The Possible Trinity: Optimal Interest Rate, Exchange Rate, and Taxes on Capital Flows in a DSGE Model for a Small Open Economy

A traditional way of thinking about the exchange rate (XR) regime and capital account openness has been framed in terms of the 'impossible trinity' or 'trilemma', in which policymakers can only have 2 of 3 possible outcomes: open capital markets, monetary independence and pegged XRs. This paper is an extension of Escude (A DSGE Model for a SOE with Systematic Interest and Foreign Exchange Policies in Which Policymakers Exploit the Risk Premium for Stabilization Purposes, 2013), which focused on interest rate and XR policies, since it introduces the third vertex of the 'trinity' in the form of taxes on private foreign debt. These affect the risk-adjusted uncovered interest parity equation and hence influence the SOE's international financial flows. A useful way to illustrate the range of policy alternatives is to associate them with the faces of a triangle. Each of 3 possible government intervention policies taken individually (in the domestic currency bond market, in the FX market, and in the foreign currency bonds market) corresponds to one of the vertices of the triangle, each of the 3 possible pairs of intervention policies corresponds to one of its 3 edges, and the 3 simultaneous intervention policies taken jointly correspond to its interior. This paper shows that this interior, or 'possible trinity' is quite generally not only possible but optimal, since the CB obtains a lower loss when it implements a policy with all three interventions. JEL E58 F38 O24


Introduction
Foreign trade and capital ows constitute the two fundamental links between the open economy and world markets. And both are affected by a host of government policies, including interest rate and exchange rate policies as well as different possible interventions in the free ow of nancial assets. A traditional way of thinking about the exchange rate regime and capital account openness has been framed in terms of the`impossible trinity' or`trilemma'. According to this view, policymakers can only have two of three possible outcomes: open capital markets, monetary independence and pegged exchange rates (see Bordo 2003). Historically, during the gold standard period there predominated open capital markets and xed exchange rates but monetary independence was lacking. Later, during the Bretton Woods period pegged exchange rates and monetary independence became possible since important capital controls prevailed. During the post-Bretton Woods period free capital mobility was again introduced, leading countries to a dif cult choice between pegged exchange rates (with the consequent loss of monetary independence) and oating exchange rates (with monetary independence). Emerging market economies that chose to have pegged exchange rates tended to have periodic debt, currency and banking crises. Hence, for some time a`bipolar view' prevailed that with free international capital markets most countries had to choose between very hard pegs and free oats. However, in many instances even very hard pegs, such as Argentina's Convertibility (that lasted 10 years), led to severe triple crises.
The 2008-9 nancial meltdown has awakened new interest in these topics. There is a renewed interest in active FX reserves management and greater concern for nancial stability due to the grave macroeconomic risks that nancial meltdowns generate. Possibly because the main developed economies were hit hard by the crisis, there has tended to be a more receptive approach to topics that were until recently frowned upon, including FX market intervention, soft capital controls in normal times and more invasive practices during the management of crises. Even the IMF is tending to accept that under some circumstances capital controls may be useful and even necessary. Ostry et al (2010), for example, point out that capital controls can help to address nancial stability concerns, at least when there are insuf cient prudential tools available. Obstfeld et al (2008) try to explain why there has been such a dramatic rise in global international reserve holdings (as a fraction of world GDP) during the post-Bretton Woods era. They argue that reserve accumulation is an important tool for managing domestic nancial instability as well as exchange rates in a world in which nancial globalization has ballooned and the domestic banking sector needs protection against different possible sources of drains ( ights from currency or deposits) by means of the central bank's role as a lender of last resort. It is these concerns, much more than the traditional trade-related needs that led to such important accumulations of international reserves.
Fratzscer (2012) investigates the motives for the use of capital controls. He uses a broad set of macroeconomic and nancial variables for 79 countries during the 1984-2009 period to assess which of four possible motives for the use of capital controls are most important (objectives related to FX policy, capital ow management, ensuring nancial stability or general macroeconomic policy). He nds that FX policy management has been a central motive for the use of capital controls. In particular,`countries with a high level of capital controls and countries actively raising existing controls are those that tend to have undervalued exchange rates and a high degree of exchange rate volatility.' He also nds that choices concerning capital ow restrictions, especially over the past decade, have been largely motivated by concerns about an overheating of the domestic economy. Recent empirical research has found that most countries, instead of choosing two of the three policy options of the`trinity', actually choose a middle ground where the three options are used without extremes. Aizenman et al (2010), for example, nd that many countries choose to have a managed exchange rate with limited nancial autonomy and controlled nancial integration (see also Aizenman 2012).
At a theoretical level, Farhi and Werning (2012) study capital control policy in a standard open economy model with xed exchange rates, building on Clarida, Gali and Gertler (2002) and Galí andMonacelli (2005, 2008). They use a nonmonetary and non-stochastic model in which there is no endogenous risk premium to study the optimal use of capital controls in response to different shocks (productivity, export demand, terms of trade, foreign interest rates and exogenous risk premia) under different pricing assumptions ( exible prices, rigid prices, oneperiod in advance price xing and Calvo price setting). They conclude that capital controls are more effective the more closed is the economy, and that they are particularly powerful in responding to uctuations in the exogenous risk premium demanded by foreign investors.
At least since the heyday of the Mundell-Fleming model, modeling the intermediate ground between a rmly pegged (or xed) exchange rate and a freely oating exchange rate has been wanting. Until very recently the dif culties involved in setting up a workable framework had not been surpassed, even with such modern developments in macroeconomic modeling such as the rational expectations revolution and DSGE modeling. In practical research, however, the analysis of the middle ground policies have not faced substantial dif culties. For example, in IMF Article 4 reports on emerging market economies, it has been traditional to study the developments in the balance sheets of the main institutional sectors (private and public nancial institutions, Central Bank (CB) and Treasury) in order to obtain insight into actual exchange and interest rate policies and their consequences. In analytical macro modeling, however, there has been resistance to explicitly modeling the nancial stocks and ows of the main sectors included in the model that would make it possible to represent such policy middle ground as managed exchange rates.
My research in the past few years has been mainly focused in this direction (Escudé 2006(Escudé , 2007(Escudé , 2009(Escudé , 2013 and has led to a workable framework in small open economy DSGE modeling where policymakers may use two policy rules in order to determine operational targets for the nominal interest rate as well as the rate of nominal currency depreciation (or, alternatively, the CB international reserves - Escudé 2006). In Escudé (2013) I show the functioning of the framework in a relatively small DSGE model that, except for a few additions, is a standard monetary New Keynesian model. The additions are basically 1) an ad hoc risk premium function that is positively dependent on household foreign debt 1 , 2) an ad hoc long-run target for the CB international reserves ratio (to GDP), 3) CB issued domestic currency bonds that are used for sterilization, 4) a careful formulation of the CB budget constraint along with the assumption that there is an institutional arrangement whereby the CB transfers ( nances) any nancial quasiscal surplus (de cit) to (with) the Treasury, thereby maintaining a constant net worth and a balance sheet structure that only changes during the transition, 5) a second policy rule where there is an operational target for the rate of nominal currency depreciation that may respond to the same variables (or gaps) as the policy rule for the nominal interest rate and, additionally, the gap between the CB international reserves ratio and its long run target, 6) the assignment of explicit instruments for the interventions in the domestic currency bond market (sales or purchases to achieve the operational target for the nominal interest rate) and the FX market (sales and purchases of international reserves to achieve the operational target for the rate of nominal currency depreciation).
