Accounting for Macrofinancial Fluctuations and Turbulence

This paper investigates the sources of macrofinancial fluctuations and turbulence within the framework of an approximate linear dynamic stochastic general equilibrium model of the world economy, augmented with structural shocks exhibiting potentially asymmetric generalized autoregressive conditional heteroskedasticity. Very strong evidence of asymmetric autoregressive conditional heteroskedasticity is found, providing a basis for jointly decomposing the levels and volatilities of key macrofinancial variables into time varying contributions from sets of shocks. Risk premia shocks are estimated to contribute disproportionately to cyclical output fluctuations and turbulence during swings in financial conditions, across the fifteen largest national economies in the world.

Abstract

This paper investigates the sources of macrofinancial fluctuations and turbulence within the framework of an approximate linear dynamic stochastic general equilibrium model of the world economy, augmented with structural shocks exhibiting potentially asymmetric generalized autoregressive conditional heteroskedasticity. Very strong evidence of asymmetric autoregressive conditional heteroskedasticity is found, providing a basis for jointly decomposing the levels and volatilities of key macrofinancial variables into time varying contributions from sets of shocks. Risk premia shocks are estimated to contribute disproportionately to cyclical output fluctuations and turbulence during swings in financial conditions, across the fifteen largest national economies in the world.

I. Introduction

In recent decades, the world economy has experienced extended periods of cyclical expansion and tranquility, occasionally disrupted by bouts of cyclical contraction and turbulence. Indeed, the Global Financial Crisis abruptly ended the extended period of cyclical expansion and tranquility known as the Great Moderation, generating cyclical output contractions and financial market turbulence across major advanced and emerging market economies. The Euro Area Sovereign Debt Crisis generated further cyclical output contractions and financial market turbulence in some major advanced economies, while the Taper Tantrum precipitated cyclical contractions and turbulence in some major emerging market economies.

These occasional bouts of cyclical output contraction and financial market turbulence may have a common cause. In a recent paper, Adrian, Boyarchenko and Giannone (2017) find that a tightening of financial conditions is associated with a reduction in the conditional mean and an increase in the conditional variance of output growth in the United States. They argue that these adverse effects on the conditional distribution of output growth are generated by financial amplification mechanisms.

To investigate the sources of macrofinancial fluctuations and turbulence, this paper augments an approximate linear dynamic stochastic general equilibrium (DSGE) model of the world economy with structural shocks exhibiting potentially asymmetric generalized autoregressive conditional heteroskedasticity (GARCH) effects. A refinement of the DSGE model documented in Vitek (2018), this model features a range of nominal and real rigidities, extensive macrofinancial linkages with both bank and capital market based financial intermediation, and diverse spillover transmission channels. Very strong evidence of asymmetric autoregressive conditional heteroskedasticity (ARCH) effects is found, providing a basis for jointly decomposing the levels and volatilities of output and financial conditions into time varying contributions from sets of shocks. Consistent with the finding of Adrian, Boyarchenko and Giannone (2017), risk premia shocks are estimated to contribute disproportionately to cyclical output fluctuations and turbulence during occasional abrupt swings in financial conditions, across the fifteen largest national economies in the world. This phenomenon struck all of the economies most affected by the Global Financial Crisis, the Euro Area Sovereign Debt Crisis, and the Taper Tantrum.

Accounting for the sources of macrofinancial fluctuations or turbulence within a DSGE framework is common. For example, Smets and Wouters (2007) decompose output growth fluctuations in the United States into contributions from various structural shocks using an approximate linear DSGE model. In another influential paper, Justiniano and Primiceri (2008) decompose output growth volatility in the United States into contributions from various structural shocks exhibiting symmetric stochastic volatility (SV) effects using an approximate linear DSGE model. Unlike these and related papers, this paper jointly analyzes the sources of macrofinancial fluctuations and turbulence in the world economy within a DSGE framework. To our knowledge, it is the first to add ARCH effects to a DSGE model. These are simpler to interpret than SV effects, as the conditional variances of the structural shocks are driven by the same innovations as their conditional means.

The organization of this paper is as follows. The next section develops the theoretical framework, while the following section describes the corresponding empirical framework. Estimation of this empirical framework is the subject of section four. Inference on the sources of macrofinancial fluctuations and turbulence is conducted in section five. Finally, section six offers conclusions and recommendations for further research.

