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# Richard Stone - MODELLING COMMODITY BALANCES IN A COMPUTABLE GENERAL EQUILIBRIUM CONTEXT

Актуальные публикации по вопросам экономики.

Источник: Economic Systems Research, 1994, Vol. 6 Issue 2, p123, 12p, 5 charts

Pyatt, Graham

MODELLING COMMODITY BALANCES IN A COMPUTABLE GENERAL EQUILIBRIUM CONTEXT

The Richard Stone Memorial Lecture, Part II

(Received June 1993; revised November 1993)

ABSTRACT

This paper is the second part of a two-part discussion which elaborates the proposition that Stone's concept of commodity technology is the appropriate assumption on which to build in modelling commodity balances. The analysis in the first part was developed from a social accounting representation of commodity balances into a formal statement of the Stone model and the conditions under which it admits an acceptable solution. This second part develops a generalization of Stone's original formulation, in the spirit that the modelling of commodity balances should be approached in a general equilibrium context which recognizes that, in the final analysis, prices and quantities are interdependent.

1. Introduction

The contribution of Sir Richard Stone to the modelling of commodity balances is best known perhaps through his exposition of the matter in the context of the UN System of National Accounts (SNA; UNSO, 1968). However, it can be argued, as in an essay to which this paper is the sequel (see Pyatt, 1994), that Stone's interest in the subject derived from an enquiry into the structural obstacles to growth, which subsequently required an analytic approach to structure in which prices and quantities could interact with each other. The Stone model of commodity balances, which is set out in the earlier essay, can be seen, therefore, as a component of a general equilibrium model: a module of a more general scheme.

In developing this module, Stone imposed some strong linearity assumptions on the structure of commodity balances. These assumptions were sufficient to permit the unique determination of commodity prices as a function of net output prices--independently of the level of activity--and the corresponding determination of gross outputs as a function of the level of final demand and otherwise as being independent of commodity prices. These strong assumptions had the particular advantage that they allowed the possibility of computing numerical solutions in an era when the power of computers was much less than it is today. Their main disadvantage derives from the loss of flexibility implicit in the strong assumptions. In particular, it was necessary for Stone to assume that the number of activities in an economy and the number of products it produced were the same (and hence that the make matrix could be inverted). He also had to assume that all imported goods were strictly complementary to goods produced domestically, in the sense that no substitution was possible, even at the margin.

These assumptions are relaxed in the present paper, which attempts to provide a flexible formulation of the modelling of commodity balances in line with current best-practice, as exemplified by general equilibrium models (CGEs). In the first instance, emphasis is placed on the modelling of the product mix of different industries in the spirit of Stone's commodity technology approach, and on the determination of imports. A generalization of the specific results obtained in these areas is then explored, which builds on the notion of a linear homogeneous intermediate technology. When this holds, it can be shown that, if the resulting model of commodity balances has a solution, then the model can be solved recursively, first for prices and then for quantities. The paper concludes by noting that Stone's commodity technology model is an interesting special case of the linear homogeneous intermediate technology model which can be obtained by imposing the condition that gross output prices should depend only on net output prices and, therefore, be independent of the scale of production.

2. An Initial Social Accounting Matrix

The social accounting matrix (SAM) presented as Table 1 reproduces Table 1 of Pyatt (1994) and provides both an initial setting and notation for the discussion to be developed in this paper. The matrix recognizes production activities as being distinct from commodities. It also distinguishes two strata of commodities, with basic commodities being the products of production activities at their point of origin, while final commodities are these same products at their point of sale. The difference, therefore, is that the final commodities are combinations of basic commodities, including the transport and distribution elements of final sales.

There are three key matrices in Table 1 and these can be referred to as the make, marketing and absorption matrices. The last of these is denoted T[sub 2a], which is expressed alternatively as p[sub 2] L[sub 2a] q[sub a] in Table 2, where P[sub 2] is the vector of level two (final) commodity prices and q[sub a] is the vector of activity gross outputs. Since the various elements of T[sub 2a] represent the total expenditure of each activity on each final commodity which is required as an intermediate input into production, it follows that each element of the matrix L[sub 2a] is a ratio which corresponds to the quantity of a particular raw material that is absorbed per unit of output by a particular activity.

Similarly, the marketing matrix, shown alternatively as T[sub 12] in Table 1 and p[sub 1] L[sub 12] q[sub 2] in Table 2, records the expenditure on domestically produced basic goods which is necessary to generate final goods: L[sub 12] is again a matrix of ratios, showing how much of each basic good is needed per unit of each final good supplied.

The third matrix, i.e. the make matrix, denoted by T[sub a1] in Table 1 and by q[sub a]S'[sub 1a]p[sub 1] in Table 2, is evidently slightly different in construction. This third matrix shows the revenue received by each activity from sales of each of the basic commodities which it produces. It follows that S[sub 1a] is again a matrix of coefficients, though these now represent the quantity of each product produced per unit output of each activity.

This brief discussion of Table I can be concluded by noting that A and A are vectors of the final sales of basic and final goods respectively. Therefore, the formulation allows (through f[sub 1]) for the final consumption of basic goods by subsistence producers. Balancing these two vectors in the SAM are two other terms, the first of which is denoted v'[sub a] in Table 1 and as q'[sub a]pi[sub a] in Table 2. The elements of this vector are the amounts of value added by each activity. Hence, pi[sub a] is a vector of net output prices or value added per unit of gross output. The remaining term is q'[sub 2]pi[sub 2], which represents commodity taxes plus the cost of any goods imported.

Since Table 1 is a SAM, it follows that corresponding row and column totals must be equal, as shown. Moreover, it is evident from the alternative version of Table 1, i.e. Table 2, that a set of row equations can be derived from the SAM by equating each row total with the sum of the elements located in that row. Similarly, a set of column equations also can be obtained. The linear model of commodity balances discussed earlier (Pyatt, 1994) is obtained directly from these row and column balance equations under the restriction that each of L[sub 2a], L[sub 12] and S[sub 1a] is a matrix of coefficients which are fixed, independent of all prices and quantities and, therefore, of the scale of production or the competitiveness of foreign goods.

3. Modelling the Make Matrix

It is apparent that the linear model represents a particularly severe restriction of the general system which is described by the row and column balance equations which can be derived from Table 1. More realistic models allow the matrices S[sub 1a], L[sub 12] and L[sub 2a] to be determined as functions of prices p[sub a], p[sub 1] and p[sub 2], and quantities q[sub a], q[sub 1] and q[sub 2]. In particular, attention can be focused on the determination of the make matrix S[sub 1a] and on the desirability of making some allowance for competitive imports. The first of these concerns is addressed in this section; the second aspect is taken up in Section 4.

The modelling of the make matrix S[sub 1a] requires a determination of how much of its various products an industry will produce, given the industry's overall level of activity. If x[sub 1], . . ., x. are the output levels for the various products, and z[sub 1], . . ., z[sub m] are the necessary inputs of raw materials and factor services, then there will be a constraint of the form

0 =F(x; z) (1)

implied by technological possibilities for producing different output mixes from given inputs. This general formulation can be simplified by assuming that equation (1) can be rewritten in the restricted form

q(x) = h(z)(2)

When this is possible, then

q = q(x)(3)

is a natural measure of output for the activity, the costs of which will depend on z and, therefore, be independent of the mix of x for any given q(x). Thus, in this special case, profit-maximizing behaviour implies that, for a given q, the output mix will be chosen so as to maximize revenue.

To formalize the implications of this approach, it can be noted that, when the prices of products p are independent of x, then the maximization of total revenue, i.e. p'x, for a given level of q, requires that

p = lambda q[sub x](4)

where lambda is a Lagrangian multiplier and q[sub x] is the vector of first-order partial derivatives of q with respect to its arguments x[sub i].

