Joint Production with Restricted Free Disposal.

Transcription

1 1 Joint Production with Restricted Free Disposal. by Christian Lager *, University Graz Abstract A single production system with constant returns can produce any level and composition of demand by appropriate intensities of the cost minimising processes. Hence, in the long run, products can never be in excess supply and there exists a system of prices of production, which is semipositive and independent of demand. These (and other) properties do not, in general, carry over to joint production systems where one or several processes produce two or more different products. The proportions in which products emerge will generally be different from those in which they are required for use. The usual approach to that problem is to apply the rule of free goods. This assumption may be applied to goods, which, if they are left where they are and as they are, cause neither costs nor benefits. But it cannot be applied to outworn machines, scrap, wastes or pollutants and is therefore not generally applicable. The present paper aims at finding conditions for the existence of cost-minimising systems for cases where this crucial assumption is either completely absent or is substituted by the assumption of restricted free disposal, i.e. by the assumption that excess production is permitted up to a certain tolerated limit. It will be proved that the conditions of the existence for cost minimising systems with free disposal carry over to systems with restricted free disposal. 1. The problem of joint production: the possibility of excess production. Most textbooks as well as most applied studies on general equilibrium concentrate on single products systems. However, joint production is not only the general case but is, as a matter of fact, very common. The importance of joint production has been illustrated by Steedman (1984) in terms * A first version of this paper was read at the Twelfth International Conference on Input-Output Techniques in May I am grateful to all attendants of that meeting and to two anonymous referees for their comments and remarks. I am also grateful to Christian Gehrke, Heinz D. Kurz and, in particular, to Neri Salvadori for comments, suggestions and for many helpful discussions. The usual disclaimers apply.

2 of several empirical examples and by consideration of some phenomena such as pollution, waste disposal, recycling of aged machines, which do not exist in single products systems. The presence of joint production introduces additional complications. Sraffa (1960, 50) recognised that if two or more different commodities are produced by one process, then the number of operated processes may fall short of the number of commodities produced: There would be more prices to be ascertained than there are processes, and therefore equations, to determine them. In order to preserve his method of solving systems of equations he assumed that in these cases there exist one or several parallel processes which produce these commodities in different proportions and suggests comparing square systems of production where n commodities are produced by n processes and therefore prices (and activity levels) are uniquely determined. But there is the problem of excess production. If commodities are not separately producible then it may happen that the proportions in which they are produced will be different from those proportions in which they are required for use. Sraffa is aware that in this case the existence of methods of producing these commodities in different proportions will be necessary but not sufficient for obtaining the required proportions. Take for example the case of two products jointly produced by each of two different methods. The possibility of varying the extent to which one or the other method is employed ensures a certain range of variation in the proportions in which the two goods may be produced in the aggregate. But this range finds its limit in the proportions in which the two goods are produced respectively by each of the two methods, so that the limits are reached as soon as one or the other method is exclusively employed. Sraffa, 1960, 53. Hence we cannot exclude the case that a system of production, which satisfies requirements for use, will exhibit unwanted quantities of some products. And, if the corresponding system of equations is solved to determine activity levels such that requirements for use are exactly met, i.e. without creating unwanted products, then some negative activity levels may result. This may be illustrated by the following example. Assume that two products are produced by two processes, α and β using labour and products. Normalize inputs and outputs such that one unit of labour is used by each process. Hence, each process can be represented as a vector of net outputs at golden rule growth, i.e. outputs minus capital inputs required for reproduction and for growth at a uniform rate which is equal to the rate of profit. Figure 1 shows that, if the demanded quantities of the

3 3 net product, denoted by C, are outside of the cone of producible combinations then there is excess production of commodity 1 and, if the system is constrained such that no commodity is produced in excess then the intensity of activity α is negative which is, however, nonsense. Commodity 1 Commodity 1 α Cone of feasible combinations of net-outputs α β β Cone of feasible combinations of net-outputs C Commodity C Commodity γ p 1 figure 1 figure Assume now that there exists an additional process, γ, which produces some quantities of product by means of some quantities of product 1 and by using one unit of labour. Note that this process could be viewed as either a productive single product process for the production of commodity or as a costly disposal process which absorbs some quantities of the unwanted product 1 and produces some quantities of product by means of labour. Hence commodity is separately producible and the demanded net output can be produced in the required proportions by a combination of process β and γ at positive activity levels. Figure envisages that in this case the relative price of commodity 1 becomes negative. This price (measured in terms of product ) is equal to the slope of the line drawn normal to the net output frontier of processes β and γ. Note that the system of processes ( βγ, ) is dominated by system (, ) αβ in the sense that the latter can produce larger net outputs by using not more labour. But the dominating system ( αβ, ) does not meet the net product in the required proportions. Hence, system ( βγ, ) must be used, and, the negative price of commodity 1 is the payment for the service of disposal of commodity 1, which is over-produced

