Fiscal Devaluation Scenarios: A Quantitative Assessment for the Italian Economy

Abstract

We study the potential impact of fiscal devaluation poon the Italian economy using IGEM, a dynamic general equilibrium model for the Italian economy developed at the Italian Department of the Treasury. The simulations show that fiscal devaluation policies are likely to produce short-run slight improvements on the external position of the economy, while the output gains seem to persist in the long run. Non-negligible distributional effects across households are also observed, since taxation on consumption tends to be regressive.

Appendices

Appendix:

This Appendix presents all the equilibrium conditions of IGEM and list all the relevant variables and parameters.

The Euler Equation of Ricardian Households

$$\begin{array}{@{}rcl@{}} {\lambda_{t}^{R}}=\beta E_{t}\lambda_{t+1}^{R}\frac{R_{t}}{\Pi_{t+1}} \end{array} $$

(1)

• λ ^R Lagrange multiplier - Ricardian households

• R Nominal interest rate factor

• π Inflation factor

• β Discount factor

The Lagrangian Multiplier of Ricardian Households

$$\begin{array}{@{}rcl@{}} {\lambda_{t}^{R}}=\frac{P_{t}}{{P_{t}^{C}}}\frac{1}{(1+{\tau_{t}^{C}})\left( {C_{t}^{R}}-h_{C^{R}}C_{t-1}^{R}\right)} \end{array} $$

(2)

• P Domestic production price level

• λ ^R Lagrange multiplier - Ricardian households

• P ^C Domestic consumption price index

• τ ^C Consumption tax (VAT)

• C ^R Consumption- Ricardian households

• h _C ^R Habit persistence parameter - Ricardian households

Note that by combining Eqs. 1 and 2 we obtain the Euler Eq. 1 of the main text.

Consumption of the Non-Ricardian Households

$$\begin{array}{@{}rcl@{}} C_{t}^{NR}=\frac{P_{t}}{\left( 1+{\tau_{t}^{C}}\right) P_{C,t}}\left[ \left( 1-\tau_{t}^{N_{A}}-\tau_{h,t}^{W^{N_{A}}}\right) WR_{t}^{N_{A}} N_{A,t}-TAX_{t}^{NR}+Tr_{t}^{NR}\right] \end{array} $$

(3)

• P Domestic production price level

• P ^C Domestic consumption price index

• τ ^C Consumption tax (VAT)

• C ^NR Consumption - non-Ricardian households

• $\tau _{h}^{SSC^{N_{A}}}$ Social contributions rate of atypical workers

• T A X ^NR Lump-sum tax, non-Ricardian households

• T r ^NR Transfers to non-Ricardian households

• τ ^N _A Average tax rate on labour income of atypical workers

• W R ^N _A Real wage of atypical workers

• N _A Employment of atypical workers

Note that this is Eq. 2 of the main text.

The Lagrangian Multiplier of Non-Ricardian Households

$$\begin{array}{@{}rcl@{}} \lambda_{t}^{NR}=\frac{P_{t}}{{P_{t}^{C}}}\frac{1}{(1+{\tau_{t}^{C}})\left( C_{t}^{NR}-h_{C^{NR}}C_{t-1}^{NR}\right)} \end{array} $$

(4)

• P Domestic production price level

• P ^C Domestic consumption price index

• τ ^C Consumption tax (VAT)

• C ^NR Consumption - non-Ricardian households

• h _C ^NR Habit persistence parameter, non-Ricardian households

• λ ^NR Lagrange multiplier, non-Ricardian households

Aggregate Consumption

$$\begin{array}{@{}rcl@{}} C_{t}=s_{NR}C_{t}^{NR}+(1-s_{NR}){C_{t}^{R}} \end{array} $$

(5)

• C Aggregate consumption

• C ^NR Consumption - non-Ricardian households

• C ^R Consumption - Ricardian households

• s _{N R} Share of non-Ricardian households

Wage Equation of Self-employed Workers

$$\begin{array}{@{}rcl@{}} &&\left(\sigma_{N_{s}}-1\right) {\lambda_{t}^{R}}\left( 1-\tau_{t}^{N_{s}}-\tau_{h}^{SSC^{N_{S}}}\right) WR_{t}^{N_{s}}N_{S,t} =\omega_{N_{s}}\sigma_{N_{s}}\left( 1-N_{S,t}\right) ^{-v_{N_{s}}}N_{S,t}+\\ &&-{\lambda_{t}^{R}}\gamma_{W^{N_{s}}}\left( \frac {WR_{t}^{N_{s}}}{indexatio{n_{t}^{W}}\times WR_{t-1}^{N_{s}}}\Pi_{t}-1\right) Y_{t}\frac{WR_{t}^{N_{s}}}{indexatio{n_{t}^{W}}\times WR_{t-1}^{N_{s}}}\Pi_{t}+\\ &&+{\beta\lambda}_{t+1}^{R}{\gamma}_{W^{N_{s}}}\left( \frac{WR_{t+1}^{N_{s}}}{indexation_{t+1}^{W}\times WR_{t}^{N_{s}}}\Pi _{t+1}-1\right) {\small Y}_{t+1}\frac{WR_{t+1}^{N_{s}}}{indexation_{t+1} ^{W}\times WR_{t}^{N_{s}}}{\Pi}_{t+1} \end{array} $$

(6)

• λ ^R Lagrange multiplier - Ricardian households

• $\sigma _{N_{s}}$ Elasticity of substitution, self-employed workers

• τ ^N _s Average tax rate on labour income of self-employed workers

• $\tau _{h}^{SSC^{N_{S}}}$ Social security contributions on self-employed workers

• W R ^N _s Real wage of self-employed workers

• N _s Employment of self-employed workers

• $\omega _{N_{s}}$ Preference parameter, self-employed workers

• $v_{N_{s}}$ Preference parameter, self-employed workers

• $\gamma _{W^{N_{s}}}$ Adjustment cost parameter on wage of self-employed

• i n d e x a t i o n ^W Wage indexation

• π Inflation factor

• Y Output

In steady state, this equation boils down to $WR^{N_{s}}=MU_{W}^{N_{S}} \frac {1+\tau ^{C}}{1-\tau ^{S}-\tau _{h}^{SSC^{N_{S}}}}MRS^{N_{S}},$where $MU_{W}^{N_{S}}=\frac {\sigma _{N_{s}}}{\sigma _{N_{s}}-1}$ and $MRS^{N_{S}} =\omega _{N_{s}}\frac {\left (1-N_{S}\right ) ^{-v_{N_{s}}}}{\lambda ^{R}}$. This corresponds to equation (3) of the main text, for the superscript S=N _S.

Wage Equation of Skilled Employed Workers

$$\begin{array}{@{}rcl@{}} &&\left( \sigma_{L_{H}}-1\right) {\lambda_{t}^{R}}\left( 1-\tau_{t}^{L_{H}}-\tau_{h}^{SSC^{L_{H}}}\right) WR_{t}^{L_{H}}L_{H,t} =\omega_{L_{H}}\sigma_{L_{H}}\left( 1-L_{H,t}\right) ^{-v_{L_{H}}}L_{H,t}+\\ &&-{\lambda_{t}^{R}}\gamma_{W^{L_{H}}}\left( \frac {WR_{t}^{L_{H}}}{indexatio{n_{t}^{W}}\times WR_{t-1}^{L_{H}}}\Pi_{t}-1\right) Y_{t}\frac{WR_{t}^{L_{H}}}{indexatio{n_{t}^{W}}\times WR_{t-1}^{L_{H}}}\Pi_{t}+\\ &&+{\beta\lambda}_{t+1}^{R}{\small \gamma}_{W^{L_{H}}}\left( \frac{WR_{t+1}^{L_{H}}}{indexation_{t+1}^{W}\times WR_{t}^{L_{H}}}\Pi _{t+1}-1\right) {Y}_{t+1}\frac{WR_{t+1}^{L_{H}}}{indexation_{t+1} ^{W}\times WR_{t}^{L_{H}}}{\Pi}_{t+1} \end{array} $$

(7)

• λ ^R Lagrange multiplier - Ricardian households

• $\sigma _{L_{H}}$ Elasticity of substitution, skilled workers

• τ ^L _H Average tax rate on labour income of permanent skilled workers

• $\tau _{h}^{SSC^{L_{H}}}$ Social security contributions on skilled workers

• W R ^L _H Real wage of skilled workers

• L _H Employment of skilled workers

• $\omega _{L_{H}}$ Preference parameter, skilled workers

• $v_{L_{H}}$ Preference parameter, skilled workers

• $\gamma _{W^{L_{H}}}$ Adjustment cost parameter on wage of permanent skilled workers

• i n d e x a t i o n ^W Wage indexation

• π Inflation factor

• Y Output

In steady state, this equation boils down to $WR^{L_{H}}=MU_{W}^{L_{H}} \frac {1+\tau ^{C}}{1-\tau ^{S}-\tau _{h}^{SSC^{L_{H}}}}MRS^{L_{H}},$ where $MU_{W}^{L_{H}}=\frac {\sigma _{L_{H}}}{\sigma _{L_{H}}-1}$ and $MRS^{L_{H}} =\omega _{L_{H}}\frac {\left (1-L_{H}\right ) ^{-v_{L_{H}}}}{\lambda ^{R}}$. This corresponds to equation (3) of the main text, for the superscript S=L _H .

