LATEC Louis De Mesnard
email: louis.de-mesnard@u-bourgogne.fr

Controlled-by-owner Firms, Mobility of Capital and Microeconomic Profit Rate Maximization

Keywords: JEL classification. L21, D21, D24, D41, D42, D43, L13. KEYWORDS. Profit Rate, Firm, Control, Coordination, Objective

When they actively control the firm, owners select the firm that has the best profit rate if the hypothesis of mobility of capital is adopted: controlled-by-owner firms are profit-rate-maximizing when sleeping-owner firms are pure-profit-maximizing. Both types are compared in monopoly, in perfect competition, in classical or in mixed duopoly. Always, controlled-by-owner firms have a lower output than comparable sleeping-owner firms. It only takes a fixed coefficient of equity capital to do that price plays no role for controlled-by-owner firms in perfect competition; in duopoly, it only takes a similar condition plus a linear demand to do that reaction functions vanish.

I. Introduction

Traditionally, the firm is assumed to maximize its pure profit. Some authors have proposed some alternative theories as Baumol's theory (1959) of maximization of the size or maximization of the revenue of the firm, the behaviorist theory of the firm [START_REF] Cyert R | A Behavioral Theory of the Firm[END_REF], the theory of separation between ownership and control [START_REF] Scherer | Industrial Market Structure and Economic Performance[END_REF], the maximization of consumer surplus, the maximization of added-value for labor managed firms [START_REF] Sertel | Workers and Incentives[END_REF]; if it is a non-profit organization, like a mutual benefit society, a hospital or a charity organization, the organization may want to equilibrate its accounts; one may quote also the maximization of consumer surplus and the maximization of the profit by unit. However, pure profit maximization remains widely adopted, because it seems to be more natural and intuitive, and more justified: as the distributed profit is in the revenue of the owner, and as the owner is also a consumer, he wishes that the firm maximizes its profit. Some papers [START_REF] Hart | On Shareholder Unanimity in Large Stock Market Economies[END_REF][START_REF] Makowski | Competitive Stocks Markets[END_REF] have tried also to prove that there is unanimity between owners under some conditions about the firm's objective.

We name objective function of the firm the quantity that the firm maximizes. [START_REF] Slade | What does an Oligopoly Maximize?[END_REF] uses the term "objective function" as the function that is maximized at the global level. For her, in the case of monopoly, both functions coincide but in any other cases, they may diverge: in competition, firms maximize profit but the market acts as a social welfare maximizing agent.

The global objective function is called ficticious-objective function by Slade and she studies the conditions of existence of such a function for an oligopoly. In the following we only use the objective function of the firm that we call shortly the objective function.

In the usual context, the capital plays a very minor role because it is fixed in the short term and cannot participate to the short run objective function of the firm. The capital as fund K serves to pay the real capital denoted K r and all the charges paid by advance by the firm. The real capital is a more popular concept inside microeconomic theory, for example in the production function. The real capital K r (machines, real estates, and so on) is usually considered in place of the capital as fund. Real capital has an incidence on production through production functions as f(K r , L) or through fixed costs. It must be distinguished from capital as fund that is used to buy real capital but also other factors (as L in the above production function).

Remember that the link between real capital and capital as fund is variable: for example, the real capital can be fixed whereas the capital as fund is variable if the prices of inputs vary. In the following, the word capital refers to the fund of capital, equity capital plus debts; when it is question of the real capital it is explicitly indicated.

The status of the opportunity cost of the equity capital is also debatable when the capital as fund (equity capital plus debts) is considered as variable, in the three cases of mobility of capital:

• Either the long run is considered, and the capital is obviously variable. An additional phenomenon could play a role: some firms may enter into the sector and the price will fall down because of the increasing competition. Here, the number of firms is assumed to remain fixed.

• Either the capital does not correspond to sunk costs: it is free ex ante and ex post because it can be resold at any moment for the same amount and without any additional costs or losses. This idea is close to those of [START_REF] Baumol W | Contestable Markets and the Theory of Industry Structure[END_REF]: when there are no sunk costs, hit-and-run raids are possible and investments can be set and removed with no cost, so the amount of capital engaged can be considered as free always. In this case, the long run is similar to the short run. Even in the long run, new firms may not be able to enter the sector if the price yet fixed at its minimum by the installed firms. Note that sunk costs are not fixed costs.

• Either a given capital yet installed corresponds to sunk costs and is not free ex post, but it can be considered nevertheless as free ex ante, before all decisions of investing are done. In this case the capital as fund is considered as variable even if it is not the long run.

If the capital is variable, the opportunity cost of the equity capital plays a role and the concept of profit considered must be the economic profit or pure profit. Denote IT^ as the profit of production or gross profit with Tl g = R -C g , where R is the firm's revenue and C g is the cost of production (g stands for "gross"). The accounting profit is equal to the gross profit minus the real financial cost t* Kd: U a = U g -td Kd, where K* is the debt and tj is the interest rate on the debt. The economic profit is equal to the accounting profit minus the opportunity cost of the equity capital: fl = Yl a -t e K e , where £ e stands for the equity capital and t e stands for the opportunity cost of one unit of equity capital of equity capital. Only opportunity costs on equity capital are considered and not other sources of opportunity costs. If the opportunity cost of one unit of capital as fund is assumed to be equal to the interest rate f, what requires the financial market to be perfect, then the cost of capital K is equal to t K. The economic profit is then II = Tig -/ K and traditionally this quantity is assumed to be maximized by the producing firm (again, ex ante or always if no sunk costs).

Note that the pure profit is not a clean and quiet, good old concept: it is problematical because is no so easy to compute. Knowing the actual structure of the debts of the firm, td can be computed. Even if in practice the mean interest rate of the firm's debt can be complicated to compute, there is no theoretical obstacle to do it. So, the problem lies on t e : it can be difficult to calculate the exact opportunity cost of equity t e K e , not because it is complicated to know the amount K e of capital as fund engaged, but because the opportunity cost by unit of equity, t e , is really a problematical idea. Does it is assumed to be equal to the real mean interest rate paid by the firm on all its debt (t e = /</)? Does it is assumed to be equal to the interest rate of the market (t e = t = td> if the financial market is perfect, considering that this capital can be lend on the financial market at the rate t)l Does it is equal to the real cost on the market of such an amount of capital (if the financial market is imperfect, and the value of t e when lending must be distinguished of the value of t e when borrowing)? Etc. In this last case, the evaluation could be impossible because the real cost of a credit can only be known if the credit is really supplied or demanded, not if it is only potentially supplied or demanded as with virtual transactions conducted only to calculate an opportunity cost. Remember that not only real transactions can be charged with transaction costs but also virtual transactions. The cost of the debt is seen from the borrower's viewpoint (the firm knows how the debt will cost) when the opportunity cost of equity is seen from the moneylender's viewpoint (the firm has a fund and tries to determine how much it can yield); the difference is all the more important that there are transaction costs. Moreover, the alternative standard investment must be well defined, perfectly known and unique. So, the evaluation of the opportunity cost is always speculative and based on hypotheses that can be contested. These critics about opportunity costs are very general when the market is not perfect, what corresponds to the reality. When you try to know the opportunity cost of an apartment that you have, you are embarrassed: either you try to compare it with other apartments around to know its rental value, either you try to rent it, just to see how it could be worth. However, in both cases, the result is not necessarily equal to the real value that you could have found in a real process. In any case, the process is costly (transaction costs) and the value found itself depends on the time spent to calculate it.

The paper will prove that, in the case of mobility of capital, the pure profit maximization is not the objective function of the a controlled-by-owner firm because the firm maximizes a ratio pure profit / equity capital, where both pure profit and equity capital are variable. Then, the two functions, profit rate maximization (for controlled-by-owner firms) and pure profit maximization (for sleeping-owner firms), will be compared for their consequences on the economics of the firm (output, coordination, etc.).

II. The firm's objective function for controlled-by-owner firms, in case of mobility of capital

I consider the case of mobility of capital, i.e., in the long term or ex ante, before all investments are done, or when it faces to no sunk costs. Pure profit maximization seems more natural and intuitive for economists. The main argument really discriminant between pure profit maximization and other objectives is the owner's behavior argument. The profit distributed by firms is either in the utility of the owner, either in its budget constraint. So, when maximizing its utility, he wishes that the firm maximizes the profit because and, as the owner's behavior influences the firm's behavior, the firm will maximize the profit. This is obviously true in the simple case when the producer is an individual entrepreneur because the profit is also the revenue of the consumer: as the profit allows the producer to buy the goods that he will consume, he might maximize its profit. For a salaried employee this is false because its revenue comes from its wages, and not from the profit, except if the system of remuneration links the salary to the output or if the salaried employee is also an owner. In other cases, it could be more difficult. Mas-Colell, Whinston and Green (1995, pp. 152-154) study the case of a unique firm, even multiproduct, owned by simple consumers. Adapted to our notations (II simply replaces p y, where y is the vector of net output), their formula for one consumer / is: max£/'(x'), s.t. px'<w' + 0'II (1) where x' stands for the consumption vector of commodities by /, p for the fixed price vector,

A. One firm

FT/

w l for the wage received by i, n for the firm's pure profit and 0' = j=r for the share of the distributed pure profit of the firm that i receives, with X, 0' = 1 (the agent has provided to the firm an equity capital equal to 0' K e , when the total equity capital is equal to K e ); note that the question of profits that are not distributed is left aside. The agent tries to increase its budget and prefers to receive the maximal Stock Exchange revenue £)' = 0' IT, what implies that the unique firm maximizes its pure profit II and the discussion could seem to be closed.

