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Implementable Rules for International Monetary Policy Coordination

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Abstract

While economic and financial integration can increase welfare, it can also complicate the policy problem, bringing about new trade-offs or magnifying the existing ones. These policy challenges can be particularly severe in the presence of large (gross) capital flows, e.g., for emerging market economies. Having a quantitative assessment of these trade-offs is key for the design of effective policy interventions. Using a Core-Periphery open-economy DSGE model with financial and nominal frictions, we offer a quantitative assessment of the implied trade-offs and of operational policy targeting rules that can bring about the highest global welfare. These rules need to trade off domestic inflation variability with foreign factors and financial imbalances. We suggest a novel approach, based on the SVAR literature, that can be used to represent targeting rules in rich models.

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Notes

  1. The empirical evidence on these effects remains ambiguous. Prasad et al. (2007) found little support for this theoretical prediction, although they point to confounding factors (governance, lack of regulation, etc.) as a key source of meager empirical evidence on the benefits of globalization.

  2. See, for example, Ball (1999), Taylor (2001), Laxton and Pesenti (2003), Kollmann (2002), Leitemo and Söderström (2005), Batini et al. (2003), Devereux and Engel (2003), Adolfson (2007), Svensson (2000), Devereux et al. (2006), Kirsanova et al. (2006), Wollmershäuser (2006), Galí and Monacelli (2005) and Pavasuthipaisit (2010).

  3. In the context of our model, two issues make this strategy impractical. First, even assuming that each country follows a three-parameter Taylor-type rule (where the interest rate responds to the lagged interest rate, the inflation rate and output growth), the optimization algorithm converges to singularity points of the welfare function, with the latter assuming unreasonably large values: Close to singularity points, the accuracy of the second-order approximation is bound to be very poor. Arbitrarily constraining the admissible range of the parameters leads to corner solutions. Second, to keep computation times within hours instead of days (on a 12 cores 2.9 GHz Intel i9 HP workstation) the numerical search of the second-order accurate welfare maximum has to be reduced to a small number of parameters (around \(3\times 2\)), compared to the simple rule we discuss in this paper (\(26\times 2\)). Using the methodology described below considerably reduces the computational burden.

  4. This section summarizes the key insights, relevant to our analysis, of Giannoni and Woodford (2003a, b) and of the SVAR literature (e.g., see Rubio-Ramirez et al. 2010, for a recent survey).

  5. This is obviously possible only if certain parameter restrictions are satisfied, so that the sum converges. This would be the case for small enough values of \(\rho\) in our example.

  6. Beyond this simple model, even eliminating all the Lagrange multipliers could leave us with a rule involving a very large number of variables. One further dimension of approximation would then consist of replacing some of the endogenous variables too. In this case, the accuracy of the approximation would inform us about which variables play a more important role in the rule (at the selected lag length).

  7. We also note that although DSGE models cannot always be represented by finite-order VARS, this is the approximation employed in empirical work. In addition, if the real-world data correspond to the model, then the misspecification should apply both to the data and the simulations generated by the model (see Cogley and Nason 1995). In our case, the “data” correspond to the simulations from the model under the optimal rule, which can then be compared to those generated using the estimated targeting rule.

  8. More details about this model can be found in Banerjee et al. (2015). The whole list of equations is provided in “Appendix 5”.

  9. In the remainder of the paper, to simplify the discussion, we will refer to capital goods financiers in both the center and peripheral countries as banks. It should be noted, however, that the key thing that distinguishes them is that they make levered investments and are subject to contract enforcement constraints. In this view, they need not be literally banks in the strict sense.

  10. This assumption is meant to capture the feature that within-country financial intermediation between savers and investors is more difficult in EMEs than in advanced economies (see, e.g., Mendoza et al. 2009). The assumption emphasizes the strong influence that core-country financial conditions exert on EME financial markets. As discussed below, EME domestic deposits act mainly to reduce the financial spillover but do not alter the qualitative properties of the cross-border transmission channel. Therefore, in the main results we assume that domestic deposits in EME are constrained to be zero. See also Cuadra and Nuguer (2018).

  11. We assume that the market for center country nominal bonds is frictionless. Adding additional frictions that limit the ability of emerging market households to invest in center country nominal bonds would just exacerbate the impact of financial frictions that are explored below.

  12. In particular, we are not assuming a special role for government debt, nor asymmetries in the degree to which the contract between depositors and core-country banks can be enforced.

  13. Home bias is adjusted to take into account of country size. In particular, for a given degree of openness \(x \le 1\), \(v^{\mathrm{e}}=1-x(1-n)\), and a similar transformation for the center country home bias parameter.

  14. Equivalently, we could assume that the bank provides risky loans to intermediate-goods producers, who use the funds to purchase capital. The only risk of this loan concerns the (real) gross return on the underlying capital stock.

  15. See, for example, Coenen et al. (2009).

  16. Note that with this approach, the model would have to be evaluated (to second order) for each draw of the parameters.

  17. To gain intuition on the sources of approximation error, we study cases with \(p>2\).

  18. More precisely in those equations, inflation appears as \(\pi _{i,{\mathrm{GDP}},t}+\varepsilon _{R_i,t}\); \(i=\{e,c\}\), where \(\varepsilon _{R_i,t}\) is an i.i.d. shock.

  19. See also Leeper et al. (1996, p. 12).

  20. Whether it emerges in practice depends on the correct specification of the lag structure too. In the basic New Keynesian model inflation and output depends only on current TFP so that perfect multicollinearity would make estimation of the tree variable VAR impossible.

  21. This is obtained rewriting the targeting rule as \(\frac{a_1 \pi _{t}^{{\mathrm{PPI}},j}+a_2 \pi _{t-1}^{{\mathrm{PPI}},j}+a_3 \pi _{t-2}^{{\mathrm{PPI}},j}}{a_1+a_2+a_3}=\frac{1}{a_1+a_2+a_3}\left( \text {rest of terms}\right)\), where \(j=\{e,c\}\) and where \(a_{1\ldots 3}\) are the coefficients of inflation.

  22. As argued above, these rules are not unique as linear combination of other variables could perfectly proxy some of the variables included in the rule.

  23. A residual change in the EME spread under these assumptions reflects roundabout flow of funds financing EME’s capital.

  24. Recently, Arias et al. (2019) exploit this fact to impose identifying restrictions consisting of priors about the way policy responds to endogenous variables.

  25. Although we are only interested in one row of the VAR, we need the full \({A}_0\) matrix.

  26. We consider \(N=\infty\) and \(N=2\).

  27. Gertler and Kiyotaki (2010) and Gertler and Karadi (2011) show that under the assumption underlying the bank problem, the value function \(J_{t}^j\), \(j=\{e,c\}\), is linear in net worth. This allows aggregation across agents and implies that \(J_{i,t}^{{\mathrm{e}}\prime }=J_{i,t}^{\mathrm{e}}N_{i,t}^{{\mathrm{e}}}.\)

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Correspondence to Charles Engel.

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This paper was prepared for the 19th Jacques Polak Annual Research Conference on “International Spillovers and Cooperation”; Washington, D.C., November 1–2, 2018. We thank Steve Wu and Nikhil Patel and the anonymous referees for useful comments and suggestions, and Emese Kuruc and Burcu Erik for collecting the data. The views expressed in this paper do not necessarily reflect the views of the Bank for International Settlements.

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Appendices

Appendix 1: Parameter Values

The values of the key parameters of the model are fixed at values prevailing in the related literature, with two exceptions. First, it turned out that in order to match the relative volatility of the investment share, a larger cost of investment than typically used in the literature was needed. We choose a value that is about twice as large as in the literature. Second, in order to improve the estimation of the VAR on simulated data, we opted for mildly autoregressive exogenous shocks and assigned a value of 0.5 to the persistence parameter (see Table 5).

