The model

The worker consumesc_e,tⁱ . With probability 1−ξⁱ_tthe match does not separate and continues intot+1. With probabilityξⁱ_t, instead, the match separates. The worker can immediately start searching for new employment. The only difference of its value to the value of a worker, who was unemployed to start with, therefore, is that the separated worker receives the sev-erance payment while the unemployed worker receives unemployment benefits (and, thus, lower consumption).V_u,tⁱ is the value of a worker who starts the period unemployed.

Value of an unemployed worker and search

An unemployed worker needs to actively search in order to find a job. All workers whose disutility of search falls below a cutoff valueι^s,it do so.⁷At the cutoff, the utility costs of search just balances with the expected gain from search:

(2.4) ι^s,it =f_tⁱβEt

h∆ⁱ_t+1i .

Here∆ⁱt =V_e,tⁱ −V_u,tⁱ marks the gain from employment and f_tⁱ marks the job-finding rate.

For tractability, the paper assumes thatF_ι(0,σ²_ι) is the logistic distribution with mean 0 and varianceσ²_ι :=π^ψ₃^2s, where a lower-caseπrefers to the mathematical constant and parameter ψs >0. Using the properties of the logistic distribution, the share of unemployed workers who search is given by

(2.5) sⁱ_t=P r ob(ι≤ι^s,i_t )=1/[1+exp{−ι^s,i_t /ψs}].

With this, the value of an unemployed worker at the beginning of the period, before the search preference shock has realized, is given by

(2.6)

V_u,tⁱ = u(c_u,tⁱ )+h +R_ι^s,i_t

−∞d F_ι(ι)+sⁱ_th

f_tⁱβEtV_e,tⁱ ₊₁+[1−f_tⁱ]βEtV_u,tⁱ ₊₁i +(1−s_tⁱ)βEtV_u,t+1ⁱ .

In the current period, the worker consumescⁱ_u,t and enjoys utility of leisureh(first row). If the worker decides to search (second row), the utility cost is ι. The term with the integral is the expected utility cost of search. With probability f_tⁱ the searching worker will find a job. In that case, the worker’s value at the beginning of the next period will beV_e,tⁱ ₊₁. With probability (1−f_tⁱ) the worker remains unemployed in the next period. If the worker does not search (third row), the worker remains unemployed.

2.2.1.2. Firms. Profits in the firm sector accrue to the workers, all of whom hold an equal amount of shares in the domestic firms. The decisions made by firms are dynamic and in-volve discounting future profits. We assume that firms discount the future using discount factorQⁱ_t,t+s, whereQⁱ_t,t+s:=β^λ_λ^it+i^s

t

, withλⁱtbeing the weighted marginal utility of the workers (the firms’ owners):

(2.7) λⁱt:=

"

eⁱ_t

u⁰(c_e,tⁱ )+ u_tⁱ u⁰(c_u,tⁱ )

#₋₁ .

7Thesinι^s,i_t stands for thesearch cutoff.

This reflects that masseⁱ_t of workers are employed at the beginning of the period and mass uⁱ_t:=1−eⁱ_tare unemployed.

Firms need a worker to produce output. A firm that enters the period matched to a worker can either produce or separate from the worker. Production entails a firm-specific resource cost,²j. This fixed cost is independently and identically distributed across firms and time with distribution functionF_²(µ²,σ²_²).F_²(·,·) is the logistic distribution with meanµ²and vari-anceσ²_² =π^ψ₃^2², withψ² >0. The firm separates from the worker (first line) whenever the idiosyncratic cost shock,²j, is larger than a state-dependent threshold²^ξ_t^,i. Using the prop-erties of the logistic distribution, conditional on the threshold, the separation rate can be expressed as

(2.8) ξⁱt=P r ob(²j≥²^ξ,i_t )=1/[1+exp{(²^ξ,i_t −µ²)/ψ²}].

Ex ante, namely, before the idiosyncratic cost shock²jis realized, the value of a firm that has a worker is given by

(2.9) J_tⁱ = −ξⁱt

hτⁱ_ξ,t+wⁱ_ti

−R_²^ξ_t^,i

−∞²jd F_²(²j)+(1−ξⁱt)h

exp{aⁱ_t}−wⁱ_t−τⁱ_J,t+EtQⁱ_t,t+1Jⁱ_t+1i .

