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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 References Acemoglu, D. (2003). Cross-country inequality trends. Journal of economic littera- ture 113, 121–149. e Arpaia, A., E. P´ rez, and K. Pichelmann (2009). 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One of the things we know that ain’t so: Is us labor’s share relatively stable? Journal of Macroeconomics 32(1). 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