wpsac)(4
POLICY RESEARCH WORKING PAPER 2068
Quitting and Labor To prevent trained workers
from quitting to open their
Turnover own businesses, firms pay
higher than market "efficiency
Microeconomic Evidence and wages" to reduce turnover.
What is the impact of
Macroeconomic Consequences macroeconomic shocks and
policy innovations, such as
Tom Krebs labor market reform, in ani
William F. Malo*ey economy where this is of
central importance?
The World Bank
Latin America and Caribbean Region
Poverty Reduction and Economic Management Sector Unit U
February 1999
l POLICY RESEARCH WORKING PAPER 2068
Summary findings
Combining microeconomic evidence with They use panel data from Mexican labor surveys to
macroeconomic theory, Krebs and Maloney present an estimate the quit function derived from the model and
integrated approach to wage and employment the results support their view that transitions from
determination in an economy where firms pay above formal salaried work to informal self-employtnent are
market "efficiency wages" to prevent trained workers quits rather than fires. (Quitting is positively related to
from quitting. the mean self-employment income and the probability of
The model offers predictions about the behavior of being rehired and negatively related to the mean formal
formal employment, labor turnover, and segmentation in salaried wage.) They then use the parameters estimated
response to formal sector productivity shocks (including from the quit function to calibrate the model economy
economic growth and tax reductions), changes in the and simulate the impacts of economic shocks and policy
desirability of self-employment (formal sector tax rates), innovations and find the impact on employment,
and the cost of training a new worker. turnover, and segmentation to be substantial.
This paper - a product of the Poverty Reduction and Economic Management Sector Unit, Latin America and Caribbean
Region - is part of a larger effort in the region to understand the functioning of developing country labor markets. Copies
of the paper are available free from the World Bank, 1818 H Street NW, Washington, DC 20433. Please contact Tania
Gomez, room 18-102, telephone 202-473-2127, fax 202-522-2119, Internet address tgomez@worldbank.org. Policy
Research Working Papers are also posted on the Web at http://www.worldbank.org/htmVdec/Publications/WVorkpapers/
home.html. The authors may be contacted at tkrebs@uiuc.edu or wmaloney@worldbank.org. February 1999. (39 pages)
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the exchange of ideas about
development issues. An objective of the series is to get the findings out quickly, even if the presentations are less than fully polished. The
papers canry the names of the authors and should be cited accordingly. The findings, interpretations, and conclusions expressed in this
paper are entirely those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors, or the
countries they represent.
Produced by the Policy Research Dissemination Center
Quitting and Labor Turnover:
Microeconomic Evidence and Macroeconomic Consequences*
Tom Krebs**
Department of Economics
University of Illinois, Urbana-Champaign
William F. Maloney***
The World Bank
Keywords: Efficiency Wages, Employment, Labor Turnover,
Macroeconomic Policy, Self-Employment..
JEL Classification Numbers: E24, E60, J41, J63.
We would like to thank Omar Arias, Patricio Aroca, Roger Koenker, Pravin Krishna, Guillermo Perry and
seminar participants at the University of Illinois for helpful comments. We also thank the Mexican National
Institute of Statistics, Geography, and Information (INEGI) for the use of the data. INEGI is in no way responsible
for the incorrect manipulation of the data or erroneous conclusions drawn from it.
Currently visiting Brown University, Department of Economics, Brown University, Robinson Hall 302-
D, 64 Waterman Street, Providence, RI 02912, tkrebs@uiuc.edu.
Poverty Reduction and Economic Management Division, Latin America and the Caribbean, The World
Bank,1818 H St. N.W., Washington, D.C. 20433, wmaloney@worldbank.org.
1. Introduction
Recent work on efficiency wage models has produced an internally consistent macro-theory of
involuntary unemployment (underemployment)' whose basic assumptions and implications have
largely been corroborated by empirical micro-studies.2 Although the microeconometric work was
motivated by macroeconomic theory, it has never been fully integrated into a macroeconomic
framework.3 In this paper we offer one version of such an integrated approach. More precisely, we
first develop an efficiency wage model with labor turnover (Phelps 1968, Stiglitz 1974, Salop 1979,
Hoon and Phelps 1992) and show that the worker's decision problem gives rise to a quit-rate
function. We then use microeconomic data to estimate this quit-rate function and to test the
specification suggested by economic theory. Finally, microeconomic evidence and macroeconomic
model are combined to evaluate the quantitative effects of changes in economic policy and other
macroeconomic shocks on the wage rate, the turnover rate, and employment in the long-run (steady
state analysis).
The efficiency wage model with labor turnover we employ is in principle applicable to any
type of movement of labor across sectors in any country (Bulow and Summers 1986). However, the
original literature on this type of efficiency wage model has mainly focused attention on
unemployment in developed countries (Phelps 1968, Salop 1979, Phelps and Hoon 1992). We, on
the other hand, test and calibrate our model using panel data on the movement of Mexican workers
between the formal salaried and the informal self-employed sector. Our choice of the data set was
motivated by the following two considerations.
'See Katz (1986) and Woodford (1994) for surveys.
2See Katz (1986) and Layard, Nickell, and Jackman (1991) for surveys.
3The quantitative papers by Danthine and Donaldson (1990,1995) and Kimball (1994) on
efficiency wage models with shirking (Shapiro and Stiglitz 1984) consider microeconomic evidence
when calibrating the macroeconomic model, but do not incorporate a microeconomic estimation equation
into the macroeconomic model as we do. In this sense, we feel that we have come closer to a full
integration of the two fields of labor economics and macroeconomics. See also Blanchard and Katz
(1997) for a statement in favor of such an integrated approach.
First, the efficiency wage model with labor tumover captures well a number of features of
LDC labor markets, and in particular the Mexican case we consider. The literature suggests that the
self-employed informal sector comprises both workers rationed out of formal salaried jobs as well
as a relatively prosperous "upper tier" that may prefer self-employment.4 In other words, the
literature is consistent with one of the central ideas of the efficiency wage model with labor
turnover, namely that at each point in time workers are voluntarily leaving their formal-sector job
for self-employment and simultaneously self-employed workers are unsuccessfully trying to reenter
the formal sector. Moreover, neither minimum wages nor unions are credible explanations for the
observed segmentation.5 Finally, Constitutional proscriptions against firing suggest quittiing as the
dominant mode ofjob separation.
Second, our data set has an important time dimension which allows us to estimate the
quitting response of individual workers to changes in macroeconomic conditions. Given our final
goal of quantitative macroeconomic analysis, this feature ofthe data set seems essential. In addition,
the data on self-employed workers offer a measure of the benefits (payoffs) to not being employed
in the formal sector that displays substantial variations over time. These variations in self-
employment benefits are important since they provide us with an additional test of the "quitting
theory" which predicts that labor turnover is positively correlated with expected benefits to self-
employment and that the quit-rate function is symmetric: the benefits-elasticity of quitting, is equal
to the negative ofthe wage-elasticity of quitting. Moreover, if the symmetry property ofthe quit-rate
function is supported by the data on self-employed workers, we may use it as a working hypothesis
(until refuting evidence is forthcoming) and apply it to unemployment. This opens the door for an
assessment of the quantitative macroeconomic effects of changes in unemployment benefits without
4 Harris and Todaro (1970) offer the canonical statement of the dualistic (rationing) viewv and
Fields (1990) discusses the "two-tier" view of the informal sector.
5Bell(1996) finds no evidence that minimum wages are binding. Maloney and Ribeiro (1998)
find evidence of union influence on employment, but none on wage setting.
2
directly estimating the elasticity of quitting with respect to unemployment benefits, usually an
impossible task given the lack of temporal variations in these benefits.
