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					         Inequality, Aggregate Demand and Crises
                 Matthieu Charpe ∗                               Stefan K¨ hn†
                                                                         u
           International Labour Organization            International Labour Organization


                                         April 23, 2012


                                               Abstract
            This paper discusses the impact of inequality on output and employment using
       a DSGE model. In particular, the model looks at the impact of lower bargaining
       power of workers on aggregate demand via the labour share of income. To address
       this issue, the model combines a search and matching model with Nash bargaining
       over income distribution with rule-of-thumb households, nominal price rigidities,
       CES production function and lower zero bound on monetary policy. The model
       features an endogenous labour share of income arising from the wage bargaining.
       Rule-of-thumb households create a transmission channel going from functional in-
       come distribution to consumption decisions, which feed back on aggregate demand
       through nominal price rigidities. Low substitution between labour and capital limits
       the increase in labour demand following a decline in wages, while the lower zero
       bound prevents an investment boom to overbalance the decline in consumption. The
       main result is that following a decline in the bargaining power of workers, the result-
       ing drop in the labour share of income lowers consumption and aggregate demand.
       It follows that downward wage rigidities such as minimum wage limit the fall in out-
       put and employment. These results stand in contrast with standard New-Keynesian
       models.
       ———————
       Keywords: Inequality, labour share, search, matching, crisis.
       JEL CLASSIFICATION SYSTEM: E24, E32, E62.




   ∗ Corresponding author, charpe@ilo.org, Bureau International du Travail, 4 route des Morillons 1211
    e
Gen` ve - CH, +41 22 799 64 43
   † kuehn@ilo.org, Bureau International du Travail, 4 route des Morillons 1211 Gen` ve - CH, +41 22 799
                                                                                   e
68 67


                                                   1
1 Introduction

This paper contributes to the literature connecting inequality and business cycle models.
It presents a DSGE model in which rising inequality leads to an economic crisis through
its negative effect on consumption and aggregate demand. Our contribution is to show
that the transmission channel between inequality and macroeconomics takes place via
aggregate demand effects rather than supply side or financial effects. This paper does not
claim that these latter effects did not contribute to the actual crisis, but that the aggregate

demand channel has been under-studied in the existing literature.

   A number of contributions have documented the increase in inequality before the
2007 crisis. Piketty and Saez (2003) as well as Atkinson et al. (2011) have described

the concentration of income in the top deciles of the population. Acemoglu (2003) and

Lemieux (2006) review evidence and factors explaining increasing wage dispersion be-

tween high skilled and low skilled workers. Looking at functional income distribution,

Blanchard (1997) and Blanchard and Giavazzi (2003) illustrate the decline in the labour

share of income. In this paper, we are interested in the functional rather than the personal

income distribution. Figure 1 shows that the labour share of income has declined in three
fourth of the 16 high income countries for which data are available over the period 1960-

2010. Section 2 further details the decline in the labour share of income in high income

countries.

   Two fundamentally different explanations for the fall in the labour share exist. The
first claims that technological progress favored capital return, while the second identifies
institutional changes as the cause of this development. All explanations require a depar-
ture from the assumption of unitary elasticity of substitution in the production function,
which is usually made in RBC models. Choi and Rios-Rull (2009) and Arpaia et al.
(2009) study the importance of the elasticity of substitution ζ between factors of produc-



                                              2
tion on their relative shares.1 Institutional factors to explain a changing labour share can
range from lower unionization rates to globalization pressure, but are usually modelled
as a fall in worker’s bargaining power.2 This paper utilizes a general constant elasticity
of substitution (CES) production function as well as a labour market featuring bargaining
over wages and employment. As such, this model allows both for technological as well
as institutional factors to affect the labour share of income.

    There are few attempts to link inequality and the crisis in the DSGE literature. In

standard New Keynesian models with a search and matching friction and bargaining over
income distribution, such as Sala et al. (2008), a decline in the labour share of income

increases output and employment. The main reason is that lower wages increase labour

demand by firms. Since the surplus from an additional match accruing to firms increases,

they have an incentives to post more vacancies. A strong supply side effect follows,

raising output. Additionally, changes in the income distribution have no effect on con-

sumption and saving decisions since the representative household receives both labour

and profit income.
    The model presented in this paper differs from standard New Keynesian models along

three dimensions. First, the model creates a channel between income distribution and

consumption / saving decisions using households’ heterogeneity. A first type of house-

hold is optimizing and makes consumption, saving and investment decisions to smooth
inter-temporal consumption based on its permanent income. Additionally, a second type
   1 Choi and Rios-Rull (2009) show that an elasticity of substitution below unity (ζ = 0.75 in their model)
is required for a DSGE model to qualitatively reproduce empirical cyclical wage share properties in re-
sponse to technology shocks. In contrast, Arpaia et al. (2009) take the presence of low- and high-skilled
labour into account when investigating the impact of technical change and changes in the relative compo-
sition in the labour force.
    2 Berthold et al. (2002) model putty-clay technology and capital, thereby featuring low substitutability

between capital and labour in the short run and high substitutability in the long run. Under these assump-
tions, a fall in workers’ bargaining power will temporarily reduce the labour share. Blanchard and Giavazzi
(2003) focus on firm entry costs and show that lower bargaining power, through raising employment, raises
the number of firms and thereby competition in the market, thus lowering the price markup. By abstracting
from aspects of technology and factor substitution, they establish that the labour share will ultimately be
related to frictional costs, like entry costs, imposed on firms.


                                                     3
of household, called rule-of-thumb household, has no access to financial markets, and
thus no saving or borrowing.3 This household relies exclusively on labour income when
employed or the replacement wage when unemployed. The presence of rule of thumb
households generates a transmission channel going from income distribution to consump-
tion decisions and aggregate demand.
    Second, the paper introduces the possibility of a liquidity trap implemented with a
lower bound on the nominal interest rate, as in Christiano et al. (2009). In a liquidity trap,

a shortage of demand, causing deflation, cannot be met by a fall in the nominal interest

rate. As a result, the real interest rate rises, further lowering consumption and investment
demand. This paper shows that the negative aggregate demand effect caused by lower

workers’ bargaining power far outweighs the positive supply effects in a liquidity trap.

This channel is especially relevant in the 2008 crisis, when the collapse of the financial

system required central banks to reduce interest rates to very low levels. Section 5 shows

that the mechanism underlined in this paper are also relevant in the absence of a lower

zero bound. However, the liquidity trap amplifies the mechanism at work.
    Third, the model assumes a CES production function, which encompasses both Cobb-

Douglas as well as Leontief production functions. Low degree of substitution between

labour and capital implies that a decline in wages does not trigger a large increase in

labour demand. This effect both produces fluctuations in the labour share of income as
well as reduces the size of the transmission channel going from wages to labour demand.
    Summarizing, this paper highlights the importance of labour income as a driver of
aggregate demand. Additionally, the relevance of this effect is underlined by conducting
an experiment where a lower floor on real wages is introduced. Such a floor can be
motivated by the downward rigidity of nominal wages, or by policy action in the form of
a minimum wage. Under such circumstances, the crisis induced by increased inequality
    3 There is a large literature on rule of thumb households, see for instance Gal´ et al. (2007) or Bosc´
                                                                                   ı                      a
et al. (2001).


                                                    4
is actually less severe since aggregate demand does not fall as strong.
   The paper conceptually closest is Kumhof and Ranciere (2010). These authors also
investigate the potential crisis-inducing impact of rising inequality. In their paper, work-

ers react to falling wage income by increasing indebtedness, which eventually can cause
an economic crisis. In contrast to this paper, Kumhof and Ranciere (2010) disregard ag-
gregate demand effects by using a highly stylized model, assuming constant employment
and abstracting from capital accumulation.

   The next section reviews empirical evidence on the time series properties of the func-
tional income distribution. Section 3 presents the mathematical derivation of the model

used. Section 4 outlines the calibration strategy used, while Section 5 presents the simu-

lation results. Finally, the last Section concludes.


