Labour Market Asymmetries in a Monetary Union

Abstract

This paper takes a first step in analysing how a monetary union performs in the presence of labour market asymmetries. Differences in wage flexibility, market power and country sizes are allowed for in a setting with both country-specific and aggregate shocks. The implications of asymmetries for both the overall performance of the monetary union and the country-specific situation are analysed. It is shown that asymmetries are not only critical for country-specific performance but also for the overall performance of the monetary union. A striking finding is that aggregate output volatility is not strictly increasing in nominal rigidities but hump-shaped. Moreover, a disproportionate share of the consequences of wage inflexibility may fall on small countries. In the case of country-specific shocks, a country unambiguously benefits in terms of macroeconomic stability by becoming more flexible, while this is not necessarily the case for aggregate shocks. There may thus be a tension between the degree of flexibility considered optimal at the country level and at the aggregate level within the monetary union.

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Notes

  1. 1.

    Seminal contributions are Obstfeld and Rogoff (1995) and Yun (1996), respectively. For monographic expositions, cf. Obstfeld and Rogoff (1996) and Woodford (2003).

  2. 2.

    Recent work has also pointed to the importance of downward nominal wage rigidities and cross-country differences in this form of rigidity, see Holden and Wulfsberg (2008).

  3. 3.

    See, for instance, Smets and Wouters (2003) for an estimated Taylor rule for the euro area, and Galí and Gertler (2007) for a recent discussion.

  4. 4.

    This means that the model could also be interpreted as a closed-economy model of a single country with I sectors, which are potentially asymmetric in terms of structures and shocks.

  5. 5.

    In order to solve the model, it is written in log-deviations from the non-stochastic steady state. Steady-state values are indicated by omission of time subscripts, and lower-case letters denote (log-)deviations from steady-state values of corresponding upper-case variables (\(x_{t}\equiv dX_{t}/X\approx \ln \left( X_{t}/X\right) \)). Throughout, aggregate log-variables are defined as weighted averages of country-specific log-variables, i.e., for any variable x, we have \(x_{t}= \sum_{i=1}^{I}v_{i}X_{it}\). In general, a log-linearization around a steady-state of \(X_{t}=\sum_{i=1}^{I}V_{i}X_{it}\) gives this average where \( v_{i}=\frac{V_{j}X_{j}}{X}\). Symmetry of the steady state implies v i  = V i .

  6. 6.

    Real money balances could be included in the utility function in order to analyse money demand. However, the central bank’s policy instrument is the interest rate, while it passively supplies the money demanded by households. Thus, as long as money enters additively separably in the utility function, nothing will change in what follows since the inclusion of money will only add a money demand relation recursively determining money demand as a function of the variables of interest. See, e.g., Woodford (2003) for a discussion.

  7. 7.

    The asset-pricing kernel is the period-t price of a claim to one unit of currency in state s t + 1 in period t + 1 divided by the probability of that state occurring conditional on period-t information, \(\Pr_{t}\left( s^{t+1}\right) \). The bond B it is a random variable paying \( B_{it}\left( s^{t+1}\right) \) units of currency in state s t + 1 in period t + 1. At time t, the household chooses the complete specification of this random variable in all states s t + 1. It follows that \(E_{t}\left[ Q_{t,t+1}B_{it}\right] \) is the allocation of resources to a portfolio of bonds.

  8. 8.

    Note that it is an implication of Eq. 15 that monetary policy affects aggregate demand in all countries symmetrically.

  9. 9.

    This differs from Eq. 10 in that implicit terms representing states where the wage to be set is not the prevailing wage are excluded.

  10. 10.

    This follows from the first-order conditions of the utility-maximisation problems as all households face the same asset-pricing kernel.

  11. 11.

    Empirically interest rate smoothing is important, we disregard it here to simplify the exposition and focus on the role of asymmetries.

  12. 12.

    The minimal state representation of the equilibrium is followed, cf. McCallum (1983; McCallum (1999).

