ISSN (Print) 0473-453X

Discussion Paper No. 1088

ISSN (Online) 2435-0982

STRUCTURAL UNEMPLOYMENT,UNDEREMPLOYMENT,AND SECULAR STAGNATION
Ken-ichi Hashimoto
Yoshiyasu Ono
Matthias Schlegl

Thirdly Revised January 2023
Secondly Revised July 2022

Revised March 2021
May 2020

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Structural unemployment, underemployment, andsecular stagnation∗
Ken-ichi Hashimoto† Yoshiyasu Ono‡ Matthias Schlegl§¶
January 2023

Abstract
In this paper, we show that underemployment and not necessarily high unemployment
becomes the main measure of economic slack in the labor market under secular stagnation.
Specifically, involuntary underemployment in the form of a persistent shortfall of working
hours occurs in the search and matching model, provided that households derive utility
from holding wealth, and quickly dominates the total employment gap under stagnation.
Conventional policy measures aimed at reducing unemployment may increase the labormarket gap through their effects on underemployment and should be used with caution. In
contrast, positive demand shocks improve unemployment and working hours, while increases
in potential output worsen both (“paradox of toil”). Our analysis provides new insights
into empirical puzzles such as Japan’s seemingly decent employment record during its lost
decades.
Keywords: Secular stagnation, underemployment, unemployment, labor market frictions, search and matching
JEL Classification: E24, E31, E44, J20, J64∗

We are grateful to Gerhard Illing, Jean-Baptiste Michau, Eiichi Miyagawa, Akihisa Shibata, Takashi Shimizu,
Mototsugu Shintani, Daishin Yasui, Kaz Yugami, Nobuyoshi Kikuchi, and to seminar participants at LMU
Munich, Kobe University, Tohoku University, Waseda University and the Tokyo Institute of Technology for helpfulcomments and suggestions. Financial supports from the Joint Usage Research Center Program at ISER and JSPS
KAKENHI Grants (19K23230, 15H05728, 20K01631, 20H05631 and 21K13261) are gratefully acknowledged.
†
Graduate School of Economics, Kobe University; Email: hashimoto@econ.kobe-u.ac.jp‡
Institute of Social and Economic Research, Osaka University; Email: ono@iser.osaka-u.ac.jp§
Department of Economics, Sophia University, Tokyo; Email: m-schlegl-4t5@sophia.ac.jp¶
Corresponding author.

1

Introduction

Macroeconomists conventionally rely on the unemployment rate as the main measure of slack in
the labor market and its implications for wage and price dynamics. However, such an analysis
is incomplete at best and potentially misleading once an economy is constrained by the effective
lower bound on the nominal interest rate and suffers from persistent stagnation. Under suchcircumstances, it can be underemployment, a state in which employed workers want to increase
working hours at the same wage but are unable to do so, rather than unemployment thatprovides a more accurate measure of labor market slack.
Japan is the prime example of an economy stuck in a stagnation equilibrium without any
natural recovery. However, Japan’s economy has performed surprisingly well in terms of its
employment record during the lost decades as the unemployment rate has remained low byinternational comparison peaking at slightly more than 5% in 2002 as illustrated in panel (a)of Figure 1. Unemployment in 2018 had reached its lowest level since the early 1990s. Yet,this decline has not resulted in significant wage or price pressures.1 The absence of widespread
unemployment is often attributed to the specific features of the Japanese labor market, mostnotably the traditional practices of long-term employment, which make unemployment in Japanless responsive to fluctuations in spending.2 Then how, if not in unemployment, did the lack ofdemand manifest itself in the labor market?
During the stagnation period the structure of Japan’s labor market has changed profoundlywith a substantial rise in underemployment in the form of part-time and non-regular employment
and declining working hours. Panel (b) of Figure 1 shows the share of part-time employment in
total employment in contrast to the OECD average. Having been less than 12% in the 1980s,
the share of part-time employees has sharply increased throughout the lost decades, primarilyreflecting the lack of alternative employment opportunities.3 By 2018, it had doubled comparedto the 1980s level whereas it only increased by 3 percentage points over all OECD countries. A
similar trend can be observed for other forms of non-regular employment (see Japan Institutefor Labour Policy and Training, 2015). Finally, the steady decline in average hours worked in
the OECD data is also indicative of the rise of underemployment in Japan since 1990.4
Rising underemployment and the absence of wage and price pressures despite low unemployment are no longer exclusively observed in Japan, but have become well-documented phenomena
in several European countries after the Great Recession as well.5 Surveying the empirical evi1
In fact, wage growth has been dismal with stagnant real and declining nominal wages. Bell and Blanchflower(2021) report that nominal wages have declined by 0.5% on average over the 2000-2016 period, while real wages
have been unchanged. In 2017, real wages actually fell by 0.2% despite the historically low unemployment rate.
2
We refer to Hashimoto and Raisian (1985) for a description of the Japanese employment system in comparison to the United States before the lost decades. Several studies show that these practices have eroded
remarkably little, at least among core workers, during the stagnation decades (see Shimizutani and Yokoyama,
2009; Kambayashi and Kato, 2011; Hamaaki et al., 2012, among others).
3
For illustration, non-regular employees consistently emphasize the “lack of regular employment opportunities”
as a major reason for their current work style in surveys conducted by the Japan Institute for Labour Policy andTraining (see Japan Institute for Labour Policy and Training, 2015).
4
The secular decline in working hours is confirmed by data from the Labour Force Survey of the Statistics
Bureau of Japan. Average working hours per week have substantially fallen from a stable level of slightly morethan 28 hours in the 1980s to less than 23 hours by the 2010s.
5Bell and Blanchflower (2021) document a substantial level of labor market underutilization in the form of1

Figure 1: Labor market developments in Japan, 1980-2018

Data sources: (a) Unemployment rate (in percent of the labor force), OECD; (b) Part-time employment rate
(in percent of total employment), OECD, Japan (solid) and OECD average (dotted line); OECD definiton:“Part-time employment is defined as people in employment (whether employees or self-employed) who
usually work less than 30 hours per week in their main job.”

dence, Blanchflower (2019, p.144) argues that “underemployment has replaced unemployment
as the main measure of labor market slack”.
In this paper, we propose a model of secular stagnation that distinguishes between structural
unemployment and demand-driven underemployment. Using this model, we show that secularstagnation leads to underemployment in the labor market but not necessarily to high unemployment, thereby providing a rationalization for the puzzling case of Japan and the theoreticalcounterpart to the empirical analysis described above. Specifically, we incorporate a preference
for wealth into the standard search and matching model of the labor market. In a setting with
infinitely-lived households, such a preference captures the bequest motive within dynasties thatis frequently incorporated in overlapping generations models. The preference for wealth createsa strong motive to save in addition to the standard consumption smoothing motive, which can
explain why empirically the saving rate of households is increasing in their wealth (see Benhabib
and Bisin, 2018; Fagereng et al., 2019).6 Importantly, the preference for wealth allows for the
possibility of a secular stagnation equilibrium as is well-know from contributions such as Ono
underemployment after the financial crisis for the United States and 26 European countries. In fact, unemployment rates have returned to their pre-crisis levels in most countries while underemployment is persistently higher.
In addition, using panel data for 19 countries from 1998-2016, the authors show that underemployment has asignificantly negative effect on wages after 2007, while the unemployment rate is insignificant. The opposite holds
before the Great Recession implying a structural break at the time the effective lower bound became binding. The
authors hence conclude that “underemployment replaces unemployment as the main influence on wages in theyears since the Great Recession”. Other studies, such as Bell and Blanchflower (2018) for the United Kingdom,
directly attribute underemployment to a shortage of aggregate demand.
6
Several recent contributions have analyzed the macroeconomic implications of a preference for wealth inmodels similar to ours. Kumhof et al. (2015), Michau et al. (2020) and Mian et al. (2021) study the dynamics
of wealth and inequality that result from endogenous differences in saving rates based on the wealth preference,
while Michau et al. (2018) analyze the dynamics of asset prices and the possibility of rational bubbles in sucha framework. Michaillat and Saez (2021) incorporate a preference for wealth (relative to the average wealthlevel) into a New Keynesian model to study business cycle fluctuations and Saez and Stantcheva (2018) employ
a similar framework for the analysis of capital taxation. For the interested reader, we refer to Zou (1994) who
provides an in-depth discussion of the idea of a preference for wealth in economic thought.

2

(1994), Michau (2018) or Schlegl (2018). High desired savings can drive the equilibrium realinterest rate into negative territory. Yet, the effective lower bound, which we model as a rate ofzero for simplicity, prevents the nominal interest rate from falling sufficiently. This causes excess
savings which depress aggregate demand. Combined with downward nominal wage rigidities inthe spirit of Schmitt-Grohé and Uribe (2016, 2017), the model economy operates in a secularstagnation equilibrium characterized by deflation and a persistent lack of demand.
In the absence of demand shortage, our model behaves similar to the standard search andmatching model except that the preference for wealth creates a new transmission channel forshocks via induced changes in the real interest rate. Structural unemployment occurs in equilibrium as vacancies and job-seekers have to match in a costly process before a position can be
filled, but conditional on being employed, households realize their potential working hours andthere is no underemployment. The tightness of the labor market affects both the intertemporalconsumption decision of the household via changes in the wealth premium and market entry andexit of firms via changes in returns on firm ownership. In equilibrium, labor market tightness
is determined such that optimal household and firm behavior are compatible with each other.
The interest rate then adjusts endogenously in the money market.
In the stagnation equilibrium, in contrast, both structural unemployment and underemployment coexist and mutually affect each other. When desired savings are sufficiently strong,
the zero lower bound on the nominal interest rate becomes a binding constraint and the realinterest rate is determined by the rate of deflation. Faced with a lack of demand and depressedsales, firms respond by cutting working hours resulting in underemployment, lower output andless consumption. The lack of demand, however, also affects the incentives for job creation.
Firm ownership becomes less attractive due to lower profits (“job creation channel”), while atthe same time holding money becomes more attractive due to increased deflation (“deflation
channel”). Both channels reduce job creation incentives for firms and result in an increase in
unemployment. The total employment gap then exceeds the unemployment rate.
We calibrate this model to match characteristics of Japan’s labor market and quantify the
relative importance of unemployment and underemployment under stagnation. For all of our
specifications, underemployment quickly dominates the total employment gap as stagnation becomes more severe. This finding is confirmed in several robustness, which alter the workings ofthe job creation and deflation channels. It is only in the case of an increasing job destruction
rate under stagnation that unemployment rises substantially. While empirical evidence points
towards a relatively stable job destruction rate in Japan, this job destruction channel is likelyof relevance in explaining labor market patterns in other economies. In addition, we examine
the effects of labor market policies and demand and supply shocks using numerical simulations.
Conventional policies aimed at reducing unemployment, specifically a reduction in unemployment benefits and the introduction of a search cost subsidy, might increase the total employment
gap under stagnation via their effects on underemployment depending on the shape of the wealthpreference. If the concavity of the preference for wealth is sufficiently low, these policies resultin movements of unemployment and underemployment in opposite directions as has been thecase in Japan during labor market liberalization policies in the early 2000s and after 2012. The

3

effects of demand and supply shocks are reversed under demand shortage. Higher demand, inthe form of government spending, increases working hours and lowers unemployment thereby
raising consumption and output. In contrast, a positive productivity shock worsens both. Thisis equivalent to the paradox of toil that is common to models of stagnation. These shocks result in co-movements of unemployment and underemployment as have been observed in Japan
during the 1990s and the financial crisis.
Our analysis concludes that secular stagnation causes demand-driven underemployment andnot necessarily high unemployment. As long as the economy is stuck in the stagnation equilibrium, wage growth and inflationary pressures do not emerge even when unemployment is
low as they are subdued by the prevalence of underemployment. It is then primarily underemployment that responds to economic disturbances and labor market policies under stagnation,
which is in line with the developments in Japan since the 1990s. These findings highlight theneed for further policy intervention in support of aggregate demand despite a seemingly decent
employment situation in terms of the unemployment rate.
Related Literature: Our paper analyzes the interactions of structural unemployment basedon search frictions and demand-driven underemployment under secular stagnation, thereby
contributing to two fields of macroeconomics that have so far been treated fairly independently.
The canonical search and matching model of the labor market (see Diamond, 1982; Mortensen,
1982; Pissarides, 1985, 2000, among others) has initiated a rich literature on structural unemployment as a consequence of frictions in the labor market.7 Variations in working hours are
introduced into this framework by Fang and Rogerson (2009) as households optimally choose
their labor supply based on the consumption versus leisure trade-off. Workers are always on
their labor supply curves and part-time and non-regular employment is purely voluntarily. Theirrise in Japan is, for instance, explained by productivity shocks in combination with labor immobility due to differences in training costs among workers (see Ariga and Okazawa, 2011) or by
an exogenous increase in part-time labor supply inducing households and firms to optimally reallocate between full-time and part-time jobs (see Kang et al., 2020). These voluntary shifts inthe composition of the labor force do not constitute underemployment as defined in this paper.8
In addition, underemployment in our model does not rely on firm or worker heterogeneity.
Other contributions, such as Manning (2003) and Ashenfelter et al. (2010), provide a microeconomic view on underemployment based on market concentration and monopsony powerof firms. From the firm’s perspective, variations in working hours are a cost-efficient way to
avoid hiring and firing costs. In contrast, firms are perfectly competitive with free market entry
and exit in our model. Underemployment is a macroeconomic phenomenon. Workers want towork additional hours at the same wage, but are constrained by a lack of demand.
Demand-driven variations in working hours are modelled in a related framework by Kudoh
et al. (2019) who analyze changes in the composition of firms’ labor demand over the business
7

Rogerson et al. (2007) survey the related literature.
In fact, we abstract from voluntary variations in working hours and assume an inelastic labor supply suchthat all variations in working hours are involuntarily. Our model can be extended to allow for fluctuations inworking hours based on the consumption versus leisure choice as well. Yet, this makes the model more complex
without substantially altering the results.
84

cycle. While labor market participation is chosen by each individual, working hours conditionalon employment are adjusting in response to demand-driven fluctuations in production, which isa features similar to our model. However, these fluctuations occur over the business cycle andare hence purely temporarily in nature, which is in stark contrast to our model which allows forthe possibility of demand-driven underemployment as a steady state phenomenon due to thebinding effective lower bound on the nominal interest rate, i.e. secular stagnation.
The secular stagnation hypothesis, first proposed by Hansen (1939), argues that an oversupply of savings at full employment permanently depresses aggregate demand as the zero lower
bound on the nominal rate prevents the real interest rate from falling sufficiently to stimulate
spending. In the presence of downward nominal wage rigidity, this results in an equilibriumwith a persistent output gap. The oversupply of savings has been modelled among others as aconsequence of demographics (Eggertsson et al., 2019), a shortage of safe assets (Caballero andFarhi, 2018) or strong liquidity preferences (Ono, 2001; Illing et al., 2018).
In this paper, we rely on a preference for wealth in line with the contributions of Ono (1994,
Chapter 11) and Michau (2018), which, however, do not consider structural unemployment
and abstract from labor market frictions.9 The labor market gap is either abstracted from or it
consists of a demand-driven shortfall of working hours only. We extend these models by allowing
for potential interactions of unemployment and underemployment in an integrated framework.
Several approaches have been proposed in the literature to model involuntary unemployment in the search and matching model, which are related to our concept of underemployment.
Michaillat (2012) shows how wage rigidities and decreasing marginal returns to labor can resultin rationing unemployment in addition to frictional unemployment. The former occurs only during recessions following negative technology shocks which reduce the marginal product of labor
below the wage to which firms optimally respond by reducing employment. While Michaillat(2012) focuses only on the supply side and business cycle variations, we complement this analysis by modelling the demand side allowing for the possibility of persistent demand shortage.
Interestingly, a similar mechanism creates spillovers from underemployment to unemployment
in our model. Underemployment occurs when the downward price rigidity becomes binding.
Then, firm profits and the marginal product of labor fall due to lower working hours resulting
in higher structural unemployment.
Similarly to our model, Michaillat and Saez (2022) also rely on a preference for wealthallowing for an equilibrium with a permanent liquidity trap. Keynesian unemployment occurs
in addition to frictional unemployment when the real interest rate, which is implicitly set by thecentral bank, is excessively high and depresses demand. Higher unemployment reduces desired
savings of households thereby restoring equilibrium. The same channel is at work in our model as
the real interest rate is excessively high under stagnation.10 However, our model also explicitly
considers firm behavior under free market entry. Then working hours and unemployment are
jointly determined to reconcile optimal household and optimal firm behavior for a given real9

In addition, Schlegl (2018) discusses secular stagnation in an economy with land while Michau (2022) carefully
analyzes the effects of helicopter drops of money under stagnation based on a preference for wealth.
10
We assume money supply targeting instead of interest rate targeting, which is why underemployment only
occurs in the stagnation equilibrium when the zero lower bound is binding in the money market. Implicitly, we
hence assume that the central bank always sets the optimal interest rate, unless it is constrained from doing so.

5interest rate. Adjustment then occurs via a combination of fewer working hours and higher
unemployment. Moreover, we show that the first effect is quantitatively more important using
extensive numerical simulations.11
This paper is organized as follows. Section 2 presents a general equilibrium model of underemployment. Section 3 analyzes the steady state equilibria. In section 4, we discuss thecalibration of the model to Japanese labor market data and then extensively study the properties of underemployment under stagnation in section 5, supported by a numerical simulationsof the model. The final section concludes. All proofs are in the mathematical appendix.

2

A general equilibrium model of underemployment

2.1

The representative household

Time is continuous and denoted by t. The infinitely-lived representative household consists ofa large number of individuals normalized to unity. Each individual is endowed with one unitof labor and one unit of time, both of which are supplied inelastically as there is no disutilityof working. Let lt ∈ [0, 1] denote the realized labor supply and xt ∈ [0, 1] realized working
hours per individual. Slack in the labor market can take two forms: Unemployment occurs
when household members are not employed (lt = 0). Let ut denote the number of unemployed
members of the household, and hence the unemployment rate, and 1−ut the number of employed
ones. Underemployment occurs when realized working hours of the employed (lt = 1) fall shortof potential working hours, i.e. xt < 1. Each household receives wage income of wt (1 − ut )xt
and unemployment benefits of zˆt ut , where wt is the real wage rate and zˆt benefit payments
per unemployed. We follow Merz (1995) and Andolfatto (1996) and assume that the household
provides perfect consumption insurance for its members so that consumption is the same for
employed and unemployed individuals.
Household assets at , in real terms, consist of interest bearing assets (bonds or equities) bt
and real money holdings mt that do not pay interest:
at = bt + mt .

