If you write down an explicitly disequilibrium model with time subscripts
differentiating commodities at different times, thus allowing for prices
of inputs and outputs to differ, one solution will be the equilibrium
prices where the inputs and outputs have the same (relative) prices. This
is also the easiest solution to analyze, and its existence is a good
indication that the equations make sense. As an historical aside, this
seems to be the approach Marx took, for example, in his work on
reproduction schemes.