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Empty sectors in a multisectoral planning model
Empty sectors in a multisectoral planning model An analytical note Sajal Lahiri This note develops and analyses a static terminal period planning model that allows for the possible fillin9 in of empty sectors. The model is a suitable extension of the standard input-output model. It is assumed that some of the commodities are not produced domestically at the base period because of indivisibilities involved in their production. The problems that this note addresses are whether or no; any such commodity should be produced domestically at the terminal period 9iven that the planners have decided to produce them domestically, the indivisibility factor permitting, and if they are to be produced, what their taroet production levels should be. Keywords: Input-output; Empty sectors; Planning
In this note, we extend the standard input-output (IO) model to include 'empty sectors' in a simple way. Before we define the concept of an empty sector, it will be useful to state the conceptual frame of our analysis. In our frame, capacities, scales and methods of production are all thought of in purely ex ante terms, as 'prospective' ones. That is. the model is defined for a definite postulated future. We shall refer to the postulated future as the terminal period, the present as the base period, and the timespan covering the two as the plannin9 horizon. In this context, empty sectors are the sectors of production with a given technology but no actual production at the base structure. Our model allows for possible 'filling in' of previous empty sectors at the terminal period by the creation of capacities with the help of suitable investments over the planning horizon. However, the planning horizon or investment does not make an explicit appearance in our frame. It is implicitly assumed that the planning horizon is sufficiently long to cover the so-called period of construction of the prospective capacities. The author is with the Department of Economics, University of Essex, Colchester CO4 3SQ, UK. The author is gratefulto ProfessorSanjit Bose- his thesis supervisor - for helpful suggestions and discussions. Final manuscript received 5 October 1985.
Detailed discussion of our model is taken up in the next section, which is followed in the final section by an analysis of the model. The
model
The problem of filling in of empty sectors is viewed here in terms of setting up capacities for producing commodities that were not produced earlier. The terms 'filling in' and 'empty sectors' are borrowed from Chakravarty ([I], chapter 7). 1 His definition of the concept and specification of the context provide the conceptual background of our discussion. The specific nature of problems discussed are, however, quite different. In the terminal period, the empty sectors may remain empty or may start producing commodities so that the filling in problem is equivalent to an investment decision problem. 2 Clearly, therefore, one is faced with a choice problem which is solved here by a set of secondary relations incorporating specific criteria of choice. Essentially, these criteria have to work back from the prospective scales of production at i See Lahiri [2] for a more detailed discussion. 2 It is assumed that when a sector is empty the demand for the product of that sector is met by imports.
0264-9993/86/030237-03 $03.00 ~ 1986 Butterworth & Co (Publishers) Ltd
237
Empty sectors in a multisectoral plannin9 model: S. Lahiri the terminal date, which in turn are governed by contemporaneous final demands. The choice criterion adopted here is simple, vis, produce the commodity if it is possible. This possibility will be seen to be restricted by indivisibility in the size of capacity-to-be installed. That is, the capacity of production in an empty sector is assumed to be of a minimum size, which is basically a reflection of the nature of technology. It follows that an empty sector will remain empty unless total demand is of this minimum size, with a once-for-all switch to production after a certain level. Basically, we are trying to answer the following question: how many of the empty sectors should be filled in by the terminal period, and what should be their production targets given that the planners have decided, from other considerations, to fill in the empty sectors, the indivisibility factor permitting? We may now turn to a formalization of the concept of 'empty sectors'. This is best done by depicting the potential IO matrix of the terminal period in a partitioned form as indicated by the left partitioned matrix below, with the corresponding technology in the base period depicted on the right.