2 Trends in the labour share of income

Figure 1 illustrates the decline in the labour share of income, which has taken place

in the majority of 16 high income countries for which data exist over the period 1960-

2010. Data are taken from the AMECO database. The adjusted wage share is defined as

percentage of GDP at current factor cost (Compensation per employee as percentage of
GDP at factor cost per person employed.) The decline has been gradual and continuous

in 11 out of 16 countries over the period 1960-2010. The largest drop took place in

Ireland, Japan and Austria with an annual growth decline of -0.54 percent, -0.38 percent
and -0.38 percent respectively. Italy, Norway and Finland displays declines around -0.30
percent annual, while the USA and Canada are close to -0.2 percent. More moderate
declines took place in France -0.17 and Sweden -0.10. Interestingly, this decline has
mainly taken place over the past two decades. In Australia and the Netherlands, the
labour share is constant but displays a large increase in the 1970’s and a long correction
thereafter. Contrastingly, in Denmark and the UK, the labour share has been fluctuating


                                              5
around a constant trend. Lastly, Belgium is the sole country in which the labour share
has been increasing at an annual rate of 0.17 percent.


                                         [Figure 1 about here.]


   These relatively large fluctuations in the labour share of income confirm existing stud-
ies. Blanchard (1997) and Blanchard and Giavazzi (2003) for instance make a similar
analysis and point to labour market rigidities as a source of these fluctuations.4 How-

ever, this result is not consensual since Kaldor predicted that the labour share is constant

around 65%. Picketty (2001) for instance makes a similar statement based on long term
time series. Contrastingly, Solow (1958) argues that while constant at the aggregate level,

the wage share displays excessive fluctuations at the sectoral level. Recently, as a similar

analysis has been conducted on US data by Young (2010).
   A reason to explain the absence of a consensus on the trends in the labour share of

income is the measurement difficulties. A first issue is the treatment of quasi public ad-

ministration and the financial sector in measuring the value added. A second issue is the

contribution of stock options and the income of the self-employed in the compensation

of labour. Askenazy (2003) shows for instance that correcting for the self-employed as
well as for quasi public administration affects the trends in the labour share significantly

in France and the USA.
   This paper does not intend to engage in the debate of the long run properties of the
labour share. However, since the under-shooting in the labour share is a short to medium
run phenomena, it raises the question of the economic consequence of this deviation on

aggregate demand and economic activity.
  4 See   Bentolila and Saint-Paul (2003) for a similar analysis.




                                                      6
3 Model
3.1 Households’ heterogeneity and aggregate quantities

There are two types of households, optimizing households denoted by subscript o and
rule of thumb households denoted by subscript r. We define the total number of house-
holds, consumption, employment and total labour endowment (labour supply) of each
household type as ϒi,t , Ci,t , Ni,t and Li,t for i = [o, r], respectively. The total aggregate
quantities are then given by the sums of these, thus Ct = Co,t +Cr,t and the equivalents.
       Consumption per household ct = Ctt is then given by
                                      ϒ


                                     ct = øc co,t + (1 − øc )cr,t ,                              (1)

               ϒo,t
where øc =     ϒt     is the share of optimizing consumers in the total population, and ci,t =
Ci,t
ϒi,t     for i = [o, r]. We assume that each household has a maximum labour endowment

of unity. We assume that rule of thumb households fully use their labour endowment,

thus Lr,t = ϒr,t . For optimizing consumers, we assume that their labour supply can be a

fraction υ , thus Lo,t = υ ϒo,t .5 This allows the model to encompass different cases. The
standard rule-of-thumb set-up as presented in Gal´ et al. (2007) corresponds to υ = 1. The
                                                 ı

polarized case where optimizing households are capitalists and rule of thumb households
are workers as in ? is given by υ = 0. The standard New Keynesian model with a single
optimizing households is achieved by assuming øc = υ = 1.
                                    Nt
       The employment rate nt =     Lt   is given by

                                     nt =øn no,t + (1 − øn )nr,t ,                               (2)

               υ øc
where øn =     øp     is the share of optimizing consumers in the workforce and ø p = 1 − (1 −
υ )øc . When υ = 1, then øn = øc and ø p = 1. The aggregate employment per household
              Nt
is given by   ϒt   = ø p nt
       5 We
         set this fraction exogenously. A model extension could have this value be determined endoge-
nously, for example as a function of wealth.


                                                   7
   Since only optimizing households hold a capital stock, per household investment and
capital stock are defined as

                                                xt =øc xo,t                                (3)

                                                kt =øc ko,t                                (4)

3.2 Labour Market Flows

In the model presented in this paper, all households face equal probabilities of finding or

loosing a job. Hence, we specify the labour market flows in aggregate quantities only.

All workers not working in a period are unemployed and looking for a job. The pool
of unemployed (relative to the labour force L) is given by ut = 1 − nt−1 . Unemployed

workers can be matched to a job and start working immediately in that period. The
                                                                                     γ 1−γ
matching function (again specified as relative to the labour force L) is mt = γm ut vt        ,

where mt are new matches, vt are posted vacancies, γ is the elasticity of matching to
unemployed workers and γm is the overall matching efficiency.

   Three definitions are used to describe the labour market: the probability of filling a

vacancy, qt = mt /vt , the job finding probability pt = mt /ut and labour market tightness
θt =    vt
       1−nt .   The model assumes quadratic employment adjustment cost ϕn,t ht following
                                                                                   mt 6
Gertler and Trigari (2009), which are specified in terms of the hiring rate ht =   nt−1 .

   Jobs separation probability is 1 − ρ . Employment at t is given by the remaining stock
of workers plus new matches.


                                            nt = ρ nt−1 + qt vt                            (6)

Thus, workers that were employed at t − 1 and who loosed their job are immediately in
  6 The   functional form used is as in Gertler and Trigari (2009)
                                                       κ 2
                                              ϕn,t =    h nt−1                             (5)
                                                       2 t




                                                       8
the pool of unemployed and are able to find a job in period t again. The probability of
filling a vacancy, qt and the job finding probability pt are given by:


                                             (              )γ
                                                 1 − nt−1
                                    qt =γm                                                  (7)
                                                    vt
                                             (              )1−γ
                                                    vt
                                    pt =γm                                                  (8)
                                                 1 − nt−1

3.3 Households

Optimizing and rule of thumb households maximize their inter-temporal utility function
                                      ∞
                         maxUi,t =   ∑ β t+ j u(ci,t+ j )    for i = [o, r],                (9)
                                     j=0

where β is the time discount factor and the period utility function u(ci,t ) is defined as

                                            c1−σ
                                                   i
                                             i,t
                               u(ci,t ) =              for i = [o, r].
                                            1−σi

Both types of households face the employment dynamics constraint.


                        ni,t = ρ ni,t−1 + pt (1 − ni,t−1 ) for i = [o, r].             (10)

3.3.1     Rule of Thumb Households

Rule of thumb households do not have access to financial markets. Therefore, their bud-

get constraint is given by their labour income plus their unemployment benefit payments
wu .

                                  cr,t ≤ wt nr,t + wu (1 − nr,t ),                     (11)


       The household maximizes its utility (9) subject to the employment and budget con-
straints, (10) and (11). The consumption of rule of thumb households is given by their




                                                   9
budget constraint (eq 11), which is always binding. Furthermore, the marginal utility of
consumption λr,t (the Lagrange multiplier on the budget constraint) is given by

                                                  λr,t = c−σ
                                                                r
                                                          r,t                                                   (12)

    The first derivative of the utility function Ur,t with respect to nr,t yields

                               Vr,t = λr,t (wt − wu ) + β Et [Vr,t+1 (ρ − pt+1 )] ,

where Vr,t is the Lagrange multiplier on the employment dynamics constraint (10), and
can thus be interpreted as the marginal utility value of a job to a household. It is useful
                                                                                                                 Vr,t
to define the value of a job in terms of a consumption good, thus we define Hnr ,t =                               λr,t .

