A General Equilibrium Model of Variable Scheduling∗
Preliminary Version
N ICK F RAZIER
Rice University
[email protected]
January, 2017
A BSTRACT: A salient feature of the U.S. labor market is the large fraction of jobs with week-to-week variability
in hours and compensation. A recent movement to limit employer control over scheduling aims to reduce
that variation and improve employee welfare. Understanding the full economic consequences of such
policies, however, requires a model of both why such jobs exist and why individuals accept them. I present a
search model with productivity shocks and worker and firm heterogeneity that generates labor contracts
that match observed patterns in variable hours and compensation. The model includes the firm’s hiring
decisions, choice over volatility in hours, and labor contracting to capture equilibrium effects of restricting
the contracting environment. I present my identification strategy and discuss the difficulty of separately
identifying the primitives of the firm and worker problems using typical data sources. I then estimate the
model using a mixture of stated preference, panel, and aggregate firm data and evaluate the consequences of
several relevant policies on employee welfare.
∗
I am grateful to Flávio Cunha and Ken Wolpin for helpful comments and suggestions.
1
Nick Frazier • 2
I.
I NTRODUCTION
Technological advances in on-demand scheduling offer firms increasingly finer control over employee
hours but have also induced a backlash against the perceived externalities created by fluctuating hours and
unpredictable work schedules. The resulting pressure has caused firms to change their policies, see Nassauer
(2016), White (2015), Tabuchi (2015), but also motivated policies that directly regulate such contracts as in
San Francisco, Seattle, and Portland. These policy changes are supported by survey evidence describing
the prevalence of such jobs in the United States and the negative consequences of variability in earnings,
hours scheduling, and employee turnover.1, 2 However, without a model describing why these jobs exist
and why individuals accept them, any characterization of the net welfare consequences of these policies
remains partial. For example, prohibiting variability in week-to-week hours seemingly makes employed
individuals better off but could also change levels of employment, hours, and compensation. These net
economic consequences may ultimately represent a welfare reduction for workers.
One important consideration motivated at least in part in the above survey evidence is how to model
the source of volatility in hours. A standard neoclassical labor supply model has the employee choosing
hours free of constraints from the firm. Such models can generate both fixed and variable hours but struggle
to replicate the mixture observed in U.S. data. The further implication that observed labor hours are then
necessarily optimal labor hours implies the new policies decrease welfare and stand at odds with survey
evidence suggesting that workers would accept pay cuts to control their hours. The literature incorporating
search frictions provides a way to generate sub-optimal contracts but often treats hours as of secondary
importance in a job contract.
I propose a general equilibrium model where heterogeneity in worker preferences interacts with firms
responding to productivity shocks to generate a mixture of jobs resembling that found in the U.S. labor
market. Similar to Bloemen (2008) my model has firms offer a combination of hours and compensation to
potential employees through meetings subject to search frictions. However, offers here are a joint distribution
of hours and wages where the choice to work is based on ex ante expected utility but the pay-offs are based
on ex post outcomes. I further incorporate directed search so that workers who differ in degrees of risk
aversion can choose whether to apply to jobs where the joint distribution is degenerate (fixed hours) or
non-degenerate (variable hours). Productivity shocks motivate firms to offer variable schedules, where
1
The prevalence of such jobs is documented in Lambert et al. (2014) using the nationally representative National Longitudinal
Survey of Youth 1997 to estimate that around three-quarters of workers in their early thirties “experience at least some fluctuation in
weekly hours.”
2
Reviews of survey evidence are Lambert et al. (2014) and Golden (2015). Works that use survey responses include Golden et al.
(2013), Henly and Lambert (2014), Ingre et al. (2012), Lambert (2008), Lambert et al. (2012), Reynolds and Aletraris (2006), Reynolds
(2003), and Stewart and Swaffield (1997).
Nick Frazier • 3
hours adjustments are possible, but heterogeneity in worker preferences tempers their use. Evidence for
this dynamic includes Mas and Pallais (2016) who use a field experiment where participants chose between
contract types found a willingness-to-pay that ranges from 0% to around 20% of wages to avoid the type
of variable contracts considered here. In estimation, I use stated preference data to identify variation in
worker preferences that resembles that used in Eriksson and Kristensen (2014) who also find a large degree
in variation in employee willingness-to-pay.
