Contents
(A) The Stationary State
(B) Surplus and Balanced Growth
The extension of the twosector Ricardian system into a multicommodity world was channeled into modern economics by John von Neumann (1937), Wassily Leontief (1936, 1941) and Piero Sraffa (1960). In this section, we shall concentrate attention on Leontief's "inputoutput" system. We should note that this system possesses some of the same structure and results of Sraffa's, with von Neumann's model being somewhat of an elaboration and generalization of both.
One of the main structural differences between Leontief's inputoutput system and the Sraffian is that the latter system adheres to the Classical theory more closely by operating on the basis of a set of data which is absent in the Leontief system  notably a "given" set of outputs.
Suppose we have a set of n capital goods. Let the output of the jth good be denoted X_{j}, so that we have an ndimensional vector X｢ = [X_{1}, X_{2}, ..., X_{n}]｢ representing outputs. Thus, our first set of data, the size and composition of output is finished. The technology is given by a set of inputrequirements. Let a_{ij} be the unit input coefficient denoting the amount of input i needed to produce a unit of good j (the order of the subscripts can be mnemonically recorded by the word "inputoutput"). Thus, to produce X_{j} units of good j, one needs a_{ij}X_{j} units of input i. Thus, letting X_{ij} be the input of i required by industry j, obviously X_{ij} = a_{ij}X_{j}, or:
a_{ij} = X_{ij}/X_{j}
Now, as noted earlier, the original Sraffa (1960) system does not presume unitinput coefficients and merely works with input requirements for a given level of output, i.e. X_{ij}, etc. are the primitive data. Assuming unit input coefficients implies we are assuming constant returns to scale, and Sraffa makes no such assumption  although Leontief does.
The advantage of setting up the program this way is that we do not need to assume a fixed set of outputs and it simplifies our notation. As a result, we can conceive of writing up a set of production functions representing Leontief's type of technology, which possesses the following properties:
(1) Constant returns to scale: doubling inputs doubles all outputs, no more and no less.
(2) Fixed proportions technology: each output is produced via a unique combination of inputs. There is no substitution among inputs which can give us the same commodity as output. There is a fixed proportions technology.
(3) Circulating capital: after the production process is completed, there will be no inputs left over. All inputs are used up in the production process.
(4) No joint production: each production process will only produce one commodity. Thus, if we have n commodities, we have, simultaneously, n production processes, each producing a single output.
Most of these assumptions are relaxed in other linear production models  notably von Neumann's (1937) and Sraffa's (1960).
In the following, we shall follow the early Leontief in formulating a "closed" inputoutput system (as introduced in Leontief, 1936), and later allow for the Leontief's "open" input output system (as used almost exclusively in later editions of Leontief's (1941) masterpiece).
Let us turn first to prices. Assume each good has a price, so that we face a vector p｢ = [p_{1}, p_{2}, ..., p_{n}]｢ . If a firm only uses capital goods as inputs, then the total costs of producing X_{j} units of good j are p_{1}X_{1j} + p_{2}X_{2j} + .... + p_{n}X_{nj} whereas the total revenue from the sale of good j is p_{j}X_{j}. Thus, for the production of X_{j} to be profitable, we need:
p_{j}X_{j} ｳ p_{1}X_{1j} + p_{2}X_{2j} + .... + p_{n}X_{nj}
Let us assume we do not have a surplus. Therefore, let us assume that total revenue is equal to total cost, i.e.
p_{j}X_{j} = p_{1}X_{1j} + p_{2}X_{2j} + .... + p_{n}X_{nj}
If this holds true for good j, it will hold true for all other goods. Thus, we obtain a system of equations:
p_{1}X_{1} = p_{1}X_{11} + p_{2}X_{21} + .... + p_{n}X_{n1}
p_{2}X_{2} = p_{1}X_{12} + p_{2}X_{22} + .... + p_{n}X_{n2}
.....................................................
