An Estimated DSGE Open Economy Model of the Indian Economy with Financial Frictions

Abstract

We develop an open economy DSGE model of the Indian economy and estimate it by Bayesian Maximum Likelihood methods. We build up in stages to a model with a number of features important for emerging economies in general and the Indian economy in particular: a large proportion of credit-constrained consumers, a financial accelerator facing domestic firms seeking to finance their investment, “liability dollarization” and incomplete exchange rate pass-through. Our estimation results support the inclusion of financial frictions in an otherwise standard small open economy model. The simulation properties of the estimated model are examined under a generalized inflation targeting Taylor-type interest rate rule with forward- and backward-looking components.

We acknowledge financial support for this research from the Foreign Commonwealth Office as a contribution to the project “Building Capacity and Consensus for Monetary and Financial Reform” led by the National Institute for Public Finance Policy (NIPFP). We also acknowledge constructive comments from an anonymous referee and Chetan Ghate. The paper has benefited from excellent research assistance provided by Rudrani Bhattachrya and Radhika Pandey, NIPFP.

Appendices

A    A Closed Economy Model

This section develops a standard New Keynesian (NK) DSGE model without any features we associate with emerging economies. To give us a preliminary insight into what is different about an emerging economy such as India. Every NK DSGE model has at its core a real business cycle model, describing the intertemporal problems facing consumers and firms and defining what would happen in the absence of the various Keynesian frictions. We first define a single-period utility for the representative agent in terms of consumption, $C_t$, and leisure, $L_t$, as

$$\begin{aligned} \Lambda _t= & {} \Lambda (C_{t}, L_t)= \frac{((C_{t}-h_CC_{t-1})^{(1-\varrho )}L_t^\varrho )^{1-\sigma }-1}{1-\sigma } \end{aligned}$$

(A.1)

In this utility function $\sigma \ge 1$ is a risk aversion parameter which is also the inverse of the intertemporal rate of substitution. $h_C$ is a habit parameter for private consumption $C_t$.¹⁴ The parameter $\varrho \in (0,1)$ defines the relative weight households place on consumption and this form of utility is compatible with a balanced growth steady state for for all $\sigma \ge 1$.¹⁵ For later use, we write down the marginal utilities of consumption and leisure as, respectively,

$$\begin{aligned} \Lambda _{C,t}= & {} (1-\varrho ) (C_{t}-h_CC_{t-1})^{(1-\varrho )(1-\sigma )-1} L_t^{\varrho (1-\sigma )} \end{aligned}$$

(A.2)

$$\begin{aligned} \Lambda _{L,t}= & {} -\varrho (C_{t}-h_CC_{t-1})^{(1-\varrho )(1-\sigma )} L_t^{\varrho (1-\sigma )-1} \end{aligned}$$

(A.3)

The value function at time t of the representative household is given by

$$\begin{aligned} \Omega _t= E_t \left[ \sum _{s=0}^\infty \beta ^s \Lambda (C_{t+s}, L_{t+s}) \right] \end{aligned}$$

(A.4)

where $\beta$ is the discount factor. In a stochastic environment, the household’s problem at time t is to choose state-contingent plans for consumption $\{C_t\}$, leisure, $\{L_t\}$ and holdings of financial savings to maximize $\Omega _t$ given its budget constraint in period t

$$\begin{aligned} B_{t+1}=B_t(1+R_t)+W_t h_t-C_t \end{aligned}$$

(A.5)

where $B_t$ is the net stock of real financial assets at the beginning of period t, $W_t$ is the real wage rate, and $R_t$ is the real interest rate paid on assets held at the beginning of period t. Hours worked are $h_t=1-L_t$ and the total amount of time available for work or leisure is normalized at unity. Government spending is financed by lump-sum nondistortionary taxes throughout. The first-order conditions for this optimization problem are

$$\begin{aligned} \Lambda _{C,t}= & {} \beta E_t\left[ (1+R_{t+1}) \Lambda _{C,t+1}\right] \end{aligned}$$

(A.6)

$$\begin{aligned} \frac{\Lambda _{L,t}}{\Lambda _{C,t}}= & {} W_{t} \end{aligned}$$

(A.7)

Equation (A.6) is the Euler consumption function: it equates the current marginal utility of consumption with the discounted marginal of consumption of a basket of goods in period $t+1$ enhanced by the interest on savings. Thus, the household is indifferent as between consuming 1 unit of income today or $1+R_{t+1}$ units in the next period. Equation (A.7) equates the marginal rate of substitution between consumption and leisure with the real wage, the relative price of leisure. This completes the household component of the RBC model.

Turning to the production side, we assume that output $Y_t$ is produced using hours, $h_t$ and beginning-of-period capital $K_t$ with a Cobb–Douglas production function

$$\begin{aligned} Y_{t}= & {} F(A_{t}, h_{t}, K_{t})=(A_{t} h_{t})^{\alpha } K_{t}^{1-\alpha } \end{aligned}$$

(A.8)

where $A_t$ is a technology parameter and $Y_t$, $h_t$, and $K_t$ are all in per capita (household) units. Assume first that capital can adjust instantly without investment costs. Then equating the marginal product of labor with the real wage and the marginal product of capital with the cost of capital (given by the real interest rate plus the depreciation rate, $R_t+\delta$), we have

$$\begin{aligned} F_{h,t}= & {} \alpha \frac{Y_t}{h_t}=W_{t}\end{aligned}$$

(A.9)

$$\begin{aligned} F_{K,t}= & {} (1-\alpha ) \frac{Y_t}{K_t}=R_{t}+\delta \end{aligned}$$

(A.10)

Let investment in period t be $I_t$. Then capital accumulates according to

$$\begin{aligned} K_{t+1}=(1-\delta ) K_t+I_t \end{aligned}$$

(A.11)

The RBC model is then completed with an output equilibrium equating supply and demand

$$\begin{aligned} Y_t=C_t+I_t+G_t \end{aligned}$$

(A.12)

where $G_t$ is government spending on services assumed to be formed out of the economy’s single good and by a financial market equilibrium. In this model, the only asset accumulated by households as a whole is capital, so the latter equilibrium is simply $B_t=K_t$. Substituting into the household budget constraint (A.5) and using the first-order conditions (A.9) and (A.10), and (A.11) we end up with the output equilibrium condition (A.12). In other words, equilibrium in the two factor markets and the output market implies equilibrium in the remaining financial market, which is simply a statement of Walras’ Law.