Note that Mas-Colell, Whinston and Green (1995, pp. 153) also indicate that, for one firm and many goods, if prices depend on the production of the firm, the result can be undetermined because the function to be maximized by the firm changes with the preferences of consumers regarding to the goods. However, this case is not essential in practice because each owner buys a very small part of the production of the firm. Anyway, if one owner has a significant power on the firm, either by its shares or by its consumption, he can influence the firm's objective. See also [START_REF] Hart | On Shareholder Unanimity in Large Stock Market Economies[END_REF] and [START_REF] Makowski | Competitive Stocks Markets[END_REF] for incomplete markets.

However, as the profit is distributed proportionally to the invested capital in a first IT A?

approximation -other forms of payment are neglected here ~ one have 0' = ^pj-= -f, but one cannot accept the hypothesis of fixity of coefficients 0'. If the 0' are assumed to be fixed this implies that varies as K e , that is to say the agent i subscribes to all new issues of firm's shares, what is very unrealistic. So, considering that both II and K e can vary simultaneously, the second influencing the first, the above program can be written as: max ^/'(x'), s.t. px'^w' + A^Tt (2)

where n = ~r~ stands for the profit rate of the firm (K e and n are controlled by the firm, when K e K[ is controlled by owner). Knowing its capital K\, the agent i wishes that the firm maximizes 7C. Note that if the owner is a sleeping partner, its wishes cannot influence the firm and consequently this one maximizes its pure profit classically, because this objective corresponds to the maximization of the net value of the firm.

Remarks.

1) The profit rate can be written 2) The profit rate fails to separate industrial decisions (determining the optimal output) to financial decisions (determining the optimal financial structure, i.e. the ratio of debts

a = -p-^rr),
what is often considered as a disadvantage in a theoretical viewpoint.

Kd + K e

However, the result is clear: the agent maximizes a ratio with the equity capital only in the denominator and not with all the capital. Moreover, the maximization of K determines the optimal output Q*, and so the optimal equity capital K e (Q*), via the function of debt K e (Q), but it determines also the optimal debt Kd(Q*), via the function of debt Kd(Q) and the whole capital

K(Q*) = K e (Q*) + Kd(Q*).
In other terms, the maximization of n determines at the same time the optimal output and the optimal financial structure. If the amount of debt is exogenously given, there is no more difficulty: simply the function of debt is a treated as a constant.

However, from the firm's viewpoint, these ratios fail to discriminate between the production decision (what level of output and how much capital as fund?) and the financing decision (what sharing between equity and debts?). The accounting profit is calculated by subtracting the cost of debts but not the cost of equities, so it depends of the main financial decision: what is the source of financing for the firm? Taking into account the "lever effect"

(the interest rate is lower than the profitability of the firm), the firm has interest to get into debt because this increases the dividend by share, what increases the satisfaction of shareholders and the value of the share. However, if all firms and markets are competitive, the profitability of firms becomes equal to the interest rate (no firms have interest to enter the industry) and the lever effect plays no role; this is the Modigliani-Miller theorem: it proves that the value of the firm is independent of the interest rate for competitive economies [START_REF] Modigliani | The Cost of Capital, Corporation Finance and the Theory of Investment[END_REF]. [START_REF] Jensen | Theory of the Firm: Managerial Behavior, Agency Costs, and Ownership Structure[END_REF] prove that there is an optimal financial structure because of agency cost: agency costs spent by moneylenders to control the firm may prevent it to reduce the part of shareholders; also bankrupt costs imply an optimal financial structure. So, a large part of the financial structure of the firm is exogenously determined: when trying to calculate the optimal output level, the manager also decides about financing, what assumes an ex ante link between production and financing; clearly it is a function of which form is partly exogenous for most firms. Figure 1 shows that the ratio of debts, a, is near zero for a small amount of output: small or new firms cannot obtain loans from banks because the risk seems high (this is why special Stock Exchange Markets have been created in some countries for the risky capitals). When the scale of production increases a little, firms can obtain loans from banks because it can prove its profitability, and the function increases. Later the function becomes more endogenously determined and the form of the function becomes less clear: firms can either go on the Stock Exchange Market, either borrow more and more, depending on their financial policy.

However, as the firm grows, its size increases and the conditions for a competitive economy become less valid: unspecification and instability increase. That this link may be complicated is not the problem: the problem is that the link can be exogenously determined for small or new firms and it can be unstable for large firms. As an illustration, a recent strategy of some companies in the Stock Market consists for a firm into buying back its own shares, to reduce the equity capital and so increase the price of the share (or to prevent this price to decrease). When a firm buys back its own shares, it modifies the form of the function oc(0, or simply it shifts the function and the optimal output is modified, or it tries to reduce the exogenous parameter a: in this last case, the output is not modified, only the profit rate is increased (but the pure profit is also not modified).

3) What is here the justification of the use the pure profit? In money terms, the firm distributes the accounting profit n a = TL g -td Kd. The opportunity cost of equity capital, t e K e is distributed (the pure profit is always distributed): it is not retained by the firm even it is an opportunity cost, because it must pay the equity capital (in the other case, the owners would support a real cost on their capital because it would be never paid). So the agent receives the dividend D l a = 6' Yl a and not D i = 9' EL So, if the owner is assumed to have no opportunity cost on its capital, he should wish the firm to maximize directly Tif=^r instead of n. The owner maximizes n if he is assumed to support an opportunity K e cost on its capital, t e Q i K e (by simplification, the same price t e is used here for the firm and for the agent), and the "pure" dividend is considered:

D^D'a-te 6' K e = 6' (Tl a -t K e ) = 0' II. •
If there is only a unique owner, pure profit maximization and profit rate maximization coincide. In other cases, if the firm is profitable, each agent will maximize K e naturally (under an eventual constraint K^K^) but he wishes that the firm maximizes n: it is only a wish, because, as there is only one firm, this firm has no incentives to do that.

B. Many firms

There should be many firms to be sure that the wish is transformed into an obligation for firms, by selecting the firm that has the best profitability. This is why it is necessary to introduce many firms in a more general approach, extrapolated from the former to introduce the question of allocation of capital. Consider many firms, each denoted j, eventually multiproduct. TV is the profit of j\ Q iJ = -is the share of the profit of firm j that i receives, with E, Q iJ = 1 (where K J e is the total equity capital invested inside firm j, and Ke is the equity capital invested by / in j). One have:

max £/'(x'), s.t. f V x'' < w i + I, Q iJ TV (3)
As above, the agent tends to prefer a budget as larger as possible, so he prefers that firms maximize their Stock Exchange revenue IT = 2 y IF = E y 0^ TV , the sum of the profits that he receives. Note that this sum is not the simple sum of firm's profits: even if the shares Q iJ are fixed and for example equal to 1 (only one shareholder for each firm), it remains a sum of profits that is not the profit of a particular firm: some may become decreasing so long as some are increasing. As above, the shares are determined by the agent and are linked to the capital that can be invested in each firm. One can write:

rr = i j id -=i j d &=i j x« Ki & (4)

TV

where n J = -stands for the profit rate of firm / and V j = -stands for the share of the capital of i that is invested in firm j. If there is at least one firm with a profitability higher than the interest rate, the agent i will select the firm j (or the firms if placed equal first) that has the best profitability (i.e., the best profit rate n J ) to maximize its II': to do this, he puts № = 1 and A/ = 0 for all j'. If all agents do the same thing (and they did if they are assumed to be similar to /), firm j must maximize its profit rate. Note that this result is obtained as the agent's capital, Kg, is fixed. To summarize, in a perfect world, the optimal behavior of the agent consists into ordering firms by decreasing order of profitability and investing on firms up to the point where the profitability becomes equal to the interest rate, depending on the possibilities authorized by the competition between owners, eventually by borrowing and lending the necessary money to the financial market. This proves that the owner must maximize the profit rate of firms. This is why I introduce in this paper the profit rate, n = -pr, as an objective of the firm.

Remarks.

1) The comparison of two pure profits has no sense, when this comparison is correct for two profit rates: profit rates are allowing to take into account the alternative possibilities of investment (Cf. the theory of investment), when pure profits cannot. However, let's take an example. A pure profit of 1 M$ for the construction of a plant with a capital of 20 M$ cannot be compared to the pure profit of $1000 to install a small itinerant pizzeria with a capital of $10000, when a profit rate of 5% can be compared always to a profit rate of 10%. The same difficulty exists inside investment theory: two intern rates of return (rate that cancels out the net actual value) can be compared always when two actual net values can be compared only if the two projects are similar in size.

2) Additionally, the capital invested by an agent can be considered also as variable. If the amount that the agent can invest exceeds the capacity of one firm /, one can have 0^ = 1 and the agent becomes the unique owner of this firm; but one can also have 0^ < 1 if there is a competition between shareholders (even if the capital of the agent is lower than the capacity of firm /, he can be obliged to find another firm is there is a competition); in both cases, the agent will be obliged to limit its investment. Either he can search another firm to invest the rest of its capital but at a lower profitability (what is acceptable only is the profitability of this second firm is itself higher than the interest rate of the market), either he can lend the exceeding money to the financial market. As the agent also can borrow the quantity of money that he wants to invest in firms if the financial market is efficient, the equity capital becomes completely variable. •

C. Ex post

Three situations have been considered: the long run, no sunk costs and ex ante. In the two first cases, all have been said, but in the third case, what happens ex post? The reasoning could be more complex. When shares have been issued, the agents have computed their investment by selecting the most profitable firm, obliging the firm to adopt profit rate maximization.

However, what happens after the creation of the firm, when K e is now considered as fixed by everybody? Obviously, if K e is fixed, the agent wishes the firm to maximize the profit; both objectives of the firm (maximizing the pure profit, n = Tl a -t e K e and maximizing the profit rate, n = -p -= t e ) 9 become equivalent to the maximization of the accounting profit Tl a .