In this framework, the CB always satis es private sector money demand and has a long run target for the in ation rate (and hence for the rate of nominal depreciation). The role of the CB balance sheet equation is simply to determine the stock of domestic currency bonds that the CB must have in its liabilities at the end of the quarter as a result of its interventions in both markets. The concept of`sterilized' FX market intervention is avoided because it implicitly subordinates exchange rate policies to interest rate policies. In principle, both policies are (equally) important and the sterilization of any unwanted monetary effects of the combined interventions in the two markets is re ected in the quarterly changes in the stock of CB bonds. The basic result in Escudé (2013) is that, leaving aside implementation costs (which remain unmodeled), it is optimal for policymakers to use the two policy rules, and consequently two operational targets and instruments. This is very intuitive since any one of the`corner' policies is obtained by introducing an additional constraint: either abstain from intervening in the FX market, which implies keeping the CB reserves constant, or abstain from intervening in the domestic currency bond market, which implies keeping the CB domestic currency bonds outstanding constant.
The present paper is a natural extension of Escudé (2013) since it introduces the third vertex of the`trinity' in the form of taxes on private foreign debt. These affect the risk-adjusted uncovered interest parity equation and hence in uence the SOE's international nancial ows. A useful way to illustrate the range of policy alternatives is to associate them with the faces of an isosceles triangle (as in Figure  1 below). Each of three possible government intervention policies taken individually (in the domestic currency bond market, in the foreign currency market, and in the foreign currency bonds market) corresponds to one of the vertices of the triangle, each of the three possible pairs of intervention policies correspond to one of the three edges of the triangle, and the three simultaneous intervention policies taken jointly correspond to the triangle's interior. This paper shows that this interior, or`possible trinity' is quite generally not only possible but optimal, since the CB obtains a lower loss when it implements a policy with three interventions. As in the parent paper, any of the boundary regimes are obtained by introducing additional constraints to the policymakers' problem when a linear-quadratic optimal control framework is used. In the parent paper there were 2 corner policies and an interior policy that combined them, and they are represented by the base of the triangle in Figure 1. In the present paper there are 6 border policies and an interior policy that combines the three possible individual interventions. To implement any of the 3 edge regimes, the instrument that corresponds to the opposite vertex must be kept constant. To implement any of the 3 vertex regimes, the instruments that correspond to opposite 2 vertices must be kept constant. The 3 instruments are 1) the stock of domestic currency bonds in the (liability side of the) CB's balance sheet, 2) the stock of foreign currency reserves in the (asset side of the) CB's balance sheet, and 3) the size of the tax or tax/subsidy scheme on household foreign currency liabilities. In this paper there are two alternative implementations for the third form of intervention: either a tax on the stock of household foreign debt or a tax (subsidy) on the increase (reduction) of household foreign debt. Obviously, there is a certain asymmetry since the rst two instruments correspond to actual nancial instruments that have a market in which the CB operates in order to obtain a desired operational target for the interest rate or the rate of nominal depreciation, whereas in the third case the instrument takes the form of a tax that must be collected (or subsidy that must be bestowed). Because most of the model is exactly as in Escudé (2013), only the extensions are detailed in the text, leaving the exposition of most of the model as well as the full set of model equations for the Appendix. The rest of the paper has the following structure. Section 2 contains the household decision problem and how it is affected by the two new forms of intervention considered. Section 3 shows the calibrations used for the model parameters and details the calibrations that are directly related to the risk premium function that foreign investors are assumed to use to determine the interest rate they demand. Section 4 speci es the alternative interest rate, exchange rate, and capital control policies that are available. Section 5 shows how the model works and illustrates the effects of the different policy regimes on the variability of the main target variables in 3 alternative frameworks available in Dynare for model solution: a) simple policy rules, b) optimal simple policy rules, and c) optimal policy under commitment and full information. Finally, Section 6 has the conclusions, Appendix A shows the parts of the model which are left out in the main body of the paper, and Appendix B lists all the model equations. Additionally, two Dynare model les are available, one for each of the two possible implementations of the tax on foreign debt. An in nitely lived representative household consumes a CES bundle of domestic and imported goods (C t ) and holds nancial wealth in the form of cash (M t ) and domestic currency denominated one period nominal bonds issued by the CB (B t ) that pay a nominal interest rate i t and are considered riskless. The household also issues one period foreign currency bonds (D t ) abroad that pay a nominal foreign currency interest rate i D t . It is assumed that foreign investors are only willing to hold the SOE's foreign currency bonds if they receive a risk premium τ D γ D t ; γ R t over the international riskless rate i t which, as a function, is exogenously given (since the Rest of the World -RW-is not modeled). This function varies directly with the SOE's aggregate foreign debt to GDP ratio γ D t and inversely with the CB's international reserves ratio γ R t (both de ned below). There is also an exogenous stochastic and time-varying component φ t of the total wedge (1 + i D t =1 + i t ) between the (foreign currency) gross interest rates that apply to the SOE and to the RW. φ t can represent general liquidity conditions in the international capital market and/or an exogenous component of the risk-premium. The foreign currency gross interest rate households face is hence: where τ D (:) is increasing and convex (τ D > 1, τ 0 D;γ D > 0 and τ 00 D;γ D > 0) in γ D t , and decreasing (τ 0 D;γ R < 0) in γ R t . The real exchange rate (RER), real foreign debt and international CB reserves (in terms of foreign prices), and corresponding foreign debt and CB reserves ratios to GDP are: where R t is the CB international reserves, S t is the nominal exchange rate, P t is the domestic goods price index, P t is the price index of the goods the SOE imports, and Y t is GDP. A second exogenous function τ M (:) represents gross transactions costs, and introduces the convenience of using cash. The household holds cash M t to economize on transaction costs because in order to purchase the consumption bundle C t it must spend τ M (:) P C t C t where P C t is the price index of the consumption bundle. The function τ M γ M t is assumed to be a decreasing and convex function (τ M > 1; τ 0 M < 0; τ 00 M > 0) of the cash/consumption ratio γ M t : where p C t is the relative price of consumption goods and m t is real cash: The representative household maximizes an inter-temporal utility function which is additively separable in (constant relative risk aversion subutility functions of) goods C t and labor N t : where β is the intertemporal discount factor, σ C , and σ N are the constant relative risk aversion coef cients for goods and labor, respectively, and ξ is a parameter. The household budget constraint in period t is: where i t is the nominal interest rate that