(1)

Let rt denote the real interest rate and Rt ≥ 0 the nominal interest rate, which are related via
the Fisher Equation by Rt = rt + πt , where πt denotes the rate of inflation. Then, real wealthevolves as:12a˙t = rt at + wt (1 − ut )xt + zˆt ut − ct − Rt mt − τt ,(2)

where τt is a real lump-sum tax, and (1 − ut )xt ∈ [0, 1] is effective employment. There are
11

Michaillat and Saez (2015) apply a matching framework to both the labor market and the product market
so that the price level and labor market tightness are jointly determined in both markets. This creates feedback
effects from demand fluctuations into the labor market. Under fixed prices, an increase in demand leads to a risein working hours and therefore lower unemployment. Yet, they also do not consider the optimal market entry
and exit decision of firms thereby not allowing for a distinction between unemployment and idle time. Landais
et al. (2015b,a) use the same framework to study the effects of labor market policies.
12
The nominal flow budget equation is Ȧt = Rt Pt bt + Wt (1 − ut )xt + Ẑt ut − Pt ct − Pt τt , where At (≡ Pt at )denotes total asset holdings. Using at = mt + bt and rt ≡ Rt − πt , where πt ≡ Ṗt /Pt , we obtain (2).

6

opportunity costs Rt when holding money due to the foregone interest earnings.
The household derives utility from consumption ct , from its holdings of liquid real money
balances mt and from its net wealth holdings. Its lifetime utility is given by

∞
U0 =

[φ(ct ) + µ(mt ) + ω(at − mSt )]e−ρt dt,

0where ρ > 0 is the discount rate and the flow utility functions satisfy φ0 > 0, φ00 < 0, µ0 > 0,
µ00 < 0 and ω 0 > 0 and ω 00 ≤ 0. We assume that there exists m̄ for which the transaction
demand for money becomes fully satiated, i.e. µ0 (mt ) = 0 for all mt ≥ m̄, to allow for aliquidity trap. Our specification of a preference for net wealth ω(at − mSt ), where mSt denotes
the real money supply, implies that the household considers the money stock a government
liability that eventually needs to be redeemed and hence not part of aggregate net wealth, eventhough each member considers his personal money holdings a part of his wealth. As there is
no heterogeneity on the household level, it is only the value of equity holdings (as bonds are inzero net supply) that affects the wealth preference in steady state.13
The household maximizes lifetime utility subject to the asset constraint (1) and the flow
budget constraint (2) for given initial wealth a0 . Optimal household behavior is described by
the Euler Equation, the money demand function and the transversality condition asηcc˙tω 0 (bt + mt − mSt )= rt − ρ +,ct
φ0 (ct )Rt =µ0 (mt )≥ 0,
φ0 (ct )lim φ0 (ct )at e−ρt = 0,

t→∞

(3)

(4)

(5)

where ηc ≡ −φ00 (ct )ct /φ0 (ct ) is the elasticity of the marginal utility with respect to consumption(and the inverse of the intertemporal elasticity of substitution), which we assume constant.14
The preference for wealth affects the intertemporal allocation of consumption (3), providing
additional incentives for saving since accumulation of wealth becomes an end in itself. A lower
level of wealth, ceteris paribus, induces the household to choose a steeper consumption pathor equivalently higher savings in order to accumulate wealth. In steady state, the preference
for wealth creates a wedge between the real interest rate and the time preference rate of thehousehold, which allows for the possibility of a negative natural real interest rate. Optimalmoney demand in (4) requires the marginal rate of substitution between money and consumption
to equal the opportunity cost of holding money. The nominal rate is at the zero lower boundwhenever mt ≥ m̄, which allows for the possibility of a liquidity trap in our framework.
13

Michau (2019) discusses and justifies the preference for net wealth in contrast to alternative specifications.
These equations are obtained from the standard current value Hamiltonian function Ht = φ(ct ) + µ(mt ) +ω(at − mSt ) + λt [rt at + wt (1 − ut )xt + zwt xt ut − ct − Rt mt − τt ], with control variables ct and mt and statevariable at and where λt is the costate variable for at .
14

7

2.2

The labor market

We assume the standard search and matching model of the labor market with free market entry
and exit and Nash bargaining between workers and firms as a mechanism for wage determination.
The government conducts two types of labor market policies. It pays benefits zˆt to unemployedworkers following the policy rulezˆt = zwt xt ,(6)

where z ∈ [0, 1) denotes the replacement rate that is proportional to labor income. In addition,
it pays a search cost subsidy s ∈ (0, 1) to support firms when posting vacancies.
2.2.1

Matching mechanism

Output is produced when firms and workers successfully match in the labor market. The matching function F (ut , vt ) describes the number of matches between vacant firms vt and unemployed
ut . F (ut , vt ) is continuously differentiable, concave, homogeneous of degree one, increasing
with respect to both ut and vt , and satisfies F (0, vt ) = 0 and F (ut , 0) = 0. Let θt denote thejobs-to-applicants ratio, i.e.,θt ≡vt∈ [0, ∞),
ut(7)

which measures the tightness of the labor market. A higher value of θt implies more vacancies
per job-seeker, which we will refer to as a tighter labor market. Firms are less likely to fillvacancies and unemployed workers are more likely to find employment the tighter the labormarket as measured by θt . Formally, the probability that a vacant firm matches with a worker
q(θt ) is simply given by the relative frequency of matches among all vacancies and satisfies
F (ut , vt )q(θt ) ≡
=F
vt




1
, 1 , q 0 (· ) < 0, q 00 (· ) > 0, q(∞) = 0.
θt(8)Similarly, the probability that a worker matches with a firm with a vacancy p(θt ) is given by
the relative frequency of matches among all unemployed and satisfiesp(θt ) ≡F (ut , vt )= F (1, θt ) = θt q(θt ), p0 (· ) > 0, p00 (· ) < 0, p(0) = 0.
ut

(9)

In each period a fraction δ of workers become unemployed, where δ constitutes an exogenous
separation rate.15 Hence, the flow into unemployment is given by δ(1 − ut ). At the same time,
the number of successful matches between vacant firms and unemployed workers is given by
F (ut , vt ) = p(θt )ut , which follows from (9). Therefore, the dynamics of the unemployment rate
are given byu̇t = δ(1 − ut ) − p(θt )ut .

(10)

15
This assumption allows us to focus on the effects of underemployment on the job creation channel as it
implies that there is no direct effect of underemployment on the job destruction rate. For the case of Japan, ourfocus on job creation is supported by empirical evidence in Ariga and Okazawa (2011) who show that hiring rates
have declined substantially during the stagnation period while separation rates appear quite stable. We discuss
the importance of this assumption for our results in detail in section 5.1.

8

2.2.2

Firm behavior under free market entry and exit

There is a large number of identical firms that produce the consumption good using labor as the
only input factor. Each firm requires exactly one worker in the production process and therefore
offers one position, which can be filled, in which case the firm is operating, or vacant. Whenoffering this position and searching for workers in the labor market, a search cost k occurs. The
government might provide a search cost subsidy at rate s ∈ [0, 1) reducing effective search costs
for the firm to (1 − s)k. Firms that successfully match hire lt = 1 workers and produce output
yt with the linear production function
yt = ȳxt lt = ȳxt ,(11)

where ȳ denotes labor productivity and xt hours per worker. The operating profit of eachproducing firm is given by (ȳ − wt )xt , which is increasing in xt .
The value functions of an operating firm Jt and a vacant firm posting a vacancy Vt , which
describe the values of participating in the labor market for firms, are hence given by the Bellman
equationsrt Jt = (ȳ − wt )xt − δ[Jt − Vt ] + J˙t ,(12)

rt Vt = −(1 − s)k + q(θt )[Jt − Vt ] + V̇t .

(13)

Firms can freely enter and exit the labor market without costs. As a consequence, a firm willenter the labor market and post a vacancy as long as its expected value is greater or equal tozero. This free entry condition therefore implies in equilibrium that
Vt = 0.

(14)

From (13), it follows immediately that the value of an operating firm Jt has to satisfy
J(θt ) =(1 − s)k, J(∞) → ∞, J 0 (· ) > 0.
q(θt )(15)

The free entry condition requires that the expected value of a filled job equals the search costs
associated with a vacancy. The tighter the labor market, the lower the probability of filling avacancy and the higher the value of an operating firm. Applying the free entry condition (14)and the equilibrium firm value (15) to (12) implies the following differential equation for labormarket tightness:ηθtJ˙t(ȳ − wt )xtθ̇t≡= rt −q(θt ) + δ,θtJt
(1 − s)k(16)

0

q (θt )
where ηθt ≡ − θtq(θ> 0 is the elasticity of the matching probability of a vacant firm witht)respect to θt . Note that ηθt is constant for a Cobb-Douglas matching function. Equation (16)represents the job creation condition of the matching model describing optimal firm behavior
for a given real wage. Intuitively, labor market tightness θt adjusts so that the real interest rate
rt equals the return on firm ownership. If the real rate exceeds the net profit yield, firms leave
the labor market leading to a drop in labor market tightness which in turn allows for capital9

gains on firm ownership over time. If the real rate is below the net profit yield, jobs are createdas firms enter the labor market leading to a rise in tightness and capital losses over time.
2.2.3

Nash bargaining

The real wage is set once workers and firms have successfully matched. While several wage
setting mechanisms are consistent with wage determination in this environment, we follow thestandard Pissarides (2000) assumption of Nash bargaining between firms and workers.
Wages are set to maximize the joint surplus of workers and firms. We derive the solution tothis problem in Appendix A. The equilibrium real wage is determined as:
wt xt = ψ (ȳxt + (1 − s)kθt ) ,(17)

where ψ ∈ (0, 1) reflects the effective bargaining power of workers.16 All else equal, the negotiated wage wt increases in workers’ bargaining power ψ, vacancy cost (1 − s)k and labor market
tightness θt , but decreases in working hours xt . Using (17), the job creation condition (16) canbe rewritten as
η θt

2.3

θ̇t
(1 − ψ)ȳxt= rt −q(θt ) + ψp(θt ) + δ.
θt
(1 − s)k(18)

Government

The government consists of a central bank and a fiscal authority. The central bank controls thenominal money supply MtS by setting the nominal money growth rate gm ≥ 0 for a given initialmoney supply M0S . The real money supply mst evolve asṁst= gm − πt ,
mst

(19)

where πt ≡ ṖPtt denotes the rate of inflation. The nominal interest rate is endogenously determined in the money market in line with (4).
The fiscal authority adjusts lump-sum taxes τt to balance the budget while providing benefitszˆt ut to the unemployed according to the policy rule in (6), subsidizing a fraction s of searchcosts k of vacant firms and financing government spending g, which is a policy parameter tomodel exogenous changes in demand, such thatτt + gm mSt = zˆt ut + skvt + g,(20)

where gm denotes the growth rate of the nominal money supply, gm mSt income from seignorage
and vt the number of vacant firms. Government bonds are in zero net supply so that interestbearing assets of the household bt solely consist of firm equity.
16

εSpecifically, ψ is defined in Appendix A as ψ ≡ 1−(1−ε)z, where ε denotes the coefficient on the surplus ofworkers in the joint surplus function and z the replacement rate in payments to unemployed workers in (6).

10

2.4

Aggregation and market clearing

Without demand shortage (xt = 1), the dynamics of the price level Pt are as in the standard
Money-in-the-Utility framework where the steady state inflation rate is determined by the money
growth rate gm and the price level adjusts to clear the money market. As a result of the search
matching mechnism real wage wt (= Wt /Pt ) is detemined, and therefore, nominal wage Wt
follows commodity price Pt .
Turning to the case with demand shortage, because the number of operating firms equals
1 − ut as each operating firm hires exactly one worker and working hours are xt (< 1), total
output is
Yt = (1 − ut )xt ȳ,(21)based on the linear production function in (11). We can interpret (1 − ut )ȳ as a measure ofpotential output in the presence of structural unemployment. Total output is reduced below
the production capacity ȳ due to structural unemployment ut and idle working hours (1 − xt )if xt < 1, and then underemployment is (1 − ut )(1 − xt ). Total output is used for private
consumption, public spending and total search costs of firms with vacancies. In the goods
market, we hence have
(1 − ut )xt ȳ = ct + g + kθt ut ,(22)

which we derive formally in Appendix B by combining the budget equations of the household
and the government. By reformulating (22), we can express xt as the ratio between effective
demand and potential output. Therefore, xt constitutes both a measure of underemployment
and a measure of effective demand shortage as reflected in the output gap.17
We assume downward price rigidity when underemployment occurs (x < 1) –i.e., there isa lower bound of the time change rate of prices Pt , which is α(xt − 1), in order to prevent
deflationary wage-price-spirals and to allow for the possibility of a stagnation steady state.18
Pt declines but its speed cannot be lower than α(xt − 1). These properties are summarized asfollows:
πt

Ṗt
=
Pt

!
= α(xt − 1) for xt < 1.

(23)

In order for the steady state to be supported, gm must be higher than α(xt − 1):
gm > α(xt − 1).

(24)

The asymmetry in the inflation process is a fundamental element of stagnation models
including among others the contributions of Schmitt-Grohé and Uribe (2016, 2017), Michau(2018), Illing et al. (2018) and Eggertsson et al. (2019) and typically results from some form of
downward nominal wage rigidity that becomes binding in case of demand shortage.
At any point in time, financial markets are in equilibrium. Bonds are in zero net supply. In
17

In the business cycle literature, output fluctuations around the natural level of output are the result ofvariations in working hours in response to various shocks. In contrast, our framework allows for the possibilityof underemployment as a steady state phenomenon, even in the absence of shocks.
18
Our model features outright deflation under stagnation to be consistent with the experience of Japan during
its lost decades when prices and wages declined on average.

11the money market, money demand of the household in (4) equals the money supply set by thecentral bank, i.e. mt = mSt ≡

MtS
Pt .

The aggregate firm value is given by (1 − ut )Jt , which inequilibrium equals the value of equity holdings of the household bt . Using (15), the net wealthof the household bt = at − mSt is therefore given by
bt = (1 − ut )(1 − s)k.
q(θt )(25)

Financial market equilibrium also satisfies the job creation condition (18) and the Fisher equation
rt = Rt − πt =µ0 (mt )− α(xt − 1) for xt < 1,
φ0 (ct )(26)

where the inflation rate is given in (23) and the nominal interest rate in (4), so that households
are indifferent between equity, bonds and money as a means of saving. In equilibrium, thereturn on money has to be the same as the return on other assets and real money holdings
evolve asṁt= gm − α(xt − 1) > 0 for xt < 1.
mt

(27)

Hence, real money holdings are constant in the absence of aggregate demand shortage but expand indefinitely in case of secular stagnation due to the effects of deflation, thereby pushing the
economy into a permanent liquidity trap as eventually mt > m̄ and the household’s transaction
demand becomes satiated.
This concludes the exposition of our model. In the next section, we analyze and contrast
the different possibilities for steady state equilibria. For simplicity, we will assume a constantnominal money supply
gm = 0,but our analysis can easily be generalized as discussed in Illing et al. (2018) or Michau (2018).

3

Occurrence of underemployment in steady state

Equations (3), (10), (18) and (27) form an autonomous dynamic system with respect to ct ,ut , θt and mt , where xt , bt and rt are derived from (22), (25) and (26) as functions of thesevariables. The equilibrium of the model is defined as follows:An equilibrium is a set of paths for quantities {ct , mt , bt , at , Yt , yt , xt , ut , vt , θt } and forprices P0 and {rt , Rt , wt , πt } such that• {ct , mt , bt , at } solves the consumer’s problem given {rt , Rt , wt , ut , θt }, a0 , m0 = M0 /P0and at = bt + mt in (1);• firms produce yt given {rt , wt , ut , θt } resulting in aggregate output Yt given by (21);• {wt , vt , ut , θt } solves the Nash bargaining problem in the labor market under free market
entry and exit of firms where vt = θt ut according to (7);12• equilibrium prices P0 and {rt , Rt , πt } are consistent with money market equilibrium, equity
market equilibrium as well as the No-Arbitrage relation between money and equities (26),while in the goods market, inflation and working hours are determined by the non-linear
Phillips curve (23) and xt ≤ 1, which hold with complementary slackness.
In a stationary steady state, consumption, the unemployment rate and labor market
tightness are constant, i.e. ċt = 0, u̇t = 0 and θ̇t = 0, while the behavior of ṁt depends on thepresence of aggregate demand shortage.
Consider first the labor market. In steady state, the number of newly unemployed equals
the number of newly filled jobs. From (10) with u̇t = 0, the unemployment rate satisfiesu(θ) =

δ, u(0) = 1, u0 (.) < 0, u00 (.) > 0.
δ + p(θ)(28)

This is the standard Beveridge curve. A tighter labor market is characterized by lower structural
unemployment due to a higher probability of finding a job, i.e. p0 (θ) > 0 in (9). There are no
direct spillovers from underemployment into unemployment or job destruction.
In contrast, underemployment directly affects the job creation incentives of firms. Under
free market entry and exit, the real interest rate equals the return on equity, which in steadystate consists of the profit yield net of the separation rate. From (18) with θ̇t = 0, it has tohold thatr=(1 − ψ)ȳx
q(θ) − ψp(θ) − δ.
(1 − s)k(29)

This relationship is illustrated by the blue curve in Figure 2 for x = 1 as a decreasing functionof θ. A tighter labor market, ceteris paribus, reduces the return on equity as both the firm
value in (15) and the negotiated real wage in (17) increase. If the real rate is above the returnon equity, firm ownership is less attractive than other assets and firms leave the labor market
resulting in a drop in labor market tightness and higher unemployment until equilibrium isrestored (and vice versa). There is a unique value θ̃ for which the return on equity is zero given
x = 1, defined byθ̃ :(1 − ψ)ȳ
q(θ̃) − ψp(θ̃) = δ.
(1 − s)k(30)

In the goods market, the supply of goods available for household consumption, i.e. total
output less entry costs of firms and government spending in (22), is given byc(θ, x) = (1 − u(θ))xȳ − kθu(θ) − g.

(31)

It shows a hump-shaped pattern with respect to labor market tightness for x = 1. Initially, total
output rises by more than search costs as unemployment decreases. At some point, however,
the increase in total search costs exceeds the increase in output, thereby lowering householdincome. Following Michaillat and Saez (2015), we refer to the former case as a “slack” steadystate and the latter one as a “tight” steady state.19
19
dc
dc
Formally, a steady state with dθ
> 0 in (31) is referred to as slack and a steady state with dθ
< 0 as tight.
Note, however, that the definition refers to the total (rather than the partial) derivative.

13

Figure 2: Steady state without underemployment

Notes: The blue curve illustrates the return on firm equity in (29) as a function of labor market
tightness θ, the black curves the natural real rate consistent with optimal household behavior in(32) for two specifications of the wealth preference. The red line represents the zero lower boundon the nominal and real (as gm = 0) interest rates. Labor market tightness is determined by theintersection of the two curves for x = 1. Such a steady state is only feasible if θf < θ̃.