Our empirical estimates of the determinants of labor flows from the salaried to the self-
employed sector strongly support the specification suggested by the quitting theory: the individual
probability ofjob separation is decreasing in the formal-sector wage (the expected payoffto staying)
and increasing in benefits to self-employment and the probability of finding a formal-sectorjob (the
expected payoff to leaving). Moreover, the above mentioned symmetry property of the quit-rate
function cannot be rejected. When the microeconomic estimates are used to calibrate the
macroeconomic model, we find the long-run effects of macroeconomic shocks on wages, labor
turnover, and (formal-sector) employment to be substantial. The strong employment response found
here stands in stark contrast to the disappointingly small unemployment effects reported by
Danthine and Donaldson (1990,1995) and Kimball (1994) who calibrate an efficiency wage model
with shirking (Shapiro and Stiglitz 1984) to US unemployment data.6
This paper can be interpreted as providing a two-stage "test" of the real world relevance of
efficiency wage models with labor turnover. In the first stage, microeconomic data are used to
estimate and test what we believe to lie at the heart of this type of efficiency wage model, namely
the quit-rate function. If the coefficients are found to be significant and of the correct sign, in a
second stage the estimated quit-rate function is incorporated into the macroeconomic model and the
calibrated model economy is used to assess the quantitative importance of efficiency wages. This
second-stage check is important since there seems to be little value in having a macroeconomic
theory of unemployment (underemployment) which is supported by microeconomic data but implies
an almost constant unemployment (underemployment) rate. In this paper we present one fully
worked out example of this two-stage procedure in the hope that it will spur interest in further
6Danthine and Donaldson (1990, 1995) explicitly consider aggregate uncertainty by solving a
stochastic dynamic general equilibrium model. Kimball (1994) studies the dynamic and steady state
effects of macroeconomic shocks in a deterministic model, but his quantitative result on employment
variations is obtained by comparing steady state equilibria.
3
applications to different countries and different sectors. Such work is likely to add an important
dimension to the existing empirical literature which has either completely focused on the micro-level
or solely relied on cross-country regressions.!
The paper is organized as follows. Section 2 develops the model. Almost all derivations,
and in particular the discussion of the worker's decision problem, is relegated to the Appendix.
Section 3 presents the empirical analysis of the panel data on Mexican workers and some additional
information on the Mexican labor market. The specification for the estimated quit-rate function is
dictated by the theory developed in Section 2. In Section 4 the macroeconomic model is calibrated
and the simulation results are presented. Section 5 concludes.
2. The Model
The model is a discrete-time, neoclassical growth model with a labor market characterized by labor
turnover, employment-adjustment costs (hiring and training costs), and wage-setting by firms. The
analysis will be confined to equilibria in which a number of economic variables grow at a constant
rate equal to the exogenous rate of technological progress (balanced growth path).
a) Workers
There is a large number of ex-ante identical, infinitely-lived workers. Workers' preferences over
random consumption sequences allow for a time-additive expected utility representation. Workers
do not participate in financial markets and therefore do not save or dissave. Hence, each worker's
7See, for example, Phelps (1994), Nickell (1997), and Phelps and Zoega (1998) for empirical
work using cross-country regressions. Blanchard and Jimeno (1995) conduct an interesting case study
comparing two countries, Spain and Portugal. The method outlined in this paper provides a fonnal
procedure for quantifying the importance of efficiency wages in explaining the different macroeconomic
experiences of two countries.
4
consumption level is equal to his current disposable income.8
In each period a worker devotes a fixed amount oftime to one of the following two activities:
working in the formal sector or working in the informal sector of the economy. Our informal-sector
data in the empirical section are taken from the self employed and we will therefore call a person
working in the informal sector a self-employed worker. In each period, a worker, regardless of his
current employment status, receives an idiosyncratic shock determining the relative attractiveness
of employment versus self-employment ("taste-shock", change in expected payoff to self-
employment). After observing the shock realization, an employed worker makes a quit/stay
decision and a self-employed worker makes a search/no-search decision. Whereas an employed
worker automatically becomes self-employed when deciding to quit a job, a self-employed worker
who decides to search for formal-sector employment receives ajob offer only with probability less
than one.
The Appendix Al discusses the Bellman equation associated with the worker's decision
problem and analyzes the resulting optimal decision rule. The optimal decision rule gives rise to a
quit-ratefunction, q. = q(w,,w,p;T,b),where q, standsforthequitrateexperiencedbyfirm i, w1
for the (growth-adjusted) wage paid by finn i, w the average (growth-adjusted) wage, p for the
probability of finding (formal sector) employment when not employed, X for the tax rate on formal-
sector labor income, and b for the average (growth-adjusted) pecuniary benefits from self
employment.
Let q(w,p;t,b) i q(w,w,p ; T,b) _be the economy-wide quit-rate fimction when all
8ln a sense, this assumption renders the model classical rather than neoclassical. It is mainly
made for tractability reasons since it trivially deterrnines the wealth distribution of workers (no wealth).
Without this assumption the wealth distribution is in general non-trivial and has to be computed as part
of the equilibrium, except when there is complete consumption insurance. The assumption of restricted
capital market participation is also made in Danthine and Donaldson (1990,1995) for the same tractability
reason. Kimball (1994) does not treat capital accumulation and therefore does not deal with wealth
effects. Phelps (1994) emphasizes wealth effects, but nowhere develops a complete general equilibrium
model with endogenize wealth distribution. We hope to dispense with this assumption in future work.
5
firms pay the same wage. Clearly, this function is identical to the quit-rate function in an economy
with only one representative firm. The function 4(.), however, is the function entering into the
profit maximization problem of an individual firm (see Appendix A2) in a many-firm economy. In
Appendix Al we show that the individual quit-rate function, q (.), satisfies
aq O; a , , as4 aq,O aq ;(l
-< 0. qeq> 0 a .0; 1
aw,. a w ap ab al
and that the economy-wide quit-rate function, q(.), satisfies
.aq ,o
aw ap
(2)
aq - laq l aq> o
ar bab -wew
The empirical analysis conducted in Section 3 tests the sign and symmetry restrictions (2) and finds
strong evidence in favor of them. Our panel data on worker transition provide no information about
the quit-rate function of an individual firm in a many-firm economy, q(.). Appendix Al, however,
shows that the two marginal quit-rate functions (approximately) satisfy
a q (w 'Wwp;,rB) Iw =W - (w,p;,r,B) aq (w,p;,r,B) 3
1 - 3bw(l -q(w,p;'r,b)) (1 -p)
=(, b - 1 - 3 (1 -q(wp,;T,b))
where 1w is the discount factor of workers. Expression (3) enables us to make quantitative
predictions about the impact of economic policy knowing only the economy-wide quit-rate :function
q(.). The parameter pt measures the difference between the reduction in the quit-rate when an
individual firm raises its own wage and the reduction in the quit-rate when all firms raise their wages
simultaneously.
6
b) Capitalists (Firms)
There are i = 1, ... ,N infinitely-lived capitalist with identical preferences who each own one firm
with identical production technology. There is one good which can be used for consumption and
investment purposes. Firm i combines capital and labor to produce output. Adjusting the amount
of labor employed is costly since there are hiring and training costs. We assume that adjustment
costs are fully paid by firms.'
Taking the economy-wide wage as given, each firm i chooses a sequence of consumption
(of owner i ), capital, investment, employment, hiring, and own-wage which maximize the capitalist'
life-time utility subject to the relevant constraints. The decision problem faced by firm (capitalist) i
is fully spelled out in Appendix A2. The resulting Euler equations for the growth-adjusted variables
are
131gP x ai( F -+-(k111 1,1+
t =c (,t+l ak( 1'' °(1 +, d 1
:.. PiC(l+g)- [i,t+l( +i,t+, (WfX +i,t+lt al It,t+l li,+1 -i T(i,t+d) (4)
X., = c t p ;Yi =XT(h.) ; Yit = ( aW(iP)
where PC and p are the capitalist' discount factor and coefficient of relative risk aversion, c.t her
growth-adjusted consumption level, kit the capital stock per efficiency unit of labor, hit the hiring
rate, li, the fraction of the labor force employed in the formal sector, and X, y,, Lagrange
multipliers associated with physical, respectively human, capital accumulation constraints. The
91n our model with credit rationed workers owning no wealth, only the firm can afford to pay
these costs. Even if workers are not credit rationed but human capital created by training is firm specific,
the firm is likely to bear the full cost of training (Salop 1979, Hoon and Phelps 1992, Phelps 1994). Of
course, we do not consider indirect means by which workers can be made to bear some of the adjustment
costs. Wages rising with tenure is one example.
