Nurfatima Jandarova – 4. Job Search

	Employed	Unemployed	Not working
Sleep	8.2	8.6	8.6
Personal care and eating	1.7	1.8	1.8
Home production, shopping, care of others	3.0	3.4	3.1
Leisure, travel, sports and socialising	5.3	8.2	7.6
Work	4.2	0.1	0.1
Study	0.1	0.1	1.1
Unspecified	0.1	0.1	0.2

Basic job search model

Workers do not know precise wage at each job, only CDF
Dynamic model with continuous time (period is )
Key components
- Assume worker receives job offer at wage in period
- If she is employed with wage , assume her utility over is
- Exogenous job destruction at rate
- Discount future at rate
Let and be discounted utility of employment and unemployment

It is customary to set up the search model in continuous time. This makes characterisation of an equilibrium easier.
In the basic model, environment remains stationary, i.e, is same at any
Workers do not observe wages of all the available jobs, instead they only see wage offered by a particular job.
- Think about job search process in the real world. Often job postings don’t have exact wage information, but is settled during interview process.
- In the model, worker knows that all wage offers are random draws from distribution
If she is employed over a period of time , she earns
Assume for simplicity no disutility of work and no risk aversion period utility exactly equals earnings
Over any period of time , worker can lose her job at rate . Hence, at she
- continues to enjoy at rate , and
- gets value of unemployment at rate
Since it is dynamic model, we also need to introduce discount factor
- Assume constant real interest rate 1 unit invested at will bring units at
- Hence, discount factor is

Basic job search model and reservation wage

Optimal job search strategy

If she receives no offer, she continues looking for a job
If she receives an offer at wage , she accepts iff

Rewrite employee expected utility

Since we assume is constant, there is

It is easy to see that here

is called reservation wage

Basic job search model and reservation wage

Expected utility
Assume job offers arrive at an exogenous rate
When unemployed, receive net income
- is unemployment benefit, and
- is total cost of job search (direct and indirect)
Can write

After simplifying and plugging in the expression for , can show that

Basic job search model and unemployment duration

Hazard rate, or exit rate from unemployment, depends on
- job offer arrival rate
- probability of offered wage being above
Hence, hazard rate is
Average duration of unemployment is

Basic job search model and comparative statics

Using implicit differentiation, can show that reservation wage is

Therefore, average duration of unemployment is

increasing in net income in unemployment:
decreasing in interest rate:
decreasing in job loss rate:
ambiguous with respect to job offer arrival rate:

Derivation steps:

Rewrite the last expression for as
Compute partial derivatives of with respect to variables of interest and plug into implicit differentiation formula

Thus, the basic model is telling us that

with more generous unemployment benefits, jobseekers stay unemployed longer

Even though the prediction is unambiguous, magnitude of this relationship is unknown. Moreover, some empirical studies show that increasing benefits can actually shorten unemployment duration among workers who received no benefits at all.
unemployment duration increases when interest rates are high

When interest rate is high, an individual cares less about the future and more about current utility. Her reservation wage is lower and, consequently, spends less time looking for a job.
people spend less time looking for a job when chance of losing the job is higher

Remember that in this model once the worker accepts the offer, she sticks to the job until exogenous event forces her to quit. Therefore, when is higher and she accepted an offer barely higher than , she knows that very quickly she’ll be looking for job again and will have a chance to get an offer at even higher wage. Conversely, when is low, once she accepts the offer, she is stuck at the job for longer period.
unemployment duration may increase or decrease with job offer arrival rate depending on how strongly the reservation wage reacts to

If reservation wage does not react to , i.e., , then average duration of unemployment decreases with . This makes sense: the more offers a worker receives, the sooner she will get one above her reservation wage. However, worker can also become picky about jobs (her reservation wage ) and spend even more time looking for best job.

