1 – Introduction
1The Leontief Input-Output model (Leontief 1986; Miller and Blair 2009) remains one the most useful in international and national accounting (BEA 2008; United Nations 2008) as well as in regional economics, even when it is used in the form of a SAM (Stone and Brown 1962; Pyatt and Thorbecke 1976; Pyatt and Round 1985; Round 2003) or CGE (Johansen 1960; Hertel 1998; Cardenete et al. 2012; Turner et al. 2012) model,  and even in environmental economics (Leontief 1970, 1972, 1973; Perese 2010; Grainger and Kolstad 2010; Kagawa 2012; Martin and Point 2012; Napoles 2012). This is why it is important to understand what is “behind” Leontief’s creation from a theoretical viewpoint. The result is rather surprising as we will see in what follows.
2The monetary model is the core of Leontief’s original contribution: it opens out onto real-world applications, amply justifying his “Nobel Prize”. It is this model which is referred to as the “Leontief model”. Indeed, one of its main innovations is that it uses monetary data (i.e., currency units) instead of physical data, which was a stroke of genius! This innovation means different commodities (e.g. “Steel” and “Energy”) can be aggregated down a column of the input-output matrix in addition to the obvious aggregation by rows and it means the number of sectors and products can be reduced by aggregation; monetary coefficients (the techno-economic coefficients) can be aggregated by column and their sum is less than one. This makes the monetary model very handy to use in applied studies: the real data extracted from national accounting are compiled in monetary terms: this point explains its success. It should be noticed that the solutions of the Leontief value model (or: Leontief model in currency units) are outputs in currencies and price indexes but not physical outputs and prices. Virtually all input-output and derived models use data in currency units (i.e., values), price indexes, and fixed coefficients defined as ratios of currency units. This type of model has either one square table as in the Leontief model (Leontief, 1936, 1986; Stone 1985; Wheale 1985; Gale, 1989; Miller and Blair, 2009) or two rectangular tables as in the Stone “Make-Use” model (1961) defined in the System of National Accounts (United Nations 2008).
3In the Leontief physical model, data are in physical quantities, fixed coefficients are defined as ratios of physical quantities (each coefficient can be greater than one), and prices are considered; the model’s solutions are outputs in physical quantities or prices. When he derived his model, Leontief began by considering a quantity model in which all units were strictly physical (number of cars produced, tons of steel, gallons of oil, etc.), reflecting actual output.  However, such a model was impossible to handle with the possibilities of computing available in the early twentieth century. Moreover, the physical model cannot readily generate applications because of the heterogeneity between commodities: even if the Leontief physical model actually existed in the form of the Material Product System (MPS) following the denomination of the United States Statistical Office, used by the USSR and China until 1993 (except for Cuba and North Korea that continue to use it), an MPS matrix had many rows and columns—potentially billions—, much more than a Leontief matrix and could be difficult to handle, even with today’s computing power. Potron’s (Abraham-Frois and Lendjel 2006; Bidard 2007, 2009) matrices were also physical matrices with some level of aggregation. While the Leontief value model is able to open out onto real-world applications, the purpose of a physical model is largely theoretical, which is completely different: a physical input-output model is unable to generate applications easily because of the heterogeneity between commodities, which means the input-output tables cannot be compiled from physical data. Notice that physical coefficients (technical coefficients) cannot be aggregated by column and they may exceed one. The physical model today serves only as a theoretical reference. This is why Leontief switched rapidly to a monetary table (Leontief, 1986: 20-22). Baumol and ten Raa (2009, p. 514) consider Leontief’s (1925) initial paper  as the first input-output study. This position is qualified by Akhabbar, relying on Spulber (1964):
“Leontief’s paper is a review of the preliminary work published in 1925 by Popov and Litoshenko on the balance of the USSR. … as emphasized by N. Spulber (1964, p. 50), in 1925 Leontief missed what makes Litoshenko and Popov’s analysis original and that was to resonate in Leontief’s subsequent work, that is to say, the principle of input-output table” [our translation]
“… Nicolas Spulber, acknowledged two statisticians and Soviet economists in particular, L.N. Popov and P.I. Litoshenko, as pioneers, without, however, considering that the invention of input-output analysis was their invention. This is this last avenue which is the most convincing one and is analyzed here.” [our translation]
7The most important thing is that Popov, Litoshenko and Leontief worked on a balance, that is, on currency units.  However, the great merit of Leontief was to recognize that double counting should be eliminated (Akhabbar 2006, p. 16; Baumol and ten Raa 2009, p. 514).