  13. 13.

    For documentation, see www.cepremap.cnrs.fr/dynare/.

  14. 14.

    Note that the decrease in prices triggers a monetary expansion which, in turn, increases activity in both countries. If the response is sufficiently strong, it is possible that output increases in both countries. In the simulations reported, the parameter values ensure that this does not happen; i.e. the direct effects of the shocks described in the text dominate.

  15. 15.

    In the next subsection, we fix the structural parameters of country 2 while allowing those of country 1 to vary in order to analyze the incentives for unilateral reform in country 1.

  16. 16.

    At the unionwide level, the standard deviations are symmetric in the degree of nominal rigidity attaining a maximum at α 1 = α 2 = 0.5. This is the point where the two countries are identical. The symmetry, of course, arises because the countries have the same size so that the restriction \(\bar{\alpha}=0.5\) implies that α 2 = 1 − α 1.

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Authors

Corresponding author

Correspondence to Torben M. Andersen.

Additional information

We are grateful for comments to previous versions made by an anonymous referee and participants at seminars at Danmarks Nationalbank, CREI at Universitat Pompeu Fabra, the Phillips-curve Conference in Kiel, the SAE conference in Granada and the Royal Economic Society Annual Conference in Warwick. Martin Seneca thanks Danmarks Nationalbank for financial support. Views expressed are those of the authors and do not necessarily reflect those of Danmarks Nationalbank or the Central Bank of Iceland.

Appendices

Appendix A: Log-linearization

Write the first-order condition (17) as

$$ W_{it}^{\ast }E_{t}\sum_{\tau =0}^{\infty }\left( \alpha _{i}\delta \right) ^{\tau }\frac{C_{it+\tau }^{-\frac{1}{\sigma }}}{P_{t+\tau }}N_{it+\tau }=E_{t}\sum_{\tau =0}^{\infty }\left( \alpha _{i}\delta \right) ^{\tau } \frac{\xi }{\left( \xi -1\right) }N_{it+\tau }^{1+\mu } $$
(35)

Taking the differential with respect to \(W_{it+\tau }^{\ast }\), C it + τ , P t + τ and N it + τ, evaluating at the steady-state values—W, C, P and N respectively—dividing through by W and rearranging gives the following log-linear approximation around the steady state:

$$ \begin{array}{lll} {\kern-12pt}\upsilon _{it}^{\ast } &=&\left( 1-\alpha _{i}\delta \right) E_{t}\sum_{\tau =0}^{\infty }\left( \alpha _{i}\delta \right) ^{\tau }\left( \mu n_{it+\tau }+\sigma ^{-1}c_{it+\tau }+p_{t+\tau }\right) \notag\\ {\kern-12pt}&=&\left( 1-\alpha _{i}\delta \right) E_{t}\sum_{\tau =0}^{\infty }\left( \alpha _{i}\delta \right) ^{\tau }\left[ \mu \left( l_{it+\tau }-\xi \left( \upsilon _{it}^{\ast }-w_{it+\tau }\right) \right) +\sigma ^{-1}c_{it+\tau }+p_{t+\tau }\right] \label{FocWageApp2} \end{array} $$
(36)

where the second equality follows by using a log-linear version of Eq. 3 to replace n it + τ. Rearranging gives

$$ \upsilon _{it}^{\ast }=\frac{\left( 1-\alpha _{i}\delta \right) }{1+\mu \xi } \left( \mu l_{it}-\mu \xi w_{it}+\sigma ^{-1}c_{it}+p_{t}\right) +\left( \alpha _{i}\delta \right) E_{t}\upsilon _{t+1}^{\ast } \label{FocWageApp3} $$
(37)

Similarly, a log-linear approximation to Eq. 25 is given by

$$ w_{it}=\alpha _{i}w_{it-1}+\left( 1-\alpha _{i}\right) \upsilon _{it}^{\ast } \label{WageLoMotionApp1} $$
(38)

