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The question is as follows: The Happyland Hospital is a monopsonist employer of nurses in the small city of Happyland. The market supply function of nurses is $S(W) = 0.1W - 100$, where $W$ is the nurses weekly wage. What is the hospital's marginal expenditure, ME? If the hospital's marginal benefit of a nurse is $2,000 per week, no matter how many nurses it hires, what is the profit-maximizing number of nurses for the hospital to hire? What will the nurse's wage be? What is the deadweight loss?

And my work so far is as follows:

$Q_s = 0.01W - 100$

$0.01W = Q + 100$

Inverse Supply: $W = 100Q + 10,000$

$ME = W+(dW/dQ)Q = (10,000 + 100Q) + 100Q$

$ME = 10,000 + 200Q$

Profit Maximization - $MB = ME$

$10,000 + 200Q = 2000$

I first derive the Inverse supply curve then use that to derive marginal revenue. Now I'm setting $MB = ME$ as this is the profit maximizing point, but the Q it yields is negative. $(-40)$. Have a feeling this is a problem point. Am I going wrong here?

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The question you are showing work out for where $S(w) = 0.01W - 100$ means that you will have no nurses who want to work unless you have a minimum wage of over 10,000 per week, and since the marginal benefit of every nurse is 2,000 per week, this hospital will not be in business. Your original question you typed at the very beginning is making you work with $S(w) = 0.1W - 100$. So let's try approaching that question instead. It also seems you need some help understanding how to get expenditure, so let's go through that too.

First, we find inverse supply.

$$W = 10S + 1000$$

Expenditure is the price of labor times quantity of labor.

$$E = WS = 10S^2 + 1000S$$

The derivative of this is your marginal expenditure.

$$\frac{dE}{dS} = 20S + 1000$$

Set up the profit maximization problem and take the derivative (which we set to zero).

$$\begin{align} \pi & = 2000S - (10S^2 + 1000S) \\ \frac{d\pi}{dS} & = 2000 - 20S - 1000 = 0\\ & = 1000 = 20S\\ \implies & \boxed{S^* = 50} \\ \end{align}$$

Then to calculate wage, you just plug in the optimal supply to your inverse supply function. To find dead-weight loss, try drawing out a graph for labor supply and demand. Labor demand should be a straight line across from $W = 2000$. Labor supply was given to you. Notice where the labor supplied equals labor demand. What would $S$ equal there and what wage would be set there? It should be different from the $S^*$ we got. That should give you enough insight to the rest of the question.

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