Analytical question

[50 points]
The setup of the problem is as follows:

• The firm has to choose how much labor to hire in order to maximize profits. The production function is $F(K,N)=AK^{\alpha}N^{1-\alpha}$. Assume that K=1, A=2, and that $\alpha=0$. The real wage (expressed in terms of the consumption good) is w.
• The agent has to choose between working (N) and enjoying leisure. The total available time is normalized to 1, so we have that N=1-L.
• The utility over consumption and leisure is defined as follows:
  
  $$U(C,L)=\gamma C+(1-\gamma)\ln(L)$$
  
  
  where $\gamma=\frac12$. The agent has no wealth and can consume only what she earns working, so:
  
  C=wN.
  
  

a) Write the firm's problem and find the first order condition. Plot the labor demand curve in a graph with N in the X axis and w in the Y axis.
b) Write the agent's problem and find the first order condition. Plot the labor supply curve in a graph with N in the X axis and w in the Y axis (be aware that $0\leq N \leq 1$).
c) Find the equilibrium in the labor market, i.e. the equilibrium w and N. d) Now assume that the utility over consumption and leisure is defined as follows:

$$U(C,L)=\left\{ \begin{array}{cc} -\infty \mbox{ for } C<\bar{... ...1-\gamma)\ln(L) \mbox{ for } C\geq \bar{C} \end{array} \right.$$

, where $\bar{C}=\frac12$ and $\gamma=\frac12$. Write the agent's problem. Find the first order condition. Plot the labor supply curve in a graph with N in the X axis and w in the Y axis. (Note that for $w\leq\frac12$ the labor supply curve is not defined.)