The combination of equations (3) and (4) now implies that

[Multiple line equation(s) cannot be represented in ASCII text] (5)

where Q[sub xx] is the symmetric matrix of second-order partial derivatives delta[sup 2]q/deltax[sub i] deltax[sub j].

Hence, we have

[Multiple line equation(s) cannot be represented in ASCII text] (6)

where

t = q'[sub x]Q[sub xx][sup -1]q[sub x] (7)

The results in equation (6) imply that the output x[sub i] of each product will be a function of prices p and the level of production q, i.e.

x[sub i] = x[sub i](p; q) (8)

Indeed, it can be shown that the function x[sub i] in equation (8) will be (a) homogeneous of degree zero in prices p; and (b) homogeneous of degree 1/p in q if q is a homogeneous function of degree p in x.

These results can be established by first noting that, if dp = mup, so that all prices change in proportion, and q remains fixed, then dq = 0. It then follows from equation (6) that

dx = mu[I - (1/t) Q[sub xx][sup -1] q[sub x][q'[sub x]]Q[sub xx][sup -1]p (9)

= lambda mu[I-(1/t) Q[sub xx][sup -1]q[sub x]q'[sub x]]Q[sub xx][sup -1]q[sub x] (10)

= 0 (11)

from equations (4) and (7). Hence, a scale increase in all prices has no effect on the vector x, i.e. the functions specified by equation (8) must be homogeneous of degree zero in prices p.

The effects of a change in q on x when prices are given can be developed by noting that, from equations (6) and (7), we have

(q'[sub x] Q[sub xx][sup -1] q[sub x]) d log x = qx[sup -1] Q[sub xx][sup -1] q[sub x] d log q (12)

However, if q is a homogeneous function of order rho in x, then

rhoq = q'[sub x] x (rho - 1) q sub x] = Q[sub xx] x (13)

Hence, it can be shown that, when q is a homogeneous function of x, the result in equation (12) implies that

delta log x/delta log q = (1/rho) i (14)

where the elements of the vector on the left-hand side are the elasticities of particular elements of x with respect to q. The result in equation (14) implies that these elasticities are the same for all products and are equal to (1/rho).

The above results have various implications, two of which are noted here. First, it follows from these results that, if the gross output price of an activity (p[sub a]) is defined as

p[sub a] = (1/q) p'x (15)

then p[sub a] is a linear homogeneous function of product prices p. Also, p[sub a] will be independent of q if output is measured as a linear homogeneous function of the quantities of products produced. The second implication to note is that the elements of the make matrix S[sub 1a] will be functions of product prices that are homogeneous of degree zero and independent of the volume of activity (q), provided that q is measured as a linear homogeneous function of x.

4. Competitive Imports

The point has been made in the Introduction that one of the least attractive features of the linear model of commodity balances is the implicit assumption embodied within it that all imports are strictly complementary inputs into domestic production or final demand. It is necessary to relax this assumption if more realistic models are to be obtained.

Historically, the main line of development has been through activity analysis, in which the market is assumed to choose between an imported good and the competing domestic good, according to the criterion of which is cheaper. Imported and domestic goods are treated as being pure substitutes in this approach, which overcomes the immediate problem posed by the linear input-output model. However, the reformulation has its own inherent weakness, which was exposed by Samuelson (1951) via the non-substitution theorem.

Specifically, the activity analysis formulation implies that countries should specialize their production of traded goods in the long run. By common consent, and at a time when direct intervention through prescriptive planning was in vogue, this implication was seen as being too extreme and, therefore, as reflecting a weakness in the activity analysis formulation. To overcome this, Armington (1969) introduced the concept of composite commodities as a vehicle for allowing the competition between domestic and imported goods to be modelled in a more flexible and less extreme way. His approach can be described with the help of Table 3.

Table 3 sets out a SAM which represents a development of Table 1, in allowing for the existence of what Armington referred to as 'composite commodities'. The essential idea here is to set to zero the entries in Table 1 which refer to intermediate and final demand for competitive imports and the domestic goods which compete with them. Instead, these competing final goods are sold to the composite commodity accounts. The role of these accounts expost is to record and, ex ante, to model the choices made by the market between those goods which compete, i.e. to determine the proportions in which competing goods will be demanded. In the extreme case, when domestic and imported goods are perfect substitutes, these proportions may be determined on an 'all-or-nothing' basis, i.e. the market will specialize on the cheaper source. This is the special case which corresponds to activity analysis. More generally, when competing goods are less than perfect substitutes, the market is likely to favour some mixture. Hence, combinations of competing goods, known as 'composite commodities', are defined. It is these combinations which are purchased as intermediate inputs or as a part of final demand.

A particular implication of this approach is that there are now two parts to the absorption matrix, i.e. T[sub 2a] and T[sub 3a]. The non-zero parts of T[sub 2a] result from intermediate purchases of non-traded goods and complementary imports, while the non-zero elements of T3a are the result of intermediate purchases of composite goods. An assumption to the effect that matrices of fixed coefficients may underly both parts of the absorption matrix (T[sub 2a] and T[sub 3a]) might now be a reasonable approximation. However, this is because the innovation of matrix T[sub 23] makes it possible to formulate the demand for competing goods as being sensitive to their relative prices.

To formalize these ideas, it can be suggested that a composite good q can be generated by combining inputs of other goods x[sub 1] , . . .,x[sub m] as required by some functional relationship

q = q(x[sub 1], . . ., x[sub m] (16)

In this case, an iso-product curve is generally assumed to be convex from the origin, in recognition of the fact that increasing quantities of one good are needed to compensate for the loss of successive units of some other good--a diminishing marginal rate of substitution.

It can now be assumed that the role of the market is to determine the quantities x[sub 1], . . ., x[sub m] is such a way that the amount of the composite commodity q which can be obtained for given cost p'x is a maximum; equivalently, the cost of obtaining a given amount of the composite commodity should be minimized. In either case, the result will be that the quantities x[sub j] will be determined as functions of prices p and the magnitude of q as

x[sub j] = x[sub j](p, q) (17)

in much the same way as the functions in equation (8) were generated earlier. Also, using an argument very similar to that which was developed previously in relation to the functions in equation (8), it can be shown that the functions x[sub j] expressed in equation (17) will be (a) homogeneous of degree zero in prices p; and (b) homogeneous of degree 1/rho in q if q is a homogeneous function of degree rho in x.

One important consequence of these results is that, if P[sub c] is the price of a composite commodity, defined by

p[sub c] = (1/q) p'x (18)

then p[sub c] will be a linear homogeneous function of the prices of those ingredients which combine together to form the composite commodity. Furthermore, p[sub c] will be independent of q if q is a linear homogeneous function of x.

5. A General System

The results in Sections 3 and 4 for modelling the make matrix and competitive imports suggest ways in which the very general system of row and column balance equations implied in Table 2 can be usefully developed, albeit at the expense of introducing an increasing number of accounts in the SAM to describe the various flows.

With Table 3 as the starting point, and introducing the new notation

T[sub 23] = p[sub 2] L[sub 23] q[sub 3], T[sub 3a] = p[sub 3] L[sub 3a] q[sub a] (19)

y[sub 3] = p[sub 3] q[sub 3], f[sub 3] = p[sub 3] phi[sub 3] (20)

an alternative SAM can be generated as in Table 4. The row and column balance equations of this new SAM define a general (non-linear) model of commodity balances which can be interpreted as an elaboration of the model implied by Table 2; this allows inter alia for the notion of composite commodities. The eight independent row and column balance equations which define this more elaborate model are set out in Table 5, together with the redundant residual balance equation.