So, there is no betrayal after the initial issue of shares if the firm maximizes the pure profit instead of the profit rate. Remember that in any case, if the firm's results are disappointing, the agent can sell its shares to another shareholder at a lower price even if in the firm's viewpoint the investment is done irrevocably; if the "game" has only one period, the firm can ignore the agent's wishes, but if it is repeated, the firm must respect its commitment to preserve the trust of shareholders for future issues.

But this is too simple. Ex ante, or even ex post if no sunk costs, as the firm can choose or adapt its production capacity to what is exactly required, the firm is always on its Viner's "long term" cost function (even it is not the long term), because the Viner's "long-term" cost is always lower or equal to the short-term cost for a same level of output. So, it is false to say that all discussions about the opportunity cost of K e -and its associated pure profit -are closed when K e is fixed: the cost function changes from the Viner's "long term" cost function to the short term cost function C£ r . So, the ex ante maximization program, max7i, will determine K(Q*) = K e (Q*) + K d (Q*\ the level of capital as fund that is required, where Q* is the ex ante output that is solution of the program. Then, K(Q*) will be fixed in the ex post short term and appears as a constraint for the ex post short term maximization program:

max nf ( 0 , s.t. Q e [Q min , Q^] (5)
where

TL g T (Q) =R-C g T
is the short term gross profit. Note that alternately, one can introduce into the gross profit, at each period, the losses caused by the potential transfer of the assets corresponding to the sunk costs: Tlg-sK 0 , where K° is the capital as fund formerly installed and s is the cost by unit of capital caused by its transfer.

Remark. A link between the output and K is assumed. The form Q->K, i.e., K(Q), is considered here, but the inverse function K -> Q, i.e., Q(K), could be also, so the implicit form f(K, 0 = 0 could be more satisfactory. Note that K>C g . The hypothesis of variability of capital ex ante is not so new. For example, see what Tirole (1989, pp. 214-217) calls the "Stackelberg-Spence-Dixit model", with an explicit function Tl(K r ), but it is the real capital that is variable here, not the capital as fund. •

The bounds Q^m and

Q m3iX depend on K(Q*), with logically Q m[n < Q* < Q m2iX .
The short term solution, denoted Q**, can be different to the ex ante solution Q* but it is not sure that

Q** > Q* or Q** < Q* at this step. Note that the output Q* is virtual, the output Q** is realized.
Even if the amount of capital is fixed when the firm maximizes the gross profit in the ex post short term, the initial profit rate maximization has determined the scale of production.

To summarize, the scale of production is determined ex ante by "maxrc" for a controlled-by-owner firm and by "max FT if the firm has sleeping owners; ex post "maxllg" determines the following optimal outputs inside the limits imposed by K(Q*), i.e., by the function of capital K(Q) and the ex ante optimal output. As the profit rate maximization theory determines the level of capital and not the real output, it can be said that it constitutes a microeconomic theory of demand of capital as fund.

In the following of this paper, I will show that the behavior of firms is largely modified if firms are assumed to maximize the profit rate instead of the pure profit. The two functions " max 7c"

and "max II" will be compared: profit rate maximization differs of pure profit maximization for its consequences on the output of the firm and on coordination between firms, in perfect and imperfect competition.

III. The economics of the profit-rate-maximizing firm

Two extreme cases will be studies: monopoly obviously but also perfect competition because firms do not interact one with other in this case. The profit-rate-maximizing firm will be always compared to the pure-profit-maximizing firm, with a variable capital as fund, in the long run, ex ante or when no sunk costs. According to the above developments, the first case, "maxn/ATe", corresponds to a firm controlled actively by its owners, the second, "maxIT', that is the more familiar, to a firm with sleeping owners.

A. The reference case, monopoly

First, we shall study a monopoly as a formal general case for the calculation of the firm's equilibrium; other cases, as perfect competition, will be derived from it. As said above, even a unique monopoly must maximize its profit rate as soon as there are many owners (if the owner is alone, both objectives coincide). In this paper, for simplicity, only the case of the monoproduct firm will be studied.

Proposition 1. The controlled-by-owner monopoly produces up to the point where marginal accounting profit is equal to average accounting profit multiplied by the relative elasticity of equity capital; or the firm produces up to the point where the relative elasticity of the accounting profit is equal to the relative elasticity of the equity capital:

Ila = e Ke/Q Tla o enjQ = e Ke /Q (6)
in which eu a /Q is the relative elasticity of the equity capital to the output, eK e /Q is the relative elasticity of the equity capital to the output and JT a = ^ is the average accounting profit.

Proof. It is immediate by deriving the profit rate ft = -p -= t e with respect to Q :

K f
Tl' a = Ua-7T^n / a = e Ke IQ Tl a • (7)

K!

Note. It is equivalent to say that TV 1 = n -«=> n 7 = eKjQ n <=> eu/Q = eKjQ, but the above formulation is adopted because the accounting profit is more familiar to firms.

We have also at optimum: U! = nK' e <=>R' = C' + nK! e (8) where C = C g +1a Kd + t e K e is the total cost (but remember that C g is formally similar to a long term function of cost because the capital is variable). The left member of this expression means that the pure profit of an extra unit of output is equal to the return of an extra unit of equity capital at the rate n (that is the value taken by the profit rate at the optimum). The right member of this expression also means: the firm equalizes its marginal receipt to its marginal cost augmented with the return of an extra unit of equity capital at the rate n. Whereas the optimum of the sleeping-owner monopoly (pure profit maximizing) is given by n ; = 0 <=> R* = C: the firm equalizes its marginal receipt with its marginal cost, what is very familiar. This correctly corresponds to ckjq = 0 when applied on the above results.

Remark. A particular case of profit rate monopoly is interesting for its simplicity: the fixed coefficient of equity capital: the controlled-by-owner monopoly, with a fixed coefficient of capital K e = kQ with k>0, produces up to the point where marginal accounting profit is equal to average accounting profit (or to the point where average accounting profit is at a maximum), H a = Yl a , because in this case eKjQ = 1. So, with a fixed coefficient of equity capital k, a controlled-by-owner monopoly produces and invests so that the pure profit contributed by an extra unit is greater than or at least equal to the average pure profit yielded by other units. Instead of comparing the pure profit brought by the last unit of output, the 72 TH , to zero, the firm compares the pure profit brought by the last unit of output to the average pure profit brought by former units, n -1, n -2,... . Note that this optimum is similar to the optimum of a monopoly that maximizes the average pure profit. Note that this is a very special case, far from an ordinary case of coefficient capital, because only the equity capital is assumed to be linearly linked to the output, not the debts or the whole capital (or the real capital). More realistically, it could be assumed that the whole capital is linearly linked to the output, 0] kQ: the results obtained with a fixed coefficient of equity capital do not remain valid. •

K(Q) = kQ, what implies K e (Q) = [1 -a(0)] K{Q) = [1 -a(
We may compare the optimum of profit rate for a controlled-by-owner monopoly with the classical microeconomic optimum (of pure profit, for a sleeping-owner monopoly). Denoting Q n as the classical pure profit equilibrium and Q n as the rate of profit equilibrium, Q n is before Q n as shown by the figure 2, for simplicity with a fixed coefficient of equity capital: Theorem 1. Consider the more probable case K' e (Q n ) > 0 , in which Q n is the optimal quantity that maximizes the rate of profit 7C. A controlled-by-owner monopoly (profit rate maximization) has a lower optimal output level than a sleeping-owner monopoly (pure profit maximization) when the financial profit rate is higher than t e at the optimum of profit rate:

n(Q n ) > 0 <=> Kj(Q n ) >t e =* (max ri) occurs before (max U)
If Kj(Q % ) < te, then (max n) occurs after (max FQ: it is less probable because in this case the firm leaves the sector. All results are reversed if K f e (Q*) < 0.

Proof. The first order condition of optimality to maximize the pure profit n is II 7 (0 = 0. The first order condition of optimality to maximize the profit rate n is (Q). Write IT' as a function of f K :

n 7 (0 -n(Q) K' e (Q) = 0. Consider the curve f n (Q) = n'(Q)-n(Q) K' e
n 7 (0= f*(0 + rc(0 K' e (Q); assume that n 7 is decreasing. Denote Qn as the point of maximization of pure profit, such as n 7 (<g n ) = 0 and denote Q n as the solution of the maximization of the profit rate, such that

f K (Q K ) = 0. 1) Suppose that K' e (Q K ) > 0 and n(Q n ) > 0. Then, Ii\Q n ) = n(Q K ) K' e (Q K ) > 0 = f n (Q K ). Near
Q n the curve f^ is under curve n 7 . Thus, Qn must be greater than Q n . At the limit, if IT 7 is vertical near Qn , n 7 will cut the x-axis at Q K and then Qn-Qn-Remark that it is sufficient that K' e (Q n ) > 0. See figure 3. 2) If n(Q n ) < 0 and K f e {Q^) > 0, the results are reverted and near to Q n , curve f* is over curve n 7 . Thus, Qn must be smaller than Q n .• Following this theorem, the monopolistic firm produces less, so at a higher price, when it maximizes the profit rate. Thus the monopoly is harsher with profit rate maximization. So monopoly is reinforced: the "max 7i" firm has a more monopolistic behavior than the "max FT firm. This favors the "max rc" theory.

It would be necessary to study second order conditions. However, the mathematical aspects (convexity, etc.) are similar in pure profit and in profit rate maximization, so this step will be left aside. Now, we will prove that coordination among firms could fail when firms are controlled by owners (i.e., profit rate maximization) instead of having sleeping owners (i.e., pure profit maximization). Some cases of coordination failures have been studied in the literature [START_REF] Heller | Coordination Failure Under Complete Markets with Applications to Effective Demand[END_REF] [START_REF] Bagwell | Advertising and Coordination[END_REF].

B. Perfect competition

Consider a single firm in a market in perfect competition.

