profit. It is necessary to compare that the first optimum with the classical optimum of the microeconomic theory, the maximisation of the pure profit. Taking into account the pure profit, and introducing a variable financial structure (under the form of a derivable function), we obtain the following fundamental theorem: Theorem 1. In the more general case of the monopoly with a function of capital, with

K'(Q) > 0 >
» compared to the profit maximisation, the maximisation of the rate of profit gives a lower optimal quantity (respectively higher) when the compensated rate of profit (the rate of profit divided by one plus the relative elasticity e a (o

) / >K{Q) °f A (Q) facing to K(Q) or divided by the relative elasticity Q K A (Q)/K{Q) °f ^A(Q) facing to K{Q)\

is higher

(respectively lower) than the interest rate.

THE RATE OF PROFIT i)

> / => max n< max n

> t ==> max n< max n or E K A (Q)/K(Q) 2) < t => max 7i>-max IT < / => max n> max fi or Q K A (Q)/K{Q)
where Q* is the optimal quantity which maximises the rate of profit Comments.

1. This theorem does not indicate the relative position of optimums in general: it requires the calculus of the optimal rate of profit. However, it is useful, because it allows a manager to know, when he has calculated the optimum of the maximisation of the rate of profit if the maximisation of the aggregate pure profit provides an optimum before or after the first optimum. Even if the capital K(Q) is fixed (K'(Q) = 0 , V0), the two points Q* P and 0* may not coincide, because in this case W(Q) = Yl' A (Q)-1 a'{Q) K{Q) and the optimum of the rate of profit X(Q) is given by Tl' A {Q) = 0. Thus, in this case, the two theories are identical if the financial structure is fixed also: a' (Q) = 0.

The first case l + e oc{Q)l'K{Q)

The hypothesis K'{Q) > 0 , VQ is more acceptable than the hypothesis K' A {Q) > 0, V0 made in [de MESNARD 1992] (see in annexe). Indeed, the sign of Q a {Q)/K(Q) is undetermined, because the part of the shareholders may increase or decrease when the amount of capital increases. In very short firms, loans often are very low and accounting capital comes from family of friends: GC{Q) is near to 1. When its size increases, the firm may loan to banks but it cannot go to the Stock Exchange: a{Q) decreases quickly. Big firms may issue bonds and may go to the Stock Exchange: a(Q) may go on decrease or may increase 12 .

6.

A large part of the argument of the pure profit is refuted: the maximisation of the rate of profit is not identical to the maximisation of the aggregate pure profit.

Proof. Before, we have: 1

+ Q a(Q)/K{Q) = Z Q K A {Q)/K{Q) anc * remember that n A (Q) = •
Next, the solution for the maximisation of the aggregate pure profit U(Q) is obtained for

™ A (Q) = t a'(Q) K{Q) +1 a{Q) K'{Q) .

Consider the curve U'{Q) = Il(g)-/ a'{Q) K{Q)-1 a{Q) K'{Q) .

The solution for the maximisation of the rate of profit n{Q) is obtained for ((?) = 4Q) K'{Q). Consider the curve Î"{Q) = Tl' A (Q) -TC{Q) rte) •

Let us compare the respective positions of W(Q) and of î K (Q). Suppose that TY{Q) is decreasing ( TI"(Q) < 0 ), SO that the optimum of the aggregate pure profit H(Q) would be a maximum. Let us write Tl'(Q) as a function of î K {Q) '

nÌQ) = n' A (Q)-n(Q)K'(Q) + 4Q) 1 + a'(Q) K(Q) cc{Q) K'(Q) <*(Q) K'{Q) + ^-'\ l + *a{Q)lK(Q)) Denote Q* p

Special case: fixity of the financial structure

Here, the parameter a is constant; thus The consequences of this result is that, in this normal case compared to the maximisation of the pure profit, the firms are smaller, there are more firms; however, the price is higher.

Q A (Q)/K{Q) = ^ ANC * = N G^<Q)'~T 0"""**)•

If -= / <=> n A

) = /, that is to say, if the rate of profit is equal to the rate of interest, the two solutions coincide.