In my model firms employ at most one worker but make decision ranging from the type of vacancy to
offer, the optimal labor contract, and production decisions in the fact of uncertainty. Workers have types
that reflect their degree of risk aversion and choose whether to apply to variable or fixed type jobs and then
whether to accept offered contracts. I estimate the model’s primitives of worker preferences, firm revenue
functions, and productivity shocks using data from a survey instrument designed to elicit preferences over
similar contracts, panel data on observed jobs, and aggregated information on firm hiring, separations, and
fluctuations in production. A detailed discussion of data requirements for identification is included.
The general equilibrium nature of the model allows us to study the economic consequences of restricting
the contracting space. A typical restriction would be limiting the amount of variability permissible. Such
a policy potentially affects the firm’s decision to offer jobs, how they set wages and hours, and overall
employment levels. Using estimated primitives I provide evidence on the resultant changes in the labor
market and provide some limited characterizations of optimal policies. These include ones similar to those
proposals described above.
The plan of the paper is as follows. In Section 2, I develop my model of variable scheduling. Section 3
discusses identification given data and details my approach. Section 4 describes my approach to estimating
degrees of worker heterogeneity. Section 5 contains the econometric framework and approach used to
estimate the general equilibrium model. In Section 6 I present the results of estimation and in Section 7 I use
these estimates to characterize the welfare costs of various policies. Section 8 concludes.
II.
M ODEL
Two types of agents populate the economy: firms and workers. Firms produce the economy’s consumption
good using the labor hours input of at most one worker. Firms act independently when making personnel
and production decisions and take aggregate equilibrium outcomes, such as the unemployment rate, as
given. They face an i.i.d. productivity shock each period that affects firm revenue such that firms prefer
to respond by adjusting labor hours. Firms are heterogeneous in the variance of this shock. Workers are
characterized by their type which determines the degree risk aversion in their preferences. The combination
Nick Frazier • 4
drives heterogeneity in the production and cross-sectional variation in observed jobs in the economy.
Firms enter a period as either matched with an worker or not. Whether they are matched and the type of
the employee if matched represent the principle state variable of the firm’s problem. Each period begins with
some economy-wide exogenous separations. Afterwards, any unmatched firm may post a vacancy to one of
two markets. The first market specifies the worker’s hours before realization of the productivity shock which
I refer to as a fixed contract. Firms may also post a vacancy for a variable contract which permits adjusting
hours to the shock. Equivalently, the variable contract specifies a menu of labor hours for all possible realized
productivity shocks. Workers face a symmetric choice to search for a specific type of vacancy. This feature
suggests the markets may differ in job-filling rates and expected types of workers and firms. Further, the
terms of the contract will depend on both the firm-specific distribution of the productivity shock and the
worker-specific degree of risk aversion.
Conceptually, both contracts can be thought of as joint distributions in hours and compensation. The
variable contract fixes a wage before production in return for working a set number of hours the random shock
determined by the shock. Thus, the contract specifies a joint distribution of labor hours and compensation,
equal to the fixed wage times hours. Fixed contracts offer degenerate distributions since both wage and
hours are fixed before realization of the productivity shock. Worker and firm types drive expected returns
and generate variation in labor contracts which in turn influences the composition of jobs in the economy.
Firms post vacancies subject to a non-convex cost structure discussed below. Upon meeting a worker, the
firm makes a take-it-or-leave-it offer to the worker but with the contract type as specified in the market where
they met. The type of contract may not change for the duration of the match and since neither the firm’s or
the worker’s individual types evolve the same contract will occur for every period of the match. Meetings
do not imply matches as a contract given the firm and worker types may not be expected profit maximizing
from the firm’s point of view. Failed meetings result in the firm and worker remaining unmatched for the
period.
Unemployed workers also choose whether to apply in the fixed or variable market. They meet firms with
posted vacancies through a search process and then choose to accept or reject an offered contract. Both parties
know the worker may lose their job in a subsequent period. Meetings are assumed to occur with probability
each period that is exogenous to the agents but determined in equilibrium. The worker accepts the job when
the expected value of working meets their participation constraint or the value of unemployment.
Mechanically, every period consists of three phases. The first phase is a labor force adjustment where
exogenous separations occur followed by the opportunity for unmatched firms and workers to participate in
the search market. Any pairing with an expected value less than the reservation value for either the firm or
worker is dissolved and both continue unmatched. The second phase is the contracting phase where labor
Nick Frazier • 5
contracts for all current matches are determined. The third and final phase is production where the random
productivity shock is realized and production occurs followed by consumption. Firms and workers discount
at the same rate between periods.