Consumption demand of the household has to be consistent with the supply of goods available for private consumption. For a given real interest rate, it follows from the Euler Equation(3) with ċt = 0 thatr =ρ−ω 0 (b(θ)),φ0 (c(θ, x))(32)

where c(θ, x) is given by (31) and b(θ) by (25) and (28) with b0 (θ) > 0 as a tighter labormarket implies both a larger number of operating firms and a higher valuation of them. In thestandard model, the steady state interest rate is simply determined by the time preference rate
ρ. In contrast, the wealth premium, i.e. the marginal rate of substitution between consumption
and wealth, creates a wedge between the time preference rate and the real interest rate as
the household saves for the purpose of accumulating wealth. We refer to the interest rate forwhich consumption demand equals the supply of goods as the “natural” real interest rate. It isrepresented by the black curves in Figure 2 for two specifications of the preference for wealth,ω 0 and ω̃ 0 , representing a strong and weak desire to save respectively.
If the real rate is above the natural rate, demand is depressed and there is excess supply in the
goods market. For x = 1, adjustment occurs via changes in θ. A drop in labor market tightness
affects the goods market via two channels. On the one hand, household wealth decreases, which
reduces consumption demand. On the other hand, the income of the household in (31) increases
in a tight steady state and falls in a slack steady state. In the former case, the wealth premiumincreases which reduces demand and increases desired savings thereby worsening the excess
supply. In the latter case, the combined effect is unclear and depends on the concavity of thewealth preference.20
20
In Michaillat and Saez (2022) a fall in labor market tightness and higher unemployment unambiguously
reduce desired savings of the household. This is because of their assumption of a preference for relative wealth,i.e. wealth relative to the average wealth across households, implying a constant marginal utility of wealth insteady state, and a slack steady state such that c0 (θ) > 0 in (31) for x = 1. As a consequence, an excessively

14

Figure 3: Steady state with underemployment

Notes: The shift in of the blue curve illustrates the effects of underemployment on the return on
firm equity in (29), the shift of the black curve the effects on the natural real rate in (32). The redline represents the rate of deflation for a given x. Underemployment and labor market tightness arejointly determined such that (33) with x < 1 holds.

In general equilibrium, unemployment and underemployment are jointly determined suchthat the optimal firm behavior in (29) is consistent with the optimal household behavior in (32).
As illustrated in Figure 2, the real interest rate is determined by the simultaneous equilibriumin goods and labor markets. The nominal rate then adjusts endogenously in the money market.
However, if the wealth preference is sufficiently strong, the steady state requires a negative realinterest rate, which is not feasible due to the zero lower bound resulting in an oversupply ofsavings and asset market disequilibrium, as holding money becomes too attractive. This occurs
when the steady state with x = 1 requires labor market tightness to be above θ̃ as defined in(30). In Figure 2, the steady state with θ2f is therefore not feasible.
Then, an involuntary drop in working hours is necessary to reconcile optimal household
and firm behavior and underemployment occurs. This stagnation steady state is characterized
by persistent deflation and expanding real money balances as is clear from (27), such thateventually, when mt > m̄, the liquidity preference of the household is satiated and the nominalinterest rate is at the zero lower bound as µ0 (m) = 0.21 The real interest rate is then determinedby the rate of deflation, represented by the red line in Figure 3, and adjustment takes place inthe labor and goods markets via both unemployment and underemployment.
In the labor market, there are fewer vacancies as firms leave the market since holding money
is more attractive than equity. This results in a drop in labor market tightness and higher unemployment. However, this adjustment is not sufficient to establish goods market equilibrium
as well, which can easily be seen in Figure 3. There is insufficient demand in the goods market,
which depresses firm sales and profits. Firms respond by reducing working hours. Underemployment occurs which reduces household income as employed members work fewer hours. Thishigh real interest rate results in demand-driven “Keynesian” unemployment, which eliminates the excess supply.
21
The transversality condition (5) needs to be satisfied as well, which requires a sufficiently low money growthrate gm and a sufficiently low rate of price adjustment α. A sufficient restriction is ρ > gm + α. Recall that we
focus on the case of gm = 0 and hence assume ρ > α.

15

in turn reduces desired savings as the wealth premium falls, which is represented by the upward
shift of the black curve in panel (b) of Figure 3 for given θ. The drop in working hours feeds backinto the labor market. As the marginal product of labor falls and firm profits drop, investment
in firms becomes less attractive resulting in further market exit. As firms leave the market,
fewer vacancies are posted and labor market tightness declines. Unemployment increases as isclear from (28).22 This is represented by the shift of the blue curve to the left in Figure 3.
In the stagnation steady state, unemployment and underemployment are determined suchthat both optimal household and firm behavior are consistent with the excessively high realinterest rate caused by deflation. This is a substantial extension of the existing literature on
involuntary unemployment where adjustment occurs through involuntary variations in unemployment only. Our analysis shows that such an adjustment is not sufficient in an integrated
framework that takes both the supply and the demand side into account.
Formally, the two possible steady states of our model are expressed in the following generalequilibrium condition combining (29), (32) and the steady state expression of (26):
 0
µ (m)


 0
if x = 1,
ω 0 (b(θ))(1 − ψ)ȳx
φ (c, x)ρ− 0=q(θ) − ψp(θ) − δ =
φ (c(θ, x))(1 − s)k α (1 − x) if x < 1.

(33)

The following lemmata summarize their existence and stability conditions:Lemma 1 The steady state without demand shortage exists ifρφ0 c(θ̃, 1) ≥ ω 0 b(θ̃) ,(34)

where b(θ) is given by (25) in combination with (28) and θ̃ is uniquely defined by (30). If thesteady state without demand shortage is unique, then it is also saddle-path stable.
At θ̃ the return on equity is zero for x = 1 and hence θ > θ̃ in steady state is not feasible as
this would require a negative real interest rate. Hence, Lemma 1 requires a sufficiently weak
wealth preference for the steady state without aggregate demand shortage to exist, which caneasily be seen in Figure 2.23 In principle, there could be multiple steady states consistent with
(33). Yet, mild technical restrictions on the functional forms of φ(.), ω(.) and q(.) are sufficientto establish uniqueness of this steady state.24 We assume that this steady state with x = 1 isunique throughout this paper.
Consider now the stagnation steady state, which is illustrated in Figure 3.
22This spillover channel from underemployment to unemployment is similar to the occurrence of rationing
unemployment in Michaillat (2012) due to the effects of negative technology shocks on the marginal product ofworkers. Yet, we do not rely on shocks as underemployment occurs endogenously in our model.
23
In addition, it needs to hold that (1−ψ)ȳq(θ) − ψp(θ) > ρ + δ > (1−ψ)ȳq(θ̄) − ψp(θ̄) where θ and θ̄ are defined(1−s)k(1−s)kby c(θ, 1) = c(θ̄, 1) = 0 in (31) with θ < θ̄. This is a purely technical restriction that ensures that creating new
vacancies is sufficiently attractive when unemployment is high. If there are no incentives to establish firms evenin case of high unemployment, existence of a steady state is not guaranteed.
24
Intuitively, uniqueness requires that an exogenous increase in wealth reduces the household’s propensity tosave via a reduction in labor market tightness θf . Not only is it the natural economic scenario to expect, it isalso technically a very mild condition. Specifically, this steady state is always unique provided that the elasticityof the unemployment rate with respect to θ is sufficiently low.

16

Lemma 2 The secular stagnation steady state with θs < θ̃ and xs < 1 exists if ρ > α and ifρφ0 c(θ̃, 1) < ω 0 b(θ̃) ,(35)

where b(θ) is given by (25) in combination with (28) and θ̃ is uniquely defined by (30). If thesecular stagnation steady state is unique, then it is also saddle-path stable.
Under stagnation, working hours xs and labor market tightness θs are jointly determined in the
goods and labor markets such that optimal household and firm behavior are consistent withthe real interest rate given by the rate of deflation. Since the return on equity at θ = θ̃ is zeroand declining in θ, it has to hold that θs < θ̃ in the stagnation steady state. Consumption isthen determined by (31) with x = xs and θ = θs .25 The condition ρ > α establishes a lower
bound on the rate of deflation thereby preventing a wage-price spiral that is not consistent withthe existence of a well-defined steady state. An excessively high rate of deflation would imply
an expansion of real money balances that violates the transversality condition (5). Lemma 2
further requires the real rate at θ̃ to be negative. This is exactly opposite to condition (34) inLemma 1.26 Again, we assume that the stagnation steady state is unique. This holds under
mild restrictions on the functional forms of the utility, production and matching functions.27
The second equality in condition (33) for x < 1 implies that firms optimally adjust working
hours and employment such that the return on equity is equalized to the rate of deflation. Thisis the modified job creation condition under stagnation and implies the equilibrium relationshipx(θ) =(1 − s)k [ψp(θ) + δ + α],
(1 − ψ)ȳq(θ) + (1 − s)kα(36)which satisfies dx(θ)
dθ > 0 and x(θ̃) = 1, with θ̃ defined in (30). For x > x(θ), the return on
equity exceeds the return on money, which makes firms enter the labor market resulting in morevacancies and lower unemployment until equilibrium is restored. Similarly, for x < x(θ), thereturn on equity is below the return on money and firms leave the labor market resulting in
fewer vacancies and higher unemployment until equilibrium is restored.
Two corollaries follow immediately from the analysis above. First, the preference for wealthreduces the steady state real interest rate relative to the standard matching model, in which
the real rate is simply given by the time preference rate ρ. In the absence of demand shortage,
the lower real rate stimulates firm creation resulting in a tighter labor market and a lower
25

To ensure non-negativity of consumption under stagnation, we impose the technical restriction on spending
g and the matching function such that c(θs , xs ) ≥ 0 exists.
26
Assuming uniqueness of the steady state with x = 1 and the one with x < 1, these steady states are mutually
exclusive. This no longer holds for gm > 0. Then both steady states are feasible for some parameter calibrations.
27
Formally, uniqueness requires that under secular stagnation an exogenous increase in wealth increases thepropensity to save of the household and therefore increases the aggregate equity value via an increase in thesteady state labor market tightness θs . Compared to the previous case, an increase in θ affects the propensity
to save via two additional channels: First, the induced decrease in the real interest rate is dampened due to theweaker response of the return on equity to increases in θ as a consequence of the prevalence of persistent deflation.
Secondly, there is a stronger effect of changes in θ on the net supply as a consequence of the induced change inrealized labor working hours. Both, higher income and a smaller sensitivity of the return on savings relative theto case of full employment support the assumption of an increase in the propensity to save in response to an
exogenous increase in wealth under stagnation.

17

unemployment rate. Second, the secular stagnation steady state necessarily has to be slack if it
is unique, i.e. an increase in labor market tightness raises steady state consumption c(θs , xs ).
This is in contrast to the steady state without demand shortage which can be slack or tightdepending on the functional form of the matching function and the preference for wealth.
Having analyzed existence and stability of the stagnation steady state, we turn to the relative importance of structural unemployment and demand-driven underemployment and their
sensitivity to labor market policies as well as demand and supply shocks. In order to get a better
understanding of the model and to quantify these effects, the theoretical analysis is supported
by a calibration and numerical simulation of the model.

4

Calibration

We set a time period to be one quarter. We assume CRRA utility for consumption and a nonhomothetic specification for the preference for wealth, which is empirically most plausible asargued by Mian et al. (2021), implying that the marginal utility of wealth decays more slowly
than the marginal utility of consumption. In our benchmark specification we implement thisfeature via a constant marginal utility, i.e. ω 0 = β. Alternatively, we consider a CARA utility
function. The parameter β reflects the weight of the wealth preference.28 We rely on a CobbDouglas matching function in our baseline specification, which implies a constant elasticityηθ = γ of the job filling rate q(θ). The scale parameter κ reflects the matching efficiency.
Specifically, we assume the following functions:φ(c) =

c1−ηc − 1,
1 − ηcω(a − ms ) = β(a − ms ) (benchmark),ω(a − ms ) = −β −ηa (a−ms )e(alternative),ηa

F (u, v) = κuγ v 1−γ .
Fifteen parameters need to be calibrated. The calibration is summarized in Table 1. We set
ηc = 2 and ηa = 0.5 to capture the difference in curvature between φ(.) and ω(.) and choose
β to generate a specific level of underemployment in steady state. The time preference rate ρrepresents an upper bound on the real rate as can be seen in (32). We assume ρ = 12% on anannual basis, resulting in a quarterly time preference rate of ρ = 0.03. Lemma 2 restricts the
speed of price adjustment to allow for the possibility of stagnation. We set α = 0.005, which
implies a maximum annual deflation rate of 2%. Without loss of generality, labor productivity
is normalized to ȳ = 1. Variations in this parameter are used to analyze the effects of supply
shocks. Our baseline calibration also assumes g = 0 and s = 0 and subsequent variations inthese parameter are used to illustrate the effects of demand shocks and the introduction of a28
Mian et al. (2021) do not directly specify the preference for wealth but the marginal utility of wealth relative
to a logarithmic specification, which reflects the utility from consumption. Their calibration implies that thismeasure is always strictly positive in (0, 1). Similarly, our benchmark specification implies that this measure isconstant at level β, which we chose in (0, 1).

18

Table 1: Parameter calibration
Parameter

Calibration

Comment

CRRA parameter in φ(.) ηc = 2

implies ηc > −bs ω 00 (bs )/ω 0 (bs )CARA parameter in ω(.) ηa = 0.5(used in the alternative specification of the wealth preference)

Scale parameter in ω(.)β

calibrated to generate a specific xs in steady stateTime preference rate

ρ = 0.03

upper bound on the real interest rate: r ≤ 0.12

Price adjustment speed

α = 0.005

maximum rate of deflation: π ≥ −0.02

Potential incomeȳ = 1

normalization, used to model supply shocks

Public spending

g=0

used to model demand shocks

Search cost subsidy

s=0

policy variable

Replacement ratez = 0.6

cf. Kitao et al. (2017), Kudoh et al. (2019)

Bargaining power

ε = 0.5cf. Mortensen and Pissarides (1994), Den Haan et al. (2000)

Bargaining power (eff.)

ψ = 0.714

ψ = ε(1 − (1 − ε)z)−1

Job separation rate

δ = 0.015

average job duration of 16.7 years

Cost of a vacancy

k = 0.431

match r = 5% (yearly) in (29) for θ̄ = 0.85 and x = 1

Elasticity of q(θ)γ = 0.5

Hosios (1990) efficiency condition, γ = εScale parameterκ = 0.449

match ū = 0.035 and θ̄ = 0.85

Note: The parameter β of the wealth preference is chosen to generate a specific output gap in steady state.
Details for the calibration of the other parameters are provided in the text.

search cost subsidy. We set the wage replacement rate for unemployed equal to z = 0.6 in (6),which follows the calibration in Kitao et al. (2017) for the U.S. and Miyamoto (2011) as wellas Kudoh et al. (2019) for Japan. In addition, the OECD reports an initial net replacement
rates of 62% for Japan (see Table 3.2 in OECD, 2006).29 Following Mortensen and Pissarides(1994) and Den Haan et al. (2000), among others, we assume symmetric bargaining power
ε = 0.5 between workers and firms. This implies an effective bargaining power of ψ = 0.714.
Contributions such as Shimer (2005), Miyamoto (2011) and Kudoh et al. (2019) have assigned
a higher bargaining power to workers of ε = 0.72 and ε = 0.6 respectively.
The matching process is calibrated to replicate empirical features of the Japanese labormarket. We set the job separation rate δ to match the average duration of a job. Miyamoto(2011) reports an average job duration rate of 18 years for Japan during the 1980 to 2010 period,
while more recent data by the Japan Institute for Labour Policy and Training is closer to 12
years. We hence choose δ = 0.015 for Japan, implying an average job duration of 16.7 years.30
Japan reports an average unemployment rate of ū = 3.5% over the 1980-2018 period (Figure1). In addition, data on the active job-openings-to-applicants ratio for the period 1980-2018 isavailable from the Japan Institute for Labour Policy and Training. We use the average value
29

It should be noted that there is substantial variation in the literature when it comes to calibrating thisparameter with values ranging between 0.2 and 0.7. Studies such as Shimer (2005) assign a slightly smaller value(z = 0.4) to the United States. Following our methodology, such a smaller value would result in a higher cost
parameter k. The results are, however, robust to this change in the calibration.
30
Japan’s job separation rate is substantially lower than the separation rate in the United States, reflecting
among others institutional features such as the traditional system of lifetime employment.

19

of θ̄ = 0.85 over this period as a measure of steady state labor market tightness. Based on thisdata, the Beveridge curve in (28) implies an average job-finding rate p(θ̄) = 0.41 on a quarterly
basis, which is in line with Miyamoto (2011), who reports a quarterly rate of 0.426 for Japan.
We calibrate the cost parameter k to yield an annual return on equity of 5% in the absence ofdemand shortage based on (29) with x = 1. This yields a vacancy cost of k = 0.431.31
As is standard in the literature (cf. Shimer, 2005; Miyamoto, 2011; Kudoh et al., 2019,
among others), we use the Hosios (1990) efficiency criterion and calibrate the parameter γ inthe matching function equal to the bargaining power parameter, i.e. γ = ε = 0.5, so that the
only inefficiencies emerge from the zero lower bound and the price rigidity. The scale parameterκ in the matching function is then calculated to match the average job finding rate p(θ). Thisimplies a value of κ = 0.449, which is in line with the values reported in Miyamoto (2011) andKudoh et al. (2019) based on a similar approach.
We conduct two robustness checks in the appendix. In robustness check I, we adjust thecalibration to match the corresponding features of the U.S. labor market with a higher separation
rate and a higher steady state unemployment rate. In robustness check II, we use a more generalmatching function with a constant elasticity of substitution between vacancies and unemployedfollowing Den Haan et al. (2000). Our main conclusions are robust to these modifications.

5

Unemployment and underemployment under stagnation

5.1

The employment gap under stagnation

In the stagnation steady state underemployment emerges in addition to structural unemployment. The total employment gap in the labor market can be decomposed as
1 − (1 − ut )xt =|{z}
Employment gap

ut
|{z}

Structural
unemployment+ (1 − ut )(1 − xt ) .
|
{z
}

(37)

Underemployment

While the unemployment rate captures the total employment gap in the standard model whenx = 1, it is an inadequate indicator of the slack in the labor market under stagnation.
How do firms optimally respond to the emergence of demand shortage? Consider the effects
of an exogenous increase in the desire to save, i.e. the marginal utility of wealth ω 0 (.), under
stagnation in (33). Higher desired savings result in excess supply in the goods market to whichfirms respond by reducing working hours and production as they are no longer able to sell
all of their products. This affects unemployment via two channels. First, the reduction in
firm profits induced by lower sales makes it less attractive for firms to enter the recruitment
market (job creation channel). Second, deflation increases, which makes money holdings more
attractive (deflation channel). Both effects imply that firms post fewer job openings, which
results in fewer vacancies and higher unemployment. Higher unemployment and lower working
31This value may be slightly higher compared to the existing literature. Miyamoto (2011) calibrate a value ofk = 0.31 for Japan. Yet, this is based on the assumption of a higher bargaining power of workers. We also can
attain lower values by assuming a higher bargaining power of workers. For instance, for ε = 0.6, the respective
value becomes k = 0.29.