The partitioning of the above matrices follows a corresponding partitioning of the set of all sectors into two, viz the set of 'producing sectors' and 'empty sectors', on the basis of whether or not a sector takes part in production at the base period. That is, at the base period each producing sector domestically produces the requirement of its good and, in contrast, the requirement of its goods corresponding to empty sectors are met by non-competitive imports. At the terminal period, however, whereas the producing sectors continue to produce their goods domestically, some of the empty sectors may enter into production activities on the basis of the choice criterion referred to above. This is the 'filling in' of empty sectors mentioned earlier. The first and second set of rows and columns in the above two partitioned matrices correspond precisely to the two sets of sectors defined above. More explicitly, in the base period the IO matrix is A, with C forming the so-called 'primary input coefficient matrix', the primary inputs being non-competitive imports. Since the empty sectors do not take part in domestic production at the base year, their input coefficients - the corresponding columns in the second partitioned matrix - are left blank. As mentioned, these sectors may start producing the goods in the terminal period" requiring non-negative input coefficients and hence the corresponding blank
238
columns may then be filled in by their input coefficients at the terminal date - these are the columns of the
ma,rix[;] We conclude this section with a formal presentation of our model. For this, let there be n producing sectors and k empty sectors and the corresponding sets be denoted by N and K respectively. The model is then represented by the following set of equations:
#:]+[:] ui=0 or u~>fii
uivi=O
ere
(1)
VieK
(2)
VieK
(3)
[:] [;]
vectors of om st,
production and final demands respectively, partitioned according to the sets N and K, v is the vector of import levels, and fii is the lower bound specified on the level of production of the ith commodity belonging to set K. Equation (1) represents the basic commodity balance equations. Equation (3) implies that any commodity belonging to the set K is either imported or domestically produced but not both. Equation (2) then guarantees that if a commodity belonging to set K i s produced domestically, its level of production is higher than the corresponding lower bound on it. Equations (2) and (3) together therefore take care of the choice criteria referred to earlier. It may be noted that when u,=O, ie when ith empty sector does not take part in production, the ith column of the matrix [ B D] becomes ineffective.
Analysis The iterative method that we propose here for computing a solution of the model developed in the last section consists of constructing four sequences of estimates. Mathematically, it constructs the sequences ~.'#j,~ ~ ~u~'~, ~_ttt, ~ and {V} by the following iterative procedure: R O~Zi) ! " - '-" u' " ,, ui~
U;~':',0)
if z~>~i otherwise
(4)
where z' and x' are given by:
qx'-'l+[q
z'J Lc DJLu'-'J LdJ ECONOMIC MODELLING
(5)
July 1986
Empty sectors in a multisectoral plannin0 model. S. Lahiri The iterative scheme is initiated by choosing such initial estimates as satisfy: A
D]Lz°J+[_d~
(6)
_o > h
(7)
That is, at the initial step, the production vector ix °, z °) is taken to be an overestimate for the given final demands as revealed in Equation (6). In addition, the production vector for set K, ie the set of empty sectors, is taken to be greater than or equal to the vector of lower bounds on it - see Equation (7). Then at each stage of the scheme, a subset of K is formed for which each sector's production level estimate is higher than the corresponding lower bound. The estimates of production levels for the remaining sectors in K are then put to zero and their estimated requirements are met by imports. Assumption 1
IA
/~C
,]
D <1
where )~(.) is the dominant characteristic root of a square matrix. Note that Assumption 1 guarantees the existence of an initial solution satisfying Equations (6) and (7). The result of this paper is stated below as a theorem. Theorem 1 The sequences {xt}, {u'} and {L,'} as defined in Equations (4)-(7) converge to a solution to Equations (1)-(3).
ECONOMIC MODELLING
July 1986
Proof. By using straightforward induction hypothesis, it can easily be proved that the sequences {x~l, {z'} and {u'l are non-increasing and {t,'} takes a jump from zero at some stage and then decreases. Since all the sequences are non-negative, using Equation (4) it at once follows that all the four sequences converge and the limiting vectors of {.x~}, {u'} and r' I solve Equations (1 }-(3). Corollary l In case of multiple solutions, the solution given by the iterative scheme Equations (4)-(7) gives the largest feasible subset of K. Proof. Since the sequences constructed are nonincreasing, it should be noted that any sector in K which goes out of the production activity at some stage of the iteration cannot have a production level greater than the corresponding lower bound in any other situation. The conclusion therefore follows. QED. We conclude this paper with a comment on our iterative scheme Equations (4)-(7). From the above corollary, it follows that our iterative scheme provides an algorithm for achieving a solution which maximizes the incidence of filling up of the empty sectors for producing the given bill of final demands.