A linear model can now be obtained by assuming tht each of S[sub 1a], L[sub 12], L[sub 2a], L[sub 23] and L[sub 3a] is a matrix of fixed coefficients. At the opposite extreme, a totally flexible model can be formulated in which each of these matrices is treated as a function of both p and q, where p is a vector of all prices p[sub a], p[sub 1], p[sub 2] and p[sub 3], and q is a vector of all quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3]

In between these extremes are a number of interesting cases which the analyst is likely to want to adopt in practice. For example, the discussion of the make matrix in Section 2 suggests that this might be written as

S[sub 1a] = S[sub 1a] (p[sub 1]) (21)

with the understanding that each element of S[sub 1a] is a homogeneous function of degree zero in the product prices p[sub 1]. An implication of this approach would be that the output of an activity (q[sub a]) is a linear homogeneous function of the quantities of the basic commodities it produces.

Similarly, the matrix L[sub 23] might be written as

L[sub 23] = L[sub 23] (p[sub 2]) (22)

with the implication that the supply of a composite commodity is a linear homogeneous function of the quantities of constituent competitive goods, and a homogeneous function of degree zero of their prices.

Given equations (21) and (22), the model specification might be completed in a particular case by assuming that, for the marketing and absorption matrices, it is indeed reasonable to postulate fixed coefficients, independent of all prices and quantities. This particular example of how a model might be formulated then illustrates the fact that, for many of the models which it might be attractive to adopt in practice, it may be quite reasonable to assume that there are constant returns to scale, in the sense that each of the matrices S[sub 1a], L[sub 12], L[sub 2a], L[sub 23] and L[sub 3a] is independent of the scale of demand and supply, as measured by the vectors q[sub a], q[sub 1], q[sub 2] and q[sub 3]. This is a limited form of scale independence, which does not exclude increasing or decreasing returns with respect to changes in the scale of factor inputs in generating net output.

Special cases of the equations in Table 5 which correspond to constant returns in this limited sense can be referred to as linear homogeneous intermediate technologies. Furthermore, situations in which the intermediate technology is linear homogeneous (or homogeneous of degree one) are interesting as a general class. One of their characteristics is to imply models which can always be solved recursively, first for prices and then for quantities.

To develop some implications of this special class of model, it can be noted that, as a general rule, each of the matrices L[sub ij] can be allowed to depend both on the prices of the various commodities being bought by a particular SAM account and on the scale on which purchases are being made. Hence, the general formulation is

L[sub 12] = L[sub 12] (p[sub 1], pi[sub 2]; q[sub 2]) (23)

L[sub 23] = L[sub 23] (p[sub 2], q[sub 3]) (24)

L[sub 2a] = L[sub 2a] (p[sub 2], p[sub 3], pi[sub a]; q[sub a]) (25)

L[sub 3a] = L[sub 3a] (p[sub 2], p[sub 3], pi[sub a]; q[sub a])(26)

Similarly, on the supply side, we have

S[sub 1a] = S[sub 1a] (p[sub 2]; q[sub a]) (27)

so that the make matrix depends on product prices and (perhaps) on the level of activity also. A linear homogeneous intermediate technology is now defined as a technology such that the expressions on the right-hand side of equations (23)-(27) are independent of all quantities.

In the general case represented by equations (23)-(27), there is no alternative other than to regard the equations in Table 5 as a set of 2(n[sub 1] + n[sub 2] + n[sub 3] + n[sub a]) independent equations in the 4n[sub 1] + 3(n[sub 2] + n[sub 3] + n[sub a]) variables p[sub a], q[sub a], pi[sub a] and pi[sub 2] plus

p[sub i], q[sub i] and phi[sub i] for i = 1, 2 and 3, where n[sub i] is the number of type i commodity accounts and n[sub a] is the number of activities. Solutions of this system of equations will normally be possible, given an appropriate set of exogenous or predetermined values for some of the variables. However, there is no evident structure to the system and, in particular, there is no possibility of solving the system for prices which are independent of scale (or of quantities).

Such a possibility does emerge, however, when intermediate technology is linear homogeneous. Specifically, in the linear homogeneous case, the price equations in

Table 5 can be written as

p[sub a] = S[sub 1a]' (p[sub a]) p[sub 1] (28)

p[sub a] = L[sub 2a]' (p[sub 2], p[sub 3], pi[sub a]) p[sub a] + L[sub 3a]' (p[sub 2], p[sub 3], pi[sub a]) p[sub 3] + pi[sub a] (29)

p[sub 2] = L[sub 12]' (p[sub 1], pi[sub 2]) p[sub 1] + pi[sub 2] (30)

p[sub 3] = L[sub 23]' (p[sub 2]) p[sub 2] (31)

which defines a set of 2n[sub a] + n[sub 1] + n[sub 2] equations in the 2n[sub a]+ n[sub 1] + 2n[sub 2] + n[sub 3] variables P[sub a], P[sub 1], P[sub 2], p[sub 3], pi[sub a] and pi[sub 2].

Some implications of equations (28)-(31) are illustrated in Figure 1, which shows the pattern of interpendence between the various price vectors, as implied by these relationships. The figure shows that there are alternative ways of evaluating P[sub a]. One of these is defined by equation (28), while the other possibility is

p[sub a] = (L[sub 2a]' L[sub 12]' + L[sub 23]' L[sub 12]') p[sub 1] + (L[sub 2a]' + L[sub 3a]' L[sub 23]') pi[sub 2] + pi[sub a] (32)

This can be obtained from equation (29) by eliminating the variables P[sub 2] and P[sub 3], according to the expressions provided by equations (30) and (31).

The important implication of equations (28) and (32) is that the model can have a solution, if and only if

(S[sub 1a]'- L[sub 2a]' L[sub 12]' - L[sub 3a]' L[sub 23]' L[sub 12]') p[sub 1) = (L[sub 2a] + L[sub 3a] L[sub 23]) pi[sub 2] + pi[sub a] (33)

This is a general result. However, there are two specific examples of it which are of particular interest.

First, if the na equations defined by the condition in equation (33) can be solved for pi[sub a], given p[sub 1] and pi[sub 2], then it will be possible, for given values of p[sub 1] and pi[sub 2], to solve recursively for all other prices in the system, i.e. for P[sub a], P[sub 2] and P[sub 3]. Also, it can be noted here that a solution of equation (33) for pi[sub a], given p[sub 1] and pi[sub 2], will always be possible when the coefficient matrices L[sub 2a] and L[sub 3a] are independent of pi[sub a].

A second situation of some interest arises when equation (33) can be solved for p[sub 1] as a function of pi[sub 2] and pi[sub a]. This is because, in this event, all the prices p[sub a], p[sub 1], P[sub 2] and P[sub 3] can be determined for given values of pi[sub a] and pi[sub 2]. The point to note in this case is that such a solution will not generally be possible. The condition in equation (33) defines a set of (non-linear) restrictions on the vector p[sub 1] for given pi[[sub a] and pi[sub 2]. However, the vector p[sub 1] has n[sub 1] elements and, as a general rule, there is no reason to expect n[sub 1] to be equal to n[sub a], or for the restrictions in equation (33) to be interdependent. Hence, a solution of equations (28)-(31) for the prices p[sub a], p[sub 1], P[sub 2] and P[sub 3], given pi[sub a] and pi[sub 2], will not normally be possible.

Given a set of prices p[sub a], p[sub 1], p[sub 2], P[sub 3], pi[sub a] and pi[sub 2] which satisfy the restrictions in equations (28)-(31), each of the matrices L[sub ij] and S[sub 1a] will be determined in the linear homogeneous intermediate technology model. Therefore, the solution of the model for quantities requires a solution of the four quantity equations

q[sub 1] = S[sub 1a]q[sub a] (34)

q[sub 1] = L[sub 12]q[sub 2] + phi[sub 1] (35)

q[sub 2] = L[sub 2a]q[sub a] + L[sub 23]q[sub 3] + phi[sub 2] (36)

q[sub 3] = L[sub 3a]q[sub a] + phi[sub 3] (37)

and these can be interpreted as linear equations in the various quantity vectors, since the coefficient matrices which appear in these equations are all predetermined by prices.