A.
Workers
Workers exist in one of two states: employed or unemployed. Employed workers supply labor and receive
compensation in accordance with their contract. Unemployed workers receive b, the guaranteed value of
home production. Workers are risk-averse and have preferences over consumption and total hours supplied
in any period. Workers vary by belonging to one of two types which captures their degree of risk-aversion.
Their type is fixed across time. Let there be πκ proportion of workers of type κ L and 1 − πκ of type κ H .
Since job contracts amount to a ex ante joint distribution of possible compensation and total labor hours, this
preference shock affects their decision to accept or reject a given job. Formally, preferences over hours and
consumption are described by the per-period utility function Z (·) where risk-aversion depends on κ
Z (C, L; κ ) =
C 1− ρ c (κ )
L 1− ρ l (κ )
−γ
.
1 − ρ c (κ )
1 − ρ l (κ )
(1)
By assumption ρ j (κ L ) < ρ j (κ H ) for j = 1, 2 so that low types are less risk averse than high types. Workers
also face a budget constraint such that total consumption in any period, C, is less than or equal to total labor
compensation, wL, or their value of home production if total hours worked, L, is zero. Equivalently,
C ≤ wL + 1[ L=0] b(κ ).
(2)
By assumption, the values of home production (b(κ L ), b(κ H )) satisfy
Z ( b (κ L ), κ L ) = Z ( b (κ H ), κ H ).
In other words, all workers receive the same utility from home production. In doing so, I abstract away
from trying pinning down heterogeneity in the utility value of home production, not the least of which is
taking a stance on whether risk-averse workers prefer home production. Notice that this assumption is
not intrinsically different than assuming a single value of home production for homogeneous workers and
affects the dynamics only in that individuals are ceteris paribus equally likely to reject a job offer.
Each period initially unemployed workers become employed with some positive probability. Let J and
U represent the value of entering the production phase as employed and unemployed, respectively. The
Nick Frazier • 6
period pay-off is a realization from the joint distribution of compensation and total labor so the ex-ante is in
expected utility and a continuation value. Specifically, the value of holding a job with a variable schedule is
J v (κ ) = E [ Z (C, L; κ )] + β E [(1 − q) J v (κ ) + qU (κ )]
(3)
where β is the discount rate and q is probability of job loss. The value of a job with a fixed contract, J f (κ ),
follows symmetrically except it has a non-random contracted pay-off such that E [ Z (C, L, κ )] = Z (C, L, κ ).
An unemployed worker may costlessly choose to search in either the fixed or variable contract market.
Both markets have equilibrium job finding rates that depend on the number of workers, U, and firms, V
participating in that market. Workers take these values as given. Let the job-finding rate for the fixed and
variable markets be φ f and φv , respectively. The firm a worker of type κ potentially meets comes from a
continuum of firm types, σi ∈ R+ , where for some sufficiently large value of σi either the worker or the firm
would prefer remaining unmatched. This behavior occurs in behavior as a worker will potentially participate
in a market if other firms can produce profitable matches. Since the worker’s type drives the cut-off I denote
the probability of this occurrence as τ (κ ).3 Consider the value of unemployment when entering the period
and participating in the fixed market,
U f (κ ) = φ f (U f , Vf )τ (κ ) E J f (κ ) + (1 − φ f (U f , Vf ) + φ f (U f , Vf )(1 − τ (κ )))( Z (b, κ ) + β E U f (κ )).
(4)
The first term is the expected return to working where integration is over the set of firm types that participate
in the fixed market. The second term is continuation value from a failed match but the discounted continuation value of unemployment and is weighted by the probability of a failed search and a failed meeting after
a successful search. Notably, the state variable does not change.
The absence of aggregate uncertainty implies a steady state equilibrium wherein if a worker applies
to jobs in the fixed market in one period they will do so in all future periods. This property is seen in the
worker’s choice between the fixed and variable markets is described by the value of unemployment
U (κ ) = max{E U f (κ ), E U v (κ )}
where since a worker’s type does not change, neither does there decision rule.
3
I derive a formal expression when discussing the matching technology below.
(5)
Nick Frazier • 7
B.