II. The Theoretical Framework

Consider a finite set of structurally isomorphic national economies indexed by i ∈ {1,…,N} which constitutes the world economy. Each of these economies consists of households, developers, firms, banks, and a government. The government in turn consists of a monetary authority, a fiscal authority, and a macroprudential authority. Households, developers, firms and banks optimize intertemporally, interacting with governments in an uncertain environment to determine equilibrium prices and quantities under rational expectations in globally integrated output and financial markets. Economy i * issues the quotation currency for transactions in the foreign exchange market.

A. The Household Sector

There exists a continuum of households indexed by h ∈ [0,1]. Households are differentiated according to whether they are credit constrained, and according to how they save if they are credit unconstrained, but are otherwise identical. Credit unconstrained households of type Z = B and measure ϕB have access to domestic banks where they accumulate deposits, and to domestic property markets where they trade real estate, where 0 < ϕB < 1. In contrast, credit unconstrained households of type Z = A and measure ϕA have access to domestic and foreign capital markets where they trade financial assets, where 0 < ϕA < 1. Finally, credit constrained households of type Z = C and measure ϕC do not have access to banks or capital markets, where 0 < ϕC < 1 and ϕB + ϕA + ϕC = 1. All households are originally endowed with one share of each domestic developer, firm and bank.

In a reinterpretation of the labor market in the model of nominal wage rigidity proposed by Erceg, Henderson and Levin (2000) to incorporate involuntary unemployment along the lines of Galí (2011), each household consists of a continuum of members represented by the unit square and indexed by (f,g) ∈ [0,1] × [0,1]. There is full risk sharing among household members, who supply indivisible differentiated intermediate labor services indexed by f ∈ [0,1], incurring disutility from work determined by g ∈ [0,1] if they are employed and zero otherwise. Trade specific intermediate labor services supplied by bank intermediated, capital market intermediated, and credit constrained households are perfect substitutes.

Consumption and Saving

The representative infinitely lived household has preferences defined over consumption Ch,i,s, housing Hh,i,s, labor supply {Lh,f,i,s}f=01, real property balances Ah,i,s+1B,H/Pi,sC, and real portfolio balances Ah,i,s+1A,H/Pi,sC represented by intertemporal utility function

Uh,i,t=Ets=tβstu(Ch,i,s,Hh,i,s,{Lh,f,i,s}f=01,Ah,i,s+1B,HPi,sC,Ah,i,s+1A,HPi,sC),(1)

where Et denotes the expectations operator conditional on information available in period t, and 0 < β < 1. The intratemporal utility function is additively separable and represents external habit formation preferences in consumption and labor supply,

u(Ch,i,s,Hh,i,s{Lh,f,i,s}f=01,Ah,i,s+1B,HPi,sC,Ah,i,s+1A,HPi,sC)=νi,sC[111/σ(Ch,i,sαCCi,s1ZϕZ)11/σ+υi,sH,H11/ς(Hh,i,s)11/ς(2)υi,sL,H01αLLf,i,s1ZϕZLh,f,i,s(gαLLf,i,s1ZϕZ)1/ηdgdf+υi,sB,H11/μ(Ah,i,s+1B,HPi,sC)11/μ+υi,sA,H11/μ(Ah,i,s+1A,HPi,sC)11/μ],

where 0 ≤ αC < 1 and 0 ≤ αL < 1. Endogenous preference shifters υi,sH,H,υi,sL,H,υi,sB,H and υi,sA,H depend on aggregate consumption or employment according to intratemporal subutility functions

υi,sH,H=νiH,H(Ci,sZϕZαCCi,s1ZϕZ)1/σ(Ci,s)1/ς,(3)
υi,sL,H=A˜i,s(Ci,sZϕZαCCi,s1ZϕZ)1/σ(Li,sαLLi,s1(Li,s/νi,sN)t)1/η,(4)
υi,sB,H=νiB,H(Ci,sZϕZαCCi,s1ZϕZ)1/σ(Ci,s)1/μ,(5)
υi,sA,H=νiA,H(Ci,sZϕZαCCi,s1ZϕZ)1/σ(Ci,s)1/μ,(6)