Proposition 2. In perfect competition the optimum of a controlled-by -owner firm is given by: Proof. We apply above results, with a price independent of Q, i.e., p = p.

For sleeping-owners firms, CKJQ = 0 and p-C.

As seen above, the output of controlled-by-owner firm is smaller than for sleeping-owner firms. So, there could be more firms in the first case (if the global supply does not change from one objective to the second); thus interaction among firms could be lower and competition greater (even if the concept of competition becomes particular).

Proposition 3. In perfect competition, it is sufficient that there is a fixed coefficient of equity capital (or, more generally, CKJQ = 1), to do that price plays no role in the equilibrium of the controlled-by-owner firm. See figure 5.

Proof. We have: eK e /o = 1. Thus C' = C at optimum: the firm produces up to the point where the average cost C is minimum. •

The following conclusions are deduced from proposition 3 for controlled-by-owner firms with coefficient of equity capital.

1) The optimum does not depend on price; thus, there is not coordination in the market. Each firm is alone, doing nothing with the price. Price is no more a signal. We need no more additional theories to explain that the output is independent to the price (like cost plus pricing, U-curves of cost, etc.). As a corollary, the output level does not vary if price varies (as long as ( 9)

the profit remains positive: when it becomes negative, the firm leaves the sector as usual).

Consequently, output level does not matter with taxation or other manipulations of prices.

2) The individual supply curve is vertical, even in perfect competition. The global supply curve is completely inelastic and the market is in global equilibrium, following a classical pattern, as shown in figure 4. This proves that a global equilibrium is compatible with a lack of coordination by price, unlike good sense tends to believe. It must be noted that, unlike in the classical pure profit maximization (where price is determined by the intercept of the global curve of demand with the global curve of supply), it is not the global equilibrium that determines individual equilibrium of the firm, but it is the individual equilibrium that determines global equilibrium by adding up: price is only a consequence of the global equilibrium, without an impact on individual level (except for the amount of firm's earnings but not for its optimal output).

Moreover, if the demand curve is also completely inelastic, there may be no equilibrium; if it is only very inelastic, the price may be very variable to slipping of demand (without any effect on the optimal firm's output, obviously).

Remarks.

1) As said before, in the ex post short run, profit rate maximization or pure profit TI ST = Ilg T -tKe both identical to gross profit maximization (see figure 5):

UST and n ST = -so maxn 5r <^max7C 57, <=>maxn| r , where n| r = i?-C| r , so at the

K optimum, R / = C s g T classically.
Here the ex post short term equilibrium is higher the ex ante profit rate equilibrium but it is lower than the ex ante pure profit equilibrium:

Q K < Q** < Q u .
However, remember that the main decision is taken ex ante: how much capital to invest. 2) Only in the case of capital coefficient, the optimum coincides with the classical point of finn's long-term equilibrium which would be situated at the minimum of C. Even if the number of firms remains fixed ex ante, before all irreversible decisions or ex post but with no sunk costs, the number of firms is variable in the long-run and p decreases as new firms enter the sector; the movement stops at p LT which is the entry point in the sector, corresponding to the minimum of C. Note that both objective functions are exactly equivalent concerning the condition of entry into the sector, that is when deciding whether the firm should or should not produce: I1>0 andrc>0<=>K/>t e (this second formulation with 7t/ seems perhaps more n -t K natural and intuitive). Also, note that as IT = Tl g -tK and n = --, the entry condition (1 -oc) K in the sector is not dependent of a. •

C. Optimal input combination and efficiency in perfect competition

Now, let's see what happens at the side of inputs. Consider n factors, used in quantity q i at a fixed price p\ The function of production is Q(q l 9 ...,q n ).

The costs of production are n C g (q\...,q n ) = p q = q\ Remind that the capital K(q\...,q") is not considered as a

/=i

factor, even if the cost of the capital (interest of debts and opportunity cost of equity) is a part of the total cost (at the interest rate i).

Classically, for a sleeping-owner firm, a "max FI" firm, each factor is paid at its net marginal productivity p Q u -tdK u d -t e K} l e : ^^O^p^^pQ^tdK^teKi (10)

K e

Thus the controlled-by-owner firm pays its factors at their marginal productivity in value, minus the marginal amount of equity capital required by this factor valorized at the rate 7C.

Proposition 4. If 7i > 0 <=> 7C/> t e (normal situation), then the remuneration of each factor is lower for a controlled-by-owner firm than for a sleeping-owner firm: each factor is no longer paid at its net marginal productivity. Therefore, less of each factor is used when the firm is controlled by owners (and

Q n < Q n ).
Proof. We provide a graphical proof with figure 6, denoting q\ as the quantity of factor / used when "max 7c" and qh as the quantity of factor / used when "max fl". •

where Q u = ^ stands for the marginal productivity of /, K lf d = stands for the marginal amount of debts required by i, K}' e = -1 stands for the marginal amount of equity capital dq l n required by i and Tl = p Q -^ p l q l -t d K d -t e K e .

1=1

Thus the sleeping-owner firm pays its factors at their gross marginal productivity in value, minus the marginal amount of capital required by this factor valorized at the rates t<i and t e . If n n Q and K are homogenous functions of degree 1, so

Q u = Q 9 ILq 1 K ll d =Kd and /=i /=i
X q l Ke = K e , the net product Q-td -t e is exhausted:

iq'P 1 ^q'IP&'-tdK^-te&A^pQ-tdKd-teKe (11) Z=l 7=1 V J
In the other hand, for a controlled-by-owner firm, a "max 7t" firm, each factor is paid at its

modified net marginal productivity p Q l/ -n K u e \ |^ = 0 <=> % =PQ if -t d K? d -t € K*-NK» =p{i -7CK» (12)
a?

PQ-ILP 1 tf-tdKd-teKe

with 71 = - Corollary. In the case where the functions Q and K are homogenous functions with degree 1, if 71 > 0, the net modified product Q = Q-^Kd--^K e --=Ke is completely exhausted:

TQ'p 1 ^^(pQ^tdK^teK^KK^^pQ-tdKd-teKe-nKe (13) 1=1 1=1 V J
The criterion of efficiency is changed for a controlled-by-owner firm: an efficient system is no more a system where "marginal cost equals marginal revenue", it becomes a system where "the average profit is maximum" (or more generally "marginal profit equals average profit multiplied by elasticity of capital"). These developments prove that it is possible to build a general equilibrium, beyond Borisov's work (1993). In all sectors of economic life, the new criterion of efficiency leads to a limited service but also fewer expenditures.

D. Duopoly

Let's take a look at the case of duopoly (competition in quantities) as the most simple form of oligopoly; the result will be generalized to oligopoly later. The demand for the sector with two firms 1 and 2 is D(Q X +Q 2 ). The inverse demand p = D~l (Q l +Q 2 ) is assumed to be decreasing. The total revenues of each firm are R'(Q l , Q 2 ) = Q l D~l (Q l +Q 2 ). The production costs of each firm are Q(£?'). The total capital (equity plus debt) of each firm is

K l d (Q) + K e (Q). The pure profits of each firm are Yl\Q\Q 2 ) = R\Q\Q 2 )
a g {Q l ) -t d K^Q 1 ) -t e Kf 9 {Qf). The profit rates are:

n\Q\Q 2 ) = Ul{QX ' Q2) (14) Bn'YO 1 Q 2 )
For the classical maximization of pure profit, we solve the equations f = 0. We obtain the following system:

AN'(G',G 2 ) dn 2 (Q\Q 2 )

dQ 2 =o = o <=> < dD ~Xf Q ? Ql) Q 2 +D-\Q X + Q 2 ) -C 2 \Q 2 ) -/, ^(2 2 ) -^(2 2 ) = 0 (16)
Generally, except special cases, the system has two reaction functions: the solution is found by the intersect of two curves (see the example later for a very simple linear demand function).

These reaction functions allow coordination between firms, by way of conjectural variations denoted v= ^~Qi' Th ere 310 ^e questions of uniqueness [START_REF] Gaudet | Uniqueness of Cournot Equilibrium: New Results From Old Methods[END_REF], existence or convergence of equilibrium [START_REF] Novshek | On the Existence of Cournot Equilibrium[END_REF], and discontinuity in reaction functions involving a non-existent equilibrium (Roberts and Sonnenchein, 1977). We do not enter the debate about the interest of reaction functions and that they are essentially concerned with competition in price (with an auctioneer). However, it must be noticed that profit rate maximization is compatible obviously with a model including capacity constraints (as seen above, it is possible to maximize profit rate with bounds on the capital). Now, assume that each firm maximizes its profit rate. The necessary conditions of optimality are:

= 0 for all i (17)

dQ 1 dn 2 (Q\Q 2 ) dQ 2 Q l n\Q\Q 2 ) -e Ke/Q 2 m Q 2 (18) ^ e K' e /Q' = dKUg) dQ i
It is easy to find counter-examples in which the reaction functions vanish with the profit rate maximization, while they do not vanish with the maximization of pure profit. Reaction functions "vanish" if the first equation excludes Q 2 , and the second equation excludes Q x .

Proposition 5. In duopoly, between controlled-by-owner firms, the reaction functions vanish if the inverse function of demand is linear,

D~x (Q x , Q 2 ) = d-a Q l -b Q 2 ,
and if the elasticity of equity capital with respect to the output level is equal to 1 for each firm, e K i elQi = 1 for all / (this occurs for a coefficient of capital: K^Q 1 ) = k l Q i for any /).

Note that only the condition on the equity capital is required, not on debts.