Firms
Firms use a production technology that converts labor hours into the consumption good. Firms are risk
neutral and consume any profits. The production technology of firm i in period t is described by
yit = ε it litα
(6)
where l is labor, ε is a productivity shock, and α is a curvature parameter to allow for diminishing returns to
scale due to abstraction away from fixed factors of production. I assume ε it ∼ LogN (1, σi2 ) such that firms
differ in the variance of their productivity shocks which I further denote as the firm’s type. Though the
productivity shock each period is an independent draw from the firm’s distribution, the firm’s type, σi , is
drawn from Fσ upon entry and fixed for the life of the firm. The firm’s type will largely determine whether it
prefers to offer a fixed or variable vacancy. Intuitively, firms with large values of σi will prefer to respond to
these shocks by their adjusting labor hours which is permitted by the terms of variable contracts but not by
fixed contracts.
Firms enter a period either unmatched or matched with a worker of particular type. The period begins
with the exogenous destruction of matches with probability δ. The probability of exogenous destruction is
independent of all firm and worker characteristics. Unmatched firms then post vacancies and hire through a
process described below. Importantly, the type of contract used in the match, whether fixed or variable, is
determined during hiring process and persists through the duration of the match. Each period and after
hiring, all matched firms contract with their workers through a take-it-or-leave-it offer with full information.
Therefore, both parties know the worker and firm type and all aggregate variables. This setting implies
workers will receive their expected reservation value in the case of variable contracts and true reservation
value for fixed contracts.
The decisions of firms are largely driven by the expected value of production. This process begins with
whether, given their type, they prefer to post a vacancy in the fixed or variable market. The decision depends
on the expected types and number of workers in that market and the expected productivity of any match.
To help build intuition I first discuss the production decision of the firm and work backwards to the hiring
decision.
B.1.
Production under Fixed Contracts
As a first case, I consider the production decision for a worker employed in a fixed contract. Both hours and
wage are set in negotiations occurring before the realization of the productivity shock. In particular, the firm
Nick Frazier • 8
seeks to maximize expected profit but because contracting occurs in a full information environment with
take-it-or-leave-it offers, the firm need only keep the worker indifferent between working and unemployment.
Formally, the wage must satisfy the following inequality,
E[ Z (wl, l; κ ) + βJ f (κ, σi )] ≥ Z (b, 0; κ ) + βU (κ )
h
i  1
1− ρ c
(1 − ρc )(b + β E U (κ ) − J f (κ, σi ) )

h
i
.
w≥
E l 1−ρc − 1−γρ l 1−ρl
l

⇒
(7)
h
i
Define H f (κ, σi ) = b + β E U (κ ) − J f (κ, σi ) which roughly represents the expected return to quitting. The
careful reader will notice that J f (κ, σi ) = J f (κ ) and H f (κ, σi ) = H f (κ )∀σi as the scale of productivity is
normalized across firms and the worker is perfectly insured against shocks.
The firm’s problem simplifies to maximizing profit subject to the worker’s participation constraint. Notice
that for a fixed contract hours worked are set independent of ε and so the constraint further simplifies to
following problem,
l f (κ ) = max E [ε i l − wl ]
α
l
s.t.
w≥
(1 − ρ c ) H f
l 1−ρc − 1−γρ l 1−ρl
l
!
1
1− ρ c
where by the convexity of the problem the constraint binds and the solution provides both a l f (κ ) such that
knowing the worker type perfectly determines fixed labor and compensation. The offer is independent on
time and the firm’s type. The wage offer, w f (κ ) is a consequence of substituting the labor demand into the
participation constraint.
One final consideration for firms operating with fixed labor contracts is the occurrence of a productivity
shock small to cause negative profit. Formally, any ε it < w f l 1f −α will generate negative profit for the firm. I
assume the firm cannot break the contract and factors in this possibility when both setting fixed contracts
and choosing between variable and fixed contracts. An analogous assumption is made about not needing
to satisfy the variable worker’s participation constraint ex-post. The reasoning behind these assumption is
discussed in Section III.
B.2.
