Back Matter
Author:
• 1 https://isni.org/isni/0000000404811396, International Monetary Fund

### Appendix I. Solution to the Investment Banks’ Maximization Problem

The monopolistic domestic investment bank maximizes its expected profits:

$\begin{array}{ll}\begin{array}{l}{\text{max}}_{{\overline{\text{ω}}}^{\text{F}},{\overline{\text{ω}}}^{\text{E}},\text{K}}\sum _{{s}^{t+1}}\text{π}\left({\text{s}}^{\text{t+1}}|{\text{s}}^{\text{t}}\right)\left[1\\ \text{ }\text{ }\text{ }\text{ }-{\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t+1}}{\text{|s}}^{\text{t}}\right)\right)\right]\left[1+{\text{R}}^{\text{F}}\left({\text{s}}^{\text{t+1}}|{\text{s}}^{\text{t}}\right)\right]\left[\text{Q}\left({\text{s}}^{\text{t}}\right){\text{K}}_{\text{i}}\left({\text{s}}^{\text{t}}\right)-{\text{N}}_{\text{i}}^{\text{E}}\left({\text{s}}^{\text{t}}\right)\right]\end{array}\hfill & \left(A1.1\right)\hfill \end{array}$

where π(st+1|st) is the probability weight for state st+1 given state st+1−1, [1 − ${\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)\right]$ represents the share of investment banks’ earnings kept by the institutions themselves, [1 + RF(st+1|st)] is the average return on the FE contracts, and [Q(st)Ki(st) − ${N}_{i}^{E}\left({s}^{t}\right)\right]$ the amount lent to entrepreneurs. This maximization is solved subject to the participation constraints of the entrepreneurs and the global banks, respectively:

$\begin{array}{cc}\begin{array}{l}\left\{\left(1-{\Gamma }_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\right\}\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)Q\left({s}^{t}\right){K}_{{j}_{i}}\left({s}^{t}\right)\\ \text{ }\text{ }\text{ }\text{ }\ge \left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right){N}_{{j}_{i}}^{E}\left({s}^{t}\right)\forall {j}_{i},{s}^{t+1}|{s}^{t}\end{array}& \left(\text{A}1.2\right)\end{array}$
$\begin{array}{cc}\begin{array}{l}\left\{{\Gamma }_{t}^{E}\left({\overline{\omega }}_{i}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)-{\mu }^{F}{G}_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)\right\}\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\left[Q\left({s}^{t}\right){K}_{i}\left({s}^{t}\right)\\ \text{ }\text{ }\text{ }\text{ }-{N}_{i}^{E}\left({s}^{t}\right)\right]\ge \left(1+R\left({s}^{t}\right)\right)\left[Q\left({s}^{t}\right){K}_{i}\left({s}^{t}\right)-{N}_{i}^{E}\left({s}^{t}\right)-{N}_{i}^{F}\left({s}^{t}\right)\right]\end{array}& \left(\text{A}1.3\right)\end{array}$

Here the focus is on the case where these participation constraints hold with equality. For the entrepreneurs, this implies that their share of returns from investing their net equity and borrowed funds are equal to what they would receive from investing only their equity. In equation (A1.2) above, $\left\{\left(1-{\Gamma }_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t-1}|{s}^{t}\right)\right)\right\}$, represents the expected share of entrepreneurial earnings kept by the entrepreneurs according to the FE contract, and ${\Gamma }_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)$, represents the gross share of entrepreneurial earnings received by the investment banks according to the same contract. The participation constraint of the global banks (equation A1.3) reflects that their net earnings from the GF contract must equal their opportunity cost of lending, (1 + R(st)). The expression $\left\{{\Gamma }_{t}^{F}\left({\overline{\omega }}_{{j}_{i}}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)-{\mu }^{F}{G}_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)\right\}\equiv {\Phi }^{F}\left({\overline{\omega }}^{F}\left({s}^{t+1}|{s}^{t}\right)$ represents the net share of investment banks’ earnings going to global banks according to the GF contract. Making use of the definition of the expected return on the loans to entrepreneurs, (1 + RF(st+1|st)) in equation (8) to rewrite the investment banks’ expected profits as:

$\begin{array}{cc}\begin{array}{l}{\mathrm{max}}_{{\overline{\text{ω}}}^{\text{F}},{\overline{\text{ω}}}^{\text{E}},\text{K}}\sum _{{s}^{t+1}}\text{π}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\left[1\hfill \\ \text{ }\text{ }\text{ }\text{ }-{\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)\right]{\Phi }_{{j}_{i,t}}^{F}\left({\overline{\omega }}_{{j}_{i}}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)\left(1\hfill \\ \text{ }\text{ }\text{ }\text{ }+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)Q\left({s}^{t}\right){K}_{i}\left({s}^{t}\right)\hfill \end{array}& \left(\text{A}1.4\right)\end{array}$

where ${\Phi }_{{j}_{i,t}}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)$ represents the net share of entrepreneurial earnings going to the investment banks according to the FE contract. The global banks’ participation constraint is rewritten by replacing (1 + RF(st+1|st)) and using ${\Phi }^{F}\left({\overline{\omega }}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)$ instead of $\left\{{\Gamma }_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)-{\mu }^{F}{G}_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)\right\}$. Hence, the Lagrangian function becomes:

$\begin{array}{c}\begin{array}{c}\begin{array}{l}\text{L}\left({\overline{\text{ω}}}^{\text{F}},{\overline{\text{ω}}}^{\text{E}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{K}\right)\\ =\sum _{{s}^{t+1}}\text{π}\left({\text{s}}^{\text{t+1}}|{\text{s}}^{\text{t}}\right)\left[1-{\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)\right]{\Phi }_{{j}_{i,t}}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)Q\left({s}^{t}\right)K{}_{i}\left({s}^{t}\right)\end{array}\\ +{\lambda }^{1}\left\{{\Phi }^{F}\left({\overline{\omega }}^{F}\left({s}^{t-1}|{s}^{t}\right){\Phi }_{{j}_{i,t}}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)Q\left({s}^{t}\right){K}_{i}\left({s}^{t}\right)-\left(1\\ +R\left({s}^{t}\right)\right)\left[Q\left({s}^{t}\right){K}_{i}\left({s}^{t}\right)-{N}_{i}^{E}\left({s}^{t}\right)-{N}_{i}^{E}\left({s}^{t}\right)\right]\right\}\\ +{\lambda }^{2}\left[\left\{1-{\Gamma }_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\right\}Q\left({s}^{t}\right)K\left({s}^{t}\right)-{N}^{E}\left({s}^{t}\right)\right]& \left(\text{A1}\text{.5}\right)\end{array}\end{array}$

The first-order-conditions of the maximization problem with regard to ${\overline{\text{ω}}}^{\text{F}},{\overline{\text{ω}}}^{\text{E}}$, and K, respectively, are given by:

$\begin{array}{cc}{\lambda }^{1}=\frac{{\sum }_{{s}^{t+1}}\text{π}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right){{\Gamma }^{\prime }}^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t+1}}|{\text{s}}^{\text{t}}\right)\right)}{{{\Phi }^{\prime }}^{F}\left({\overline{\omega }}^{F}\left({s}^{t+1}|{s}^{t}\right)}& \left(\text{A1}\text{.6}\right)\end{array}$
$\begin{array}{cc}\begin{array}{c}\sum _{{\text{s}}^{t+1}}\text{π}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\left[1-{\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t+1}}|{\text{s}}^{\text{t}}\right)\right)\right]{{\Phi }^{\prime }}_{{j}_{i,t}}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{\text{t}+1}|{s}^{t}\right)\right)\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\\ +{\lambda }^{1}{\Phi }^{F}\left({\overline{\omega }}^{F}\left({s}^{t+1}|{s}^{t}\right){{\Phi }^{\prime }}_{{j}_{i,t}}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\\ -{\lambda }^{2}{{\Gamma }^{\prime }}^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)=0\end{array}& \left(\text{A1}\text{.7}\right)\end{array}$
$\begin{array}{cc}\begin{array}{c}\begin{array}{l}\sum _{{\text{s}}^{t+1}}\text{π}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\left[1-{\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)\right]{\Phi }_{{j}_{i},t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\\ +{\lambda }^{1}\left[{\Phi }^{F}\left({\overline{\omega }}^{F}\left({s}^{t+1}|{s}^{t}\right){\Phi }_{{j}_{i,t}}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)-\left(1+R\left({s}^{t}\right)\right)\right]\end{array}\\ +{\lambda }^{2}\left[1-{\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t+1}}|{\text{s}}^{\text{t}}\right)\right)\right]=0\end{array}& \left(\text{A1}\text{.8}\right)\end{array}$

Inserting (A1.6) into (A1.7) and simplifying yields:

$\begin{array}{c}{\lambda }^{2}=\\ \begin{array}{cc}\frac{{\sum }_{{\text{S}}^{\text{t}+1}}\text{π}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\left[1-{\Gamma }^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)\right]{\Phi }_{{j}_{i,t}}^{\text{'}E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)}{{\Gamma }^{\text{'}\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)}& \left(\text{A1}\text{.9}\right)\end{array}\\ +\frac{{\sum }_{{\text{S}}^{t+1}}\text{π}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right)\right]{{\Gamma }^{\prime }}^{\text{F}}\left({\overline{\text{ω}}}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right){\Phi }^{F}\left({\overline{\omega }}^{F}\left({s}^{t+1}|{s}^{t}\right){\Phi }_{{j}_{i,t}}^{\text{'}E}\right)\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\left(1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)}{{\Gamma }^{\text{'}\text{F}}\left({\overline{\omega }}^{\text{F}}\left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\right){{\Phi }^{\prime }}^{F}\left({\overline{\omega }}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)}\end{array}$

Inserting into (A1.8) yields equation (14) in the main text, i.e.:

$\begin{array}{cc}\begin{array}{c}0=\sum \pi \left({\text{s}}^{\text{t}+1}|{\text{s}}^{\text{t}}\right)\\ \left\{\left[1-{\Gamma }_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)\right]{\Phi }_{i,t}^{F}\left({s}^{t+1}|{s}^{t}\right){\Phi }_{i,t}^{E}\left({s}^{t+1}|{s}^{t}\right)\left[1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right]\right\}\\ +\frac{{{\Gamma }^{\prime }}_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)}{{{\Phi }^{\prime }}_{i,t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)}{\Phi }_{i,t}^{F}\left({s}^{t+1}|{s}^{t}\right){\Phi }_{i,t}^{E}\left({s}^{t+1}|{s}^{t}\right)\left[1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right]\\ -\frac{{{\Gamma }^{\prime }}_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)}{{{\Phi }^{\prime }}_{i,t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)}\left[1+R\left({s}^{t}\right)\right]\\ +\frac{\left[1-{\Gamma }_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right)\right]{{\Phi }^{\prime }}_{i,t}^{E}\left({s}^{t+1}|{s}^{t}\right)}{{{\Gamma }^{\prime }}_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)}\left[1-{\Gamma }_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\right]\left[1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right]\\ +\frac{{{\Gamma }^{\prime }}_{t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right){\Phi }_{i,t}^{F}\left({s}^{t+1}|{s}^{t}\right){{\Phi }^{\prime }}_{i,t}^{E}\left({s}^{t+1}|{s}^{t}\right)}{{{\Phi }^{\prime }}_{i,t}^{F}\left({\overline{\omega }}_{i}^{F}\left({s}^{t+1}|{s}^{t}\right)\right){{\Gamma }^{\prime }}_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)}\left[1-{\Gamma }_{t}^{E}\left({\overline{\omega }}_{{j}_{i}}^{E}\left({s}^{t+1}|{s}^{t}\right)\right)\right]\left[1+{R}^{E}\left({s}^{t+1}|{s}^{t}\right)\right],\forall {j}_{i}.\end{array}& \left(\text{A1}\text{.10}\right)\end{array}$

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I would like to thank Mr. Ásgeir Daníelsson, Central Bank of Iceland and Mr. Karl Walentin, Sveriges Riksbank for their helpful comments and suggestions, as well as the Norges Bank and my previous employer, the Central Bank of Iceland.

Shin (2013), Turner (2014), and Chui et al. (2016) analyze the so-called second phase of global liquidity where bond flows replaced banking flows. The model developed in this paper is silent on such developments.

Gertler, Gilchrist and Natalucci (2007), Devereux et al. (2006), Elekdag et al. (2004, 2006), Céspedes et al. (2004), and Aghion et al. (2004) extended the traditional financial accelerator framework to the open economy.

A historical account of Icelandic banks’ reliance on foreign funding and its financial and macroeconomic implications is provided in Einarsson et al. (2015, 2016).

This difference is still present if countries experiencing a systemic banking crisis is excluded from both groups.

As is common in models with financial frictions, this model does neither allow for substitution between bank credit and bonds, nor maturity transformation in financial intermediation.

Analytical expressions for the variables appearing in equation (14) are available in Dynare and are listed in Hirakata et al. (2009).

Cross-Border Credit Intermediation and Domestic Liquidity Provision in a Small Open Economy
Author: Thorvardur Tjoervi Olafsson