Subtracting w it from both sides of Eq. 36 and using Eq. 37 to eliminate \(\upsilon _{it}^{\ast }\) gives

$$ \omega _{it}=\frac{\left( 1-\alpha _{i}\right) \left( 1-\alpha _{i}\delta \right) }{\alpha _{i}\left( 1+\mu \xi \right) }\left[ \mu l_{it}+\sigma ^{-1}c_{it}-\left( w_{it}-p_{t}\right) \right] +\delta E_{t}\omega _{it+1} \label{WageApp1} $$
(39)

where ω it  = w it  − w it − 1.

Appendix B: Flexible wage equilibrium

Suppose wages are flexible as well as prices. In this case, the wage equation becomes

$$ w_{it}=p_{t}+\mu l_{it}+\sigma ^{-1}c_{t} $$
(40)

Substituting out l it by a linear version of the production function (5) gives

$$ w_{it}=p_{t}+\mu \left( \frac{1}{\gamma }y_{it}-\frac{1-\gamma }{\gamma } u_{it}\right) +\sigma ^{-1}c_{t} $$
(41)

implying

$$ w_{t}=p_{t}+\mu \left( \frac{1}{\gamma }y_{t}-\frac{1-\gamma }{\gamma } u_{t}\right) +\sigma ^{-1}y_{t} $$
(42)

Inserting this in aggregated supply

$$ y_{t}=\beta \left( p_{t}-w_{t}\right) +u_{t} $$
(43)

gives

$$ \bar{y}_{t}=\beta \left( -\mu \left( \frac{1}{\gamma }\bar{y}_{t}-\frac{ 1-\gamma }{\gamma }u_{t}\right) -\sigma _{t}^{-1}\bar{y}\right) +u_{t} $$
(44)

or

$$ \bar{y}_{t}=\frac{1+\mu }{1+\mu \left( 1+\beta \right) +\beta \sigma ^{-1}} u_{t} $$
(45)

Appendix C: Sticky wage equilibrium

Imposing the equilibrium condition means c it  = c t  = y t . Using this, the wage equation Eq. 19 and the aggregate supply relation (8) can be combined to give the ’AS’ relation

$$ \begin{array}{lll} &&\pi _{it}+\beta ^{-1}\left[ \left( u_{it}-u_{it-1}\right) -\left( y_{it}-y_{it-1}\right) \right] \notag \\ &&{\kern1pc}=\Lambda _{i}\left[ \left( \mu +\left( 1+\mu \right) \beta ^{-1}+\theta ^{-1}\right) y_{it}+\left( \sigma -\theta ^{-1}\right) y_{t}-\left( 1+\mu \right) \beta ^{-1}u_{it}\right] \\ &&{\kern2pc}+\delta \left( E_{t}\pi _{it+1}+\beta ^{-1}\left[ \left( \rho _{u}-1\right) u_{it}-\left( E_{t}y_{it+1}-y_{it}\right) \right] \right) \label{AS} \end{array} $$
(46)

where a log-linear version of Eq. 5 has been used to substitute out l it . Similarly, combining the Euler equation Eq. 23, the Taylor rule Eq. 26 and the intratemporal demand function (24) gives the ’IS’ relation

$$ \begin{array}{lll} \label{IS} y_{it}-y_{it-1} &=&-\theta \left( \pi _{it}-\pi _{t}\right) +E_{t}y_{t+1} \\ &&-\sigma \left( k_{\pi }\pi _{t}+k_{y}\left( y_{t}-\Xi u_{t}\right) -E_{t}\pi _{t+1}\right) -y_{t-1} \end{array} $$
(47)

Hence, two equations summarise the dynamics of output and inflation for each country. Disturbances to the system follow from the stochastic process

$$ u_{it}=\rho _{u}u_{it-1}+\varepsilon _{it} $$
(48)

