A follow-up note on the plausibility of the Leontief and Ghosh closed models

ABSTRACT Herein we consider Leontief and Ghosh models that partly endogenize both part of final demand and part of value-added. We use Osterhaven's [(2012) Adding Supply-driven Consumption Makes the Ghosh Model Even More Implausible. Economic Systems Research, 24, 101–111] numerical three-sector example to show that anomalies of the sort he finds for a Ghosh closed model can also be found in the closed version of a Leontief model. By assuming, as Oosterhaven did, that aggregate exogenous resources are fixed, we obtain mirror results to his in a Ghosh setting, albeit in the more-traditional Leontief instance. Such numerical anomalies for the three-sector case turn out to be generic to both partially closed models for any 2×2 input–output model. A proof for the general n×n case remains to be uncovered.


Introduction
extend Ghosh's model to include the income feedbacks between valueadded and consumption demand in a manner which is, mathematically speaking, fully equivalent to Leontief's closed model when private consumption is made endogenous. In Guerra and Sancho (2011) value-added is seen to grow in all sectors along with their output, which overcomes the usual implausibility found in such model. In turn, Oosterhaven (2012) uses the same numerical 3x3 input-output table from Guerra and Sancho (2011) and introduces an additional rule in the exogenous variables (i.e. value-added) of the closed Ghosh model. This rule consists in performing a neutral redistribution among sectors of the fixed exogenous aggregate resources, which we can interpret as the implementation of some policy directive. Oosterhaven (2012) solves the model under the neutral redistribution and finds an inverse relationship between value-added and output: less value-added may go hand in hand with more output, both at the sectorial and aggregate levels. From this result, Oosterhaven (2012) concludes that the closed model is even more implausible than the standard Ghosh model.
The purpose of this note is to show that the same type of anomalous economic behavior that affects the Ghosh closed also affects the Leontief closed model. We also use a 3x3 input-output table, apply the same neutral redistribution rule to Leontief's closed model-in this case to final demand-and end up observing an inverse relationship between output and endogenous consumption demand, at the sectorial and aggregate levels. That is, less final demand in the economy may also go hand by hand with more output. We may also refer to this mismatch between changes in demand and changes in output as an alleged anomaly, or implausible behavior, of the Leontief closed model.
These findings are not surprising if we take into account that both models, once we close them in relation to households' final demand and value-added ownership, share a common mathematical structure. In fact, in Manresa and Sancho (2013) we show that these alleged anomalies of both models, under the neutrality rule, will always happen in a 2x2 sector Input-Output framework, except in very special cases of the base input-output data that are quite unlikely in empirical terms 1 . If we categorize these models' behavior as implausible or anomalous, this concern requires further scrutiny, and this is what we do in this short note. We believe it is important for input-output researchers and practitioners to be aware of the possibilities, limitations and issues that may affect the models we so much cherish and use. Inasmuch as we keep using these models, we feel it is essential to know and report all their properties, positive or negative. In Section 2 we briefly describe the main features of both extended models. In Section 3 we discuss the nature and the implications of some numerical examples to shed some light on this issue. Section 4 ends the note with some general concluding remarks.

Brief conceptual reminder of the closed Leontief and Ghosh models.
Let us start from an n sector input-output economy described by observed data (x, f, v, Z) where vector x stands for gross output, vector f is final demand, vector v is value-added, and matrix Z stands for intermediate transactions 2 . Using the data and assuming the usual proportionality relationships, we define technical and sales coefficients matrices A=(aij) and B=(bij), respectively, by way of: (1) In the simple Leontief model matrix A links final demand f with output x through the linear relationship: (2) Similarly, in the simple Ghosh model we would have matrix B linking value-added v with output x by: (3) Notice that value-added v is not present in expression (2) and hence plays no role in output determination; similarly for f in (3). This endogeneity gap can be closed through the closed models 3 . However, since no economic model can be completely closed 4 , we will distinguish within f and v an endogenous part and an exogenous remainder, say 'private' (endogenous) and 'public' (exogenous) final demand and value-added. Therefore, we make , and , with index '1' standing for 'private' and '2' for 'public'. In Leontief's closed model, private demand f 1 is made endogenous in response to changes in v triggered by changes in x, whereas in the Ghosh closed model v 1 is made endogenous and dependent on changes in f, which are in turn triggered by any changes happening in x. Let us denote these relationships by in Leontief, and in Ghosh. Substituting them in (2) and (3) we obtain: The notational convention is that y represents a column vector and yʹ a row vector.
3 See Leontief (1941Leontief ( , 1953 for a description of the closed model. 4 Leontief (1986) concludes that we cannot truly close static models, hence the need for some exogenous driving force such as investment, government demand or net exports.
We can solve these two expressions for equilibrium solutions in both closed models and compare initial and counterfactual allocations.