Now let us introduce investment costs. It is convenient to introduce capital producing firms that at time t convert $I_t$ of output into $(1-S(X_t))I_t$ of new capital sold at a real price $Q_t$ and at a cost (that was absent before) of $S(X_t)$. Here, $X_t\equiv \frac{I_t}{I_{t-1}}$ and the function $S(\cdot )$ satisfies $S',\, S'' \ge 0\,;\,\,S(1+g)=S'(1+g)=0$. Thus, investment costs are convex and disappear along in the balanced growth steady state. They then maximize expected discounted profits

$$\begin{aligned} E_t \sum _{k=0}^\infty D_{t,t+k} \left[ Q_{t+k} (1-S\left( I_{t+k}/I_{t+k-1}\right) )I_{t+k} -I_{t+k} \right] \end{aligned}$$

where $D_{t,t+k} \equiv \beta \frac{ \Lambda _{C,t+k}}{ \Lambda _{C,t}}$ is the real stochastic discount rate and

$$\begin{aligned} K_{t+1}= & {} (1-\delta ) K_t+(1-S(X_t))I_t \end{aligned}$$

(A.13)

This results in the first-order condition

$$\begin{aligned} Q_t (1-S(X_t)-X_t S'(X_t))+E_t \left[ \frac{1}{(1+R_{t+1})}\, Q_{t+1} S'(X_{t+1}) \frac{I_{t+1}^2}{I_t^2} \right] =1 \end{aligned}$$

(A.14)

Demand for capital by firms must satisfy

$$\begin{aligned} E_t[(1+R_{t+1}) {RPS}_{t+1}] =\frac{E_t\left[ (1-\alpha ) \frac{P_{t+1}^W Y_{t+1}}{K_{t+1}}+(1-\delta ) Q_{t+1}\right] }{Q_{t}} \end{aligned}$$

(A.15)

In (A.15) the right-hand side is the gross return to holding a unit of capital in from t to $t+1$. The left-hand side is the gross return from holding bonds, the opportunity cost of capital and includes an exogenous risk premium shock ${RPS}_t$, which, for now, we leave unmodeled. We complete the setup with investment costs by defining the functional form

$$\begin{aligned} S(X)=\phi _X (X_t-(1+g))^2 \end{aligned}$$

(A.16)

where g is the balanced growth rate. The RBC model we have set out defines a equilibrium in output, $Y_t$, consumption $C_t$, investment $I_t$, capital stock $K_t$ and factor prices, $W_t$ for labor and $R_t$ for capital, and the price of capital $Q_t$, given exogenous processes for technology $A_t$, government spending $G_t$ and the risk premium shock ${RPS}_t$.

The NK framework combines the DSGE characteristics of RBC models with frictions such as monopolistic competition—in which firms produce differentiated goods and are price setters, instead of Walrasian determination of prices—and nominal rigidities, in which firms face constraints on the frequency with which they are able to adjust their prices. Therefore, we now introduce a retail sector that uses a homogeneous wholesale good to produce a basket of differentiated goods for consumption

$$\begin{aligned} C_{t}=\left( \int _{0}^{1}C_{t}(m)^{(\zeta -1)/\zeta }{d}k\right) ^{\zeta /(\zeta -1)} \end{aligned}$$

(A.17)

where $\zeta$ is the elasticity of substitution. This implies a set of demand equations for each intermediate good m with price $P_{t}(m)$ of the form

$$\begin{aligned} C_{t}(m)=\left( \frac{P_{t}(m)}{P_{t}}\right) ^{-\zeta }C_{t} \end{aligned}$$

(A.18)

where $P_{t}=\left[ \int _0^1 P_{t}(m)^{1-\zeta } {d}m \right] ^{ \frac{1}{1-\zeta }}$. $P_{t}$ is the aggregate price index.

Conversion of good m from a homogeneous output requires a cost $c Y_t^W (m)$ where wholesale production uses the production technology (A.8). Thus

$$\begin{aligned} Y_t(m)= & {} (1-c) Y_t^W (m)\end{aligned}$$

(A.19)

$$\begin{aligned} Y_t^W= & {} (A_{t} h_{t})^{\alpha } K_{t}^{1-\alpha } \end{aligned}$$

(A.20)

To introduce price stickiness, we assume that there is a probability of $1-\xi$ at each period that the price of each intermediate good m is set optimally to $P_{t}^{0}(m)$. If the price is not reoptimized, then it is held fixed.¹⁶ For each intermediate producer m the objective is at time t to choose $\{P_{t}^0(m)\}$ to maximize discounted profits

$$\begin{aligned} E_{t}\sum _{k=0}^{\infty } \xi ^k D_{t,t+k}Y_{t+k}(m)\left[ P_{t}^{0}(m)-P_{t+k} {MC}_{t+k} \right] \end{aligned}$$

(A.21)

subject to (A.18), where $D_{t,t+k}$ is now the nominal stochastic discount factor over the interval $[t,t+k]$. The solution to this is

$$\begin{aligned} E_{t}\sum _{k=0}^{\infty } \xi ^k D_{t,t+k}Y_{t+k}(m)\left[ P_{t}^{0}(m)-\frac{1}{ (1-1/\zeta ) }P_{t+k} {MC}_{t+k} {MS}_{t+k}\right] =0 \end{aligned}$$

(A.22)

In (A.22) we have introduced a markup shock $MS_t$ to the steady-state markup $\frac{1}{ (1-1/\zeta ) }$. By the law of large numbers, the evolution of the price index is given by

$$\begin{aligned} P_{t+1}^{1-\zeta }=\xi P_{t}^{1-\zeta }+(1-\xi )(P_{t+1}^{0})^{1-\zeta } \end{aligned}$$

(A.23)

In setting up the model for simulation and estimation, it is useful to represent the price dynamics as difference equations. Using the fact that for any summation $S_t \equiv \sum _{k=0}^\infty \beta ^k X_{t+k}$, we can write

$$\begin{aligned} S_t= & {} X_t+\sum _{k=1}^\infty \beta ^k X_{t+k}=X_t+\sum _{k'=0}^\infty \beta ^{k'+1} X_{t+k'+1} \text{ putting } k'=k+1\nonumber \\= & {} X_t+\beta S_{t+1} \end{aligned}$$

(A.24)

and defining here the nominal discount factor by $D_{t,t+k} \equiv \beta \frac{ \Lambda _{C,t+k}/P_{t+k}}{ \Lambda _{C,t}/P_t}$, inflation dynamics are given by

$$\begin{aligned} \frac{P_t^0}{P_t}= & {} \frac{H_t}{J_t}\end{aligned}$$

(A.25)

$$\begin{aligned} H_{t}-\xi \beta E_t [\Pi _{t+1}^{\zeta -1} H_{t+1}]= & {} Y_{t} \Lambda _{C,t}\end{aligned}$$

(A.26)

$$\begin{aligned} J_{t}-\xi \beta E_t [\Pi _{t+1}^{\zeta }J_{t+1}]= & {} \left( \frac{1}{1-\frac{1}{\zeta }}\right) Y_{t} \Lambda _{C,t} {MC}_{t}{MS}_t \end{aligned}$$