8Anyway, both models—physical and monetary—should be comparable from a theoretical point of view. If the definition and the number of sectors and products are the same in both models, they should always reflect the same reality, that is, they should provide the same results after multiplying quantities by prices (while they should be the same if prices are fixed as underlined by Leontief (1986)). We say that the Leontief value model is coherent only if the solutions of both models are similar for the same definition of sectors and same number of sectors: same outputs—after multiplication by prices for the physical outputs—, and same prices. Consistency is obviously required if we are to have an economically meaningful model. This is why one could wonder under what conditions consistency can be achieved from a theoretical viewpoint (but not from a national accounting viewpoint). We will see that, while consistency is obviously required for outputs, this is not so for prices for which it is conditional.
9We emphasize that we discuss the Leontief model, and not the so-called Ghosh model. The Ghosh model is based on output coefficients. Even if the reader may think that there is an analogy between this approach and that adopted by de Mesnard (2009), the Ghosh model is completely different from the Leontief model. De Mesnard (2009) shows conclusively that the Ghosh model should be rejected. 
10The paper is organized as follows. Section 2 formally derives the Leontief value model and the corresponding Leontief physical model. Section 3 examines under what conditions the two models coincide. Section 4 concludes and section 1 is this introduction. Appendix (5.1) will help the reader by providing a catalog of the main equations used in the paper.
2 – Formal derivation of the models
2.1 – The accounting balances
2.1.1 – Physical closure
11The Leontief model should be physically closed. This is achieved by an accounting equation:  the total of sales is equal to the total output, in physical quantities. Therefore, for each commodity i and each seller i, the total of what is sold to the buyers (including i) is equal to the total of what can be sold. In matrix terms, this is:
13where s is the sum vector (i.e., s′ = (1, …, 1), the prime indicating the transposition operation), Z̅ is the flow matrix (or matrix of intermediary consumptions) in physical terms, xij being the quantity of input i (produced by sector i), absorbed by sector j, f̅ is the final demand vector in physical terms, and x̅ is the output vector in physical terms.
14Closeness is not a strong assumption: it is sufficient to consider that each buyer i buys all that has not been bought by the others. It can be deduced from (1) that x̅ > 0.
2.1.2 – Money closure
15Even if the model is physical, commodities and labor in each sector j have prices, respectively denoted p and pν.  The model is doubly closed. First, for each seller i, the total value of what is sold to all buyers (including i) is equal to the total value of what can be sold, which is the same as the physical closure. In matrix terms, this is p̂Z̅s + p̂f̅ = p̂x̅ ⇔ (1), which is equivalent to
17where p denotes the price vector, Z ≡ p̂Z̅ is the flow matrix in value (i.e., in currency units) and Z ≡ p̂f̅ is the final demand vector in value, the hat (e.g. in p̂) denoting the diagonal matrix formed from a vector (e.g. p), that is,
19Second, for each buyer j, the total value of what is bought from all sellers (including j) is equal to the total value of what can be bought. In matrix terms, this is , which is equivalent to
21where v̅ is the vector of the physical quantities of labor (or of any other factor) used by each sector and v = p̂νv̅.  Any model, Leontief value model or Leontief physical model of similar structure, should follow the balances (2) and (3) (the physical balance (1) is redundant with (2)). Although it might seem more natural to present the Leontief value model first, it is in practice much easier to present the Leontief physical model first.
2.2 – The Leontief physical model
22It is necessary to recall how the Leontief physical model that corresponds to the monetary model is derived (with final demand and only one factor of production and a single technique with complementary inputs, not a general production prices model à la Sraffa). In this model, quantities and prices are explicitly considered.
2.2.1 – The production coefficients
2.2.2 – The model’s equations
25In the primal, when (4) is included, the balance (2) turns out to be
27Notice that the solution
29is expressed in physical terms.
