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FEDERAL RESERVE BANK OF SAN FRANCISCO
WORKING PAPER SERIES
Dynamic Labor Reallocation with Heterogeneous
Skills and Uninsured Idiosyncratic Risk
Ester Faia
Goethe University Frankfurt and CEPR
Ekaterina Shabalina
Goethe University Frankfurt
Marianna Kudlyak
Federal Reserve Bank of San Francisco
June 2021
Working Paper 2021-16
https://www.frbsf.org/economic-research/publications/working-papers/2021/16/
Suggested citation:
Faia, Ester, Ekaterina Shabalina, Marianna Kudlyak. 2021 “Dynamic Labor
Reallocation with Heterogeneous Skills and Uninsured Idiosyncratic Risk,” Federal
Reserve Bank of San Francisco Working Paper 2021-16.
https://doi.org/10.24148/wp2021-16
The views in this paper are solely the responsibility of the authors and should not be interpreted
as reflecting the views of the Federal Reserve Bank of San Francisco or the Board of Governors
of the Federal Reserve System.
Dynamic Labor Reallocation with Heterogeneous
Skills and Uninsured Idiosyncratic Risk∗
Ester Faia†
Goethe University Frankfurt and CEPR
Marianna Kudlyak‡
Ekaterina Shabalina
§
Federal Reserve Bank of San Francisco and CEPR
Goethe University Frankfurt
June 10, 2021
Abstract
Occupational specificity of human capital motivates an important role of occupational
reallocation for the economy’s response to shocks and for the dynamics of inequality.
We introduce occupational mobility, through a random choice model with dynamic
value function optimization, into a multi-sector/multi-occupation Bewley (1980)Aiyagari (1994) model with heterogeneous income risk, liquid and illiquid assets, price
adjustment costs, and in which households differ by their occupation-specific skills.
Labor income is a combination of endogenous occupational wages and idiosyncratic
shock. Occupational reallocation and its impact on the economy depend on the
transferability of workers’ skills across occupations and occupational specialization of
the production function. The model matches well the statistics on income and wealth
inequality, and the patterns of occupational mobility. It provides a laboratory for
studying the short- and long-run effects of occupational shocks, automation and task
encroaching on income and wealth inequality. We apply the model to the pandemic
recession by adding an SIR block with occupation-specific infection risk and a ZLB
policy and study the impact of occupational and aggregate labor supply shocks. We
find that occupational mobility may tame the effect of the shocks but amplifies earnings
inequality, as compared to a model without mobility.
JEL: J22, J23, J31, J62, E21, D31.
Keywords: Occupational Mobility, Heterogeneous Agents, Skills, Income and Wealth
Inequality, Discrete Choice Optimization.
∗
†
‡
§
Any opinions expressed are those of the authors and do not reflect those of the Federal Reserve Bank of
San Francisco or the Federal Reserve System. Ester Faia gratefully acknowledges support from the DFG
grant FA-1022-2. The authors thank Ludwig Straub for his generous help with the solution algorithm
and Benjamin Moll and Ricardo Reis for helpful comments. The authors thank Max Mayer for excellent
help with the k-means algorithm.
faia@wiwi.uni-frankfurt.de
marianna.kudlyak@sf.frb.org
ekaterina.pyltsyna@gmail.com
1
Contents
1 Introduction
3
2 Related Literature
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3 Heterogenous Agents Model with Dynamic Occupational Choice and
Idiosyncratic Income Risk
3.1 Consumption-Saving and Occupational Choice . . . . . . . . . . . . . . . . .
3.2 Heterogeneous Firms: Labor Demand, Capital Demand, and Pricing Decisions
3.3 Asset Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Monetary and Fiscal Policy . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 SIR Block . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Market Clearing: Labor, Goods and Asset Markets . . . . . . . . . . . . . .
3.7 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4 Analytical Results
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5 Model Parametrization and Solution Method
5.1 Skills, Occupations and Skills Distribution . .
5.2 Sectors and Production Function . . . . . . .
5.3 Other Parameters . . . . . . . . . . . . . . . .
5.4 Solution Method . . . . . . . . . . . . . . . .
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Policy Shock
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Application of the Model to the Pandemic Recession
Occupational Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Role of Monetary Policy . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Role of Job Specialization and Skill-Transferability . . . . . . . . . . . .
Sectoral Productivity Shocks in a Two-Sector Model . . . . . . . . . . . . .
Aggregate Labor Supply Shock in an One-Sector Model . . . . . . . . . . . .
Dynamic Response to Combined Aggregate Supply and Occupational Shocks
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6 Quantitative Results
6.1 Steady State Results . . . . . . . . . . . .
6.1.1 Occupational Sorting . . . . . . . .
6.1.2 Income and Wealth Distributions .
6.2 Dynamic Responses to Standard Monetary
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7.1
7.2
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8 Conclusions
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A Derivations of Analytical Results
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B Computation Method
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C Constructing Occupational Clusters Using k-means Algorithm
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D Appendix Tables
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E Other Quantitative Results
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F Wealth Distribution Graphs
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2
1. Introduction
The role of worker and job reallocation in an economy’s response to shocks and its impact on
inequality is one of the fundamental questions in economics (Schumpeter (1942)). The focus
has been on sectoral shocks and the ensuing reallocation across sectors (Lilien (1982)). The
shocks, however, often fail to generate large aggregate disturbances as there is little barrier to
sectoral labor reallocation. Instead, the literature finds that workers’ skills or, broadly, human
capital are closely related to occupational tasks (Gathmann and Schönberg (2010), Guvenen,
Kuruscu, Tanaka and Wiczer (2020)). Consequently, the degree of skill transferability and
substitutability of occupational hours in the production function play a central role for worker
and job reallocation. Given this technological side, reallocation is a result of an interplay of
the type of shocks, skills, wages and wealth, whereby the latter affects workers’ decision of
shifting labor supply across occupations. Occupational reallocation is particularly important
for understanding the effects of technological shocks that change occupational tasks (see,
for example, Autor, Levy and Murnane (2003)). Studying the interplay of these forces and
the impact of occupational reallocation on the macroeconomy and inequality requires a
dynamic macro model with optimal occupational decision in an environment where agents
are heterogeneous in terms of their skills and wealth accumulation, that is, uninsurable
idiosyncratic risk.
We introduce endogenous occupational mobility, through a discrete choice with value
function optimization based on workers’ comparative advantage in Roy (1951)’s tradition, in
a general equilibrium multi-sector and multi-occupation Bewley (1980)-Aiyagari (1994)-style
model with idiosyncratic income shocks, liquid and illiquid assets and in which agents differ
with respect to their occupational-specific skills. In the model, occupational choice depends
on skills as well as wealth and income risk. Our dynamic discrete occupational choice set-up
with Gumbel-distributed taste shocks follows Rust (1987). Taste shocks capture the role of
compensating differentials and other non-pecuniary aspects such as job amenities (see Taber
and Vejlin (2020)). Income inequality in the model is linked to the occupational choice and
depends both on idiosyncratic income shocks (luck) and skills (talent or human capital). The
3
model also features sticky prices to examine the role of reallocation in the short-run, while
taking into account the role of stabilization policies.
Workers in the model belong to one of a discrete set of types that differ in their skills and,
therefore, efficiency units that they can provide in different occupations. The skill distribution
matrix of worker types across occupations captures workers’ comparative advantage and,
hence, in the Roy tradition, is a key determinant of occupational choice and mobility. Each
period, workers experience idiosyncratic productivity shocks. They choose which occupation
to work at or whether to stay out of employment, an option which provides a fixed flow value.
We model occupational decision through a random discrete choice problem in which workers
compare value functions in each occupation. These value functions depend on workers’ skill,
wealth and idiosyncratic income shocks. Imperfect skill-transferability or job specialization
govern the extent of mobility. Our model with endogenous occupational choice complements
recent work by Eeckhout and Sepahsalari (2020) who built a directed search model with
uninsurable risk that links wealth accumulation and occupational sorting.
Labor income in our model includes both an idiosyncratic exogenous component and
occupational wages, which are endogenously determined via labor market clearing in each
occupation. Wages depend on the skill distribution, which characterizes worker’s skills
transferability across occupations, and the job specialization, which characterizes imperfect
substitutability of occupations in the sectoral production function. Hence, our model
endogenizes the cross-sectional distribution in income, i.e., income inequality.
We derive a set of analytical results focusing on the technological side of reallocation,
that is, on the extent to which aggregate and cross-sectional changes in labor demand affect
reallocation. We show that allocation of labor across occupations and sectors depends on
the primitives which affect labor demand, i.e., the occupational weights and the degree of
occupational substitutability in the production function, and labor supply, i.e., the skill
distribution in the population. We also show that changes in occupational substitutability
affect the elasticity of occupational labor demand with respect to wages and the aggregate
capital-labor substitution. Finally, lower job substitutability, by raising job specialization,
4
raises occupational rents (i.e., mark-ups) and serves a role akin to occupational mobility
costs.
We solve the model using the sequence-space Jacobian method developed by Auclert,
Bardóczy, Rognlie and Straub (2019) for general equilibrium heterogeneous agent models in
discrete time with expectations. We extend the method by adding a guess-and-verify procedure
for solving the two-stage households’ decisions problem, which involves a consumption-saving
and a discrete-choice occupational decisions. Under the extreme type I taste shocks, the model
delivers a closed-form solution for the occupational choice probabilities. In our application of
the model to the pandemic, we adopt the fully non-linear routine from Auclert et al. (2019)’s
algorithm to accommodate the transitional dynamics between steady states.
We parameterize novel elements of the model using micro data and employ the standard
parameter values from the literature on heterogeneous agents models for the remaining
parameters. One of the important primitives of the model is the skill distribution matrix of
worker types across occupations. To construct the skill matrix, we group detailed occupations
into broad occupational categories based on the proximity of the occupational tasks described
in O-NET using a k-means machine learning algorithm (see also Bonhomme, Lamadon
and Manresa (2017) and Grigsby (2020)). We then measure transferability of skills across
the occupational categories based on the differences in occupational tasks using data from
O-NET (see Poletaev and Robinson (2008) or Gathmann and Schönberg (2010)), which
captures horizontal differentiation, and differences in the average educational attainment in
occupations using data from the BLS, which captures vertical specialization.
Our parameterized model in the steady state matches well various statistics on asset
and income inequality using data from the Survey of Consumer Finance. Examining the
determinants of long-run occupational sorting, we find that workers are more likely to choose
occupations in which they have highest comparative advantage and that pay high wages.
Wealth also matters for occupational sorting - asset-rich individuals are indifferent between
employment and non-employment, while asset-poor individuals choose occupations according
to their skills and wages. The income and asset distributions in the model exhibit similar
5
degrees of dispersion and skeweness as those in the data, with larger concentration in the top
percentiles.
We first examine the dynamic response of our model with occupational mobility to a
monetary policy shock and compare our model’s response to the response from a model
without occupational mobility. As in the existing models with uninsurable risk, in response to
a contractionary monetary policy shock, consumption declines due to precautionary savings
motive, investment declines due to the paradox of thrift, and assets dynamics exhibit BaumolTobin flight to liquidity (see Bayer, Lütticke, Pham-Dao and Tjaden (2019)). The monetary
transmission is characterized by the distributional channels, namely earnings-heterogeneity
channel, the Fisher effect and the interest rate exposure channel (see Kaplan, Moll and
Violante (2018), Auclert (2019)) and the expectational ones (see Galı́ and Gertler (2007)).
Two novel channels arise from occupational reallocation. First, workers reallocate across
occupations. Second, some workers choose not to work. If the reallocation channel prevails,
the effect of the contractionary shock is tamed through reallocation. Which of the channels
prevails depend on the type and the magnitude of the shock, monetary policy stance and
other structural characteristics of the economy. Finally, while the model with occupational
mobility may tame the contractionary effect of a shock, it generally raises earnings inequality.
The occupational reallocation mechanism is best demonstrated in a setting with a large
recessionary shock that hits different occupations unevenly. We therefore apply the model
to the recent pandemic recession. The pandemic recession featured a marked reallocation
channel through a combination of two kinds of shocks: (1) uneven and persistent occupational
or sectoral shocks through the government-mandated shutdowns of non-essential businesses
(see, for example, Barrero, Bloom and Davis (2020)), and (2) a large aggregate labor supply
shock through infections. The pandemic featured other forces and channels. Our goal is not
to account for all the shocks during the pandemic, but to highlight the role of reallocation and
its impact on the economy in response to shocks and policies. Furthermore, the shocks that
unevenly hit different occupations extend beyond government-mandated stringency measures
in the pandemic and are related to the ongoing automation, acceleration in task-encroaching,
6
and the disappearance of certain tasks (Autor and Reynolds (2020), Chernoff and Warman
(2020)).
We introduce occupational shocks in the model by reducing labor demand in certain
occupations (retail sales) and raising it in others (healthcare) via shocks to the occupational
shares in the production function. As in the models with uninsurable risk, precautionary
savings motive induces decline in consumption and the flight to liquidity, and the paradox
of thrift rationalizes the fall in investment.1 The decline in asset prices, following asset
liquidation, reduces wealth inequality by inflicting larger losses for those in the top-percentiles
of the wealth distribution.2 In the model with occupational mobility, workers reallocate
from shrinking to expanding occupations. The increase in income dispersion enhances
precautionary savings and, following a large decline in wages, some workers opt out of work.
The reallocation and opting out of work exacerbate the recession. The zero-lower-bound
policy as compared to the Taylor rule has a dual effect. By equalizing returns on wealth, the
zero-lower-bound policy curbs reallocation incentives. On the other hand, it reduces the cost
of investment, which in turn raises labor income and demand. Additionally, we show that the
extent of the recession as well as the patterns of income inequality largely depend on the type
of occupation hit by the shock: a negative shock to a high-wage occupation as compared to
a low-wage occupation, worsens the recession and increases the earnings inequality. Finally,
a shock to sectoral productivity produces qualitatively similar dynamics; the magnitude of
the impact of the sectoral shock depends on the composition of the occupation-specific labor
in the shrinking sector.
We introduce the aggregate labor supply shock due to the pandemic infection by augmenting the model with a standard SIR block that features differential infection risks across
occupations. In the model, forward-looking optimizing agents respond to the infection
risk through occupational mobility and opting out of work. The aggregate labor supply
shock induces a fall in labor hours in all occupations, more so in occupations with high
risk. Reallocation across occupations is minimal and most of the dynamics is driven by an
increased share of workers in non-employment due to the fall in wages and the increase in
1
2
See earlier work by Aiyagari and Gertler (1991) for impact of flight to liquidity on asset prices.
Atkinson, Piketty and Saez (2011) Piketty and Saez (2014), and Scheidel (2020) argue that loss of asset
values is one of the reasons behind the decline in inequality at the beginning of the 20th century.