where ι > 0. The intratemporal utility function is strictly increasing with respect to consumption if and only if serially correlated consumption demand shock νi,sC satisfies νi,sC>0. Given this parameter restriction, this intratemporal utility function is strictly increasing with respect to housing if and only if νiH,H>0, is strictly decreasing with respect to labor supply if and only if serially correlated labor supply shock νi,sN satisfies νi,sN>0, is strictly increasing with respect to real property balances if and only if νiB,H>0, and is strictly increasing with respect to real portfolio balances if and only if νiA,H>0. Given these parameter restrictions, this intratemporal utility function is strictly concave if σ > 0, ς > 0, η > 0 and μ > 0. In steady state equilibrium, νiB,H equates the marginal rate of substitution between real property balances and consumption to one, while νiA,H equates the marginal rate of substitution between real portfolio balances and consumption to one.

The representative household enters period s in possession of previously accumulated property balances Ah,i,sB,H which yield return ih,i,sAB,H, and portfolio balances Ah,i,sA,H which yield return ih,i,sAA,H. Property balances are distributed across the values of bank deposits Bh,i,sD,H which bear interest at deposit rate ii,s1D, and a real estate portfolio Sh,i,sH,H which yields return ih,i,sSH,H. It follows that (1+ih,i,sAB,H)Ah,i,sB,H=(1+ii,s1D)Bh,i,sD,H+(1+ih,i,sSH,H)Sh,i,sH,H. The value of this real estate portfolio is in turn distributed across the values of developer specific shares {Vi,e,sHSh,i,e,sH,H}e=01, where Vi,e,sH denotes the price per share. It follows that (1+ih,i,sSH,H)Sh,i,sH,H=01(i,e,sH+Vi,e,sH)Sh,i,e,sH,Hde, where i,e,sH denotes the dividend payment per share. Portfolio balances are distributed across the values of internationally diversified short term bond Bh,i,sS,H, long term bond Bh,i,sL,H and stock Sh,i,sF,H portfolios which yield returns ih,i,sBS,H,ih,i,sBL,H, and ih,i,sSF,H, respectively. It follows that (1+ih,i,sAA,H)Ah,i,sA,H=(1+ih,i,sBS,H)Bh,i,sS,H+(1+ih,i,sBL,H)Bh,i,sL,H+(1+ih,i,sSF,H)Sh,i,sF,H. The values of these internationally diversified short term bond, long term bond and stock portfolios are in turn distributed across the domestic currency denominated values of economy specific short term bond {i,j,sBh,i,j,sS,H}j=1N, long term bond {i,j,sBh,i,j,sL,H}j=1N and stock {i,j,sSh,i,j,sF,H}j=1N portfolios, where nominal bilateral exchange rate i,j,s measures the price of foreign currency in terms of domestic currency. It follows that (1+ih,i,sBS,H)Bh,i.sS,H=j=1Ni,j,s(1+ij,s1S)Bh,i.j,sS,H where ij,s1S denotes the economy specific yield to maturity on short term bonds, (1+ih,i,sBL,H)Bh,i,sL,H=j=1Ni,j,s(1+ih,i,j,sBL,H)Bh,i,j,sL,H where ih,i,j,sBL,H denotes the economy specific return on long term bonds, and (1+ih,i,sSF,H)Sh,i,sF,H=j=1Ni,j,s(1+ih,i,j,sSF,H)Sh,i,j,sF,H where ih,i,j,sSF,H denotes the economy specific return on stocks. The local currency denominated values of economy specific long term bond portfolios {Bh,i,j,sL,H}j=1N are in turn distributed across the values of economy and vintage specific long term bonds {{Vj,k,sBBh,i,j,k,sL,H}k=1s1}j=1N, where Vj,k,sB denotes the local currency denominated price per long term bond, with Vj,k,kB=1. It follows that (1+ih,i,j,sBL,H)Bh,i,j,sL,H=k=1s1(Πj,k,sB+Vj,k,sB)Bh,i,j,k,sL,H, where Πj,k,sB=(1+ij,kLωB)(ωB)skVj,k,kB denotes the local currency denominated coupon payment per long term bond, and ij,kL denotes the economy and vintage specific yield to maturity on long term bonds at issuance. In parallel, the local currency denominated values of economy specific stock portfolios {Sh,i,j,sF,H}j=1N are distributed across the values of economy, industry and firm specific shares {{{Vj,k,l,sSSh,i,j,k,l,sF,H}l=01}k=1M}j=1N, where Vj,k,l,sS, denotes the local currency denominated price per share. It follows that (1+ih,i,j,sSF,H)Sh,i,j,sF,H=k=1M01(Πj,k,l,sS+Vj,k,l,sS)Sh,i,j,k,l,sF,Hdl, where Πj,k,l,sS denotes the local currency denominated dividend payment per share. During period s, the representative household receives profit income from banks Πi,sC, and supplies differentiated intermediate labor services {Lh,f,i,s}f=01, earning labor income at trade specific nominal wages {Wf,i,s}f=01. The government levies a tax on household labor income at rate τi,sL, and remits household type specific lump sum transfer payments Th,i,sZ. These sources of wealth are summed in household dynamic budget constraint:

Ah,i,s+1B,H+Ah,i,s+1A,H=(1+ih,i,sAB,H)Ah,i,sB,H+(1+ih,i,sAA,H)Ah,i,sA,H+i,sC+(1τi,sL)01Wf,i,sLh,f,i,sdf+Th,i,sZPi,sCCh,i,sιi,sHHh,i,s.(7)

According to this dynamic budget constraint, at the end of period s, the representative household holds property balances Ah,i,s+1B,H and portfolio balances Ah,i,s+1A,H. Property balances are allocated across the values of bank deposits Bh,i,s+1D,H and the real estate portfolio Sh,i,s+1H,H, that is Ah,i,s+1B,H=Bh,i,s+1D,H+Sh,i,s+1H,H. The value of this real estate portfolio is in turn allocated across the values of developer specific shares {Vi,e,sHSh,i,e,s+1H,H}e=01 subject to Sh,i,s+1H,H=01Vi,e,sHSh,i,e,s+1H,Hde. Portfolio balances are allocated across the values of internationally diversified short term bond Bh,i,s+1S,H, long term bond Bh,i,s+1L,H and stock portfolios Sh,i,s+1F,H, that is Ah,i,s+1A,H=Bh,i,s+1S,H+Bh,i,s+1L,H+Sh,i,s+1F,H. The values of these internationally diversified short term bond, long term bond and stock portfolios are in turn allocated across the domestic currency denominated values of economy specific short term bond {i,j,sBh,i,j,s+1S,H}j=1N, long term bond {i,j,sBh,i,j,s+1L,H}j=1N and stock {i,j,sSh,i,j,s+1F,H}j=1N portfolios subject to Bh,i,s+1S,H=j=1Ni,j,sBh,i,j,s+1S,H,Bh,i,s+1L,H=j=1Ni,j,sBh,i,j,s+1L,H and Sh,i,s+1F,H=j=1Ni,j,sSh,i,j,s+1F,H, respectively. The local currency denominated values of economy specific long term bond portfolios {Bh,i,j,s+1L,H}j=1N are in turn allocated across the local currency denominated values of economy and vintage specific long term bonds {{Vj,k,sBBh,i,j,k,s+1L,H}k=1s}j=1N subject to Bh,i,j,s+1L,H=k=1sVj,k,sBBh,i,j,k,s+1L,H. In parallel, the local currency denominated values of economy specific stock portfolios {Sh,i,j,s+1F,H}j=1N are allocated across the local currency denominated values of economy, industry and firm specific shares {{{Vj,k,l,sSSh,i,j,k,l,s+1F,H}l=01}k=1M}j=1N subject to Sh,i,j,s+1F,H=k=1M01Vj,k,l,sSSh,i,j,k,l,s+1F,Hdl. Finally, the representative household purchases final private consumption good Ch,i,s at price Pi,sC, and rents final housing service Hh,i,s at price ιi,sH.

