<p>The profit function is an <em>optimal</em> relation. So it must satisfy the first-order conditions for a maximum:</p>

<p>$$ \text{f.o.c} : w^*: PL' - L - w^*L' =0 \implies L\left[ (P-w^*) \frac {L'}{L}-1\right] = 0$$</p>

<p>In order for this condition to be satisfied no matter what the wage is, it must be the case that, for all wage-levels we have </p>

<p>$$(P-w) \frac {L'}{L}-1 = 0 \;\;\forall w \implies L' = \frac{1}{P-w}L \;\;\forall w$$</p>

<p>Of course this looks exactly like the first-order condition - <strong>the critical difference is that it now must hold for all $w$</strong>, and so it is not any more a condition that can hold for a specific $w$, but a differential equation in $w$, that falls in the general category $y'_x = f(x)\cdot y$. It can be solved (it is "separable").</p>

<p>I guess you can finish this.</p>

CC BY-SA 3.0
2016-02-19T01:43:14.360Z
2016-02-19T02:03:42.737Z
2016-02-19T02:03:42.737Z
Alecos Papadopoulos
<p>Suppose a monopsonistic firm has the following simple profit function:
$$\pi (w) = PL(w) - wL(w) $$
where $L(w)$ is the labor supply, $P$ is the constant output price, and it has a linear production technology.</p>

<p>What is a labor supply schedule $L(w)$ that would make the profit function insensitive to the wage? </p>

CC BY-SA 3.0
2016-02-18T21:10:27.333Z
This ends up requiring solving a not-totally-trivial non-linear differential equation. Are you familiar with the subject?
Alecos Papadopoulos
CC BY-SA 3.0
2016-02-19T01:21:55.217Z
yes, familiar with diff eq, could you point me along further with what you mean? i am having trouble with setting this up.
user104007
CC BY-SA 3.0
2016-02-19T01:32:14.920Z
Sorry should have tagged you @Alecos Papadopoulos
user104007
<microeconomics><labor-economics><theory>
Static wage posting model and indifference to wage levels
user104007
# Static wage posting model and indifference to wage levels

(Asked by: username_0 on 2016-02-18)

Suppose a monopsonistic firm has the following simple profit function:
$$\pi (w) = PL(w) - wL(w) $$
where $L(w)$ is the labor supply, $P$ is the constant output price, and it has a linear production technology.

What is a labor supply schedule $L(w)$ that would make the profit function insensitive to the wage?

**Tags:** microeconomics, labor-economics, theory
> **Comments:**
> **username_1**: This ends up requiring solving a not-totally-trivial non-linear differential equation. Are you familiar with the subject?  
> **username_0**: yes, familiar with diff eq, could you point me along further with what you mean? i am having trouble with setting this up.  
> **username_0**: Sorry should have tagged you @Alecos Papadopoulos  

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## 1 answers

### Answer by username_1 on 2016-02-19. Score: 1 | Accepted answer

The profit function is an *optimal* relation. So it must satisfy the first-order conditions for a maximum:

$$ \text{f.o.c} : w^\*: PL' - L - w^\*L' =0 \implies L\left[ (P-w^\*) \frac {L'}{L}-1\right] = 0$$

In order for this condition to be satisfied no matter what the wage is, it must be the case that, for all wage-levels we have

$$(P-w) \frac {L'}{L}-1 = 0 \;\;\forall w \implies L' = \frac{1}{P-w}L \;\;\forall w$$

Of course this looks exactly like the first-order condition - **the critical difference is that it now must hold for all $w$**, and so it is not any more a condition that can hold for a specific $w$, but a differential equation in $w$, that falls in the general category $y'\_x = f(x)\cdot y$. It can be solved (it is "separable").

I guess you can finish this.


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