The Dollar’s Imperial Circle

Abstract

In this paper, we highlight a new channel through which dollar fluctuations can become a self-fulfilling pro-cyclical force. We call this mechanism Imperial Circle as it makes the dollar the dominant macroeconomic variable in the context of the current international monetary system. At the core of it, there is a fundamental asymmetry between the shrinking exposure of the “real” US economy to global developments versus the growing global role of the US dollar. Dollar appreciation leads to a decline in global economic activity, which in turn benefits, in relative terms, the dollar itself, reinforcing the initial appreciation and its effects.

Appendices

Appendix

Trade Parameter Values

See Table 6.

Table 6 Openness parameter descriptions and values

Full Set of Equilibrium Conditions

Home country equations:

$$\begin{aligned} C_t^{\frac{-1}{\sigma }}= & {} \beta {\mathbb {E}}_{t} \left[ C_{t+1}^{\frac{-1}{ \sigma }} \frac{R_t^{n}}{\pi ^{C}_{t+1}} \right] \end{aligned}$$

(B.1)

$$\begin{aligned} w_t= & {} \chi _o L_t^\chi C_t^{\frac{1}{\sigma }} \end{aligned}$$

(B.2)

$$\begin{aligned} C_{Ht}= & {} \nu _H^H \left( p_{HT,t} \right) ^{-\rho _c} C_t^T \end{aligned}$$

(B.3)

$$\begin{aligned} C_{Ft}= & {} \nu _F^H \left( \frac{p_{Ft}}{p_{Tt}} \right) ^{-\rho _c} C_t^T \end{aligned}$$

(B.4)

$$\begin{aligned} C_{Wt}= & {} \nu _W^H \left( {\mathcal {T}}_H^W p_{HT,t} \right) ^{-\rho _c} C_t^T \end{aligned}$$

(B.5)

$$\begin{aligned} C_{t}^T= & {} \omega _s \left( p_{T,t} \right) ^{-\rho _s} C_t \end{aligned}$$

(B.6)

$$\begin{aligned} C_{t}^N= & {} (1-\omega _s) \left( p_{N,t} \right) ^{-\rho _s} C_t \end{aligned}$$

(B.7)

$$\begin{aligned} M^T_{Nt}= & {} {\nu _{pN}^H} \left( \frac{p_{N}p_{HM,t} }{p_{T,t}p_{HT,t}} \right) ^{-\rho _m} M^T_t \end{aligned}$$

(B.8)

$$\begin{aligned} M^T_{Ht}= & {} {\nu _{pH}^H} \left( p_{HM,t}\right) ^{-\rho _m} M^T_t \end{aligned}$$

(B.9)

$$\begin{aligned} M^T_{Ft}= & {} {\nu _{pF}^H} \left( \frac{p_{Ft}}{p_{Tt}} \frac{p_{HM,t}}{p_{HT,t}}\right) ^{-\rho _m} M^T_t \end{aligned}$$

(B.10)

$$\begin{aligned} M^T_{Wt}= & {} {\nu _{pW}^H} \left( {\mathcal {T}}_H^W p_{HM,t} \right) ^{-\rho _m} M^T_t\end{aligned}$$

(B.11)

$$\begin{aligned} M^N_{Nt}= & {} {\nu _{pN}^N} \left( p_{NM,t} \right) ^{-\rho _m} M^N_t \end{aligned}$$

(B.12)

$$\begin{aligned} M^N_{Ht}= & {} {\nu _{pH}^N} \left( \frac{p_{T,t}p_{HT,t}}{p_{N,t}}p_{NM,t}\right) ^{-\rho _m} M^N_t \end{aligned}$$

(B.13)

$$\begin{aligned} M^N_{Ft}= & {} {\nu _{pF}^N} \left( \frac{p_{Ft}}{p_{Nt}}p_{NM,t} \right) ^{-\rho _m}M^N_t \end{aligned}$$

(B.14)

$$\begin{aligned} M^N_{Wt}= & {} {\nu _{pW}^N} \left( \frac{p_{T,t}p_{HT,t}}{p_{N,t}}{\mathcal {T}}_{H,t}^W p_{NM,t} \right) ^{-\rho _m} M^N_t \end{aligned}$$

(B.15)

$$\begin{aligned} p_{HT,t}= & {} \left[ \frac{1-\nu _F^H (p_{F,t}/p_{T,t})^{1-\rho _c}}{\nu _H^H+\nu _W^H ( {\mathcal {T}}_H^W )^{1-\rho _c}} \right] ^{\frac{1}{1-\rho _c}} \end{aligned}$$

(B.16)

$$\begin{aligned} p_{N,t}= & {} \left[ \frac{1-\omega _s (p_T)^{1-\rho _s}}{1-\omega _s}\right] ^{\frac{1}{1-\rho _s}} \end{aligned}$$

(B.17)

$$\begin{aligned} {p_{HM,t}}= & {} \left[ \nu _{pN}^H \left( \frac{p_{N} }{p_{T,t}p_{HT,t}} \right) ^{1-\rho _m}+ \nu _{pH}^H + \nu _{pF}^H\right. \nonumber \\{} & {} \left. \left( \frac{p_{Ft}}{p_{Tt} p_{HT,t}} \right) ^{1-\rho _m} + \nu _{pW}^H ({\mathcal {T}}_H^W)^{1-\rho _m}\right] ^{-\frac{1}{1-\rho _m}} \end{aligned}$$

(B.18)

$$\begin{aligned} p_{NM,t}= & {} \left[ \nu _{pN}^N +\nu _{pH}^N \left( \frac{p_{T,t}p_{HT,t}}{ p_{N,t}} \right) ^{1-\rho _m} + \nu _{pF}^N\left( \frac{p_{Ft}}{p_{Nt}} \right) ^{1-\rho _m}\right. \nonumber \\{} & {} \left. + \nu _{pW}^N\left( \frac{p_{T,t}p_{HT,t}}{p_{N,t}}{\mathcal {T}}_{H,t}^W \right) ^{1-\rho _m} \right] ^{-\frac{1}{1-\rho _m}} \end{aligned}$$

(B.19)

$$\begin{aligned} Y^T_t= & {} \overline{K^T}^\alpha (Z_t^T L_t^T)^{\gamma _p} (M^T_t)^{1-\alpha -\gamma _p} \end{aligned}$$

(B.20)

$$\begin{aligned} w_t= & {} \frac{\gamma _p}{\alpha }\frac{\overline{K^T}}{L_t^T}r_{K,t}^T \end{aligned}$$

(B.21)

$$\begin{aligned} p_{HT,t}p_{T,t}/p_{HM,t}= & {} \frac{1-\alpha -\gamma _p}{\alpha }\frac{K^T}{M^T_t}r_{K,t}^T \end{aligned}$$

(B.22)

$$\begin{aligned} mc_t^T= & {} \left( \frac{w_t/Z_t^T}{\gamma _p} \right) ^{\gamma _p} \left( \frac{r_{K,t}^T}{\alpha } \right) ^\alpha \left( \frac{p_{HT,t}p_{T,t}/p_{HM,t}}{ 1-\alpha -\gamma _p}\right) ^{1-\alpha -\gamma _p} \end{aligned}$$

(B.23)

$$\begin{aligned} {\pi _{t}}^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) ({\pi _{t}}^o)^{-\frac{1}{ \theta _p}} + \xi _p(\pi _{t-1})^{-\frac{\iota _p}{\theta _p}} \end{aligned}$$

(B.24)

$$\begin{aligned} {\pi _{t}}^o= & {} (1+\theta _p) \frac{x_{1T,t}}{x_{2T,t}} \pi _t \end{aligned}$$

(B.25)

$$\begin{aligned} x_{1T,t}= & {} \omega _s^{\frac{1}{\rho _s}} C_t^{\frac{1}{\rho _s}-\frac{1}{\sigma }} C_{T,t}^{-\frac{1}{\rho _s}} mc_t^T Y_{t}^T \nonumber \\{} & {} + \beta \xi _p {\pi _t}^{-\iota _p\frac{1+\theta _p}{\theta _p}} {\mathbb {E}}_t \left\{ x_{1T,t+1} {\pi _{t+1}}^{\frac{1+\theta _p}{\theta _p}} \right\} \end{aligned}$$

(B.26)

$$\begin{aligned} x_{2T,t}= & {} \omega _s^{\frac{1}{\rho _s}} C_t^{\frac{1}{\rho _s}-\frac{1}{\sigma }} C_{T,t}^{-\frac{1}{\rho _s}} p_{HT,t} p_{T,t} Y_t^T \nonumber \\{} & {} + \beta \xi _p {\pi _t}^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p}\right) } {\mathbb {E}}_t\left\{ x_{2T,t+1} {\pi _{t+1}}^{\frac{1+\theta _p}{\theta _p} - 1}\right\} \end{aligned}$$

(B.27)

$$\begin{aligned} Y^N_t= & {} \overline{K^N}^\alpha (Z_t^N L_t^N)^{\gamma _p}(M^N_t)^{1-\alpha -\gamma _p} \end{aligned}$$

(B.28)

$$\begin{aligned} w_t= & {} \frac{\gamma _p}{\alpha }\frac{\overline{K^N}}{L_t^N}r_{K,t}^N \end{aligned}$$

(B.29)

$$\begin{aligned} p_{N,t}/p_{NM,t}= & {} \frac{1-\alpha -\gamma _p}{\alpha }\frac{K^N}{M^N_t} r_{K,t}^N \end{aligned}$$

(B.30)

$$\begin{aligned} mc^N_t= & {} \left( \frac{w_t/Z_t^N}{\gamma _p} \right) ^{\gamma _p} \left( \frac{r_{K,t}^N}{\alpha } \right) ^\alpha \left( \frac{p_{N,t}/p_{NM,t}}{1-\alpha -\gamma _p}\right) ^{1-\alpha -\gamma _p} \end{aligned}$$

(B.31)

$$\begin{aligned} {\pi _{t}^N}^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) ({\pi _{t}^N}^o)^{-\frac{1}{\theta _p}} + \xi _p(\pi _{t-1}^N)^{-\frac{\iota _p}{\theta _p}} \end{aligned}$$

(B.32)

$$\begin{aligned} {\pi _{t}^N}^o= & {} (1+\theta _p) \frac{x_{1N,t}}{x_{2N,t}} \pi _t^N \end{aligned}$$

(B.33)

$$\begin{aligned} x_{1N,t}= & {} (1-\omega _s)^{\frac{1}{\rho _s}} C_t^{\frac{1}{\rho _s}-\frac{1}{\sigma }} C_{N,t}^{-\frac{1}{\rho _s}} mc_t^N Y_{t}^N \nonumber \\{} & {} + \beta \xi _p {\pi _t^N}^{-\iota _p \frac{1+\theta _p}{\theta _p}} {\mathbb {E}}_t \left\{ x_{1N,t+1} {\pi _{t+1^N}}^{\frac{1+\theta _p}{\theta _p}} \right\} \end{aligned}$$

(B.34)

$$\begin{aligned} x_{2N,t}= & {} (1-\omega _s)^{\frac{1}{\rho _s}} C_t^{\frac{1}{\rho _s}-\frac{1}{\sigma }} C_{N,t}^{-\frac{1}{\rho _s}} p_{N,t} Y_t^N \nonumber \\{} & {} + \beta \xi _p {\pi _t^N}^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p}\right) } {\mathbb {E}}_t \left\{ x_{2N,t+1} {\pi _{t+1}^N}^{\frac{1+\theta _p}{\theta _p} - 1} \right\} \end{aligned}$$

(B.35)

$$\begin{aligned} \pi ^C_t= & {} \pi _t \frac{p_{HT,t-1}p_{T,t-1}}{p_{HT,t}p_{T,t}} \end{aligned}$$

(B.36)

$$\begin{aligned} \pi ^C_t= & {} \pi _t^N\frac{p_{Nt-1}}{p_{N,t}} \end{aligned}$$

(B.37)

$$\begin{aligned} \pi ^C_t= & {} \pi _{Ft} \frac{p_{Ft-1}}{p_{Ft}} \end{aligned}$$

(B.38)

$$\begin{aligned} \pi ^C_t= & {} \pi ^W_{Ft} \frac{p^W_{Ft-1}}{p^W_{Ft}} \end{aligned}$$

(B.39)

$$\begin{aligned} Y_{t}^T= & {} C_{Ht} + M^T_{Ht}+M^N_{Ht}+ \frac{n_F}{n_H}\left( C_{Ht}^* + {M^T_{Ht}}^*+{M^N_{Ht}}^*\right) \nonumber \\{} & {} + \frac{n_W}{n_H}\left( C_{Ht}^{**} + {M^T_{Ht}}^{**}+{M^N_{Ht}}^{**}\right) \end{aligned}$$

(B.40)

$$\begin{aligned} L_t= & {} L^T_t + L^N_t \end{aligned}$$

(B.41)

$$\begin{aligned} Y^N_t= & {} C^N_t + M^N_{Nt} + M^T_{Nt} \end{aligned}$$

(B.42)

$$\begin{aligned} R_t^n= & {} \beta ^{-1} {\pi _t}^{\gamma _{\pi }} \end{aligned}$$

(B.43)

where $p_{HT,t} \equiv \frac{P_{H,t} }{P^T_{t}}$, $p_{HM,t} \equiv \frac{ P_{H,t} }{P^T_{M,t}}$, $p_{NM,t} \equiv \frac{P^N_{t} }{P^N_{M,t}}$, $p_{N,t} \equiv \frac{P^N_t}{P_{C,t}}$, $p_{Ft} \equiv \frac{P_{Ft}}{P_{Ct}}$ , $p^W_{Ft} \equiv \frac{P^W_{Ft}}{P_{Ct}}$, $\pi ^C_{t} \equiv \frac{P_{C,t} }{P_{C,t-1}}$, $\pi _{t} \equiv \frac{P_{H,t}}{P_{H,t-1}}$, $\pi ^N_{t} \equiv \frac{P^N_{t}}{P^N_{t-1}}$, $\pi _{Ft} \equiv \frac{P_{Ft}}{P_{Ft-1}}$, and $\pi ^W_{Ft} \equiv \frac{P^W_{Ft}}{P^W_{Ft-1}}$.

Foreign country equations:

$$\begin{aligned} {C_t^*}^{\frac{-1}{\sigma }}= & {} \beta ^* {\mathbb {E}}_{t} \left[ {C^*_{t+1}}^{\frac{-1}{\sigma }} \frac{{R_t^*}^{n}}{{\pi ^*}^{C}_{t+1}} \right] \end{aligned}$$

(B.44)

$$\begin{aligned} w^*_t= & {} \chi _o {L_t^*}^\chi {C_t^*}^{\frac{1}{\sigma }} \end{aligned}$$

(B.45)

$$\begin{aligned} C_{Ht}^*= & {} \nu _H^F \left( {\mathcal {T}}_F^H p_{FT,t}^* \right) ^{-\rho _c} {C_t^T}^* \end{aligned}$$

(B.46)

$$\begin{aligned} C_{Ft}^*= & {} \nu _F^F \left( p_{FT,t}^*\right) ^{-\rho _c} {C_t^T}^* \end{aligned}$$

(B.47)

$$\begin{aligned} C_{Wt}^*= & {} \nu _W^F \left( {\mathcal {T}}_F^Wp_{FT,t}^{*} \right) ^{-\rho _c} {C_t^T}^* \end{aligned}$$

(B.48)

$$\begin{aligned} {C_{t}^T}^*= & {} \omega _s^* \left( p_{T,t}^* \right) ^{-\rho _s} C_t^* \end{aligned}$$

(B.49)

$$\begin{aligned} {C_{t}^N}^*= & {} (1-\omega _s^*) \left( p_{N,t}^* \right) ^{-\rho _s} C_t^* \end{aligned}$$

(B.50)

$$\begin{aligned} {M^T_N}^*= & {} {\nu _{p{N}}^F}\left( \frac{p_{N}^*p_{FM,t}^* }{p_{T,t}^*p_{FT,t}^*} \right) ^{-\rho _m} {M^T}^* \end{aligned}$$

(B.51)

$$\begin{aligned} {M^T_H}^*= & {} {\nu _{pH}^F}\left( {\mathcal {T}}_F^H p_{FM,t}^* \right) ^{-\rho _m}{M^T}^* \end{aligned}$$

(B.52)

$$\begin{aligned} {M^T_F}^*= & {} {\nu _{pF}^F} \left( p_{FM,t}^* \right) ^{-\rho _m} {M^T}^* \end{aligned}$$

(B.53)

$$\begin{aligned} {M^T_W}^*= & {} {\nu _{pW}^F} \left( {\mathcal {T}}_F^Wp_{FM,t}^* \right) ^{-\rho _m} {M^T}^* \end{aligned}$$

(B.54)

$$\begin{aligned} {M^N_N}^*= & {} {\nu _{p{N}}^{N^*}}\left( p_{NM,t}^* \right) ^{-\rho _m} {M^N}^* \end{aligned}$$

(B.55)

$$\begin{aligned} {M^N_H}^*= & {} {\nu _{p{H}}^{N^*}}\left( \frac{p_{T,t}^*p_{FT,t}^*}{p_{N,t}^*}{\mathcal {T}}_F^H p_{NM,t}^*\right) ^{-\rho _m} {M^N}^* \end{aligned}$$

(B.56)

$$\begin{aligned} {M^N_F}^*= & {} {\nu _{p{F}}^{N^*}}\left( \frac{p_{T,t}^*p_{FT,t}^*}{p_{N,t}^*}p_{NM,t}^* \right) ^{-\rho _m} {M^N}^* \end{aligned}$$

(B.57)

$$\begin{aligned} {M^N_W}^*= & {} {\nu _{p{W}}^{N^*}}\left( \frac{p_{T,t}^*p_{FT,t}^*}{p_{N,t}^*}{\mathcal {T}}_F^W p_{NM,t}^*\right) ^{-\rho _m} {M^N}^* \end{aligned}$$

(B.58)

$$\begin{aligned} {p_{FT,t}^*}= & {} \left[ \nu _H^F \left( {\mathcal {T}}_F^H \right) ^{1-\rho _c} +\nu _F^F +\nu _W^F\left( {\mathcal {T}}_F^W \right) ^{1-\rho _c} \right] ^{-\frac{1}{1-\rho _c}} \end{aligned}$$

(B.59)

$$\begin{aligned} p_{N,t}^*= & {} \left[ \frac{1-\omega _s^* (p_T^*)^{1-\rho _s}}{1-\omega _s^*}\right] ^{\frac{1}{1-\rho _s}} \end{aligned}$$

(B.60)

$$\begin{aligned} {p_{FM,t}^*}= & {} \left[ \nu _{pN}^F \left( \frac{p_{N}^* }{p_{T,t}^*p_{FT,t}^*}\right) ^{1-\rho _m} \right. \nonumber \\{} & {} \left. +\nu _{pH}^F \left( {\mathcal {T}}_F^H \right) ^{1-\rho _m} +\nu _{pF}^F +\nu _{pW}^F\left( {\mathcal {T}}_F^W \right) ^{1-\rho _m} \right] ^{-\frac{1}{1-\rho _m}} \end{aligned}$$

(B.61)

$$\begin{aligned} p_{NM,t}^*= & {} \left[ \nu _{p{N}}^{N^*} + \nu _{pH}^{N^*} \left( \frac{p_{T,t}^*p_{FT,t}^*}{p_{N,t}^*} {\mathcal {T}}_F^H \right) ^{1-\rho _m} +\nu _{pF}^{N^*}\left( \frac{p_{T,t}^*p_{FT,t}^*}{p_{N,t}^*} \right) \right. \nonumber \\{} & {} \left. +\nu _{pW}^{N^*}\left( \frac{p_{T,t}^*p_{FT,t}^*}{p_{N,t}^*} {\mathcal {T}}_F^W\right) ^{1-\rho _m} \right] ^{-\frac{1}{1-\rho _m}} \end{aligned}$$

(B.62)

$$\begin{aligned} {Y^T}^*= & {} \overline{{K^T}^*}^\alpha ({Z_t^T}^* {L_t^T}^*)^{\gamma _p} ({M^T}^*)^{1-\alpha -\gamma _p} \end{aligned}$$

(B.63)

$$\begin{aligned} w^*= & {} \frac{\gamma _p}{\alpha } \frac{{{\overline{K}}^T}^*}{{L^T_t}^*}\frac{{r_{K,t}^T}^* }{1+(R^n_{risk,ft}-1) \theta _w} \end{aligned}$$

(B.64)

$$\begin{aligned} p_{FT,t}^*p_{T,t}^*/p_{FM,t}^*= & {} \frac{1-\alpha -\gamma _p}{\alpha }\frac{{{\overline{K}}^T}^*}{{M_t^T}^*} \frac{{r_{K,t}^T}^* }{1+(R^n_{risk,ft}-1)\theta _w} \end{aligned}$$

(B.65)

$$\begin{aligned} {mc^T_t}^*= & {} \left( \frac{w_t^*/{Z_t^T}^*[1+(R^n_{risk,ft}-1)\theta _w]}{\gamma _p} \right) ^{\gamma _p} \left( \frac{{r_{K,t}^T}^*}{\alpha }\right) ^\alpha \nonumber \\{} & {} \left( \frac{p_{FT,t}^*p_{T,t}^*/p_{FM,t}^*[1+(R^n_{risk,ft}-1)\theta _w]}{1-\alpha -\gamma _p}\right) ^{1-\alpha -\gamma _p} \end{aligned}$$

(B.66)

$$\begin{aligned} {\pi _{t}^*}^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) ({\pi _{t}^*}^o)^{-\frac{1}{\theta _p}} + \xi _p(\pi _{t-1}^*)^{-\iota _p \frac{1}{\theta _p}} \end{aligned}$$

(B.67)

$$\begin{aligned} {\pi _{t}^*}^o= & {} (1+\theta _p) \frac{x_{1T,t}^*}{x_{2T,t}^*} \pi _t^* \end{aligned}$$

(B.68)

$$\begin{aligned} x_{1T,t}^*= & {} {\omega _s^*}^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^*}^{-\frac{1}{\rho _s}} {mc_t^T}^* {Y_{Dt}^T}^* \nonumber \\{} & {} +\beta ^* \xi _p {\pi _t^*}^{-\iota _p\frac{1+\theta _p}{\theta _p}} {\mathbb {E}}_t\left\{ {(\pi _{t+1}^*)}^{\frac{1+\theta _p}{\theta _p}} x_{1T,t+1}^* \right\} \end{aligned}$$

(B.69)

$$\begin{aligned} x_{2T,t}^*= & {} {\omega _s^*}^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^*}^{-\frac{1}{\rho _s}} p_{FT,t}^* p_{T,t}^* {Y_{Dt}^T}^* \nonumber \\{} & {} + \beta ^* \xi _p {\pi _t^*}^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p} \right) }{\mathbb {E}}_t \left\{ {(\pi _{t+1}^*)}^{\frac{1+\theta _p}{\theta _p} - 1} x_{2T,t+1}^* \right\} \end{aligned}$$

(B.70)

$$\begin{aligned} {Y^T_D}^*= & {} C_F^* + {M_F^T}^* + {M_F^N}^* \end{aligned}$$

(B.71)

$$\begin{aligned} {\ \pi _{Ft} }^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) {\pi _{Ft}^{o}}^{-\frac{1}{\theta _p}} + \xi _p (\pi _{Ft-1})^{-\iota _p \frac{1}{\theta _p}} \end{aligned}$$

(B.72)

$$\begin{aligned} \pi _{Ft}^{o}= & {} (1+\theta _p) \frac{z_{1t}}{z_{2t}} \pi _{Ft} \end{aligned}$$

(B.73)

$$\begin{aligned} z_{1t}= & {} {\omega _s^*}^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^*}^{-\frac{1}{\rho _s}} N_t {mc^T}^*_t \nonumber \\{} & {} + \beta ^* \xi _p(\pi _{Ft})^{-\iota _p \frac{1+\theta _p}{\theta _p}}{\mathbb {E}}_t \left\{ (\pi _{Ft+1})^{\frac{1+\theta _p}{\theta _p}}z_{1t+1} \right\} \end{aligned}$$

(B.74)

$$\begin{aligned} z_{2t}= & {} {\omega _s^*}^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^*}^{-\frac{1}{\rho _s}} N_t \frac{p_{Ft}}{{\mathcal {Q}}^H_{Ft}} \nonumber \\{} & {} + \beta ^* \xi _p (\pi _{Ft})^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p}\right) }{\mathbb {E}}_t \left\{ (\pi _{Ft+1})^{\frac{1+\theta _p}{\theta _p}-1} z_{2t+1} \right\} \end{aligned}$$

(B.75)

$$\begin{aligned} N_t= & {} C_{Ft}+M_{Ft}^T + M_{Ft}^N \end{aligned}$$

(B.76)

$$\begin{aligned} {\ \pi ^W_{Ft} }^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) {\pi _{Ft}^{Wo}}^{-\frac{1}{\theta _p}} + \xi _p (\pi ^W_{Ft-1})^{-\iota _p \frac{1}{\theta _p}} \end{aligned}$$

(B.77)

$$\begin{aligned} \pi _{Ft}^{Wo}= & {} (1+\theta _p) \frac{z^W_{1t}}{z^W_{2t}} \pi ^W_{Ft} \end{aligned}$$

(B.78)

$$\begin{aligned} z^W_{1t}= & {} {\omega _s^*}^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^*}^{-\frac{1}{\rho _s}} N^W_t {mc^T}^*_t \nonumber \\{} & {} + \beta ^* \xi _p(\pi ^W_{Ft})^{-\iota _p \frac{1+\theta _p}{\theta _p}}{\mathbb {E}}_t \left\{ (\pi ^W_{Ft+1})^{\frac{1+\theta _p}{\theta _p}}z^W_{1t+1} \right\} \end{aligned}$$

(B.79)

$$\begin{aligned} z^W_{2t}= & {} {\omega _s^*}^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^*}^{-\frac{1}{\rho _s}} N^W_t \frac{p^W_{Ft}}{{\mathcal {Q}}^H_{Ft}} \nonumber \\{} & {} + \beta ^* \xi _p (\pi ^W_{Ft})^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p}\right) }{\mathbb {E}}_t \left\{ (\pi ^W_{Ft+1})^{\frac{1+\theta _p}{\theta _p}-1} z^W_{2t+1} \right\} \end{aligned}$$

(B.80)

$$\begin{aligned} N_t^W= & {} C_{Ft}^{**}+{M_{Ft}^T}^{**} + {M_{Ft}^N}^{**} \end{aligned}$$

(B.81)

$$\begin{aligned} {Y^N}^*= & {} \overline{{K^N}^*}^\alpha (Z_t^* {L_t^N}^*)^{\gamma _p}({M^N}^*)^{1-\alpha -\gamma _p} \end{aligned}$$

(B.82)

$$\begin{aligned} w^*= & {} \frac{\gamma _p}{\alpha } \frac{{{\overline{K}}^N}^*}{{L^N_t}^*}\frac{{r_{K,t}^N}^* }{1+(R^n_{risk,ft}-1) \theta _w} \end{aligned}$$

(B.83)

$$\begin{aligned} p_{N,t}^*/p_{NM,t}^*= & {} \frac{1-\alpha -\gamma _p}{\alpha }\frac{{{\overline{K}}^N}^*}{{M_t^N}^*} \frac{{r_{K,t}^N}^* }{1+(R^n_{risk,ft}-1) \theta _w} \end{aligned}$$

(B.84)

$$\begin{aligned} {mc^N_t}^*= & {} \left( \frac{w_t^*/{Z_t^N}^*[1+(R^n_{risk,ft}-1)\theta _w]}{\gamma _p} \right) ^{\gamma _p} \left( \frac{{r_{K,t}^N}^*}{\alpha }\right) ^\alpha \nonumber \\{} & {} \left( \frac{p_{N,t}^*/p_{NM,t}^*[1+(R^n_{risk,ft}-1)\theta _w]}{1-\alpha -\gamma _p}\right) ^{1-\alpha -\gamma _p} \end{aligned}$$

(B.85)

$$\begin{aligned} {{\pi _{t}^N}^*}^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) ({{\pi _{t}^N}^*}^o)^{-\frac{1}{\theta _p}} + \xi _p({\pi _{t-1}^N}^*)^{-\frac{\iota _p}{\theta _p}} \end{aligned}$$

(B.86)

$$\begin{aligned} {{\pi _{t}^N}^*}^o= & {} (1+\theta _p) \frac{x_{1N,t}^*}{x_{2N,t}^*} {\pi _t^N}^*\end{aligned}$$

(B.87)

$$\begin{aligned} x_{1N,t}^*= & {} (1-\omega _s^*)^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{N,t}^*}^{-\frac{1}{\rho _s}} {mc_t^N}^* {Y_{t}^N }^*\nonumber \\{} & {} +\beta ^* \xi _p {{\pi _t^N}^*}^{-\iota _p \frac{1+\theta _p}{\theta _p}} {\mathbb {E}}_t \left\{ x_{1N,t+1}^* {{\pi _{t+1^N}}^*}^{\frac{1+\theta _p}{\theta _p}}\right\} \end{aligned}$$

(B.88)

$$\begin{aligned} x_{2N,t}^*= & {} (1-\omega _s^*)^{\frac{1}{\rho _s}} {C_t^*}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{N,t}^*}^{-\frac{1}{\rho _s}} p_{N,t}^* {Y_t^N}^* \nonumber \\{} & {} +\beta ^* \xi _p {{\pi _t^N}^*}^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p}\right) } {\mathbb {E}}_t \left\{ x_{2N,t+1}^* {{\pi _{t+1}^N}^*}^{\frac{1+\theta _p}{\theta _p} - 1}\right\} \end{aligned}$$

(B.89)

$$\begin{aligned} {\pi _t^*}^C= & {} \pi _t^* \frac{p_{FT,t-1}^*p_{T,t-1}^*}{p_{FT,t}^*p_{T,t}^*}\end{aligned}$$

(B.90)

$$\begin{aligned} {\pi _t^*}^C= & {} {\pi _t^N}^*\frac{p_{Nt-1}^*}{p_{N,t}^*} \end{aligned}$$

(B.91)

$$\begin{aligned} {Y^T}^*= & {} C_{Ft}^* + {M^T_{Ft}}^*+ {M^N_{Ft}}^* +\frac{n_H}{n_F} \left( C_{Ft}+M_{Ft}^T + M_{Ft}^N \right) \nonumber \\{} & {} + \frac{n_W}{n_F} \left( C_{Ft}^{**}+{M_{Ft}^T}^{**} + {M_{Ft}^N}^{**}\right) \end{aligned}$$

(B.92)

$$\begin{aligned} L^*= & {} {L^T}^* + {L^N}^* \end{aligned}$$

(B.93)

$$\begin{aligned} {Y^N}^*= & {} {C^N}^*+ {M^N_N}^* + {M^T_N}^* \end{aligned}$$

(B.94)

$$\begin{aligned} {R_t^*}^n= & {} {\beta ^*}^{-1} {{\pi ^C_t}^*}^{\gamma _{\pi }} \end{aligned}$$

(B.95)

where $p_{FT,t}^* \equiv \frac{P_{F,t}^* }{{P^T_{t}}^*}$, $p_{FM,t}^* \equiv \frac{{P_{F,t}}^* }{{P^T_{M,t}}^*}$, $p_{NM,t}^* \equiv \frac{{P^N_{t}}^* }{{ P^N_{M,t}}^*}$, $p^*_{N,t} \equiv \frac{{P^N_t}^*}{P_{C,t}^*}$, ${\pi ^C_{t}} ^* \equiv \frac{P_{C,t}^*}{P_{C,t-1}^*}$, $\pi ^*_{t} \equiv \frac{P_{F,t}^*}{ P_{F,t-1}^*}$, and ${\pi ^N_{t}}^* \equiv \frac{{P^N_{t}}^*}{{P^N_{t-1}}^*}$.

World country equations:

$$\begin{aligned} {C_t^{**}}^{\frac{-1}{\sigma }}= & {} \beta ^{**} {\mathbb {E}}_{t} \left[ {C^{**}_{t+1}}^{\frac{-1}{\sigma }} \frac{{R_t^{**}}^{n}}{{\pi ^{**}}^{C}_{t+1}}\right] \end{aligned}$$

(B.96)

$$\begin{aligned} w^{**}_t= & {} \chi _o {L_t^{**}}^\chi {C_t^{**}}^{\frac{1}{\sigma }} \end{aligned}$$

(B.97)

$$\begin{aligned} C_{Ht}^{**}= & {} \nu _H^W \left( \frac{1}{{\mathcal {T}}_H^W} p_{WT,t}^{**}\right) ^{-\rho _c} {C_t^T}^{**} \end{aligned}$$

(B.98)

$$\begin{aligned} C_{Ft}^{**}= & {} \nu _F^W \left( \frac{p_F^W}{p_T{**}}\frac{1}{{\mathcal {Q}}_W^H}\right) ^{-\rho _c} {C_t^T}^{**} \end{aligned}$$

(B.99)

$$\begin{aligned} C_{Wt}^{**}= & {} \nu _W^W \left( p_{WT,t}^{**} \right) ^{-\rho _c} {C_t^T}^{**}\end{aligned}$$

(B.100)

$$\begin{aligned} {C_{t}^T}^{**}= & {} \omega _s^{**} \left( p_{T,t}^{**} \right) ^{-\rho _s}C_t^{**} \end{aligned}$$

(B.101)

$$\begin{aligned} {C_{t}^N}^{**}= & {} (1-\omega _s^{**}) \left( p_{N,t}^{**} \right) ^{-\rho _s}C_t^{**} \end{aligned}$$

(B.102)

$$\begin{aligned} {M^T_N}^{**}= & {} {\nu _{p{N}}^W}\left( \frac{p_{N}^{**}p_{WM,t}^{**} }{p_{T,t}^{**}p_{WT,t}^{**}}\right) ^{-\rho _m} {M^T}^{**} \end{aligned}$$

(B.103)

$$\begin{aligned} {M^T_H}^{**}= & {} {\nu _{pH}^W}\left( \frac{1}{{\mathcal {T}}_H^W} p_{WM,t}^{**}\right) ^{-\rho _m} {M^T}^{**} \end{aligned}$$

(B.104)

$$\begin{aligned} {M^T_F}^{**}= & {} {\nu _{pF}^W} \left( \frac{p_{Ft}^W}{{\mathcal {Q}}_{Wt}^H}\frac{p_{WM}^{**}}{p_{WT,t}^{**}p_{T,t}^{**}} \right) ^{-\rho _m} {M^T}^{**} \end{aligned}$$

(B.105)

$$\begin{aligned} {M^T_W}^{**}= & {} {\nu _{pW}^W} \left( p_{WM,t}^{**} \right) ^{-\rho _m} {M^T}^{**} \end{aligned}$$

(B.106)

$$\begin{aligned} {M^N_N}^{**}= & {} {\nu _{p{N}}^{N^{**}}}\left( p_{NM,t}^{**} \right) ^{-\rho _m} {M^N}^{**} \end{aligned}$$

(B.107)

$$\begin{aligned} {M^N_H}^{**}= & {} {\nu _{p{H}}^{N^{**}}}\left( \frac{p_{T,t}^{**}p_{WT,t}^{**}}{p_{N,t}^{**}}\frac{p_{NM,t}^{**}}{{\mathcal {T}}_{H,t}^W} \right) ^{-\rho _m} {M^N}^{**} \end{aligned}$$

(B.108)

$$\begin{aligned} {M^N_F}^{**}= & {} {\nu _{p{F}}^{N^{**}}}\left( \frac{p_{Ft}^W}{{\mathcal {Q}}_{Wt}^H}\frac{p_{NM,t}^{**}}{p_{N,t}^{**}} \right) ^{-\rho _m} {M^N}^{**} \end{aligned}$$

(B.109)

$$\begin{aligned} {M^N_W}^{**}= & {} {\nu _{p{W}}^{N^{**}}}\left( \frac{p_{T,t}^{**}p_{WT,t}^{**}}{p_{N,t}^{**}} p_{NM,t}^{**} \right) ^{-\rho _m} {M^N}^{**} \end{aligned}$$

(B.110)

$$\begin{aligned} p_{WT,t}^{**}= & {} \left[ \frac{1-\nu _F^W {\left( \frac{p_F^W}{p_T{**}}\frac{1}{{\mathcal {Q}}_W^H}\right) }^{1-\rho _c}}{\nu _H^W {\left( \frac{1}{{\mathcal {T}}_H^W} \right) }^{1-\rho _c} +\nu _W^W} \right] ^{\frac{1}{1-\rho _c}} \end{aligned}$$

(B.111)

$$\begin{aligned} p_{N,t}^{**}= & {} \left[ \frac{1-\omega _s^{**} (p_T^{**})^{1-\rho _s}}{1-\omega _s^{**}}\right] ^{\frac{1}{1-\rho _s}} \end{aligned}$$

(B.112)

$$\begin{aligned} {p_{WM,t}^{**}}= & {} \left[ \nu _{pN}^W \left( \frac{p_{N}^{**} }{p_{T,t}^{**}p_{WT,t}^{**}}\right) ^{1-\rho _m} +\nu _{pH}^W \left( \frac{1}{{\mathcal {T}}_{Ht}^W}\right) ^{1-\rho _m} \right. \nonumber \\{} & {} \left. + \nu _{pF}^W \left( \frac{p_{Ft}^W}{{\mathcal {Q}}_{Wt}^H p_{WT,t}^{**}p_{T,t}^{**}}\right) ^{1-\rho _m}+\nu _{pW}^W\right] ^{-\frac{1}{1-\rho _m}} \end{aligned}$$

(B.113)

$$\begin{aligned} p_{NM,t}^{**}= & {} \left[ \nu _{p{N}}^{N^{**}} + \nu _{pH}^{N^{**}} \left( \frac{p_{T,t}^{**}p_{WT,t}^{**}}{p_{N,t}^{**}{\mathcal {T}}_{H,t}^W}\right) ^{1-\rho _m} \right. \nonumber \\{} & {} \left. + \nu _{pF}^{N^{**}}\left( \frac{p_{Ft}^W}{{\mathcal {Q}}_{Wt}^H p_{N,t}^{**}} \right) ^{1-\rho _m} + \nu _{pW}^{N^{**}}\left( \frac{p_{T,t}^{**}p_{WT,t}^{**}}{p_{N,t}^{**}} \right) ^{1-\rho _m} \right] ^{-\frac{1}{1-\rho _m}} \end{aligned}$$

(B.114)

$$\begin{aligned} {Y^T}^{**}= & {} \overline{{K^T}^{**}}^\alpha ({Z_t^T}^{**} {L_t^T}^{**})^{\gamma _p} ({M^T}^{**})^{1-\alpha -\gamma _p} \end{aligned}$$

(B.115)

$$\begin{aligned} w^{**}= & {} \frac{\gamma _p}{\alpha } \frac{{{\overline{K}}^T}^{**}}{{L^T_t}^{**}}\frac{{r_{K,t}^T}^{**} }{1+(R^n_{t}-1) \theta _w} \end{aligned}$$

(B.116)

$$\begin{aligned} p_{WT,t}^{**}p_{T,t}^{**}/p_{WM,t}^{**}= & {} \frac{1-\alpha -\gamma _p}{\alpha }\frac{{{\overline{K}}^T}^{**}}{{M_t^T}^{**}} \frac{{r_{K,t}^T}^{**} }{1+(R^n_{t}-1) \theta _w} \end{aligned}$$

(B.117)

$$\begin{aligned} {mc^T_t}^{**}= & {} \left( \frac{w_t^{**}/{Z_t^T}^{**}[1+(R^n_{t}-1)\theta _w]}{\gamma _p} \right) ^{\gamma _p} \left( \frac{{r_{K,t}^T}^{**}}{\alpha } \right) ^\alpha \nonumber \\{} & {} \left( \frac{p_{WT,t}^{**}p_{T,t}^{**}/p_{WM,t}^{**}[1+(R^n_{t}-1)\theta _w]}{1-\alpha -\gamma _p}\right) ^{1-\alpha -\gamma _p} \end{aligned}$$

(B.118)

$$\begin{aligned} {\pi _{t}^{**}}^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) ({\pi _{t}^*}^o)^{-\frac{1}{\theta _p}} + \xi _p(\pi _{t-1}^*)^{-\frac{\iota _p}{\theta _p}} \end{aligned}$$

(B.119)

$$\begin{aligned} {\pi _{t}^{**}}^o= & {} (1+\theta _p) \frac{x_{1T,t}^{**}}{x_{2T,t}^{**}}\pi _t^{**} \end{aligned}$$

(B.120)

$$\begin{aligned} x_{1T,t}^{**}= & {} {\omega _s^{**}}^{\frac{1}{\rho _s}} {C_t^{**}}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^{**}}^{-\frac{1}{\rho _s}} {mc_t^T}^{**} {Y_{t}^T}^{**} \nonumber \\{} & {} + \beta ^{**} \xi _p {\pi _t^{**}}^{-\iota _p \frac{1+\theta _p}{\theta _p}} {\mathbb {E}}_t \left\{ x_{1T,t+1}^{**} {\pi _{t+1}^{**}}^{\frac{1+\theta _p}{\theta _p}}\right\} \end{aligned}$$

(B.121)

$$\begin{aligned} x_{2T,t}^{**}= & {} {\omega _s^{**}}^{\frac{1}{\rho _s}} {C_t^{**}}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{T,t}^{**}}^{-\frac{1}{\rho _s}} p_{WT,t}^{**}p_{T,t}^{**} {Y_t^T}^{**} \nonumber \\{} & {} + \beta ^* \xi _p {\pi _t^{**}}^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p}\right) } {\mathbb {E}}_t \left\{ x_{2T,t+1}^{**} {\pi _{t+1}^{**}}^{\frac{1+\theta _p}{\theta _p} - 1}\right\} \end{aligned}$$

(B.122)

$$\begin{aligned} {Y^N}^{**}= & {} \overline{{K^N}^{**}}^\alpha (Z_t^{**} {L_t^N}^{**})^{\gamma _p}({M^N}^{**})^{1-\alpha -\gamma _p} \end{aligned}$$

(B.123)

$$\begin{aligned} w^{**}= & {} \frac{\gamma _p}{\alpha } \frac{{{\overline{K}}^N}^{**}}{{L^N_t}^{**}}\frac{{r_{K,t}^N}^{**} }{1+(R^n_{risk,wt}-1) \theta _w} \end{aligned}$$

(B.124)

$$\begin{aligned} p_{N,t}^{**}/p_{NM,t}^{**}= & {} \frac{1-\alpha -\gamma _p}{\alpha }\frac{{{\overline{K}}^N}^{**}}{{M_t^N}^{**}} \frac{{r_{K,t}^N}^{**} }{1+(R^n_{risk,wt}-1) \theta _w} \end{aligned}$$

(B.125)

$$\begin{aligned} {mc^N_t}^{**}= & {} \left( \frac{w_t^{**}/{Z_t^N}^{**}[1+(R^n_{risk,wt}-1)\theta _w]}{\gamma _p} \right) ^{\gamma _p} \left( \frac{{r_{K,t}^N}^{**}}{\alpha } \right) ^\alpha \nonumber \\{} & {} \left( \frac{p_{N,t}^{**}/p_{NM,t}^{**}[1+(R^n_{risk,wt}-1)\theta _w]}{1-\alpha -\gamma _p}\right) ^{1-\alpha -\gamma _p} \end{aligned}$$

(B.126)

$$\begin{aligned} {{\pi _{t}^N}^{**}}^{-\frac{1}{\theta _p}}= & {} (1-\xi _p) ({{\pi _{t}^N}^{**}}^o)^{-\frac{1}{\theta _p}} + \xi _p({\pi _{t-1}^N}^{**})^{-\frac{\iota _p}{\theta _p}} \end{aligned}$$

(B.127)

$$\begin{aligned} {{\pi _{t}^N}^{**}}^o= & {} (1+\theta _p) \frac{x_{1N,t}^{**}}{x_{2N,t}^{**}} {\pi _t^N}^{**} \end{aligned}$$

(B.128)

$$\begin{aligned} x_{1N,t}^{**}= & {} (1-\omega _s^{**})^{\frac{1}{\rho _s}} {C_t^{**}}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{N,t}^{**}}^{-\frac{1}{\rho _s}} {mc_t^N}^{**} {Y_{t}^N }^{**}\nonumber \\{} & {} + \beta ^{**} \xi _p {{\pi _t^N}^{**}}^{-\iota _p \frac{1+\theta _p}{\theta _p}} {\mathbb {E}}_t \left\{ x_{1N,t+1}^{**} {{\pi _{t+1^N}}^{**}}^{\frac{1+\theta _p}{\theta _p}} \right\} \end{aligned}$$

(B.129)

$$\begin{aligned} x_{2N,t}^{**}= & {} (1-\omega _s^{**})^{\frac{1}{\rho _s}} {C_t^{**}}^{\frac{1}{\rho _s}-\frac{1}{\sigma }} {C_{N,t}^{**}}^{-\frac{1}{\rho _s}} p_{N,t}^{**} {Y_t^N}^{**} \nonumber \\{} & {} + \beta ^{**} \xi _p {{\pi _t^N}^{**}}^{\iota _p \left( 1-\frac{1+\theta _p}{\theta _p}\right) } {\mathbb {E}}_t \left\{ x_{2N,t+1}^{**} {{\pi _{t+1}^N}^{**}}^{\frac{1+\theta _p}{\theta _p} - 1} \right\} \end{aligned}$$

(B.130)

$$\begin{aligned} {\pi _t^{**}}^C= & {} \pi _t^{**} \frac{p_{WT,t-1}^{**}p_{T,t-1}^{**}}{p_{WT,t}^{**}p_{T,t}^{**}} \end{aligned}$$

(B.131)

$$\begin{aligned} {\pi _t^{**}}^C= & {} {\pi _t^N}^{**}\frac{p_{Nt-1}^{**}}{p_{N,t}^{**}} \end{aligned}$$

(B.132)

$$\begin{aligned} {Y_{t}^T}^{**}= & {} C_{Wt}^{**} + {M^T_{Wt}}^{**} + {M^N_{Wt}}^{**} + \frac{n_H}{n_W}\left( C_{Wt} + M^T_{Wt} + M^N_{Wt} \right) \nonumber \\{} & {} + \frac{n_F}{n_W}\left( C_{Wt}^{*} + {M^T_{Wt}}^{*}+ {M^N_{Wt}}^{*}\right) \end{aligned}$$

(B.133)

$$\begin{aligned} L^{**}= & {} {L^T}^{**} + {L^N}^{**} \end{aligned}$$

(B.134)

$$\begin{aligned} {Y^N}^{**}= & {} {C^N}^{**} + {M^N_N}^{**} + {M^T_N}^{**} \end{aligned}$$

(B.135)

$$\begin{aligned} {R_t^{**}}^n= & {} {\beta ^{**}}^{-1} {{\pi ^C_t}^{**}}^{\gamma _{\pi }} \end{aligned}$$

(B.136)

where $p_{WT,t}^{**} \equiv \frac{P_{W,t}^{**} }{{P^T_{t}}^{**}}$, ${ p^T_{WM,t}}^{**} \equiv \frac{{P_{W,t}}^{**} }{{P^T_{M,t}}^{**}}$, ${ p^T_{NM,t}}^{**} \equiv \frac{{P^N_{t}}^{**} }{{P^N_{M,t}}^{**}}$, $p^{**}_{N,t} \equiv \frac{{P^N_t}^{**}}{P_{C,t}^{**}}$, ${\pi ^C_{t}}^{**} \equiv \frac{P_{C,t}^{**}}{P_{C,t-1}^{**}}$, $\pi ^{**}_{t} \equiv \frac{ P_{W,t}^{**}}{P_{W,t-1}^{**}}$, and ${\pi ^N_{t}}^{**} \equiv \frac{{P^N_{t}} ^{**}}{{P^N_{t-1}}^{**}}$.

Equations Common to all countries :

$$\begin{aligned} {\mathcal {Q}}_{Ft}^H= & {} \frac{1}{{\mathcal {T}}_{Ft}^H} \frac{p_{HT,t}p_{T,t}}{p_{FT,t}^* p_{T,t}^*}\end{aligned}$$

(B.137)

$$\begin{aligned} {\mathcal {Q}}_{Wt}^H= & {} {\mathcal {T}}_{Ht}^W \frac{p_{HT,t}p_{T,t}}{p_{WT,t}^{**}p_{T,t}^{**}} \end{aligned}$$

(B.138)

$$\begin{aligned} {\mathcal {T}}_{Ft}^W= & {} \frac{{\mathcal {Q}}_{Wt}^H}{{\mathcal {Q}}_{Ft}^H}\frac{p_{WT,t}^{**} p_{T,t}^{**}}{p_{FT,t}^* p_{T,t}^*} \end{aligned}$$

(B.139)

$$\begin{aligned} \Lambda _{t,t+1}= & {} \Lambda _{t,t+1}^* \frac{{\mathcal {Q}}_{Ft}^H}{{\mathbb {E}}_t {\mathcal {Q}}_{Ft+1}^H} \phi _{Ht}^* \end{aligned}$$

(B.140)

$$\begin{aligned} \Lambda _{t,t+1}= & {} \Lambda _{t,t+1}^{**} \frac{{\mathcal {Q}}_{Wt}^H}{{\mathbb {E}}_t {\mathcal {Q}}_{Wt+1}^H} \phi _{Ht} ^{**} \end{aligned}$$

(B.141)

$$\begin{aligned}&p_{FT,t}^* p_{T,t}^* \left[ {\mathcal {T}}_{Ft}^H (C_{Ht}^*+{M^T_{Ht}}^* +{M^N_{Ht}} ^*) + {\mathcal {T}}_{Ft}^W (C_{Wt}^*+{M^T_{Wt}}^*+{M^N_{Wt}}^*)\right] \nonumber \\&\qquad -\frac{1}{ {\mathcal {Q}}_{Ft}^H} \left( \frac{n_H}{n_F} p_{Ft} N_t + \frac{n_W}{n_F}p^W_{Ft} N^W_t\right) \nonumber \\&\quad =\frac{1}{{\mathcal {Q}}_{Ft}^H} \left( \frac{B_{t-1}^*}{\pi _t^C} - \frac{ B_{t}^* }{R^n_{t} \phi _{Ht}^* }\right) - (R^n_{risk,ft}-1) \theta _w \left( \frac{ p_{FT,t}^*p_{T,t}^*}{p_{FM,t}^*} {M_t^T}^*+\frac{p_{N,t}^*}{p_{NM,t}^*} { M_t^N}^*+w_t^* L_t^* \right) \end{aligned}$$

(B.142)

$$\begin{aligned}&p_{WT,t}^{**} p_{T,t}^{**} \left[ \frac{1}{{\mathcal {T}}_{Ht}^W} (C_{Ht}^{**} + {M^T_{Ht} }^{**}+ {M^N_{Ht}}^{**}) - \frac{n_H}{n_W}( C_{Wt} + M^T_{Wt}+ M^N_{Wt})\right. \nonumber \\&\qquad \left. -\frac{n_F}{n_W} (C_{Wt}^{*} + {M^T_{Wt}}^{*}+ {M^N_{Wt}}^{*}) \right] + \frac{ p^W_{F,t}}{{\mathcal {Q}}_{Wt}^H } N^W_t \nonumber \\&\quad = \frac{1}{{\mathcal {Q}}_{Wt}^H} \left( \frac{B_{t-1}^{**}}{\pi _t^C} - \frac{ B_{t}^{**}}{R^n_{t} \phi _{Ht}^{**} }\right) - (R^n_{t}-1) \theta _w \left( \frac{p_{WT,t}^{**}p_{T,t}^{**}}{p_{WM,t}^{**}} {M_t^T}^{**}+ \frac{p_{N,t}^{**}}{p_{NM,t}^{**}} {M_t^N}^{**}+w_t^{**} L_t^{**} \right) \end{aligned}$$

(B.143)

where $\Lambda _{t,t+1} \equiv \beta \left( \frac{{\mathbb {E}}_t C_{t+1}}{C_t} \right) ^{- \frac{1}{\sigma }}$, $\Lambda _{t,t+1}^* \equiv \beta ^*\left( \frac{{\mathbb {E}}_t C_{t+1}^*}{ C_t^*} \right) ^{-\frac{1}{\sigma }}$, $\Lambda _{t,t+1}^{**} \equiv \beta ^{**}\left( \frac{{\mathbb {E}}_t C_{t+1}^{**}}{C_t^{**}} \right) ^{-\frac{1}{\sigma }}$, $\phi _{H,t}^* = exp \left( -\nu _H^*\frac{B_{t}^*}{{\mathcal {Q}}_{Ft}^H {Y_t}^* } \right)$, $\phi _{H,t}^{**} = exp \left( -\nu _H^{**}\frac{B_{t}^{**}}{\mathcal { Q}_{Wt}^H {Y_t}^{**} } \right)$, $R^n_{risk,ft} = R^n_t + \left( \frac{S_{F,t-1}^H}{S_{Ft}^H}-1\right)$, and ${\mathcal {Q}}_{Ft}^H \equiv \frac{ S_{Ft}^H P_{C,t}^* }{P_{C,t}}$, ${\mathcal {Q}}_{Wt}^H \equiv \frac{S_{Wt}^H P_{C,t}^{**} }{P_{C,t}}$, ${\mathcal {T}}_{Ft}^H \equiv \frac{P_{Ht}^*}{P_{Ft}^* }$, ${\mathcal {T}}_F^W \equiv \frac{P_W^*}{P_F^*}$, ${\mathcal {T}}_{Wt}^H \equiv \frac{P_{Wt}}{P_{Ht}}\equiv \frac{P_{Wt}^{**}}{P_{Ht}^{**}}$, $B_t^* = \frac{ B_{Ht}^*}{P_{C,t}}$, and $B_t^{**} = \frac{B_{Ht}^{**}}{P_{C,t}}$.

List of Variables (143): $C_t$, $C_t^*$, $C_t^{**}$, $C_t^T$, ${ C_t^T}^*$, ${C_t^T}^{**}$, $C_t^N$, ${C_t^N}^*$, ${C_t^N}^{**}$, $C_{H,t}$, $C_{H,t}^*$, $C_{H,t}^{**}$, $C_{F,t}$, $C_{F,t}^*$, $C_{F,t}^{**}$, $C_{W,t}$ , $C_{W,t}^*$, $C_{W,t}^{**}$, $M^T_t$, ${M^T_t}^*$, ${M^T_t}^{**}$, $M^N_t$ , ${M^N_t}^*$, ${M^N_t}^{**}$, ${M^T_{N,t}}$, ${M^T_{N,t}}^*$, ${M^T_{N,t}} ^{**}$, ${M^T_{H,t}}$, ${M^T_{H,t}}^*$, ${M^T_{H,t}}^{**}$, ${M^T_{F,t}}$, ${ M^T_{F,t}}^*$, ${M^T_{F,t}}^{**}$, ${M^T_{W,t}}$, ${M^T_{W,t}}^*$, ${ M^T_{W,t}}^{**}$, ${M^N_{N,t}}$, ${M^N_{N,t}}^*$, ${M^N_{N,t}}^{**}$, ${ M^N_{H,t}}$, ${M^N_{H,t}}^*$, ${M^N_{H,t}}^{**}$, ${M^N_{F,t}}$, ${M^N_{F,t}} ^*$, ${M^N_{F,t}}^{**}$, ${M^N_{W,t}}$, ${M^N_{W,t}}^*$, ${M^N_{W,t}}^{**}$, $Y_t^T$, ${Y_t^T}^*$, ${Y_t^T}^{**}$, $Y_t^N$, ${Y_t^N}^*$, ${Y_t^N}^{**}$, ${Y^T_{D,t}}^*$, $N_t$, $N_t^W$, $L_t$, $L_t^*$, $L_t^{**}$, $L_t^T$, ${L_t^T} ^*$, ${L_t^T}^{**}$, $L_t^N$, ${L_t^N}^*$, ${L_t^N}^{**}$, $w_t$, $w_t^*$, $w_t^{**}$, $r_{K,t}^T$, ${r_{K,t}^T}^*$, ${r_{K,t}^T}^{**}$, $r_{K,t}^N$, ${ r_{K,t}^N}^*$, ${r_{K,t}^N}^{**}$, $mc_t^T$, ${mc_t^T}^*$, ${mc_t^T}^{**}$, $mc_t^N$, ${mc_t^N}^*$, ${mc_t^N}^{**}$, $p_{HT,t}$, $p_{FT,t}^*$, $p_{WT,t}^{**}$, $p_{HM,t}$, $p_{FM,t}^*$, $p_{WM,t}^{**}$, $p_{NM,t}$, $p_{NM,t}^*$, $p_{NM,t}^{**}$, $p_{T,t}$, $p_{T,t}^*$, $p_{T,t}^{**}$, $p_{N,t}$, $p_{N,t}^*$, $p_{N,t}^{**}$, $p_{F,t}$, $p^W_{F,t}$, $R^n_t$, ${ R^n_t}^*$, ${R^n_t}^{**}$, $\pi ^C_t$, ${\pi ^C_t}^*$, ${\pi ^C_t}^{**}$, $\pi _t$, ${\pi _t}^*$, ${\pi _t}^{**}$, $\pi _t^N$, ${\pi _t^N}^*$, ${\pi _t^N} ^{**}$, $\pi _{F,t}$, $\pi ^W_{F,t}$ $\pi ^0_t$, ${\pi ^0_t}^*$, ${\pi ^0_t}^{**}$ , ${\pi ^N}^0_t$, ${{\pi ^N}^0_t}^*$, ${{\pi ^N}^0_t}^{**}$, $\pi _{F,t}^0$, $\pi _{F,t}^{0W}$, $x_{1T,t}$, $x_{1T,t}^*$, $x_{1T,t}^{**}$, $x_{1N,t}$, $x_{1N,t}^*$, $x_{1N,t}^{**}$, $z_{1,t}$, $z^W_{1,t}$, $x_{2T,t}$, $x_{2T,t}^*$, $x_{2T,t}^{**}$, $x_{2N,t}$, $x_{2N,t}^*$, $x_{2N,t}^{**}$, $z_{2,t}$, $z^W_{2,t}$, ${{\mathcal {T}}_t}_F^H$, ${{\mathcal {T}}_t}_F^W$, ${ {\mathcal {T}}_t}_H^W$, ${\mathcal {Q}}_{Ft}^H$, ${\mathcal {Q}}_{Wt}^H$, $B_t^*$, $B_t^{**}$.