7
infection risk. This experiment provides insights into the impact of other aggregate labor
supply shocks that trigger decline in labor force participation.
The paper proceeds as follows. Section 2 reviews the related literature. Section 3 presents
the model. Section 4 presents analytical results. Section 5 explains data used in the
calibration. Section Section 6 shows steady state results and the model’s responses to a
standard monetary policy shock. Section 7 presents and discusses simulation results under
various shocks. Section 8 concludes. Appendices follow.
2. Related Literature
The role of sectoral shocks and reallocation for the macroeconomy has a long tradition in
economics (Schumpeter (1942), Lilien (1982), Abraham and Katz (1986) and, more recently,
Chodorow-Reich and Wieland (2020)). The literature focuses on sectoral reallocation, not
occupational mobility. It also generally does not emphasize the link between reallocation
and inequality.
Our focus on occupational mobility is motivated by an idea that skills or, more broadly,
human capital are closely related to the occupational composition of tasks (Gathmann and
Schönberg (2010), Guvenen et al. (2020)). We model occupational choice following a seminal
work by Roy (1951), empirical underpinnings of which can be found in Heckman and Honore
(1990) and Buera (2006). Recently, Grigsby (2020) estimates a static Roy model to account
for the wage stagnation following the financial crisis; Yamaguchi (2012) estimates a dynamic
frictionless Roy model with a posited wage function; and Jaimovich, Saporta-Eksten, Siu and
Yedid-Levi (2020) studies the role of automation in a general equilibrium Roy model. Our
model features a dynamic occupational choice (as in Boskin (1974), Rust (1987) and Keane
and Wolpin (1997)) and its interaction with uninsurable risk and wealth. Card, Cardoso,
Heining and Kline (2018) applies discrete choice model to studying labour supply. Berger,
Herkenhoff and Mongey (2019) discusses an equivalence between nested-logit and nested-CES
used in oligopsony — this conceptually parallels the occupational rents/mark-ups which we
highlight in our model.
8
A paper related to ours is Eeckhout and Sepahsalari (2020). They developed a model
with heterogeneous agents, uninsurable risk and directed search that provides a link between
inequality and labor search behaviour. In their model, sorting arises from a monotone
matching condition, in our shifting probabilities result from a value function optimization. In
this respect, the directed search and the Roy approach provide complementary insights. In
our model imperfect skill-transferability and the elasticity of substitution across occupations
act as frictions to mobility.
A growing literature builds general equilibrium models with heterogeneous uninsurable
risk and nominal rigidities. First insights on the different transmission of shocks between the
representative agent and the heterogenous agents model came from models featuring liquidity
constrained and unconstrained agents (Bilbiie (2008) or Debortoli and Galı́ (2017)). Our
model highlights how inequality can affect also labour supply and occupational reallocation
on top and above the consumption decisions. A growing literature has further developed
models with heterogenous agents and nominal rigidities (McKay, Nakamura and Steinsson
(2016), McKay and Reis (2016), Guerrieri and Lorenzoni (2017), Gornemann, Kuester and
Nakajima (2016),Kaplan et al. (2018), Bayer et al. (2019), Ravn and Sterk (2017), Auclert et
al. (2019), Auclert and Rognlie (2018), Auclert, Rognlie and Straub (2018), (Auclert, Rognlie
and Straub 2020) or Dupor, Karabarbounis, Kudlyak and Mehkari (2018), among others).
These papers do not study the role of skill heterogeneity, human capital or occupational
choice. Our model endogenizes the cross-sectional variation in income and provides a two-way
link between inequality and occupational mobility.
The role of skills for inequality is also studied in Keane and Wolpin (1997) and Huggett,
Ventura and Yaron (2011). Early work on the link between inequality and occupational choice
include Galor and Zeira (1993) and Banerjee and Newman (1993). The impact of education
on wage inequality in an incomplete-market life-cycle model is in Heathcote, Storesletten
and Violante (2010). Reallocation channels are featured in Lise and Postel-Vinay (2020), in
experimentation models such as Jovanovic (1979) and Antonovics and Golan (2012) and in
models with rest unemployment (see Alvarez and Shimer (2011)). They do not endogeneize
the shifting probabilities (both across occupations and in and out of employment) and do
9
not consider the role of uninsurable risk and the interaction between occupational mobility
and inequality.
Our model features jointly the choice of labor hours (intensive), labor participation and
of the type of occupation (extensive margin and mobility choice). Chang and Kim (2006)
and Chang, Kim, Kwon and Rogerson (2019) embed the choice of heterogenous labour
hours in models with uninsurable risk and Heathcote, Perri and Violante (2020) add a
non-participation margin.
Our paper is also related to the literature studying the impact of the pandemic shock
on the severity of the recession and inequality (Kaplan, Moll and Violante (2020), Bayer,
Born, Luetticke and Müller (2020), Eichenbaum, Rebelo and Trabandt (2020a), Krueger,
Uhlig and Xie (2020) and Guerrieri, Lorenzoni, Straub and Werning (2020)). Guerrieri
et al. (2020) study sectoral reallocation during the pandemic, while our paper examines
occupational reallocation. Our application that considers the differential impact of supply
and sector-demand shocks relates our paper to Baqaee and Farhi (2020).
3. Heterogenous Agents Model with Dynamic
Occupational Choice and Idiosyncratic Income Risk
The key novelty of our work is embedding endogenous occupational mobility in a multi-sector
multi-occupation Bewley (1980)-Aiyagari (1994) model with liquid and non-liquid assets
and sticky prices. In the model, risk-averse infinitely lived households are heterogeneous
- they differ by skill. The household’s skill-type determines the effective labor units that
the household can supply in each occupation. The household can also choose to remain
non-employed, receiving a flow value of non-employment. Households’ optimization can be
summarized as a two-stage problem. In the first stage, households choose whether to work and
which occupation to work at by solving a discrete choice problem comparing value functions
across all possible choices. We obtain a closed-form solution for the occupational choice
probabilities, under the assumption of extreme value type I taste shocks. In equilibrium, the
occupational choice probabilities depend on the value functions, which in turn depend on
10
wages, wealth and skills. Given the occupational choice, in the second stage, households solve
a consumption-saving problem and choose consumption and investment in liquid and nonliquid assets. The cost of liquidating the non-liquid asset impairs households’ precautionary
saving ability; consequently, any inequality in wages entrenches into wealth inequality.
Aggregate output is a CES bundle of sectoral varieties. The production block features
S sectors, each one employing each of the O occupations, according to the sector-specific
shares, αs,o . Sectoral output is produced by bundling the effective units of labor from all
occupations using a sector-specific production function. Each occupation is characterized by
its own market clearing condition.
In this section, we describe heterogeneous households and heterogeneous firms problems
and define the equilibrium.
3.1. Consumption-Saving and Occupational Choice
Households can be employed in one of many different occupations indexed by o ∈ {1, ..., O}
or non-employed. When we discuss occupational choice, we refer to the choice out of O + 1
options, where option O + 1 is non-employment. These options differ by the associated flow
values and the taste shocks.
Skill heterogeneity. Households differ in their skills and, hence, efficiency units of labor
that they can provide in each occupation. The household’s skill-type (hereafter, type) is
indexed by j ∈ {1, ..., J}. There is the time-invariant mass mj of each type j. Each type
j is characterized by vector γj , which describes the efficiency units that the type j can
provide in each occupation o ∈ {1, ..., O}. Γ is the J × O matrix that stacks the (transposed)
vectors for all J types. The matrix captures the way in which household’s skill heterogeneity
relates to occupations and plays a crucial role in determining the degree of reallocation across
occupations. Each matrix element γjo characterizes the comparative advantage of worker
of type j in performing the tasks in occupation o. The skill-matrix is fixed over time. It
is possible to extend the model to allow for the matrix to change in response to persistent
occupational shocks.
Non-employment state. The menu of possible household’s activity choices includes
11
non-employment. In the non-employment state, which encompasses out of the labor force
and non-employment, individuals receive the flow value of non-work expressed in real income
units, h. The flow value of non-work encompasses home-production and self-employment.3
The flow value of non-work, h, is the same for all household types.
Idiosyncratic productivity risk. At the end of each period t, employed and nonemployed households experience idiosyncratic productivity shock et . The shock follows
an ne -state Markov process with transition matrix P (et+1 |et ) and stationary distribution
π(e). The shock is independent of the household’s skill type or current occupation (or the
non-employment state).
o
Labor income. Real after-tax labor income ξj,t
of individual of type j at time t in
occupation o is a function of efficiency units of type j in occupation o, γjo , idiosyncratic
productivity state et , real wage in occupation o wto , labor hours not , and tax rate τt :
o
ξj,t
= (1 − τt )et wto γjo not
(1)
Heterogeneity enters labor income through three components. The first component is the
time-invariant efficient labor units supplied by household of type j in occupation o. The
second component is the occupational wage, which is endogenously determined in the labor
market. The third component is the exogenous idiosyncratic productivity shock, common
to all heterogeneous agents models with uninsurable risk. In the model, occupational wage
is the same in each sector. Therefore, workers are indifferent across sectors, conditional on
their occupational choice. This is supported by the empirical evidence that human capital is
occupation-specific.
Let ξj,t denote the (O + 1)-dimensional vector of the real after-tax labor income of type j
o
individuals in time t in all possible occupations o ∈ O, which consists of elements ξj,t
, and
O+1
the real after-tax income-equivalent in the non-employed state, ξj,t
= h for all j ∈ J.
Taste shocks. Each period, households draw idiosyncratic occupational-preference shock
φot for all o ∈ O + 1 (in terms of utility). The taste shock is assumed to be i.i.d. across workers
3
An alternative formulation is the one adopted in Grigsby (2020) who assumes a zero flow value of
non-employment, but subtracts a constant opportunity costs from the income in each occupation.
12
and occupations and follows a type 1 extreme value distribution, with density f (φ) and
cumulative distribution function F (φ). The distributional assumptions follow the tradition
of discrete choice models and generate a tractable form for the choice probabilities (see
McFadden and Zarembka (1974)). The occupation-specific taste shock serves as a smoothing
device in discrete choice models (see Rust (1987); Iskhakov, Jørgensen, Rust and Schjerning
(2015)).
Sequence of decisions. Each period a household of type j decides on the occupational
choice or non-employment o ∈ {1, ..., O, O + 1}, their consumption and savings. Hours are
equal across occupations no = n for all o and are chosen by representative heads. The state
in period t is (et , at−1 , bt−1 , φt ) where et is the idiosyncratic productivity shock, at are the
illiquid assets, bt are the liquid assets, and φt is the (O + 1)-vector of the taste shocks across
all occupations and the non-employment state. Household of type j maximizes the following
value function, given the state:
Vj (et , at−1 , bt−1 , φt ) = max u(ct , not ) + φot + βEφ Ee Vj (et+1 , at , bt , φt+1 )
ot ,ct ,at ,bt
o
s.t. ct + at + bt = ξj,t
+ (1 + rta )at−1 + (1 + rtb )bt−1 − Φ(at , at−1 )
at ≥ 0,
(2)
bt ≥ b
o
where ξj,t
is the income in occupation o, rta and rtb are the interest rates on illiquid and
liquid assets, respectively, Φ(.) is the convex portfolio adjustment costs defined below, and
β is the time discount factor. The instantaneous utility takes the standard separable form
u(c, n) =
c1−σ
1−σ
−
n1+ρ
.
1+ρ
For liquid assets, equation bt ≥ b acts as a borrowing constraint. The
o
value function depends on the idiosyncratic income shock, et , through ξj,t
.
The household’s problem can be broken into two stages and solved backwards as follows.
First, households make their consumption-saving decision, conditional on the occupation
choice at time t and taking into account all possible occupations that they can work at
in the future. Second, after substituting the consumption-saving policy functions for each
occupation into the value functions, the household determines the probabilities of working in
each occupation by comparing value functions across occupations.
Consumption-saving decision. Consider consumption-saving decision of a household
13
that optimally chooses occupation o. Conditional on occupational choice o, each household
of type j in period t chooses consumption, ct , savings in liquid, bt , and illiquid, at , assets to
maximize the following value function:
Vjo (et , at−1 , bt−1 ) = max u(ct , nt ) + βEφ Ee Vj (et+1 , at , bt , φt+1 )
ct ,at ,bt
o
s.t. ct + at + bt = ξj,t
+ (1 + rta )at−1 + (1 + rtb )bt−1 − Φ(at , at−1 )
at ≥ 0,
(3)
bt ≥ b
where Vjo (et , at−1 , bt−1 ) is the value function in time t of type j conditional on occupation o
and φot evaluated at 0.
The portfolio adjustment costs take the following functional form:
χ1 at − (1 − rta )at−1
Φ(at , at−1 ) =
χ2 (1 + rta )at−1 + χ0
χ2
[(1 + rta )at−1 + χ0 ]
(4)
with χ0 > 0, χ1 > 0 and χ2 > 1. Note that Φ(at , at−1 ) is bounded, differentiable, and convex
in both arguments. Consumption-savings and portfolio decisions follow the standard Euler
conditions (shown in Appendix B).
Labor hours are chosen by the family as a whole and are determined through the intratemporal first order condition:
ϕnρt
=
O X
J
X
Z
mj
uc (et , at−1 , bt−1 , ot )θj (o|et , at−1 , bt−1 )
o=1 j=1
o
∂ξj,t
dDj (et , at−1 , bt−1 )
∂not
(5)
where θj (o|et , at−1 , bt−1 ) are optimal probabilities of each occupation (see derivation below).
The solution to the optimization problem is a collection of type- and occupation-specific
policy functions coj (et , at−1 , bt−1 ), aoj (et , at−1 , bt−1 ) and boj (et , at−1 , bt−1 ) that depend on the
path rsa , rsb , τs s>t , taken as given by households.
Occupational decision. Following the tradition of random discrete choice models, we
model occupational choice as a comparison of the value functions, which is an inherently
14
dynamic problem. Household of type j chooses occupation o by comparing value functions
from each choice, evaluated at the optimal consumption-saving policies:
o = argmax[1,.....,O,O+1] [Ṽjo + φot ]
(6)
where Ṽjo is the value function Vjo (et , at−1 , bt−1 ) evaluated at the optimal consumption-saving
policy coj (et , at−1 , bt−1 ), aoj (et , at−1 , bt−1 ) and boj (et , at−1 , bt−1 ). We skip the dependence of the
value function on the state for notational convenience. Note that the taste shock, φot , only
affects preferences for occupation and does not influence the consumption-saving decision.
The solution o to the occupational choice problem (6) satisfies
0
0
F (Ṽjo + φot ≥ Ṽjo + φot )∀o0 6= o ∈ O + 1
(7)
where F (.) is the cumulative density of the taste shock.