CB bonds B t pay each quarter, W t is the nominal wage rate, Π t is nominal pro ts, Tax t is nominal lump sum taxes net of lump sum transfers and Tax DCol t is the government collection from a tax (or subsidy) related to the household's foreign debt. The latter is the main innovation in this paper with respect to Escudé (2013): here the government implements either a tax or a tax/subsidy scheme to in uence capital ows from/to the RW. Let tax D t be a tax rate related to household foreign debt (i.e., foreign currency liabilities of domestic residents that are assets of residents in the RW). Two different foreign debt related concepts are considered: the rst is simply a tax on the level of household foreign debt (and in this case I use tax D t for notation); the second is more complicated, since it is a tax on increases in the level of foreign debt and, symmetrically, a subsidy on foreign debt cancellations (and in this case I use the notation taxsub D t to distinguish the fact that in this case it is also a subsidy rate). In the terminology used below, increases (reductions) in the level of foreign debt are referred to as capital in ows (out ows). This should pose no ambiguity as, for simplicity, households in this paper have no access to foreign assets. Consequently, there are two different possible forms for Tax DCol t in the household (nominal and real, respectively) budget constraint of period t: Form 1 (level): Form 2 (change in level): where imported goods in ation π t is de ned as π t P t P t 1 : Introducing (1) in (6) and dividing by P t , the real budget constraint is: where tax DCol t can adopt one of the two possible forms ((7) or (8)), and real CB bonds, the real wage, real lump sum taxes, and domestic in ation, are de ned as: The tax DCol t term in (9) is the only change in the household real budget constraint with respect to the parent model in Escudé (2013). To simplify, it is assumed that there is no imported goods in ation in the non-stochastic steady state (NSS) (π = 1). In the case of taxsub D t , if a shock makes the household foreign debt d t temporarily increase and at some point it begins to decrease (until it again reaches its long run or non-stochastic steady state (NSS) value), the government rst collects the (distortionary) tax during some time and at some point begins to return it as a subsidy. This makes the de cit closing lump-sum tax decline during the initial phase, and increase during the second phase to compensate for the subsidy. It is assumed in this paper that in the NSS tax D or taxsub D are positive and less than unity. 2 The household chooses the sequence C t+ j ; m t+ j ; b t+ j ; d t+ j ; N t+ j j=0;:::;∞ that maximizes (5) subject to its sequence of budget constraints (9) (and initial values for the predetermined variables). In the taxsub D t case, the Lagrangian is hence: ; where (9) has been rearranged by gathering the terms in d t and d t 1 , respectively (after inserting (8)). β j λ t+ j are the Lagrange multipliers, and can be interpreted as the marginal utility of real income. 3 In the simple tax D t case the only change is that (apart from substituting tax D t for taxsub D t ) the tax term within the square bracket disappears.
For both forms of tax on foreign debt, the rst order conditions for an optimum for variables C, m, b, and N, are exactly the same as in the parent paper: Only the rst order condition for d t is affected by the introduction of the control on capital ows and it differs for the two cases: Form 1 (level): Form 2 (change in level): In (10) and the last two expressions, the auxiliary functions ϕ M and ϕ D that have been introduced for convenience are de ned as: Escudé (2013), combining (11) and (12) gives the demand function for cash: where L (:) is de ned as: and is strictly decreasing, Under the assumption that the CB always satis es cash demand, (17) is henceforth the`cash market clearing condition'.
Using (10) to eliminate λ t from (12) and (13) yields the Euler equation and the household's labor supply, respectively: where in the rst of these π C t P C t =P C t 1 is the gross rate of in ation of the basket of consumption goods and the identity p C t =p C t 1 = π C t =π t is used. Finally, the de nition of the RER in (2) gives the identity e t =e t 1 = δ t π t =π t , where δ t S t =S t 1 is the rate of nominal depreciation of the domestic currency. Hence, (15) may be written as: Eliminating β using (12) yields: : www.economics-ejournal.org Using the fact that the expected value of the product of two random variables is the product of the expected values plus the covariance of the two variables, gives Therefore, to a rst order approximation the covariance term can be ignored and the risk-adjusted UIP equation is simply: where in the second equality ϕ D (:) 1 + ϕ D (:) is used. Notice that an increase in taxsub D t has the effect of increasing the domestic interest rate (ceteris paribus), while an expected increase in the next period has the opposite effect. Hence, if taxsub D t increases initially and is subsequently expected to fall, both have the effect of increasing the domestic interest rate (ceteris paribus). In the case of the simple tax on the level of debt, to a rst order approximation the UIP equation is:

The public sector
The public sector includes the Government and the CB. The CB issues currency (M t ) and domestic currency bonds (B t ), and holds international reserves (R t ) in the form of foreign currency denominated riskless bonds issued by the RW. The CB supplies whatever amount of cash is demanded by households, and can in uence these supplies by changing R t or B t , i.e. intervening in the foreign exchange market or in the domestic currency bond market. It is assumed that CB bonds are only held by domestic residents and that the CB transfers its quasi-scal surplus to (or has its quasi-scal de cit nanced by) the Government each period, maintaining its net worth at zero each period. 4 Hence, the CB balance, for all t, is: The Government spends on goods, receives the quasi-scal surplus (or nances the quasi-scal de cit) of the CB, and collects taxes. It is assumed that scal policy consists of an exogenous autoregressive path for real government expenditures as a (gross) fraction (G t ) of private consumption τ M (:) p C t C t , collecting the tax on private capital ows, and collecting whatever lump-sum taxes are needed to balance the budget each period. The Public Sector real ow budget constraint is hence: where the real quasi-scal surplus includes interests on CB assets and capital gains or losses on CB international reserves: and the real domestic currency value of the tax collection related to capital ows is either (7) or (8).
Since the description of the rest of the model is exactly the same as in Escudé (2013) it is relegated to Appendix A.

Functional forms for auxiliary functions
The functional forms used for the endogenous risk premium and transaction costs functions are the same as in Escudé (2013): For convenience, de ne the respective net functions as: The partial elasticities of τ D and τ M (used below in calibrations) are, respectively: Finally, the liquidity preference function (18) that results from (28) is:

Calibration of parameters and the non-stochastic steady state
In this section the calibrated parameters that are used in the exercises below are shown and the calibration procedure used is only detailed inasmuch as it differs from that of the parent paper. Since the only expansion in this paper is that there is either a tax or a tax/subsidy scheme related to foreign debt (even in the NSS), the rest of the calibrations are the same as in Escudé (2013), which the interested reader can consult. It is convenient to stress that, although Argentine data has been used for some of the calibrations, the main objective has been to have a calibrated SOE economy similar in many respects to some of those most cited in the literature (e.g., Galí and Monacelli 2005 and De Paoli 2006) but endowed with the innovations that allow for the systematic and simultaneous use of interest and nominal depreciation policy rules.