7
multiplier yt is the (utility) value of a trained employee to the firm (the analog to Tobin's q).
Further, Sis the depreciation rate of physical capital, F(.) a standard neoclassical production
function, and T(.) the adjustment cost function.
c) The Government
The government collects taxes and spends the tax receipts on consumption goods. We assume that
tax receipts are equal to outlays in each period and that therefore the fiscal budget is always
balanced. For simplicity, we assume that informal-sector workers and capitalists are not taxed so
that the tax revenue, which is equal to fiscal spending, is (Twl)N.
d) Steady State Equilibrium
We are interested in symmetric steady state (balanced growth) equilibria. Such an equilibrium is
defined as a list of (growth-adjusted, per-capitalist) values for output, y *, capital k *, capitalists'
consumption c*, (fornal-sector) employment , 1*, real wage, w *, and hiring (quitting) rate,
h ' = q *, such that:
i)Given the quit-rate function, q(.), and expected labor market conditions, (w ,p*),
y * , k * , I *, h *, c *, w * are the solution to the capitalists' (firms') optimization problem if the
initial capital stock and stock of trained employees is (k * * ).
ii)The quit-rate function, q(.), is the solution to the worker's optimization problem.
iii)Expectations are fulfilled, that is, expected values are equal to actual values.
Because of the assumption of constant returns to scale, the marginal products only depend
onthecapitaltolaborratio: (k,l) = f'(E) and -(k 1) = f(E) -f'(k)k with
a k a I
k = k/ I and f(k) = F(k, 1). Using the definition of a steady state equilibrium and Equation (4),
we immediately derive the following characterization of a steady state equilibrium. The growth-
adjusted capital stock per employed worker is
8
For given k, the equilibrium wage rate, w and the equilibrium probability of finding aj ob, p *,
are the solution to
f~(k )-f(k*)k* -w - T(Q(w ,p )) [ic(1 +g)yP jT(q(w*,p )) = 0
(6)
T'(q(w*p*))p + 1 0= O (
aq
Finally, the equilibrium values of the remaining variables are determined by
h= q = q(w*,p*) ; 1* = (I + h- -1 k* =P*
p*v(l-q*) (7
c*= F(k*,l*) - Sk* - w*l* - T(h*)l*
where v (1 - q *) is the fraction of self-employed workers searching for a formal-sector job. The
parameter v is a constant which is discussed in more detail in Appendix Al.
Evidently, the two equations (6) determining the equilibrium values w* and p* are
equivalent to
Z - w- T(q(w*,p*)) - rT(q(w*,p*)) = 0
(8)
T(q(w',p) + =0
R (w *sp *) '( p*)
'0The consumption of a worker if employed is w * (1 -T) and if self-employed is b. The
government consumes (Tw *l *)N.
9
where Z = f(k*) +f (k*)k* is the marginal (revenue) product of labor and r = 1 - C (1 +g) P
the real interest rate.
It is often useful to depict the solution to (8) as the intersection of a downward-sloping "labor
demand curve" and an upward-sloping efficiency (incentive) wage curve. To the extend that the
first equation in (8) expresses the optimal employment choice by firmns and the second equation
represents the optimal wage setting by firms, this terminology seems justified. The first equation
in (6) implicitly defines a function w = w d(p) with
dwd (T' +rT") q
dp 1 + (T' + r T")) (9)
6w
Since aq < 0, aq > 0, T' > 0,and T" 2 0,thecurve wd(.) isdownwardslopingifandonly
aw a
if I aq I (T' + rT") < 1. For small rT" this condition is always satisfied since
6w
q I T'= < 1 (this follows fromthe second equationin(8)). Note also thatifthereal interest
rate is non-negative, 1. > 1 is a necessary (and if r T" 0 O a sufficient) condition for the w d_
schedule to be downward sloping. Thus, in order to have a labor demand function with a negative
slope, the reduction in the quit-rate due to the increase of the individual firm's wage must be larger
than the reduction in the quit-rate when all firms simultaneously increase their wage, wvhich is
generally true in the model considered here.
The second equation in (8) implicitly defines a function with
T - ( 1 aq -2 a2_ ag aq
dw_ alp k ajwJ 8p6 aw ap aw 1
dp T" Iaq_ ( ± 2§ a2q +6 q (0 )
a3W VA aw) aw2 aw aw
10
There is no straightforward way of signing the expression (9).) Our empirical results suggest that
the quadratic and cross-derivative terms are zero. If in addition T" = 0, we have
dp p 8w
which turns out to be positive for the range of parameter values considered in this paper. Hence, the
efficiency wage curve is upward-sloping.
e) An Extension: Vacancies and Matching
The efficiency wage model developed so far shares one important feature with traditional search
models,"2 namely an endogenously determined labor turnover rate. However, it also differs from
those models in important ways. First, in the efficiency wage model firms set wages in contrast to
the Nash bargaining solution usually deployed by the search literature. In a sense, the efficiency
wage model is the limit case of the bargaining model in which firms have all the bargaining power
and are therefore in a position to make a take-it-or-leave-it offer to workers. Second, the search
literature deploys a matching (hiring) function relating the number of hires, hl, to the number of
vacancies, v, and the number of unemployed I - 1. This matching function describes the efficiency
of the job reallocation process. In this section, we briefly discuss the possibility of incorporating a
matching function into the efficiency wage model developed here.
For the sake of concreteness, consider the case in which the matching function is of the
Cobb-Douglas type, h I = va (1 - 1)1 -a where we assumed for simplicity that all unemployed (self-
employed) workers are searching for a formal-sectorjob.'3 Suppose further that the only adjustment
cost is the cost of posting a vacancy and that the unit cost of a vacancy is a constant, c. Hence, the
total cost of changing the employment level is T = c v. Eliminating the number of vacancies, v,
yields a total adjustment cost T = c (hl)l1a ( 1 - 1)(1 -1/a) and a cost of adjustment per employee of
"The sufficiency conditions for the firm's optimization problems do not guarantee an upward-
sloping curve.
12See Blanchard and Katz (1997) and Rogerson (1997) for recent surveys.
"3Recall that in our notation h is the hiring rate and hI is total hires of one firm. For simplicity,
we have set the number of firms to one: N = 1.
11
T = c hi/a ( 1/i) Thus, we have an adjustment cost function which is convex in the hiring
rate, h, but also depends on the employment level, 1. This dependence of the "training and hiring
costs" on the employment level is the only difference to the previous model formulation. It is,
however, straightforward to incorporate the more general adjustment cost function into the efficiency
wage model resulting into three steady state equations in the three unknowns w * ,p * , *. We plan
to conduct quantitative policy analysis for such an extended model with a general matching
technology in the future. The efficiency wage model developed in the last sections can be thought
of as the special case in which a = 1 since then we have T = c h, that is, an adjustment cost function
which only depends on the hiring rate. Incidentally, this function is linear, a property we assume in
the quantitative section 4.
3. Microeconomic Data Analysis
This section uses micro-economic data from Mexico to estimate the quit-rate function and to test the
model's predictions about the determinants of quitting behavior. The quit-rate function estimated
here is used in Section 4 for macroeconomic policy analysis.
a) Data Description
The National Urban Employment Survey (ENEU) conducts extensive quarterly household interviews
in the 16 major metropolitan areas and is available from 1987 to 1993. The sample is selected to be
geographically and socio-economically representative. The statistical agency (INEGI) expanded it
significantly over the period by adding municipalities, however, we include only those present in
every year of the survey to prevent changes in composition. The questionnaire is extensive in its
coverage of participation in the labor market, wages, hours worked, etc. that are traditionally found
in such employment surveys. INEGI's treatment of sample design, collection, and data cleaning is
careful. Surveys and documentation of methodology are available on request.