Table 5 Parameter values

Appendix 2: Data Moments

All the structural parameters of our model are fixed at values prevailing in the related literature. In order to ensure that the relative volatility of key variables of the model is commensurate with their empirical counterpart, we estimate the standard deviations of the three key shocks underlying our experiments (productivity, financial and policy shocks) using an SMM approach. We use various data sources. For US credit spreads, we use the “GZ” spreads computed by Gilchrist and Zakrajšek (2012) (see also the Fed’s updates of these series; Favara et al. 2016). The standard deviation of these spreads hovers around 100 basis points, depending on frequency (they are available at monthly and quarterly frequency) and on the sample period. For EMEs, we use similar series constructed by Caballero et al. (2019). The standard deviations of these variables are reported in Table 6.

For the remaining macroeconomic variables, we built a panel of quarterly data (1990Q1–2019Q2) from various sources as reported in Table 7. For comparison, we report also standard deviations for the euro area, although we use only the USA as a representative AE when estimating the shocks. Since interest rates and inflation display a persistent downward trend, we subtract a linear trend from these series before computing standard deviations. Since EMEs displayed periods of heightened volatility, for robustness we controlled for outliers, but did not find economically significant differences in the moments of interest.

Since EMEs appear to be more heterogeneous, limiting attention to the median (or average) standard deviations would be inappropriate. We thus compute three percentiles (10, 50 and 90%) as well as the mean and two measures of dispersion (standard deviations and interquartile range).

We use these data to estimate the standard deviation of the three sets of shocks. Table 8 compares the moments produced by the model at the estimated parameters with the empirical counterparts. (The values of the standard deviations of the structural shocks are shown in Table 9.) Considering that we estimate six parameters using 11 moments (while keeping all other parameters constant), the model performs relatively well. For the EME, the model generates standard deviations below the tenth percentile only the real exchange rate and for interest rates, albeit less significantly. Inflation and the policy rate are also poorly matched for the AE. Fine-tuning the policy rule could further improve on this margin. Nevertheless, as the main focus of our paper is normative, we consider these results sufficiently satisfactory.

Table 6 Standard deviations of EMEs credit spreads (basis points).
Table 7 Data: standard deviations.
Table 8 Model versus data: standard deviations (%)
Table 9 Estimated standard deviation of shocks (in percent)

Appendix 3: Identification Via IRFs Matching

Let us recall that in order to determine the policy equation (rule) in the VAR, we need to identify the monetary policy shock (Leeper et al. 1996, p. 9).Footnote 24

We achieve identification by minimizing the distance between the “true” IRFs to a policy shock (based on the DSGE under optimal policy) and the IRFs to a policy shock obtained from the VAR estimated on simulated data. Our IRFs matching (IRM) approach can be seen as an extension of the sign-restriction identification method used in the empirical VAR literature (e.g., Uhlig 2005; Rubio-Ramirez et al. 2010). In this section, we only explain the main differences with respect to the standard sign-restriction (SR) approach.

Both the IRM and SR approaches start from an estimated VAR equation

$$\begin{aligned} y_t=A y_{t-1}+u_t \end{aligned}$$
(9.1)

where \(u_t\) is a vector of reduced-form shocks.

Both approaches seek to identify one or more structural shocks (elements) in the vector \(\varepsilon _t\) such that \(u_t=A_0^{-1}\varepsilon _t\), where the (contemporaneous coefficient) matrix \(A_0\) is unknown. The first key step in the identification amounts to finding the matrix \(A_0\). The second step amounts to identifying which element in \(\varepsilon _t\) corresponds to the shock of interest. To this aim, note that a factorization of the covariance matrix or the residual shocks can be written as

$$\begin{aligned} E(u_t u_t^\prime )\equiv \Sigma _{u}=PQQ^\prime P^\prime \end{aligned}$$
(9.2)

where P is a factorization of \(\Sigma _{u}\), (e.g., Cholesky) and Q is an orthonormal rotation matrix such that \(QQ^\prime =I\) (e.g., Canova and De-Nicoló 2002).

\(\Sigma _{u}\) offers a set of \(\frac{n\left( n+1\right) }{2}\) independent parameters, while the matrix \({A}_0^{-1}\equiv PQ\) has \(n^2\) parameters. In order to determine the remaining \(n^2-\frac{n\left( n+1\right) }{2}\) parameters, we need \(\frac{n\left( n-1\right) }{2}\) restrictions (order condition).Footnote 25

The standard SR approach finds the missing restrictions by imposing that the sign of the VAR IRFs (on impact or over a longer horizon) matches the sign of IRFs suggested, e.g., by theory. Here is where our IRM approach differs. We know exactly the value of the IRFs of each variable and for any horizon to a monetary policy shock: Our DGP is the DSGE. We can then obtain the missing \(\frac{n\left( n-1\right) }{2}\) restrictions by minimizing the distance of the VAR IRFs to the “true” IRFs to a monetary policy shock.

Since we have n variables, with h IRFs periods per variable we have \(n\times h\) IRF-gap points. Then, we need \(h^*=n/2 - 1/2\) periods per variable for an exact identification (i.e., h such that \(n\times h=\frac{n\left( n-1\right) }{2}\)). Thus, for example, with nine variables we can reach exact identification with four periods per variable. These points do not need to be contiguous. For example, with nine variables, we choose to match the first two periods together with the fourth and sixth periods.

Let’s define the difference between IRFs as

$$\begin{aligned} \nu _h\equiv \left[ \left( x_{1}-{\widetilde{x}}_{1}\right) , \ldots , \left( x_{h}-{\widetilde{x}}_{h}\right) \right] \end{aligned}$$
(9.3)

so that \(\nu _h\) is a \(n\times h\) matrix, \(x_{t}\) is the \(n\times 1\) vector of VAR IRFs and \({\widetilde{x}}_{t}\) is the \(n\times 1\) vector of target IRFs (i.e., from the DGP). If \(h=h^*\), \(\nu _h\) contains exactly \(\frac{n\left( n-1\right) }{2}\) elements: the missing restrictions.

Since an exact solution (\(\nu _h=0\)) is unlikely to exist, due to omitted variables in the VAR, we look for a solution that minimizes the IRFs gap, i.e.,

$$\begin{aligned} Q=\arg \min _{Q} \left\| {\mathbb {W}}\nu _h\right\| _{N} \end{aligned}$$
(9.4)

subject to

$$\begin{aligned} PP^\prime =\Sigma _{u} \end{aligned}$$
(9.5)

where \(\Vert \cdot \Vert _N\) is the N norm of a vector and \({\mathbb {W}}\) is a weighting matrix used to give more prominence to particular horizons.Footnote 26 We choose \({\mathbb {W}}\) to penalize sign discrepancies in the first two periods.

The matrix P can be obtained in different ways. The two most common consist of (i) the Cholesky factorization of the estimated variance of the residuals \(u_t\) and (ii) the spectral decomposition (\(E\left( u_t u_t^\prime \right) =PDP^\prime\), where P is the eigenvector matrix and D the diagonal eigenvalue matrix). See Rubio-Ramirez et al. (2010) for details.

We choose the Cholesky factorization, such that the whole set of \(n^2\) restrictions would be

$$\begin{aligned} \mathrm{Restrictions}=\left\{ \begin{matrix} \arg \min _{Q} \left\| {\mathbb {W}}\nu _h\right\| _{N}\\ P=chol(\Sigma _{u}) \end{matrix}\right. \end{aligned}$$
(9.6)

This implies, for example, that the restrictions we are looking for amount to \(\frac{n\left( n-1\right) }{2}\) possible pairwise rotations of the rows of a Givens (Jacobi) matrix Q, for a given rotation angle \(\theta\).

