Chapter 2: Labor Market Policy · Chapter 2: Labor Market Policy Literature: Pierre Cahuc, St´ephane Carcillo, and Andr´e Zylberberg: Labor Economics Chapters 12.2, 13.1-2, and

Labor Economics Introduction

Chapter 2:

Labor Market Policy

Literature:

Pierre Cahuc, Stéphane Carcillo, and André Zylberberg: Labor Economics

Chapters 12.2, 13.1-2, and 14.2.4

Christopher Pissarides: Equilibrium Unemployment

Chapter 2

Prof. Dr. Christian Holzner Page 92

Labor Economics Introduction

Content:

2. Labor Market Policy

2.1 Minimum Wage

2.2 Employment Protection Legislation

2.3 Optimal Unemployment Insurance

2.4 Counselling and Wage Subsidies


Labor Economics Minimum Wage

2.1 Minimum Wage

Idea:

A binding minimum wage (above the market wage) decreases firms’ profits and va-cancy creation and therefore increases unemployment. However, a certain minimumwage might be socially optimal, if workers’ bargaining power is too low (Hosioscondition).

A minimum wage can also increase the gains from searching for a job and thereforeincrease the search intensity.

) The overall e↵ect depends on labor market circumstances.



Simple DMP-model and the Hosios condition:

The expected utility of employed persons and that of unemployed persons are:

rVe = w + q (Vu � Ve)rVu = z + ✓m(✓) (Ve � Vu)

Profits expected from a filled job and a vacant one satisfy:

r⇧e = y � w + q(⇧v � ⇧e)r⇧v = �h +m(✓)(⇧e � ⇧v)

With the free entry condition, these two equalities yield the job creation condition:

h

m(✓)=

y � wr + q



The expected utility of an unemployed workers is:

rVu =(r + q)z + ✓m(✓)w

r + q + ✓m(✓)

Using the job creation condition , we can eliminate w

rVu =✓m(✓)y + (r + q)z � ✓(r + q)h

r + q + ✓m(✓)

The value of the labor market tightness that maximizes the utility, satisfies:

[1� ⌘(✓)] (y � z)r + q + ⌘(✓)✓m(✓)

=

h

m(✓),

where ⌘(✓) = �✓m0(✓)/m(✓) is the elasticity of the matching function.

) The expected utility of unemployed workers is maximized when the mini-mum wage equals the wage level of the decentralized economy in which the bargai-ning power parameter satisfies the Hosios condition.



Implication:

• The level of the bargaining wage, when the Hosios condition (� = ⌘(✓)) issatisfied, is denoted w⇤.

• If w < w⇤, any increase in the minimum wage increases the utility of beingunemployed. This triggers entry into the labor market, i.e., increases participation,and the unemployment rate, but has an ambiguous impact on employment.

- In consequence, when the bargaining power of workers is too low to satisfythe Hosios condition, an increase in the minimum wage improves the welfareof unemployed workers.

- Thus, minimum wage hikes can improve labor market e�ciency.

• If w � w⇤, any increase in the minimum wage entails a decline in labor marketparticipation and an increase in unemployment, which necessarily leads to a fallin employment.



The influence of the minimum wage on search e↵ort and unemployment:

We observed that the minimum wage can increase labor market participation in theDMP-matching model.

However, in the model just presented, workers’ e↵ort was exogenous.

With endogenous job search e↵ort, the expected discounted utilities of a job seekerand a job holder are:

rVe = w + q (Vu � Ve)rVu = max

ez � �(e) + ↵e (Ve � Vu)

- e denotes the intensity of the job search,

- ↵ the exogenous arrival rate of job o↵ers per unit of search intensity,

- z � �(e) the instantaneous utility of a job seeker.



The optimal search e↵ort is:

�0(e) = ↵ (Ve � Vu)

Ve � Vu increases with the minimum wage. Therefore, this equation implies thatsearch e↵ort increases with the minimum wage (due to the exogenous ↵).

In steady state, the value of the unemployment rate u is:

u =q

q + ↵e

The hike in the minimum wage, which increases the search e↵ort, decreases theunemployment rate.

) To sum up, the minimum wage can improve employment and decrease the unem-ployment rate when the minimum wage is su�ciently low.



Framework with endogenous search intensity and matching rate:

We use a similar framework as in Fredriksson and Holmlund (2001) with a uniformunemployment benefit, i.e.,

Workers are risk averse, i.e., their instantaneous utility is given by a concave utilityfunction u (x) = ln x.

Unemployment benefits are proportional to wages, i.e., z = bw.

The matching probability of an unemployed worker is given by her individual searchintensity s and aggregate matching probability, i.e., s✓m (✓), where the markettightness is given by ✓ = v/su.

Individuals chose their search intensity according to a cost function c (s) = � ln (1� s).



Firm’s vacancy creation condition:

Free entry implies that the value of a vacancy is equal to zero, i.e., ⇧v = 0.

Thus, firms create vacancies until the cost of recruiting a worker equals theexpected discounted profit of employing a worker, i.e.,

h

m (✓)=

y � wr + q

. (1)

Implication of a minimum wage:

A binding minimum wage, i.e., w > w, increases the wage cost and decreases profitsand vacancy creation. Thus, a binding minimum wage decreases the market tightness.



Search intensity and gains from search:

Unemployed workers choose their search intensity si in order to maximize their life-time utility, i.e.,

rVu = maxs

[ln bw (1� s) + s✓m (✓) [Ve � Vu]]

Worker trade o↵ the marginal cost of search 1/ [1� s] with the additional expectedgains from searching ✓m (✓) [Ve � Vu], i.e.,

1

1� s = ✓m (✓) [Ve � Vu] .

Using the value of being employed, i.e., rVe (w) = lnw + q [Vu � Ve], allows us towrite

[Ve (w)� Vu] =lnw � rVu

r + q



Search intensity and the minimum wage:

A binding minimum wage, i.e., w > w, has two opposite e↵ects on the searchintensity of unemployed workers,

1. it increases the surplus of becoming employed, i.e.,

[Ve (w)� Vu] > [Ve (w)� Vu] ,

for a given value of being unemployed rVu. This increases the search intensity.