3.

The results of [ de MESNARD 1992 ] are retrieved for a = 1 (except for the theorem Ibis p. 112-113): see annexe.

Result 1. The optimum is at the same place whatever be a and t. Thus, if the condition > / <z> K a \Q K ) > t is valid for one value of a (respectively /), it is valid for any other value of a (respectively t).

Comment. This result shows that taking into account of the financial structure is a little trivial if this structure remains fixed. The interesting result is obtained when the financial structure is variable.

Proof It is obvious: ^{Q) = n[j {Q) is independent of a and /.

Example

We keep the same example as above: The values of n are given in percent.

C(Q) = 5 Q 3 +10 Q 2 -5 Q + 20, />(Q) = -5 0 +100, = pQ, K(Q) =

13

A logistical function is well adapted to such a phenomenon: a slow decrease, followed by a fast decrease and a slow decrease: when a firm is developing, it has some difficulties to find loaned funds in the beginning; after, things are more easy; at the end, the part of loans must not exceed 1. 0.5 1.5 2.5 3.5

M U2 7t0 7C2 7C3 7l4 7l5 7t6 7t7 7C8

Maximisation of the pure profit compared to the maximisation of the rate of profit

We see that the optimum of n\ and /r3 are to the left of the optimum of the pure profit maximisation; the optimum of nl and n\ are to the right; the optimum of n5 and n6 also are to the right, but the optimum of 7r5 is quasi degenerated because the curve is quasi horizontal in a wide part (and a local optimum); this is the effect of the point of inflexion; the important fact is that we obtain a quasi horizontal optimal section of curve for the objective function without supposing that the cost curve is itself horizontal (U curve) as it is made in the classical literature.

The optimum of 7r3 is degenerated: an horizontal optimal section for the objective curve is possible if the part of shareholders increases but also if it decreases.

Note that the curves >T0, nl and /r8 are the limit curves.

For a constant part of shareholders and a variable interest rate, the optimum of the maximisation of the rate of profit is unchanged (7z0,7r7,7i&) when the interest rate increases; this optimum goes toward the left if the part of shareholders is decreasing ( n\ and ;r2), and goes to the right if the part of shareholders is increasing ( ^5 and 7i6).

When t= 0,40 the aggregate pure profit is negative, while the rate of profit is sometimes positive (curves 7i0,7t2,7t6).

The optimum of curves 7t5 and n& correspond to a large production but with a low rate of profit. The optimum of curves n\ and n2 correspond to a large rate of profit but with a low production. Note that the optimum of the rate of profit may be realisable even if the optimum of the pure profit is not realisable (curves n2,7t6 and 112).

The following graph shows the classical curves when in the case of fixed coefficient of capital and constant part of shareholders (for other cases as variable coefficient of capital or variable part of shareholders, the graph may be not so simple):

Classical curves: monopoly with fixed coefficient of capital and constant part of shareholders CM is the average cost, CAM is the average accounting cost and CA ? is the marginal accounting cost.

Dynamics

The question of dynamics is: does the optimum of the rate of profit tends toward the optimum of the profit? In the above example, starting from the optimum of the rate of profit, the rate of profit is decrasing when the production increases15 . Thus, the firm may not tend by itself to increase is scale of its production.

In fact, a dynamical evolution from one optimum to another point requires to introduce another reflections. For example, the demand function moves in a long term basis or another firms enter the sector.

As in the classical microeconomic theory, even if there is maximisation, the optimum may be unacceptable. In the classical case the optimum of pure profit may be negative. In the theory of the maximisation of a ratio of profit, the optimum may be inferior to the interest rate. The point were the optimum becomes acceptable is the entry point in the sector.

