# Interpreting wedge between MP and wage

I am an econ master student and I need a bit of help understanding an issue, for my thesis. I have two related questions.

Assume a firm producing a good with labor and capital, $Y=F(L,\bar K)$. Capital is fixed, so the firm optimizes only labor. Consider two alternative scenarios:

1) Workers have monopsonistic power in the labor market: the supply of labor is $w(L)$, upward slopping, with elasticity $\eta$. In this case, profits are:

$$\Pi(L) = pY - w(L)L - rK$$

It can be shown that optimal labor is given by:

$$\frac{\partial Y}{\partial L} = \left(1+\frac{1}{\eta}\right)\frac{w}{p}$$

If the elasticity of supply is infinity (competitive market), there is no wedge between MP and wages. If the elasticitity is below infinity (for example 2), there is a positive wedge (term is 1.5) so workers are paid LESS than their competitive wage. This is counterintuitive. Why can the firm pay LESS than competitive wage, if workers are the ones with the power? This is question 1. It is also strange that if supply is perfectly inelastic ($\eta=0$), the term is infinity, which means competitive wage is zero? I am lost. Now to question 2.

2) Firms have monopolistic power in the product market: the demand for output is $p(Y)$, downward sloppoing, with elasticity $\epsilon$ (which is negative)

$$\Pi(L) = p(Y)Y - wL - rK$$

It can be shown that optimal labor is given by:

$$\frac{\partial Y}{\partial L} = \frac{1}{\left(1+\frac{1}{\epsilon}\right)}\frac{w}{p}$$

Here I am very very lost. So, in the case of competitive markets ($\epsilon=-\infty$), workers are paid their MP. If the elasticity is between negative infinity and -1, the wedge is positive. However, if the elasticity is $\epsilon=-1$, the wedge is infinity. What does this mean? What is so special about $\epsilon=-1$? Even worse, if the elasticity is between 0 and -1, the wedge is negative!! What the heck does this mean?

Thanks for taking the time to read this question. If you have any further question, let me know.

Bob.

You are right that for $|\epsilon|\leq1$, you would have issues with the wedge. However, this cannot occur in equilibrium, so it is not a problem.
First note, elasticity is not constant (except in the rare case of a constant elasticity of demand function). The elasticity of demand varies across the points on the demand function. In some regions it is $|\epsilon|>1$ and in others it is $|\epsilon|\leq1$.
This means that $|\epsilon|>1$ always holds for a profit maximizing monopolist and the wedge is always positive.