4 4 by the dominating system ( αβ, ). Cf. Salvadori and Steedman (1988, p.180). For the concept of dominance see Filippini and Filippini (198). Sraffa was, however, aware of the fact that some prices of production may be negative at some feasible level of the wage rate. But he suggested that such negative prices cannot persist in the long run: This being unacceptable, those among the methods of production that give rise to such results would be discarded to make room for others which in the new situation were consistent with positive prices. Sraffa (1960, 70). Sraffa s suggestion has been interpreted by Salvadori (1985, p. 163) in the sense that the force pushing the economy to positive prices in actual fact is competition, i.e. choice of technique but demonstrated in the same article that, given a set of existing 1 technological alternatives, (a) the unique cost-minimising system may have negative prices, even if another system exists which is consistent with positive prices, and that (b) a cost-minimising system may not exist, even if systems of production happen to exist. The latter is the case if, for example, two systems of production exist and one system is cost minimising at prices of the other system and vice versa. Hence there is always an incentive to switch from one system to the other one and, therefore, a long run position is not feasible and there is no cost minimising system. Salvdori (1985) concluded that fact (a) implies that Sraffa s view, that the choice of technique is sufficient to ensure positive long-run prices cannot be sustained in general and that fact (b) is even more remarkable as it implies that either the long-period theory or Sraffa s formalisation of joint production must be abandoned. 1 One may carry Salvadori s concept of existing technological alternatives a little further and argue that there is a tendency towards finding new methods which minimize excess production. In this sense Sraffa s claim can be traced back to Marx who stressed the capitalist s fanatical insistence on economising the means of production such that by prevention as well as by recycling activities excretions of production are minimised. This argument is used by Schefold (1987, p. 1033) for a rationalisation of square systems. For a comprehensive discussion of Classical and Early Neoclassical Economists on Joint Production see: Kurz (1986).

5 5. A solution to the problem: The rule of free goods Following Sraffa s method, much of the literature on joint production is concerned with square systems, and some justification for Sraffa s square approach has been provided for the special case that (1) consumption patterns do not depend on prices and incomes, that () the economy grows with a rate which is equal to rate of profit and, that (3) the rule of free goods is applied to commodities which are produced in excess (See for instance: Steedman, 1976; Schefold, 1978 or Bidard, 1986). Salvadori (198, 1985) suggested to introduce a new formalisation of the joint production model which comes close to the concept of von Neumann (1937). It consists in determining a costminimising system in such a way that (i) (ii) (iii) (iv) processes are operated such that given requirements for use are satisfied, no process obtains extra profits, processes which incur extra costs will not be operated, and the prices of those commodities which are produced in excess are set equal to zero. Condition (i) rules out excess demand for reproducible goods and is, therefore, a precondition for a long-run position in the classical sense. If conditions (ii) and (iii) hold then prices are determined in such a way that revenues of processes, which are actually operated, cover just costs of production including an ordinary return at a uniform rate, i.e. prices are taken to fulfil the condition of reproduction. Hence there is, in that respect, no disagreement between Sraffa and von Neumann. There are, however, two essential differences: First, in contrast to von Neumann, Sraffa s analysis is based on equalities and the comparison of some square systems of production and, second, and even more essentially, von Neumann adopted assumption (iv), i.e. the rule of free goods, which is never mentioned in Sraffa. The method suggested by Salvadori can be scetched as follows. Assume that the exogenous data, which are (A) the m methods of production available to produce n commodities, defined by the triplet { B, A, l}, where n m B and R + n m A are semipositive matrices of outputs and inputs and R + l R + m is a row vector of labour inputs, (B) the requirements for consumption purposes denoted by column vector f R n, and

6 6 (C) the rate of growth and the rate of profit, denoted by scalars g and r respectively, are given. The problem is to determine the endogenous variables, which are (a) intensities of processes, denoted by column vector x R m, (b) the wage rate, w, (c) prices of products, denoted by row vector p R m such that the conditions (i) to (iv) are met. Expressing prices in terms of labour commanded, i.e. w = 1, these conditions can be formalised as follows: ( ) ( 1 r ) (1.i) ( 1 g ) B A + x > f p B A + < l (1.ii) ( ) (1.iii) ( ) k ( ) k k ( ( )) k: p B A 1+ r e < l x = 0,i.e. p B A 1 + r x= lx, x 0 (1.iv) ( ) ( ) ( ( )) i: ei B A 1+ g x> fi pi = 0,i.e. p B A 1 + g x = pf, p 0 where e i (e k ) are row (column) vectors with elements being zero except the i-th (k-th) element which is equal to one. Note that, if g 0 then vectors f and x cannot be considered as vectors of constant levels but are to be conceived as vectors of constant proportions. It has been proved that if (1) commodities are consumed in constant proportions and if () the rate of proportional growth does not exceed the rate of profit, then a sufficient condition for the existence of a costminimising system (i) to (iv) is: it is feasible to operate some processes in such a way that consumption and investment requirements for proportional growth at a rate which is equal to the rate of profit are satisfied. Furthermore, if under these conditions a cost-minimising system exists then the number of operated processes equals the number of commodities which are not in excess supply. The application of the rule of free goods for products produced in excess, i.e. condition (iv), is crucial and can be viewed in several ways. One may assume free disposal, i.e. to presume the

