Optimal Monetary Policy with heterogeneity in wage stickiness

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Optimal Monetary Policy with

Heterogeneity in Wage Stickiness

Rutger Jansen (10868070)

Supervisor:

Christian A. Stoltenberg Master Thesis Economics

Specialisation: Monetary policy and banking July, 2016

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Rutger Jansen

Abstract

This thesis studies optimal monetary policy in Europe. The thesis’ key objective is to analyse if differences in labour market dynamics across countries in Europe have implications for opti-mal monetary policy. These labour market differences are quantified by the difference in wage rigidities, to make it possible to implement them in a DSGE model. This thesis compares the welfare implications of different monetary policy rules. It applies these different Taylor rules to multiple countries in Europe and analyses if the heterogeneity in wage rigidity has implications for optimal monetary policy. The key findings are that the differences in wage rigidity have important implications for optimal monetary policy. For countries with high wage rigidity like Italy it is optimal to have a lower interest rate reaction coefficient, φ, meaning that the mone-tary policy targeting rule in those countries should be less strict. This is caused by the positive relation between the wage rigidity and the variance in output gap and therefore the welfare loss. Welfare-theoretic losses were calculated and strict price inflation targeting generates relatively large welfare losses, whereas other simple policy rules perform better.

Statement of Originality

This document is written by Student Rutger Jansen who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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TABLE OF CONTENTS Rutger Jansen

1 Introduction

The desirability of alternative monetary policies has long been and continues to be one of the most analysed and debated issues in macroeconomics. One element of the debate is whether some countries in the Eurozone would be better off conducting their own monetary policy. Naturally, the characteristics of the economies of countries in the Eurozone differ. But does this have implications for monetary policy and does it mean that their optimal monetary policy also differs? The question is: does one policy fit all? This thesis will focus on differences in the labour market characteristics. As will be shown in next section, the biggest differences can be found there. Specifically, the rigidity of wages will be investigated. The evidence on the flexibility of wages across countries is reviewed and the implications of these differences for optimal policy choices are analysed. Hereby, the focus lies on countries that are characterised by the same amount of flexibility in goods prices but large differences with respect to the flexibility of wages: Italy, Portugal and Poland.

This thesis studies optimal monetary policy in the presence of nominal rigidities using the Smets & Wouters (2003)-model. The consequences of nominal rigidities on the optimal design of monetary policy have been analysed before and were first examined assuming that only prices were sticky by Clarida, Gali & Gertler (1999). Such analysis was extended to the case of economies where both prices and wages were sticky by Erceg, Henderson & Levin (2000).

We know that wage rigidities complicate monetary policy in at least two ways by Erceg et al. (2000) and Christiano, Eichenbaum & Evans (2001). First, wage rigidity increases the cost of stabilising inflation in the face of cost-push and other shocks. With sticky wages, only a part of the people can adjust their wage to a decrease in output after a positive monetary policy shock, which implies a smaller effect of output fluctuation on marginal cost and further to the inflation. Hence if there are differences between wage rigidities across countries in Europe this could result in different optimal monetary policy for different countries.

Evidence from the Wage Dynamics Network (WDN) shows that there in fact exists a significant heterogeneity in wage rigidity in Europe. This thesis will therefore analyse the effect of this het-erogeneity on monetary policy. This thesis will answer two questions. What are the implications of differences in wage rigidity in Europe for optimal monetary policy and secondly does the fact that there are significant rigidities increase the desirability of alternative monetary policies. Normally the monetary authority only targets price inflation but would it be desirable to also target wage inflation for instance?

It requires a profound understanding of the size and character of the rigidities and implies the incorporation of these rigidities in macro modelling to analyse monetary policy. The next section will, therefore, describe an extensive research done by the WDN and the Inflation Persistence Net-work (IPN). They created data from surveys on price and wage rigidities for different countries in Europe. When it comes to implementing these rigidities into the model there are in general two op-tions. The two most widely applied mechanisms to include rigidities in theoretical models are based on Taylor (1980) and Calvo (1983). Whereas the Taylor model is based on fixed price durations, the Calvo model includes some uncertainty in a model, as it sets a probability to determine the

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1 Introduction Rutger Jansen

rate of rigidity. This thesis will use a Calvo price and wage rigidity model. It aims at contributing to this research field by using a widely applied and more general DSGE model and applying it to several countries in Europe. To summarise this thesis will conduct a multi-country analysis of the effect heterogeneity in wage and price rigidity on monetary policy.

The key findings are that the differences in wage rigidity have important implications for optimal monetary policy. For countries with high wage rigidity like Italy it is optimal to have a lower interest rate reaction coefficient, φ, meaning that the monetary policy targeting rule in those countries should be less strict. This is caused by the positive relation between the wage rigidity and the variance in output gap and therefore the welfare loss. Welfare-theoretic losses were calculated and strict price inflation targeting generates relatively large welfare losses, whereas other simple policy rules perform better.

The rest of the thesis is organised as follows. The data on nominal rigidities for different countries in Europe is described in section 2. The Smets & Wouters (2003)-model, that is used, is described in section 3. The whole model will be explained here and the resulting impulse response functions will be analysed. Section 4 is devoted to the monetary policy analysis. Finally, section 5 reviews the main conclusions of the thesis.

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2 Data on rigidities Rutger Jansen

2 Data on rigidities

To quantify the differences in flexibility of the labour market in different countries across Europe, data on the flexibility of wages is used. The WDN has analysed the frequency of wage changes for around 17.000 firms in Europe by doing surveys. About 60 percent of the 17.000 firms surveyed, report they typically change wages once a year. The average duration of wage spells was about 15 months and the average duration of price spells was only 9.5 months. So prices are on average changed more often than wages. This survey evidence is confirmed by the analysis of micro data in a few countries for which quarterly wage data is available. By using the survey data on the frequency of wage and price changes, estimates of the average duration of price and wage spells are made. The table below presents data on estimated average duration of price and wage spells for different countries in Europe (Bertola, Dabusinskas, Hoeberichts, Izquierdo, Kwapil, Montorn`es & Radowski 2012).

Table 1: Survey evidence on nominal rigidities

Country Price rigidity(in months) Wage rigidity(in months)

Austria (AUT) 9.1 12.5 Belgium (BEL) 9.9 12.6 Spain (ESP) 9.7 11.9 France (FRA) 10.1 12 Greece (GRC) 10.2 11.9 Ireland (IRL) 8.5 12.8 Italy (ITA) 9.5 20.3 Netherlands (NLD) 9.1 13.9 Portugal (PRT) 9.5 12.9

Czech Republic (CZE) 9.7 14.6

Estonia (EST) 10 12.7

Hungary (HUN) 10.7 13.8

Lithuania (LTU) 8.4 11.4

Poland (POL) 9.5 15.4

Slovenia (SVN) 9.6 11.8

Euro area (EUR) 9.6 15.0

Total (ALL) 9.6 14.9

Note: The numbers contained in this table are primarily based on results from the WDN survey as presented in Druant, Fabiani, Kezdi, Lamo, Martins & Sabbatini (2012). The information in columns price rigidity and wage rigidity stems from their Table. The values are weighted by employment weights, rescaled excluding non-responses.

The degree of the price and wage rigidity determines the speed of adjustment of the economy to shocks and the magnitude of the adjustment costs. From the literature we know that the more rigid prices are, the less responsive inflation becomes to changes in its proximate determinants,

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2 Data on rigidities Rutger Jansen

inertia of the output gap itself. The IPN survey shows that the response of inflation becomes flatter with more sticky prices (Altissimo, Ehrmann & Smets 2006). In other words, it takes longer for inflation to return to the value it had before the shock occurred. It is important to note that this positive relationship between the degree of inflation persistence and the degree of price stickiness also depends on the degree to which inflation depends on its own past and on the behaviour of the policy-makers.