The following are immediately obtained from the NSS versions of various equations: 5 er=Y = γ R ; π = π C = π T ; δ = π T =π ; π = π X = p = 1; 1 + i = π T =β : The standard errors and persistence parameters used for the six shock variables are given in Table 2. They were calibrated taking into account the available time series for Argentina and the RW during the period 1994.1-2009.2: public consumption to GDP in the case of σ G , imported and exported goods in ation as they conform Argentina's XTT, in the cases of σ π and σ π , Libor 3 months in the case of σ i , and balance of payments information on private sector foreign debts and interest payments as well as the author's calculation of the spread over Libor 3 months, in the case of σ φ . The only cases in which the standard deviations were taken exactly according to the data are the cases of σ i ; σ π , and σ π . The rest were calibrated taking both the data (except for σ ε ) and the resulting theoretical standard deviation and variance decomposition for GDP with a baseline calibration of (38) and (39): h 1 = 0:8, h 2 = 0:8, k 4 = 0:8, and the rest of the coef cients equal to zero. This implied diminishing the observed standard deviation of G (from 0.054 in a simple AR(1) estimation from which the value for the persistence parameter ρ G was taken), which seemed to weigh too heavily in the volatility of Y , and increasing the standard deviation of φ (from 0.0034), which seemed not to weigh enough. Table 2 Calibration of shock variables Standard deviations Persistence parameters σ ε 0:01 Aside from the introduction of the new simple policy rule ((40) below) and the tax equation (either (7) or (8)), the only equation that changes with respect to the previous paper is the  or (23)) which, using (29) and (33) and manipulating, gives at the NSS: Tax on level of debt: Tax on change of debt: The risk function parameters α 1 , α 2 and α 3 in (27) need to be calibrated. First, note that (31) gives directly which, given calibrated values for the elasticities and the great ratios, yield: Equations (36) and (37) also imply Therefore, using either (34) or (35) the value of α 1 takes two different forms, according to the assumption on the capital control tax: Tax on level of debt: Tax on change of debt: In order to be able to reasonably compare the two alternative forms of capital controls, their NSS values can be calibrated so that the non-tax variables have the same NSS values. In particular, this requires that the risk premium have the same value in the NSS. Looking at (34) and (35), a little algebra shows that for this to be the case the following relation between tax D and taxsub D must hold: 4 Interest rate, exchange rate, and capital control policies As in Escudé (2013), in this paper the CB uses to stabilize the SOE's macroeconomy either I) simple policy rules or II) optimal control under commitment and full information. The simple rules may be Ia) simple and with exogenous coefcients, or Ib) simple and with optimal coef cients. In case Ia), the simple interest rate rule is a feedback rule, and the simple rules for nominal depreciation and the tax/subsidy on capital ows may or may not involve feedback. In case Ib), the CB is assumed to obtain the values of the coef cients in the policy rules by minimizing a weighted average of the squared deviations of certain target (endogenous) variables from their NSS values. When the CB uses II) (i.e., optimal policies under commitment and full information), the simple policy rules disappear and the CB obtains the trajectories for the intermediate targets (nominal interest rate, nominal rate of depreciation) and the tax rate (or tax/subsidy rate) by minimizing an expected discounted intertemporal quadratic loss function of the target variables.
In Escudé (2013) it was shown that when the target variables are the in ation rate, GDP, and the RER, it is`always' better to use two policy rules instead of one of the two`corner' regimes of Floating Exchange Rate -FER-or Pegged Exchange Rate -PER-(which can equivalently be called Floating Interest Rate -FIRregime). In the FER regime the CB abstains from intervening in the FX market and has an intermediate target for the nominal interest rate, while in the PER regime the CB abstains from intervening in the domestic currency bond market and has an intermediate target for the rate of nominal depreciation. In the Managed Exchange Rate -MER-regime, on the other hand, there are two simple rules: one for the nominal rate of interest and an another for the rate of nominal currency depreciation. It turned out that it was`always' better two use the MER regime, in the sense that the CB obtains lower losses under this regime for any set of CB preferences for in ation, GDP, or RER stabilization. The reason for this gain in using two rules is that the CB can thus better exploit private capital ows for its stabilization purposes, given the fact that these ows are (mainly) jointly determined by the risk-adjusted Uncovered Interest Parity condition (UIP) and its policy rules. Determining both ends of the UIP equation by means of the operational targets in the two policy rules has a crucial effect on the foreign-debt to GDP ratio, which is assumed to determine the risk assessment of foreign investors and hence the wedge between the domestic interest rate and the expected rate of depreciation of the currency. This paper starts from that point and explores the effects of adding an additional policy rule: one that involves the particular form of`capital control' device (tax D t or taxsub D t ) introduced in the previous sections. As shown above, a tax on foreign debt or a tax/subsidy scheme on private capital ows becomes an integral part of the risk-adjusted UIP equation. Hence, if the CB has a policy rule for determining the level of these taxes or taxes/subsidies, it has an additional instrument that can affect private capital ows. With the same methods used in the previous paper, this paper shows that it is generally optimal to use 3 policy rules instead of either 2 or 1, and for the same reasons. Hence, the Trinity of interest rate, exchange rate, and capital control policies in the SOE is not only Possible, but is also Optimal. 7

Figure 2
The 7 interest rate, depreciation rate, and capital control regimes vertices: 1 rule and instrument edges: 2 rules and instruments interior: 3 rules and instruments In the MER regime, the CB, through its regular and systematic interventions in the domestic currency bond (or`money') and foreign exchange markets, aims for the achievement of two operational targets: one for the interbank interest rate i t ; and another for the rate of nominal depreciation δ t . When there are simple policy rules, the CB can uses operational targets for i t and δ t that respond to deviations of the consumption in ation rate (π C t ), GDP (Y t ) and/or the RER (e t ) from their respective NSS levels. The rate of nominal depreciation can additionally respond to deviations of the CB's international reserves (IRs) to GDP ratio from a long run target (γ R ). In this paper, there is an additional policy rule that determines the tax or tax/subsidy related to foreign debt, which in principle can respond to the same three basic target variables as the preceding rules. But it has been deemed preferable to be a little more speci c by making it also respond to the deviations in the exogenous risk/liquidity shock φ t , given that this shock variable directly affects the UIP equation and that such shocks are empirically important for emerging market economies. There may also be history dependence (or inertia) in any of the feedback rules through the presence of the lagged operational target variable. Hence, the simple rules in the case of tax D t are the following: 8 Any one (or any two) of these simple rules can be replaced by a corresponding equation that simply maintains a corresponding endogenous variable at its NSS value. As in Escudé (2013), in the case of the rst two of these policy rules the instrument (b t and r t ) that the CB uses (at high frequency) to achieve the respective operational target (for i t or δ t ) are endogenous variables. Hence, when the CB abstains from intervening in the bond or FX market it keeps the corresponding instrument constant. Speci cally, when there is a FER regime, the second of the above simple rules must be replaced by an equation that keeps the stock of CB foreign currency reserves constant at the NSS level (r t = r). And when there is a PER regime, the rst of the above simple rules is replaced by an equation that keeps the stock of CB domestic currency bonds constant at the NSS level (b t = b). In the present paper, there is the additional possibility that the government (assumed to generally coordinate with the CB) abstain from actively using capital controls. In that case, the third simple policy rule above is replaced by the equation that keeps tax D t (or, taxsub D t , which for succinctness is not repeated below) at its NSS level. Hence, the three possible substituting equations are, respectively: As Figure 2 illustrates, there are seven possible`policy regimes', corresponding to the seven`faces' of the triangle (or 2-simplex). The three vertices (0-faces) of the triangle represent the`pure' policies in which there is only one rule, and hence the other two are replaced by their substitutes. The three edges (1-faces) of the triangle represent the three policy regimes in which two of the rules operate and the third is replaced by its substitute. And the interior (2-face) of the triangle represents the case in which all three rules are used (the Possible Trinity). This latter policy regime is denominated Managed Exchange Rate with Capital Control (MER+CC).