p_{n}X_{n} = p_{1}X_{1n} + p_{2}X_{2n} + .... + p_{n}X_{nn}
This is Sraffa's (1960) initial set of equations. To transform this into Leontief's (1941) system, we divide each equation by the associated quantity:
p_{1} = p_{1 }(X_{11}/X_{1}) + p_{2}(X_{21}/X_{1}) + .... + p_{n}(X_{n1}/X_{1})
p_{2} = p_{1 }(X_{12}/X_{2}) + p_{2}(X_{22}/X_{2}) + .... + p_{n}(X_{n2}/X_{2})
.....................................................
p_{n} = p_{1 }(X_{1n}/X_{n}) + p_{2}(X_{2n}/X_{n}) + .... + p_{n}(X_{nn}/X_{n})
or, in the more familiar form of unit input coefficients where a_{ij} = X_{ij}/X_{j} for any input i and output j:
p_{1} = p_{1 }a_{11} + p_{2}a_{21} + .... + p_{n}a_{n1}
p_{2} = p_{1}a_{12} + p_{2}a_{22} + .... + p_{n}a_{n2}
...............................................
p_{n} = p_{1}a_{1n} + p_{2}a_{2n} + .... + p_{n}a_{nn}
or simply:
p_{1} 
a_{11} 
a_{21} 
... 
a_{n1} 
p_{1} 

p_{2} 
= 
a_{12} 
a_{22} 
... 
a_{n2} 
p_{2} 

.... 
... 
... 
... 
... 
... 

p_{n} 
a_{1n} 
a_{2n} 
... 
a_{nn} 
p_{n} 
(we can't get this darn program to write up matrix borders, so we differentiate with colors), or, letting A be the nｴ n matrix of unit input coefficients and p the ndimensional price vector, we can rewrite this as:
p = Ap
This is the "no surplus" pricecost equality. This can be rewritten as:
(I  A)p = 0
where we now have a homogeneous system. For the solution values (p) not to be zero, then the determinant of (I  A) must vanish, i.e.
I  A = 0
If true, then the system can be readily solved. As I  A = 0, we know that the homogeneous system (I  A)p = 0 will have a nontrivial solution p  in fact, it will have an infinite number of nontrivial solutions. However, even though we cannot determine the absolute levels of p that solves this, we can determine their proportionality. For instance, in a 2 ｴ 2 system, it can be easily shown that, from (I  A)p = 0, or:
(1a_{11}) 
a_{21} 
p_{1} 
0 

= 

a_{12} 
(1a_{22}) 
p_{2} 
0 
we obtain:
(1a_{11})p_{1}  a_{21}p_{2} = 0
a_{12}p_{1} + (1a_{22})p_{2} = 0
As the first equation is linearly related to the second (due to the nonsingularity of I  A), then:
p_{1}/p_{2} = a_{21}/(1a_{11}) = (1a_{22})/a_{12}
the different expressions for p_{1}/p_{2} are, in fact, the same due to linear dependence (notice that the equality implies that (1a_{11})(1a_{22})  a_{12}a_{21} = 0, the vanishing determinant of (I  A)). Thus, it is quite simple to calculate p_{1}/p_{2} if we know the values of the technological coefficients.
What about quantities? In the closed Leontief system, we must assume we have a fully "selfreplacing" economy, i.e. an economy which produces at least enough of a commodity as is demanded by other industries as an input. Thus, if X_{i} is the output of commodity i, then:
X_{i} ｳ a_{i1}X_{1} + a_{i2}X_{2} + .... + a_{in}X_{n}
where the terms on the right represent the sum of demands for commodity i by the other industries. If we assume stationarity, or perfect selfreplacement, then the economy produces just enough of that commodity to fulfill input demands, and no more. Thus, our inequality becomes an equality  one may wish to think of this as a marketclearing condition (so total supply equals total demand for each commodity) or, as Dorfman, Samuelson and Solow (1958: 246) prefer to characterize it as "a commoditybycommodity Say's Law". As this is true for all commodities i, we can write out the system:
X_{1} = a_{11}X_{1} + a_{12}X_{2} + .... + a_{1n}X_{n}
X_{2} = a_{21}X_{1} + a_{22}X_{2} + .... + a_{2n}X_{n}
...............................................