Wage Equation of Unskilled Employed Workers

$$\begin{array}{@{}rcl@{}} &&\left( \sigma_{L_{L}}-1\right) {\lambda_{t}^{R}}\left( 1-\tau _{t}^{L_{L}}-\tau_{h}^{SSC^{L_{L}}}\right) WR_{t}^{L_{L}}L_{L,t} =\omega_{L_{L}}\sigma_{L_{L}}\left( 1-L_{L,t}\right) ^{-v_{LL}}L_{L,t}+\\ &&-{\lambda_{t}^{R}}\gamma_{W^{L_{L}}}\left( \frac{WR_{t}^{L_{L}}} {indexatio{n_{t}^{W}}\times WR_{t-1}^{L_{L}}(h_{L_{L}})}\Pi_{t}-1\right) Y_{t}\frac{WR_{t}^{L_{L}}}{indexatio{n_{t}^{W}}\times WR_{t-1}^{L_{L}}}\Pi_{t}+\\ &&+{\beta\lambda}_{t+1}^{R}\gamma_{W^{L_{L}}}\left( \frac{WR_{t+1}^{L_{L}}} {indexation_{t+1}^{W}\times WR_{t}^{L_{L}}}\Pi_{t+1}-1\right) Y_{t+1} \frac{WR_{t+1}^{L_{L}}}{indexation_{t+1}^{W}\times WR_{t}^{L_{L}})}\Pi_{t+1} \end{array} $$

(8)

• λ ^R Lagrange multiplier - Ricardian households

• $\sigma _{L_{L}}$ Elasticity of substitution, unskilled workers

• τ ^L _L Average tax rate on labour income of permanent unskilled workers

• $\tau _{h}^{SSC^{L_{L}}}$ Social security contributions on unskilled workers

• W R ^L _L Real wage of unskilled workers

• L _L Employment of unskilled workers

• $\omega _{L_{L}}$ Preference parameter, unskilled workers

• $v_{L_{L}}$ Preference parameter, unskilled workers

• $\gamma _{W^{L_{L}}}$ Adjustment cost parameter wage of permanent unskilled workers

• i n d e x a t i o n ^W Wage Indexation

• π Inflation factor

• Y Output

In steady state, this equation boils down to $WR^{L_{L}}=MU_{W}^{L_{L}} \frac {1+\tau ^{C}}{1-\tau ^{S}-\tau _{h}^{SSC^{L_{L}}}}MRS^{L_{L}},$where $MU_{W}^{L_{L}}=\frac {\sigma _{L_{L}}}{\sigma _{L_{L}}-1}$ and $MRS^{L_{L}} =\omega _{L_{L}}\frac {\left (1-L_{L}\right ) ^{-v_{L_{L}}}}{\lambda ^{R}}$. This corresponds to Eq. 3 of the main text, for the superscript S=L _L .

The Supply of Atypical Labour Services

$$\begin{array}{@{}rcl@{}} \frac{1}{\lambda_{t}^{NR}}=WR_{t}^{N_{A}}\left( 1-N_{A,t}\right) ^{v_{N_{A}} }\frac{1-\tau_{t}^{N_{A}}-\tau_{h,t}^{W^{N_{A}}}}{\omega_{N_{A}}} \end{array} $$

(9)

• λ ^NR Lagrange multiplier - non-Ricardian households

• W R ^N _A Real wage of atypical workers

• N _A Employment of atypical workers

• $v_{N_{A}}$ Preference parameter, atypical workers

• τ ^N _A Average tax rate on labour income of atypical workers

• $\tau _{h}^{W^{N_{A}}}$ Social security contributions rate on atypical workers

• $\omega _{N_{A}}$ Preference parameter, atypical workers

In steady state, this equation boils down to $WR^{N_{A}}=\frac {1+\tau ^{C}} {1-\tau ^{S}-\tau _{h}^{SSC^{N_{A}}}}MRS^{N_{A}},$where $MRS^{N_{A}} =\omega _{N_{A}}\frac {\left (1-N_{A}\right ) ^{-v_{N_{A}}}}{\lambda ^{NR}}$. This corresponds to equation (3) of the main text, for the superscript S=N _A and $MU_{W}^{N_{A}}=1$

Aggregate Employment

$$\begin{array}{@{}rcl@{}} LN_{t}=s_{L_{L}}L_{L,t}+s_{L_{H}}L_{H,t}+s_{N_{S}}N_{S,t}+s_{N_{A}}N_{A,t} \end{array} $$

(10)

• LN Total employment

• $s_{L_{L}}$ Share of unskilled employed

• L _L Employment of unskilled workers

• $s_{L_{H}}$ Share of skilled employed

• L _H Employment of skilled workers

• $s_{N_{S}}$ Share of self-employed workers

• N _S Employment of self-employed workers

• $s_{N_{A}}$ Share of atypical workers

• N _A Employment of atypical workers

Production Function of Intermediate-goods Producers

$$\begin{array}{@{}rcl@{}} Y_{t}=A_{t}\left( L_{CES,t}-O{H_{t}^{L}}\right) ^{\alpha_{L}}\left( N_{CES,t}-O{H_{t}^{N}}\right) ^{\alpha_{N}}\left( {u_{t}^{K}}K_{t}\right) ^{1-\alpha_{L}-\alpha_{N}} \end{array} $$

(11)

• Y Output

• A Total factor productivity

• L _{C E S} CES aggregator of L _H andL _L

• N _{C E S} CES aggregator of N _S andN _A

• O H ^L Overhead labour

• O H ^N Overhead labour

• K Capital

• α _L Production function parameter, L _L andL _H workers

• α _N Production function parameter, N _S andN _A workers

• u ^K Capacity utilization of capital

This is equation (4) of the main text.

Permenat Workers CES Aggregate

$$\begin{array}{@{}rcl@{}} L_{CES,t}=\left[ sy_{L_{L}}^{^{\frac{1}{\sigma_{L}}}}\left( ef_{L_{L}} LY_{L,t}\right) ^{\frac{\sigma_{L}-1}{\sigma_{L}}}+sy_{L_{H}}^{^{\frac {1}{\sigma_{L}}}}\left( ef_{L_{H}}LY_{H,t}\right) ^{\frac{\sigma_{L} -1}{\sigma_{L}}}\right] ^{\frac{\sigma_{L}}{\sigma_{L-1}}} \end{array} $$

(12)

• L _{C E S} CES aggregator of L _L andL _H

• $sy_{L_{L}}$ Share of unskilled workers

• $ef_{L_{L}}$ Efficiency of unskilled employed workers

• L Y _L Inputs of permanent unskilled workers

• σ _L Elasticity of substitution, employed workers

• $sy_{L_{H}}$ Share of skilled workers

• $ef_{L_{H}}$ Efficiency of skilled employed workers

• L Y _H Inputs of permanent of permanent skilled workers

This is equation (5) of the main text.

Self-employed and Atypical Labour CES Aggregate

$$\begin{array}{@{}rcl@{}} N_{CES,t}=\left[ sy_{N_{S}}^{^{\frac{1}{\sigma_{N}}}}\left( ef_{N_{S}} NY_{S,t}\right) ^{\frac{\sigma_{N}-1}{\sigma_{N}}}+sy_{N_{A}}^{\frac {1}{\sigma_{N}}}\left( ef_{N_{A}}NY_{A,t}\right) ^{\frac{\sigma_{N} -1}{\sigma_{N}}}\right] ^{\frac{\sigma_{N}}{\sigma_{N-1}}} \end{array} $$

(13)

• N _{C E S} CES aggregator of N _S andN _A

• $sy_{N_{s}}$ Share of self-employed workers

• $ef_{N_{s}}$ Efficiency of self-employed workers

• N Y _S Inputs of self-employed workers

• σ _N Elasticity of substitution, self-employed and atypical workers

• $sy_{N_{A}}$ Share of atypical workers

• $ef_{N_{A}}$ Efficiency of atypical workers

• N Y _A Inputs of atypical workers

This is equation (6) of the main text.