Proof. With these conditions, the system of reaction functions becomes:

aQ x +bv x Q x +C x g '(Q x ) + t d K l J(Q x ) + t e K X /(Q X ) = C x JQ x ) + t d K x d (Q x )^t e K x e (Q x ) (19) a v 2 Q 2 + b Q 2 + Cf(Q 2 ) + t d K d (Q 2 ) + t e K?(Q 2 ) = c2(fi 2 )+/^}(0 2 )+/./^(e 2 )
and if there is a true coefficient of capital {K} e {Q l ) = k l Q i for any /):

a Q x + b v 1 Q x + Ci\Q l ) + t d K l J(Q x ) = C\ (Q x ) + t d K l d (Q x ) av 2 Q 2 + bQ 2 + C 2 '{Q 2 ) + t d K v d (Q 2 ) = C 2 JQ 2 ) + t d K 2 d {Q 2 ) (20)
As each equation is independent to the other, there are no more reaction functions. Thus, Q x does not depend on Q 2 and reciprocally. •

In fact, reaction functions continue to exist: they are vertical and with no interest because each firm does not take into account the other firm to determine its own output (see figure 7). This is an interesting case because all ordinary models of duopoly have a reaction function, even though it is known that duopoly does not always have a stable solution (and even if dynamics may be complicated or chaotic [START_REF]Exotic Phenomena in Games and Duopoly Models[END_REF] [START_REF] Piatecki | Sources du chaos dans les interactions stratégiques[END_REF]) or even this solution can be non-unique or cannot exist. Here are some counter-examples of non-existence when maximizing profit rate. Note that for a pure-profit-maximizing duopoly, the reaction functions do not vanish, even if demand is linear with a coefficient of equity capital:

-aQ^bv x Q^d-aQ^bQ 2 -C\\Q x )-t d K x ( !{Q^t e k^0 1 -av 2 Q 2 -bQ 2 +d-aQ l -bQ 2 -C 2 \Q 2 )-t d K 2 J(Q 2 )-t e k 2 =0 J 1 ;
When reaction functions are canceled, that is when the solution is not found by the intersect of two functions, each firm acts independently of the other to find the optimal solution (even it has an opinion about the other firm). In a dynamic perspective, each firm never observes the other firm and there is no iterative adjustment ("immediate" solution). Finally, there is no coordination among firms and no game. We can say that duopoly is degenerated. Nash equilibrium exists anyway but it is special. As the slope of reaction functions is vertical it does not matter to know if firms are strategic substitutes or strategic complements to the mind of [START_REF] Bulow | Multimarket Oligopoly: Strategic Substitutes and Complements[END_REF].

The monopoly can be mixed, that is it can a duopoly where each firm may have a different objective, depending on it is controlled or not by owners. In itself, the idea of a mixed duopoly or oligopoly, with one profit-maximizing firm and one revenue-maximzing firm for example, is not new (De Fraja and Delbono, 1990), (Barros, 1994 and[START_REF] Mas-Colell A | Microeconomic Theory[END_REF]. Barros insists on contracts and incentives on managers. Other types of mixed duopoly can be considered. For example, following Baumol one a firm may want to maximize its revenue or want to equilibrate its budget when the other is profit rate-rate maximizing, or want to maximize the consumer surplus. For example, one firm will be controlled by owners (profit-rate-maximizing, assume that it is firm 1) and the second will have sleeping owners (pure-profit-maximizing, assume that it is firm 2):

dD-\Q l +Q 2 ) a E 2
Following theorem 1, the controlled-by-owner firm 1 has a lower output than if is a sleeping-owner firm (the left side of the first equation is decreasing by respect to Q x , the left side of the first equation is decreasing by respect to Q 2 , the right side of the first equation is positive when the right side of the second equation is zero). So, if firm 1 is controlled-by-owner instead of sleeping-owner when firm 2 remains sleeping-owner in any case, firm 1 has a lower output and firm 2 has a higher output (see figure 8).

Figure 8. Reaction functions of a mixed duopoly

If the demand is linear and e K i elQi = 1, one have:

a Q x + b v l Q l + C l g '(Q l ) + t d K l d \Q l ) + t e K l /(Q l ) = Cl(Q x ) + t d k^Q x ) + t e kl(Q x ) aQ l +(av 2 + 2b)Q 2 -d+ C 2 '(Q 2 ) + t d K 2 J(Q 2 ) + t e K 2/ (Q 2 ) = 0 (23)
and the reaction function of firm 1 is vanishing (i.e., it is vertical) but the reaction function of firm 2 remains. It is true even if both firms have a coefficient of equity capital:

aQ l +bv x Q x +Cl'(Q l ) + t d K l J(Q x ) = C\{Q x ) + t d K x d {Q x ) aQ l +(av 2 +2b)Q 2 -d+ Cf(Q 2 ) + t d K 2 J(Q 2 ) + t e k 2 =0
(24)

One question remains. Does collusion is possible with profit rate maximization? If yes, a certain form of coordination remains possible. Classically, the collusion equilibrium is found by the maximization of the sum of the pure profit functions: max 11(0 1 +Q 2 ), where, Firm 1 will try to maximize 71(0* + Q 2 ) when firm 2 will try to maximize H{Q X + Q 2 )

IV. Conclusion

In this paper, the usual theory of the firm is reexamined by considering the capital as variable, either in the long run, or ex ante, or even ex post if no sunk costs. After reminding in introduction that pure profit maximization is unrealistic because it is very difficult to evaluate the exact opportunity cost of equities, in the first part of this paper it is shown that the owner wishes the firms to maximize their profit rate instead of their pure profit. So, in these With the profit rate maximization, one calculate max n(Q l + Q 2 ) with,

D-l (Q l +Q 2 ) (Q l +Q 2 )-Cl(Q l )-Cj(Q 2 )-t d [K№) + K 2 (Q 2 )]^ (26)
and one solve +Q ) _ Q ^or a y ^ Mixed collusion, i.e., collusion between a controlled-by-owner firm and a sleeping-owner firm is not possible. Assume that firm 1 is controlled-by-owner and firm 2 is sleeping-owner:

n(Q l +Q 2 ) = ^-1 (0 l +fi 2 )(fi , +0 2 )-Ci(E 1 )-C}(fi 2 )-frf[A:l(fi I ) + I:i(0 2 )]-/. [Kl(Q l ) + f^{Q 2 N^1 + 2 2 ) = n 1 (0 1 )+n 2 (£ 2 ) = situations, the firm's output, and above all the scale of production via the firm's demand of capital, depends on a profit rate maximization program, and not on a pure profit maximization program, when the short run with a fixed capital remains as usual, gross profit maximizing.

In a second part, the two objective functions, profit rate maximization for a firm controlled by its owners and pure profit maximization for a firm with sleeping owners, were compared for their consequences on the main models of industrial organization theory: monopoly, perfect competition, classical or mixed duopoly. Even if the long term conditions of entry into the sector are similar for both objectives, the optimal output level is always lower for controlled-by-owner firms than comparable for sleeping-owner firms. This has obvious consequences on the results of the models of industrial organization from the moment that the hypothesis of mobility of capital is adopted.

If a modified definition of marginal productivity is adopted by subtracting the marginal amount of equity capital required by the factor valorized at the equilibrium profit rate, then factors remain paid at their modified marginal productivity. Even if fewer factors are used, the net modified product remains exhausted for homogenous functions with degree 1. The duopoly can be mixed, with one firm controlled by owners and another firm with sleeping owners.

However collusion remains always possible.

We have proved that coordination among controlled-by-owner firms may fail by contrast to sleeping-owner firms. It only takes a fixed coefficient of equity capital to do that price plays no role for controlled-by-owner firms in perfect competition: in this case, the price signal plays no role and firms are not coordinated even if they are smaller than in the classical model; there remains a global equilibrium but with an inelastic supply curve. In imperfect competition, for a duopoly in quantities between controlled-by-owner firms, it only takes a fixed coefficient of equity capital and a linear demand function to do that reaction functions vanish: in this case, controlled-by-owner firms do not consider the other firms to determine their optimal output.

Figure 1 .

 1 Figure 1. Financial structure: ratio of debts by total capital

Figure 2 .

 2 Figure 2. Monopoly (here, with fixed coefficient of equity capital)

Figure 3 .

 3 Figure 3. Mathematical comparison of optima

Figure 4 .

 4 Figure 4. Global equilibrium in short-run perfect competition

Figure 5 .

 5 Figure 5. Equilibrium of the controlled-by-owner firm in perfect competition with

Figure 6 .

 6 Figure 6. Optimal quantities of factors

Figure 7 .

 7 Figure 7. Reaction functions of a profit-rate-maximizing duopoly with linear demand

Figure 1 .FigureFigure 7 .

 17 Figure 1. Financial structure: ratio of debts by total capital

71 = -]-= ^-^ -%f-t e . K/= stands for the K e K e Ke financial

  profit rate which consists into comparing the accounting profit Yl a = Tl g -td Kd to the equity capital: it is also referred as return on equity, what is familiar to shareholders who use it to measure profitability. Even if the expression of the profit rate incorporates t e , the opportunity cost of the equity capital plays no role in the maximization of it as long as t e is fixed: knowing the difficulty to determine t e , this is a great advantage on the pure profit maximization.


[START_REF] Stone R | A long-term growth model for the British economy[END_REF], followed by others, as [START_REF] Paelinck | Etude empirique sur l'évolution de coefficients «input-output». Essai d'application de la procédure «RAS» de Cambridge au tableau interindustriel belge[END_REF], give an interpretation of the left and right multipliers, that are found when a RAS method is used to update a matrix, in terms of absorption effect and fabrication effect I begin by a presentation in conformity with Input-Output analysis but an extension apart of this field (transportation flows, demographic flows, etc.) will be presented.

Consider two matrices of technical coefficients, for two different dates t = 0 and t= 1, A evaluated at the prices of t = 1: A0 and A 1 .Then the projection A of A0 on the year t = 1 is the transformation of A0 such as this matrix gets the same margins than A 1; this is given by: A = RAS(A°, A 1) = R A0 S, where RAS( ) is the operator well known as the "RAS method".

