Model

production function is linear, y_j,t|

j,t>_t=e^ztA_t+_j,t, (2.3) wheree^ztA_tdenotes aggregate productivity and_j,tis a match’s idiosyncratic productivity drawn from a logistic distribution with mean −µ_,t and vari-ance ^ψ

,t2 π²

3 . A firm and a worker jointly decide whether to produce or to separate the match using a cut-offrule_t.If the idiosyncratic shock is lower than the cut-offpoint, the match separates. The cut-offrule is an outcome of the bargaining discussed below. In case of a separation, the worker be-comes unemployed and the firm incurs wasteful separation costsτ_eu,t. Using properties of the logistic distribution we obtain the expected output of a match,

Ye_t = Z ^∞

(e^ztA_t+_j,t)df(_j,t) + Z

−∞

−τ_eu,tdf(_j,t)

= (1−π_eu,t)(e^ztA_t−µ_,t) +Ψ_t (2.4) Ψ_t = −ψ_,t[(1−π_eu,t) log(1−π_eu,t) +π_eu,tlogπ_eu,t]−π_eu,tτ_eu,t, (2.5) whereΨ_t denotes the option value of separation. The probability of separa-tion before the revelasepara-tion of_j,treads

π_eu,t=P(_j,t< ) =







1 + exp

−_t−µ_,t ψ_,t









−1

.

The economy’s total output, available for consumption and investment is given by

Y_t=l_tYe_t. (2.6)

Assume the following process for exogenous productivity witha_t≡log(A_t):

a_t+1−a_t = g_a+x_t+σ_aa,t+1 x_t+1 = ρ_xx_t+σ_xσ_ax,t+1

zt+1 = ρ_zz_t+σ_zz,t+1 (2.7)

where_a,t+1,_x,t+1and_z,t+1areiid standard normal variables. Productivity growth has a deterministic mean componentg_aand two stochastic compo-nents: _a,t+1 is a transitory shock to the technology growth rate,_x,t+1 is a shock to the persistent component of the growth rate.

Vacancy posting and matching The timing in a period is as follows. At the beginning of a period,lt matches exist. The aggregate states of productivity and the idiosyncratic productivities are revealed. Firms and workers bargain over wages or jointly decide to separate a match. Next, the l_t(1−π_eu,t) non-separated matches produce output which is used for consumption and investment. Finally, vacancies and unemployed workers are matched in a frictional market. The sum of old and new matches determines next period’s l_t+1.

The representative firm, which is the mutual fund that owns all matches, discounts dividends with the representative agent’s stochastic discount factor (2.2). It takes the probability of a matchqt as given and chooses vacancies and employment to maximize its cum-dividend stock price,

P_t^c = max

{v_t+τ,l_t+τ}

E_t

∞

X

τ=0

M_t+τD_t+τ s.t.l_t+1 = l_t(1−π_eu,t) +q_tv_t

D_t = l_t

(1−π_eu,t)(e^ztA_t−µ_,t−W_t) +Ψ_t−κ_1,tv_t−q_tv_tκ_2,t v_tq_t ≥ 0.

The firm’s value of a match at the beginning of a period is the derivative of P_t^c with respect tol_t,

J_t≡(1−π_eu,t)(e^ztA_t−µ_,t) +Ψ_t−(1−π_eu,t)W_t+ (1−π_eu,t)E_tM_t+1J_t+1, (2.8) whereW_t denotes the worker’s wage. Firms post vacancies at a costκ_1,t and pay training costs,κ_2,t, if the vacancy is filled. Via the first-order conditions of the problem, we derive the free-entry condition

κ_1,t−Θ_t=q_t(E_tM_t+1J_t+1−κ_2,t). (2.9)

Investment equals the resources spent on the matching process, κ_1,tv_t+q_tv_tκ_2,t.

I follow Petrosky-Nadeau and Zhang (2017) and impose a non-negativity constraint on posted vacancies,v_t≥0. The constraint’s Lagrange multiplier is Θ_t. See Appendix 2.B.2 for a detailed solution of the firm’s dynamic problem.

Assume that firms post vacanciesv_t and find workers according to the matching function

Ξ_m(u_t, v_t) = utvt

(u_t^ι+v_t^ι)^1ι

ι >0 (2.10)

borrowed from den Haan et al. (2000). This function has the advantage that transition rates stay within [0,1]. This improves the computational stability compared to a standard Cobb-Douglas matching function.³ Denoting labour market tightnessθ_t≡ ^vt

u_t, the job-finding rateπ_ue,t and vacancy-filling rateq_t are given by

π_ue,t = (1 +θ_t^−ι)^−1ι (2.11) q_t = (1 +θ_t^ι)^−1ι. (2.12) Finally, labour’s law of motion reads

l_t+1=l_t(1−π_eu,t) +π_ue,t(1−l_t).

Bargaining and separation The representative family earns wages for each productive match and replacement incomebtper unemployed worker.