We then obtain
                                                 [ r                         ]
                           Hnr ,t =wt − wu + β Et Λt,t+1 (ρ − pt+1 ) Hnr ,t+1 ,                                 (13)

                      λr,t+1
where Λt,t+1 =
       r
                       λr,t    is the stochastic discount factor for rule of thumb consumers.

3.3.2    Optimizing Consumers

Like rule of thumb households, optimizing households also earn labour income and un-

employment benefits. These quantities have to be scaled by the relative labour market

participation υ when expressing in per-household terms. Additionally, they can invest in
bonds paying a gross nominal interest rate Rn,t . When Bo,t is the total nominal quantity
                                                                    Bo,t
of bonds held by optimizing households, then bo,t =                 Pt ϒo    is the real stock of bonds per
optimizing household. Finally, they can accumulate physical capital ko,p,t subject to the
accumulation function

                                  ko,p,t = (1 − δ )ko,p,t−1 + xo,t (1 − ϕk,t ),                                 (14)

where ϕk,t are capital adjustment costs.7
                                                                             (               )2
                                                                        ηk        xo,t
   7 Capital    adjustment costs follow the usual specification ϕk,t =   2        xo,t−1   − 1 , so that ϕk = 0 at the
steady state.


                                                       10
   Optimizing households are allowed to vary the usage of physical capital by the fac-
tor uk,t , to earn a return uk,t rk,t on their physical capital stock. There is a cost ℑ(uk,t )
associated with capacity over- or under-utilization.8 Actual capital is determined by

                                                      ko,t =uk,t ko,p,t−1                                          (15)


   The budget constraint of optimizing households is given by

      co,t + xo,t + bo,t + ℑ(uk,t )ko,p,t−1
                                                                              Rn,t−1
             ≤ wt υ no,t + wu υ (1 − no,t ) + rk,t uk,t ko,p,t−1 +                   bo,t−1 − τo,t + Πt ,          (16)
                                                                                πt

where Πt are profit receipts from firms, πt =                        Pt
                                                                  Pt−1   is the gross price inflation rate and Pt is

the aggregate price level.
   The household maximizes its utility (9) subject to the employment dynamics con-

straint (10), the capital accumulation (14) and the budget constraint (16). We define the

Lagrange multipliers on the employment constraint as Vo (thus the marginal value of

a job), the budget constraint as λo (thus the marginal utility of consumption), and the
capital accumulation constraint as λk (thus the marginal utility value of a unit of capital).
                        λk,t
   Defining φt =         λo,t   (Tobin’s q), the first order conditions are given by

                     λo,t = co,t −σ
                                       o
                                                                                                                   (17)
                            1 πt+1
                  Λt,t+1 =
                   o
                                                                                                                   (18)
                            β Rn,t
                                 ( o [                                           ])
                      φt = β Et Λt,t+1 rk,t+1 uk,t+1 − ℑ(uk,t+1 ) + φt+1 (1 − δ )                                  (19)
                                  {           ( o )2 (          )}
                                               xt+1     xt+1
                         1 − β Et φt+1 Λt,t+1 xo
                                        o
                                                         xt − 1
                                                  t
                   φt =             (       o
                                                (         ))                                                       (20)
                                1 − ϕt + xxt ηk xt−1 − 1
                                          o
                                                    xt
                                                       t−1
                                   ψ (uk,t −1)
                     rk,t = rk e                                                                                   (21)

                     Vo,t = λo,t υ (wt − wu ) + β Et [Vo,t+1 (ρ − pt+1 )] ,
                                                 (                   )
                                                                                               ∂ ℑ(uk,t )
   8 The   functional form is ℑ(uk,t ) =   rk
                                           ψ         eψ (uk,t −1) − 1 , so that ℑ(1) = 0 and     ∂ uk,t     > 0.


                                                               11
                      λo,t+1
where Λt,t+1 =
       o
                       λo,t    is the stochastic discount factor for optimizing households.

    Similarly to rule of thumb households, we define the value of a job in terms of a
                                        Vo,t
consumption good Hno ,t =               λo,t .   We then obtain

                                                  [ o                         ]
                        Hno ,t =υ wt − υ wu + β Et Λt,t+1 (ρ − pt+1 ) Hno ,t+1 ,

3.4 The Wholesale Good Firm

Wholesale good firms produce output using capital and labour using a production func-

tion of the form Ytw = F(Kt , Nt ). We specify a CES production function, which is homo-

geneous of degree one. Therefore, output per household can be expressed as
                                      [                                       ] ζ
                                                 ζ −1                     ζ −1 ζ −1
                               ytw   = α (Bk kt ) ζ + (1 − α )(Bn ø p nt ) ζ        ,         (22)


where ζ is the elasticity of substitution, Bk and Bn are technology (scaling) parameters

and α is a share parameter. The Cobb-Douglas case occurs when ζ = 1.
    The firm maximizes its value Ft , expressed as per household, by selling output at the

real price ptw ,9 renting capital kt at price rk,t , and hiring labour nt at price wt , subject

to the dynamic equation governing employment as well as the quadratic employment
adjustment cost. The value is given by

                                               κ                        [ o         ]
                     Ft = ptw ytw − wt ø p nt − ht2 nt−1 − rtk kt + β Et Λt,t+1 Ft+1 ,        (23)
                                               2

where Λt,t+1 is also the firms’ discount factor as they are owned by optimizing house-
       o


holds. The first order conditions with respect to k, h and n (where we do not evaluate
   9 Section   3.7 specifies ptw .




                                                           12
∂ h/∂ n as each firm is small) are given, in that order, by
                                             (         )1
                                      ζ −1       ytw    ζ
                 rk =   ptw α (Bk )     ζ                                             (24)
                                                 kt
               κ ht = Jt                                                              (25)
                                              [       κ 2 ]      [ o          ]
                Jt = ptw atn − ø p wt + β Et Λt,t+1 ht+1 + β ρ Et Λt,t+1 Jt+1
                                                  o
                                                                                      (26)
                                                      2
                                                     ( w)1
                               (         ν
                                               ) ζ −1 yt ζ
                atn = (1 − α ) zn,t Bn kg,t ø p ζ                                     (27)
                                                       nt

The marginal productivity of labour is given by an . Jt is the Lagrange multiplier on

the ”budget” constraint of employment dynamics (6), and thus can be interpreted as the

marginal value to the firm of having another worker.

3.5 Bargaining

Firms and workers engage in Nash Bargaining over the joint surplus, the outcome of

which is the wage wt∗ . ηt is the workers relative bargaining power and is time dependant
since the experiment considered in this paper is a temporary shock on ηt .

                                      {                }
                                            ηt    1−ηt
                           wt∗   ≡ max (Ht ) (Jt )       ,       0 < ηt < 1           (28)

    The bargaining solution implies ηt Jt = (1 − ηt )Ht , where the aggregate worker sur-

plus is given as a weighted average of the individual surpluses according to their share in
the labour force, Ht = øn Hno ,t + (1 − øn )Hnr ,t .
    The bargaining set, the total surplus, is given by St = wt − wt , where wt is the max-
                                                            ¯               ¯
imum wage when firms’ surplus Jt = 0, and wt is the minimum wage when workers
surplus Ht = 0. The negotiated wage is the weighted average of these reservation wages,
wt∗ = ηt wt + (1 − ηt )wt . By substituting Jt = κ nt−1 , we obtain
         ¯                                         qt vt




                                                            13
                1 w n                         1     {                }
        wt∗ =ηt    pt at + (1 − ηt ) wu + ηt β Et Jt+1 Λt,t+1 pt+1
                                                            o
               øp                            øb
                           {                  }        (           )
                   1               κ        2              1     1      [ o         ]
              + ηt β Et Λt,t+1 (ht+1 ) + ηt ρβ
                              o
                                                              −      Et Λt,t+1 Jt+1
                   øp              2                      ø p øb
                            1 − øn   {                                      }
              + (1 − ηt ) β        Et Hnr ,t+1 (ρ − pt+1 )(Λt,t+1 − Λt,t+1 )
                                                             o       r
                                                                                         (29)
                              øb

   Hall (2005) demonstrates that real wage stickiness greatly improves the ability of a
search and matching model to match empirical employment dynamics. For this reason,
we follow him by utilizing the following wage rule


                                wt =ρw wt−1 + (1 − ρw )wt∗                               (30)


The actual wage is a weighted average between the Nash bargained wage and the past

period’s wage.