30In the dual, when (4) is included, the balance (3) turns out to be
32where the matrix is the diagonal matrix of labor coefficients . Equation (7) is the dual physical Leontief model, i.e., the physical Leontief price model. Therefore, the prices come naturally as a function of the input labor coefficients multiplied by the factor price:
36is the interindustry matrix of direct and indirect quantities of labor.
37Definition The term j of the row vector
39is the direct and indirect quantity of labor (measured in physical units) incorporated per unit of physical output produced in sector j. This concept is called the vertically integrated coefficients of labor, which for Pasinetti (1977, chap. 5, subsection 2 of the appendix) correspond to the “Marxian values”.
2.3 – The Leontief value model
40In the Leontief value model, physical quantities are never introduced. This model corresponds to the original Leontief model (1936, 1986): after defining a table in physical quantities, Leontief switches rapidly to a table in money (Leontief, 1986: 20-22). We assume that the set of commodities is the same in both physical and monetary models.  The balances (2) and (3) still hold. Technical coefficients are defined in monetary terms (i.e., in currency units) and assumed to be stable (Leontief 1970, 1986; Miller and Blair 2009):
2.3.1 – The primal
42Solving the primal—by rows—generates no difficulties. Plugging (11) into equation (2) implies
44The outputs in currency units are now deduced from final demands in currency units, that is,
46Equations (6) and (13) are the same after premultiplying both sides of (6) by p̂. This is not the case of the dual.
2.3.2 – The dual
47The dual price model should use price indexes instead of prices as it is nonsense to multiply prices by values (i.e., quantities in currency units): we may expect a price model similar to
49There arises here a considerable difficulty. Denote by L = v̂[x̂]–1 the diagonal matrix of labor coefficients , in currency units. If we directly introduce the technical coefficients (11) into the balance (3), this turns out to be an identity: for any j, that is,
51Hence, the dual Leontief value model cannot be derived from (3): this operation ends in deadlock.  In order to get out of this difficulty, prices should be considered. However, from (11), we deduce that . Therefore, the product for any j, i.e., p′A, turns out to be p′p̂A̅p̂–1 = s′[p̂]2A̅p̂–1, where p̂2, i.e., the diagonal matrix of terms p2i, is obviously an economic nonsense. We deduce that it is prohibited to apply prices over technical coefficients.
52Fortunately, there is a classical solution, in dynamics. This consists in considering two distinct periods, the base period (denoted 0) and the current period (denoted T). The variation of prices Δp(0,T) is calculated between period 0 and period T.
53We write the physical Leontief price model, i.e., equation (7), for the base period 0, that is,
55which yields the solution
57A variation Δp(0,T)ν in the prices of labor between the base period and the current period generates a variation in the output prices Δp(0,T) :
59Adding (17) to (18) yields:
65We call p̃(T) Leontief prices (and p̃(T)ν labor Leontief prices). Equation (19) corresponds to the model
67that is, the current Leontief prices p̃(T) are derived from the base prices of labor p̃(T)ν and from the physical coefficients of the current period, i.e., production structure A̅(0) and physical structure of labor costs L̅(0). 
68Proposition 1 The Leontief price model (14) turns out to be
70which is similar to (14) but where
73Equation (22) is the Leontief price model where index prices multiply technical coefficients and labor coefficients, both in currency units.
74Proof. By rewriting equation (21) as
76we obtain (22).
77With price indexes the product π̃′A(0) becomes meaningful. From (22), as A(0) and L(0) are assumed to be stable, one deduces that the Leontief price indexes π̃ are formed from the Leontief labor price index, π̃ν, but by using the monetary production structure and the monetary labor coefficients of the base period: π̃′ = π̃′νL(0) [I – A(0)]–1. 
3 – Dual Leontief value model vs. dual Leontief physical model
79We compare the dual of the Leontief value model, (21) and (22), to the dual of a Leontief physical model based on a physical matrix for period T. We rewrite (7), for the current period:
81which yields the solution
83Equation (25) is similar to (8) for the current period T and defines the true (current) prices. These prices can be compared to the Leontief prices given by (17). Notice that equation (25) only differs from (17) by the fact that A̅(T) and L̅(T) replace A̅(0) and L̅(0) respectively.
