unemployment: shapiro-stiglitz · 2019. 11. 29. · assumptions...

Unemployment: Shapiro-Stiglitz

Myungun Kim

Myungun Kim Unemployment: Shapiro-Stiglitz

Motivation

Shapiro-Stiglitz (1984)I Firms’ limited monitoring abilitiesI How to incentivise workers to exert effort?


Assumptions: Workers

I A large number of workers LI A large number of firms NI Three states : E (Employed), S (Employed but Shirk), U

(Unemployed)


Assumptions

I. The representative worker’s lifetime utility

U =∫ ∞

t=0e−ρtu(t)dt (1)

where ρ is the discount rate and u(t) is instantaneous utility attime t

u(t) ={

w(t)− e(t) if employed0 if unemployed

(2)

e is either e (work) or 0 (shirk)



II. The process of job breakup occurs with probability b per unittime (E → U)

III. Firm’s detection of workers who are shirking occurs withprobability q per unit time (S → U)

IV. Unemployed workers find employment ar rate a per unit time(U → E )



(Optional):

Probabilities are expressed in ‘per unit time’ rate.

Still being employed after a year - (1− b)Still being employed at the middle of the year - (1− 1

2b)2

Still being employed at end of any month - (1− 112b)12

So, continuous form limn→∞(1− 1nb)n = e−b

Probability that someone starting in E still in E after ∆t : e−b∆t

Probability that someone starting in S still in S after ∆t : e−(b+q)∆t

Probability that someone starting in U still in U after ∆t : e−a∆t (Eachworker takes a as given, but a is determined endogenously in the wholeeconomy.)



(Optional):

The probability that the worker is still employed in the job at somelater time t is given by

P(t) = e−b(t−t0) (3)

Using this, what is the probability that, given that the worker isstill employed at t, he or she is still employed at t + τ?

P(t + τ |t) =P(t + τ)

P(t)= e−b(t+τ−t0)

e−b(t−t0)= e−bτ

which does not depend on how long the worker has already beenemployed (no need to keep track of how long workers have been intheir jobs).



(Optional):

If a worker is employed but shirking, what is the probability that heor she is still employed time τ later?

P[Not caught and still employed]= P[Not caught]× P[Still employed]= e−bτ e−qτ

(4)

where the first equality holds because of independence of twoevents.


Assumptions: Firms

π(t) = F (eL(t))− w(t)[L(t) + S(t)] (5)

where L workers and S shirkers. The firm’s decisions affect profitsonly at that date. No need to analyse the present value of profits.


Expected Value of Discounted Lifetime Utility

Value of being in state i is denoted as follows

Vi , i = E ,S,U

Vi denote an expected discounted lifetime utility of a worker instate i . As the model focuses on the steady state, Vis are constant.

How are these values determined?



(Optional):

Consider a worker who is employed and exerts effort at time t = 0.Suppose that time is divided into intervals of length ∆t. LetVE (∆t) denote the value of employment as of the beginning of aninterval

VE (∆t) =∫ ∆t

t=0e−bte−ρt(w − e)dt

+ e−ρ∆t [e−b∆tVE (∆t) + (1− e−b∆t)VU(∆t)](6)



(Optional):

Solving the integral and applying l’Hopital’s rule,

VE = 1ρ+ b [(w − e) + bVU ] (7)



Using this approach, Vis are derived as follows

ρVE = w − e − b(VE − VU) (8)

ρVS = w − (b + q)(VS − VU) (9)

ρVU = a(VE − VU) (10)



Assuming the risk-neutral investor, the intuitive explanation of theequation (8) is as follows

ρVE = w − e − b(VE − VU)

LHS : Utility return on being in state E and ρ is the discount rate(similar to rate of return times the capital value of an asset (VE )).