Since the price rigidity has such a proclaimed effect on monetary policy, it is also interesting to analyse the consequences of the wage rigidity on monetary policy. Especially as the data shows the wage rigidity is even bigger across Europe. Studies conducted within the euro area by the IPN revealed a considerable amount of price stickiness in Eurozone countries as measured by the frequency of price changes (Altissimo, Ehrmann & Smets 2006). The wage rigidity, however, is even bigger than the price rigidity, as can be seen in table 1 above. More importantly, the heterogeneity or dispersion across countries is also bigger in wage rigidity. As table 2 shows that the standard deviation of wage rigidities is higher than price rigidities. This means that heterogeneity across countries is bigger for wage rigidity than for price rigidity (Bertola et al. 2012). It is the heterogeneity that complicates monetary policy and the fact that there is only a single monetary policy in the Eurozone. Since heterogeneity is bigger in wage rigidity, the focus of this thesis will be on wage rigidity.

Table 2: Dispersion statistics

Variable Mean Std. Dev. Min Max

Price rigidity (ξp) 0.7285714 0.0148516 0.7 0.756

Wage rigidity (ξw) 0.7990714 0.0249691 0.769 0.863

Note: This table shows the dispersion statistics of wage and price rigidity.

The frequency of price and wage changes provides a measure of the extent of nominal rigidities which are an essential ingredient in the calibration of standard DSGE models with staggered ad-justment mechanisms that are widely used for monetary policy analysis. This thesis will also make use of the data on the frequency of price and wage changes. To model this data, the wage and price spells need to be expressed into something that can be used in the model. The transformation of price or wage spells expressed in months needs to be transformed into a Calvo parameter. This can be done by a formula to transform the duration of price spells to a Calvo parameter:

d = −1

ln(1 − θ) (1)

where d denotes the duration of price spells in the applied unit of time (quarter years in this model) and (1 − θ) states the corresponding Calvo parameter (Carvalho 2006). Given the fact that this is a quarterly model the Calvo parameter for price and wage changes once a year will become θ = 0.2211. So the Calvo parameter 1 − θ will be 0.778. Based on this approach, the Calvo statistics were computed and are presented in Table 3 below .1

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3 The model Rutger Jansen

Table 3: Calvo parameters

(θ) (ξw)

Country Price parameters Wage parameters

Austria (AUT) 0.719 0.787 Belgium (BEL) 0.739 0.788 Spain (ESP) 0.734 0.777 France (FRA 0.743 0.779 Greece (GRC) 0.745 0.777 Ireland (IRL) 0.703 0.791 Italy (ITA) 0.729 0.863 Netherlands (NLD) 0.719 0.806 Portugal (PRT) 0.729 0.793

Czech Republic (CZE) 0.734 0.814

Estonia (EST) 0.741 0.790 Hungary (HUN) 0.756 0.805 Lithuania (LTU) 0.700 0.769 Poland (POL) 0.729 0.823 Slovenia (SVN) 0.732 0.776 Total (ALL) 0.732 0.818 Eurozone(EUR) 0.732 0.818

Note: In the table above the data on price and wage adjustment frequency is converted to a Calvo parameter that can be used in the DSGE model. The data from the table above is based on real survey data of different countries in Europe and will be used in the model to analyse the effect of the rigidities on monetary policy. In the next section, the DSGE model that has been used will be fully explained.

3 The model

The model presented in this thesis is based on the Smets & Wouters (2003)-model. Price stickiness is introduced via a Calvo parameter and the model’s elaboration allows to examine its consequences to a greater extent. Sticky wages are also included via a Calvo parameter, but a possible heterogeneity in rigidity due to for example different professions or labour unions is outside the scope of this thesis and so it is not included in the model (Smets & Wouters 2003). There is, however, heterogeneity in wage rigidity across countries. As we have seen earlier especially the wage rigidity differs a lot across countries in Europe. In this section, the main building blocks of the model, that are used to analyse the effect of this heterogeneity in wage and price rigidity, are presented.

The τ th household makes a sequence of decisions during each period. Households maximise a utility function with three arguments (consumption, money and leisure) over an infinite life horizon. There also is a time-varying habit variable added to the utility function and labour is supplied in

Calvo parameter for this model the calculation of Carvalho (2006) needs to be adjusted slightly. Because there the chance to update your price or wage is equal to (1 − θ) and (1 − ξw) respectively.

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3.1 The linearised model Rutger Jansen

a monopolistic competition market. This type of competition in the labour market comes from the fact that the labour skills of every household differ slightly and the households, therefore, can exert some monopoly power over their type of labour. The monopoly power is however limited by the degree of substitutability and the households set their wages only under terms of a Calvo parameter reducing their abilities to react to changes in the environment in a timely manner.

Capital is also accumulated by the households and rented to firms. An increase in the capital stock leads to adjustment costs, which can be avoided by modifying the capital utilisation of already existing capital. But this modification also bears costs in the form of foregone consumption.

On the production side, the market is divided in an intermediate market with a continuum of intermediate goods firms in a monopolistic competition and a perfectly competitive final goods market. The monopolistic power in this intermediate market is caused by the small diversity in products and limited by the degree of substitutability, just like the wages. Prices are set in a function of current and expected marginal costs but are also determined by the past inflation rate. However prices can only be set by a part of the firms which is in accordance with a Calvo mechanism and the only inputs in this part of the market are labour and capital provided by the households. Firms that may not modify their price in a certain period are assumed to index their prices according to the prior inflation rate. This leads to a backward-looking component in the inflation equation.

Lastly, a full set of structural shocks is introduced to the various structural equations. Next to two supply shocks, a productivity/technology shock and a labour supply shock, Smets and Wouters added three demand shocks (a preference shock, a shock to the investment adjustment cost function, and a government consumption shock), three cost-push shocks (modelled as shocks to the markup in the goods and labour markets and a shock to the required risk premium on capital) and two monetary policy shocks. In the end, only the log-linearised version of the model is used, which can be found in the section below. The full derivation of the model can be found in appendix A.

3.1 The linearised model

For the monetary policy analysis later on, the model has to be linearised around the non-stochastic steady state. Below the log-linearised model is presented. The equations are log-linearised around their non-stochastic steady state according to Xt = ¯Xexˆt where ˆxt = log(Xt/ ¯X) The ∧ above a

variable denotes it’s a log deviation from steady state. Variables dated at time t + 1 refer to the rational expectation of those variables.

The consumption equation with external habit formation is given by: ˆ Ct= h 1 + hCt−1ˆ + 1 1 + hCt+1ˆ − 1 − h (1 + h)σc ( ˆRt− ˆπt+1) + 1 − h (1 + h)σc ˆ b_t (2)

Where ˆb_t is a shock to the preferences of people. It affects the inter temporal substitution.

A habit component h is added. If h = 0, the habit term has no effect. The equation above then reduces to the traditional forward-looking consumption equation. If h 6= 0, consumption depends

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on a weighted average of past and expected future consumption. Note that in this case, the interest elasticity of consumption depends not only on the intertemporal elasticity of substitution but also on the habit persistence parameter. A high degree of habit persistence will tend to reduce the impact of the real interest rate on consumption for a given elasticity of substitution. The investment equation is given by: ˆ It= 1 1 + β ˆ It−1+ β 1 + βIt+1+ ϕ 1 + β ˆ Qt+ ˆIt (3)

where ˆI_t is a shock to investment and where φ = 1/ ¯S”. A shock to the investment cost function, which is assumed to follow a first-order autoregressive process with an IID-Normal error term: ˆ

I_t = ρIˆIt−1+ ηtI. As discussed in Christiano et al. (2001), modelling the capital adjustment costs

as a function of the change in investment rather than its level introduces additional dynamics in the investment equation, which is useful in capturing the hump-shaped response of investment to various shocks including monetary policy shocks. A positive shock to the adjustment cost function, ˆ

I_t, temporarily reduces investment. The log-linearised equation for Q becomes: ˆ Qt= −( ˆRt− ˆπt+1) + 1 − δ 1 − δ + ¯rk_Qˆ t+1 + r¯ k 1 − δ + ¯rkrˆ k t+1+ η Q t (4) where β = 1/(1 − δ + ¯rk).