The bottom edge of the triangle (including its two vertices) represents the three policy regimes studied in Escudé (2013) (in which there were no capital controls). These three regimes keep tax D t constant at its NSS level (which in the parent paper was not de ned but in this paper may be zero or positive). The MER regime uses the rst two of the above simple policy rules and replaces the third policy rule by tax D t = tax D . Parting from the MER regime, the Floating Exchange Rate regime (FER) additionally replaces the second policy rule by r t = r, and the Pegged Exchange Rate regime (PER) instead additionally replaces the rst policy rule by b t = b. The upper left edge of the triangle (FER+CC regime) adds the capital control rule to the interest rate rule, and the upper right edge of the triangle (PER+CC regime) adds the capital control rule to the nominal rate of depreciation rule.
The top vertex of the triangle is the policy which only uses the capital control rule and keeps the two usual instruments constant at their NSS levels. It may come as a surprise to many that such a policy rule easily makes the model satisfy the Blanchard-Kahn conditions for stability and determinacy. In fact, using the calibrations detailed in Escudé (2013) and section 3 above, and starting from the baseline simple policy rule de ned in the second column of Table 3, each coefcient can vary individually within the (very wide) intervals given by the third column without impairing the Blanchard-Kahn conditions 9 : So far, simple policy rules have been considered, whether their coef cients are exogenously given or optimal in the sense that they represent the minimum of an ad-hoc CB loss function. The latter case is handled by means of the`osr' (optimal simple rule) Dynare command. In the case of optimal policy under commitment there are no simple policy rules, the corresponding equations disappear, and hence there are more endogenous variables than system equations. The paths for the endogenous variables that lack an equation (the set or a non-empty subset of the three intermediate targets) are obtained as solutions to the optimal control problem in which the CB minimizes the expected discounted sum of all (present and) future losses. This case is handled by means of the`ramsey' Dynare command. The appropriate combination of`instruments' (i.e., the control variables whose paths are obtained as optimum for the optimal control problem) must be chosen, and the corresponding substitute equation(s) must be introduced for those of the three possible`instruments' variables that are not used as such. For example, for solving the model for the MER+CC regime under`ramsey' using Dynare, the optioǹ instruments=(ii,delta,taxsubD)' for the`ramsey' command must be used and the three simple rules are simply eliminated (with no substitute equation). But for the remaining 6 policy regimes at least one of these instruments is not used. In particular, for the FER+CC regime the option to use is`instruments=(ii,taxsubD)', the rst and third simple rules must be eliminated (with no substitute equation) and the second policy rule must be replaced by r t = r. Analogously, for the PER+CC regime, the option to use is`instruments=(delta,taxsubD)', the second and third simple rules must be eliminated (with no substitute equation) and the rst policy rule must be replaced by b t = b. As an example of the three policy regimes in which only one instrument is used let us take the case of the CC regime. In this case, the option to use is`instruments=(taxsubD)', the third simple rule is eliminated (with no substitute equation) and the rst two policy rules are substituted by b t = b and r t = r.
It should be clear that in the case of the Ramsey problem, the optimal policy under any one of the six`boundary' regimes cannot dominate the optimal rule under the MER+CC regime due to the fact that in any of the latter the government imposes at least one additional restriction on itself (`ties its hands'), hence relinquishing its use of one or more of its potential`control' variables and using instead one or more of the possible substitute equations. For the same reason, the optimal loss for a`vertex' policy (one of the vertices of the triangle) cannot be greater than the optimal loss for an`edge' policy (one of the sides of the triangle) that has that vertex as one of its extremes. Hence, there is a clear hierarchy here: the optimal loss for the MER+CC regime is less than or equal to the optimal loss of the MER, FER+CC, or PER+CC regimes (its edges), the optimal loss of the MER regime is less than or equal to the optimal loss of the FER or PER regimes (its vertices), the optimal loss of the FER+CC regime is less than or equal to the optimal loss of the FER or CC regimes (its vertices), and the optimal loss of the PER+CC regime is less than or equal to the optimal loss of the PER or CC regimes (its vertices). What is of interest is the extent to which an additional instrument reduces the loss, the ranking of the losses within the three edges and within the three vertices, and how the relative losses vary with different CB preferences (or styles').