20

Table 2: Steady state with underemployment
Shortfall of working hours 1 − xs (targeted)

2%

5%

10 %

20 %

Unemployment rate under stagnation us

3.47

3.52

3.62

3.84

Total employment gap in equation (37)

5.40

8.34

13.26

23.07

Underemployment share in employment gap

35.7

57.8

72.7

83.4

Difference in unemployment rates us − uf

0.04

0.11

0.22

0.49

Shortfall of consumption cs /cf − 1

-2.0

-5.1

-10.2

-20.5

Shortfall of production Y s /Y f − 1

-2.0

-5.1

-10.1

-20.4

Rows: (1) Targeted shortfall of working hours below potential, i.e. 1 − xs . Generated byadjusting the scale parameter β in the marginal utility of wealth. (2) Unemployment rate
(in %). (3) Total employment gap (in %), see equation (37). (4) Share of underemployment
in the total employment gap (in %). (5) Difference of unemployment rate relative tostandard model without demand shortage (in %). (6) Consumption shortfall relative tothe standard model without demand shortage (in %). (7) Consumption shortfall relative
to the standard model without demand shortage (in %).

hours reduce household income, the fall of which has to be sufficiently strong to reduce thehigher desired savings. In addition, the real wage, consumption and output fall. The following
proposition summarizes these effects.
Proposition 1 As stagnation worsens, due to an exogenous increase in the marginal utility of
wealth, both unemployment and underemployment increase while total output, household consumption and the real wage decline in steady state.
The employment gap widens as stagnation worsens due to rising unemployment and underemployment. The relative response of working hours and unemployment to worsening stagnation
depends on the slope of the Beveridge curve in (28) relative to the slope of the job creation
condition (36) as neither is directly affected by variations in the wealth premium. The steeper
the Beveridge curve and the flatter the job creation curve under stagnation, the stronger the
response of unemployment to demand shortage:dus
u0 (θ)=<0dxs
x0 (θ)(38)

Table 2 summarizes the properties of the stagnation steady state for the baseline calibration.32 The parameter β is calibrated to target a specific shortfall of working hours, which
is illustrated in the first row.33 All other properties follow from this target. Rows (2) to (4)show the unemployment rate, the total employment gap and the share of underemployment.
Rows (5) to (7) contrast the stagnation case to the standard case with x = 1 assuming that the32
These values are in fact barely affected by the specification of the matching function and the wealth preference.
This is in stark contrast to the effects of parameter variations discussed later.
33
Note that this is equivalently to specifying a negative value for the natural real interest rate for x = 1 (whichcannot be obtained due to the binding lower bound on the nominal interest rate).

21

Figure 4: Employment gap under stagnation

Note: This figure shows the decomposition of the employment gap 1 − (1 − u)x among the unemployment
rate u (dashed line) and underemployment (1 − u)(1 − x) (shaded area) as a function of the degree of
demand shortage (1 − x) (generated by variations in the scale parameter β of the wealth preference). The
baseline calibration is described in the main text, robustness checks I (based on U.S. data) and II (basedon a CES matching function) are discussed in the appendix.

negative real interest rate could be attained.34 Row (5) shows the difference in unemployment
rates, while rows (6) and (7) contain the shortfall of consumption and production.
As demand shortage worsens, the total employment gap widens considerably with bothunemployment and underemployment increasing. This is in stark contrast to the standard
case where a stronger desire to save reduces unemployment. However, the spillovers fromdemand shortage to the unemployment rate are quantitatively weak. Even at a 20% shortfall ofworking hours, structural unemployment is only about half a percentage point higher than in
the alternative steady state. Underemployment quickly dominates in the total employment gap
accounting for already one third of the gap at a 2 % shortfall of working hours and more than half
at a 5 % shortfall. In addition, we document strong spillover effects from underemployment in
the labor market into macroeconomic variables in the goods market. It is the shortfall of working
hours rather than the increase in the unemployment rate that accounts almost completely forthe shortfall in consumption and production under stagnation.
For further illustration, Figure 4 shows the decomposition of the steady state employment
gap into unemployment and underemployment (shaded area) for the baseline calibration andtwo alternative calibrations. These are based on U.S. labor market data (I), which most notably
features a higher job separation rate δ, and a different specification of the matching function (II),described in detail in the appendix. In all cases, underemployment rises steeply once stagnation
occurs and quickly dominates in the employment gap. In contrast, the response of structural
unemployment is weak, albeit spillover effects from demand shortage into unemployment areslightly stronger using the calibration based on the U.S. labor market. It is demand-driven
underemployment that provides a more accurate measure of the slack in the labor and goods
markets under stagnation.
We next analyze the robustness of these findings. For a given degree of demand shortage
34
This is the appropriate benchmark for comparisons since changes in the strength of the wealth preference
also affect the standard steady state. A stronger wealth preference, i.e. a stronger desire to save, reduces the realinterest rate, which increases labor market tightness and results in a lower unemployment rate.

22

Table 3: Comparative statics for the steady state unemployment rate
Calibration:

Baseline

k = 0.2

ψ = 0.5

γ = 0.2

γ = 0.8

δ = 0.03

α = 0.025

Unemployment rate us

3.52

2.41

2.25

3.71

3.35

6.96

3.52

Total employment gap

8.3

7.3

7.1

8.5

8.2

11.6

8.3

Underemployment share

57.7

67.0

68.5

56.5

59.1

40.1

57.7

Increase in unemployment

0.10

0.07

0.06

0.17

0.04

0.19

0.10

Parameter variations are in comparison to the baseline calibration in Table 1 for a steady state with a targeted
5% shortfall of working hours below potential. Rows: (1) Unemployment rate (in %). (2) Total employment
gap (in %), see equation (37). (3) Share of underemployment in the total employment gap (in %). (4) Increasein unemployment rate (in %) when shortfall of working hours increases to 10%.

1 − x, equilibrium labor market tightness and unemployment are affected by the parameters in
job creation condition (36) and the Beveridge curve (28) as well as the shape of the matching
function, but independent of the wealth preference. Table 3 summarizes the effects of someparameter changes in comparison to our baseline calibration for a given 5% shortfall of working
hours (corresponding to the second column in Table 2). For instance, higher firm productivityȳ or lower vacancy cost k are associated with a tighter labor market and hence a lower unemployment rate as, all else equal, they imply a higher return on firm ownership and hence more
incentives for posting job openings.35 Similarly, a weaker bargaining power of workers ψ results
in a lower unemployment rate via the same channel. The curvature parameter γ in the matching function affects the unemployment rate for given working hours via both the job creation
condition and the Beveridge curve, with lower values of γ resulting in a higher unemployment
rate compared to our baseline calibration (and vice versa).36 A higher job separation rate δ,
while reducing the return on equity, has the additional effect of shifting the Beveridge curve,
thereby directly increasing the unemployment rate. This is why variations in δ lead to substantial changes in the level of unemployment. Finally, more rapid price adjustment implies ahigher rate of deflation, which makes holding money more attractive relative to firm ownership.
Hence, unemployment increases though the magnitude is quantitatively discernible.
While variations in these parameters can substantially alter the steady state level of unemployment, they barely affect the magnitude of the spillover from demand-driven underemployment into the unemployment rate. This is illustrated in the last row of Table 3, which shows
the increase in the unemployment rate under worsening stagnation, specifically under the assumption that the shortfall of working hours increases from 5% to 10%. None of the parametervariations generates a substantial response of unemployment, implying that the lack of demandis primarily reflected in variations in underemployment.
Next, we modify the workings of the transmission channels that determine the spillovers
35
In fact, only the ratio of these parameters, i.e. effective vacancy cost relative to labor productivity, affects
the composition of the steady state employment gap of our model, which is why only one parameter variation isshown in Table 2.
36
In addition, the matching efficiency κ affects the equilibrium level of unemployment. However, by rewriting
the job creation condition and the Beveridge curve in terms of matching efficiency units, it is easy to show thatan increase in κ is equivalent to a proportional reduction in δ and α.

23

1Figure 5: Transmission channels
0.9

0.8

0.7

0.6

0.5

0.4

from underemployment into unemployment. These are illustrated by the solid arrows in Figure

0.3

5. On the one hand, an increase in underemployment following a stronger desire to save (via
the wealth preference) reduces the return on equity as firm profits decline, which reduces job
openings resulting in higher unemployment. This is the job creation channel. On the other

0.2

hand, underemployment worsens deflation which makes holding money more attractive further
reducing the incentives for job creation. This is the deflation channel. While we provide a

0.1

microfoundation for the job creation channel, the deflation channel is based on the reducedform specification of the Phillips curve in (23).
00four robustness
the sensitivityof our0.6results to the0.7specifi0.1We conduct0.2
0.3 checks to analyse
0.4
0.5
cation of the Phillips curve. These are summarized in part A of Table 4, which reports theunemployment rate for a given degree of underemployment under stagnation in comparison to
our baseline specification. Our first modification simply assumes a constant rate of deflation
under stagnation, effectively shutting down the deflation channel so that unemployment is only
affected via the job creation channel.37 This results in a slightly weaker response of the unemployment rate to worsening stagnation, even though the effect is barely noticeable. In turn, this
implies that our baseline results are not driven by the deflation but rather by the job creation
channel. We then strengthen the deflation channel by assuming a stronger response of deflation
to underemployment with a Phillips curve coefficient of α = 0.025, which is a fivefold increase
relative to the baseline calibration. While the stronger response of deflation to underemployment results in a stronger spillover into unemployment, the effect is quantitatively small yielding
only an additional 0.03 percentage points increase in unemployment despite a 20% shortfall ofworking hours.38 Next, we drop our assumption that deflation is driven by underemployment
37

In Table 4, we assume price stability under stagnation to avoid a discontinuity at x = 1. Assuming a positive,
but constant rate of deflation results in a higher steady state level of unemployment, but does not yield stronger
spillover effects since the deflation channel still remains shut down.
38
We could consider even larger values of α although this requires some adjustments of other parameters toensure that the existence conditions in Lemma 2 are met. Our simulations show that even unrealistically large

24

0.8

Table 4: Transmission channels
Shortfall of working hours 1 − xs (targeted)

2%

5%

10%

20%

Unemployment rate with baseline calibration

3.47

3.52

3.62

3.84

A. Deflation channel: −πt = α1 (1 − xt ) + α2 (ut − ũ)Constant deflation rate: α1 = 0, α2 = 0

3.47
(-0)

3.52
(-0)

3.62
(-0)

3.83
(-0.01)

Steeper Phillips curve: α1 = 0.025, α2 = 0

3.47
(+0)

3.53
(+0.01)

3.63(+0.01)3.87(+0.03)

Unemployment-based: α1 = 0, α2 = 1

3.47
(+0)

3.53
(+0)

3.63
(+0.01)3.86(+0.02)

Based on employment gap: α1 = 0.025, α2 = 1

3.47
(+0)

3.53
(+0.01)3.64(+0.02)3.90(+0.06)

Weak spillover effects: y0 = 1, y1 = 0.1

3.47
(+0)

3.53
(+0.01)3.64(+0.02)3.88(+0.04)

Strong spillover effects: y0 = 1, y1 = 1

3.50(+0.03)3.61(+0.09)3.81(+0.19)4.29(+0.45)

Weak spillover effects: δ0 = 0.015, δ1 = 0.1

3.92(+0.46)

4.68(+1.16)

5.99(+2.37)

8.81(+4.97)

Strong spillover effects: δ0 = 0.015, δ1 = 1

7.97(+4.50)14.66(+11.14)

25.43(+21.81)44.88(+41.04)

B. Job creation channel: ȳ = y0 − y1 (1 − xt )C. Job destruction channel: δ = δ0 + δ1 (1 − xt )Note: Numbers in brackets show the change in the unemployment rate relative to the baseline case.
ũ is the unemployment rate when θ = θ̃ and x = 1.

and instead assume a traditional Phillips curve with a (strong) response of deflation to unemployment. This specification creates a feedback loop from unemployment into the attractivenessof holding money and thereby into the job creation incentives of firms resulting in further unemployment as represented by the upper dotted arrow in Figure 5. Despite its appeal, thequantitative contribution of this channel is limited even under the extreme assumption of aone-on-one response of deflation to the unemployment rate. Our final specification combines
the two previous ones allowing for a strong response of deflation to both underemployment
and unemployment. Naturally, spillovers are strongest under this specification. However, theirquantitative importance is still limited, further highlighting the robustness of our results to thespecification of the Phillips curve.
When it comes to the job creation channel, our baseline specification assumes a linear
production function with constant labor productivity ȳ in (11). Yet, the literature suggests
values of α add quantitatively little to the response of unemployment. For instance, a value of α = 0.1 only adds
another 0.1 percentage points of unemployment for a 20% shortfall of working hours.

25that there might be increasing returns to working hours. One factor, put forward among othersby Bewley (1999), is the negative effect of shorter hours on workers’ morale. If involuntary
underemployment reduces workers’ motivation and lowers their productivity, firms will be morereluctant to cut working hours resulting in a potentially stronger response of unemployment
instead of underemployment by reinforcing the job creation channel. This is illustrated by thedotted arrow in the center of Figure 5. While we do not provide a full microfoundation forthis effect, a reduced-form analysis shows that the additional effect of such a mechanism on
unemployment under stagnation is relatively weak. Specifically, part B of Table 4 reports theunemployment rate for a given degree of underemployment under the assumption that laborproductivity is linearly decreasing with a shortfall of working hours. We consider the casesof a weak response of productivity (0.1 percentage point reduction for each percentage point
shortfall of working hours) and a strong response (1 percentage point drop). Yet, even in thelatter case, the additional response of unemployment is substantially smaller compared to theincrease in underemployment, even for a substantial degree of demand shortage.
The main reason why these modifications of the deflation and job creation channels have notresulted in a significantly stronger response of the unemployment rate is that their effects on
unemployment work only indirectly through changes in labor market tightness. Put differently,
the degree of stagnation does not affect the steady state Beveridge curve. Instead, the economy
moves to a new steady state on the same Beveridge curve as stagnation worsens.
In our framework, the stability of the Beveridge curve is a consequence of the assumption of aconstant job separation rate. As our primary motivation is to explain the absence of unemployment during Japan’s stagnation period, this assumption is intended to reflect the institutional
structure of the Japanese labor market, in particular the system of lifetime employment forcore workers and the seniority-based system of remuneration and promotions. Houseman andOsawa (1998) and Ariga and Okazawa (2011) provide empirical support for the relative stabilityof Japan’s job destruction rate over time, particularly in comparison to the U.S. economy.
In economies with more flexible firing practices, job separations are likely to increase under
stagnation. This affects the unemployment rate in two ways as illustrated by the dashed arrows
in the lower part of Figure 5. A higher job destruction rate δ reduces firm profits, thereby
indirectly raising the unemployment rate via fewer job openings. In addition, the higher inflow
into unemployment shifts the Beveridge curve directly contributing to higher unemployment.
Relying again on a reduced-form specification, part C of Table 4 shows the spillovers effects associated with such a systematic variation in the job separation rate. Starting from the baseline
calibration of δ = 0.015 in the absence of stagnation, the two scenarios assume a sensitivityof the separation rate of 0.1 (weak scenario) and 1 (strong scenario) with respect to underemployment implying job separation rates of 0.025 and 0.115 for a 10% shortfall of work hours
respectively. Even in the weak scenario, the additional effect on the unemployment rate is
substantial, adding 2.4 percentage point additional unemployment at the 10% shortfall levelcompared to the baseline model.
Hence, while this model offers an explanation of the seemingly paradox case of Japan’slabor market during its lost decades, its application to other advanced economies requires more26

caution. The assumption of an exogenous separation rate seems to be well-justified considering
the institutional structure of the Japanese labor market and allows for the isolation of the effectsof persistent stagnation on the job creation incentives of firms. Yet, its potential interactions
with job destruction through shifts of the Beveridge curve, whose microfoundation is well beyond
the scope of the analysis in this paper, are likely to be of importance when applying thisframework to economies with a different institutional structure of the labor market.

5.2

Comparative statics: Labor market policies

The following proposition summarizes the steady state effects of two traditional labor market
policies, variations in the replacement rate z and the introduction of a search cost subsidy s.
Proposition 2 Labor market policies have the following effects on labor market tightness θ,
the unemployment rate u, working hours x, output Y and consumption c in steady state:
Stagnation steady state:

θ

u

x

Y

c

Replacement rate z

-

+

+/-

+/-

+/-

Search cost subsidy: s

+/-

+/-

+/-

+/-

+/-

Standard steady state:

θ

u

x

Y

c

Replacement rate z

-

+

0

-

+/-

Search cost subsidy s

+

-

0

+

+/-

Consider first the standard case. A lower replacement rate z increases firm profits as theeffective bargaining power of workers ψ falls. Firms enter the labor market and unemployment
is reduced. This is the job creation channel. The tighter labor market affects both household
wealth due to a higher firm valuation in (15) and net income in (31) creating a novel transmission channel through changes in the wealth premium. The working of the wealth channel canreinforce the job creation channel or work opposite to it depending on the shape of the wealthpremium. Yet, it is the job creation channel that dominates in the absence of demand shortage.
This can easily be seen graphically in Figure 2 as the lower replacement rate shifts the returnon equity upwards without affecting optimal household behavior directly. A search cost subsidy
has the additional effect of directly increasing the firm value, thereby stimulating consumption
demand.39 Both policies result in a tighter labor market with a lower unemployment rate.
Lower unemployment in turn increases output as is clear from (21), whereas the overall effecton consumption is undetermined due to the hump-shaped patterns of (31).40
39

Graphically, introducing a cost subsidy leads to an upwards shift in both the return on equity and the naturalrate in Figure 2, but the shift in the former always dominates the shift in the latter.
40
Michaillat and Saez (2022) analyze business cycle fluctuations in the slack equilibrium in a model with awealth preference. This focus seems natural since it implies that an increase in labor market tightness raises
total final output (ȳ(1 − u)) by more than total search costs (kv = kθu). Pissarides (2000, Chapter 8) provides
a general discussion of the interactions of labor market tightness and consumption in a matching model.

