For brevity I have deleted much of the original post and paraphrased
parts:

>               [ I - A ] Q = Y.


[where A is a "technique" matrix, Q is the vector of gross outputs,
Y is the vector of net outputs]

>                       -1
>          Q = [ I - A ]   Y.


.....

>                           -1
>      v = L Q = L [ I - A ]   Y.


[where v is a vector of employment levels for different types of labor,
and L is another "technique" matrix]

....
                        T
Numeraire assumption:  p Y = 1

where p is the vector of stationary output prices

>     p(1) Y(1) + p(2) Y(2) + ... + p(n) Y(n) = 1.



Rate of profit for process j:

>          r = [ p(j) - In(j) ]/In(j).


[Assumed constant across processes]

> If two processes operate with different rates of profit, firms will tend
> to discontinue the process with the lower rate and enter the one with
> the higher rate.  Classical equilibrium exists when the tendency for
> firms to seek the highest profit has leveled the rate of profit to an
> equal value in all processes.  This yields the following system of
> matrix equations, which defines the "prices of production" obtaining in
> an equilibrium system for any given physical list of inputs and outputs:

>           T      T               T
>        ( p  A + w  L )(1 + r) = p .


[where w is the vector of wage rates.]

Actually classical equilibrium is based on the two notions:

1)  Firms will move out of processes with negative rates of profit
2)  Firms will move into processes with positive rates of profit

These together require that in equilibrium the rate of profit will be
zero for any process with positive output, but may be negative for a
process with zero production.  So we have equal rates of profit in all
processes if and only if r = 0.


> With a little manipulation one obtains:

>      T                    -1  T
>     p  = [ I - (1 + r) A ]   w  L (1 + r).


Check your linear algebra!  This should be:

      T    T                            -1
     p  = w  L (1 + r) [ I - (1 + r) A ]

> Multiplying both sides by net output, Y, gives:

>                          -1  T
>     1 = [ I - (1 + r) A ]   w  L Y (1 + r),


which should actually be

          T                            -1
     1 = w  L (1 + r) [ I - (1 + r) A ]   Y

          T            -1       T        T
or   1 = w  L [ I - A ]   Y =  w  L Q = w  v   if r = 0.

> which implicitly defines a function specifying the rate of profit, r, as
> a function of the various wage rates. 


and also of the equilibrium level of output that has not yet
been determined.

> This function has some interesting properties.  First, if all wages are
> zero, the equal profit rate system defines a maximum rate of profit,
> which, usually, will be finite.


If all wages are zero then the above equation reduces to 1 = 0.  All
wages cannot be zero if we are to accept your framework.

>   Second, if all wages but one are fixed,
> increasing that one wage will result in a lower rate of profit.


This is simple accounting.  You have fixed real output and the real
price of all inputs except one.  If you raise the price of the
remaining input, profit rates must decline somewhere, hence everywhere
by your equal profit rate assumption.

>   Third,
> if the rate of profit and all wages but two, are fixed, increasing one
> of these wages will lower the remaining wage.  Third, if all wages but
> one are fixed, that wage will have a maximum value corresponding to a
> zero rate of profit.


These are equally obvious, for the same reasons.

> These relationships can be interpreted as
> reflecting the distribution of a physically specified surplus in money
> terms.  And they are not as obvious as they may seem, for in a slightly
> more general model the rate of profit may increase with an increasing
> wage.


You have not told us what you are assuming about labor supply.  Are
you adopting the assumption of perfectly inelastic supply commonly
used in neoclassical models?  If so then we must add the constraints
that employment does not exceed the supply of labor and also that
wages are positive only for labor that is fully employed.  Your model
of production cannot be closed without some kinds of restrictions like
these on factor markets.


> 4.0 Choice of Technique

> The above has shown how, given a physical specification of a technique
> by matrices A and L, and given the wages w of all types of labor, one
> can determine the rate of profit in the corresponding equilibrium
> "prices of production" system.  Given two or more alternative
> techniques, (A1, L1), (A2, L2), ..., one can find the rate of profit
> corresponding to each system of wages.  Profit maximizing will lead to
> the technique with the highest rate of profit becoming the equilibrium
> technique corresponding to the given wages and net outputs.  In general,
> this technique will vary with wages. 


I don't see how you can compute equilibrium prices and allocations at
all.  Given the neoclassical assumptions, if all goods are being
produced then we must have a zero rate of profit in each sector.
Other values for the rate of profit are evidence that we are not
looking at a neoclassical equilibrium.

So we must have some other notion of equilibrium at work here.  What
do you have in mind?

> 5.0 Conclusions

> This note has presented a classical model of equilibrium demonstrating
> that the effects on the profitability of alternate techniques associated
> with a change in wages provides no basis for beliefs in a well-behaved
> demand curve for labor such that lower wages are associated with a
> greater demand for labor. 


I'm afraid that your claim to a "classical model of equilibrium" is
invalid.

> This model has other destructive consequences for Neoclassical beliefs. 
> In particular, examples can be constructed in which two viable
> techniques are such that the labor requirements of one technique are
> less than the requirements of the other technique for all types of
> labor.  Since these two techniques are assumed to produce the same net
> output, efficiency demands that the technique with the lower labor
> requirements be used.


Not at all.  Your productions processes have inputs other than labor.
There is no reason to expect that a technique will be more efficient
simply because it uses less of all kinds of labor.  The labor
requirements might be offset by increased requirements for other kinds
of inputs.

>   But, for some combinations of wages, the
> technique that maximizes the rate of profit may very well be the other
> technique.  In this model of competition, maximizing the rate of profit
> may very well lead to the choice of a socially inefficient technique. 

> Finally, this model suggests that the marginal productivity theory of
> distribution "is all bosh" (Joan Robinson).  No physically specified
> unit of measurement of capital can be found such that the marginal
> product of capital equals the rate of profits.  Likewise for labor(s).


This model does not have anything resembling capital in it!
Furthermore, nothing in neoclassical theory suggests that the "rate of
profit" as you have defined it should have anything to do with the
marginal product of capital.

> So one can perfectly well deny the vision underlying Neoclassical
> theory of how prices function, including factor prices, and still
> investigate rigorous economic theory.


Perhaps, but this doesn't quite make it.