5 The role of a tax/subsidy in capital in ows/out ows in stabilization

Preliminary illustration of the effects of introducing capital controls through simple policy rules
First, let us illustrate how the introduction of a variable tax/subsidy scheme on capital ows can achieve stabilization objectives, by which is meant a reduction in the standard deviation (s.d.) of certain target variables. Let us assume that initially there is a MER regime with the simple policy rules de ned in the rst four columns of Table 4. Running the model gives the s.d. shown in the last column for some of the typical target variables (π C t ; Y t ; e t ), intermediate target variables (i t ; δ t ), and three variables related to household foreign debt ( d, γ D , ϕ D ). Since taxsubD is not used as an instrument in the MER regime its s.d. is zero. Next the model was run using a MER+CC regime which has the same two policy rules as in the MER regime and an additional simple policy rule for the tax/subsidy scheme shown in columns 5 and 6 of Table 5. this table shows that the s.d. for consumption in ation has been reduced by 37.4%, whereas the s.d. for GDP and the RER have both increased by 5.2%. The s.d. for the two operational targets in the MER regime (the nominal interest rate and the rate of nominal depreciation) have been reduced by 22.9% and 26.1%, respectively. Of course, this has been achieved by increasing the s.d. for taxsubD in nitely (since it was null under the MER regime and now it is positive), and the s.d. for d, γ D and ϕ D by 42.5%, 58.6% and 58.3%, respectively.  Figures 3 and 4 show the Impulse Response Functions (IRFs) corresponding to a surprise reduction in the exogenous risk/liquidity premium φ . Under the MER regime, the liquidity shock induces households to take advantage of the cheaper funds and thereby increase their foreign debt on impact and increase their consumption. However, the shock also generates real appreciation, making exports fall. This negative effect predominates over the increase in consumption, so GDP falls. Figure 4 shows the IRFs after the tax/subsidy scheme has been introduced (and hence there is a MER+CC regime). The behavior of households is seen to be quite different in the initial quarters. Instead of initially increasing their foreign debt they reduce it (thereby obtaining a subsidy) and instead of increasing their consumption, they reduce it. Instead of a real appreciation, there is now an initial real depreciation, increasing exports. The latter effect neutralizes the fall in consumption, so there is no initial effect on GDP but subsequently it rises for a few quarters, since consumption recovers faster than exports start to fall. In essence, the introduction of the tax/subsidy scheme generates during the initial quarters a substitution of expensive foreign funds for cheap government funds (that pay no interest). It can be concluded that governments that use a MER regime with the coef cients shown above in a SOE prone to signi cant RW liquidity shocks and have a stronger preference for stabilizing consumption in ation than GDP or the RER have something to gain by introducing a tax/subsidy scheme on capital ows. This exercise is only an illustration of how the tax/subsidy scheme can change the dynamic paths of variables that typically interest policymakers most. One should bear in mind that the IRFs illustrate the effect of the introduction of the tax/subsidy scheme on the (deterministic) dynamics of the model when there is a shock to the exogenous risk/liquidity premium, whereas the standard deviations of the endogenous variables shown in Tables 4 and 5 above illustrate the stochastic properties of the whole set of shock variables. Table 6 below show the variance decompositions of the variables corresponding to the two exercises. They show that the four really signi cant shocks in the model are those that hit public expenditures (G), the exogenous risk/liquidity premium (φ -phiStar), and the in ation rates for imports (π -piStar) and exports (π -piStarX), whereas the shocks on productivity (ε-epsilon) and the world interest rate (i -iStar) individually account for at most 5.7% of the variances of the target variables. In order to implement optimal simple policy rules, in this subsection policymakers are assumed to minimize an ad hoc loss function that is a weighted average of the variances of certain target variables and the change in the intermediate target variables (∆i t ,∆δ t ) . This operationalizes our assumption that policymakers have preferences for the relative stabilization of target variables which are de ned by the weights they attach to the variance of the different potential target variables (ination, GDP, RER). In order to additionally re ect a preference for the avoidance of too much policy activism, the variance of the changes in the interest rate and the nominal depreciation rate are also included in the loss function. Speci cally, it is assumed that, given the weights ω k , policymakers search for the coef cients in the simple policy rules h i ; k i ; j i that minimize the loss function within the curly brackets below, that is simply a weighted average of the variances of certain variables. arg min Notice that in addition to the usual terms (with weights ω π , ω Y , ω ∆i ), this loss function also allows for CB preferences with respect to the variances of the RER and the changes in the rate of nominal depreciation (with weights ω e , ω ∆δ , respectively). Four different CB styles (A-D) are de ned in Table 7 according to the combinations of weights in each. All these styles are de ned using the same weights for the variances of the changes in each of the operational targets (50) because it is the different preferences with respect to the variances of the target variables that is the main concern here. Also, zeros have been avoided by giving a weight of 1 to target variables with very little importance. Hence, these CB preferences can be summarized by saying that in style A only in ation matters, in style B only GDP matters, both in ation and GDP matter equally in style C, and the RER matters as much as in ation and GDP in style D. The`osr' (`optimal simple rules') command in Dynare calculates the loss for an initial set of values for the coef cients h i , k i , j i , and then follows an algorithm that searches for lower losses by changing the values of these coef cients. There are several options for the algorithm. The second has been used, which is the default option. For simplicity, the model was run assuming that the risk premium does not respond to the CB international reserves (ε τ D;2 = 0). Also, only the exercise for the tax/subsidy case was done in its entirety, since this procedure is very time consuming and the differences with the tax case did not seem too signi cant or particularly interesting. It should be mentioned that there is no guarantee that the optimal coef cients found in any run are a global optimum: different initial values can and usually do lead to different nal coef cients and (locally) optimal losses. Hence, in all cases several different initial sets of values (at least 5 and sometimes as many as 10) have been used, and the set that achieved the lowest loss was chosen. In some cases, this implied obtaining extremely high absolute values for some of the coef cients. Nevertheless, the lowest loss obtained in any of the runs, along with the corresponding policy rule coef cients, is reported in the tables below. Table 8 below reports the losses thus obtained for each regime and each CB style, as well as the losses relative to the MER+CC regime and the ranking of the regimes for each CB style. For all the CB styles, the lowest loss was obtained with the MER+CC and the second lowest loss was obtained with the MER regime, with an increase in loss as little as 5% (for style A) and as high as 40% (for style B). The third and fourth lowest losses, however, vary with the CB style. Style A gave the PER+CC regime as third in the ranking, and the FER+CC regime fourth. This order is inverted for styles B, C, and D. Hence, giving up either the interest rate or the nominal depreciation rule, implies increases in loss of at least 50% and as high as 570%. For all styles the simple CC regime (in which the only policy rule is the tax/subsidy scheme) obtained the fth highest loss. The last two places in the ranking are obtained by the FER and PER regimes: the FER regime is last in the ranking for styles A, C and D whereas the PER regime is last for style B. Table 9 shows the optimal coef cients in the simple policy rules obtained for each regime and style. To obtain some intuition on why the losses are lower in the MER+CC regime than in any of the six`boundary regimes', let us take the log-linear approximations of the simple policy rules equations and the UIP equation:  Leading the second and third equations and eliminating b i t , E t b δ t+1 , and E t t b ax ∆d t+1 from the fourth gives the following: On the l.h.s. is the log-linear deviation (from the NSS) of the UIP risk/liquidity premium. On the r.h.s. is a complex term that exclusively depends on the loglinear deviations of the (contemporary and expected) potential target variables the CB may use for its policy rules (including the CB international reserves) and the policy instruments. All the subterms on the r.h.s. are multiplied by a policy rule coef cient. 10 The CB policy rules have the effect of modifying the effects that the UIP risk/liquidity premium has on some important variables as a response to shocks. By making some of these coef cients equal to zero, the constraints that the respective 6`boundary' regimes impose imply that the CB has less leeway to affect international capital ows in the direction that may help it stabilize the economy according to its style (or preferences). For example, under the FER regime, in which all the k s and j s as well as ta b xsub D t are zero, the equation reduces to Clearly, with such zero constraints policymakers have less ability to in uence the debt ratio (and hence foreign debt and capital ows) in the direction convenient for stabilizing the target variables than in the general case.