27Under stagnation, the job creation channel no longer necessarily dominates the wealth channel, which is why the effects of labor market policies are ambiguous. In fact, the relative strengthof the transmission channels depends on the behavior of the wealth premium, which is related
to the curvature of the wealth preference and the shape of the consumption function in (31).
As before, a lower replacement rate z increases firm profits leading to market entry, a tighterlabor market and less unemployment for given working hours. These affect the goods market
via changes in the wealth premium. On the one hand, consumption demand is stimulated by
the higher firm valuation.41 On the other hand, lower unemployment and a larger number offirms affect the amount of production allocated to consumption, which can increase or decreasedepending on the shape of (31) for a given x.
Assume the extreme case of a linear wealth preference, a low degree of demand shortage
and a slack steady state. In this case, the increase in labor market tightness results in a higher
wealth premium, which increases desired savings and reduces consumption demand. Firms
respond to the lower demand by cutting production, which is why working hours drop andunderemployment rises. The reduction in working hours reduces desired savings and thereby
eliminates the excess supply.42 It also mitigates the initial rise in labor market tightness.
Deflation also increases and so does the real interest rate, which reduces consumption.
Assume instead a concave wealth preference or a high degree of demand shortage or a tightsteady state. In this case, the increase in labor market tightness has the opposite effect as it
reduces the wealth premium. This stimulates consumption demand and allows firms to increase
production and sales, which is why working hours increase and underemployment declines.
The increase in working hours reduces excess demand and reinforces the initial rise in labormarket tightness as holding money becomes less attractive due to lower deflation. Household
consumption rises substantially in response to both a lower real interest rate and higher wealth.
Table 5 shows the effects of an increase in the replacement rate z by 5% as well as the effects
of introducing a s = 10% search cost subsidy for two specifications of the wealth preference.
The first column for each policy shows the results for a linear wealth preference, corresponding
to the first scenario above, the second column refers to the CARA specification, reflecting thesecond scenario. We assume 5% underemployment in the steady state before the policy.43
All parameter variations in Table 5 have the same qualitative effects on labor market tightness and unemployment as in the standard case. Their effects on unemployment under stagnation are also quantitatively similar to the case without demand shortage. A higher replacement
rate reduces labor market tightness and increases structural unemployment under stagnation.
As the cost subsidy also affects the firm value in (15) and hence the wealth premium and saving
41
The strength of this channel depends on the strength and curvature of the wealth preference and the shapeof the matching function, which affects the response of wealth to changes in labor market tightness. This channel
does not operate in the extreme case of a linear wealth preference and hence a constant marginal utility of wealth.
42
Strictly speaking, this only holds for a sufficiently small intertemporal elasticity of substitution for consumption ηc−1 . Specifically, ηc > αρ is a sufficient (but not necessary) condition for an increase (decrease) in working
hours to result in excess supply (demand) in the goods market for a given θ as shown in Appendix F. This is aweak restriction and we will assume that it holds throughout our analysis.
43
Tables 5 and 6 quantify the effects of parameter variations for our baseline calibration. The respective tables
for the alternative calibrations can be found in the appendix. All effects are shown for the stagnation steadystate with xs = 0.95 before the parameter variation.

28

Table 5: Steady state effects of labor market policies
Benefits z ↑Change in:

Subsidy s

LinearCARA

Linear

CARA

-0.15

1.43

0.11

1.27

0.23

0.27

-0.18

-0.16

0.07

1.63

-0.06

1.08

Consumption ∆c /c (in %)

0.01

-1.70

-0.01

-1.27

Production ∆Y s /Y s (in %)

-0.08

-1.78

0.07

-1.18

Unemployment ∆uf

0.23

0.22

-0.17

-0.18

Consumption ∆c /c (in %)

-0.15

-0.15

0.10

0.12

Production ∆Y f /Y f (in %)

-0.24

-0.23

0.18

0.18

Underemployment ∆(1 − xs )sUnemployment ∆u

Employment gap in (37)
s

f

s

f

Policies: (i) Increase in the replacement rate from z = 0.6 to z = 0.65. (ii)Introduction of a search cost subsidy of s = 10%.
Rows: (1) Change in underemployment (in pp). (2) Change in unemployment
rate (in pp). (3) Change in total employment gap (in pp). (4) Change inprivate consumption (in %). (5) Change in production (in %). (6) Change inunemployment rate in the absence of demand shortage (in pp). (7) Change
in consumption in the absence of demand shortage (in %). (8) Change inproduction in the absence of demand shortage (in %).

patterns of the household, the combined effect of this policy is unclear and labor market tightness could in principle drop, following lower costs. However, this requires a sufficiently strong
wealth channel and does not occur in our simulations. It could, however, result from a sufficiently strong curvature of the wealth utility in combination with substantial underemployment
and a low degree of complementarity of vacancies and unemployed in the matching function.
While the effects of these policies on the unemployment rate under stagnation are quantitatively and qualitatively similar to those of the standard model, their implications for underemployment depend critically on the shape of the wealth premium as explained above. In ourbaseline specification with a constant marginal utility of wealth, the response of underemployment is in stark contrast to the unemployment rate with underemployment declining in response
to higher unemployment benefits and increasing following the introduction of a cost subsidy.
Hence, unemployment and underemployment move in opposite directions and the response of
the total employment gap is indeterminate. In contrast, our alternative specification of a CARA
wealth preference results in co-movements of underemployment and unemployment.
In general, we document co-movements of underemployment and unemployment when the
degree of stagnation is small, vacancies and unemployed are closer substitutes in the matchingprocess and for a stronger curvature of the utility from wealth. In contrast, underemployment
and unemployment move in opposite directions following wage and costs shocks when stagnation
is sufficiently severe, for a higher degree of complementarity of vacancies and unemployed and fora lower concavity of the wealth preference.44 The effects of wage and cost shocks on consumption
44
Note that the steady state without aggregate demand shortage is slack in our simulation so that x increases
following higher wages in case of a constant marginal utility of wealth in Table 5.

29and output follow directly from this analysis. In fact, for a constant marginal utility of wealth,consumption and working hours have to change in the same direction.45 The spillover effects
from the labor market into consumption and output are weak in our baseline specification but
can be quite substantial in the alternative case of a CARA specification.
To conclude, the effects of traditional labor market policies under stagnation are ambiguous
and depend on several factors related to the functional form of the wealth preference and thematching function. Policies that focus exclusively on reducing the unemployment rate might infact increase the total employment gap via their effect on underemployment and should only be
employed with great caution.

5.3

Comparative statics: Demand and supply shocks

The following proposition summarizes the steady state effects of demand and supply shocks inthe form of variations in government spending g and labor productivity ȳ.
Proposition 3 Exogenous variations in demand and supply have the following effects on labormarket tightness θ, the unemployment rate u, working hours x, output Y and consumption c insteady state:
Stagnation steady state:

θ

u

x

Y

c

Supply shock ȳ

-

+

-

-

-

Demand shock g

+

-

+

+

+

Standard steady state:

θ

u

x

Y

c

Supply shock ȳ

+

-

0

+

+/-

Demand shock g

-

+

0

-

+/-

Supply and demand shocks unambiguously result in co-movements of structural unemployment and underemployment under stagnation. These shocks have opposite effects on both labormarket variables and macroeconomic variables under stagnation compared to the standard case.
Their qualitative effects are independent of the shape of the wealth preference and the matching
function. The reversal of the effects compared to the standard model is a direct consequence ofthe shortage of aggregate demand under stagnation.
Higher labor productivity ȳ raises household income and the desire to save in the goods
market resulting in lower demand. Firms respond to reduced sales by cutting working hours.
In fact, the decline in working hours overcompensates the effect of the increase in productivity
so that firm profits drop and firms exit the market. Labor market tightness falls and bothunemployment and underemployment increase. This reduces household consumption and total
output, thereby overcompensating the effect of the higher productivity. This “paradox of toil” isa common occurrence in stagnation models and in stark contrast to the responses of consumption
45

Any increase (decrease) in working hours directly lowers (increases) the real interest rate, which requires anincrease (a reduction) in the wealth premium. With ω 0 (b) = β, consumption has to increase (decrease).

30

Table 6: Steady state effects of supply and demand shocks
Productivity ȳ ↑Change in:

Spending g ↑LinearCARA

Linear

CARA

1.02

1.03

-1.11

-1.26

(+)0

(+)0

-0.02

-0.02

0.99

1.00

-1.09

-1.24

Consumption ∆c /c (in %)

-0.09

-0.10

0.09

0.26

Production ∆Y s /Y s (in %)

-0.09

-0.10

1.19

1.36

Unemployment ∆uf

-0.02

-0.02

(+)0

(+)0

Consumption ∆c /c (in %)

1.03

1.03

-1.05

-1.05

Production ∆Y f /Y f (in %)

1.02

1.02

(-)0

(-)0

Underemployment ∆(1 − xs )sUnemployment ∆u

Employment gap in (37)
s

f

s

f

Shocks: (i) 1% increase in productivity (or potential output) ȳ. (ii) Increase inpublic spending from g = 0 to g = 0.01 (i.e. 1% of potential output).
Rows: (1) Change in underemployment (in pp). (2) Change in unemployment
rate (in pp). (3) Change in total employment gap (in pp). (4) Change inprivate consumption (in %). (5) Change in production (in %). (6) Change inunemployment rate in the absence of demand shortage (in pp). (7) Change
in consumption in the absence of demand shortage (in %). (8) Change inproduction in the absence of demand shortage (in %).

and output in the standard model where higher productivity increases firm profits making
market entry more attractive to firms.
Higher demand via higher government spending g results in a tighter labor market with botha lower unemployment rate and higher realized working hours. The reason is as follows. Higher
public spending creates additional demand to which firms respond by increasing production.
Higher firm profits and less deflation make investment in firms more attractive. As more firms
enter the market, labor market tightness increases and unemployment declines. This in turnstimulates consumption. Hence, there is a crowding-in effect of government spending.46 In
contrast, the preference for wealth creates a crowding out effect of government spending in thestandard model as higher government spending increases the real interest rate, which reduces
the number of vacancies as firms leave the labor market and increases unemployment.47
Table 6 summarizes the effects of supply and demand shocks, specifically a 1% increase inproductivity ȳ and an increase in government spending to g = 0.01 (i.e. 1% of potential output).
Quantitatively, these macro shocks cause substantial variations in working hours, consumption and output. However, their spillover effects into the unemployment rate are weak as
fluctuations in the total employment gap are overwhelmingly driven by changes in underemployment, further highlighting the insufficiency of the unemployment rate to capture the total
slack in the labor and goods market of a stagnating economy.
46
Note that this is not related to the notion of deficit spending. In fact, any increase in government spendingis budget neutral as the government runs a balanced budget each period by equation (20).
47Note that this is a novel transmission channel for shocks which extends the standard search and matchingmodel, where demand variations have no effect on firm creation in steady state as the real interest rate isdetermined by the time preference rate ρ.

31

6Conclusion

We have presented a model of secular stagnation that distinguishes between structural unemployment and demand-driven underemployment as a manifestation of economic slack in thelabor market. In a permanent liquidity trap, the unemployment rate becomes an insufficientand potentially misleading indicator of the extent of the aggregate demand problem leading toinadequate policy conclusions. Under stagnation, the lack of demand can express itself in various forms of underemployment, such as part-time or non-regular employment. These provide
better indicators of the stance of the economy and the need for policies in support of demand.
Our findings also suggest that traditional labor market policies aiming to improve the supply
side might still succeed in lowering unemployment, but at the same time could contributeto more widespread underemployment in a stagnating economy. Put differently, albeit thesepolicies can succeed in creating new jobs, these are non-regular ones such as part-time jobs.
As a consequence, increases in the employment rate are not reflected in an expansion in total
output but might in fact be hurtful for both output and consumption. This insight helps
to explain the continued sluggishness in the Japanese economy despite the seemingly decent
employment record. Examples of such policies include reductions in unemployment benefits,
job-creation subsidies to firms or policies aimed at reducing the bargaining power of workers inwage negotiations. These policies should therefore be used only with great caution. In contrast,
our model highlights the need for policies that improve aggregate demand as such measuresboth create new jobs and decrease the degree of underemployment.

References
Andolfatto, David, “Business Cycles and Labor-market Search,” American Economic Review,
1996, 86 (1), 112–132.
Ariga, Kenn and Ryosuke Okazawa, “Labor Immobility in Japan: Its Causes and Consequences,” in K. Hamada, A. Kashyap, and D. E. Weinstein, eds., Japan’s Bubble, Deflation,and Long-term Stagnation, MIT Press, Cambridge, 2011, chapter 8, pp. 263–308.
Ashenfelter, Orley C., Henry Farber, and Michael R. Ransom, “Labor Market Monopsony,” Journal of Labor Economics, 2010, 28 (2), 203–210.
Bell, David N. F. and David G. Blanchflower, “Underemployment and the Lack of Wage
Pressure in the UK,” National Institute Economic Review, 2018, 243 (1), R53–R61.
and , “Underemployment in the United States and Europe,” ILR Review, 2021, 74 (1),56–94.
Benhabib, Jess and Alberto Bisin, “Skewed Wealth Distributions: Theory and Empirics,”
Journal of Economic Literature, 2018, 56 (4), 1261–1291.
Bewley, Truman F., Why Wages Don’t Fall During a Recession, Harvard University Press,
1999.
Blanchflower, David G., Not Working: Where Have All the Good Jobs Gone?, Princeton
University Press, 2019.

32

Caballero, Ricardo J. and Emmanuel Farhi, “The Safety Trap,” Review of Economic
Studies, 2018, 85 (1), 223–274.
Diamond, Peter A., “Wage Determination and Efficiency in Search Equilibrium,” Review of
Economic Studies, 1982, 49 (2), 217–227.
Eggertsson, Gauti B., Neil R. Mehrotra, and Jacob Robbins, “A Model of Secular
Stagnation: Theory and Quantitative Evaluation,” American Economic Journal: Macroeconomics, 2019, 11 (1), 1–48.
Fagereng, Andreas, Martin Blomhoff Holm, Benjamin Moll, and Gisle Natvik, “Saving Behavior Across the Wealth Distribution: The Importance of Capital Gains,” Working
Paper, Oslo University and Princeton University 2019.
Fang, Lei and Richard Rogerson, “Policy Analysis in a Matching Model with Intensive andExtensive Margins,” International Economic Review, 2009, 50 (4), 1153–1168.
Haan, Wouter J. Den, Garey Ramey, and Joel Watson, “Job Destruction and Propagation of Shocks,” American Economic Review, 2000, 90 (3), 482–498.
Hamaaki, Junya, Masahiro Hori, Saeko Maeda, and Keiko Murata, “Changes in theJapanese Employment System in the Two Lost Decades,” Industrial and Labor Relations
Review, 2012, 65 (4), 810–846.
Hansen, Alvin, “Economic Progress and Declining Population Growth,” American Economic
Review, 1939, 29 (1), 1–15.
Hashimoto, Masanori and John Raisian, “Employment Tenure and Earnings Profiles inJapan and the United States,” American Economic Review, 1985, 75 (4), 721–735.
Hosios, Arthur J., “On the Efficiency of Matching and Related Models of Search and Unemployment,” Review of Economic Studies, 1990, 57 (2), 279–298.
Houseman, Susan and Machiko Osawa, “What is the Nature of Part-Time Work in
the United States and Japan?,” in Colette Fagan and Jacqueline O’Reilly, eds., Part-Time
Prospects: An International Comparison of Part-time Work in Europe, North America andthe Pacific Rim, Routledge, 1998, chapter 12, pp. 232–251.
Illing, Gerhard, Yoshiyasu Ono, and Matthias Schlegl, “Credit Booms, Debt Overhang
and Secular Stagnation,” European Economic Review, 2018, 108, 78–104.
Japan Institute for Labour Policy and Training, “Trends in Non-regular Employment
in Japan and Analysis of Several Related Themes,” in “Labor Situation in Japan and Its
Analysis: Detailed Exposition 2014/2015,” Japan Institute for Labour Policy and Training,
2015, chapter 1, pp. 1–21.
Kambayashi, Ryo and Takao Kato, “The Japanese Employment System after the Bubble
Burst: New Evidence,” in K. Hamada, A. Kashyap, and D. E. Weinstein, eds., Japan’s Bubble,
Deflation and Long-term Stagnation, MIT Press, Cambridge, 2011, chapter 7, pp. 217–261.
Kang, Hyunju, Jaevin Park, and Hyunduk Suh, “The Rise of Part-time Employment in
the Great Recession: Its Causes adn Macroeconomic Effects,” Journal of Macroeconomics,
2020, 66.
Kitao, Sagiri, Lars Ljungqvist, and Thomas J. Sargent, “A Life-cycle Model of TransAtlantic Employment Experiences,” Review of Economic Dynamics, 2017, 25, 320–349.

33

Kudoh, Noritaka, Hiroaki Miyamoto, and Masaru Sasaki, “Employment and Hours
over the Business Cycle in a Model with Search Frictions,” Review of Economic Dynamics,
2019, 31, 436–461.
Kumhof, Michael, Romain Rancière, and Pablo Winant, “Inequality, Leverage, andCrises,” American Economic Review, 2015, 105 (3), 1217–1245.
Landais, Camille, Pascal Michaillat, and Emmanuel Saez, “A Macroeconomic Approach
to Optimal Unemployment Insurance: Application,” American Economic Journal: Economic
Policy, 2015, 10 (2), 182–216.
, , and , “A Macroeconomic Approach to Optimal Unemployment Insurance: Theory,”
American Economic Journal: Economic Policy, 2015, 10 (2), 152–181.
Manning, Alan, Monopsony in Motion, Princeton University Press, 2003.
Merz, Monika, “Search in the Labor Market and the Real Business Cycle,” Journal of Monetary Economics, 1995, 36 (2), 269–300.
Mian, Atif, Ludwig Straub, and Amir Sufi, “Indebted Demand,” Quarterly Journal of
Economics, 2021, 136 (4), 2243–2307.
Michaillat, Pascal, “Do Matching Frictions Explain Unemployment? Not in Bad Times,”
American Economic Review, 2012, 102 (4), 1721–1750.
and Emmanuel Saez, “Aggregate Demand, Idle Time, and Unemployment,” The Quarterly
Journal of Economics, 2015, 130 (2), 507–569.
and , “Resolving New Keynesian Anomalies with Wealth in the Utility Function,” The
Review of Economics and Statistics, 2021, 103 (2), 197–215.
and , “An Economical Business-Cycle Model,” Oxford Economic Papers, 2022, 74 (2),382–411.
Michau, Jean-Baptiste, “Secular Stagnation: Theory and Remedies,” Journal of Economic
Theory, 2018, 176, 552–618.
, “The Preference for Net Wealth,” 2019. Working Paper, Ecole Polytechnique.
, “Helicopter Drops of Money under Secular Stagnation,” 2022. Working Paper, Ecole Polytechnique.
, Yoshiyasu Ono, and Matthias Schlegl, “Wealth Preference and Rational Bubbles,”CESifo Working Paper No. 7148 2018.
,

, and

, “Wealth Preference and Inequality,” 2020. unpublished manuscript.

Miyamoto, Hiroaki, “Cyclical Behavior of Unemployment and Vacancies in Japan,” Japanand the World Economy, 2011, 23 (3), 214–225.
Mortensen, Dale T., “The Matching Process as a Noncooperative Bargaining Game,” in “TheEconomics of Information and Uncertainty” NBER Chapters, National Bureau of Economic
Research, Inc, May 1982, pp. 233–258.
and Christopher A. Pissarides, “Job Creation and Job Destruction in the Theory of
Unemployment,” Review of Economic Studies, 1994, 61 (3), 397–415.
OECD, “OECD Employment Outlook,” Technical Report, Organization for Economic Cooperation and Development 2006.
34

Ono, Yoshiyasu, Money, Interest, and Stagnation - Dynamic Theory and Keynes’s Economics,
Oxford: Clarendon Press, 1994.
, “A Reinterpretation of Chapter 17 of Keynes’s General Theory: Effective Demand Shortage
under Dynamic Optimization,” International Economic Review, 2001, 42 (1), 207–236.
Pissarides, Christopher A., “Short-run Equilibrium Dynamics of Unemployment Vacancies,
and Real Wages,” American Economic Review, 1985, 75 (4), 676–690.
, Equilibrium Unemployment Theory, 2nd ed., MIT Press, Cambridge, MA, 2000.
Rogerson, Richard, Robert Shimer, and Randall Wright, “Search-theoretic Models of
the Labor Market: A Survey,” Journal of Economic Literature, 2007, 43, 959–988.
Saez, Emmanuel and Stefanie Stantcheva, “A Simpler Theory of Optimal Capital Taxation,” Journal of Public Economics, 2018, 162, 120–142.
Schlegl, Matthias, “Secular Stagnation in an Economy with Land,” ISER Discussion Paper
No. 1032, Institute of Social and Economic Research, Osaka University 2018.
Schmitt-Grohé, Stephanie and Martı́n Uribe, “Downward Nominal Wage Rigidity, Currency Pegs, and Involuntary Unemployment,” Journal of Political Economy, 2016, 124 (5),1466–1514.
and , “Liquidity Traps and Jobless Recoveries,” American Economic Journal: Macroeconomics, 2017, 9 (1), 165–204.
Shimer, Robert, “The Cyclical Behavior of Equilibrium Unemployment and Vacancies,”
American Economic Review, 2005, 95 (1), 25–49.
Shimizutani, Satoshi and Izumi Yokoyama, “Japan’s Long-term Employment PracticeSurvived? Developments since the 1990s,” Industrial and Labor Relations Review, 2009, 62(3), 313–326.
Zou, Heng-fu, “‘The Spirit of Capitalism’ and Long-Run Growth,” European Journal of Political Economy, 1994, 10 (2), 279–293.