The current value of the capital stock depends negatively on this period’s real interest rate and positively on the expected future value of the capital stock and the expected rental rate. The intro-duction of a shock to the required rate of return on equity investment, ηQ_t , is meant as a shortcut to capture changes in the cost of capital that may be due to stochastic variations in the external finance premium. This is the only shock that is not directly related to the structure of the economy. In a fully-fledged model, the production of capital goods and the associated investment process could be modelled in a separate sector. In such a case, imperfect information between the capital producing borrowers and the financial intermediaries could give rise to a stochastic external finance premium. For example, in Bernanke, Gertler & Gilchrist (1999), the deviation from the perfect capital market assumptions generates deviations between the return on financial assets and equity that are related to the net worth position of the firms in their model. Here, Smets and Wouters implicitly assume that the deviation between the two returns can be captured by a stochastic shock, whereas the steady-state distortion due to such informational frictions is zero (Smets & Wouters 2003).

The capital accumulation equation: ˆ

Kt= (1 − δ) ˆKt−1+ δ ˆIt−1 (5)

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new Keynesian Phillips curve:2 ˆ πt= β 1 + βγp ˆ πt+1+ γp 1 + βγp ˆ πt−1+ 1 1 + βγp (1 − βθ)(1 − θ) θ [αˆr k t + (1 − α) ˆwt− ˆαt] + η p t (6) ˆ

α_t represents a shock to the production function. So this is a technology or productivity shock. The wage equation becomes:3

ˆ wt= β 1 + βwˆt+1+ 1 1 + βwˆt−1+ β 1 + βπˆt+1− 1 + βγw 1 + β πˆt+ γw 1 + βπˆt−1 − 1 1 + β (1 − βξw)(1 − ξw) (1 +(1+λw)σL λw )ξw ˆ wt− σLLˆt− σc 1 − h( ˆCt− h ˆCt−1) + ˆ L t + η_tw (7)

The labour demand equation becomes: ˆ

Lt= − ˆwt+ (1 + ψ)ˆrtk+ ˆKt (8)

The goods market equilibrium condition:4 ˆ

Yt= cyCˆt+ τ kyIˆt+ ˆtG= φˆat + φα ˆKt−1+ φαψˆrkt + φ(1 − α) ˆLt (9)

Finally, the model is closed by adding the following empirical monetary policy reaction function. There are three simple Taylor rules applied to the model. These monetary policy rules are explained in section 4. For the impulse response functions in the section below, only the regular price inflation targeting monetary policy rule is used. Which looks like:

ˆ

Rt= 1.5ˆπt+ 0.9 ˆRt−1

The stochastic behaviour of the system of linear rational expectations equations is driven by ten exogenous shock variables. Not all the shocks that are incorporated in the Smets and Wouters model are analysed, however. A summary of the reviewed shocks is given in the table below.

2_ηp

t is a shock to the price markup., ηp doesn’t depend on _1+βγ1 _p(1−βθ)(1−θ)_θ like in Smets and Wouters 3

Smets & Wouters (2003) have given the labour supply shock ˆLt a negative sign, this should be positive. ηtw, a

wage markup shock is outside the brackets unlike in Smets & Wouters (2003)

4

ˆ

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3.2 Parameter values Rutger Jansen

Table 4: Summary of shocks Technology and preferences Cost-push

AR(1) IID ˆ I_t = ρIIt−1+ ηIt λp,t= λp+ ηpt ˆ _ta= ρaat−1+ ηta λw,t= λw+ ηwt ˆ b_t = ρbbt−1+ ηtb ˆ tL= ρLLt−1+ ηLt

Note: This table summarises all the shocks that are in the DSGE model and are analysed

3.2 Parameter values

A number of parameters were taken directly from the Smets & Wouters (2003)-paper. The discount factor, β is calibrated to be 0.99, which implies an annual steady-state real interest rate of 4 percent. The depreciation rate, τ , is set equal to 0.025 per quarter, which implies an annual depreciation of capital equal to 10 percent. The thesis sets α= 0.30, which roughly implies a steady-state share of labour income in total output of 70 percent. The share of steady-state consumption in total output is assumed to be 0.6, while the share of steady-state investment is assumed to be 0.22. λw is set

equal to 0.5, which is somewhat larger than the findings in the micro-econometric studies by Griffin (1996) based on U.S. data (Smets & Wouters 2003). The value for price indexation is set equal for all countries and is set to be 0.469. The value for wage indexation is also the same for all countries and set equal to 0.763. These values are estimated by Smets & Wouters (2003).

Their estimate of the intertemporal elasticity of substitution _σ1 is less than one and close to the assumption made in much of the RBC literature which assumes an elasticity of substitution between a half and one. However, one needs to be careful when making such comparisons, as I have assumed external habit formation which turns out to be significant. The external habit stock is estimated to be 55 percent of past consumption, which is somewhat smaller than the estimates reported in Christiano et al. (2001).

The estimates of the elasticity of substitution and habit formation imply that an expected one percent increase in the short-term interest rate for four quarters has an impact on consumption of about 0.28.

The other parameters are taken from column 5 of table 1 of the Smets & Wouters (2003)-paper. This column contains the estimated posterior mode of the parameters, which is obtained by directly maximising the log of the posterior distribution with respect to the parameters, and an approximate standard error based on the corresponding Hessian. Uhlig (2014) used the same parameters when he recreated the Smets & Wouters (2003)-model. In the table below all the parameters and their values are described.

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3.3 Impulse response analysis Rutger Jansen

Table 5: Parameter values

Parameter Value Description

α 0.30 Steady state share of Capital income in total output

β 0.99 Discount factor

τ 0.025 Depreciation rate of capital

cy 0.6 The share of steady-state consumption in total output

invy 0.22 The share of steady-state investment in total output

gy 0.18 Government expenditure share in total output

λw 0.5 Markup in wage setting

ϕ 6.771 Investment adjustment cost

σc 1.353 Coefficient of the relative risk aversion of the household

¯

rk 1/(β − 1 + τ ) Steady state return on capital

h 0.573 Habit portion of past consumption

κy 8.8 Capital output ratio

ξw See table 2 Calvo wage stickiness in the sticky system

σl 2.400 Inverse of elasticity of labour supply

ξp See table 2 Calvo price stickiness in the sticky system

γw 0.763 Degree of partial indexation of wage

γp 0.469 Degree of partial indexation of price

ψ 0.169 Elasticity of the capital utilization cost function

φ 1.408 1 + Share of fixed cost in production

ρa 0.823 Autoregressive parameter of productivity shock

ρb 0.855 Autoregressive parameter of preference shock

ρl 0.889 Autoregressive parameter of labour supply shock

ρi 0.927 Autoregressive parameter of investment shock

σL 3.52 Standard deviation of the labour supply shock

σb 0.336 Standard deviation of the preference shock

σa 0.598 Standard deviation of the technology shock

σi 0.085 Standard deviation of the investment shock

σλw 0.289 Standard deviation of the wage markup shock

σλp 0.16 Standard deviation of the price markup shock

Note: This table shows the values for all the parameters used in the DSGE model.

3.3 Impulse response analysis

Six shocks are added to the model. They will be analysed one by one in this section. For Italy and Portugal the impulse response functions for the output gap, the nominal interest, wage inflation and the price inflation are shown in appendix B. Since Portugal and Italy have the same price rigidity, the only difference is the wage rigidity. So all the differences in the impulse response functions will be caused by the difference in wage rigidity. 5 In the section below the impulse response functions for the different types of shocks will be analysed.