Bank Intermediated Households

The representative bank intermediated household has a capitalist spirit motive for holding real property balances, independent of financing deferred consumption, motivated by Weber (1905). It also has a diversification motive over the allocation of real property balances across alternative assets which are imperfect substitutes, motivated by Tobin (1969). The set of assets under consideration consists of bank deposits and domestically traded real estate. Preferences over the real values of bank deposits Bh,i,s+1D,H/Pi,sC and the real estate portfolio Sh,i,s+1H,H/Pi,sC are represented by constant elasticity of substitution intratemporal subutility function

Ah,i,s+1B,HPi,sC=[(1ϕH)1ψH(Bh,i,s+1D,HPi,sC)ψH1ψH+(ϕH)1ψH(1νi,sHSh,i,s+1H,HPi,sC)ψH1ψH]ψHψH1,(8)

where serially correlated housing risk premium shock νi,sH satisfies νi,sH>0, while 0 < ϕH < 1 and ψH > 0. Preferences over the real values of developer specific shares {Vi,e,sHSh,i,e,s+1H,H/Pi,sC}e=01 are in turn represented by constant elasticity of substitution intratemporal subutility function:

Sh,i,s+1H,HPi,sC=[01(Vi,e,sHSh,i,e,s+1H,HPi,sC)ψH1ψHde]ψHψH1.(9)

In the limit as νiB,H0 there is no capitalist spirit motive for holding real property balances, while in the limit as ψH → ∞ there is no diversification motive over the allocation of real property balances across alternative assets which in this case are perfect substitutes.

In period t, the representative bank intermediated household chooses state contingent sequences for consumption {Ch,i,s}s=t, housing {Hh,i,s}s=t, labor force participation {{Nh,f,i,s}f=01}s=t, property balances {Ah,i,s+1B,H}s=t, bank deposit holdings {Bh,i,s+1D,H}s=t, and real estate holdings {{Sh,i,e,s+1H,H}e=01}s=t to maximize intertemporal utility function (1) subject to dynamic budget constraint (7), the applicable restrictions on financial asset holdings, and terminal nonnegativity constraints Bh,i,T+1D,H0 and Sh,i,e,T+1H,H0 for T → ∞. In equilibrium, abstracting from the capitalist spirit motive for holding real property balances, the solutions to this utility maximization problem satisfy intertemporal optimality condition

EtβuC(h,i,t+1)uC(h,i,t)Pi,tCPi,t+1C(1+ih,i,t+1AB,H)=1,(10)

which equates the expected present value of the gross real property return to one. In addition, these solutions satisfy intratemporal optimality condition

uH(h,i,t)uC(h,i,t)=ιi,tHPi,tC,(11)

which equates the marginal rate of substitution between housing and consumption to the real rental price of housing. Furthermore, these solutions satisfy intratemporal optimality condition

uLf(h,f,i,t)uC(h,i,t)=(1τi,tL)Wf,i,tPi,tC,(12)

which equates the marginal rate of substitution between leisure and consumption for the marginal trade specific labor force participant to the corresponding after tax real wage. Abstracting from risk premium shocks, the expected present value of the gross real property return satisfies intratemporal optimality condition

(1ϕH){1+EtβuC(h,i,t+1)uC(h,i,t)uC(h,i,t)uAB(h,i,t)Pi,tCPi,t+1C[(1+ih,i,t+1AB,H)(1+ii,tD)]}1ψH(13)+ϕH01{1+EtβuC(h,i,t+1)uC(h,i,t)uC(h,i,t)uAB(h,i,t)Pi,tCPi,t+1C[(1+ih,i,t+1AB,H)Πi,e,t+1H+Vi,e,t+1HVi,e,tH]}1ψHde=1,

which relates it to the expected present values of the gross real returns on bank deposits and real estate. Finally, abstracting from the portfolio diversification motive over the allocation of real property balances these solutions satisfy intratemporal optimality condition

EtβuC(h,i,t+1)uC(h,i,t)Pi,tCPi,t+1C[(1+ii,tD)Πi,e,t+1H+Vi,e,t+1HVi,e,tH]=uAB(h,i,t)uC(h,i,t)(11νi,tH),(14)

which equates the expected present values of the gross real risk adjusted returns on bank deposits and real estate. Provided that the intertemporal utility function is bounded and strictly concave, together with other optimality conditions, and transversality conditions derived from necessary complementary slackness conditions associated with the terminal nonnegativity constraints, these optimality conditions are sufficient for the unique utility maximizing state contingent sequence of bank intermediated household allocations.

