Fiscal and Macroprudential Policies in a Monetary Union

JOSÉ E. BOSCÁ

JAVIER FERRI

MARGARITA RUBIO

Documento de Trabajo 2022/08

octubre de 2022

fedea

Las opiniones recogidas en este documento son las de sus autores y no coinciden necesariamente con las de Fedea.

JosÈ E Bosc·y

Javier Ferriz

University of Valencia and Fedea

University of Valencia and Fedea

Margarita Rubiox

University of Nottingham

October 2022

Abstract

In the European Monetary Union (EMU), monetary policy is decided by the European Central Bank (ECB). This can create some imbalances that can potentially be corrected by national policies. So far, Öscal policy was the natural candidate to adjust those imbalances. Nevertheless, after the global Önancial crisis (GFC), a new policy candidate has emerged, namely national macroprudential policies, with the mission of reducing Önancial risks. This issue gives rise to an interesting research question: how do macroprudential and Öscal policies interact? By a§ecting real interest rates and the level of activity, a discretionary macroprudential policy alters the evolution of public debt and can impose a Öscal cost when the government is forced to increase tax rates to stabilize the public debt-to-GDP ratio. In a monetary union, a domestic macroprudential shock creates substantial cross border Önancial e§ects and also ináuences the foreign country Öscal stance. Moreover, a discretionary government spending policy a§ects housing prices, so the strenght with which macroprudential policy reacts to a change in the price of houses has an impact on the Öscal multiplier.

Keywords: Monetary union, macroprudential policy, Öscal policy, monetary policy

This paper is part of the project PID2020-116242RB-I00 funded by MCIN/AEI/ 10.13039/501100011033. We thank to conference and seminar participants at the ASSA Annual Meeting, 2021, IX Meeting on International Economics, IFABS, 46th SAEe, 14th Annual Meeting of the Portuguese Economic Journal, ICMAIF 2021, XXIII Encuentro de Economia Aplicada, University of Paris Nanterre, Banque de France, Tor Vergata University of Rome, Banca díItalia, Eberhard Karl University of T¸bingen, 37th meeting of the European Economic Association, 16th annual Dynare Conference and 53rd Annual Conference of the Money, Macro and Finance Society. Special thanks to Stefan Gebauer for his very useful suggestions. Jose E. Bosca and Javier Ferri acknowledge the Önancial support of the Generalitat Valenciana grant GVPROMETEO2020-083, Fundacion Rafael del Pino and BBVA Research.

yUniversity of Valencia

zUniversity of Valencia

xUniversity of Nottingham, University Park, Sir Clive Granger Building, Nottingham. e-mail. margarita.rubio@nottingham.ac.uk.

JEL ClassiÖcation: E32, E44, F45

1 Introduction

The global Önancial crisis (GFC) introduced new challenges for the implementation of macro-Önancial policies. Risks to Önancial stability made it clear the necessity of policies to stabilize the Önancial system, namely macroprudential policies. However, these kind of policies need to coexist with existing policies such as monetary and Öscal policy, making their coordination an important topic of research. This issue becomes especially interesting in the context of a monetary union, in which monetary policy is centralized by a common central bank and Öscal policy is decentralized to national authorities. How to set up macroprudential policies in this kind of setting has been a question of concern.

In the European Union (EU), after a long policy debate, a new macroprudential framework has emerged. Although some macroprudential tools such as capital requirements are under the umbrella of the Basel Committee and are set up internationally, other tools such as the loan-to-value (LTV) are decided at a national level under the supervision of the European Systemic Risk Board (ESRB). These national tools are left at discretion of national authorities, which need to decide on their setting, depending on both national and European Önancial markets developments.

In the European Monetary Union (EMU), monetary policy is decided by the European Central Bank (ECB). Countries have lost their capacity to decide on monetary policy at a national level plus the exchange rate is no longer a tool of adjustment. This can create some imbalances than can potentially be corrected by national policies. So far, Öscal policy was the natural candidate to adjust those imbalances. Nevertheless, after the GFC, a new policy candidate has emerged, namely national macroprudential policies. This issue gives rise to important research questions: what is the Öscal impact of macroprudential policies? Do national macroprudential policies impose a Öscal cost on foreign countries - and hence provide an additional argument for cross-country coordination? How can coordination between macroprudential, Öscal and monetary discretionary policies within a country improve macroeconomic outcomes? How should macroprudential rules be designed in order to optimize the impact of Öscal and monetary surprises? Are trade-o§s present in the design of the macroprudential rules according to targets? Which are the existing trade-o§s among di§erent shocks?

In the present paper, we develop a simple two country DSGE model in a monetary union. We calibrate these two countries to match the features of Spain and Germany. There are two types of agents in each economy. These two agents di§er in their temporal discount rate, becoming borrowers and lenders. We allow lenders to have di§erent preferences for national and foreign borrowers debt which illustrates an important empirical feature in an open economy; the Önancial transactions between the agents inside the economy and between those across economies. In doing so, we follow Sargent (1987) or, more recently Krishnamurthy and Vissing-Jorgensen (2015) or Reis (2020). Our utility-maximizer individuals have debt in the utility function. Moreover, the individuals with lower temporal discount rate have a positive asset position. This structure allows us to introduce country-speciÖc bond preferences that summarizes imperfect Önancial integration. The bond market consists of four kind of bonds: lenders in both countries have access to (public and private) national and foreign bonds. Our model generates a downward-sloping demand for di§erent types of bonds.

We also consider an endogenous risk premium that varies over time, as in Blagov (2013) and Hansen (2015). In our model, the risk premium arises as an external e§ect that creates an additional wedge between domestic and foreign issued bonds. Borrowers from relatively more indebted countries will have to pay higher interest rates.

The paper is related to the general equilibrium literature that allows for international lending, like in St‰hler and Thomas (2012) or Kolasa (2009). In those papers, one country becomes a net borrower. In our paper, the borrowing country displays agents who are lending home and abroad, as also there are agents in the lending country who borrow from the foreign borrowing country. Our paper is also related to research that looks at policy interactions, i.e. monetary-Öscal (some examples on this link are Beetsma and Jensen, 2005; Ferrero, 2009, Farhi and Werning, 2017; Demid, 2018; or Bonam and Lukkezen, 2019) and monetary-macroprudential (see Farhi and Werning, 2016 for a uniÖed approach and BussiËre et al., 2020, for a recent survey).

The most closely related work to ours is Malmierca (2022), which combines the presence of international lending in a monetary union with the analysis of monetary-Öscal-macroprudential policy interactions. The key question it addresses is whether macroprudential-Öscal policy interaction is relevant for stabilization in a monetary union where national economies cannot use monetary policy. Following a credit risk shock originating in the net borrower country, the paper analyzes the response of key macroeconomic variables under two alternative implementations of macroprudential policy, country-targeted and supranational, interacting with three di§erent combinations of Öscal and monetary policy based on the active/passive deÖnitions introduced by Leeper (1991). In the present paper, the focus of the analysis is, however, di§erent. We study the transmission channels of symmetric or asymmetric dis cretionary policies (macroprudential, Öscal and monetary) from one country to another in a monetary union. We also examine the magnitude of the e§ects of such policies, depending on the tightness of

Öscal, monetary or macroeconomic rules. In fact, our work has a closely related approach to Reis (2020) who has explored the Öscal footprint of macroprudential policies. He identiÖes three channels through which this e§ect travels: Örst, an increase in the price of government bonds which eases the government budget constraint; second, a reduction on activity and, hence, on government revenues, which makes the government constraint more tight; third, a lower bailout costs. In our paper we present results with a fully developed dynamic general equilibrium model. The model accounts for the Örst two channels adding a cross-border lending/borrowing channel, but abstracting from defaults and bailouts.

Results show that a discretionary macroprudential policy in one country produces substantial crossborder e§ects on Önancial assets, real activity and the Öscal footprint. However, the spillover e§ects of the Öscal policy are much more limited. We also Önd that for symmetric macroprudential shocks a§ecting both countries, there is a monetary surprise that can replicate the results, up to the e§ects on private debt. That means that a simultaneous discretionary expansionary monetary policy by the central bank can neutralize three gaps that arise after a macroprudential policy (in output, ináation and taxes) without killing the e§ect on private debt. When the reduction of ináation becomes an objective itself, coordination between macroprudential and Öscal discretionary policies can give rise to better outcomes by improving output results, without killing the ináation drop or imposing a cost in terms of higher taxes in the short run. In terms of the design of policy rules, we conclude that a combination of tight macroprudential policy and loose monetary policy maximizes the e§ects of Öscal policy, but minimizes the impact of technology shocks. Moreover, a tight macroprudential rule helps Öscal policy to achieve the target of a larger output, and monetary policy to reduce ináation. After a productivity shock, however, a tight macroprudential rule diminishes the positive impact on GDP, but reinforces the negative impact on ináation.

The remainder of the paper goes as follows. Section 2 presents the model setup. Section 3 shows the Önancial and Öscal impact of macroprudential policy. Section 4 studies the e§ects of Öscal policy. Section 5 compares the e§ects of coordinated discretionary macroprudential, monetary and Öscal policies. Section 6 discusses how committing to a looser or tighter rule by the di§erent policy institutions can alter the results of discretionary policies. Section 7 concludes.

2 The Model

We consider two countries; A (domestic) and B (foreign), that trade consumption and bonds in a monetary union. Agents in each country have the option to choose between di§erent assets. Bonds are tradable, whereas houses are non tradable. The di§erence between the discount rates among households endogenously divides consumers into borrowers and savers in each economy. Governments can also borrow. Both, public and private borrowers are able to borrow from national or foreign lenders. By the same token, lenders choose between lending to national or foreign (public or private) borrowers. Therefore, there are four di§erent bonds in the monetary union. We add country-speciÖc preferences for bonds to account for bond-speciÖc demand functions. Borrowing entails an external e§ect in the form of a risk premium, which evolves according to relative total debt-to-GDP.

Households also buy domestic and foreign goods. However, for simplicity, we do not consider home bias in this market. Labor hired in a competitive market is the only factor used in production. Prices of Önal goods are sticky, in a Calvo fashion (Calvo, 1983).