The relative ef ciency of the different policy regimes with optimal policy under commitment
In this subsection Dynare's`ramsey' command is used to obtain the optimal policy under commitment, i.e., the paths for the control variables (in the sense of optimal control theory, but in our terminology the intermediate targets) that yield the minimum expected value, conditional on the information at t = t 0 , of a discounted ad hoc loss function: where L t is given by: given initial values for the predetermined values, and subject to all the non-policy model equations. Notice that two extra terms have been added in the period loss function. The rst one implies a preference (measured by ω r ) for CB international reserves stability. The reason for this is that, under a Ramsey optimal policy, system stability requires that ω r > 0. Since there is no speci c interest here in evaluating the effects of preferences for reserves stability, it has simply been assumed that ω r = 1 (a very low preference) and (aside from this addition) the same de nition of CB styles as in the previous section has been maintained. There is also a term that implies a penalty (measured by ω taxsubD ) for using the tax/subsidy on capital ows. The reason for this is that if ω taxsubD = 0 it is necessary to increase the policymakers' discount factor above the assumed household intertemporal discount factor β = :99, to, say, planner_discount=0.999, to achieve convergence.
Also, if ω taxsubD = 0, an excessively intense (and perhaps quite unrealistic) use of this instrument is sometimes obtained as well as a consequent excessive inertia in some of the endogenous variables. 11 Nevertheless, to see what the unrestrained use of this policy instrument and a much more restrained use imply in the ranking of regimes, the losses for ω taxsubD = 0 and ω taxsubD = 10 are also reported below. Table 10 below reports the losses, the relative losses, and the ranking of the alternative regimes for each CB style. This time, however, there is no ambiguity with respect to the optimum, since the linear-quadratic framework ensures that a global optimum is achieved in each of the 28 cases. Ranking Loss Regime

Relative loss
11 If instead of a quadratic penalty for deviations from the NSS one proceeds as with the two operational targets and assumes a preference for small changes in the tax/subsidy, the model delivers a unit root.  Ranking Loss Regime

Relative loss
As expected, the MER+CC regime always dominates the six`boundary' regimes. In this case, however, the loss with the FER+CC regime is always less than 1% above the loss with the MER+CC regime. Hence, although the FER+CC regime is clearly second in rank, adding as control variable the rate of nominal depreciation when there one is already using the nominal interest rate and the tax/subsidy scheme does not signi cantly increase the realization of stabilization objectives. Also, the PER+CC regime is always third in rank. In this case, the loss under CB styles A, C, and D are always less than 1.4% higher than with the MER+CC regime. But the increase in loss is more than 3% higher with CB style B and as much as 10% higher in the unrealistic case in which no penalty is imposed on the use of the tax/subsidy (ω taxsubD = 0). The MER regime (where policymakers abstain from using the tax/subsidy on capital ows) is at least 29% more costly than in the MER+CC regime and as much as 870% higher (for style C and ω taxsubD = 0). The three`vertex' regimes (FER, PER, CC) have losses that are much higher than in the MER+CC regime (ranging from 30% to 1200% higher). The MER regime is fourth in ranking for all CB styles for ω taxsubD = 1 and 10, but for ω taxsubD = 0 it is only fourth for CB style B (only GDP matters) and is fth for the rest of the styles, being replaced in the fourth position by regime CC. The ranking among the three`vertex' regimes vary from style to style. For ω taxsubD = 1 and 10 the CC regime is always last in the ranking, but for ω taxsubD = 0 it is only in the last position for CB style B and the FER regime is last for the other styles. More generally, in the loss comparison between an edge and its opposite vertex (see Figure 2), in all the cases reported the loss in the FER+CC regime is less then the loss in the PER regime and the loss in the PER+CC regime is less than the loss in the FER regime. However, as already mentioned, in the loss comparisons between the MER and CC regimes, while the loss of the MER regime is lower for ω taxsubD = 1 and 10, it is higher in the ω taxsubD = 0 case for all styles except B. Table 11 below reports the same exercises for the case in which there is a tax on the level of debt (no subsidy). Most of what has been said about the tax/subsidy scheme is valid also for this case. For this reason, only the main differences will be mentioned. First, note that when ω taxD = 0 all the regimes in which there is a tax on foreign debt (and hence CC appears in the regime nomenclature) the losses are exactly the same as in the tax/subsidy scheme. This means that when there is no penalty the same path for debt related tax collection can be achieved with either form of implementation. Hence, only for the MER, FER, and PER regimes are there any differences in loss, and they are quite minor (less then 0.66% in absolute value), with no change in the ranking. In the vertex versus opposite edge comparison, even for ω taxD = 1, or ω taxD = 10 there are styles in which the CC regime has a lower loss than the MER regime.   Loss Regime

Conclusion
The parent paper Escudé (2013) tries to bridge the gap between the fact that many central banks systematically intervene in the foreign exchange market and the absence of any generally accepted model for the representation of this practice. It presents a model and a policy framework in which the CB can simultaneously intervene in the foreign exchange and domestic currency bond markets, varying its outstanding bond liabilities and reserve assets in order to achieve two operational targets: one for the interest rate and another for the rate of nominal depreciation. To make this possible, the DSGE model includes nancial variables and institutional practices (`nuts and bolts' of central banking) that are left out of the modeling when only the usual`corner' policy regimes of a pure oat or a pure peg are considered, but cannot be left out when trying build a more general model. In the present paper I have extended this policy framework to include the third corner of the traditional`impossible trinity' by including the use of a tax on household foreign currency liabilities or a tax/subsidy scheme that taxes increases in such liabilities and subsidizes their reductions. In the present context, however, there is à possible trinity' since all three forms of intervention can be used simultaneously. Furthermore, when used in the optimal control framework this`trinity' is not only possible but also optimal in the sense that any of the 6 boundary regimes (the 3 edges and the 3 vertices of the policy triangle) cannot attain a lower loss than the simultaneous use of 3 instruments and 3 (operational) targets due to the obvious fact that they are derived from adding one or two additional constraints that re ect the abstention from using one or two of the 3 possible instruments (interventions). As in the parent paper, three frameworks are considered: 1) simple policy rules, 2) optimal simple policy rules, where the coef cients of the simple rules are derived by minimizing an ad hoc loss function that is a weighted average of the variances of the target variables (in ation, GDP and the RER), 3) optimal policy under commitment and full information, where policymakers minimize an ad hoc discounted intertemporal loss function related to the same target variables. The main interest in 2) and 3) is in seeing whether the capital controls can achieve a signi cant reduction in loss to the already potent simultaneous interest and exchange rate policies, and in obtaining the ranking in the losses obtained for the 7 possible policy regimes (the use of all 3 policies and the 6 additional possibilities derived from eliminating either one or two of these policies).