By analysing the impulse response functions it is possible to also investigate the monetary

5

The difference in wage indexation is beyond the scope of this thesis to fully focus on the wage rigidity. γw the

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3.3 Impulse response analysis Rutger Jansen

policy. For instance, the thesis can look at the differences between the two countries and see how the differences in wage rigidity are reflected in the impulse response functions. In the section below the impulse response functions for the different types of shocks will be analysed.

The first shock, figure 8, is a technology shock. Due to the rise in productivity, the marginal cost falls on impact. As monetary policy does not respond strongly enough to offset this fall in marginal cost, inflation falls. The drop in the marginal costs caused by the supply shock makes that inflation falls and due to the high estimated labour supply elasticity employment also falls, therefore, the real wages rise. Resulting in a short-run trade-off between unemployment and inflation, the known Phillips curve. This result is similar to what has been found in the literature. As pointed out by Galı & Gertler (1999) the fall in employment is consistent with estimated impulse responses of identified productivity shocks in the US and is in contrast to the predictions of the standard RBC model without nominal rigidities. The technology shock which practically acts as a cost-push shock in a model with real wage rigidities, and to which the real wage and prices react in opposite direction: the initial decrease in inflation leads to more real wage rigidity and amplifies the hiring incentives of firms, thus amplifying employment and output (Babeck`y, Du Caju, Kosma, Lawless, Messina & R˜o˜om 2010).

The second shock, figure 9, is a preference shock. The preference shock causes the marginal utility of consumption to go up. After a positive change in preference, consumption goes up and therefore output grows significantly. Meanwhile, investment declines sharply. The fall in investment is due to the crowding out effect on investment. The increase in capacity necessary to satisfy increased demand is delivered by an increase in the utilisation of installed capital and an increase in employment. Increased consumption demand puts pressure on the prices of production factors: both the rental rate on capital and the real wage rise, putting upward pressure on the marginal cost and inflation (Smets & Wouters 2003).

The third shock, figure 10, is an investment shock. Monopolistic competition with sticky prices and wages is the fundamental mechanism for the transmission of these shocks. This friction breaks the intratemporal condition, by driving an endogenous wedge between the marginal product of labour and the marginal value of time. Intuitively, a positive shock to the marginal productivity of investment increases the real interest rate, giving households an incentive to save more and postpone consumption. With lower consumption, the marginal utility of income increases, shifting labour supply to the right to an intertemporal substitution effect. Along an unchanged labour demand schedule, this supply shift raises hours and output but depresses wages and labour productivity Justiniano, Primiceri & Tambalotti (2010).

The fourth, figure 11, is a labour supply shock. This causes the wage to go down and even though Smets and Wouters got the sign wrong in their wage equation their impulse response function looks the same. The real interest rate is not significantly affected due to the higher persistence of the labour supply shock, the real wage however is affected. As said earlier the real wage falls significantly. It is this significant fall in the real wage that leads to a fall in the marginal cost and a fall in inflation (Smets & Wouters 2003).

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3.4 Differences between Portugal and Italy Rutger Jansen

The fifth shock, figure 12, is a shock to the price markup. This is called a markup shock because λp,t is the desired markup of price over marginal cost for intermediate firms. This shock causes

the inflation to jump up instead of the real wage. This shock is very similar to the wage markup shock but the main difference is found in the wage inflation. With a wage markup shock, the wage inflation jumps up and then goes down again and with the price markup shock it is exactly reversed. The last shock, figure 13, is a shock to the wage markup. In the original Smets & Wouters(2003)-model, a shock to wage markup causes the wage to jump up. This is also the case in the adjusted version of their model used in this thesis. Both the price inflation and the nominal interest rate also rise. The nominal interest rate increases more than the inflation resulting in the real interest rate to go up. This rise in real interest rate makes that people postpone consumption and hence the output gap becomes negative.

In this section, the shocks were analysed one by one but there was no heterogeneity in wage rigidity yet. Now that the transmission mechanisms of the different shocks are mapped, the impli-cations of the difference in wage rigidity can be analysed. So in the next section heterogeneity in wage rigidity will be introduced by analysing more than one country.

3.4 Differences between Portugal and Italy

In the sections above all the different shocks were discussed and their corresponding impulse response functions are analysed. But that analysis alone doesn’t learn us anything about the monetary policy implications of wage rigidity differences. Since the main focus of this thesis is on the wage rigidity and how it affects the monetary policy, it will now examine using the impulse response functions how the differences in wage rigidity influence the behaviour of the economy after a shock. Because if there are major differences between countries with different wage rigidities it means that wage rigidity has implications for optimal monetary policy.

The same six shocks that were mentioned earlier will be reviewed but now the main focus will be on the differences between the two countries. There are three variables that are looked at in particular. The real wage, output and inflation. This is because normal monetary policy focuses on inflation and since there is price rigidity inflation is not optimal. This also is the case for wage inflation since there is wage rigidity as well. Variance in output is also bad for welfare since every deviation from the steady state level of output is also not optimal.

The same six shocks as in the section above will now be reviewed. The first shock, a technology shock causes wages to jump up as has been analysed in the earlier section. The difference in wage rigidity should, in theory, result in a smaller jump in wage inflation for the country with the lower wage rigidity. This is confirmed by figure where 1 can be seen that the wage inflation reacts stronger initially in Portugal than in Italy. Following this reaction in wage inflation, the employment goes down. Since fewer people can update their wage in Italy the real wage increase is smaller and thus the level of employment stays higher. The higher level of employment causes the output gap to increase more in Italy. Finally, the interest rate is one for one with price inflation so the interest rate in Italy drops down more.

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3.4 Differences between Portugal and Italy Rutger Jansen

The preference shock causes the marginal utility of consumption to enlarge and the marginal disutility of labour to increase as well. This way it affects the intertemporal substitution of house-holds resulting in a higher level of consumption. The difference here between Italy and Portugal is caused by the difference in wage inflation. The increased consumption creates wage inflation which by its turn raises prices. The rise in prices is bigger in Portugal because the rise in wage inflation is higher there. Ultimately resulting in a higher real interest rate in Portugal. People postpone consumption because of this high real interest rate. Resulting in a bigger change in output gap in Italy.

The third shock, an investment shock is almost the same for the two countries. The small differences are caused by the difference in real interest rate which is caused by the different labour market dynamics because of the different wage rigidities. So the focus won’t be on the investment shocks as wage rigidity affects the impulse response functions very little.

The fourth shock, a labour supply shock does show interesting differences. The labour supply shock increases the disutility people get from labour. This has a negative effect on the real wage. So the wage inflation is negative first. It is this fall in the real wage that leads to a fall in the marginal cost and a fall in inflation. The bigger wage rigidity in Italy makes that the initial wage fall is bigger in Portugal. This results in a smaller rise in demand for labour which is reflected in the bigger negative output gap in Italy.

The shock to the price markup shows almost no difference in wage inflation or price inflation but does show a difference in the output gap. Italy’s output gap reacts more to the shock. The first is because wage rigidities are relatively irrelevant for inflation dynamics since the contributions to the marginal cost of unit labour costs and the frictional component largely offset each other (Faccini, Millard & Zanetti 2011). The reaction in output gap can be explained by the fact that wage rigidities affect the behaviour of employment in the short run: with wage rigidities, the responses of employment and vacancies become positive after a couple of quarters while they remain negative with flexible wages (Faccini et al. 2011). This causes the differences in the output gap.

The final shock, a wage markup shock shows little difference in wage inflation. There are big differences between Italy and Portugal though. Italy shows a bigger response in both the output gap and the price inflation. A positive wage markup shock represents a negative shift of labour supply. When wages are sticky, the labour supply schedule is relatively flat. Hence, the drop in equilibrium hours and output can be moderate. On the other hand, with flexible wages, the labour supply curve is substantially steeper. This explains the differences in the impulse response functions. Italy has a relatively flatter labour supply curve (Justiniano & Primiceri 2008).

