2.1. Households

2.1.1. Unconstrained households

Unconstrained households are denoted by $i$ and their utility function writes:

(1) \begin{equation} E_{t}\sum _{k=0}^{\infty }\beta ^{k}\left ( \epsilon _{t}^{z}\ln c_{i,t+k}- \frac {l_{i,t+k}^{1+\eta }}{1+\eta }\right ) \end{equation}

with $\beta ^{k}$ the discount factor of households and $\eta$ the inverse of Frish elasticity of work effort and $\epsilon _{t}^{z}$ is a preference shock that affects consumption and follows an AR(1) process detailed in section 2.6. The utility depends positively on individual consumption $c_{i,t}$ and negatively on labor supply $l_{i,t}$ .

Unconstrained households face the following budget constraint:

(2) \begin{eqnarray} &&\left ( 1+\tau _{t}^{c}\right ) c_{i,t}+b_{i,t}+I_{g,t}^{s}\nonumber \\[6pt] &=&\left ( 1-\tau _{t}^{W}\right ) w_{i,t}l_{i,t}+\psi _{t}^{g}\frac { R_{t-1}^{ea}}{\pi _{c,t}}b_{i,t-1} \end{eqnarray}

where $w_{i,t}$ is the real wage. $I_{g,t}^{s}$ is the investment in public capital stock.Footnote ⁶ $\tau _{t}^{c},\tau _{t}^{W}$ are respectively taxes on consumption and wage and they are assumed to be exogenous and described in section 2.6. $b_{i,t}$ is the portfolio of sovereign bonds held by households. $\ R_{t}^{ea}$ is the risk free interest rate, $ \psi _{t}^{g}$ is the global risk premium attached to the households’ portfolio and $\psi _{t}^{g}\frac {R_{t-1}^{ea}}{P_{c,t}}$ can be interpreted as the global real return on portfolio of sovereign bonds.

When the recovery plan is European, such that each country borrows at the same rate, we assume that $\psi _{t}^{g}=1.$

When the recovery plan is national, we introduce a country-specific risk premium such that $\psi _{t}^{g}\neq 1$ , following the methodology outlined in Badarau et al. (Reference Badarau, Huart and Sangar2021).

In this specification, we assume that households hold a portfolio of sovereign securities composed of a proportion $b_{1}$ of domestic bonds ( $ b_{i,t}^{d}$ ) and $\left ( 1-b_{1}\right )$ of foreign bonds ( $b_{i,t}^{\ast }$ ).

(3) \begin{equation} b_{i,t}=\left ( b_{1}^{\frac {1}{\theta _{b}}}\left ( b_{i,t}^{d}\right ) ^{ \frac {\theta _{b}-1}{\theta _{b}}}+(1-b_{1})^{\frac {1}{\theta _{b}} }b_{i,t}^{\ast ^{\frac {\theta _{b}-1}{\theta _{b}}}}\right ) ^{^{\frac {\theta _{b}}{\theta _{b}-1}}} \end{equation}

$\theta _{b}\gt 0$ is the constant elasticity of substitution between domestic and foreign bonds.

For the sake of simplicity, we assume that the linear combination of bond quantities leads to the same linear combination of risk. The global risk premium of the portfolio is therefore an aggregation of domestic ( $\psi _{t}^{d}$ ) and foreign risk premia ( $\psi _{t}^{\ast }$ ).Footnote ⁷

(4) \begin{equation} \psi _{t}^{g}=\left ( b_{1}\left ( \psi _{t}^{d}\right ) ^{1-\theta _{b}}+(1-b_{1})\left ( \psi _{t}^{\ast }\right ) ^{1-\theta _{b}}\right ) ^{ \frac {1}{1-\theta _{b}}} \end{equation}

The optimization problem of unconstrained households with respect to $ c_{i,t}$ , $b_{i,t}$ and $l_{i,t}$ gives the following first-order conditions:

(5) \begin{equation} \lambda _{i,t}=\frac {\epsilon _{t}^{z}}{c_{i,t}\left ( 1+\tau _{t}^{c}\right ) } \end{equation}

(6) \begin{equation} \lambda _{i,t}=\frac {\beta E_{t}\lambda _{i,t+1}\psi _{t}^{g}R_{t}^{ea}}{\pi _{c,t+1}} \end{equation}

(7) \begin{equation} w_{i,t}=\frac {l_{i,t}^{\eta }}{\lambda _{i,t}\left ( 1-\tau _{t}^{w}\right ) } \end{equation}

where $\lambda _{i,t}$ is the Lagrangian multiplier of unconstrained households optimization problem. Note that equation 6 represents the uncovered interest parity condition.

2.1.2. Liquidity constrained households

Liquidity constrained households are denoted by $j$ and their utility function writes:

(8) \begin{equation} E_{t}\sum _{k=0}^{\infty }\beta ^{k}\left ( \epsilon _{t}^{z}\ln c_{j,t+k}- \frac {l_{j,t+k}^{1+\eta }}{1+\eta }\right ) \end{equation}

The utility depends positively on individual consumption $c_{j,t}$ and negatively on labor supply $l_{j,t}$ . Liquidity constrained households face the following budget constraint:

(9) \begin{equation} \left ( 1+\tau _{t}^{c}\right ) c_{j,t}=\left ( 1-\tau _{t}^{W}\right ) w_{j,t}l_{j,t} \end{equation}

where $w_{j,t}$ is the real wage.