Production under Variable Contracts
The production decision of a firm with a worker employed in a variable contract has a slightly different
structure as the wage is determined separately from hours. To illustrate, given a contracted wage w, the
Nick Frazier • 9
profit maximizing labor demand is the solution to
łv (κ, ε it ) = max ε it l α − wl
l
⇒ lv =
αε it
1
1− α
w
where intuitively conditional labor demand is increasing in the productivity shock and decreasing in the
wage. Notably, for ε > 0 the firm always produces a positive amount so that there would never be a
reason for the firm to terminate a match during production. Furthermore, even if the worker’s participation
constraint is not satisfied by the ex-post wage and labor demand combination, I make the assumption that
the worker cannot break the contract. This assumption is symmetric to that made in the section above.
Though the firm sets labor optimally given the realization of ε, the wage is set during the contracting
phase before its realization with regard to expected production. Since contracting occurs in a full information
environment with take-it-or-leave-it offers, the firm need only keep the worker indifferent between working
and unemployment as specified by (7). Here the wage is set to expected hours for which I substitute in the
labor demand conditional on κ and ε to yield


1
1− ρ c
 (1 − ρc )(b + β E [U (κ ) − J v (κ, σi )]) 

w≥
1− ρ
1− ρ c


γ
αε it 1−α
αε it 1−αl
E
− 1− ρ w
w
(8)
l
which has an implicit solution for w. Notably, the expected total labor depends on the firm’s type, σi , which
appears in the integration in the denominator. The worker’s type enters the determination through ρc , ρl , U,
and J v which are all functions of κ. Here the constraint also binds and the above equation determines
wv (κ, σi ) for any firm worker pair. The implication of a matched worker/firm pair in a variable contract is a
series of realizations that resemble the week-to-week variation in worker schedules observed in the data.
B.3.
Determining Employment
Firms enter each period as either matched, and with employee’s type or as unmatched. Before the adjustment
phase jobs are exogenously destroyed at rate δ. This feature replicates structural unemployment and affects
variable and fixed jobs equally by assumption. Next, all unmatched workers and firms participate in a
matching process where firms post a vacancy to either the fixed or variable market and workers apply to one
of the two markets. The cost to a firm of a vacancy is k and vacancies must be re-posted each period. Any
filled vacancy is characterized by a pairing {σi , κ } which is fixed for the duration of the match.
The existence of two search markets requires that they have separately determined aggregate conditions.
Nick Frazier • 10
I make the assumption that they are subject to the same technology of matching. Let Uv and U f be the
number of workers applying to the variable and fixed markets respectively. Let Vv and Vf be the number of
vacancies firms post to the variable and fixed markets, respectively.
The existence of search frictions imply meetings happen with some probability. Since both workers
and firms vary in their type, the meeting of any two pairings is also probabilistic. Let ψv and ψ f be the
probability of meeting a worker in the variable and fixed markets, respectively. Though I have only two
types of workers, firms are continuous in type, and so given a large value of σi some firms may meet a
worker of type κ high enough that they would prefer to continue unmatched. This probability depends on
σi and density of types that participate in the market which is constant over time in steady state. Let the
probability of such a successful meeting be η (σi ). Then the value of posting a variable vacancy to a firm is
h i
Qv (σi ) = −k + ψv (Uv , Vv )ηv (σi ) E V f + (1 − ψv (Uv , Vv ) + ψv (Uv , Vv )(1 − ηv (σi ))) E [ Qv ] .
(9)
The expectation is taken over both ε and the distribution of worker types participating in the market
conditional on the meeting being successful. The continuation value of a failed search is Qv as without an
evolution of the state variable and a steady state market the firm will always choose the same market. The
value of posting a fixed vacancy is analogous. When making decisions firms take labor market conditions as
given.
The firm’s decision to post a vacancy in one market over the other is driven by expected return. Formally,
the value of posting a vacancy is
Q(σi ) = max{E Qv (σi ), E Q f (σi )}
(10)
where because of the equal costs the problem reduces to the expected return as a function of aggregate
vacancy-filling rates and the expected return to production for the two types of contracts. The firm also takes
as given the behavior of workers who also choose which market to participate in given their own type.
The worker may costlessly apply to either market and chooses the highest expected return given their
type. Similar to the discussion in A, the value of applying to the variable market for an unmatched worker is
U v (κ ) = φv (Uv , Vv )τ (κ ) E J v (κ ) + (1 − φv (Uv , Vv ) + φv (Uv , Vv )(1 − τ (κ )))( Z (b, κ ) + β E U v (κ )).