Taking expectation with respect to the current period taste shocks of the household value
function in (2) and evaluating at the optimal consumption-saving choice yields
h
i
X
Eφ Ṽj (et+1 , at , bt , φt+1 ) = Eφ max Ṽjo (et , at−1 , bt−1 ) + φot = ln
exp(Ṽjo (et , at−1 , bt−1 ))
ot
ot
(8)
where the last equality follows from the properties of the Gumbel distribution.
Combining (6) and (8) yields the following closed-form solution for the occupational choice
probabilities:
exp(Ṽjo (et , at−1 , bt−1 ))
θj (o|et , at−1 , bt−1 ) = P
.
o
o exp(Ṽj (et , at−1 , bt−1 ))
(9)
Equation 9 shows that the probabilities of moving among occupations and the non-work
state depends on the households’ value functions. This implies that the wage distribution,
the wealth accumulation and the idiosyncratic income shocks have an impact on occupational
mobility. We solve the two-stage optimization problem numerically through a guess-and-verify
procedure described in Appendix B.
15
Household aggregates. We obtain the aggregate policy functions for consumption and
asset holdings by integrating occupation-specific policy functions weighted by occupation
choice probabilities over the measure of households in state et that own assets in sets A− and
B− at the start of date t, which is given by D(et , A− , B− , ) = P r(e = et , at−1 ∈ A− , bt−1 ∈ B− ),
and aggregating across all occupations o ∈ O + 1 and across all household types j ∈ J.
The aggregate policy functions for households for every period i ≥ t are:
At (ria , rib , τi , Ni )
=
J
X
mj
O+1
XZ
aoj (et , at−1 , bt−1 )θj (o|et , at−1 , bt−1 )dDj (et , at−1 , bt−1 )
o=1
j=1
(10)
Bt (ria , rib , τi , Ni )
=
J
X
mj
O+1
XZ
boj (et , at−1 , bt−1 )θj (o|et , at−1 , bt−1 )dDj (et , at−1 , bt−1 )
o=1
j=1
(11)
Ct (ria , rib , τi , Ni )
=
J
X
mj
O+1
XZ
coj (et , at−1 , bt−1 )θj (o|et , at−1 , bt−1 )dDj (et , at−1 , bt−1 )
o=1
j=1
(12)
where Ni is total hours (in employment and non-employment).
Total labor hours supplied in each occupation o is given by:
Nto
=
not ζto
J
X
Z
mj
θj (o|et , at−1 , bt−1 )dDj (et , at−1 , bt−1 )
(13)
j=1
where not is the labor hours per occupation. ζ is the coefficient that determines the decrease
in labor supply due to an increase in the number of infected people in response to an
infection shock, i.e. ζto = 1 − It · ϑo , where It is the number of infected people and ϑo is the
occupation-specific constant.
The total effective labor supply in occupation o is given by the efficiency-units-weighted
and idiosyncratic-shock-weighted employment in each occupation:
Lo,Supply
t
=
not ζto
J
X
mj γjo
Z
et θj (o|et , at−1 , bt−1 )dDj (et , at−1 , bt−1 )
j=1
16
(14)
3.2. Heterogeneous Firms: Labor Demand, Capital Demand, and
Pricing Decisions
There is a set of sectors (i.e., industries) s ∈ 1, ..., S, each producing sector-specific (i.e.,
variety) output ys,t using a sector-specific production function. A representative competitive
firm aggregates different varieties from all sectors in the economy through a CES aggregate
consumption bundle:
Yt =
S
X
1
η
η−1
η
η
! η−1
fs ys,t
,
(15)
s=1
where η is the elasticity of substitution across varieties and fs are the sector weights that
capture the size of each sector in the economy. This implies that optimal demand equation
η1
Yt
for each variety is given by ps,t = fs ys,t pt , where pt is the aggregate price index and is
normalized to 1 in the steady state.
Each sector employs a sector-specific combination of occupations, in terms of occupationspecific effective labor hours ls,o,t , with different shares αs,o and elasticity of substitution σs .
Total labor demand by sector s is
Ls,t =
O
X
! σ1
s
σs
αs,o ls,o,t
(16)
o=1
The elasticity of substitution across occupations,
1
,
σs
plays a crucial role in determining the
extent of reallocation across occupations as we show in the analytical results in Section 4. It
also determines the elasticity of substitution of labor demand with respect to wages; hence,
it captures the degree of occupational specialization.
Sectors produce output by combining total labor input and capital using a nested CES
production function:
νs
s
ys,t = zs,t ks,t−1
L1−ν
s,t
(17)
where ys,t is the variety, zs,t is sector total factor productivity and νs is the capital share.
Firms in each sector s choose labor demand, ls,o,t , for each occupation o ∈ O, capital
17
demand, ks,t , and investment, Is,t , to maximize the sum of future discounted real profits,
which recursively reads as follows:
(
Js,t (ks,t−1 ) =
max
ps,t ,ks,t ,Is,t ,ls,o,t
s.t.
O
X
1
ps,t
ys,t −
wto ls,o,t − Is,t −
pt
2κεI
o=1
)
η
Jt+1 (ks,t )
2
− ln(1 + πs,t ) Yt +
2κ
1 + rt+1
ks,t − ks,t−1
ks,t−1
2
ks,t−1
(18)
ks,t = (1 − δ)ks,t−1 + Is,t
1
Yt η
ps,t = fs
pt
ys,t
(19)
ν
1−ν
ys,t = zs,t ks,t−1
Ls,t
(21)
(20)
where equation (20) is the capital accumulation equation, δ is the depreciation rate of capital,
η
2κ
ln(1 + πs,t )2 Yt is the quadratic price adjustment cost. Firms face quadratic adjustment
2
ks,t −ks,t−1
1
costs on physical capital, with a standard functional form of 2κε
ks,t−1 , which
ks,t−1
I
and
leads to a variable price of capital.
The first order conditions for labor demand is:
ls,o,t =
where µp =
η
.
η−1
ps,t (1 − νs )αs,o
µp pt wto
1
1−σ
s
νs
zs,t ks,t−1
σs
(1−ν)(1−σ
σs −1+νs
s)
(1−νs )(σs −1)
ys,t
,
(22)
Defining qs,t as the Lagrange multiplier on the capital accumulation in
sector s and, hence, as the shadow price of capital, the firm’s first order condition for the
capital stock in sector s is:
Ls,t+1
ks,t
1−νs
(1 + rt+1 )qs,t = νs zs,t+1
mcs,t+1 −
"
#
2
ks,t+1
1
ks,t+1 − ks,t
ks,t+1
−
− (1 − δ) +
+
qs,t+1
ks,t
2κεI
ks,t
ks,t
18
(23)
where mcs,t+1 is the Lagrange multiplier on the production constraint and represents real
marginal cost. The first order condition with respect to investment is
qs,t
1
=1+
κεI
ks,t − ks,t−1
ks,t−1
(24)
The first order condition with respect to prices leads to the sector-specific Phillips curve:
log(1 + πs,t ) = κ
ps,t
pt
−η
(mcs,t −
Yt+1
1
1 ps,t
)+
log(1 + πs,t+1 )
µp pt
Yt
1 + rt+1
(25)
Note that our model features heterogeneous households and heterogeneous firms. Since firms
in each sector use a different production function, they also choose different capital and
investment inputs and different price of capital, qs,t , and, hence, different equity price and
dividends. Aggregation is obtained by summing profits and inputs across different sectors.
3.3. Asset Returns
Let rt denote the ex-post return on government bonds. Let vs,t be the ex-dividend price
of equity and ds,t+1 be the dividend in sector s. The real return on equity is
The no-arbitrage condition for each sector s is: vs,t =
ds,t+1 +vs,t+1
.
1+rt+1
ds,t+1 +vs,t+1
.
vs,t
The combined return on
illiquid assets is:
(1 +
rta )
=
S
X
vs,t ds,t + vs,t
s=1
At
vs,t−1
+
B g − Bt
(1 + rt )
At
(26)
Government does not issue new debt and pays interest on the constant level B g .
To obtain the return on liquid assets, we introduce a representative competitive financial
intermediary in the model which transforms illiquid into liquid assets through a technology
that operates at a proportional cost ψ. Arbitrage between the two assets coupled with a zero
profit condition on the intermediary leads to the following expression for the return on liquid
assets: rtb = rt − ψ.
19
3.4. Monetary and Fiscal Policy
Fiscal policy Gt follows a balanced-budget policy:
τt
O X
S
X
wto ls,o,t = rt B g + Gt ,
(27)
o=1 s=1
Monetary policy follows a simple Taylor-type rule:
it = rt∗ + φπ πt + φy (Yt − Yss )
(28)
where it is the monetary policy interest rate, φπ is the weight on inflation πt , φy is the weight
on output gap, rt is the real interest rate, rt∗ is the natural interest rate, which is equal to
the real interest rate in the steady state, and 1 + rt =
1+it−1
.
1+πt
Stabilization policy, by responding to the dynamics of macroeconomic variables, can
alter the role of shocks and other channels in the model. Hence, to fully assess the role of
occupational reallocation for the macroeconomic dynamics, the endogenous response of the
policy stance shall be taken into account.
3.5. SIR Block
We use the standard SIR model.4 There are three possible states in which each individual can
be: susceptible (S), infected (I), and recovered (R). The disease spreads when susceptible
and infected get in contact with each other. The number of susceptible decreases because they
get infected. The number of infected declines as they recover. Recovered people are immune
to the disease. The only salient modification is the inclusion of occupation-specific risks, with
4
See, for example, Heathcote et al. (2020).
20
face-to-face occupations featuring higher infection risk. The mathematical representation of
the model is as follows:
St It
− εpandemic , S0 = 1
Nt
St It
− γsir It + εpandemic , I0 = 0
= It + βsir
Nt
St+1 = St − βsir
(29)
It+1
(30)
Rt+1 = Rt + γsir It ,
R0 = 0
(31)
Nt = St + It + Rt
(32)
where βsir is the contact and infection rate, γsir is the probability of recovery within the
unit of time, and
βsir
γsir
is the average number of contacts the infected person has during the
infectious period.
The SIR block connects to the HANK model through the aforementioned decrease in the
supply of labor hours – via ζto as described in eq. (14). Our model allows accounting for the
households behavioural response to the infection shocks. The model features an infection
risk that differs by occupation. In response to the infection shock, households adjust their
consumption-saving and occupational decisions, which in turn affects the pace of infection
spread.5
3.6. Market Clearing: Labor, Goods and Asset Markets
Wages are determined in equilibrium by equating labor supply and demand for each occupation,
hence:
Lo,Demand
t
=
S
X
ls,o,t =
not ζto
s=1
|
5
{z
Demand
J
X
Z
mj
et θj (o|et , at−1 , bt−1 )dDj (et , at−1 , bt−1 ) = Lo,Supply
(33)
t
j=1
}
|
{z
Supply
}
See Boppart, Harmenberg, Hassler, Krusell and Olsson (2020) on the time use choice during pandemics,
Farboodi, Jarosch and Shimer (2020), Alfaro, Faia, Lamersdorf and Saidi (2020), and Engle, Keppo,
Kudlyak, Smith and Wilson (2020) on the importance of accounting for the behavioural response in in
the pandemic path, Barry (2020) and Barro, Ursúa and Weng (2020) on evidence on absenteeism and low
participation in pandemics.
21
Market clearing takes place at the occupation level. As noted earlier, workers are indifferent
among sectors, hence, there is full mobility and wage equalization across sectors.
Aggregate good supply is equal to aggregate good demand, hence:
Yt + hLO+1
= Ct + Gt + It + ψBt + Φt ,
t
(34)
where ψBt is the resource cost from liquidating assets, LO+1
is the number of individuals in
t
the non-employment state, and where all variables above are aggregates obtained through
the joint distribution, Dt , defined above.
Finally, asset markets clearing implies
At + Bt =
S
X
vs,t + B g ,
(35)
s
where again aggregation is obtained through the joint distribution Dt .
3.7. Equilibrium
Definition 1: Competitive Equilibrium
A Competitive Equilibrium of the economy satisfies the following definition:
• The sequence [ct , at , bt ]∞
t=0 solves households’ consumption-saving decision in eq. (3),
given the distribution of idiosyncratic shocks, P (et+1 |et ) and the sequence of prices
rta , rtb , rt , wt .
• The sequence of probabilities [θt ]∞
t=0 solves households’ occupational choice problem in
(6).
• Aggregate assets holdings and consumption of the households are equal to the product
of the individual optimal functions and the distribution of households across occupations
and assets.
• Firms choose labor demand for each occupation, ls,o,t , and capital inputs, ks,t to solve
discounted profits optimization, given in section 3.2.
22
• The fraction of the population that is infected, susceptible or recovered is determined
by the system of difference equations in eq. (30).
• Market clearing and the aggregate resource constraints are satisfied.
• Monetary policy determines the short term interest rate according to eq. (28) and fiscal
policy follows a balanced budget rule as in eq. (27).
4. Analytical Results
In this section, we provide some analytical results that highlight the role of technological
parameters (sectoral output shares, sectoral occupational shares, job specialization) on the
extent of reallocation, on job rents and on aggregate output. Our focus here is mostly on how
the technological parameters affect reallocation through cross-sectional and overall changes
in labor demand. For analytical tractability the focus here is on long run results.
Proposition 1. Determinants of Allocation of Labor across Occupations and
νs
s
L1−ν
Sectors. Consider an economy in which sectoral output is given by ys,t = zs,t Ks,t−1
s,t ,
η1
sectoral prices follow ps,t = fs yYs,tt
pt , labor aggregation across occupations is given by
P
σ1
s
O
σs
α
l
Ls,t =
and aggregate labor input demand is given by Equation (A.10). For
s,o
s,o,t
o=1
given wage distribution, the allocation of effective units of labor across occupations depends
on the following primitives: αs,o across all s and o and σs . The allocation of labor shares
across sectors depend on the following primitives: αs,o , σs , νs , zs,t and fs across all s.
Proof . See Appendix A.
The allocation of labor shares is determined by labor demand and its primitives. Those
are the weights in the production function, αs,o , νs and fs , the sectoral productivity, zs,t , and
the job substitution of specialization, σs . More specialized jobs reduce the elasticity of labor
demand to wage. In Appendix A labor shares across occupations are shown to depend also
on the wage distribution, with the latter being affected by labor supply primitives, namely Γ.
Hence, the result for occupational labor shares in Proposition 1 is partial equilibrium holding
for a given labor supply.