The solution of the optimization problem of these households with respect to $c_{j,t}$ and $l_{j,t}$ gives the following first order conditions

(10) \begin{equation} \lambda _{j,t}=\frac {\epsilon _{t}^{z}}{c_{j,t}\left ( 1+\tau _{t}^{c}\right ) } \end{equation}

(11) \begin{equation} w_{j,t}=\frac {l_{j,t}^{\eta }}{\lambda _{j,t}\left ( 1-\tau _{t}^{w}\right ) } \end{equation}

where $\lambda _{j,t}$ is the Lagrangian multiplier of liquidity constrained households optimization problem.

The presence of liquidity constrained households helps to capture Keynesian effects of fiscal policy as the economy is populated by agents for which the Ricardian equivalence does not hold. We anticipate that the effect of fiscal policy will be larger when the share of liquidity constrained households in the economy $\left ( 1-\mu \right )$ is itself larger.

2.2. Firms

2.2.1. Intermediate firms

Entrepreneurs produce intermediate goods under perfect competition regime. Each domestic intermediate goods firm produces goods $y_{t}$ that are sold domestically and abroad following a production function close to Leeper et al. (Reference Leeper, Walker and Yang2010):

(12) \begin{equation} y_{t}=T_{t}\left ( k_{t-1}^{\alpha }l_{f,t}^{1-\alpha }k_{g,t-1}^{\alpha _{g}}\right ) \end{equation}

where $k_{t-1}$ is productive capital rented by capital producer, $l_{f,t}$ is the aggregated labor demand such that $l_{f,t}=l_{i,t}^{\mu }l_{j,t}^{1-\mu }$ with $\mu$ the share of unconstrained households in the production function. $k_{g,t-1}$ is the aggregated public capital, $\alpha$ is the share of private capital in the production function and $\alpha _{g}$ is the elasticity of output with respect to public capital. $T_{t}$ is a permanent technology shock that follows an AR(1) process detailed in section 2.6.

Entrepreneurs denoted by $e$ , maximize their utility, which depends only on consumption, according to the following objective function

(13) \begin{equation} \mathbb{E}_{0}\sum _{k=0}^{\infty }\beta _{e}^{k}\ln \left ( {c_{e,t+k}} \right ), \end{equation}

where $c_{e,t}$ denotes entrepreneurs’ consumption, and $\beta _{e}$ the entrepreneurs’ discount factor. Entrepreneurs are assumed to discount the future more heavily than households such that their discount factor $\beta _{e}$ is lower than the households’ ( $\beta _{e}\lt \beta$ ).

Entrepreneurs’ decisions are taken according to the following budget constraint

(14) \begin{equation} c_{e,t}+w_{j,t}l_{j,t}+w_{i,t}l_{i,t}+R_{k,t}k_{t}=\frac {y_{t}}{x_{t}} +R_{k,t}\left ( 1-\delta _{k}\right ) k_{t-1}, \end{equation}

where $R_{k,t}$ is the nominal market price of capital and $x_{t}$ the markup of final over intermediate goods and $\delta _{k}$ is the depreciation rate of capital.

The optimality conditions of the entrepreneurs after maximization of their utility subject to their budget constraint and production function are:

(15) \begin{equation} \frac {1}{c_{e,t}}R_{k,t}=\beta _{e}\mathbb{E}_{t}\left [ \frac {1}{c_{e,t+1}} \left ( \alpha \frac {y_{t+1}}{x_{t+1}k_{t}}+R_{k,t+1}\left ( 1-\delta _{k}\right ) \right ) \right ] \end{equation}

(16) \begin{equation} w_{i,t}=\frac {\mu \left ( 1-\alpha \right ) }{l_{i,t}}\frac {y_{t}}{x_{t}}, \end{equation}

(17) \begin{equation} w_{j,t}=\frac {\left ( 1-\mu \right ) \left ( 1-\alpha \right ) }{l_{j,t}}\frac { y_{t}}{x_{t}}. \end{equation}

2.2.2. Capital good producer

Capital producers are in a competitive market. Their aim is to produce new capital and to sell it to entrepreneurs at the nominal market price $R_{k}$ .

Following Gerali et al. (Reference Gerali, Neri, Sessa and Signoretti2010), capital producers buy an amount $I_{t}$ of final good at the beginning of each period and the stock of old non-depreciated capital $\left ( 1-\delta _{k}\right ) k_{t-1}$ from entrepreneurs. Old capital can be converted one to one into new capital. We assume quadratic adjustment costs. Finally, the amount that capital good producers can produce is:

(18) \begin{equation} k_{t}=(1-\delta _{k})k_{t-1}+\left [ 1-\frac {\kappa _{k}}{2}\left ( \frac { \epsilon _{t}^{qk}i_{t}}{i_{t-1}}-1\right ) ^{2}\right ] I_{t} \end{equation}

where $I_{t}$ is the investment, $\kappa _{k}$ is the adjustment cost of a change in investment and $\epsilon _{t}^{qk}$ is a shock to the efficiency of investment, which follows an AR(1) process, detailed in section 2.6.