3.1 Appendix 3.1: Using Optimization Algorithms to Match IRFs

A novel methodological contribution of this paper is to use optimization algorithms to find the optimal rotation matrix in the IRFs matching exercise. By using Givens rotations (Golub and Van Loan 1996), we can parametrize the optimization problem (IRFs matching) in terms of the vector of rotation angles \(\theta\). This vector has length \(n_\theta \equiv \frac{n\left( n-1\right) }{2}\), since there are exactly \(n_\theta\) possible permutations of the elements of the innovation vector, and there is one angle per permutation. The matching function in our case appears smooth and continuous along each element of \(\theta\), but it is highly nonlinear moving across angles, with a large number of local minima. We use a suite of optimization algorithms to find the optimal rotation matrix. In particular, we start the search by using a simulated annealing optimization method (implemented in the GenSA package in R by Xiang et al. 2013), followed by a genetic algorithm method (implemented in the GA package in R by Scrucca 2013), and refine the search using a gradient-based method.

Appendix 4: Deterministic Steady State

In order to find the rational-expectation solution of our model using perturbation methods, we need to find the solution of the deterministic steady state of the nonlinear model. Most, but not all, of the variables can be solved recursively. We adopt the following strategy, we search numerically for four variables—labor supply in both countries (\(H^{\mathrm{e}}\) and \(H^{{\mathrm{c}}}\)), the price index of EME goods relative to EME’s CPI (\(P^{{\mathrm{c}}}\)) and the cost of funds for EME banks \(R_{c}\)—that can satisfy four residual conditions. The other variables are solved recursively. The omission of the time (t) subscript denotes steady-state values.

Let’s start with the EME banks. It is convenient to express the value function of the bank relative to its net worth. Note that from the vantage point of the banks existing at time t net worth at time t is total net worth (i.e., \(N^{{\mathrm{e}}}\) sum of surviving banks’ net worth and transfer), while the relevant net worth in the bank surviving in the future is the surviving banks’ net worth (\(N_{i}^{{\mathrm{e}}}\)), i.e.,

$$\begin{aligned} \begin{array}{ll} v_{i}^{{\mathrm{e}}}=&E_t\Lambda ^{{\mathrm{e}}}\left\{ \left( 1-\theta \right) \frac{N_{i}^{{\mathrm{e}}}}{N^{{\mathrm{e}}}} +\theta \frac{N_{i}^{{\mathrm{e}}}}{N^{{\mathrm{e}}}}v_{i}^{{\mathrm{e}}}\right\} \end{array} \end{aligned}$$
(10.1)

where \(v_{i}^{{\mathrm{e}}}=\frac{J_{i}^{{\mathrm{e}}}}{N^{{\mathrm{e}}}}\).

In the steady state, we have

$$\begin{aligned} v_i^{{\mathrm{e}}}=\frac{\beta \left( 1-\theta \right) \frac{N_{i}^{{\mathrm{e}}}}{N^{{\mathrm{e}}}}}{1-\beta \theta \frac{N_{i}^{{\mathrm{e}}}}{N^{{\mathrm{e}}}}} \end{aligned}$$
(10.2)

where

$$\begin{aligned} N^{{\mathrm{e}}}=\frac{\theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}}{1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}}K^{{\mathrm{e}}} \end{aligned}$$
(10.3)

where \(Rx^{{\mathrm{e}}}\equiv R_{k}^{{\mathrm{e}}}-\frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\), and

$$\begin{aligned} N_i^{{\mathrm{e}}}=\frac{N^{{\mathrm{e}}}-\delta _T^{{\mathrm{e}}} K^{{\mathrm{e}}}}{\theta } \end{aligned}$$
(10.4)

so that, using Eq. (10.3) have

$$\begin{aligned} \frac{N_i^{{\mathrm{e}}}}{N^{{\mathrm{e}}}}=\frac{1-\delta _T^{{\mathrm{e}}} \frac{1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}}{\theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}}}{\theta }=\frac{{ Rx^{{\mathrm{e}}} }+\delta _T^{{\mathrm{e}}} \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}}{\left( \theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}\right) } \end{aligned}$$
(10.5)

Assuming that the ICC is binding, and using Eqs. (10.3) and (10.2), we can write Eq. (11.4) as

$$\begin{aligned} \kappa ^{{\mathrm{e}}} \frac{1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}}{\theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}} = \frac{\beta \left( 1-\theta \right) }{\frac{N^{{\mathrm{e}}}}{N_{i}^{{\mathrm{e}}}}-\beta \theta } \end{aligned}$$
(10.6)

and by using Eq. (10.5) we can further simplify the latter as

$$\begin{aligned} \kappa ^{{\mathrm{e}}} \frac{1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}}{\theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}} = \frac{\left( 1-\theta \right) \beta \left( { Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}} \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}}\right) }{{\left( \theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}\right) }-\beta \theta \left( { Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}} \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}}\right) } \end{aligned}$$
(10.7)

or

$$\begin{aligned} \frac{\kappa ^{{\mathrm{e}}} \left( 1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) }{{\beta \left( 1-\theta \right) }}\left( \theta \left( 1-\beta \right) Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}\left( 1-\beta \theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) \right) = \left( \theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}\right) \left( Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}} \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) \end{aligned}$$
(10.8)

or

$$\begin{aligned}&\frac{\kappa ^{{\mathrm{e}}} \left( 1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) }{\beta \left( 1-\theta \right) }\left( \theta \left( 1-\beta \right) Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}\left( 1-\beta \theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) \right) = \theta Rx^{{\mathrm{e}}}{}^2 \nonumber \\&\quad +\delta _T^{{\mathrm{e}}}\left( 1+\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}{}^2 \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}} \end{aligned}$$
(10.9)

which is a quadratic equation in the credit spread \(Rx^{{\mathrm{e}}}\) with solution

$$\begin{aligned} Rx^{{\mathrm{e}}}_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \end{aligned}$$
(10.10)

where

$$\begin{aligned} a=\,&\theta \end{aligned}$$
(10.11)
$$\begin{aligned} b=\,&\delta _T^{{\mathrm{e}}}\left( 1+\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) -\frac{\kappa ^{{\mathrm{e}}} \left( 1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) }{\beta \left( 1-\theta \right) }\theta \left( 1-\beta \right) \end{aligned}$$
(10.12)
$$\begin{aligned} c=\,&\delta _T^{{\mathrm{e}}}{}^2 \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}-\frac{\kappa ^{{\mathrm{e}}} \left( 1-\theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) }{\beta \left( 1-\theta \right) }\delta _T^{{\mathrm{e}}}\left( 1-\beta \theta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) \end{aligned}$$
(10.13)

We can then use Eqs. (11.10) and (11.11) evaluated in the steady state, i.e.,

$$\begin{aligned} \gamma ^{{\mathrm{e}}}=\beta \left( 1-\theta +\theta J^{{\mathrm{e}}\prime }_i\right) \frac{Rx^{{\mathrm{e}}}}{\kappa _{{\mathrm{e}}}}, \end{aligned}$$
(10.14)

and

$$\begin{aligned} J_{i}^{{\mathrm{e}}\prime }\left( \gamma ^{{\mathrm{e}}}-1\right) +\beta \left( 1-\theta +\theta J^{{\mathrm{e}}\prime }_i\right) \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}=0, \end{aligned}$$
(10.15)

respectively, to obtain a quadratic equation in \(J_{i}^{{\mathrm{e}}\prime }\), i.e.,

$$\begin{aligned} \theta \beta \frac{Rx^{{\mathrm{e}}}}{\kappa _{{\mathrm{e}}}}J^{{\mathrm{e}}\prime 2}_i-\left( 1-\left( 1-\theta \right) \beta \frac{Rx^{{\mathrm{e}}}}{\kappa _{{\mathrm{e}}}}-\theta \beta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}\right) J_{i}^{{\mathrm{e}}\prime }+\left( 1-\theta \right) \beta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}=0, \end{aligned}$$
(10.16)

with solution

$$\begin{aligned} J^{{\mathrm{e}}\prime }_i{}_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a}; \end{aligned}$$
(10.17)

where

$$\begin{aligned} a&=\theta \beta \frac{Rx^{{\mathrm{e}}}}{\kappa _{e}} \end{aligned}$$
(10.18)
$$\begin{aligned} b&=\left( 1-\theta \right) \beta \frac{Rx^{{\mathrm{e}}}}{\kappa _{{\mathrm{e}}}}+\theta \beta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}-1\end{aligned}$$
(10.19)
$$\begin{aligned} c&=\left( 1-\theta \right) \beta \frac{{\widetilde{R}}_{v}}{\psi_{\mathrm{D}}}. \end{aligned}$$
(10.20)

We can then use Eq. (10.14) to obtain \(\gamma ^{{\mathrm{e}}}\).