23
Proposition 2. Occupational Specialization and Rents. Consider the economy
described Proposition 1. The elasticity of labor demand with respect to wages in occupation
o
o is εol,w = σs1−1 1 − PO ls,olw 0 wo0 and declines with σs (εol,w = σs1−1 with infinite number of
o0 =1 s,o
occupations) and the mark-up or rent extracted by each occupation are given by the inverse of
ls,o wo
l
w0
o0 =1 s,o0 o
ls,o wo
σs − PO
l
w0
o0 =1 s,o0 o
1− PO
µw =
(with infinite number of occupations µw =
εl,w
εl,w +1
=
1
).
σs
Proof . See Appendix A.
The above proposition highlights the role of job substitutability,
1
,
σ
on reallocation as
driven by cross-sectional changes in labor demand. Higher σ increases substitutability of
occupations for firms and the elasticity of labor demand to wages. This is the sense in
which lower substitability represents a mobility cost (see Kennan and Walker (2011)). This
implies that any cross-sectional variation in wages triggers less cross-sectional changes in
labor demand, hence less reallocation (see also our simulation results in section 7.1). It also
implies that, since job specialization decreases, job rents or mark-up decrease too.
Proposition 3. The Role of Reallocation for Output. In the framework described
P
in Proposition 1, given the normalization O
o=1 αs,o = 1 ∀s, a rise in σ, raises aggregate
output.
Proof . See Appendix A.
The results from Proposition 3 target overall changes in labor demand following changes
in σ. Changes in the latter also modify the capital-labor complemetarities or substitutability,
and through the latter changes in aggregate labor demand.
5. Model Parametrization and Solution Method
5.1. Skills, Occupations and Skills Distribution
A key element of the reallocation channel is the matrix Γ of skills distribution. Each vector
of this matrix provides the set of talents that each household type J has for each occupation.
Hence, each element of the matrix, γjo , provides the comparative advantage, in terms of
24
efficiency units of labor, of each household j for occupation o. O-NET data are employed to
calibrate this matrix, as it is the best suited to quantify the concept of task-specific human
capital (Gathmann and Schönberg (2010)). First, to reduce the curse of dimensionality the
3-digit detailed occupations are grouped into 8 with similar job requirements using k-means
clustering. Second, the entries of the matrix Γ are filled using the skill-requirements vectors
provided in O-NET.
Specifically, first occupations are clustered using O-Net information on the importance of
detailed skill, ability and knowledge requirements for 968 unique occupations. Each detailed
requirement is ranked by its importance on a scale from 0 to 7.6 A k-means clustering
algorithm is used to identify clusters of 3-digit occupations with similar requirements.
Mathematically the k-mean algorithm can be summarized as follows. Let M be distinct
elements in the vector of job requirements reported by O-Net. Let G denote a set of detailed
occupations in O-Net, each characterized by the M-dimensional vector of requirements hg .
Given the set of observations (h1 ,h2 , ..., hG ), k-means clustering aims to partition the G
observations into k (≤ G) sets O = (O1 , O2 , ..., Ok ) so as to minimize the within-cluster sum
of squares (i.e. variance). Formally, the objective is to find:
arg min
O
k X
X
kh − mi k2
(36)
i=1 h∈Oi
where mi is the mean of points in Si . k is set to 8. The resulting 8 occupational clusters
are summarised as follows: (1) Manual trade occupations, (2) Management and supervisory
occupations, (3) Machine operators, (4) Engineering occupations, (5) Healthcare and community occupations, (6) Personal services, (7) Technical-support occupations, and (8) Office
and administrative support.
Second, the entries of the Γ matrix are filled with a comparative advantage index constructed
from O-NET skill-requirements. Specifically, the number of household skill-types is set equal
to the number of occupations and their related required skills. Each household j holds
an absolute advantage for the occupation along the diagonal. Next, a measure of skill6
For example, O-Net contains information on 52 different elements within abilities - such as ”Problem
Sensitivity”, ”Mathematical Reasoning” or ”Manual Dexterity”. Skill requirements are ”Management of
Financial Resources”, ”Social Perceptiveness” or ”Complex Problem Solving” etc.
25
transferability is constructed as the Euclidean distance between skill of household j (required
in occupation j = o) and skill required in occupation o0 . In the data the skill distance between
occupational clusters o and o0 is the sum of the weighted (by the employment shares in 2019)
absolute difference between the elements of Ho and Ho0 :
γjo =
M
X
abs hjm − hom ,
(37)
m
This implies that the diagonal entries are the smallest.7 Next, to translate the Euclidean
distance into a comparative advantage, we re-normalize every row so that the diagonal entries
become all ones. This implies that in the final matrix the higher the entry value, the highest
is the household comparative advantage for that specific occupation. The characterization
done so far captures the concept of horizontal or core-task specialization. In the data
however occupations are also hierarchical. To capture also vertical specialization, each entry
is multiplied by the share of occupations in cluster o with the same or higher educational
attainment. The latter is computed as the mode of educational attainment for all occupations
in the cluster using the 2018 BLS data and considering three levels: (1) less than college,
(2) college, and (3) more than college. While the above procedure is used to calibrated
the baseline matrix, our simulations also consider alternatives to that. This is done both
for robustness reasons and also to assess the role of counterfactual skill-distribution on the
reallocation channel. Our matrix Γ in the baseline calibration looks as following:
Table 1: Skill-distribution Γ matrix. Each row represents one type of households. Each column
represents skills that different types of households have. The higher the entry in the matrix, the more skilled
is the type for that occupation. Note that the diagonal has been normalized to one so as to make all other
entries comparable in relative terms.
7
Type of households
Occ. 1 Occ. 2
Occ. 3
Occ. 4
Occ. 5
Occ. 6
Occ. 7 Occ. 8
Type
Type
Type
Type
Type
Type
Type
Type
1
0.022
0.38
0.018
0.037
0.14
0.18
0.083
0.26
0.019
1
0.014
0.03
0.11
0.23
0.085
0.33
0.27
0.28
1
0.27
0.49
0.32
0.42
0.38
0.57
0.34
0.45
1
0.58
0.42
0.42
0.54
0.12
0.48
0.13
0.25
1
0.45
0.37
0.19
0.029
0.35
0.017
0.038
0.12
1
0.14
1
2
3
4
5
6
7
8
0.34
1
0.32
0.59
0.58
0.57
0.46
0.68
0.21
0.097
0.24
0.07
0.12
0.22
0.35
1
Note that the entries are never zeros: since those are grouped occupations the set of skills is an average
across all the occupations in the cluster.
26
At last, in the baseline parametrization, the shares of households with skill j is distributed
uniformly, i.e., µj = 1/J = 1/8 for all j ∈ J.
5.2. Sectors and Production Function
Industries are aggregated into 8 broad NACE sectors. The weights on the sectoral output in
the representative consumption bundle, fs , are set according to the share of sectoral output
in the total output using data from KLEMS. Their values are in Table 2. The sectoral capital
and labor shares are also obtained from KLEMS using the long-run averages from 1987 - 2015.
Table 8 in the Appendix provides the list of sectors and their capital and labor shares. In
the model capital and labor shares are used to calibrate the parameter νs . In the one-sector
model νs is calibrated so as to deliver a capital share of 0.40. The latter is obtained from
the average of the sectoral shares weighted by fs . The baseline σs is set to 0.2 for all s.
The parameter is then varied in the simulations to assess the impact of job specialization
νs 1−νs
on reallocation. The αs shares are calibrated as follows. Production, ys,t = Ks,t
Ls,t , and
P
σ1
s
O
σs
aggregate labor demand, Ls,t =
, are substituted in the firms’ first order
o=1 αs,o ls,o,t
conditions for labor:
1
νs −νs
(1 − νs )Ks,t
Ls,t αs,o
σs
Then taking the ratios,
αs,o
αs,o0
=
ls,o,t
ls,o0 ,t
1
αs,o0
1−σs
O
X
! σ1 −1
s
σs
αs,o ls,o,t
σs −1
σs ls,o,t
= wo
(38)
o=1
wo
wo0
and summing up delivers:
1−σs
O
X
ls,o,t
wo
=
ls,o0 ,t
wo0
o=1
(39)
From the BLS Occupational Employment Statistics (OES) for the 2014-2019, data on wages
for each of our occupational clusters are extracted and, once weighted by employment in each
occupations, they are used to calibrate the steady state real wages in occupational clusters
1-8. See their value in Table 2. The same employment shares and wages are then used in
equation 39 to calibrate the αs,o .
27
Table 2: Sector and occupation share. Steady State Wage Distribution is calibrated on OES-BLS.
Sector shares are calibrated on KLEMS data. Occupation weights α are obtained through steady state
solution of model’s equation and by using data for wages in each occupation.
Name
Symbol
Data
Sector shares
Occupation shares
Wage distribution
fs
αs,o
wo
0.19, 0.18, 0.03, 0.15, 0.24, 0.03, 0.16, 0.02
0.078, 0.254, 0.092, 0.280, 0.035, 0.085, 0.070, 0.106
14.1, 40.8, 16.7, 46.3, 6.2, 14.9, 12.8, 18.9
5.3. Other Parameters
Time is in quarters. Most of the other parameters are set according to values generally used
in models with heterogeneous agents and uninsurable risk. In the most cases values are set
to reach certain targets for asset distributions. The steady state interest rate on government
bonds is calibrated to 0.0125, close to the 5-year average of the Effective Federal Funds Rate
(see Auclert et al. (2019)). The time discount factor, β, is then obtained from the asset
market clearing condition. The portfolio adjustment cost parameter χ1 is set to 6.19 to
match the quarterly interest rate of 0.0125 and the debt to output ratio of 1.04. Following
Auclert et al. (2019) the portfolio adjustment costs parameters are set as follows χ0 = 0.25
and χ2 = 2 and the steady state mark-up is set to 1.015 in order to match the steady state
sum of liquid and non-liquid assets, which in our model is 16.26. The steady state level of
government spending is set to 0.2 of GDP (a widely used ”great ratio” from the literature).
The value of taxes is then obtained from the fiscal policy rule and is equal to 0.4. In the
baseline we fix the interest rate. Impulse responses are also shown under the Taylor rule
with 1.5 for the weight on the inflation gap and 0 for the weight on the output gap. The
slope of the Phillips curve, the capital depreciation rate and the capital adjustment costs are
calibrated to 0.1, 0.02, and 4, respectively, following both Kaplan et al. (2018) and Auclert
et al. (2019).
The flow value of non-employment, ht , is calibrated using data from Chodorow-Reich and
Karabarbounis (2016) who estimate the opportunity cost of employment using a wide range
of data sources (Consumer Expenditure Survey (CES), the Panel Study of Income Dynamics
(PSID) and the Current Population Survey (CPS)). Under separable preferences, which is
the case considered in our model, they report a ratio of opportunity cost of employment to
28
Table 3: Parameter Values, Description and Source
Parameter
Description
Skills and Occupations
O
J
mj
Γ
Final Composite Good
S
Pt
Value and source
Number of occupations
Number of skill types
Distribution of skill types
Skill transferability matrix
8, clustered by k-means
8, clustered from O*NET
1/J (uniform across J)
See Section 5.1
Number of sectors
Aggregate Price
1 (2 in 7.4)
Normalized to 1 in the
steady state.
0.2 (baseline)
0.4, KLEMS, Section 5.2
OES-BLS, see Section 5.2
I
Elasticity of substitution between occupations
Capital share
steady state wage per efficiency unit in occupation o
Capital depreciation
Capital adj. parameter
β
χ0
χ1
Time discount factor
Portfolio adj. cost pivot
Portfolio adj. cost scale
χ2
σ
ρ
ρz
σz
ht
Portfolio adj. cost curvature
EIS
Inverse Frisch elasticity
Autocorrelation of earnings
Cross-sectional std of log earnings
Flow value of non-employment
ϕ
Dis-utility parameter
0.979, see Section 5.3
0.25, Auclert et al. (2019)
6.19 (target Bh = 1.04Y , Auclert et al. (2019))
2, Auclert et al. (2019)
0.5 Auclert et al. (2019)
1 Auclert et al. (2019)
0.966, Auclert et al. (2019)
0.92, Auclert et al. (2019)
47% of average income, see
Section 5.3 Auclert et al.
(2019)
1.71 (target n = 1)
Real interest rate
Liquidity premium
steady state markup
0.0125, Auclert et al. (2019)
0.005, Auclert et al. (2019)
1.015, Auclert et al. (2019)
Coefficient on inflation in Taylor rule
Coefficient on output gap in Taylor rule
Tax rate
Bond supply
Slope of the Phillips curve
1.5, Auclert et al. (2019)
0, Auclert et al. (2019)
0.401, Auclert et al. (2019)
2.8, Auclert et al. (2019)
0.1, Auclert et al. (2019)
Contact rate
1.5 Eichenbaum, Rebelo and
Trabandt (2020b)
0.8
See Section 5.3
Production Function
σs
νs
wo
δ
0.02, Auclert et al. (2019)
4, Auclert et al. (2019)
Households
Asset Markets
r
ψ
µp
Monetary and Fiscal Policy
φ
φy
τ
Bg
κ
SIR Block
βSIR
γSIR
ϑo
Inverse of the infectious period
Occupation infection risk
29
the marginal product of employment (47%). This value is then multiplied by our average
steady state income for all employed households and delivers a value of 0.16. For comparison,
households’ steady state income (see eq. (1)) varies in our model between 0.06 and 1.15. This
opportunity cost includes unemployment insurance and other benefits like Medicaid as well
as foregone value of non-working time expressed in units of consumption.
Finally the SIR block is calibrated as in Eichenbaum et al. (2020b). Their basic reproduction
number, which is set to 1.5, is used to calibrate βSIR and γSIR , which is then set to 0.8.
At last, ϑo is parameterized according to the average infectiousness of the occupations in
our broad occupational clusters. Infectious risk is measured using O*NET data and is as
follows: 0.037, 0.031, 0.034, 0.029, 0.043, 0.036, 0.039, 0.033. Table 3 contains the list of
those parameters.
At last, shocks in the numerical application are calibrated as follows. The infection shock
is initiated through an increase in εpandemic of 10−4 . The occupational shock is done by
shrinking the occupational weight in occupation 7 (retail, sales, etc.) by 60% and by raising
the occupational weight in occupation 5 (health) by 10%. The persistence of the shocks is
set to 0.7.
5.4. Solution Method
We solve the model using the Sequence Spaced Jacobian Solution Method built by Auclert et
al. (2019). The method accounts for the impact of shocks on expectations. This is important
in our model because it features (1) an endogenous policy response and (2) a SIR block, which
brings the economy from one steady state to another. We extend the solution method to
include endogenous occupational choice. To this purpose we add a guess-and-verify procedure
for solving the households’ two-stage optimization problem. Specifically, using a guessed
value function, we obtain the policy functions for the consumption-saving problem for each
occupation. The latter are then substituted into the value functions to solve for the optimal
discrete occupational choice. The guess is updated until convergence. Appendix B contains a
detailed description of our algorithm. Note that the equilibrium is not locally indeterminate
30
based on the winding number check introduced by Auclert et al. (2019), which is analogous
to the Blanchard-Kahn condition.