We guess that output and inflation in country i take the forms:

$$ y_{it}=\sum_{j}v_{\!j}b_{0}^{ij}u_{\!jt}+\sum_{j}v_{\!j}b_{1}^{ij}u_{\!jt-1}+ \sum_{j}v_{\!j}b_{2}^{ij}y_{\!jt-1} $$
(49)

and

$$ \pi _{it}=\sum_{j}v_{\!j}c_{0}^{ij}u_{\!jt}+\sum_{j}v_{\!j}c_{1}^{ij}u_{\!jt-1}+ \sum_{j}v_{\!j}c_{2}^{ij}y_{\!jt-1} $$
(50)

These conjectures imply the following expressions for aggregate output and inflation:

$$ y_{t} =\sum_{i}\sum_{j}v_{i}v_{\!j}b_{0}^{ij}u_{\!jt}+\sum_{i} \sum_{j}v_{i}v_{\!j}b_{1}^{ij}u_{\!jt-1}+\sum_{i} \sum_{j}v_{i}v_{\!j}b_{2}^{ij}y_{\!jt-1} $$
(51)
$$ \pi _{t} =\sum_{i}\sum_{j}v_{i}v_{\!j}c_{0}^{ij}u_{\!jt}+\sum_{i} \sum_{j}v_{i}v_{\!j}c_{1}^{ij}u_{\!jt-1}+\sum_{i} \sum_{j}v_{i}v_{\!j}c_{2}^{ij}y_{\!jt-1} $$
(52)

In addition, expectations become

$$ E_{t}y_{it+1} =\sum_{j}v_{\!j}\left( b_{0}^{ij}\rho _{u}+b_{1}^{ij}\right) u_{\!jt}+\sum_{j}v_{\!j}b_{2}^{ij}y_{\!jt} $$
(53)
$$ E_{t}\pi _{it+1} =\sum_{j}v_{\!j}\left( c_{0}^{ij}\rho _{u}+c_{1}^{ij}\right) u_{\!jt}+\sum_{j}v_{\!j}c_{2}^{ij}y_{\!jt} $$
(54)
$$ E_{t}y_{t+1} =\sum_{i}\sum_{j}v_{i}v_{j}\left( b_{0}^{ij}\rho _{u}+b_{1}^{ij}\right) u_{\!jt}+\sum_{i}\sum_{j}v_{i}v_{\!j}b_{2}^{ij}y_{\!jt} $$
(55)
$$ E_{t}\pi _{t+1} =\sum_{i}\sum_{j}v_{i}v_{\!j}\left( c_{0}^{ij}\rho _{u}+c_{1}^{ij}\right) u_{\!jt}+\sum_{i}\sum_{j}v_{i}v_{\!j}c_{2}^{ij}y_{\!jt} $$
(56)

To verify our conjectures, we find values of the coefficients \(\Big(b_{0}^{ij},b_{1}^{ij},b_{2}^{ij},\) \(c_{0}^{ij},c_{1}^{ij},c_{2}^{ij}\Big)\) that satisfy the restrictions imposed by the log-linear model. Inserting the conjectures in Eq. 45 gives