(A.27)

$$\begin{aligned} \Pi _{t}:\quad 1= & {} \xi \Pi _{t} ^{\zeta -1}+(1-\xi )\left( \frac{J_{t}}{H_{t}}\right) ^{1-\zeta } \end{aligned}$$

(A.28)

Real marginal costs are no longer fixed and are given by

$$\begin{aligned} {MC}_t=\frac{P_t^W}{P_t} \end{aligned}$$

(A.29)

Nominal and real interest rates are related by the Fischer equation

$$\begin{aligned} E_t[1+R_{t+1}]=E_t \left[ \frac{1+R_{n,t}}{\Pi _{t+1}} \right] \end{aligned}$$

(A.30)

where the nominal interest rate is a policy variable, typically given in the literature by a standard Taylor-type rule:

$$\begin{aligned} \log \left( \frac{1+R_{n,t}}{1+R_n}\right) =\rho \log \left( \frac{1+R_{n,t-1}}{1+R_n}\right) +\theta _{\pi } \log \left( \frac{\Pi _{t}}{\Pi }\right) + \theta _{y} \log \left( \frac{Y_{t}}{Y}\right) \end{aligned}$$

(A.31)

In fact, we will model monetary policy in a more general way by formulating a Calvo-type forward/backward interest rate rule in inflation targets as in Levine et al. (2007) and Gabriel et al. (2009). This is defined by

$$\begin{aligned} \log \left( \frac{1+R_{n,t}}{1+R_n}\right)= & {} \rho \log \left( \frac{1+R_{n,t-1}}{1+R_n}\right) +\theta _{\pi } \log \frac{\Theta _{t}}{\Theta } +\phi _{\pi } \log \frac{\Phi _t}{\Phi }+\theta _{y} \log \left( \frac{Y_{t}}{Y}\right) +\epsilon _{{MPS},t}\nonumber \\ \end{aligned}$$

(A.32)

where $\epsilon _{{MPS},t}$ is a monetary policy shock and

$$\begin{aligned} \log \Phi _{t}= & {} \log \Pi _t +\tau \log \Phi _{t-1} \end{aligned}$$

(A.33)

$$\begin{aligned} \varphi E_t [\log \Theta _{t+1}]= & {} \log \Theta _t-(1-\varphi )\log (\Pi _t) \end{aligned}$$

(A.34)

The Calvo rule can be interpreted as a feedback from expected inflation (the $\theta \log \frac{\Theta _{t}}{\Theta }$ term) and past inflation (the $\phi \log \frac{\Phi _t}{\Phi }$ term) that continues at any one period with probabilities $\varphi$ and $\tau$, switching off with probabilities $1-\varphi$ and $1-\tau$. The probability of the rule lasting for h periods is $(1-\varphi )\varphi ^{h}$, hence the mean forecast horizon is $(1-\varphi )\sum _{h=1}^{\infty }h\varphi ^{h}=\varphi /(1-\varphi )$. With $\varphi =0.5$, for example, we would have a Taylor rule with one-period lead in inflation ($h=1)$. Similarly, $\tau$ can be interpreted as the degree of backward-lookingness of the monetary authority.

This rule can also be seen as a special case of a Taylor-type rule that targets h-step-ahead (back) expected rates of inflation and past inflation rates (with $h=1,2,\ldots ,\infty )$

$$\begin{aligned} i_{t}=\rho i_{t-1}+\theta _{0}\pi _{t}+\theta _{1}E_{t}\pi _{t+1}+\theta _{2}E_{t}\pi _{t+2}+\cdots +\gamma _{1}\pi _{t-1}+\gamma _{2}\pi _{t-2}+\cdots , \end{aligned}$$

(A.35)

albeit one that imposes a specific structure on the $\theta _{i}$’s and $\gamma _{i}$’s (i.e., a weighted average of future and past variables with geometrically declining weights). This has an intuitive appeal and interpretation, reflecting monetary policy in an uncertain environment: the more distant the h-step-ahead forecast, the less reliable it becomes, hence the less weight it receives. In turn, past inflation has a typical Koyck-lag structure.

Note that we are approximating the behavior of the central bank with an instrument rule, rather than assuming that the monetary authority optimizes a specific loss function. Despite the lack of a substantial body of evidence for the Indian case, the forward/backward-looking Calvo-type formulation can be useful to analyze the RBI’s interest rate setting behavior. Bhattacharya et al. (2010), using VAR methods, find monetary policy in India to have weak transmission channels. On the other hand, however, Virmani (2004) reports on the potential forward/backward-looking behavior of the RBI using instrumental rules, suggesting that a backward-looking rule explains the data well. Our proposal nests both types of behavior and can therefore shed light on their relative importance.

The structural shock processes in log-linearized form are assumed to follow AR(1) processes

$$\begin{aligned} \log A_{t}-\log \bar{A}_{t}= & {} \rho _{A} (\log A_{t-1}-\log \bar{A}_{t-1})+\epsilon _{A,t}\\ \log G_t-\log \bar{G}_t= & {} \rho _G (\log G_{t-1}-\log \bar{G}_{t-1})+\epsilon _{G,t}\\ \log {MS}_t-\log MS= & {} \rho _{MS} (\log {MS}_{t-1}-\log {MS})+\epsilon _{{MS},t}\\ \log {RPS}_t-\log {RPS}= & {} \rho _{{RPS}} (\log {RPS}_{t-1}-\log {RPS})+\epsilon _{{RPS},t} \end{aligned}$$

where ${MS}={RPS}=1$ in the steady state (so $\log {MS}=\log {RPS}=0$), while the monetary policy shock $\epsilon _{{MPS},t}$ is assumed to be i.i.d with zero mean. This completes the specification of the benchmark NK model.

B    Summary of Closed Economy Model

The following summarizes the dynamic model for the closed economy which applies to the foreign bloc. Note that the baseline model in Appendix A puts $\lambda =0$ and shuts down the financial accelerator.