2. it decreases the market tightness and therefore the matching probability of aworker, i.e., ✓m (✓) decreases. Thus, a unit of search intensity s is less likely tolead to a job contact. This decreases the search intensity.



Wage bargaining (without the minimum wage):

w = argmaxw

(Ve � Vu)� (⇧e � ⇧v)(1��)

The bargaining outcome implies the following gross wage, i.e.,

w =�y

(1� �) [Ve � Vu] (r + q) + �(2)

Intuition:

The gross wage w increases with the market tightness ✓, since the value of beingunemployed increases with ✓, which increases the value of being unemployed and thusdecreases the worker’s surplus of becoming employed, i.e., @ [Ve � Vu] /@✓ < 0.



Figure 2.1: Market tightness and gross wage in equilibrium



Figure 2.2: Market tightness and a binding minimum wage



Wages and workers’ employment surplus:

Using the Bellman equations for unemployed and employed workers implies the fol-lowing equation for the surplus of a worker, i.e.,

[Ve � Vu] =lnw � ln bw � ln (1� s)

r + q + s✓m (✓)

= � ln [b (1� s)]r + q + s✓m (✓)

The search intensity of an unemployed worker depends on relative incomegain between unemployment and employment, i.e., on the proportional unemploy-ment benefit rate b, and not on the level of the wage.



Market tightness ✓ and search intensity s:

The optimal search decision in equilibrium is, therefore, given by

1

1� s = �✓m (✓)ln [b (1� s)]

r + q + s✓m (✓)

Using the implicit function theorem, i.e.,

G = r + q + s✓m (✓) + (1� s) ✓m (✓) ln [b (1� s)] = 0,we can determine, whether the search intensity of an unemployed worker increaseswith the market tightness, i.e.,

ds

d✓= �m (✓) [1� ⌘ (✓)] [s + (1� s) ln [b (1� s)]]�✓m (✓) ln [b (1� s)]

= m (✓) [1� ⌘ (✓)] 1� sr + q + s✓m (✓)

r + q

✓m (✓)> 0

Intuition:A higher market tightness increases the matching probability of a worker per searchunit and therefore the return to search, i.e., ✓m (✓) [Ve � Vu].



Equilibrium unemployment:

The unemployment rate (Beveridge curve) is given by

u =q

q + s✓m (✓). (3)

The introduction of a binding minimum wage

• decreases the market tightness ✓, since higher wage costs reduce profits andvacancy creation,

• decreases the search intensity of unemployed workers, since it decreases the returnto search ✓m (✓) [Ve � Vu],

• and thus unambiguously increases unemployment.



Figure 2.3: Unemployment and a binding minimum wage


Labor Economics Employment Protection Legislation

2.2 Employment Protection Legislation

Idea:

The government implements employment protection in order to influence the layo↵decision of firms.

The Mortensen-Pissarides Model needs to be adjusted to allow for an endogenousseparation rate.

Question: How does employment protection influence layo↵s and hirings?



Employment protection in Germany:

• Extraordinary dismissal (stealing, sexual harassment, ...)

• Ordinary dismissal (rules apply only for firms with more than 10 employees)

a) Personal reasons (long lasting diseases which lead to work-inability),

b) Behavioral reasons (refusing to work, using business equipment privately, ...),

c) Business reasons (restructuring, insolvency, ...). Layo↵s due to business reasonshave to obey certain social criteria, i.e., certain groups are last to be laid o↵(disabled, with family obligations, old age, long tenure). If an employee doesnot pursue an employment lawsuit against a wrongful dismissal, an employercan o↵er a layo↵ compensation of 0.5 of monthly earnings per year of tenure.

• Notice period

There is no period of notice in case of an extraordinary dismissal,

The notice period increases with tenure from one month up to seven months.



2.2.1 Endogenous separation rate, if wages are constant

Idea:

A worker and a firm are not able to decrease the wage unilaterally, e.g. due to unions.

A worker might prefer a lower wage instead of being laid o↵. This will lead to invo-luntary layo↵s.

Output is lost, if wages cannot be negotiated downward.



Framework:

Like in the Mortensen-Pissarides model, except:

A worker’s productivity changes at rate �.

The productivity is drawn randomly from the distribution G ("), where " 2 ]�1, "].

The wage level w is exogenously given.

Firing costs:

• Tax component fa: Notice period, legal costs, cost imply by social criteria, ...• Severance payment (transfer) fe: Notice period, layo↵ compensation.



Firms’ layo↵ decision at constant wages:

At a productivity " the value of employing a worker is given by

r⇧e (") = "� w + � [⇧� � ⇧e (")] , (4)

where ⇧� equals the expected continuation value of employing a worker, if achange in productivity occurs.

If a firm lays o↵ a worker, it incurs the firing cost f = fa + fe. It will then searchfor a new worker by opening a vacancy, i.e., ⇧v.

A firm lays o↵ a worker, if the value of employing the worker is less than the cost oflayo↵ and value of opening a new vacancy, i.e., ⇧e (") < �f + ⇧v.

At the reservation productivity "d the firm is indi↵erent between keeping theworker or laying the worker o↵, i.e.,

⇧e ("d) = �f + ⇧v. (5)



Reservation productivity:

Using the free entry condition, i.e., ⇧v = 0, and evaluating equation (4) at "dimplies

"d = w � (r + �) f � �⇧�. (6)

Using equations (4) and (6) allows us to write the value of employing a worker asfollows, i.e.,

⇧e (") ="� "dr + �

� f . (7)

The expected continuation value of a productivity change equals the firing cost�f , if the productivity " is below the reservation productivity "d and the value ofemploying a workers for productivities " above the reservation productivity, i.e.,

⇧� =

Z "d

�1�fdG (") +

Z "

"d

⇧e (") dG (") . (8)



Substituting ⇧e (") into equation (8) gives an expression for the expected conti-nuation value

⇧� = �f +1

r + �

Z "

"d

("� "d) dG (") .