The profit maximization of the capital good producers delivers a dynamic equation for the real price of capital similar to Smets and Wouters (Reference Smets and Wouters2003, Reference Smets and Wouters2007):

(19) \begin{eqnarray} 1 &=&q_{k}\left ( 1-\frac {\kappa _{k}}{2}\left ( \frac {\epsilon _{t}^{qk}I_{t} }{I_{t-1}}-1\right ) ^{2}\right ) -\kappa _{k}\left ( \frac {\epsilon _{t}^{qk}I_{t}}{I_{t-1}}-1\right ) \left ( \frac {\epsilon _{t}^{qk}I_{t}}{ I_{t-1}}\right ) \notag \\[6pt] && +\, \beta _{e}\left ( \frac {c_{e,t+1}}{c_{e,t}}\right ) ^{-\sigma _{e,t}}R_{k,t+1}\kappa _{k}\left ( \frac {\epsilon _{t+1}^{qk}I_{t+1}}{I_{t}} -1\right ) \left ( \frac {\epsilon _{t+1}^{qk}I_{t+1}}{I_{t}}\right ) ^{2} \end{eqnarray}

2.2.3. Retailers

To motivate sticky prices, we assume implicit cost of adjusting nominal prices and monopolistic competition at retail level.

Retailers in a one-country model

A continuum of retailers of mass 1, indexed by $z$ buy intermediate goods $ y_{t}$ from entrepreneurs at a wholesale price $P_{t}^{w}$ , differentiate them at no cost and sell differentiated goods at a retail price $P_{t}$ .

Under this assumption, $x_{t}=P_{t}/P_{w,t}$ denotes the markup of final goods over that of intermediate goods.

Retailers $z$ bundle intermediate goods $y_{t}$ according to the following CES technology

(20) \begin{equation} y_{t}=\left [ \int _{0}^{1}y_{t}\left ( z\right ) ^{\frac {\epsilon _{t}-1}{ \epsilon _{t}}}dz\right ] ^{\frac {\epsilon _{t}}{\epsilon _{t}-1}}, \end{equation}

where $\epsilon _{t}$ is the elasticity of substitution between intermediate goods, detailed in Section 2.6.

Given the aggregate output index (Eq. 20), the price index is $P_{t}$ is

(21) \begin{equation} P_{t}=\left [ \int _{0}^{1}P_{t}\left ( z\right ) ^{1-\epsilon _{t}}dz\right ] ^{ \frac {1}{1-\epsilon _{t}}}, \end{equation}

such that each retailer faces an individual demand curve:

(22) \begin{equation} y_{t}\left ( z\right ) =\left ( \frac {P_{t}\left ( z\right ) }{P_{t}}\right ) ^{-\epsilon _{t}}y_{t}. \end{equation}

Retailers choose $P_{t}\left ( z\right )$ to maximize

(23) \begin{equation} \mathbb{E}_{t}\sum _{k=0}^{\infty }\Lambda _{t,k}\left [ P_{t}\left ( z\right ) y_{t}\left ( z\right ) -P_{t}^{w}y_{t}\left ( z\right ) -\frac {\kappa _{p}}{2} \left ( \frac {P_{t}\left ( z\right ) }{P_{t-1}\left ( z\right ) }-\pi _{t-1}^{\iota _{p}}\pi ^{1-\iota _{p}}\right ) ^{2}P_{t}y_{t}\right ], \end{equation}

where $\Lambda _{t,k}=\beta _{p}U_{c,t+k}/U_{c,t}$ denotes the stochastic discount factor, assuming the demand curve (Eq. 22) and wholesale price $P_{t}^{w}$ as given.

The optimality condition associated with the retailers’ problem is

(24) \begin{equation} \begin{array}{c} 1-\epsilon _{t}+\frac {\epsilon _{t}}{X_{t}}-\kappa _{p}\left ( \pi _{t}-\pi _{t-1}^{\iota _{p}}\pi ^{1-\iota _{p}}\right ) \pi _{t} \\[10pt] +\beta _{p}\left ( \frac {\epsilon _{z,t+1}}{\epsilon _{z,t}}\frac {c_{i,t}}{ c_{i,t+1}}\right ) \kappa _{p}\left ( \pi _{t+1}-\pi _{t}^{\iota _{p}}\pi ^{1-\iota _{p}}\right ) \pi _{t+1}\frac {y_{t+1}}{y_{t}}=0. \end{array} \end{equation}

In our model, a positive shock to $\epsilon _{t}$ leads to a decrease in the optimal value of markups, which can be interpreted as a negative price markup shock.

Retailers in a two-country model

A continuum of retailers of mass 1, indexed by $z$ buy intermediate goods $ y_{t}$ from entrepreneurs at a wholesale price $P_{t}^{w}$ , differentiate them at no cost and sell differentiated goods at a retail price $P_{h,t}(z),$ $P_{x,t}(z),$ $P_{im,t}(z)$ .

We assume that domestic and foreign retailers sell their differentiated good in both domestic and foreign markets with the same structure of price such that we get the same pricing equation with subscript $h$ , $x$ and $im$ for each differentiated goods ( $h_{t}(z),$ $EX_{t}(z),$ $IM_{t}(z)$ )

The price setting behavior of intermediate good firm follows an adjustment à la Calvo implying that each period, a proportion $\left ( 1-\zeta _{h}\right )$ of firms receive permission to optimally reset their prices and choose $\widetilde {P}_{h,t}$ . Retailers that receive permission to reset prices choose $\widetilde {P}_{h,t}$ that solve the maximization problem taking as given $P_{t}^{w}$ and demand of differentiated good $ h_{t}(z),$ $EX_{t}(z),$ $IM_{t}(z)$ (demand is given in section 2.2.3)

The FOC obtained from the optimal pricing behavior is:

(25) \begin{equation} \sum _{k=0}^{\infty }\zeta _{h}E_{t}\left [ \Lambda _{t+k}\left ( \frac { \widetilde {P}_{h,t}}{P_{h,t+k}}-\frac {x}{x_{t+k}}\right ) h_{t+k}\left ( z\right ) \right ] =0 \end{equation}

where $x_{t}$ is the markup $\left ( P_{h,t}/P_{t}^{w}\right )$ which in steady state is equal to $x_{t}=\epsilon /(\epsilon -1)$ , with $\epsilon$ the elasticity of substitution between intermediate goods.