Demand of Permanent Skilled Labour

$$\begin{array}{@{}rcl@{}} WR_{t}^{L_{H}}\left( 1-sub_{t}^{L_{H}}+\tau_{f,t}^{SSC^{L_{H}}}\right) &=&\alpha_{L}MC_{t}\frac{Y_{t}}{L_{CES,t}-OH^{L}}sy_{L_{H}}^{\frac{1}{\sigma _{L}}}\left( ef_{L_{H}}\right) ^{\frac{\sigma_{L}-1}{\sigma_{L}}}\left( \frac{L_{CES,t}}{LY_{H,t}}\right) ^{\frac{1}{\sigma_{L}}}\\ && -\gamma_{L_{H}}\left( \frac{LY_{H,t}}{LY_{H,t-1}}-1\right) Y_{t}\frac {1}{LY_{H,t-1}}\\ && +\beta\frac{\lambda_{t+1}^{R}}{{\lambda_{t}^{R}}}\gamma_{L_{H}}\left( \frac{LY_{H,t+1}}{LY_{H,t}}-1\right) Y_{t+1}\frac{LY_{H,t+1}}{LY_{H,t}^{2}} \end{array} $$

(14)

• W R ^L _H Real wage of skilled workers

• L Y _H Inputs of permanent skilled workers

• $sub_{t}^{L_{H}}$ Wage subsidy, skilled

• $\tau _{f}^{SSC^{L_{H}}}$ Social security contributions levied on firms, skilled

• O H ^L Overhead labour

• α _L Production function parameter,

• L _L and L _H workers

• MC Marginal cost

• Y Output

• L _{C E S} CES aggregator of L _L andL _H

• $sy_{L_{H}}$ Employment share of skilled workers

• $ef_{L_{H}}$ Efficiency of skilled employed workers

• λ ^R Lagrange multiplier - Ricardian households

• σ _L Elasticity of substitution, employed workers

• $\gamma _{L_{H}}$ Adjustment costs parameter, skilled employed workers

• β Discount factor

In steady state this equation reads $WR^{L_{H}}\left (1-sub^{L_{H}}+\tau _{f}^{SSC^{L_{H}}}\right ) \frac {1} {MC}=\alpha _{L}\frac {Y_{t}}{L_{CES}-OH^{L}}sy_{L_{H}}^{\frac {1}{\sigma _{L}}} \left (ef_{L_{H}}\right ) ^{\frac {\sigma _{L}-1}{\sigma _{L}}}\left (\frac {L_{CES}}{LY_{H}}\right ) ^{\frac {1}{\sigma _{L}}}$

which is equivalent to equation (7) of the main text,

s u b ^L _H =0,M U _P =M C ⁻¹ and $MPL^{L_{H}}=\alpha _{L} \frac {Y_{t}}{L_{CES}-OH^{L}}sy_{L_{H}}^{\frac {1}{\sigma _{L}}}\left (ef_{L_{H}}\right ) ^{\frac {\sigma _{L}-1}{\sigma _{L}}}\left (\frac {L_{CES}} {LY_{H}}\right ) ^{\frac {1}{\sigma _{L}}},$ for S=L _H .

Demand of Unskilled Employed Labour

$$\begin{array}{@{}rcl@{}} WR_{t}^{L_{L}}\left( 1-sub_{t}^{L_{L}}+\tau_{f,t}^{SSC^{L_{L}}}\right) &=&\alpha_{L}MC_{t}\frac{Y_{t}}{L_{CES,t}-OH^{L}}sy_{L_{L}}^{\frac{1}{\sigma _{L}}}\left( ef_{L_{L}}\right) ^{\frac{\sigma_{L}-1}{\sigma_{L}}}\left( \frac{L_{CES,t}}{LY_{L,t}}\right) ^{\frac{1}{\sigma_{L}}}\\ && -\gamma_{L_{L}}\left( \frac{LY_{L,t}}{LY_{L,t-1}}-1\right) Y_{t}\frac {1}{LY_{L,t-1}}\\ && +\beta\frac{\lambda_{t+1}^{R}}{{\lambda_{t}^{R}}}\gamma_{L_{L}}\left( \frac{LY_{L,t+1}}{LY_{L,t}}-1\right) Y_{t+1}\frac{LY_{L,t+1}}{LY_{L,t}^{2}} \end{array} $$

(15)

• W R ^L _L Real wage of skilled workers

• L Y _L Inputs of permanent unskilled workers

• $sub_{t}^{L_{L}}$ Wage subsidy, unskilled

• $\tau _{f}^{SSC^{L_{L}}}$ Social security contributions levied on firms, unskilled

• O H ^L Overhead labour cost

• α _L Production function parameter, L _L and L _H workers

• MC Marginal cost

• Y Output

• L _{C E S} CES aggregator of L _L and L _H

• $sy_{L_{L}}$ Employment share of skilled workers

• $ef_{L_{L}}$ Efficiency of skilled employed workers

• λ ^R Lagrange multiplier - Ricardian households

• σ _L Elasticity of substitution, employed workers

• $\gamma _{L_{L}}$ Adjustment costs parameter, skilled employed workers

• β Discount factor

In steady state this equation reads

$WR^{L_{L}}\left (1-sub^{L_{L}}+\tau _{f}^{SSC^{L_{L}}}\right ) \frac {1} {MC}=\alpha _{L}\frac {Y_{t}}{L_{CES}-OH^{L}}sy_{L_{L}}^{\frac {1}{\sigma _{L}}} \left (ef_{L_{L}}\right ) ^{\frac {\sigma _{L}-1}{\sigma _{L}}}\left (\frac {L_{CES}}{LY_{L}}\right ) ^{\frac {1}{\sigma _{L}}}$

which is equivalent to equation (7) of the main text,

s u b ^L _L =0,M U _P =M C ⁻¹and $MPL^{L_{L}}=\alpha _{L} \frac {Y_{t}}{L_{CES}-OH^{L}}sy_{L_{L}}^{\frac {1}{\sigma _{L}}}\left (ef_{L_{L}}\right ) ^{\frac {\sigma _{L}-1}{\sigma _{L}}}\left (\frac {L_{CES}} {LY_{L}}\right ) ^{\frac {1}{\sigma _{L}}},$ for S=L _L .

Demand of Self-employed Labour

$$\begin{array}{@{}rcl@{}} WR_{t}^{N_{S}} & =&\alpha_{N}MC_{t}\frac{Y_{t}}{N_{CES,t}-OH^{N}}sy_{N_{S}} ^{\frac{1}{\sigma_{N}}}\left( ef_{N_{S}}\right) ^{\frac{\sigma_{N} -1}{\sigma_{N}}}\left( \frac{N_{CES,t}}{NY_{S,t}}\right) ^{\frac{1} {\sigma_{N}}}\\ & &-\gamma_{N_{S}}\left( \frac{NY_{S,t}}{NY_{S,t-1}}-1\right) Y_{t}\frac {1}{NY_{S,t-1}}\\ && +\beta\frac{\lambda_{t+1}^{R}}{{\lambda_{t}^{R}}}\gamma_{N_{S}}\left( \frac{NY_{S,t+1}}{NY_{S,t}}-1\right) Y_{t+1}\frac{NY_{S,t+1}}{NY_{S,t}^{2}} \end{array} $$

(16)

• W R ^N _S Real wage of self-employed workers

• N Y _S Inputs of self-employed workers

• α _N Production function parameter, N _S and N _A workers

• O H ^N Overhead labour cost

• MC Marginal cost

• Y Output

• N _{C E S} CES aggregator of N _S and N _A

• $sy_{N_{s}}$ Employment share of self-employed workers

• $ef_{N_{s}}$ Efficiency of self-employed workers

• λ ^R Lagrange multiplier - Ricardian households

• σ _N Elasticity of substitution, N _S and N _A workers

• $\gamma _{N_{S}}$ Adjustment costs parameter, self-employed workers

• β Discount factor

In steady state this equation reads

$WR^{N_{S}}\frac {1}{MC}=\alpha _{N}MC\frac {Y}{N_{CES}-OH^{N}}sy_{N_{S}} ^{\frac {1}{\sigma _{N}}}\left (ef_{N_{S}}\right ) ^{\frac {\sigma _{N}-1} {\sigma _{N}}}\left (\frac {N_{CES}}{NY_{S}}\right ) ^{\frac {1}{\sigma _{N}}}$

which is equivalent to equation (7) of the main text,

M U _P =M C ⁻¹and $MPL^{N_{S}}=\alpha _{N}MC\frac {Y}{N_{CES}-OH^{N}} sy_{N_{S}}^{\frac {1}{\sigma _{N}}}\left (ef_{N_{S}}\right ) ^{\frac {\sigma _{N}-1}{\sigma _{N}}}\left (\frac {N_{CES}}{NY_{S}}\right ) ^{\frac {1}{\sigma _{N}}},$ for S=N _S .