The reservation productivity is therefore given by substituting ⇧� into equation (6),i.e.,

"d = w � rf ��

r + �

Z "

"d

("� "d) dG (") . (9)

Intuition:

The reservation productivity is lower than the wage, because

- a layo↵ causes firing costs rf , that can be avoided, if the worker is not laid o↵,

- there exists a probability, that the worker’s productivity will be above the wage infuture periods (expected continuation value).



Figure 2.4: The reservation productivity



Vacancy creation condition:

Assumption: A newly employed worker has always the highest productivity, i.e.," = ".

Using the free entry condition, i.e., ⇧v = 0, implies the following vacancy crea-tion condition, i.e.,

h

m (✓)= ⇧e (") =

"� "dr + �

� f, (10)

where the last equality follows from equation (7).

The wage level has only an indirect e↵ect on the vacancy creation condition byinfluencing the reservation productivity, i.e.,

h

m (✓)=

"� wr + �

� �r + �

f +�

(r + �)2

"Z

"d

("� "d) dG (") .



Vacancy creation curve:

A higher reservation productivity "d implies that a worker is more likely to be laido↵. This implies that the expected time of profitable production decreases, if thereservation productivity increases. This reduces the number of vacancies that arecreated. Thus, the market tightness decreases with a higher reservation productivity.



Figure 2.5: The vacancy creation curve



Equilibrium market tightness and reservation productivity:

Since wages are exogenously given, the reservation productivity is independentof the market tightness and given by equation (9), i.e.,

"d = w � rf ��

r + �

Z "

"d

("� "d) dG (") .

The vacancy creation condition is given by equation (10), i.e.,

h

m (✓)= ⇧e (") =

"� "dr + �

� f.

These two equations determine the equilibrium market tightness ✓ and the equi-librium reservation productivity "d.



Beveridge curve:

The separation rate q is now endogenously given by the firing rate �G ("d), i.e.,

• � is the shock arrival rate,

• G ("d) is the probability that the new productivity is below the reservation pro-ductivity.

Equilibrium unemployment is in steady state, if the inflows into unemployment, i.e.,�G ("d) [1� u], are equal to the outflows, i.e., ✓m (✓)u. Rearranging implies

u =�G ("d)

✓m (✓) + �G ("d).



2.2.2 Endogenous separation rate and individual wage bargaining

Idea:

Wages can be renegotiated every time the productivity changes.

Workers will accept lower wages in order to avoid being laid o↵, as long as the valueof being employed exceeds the value of being unemployed.

Layo↵s are e�cient, since workers and firms will only separate, if the surplus of beingmatched is zero.



Bellman equations for the firm:

The value of employing a worker depends on the productivity ", i.e.,

r⇧o = "� wo + � [⇧� � ⇧o] ,

r⇧e (") = "� w (") + � [⇧� � ⇧e (")] ,

where ⇧o equals the value of employing an outsider, i.e., a new worker.

The expected continuation value of a productivity change equals,

⇧� =

Z "d

�1� (fa + fe) dG (") +

Z "

"d

⇧e (") dG (") . (11)



Surplus for the firm:

Surplus of employing an outsider,

SFo = ⇧o � ⇧v. (12)

An outsider will become an insider, once he is employed. From this point onward, thefirm has to pay the firing cost (fa + fe), if the worker is laid o↵.

Surplus of keeping an insider employed, who is entitled to severance paymentfe, if no bargaining agreement is reached. The later also causes fixed firing costs fa,i.e.,

SF (") = ⇧e (")� [⇧v � (fa + fe)] . (13)



Bellman equations of the worker:

The value of being unemployed is given by

rVu = z + ✓m (✓) [Vo � Vu] .

The value of being employed as an outsider equals, i.e.,

rVo = wo + � [V� � Vo] ,

where wo equals the wage that is obtained in the wage bargaining as an outsider.

The value of being employed as an insider is given by

rVe (") = w (") + � [V� � Ve (")] .



Expected continuation payo↵ for a worker:

If a productivity shock occurs, which happens at rate �, and productivity " changes,then wages are renegotiated or the worker is laid o↵.

The expected continuation payo↵ for a worker equals the severance paymentfe plus the value of being unemployed rVu, if the new productivity " is below thereservation productivity "d and the value of being employed rVe ("), if the productivity" is above the reservation productivity, i.e.,

V� =

Z "d

�1(Vu + fe) dG (") +

Z "

"d

Ve (") dG (") , (14)



Surplus for the worker:

Surplus of becoming employed as an outsider,

SLo = Vo � Vu (15)

Once employed the worker becomes an insider and is entitled to severance paymentfe , if he is laid o↵.

Surplus of staying employed as an insider,

SL (") = Ve (")� [Vu + fe] . (16)



Total match surplus:

Total surplus generated by forming a match with an outsider:

So = ⇧o � ⇧v + Vo � Vu

(r + �)So = " + � [⇧� + V�]� (r + �)Vu

Total surplus generated by keeping an insider employed:

S (") = ⇧e (")� [⇧v � fa � fe] + Ve (")� Vu � fe

(r + �)S (") = " + � [⇧� + V�]� (r + �) [Vu � fa]



Wage bargaining outcome:

Wage bargaining leads to the following surplus splitting rules:

Outsiders:

⇧o � ⇧v = (1� �)So,

Vo � Vu = �So,

Insiders:

⇧e (")� [⇧v � f ] = (1� �)S (")

Ve (")� (Vu + fe) = �S (") .



E�cient separations:

A firm lays o↵ a worker, if the surplus is negative, i.e.,

SF (") < 0 () ⇧e (") < ⇧v � f.

The surplus splitting rule implies that the worker’s surplus is also negative, if thefirm’s surplus is negative, i.e.,

SF (") = (1� �)S (") = 1� ��

SL (") =) SF (") < 0 () S (") < 0.

The reservation productivity equals the productivity at which the common surplusis zero, i.e.,

S ("d) = 0.