$$\begin{aligned} \Lambda _{2,t}= & {} \Lambda (C_{2,t}, L_t)= \frac{(C_{2,t}^{(1-\varrho )}L_{2,t}^\varrho )^{1-\sigma }-1}{1-\sigma }\\ \Lambda _{C_2,t}= & {} (1-\varrho ) C_{2,t}^{(1-\varrho )(1-\sigma )-1} (1-h_{2,t})^{\varrho (1-\sigma )} ) \\ \Lambda _{L_2,t}= & {} \varrho C_{2,t}^{(1-\varrho )(1-\sigma )} L_{2,t}^{\varrho (1-\sigma )-1}\\ \Lambda _{C_2,t}= & {} \beta E_t\left[ (1+R_{t+1}) \Lambda _{C_2,t+1}\right] \\ \frac{\Lambda _{L_2,t}}{\Lambda _{C_2,t}}= & {} \frac{W_{t}}{P_t}\\ L_{2,t}\equiv & {} 1-h_{2,t}\\ h_{1,t}= & {} 1-\rho \\ C_{1,t}= & {} \frac{W_t h_{1,t}}{P_t}\\ h_{t}= & {} \lambda h_{1,t}+ (1-\lambda ) h_{2,t}\\ C_t^e= & {} \frac{1-\xi _e}{\xi _e}N_t\\ C_t= & {} \lambda C_{1,t}+(1-\lambda ) C_{2,t}+C_t^e\\ Y_{t}^W= & {} F(A_{t}, h_{t}, K_{t})=(A_{t} h_{t})^{\alpha } K_{t}^{1-\alpha }\\ Y_{t}= & {} (1-c)Y_{t}^W\\ \frac{P_{t}^W}{P_t} F_{h,t}= & {} \frac{P_{t}^W}{P_t}\frac{\alpha Y_t^W}{h_t} = \frac{W_{t}}{P_t}\\ P_{t}= & {} \frac{1}{1-\frac{1}{\zeta }}P_{t}^W \\ K_{t+1}= & {} (1-\delta ) K_t+(1-S(X_t))I_t\\ X_t\equiv & {} \frac{I_t}{I_{t-1}}\\ Q_t (1-S(X_t)-X_t S'(X_t))+ & {} E_t \left[ \frac{1}{(1+R_{t+1})}\, Q_{t+1} S'(Z_{t+1}) \frac{I_{t+1}^2}{I_t^2} \right] =1\\ E_t [(1+R_{t+1})\Theta _{t+1}]= & {} E_t [1+R_{k,t+1}] \\ 1+R_{k,t}= & {} \frac{ (1-\alpha _I) \frac{P_{t}^W}{P_{t}}\frac{Y_{t}^W}{K_{t}}+(1-\delta ) Q_{t}}{Q_{t-1}}\\ \Theta _t= & {} s \left( \frac{N_t}{Q_{t-1} K_t}\right) RP_t=k\, \left( \frac{N_t}{ Q_{t-1}K_t}\right) ^{-\chi }{RPS}_t \\ N_{t+1}= & {} \xi _e V_t +(1-\xi _e) D_t^{e}\\ D_t^e= & {} \bar{D}_t^e \text{(BGP } \text{ steady } \text{ state) } \\ V_t= & {} (1+ R_{k,t}) Q_{t-1} K_{t}-\Theta _{t}(1+R_{t})(Q_{t-1}K_{t}-N_t) \end{aligned}$$

$$\begin{aligned} S(X_t)= & {} \phi _X (X_t-(1+g))^2\\ Y_{t}= & {} C_{t}+G_t+I_t\\ H_{t}-\xi \beta E_t [\Pi _{t+1}^{\zeta -1} H_{t+1}]= & {} Y_{t} \Lambda _{C,t}\\ J_{t}-\xi \beta E_t [\Pi _{t+1}^{\zeta }J_{t+1}]= & {} \left( \frac{1}{1-\frac{1}{\zeta }}\right) Y_{t} \Lambda _{C,t}{MS}_t {MC}_{t} \\ 1= & {} \xi \Pi _{t} ^{\zeta -1}+(1-\xi )\left( \frac{J_{t}}{H_{t}}\right) ^{1-\zeta } \\ {MC}_t= & {} \frac{P_t^W}{P_t}\\ 1+R_{t}= & {} \frac{1+R_{n,t-1}}{\Pi _{t}} \\ \log A_{t}-\log \bar{A}_{t}= & {} \rho _{A} (\log A_{t-1}-\log \bar{A}_{t-1})+\epsilon _{A,t}\\ \log G_t-\log \bar{G}_t= & {} \rho _G (\log G_{t-1}-\log \bar{G}_{t-1})+\epsilon _{G,t}\\ \log {MS}_t-\log {MS}= & {} \rho _{MS} (\log {MS}_{t-1}-\log {MS})+\epsilon _{MS,t}\\ \log {RPS}_t-\log {RPS}= & {} \rho _{{RPS}} (\log {RPS}_{t-1}-\log {RPS})+\epsilon _{{RPS},t}\\ \log \left( \frac{1+R_{n,t}}{1+R_n}\right)= & {} \rho \log \left( \frac{1+R_{n,t-1}}{1+R_n}\right) +\theta \log \frac{\Theta _{t}}{\Theta } +\phi \log \frac{\Phi _t}{\Phi }+\epsilon _{{MPS},t}\\ \log \Phi _{t}= & {} \log \Pi _t +\tau \log \Phi _{t-1} \\ \varphi E_t [\log \Theta _{t+1}]= & {} \log \Theta _t-(1-\varphi )\log (\Pi _t) \end{aligned}$$

The steady state is given by the following:

$$\begin{aligned} \bar{N}_t= & {} \frac{(1-\xi _e) \bar{D}_t}{(1-\xi _e (1+R_k))} \end{aligned}$$

(B.36)

$$\begin{aligned} 1+R_k= & {} (1+R)\,\, s\left( \frac{\bar{N}_t}{\bar{K}_t}\right) \end{aligned}$$

(B.37)

$$\begin{aligned} \frac{\bar{K}_{t}}{\bar{Y}_{t}^W}= & {} \frac{1-\alpha }{R_k+\delta } \end{aligned}$$

(B.38)

Choose a functional form

$$\begin{aligned} s \left( \frac{\bar{N}_t}{Q \bar{K}_t}\right) =k \left( \frac{\bar{N}_t}{ Q\bar{K}_t}\right) ^{-\chi } \end{aligned}$$

We obtain $\chi$ from econometric studies and we have data on the risk premium $\Theta =\frac{1+R_k}{1+R}$ and leverage (= borrowing/net worth)

$$\begin{aligned} \ell = \frac{QK-N}{N}=\frac{QK}{N}-1=\frac{1}{n_k}-1 \end{aligned}$$

defining $n_k \equiv \frac{N}{Q K}$. Then we can set the scaling parameter k from (B.37) as

$$\begin{aligned} k=\Theta n_k^\chi \end{aligned}$$

Then in the baseline steady state used to calibrate parameters, we put $\bar{N}_t=n_k \bar{K}_t$ and calibrate $\bar{D}^e$ from (B.36). The nonzero-inflation steady state and the calibrated k are given by

$$\begin{aligned} 1+R= & {} \frac{(1+g)^{1+(\sigma -1)(1-\varrho )}}{\beta }\\ 1+R_n= & {} \Pi (1+R)\\ Q= & {} 1\\ \bar{Y}_{t}= & {} (1-c) (h_t \bar{A}_{t})^{ \alpha } \bar{K}_{t}^{1-\alpha } \\ \frac{\varrho \bar{C}_{2,t}}{(1-\varrho )(1-h)}= & {} \bar{W}_{t} \\ \bar{C}_{1,t}= & {} \bar{W}_{t} h \\ \frac{\alpha P^W \bar{Y}_{t}^W}{P h}= & {} \bar{W}_{t}\\ \frac{\bar{K}_{t}}{\bar{Y}_{t}^W}= & {} \frac{1-\alpha }{R_k+\delta }\\ 1+R_k= & {} (1+R)\Theta \\ \Theta= & {} k\,n_k^{-\chi }=k\left( \frac{\bar{N}_t}{Q \bar{K}_t}\right) ^{-\chi }\\ \bar{I}_t= & {} (\delta +g) \bar{K}_{t}\\ \bar{Y}_{t}= & {} \bar{C}_{t}+ \bar{I}_t+\bar{G}_t \\ 1= & {} \frac{1}{1-\frac{1}{\zeta }}\frac{P^W}{P} \\ \bar{N}_t= & {} n_k \bar{K}_{t} = \frac{(1-\xi _e) \bar{D}_t^e}{(1-\xi _e (1+R_k))} \text{(determines } \bar{D}_t^ e)\\ \end{aligned}$$