Oosterhaven's neutrality rule
For the Ghosh closed model Guerra and Sancho (2011) show that changes in the exogenous part of value-added imply not only that output increases everywhere-as expected-but that total value-added increases in all sectors as well, thus breaking the implausibility of the simple Ghosh model regarding unresponsive value-added to changes in output. In differential terms this result could be expressed as: implies and (4) That is, any positive exogenous change in sector k value-added will generate non-negative increases in output and total value-added in all sectors.
In the example of Oosterhaven (2012) for the Ghosh model, the use of the neutral reshuffling rule keeps constant the level of aggregate exogenous value-added. He takes two sectors, say sectors n and m, and assumes that sector n has 5 less units of the exogenous resource whereas sector m has 5 more units. Total use is therefore constant in the aggregate. Notice the possible policy interpretation of these changes. A direct application of Guerra and Sancho's (2011) result from expression (4) yields: (4′) If we add up the individually correct implications for sectors n and m, we can check that the overall combined effects do not have any definite sign; anything goes. Observe, however, that all that is taking place here is the summation (thanks to the separability implicit in the linear models) of two independent general equilibrium adjustments that work in the opposite direction of each other. The implausible behavior that Oosterhaven (2012) describes comes from the interplay, within the structure of the linear model, of two equilibrium shocks working in opposite directions under the constraint of constant aggregate resources.
The same would occur verbatim in the Leontief extended model. It is mathematically straightforward to derive the counterpart of Property (4) for the Leontief closed model version: For the same type of neutral rule, if we apply it to the exogenous part of final demand in sectors n and m and we use identical values as above we will now conclude:  Notice from expression (5′) that, once again, the application of the neutral rule can give rise, a priori, to any sign in Δxi and in Δfi. As long as signs are concerned, therefore, we may expect that the same type of implausible behavior of the endogenous variables could take place in the close Leontief model-and for the exact same formal reasons-as in the closed Ghosh model.
We now move to the numerical examples constructed using the neutrality rule.

Numerical examples based on the Guerra and Sancho I-O table and its transposed
The highlights of this section are that, under the neutrality rule, we will observe in the Ghosh closed model that an increase in total production in one sector goes along with a decrease in its total value-added. Similarly, we will observe in the Leontief closed model sectors where an increase in total production is accompanied by a decrease in final demand. (2011) Table 2) calculates his results, which we reproduce as Table 1 here. These numerical results lay down the foundations of his criticism to the closed Ghosh model. The results of a change on production, ∆x, value-added, ∆v, and final demand, ∆f, stem from the implementation of the neutral resource shift in two of the sectors. He concludes that the equilibrium magnitudes of the closed Ghosh model present an anomalous behavior. Namely, that output and value-added move in different directions. For instance, the addition of five units of the public input in sector 2 together with a reduction of five units in sector 3 yields Oosterhaven's last block of columns in Table 1. In this block of numbers, we clearly see that at the sectorial level, more output goes along with less value-added in sectors 1 (+0.63 vs. -0.17) and 2 (+4.82 vs. -0.09), and the same happens at the aggregate level (+1.92 vs. -0.60). From this particular anomaly in this simulation, Oosterhaven (2012) concludes that the closed model of Ghosh is even more implausible than the simple Ghosh model.

Using Guerra and Sancho
[ Table 1 around here]

Implausible behavior in the Leontief closed model.
Following the same experiment performed by Oosterhaven for the Ghosh closed model, we now focus on our symmetrical reasoning as applied to Leontief's closed model. We build the results in Table 2 using the transposed of the input-output table of Guerra and Sancho (2011) (see the Appendix). We then use the same neutral redistribution rules as before and we obtain (by simple duality) the same numbers as in Table 1 with no need for actual new computations.
Looking at the second block of numbers in Table 2 (from the left) gives us the Leontief counterpart to the data for Ghosh in Table 1. We can see the same numbers as in the Ghosh case, simply because of the transposition of the I-O table, but they are now in two different columns. We see that final demand for goods 1 and 2 goes down (-0.17 and -0.09) whereas their respective sectorial output goes up (+0.63 and +4.82). Even worse, total output increases (+1.92) even though aggregate demand goes overall down (-0.60), along with the same happening in aggregate value-added (-0.60). There is less demand for goods in general (-0.60) and less value-added (-0.60) around but, even so, the economy ends up producing more goods (+1.92). The behavior of the aggregate variables in the Leontief closed model mimics what we observe happens as well in Ghosh's closed model. For the same reasons, we may therefore conclude that such behavior is implausible as well. [

Final remarks
Remark 1. We show that the anomalous economic behavior reported by Oosterhaven (2012) for the closed Ghosh model has a parallel counterpart in the Leontief closed model. In the model of Ghosh, we see that more aggregate, or sectoral, output may go hand in hand with less value-added. Correspondingly, in the closed Leontief model we find that a higher level of aggregate, or sectoral, output may go along with less final demand. The question is if this behavior happens by chance and it is just a reflection of a particular configuration of the I-O data. The tentative answer seems to be no, as the general results for the 2×2 input-output case shows and the 3x3 examples discussed here suggest. For the n×n general I-O framework, we need to develop further research either by generalizing (or eventually disproving) the 2×2 results, or else by systematic numerical simulations that would allow to ascertain the nature and prevalence of these types of effects.
Remark 2. These results, however provisional, give us a warning since they pinpoint some of the limitations of the research that uses these models either for theoretical or empirical purposes. Of particular relevance are the possible consequences for public policy design. Think of the government implementing a new policy under some aggregate resource restriction, like redirecting public expenditure from sector n to sector m under a fixed budget constraint, for whatever reason.
Our results as well as Oosterhaven's indicate that policy makers should be aware of the unintended and possible perverse outcomes, contrary to the purported policy goals, that may ensue.  Source: From "transposing" the I-O data from Table 2a in Guerra and Sancho (2011).