7 7 existence of a disposal process by means of which a product can be freely destroyed by magic (Kurz, 1986, p. 4). Alternatively, it may be supposed, that for each process producing commodity i jointly with other commodities there exists another process with the same outputs and the same inputs except that commodity i is not produced (Kurz and Salvadori, 1995, p. 8). These alternative interpretations of the rule of free goods can be envisaged by figures 3 and 4 respectively. Commodity 1 Commodity 1 β β C C Commodity Commodity δ figure 3 (Free disposal for commodity 1) β figure 4 (Commodity is separately producible) Figure 3 represents the assumption of free disposal in the sense that there exists a process δ which uses (absorbs) some amounts of commodity 1 and produces nothing. That process uses no labour nor inputs of commodity. Hence it may be represented by the negative branch of the commodity 1 axis. Figure 4 illustrates the assumption that there exists a process ß which uses the same amounts of inputs and produces the same amounts of outputs as process ß with the exception that commodity 1 is not produced. In both cases the price of commodity 1 in terms of commodity is equal to the slope of the commodity axis which is, however, equal to zero. Both interpretations of (iv) are extremely restrictive. The latter requires that all commodities are, in principle, separately producible and consequently presupposes that there exists at least one system of production which is all-productive. Furthermore, the proposition that two processes exist which have the same inputs and the same outputs except that one of the joint products is not produced, would involve two different laws of nature (Steedman, 1987, p. 40). However restrictive this For a proof see: Kurz and Salvadori (1995, p. 3-38)

8 8 assumption is, it is common to most fixed capital models because it facilitates the analysis of the optimal truncation period. Let us now turn our attention to the assumption of free disposal. This assumption supposes the existence of a process to which that commodity is the sole input and from which there is no output. This proposal is not simply an unrealistic assumption but denies one of the most fundamental physical concepts - the principle of conservation of mass-energy (Steedman, 1987, p. 40, Salvadori and Steedman, 1988, p. 180). Note that the rule of free goods may also be applied for non producible goods such as land which cannot be disposed off. Hence, this rule cannot always be associated with the assumption of free disposal. The crucial assumption (iv) can be viewed in a third way 3 by assuming that the presence of some quantities of a product or of a factor which is available in excess does not cause any costs or benefits. This implies that the good is either left where it is and as it is or if it moves or changes its form, that must be the result of non-human activities such as the transport by wind or water. For some or even most products this assumption cannot be sustained. An old machine which is worn out and which is of no use anymore would occupy costly space and may hinder the process of production if it is left where it is. Hence it might be cost minimising to dismantle the machine and to throw the parts into the near river. But this disposal process would require labour and other inputs such as transport equipment. If therefore disposal is costly the price of the outworn machine would be negative. If a product which, if it is available in excess, causes costs for the producer of that commodity, then the producer has three options: (i) the producer leaves that product where it is and as it is and carries the cost of the existence of that product; (ii) the producer may avoid excess production of that commodity by making use of another method and (iii) the producer uses a disposal process which transforms that product into another product which is either usefull and can be sold (e.g. scrap), or does not create any further costs for the producer (scrapped machine which is thrown away). However, for some joint products such as dust or liquid and gaseous substances there might be no such internal cost of disposal if complete absence of property rights of the environment is assumed. In this case the price of these products will be zero. This does, however, not imply that the existence of such products does not affect the system of relative prices. Emissions will, in general, have some influence on the production possibility set of other firms and therefore will also influence the system of relative prices. If the concentration of sulphur dioxide in the air or that of acid in the water exceeds a tolerable limit then the harvest of timber or the catch of fish will be 3 For a comprehensive discussion of the free disposal assumption see: Steedman (1987, p. 40)

9 9 reduced and prices for these products will go up. Hence even in those cases where the price of a product is zero we cannot exclude that product from the set of commodities under consideration unless it can be reasonably assumed that the excess product does not cause any costs nor generate any benefits. 3. A generalisation: restricted free disposal However the free disposal assumption is viewed, it is an assumption which cannot generally be sustained. In the following we will therefore try to find conditions for the existence of cost-minimising systems where this crucial assumption is either completely abandoned or, at least, relaxed. Increasingly the emphasis is on internalising external effects through the allocation of property rights (Perrings, 1987, p. ). The creation of excess products is always associated with the use of particular locations or, more general, with particular parts of the environment, where these unwanted products can be disposed of. Plots of land or lakes can be either used for the production of crops and for the breeding of trouts respectively or for the disposal of some wastes. If these parts of the environment are owned by profit maximising agents or agencies then the use of these factors creates costs. Ricardo ( , Vol. I, p. 75) suggested that if some environmental resources... were of various qualities; if they could be appropriated, and each quality existed only in moderate abundance, they, as well as the land, would afford a rent, as the successive qualities were brought into use. Following this proposal it is assumed that some parts of the environment or the rights for its utilisation are privately owned. Other parts of the environment are assumed to be controlled by an environmental agency which imposes restrictions to the utilisation of the environment under control by proclamation of certain maximal levels for the annual disposal of some products which are considered as dangerous or damaging if the issued limits are exceeded 4. These products, such as some chemicals, may be useful and may be absorbed in some production processes or by some household activities. But if these products are produced in excess and if the annual excess product exceeds an acceptable limit then it is considered as damaging. Note that we do not know a priori 4 Note that these limits are not defined as stocks but as annual flows. This implies that it is accepted that some quantities of products produced in excess can be disposed of every year and that the accumulated stocks of existing bads may grow at a rate which depends on the annual rate of disposal but also on the capacity of the environment to absorbe the stocks of bads.