The aggregate price index $P_{h,t}$ evolves according to:

(26) \begin{equation} P_{h,t}^{1-\epsilon }=\left ( 1-\zeta _{h}\right ) \widetilde {P} _{h,t}^{1-\epsilon }+\zeta _{h}\left ( \pi _{h,t-1}\right ) ^{1-\epsilon } \end{equation}

Final good firms

Final good are obtained by aggregating intermediate goods through a CES aggregation tool:

(27) \begin{equation} h_{t}=\left ( \int \limits _{0}^{1}h_{t}\left ( z\right ) ^{\frac {\epsilon -1}{ \epsilon }}dz\right ) ^{\frac {\epsilon }{\epsilon -1}} \end{equation}

The demand equation of differentiated good are determined by choosing the optimal use of differentiated goods minimizing the expenditure for the bundles of differentiated goods subject to aggregation constraint:

(28) \begin{equation} h_{t}\left ( z\right ) =\left ( \frac {P_{h,t}(z)}{P_{h,t}}\right ) ^{-\epsilon }h_{t} \end{equation}

and aggregate price index are:

(29) \begin{equation} P_{h,t}=\left ( \int \limits _{0}^{1}P_{h,t}(z)^{1-\epsilon }dz\right ) ^{\frac {1 }{1-\epsilon }} \end{equation}

Final goods firms combine purchases of the domestically produced intermediate goods with the imported intermediate goods. They combine it into three distinct non-tradable goods, namely final consumption good $q_{c}$ , investment good $q_{i}$ and public consumption good $g$ . They are characterized symmetrically such that we use the same equation for all final goods, only modifying the subscript.

The representative firm produces non-tradable final private consumption good $q_{c,t}$ , combining the purchase of a bundle of domestically produced intermediate goods $h_{c,t}$ and a bundle of foreign imported goods $ im_{c,t}$ using a constant return to scale CES technology:

(30) \begin{equation} q_{c,t}=\left [ \nu _{c,t}^{\frac {1}{\mu _{c}}}\left ( h_{c,t}\right ) ^{1- \frac {1}{\mu _{c}}}+\left ( 1-\nu _{c,t}\right ) ^{\frac {1}{\mu _{c}}}\left ( im_{c,t}\right ) ^{1-\frac {1}{\mu _{c}}}\right ] ^{\frac {\mu _{c}}{\mu _{c}-1}} \end{equation}

where $\mu _{c}$ denotes the intratemporal elasticity of substitution between the bundles of domestic and foreign intermediate goods and $\nu _{c,t}$ measures the weight of domestic tradable intermediate goods.

The final goods firm chooses a combination of domestic $h_{c,t}$ and imported goods $im_{c,t}$ that minimizes expenditure $ P_{h,t}h_{c,t}+P_{im_{c},t}im_{c,t}$ , subject to technology constraint Eq. 30 and taking the input price indices $P_{h,t}$ and $ P_{im_{c},t}$ as given. Prices evolve according to:

(31) \begin{equation} P_{c,t}=\left [ \nu _{c,t}P_{h,t}^{1-\mu _{c}}+\left ( 1-\nu _{c,t}\right ) P_{im_{c},t}^{1-\mu _{c}}\right ] ^{\frac {1}{\mu _{c}-1}} \end{equation}

2.3. Monetary policy

As in Christoffel et al. (Reference Christoffel, Coenen and Warne2008), the common monetary authority sets the nominal interest rate according to the following log-linear Taylor rule:

(32) \begin{equation} R_{t}^{ea}=\phi _{R}R_{t-1}^{ea}+\left ( 1-\phi _{R}\right ) \left [ R^{ea}+\phi _{\pi }\left ( \pi _{c,t}^{ea}-\pi _{c}^{ea}\right ) +\phi _{gy}\left ( y_{gr,t}^{ea}-1\right ) \right ] +\epsilon _{R,t} \end{equation}

where $y_{gr,t}^{ea}=\frac {Y_{t}^{ea}}{Y_{t-1}^{ea}}$ denotes the euro area output growth, $P_{c,t}^{ea}$ is the euro area gross inflation rate and $\pi _{c}^{ea}$ is the euro area gross inflation at its steady state value, $\phi _{R}$ captures the degree of interest rate smoothing, $\phi _{\pi }$ and $ \phi _{gy}$ are policy coefficients reflecting the weights of inflation and the output gap, respectively and $\epsilon _{R,t}$ is an exogenous monetary policy shock identical for all European countries, following an AR(1) process detailed in section 2.6.

