I'm trying to see the effect of a restriction on production in a model where one factor of production is perfectly elastic and the other is fixed.
Specifically, suppose the production function is Cobb-Douglas: $y = k^\alpha l^{1-\alpha}$. Then cost minimisation gives \begin{align} k &= \left(\frac{\alpha}{R} \frac{W}{1-\alpha} \right)^{1-\alpha} y \quad \quad \quad (1) \\ l &= \left(\frac{1-\alpha}{W} \frac{R}{\alpha} \right)^\alpha y \quad \quad \quad (2) \end{align} where $R$ is the price of $k$ and $W$ is the price of $l$. The goods market is perfectly competitive, so price equals marginal cost: $$ p = \left(\frac{R}{\alpha}\right)^\alpha \left(\frac{W}{1-\alpha}\right)^{1-\alpha}. \quad \quad \quad (3) $$ Demand for the good is given by $$ y = \frac{\phi}{p} \quad \quad \quad (4) $$ where $\phi$ is a parameter. Equations (1) - (4) give four equations in the six variables $y, k, l, p, R, W$. Now suppose $k$ is perfectly elastic and $l$ is perfectly inelastic, i.e. $$ R = \bar{R} \quad \quad \quad (5) \\ l = 1 \quad \quad \quad (6). $$ I can solve these six equations in six variables. But now suppose I want to introduce a quantity restriction, $y \leq \bar{y}$ (where $\bar{y}$ is sufficiently low as to be binding). To avoid an overdetermined system, which of the six original equations do I drop? Alternatively, do I redo an optimisation equation, including this extra constraint, so that there is an additional Lagrange multipler variable?
Intuitively, I think that the price $W$ of the inelastic factor of production $l$ will be bid up, but I'm not sure how this mathematically arises.
Thanks!