In the open Leontief model described above, given exogenous final demands, we can solve for quantities uniquely; similarly, given exogenous primary input costs, we can solve for prices uniquely. Thus, prices and quantities seem to be solvable quite independently of each other - a fundamental feature of the Classical system and one quite contrary to the Neoclassical.

However, in the Leontief system final demand **c** and unit factor
returns **Bw** are given exogenously and may not seem related to one another. If we
impose the logical condition that **w**｢** B**｢** x** = **p**｢** c**, where **w**｢** B**｢** x** are the total
factor payments in the economy and **p**｢** c** is the
value of the demanded bundle, then we obtain the intuitively plausible idea that owners of
factors use their factor returns to make final demands. In this way, then, the quantity
side and the price side will not be unrelated to one another, even if one can determine
prices independently of quantities and vice-versa.

If we were to rewrite the Leontief system as an appropriate linear
programming problem, we actually obtain the result that **w**｢**B**｢**x** = **p**｢**c** (Dorfman, Samuelson
and Solow, 1958: Ch.9). However, to construct a
linear programming problem we must relax the equality restraints imposed earlier and
specify an appropriate objective function. Consider the case, then, when we allow an
inequality in the quantity restrictions so that:

xｳAx+c

so that we can allow commodities to be produced in *excess* of input
demands and final demand, then it seems we are no longer restricted to a single
determinate solution. Supposing our old two-sector case, then we have:

(1-a

_{11})X_{1}- a_{12}X_{2}ｳ C_{1}

-a

_{21}X_{1}+ (1-a_{22})X_{2}ｳ C_{2}

Thus, in Figure 1, the inequality of the quantity equations implies that a
solution (X_{1}*, X_{2}*) need no longer be at point X, the intersection
of L_{1} and L_{2}. Instead, we must allow for the possibility of the
strict inequality holding for any of the equations in our system. Thus, the inequality in
the first equation allows all output combinations in the area *below* L_{1}
are allowable and the inequality in the second equation implies that all areas *above*
L_{2} are allowable. The shaded area in Figure 1, thus, represents the output
combinations which are feasible given the inequalities.

Figure 1- Quantity Determination

However, we still want to obtain (X_{1}*, X_{2}*) as the
solution, thus we must specify an objective function which yields this. Consider the
minimization of total factor payments, **w**｢**B**｢**x**. This is, of course, merely a weighted sum of outputs,
i.e. in the two sector case:

w｢B｢x= wb_{01}X_{1}+ wb_{02}X_{2}

We can represent this in Figure 1 by the negatively-sloped locus B with
horizontal intercept (**w**｢**B**｢**x**)/wb_{01
}> 0 and slope -b_{01}/b_{02} < 0. B represents the
combinations of outputs X_{1} and X_{2 }which yield the same factor
returns, **w**｢**B**｢**x**.
Shifting B rightwards increases total factor payments, whereas shifting it left decreases
factor payments. Thus, by minimizing **w**｢**B**｢**x**, then we shift B leftwards to B* where B* intersects the
equilibrium point X at (X_{1}*, X_{2}*).

As a consequence, with inequalities, we can view the problem of solving for quantities as a linear programming problem of the following sort:

min

w｢B｢x

s.t.

xｳA｢x+c

xｳ 0

Thus, we seek to choose non-negative output levels (**x**) that
minimize factor payments (**w**｢** B**｢** x**) subject to the constraint that supply of outputs (**x**)
not exceed demand for outputs (**A**｢** x** + **c**).
Thus, given technology (**A**, **B**), the level and composition of final demand (**c**)
and the historically-given real wage (**w**), we ought to be able to solve for
quantities, **x*** ｳ 0. Notice the important conclusion
that, set up this way, output prices do *not* enter into the determination of
quantities.

In linear programming, every primal problem has a dual problem. In this case, we wish to consider the prices-cost equations as inequalities in the following form:

p｣Ap+Bw

where price cannot exceed cost of production. In a two-sector system with a single primary input, this implies:

(1-a

_{11})p_{1}- a_{21}p_{2}｣ wa_{01}

-a

_{12}p_{1}+ (1-a_{22})p_{2}｣ wa_{02}

which implies, as shown in Figure 2, that we no longer are restricted to
solution (p_{1}*, p_{2}*) at the intersection of the M_{1} and M_{2}
curves (point P). Rather, the first inequality allows us to consider the area *above*
M_{1} and the second inequality allows combinations in the area *below* M_{2}.
Thus, any price combination in the shaded area in Figure 2 becomes feasible.

Figure 2- Price Determination

The appropriate objective function now is to maximize the value of final
demand, i.e. maximize **p**｢** c**. In a two-sector
system, this is written:

p｢c= p_{1}C_{1}+ p_{2}C_{2}

Thus, we can superimpose a negatively-sloped locus C in figure 2
representing the combinations of p_{1} and p_{2} which yield the same
value of final demand, **p｢ c** for a given C_{1}
and C_{2}. Thus, the locus C has horizontal intercept **p**｢**
c**/C_{1} > 0 and slope -C_{1}/C_{2} < 0. Shifting the C
locus to the left decreases the value of final demand, whereas shifting it to the left
increases the value of final demand. Thus, maximizing **p｢ c**,
we move to the highest locus C* and thus obtain the equilibrium solution (p_{1}*,
p_{2}*) at P, the intersection of the M_{1} and M_{2} curves.