Intuition for e�cient separations:

• If the worker’s surplus is positive, while the firm’s surplus is negative, then theworker prefers to stay employed at a lower wage (as long as the new wage gua-rantees Ve (")� [Vu + fe] � 0).

• The worker will therefore o↵er the firm a lower wage, if the firm keeps himemployed.

• The firm will agree to the lower wage, if the surplus of the firm at the lower wageis not negative, i.e., ⇧e (")� [⇧v � f ] � 0.

• Only if both parties agree that there is no surplus left, then the worker will belaid o↵.



Total match surplus:

The total surplus of keeping a worker employed is

(r + �)S (") = " + � [⇧� + V�]� (r + �) [Vu � fa]

substituting the continuation surplus

S� = ⇧� + V� =

Z "d

�1[Vu � fa] dG (") +

Z "

"d

[⇧e (") + Ve (")] dG (")

=

Z "d

�1[Vu � fa] dG (") +

Z "

"d

[Vu � fa] dG (") +Z "

"d

S (") dG (")

= [Vu � fa] +Z "

"d

S (") dG (")

implies

(r + �)S (") = "� rVu + rfa + �Z "

"d

S (") dG (") . (17)



Reservation productivity:

Evaluating equation (17) at "d in order obtain (r + �) [S (")� S ("d)] = "� "d, andnoting that the surplus is zero at the reservation productivity, i.e., S ("d) = 0, implies

S (") ="� "dr + �

(18)

Substituting S (") in the integral of equation (17) and evaluating at "d gives thereservation productivity, i.e.,

"d = rVu � rfa ��

r + �

Z "

"d

("� "d) dG (") . (19)

Intuition:

The firm and the worker are indi↵erent between employment and non-employment,if the value of continued production "d equals the worker’s outside option r [Vu + fe]minus the firm’s direct cost r [fa + fe] and opportunity cost of waiting for anotherproductivity shock that might lead to profitable production.



Vacancy creation condition:

Using the free entry condition, i.e., ⇧v = 0, implies the following vacancy crea-tion condition, i.e.,

h

m (✓)= ⇧o = (1� �)So,

where wage bargaining implies the last equality.

The surplus of employing an outsider So is identical to the surplus of keeping aninsider employed at the highest productivity, i.e., S ("), minus the fixed firing costfa, since not achieving an agreement with an outsider implies – in contrast to thebargaining with an insider – no firing costs (the severance payment fe is not lost,since it implies a transfer between the firm and the worker),

So = S (")� fa.Using equation (18) implies the following vacancy creation condition:

h

m (✓)= (1� �)

"� "dr + �

� fa�. (20)



Vacancy creation curve:

Slope:

A higher reservation productivity "d implies that a worker is more likely to be laido↵. This implies that the expected time of profitable production decreases, if thereservation productivity increases. This reduces the number of vacancies that arecreated. Thus, the market tightness decreases with a higher reservation productivity.

Intuition:

• Firms create vacancies as long as the cost of recruiting an additional workerh/m (✓) equals the expected value of employing an outsider.

• The expected value of employing an outsider equals the fraction (1� �) of thetotal surplus of the match.

• The surplus of employing an outsider contains the fixed firing cost fa, since bothparties anticipate that they will have to pay this fixed cost in the future.

• The surplus is independent of the wages paid, since wages are only a transferbetween the firm and the worker.



Figure 2.6: Vacancy creation curve with endogenous wages



Job destruction curve:

Substituting the value of being unemployed into equation (19), i.e.,

rVu = z + ✓m (✓) (Vo � Vu) = z + ✓m (✓) �So

= z + ✓m (✓) �

"� "dr + �

� fa�= z +

�

1� �h✓

determines the job destruction curve, i.e., the reservation productivity as a func-tion of the market tightness,

"d = z +�

1� �h✓ � rfa ��

r + �

Z "

"d

("� "d) dG (") .

Intuition for the slope of the job destruction curve

A higher market tightness increases the value of being unemployed. This decreasesthe total surplus of a match and implies that the productivity at which the surplus iszero is higher. Thus, the reservation productivity increases with the market tightness.



Figure 2.7: Job destruction curve with endogenous wages



Equilibrium with endogenous wages:

The equilibrium is determined by the market tightness ✓, the reservation productivity"d, wages for in- and outsiders w (") and wo, and the unemployment rate u.

Job creation curve:

"d = "� (r + �) fa �r + �

1� �h

m (✓)

Job destruction curve:

"d = z +�

1� �h✓ � rfa ��

r + �

Z "

"d

("� "d) dG (")

Given the market tightness ✓⇤ and the reservation productivity "⇤d in equilibrium, theunemployment rate is given by the

Beveridge curve:

u =�G ("⇤d)

✓⇤m (✓⇤) + �G ("⇤d)



Wages for outsiders:

The surplus splitting rule for outsider implies

(r + �)⇧o = (r + �) (1� �) [S (")� fa]

= (1� �)✓"� rVu + rfa + �

Z "

"d

S (") dG (")

◆� (r + �) (1� �) fa

= (1� �) ("� rVu) + �Z "

"d

⇧ (") dG (")� (1� �)�fa

The value of employing a new worker is given by

(r + �)⇧o = "� wo + �⇧�

= "� wo + �Z "

"d

⇧ (") dG (")� � (fa + fe)

Substituting (r + �)⇧o and rearranging implies

wo = (1� �) rVu + �"� � (fa + fe) + (1� �)�faSubstituting rVu by z + �/ (1� �)h✓ implies the wage equation for outsiders, i.e.,

wo = (1� �) z + � (" + h✓)� � (fe + �fa) , (21)



Wages for insiders:

The surplus splitting rule for outsider implies

(r + �)⇧ (") = (r + �) (1� �)S (")

= (1� �) ("� rVu) + �Z "

"d

⇧ (") dG (") + (1� �) rfa

The value of keeping a worker employed is given by

(r + �)⇧ (") = "� w (") + �⇧�

= "� w (") + �Z "

"d

⇧ (") dG (")� � (fa + fe)

Substituting (r + �)⇧ (") and rVu implies the following wage equation for insiders,i.e.,

w (") = (1� �) z + � (" + h✓) + r (fe + �fa) , (22)

The wage di↵erence between equally productive in- and outsiders is given by

w (")� wo = (r + �) (fe + �fa) .