A condition for the existence of a solution to equations (34)-(37) must be that the expression for q[sub 1] provided by equation (34) is consistent with the alternative expression provided by equation (35) after substituting out q[sub 2] and q[sub 3] according to equations (34) and (37). Hence, a necessary condition for a solution is that

(S[sub 1a] - L[sub 12]L[sub 2a] - L[sub 12]L[sub 23]L[sub 3a]) q[sub a] = phi[sub 1] + L[sub 12]phi[sub 2] + L[sub 12]L[sub 23]phi[sub 3] (38)

This condition can always be satisfied in the sense that, for given prices and, therefore, for given coefficient matrices, the vector phi[sub 1] can always be determined for given values of the vectors q[sub a], phi[sub 2] and phi[sub 3]. Hence, the quantity side of the linear homogeneous intermediate technology model can always be solved for quantities q[sub 1], q[sub 2], q[sub 3] and phi[sub 1], given prices which satisfy the condition in equation (33) and predetermined values of the vectors q[sub a], phi[sub 2] and phi[sub 3].

Similarly, if the condition in equation (38) can be solved for q[sub a], given prices and the vectors phi[sub 1], phi[sub 2] and phi[sub 3], then it is straightforward to show that the model can be solved for all the quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3]. The necessary condition in this case, with the coefficient matrices predetermined, is simply that the matrix

[(S[sub 1a] - L[sub 12] L[sub 2a] - L[sub 12] L[sub 23] L[sub 3a])[sup -1]] (39)

should exist for all admissible constellations of prices.

Finally, to complete the analysis, it can be noted that it is possible to formulate a model in which not only are the prices p[sub a], p[sub l], P[sub 2] and P[sub 3] uniquely determined by pi[sub a] and pi[sub 2], independent of all quantities, but also the quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3] are uniquely determined, independent of prices, for given phi[sub 1], phi[sub 2] and phi[sub 3]. It is evident from Table 5 and equations (23)-(27) that such a model must be a special case of the linear homogeneous intermediate technology model, in which each of the matrices L[sub ij]. and S[sub 1a] is fixed independent of prices (and quantities), i.e. a linear model. Moreover, this linear model must be such that the inverse matrix defined by (39) exists. In addition, from the condition in equation (3), it follows that, in such circumstances, it will always be possible to solve for p[sub 1] as a function of pi[sub a] and pi[sub 2]; hence, it will be possible to solve for all the prices P[sub a], P[sub 1], P[sub 2] and P[sub 3] as a function of pi[sub a] and pi[sub 2].

Since the Stone model can be interpreted as a reduced form of the linear model corresponding to the SAM shown in Table 4, it follows that Stone's model is the most general model of commodity balances that is consistent with the commodity technology assumption and allows for all the prices P[sub a], P[sub 1], P[sub 2] and P[sub 3], and all the quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3] tO be determined independently of each other as functions of pi[sub a] and pi[sub 2], on the one hand, and phi[sub 1], phi[sub 2] and phi[sub 3] on the other hand.

Table 1. A basic social accounting matrix Commodities Activities Basic Final All other accounts Totals

Activities 0 T[sub a1] 0 0 y[sub a]

Commodities

Basic 0 0 T[sub 12] f[sub 1] y[sub 1]

Final T[sub 2a] 0 0 f[sub 2] y[sub 2]

All other accounts v'[sub a] 0 v'[sub 2] -- v=f

Totals y'[sub a] y'[sub 1] y'[sub 2] f=v

Table 2. A developed version of Table 1

Information is presented in the following order: Activities; Commodities, Basic; Commodities, Final; Commodities, Composite; All other accounts; Totals

Activities; 0; q[sub a]S'[sub 1a]p[sub 1]; 0; 0; p[sub a]q[sub a]

Commodities

Basic; 0; 0; p[sub 1]L[sub 12]q[sub 2]; p[sub 1]phi[sub 1]; p[sub 1]q[sub 1]

Final; p[sub 2]L[sub 2a]q[sub a]; 0; 0; p[sub 2]phi[sub 2]; p[sub 2]q[sub 2]

All other accounts; q'[sub a]pi[sub a]; 0; q'[sub a2]pi[sub 2]; --; --

Totals; q'[sub a]p[sub a]; q'[sub 1]p[sub 1]; q[sub 2]p[sub 2]; --

Table 3. An extended version of Table 1, introducing composite commodity accounts

Information is presented in the following order: Activities; Commodities, Basic; Commodities, Final; Commodities, Composite; All other accounts; Totals

Activities; 0; T[sub a1]; 0; 0; 0; y[sub a]

Commodities

Basic; 0; 0; T[sub 12]; 0; f[sub 1]; y[sub 1]

Final; T[sub 2a]; 0; 0; T[sub 23]; f[sub 2]; y[sub 2]

Composite; T[sub 3a]; 0; 0; 0; f[sub 3]; y[sub 3];

All other accounts; v'[sub a]; 0; v'[sub 2]; 0; --

Totals; y'[sub a]; y'[sub 1]; y'[sub 2]; y'[sub 3]; --

Table 4. An alternative version of Table 3, based on the commodity technology assumption and the law of one price

Information is presented in the following order: Activities; Commodities, Basic; Commodities, Final; Commodities, Composite; All other accounts; Totals

Activities; 0; q[sub a]S'[sub 1a]p[sub 1]; 0; 0; 0; p[sub a]q[sub a]

Commodities

Basic; 0; 0; p[sub 1]L[sub 12]q[sub 2]; 0; p[sub 1]phi[sub 1]; p[sub 1]q[sub 1]

Final; p[sub 2]L[sub 2a]q[sub a]; 0; 0; p[sub 2]L[sub 23]q[sub 3]; p[sub 2]phi[sub 2]; p[sub 2]q[sub 2]

Composite; p[sub 3]L[sub 3a]q[sub a]; 0; 0; 0; p[sub 3]phi[sub 3]; p[sub 3]q[sub 3]

All other accounts; q'[sub a]pi[sub a]; 0; q'[sub 2]pi[sub 2]; 0; --; --

Totals; q'[sub a]p[sub a]; q'[sub 1]p[sub 1]; q'[sub 2]p[sub 2]; q'[sub 3]p[sub 3]; --

Table 5. The balance equations implied by Table 4

Information is presented in the following order: Account; Row equation; Column equation

Activities; p[sub a] = S'[sub 1a]p[sub 1]; p'[sub a] = p'[sub 2]L[sub 2a] + p'[sub 3]L[sub 3a] + pi'[sub a]

Commodities

Basic; q[sub 1] = L[sub 12]q[sub 2] + phi[sub 1]; q'[sub 1] = q'[sub a]S'[sub 1a]

Final; q[sub 2] = L[sub 2a]q[sub a] + L[sub 23]q[sub 3] + phi[sub 2]; p'[sub 2] = p'[sub 1]L[sub 12] + pi'[sub a]

Composite; q[sub 3] = L[sub 3a]q[sub a] + phi[sub 3]; p'[sub 3] = p'[sub 2]L[sub 23]

All other accounts; q'[sub a]pi[sub a] + q'[sub 2]pi[sub 2] = p'[sub 1]phi[sub 1] + p'[sub 2]phi[sub 1] + p'[sub 2]phi[sub 2] + p'[sub 3]phi[sub 3]

References

Armington, P. (1969) A theory of demand for products distinguished by place of production, IMF Staff Papers, 16, pp. 159-178.

Pyatt, G. (1994) Modelling commodity balances: a derivation of the Stone model, Economic Systems Research, 6, pp. 5-20.

Samuelson, P. A. (1951) Abstract of a theorem concerning substitution in open Leontief models, in: T. C. Koopmans (ed.) Activity Analysis of Production and Allocation, Cowles Commission Monograph 13 (Wiley, New York).