After analysing all the shocks, the conclusion is that the difference in wage rigidity is relatively irrelevant for inflation dynamics but has a strong effect on labour market dynamics. This is reflected in the impulse response functions of the output gap where is shown that the wage rigidity definitely has the largest effect on the variance of the output gap. As can be seen by the fact that Italy has the highest deviation in output gap for almost all shocks, as is highlighted in figure 3. Since monetary policy can only stabilise either the wage inflation, the price inflation or the output gap.

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4 Optimal monetary policy Rutger Jansen

The conclusion is that wage rigidity will definitely have a significant effect on the optimal monetary policy. So this type of heterogeneity in wage rigidity increases the desirability of monetary policy arrangements. In the next section, this thesis will, therefore, investigate how much and in what way the wage rigidity affects the optimal monetary policy.

4 Optimal monetary policy

Using the Smets & Wouters (2003)-model this thesis will now analyse the monetary policy for different countries in Europe. Simple Taylor rules are used as monetary policy rules and they are compared on how they perform for different countries in Europe. We have seen in section 2 that different countries have different nominal wage and price rigidities. This was also reflected in the impulse response functions that are simulated for this model. First, the Taylor rules that are used will be discussed below.

The first is an ordinary inflation Taylor rule. Which takes the following form: ˆ

Rt= 1.5ˆπt+ 0.9 ˆRt−1

Given the fact that the ECB has inflation targeting as the clear and only objective, you would expect there to be a direct relation between inflation and the interest rate, like the Taylor rule above suggests.

The second Taylor rule focuses on wage inflation. Since the Smets and Wouters model has wage rigidity next to price rigidity, wage inflation can also cause inefficient distortions. So wage inflation targeting might also be interesting. The wage targeting Taylor rule takes the following form:

ˆ

Rt= 1.5ˆπtw+ 0.9 ˆRt−1

where the wage inflation is set to be equal to ˆ

πw_t = ( ˆwt− ˆwt−1) + ˆπt

Third and lastly a compound targeting Taylor rule is used. Compound meaning that the inflation that is targeted in the interest rate rule is a combination of price and wage inflation. The Taylor rule then becomes:

ˆ

Rt= 1.5ˆπtcomp+ 0.9 ˆRt−1

where

ˆ

π_tcomp = 0.23ˆπt+ 0.77ˆπtw6

These three Taylor rules are then applied to the model described in the earlier sections. This has been done for the following countries: Portugal and Italy and Poland. All these countries have the same price rigidity but different levels of wage rigidity. Poland is added to this analysis to increase

6

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the significance of the results.

To analyse the different rules and to see what the best rule is, OSR (Optimal Simple Rule) is used in dynare. OSR minimalises a given loss function. Next to the usual output gap and inflation the loss function also depends on wage inflation because there is also wage rigidity. The variances of the variables in the loss functions are used to compute an approximation for the welfare loss. The loss function takes the following form:

L = ˆπ_t2+ ˆπwt2_{+ ˆ}_x2

t

The choice for this loss function is based on the literature about monetary policy and loss functions. It has been used before by Debortoli, Kim, Lind´e & Nunes (2015) and Gal´ı (2015). It is not a fully-fledged welfare analysis but an ad hoc loss function because the relative weights between the stabilization goals is exogenously set. Using this welfare-theoretic loss function the monetary policy of different countries will be analysed. It is optimal to fully stabilise all variables in this loss function. But since it is not possible to fully stabilise all three variables the second best is to minimise the variance of all the variables in the loss function. There exists a trade-off between stabilising the output gap, the price inflation, and the wage inflation rate. It is impossible for more than one of the three variables to have zero variance (Erceg et al. 2000). So optimal monetary policy is achieved at the point where the welfare loss approximation is minimised. To find this point OSR is used. Given the following interest rate rule:

ˆ

Rt= φˆπt+ 0.9 ˆRt−1

OSR finds the optimal value for φ, the reaction coefficient. The OSR is used to find the optimal value for each country. The optimal values are written down in table 6 below. Besides computing the optimal values using OSR the thesis has also performed a welfare analysis for different values of φ, the interest rate reaction coefficient. The results can be found in table 7 in appendix B. For the wage inflation, the output gap and the price inflation the variances are computed for different values of φ. 2, 1.5, 1, 0.5 and 0.15 are used as values for φ. The optimal values for φ in table 7 correspond with the optimal values in table 6.

To make the results clearer, the results that are shown in table 7 will be presented in graphs below. Figures 1 - 7 highlight the main results of the welfare analysis of this thesis. Figure 1 shows the welfare loss for the different targeting rules: price inflation, wage inflation and compound inflation for Poland, Portugal and Italy. Poland was added here to see if it would fall in between Portugal and Italy as would be expected as Poland has lower wage rigidity than Italy and higher than Portugal. Figures 2 - 4 show the relationship between the variance of the output gap, wage inflation and price inflation with the interest reaction coefficient φ. Finally, figures 5- 7, in appendix B show the difference in welfare loss for different monetary policy targeting rules. Here can be seen that the wage inflation and compound targeting rules are superior to the price inflation targeting rule. The wage inflation and compound targeting rule are almost identical.

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Table 6: Optimal values

Country Price Wage Compound

Portugal 0,364 0,427 0,409

Italy 0,188 0,209 0,205

Poland 0,283 0,322 0,317

Note: The numbers in the table represent the optimal values for φ, the reaction coefficient of the interest rate. They are calculated using OSR in dynare. 0,364 hence means that for the price inflation targeting rule:

ˆ

Rt= φˆπt+ 0.9 ˆRt−1

the welfare loss minimizing value of φ is 0,364.

Figure 1: Welfare loss Poland - Portugal - Italy

Note: The figure above highlights the welfare loss for wage, price or compound targeting rules in Italy, Poland and Portugal. These countries have different levels of wage rigidity. The variance or welfare loss is plotted on the y-axis and φ on the x-axis. Hereby the relation between the welfare loss and φ is shown. But the graph also shows that the welfare loss increases with respect to wage rigidity. As Italy has the highest welfare loss and Poland the second highest.

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Figure 2: Variance of output gap

Figure 3: Variance of wage inflation

Figure 4: Variance of price inflation

Note: The figures above highlight the variance of the wage, price and compound targeting rules in Italy and Portugal. These two countries have different levels of wage rigidity. The variance is plotted on the y-axis and φ on the x-axis. Hereby the relation between the variance and φ is shown and it is shown that this relation is different for inflation and output gap.

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After analysing the table above and the figures 1 – 7. Three important points about monetary policy are found.

The first point is that increasing the interest rate reaction is not always better for welfare. The monetary authority must make a trade-off between stabilising wage inflation and price inflation on the one hand and the output gap on the other hand. As can be seen in figures 3 and 4 the price and wage inflation are best stabilised by increasing the interest rate response to deviations in wage or price inflation. For the output gap, however, the relation is different. First increasing the reaction coefficient φ improves the welfare. But this only until a certain point. After the optimal point is reached increasing the reaction further will increase the welfare loss as can be seen in figure 2. To state it differently, the variance in output gap is increasing with respect to the reaction coefficient and the variance of the wage and price inflation is decreasing with respect to the reaction coefficient. The second point is that higher wage rigidity increases the welfare loss substantially as is high-lighted in figure 1. This is mostly caused by an increase in output gap variance. Which also was concluded after analysing the impulse response functions for Italy and Portugal. Since this out-put gap variance increases with respect to an increase in the reaction coefficient (φ), higher wage rigidity favours a relatively lower reaction coefficient. Table 6 shows that the optimal values are substantially higher for Portugal than for all the different targeting rules. So Portugal favours a higher reaction coefficient φ since it has lower wage rigidity. The fact that Poland is in between Portugal and Italy strengthens the conclusion that there is a positive relationship between wage rigidity and welfare loss.