So far we have then used five Eqs. (10.6), (10.2), (10.5), (10.14) and (10.15), to obtain the five variables \(v_i^{{\mathrm{e}}}\), \(\frac{N_i^{{\mathrm{e}}}}{N^{{\mathrm{e}}}}\), \(Rx^{{\mathrm{e}}}\), \(J^{{\mathrm{e}}\prime }_i\) and \(\gamma ^{{\mathrm{e}}}\), given a guess (draw) for \(R_v\).

In order to address the AE banks, we need to solve for the level of capital in EME and, thus, the amount of EME banks’ borrowing.

With a guess for \(P^{\mathrm{e}},\) we can use the CPI indexes to solve for \(P^{\mathrm{c}}\) and RER, i.e.,

$$\begin{aligned} {\mathrm{RER}}=\left[ \frac{2v-1}{(1-v)}\left( \frac{v}{2v-1}- {\tilde{P}}_{{\mathrm{e}}}^{{1-\eta }}\right) \right] ^{\frac{1}{{1-\eta }}} \end{aligned}$$
(10.21)

and

$$\begin{aligned} {\tilde{P}}_{{\mathrm{c}}}=\left( \frac{v}{2v-1}-\frac{(1-v)}{2v-1}{\mathrm{RER}}^{\eta -1} \right) ^{\frac{1}{1-\eta }} \end{aligned}$$
(10.22)

Since we have a guess for \(H^{\mathrm{e}}\) too, we can use the firms’ demand for capital to obtain the EME capital stock

$$\begin{aligned} K^{\mathrm{e}}=\left( \frac{P^{\mathrm{e}}\alpha }{R_{k}^{\mathrm{e}}-\left( 1-\delta \right) } \right) ^{\frac{1}{1-\alpha }}H^{\mathrm{e}} \end{aligned}$$
(10.23)

and

$$\begin{aligned} N^{{\mathrm{e}}}=\frac{\theta Rx^{{\mathrm{e}}} +\delta _T^{{\mathrm{e}}}}{1-\theta R_v}K^{{\mathrm{e}}} \end{aligned}$$
(10.24)

With these, we can solve for the steady-state EME banks’ borrowing

$$\begin{aligned} V^{\mathrm{e}}=\frac{K^{\mathrm{e}}-N^{\mathrm{e}}}{{\mathrm{RER}} \psi_{\mathrm{D}}}. \end{aligned}$$
(10.25)

The return on AE capital can be easily derived from the arbitrage condition

$$\begin{aligned} R_k^{{\mathrm{c}}}=R_v. \end{aligned}$$
(10.26)

Having guessed a value for \(H^{\mathrm{c}},\) we can thus derive the capital stock in AE from firms’ demand for capital

$$\begin{aligned} K^{\mathrm{c}}=\left( \frac{P^{\mathrm{c}}\alpha }{R_{k}^{\mathrm{c}}-1}\right) ^{\frac{1}{1-\alpha }}H^{\mathrm{c}}. \end{aligned}$$
(10.27)

We can now compute the following AE banks’ variables

$$\begin{aligned} N^{\mathrm{c}}&=\frac{\theta \left( \left( R_{k}^{\mathrm{c}}-\beta ^{-1}\right) \left( V^{{\mathrm{e}}}+K^{{\mathrm{c}}}\right) \right) +\delta _{T}K^{\mathrm{c}}}{1-\theta \beta ^{-1}} \end{aligned}$$
(10.28)
$$\begin{aligned} N^{\mathrm{c}}_i&=\frac{N^{\mathrm{c}}-\delta _{T}K^{\mathrm{c}}}{\theta } \end{aligned}$$
(10.29)

and the banks’ value relative to net worth

$$\begin{aligned} v_{c}=\beta \frac{N^{\mathrm{c}}_i}{N^{\mathrm{c}}}\left[ (1-\theta )+\theta v_{{\mathrm{c}}}\right] , \end{aligned}$$
(10.30)

where we have used the fact that \(R^*=\beta ^{-1}\).

We can then use the FOC of the AE banks with respect to \(V^{{\mathrm{e}}}\) and the envelope condition to solve for \(J_{i}^{{\mathrm{c}}\prime }\) and \(\gamma ^{{\mathrm{c}}}\). Analogously to the EME banks’ problem, this entails a quadratic equation:

$$\begin{aligned} J_{i}^{{\mathrm{c}}\prime }{}_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \end{aligned}$$
(10.31)

where

$$\begin{aligned} a&=\frac{\beta }{\kappa ^{{\mathrm{c}}}}\theta (R_k^{{\mathrm{c}}}-R^*); \end{aligned}$$
(10.32)
$$\begin{aligned} b&=\frac{\beta }{\kappa ^{{\mathrm{c}}}}(1-\theta )(R_k^{{\mathrm{c}}}-R^*)-(1-\beta \theta R^*); \end{aligned}$$
(10.33)
$$\begin{aligned} c&=\beta (1-\theta ) R^*; \end{aligned}$$
(10.34)

and

$$\begin{aligned} \gamma ^{{\mathrm{c}}}=\frac{\beta }{\kappa ^{{\mathrm{c}}}}(R_k^{{\mathrm{c}}}-R^*)\left( 1-\theta +\theta J^{{\mathrm{c}}}_i{}^\prime \right) . \end{aligned}$$
(10.35)

The rest of the variables can be derived recursively from the steady-state version of the respective equations listed in “Appendix 5”.

The four residual conditions necessary to verify our guesses are EME households’ budget constraint (11.49), the resource constraints in each economy (11.50) and (11.51) and the ICC constraint of AE banks (11.21).

Appendix 5: Full List of Equations Used in the Simulation

In this appendix, we list all the equations of the model used in the derivation of the numerical results. For simplicity, we omit the expectation operator, with the understanding that expressions involving \(t+1\) variables hold in expectation. We start by giving some more details about the derivation of the bank’s first-order conditions, before listing the equations describing households’ and firms’ choices and equilibrium conditions.

5.1 Appendix 5.1: EME Banks

There are an infinite number of identical banks indexed by the subscript i. The representative EME bank’s optimal value is

$$\begin{aligned} {J_{i,t}^{\mathrm{e}}} =\max _{[{K_{i,t+1}^{\mathrm{e}},V_{i,t}^{\mathrm{e}},D_{i,t}^{{\mathrm{e}}}}]} E_{t}\Lambda _{t+1|t}^{\mathrm{e}}\left[ (1-\theta )N_{i,t+1}^{{\mathrm{e}}}+\theta J_{i,t+1}^{\mathrm{e}}\right] , \end{aligned}$$
(11.1)

where

$$\begin{aligned} N_{i,t}^{{\mathrm{e}}}=R_{k,t}^{{\mathrm{e}}}Q_{t-1}^{{\mathrm{e}}}K_{i,t-1}^{{\mathrm{e}}}-{\mathrm{RER}}_t\frac{R_{v,t-1}}{\pi _{t}^{{\mathrm{c}}}} V_{i,t-1}^{{\mathrm{e}}}-\frac{R^{{\mathrm{e}}}_{t-1}}{\pi _{t}^{{\mathrm{e}}}}D_{i,t-1}^{{\mathrm{e}}}, \end{aligned}$$
(11.2)

subject to the balance sheet

$$\begin{aligned} N_{i,t}^{\mathrm{e}}+{\mathrm{RER}}_{t}V_{i,t}^{{\mathrm{e}}}+D_{i,t}^{{\mathrm{e}}}=Q_{t}^{{\mathrm{e}}}K_{i,t}, \end{aligned}$$
(11.3)

and incentive compatibility constraint

$$\begin{aligned} J_{i,t}^{{\mathrm{e}}}\ge \kappa ^{{\mathrm{e}}}_t Q_t^{{\mathrm{e}}} K_{i,t}^{{\mathrm{e}}}. \end{aligned}$$
(11.4)

Domestic deposit constraint

$$\begin{aligned} D_{i,t}^{{\mathrm{e}}}\le \left( \psi_{\mathrm{D}} -1 \right) {\mathrm{RER}}_{t}V_{i,t}^{{\mathrm{e}}} \end{aligned}$$
(11.5)

where \(\psi_{\mathrm{D}} \ge 1\). Since \(E_t\left( \frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}}\frac{R_{v,t}}{\pi _{t+1}^{{\mathrm{c}}}}-\frac{R^{{\mathrm{e}}}_{t}}{\pi _{t+1}^{{\mathrm{e}}}}\right) \ge 0\)—which is due to the presence of credit spreads in combination with the international asset pricing condition of the households—then constraint (11.5) is always binding and we can rewrite the balance sheet equation (11.3) as

$$\begin{aligned} N_{i,t}^{\mathrm{e}}+\psi_{\mathrm{D}} {\mathrm{RER}}_{t}V_{i,t}^{{\mathrm{e}}}=Q_{t}^{{\mathrm{e}}}K_{i,t}, \end{aligned}$$
(11.6)

and the net-worth equation (11.2)

$$\begin{aligned} N_{i,t}^{{\mathrm{e}}}=R_{k,t}^{{\mathrm{e}}}Q_{t-1}^{{\mathrm{e}}}K_{i,t-1}^{{\mathrm{e}}}-{\widetilde{R}}_{v,t-1} {\mathrm{RER}}_t V_{i,t-1}^{{\mathrm{e}}}, \end{aligned}$$
(11.7)

where \({\widetilde{R}}_{v,t-1}=\left( \frac{R_{v,t-1}}{\pi _{t}^{{\mathrm{c}}}} +\frac{R^{{\mathrm{e}}}_{t-1}}{\pi _{t}^{{\mathrm{e}}}}\left( \psi_{\mathrm{D}} -1 \right) \right)\)

Using Eq. (11.6) to replace \(V_{i,t}^{{\mathrm{e}}}\) in Eq. (11.7) yields

$$\begin{aligned} N_{i,t+1}^{{\mathrm{e}}}=\left( R_{k,t+1}^{{\mathrm{e}}}-\frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}} \frac{{\widetilde{R}}_{v,t}}{\psi_{\mathrm{D}}}\right) Q_{t}^{{\mathrm{e}}}K_{i,t}^{{\mathrm{e}}} +\frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}}\frac{{\widetilde{R}}_{v,t}}{\psi_{\mathrm{D}}}N_t^{{\mathrm{e}}}. \end{aligned}$$
(11.8)

And aggregating across banks yields

$$\begin{aligned}&\int N_{i,t+1}^{{\mathrm{e}}} \text {d}i = N_{t+1}^{\mathrm{e}}=\theta \left( \left( R_{k,t+1}^{{\mathrm{e}}}-\frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}}\frac{{\widetilde{R}}_{v,t}}{\psi_{\mathrm{D}}}\right) Q_{t}^{{\mathrm{e}}}K_{t}^{{\mathrm{e}}}+\frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}}\frac{{\widetilde{R}}_{v,t}}{\psi_{\mathrm{D}}}N_t^{{\mathrm{e}}}\right) \nonumber \\&\quad +\delta _T Q_{t+1}^{\mathrm{e}}K_{t}^{\mathrm{e}}. \end{aligned}$$
(11.9)

The first-order conditions with respect to \(K_{i,t}^{{\mathrm{e}}}\) is

$$\begin{aligned} K_{i,t}^{{\mathrm{e}}}:\, E_t\Omega _{t+1|t}^{{\mathrm{e}}}\left( R_{k,t+1}^{{\mathrm{e}}}-\frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}}\frac{{\widetilde{R}}_{v,t}}{\psi_{\mathrm{D}}}\right) Q_{t}^{{\mathrm{e}}}-Q_{t}^{{\mathrm{e}}}\kappa _{{\mathrm{e}},t}\gamma _t^{{\mathrm{e}}}=0, \end{aligned}$$
(11.10)

the envelope condition is

$$\begin{aligned} N_{i,t}^{\mathrm{e}}:\,-J_{i,t}^{{\mathrm{e}}\prime }\left( 1-\gamma _t^{{\mathrm{e}}}\right) +E_t\Omega _{i,t+1|t}^{{\mathrm{e}}}\frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}}\frac{{\widetilde{R}}_{v,t}}{\psi_{\mathrm{D}}}=0, \end{aligned}$$
(11.11)

and the complementary slackness conditions is

$$\begin{aligned} \gamma _t^{{\mathrm{e}}}\ge 0;\,\left( J_{i,t}^{{\mathrm{e}}}- \kappa ^{{\mathrm{e}}}_t Q_t^{{\mathrm{e}}} K_{i,t}^{{\mathrm{e}}}\right) \gamma _t^{{\mathrm{e}}}=0; \end{aligned}$$
(11.12)

where \(\Omega _{i,t+1|t}^{{\mathrm{e}}}\equiv \Lambda _{t+1|t}\left[ \left( 1-\theta \right) +\theta J_{i,t+1}^{{\mathrm{e}}\prime } \right]\) is the discount factor of the bank, and \(\gamma _t^{{\mathrm{e}}}\) is the multiplier associated with constraint (11.4).Footnote 27

5.2 Appendix 5.2: AE Banks

There are an infinite number of identical banks indexed by the subscript j. The AE representative bank’s value function is

$$\begin{aligned} J_{j,t}^{\mathrm{c}}=E_{t}\max _{K_{j,t+1}^{\mathrm{c}},V_{j,t}^{\mathrm{e}},D_{t}^{\mathrm{c}}} \Lambda _{t+1|t}^{\mathrm{c}}\left[ (1-\theta )N_{j,t+1}^{{\mathrm{c}}}+\theta J_{j,t+1}^{\mathrm{c}}\right] , \end{aligned}$$
(11.13)

subject to the balance sheet constraint

$$\begin{aligned} \frac{n}{1-n}V_{j,t}^{\mathrm{e}}+Q_{t}^{\mathrm{c}}K_{j,t}^{\mathrm{c}}=N_{j,t}^{\mathrm{c}}+D_{t}^{\mathrm{c}}, \end{aligned}$$
(11.14)

and the incentive compatibility constraint

$$\begin{aligned} J_{j,t}\ge \kappa ^{{\mathrm{c}}}_t\left( \frac{n}{1-n}V_{j,t}^{\mathrm{e}}+Q_{{\mathrm{c}},t}K_{j,t}^{\mathrm{c}}\right) ; \end{aligned}$$
(11.15)

where the net worth is defined as

$$\begin{aligned} N_{j,t+1}^{\mathrm{c}}=R_{k,t+1}^{\mathrm{c}}Q_{t}^{{\mathrm{c}}}K_{j,t}^{{\mathrm{c}}}+ \frac{R_{v,t}}{\pi _{t+1}^{{\mathrm{c}}}} \frac{n}{1-n}V_{j,t}^{{\mathrm{e}}}- {\frac{R_{t}^{\mathrm{c}}}{\pi _{t+1}^{{\mathrm{c}}}}}D_{j,t}^{\mathrm{c}}. \end{aligned}$$
(11.16)

Using Eq. (11.14) to eliminate deposits \(D_{j,t}^{{\mathrm{c}}}\) from Eq. (11.16) we obtain

$$\begin{aligned} N_{j,t+1}^{\mathrm{c}}=\left( R_{k,t+1}^{\mathrm{c}}- {\frac{R_{t}^{\mathrm{c}}}{\pi _{t+1}^{{\mathrm{c}}}}}\right) Q_{t}^{{\mathrm{c}}}K_{j,t}^{{\mathrm{c}}}+ \left( \frac{R_{v,t}}{\pi _{t+1}^{{\mathrm{c}}}} -\frac{R_{t}^{\mathrm{c}}}{\pi _{t+1}^{{\mathrm{c}}}}\right) \frac{n}{1-n}V_{j,t}^{{\mathrm{e}}}+ {\frac{R_{t}^{\mathrm{c}}}{\pi _{t+1}^{{\mathrm{c}}}}}N_{j,t}^{\mathrm{c}}. \end{aligned}$$
(11.17)

The first-order conditions of the banks’ problem are

$$\begin{aligned} K_{j,t}^{{\mathrm{c}}}:\,&\Omega _{j,t+1|t}^{{\mathrm{c}}}\left( R_{k,t+1}^{\mathrm{c}}- { \frac{R_{t}^{\mathrm{c}}}{\pi _{t+1}^{{\mathrm{c}}}}}\right) Q_{t}^{{\mathrm{c}}}-Q_{t}^{{\mathrm{c}}}\kappa _{{\mathrm{c}},t}\gamma _t^{{\mathrm{c}}}=0 \end{aligned}$$
(11.18)
$$\begin{aligned} V_{j,t}^{{\mathrm{c}}}:\,&\Omega _{j,t+1|t}^{{\mathrm{c}}}\left( \frac{R_{v,t}}{\pi _{t+1}^{{\mathrm{c}}}}- {\frac{R_{t}^{\mathrm{c}}}{\pi _{t+1}^{{\mathrm{c}}}}}\right) -\kappa _{{\mathrm{c}},t}\gamma _t^{{\mathrm{c}}}=0, \end{aligned}$$
(11.19)

the envelope condition is

$$\begin{aligned} N_{j,t}^{\mathrm{c}}:\,-J_{j,t}^{c\prime }\left( 1-\gamma _t^{{\mathrm{c}}}\right) +E_t\Omega _{j,t+1|t}^{{\mathrm{c}}}\frac{{\mathrm{RER}}_{t+1}}{{\mathrm{RER}}_{t}}\frac{R_{t}^{\mathrm{c}}}{\pi _{t+1}^{{\mathrm{c}}}}=0, \end{aligned}$$
(11.20)

and the complementary slackness conditions is

$$\begin{aligned} \gamma _t^{{\mathrm{c}}}\ge 0;\,\left[ J_{j,t}^{{\mathrm{c}}}- \kappa ^{{\mathrm{c}}}_t \left( Q_t^{{\mathrm{c}}} K_{j,t}^{{\mathrm{c}}}+\frac{n}{1-n}V_{j,t}^{{\mathrm{e}}}\right) \right] \gamma _t^{{\mathrm{c}}}=0; \end{aligned}$$
(11.21)

where the notation is symmetric to that used for the EME bank.

5.3 Appendix 5.3: Households and Firms

Households preferences:

$$\begin{aligned}&U^{\mathrm{e}}_{t}=\left( \frac{\left( C^{\mathrm{e}}_{t}\right) ^{1-\sigma }}{1-\sigma }-\frac{\chi \, \left( H^{\mathrm{e}}_{t}\right) ^{1+\psi }}{1+\psi }\right) \end{aligned}$$
(11.22)
$$\begin{aligned}&\quad U^{\mathrm{c}}_{t}\left( \frac{\left( C^{\mathrm{c}}_{t}\right) ^{1-\sigma }}{1-\sigma }-\frac{\chi \, \left( H^{\mathrm{c}}_{t}\right) ^{1+\psi }}{1+\psi }\right) \end{aligned}$$
(11.23)

Price-dispersion measures:

$$\begin{aligned} \left( \Delta ^{\mathrm{e}}_{t}\right) =\varsigma \, \left( \Delta ^{\mathrm{e}}_{t-1}\right) \, \left( \frac{\pi ^{{\mathrm{PPI,e}}}_{t}}{\pi ^{{\mathrm{PPI,e}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}}+\left( 1-\varsigma \right) \, \left( \Pi ^{*,e}_{t}\right) ^{\left( -\sigma _{\mathrm{p}}\right) } \end{aligned}$$
(11.24)
$$\begin{aligned} \left( \Delta ^{\mathrm{c}}_{t}\right) =\varsigma \, \left( \Delta ^{\mathrm{c}}_{t-1}\right) \, \left( \frac{\pi ^{{\mathrm{PPI,c}}}_{t}}{\pi ^{{\mathrm{PPI,c}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}}+\left( 1-\varsigma \right) \, \left( \Pi ^{*,c}_{t}\right) ^{\left( -\sigma _{\mathrm{p}}\right) } \end{aligned}$$
(11.25)

where \(\chi _\pi =\{0,1\}\) is a Boolean parameter capturing price indexation to steady-state inflation. In the reported simulation, it takes the value 1.

Optimal price set by price-changing firms:

$$\begin{aligned} \left( \Pi ^{*,e}_{t}\right) =\frac{\left( F^{\mathrm{e}}_{t}\right) }{\left( G^{\mathrm{e}}_{t}\right) } \end{aligned}$$
(11.26)
$$\begin{aligned} \left( \Pi ^{*,c}_{t}\right) =\frac{\left( F^{\mathrm{c}}_{t}\right) }{\left( G^{\mathrm{c}}_{t}\right) } \end{aligned}$$
(11.27)

where

$$\begin{aligned} \left( F^{\mathrm{e}}_{t}\right)= \left( Y^{\mathrm{e}}_{t}\right) \, \left( {\mathrm{MC}}^{\mathrm{e}}_{t}\right) +\varsigma \, \left( \Lambda ^{{\mathrm{e}}}_{t+1|t}\right) \, \left( \frac{\pi ^{{\mathrm{PPI,e}}}_{t+1}}{\pi ^{{\mathrm{PPI,e}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}}\, \left( F^{\mathrm{e}}_{t+1}\right) \end{aligned}$$
(11.28)
$$\begin{aligned} \left( F^{\mathrm{c}}_{t}\right)= \left( Y^{\mathrm{c}}_{t}\right) \, \left( {\mathrm{MC}}^{\mathrm{c}}_{t}\right) +\varsigma \, \left( \Lambda ^{{\mathrm{c}}}_{t+1|t}\right) \, \left( \frac{\pi ^{{\mathrm{PPI,c}}}_{t+1}}{\pi ^{{\mathrm{PPI,c}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}}\, \left( F^{\mathrm{c}}_{t+1}\right) \end{aligned}$$
(11.29)
$$\begin{aligned} \left( G^{\mathrm{e}}_{t}\right)= \frac{\left( Y^{\mathrm{e}}_{t}\right) \, \left( P^{\mathrm{e}}_{t}\right) }{\frac{\sigma _{\mathrm{p}}}{\sigma _{\mathrm{p}}-1}}+\varsigma \, \left( \Lambda ^{{\mathrm{e}}}_{t+1|t}\right) \, \left( \frac{\pi ^{{\mathrm{PPI,e}}}_{t+1}}{\pi ^{{\mathrm{PPI,e}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}-1}\, \left( G^{\mathrm{e}}_{t+1}\right) \end{aligned}$$
(11.30)
$$\begin{aligned} \left( G^{\mathrm{c}}_{t}\right)= \frac{\left( Y^{\mathrm{c}}_{t}\right) \, \left( P^{\mathrm{c}}_{t}\right) }{\frac{\sigma _{\mathrm{p}}}{\sigma _{\mathrm{p}}-1}}+\varsigma \, \left( \Lambda ^{{\mathrm{c}}}_{t+1|t}\right) \, \left( \frac{\pi ^{{\mathrm{PPI,c}}}_{t+1}}{\pi ^{{\mathrm{PPI,c}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}-1}\, \left( G^{\mathrm{c}}_{t+1}\right) \end{aligned}$$
(11.31)

where MC is the marginal cost.

Relative PPI price dynamics:

$$\begin{aligned} 1= \varsigma \, \left( \frac{\pi ^{{\mathrm{PPI,e}}}_{t}}{\pi ^{{\mathrm{PPI,e}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}-1}+\left( 1-\varsigma \right) \, \left( \Pi ^{*,e}_{t}\right) ^{1-\sigma _{\mathrm{p}}} \end{aligned}$$
(11.32)
$$\begin{aligned} 1= \varsigma \, \left( \frac{\pi ^{{\mathrm{PPI,c}}}_{t}}{\pi ^{{\mathrm{PPI,c}}}{}^{\chi _{\pi }}}\right) ^{\sigma _{\mathrm{p}}-1}+\left( 1-\varsigma \right) \, \left( \Pi ^{*,c}_{t}\right) ^{1-\sigma _{\mathrm{p}}} \end{aligned}$$
(11.33)

Euler equation for foreign bonds:

$$\begin{aligned} \frac{\frac{\left( \Lambda ^{{\mathrm{e}}}_{t+1|t}\right) \, \left( R^*_{t}\right) \, }{\left( \pi ^{\mathrm{c}}_{t+1}\right) }\, \left( {\mathrm{RER}}_{t+1}\right) }{\left( {\mathrm{RER}}_{t}\right) }-1=0 \end{aligned}$$
(11.34)

Consumption Euler equation for EME

$$\begin{aligned} \left( R^{\mathrm{e}}_{t}\right) {\frac{\left( \Lambda ^{{\mathrm{e}}}_{t+1|t}\right) }{\left( \pi ^{\mathrm{e}}_{t+1}\right) }}= 1 \end{aligned}$$
(11.35)

Center country consumption Euler equation:

$$\begin{aligned} \frac{\left( \Lambda ^{{\mathrm{c}}}_{t+1|t}\right) \, \left( R^*_{t}\right) }{\left( \pi ^{\mathrm{c}}_{t+1}\right) }-1=0 \end{aligned}$$
(11.36)

Gross return on capital:

$$\begin{aligned} \left( R^{k,e}_{t}\right) \, \left( Q^{\mathrm{e}}_{t-1}\right)= \left( \left( {\mathrm{MC}}^{\mathrm{e}}_{t}\right) \, \left( A^{\mathrm{e}}_{t}\right) \, \alpha \, \left( H^{\mathrm{e}}_{t}\right) ^{1-\alpha }\, \left( K^{\mathrm{e}}_{t-1}\right) ^{\alpha -1}+\left( 1- \delta \right) \, \left( Q^{\mathrm{e}}_{t}\right) \right) \end{aligned}$$
(11.37)
$$\begin{aligned} \left( R^{k,c}_{t}\right) \, \left( Q^{\mathrm{c}}_{t-1}\right)= \left( \alpha \, \left( {\mathrm{MC}}^{\mathrm{c}}_{t}\right) \, \left( A^{\mathrm{c}}_{t}\right) \, \left( H^{\mathrm{c}}_{t}\right) ^{1-\alpha }\, \left( K^{\mathrm{c}}_{t-1}\right) ^{\alpha -1}+\left( 1- \delta \right) \, \left( Q^{\mathrm{c}}_{t}\right) \right) \end{aligned}$$
(11.38)

Aggregate price indexes (in units of consumption aggregator):

$$\begin{aligned} 1= \nu ^{{\mathrm{e}}} \, \left( P^{\mathrm{e}}_{t}\right) ^{1-\eta _p }+\left( 1-\nu ^{{\mathrm{e}}} \right) \, \left( \left( P^{\mathrm{c}}_{t}\right) \, \left( {\mathrm{RER}}_{t}\right) \right) ^{1-\eta _p } \end{aligned}$$
(11.39)
$$\begin{aligned} 1= \nu ^{{\mathrm{c}}} \, \left( P^{\mathrm{c}}_{t}\right) ^{1-\eta _p }+\left( 1-\nu ^{{\mathrm{c}}} \right) \, \left( \frac{\left( P^{\mathrm{e}}_{t}\right) }{\left( {\mathrm{RER}}_{t}\right) }\right) ^{1-\eta _p } \end{aligned}$$
(11.40)

CPI inflation measures:

$$\begin{aligned} \left( P^{\mathrm{e}}_{t}\right) \, \left( \pi ^{\mathrm{e}}_{t}\right)= \left( P^{\mathrm{e}}_{t-1}\right) \, \left( \pi ^{{\mathrm{PPI,e}}}_{t}\right) \end{aligned}$$
(11.41)
$$\begin{aligned} \left( P^{\mathrm{c}}_{t}\right) \, \left( \pi ^{\mathrm{c}}_{t}\right)= \left( P^{\mathrm{c}}_{t-1}\right) \, \left( \pi ^{{\mathrm{PPI,c}}}_{t}\right) \end{aligned}$$
(11.42)

Labor market-clearing condition:

$$\begin{aligned} \left( A^{\mathrm{e}}_{t}\right) \, \left( {\mathrm{MC}}^{\mathrm{e}}_{t}\right) \, \left( 1-\alpha \right) \, \left( H^{\mathrm{e}}_{t}\right) ^{\left( -\alpha \right) }\, \left( K^{\mathrm{e}}_{t-1}\right) ^{\alpha }\, \left( C^{\mathrm{e}}_{t}\right) ^{\left( -\sigma \right) }= \chi \, \left( H^{\mathrm{e}}_{t}\right) ^{\psi } \end{aligned}$$
(11.43)
$$\begin{aligned} \left( A^{\mathrm{c}}_{t}\right) \, \left( {\mathrm{MC}}^{\mathrm{c}}_{t}\right) \, \left( 1-\alpha \right) \, \left( H^{\mathrm{c}}_{t}\right) ^{\left( -\alpha \right) }\, \left( K^{\mathrm{c}}_{t-1}\right) ^{\alpha }\, \left( C^{\mathrm{c}}_{t}\right) ^{\left( -\sigma \right) }= \chi \, \left( H^{\mathrm{c}}_{t}\right) ^{\psi } \end{aligned}$$
(11.44)

Accumulation law for capital:

$$\begin{aligned} \left( K^{\mathrm{e}}_{t}\right)= \left( \left( I^{{\mathrm{e}}}_{t}\right) +\left( K^{\mathrm{e}}_{t-1}\right) \, \left( 1- \delta \right) \right) \end{aligned}$$
(11.45)
$$\begin{aligned} \left( K^{\mathrm{c}}_{t}\right)= \left( \left( I^{{\mathrm{c}}}_{t}\right) +\left( 1- \delta \right) \, \left( K^{\mathrm{c}}_{t-1}\right) \right) \end{aligned}$$
(11.46)

Investment Euler equations:

$$\begin{aligned} \left( Q^{\mathrm{e}}_{t}\right)&=& 1+\frac{\eta }{2}\, \left( \frac{\left( I^{{\mathrm{e}}}_{t}\right) }{\left( I^{{\mathrm{e}}}_{t-1}\right) }-1\right) ^{2}+\left( \frac{\left( I^{{\mathrm{e}}}_{t}\right) }{\left( I^{{\mathrm{e}}}_{t-1}\right) }-1\right) \, \frac{\left( I^{{\mathrm{e}}}_{t}\right) \, \eta }{\left( I^{{\mathrm{e}}}_{t-1}\right) } \nonumber \\&&-\left( \Lambda ^{{\mathrm{e}}}_{t+1|t}\right) \, \eta \, \left( \frac{\left( I^{{\mathrm{e}}}_{t+1}\right) }{\left( I^{{\mathrm{e}}}_{t}\right) }\right) ^{2}\, \left( \frac{\left( I^{{\mathrm{e}}}_{t+1}\right) }{\left( I^{{\mathrm{e}}}_{t}\right) }-1\right) \end{aligned}$$
(11.47)
$$\begin{aligned} \left( Q^{\mathrm{c}}_{t}\right)&=&1+\frac{\eta }{2}\, \left( \frac{\left( I^{{\mathrm{c}}}_{t}\right) }{\left( I^{{\mathrm{c}}}_{t-1}\right) }-1\right) ^{2}+\left( \frac{\left( I^{{\mathrm{c}}}_{t}\right) }{\left( I^{{\mathrm{c}}}_{t-1}\right) }-1\right) \, \frac{\left( I^{{\mathrm{c}}}_{t}\right) \, \eta }{\left( I^{{\mathrm{c}}}_{t-1}\right) }\nonumber \\&&-\left( \Lambda ^{{\mathrm{c}}}_{t+1|t}\right) \, \eta \, \left( \frac{\left( I^{{\mathrm{c}}}_{t+1}\right) }{\left( I^{{\mathrm{c}}}_{t}\right) }\right) ^{2}\, \left( \frac{\left( I^{{\mathrm{c}}}_{t+1}\right) }{\left( I^{{\mathrm{c}}}_{t}\right) }-1\right) \end{aligned}$$
(11.48)

Aggregate budget constraint for EME households:

$$\begin{array}{l} \left( C^{\mathrm{e}}_{t}\right) +B^{\mathrm{e}}_{t}\, \left( {\mathrm{RER}}_{t}\right)= \left( Y^{\mathrm{e}}_{t}\right) \, \left( P^{\mathrm{e}}_{t}\right) -\left( K^{\mathrm{e}}_{t-1}\right) ^{\alpha }\, \left( H^{\mathrm{e}}_{t}\right) ^{1-\alpha }\, \left( {\mathrm{MC}}^{\mathrm{e}}_{t}\right) \, \left( A^{\mathrm{e}}_{t}\right) \, \alpha \nonumber \\ \qquad \qquad \qquad \qquad +\,\left( Q^{\mathrm{e}}_{t}\right) \, \left( I^{{\mathrm{e}}}_{t}\right) -\left( I^{{\mathrm{e}}}_{t}\right) \, \left( 1+\frac{\eta }{2}\, \left( \frac{\left( I^{{\mathrm{e}}}_{t}\right) }{\left( I^{{\mathrm{e}}}_{t-1}\right) }-1\right) ^{2}\right) \nonumber \\ \qquad \qquad \qquad \qquad +\, \left( 1-\theta \right) \,N^{\mathrm{e}}_{i,t} -\left( K^{\mathrm{e}}_{t-1}\right) \, \left( Q^{\mathrm{e}}_{t}\right) \, \delta _T + \left( {\mathrm{RER}}_{t}\right) \, \frac{\left( R^*_{t-1}\right) }{\left( \pi ^{\mathrm{c}}_{t}\right) }\, B^{\mathrm{e}}_{t-1} \end{array}$$
(11.49)

Goods market-clearing conditions:

$$\begin{aligned} \left( K^{\mathrm{e}}_{t-1}\right) ^{\alpha }\, \left( A^{\mathrm{e}}_{t}\right) \, \left( H^{\mathrm{e}}_{t}\right) ^{1-\alpha }= \left( \Delta ^{\mathrm{e}}_{t}\right) \, \left( Y^{\mathrm{e}}_{t}\right) \end{aligned}$$
(11.50)
$$\begin{aligned} \left( K^{\mathrm{c}}_{t-1}\right) ^{\alpha }\, \left( A^{\mathrm{c}}_{t}\right) \, \left( H^{\mathrm{c}}_{t}\right) ^{1-\alpha }= \left( \Delta ^{\mathrm{c}}_{t}\right) \, \left( Y^{\mathrm{c}}_{t}\right) \end{aligned}$$
(11.51)

Aggregate demand for domestic and foreign goods:

$$\begin{array}{l} \left( Y^{\mathrm{e}}_{t}\right)= \nu ^{{\mathrm{e}}} \, \left( P^{\mathrm{e}}_{t}\right) ^{\left( -\eta _p \right) }\, \left( \left( C^{\mathrm{e}}_{t}\right) +\left( I^{{\mathrm{e}}}_{t}\right) \, \left( 1+\frac{\eta }{2}\, \left( \frac{\left( I^{{\mathrm{e}}}_{t}\right) }{\left( I^{{\mathrm{e}}}_{t-1}\right) }-1\right) ^{2}\right) \right) \nonumber \\ \qquad \qquad+\,\left( 1-\nu ^{{\mathrm{c}}} \right) \, \frac{1-n}{n}\, \left( \frac{\left( P^{\mathrm{e}}_{t}\right) }{\left( {\mathrm{RER}}_{t}\right) }\right) ^{\left( -\eta _p \right) }\, \left( \left( C^{\mathrm{c}}_{t}\right) +\left( I^{{\mathrm{c}}}_{t}\right) \, \left( 1+\frac{\eta }{2}\, \left( \frac{\left( I^{{\mathrm{c}}}_{t}\right) }{\left( I^{{\mathrm{c}}}_{t-1}\right) }-1\right) ^{2}\right) \right) \end{array}$$
(11.52)
$$\begin{aligned} \left( Y^{\mathrm{c}}_{t}\right)&=& \left( \left( C^{\mathrm{e}}_{t}\right) +\left( I^{{\mathrm{e}}}_{t}\right) \, \left( 1+\frac{\eta }{2}\, \left( \frac{\left( I^{{\mathrm{e}}}_{t}\right) }{\left( I^{{\mathrm{e}}}_{t-1}\right) }-1\right) ^{2}\right) +\right) \, \left( 1-\nu ^{{\mathrm{e}}} \right) \, \frac{n}{1-n}\, \left( \left( P^{\mathrm{c}}_{t}\right) \, \left( {\mathrm{RER}}_{t}\right) \right) ^{\left( -\eta _p \right) }\nonumber \\&&+\,\left( \left( C^{\mathrm{c}}_{t}\right) +\left( I^{{\mathrm{c}}}_{t}\right) \, \left( 1+\frac{\eta }{2}\, \left( \frac{\left( I^{{\mathrm{c}}}_{t}\right) }{\left( I^{{\mathrm{c}}}_{t-1}\right) }-1\right) ^{2}\right) \right) \, \nu ^{{\mathrm{c}}} \, \left( P^{\mathrm{c}}_{t}\right) ^{\left( -\eta _p \right) } \end{aligned}$$
(11.53)

Appendix 6: Different Degrees of Financial Dependence

In the main text, we have assumed that the EME banks cannot borrow domestically. This assumption aims to maximize the degree of financial dependence of the periphery country. Allowing for domestic deposits reduces the cost of borrowing for EME banks, as the deposit rate lies below the cost of cross-border loans. Figures 19 and 20 compare the response of a subset of variables to TFP and financial shocks in the center country under two scenarios: (i) no domestic deposits and (ii) liabilities equally split between domestic and foreign loans. Importantly, this comparison is performed using the baseline parameter values, and in particular using the baseline severity of the agency problem. As it is apparent from the figures, the degree of financial dependence is virtually irrelevant to the larger economy (AE). On the other hand, and as expected, the EME is more insulated from foreign shocks when it can use domestic funds to finance banks and thus production.

Fig. 19
figure 19

Role of domestic deposits: AE TFP shock

Fig. 20
figure 20

Role of domestic deposits: AE financial shock

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Devereux, M.B., Engel, C. & Lombardo, G. Implementable Rules for International Monetary Policy Coordination. IMF Econ Rev 68, 108–162 (2020). https://doi.org/10.1057/s41308-019-00104-1

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