The ENEU is structured so as to track a fifth of each sample across a five quarter period. To
construct the panels, workers were matched by position in an identified household, level of
education, age and sex to ensure against generating spurious transitions. Using just the first
12
variables to concatenate and following changes in sex across the panel led to mismatching (or
misreporting) of under .5 percent. Taken together, we have 24 complete panels of 5 periods
spanning a total of 28 quarters where transitions could occur across a seven year period which
includes a time of recession (1987-88), recovery (1989-91), and then stagnation (1992-1993).
Though the model deals with decisions to be self-employed generally, we further narrow the
population by only considering self-employed workers in the "informal" sector. We use the term
'informal' here to refer to those unprotected by labor law, more specifically, owners of firms under
16 employees who do not have social security or medical benefits. In fact, under 1% of these firm
have more than 5 employees so the definition corresponds closely to that commonly used in the
development literature. Formal salaried workers are defined as those in firms of over 16 workers
who enjoy labor protections. To eliminate the "self-employed" in consulting firms or other high end
activities, we analyze male workers with a high school education or less between the ages of 16-65.
The dependent variable is a 0, 1 index that captures whether the worker moved during a
particular quarter. If he should move back to salaried employment and then move again to self-
employment, this is counted as a second quit. The macro wage and benefit variables employed in
the regression analysis, logw1 and logb,, are the median for the entire sample (spanning five
panels) for each quarter. The probability of being hired, p,,is the number ofthe self-employed who
transition to the salaried sector, as a fraction of those looking for a salaried job. Vhile we know the
total number of self-employed, we do not observe the share searching in each period which theory
predicts should vary with macro-shocks. We assume that this fraction is proportional to, or at least
highly correlated with, the standard measure of search, the unemployment rate. The probability p,
is therefore proxied by Pt, the number of the self-employed moving into salaried work divided by
the number of unemployed. Even though we thereby avoid the need for time series data on the
search intensity, we still require information about the sample average of this variable in order to
rescale the estimated coefficient appropriately when conducting the macroeconomic simulations in
Section 4. To this end, we use the survey response from the National Micro-Enterprise Survey
(ENAMIN) which in 1992 re-interviewed roughly 11,000 of those in the 1991:4 ENEU who
13
declared themselves self-employed. In particular, it asks the motivation for opening the business and
offers eight non-exclusive responses. 13.5 percent responded that they could not find work as a
salaried worker and another 3.2 percent responded they were laid off at their previous job. We use
the rounded up number of 20% in our baseline model as an estimate for the average fraction of self-
employed workers searching for a formal-sector job (see Section 4).
The last two columns of table 1 present the summary statistics for the variables used in the
regression analysis. On average 2.5% of salaried workers transit into self-employment in one
quarter. The standard deviations also suggest substantial variation in the macro variables across the
period.
b) Econometric Specification and Results
We are interested in implementing an empirical procedure allowing us to estimate the quit-rate
function q = q (w,p, b). In the Appendix we show that if workers' utility is logarithmic and if the
optimal decision rule is approximtated by a first-order Taylor expansion, then we can write
q = (D(a% + a- logw + a P + a, logb + oC) j_1)
where PD(.) is the distribution function of an unobserved, worker-specific shock variable '9'. The
random variables { C0', are identically distributed and capture all differences in taste and ability
across workers influencing their quitting decision. The relevant derivatives ofthe quit-rate functions
are then given by I aq = 0, b ab = a, (p(x), and aq = a p(x), where p(x)is the
b ab ~~ap P
probability density evaluated at x = ao + a W + a D + a. b. Hence, the inequalities (2) expressing
the implications of the theory now read a < 0, a > 0, and ab> Oand the symmetry relation says
14
aw = -a,. fThere are two difficulties with implementing (11) econometrically.14
First, the discussion in the Appendix only deals with the case of deterministic changes in the
macroeconomic variables w , b , and p whereas the data are better described by a stochastic process
{(w,,pt,bt)}70. If, however, the process of macro variables,{(w,,p,,b,)}0, is Markovian, a
straightforward extension of the analysis shows that the optimal decision rule of workers still gives
rise to a quit-rate function
qt = (DaO + al loew± + a p + a, logb, +). (12)
since the current macro state (w,,pt, b,) is a sufficient statistic for the future evolution ofthe relevant
macro variables (we again assumed a first-order Taylor approximation). With some weak additional
assumptions, it can also be shown that the restriction imposed upon the signs of the coefficients
a, a " al, still hold. From a quantitative point of view, the quit-rate function derived in the
Appendix corresponds to the quit-rate function when macroeconomic variables follow a random
walk (highly persistent macroeconomic shocks). Given our short sample period (28 observations
over seven years), we do not attempt to test the random walk assumption.
Second, if we only use the macro variables w ,pt, b, as right-hand-side variables, we
certainly loose a large amount of information about economic variables influencing the decision of
an individual worker since any observed difference in behavior is automatically attributed to the
unobserved idiosyncratic shock, 0'. On the other hand, if we use the observed wage of individual
workers on the right-hand-side (plus additional human capital variables), as is done by previous
14 There is an extensive literature on structural estimation of Markov decision processes (See
Eckstein and Wolpin(1989) for a review and Daula and Moffitt (1995) in the labor literature). In contrast
to most of this literature, we do not recover the structural (preference) parameters from the estimated
coefficients aqlax since our macroeconomic policy experiments can be conducted without it.
Estimates of preference parameters, however, would be essential for a quantitative welfare analysis.
15
cross-sectional studies estimating quitting equations,"5 we mix together (transitory) micro- and
(permanent) macro-shocks since both types of shocks cause the individual wage to change.. In this
paper, however, we are mainly interested in the quitting response of individual workers to permanent
macro shocks. These difficulties with using the currently observed individual wage are also the
reason why we do not attempt to estimate the quit-rate function of an individual firm,
q, = q(wi ,w ,p, b). Our approach to this problem is to add to the right-hand-side of (12) additional
individual-specific human capital variables:'6
q= = DaO +aogw+a + a loab, + Y>a.H{i +(13)
The particular specification (13) with 0t satisfying standard error-term assumptions could,
for example, arise as the linearized solution of the worker's dynamic programming problem under
the following conditions. Suppose we decompose the individual wage and the individual self-
employment benefits as follows
logwi = logw, + logij-
(14)
logb/ = logb, + logn,,
where the macroeconomic variables, logw1, log b,, follow a Markov process, or more specifically
a random walk. Moreover, assume that the evolution of the idiosyncratic wage and benefit
component is given by
logrl'> = YS H + 8w
(15)
log ,bt = Ek Ybkkt + (1bt)
5See, for example, Pencavel (1972), Krueger and Summers (1988), and Campbell (1993).
"6Though in the empirical analysis we do not growth adjust either w or b, in the log-formulation
such an adjustment would be entirely captured in the intercept term, a0.
16
wherethe ,,* are cons,nthes humancapitalvariables Hk, followamovprocess, foreach t = 0,1,...
the random variables { e',4 and { erbt }j., are identically and independently distributed, and for
each j c J the processes { e,4. and { elb } are serially uncorrelated. In principle, these
assumptions are not too restrictive as long as the observed variables, HkJ exhaust the list of relevant
idiosyncratic variables. In practice, however, there are always unobserved variables relevant for the
worker's quitting decision whose existence causes a violation of our error-term assumption. We
employ a random effects probit routine with estimates of Huber-While robust standard errors (see
below) to ameliorate this problem.