The third point is that of the three Taylor rules that are used, the wage targeting rule does the best, as is shown in figures 5, 6 and 7. The wage targeting rule has the lowest approximated welfare loss of all reaction coefficients. The monetary authority should also respond to changes in wage inflation. Since minimising the welfare loss should be the objective of the monetary authorities and the fact that the wage targeting rule does this the best. The conclusion done by Levin, Onatski, Williams & Williams (2006) that, under Calvo-style wage setting, a wage inflation targeting rule performs well compared to a price inflation targeting rule is therefore confirmed by my analysis.

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5 Conclusion Rutger Jansen

5 Conclusion

This thesis was about analysing the effect of heterogeneity in wage rigidity across countries in Europe on monetary policy. First it was shown that in fact there is a significant degree of heterogeneity in wage rigidity across countries in Europe. Using the Smets & Wouters (2003) model the impulse response functions of different countries in Europe were created and analysed. Six shocks were added to the model. So six impulse response functions were analysed. For Italy and Portugal the impulse response functions for the output gap, the nominal interest, wage inflation and the price inflation are shown in appendix B. Since Portugal and Italy have the same price rigidity, the only difference is the wage rigidity. The impulse response functions look different and thereby give an opportunity to analyse monetary policy. After analysing all the shocks, the conclusion is that the difference in wage rigidity is relatively irrelevant for inflation dynamics but has a strong effect on labour market dynamics. This is reflected in the impulse response function of the output gap. The wage rigidity definitely has the largest effect on the variance of the output gap.

The key finding of the first section was that differences in wage rigidity have important impli-cations for optimal monetary policy and increase the desirability of monetary policy arrangements. This claim is supported by the difference in optimal values in table 6. The only difference between the countries is the wage rigidity and this causes the optimal values of φ to change.

Optimal monetary policy was also analysed. Some interesting things were found. The thesis compared alternative monetary policy rules on the basis of welfare approximation analysis. Three different Taylor rules were analysed and strict price inflation targeting generates relatively large welfare losses, whereas other simple policy rules, like wage inflation targeting, perform better. So the monetary authority should also respond to changes in wage inflation. Further it can be concluded from table 6 that countries with high wage rigidity like Italy prefer a lower reaction coefficient, φ, meaning that the targeting rule in those countries should be less strict. Hence for Portugal, which has relatively low wage rigidity, it is ceteris paribus optimal to have a stricter targeting rule.

Overall can be concluded that wage rigidity complicates the task of the monetary authority. To answer the questions raised in the introduction. Are some countries better of performing their own monetary policy? There definitely is evidence that the differences in wage rigidity in Europe result in different optimal monetary policy for countries. So one policy does not fit all and some countries would be better off performing their own monetary policy. Further research on monetary policy should be conducted to specify which countries would especially be better off conducting their own monetary policy in the Eurozone. As this is beyond the scope of this thesis.

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BIBLIOGRAPHY Rutger Jansen

Bibliography

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Altissimo, F., Ehrmann, M. & Smets, F., 2006. ‘Inflation persistence and price-setting behaviour in the euro area-a summary of the ipn evidence’, ECB Occasional Paper (46).

Babeck`y, J., Du Caju, P., Kosma, T., Lawless, M., Messina, J. & R˜o˜om, T., 2010. ‘Downward nominal and real wage rigidity: Survey evidence from european firms*’, The Scandinavian Journal of Economics 112(4), 884–910.

Bernanke, B. S., Gertler, M. & Gilchrist, S., 1999. ‘The financial accelerator in a quantitative business cycle framework’, Handbook of macroeconomics 1, 1341–1393.

Bertola, G., Dabusinskas, A., Hoeberichts, M., Izquierdo, M., Kwapil, C., Montorn`es, J. & Rad-owski, D., 2012. ‘Price, wage and employment response to shocks: evidence from the wdn survey’, Labour Economics 19(5), 783–791.

Calvo, G. A., 1983. ‘Staggered prices in a utility-maximizing framework’, Journal of monetary Economics 12(3), 383–398.

Carvalho, C., 2006. ‘Heterogeneity in price stickiness and the real effects of monetary shocks’, Frontiers in Macroeconomics 6(3).

Christiano, L. J., Eichenbaum, M. & Evans, C., 2001. Nominal rigidities and the dynamic effects of a shock to monetary policy, Technical report, National bureau of economic research.

Clarida, R., Gali, J. & Gertler, M., 1999. The science of monetary policy: a new keynesian perspective, Technical report, National bureau of economic research.

Debortoli, D., Kim, J., Lind´e, J. & Nunes, R. C., 2015. ‘Designing a simple loss function for the fed: does the dual mandate make sense?’.

Druant, M., Fabiani, S., Kezdi, G., Lamo, A., Martins, F. & Sabbatini, R., 2012. ‘Firms’ price and wage adjustment in europe: Survey evidence on nominal stickiness’, Labour Economics 19(5), 772– 782.

Erceg, C. J., Henderson, D. W. & Levin, A. T., 2000. ‘Optimal monetary policy with staggered wage and price contracts’, Journal of monetary Economics 46(2), 281–313.

Faccini, R., Millard, S. & Zanetti, F., 2011. ‘Wage rigidities in an estimated dsge model of the uk labour market’.

Gal´ı, J., 2015. Monetary policy, inflation, and the business cycle: an introduction to the new Key-nesian framework and its applications, Princeton University Press.

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BIBLIOGRAPHY Rutger Jansen

Galı, J. & Gertler, M., 1999. ‘Inflation dynamics: A structural econometric analysis’, Journal of monetary Economics 44(2), 195–222.

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Justiniano, A. & Primiceri, G., 2008. ‘Potential and natural output’, Manuscript, Northwestern University .

Justiniano, A., Primiceri, G. E. & Tambalotti, A., 2010. ‘Investment shocks and business cycles’, Journal of Monetary Economics 57(2), 132–145.

Kollmann, R., 2001. ‘The exchange rate in a dynamic-optimizing business cycle model with nominal rigidities: a quantitative investigation’, Journal of International Economics 55(2), 243–262. Levin, A. T., Onatski, A., Williams, J. & Williams, N. M., 2006. Monetary policy under uncertainty

in micro-founded macroeconometric models, in ‘NBER Macroeconomics Annual 2005, Volume 20’, MIT Press, pp. 229–312.

Smets, F. & Wouters, R., 2003. ‘An estimated dynamic stochastic general equilibrium model of the euro area’, Journal of the European economic association 1(5), 1123–1175.

Taylor, J. B., 1980. ‘Aggregate dynamics and staggered contracts’, The Journal of Political Economy pp. 1–23.

Uhlig, H., 2014. ‘The smets-wouters model. monetary and fiscal policy’, Humboldt-Universit¨at zu Berlin. Accessed 25.

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6 Appendices Rutger Jansen

6 Appendices

Appendix A

The Household Sector

Households supply a differentiated type of labour. This means that each household has a small monopoly power over the supply of its labour. Since it has monopoly power, the households will change their supply to meet the demand for their labour supply. Each household τ maximizes an intertemporal utility function given by :

Ut= E0 ∞ X t=0 βtˆb_t (C t t− Ht)1 − σc 1 − σc − ˆL_t (l τ t)1+σl 1 + σl + ˆ M t 1 − σm Mτ t Pt 1−σm! (10) Cτ

t refers to household τ ’s consumption and consumption has a positive relationship with utility. lτt

is the amount of labour the household provides and labour has a negative influence on utility. This is often referred to as the disutility of labour. Ht defines the external habit and depends on the

aggregated consumption in the prior period Ht= hCt−1. This doesn’t show up in the derivative for

consumption because people don’t take into account aggregate consumption in their decision. The habit consists of lagged aggregate consumption, which is not affected by one agent’s consumption decision (Abel 1990). The shock b

t affects the intertemporal substitution and Lt the labour supply

decision. σc represents the relative risk aversion of households or the inverse of the intertemporal

elasticity of substitution. σl represents the inverse of the elasticity of work effort with respect to

real wage. Households maximise their utility subject to the budget constraint which is: M_tτ Pt + bt Bτ_t Pt = M τ t−1 Pt + B τ t−1 Pt + Y_tτ − C_tτ− I_tτ (11) Where Y_tτ= w_tτlτ_t + Aτ_t +rk_tzτ_tK_t−1τ − Ψ(zτ t)Kt−1τ + Divτ_t (12)

Households hold their wealth in the form of cash balances Mt and bonds Bt. Bonds have a price

of bt. The households’ income exists of three components. First, they get a wτtlτt wage for the

hours they worked. The second term between brackets is the return on real capital minus the costs associated with capital utilisation. z_tτ here is the utilisation of capital. Div_tτ, Aτ_t are respectively the dividends derived from the imperfect competitive firms and the net cash inflow from participating in state-contingent security markets at time t. The state-contingent securities insure the households against variations in household specific labour income. As a result (wτ_tlτ_t + Aτ_t) will be equal to aggregate labour income and the marginal utility of wealth will be identical across different types of households.