There is a progressive tax scheme, with a áat rate and an exempt labor income. The Öscal authority moves the áat rate in order to stabilize the government debt-to-GDP ratio. Macroprudential institutions monitor housing prices and decide the LTV. The central bank chooses the reference interest rate on the basis of the monetary union ináation.

Both countries are symmetrical in terms of the equations characterizing their economies. The relative weights of the population between countries are represented by ! and (1 !) for country A and B; respectively. Within each country the share of borrowers is represented by and , respectively. That is, if we designate by N the total population in the monetary union, then , where and stand for population in economy A and B; respectively. Then, we deÖne and If we assume that the patient and impatient population in economy A are and then and . Similarly for country B; and

Next, we sketch the model, introducing the most relevant equations and leaving the rest to Appendix 1. We also omit the equivalent equations of country B.

2.1 Households

2.1.1 Patient Households

Patient households discount the future at a lower rate than impatient households. This fact drives them to be the lenders in the economy because they value relatively more future consumption than the group of borrowers in the population.

In general, lower-case letters stand for real variables whereas capital letters are reserved for nominal variables. Patient households solve the following optimization problem, maximizing their utility function:

\[U ^ {l} = E _ {0} \sum_ {i = 0} ^ {\infty} \beta^ {l i} \left( \begin{array}{c} \ln c _ {t + i} ^ {l} + \gamma_ {h} \ln h _ {t + i} ^ {l} + \\ + \gamma_ {b _ {A t}} (\bullet) \ln (b _ {A t} ^ {l}) + \chi_ {B} \ln (b _ {B t} ^ {l}) + \\ \gamma_ {b _ {A t}} ^ {g} (\bullet) \ln (b _ {A t} ^ {g}) + \chi_ {B} ^ {g} \ln (b _ {B t} ^ {g}) - \frac {(n _ {t + i} ^ {l}) ^ {1 + \eta}}{1 + \eta} \end{array} \right).\]

Variables are written relative to the population of lenders that is, can be interpreted as total consumption of lenders divided by the total amount of lenders in economy A. In the same vein, stands for per capita working hours and represents the stock per capita of houses owned by lenders. Parameters ; relate to the elasticity of labor supply with respect to wages and the preferences for housing, respectively. The utility function distinguishes between four types of bonds that capture di§erent Önancial alternatives for lenders. Bonds issued by national borrowers in hands of national lenders are denoted by where and are nominal bonds, which pay a gross nominal interest rate . Similarly, bonds bought by national lenders from the home government are called whereas and stand for bonds issued by foreign households and government.

Preferences di§er between public and private bonds because of di§erences in safety and/or liquidity, and also di§er between domestic and foreign bonds, due to imperfect Önancial market integration. The assumption here is that di§erent assets o§er non-pecuniary services to households, a generalization of the modelís utility function in Sargent (1987) or, more recently, in Krishnamurthy and Vissing-Jorgensen (2015) or Reis (2020). In this framework, as the total debt-to-output ratio in economy A increases with respect to economy B, bonds in economy B become more attractive. In particular, preferences for bonds change over time as a function of the ratio of total debt-to-output in economy A with respect to economy B. That is,

\[\gamma_ {b _ {A t}} (\bullet) = \chi_ {A} + \vartheta \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right),\]

and

\[\gamma_ {b _ {A t}} ^ {g} (\bullet) = \chi_ {A} ^ {g} + \vartheta \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right),\]

where and represent aggregate debt (to be deÖned more precisely below) in country A and B, respectively, and # is a positive parameter. Notice that the di§erence captures the existence of a public-private bond bias, whilst the presence of a country bias would be reáected by and

We assume that patient households do not internalize the e§ect that their lending decision has on the terms and , i.e. they do not consider how their decided amount of lending may a§ect the aggregates and .

The budget constraint, in real terms, can be written as follows:

\[\begin{array}{r l} & c _ {A t} ^ {l} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {l} + q _ {t} (h _ {t} ^ {l} - h _ {t - 1} ^ {l}) + b _ {A t} ^ {l} + \frac {P _ {B t}}{P _ {A t}} b _ {B t} ^ {l} + b _ {A t} ^ {g} + \frac {P _ {B t}}{P _ {A t}} b _ {B t} ^ {g} \\ & \leq (1 - x _ {t} ^ {l}) w _ {t} n _ {t} ^ {l} + \frac {R _ {A t - 1}}{\pi_ {A t}} b _ {A t - 1} ^ {l} + \frac {P _ {B t}}{P _ {A t}} \frac {R _ {B t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {l} \\ & + \frac {R _ {A t - 1} ^ {g}}{\pi_ {A t}} b _ {A t - 1} ^ {g} + \frac {P _ {B t}}{P _ {A t}} \frac {R _ {t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {g} + d _ {t}. \end{array}\tag{1}\]

and are, respectively, per capita consumption of home and foreign produced goods. and are the producer price indexes in countries A and B. The numeraire is ; which we use to deáate all the nominal variables, including loans, wages and proÖts. is the inverse of the terms of trade and the real price of houses, . A positive value for and corresponds to a lending amount, whereas a negative value implies a borrowing amount. For patient households both are positive. Bonds yield an interest rate, which depends on each particular type, with bonds issued by government in country B yielding an interest rate that coincides with the policy rate. are proÖts stemming from the monopolistically competitive Örms that lenders own. Finally, we assume that labor income , i.e. wages times per capita working hours) is taxed, while, for the sake of simplicity, proÖts and bond returns are not taxed. The average tax rate on lenders labor income is deÖned as , where represents the per capita amount of taxes paid by these households,

\[t _ {A t} ^ {l} = m _ {A t} (w _ {t} n _ {t} ^ {l} - \overline {{t _ {A}}}).\tag{2}\]

The previous expression assumes a progressive tax scheme by means of the introduction of a per capita tax-exempt income, equal to ; and a áat tax rate on labor income, . Hence, the average tax rate increases with income, helping to strengthen the automatic stabilizer feature of the tax. As will be evident below, is going to be the instrument used by the Öscal authority to keep the ratio of public debt-over-GDP constant in the long run. A higher implies a more progressive tax scheme.

The ináation rate on the domestically produced goods, , and foreign goods, , are deÖned as

\[\pi_ {A t} = \frac {P _ {A t}}{P _ {A t - 1}},\tag{3}\]

\[\pi_ {B t} = \frac {P _ {B t}}{P _ {B t - 1}}.\tag{4}\]

The consumption basket for lenders, in terms of utility, is deÖned as

\[c _ {t} ^ {l} = (c _ {A t} ^ {l}) ^ {\omega} (c _ {B t} ^ {l}) ^ {1 - \omega}\tag{5}\]

Patient households maximize with respect to and (leading to standard Örst order conditions, as can be seen the Appendix 1), and also over the four Önancial assets ; and . Optimal decisions regarding bonds should satisfy the following conditions:

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {A t}}{\pi_ {A t + 1}} \right] + \frac {\chi_ {A}}{b _ {A t} ^ {l}} + \frac {\vartheta}{b _ {A t} ^ {l}} \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right),\tag{6}\]

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {B t}}{\pi_ {A t + 1}} \right] + \frac {P _ {A t}}{P _ {B t}} \frac {\chi_ {B}}{b _ {B t} ^ {l}},\tag{7}\]

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {A t} ^ {g}}{\pi_ {A t + 1}} \right] + \frac {\chi_ {A} ^ {g}}{b _ {A t} ^ {g}} + \frac {\vartheta}{b _ {A t} ^ {g}} \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right),\tag{8}\]

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {t}}{\pi_ {A t + 1}} \right] + \frac {P _ {A t}}{P _ {B t}} \frac {\chi_ {B} ^ {g}}{b _ {B t} ^ {g}},\tag{9}\]

where is the Lagrangian multiplier associated with the restriction (1). From these conditions, we can obtain non-arbitrage conditions among the four assets. Di§erences in interest rates between di§erent bonds depend on three factors. First, the amount of bonds held by households, as captured by the second term in the right hand side of the above expressions. Other things equal, an increase in the amount of one type of bonds reduces its price and rises its interest rate vis-‡-vis other bonds. Thus, there is a downward-sloping demand for bonds. Second, the term related to the endogenous risk premium. Third, a factor capturing the terms of trade, . An increase in the last two factors augments the yield on domestic bonds vis-‡-vis foreign bonds. From equations (8) and (9) let us deÖne this risk premium term as

\[\phi_ {t} \equiv - \frac {\pi_ {A} \vartheta}{b _ {A} ^ {g} \beta^ {l} \lambda^ {l}} \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right),\tag{10}\]

which represents the increase in to a given change in :Notice that, caeteris paribus, it increases with the ratio of total debt-over-output in country A. In (10) we keep some variables constant at their initial steady-state value to emphasize the contribution of changes in the ratios of household debt-to-GDP.

Together with the demand for bonds from foreign households, the above expressions produce induced e§ects of economic policies on the desired composition of bonds in the portfolio of the lenders households, which translate into changes in the decisions of private/public and national/ foreign borrowers.

2.1.2 Impatient Households

Impatient households have a higher discount rate than patient households , which drives them to be the borrowers of the economy. These agents sell bonds to the lenders of both economies, paying for these the corresponding interest rates. Impatient households solve the following optimization problem, maximizing the utility function

\[U _ {0} ^ {r} = E _ {0} \sum_ {i = 0} ^ {\infty} \beta^ {r i} \left(\ln c _ {t + i} ^ {r} + \gamma_ {h} \ln h _ {t + i} ^ {r} - \frac {(n _ {t + i} ^ {r}) ^ {1 + \eta}}{1 + \eta}\right),\]

subject to

\[\begin{array}{l} c _ {A t} ^ {r} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {r} + q _ {t} (h _ {t} ^ {r} - h _ {t - 1} ^ {r}) + \frac {R _ {A t - 1}}{\pi_ {A t}} b _ {t - 1} ^ {r} \\ \leq (1 - x _ {t} ^ {r}) w _ {t} n _ {t} ^ {r} + b _ {t} ^ {r}. \end{array}\tag{11}\]

can be interpreted as total consumption of borrowers divided by the total amount of borrowers in economy A (N_r). expresses total real private borrowing in country A from domestic and foreign lenders, and is deÖned as a positive variable. Similarly to impatient households, is deÖned as

, where

\[t _ {A t} ^ {r} = m _ {A t} (w _ {t} n _ {t} ^ {r} - \overline {{t _ {A}}}).\tag{12}\]

Hence, the average tax rate will di§er between borrowers and lenders given that, although wages will be common across households, working hours may be di§erent.