Two forms of (soft) capital controls are considered: 1) a tax on the level of household foreign debt, and 2) a tax/subsidy scheme in which increases in household foreign debt are taxed and reductions are subsidized. The results show that the losses obtained are not very different for the two forms of capital controls. A point of interest lies in gauging the extent to which forfeiting one or two of the possible instruments reduces the loss for different central bank preferences (de ned by the weights in the ad hoc loss function). The results show that increases in the loss function due to forfeiting intervention in the FX market or in the CB bond market (starting from the`possible trinity' are much higher in the case of optimal simple rules than in the case of optimal policy under commitment. But since there is no assurance that a global minimum is reached when using optimal simple policy rules with Dynare, the results for optimal policy under commitment are a welcome complement. In the latter context it was found that there is hardly any additional loss in forfeiting FX policy (and letting the exchange rate oat) when starting from the use of the 3 instruments. The increase in loss is signi cantly higher but not very high when it is the interest rate policy that is eliminated (and the interest rate is allowed to oat). In most cases, doing away with the capital controls implies a large increase in loss. A surprising result is that, depending on the CB preferences, the Managed Exchange Rate regime may be superior or inferior to the simple Capital Control (CC) regime in which both interest rate and FX policies are eliminated and only the CC rule stabilizes the economy. The production side of the economy is exactly as in Escudé (2013). There is perfect competition in the production (or bundling) of nal domestic output Q t , using as inputs the output of a continuum of monopolistically competitive rms. A representative nal domestic output rm uses the following CES technology: where Q t (i) is the output of the intermediate domestic good i and θ is the elasticity of substitution between any two varieties of goods. Pro t maximization yields the demand for each type of domestic good as an input and the domestic goods price index: The production function of each rm is proportional to its use of labor: Q t (i) = ε t N t (i), where ε t is an industry-wide transitory productivity shock. Hence, each rm's real marginal cost (in terms of domestic goods) is mc t = w t =ε t . Since N t (i) is rm i's labor demand, using (45) and integrating yields aggregate labor demand: where ∆ t is a measure of price dispersion at period t de ned as: Equating labor supply (20) and demand (46) gives the labor market equilibrium real wage: Firms make pricing decisions taking the aggregate price and quantity indexes as parametric. Every period, each rm has a probability 1 α of being able to set the optimum price for its speci c type of good. The rms that can't optimize must leave the same price they had last period. The pricing problem of rms that get to optimize is: subject to the demand they will face until they can again optimize: Q t+ j (i) = Q t+ j P t (i) P t+ j θ : where the pricing kernel of rms is: In Escudé (2013) it is shows that the rm's rst order condition is: (β α) j Q t+ j p C t+ j C σ C t+ j P t+ j P t θ e p t P t P t+ j θ θ 1 mc t+ j : where e p t e P t =P t is the relative price of rms that optimize with respect to the general price level. Also, the price index in (45) implies the following law of motion for the aggregate domestic goods price index: P 1 θ t = α (P t 1 ) 1 θ + (1 α) e P 1 θ t : Dividing through by P 1 θ t 1 and rearranging yields the relative price of optimizers as an increasing function of the in ation rate: The Phillips equation (51) can be expressed in recursive form by de ning: (β α) j Q t+ j = p C t+ j C σ C t+ j P t+ j P t θ 1 (54) (β α) j Q t+ j = p C t+ j C σ C t+ j P t+ j P t θ mc t+ j www.economics-ejournal.org domestic and imported consumption in total consumption expenditures. It is assumed that there is a bias for domestic goods, i.e., a D > 1=2 > a N , and that θ C > 1. Minimization of total consumption expenditure P C t C t = P t C D t +S t P t C N t subject to (56) for a given C t , yields the following relations: Introducing these in (56) yields the consumption price index: and dividing through by P t yields a relation between the relative prices of consumption and imported goods and the RER: Relations (58) can used to eliminate C D t and C N t : And the shares of domestic and imported goods in total expenditure can be expressed in terms of e t and p C t as:

Export rms
Firms in the export sector use domestic goods and the composite of goods that de nes GDP. I assume that the export good is a single homogenous primary good (a commodity). Firms in this sector sell their output in the international market at the foreign currency price P X t . They are price takers in factor and product markets. The price of primary goods in terms of the domestic currency is merely the exogenous international price multiplied by the nominal exchange rate: S t P X t : Let the production function employed by rms in the export sector be the following: where Q X t is the amount of domestic goods used as input in the export sector and Y t is real GDP. These rms maximize pro t S t P X t X t P t Q X t subject to (62). In terms of domestic goods, they maximize: where the SOE's external terms of trade (XTT) is de ned as p t P X t =P t , where P t is the price index of the foreign currency price of the SOE's imports. Notice that the XTT is a ratio of two price indexes determined in the RW. Hence, the follow identity relates the rates of foreign in ation of exported and imported goods to the XTT (giving the dynamics of the XTT): where π X t P X t P X t 1

:
The rst order condition for pro t maximization yields the export sector's (factor) demand for domestic goods: Inserting this in (62) shows that the real value of exports in terms of domestic goods is: X t = S t P X t X t P t = e t p t X t = e t p t b A e t p t where b X 1 b A 1 and κ X b A b A =(1 b A ) are introduced for simplicity of notation.

Domestic output, GDP, and the balance of payments
Government expenditure is assumed to be a time-varying and stochastic fraction G t of private consumption expenditure. De ne the gross government expenditure fraction as: G t 1 + G t . For simplicity, it is assumed that the government must pay the same transaction costs as the private sector when it purchases domestic ρ π π X 1 ρ π exp σ π ε π t π t = π t 1 ρ π (π ) 1 ρ π p t 1 α π exp σ π ε π t : In the implementation in Dynare, the 6 shock variables have been expressed in logs. For example, the rst of the shock equations is ε t = ρ ε ε t 1 + σ ε ε ε t (where changing the name of the variable to e.g. lε t has been avoided) and every time ε t appears in the (non-shock) model equations above it is replaced by exp(ε t ) to eliminate the log transformation. Additionally, in Dynare the standard deviation of the stochastic shocks is de ned in the`shocks' block. For example, in the mod le the rst shock equation is actually ε t = ρ ε ε t 1 + ε ε t . Because leads and lags of variables are forbidden within the Dynare`plan-ner_objective' command, in the case of optimal policies under commitment it is necessary to introduce two new variables and equations: di t = i t i t 1 , dδ t = δ t δ t 1 :