Demand of Atypical Labour

$$\begin{array}{@{}rcl@{}} WR_{t}^{N_{A}}\left( 1-sub_{t}^{N_{A}}+\tau_{f,t}^{SSC^{N_{A}}}\right) &=&\alpha_{N}MC_{t}\frac{Y_{t}}{N_{CES,t}-OH^{N}}sy_{N_{A}}^{\frac{1}{\sigma _{N}}}\left( ef_{N_{A}}\right) ^{\frac{\sigma_{N}-1}{\sigma_{N}}}\left( \frac{N_{CES,t}}{NY_{A,t}}\right) ^{\frac{1}{\sigma_{N}}}\\ && -\gamma_{N_{A}}\left( \frac{NY_{A,t}}{NY_{A,t-1}}-1\right) Y_{t}\frac {1}{NY_{A,t-1}}\\ && +\beta\frac{\lambda_{t+1}^{R}}{{\lambda_{t}^{R}}}\gamma_{N_{A}}\left( \frac{NY_{A,t+1}}{NY_{A,t}}-1\right) Y_{t+1}\frac{NY_{A,t+1}}{NY_{A,t}^{2}} \end{array} $$

(17)

• W R ^N _A Real wage of atypical workers

• N Y _A Inputs of atypical workers

• α _N Production function parameter, N _S and N _A workers

• O H ^N Overhead labour cost

• s u b ^N _A Wage subsidy, atypical

• MC Marginal cost

• Y Output

• $\tau _{f}^{SSC^{N_{A}}}$ Social security contributions levied on firms, atypical

• N _{C E S} CES aggregator of N _S and N _A

• $sy_{N_{A}}$ Employment share of atypical workers

• $ef_{N_{A}}$ Efficiency of atypical workers

• λ ^R Lagrange multiplier - Ricardian households

• σ _N Elasticity of substitution, N _S and N _A workers

• $\gamma _{N_{A}}$ Adjustment costs parameter, atypical workers

• β Discount factor

Insteady state this equation reads $WR^{N_{A}}\frac {1}{MC}=\alpha _{N} MC\frac {Y}{N_{CES}-OH^{N}}sy_{N_{A}}^{\frac {1}{\sigma _{N}}}\times \left (ef_{N_{A}} \right ) ^{\frac {\sigma _{N}-1}{\sigma _{N}}}\left (\frac {N_{CES}}{NY_{A}} \right ) ^{\frac {1}{\sigma _{N}}}$, which is equivalent to equation (7) of the main text, where s u b ^N _A =0, M U _P =M C ⁻¹ and $MPL^{N_{S}}=\alpha _{N}MC\frac {Y}{N_{CES}-OH^{N}}sy_{N_{A}}^{\frac {1}{\sigma _{N}}}\times \left (ef_{N_{A}}\right ) ^{\frac {\sigma _{N}-1}{\sigma _{N}}}\left (\frac {N_{CES}} {NY_{A}}\right ) ^{\frac {1}{\sigma _{N}}}$, for S=N _A .

Equilibrium in the Labour Market, Unskilled Employed Workers

$$\begin{array}{@{}rcl@{}} LY_{L,t}=s_{L_{L}}L_{L,t} \end{array} $$

(18)

• L Y _L Inputs of unskilled workers

• $s_{L_{L}}$ Population share of unskilled employed workers

• L _L Unskilled employment

Equilibrium in the Labour Market, Skilled Employed Workers

$$\begin{array}{@{}rcl@{}} LY_{H,t}=s_{L_{H}}L_{H,t} \end{array} $$

(19)

• L Y _H Inputs of skilled workers

• $s_{L_{H}}$ Population share of skilled employed workers

• L _H Skilled employment

Equilibrium in the labour Market, Self-employed Workers

$$\begin{array}{@{}rcl@{}} NY_{S,t}=s_{N_{S}}N_{S,t} \end{array} $$

(20)

• N Y _S Inputs of self-employed workers

• $s_{N_{S}}$ Population share of self-employed workers

• N _S Self-employed employment

Equilibrium in the labour Market, Atypical Workers

$$\begin{array}{@{}rcl@{}} NY_{A,t}=s_{N_{A}}N_{A,t} \end{array} $$

(21)

• N Y _A Inputs of atypical workers

• $s_{N_{A}}$ Population share of atypical workers

• N _A Atypical employment

Physical Capital Accumulation Equation

$$\begin{array}{@{}rcl@{}} K_{t+1}=\Psi_{0,t}\left( \left( 1-\delta_{K}\right) K_{t}+I_{t}\right) \end{array} $$

(22)

• K Capital

• δ _K Depreciation rate of K

• I Investments

• Ψ₀ Capital quality

Investment Equation

$$\begin{array}{@{}rcl@{}} \Psi_{0,t}q_{t}-1=\gamma_{I}\left( \frac{I_{t}}{K_{t}}-\delta_{K}\right) -tc{r_{t}^{K}} \end{array} $$

(23)

• K Capital

• δ _K Depreciation rate of K

• I Investments

• q Tobin’s marginal q

• γ _I Adjustment costs parameter, investments

• t c r ^K Tax credit

• Ψ₀ Capital quality

Tobin’s q

$$\begin{array}{@{}rcl@{}} q_{t} & =&\beta E_{t}\frac{\lambda_{t+1}^{R}}{{\lambda_{t}^{R}}}\frac{\Pi _{t+1}^{I}}{\Pi_{t+1}}\left[ (1\!-\!\tau_{t+1}^{K})r_{t+1}^{K}u_{t+1}^{K} +\tau_{t+1}^{K}u_{t+1}^{K}\delta_{K}+\Psi_{0,t+1}q_{t+1}\left( 1-\delta _{K}\right) \right] \!\!-\beta E_{t}\frac{\lambda_{t+1}^{R}}{{\lambda_{t}^{R}}}\frac{\Pi_{t+1}^{I}} {\Pi_{t+1}}\\ & &\times \left[ \frac{\gamma_{I}}{2}\left( \frac{I_{t+1}}{K_{t+1}} -\delta_{K}\right) ^{2}-\gamma_{I}\left( \frac{I_{t+1}}{K_{t+1}}-\delta _{K}\right) \frac{I_{t+1}}{K_{t+1}}+\gamma_{{u_{1}^{K}}}\left( u_{t+1} ^{K}-1\right) +\frac{\gamma_{{u_{2}^{K}}}}{2}\left( u_{t+1}^{K}-1\right) ^{2}\right] \end{array} $$

• λ ^R Lagrange multiplier - Ricardian households

• β Discount factor

• δ _K Depreciation rate of capital

• I Investments

• π Inflation factor

• K Capital

• q Tobin’s marginal q

• τ ^K Capital tax rate

• γ _I Adjustment costs parameter, investments

• $\gamma _{{u_{2}^{K}}}$ Adjustment costs parameter, Capacity utilization of capital

• u ^K Capacity utilization of capital

• t c r ^K Tax credit

• r ^K Rental rate of K

• π^I Investment goods inflation

• Ψ₀ Capital quality

The Demand of Capital

$$\begin{array}{@{}rcl@{}} {r_{t}^{k}}{u_{t}^{K}}=\frac{P_{t}}{{P_{t}^{I}}}\left( 1-\alpha_{L}-\alpha _{N}\right) MC_{t}\frac{Y_{t}}{K_{t}} \end{array} $$

(25)

• r ^k Rental rate of K

• P Domestic production price level

• P ^I=P ^C Investments price index

• α _L Production function parameter, L _L and L _H workers

• α _N Production function parameter, N _S and N _A workers

• u ^K Capacity utilization of capital

• MC Marginal cost

• Y Output

• K Capital

Inflation Equation (New Keynesian Phillips Curve)

$$\begin{array}{@{}rcl@{}} &&Y_{t}-\gamma_{P}\left( \frac{\Pi_{t}}{indexatio{n_{t}^{P}}}-1\right) Y_{t}\frac{\Pi_{t}}{indexatio{n_{t}^{P}}}+\notag\\ &&+\beta\gamma_{P}E_{t}\frac{\lambda_{t+1}^{R}}{{\lambda_{t}^{R}}}\left( \frac{\Pi_{t+1}}{indexation_{t+1}^{P}}-1\right) Y_{t+1}\frac{\Pi_{t+1}} {indexation_{t+1}^{P}}=\left( 1-MC_{t}\right) \theta_{Y}Y_{t}\\ \end{array} $$

(26)