Capital Market Intermediated Households

The representative capital market intermediated household has a capitalist spirit motive for holding real portfolio balances, independent of financing deferred consumption, motivated by Weber (1905). It also has a diversification motive over the allocation of real portfolio balances across alternative financial assets which are imperfect substitutes, motivated by Tobin (1969). The set of financial assets under consideration consists of internationally traded and local currency denominated short term bonds, long term bonds, and stocks. Short term bonds are discount bonds, while long term bonds are perpetual bonds with coupon payments that decay exponentially at rate ωB where 0 < ωB < 1, following Woodford (2001). Preferences over the real values of internationally diversified short term bond Bh,i,s+1S,H/Pi,sC, long term bond Bh,i,s+1L,H/Pi,sC and stock Sh,i,s+1F,H/Pi,sC portfolios are represented by constant elasticity of substitution intratemporal subutility function

Ah,i,s+1A,HPi,sC=[(ϕMA)1ψA(Bh,i,s+1S,HPi,sC)ψA1ψA+(ϕBA)1ψA(1υi,sBBh,i,s+1L,HPi,sC)ψA1ψA+(ϕSA)1ψA(1υi,sSSh,i,s+1F,HPi,sC)ψA1ψA]ψAψA1,(15)

where internationally and serially correlated duration risk premium shock satisfies υi,sB satisfies υi,sB>0 and internationally and serially correlated equity risk premium shock υi,sS satisfies υi,sS>0, while 0ϕMA1,0ϕBA<1,0ϕSA<1,ϕMA+ϕBA+ϕSA=1 and ψA > 0. Preferences over the real values of economy specific short term bond {i,j,sBh,i,j,s+1S,H/Pi,sC}j=1N, long term bond {i,j,sBh,i,j,s+1L,H/Pi,sC}j=1N and stock {i,j,sSh,i,j,s+1F,H/Pi,sC}j=1N portfolios are in turn represented by constant elasticity of substitution intratemporal subutility functions

Bh,i,s+1S,HPi,sC=[j=1N(ϕi,jB)1ψA(1νi,si,j,sBh,i,j,s+1S,HPi,sC)ψA1ψA]ψAψA1,(16)
Bh,i,s+1L,HPi,sC=[j=1N(ϕi,jB)1ψA(1νi,si,j,sBh,i,j,s+1L,HPi,sC)ψA1ψA]ψAψA1,(17)
Sh,i,s+1F,HPi,sC=[j=1N(ϕi,jS)1ψA(1νi,si,j,sSh,i,j,s+1F,HPi,sC)ψA1ψA]ψAψA1,(18)

where serially correlated currency risk premium shocks νj,s satisfy νj,s>0, while 0ϕi,jB1,j=1Nϕi,jB=1,0ϕi,jS1 and j=1Nϕi,jS=1. Finally, preferences over the real values of economy and vintage specific long term bonds {{i,j,sVj,k,sBBh,i,j,k,s+1L,H/Pi,sC}k=1s}j=1N and economy, industry and firm specific shares {{{i,j,sVj,k,l,sSSh,i,j,k,l,s+1F,H/Pi,sC}l=01}k=1M}j=1N are represented by constant elasticity of substitution intratemporal subutility functions

i,j,sBh,i,j,s+1L,HPi,sC=[k=1s(ϕi,j,k,sB)1ψA(i,j,sVj,k,sBBh,i,j,k,s+1L,HPi,sC)ψA1ψA]ψAψA1,(19)
i,j,sSh,i,j,s+1F,HPi,sC=[k=1M(ϕi,j,kS)1ψA01(i,j,sVj,k,l,sSSh,i,j,k,l,s+1F,HPi,sC)ψA1ψAdl]ψAψA1,(20)

where 0ϕi,j,k,sB1,k=1sϕi,j,k,sB=1,0ϕi,j,kS1 and k=1Mϕi,j,kS=1. In the limit as νiA,H0 there is no capitalist spirit motive for holding real portfolio balances, while in the limit as ψA → ∞ there is no diversification motive over the allocation of real portfolio balances across alternative financial assets which in this case are perfect substitutes.