10 10 whether there is excess production nor do we know that it exceeds the accepted level. This will be a result of the analysis. Hence these products are referred to as potential bads. The acceptable limits for excess production are not defined with respect to the excess product of the single firm but with respect to the total excess production of the entire economic system. The environmental policy instrument might be envisaged as a permit system where an agency issues limited numbers of tradable certificates which entitle the right to dispose of a certain quantity of some products each year. Someone who produces potential bads has several options: (i) he may find somebody who takes the product, (ii) he may buy costly permits and use the right to dispose of excess production, or, if that is technically feasible, he may (iii) use an alternative method which is more expensive but by which excess production can be avoided or, at least, reduced. Assume that the potential bad is a useful product, which can be utilised as an input for other processes or which is demanded as a consumer good. In this case and, if its total supply does not exceed the total demand, it can be sold at a positive price. If, on the other hand, that product is produced in excess of the quantities demanded, the difference must be disposed of. If total excess production does not exceed the tolerated level then permits are free and the excess products can be freely disposed off and its price is equal to zero. In that case option (iii), i.e. to use the alternative method, is not profitable because it will incur extra costs. But if excess production is such that the tolerated limit of disposal is met then the competition amongst producers of these potential bads will increase the price of the permits. Case (i), i.e. to find somebody who takes the bad may be still a possible option. But now things have changed and the producers of bads have to pay for delivery and the users of the bad obtain revenues for taking the unwanted product, i.e. the price of the bad is negative. But there is also the option (iii), i.e. to avoid or to reduce production of these bads by using an alternative process. This process create comparably higher costs for reducing or avoiding the production of bads. Hence the price of bads is again negative. It is up to the cost minimising producers of bads to choose among alternatives (i) to (iii). Therefore the price of permits will increase until it equals the negative price of reducing the excess product of the bad by making use of either one of the two other options, (i) or (iii). However, we cannot decide in advance whether a potential bad is in actual fact a bad and whether its price is positive, zero or negative. The policy proclaimed by the environmental agency is subject to various causes and may differ by the state of development and the views and aims of the society as well as by the quality of the

11 11 environment left to the society. Hence, the list of products considered as potential bads as well as their maximal levels of excess production are predetermined by causes which are not considered here. Consequently, the classification which subdivides products into goods and potential bads as well as the tolerated levels for excess production of the latter are assumed to be part of the exogenous data. Note that for the special case that there are no pontential bads, we are back to the assumption of free disposal. The formal analysis is based on the following definitions: (D.1) The set of all commodities can be subdivided into two subsets [ ] C, C : C C = C; C C =. For commodities i C 1, called goods, there is free disposal and we can preclude that prices become negative in cost minimising systems. For products j C, called potential bads, there is restricted free disposal in the sense that total excess production cannot exceed a certain tolerated level t j 0. (D.) Given the set of available processes, T, we may define two subsets (which may be empty). Processes in the first subset, T 1, do not use products j C. Processes in the second set, T, do not produce products j C. Note that given these definitions we have not yet restricted the analysis by any assumptions. We have neither assumed that, C 1, i.e. that there are any products which can be freely disposed of, nor have we excluded the case that the tolerated limits may be zero for some or for all products belonging to C. Furthermore (D.) does not presume that there are any processes in T 1 or in T or, if there are any such processes, that the intersection of the two sets is zero. In order to analyse the system of production in terms of long run positions the following assumptions are made: (A.1) All firms have access to all known methods of production and the set of these methods available to the firm is (i) constant (no technical progress) and (ii) independent of the size of the firm itself (constant returns within the single firm). (A.) Commodities are consumed in given constant proportions. (A.3) All quantities (including the tolerated quantities of excess production) grow at a uniform rate which does not exceed the rate of profit, i.e. g r.

12 1 (A.4) Natural resources other than homogenous human labour are set aside. Assumption (A.1), and in particular (ii) is a precondition for the existence of free competition. 5 (A.) and (A.4) are made to concentrate on the topic, i.e. joint production. (A.) assumes away that consumption may vary with prices and/or with the distribution of income. For models which are, in that respect, more general see: Kurz and Salvadori (1995, pp and pp ). (A.4) allows to abstract from the utilisation of any natural factors other than labour. Later on it will be shown that scarcity of non exhaustible resources (Ricardian land) and, under certain conditions, renewable resources can be considered without changing the basic features of the approach. (A.3) is crucial. Given the set of available methods of production, i.e. setting technical progress aside, the assumption that limits to excess production grow at the same rate as all other quantities can hardly be sustained. Note that assumption (A.3) is made for analytical purpose, i.e. to compare the model with restricted free disposal to the model with free disposal and to show that the conditions for existence of cost minmizing systems carry over from the latter to the former. If one is not willing to accept that assumption (A.3) then it may either be substituted by the proposition of a stationary economy, i.e. g = 0, or it might be assumed that tolerated excess production is zero for all commodities. Assume the exogenous data (A), (B), (C) as defined in section and, in addition, (D) a preconceived classification with respect to goods and potential bads and accepted limits for excess production for each commodity j C denoted by a non-negative column vector t R + n to be given. Note that, if g 0 then, by assumption (A.3), the accepted limits for excess production are not constant but are taken to grow at rate g. Hence, if g 0 and if t 0, vector t refers to constant proportions and not to constant levels. The problem is to determine the endogenous variables, x, p, w, as defined in section, such that the conditions (i) to (iii), specified in section, i.e. (i) (ii) (iii) processes are operated such that given requirements for use are satisfied, no process obtains extra profits, processes which incur extra costs will not be operated, 5 For a comprehensive discussion of the classical notion of free competition and its prerequisites see: Kurz and Salvadori (1995, chapter 1).