Additionally, these consumers face a borrowing constraint of the form

\[E _ {t} \left[ \frac {R _ {A t}}{\pi_ {A t + 1}} b _ {t} ^ {r} \right] \leq E _ {t} \left[ k _ {A t} q _ {t + 1} h _ {t} ^ {r} \right],\tag{13}\]

where can be interpreted as a loan-to-value ratio (LTV) and will be the instrument for the macroprudential policy. Notice that keeping constant the rest of variables, a rise in the price of fall in will increase the supply of private bonds.

The consumption basket for borrowers is deÖned as

\[c _ {t} ^ {r} = (c _ {A t} ^ {r}) ^ {\omega} (c _ {B t} ^ {r}) ^ {1 - \omega}.\tag{14}\]

The impatient household maximizes with respect to ; and . The derivative with respect to yields

\[\lambda_ {t} ^ {r} = \beta^ {r} E _ {t} \left[ \lambda_ {t + 1} ^ {r} \frac {R _ {A t}}{\pi_ {A t + 1}} \right] + \xi_ {t} R _ {A t},\tag{15}\]

where is the Lagrangian multiplier associated with the restriction (11). Borrowers in country A will pay the interest rate , regardless who provides the funds (domestic or foreign lenders).

2.2 Firms

We have J Örms of mass 1. Each Örm produces a di§erentiated good and takes decisions subject to three constraints: a constant returns production technology; a downward sloping demand curve, and a perfect competition labor market. In all that follows all variables are represented in per capita terms of total population in the economy. The optimization problem can be written as:

\[\min W _ {t} n _ {t} (j),\]

subject to:

\[y _ {t} (j) = z _ {t} n _ {t} (j),\tag{16}\]

\[y _ {t} (j) = \left(\frac {P _ {A t} (j)}{P _ {A t}}\right) ^ {- \varepsilon} y _ {t},\tag{17}\]

where is the nominal wage and is the technical level, both common to all Örms.

Optimization with respect to employment yields the following standard labor demand in real terms:

\[w _ {t} = m c _ {t} \frac {y _ {t}}{n _ {t}}.\tag{18}\]

Optimal prices are obtained assuming a Calvo scheme:

\[\max _ {p _ {A t} (j)} \Pi_ {0} = E _ {t} \sum_ {i = 0} ^ {\infty} \lambda_ {t + i} ^ {l} (\beta^ {l} \theta) ^ {i} \left[ \left(\prod_ {r = 1} ^ {i} \frac {(\pi_ {A t + r - 1}) ^ {\zeta}}{\pi_ {A t + r}} p _ {A t} (j) - m c _ {t + i}\right) y _ {t + i} (j) \right],\]

subject to the variety demand function,

\[y _ {t + i} (j) = \left(\prod_ {r = 1} ^ {i} \frac {(\pi_ {A t + r - 1}) ^ {\zeta}}{\pi_ {A t + r}} p _ {A t} (j)\right) ^ {- \varepsilon} y _ {t + i}.\tag{19}\]

A proportion of Örms do not reset prices optimally at t and adjust them according to a simple indexation rule to catch up with lagged ináation: : We are assuming that Örms that are not allowed to change prices optimally reset prices each period according to ináation. stands for the relative price : Taking into account that all Örms will set the same optimal price, the solution to the above problem renders the following New Keynesian Phillips curve,

\[\left(\frac {1 - \theta (\pi_ {A t}) ^ {\varepsilon - 1}}{1 - \theta}\right) ^ {\frac {1}{1 - \varepsilon}} = \frac {\varepsilon}{\varepsilon - 1} \frac {E _ {t} \sum_ {i = 0} ^ {\infty} \lambda_ {t + i} ^ {l} (\beta^ {l} \theta) ^ {i} m c _ {t + i} y _ {t + i} \left(\prod_ {r = 1} ^ {i} \frac {(\pi_ {A t + r - 1}) ^ {\zeta}}{(\pi_ {A t + r})}\right) ^ {- \varepsilon}}{E _ {t} \sum_ {i = 0} ^ {\infty} \lambda_ {t + i} ^ {l} (\beta^ {l} \theta) ^ {i} y _ {t + i} \left(\prod_ {r = 1} ^ {i} \frac {(\pi_ {A t + r - 1}) ^ {\zeta}}{(\pi_ {A t + r})}\right) ^ {1 - \varepsilon}}.\tag{20}\]

2.3 Aggregation

There are some restrictions linking debt. In the domestic economy, bonds issued by domestic borrowers (total private debt) may be in hands of either domestic or foreign lenders:

\[b _ {t} ^ {r} = \frac {(1 - \tau)}{\tau} b _ {A t} ^ {l} + \frac {(1 - \omega)}{\omega} \frac {(1 - \tau^ {*})}{\tau} b _ {A t} ^ {* l},\tag{21}\]

where is economy Bís lenders holdings of bonds issued by economy Aís borrowers.

With respect to public debt, in the domestic economy:

\[b _ {t} ^ {g} = (1 - \tau) b _ {A t} ^ {g} + (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} b _ {A t} ^ {* g}.\tag{22}\]

where is economy Bís holdings of bonds issued economy Aís government.

Total debt, i.e. the sum of public and private debt, in economy A can be deÖned as:

\[b _ {t} = b _ {t} ^ {g} + \tau b _ {t} ^ {r}\tag{23}\]

Aggregation over housing results in:

\[\tau h _ {t} ^ {r} + (1 - \tau) h _ {t} ^ {l} = h,\tag{24}\]

where h is an exogenous variable representing the per capita stock of housing in economy A.

Consumption can also be aggregated using the shares of lenders and borrowers in each economy. Consumption in country A of goods produced in country A:

\[c _ {A t} = \tau c _ {A t} ^ {r} + (1 - \tau) c _ {A t} ^ {l}.\tag{25}\]

Consumption in country A of goods produced in country B (imports (exports) by country A (B)):

\[c _ {B t} = \tau c _ {B t} ^ {r} + (1 - \tau) c _ {B t} ^ {l}.\tag{26}\]

Total consumption in country A can be deÖned as

\[c _ {t} = \tau c _ {t} ^ {r} + (1 - \tau) c _ {t} ^ {l} = c _ {A t} + \frac {P _ {B t}}{P _ {A t}} c _ {B t},\tag{27}\]

where

\[c _ {t} ^ {l} = c _ {A t} ^ {l} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {l},\tag{28}\]

and

\[c _ {t} ^ {r} = c _ {A t} ^ {r} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {r}.\tag{29}\]

Employment is aggregated as

\[n _ {t} = \tau n _ {t} ^ {r} + (1 - \tau) n _ {t} ^ {l}.\tag{30}\]

Total government revenues are characterized by,

\[t _ {t} = (1 - \tau) t _ {A t} ^ {l} + \tau t _ {A t} ^ {r} = m _ {A} (w _ {t} n _ {t} - t _ {A t}).\tag{31}\]

Finally, the aggregate production function for the domestic economy is,

\[y _ {t} = z _ {t} n _ {t}.\tag{32}\]

2.4 Fiscal, Monetary and Macroprudential Rules

We assume an exogenous amount of non-productive government consumption for the domestic economy and for economy B. Public debt evolves according to

\[b _ {t} ^ {g} = \frac {R _ {A t - 1} ^ {g}}{\pi_ {A t}} b _ {t - 1} ^ {g} + (g _ {t} - t _ {t}).\tag{33}\]

\[b _ {t} ^ {* g} = \frac {R _ {t - 1}}{\pi_ {B t}} b _ {t - 1} ^ {* g} + (g _ {t} ^ {*} - t _ {t} ^ {*}).\tag{34}\]

Common monetary policy is characterized by a simple Taylor rule. The central bank takes into account the weighted average of the monetary union countriesíináation:

\[R _ {t} = R _ {t - 1} ^ {\rho} \left[ \left(\pi_ {A t} ^ {\omega} \pi_ {B t} ^ {1 - \omega}\right) ^ {\Phi} \overline {{R}} \right] ^ {1 - \rho}.\tag{35}\]

Each country uses the áat tax as the instrument to stabilize the ratio of total public debt-over-GDP in the long run. Then, the Öscal policy rule can be represented as:

\[m _ {A t} = m _ {A t - 1} + \psi_ {1} f _ {t} \left(\frac {b _ {t} ^ {g}}{y _ {t}} - \overline {{\left(\frac {b ^ {g}}{y}\right)}}\right) + \psi_ {2} f _ {t} \left(\frac {b _ {t} ^ {g}}{y _ {t}} - \frac {b _ {t - 1} ^ {g}}{y _ {t - 1}}\right),\tag{36}\]

where the parameter captures the speed of adjustment from the current ratio to the desired ratio, and is a dummy variable that controls for the time period in which the Öscal rule is initially inactive.

Similarly, for country B,

\[m _ {B t} = m _ {B t - 1} + \psi_ {1} ^ {*} f _ {t} ^ {*} \left(\frac {b _ {t} ^ {* g}}{y _ {t} ^ {*}} - \overline {{\left(\frac {b ^ {* g}}{y ^ {*}}\right)}}\right) + \psi_ {1} ^ {*} f _ {t} ^ {*} \left(\frac {b _ {t} ^ {* g}}{y _ {t} ^ {*}} - \frac {b _ {t - 1} ^ {* g}}{y _ {t - 1} ^ {*}}\right).\tag{37}\]

As an approximation for a realistic macroprudential policy, we consider a Taylor-type rule for the loan-to-value ratio, in the spirit of a Taylor rule for monetary policy. In standard models, the LTV ratio is a Öxed parameter which is not a§ected by economic conditions. However, we can think of regulations of LTV ratios as a way to moderate credit booms. When the LTV ratio is high, the collateral constraint is less tight. And, since the constraint is binding, borrowers will borrow as much as they are allowed to. Lowering the LTV tightens the constraint and therefore restricts the loans that borrowers can obtain. Recent research on macroprudential policies has proposed Taylor-type rules for the LTV ratio so that it reacts inversely to variables such that the growth rates of GDP, credit, the credit-to-GDP ratio or house prices. These rules can be a simple illustration of how a macroprudential policy could work in practice. We consider a decentralized macroprudential policy in which each country can implement its own rule for the LTV, as it is the case within the EU:

\[k _ {A t} = k _ {S S A} \left(\frac {q _ {t}}{\overline {{q}}}\right) ^ {- \phi_ {A q} ^ {k}},\tag{38}\]

\[k _ {B t} = k _ {S S B} \left(\frac {q _ {t} ^ {*}}{\overline {{q ^ {*}}}}\right) ^ {- \phi_ {B q} ^ {k}},\tag{39}\]

where and are the steady-state values for the loan-to-value ratio and house prices in country A. measures the response of the loan-to-to value to house prices in country A. This kind of rule would deliver a lower LTV ratio in booms, when house prices are high, therefore restricting the credit in the economy and avoiding a credit boom derived from good economic conditions.