C    Summary of Standard Open Economy Model

For the small open economy as $\nu \rightarrow 0$ and $\text{ w }_C^*\rightarrow 1$, from (5) we have that $\frac{1-\nu }{\nu } (1-\text{ w }_C^*) \rightarrow 1-\omega _C^*$. Similarly, $\frac{1-\nu }{\nu } (1-\text{ w }_I^*) \rightarrow 1-\omega _I^*$.

$$\begin{aligned} \Lambda _{C,t}:\quad \frac{1}{1+R_{n,t}}= & {} \beta E_t \left[ \frac{\Lambda _{C,t+1}}{\Lambda _{C,t}\Pi _{t+1}} \right] \end{aligned}$$

(C.39)

$$\begin{aligned} \frac{W_t}{P_{C,t}}= & {} \frac{\Lambda _{L,t}}{\Lambda _{C,t}}= - \frac{\Lambda _{h,t}}{\Lambda _{C,t}}\end{aligned}$$

(C.40)

$$\begin{aligned} C_{2,t}: \quad \Lambda _{C,t}= & {} (1-\varrho )C_{2,t}^{(1-\varrho )(1-\sigma )-1} (1-h_t)^{\varrho (1-\sigma )}\end{aligned}$$

(C.41)

$$\begin{aligned} \lambda _{h,t}= & {} -C_{2,t}^{(1-\varrho )(1-\sigma )} \varrho (1-h_t)^{\varrho (1-\sigma )-1} \end{aligned}$$

(C.42)

$$\begin{aligned} C_{1,t}= & {} \frac{W_{t}h_{t}}{P_{C,t}}\end{aligned}$$

(C.43)

$$\begin{aligned} C_{t}= & {} \lambda C_{1,t}+(1-\lambda )C_{2,t} \end{aligned}$$

(C.44)

$$\begin{aligned} \left( \frac{P_{F,t}}{P_{C,t}}\right) :\quad 1= & {} \left[ \text{ w }_C \left( \frac{P_{H,t}}{P_{C,t}}\right) ^{1-\mu _C }+(1-\text{ w }_C)\left( \frac{P_{F,t}}{P_{C,t}}\right) ^{1-\mu _C }\right] ^\frac{1}{1-\mu _C}\end{aligned}$$

(C.45)

$$\begin{aligned} \frac{P_{H,t}}{P_{C,t}}= & {} \frac{1}{[\text{ w }_C+(1-\text{ w }_C)\mathcal {T}_t^{1-\mu _C}]^\frac{1}{1-\mu _C}}\end{aligned}$$

(C.46)

$$\begin{aligned} \text{ where } \mathcal {T}_t\equiv & {} \frac{P_{F,t}}{P_{H,t}}\nonumber \\ C_{F,t}= & {} \text{ w }_C \left( \frac{P_{H,t}}{P_{C,t}}\right) ^{-\mu _C} C_{t}\end{aligned}$$

(C.47)

$$\begin{aligned} C_{F,t}= & {} (1-\text{ w }_C)\left( \frac{P_{F,t}}{P_{C,t}}\right) ^{-\mu _C} C_{t}\end{aligned}$$

(C.48)

$$\begin{aligned} {C_{H,t}}^*= & {} (1-\omega _C^*) \left( \frac{P_{H,t}}{P_{C,t} {RER}_{C,t}}\right) ^{-\mu _C^*} {C_{t}}^*\end{aligned}$$

(C.49)

$$\begin{aligned} H_t: \quad H_t-\xi _H \beta E_t [\Pi _{H,t+1}^{\zeta -1} H_{t+1}]= & {} Y_t \Lambda _{C,t}\end{aligned}$$

(C.50)

$$\begin{aligned} J_t: \quad J_t-\xi _H \beta E_t [\Pi _{H,t+1}^{\zeta }J_{t+1}]= & {} \frac{1}{1-\frac{1}{\zeta }}MS_t Y_t \Lambda _{C,t} {MC}_t \end{aligned}$$

(C.51)

$$\begin{aligned} \Pi _{H,t}: \quad 1= & {} \xi _H \Pi _{H,t} ^{\zeta -1}+(1-\xi _H)\left( \frac{J_t}{H_t}\right) ^{1-\zeta } \end{aligned}$$

(C.52)

$$\begin{aligned} {MC}_t= & {} \frac{P_{H,t}^W}{P_{H,t}}=\frac{P_{H,t}^W/P_{C,t}}{P_{H,t}/P_{C,t}}=\frac{\frac{W_t}{P_{C,t}}h_t}{\alpha Y_t\frac{P_{H,t}}{P_{C,t}}}\end{aligned}$$

(C.53)

$$\begin{aligned} h_t: \quad Y_{t}^W= & {} (A_{t} h_t)^\alpha K_{t}^{1-\alpha } \end{aligned}$$

(C.54)

$$\begin{aligned} Y_t= & {} (1-c) Y_t^W\end{aligned}$$

(C.55)

$$\begin{aligned} \frac{P_{H,t}^W}{P_{C,t}}:\quad \frac{P_{H,t}^W}{P_{C,t}}= & {} {MC}_t\frac{P_{H,t}}{P_{C,t}}\end{aligned}$$

(C.56)

$$\begin{aligned} Q_t: E_t \left[ 1+R_{t+1}\right]= & {} \quad \frac{E_t\left[ \frac{P_{H,t+1}^W}{P_{t+1}}(1-\alpha ) \frac{Y_{t+1}}{K_{t+1}}+(1-\delta ) Q_{t+1}\right] }{Q_{t}} \end{aligned}$$

(C.57)

$$\begin{aligned} R_t: \quad 1+R_{t}= & {} \frac{1+R_{n,t-1}}{1+\Pi _{t}}\end{aligned}$$

(C.58)

$$\begin{aligned} K_{t+1}= & {} (1-\delta ) K_t+(1-S(X_t))I_t\nonumber \\\,\,\,\, S',\, S''\ge & {} 0\,;\,\,S(1+g)=S'(1+g)=0\end{aligned}$$