13 13 and the following conditions, which defines the restricted free disposal assumption, are met: (iv) If a product i C 1 is available in excess then the price of that products is zero, i.e. free disposal for goods. (v) Excess production of products j C, i.e. bads, do not exceed the tolerated limits. (vi) If a product j C is available in excess then the price for that product is not positive. (vii) If excess production for a product j that products is zero. C is lower than the tolerated limit, then the price of Conditions (i) to (iii) define the long run position, i.e. demand is satisfied, producers use the cost minimising methods of production, stocks of capital goods available for production are adjusted to demand, and therefore no production activity obtains extra profits. Hence a uniform rate of profit prevails and prices cover just costs of production including the normal profits. (iv) is the free disposal assumption for goods. Conditions (v) to (vii) refer to potential bads. (v) defines the environmental constraint for these products. (vi) says that the prices of such products, if they are available in excess, are not positive. These prices are equal to zero if excess production does not meet the accepted boundaries (vii), otherwise, i.e. if the tolerated limits are met, they are negative. Note that the assumption of free disposal, is now restricted in the sense that it is applied to all products i and to products j C as far as the excess production of the latter is tolerable. C 1 Since conditions (iv) to (vii) refer to some subsets of the commodity set we may, by appropriate ordering of the rows, represent matrices A, B and the vectors f and p by partitioned matrices or vectors where index 1 refers to goods and index refers to potential bads. Prices of potential bads may be either positive, zero or negative. Therefore we may express the subvector of prices for potential bads by the sum of two vectors one being nonnegative and the second is non-positive, i.e. + + ( ) : [ p 0 p 0] p = p + p ;. Expressing all prices in terms of labour commanded, i.e. w = 1, we may represent the conditions (i) to (vii) by the system (.i) to (.vii): ( ) ( 1 r ) (.i) ( 1 g ) B A + x > f p B A + < l (.ii) ( )

14 14 (.iii) ( ) k ( ) k k ( ( )) k: p B A 1+ r e < l x = 0,i.e. p B A 1 + r x= lx, x 0 (.iv) ( ) ( ) ( ( )) i C : e B A 1+ g x > f p = 0,i.e. p B A 1 + g x = pf, p 0 1 i i i where B 1, A 1 and f 1 are sub-matrices of outputs and inputs and the sub-vector of consumer demand referring to products i ( ) (.v) ( 1 g ) B A + x f t C 1. where B, A are sub-matrices of outputs and inputs referring to products j C. The positive elements of sub-vector f represent consumer demand for potential bads while the negative elements refer to the generation of these products by households. ( ) (.vi) ( ) j C : 1 0 e j B A + g x > f j p + j =, ( ) i.e. ( g ) p B A 1 + x = pf, p 0 ( ) (.vii) ( ) j C : 1 0 j g fj tj p e B A + x < + j =, ( ) ( ) i.e. ( g ) p B A 1 + x = p f + t, p 0 The following theorems, which are proved in the appendix, provide conditions for the existence of cost minimising systems.the first theorem is based on the golden rule assumption of a rate of growth which equals the rate of profit 6. All other theorems relax that assumption but require additional conditions concerning the set of processes. Theorem 1: If assumptions A.1, A., A.3 and A.4 hold and if the rate of profit equals the rate of growth then system (.i) to (.vii) has a solution if and only if there exists a vector x 0 which satisfies inequalities (.i) and (.v), i.e: [ 0 ( ( 1 )) ( ( 1 )) ] x : x ; B A + g x f; B A + g x f + t. 6 For a proof of Theorem 1 that utilizes other mathematical tools see Hosoda (1998).