X_{n} = a_{n1}X_{1} + a_{n2}X_{2} + .... + a_{nn}X_{n}
or, in matrix form:
X_{1} 
a_{11} 
a_{12} 
... 
a_{1n} 
X_{1} 

X_{2} 
= 
a_{21} 
a_{22} 
... 
a_{2n} 
X_{2} 

.... 
... 
... 
... 
... 
... 

X_{n} 
a_{n1} 
a_{2n} 
... 
a_{nn} 
X_{n} 
where, note, the matrix of unit input coefficients here is merely the transpose of our earlier matrix, A. Thus, letting x = [X_{1}, X_{2}, ..., X_{n}]｢ , then we can rewrite this as:
x = A｢ x
where A｢ is the transpose of our earlier A. Of course, this can be rewritten as:
(I  A｢ )x = 0
which is again a homogeneous system. The linear dependence which guaranteed a vanishing determinant for the first case will guarantee, once again, that we have a vanishing determinant I  A｢  = 0. Thus, for a 2 ｴ 2 system, from (I  A｢ )x = 0, we have:
(1a_{11}) 
a_{12} 
X_{1} 
0 

= 

a_{21} 
(1a_{22}) 
X_{2} 
0 
or:
(1a_{11})X_{1}  a_{12}X_{2} = 0
a_{21}X_{1} + (1a_{22})X_{2} = 0
which, again, because of linear dependence then:
X_{1}/X_{2} = a_{12}/(1a_{11}) = (1a_{22})/a_{21}
which is again easy to calculate. However, to obtain absolute levels of p_{1}, p_{2} or X_{1}, X_{2}, we must normalize the system (e.g. we can assume p_{1} = 1, or x_{1}^{2} + x_{2}^{2} = 1, etc.). For general ndimensional cases, however, we must appeal to the PerronFrobenius theorem. To deal with this, we shall await the introduction of a surplus and balanced growth, of which the stationary, nosurplus economy is but a special case.
(B) Surplus Economies and Balanced Growth
Let us now move to the world where there is a surplus. In this case, price is not equal to cost of production but rather is marked up above it: there is a surplus that accrues to the owners of the outputs. Beginning with our system of equations:
p = Ap
adding the surplus is an easy affair. Since we have assumed away fixed capital, then the surplus must be written in the form of circulating capital. For the ith equation, this would give us:
p_{i} = (1+r)(p_{1}a_{1i} + p_{2}a_{2i} + .... + p_{n}a_{ni})
where r can be interpreted as rate of profit in the sense of a markup over unit costs. This rate of profit, r, is assumed uniform for all outputs. This complies with Classical understanding and the reasoning can be recalled by referring to the idea of the longrun as when there are no excess profits in any sector (uniform rate of profit). To our old system we just multiply the matrix A by the scalar (1+r) so:
p = (1+r)Ap
The quantity side of the story we shall ignore for the moment since it will involve growth. The solution procedure for prices and profit is as simple as before. All we do is recognize that p = (1+r)Ap implies that:
(I  (1+r)A)p = 0
so that, for a nontrivial solution, I  (1+r)A = 0, i.e. the determinant of the new technology matrix must disappear. Now, deriving a solution for a 2 ｴ 2 system is straightforward enough, as I  (1+r)A = 0 yields a seconddegree polynomial for which we can find a pair of solutions for (1+r), one of which, by Descartes's rule of signs, will be positive and the other negative. For a ndimensional system, we face a larger polynomial and thus things get a bit more complicated, but in fact we obtain an analagous result.
To accomplish this, we need to take refuge in the PerronFrobenius theorems on nonnegative square matrices. The theorems can be divided into those for general nonnegative square matrices and for irreducible nonnegative square matrices. By the assumptions of unit input coefficients, a_{ij} ｳ 0 for all i, j = 1, .., n, so that A is definitely a nonnegative square matrix. To make it irreducible, we must appeal to the assumption that all goods are "basic". Sraffa (1960) distinguished between "basic" and "nonbasic" goods, where a basic good enters into the production of every other good, either directly or indirectly, and a nonbasic does not. Thus, if we assume all goods are basic, then the matrix A is an irreducible, nonnegative square matrix.