In this case the diagonal matrix R is interpreted by the above authors as an absorption (or also substitution) effect: factors r, affect a row of A0 and reflect the modification of the outlet of a commodity. The diagonal matrix S as a fabrication (or also transformation) effect: factors Sj affect a column of A0 and reflect the modification in the so-called "degree of fabrication". A similar interpretation can be done for other field than Input-Output analysis: when you consider transportation or demographic flows, for example, in a word any matrix of exchange, a matrix Z is computed as Z = R Z° S such as Z has the same margins than another matrix Z 1 and you have also a row effect for the left diagonal matrix R and a column effect for the right diagonal matrix S 2.

I. Introduction 1

This is exposed also by [START_REF] Snower | New methods of updating input-output matrices[END_REF].

2

Note that, if Z° is the flow matrix that corresponds to A0 and A1 to Z1, then it is demonstrated [START_REF] De | Unicity of Biproportion[END_REF] that RAS(A0, A1) = RAS(Z°, A1), but both are not necessarily However, I will show that technical difficulties prevent them to be relevant of such interpretation.

II. Non identification of terms R and S

Even if this interpretation is commonly accepted, as the terms r, and Sj are not identified 3, they cannot be interpreted for themselves as a absorption effect and a fabrication effect. If all the coefficients r, are multiplied by X then all the coefficients Sj are multiplied by ^ and conversely: multiplying fabrication effects by X will divide absorption effects by A,, and conversely. This removes all signification to R and S in terms of fabrication or absorption effects. It is very simple to prove it. By commodity, the demonstration will be based on the algorithm presented by Bachem and Korte ( 1979)4:

(1) r, = m a '*-for all i, and Sj = ----for all j X S j a J X rt a°g

j= 1 /= 1
This type of algorithm has an iterative numerical solution (k is an index of iteration), for example:

1 a \ (2) r i ( k + 1) = -------------
for all i, and S j ( k + 1) = --------------for all j X Sj(k) a® -'L r i(k+ 1) aj >i <=i equal to RAS(Z°, Z1).

3

I use this term by reference with econometrics, even if nothing is stochastic here.

4

Bachem and Korte (1979) have proved that their algorithm is equivalent to the RAS algorithm.

1

(3) r) = ----for all i, and sj = ----for all j E * 0

V * o Sj a v /=1 i=l
Now, assume that all sj(0) are replaced by Sj(0) = Xsj(0). Then, r,(l) will be replaced by r,(D:

(4) r ,(1) = ! ------= *--------= T r/(l), for all i X sj{0) 4 Z ls j( 0 ) 4 j= i p l and ( 5) Remark. Note that the products r, Sj, for all (/, j), are identified, i.e., r* s* = r* s* for all (i, j ) , so it remains allowed to conduct a decomposition of change over time (Mesnard, 1990[START_REF]A biproportional filter to compare technical and allocation coefficient variations[END_REF][START_REF] Van Der | The nature of changes in the EU cost structure of production 1965-85: an RAS approach[END_REF] because this one is based on the 5 It is not the aim of this paper to discuss about the properties of this equilibrium, and more generally, of biproportional algorithms; for further information, see (Bacharach, 1970), (Bachem and Korte, 1979), (Balinski and Demange, 1989), [START_REF] De | Unicity of Biproportion[END_REF]. Remember that it is demonstrated that all algorithms used to compute a biproportion, RAS or another, are equivalent and give the same result [START_REF] De | Unicity of Biproportion[END_REF].

Sj( 1) = -j-^------= A , 5y (

6

A similar property can be found with other methods as soon as they are based on an iterative algorithm; for example, the bicausative method (Mesnard, 2000).

computation of the product R A0 S to be compared to A 1, or R Z° S to be compared to Z 1, and not on the particular interpretation of R or S. ■

III. About a tentative of correction

Van der Linden and Dietzenbacher have proposed to normalize the substitution effect vector because the substitution effect is equal to zero for the whole economy (van der Linden and Dietzenbacher, 1995, p. 129, formula 13):

(6) x ! x) i l Z iXj i ,
where xj is the output of sector j at year 1; that is:

(7) X r* x 4 sj x} = X X 4 s j X1 or X X ay xj = X X a\ sj x) ' j i j i j i j
The idea is interesting and clever because it seems to remove the unique level of underidentification, but some arguments can be introduce against it. First, this normalization is based on a certain interpretation of r terms: this interpretation is economic, not mathematical.

If you choose another economic interpretation of the r terms, you have another formula and you cannot justify the normalization as it is.

Second, the normalization does not remove all degrees of freedom that cause non-identification. To understand it, you have to see that r and s terms are not independent, but linked by formula (1). For example, in formula (6) the terms s* have to be replaced by their expression in formula ( 1), what gives a relation between the r* only, but not linear: the terms r* completely disappear from the right member of formula ( 6) what becomes:

The left member of this expression simplifies as a constant, when the right member is hyperbolic (see the simple example in Annex 1). As it includes now the definition of the r terms, normally this expression would define exactly these terms (i.e., r* = est, for all i), but it is not the case (i.e., r* =f (rj, ..., r*_,) for example): if the r terms are « in a «-dimension space, it leaves n-1 degrees of freedom. Normalization fails to fix completely the r terms and non-identification stays.

Remark. This is caused by the fact that there are more than one degree of non identification. When non-identification was exposed above, all s terms were assumed to be multiplied by the same constant X, what implied that all the r terms were divided by the same constant X. However, it is not the general case. In the general case, each Sj term could be multiplied by its own value of Xj so there are m degrees of non-identification (or n degrees, if the initialization begins by the r,), what increases the difficulty of the problem. ■

Third, even if it is admitted that the normalization of substitution effects leads to a solution and allows to remove the non-identification, it could bring some additional difficulties because if the global substitution effect is zero, the formula shows that some terms could be positive and some negative! When r or s terms are not positive, the interpretation of fabrication effects becomes difficult and you could obtain negative terms in the product matrix R A S when you accept negative terms inside matrices R or S, what cannot be correctly interpreted 1. So, 7

Remember that the properties of convergence of the RAS method (Bacharach, 1970) (Mesnard 1994) are granted only if all terms of the A matrix are positive. If all s terms are positive, all r terms are positive, so if you choose all fabrication effects to be positive, you positivity has to be added as two additional constraints: r, > 0 for all i and s* > 0 for all j (or even r* e 9l+ and S j e 9t+).

Four, the normalization condition does not take into account the fact that the equilibrium value of r and s terms must be found only iteratively: the condition has to be set not only for the formula at equilibrium but also for the formula at each step of the iterative computation of r and s terms:

(9) 5 > ,(* + l) X a\ Sj(k) Xj = £ X <4 Sj(k) x) * J i J
C l where S j ( k ) can be replaced by S j ( k ) = ----------, knowing that r and 5 terms are calculated £ n(k) a\

7=1

by the iterative formula (2), under two additional constraints of positivity, r,(£) > 0 for all i and k, and sj{k) > 0 for all j and k:

'L a l r i(k+ l) (10) T a lj xj -n------------= £ a\j xj -------- 1 E «J r,(k) J X aj r,(k) t= 1 i= 1
The left member is no more a constant as in formula ( 8) and the result is not so simple because ri(k+ 1) is not equal to /*/(£). Then, to be rigorous, one has to demonstrate that there is a path -that respect the above constraints at each step k -from the initialization to the equilibrium values of r and ^ terms 8. Even when it is assumed for a particular matrix, A, that the solution exists, it has to be demonstrated that this solution can be reached without passing by some negative values of the terms r,(£) or S j ( k ) , or it is necessary to impose the two additional constraints r^k) e 9?+ and Sj(k) e Knowing an acceptable solution, it could seem have all substitution effects all positive also.

attractive to find a correct initialization of the process that corresponds to this rule. However, it is impossible to do this because the problem is transcendent9.

Remark. In formual (10), if the terms r,(k) are considered by approximation as constant by respect to the terms r t{ k + 1), the above relation becomes linear at each step by respect to ther,(A:+l).

Five, last but not least, the r and s terms are found as Lagrange multipliers of an optimization process as the minimization of the quantity of information (see Annex 2):

A (11) min X X ¿y lo g ^J, ' • J a a under, (12)
X ¿y = a), for all / and X ay = a 1 ,.-for all j.

The Lagrangian is: I = X X ay l o g + X Xt J J au ' '

X ay-a).

. J + X M -y j X ày -a\j maximization gives:

(13) --= 0 « atj = exp-(l + X() a« exp-|xy, for all i j d ay

Combined into the constraints, this gives:

i (14) exp-(l + Xi) = --7-1 -------for all i 2j ay exp-n,

Its

There is a parallel with the computation of a general equilibrium: one has to demonstrate, knowing the initial conditions, that the equilibrium exists and that it can be reached by a feasible path, with no negative outputs.

IV. Conclusion

In the RAS method commonly used in Input-Output analysis, but also in other fields where exchange matrices are used (transportation flows, demographic flows, etc.), factors R and S have no signification at all by themselves because they are not identified. The interesting tentative of correction consisting into a normalization of substitution effects, following the argument that the total substitution effect is zero (van der Linden and Dietzenbacher, 1995), raises some important difficulties that prevent to consider it as acceptable. So, even if the RAS method and biproportional methods can be credited of some technical qualities 10 and even if RAS and biproportion can be seen as a simple generalization of "naive" approaches (as the 10 As the non negativity of the projected matrix, or as the characteristic of the projected matrix to be the closer to the initial matrix in terms of information theory or entropy. comparison of technical coefficient matrices), their economic interpretation remains difficult except if they are used to conduct an analysis of change over time.

