Households
$$ {C}_t^{OLG}= MP{C}_t\left( In{c}_t+\left(1+{r}_{t-1}\right){B}_{t-1}+\left(1+{r}_{t-1}^{\ast}\right)\frac{REE{R}_t}{REE{R}_{t-1}}{B}_{t-1}^{\ast}\right) $$
$$ In{c}_t=\left(1-{\tau}_t^L\right){w}_t\psi {S}_t+ Profi{t}_t-{T}_t+\left(1-\omega \right){E}_t\frac{In{c}_{t+1}}{1+{r}_t}\frac{1}{1+{g}_{t+1}^{N.}} $$
Capital producers
$$ {K}_t= In{v}_t+\left(1-\delta \right){K}_{t-1} $$
$$ \frac{Q_t}{p_t^{Inv}}=1+S\left(\frac{Inv_t}{Inv_{t-1}}\right)+{S}^{\prime}\left(\frac{Inv_t}{Inv_{t-1}}\right)\frac{Inv_t}{Inv_{t-1}}-\frac{1-\omega }{1+{r}_t}{E}_t\frac{p_{t+1}^{Inv}}{p_t^{Inv}}{S}^{\prime}\left(\frac{Inv_{t+1}}{Inv_t}\right){\left(\frac{Inv_{t+1}}{Inv_t}\right)}^2 $$
$$ re{t}_{t+1}=\frac{E_t\left(1-{\tau}_{t+1}^K\right){r}_{t+1}^K+{\tau}_{t+1}^K\delta {Q}_{t+1}+{Q}_{t+1}\left(1-\delta \right)}{Q_t} $$
Banks
$$ {N}_t=\theta \left[\;\left( re{t}_t-\left(1+{r}_{t-1}\right)\left(1+{\xi}_{t-1}^{EP}\right)\right)\frac{\eta_{t-1}}{\lambda -{v}_{t-1}}+\left(1+{r}_{t-1}\right)\left(1+{\xi}_{t-1}^{EP}\right)\right]{N}_{t-1}+{\omega}^{Bank}{Q}_t{K}_{t-1} $$
$$ {Q}_t{K}_t=\frac{1}{1-{\psi}_t}\frac{\eta_t}{\lambda -{v}_t}{N}_t $$
Intermediate firms
$$ \frac{\varphi }{\varphi -1}{mc}_t=1+\frac{1}{\varphi -1}R\left(\frac{1+{\hat{\pi}}_t}{{\left(1+{\hat{\pi}}_{t-1}\right)}^{\gamma }}\right)+\frac{1}{\varphi -1}{R}^{\prime}\left(\frac{1+{\hat{\pi}}_t}{{\left(1+{\hat{\pi}}_{t-1}\right)}^{\gamma }}\right)\frac{1+{\hat{\pi}}_t}{{\left(1+{\hat{\pi}}_{t-1}\right)}^{\gamma }}+\left(1-\omega \right){E}_t\frac{1}{\varphi -1}\frac{Y_{t+1}}{Y_t}\frac{R^{\prime}\left(\frac{1+{\hat{\pi}}_{t+1}}{{\left(1+{\hat{\pi}}_t\right)}^{\gamma }}\right)\left(\frac{1+{\hat{\pi}}_{t+1}}{{\left(1+{\hat{\pi}}_t\right)}^{\gamma }}\right)}{1+{r}_t} $$
Retailer firms
j ∈ {
C,
INv,
Gov,
X}:
$$ m{c}_t^j={\left[\mu +\left(1-\mu \right){\left( REE{R}_t{\tilde{p}}_t^{M,j}\right)}^{1-\kappa}\right]}^{\frac{1}{1-\kappa }} $$
$$ {Y}_t^j=\mu {\left(\frac{p_t^Y}{m{c}_t^j}\right)}^{-\kappa }{Z}_t^j $$
$$ \frac{\varphi_j}{\varphi_j-1}\frac{mc_t^j}{p_t^j}=1+\frac{1}{\varphi_j-1}R\left(1+{\hat{\pi}}_t^j\right)+\frac{1}{\varphi_j-1}{R}^{\prime}\left(1+{\hat{\pi}}_t^j\right)\frac{1+{\hat{\pi}}_t^j}{{\left(1+{\hat{\pi}}_{t-1}^j\right)}^{\vartheta }}-{E}_t\frac{1}{\varphi_j-1}\frac{1-\omega }{1+{r}_t}\frac{p_{t+1}^j}{p_t^j}\frac{Z_{t+1}^j}{Z_t^j}{R}^{\prime}\left(1+{\hat{\pi}}_{t+1}^j\right)\frac{1+{\hat{\pi}}_{t+1}^j}{{\left(1+{\hat{\pi}}_t^j\right)}^{\vartheta }} $$
$$ Profi{t}_t^j={p}_t^j{Z}_t^j-{Y}_t^j- REE{R}_t{M}_t^j\left(1+G\left(\cdotp \right)\right)-{p}_t^j{Z}_t^jR\left(\cdotp \right) $$
Government
$$ \mathit{\operatorname{Re}}{v}_t={\tau}_t^C{C}_t+\left({\tau}_t^L+{\tau}_t^S\right){w}_t{L}_t\left({\tau}_t^K{r}_t^K-{\tau}_t^K\delta {Q}_t\right){K}_{t-1}+{T}_t+ EU{F}_t $$
$$ \mathrm{E}x{p}_t=T{R}_t+{p}_t^{Gov} Go{v}_t+{p}_t^{Gov} In{v}_t^{Gov}+{\psi}_t{Q}_t{K}_t $$
Monetary policy
$$ 1+{i}_t={\left(1+{i}_{t-1}\right)}^{\rho_i}{\left[\left(1+i\right){\left(\frac{1+{\pi}_{t+1}^C}{1+{\pi}_{t+1}^{C, tar}}\right)}^{\phi_{\pi }}\right]}^{1-{\rho}_i}{e}^{\epsilon_t^i} $$
Foreign trade
$$ {X}_t={\left(\frac{p_t^X}{REE{R}_t}\right)}^{-\theta } GD{P}_t^{\ast } $$
$$ T{B}_t={p}_t^X{X}_t- REE{R}_t{M}_t $$
$$ T{B}_t={B}_t^{\ast }-\left(1+{r}_{t-1}^{\ast}\right)\frac{REE{R}_t}{REE{R}_{t-1}}{B}_{t-1}^{\ast }- EU{F}_t $$