2.4. Government

Government expenditures are split between government consumption $g_{t}$ (supposed constant unless a shock occurs) and investment in public capital $ I_{g,t}$ . Government expenditures are financed by a set of fiscal instruments allowing to collect taxes on consumption and on labor income. Government expenditures are also financed by debt contracted with households $b_{i,t}$ . The government real budget constraint writes:

(33) \begin{equation} taxes_{t}+b_{i,t}=g_{t}+I_{g,t}+\frac {\psi _{t}R_{t-1}^{ea}}{\pi _{c,t}} b_{i,t-1} \end{equation}

where $taxes_{t}$ denotes the total tax revenues:

(34) \begin{equation} taxes_{t}=\tau _{t}^{C}c_{t}+\tau _{t}^{w}\left ( w_{i,t}l_{i,t}+w_{j,t}l_{j,t}\right ) \end{equation}

$g_{t}$ is assumed to be exogenous such that:

(35) \begin{equation} g_{t}=g+\nu _{t}^{G} \end{equation}

with $\nu _{t}^{G}$ an exogenous shock of current expenditure described in section 2.6.

The borrowing rate associated with the issuance of sovereign debt is elastic to the level of current debt. Thus, the risk premium depends on the deviation of the debt-to-GDP ratio from its target:

(36) \begin{equation} \psi _{t}=\exp \left ( \psi _{n}\left ( b_{i,t-1}-\overline {b}p_{y}y\right ) \right ) \end{equation}

where $\psi _{n}$ is the sensitivity coefficient of the risk premium to public indebtedness and $\overline {b}$ is the steady-state target for the stock of real debt in proportion to GDP (Badarau et al. Reference Badarau, Huart and Sangar2021).

2.4.1. Fiscal policy

We assume that the government fulfills a fiscal rule that forces the primary structural balance to adjust in order to stabilize the long run debt-to-GDP ratio. The government’s primary structural balance ( $S_{Struct}$ ) is defined as:

(37) \begin{equation} Sstruc_{t}=\tau _{t}^{C}c+\tau _{t}^{w}\left ( w_{i}l_{i}+w_{j}l_{j}\right ) -g-I_{g} \end{equation}

where variables without time subscript relate to their steady state values. Shifts to public investment and public consumption are interpreted as structural shocks to public finances. Meanwhile, we assume, following LeMoigne et al. (Reference LeMoigne, Saraceno and Villemot2016), that the government is tied to the following golden rule of public finance:

(38) \begin{equation} Sstruc_{t}-\left ( \left ( \frac {\psi _{t}R_{t-1}^{ea}}{\pi _{c,t}}-1\right ) b_{i,t-1}\right ) =\phi _{b}\left ( b_{i,t-1}-\overline {b}p_{y}y\right ) -\epsilon _{t}^{G} \end{equation}

where $\phi _{b}$ is the sensitivity of lump-sum taxes to debt-to-output ratio and $\epsilon _{t}^{G}$ is an exogenous shock allowing for a temporary deviation from the fiscal rule.

2.4.2. Public investment

In order to align with the principles of NGEU, the European economic recovery plan centered around investment, our model must incorporate public investment. To achieve this, we adopt the framework introduced by Leeper et al. (Reference Leeper, Walker and Yang2010), which involves categorizing public expenditures into two distinct components: consumption and investment in public capital ( $I_{g,t}$ ).

Additionally, and still following Leeper et al. (Reference Leeper, Walker and Yang2010), $I_{g,t}$ stands for the implemented government investment which is different from authorized government investment $A_{t}.$ Leeper et al. (Reference Leeper, Walker and Yang2010) assume that government investment turns into public capital through a time-to-build process, such that the authorized government spending is not immediately implemented and productive.

The law of motion for public capital is:

(39) \begin{equation} k_{g,t}=(1-\delta _{g})k_{g,t-1}+A_{t-N} \end{equation}

where $N$ is the number of quarters between granting budget authority and completing a project, $\delta _{g}$ the depreciation rate of public capital, and $A_{t-N}$ the authorized government investment (the stock). The law of motion entails that public capital accumulates on a much longer scale of time (n) than private capital because the stock of public capital is useful only once the project is fully completed.

We add another hypothesis of Leeper et al. (Reference Leeper, Walker and Yang2010): spending outlays authorized by the government occur over several periods. Thus, we consider the spending rates $\{\phi _{0},\phi _{1},\ldots, \phi _{N-1}\}$ from the date the investment is authorized ( $t=0$ ) to the period before project completion ( $t=N-1$ ). Thus we have:

(40) \begin{equation} I_{g,t+1}=\sum _{n=0}^{N-1}\phi _{n}A_{t-n} \end{equation}

with $A_{t}$ decided by the government at each period, which follows an AR(1) process described in section 2.6.

Moreover, following LeMoigne et al. (Reference LeMoigne, Saraceno and Villemot2016), we take into account the possibility that private investment will crowd-in the projects. In order to model this, we force part of the Ricardian (unconstrained) agents’ savings into public capital stock in the form of a lump-sum tax ( $I_{g,t}^{s}$ ):

(41) \begin{equation} (1-\mu )I_{g,t}^{s}=Lp\times \left ( I_{t}^{S}-I^{S\ast }\right ) \end{equation}

with $Lp$ the amount of the private leverage effect.

