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.

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 $\{b_{0}^{ij},c_{0}^{ij}\}_{ij}$.