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The Firms

Final Good Firms

As in the Smets & Wouters (2003)-model, there is a single final good produced and a continuum of intermediate goods indexed by j where j is distributed over the unit interval. The final-good sector is perfectly competitive. The final good is used for consumption and investment by the households.

Yt= " Z 1 0 (y_tj)1/(1+λp,t)dj #1+λp,t (13)

Equation (5) represents the production function of the good final good at date t. λp,t in the

production function, is a stochastic variable which determines the markup in the goods market. Shocks to this parameter will be interpreted as a “cost-push” shock to the inflation equation. The cost minimization condition in the final good sector can be written in the form of a demand function for the intermediate good Y_tj

y_tj = p j t Pt !−1+λp,t λp,t Yt (14)

This equation is obtained by solving the following cost minimization problem:

min(yj_t) Z 1 0 P_tjY_tjdj s.t. Z 1 0 (Y_tj) 1 1+λp_t_dj 1+λ p t ≥ Y_t

Ptrepresents the minimum cost of producing one unit of the final-goods bundle Yt, which, because of

the constant-returns-to-scale assumption, is independent of the quantity produced. For this reason I interpret Pt, the Lagrange multiplier, as the aggregate price index. The intuition here is that Pt

tells us at the optimum what the value for the firm is, in dollar terms, of relaxing the constraint on production with one unit. The lagrangian becomes:

L = Z 1 0 P_tjY_tjdj + Pt Yt− Z 1 0 (Y_tj) 1 1+λp_t_dj 1+λ p t! (15) Because of perfect competition the profits will be zero and all firms will ask the same price. Hence the aggregate price index is the same as the price of every individual final good and can be written as: Pt= Z 1 0 pj_t−1/(λp,t)dj !λp,t (16)

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Intermediate Good Firms

The intermediate goods sector is made by a continuum of monopolistically competitive firms owned by consumers, indexed by j ∈ (0, 1). Each intermediate good is produced by a single firm. They hire labour and capital form households, paying wages Wtand capital return Rkt (Smets & Wouters

2003).

y_tj = ˆa_tK˜_j,tαL1−α_j,t − Φ (17)

These firms minimize cost subject to the production function above. The Lagrangian therefore becomes: min ( ˜Kj,t−1,Ljt) Wt Pt Lj_t+ Rk_tK˜j,t−1+ ζt Y_tj− ˆa_tK˜_j,tαL1−α_j,t (18) where ˆa_t is a productivity shock, ˜Kj,t is the effective utilization of the capital stock given by the

utilization of capital times last periods capital stock: ˜Kj,t = ztK˜j,t−1. Lj,t is an index of different

types of labour used by the firm given by equation (6). If you work out the first order condition you get the following optimality condition:

WtLj,t

rk tK˜j,t

= 1 − α

α (19)

Equation (10) implies that the capital-labour ratio will be identical across intermediate goods pro-ducers and hence the aggregate capital-labour ratio will be equal. The firms’ marginal costs are given by: M Ct= 1 ˆ a_tW 1−α t rtk α (α−α(1 − α)(1−α)) (20)

This implies that there are constant returns to scale as neither capital nor labour show up in the marginal cost function. Given the expression of marginal costs above, profits of the intermediate firms are defined as:

πj_t = (pj_t − M Ct) pj_t Pt − 1+λp,t λp,t (Yt) − M CtΦ (21)

Consumption, Investment and Capital Accumulation

The maximization of the utility function of the households, equation (2), subject to the budget constraint (3) with respect to consumption yields the marginal utility of consumption:

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If you combine the first order condition for consumption and bonds you get the Lucas asset pricing equation for bonds. Which takes the following form:

Et " Bλt+1 λt RtPt Pt+1 # = 1 (23)

Where Pt states the aggregate price level, calculated as in equation (8).

Households own the capital stock, which the intermediate firms use as a production factor, see equation (9). The intermediate firms pay a rental rate rk_t to the households. The households can increase the supply of rental services from capital either by investing in additional capital It, which

takes one period to be installed or by changing the utilisation rate of already installed capital zt. The

households have to find the optimal balance between consumption, investment and the utilisation rate of capital. Both an increase in investment and the utilisation rate of capital are costly in terms of foregone consumption, see the intertemporal budget constraint (3) and (4). Households therefore choose the capital stock, investment and the utilization rate in order to maximize their intertemporal objective function subject to the intertemporal budget constraint and the capital accumulation equation, which is given by:

Kt= Kt−1[1 − δ] + [1 − S(

ˆ I_tIt

It−1

)]It (24)

In the capital accumulation equation It stands for gross investment, δ is the depreciation rate and

the adjustment cost function S(.) is a positive function of changes in investment. S(.) equals zero in steady state with a constant investment level. In addition, I assume that the first derivative also equals zero around equilibrium, so that the adjustment costs will only depend on the second-order derivative as in Christiano et al. (2001). Also a shock to the investment cost function is introduced, which is assumed to follow a first-order autoregressive process with an IID-Normal error term: ˆ

I_t = ρIˆIt−1+ ηtI. The first-order conditions result in the following equations for the real value of

capital, investment and the rate of capital utilization:

Qt= Et " βλt+1 λt Qt+1(1 − δ) + zt+1rkt+1− Ψ(zt+1) # (25) QtS 0 (ˆ I tIt It−1 )ˆ I tIt It−1 + BEtQt+1 λt + 1 λt S0(ˆ I t+1It+1 It )(ˆ I t+1It+1 It )It+1 It = 1 (26) r_tk= Ψ0(zt) (27)

Equation (17) states that the real value of capital depends on the expected future value taking into account the depreciation rate and the expected future return as captured by the rental rate times the expected rate of capital utilisation. The first order condition for the utilisation rate (19) equates the cost of higher capital utilisation with the rental price of capital services. As the rental

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rate increases, it becomes more profitable to use the capital stock more intensively up to the point where the extra gains match the extra output costs.7

Wage setting

Each households offers a type of labour that is heterogenetic and no perfect substitute for other household’s labour. The aggregate labour supply looks like the following:

Lt= Z 1 0 lτ_t1/(1+λw,t)_dτ !1+λw,t (28)

λw,tis a IID shock. wtτ stands for the wage at which households supply labour at the market clearing

quantity. Their ability to adjust to shocks is limited by the Calvo staggered wages mechanism. The probability to not be able to adjust one’s wage is equal to the Calvo parameter ξw. If people cannot

adjust their wages, wages are modified with respect to prior inflation πt−1 following this formula:

w_tτ = (πt−1)γwwτt−1 (29)

Households act as price-setters in the labour market. Following Kollmann (2001) and Erceg et al. (2000), Smets and Wouters assumed that wages can only be optimally adjusted after some random ”signal” is received. The probability that a particular household can change its nominal wage in period t is constant and equal to 1 − ξw. A household τ which receives such a signal in period t will

thus set a new nominal wage, ˜wt

t, taking into account the probability that it will not be re-optimised

in the near future. Households set their nominal wages to maximise their intertemporal objective function subject to the intertemporal budget constraint and the demand for labour. To do that each household solves the following maximisation problem:

L = Et ∞ X i=0 Biξi_wUt+i− λt+i − wτ_tlτ_t − Aτ_t −r_tkz_tτK_t−1τ − Ψ(zτ_tK_t−1τ ) − Divτ_t + C_tτ+ I_tτ (30)

subject to: equation (19) and (20). Solving this problem gives you the following first order condition: Et ∞ X i=0 Biξ_wi w˜ t t Pt (Pt/Pt−1)γw Pt+i/Pt+1+i ! lτ_t+iUc,t+i 1 + λw,t+i = Et ∞ X i=0 Biξ_wilτ_t+iUl,t+i (31)

This equation states that the wage is set in such a way that that the present value of the marginal return to working is a mark-up over the present value of marginal cost of working. When wages are perfectly flexible ξw = 0 the real wage will be a mark-up equal to 1 + λw,t over the current ratio

of the marginal disutility of labour and the marginal utility of an additional unit of consumption. Given the above optimal wage setting rule, the law of motion of the aggregate wage index will be:

7

An implication of variable capital utilisation is that it reduces the impact of changes in output on the rental rate of capital and therefore smoothens the response of marginal cost to fluctuations in output.

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6 Appendices Rutger Jansen (Wt)−1/λw,t = ξ Wt−1 Pt−1 Pt−2 γw!−1/λw,t + (1 − ξ)( ˜wt_t)−1/λw,t (32) Price setting

Each intermediate firm has market power so it can set its own prices. Each firm chooses the price level which maximises the sum of expected profits. So they maximise equation (13) taking into account the discount factor and the fact that there is a probability that they may not always set their prices. Firms can only set their prices when they receive a signal. Prices of firms that do not receive a price signal are indexed to last period’s inflation rate. Profit optimisation by producers that are “allowed” to re-optimize their prices at time t results in the following first-order condition:

Et ∞ X i=0 Biθ_piy_t+ij p˜ j t Pt (Pt−1+i/Pt−1) Pt+i/Pt !γp − (1 + λ_p,t+i)M Ct+i= 0 (33)

Equation (25) shows that the price set by firm j , at time t, is a function of expected future marginal costs. The price will be a markup over these weighted marginal costs. If prices are perfectly flexible θ = 0 then the markup in period t will be equal to 1 + λp,t. With sticky prices, the markup becomes

variable over time when the economy is hit by exogenous shocks. A positive demand shock lowers the markup and stimulates employment, investment and real output.

The definition of the price index in equation (26) implies that the law of motion for the price level is given by (Pt)−1/λp,t = θ Pt−1( Pt−1 Pt−2 )γp −1/λp,t + (1 − θ)(˜pj_t)−1/λp,t (34)

Market equilibrium conditions

For the goods market to be in equilibrium the total production has to equal demand by households for consumption and investment and the government:

Yt= Ct+ Gt+ It+ Ψ(zt)Kt−1 (35)

The capital rental market is in equilibrium when the demand for capital by the intermediate goods producers equals the supply by the households. The labour market is in equilibrium if the firms’ demand for labour equals labour supply at the wage level set by households. The interest rate is determined by a reaction function that describes monetary policy decisions. This rule will be discussed in the following sections of the thesis. In order to maintain money market equilibrium, the money supply adjusts endogenously to meet the money demand at those interest rates.

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Appendix B

Figure 5: Welfare loss Italy price - wage - compound

Figure 6: Welfare loss Portugal price - wage - compound

Figure 7: Welfare loss Poland price - wage - compound

Note: The figures above highlight the variance of the wage, price or compound targeting rules in Italy, Portugal and Poland. These countries have different levels of wage rigidity. The variance or welfare loss is plotted on the y-axis and φ on the x-axis. Hereby is highlighted which targeting rule is the best for each country. These graphs are a clarification of figure 1. Where it is to hard to tel which targeting rule is best.

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Table 7: Variances of different targeting rules for Poland, Portugal and Italy

Poland Poland Poland Italy Italy Italy Portugal Portugal Portugal Interest rate coefficient (φ) Price Wage Compound Price Wage Compound Price Wage Compound Wage inflation 2.00 0.2440 0.2327 0.2337 0.2564 0.2504 0.25 0.2374 0.2215 0.2237 1,5 0.2490 0.2419 0.2423 0.26 0.2586 0.2587 0.2418 0.2311 0.2328 1.00 0.2582 0.2558 0.2554 0.2719 0.2712 0.2719 0.2506 0.2457 0.2467 0,5 0.2810 0.2837 0.2825 0.2946 0.2969 0.2982 0.2737 0.2754 0.276 Optimal 0.3093 0.3062 0.3056 0.3478 0.3448 0.3451 0.2889 0.2835 0.2864 0,15 0.3722 0.3642 0.368 0.3689 0.3722 0.3739 0.3662 0.3733 0.3756 Price inflation 2.00 0.0701 0.0918 0.0863 0.077 0.0915 0.0964 0.0659 0.0891 0.0836 1,5 0.0774 0.0981 0.093 0.085 0.0991 0.1037 0.0727 0.095 0.0899 1.00 0.0892 0.1086 0.1038 0.0979 0.1112 0.1156 0.084 0.1048 0.1004 0,5 0.1147 0.132 0.1277 0.125 0.1369 0.1409 0.1091 0.1276 0.1244 Optimal 0.1434 0.1518 0.1486 0.1826 0.1852 0.1873 0.124 0.1341 0.1333 0,15 0.2127 0.1964 0.2055 0.2047 0.2127 0.2158 0.1952 0.2096 0.2099 Output Gap 2.00 13.3498 10.6597 10.8838 25.7973 21.9312 21.4555 8.7756 6.8468 7.1553 1,5 11.2412 9.1379 9.3433 21.6905 18.6298 18.1948 7.4215 5.9327 6.2131 1.00 8.9664 7.4913 7.6635 17.0869 14.8959 14.5301 6.00082 4.9824 5.2166 0,5 6.6698 5.8601 5.9811 11.9246 10.6718 10.4132 4.7324 4.1859 4.3425 Optimal 6.0596 5.5367 5.6318 9.2795 8.6442 8.5032 4.5727 4.145 4.2793 0,15 8.9011 6.7733 6.5332 9.4011 8.9011 8.7707 5.6369 5.454 5.5016 Total 2.00 13.6639 10.9842 11.2038 26.1307 22.2731 21.8019 9.0789 7.1574 7.4626 1,5 11.5676 9.4779 9.6786 22.0355 18.9875 18.5572 7.736 6.2588 6.5358 1.00 9.3138 7.8557 8.0227 17.4567 15.2783 14.9176 6.33542 5.3329 5.5637 0,5 7.0655 6.2758 6.3913 12.3442 11.1056 10.8523 5.1152 4.5889 4.7429 Optimal 6.5123 5.9947 6.086 9.8099 9.1742 9.0356 4.9856 4.5626 4.699 0,15 9.486 7.3339 7.1067 9.9747 9.486 9.3604 6.1983 6.0369 6.0871

Note: This table represents the theoretical moments calculated with dynare. The variances are calculated for the wage and price inflation as well as the output gap and the total variance which equals the welfare loss.

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6 Appendices Rutger Jansen Figure 8: T ec hnology sho c k with price targeting

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6 Appendices Rutger Jansen Figure 9: Preference sho c k with price targeting

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6 Appendices Rutger Jansen Figure 10: In v estmen t sho c k with price targeting

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6 Appendices Rutger Jansen Figure 11: Lab our sho c k with price targeting

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6 Appendices Rutger Jansen Figure 12: Pri ce markup sho c k with price targeting

Optimal Monetary Policy with heterogeneity in wage stickiness