The technology for estimating mover-stayer models in a rotating panel context is not
presently developed."7 As Maddala(1987) notes, simply pooling the data and then using standard
probit techniques would yield consistent but inefficient estimates because the correlation across
observations is not being exploited. As an internediate approach, we maintain the integrity of each
of the 24 panels, but "pool" them to form one large four-period panel (the first quarter is used to
establish the worker in the formal sector) of 100,978 observations. Each observation includes the
human capital variables and move-stay index particular to the individual, as well as the two macro-
earnings variables and the proxy for the probability of finding a job corresponding to the potential
move's location in the span of 28 periods. We then estimate the transition equation using STATA's
panel probit routine.18 This permits estimating Huber-While robust standard errors as a measure to
ameliorate violations of the assumptions on a' and made above.
The results presented in the first two columns of table 2 strongly support the specification
17A substantial literature exists on limited dependent models in a panel context (see Maddala
1983 and Baltagi 1996 for overviews) and continuous dependent variables in an incomplete or rotating
panel context (see Nijman, Verbeek and van Soest 1991 for a recent overview), but there is no literature
analyzing limited dependent variables in a rotating panel context.
18 As Maddala(1987) notes, random effects probit estimators are consistent while the fixed effect
estimators are not, in additional to being computationally difficult. We find it unlikely that there is any
correlation between the random individual effects and the macro explanatory variables that would require
use of a fixed effects estimator.
17
suggested by our theory. The two earnings elasticities enter of predicted sign and significantly at
the .1% level. A rise in self-employed earnings relative to formal-sector earnings leads to more
workers frying their hand at self-employment. Our proxy,fl5, for the probability of finding a formal
sector job also enters as expected and very significantly offering direct support to the efficiency
wage dynamic postulated here. The higher the probability of finding another job in the formal
sector, the more likely a worker will risk starting his own business. A more complete model with
squares and cross terms of the macro variables was also estimated but none of these additions were
significant and the results are not reported. Also consistent with the model, a x2 test cannot reject
the symmetry of the two income related elasticities at the 5% level. The next two columns present
the results when this constraint is imposed. As expected, they are very similar and are used in the
simulations.
The human capital variables enter significantly in both regressions. Since in the present
specification they explain the idiosyncratic component of the wage-benefit differential, the signs
of the combined effects (evaluated at the mean) are plausible. Increased schooling may plausibly
lead to a larger differential, and hence a lower propensity to move, because of a greater demand for
skills in the formal salaried sector. Conditioning on schooling, more experience may raise the
probability of success in the self-employed sector, lower the differential and hence raise the
probability of moving.
The regressions were also run with an alternative dependent variable that dropped
observations after the first quit and yielded similar results. We also compared the results to the
theoretically consistent standard probit techniques on the pooled sample and found them very
similar (available on request).
Our estimation results confirm the most basic implications of the "quitting theory". Even
though this is not the place for a detailed analysis of the full range of alternative views of job-
separation, we point out that the estimation results are inconsistent with the simplest "firing theory",
18
that is, the theory that termination of existing worker-firm matches is mainly a decision made by
firms. To see this, suppose that the quit-rate is a constant independent of any economic factors.
Suppose further that the firm's decision problem is identical to the one discussed in our efficiency
wage model with the only exception that the wage is determined in a competitive labor market, the
labor supply function being derived from a standard labor/leisure choice of workers. Consider now
the response of the economy, which is initially in steady state, to a negative productivity (demand)
shock. The dynamic response is likely to be an increase in firing and a reduction in hiring by firms
since the new optimal employment level is lower. In addition, reduced demand for labor can be
expected to decrease the wage. Hence, firing (the left-hand-side variable) is negatively correlated
with the wage and the probability of finding a job (hiring) and this theory therefore predicts a < O
and a < 0. If formal-sector and informal-sector productivity shocks are uncorrelated or positively
p
correlated, which is a reasonable assumption, we have in addition the prediction ab, 0. Thus, two
of the three inequalities are rejected by the data. We should also mention that if we consider an
exogenous change in the wage (change in minimum wage) for constant productivity, similar
reasoning yields a > 0 . a < 0 . al = 0, which is even more strongly rejected by the data.
4. Macroeconomic Policy Analysis
Section 2 developed a macroeconomic model with labor turnover. The previous section estimated
an aggregate quit-rate function q (w,p;,r, b) using microeconomic data. In this section, we combine
the theoretical analysis with the empirical estimates in order to assess the quantitative effects of
different macroeconomic shocks on the wage rate, the turnover rate, and formal-sector employment.
This is done by implicitly differentiating Equation (8) with respect to the parameters under
consideration and thereby calculating the effect on the wage rate and the probability of finding ajob.
The effect on the labor turnover rate and formal-sector employment is then calculated using
dh = dq = -dw + q dp + aq d
aw ap a'r
(16)
dl = 1 + h I h d - d
v(l-q*) p p*v(1-q* P
19
for the case of a tax reduction programn. Analogous expressions are used for changes in other
parameters.
a) Calibration
In order to make quantitative statements, we have to assign values to several parameters. We first
discuss a baseline version of the model and then proceed to explore the sensitivity of our results to
variations in the parameter values. To be consistent with the empirical work, we assume that the
entire labor force consists of formal-sector workers and self-employed individuals.
An essential part of the model is the adjustment cost function T(.). The comparative statics
result depend on T' and T". The value of T' is determined by theory through the second equation
in (10). Since we have no data allowing us to estimate T", we assume T" = O, that is, we assume
that convexities in adjustmnent costs are weak (Section 2e provides one scenario in which this is the
case). This assumption has the further advantage that the initial level of the equilibrium real interest
rate is irrelevant for the quantitative impact of macroeconomic shocks.
Another building block of the model is the economy-wide quit-rate function q(.)whose
properties determine the (local) effectiveness of economic policy. We take as values for the first-
order derivatives the point estimates of Section 3 and assume, in accordance with the empirical
results, that all second-order derivatives are zero.9
The function p (.)entering the second equation of (8) is calculated using the expression (3)
withtevalues,h = = 0.02448,p* = 0.2148,andW = O.9.ThequarterlyseparationrateofO.02448
is directly taken from the data and is the averaged, quarterly fraction of formal-sector employees
19Instead of rewriting equation (8) in terms of log-wages, we simply normalize the equilibrium
wage before the policy change to one. Thus, we have aqlaw = -aqlaT. The estimates of aqlax are
constructed by multiplying the estimated coefficients an by p (x). In the case of aq/ap, the estimate
of aq/apf is further multiplied by .2148/6.313 yielding an estimate for aqlap.
20
leaving for self-employment. The probability of finding ajob, p x = 0.2148, is calculated using the
formula p* = h * with a hiring (quitting) rate h* = q = 0.02448, a formal-sector
employment level I * = 0.637, and a fraction of self-employed workers searching for a formal-
sector job,v (1 - q *), taken to be 0.2 (see the discussion in Section 3 for the value 0.2). The implied
value of v is 0.21. Assigning a value to 13 is complicated by the assumption that workers do not
participate in capital markets, which implies that we cannot simply follow the real business cycle
literature and infer the discount factor from the long-run real interest rate (rate of return to capital).
Of course, if workers' discount factor were equal to the discount factor of capitalists, then this
procedure would still be valid. But one reason why workers have only a small amount of wealth
might be exactly their impatience relative to capitalists. We decide to use a discount factor
13 = 0.9 for the baseline model, which is considerably lower than the 0.96 used in the real business
cycle literature.20 Table 2 summarizes the choice of parameter values for the baseline model.
b) Simulation
Consider first a tax reduction program which lowers the tax on formal-sector labor income, r, by
1% but leaves the (after-tax) earnings of self-employed workers, b, unchanged. For simplicity, we
also assume that any change in government revenues is met by a corresponding change in
government spending so that the fiscal budget is balanced before and after the tax reform. As shown
in table 4, the quantitative effects of such a tax reform on all three variables of main interest is
substantial: the before-tax wage increases by 0.68%, the labor-turnover rate decreases by 2.63% (of
its initial value of .02448), and the formal-sector employment level rises by 0.91% (of its original
level of 0.637).
