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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.

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Notes

  1. After the seminal papers by Smets and Wouters (2003, 2007) and Christiano et al. (2005), which have shown how the DGE modelling approach can be used as a fruitful tool for policy analysis, a number of central banks, ministries and international institutions have developed their own DGE model for the analysis of structural and fiscal reforms. Some examples include the International Monetary Fund’s Global Economy Model - GEM (see Bayoumi 2004), the New Area Wide Model - NAWM - of the European Central Bank (see Christoffel et al. 2008), the Euro Area and Global Economy - EAGLE - model (see Gomes et al. 2012), the variants of the QUEST model of the European Commission (e.g. Roeger et al. 2008, 2009)

  2. In general, it should be stressed that in the last years a greater attention has been addressed to the study of fiscal policy issues and of the related welfare implications for households in the context of open economies. In this respect, see Evers (2012), Kumhof and Laxton (2013), Hohberger et al. (2014) and Vogel et al. (2013). In particular, Hohberger et al. (2014) evaluate the stabilizing performance of rules adjusting the composition of public spending on tradable and non-tradable goods and compare the results with those obtained under a fiscal-devaluation feedback rule.

  3. In 2013 Italy recorded a total tax wedge as a percentage of labour costs of 47.8 %, with the share of employers’s SSC accounting for more than 50 % of this wedge. The average tax wedge for the OECD countries is about 35 %. See OECD (2014). The so-called atypical workers, belonging to the secondary market, accounts for more than 20 % of the employed.

  4. With the advent of the euro, similarly to Italy, all these countries experienced a sustained appreciation of their real exchange rates with only a slight correction with the recession (see IMF 2013b, 2013c, 2013d), and a deterioration of their external balance positions, with persistent current account deficits and increasing external liabilities. In addition, in all these economies the total tax wedge on labour cost is above 40 % (41.6 % for Greece, 41.1 % for Portugal, 40.7 % for Spain). In 2013 the share of employers’ SSC is above 20 % for Greece (21.5 %) and Spain (23 %), while for Portugal is 19.3 %. See OECD (2014). At the of 2013 according to the European Labour Force Survey, in EU28 74 % of workers can be defined as permanent, while in the countries of interest this share is much lower: 60 % for Greece, which is characterized by a high share of self-employed workers, 63 % for Spain and 66 % for Portugal.

  5. This point is relevant for those countries that show a high share of self-employed workers, as Greece, for instance, where the self-employed workers account for more than 30 % of the employed workforce.

  6. In the Technical Appendix we also report our results under the assumption that the conduct of monetary policy is described by a Taylor-type rule (see e.g. Taylor (1993) and Clarida et al. (1999)), allowing for a certain degree of inertia of the interest rate response to inflation and output and calibrated according to the fact that Italy belongs to a monetary union.

  7. The estimation is done using Dynare. For details, see http://www.cepremap.cnrs.fr/dynare/ and Adjemian et al. (2011).

  8. Non-stationary series have been detrended by taking first difference.

  9. In the Technical Appendix we show that the habit persistence parameters are crucial in shaping shape the response of the economy in the first quarters of our experiments. We find that initially the effects on aggregate consumption and, in some respect also on output, are slightly larger for a high habit persistence of non-Ricardian households.

  10. Deterministic simulations are generally carried out when studying the effects of structural and/or fiscal reforms involving permanent changes in some structural parameters and/or tax rates. For several examples of reform packages simulated adopting this solution method, see Roeger et al. (2008).

  11. This is usually the preferred strategy when dealing with large scale models. See Roeger et al. (2008) and (2009) for the QUEST III model.

  12. Notice that in this analysis we are comparing the initial baseline level of welfare with the one achieved in the long run following the fiscal devaluation reform. For an analysis of the stabilizing properties of a fiscal-devaluation feedback rule in response to fluctuations in trade balance, see Hohberger et al. (2014).

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Acknowledgments

We are very grateful to the editor of this journal, George S. Tavlas, and to three anonymous referees for their very valuable comments and suggestions. We would also like to thank Filippo Pericoli for helpful discussions and an anonymous referee of the working paper series of the Department of the Italian Treasury for their very useful comments on an earlier version of this paper. The usual disclaimer applies. The simulation tool used in this paper, IGEM, is currently managed at Sogei S.p.A. - IT Economia - Modelli di Previsione ed Analisi Statistiche. The views expressed herein are those of the authors and not necessarily reflect those of Sogei S.p.A. and of the Italian Ministry of Economy and Finance. A Technical Appendix is available on the corresponding author’s webpage.

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Correspondence to Barbara Annicchiarico.

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

  • πI 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,πIMPEXP 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)
  • πC 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

  • πIMP 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

  • πEXP 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)
  • πIMP 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)
  • πEXP 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

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Annicchiarico, B., Dio, F.D. & Felici, F. Fiscal Devaluation Scenarios: A Quantitative Assessment for the Italian Economy. Open Econ Rev 26, 731–785 (2015). https://doi.org/10.1007/s11079-014-9335-7

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  • DOI: https://doi.org/10.1007/s11079-014-9335-7

Keywords

  • Fiscal Devaluation
  • DGE
  • Fiscal Reforms
  • Italy

JEL Classification

  • E10
  • C50
  • E60