2.5 GDP and Balance of Payments

The total resource constraint should satisfy the condition that total production should be equal to the sum of factor incomes or total Önal demand in the economy. That is,

\[y _ {t} = w _ {t} n _ {t} + (1 - \tau) d _ {t},\tag{40}\]

\[y _ {t} = c _ {A t} + \frac {(1 - \omega)}{\omega} c _ {A t} ^ {*} + g _ {t}.\tag{41}\]

To Önd an expression for aggregate Örmís proÖts in economy A, we can combine expressions (40) and (18) to obtain:

\[(1 - \tau) d _ {t} = \left(\frac {1}{m c _ {t}} - 1\right) w _ {t} n _ {t}.\tag{42}\]

To obtain an expression for the balance of payments, Örst, we multiply the household budget constraints (1) and (11) by their respective shares in population (1 ) and and aggregate them. Then, we substitute into the previous expression to reach:

\[\begin{array}{r l r} & & {(1 - \tau) \frac {P _ {B t}}{P _ {A t}} \left(b _ {B t} ^ {l} + b _ {B t} ^ {g}\right) + (1 - \tau) b _ {A t} ^ {g} - (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} b _ {A t} ^ {* l} =} \\ & & {\left(\frac {(1 - \omega)}{\omega} c _ {A t} ^ {*} - \frac {P _ {B t}}{P _ {A t}} c _ {B t}\right) + (g _ {t} - t _ {t}) +} \\ & & {+ (1 - \tau) \frac {P _ {B t}}{P _ {A t}} \left(\frac {R _ {B t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {l} + \frac {R _ {t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {g}\right) + (1 - \tau) \frac {R _ {A t - 1} ^ {g}}{\pi_ {A t}} b _ {A t - 1} ^ {g}} \\ & & {- (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} \frac {R _ {A t - 1}}{\pi_ {A t}} b _ {A t - 1} ^ {* l},} \end{array}\tag{43}\]

with . Notice that all the previous variables are in terms of total population in economy A (they are divided by

A clear intuition of the balance of payment condition can be grasped if we obtain the steady state version of (43)

\[\begin{array}{c} (1 - \tau) \frac {P _ {B}}{P _ {A}} b _ {B} ^ {l} \left(1 - \frac {R _ {B}}{\pi_ {B}}\right) - (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} b _ {A} ^ {* l} \left(1 - \frac {R _ {A}}{\pi_ {A}}\right) \\ + (1 - \tau) \frac {P _ {B}}{P _ {A}} b _ {B} ^ {g} \left(1 - \frac {R}{\pi_ {B}}\right) - (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} b _ {A} ^ {* g} \left(1 - \frac {R _ {A} ^ {g}}{\pi_ {A}}\right) + (t - g) = \\ \left(\frac {(1 - \omega)}{\omega} c _ {A} ^ {*} - \frac {P _ {B}}{P _ {A t}} c _ {B}\right), \end{array}\tag{44}\]

where the the LHS in expression (44) represents the variation in the net foreign asset position, as the di§erence between bonds owned abroad (both private and public) and domestic bonds (private and public) owned by foreigners. The RHS is the current account balance.

2.6 Calibration

The values assigned to the parameters of the model use information for Spain (country A) and Germany (country B). We impose some parameters and use the equations of the steady state of the model to calibrate the rest. In this way, we are able to reproduce relevant observable economic facts of the two countries. Table 1 reáects the parameters that have been initially set and the sources used, while Table 2 reproduces the calibrated parameters and the targets obtained with the model.

The weight of Spain, has been set according to Eurostat population statistics. The parameter the reaction of the risk premium to changes in the ratio of debt-to-GDP, takes a value of 0:09. It comes from an estimation of a 4:5 basis points increase in the risk premium (the di§erence between interest rates on public debt in Spain and Germany) for every one percentage point increase in the debt- To obtain the value of 0:09 we use equation (10). The Taylor rule parameters and are the standard ones. The Örst value is consistent with the original parameters proposed by Taylor in 1993. The latter value reáects a realistic degree of interest-rate smoothing (see McCallum, 2001). They are also consistent with recent estimations of Taylor rules (see for example, Sauer and Sturm, 2007). The inverse of Frisch elasticities, the Calvo price probabilities, and the ináation indexation are the ones estimated for Spain and Germany by Casares and V·zquez (2018). As regards the share of credit constrained consumers, we guess a value of for To set we correct using the relationship between the ratio of household debt-over-GDP in Germany and the same ratio in Spain. The value of the elasticity of substitution among di§erent goods in the German monopolistic sector has been calculated for a price-cost margin of 39%, according to the Deutsche Bundesbank (2017). We assume that the price-cost margin in Spain is halfway between the estimates for Germany and Italy in this study. We use the European Mortgage Federation information (Westig and Bertalot, 2017) to set the value for the LTV. Also, we assume the same intensity of reaction between Spain and Germany in the Öscal rule.

Table 2 shows the values of some parameters related to the targeting of di§erent empirical facts. In our model, steady-state interest rates depend on the discount rate and preference parameters for di§erent types of bonds. We chose so that, given the value of the bond preference parameters, the annual interest rate of the German government bond is 1:9%:

Then, we set the impatient discount rate for Spanish lenders at the same Ögure. Notice that this 1 F C (2018) 1 IME (2017)

See European Commission (2018) and IMF (2017).

² GalÌ, LÛpez-Salido and VallÈs (2004), in a model with only Ricardian and rule-of-thumb consumers, consider that the best guess for the share of non Ricardian consumers is in the neighbourhood of

does not imply that the interest rate of the Spanish public debt is the same as that of Germany, due to the di§erent country preferences for bonds and the existence of a risk premium. We consider that the borrowerís discount rate is 2 percentage points lower, in the range of values observed in the literature (see Iacoviello, 2005 for a discussion on the calibration of this parameter). There are four preference parameters that a§ect the demand for bonds in Spain and the respective ones for Germany . We calibrate these values so that the steady-state model solution reproduces two sets of facts: (a) a set of interest rates (the 1 to 2 year Spanish government bond, and the mortgage interest rates in Spain and Germany, and ; (b) a set of debt ratios (the ratio of Spanish public debt in the hands of Spanish households, the ratio of German public debt in German householdsíhands, the ratio of German private debt in German householdsíhands, total Spanish debt over-GDP, and total German debt-over-GDP). The housing preference parameter for Spain has been chosen to obtain the residential stock-over-GDP according to Bank of Spain (BdE). Following Deloittte (2016) report, the number of dwellings per citizen in Spain is roughly the same as in Germany. Thus, we set the value of this parameter for Germany to replicate the same ratio as in Spain. The values for tax-exempt income and guarantee that the áat rates for Spain and Germany are 0:31 and 0:37, respectively.³

As shown in Table 3, we normalize the steady-state per capita aggregate income in Spain to 1, and the German per capita income to 1:3, according to Eurostat. Government consumption represents 18:5% of GDP in Spain and 18:8% in Germany. The targeted long-run public debt-over-GDP is set to an annual 60 and 85 per cent in Germany and Spain, respectively. The German Ögure is in accordance with the the EUís Stability and Growth Pact, while the Spanish one is close to the average along the last decade. Table 4 shows the model steady-state values for aggregate demand and bonds.

³ These tax rates are approximated using data on taxation from the European Commission.

Table 1. Parameters imposed in the model

Table 2. Parameters calibrated from model equations

Table 3. Normalizations and exogenous variables

Table 4. Steady state (in country per capita values on a quarterly basis)

3 The Financial and Fiscal Impact of Macroprudential Policy

Macroprudential policy aims at stabilizing Önancial risks, mainly a§ecting the amount of private debt in the economy. However, by doing so, it can also a§ect public debt and taxes. In this section, we present the dynamics shown by the model after a macroprudential policy consisting of permanently lowering the LTV in Spain by two percentage points, from 0:80 to 0:78. Although this is an asymmetric policy intervention, we also study cross-border e§ects on the German economy. We assume here that the macroprudential rule is not active, by setting

Figures 1 to 4 show the dynamics of some macroeconomic variables, in percentage deviations with respect to their steady state, following the policy. The macroprudential policy reduces largely the amount of private debt in the economy (Figure 1). A less indebted economy generates a reduction in the risk premium (Figure 2), pushing up the demand for public and private Spanish bonds. Figure 1 illustrates the e§ects on bond holdings. While German holdings of Spanish government bonds boost, this is no the case for Spanish lenders, who Önd it more proÖtable to invest in housing. Figure 3 shows that there is a diversion from borrowersíhousing demand, which is heavily punished by a tighter collateral constraint, to housing demand by lenders. It induces a change in the Spanish lenders portfolio of assets, from bonds to houses. Houses are non-tradable goods, and this housing e§ect is absent in Germany. The reduction in the supply of Spanish private bonds a§ects negatively the equilibrium interest rate of household bonds, as is evident in equation (13). Overall, Figure 2 reveals that these movements in the supply and demand of Spanish bonds lowers the private bonds interest rate by 26 basis points (bp) on impact and the government bond interest rate by 33 bp. Figure 2 also displays a reduction in the ináation rate in both countries, which is due to the fall in GDP (Figure 3) provoked by the macroprudential policy.