(11)
where the expectation for J v is taken over both firm type conditional on an accepted match and future
possible shocks given σi . Again, a worker who applies to the variable market in one period would always
apply to the same market in all future periods.
Nick Frazier • 11
In the absence of aggregate shocks the economy rests in a steady state across periods. I therefore assume
a simple matching function that provides the same equilibrium contact rate in every period. The vacancy
filling rate for market k, ψk (Uk , Vk ), depends on θv =
Vk
Uk
which is a state of the aggregate labor market
variable that is determined in equilibrium but taken as given by all agents when making decisions. Giving
the set-up, unemployment in each market is constant in equilibrium and defined by the flows in
Uk0 = (1 − Uk )q(Uk , Vk ) + (1 − ψ(Uk , Vk ))Uk .
(12)
I also assume a matching function with a Cobb-Douglas technology with a constant returns to scale specification. Formally, matches in any period are equal to
M(Uk , Vk ) = µUkν Vk1−ν
(13)
where Uk is the level of unemployment, Vk is the number of vacancies, q is the separation rate equal to quits
plus fires, φk = M(Uk , Vk )/Uk is the job finding rate, and ψk = M(Uk , Vk )/Vk represents the vacancy filling
rate.
C.
Equilibrium
The ingredients for equilibrium in this economy are the optimization behavior of firms and workers and
worker-flow conditions. Firm behavior must specify an optimal labor contract for hours and wage for the
fixed market subject to the worker’s type-specific participation constraint. There must be an equilibrium
state-contingent hours schedule which solves the profit maximization problem for variable contracts and
an attendant wage pairing that satisfies the worker’s type-specific participation constraint for all feasible
matches in the variable market. Unmatched firms and unmatched workers must have a decision rule for
which market they will post and apply to, respectively. Workers must also have a decision rule on whether
to accept or reject any contract.
The consistency of the worker flow conditions requires that the vacancy filling rate, taken as given in the
firm’s problem, be consistent with the aggregate market tightness in equilibrium. Steady state equilibrium
also requires that unemployment follow (12).
D.
Discussion
Firm heterogeneity in the variance of the productivity shock drives variation in the desirability of adjusting
labor to the shock. Heterogeneity in worker preferences over degree of risk aversion tempers observed
Nick Frazier • 12
contracts so that some firms will prefer to offer fixed hour contracts at a lower wage. Search frictions induce
mis-matches in the sense of not all high variance firm types will match with less risk-averse workers and
vice versa. Since wages are set before production and ex post observed hours vary for some jobs the model
generates both a mixture of jobs with fixed hours and compensation as well as variation in hours and
compensation conditional on variable contracts. Increasing the heterogeneity in worker types would provide
for a greater range of observed wages for both types. Increasing the heterogeneity of firm types also produces
more variation in wages for variable contracts. In estimation both of these types would likely be continuous.
The model further generates cross-sectional variation across workers and firm through productivity
shocks. Even identical variable matches, up to firm and worker type, may produce different worker outcomes
through productivity shocks. Identical fixed matches will produce the same observed compensation and
hours.
III.
I DENTIFICATION
Given my intent to evaluate the general equilibrium consequences of policies that restrict contractual
variation in hours I require a model of optimizing firms and workers. I choose to model the source of
varying hours as the result of firm-specific productivity shocks and worker heterogeneity. These allow for an
exogenous driver of variation in hours while allowing for the possibility of mis-match where some workers
have variable contracts despite preference for fixed contracts. The gains, then, from the counter factual
policies I consider will often hinge on the degree of mis-match which is turn driven by general equilibrium
effects of search frictions.
Given my premise that observed hours are often not optimal hours from the perspective of the employee,
I require data beyond the typical panel of observed jobs familiar to the literature. This conflict results from
the inability, even given the full set of characteristics for observed jobs, to separate the influence of firm
optimizing behavior from worker optimizing behavior. Firm’s maximize profit given available technology
through production and labor force adjustment subject to stochastic productivity and search frictions which
generate the desirability of variable contracts from the firm’s perspective. Workers maximize utility but
through search frictions do not have direct control over the details of their hours schedule or compensation.
Thus, an observed job with high variability could be from a strong demand from the firm or relatively low
disinclination from the worker. Clearly, separating the role of these mechanisms has strong implications for
the overall effect on welfare.