The dual of this problem, then, is stated as the following:

max

p｢c

s.t.

p｣Ap+Bw

pｳ 0

thus we seek to choose the non-negative prices (**p**) that maximize
the value of final demand (**p｢ c**) given that prices do
not exceed unit cost of production **p** ｣ **Ap** + **Bw**.
Thus, given technology (**A**, **B**), the level and composition of final demand (**c**)
and the historically-given real wage (**w**), we can solve for prices, **p*** ｳ 0. Notice that here we have the important condition that
quantities of output (**x**) do *not* enter into the determination of prices.

Having specified the appropriate linear programming problems, let us now turn to the solution. We can rewrite this primal problem of minimizing factor payments as a Lagrangian of the following sort (cf. Takayama, 1974):

L(

x, l ) =w｢B｢x + l((I-A｢)x-c)

where l is a vector of Lagrangian multipliers.
The solution to this problem is a pair (**x***, l *) such
that:

(i)

w｢B｢x* ｣w｢B｢xfor allxｳ 0, (I-A｢)xｳc(ii) l *((

I-A｢)x-c) = 0 where l * ｳ 0

The first condition merely states that **x*** minimizes the objective
function; the second condition is the complementary slackness condition: at the solution **x***,
either a particular constraint is binding (i.e. net outputs equal consumption demand for a
particular good j, so that so that (**I**_{j} - **a**_{j}｢)x_{j} = c_{j} where **I**_{j} and **a**_{j}｢_{ }are the jth rows of **I** and **A**｢** **respectively) or, if not binding ((**I**_{j} - **a**_{j}｢_{ })x_{j} > c_{j}, thus overproduction
of good j), then the associated multiplier will be zero (l _{j}*
= 0).

Let us now turn to the dual problem of maximizing the value of final demand. Rewriting this into a Lagrangian form:

L(

p, m ) =p｢c+ m [Bw- (I-A)p]

where m is a vector of Lagrangian multipliers
associated with this problem. The solution to the dual programming problem can be
characterized as a pair (**p***, m *):

(i)

p｢*cｳp｢cfor allpｳ 0,Bwｳ (I-A)p

(ii) m *(

Bw- (I-A)p*) = 0 where m * ｳ 0

Now, the first condition states that the solution **p*** must mazimize
the value of net output. The second condition is complementary slackness again: at the
solution **p***, either a particular constraint is binding (i.e. price equals unit cost
of production for a given particular good i, so that **b**_{i}**w - **(**I**_{i}
- **a**_{i})p_{i} = 0 where **b**_{i}, **a**_{i}
and **I**_{i}** **are the ith rows of **B**, **A** and **I **respectively)
or, if not binding (**b**_{i}**w - **(**I**_{i} - **a**_{i})p_{i}
> 0), then the associated multiplier will be zero (m _{i}*
= 0).

Appealing to the duality theorem of linear programming, we can observe the following:

(i) m

_{j}* = x_{j}* for all j = 1, .., n.

(ii) l

_{i}* = p_{i}* for all i = 1, .., m.

(iii)

p｢*c=w｢B｢x*

Condition (i) states that the multiplier in the dual (m
_{j}*) is equivalent to the solution to the primal problem (x_{j}*) and
(ii) states that the multiplier in the primal problem (l _{i}*)
is equal to the solution the dual problem (w_{i}*). Thus, the first Lagrangian
could actually be rewritten as:

L(

x,w) =p(I-A｢)x + w(v-B｢x)

where we have substituted **w** for the multiplier l
, whereas the second Lagrangian can be rewritten:

L(

w,x) =w｢v+x[Bw- (I-A)p]

where we have substituted **x** for the multiplier m
. Thus, the complementary slackness conditions become:

p*((I-A｢ )x-c) = 0

x*(Bw- (I-A)p*) = 0

where the first implies the familiar "free goods" assumption (if
the net output of any good exceeds its consumption demand, then the price of that good is
zero) and the second implies the "excess cost" condition (if price falls below
unit cost of production for any good, then that good will not be produced). With
sufficient assumptions on the Leontief system (e.g. indecomposability of **A**), the
free goods and excess cost conditions will not be applied as all goods will be produced
and they will have a positive price.

Condition (iii) is quite interesting since it states that, in equilibrium,
factor payments will be equal to final demand - thus confirming our earlier statements.
More interesting is that this does not implicate our assumptions about the independence of
the determination of quantities and prices. We earlier stated that (**A**, **B**, **w**,
**c**) enter into each of the programming problems. However, as it turns out, **B**
and **w **turn out to be irrelevant to the determination of equilibrium quantities and **c**
turns out to irrelevant to the determination of equilibrium prices.

We can see this diagramatically in Figures 1 and 2. In Figure 1, we can
see that as long as **w** ｳ 0 and **B** ｳ 0, then a locus B with *any* slope will, when minimized,
yield the unique solution point X. This is true regardless of the values of the weights on
the locus, **w** and **B**. Thus, in effect, only **A** and** c** really
matters to the solution, **x***. We are thus back to the regular Leontief equation,
where **x** = **A**｢** x** + **c** and factor
input requirements and factor payments do not affect the solution quantities, **x*** ｳ 0. Similarly, in Figure 2, we can see that as long as **c** ｳ 0, then a locus C with *any* slope will, upon maximization,
yield the unique solution point P. Thus, the size and composition of final demands, **c**,
do *not* affect the determination of equilibrium prices; only **w**, **A** and **B**
matter for the determination of equilibrium prices, the values of C_{1} and C_{2}
are quite irrelevant. Thus, the Leontief assertion, **p** = **Ap** + **Bw**,
remains and demands do not affect the equilibrium prices, **p*** ｳ
0. In short, the Classical conclusions of the
independence of the determination of prices and quantities maintain themselves even when
expressed in linear programming form.