This lump-sum tax is taken directly from the Ricardian agents’ savings, thus enter their budget constraint (Eq. 2) as well as the goods’ market clearing condition. It is directed into the public capital stock:

(42) \begin{equation} k_{g,t}=(1-\delta _{g})k_{g,t-1}+A_{t-(N-1)}+Lp\times \left ( A_{t-(N-1)}-A^{\ast }\right ) \end{equation}

2.5. Equilibrium and market clearing

Market clearing conditions write:

(43) \begin{equation} y_{t}=c_{,t}+I_{t}+g_{t}+I_{g,t}+I_{g,t}^{s}+\frac {1-s}{s}EX_{t}-IM_{t} \end{equation}

(44) \begin{equation} q_{c,t}=c_{t} \end{equation}

(45) \begin{equation} q_{i,t}=I_{t} \end{equation}

Aggregation equations are

(46) \begin{equation} w_{t}=\left ( 1-\mu \right ) w_{j,t}+\mu w_{i,t} \end{equation}

(47) \begin{equation} c_{t}=\left ( 1-\mu \right ) c_{j,t}+\mu c_{i,t}+c_{e,t} \end{equation}

(48) \begin{equation} h_{t}=h_{c,t}+h_{i,t}+g_{t}+I_{g,t}+I_{g,t}^{s} \end{equation}

(49) \begin{equation} im_{t}=im_{c,t}+im_{i,t} \end{equation}

2.6. Stochastic structure

The structural shocks are assumed to follow a first-order autoregressive functional form:

(50) \begin{equation} X_{t}=\left ( 1-\rho _{X}\right ) \overline {X}+\rho _{X}X_{t-1}+\eta _{t}^{X}, \end{equation}

where $X_{t}\in \left \{ \epsilon _{t}^{z},T_{t},\tau _{t}^{c},\tau _{t}^{W},\epsilon _{t}^{G},A_{t},\nu _{t}^{G}\right \}$ , $\overline {X}$ is the steady-state value of $X_{t}$ , $\rho _{X}\in \left [ 0,1\right [$ is the first-order autoregressive parameter of the shock $X_{t}$ and the innovation $\eta _{t}^{X}$ is an $i.i.d$ normal error term with zero mean and standard deviation $\sigma _{X}$ .

5.1. Public consumption shock

Whether the initial shock arises in the core (Figure 2) or the periphery (Figure 3), the rise in public consumption leads to an increase in output relative to its long-term equilibrium level. However, the effect on output is not persistent. Comparing the different funding frameworks shows that a recovery financed by grants (dotted line) has a larger effect on production than one financed by EU loans (dashed line) or national funding (solid line) in the country hit by the shock. Actually, EU or national loans require an increase in distortionary taxation which is absent if funding stems from a grant.

Figure 2.

Public consumption shock in the core—model with national stimulus, EU loans or grants.

Figure 3.

Public consumption shock in the periphery—model with national stimulus, EU loans or grants.

The growth effect after a fiscal stimulus funded with an EU grant increases tax receipts and leads to a reduction in public debt. According to the fiscal rule (Eq. 38), tax rates can decline and generate a crowding-in effect between public consumption and private investment in the country hit by the shock, despite rising policy rate. Conversely, the country which is not directly hit by the shock sees a reduction of private investment because of the rise in the policy rate following the shock in the neighbor country.

The comparison of the origin of the shock (Figure 2 vs. 3) reveals that the short-term growth impact of a public consumption shock is twice higher when it occurs in the core than in the periphery. A shock in the core shows an initial positive effect on the periphery that rapidly disappears. In contrast, a shock in the periphery immediately weighs on the core. With funding via grants, the lower public debt in the periphery limits the wealth effect in the core.

5.2. Public investment shock

The macroeconomic effects after a public investment shock (either in the core, see Figure 4, or in the periphery, see Figure 5) are similar to those detailed in the closed-economy framework of LeMoigne et al. (Reference LeMoigne, Saraceno and Villemot2016). By construction and in contrast with the shock on public consumption, the investment stimulus plan has persistent effects. The simulation of the public investment shock shows two phases. The first one corresponds to the time-to-build period of public capital. During this phase, public investment leads to an increase of aggregate demand, up to respectively $0.6\%$ (shock on the core) and $0.2\%$ (shock on the periphery) above the long-run equilibrium level, and to an increase of inflation. It triggers a reaction of monetary policy that hikes the nominal interest rate. This in turn dampens the expansionary effect of the initial shock. These effects explain the U-shaped GDP reaction function between quarters 1 and 12 following the shock. In the second phase of the shock, the investment effect disappears (explaining the sharp drop at $t=12$ , when GDP shortly becomes negative); however, the new infrastructures become productive, leading to a positive supply effect. Production remains for a very long time above its steady state value, by $0.2\%$ (regardless of the origin of the shock) as long as the new public capital is not completely depreciated. The increase in aggregate supply also leads to lower inflation, and therefore to a softened monetary policy.

Figure 4.

Public investment shock in the core—model with national stimulus, EU loans or grants.

Figure 5.

Public investment shock in the periphery—model with national stimulus, EU loans or grants.

In contrast with a shock on public consumption, the funding of a shock on public investment does not matter much, at least in the first phase. Actually, the dynamic responses of the output are very similar. The main difference lies in the evolution of public debt: by construction, it increases under the EU common loan framework whereas it declines under the grant framework. Overall, these results highlight how intricate the wealth effect might be with distortionary taxation. Under the loans framework, the rise of public debt which paves the way for the wealth effect is limited by the rise in distortionary taxation. The results also show that a national stimulus package will have smaller effects than a European stimulus package funded via NGEU. Since public investment is financed by new issuances of national public debt, risk premia increase and crowd-out private consumption and investment.