Without the primitives of both the firm’s and the worker’s problem any evaluation of the general
equilibrium effect of these policies. The model permits adjustments by firms on the extensive and intensive
Nick Frazier • 13
margins of labor scheduling while still allowing estimation of welfare impact. To separately identify worker
preferences, I propose using a purpose made survey instrument to elicit preferences from a sample of workers
for whom the kinds of jobs I consider would be relevant. I can then further re-weight my non-representative
sample using the representative sample from the National Longitudinal Survey of Youth 1997 (NLSY97).
This matching is facilitated by the collection of demographic and employment information similar to that
solicited by the NLSY97. As discussed below, these survey instruments can be used to trace out preferences
over the kinds of jobs and contracts used in my model. A similar approach is used in Eriksson and Kristensen
(2014). The controlled environment of stated preferences data even allows us to estimate the compensating
differentials workers would require between various contracts.
Given both relative values and the distribution of preferences, I then estimate the general equilibrium
model using the (1) characteristics from pertinent jobs in the NLSY97, which notably collects information
on the distribution of usual hours, and aggregate data on firm behavior such as that available in JOLT. The
observed jobs will be used to pin down the hours, wages, and compensation of jobs in my economy. The
data on hiring, firing, and job openings from sources like JOLT will discipline the labor force adjustment
technology where correlations reveal fixed costs and movements determine structural and transitory flows.
This largely leaves the firm’s production technology and the economy’s stochastic productivity shock process
to resolve remaining variation in observed jobs and firm outcomes with the intention that the primitives
be estimated with as little structure as possible given their importance to the consequences of the policy
changes under consideration.
IV.
E STIMATION
I would prefer matched employer-employee data but the countries that have this don’t have these types of
jobs.
Estimation of the model closely follows the identification argument. I first estimate preferences using
the functional forms supplied above while allowing for heterogeneity in risk aversion and controlling for
other job attributes. The second step estimates the general equilibrium model using the simulated method of
moments taking preferences as given. The procedure simulates the economy and attempts to match a set of
chosen moments that replicate patterns in the data. In practice this may also require some parameters to be
calibrated.
The survey was conducted in January 2016 with estimation using the above specifications currently
underway.
Nick Frazier • 14
V.
E STIMATION OF P REFERENCES
Each individual n ∈ N at choice set j ∈ J has a utility specified for alternative i ∈ {1, 2, 3} such that utility
Z (·) is given by
Znji =
C 1− ρ n
L1−τn
−γ
+ φXnji + ε nji
1 − ρn
1 − τn
(14)
where ln ρn = ρ + κn and ln τn = ρ + κn for κn ∼ N (0, σκ2 ). This induces correlation between risk aversion in
hours and consumption while maintaining a reasonably parsimonious three parameters.
• ρ: mean risk-version in consumption
• τ: mean risk-version in hours
• Xnji : vector of other characteristic for alternative
– Fixed effect for different degrees
– degree of flexibility
– degree of advanced notice
• ρ: vector of valuations for other job characteristics
• κn : type of person n
Mixed Logit Model
ε nji − ε njk < Znjk − Znji
• ε ∼ i.i.d extreme value type 1
⇒ Znjk − Znji ∼ logistic distribution
• κn allows arbitrary correlation across j and i
– and correlation for risk-aversion in consumption and hours
A.
Data Description
Survey Data
• Commission a survey targeted at recovering preferences
• Vignette: 2 job offers and an unemployment value
• Jobs the 6 attributes in a discrete set of values
– Hours
– Variation in hours
– Wage
– Variation in wage (productivity pay)
– Scheduling flexibility
– Advanced notice
Dimension
Values
Notes
Advanced Notice
≤ 1 week
1–2 Weeks
3+ weeks
You will learn your schedule each week
You will learn your schedule 1-2 weeks before
You will learn your schedule at least 3 weeks in advance
Scheduling Flexibility
None.
A lot
Starting and finishing times are decided by your employer and
you cannot change them on your own
Starting and finishing times are decided by your employer but
with your input
You can decide the time you start and finish work, within certain
limits
You are entirely free to decide when you start and finish work
Hours
10
20
30
40
hours per week
hours per week
hours per week
hours per week
Variance in Hours
Fixed
±20%
±50%
e.g. always h hours/week
Hours vary between .8h, h, 1.2h
Hours vary between .5h, h, 1.5h
Pre-Tax Wage
Low
Mid1
Mid2
High
0.9 × wi,p + 0.1 × wi,r Pessimistic
0.6 × wi,p + 0.4 × wi,r Low Mixture
0.4 × wi,r + 0.6 × wi,o High Mixture
0.1 × wi,r + 0.9 × wi,o Optimistic
Variance in Pay
Fixed
±10%
Always w per hour.