Figure 2.8: Equilibrium market tightness and reservation wage



Figure 2.9: Equilibrium unemployment rate



Unemployment in- and outflows:



Unemployment rate and long-term unemployment rate:



Summary of constant wages versus endogenous wages:

At constant wages the severance payment fe as well as the tax component fa make itmore costly to layo↵ workers. Thus, f = fe+fa decreases the reservation productivity.Under endogenous wages severance payments fe constitute only a transfer betweenworkers and firms and therefore do not influence the total surplus and the separationdecision. The tax component fa, however, makes a layo↵ more costly and reducesthe reservation productivity.

At constant wages workers will be laid o↵ although they are willing to decreasetheir wages (involuntary layo↵s). If wages are endogenous, workers agree to wagecuts as long as their surplus is no less than the value of being unemployed, i.e., allseparations are voluntary. This implies that a worker and a firm only separate, if thereis no surplus, i.e., surplus is negative.

At constant wages severance payments fe and the tax component fa reduce thesurplus of a firm and therefore decrease firms’ profits and vacancy creation. Underendogenous wages only the tax component fa reduces the total surplus of a matchand therefore decrease the vacancy creation.


Labor Economics Optimal Unemployment Insurance

2.3 Optimal Unemployment Insurance

Idea:

Public unemployment insurance systems were created in many European countries atthe beginning of the twentieth century.

The purpose of state intervention is to insure workers against the risk of unemploy-ment, a burden the state or trade unions were forced to assume because imperfectinformation hinders the creation of private insurance systems providing compensationfor job loss.

The state also intervenes to provide social assistance, redistributing income in favorof the most disadvantaged workers.

The idea of compensating for job loss has critiques since benefit payments mayincrease the duration of unemployment by reducing the e↵ort for job search.



An overview of unemployment systems

The key parameter of unemployment insurance is the replacement ratio (replacementratio = benefit payment/last wage earned).

A fraction of people, who lose their jobs do not receive benefits from unemploymentinsurance system, because they have not paid enough contributions to get the benefit.They receive unemployment assistance:

• Unemployment insurance systems pay benefits for a limited period to persons,who have already been employed and paid unemployment benefit contributions.

• Unemployment assistance are payments made by the social assistance fund fi-nanced through general taxation. Its benefits are conditional upon job search andavailability for work. But they are generally unlimited in time and independent ofpast earnings.



Unemployment insurance system in selected countries:

Country Maximum Payment rate (% of earnings base) Min benefit Max benefitduration(months)

Initial rateat end of legal

entitlement periodas a % of AW as a % of AW

France 24 57-75 28.1 227.5Germany 12 60 – 91.7Spain 24 70 60 (after 6 months) 24.1 52.6Sweden 35 80 70 (after 9 months) 22.6 48.0United Kingdom 6 Fixed amount (9.9% of AW) – –



Unemployment assistance system in selected countries:

CountryDuration(months)

Payment rate Maximum benefit Test on

as a % of AW Assets IncomeFrance 6 months (renewable) Fixed amount 15.6 FamilyGermany Unlimited Fixed amount 10.2 Yes FamilySpain 30 Fixed amount 20.6 FamilySweden 14 Fixed amount 22.6 IndividualUnited Kingdom Unlimited Fixed amount 9.9 Yes Family



2.3.1 Unemployment insurance financed by social security contributions

Idea:

The unemployment insurance payments are financed through contributions (taxes).

Budget externality:

A firm does not take into account that creating a vacancy decreases the unemploy-ment insurance payments and increases the tax revenue.

Firms create not enough vacancies compared to the social optimum.



Framework:

Like in the Mortensen-Pissarides model, except

Workers finance the unemployment insurance scheme through contributions c.

The net wage is given by

wn = w � c.

(since wages are unique, we can write contributions as lump sum contributions).

Government budget equation:

All employed workers have to pay contributions c in order to finance the unemploy-ment insurance benefits z for unemployed workers, i.e.,

c (1� u) = zu () c = z u1� u.

Note, that the contributions increase with the number of unemployed workers.



Worker’s surplus:

Value of being unemployed is given by

rVu = z + ✓m (✓) (Ve � Vu) .

The value of being employed is given by

rVe = w � c + q (Vu � Ve) .

The surplus of becoming employed is given by

SL = Ve � Vu =w � c� rVu

r + q.



Firm’s vacancy creation condition and firm’s surplus:

Free entry implies that the value of a vacancy is equal to zero, i.e., ⇧v = 0.

The value of employing a worker (firm’s match surplus) equals,

SF = ⇧e � ⇧v =y � wr + q

.

Thus, firms create vacancies until the cost of recruiting a worker equals theexpected discounted profit of employing a worker, i.e.

h

m (✓)=

y � wr + q

. (23)



Wage determination:

Total surplus of a match:

S = ⇧e � ⇧v + Ve � Vu =y � c� rVu

r + q

Nash-Bargaining outcome (surplus splitting rule):

⇧e � ⇧v = (1� �)S and Ve � Vu = �S

Wage equation:

w = �y + (1� �) rVu + (1� �) c.



Gross wage curves:

The gross wage can be obtained by substituting the value of being unemployed,

rVu = z +�

1� �h✓.

and the unemployment insurance contributions,

c = zu

1� u = zq

✓m (✓)

into the gross wage equation, i.e.

w = �y + (1� �)z +

�

1� �h✓�+ (1� �) z q

✓m (✓)

= (1� �) z✓1 +

q

✓m (✓)

◆+ � [y + h✓] .



Slope and intuition for the gross wage curve:

The gross wage is a non-linear function of the market tightness, i.e.,

@w

@✓= (1� �) z

✓�q [✓m (✓)]

0

[✓m (✓)]2

◆+ �h.