There are various PerronFrobenius theorems. For general nonnegative square matrices, we have the following:
(G.1) A has at least one nonnegative eigenvalue.
(G.2) If l is a eigenvalue of A, then l  ｣ l * (the largest nonnegative eigenvalue, called the "Frobenius root" and denoted l *, is greater than the absolute value of any other eigenvalue)
(G.3) If A and B are two nonnegative square matrices, where B ｳ A ｳ 0, then l *_{A} ｳ l _{B}*.
(G.4) If there is a r > l *, then (r I  A)^{1} exists and is nonnegative.
(G.5) min_{j} ・/font> _{i} a_{ij} ｣ l * ｣ max_{j} ・/font> _{i} a_{ij }(the value of the Frobenius root lies somewhere between the greatest and the smallest of the values of the row sums of the matrix A).
If the matrix is also irreducible, then the following also hold:
(I.1) The Frobenius root l * > 0 and any associated right eigenvector x* > 0 (the same applies for the left eigenvector).
(I.2) If Ax = l x for some l ｳ 0 and x ｳ 0, then l = l * (i.e. for any eigenvalue l other than Frobenius root, the associated eigenvector x will have at least one negative element).
(For details and proofs, consult.Debreu and Herstein (1953), Morishima (1964), Murata (1977), Nikaido (1960), Pasinetti (1975), Takayama (1974) and Kurz and Salvadori (1995).
The implications of the PeronFrobenius theorems for our system will be made clear as we go along. Firstly, we must recognize that from (I  (1+r)A)p = 0, then
(1/(1+r)p = Ap
where, letting l = (1/(1+r)), then the system becomes the eigenvalue form:
l p = Ap
where l is an eigenvalue of A and p an associated right eigenvector (recall that eigenvectors of a given eigenvalue are linearly dependent, thus there is a unique p in ratio terms, but not in absolute terms, associated with any eigenvalue). As A is a nonnegative square matrix, then by (G.1) of the PerronFrobenius theorems, there will be a nonnegative maximum eigenvalue, the Frobenius root, l *. By (G.2), this eigenvalue will be greater than or equal to the absolute value of any other eigenvalue of the system, i.e. l * ｳ l  for all other eigenvalues l . If we further assume that A is irreducible, by assuming a basic system for instance, then by PerronFrobenius theorem (I.1), the Frobenius root will be strictly positive and any associated right eigenvector (call it p*) will also be strictly positive, i.e. p* > 0.
Now for the rattlesnake juice: by PerronFrobenius theorem (I.2), if there is another eigenvalue l with an associated right eigenvector p (i.e. l p = Ap), then there will be at least one negative element in p. What this implies is that any eigenvalue other than the Frobenius root will have an economicallymeaningless set of prices, p  as prices cannot be negative. Thus, we can establish already that for an irreducible Leontief system to make economic sense, there is only one candidate for solution: namely, the Frobenius root l * and its associated eigenvector p* as this is the only one which has nonnegative prices (or rather, price ratios). Even stronger is the result from (I.1) that all price ratios in p* will be strictly positive  thus we are disallowing free goods.
What about the rate of profit? As we made l = 1/(1+r), then as we only have one candidate for l (i.e. the Frobenius root l *), then:
r = (1l *)/l *
As l * is unique, then r is uniquely determined. However, are we guaranteed that the rate of profit is positive, i.e. is r > 0? For r > 0, then the Frobenius root must be less than one, i.e. l ^{*} < 1. Is this true? To establish this we must appeal to the idea of "productive matrices". The matrix A is defined as "productive" if p > Ap, i.e. sale price is strictly greater than unit cost of production. This, of course, is necessary if we are to generate a surplus. We now prove the following:
Theorem: If p* > Ap*, then r > 0.