V. Annexes

A. Annex 1

1. Computation of the constraints r terms at equilibrium 0.4 0.1 0.2 ,A! = 0.9 _ 0.3 0 1 0.5 0.8

In the normalization expression, -

20 + 0.2 r\ 0.2 r\ + 0.1 r*2 t 20 + 0.2 20 + 0.1 r*2 0.2 r x +U.1 r2 0.2 r x +u.i r2 +0 3 ------^------ ' 0.1 r\ +0.3

Iterative computation of the constraints r terms

The normalization expression is: • Kullback and Liebler minimization of information [START_REF] Liebler | On information and sufficiency[END_REF], [START_REF] Kullback | Information, Theory and Statistics[END_REF], [START_REF] Snickars | A Minimum Information Principle. Theory and Practice[END_REF]: min X X Zy l°g-f • < ■ J zu

• The minimization of interactions of [START_REF] Watanabe | Knowing and guessing. A quantitative study of inference and information[END_REF] and [START_REF] Guiasu | An entropic measure of connection and interdependence between the subsystems of a given large system[END_REF], etc.

Stone's empirical method RAS respects the conditions. Historically, it was developed by Stone but the concept of biproportion was first formalized by Bacharach (1970).

The following algorithm is also correct (Bachem and Korte, 1979):

(20)

r, = 7=1 for all i z.i sj = - X Ui 4
for all j

¿=1

Several algorithms respect the two conditions of a biproportion. And it is demonstrated [START_REF] De | Unicity of Biproportion[END_REF]) that all algorithms that respect the conditions of a biproportion lead necessarily to the same mathematical results 11. Bacharach (1970) gives a demonstration of the uniqueness of the solution of RAS. As a biproportion can be deduced the minimization of an information function,

A min I = X X Zjj log ^ under X zy = z\j and X zy = z\.

z ij i J Z i j i J
and as this function, SR, of the n x m terms zy is continuous, derivable, on its compact interval of variation, the function -I is convex and it has a unique maximum on its interval of variation. Ai't-Sahalia, Balinski and Demange (1988) have also established that the

11

Even if there can be some differences for computation speed and for the effect of successive rounds on the precision of computation, as studied by Bachem and Korte.

matrix that minimizes the information criteria is unique n. In conclusion, the computation of a biproportion is a "safe" operation, even it cannot be computed analytically.

Other methods to compute a biproportion

A Also, there exists other methods to found a matrix Z, the nearest to a matrix Z° that respects the margins of another matrix Z 1 (i.e. under constraints of margins: X zy = X z\ and 7 7

Z z y = Z Zff):

j j
• the minimization of the quadratic deviation (Frobenius norm of the difference matrix):

min > i j V J
• the minimization of the absolute differences: min X X \zy ~z \ \, what is not continuously i y derivable,

• the minimization of the Holder norm at the power p: min X X \zv -4 \ p , knowing that Balinski and Demange (1989) have studied the axioms of biproportion in real numbers and in integers; see also (Ai't-Sahalia, Balinski and Demange, 1988). This is applied to voting problems; see also [START_REF] Michel | Apportionment[END_REF] and Balinski and Gonzalez (1996). See also [START_REF] Toh Mun-Heng | The RAS Approach in Updating Input-Output Matrices: An Instrumental Variable Interpretation and Analysis of Structural Change[END_REF].

\ zÿ ~zij • Neyman's % 2 ' ■ X X -t . -• i j z ÿ
But generally these methods lead to various problems:

• non-linearities or non-differentiabilities in the found system as for Neyman or absolute differences, A

• or negative terms in Z as for the minimization of the Frobenius norm. Negative terms are impossible to explain in an input-output context: if Z° has no negative terms, how to justify in an economic view point, the existence of some negative terms inside the projected matrix Z?

VI. Bibliographical references

  * is replaced by r) = r* and s* by s* = X s*. Similar results are found by reverting the role of the r and s terms. The same result holds for any exchange matrices, Z° and Z 1 instead of A0 and A 1. So, r and s terms are not identified6.

i

  After a changing of variables, r, = exp-(l + X,) for all i and Sj = exp-|iy for ally, this gives:As the r and s terms are a simple transformation of the multipliers, the normalization appears to be a constraint on the multipliers, what is unusual and in contradiction with the concept of Lagrange multiplier (or Kuhn-Tucker multiplier).

  r\ 0.1 s\ 20 + r\ 0.2*2 25 + r*2 0.3 s\ 20 + r*2 0.1 25 = 0.1 s\ 20 + 0.2 si 25 + 0.3 s\ 20 + 0.1 ^ 25 the terms s\ = ____ U.D____ j * _ ____ U.o____ 0.1 r\ +0.3 r*2 2 ~ 0.2 r\ +0.1 r\ -have to be inserted:

[

  O.l ri(*+l) + 0.3r2(*+l)] Ji(*)20 + [0.2n(jfc+l) +0.1 r2{k+ 1)] s2(k) 25 = 0.4 s\(k) 20 + 0.3 j 2

  AÏT-SAHALIA Yacine,Michel L. BALINSKI and Gabrielle DEMANGE, 1988. "Le redressement des tables de contingence: deux nouvelles approches", in Mélanges Economiques, Essais en l'honneur de Edmond Malinvaud, Economica -Editions EHESS. BACHARACH, Michael, 1970. Biproportional Matrices and Input-Output Change. Cambridge University Press, Cambridge. BACHEM, Achim and Bernhard KORTE, 1979. "On the RAS-Algorithm," Computing 23, 189-198.

This was demonstrated(Bacharach, 1970) in the case without on the r or s terms.

Replacing the terms s in the normalization expression gives: i n 0.1 n ( * + l ) + 0.3r2(;fc+l) ^0.

B. Annex 2 1. Remind about the computation of a biproportion

A Consider two non-negative matrices Z ° and Z 1. The result of a biproportion, Z = K(Z°, Z 1), is equal to RZ° S, where R and S are diagonal matrices. A biproportion must respect two conditions:

1) Z must have the same row and column margins than Z 1:

The R and S terms are there to guarantee the respect of this condition.

A 2) Z is the matrix the nearest to Z° following a certain criterion.

This criterion can be:

• The maximization of entropy [START_REF] Wilson | Entropy in Urban and Regional Modeling[END_REF]: max -X X Zjj logZy , under the constraint ' J c = X X ^ Cy where C is the total cost and c,y is a cost, that can be considered as


Keywords: Production function, factors of production, firm, conflicts

Considering sub-factors, like different categories of capital or different categories of work, a production function is proposed: each sub-factor of one factor may be combined with each sub-factor of the other factor. Different types of programs exists at different levels. The conditions to obtain identical solution for these programs are studied. Considering that there is one responsible manager by program, conflicts between managers of each program are possible because these programs give non identical solutions in general. The important fact is that these intern conflicts appear naturally without extra economical considerations.

INTRODUCTION

In the neo-classical production functions, there are some factors, capital, labour, etc. It is the case of the Cobb-Douglas and CES production functions). Both of these functions can be generalised to an undetermined number of factors. However, the factors are globally combined, and are not combined separately, one with another, that is to say by couple.

DEFINITION OF THE PRODUCTION FUNCTION WITH COMBINING SUB-FACTORS

We call this production function the Production Function with Combining Sub-Factors; its name is summarised by the acronym CSF.

The production function

In Remarks.

• The CSF remains neo-classical.

• If J = 1 and / = 1, we have a simple neo-classical production function.

• For the production function of many outputs, or for a multistage process of production, we have many matrices (for two factors).

• The CSF may be applied to macroeconomics. Sub-factors are aggregates, like individual firms, small firms, great firms, banks, etc, or like households, rentals, etc.

Generalisation

It is possible to generalise it to the case of three or more factors, strongly increasing the number of inter conflicts. This is specially suited for macroeconomics.

For three factors, we have:

Q=Z Z Z% = Z Z Z^y*^* >-<>*) =£ £ £ q, -£ cf + £ ^. £ £ £ (," +yiJk p> +P i) /=1 y=lX:=l /=1
j=\ i=\ j=\k=\

MAXIMISATION OF PRODUCTION UNDER A CONSTRAINT OF COST

The isoquants

We have three great types of isoquants:

• An elementary isoquant is the set of points obtained when xy and yy vary, Qy remaining fixed.

• A partial isoquant is the set of points obtained when xy and yy vary, Q* or Q y . remaining fixed. There are two sets of partial isoquants.

• A global isoquant is the set of points obtained when xy and yy vary, Q remaining fixed. Note that, these three categories of isoquants are confused in a simple production function with .7 = 1 and / = 1.

The costs

We have three great types of costs:

• The elementary costs, that is to say the cost attached to the contribution of the couple of sub-factors is:

where p* is the price of the sub-factor / of type x and p y is the price of the sub-factor j of type>>.

• The total cost attached to the combination of / with ally i.e. the cost attached to the total contribution of the sub-factor / (/ belonging to the factor x) is:

and the total cost attached to the combination of j with all / i.e. the cost attached to the total contribution of the sub-factor/ (y belonging to the factory) is:

/ = 1
We call these the partial costs.

• The total cost of production is:

/=ly=l /=1 y = 1 /=ly=l
Remark. The cost in each sub-factor may be defined: for example, the cost in sub-factor /, that / J is to say X x ij P* or X y*j P y ^^us some ot her type of programs may be defined.

#=i 7 y=i J

Maximisation

The different types of costs and the different type of isoquants may be combined to calculate an optimum, giving sixteen great types of possible substitutions. The first two programs provide identical solutions when However, the production is not a simple generalisation of the ordinary Cobb-Douglas function.

