Leontief versus Ghoshian price and quantity models.

• Author: Oosterhaven, Jan

1. Introduction

   The classical, "demand-driven" input-output model dates back to Leontief [24; 25]. Its economically complete opposite, the so-called "supply-driven" input-output model was developed almost forty years ago by Ghosh [16], whereas its first application came more than ten years later by Augustinovics [1], followed by Giarratani [18]. The first theoretical reservations were formulated another ten years later by Giarratani [19] and Oosterhaven [31, 140-1].

   Despite these reservations, the model came more or less en vogue in the 1980s. Several applications appeared; at best paying only lip service to this earlier critique [10; 5]. Moreover, without much reservations the model also entered the major input-output textbook [28, 317-22], while uncritical generalizations and reiterations appeared in the theoretical literature [8; 3; 9]. Then a heavy discussion about the economic underpinning of the supply-driven quantity model was initiated by Oosterhaven [33; 36; 21; 34]. With that debate the enthusiasm to perform downright applications of the supply-driven model seems to have diminished.

   Literally in the footnotes of that debate, the (non-)existence, the mathematics and the possible economic interpretation of the dual, price version of the supply-driven model was discussed [33; 34]. Independently of this discussion, Davar [9] presented the dual model in extension, but did not discuss the economic plausibility of the underlying economic assumptions nor their implications. In view of the misinterpretations and misuse of the quantity version of the supply-driven model, this contribution aims at an early evaluation of its price version that is as complete and as balanced as possible.

   It is well known that the dual version of the regular input-output model is used to simulate cost-push inflationary processes [26, 188-201; 7, 246]. Other applications relate to, for instance, the price effects of pollution abatement [17], effects of rising energy cost [29], and energy price effects in multiregional or in extended interregional models [35; 32]. Of course, this dual, price version of the Leontief model is based on the same stringent, standard assumptions of the quantity version and on some heavy, additional assumptions that are made for the price version [37].

   Naturally, the potential use of the dual of the supply-driven model lies in the simulation of demand-pull inflationary processes as opposed to the cost-push applications of the dual Leontief model. Hence, part of our evaluation will be directed toward the question whether such applications are justified from an economic point of view or not.

   Our discussion will start in the next section with a brief summary of the Leontief quantity and price models and their mutual relationship. Next, both Ghoshian models will be presented, followed by a discussion of their economic plausibility. Our essentially negative verdict is combined with some concluding remarks in the last section.

2. The Leontief Quantity and Price Model in Brief

   For both the Leon fief and the Ghoshian input-output models we assume a closed economy with I intermediate sectors (i.e., industries), Q final sectors (i.e., final demand categories) and P primary sectors (i.e., value added categories). Both models start from the same base year input-output table on which their accounting identities are based.

   The mathematics of the well known Leontief model may be summarized in two sets of equations. First, the base year accounting identities for total output, across the rows of the i-o table, with all (index) prices standardized at one:

   x = Zi + Yi = Zi + y (1)

   and, second, the behavioral relations with fixed intermediate and primary input coefficients, derived from the columns of the i-o table:

   [Mathematical Expression Omitted]

   where:

   x = I-vector with total output (= total input) per industry,

   Z = I x I-matrix with intermediate outputs (= intermediate inputs),

   Y = I x Q-matrix with final outputs,

   V = P x I-matrix with primary inputs,

   i = vector of the appropriate size with ones, i.e., a summation vector,

   ^ = diagonal matrix made from the corresponding vector,

   A = I x I-matrix with fixed intermediate input coefficients,

   C = P x I-matrix with fixed primary input coefficients.

   The demand-driven model (1)-(2) with its exogenous final output has the following well known solutions for endogenous total output:

   x = [(I - A).sup.-1]y = Ly (3)

   and endogenous intermediate and primary inputs:

   [Mathematical Expression Omitted]

   where: L = I x I so-called Leontief-inverse.

   The price version of the Leontief model is almost equally well known [7; 37]. We will call it the cost-push i-o price model in order to make a clear distinction with its demand-pull opposite.(1) The cost-push price model is based on the following accounting identity for the values of sectoral total inputs (i.e., total cost):

   [Mathematical Expression Omitted]

   where:

   p = I-vector with (index)prices for total sectoral output,

   [p[prime].sub.v] = P-vector with (index)prices for primary inputs per category.

   Next, the price version, just like the quantity version, assumes fixed input coefficients. Substitution of (2) in (5) and post-multiplication by [Mathematical Expression Omitted] gives the unit costs per sector (which under perfect competition equals the prices for total output) as the weighted average of the prices for the intermediate and primary inputs:

   p[prime] = p[prime]A + [p[prime].sub.v]C. (6)

   Note that the sum of the weights for each industry equals one: i[prime]A + i[prime]C = i[prime], as the input accounting identities in (9) are measured in unit price-indices of the base-year. Hence, for this base-year: p[prime] = i[prime] = [p[prime].sub.v].

   The essence of the Leontief price model lies in the additional assumptions regarding the causal relationships between these prices. The primary input prices (homogeneous per row) are assumed to be exogenous, whereas the I prices for the single, homogeneous outputs are determined by the solution of the model:

   p[prime] = [p[prime].sub.v]C[(I - A).sup.1] = [p[prime].sub.v]CL. (7)

   The economic interpretation of (7) is simple. Primary input prices [p[prime].sub.v] may change exogenously. The size of the direct effect of such changes on endogenous unit cost (= output prices p[prime]) is determined by the fixed primary input coefficients C. Next, the prices of intermediate inputs rise as firms compensate their unit cost increases in their output prices. The subsequent endogenous increases in intermediate input prices again cause...