(C.59)

$$\begin{aligned} X_t= & {} \frac{I_t}{I_{t-1}}\end{aligned}$$

(C.60)

$$\begin{aligned} S(X_t)= & {} \frac{\phi _I}{2}(X_t-(1+g))^2\end{aligned}$$

(C.61)

$$\begin{aligned} I_t:\quad \frac{P_{I,t}}{P_{C,t}}= & {} Q_t (1-S(X_t)-X_t S'(X_t))+E_t \left[ \frac{Q_{t+1} S'(X_{t+1})}{(1+R_{t+1})}\, \frac{I_{t+1}^2}{I_t^2} \right] \qquad \qquad \end{aligned}$$

(C.62)

$$\begin{aligned} I_{H,t}= & {} \text{ w }_I \left( \frac{P_{H,t}/P_{C,t}}{P_{I,t}/P_{C,t}}\right) ^{-\mu _I} I_t\end{aligned}$$

(C.63)

$$\begin{aligned} I_{F,t}= & {} (1-\text{ w }_I) \left( \frac{P_{F,t}/P_{C,t}}{P_{I,t}/P_{C,t}}\right) ^{-\mu _I} I_t \end{aligned}$$

(C.64)

$$\begin{aligned} I^*_{H,t}= & {} (1-\omega _I^*) \left( \frac{P_{H,t}/P_{C,t}}{P_{I,t}/P_{C,t} {RER}_{I,t}}\right) ^{-\rho ^*_I} I^*_t\end{aligned}$$

(C.65)

$$\begin{aligned} \frac{P_{I,t}}{P_{C,t}}= & {} \left[ \text{ w }_I \left( \frac{P_{H,t}}{P_{C,t}}\right) ^{1-\mu _I}+(1-\text{ w }_I) \left( \frac{P_{F,t}}{P_{C,t}}\right) ^{1-\mu _I}\right] ^{\frac{1}{1-\mu _I}}\end{aligned}$$

(C.66)

$$\begin{aligned} Y_t: \quad Y_t= & {} C_{H,t}+I_{H,t}+C_{H,t}^*+I_{H,t}^*+ G_t\end{aligned}$$

(C.67)

$$\begin{aligned} \frac{S_t}{S_{t-1}}= & {} \frac{{RER}_{C,t} \,\Pi _t}{\,\,\,{RER}_{C,t-1}\, \Pi _{t}^*}\end{aligned}$$

(C.68)

$$\begin{aligned} \Pi _{F,t}: \quad \frac{\mathcal {T}_t}{\mathcal {T}_{t-1}}= & {} \frac{\Pi _{F,t}}{\Pi _{H,t}}\end{aligned}$$

(C.69)

$$\begin{aligned} \mathcal {T}_t: \quad {RER}_{C,t}= & {} \frac{1}{\left[ 1-\text{ w }_C+\text{ w }_C \mathcal {T}_t^{\mu _C-1}\right] ^\frac{1}{1-\mu _C} }\end{aligned}$$

(C.70)

$$\begin{aligned} {RER}_{I,t}= & {} \frac{1}{\left[ 1-\text{ w }_I+\text{ w }_I \mathcal {T}_t^{\mu _I-1}\right] ^\frac{1}{1-\mu _I} }\end{aligned}$$

(C.71)

$$\begin{aligned} \Pi _t= & {} [\text{ w }_C (\Pi _{H,t})^{1-\mu _C}+(1-\text{ w }_C) (\Pi _{F,t})^{1-\mu _C}]^\frac{1}{1-\mu _C}\end{aligned}$$

(C.72)

$$\begin{aligned} \log (1+R_{n,t})/(1+R_n)= & {} \rho _r \log (1+R_{n,t-1})/(1+R_n)+(1-\rho _r)(\theta _\pi E_t[\log \Pi _{t+1}]/\Pi \nonumber \\&+\,\theta _s \log S_t/S)+\epsilon _{r,t+1}\end{aligned}$$

(C.73)

$$\begin{aligned} {RER}_t^r= & {} \frac{\Lambda _{C,t}^*}{\Lambda _{C,t}}\end{aligned}$$

(C.74)

$$\begin{aligned} 1+R^*_t= & {} \frac{1+R^*_{n,t-1}}{1+\Pi ^*_{t}}\end{aligned}$$

(C.75)

$$\begin{aligned} \frac{1}{(1+R_{n,t}^*) \phi (\frac{S_t B_{F,t}^*}{P_{H,t} Y_t})} S_t B_{F,t}^*= & {} S_{t} B_{F,t-1}^*+TB_t\end{aligned}$$

(C.76)

$$\begin{aligned} \phi (\frac{S_t B_{F,t}^*}{P_{H,t} Y_t})= & {} \exp \left( \frac{\phi _B S_t B_{F,t}^*}{P_{H,t} Y_t}\right) \,;\,\, \phi _{B}<0\end{aligned}$$

(C.77)

$$\begin{aligned} TB_t= & {} P_{H,t}Y_t-P_{C,t} C_t -P_{I,t} I_t- P_{H,t} G_t \end{aligned}$$

(C.78)

Then the real exchange rate is given by

$$\begin{aligned} {RER}_{C,t}= & {} {RER}_t^d {RER}_t^r\\ {RER}_t^d: 0= & {} \quad E_t \left[ \frac{\Lambda _{C,t+1}}{\Lambda _{C,t}}\frac{{RER}_{t+1}^r}{{RER}_t^r} \frac{1}{\Pi _{t+1}^*} \left( \frac{1}{\phi (\frac{S_t B_{F,t}^*}{P_{H,t} Y_t})\exp (\epsilon _{UIP,t+1})}-\frac{{RER}_{t+1}^d}{{RER}_t^d}\right) \right] \nonumber \end{aligned}$$

(C.79)

Shocks:

$$\begin{aligned} \log \frac{A_{t+1}}{A}= & {} \rho _a \log \frac{A_{t}}{A}+\epsilon _{a,t+1} \end{aligned}$$

(C.80)

$$\begin{aligned} \log \frac{G_{t+1}}{G}= & {} \rho _g \log \frac{G_{t}}{G}+\epsilon _{g,t+1} \end{aligned}$$

(C.81)

$$\begin{aligned} \log \frac{{MS}_{t+1}}{{MS}}= & {} \rho _{{ms}} \log \frac{{MS}_{t}}{{MS}}+\epsilon _{{ms},t+1} \end{aligned}$$

(C.82)

$$\begin{aligned} \log \frac{{UIP}_{t+1}}{{UIP}}= & {} \rho _{{UIP}} \log \frac{{UIP}_{t}}{{UIP}}+\epsilon _{{uip},t+1} \end{aligned}$$

(C.83)