6. Quantitative Results
In this section our model is simulated to assess its quantitative properties. We start with
the steady state results, the goal being of assessing the empirical fit of the model to the
long-run cross-sectional distributions, with a focus on occupational sorting, and its link to
inequality, and the matching of the model-generated income and wealth distributions to the
data. Second, we simulate the dynamic response of the model under a standard 25 basis
point contractionary monetary policy shock and compare the results to the ones from a model
without occupational mobility. Our goal is to describe the novel channels in our model.
6.1. Steady State Results
6.1.1. Occupational Sorting
Table 4 shows effective labor hours in the steady state supplied in each occupation by
each household type. The rows correspond to different household types, and the columns
correspond to different occupations. Each entry equals the effective labor hours supplied
in each occupation aggregated across idiosyncratic income shocks and asset holdings, i.e.
R o
γj et θj (o|et , at−1 , bt−1 )noj,t dDj,t (et , at−1 , bt−1 ). We find that, first, households supply most
hours in the occupation in which they have the highest comparative advantage (i.e., in each
row, the entry on the diagonal of the matrix is the highest). Second, across occupations,
effective labor supply is higher in occupations with higher wages, namely occupation 2
(Management) and occupation 4 (Engineering).
The probability of choosing an occupation depends on skills and wealth. In our model,
in steady state asset-rich individuals are effectively indifferent between non-employment
or work in any of the eight occupations. In contrast, we find a much tighter link between
labor income and occupational choice for the asset-poor individuals. Table 5 shows the
steady state probabilities of choosing an occupation for households in the lower ten percent
31
Table 4: Effective Labor Supply by Occupation and Household Type, in the Steady State.
Each matrix entry shows the effective labor hours supplied inReach occupation in the steady state aggregated
across idiosyncratic income shocks and asset holdings, i.e. γjo et θj (o|et , at−1 , bt−1 )noj,t dDj,t (et , at−1 , bt−1 ).
Each row represents a household’s skill type. Each column represents an occupation. The diagonal entries
represent households’ types with highest level of occupational skills. See text for the definition of households’
types and how they relate to the occupations. The last row presents the sum across household types.
Household Type
Occ. 1 Occ. 2
Occ. 3
Occ. 4
Occ. 5
Occ. 6
Occ. 7
Occ. 8
Type
Type
Type
Type
Type
Type
Type
Type
0.151
0.109
0.116
0.109
0.109
0.110
0.111
0.110
0.123
0.163
0.123
0.133
0.135
0.133
0.128
0.139
0.114
0.110
0.152
0.110
0.110
0.111
0.113
0.111
0.125
0.122
0.123
0.164
0.124
0.130
0.124
0.127
0.113
0.121
0.112
0.116
0.146
0.121
0.114
0.115
0.123
0.111
0.120
0.111
0.113
0.151
0.119
0.116
0.111
0.109
0.115
0.109
0.109
0.110
0.150
0.110
0.113
0.112
0.114
0.111
0.111
0.113
0.117
0.153
0.926
1.076
0.931
1.039
0.959
0.963
0.922
0.945
1
2
3
4
5
6
7
8
Sum across types
of the wealth distribution for three different realizations of the idiosyncratic income shock.
First, regardless of the realization of the income shock, the households choose occupations
with the highest wages (occupations 2 and 4). For example, Böhm, Gaudecker and Schran
(2019) document the empirical link between high wages and expanding occupations. Second,
households with the low income realization are more likely to choose non-employment than
the households with high or medium realization of income shock. Third, for the households
with the high income realization skills play a more important role in their occupational
choice as compared to the households with lower income realization, who choose occupations
with the highest wage. This is consistent with Gathmann and Schönberg (2010) who find
that occupational switching is more likely to occupations with higher wage growth or higher
skill price. Finally, the patterns featured in our model are consistent with the statistics on
employment patterns across the wealth distribution as reported in Mustre-del Rio (2015)
from the National Longitudinal Survey of the Youth (NLSY) data. First, heterogeneity in
labour shares in Table 5 is not all driven by wages, but is also associated with skills and
wealth. Second, the income-rich households in Table 5 have larger employment shares (lower
non-employment shares) than the income-poor individuals. On the other hand, the assetrich individuals in our model, due to their better ability to smooth consumption, are more
inclined to choose non-employment than the asset-poor individuals.
32
Table 5: Occupational Choice of Asset-Poor Individuals, in the Steady State. The table shows
the probabilities of choosing occupations and the non-employment state for the asset-poor households,
aggregated across all skill types. The asset-poor are those at the bottom 10% of both the liquid assets
distribution and the illiquid assets distribution.
Income shock
Low
Medium
High
Occ. 1 Occ. 2
Occ. 3
Occ. 4
Occ. 5
Occ. 6 Occ. 7 Occ. 8 Non-employment
0.095
0.107
0.114
0.100
0.117
0.116
0.172
0.140
0.129
0.086
0.084
0.096
0.097
0.113
0.115
0.167
0.138
0.127
0.093
0.102
0.114
0.105
0.121
0.117
0.086
0.078
0.072
6.1.2. Income and Wealth Distributions
Table 6 shows the income and asset distribution statistics in the model and in the data.
We compute the empirical statistics from the Survey of Consumer Finance (SCF). Liquid
assets consist of the following categories: transactions accounts, directly held bonds, directly
held stocks, and credit card balances. Non-liquid assets consist of certificate of deposits,
savings bonds, cash value of insurance, other managed assets, retirement accounts, stock
holdings, and primary residence net of mortgage home loans. To facilitate the comparison
with the steady state numbers in the model, we average the SCF statistics across all available
years, 1989-2019. We compute the average asset holdings by different wealth and income
percentiles.
The model captures well the average holdings of liquid and illiquid assets (the first two
rows in Table 6). The model’s Gini coefficients compare well to the empirical Gini coefficients
reported in the World Inequality Database and Kaplan et al. (2018). Next, the model matches
fairly well the mean assets holdings by different percentiles of the income distribution. The
main drawback lies in the top 10% for illiquid asset holdings, which is larger in the data.
Overall the matching is encouraging given that the labor income distribution is endogenous
in the model. The SCF data also contain information on income and assets by broad
occupational groups. We compare the SCF group “Managers” to our occupations 2 and 4,
and the SCF group “Technical, Sales, Services” to our occupations 5, 6 and 7. The ratios of
mean income to liquid assets in the model compare well to the data.
Appendix F contains plots of wealth and wage distributions. The wealth distributions are
skewed. The distribution of liquid assets exhibits a peak at zero, which corresponds to the
33
Table 6: Income and Asset Distributions, Model’s steady state and Data. The model statistics
are calculated at the steady state. Income statistics from the model includes pre-tax labor income and
capital gains but do not include inactivity benefits. Data on income, liquid and non-liquid assets are from
the Survey of Consumer Finance, averages over 1989-2019. Liquid assets consist of transactions accounts,
directly held bonds, directly held stocks, and credit card balances. Non-liquid assets consist of certificate of
deposits, savings bonds, cash value of insurance, other managed assets, retirement accounts, stock holdings,
and primary residence net of mortgage home loans. The deciles for illiquid assets holdings are based only on
the value of primary residence. The data on Gini coefficients are from the World Inequality Database and
Kaplan et al. (2018).
Statistics
Data
Wealth distribution
Mean Liquid Assets/GDP
Median Illiquid/GDP
Gini coefficients
Income
Liquid assets
Illiquid assets
Shares of liquid assets per income percentile
less than 20th percent.
20th-40th percent.
40th-60th percent.
60th-80th percent.
80th-100th percent.
Shares of illiquid assets per income percentile
less than 20th percent.
20th-40th percent.
40th-60th percent.
60th-80th percent.
80th-100th percent.
Income/Liquid Assets, by Occupation
Managers and Professionals
Technical, Sales and Services
34
Model
0.26
2.92
0.26
3.80
0.52
0.98
0.81
0.39
0.71
0.50
0.05
0.10
0.08
0.13
0.63
0.04
0.13
0.12
0.21
0.39
0.07
0.09
0.11
0.15
0.57
0.06
0.07
0.15
0.29
0.28
1.80
2.74
1.48
3.92
share of the hand-to-mouth households in the model. In our simulations at the recession’s
trough, the upper tail becomes thinner and the lower tail fatter. This is even more so for the
non-liquid assets distribution.
6.2. Dynamic Responses to Standard Monetary Policy Shock
In this section, we simulate the model’s response to a standard 25 basis point contractionary
monetary policy shock. We then compare our model to a model without mobility, where the
occupational choice remains fixed at the steady-state values. Our goal is to characterize the
channels that arise in the model with mobility as compared to a model without mobility.
Figure 1 displays the impulse responses for selected variables, namely GDP, investment,
consumption, policy rule, aggregate average wage (weighted by shares in each occupation),
total effective labor hours, inflation, the wage Gini, effective labor hours across occupations
(only in the model with occupational mobility), illiquid and liquid assets. As expected the
shock is contractionary. The baseline channels noted elsewhere in the literature studying
heterogeneous agents models with uninsurable risk and with nominal rigidities are operative
here as well. Aggregate consumption falls due to precautionary saving. The latter also induces
a flight to liquidity, as households substitute non-liquid with liquid assets. Investment falls due
to the paradox of thrift – as households save more, demand and output decline. The monetary
transmission is characterized by the typical three channels — the earnings heterogeneity
channel, the Fisher effect and the interest rate exposure channel (see Auclert (2019)). The
first channel implies that households reduce consumption in response to the decline in income
or changes in interest rates, the more so the higher their marginal propensity to consume.
The second channel accounts for changes in demand induced by deflationary/inflationary
expectations. The third channel produces changes in the asset distribution, which vary across
households according to their interest rate exposure.
The model with occupational mobility is characterized by two additional channels. First,
the decline in wages induces some workers to opt out of employment and this may worsen the
contraction. This channel is playing an increasing role particularly in recent recessions, which
were followed by jobless recoveries. While the size of the shock considered here is rather
35
Figure 1: Impulse Responses to a Monetary Policy Shock. The figure shows the percentage
deviations of the specified series from the steady state in response to a monetary policy shock. The shock is a
25 b.p. increase in the Central Bank’s interest rate; the shock follows an AR(1) process with the persistence
parameter 0.7, such that the shock almost dissipates by the 13th quarter. The figure shows the impulse
responses from the model with the occupational mobility (in blue) and from the model without it (in orange).
Central Bank follows the Taylor rule. The average wage is the weighted average wage across occupations
where the weights are occupational employment shares in terms of efficiency units. The effective labor hours
is the sum of the labor hours across occupations. Wage Gini is presented in absolute values. Occupational
labor dynamics are shown in the last panel, where each line corresponds to the effective labor hours in the
specified occupation. X-axis shows the time in quarters.
36
small, this channel may play a more substantial role in larger recessions. Second, workers
can reallocate from shrinking to expanding occupations. The reallocation channel can tame
a recession. In this particular shock the second channel seems to prevail. However, more
generally whether the first or the second channel prevail may vary (as we will see below),
depending on the type of shock and its magnitude, the policy stance and other structural
characteristics of the economy. In particular, the reallocation channel is amplified under
sectoral or occupational shocks like the ones we consider below in the pandemic application.
Note that even in the model with fixed probabilities some changes in labor hours take
place due to changes in labor demand and in labor supply, through the impact on labor
dis-utility. Still, in our model changes occur more realistically through the extensive margin
and, despite the small size of the shock, those matter in driving the macro dynamic.
The reallocation channel has further implications for earnings inequality. The wage Gini
(first panel, last row) fluctuates more in the model with mobility. Generally speaking, while
the recession in the model with reallocation may be tamed, the income inequality tends to
worsen.
The aggregate shock described above is not the ideal setting in exploiting the properties of
our model: it is an aggregate small shock which evenly hits all sectors. Our occupational
reallocation mechanism can be best exploited in a setting with a large recessionary shock,
pushing many workers out of employment, and unevenly hitting the economy. For this reason
the model is applied next to the recent pandemic recession, with the latter features a marked
reallocation channel through a unique combination of uneven and persistent occupational or
sectoral shocks and aggregate labor supply shocks.
7. An Application of the Model to the Pandemic
Recession
Our model provides a laboratory for studying the role of occupational reallocation in
the economy’s response to shocks. We apply the model to the pandemic recession. The
pandemic recession involves two types of shocks - a negative aggregate labour supply shock
37
due to the infections and a demand shock triggered by government-mandated lockdown.
The role of occupational mobility differs under these two shocks. Under the infection
shock, all occupations are hit, and little occupational reallocation takes place. Under the
government-mandated lockdown, occupations are hit differentially, which triggers occupational
reallocation.
We, first, examine the effect of the occupational demand shocks and the aggregate supply
shock separately, and then jointly. Studying model under each shock separately helps
demonstrate the model’s mechanism.
To investigate the role of skill mismatch and skill transferability, we simulate the model
under different degrees of job specialization, which is captured by the elasticity of substitution
across occupations in the production function, σ, and under different degrees of workers’
skill transferability across occupations, which is captured by the skill-distribution matrix, Γ.
This analysis helps understand how the economies with different distribution of skills and
educational attainment or with different production structure respond to various shocks.
While sticky prices are not essential for studying occupational reallocation and the economy’s response in the long-run, price rigidity affects the economy’s short-run response to
shocks and stabilization policy. To dissect the role of policies, we, additionally, examine the
model’s response under different monetary policies - the standard Taylor rule and the zero
lower bound policy.
We begin our investigation by studying reallocation in an eight-occupation one-sector
model. We then simulate an eight-occupation two-sector model, which allows examining the
effect of sector-specific in addition to the occupational shocks. To demonstrate the role of
occupational mobility, we compare the results from our model with occupational mobility to
a HANK model without occupational mobility.
7.1. Occupational Shocks
The government-mandated lockdown during the pandemic of nonessential sectors resulted
in a positive demand shock for some occupations and a negative demand shock for others.
While the lockdown concerned sectors and not occupations, the resulting shock manifested
38
itself through occupations. For example, the shutdown of dining-in restaurants resulted
in a negative demand shock to waiters but a positive demand shock for delivery workers.