If the rate of gross profit is inferior to the interest rate, the firm will never enter in the sector whatever be its financial structure (the pure profit is negative). If 7t(j{Q)>t, the firm may enter, depending on the ratio it choose as an objective function. If the firm chooses the rate of profit, it enter if 7i(Q)>t; if the firm chooses the rate of accounting profit, it enters if n A {Q)>t. 1 is e a(Q)/K(Q)

>0

(remember a(Q*)<\) which is not the more realistic case; thus generally He;)] cc(Q)/K(Q)

Q=Ql < 1 and the very long term optimum is to the left of the classical long term optimum. Thus, the question which remains is: does the market clears?

THE RATE OF ACCOUNTING PROFIT WITH VARIABLE FINANCIAL STRUCTURE

The case studied in [de MESNARD 1992 ], the optimum of the maximisation of the rate of accounting profit, is formally retrieved putting a ~ 1. However, we give the general solution of the maximisation of the rate of profit.

The solution for the maximisation of the aggregate pure profit Tl(Q) is obtained for:

tcc'(Q)K(Q) + ta(Q)K'(Q) K'A ((?) = * cc'(Q) K(Q) + ta(Q) K'{Q) = / <Z(Q) K{Q)- a{Q) K{Q) c*n> A (Q) = ta(Q)K(Q) a'{Q) , K'(Q) . cc{Q) K{Q) Consider the curve U'{Q) = IT^ [Q)-t CC{Q) K(Q) <x'{Q) i K'(Q)' a{Q) K{Q)
The solution for the maximisation of the rate of accounting profit XA(Q) * s obtained for:

K f io\ K'A (Q) = (Q)/Q n A (Q) «• ™ A (Q) = U A (Q) AQ) a{Q) K{Q) Ua[Q) ( a'(Q) , K'(Q) + a(Q) K{Q) Consider the curve f"{Q) = W A (Q)-U A (Q)
Let us write TI'(Q) as a function of f"{Q) :

N'(E) = f"(Q) + [U A {Q)-1 a{Q) K{Q)] aJQ) , K '(Q) <*{Q) K{Q) a'{Q) , K'{Q) «((?) K{Q) »N-(E)=FF(E). <*{Q) K{Q) [a'{Q) K{Q) + a{Q)K'{Q)} O U'{Q) = f K (Q) + [^ (<?)-/ ]^ {Q)
Thus, to study the respective position of the optimum of the maximisation of the pure profit and the maximisation of the rate of accounting profit we must compare n A (Q) and t, depending on the signlof K' A {Q); this is the condition of [de MESNARD 1992]. However, the sign of

K' A {Q) or G'(G) , K'(Q)' <*{Q) K{Q)
is not determined clearly when there is a financial structure. Thus, the optimum of the maximisation of the rate of profit is to the left of the optimum of the maximisation of the aggregate pure profit if

x A \Q)>t and K A (Q)>Q, or n A {Q)<t and K A (Q) < °> and * is to the right if n A (Q) > t and K' A (Q) < 0 or n A (Q) < t and K' A (Q) > 0.
Maximisation of the pure profit compared to the maximisation of the rate of accounting profit A second argument is that it may be relatively difficult to find shareholders: it requires to go at the Stock Exchange. For many firms it is impossible because they are too much small (curve /r4 of the above example, giving a decreasing part of shareholders); even for big firms, it is not always possible to issue a big amount of shares or to make repeated issues. That is why many firms loan, by bonds or with banks.

A third argument is that the loans were supposed to be perpetual implicitly. If not, shareholders must be paid back. This pay back is a deduction over the cash and then on the accounting profit distributed to shareholders (and appears only in the balance sheet). Thus the effective rate of remuneration of the accounting capital may be lower than the maximum possible: often, managers decide to pay a dividend giving a normal remuneration inferior to the rate of accounting profit (for example, a rate equal to the rate of profit), and they use the rest of the accounting profit to pay back loans.

These simple financial and accounting arguments show that the theory of the maximisation of the rate of profit must be understood in that way. It is not a financial theory; it is a microeconomic theory. The part of shareholders a{Q), that is to say the financial structure, must be seen as either a parameter, either an endogenous variable, but not as an exogenous variable to be optimised. To find the optimal value of the production which maximises the rate of profit, we 17

The contrary and reasonable strategy, decrease the part of shareholder to increase the rate of accounting profit, is the well known financial shift effect. must fix the value of the parameter a then see how the optimal production changes if a changes, or we must suppose a law of evolution for a{Q) then see how the optimal production changes at the inflexion point (or the law itself) changes, and we must not try to find the optimal value of a{Q) which maximises the rate of profit for a given level of production.