3.6 The Final Goods Firm

The final good (expressed per household), yt , is produced in a competitive market ac-

cording to the following CES technology:
                                      (∫     1    1
                                                        )µ
                                                  µ
                               yt =              yi,t di            µ ≥1                 (31)
                                         0

                                                                            1
where each input yi,t is a differentiated intermediate good. The term      1−µ   indicates the
price elasticity of the demand for any intermediate good i. Each period, final goods
producers choose a continuum of differentiated intermediate goods, yi,t at price Pi,t , to
maximize their profits subject to the CES technology (31). The demand function for

intermediate goods can be derived as follow:
                                                 (        )    µ
                                             Pi,t             1−µ
                                      yi,t =                        yt                   (32)
                                             Pt



                                                     14
3.7 Intermediate Good Firms

Intermediate good firms purchase homogeneous goods from the wholesale sector and

relabel them to produce differentiated goods. These differentiated goods are then sold
in a monopolistic competitive market to the final good firms. Furthermore, we assume

that intermediate good firms are subject to price stickiness, whereby a fraction χ1 cannot
reset its price in a certain period and set price Pt−1 . Of the fraction 1 − χ1 that is able to
reset its price, only a fraction χ2 performs a full optimization of its reset price, setting Pt∗ ,
while a fraction 1 − χ2 resets its price according to the simple rule Pb,t = Pt−1 πt−1 .

    The aggregate price Pt is given by Pt = χ1 Pt−1 + (1 − χ1 )Pt , where Pt = χ2 Pt∗ + (1 −
                                                                ˜          ˜

χ2 )Pb,t is the aggregate reset price. Normalizing these equations by Pt , we get:

                                            1 =χ1 πt−1 + (1 − χ1 ) pt
                                                                   ˜                                    (33)
                                                                       πt−1
                                         pt =χ2 pt∗ + (1 − χ2 )
                                         ˜                                                              (34)
                                                                        πt
                  Pt∗
    where pt∗ =   Pt    is the ”real” optimized reset price.
    Firms being able to optimize choose price pt∗ by maximizing their discounted stream

of real profits.
                                        ∞                   [              ]
                                                                 Pt∗
                                        ∑
                            max Et (χ1 β )s Λt,t+s
                             Pt∗
                                             o
                                                                Pt+s
                                                                     − pt+s yi,t+s
                                                                        w
                                                                                                        (35)
                                  s=0

subject to the demand equation (32). ptw represents the (real) purchasing price of whole-

sale goods, and thus the marginal costs.
    The first order condition is
                           1
                  f1,t =     f2,t                                                                       (36)
                           µ
                  where
                                                                         (           )    µ
                                    µ
                                                           1−µ −1
                                                                  −µ
                                                                               pt∗       1−µ
                  f1,t = (pt∗ ) 1−µ yt ptw + Λt,t+1 χ1 β πt+1                  ∗               f1,t+1   (37)
                                                                              pt+1
                                                                  (           ) 1
                                    1
                                                            −µ
                                                                        pt∗    1−µ
                  f2,t = (pt∗ ) 1−µ yt   + Λt,t+1 χ1 β πt+1 1−µ
                                                                        ∗            f2,t+1             (38)
                                                                       pt+1

                                                       15
    Firms set their price not at the current optimal level but at the level they deem optimal
over the expected lifetime of their set price. In the presence of inflation, this means that
firms having reset their price earlier will have a lower relative price than firms that just
reset their price, and will therefore have a higher share of aggregate demand. This means
that marginal products are not equal across firms, but that there will be inefficiencies due
to price dispersion, denoted with the symbol st . This means that the quantity available for
aggregate demand, yt , is not necessarily equal to the quantity from the per firm production
                                     1
function ytw , but only a fraction   st   of it. Hence, we have the relationships

                             ytw =st yt                                                        (39)
                                                  − 1−µ
                                                     1            1
                              st =(1 − χ1 ) pt
                                            ˜             + χ1 πt1−µ st−1                      (40)

                                                                                    1
    In a zero inflation steady state the optimal reset price will be given by        µ   = pw . Thus,

firms set price as a mark-up on nominal marginal costs.

3.8 Policies and resource constraint

Due to the lower zero bound on monetary policy, the interest rate set by the Central Bank

is the maximum of the interest rate as determined by a Taylor rule Rtn∗ and zero.

                                           Rtn = max [Rtn∗ , 0]                                (41)


The implementation of the zero bound is presented in Appendix B.
    The Taylor rule sets the interest rate according to a criteria of interest rate smoothing,
and measures of inflation and output. ϕm is the parameter driving the Taylor rule inertia,

while ϕπ and ϕy are the parameters setting the response of the interest rate to inflation
and output.
                                { n∗ }ϕm {( ) ( )ϕy }1−ϕm
                         Rtn∗    Rt−1      πt ϕπ yt
                              =                                                                (42)
                         Rn       Rn       π     y



                                                   16
      The government pays unemployment benefits and finances these using lump sum
taxes on optimizing households, which is thus equivalent to debt financing.10 Therefore,
rule of thumb households are not subjected to cyclical tax fluctuations. The resource
constraint is given by summing the budget constraints of both type of households (11),
(16) as well as the profit equation of firms.
                                                       κ qt2 vt2
                                     yt =ct + xt +               + ℑ(uk,t )øc ko,p,t−1                           (43)
                                                       2 nt−1
      Finally, the exogenous process subjected to a shock in this paper is ηt , which evolves

according to the autoregressive process

                                                ηt = (1 − ρη )η + ρη ηt−1                                        (44)

3.9 Equilibrium

The stationary equilibrium consists in processes for the flow variables [y, yw , c, co , cr , x, xo , an ],

the stock variables [n, no , nr , k, ko , ko,p ], the prices [Rn , rk , φ , uk , w, w∗ , pw , π , p, p∗ , f 1 , f 2 , s],
                                                                                                  ˜

the labour market rates [q, v, p] and the utility and discount rates [J, Hr , λo , Λo , λr , Λr ]

, given the structural parameters [øc , υ , σ o , σ r , β , δ , ψ , ηk ], the labour market param-

eters [κ , ρ , γm , γ , ρw ], the production parameters [α , Bk , Bn , ζ ], the pricing parame-
ters [µ , χ1 , χ2 ], the policy parameters [wu , ρm , ϕπ , ϕy , τr , τo ] and the exogenous process
[η , ρη ] satisfying the equilibrium conditions given by equations (1), (2), (3), (4), (6),

(10), (7), (8), (11), (12), (13), (14), (15), (17), (18), (19), (20), (21), (22), (24), (25),
(26), (27), (29), (30), (33), (34), (36), (37), (38), (39), (40), (42), (43) and (44) and the
                                λi,t+1
definitions Λt,t+1 =
            i
                                 λi,t    for i = [o, r].


4        Steady State and Calibration

The steady state of the model is given when all variables are constant over time. In
principle, the steady state can be solved given all structural parameters. In practice, it
    10 τ      = wu (1 − nr,t + υ (1 − no,t ))
        o,t


                                                            17
is usual procedure to calibrate target values for certain variables and derive structural
parameters from these.
   First, this paper calibrates a zero inflation steady state and normalizes the price level
to unity. Next, this paper calibrates the job separation rate ρ , the job finding rate p and
labour market tightness θ to match empirically observed values, and uses these to derive
the structural parameters γm and κ . Furthermore, it is useful with CES production func-
tions to normalize steady state output to unity. This requires the technology parameters

Bk and Bn to be computed as the inverse of the steady states of the factors of production,

k and n. These steady state values are easily derived using knowledge of the real interest
rate (given the discount parameter β ) and the job separation and finding rate.

