1. Construct the Bellman equation for the problem faced by
(i) a type i individual who is unemployed at the beginning of the period,
(ii) a type i individual who is employed at the beginning of the period,
(iii) a firm attached to an employed individual of type i, and
(iv) a firm with an unfilled vacancy. Do this fori = C, M.
2. Write down the Nash Bargaining problem for a type i worker-firm pair.
3. Define an equilibrium for the aggregate labour market.
4. Find the steady state expressions that characterize the equilibrium wage, market tightness and unemployment rate for both types of workers, that is, (wi, ?i, ui),i = C, M.
5. Using the parameter values in Table 1, solve for the steady state of the model. In doing so, find a value for z such that the steady state unemployment rate is 5%.
6. Construct the Markov Chain for the cost of a vacancy using the rouwenhorst.m function. In this part of the project, given the steady state vacancy cost is z, let “zgrid” be the grid of points returned by the rouwenhorst.
m function such that [zgrid, PI] = rouwenhorst(rhoZ,seZ, nstatez)where nstatez = nz, seZ = ?z and rhoZ = ?z are the Matlab variable name equivalents of the variables in Table 1.
The rouwenhort.m function returns the nstatez × nstatez transition probability matrix for the vacancy costs, zi, for i = 1, ..., nz. Construct a grid for z using the command:
ezt = exp(zgrid) ? z (you can label z as “zbar” in your code). This will give you a nz × 1 array of values for zi > 0 sorted in ascending order and centered around the steady state value of z. (See class notes about the construction of Markov Chains.)
7. Solve for the equilibrium values of ?C(z) and ?M(z) in the stochastic labour market economy.
8. Simulate a 48 period impulse response functions of your economy for the experiment in which the economy initially is in its steady state, then the vacancy cost rises from its median state to state z20 for 10 periods and then drops straight back to its median value.
In a 2 ×3 subplot, plot the impulse responses for the following:
(a) Plot 1: Vacancy Costs
(b) Plot 2: The unemployment rate for i = C, M as well as the aggregate unemployment rate
(c) Plot 3: The job finding probability for i = C, M
(d) Plot 4: The job filling probability for i = C, M
(e) Plot 5: The wage for i = C, M
(f) Plot 6: Type C workers’ share of total employmentIn any plot that consists multiple lines, include a legend and ensure that each line has a distinct colour.