15 15 Note that for Theorem 1 no assumptions are required concerning the content of the commodity sets C 1 and C. Hence the theorem holds for the special case of free disposal for all goods, i.e. C = and C = C 1 as well as for the case, C 1 = and C = C, where arbitrary limits for excess production are imposed for all commodities. In this latter case B = B, A = A and f = f, and the sufficient and necessary condition becomes [ ( ( )) ] x : x 0; f B A 1 + g x f + t. If the free disposal assumption is completely abandoned for all products, i.e. if t = 0, then that condition becomes [ ( ( )) ] x : x 0; B A 1 + g x = f, i.e. it is feasible to produce requirements for use without creating any excess production. Theorem 1 is based on the equality of the rate of growth and the rate of profit. If the rate of profit is positive then the system is assumed to grow at that positive rate. Long run growth is incompatible with constant levels of abatement. Hence the following theorems relax the golden rule assumption. Theorem : If the assumptions A.1, A., A.3, and A.4 hold then system (.i) to (.vii) has a solution if the following additional conditions are met: (A.5) (A.6) There exists a semipositive vector of intensities x * which satisfies [ 0 ( ( 1 r) ) ( ( 1 r )) ] x * : x * ; B A + x * f; B A + x * f + t If there are some commodities for which there is restricted free disposal, i.e: if C, then no process use these products, i.e. T = 1 T and therefore A = 0. The viability condition (A.5) states that it is possible to produce goods for consumption and investment purposes such that golden rule growth is potentially feasible, i.e. that the system may grow

16 16 at a rate which is equal to the rate of profit without exceeding the constraints for excess production given by vector t if these accepted boundaries would also grow at the rate r. Note that this does not mean that the quantities or that the accepted limits do actually grow at rate r. If the system is constrained by constant accepted limits of excess production it is stationary. The viability condition requires that technology is such that growth with rate r is potentially feasible provided that the accepted limits for disposal of excess products of the latter would grow at that rate. Since the industrial use of potential bads is ruled out by condition (A.6), condition (A.5) becomes [ 0 ( 1 1( 1 r )) 1 ] x * : x * ; B A + x * f ; f B x * f + t It has been mentioned above that the assumption of growing levels of accepted pollution is crucial and that one may substitute that assumption by assuming either a stationary economy, i.e. g = 0, or by assuming no excess production of potential bads at all. In the latter case, i.e. if t = 0, (A.5) is feasible and a cost minimising system exists if and only if it is possible to produce goods without generating any potential bads in excess, i.e. if and only if there exist activity levels x *, such that B x * = f 0. This requires that (growing) outputs of potential bads are totally absorbed by (growing) consumer demand for these products and that no potential bads are generated by households. If bads are not demanded at all, then the viability condition implies that there exist processes which satisfy demand for goods and do not produce any bads. Assume now that g = 0 and that potential bads are neither demanded nor generated by households, i.e. f = 0. It has been argued that the constraint on excess production imposed by the environmental agency, reflected by vector t, could be viewed as the amounts of available certificates which stand for the right to dispose excess products of. Denote the prices of permits by vector y 0 and recall that the negative price of a bad, i.e. the cost of avoiding the production of that bad equals the price of a permit which entitles the right to dispose of one unit of the excess product. Hence y p and because A = 0 inequality (.ii) and equation (.iii), which refer to the no extra cost and to the no extra profit condition, can be written as + (.ii ) ( 1 ) p B + p B p A + r + yb + l and

17 17 + (.iii ) ( ) = ( 1 1( 1 + r ) + + ) p B p B x p A yb l x where the right hand side shows revenues and the left hand side refers to cost of production including cost of permits required. That demonstrates that the output matrix of potential bads, i.e. B, is equivalent to the matrix representation of input requirements for permits. Accordingly the environmental constraint (.v) becomes (.v ) B x t and can be interpreted as the condition that the industrial use of permits cannot exceed the amounts of these certificates supplied by the environmental authority. Permits which are issued in limited amounts and natural resources which are available in given and constant quantities are, in principle, the same thing. Hence, if it is assumed that bads are not demanded, i.e. if f = 0 and A = 0, then the system (.1) to (.4) is formally equivalent to a general joint production model with free disposal and with Ricardian land where vector t and matrix B denote given amounts of different types of land and inputs of these lands respectively and y is the vector of the rental rates for the different lands determined by the Ricardian concept of internal and external differential rent. The existence condition for such a model provided by Kurz and Salvadori (1995, p. 99) are equivalent to the viability condition (A.5) of theorem. 7 Condition (A.6) defines the industrial use of potential bads away and therefore rules the existence of abatement processes out. The only possibility to adjust to the accepted levels of excess production is to utilise methods which generate less potential bads or avoid the production of these products at all. This assumption cannot be sustained. But it is possible to obtain a theorem which is based on less restrictive assumptions. Theorem 3: Assumption (A.6) of theorem can be substituted by the following assumption: (A.7) If there are some commodities for which there is restricted free disposal, i.e. if C 0, and if there are some processes which use these products, i.e. if T 0, 7 For a straightforward application of the Ricardian theory of differential rent to the determination of the prices of goods and bads see Lager (1998).