$$ \begin{array}{lll} &&\sum_{\!j}v_{\!j}c_{0}^{ij}u_{\!jt}+\sum_{\!j}v_{\!j}c_{1}^{ij}u_{\!jt-1}+ \sum_{\!j}v_{\!j}c_{2}^{ij}y_{\!jt-1} +\beta ^{-1}\left( u_{it}-u_{it-1}\right) \\ &&{\kern1pc}-\beta ^{-1}\left( \sum_{\!j}v_{\!j}b_{0}^{ij}u_{\!jt}+\sum_{\!j}v_{\!j}b_{1}^{ij}u_{\!jt-1}+ \sum_{\!j}v_{\!j}b_{2}^{ij}y_{\!jt-1}-y_{it-1}\right) \\ &&{\kern1pc}=\!\Lambda _{i}\left( \mu \!+\!\left( 1\!+\!\mu \right) \beta ^{-1}+\theta ^{-1}\right) \!\left( \sum_{\!j}v_{\!j}b_{0}^{ij}u_{\!jt}+\sum_{\!j}v_{\!j}b_{1}^{ij}u_{\!jt-1}\!+\! \sum_{\!j}v_{\!j}b_{2}^{ij}y_{\!jt-1}\!\right) \\ &&{\kern2pc}+\!\Lambda _{i}\!\left( \sigma \!-\!\theta ^{-1}\right) \!\!\left(\! \sum_{n}\!\sum_{\!j}v_{n}\!v_{\!j}b_{0}^{nj}u_{\!jt}\!+\!\!\sum_{n} \!\sum_{\!j}\!v_{n}\!v_{\!j}b_{1}^{nj}u_{\!jt-1}\!+\!\sum_{n} \!\sum_{\!j}\!v_{n}v_{\!j}b_{2}^{nj}y_{\!jt-1}\!\right) \\ &&{\kern2pc}-\left( 1+\mu \right) \beta ^{-1}\Lambda _{i}u_{it}+\delta \sum_{\!j}v_{\!j}\left( c_{0}^{ij}\rho _{u}+c_{1}^{ij}\right) u_{\!jt} \\ &&{\kern2pc}+\delta \sum_{n}v_{n}c_{2}^{in}\left( \sum_{\!j}v_{\!j}b_{0}^{nj}u_{\!jt}+\sum_{\!j}v_{\!j}b_{1}^{nj}u_{\!jt-1}+ \sum_{\!j}v_{\!j}b_{2}^{nj}y_{\!jt-1}\right) \\ &&{\kern2pc}+\delta \beta ^{-1}\left( \rho _{u}-1\right) u_{it}-\delta \beta ^{-1}\sum_{\!j}v_{\!j}\left( b_{0}^{ij}\rho _{u}+b_{1}^{ij}\right) u_{\!jt} \\ &&{\kern2pc}-\delta \beta ^{-1}\sum_{\!j}v_{\!j}b_{2}^{ij}\left( \sum_{n}v_{n}b_{0}^{jn}u_{nt}+\sum_{n}v_{n}b_{1}^{jn}u_{nt-1}+ \sum_{n}v_{n}b_{2}^{jn}y_{nt-1}\right) \\ &&{\kern2pc}+\delta \beta ^{-1}\left( \sum_{\!j}v_{\!j}b_{0}^{ij}u_{\!jt}+\sum_{\!j}v_{\!j}b_{1}^{ij}u_{\!jt-1}+ \sum_{\!j}v_{\!j}b_{2}^{ij}y_{\!jt-1}\right) \end{array} $$
(57)

Equating coefficients on u it gives the restriction

$$ \begin{array}{lll} v_{i}c_{0}^{ii}\!+\!\beta ^{-1}\!-\!\beta ^{-1}v_{i}b_{0}^{ii} &=&\Lambda _{i}\left( \mu \!+\!\left( 1\!+\!\mu \right) \beta ^{-1}\!+\!\theta ^{-1}\right) v_{i}b_{0}^{ii}\!+\!\Lambda _{i}\left( \sigma \!-\!\theta ^{-1}\right) \sum_{j}v_{j}v_{i}b_{0}^{ji} \\ &&-\left( 1+\mu \right) \beta ^{-1}\Lambda _{i}+\delta v_{i}\left( c_{0}^{ii}\rho _{u}+c_{1}^{ii}\right) +\delta \sum_{j}v_{j}v_{i}c_{2}^{ij}b_{0}^{ji} \\ &&+\delta \beta ^{-1}\left( \rho _{u}-1\right) -\delta \beta ^{-1}v_{i}\left( b_{0}^{ii}\rho _{u}+b_{1}^{ii}\right) \\ &&-\delta \beta ^{-1}\sum_{j}v_{j}v_{i}b_{2}^{ij}b_{0}^{ji}+\delta \beta ^{-1}v_{i}b_{0}^{ii} \label{R1} \end{array} $$
(58)