84Now, we have the tools with which to answer the following question: Under what conditions is the Leontief value model a Leontief physical model? That is, under what conditions are the Leontief prices equal to the true current prices? We assume that the vector of labor prices is the same in both models, i.e., ; the price of labor being exogenous in the context of the input-out model, this is a normal hypothesis: from (20), it simply means that we are able to correctly forecast the variation Δp(0,T)ν of the price of labor from its value p(0)ν in the base period.
85Remark The matrix L̅(t) [I – A̅(t)]–1 is called the interindustry matrix of direct and indirect quantities of labor (measured in physical units) incorporated per unit of physical output because it holds that
90Definition The dual Leontief value model is coherent if it is the same as the dual Leontief physical model, that is, if the Leontief prices are equal to the current prices: p̃(T) = p(T).
91Theorem 1 The Leontief value model is coherent if and only if the interindustry matrix of direct and indirect quantities of labor is stable over time, that is:
93Proof. From (26) we deduce that
95As labor is exogenous in the Leontief value model, we can posit the assumption that p(0)ν = p̃(0)ν. Therefore (27) turns out to be
99Corollary 1 If the Leontief value model is coherent, then the vertically integrated coefficients of labor are stable over time, that is,
101Proof. The proof is obvious by premultiplying (26) by s′.
102The stability over time of the vertically integrated coefficients of labor means that any sector j needs as much labor to produce one physical unit of commodity in the current period as in the base period: there are no productivity gains with respect to labor. The Leontief production function is linear but nothing would a priori prevent the evolution of the coefficients from reflecting such productivity gains; corollary 1 excludes them. Notice that assuming or ensuring the matrix L̅(t) [I – A̅(t)]–1 is stable is much stronger than assuming the vector s′L̅(t) [I – A̅(t)]–1 is stable. Indeed, L̅(t) [I – A̅(t)]–1 being stable implies that s′L̅(t) [I – A̅(t)]–1 is stable but the reciprocal proposition is false.
103The cases where equation (26) holds can be qualified as pseudo physical stability. However, true physical stability, that of A̅ and L̅, implies pseudo physical stability, as shown by the following corollary:
104Corollary 2 If two of the three following properties hold, the third one holds also:
106Proof. If L̅(0) = L̅(T) and A̅(0) = A̅(T) hold simultaneously, then (26) holds. If L̅(0) = L̅(T) and (26) hold simultaneously, then
108If A̅(0) = A̅(T) and (26) hold simultaneously, then L̅(0) = L̅(T).
109Corollary 3 p̃(T) = p(T) holds if all physical coefficients L̅ and A̅ are stable:
111Proof. The proof follows directly from corollary 2.
112Assuming that the physical production coefficients are stable over time, i.e., L̅(0) = L̅(T), is a very strong and rather unrealistic hypothesis. In this case, the possible study of the structural change in the production matrix (i.e., detecting which coefficients have changed between years 0 and T) becomes irrelevant: the physical structural change becomes equal to zero (A̅(0) = A̅(T) by hypothesis) and the structural change in currency units turns out to be merely a price effect.  Conversely, the following proposition can be derived:
113Proposition 2 When the distance between the base period and current period comes to zero then the Leontief value model always turns out to be coherent.
114Proof. When T → 0, the coefficients do not change: A̅(0) = A̅(T) and L̅(0) = L̅(T), which implies that L̅(0) [I – A̅(0)]–1 = L̅(t) [I – A̅(t)]–1.
115Moreover, the prices do not change when T → 0 : Δp(0,T)ν → 0, which implies that p̃(T)ν = p(0)ν. Therefore, Δp(0,T)ν → 0 and p̃(T) = p(0).
4 – Conclusion
116The Leontief input-output model remains very important today (it is the basis of national accounting, SAM and CGE models, and most environmental analyses). After recalling the accounting equations that guarantee the physical and money closure of the economy, we have first recalled (i) what the physical Leontief model—with its production coefficients”, its “primal” and its “dual” price model—is, and (ii) what the Leontief value model—with its famous technical coefficients, its equally famous “primal” model and its “dual” price model—is. As the physical model requires too much data, it is not realistic to use it. Wisely, Leontief chose to use the monetary model. However, the “dual” price model should use price indexes instead of prices as it is nonsense to multiply prices by values (i.e., quantities in currency units), which generates a considerable difficulty.