35

Appendix: Robustness checks
We summarize the result for two alternative calibrations in this appendix. In robustness checkI, we match the model to the U.S. labor market data, using the same approach as in the maintext. For robustness check II, we employ a different specification of the matching function,
specifically a general CES function following Den Haan et al. (2000). The associated changes in
the calibration for both robustness checks are summarized in Table 7. Otherwise the calibrationin Table 1 remains unchanged.
Table 7: Changes in parameter calibration
Robustness check I: Calibration based on U.S. data
Job separation rate

δ = 0.1

average job duration of 2.5 years

Cost of a vacancy

k = 0.361

match r = 5% (yearly) in (29) for θ̄ = 1 and x = 1

Scale parameterκ = 1.487

match ū = 0.063 and θ̄ = 1

Robustness check II: CES matching function
CES parameterγ = 1.27

following Den Haan et al. (2000)

Scale parameterκ = 0.778

match ū = 0.035 and θ̄ = 0.85

Robustness check I: U.S. labor market data
First, we calibrate the matching process based on U.S. data. Shimer (2005) estimates anaverage job duration of 2.5 years in the U.S. for the period 1951 to 2003, which is substantiallyshorter than in the respective value for Japan. We use this estimate to calibrate a quarterly
separation rate of δ = 0.1. Based on the average unemployment rate in the United States overthe 1980-2018 period of ū = 6.3%, the Beveridge curve in (28) then allows us to calculate theaverage job-finding rate p(θ) implied by the model. The implied value of 1.49 (on a quarterly
basis) is close to the ones in Shimer (2005), who estimates the monthly job-finding rate to beequal to 0.45, which implies an average quarterly value of 1.35.
In contrast to Japan, there is no comparable data on labor market tightness available for theUnited States over an equally long time period. We therefore follow Shimer (2005) and target
a steady state value of θ̄ = 1, which we use to calibrate the scale parameter κ = 1.487 in thematching function to be in line with the average job finding rate p(θ).
Finally, the cost parameter k = 0.361 is chosen to yield an annual return on equity of 5%in the absence of aggregate demand shortage using the steady state expression (29) with x = 1
and the average labor market tightness value.
Table 8 summarizes the properties of the stagnation steady state for different targeted valuesof the working hour shortfall, which are generated by variations in the strength of the wealthpreference. Tables 9 and 10 illustrate the effects of parameter variations on underemployment
and the unemployment in the stagnation steady state.
This calibration results in a higher steady state unemployment rate and slightly stronger

36

Table 8: Steady state with underemployment
Shortfall of working hours 1 − xs (targeted)

2%

5%

10 %

20 %

Unemployment rate under stagnation us

6.33

6.43

6.6

6.99

Total employment gap in equation (37)

8.2

11.1

16.0

25.6

Underemployment share

22.8

42.1

58.6

72.7

Difference in unemployment rates us − uf

0.07

0.17

0.35

0.77

Shortfall of consumption cs /cf − 1

-2.1

-5.2

-10.5

-20.9

Shortfall of production Y s /Y f − 1

-2.1

-5.2

-10.4

-20.7

Rows: (1) Targeted shortfall of working hours below potential, i.e. 1 − xs . Generated byadjusting the scale parameter β in the marginal utility of wealth. (2) Unemployment rate
(in %). (3) Total employment gap (in %), see equation (37). (4) Share of underemployment
in the total employment gap (in %). (5) Difference of unemployment rate relative tostandard model without demand shortage (in %). (6) Consumption shortfall relative tothe standard model without demand shortage (in %). (7) Consumption shortfall relative
to the standard model without demand shortage (in %).

Table 9: Steady state effects of labor market policies
Benefits z ↑Subsidy s

Change in:LinearCARA

LinearCARA

Underemployment ∆(1 − xs )

-0.28

0.11

0.2

0.46

Unemployment ∆us

0.41

0.43

-0.32

-0.31

Employment gap in (37)

0.13

0.51

-0.12

0.14

Consumption ∆cs /cs (in %)

0.02

-0.40

-0.02

-0.31

s

Production ∆Y /Y (in %)

-0.15

-0.57

0.13

-0.16

Unemployment ∆uf

0.41

0.41

-0.31

-0.32

Consumption ∆cf /cf (in %)

-0.28

-0.27

0.20

0.20

-0.44

-0.44

0.34

0.34

s

f

Production ∆Y /Yf(in %)

Policies: (i) Increase in the replacement rate from z = 0.6 to z = 0.65. (ii)Introduction of a search cost subsidy of s = 10%.
Rows: (1) Change in underemployment (in pp). (2) Change in unemployment
rate (in pp). (3) Change in total employment gap (in pp). (4) Change inprivate consumption (in %). (5) Change in production (in %). (6) Change inunemployment rate in the absence of demand shortage (in pp). (7) Change
in consumption in the absence of demand shortage (in %). (8) Change inproduction in the absence of demand shortage (in %).

spillovers from demand shortage into the labor market as also illustrated in Figure 4. Yet,underemployment continues to be the decisive factor in explaining variations in the total employment gap as well as the shortfall of consumption and production relative to the steady state
without demand shortage.

37

Table 10: Steady state effects of supply and demand shocks
Productivity ȳ ↑Change in:

Spending g ↑LinearCARA

Linear

CARA

1.02

1.02

-1.13

-1.17

(+)0

(+)0

-0.04

-0.04

0.95

0.96

-1.10

-1.13

Consumption ∆c /c (in %)

-0.09

-0.09

0.10

0.14

Production ∆Y s /Y s (in %)

-0.09

-0.09

1.23

1.28

Unemployment ∆uf

-0.03

-0.03

(+)0

(+)0

Consumption ∆c /c (in %)

1.05

1.05

-1.10

-1.10

Production ∆Y f /Y f (in %)

1.04

1.04

(-)0

(-)0

Underemployment ∆(1 − xs )sUnemployment ∆u

Employment gap in (37)
s

f

s

f

Shocks: (i) 1% increase in productivity (or potential output) ȳ. (ii) Increase inpublic spending from g = 0 to g = 0.01 (i.e. 1% of potential output).
Rows: (1) Change in underemployment (in pp). (2) Change in unemployment
rate (in pp). (3) Change in total employment gap (in pp). (4) Change inprivate consumption (in %). (5) Change in production (in %). (6) Change inunemployment rate in the absence of demand shortage (in pp). (7) Change
in consumption in the absence of demand shortage (in %). (8) Change inproduction in the absence of demand shortage (in %).

In addition, our conclusions on the effects of labor market policies and supply and demand
shocks continue to hold. The latter result in co-movements of unemployment and underemployment, while the effects of the former depend on the strength and curvature of the wealthpreference as well as the shape of the matching function.
Robustness check II: CES matching function
For the second robustness check, we employ the following general CES matching function
F (u, v) = κ u−γ + v −γ

−1/γ

,

where we use the same calibration as in Den Haan et al. (2000) with an elasticity parameter of
γ = 1.27.48 Using this specification, we pursue the same approach as in the main text.
The steady state with this alternative matching function is almost identical to the one
depicted in Table 2 on therefore omitted here. Tables 11 and 12 illustrate the effects of parametervariations on underemployment and the unemployment in the stagnation steady state.
Compared to Tables 5 and 6 in the main text, there are only minor quantitative and noqualitative changes of the responses of the model variables to various parameter variations
emphasizing again that it is predominantly the functional form of the wealth preference that
determines the response to “micro” shocks. Demand shortage manifests itself predominantly inunderemployment with few spillovers to the unemployment rate.
48
Note that the elasticity of substitution between vacancies and unemployed in the matching function is givenby (1 + γ)−1 . Higher values of γ imply that vacancies and unemployed are complements in producing successful
matches, while lower values imply a higher degree of substitutability.

38

Table 11: Steady state effects of labor market policies
Benefits z ↑Change in:

Subsidy s

LinearCARA

Linear

CARA

-0.21

1.13

0.13

1.38

0.27

0.30

-0.19

-0.17

0.06

1.37

-0.06

1.17

Consumption ∆c /c (in %)

0.02

-1.43

-0.01

-1.36

Production ∆Y s /Y s (in %)

-0.06

-1.50

0.06

-1.28

Unemployment ∆uf

0.26

0.25

-0.18

-0.18

Consumption ∆c /c (in %)

-0.19

-0.18

0.11

0.11

Production ∆Y f /Y f (in %)

-0.27

-0.26

0.19

0.19

Underemployment ∆(1 − xs )sUnemployment ∆u

Employment gap in (37)
s

s

f

f

Policies: (i) Increase in the replacement rate from z = 0.6 to z = 0.65. (ii)Introduction of a search cost subsidy of s = 10%.
Rows: (1) Change in underemployment (in pp). (2) Change in unemployment
rate (in pp). (3) Change in total employment gap (in pp). (4) Change inprivate consumption (in %). (5) Change in production (in %). (6) Change inunemployment rate in the absence of demand shortage (in pp). (7) Change
in consumption in the absence of demand shortage (in %). (8) Change inproduction in the absence of demand shortage (in %).

Table 12: Steady state effects of supply and demand shocks
Productivity ȳ ↑Spending g ↑Change in:LinearCARA

LinearCARA

Underemployment ∆(1 − xs )

1.02

1.03

-1.11

-1.24

Unemployment ∆us

(+)0

(+)0

-0.02

-0.03

Employment gap in (37)

0.99

1.00

-1.09

-1.22

Consumption ∆cs /cs (in %)

-0.09

-0.10

0.10

0.24

s

Production ∆Y /Y (in %)

-0.09

-0.10

1.19

1.34

Unemployment ∆uf

-0.02

-0.02

(+)0

(+)0

Consumption ∆c /c (in %)

1.03

1.03

-1.05

-1.05

Production ∆Y f /Y f (in %)

1.02

1.02

(-)0

(-)0

s

f

f

Shocks: (i) 1% increase in productivity (or potential output) ȳ. (ii) Increase inpublic spending from g = 0 to g = 0.01 (i.e. 1% of potential output).
Rows: (1) Change in underemployment (in pp). (2) Change in unemployment
rate (in pp). (3) Change in total employment gap (in pp). (4) Change inprivate consumption (in %). (5) Change in production (in %). (6) Change inunemployment rate in the absence of demand shortage (in pp). (7) Change
in consumption in the absence of demand shortage (in %). (8) Change inproduction in the absence of demand shortage (in %).

39

For Online Publication: Mathematical appendix
Appendix A: Derivation of the wage rate (17) under Nash bargaining
Let Et and Ut denote the value functions of employed and unemployed household members.
The respective Bellman equations are given by
rt Et = wt xt − δ[Et − Ut ] + Ėt ,(A.1)

rt Ut = zˆt + θt q(θt )[Et − Ut ] + U̇t .

(A.2)

We assume that the household takes unemployment payments zˆt as given and does not internalize the policy rule (6) of the government. Firms and workers set wages to maximize theirjoint surplus given by(Et − Ut )ε (Jt − Vt )1−ε ,(A.3)
where ε ∈ (0, 1) denotes the bargaining power of workers. Using the Bellman equations for
operating firms (12) and employed workers (A.1) in (A.2) and taking the derivative with respect
to the real wage, the optimal sharing rule of the total matching surplus is given by(1 − ε)(Et − Ut ) = ε(Jt − Vt ).

(A.4)

From the Bellman equations (A.1) and (A.2), we have˙ Ut ].
(rt + δ + θt q(θt )) [Et − Ut ] = wt xt − zˆt + [Et −(A.5)

Using the optimal sharing rule (A.4), the above equation is rewritten as(rt + δ + θt q(θt ))εε ˙
Jt = wt xt − zˆt +
Jt .
1−ε
1−ε

(A.6)

Using the Bellman equation (12) and the free entry condition (14), the value of an operating
firm is given as
J˙t = (rt + δ)Jt − (ȳ − wt )xt .
(A.7)
Substituting this equation for J˙t into (A.6) givesθt q(θt )εεJt = wt xt − zˆt −(ȳ − wt )xt .
1−ε
1−ε

(A.8)

Using expression (15) for the value of an operating firm Jt and the policy rule (6) for zˆt , thisequation gives the wage rate under Nash bargaining as
wt x t =ε[ȳxt + (1 − s)kθt ] .
1 − (1 − ε)z

(A.9)

ε
Define ψ ≡ 1−(1−ε)z
as the effective bargaining power of workers and (A.9) becomes equation(17) for the negotiated wage in the main text.

Appendix B: Derivation of the goods market clearing condition (22)Substituting the dynamics of the money supply (27) and the government budget constraint (20)
into the flow of funds constraint of the household (2) givesḃt = rt bt + wt xt (1 − ut ) − ct − skvt − g.

40

(B.1)

Differentiating bt = Jt (1 − ut ) from the asset market clearing condition (25) with respect totime, we also haveḃt = J˙t (1 − ut ) − Jt u̇t .
(B.2)
Substituting (B.2) and bt = Jt (1 − ut ) into equation (B.1) givesJ˙t (1 − ut ) − Jt u̇t = rt Jt (1 − ut ) + wt xt (1 − ut ) − ct − g − skt vt .

(B.3)

Using the dynamic equations (10) and (A.7) to substitute for u̇t and J˙t in (B.3), we get(1 − ut )ȳxt = Jt p(θt )ut + ct + g + skt vt .

(B.4)

Finally, using expression (15) for the value of an operating firm Jt and the identities vt ≡ θt ut
and p(θt ) ≡ θt q(θt ), equation (B.4) gives the goods market clearing condition as equation (22)in the main text.

Appendix C: Proof of Lemma 1
For notational brevity, from (15) and (17), define the dividend yield (ȳ − wt )xt /Jt asϑ(θt , xt ) ≡(1 − ψ)ȳxt(ȳ − wt )xt=
q(θt ) − ψp(θt ).
Jt(1 − s)k(C.1)

Let ϑ(θt , 1) denote the dividend yield when there is no underemployment and ϑθ (θt , 1) its partial
derivative with respect to θt . As ȳ > z, we have ϑ(0, 1) > 0, ϑ(∞, 1) < 0 and ϑθ (θt , 1) < 0.
Derivation of the dynamic system when xt = 1:
Without aggregate demand shortage, we have xt = 1 and ẋt = 0. The goods market clearing
condition (31) then defines consumption at any time t as a function of the unemployment rate
ut and the labor market tightness θt as
ct (θt , ut ) = (1 − ut )ȳ − kθt ut − g,(C.2)

where the partial derivatives are given by cu = −(ȳ+kθ) < 0 and cθ = −kut < 0. Differentiating
this function with respect to t impliesċt = −u̇t (ȳ + kθt ) − kut θ̇t .

(C.3)

Using (3), (10) and (18) to substitute for ċt , u̇t and θ̇t , we derive the following expression for
the real interest rate rt at any time t:ctω 0 (bt )kut θtrt − ρ + 0=−[rt − ϑ(θt , 1) + δ] − (ȳ + kθt )[δ(1 − ut ) − p(θt )ut ],ηcφ (ct )ηθt
ω 0 (bt )rt (θt , ut ) = ϕt ρ − 0+ (1 − ϕt ) [ϑ(θt , 1) − δ] − ϕ̃t [δ(1 − ut ) − p(θt )ut ] ,(C.4)
φ (ct )
whereϕt =ct /ηcct /ηc + kut θt /ηθt

,

ϕ̃t =

ȳ + kθt.
ct /ηc + kut θt /ηθt

From (C.2), ct is a function of θt and ut . Hence, the real rate rt is also a function of θt andut as bt depends on the same variables by (A.4). Using these expressions, we can rewrite the41model as a system of three differential equations in mt , ut and θt :
µ0 (mt )ṁt = gm − 0+ rt (θt , ut ) mt ,φ (ct (θt , ut ))(C.5)

u̇t = (1 − ut )δ − p(θt )ut ,

(C.6)

θt
[rt (θt , ut ) − ϑ(θt , 1) + δ].
η θt

(C.7)

θ̇t =
Existence:

Reformulate the first equation of (33) to define the function H(θ) as
0 (1 − u(θ)) (1−s)kωq(θ)(1 − ψ)ȳ
q(θ) − 0.
H(θ) = ρ + δ + ψp(θ) −(1 − s)kφ ((1 − u(θ))ȳ − kθu(θ) − g)

(C.8)

The function c(θ, 1) = (1 − u(θ))ȳ − kθu(θ) − g in (31) with x = 1 defines consumption asa continuous function of θ with a hump-shaped pattern that attains a maximum at a strictly
positive value.49 Define θ and θ̄ by c(θ, 1) = c(θ̄, 1) = 0 with θ < θ̄. Note that θ = 0 for g = 0.
Then c(θ, 1) ≥ 0 and H(θ) is a continuous function of θ for θ ∈ (θ, θ̄). As ω 0 (.) < ∞, we haveH(θ) = ρ + δ + ψp(θ) −(1 − ψ)ȳ
q(θ) = ρ + δ − ϑ(θ, 1),(1 − s)kH(θ̄) = ρ + δ + ψp(θ̄) −(1 − ψ)ȳ
q(θ̄) = ρ + δ − ϑ(θ̄, 1).
(1 − s)k

Note that H(θ) < H(θ̄) since ϑθ (θ, 1) < 0. Without any other restrictions, a steady state withH(θf ) = 0 always exists for θf ∈ (θ, θ̄) if H(θ) < 0 < H(θ̄) or equivalently if
ϑ(θ, 1) − δ > ρ > ϑ(θ̄, 1) − δ.