   Table 1 shows the parameter calibration used for the numerical simulations carried

out further below. The parameters are essentially taken from Gertler and Trigari (2009),

who estimated a similar model for the US economy. The relative risk aversion is identical

for both households σ o = σ r and is set at 1. It follows that the utility function takes the
form of a logarithmic function. The time discount factor β is set at 0.992, generating

an annual interest rate of 3.2%. Capital depreciates at a rate of 2.5% per quarter, which
corresponds to 10% annual rate of depreciation. The cost of capital adjustment ηk is 3

and the cost of capacity utilization is set at 0.7 following the estimation made by Sala
et al. (2008) for the US economy.

   The parameters of the labour market are conventional and taken from Shimer (2005).
The job surviving rate ρ is set at 90%, while the job finding probability p and the labour
market tightness are equal to 0.95 and 0.5 at the steady state respectively. The elasticity

of matching to unemployed workers γ is 0.5. An important parameter in search and
matching models is the replacement ratio ω . In models without strong wage stickiness,
a high value is needed to generate realistic employment fluctuations. Gertler and Trigari

(2009) estimate this value to be 0.72 in a model with wage stickiness and 0.98 in a model


                                             18
without wage stickiness. We choose an intermediate value ω = 0.9. Since restrictions
are placed on two variables p and θ , the steady states for labour market variables are
found by solving endogenously for the two parameters γm , the efficiency of the matching

function, and κ , the employment adjustment cost. They are respectively equal to 1.345
and 0.6572. These parameters produces an employment rate n of nearly 90% at the steady
state. Finally, wage rigidity is moderate with ρw = 0.3.


                                   [Table 1 about here.]


   The capital share α is set at 0.3 and µ is set at 1.11 for a mark up of 11%, generating a
labour share of income of 63% at the steady state. The coefficient Bk and Bn are equal to

0.1223 and 1.3003 respectively, which corresponds to the inverse of the steady state value
of the capital stock and employment in order to normalize the CES production function.

The set of parameters related to nominal price rigidities is conventional. 75% of firms are
unable to adjust their price to the optimal price every period. There is no price indexation

χ2 = 1. Monetary policy inertia ρm is set at 0.8, while the reaction of the interest rate
to inflation and output are 1.7 and 0.2 respectively. Lastly, the experiment undertaken

consists in a negative shock on the bargaining power of workers η , which is equal to 0.5
at the steady state. The shock is given by ε = 0.05η and produces a 1.5% decline in the
labour share of income in the baseline calibration. The persistence of the shock ρη is 0.9.

   We define the baseline model when optimizing households’ participation to the labour
market is given by υ = 0.5. The share of rule of thumb households (1 − øc ) is 70%
         ı
as in Gal´ et al. (2007). Furthermore, the production function assumes a lower degree

of substitution between labour and capital, ζ = 0.6, an intermediate value between the

Cobb-Douglas case ζ = 1 and the Leontief case ζ = 0. To simulate a standard New Key-
nesian model, we set the share of optimizing households and the elasticity of substitution

between capital and labour to unity (øc = 1, υ = 1, ζ = 1). We also consider another


                                            19
extreme case where optimizing households are identified as capitalists, thus not earning
labour income (υ = 0). All intermediate calibrations are possible.


5       Results
5.1      Baseline results

This section presents the simulated results of a fall in worker’s bargaining power using

the model described in this paper. The solid line in Figure 2 shows the baseline cali-
bration, while the dashed line shows a standard New Keynesian model with search and
matching in the labour market.11 The figures for output, consumption and investment

below represent percentage point deviations in terms of GDP, which in turn is normal-

ized to one in steady state. Inflation, employment and the labour share are represented as

percentage point deviations.

                                       [Figure 2 about here.]

     The fall in bargaining power causes a fall in the labour share for three reasons. First,

workers can only bargain for lower wages, leading to a fall in real wages. Second, em-

ployment adjusts slowly with search and matching in the labour market. Third, with low

substitution between capital and labour, the price effect from a fall in the wage is not

countered fully by the quantity effect from an eventual rise in employment. In the stan-
dard New Keynesian model, only the first two effects are present, explaining why the
labour share quickly returns to its baseline value. Contrastingly, the labour share falls
by more than 1% on impact and stay below steady state for more than 15 quarters in the
baseline calibration.
    11 Thebaseline calibration includes rule of thumb households øc = 0.3 and full participation of opti-
mizing households in the labour market υ = 0.5. The production function has low degree of substitution
between labour and capital ζ = 0.6. The parameters are presented in Table 1. The standard New Keyne-
sian calibration assumes away rule of thumb households øc = 1, while optimizing households participate
in the labour market υ = 1. The production function is Cobb-Douglas ζ = 1, while there is no bound on
the interest rate and no wage rigidity ρw = 0. Remaining parameters are identical to the parameters of the
baseline model.


                                                   20
   The experiment of lowering workers’ bargaining power usually raises output, con-
sumption, investment and employment in a Standard New Keynesian model. Firms’
surplus from employment relationships rises, thus increasing vacancies, the number of
matches, employment and output. This increases both the marginal product of capital
and aggregate saving, thus raising investment. Consumption rises since permanent in-
come increases. Inflation falls since the increase in employment and the fall in wages
lowers marginal costs.

   The baseline calibration additionally takes several aspects into account that are ig-
nored by the standard New Keynesian model. First, the presence of rule of thumb con-

sumers makes the income distribution an important driver of aggregate demand. This

channel is absent in the standard model since a representative household earns all in-

come. Second, a lower elasticity of substitution between capital and labour implies a

larger fall in the labour share due to a fall in bargaining power since the price and sub-

stitution effects from a fall in wages do not cancel. Third, the baseline simulation allows

for the presence of a liquidity trap. The fall in aggregate demand lowers inflation, in re-

sponse to which the central bank should lower the nominal interest rate. In the presence

of a lower bound on the nominal interest rate, the fall in inflation actually raises the real

interest rate. This has a threefold effect. Vacancy posting falls since the surplus from
a match decreases, investment demand falls, and consumption demand by optimizing
households falls as well.
   Summarizing, the baseline simulation allows for strong aggregate demand effects
in response to a falling bargaining power of workers to occur. The solid line in Fig-
ure 2 shows that under such circumstances a fall in workers’ bargaining power leads to
long-lasting reductions in important macroeconomic variables. Aggregate consumption
declines by 1.2 percent on impact. Output declines by 2.3 percent and employment by
1.7 percent. Additionally, investment also falls for 3 quarters. It follows that a reduction


                                            21
in worker’s bargaining power in a situation of economic recession with low interest rates
further depresses economic activity on impact.

5.2 Sensitivity analysis

This section presents some sensitivity analysis to illustrate the importance of the different
transmission channels at work in the baseline calibration. Figure 3 presents the sensitiv-
ity analysis concerning the impact of the income distribution as a driver of aggregate

demand. The solid line shows the baseline calibration as presented above. The dashed

line shows a calibration of limited labour market participation, where optimizing house-

holds behave purely as capitalists and do not participate in the labour market υ = 0. As
a result, a larger share of total consumption cannot be smoothed intertemporally.12 A
fall in bargaining power therefore leads to a larger fall in aggregate demand, and con-

sequently to a more severe depression of economic activity. Furthermore, the model is
moved closer to an instability region, thus producing a kink in the dynamic path of the

variables.13

   The dashed-dotted line represents the case where there are no rule-of-thumb con-

sumers, thus there is no demand effect from changes in the functional income distribu-

tion (øc = 1 and υ = 1). However, the case still allows for the economy to be facing a
liquidity trap. In this case, the increase in aggregate saving described above, combined
with the fall in inflation, do not boost investment but cause the lower bound on the nom-
inal interest rate to be binding. Compared to the baseline model, the absence of income

distribution effects on aggregate demand cancels the importance of the lower bound.