Whether implemented by the core or by the periphery, the investment stimulus plan has a positive overall effect on aggregate GDP. However, thanks to trade spillover effects, a recovery through investment in the periphery has beneficial effects in the short term in the core. Peripheral countries, on the other hand, have to wait longer to see the beneficial effects of an investment stimulus in the core, but then benefit from higher spillovers. This owes to the (realistic) assumption that the degree of openness of the core is smaller than that of the periphery.

5.3. The multiplier effects of the stimulus

The former sub-sections have highlighted the macroeconomic mechanisms triggered by the fiscal shocks. Now, a numerical analysis of fiscal multipliers helps to discriminate between the different features of a fiscal stimulus. The (undiscounted) cumulated multipliers of additional GDP after the different stimulus packages are reported in Tables 5 and 6 for both countries (core and periphery). Table 5 (resp. 6) reports a shock originating in the core (resp. the periphery). Hence, we not only report a proxy for the effectiveness of the fiscal shock on the country where the shock occurred but also the spillovers on the output of the other country.

Table 5.

Fiscal multipliers by type of stimulus and by type of funding—Shock on the core

Table 6.

Fiscal multipliers by type of stimulus and by type of funding—Shock on the periphery

While the multiplier effects might seem very large, they are not unusual in the literature. On impact, the cumulative multiplier effects are consistent with those of Pappa and Canova (Reference Pappa and Canova2021) after an investment shock funded via the European Regional Development Fund. After one year, they are close to those found by Bouakez et al. (Reference Bouakez, Guillard and Roulleau-Pasdeloup2017) in the US following the American Recovery and Reinvestment Act of 2009. In the longer run, Ramey (Reference Ramey2020) shows multiplier effects close to 9 or even 10 in the literature using New-Keynesian models. Pfeiffer et al. (Reference Pfeiffer, Varga and in ’t Veld2023) find lower multiplier effects than those reported in the table below because they include a third region to the model, the rest of the world. Hence, they report higher import leakages from a European fiscal stimulus than we do.

Reporting undiscounted fiscal multipliers has a possible important advantage in terms of the political economy of reforms. The use of a discount factor is plagued with uncertainty, a point well known and for long in the literature about investments with delayed effects in the area of climate policy (see, e.g., Newell and Pizer Reference Newell and Pizer2004). Unless the discount factor increases with the investment horizon (see, e.g., Gollier Reference Gollier2002), the use of a constant discount factor may act as a drag on (public) investment and may give incentive to prefer public consumption with more immediate effects.

From Tables 5 and 6, there are 6 main outcomes which hold similarly for both countries (core and periphery).

First, public investment shocks have a more substantial impact on GDP than public consumption shock. It appears so on impact, one year after the shock, and in the longer run when the time-to-build process has boosted the productive capacities of the different economies. This highlights the mixture of demand-driven and supply-driven features of public investment in contrast with public consumption and also the cost of distortionary taxation arising from a fiscal impetus on public consumption.

Second, after a shock on public investment, the impact on GDP is the highest for the country where the shock occurred if it is funded by EU loans until the "time-to-build” has occurred. After that, beyond 3 years, the impact on GDP is the highest when the fiscal impetus is funded by grants. During the first 3 years after the shock, the wealth effect from Ricardian consumers positively contributes to GDP.

Third, maximum spillovers from the fiscal impetus are achieved 3 years after the shock and when it is funded by EU loans. This confirms that wealth effects matter: EU loans, but not grants, lead to a rise in public debts that in turn generates some wealth effects for Ricardian households.

Fourth, the simulations give an appraisal of the real advantage of grants vs. loans. Actually, grants permit to escape a rise in taxes to fund the increase in public consumption and limit that of public debt. The opportunity costs of raising taxes after a shock to public consumption can be proxied by the difference between the multiplier effects when the shock is funded via grants or loans. For the aggregate area, it amounts to + 0.7 percentage points of GDP after 1 quarter if the shock has hit the core, and to + 0.4 pp if it has hit the periphery. It will climb to + 0.9 pp after 20 years in the first case, but decrease to + 0.25 pp in the second case.

Fifth, public investment multipliers also show that the NGEU program would give GDP more traction in both the short and long run compared to a similar fiscal shock financed by domestic debt. This is due to the effect of risk premia that adjust as a result of the increase in public debt. Despite the wealth effect, the rise in the risk premia and in debt after a domestic shock produces a decline in private investment that weighs on aggregate demand. The superiority of a European investment shock on a domestic investment shock shows only one exception: after 1 year, a shock on the periphery has a higher GDP spillover on the core if the shock has been a domestic one. This owes to the higher rise of peripheral debt that generates a higher wealth effect in the core, in contrast to a European-funded shock in the periphery. This effect is limited in time, though.

Sixth, NGEU grants are more important for public consumption than for public investment. Comparing cumulative fiscal multipliers after a shock on public consumption shows that they are much higher when spending is funded by grants rather than loans: while the short-term positive demand effect of the shock funded by a grant gives rise to a sharp increase in the nominal interest rate, the lower debt overturns the latter (the risk premium declines) which transmits into a lower real interest rate and a rise in private investment. In contrast, the rise in distortionary taxes when EU loans are used dampens the impact on output. The choice between loans and grants after a public investment shock appears less crucial (cumulative fiscal multipliers are very close): the mechanism of the golden rule of public finance which prevails when public investment increases prevents the rise in distortionary taxation that follows a rise in public consumption. The respective impacts on GDP of the wealth effect and higher taxation will somewhat neutralize: public debt and the wealth effect will be a bit higher under EU loans financing than under grants; meanwhile, higher taxation will also be a bit higher. All in all, whatever the EU financing instrument, a shock on public investment will produce almost the same monetary reaction. In contrast to a shock on public consumption, the rise in the nominal interest rate is gradual.