(C.9)

Money market equilibrium requires a positive real interest rate in steady state. Define θ̃ by
ϑ(θ̃, 1) = δ as in (30), which implies a zero net return on equity and hence a real interest rateof zero. Since ϑθ (θ, 1) < 0, equilibrium requires θf ≤ θ̃. The steady state without aggregate
demand shortage exists in (θ, θ̃) if the following condition holds:50
(C.10)
H(θ̃) ≥ H(θf ) = 0 ↔ ρφ0 c(θ̃, 1) ≥ ω 0 b(θ̃) .
Conditions (C.9) and (C.10) are the necessary existence conditions of Lemma 1. Uniquenessof the steady state requires H 0 (θ) > 0 for all θ with H(θ) = 0. This derivative is given by:ω 00 (b)φ0 (c)
ω 0 (b) φ00 (c) dc(θ, 1)
H 0 (θ) = 0− 00 ϑθ (θ, 1) − b0 (θ) + 00.
(C.11)φ (c)ω (b)ω (b) φ0 (c) dθWe assume that H 0 (θf ) > 0 so that the steady state with x = 1 is unique.
49

We ensure this by choosing the parameters g and k sufficiently low relative to ȳ.
We assume a constant nominal money supply and hence zero inflation, i.e. gm = 0. For gm > 0, θ̄ becomesa function of gm as the real interest rate cannot fall below −gm .
50

42

Stability:
The dynamic system is given by (C.5), (C.6) and (C.7). The unemployment rate ut is a predetermined state variables, while labor market tightness θt (via vacancies vt ) and the money
supply (for x = 1) are control variables and can jump. Stability of the dynamic system therefore
requires one negative and two positive eigenvalues. The Jacobian of this system evaluated at
the steady state with x = 1 is given by
h 0
i
h 0
i


00
µ (m)φ00 (c)cuµ (m)φ00 (c)cθm− µφ0(m)
+
r
m
+
r
m
u
θ
(c)
φ0 (c)2
φ0 (c)2

(C.12)
,0−[δ + p(θ)]−p0 (θ)u
θ
θ
0
ηθ ru
ηθ [rθ − ϑθ (θ, 1)]
where rχ denotes the partial derivative of the real rate defined in (C.2) with respect to variable
χ. The eigenvalues of this system solve the following characteristic equation:
 00
−[δ + p(θ)] − λ−p0 (θ)uµ (m)Ω(λ) = − 0
m−λ ·
= 0.
(C.13)
θ
θ
φ (c)
ηθ ru
ηθ [rθ − ϑθ (θ, 1)] − λIt is clear that λ1 = −µ00 (m)m/φ0 (c) > 0 is one solution. The other two eigenvalues solve


θ
θ
θ
2
[rθ − ϑθ (θ, 1)] − δ − p(θ) λ + ru p0 (θ)u − [δ + p(θ)](rθ − ϑθ (θ, 1)) = 0. (C.14)
λ −
ηθ
ηθ
ηθ
We can recover the sign of the eigenvalues fromλ2 λ3 =


θ  0
ru p (θ)u − [δ + p(θ)](rθ − ϑθ )(θ, 1) .
ηθ

(C.15)

From (C.4), we get the following expression for the partial derivative of rt with respect to
ut and θt in steady state: 00ω (b)φ0 (c)bu − ω 0 (b)φ00 (c)cu
ru = −ϕ+ ϕ̃[δ + p(θ)],φ0 (c)2
 00
ω (b)φ0 (c)bθ − ω 0 (b)φ00 (c)cθrθ = −ϕ+ (1 − ϕ)ϑθ (θ, 1) + ϕ̃p0 (θ)u.
φ0 (c)2
Also note that the unemployment rate in steady state is a function of labor market tightness
from equation (28) with du/dθ ≡ u0 (θ) = −p0 (θ)u(θ)/(δ + p(θ)). Using this expression, we canrelate the partial derivatives cu and cθ to the total derivative dc(θ, 1)/dθ = c0 (θ, 1) in steadystate as follows (and equivalently for b(θ)):51
dc(θ, u)dc(θ, 1)p0 (θ)u(θ)≡= cθ + cu u0 (θ) = cθ − cu,dθdθ
δ + p(θ)
db(θ, u)p0 (θ)u(θ)≡ b0 (θ) = bθ + bu u0 (θ) = bθ − bu.
dθ
δ + p(θ)
Using these properties, we rewrite the above equation as

 00

θ
ω (b)φ0 (c)bu − ω 0 (b)φ00 (c)cu 0
λ2 λ3 =
−ϕp (θ)u + ϕ̃[δ + p(θ)]p0 (θ)uηθ
φ0 (c)2
51

Note that dϑ(θ, 1)/dθ ≡ ϑθ (θ, 1) = ϑθ in the absence of aggregate demand shortage.

43




 00
ω (b)φ0 (c)bθ − ω 0 (b)φ00 (c)cθ0+ (1 − ϕ)ϑθ (θ, 1) + ϕ̃p (θ)u − ϑθ ,−[δ + p(θ)] −ϕφ0 (c)2 00
θω (b)φ0 (c)bu − ω 0 (b)φ00 (c)cu 0=−ϕp (θ)uηθ
φ0 (c)2  00ω (b)φ0 (c)bθ − ω 0 (b)φ00 (c)cθ+ ϕϑθ (θ, 1) ,+[δ + p(θ)] ϕ
φ0 (c)2





 00
θ
p0 (θ)up0 (θ)uω 0 (b)φ00 (c)ω (b)= ϕ[δ + p(θ)]
bθ − bu−cθ − cu+ ϑθ (θ, 1) ,ηθ
φ0 (c)
δ + p(θ)φ0 (c)2
δ + p(θ)
 00θω (b) 0
ω 0 (b)φ00 (c) dc(θ, 1)= ϕ[δ + p(θ)]
b (θ) −+ ϑθ (θ, 1) ,ηθ
φ0 (c)
φ0 (c)2dθ
φ0 (c)
θω 00 (b)ω 0 (b) φ00 (c) dc(θ, 1)
0− 00 ϑθ (θ, 1) − b (θ) + 0.
= − ϕ[δ + p(θ)] 0
ηθ
φ (c)
ω (b)
φ (c) ω 00 (b) dθ
The term in the brackets of the last expression above is identical to expression (C.11) above.
Hence, we can rewrite this equation asλ2 λ3 = −θϕ[δ + p(θ)]H 0 (θ) < 0.
ηθ

(C.16)

If the steady state is unique, it holds that H 0 (θ) > 0. It then follows that λ2 λ3 < 0, which
implies one negative and one positive eigenvalue. Together with λ1 > 0, the dynamic system has
one negative and two positive eigenvalues and therefore exhibits saddle-path stability aroundthe steady state without demand shortage.

Appendix D: Proof of Lemma 2
Existence:
Reformulate the second equation of (33) to define the function G(θ) asω 0 (1 − u(θ)) (1−s)kq(θ)G(θ) = ρ − 0− α(1 − x(θ)),φ (x(θ)(1 − u(θ))ȳ − kθu(θ) − g)

(D.1)

where x(θ) is the increasing part of the asset market equilibrium curve given by (36). A welldefined steady state requires non-negative consumption. Define θs as the smallest value of θsuch that c(θs , x(θs )) = 0, where c(θ, x(θ)) ≡ x(θ)(1 − u(θ))ȳ − kθu(θ) − g. Note that θs = θ = 0
for g = 0, where θ is defined in the proof of Lemma 1.52 In addition, let θ̃ by defined as beforeby ϑ(θ̃, 1) = δ in (30), which implies x(θ̃) = 1 in (36). We then haveG(θs ) = ρ − α(1 − x(θs )) = ρ − αϑ(θs , 1) − δsα + (1−ψ)ȳ(1−s)k q(θ )

> 0,



ω 0 (1 − u(θ̃)) (1−s)k
q(θ̃)



ω 0 b(θ̃)
 =ρ− .
G(θ̃) = ρ − 
φ0 (1 − u(θ̃))ȳ − k θ̃u(θ̃) − g
φ0 c(θ̃, 1)
52

We ensure non-negativity of consumption in steady state by choosing the parameters g and k sufficiently low
relative to ȳ.

44Note that G(θs ) > 0 if ρ > α since the ratioϑ(θs ,1)−δ
is always strictly smaller than unity.
(1−ψ)ȳα+ (1−s)k q(θs )Existence of the secular stagnation steady state then requires G(θ̃) < 0 or equivalentlyρφ0 c(θ̃, 1) < ω 0 b(θ̃) .

(D.2)

Whenever full employment is not feasible, we have G(θ̃) = H(θ̃) < 0 by Lemma 1. It thenfollows that there exists θs such that G(θs ) = 0.
Uniqueness of the steady state requires G0 (θ) < 0 for all θ with G(θ) = 0. This derivative is
given by
ω 00 (b)ω 0 (b) φ00 (c) dc(θ, x(θ))φ0 (c) dx(θ)
G0 (θ) = 0−b0 (θ) + 00
.
(D.3)
+
α
φ (c)
ω (b) φ0 (c)dθω 00 (b)dθThroughout this paper, we assume that G0 (θs ) < 0 so that the steady state is unique. Uniqueness in turn implies that the secular stagnation steady state is slack. Reformulating G0 (θ) < 0
impliesω 0 (b) φ00 (c) dc(θ, x(θ))φ0 (c) dx(θ)
0
>
b
(θ)
−
α
> 0,
(D.4)
ω 00 (b) φ0 (c)dθω 00 (b) dθdc(θ,x(θ))> 0.
since dx(θ)
dθ > 0 in steady state, which can be seen in (36). It follows thatdθ

Stability:
In the stagnation steady state, the liquidity preference of the household is satiated such that
Rt = 0 and hence rt = Rt − πt = α(1 − xt ). The goods market clearing condition (31) defines
realized working hours xt at any time t as a function of ct , θt and ut as
xt =ct + kθt ut + g,(1 − ut )ȳ(D.5)

where the partial derivatives satisfy xc > 0, xθ > 0 and xu > 0. Define m̃t ≡ 1/mt with m̃ = 0
in steady state. The dynamic system is given by the following differential equations for m̃t , ct ,θt and ut :˙ t = α (xt − 1) m̃t ,
m̃
(D.6)


ω 0 (b(θt , ut )) ct
c˙t = α(1 − xt ) − ρ +,(D.7)
φ0 (ct )ηcθ̇t = [α(1 − xt ) − ϑ(θt , xt ) + δ]

θt
,
ηθt

(D.8)

u̇t = δ(1 − ut ) − p(θt )ut ,(D.9)

where b(θt , ut ) and ϑ(θt , xt ) are given by (25) and (C.1) respectively. Let ϑθ and ϑx denotethe partial derivatives of ϑ(θt , xt ) with respect to θt and xt . Since the price level cannot adjust
freely, the money supply grows with the rate of deflation and is hence a predetermined variable.
In addition, the unemployment rate is a state variable whereas consumption and labor market
tightness are jump variables. Stability of the system therefore requires two positive and two
negative eigenvalues. The Jacobian of this system evaluated at the secular stagnation steadystate is as follows:α(x − 1)
αxc m̃
αxθ m̃
αxu m̃hihihiω 0 (b)φ00 (c)ω 00 (b)bθω 00 (b)bu 

c
c
c
0


ηc −αxc − φ0 (c)2ηc −αxθ + φ0 (c)ηc −αxu + φ0 (c) . (D.10)
θ
θ
θ

0
[−αxc − ϑx xc ][−αxθ − ϑx xθ − ϑθ ][−αxu − ϑx xu ] 
ηθ

0

ηθ

0

−p0 (θ)u45ηθ−[δ + p(θ)]

The eigenvalues of this system solve the following characteristic equation:

c
ηc

h

i
0
00 (c)−αxc − ω (b)φ
−λ
0
2
φ (c)
θ
ηθ [−αxc − ϑx xc ]Ω(λ) = [α(x − 1) − λ] ·hihiω 00 (b)bθω 00 (b)bu
c
c
−αx
+
−αx
+
0
u
0
θ
ηc
ηc
φ (c)
φ (c)
θ
ηθ [−αxθ − ϑx xθ − ϑθ ] − λ−p0 (θ)u0θηθ [−αxu − ϑx xu ]= 0. (D.11)−[δ + p(θ)] − λIt is clear that λ1 = α(x − 1) < 0 is one solution. The other eigenvalues solveihhiω 0 (b)φ00 (c)ω 00 (b)bu
c
c
−
λ
−
−
αx
+
αx
−
c
u
ηc
ηc
φ0 (c)
φ0 (c)2Ω̃(λ) = p0 (θ)u ·θθ− ηθ [α + ϑx ] xc
− ηθ [α + ϑx ] xu−[δ + p(θ) + λ] ·hi
000 (c)− ηcc αxc + ω (b)φ−λ
φ0 (c)2ih00
θ− ηcc αxθ − ωφ0(b)b(c)− ηθθ [α + ϑx ] xc

− ηθθ [αxθ + ϑx xθ + ϑθ ] − λ= 0.

(D.12)We rewrite this expression asΩ̃(λ) = −λ3 + A2 λ2 − A1 λ + A0 = 0,
where

(D.13)
ω 0 (b)φ00 (c)
θ
αxc +
−
(αxθ + ϑx xθ + ϑθ ) − δ − p(θ),
0
2
φ (c)
ηθ
 


θ
c
ω 0 (b)φ00 (c)θ0A1 = −p (θ)u (α + ϑx ) xu + [δ + p(θ)]αxc ++(αxθ + ϑx xθ + ϑθ )ηθηc
φ0 (c)2
ηθ




c θ
c θ
ω 00 (b)bθω 0 (b)φ00 (c)+(αxθ + ϑx xθ + ϑθ ) −(α + ϑx ) xc αxθ − 0
,
αxc +
ηc ηθ
φ0 (c)2
ηc ηθ
φ (c)



 
ω 0 (b)φ00 (c)ω 00 (b)buc θ0(α + ϑx ) αxc +xu − αxu −xc
A0 = p (θ)uηc ηθφ0 (c)2
φ0 (c)
ω 0 (b)φ00 (c)ω 00 (b)bθ
c θ
αxc +
(αxθ + ϑx xθ + ϑθ ) − (α + ϑx ) xc αxθ − 0.
−[δ + p(θ)]ηc ηθ
φ0 (c)2φ (c)cA2 = −
ηc

Noting that u0 (θ) = −p0 (θ)u/[δ + p(θ)] and b0 (θ) = bθ + bu u0 (θ), we can modify A0 as follows:
 0


c θ
p (θ)uω 0 (b)φ00 (c)ω 00 (b)bu
A0 = [δ + p(θ)](α + ϑx ) αxc xu +xu − αxu xc +xcηc ηθ δ + p(θ)φ0 (c)2
φ0 (c)
ω 0 (b)φ00 (c)ω 00 (b)bθ(αxθ + ϑx xθ + ϑθ ) + (α + ϑx ) xc αxθ − 0− αxc +
φ0 (c)2φ (c)c θ
ω 0 (b)φ00 (c)ω 00 (b)buω 00 (b)bθ00= [δ + p(θ)](α + ϑx ) −
x
u
(θ)
−
x
u
(θ)
−
xc + xc αxθ
u
c
ηc ηθ
φ0 (c)2
φ0 (c)
φ0 (c)ϑθω 0 (b)φ00 (c)ϑθ−αxc xθ +−xθ +α + ϑxφ0 (c)2
α + ϑxc θ
ω 0 (b)φ00 (c) xu 0xθϑθu (θ) ++= [δ + p(θ)](α + ϑx ) xc −ηc ηθφ0 (c)2xc
xc(α + ϑx ) xc

ω 00 (b) 0αϑθb (θ) −− 0.
φ (c)α + ϑx46

Using the partial derivatives of x defined in (D.5), we can rewrite the term in brackets of thelast equation above as follows:
xu 0
xθ
ϑθ
−ϑθ
u (θ) ++= (ȳx + kθ)u0 (θ) + ku − (1 − u)ȳxcxc
(α + ϑx ) xc
(α + ϑx )= (ȳx + kθ)u0 (θ) + ku − (1 − u)ȳdc(θ, x(θ))
dx(θ)
≡
,
dθ
dθ

ϑθ
where dx(θ)
dθ = − (α+ϑx ) follows from the No-Arbitrage condition (36) and c(θ, x(θ)) is defined in(31). It then follows that A0 can be rewritten asc θ
ω 0 (b) φ00 (c) dc(θ, x(θ))ω 00 (b)φ0 (c) αϑθ0A0 = [δ + p(θ)]−b (θ) + 00.
(α + ϑx ) xc 0
− 00
ηc ηθ
φ (c)
ω (b) φ0 (c)dθω (b) α + ϑx

Using the expressions for ϑx , ϑθ and dx(θ)
dθ , it is easy to see that the last part of this term isidentical to expression (D.3). Hence, we can rewrite this equation as
A0 = [δ + p(θ)]

c θ(α + ϑx ) xc G0 (θ).
ηc ηθ

(D.14)

If the secular stagnation steady state is unique, it holds that G0 (θ) < 0 and hence A0 < 0. Since
A0 can be rewritten as the product of the three remaining eigenvalues, i.e. A0 = λ2 λ3 λ4 < 0,
we either have one or three additional negative eigenvalues.
Stability requires two positive and two negative eigenvalues. Since A2 = λ2 + λ3 + λ4 andA1 = λ2 λ3 + λ2 λ4 + λ3 λ4 , we require either A2 > 0 and/or A1 < 0.
Suppose A2 > 0, then stability immediately follows. Suppose instead A2 ≤ 0. This implies


c
θ
θ
ω 0 (b)φ00 (c)−−(α + ϑx ) xθ ≤ δ + p(θ) + ϑθ ,(D.15)
αxc +
0
2
ηc
φ (c)
ηθ
ηθ
which we use to show that A1 < 0 in this case, from which stability follows. Reformulate A1 as
 0



−p (θ)uϑθcω 0 (b)φ00 (c)θxu + xθ ++ [δ + p(θ)]αxc +A1 = [δ + p(θ)] (α + ϑx )ηθ
δ + p(θ)α + ϑxηc
φ0 (c)2






c θ
ω 0 (b)φ00 (c)ω 0 (b)φ00 (c)ω 00 (b)bθ
+
αxc +
ϑ
+
(α
+
ϑ
)
αx
+
x
−
x
αx
−
x
c
c
θ
θ
θ
ηc ηθ
φ0 (c)2
φ0 (c)2
φ0 (c)




xu 0
θ
xθ
ϑθ
c
ω 0 (b)φ00 (c)= [δ + p(θ)] (α + ϑx ) xc
u (θ) +++ [δ + p(θ)]
αxc +
ηθ
xc
xc
xc (α + ϑx )ηc
φ0 (c)2
c θ
ω 0 (b)φ00 (c)c θ
ω 0 (b)φ00 (c)ω 00 (b)bθ
+
αxc +
ϑθ +
(α + ϑx )xθ + 0xc
ηc ηθ
φ0 (c)2ηc ηθ
φ0 (c)2φ (c)θdc(θ, x(θ))
cθ= −[δ + p(θ)] (α + ϑx ) xc
+ αxc δ + p(θ) + ϑθ
ηθ
dθ
ηc
ηθ


 0

0
00
00
c ω (b)φ (c)θc θ
ω (b)φ (c)ω 00 (b)bθ+δ + p(θ) + ϑθ +(α + ϑx )xθ + 0xc
ηc φ0 (c)2ηθηc ηθφ0 (c)2
φ (c)



θ
dc(θ, x(θ))θ
cω 0 (b)= −[δ + p(θ)] (α + ϑx ) xc
+ δ + p(θ) + ϑθ
αxc − 0
ηθ
dθ
ηθ
ηc
φ (c)
 0

c θ
ω (b)φ00 (c)ω 00 (b)bθ+(α + ϑx )
x
+
xc .
θ
ηc ηθ
φ0 (c)2
φ0 (c)

47



0 (b)
For sufficiently high values of ηc , specifically for all ηc > αρ , it holds that ηcc αxc − ωφ0 (c)< 0.
It then follows that for A2 ≤ 0, we havedc(θ, x(θ))ω 0 (b)ω 0 (b)θcθc
αxc − 0αxc − 0A1 ≤ −[δ+p(θ)] (α + ϑx ) xc−+
(α + ϑx ) xθηθdθ
ηc
φ (c) ηθ
ηc
φ (c)

 0
c θ
ω 00 (b)bθω (b)φ00 (c)+xθ + 0xc(α + ϑx )ηc ηθ
φ0 (c)2φ (c)θω 0 (b) 2
dc(θ, x(θ))
cθc= −[δ + p(θ)] (α + ϑx ) xc
αxc − 0−−(α + ϑx ) xθ αxc
ηθ
dθ
ηc
φ (c)
ηθ
ηc
θ
ω 0 (b)θω 0 (b)c θω 00 (b)bθ(α + ϑx ) xθ 0−
(α + ϑx ) 0 xθ +(α + ϑx ) 0xcηθφ (c) ηθ
φ (c)
ηc ηθ
φ (c)




θ
ω 0 (b) 2 c θω 00 (b)bθdc(θ, x(θ))
c= −[δ+p(θ)] (α + ϑx ) xc
αxc − 0−.
−(α + ϑx ) xc xθ − 0
ηθ
dθ
ηc
φ (c)
ηc ηθ
φ (c)
+

Since dc(θ,x(θ))> 0 in the stagnation steady state, which follows from the uniqueness condition,dθ
all three terms are negative and we have A1 < 0. Therefore, we cannot have three negative
eigenvalues, which together with A0 < 0 and λ1 > 0 implies that there are exactly two positive
and two negative eigenvalues. It follows that the dynamic system is saddle path stable aroundthe stagnation steady state.