Pay varies between .9w, w, 1.1w
A little.
Some.
Table 1: A full description of the attributes for each job used in my vignettes.
Nick Frazier • 16
Figure 1: An example of a vignette from my survey instrument used to elicit preferences.
Nick Frazier • 17
B.
Estimation of General Equilibrium Model
SMM or indirect inference similar to Cooper et al. (2007). Using observed jobs and moments for aggregate
workforce.
Nick Frazier • 18
R EFERENCES
Bloemen, Hans G., “Job Search, Hours Restrictions, and Desired Hours of Work,” Journal of Labor Economics,
2008, 26 (1), pp. 137–179.
Cooper, Russell, John Haltiwanger, and Jonathan L. Willis, “Search frictions: Matching aggregate and
establishment observations,” Journal of Monetary Economics, 2007, 54, pp. 56–78.
Eriksson, Tor and Nicolai Kristensen, “Wages or Fringes? Some Evidence on Trade-Offs and Sorting,”
Journal of Labor Economics, 2014, 32 (4), pp. 899–928.
Golden, Lonnie, “Irregular Work Scheduling and Its Consequences,” Economic Policy Institute, 2015, Briefing
Paper No. 394.
, Julia R. Henly, and Susan Lambert, “Work Schedule Flexibility: A Contributor to Happiness?,” Journal
of Social Research and Policy, 2013, 4 (2).
Henly, Julia R. and Susan J. Lambert, “Unpredictable Work Timing in Retail Jobs: Implications for Employee
Work-Life Outcomes,” Industrial and Labor Relations Review, 2014, 67 (3), pp. 986–1016.
Ingre, Michael, Torbjörn Åkerstedt, Mirjam Ekstedt, and Göran Kecklund, “Periodic self-rostering in
shift work: correspondence between objective work hours, work hour preferences (personal fit), and work
schedule satisfaction,” Scandinavian Journal of Work, Environment & Health, 2012, 38 (4), pp. 327–336.
Lambert, Susan J., “Passing the Buck: Labor Flexibility Practices that Transfer Risk onto Hourly Workers,”
Human Relations, 2008, 61 (9), pp. 1203–27.
, Anna Haley-Lock, and Julia R. Henly, “Schedule Flexibility in Hourly Jobs: Unanticipated Consequences and Promising Directions,” Community, Work and Family, 2012, 15 (3), pp. 239–315.
, Peter J. Fugiel, and Julia R. Henly, “Precarious Work Schedules among Early-Career Employees in the
US: A National Snapshot ,” Technical Report 2014.
Mas, Alexandre and Amanda Pallais, “Valuing Alternative Work Arrangements,” Working Paper 22708,
National Bureau of Economic Research 2016.
Nassauer, Sarah, “Wal-Mart Rolls Out a New Worker Scheduling System,” Wall Street Journal, August 2016.
Reynolds, Jeremy, “You Can’t Always Get the Hours You Want: Mismatches between Actual and Preferred
Work Hours in the U.S.,” Social Forces, 2003, 81 (4), pp. 1171–1199.
and Lydia Aletraris, “Pursuing Preferences: The Creation and Resolution of Work Hour Mismatches,”
American Sociological Review, 2006, 71 (4), pp. 618–638.
Rosen, Sherwin, “The Theory of Equalizing Differences,” in O. Ashenfelter and R. Layard, eds., Handbook of
Labor Economics, Vol. 1 of Handbook of Labor Economics, Elsevier, 1987, chapter 12, pp. pp. 641–692.
Stewart, Mark B. and Joanna K. Swaffield, “Constraints on the Desired Hours of Work of British Men,” The
Economic Journal, 1997, 107 (441), pp. 520–535.
Tabuchi, Hiroko, “Abercrombie & Fitch to End On-Call Shifts for Workers,” New York Times, August 2015.
White, Gillian B., “The Very Real Hardship of Unpredictable Work Schedules,” The Atlantic, April 2015.
Figure 2: Taken from Lambert et al. (2014). Highlights the high degree of week-to-week variability in hours.