117We show that the monetary price model cannot be derived unless we consider two periods of time, a base period and a current period. This forces us to redefine the prices as what we call Leontief prices, i.e., the base prices inflated by the variation in prices between the base period and the current period. This leads to a model in which price indexes replace prices: the true Leontief price model uses price indexes—defined as the variation of prices with respect to the base period—along with the technical coefficients of the base period. 
118However, the true Leontief price model is generally not coherent, unless a very strong, quite classical, assumption is made: the interindustry matrix of direct and indirect quantities of labor incorporated per unit of physical output is stable over time. This assumption is satisfied when two of the three following very strong conditions hold: (i) the vertically integrated coefficients of labor are stable, (ii) the production coefficients in physical terms are stable, (iii) the physical coefficients of labor are stable over time. This implies that, if we want the Leontief model to be coherent, we probably have to abandon most of the analyses usually conducted such as technical change.
119There lies the “trick”: if “magic Leontief” had a brilliant idea when he created his “Leontief model” (i.e., the Leontief value model), his idea involves a “trick” that nobody has spotted until now. Leontief never revealed his trick: one may say that it is unfair but a magician never reveals what is really placed in the hat and where the rabbit comes from. Was Leontief aware of the existence of his trick? It is unfortunately too late to ask him but if the answer is “no”, Leontief can be compared to a poet. A poet writes beautiful verse that we love to read but often he is unaware of the message he really wants to put across: the poem is enchanting; that in itself is sufficient for our eyes and ears.
5 – Appendix
5.1 – List of equations
5.1.1 – Physical model
120• Equation (1): physical closure
122• Equation (2): first monetary closure
124• Equation (3): second monetary closure
126• Equation (4): matrix of production coefficients defined in physical terms
128• Equation (5): primal physical Leontief model
130• Equation (6): solution of the primal physical Leontief model
132• Equation (7): dual physical Leontief model, i.e. physical Leontief price model
134• Equation (8): solution of the physical Leontief price model
136• Equation (9): interindustry matrix of direct and indirect quantities of labor
138• Equation (10): vertically integrated coefficients of labor
5.1.2 – Value model
140• Equation (11): technical coefficients
142• Equation (12): traditional Leontief model (i.e., primal, in currency units)
144• Equation (15): identity found from the dual Leontief model
146• Equation (21): Leontief dual model with prices
148• Equation (22): so-called Leontief price model, i.e., Leontief dual model with price indexes.
150• Equation (23): diagonal matrix of labor coefficients in currency units for the base year
152• Equation (24): Leontief physical price model for period T
5.2 – About the Ghosh model
154In de Mesnard (2009) we stated four propositions:
(1) Only value outputs [i.e., output in currency units] are found in the Ghosh physical model, not quantities or prices, while in the Leontief physical model, outputs in physical quantities are found in the primal and prices in the dual.
(2) What is considered as the equation of the primal Ghosh value model, that is, x′B + v′ = x′, is actually that of the Ghosh physical model.
(4) The Leontief value model and the Ghosh value model provide the same price indexes.
159We added (p. 369):
Consequently, the Ghosh value model is redundant because the dual of the Leontief value model … does the job as well in a simpler and more natural way (the Leontief model is a production prices model!).
(i) The Ghosh model offers solutions of limited interest, being incapable of separating quantities and prices or values[i.e., quantities in currency units] and price indexes: we have to forget any idea of prices or price indexes with the Ghosh model.
(ii) There is a misunderstanding about the so-called equation of the Ghosh value model: it is actually that of the Ghosh physical model. Hence, it is wrong to compare the primal of the Leontief [value] model, Ax + f = x, to the Ghosh value model, x′B + v′ = x′.
(iii) … our methodology—introducing prices or price indexes by returning to accounting equations—demonstrates that the familiar dual Leontief model of production prices performs the same task [studying propagation of cost variations upon prices] in a much simpler way.