                                          [Figure 3 about here.]

   Figure 4 shows that the mechanisms introduced by this paper, the importance of the
income of workers to support aggregate demand, induces a fall in output and employment
  12 The   wage share is more or less fixed due to our calibration strategy.
  13 Increasing  price stickiness or decreasing ζ moves the economy closer to the instability.


                                                     22
after a fall in bargaining power even in the absence of a liquidity trap. The calibration
has been changed to have a low elasticity of substitution (ζ = 0.05), no lower bound on
monetary policy, while the share of optimizing households remains constant at υ = 0.5.

                                            [Figure 4 about here.]

   The solid line in Figure 4 shows that output and employment fall on impact. Due to

the low elasticity of substitution, the fall in bargaining power has a strong effect on the
labour share, which, coupled with price stickiness and the impossibility of consumption
smoothing, induces a strong negative demand effect. Nevertheless, the fall of the real

interest rate combined with the saving shock increase investment, which causes a quick

rebound of output. Although consumption falls on impact and stays below zero for 4

quarters, output increases. In the absence of a liquidity trap, the nominal interest rate

falls with inflation leading to a decline in the real interest rate. It follows that both labour

demand and investment react positively. The speed at which output recovers is partly de-

termined by the existence of capital adjustment costs, which delays investment decisions.

Monetary policy shortens the recession by stimulating both supply and demand channels.
   This simulation assumes very low elasticity of substitution between labour and capi-

tal. The distinction between Leontief and Cobb-Douglas production function is a central

element of the literature discussing changes in the labour share. Following a change in
wages, the degree to which firms substitute capital and labour affects the level of employ-
ment. The labour share drops by 0.37 percentage point for an elasticity of substitution of
0.05 (in the Figure 4).14 The impact on output and employment is however quasi null,
given the strength of the other transmission channels.
   The dashed line in Figure 4 corresponds to a case in which there is no participation

of optimizing households to the labour market υ = 0. This polarized distribution of in-
come between workers and optimizing households amplifies the wage-aggregate demand
  14 The   elasticity of substitution is 0.6 in the baseline case in Figure 2.


                                                        23
channel. Output declines by 0.5 percentage point and stays negative for 2 quarters. The
fall in the labour share is also more pronounced at 0.5 percentage point.
   The dashed doted line in Figure 4 shows the importance of nominal price rigidities.
The transmission channel between lower consumption and output depends on the pres-
ence of price stickiness. Increasing price rigidity from 4 quarters to 8 quarters magnifies
the demand side effects (dashed line). The fall in worker’s bargaining power lowers the
labour share further producing a larger drop in output and employment. Output and em-

ployment drop by 0.6 percentage point and 0.5 percentage point on impact respectively,

while they both stay negative for 3 quarters.

5.3 Minimum wage as a lower bound on wage

Figure 2 has illustrated the importance of labour income for aggregate demand in the

proximity of the lower zero bound in monetary policy. The transmission channel going

from labour income to aggregate demand modifies the traditional views on minimum

wage. In a standard New-Keynesian model, the minimum wage is seen as hampering the

downward adjustment in wages. This in turn limits labour demand of firms and amplifies

business cycle fluctuations. Contrastingly, in the present model, the minimum wage sets

a lower floor on labour income, which sustains consumption and aggregate demand. The

direct negative effect of the minimum wage on labour demand is balanced by its positive
impact on aggregate demand.
   In this section, minimum wage is modelled in a similar way than the lower zero bound
in monetary policy. The actual wage is the maximum between the wage rule (eq 30) and

the minimum wage.



                                   wt = max [wtr , wmin ]                            (45)

with wt the actual wage, wtr the wage according to the wage rule in eq 30 and wmin the

                                            24
minimum wage.
   Figure 5 Panel A reproduces the baseline simulation with (dashed line) and without
a minimum wage (solid line). The calibration is similar to the baseline model presented

in Figure 2. The main result is that the minimum wage reduces the size of the recession
following a decline in the bargaining power of workers. The minimum wage reduces the
drop in output from 2.2 percentage point to 1.5 percentage point on impact. The drop in
the labour share of income is also smaller from -1 percentage point to -0.67 percent on

impact. It follows that the drop in consumption is significantly lower than in the absence

of a lower bound on wages, sustaining aggregate demand. A secondary effect is related
to the adjustment in price. Since inflation declines less in the presence of the minimum

wage, the increase in the real interest rate is more moderate, which is less detrimental to

investment and labour demand. The adjustment in the markup also sustains the surplus

from an additional match, which affects the hiring decisions of firms.
   In Figure 5 Panel B, a similar mechanism takes place although the minimum wage

is applied to the set of parameters described in Figure 4. The dashed line shows that

a lower bound on the real wage also works in the absence of a liquidity trap. Income

and consumption of rule of thumb consumers is supported, which stabilizes aggregate

demand and lessens the negative impact of the fall in bargaining power.

                                  [Figure 5 about here.]


6 Conclusion

The model presented in this paper shows that under certain conditions a change in income
distribution in favor of capital leads to lower employment and output. This result stands
in contrast with the conclusion from a standard New Keynesian model, which finds virtue
to wage moderation. To reach this result, the modelling strategy has been to reinforce the
transmission channel from income distribution to consumption decisions by combining

                                            25
rule-of-thumb households and nominal price rigidities. This transmission is strengthened
in the presence of a lower zero bound in monetary policy.
   In the New-Keynesian model, the main transmission channel goes through labour
demand. The increase in consumption and investment follows from the increase in em-
ployment and permanent income. Contrastingly, in the present model, consumption drops
due to the presence of rule-of-thumb households, while investment is negatively affected
by the increase in the real interest rate. The lower zero bound also affects labour demand

negatively. It follows that lower bargaining power leads to lower output and employ-

ment in the presence of a liquidity trap. An important implication of this model is that
downward wage rigidity sustains aggregate demand and reduces the fall in output.

   Two extensions to the present paper can be envisioned. The first is to allow workers to

have some access to financial markets, and thus engage in some limited borrowing. This

allows the study of the effect of inequality on household indebtedness, thereby following

Kumhof and Ranciere (2010). Second, the extension to a two country model allows

to study a number of research questions present on the current political agenda. In an

open economy, a falling wage will additionally raise export demand, depending on the

exchange rate regime. However, such a policy could be a beggar-thy-neighbor policy

by raising unemployment in the foreign country. Furthermore, international imbalances
might result. Given the results obtained in this paper, an interesting addition to the policy

debate is likely to result from these extensions.




                                             26
A Steady State

In a zero inflation steady state with π = 1, we can normalize the price level to unity, and
                                  1
thus obtain p = 1, s = 1 and pw = µ . Furthermore, capital utilization is at unity uk = 1,
            ˜
while Tobin’s q is unity φ = 1. Therefore, we can derive
                                              1
                                       Rn =                                            (46)
                                              β
                                              (                  )
                                            1
                                       rk =   −1+δ                                     (47)
                                            β
    Given ρ and p, and calibrating γ , we can derive
                                                    p
                                        γm =                                           (48)
                                                  θ 1−γ
                                         q =γm θ −γ                                    (49)
                                                    p
                                         n=                                            (50)
                                                  1−ρ + p
                                         v =θ (1 − n)                                  (51)

Since the replacement wage is the same for both types of households, the employment

ratio will also be the same in steady state, thus n = no = nr . The number of optimizing

consumers in employment approaches zero as υ → 0.
    When using a CES production function, it is useful to normalize output at unity, thus
y = 1. This requires to set the technology parameters multiplying the factors at the inverse

of the factor’s steady state values.
                                    rk
                               Bk =                                                    (52)
                                   α pw
                                    1
                              Bn =                                                     (53)
                                   ø pn
                                                                 ( )1
                                                          ζ −1    1 ζ
                               a =(1 − α ) (Bn ø p )
                                n                           ζ                          (54)
                                                                  n
Using this procedure, α actually represents the steady state factor share of capital. The
steady state value of capital is thus k = 1/Bk , while the steady state value of investment

is x = δ k.