References 1 S. Chakravarty, The Looic of hwestment Planning, North Holland, Amsterdam, 1959. 2 S. Lahiri, Theoretical lnput4)utput Analysis: Three Generalizations, unpublished PhD thesis, Indian Statistical Institute, Calcutta, 1976.
239
In this note, we extend the standard input-output (IO) model to include 'empty sectors' in a simple way. Before we define the concept of an empty sector, it will be useful to state the conceptual frame of our analysis. In our frame, capacities, scales and methods of production are all thought of in purely ex ante terms, as 'prospective' ones. That is. the model is defined for a definite postulated future. We shall refer to the postulated future as the terminal period, the present as the base period, and the timespan covering the two as the plannin9 horizon. In this context, empty sectors are the sectors of production with a given technology but no actual production at the base structure. Our model allows for possible 'filling in' of previous empty sectors at the terminal period by the creation of capacities with the help of suitable investments over the planning horizon. However, the planning horizon or investment does not make an explicit appearance in our frame. It is implicitly assumed that the planning horizon is sufficiently long to cover the so-called period of construction of the prospective capacities. The author is with the Department of Economics, University of Essex, Colchester CO4 3SQ, UK. The author is gratefulto ProfessorSanjit Bose- his thesis supervisor - for helpful suggestions and discussions. Final manuscript received 5 October 1985.
Detailed discussion of our model is taken up in the next section, which is followed in the final section by an analysis of the model. The
model
The problem of filling in of empty sectors is viewed here in terms of setting up capacities for producing commodities that were not produced earlier. The terms 'filling in' and 'empty sectors' are borrowed from Chakravarty ([I], chapter 7). 1 His definition of the concept and specification of the context provide the conceptual background of our discussion. The specific nature of problems discussed are, however, quite different. In the terminal period, the empty sectors may remain empty or may start producing commodities so that the filling in problem is equivalent to an investment decision problem. 2 Clearly, therefore, one is faced with a choice problem which is solved here by a set of secondary relations incorporating specific criteria of choice. Essentially, these criteria have to work back from the prospective scales of production at i See Lahiri [2] for a more detailed discussion. 2 It is assumed that when a sector is empty the demand for the product of that sector is met by imports.
0264-9993/86/030237-03 $03.00 ~ 1986 Butterworth & Co (Publishers) Ltd
237
Empty sectors in a multisectoral plannin9 model: S. Lahiri the terminal date, which in turn are governed by contemporaneous final demands. The choice criterion adopted here is simple, vis, produce the commodity if it is possible. This possibility will be seen to be restricted by indivisibility in the size of capacity-to-be installed. That is, the capacity of production in an empty sector is assumed to be of a minimum size, which is basically a reflection of the nature of technology. It follows that an empty sector will remain empty unless total demand is of this minimum size, with a once-for-all switch to production after a certain level. Basically, we are trying to answer the following question: how many of the empty sectors should be filled in by the terminal period, and what should be their production targets given that the planners have decided, from other considerations, to fill in the empty sectors, the indivisibility factor permitting? We may now turn to a formalization of the concept of 'empty sectors'. This is best done by depicting the potential IO matrix of the terminal period in a partitioned form as indicated by the left partitioned matrix below, with the corresponding technology in the base period depicted on the right.