Another important feature of the occupation-specific shocks during the pandemic is the
differential persistence of these shocks across occupations. While the lockdowns were shortterm, the resulting occupational demand shocks might be persistent and the demand for
some occupations might not return to their initial levels. Firms might automate, employ
new technologies, and reorganize their production. For example, Autor and Reynolds (2020)
argue that the pandemic has poised to reshape labor markets due to telepresence, urban
de-densification, employment concentration in large firms, and general automation forcing,
and that those trends will induce disappearance of certain occupations, mostly face-to-face
ones.8 Our model allows us studying the economy’s response to the various occupational
shocks with differential persistence.9
We model the occupational shock as an exogenous and persistent change in occupationspecific shares in the production function, αs,o . Our baseline occupational shock is a
combination of shocks to two occupations: a 10% increase in the production share of
occupation 5 (healthcare) and a 60% decline in the production share of occupation 7 (which
includes service-related and sales occupations). The large close down of one occupation
is meant to mimic the stringency measures that applied to most nonessential services. In
most countries about 60% of the economic activity had been restricted. We model each
occupational shock as an AR(1) process with persistence 0.7, so that the shock dissipates
after 13 quarters. We simulate the shock in the economy with 8 occupations and one sector.
For our baseline results, the monetary policy has a fixed interest rate (which we would refer
as the zero lower bound case), as this was the scenario enacted in response to the pandemic
recession.
Figure 2 shows the dynamics for selected aggregate variables, inequality measures and
occupational mobility. Figure 2 shows GDP, investment, consumption, the policy rate,
8
9
Chernoff and Warman (2020) examine regional and country variation in jobs concentrated in high-risk
and highly-automatable occupations and find that these occupations exhibit the highest likelihood of
disappearing post-pandemic.
Moll, Rachel and Restrepo (2021) studies the effect of automation on wealth inequality in an Aiyagari-style
model but without occupational choice. Our model allows studying the effect of automation on income
and wealth inequality allowing workers to change their occupation when their jobs get automated.
39
aggregate wage (average wage across occupations, weighted by the occupational labor shares),
total effective labor hours, the share of population in the non-employment state, the wage Gini,
dynamics of wages in different occupations, liquid assets and non-liquid assets, occupational
choice probabilities and the share in the non-employment state. All series (except the
occupational choice probabilities and the wage Gini coefficients) are in the percentage
deviations from their steady state levels. The figure shows the impulse responses from the
model with occupational mobility (blue lines) and without occupational mobility (orange
lines).
First, the fall in the GDP is sizable. In most economies the fall in GDP was larger, but it
was due to a combination of recessionary shocks or to stricter stringency measures than the
ones simulated here. Beyond that, the fall in GDP, observed in various countries, was also
likely driven by which sectors were hit by stringency measures and how large are those sectors.
For instance Figure 10 in Appendix E shows the impact of a 60% closure of occupations
in the second cluster, this includes sales, education, other business, but also managerial
positions in finance, hence also other likely sectors which were hit by the lock down measures.
As this sector accounts for a large share of the economy, a similarly sized shock as in the
baseline, but in occupation 2, has a much bigger impact on GDP. The decline in this case is
around 18%. To sum up, the exact decline of output is likely to depend on many factors
which may vary across economies and that can be accommodated within our model.
Second, as before the standard channels of models with uninsurable risk are also at work
here. Precautionary savings reduces consumption and induces a flight to liquidity, the paradox
of thrift induces the investment decline. Beyond that, the occupational shock induces a
reallocation from shrinking to expanding occupation, which should tame the recession. On the
other side, the increase in income dispersion may deepen the fall in demand, by sharpening
precautionary saving motives, and the large fall in wages induces some workers to opt of the
labor force.10 The last two forces prevail, and the recession is deeper as compared to the
model without mobility.
Two observations are worth noting here in relation to the reallocation mechanism.
10 The model without mobility also features a change in labor hours via intensive labor hours choice.
40
First, the dynamics of the earnings inequality depend on the occupations that are hit. We
analyze the negative shock to occupation 7 (retail, sales, etc.) which has a low wage. On
the other hand, if high-wage occupations face negative shocks, we see a decrease in the wage
Gini (for example, if the shock occurs in the occupation 2 (Management)). Figure 11 in
Appendix E for instance show that while inequality raises (wage Gini raises) when the shock
hits a high wage occupation (occupation 2 employing mainly managerial services in various
sectors), the opposite is true when the shock hit a low-wage occupation (occupation 1 for
instance, namely construction). While the nature of this exercise is purely suggestive, its
implications may be relevant for other applications of our model. For instance, the raise in
inequality associated with the 2007-2008 financial crisis is likely induced, through the lens of
our model, by the prevalence of the shocks in low-wage occupations.
Second, reallocation induces a cost proportional to the distance in skill-requirements
between past and new occupations. To fix ideas consider workers, whose talents are closer to
the skill-requirements of occupation 7. The closing of the latter forces them to reallocate
to a new occupation, whose skill-requirement is most likely be further away from their own
abilities, or to non-employment. In both cases they will face a wage cut.
Interestingly, simulations of the same shock under a Taylor rule (see Appendix E) show
that, under this alternative policy stance, the reallocation across occupations is greater. As
this channel prevails over the opt-out of employment channel, in this case the model with
mobility tames the recession compared to the one without. In the absence of the sheltering
provided by the expansionary policy workers reallocate more actively. This provides an
interesting exchange between a policy, the ZLB, that fosters demand by enhancing income
capacity, and a policy, the Taylor rule, that lubricates the labor market by activating mobility.
7.2. The Role of Monetary Policy
In our previous simulations, the monetary policy was held at the zero lower bound. This was
the policy enacted soon after the start of the pandemic.
To discuss the role of policy, Figure 3 compares the impulse responses of selected variables
for the model with occupational mobility under different monetary policy stances and in
41
Figure 2: Impulse Responses to the Occupational Shock. The figure shows the percentage deviations
of the selected variables from the steady state in response to an occupational shock. The occupational shock
is a 10% increase in α5 and a 60% decrease in α7 ; each shock follows an AR(1) process with the persistence
parameter 0.7, such that the shock almost dissipates by the 13th quarter. The figure shows the impulse
responses from the model with the occupational mobility (in blue) and from the model without it (in orange).
The average wage is the weighted average wage across occupations where the weights are occupational
employment shares in terms of efficiency units. The effective labor hours is the sum of the labor hours
across occupations. The non-employment share shows the share of households that choose non-employment.
Labor aggregator is Ls and shows the aggregated labor used in the production. Wage Gini is presented
in absolute values. Occupational wage dynamics are shown in the last row, where each line corresponds
to the wage in the specified occupation. Mobility panel shows probability (aggregated across households
and occupations) of being employed and probability of being in the non-employed state (aggregated across
households). Probability of being in the non-employed state is shown on the right axis. X-axis shows the
time in quarters.
42
response to the occupational shock. In each panel, the blue line represents the model under
the ZLB and the yellow line the model under a Taylor rule.
First, since as explained earlier the ZLB curbs the reallocation across occupations, more
workers opt out of employment (third panel, second row). Still, the zero rates have several
effects. They tame the decline in consumption demand, by raising the marginal propensity
to consume. As well understood by now, in heterogeneous agents models monetary policy
has a differential impact on agents depending on their income distribution and on their asset
allocation. Hence, the observed macroeconomic impact results from the aggregation of the
differential effects.
Investment declines less under the ZLB. Investment declines when interest rate increases
and when the marginal productivity of labor declines. While the second effect is present under
both monetary policies, the first is absent under the ZLB. The lower decline in investment
reduces the decline in production, which in turn reduces the decline in labor income in
all occupations (first panel, second row). This raises the overall number of labor hours
supplied under the ZLB compared to the Taylor rule. The increased earnings capacity raises
consumption demand under the ZLB relatively to the Taylor rule. Although workers are less
active in reallocating, the ZLB tames the recession due to the boost in demand. Interestingly,
the ZLB stance also raises earnings’ inequality (wage Gini in the first panel, the last row):
since workers reallocate less across jobs, the differences in labor demand triggered by the
occupational shocks are compensated by larger cross-sectional variation in wages.
7.3. The Role of Job Specialization and Skill-Transferability
As shown in propositions 2 and 3, the job substitutability,
1
,
σs
affects the extent of reallocation,
both in and out of employment and across occupations. To this purpose figure 4 shows the
dynamics of the selected variables in response to the occupational shock comparing the model
under σs = 0.2 versus the model with σs = 0.5. On the one side, and compatibly with our
analytical derivations from Proposition 2, higher σs implies for firms higher substitutability
of labor or lower elasticity of labor demand to wages. This dampens reallocation across jobs.
On the other side, the capital labor substitution declines as shown in Proposition 3. This
43
Figure 3: Impulse Responses to the Occupational Shock under the Zero Lower Bound (Baseline) and the Taylor Rule. The figure shows the percentage deviations of the selected variables from the
steady state in response to an occupational shock. The occupational shock is a 10% increase in α5 and a 40%
decrease in α7 ; each shock follows an AR(1) process with the persistence parameter 0.7, such that the shock
almost dissipates by the 13th quarter. The figure shows the impulse responses from the model with the ZLB
(in blue) and from the model where the monetary policy is following the Taylor rule (in orange). The average
wage is the weighted average wage across occupations where the weights are occupational employment shares
in terms of efficiency units. The effective labor hours is the sum of the labor hours across occupations. The
non-employment share shows the share of households that choose non-employment. Labor aggregator is Ls
and shows the aggregated labor used in the production. Wage Gini is presented in absolute values. Mobility
panel shows probability (aggregated across households and occupations) of being employed and probability
of being in the non-employed state (aggregated across households). Probability of being in the non-employed
state is shown on the right axis. X-axis shows the time in quarters.
44
tames the fall in wages, reduces the fraction of workers that leave the labor force and reduces
the decline in demand. The second effects prevails so that, for this shock, the recession is
tamed under higher σs .
Another model’s primitive that affects the extent of occupational reallocation in the model
is the degree of workers’ skill transferability across occupations. This is captured by the
skewness of the skill distribution matrix, Γ.
Figure 5 compares impulse responses of selected variables to the occupational shock under
the model with the baseline skill distribution matrix and with one characterized by the lower
variance.
A more even skill distribution induces more reallocation in response to uneven shocks,
as tasks are more substitutable for workers. On the other hand, income dispersion (the
effect discussed above) increases even more (see wage Gini). While it is easier for firms
to substitute workers, the wages go down more and hence consumption also declines more.
Overall, with more even skill distribution, the effects that deepen recession are amplified,
though quantitative magnitudes are relatively small.
7.4. Sectoral Productivity Shocks in a Two-Sector Model
Our model features reallocation both across occupations and across sectors. So far the focus
has been on occupational reallocation. In this subsection, we consider a two-sector variant of
the model and study the impulse responses to a sectoral productivity shock. In proposition 1
we have shown that the sectoral allocation of labor shares depends, among other things, on
the weights fs . The sectoral shock is therefore modelled as a negative shock to fs in one sector.
Compatibly with the pandemic application, the latter is chosen to be ”Accommodation and
Food Service Activities, Arts, Entert., Rec. and Other Services”. Most of labor in this sector
is employed in occupations 6 and 7 (about 30% in each). Therefore, the shock translates
into a negative labor demand shock to occupations 6 and 7. Figure 6 shows dynamics of the
usual macro, mobility and inequality variables. The effective labor hours and wages in the
occupations involved by the closure go down. Overall the shock is recessionary triggering
declines in consumption, investment and GDP. However, its negative impact is much smaller
45
Figure 4: Impulse Responses to the Occupational Shock under Different Substitutability of
Labor. The figure shows the percentage deviations of the selected variables from the steady state in
response to an occupational shock. The occupational shock is a 10% increase in α5 and a 60% decrease in
α7 ; each shock follows an AR(1) process with the persistence parameter 0.7, such that the shock almost
dissipates by the 13th quarter. The figure shows the impulse responses from the model with the baseline
substitutability of labor (with σs = 0.2 in blue) and from the model with a higher substitutability of labor
(with σs = 0.5 in orange). The average wage is the weighted average wage across occupations where the
weights are occupational employment shares in terms of efficiency units. The effective labor hours is the
sum of the labor hours across occupations. The non-employment share shows the share of households that
choose non-employment. Labor aggregator is Ls and shows the aggregated labor used in the production.
Wage Gini is presented in absolute values. Mobility panel shows probability (aggregated across households
and occupations) of being employed and probability of being in the non-employed state (aggregated across
households). Probability of being in the non-employed state is shown on the right axis. X-axis shows the
time in quarters.
46
Figure 5: Impulse Responses to the Occupational Shock under the Baseline and a Lower
Variance of the Skill Distribution. The figure shows the percentage deviations of the selected variables
from the steady state in response to an occupational shock. The occupational shock is a 10% increase in α5
and a 60% decrease in α7 ; each shock follows an AR(1) process with the persistence parameter 0.7, such that
the shock almost dissipates by the 13th quarter. The figure shows the impulse responses from the model
with the baseline degree of skill transferability across occupations (in blue), and from the model with a
higher degree of skill transferability (in orange). γ in the baseline model is obtained by using exp(γ)/(1 + γ)
2
transformation, γ in the version with the lower variation in gamma is obtained as exp(γ)/(1 + γ + γ2 ). The
latter γ specification delivers four times smaller standard deviation (the skewness and kurtosis in the two
specification are close to each other, with a bit larger values for the second specification, 1.7 vs. 2 for skewness
and 4.5 vs. 5.3 for kurtosis). ). The average wage is the weighted average wage across occupations where the
weights are occupational employment shares in terms of efficiency units. The effective labor hours is the
sum of the labor hours across occupations. The non-employment share shows the share of households that
choose non-employment. Labor aggregator is Ls and shows the aggregated labor used in the production.
Wage Gini is presented in absolute values. Mobility panel shows probability (aggregated across households
and occupations) of being employed and probability of being in the non-employed state (aggregated across
households). Probability of being in the non-employed state is shown on the right axis. X-axis shows the
time in quarters.
47
than that of the occupational shock. The reason is that only a fraction of the occupations in
the closed sectors is hit by the shock.
7.5. Aggregate Labor Supply Shock in an One-Sector Model
This section examines the effects of the aggregate infection shock on labor supply. For the
ease of the exposition, we simulate the shock in the one sector model.
We initiate the shock through an increase in εpandemic of 10−4 . Since the SIR model is
non-linear and brings the economy to a new steady state, we use the non-linearized solution
from Auclert et al. (2019). We extend the solution to include the possibility of the two steady
states.11
The shock reduces labor supply on impact, more so in occupations with high infection
risk, ϑo (see equation 13). In our baseline calibration, the infection shock is long-lived so
that the peak of the infection is reached after five years. The SIR reproduction number of
1.5 determines the length of the shock. The model abstracts from vaccinations and other
measures which may alter its duration.12 This is so since agents internalize the effect of the
shock and reduce the labor hours in risky occupations. Still, faster spread of the disease
would magnify the impact of the labor supply shock.