• γ _P Adjustment costs parameter, price

• π Inflation factor

• i n d e x a t i o n ^P Price indexation

• β Discount factor

• λ ^R Lagrange multiplier - Ricardian households

• MC Marginal cost

• Y Output

• 𝜃 _Y Elasticity of substitution intermediate goods

Real Profits

$$\begin{array}{@{}rcl@{}} PRO_{t} & =&Y_{t}-WR_{t}^{L_{L}}\left( 1-sub_{t}^{L_{L}}+\tau_{f,t} ^{SSC^{L_{L}}}\right) LY_{L,t}-WR_{t}^{L_{H}}\left( 1-sub_{t}^{L_{H}} +\tau_{f,t}^{SSC^{L_{h}}}\right) LY_{H,t}\\ & &-WR_{t}^{N_{S}}NY_{S,t}-WR_{t}^{N_{A}}\left( 1-sub_{t}^{N_{A}}+\tau _{f,t}^{SSC^{N_{A}}}\right) NY_{A,t}\\ & &-\frac{{P_{t}^{I}}}{P_{t}}{r_{t}^{K}}{u_{t}^{K}}K_{t}-\frac{\gamma_{P}}{2}\left( \frac{\Pi_{t}}{indexatio{n_{t}^{P}}}-1\right) ^{2}Y_{t}\\ && -\frac{\gamma_{L_{H}}}{2}\left( \frac{LY_{H,t}}{LY_{H,t-1}}-1\right) ^{2}Y_{t}-\frac{\gamma_{L_{L}}}{2}\left( \frac{LY_{L,t}}{LY_{L,t-1}} -1\right) ^{2}Y_{t}\\ && -\frac{\gamma_{N_{S}}}{2}\left( \frac{NY_{S,t}}{NY_{S,t-1}}-1\right) ^{2}Y_{t}-\frac{\gamma_{N_{A}}}{2}\left( \frac{NY_{A,t}}{NY_{A,t-1}} -1\right) ^{2}Y_{t} \end{array} $$

(27)

• Y Output

• W R ^L _L Real wage of unskilled workers

• W R ^L _H Real wage of skilled workers

• W R ^N _S Real wage of self-employed workers

• W R ^N _A Real wage of atypical workers

• r ^K Nominal rental rate of K

• K Capital

• $sub_{t}^{L_{L}}$ Wage subsidy, unskilled

• $\tau _{f}^{SSC^{L_{L}}}$ Social security contributions levied on firms, skilled

• $sub_{t}^{L_{H}}$ Wage subsidy, skilled

• $\tau _{f}^{SSC^{L_{H}}}$ Social security contributions levied on firms, unskilled

• s u b ^N _A Wage subsidy, atypical

• $\tau _{f}^{SSC^{N_{A}}}$ Social security contributions levied on firms, atypical

• γ _{p x} Adjustment costs parameter, price

• u ^K Capacity utilization of capital

• P ^I=P ^C Investments price index

• P Domestic production price level

• π Inflation factor

• $\gamma _{L_{H}}$ Adjustment costs parameter, skilled employed workers

• $\gamma _{L_{L}}$ Adjustment costs parameter, unskilled employed workers

• $\gamma _{N_{S}}$ Adjustment costs parameter, self-employed workers

• $\gamma _{N_{A}}$ Adjustment costs parameter, atypical workers

• L Y _H Inputs of skilled workers

• L Y _L Inputs of unskilled workers

• N Y _S Inputs of self-employed workers

• N Y _A Inputs of atypical workers

• i n d e x a t i o n ^P Price indexation

• PRO Profits, total

Accumulation of Public Capital

$$\begin{array}{@{}rcl@{}} KG_{t+1}=IG_{t}+\left( 1-\delta_{KG}\right) KG_{t} \end{array} $$

(28)

• KG Public capital

• IG Public investments

• δ _{K G} Depreciation rate of KG

Flow Budget Constraint of the Government in Real Terms

$$\begin{array}{@{}rcl@{}} BR_{t} & =&\frac{R_{t-1}}{\Pi_{t}}BR_{t-1}+\frac{{P_{t}^{C}}}{P_{t}}G_{t} +\frac{{P_{t}^{I}}}{P_{t}}IG_{t}+Tr_{t}+\\ && -TAX_{t}-\left( LTAX_{t}+TVAT_{t}+KTAX_{t}+PROTAX_{t}\right) +\\ & &+SUB_{t} \end{array} $$

(29)

• BR Real public debt

• R Nominal interest rate factor

• π Inflation factor

• P ^C Domestic consumption price index

• P Domestic good price level

• G Public expenditure

• IG Public Investment

• Tr Transfers

• TAX Lump-sum tax revenues

• LTAX Total labour tax revenues

• TVAT Total VAT revenues

• KTAX Total capital tax revenues

• PROTAX Tax on profits

• SUB Subsidies, Total

Transfers

$$\begin{array}{@{}rcl@{}} Tr_{t}=s_{NRTR}Tr_{t}^{NR}+(1-s_{NRTR})T{r_{t}^{R}} \end{array} $$

(30)

• Tr Transfers

• s _{N R} Share of non-Ricardian households

• T r ^NR Transfers to non-Ricardian households

• T r ^R Transfers to Ricardian households

Labour Taxes and Social Security Contributions

$$\begin{array}{@{}rcl@{}} LTAX_{t} & =s_{L_{L}}L_{L_{,t}}WR_{t}^{L_{L}}\left( \tau_{t}^{L_{L}} +\tau_{h,t}^{SSC^{L_{L}}}+\tau_{f,t}^{SSC^{L_{L}}}\right) +s_{L_{H}} L_{_{H},t}WR_{t}^{L_{H}}\left( \tau_{t}^{L_{H}}+\tau_{h,t}^{SSC^{L_{H}}} +\tau_{f,t}^{SSC^{L_{H}}}\right) \\ & +s_{N_{S},t}N_{_{S},t}WR_{t}^{N_{S}}\left( \tau_{t}^{N_{S}}+\tau _{h,t}^{SSC_{N_{s}}}\right) +s_{N_{A}}N_{_{A,t}}WR_{t}^{N_{A}}\left( \tau_{t}^{N_{A}}+\tau_{h,t}^{SSC_{N_{A}}}+\tau_{f}^{SSC_{N_{A}}}\right) \end{array} $$

(31)

• LTAX Total labour tax revenues

• $s_{L_{L}}$ Population share of unskilled employed workers

• L _L Employment of unskilled workers

• W R ^L _L Real wage of skilled workers

• τ ^L _L Average tax rate on labour income of permanent unskilled workers

• $\tau _{h}^{SSC^{L_{L}}}$ Social security contributions on unskilled workers

• $\tau _{f}^{W^{L_{L}}}$ Social security contributions levied on firms, atypical, unskilled

• $s_{L_{H}}$ Population share of skilled employed workers

• L _H Skilled employment

• W R ^L _H Real wage of skilled employed workers

• τ ^L _H Average tax rate on labour income of permanent skilled workers

• $\tau _{h}^{SSC^{L_{H}}}$ Social security contributions on skilled workers

• $\tau _{f}^{SSC^{L_{H}}}$ Social security contributions levied on firms, atypical, skilled

• $s_{N_{S}}$ Population share of self-employed workers

• N _S Employment of self-employed workers

• W R ^N _S Real wage of self-employed workers

• τ ^N _s Average tax rate on labour income of self-employed workers

• $\tau _{h}^{SSC^{N_{S}}}$ Social security contributions on self-employed workers

• $s_{N_{A}}$ Population share of atypical workers

• N _A Employment of atypical workers

• W R ^N _A Real wage of atypical workers

• $\tau _{h}^{SSC_{N_{A}}}$ Social contributions rate of atypical workers

• $\tau _{f}^{SSC_{N_{A}}}$ Social security contributions levied on firms, atypical, atypical

• τ ^N _A Average tax rate on labour income of atypical workers

Consumption Taxes

$$\begin{array}{@{}rcl@{}} TVAT_{t}={\tau_{t}^{C}}\frac{{P_{t}^{C}}}{P_{t}}\left[ s_{NR}C_{t}^{NR} +(1-s_{NR}){C_{t}^{R}}\right] \end{array} $$

(32)

• TVAT Total VAT revenues

• τ ^C Consumption tax (VAT)

• s _{N R} Share of non-Ricardian households

• C ^NR Consumption- non-Ricardian households

• C ^R Consumption- Ricardian households

• P ^C Domestic consumption price index

• P] Domestic production price level

Capital Taxes Nnet of Tax Credit

$$\begin{array}{@{}rcl@{}} KTAX_{t}=\frac{{P_{t}^{I}}}{P_{t}}{\tau_{t}^{K}}\left( {r_{t}^{K}}-\delta ^{K}\right) {u_{t}^{K}}K_{t}-tc{r_{t}^{K}}\frac{{P_{t}^{I}}}{P_{t}}I_{t} \end{array} $$

(33)

• KTAX Total capital tax revenues

• τ ^K Capital tax rate

• K Capital

• r ^K Nominal rental rate of K

• δ ^K Depreciation rate of K

• u ^K Capacity utilization of capital

• P ^I=P ^C Investments price index

• P Domestic good price level

• I Investments

• t c r ^K Tax credit

Fiscal Rule

$$\begin{array}{@{}rcl@{}} TAX_{t}=\overline{TAX}+T_{B}(BR_{t}-\overline{BR})+T_{D}(DR_{t}-\overline {DR})+T_{Y}\left( Y_{t}-Y_{t-1}\right) \end{array} $$

(34)

• TAX Lump-sum tax revenues

• $\overline {TAX}$ Tax rule shock variable

• D B R Difference variable, Real public debt

• T _B Tax rule parameter, public debt

• BR Real public debt

• $\overline {BR}$ Real public debt target

• T _D Tax rule parameter, public deficit

• DR Real public deficit

• $\overline {DR}$ Real public deficit target

• T _Y Tax rule parameter, output

• Y Output

which corresponds to Eq. 7 of the paper.

