Master APE  Labor Economics Topics for homework

Master APE Labor Economics Topics for homework

François Langot

November, 2019

Rules

There are 5 topics. Each of them must be solved by one group of two students. A topic can be chosen by only one group of two students. Therefore, after a coordination between you, I will ask you to send me your choice for the topic and the name of your partner. Each student group must send me its homework before April the 1th at 12 pm. You must send me the PDF le of your report, the matlab/octave le for the programs and the excel le for the data. Mail address flangot@univ-lemans.fr.

Topic 1: The Labor Gap

• Find OECD data (frequency = annual) for total hours (H_t, aggregate consumption (C_t) and output (Y_t) for the US, France, the UK, Italy and Germany. You must retain the largest sample.

• Using these data, compute the annual gap between the marginal rate of substitution and the marginal productivity of hours. You will use the following functional forms

U(C_t,1−h_t) = log(C_t) + Θ log(1−h_t) and Y_t=A_tK_t^αH_t^1−α with α = 0.4 and Θ = 2.

• Provide some foundations to the calibrations α and Θ.

• How to interpret these results? Find time series that can explain these gaps in all these countries.

∗Le Mans Université (Gains-TEPP & IRA) & Senior Member of the Institut Universitaire de France &

Paris School of Economics & Cepremap (ENS-Paris)

• Find the time series for Θ_i,t, for each country i and for each periodt that insure that labor gap is equal to zero ∀i, t. Comment.

• Robustness analysis. Reproduce the same exercise with the utility function U(C_t,1−h_t) = C_t^θ(1−Ht)^1−θ1−σ

−1 1−σ

where you will vary the values for σ and θ. Does these alternative scenario change signicantly the shape of the labor gap in each country? You will give an answer of this question for reasonable values of σ and θ. For the time-varying exercise, you will consider time variations within each country of the parameter θ_i,t. Comment.

Topic 2. Transitory vs. permanent changes in government spending

Using the basic RBC model, you will analyze the permanent vs. transitory impact of scal policy. The representative agent maximizes the utility function:

csmax,ls,ks+1Et +∞

X

s=t

β^s−t

log(c_s)− l_s^1+χ 1 +χ

subject to the constraint

c_t+k_t+1 ≤(1−δ)k_t+y_t−t_t ∀ t = 1,· · · ,+∞

where k_t, l_t, c_t and y_t are respectively the capital stock, the number of hours worked, the consumption and the production of the agent. The parameters δ ∈ [0,1), β ∈ [0,1) and χ ∈[0,∞) are respectively the depreciation rate, the discount factor and the inverse of the elasticity of the labor supply. t_t is the lump sum tax paid by the agent, at each date t.

Each agent has access to a production technology allowing him to produce yt: y_t=s_tk_t^ml^1−m_t

where s_t is the technological shock, driven by the following process:

log(s_t+1) =ρ_slog(s_t) +_s,t+1

Assuming that the size of the population is normalized to unity, the budgetary constraint of the government is:

Solve this problem numerically by developing the Matlab or Octave code. During the calibration, it is required that in the steady state g/y = 0.2. You must provide some foundations to your calibration choice.

Using the IRF, you will explain why a permanent increase ing_t does not have the same impact that a transitory shock on g_t.

Topic 3. Unemployment benets and unemployment du- ration

In the model of job search, it is assumed that unemployment benets b(t) depends on the time spent unemployed t. When the worker is employed, her utility is the salary w, whereas when she is unemployed her utility is b −ψ ×a(t) with ψ > 0. ψ ×a(t) represents the disutility of search eort a(t) after t periods of unemployment. The suppliers of labor have a probability π(a(t)) = 1−exp(−φ ×a(t)) to draw an oer. They then have access to a unique wage w. Each job is destroyed with probability s. We denote β the discount factor.

• Write the agent program, assuming that we haveb(t) =bfort = 1, ..., nandb(t) =b/2 for t > n. For a numerical application, taken = 12 months.

• Write the agent program, assuming that we have b(t) =b×(t^−µ).

• Write program for this model (numerical analysis with Matlab or Octave) with b(t) = b×(t^−µ) and for the following calibration parameters:

Table 1: Parameters

ψ s β w b µ

1 0.2/12 0.995 100 75 0.25

In order to complete the calibration, determine the value of φ to obtain an average duration of unemployment of 13 months in an economy whereb(t)is constant∀t, with b = 50.

• In an economy with a deceasing unemployment benets with the unemployment spell and with the previous calibration (b(t) = b×(t^−µ)), analyze the impact of a change of b (the scale of the unemployment benets) and µ(the shape of the unemployment benets) on the results. Explain.

Topic 4. Minimum wage, heterogenous productivity and inequalities

We present an equilibrium on the labor market where there are two types of rms i = 1,2, characterized by their levels of productivity: y₁ < y₂. The exogenous fraction of rms with low productivity level is denoted σ. λ is the probability of obtaining a wage oer.

The prots of each type of rms is:

π = sλ

(s+λ[1−F(w)])²(y_i−w) (1) We dene the distribution function oers pay F(.) as the weighted average salary oer made by the two types of company:

F(w) = σF₁(w) + (1−σ)F₂(w)

where F_i(.) is the distribution function of wage oers posted by the rm-type i.

• Determine the conditions dening the equilibrium, ie the distributions of wage oers and rm prots for each level of productivity, (F1, F2, π1, π2).

• For what value of the minimum wage, rms of type 1 are excluded from the market?

• Write the code le (for Matlab or another tools) of this model for the following param- eters:

Table 2: Parameters

mw s λ y1 y2 σ

0.8 0.287 0.142 2 2.5 0.25

• Explain the impact of a wage increase in wage inequality that is measured by the dierence between the average wage and the mw.

To do this, you compare the impact of this policy in two economies, one whereσ= 0.25 and the other where σ = 0.75, then two other economies, where a{y1, y2} ={2,2.25}, the other where {y₁, y₂}={2,3}. Comment on your results.

Topic 5. Minimum wage, heterogenous outside opportu-

with external opportunities equal to b₁ (b₂). The total population is thenm =m₁+m₂ and the aggregate unemployment rateu=u₁+u₂. λis the probability of obtaining a wage oer.

• Determine the reservation wage R₁ and R₂ of the two types of agent.

• Determine the equilibrium between the Ins and Outs of unemployment for the suppliers of labor of type i.

• Determine the equilibrium ow for a given wage level wage (w).

• Determine the distribution function of wages G(w).

• Show that the number of employees per rm is

for w≥R₂, l(w|R₁, R₂, F) = λ sm

(s+λ[1−F(w)])² for w < R2, l(w|R1, R2, F) = λ sm₁

(s+λ[1−F(w)])²

• The optimal strategy for xing wages is a solution of:

π = max

w (y−w)l(w|R₁, R₂, F)

Show that the introduction of heterogeneity of suppliers of labor can lead to a wage oer distribution with discontinuous support. Explain.

• Write the code le (for Matlab or another tools) of this model for the following param- eters:

Table 3: Parameters

y s λ b₁ b₂ m₁/m₂ 2 0.287 0.142 .6 .8 1

• How the introduction of a minimum wage can reduce ineciencies in the labor market.

What is the threshold level of the minimum wage to reduce unemployment. Explain.