Figure 7 shows the dynamics for selected macro, mobility inequality variables in response
to an infection shock. Figure 7 shows macro variables: GDP, investment, consumption, the
policy rate, aggregate wage (average wage across occupations, weighted by the occupational
labor shares), total effective labor hours, the share of population in the non-employment
state, and the labor aggregator, the wage Gini, non-liquid assets, liquid assets and infections.
All series are in the percentage deviations from their steady state levels (except infection
share and wage Gini).
The extent of reallocation across occupations under this aggregate shock is modest. Some
reallocation occurs between high infection risk and low infection risk occupations. The bulk
11 Numerically our algorithm converges for shocks that are limited in size. This constrains the extent to
which we can draw quantitative implications from the infection shock.
12 We do not attempt to make exact predictions on the unfolding of the infection, which depends on a
combination of local geography, density, behavioural factors and social preferences.
48
Figure 6: Impulse Responses to a Sectoral Shock in a Two-Sector Model. The figure shows the
percentage deviations of the specified series from the steady state in response to a sectoral shock. The
sectoral shock is a 90% decline in f2 (f2 is 2.7% of the total economy); the shock follows an AR(1) process
with the persistence parameter 0.7, such that the shock almost dissipates by the 13th quarter. The figure
shows the impulse responses from the model with the occupational mobility (in blue). The average wage
is the weighted average wage across occupations where the weights are occupational employment shares in
terms of efficiency units. The effective labor hours is the sum of the labor hours across occupations. The
non-employment share shows the share of households that choose non-employment. Labor aggregator is Ls
and shows the aggregated labor used in the production. Wage Gini is presented in absolute values. Mobility
panel shows probability (aggregated across households and occupations) of being employed and probability
of being in the non-employed state (aggregated across households). Probability of being in the non-employed
state is shown on the right axis. X-axis shows the time in quarters.
49
Figure 7: Impulse Responses to the Aggregate Labor Supply Shock. The figure shows the
percentage deviations of the specified series from the steady state in response to the infection shock. The
shock is an increase in εpandemic by 10−4 . The figure shows the impulse responses from the model with the
occupational mobility (in blue) and from the model without it (in orange). The average wage is the weighted
average wage across occupations where the weights are occupational employment shares in terms of efficiency
units. The effective labor hours is the sum of the labor hours across occupations. The non-employment
share shows the share of households that choose non-employment. Labor aggregator is Ls and shows the
aggregated labor used in the production. Wage Gini is presented in absolute values. The infected share is
the share of the currently infected people in the population. X-axis shows the time in quarters.
50
of the transmission is channelled through the share of workers in the non-employment state
due to the fall in wages and the increase in the infection risk.
The wage Gini declines. Wages in high risky occupations, which were also low-wage
occupations from the start, raise. This reduces income dispersion. Scheidel (2020) argues for
a similar mechanism characterizing pandemics through history.
7.6. Dynamic Response to Combined Aggregate Supply and
Occupational Shocks
Examining each shock separately helps to transparently discuss the model-channels. The
occupational shocks highlight the extent of reallocation across occupations, the labor supply
shock fine tunes the extent of reallocation in and out of employment.
The pandemic featured a rare combination of aggregate supply and occupational shocks.
For this reason and to appreciate the combined effect of the two shocks, figure 8 plots the
dynamic responses to both.
Overall, the recession depicts a more realistic V-shape (see panels 1, 2 and 3 in the first
row). Occupational reallocation in here is triggered both by the occupational shocks and by
the varying infection risk across occupations. Both of them also reduce labor participation.
51
Figure 8: Impulse Responses to the Aggregate Labor Supply and Occupational Shocks.. The
figure shows the percentage deviations of the selected variables from the steady state in response to the
occupational shocks and to the infection shock. The occupational shock is a 2.5% increase in α5 and a 5%
decrease in α7 ; each shock follows an AR(1) process with the persistence parameter 0.7, such that the shock
almost dissipates by the 13th quarter. The infection shock is an increase in εpandemic by 10−4 . The figure
shows the impulse responses from the model with the occupational mobility (in blue). The average wage
is the weighted average wage across occupations where the weights are occupational employment shares in
terms of efficiency units. The effective labor hours is the sum of the labor hours across occupations. The
non-employment share shows the share of households that choose non-employment. Labor aggregator is Ls
and shows the aggregated labor used in the production. Wage Gini is presented in absolute values. The
infected share is the share of currently infected people in the population. The occupational shocks presented
in the last panel are shown on the right axis. X-axis shows the time in quarters.
52
8. Conclusions
Inequality depends on idiosyncratic income shocks and skills. Occupational reallocation plays
an important role of the economy’s response to shocks and in shaping the inequality.
We propose a multi-sector and multi-occupation model in which agents are heterogeneous
in skills and are exposed to idiosyncratic income risk, along the Bewley (1980)-Aiyagari
(1994) tradition. We introduce Roy-type occupational choice driven by skill comparative
advantage in the model and use the model to study economies response to shocks and the
impact of shocks on inequality.
We summarize the contribution of our wok as follows. First, embedding an optimizing
occupational mobility choice into a traditional heterogeneous agents model allows us to
endogenize cross-sectional variation in earnings and to tie it to labor market conditions. Our
model offers a two-way link between inequality and occupational choice: the position of the
households on the wealth distribution affects, jointly with the skills, their mobility decisions.
While the role of reallocation has been largely recognized as an indicator of economic resilience
to shocks, the literature has so far focused on sectoral reallocation while neglecting the role
of occupational reallocation. But worker skills are largely occupation- and not sector-specific.
In addition, technological change emphases the importance of changes to occupational tasks.
Our model matches well long-run inequality and mobility distributions. Therefore, it
provides a good laboratory economy to study the role of reallocation, particularly in response
to large shocks. We apply the model to the recent pandemic recession, which featured a
combination of aggregate labor supply and occupational labor demand shocks. We find that
the occupational shocks foster reallocation, but deepen the recession and boost wage inequality.
The aggregate labor supply shock (triggered by infection) triggers an increase in the number
of people out of work but triggers little reallocation across occupations. Comparing our
model with mobility to a model without mobility, we find that the model without mobility
could underestimate the extent of the recessions and the changes in inequality over time.
Finally, we show that the role of job specialization and dispersion in the skill-distribution,
which captures workers’ comparative advantage, affect the extent of reallocation, hence the
dynamics of the recession and inequality.
53
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60
A. Derivations of Analytical Results
Proposition 1 From the firm’s optimization problem:
ls,o,t =
ps,t (1 − νs )αs,o
µp pt wo,t
1
1−σ
s
νs
zs Ks,t−1
σs
(1−ν)(1−σ
σs −1+νs
s)
(1−νs )(σs −1)
ys,t
(A.1)
Dividing the effective labor across occupations in the same sector delivers (from now on time
sub-index is neglected):
ls,o
=
ls,o0
αs,o0 wo
αs,o wo0
−1
1−σ
s
(A.2)
The above shows that labor shares across occupations depend, for given wage distribution,
on the following primitives: αs,o , σs . Note that the wage distribution is not a primitive in
the model as it arises from the market clearing, hence is affected by labor supply, whose
primitives include the Γ matrix.
Next, the labor shares across sectors are derived. Using eq. (A.2), the definition of the
production function and the definition of occupation aggregator the production function in
each sector can be written as:
ys = zs Ksνs
O
X
σs
αs,o0 ls,o
o0 =1
1−νs
zs Ksνs ls,o
wo
αs,o
αs,o0 wo
αs,o wo0
1−νs
1−σ
O
X
s
α
1−νs
σs ! σ
1−σ
s
1
1−σs
s,o0
s
=
w
−σs
1−σs
o0
(A.3)
s
! 1−ν
σ
s
(A.4)
o0 =1
1
The prices in each sector are given by ps = (fs yYs ) η p, therefore:
1
σs −1+νs
1
1
η(1−νs −σs )−1+νs
η(1−νs )(1−σs )
(1−νs )(σs −1)
ps1−σs ys,t
= (fs Y ) η(1−σs ) p 1−σs ys
61
.
(A.5)
The last equation can be substituted into eq. (A.1) and, after using eq. (A.4), this delivers:
1
1
η(1−νs −σs )−1+νs
σs
1
1
(1 − νs ) 1−σs αs,o 1−σs
=
(zs Kzνs ) (1−νs )(1−σs ) (fs Y ) η(1−σs ) p 1−σs ys η(1−νs )(1−σs )
µp p
wo
1
1−σ
1
s
σs
1
1
(1 − νs )
αs,o 1−σs
=
(zs Kzνs ) (1−νs )(1−σs ) (fs Y ) η(1−σs ) p 1−σs ·
µp p
wo
s −σs )−1+νs
η(1−ν
s
! 1−ν
η(1−νs )(1−σs )
1−νs
1−σ
σs
O
−σs
X 1
s
wo
1−σs
1−σs
1−νs
· zs Ksνs ls,o
αs,o
0 w o0
αs,o
o0 =1
ls,o
(A.6)
Collecting terms from eq. (A.6) delivers:
"
ls,o =
(1 − νs )
µp p
O
X
·
α
1
1−σs
s,o0
1
1−σ
w
s
1
η(1−σs )
(fs Y )
−σs
1−σs
o0
p
s )−ησs
! (η−1)(1−ν
ησ (1−σ )
s
Setting Ms =
o0 =1
α
αs,o
wo
ηνs +1−ν2s
η(1−σs )
(η−1)(1−νs )
(zs Ksνs ) η(1−νs )(1−σs ) ·
(A.7)
s)
ηνη(1−σ
+1−ν
s
s
s
o0 =1
PO
1
1−σs
1
1−σs
s,o0
w
−σs
1−σs
o0
s )−ησs
(η−1)(1−ν
ησ (1−σ )
s
s
and dividing by effective labor in the same
occupation, but in different sectors by each other, delivers:
"
ls0 ,o
=
ls,o
1
(1 − νs0 ) 1−σs0 f
1
1
η(1−σs0 )
s0
1
η(1−σs )
(1 − νs ) 1−σs fs
αs0 ,o
wo
ηνs0 +1−ν2s0
αs,o
wo
η(1−σs0 )
ηνs +1−ν2s
η(1−σs )
ν
zs0 Ks0s0
(zs Ksνs )
(η−1)(1−νs0 )
η(1−νs0 )(1−σs0 )
(η−1)(1−νs )
η(1−νs )(1−σs )
# ηνη(1−σ
s0 )
+1−ν
s0
s0
Ms0
s)
ηνη(1−σ
+1−ν
s
(A.8)
s
Ms
If σs = σs0 and νs = νs0 then the last ratio simplifies:
η(1−σ )
ls0 ,o
=
ls,o
f s0
fs
1
η(1−σ
)
s
αs0 ,o
αs,o
ηνs +1−ν2s
η(1−σs )
(η−1)(1−νs )
zs0 Ksν0s η(1−νs )(1−σs )
zs Ksνs
Ms0
Ms
s
! ηνs +1−ν
s
(A.9)
The above shows that the labor shares across sectors depend upon the following primitives:
αs,o , σs , νs , fs , and zs .
62
Proposition 2 From the intermediate firm’s problem, the first-order condition for the
demand for labor in each occupation in each sector reads as follows:
ls,o,t =
ps,t (1 − νs )αs,o
µp pt wo,t
1
1−σ
s
νs
zs Ks,t−1
σs
(1−ν)(1−σ
σs −1+νs
s)
(1−νs )(σs −1)
ys,t
(A.10)
Dividing the labor demand in one occupation by the labor demand in another, but for the
−1
αs,o0 wo 1−σs
, which implies that:
same sector, delivers: lls,o0 = αs,o
w 0
s,o
o
w
σs
σs −1
o0
−1
σs σs −1
αs,o0
ls,o
=
wo
αs,o
σ σ−1
s
s
σs
αs,o0 ls,o
0
(A.11)
which upon expressing ls,o in terms of the other variables and upon some re-shuffling, it can
be summed over both sides by o0 to obtain: Ls :
σs
ls,o
O
X
w
σs
σs −1
o0
α
−1
σs −1
s,o0
=
o0 =1
Isolating ls,o from the A.12 delivers ls,o =
wo
αs,o
wo
αs,o
σ σ−1
s
s
Lσs s
(A.12)
! σ1
σ 1−1
s
s
1
σs
PO
o0 =1
−1
σs −1
woσ0s −1 αs,o
0
Ls , which, after
using Equation (A.10), can be written as:
ls,o =
where Fs = (1 −
s Y
νs )L−σ
y
s
µp s
wo
αs,o
1
1−σ
s
σ 1−1
! σ1
s
Fs
s
PO
o0 =1
wo0 ls,o0
Ls
(A.13)
. Using Equation (A.13), the elasticity of labor demand
with respect to wages, with a continuum of occupations, is:
εl,w =
while it reads as follows: εl,w =
1
σs −1
∂ls,o wo
1
=
∂wo ls,o
σs − 1
1−
ls,o wo
0
o0 =1 ls,o0 wo
PO
(A.14)
under a finite number of occupations.
The markup is then the inverse of the following expression (under an infinite number of
occupations):
µw =
εll,w
1
=
εl,w + 1
σs
63
(A.15)
and the inverse of the following expression with finite number of occupations: µw =
ls,o wo
l
w0
0
o =1 s,o0 o
ls,o wo
σs − PO
l
w0
o0 =1 s,o0 o
1− PO
. This proves part (1). When σs = 1 the mark-up is also one and the la-
bor aggregator is Ls =
PO
αs,o ls,o,t , hence occupations are perfectly substitutable. This
o=1
proves part (2). When αs,o = 1 for every s, then the total labor demand in each sector s is
equal to
O
X
1
PO
ls,o
o=1
σs −1
o=1 wo
=
Ls
σs 1
PO
σs
σs −1
o0 =1 wo0
(A.16)
When σs = 1 it follows that: ls,o = I(wo = mino (wo ))Ls = w1o pps η−1
(1−νs )ys I(wo = mino (wo ))
η
P
with O
o=1 ls,o = Ls and where I(wo = mino (wo )) is an indicator function that equals 1 for
the occupation with the lowest wage.
Proposition 3 The production function is as follows:
ys =
s
zs Ksνs L1−ν
s
=
zs Ksνs
O
X
s
! 1−ν
σ
s
σs
αs,o ls,o
(A.17)
o=1
Aggregating sectoral production functions into total output delivers:
Y =
S
X
η−1
η
1
η
η
! η−1
fs y s
S
1
X
=
fsη zs Ksνs
s=1
s=1
O
X
η−1
s
! 1−ν
η
σ
η
η−1
s
σs
αs,o ls,o
o=1
(A.18)
The derivative of the total output with respect to σs in a specific sector is:
O
X
η−1 (η − 1)
1
∂Y
η
=
Y η fs (zs Ksνs ) η
∂σs
η−1
η
1
η
Y
1
η
1
η
fs (zs Ksνs )
η−1
η
O
X
s
o=1
+
1 − νs
1
P
O
σs
σs
α
l
s,o
s,o
o=1
s
σs
αs,o ls,o
o=1
s)
! −(1−ν
ησ
σs
αs,o ls,o
s)
! −(1−ν
ησ
O
X
s "
! 1−ν
σ
s
σs
αs,o ls,o
o=1
O
X
!