The lower ináation rates and a weaker level of activity move government debt up at period t (Figure 1). The increase in the government-debt-to-GDP ratio is even higher, so the Öscal authority reacts by rising the marginal tax according to its Öscal rule. Hence, using Reisí semantics (Reis, 2020), the macroprudential policy intervention makes a positive Öscal footprint in our model economy. Interestingly, the policy produces substantial cross-border e§ects not only in terms of Önancial assets, as we have seen, but also on real activity and Öscal outcomes. German GDP falls almost as much as it does in Spain during 5 quarters, but contrary to Spain, GDP recovers gradually afterwards. The main reason for that is the pronounced drop in German exports (Figure 3). The rise in public-debt-to-GDP compels the German government to lift the tax rate (Figure 4), a policy which is relatively more harmful in terms of income for borrowers than for lenders.

The tightening of macroprudential policy in Spain triggers tighter Öscal policies in response to higher debt to GDP ratios in both countries. So Öscal policy is not acting countercyclically, but procyclically in this scenario. This result seems di§erent from standard Öndings of other papers on, for example, monetary/Öscal policy coordination, which usually Önd that a contractionary shock of one policy might trigger some countercyclical response by the other. In fact, this is what happens also in our model when Öscal policy is tightened, provoking lower output and ináation and a monetary countercyclical response by lowering interest rates via the Taylor rule.

The fact that using Spanish macroprudential policy to reduce the stock of private debt alters in such a way the macroeconomic outcomes of the other country in the monetary union calls for a supranational coordination of these policies. Our results indicate that macroeconomic policy in Spain does not only cause a positive Öscal footprint in Spain but also in Germany, imposing thus an economic and possibly political cost on the neighboring country. If the German government is aware of this circumstance it could decide to react in two ways. First, it could partially abate borrowing constraints, which in our model consists of rising the LTV. In this way, it would penalize Önancial stability to relieve the Öscal burden. Second, it might decide to punish Spanish government through a tighter macroprudential policy in a similar game to a trade war, in which case both economies would end up with a much higher tax rate.

Figure 1: Macroprudential policy in Spain: bonds

Figure 2: Macroprudential policy in Spain: interest rates

Figure 3: Macroprudential policy in Spain: real activity

Figure 4: Macroprudential policy in Spain: taxes

4 Discretionary Symmetric Policies

In this section, we study the possibilities that arise in our model for policy coordination among policy makers in both countries. To start with, we assume that governments in the two countries coordinate their actions in such a way that the direction and intensity in the use of macroprudential and Öscal policy in both countries is the same. We call this a symmetric macroprudential and Öscal policy intervention. Also, we introduce a discretionary monetary policy conducted by the central bank.

The left-hand column of Figure 5 shows the response after a permanent 2 percentage point drop in LTV in Germany and Spain, which is the symmetrical equivalent of the macroprudential policy in Spain studied above. Compared to the one-country policy, the simultaneous intervention of macroprudential authorities in both economies produces a non-linear increment in the Öscal footprint of both countries, understood as the increase in the tax rate necessary to accomplish the Öscal rule. This result can be explained by cross spillover e§ects.

In the middle column we show the response of macroeconomic variables to a contractionary monetary policy shock. More speciÖcally, we assume that the central bank changes the interest rate in away that replicates the same impact e§ect as macroprudential policy on the Spanish GDP.⁴ As compared to macroprudential policy, the monetary shock generates very similar results for GDP, ináation and government bonds. It also produces an almost identical Öscal footprint (increase in the tax rate), the main di§erence being in the size of the e§ects on private bonds, which is more pronounced in the case of a macroprudential symmetric intervention.

⁴ This shock is exactly a 44 bp positive surprise in the interest rate lasting three years. However, due to the perfect foresight assumption we use to solve the model, interest rates actually fall on impact.

Figure 5: Symmetric shocks: macroprudential, monetary and Öscal

In the right column, we represent a contractionary Öscal policy.⁵ For the same drop in GDP, Öscal policy entails a smaller ináation reduction, a more modest e§ect on the tax rate, which soon turns negative, and virtually no e§ect on private debt. The analogies and contrasts in the e§ects among the three policies bring the possibility of using a di§erent policy mix for the achievement of di§erent combinations of targets. Consider that policy-makers pursue four targets. Add to the three economic goals (output, ináation and Önancial stability) one pure political target, and assume that governments dislike the increase in the tax rate to which they are forced when government debt to output goes up. Assume that macroprudential institutions detect a need for a Önancial correction and lower the LTVs in the two countries. This comes at the cost of lower output, a downward deviation of ináation rates targets, and higher taxes. A simultaneous discretionary expansionary monetary policy by the central bank can neutralize the three gaps (output, ináation and taxes) without killing the e§ect on private debt.

⁵ The permanent drop on government spending amounts to 0.5 pertentage points of GDP in both countries.

Assume now that, in addition to the Önancial correction, the central bank sees as necessary to reduce the ináation rate. The central bank may consider a no-surprise policy, letting the monetary rule and the macroprudential shock to work. This will reproduce again the e§ects of the Örst column. The central bank fulÖlls the target of a lower ináation, the macroprudential authority gets the goal of less private debt, and the cost is expressed in terms of weaker output and a stronger tax bit. However, a coordination with governments can improve the outcomes if they carry out an expansionary Öscal policy. A simultaneous macroprudential-Öscal policy mix in this way can improve output without killing the ináation drop or imposing a cost in terms of taxes in the short run.

The takeaway message of the previous discussion is that the di§erent policy makers (macroprudential, Öscal and monetary authorities) can take advantage of the discretionary margin they have to coordinate their actions in order to improve macroeconomic outcomes.

5 Policy Rules and Policy Interplay

In all the discussion so far we have kept constant the policy rule parameters at their benchmark values. This implies that the Öscal and monetary policy rules were operative, but the macroprudential rule was not playing any role given that benchmark values are . In this section we relax these assumptions and study how committing to a looser or tighter rule by the di§erent policy institutions can alter the results of discretionary policies. Figure 10 shows the results.

Figure 10 depicts impact e§ects in Spain under di§erent parameterizations of monetary, macroprudential and Öscal rules. A higher value of the parameters mean, respectively, a more intense reaction in the interest rate, given a change in ináation, a bigger movement in the LTV for a given variation in the price of housing, or a greater change in the tax rate after deviations in the public debt over GDP from its long-run target. Thus, larger values of , , and are associated with tighter monetary,

macroprudential and Öscal rules.

LTV shock

Figure 10: Policy shocks and policy rules: impact e§ects

Figure 10 studies the e§ectiveness of Öscal, macroprudential and monetary discretionary policies to ináuence their main targets: GDP, private debt, and ináation. More particularly, we analyze the interaction between a given discretionary policy and the policy rules related with the other two policies. In all three cases we compare the results with those we observe when the economy is a§ected by a permanent increase in productivity of one percentage point. Both discretionary policies or productivity increases, as well as induced changes in rules, only occur in Spain.

For example, in the case of a government consumption policy in Spain (Örst row in Figure 10), both monetary and macroprudential rules interact with the e§ect of the policy or the productivity shock. After a permanent increase in government spending of one percentage point of GDP, there is an increase in output, but the multiplier is less than one. The monetary and the macroprudential rule interact with the e§ect of the policy. A combination of a loose monetary policy and a tight macroprudential rule maximizes the Öscal multiplier. In our plot, the multiplier goes up by 9 percent with respect to the minimum value that is reached when the macroprudential rule is inactive and the monetary rule is the tightest of all those considered. However, the combination of policies that maximizes the e§ect of a Öscal shock is the same that minimizes the impact e§ect of a productivity shock on GDP. Therefore, there is a kind of trade-o§ between the policy-mix that would be preferred by the Öscal authority in order to enlarge the e§ectiveness of its policy, and the production beneÖts the economy would receive from a technology shock with that combination.

In the next row, we examine the impact of a discretionary macroprudential policy (consisting of a 1 percentage point cut in the Spanish LTV) on private debt. In this case, we focus on the interplay with Öscal and monetary policies, as reáected in their policy rules. As regards private debt, the LTV shock does not interact with the Öscal rule, and mildly interacts with the monetary rule. Actually, the di§erence in the e§ectiveness of the policy is about 5 percent, depending whether the monetary policy is loose (the maximum e§ect) or tight (the minimum one). In this case the preferred policy-mix by the macroprudential institution is the same that maximizes the impact on private debt of a technology shock.

Finally, we inspect the e§ectiveness of a monetary surprise of 50 basis point to reduce ináation. We confront this policy with the Öscal and macroprudential rules. Again, the strength of the reaction of the tax rate measured by the Öscal rule only plays a marginal role in shaping the impact on ináation. More important is the task of the macroprudential policy. In our model, a tighter macroprudential policy may contribute to increase the success of the monetary surprise to lower ináation by 13 percent. This policy is also aligned with the most adequate one to take advantage in terms of ináation of a productivity shock.

From the above discussion we can conclude that a tight macroprudential rule seems to help in achieving the targets of discretionary Öscal and monetary policies. After a productivity shock, however, a tight macroprudential rule diminishes the positive impact on GDP and reinforces the negative impact on ináation. The role of the Öscal rule to interact with other policies and change their outcomes is only marginal.

6 Conclusions

In this paper, we develop a simple two-country general equilibrium model in a monetary union with borrowers and lenders in each economy. Public and private borrowers face a downward sloping bond-speciÖc demand function. We calibrate the two countries to proxy Spain and Germany, paradigmatic examples of periphery and core countries. We use the model to study the interactions between macroprudential and Öscal policies.

After a macroprudential policy in Spain there is a change in the Spanish lenders portfolio of assets, from bonds to houses. There is also a deviation of Spanish bond holdings from national to foreign investors. Lower ináation rates and a weaker level of activity move government debt up. The increase in the government-debt-to-GDP ratio is even higher, so the Öscal authority reacts by rising the marginal tax according to a Öscal rule. German exports fall due to weaker Spanish demand, and so does aggregate output. The rise in public-debt-to-GDP compels the German government to lift the tax rate Thus, an asymmetric macroprudential intervention creates a non trivial positive footprint in the whole monetary union.

The e§ects of a symmetric macroprudential intervention in the monetary union, including the Öscal footprint, can be replicated, up to the e§ect on private debt, by means of a positive shock on the interest rates by the European Central Bank. It provides an argument for a coordination between macroprudential and monetary policy. A European coordinated macroprudential and monetary intervention allows for Önancial stabilization but neutralizes the Öscal footprint in the European countries and unwanted e§ects on output and ináation.

