Quantity restriction in model with fixed factor of production 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 &amp;= \left(\frac{\alpha}{R} \frac{W}{1-\alpha} \right)^{1-\alpha} y \quad \quad \quad (1) \\
l &amp;=  \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!
Summary The equilibrium wage will decrease. The reasoning is that, due to the output constraint, total labour demand will go down. Then because of the fixed labour supply (and the fixed price of capital), this makes it that the equilibrium wage falls.
Derivation
We have the first order condition for the (aggregate) labor demand.
$$
l = (1-\alpha)^\alpha W^{-\alpha} R^\alpha \alpha^{-\alpha} y \tag{1}
$$
Take first the case, where there is perfect competition and no output restrictions. Then equalising (inverse) demand and (inverse) supply for the output market gives:
$$
\begin{align*}
&amp;\frac{\phi}{y} = p = R^\alpha \alpha^{-\alpha} W^{1- \alpha} (1-\alpha)^{1- \alpha}. \\
\to &amp;y = \phi R^{-\alpha} \alpha^\alpha W^{\alpha-1} (1-\alpha)^{\alpha-1} \tag{2}
\end{align*}
$$
Subsituting $(2)$ in $(1)$ gives:
$$
l = (1-\alpha)^{2 \alpha-1} W^{-1} \phi
$$
This is the (aggregate) demand for $l$. Labour supply equals $1$ so equilibrium on the labour market requires that:
$$
\begin{align*}
&amp;1 = (1-\alpha)^{2 \alpha - 1} W^{-1} \phi,\\
\to &amp;W = (1-\alpha)^{2 \alpha - 1} \phi. \tag{3}
\end{align*}
$$
This is the equilibrium wage.
Plugging $(3)$ back into $(2)$ gives the equilibrium output.
$$
\begin{align*}
y &amp;= \phi R^{-\alpha} \alpha^\alpha (1-\alpha)^{\alpha-1} (1-\alpha)^{(2 \alpha-1)(\alpha-1)}\phi^{\alpha-1},\\
&amp;= \phi^\alpha R^{-\alpha} \alpha^\alpha (1-\alpha)^{2\alpha(\alpha-1)}
\end{align*}
$$
Notice that we do not need to consider ourselves with the equilibrium on the capital market as $R$ is fixed.
Now the alternative with a fixed supply (use subscripts $f$ for the equilibrium prices). If $\overline{y} &lt; y$ then the price on the output market will be given by:
$$
p_f = \frac{\phi}{\overline{y}}.
$$
The price will no longer equal marginal costs but will be higher. Then aggregate labour demand will be equal to:
$$
l = (1- \alpha)^\alpha W_f^{-\alpha} R^\alpha \alpha^{-\alpha} \overline{y}.
$$
Setting this equal to aggregate supply (which equals 1) gives the equilibrium wage if output is restricted:
$$
W_f = (1-\alpha) R \alpha^{-1} \overline{y}^{1/\alpha}.
$$
Then:
$$
W_f &lt; W,\\
\iff (1-\alpha) R \alpha^{-1} \overline{y}^{1/\alpha} &lt; (1-\alpha)^{2 \alpha-1} \phi,\\
\iff \overline{y} &lt; \phi^\alpha (1-\alpha)^{2\alpha(\alpha-1)}R^{-\alpha} \alpha^\alpha = y
$$
So the wage on the restricted market is lower as we assumed that $\overline{y} &lt; y$.