Job search with benefit eligibility

Two types of unemployed: eligible () and non-eligible ()
- Non-eligible earns income
- All workers who have worked at least once are eligible
Expected utility of accepting offer same:
Reservation wage of non-eligible worker satisfies
Expected utility of non-eligible unemployed worker is
Using all of the above, we can write as
Can show that and

For example, in Finland, there three types of unemployment benefits

labour market subsidy: no work requirement, but assess need for the subsidy
basic unemployment allowance: 12 months of work history and salary above certain level; fixed amount irrespective of prior wages
earnings-related unemployment allowance: 12 months of work history and salary above certain level; benefit amount depends on prior wages

Model

Consider a non-eligible unemployed worker that received a wage offer . Her expected utility of accepting the job offer is same as that of unemployed person and consists of instantenous utility and weighted drop in utility if she looses this job . Since she has already been employed, even after losing this job, she will be unemployed worker eligible for benefits the expected utility depends on (and not on ).

The expression for expected utility of non-eligible unemployed worker is derived as follows. Notice that while she is unemployed, she earns . With probability she receives an offer that she can either reject or accept and with probability she remains a non-eligible unemployed worker. The optimal strategy is still the same: accept offers above reservation wage and reject the rest. The only difference is that now she has her own reservation wage , which may be different than of eligible unemployed worker. Therefore, the expression for her discounted expected value can be written as

After following similar steps as before, this expression can be simplified to

Now, we can use and to substitute on the left-hand side and on the right-hand side of the above equation. The result relates and as follows

Hence, reservation wage of non-eligible accounts for the fact that unemployed worker gets some basic income , can get and accept an offer above her reservation wage, and faces the chance of not being able to “upgrade” to eligible worker (and her ).

Again, using implicit function theorem, we can show that since as shown before. Similarly, time spent unemployed for non-eligible worker is . Hence, .

Other extensions of basic job search model

Job search model with participation choice

Let be net income of non-participation. Then

If , worker decides to not participate
If , but offered wage ,the worker decides to keep looking for jobs
If , the worker accepts the job offer

Can be useful to study discouraged workers

Other extensions of basic job search model

Job search model with participation choice

On-the-job search

Workers can keep searching for jobs while employed
Let be offer arrival rate for employed and - for the unemployed
Reservation wage is increasing in and decreasing in

Other extensions of basic job search model

Job search model with participation choice

On-the-job search

Effort of job search

Job arrival rate is endogenous and depends on
- state of the labour market
- worker’s search effort
Reservation wage is increasing in and
Optimal effort is increasing in and decreasing in

See Section 2.2.3 of Cahuc, Carcillo, and Zylberberg (2014) for the step-by-step analysis of this model.

The key difference is that job offer arrival parameter is no longer exogenous, but depends on individual effort and state of the labour market . Thus, optimal strategy of the worker is described by reservation wage and optimal effort . The comparative statics of interest show that

(both optimal search effort and reservation wage are increasing in ) and that

(reservation wage is increasing in , while optimal effort is decreasing in ). This suggests that when unemployment benefits shrink AND state of labour market worsens (fewer vacancies), the direction of change in optimal search effort is ambiguous. It can decrease, increase or not change. As a result, also, the effect of both and on average duration of unemployment spell is unknown.

Other extensions of basic job search model

Job search model with participation choice

On-the-job search

Effort of job search

Job search and wealth

Workers’ value functions depend on their wealth
Reservation wage increases with wealth
Optimal search effort decreases with wealth

Other extensions of basic job search model

Job search model with participation choice

On-the-job search

Effort of job search

Job search and wealth

Job search and benefit sanctions

Unemployment benefits often require active job search
If fail to meet those conditions, benefits are sanctioned
Workers that get sanctioned increase their search effort
Threat of sanctions makes unsanctioned workers increase search effort

Other extensions of basic job search model

Job search model with participation choice

On-the-job search

Effort of job search

Job search and wealth

Job search and benefit sanctions

Job search and non-stationary environment

Labour market environment is not same at every
Example: benefits often decrease or stop after some time
- reservation wage is decreasing over time
Example: unemployed workers may receive fewer offers over time
- reservation wage is decreasing over time
- dynamics of average duration of unemployment is unknown

In general, the environment of labour market may be non-stationary for a number of reasons.

aggregate shocks can change the distribution of wages over time
financial constraints may not allow one to smooth consumption over the entire “optimal” length of unemployment
job offer arrival rate can diminish the longer one is unemployed
employment benefits may decrease or stop after a certain period of time

See Section 2.2.6 of Cahuc, Carcillo, and Zylberberg (2014) for the example of the model where unemployment benefits reduce over time.