Replacement income is financed with lump-sum taxes, denoted T_t. The

3When the discounted firm surplus is low, firms post few (or even no) vacancies. With a Cobb-Douglas matching function, a reduction of vacancies increases job-finding without bound, jeopardizing stability. Instead of the new matching function, one could restrict the vacancy-filling rate, but this introduces a kink which again jeopardizes stability. The den Haan et al. (2000) function restricts the elasticity with respect to job-seekers, _1+θ¹⁻ι∈[0,¹₂]: i) Unless the job-finding rate exceeds the vacancy-filling rate, labour market tightnessθ=^π_q^ue is smaller than one. Hence, the upper bound for the elasticity of the matching function with respect to unemployment is ¹₂achieved atι→0. Sedl´aˇcek (2016) and others estimate this elasticity to exceed ¹₂ (see their Table 1). ii) If one assumes Nash bargaining and the Hosios condition at steady state values, the bargaining power of job-seekers cannot exceed¹₂ for the same reasons.

mutual fund’s sharess_t trade at priceP_t. BondsBe_t return a risk-free interest rateR^f_t. The family’s consumption reads

C_t=W_tl_t(1−π_eu,t)−T_t+b_t(1−l_t+π_eu,tl_t) +s_t(D_t+P_t)−s_t+1P_t+Be_t− 1 R^f_t+1

eB_t+1.

The family’s value of an additional worker in terms of consumption goods reads

∆_t≡

∂V_t

∂l_t

(1− ¹

ψ)C

−¹

ψ

t

= [(W_t−b_t)(1−π_eu,t)] +E_tM_t+1∆_t+1(1−π_eu,t−π_ue,t).

Define the joint surplus of a matchΣ_t ≡∆_t+J_t and denote the worker’s bargaining power with%_t. Nash bargaining determines the wage and the separation rate,

(π_eu,t, W_t) = arg max

πeu,t,Wt

∆^%_t^tJ_t^1−%t. (2.13) The first-order condition of (2.13) yields the efficient reservation produc-tivity_t=b_t−e^ztA_t−τ_eu,t−E_tM_t+1Σ_t+1. If the idiosyncratic productivity of a match falls below this cut-offvalue, a match is separated. By property of the logistic distribution the separation rate reads

π_eu,t =

"

1 + exp E_tM_t+1Σ_t+1+e^ztA_t−µ_,t+τ_eu,t−b_t ψ,t

!#⁻1

(2.14) Via the free-entry condition (2.9) and the first-order conditions of (2.13), obtain the Nash wage (see Appendix 2.B.3)

W_t=

(1−π_eu,t) ⁻1

%_t

(1−π_eu,t)(e^ztA_t−µ_,t) +Ψ_t+ (1−π_eu,t)E_tM_t+1J_t+1 + (1−%t)

bt(1−πeu,t)−E_tMt+1∆_t+1(1−πeu,t−πue,t)

. (2.15) To introduce wage rigidity, I follow Jung and Kuester (2015) and assume that the worker’s bargaining power decreases in the persistent component of

the technology growth ratex_tand TFP’s transitory componentz_t,

%_t= ¯%e^{−xtαx−ztαz}. (2.16) This turns the shocks_z and_xinto joint shocks to wages and productivity, or surplus shocks. In Section 2.4, I describe why this assumption is useful in the context of a long-run risk model. In short: when the worker’s outside option and vacancy-posting costs are proportional to productivity, the fun-damental surplus is not volatile enough to solve the Shimer puzzle, even if the fundamental surplus is calibrated to be small.

Aggregation and productivity adjustment Bonds are in zero net supply, Be_t = 0 ∀t, and the representative family owns the mutual fund, s_t = 1 ∀t.

Taxes are used to pay unemployment benefits, so T_t = b_t(1−l_t+π_eu,tl_t).

Together, aggregate consumption reduces to output minus investment, C_t=W_tl_t(1−π_eu,t) +D_t=Y_t−κ_1,tv_t−q_tv_tκ_2,t.

Finally, the return of the risk-free bond reads 1

R^f_t+1

=E_tM_t+1. (2.17)

See Appendix 2.B.2 for a derivation of stock returns and the stock price, P_t= κ_1,t+q_tκ_2,t

q_t −Θ_t

!

| {z }

E_tM_t+1J_t+1

l_t+1. (2.18)

The term in parentheses is Tobin’s marginal q, the shadow price of employ-ment in the firm’s optimization problem.

I solve the stationary version of the model (Appendix 2.B.1). In short, Atscales the constant parameters{µ, ψ, κ1, κ2, τeu, b}, e.g.bt=bAt. Define the productivity-adjusted variablesc_t= ^C_A^t

t,w_t= ^W_A^t

t d_t= ^D_A^t

t,p_t=_A^Pt

t,Ψe_t= ^Ψ_A^t

t, eΣ_t= ^Σ_A^t

t. The model is solved globally with the projection method outlined in Appendix 2.F.1 and estimated with the method of simulated moments outlined in Appendix 2.F.2. The next two sections parametrize the models:

First, Section 2.3 parametrizes the RBC model with cyclical fluctuation z_t. Next, Section 2.4 parametrizes the LRR model with a small, persistent component of growthx_t. In each section, I simulate the model and match the parametrized model to unemployment and output time series to examine the models’ predictive power for other variables.

Im Dokument Inheritance taxation, unemployment, and asset pricing in frictional labour markets (Seite 81-88)