Endogenous variables’ equations

Producers

Production (GDP) \[Y = CH + I + G \tag{1}\]

Notional labor demand \[L^{n} = \left( \frac{Y}{PROG^{L}} \right) . \left( \left( \varphi^{L} \right) ^ {\rho^{KL}} \right) . \left( \frac{\left( \frac{w}{PROG^{L}} \right)}{c^{Y}} \right) ^ {\left( -\rho^{KL} \right)} \tag{2}\]

Notional capital demand \[K^{n} = \left( \frac{Y}{PROG^{K}} \right) . \left( \left( \varphi^{K} \right) ^ {\rho^{KL}} \right) . \left( \frac{\left( \frac{c^{K}}{PROG^{K}} \right)}{c^{Y}} \right) ^ {\left( -\rho^{KL} \right)} \tag{3}\]

Investment \[\varDelta \left(\operatorname{log} I\right) = \alpha^{I,Kn} . \varDelta \left(\operatorname{log} K^{n}\right) + \alpha^{I,I1} . \varDelta \left(\operatorname{log} I_{t-1}\right) + \alpha^{I,KnK1} . \operatorname{log} \frac{K^{n}_{t-1}}{K_{t-1}} - \alpha^{I,rK} . \varDelta \left(r - \frac{\varDelta \left(p\right)}{p_{t-1}}\right) \tag{4}\]

Capital stock \[K = K_{t-1} \; \left( 1 - \delta \right) + I_{t-1} \tag{5}\]

Notional production price \[p^{n} . Y = c^{Y} . Y . \left( 1 + m^{up} \right) \tag{6}\]

Notional mark-up \[\varDelta \left(\operatorname{log} \left(1 + m^{up,n}\right)\right) = \rho^{mupn,Ln} . \varDelta \left(\operatorname{log} \frac{L^{n}}{L}\right) + \rho^{mupn,Kn} . \varDelta \left(\operatorname{log} \frac{K^{n}}{K}\right) \tag{7}\]

Notional unit cost production cost \[c^{Y} . Y = w . L^{n} + c^{K} . K^{n} \tag{8}\]

Capital cost \[c^{K} . K = p^{K}_{t-1} \; K_{t-1} \; \left( \delta + r^{K}_{t-1} \right) \tag{9}\]

Average price of the accumulated capital stock \[p^{K} . K = p^{K}_{t-1} \; K_{t-1} \; \left( 1 - \delta \right) + p_{t-1} \; I_{t-1} \tag{10}\]

Average interest rate paid on the debt \[r^{K} . p^{K} . K = r^{K}_{t-1} \; p^{K}_{t-1} \; K_{t-1} \; \left( 1 - \delta \right) + p_{t-1} \; I_{t-1} \; r_{t-1} \tag{11}\]

Profit of firms (before investment) \[PROF = p . Y - w . L - c^{K} . K \tag{12}\]

Firms’ savings \[SAV^{F} = PROF - DIV - p . I \tag{13}\]

Firms’ bank debt \[DEBT^{F} = DEBT^{F}_{t-1} \; \left( 1 - \varphi^{RD^{F}}_{t-1} \right) - SAV^{F} \tag{14}\]

Households

Notional households consumption \[CH^{n} . p = \left( 1 - \sigma \right) . INC . \left( 1 - t^{inc} \right) \tag{15}\]

Households’ income \[INC = \left( w . L + DIV \right) \tag{16}\]

Notional dividend for households \[DIV^{n} = PROF \tag{17}\]

Notional propensity to save equation \[\varDelta \left(\operatorname{log} \left(1 - \sigma^{n}\right)\right) = \rho^{\sigma,U} . \varDelta \left(U\right) - \rho^{\sigma,r} . \varDelta \left(r\right) + \rho^{\sigma,p} . \varDelta \left(\frac{\varDelta \left(p\right)}{p_{t-1}}\right) - \rho^{\sigma,DEBT} . \varDelta \left(\operatorname{log} \left(\frac{DEBT^{G}}{\left( p . Y \right)}\right)\right) \tag{18}\]

Households’ savings \[SAV^{H} = INC . \left( 1 - t^{inc} \right) - p . CH \tag{19}\]

Households’ total wealth \[WEALTH = WEALTH_{t-1} + SAV^{H} \tag{20}\]

Government and Central Bank

Notional interest rate of the Central Bank (Taylor reaction function) \[\varDelta \left(r^{n}\right) = \rho^{rn,p} . \varDelta \left(\frac{\varDelta \left(p\right)}{p_{t-1}}\right) - \rho^{rn,U} . \varDelta \left(U\right) \tag{21}\]

Notional income tax rate \[\varDelta \left(t^{inc,n}\right) = \rho^{tinc,debt} . \varDelta \left(\frac{DEBT^{G}}{\left( p . Y \right)}\right) \tag{22}\]

Government’s savings \[SAV^{G} = t^{inc} . INC - p . G - DEBT^{G}_{t-1} \; \left( \varphi^{RD^{G}}_{t-1} + r^{DEBT,G}_{t-1} \right) \tag{23}\]

Average interest rate paid on the total Government’s debt \[\varDelta \left(r^{DEBT,G}\right) = \varDelta \left(r\right) \tag{24}\]

Total Government’s debt \[DEBT^{G} = DEBT^{G}_{t-1} \; \left( 1 - \varphi^{RD^{G}}_{t-1} \right) - SAV^{G} \tag{25}\]

Labor market

Notional wage (WS or Phillips curve) \[\varDelta \left(\operatorname{log} w^{n}\right) = \rho^{wn} + \rho^{wn,pe} . \varDelta \left(\operatorname{log} p^{e}\right) + \rho^{wn,PROGL} . \varDelta \left(\operatorname{log} PROG^{L}\right) - \rho^{wn,U} . U - \rho^{wn,dU} . \varDelta \left(U\right) \tag{26}\]

Unemployment rate \[U = 1 - \frac{L}{LF} \tag{27}\]

Adjustments

Wage \[\varDelta \left(\operatorname{log} w\right) = \alpha^{W,Wn} . \varDelta \left(\operatorname{log} w^{n}\right) + \alpha^{W,W1} . \varDelta \left(\operatorname{log} w_{t-1}\right) - \alpha^{W,W1Wn1} . \operatorname{log} \frac{w_{t-1}}{w^{n}_{t-1}} \tag{28}\]

Production price \[\operatorname{log} p = \alpha^{P,Pn} . \operatorname{log} p^{n} + \left( 1 - \alpha^{P,Pn} \right) . \left( \operatorname{log} p_{t-1} + \varDelta \left(\operatorname{log} p^{e}\right) \right) \tag{29}\]

Expected production price inflation \[\varDelta \left(\operatorname{log} p^{e}\right) = \alpha^{Pe,Pe1} . \varDelta \left(\operatorname{log} p^{e}_{t-1}\right) + \alpha^{Pe,P1} . \varDelta \left(\operatorname{log} p_{t-1}\right) + \alpha^{Pe,Pn} . \varDelta \left(\operatorname{log} p^{n}\right) \tag{30}\]

Households final consumption \[\operatorname{log} CH = \alpha^{CH,CHn} . \operatorname{log} CH^{n} + \left( 1 - \alpha^{CH,CHn} \right) . \left( \operatorname{log} CH_{t-1} + \varDelta \left(\operatorname{log} CH^{e}\right) \right) \tag{31}\]

Expected households final consumption growth \[\varDelta \left(\operatorname{log} CH^{e}\right) = \alpha^{CHe,CHe1} . \varDelta \left(\operatorname{log} CH^{e}_{t-1}\right) + \alpha^{CHe,CH1} . \varDelta \left(\operatorname{log} CH_{t-1}\right) + \alpha^{CHe,CHn} . \varDelta \left(\operatorname{log} CH^{n}\right) \tag{32}\]

Labor \[\operatorname{log} L = \alpha^{L,Ln} . \operatorname{log} L^{n} + \left( 1 - \alpha^{L,Ln} \right) . \left( \operatorname{log} L_{t-1} + \varDelta \left(\operatorname{log} L^{e}\right) \right) \tag{33}\]

Expected labor growth \[\varDelta \left(\operatorname{log} L^{e}\right) = \alpha^{Le,Le1} . \varDelta \left(\operatorname{log} L^{e}_{t-1}\right) + \alpha^{Le,L1} . \varDelta \left(\operatorname{log} L_{t-1}\right) + \alpha^{Le,Ln} . \varDelta \left(\operatorname{log} L^{n}\right) \tag{34}\]

Dividend for households \[\operatorname{log} DIV = \alpha^{DIV,DIVn} . \operatorname{log} DIV^{n} + \left( 1 - \alpha^{DIV,DIVn} \right) . \left( \operatorname{log} DIV_{t-1} + \varDelta \left(\operatorname{log} DIV^{e}\right) \right) \tag{35}\]

Expected dividend for households \[\varDelta \left(\operatorname{log} DIV^{e}\right) = \alpha^{DIVe,DIVe1} . \varDelta \left(\operatorname{log} DIV^{e}_{t-1}\right) + \alpha^{DIVe,DIV1} . \varDelta \left(\operatorname{log} DIV_{t-1}\right) + \alpha^{DIVe,DIVn} . \varDelta \left(\operatorname{log} DIV^{n}\right) \tag{36}\]

Interest rate of the Central Bank \[r = \alpha^{r} . r^{n} + \left( 1 - \alpha^{r} \right) . r_{t-1} \tag{37}\]

Propensity to save \[\sigma = \alpha^{\sigma} . \sigma^{n} + \left( 1 - \alpha^{\sigma} \right) . \sigma_{t-1} \tag{38}\]

Mark-up \[m^{up} = \alpha^{m,up} . m^{up,n} + \left( 1 - \alpha^{m,up} \right) . m^{up}_{t-1} \tag{39}\]

Income tax rate \[t^{inc} = \alpha^{t,inc} . t^{inc,n} + \left( 1 - \alpha^{t,inc} \right) . t^{inc}_{t-1} \tag{40}\]