Proof: Suppose r ｣ 0. Then, it must be that l * ｳ 1. Following the PerronFrobenius theorem (I.1), Ap* = l *p* where p* > 0. Thus, by (G.2) for any other l , we know l  ｣ l *. This implies that Ap* ｳ l p* for all l ｹ l *. Consider now c > l *. Then, Ap* < cp*. Suppose c = 1. Then, Ap* < p*. But c = 1 implies l * < 1. Thus, Ap* < p* implies l * < 1 which implies, in turn, that r > 0.ｧ
What conditions guarantee that the economy is productive, i.e. that p > Ap? Here we must appeal to the HawkinsSimon condition (from Hawkins and Simon, 1949; GeorgescuRoegen, 1951).. This condition is simply the following:
HawkinsSimon: a square matrix B is "productive" or fulfills the "HawkinsSimon conditions" if all the successive principal leading minors of A are positive.
Consider now the matrix (I  A). Then, we can see immediately that for the HawkinsSimon conditions to hold, it must be that:
(i) (1a_{11}) > 0
(1a_{11}) a_{21}
(ii) a_{12} (1a_{22}) > 0
(1a_{11}) a_{21} a_{31}
(iii) a_{12} (1a_{22}) a_{32} > 0
a_{13} a_{23} (1a_{33})
and so on for successive leading minors. For these to hold, assumptions do not have to be outlandish. Recall that (I  A) is a matrix with nonpositive offdiagonal elements (off diagonal elements of (I  A) are a_{ij} where i ｹ j and a_{ij} ｳ 0) and the diagonal elements are (1a_{ii}) for i = 1, .., n. Now, the first step of (i) is that (1a_{11}) > 0, which is reasonable. The second step (ii) implies that:
(1a_{11})(1a_{22}) > a_{21}a_{12}
or, as a_{21}, a_{12} ｳ 0 and as, by (i), (1a_{11}) > 0, then it must be that (1a_{22}) > 0. For (iii), it must be that:
a_{31} [a_{12}a_{23} + (1a_{22})a_{13}] + a_{32}[(1a_{11})a_{23}  a_{21}a_{13}] + (1a_{33})[(1a_{11})(1a_{22})  a_{21}a_{12}] > 0
which may be fulfilled.
The implications of the HawkinsSimon conditions can be seen readily. By the HawkinsSimon conditions, then (I  A) > 0, which implies that (I  A)p* > 0 as the eigenvector p* associated with the Frobenius root is strictly positive, i.e. HawkinsSimon conditions imply that p* > Ap*. Thus, we see that HawkinsSimon conditions are sufficient to guarantee that p* > Ap* and, subsequently, that l * < 1 and thus r > 0. Furthermore, by PerronFrobenius (G.4), if l * < 1, then we can see immediately that (I  A)^{1} exists and is nonnegative. We refer to Nikaido (1960, 1968) and Takayama (1974) for the proofs that HawkinsSimon conditions on (IA) are both necessary and sufficient to prove that (I  A)^{1} exists.
In sum, let us recall what we have demonstrated: namely, that in a basic system with a surplus, where p = (1+r)Ap, we can find a unique rate of profit, r > 0 and a unique set of price ratios p which solves this system and that these prices are all strictly positive. In short, we can find equilibrium prices and rate of profit on the basis of technological coefficients (A) which include, within them, the subsumed real wage (part of input coefficients).