United Nations, Statistical Office (1968) A System of National Accounts, Studies in Methods, Series F, No. 2, Rev. 3 (United Nations, New York).

Pyatt, Graham

MODELLING COMMODITY BALANCES IN A COMPUTABLE GENERAL EQUILIBRIUM CONTEXT

The Richard Stone Memorial Lecture, Part II

(Received June 1993; revised November 1993)

ABSTRACT

This paper is the second part of a two-part discussion which elaborates the proposition that Stone's concept of commodity technology is the appropriate assumption on which to build in modelling commodity balances. The analysis in the first part was developed from a social accounting representation of commodity balances into a formal statement of the Stone model and the conditions under which it admits an acceptable solution. This second part develops a generalization of Stone's original formulation, in the spirit that the modelling of commodity balances should be approached in a general equilibrium context which recognizes that, in the final analysis, prices and quantities are interdependent.

1. Introduction

The contribution of Sir Richard Stone to the modelling of commodity balances is best known perhaps through his exposition of the matter in the context of the UN System of National Accounts (SNA; UNSO, 1968). However, it can be argued, as in an essay to which this paper is the sequel (see Pyatt, 1994), that Stone's interest in the subject derived from an enquiry into the structural obstacles to growth, which subsequently required an analytic approach to structure in which prices and quantities could interact with each other. The Stone model of commodity balances, which is set out in the earlier essay, can be seen, therefore, as a component of a general equilibrium model: a module of a more general scheme.

In developing this module, Stone imposed some strong linearity assumptions on the structure of commodity balances. These assumptions were sufficient to permit the unique determination of commodity prices as a function of net output prices--independently of the level of activity--and the corresponding determination of gross outputs as a function of the level of final demand and otherwise as being independent of commodity prices. These strong assumptions had the particular advantage that they allowed the possibility of computing numerical solutions in an era when the power of computers was much less than it is today. Their main disadvantage derives from the loss of flexibility implicit in the strong assumptions. In particular, it was necessary for Stone to assume that the number of activities in an economy and the number of products it produced were the same (and hence that the make matrix could be inverted). He also had to assume that all imported goods were strictly complementary to goods produced domestically, in the sense that no substitution was possible, even at the margin.

These assumptions are relaxed in the present paper, which attempts to provide a flexible formulation of the modelling of commodity balances in line with current best-practice, as exemplified by general equilibrium models (CGEs). In the first instance, emphasis is placed on the modelling of the product mix of different industries in the spirit of Stone's commodity technology approach, and on the determination of imports. A generalization of the specific results obtained in these areas is then explored, which builds on the notion of a linear homogeneous intermediate technology. When this holds, it can be shown that, if the resulting model of commodity balances has a solution, then the model can be solved recursively, first for prices and then for quantities. The paper concludes by noting that Stone's commodity technology model is an interesting special case of the linear homogeneous intermediate technology model which can be obtained by imposing the condition that gross output prices should depend only on net output prices and, therefore, be independent of the scale of production.

2. An Initial Social Accounting Matrix

The social accounting matrix (SAM) presented as Table 1 reproduces Table 1 of Pyatt (1994) and provides both an initial setting and notation for the discussion to be developed in this paper. The matrix recognizes production activities as being distinct from commodities. It also distinguishes two strata of commodities, with basic commodities being the products of production activities at their point of origin, while final commodities are these same products at their point of sale. The difference, therefore, is that the final commodities are combinations of basic commodities, including the transport and distribution elements of final sales.

There are three key matrices in Table 1 and these can be referred to as the make, marketing and absorption matrices. The last of these is denoted T[sub 2a], which is expressed alternatively as p[sub 2] L[sub 2a] q[sub a] in Table 2, where P[sub 2] is the vector of level two (final) commodity prices and q[sub a] is the vector of activity gross outputs. Since the various elements of T[sub 2a] represent the total expenditure of each activity on each final commodity which is required as an intermediate input into production, it follows that each element of the matrix L[sub 2a] is a ratio which corresponds to the quantity of a particular raw material that is absorbed per unit of output by a particular activity.

Similarly, the marketing matrix, shown alternatively as T[sub 12] in Table 1 and p[sub 1] L[sub 12] q[sub 2] in Table 2, records the expenditure on domestically produced basic goods which is necessary to generate final goods: L[sub 12] is again a matrix of ratios, showing how much of each basic good is needed per unit of each final good supplied.

The third matrix, i.e. the make matrix, denoted by T[sub a1] in Table 1 and by q[sub a]S'[sub 1a]p[sub 1] in Table 2, is evidently slightly different in construction. This third matrix shows the revenue received by each activity from sales of each of the basic commodities which it produces. It follows that S[sub 1a] is again a matrix of coefficients, though these now represent the quantity of each product produced per unit output of each activity.

This brief discussion of Table I can be concluded by noting that A and A are vectors of the final sales of basic and final goods respectively. Therefore, the formulation allows (through f[sub 1]) for the final consumption of basic goods by subsistence producers. Balancing these two vectors in the SAM are two other terms, the first of which is denoted v'[sub a] in Table 1 and as q'[sub a]pi[sub a] in Table 2. The elements of this vector are the amounts of value added by each activity. Hence, pi[sub a] is a vector of net output prices or value added per unit of gross output. The remaining term is q'[sub 2]pi[sub 2], which represents commodity taxes plus the cost of any goods imported.

Since Table 1 is a SAM, it follows that corresponding row and column totals must be equal, as shown. Moreover, it is evident from the alternative version of Table 1, i.e. Table 2, that a set of row equations can be derived from the SAM by equating each row total with the sum of the elements located in that row. Similarly, a set of column equations also can be obtained. The linear model of commodity balances discussed earlier (Pyatt, 1994) is obtained directly from these row and column balance equations under the restriction that each of L[sub 2a], L[sub 12] and S[sub 1a] is a matrix of coefficients which are fixed, independent of all prices and quantities and, therefore, of the scale of production or the competitiveness of foreign goods.

3. Modelling the Make Matrix

It is apparent that the linear model represents a particularly severe restriction of the general system which is described by the row and column balance equations which can be derived from Table 1. More realistic models allow the matrices S[sub 1a], L[sub 12] and L[sub 2a] to be determined as functions of prices p[sub a], p[sub 1] and p[sub 2], and quantities q[sub a], q[sub 1] and q[sub 2]. In particular, attention can be focused on the determination of the make matrix S[sub 1a] and on the desirability of making some allowance for competitive imports. The first of these concerns is addressed in this section; the second aspect is taken up in Section 4.

The modelling of the make matrix S[sub 1a] requires a determination of how much of its various products an industry will produce, given the industry's overall level of activity. If x[sub 1], . . ., x. are the output levels for the various products, and z[sub 1], . . ., z[sub m] are the necessary inputs of raw materials and factor services, then there will be a constraint of the form

0 =F(x; z) (1)

implied by technological possibilities for producing different output mixes from given inputs. This general formulation can be simplified by assuming that equation (1) can be rewritten in the restricted form

q(x) = h(z)(2)

When this is possible, then

q = q(x)(3)

is a natural measure of output for the activity, the costs of which will depend on z and, therefore, be independent of the mix of x for any given q(x). Thus, in this special case, profit-maximizing behaviour implies that, for a given q, the output mix will be chosen so as to maximize revenue.

To formalize the implications of this approach, it can be noted that, when the prices of products p are independent of x, then the maximization of total revenue, i.e. p'x, for a given level of q, requires that

p = lambda q[sub x](4)

where lambda is a Lagrangian multiplier and q[sub x] is the vector of first-order partial derivatives of q with respect to its arguments x[sub i].

The combination of equations (3) and (4) now implies that

[Multiple line equation(s) cannot be represented in ASCII text] (5)

where Q[sub xx] is the symmetric matrix of second-order partial derivatives delta[sup 2]q/deltax[sub i] deltax[sub j].