The implications of this model are intuitive: the reservation wage is decreasing over time. Therefore, average duration of unemployment spell is also decreasing over time, since the worker is willing to accept wider set of jobs. However, as noted in the book, if the model had allowed for diminishing job offer arrival rate, the result on average duration of unemployment would be ambiguous: worker is still willing to accept wider set of jobs, but she’s also receiving fewer and fewer jobs.

Other extensions of basic job search model

Job search model with participation choice

On-the-job search

Effort of job search

Job search and wealth

Job search and benefit sanctions

Job search and non-stationary environment

See Section 2 in Chapter 5 of Cahuc, Carcillo, and Zylberberg (2014) for discussion of these extensions and references

Equilibrium search models

Basic model is partial: distribution of wages is exogenous
Add labour demand to the picture to get endogenous
Main implications:
- With on-the-job search, wages rise when switching jobs
- No wage offered below reservation wage
- Wage dispersion with more competition for workers (high )
See Section 4 in Chapter 5 of Cahuc, Carcillo, and Zylberberg (2014)!

With basic job search, the equilibrium wage is simply equal to the reservation wage
- There is no reason for firms to offer wages above as it would only decrease their profits without increasing output
- Hence, equilibrium collapses to a degenerate distribution with point mass at
- This is called Diamond paradox
On-the-job search is necessary to circumvent this issue and have non-degenerate equilibrium
Since employed worker would only accept wage offers above her current wage, her observed wage necessarily increases if she switches jobs
Moreover, her wages are rising over time since she’s had longer time to receive better offers
In this model, there is no reason for a firm to offer wage below reservation wage lowest wage
- If the firm did offer wage below , then neither employed nor unemployed workers would accept the offer
Wage dispersion increases when competition is high
- Wage dispersion here is measured as the ratio of mean wages to min wages
- This ratio is increasing in
- That is, when job offer arrival rate is increases or when job separation rate decreases, firms will need to offer substantially higher wages to convince a worker to switch the job.
- You can also think that when , each employed worker receives more offers. That is, more firms are competing for each worker and workers get to be more picky (they have higher bargaining power).

	Age		Age
	Low experience	High experience	Low experience	High experience
Low earning	eRR	eRR	eRR	ePBD-RR
High earning	Control	Control	Control	ePBD

	Before 1989	After 1989	Change over time	DiD
ePBD	16.250	18.670	2.420	1.130
	(0.080)	(0.090)	(0.120)	(0.180)
Obs.	48 294	51 110
eRR	17.790	20.030	2.240	0.960
	(0.120)	(0.160)	(0.200)	(0.240)
Obs.	17 160	15 310
ePBD-RR	19.010	23.550	4.530	3.250
	(0.170)	(0.240)	(0.200)	(0.240)
Obs.	11 992	9 182
Control	15.240	16.520	1.290
	(0.080)	(0.090)	(0.130)
Obs.	33 815	38 958

	Dependent variable
	Breadth of applications	N applications	N interviews	N offers
* p < 0.1, p < 0.05, * p < 0.01
Treatment	0.030	0.010	0.440*	-0.130
	(0.200)	(0.090)	(0.280)	(0.250)
Treatment x init. broad	-0.430*	-0.050	-0.070
	(0.220)	(0.120)	(0.240)
Treatment x init. narrow	0.490*	0.080	1.030***	0.100
	(0.290)	(0.130)	(0.550)	(0.560)
Num.Obs.	305	487	464	253