Lump-sum Taxes Levied on Ricardian Households

$$\begin{array}{@{}rcl@{}} TA{X_{t}^{R}}=\left( 1-s_{TAX}^{NR}\right) TAX_{t} \end{array} $$

(35)

• T A X ^R Lump-sum tax revenues levied on Ricardian households

• $s_{TAX}^{NR}$ Lump-sum tax share levied on non-Ricardian households

• TAX Lump-sum tax revenues

Lump-sum Taxes Levied on Non-Ricardian Households

$$\begin{array}{@{}rcl@{}} TAX_{t}^{NR}=s_{TAX}^{NR}TAX_{t} \end{array} $$

(36)

• T A X ^NR Lump-sum tax revenues levied on non-Ricardian households

• $s_{TAX}^{NR}c$ Lump-sum tax share levied on non-Ricardian households

• TAX Lump-sum tax revenues

Real Deficit

$$\begin{array}{@{}rcl@{}} DR_{t}&=&\frac{R_{t-1}-1}{\Pi_{t}}BR_{t-1}+\frac{{P_{t}^{C}}}{P_{t}}G_{t} +\frac{{P_{t}^{I}}}{P_{t}}IG_{t}+Tr_{t}-TAX_{t}\\&&-\left(LTAX_{t}+CTAX_{t} +KTAX_{t}+PROTAX_{t}\right) +SUB_{t}\\ \end{array} $$

(37)

• DR Real public deficit

• R Nominal interest rate factor

• π Inflation factor

• BR Real public debt

• ${P_{t}^{I}}=P^{C}$ Domestic consumption price index

• P Domestic production price level

• G Public expenditure

• IG Public Investment

• Tr Transfers

• TAX Lump-sum tax revenues

• LTAX Total labour tax revenues

• TVAT Total VAT revenues

• KTAX Total capital tax revenues

• PROTAX Tax on profits

• SUB] Subsidies, Total

Resource Constraint of the Economy

$$\begin{array}{@{}rcl@{}} Y_{t} & \!=\!&\frac{{P_{t}^{C}}}{P_{t}}\left( G_{t}+C_{t}\right) +\frac{P_{t} ^{I}}{P_{t}}\left( I_{t}+IG_{t}\right) +\frac{S_{t}P_{X,t}}{P_{t}} EXP_{t}-\frac{P_{M,t}}{P_{t}}IMP_{t}\\ & &+\frac{\gamma_{px}}{2}\left( \frac{\Pi_{t}}{indexatio{n_{t}^{P}}}-1\right) ^{2}Y_{t}+\frac{\gamma_{I}}{2}\frac{{P_{t}^{I}}}{P_{t}}\left( \frac{I_{t}} {K_{t}}-\delta_{K}\right) ^{2}K_{t}+\frac{\gamma_{L_{H}}}{2}\left( \frac{LX_{H,t}}{LX_{H,t-1}}-1\right) ^{2}X_{t}\\ & &+\frac{\gamma_{L_{L}}}{2}\left( \frac{LX_{L,t}}{LX_{L,t-1}}-1\right) ^{2}Y_{t}+\frac{\gamma_{N_{S}}}{2}\left( \frac{NX_{S,t}}{NX_{S,t-1}} -1\right) ^{2}Y_{t}+\frac{\gamma_{N_{A}}}{2}\left( \frac{NX_{A,t}} {NX_{A,t-1}}-1\right) ^{2}Y_{t}\\ & &+\frac{\gamma_{W^{L_{L}}}}{2}\!\left( \frac{WR_{t}^{L_{L}}}{indexation_{t} ^{W}\times WR_{t-1}^{L_{L}}}\Pi_{t}\!-\!1\right) ^{2}Y_{t}\!+\!\frac{\gamma _{W^{L_{H}}}}{2}\!\left( \frac{WR_{t}^{L_{H}}}{indexatio{n_{t}^{W}}\!\times\! WR_{t-1}^{L_{H}}}\Pi_{t}\!-\!1\right) ^{2}Y_{t}\\ & &+\frac{\gamma_{W^{N_{S}}}}{2}\left( \frac{WR_{t}^{N_{S}}}{indexation_{t} ^{W}\times WR_{t-1}^{N_{S}}}\Pi_{t}-1\right) ^{2}Y_{t}\!+\!\frac{\gamma_{IMP}} {2}\frac{P_{M,t}}{P_{t}}\left( \frac{\Pi_{t}^{IMP}}{indexation_{t}^{IMP}} \!-\!1\right) ^{2}IMP_{t}\\ & &+\frac{\gamma_{EXP}}{2}\frac{S_{t}P_{X,t}}{P_{t}}\!\left( \frac{\Pi _{t}^{EXP}}{indexation_{t}^{EXP}}-1\right) ^{2}EXP_{t}\!+\!\frac{{P_{t}^{I}}} {P_{t}}\left[ \gamma_{{u_{1}^{K}}}\left( {u_{t}^{K}}-1\right) +\frac {\gamma_{{u_{2}^{K}}}}{2}\left( {u_{t}^{K}}-1\right) ^{2}\right] K_{t} \\\end{array} $$

(38)

• Y Output

• σ _c Elasticity of substitution between domestic and foreign goods

• G Public expenditure

• IG Public Investment

• C Aggregate consumption

• I Investments

• EXP, IMP Exports, Imports

• π,π^IMP,π^EXP Inflation factors

• δ _K Depreciation rate of K

• γ _{p x} Adjustment costs parameter, price

• ${P,P}_{M}{,P}^{I}{,P}^{C}{,P}_{X}$ Prices

• ${\gamma }_{L_{H}}{,\gamma }_{L_{L}}$ Adjustment costs parameter, skilled and unskilled employed workers

• ${WR}^{N_{s}},{WR}^{N_{A}}$ Real wage of self-employed and atypical workers

• ${\gamma }_{N_{S}}{,\gamma }_{N_{A}}$ Adjustment costs parameter, self-employed and a typical workers

• K Capital

• γ _I Adjustment costs parameter, investments

• ${\gamma }_{{u_{1}^{K}}}{,\gamma }_{{u_{2}^{K}}}$ Adjustment costs parameters, Capacity

• ${\gamma }_{W^{L_{L}}}{,\gamma }_{W^{L_{H}}}$ Adjustment costs parameter, wage of unskilled and skilled employed workers

• S Nominal exchange rate

• γ _{I M P} ,γ _{E X P} Adjustment costs parameter, import and export

• u ^K Capital capacity utilization

• ${WR}^{L_{L}}{,WR}^{L_{H}}$ Real wage of unskilled and skilled employed workers

• L Y _H ,L Y _L Inputs of skilled and unskilled workers

• N Y _S ,N Y _A Inputs of self-employed and atypical workers

• i n d e x a t i o n ^EXP,i n d e x a t i o n ^IMP Export and Import Indexation

• i n d e x a t i o n ^W Wage Indexation

Taylor Rule

$$\begin{array}{@{}rcl@{}} \frac{R_{t}}{\overline{R}}=\left( \frac{R_{t-1}}{\overline{R}}\right) ^{\iota_{r}}\left[ \left( \frac{\Pi_{t}}{\overline{\Pi}}\right) ^{\iota_{\pi}}\left( \frac{Y_{t}}{Y_{t-1}}\right) ^{\iota_{y}}\left( \frac{S_{t}}{\overline{S}}\right) ^{\iota_{s}}\right] ^{1-\iota_{r}} \end{array} $$

(39)

• R Nominal interest rate factor

• $\overline {R}$ Logn-run interest rate factor

• π Inflation factor

• $\overline {\Pi }$ Target inflation

• Y Output

• S Nominal exchange rate

• $\overline {S}$ Nominal exchange rate target

• ι _r Taylor rule parameter, interest

• ι _π Taylor rule parameter, inflation

• ι _Y Taylor rule parameter, output

• ι _s Taylor rule parameter, exchange rate

Indexation - Prices

$$\begin{array}{@{}rcl@{}} indexatio{n_{t}^{P}}=\Pi_{t-1}^{\kappa_{p}}\overline{\Pi}^{1-\kappa_{p}} \end{array} $$

(40)