Obviously, the theory of the maximisation of the rate of gross profit does not bring about these problems because it does not depend on the financial structure. The theory of the maximisation of the rate of accounting profit directly take into account the profitability of the accounting capital and reflects the influence of the shareholders over the firm.

CONCLUSION

In this paper, the question is: what is the objective function of the firm? To answer, we proposed a typology of the ratios of profit. Taking into account the financial structure (by a coefficient of financial structure: the accounting capital over the whole capital), in addition to the rate of accounting profit (seen in [de MESNARD 1992 ]) we studied the rate of gross profit (gross profit over capital) and the rate of profit (accounting profit over capital).

The main result concerns the optimal production when the rate of profit is maximised instead of the aggregate pure profit: it is lower when the compensated rate of profit (the ratio of profit over accounting capital, divided by the relative elasticity of the accounting capital with regard to the whole capital) is higher than the interest rate. This result may be particularise to the case where the financial structure is fixed (the production obtained with the rate of profit maximisation instead of the maximisation of the aggregate pure profit is lower when the rate of profit is higher than the interest rate) and it generalises the preceding results.

One of the more interesting consequences of the theory of the maximisation of the rate of profit is that, as each firm is smaller often, there are more firms; thus the interaction between firms is lower and competition is greater. Moreover, the clearing of the market must be proved again.

An argument in favour of the theory of the maximisation of a ratio of profit is that its thought processes are near the operational economic calculus. Within firms, managers thinks in terms of ratio of profit, that is to say in terms of profitability, more than in terms of opportunity costs and more than in terms of maximisation of an aggregate profit, even it is a pure profit. The question rests into the choice among the three ratios studied: rate of accounting profit, rate of profit, rate of gross profit. However, the result of the maximisation of a ratio of profit remains different to the result of the maximisation of the aggregate pure profit. This may provide a program of research with the following questions. What become the classical models when a ratio of profit is chosen instead of the aggregate pure profit, whatever be this ratio? What ratio of profit must be chosen? What happens when one firm choose one ratio, and one another firm choose another ratio?

  For example, in the Lerner-Carleton model, the firm maximises its present actualised value, under a constraint of financing, which indicates the relation the rate of profit and the value of the capital [LERNER and CARLETON 1966]. The present actualised value of the firm is the (l -r) N / = (l -r) N 0 (l + g)*~], where N is the profit and r is the share of benefits kept by the firm. This gives the Gordon-Shapiro formula (when n->co): V ---. And the rate of growth depends on the rate of profit n by the trivial relation g = nr . At this point, the financial constraint describes what level of the rate of profit is obtained with each level of capital: n=f(K) 7 .

5

  For example R. Lantner in a short quotation in the educational review DEES [ LANTNER 1987, p. 71 ]:

  is well adapted to such a phenomenon: a slow decrease, followed by a fast decrease and a slow decrease. When a firm is developing, it has some difficulties to find loaned funds in the beginning; after, things are more easy; at the end, the part of loans must not exceed 1. of the maximisation of a general ratio of profit In [ de MESNARD 1992 ] three cases of maximisation of the rate of accounting profit were studied. Consider the more general case, the monopoly with function of capital, and suppose that the first derivative of the function of capital is positive (that is to say eK(O)/Q > ® )• Consider a ratio of profit: ^U(Q) = ^~~' where UIJ(Q) = R{Q)-C(J{Q) is an undefined expression of the K\Q) profit and 71U{Q) is the corresponding ratio of profit. The maximisation of this ratio ^ * S * C relat * ve eIast * cit y °f * e ca P ltal t0 the production and n^y (0) = ^JTJC?) j s t h e average profit. The firm produces up to the point where the marginal accounting profit be equal to the average accounting profit multiplied by the relative elasticity of the capital to the production At the optimum, the relative elasticity of the capital to the production is equal to 1. Indeed, the above result could be written ^^H^J = 1 0 e p{Q)/K{Q) = 1 . As a particular case, [de MESNARD 1992 ] studied the monopoly with fixed coefficient of capital: K{Q) = k 0 and ^K(Q)/Q = Thus ^U(Q) = R[/(Q) : the firm produces up to the point where the marginal accounting profit is equal to the average accounting profit (or to the point where the average profit is maximal). As said in [ de MESNARD 1992 ], this corresponds to an observed behaviour of the firms: a firm produces and invest as much as the profit brought by a supplementary unit is superior or at least equal to the average profit brought by other units. And this point is not the point of long term equilibrium of the traditional monopoly (R'iQ) = CJJ(Q) at a long term) and do not correspond to the point of the minimum of the average cost. If the marginal accounting profit is decreasing (condition of diminishing returns, in order to obtain a maximum of the accounting profit), the optimum is to the left to the point which maximises the accounting profit H ; however, this does not take into account the question of the pure profit and then it does not proof that, always, the optimum of the maximisation of the rate of profit is to the left of the classic microeconomic optimum (maximisation of the pure profit): it must be done further. As another particular case [ de MESNARD 1992 ] studied the case of perfect competition with fixed coefficient of capital: p{Q)~ p and e K{Q)/Q ™ s ê* ves Q/(ô) = Q/(ô)-The competitive firm produces up to the minimum point of the average accounting cost. In [ de MESNARD 1992 ] two consequences are brought out.

  e a{Q)/K{Q) to a lower size of firms. This is very important. < 1 . Then, the standard case corresponds He;)] l + e a{Q)/K{Q) *'KA{Q)/K(Q)\Q = Q* the two points Qp and 0* coincide.

  the point of maximisation of the aggregate pure profit, such as n'((?p ) = 0 . Denote Q* the solution of the maximisation of the rate of profit, such as (curve f"(Q) is under the curve Tl'{Q). Thus, Q* P must be higher than Q* At the limit, if U'{Q) is vertical near to Q* P , Then, near to £>* , the curve f K {Q) is over the curve n'{Q) ^ , W(Q) is horizontal near to £>* and Q* P is rejected to infinite. ta(Q*J 2. Even if the optimum of the pure profit is unique, the optimum of the rate of profit may be non unique: it would be necessary to study the case of the curves Yl'(Q) and f"(Q) with multiples zeros. • Remark. 1. If the elasticity of the capital regarding to the production is negative, the result are inverse: this case is not realistic, because it implies that less capital is used when production increases. If the elasticity is nil, the two solutions are identical.

However, as we

  have n A (Q) = KQ (Q) if and only if UQ {Q) = t , the condition ICQ (Q) > t is equivalent to the condition 7t A {Q)>t . As fi(Q)< XG(Q)> ^ entr y point when the firm maximises the rate of profit is to the left of the entry point when the firm maximises the rate of gross profit or the rate of accounting profit: the pure profit is not necessary nil.Nevertheless, the rate of profit is not a good indicator of entry in the sector. Suppose that n{Q)<t with a positive accounting profit and TCQ (Q)>t ; then the firm does not enter in the sector; however the pure profit is positive and an inferior positive profit may be realised by lending the accounting capital at the monetary market; thus it is better for the firm to enter even if 7t{Q) < t. This proves that the good indicators of entry are the rate of gross profit or the rate of accounting profit, even it may be used for the maximisation.The model of maximisation of a ratio of profit is a long term model because capital varies. It leads to consider a very long term, where the number of firms varies (this very long term is analogous to the long term of the neoclassical microeconomic theory).In perferct competition, the classical reasoning is that if the pure profit is positive, other firms may want to enter in the sector; thus the pure profit of each firm decreases until zero where the number of firms is maximal; this point corresponds to the minimum point of the curve of long term cost16 . In monopoly, the demand function moves to the point where the profit is nil (tangency between the demand function and the average cost function).. Analogously, when the rate of profit is superior to the interest rate, another firms may want to enter in the sector because profitability is higher there; thus the rate of profit of each firm decreases toward the interest rate where the number of firms is maximal. The entry point is the point of very long term equilibrium. Thus,• As seen before, when firms maximise the rate of accounting profit, the point of very long3.4.2. THE VERY LONG TERMcorresponds to the point where the pure profit is nil and 16 Note that this reasoning seems contradictory with the general equilibrium theory where the number of agents is given. maximum. This result was not clearly stated in [ de MESNARD 1992 ] and it is the same with the rate of gross profit. However, when firms maximise the rate of profit, it must be pointed out that the point of very long term equilibrium ) -> t does not correspond to the point where the pure profit is nil. And when the financial structure is variable, this point does not coincide with the