σs
αs,o ls,o
ln(ls,o )
o=1
64
∂
∂σs
O
X
s
! 1−ν
σ
s
σs
αs,o ls,o
=
o=1
(A.19)
!
O
X
1 − νs
σs
− 2 ln
αs,o ls,o
σs
o=1
The derivative is positive if the expression in the square brackets is positive. The latter reads
as follows:
O
X
− 1 − νs ln
σs2
!
σs
αs,o ls,o
+
o=1
1 − νs
1
P
O
σs
σs
α
l
s,o
s,o
o=1
O
X
!
σs
ln(ls,o ) ≥ 0
αs,o ls,o
(A.20)
o=1
Collecting the terms delivers:
O
X
1
P
O
o=1
σs
αs,o ls,o
!
σs
ln(ls,o )
αs,o ls,o
o=1
O
X
1
σs
≥ ln
αs,o ls,o
σs
o=1
!
(A.21)
Putting σs on the left-hand side and then into the logarithm delivers:
O
X
!
σs
σs
)
ln(ls,o
αs,o ls,o
≥
PO
!
σs
αs,o ls,o
ln
O
X
!
σs
αs,o ls,o
(A.22)
o=1
o=1
o=1
Given
O
X
o=1
αs,o = 1, the last inequality holds for any values of σs and ls,o due to Jensen’s
13
, which states that for a real convex function ϕ and λ1 + ... + λn = 1 if ∀i λi ≥ 0
inequality
and xi ≥ 0 then :
ϕ
n
X
!
g(xi )λi
≤
i=1
n
X
ϕ(g(xi ))λi
(A.23)
i=1
σs
Equation A.22 corresponds to a Jensen’s inequality for ϕ(x) = xlnx, λi = αs,o , g(xi ) = ls,o
,
n = O and i = o.
B. Computation Method
The model is simulated by adapting the algorithm developed in Auclert et al. (2019). The
underlying rationale of the method lies in finding a solution in sequence space, in which all
operations like summation and multiplication are defined for sequences, see, for example,
Dunford and Schwartz (1958)). Any variable in the model is therefore an infinite sequence.
Multivariate Newton method is used to detect, for any variable, sequences such that the
13 https://encyclopediaofmath.org/index.php?title=Jensen_inequality, accessed 17.02.21
65
initial conditions are satisfied, namely the model delivers a deterministic steady state, and
all variables’ dynamic paths are generated by the model equations given specified shock
sequences. The solution relies on an approximation of the infinite sequences with finite
ones. Auclert et al. (2019) show that such an approximation is good enough when the finite
sequences considered are long enough.
The algorithm offers a log-linear approximated solution, whereby the sequences return to
the same steady state, and a fully non-linear solution, which can accommodate the possibility
of transitioning across steady states. Our mode features a SIR block, which implies a
transition to a new steady state. For this reason the non-linear solution is adopted and
adjusted to accommodate transition between the two steady states. Technically, initial and
final steady states are inserted into the Newton method. The Jacobian and errors are then
computed accordingly. The steady states are obtained from the deterministic version of the
model equations and by normalizing aggregate output to 1. The code is written in Python
3.7.
Adapted Solution Algorithm with Fixed Point Routine The household optimization problem in the main text 3.1 is solved in two-stages. Numerically this has been
implemented through a guess and verify procedure with the following steps:
1. Guess future value function, Wa (zj,t , at , bt , φo,t+1 ),
Wb ((zj,t , at , bt , φo,t+1 ) where W (zj,t , at , bt , φo,t+1 ) =
PO
o=1 V (zj,t , at , bt , φo,t+1 )p(ot+1 |at , bt , zj,t )
2. For each occupation compute policy functions, i.e. calculate a(zj,t , bt−1 , at−1 ) and
b(zj,t , bt−1 , at−1 ) ∀o (using F.O.C. s from the optimization problem in Equation (3))
3. Feed policy functions into the value and update: V o (zj,t , at−1 , bt−1 )
4. Compute occupation choice probabilities:
δ(ot |zj,t , bt−1 , at−1 ) =
exp(V o (zj,t ,bt−1 ,at−1 )
P
o
o0 exp(V (zj,t ,bt−1 ,at−1 )
t
5. Update guesses of Wa (zj,t , at , bt , φo,t+1 ), Wb (zj,t , at , bt , φo,t+1 ) using envelope conditions
and probabilities from step 4.
Aggregation is done using steady state distribution of the idiosyncratic income shocks,
66
liquid and illiquid assets and the probabilities of being in each occupation which we assume
we can handle as masses of our population. Mathematically the guess is used for the value
function that reads as:
V j (et , at−1 , bt−1 ) = log[
X
o
+ (1 + rta )at−1 + (1 + rtb )bt−1 −
exp(max
u(ξj,t
o o
at ,bt
ot
Φ(aot , at−1 ) − aot − bot ) + λt (bot − b) + µt aot + βEet+1 V j (et , aot , bot ))] (B.1)
where V j (et , at−1 , bt−1 ) = Eφ Vj (et , at−1 , bt−1 , φt ). Using the above we compute the F.O.C.s
with respect to liquid and illiquid assets as following:
uc (ct |o) = λt + βEe ∂b V j (et+1 , aot , bot )
(B.2)
uc (ct |o)[1 + Φ1 (at , at−1 )] = µt + βEe ∂a V j (et+1 , aot , bot )
(B.3)
Envelope conditions:
P
∂b V t (et , at−1 , bt−1 )
=
P
∂a V t (wt , at−1 , bt−1 )
=
ot
ot
exp(Ṽto (et , bt−1 , at−1 ))u0 (c∗t |ot )(1 + rtb )
exp(V t (et , at−1 , bt−1 ))
exp(Ṽto (et , bt−1 , at−1 ))u0 (c∗t |ot )(1 + rta − Φ2 (a∗t , at−1 ))
exp(V t (et , at−1 , bt−1 ))
(B.4)
(B.5)
where uc (ct |o) is the marginal utility of consumption in occupation o. The above operators
are inserted in the computational method for heterogeneous agents by Auclert et al. (2019).
67
C. Constructing Occupational Clusters Using k-means
Algorithm
The 8 clusters obtained through k-means algorithm can be summarized as follows:
1. Cluster1, Manual trade occupations, predominantly includes occupations from the Construction and extraction occupations, Installation, maintenance and repair occupations
and Production occupations.
2. Cluster 2, Management and supervisory occupations, includes Management occupations,
Business and financial operations occupations, Education, training and library occupations and Sales and related occupations., occupations of sales agents or supervisory
workers from the Sales and related occupations group. Finally this cluster includes
occupations from the social science field.
3. Cluster 3, Machine operators, includes residential construction occupations such as like
carpet or drywall installers or masons, some occupations from Production occupations
and others, such as machine operator from the Transportation and material moving
occupations category.
4. Cluster 4, Engineering occupations, includes Management occupations, Computer and
mathematical science occupations, Life, physical and social science occupations and
Architecture and engineering occupations. The cluster also includes occupations, such
as cashier and counter clerks from the Sales and related occupations group, but also
scientific occupations in life and physical science (occupations contain the key word
scientist)
5. Cluster 5, Healthcare and community occupations, is predominantly populated by
Healthcare practitioner and technical occupations and almost all Community and social
service occupations.
6. Cluster 6, Personal service occupations, mainly technical-support occupations from
the larger Architecture and engineering occupations, Life, physical and social science
occupations and Computer and mathematical science occupations groups.
7. Cluster 7, Technical-Support occupations, is mainly populated with occupations Food
68
Table 7: Occupational Clusters and Major Occupation Groups
Occupation Clusters
Detailed Occupation Groups
1
2
3
4
5
6
7
8
Total
Management occupations
Business and financial operations occupations
Computer and mathematical science occupations
Architecture and engineering occupations
Life, physical, and social science occupations
Community and social service occupation
Legal occupations
Education, training, and library occupations
Arts, design, entertainment, sports, and media occupations
Healthcare practitioner and technical occupations
Healthcare support occupations
Protective service occupations
Food preparation and serving related occupations
Building and grounds cleaning and maintenance occupations
Personal care and service occupations
Sales and related occupations
Office and administrative support occupations
Farming, fishing, and forestry occupations
Construction and extraction occupations
Installation, maintenance, and repair occupations
Production occupations
Transportation and material moving occupations
0
0
0
0
0
0
0
0
0
0
0
2
0
0
0
0
0
3
20
30
10
6
19
17
4
0
7
3
2
7
5
1
0
1
0
0
2
7
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
2
1
0
3
2
19
6
56
20
4
2
6
18
9
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
1
0
0
0
2
0
0
0
23
2
7
0
0
1
0
0
0
0
0
0
0
2
0
1
3
7
0
0
0
6
2
1
2
2
2
1
0
2
2
1
1
5
5
0
0
0
0
0
0
0
3
3
0
3
4
9
2
12
5
10
1
0
0
4
2
0
5
0
0
0
0
2
1
3
1
0
0
0
0
1
4
30
0
0
0
0
0
26
25
11
21
23
5
4
11
17
27
6
16
12
6
18
17
48
8
40
37
75
33
Total
71 78 11040 37 45 58 47 486
preparation and serving related occupations, Sales and related occupations, Healthcare
support occupations and Personal care and service occupations.
8. Cluster 8, Office and administrative support occupations, consists mainly of Office and
administrative support occupations.
Table 7 cross-tabulates the major occupation group and our assignment to one of the 8
clusters for the 3-digit occupations.
69
D. Appendix Tables
Table 8: Sectors: Capital and labor Shares,
KLEMS Data, Averages 1987-2015
Sectors Name
Capital Share
labor Share
Accommodation and Food Service
Community, Social and Pers. Services
Construction
Electr., Gas and Water Supply
Financial and Insurance
Information and Communication
Manufacturing
Mining and Qarrying
Scien., Tech., Admin. and Sup. Service
Real Estate Activities
Transportation and Storage
Wholesale and Retail Trade
Total
0.35
0.19
0.17
0.70
0.41
0.50
0.44
0.68
0.18
0.94
0.27
0.42
0.44
0.65
0.81
0.83
0.30
0.59
0.50
0.56
0.32
0.82
0.06
0.73
0.58
0.56
E. Other Quantitative Results
Figure 9 shows the dynamics of selected variables to an occupational shock under the Taylor
rule. Reallocation across occupations (second panel, third row) appears enhanced compared
to the case of the ZLB policy presented in the main text. As a result less workers opt out of
the labor force, but rather reallocate to other jobs. As cross-occupational reallocation prevails,
under this policy stance the model with mobility tames the recession compared to the one
without. In absence of the sheltering provided by the expansionary policy workers reallocate
more actively. This provides an interesting exchange between a policy, the ZLB, that fosters
demand by enhancing income capacity, and a policy, the Taylor rule, that lubricates the
labor market by activating mobility.
Figure 10 below presents impulse responses of the usually selected variables to an adverse
occupational shock to occupation 2, which includes sales, education, but also managerial
positions in business and finance.
70
Figure 9: Impulse Responses to an Occupational Shock. The figure shows the percentage deviations
of the selected variables from the steady state in response to an occupational shock. The occupational
shock is a 10% increase in α5 and a 60% decrease in α7 ; each shock follows an AR(1) process with the
persistence parameter 0.7, such that the shock almost dissipates by the 13th quarter. The figure shows the
impulse responses from the model with the occupational mobility (in blue) and from the model without it
(in orange). Central Bank follows the Taylor rule. The average wage is the weighted average wage across
occupations where the weights are occupational employment shares in terms of efficiency units. The effective
labor hours is the sum of the labor hours across occupations. The non-employment share shows the share
of households that choose non-employment. Labor aggregator is Ls and shows the aggregated labor used
in the production. Wage Gini is presented in absolute values. Occupational labor dynamics are shown in
the last row, where each line corresponds to the effective labor hours in the specified occupation. Mobility
panel shows probability (aggregated across households and occupations) of being employed and probability
of being in the non-employed state (aggregated across households). Probability of being in the non-employed
state is shown on the right axis. X-axis shows the time in quarters.
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Figure 10: Impulse Responses to an Occupational Shock in Occupation 2 (Business, Sales,
Education.). The figure shows the percentage deviations of the selected variables from the steady state in
response to an occupational shock. The occupational shock is a 60% decrease in α2 ; the shock follows an
AR(1) process with the persistence parameter 0.7, such that the shock almost dissipates by the 13th quarter.
The figure shows the impulse responses from the model with the occupational mobility (in blue) and from
the model without it (in orange). The average wage is the weighted average wage across occupations where
the weights are occupational employment shares in terms of efficiency units. The effective labor hours is the
sum of the labor hours across occupations. The non-employment share shows the share of households that
choose non-employment. Labor aggregator is Ls and shows the aggregated labor used in the production.
Wage Gini is presented in absolute values. Occupational wage dynamics are shown in the last row, where
each line corresponds to the wage in the specified occupation. Mobility panel shows probability (aggregated
across households and occupations) of being employed and probability of being in the non-employed state
(aggregated across households). Probability of being in the non-employed state is shown on the right axis.
X-axis shows the time in quarters.
Figure 11 below shows the responses of the wage Gini to occupational shocks hitting
high-wage occupations (right panel) or low-wage occupations (left panel).
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Figure 11: Gini Coefficients for Different Occupational Shocks. The figure shows the wage Gini
coefficients for the occupational shock to occupation 1 (Construction) and for the shock to occupation 2
(Management and Financial). The shock is a 60% decrease in α1 or α2 respectively; each shock follows an
AR(1) process with the persistence parameter 0.7, such that the shock almost dissipates by the 13th quarter.
The figure shows wage Gini coefficients from the model with the occupational mobility (in blue) and from
the model without it (in orange). Wage Gini coefficients are presented in absolute values.
F. Wealth Distribution Graphs
Figure 12: Distributions in the Steady State. The figure shows the distribution of total wealth,
liquid/non-liquid assets or wages. X-axis represents wealth/wage values and Y-axis represents probability
densities. The wage distribution plots wages for each of the eight occupations (x-axis) against the share of
total population working in each occupation. The non-liquid and the wealth distributions are cut at the
upper tail (the total probability of the cut parts is 10−13 ).
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@FedFRASER