5.4. Sensitivity analysis

Results remain consistent across a range of plausible values for parameters. For instance, robustness analyses show that the values of the public investment multiplier are sensitive to the level of leverage, set at 2 in the benchmark case (see Table 7). This level could well be lower or higher, making the cumulative multiplier effects discussed so far either over- or under-estimated.

Table 7.

Fiscal multipliers of public investment financed by loans, by level of leverage

Moreover, fiscal multipliers are sensitive to the assumption about how productive public capital may be. A lower capital productivity reduces the fiscal multiplier of public investment in the long run (Table 8). Symmetrically, a lower time-to-build period of public investment increases the fiscal multiplier.

Table 8.

Fiscal multipliers of public investment financed by loans, for alternative calibrations

6.1. Fiscal stimulus, ZLB and loans

We analyze the path of nominal interest rate (Figure 6) and GDP (resp. Figure 7 on core and Figure 8 on periphery) when the economy is plunged into a post-Covid situation under three scenarios. In the first scenario (dotted line), there are no fiscal plans (or no fiscal reactions to the shocks). The second scenario (dashed line) implies a public investment recovery plan while the third scenario (solid line) is a public consumption recovery plan. In these simulations, all shocks are assumed to affect the core and the periphery in the same proportion and fiscal plans are financed by loans.Footnote ¹⁵

Figure 6.

Impact on interest rates of a fiscal shock in the context of a ZLB—Financing by loans.

The stimulus package, whether it is an increase in consumption or investment, leads to an increase in aggregate demand which contributes to increasing inflation and therefore triggers an intervention from monetary authorities via rising policy rates. The investment-led recovery allows for a larger and more long-lasting exit from the ZLB than a consumption-led policy since the latter also generates a rise in distortionary taxation.

Figure 7.

Impact on core GDP of a fiscal shock in the context of a ZLB—Financing by loans.

Whatever the nature of the EU recovery plan (public investment- or consumption-related), its implementation closes the output gap fairly quickly. However, at the end of the time-to-build period, or once public consumption spending is completed, GDP falls significantly. In the long run, GDP remains below its steady state if the recovery is carried out by public consumption. On the other hand, it is durably above its steady state with a recovery carried out by an investment stimulus. This latter outcome is consistent with Bouakez et al. (Reference Bouakez, Guillard and Roulleau-Pasdeloup2020).

Figure 8.

Impact on periphery GDP of a fiscal shock in the context of a ZLB—Financing by loans.

6.2. Fiscal stimulus, ZLB and grants

We run the same simulations as in the former subsection but we analyze the path of nominal interest rate (Figure 9) and GDP (resp. Figure 10 on core and Figure 11 on periphery) focusing on grants. The real impacts on the core and the periphery share similar shapes as those reported when the fiscal shock was funded by loans. However, a public consumption shock financed by a grant now shows a larger impact on GDP in the short run than a shock on public investment. It also shows a larger impact than the same shock funded by loans. The strong and immediate impact of public consumption coupled with the free-lunch opportunity offered by a grant triggers a sharper rise in the policy rate. The debt-economy provided by grants thus limits recourse to taxes while the ZLB is more rapidly overcome. This illustrates again the opportunity costs of loans vs. grants. In the longer run, a shock on public investment funded by grants has a higher impact on GDP, in comparison with a shock on public consumption: this highlights the supply-driven property of public investment in the long run.

Figure 9.

Impact on interest rates of a fiscal shock in the context of a ZLB—Financing by grants.

Figure 10.

Impact on core GDP of a fiscal shock in the context of a ZLB—Financing by grants.

6.3. Fiscal stimulus, the EU and the ZLB: a summary of results

Table 9 summarizes the main outcomes of the simulations of policy shocks under an extended-path approach. For each type of fiscal shock (on public investment or on public consumption) and each type of funding (loans or grants), Table 9 shows the extra-gain of the policy, including cross-spillovers, vis-à-vis no policy at all. In most cases, an EU-wide fiscal reaction to the Covid-like crisis brings positive and large output effects. Exception to this occurs when the core and peripheral countries target public consumption expenditures and fund fiscal policy via loans. The early rise in the nominal interest rate coupled with raising taxes slows down the pace of the recovery: the output effect peaks at + 0.3–0.4 pp after 3 years and turns negative in the very long run. In contrast, grants are shown to be superior to loans after the Covid-like crisis when they fund a shock on public consumption. In this case, output effects are larger than with EU loans. This result is consistent with those reported with stochastic simulations and a linear Taylor-type rule for monetary policy.

Table 9.

Cumulative effect on output of a European fiscal stimulus, in comparison with no fiscal stimulus, in percentage points of GDP

Figure 11.

Impact on periphery GDP of a fiscal shock in the context of a ZLB—Financing by grants.

There is a new result when the extended path approach is used and a ZLB/Covid-like situation is simulated. Now grants are surpassed by loans to fund public investment: whatever the horizon, the cumulative fiscal multipliers after a shock on public investment are always larger when they are backed by loans. In this situation, the wealth effect brings some additional traction on economic activity.