There is also a link between discretionary Öscal policy and macroprudential policy, as captured by a rule. A negative public spending policy positively a§ects housing prices, so when macroprudential policy is tight, private debt is more a§ected and borrowersíconsumption falls more. Thus, a tight macroprudential policy increases the Öscal multiplier. Moreover, a tight macroprudential rule also helps to a monetary surprise to achieve a higher impact on ináation. After a permanent productivity shock, however, a tight macroprudential rule diminishes the positive impact of productivity on GDP and reinforces the negative impact on ináation.

References

1. [1] Beetsma R. and H. Jensen (2005), "Monetary and Öscal policy interactions in a micro-founded model of a monetary union", Journal of International Economics, 67(2), 320-352.
2. [2] Blagov, B. (2013), "Financial crises and time-varying risk premia in a small open economy: a Markov-Switching DSGE model for Estonia," Bank of Estonia WP2013-8.
3. [3] Bonam D. and J. Lukkezen (2019), "Fiscal and monetary policy coordination, macroeconomic stability, and sovereign risk premia", Journal of Money Credit and Banking, 51 (2-3), 581-616.
4. [4] BussiËre, M., Cao, J., de Haan, J., Hills, R., Lloyd. S., Meunier, B., Pedrono, J. Reinhardt, D., Sinha, S., Sowerbutts, R. and Konstantin Styrin (2020), "The interaction between macroprudential policy and monetary policy: Overview", Review of International Economics, DOI: 10.1111/roie.12505
5. [5] Calvo, G.A. (1983), "Staggered prices in a utility-maximizing framework," Journal of Monetary Economics, 12 (3), 383-398.
6. [6] Casares, M. and J. V·zquez (2018), "Why are labor markets in Spain and Germany so di§erent?," Economic Modelling, 75 (C), 320-335.
7. [7] Clarida, R., Gali, J. and M. Gertler (1999), "The science of monetary policy: a new Keynesian perspective", Journal of Economic Literature, 3 (4), 1661ñ1707.
8. [8] Clarida, R., J. GalÌ and M. Gertler (2000), "Monetary policy rules and macroeconomic stability: evidence and some theory," Quarterly Journal of Economics, 115, 147-180.
9. [9] Deloitte (2016), "Overview of European Residential Market," Property Index, 5th edition, July.
10. [10] De Walque, G., Smets, F. and R. Wouters (2005), "An estimated two-country DSGE model for the Euro area and the US economy," European Central Bank, mimeo.
11. [11] Demid E. (2018), "Fiscal and monetary policy: coordination or conáict?", International Economic Journal, 32(4), 547-571.
12. [12] Deutsche Bundesbank (2017), "Mark-ups of Örms in selected European countries," Monthly Report, December, 53.
13. [13] Engel, C. and A. Matsumoto (2006), "Portfolio choice in a monetary open-economy DSGE model," National Bureau of Economic Research.
14. [14] Farhi, E. and I. Werning (2016), ìA theory of macroprudential policies in the presence of nominal rigiditiesî, Econometrica, 84, 1645-1704.
15. [15] Farhi, E. and I. Werning (2017), "Fiscal Unions", American Economic Review, 107(12), 3788-834.
16. [16] Ferrero, A. (2009), ìFiscal and Monetary Rules for a Currency Union.î, Journal of International Economics, 77 (1), 1ñ10.
17. [17] European Commission (2018), Quarterly Report on the Euro Area, 17, 4, Ch I.
18. [18] GalÌ, J. and M. Gertler (2007), "Macroeconomic modeling for monetary policy evaluation," Journal of Economic Perspectives, 21 (4), 25-46.
19. [19] Gali, J. and T. Monacelli (2005), "Monetary policy and exchange rate volatility in a small open economy," The Review of Economic Studies, 72 (3), 707-734.
20. [20] GalÌ, J, LÛpez-Salido, D. and J. VallÈs (2004), "Rule-of-Thumb Consumers and the Design of Interest Rate Rules," Journal of Money, Credit and Banking, 36 (4), 739-763.
21. [21] Hansen, G.D. and S. Imrohoro ^· glu, S. (2016), "Fiscal reform and government debt in Japan: Aº neoclassical perspective," Review of Economic Dynamics, 21, 201-224.
22. [22] Iacoviello, M. (2005), "House prices, borrowing constraints, and monetary policy in the business cycle," American Economic Review, 95 (3), 739-764.
23. [23] IMF (2017), "Greece: Request for Stand-By Arrangement", Press Release, Sta§ Report, and Statement by the Executive Director for Greece.
24. [24] Kolasa, M. (2009), "Structural heterogeneity or asymmetric shocks? Poland and the euro area through the lens of a two-country DSGE model," Economic Modelling, 26 (6), 1245-1269.
25. [25] Krishnamurthy, A. and A. Vissing-Jorgensen (2015), "The Impact of Treasury Supply on Financial Sector Lending and Stability," Journal of Financial Economics, 118 (3), 571ñ600.
26. [26] Leeper, E. (1991), "Equilibria under ëactiveíand ëpassiveímonetary and Öscal policies", Journal of Monetary Economics, 27(1), 129ñ147.
27. [27] Malmierka, M. (2022), "Stabilization and the policy mix in a monetary union", The Quarterly Review of Economics and Finance, 83, 92-118.
28. [28] McCallum, B. (2001), "Should Monetary Policy Respond Strongly To Output Gaps?," American Economic Review, 91 (2), 258ñ262.
29. [29] Monacelli, T. (2009), "New Keynesian models, durable goods, and collateral constraints," Journal of Monetary Economics, 56 (2), 242-254.
30. [30] Monacelli, T. (2008), "Optimal monetary policy with collateralized household debt and borrowing constraints," NBER Chapters, in: Asset Prices and Monetary Policy, 103-146.
31. [31] Ratto, M., Roeger, W. and J. inít Veld (2009), "QUEST III: An estimated open-economy DSGE model of the euro area with Öscal and monetary policy," Economic Modelling, 26 (1), 222-233.
32. [32] Reis, R. (2020), "The Fiscal Footprint of Macroprudential Policy," LSE unpublished manuscript.
33. [33] Rubio, M (2014), "Housing-market heterogeneity in a monetary union," Journal of International Money and Finance, 40, 163-184.
34. [34] Sargent, T.J. (2009), "Dynamic macroeconomic theory," Harvard University Press.
35. [35] Sargent, T.J. (1987), "Macroeconomic theory," Academic Press New York.
36. [36] Sauer, S. and J. E. Sturm (2007), "Using Taylor Rules toUnderstand European Central Bank Monetary Policy," German Economic Review, 8 (3), 375ñ398.
37. [37] Schmitt-GrohÈ, S. and M. Uribe (2003), "Closing small open economy models," Journal of International Economics, 61 (1), 163-185.
38. [38] Solow, R. (2010), "Prepared Statement Robert Solow Professor Emeritus, House Committee on Science and Technology Subcommittee on Investigations and Oversight," Building a Science of Economics for the Real World, 20.
39. [39] St‰hler, N. and C. Thomas (2012), "FiModó A DSGE model for Öscal policy simulations," Economic Modelling, 29 (2), 239-261.
40. [40] Yun, T. (2011), "Transmission mechanisms of the public debt," Manuscript, Seoul Na.
41. [41] Westig, D. and C. Bertalot (2017), "Hypostat 2017: A Review of Europeís Mortgage and Housing Markets," European Mortgage Federation

Appendices

Appendix 1

In this Appendix, we show all the equations of the model

National economy households equations

The Patient Households

\[\gamma_ {b _ {A t}} (\bullet) = \chi_ {A} + \vartheta \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{45}\]

\[\gamma_ {b _ {A t}} ^ {g} (\bullet) = \chi_ {A} ^ {g} + \vartheta \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{46}\]

\[\begin{array}{r l} & c _ {A t} ^ {l} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {l} + q _ {t} (h _ {t} ^ {l} - h _ {t - 1} ^ {l}) + b _ {A t} ^ {l} + \frac {P _ {B t}}{P _ {A t}} b _ {B t} ^ {l} + b _ {A t} ^ {g} + \frac {P _ {B t}}{P _ {A t}} b _ {B t} ^ {g} \\ & \leq (1 - x _ {t} ^ {l}) w _ {t} n _ {t} ^ {l} + \frac {R _ {A t - 1}}{\pi_ {A t}} b _ {A t - 1} ^ {l} + \frac {P _ {B t}}{P _ {A t}} \frac {R _ {B t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {l} \\ & + \frac {R _ {A t - 1} ^ {g}}{\pi_ {A t}} b _ {A t - 1} ^ {g} + \frac {P _ {B t}}{P _ {A t}} \frac {R _ {t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {g} + d _ {t}, \end{array}\tag{47}\]

\[t _ {A t} ^ {l} = m _ {A t} (w _ {t} n _ {t} ^ {l} - \overline {{t _ {A}}})\tag{48}\]

\[x _ {t} ^ {l} = \frac {t _ {A t} ^ {l}}{w _ {t} n _ {t} ^ {l}}\tag{49}\]

\[\pi_ {A t} = \frac {P _ {A t}}{P _ {A t - 1}}\tag{50}\]

\[c _ {t} ^ {l} = c _ {A t} ^ {l} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {l}\tag{51}\]

\[\lambda_ {t} ^ {l} = \frac {\omega}{c _ {A t} ^ {l}}\tag{52}\]

\[\frac {c _ {A t} ^ {l}}{c _ {B t} ^ {l}} = \frac {\omega P _ {B t}}{(1 - \omega) P _ {A t}}\tag{53}\]

\[\frac {\gamma_ {h}}{h _ {t} ^ {l}} = \lambda_ {t} ^ {l} q _ {t} - \beta E _ {t} \left[ \lambda_ {t + 1} ^ {l} q _ {t + 1} \right]\tag{54}\]

\[w _ {t} = \frac {c _ {A t} ^ {l}}{\omega} \left(n _ {t} ^ {l}\right) ^ {\eta}\tag{55}\]