Intuition:

1. The first term is negative, because a higher the market tightness decreases the un-employment rate. This reduces the cost of financing the unemployment insurancesystem and therefore reduces the contribution c. The gross wage (wage cost tothe employer) decreases by (1� �) z, i.e., decreases the wage according to thefraction (1� �) (firm’s bargaining power) of unemployment insurance paymentsz.

2. The second term is positive, because a higher market tightness increases theworker’s outside option (value of being unemployed). This increases the wagethat the worker gets.



Net wage curve:

wn = z

✓(1� �)� � q

✓m (✓)

◆+ � [y + h✓]

The net wage increases with the market tightness for two reasons:

1. The first term is positive, because a higher the market tightness decreases the un-employment rate. This reduces the cost of financing the unemployment insurancesystem and therefore reduces the contribution c. The gross wage (wage cost tothe employer) increases by �z.

2. The second term is positive, because a higher market tightness increases theworker’s outside option (value of being unemployed). This increases the wagethat the worker gets.



Figure 2.10: Gross and net wage curves



Gross wage and labour market tightness in equilibrium:

Job creation curve:

w = y � hm (✓)

(r + q)

Gross wage curve:

w = (1� �) z✓1 +

q

✓m (✓)

◆+ � [y + h✓]

The non-monotonicity of the gross wage curve implies that multiple equilibriacan exists.



Figure 2.11: Multiple equilibria (in market tightness and gross wages) due to thebudget externality



Figure 2.12: Multiple equilibria (unemployment rates) due to the budget externality



Intuition for the existence of multiple equilibria:

The reason for the existence of multiple equilibria is in the budget externality, i.e.,the fact that firms do not consider the e↵ect vacancy creation has on the numberof unemployed and employed workers and therefore on tax revenues and the cost offinancing unemployment insurance.

If firms were to take the budget externality into account, they would create morevacancies in order to reduce the cost of the unemployment insurance system.



A good and a bad equilibrium exists:

1. In the good equilibrium firms create many vacancies. This decreases unem-ployment insurance payments and therefore leads to lower contribution rates.The reduced contributions increase the surplus of a match, lower the gross wageand therefore increase the profit of firms. The high profits triggers more vacancycreation such that a good equilibrium can exist.

2. In the bad equilibrium firms create few vacancies. This increases unemploy-ment insurance payments and therefore leads to higher contribution rates. Thehigh contributions decrease the surplus of a match, increases the gross wage andtherefore decrease the profit of firms. The low profits prevents firms from creatingmore vacancy creation such that a bad equilibrium prevails.



Deriving the social optimum:

The social planner maximizes aggregate welfare, i.e.

max

✓

Z 1

0[(y � c) (1� u) + zu� h✓u] e�rtdt

subject to the budget constraint

zu = c (1� u)

and the constraint implied by matching frictions, i.e.

u̇ = q (1� u)� ✓m (✓)u



Hamiltonian (after substituting the budget constraint):

H = [y (1� u)� h✓u] e�rt + µ [q (1� u)� ✓m (✓)u]

FOC:

@H

@✓= 0 () he�rt = �µm (✓)

1 +

✓m0 (✓)

m (✓)

�(24)

@H

@u= �µ̇ () [�y � h✓] e�rt � µ [q + ✓m (✓)] = �µ̇ (25)

Transversality condition:

lim

t!1µ u = 0



Di↵erentiating equation (24) with respect to time t implies: µ̇ = �rµ

Substituting µ using equations (24) and (25) implies the following condition forthe optimal labour market tightness ✓, i.e.,

h

m (✓)= [y + h✓]

1� ⌘ (✓)r + q + ✓m (✓)

orh

m (✓)=

(1� ⌘ (✓)) yr + q + ⌘ (✓) ✓m (✓)

where ⌘ (✓) equals the elasticity of the matching function with respect to the unem-ployment rate u, i.e.,

⌘ (✓) = �✓m0(✓)

m (✓)



Comparing the social planner’s solution

h

m (✓)=

(1� ⌘ (✓)) yr + q + ⌘ (✓) ✓m (✓)

with the decentralized market solution

h

m (✓)=

(1� �)y � z

✓1 +

q

✓m (✓)

◆�

r + q + �✓m (✓)

implies that the government can implement the social planner’s solution bychoosing unemployment benefits proportional to productivity, i.e., z = by,such that the social planner’s solution is identical to the decentralized market solution,i.e.,

b = [1� u]1� 1� ⌘ (✓)

1� �r + q + �✓m (✓)

r + q + ⌘ (✓) ✓m (✓)

�,

which is only feasible, if ⌘ (✓) > �.



2.3.2 Optimal unemployment insurance profile

Question:

What is the optimal profile of benefits over the unemployment spell?

Answer:

Research suggests that a time profile in which the amount of benefit decreases withthe duration of unemployment may be optimal, since it ensure that unemployed havean incentive to search actively for a job.



Framework:

• Hopenhayn and Nicolini (2009) assume that the optimal contract should minimizethe average cost of a jobless person while o↵ering him a given exogenous level ofexpected utility at the same time V tu .

• Individuals are risk averse, which we model by a concave utility function v (.) withv0 > 0 and v00 < 0.

• The government proposes a contract specifying the values bt of the unemploymentbenefit and the value g of the transfer received while being employed.

• We assume that e↵ort is not verifiable. All jobs o↵er the same exogenous con-stant wage w and jobs are never destroyed. If a job seeker finds work after anunemployment spell of t periods, she receives a net wage of (w + g) and keepsher new job indefinitely.

• Denoting � 2 [0, 1] the discount factor, the discounted expected utility of a jobseeker after t periods of unemployment is:

V te =v (w + g)

1� � .



Incentive constraint:

The evolution of the expected utility of a job seeker making an e↵ort a is,

V tu = v(bt)� a + �⇥pV t+1e + (1� p)V t+1u

⇤. (26)

where p the probability he can find a job that starts at period t + 1.

