In the "closed" Leontief system, all inputs into production are produced and all outputs exist merely to serve as inputs. Labor requirements, as noted earlier, were "subsumed" in corn requirements via a subsistence wage and thus labor itself became effectively a "produced" good. In the "open" Leontief system, used in Leontief (1941), these assumptions are relaxed and we allow for discretionary consumption and non-produced factors. In other words, there is final consumption demand that has to be met and factor endowments (e.g. labor) that have to be paid. The structure of production must consequently be modified.

Let us begin with the open Leontief model without surplus or growth, thus
we have a static system. Let us refer to the final demand or consumption of the ith good
as C_{i}. Then, given n goods, we have a n ｴ 1 column
vector **c**｢** **= [C_{1}, C_{2}, ...,
C_{i}, ..., C_{n}] representing final demands in the economy. Now,
previously we had the output of the jth good being entirely demanded as inputs into other
industries. However, now, not all output of good j is demanded as a factor input, but some
part of it is demanded for final consumption. Therefore, for the jth industry, we have the
condition that:

X

_{j}= a_{j1}X_{1}+ a_{j2}X_{2}+ ... + a_{jn}X_{n}+ C_{j}

i.e. output of good j must cover input demands by firms and final demand. Doing so for all industries, j = 1, .., n, we then have the system:

X

_{1}= a_{11}X_{1}+ a_{12}X_{2}+ .... + a_{1n}X_{n}+ C_{1}

X_{2}= a_{21}X_{1}+ a_{22}X_{2}+ .... + a_{2n}X_{n}+ C_{2}

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

X_{n}= a_{n1}X_{1}+ a_{n2}X_{2}+ .... + a_{nn}X_{n}+ C_{n}

or, in matrix form:

X |
a |
a |
... |
a |
X |
C |
||||

X |
= |
a |
a |
... |
a |
X |
+ |
C |
||

.... |
... |
... |
... |
... |
... |
... |
||||

X |
a |
a |
... |
a |
X |
C |

which can be rewritten as:

x=A｢x+c

This can be rewritten as:

(

I-A｢)x=c

which is, as we can see immediately, no longer a homogeneous system of
equations. Thus, *given* a vector of final demands **c**, we can obtain a
determinate solution for **x **by inversion:

x= (I-A｢)^{-1}c

where (**I** - **A｢ **)^{-1} is
the inverse of the original technology matrix.

However, before we do anything else, we must know for a fact that (**I**
- **A｢ **)^{-1} exists, i.e. (**I** - **A｢ **) is non-singular. This can be verified by invoking the Hawkins-Simon conditions. If (**I** - **A｢ **) fulfill the Hawkins-Simon conditions, so that its principal
leading minors are positive, then an inverse, (**I** - **A**)^{-1} exists
and is non-negative (cf. Nikaido, 1960, 1968;
Takayama, 1974). This can in fact be found by using the matrix power series:

(

I-A｢)^{-1}=c+A｢ c+A｢^{2}c+ .... = ・/font>_{k=0}^{･}_{ }A^{k}c

This result implies that if (**I** - **A**｢
) fulfills the Hawkins-Simon condition then for any given **c** ｳ
0, there exists an **x** ｳ 0 that solves the system **x**
= **A｢ x** + **c**.

To understand this, it is worthwhile examining a two-sector case
graphically. As we know, with two sectors, the condition (**I** - **A**｢** **)**x** = **c** can be written:

(1-a |
-a |
X |
C |
||

= |
|||||

-a |
(1-a |
X |
C |

or:

(1-a

_{11})X_{1}- a_{12}X_{2}= C_{1}

-a

_{21}X_{1}+ (1-a_{22})X_{2}= C_{2}

Thus, we have two linear equations. Given C_{1} and C_{2},
we can draw two lines in a (X_{1}, X_{2}) space (denoted L_{1} and
L_{2}) for the system of equations as in Figure 1. Frontier L_{1} maps the
levels of X_{1} and X_{2} that satisfy the first equation and L_{2}
maps the levels which satisfy the second equation. As per the first equation, line L_{1}
has vertical intercept C_{1}/(-a_{12}) < 0, horizontal intercept C_{1}/(1-a_{11})
> 0 and slope (1 - a_{11})/a_{12}. From the second equation, line L_{2}
has vertical intercept C_{2}/(1-a_{22}) > 0, horizontal intercept C_{2}/(-a_{21})
< 0 and slope a_{21}/(1 - a_{22}) > 0. Thus, the equilibrium values
of X_{1} and X_{2} which satisfy both equations must be at the
intersection of the two locii L_{1} and L_{2}. This is shown in Figure 1
as the points (X_{1}^{*}, X_{2}^{*}).

Figure 1- Quantity Determination

It is easy to notice that an intersection is guaranteed only if the slope
of L_{1} is greater than the slope of L_{2}, i.e.

(1-a

_{11})/a_{12}> a_{21}/(1-a_{22})

or:

(1-a

_{11})(1-a_{22}) - a_{12}a_{21}> 0

which, it must be noticed, merely states that the determinant of the
matrix (**I** - **A**｢** **) is positive. This is *precisely*
the Hawkins-Simon condition applied to the two-sector case. If, on the other hand, **|I**
- **A**| < 0, then notice that this would imply that (1-a_{11})/a_{12}
< a_{21}/(1-a_{22}) so that the slope of L_{1} would be smaller
than the slope of L_{2} which, as we can immediately see diagramatically, implies
that L_{1} and L_{2} will not intersect - i.e. there is no non-negative
solution X_{1}*, X_{2}*.

Finally, let us note that the intercepts of the locii contain C_{1}
and C_{2} in them - thus a rise in C_{1} will obviously shift the locus L_{1}
to the right whereas a rise in C_{2} will shift the locus L_{2} to the
left. Thus, rises in either C_{1} or C_{2} or both will lead to rises in
both equilibrium X_{1}* and X_{2}*. Similarly, a fall in the own demand
(i.e. a fall in a_{11} or a_{22} coefficients) will result in lower
equilibrium outputs whereas a fall in cross-demand (a_{12} and a_{21})
will result also in lower equilibrium outputs. This can be observed from the slope and
intercept conditions.

Let us now turn to the dual problem - namely, prices. We have assumed no
surplus, thus we now must ask where the "final demand" in the quantity side
arises from. The simplest explanation would be to presume that there are primary inputs
which receive returns which they then use to make final demands on output. A "*primary
input*" is a non-produced factor of production - for instance, land or labor. An
example, would be workers who sell their labor to firms for a wage and, from the proceeds,
demand consumption goods. In this case, let a_{0j} be the amount of labor
necessary to produce a unit of good j. Let workers receive a uniform wage rate, w. Thus,
for production to be viable, the sale price of good j must not fall below the cost of
producing a single unit of good j:

p

_{j}ｳ p_{1}a_{1j}+ p_{2}a_{2j}+ .... + p_{n}a_{nj}+ wa_{oj}

So, sale price must cover unit costs of purchasing inputs from other firms and paying labor its wages. As there is no surplus, we can assume an equality for this and all other sectors, so:

p

_{1}= p_{1 }a_{11}+ p_{2}a_{21}+ .... + p_{n}a_{n1}+ wa_{01}

p_{2}= p_{1}a_{12}+ p_{2}a_{22}+ .... + p_{n}a_{n2}+ wa_{02}

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

p_{n}= p_{1}a_{1n}+ p_{2}a_{2n}+ .... + p_{n}a_{nn}+ wa_{on}

is our system of equations, or in matrix form:

p |
a |
a |
... |
a |
p |
a |
|||

p |
= |
a |
a |
... |
a |
p |
a |
||

.... |
... |
... |
... |
... |
... |
+ |
... |
||

p |
a |
a |
... |
a |
p |
a |

or, thinking of **a**_{0 }= [a_{01}, ..., a_{0n}]｢ as a column vector of unit labor requirements, then:

p=Ap+ wa_{0}

We can rewrite this as:

(

I-A)p= wa_{0}

which is also a non-homogeneous system. Thus a solution **p** can be
found by inversion, namely:

p= (I-A)^{-1}wa_{0}

The existence of the inverse (**I** - **A**)^{-1} is
effectively solved by the same manner as before. If the Hawkins-Simon conditions are
fulfilled for (**I** - **A｢ **), they will also be
fulfilled for (**I** - **A**), and thus an inverse will exist and will be
non-negative. Thus for any given non-negative set of unit wage-bills, w**a**_{0}
ｳ 0, there is a set of non-negative prices **p** ｳ 0 which solve the system **p** = **Ap** + w**a**_{0}.

A diagramatic example demonstrating the necessity of the Hawkins-Simon
conditions for the existence of a solution **p** can also be drawn for the two-sector
case and will be analogous to before. Namely, taking the two-sector model, then (**I**
- **A**)**p** = w**a**_{0}

(1-a |
-a |
p |
wa |
||

= |
|||||

-a |
(1-a |
p |
wa |

or:

(1-a

_{11})p_{1}- a_{21}p_{2}= wa_{01}

-a

_{12}p_{1}+ (1-a_{22})p_{2}= wa_{02}

We can thus depict these two equations as in Figure 2 via the lines M_{1}
and M_{2}. The line M_{1} represents the levels of prices which fulfill
the first equation, thus it has vertical intercept wa_{01}/(-a_{21}) <
0, horizontal intercept wa_{01}/(1-a_{11}) > 0 and slope (1-a_{11})/a_{21}
> 0. Similarly, M_{2} represents the values fulfilling the second equation,
thus it has vertical intercept wa_{02}/(1-a_{22}) > 0, horizontal
intercept wa_{02}/(-a_{12}) < 0 and slope a_{12}/(1-a_{22})
> 0. The solution values p_{1}* and p_{2}* are given by the
intersection of the two curve M_{1} and M_{2}. Notice that if the primary
factor payments increase, then M_{1} and M_{2} shift accordingly.
Specifically, if w increases, M_{1} shifts to the right and M_{2} shifts
to the left. A rise in labor input requirements, a_{01} and a_{02} will
have analogous effects. The necessity of the Hawkins-Simon conditions for the existence of
a non-negative p_{1}*, p_{2}* can be noted by referring to the fact that M_{1}
must have a greater slope than M_{2}, i.e. that (1-a_{11})/a_{21}
> a_{12}/(1-a_{22}), which translates simply into (1-a_{11})(1-a_{22})
- a_{21}a_{12} > 0.

Figure 2- Price Determination

Luigi Pasinetti (1975) sees the
condition that **p** = (**I** - **A**)^{-1}w**a**_{0} as
insinuating the Classical "*labor theory
of value*". If w = 1, we can see that (**I** - **A**)^{-1}**a**_{0}
is a vector of vertically integrated labor coefficients which "represents the
quantities of labor directly and indirectly "embodied" in each physical unit of
the commodities of the net product of the economic system (what Marx called "values" by definition)"
(Pasinetti, 1975: p.76). Thus, prices **p** are proportional to the physical quantities
of embodied labor - the essence of the labor theory of value.

Of course, we need not assume that labor is the only primary input. We
could have several primary inputs at once. For instance, suppose we have m primary inputs,
letting b_{kj} denote the unit input coefficient of primary input k into the
production of the jth output. Each primary input receives a return w_{k} and thus,
for the jth output, the price-cost equality becomes:

p

_{j}= p_{1}a_{1j}+ p_{2}a_{2j}+ .... + p_{n}a_{nj}+ w_{1}b_{1j}+ w_{2}b_{2j}.... + b_{mj}

so that we obtain a system of equations in the following form:

p=Ap+Bw

where **w** is a (m ｴ 1) column vector of
primary input returns and **B** is an (n ｴ m) matrix of
unit input coefficients for the primary factors. Assuming **w** and **B** are given,
then we can solve this for **p** by inversion again:

p= (I-A)^{-1}Bw

which again, is determinate by the Hawkins-Simon conditions.