If the ROW is not modeled explicitly we close the model with exogenous AR(1) shocks

$$\begin{aligned} \log (1+R_{n,t}^*)/(1+R_n^*)= & {} \rho _r^* \log (1+R_{n,t-1}^*)/(1+R_n^*)+\epsilon _{r,t+1}^* \end{aligned}$$

(C.84)

$$\begin{aligned} \log \frac{\Pi ^*_{t+1}}{\Pi ^*}= & {} \rho _\pi ^* \log \frac{\Pi _{t}^*}{\Pi ^*}+\epsilon _{\pi ,t+1}^* \end{aligned}$$

(C.85)

$$\begin{aligned} \log \frac{C_{t+1}^*}{C^*}= & {} \rho _c^* \log \frac{C_{t}^*}{C^*}+\epsilon _{c,t+1}^* \end{aligned}$$

(C.86)

$$\begin{aligned} \log \frac{I_{t+1}^*}{I^*}= & {} \rho _i^* \log \frac{I_{t}^*}{I^*}+\epsilon _{i,t+1}^* \end{aligned}$$

(C.87)

$$\begin{aligned} \log \frac{\Lambda _{t+1}^*}{\Lambda ^*}= & {} \rho _\Lambda ^* \log \frac{\Lambda _{t}^*}{\Lambda ^*}+\epsilon _{\lambda ,t+1}^* \end{aligned}$$

(C.88)

Otherwise $R_{n,t}^*$, $\Pi _t^*$, $C_t^*$ and $I_t^*$ are modeled as before. First, assume zero growth in the steady state: $g=g^*=0$ and nonnegative inflation. Then we have

$$\begin{aligned} R_n : \quad 1+R_n= & {} (1+R_{n}^*)\phi \left( \frac{S B}{P}\right) \end{aligned}$$

(C.89)

$$\begin{aligned} \frac{W}{P}= & {} -\frac{U_{L}}{U_{C}} \end{aligned}$$

(C.90)

$$\begin{aligned} U_{C}= & {} (1-\varrho )C_{2}^{(1-\varrho )(1-\sigma )-1} (1-L)^{\varrho (1-\sigma )} \end{aligned}$$

(C.91)

$$\begin{aligned} U_{L}= & {} -C_{2}^{(1-\varrho )(1-\sigma )} \varrho (1-L)^{\varrho (1-\sigma )-1} \end{aligned}$$

(C.92)

$$\begin{aligned} C_{1}= & {} \frac{WL}{P_C} \end{aligned}$$

(C.93)

$$\begin{aligned} C= & {} \lambda C_{1}+(1-\lambda )C_{2} \end{aligned}$$

(C.94)

$$\begin{aligned} P_F/P_C : \quad 1= & {} \left[ \text{ w }_C \left( \frac{P_{H}}{P_C}\right) ^{1-\mu _C }+(1-\text{ w }_C)\left( \frac{P_{F}}{P_C}\right) ^{1-\mu _C }\right] ^\frac{1}{1-\mu _C} \end{aligned}$$

(C.95)

$$\begin{aligned} \frac{P_{H}}{P_C}= & {} \frac{1}{[\text{ w }_C+(1-\text{ w }_C)\mathcal {T}^{1-\mu _C}]^\frac{1}{1-\mu _C}} \end{aligned}$$

(C.96)

$$\begin{aligned} C_{H}= & {} \text{ w }_C \left( \frac{P_{H}}{P_C}\right) ^{-\mu _C} C \end{aligned}$$

(C.97)

$$\begin{aligned} C_{F}= & {} (1-\text{ w }_C)\left( \frac{P_{F}}{P_C}\right) ^{-\mu _C} C \end{aligned}$$

(C.98)

$$\begin{aligned} {C_{H}}^*= & {} (1-\omega _C^*) \left( \frac{P_{H}}{P_C {RER}_C}\right) ^{-\mu _C^*} {C}^* \end{aligned}$$

(C.99)

$$\begin{aligned} H(1-\xi _H \beta )= & {} YU_{C} \end{aligned}$$

(C.100)

$$\begin{aligned} J (1-\xi _H \beta )= & {} \frac{1}{1-\frac{1}{\zeta }}Y U_{C} {MC} \end{aligned}$$

(C.101)

$$\begin{aligned} {MC} : \quad H= & {} J \end{aligned}$$

(C.102)

$$\begin{aligned} {MC}= & {} 1-\frac{1}{\zeta }=\frac{C_2}{\alpha Y\frac{P_H}{P_C}} \end{aligned}$$

(C.103)

$$\begin{aligned} Y= & {} (1-c)(A L)^\alpha K^{1-\alpha } \end{aligned}$$

(C.104)

$$\begin{aligned} \frac{P_{H}^W}{P_C}= & {} {MC}\frac{P_{H}}{P_C} \end{aligned}$$

(C.105)

$$\begin{aligned} K= & {} \frac{(1-\alpha ){MC}\frac{P_{H}}{P_C}Y}{(R+\delta )Q} \end{aligned}$$

(C.106)

$$\begin{aligned} 1+R= & {} \frac{1+R_n}{\Pi } \end{aligned}$$

(C.107)

$$\begin{aligned} I= & {} (g+\delta )K \end{aligned}$$

(C.108)

$$\begin{aligned} X= & {} 1 \end{aligned}$$

(C.109)

$$\begin{aligned} S(X)= & {} S'(X)=0 \end{aligned}$$

(C.110)

$$\begin{aligned} Q= & {} \frac{P_I}{P_C} \end{aligned}$$

(C.111)

$$\begin{aligned} I_{H}= & {} \text{ w }_I \left( \frac{P_{H}/P_C}{P_{I}/P_C}\right) ^{-\mu _I} I \end{aligned}$$

(C.112)

$$\begin{aligned} I_{F}= & {} (1-\text{ w }_I) \left( \frac{P_{F}/P_C}{P_{I}/P_C}\right) ^{-\mu _I} I \end{aligned}$$

(C.113)

$$\begin{aligned} I^*_{H}= & {} (1-\omega _I^*)\left( \frac{P_{H}}{P {RER}}\right) ^{-\mu ^*_I} I^* \end{aligned}$$

(C.114)

$$\begin{aligned} \frac{P_I}{P_C}= & {} \left[ \text{ w }_I \left( \frac{P_H}{P_C}\right) ^{1-\mu _I }+(1-\text{ w }_I) \left( \frac{P_F}{P_C}\right) ^{1-\mu _I} \right] ^{\frac{1}{1-\mu _I}} \end{aligned}$$

(C.115)

$$\begin{aligned} Y= & {} C_{H}+I_{H}+ {EX}_C + {EX}_I+ G_t \end{aligned}$$

(C.116)

$$\begin{aligned} EX_{C}= & {} C_{H,t}^*=(1-\omega _{C,t}^*)\left( \frac{P_{H}}{P_{C} {RER}_{C}}\right) ^{-\mu _C^*} C^* \end{aligned}$$