5.3 – Inversion of periods 0 and T
163When we reverse the role of periods 0 and T, (17) turns out to be
165A variation Δp(T,0)ν in the prices of labor generates a variation in the output prices Δp(T,0) :
167Adding (28) to (29) yields:
169by denoting p̃(0) = p(T) + Δp(T,0) and p̃(0)ν = p(T)ν + Δp(T,0)ν. That is, the current Leontief prices p̃(0) are derived from the current prices of labor p̃(T)ν but also from the physical coefficients of the base period, i.e., production structure A̅(T) and physical structure of labor costs L̅(T). This leads to the following model
171And we also obtain
173by denoting for commodities and for labor. That does not change Theorem 1 and Corollaries 1, 2, and 3, or Proposition 2.
I am grateful to an anonymous referee for comments on the initial submission of this paper.
Univ. Bourgogne Franche-Comté, CREGO (EA 7317); IAE Dijon, 2 Bd Gabriel, 21000 DIJON, FRANCE. E-mail: firstname.lastname@example.org.
SAM stands for Social Accounting Matrix and CGE for Computable General Equilibrium.
This version of the Leontief model can be viewed as a simplified avatar of the classical model of “production prices” (Ricardo, Marx, Sraffa, etc.; see Pasinetti 1977).
See also Leontief (1977).
See also Kohli (no date).
See Appendix 5.2.
This physical closure is not to be confused with what Leontief names his “closed” model. We limit all the discussions to the familiar “open” model in which the final demand is exogenous and there is only one exogenous factor. Situations of more than one factor are not considered. However, the discussions can easily be transposed to the “closed” model.
For what concerns prices, we do not get into national accounting considerations such as basic prices, purchasers’ prices, basic point prices (or delivered prices), etc.
As the price of labor varies across sectors, there is implicitly either an assumption that labor is heterogeneous (the labor employed by one sector not being the same as the labor employed by another sector), or an assumption that labor is immobile (otherwise the price of labor would be uniform among sectors because it would transfer from low-paying to high-paying sectors), or an assumption that working conditions across different sectors are heterogeneous. Anyway, it is more general (even if it is a little more complicated) to consider that the price of labor varies across sectors: a uniform price can always be retrieved by setting for all j.
Remember that nothing prevents the coefficients from exceeding one: their magnitude depends on the scale chosen but this has no impact on the solution as the determinant |I – A̅| is scale-independent, as is the result, after appropriate conversion of scale.
Obviously, this cannot be taken for granted in practice. On the one hand, there could be more commodities in monetary input-output tables because some immaterial commodities are impossible to capture in physical terms. On the other hand, there could be more commodities in physical input-output tables because it is impossible to aggregate commodities easily. Generally, the old-fashioned attempts, such as the MPS, to build an input-output table in physical terms lead to many more commodities than in the Leontief value model. However, we neglect this question because we want to discuss the theoretical foundations of the Leontief value model and not national accounting.
In other words, the column sums of L[I – A]–1 are identically equal: from the balance (15), it holds that s′L[I – A] ≡ s′. This property means that it always takes, directly and indirectly, $100 of labor to produce $100 of any commodity.
The model remains static, with two periods 0 and T; it is absolutely not dynamic in the sense of the so-called dynamic Leontief value model (1970), where the primal equation in monetary terms is x(t) – A(t)x(t) – B(t) (x(t+1) – x(t)) = f(t), B(t) being the matrix of capital coefficients.
Laspeyres or Paasche index prices are more sophisticated, taking into account not only the evolution of true prices but also of physical quantities by introducing weights (Fisher and Shell 1997).
It is obvious that, at the base period (t = 0), the Leontief price indexes are identically equal to 1: π̃′ = s′νL(0) [I – A(0)]–1 ≡ s′ because of the balance equation (15).
See Appendix 5.3. This relates to the problematics of Laspeyres and Paasche price indexes (Fisher and Shell 1997).
Provided that the economy is productive (see the Hawkins-Simon conditions).
Remember that two polynomial expressions are equal if and only if their coefficients are equal.
This analysis could be extended to the case of multiple factors even if the Leontief value model typically considers only one factor, labor.
Respectively, Leontief prices, are the current prices deflated by the variation in prices between the current and base periods. Thus, the true Leontief price model uses price indexes—defined as the variation in prices with respect to the current period—along with the technical coefficients of the current period.
We do not cite the fourth conclusion about Cronin’s model.