Equating coefficients on u jt where j ≠ i gives

$$ \begin{array}{lll} c_{0}^{ij}\!-\!\beta ^{-1}b_{0}^{ij} &=&\Lambda _{i}\left( \mu \!+\!\left( 1\!+\!\mu \right) \beta ^{-1}\!+\!\theta ^{-1}\right) b_{0}^{ij}\!+\!\Lambda _{i}\left( \sigma \!-\!\theta ^{-1}\right) \sum_{n}v_{n}b_{0}^{nj} \\ &&+\delta \left( c_{0}^{ij}\rho _{u}+c_{1}^{ij}\right) +\delta \sum_{n}v_{n}c_{2}^{in}b_{0}^{nj} \\ &&-\delta \beta ^{-1}\left( b_{0}^{ij}\rho _{u}+b_{1}^{ij}\right) -\delta \beta ^{-1}\sum_{n}v_{n}b_{2}^{in}b_{0}^{nj}+\delta \beta ^{-1}b_{0}^{ij} \label{R2} \end{array} $$
(59)

Equating coefficients on u it − 1:

$$ \begin{array}{lll} v_{i}c_{1}^{ii}-\beta ^{-1}-\beta ^{-1}v_{i}b_{1}^{ii} &=&\Lambda _{i}\left( \mu +\left( 1+\mu \right) \beta ^{-1}+\theta ^{-1}\right) v_{i}b_{1}^{ii} \\ &&+\Lambda _{i}\left( \sigma -\theta ^{-1}\right) \sum_{n}v_{n}v_{i}b_{1}^{ni}+\delta \sum_{n}v_{n}v_{i}c_{2}^{in}b_{1}^{ni} \\ &&-\delta \beta ^{-1}\sum_{n}v_{n}v_{i}b_{2}^{in}b_{1}^{ni}+\delta \beta ^{-1}v_{i}b_{1}^{ii} \label{R3} \end{array} $$
(60)

Equating coefficients on u jt − 1where j ≠ i:

$$ \begin{array}{lll} c_{1}^{ij}-\beta ^{-1}b_{1}^{ij} &=&\Lambda _{i}\left( \mu +\left( 1+\mu \right) \beta ^{-1}+\theta ^{-1}\right) b_{1}^{ij} \\ &&+\Lambda _{i}\left( \sigma -\theta ^{-1}\right) \sum_{n}v_{n}b_{1}^{nj}+\delta \sum_{n}v_{n}c_{2}^{in}b_{1}^{nj} \\ &&-\delta \beta ^{-1}\sum_{n}v_{n}b_{2}^{in}b_{1}^{nj}+\delta \beta ^{-1}b_{1}^{ij} \label{R4} \end{array} $$
(61)

Equating coefficients on y it − 1 gives

$$ \begin{array}{lll} v_{i}c_{2}^{ii}-\beta ^{-1}\left( v_{i}b_{2}^{ii}-1\right) &=&\Lambda _{i}\left( \mu +\left( 1+\mu \right) \beta ^{-1}+\theta ^{-1}\right) v_{i}b_{2}^{ii}\\ &&+\Lambda _{i}\left( \sigma -\theta ^{-1}\right) \sum_{n}v_{n}v_{i}b_{2}^{ni} \\ &&+\delta \sum_{n}v_{n}v_{i}c_{2}^{in}b_{2}^{ni}-\delta \beta ^{-1}\sum_{n}v_{n}v_{i}b_{2}^{in}b_{2}^{ni}\\ &&+\delta \beta ^{-1}v_{i}b_{2}^{ii} \label{R5} \end{array} $$
(62)

and on y jt − 1where j ≠ i:

$$ \begin{array}{lll} v_{\!j}c_{2}^{ij}-\beta ^{-1}v_{\!j}b_{2}^{ij} &=&\Lambda _{i}\left( \mu +\left( 1+\mu \right) \beta ^{-1}+\theta ^{-1}\right) v_{j}b_{2}^{ij}+\Lambda _{i}\left( \sigma -\theta ^{-1}\right) \sum_{n}v_{n}v_{\!j}b_{2}^{nj} \\ &&+\delta \sum_{n}v_{n}v_{\!j}c_{2}^{in}b_{2}^{nj}-\delta \beta ^{-1}\sum_{n}v_{n}v_{j}b_{2}^{in}b_{2}^{nj}+\delta \beta ^{-1}v_{\!j}b_{2}^{ij} \label{R6} \end{array} $$
(63)

Inserting conjectures in Eq. 45 gives

$$ \begin{array}{lll} &&\sum_{j}v_{j}b_{0}^{ij}u_{\!jt}+\sum_{j}v_{j}b_{1}^{ij}u_{\!jt-1}+ \sum_{j}v_{j}b_{2}^{ij}y_{jt-1}-y_{it-1} \\ &&{\kern1pc} =-\theta \left( \sum_{j}v_{j}c_{0}^{ij}u_{jt}+\sum_{j}v_{j}c_{1}^{ij}u_{jt-1}+ \sum_{j}v_{j}c_{2}^{ij}y_{jt-1}\right) \\ &&{\kern2pc}+\!\left( \theta \!-\!\sigma k_{\pi }\right)\! \left(\! \sum_{n}\!\sum_{j}v_{n}v_{\!j}c_{0}^{nj}u_{\!jt}\!+\!\sum_{n}\! \sum_{j}v_{n}v_{\!j}c_{1}^{nj}u_{\!jt-1}\!+\!\sum_{n}\! \sum_{j}\!v_{n}v_{\!j}c_{2}^{nj}y_{\!jt-1}\!\right) \\ &&{\kern2pc}+\sum_{n}\sum_{j}v_{n}v_{\!j}\left( b_{0}^{nj}\rho _{u}+b_{1}^{nj}\right) u_{\!jt} \\ &&{\kern2pc}+\sum_{m}\sum_{n}v_{m}v_{n}b_{2}^{mn}\left( \sum_{j}v_{\!j}b_{0}^{nj}u_{\!jt}+\sum_{j}v_{\!j}b_{1}^{nj}u_{jt-1}+ \sum_{j}v_{\!j}b_{2}^{nj}y_{\!jt-1}\right) \\ &&{\kern2pc}-\sigma k_{\!y}\!\left( \!\sum_{n}\sum_{j}v_{n}v_{\!j}b_{0}^{nj}u_{\!jt}+\sum_{n} \sum_{j}v_{n}v_{\!j}b_{1}^{nj}u_{\!jt-1}+\sum_{n} \sum_{j}v_{n}v_{\!j}b_{2}^{nj}y_{\!jt-1}\right)\\ &&{\kern2pc}+\sigma k_{\!y}\Xi \sum_{\!j}v_{j}u_{\!jt} +\sigma \sum_{n}\sum_{j}v_{n}v_{\!j}\left( c_{0}^{nj}\rho _{u}+c_{1}^{nj}\right) u_{\!jt} \\ &&{\kern2pc}+\sigma \sum_{m}\sum_{n}v_{m}v_{n}c_{2}^{mn}\left( \sum_{j}v_{\!j}b_{0}^{nj}u_{\!jt}+\sum_{\!j}v_{\!j}b_{1}^{nj}u_{\!jt-1}+ \sum_{j}v_{j}b_{2}^{nj}y_{\!jt-1}\right)\\ &&{\kern2pc}-\sum_{j}v_{\!j}y_{\!jt-1} \end{array} $$
(64)