This is the Leontieftype generalization of Ricardo's system to n commodities. Sraffa (1960) solution to the original Ricardian system also appeals to PerronFrobenius, but assumes a given size and composition of output  thus it does not assume constant returns, etc. However, recall that Ricardo and Sraffa ask about wageprofit tradeoffs. We can also see this here. Assuming that the real wage is subsumed under, say, corn inputs, then raising the corn input requirement is equivalent to increasing the real wage. Now, suppose we have a matrix A_{1} which corresponds to the unitinput coefficients before the wage hike and let A_{2} be the matrix which corresponds to the coefficients after the wage hike. As this is equivalent to raising corn requirements (all other coefficients remaining equal), then obviously 0 ｣ A_{1} < A_{2}. By PerronFrobenius theorem (G.3), then the Frobenius root associated with the second matrix (call it l _{2}*) is greater than the Frobenius root associated with the first (l _{1}*), i.e.
l _{1}* < l _{2}*
As r_{1} = (1l _{1}*)/l _{1}* and r_{2} = (1l _{2}*)/l _{2}*, then this implies that:
r_{1} > r_{2}
the rate of profit associated with A_{1} is greater than the rate associated with A_{2}. In other words, the increase in the wage rate (through the corn coefficient) has led to a decline in the rate of profit. Consequently, there is a wageprofit tradeoff as Ricardo and Sraffa stipulated.
What Sraffa did not ask, and the Ricardians did, was about the consequences of profit upon economic growth. The Leontief system permits us to answer this by turning to balanced growth. Recall that the nosurplus stationary economy faced the following:
x = A｢ x
where A｢ is the transpose of the earlier matrix A. With positive surplus, we now have it that:
x > A｢ x
more outputs are produced than is needed to meet input requirements. These are eventually allocated as profits that accrue to capitalists. If capitalists decide to reinvest it, then output grows. Thus, we must impose a new condition if we want to keep prices constant (and thus in steadystate equilibrium) over time: namely, we must assume balanced growth, or that the quantity of every good increases at a uniform proportional rate, g. This means, effectively, that capitalists desire to expand the output of every commodity at the rate g  thus they expand input demands for every commodity at the rate g. Thus, production in any period must be enough to meet not only the industry input demands for pure replication, A｢ x, but also an additional input demand for industry growth, g(A｢ x). Thus:
x ｳ (1+g)A｢ x
Assuming then that just enough is produced to meet input demands, then the inequality becomes an equality:
x = (1+g)A｢ x
This is now our balanced growth condition. Now, let us proceed as we did earlier. Dividing through by (1+g), we can write:
(1/(1+g))x = A｢ x
letting l = 1/(1+g), we can rewrite this as:
l x = A｢ x
which is again a characteristic equation. Now, as A｢ is merely the transpose of A we can think of x as the left eigenvector of A. Now, assuming irreducibility and as A｢ is a nonnegative, square matrix, then by FrobeniusTheorems (G.1, G.2, I.1), there is a Frobenius root l * > 0 which yields a left eigenvector x* such that:
l *x* = A｢ x*
where, by (I.1) all elements of x* are strictly positive, i.e. x* > 0. Naturally, as x* is an eigenvector, than absolute levels are not determinate, albeit ratios between elements of x* are  thus, x* establishes the quantity ratios that yield balanced growth. By the same arguments as before, this is the unique solution to our system as any eigenvalue other than the Frobenius root will yield an eigenvector x which contains at least one negative element  and negative quantities, in this context, are economicallymeaningless. Thus, there is a unique rate of growth, g = (1l *)/l * and a unique set of quantity ratios, x*, that solve our system  and, furthermore, all quantities will be positive (thus, no good goes unproduced).
Once again, by the same reasoning as before, a necessary condition for positive growth, g > 0, is that l * < 1, and that this is guaranteed in a productive system where x > A｢ x. A necessary and sufficient condition for this last to be fulfilled is that the HawkinsSimon conditions apply to (IA｢ )  and if they applied for (I  A), they will apply for (I  A｢ ) as well. Thus, note that this means that x > A｢ x implies p > Ap and viceversa (if all profit is saved and reinvested). A more interesting connection is the the Frobenius root l * associated with A is the same as the Frobenius root associated with A｢ , thus implying that:
g = r = (1l *)/l *
or that the rate of growth is equal to the rate of profit  better known as the "Golden Rule" of economic growth. Subsequently, the wageprofit tradeoff will also imply, by analogous reasoning, a wagegrowth tradeoff. Thus, Classical concerns with the implications of income distribution on economic growth dovetail nicely into this multisectoral system.