  of the firm, the maximisation of the rate of profit, is realistic, because it corresponds to an economical idea of the capital: that is why we have logically supposed K'iQ)> 0 And the firm knows how many funds it engaged in the capital: the accounting capital plus the reserved plus the long term debts. However, there are some problems. Consider the following reasoning. When the part of shareholders is taken as an increasing parameter (that is to say when Q is given), the rate of profit increases: if a = 0 then n-7TQ -t and if a = 1 then n= UQ . Thus the firm may want to increase the part of shareholders in order to increase the rate of profit. This strategy decreases the rate of accounting profit: if or = 0 then n A -> °o and if a = 1 then n A = UQ 17 . Thus, there is an incompatibility between the objective function of the firm and the objective function of the shareholders 18 . This reasoning does not take into account one parameter: the variability of Q. To change the part of shareholders a{Q), it may be necessary to change the optimal level of production Q . Thus UQ (Q) may vary and the comparison is not possible. In the above example, the curve u5 shows an increasing part of shareholders which leads to a quasi non increasing rate of profit, and it shows a considerable increase of the production to go from a low level to a high level of a(Q) (from nl to nO in the following example).

18

  This problem does not occur in [ de MESNARD 1992] because the objective function was the maximisation of the rate of accounting profit.

• (accounting profit) -(opportunity costs per unit of capital). (capital) > 0 (accounting profit) , . n . lx • or - > (opportunity costs per unit of capital) . (capital)

Theorem 1 becomes: Theorem V. Compared to the profit maximisation, the maximisation of the rate of profit gives a lower optimal quantity (respectively higher) when the rate of profit is higher (respectively lower) than the interest rate: > t => max K< max n or

n A \Q* N ) > t max n< max n 2.

< t => max n>-max n or n A ((?* ] < t => max n>-

max n where 0* is the optimal quantity which maximises the rate of profit n{Q). Comments. 1. As in [ de MESNARD 1992 ]the first case correspond to a standard case (only when the financial structure is fixed) because the condition

	> t <=> n A \Q n )>t	is normally
	respected (

if there exists some alternative investments): if it is not, an alternative investment would have be chosen. Thus, the maximisation of the rate of profit normally leads toward a lower optimal production than the maximisation of the aggregate pure profit does. Another large part of the argument of pure profit is out: the maximisation of the aggregate pure profit is not the more careful strategy because the maximisation of the rate of profit normally leads toward a lower investment.

In any case, the amount of funds to be invested must be limited. If the amount is not limited, the aggregate pure profit may be maximised, even if there are alternative possibilities of investment (however, in this case, as the capital has no price, the opportunity cost of the accounting capital is nil: the pure profit is equal to the accounting profit).

Lerner and Carleton use a simple function of capital like a coefficient of capital indicating the amount of capital for each level of production: as the rate of profit is linked up to the amount of production by the demand function and by the cost function, this rate is linked up to the capital.

The argument were: loans are included into costs and are not taken into account in the calculus of the opportunity cost of accounting capital.

See the discussion farther.

An increasing part of shareholders is less realistic. However, it is temporally possible and the case is show in a didactic way.

Note that the rate of profit is not always decreasing when the production is increasing, as in the Lerner-Carleton model.