                                                   27
   Furthermore, it is useful to define the replacement wage as a fraction ω of the steady
state real wage, thus wu = ω w. Using this equation as well as the steady state versions of
the labour market equations (25), (26), and (29), we can derive the steady state values of

w, and the parameter κ required for the labour market to solve to the equilibrium values
calibrated above. The system of equations to be solved is given by
                                             [                   ]
                       qv                       qv     κ [ qv ]2
                     κ =p a − ø p w + β κ ρ +
                             w n
                                                                                      (55)
                        n                       n      2 n
                              1 w n                      1     qv
                       w∗ =η p a + (1 − η ) wu + η β κ p
                              øp                        øb     n
                                                 (           )
                                 1 κ                1      1     qv
                            + η β h2 + ηρβ             −       κ                      (56)
                                 øp 2               ø p øb        n

w is be larger the smaller is ø p . The intuition is that we calibrate output per household

to unity, while the labour share in the production function is α . A smaller labour force
participation (as is implied by smaller ø p ) requires a larger real wage. This will also
increase the consumption share of rule-of-thumb households.

   We then get the steady state for Hnr

                                             (1 − ω )w
                                   Hnr =                                              (57)
                                           1 − β (ρ − p)

   The steady state of aggregate consumption is a residual of the resource constraint:

                                                  κ q2 v2
                                    c =y − x −                                        (58)
                                                  2 n

Lastly, consumption steady states of both type of households are defined as follow:

                                cr =wn + wu (1 − n) − τr                              (59)
                                       c − (1 − øc )cr
                                co =                                                  (60)
                                             øc

B Zero Bound

The procedure for the introduction of a lower bound on a variable into a stochastically
simulated model in Dynare is described in Holden (2011). To introduce a bound on a

                                             28
variable, a sequence of non-autoregressive, one period, pre-known shocks to the variable
in question are introduced into the model. For each of these shocks, the impulse response
functions for all variables can be computed. Then, a linear combination of these shocks
is computed using an optimization procedure such that the added impulse response func-
tions from all these individual shocks plus the IRF of the original shock allow the bound
to hold on the variable in question. Finally, the IRF for all variables can be computed
given the linear combination of the lower bound shocks.


C Detailed Derivations
C.1 Household heterogeneity and aggregate quantities

The derivation of the rates øn and ø p surrounding (2) is given by


                            Lt =Lo,t + Lr,t = υ ϒo,t + ϒr,t                         (61)
                                 Nt     Lo,t        Lr,t
                            nt =    =        no,t +      nr,t                       (62)
                                 Lt      Lt         Lt
                                    υ ϒo,t             ϒr,t
                            nt =               +                                    (63)
                                 υ ϒo,t + ϒr,t υ ϒo,t + ϒr,t
                                      ϒo,t               ϒr,t
                                     υ  ϒt               ϒt
                            nt = υ ϒ +ϒ          +   υ ϒo,t +ϒr,t
                                                                                    (64)
                                    o,t    r,t
                                     ϒt                  ϒt
                                        υ øc        1 − øc
                            nt =                +                                   (65)
                                   υ øc + 1 − øc υ øc + 1 − øc

which equals the equations in the main text. Furthermore, we show that

                                    Nt Lt
                                      = nt
                                    ϒt ϒt
                                        υ ϒo,t + ϒr,t
                                      =               nt
                                             ϒt
                                         =(υ øc + 1 − øc )nt                        (66)




                                                 29
C.2 Optimizing households

The first order condition w.r.t. investment is
                             ∂ ϕk,t                  ∂ ϕk,t+1
 λo,t =λk,t (1 − ϕk,t ) − λk,t xo,t  − λk,t+1 xo,t+1
                              ∂ xo,t                  ∂ xo,t
         (                     (            ))                    (       ) (          )
                      xo,t         xo,t                             xo,t+1 2 xo,t+1
    1 =φt 1 − ϕk,t −        ηk          −1        + φt+1 Λt,t+1 β
                                                            o
                                                                                    −1
                     xo,t−1      xo,t−1                              xo,t     xo,t
                                                                                      (67)
All remaining foc are straight forward.

C.3 Wholesale Good Firm

The production function of the wholesale good firm is Ytw = F(Kt , Nt ), which can be

specified as
                                               (           )
                                      Ytw          Kt Nt
                                          =F         ,                                 (68)
                                      ϒt           ϒt ϒt
    The ”constraint” of employment dynamics can be written as

                                      nt = ρ nt−1 + ht nt−1                            (69)

The Lagrange multiplier on this constraint is defined as Jt . Given this, the derivation of

the foc w.r.t. nt yields
                            κ [ o          ]      [ o              ]        [ o          ]
   Jt = ptw atn − ø p wt − β Et Λt,t+1 ht+1 + β Et Λt,t+1 ht+1 Jt+1 + β ρ Et Λt,t+1 Jt+1
                                        2
                            2
                                                                                        (70)
which solves to (26) when using Jt+1 = κ ht+1 .

C.4 Bargaining

The maximum and minimum wage in bargaining, wt when Jt = 0 and wt when Ht = 0,
                                            ¯
are given by
              1 w n 1       [       κ 2 ] 1             [ o          ]
         wt =
         ¯      pt at + β Et Λt,t+1 ht+1 + β ρ Et Λt,t+1 Jt+1
                              o
                                                                                       (71)
             øp        øp           2         øp
                   1    [          (                                             )]
         wt =wu − β Et (ρ − pt+1 ) øn Λt,t+1 Hno ,t+1 + (1 − øn )Λt,t+1 Hnr ,t+1
                                       o                          r
                                                                                       (72)
                   øb

                                               30
                                                                                                               ηt
where we define øb = υ øn + 1 − øn . Using the bargaining solution Ht =                                        1−ηt Jt   as well as

the definition Ht = øn Hno ,t + (1 − øn )Hnr ,t , we can substitute for Hno ,t
                                                     ηt 1         1 − øn
                                       Ho,t =                Jt −        Hnr ,t                                              (73)
                                                   1 − ηt øn        øn
    Applying wt∗ = ηt wt + (1 − ηt )wt , we obtain
                      ¯
                   1 w n                        1     {                }
         wt∗ =ηt      pt at + (1 − ηt ) wu + ηt β Et Jt+1 Λt,t+1 pt+1
                                                              o
                  øp                           øb
                              {                 }        (           )
                      1               κ       2              1     1      [ o         ]
                 + ηt β Et Λt,t+1 (ht+1 ) + ηt ρβ
                                 o
                                                                −      Et Λt,t+1 Jt+1
                      øp              2                     ø p øb
                               1 − øn {                                       }
                 + (1 − ηt ) β       Et Hnr ,t+1 (ρ − pt+1 )(Λt,t+1 − Λt,t+1 )
                                                               o       r
                                                                                                                             (74)
                                 øb
                                         qt vt                qt vt
Finally, the substitutions ht =          nt−1      and Jt = κ nt−1 are made.