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {A t}}{\pi_ {A t + 1}} \right] + \frac {\chi_ {A}}{b _ {A t} ^ {l}} + \frac {\vartheta}{b _ {A t} ^ {l}} \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{56}\]

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {B t}}{\pi_ {A t + 1}} \right] + \frac {P _ {A t}}{P _ {B t}} \frac {\chi_ {B}}{b _ {B t} ^ {l}}\tag{57}\]

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {A t} ^ {g}}{\pi_ {A t + 1}} \right] + \frac {\chi_ {A} ^ {g}}{b _ {A t} ^ {g}} + \frac {\vartheta}{b _ {A t} ^ {g}} \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{58}\]

\[\lambda_ {t} ^ {l} = \beta^ {l} E _ {t} \left[ \lambda_ {t + 1} ^ {l} \frac {R _ {t}}{\pi_ {A t + 1}} \right] + \frac {P _ {A t}}{P _ {B t}} \frac {\chi_ {B} ^ {g}}{b _ {B t} ^ {g}}\tag{59}\]

\[\phi_ {t} \equiv - \vartheta \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{60}\]

The Impatient Households

\[c _ {A t} ^ {r} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {r} + q _ {t} (h _ {t} ^ {r} - h _ {t - 1} ^ {r}) + \frac {R _ {A t - 1}}{\pi_ {A t}} b _ {t - 1} ^ {r}\tag{61}\]

\[\leq \left(1 - x _ {t} ^ {r}\right) w _ {t} n _ {t} ^ {r} + b _ {t} ^ {r}.\]

\[t _ {A t} ^ {r} = m _ {A t} (w _ {t} n _ {t} ^ {r} - \overline {{t _ {A}}})\tag{62}\]

\[x _ {t} ^ {r} = \frac {t _ {A t} ^ {r}}{w _ {t} n _ {t} ^ {r}}\tag{63}\]

\[E _ {t} \left[ \frac {R _ {A t}}{\pi_ {A t + 1}} b _ {t} ^ {r} \right] \leq E _ {t} \left[ k _ {A t} q _ {t + 1} h _ {t} ^ {r} \right],\tag{64}\]

\[c _ {t} ^ {r} = c _ {A t} ^ {r} + \frac {P _ {B t}}{P _ {A t}} c _ {B t} ^ {r}\tag{65}\]

\[\lambda_ {t} ^ {r} = \frac {\omega}{c _ {A t} ^ {r}}\tag{66}\]

\[\frac {c _ {A t} ^ {r}}{c _ {B t} ^ {r}} = \frac {\omega P _ {B t}}{(1 - \omega) P _ {A t}}\tag{67}\]

\[\frac {\gamma_ {h}}{h _ {t} ^ {r}} = q _ {t} \lambda_ {t} ^ {r} - E _ {t} \left[ \beta^ {r} q _ {t + 1} \lambda_ {t + 1} ^ {r} + \xi_ {t} k _ {A t} q _ {t + 1} \pi_ {A t + 1} \right]\tag{68}\]

\[w _ {t} = \frac {c _ {A t} ^ {r}}{\omega} (n _ {t} ^ {r}) ^ {\eta}\tag{69}\]

\[\lambda_ {t} ^ {r} = \beta^ {r} E _ {t} \left[ \lambda_ {t + 1} ^ {r} \frac {R _ {A t}}{\pi_ {A t + 1}} \right] + \xi_ {t} R _ {A t}\tag{70}\]

The foreign countryís households

The Patient Households

\[\gamma_ {b _ {A t}} ^ {*} (\bullet) = \chi_ {A} ^ {*} + \vartheta \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{71}\]

\[\gamma_ {b _ {A t}} ^ {* g} (\bullet) = \chi_ {A} ^ {* g} + \vartheta \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{72}\]

\[\begin{array}{r l} & {\frac {P _ {A t}}{P _ {B t}} c _ {A t} ^ {* l} + c _ {B t} ^ {* l} + q _ {t} ^ {*} \left(h _ {t} ^ {* l} - h _ {t - 1} ^ {* l}\right) + \frac {P _ {A t}}{P _ {B t}} b _ {A t} ^ {* l} + b _ {B t} ^ {* l} + \frac {P _ {A t}}{P _ {B t}} b _ {A t} ^ {* g} + b _ {B t} ^ {* g}} \\ & {\leq \left(1 - x _ {t} ^ {* l}\right) w _ {t} ^ {*} n _ {t} ^ {* l} + \frac {P _ {A t}}{P _ {B t}} \frac {R _ {A t - 1}}{\pi_ {A t}} b _ {A t - 1} ^ {* l} + \frac {R _ {B t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {* l}} \\ & {+ \frac {P _ {A t}}{P _ {B t}} \frac {R _ {A t - 1} ^ {g}}{\pi_ {A t}} b _ {A t - 1} ^ {* g} + \frac {R _ {t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {* g} + d _ {t} ^ {*}} \end{array}\tag{73}\]

\[t _ {B t} ^ {l} = m _ {B t} (w _ {t} ^ {*} n _ {t} ^ {* l} - \overline {{t _ {B}}})\tag{74}\]

\[x _ {t} ^ {* l} = \frac {t _ {B t} ^ {l}}{w _ {t} ^ {*} n _ {t} ^ {* l}}\tag{75}\]

\[c _ {t} ^ {* l} = \frac {P _ {A t}}{P _ {B t}} c _ {A t} ^ {* l} + c _ {B t} ^ {* l}\tag{76}\]

\[\pi_ {B t} = \frac {P _ {B t}}{P _ {B t - 1}}\tag{77}\]

\[\lambda_ {t} ^ {* l} = \frac {1 - \omega}{c _ {B t} ^ {* l}}\tag{78}\]

\[\frac {c _ {A t} ^ {* l}}{c _ {B t} ^ {* l}} = \frac {\omega}{(1 - \omega)} \frac {P _ {B t}}{P _ {A t}}\tag{79}\]

\[\frac {\gamma_ {h} ^ {*}}{h _ {t} ^ {* l}} = \lambda_ {t} ^ {* l} q _ {t} ^ {*} - \beta^ {* l} E _ {t} \left[ \lambda_ {t + 1} ^ {* l} q _ {t + 1} ^ {*} \right]\tag{80}\]

\[w _ {t} ^ {*} = \frac {c _ {B t} ^ {* l}}{1 - \omega} \left(n _ {t} ^ {* l}\right) ^ {\eta^ {*}}\tag{81}\]

\[\lambda_ {t} ^ {* l} = \beta^ {* l} E _ {t} \left[ \lambda_ {t + 1} ^ {* l} \frac {R _ {A t}}{\pi_ {B t + 1}} \right] + \frac {P _ {B t}}{P _ {A t}} \frac {\chi_ {A} ^ {*}}{b _ {A t} ^ {* l}} + \frac {\vartheta}{b _ {A t} ^ {* l}} \frac {P _ {B t}}{P _ {A t}} \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{82}\]

\[\lambda_ {t} ^ {* l} = \beta^ {* l} E _ {t} \left[ \lambda_ {t + 1} ^ {* l} \frac {R _ {B t}}{\pi_ {B t + 1}} \right] + \frac {\chi_ {B} ^ {*}}{b _ {B t} ^ {* l}}\tag{83}\]

\[\lambda_ {t} ^ {* l} = \beta^ {* l} E _ {t} \left[ \lambda_ {t + 1} ^ {* l} \frac {R _ {A t} ^ {g}}{\pi_ {B t + 1}} \right] + \frac {P _ {B t}}{P _ {A t}} \frac {\chi_ {A} ^ {* g}}{b _ {A t} ^ {* g}} + \frac {P _ {B t}}{P _ {A t}} \frac {\vartheta}{b _ {A t} ^ {* g}} \left(\frac {b _ {t} ^ {*}}{b _ {t}} \frac {y _ {t}}{y _ {t} ^ {*}} - 1\right)\tag{84}\]

\[\lambda_ {t} ^ {* l} = \beta^ {* l} E _ {t} \left[ \lambda_ {t + 1} ^ {* l} \frac {R _ {t}}{\pi_ {B t + 1}} \right] + \frac {\chi_ {B} ^ {* g}}{b _ {B t} ^ {* g}}\tag{85}\]

Impatient Households

\[\begin{array}{r l} & {\frac {P _ {A t}}{P _ {B t}} c _ {A t} ^ {* r} + c _ {B t} ^ {* r} + q _ {t} ^ {*} \left(h _ {t} ^ {* r} - h _ {t - 1} ^ {* r}\right) + \frac {R _ {B t - 1}}{\pi_ {B t}} b _ {t - 1} ^ {* r}} \\ & {= (1 - x _ {t} ^ {* r}) w _ {t} ^ {*} n _ {t} ^ {* r} + b _ {t} ^ {* r}} \end{array}\tag{86}\]

\[t _ {B t} ^ {r} = m _ {B t} (w _ {t} ^ {*} n _ {t} ^ {* r} - \overline {{t _ {B}}})\tag{87}\]

\[x _ {t} ^ {* r} = \frac {t _ {B t} ^ {r}}{w _ {t} ^ {*} n _ {t} ^ {* r}}\tag{88}\]

\[E _ {t} \left[ \frac {R _ {B t}}{\pi_ {B t + 1}} b _ {t} ^ {* r} \right] \leq E _ {t} \left[ k _ {B t} q _ {t + 1} ^ {*} h _ {t} ^ {* r} \right]\tag{89}\]

\[c _ {t} ^ {* r} = \frac {P _ {A t}}{P _ {B t}} c _ {A t} ^ {* r} + c _ {B t} ^ {* r}\tag{90}\]

\[\lambda_ {t} ^ {* r} = \frac {1 - \omega}{c _ {B t} ^ {* r}}\tag{91}\]

\[\frac {c _ {A t} ^ {* r}}{c _ {B t} ^ {* r}} = \frac {\omega P _ {B t}}{(1 - \omega) P _ {A t}}\tag{92}\]

\[\frac {\gamma_ {h} ^ {*}}{h _ {t} ^ {* r}} = q _ {t} ^ {*} \lambda_ {t} ^ {* r} - E _ {t} \left[ \beta^ {* r} q _ {t + 1} ^ {*} \lambda_ {t + 1} ^ {* r} + \xi_ {t} ^ {*} k _ {B t} q _ {t + 1} ^ {*} \pi_ {B t + 1} \right]\tag{93}\]

\[w _ {t} ^ {*} = \frac {c _ {B t} ^ {* r}}{(1 - \omega)} (n _ {t} ^ {* r}) ^ {\eta^ {*}}\tag{94}\]

\[\lambda_ {t} ^ {* r} = \beta^ {* r} E _ {t} \left[ \lambda_ {t + 1} ^ {* r} \frac {R _ {B t}}{\pi_ {B t + 1}} \right] + \xi_ {t} ^ {*} R _ {B t}\tag{95}\]