Figure 1 shows the effect of the tax reduction program as shifts in the labor demand and
efficiency wage curve. The labor demand curve shifts to the right: for given before-tax wage, w,
200f course, the value of 0.96 is derived by considering US data.
21
a tax reduction increases the after-tax wage reducing quitting which in turn increases the demand for
labor and therefore p. The efficiency wage curve shifts to the left. Taken together, we conclude that
the wage unambiguously rises, but the effect onp is ambiguous. The decrease in p shows that the
change in the efficiency wage curve dominates. Even thoughp decreases, formal-sector employment
increases because of the pronounced reduction in equilibrium hiring.
We can also evaluate the welfare consequences of the tax reform. Clearly, workers gain in
the ex-ante and in the ex-post sense since the labor income (and therefore consumption) of iormal-
sector workers increases and the earnings of informal-sector workers has not changed. The
consumption of capitalists might increase or decrease depending on the shape of the production and
training cost function as can be inferred from the expression for equilibrium consumption of
capitalists (7). Finally, the effect on government revenue, (twl)N, is in general ambiguous since
there is a direct effect (the decrease in x) which tends to reduce revenues but two indirect effects
which tend to increase revenues (the increase in w and 1). For the Mexican case considered here the
average tax rate is a very small 8% which implies that the direct effect dominates and the tax
revenues are reduced. As mentioned above, in our simulation we assumed that this downfall in
receipts is matched by an equal reduction in outlays. Finally, it is worth mentioning that output of
the formal-sector economy is always increased simply because more workers are employed in the
formal sector. If the productivity of the formal sector is higher than the productivity of self-
employed workers, then the reallocation of workers across sectors unambiguously raises total output.
Hence, in principle both workers and capitalists can always be made better off if the government
redistributes (in a non-distortionary fashion) some of the gains of formal-sector employees to
capitalists.
Consider now a decrease in the earnings of self-employed workers, b, by 1% of its original
value.2" One possible reason for such a decrease in b is improved tax collection in the informal
21For simplicity and for lack of precise data, we assume that initially w = b = 1 .
22
sector. Alternatively, one can think a drop in the demand for goods produced in the informal-sector
(non-tradable goods). Clearly, if .a! = -! (recall b = 1), a hypothesis suggested by theory and
atr ab
supported by our empirical evidence, then a 1% decrease in b is equivalent to a 1% increase in r
except that in the former case the after-tax income of formal-sector employees increases by 1.68%
and in the latter case by 0.68%. Notice, however, that the ratio w(l -')/b, which measures the
pecuniary reward of formal-sector work relative to informal-sector work (earnings gap), rises in
both cases by the same amount, namely 1.68%.
Let us now turn to the discussion of events shifting the labor demand curve only. First,
suppose that the growth-adjusted marginal (revenue) product of formal-sector work, Z, increases by
1 %.22 Such an increase in Z could arise, for example, if along the development path the ratio of
formal-sector productivity to informal-sector productivity increases because secular technological
progress disproportionally favors the formal sector. As a second example, one can think of product
and/or labor market deregulation as well as a reduction in labor taxes paid by firms. It also follows
from equation (8) that a reduction in the average training cost (for constant T' and T" ) is equivalent
to an increase in Z by the same amount.23 The qualitative impact of any of these events is illustrated
in Figure 2. For given wage w, a higher marginal product of labor increases the demand for labor
which in turn increases hiring and therefore the probability of finding ajob -- the labor demand curve
shifts to the right. Since the efficiency wage curve is unchanged but downward sloping, the wage
rises but the probability of finding a job decreases in the new equilibrium.
The second line in table 4 summarizes the quantitative effects of a 1% increase in Z which
are very similar to the previous case: the before-tax wage increases by 1.56%, the turnover rate
22For simplicity, we set Z = w = 1, that is, we assume that training cost are "small" compared to
wage cost.
23Observe, however, that an increase in Z by 1% of its initial value is very different from a
decrease in training cost by 1% of total training cost as long as T(average training cost) is considerably
smaller than Z
23
declines by 2.19%, and the employment level rises by 0.93%.
Finally, we turn to changes in the real interest rate, r. It follows immediately from Eq. (8) that
a 1% (100 basis points) drop in r is equivalent to an increase in Z by T'%. Since
T' = (p k -y = 10.53 in the baseline model, the impact on all endogenous variables is very large.
The linear approximation used here, however, is certainly not appropriate for changes of such a
magnitude and the implied changes of the endogenous variables therefore appear unrealistically
large. On the other hand, our analysis indicates that the real interest rate is one ofthe most important
variables, a result consistent with the finding by Phelps and Zoega (1998) using unemployment
cross-country regressions. Notice also that even though fiscal policy has no effect on ithe real
interest rate in the basic model developed in this paper, leaving changes in the discount factor of
capitalists and/or changes in productivity growth as the only sources of interest rate charnges, an
OLG version of the model along the lines developed by Blanchard (1985) and used by Hoon and
Phelps (1992) would deliver a link between fiscal policy and the real interest rate. Moreover, the
equation system (8) can also be viewed as representing the equilibrium conditions of a small open
economy version of the model with exogenous world interest rate r.
c) Sensitivity analysis
This section discusses how parameter uncertainty affects the simulation results. Our approach is to
change one parameter value at a time and to report the effects of an increase in Z for the new
parameter constellation (results for changes in Tare similar). First we decrease the discount factor
of workers, %, from its original value of 0.90 to 0.80. In another experiment, we change our
estimate of the fraction of self-employed workers searching for a formal-sector job, v (1 -q "), from
its original value of 0.20 to 0.30. Finally, we also consider decreases in the absolute value of the
quit-rate elasticities by one standard error. Table 5 summarizes the results and reveals that the
parameter variations do not change our main conclusions. Indeed, the response of the economy is
surprisingly robust to changes in parameter values.
24
5. Conclusion
In this paper we incorporated microeconomic evidence about Mexican workers moving into self-
employment into a macroeconomic efficiency model with labor turnover and used the calibrated
model economy to evaluate the quantitative effects of changes in economic policy and other
macroeconomic shocks. As we have already mentioned in the Introduction, an application of this
method to different countries and unemployment is high on the agenda for future research. In this
respect, an interesting question is whether the calibrated model economy generates a wage curve
(Blanchflower and Oswald 1994), that is, a negative relationship between wages and unemployment.
A glance at table 3 confirms that at least for the case considered here, a wage curve does indeed
arise, that is, regardless of the type of macroeconomic shock there is always a negative association
between the formal-sector wage and self-employment.
There are several extensions of the model analyzed in this paper which seem promising.
First, allowing for a more realistic matching technology (see Section 2e) establishes an interesting
link between search intensity and efficiency wages. Second, a comparison of the market outcome
with the (constrained) efficient allocation could offer valuable theoretical insights. Finally, a
stochastic version of the model with aggregate uncertainty would permit the study of business
cycles. We plan to study these extensions in future work.
25
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28
Appendix.
Al. Worker's decision problem
a) Representative firm economy
There is a continuum of ex-ante identical, infinitely-lived workers whose total probability mass is normalized
to N (the number of firms). Each worker has preferences over random consumption sequences, { C
that are time- and state-additive with logarithmic one-period utility function24
U UC , ) = 10r (C (A1)
1. .v& . T-. t=
where P. stands for the worker's pure discount factor. Workers are either employed in the formal sector
(employed) or work in the informal sector (self-employed). Workers do not participate in capital markets
and their consumption level is therefore equal to their current disposable income. Hence, if we denote the
formal sector wage by W,, the tax rate on labor income from formal-sector work by T, and the net pecuniary
benefit to self-employment by B, ,, we have
I WI (1 --l) if employed in formal sector in period t
Bt , t if self-employed in period t (A2)
Here B is the average benefit from self-employment and 0, a worker-specific (random) component. For
given wage rate and benefits, a worker chooses a quit/search rule which maximizes (Al) subject to the budget
constraint (A2).