Hence, we have

[Multiple line equation(s) cannot be represented in ASCII text] (6)

where

t = q'[sub x]Q[sub xx][sup -1]q[sub x] (7)

The results in equation (6) imply that the output x[sub i] of each product will be a function of prices p and the level of production q, i.e.

x[sub i] = x[sub i](p; q) (8)

Indeed, it can be shown that the function x[sub i] in equation (8) will be (a) homogeneous of degree zero in prices p; and (b) homogeneous of degree 1/p in q if q is a homogeneous function of degree p in x.

These results can be established by first noting that, if dp = mup, so that all prices change in proportion, and q remains fixed, then dq = 0. It then follows from equation (6) that

dx = mu[I - (1/t) Q[sub xx][sup -1] q[sub x][q'[sub x]]Q[sub xx][sup -1]p (9)

= lambda mu[I-(1/t) Q[sub xx][sup -1]q[sub x]q'[sub x]]Q[sub xx][sup -1]q[sub x] (10)

= 0 (11)

from equations (4) and (7). Hence, a scale increase in all prices has no effect on the vector x, i.e. the functions specified by equation (8) must be homogeneous of degree zero in prices p.

The effects of a change in q on x when prices are given can be developed by noting that, from equations (6) and (7), we have

(q'[sub x] Q[sub xx][sup -1] q[sub x]) d log x = qx[sup -1] Q[sub xx][sup -1] q[sub x] d log q (12)

However, if q is a homogeneous function of order rho in x, then

rhoq = q'[sub x] x (rho - 1) q sub x] = Q[sub xx] x (13)

Hence, it can be shown that, when q is a homogeneous function of x, the result in equation (12) implies that

delta log x/delta log q = (1/rho) i (14)

where the elements of the vector on the left-hand side are the elasticities of particular elements of x with respect to q. The result in equation (14) implies that these elasticities are the same for all products and are equal to (1/rho).

The above results have various implications, two of which are noted here. First, it follows from these results that, if the gross output price of an activity (p[sub a]) is defined as

p[sub a] = (1/q) p'x (15)

then p[sub a] is a linear homogeneous function of product prices p. Also, p[sub a] will be independent of q if output is measured as a linear homogeneous function of the quantities of products produced. The second implication to note is that the elements of the make matrix S[sub 1a] will be functions of product prices that are homogeneous of degree zero and independent of the volume of activity (q), provided that q is measured as a linear homogeneous function of x.

4. Competitive Imports

The point has been made in the Introduction that one of the least attractive features of the linear model of commodity balances is the implicit assumption embodied within it that all imports are strictly complementary inputs into domestic production or final demand. It is necessary to relax this assumption if more realistic models are to be obtained.

Historically, the main line of development has been through activity analysis, in which the market is assumed to choose between an imported good and the competing domestic good, according to the criterion of which is cheaper. Imported and domestic goods are treated as being pure substitutes in this approach, which overcomes the immediate problem posed by the linear input-output model. However, the reformulation has its own inherent weakness, which was exposed by Samuelson (1951) via the non-substitution theorem.

Specifically, the activity analysis formulation implies that countries should specialize their production of traded goods in the long run. By common consent, and at a time when direct intervention through prescriptive planning was in vogue, this implication was seen as being too extreme and, therefore, as reflecting a weakness in the activity analysis formulation. To overcome this, Armington (1969) introduced the concept of composite commodities as a vehicle for allowing the competition between domestic and imported goods to be modelled in a more flexible and less extreme way. His approach can be described with the help of Table 3.

Table 3 sets out a SAM which represents a development of Table 1, in allowing for the existence of what Armington referred to as 'composite commodities'. The essential idea here is to set to zero the entries in Table 1 which refer to intermediate and final demand for competitive imports and the domestic goods which compete with them. Instead, these competing final goods are sold to the composite commodity accounts. The role of these accounts expost is to record and, ex ante, to model the choices made by the market between those goods which compete, i.e. to determine the proportions in which competing goods will be demanded. In the extreme case, when domestic and imported goods are perfect substitutes, these proportions may be determined on an 'all-or-nothing' basis, i.e. the market will specialize on the cheaper source. This is the special case which corresponds to activity analysis. More generally, when competing goods are less than perfect substitutes, the market is likely to favour some mixture. Hence, combinations of competing goods, known as 'composite commodities', are defined. It is these combinations which are purchased as intermediate inputs or as a part of final demand.

A particular implication of this approach is that there are now two parts to the absorption matrix, i.e. T[sub 2a] and T[sub 3a]. The non-zero parts of T[sub 2a] result from intermediate purchases of non-traded goods and complementary imports, while the non-zero elements of T3a are the result of intermediate purchases of composite goods. An assumption to the effect that matrices of fixed coefficients may underly both parts of the absorption matrix (T[sub 2a] and T[sub 3a]) might now be a reasonable approximation. However, this is because the innovation of matrix T[sub 23] makes it possible to formulate the demand for competing goods as being sensitive to their relative prices.

To formalize these ideas, it can be suggested that a composite good q can be generated by combining inputs of other goods x[sub 1] , . . .,x[sub m] as required by some functional relationship

q = q(x[sub 1], . . ., x[sub m] (16)

In this case, an iso-product curve is generally assumed to be convex from the origin, in recognition of the fact that increasing quantities of one good are needed to compensate for the loss of successive units of some other good--a diminishing marginal rate of substitution.

It can now be assumed that the role of the market is to determine the quantities x[sub 1], . . ., x[sub m] is such a way that the amount of the composite commodity q which can be obtained for given cost p'x is a maximum; equivalently, the cost of obtaining a given amount of the composite commodity should be minimized. In either case, the result will be that the quantities x[sub j] will be determined as functions of prices p and the magnitude of q as

x[sub j] = x[sub j](p, q) (17)

in much the same way as the functions in equation (8) were generated earlier. Also, using an argument very similar to that which was developed previously in relation to the functions in equation (8), it can be shown that the functions x[sub j] expressed in equation (17) will be (a) homogeneous of degree zero in prices p; and (b) homogeneous of degree 1/rho in q if q is a homogeneous function of degree rho in x.

One important consequence of these results is that, if P[sub c] is the price of a composite commodity, defined by

p[sub c] = (1/q) p'x (18)

then p[sub c] will be a linear homogeneous function of the prices of those ingredients which combine together to form the composite commodity. Furthermore, p[sub c] will be independent of q if q is a linear homogeneous function of x.

5. A General System

The results in Sections 3 and 4 for modelling the make matrix and competitive imports suggest ways in which the very general system of row and column balance equations implied in Table 2 can be usefully developed, albeit at the expense of introducing an increasing number of accounts in the SAM to describe the various flows.

With Table 3 as the starting point, and introducing the new notation

T[sub 23] = p[sub 2] L[sub 23] q[sub 3], T[sub 3a] = p[sub 3] L[sub 3a] q[sub a] (19)

y[sub 3] = p[sub 3] q[sub 3], f[sub 3] = p[sub 3] phi[sub 3] (20)

an alternative SAM can be generated as in Table 4. The row and column balance equations of this new SAM define a general (non-linear) model of commodity balances which can be interpreted as an elaboration of the model implied by Table 2; this allows inter alia for the notion of composite commodities. The eight independent row and column balance equations which define this more elaborate model are set out in Table 5, together with the redundant residual balance equation.

A linear model can now be obtained by assuming tht each of S[sub 1a], L[sub 12], L[sub 2a], L[sub 23] and L[sub 3a] is a matrix of fixed coefficients. At the opposite extreme, a totally flexible model can be formulated in which each of these matrices is treated as a function of both p and q, where p is a vector of all prices p[sub a], p[sub 1], p[sub 2] and p[sub 3], and q is a vector of all quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3]

In between these extremes are a number of interesting cases which the analyst is likely to want to adopt in practice. For example, the discussion of the make matrix in Section 2 suggests that this might be written as

S[sub 1a] = S[sub 1a] (p[sub 1]) (21)

with the understanding that each element of S[sub 1a] is a homogeneous function of degree zero in the product prices p[sub 1]. An implication of this approach would be that the output of an activity (q[sub a]) is a linear homogeneous function of the quantities of the basic commodities it produces.