The partitioning of the above matrices follows a corresponding partitioning of the set of all sectors into two, viz the set of 'producing sectors' and 'empty sectors', on the basis of whether or not a sector takes part in production at the base period. That is, at the base period each producing sector domestically produces the requirement of its good and, in contrast, the requirement of its goods corresponding to empty sectors are met by non-competitive imports. At the terminal period, however, whereas the producing sectors continue to produce their goods domestically, some of the empty sectors may enter into production activities on the basis of the choice criterion referred to above. This is the 'filling in' of empty sectors mentioned earlier. The first and second set of rows and columns in the above two partitioned matrices correspond precisely to the two sets of sectors defined above. More explicitly, in the base period the IO matrix is A, with C forming the so-called 'primary input coefficient matrix', the primary inputs being non-competitive imports. Since the empty sectors do not take part in domestic production at the base year, their input coefficients - the corresponding columns in the second partitioned matrix - are left blank. As mentioned, these sectors may start producing the goods in the terminal period" requiring non-negative input coefficients and hence the corresponding blank
238
columns may then be filled in by their input coefficients at the terminal date - these are the columns of the
ma,rix[;] We conclude this section with a formal presentation of our model. For this, let there be n producing sectors and k empty sectors and the corresponding sets be denoted by N and K respectively. The model is then represented by the following set of equations:
#:]+[:] ui=0 or u~>fii
uivi=O
ere
(1)
VieK
(2)
VieK
(3)
[:] [;]
vectors of om st,
production and final demands respectively, partitioned according to the sets N and K, v is the vector of import levels, and fii is the lower bound specified on the level of production of the ith commodity belonging to set K. Equation (1) represents the basic commodity balance equations. Equation (3) implies that any commodity belonging to the set K is either imported or domestically produced but not both. Equation (2) then guarantees that if a commodity belonging to set K i s produced domestically, its level of production is higher than the corresponding lower bound on it. Equations (2) and (3) together therefore take care of the choice criteria referred to earlier. It may be noted that when u,=O, ie when ith empty sector does not take part in production, the ith column of the matrix [ B D] becomes ineffective.
Analysis The iterative method that we propose here for computing a solution of the model developed in the last section consists of constructing four sequences of estimates. Mathematically, it constructs the sequences ~.'#j,~ ~ ~u~'~, ~_ttt, ~ and {V} by the following iterative procedure: R O~Zi) ! " - '-" u' " ,, ui~
U;~':',0)
if z~>~i otherwise
(4)
where z' and x' are given by:
qx'-'l+[q
z'J Lc DJLu'-'J LdJ ECONOMIC MODELLING
(5)
July 1986
Empty sectors in a multisectoral plannin0 model. S. Lahiri The iterative scheme is initiated by choosing such initial estimates as satisfy: A
D]Lz°J+[_d~
(6)
_o > h
(7)
That is, at the initial step, the production vector ix °, z °) is taken to be an overestimate for the given final demands as revealed in Equation (6). In addition, the production vector for set K, ie the set of empty sectors, is taken to be greater than or equal to the vector of lower bounds on it - see Equation (7). Then at each stage of the scheme, a subset of K is formed for which each sector's production level estimate is higher than the corresponding lower bound. The estimates of production levels for the remaining sectors in K are then put to zero and their estimated requirements are met by imports. Assumption 1
IA
/~C
,]
D <1
where )~(.) is the dominant characteristic root of a square matrix. Note that Assumption 1 guarantees the existence of an initial solution satisfying Equations (6) and (7). The result of this paper is stated below as a theorem. Theorem 1 The sequences {xt}, {u'} and {L,'} as defined in Equations (4)-(7) converge to a solution to Equations (1)-(3).
ECONOMIC MODELLING
July 1986
Proof. By using straightforward induction hypothesis, it can easily be proved that the sequences {x~l, {z'} and {u'l are non-increasing and {t,'} takes a jump from zero at some stage and then decreases. Since all the sequences are non-negative, using Equation (4) it at once follows that all the four sequences converge and the limiting vectors of {.x~}, {u'} and r' I solve Equations (1 }-(3). Corollary l In case of multiple solutions, the solution given by the iterative scheme Equations (4)-(7) gives the largest feasible subset of K. Proof. Since the sequences constructed are nonincreasing, it should be noted that any sector in K which goes out of the production activity at some stage of the iteration cannot have a production level greater than the corresponding lower bound in any other situation. The conclusion therefore follows. QED. We conclude this paper with a comment on our iterative scheme Equations (4)-(7). From the above corollary, it follows that our iterative scheme provides an algorithm for achieving a solution which maximizes the incidence of filling up of the empty sectors for producing the given bill of final demands.
References 1 S. Chakravarty, The Looic of hwestment Planning, North Holland, Amsterdam, 1959. 2 S. Lahiri, Theoretical lnput4)utput Analysis: Three Generalizations, unpublished PhD thesis, Indian Statistical Institute, Calcutta, 1976.
239