A job seeker putting an e↵ort a attains the utility level v(bt)� a.The unemployed person has the opportunity to “cheat”by making no e↵ort andreceiving unemployment insurance benefits. The utility obtained by cheating is,

V ts = v(bt) + �Vt+1u

The incentive compatibility constraint is,

��V t+1e � V t+1u

�� a

p. (27)

The need to give an incentive obliges the principal to pay a “rent” to workers findinga job, which equals at least a/p.



Government’s perspective:

The discounted cost of the transfer gt to the principal (government) is,

Cte =gt

1� � .

The cost associated with an unemployed job seeker, who actively searches is,

Ctu = bt + �⇥pCt+1e + (1� p)Ct+1u

⇤.

The government minimizes the expected cost,

Ctu�V tu

�= min

bt,Vt+1e ,V

t+1u

bt + �⇥pCt+1e

�V t+1e

�+ (1� p)Ct+1u

�V t+1u

�⇤

subject to the guaranteed utility V tu in equation (26) and the incentive compatibilityconstraint (27).



Lagrangian:

L = bt + �⇥pCt+1e

�V t+1e

�+ (1� p)Ct+1u

�V t+1u

�⇤

+µ⇥v(bt)� a� V tu + �

⇥pV t+1e + (1� p)V t+1u

⇤⇤

+�⇥�p

�V t+1e � V t+1u

�� a

⇤.

The first order conditions give,@L@bt

= 0 () µv0(bt) = �1, (28)

@L@V t+1u

= 0 ()@Ct+1u

�V t+1u

�

@V t+1u= �

p

1� p � µ = �p

1� p +1

v0(bt), (29)

@L@V t+1e

= 0 ()@Ct+1e

�V t+1e

�

@V t+1e= �� µ = �� + 1

v0(bt). (30)

As V tu is considered a choice parameter by the government, the evolope theoremimplies,

@Ctu (Vtu)

@V tu= �µ = 1

v0(bt)for all t. (31)



Optimal unemployment insurance profile:

Applying equation (31) at t + 1 and substituting it into the FOC (29) implies

1

v0(bt+1)� 1

v0(bt)= �

p

1� p. (32)

We can show that � < 0 by contradiction. If � � 0 we get bt+1 � bt. Furthermore,since

Cte =g

1� � and Vte =

v (w + g)

1� �@Cte (V

te )

@V te=

1

1� �dg

dV te=

1

v0(w + g),

implies bt � w + g due to FOC (30), i.e.,1

v0(w + g)� 1

v0(bt)= ��,

unemployed workers have no incentive to search actively (incur cost a). This violatesthe incentive compatibility constraint (27) since V t+1u > V

t+1e .



Optimal unemployment insurance profile:

With � < 0 equation (32) implies that benefits have to decrease with unemploymentduration, i.e.,

bt+1 < bt

Intuition:

• Unemployment benefits in period 1 have no incentive e↵ect, since they are paidregardless of whether a worker finds a job or not.

• Unemployment benefits in later periods are only paid, if workers did not find ajob. They therefore influence the search intensity of workers.

• With increasing unemployment benefits in later periods unemployed workers haveno incentive to search actively for a job.

• Decreasing unemployment benefits in later periods and increasing them in earlierperiods to keep total expenditure constant, increases the incentive for unemployedto find a job sooner than later.



Results:

• The agency in charge of the insurance minimizes its own costs, guaranteeing acertain level of utility to the job seeker and incentivizing unemployed to searchactively.

• Its goal is to minimize the cost needed to an entry job seeker, by choosing optimalvalues bt and g, and respecting the incentive constraint and the participationconstraint.

• Hopenhayn and Nicolini (1997, 2009) find that the optimal profile of benefitsought to decrease with the duration of unemployment when individuals are con-suming their whole per period income.

• If transfers to those who become employed are allowed, the rate at which benefitpayments tail o↵ becomes very weak, and the replacement rate very high.



The optimal profile of unemployment benefit with moral hazard

System with tax on wages System without taxWeeks Replacement Tax on Replacement rate

of unemployment rate (%) wages (%) without tax on wages (%)1 99.0 -0.5 85.82 98.9 -0.4 80.83 98.8 -0.3 76.34 98.7 -0.2 72.15 98.6 -0.1 68.26 98.5 0.0 64.77 98.4 0.1 61.48 98.3 0.2 58.412 97.9 0.6 48.216 97.5 1.0 40.526 96.5 2.0 27.752 94.0 4.5 13.4

Source : Hopenhayn and Nicolini (1997, p. 426).



Optimal unemployment insurance and the business cycle

• Kroft and Notowidigdo (2011) find that the moral hazard cost of unemploymentbenefit is pro-cyclical while the consumption-smoothing term is acyclical.

) We ought to conclude that optimal unemployment benefit should be contra-cyclical i.e. higher benefits in bad times.

• Similarly, Landais (2013) also concludes that the labor supply response to un-employment benefit is (weakly) pro-yclical: Increases in the unemployment rateare associated with a slight decrease in this estimated elasticity of unemploymentduration with respect to benefits.

• Jung and Kuester (2011) have integrated this dimension into a search and mat-ching model and find that hiring subsidies, lay-o↵ taxes and the replacement rateat which insurance is paid, should all rise in recessions.


Labor Economics Counselling and Wage Subsidies

2.4 Counselling and Employment Subsidies

Idea:

The public employment agency counsels only some unemployed workers or subsidiesonly part of the jobs.

Counselling only some workers has a negative congestion externality on non-counselled workers. The overall e↵ect on unemployment depends on the labor marketcircumstances.

Employment subsidies increase the value of employing a worker and leads thereforeto additional vacancy creation.



2.4.1 Counselling some unemployed workers

Framework:

A matching model with counselled and non-counselled unemployed:

• Workers are identical and can be employed, unemployed and counselled, or un-employed and not counselled.

• We denote u and eu the number of non-counselled and counselled unemployedworkers.