• $indexatio{n_{t}^{P}}$ price indexation

• κ _p Backward indexation, prices

• π Inflation factor

• $\overline {\Pi }$ Target inflation

Indexation - Wages

$$\begin{array}{@{}rcl@{}} indexatio{n_{t}^{W}}=\Pi_{t-1}^{\kappa_{W}}\overline{\Pi}^{1-\kappa_{W}} \end{array} $$

(41)

• i n d e x a t i o n ^W Wage indexation

• κ _W Backward indexation, wage

• π Inflation factor

• $\overline {\Pi }$ Target inflation

Welfare Function of Ricardian Households

$$\begin{array}{@{}rcl@{}} Welfar{e_{t}^{R}} & =&\log\left( {C_{t}^{R}}-h_{C^{R}}\overline{C}_{t-1} ^{R}\right) +\frac{s_{N_{S}}}{1-s_{NR}}\frac{\omega_{N_{s}}}{1-v_{N_{s}}} \left( 1-N_{S,t}\right) ^{1-v_{N_{s}}}\notag\\ &&+\frac{s_{L_{L}}}{1-s_{NR}} \frac{\omega_{L_{L}}}{1-v_{L_{L}}}\left( 1-L_{L,t}\right) ^{1-v_{L_{L}}} +\frac{s_{L_{H}}}{1-s_{NR}}\frac{\omega_{L_{H}}}{1-v_{L_{H}}}\left( 1-L_{H,t}\right) ^{1-v_{L_{H}}}\\ & &+\beta E_{t}Welfare_{t+1}^{R} \end{array} $$

(42)

• $Welfar{e_{t}^{R}}$ Welfare Ricardian households

• C ^R Consumption - non-Ricardian households

• $\omega _{N_{S}},\omega _{L_{L}},\omega _{L_{H}}$ Preference parameters

• N _S ,L _L ,L _H Employment

• $v_{N_{S}},v_{L_{L}},v_{L_{H}}$ Preference parameters

• h _C ^R Habit parameter, Ricardian households

• β Discount factor

• $s_{N_{S}}$ Share of self-employed workers

• $s_{L_{L}}$ Share of unskilled employed

• s _{N R} Share of non-Ricardian households

• $s_{L_{H}}$ Share of skilled employed

Welfare Function of Non-Ricardian Households

$$\begin{array}{@{}rcl@{}} Welfare_{t}^{NR}\!=\!\log(C_{t}^{NR}\!-\!h_{C^{NR}}\overline{C}_{t-1}^{NR} )+\frac{s_{N_{A}}}{s_{NR}}\frac{\omega_{N_{A}}}{1-v_{N_{A}}}\left( 1\!-\!N_{A,t}\right) ^{1-v_{N_{A}}}+\beta E_{t}Welfare_{t+1}^{NR} \end{array} $$

(43)

• W e l f a r e ^NR Welfare non-Ricardian households

• $C_{t}^{NR}$ Non-Ricardian consumption

• h _C ^NR Habit parameter, non-Ricardian households

• $v_{N_{A}}$ Preference parameter, atypical workers

• N _A Employment of atypical workers

• $\omega _{N_{A}}$ Preference parameter, atypical workers

• β Discount factor

• s _{N R} Share of non-Ricardian households

• $s_{N_{A}}$ Share of atypical workers

Total welfare

$$\begin{array}{@{}rcl@{}} Welfare_{t}=s_{NR}Welfare_{t}^{NR}+(1-s_{NR})Welfar{e_{t}^{R}} \end{array} $$

(44)

• Welfare Total welfare

• W e l f a r e ^R Welfare Ricardian households

• s _{N R} Share of non-Ricardian households

• W e l f a r e ^NR Welfare non-Ricardian households

Imports Demand

$$\begin{array}{@{}rcl@{}} IMP_{t}=\alpha_{IMP}\left( \frac{{P_{t}^{M}}}{{P_{t}^{C}}}\right) ^{-\sigma _{IMP}}(C_{t}+I_{t}+G_{t}+IG_{t}) \end{array} $$

(45)

• IMP Imports

• α _{I M P} Share of foreign goods in total consumption

• σ _{I M P} Elasticity of substitution between domestic and foreign goods

• P ^M Import price level

• P ^C Domestic consumption price index

• G Public expenditure

• IG Public Investment

• C Aggregate consumption

• I Investments

Exports Demand

$$\begin{array}{@{}rcl@{}} EXP_{t}=\alpha_{EXP}\left( \frac{{P_{t}^{X}}}{P_{t}^{C^{\ast}}}\right) ^{-\sigma_{EXP}}WD_{t} \end{array} $$

(46)

• EXP Export demand

• α _{E X P} Share of foreign goods in total consumption for the rest of the world

• P ^X Export price level

• $P_{t}^{C^{\ast }}$ Foreign consumption price index

• S Nominal exchange rate

• σ _{E X P} Elasticity of substitution between domestic and foreign goods in the rest of the world

• WD World demand

Imported Good Price Level

$$\begin{array}{@{}rcl@{}} {P_{t}^{M}}=\Pi_{t}^{IMP}P_{t-1}^{M} \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} P^{M} && \text{Import price level}\\ \Pi^{IMP} && \text{Nominal exchange rate} \end{array} $$

Domestic Production Price Level

$$\begin{array}{@{}rcl@{}} P_{t}=\Pi_{t}P_{t-1} \end{array} $$

(48)

• P Domestic good price level

• π Inflation factor

Foreign Production Price Level

$$\begin{array}{@{}rcl@{}} P_{t}^{\ast}=\Pi_{t}^{\ast}P_{t-1}^{\ast} \end{array} $$

(49)

• P ^∗ Foreign production price level

• π^∗ Foreign Inflation factor

Foreign Consumption Price Index

$$\begin{array}{@{}rcl@{}} P_{t}^{C^{\ast}}=\Pi_{t}^{C^{\ast}}P_{t-1}^{C^{\ast}} \end{array} $$

(50)

• $P^{C^{\ast }}$ Foreign consumption price index

• $\Pi ^{C^{\ast }}$ Foreign consumption Inflation factor

Domestic Consumption Price Index

$$\begin{array}{@{}rcl@{}} P_{C,t}\equiv\left[ (1-\alpha_{IMP})P_{t}^{1-\sigma_{IMP}}+\alpha_{IMP} {}\left( {P_{t}^{M}}\right) ^{1-\sigma_{IMP}}\right] ^{\frac{1} {1-\sigma_{IMP}}} \end{array} $$

(51)

• P ^C Domestic consumption price index

• P Domestic production price level

• α _{I M P} Share of foreign goods in total consumption

• P ^M Import price level

• σ _{I M P} Elasticity of substitution between domestic and foreign goods

Euler Equation on Foreign Assets

$$\begin{array}{@{}rcl@{}} S_{t}{\lambda_{t}^{R}}=\beta E_{t}\lambda_{t+1}^{R}\frac{R_{t}^{\ast}+rpbrf_{t}} {\Pi_{t+1}}S_{t+1} \end{array} $$

(52)

• λ ^R Lagrange multiplier - Ricardian households

• S Nominal exchange rate

• π Inflation factor

• R ^∗ Nominal interest rate on foreign assets

• β Discount factor

• rpbrf Risk premium on external debt

Trade Balance (% Output)

$$\begin{array}{@{}rcl@{}} TBY_{t}=\left( \frac{S_{t}{P_{t}^{X}}}{P_{t}}EXP_{t}-\frac{{P_{t}^{M}}}{P_{t}} IMP_{t}\right) /Y_{t} \end{array} $$

(53)

• TBY Trade balance as % of GDP

• P ^M Import price level

• P ^X Export price level

• S Nominal exchange rate

• EXP Exports

• IMP Imports

• P Domestic production price level

• Y Output

Terms of Trade

$$\begin{array}{@{}rcl@{}} TOT_{t}=\frac{S_{t}{P_{t}^{X}}}{{P_{t}^{M}}} \end{array} $$

(54)

• TOT Terms of trade

• P ^M Import price level

• P ^X Export price level

• S Nominal exchange rate

Current Account

$$\begin{array}{@{}rcl@{}} CA_{t}=\frac{S_{t}{P_{t}^{X}}}{P_{t}}EXP_{t}-\frac{{P_{t}^{M}}}{P_{t}} IMP_{t}+\frac{\left( R_{t-1}^{\ast}-1+rpbrf_{t-1}\right)} {\Pi_{t}} \frac{S_{t}}{S_{t-1}}BR_{t-1}^{F} \end{array} $$

(55)