We are interested in an equilibrium growth path along which consumption,C,, and wage, WF, grow
at a constant rate equal to the growth rate of technological progress,g. Thus, we introduce the following
growth-adjusted variables: c C IA w = W/A, with A, = Ao (1+g)' being the exogenous
parameter of labor efficiency. We also assume that average self-employment benefits, B,, grow at the rate
g and define: b = Bt /At. We have dropped the time-subscript on growth-adjusted wage and benefit
payments to indicate that these variables are expected to be constant, an expectations turning out to be correct
24The log-utility assumption is not essential (CRRA-utility would suffice), but provides a direct
link to the empirical section in which log-wages are used in the regressions.
29
in equilibrium. To ensure that the worker's optimization problem is well-defined, we assume
iIn (1 +) < 1 In this case, maximization of(A1) subject to (A2) is equivalent to solving:
MaKxlm E [ Iftlo_j A3)
T- - t=O |(3
w (I -r) if employed in formal sector in period t
subject to: c - i
There are two sources of idiosyncratic uncertainty: uncertainty about the idiosyncratic component
of self-employment benefits and uncertainty about the individual employment status. We assume that the
sequence of shocks , { (O })7r with Ot = log , is a sequence of identically and independently distributed
random variables with distribution function ( .)and density function (.) .2 Further, the probability of
finding a formal sector job when searching is a constant, p. The decision problem (A3) therefore displays
a recursive structure and the corresponding Bellman equation reads
Ve(O,x) = max{{log(w(O-t)) + r3fVe(OI.x)db(0')I (
{ log b + 0 + f vs(O'. x, d1(0') } (A4
V'(0,x) = max { logb + 0 + [ f Vs(O'.x) dD('0') }
{p(log(w(l -T)) + pLfVe(O'.x)df(O')) +(l -i)(logb +o+8f V5'x)&d(01)}}.
where x - (wp, r,b). Further, V e(O,x), respectively VS(0,x), denotes the (utility) value of pursuing
the optimal policy when employed, respectively self-employed, when the macroeconomic state is x.
25Although most of the properties of the quit-rate function also hold for general Markov
processes, the particular expression for p (see A9) would of course change.
30
It follows from (A4) that the optimal strategy is to define a cut-off value, Oc = O0(x), and to quit
a formal-sectorjob if 0 2 0 and to search for a formal-sectorjob if 0 0. Moreover, the optimal cut-off
value is implicitly defined by the following equation
oc - log(w(1-,r)/b) - f3 EAV(0 .x) = 0
1- [()o((1t,) " (A5)
EAV(0,CX) - 1- V( (D¢)log(w(1-T)/b) - f 0'dOf)
The economy-wide quit-rate function is then simply given by q(x) - 1 - (D(0 (x)), which establishes
the link between the solution to the worker's decision problem and the quit-rate function entering into the
firm's decision problem. Using a probit model in Section 3 amounts to assuming a normally distributedO
and a linear approximation for the function 0 (x). Finally, the fraction of self-employed workers searching
(applying) for a formal sector job is 4D (0 (x)).
ae
Since a = - p(0 ) a for x = w,p,,r, b, the properties aboutthe partial derivatives of the quit-
ax C axn
rate function stated in Eq. (2) follow from implicitly differentiating (A5) and signing the result. In order to
calculate the function ,u(.), we use the following expression:
1 aEAV
aw aEAV
c (A6)
+ P,(l-p)P(D(O
1 l-pw(lw-P)td)
w 1 + P(O) gl(0,x)
where g, (0 ,x) is a function independent of the probability density p (0)
Suppose now that there is a (growth-adjusted) fixed cost of searching, f = log F. In this case, the
optimal policy is to define to cut-off values, 0 and 0 and to quit a formal-sector job if 0 2 0c2 and to
start searching if 0 < O . Moreover, we have 0 c2 = 0c2 (x) implicitly defined by an equation analogous
31
to (A5) and O.,= OC (X) = Oc2(x) - flp. For the quantitative analysis both the derivatives ofthe quit-rate
function, 2a - - ,and the derivatives of the search-rate function, s cl
Ox,, ""c2~ Ox" 'axn aq C ax as
important role. The empirical analysis provides us with detailed information aboutq, but good d on s
important ~~~~~~~ ~ ~~~~~~~~~~~abot- daaxon
are lacking. However, since we have 0 c (x) = 0 2(X) - flp, the following expressions relate the search-
rate function to the (observed) quit-rate function: aS - _ ¢(02 afqP) a and
as (P(0-fiP) aq O ax" (0c2) OaX,
ap 9(0,2) L ap p .
In principle, the above relationship in conjunction with information on search cost, c, and the
distributional characteristics of the idiosyncratic shocks, 0, could be used to calculate the response of the
search intensity to aggregate shocks. However, the expression also makes clear that the quantitative answer
crucially depends on distributional assumptions determining the ratio of probability densities evaluated at the
cut-off value, . In this paper, we choose not to make assumptions about the distributional
characteristics of 0 .26 Instead, we impose simplifying assumptions rendering all results independent of the
probability density evaluated at the respective cut-off value. More specifically, we deal with the problem of
unobserved search intensity by assuming that search costs are small, that is, f - 0. In this case, the
derivatives of the search intensity function are simply given by the negative ofthe derivatives of the quit-rate
fimction:
Os -aq
ax ax
The assumption of small search cost seems to create a problem by itself since in this case the
quitting/search model developed so far implies that s = 1 - q, which is not the case for the data set we
consider. However, even if there are no search costs, the above relationship need not hold if the idiosyncratic
shock process exhibits serial correlation. In the case of serial correlation, there is still one common cut-off
value for employed and self-employed, but the (stationary) distributions are different because of self-
selection, that is, the quit rate is 1 - O ( 0)but the search intensity is ( (0 ) and therefore s 1 - q. Since
we only have a very limited knowledge of the serial correlation properties of this idiosyncratic process, we
only deal with the extreme case in which there are two types of workers. The behavior of type I workers is
correctly described by the Bellman equation (A4) (no serial correlation). Type two workers, on the other
26Especially when 0 is interpreted as "taste-shock", such a procedure seems very questionable.
32
hand, never move (no stochastic shock) and therefore stay either in the formal or the informal sector forever
(for the time span covered by our data). If a fraction v of the total labor force is correctly characterized by
the Bellman equation (A4) (type I), then the search rate is given by s(x) = v(1 -q(x)), which is the
expression used in the text.2' Notice that v is a number independent of x.
b) Many Firms
We consider now an economy in which i = 1, ... ,N identical firms set wages in an uncoordinated fashion
(they play a Nash game). Except for this modification, the economy is identical to the one considered in part
a). For simplicity, we again consider the case of no search cost. The Bellman equation of a worker who is
currently employed by firm i paying wage w. reads
rVe(O(W x) = max{{log(w!(-0)) + 0 [fl`(O.w.,x) dO(I')}j (A7)
{ log b + 0 + Vs(O', x) dq(Ot)}}
where V'(.) is the value function of the Bellman equation (A4) (assuming all other firms are paying wage
w). The optimal strategy is again to define a cut-off value, Oci = 0 (w.,x), and to quit a formal-sectorjob
if 0 2 O and to search for a formal-sectorjob if 0 < O... Moreover, the optimal cut-off value is implicitly
defined by the following equation
H - log(wP(1-t)/b) - f EA V(w;,0.,x) = 0
EAV(wi,0c,x) log(w(-T))-pog(w(l-))-(-p)logb-pP EAVI
1 -plog(wlog ) -(PE) AV4(Ag
1 - f(D10 d'(0
The quit-rate function of firm i isthengivenby q(w.,x) -I -4 (0C(wP,x)).
a (p(O~~ a- a _ _
Since - 0 = ) -c forx = w,p,T,band- = -(P0(c ,the PrP
ax n te roerie
27This formula changes once we allow for the possibility that the fraction of formal-sector-only
workers is different from the fraction of informal-sector-only workers.