C.5 Intermediate Good Firms

Maximization of the profit function (35) w.r.t. the demand function (32) yields
       ∗ µ       ∞                                                ∗ µ −1 ∞                         µ
  1                                      −     1
                                                             µ                                  − 1−
 1−µ
     Pt 1−µ      ∑ (χ1β )sΛt,t+sPt+s µ yt+s =
                           o       1−
                                                            1−µ
                                                                Pt 1−µ ∑ (χ1 β )s Λt,t+s pt+1 Pt+s µ yt+s
                                                                                   o      w
                 s=0                                                    s=0
         ∗ 1−µ
            1     ∞                      −     1
                                                                ∗ 1−µ ∞
                                                                   µ
                                                                                                          −    µ
     1
       P
     µ t         ∑ (χ1β )sΛt,t+sPt+s µ yt+s = Pt
                           o       1−
                                                                       ∑ (χ1β )sΛt,t+s pt+1Pt+s µ yt+s
                                                                                 o      w     1−
                                                                                                                             (75)
                 s=0                                                   s=0
                           f2,t                                                            f1,t

Deriving f2,t
                       ∗ 1−µ ∞
                          1
                                                        −   1
           f2,t =Pt            ∑ (χ1β )sΛt,t+sPt+s µ yt+s
                                         o       1−

                           s=0
                     (   ∗)
                                1
                   P           1−µ           ∗ 1−µ ∞
                                                1
                                                                                 −     1
                 = t
                   Pt
                                     yt + Pt          ∑ (χ1β )sΛt,t+sPt+s µ yt+s
                                                                o       1−

                                                     s=1
                           1                            ∗ 1−µ ∞
                                                           1
                                                                                                  −   1
                 =(pt∗ ) 1−µ yt + χ1 β Λt,t+1 Pt
                                        o
                                                                 ∑ (χ1β )sΛt+1,t+1+sPt+1+syt+1+s
                                                                           o           1−µ

                                                                s=0
                                                       (          )    1
                           1                                Pt∗       1−µ
                 =(pt∗ ) 1−µ yt      + χ1 β Λt,t+1
                                             o
                                                             ∗              f2,t+1
                                                           Pt+1
                                                                  (          )    1
                           1                        − 1−µ
                                                       1
                                                                       pt∗       1−µ
                 =(pt∗ ) 1−µ yt      + χ1 β Λt,t+1 πt+1
                                             o
                                                                       ∗               f2,t+1                                (76)
                                                                      pt+1
An equivalent derivation holds for f1 ,t.

                                                             31
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                                            34
Figures




                                                                      Figure 1: (adjusted) labour share of income (at factor cost)
                                                                Australia                                  Austria                                    Belgium                                     Canada
                                          60 65 70 75




                                                                                                                              60 65 70 75




                                                                                                                                                                           60 65 70 75
                                                                                    80
 labour share of income (in percentage)




                                                                                    60




                                                        1960   1980   2000   2020                 1960   1980   2000   2020                 1960    1980   2000     2020                 1960   1980    2000    2020

                                                                Denmark                                    Finland                                    France                                       Ireland
                                          65 70 75




                                                                                    6065707580




                                                                                                                              65 70 75 80




                                                                                                                                                                           50 60 70 80




                                                        1960   1980   2000   2020                 1960   1980   2000   2020                 1960    1980   2000     2020                 1960   1980    2000    2020

                                                                  Italy                                    Japan                                    Netherlands                                   Norway
                                          60 65 70 75




                                                                                    65 70 75 80




                                                                                                                                                                           5055606570
                                                                                                                              60 65 70 75




                                                        1960   1980   2000   2020                 1960   1980   2000   2020                 1960    1980   2000     2020                 1960   1980    2000    2020

                                                                 Spain                                    Sweden                                   United Kingdom                               United States
                                          60 65 70 75




                                                                                                                                                                           6466687072
                                                                                    65 70 75 80




                                                                                                                              7072747678




                                                        1960   1980   2000   2020                 1960   1980   2000   2020                 1960    1980   2000     2020                 1960   1980    2000    2020

                                                                                                                        year
                                          Graphs by country




                                                                                                                              35
            Figure 2: Standard New Keynesian vs baseline model

              Output                                         Consumption
                                             0.5%
  0%
                                              0%

 −1%                                     −0.5%

                                             −1%
 −2%
                                         −1.5%
        0     5           10      15                0           5          10      15

             Investment                                       Employment
 0.5%                                        0.5%
                                              0%
                                         −0.5%
  0%
                                             −1%
                                         −1.5%
−0.5%                                        −2%
        0     5           10      15                0           5          10      15

            Labour Share                                       Inflation
  0%                                          0%


−0.5%
                                         −0.5%
 −1%


−1.5%                                        −1%
        0     5           10      15                0           5          10      15
                   Baseline Model with lzb              Standard New Keynesian Model




                                       36
            Figure 3: Sensitivity Analysis: Income Distribution

             Output                                       Consumption
                                            0.5%
  0%
                                             0%
 −1%
                                        −0.5%
 −2%                                        −1%
 −3%                                    −1.5%
 −4%                                        −2%
        0    5           10       15               0        5           10      15

            Investment                                    Employment
 0.5%
                                             0%

                                            −1%
  0%
                                            −2%

−0.5%                                       −3%
        0    5           10       15               0        5           10      15

            Labour Share                                    Inflation
  0%                                         0%


−0.5%                                   −0.5%


 −1%                                        −1%


−1.5%                                   −1.5%
        0    5           10       15            0            5         10        15
                                            Baseline Model          Baseline Model
             Baseline Model with lzb
                                            no participation        no RoT consumers




                                       37
            Figure 4: Sensitivity Analysis: CES and Price Stickiness

               Output                                          Consumption
 0.5%                                           0.5%


  0%
                                                  0%
−0.5%


 −1%                                        −0.5%
        0       5           10       15                0           5           10          15

               Investment                                      Employment
 0.5%                                           0.5%
 0.4%
                                                  0%
 0.3%
 0.2%
                                            −0.5%
 0.1%
  0%                                             −1%
        0       5           10       15                0           5           10          15

              Labour Share                                         Inflation
  0%                                              0%
                                            −0.1%
                                            −0.2%
−0.5%
                                            −0.3%
                                            −0.4%
 −1%                                        −0.5%
        0       5         10          15               0           5           10          15
                Baseline no lzb
                                                no participation          High price rigidity
                + low substitution CES




                                           38
                      Figure 5: Minimum wage

             Output                                   Consumption
 0.5%                               0.5%

  0%
                                       0%
−0.5%

 −1%                               −0.5%

−1.5%
                                       −1%
 −2%

−2.5%                              −1.5%
        0    5         10    15              0           5         10         15


            Labour Share
  0%



                                         Baseline Model
−0.5%                                                               With minimum wage
                                         with lzb



 −1%




−1.5%
        0    5         10    15




                            (a) Panel A

             Output                                   Consumption
 0.5%                               0.5%




  0%

                                       0%

−0.5%




 −1%                               −0.5%
        0    5         10    15              0           5         10         15


            Labour Share
  0%


−0.2%
                                             High price rigidity        With minimum wage
−0.4%


−0.6%


−0.8%


 −1%
        0    5         10    15




                            (b) Panel B


                                  39
Tables

                      Table 1: Calibration: baseline model

    Structural parameters
    Share of Optimizing Consumers                              øc = 0.3
    Labour market participation of optimizing consumers        υ = 0.5
    Relative risk aversion parameters                        σo = σr = 1
    Discount factor                                           β = 0.992
    Capital depreciation rate                                 δ = 0.025
    Capital adjustment cost                                     ηk = 3
    Capital utilization cost                                   ψ = 0.7
    Labour market parameters
    Exogenous job loss probability                           1 − ρ = 0.1
    Target job finding probability                              p = 0.95
    Labour market tightness                                    θ = 0.5
    Matching elasticity                                        γ = 0.5
    Implied matching function parameter                      γm = 1.345
    Implied employment adjustment cost                       κ = 0.6605
    Implied employment rate                                  n = 0.9048
    wage rigidity                                             ρw = 0.3
    Production parameters
    Capital share                                              α = 0.3
    Elasticity of substitution                                 ζ = 0.6
    Capital technology                                       Bk = 0.1223
    Labour technology                                        Bn = 1.3003
    Pricing parameters
    Demand elasticity                                         µ = 1.11
    Price stickiness                                          χ = 0.75
    Price indexation                                           χ1 = 1
    Policy parameters
    Replacement rate                                          ω = 0.9
    Interest rate smoothing                                   ρm = 0.8
    Inflation response                                         ϕπ = 1.7
    Output response                                           ϕy = 0.2
    Bargaining power                                          η = 0.5
    Bargaining power auto-regressive coefficient               ρη = 0.9




                                      40

				
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