Equating coefficients on u jt gives

$$ \begin{array}{lll} b_{0}^{ij} &=&-\theta c_{0}^{ij}+\left( \theta -\sigma k_{\pi }\right) \sum_{n}v_{n}c_{0}^{nj}+\sum_{n}v_{n}\left( b_{0}^{nj}\rho _{u}+b_{1}^{nj}\right) \\ &&+\sum_{m}\sum_{n}v_{m}v_{n}b_{2}^{mn}b_{0}^{nj}-\sigma k_{y}\sum_{n}v_{n}b_{0}^{nj}+\sigma k_{y}\Xi v_{j}+\sigma \sum_{n}v_{n}\left( c_{0}^{nj}\rho _{u}+c_{1}^{nj}\right) \\ &&+\sigma \sum_{m}\sum_{n}v_{m}v_{n}c_{2}^{mn}b_{0}^{nj} \label{R7} \end{array} $$
(65)

On u jt − 1:

$$ \begin{array}{lll} b_{1}^{ij} &=&-\theta c_{1}^{ij}+\left( \theta -\sigma k_{\pi }\right) \sum_{n}v_{n}c_{1}^{nj}+\sum_{m}\sum_{n}v_{m}v_{n}b_{2}^{mn}b_{1}^{nj} \\ &&-\sigma k_{y}\sum_{n}v_{n}b_{1}^{nj}+\sigma \sum_{m}\sum_{n}v_{m}v_{n}c_{2}^{mn}b_{1}^{nj} \label{R8} \end{array} $$
(66)

y it − 1:

$$ \begin{array}{lll} v_{i}b_{2}^{ii}-1 &=&-\theta v_{i}c_{2}^{ii}+\left( \theta -\sigma k_{\pi }\right) \sum_{n}v_{n}v_{i}c_{2}^{ni}+\sum_{m} \sum_{n}v_{m}v_{n}b_{2}^{mn}v_{i}b_{2}^{ni} \\ &&-\sigma k_{\!y}\sum_{n}v_{n}v_{i}b_{2}^{ni}+\sigma \sum_{m}\sum_{n}v_{m}v_{n}c_{2}^{mn}v_{i}b_{2}^{ni}-v_{i} \label{R9} \end{array} $$
(67)

and finally, y jt − 1 where j ≠ i:

$$ \begin{array}{lll} b_{2}^{ij} &=&-\theta c_{2}^{ij}+\left( \theta -\sigma k_{\pi }\right) \sum_{n}v_{n}c_{2}^{nj}+\sum_{m}\sum_{n}v_{m}v_{n}b_{2}^{mn}b_{2}^{nj} \\ &&-\sigma k_{\!y}\sum_{n}v_{n}b_{2}^{nj}+\sigma \sum_{m}\sum_{n}v_{m}v_{n}c_{2}^{mn}b_{2}^{nj}-1 \label{R10} \end{array} $$
(68)

The restrictions (55)–(60) and (62)–(65) constitute a system of 6I 2 equations determining the 6I 2 coefficients in the conjectures. Indeed, this system is recursive. The 2I 2 restrictions from equating coefficients on y it − 1 may be combined to solve for \( \{b_{2}^{ij},c_{2}^{ij}\}_{ij}\,\), which may then be used in the 2I 2 restrictions from u it − 1 to solve for {\(b_{1}^{ij},c_{1}^{ij}\}_{ij}\). Finally, these coefficients may be used in the restrictions from equation coefficients on u it to find the remaining 2I 2 coefficients \( \{b_{0}^{ij},c_{0}^{ij}\}_{ij}\).

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Andersen, T.M., Seneca, M. Labour Market Asymmetries in a Monetary Union. Open Econ Rev 21, 483–514 (2010). https://doi.org/10.1007/s11079-009-9136-6

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Keywords

  • Wage formation
  • Nominal wage rigidity
  • Staggered contracts
  • Monetary policy
  • Monetary union
  • Business cycles
  • Shocks

JEL Classification

  • E30
  • E52
  • F41