33
about the partial derivatives of the quit-rate function stated in Eq. (1) follow from implicitly differentiating
(AS) and signing the result. In order to calculate the function 1i (.), we need an explicit expression of the
following derivative:
1 aEAI2
w ow. *wlW-
awl aEA V
Ci (A9)
1 +
1 1 - f3d(O)
w 1 +2es)(c
where g2 (.) is a function independent of the density (p (0c).
We are interested in the ratio
aq AOC
aw. iz" OCW.-(A)
dwz wi~ ow, I- (A10)
aq ae
aw aw
Expression (Al 0) depends on the probability density evaluated at the cut-off value, q(p0), which is difficult
to pin down without a very detailed knowledge of the distributional characteristics of the random variable
0. Evoking again the principle of "independence of results from irrelevant distributional details", we ask for
an approximation yielding an expression for ,uwhich is independent of (p (0). One such approximation is
to take the limit p(Oc) - 0, that is, to evaluate ,u at small values of (p(O). In this case, (Al 0) becomes
Equation (5) used in the text (observing that O(0) = 1 - q).
A2. Capitalist' Decision Problem
There are i = 1,... ,N identical, infinitely-lived capitalist with time-additive preferences over conisumption
sequences, {C1.}g=O:
34
cl-P -I
U({C. o) = P I l (All)
In (All) p stands for the pure discount factor of capitalists and p for their coefficient of relative risk
aversion.
Each capitalist owns one firm. Each firm i (owned by capitalist i ) combines capital Kit and labor L,
to produce output Y . Changes in the employment level create adjustment cost. Hiring new workers is
associated with recruitment and training cost. In the case of training cost, we assume that training takes one
period and that the human capital created through training is destroyed once the worker quits (match specific
human capital), that is, newly hired workers always have to be trained regardless of their previous work
experience. Following Phelps and Hoon (I1992), whose specification of employment adjustment cost derives
from Hayashi's treatment of capital adjustment cost (Hayashi 1982), we assume that the total cost of training
and hiring is independent of the capital stock and that net output produced is linear homogenous in
(K.,, A L., A H.,), where A, stands for the (common) efficiency of labor and Ht for the number of
U9 t it, I I t ~~~~~~~~~~~~~~~~~~~~~~~~~~~~It
workers hired by firm i. More precisely, we assume that output net of adjustment cost is equal to
F(K1 , A, Lit) - T(A H.1, A Lit), where F(.,.) is a standard neoclassical production function and
T(.,.)is a linear homogenous function. Moreover, the adjustment cost function satisfies: a?ax,y) > o,
ax
a?x,y) < O, a i(,y) 2 0. Notice that we permit the case of linear adjustment costs, ay) = O, for
ay ~~ax2 ax2
which adjustment to the optimal employment level is instantaneous.
Each firm employs a large number of workers. Firm i chooses sequences of consumption (of
owner i ), capital, investment, employment, and hiring as well as a wage rate28 which maximize (Al1) subject
to the constraints29
28We only allow the firm to choose one wage rate, not an entire sequence of wage rates. Deriving
the quit-rate function from the worker's Bellman equation when the firn can choose arbitrary wage
sequences { Wf }t O is a nightmare.
29We assume that each self-employed worker produces what he consumes, namely b. If b is
interpreted as unemployment benefits, then it should be added to the right-hand-side of the last last
constraint .
35
K. ttl=( 1 -8) Kit + 1.
= (1 -q(w,,x)) Lit + Ht
Y, = F(KO,A1L.1) (A12)
Yt =C. + 1 + W A L + T(AL ,A H.)
I1 It It i t it l it, ItI
Kio , L.0 given -
Introducethefollowinggrowth-adjusted: k.t = K IAO, i. = I.A, c.1 = C .IA,,y iA
(recall that the size of the labor force is normalized to one). Introduce further the hiring rateh = H /IL
and let lit = Lit . Clearly, maximizing (All) subject to (A12) is equivalent to solving the following
optimization problem:
max E fit tt
C
subject to : k11t+ = -+g [(1-a)k,1 + ii,] (A13)
II I+ =[1 + h - 4(W.,X) J lit
F(k.t1l.) = I+ i + WI i + T(h.) i
kio 0 Ii given,
with T(h.,) - T(h,t, 1) and the growth-adjusted discount factor PC= (1 +g)'7. To render the
capitalist' maximization problem well-defined, we require f3 < 1. Writing down the Euler equations
associated with the optimization problem (A13) leads to Equation (4).3°
30The sufficiency conditions require certain assumptions about the second-order derivatives of
the quit-rate function. We do not attempt to derive these properties from the Bellman equation (A4) and
(A7).
36
Table 1.
Summary Statistics l
Var Nobs Mean Std. Dev. Min Max
move 100978 .02448 .1545 0 1
hire 100978 .6313 .1866 .34 1.13
mwl 100978 1.674 .05319 1.51 1.76
mw3 100978 1.856 .1060 1.54 1.94
exp 100978 21.61 13.01 0 62
exp2 100978 636.3 684.9 0 3844
sch 100978 7.302 2.808 0 11
| sch2 100978 61.2 0 37.02 0 121
Table 2.
- _____________Probit Regression
Unconstrained Symmetry Imposed Summary Statistics
Coeff. Std. Err. Coeff. Std. Err. Mean Std. Err.
move .0245 .1545
log w -1.3153 .3895 1.674 .05319
log b .8879 .1965 1.857 .1060
log w- log b -.6436 .1522 -.183 .06625
P .3122 .05647 .3650 .04801 .6310 .1866
school -.08249 0.01231 -.08253 0.01231 7.302 2.808
(school)2 .004241 0.000968 .004232 0.000978 61.204 37.02
experience .03200 .002887 .03200 .002886 21.612 13.00
(experience)2 -.000535 .0000556 -.000536 .0000556 636.327 683.9
constant -1.6167 .3860 -2.3195 .06131
Nobs 100978 100978
Significance X2(6)=440.8 p=O.OO X2(5)=440.7 p=0.00
Test of a.=-ab x _2(1)=3.41 p=.065
37
Table 3.
Parameter Values for Baseline Model
Quit-rate elasticity aqiaw = - aqlab - aqla| -.03737
Quit-rate elasticity aqlap .06231
Formal-sector employment I* .637
Average turnover rate q = h .02448
Fraction of self-employed searching for a job V (1 -q_) .20
Workers' discount factor P.r .90
Implied average probability of finding ajob p .2148
Implied parameter pi 2.54
Implied parameter v 0.21
Table 4.
Policy Effects: Baseline Model 1
Tax reduction Ar = - .01 MPL increase AZ = .01
Wage change Aw .0068 [0.68%] .0156 [1.56%]
Probability change Ap -.000281 [-0.13%] .000776 [.36%]
Turnover rate change Aq -.000643 [-2.63%] -.000535 [- 2.19%]
Employment change Al .00577 [0.91%] .00589 [.93%]
Table 5.
|________________ Sensitivity Analysis: Effects of AZ = .01 =
____________ f3=.8 v(1 -q *) = .3 aqlaw = -.0285 aqlap = .0541
Aw .0222 .0183 .0156 .0157
Ap .002407 .000579 .000593 .0007389
Aq -.000803 -.000628 -.000409 -.000544
Al .00802 .00687 .00449 .00599
38
Figure 1. A reduction in labor-income tax.
NAEW' EW
wIN
1W'\N
I I ~~~~D )
p
Figure 2. An increase in the marginal revenue product of labor
j ~~~EW
X~~~~ N
D
p
39
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