The National Firms

\[w _ {t} = m c _ {t} \frac {y _ {t}}{n _ {t}}\tag{96}\]

\[\left(\frac {1 - \theta \left(\frac {\pi_ {A t}}{\pi_ {A t - 1} ^ {\zeta}}\right) ^ {\varepsilon - 1}}{1 - \theta}\right) ^ {\frac {1}{1 - \varepsilon}} = \frac {\varepsilon}{\varepsilon - 1} \frac {V _ {t}}{F _ {t}}\tag{97}\]

\[V _ {t} = \lambda_ {t} ^ {l} m c _ {t} y _ {t} + E _ {t} (\beta^ {l} \theta) \left(\frac {\pi_ {A t} ^ {\zeta}}{\pi_ {A t + 1}}\right) ^ {- \varepsilon} V _ {t + 1}\tag{98}\]

\[F _ {t} = \lambda_ {t} ^ {l} y _ {t} + E _ {t} (\beta^ {l} \theta) \left(\frac {\pi_ {A t} ^ {\zeta}}{\pi_ {A t + 1}}\right) ^ {1 - \varepsilon} F _ {t + 1}\tag{99}\]

The Foreign Firms

\[w _ {t} ^ {*} = m c _ {t} ^ {*} \frac {y _ {t} ^ {*}}{n _ {t} ^ {*}}\tag{100}\]

\[\binom{1 - \theta \left(\frac {\pi_ {B t}}{\pi_ {B t - 1} ^ {\zeta}}\right) ^ {\varepsilon^ {*} - 1}}{\hline 1 - \theta} ^ {\frac {1}{1 - \varepsilon^ {*}}} = \frac {\varepsilon^ {*}}{\varepsilon^ {*} - 1} \frac {V _ {t} ^ {*}}{F _ {t} ^ {*}}\tag{101}\]

\[V _ {t} ^ {*} = \lambda_ {t} ^ {* l} m c _ {t} ^ {*} y _ {t} ^ {*} + E _ {t} (\beta^ {* l} \theta^ {*}) \left(\frac {\pi_ {B t} ^ {\zeta}}{\pi_ {B t + 1}}\right) ^ {- \varepsilon} V _ {t + 1} ^ {*}\tag{102}\]

\[F _ {t} ^ {*} = \lambda_ {t} ^ {* l} y _ {t} ^ {*} + E _ {t} (\beta^ {* l} \theta^ {*}) \left(\frac {\pi_ {B t} ^ {\zeta}}{\pi_ {B t + 1}}\right) ^ {1 - \varepsilon} F _ {t + 1} ^ {*}\tag{103}\]

Aggregation

\[b _ {t} ^ {r} = \frac {(1 - \tau)}{\tau} b _ {A t} ^ {l} + \frac {(1 - \omega)}{\omega} \frac {(1 - \tau^ {*})}{\tau} b _ {A t} ^ {* l},\tag{104}\]

\[b _ {t} ^ {* r} = \frac {\omega}{(1 - \omega)} \frac {(1 - \tau)}{\tau^ {*}} b _ {B t} ^ {l} + \frac {(1 - \tau^ {*})}{\tau^ {*}} b _ {B t} ^ {* l},\tag{105}\]

\[b _ {t} ^ {g} = (1 - \tau) b _ {A t} ^ {g} + (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} b _ {A t} ^ {* g}.\tag{106}\]

\[b _ {t} ^ {* g} = (1 - \tau) \frac {\omega}{(1 - \omega)} b _ {B t} ^ {g} + (1 - \tau^ {*}) b _ {B t} ^ {* g},\tag{107}\]

\[b _ {t} = b _ {t} ^ {g} + \tau b _ {t} ^ {r}\tag{108}\]

\[b _ {t} ^ {*} = b _ {t} ^ {* g} + \tau^ {*} b _ {t} ^ {* r}\tag{109}\]

\[\tau h _ {t} ^ {r} + (1 - \tau) h _ {t} ^ {l} = h\tag{110}\]

\[\tau^ {*} h _ {t} ^ {* r} + (1 - \tau^ {*}) h _ {t} ^ {* l} = h ^ {*}\tag{111}\]

\[c _ {A t} = \tau c _ {A t} ^ {r} + (1 - \tau) c _ {A t} ^ {l}\tag{112}\]

\[c _ {B t} ^ {*} = \tau^ {*} c _ {B t} ^ {* r} + (1 - \tau^ {*}) c _ {B t} ^ {* l}\tag{113}\]

\[c _ {B t} = \tau c _ {B t} ^ {r} + (1 - \tau) c _ {B t} ^ {l}\tag{114}\]

\[c _ {A t} ^ {*} = \tau^ {*} c _ {A t} ^ {* r} + (1 - \tau^ {*}) c _ {A t} ^ {* l}\tag{115}\]

\[c _ {t} = \tau c _ {t} ^ {r} + (1 - \tau) c _ {t} ^ {l} = c _ {A t} + \frac {P _ {B t}}{P _ {A t}} c _ {B t}\tag{116}\]

\[c _ {t} ^ {*} = \tau c _ {t} ^ {* r} + (1 - \tau) c _ {t} ^ {* l} = c _ {B t} ^ {*} + \frac {P _ {A t}}{P _ {B t}} c _ {A t} ^ {*}\tag{117}\]

\[n _ {t} = \tau n _ {t} ^ {r} + (1 - \tau) n _ {t} ^ {l}\tag{118}\]

\[n _ {t} ^ {*} = \tau^ {*} n _ {t} ^ {* r} + (1 - \tau^ {*}) n _ {t} ^ {* l}\tag{119}\]

\[t _ {t} = (1 - \tau) t _ {A t} ^ {l} + \tau t _ {A t} ^ {r}\tag{120}\]

\[t _ {t} ^ {*} = (1 - \tau^ {*}) t _ {B t} ^ {l} + \tau^ {*} t _ {B t} ^ {r}\tag{121}\]

\[y _ {t} = z _ {t} n _ {t}\tag{122}\]

\[y _ {t} ^ {*} = z _ {t} ^ {*} n _ {t} ^ {*}\tag{123}\]

Fiscal, monetary and macroprudential policies

\[b _ {t} ^ {g} = \frac {R _ {A t - 1} ^ {g}}{\pi_ {A t}} b _ {t - 1} ^ {g} + (g _ {t} - t _ {t})\tag{124}\]

\[b _ {t} ^ {* g} = \frac {R _ {t - 1}}{\pi_ {B t}} b _ {t - 1} ^ {* g} + (g _ {t} ^ {*} - t _ {t} ^ {*})\tag{125}\]

\[R _ {t} = R _ {t - 1} ^ {\rho} \left[ \left(\pi_ {A t} ^ {\omega} \pi_ {B t} ^ {1 - \omega}\right) ^ {\Phi} \overline {{R}} \right] ^ {1 - \rho}\tag{126}\]

\[m _ {A t} = m _ {A t - 1} + \psi f _ {t} \left(\frac {b _ {t} ^ {g}}{y _ {t}} - \overline {{\left(\frac {b ^ {g}}{y}\right)}}\right)\tag{127}\]

\[m _ {B t} = m _ {B t - 1} + \psi^ {*} f _ {t} ^ {*} \left(\frac {b _ {t} ^ {* g}}{y _ {t} ^ {*}} - \overline {{\left(\frac {b ^ {*} g}{y ^ {*}}\right)}}\right)\tag{128}\]

\[k _ {A t} = k _ {S S A} \left(\frac {q _ {t}}{\overline {{q}}}\right) ^ {- \phi_ {A q} ^ {k}}\tag{129}\]

\[k _ {B t} = k _ {S S B} \left(\frac {q _ {t} ^ {*}}{\overline {{q}} ^ {*}}\right) ^ {- \phi_ {B q} ^ {k}}\tag{130}\]

GDP and balance of payments restriction

\[\begin{array}{r l r} & & {(1 - \tau) \frac {P _ {B t}}{P _ {A t}} \left(b _ {B t} ^ {l} + b _ {B t} ^ {g}\right) + (1 - \tau) b _ {A t} ^ {g} - (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} b _ {A t} ^ {* l} =} \\ & & {\left(\frac {(1 - \omega)}{\omega} c _ {A t} ^ {*} - \frac {P _ {B t}}{P _ {A t}} c _ {B t}\right) + (g _ {t} - t _ {t}) +} \\ & & {+ (1 - \tau) \frac {P _ {B t}}{P _ {A t}} \left(\frac {R _ {B t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {l} + \frac {R _ {t - 1}}{\pi_ {B t}} b _ {B t - 1} ^ {g}\right) + (1 - \tau) \frac {R _ {A t - 1} ^ {g}}{\pi_ {A t}} b _ {A t - 1} ^ {g}} \\ & & {- (1 - \tau^ {*}) \frac {(1 - \omega)}{\omega} \frac {R _ {A t - 1}}{\pi_ {A t}} b _ {A t - 1} ^ {* l}.} \end{array}\tag{131}\]

\[y _ {t} = c _ {A t} + \frac {(1 - \omega)}{\omega} c _ {A t} ^ {*} + g _ {t}\tag{132}\]

\[(1 - \tau) d _ {t} = \left(\frac {1}{m c _ {t}} - 1\right) w _ {t} n _ {t}\tag{133}\]

\[y _ {t} ^ {*} = c _ {B t} ^ {*} + \frac {\omega}{(1 - \omega)} c _ {B t} + g _ {t} ^ {*}\tag{134}\]

\[\left(1 - \tau^ {*}\right) d _ {t} ^ {*} = \left(\frac {1}{m c _ {t} ^ {*}} - 1\right) w _ {t} ^ {*} n _ {t} ^ {*}\tag{135}\]