• CA Current account

• EXP Exports

• IMP Imports

• P ^M Import price level

• P ^X Export price level

• P Domestic production price level

• rpbrf Risk premium on external debt

• S Nominal exchange rate

• R ^∗ Nominal interest rate on foreign assets

• π Inflation factor

• B R ^F Real foreign assets

Foreign Assets net Position in Real Terms

$$\begin{array}{@{}rcl@{}} B{R_{t}^{F}}=\frac{R_{t-1}^{\ast}+rpbrf_{t-1}}{\Pi_{t}}\frac{S_{t}}{S_{t-1}} BR_{t-1}^{F}+\frac{S_{t}{P_{t}^{X}}}{P_{t}}EXP_{t}-\frac{{P_{t}^{M}}}{P_{t} }IMP_{t} \end{array} $$

(56)

• B R ^F Real foreign assets

• EXP Exports

• IMP Imports

• P ^M Import price level

• P Domestic production price level

• rpbrf Risk premium on external debt

• S Nominal exchange rate

• P ^X Export price level

• R ^∗ Nominal interest rate on foreign assets

• π Inflation factor

• P _X Export good price level

Risk Premium

$$\begin{array}{@{}rcl@{}} rpbrf_{t}=-\varphi^{F}\left[ e^{\left( B{R_{t}^{F}}-BR^{F}\right)} -1\right] \end{array} $$

(57)

• rpbrf Risk premium on external debt

• B R ^F Real foreign assets

• φ ^F Risk parameter on external debt

CPI Inflation

$$\begin{array}{@{}rcl@{}} {\Pi_{t}^{C}}=\frac{{P_{t}^{C}}}{P_{t-1}^{C}} \end{array} $$

(58)

• π^C Consumption Inflation factor

• P ^C Domestic consumption price index

Investment Goods Price Level

$$\begin{array}{@{}rcl@{}} {P_{t}^{I}}={P_{t}^{C}} \end{array} $$

(59)

• ${P_{t}^{I}}$ Investment goods price level

• P ^C Domestic consumption price index

Investment Goods Inflation

$$\begin{array}{@{}rcl@{}} {\Pi_{t}^{I}}=\frac{{P_{t}^{I}}}{P_{t-1}^{I}} \end{array} $$

(60)

• ${\Pi _{t}^{I}}$ Investment goods inflation

• ${P_{t}^{I}}$ Investment goods price level

Export Price Level

$$\begin{array}{@{}rcl@{}} {P_{t}^{X}}=\Pi_{t}^{EXP}P_{X,t-1} \end{array} $$

(61)

• P ^X Export price level

• P i ^EXP Export inflation factor

Import Price Inflation

$$\begin{array}{@{}rcl@{}} {\lambda_{t}^{R}}\left[ \left( 1-\theta_{IMP}\right) \frac{{P_{t}^{M}}}{P_{t}} IMP_{t}-\gamma_{IMP}\left( \frac{\Pi_{t}^{IMP}}{indexation_{t}^{IMP}} -1\right) IMP_{t}\frac{\Pi_{t}^{IMP}}{indexation_{t}^{IMP}}\right.\\ \left.+\frac{S_{t} P_{t}^{\ast}}{P_{t}}\theta_{IMP}IMP_{t}\right] +\beta\gamma_{IMP}E_{t}\lambda_{t+1}^{R}\left( \frac{\Pi_{t+1}^{IMP}} {indexation_{t+1}^{IMP}}-1\right) IMP_{t+1}\frac{\Pi_{t+1}^{IMP}} {indexation_{t+1}^{IMP}}=0\\ \end{array} $$

(62)

• λ ^R Lagrange multiplier - Ricardian households

• 𝜃 _{I M P} Elasticity of import demand

• P ^M Import price level

• P Domestic production price level

• IMP Imports

• γ _{I M P} Adjustment costs parameter, import

• π^IMP Import goods inflation

• i n d e x a t i o n ^IMP Import indexation

• S Nominal exchange rate

• P ^∗ Foreign production price level

• β Discount factor

Export Price Inflation

$$\begin{array}{@{}rcl@{}} {\lambda_{t}^{R}}\left[\left( 1-\theta_{EXP}\right) \frac{S_{t}{P_{t}^{X}}} {P_{t}}EXP_{t}-\gamma_{EXP}\left( \frac{\Pi_{t}^{EXP}}{indexation_{t}^{EXP}} -1\right) EXP_{t}\frac{\Pi_{t}^{EXP}}{indexation_{t}^{EXP}}\right.\\\left.+\theta _{EXP}EXP_{t}{\vphantom{\left( 1-\theta_{EXP}\right)\frac{S_{t}{P_{t}^{X}}} {P_{t}}EXP_{t}-\gamma_{EXP}}}\right] +\beta E_{t}\lambda_{t+1}^{R}\gamma_{EXP}\left( \frac{\Pi_{t+1}^{EXP}} {indexation_{t+1}^{EXP}}-1\right) EXP_{t+1}\frac{\Pi_{t+1}^{EXP}} {indexation_{t+1}^{EXP}}=0\\ \end{array} $$

(63)

• λ ^R Lagrange multiplier - Ricardian households

• 𝜃 _{E X P} Elasticity of export demand

• P ^X Export price level

• P Domestic production price level

• EXP Exports

• γ _{E X P} Adjustment costs parameter, export

• π^EXP Export goods inflation

• i n d e x a t i o n ^EXP Export indexation

• S Nominal exchange rate

• P ^∗ Foreign final good price level

• β Discount factor

Import Price Indexation

$$\begin{array}{@{}rcl@{}} indexation_{t}^{IMP}=\left( \Pi_{t-1}^{IMP}\right) ^{\kappa_{IMP}}\left( \overline{\Pi}^{\ast}\right) ^{1-\kappa_{IMP}} \end{array} $$

(64)

• π^IMP Import goods inflation

• i n d e x a t i o n ^IMP Import indexation

• κ _{I M P} Backward indexation, export

• $\overline {\Pi }^{\ast }$ Foreign inflation factor

Export Price Indexation

$$\begin{array}{@{}rcl@{}} indexation_{t}^{EXP}=\left( \Pi_{t-1}^{EXP}\right) ^{\kappa_{EXP}}\left( \overline{\Pi}\right) ^{1-\kappa_{EXP}} \end{array} $$

(65)

• π^EXP Export goods inflation

• i n d e x a t i o n ^EXP Export indexation

• κ _{E X P} Backward indexation, export

• $\overline {\Pi }$ Target inflation

Capacity Utilization of Capital

$$\begin{array}{@{}rcl@{}} (1-{\tau_{t}^{K}}){r_{t}^{K}}+{\tau_{t}^{K}}\delta_{K}-\gamma_{{u_{1}^{K}}} -\gamma_{{u_{2}^{K}}}\left( {u_{t}^{K}}-1\right) =0 \end{array} $$

(66)

• τ ^K Capital tax rate

• r ^K Rental rate of K

• u ^K Capacity utilization of capital

• ${\gamma }_{{u_{1}^{K}}}$ Adjustment costs parameter, Capacity utilization of capital

• ${\gamma }_{{u_{2}^{K}}}$ Adjustment costs parameter, Capacity utilization of capital

• δ _K Depreciation rate of capital

Subsidies

$$\begin{array}{@{}rcl@{}} {}SUB_{t}\!=\!sub_{t}^{L_{L}}s_{L_{L}}L_{L,t}WR_{t}^{L_{L}}\!+\!sub_{t}^{L_{H}}s_{L_{H}} L_{H,t}WR_{t}^{L_{H}}\!+\!sub_{t}^{N_{A}}s_{N_{A}}N_{_{A},t}WR_{t}^{N_{A}} \end{array} $$

(67)

• SUB Subsidies, Total

• $s_{L_{L}}$ Population share of unskilled employed workers

• L _L Employment of unskilled workers

• W R ^L _L Real wage of skilled workers

• s u b ^L _L Wage subsidy, unskilled

• s u b ^L _H Wage subsidy, skilled

• $s_{L_{H}}$ Population share of skilled employed workers

• L _H Skilled employment

• W R ^L _H Real wage of skilled employed workers

• $s_{N_{A}}$ Population share of atypical workers

• N _A Employment of atypical workers

• W R ^N _A Real wage of atypical workers

• s u b ^N _A Wage subsidy, atypical

Tax on Profits

$$\begin{array}{@{}rcl@{}} PROTAX_{t}=\tau_{t}^{\Pr o}PRO_{t} \end{array} $$

(68)

• PROTAX Tax on profits, total

• $\tau ^{\Pr o}$ Tax rate on profits

• P R O Profits, total

Share of Non-Ricardian Households

$$\begin{array}{@{}rcl@{}} s_{NR}=s_{N_{A}} \end{array} $$

(69)

• s _{N R} Share of non-Ricardian households

• $s_{N_{A}}$ Share of atypical workers

Real exchange Rate

$$\begin{array}{@{}rcl@{}} RER_{t}=\frac{S_{t}P_{t}^{\ast}}{P_{t}} \end{array} $$

(70)

• RER Real exchange rate

• P Domestic production price level

• S Nominal exchange rate

• P ^∗ Foreign final good price level