RHS : w − e is the dividend earned while in state E plus theexpected capital gain (b is the probability of changing between thestates)


No Shirking Condition (NSC)

The wages must be such that

VE ≥ VS (11)

The firm would choose w so that

VE = VS (12)



Using VE = VS , the equations (8) and (9) can be combined,

w − e − b(VE − VU) = w − (b + q)(VE − VU) (13)

Hence,

VE − VU = eq (14)



From (8) and (9),

VE = 1ρ+ b [(w − e) + bVU ] (15)

VS = 1ρ+ b + q [(w − e) + (b + q)VU ] (16)

Using the condition that VE = VS ,

1ρ+ b [(w − e) + bVU ] = 1

ρ+ b + q [(w − e) + (b + q)VU ] (17)

Solving for w ,

w = eq (b + ρ+ q) + ρVU (18)



From (10) and (14),

w = eq (b + ρ+ q) + a e

q

= e + (b + ρ+ a) eq

(19)

It is briefly mentioned above that a is endogenously determined inthe economy. Since the economy is in steady state,

bNL = a(L− NL) (20)

Solving for a,

a = NLL− NL

b (21)



The NSC condition is

w = e + (ρ+ LL− NL

b) eq (22)



Note: No shirking condition (NSC) in Pontus’ lecture slides is

w ≥ b + eπ(1− p)

= b + eπu

where p the probability of finding another job. π denotes theprobability of shirkers getting caught.

In this slide, there is no discount factor ρ, as he did not considerthe lifetime utility, but only the present utility.


Comparative Statistics



Downward sloping LD

I From firm’s profit maximisation, eF ′(eL) = wI The demand curve is downward sloping

Vertical LS

I There are L identical workers that supply one unit of labourinelastically



(a) An increase in workers’ discount rate ρ

If the discount rate increases and workers become less patient,NSC will shift upward. This increases the equilibrium levels ofunemployment and the real wage.

Intuitively, workers discount the future more, so they are lessdeterred by the prospect of becoming unemployed in the futurewhen caught shirking. To prevent shirking, higher real wages areneeded to increase the loss when the worker is fired.



(b) a recession (negative productivity shock)

Let’s assume that a productivity is multiplicatively included in theproduction function in the following from

AF (L)

.Consider a decrease in A (negative productivity shock)

Holding everything constant, a reduction in productivity will lowerwage level. This results in a shift down of labour demand curve ata given level of employment. This results in a reduction in wagenecessary to induce workers to exert effort.



(b) a decline in the job breakup rate b

w = e + (ρ+ LL− NL

b) eq (23)

A decline in the rate of job destruction b will shift NSC downward,reducing the real wage and decreasing the level of unemployment.

Intuitively, since workers are less likely to lose their job in theabsence of shirking, the value of being employed increases. Thisreduces the real wage and the level of unemployment that isneeded to deter shirking.



(d) a wave of immigration increases the labour force

w = e + (ρ+ LL− NL

b) eq (24)

eF ′(eL) = w (25)

If the labour force L increases, the vertical portion of the laboursupply curve shifts to the right. The labour demand curve isunaffected. The NSC curve shifts down.



Intuitively, at a given NL, L− NL is now higher. Thus, at a givenlevel of employment, if workers become unemployed, they are likelyto stay unemployed longer.

Thus, at a given level of employment, the cost of shirking isgreater for a worker and thus firms can get away with paying alower wage to deter shirking.

This reduces the real wage and it increases the level of employmentbut less than the increase in the labour force (see a figure in theprevious slide). Hence, the level of unemployment increases.



(e) there is no turnover (b = 0). Once unemployed, theunemployment spell lasts forever.

w = e + ρeq (26)

Employed workers are never hired if there is no turnover.

Intuition : the gain from shirking e. The cost is a probability q ofbecoming permanently unemployed and losing the discountedsurplus from the job q w−e

ρ . Hence, w = e + ρ eq .


Implications of the Model

Perfect information about shirkers

The unemployment is involuntary in the model. The unemployedworkers are willing to work at a prevailing wage or lower, despitenot being able to promise not to shirk at such wage.

The model relies on firms’ inability to monitor workers.

If they have perfect information about shirkers (q →∞),equilibrium occurs at the point where Ls = Ld (∵ NSC curvebecomes backward L)



Welfare improving policies?

The first-best allocation is for everyone to be employed and exerteffort.

The government cannot bring this about by dictating that firmsmove down the labour demand curve until everyone becomesemployed. This policy will cause workers to shirk, leading to zerooutput.

Shapiro and Stiglitz suggest wage subsidies financed by lump-sumtaxes or profits taxes, which will shift up the labour demand curve.



Wage rigidity in the model

A negative productivity shock results in a lower demand for labour,leading to a reduction in wage. However, a fall in wage is less thanin the case where we do not consider no-shirking condition.


unemployment: shapiro-stiglitz · 2019. 11. 29. · assumptions...

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