Upon separation, the firm is mandated to pay layoff taxτⁱ_ξ,t to the government and a sever-ance payment of a period’s wagew_tⁱto the worker. Instead, if²j does not exceed the thresh-old, the firm will not separate, the firm will pay the resource cost, and the match will produce (second line). aⁱ_t is a member-state specific labor-productivity shock. This is the source of ex-post heterogeneity of member states. The firm produces exp{aⁱ_t} units of the good and pays the wagewⁱ_tto the worker. In addition, the firm pays a production taxτⁱ_J,t. A match that produces this period continues into the next.

The labor-productivity shock,aⁱ_t, evolves according to

aⁱ_t=ρaaⁱ_t−1+εⁱa,t, ρa∈[0, 1),εⁱa,t∼N(0,σ²a).

An employment-services firm that does not have a worker can post a vacancy. If the firm finds a worker, the worker can start producing from the next period onward. Accounting for subsidies by the member state, the cost to the firm of posting a vacancy isκv(1−τⁱ_v,t).

κv >0 marks a resource cost, andτⁱ_v,t the government’s subsidy for hiring. In equilibrium, employment-services firms post vacancies until the after-tax cost of posting a vacancy equals the prospective gains from hiring:

(2.10) κv(1−τⁱv,t)=qⁱ_tEt

hQ_t,tⁱ ₊₁J_tⁱ₊₁i , whereq_tⁱis the probability of filling a vacancy.

Letvⁱ_tbe the mass of vacancies posted. Matchesmⁱ_t evolve according a constant-returns matching function:

(2.11) mⁱ_t=χ·h

v_tⁱi_γ

·h

[ξⁱteⁱ_t+1−e_tⁱ]sⁱ_ti_1−γ

,γ∈(0, 1).

Here,χ>0 is match efficiency. The mass of workers who potentially search isξⁱ_teⁱ_t+1−eⁱ_t, withξⁱteⁱ_tbeing workers separated at the beginning of the period.sⁱ_tis the share of those who

do actually search. With this, employment evolves according to (2.12) e_tⁱ₊₁=[1−ξⁱt]·eⁱ_t+mⁱt.

Total production of output is given by

(2.13) yⁱ_t=eⁱ_t(1−ξⁱt) exp{aⁱ_t},

whereeⁱ_t(1−ξⁱt) is the mass of existing matches that are not separated int.

For subsequent use, define labor-market tightness asθⁱ_t:=v_tⁱ/([ξⁱ_teⁱ_t+1−eⁱ_t]s_tⁱ), the job-finding rate as f_tⁱ :=mⁱ_t/([ξⁱteⁱ_t+1−eⁱ_t]sⁱ_t)=χ[θⁱ]^γ, and the job-filling rate asq_tⁱ:=mⁱ_t/v_tⁱ= χ[θⁱ]^γ−1=f_tⁱ/θⁱt. Also, define unemployment asuⁱ_t:=1−eⁱ_t.

Dividends

Dividends in each member state arise from the profits generated by the member state’s firms, namely,

(2.14) Πⁱt = −eⁱ_th R_²^ξ_t^,i

−∞²d F_²(²)i

+eⁱ_t(1−ξⁱt)h

exp{aⁱ_t}−wⁱ_t−τⁱ_J,ti

−eⁱ_tξⁱt

h

w_tⁱ+τⁱ_ξ,ti

−[κv−τⁱ_v,t]vⁱ_t.

2.2.1.3. Bargaining between firm and worker. At the beginning of the period, matched workers and firms observe the aggregate shock,aⁱ_t. Conditional on this, andpriorto observ-ing match-specific cost shock²j, firms and workers bargain over the wage and the severance payment as well as over a state-contingent plan for separation. Anticipating that the firm will insure the risk-averse worker against the idiosyncratic risk associated with²jso that the wage,wt, is independent of the realization of²jand that the severance payment equals the wage, firm and worker solve

(2.15) (wⁱ_t,²^ξ_t)=arg max_wi

t,²^ξ,it (∆ⁱt)^1−ηt(J_tⁱ)^ηt,

whereηt measures the bargaining power of the firm. We shall assume thatηt is linked to productivity asηⁱ_t=η·exp{γw·a_tⁱ}, withγ≥0. Ifγw>0, the bargaining power of firms is low in recessions and high in booms.