Costs

7=1

and if Pjj + yy -1 for every / and j (first degree homogeneity), when

O C _ _ _ C _ C x ±y-= -2-e> C u = C x denoting C" = and C, x = . Of C x ,J ' " Q,j ' Q x
The two last program provide identical solutions when

7=1 = S_ /=17=1
and if Py+ yy = 1 for every / and j, when

O x C x _ _ _ r = -*-<=> C* = C denoting C = -.
The first and the last program provide identical solutions when

(fiij + rij)Qij _Cjj 1 J i \ C /=17=1
and if Py + Yij = 1 f°r every / and j, when

The managers

We may suppose that there is a manager by program inside the firm, responsible for the correct maximisation of the production under the constraint of costs. Each program looks like a "center of cost". In the organisational practice, often people consider "centers of profit".

There is a slight difference: a center of profit has under its control its receipts and its costs (even if its receipts are theoretical when the center of profit sells to another service into the firm).

There

Possibility of conflicts

We say that there is a conflict when the program of one manager is not compatible with the program of one another manager.

The conditions to have no hierarchical conflicts are clear (with Cobb-Douglas form and first degree homogeneity): all mean cost (elementary, partial and total) must be equal. These conditions seem to be very difficult to reach in practical situations. We summarise it into the following table (with Cobb-Douglas form and first degree homogeneity): appear inside it. This is very important, the combination of inputs is no more a simple problem, but a complex problem, into which there are possibilities of contradictory choices.

Remarks.

1. If we think in terms of duality, that is to say if the cost are minimised under a constraint of production (i.e. on an isoquant), we have sixteen types of programs, but programs in a same row are identical. This shows that with CSF, to maximise production under a constraint of cost is not identical to minimise cost under a constraint of production: conflicts will be not the same.

2. If there is no manager for a program, there is no possibility of conflict with this program.

It is not a difficulty. For example, we may think that there is no manager for the labor subfactors.

3. In macroeconomics, conflicts between sub-aggregates may be seen as conflicts between social classes (middle class against workers, rentals against active capitalists, great firms against small firms, etc.).

CONCLUSION

The The results may be enlarged to another production function like CES and all this may be extended to macroeconomics, with the possibility of a conflicts between aggregates.

ANNEXE

Maximisation of production under elementary cost

It is obviously the simple case of a simple production function.

Max Qy = ay x&J y Yij

Maximisation of production under partial cost

For the partial isoquants Q x , the problem is:

  For example, in the production function of Uzawa [ UZAWA 1962 ], there are groups of factors. Into the groups, the function is CES (there is a different function by group), between the groups, the function is Cobb-Douglas. Into a group, the combination of factors remains global, between the groups, the combination of groups remains global. It is the same thing for the production function of Sato [ SATO 1967 ]. We have groups of factors; into each groups, the factors are globally combined by a CES function (there is a different function by group), and between the groups, the groups are combined by a CES function. It is interesting to point out that these two production functions were created to introduce an elasticity of substitution variable from one couple of factors to one another couple of factors, in the CES (were the elasticity of substitution is normally constant), and not to introduce couple combination of factors. In these production functions, factors remain combined globally into one group. Factors are not combined two by two: if one factor is present into one group, it is not present into any other group. Moreover, the logic of division of factors into groups is not clear. Are they subprocesses of production? For the following, we will think in terms of factors and sub-factors. Factors are classical: labour, capital, and so on. Sub-factors of a factors are some type of this factor: non-qualified labour and qualified labour are sub-factors of the factor labour, machines and immaterial capital are sub-factors of the factor capital. In the classical approach, we cannot combine sub-factors of one type with sub-factors of one another type, because every sub-factors are on the same set. For example, if we decompose labour into non-qualified labour and qualified labour, and capital into machines and immaterial capital, we have one single set of factors: { non-qualified labour, qualified labour, machines, immaterial capital }. Each of these sub-factors contributes to the production in the same way: the manner that factors are combined does not play any role. It is the principle of the Leontief approach: for example, we have many sectors of services (to firms, to households, etc.) instead of one sector of services. It would be better to have a production functions allowing to combine sub-factors two by two.For example, we may want to combine non-qualified labour with machines, non-qualified labour with immaterial capital, qualified labour with machines and qualified labour with immaterial capital. That is to say, we may want to form a type of matrix:

  our production function, we consider a function of production of a single output, in a single process of production, with two types of factors, for example, machines and labour. Each machine uses any types of labour, any type of labour is used on any machine. Two tables will indicate how each sub-factor of one factor uses each sub-factor of the other factor. Consider two types of factors. The first type of factor has / sub-factors noted /. The second type of factors has J sub-factors noted j. Denote x^ the quantity of sub-factor /, of the first type, combined with the sub-factor j, of the second type, measured in unit of sub-factor /. Denote yy the quantity of sub-factor j , of the second type, combined with the sub-factor /, of the first type, measured in unit of sub-factor j. • To produce, we combine each sub-factor of one factor with each sub-factor of the other factor, we form every couple of sub-factors. When the sub-factor / and the subfactor j are combined, we put x^ units of / and y t j units of j. These two quantities of factors contribute to the production of the single output for an amount: Qij = f(xij>yij) • We call it the contribution of the couple of sub-factors {/,>} .• We do the hypothesis that the total production is the sum of the partial production, i.that there are many other ways to combine elementary productions.• Denote Q* = ^Qy = Xn^y^/yJ ^ ^e contribution of the sub-factor / Q y -= ^Qy = XK^//»^) as contribution of the sub-factor j (of the second

  are the general manager for the general program, a sub-factor manager for each contribution of sub-factor Q* and Q y . , a manager of each contribution of couple of subfactors Qjj . When indices are corresponding, for example a couple {/,/} and a sub-factor / , managers are in a hierarchy. Naturally, when indices are not corresponding, for example a couple {i 9 j} and a sub-factor /', managers are not hierarchically placed but transversally. sub-factor j of factor 2, the conditions are --= --. pi PJ With this type of production function, it appears the possibility of conflicts between the managers. The firm is no more a black box, a single point: agents and conflicts naturally

  classical microeconomic firm looks punctual: it is a black box, we do not know what is inside it. The classical microeconomic production function put each factor in the same plan: when there are more than two factors, only the global combination of factors are taken into account, and not the combinations between factors taken two by two. Thus, we propose a type of production function, the Production Function with Combining Sub-Factors (CSF), allowing to combine sub-factors two by two. For example, if factors are work and capital, sub-factors are different categories of work and different categories of capital. In itself, it is interesting to know the consequences of the combination of different categories of factors taken two by two. However, the principal message of this paper is not there. Indeed, many type of programs are possible, like maximise production under total cost, maximise production under the cost of the combination of one sub-factor (of one factor) with every sub-factors (of the other factor), maximise production under the cost of the combination of one sub-factor (of one factor) with one sub-factor (of the other factor).Conditions to obtain the same optimum for each program are studied: these conditions are often unattainable (in the simplest cases, the optimal mean cost of production must be equal in each programs).Thus it appear possibilities of conflicts between managers, assuming that there is a manager responsible of each program. Intern conflicts are not artificially introduced, they come naturally without extra-economical or sociological considerations. This is very important because it make a bridge between the theories of intern organisation of the firm (Chandler, etc.) and the microeconomic theory.

/

  Q? = £ a 0 x PiJ y Yii , or Max (? = X Z a u x ^ ) j=\ i=ly=l under Cy = xy p* + y i} p y j Cy -Xjj p] + y y p y = 0 => À = ( py, + y y )

  Qy = ay x Pij y Yij , or Max (? = X Z a U x V** )

Table 1 . The combinations Note that couple combination of inputs may exists in the literature. For example, in the function of Gutenberg [ KRELLE 1969 and 1970], a production function with complementary factors which is a generalisation of the Leontief production function, there are aggregate corresponding to stock factors. Each aggregate, like welding, provide services, like a certain number of welds. As each aggregate needs some consumer inputs, like work, raw materials, etc., we have a matrix aggregates / consumer inputs. Even if each consumer input provide services to each agregate, there is no symmetry of treatment between aggregates and consumer inputs. The aim of Gutenberg is to introduce stock mean of production: its production function is suitable to take into account only these two special categories of factors, and not to take into account every category of factors.

 1

Table 2 . Typology of programs In fact, programs in a same column are identical: there are four different types of program only, that is to say

 2

I J + I + J + \ programs:

if we have only two sub-factors by factors, we have the amount of nine programs to solve! 4. MANAGERS AND CONFLICTS 4.1. Comparison of programs What are the conditions to obtain identical solutions for the different programs? For example, let us study three programs, an elementary program (the first in the table 2), a partial program (a program in the middle of the table 2), the global program (the last in the table 2). If elementary contribution to production are in a Cobb-Douglas form the annexe shows that the solutions are:

	. Max Qjj = a 0 x^ y r>J under Cy = xy pf + y y p y :
			, for every / and j
	Max Q 7=1		7=1
	'J	J		X
	iL^ij + r^Qij	P '
	J			, for every / and j
		Yij Qij	cf
	»J= J \-J
	7 = 1
	Max Q = Z 2>7 x*' /* under C = £ Z ( 7/ = ZZU	P* + y>j Pj) :
		1=1>=i		1=17=1	/=17=1
		A/ (?//	c:
	i/ ~ I J		x
	Y,Z(n,	+ r 0 )Qy	p '
				, for every / and /
		Yij Qij	c
	y 'j~	[ J		y
	/=17=1

x = Yj a ij x 0 y YiJ > for ever y '> under c i = ZU7 Pi + y>j P]) '•

=

-X p y = 0 => y.. = ---

, for every / and / p i

7=1

Obviously, the problem is symmetrical for Q y . and C y .

Maximisation of production under total cost

The problem is:

(or Max Qy = «, y /* , or Max (?* = J a,-,-y r,J )

The Lagrangian is

and at equilibrium: