2.1 Economic environment

The main feature defining whether a country is an emerging economy is that its financial sector has a limited intermediation capacity, meaning it is unable to issue deposit claims for their households to some extent. As a consequence, it will have to resort to the international financial banking sector to make up for the difference and being able to meet their firms’ funding needs. This environment is depicted in Figure 1, where the red arrows represent financial flows.

Figure 1.

Financial flows environment in the model.

Note: All arrows denote financial flows. The blue arrows, in addition, refer to flows that are paid to the banks by their borrowers. This latter type of flow—or specifically the associated rate of return perceived by financial intermediaries—is the one affected by the prudential regulations in the model.

Such structure implies that the emerging economies are financially dependent on the funding from center banks, and in an environment of imperfect information in the lending contracts, this could imply a double layer of agency frictions in the economy: that between center households and banks and another one between global banks and emerging country banks. We also assume the friction is more accentuated in the peripheries.Footnote ¹²

For simplicity, the real sector will consist only of one consumption good, and there will be no deviations from the law of one price. Preferences are identical between agents, implying the parity or purchasing power holds, and the real exchange rate will be constant (equal to one), playing no role in this version of the model. Additionally, the households will have access to an international market of non-contingent bonds. This is relevant as it implies that, despite the limited capacity to hold deposits, the saving decisions of emerging economies’ households are not curtailed in any way once they trade these assets.

Finally, the lending relationships are subject to a limited enforceability friction which induces an external finance premium and augments the scale of intermediation and credit cycles. The external premium takes the form of an increased return rate for the banks which raises their—expected and eventual—revenues. Such revenues will be targeted by the macroprudential regulation, meaning it will attack the financial friction at its origin.

2.2 Timing and countries setup

The world consists of three economies that live for two periods $t=1,2$ . The economies are indexed by $i={a,b,c}$ , where the first two will be emerging countries ( $a$ and $b$ ) and the third one is a developed economy that acts as financial center (c). The relative population sizes of the economies are $n_i$ with $1-(n_a+n_b) \geq \tfrac {1}{2}$ . Each economy has five types of agents: households, final consumption good producers, capital producers, banks, and a government sector.

As mentioned before, preferences across countries’ households are identical, and there is only one final consumption good worldwide that is freely traded and produced in all locations. In terms of notation, superindexes denote the country, while subindexes refer to other features such as the sector of the economy and time periods. Additionally, if a superindex is omitted, it normally means that the variable or equation applies to the three countries.

2.3 Investors

For simplicity, the investment decision is separated from the other household decisions and will be subject to adjustment costs. Physical capital is produced in a competitive market by using old capital and investment. The investment will be subject to convex adjustment costs, with the total cost of investing $I_1$ being:

\begin{equation*} C(I_1) = I_1\left (1 + \frac {\zeta }{2} \left (\frac {I_1}{\bar I}-1\right )^2 \right ), \end{equation*}

where $\bar I$ represents the reference level for defining the adjustment cost; The reference level is usually set at the steady state, the previous level of investment, or a combination. In any case, it must hold that $C(0) = 0, \ C''(\! \cdot \!) \gt 0$ . The capital-producing firms (investors) buy back the old capital stock from the firms at price $Q_1$ and produce new capital subject to the adjustment costs (proportional to the parameter $\zeta$ ).

The investor solves:Footnote ¹³

\begin{equation*} \max _{I_1} \ Q_1 I_1 - I_1\left (1 + \frac {\zeta }{2} \left (\frac {I_1}{\bar I}-1\right )^2 \right )\!, \end{equation*}

the optimality condition (F.O.N.C.) is,

(1) \begin{equation} [I_1] \, : \qquad Q_1 = 1 + \frac {\zeta }{2} \left (\frac {I_1}{\bar I}-1\right )^2 + \zeta \left (\frac {I_1}{\bar I}-1\right ) \frac {I_1}{\bar I}, \end{equation}

2.4 Firms

Each period, the firms will operate with a Cobb–Douglas technology that aggregates capital predetermined at the end of the period before. This technology of aggregation is given by $Y_t = A_t(\xi _tK_{t-1})^\alpha$ , where $A_t$ is the aggregate productivity and $\xi$ is a capital-specific productivity or quality term. The capital in $t=1$ follows standard dynamics with depreciation as,

(2) \begin{equation} K_1 = I_1 + (1-\delta ) \xi _1 K_0. \end{equation}

The capital in the initial period will be provided directly by the households in the quantity $K_0$ . However, after that, the firm funds physical capital acquisitions for future production ( $K_1$ ) using lending from the banking sector. Given the model’s timing, there is only one period of intermediation ( $t=1$ ) when lending is extended to acquire capital for production in the final period ( $t=2$ ).

In this setup, the firms solve a slightly different problem each period. First, they decide how much capital to rent from households:

\begin{align*} \max _{K_{0}} \ \pi _{f,1} &= Y_1 - r_1 K_{0}, \\[2pt] s.t. \quad & Y_1 = A_1(\xi _1 K_{0})^\alpha , \end{align*}

where $r_1$ is the rental rate of capital, which, from the optimality condition, is $r_1 = \alpha A_1\xi _1^\alpha K_{0}^{\alpha -1}$ . For the second period, the firms take into account the cost of funding and the revenue of selling the remaining capital stock to capital good producers that carry out the necessary investment to build the capital stock for the next period. Thus, in the second period, the firm will solve:

\begin{align*} \max _{K_{1}} \ \pi _{f,2} &= Y_2 + Q_2 (1-\delta ) \xi _2 K_1 - R_{k,2} Q_1 K_1, \\[2pt] s.t. \quad & Y_2 = A_2(\xi _2 K_{1})^\alpha . \end{align*}

With F.O.N.C.,

\begin{equation*} [K_{1}]\,: \qquad \alpha A_2 \xi _2^\alpha K_1^{\alpha -1} + (1-\delta ) \xi _2 Q_2 = R_{k,2} Q_1. \end{equation*}

To facilitate the model notation, we follow the same definition for $r_2$ , that is, $r_2 = \alpha A_2 \xi _2^\alpha K_{1}^{\alpha -1}$ .

Substituting in the optimality condition for $K_1$ , we obtain that the rate paid to the banks by the firms is given by $\tilde R_{k,2} = \tfrac {r_2 + (1-\delta )\xi _2Q_2}{Q_1}$ . Moreover, by taking into account the possibility of a macroprudential tax on the marginal return on capital, such as in Agénor et al. (Reference Agénor, Jackson, Kharroubi, Gambacorta, Lombardo and Silva2021), we have that the effective rate obtained by the banks, that is, after paying the macroprudential taxes ( $ \tau r_2 K_1$ ) to the government is given by:

(3) \begin{equation} R_{k,2} = \frac {(1-\tau )r_2 + (1-\delta ) \xi _2 Q_2}{Q_1}. \end{equation}

For the sake of clarity, it is important to notice that the firms will pay the pre-taxes banking rate. Only afterward, the banks will consider the effect of the taxes in their profits.Footnote ¹⁴ We elaborate on the policy tool and the role of this return rate in later subsections.Footnote ¹⁵

2.4.1 Capital dynamics and ownership

The dynamics of the model will be driven (within and cross-country) by the capital flows. For that reason, it is relevant to clarify how capital is held, and profited from, by several types of agents in a single period.

Figure 2.

Capital ownership within a period.

Note: This figure describes the ownership of capital across the agents of the model for a generic period $t$ . In terms of our baseline model $t = 1$ ; similarly, $t={1,2}$ for the second setup with two periods of intermediation.

There is only one period of capital accumulation ( $t = 1$ ). The initial capital will be given for that period as $K_0$ . Then, by the end of the accumulation period, the capital in the economy will be given by $K_1$ . That capital will be used for the following period’s production. The capital ownership between agents throughout each period is shown in Figure 2, which explains a typical period with intermediation.

It should be noticed that the capital used for production in the period $t = 1$ cannot be subject to intermediation since there are no banks before the rest of the agents exist (the banks themselves are owned household agents). Therefore, the pre-existing capital stock ( $K_0$ ) will be provided directly from households to firms without explicit financial intermediation.Footnote ¹⁶

2.5 Banks

This is the target sector of the macroprudential policies. The set up is largely based on Gertler and Karadi (Reference Gertler and Karadi2011). There is a financial intermediation sector in the first period that facilitates funding for firms at the local level. In addition, the bank at the center is also a global creditor and extends loans to banks in other locations. In terms of its functioning, the bank receives a start-up capital by their owner household and will try to maximize the value of the banking activities, given by the present value of its profits. Finally, at the end of its life, the bank will give back their net worth to the households as profits.

There will be a costly enforcement agency friction where it is possible for the banks to divert a portion of the assets they intermediate. The eventual implication of this is the imposition of an external finance premium to the banking revenue rates, which is imposed to prevent the banks from absconding assets and to align their incentives with those of the assets’ owners. This is the financial friction in this environment that augments the credit cycles.

Starting from this section, it will be useful in some cases to use a super index $e$ for denoting variables from emerging economies ( $e=\{a,b\}$ ), whereas as before, variables that apply for all countries are either left without a superindex or labeled with an index $i$ when necessary (e.g., when the same expression involves variables from various locations).

2.5.1 Emerging countries

The financial system of the emerging countries will have a limited capacity of intermediation of deposits from local households. For simplicity, we assume that there are not any local deposits in these economies, implying that they rely almost entirely on foreign lending from the center banks for providing funding to firms for production. Therefore, the balance sheet of the bank includes, on the asset side, the lending provided to firms, and on the liability and equity side, the foreign lending from center banks and a start-up capital they receive from local households.

The lending relationship between foreign and local banks will be subject to agency frictions, arising from the fact that creditor banks could default on their debt repayment and divert a portion $\kappa$ of their intermediated assets.Footnote ¹⁷ In either case (default or not), the gross return from intermediation for the bank is $R_{k,2}$ as defined in equation (3).

The emerging market bank maximizes its franchise value in the period 1 ( $J_1$ ):

(4) \begin{align} \max _{F_1^e,L_1^e} J_1^e \,= \, \mathbb{E}_1\Lambda _{1,2}^e\pi _{b,2}^e &= \mathbb{E}_1\Lambda _{1,2}^e\big(R_{k,2}^e L_1^e - R_{b,1}^eF_1^e\big), \nonumber \\[4pt] s.t. \quad L_1^e &= F_1^e + \delta _bQ_1^eK_0^e, \\[-2pt] \nonumber \end{align}

(5) \begin{align} J_1^e &\geq \kappa \mathbb{E}_1\Lambda _{1,2}^e R_{k,2}^eL_1^e, \\[-10pt] \nonumber \end{align}

where $L_1^e = Q_1^eK_1^e$ is the total intermediated lending, $F_1^e$ is the foreign interbank lending borrowed from the center bank at a gross rate $R^e_{b,1}$ , $\delta _bQ_1^eK_0^e$ is the start-up capital received from households, and $\Lambda _{1,2}^i = \beta u'(C_2^i)/u'(C_1^i)$ is the stochastic discount factor for a household in country $i$ . At the same time, the constraints correspond to the balance sheet of the bank and an incentive compatibility constraint (ICC) imposing that the value of the bank equals or exceeds the value from defaulting.

The F.O.N.C. with respect to the foreign debt is:

(6) \begin{align} [F_{1}]\,: \qquad (1+\mu ^e)\mathbb{E}_1\Lambda _{1,2}^e\big(R_{k,2}^e-R_{b,1}^e\big) = \mu ^e\mathbb{E}_1 \kappa \Lambda _{1,2}^e R_{k,2}^e, \end{align}

where $\mu ^e$ is the Lagrange multiplier of the ICC. This expression is already informative about some implications of the frictions that we explore in propositions at the end of this section.Footnote ¹⁸

2.5.2 Advanced economy

To simplify, we assume there is no agency problems at the center.Footnote ¹⁹ Then, the center bank solves:

(7) \begin{align} \max _{F_1,L_1,D_1} J_1 = \mathbb{E}_1\Lambda _{1,2} \pi _{b,2}^c &= \mathbb{E}_1\Lambda _{1,2}\big(R_{b,1}^aF_1^a + R_{b,1}^bF_1^b +R_{k,2}^c L_1^c - R_{D,1}D_1\big), \notag \\[2pt] s.t. \quad &F_1^a + F_1^b + L_1 = D_1 + \delta _bQ_1^cK_0^c. \end{align}

The only restriction will be the balance sheet of the bank that now counts with the foreign interbank flows on the asset side and the local center deposits on the liability side ( $D_1$ ). Additionally, the deposits from households are subject to a gross rate $R_{D,1}$ .

The associated F.O.N.C.s are:

\begin{align*} \big[F_1^a\big]\,: \quad \mathbb{E}_1\big(R_{b,1}^a - R_{D,1}\big) = 0, \qquad \big[F_1^b\big]\,: \quad \mathbb{E}_1\big(R_{b,1}^b - R_{D,1}\big) = 0,\qquad \big[L_1^c\big]\,: \quad \mathbb{E}_1\big(R_{k,2}^c - R_{D,1}\big) = 0. \end{align*}

An important consequence of these optimality conditions is that a policy that affects the revenue rate $R_{k,2}^c$ will have general equilibrium effects and inadvertently lower the cost of debt for debtor economies ( $R_{b,1}^a, R_{b,1}^b$ ). This implies an interaction between the credit spreads and financial frictions between countries that is overlooked by nationally oriented planners.Footnote ²⁰

2.6 Macroprudential policy and public budget

Among the number of possible prudential policiesFootnote ²¹ (VaR regulations, leverage caps, loan/value ratios, etc.) we consider a general type of policy that, as explained by Agénor et al. (Reference Agénor, Jackson, Kharroubi, Gambacorta, Lombardo and Silva2021), encompasses a broad set of macroprudential regulations: a tax ( $\tau$ ) on the return to capital ( $R_{k2} = [(1-\tau ^i)r_t+(1-\delta )\xi _2Q_2]/Q_1$ ). This will be a tax levied on the banking sector, as shown in Equation (3).

Although prudential in nature—as it is implemented on the intermediation sector—the policy tool can also be thought in practice as a device to impose controls on capital flows. This is the case because the tax has the advantage of affecting directly the wedge between the return on capital and borrowing rate (cost of funds for the bank), that is, the credit spread, which in turn drives financial flows at the interbank level. Thus, we are taxing the source of inefficiencies directly.Footnote ²²

On the public budget level, this is reflected as a distortionary tax funded with lump sum taxes ( $T$ ) in each period; that is, we assume a balanced fiscal budget,

\begin{equation*} \tau ^i r_2 K_{1} + T = 0. \end{equation*}

When setting the taxes optimally, each social planner might consider whether to maximize her national welfare or to join cooperative arrangements which would dictate policy centrally.Footnote ²³ We explore these cases as an additional exercise in Section 6.

2.7 Households

The households derive utility from consumption and its lifetime utility is given by $U = u(C_1) + \beta u(C_2)$ with $u(C) = \tfrac {C^{1-\sigma }}{1-\sigma }$ . The budget constraints in each period are the following:

Emerging markets:

(8) \begin{align} C_1^e + \frac {B_1^{e}}{R_1^e} &= r_1^e K_0^e + \pi _{f,1}^e + \pi _{inv,1}^e - \delta _b Q_1^e K_0^e, \\[-10pt]\nonumber \end{align}

(9) \begin{align} C_2^e &= \pi _{f,2}^e + \pi _{b,2}^e + B_1^e - T^e, \quad for \ e=\{a,b\}, \\[-2pt] \nonumber \end{align}

where $C$ is the final consumption good, $B$ a non-contingent internationally traded bond, $r_1$ the rental rate of capital, $Q$ the relative price of capital, $K$ the capital stock, and $T$ is a lump sum tax. Additionally, $\pi$ stands for profits which can come from production activities in final goods ( $f$ ), capital goods ( $inv$ ), or banking services ( $b$ ).

Advanced Economy:

(10) \begin{align} C_1^c + \frac {B_1^{c}}{R_1^c} + D_1 &= r_1^c K_0^c + \pi _{f,1}^c + \pi _{inv,1}^c - \delta _b Q_1^c K_0^c, \\[-10pt] \nonumber \end{align}

(11) \begin{align} C_2^c &= \pi _{f,2}^c + \pi _{b,2}^c + B_1^c + R_{D,1}D_1 - T^c, \\[12pt] \nonumber \end{align}

where the advanced economy also includes local deposits $D$ in the budget constraint as these are intermediated by their banks. Additionally, the profits are given by:Footnote ²⁴

\begin{align*} \pi _{f,1} &= A_1 \xi _1^\alpha K_0^{\alpha } - r_1 K_{0}\\[2pt] \pi _{f,2} &= A_2 \xi _2^\alpha K_1^{\alpha } + Q_2 (1-\delta )\xi _2 K_1 - R_{k,2} Q_1 K_1\\[2pt] \pi _{inv,1} &= Q_1 I_1 - I_1 \left (1+\frac {\zeta }{2} \left ( \frac {I_1}{\bar I} -1 \right )^2 \right )\\[2pt] \pi _{b,2}^e &= R_{k,2}^e Q_1^e K_1^e - R_{b,1}^e F_1^e, \quad for \ e=\{a,b\} \\[2pt] \pi _{b,2}^c &= R_{b,1}^a F_1^a + R_{b,1}^b F_1^b + R_{k,2}^c Q_1^c K_1^c - R_{D,1} D_1 \end{align*}

In the first period, households maximize their lifetime utility stream subject to the budget constraints for the first and second periods. The F.O.N.C. for the three countries’ households is:

(12) \begin{align} u'(C_1) \,=\, \beta R_1 \mathbb{E}_1 [u'(C_2)], \qquad u'\big(C_1^c\big) \,=\, \beta R_{D,1} \mathbb{E}_1 \big[u'\big(C_2^c \big)\big], \end{align}

where the first equation is the Euler equation for bonds and applies to the three economies, while the second is the Euler equation for local deposits and holds only for country $c$ .

2.8 Market clearing

At the world level, bonds are characterized by zero-net-supply,

(13) \begin{equation} n_a B_1^a + n_b B_1^b + n_c B_1^c = 0. \end{equation}

The goods market clearing conditions for each period are

\begin{align*} n_a \left ( C_1^a + I_1^a \left (1+\frac {\zeta }{2} \left ( \frac {I_1^a}{\bar I} -1 \right ) \right ) \right ) + n_b \left ( C_1^b + I_1^b \left (1+\frac {\zeta }{2} \left ( \frac {I_1^b}{\bar I} -1 \right ) \right ) \right )& \\[2pt] + n_c \left ( C_1^c+ I_1^c \left (1+\frac {\zeta }{2} \left ( \frac {I_1^c}{\bar I} -1 \right ) \right ) \right ) = n_a Y_1^a &+ n_b Y_1^b + n_c Y_1^c,\\[2pt] n_a C_2^a + n_b C_2^b + n_c C_2^c = n_a Y_2^a + n_b Y_2^b + n_c Y_2^c \end{align*}

Finally, given that there is only one final good and the law of one price holds (so that the real exchange rate in all cases is one), we have by an uncovered interest rate parity argument that: $R_1^a = R_1^b = R_1^c = R_1$ , where $R_1$ denotes the world interest rate on bonds in period $1$ .Footnote ²⁵

2.9 Equilibrium

Given the policies $\{\tau ^a, \tau ^b, \tau ^c\}$ , the equilibrium consists of prices $\{Q_t^i\}$ , rates $\{R_1, R_{k,2}^e\}$ and quantities $\{B_1^i, K_1^i, F_1^e, D, C_t^i, I_t^i\}$ for $t=\{1,2\}$ , with $i=\{a,b,c\}$ , $e=\{a,b\}$ , such that the households solve their utility maximization problem, the firms solve their profits maximization problems, banks maximize their franchise value, and the goods and bonds markets clear. This allocation is characterized by the solution to the system of equations (1)–(13) where some equations apply for the three economies, other show up by country type, and the clearing condition is considered once (26 equations and variables in total). The simplified system of equations we use to solve the model is summarized in Table A1 in Appendix A.

2.10 Some relevant implications of the model

From this setup, we can already derive a number of important results that can be helpful in understanding the policy implications of the model. First, we can link the extent of the inefficiency—captured by the credit spread—and financial friction to a specific parameter ( $\kappa$ ):

Proof. W.L.O.G. we will work in a perfect foresight setup, otherwise the same result applies to the expected credit spread. From the F.O.N.C. in equation (6), we can obtain:

\begin{equation*} R_{k,2}^e = \underset {\Phi }{\underbrace {\frac {1+\mu ^e}{1+(1-\kappa )\mu ^e}} }R_{b,1}. \end{equation*}

$\Phi \gt 1$ represents the proportionality scale between $R_{k,2}$ and $R_{b,1}$ and guarantees the credit spread is positive in the model. The larger $\Phi$ , the greater the spread. At the same time, $\mu \gt 0$ by definition of the ICC (and the fact that it binds). Hence, it follows that,

\begin{align*} \frac { \partial \Phi }{\partial \kappa } = \frac {\mu (1+\mu )}{(1-(1-\kappa )\mu )^2}\gt 0 \\[-30pt] \end{align*}

The results and exercises in later sections will exploit extensively this result to draw lessons on the role of the extent of financial frictions in shaping the policy leakages of prudential regulations.Footnote ²⁶

On the other hand, we can derive another result to elaborate on how general is our framework in representing the prudential toolkit in real life:

Proof. W.L.O.G. we will work in a perfect foresight setup, otherwise the same result applies to the expected value of the leverage. In the ICC (binding) we substitute the total foreign lending $F_1^e = Q_1^eK_1^e - \delta _B Q_1^eK_0^e$ for any emerging economy $e = \{a,b\}$ and solve for the total assets $L_1^e = Q_1^eK_1^e$ in terms of the initial net worth of banks:

\begin{equation*} L_1 \, = \, \underset {\phi _L: \ \text{leverage ratio}}{\underbrace {\frac {R_{b,1}^e}{R_{b_1}^e-(1-\kappa ^e)R_{k,2}}}}\delta _BQ_1^eK_0^e, \end{equation*}

We can substitute $R_{k,2}^e = [(1-\tau ^e)r_2^e - (1-\delta )\xi ^e_2 Q_2]/Q_1$ and differentiate with respect to $\tau ^e$ :

\begin{equation*} \frac {\partial \phi _L}{\partial \tau ^e} \, = \, -\frac {(1-\kappa ^e)R_{b,1}^e(r_2^e)}{\big(R_{b,1}^e-(1-\kappa ^e)R_{k,2}^e\big)^2Q_1^e} \lt 0 \end{equation*}

This result takes into account that the denominator is never zero given the ICC is binding and the credit spread is positive.

A direct implication of this result is that, as mentioned above, the tool we assume has analogous implications in terms of the standard macroprudential policy toolkit (e.g., leverage ratios).Footnote ²⁷

3.1 Domestic effects of policy

The direct—or domestic—welfare effect of the tax for the emerging economies is given by,

\begin{align*} \frac {dW^a}{d\tau ^a} = \beta \lambda _2^a \left \{ R_1 I_1^a \frac {dQ^a_1}{d\tau ^a} + \frac {B^a_1}{R_1} \frac {dR_1}{d\tau ^a} + \left ( \phi (\tau ^a) r_2^a + \kappa ^a(1-\delta )\xi _2^a Q_2^a \right ) \frac {dK^a_1}{d\tau ^a} + \alpha (1-\kappa ^a) Y_2 \right \}, \end{align*}

where the $d$ is the total derivative operator. The same functional form applies for country $b$ . Each term in this expression is associated with a source of variations on welfare:Footnote ²⁸

Changes in investment profits: The first term corresponds to changes in the investment profits, and its sign depends on whether the country is investing above or below the reference level in the adjustment cost function. For our parameters and initial state values, the sign is positive.

Changes in external assets position: The second term reflects the welfare effects from changes in the international debt position. $\tfrac {dR_1}{d\tau ^a}$ is negative as there is a lower demand for funds by the levied banks. The sign of the whole term, however, depends on the sign of $\tfrac {B_1^a}{R_1}$ (net foreign assets) which is positive for emerging markets (and negative for the center).

Change in welfare by distorting capital accumulation: The third term reflects the change in welfare after hindering capital accumulation; hence, it will be proportional to the change in physical capital holdings and to the sources of profit from holding capital, that is, the marginal product of capital as well as its after-depreciation resale value. The sign of this term is negative as capital accumulation lowers with a tax raise.

Finally, the last term reflects the direct effect of the policy tool on welfare. Even from a planners’ perspective, this effect will not cancel out for the emerging markets (as in the center) because of the presence of a binding ICC for these economies. Its sign is positive. Importantly, we can see there are offsetting welfare effects in the entire expression, and at the same time, the signs and magnitudes depend on the reference point and scale of the policy change that each country planner would plan to implement.Footnote ²⁹

For the center economy, the effect is:

\begin{align*} \frac {dW^c}{d\tau ^c} = \beta \lambda _2^c \left \{ R_1 I_1^c \frac {dQ^c_1}{d\tau ^c} + {\frac {B^c_1}{R_1} \frac {dR_1}{d\tau ^c} }+{ \left ( r_2^c + (1-\delta )\xi _2^{c}Q_2^c \right )}\frac {dK_1^c}{d\tau ^c} + R_{b,1}^{e} {\left ( \frac {dF_1^a}{d\tau ^c} + \frac {dF_1^b}{d\tau ^c}\right ) }+ \frac {dR_{b,1}^{e}}{d\tau ^{c}} F_1^{ab} \right \}, \end{align*}

where $F_1^{ab} = F_1^a + F_1^b$ is the total intermediation to emerging economies, and $R_{b,1}^e$ is the interest rate paid by emerging banks (these equalize in equilibrium). The interpretations for the first three terms are analogous to those of the emerging country mentioned above. The final two terms correspond to:

Welfare effect from changes in intermediation profits: this is an effect coming from the change of the tax on the funding quantities or gross rates related to cross-border lending. In the context of the model, this is also related to the scale of aggregate intermediation. This scale affects the centers, as the latter contains the creditor banks for global markets. Notice the emerging markets can also be affected by the dynamics of financial intermediation, but mostly through their implications for their capacity to fund physical capital.

To the risk of being repetitive, it is still important to reiterate that the signs of these effects are not trivial and may lead to varied—and potentially conflicting— welfare effects. For example, depending on the debt position, the country may benefit from higher taxes, which in itself may provide incentives to national policymakers to alter their policy setup to induce changes in the interest rate that improve the financial position of their economy. This in itself may lead to an increased regulatory activity that may disrupt financial stability.

3.2 Cross-border policy effects

The welfare effect between emerging countries is,

\begin{align*} \frac {dW^a}{d\tau ^b} = \beta \lambda _2^a \left \{ R_1 I_1^a \frac {dQ^a_1}{d\tau ^b} + \frac {B^a_1}{R_1} \frac {dR_1}{d\tau ^b} + \left ( \phi (\tau ^a) r_2^a + \kappa ^a(1-\delta )\xi _2^{a}Q_2^a \right ) \frac {dK^a_1}{d\tau ^b} \right \}\!, \end{align*}

with an analogous counterpart following for the effect in $W^b$ when $\tau ^a$ is changed. Notice this expression is similar to the within-country effect of their own tax. Although, in contrast, the last term is absent given there is not a direct welfare effect from a tax at the cross-country level.

The emerging country welfare effect from a change in the center country tax is,

\begin{align*} \frac {dW^a}{d\tau ^c} = \beta \lambda _2^a \left \{R_1 I_1^a \frac {dQ^a_1}{d\tau ^c} + \frac {B^a_1}{R_1} \frac {dR_1}{d\tau ^c} + \left ( \phi (\tau ^a) r_2^a + \kappa ^a(1-\delta )\xi _2^{a}Q_2^a \right ) \frac {dK^a_1}{d\tau ^c} \right \}\!. \end{align*}

On the other hand, the effect of a change in an emerging tax in the welfare of the center is,

\begin{align*} \frac {dW^c}{d\tau ^e} = {\beta}{ \lambda} _2^c \left \{R_1 I_1^c \frac {dQ^c_1}{d\tau ^e} + \frac {B^c_1}{R_1} \frac {dR_1}{d\tau ^e} + \left (r_2^c + (1-\delta ){\xi _2^{c}}Q_2^c \right )\frac {dK_1^c}{d\tau ^e} + {R_{b,1}^{e}} \left ( \frac {dF_1^a}{d\tau ^e} + \frac {dF_1^b}{d\tau ^e}\right ) +{ \frac {dR_{b,1}^{e}}{d\tau ^a}} {F_1^{ab} }\right \}\!, \end{align*}

where as before $F^{ab}_1$ is the total intermediation to the emerging economies, and $R_{b,1}^e = R_{b,1}^a = R_{b,1}^b$ is the interest rate paid by emerging banks to the center intermediary. The interpretations of each term follow analogous intuitions to those explained in Subsection 3.1.

3.2.1 Optimal toolkit and its drivers

We can use these effects expressions as first-order conditions for national planners and derive the optimal taxes (i.e., setting $dW^i/d\tau ^i = 0$ and solve for $\tau ^i$ ). The optimal emerging tax would be:

\begin{equation*} \tau ^{e \ *} {=}{\frac {-1}{\alpha ({1} - {\kappa}^{e})}} \left \{ \frac {1}{r_2^e} \left [ \left ( R_1 I^e_1 {\frac {dQ^e_1}{dK^e_1} }+ {\frac {B^e_1}{R_1}} {\frac {dR_1}{dK^e_1}} \right ) + {\kappa ^a} (1-\delta){\xi _2^{e}}Q_2 \right ]\! + {1} {+} {\alpha} {(\kappa ^e - 1)} \right \}\!, \,\,\, {\text{for }} e= \{a,b\}. \end{equation*}

Similarly, for the financial center ( $c$ ):

\begin{align*} \tau ^{c \ *} {=} \frac {Q_1^c}{r_2^c} \left \{ R_1 I_1^c \frac {dQ_1^c}{dF^{ab}_1} {+} \frac {B_1^c}{R_1} \frac {dR_1}{dF^{ab}_1} {+} \big (r_2^c {+} (1-\delta )\xi _2^{c}Q_2\big)\frac {dK^c_1}{dF^{ab}_1} +\big(F_1^a + F_1^b\big) \frac {dR_{b,1}^{e}}{dF^{ab}_1} + (1 - \delta )\xi _2^{c}\frac {Q_2}{Q_1^c} \right \} {+} 1, \end{align*}

with $dF^{ab}_1 = dF_1^a + dF_1^b$ .

From these expressions, we get an idea about the effects driving the optimal taxes. The peripheral tax depends on the effect on prices and interest rates from changes in the capital stock, which is proportional to the investment and foreign bonds position. Other relevant features are the resale price of capital and the marginal product of capital whose increases lead to lower tax values. The intuition here is that, if capital becomes more productive, it is better to distort the economy by less. We will see in later sections that this is a feature distinguishing policy regimes with different levels of decentralization: The more internationally centralized—or coordinated—policies can achieve the same effects with lower interventionism.

Here is useful to remember that, in equilibrium the marginal product of capital is directly taxed by the tool. As a result, we could interpret that in order to have a meaningful effect, the tax (or subsidy) will have to be set more strongly in countries with lower marginal product of capital. Finally, it is noticeable that the extent of the financial distortion ( $\kappa ^e$ ) plays an amplifying role—for a stronger financial friction, a more stringent policy stance would have to be implemented.

On the other hand, the financial center optimal tool is driven by the effect of the changed aggregate international intermediation ( $F_1^{ab}$ ) on both the sources of revenue for the banking sector (prices and revenue rate), and on domestic capital intermediation. ( $K^c$ ). Both features reflect the global creditor role of the center; on one side the former—international lending volume effect—leads to direct changes in profits, but the latter effects reflect a substitution of local for global intermediation as more resources that would go to domestic firms are instead flowing to other locations. In either case, notice how the effects of policy, both at center and peripheries, are pinned down at first by the effect on interbank intermediation and later by how this affects each banks’ profitability.Footnote ³⁰

5.1 Banks

EME-Banks. The problem of the bank is extended to account for the probability of continuation in the intermediation activities. This is also reflected in the constraints that now include the balance sheet period of future periods, which is affected by the net worth of the bank that now includes the profits from previous periods.Footnote ³²

In the first period of intermediation (end of t = 1), the bank aims to maximize its expected franchise value, given by $J_1$ , and solves:

\begin{align*} J_1^e &= \max _{F_1^e,L_1^e} \mathbb{E}_1 \{(1-\theta )\Lambda _{1,2}^e \big(R_{k,2}^e L_1^e - R_{b,1}^e F_1^e\big) + {\theta \Lambda _{1,3}^e\big(R_{k,3}^eL_2^e - R_{b,2}^eF_2^e\big)} \}, \end{align*}

\begin{align*} \qquad \qquad s.t &\quad L_1^e = F_1^e + \delta _BQ_1^eK_0^e,\qquad \qquad& {\text{[Balance sheet in t=1]}} \\[3pt]\quad &\quad {L_2^e = F_2^e + \delta _BQ_2^eK_1^e + \theta \big[R_{k,2}^eL_1^e - R_{b,1}^eF_1^e \big]},\qquad \qquad& {\text{[Balance sheet in t=2]}} \\[3pt]\quad & \quad J_1^e \geq \kappa Q_1^e K_1^e,\qquad \qquad \quad &{\text{[ICC, t=1]}} \end{align*}

where the country index for emerging economies is $e$ with $e = \{a,b\}$ , $L_t = Q_tK_t$ is the total lending intermediated with the local firms, $F_t$ is the cross-border borrowing they obtain from the center, $R_{k,t}$ is the gross revenue rate of the banking services, paid by the firms, $R_{b,t}$ is the interbank borrowing rate for the banks, $Q_t$ is the price of capital, $\delta _B Q_t K_{t-1}$ a start-up capital the bankers get from their owner households, and $\Lambda _{t,t+j}$ is the stochastic discount factor between periods $t$ and $t+j$ . It can be noted that the last term in the objective function, and the second constraint are the new terms relative to the previous setup of the bank’s problem while the third constraint is the ICC, imposed to align the incentives of banks with lenders in a way that the former doesn’t abscond assets. This friction will lead to amplified credit spreads.

The present value of the bank will be given by the expected profits in the next period, which now include the possibility of exit from the banking business, with an associated probability of survival $\theta$ .Footnote ³³ Thus, with probability ( $1-\theta$ ) the bank will fail and transfer back its profits to the household, and with probability $\theta$ the bank will be able to continue and pursue future profits.

In this new setup, a key property is that of profits retention. That is, the banks will retain any profits and reinvest them for as long as they remain in business. They continue doing this until they exit the business and report the accumulated profits to the households. This feature will boost the effects of policy in these economies because now a prudential tool has a longer lasting effect on the balance sheets of surviving banks.Footnote ³⁴

In the second period, the banks solve a simpler problem as their objective will not depict a continuation value, making their decisions static:

\begin{align*} J_2^e &= \max _{F_2^e,L_2^e} \mathbb{E}_2 \big\{\Lambda _{2,3}^e \big(R_{k,3}^eL_2^e - R_{b,2}^eF_2^e \big) \big \}, \\[3pt] s.t. &\quad L_2^e = F_2^e + \delta _BQ_2^eK_1^e + {\theta \big[R_{k,2}^eL_1^e - R_{b,1}^eF_1^e \big]}, \\[3pt] & \quad J_2^e \geq \kappa Q_2^e K_2^e. \end{align*}

It can be noticed the problem they solve is not entirely analogous to the simpler model—even if also static—as the resources of the bank are now affected by their previous intermediation decisions, because the balance sheet constraint includes retained profits from the last period.

From these two problems, we can obtain the following first-order conditions:

\begin{align*} \big[F_{1}^e\big]\,: \quad \mathbb{E}_1\Omega _1^e \big(1+\mu _1^e \big) \big(R_{k,2}^e - R_{b,1}^e\big) = \kappa \mu _1^e, \qquad \big[F_{2}^e\big]\,: \quad \mathbb{E}_2 \big(1+\mu _2^e \big) \big(R_{k,3}^e - R_{b,2}^e\big) = \kappa \mu _2^e, \end{align*}

where $\mu _t^e$ is the Lagrange multiplier of the ICC of the bank in country $e$ in each period and $\Omega _1^e = (1-\theta )\Lambda _{1,2}^e+\theta ^2R_{k,3}^e\Lambda _{1,3}^e$ is the effective stochastic discount factor of the bankers that accounts for the probability of a bank failure in the future. With these conditions, the results of Proposition1 also apply here; that is, a binding ICC leads to a positive credit spread that grows with the extent of the friction $\kappa$ .Footnote ³⁵

Center-banks. In $t=1$ the center bank solves:

\begin{align*} \begin{split} J_1^c = \max _{F_1^a,F_1^b,L_1^c,D_1} \mathbb{E}_1 \big \{(1-\theta )\Lambda _{1,2}^c \big(R_{k,2}^cL_1^c + R_{b,1}^aF_1^a + R_{b,1}^bF_1^b - R_{D,1}D_1\big) \big. \\ \big. \qquad+ {\Lambda _{1,3}^c\theta \big(R_{k,3}^cL_2^c + R_{b,2}^aF_2^a + R_{b,2}^bF_2^b - R_{D,2}D_2\big) } \big \}, \end{split} \end{align*}

\begin{align*} \qquad s.t &\quad L_1^c + F_1^a + F_1^b = D_1 + \delta _BQ_1^cK_0^c, \qquad \qquad & {\text{[Balance sheet in t = 1]}} \\[2pt]&\quad L_2^c + F_2^a + F_2^b = D_2 + \delta _BQ_2^cK_1^c \\[2pt]& \quad\quad\,\,+ \theta \big[R_{k,2}^cL_1^c + R_{b,1}^aF_1^a + R_{b,1}^bF_1^b - R_{D,1}D_1 \big],\qquad \qquad & {\text{[Balance sheet in t = 2]}} \end{align*}

this problem is dynamic as it accounts for the potential profits and balance sheets of every intermediation period. These profits also reflect that the bank is a global creditor. In contrast, in the next period the bank will solve a simpler (static) problem consisting of maximizing the profits of a single—terminal intermediation—period,

\begin{align*} J_2^c &= \max _{F_2^a,F_2^b,L_2^c,D_2} \mathbb{E}_2 \big \{\Lambda _{2,3}^c \big(R_{k,3}^cL_2^c + R_{b,2}^aF_2^a + R_{b,2}^bF_2^b - R_{D,2}D_2 \big) \big \}, \\[3pt] s.t. &\quad L_2^c + F_2^a + F_2^b = D_2 + \delta _BQ_2^cK_1^c + {\theta \big[R_{k,2}^cL_1^c + R_{b,1}^aF_1^a + R_{b,1}^bF_1^b - R_{D,1}D_1 \big]}. \end{align*}

As in the baseline model, the resulting first-order conditions just reflect that the expected credit spread is zero for all of the assets considered by the center ( $F_2, L_2, D_2$ ). By using that result and the perfect foresight assumption, we can drop the borrowing cross-border rates ( $R_{b,t}$ ) as they are all equal to the rate for deposits ( $R_{D,t}$ ).

5.2 Macroprudential policy

The policy setup is analogous to the baseline setup. The effective revenue rate perceived by the banks after paying their taxes is $R_{k,t} = \tfrac {(1-\tau _t)r_t + (1-\delta )Q_t}{Q_{t-1}}$ , where $\tau _t$ is the macroprudential tax. What differs now, however, is that $\tau _2$ affects directly the first intermediation period when the banks’ decisions are forward-looking, and $\tau _3$ the terminal period where the decisions are static. Hence, it follows that $\tau _2$ and $\tau _3$ are respectively a forward-looking and a static policy tool.Footnote ³⁶

Analytical welfare effects. We can derive the analytical welfare effects in this case using an analogous procedure based on Davis and Devereux (Reference Davis and Devereux2022). However, a key difference here is that we can track the effect of one more tax, namely, the tool with persistent effects on the balance sheets, which depicts dynamic welfare effects too.

A social planner will consider the following welfare expressions.

\begin{align*} \begin{split} W_{0}^{a}=u\left (C_{1}^{a}\right ) + \beta u\left (C_{2}^{a}\right ) + \beta ^{2} u\left (C_{3}^{a}\right ) + \lambda _{1}^a\left \{A_{1}^{a} K_{0}^{a \ \alpha }+Q_{1}^{a} I_{1}^{a} - C(I_{1}^{a}, I_{0}^{a}) - \delta _{B} Q_{1}^{a} K_{0}^{a} - C_{1}^{a} - \tfrac {B_{1}^{a}}{R_{1}}\right \} & \\[3pt] +\beta \lambda _{2}^{a}\left \{\varphi (\tau _2^a) A_{2}^{a} K_{1}^{a \ \alpha } + Q_{2}^{a} I_{2}^{a} - C(I_{2}^{a}, I_{1}^{a}) - \delta _{B} Q_{2}^{a} K_{1}^{a}+ \kappa \left (\tfrac {Q_{1}^{a} K_{1}^{a}}{\Lambda _{12}} - \Lambda _{23} \theta Q_{2}^{a} K_{2}^{a}\right ) +B_{1}^{a}-C_{2}^{a} - \tfrac {B_{2}^{a}}{R_{2}}\right \} & \\[3pt] +\beta ^{2} \lambda _{3}^{a}\left \{\left (1-\alpha \left (1-\tau _{3}^{a}\right )\right ) A_{3}^{a} K_{2}^{a \ \alpha }+\kappa \tfrac {Q_{2}^{a} K_{2}^{a}}{\Lambda _{12}}+B_{2}^{a}-C_{3}^{a}\right \}\!,& \end{split} \end{align*}

where $\varphi (\tau ) = \left (1-\alpha \left (1-\tau \right )\right )$ and with an analogous expression for the economy $b$ , and

\begin{align*} \begin{split} W_{0}^{c}=u\left (C_{1}^{c}\right ) + \beta u\left (C_{2}^{c}\right ) + \beta ^{2} u\left (C_{3}^{c}\right ) +\lambda _{1}^{c}\left \{A_{1}^{c} K_{0}^{c \ \alpha } +Q_{1}^{c} I_{1}^c-C(I_{1}^{c}, I_{0}^{c}) - \delta _{B} Q_{1}^{c} K_{0}^{c} - C_{1}^{c} - \tfrac {B_{1}^{c}}{R_{1}} - D_{1}\right \} &\\[3pt] +\beta \lambda _{2}^{c}\bigg \{ \left (1-\alpha \theta \left (1-\tau _{2}^c\right )\right ) A_{2}^{c} K_{1}^{c \ \alpha }+Q_{2}^{c} I_{2}^{c}-C\left ( I_{2,}^{c} I_{1}^{c}\right ) \bigg . \\[3pt] \left . +(1-\theta )\left ((1-\delta ) Q_{2}^{c} K_{1}^{c}+R_{b 1}^{a} F_{1}^{a}+R_{b 1}^{b} F_{1}^{b}\right ) -\theta R_{1} D_{1} -\delta _{B} Q_{2}^c K_{1}^{c}+B_{1}^{c}-C_{2}^{c}-\tfrac {B_{2}^{c}}{R_{2}}-D_{2} \right \}& \\[3pt] +\beta ^{2} \lambda _{3}^{c}\big \{A_{3}^{c} K_{2}^{c \ \alpha }+(1-\delta ) Q_{3} K_{2}^{c}+R_{b 2}^{a} F_{2}^{a}+R_{b 2}^{b} F_{2}^b+B_{2}-C_{3}^{c}\big \}.& \end{split} \end{align*}

These expressions are obtained by setting the welfare plus the budget constraints in each period and imposing the private equilibrium conditions. These are equivalent to the usual welfare as the constraints are binding; however, this setup allows to gauge the effects of policy more broadly.

As before, we can obtain the welfare effects from changing the taxes. Here, a planner setting the tax in the last period takes the taxes and variables from the previous period as given, and hence, we just need to differentiate with respect to $R_2, Q_2, I_2. K_2$ for both types of countries plus $R_{b,2}, F_2$ for the center. In contrast, we must also consider the lagged versions of these variables for the first period.Footnote ³⁷

The welfare effects of the taxes are:

For the EMEs’ instruments,

\begin{equation*} \frac {d W^{a}_0}{d \tau _{2}^{a}}=\beta \lambda _{2}^{a} \bigg \{ \overset {\text{static effects}}{ \overbrace {\alpha _1(\kappa ) \frac {d K_{1}^{a}}{d \tau _{2}^{a}} + \alpha _2(\kappa )\frac {d Q_{1}^{a}}{d \tau _{2}^{a}} + \frac {B_{1}^{a}}{R_{1}} \frac {d R_{1}}{d \tau _{2}^{a}} + \alpha Y_2^a }} + \overset {\text{dynamic effects}}{\overbrace { \alpha _3(\kappa ) \frac {d K_{2}^{a}}{d \tau _{2}^a} + \alpha _4(\kappa ) \frac {d Q_{2}^{a}}{d \tau _{2}^{a}} + \frac {B_{2}^{a}}{(R_{2})^2} \frac {d R_{2}}{d \tau _{2}^{a}} }} \bigg \}, \end{equation*}

\begin{equation*} \frac {d W^{a}_0}{d \tau _{3}^a}= \left . \beta \lambda _{2}^{a} \bigg \{ \overset { \text{(only) static effects} }{ \overbrace { \alpha _5(\kappa ) \frac {d K_{2}^{a}}{d \tau _{3}^a} + \alpha _4(\kappa ) \frac {d Q_{2}^{a}}{d \tau _{3}^{a}} + \frac {B_{2}^{a}}{(R_{2})^2} \frac {d R_{2}}{d \tau _{3}^{a}} + \alpha \frac {Y_3^a}{R_2} } } \bigg \}, \right . \end{equation*}

with $\alpha _1(\kappa ) = \kappa R_{1} Q_{1}^{a}+\varphi \left (\tau _{2}^{a}\right ) r_2^a$ , $\alpha _2(\kappa ) = R_1\left (I_{1}^{a}+\kappa K_{1}^{a}\right )$ , $\alpha _3(\kappa ) = \kappa \left (1-\theta \Lambda _{23}\right ) Q_{2}^{a} + \varphi \left (\tau _{3}^{a}\right )\Lambda _{12} r_3^a$ , $\alpha _4(\kappa ) = I_{2}^{a}+\kappa \left (1-\theta \Lambda _{23}\right )K_{2}^{a}$ , $\alpha _5(\kappa )=\kappa \left (1-\theta \Lambda _{23}\right ) Q_{2}^{a}+\varphi \left (\tau _{3}^a\right ) \Lambda _{23} r_3^a$ , and $\tfrac {\partial \alpha _s}{\partial \kappa } \gt 0$ for $s = \{1, \ldots ,5\}$ .

And for the center’s tools,

\begin{align*} \begin{split} \frac {d W^{c}_0}{d \tau _{2}^c} = \overset {\text{static effects}}{ \overbrace {\beta \lambda _{2}^c \left \{ \gamma _1 \frac {d K_{1}^{c}}{d \tau _{2}^{c}}+\left (\tfrac {B_{1}^{c}}{R_{1}}-\theta D_{1}\right ) \frac {d R_{1}}{d \tau _{2}^{c}} + \tfrac {K_{1}^{c}}{R_1} \frac {d Q_{1}^{c}}{d \tau _{2}^{c}} + \alpha \theta Y_2^c + (1-\theta )\left (F_{1}^{ab} \frac {d R_{b,1}^{eme}}{d \tau _{2}^c} + R_{b,1}^{eme}\frac {d F_{1}^{ab}}{d \tau _{2}^{c}}\right )\right \} }}& \\[3pt] \underset {\text{dynamic effects}}{\underbrace {+\beta ^{2} \lambda _{3}^{c} \left \{ \gamma _2 \frac {d K_{2}^{c}}{d \tau _{2}^{c}} + \tfrac {B_{2}^{c}}{R_{2}} \frac {d R_{2}}{d \tau _{2}^{c}} + \gamma _3 \frac {d Q_{2}^{c}}{d \tau _{2}^{c}} + F_{2}^{ab} \frac {d R_{b,2}^{eme}}{d \tau _{2}^{c}} +R_{b,2}^{eme} \frac {d F_{2}^{ab}}{d \tau _{2}^{c}} \right \}}},& \end{split} \end{align*}

\begin{align*} \frac {d W^{c}_0}{d \tau _{3}^{c}}=\beta ^{2} \lambda _{3}^{c} \left \{ \gamma _2 \frac {d K_{2}^{c}}{d \tau _{3}^{c}} + \frac {B_{2}^{c}}{R_{2}} \frac {d R_{2}}{d \tau _{3}^{c}} + \gamma _3 \frac {d Q_{2}^{c}}{d \tau _{3}^{c}} + F_{2}^{ab} \frac {d R_{b, 2}^{eme}}{d \tau _{3}^{c}} + R_{b, 2}^{eme}\frac {d F_{2}^{ab}}{d \tau _{3}^{c}} \right \}\!, \end{align*}

with $\gamma _1 = \left (1-\alpha \theta \left (1-\tau _{2}^{c}\right )\right ) r_2^c+(1-\theta )(1-\delta ) Q_{2}^{c}$ , $\gamma _2 = \left (r_3^c+(1-\delta ) Q_{3}\right )$ , $\gamma _3 = R_2 \big (I_{2}^{c} +(1-\theta ) (1-\delta ) K_{1}^{c}\big )$ , and $F^{ab}_t = F^a_t + F^b_t$ .

The interpretation of these effects goes as follows: First, we can see that there are more sources of variations for taxes that are forward-looking in nature ( $\tau _2$ ), whereas for the terminal taxes, we only get the static drivers—described in the simpler baseline; this alone might explain why the former instruments may have stronger welfare effects than the latter.

On the other hand, there are four drivers of the static welfare effects of the tax, as pointed out in previous sections: these are changes in welfare from (i) hindering capital accumulation; (ii) changes in the global interest rate, which are proportional to the net foreign asset position; (iii) changes in the prices of capital; and, in addition for the center, (iv) changes in the cross-border lending rates and quantities. The welfare effects (i) and (iv) are negative and capture a halting in banking intermediation, while the sign of (ii) and (iii) depends, respectively, on whether an economy is a net creditor or on the investment growth. We expect (ii) to be positive for an emerging economy and negative for the center.

To reflect on the compared effects relative to our baseline, we can see that the dynamic toolkit effects will have similar drivers but also include effects on future variables, for instance, (i) would include the effect on future capital accumulation and (ii) on the future net assets position. The signs for the dynamic effects may not be as straightforward as we may expect similar signs but with potential corrections, for example, when tighter initial taxes imply delaying investment or capital accumulation plans for future periods. Simultaneously, and similar to the static case, it can be noticed that the welfare effects interact with the extent of the financial frictions (captured by $\kappa$ ), and, as before, the effects are stronger for a larger extent of the frictions. This can be seen by checking that $\alpha _j(\kappa )$ increases in $\kappa$ for all $j = \{1,2,3,4,5\}$ .

Optimal taxes. We can obtain expressions for the optimal taxes by taking these welfare effects as first-order conditions for the planner as in prior sections. The features driving each tool are analogous to the ones described in the static baseline. As before, we have that regulators at the center trade-off local intermediation for global lending, a relevant feature for understanding the importance of the center’s instrument in generating cross-border policy leakages and welfare effects abroad. At the same time, and in addition to the previous findings, now we have that the forward-looking taxes are driven by the changes in future variables, for example, capital accumulation after changes in the level of banking intermediation. The expressions for these optimal taxes are shown in Appendix E.

Finally, unlike the static version of the optimal tool, in this case is not as straightforward to determine if a larger extent of the friction calls for a more stringent policy setting. On top of the static amplification effect, the dynamic effect takes into account the expected relative performance of the economy in future periods, which is captured by the interaction between stochastic discount factors on different dates. In that sense, if the friction is such that intermediation implies stronger economic fluctuations (current or future) these additional effects activate.

6.1 Welfare effects in different policy regimes

Before setting the planning problem and solving for the tools, it is useful to understand the welfare effect of the taxes on the policy objective of the planners. For the standard case of a planner that takes decisions at the central level—or a nationally oriented planner—the domestic welfare effect dictates the total effect on her objective function. On the other hand, as we are dealing with several planners, we could also consider that these decide to form coalitions and set their policies with different levels of centralization. The possible combination of cases we consider, the effect of policy changes on their objectives, and the toolkit each planner has at hand are shown in Table 2.

Table 2.

Welfare spillovers in the model

Notes: $i$ denotes the country index that also establishes the policy jurisdiction of each tool. For example, $\tau ^{i}$ with $i=c$ denotes the policy tool set in country $c$ that affects the financial intermediation activities of banks operating in such economy. Additionally, in general, $i = {a,b,c}$ as the effect on welfare may originate in any economy and affect welfare through their local or international effects.

Table 2 summarizes the effect of any policy change on the objective of each type of planner. To understand the possible effects, we can consider the example of a coalition of two countries, the associated policymakers may decide to cooperate and set their toolkit jointly, and in that case, the policy objective function would be a combination of the welfare of both economies.

With no individual null effects, we have that the total spillover effects between Nash and centralized (or cooperative) cases will differ. As a result, when solving the Ramsey Planning Problems we should obtain different optimal tool levels across policy setups.Footnote ³⁹

The associated Ramsey planner problem is solved for each of the planners in the four cases. This policy problem consists of maximizing a welfare objective subject to the conditions characterizing the private equilibrium of other agents. The objective and problem to solve in each regime are explained in detail in Appendix C. Similar to the private equilibrium case, the numerical solutions reported here are solved nonlinearly, but with a system of equations accounting for the first-order necessary conditions of the involved policymakers in each regime. With this, now we can also solve for the taxes as these are no longer taken as given.

6.2 Implied optimal policies

The results, shown in Table 3, reflect the policy trade-off the planners face: they can implement a tax to undo the financial friction or increase financial intermediation and production by subsidizing the banking sector. In the baseline or nationally oriented case, the emerging planners focus on undoing the friction with a tax. The same is true for the center planner. However, the latter taxes the local banking sector heavily to favor intermediation abroad—where its banks could profit at a higher rate—rather than mitigating the friction, a pattern that aligns with the assumption that the friction is present mostly in peripheral countries, as specified in our baseline setup.Footnote ⁴⁰

Table 3.

Ramsey-optimal taxes under each policy setup

Units: Proportional tax on banking rate of return.

Notes: The solution is obtained using a nonlinear numerical solver of the system of equations characterizing the Ramsey problem allocation of each regime.

When allowing for different levels of cooperation, or of centralization of the policies, we see (from the absolute value of the instruments) that cooperation allows the planner to regulate with more conservative taxes to deliver—as will be shown below—a comparable effect. Interestingly, by internalizing the effect of domestic policies to other locations, a globally cooperative arrangement gives space to subsidize intermediation in emerging economies, while the center taxes are set more loosely which indirectly mitigates the extent of the friction at the peripheries.Footnote ⁴¹ Thus, in a fully cooperative case, each country-specific tool is designed with a greater leaning toward generating prosper-thy-neighbor effects. The intuition in this case is that, as long as the frictions are attended in any way (with any country’s toolkit), countries can benefit from higher levels of global intermediation in a similar fashion to how money expansions can be welfare improving for other countries in Obstfeld and Rogoff (Reference Obstfeld and Rogoff1995) and Corsetti and Pesenti (Reference Corsetti and Pesenti2001).

6.3 Effects of policy

We can compare the regimes in terms of other economic outcomes. For example, how effective they are at mitigating the frictions, and the implied welfare they deliver. For that, in Table 4 we show the equivalent compensation changes—in terms of consumption increases—that agents undergo from transitioning from a benchmark allocation to one of the regimes (with optimal policies). For example, if the number is $\phi \gt 1$ and the benchmark is the no-policy equilibrium, we say that agents benefit from the policy in a way that would allow them to expand consumption by $(\phi -1)\times 100\%$ .Footnote ⁴²

Table 4.

Welfare comparison across policy schemes with respect to the first-best allocation (left panel) and with respect to the no-policy equilibrium (right panel)

Units: Proportional steady-state consumption increase in the benchmark model. That is, by how much consumption in the benchmark should be scaled to match welfare in the column’s regime.

The results indicate that all regimes are capable of mitigating the financial frictions. We can see this in the fact that all countries can improve welfare relative to the no-policy equilibrium. Moreover, they can fully undo the effect of the frictions since the welfare improvement is such that the policies can mimic the first-best allocation (equilibrium in the absence of financial frictions).

A second salient result is that all regimes deliver similar welfare outcomes even if they imply different combinations of prudential tools. This could be deemed surprising given the interpretations provided before. However, this can be the case, as explained by Korinek (Reference Korinek2016), because the conditions for a first-welfare theorem of financial regulations are met. That is, either the cross-border policy spillovers are efficient, the policies are too flexible, or the costs of regulations are trivial. Our setup has at least the two last properties as we don’t constrain the range of possible taxes and the cost of excessive regulation does not affect the policy objective directly.Footnote ⁴³ To explore this we also carry a calculation of the welfare effects in the presence of policy costs in the spirit of Dedola et al. (Reference Dedola, Karadi and Lombardo2013) and Agénor et al. (Reference Agénor, Jackson and Jia2021).

6.4 Policy costs of prudential interventions

To consider the case of costly interventions, we solve the modified Ramsey problems where we include a convex cost of policy implementation. The objective function of the planner becomes

\begin{gather*} \max _{\mathbf{x_t},\tilde \tau _t} \quad W^{objective}_t = f \big(\alpha ^i,W^i_t\big) {- \Gamma (\tau ^i)}, \\[2pt] s.t. \qquad \mathbb{E}_tF(\mathbf{x_{t-1}}, \mathbf{x_t}, \mathbf{x_{t+1}}, \tau _t, \theta ), \end{gather*}

with $ \tilde \tau \subseteq \tau$ and welfare weights $\alpha ^i \geq 0$ . Here, $f(\alpha ^i,W^i_t)$ corresponds to the same objective functions considered before and $\Gamma (\tau ^i) = \psi (\tau ^i)^2$ denotes a quadratic policy implementation cost.Footnote ⁴⁴

The results, reported in Table 5, suggest the presence of gains from policy centralization for every country and globally. In addition, the high cost of policy implementation leads the countries to set their tools much more conservatively compared to the baseline. Finally, every cooperative setup matches the first best.Footnote ⁴⁵ Put in perspective, these results imply that if regulation is costly, the nationally oriented policies (non-cooperative) can mitigate only about half of the welfare cost of financial frictions that in our baseline amounted to about $4\%$ of consumption losses per period. In contrast, the cooperative regimes can bring the economy even closer to the first-best allocation, effectively undoing the remaining welfare cost implied by the friction.

Table 5.

Welfare comparison across policy schemes with respect to the non-cooperative Nash equilibrium and policy implementation costs

Units: Proportional steady-state consumption increase in the baseline non-cooperative regime.

Taking stock, a preliminary view to the regimes’ outcomes may hint we should favor the idea that coordination gains are nil. However, the welfare equivalency result is overturned once we introduce regulatory costs. The prudential regulation costs can be easily rationalized and are subject of recent studies (e.g., Richter et al. Reference Richter, Schularick and Shim2019; Boar et al. Reference Boar, Gambacorta, Lombardo and da Silva2017). Furthermore, the inclusion of these is meaningful in our framework, as the simplifications—finite horizon and perfect foresight—may undermine the welfare cost of volatile regulations. Due to this, a comprehensive welfare accounting of these gains goes beyond the scope of this paper that instead focuses on the analytical and numerical exploration of the prudential leakages. Nonetheless, although only indicative, the presence of gains based on the higher interventionism of decentralized regulations still aligns with the findings of studies (for alternative instruments and environments) such as Davis and Devereux (Reference Davis and Devereux2022), Jin and Shen (Reference Jin and Shen2020), and Agénor et al. (Reference Agénor, Jackson and Jia2021), among others.Footnote ^{46 ,} Footnote ⁴⁷

More importantly for our main research question, the presence of non-trivial policy leakages leads to interdependencies between policymakers in different locations that allow for a wide menu of regulatory combinations to manage the trade-off between undoing financial frictions and curtailing financial intermediation given the costs of regulation. Clearly, and in line with empirical studies, these policies do leak beyond their jurisdiction which can have consequences for policy design adjustments.