The first-order condition for the wage is as follows (2.16) (1−ηⁱt)J_tⁱ=ηⁱt

∆ⁱ_t u⁰(c_e,tⁱ ).

It states that after adjusting for the bargaining weights, the value of the firm equals the surplus of the worker from working expressed in units of consumption when employed.

The first-order condition for the separation cutoff yields

²^ξ,it = h

exp{aⁱ_t}−τⁱJ,t+τⁱ_ξ,t+EtQⁱ_t,t+1Jⁱ_t+1i

+βEt∆ⁱ_u,t+1+ψslog(1−sⁱ_t)−h u⁰(c_e,tⁱ ) . (2.17)

2.2.1.4. Federal RI scheme and market clearing. As for the federal unemployment rein-surance scheme, let BF

³uⁱ_t;u^{i,av g}_t ´

mark transfers of final goods from the federal level to the governments of member states. These transfers condition on the member state’s cur-rent unemployment,uⁱ_t. They may also condition on a moving index of past unemployment,

u^{i,av g}_t :=δu^{i,av g}_t−1 +(1−δ)uⁱ_t−1, withδ∈(0, 1). All member states are subject to the same struc-ture of the federal RI scheme. LetτFmark a flat, time-independent contribution toward the federal RI scheme, paid by each member state.

Anticipating that member-state governments do not have access to international bor-rowing or lending (see Section 2.2.2.2), goods market clearing in each member state requires that in each of them

(2.18) y_tⁱ+BF

³u_tⁱ;u^{i,av g}_t ´

−τF=eⁱ_tc_e,tⁱ +uⁱ_tc_u,tⁱ +eⁱ_t Z _²^ξ_t^,i

−∞²d F_²(²)+κvvⁱ_t.

The left-hand side has goods produced in the member state plus the net transfers received under the federal RI scheme. In equilibrium, goods are used for either consumption (the first two terms on the right-hand side), for production costs, or for vacancy-posting costs. Once markets clear in all the member states, they also clear for the union as a whole.

2.2.2. Government sector. There are two levels of government, the federal level and the local (member-state) level. At the beginning of periodt=0, before idiosyncratic shocks to member states have materialized, the federal government can set up a federal unemploy-ment reinsurance (“RI”) scheme, knowing the initial distribution of member states in the state space, and anticipating the response by member states and households. Periodt=0 is the first period in which the scheme will make payouts and collect contributions. The fed-eral government is a first mover. Member-state governments, when choosing labor-market policies, take the federal RI scheme as given. The federal RI scheme is implemented in a per-manent manner and under full commitment. This choice of timing seems a reasonable first pass for many countries; even more so for the European Monetary Union/European Union, we believe, where changes to binding agreements often require unanimity. We describe each government level in turn.

2.2.2.1. The federal government’s problem. Letµtmark the distribution of member states across the possible states of the economy, in periodt. Let ˜µtbe the induced distribution over the payout-relevant characteristics (uⁱ_t,u^{i,av g}_t ). Note that, once conditioning on the initial distribution of member states at the beginning of time, by the law of large numbers, bothµt

and ˜µt,t=0, 1, ..., are measurable at the beginning of periodt=0.⁸

The federal government has access to international borrowing and lending at a fixed gross interest rate 1+r =1/β. The federal RI scheme has to be self-financing in the sense that payouts or any debt be financed completely by the federal RI taxes. Assuming that there is no initial debt, this is the case if

(2.19)

X∞ t=0

(1+r)^−t Z

¡BF(ut;u_t^{av g})−τF¢

dµ˜t=0.

Here, the integral is over the distribution of (uⁱ_t,u^{i,av g}_t ) in all member states in the respective periodt.

Weighting all households in a member state equally, using (2.1) and the logistic distribu-tion, after shocks have realized in periodta member state’s utilitarian welfare function can

8The position of each member stateiin the distribution is random. Since all risk is idiosyncratic, though, and member states are given equal weight in the federal planner’s welfare function, it does not matter which member state is in what position of the distribution.

be written as (see Jung and Kuester 2015 for a derivation) (2.20) W_tⁱ:=Et

X∞ k=t

β^kh

eⁱ_ku(c_e,kⁱ )+u_kⁱu(c_u,kⁱ )+(eⁱ_kξⁱ_k+uⁱ_k)(Ψs(s_kⁱ)+h)i .