Similarly, the matrix L[sub 23] might be written as

L[sub 23] = L[sub 23] (p[sub 2]) (22)

with the implication that the supply of a composite commodity is a linear homogeneous function of the quantities of constituent competitive goods, and a homogeneous function of degree zero of their prices.

Given equations (21) and (22), the model specification might be completed in a particular case by assuming that, for the marketing and absorption matrices, it is indeed reasonable to postulate fixed coefficients, independent of all prices and quantities. This particular example of how a model might be formulated then illustrates the fact that, for many of the models which it might be attractive to adopt in practice, it may be quite reasonable to assume that there are constant returns to scale, in the sense that each of the matrices S[sub 1a], L[sub 12], L[sub 2a], L[sub 23] and L[sub 3a] is independent of the scale of demand and supply, as measured by the vectors q[sub a], q[sub 1], q[sub 2] and q[sub 3]. This is a limited form of scale independence, which does not exclude increasing or decreasing returns with respect to changes in the scale of factor inputs in generating net output.

Special cases of the equations in Table 5 which correspond to constant returns in this limited sense can be referred to as linear homogeneous intermediate technologies. Furthermore, situations in which the intermediate technology is linear homogeneous (or homogeneous of degree one) are interesting as a general class. One of their characteristics is to imply models which can always be solved recursively, first for prices and then for quantities.

To develop some implications of this special class of model, it can be noted that, as a general rule, each of the matrices L[sub ij] can be allowed to depend both on the prices of the various commodities being bought by a particular SAM account and on the scale on which purchases are being made. Hence, the general formulation is

L[sub 12] = L[sub 12] (p[sub 1], pi[sub 2]; q[sub 2]) (23)

L[sub 23] = L[sub 23] (p[sub 2], q[sub 3]) (24)

L[sub 2a] = L[sub 2a] (p[sub 2], p[sub 3], pi[sub a]; q[sub a]) (25)

L[sub 3a] = L[sub 3a] (p[sub 2], p[sub 3], pi[sub a]; q[sub a])(26)

Similarly, on the supply side, we have

S[sub 1a] = S[sub 1a] (p[sub 2]; q[sub a]) (27)

so that the make matrix depends on product prices and (perhaps) on the level of activity also. A linear homogeneous intermediate technology is now defined as a technology such that the expressions on the right-hand side of equations (23)-(27) are independent of all quantities.

In the general case represented by equations (23)-(27), there is no alternative other than to regard the equations in Table 5 as a set of 2(n[sub 1] + n[sub 2] + n[sub 3] + n[sub a]) independent equations in the 4n[sub 1] + 3(n[sub 2] + n[sub 3] + n[sub a]) variables p[sub a], q[sub a], pi[sub a] and pi[sub 2] plus

p[sub i], q[sub i] and phi[sub i] for i = 1, 2 and 3, where n[sub i] is the number of type i commodity accounts and n[sub a] is the number of activities. Solutions of this system of equations will normally be possible, given an appropriate set of exogenous or predetermined values for some of the variables. However, there is no evident structure to the system and, in particular, there is no possibility of solving the system for prices which are independent of scale (or of quantities).

Such a possibility does emerge, however, when intermediate technology is linear homogeneous. Specifically, in the linear homogeneous case, the price equations in

Table 5 can be written as

p[sub a] = S[sub 1a]' (p[sub a]) p[sub 1] (28)

p[sub a] = L[sub 2a]' (p[sub 2], p[sub 3], pi[sub a]) p[sub a] + L[sub 3a]' (p[sub 2], p[sub 3], pi[sub a]) p[sub 3] + pi[sub a] (29)

p[sub 2] = L[sub 12]' (p[sub 1], pi[sub 2]) p[sub 1] + pi[sub 2] (30)

p[sub 3] = L[sub 23]' (p[sub 2]) p[sub 2] (31)

which defines a set of 2n[sub a] + n[sub 1] + n[sub 2] equations in the 2n[sub a]+ n[sub 1] + 2n[sub 2] + n[sub 3] variables P[sub a], P[sub 1], P[sub 2], p[sub 3], pi[sub a] and pi[sub 2].

Some implications of equations (28)-(31) are illustrated in Figure 1, which shows the pattern of interpendence between the various price vectors, as implied by these relationships. The figure shows that there are alternative ways of evaluating P[sub a]. One of these is defined by equation (28), while the other possibility is

p[sub a] = (L[sub 2a]' L[sub 12]' + L[sub 23]' L[sub 12]') p[sub 1] + (L[sub 2a]' + L[sub 3a]' L[sub 23]') pi[sub 2] + pi[sub a] (32)

This can be obtained from equation (29) by eliminating the variables P[sub 2] and P[sub 3], according to the expressions provided by equations (30) and (31).

The important implication of equations (28) and (32) is that the model can have a solution, if and only if

(S[sub 1a]'- L[sub 2a]' L[sub 12]' - L[sub 3a]' L[sub 23]' L[sub 12]') p[sub 1) = (L[sub 2a] + L[sub 3a] L[sub 23]) pi[sub 2] + pi[sub a] (33)

This is a general result. However, there are two specific examples of it which are of particular interest.

First, if the na equations defined by the condition in equation (33) can be solved for pi[sub a], given p[sub 1] and pi[sub 2], then it will be possible, for given values of p[sub 1] and pi[sub 2], to solve recursively for all other prices in the system, i.e. for P[sub a], P[sub 2] and P[sub 3]. Also, it can be noted here that a solution of equation (33) for pi[sub a], given p[sub 1] and pi[sub 2], will always be possible when the coefficient matrices L[sub 2a] and L[sub 3a] are independent of pi[sub a].

A second situation of some interest arises when equation (33) can be solved for p[sub 1] as a function of pi[sub 2] and pi[sub a]. This is because, in this event, all the prices p[sub a], p[sub 1], P[sub 2] and P[sub 3] can be determined for given values of pi[sub a] and pi[sub 2]. The point to note in this case is that such a solution will not generally be possible. The condition in equation (33) defines a set of (non-linear) restrictions on the vector p[sub 1] for given pi[[sub a] and pi[sub 2]. However, the vector p[sub 1] has n[sub 1] elements and, as a general rule, there is no reason to expect n[sub 1] to be equal to n[sub a], or for the restrictions in equation (33) to be interdependent. Hence, a solution of equations (28)-(31) for the prices p[sub a], p[sub 1], P[sub 2] and P[sub 3], given pi[sub a] and pi[sub 2], will not normally be possible.

Given a set of prices p[sub a], p[sub 1], p[sub 2], P[sub 3], pi[sub a] and pi[sub 2] which satisfy the restrictions in equations (28)-(31), each of the matrices L[sub ij] and S[sub 1a] will be determined in the linear homogeneous intermediate technology model. Therefore, the solution of the model for quantities requires a solution of the four quantity equations

q[sub 1] = S[sub 1a]q[sub a] (34)

q[sub 1] = L[sub 12]q[sub 2] + phi[sub 1] (35)

q[sub 2] = L[sub 2a]q[sub a] + L[sub 23]q[sub 3] + phi[sub 2] (36)

q[sub 3] = L[sub 3a]q[sub a] + phi[sub 3] (37)

and these can be interpreted as linear equations in the various quantity vectors, since the coefficient matrices which appear in these equations are all predetermined by prices.

