Labour Market: simple macroeconomic modeling

Question

I just started reviewing macroeconomic modeling for an upcoming entry exam, and I'm not sure about my answers in this application:

Suppose that the economy of a country has a representative firm with the following production function $y = f(N) = 2N - (1/2)N^2$ knowing that $N$ is the size of employment.

1) Write the profit of this firm knowing that the price of the good is $P$ and the wage is $W$.
2) Determine the demand for labour which is the optimal size of this firm. Comment on your results.
3) From the last question, deduce the optimal size of production which is the optimal supply in volume of the good produced by this firm. Comment on your results.
4) Suppose the total supply of labour is exogenous, and is given by: $N^o = (W/P)$. Define the notion of labour market equilibrium, and calculate the equilibrium real wage of this economy.
5) At this general equilibrium, calculate total production (in volume), the total real payroll, total real profits (profits deflated by the price), and comment on the distribution in this economy.

Admittingly, I'm still quite weak when it comes to macroeconomic modeling, I haven't yet found a good enough source so all my knowledge comes from scattered sources all over the web.

But here's my attempt:

1) I know profits are
$$\pi = Revenue -Cost \equiv P.y - W.N$$ Where P is the price of the good, y the quantity produced, W the wage and N the size of employees.

Since the quantity produced is given as a function of N I re-wrote that expression into: $$ P.y - W.N \equiv P. 2N-(1/2)N^2 - W.N$$

Which can simplify to: $$\pi = P.N(\dfrac{-N+4}{2}-w)$$

Not sure if this is what they mean but I stopped here.

2) I've read somewhere that optimal demand for labour is reached when $Marginal Benefit = Marginal Cost$

Marginal Benefit is defined as $Marginal Product \times Price$ so in our case that should be the derivative of the Production Function (marginal product) times price, or $MB = (2-N)\times P$ I've read that Marginal Cost is the real wage $\dfrac{W}{P}$
So at equilibrium $(2-N)P = \dfrac{W}{P}$ which after some computation we get $$N=-\dfrac{W-2P^2}{P^2}$$ which should be the demand for labour at equilibrium.

3) Since N we found is (supposedly) the optimal demand for labour, and production of the good is a function of N alone, I thought we'd get the optimal size of production by inputting the expression of $N$ we just found in the production function $y$ that was given to us. After some long computations, I got: $$y=-\dfrac{W^2}{2P^4}+2$$

4) The real wage at equilibrium is obtained when Demand for Labour equals Supply for Labour. Using the exogenous expression for Labour Supply we were given this means: $$\dfrac{W}{P}=-\dfrac{W-2P^2}{P^2}$$

Which when solved get $$p=1, w=1$$ and thus real wage would also be equal to $1$

5) Replacing values in various expressions we derived in prior questions I got: $$N=1$$ $$y=\dfrac{3}{2}$$ $$payroll=\dfrac{W}{P}.N = 1$$ $$\pi =\dfrac{1}{2}$$

Now I'd be very surprised if I had any answer right since I'm very new to this, so I'd appreciate some feedback on my answers. Also, I'd appreciate some help with the comments on the answers especially the one about the distribution.

Thanks.

Answer 1

To find demand for labor, solve the firm's profit maximization problem : \begin{eqnarray*} \max_N & \ P(2N - 0.5N^2) - WN\end{eqnarray*} Differentiating the objective with respect to $N$ and setting it equal to 0 yields the demand function for labor : \begin{eqnarray*} N^d = 2 - \frac{W}{P}\end{eqnarray*} Corresponding supply of output is

\begin{eqnarray*} Y = f\left(2 - \frac{W}{P}\right) = \left(2 - \frac{W}{P}\right)\left(1 + \frac{W}{2P}\right) = 2 - \frac{1}{2}\left(\frac{W}{P}\right)^2\end{eqnarray*}

Given the supply of labor is \begin{eqnarray*} N^s = \frac{W}{P}\end{eqnarray*}

Equilibrium in the labor market occurs where $N^d = N^s$. Solving it, we get equilibrium employment, real wage, output, profits as follows : \begin{eqnarray*} N^* & = & 1 \\ \left(\frac{W}{P}\right)^* & = & 1 \\ Y^* & = & 1.5 \\ \pi^* & = & 0.5 \end{eqnarray*}