Appendix E: Proof of Proposition 1
Use the steady state conditions (33) with x < 1 to define G(θ), where x(θ) is given by (36), as
follows:
ω 0 (.)
G(θ) = ρ − 0− α(1 − x(θ)).
(E.1)
φ (c(θ, x(θ)))
We model worsening stagnation by exogenous increases in the steady state marginal utility of
wealth ω 0 (.), e.g. via modifying the scale parameter of the wealth preference relative to thepreference for consumption. This implies a stronger desire to save and a decline in the natural
real interest rate. Recall from before that G0 (θ) < 0 in steady state.
The effects of worsening stagnation on labor market tightness is given bydθω 0 G0 (θ)
=
<0.
dω 0
φ0 (c(θ, x(θ)))(E.2)

It then follows immediately from (28), (36) and (21) that
du
dx
dY
>0,
<0,
<0.
dω 0
dω 0
dω 0

(E.3)

dx
The effect on consumption follows from (E.1). In steady state, G(θ) = 0. As dω
0 < 0, itnecessarily follows that
dc
<0.
(E.4)
dω 0
Finally, from (17) and (36), the real wage can be rewritten as(1 − ψ)ȳp(θ) + (1 − s)kαθ
wt = ψ ȳ +,(E.5)
ψp(θ) + δ + α

48

which givesψ(δ + α)[(1 − ψ)ȳp0 (θ) + (1 − s)kα] − (1 − s)kαψ 2 θ2 q 0 (θ) dθ
dw
=
<0,
dω 0
[ψp(θ) + δ + α]2
dω 0

(E.6)

dθ
where the negative sign results from p0 (θ) > 0 in (9), q 0 (θ) < 0 in (10) and dω
0 < 0 in (E.2).
Expressions (E.2), (E.3), (E.4) and (E.6) establish Proposition 1.

Appendix F: Effects of exogenous variations in x
We want to derive the conditions under which an exogenous increase (decrease) in x creates
excess supply (demand) in the goods market for a given θ, i.e.
Υ(θ) ≡ −ω 0 (b)φ00 (c)(1 − u)ȳ
− α > 0.
φ0 (c)2

Consider the following modifications using the definition of ηc and ccω 0 (b) (1 − u)ȳηc(1 − u)ȳρηc − α(1 − x)ηc − αΥ(θ) = 0
−α=
φ (c)
c
c
(1 − u)ȳ(1 − u)ȳ(1 − u)ȳx − kuθ − g=ρηc − α(1 − x)ηc − αc(1 − u)ȳ(1 − u)ȳkuθ + g=(ρ − α)ηc + αx(ηc − 1) + α.
c(1 − u)ȳ

(F.1)

(F.2)

It is easy to see that Υ(θ) > 0 for ηc ≥ 1. Suppose ηc < 1, then x = 1 establishes a lower boundon the right hand side:(1 − u)ȳkuθ + g(1 − u)ȳΥ(θ) >
ρηc − α + α>[ρηc − α] .
(F.3)
c(1 − u)ȳc
It follows directly that ηc > αρ is a sufficient condition for Υ(θ) > 0 or−

ω 0 (b)φ00 (c)(1 − u)ȳα> α if ηc > .
φ0 (c)2
ρ(F.4)

Appendix G: Proof of Proposition 2 and Proposition 3Standard steady state:
Use (33) with x = 1 to define the function H(θ, χ), where χ denotes any parameter in themodel, as follows:
0 (1 − u(θ)) (1−s)kωq(θ)(1 − ψ)ȳH(θ, χ) = ρ + δ + ψp(θ) −q(θ) −,(G.1)
0(1 − s)kφ (c(θ, 1))with H(θf ) = 0 and H 0 (θf ) > 0 in a unique steady state where Hθ denotes the derivative withrespect to θ evaluated in steady state. c(θ, 1) is defined in (31). The effects of changes in aparameter χ on the labor market tightness θ can be recovered from (G.1) as
Hχ
dθ
=−
.
dχ
Hθ

49

(G.2)

It holds that
Hz =
Hs =
Hȳ =

(ȳq(θ) + (1 − s)kp(θ) ∂ψ> 0,(1 − s)k∂zω 00 (b)(1 − u(θ))k (1 − ψ)ȳq(θ)−< 0,
φ0 (c)q(θ)(1 − s)2 k

ω 0 (b)φ00 (c)(1 − u(θ)) (1 − ψ)q(θ)−< 0,
φ0 (c)2(1 − s)k
Hg = −

ω 0 (b)φ00 (c)> 0.
φ0 (c)2

This implies the following relationship between the steady state labor market tightness and themodel parameters:(G.3)
θf = θ(|{z}z , |{z}s , ȳ , g ).
|{z} |{z}
−

+

+

−

From the Beveridge curve in (28), it immediately follows that the effects on the unemployment
rate are opposite to those on the labor market tightness, i.e.
uf = u(|{z}z , |{z}s , ȳ , g ).
|{z} |{z}
+

−

−

(G.4)

+

In addition, total output Y = (1 − u(θ))ȳ is affected by the model parameters in the same way
as the labor market tightness, except that changes in productivity are reinforced, i.e.
Y f = Y (|{z}z , |{z}s , ȳ , g ).
|{z} |{z}
−

+

+

(G.5)

−

Finally, the effects on household consumption can be derived from the goods market clearing
condition (31). Specifically, it holds thatdc(θ, 1)dc(θ, 1) dθ∂c(θ, 1)
=
+
,
dχ
dθ dχ
∂χ

(G.6)

∂c
∂c
∂c
where ∂χ= 0 for χ = z, s, ∂χ= 1 − u for χ = ȳ and ∂χ= −1 for χ = g. Moreover,0dc(θ, 1)/dθ = −u (θ)(ȳ +kθ)−ku(θ) and its sign is not uniquely determined. For dc(θ, 1)/dθ > 0(“slack” steady state), the sign of the effects of all parameter changes are identical to those of
the labor market tightness. For dc(θ, 1)/dθ < 0 (“tight” steady state), the sign of the effects
of changes in ε, z and s is opposite to those on θ, while the effects of variations in ȳ and g areindeterminate.

Stagnation steady state:(i) Effects on labor market tightness and unemployment: Use the steady state conditions (33) with x < 1 and (36) to define G(θ, χ), where x(θ) is given by (36) as x < 1 and χ
denotes any parameter as follows:ω 0 (1 − u(θ)) (1−s)kq(θ)
G(θ, χ) = ρ −− α(1 − x(θ)),(G.7)
0
φ (c(θ, x(θ)))
with G(θs , χ) = 0 and Gθs < 0 in steady state where Gθs denotes the total derivative withrespect to θ evaluated in steady state. The negative sign follows from the uniqueness of thesteady state. c(θ, x(θ)) is given by (31) with x = x(θ). The effects of changes in a parameter χ50

on the labor market tightness θ can be recovered from (G.7) as
Gχ
dθ
=−
.
dχ
Gθ s

(G.8)

In addition, we derive the following partial derivatives from x(θ, χ) in (36):xz > 0 , xs < 0 , xȳ < 0 , xg = 0 , x + ȳxȳ > 0.

(G.9)

Using (G.7) to (G.9) and assuming that ηc > αρ (see Appendix F), we derive the following
properties:xzω 0 (b)φ00 (c)(1 − u)ȳ
dθ
+α
< 0,
=−
dz
φ0 (c)2
Gθ s

 0
dθ
xs
ω 00 (b)b(θ) 1
ω (b)φ00 (c)(1 − u)ȳ
+
α
=−
−
,
0
0
2
ds
(1 − s)φ (c) Gθs
φ (c)
Gθ s

(G.10)(G.11)

ω 0 (b)φ00 (c)(1 − u) x + ȳxȳ
xȳ
dθ
=−
−α
< 0,
dȳ
φ0 (c)2
Gθ s
Gθ s(G.12)dθω 0 (b)φ00 (c) 1
=
> 0.
dg
φ0 (c)2 Gθs(G.13)

Expressions (G.10) to (G.13) imply the following response of labor market tightness toparameter variations. The response of the unemployment rate is opposite as is clear from theBeveridge curve in (28):θs = θ(|{z}z , |{z}s , ȳ , g ),(G.14)|{z} |{z}
−

?

−

+

s

u = u(|{z}z , |{z}s , ȳ , g ).
|{z} |{z}+?

+(G.15)−(ii) Effects on realized working hours: From the asset market equilibrium curve (36), we
recover the effects of parameter changes on realized working hours as follows:dx(θ, χ)dx(θ) dθdx(θ) Gχdx(θ) Gχ xχ
= xχ += xχ −= Gθ s −,(G.16)dχdθ dχdθ Gθsdθ xχ Gθswhere dx(θ)
dθ > 0 and Gχ are the partial derivatives of G(θ, χ) with respect to any parameter χ.
dθ0Using the expressions above for dχand dc(θ,x(θ))= (1 − u)ȳ dx(θ)
dθdθ − (ȳx(θ) + kθ)u (θ) − ku from(31), we get the following results:−ω 00 (b)b0 (θ) ω 0 (b) φ00 (c)
xz
dx
0
=
− 0
[ku + (ȳx + kθ)u (θ)],
dz
φ0 (c)
φ (c) φ0 (c)
Gθ s
 00



ω (b)
dx(θ) b(θ)
ω 0 (b) φ00 (c)
xs
dx
0
0
=
−
−
b
(θ)
−
[ku
+
(ȳx
+
kθ)u
(θ)]
,
ds
φ0 (c)dθ xs (1 − s)
φ0 (c) φ0 (c)
Gθ s



dx
ω 00 (b)b0 (θ) ω 0 (b)φ00 (c)dx(θ) (1 − u)x(θ)
xȳ
0
= −
−
ku
+
(ȳx
+
kθ)u
(θ)
+
0
0
2
dȳ
φ (c)
φ (c)
dθ
xȳ
Gθ s



00
0
0
00
ω (b)b (θ) ω (b)φ (c) dx(θ) (1 − u)dc(θ, x(θ))
xȳ
= −
−
(x + ȳxȳ ) −
.
0
0
2
φ (c)
φ (c)
dθ
xȳ
dθ
Gθ s

From Gθs < 0, we derive the following upper bound for the first term in (G.19):−

ω 00 (b)b0 (θ)ω 0 (b)φ00 (c) dc(θ, x(θ))
dx(θ)
<
−
−α
.
0
0
2
φ (c)
φ (c)
dθ
dθ

51

(G.17)(G.18)(G.19)

Since xȳ < 0 and Gθs < 0, from (G.19) it follows that



dx
ω 0 (b)φ00 (c) dc(θ, x(θ))dx(θ) ω 0 (b)φ00 (c) dx(θ) (1 − u)dc(θ, x(θ))
xȳ
< −
−α
−
(x + ȳxȳ ) −
0
2
0
2
dȳ
φ (c)
dθ
dθ
φ (c)
dθ
xȳ
dθ
Gθ s


ω 0 (b)φ00 (c) (1 − u)
xȳ
dx(θ)
< 0,
(G.20)
α+(x + ȳxȳ )=−dθ
φ0 (c)2xȳ
Gθ sdx
dx(θ) dθ
=
> 0.
dg
dθ dg

(G.21)

Taken together, we have derived the following effects of parameter changes in (G.17) to (G.21):
xs = x(|{z}z , |{z}s , ȳ , g ).
|{z} |{z}?

?

−(G.22)+(iii) Effects on total output: The effects of parameter changes on total output follow fromthe total differential of Y (θ) = (1 − u(θ))x(θ)ȳ asdY
dudx= −x(θ)ȳ+ (1 − u)ȳ
+ Yχ ,dχdχ
dχ

(G.23)

where Yχ = (1 − u)x for χ = ȳ and zero otherwise. It then follows that these effects are given
by


dx(θ)
dY
dθ
= (1 − u)ȳxz + (1 − u)ȳ− xȳu0 (θ),| {z }
dz
dθ
dz
|
{z
}
+

(G.24)

−



dθ
dY
dx(θ)
= (1 − u)ȳxs + (1 − u)ȳ− xȳu0 (θ),| {z }
ds
dθ
ds
|
{z
}
−

(G.25)

+/−

dY
dx
dθ
= (1 − u)ȳ− xȳu0 (θ)+ (1 − u)x(θ)
dȳ
dȳ
dȳ





dx(θ) 1 − ux ω 00 (b)b0 (θ)
0
+
αx(θ)
ȳu
(θ)
+
= −(1 − u) ȳ +xȳφ0 (c)dθxȳω 0 (b)φ00 (c)xxȳ−(1 − u)ȳ +.
(ku + kθu0 (θ))φ0 (c)2
xȳ
Gθ s
00

0

(θ)
As above, we apply an upper bound on − ω φ(b)bthat results from Gθs < 0. Since xȳ < 0 and
0 (c)Gθs < 0, it follows that



 0



x(θ)
ω (b)φ00 (c) dc(θ, x(θ))dx(θ)dx(θ) 1 − udY0< −(1 − u) ȳ ++α+ αx ȳu (θ) +dȳxȳφ0 (c)2
dθ
dθ
dθ
xȳ



ω 0 (b)φ00 (c)
x(θ)
xȳ
0
ȳ +
−(1 − u)(ku + kθu (θ))φ0 (c)2
xȳ
Gθ s





x(θ) ω 0 (b)φ00 (c)dx(θ)
0= −(1 − u) ȳ +
ȳ
(1
−
u)
−
u
(θ)x(θ)xȳ
φ0 (c)2
dθ


dx(θ)
xȳ
−αȳ (1 − u)− x(θ)u0 (θ)
dθ
Gθ s
 

 0


x(θ) ω (b)φ00 (c)
xȳ
dx(θ)
0
= − (1 − u)
− u (θ)x(θ) ȳ (1 − u) ȳ ++α< 0,(G.26)
0
2
dθ
xȳ
φ (c)
Gθ s
|
{z
} |
{z
} |{z}
+

+

52

+

 0

dY
p (θ)
dx(θ)
p(θ)ȳ dθ=ux(θ) +
> 0.
dg
p(θ)
dθ
δ + p(θ) dg(G.27)

Taken together, we have derived the following effects of parameter changes from (G.24) to(G.27):(G.28)
Y s = Y (|{z}z , |{z}s , ȳ , g ).
|{z} |{z}
?

?

−

+

(iv) Effects on consumption: Finally, the effects on consumption are given by the differential of (31) with x = x(θ) given by (36) using the expressions above as

 dθ
dc
dx
dc(θ, x(θ)) dθ= −[ȳx + kθ]u0 (θ) − ku
+ (1 − u)ȳ+ cχ =+ (1 − u)ȳxχ + cχ , (G.29)
dχ
dχ
dχ
dθ
dχ
where dc(θ,x(θ))> 0, cz = cs = 0, cȳ = (1 − u)x > 0 and cg = −1 < 0.
dθdc(θ, x(θ)) dθω 00 (b)b0 (θ) xz
dc
0=+ (1 − u)ȳxz = [(ȳx + kθ)u (θ) + ku] α − (1 − u)ȳ,dz | dθ{z dz} | {z }φ0 (c)
Gθ s(G.30)+−dc
dc(θ, x(θ)) dθ=+ (1 − u)ȳxs ,ds | dθ{z ds} | {z }(G.31)

−

+/−


 

ω 0 (b)φ00 (c) dc(θ, x(θ))ω 00 (b)b0 (θ) xȳdc0= (1 − u)x 1 − 0 2+ [(ȳx + kθ)u (θ) + ku(θ)] α − (1 − u)ȳdȳ
φ (c) Gθsdθφ0 (c)
Gθ s
a



φ02 α dxdθ(θ) − ω 00 (b)b0 (θ)φ0 (c)dc(θ, x(θ))dx(θ)ω 00 (b)b0 (θ) xȳȳ−= (1 − u)x
+
(1
−
u)
α
−
(1
−
u)ȳ
φ0 (c)2 Gθsdθdθφ0 (c)
Gθ s
 


a
00
0
dx (θ)dc(θ, x(θ))ω (b)b (θ) 1= α (1 − u)(ȳxȳ + x) −xȳ − (1 − u) (ȳxȳ + x)dθdθφ0 (c)
Gθ s





dc(θ, x(θ))
dxa (θ) ω 00 (b)b0 (θ)  1 < 0,xȳ + (1 − u) (ȳxȳ + x) α−= −α(G.32)
 Gθ s
0 (c)
dθ
dθ
φ
|
{z
}
|
{z
}
+ as xȳ <0+ as ȳxȳ +x>0dcdxa (θ) ω 00 (b)b0 (θ) 1
=− α
−
> 0.
dg
dθ
φ0 (c)
Gθ s(G.33)

Taken together, (G.30) to (G.33) imply the following effects of parameter changes:
c = c(|{z}z , |{z}s , ȳ , g ).
|{z} |{z}
?

?

53

−

+

(G.34)