Equilibrium conditions
$$ {Y}_t={Y}_t^C+{Y}_t^{Inv}+{Y}_t^G+{Y}_t^X+{Y}_tR\left(\cdotp \right)+{p}_t^{Inv} In{v}_tS\left(\cdotp \right)+w{c}_t{L}_tR\left(\cdotp \right) $$
$$ {M}_t={M}_t^C\left(1+G\left(\cdotp \right)\right)+{M}_t^{Inv}\left(1+G\left(\cdotp \right)\right)+{M}_t^G\left(1+G\left(\cdotp \right)\right)+{M}_t^X\left(1+G\left(\cdotp \right)\right) $$
$$ GD{P}_t={p}_t^C{C}_t+{p}_t^{Inv} In{v}_t+{p}_t^G{G}_t+{p}_t^X{X}_t- REE{R}_t{M}_t $$
Parameters
-
B
∗—Foreign bond
-
K
t—Capital
-
MPC
t—Marginal propensity to consume
-
REER
t—Real effective exchange rate
-
W—Nominal wage
-
i
t—Nominal interest rate
-
r
∗—Foreign interest rate
-
h—Technical parameter for households’ behaviour (habit parameter)
-
p—Nominal prices
-
B—Domestic bond
-
C—Consumption
-
Debt—Government debt
-
E—Expectations
-
EUF
t—EU funds
-
Exp—Expenditures of the government
-
G—Adjustment function
-
GB—Balance for the government budget
-
Inc—Total income for households
-
Inv—Investments
-
L—Labour force
-
M—Import
-
N—Net value
-
Profit—Profit for final producer
-
Q—Tobin’s Q
-
R—Rotemberg’s cost function
-
Rev—Revenues of the government
-
T—Taxes
-
TB—Trade Balance
-
TC—Total cost
-
TR—Transfers
-
X—Export
-
Y—Total output
-
g—Growth rate
-
mc
t—Marginal cost
-
r—Interest rate
-
ret—Return
-
v—Technical parameter for financial sector
-
α—Technical parameter for production
-
β—Technical parameter for households’ behaviour
-
δ—Amortization rate
-
η—Technical parameter for financial sector
-
θ—Technical parameter for financial sector
-
λ—Technical parameter for financial sector
-
μ—Import share in production (technical parameter)
-
ξ—Yield spread between risk-free (government bond) and risky (corporate bond)
-
π—Inflation
-
τ—Taxation rate
-
φ—Technical parameter for pricing
-
ψ—Technical parameter for households
-
ω—Technical parameter for household
-
ϕ—Technical parameter for monetary policy
B
∗—Foreign bond
K
t—Capital
MPC
t—Marginal propensity to consume
REER
t—Real effective exchange rate
W—Nominal wage
i
t—Nominal interest rate
r
∗—Foreign interest rate
h—Technical parameter for households’ behaviour (habit parameter)
p—Nominal prices
B—Domestic bond
C—Consumption
Debt—Government debt
E—Expectations
EUF
t—EU funds
Exp—Expenditures of the government
G—Adjustment function
GB—Balance for the government budget
Inc—Total income for households
Inv—Investments
L—Labour force
M—Import
N—Net value
Profit—Profit for final producer
Q—Tobin’s Q
R—Rotemberg’s cost function
Rev—Revenues of the government
T—Taxes
TB—Trade Balance
TC—Total cost
TR—Transfers
X—Export
Y—Total output
g—Growth rate
mc
t—Marginal cost
r—Interest rate
ret—Return
v—Technical parameter for financial sector
α—Technical parameter for production
β—Technical parameter for households’ behaviour
δ—Amortization rate
η—Technical parameter for financial sector
θ—Technical parameter for financial sector
λ—Technical parameter for financial sector
μ—Import share in production (technical parameter)
ξ—Yield spread between risk-free (government bond) and risky (corporate bond)
π—Inflation
τ—Taxation rate
φ—Technical parameter for pricing
ψ—Technical parameter for households
ω—Technical parameter for household
ϕ—Technical parameter for monetary policy
