Activity 1.1 : problem set 1

Consider a continuum of workers and rms. Each rm has only one job. The total mass of the population is normalized to one, which implies that the unemployment rate is equal to the level of unemployment. Let be the number of vacancies, the matching function is given by:

1. Define the transition rate for a worker from unemployment to employment

as a function of the labor market tightness . De ne also the transition

rate for a vacancy position as a function of .

2. Consider an exogenous rate of job destruction , and compute the steady-state unem-ployment rate as a function of and .

3. Let denote the wage, productivity of a job-worker pair, and the

recruitment cost, by using Bellman equations, write down (i) the steady-state value of a rm with a lled position (denoted ), (ii) the steady-state value of a vacant job (denoted ).

4. Write the free entry condition which characterizes as a function of , , ,

, .

5. Consider now that the wage equation solves , write down the two

dimensional system of equations which de ne steady-state equilibrium

values of labor market tight-ness and unemployment rate .

6. The government introduces an employment subsidy paid to the employee

such that wage equation is now de ned by . What is the impact

of on equilibrium labor market tightness and unemployment rate? Draw a picture of the equilibrium with and without the employment subsidy.

I. Activity 1.2 : problem set 2

We consider the following matching model. The matching function is given by

There is free entry in vacancy posting and the cost of posting a vacancy per unit of time is equal to . Wages are set by intertemporal Nash bargaining so that

with the surplus where , , , are the intertemporal values

for employed, unemployed, filled job and vacant job. The job destruction rate is and the interest rate is :

1. Define the contact rate for firm, , with . Check that has

constant returns to scale and is concave in each of its arguments. What is the elasticity of with respect to as a function of ?

2. Show that the Beveridge curve has the following equation:

3. Write down the value functions , , , denoting the productivity and the unemployed

income.

The basic matching model

4. Show that the equilibrium value of must satisfy:

5. What is the steady-state unemployment rate ? How does it depend on , , , and ? Why?

6. Show that the following relationship exists between the wage and :

7. How do wages depend on , , c, and ?

8. Consider . Write down the central planner's problem, the Hamiltonian,

and the

corresponding first-order conditions.

9. Show that in steady state the optimal value of must satisfy

10. What relationship must hold among the model's parameters for the optimal value of

to be the same as the equilibrium value of ?

11.Check that this is equivalent to

with being the common solution to the two allocations.

12. How does this last condition relate to the Hosios efficiency condition?

Consider a continuum of workers and rms. Each rm has only one job. The total mass of the population is normalized to one, which implies that the unemployment rate is equal to the level of unemployment. Let be the number of vacancies, the matching function is given by:

1. Define the transition rate for a worker from unemployment to employment

as a function of the labor market tightness . De ne also the transition

rate for a vacancy position as a function of .

2. Consider an exogenous rate of job destruction , and compute the steady-state unem-ployment rate as a function of and .

3. Let denote the wage, productivity of a job-worker pair, and the

recruitment cost, by using Bellman equations, write down (i) the steady-state value of a rm with a lled position (denoted ), (ii) the steady-state value of a vacant job (denoted ).

4. Write the free entry condition which characterizes as a function of , , ,

, .

5. Consider now that the wage equation solves , write down the two

dimensional system of equations which de ne steady-state equilibrium

values of labor market tight-ness and unemployment rate .

6. The government introduces an employment subsidy paid to the employee

such that wage equation is now de ned by . What is the impact

of on equilibrium labor market tightness and unemployment rate? Draw a picture of the equilibrium with and without the employment subsidy.

I. Activity 1.2 : problem set 2

We consider the following matching model. The matching function is given by

There is free entry in vacancy posting and the cost of posting a vacancy per unit of time is equal to . Wages are set by intertemporal Nash bargaining so that

with the surplus where , , , are the intertemporal values

for employed, unemployed, filled job and vacant job. The job destruction rate is and the interest rate is :

1. Define the contact rate for firm, , with . Check that has

constant returns to scale and is concave in each of its arguments. What is the elasticity of with respect to as a function of ?

2. Show that the Beveridge curve has the following equation:

3. Write down the value functions , , , denoting the productivity and the unemployed

income.

The basic matching model

4. Show that the equilibrium value of must satisfy:

5. What is the steady-state unemployment rate ? How does it depend on , , , and ? Why?

6. Show that the following relationship exists between the wage and :

7. How do wages depend on , , c, and ?

8. Consider . Write down the central planner's problem, the Hamiltonian,

and the

corresponding first-order conditions.

9. Show that in steady state the optimal value of must satisfy

10. What relationship must hold among the model's parameters for the optimal value of

to be the same as the equilibrium value of ?

11.Check that this is equivalent to

with being the common solution to the two allocations.

12. How does this last condition relate to the Hosios efficiency condition?

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