A condition for the existence of a solution to equations (34)-(37) must be that the expression for q[sub 1] provided by equation (34) is consistent with the alternative expression provided by equation (35) after substituting out q[sub 2] and q[sub 3] according to equations (34) and (37). Hence, a necessary condition for a solution is that

(S[sub 1a] - L[sub 12]L[sub 2a] - L[sub 12]L[sub 23]L[sub 3a]) q[sub a] = phi[sub 1] + L[sub 12]phi[sub 2] + L[sub 12]L[sub 23]phi[sub 3] (38)

This condition can always be satisfied in the sense that, for given prices and, therefore, for given coefficient matrices, the vector phi[sub 1] can always be determined for given values of the vectors q[sub a], phi[sub 2] and phi[sub 3]. Hence, the quantity side of the linear homogeneous intermediate technology model can always be solved for quantities q[sub 1], q[sub 2], q[sub 3] and phi[sub 1], given prices which satisfy the condition in equation (33) and predetermined values of the vectors q[sub a], phi[sub 2] and phi[sub 3].

Similarly, if the condition in equation (38) can be solved for q[sub a], given prices and the vectors phi[sub 1], phi[sub 2] and phi[sub 3], then it is straightforward to show that the model can be solved for all the quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3]. The necessary condition in this case, with the coefficient matrices predetermined, is simply that the matrix

[(S[sub 1a] - L[sub 12] L[sub 2a] - L[sub 12] L[sub 23] L[sub 3a])[sup -1]] (39)

should exist for all admissible constellations of prices.

Finally, to complete the analysis, it can be noted that it is possible to formulate a model in which not only are the prices p[sub a], p[sub l], P[sub 2] and P[sub 3] uniquely determined by pi[sub a] and pi[sub 2], independent of all quantities, but also the quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3] are uniquely determined, independent of prices, for given phi[sub 1], phi[sub 2] and phi[sub 3]. It is evident from Table 5 and equations (23)-(27) that such a model must be a special case of the linear homogeneous intermediate technology model, in which each of the matrices L[sub ij]. and S[sub 1a] is fixed independent of prices (and quantities), i.e. a linear model. Moreover, this linear model must be such that the inverse matrix defined by (39) exists. In addition, from the condition in equation (3), it follows that, in such circumstances, it will always be possible to solve for p[sub 1] as a function of pi[sub a] and pi[sub 2]; hence, it will be possible to solve for all the prices P[sub a], P[sub 1], P[sub 2] and P[sub 3] as a function of pi[sub a] and pi[sub 2].

Since the Stone model can be interpreted as a reduced form of the linear model corresponding to the SAM shown in Table 4, it follows that Stone's model is the most general model of commodity balances that is consistent with the commodity technology assumption and allows for all the prices P[sub a], P[sub 1], P[sub 2] and P[sub 3], and all the quantities q[sub a], q[sub 1], q[sub 2] and q[sub 3] tO be determined independently of each other as functions of pi[sub a] and pi[sub 2], on the one hand, and phi[sub 1], phi[sub 2] and phi[sub 3] on the other hand.

Table 1. A basic social accounting matrix Commodities Activities Basic Final All other accounts Totals

Activities 0 T[sub a1] 0 0 y[sub a]

Commodities

Basic 0 0 T[sub 12] f[sub 1] y[sub 1]

Final T[sub 2a] 0 0 f[sub 2] y[sub 2]

All other accounts v'[sub a] 0 v'[sub 2] -- v=f

Totals y'[sub a] y'[sub 1] y'[sub 2] f=v

Table 2. A developed version of Table 1

Information is presented in the following order: Activities; Commodities, Basic; Commodities, Final; Commodities, Composite; All other accounts; Totals

Activities; 0; q[sub a]S'[sub 1a]p[sub 1]; 0; 0; p[sub a]q[sub a]

Commodities

Basic; 0; 0; p[sub 1]L[sub 12]q[sub 2]; p[sub 1]phi[sub 1]; p[sub 1]q[sub 1]

Final; p[sub 2]L[sub 2a]q[sub a]; 0; 0; p[sub 2]phi[sub 2]; p[sub 2]q[sub 2]

All other accounts; q'[sub a]pi[sub a]; 0; q'[sub a2]pi[sub 2]; --; --

Totals; q'[sub a]p[sub a]; q'[sub 1]p[sub 1]; q[sub 2]p[sub 2]; --

Table 3. An extended version of Table 1, introducing composite commodity accounts

Information is presented in the following order: Activities; Commodities, Basic; Commodities, Final; Commodities, Composite; All other accounts; Totals

Activities; 0; T[sub a1]; 0; 0; 0; y[sub a]

Commodities

Basic; 0; 0; T[sub 12]; 0; f[sub 1]; y[sub 1]

Final; T[sub 2a]; 0; 0; T[sub 23]; f[sub 2]; y[sub 2]

Composite; T[sub 3a]; 0; 0; 0; f[sub 3]; y[sub 3];

All other accounts; v'[sub a]; 0; v'[sub 2]; 0; --

Totals; y'[sub a]; y'[sub 1]; y'[sub 2]; y'[sub 3]; --

Table 4. An alternative version of Table 3, based on the commodity technology assumption and the law of one price

Information is presented in the following order: Activities; Commodities, Basic; Commodities, Final; Commodities, Composite; All other accounts; Totals

Activities; 0; q[sub a]S'[sub 1a]p[sub 1]; 0; 0; 0; p[sub a]q[sub a]

Commodities

Basic; 0; 0; p[sub 1]L[sub 12]q[sub 2]; 0; p[sub 1]phi[sub 1]; p[sub 1]q[sub 1]

Final; p[sub 2]L[sub 2a]q[sub a]; 0; 0; p[sub 2]L[sub 23]q[sub 3]; p[sub 2]phi[sub 2]; p[sub 2]q[sub 2]

Composite; p[sub 3]L[sub 3a]q[sub a]; 0; 0; 0; p[sub 3]phi[sub 3]; p[sub 3]q[sub 3]

All other accounts; q'[sub a]pi[sub a]; 0; q'[sub 2]pi[sub 2]; 0; --; --

Totals; q'[sub a]p[sub a]; q'[sub 1]p[sub 1]; q'[sub 2]p[sub 2]; q'[sub 3]p[sub 3]; --

Table 5. The balance equations implied by Table 4

Information is presented in the following order: Account; Row equation; Column equation

Activities; p[sub a] = S'[sub 1a]p[sub 1]; p'[sub a] = p'[sub 2]L[sub 2a] + p'[sub 3]L[sub 3a] + pi'[sub a]

Commodities

Basic; q[sub 1] = L[sub 12]q[sub 2] + phi[sub 1]; q'[sub 1] = q'[sub a]S'[sub 1a]

Final; q[sub 2] = L[sub 2a]q[sub a] + L[sub 23]q[sub 3] + phi[sub 2]; p'[sub 2] = p'[sub 1]L[sub 12] + pi'[sub a]

Composite; q[sub 3] = L[sub 3a]q[sub a] + phi[sub 3]; p'[sub 3] = p'[sub 2]L[sub 23]

All other accounts; q'[sub a]pi[sub a] + q'[sub 2]pi[sub 2] = p'[sub 1]phi[sub 1] + p'[sub 2]phi[sub 1] + p'[sub 2]phi[sub 2] + p'[sub 3]phi[sub 3]

References

Armington, P. (1969) A theory of demand for products distinguished by place of production, IMF Staff Papers, 16, pp. 159-178.

Pyatt, G. (1994) Modelling commodity balances: a derivation of the Stone model, Economic Systems Research, 6, pp. 5-20.

Samuelson, P. A. (1951) Abstract of a theorem concerning substitution in open Leontief models, in: T. C. Koopmans (ed.) Activity Analysis of Production and Allocation, Cowles Commission Monograph 13 (Wiley, New York).

United Nations, Statistical Office (1968) A System of National Accounts, Studies in Methods, Series F, No. 2, Rev. 3 (United Nations, New York).

*Опубликовано 19 декабря 2004 года*

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