# 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 &= \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!

• Your equations (1) and (2) are not correct as the cost minimizing levels of $k$ and $l$ should depend on output $y$ (you are minimizing costs conditional on some level of output). Next, you can't answer your question without specifying the market structure for the output (perfect competition, monopoly, oligopoly...). Finally, a monopolist never sells on the inelastic part of the demand curve (you have a unit elastic demand curve). Your revenue equals $\phi$ (constant). As marginal costs are also constant a monopolist will produce $0$ at infinite price making $\phi$ profits.
– tdm
May 24, 2021 at 16:03
• I accidentally left out the $y$ in eqs (1) and (2). I have fixed them now - thanks! Re your other points, I specified that the goods market is perfectly competitive - that's where the price = marginal cost in (3) comes from. Did you have some other condition in mind?
– John
May 24, 2021 at 16:43

## 1 Answer

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*} &\frac{\phi}{y} = p = R^\alpha \alpha^{-\alpha} W^{1- \alpha} (1-\alpha)^{1- \alpha}. \\ \to &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*} &1 = (1-\alpha)^{2 \alpha - 1} W^{-1} \phi,\\ \to &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 &= \phi R^{-\alpha} \alpha^\alpha (1-\alpha)^{\alpha-1} (1-\alpha)^{(2 \alpha-1)(\alpha-1)}\phi^{\alpha-1},\\ &= \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 $$\overline{y} < y$$ 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: $$W_f < W,\\ \iff (1-\alpha) R \alpha^{-1} \overline{y}^{1/\alpha} < (1-\alpha)^{2 \alpha-1} \phi,\\ \iff \overline{y} < \phi^\alpha (1-\alpha)^{2\alpha(\alpha-1)}R^{-\alpha} \alpha^\alpha = y$$ So the wage on the restricted market is lower as we assumed that $$\overline{y} < y$$.

• I can see how your response follows from dropping equation (3), i.e. the 'price = marginal cost' condition. But isn't dropping this condition inconsistent with the perfectly competitive market structure? For example, any firm could cut its price slightly, capture the entire market share, and make greater profits.
– John
May 25, 2021 at 0:38
• @John. You assume perfect competition which means that firms are price takers and do not consider what will happen to their individual demand if one of them drops its price (i.e. they set quantities and consider prices as exogenous). What you are describing is Bertrand competition. Bertrand competition will indeed lead to price = marginal costs.
– tdm
May 25, 2021 at 5:16