(C.117)

$$\begin{aligned} EX_{I}= & {} I_{H,t}^*=(1-\omega _{I,t}^*)\left( \frac{P_{H}}{P_{I} {RER}_{I}}\right) ^{-\mu _I^*} I^* \end{aligned}$$

(C.118)

$$\begin{aligned} {RER}_C= & {} \frac{1}{\left[ 1-\text{ w }_+\text{ w }_C \mathcal {T}^{\mu _C-1}\right] ^\frac{1}{1-\mu _C} } \end{aligned}$$

(C.119)

$$\begin{aligned} {RER}_I= & {} \frac{1}{\left[ 1-\text{ w }_I+\text{ w }_I \mathcal {T}^{\mu _I-1}\right] ^\frac{1}{1-\mu _I} }\end{aligned}$$

(C.120)

$$\begin{aligned} R^*_n : \quad 1= & {} \beta (1+R^*_{n})\end{aligned}$$

(C.121)

$$\begin{aligned} 1+R^*= & {} \frac{1+R_n^*}{\Pi ^*} \end{aligned}$$

(C.122)

The model is complete if we pin down the steady state of the foreign assets or equivalently the trade balance (TB). In other words, there is a unique model associated with any choice of the long-run assets of our SOE.¹⁷ The trade balance is

$$\begin{aligned} TB=P_{H}Y- P_C C -P_{I} I- P_{H} G=\underbrace{P_H {EX}_C-(P_C C-P_H C_H)}_{\text{ Net } \text{ Exports } \text{ of } \text{ C-goods }} +\underbrace{P_H {EX}_I-(P_I I-P_H I_H)}_{\text{ Net } \text{ Exports } \text{ of } \text{ I-goods }} \end{aligned}$$

(C.123)

using (C.116), for some choice of TB, say zero.

The problem now is that we need to force the nonlinear model to this steady state even when the latter may not be completely accurate. A way of doing this is to add a term $\theta _{tb} \log (TB_t/TB)$ to the Taylor rule with a very small $\theta _{tb}>0$ so that when there is a trade surplus the rule makes the nominal exchange rate appreciate slightly.

Finally, we calibrate $\omega _C$ and $\omega _I$ using trade data. From (C.123) we have

$$\begin{aligned} cs_{imp}\equiv & {} \frac{\text{ C-imports }}{\text{ GDP }}=\frac{C_F}{Y}=c_y(1-\text{ w }_C)\left( \frac{P_{F}}{P_C}\right) ^{-\mu _C}\end{aligned}$$

(C.124)

$$\begin{aligned} is_{imp}\equiv & {} \frac{\text{ I-imports }}{\text{ GDP }}=\frac{I_F}{Y}=i_y(1-\text{ w }_I)\left( \frac{P_{F}}{P_I}\right) ^{-\mu _I}\end{aligned}$$

(C.125)

$$\begin{aligned} cs_{exp}\equiv & {} \frac{\text{ C-exports }}{\text{ GDP }}=(1-\omega _C^*)\left( \frac{P_{H}}{P_C {RER}_C}\right) ^{-\mu _C^*} c_y^* \frac{Y^*}{Y}=\frac{C_H^*}{Y}\end{aligned}$$

(C.126)

$$\begin{aligned} is_{exp}\equiv & {} \frac{\text{ I-exports }}{\text{ GDP }}=(1-\omega _I^*)\left( \frac{P_{H}}{P_I {RER}_I}\right) ^{-\mu _I^*} i_y^* \frac{Y^*}{Y}=\frac{I_H^*}{Y} \end{aligned}$$

(C.127)

Hence using data for shares $cs_{imp}$, $is_{imp}$, $cs_{exp}$ and $is_{exp}$, we can calibrate $\omega _C$ and $\omega _I$. Use data for India: $cs_{imp}=0.10$, $is_{imp}=0.15$, $cs_{exp}=0.23$, and $is_{exp}=0.02$ for $TB=0$. With balanced steady-state growth, the balanced growth steady-state path of the model economy with or without investment costs is given by $Q=1$ and

$$\begin{aligned} \frac{\bar{\Lambda }_{C,t+1}}{\bar{\Lambda }_{C,t}}\equiv 1+ g_{\Lambda _C}=\left[ \frac{\bar{C}_{t+1}}{\bar{C}_t} \right] ^{(1-\varrho )(1-\sigma )-1)} =(1+g)^{((1-\varrho )(1-\sigma )-1)} \end{aligned}$$

(C.128)

Thus from (C.39)

$$\begin{aligned} 1+R=\frac{(1+g)^{1+(\sigma -1)(1-\varrho )}}{\beta } \end{aligned}$$

(C.129)

Similarly for the foreign bloc

$$\begin{aligned} 1+R^*=\frac{(1+g^*)^{1+(\sigma ^*-1)(1-\varrho ^*)}}{\beta ^*} \end{aligned}$$

(C.130)

It is then possible to have different preferences, inflation and growth rates provided

$$\begin{aligned} \frac{1+R_n}{1+R_n^*}= \phi \left( \frac{S B}{P}\right) = \frac{\Pi (1+R)}{\Pi ^*(1+R^*)} =\frac{\Pi \beta ^*}{\Pi ^* \beta }\frac{(1+g)^{1+(\sigma -1)(1-\varrho )}}{(1+g^*)^{1+(\sigma ^*-1)(1-\varrho ^*)}} \end{aligned}$$

(C.131)

which pins down the assets in the steady state.

D    Summary of Open Economy Model with Financial Frictions : Complete Exchange Rate Pass-Through

Note that there are already two financial frictions in the previous model: Ricardian households pay a risk premium for their international borrowing there are liquidity-constrained households. To complete the model we add a financial accelerator consisting of

$$\begin{aligned} E_t [1+R_{k,t+1}]= & {} E_t\left[ \Theta _{t+1} \left( \varphi E_t \left[ (1+R_{t+1}) \right] + (1-\varphi ) E_t \left( (1+R_{t+1}^*) \frac{{RER}_{C,t+1}}{\,\,{RER}_{C,t}} \right) \right) \right] \end{aligned}$$

(D.132)

$$\begin{aligned} \Theta _t= & {} k\left( \frac{N_t}{Q_{t-1} K_t}\right) ^{-\chi }\end{aligned}$$

(D.133)

$$\begin{aligned} N_{t+1}= & {} \xi _e V_t +(1-\xi _e) D_t^{e}\end{aligned}$$

(D.134)

$$\begin{aligned} V_t= & {} (1+ R_{k,t}) Q_{t-1} K_{t}-\Theta _t \left[ \varphi (1+R_{t}) + (1-\varphi ) (1+R_{t}^*) \frac{{RER}_{C,t}}{\,\,{RER}_{C,t-1}}\right] (Q_{t-1}K_{t}-N_t)\nonumber \\ \end{aligned}$$

(D.135)