Let's consider a production function \mathbf{f}(\mathbf{X}, \mathcal{L}). Assuming the existence of a nonnegative profit rate \pi, we have the following equation:
\mathbf{p} \cdot \mathbf{f}(\mathbf{X}, \mathcal{L}) = (1 + \pi)(\mathbf{p} \mathbf{X} + w \mathcal{L})
where \mathbf{p} and $\mathcal{L}$ are 1 \times n commodity price vector and labor inputs vector, respectively.
If we assume the production function is of the Leontief type, that is, \mathcal{L} = \mathbf{L} \cdot \mathbf{y} and \mathbf{y} = \mathbf{f}(\mathbf{X}, \mathbf{L} \cdot \mathbf{y}) = A^{-1} \mathbf{X}, we can derive Sraffa's equation for prices:
\mathbf{p} = (1 + \pi)(\mathbf{p} A + \mathbf{L})
By canceling \mathbf{y} from both sides of the equation, as it holds for any possible \mathbf{y}.
However, it is generally observed that we cannot conclude that the price can be determined separately from the production quantity \mathbf{y} or the intermediate input \mathbf{X} and labor input \mathcal{L}. Thus, it is the linearity or the property of fixed input ratios of the production function that allows for the separate determination of price and quantity.
I'm wondering whether there is any in-depth research on this topic. Any recomendations for relevant studies or your own opinions are more than welcomed.