The first term is the consumption-related utility of employed workers. The second term is the consumption-related utility of unemployed workers. The third term refers to the value of leisure and the utility costs of search.⁹

The federal government’s problem is to (2.21)

max_{BF(·;·),τF} R

W₀dµ0

s.t. member states’ policy response (see Section 2.2.2.2)

induced law of motion of member states’ economies (earlier sections) financing constraint (2.19),

where the maximization overBF(·;·) indicates that the federal government chooses the shape of the payout function of the federal RI scheme. Throughout, we assume thatBF(·;·) will be continuously differentiable. In choosing the payout function, the federal planner anticipates the response to the scheme of both the member member states’ governments and of the constituents of each member state. Last, the federal RI scheme is restricted to break even.

2.2.2.2. The member-state government’s problem. The member-state government does not have access to international financial markets, nor does it issue debt to local residents.

The member-state government faces the budget constraint (2.22) eⁱ_t(1−ξⁱt)τⁱ_J,t+eⁱ_tξⁱtτⁱ_ξ,t+BF

³uⁱ_t;u^{i,av g}_t ´

=uⁱ_tbⁱ_t+κvτⁱv,tvⁱ_t+τF.

The left-hand side shows the revenue from the production and layoff taxes, and the transfers received under the federal RI scheme. The right-hand side has unemployment benefits and vacancy subsidies paid by the member state, as well as the federal RI contribution.

We model the member-state government as a utilitarian Ramsey planner, that acts in the interest of its own constituency. It chooses unemployment benefits, Pigouvian layoff taxes, and hiring subsidies (or a subset of the three thereof ), next to taxes on production. The ex-position below assumes that the member state chooses all labor-market policy instruments.

The member-state’s problem is analogous when the member state only has access to some of the instruments, a case that we explore in the numerical analysis in Section 2.5.

The member state sets policies in period 0, before shocks have realized, but after the federal level has announced the shape of its federal RI scheme. We look at two different scenarios. Either, the member state chooses fixed levels of policies once and for all, or the

9HereΨs(s_kⁱ) := −ψs h

(1−s_kⁱ) log(1−s_kⁱ)+s_kⁱlog(sⁱ_k) i

.Ψ_ξ(ξⁱ_k), which is used further below, is defined in an anal-ogous manner.

member state chooses state-contingent policies. In the former case, the member-state gov-ernment’s problem is to choose state-and-time-independent labor-market policies, with pro-duction taxes balancing the budget

(2.23) max

{τⁱv,τⁱ_ξ,bⁱ,τⁱ_J,t}

RW0dµ0

s.t. a given federal RI schemeBF,τF

the induced law of motion of the member state’s economy (earlier sections) the member-state government’s budget constraint (2.22),

whereW0is given by (2.20). Being atomistic, the member-state planner takes the federal RI scheme as given. Also, there is no strategic interaction with other member states’ govern-ments. Each member state’s government does anticipate, however, how its choice of labor-market instruments affects the local economy. We model a one-time choice of labor-labor-market instruments, with commitment to these values afterward.

An alternative scenario for the member-state government, that we also look at, is that the member state chooses state-contingent labor-market policies.

(2.24) max

{τⁱv,t,τⁱ_ξ,t,bⁱ_t,τⁱJ,t}

RW0dµ0

s.t. a given federal RI schemeBF,τF

the induced law of motion of the member state’s economy (earlier sections) the member-state government’s budget constraint (2.22).

The main difference between the problems in (2.23) and (2.24) is that in the latter all instru-ments are chosen in a state-contingent way. That is the value of instruinstru-ments changes with the state of the economy. What remains fixed over time and states of nature, however, is the shape of the federal UI scheme, which the federal government sets in at the beginning of periodt=0. In evaluating welfare in this scenario, we restrict ourselves to optimal Ramsey policies from a time-less perspective.

2.2.3. GDP. We view the resources spent retaining the match as intermediate goods, such that the definition of GDP is

(2.25) g d pⁱ_t=yⁱ_t−eⁱ_t Z _²^ξ,i_t

−∞²d F_²(²).

Market clearing expressed in units of GDP then is

(2.26) g d pⁱ_t+BF

³ uⁱ_t;zⁱ_t´

−τF=eⁱ_tc_e,tⁱ +uⁱ_tc_u,tⁱ +κvvⁱ_t.

Im Dokument Essays on the Macroeconomics of Labor Markets (Seite 57-63)