• Counselled unemployed workers are assumed to produce a di↵erent number ofe�ciency units of search, denoted by � > 1.

• In this setting, the number of e�ciency units of job search per unit of timeamounts to s = u + �eu.

• The market tightness is therefore given by ✓ = v/s.



Value of a vacancy:

A firm that creates a vacant job can fill it with a counselled or a non-counselledworker.

Then, the value of a vacant job is,

r⇧v = �h +m(✓)h↵e⇧e + (1� ↵)⇧e � ⇧v

i.

• h is the search cost,

• m(✓)↵ = m(✓)�eu/s is the probability of meeting a counselled worker,

• e⇧e is the value of a job filled with a counselled worker,

• ⇧e is the value of a job filled with a non-counselled worker.



Job creation curve:

Assuming that jobs are destroyed at the exogenous rate q, the asset value of a jobsatisfies,

re⇧e = y � ew + q(⇧v � e⇧e),r⇧e = y � w + q(⇧v � ⇧e).

• y is the productivity of jobs,• ew and w are respectively the wage of counselled and uncounselled workers,

The free entry condition ⇧v = 0 implies that,

h

m(✓)= ↵e⇧e + (1� ↵)⇧e = y � [↵ ew + (1� ↵)w] .



Workers’ Bellman equations:

Wages are assumed to be negotiated.

A non-counselled worker has a probability µ to enter counselling, the value of jobsearch and of a job for a non-counselled worker are,

rVu = z + µ(eVu � Vu) + ✓m(✓)(Ve � Vu),rVe = w + q(Vu � Ve).

The value of job search and of a job for a counselled worker are,

reVu = z + �✓m(✓)(eVe � eVu),reVe = ew + q(Vu � eVe).



Surplus and Bargaining:

The surplus of a job filled by a previously counselled worker and that of a non-counselled worker are,

eS = (eVe � eVu) + (e⇧e � ⇧v) and S = (Ve � Vu) + (⇧e � ⇧v) .

The sharing rule according to Nash-Bargaining is,

eVe � eVu = � eS and Ve � Vu = �S. (33)

We arrive at the value of surpluses S and eS as a function of ✓,

(r + q) eS = y � z � �✓m(✓)� eS + q✓m(✓)�(�eS � S)

r + µ,

(r + q)S = y � z � ✓m(✓)�S + µ✓m(✓)�(�eS � S)

r + µ.



Market tightness:

Using the free entry condition and the sharing rules (33), one gets a third relationbetween S, eS and ✓,

h

m(✓)= (1� �)

h↵eS + (1� ↵)S

i(34)

Equation (34) defines an increasing relationship between the surpluses and labormarket tightness.

Since ↵ = �eu/s with s = u+�eu is endogenously determined by the market tightness,we need two further equations to pin down the equilibrium.



Beveridge curve:

The laws of motion of unemployment for the two categories of unemployed workersare:

deudt

= µu� �✓m(✓)eu and dudt

= q(1� u� eu)� µu� ✓m(✓)u

Considering the steady state, the equilibrium value of total unemployment is,

u⇤ = u + eu = q [µ + �✓m(✓)]q [µ + �✓m(✓)] + �✓m(✓) [µ + ✓m(✓)]

(35)

Equation (35) is the equation for the Beveridge curve.



Equilibrium:



Comparative Statics:

There are 3 consequences of an increase in the proportion of counselled workers:

1. The composition e↵ect reduces the probability that non-counselled workers get ajob o↵er and the expected profits of filled jobs and then induces firms to createfewer job vacancies.=) Thus, the job creation curve (JC) moves to right.

2. The wage e↵ect contributes to reduce expected profits and hence to reduce labormarket tightness.=) It also decreases the value of the surplus of jobs filled with non-counselledworkers because it improves their outside option.

3. The direct e↵ect: The value of the surplus of jobs filled with counselled workersincreases when there is more counselling because theses jobs are filled more rapidlythanks to higher search intensity



2.4.2 Employment Subsidies

Framework:

Let us consider a set of ex ante identical firms. Only a fraction ↵ of firms can benefitfrom a subsidy s granted by the government.

The value of jobs for a firm receiving subsidies e⇧e and the corresponding value ⇧efor firms not receiving them are:

re⇧e = y � ew + s + q(⇧v � e⇧e) and r⇧e = y � w + q(⇧v � ⇧e)

where ew is the wage of subsidized jobs.



Surplus:

The value of being unemployed is given by,

rVu = z + ✓m(✓)(↵eVe + (1� ↵)Ve � Vu).

The expected utilities of an employee who occupies a subsidized job eVe and or anunsubsidized one Ve are given by,

reVe = ew + q(Vu � eVe) and rVe = w + q(Vu � Ve).

The surpluses of subsidized ˜S and nonsubsidized S jobs are,

eS = y + s� rVu � ⇧vr + q

and S =y � rVu � ⇧v

r + q.



Wages:

Wage bargaining determines the negotiated wage such as,

ew = �(y + s) + (1� �)rVu,w = �y + (1� �)rVu = ew � �s.

The expression of wage of subsidized jobs shows that the subsidy raises the wage,because bargaining implies that workers and employers share the increased surplus.



Market tightness:

Similar to the basic matching model, the equilibrium value of labor market tightnessis given by,

(1� �)(y + ↵s� z)r + q + �✓m(✓)

=

h

m(✓).

Comparative statics:

This equation shows that labor market tightness always increases with the amount ofthe employment subsidies and with the share of firms that benefit from the subsidy.



Equilibrium unemployment:

The Beveridge curve remains the same as in the simple DMP-model, i.e.,

u =q

q + ✓m(✓).

=) Employment subsidies therefore reduce unemployment.

Critique:

Subsidies to declining industries slow down the necessary adjustment.

Taxes to pay of employment subsidies create welfare costs.


Chapter 2: Labor Market Policy · Chapter 2: Labor Market Policy Literature: Pierre Cahuc, St´ephane Carcillo, and Andr´e Zylberberg: Labor Economics Chapters 12.2, 13.1-2, and

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