18 18 then these processes which use potential bads do not produce any potential bad, i.e. T1 T = 0. From (A.7) it follows that we obtain, by appropriate ordering of processes and products, A A A = A and B B B = 11 1 B 1 0 where the first (last) rows represent inputs or outputs of products i C 1 ( j C ) for which there is (restricted) free disposal and the first (last) columns represent processes belonging to set T 1 (T ). The processes which use potential bads may be considered as disposal processes which absorb bads and produce goods which are either useful or at least harmless. 8 Note that these processes are defined away by (A.6) where it is assumed that T = 0. Hence (A.6) is more restrictive than (A.7) and theorem 3 is a generalisation of theorem. It has been argued that growing levels of accepted excess production do not make much sense. Therefore, the relevant cases are either (i) g > 0 and t = 0 or (ii) g = 0 and t 0. In the former case the viability condition becomes ( ( 1 r) ) ( 1 r) ( ) B A + x + B A + x f ; * * ( 1 r) B x A + x = f * * 1 1 It follows that growth with rate g ( r g ) : > 0 can be feasible even if f = 0 and t=0 and even if it is not possible to produce goods without generating some bads, provided there exist some clean up processes which transform unwanted bads into useful goods which are either demanded or which are at least harmless. The theorems given above are based on the assumption that production processes use products and labour only. It was mentioned above that for the case that no potential bads are used in any industrial process and if the economy is stationary then the system is formally equivalent to a model with Ricardian land. Hence, it seems to be straightforward to abstain from the restrictive assumption (A.4) and allow for the utilisation of Ricardian land and, assuming certain conditions, for the use of renewable resources. The concept of long run positions presupposes that the set of available methods is given and does not change. Growing economic activity implies growing intensities of the

19 19 use of the lands and/or successive use of lands of lower quality. In both cases the set of methods employed will change. However, a long run position with Ricardian land can be determined for a stationary economy. This cannot be generally presumed if renewable resources are taken into account. Processes which utilise these natural factors will, in general, depend on the stocks of resources available. The catch of fish, for instance, depend on the amounts of fish available and the fishery processes will change with the population of fish. Therefore the constancy of the stocks of renewable resources is a precondition for the application of the classical method of long run positions. A comprehensive discussion of a model with renewable resources can be found in Kurz and Salvadori (1995, pp ). It is demonstrated that under the usual assumptions for the renewal of resources some stationary points exist where the stocks of resources do not change, and some of these points are locally stable and might be considered as long period positions. One of these stationary and stable points reflect a doomsday scenario where the stocks of resources have been totally depleted. Here it is assumed that the exploitation of these resources, such as the catch of fish or the harvest of timber in a virgin forest, is neither to large nor to small but is such that it compensates the natural renewal of the resources. Hence the stocks of resources such as the population of fish or the quantity of timber available remains constant over time and the economic system is able for reproduction under constant technological conditions. The constant rate of reproduction of renewable resources is assured by environmental policies which set proper upper limits for the exploitation of these resources such that a stable stationary point with a feasible constant rate of utilisation of these resources is attained. Given that assumption renewable resources can be dealt with like Ricardian land. The use of exhaustible resources is, however, excluded. Substitute (A.4) by assumption (A.4*) Commodities are produced by means of commodities, by homogenous labour, by non exhaustible natural resources (Ricardian land) and by renewable resources and assume that the latter are utilised in such a way, that the set of feasible methods of production does not change. Let A R h m R + be the matrix of inputs of h different qualities of land or renewable resources and let the vectors t R h h R + and y R R + denote available amounts of lands or upper limits of exploitation 8 In Leontief s pollution model (Leontief, 1970) it is assumed that abatement processes absorb bads (or produce clean up services) by means of labour and goods but produce nothing. That assumption cannot be sustained. It is a fundamental physical principle that any process which use some materials must have some material outputs.

20 0 of renewable resources and the corresponding rental rates respectively. A cost minimising system with land and renewable resources exploited at a feasible constant rate is determined by the following system: (3.i) ( B A) x f ( 1+ r ) + ( ( 1+ r) ) R R ( 1+ r) ( 1 r ) ( ( 1 r) ) ( 1 r) p B A p B A y A l (3.ii) ( ) p1 B1 A1 + x + p B A + x yra R x + = lx and x 0 (3.iii) ( ) p B A x = p f and p 1 0, (3.iv) ( ) (3.v) ( ) B A x f t + + p B A x = p f and p + 0 (3.vi) ( ) (3.vii) p ( B A ) x p ( f t) (3.viii) A x t = + and p 0 R R (3.ix) y A x = y t and y R R R R R 0, System (3) can be derived from system () by setting g = 0 and by adding some constraints concerning the utilisation of resources. (3.viii) states that the total use of land cannot exceed the quantities of land available and the utilisation of renewable resources cannot exceed the tolerable level. (3.ix) says that the rental rates of land (renewable resources) are either positive, if the land is fully utilised (if the level of tolerable exploitation is met), or equal to zero for those resources which are not fully utilised. The conditions for no extra profits (3.ii) and no extra costs (3.iii) are adjusted to take account of the cost of the rents payable post factum for the utilisation of the natural resources. It is shown in the annex that the conditions for the existence of a cost minimising technique provided in theorems and 3 carry over to the following theorems. Theorem 4: If assumptions A.1, A., A.3 and A.4* hold then system (3.i) to (3.ix) has a solution if the following additional conditions are met:

21 1 (A.5*) There exists a semipositive vector of intensities x * which satisfies x * 0; B A 1 + r x * f; B A 1 + r x * f + t; A 1 + r x * t ; [ ( ( )) ( ( )) R( ) R ] (A.6) If there are some commodities for which there is restricted free disposal, i.e: if C, then no process uses these products, i.e: T = 1 T and therefore A = 0. Theorem 5: Assumption (A.6) of theorem 4 can be substituted by assumption (A.7) i.e. if there are some commodities for which there is restricted free disposal, i.e. if C 0, and if there are some processes which use these products, i.e. if T 0, then these processes which use potential bads do not produce any potential bad, i.e. T1 T = Conclusions Whenever there are some processes which produce jointly two or more different products then the problem of excess production and the related problem of some negative prices of production cannot be avoided. The usual approach is to circumnavigate these difficulties by assuming free disposal. It has been argued that this assumption cannot be sustained in general. In this article it has been proposed to substitute that assumption by assuming that (i) free disposal is permitted for some products which are considered as harmless and that (ii) for other products there is restricted free disposal in the sense that excess production cannot exceed given tolerated levels. It is shown that the conditions for the existence of cost minimising joint production systems with free disposal, in particular the viability conditions, carry over to joint production systems with restricted free disposal provided that potential bads are either not used at all or, if they are used by some processes, then these processes do not produce potential bads. The latter condition assures that the absorption of unwanted products does not go with the creation of other potential bads. The assumption of free disposal is strictly stronger than the assumption of restricted free disposal. Hence, the theorems presented in this paper are a generalisation of some important results which can be found in the modern classical literature on joint production.

22 Appendix The problem of concern is similar to the problem of computing an optimal invariant capital stock studied by Dantzig and Manne (1974) and by Jones (198, 1986). Therefore, following Kurz and Salvadori (1995), the conditions for the cost minimising systems will be formulated in terms of a Linear Complementary Problem (LCP). Given a matrix M R n n and a vector q R n the LCP(q, M) consists in finding a solution [ = ] z: z ; Mz q; z M z q 0. The LCP(q, M) is feasible if [ ] z : z ; Mz q 0. The proofs of the theorems are based on two lemmata concerning the existence of solutions to the LCP(q, M) provided by Cottle, Pang and Stone, 199. Lemma A: Let M R n n be positive semi-definite, i.e: z Mz 0 for all z R n. If the LCP(q, M) is feasible, then it is solvable. (For the proof see Cottle, Pang and Stone, 199, pp 139) Lemma B: Let M R n n be copositive, i.e: z Mz 0 for all z R n +. Let q R n. If the implication [ y 0 My 0 y My = 0],, [ y q 0 ] is valid, then a solution for LCP(q, M) exists. (For the proof see Cottle, Pang and Stone, 199, pp 179) We shall first proof the theorems which are based on the assumption (A.4). Hence system (.i) to (.vii) is concerned.

23 3 Define matrices B C = B A A ( 1+ g) A ( 1+ g) ( 1+ g) B 1 1, B D = B A A ( 1 + r ) A ( 1 + r) ( 1 + r) B 1 1, p 1 + and vectors p = p, p f1 d = f + ( f t). Hence system (.i) to (.vii) is equivalent to system (I.1) Cx d (I.) D p l (I.3) p Cx = p d (I.4) x D p = x l (I.5) p 0; x 0 which corresponds to the LCP (II.1) (II.) (II.3) 0 C p d D 0 x l p 0 C p p d x D 0 x = x l p 0 x Proof of Theorem 1: The only if part of theorem 1 is trivial. Hence it suffices to proof the existence of a solution. From r = g it follows that C = D. Hence the matrix M of LCP(q, M) is positive semi-definite. [ ] x : x 0 ; Cx d is given by assumption. Hence the LCP has a feasible solution for z 0 =. The existence of a solution for the LCP follows from Lemma A. x Q.E.D.

24 4 The proofs of all other theorems are based on a corollary which follows from Lemma C: Let S R n n be skew symmetric. Hence S + S = 0. Let C R n n (S+C, q) has a solution if there exists a vector z * such that ( ) be copositive. The LCP [ z * 0, S C z * q] Proof: From S + S = 0 it follows that S is copositive and consequently S+C is also copositive. Hence it suffices to proof that the implication of lemma B holds. If that vector z * 0 exists, then, because S S Hence, by Farkas Lemma, there is no y: ( ) S C z * = S + C z * q. =, it satisfies ( ) ( ) [ y 0, S + C y 0 y q > 0] [ ],. Consequently the implication of lemma B, i.e. y 0, ( S + C) y 0, y ( S + C) y = 0 [ y q ] Q.E.D. 0, holds. The following corollary is an implication of lemma C. Let M 0 U a b a b = ; U R ; V R, V 0 q and let q = q q 1 ; 1 R ; R q a b. The LCP(M, q) has a solution if (i) U V, (ii) q 0 and * * * (iii) z1 [ z1 0 V z1 q1] : ;. 0 U 0 0 Proof: M = + can be represented as a sum of two matrices, one being U 0 U V 0 skew symmetric and the other one is copositive by assumption (i). From lemma C it follows that 0 U 0 U V there is a solution to the LCP if q1 z*: z*. That vector exists U q * z by assumptions (ii) and (iii) being z* = 1. Hence there is a solution to the LCP. 0 Since theorem 3 is a generalisation of theorem it suffices to proof the former. Q.E.D.