In period t, the representative capital market intermediated household chooses state contingent sequences for consumption {Ch,i,s}s=t, housing {Hh,i,s}s=t, labor force participation {{Nh,f,i,s}f=01}s=t, portfolio balances {Ah,i,s+1A,H}s=t, short term bond holdings {{Bh,i,j,s+1S,H}j=1N}s=t, long term bond holdings {{{Bh,i,j,k,s+1L,H}k=1t}j=1N}s=t, and stock holdings {{{{Sh,i,j,k,l,s+1H}l=01}k=1M}j=1N}s=t to maximize intertemporal utility function (1) subject to dynamic budget constraint (7), the applicable restrictions on financial asset holdings, and terminal nonnegativity constraints Bh,i,j,T+1S,H0,Bh,i,j,k,T+1L,H0 and Sh,i,j,k,l,T+1H0 for T → ∞. In equilibrium, abstracting from the capitalist spirit motive for holding real portfolio balances, the solutions to this utility maximization problem satisfy intertemporal optimality condition

EtβuC(h,i,t+1)uC(h,i,t)Pi,tCPi,t+1C(1+ih,i,t+1AA,H)=1,(21)

which equates the expected present value of the gross real portfolio return to one. In addition, these solutions satisfy intratemporal optimality condition

uH(h,i,t)uC(h,i,t)=ιi,tHPi,tC,(22)

which equates the marginal rate of substitution between housing and consumption to the real rental price of housing. Furthermore, these solutions satisfy intratemporal optimality condition

uLf(h,fi,t)uC(h,i,t)=(1τi,tL)Wf,i,tPi,tC,(23)

which equates the marginal rate of substitution between leisure and consumption for the marginal trade specific labor force participant to the corresponding after tax real wage. Abstracting from risk premium shocks, the expected present value of the gross real portfolio return satisfies intratemporal optimality condition

ϕMAj=1Nϕi,jB{1+EtβuC(h,i,t+1)uC(h,i,t)uC(h,i,t)uAA(h,i,t)Pi,tCPi,t+1C[(1+ih,i,t+1AA,H)(1+ij,tS)i,j,t+1i,j,t]}1ψA+ϕBAj=1Nϕi,jBk=1tϕi,j,k,tB{1+EtβuC(h,i,t+1)uC(h,i,t)uC(h,i,t)uAA(h,i,t)Pi,tCPi,t+1C[(1+ih,i,t+1AA,H)Πj,k,t+1B+Vj,k,t+1BVj,k,tBi,j,t+1i,j,t]}1ψA(24)+ϕSAj=1Nϕi,jSk=1Mϕi,j,kS01{1+EtβuC(h,i,t+1)uC(h,i,t)uC(h,i,t)uAA(h,i,t)Pi,tCPi,t+1C[(1+ih,i,t+1AA,H)Πj,k,l,t+1S+Vj,k,l,t+1SVj,k,l,tSi,j,t+1i,j,t]}1ψAdl=1,

which relates it to the expected present values of the gross real returns on domestic and foreign short term bonds, long term bonds, and stocks. In addition, abstracting from the portfolio diversification motive over the allocation of real portfolio balances these solutions satisfy intratemporal optimality condition

EtβuC(h,i,t+1)uC(h,i,t)Pi,tCPi,t+1C[(1+ii,ts)(1+ij,ts)i,j,t+1i,j,t]=uAA(h,i,t)uC(h,i,t)(1νi,t1νj,t),(25)

which equates the expected present values of the gross real risk adjusted returns on domestic and foreign short term bonds. Furthermore, abstracting from the portfolio diversification motive over the allocation of real portfolio balances these solutions satisfy intratemporal optimality condition

EtβuC(h,i,t+1)uC(h,i,t)Pi,tCPi,t+1C[(1+ii,tS)Πi,k,t+1B+Vi,k,t+1BVi,k,tB]=uAA(h,i,t)uC(h,i,t)1νi,t(11υi,tB),(26)

which equates the expected present values of the gross real risk adjusted returns on domestic short and long term bonds. Finally, abstracting from the portfolio diversification motive over the allocation of real portfolio balances these solutions satisfy intratemporal optimality condition

EtβuC(h,i,t+1)uC(h,i,t)Pi,tCPi,t+1C[(1+ii,tS)Πi,k,l,t+1S+Vi,k,l,t+1SVi,k,l,tS]=uAA(h,i,t)uC(h,i,t)1νi,t(11υi,tS),(27)

which equates the expected present values of the gross real risk adjusted returns on domestic short term bonds and stocks. Provided that the intertemporal utility function is bounded and strictly concave, together with other optimality conditions, and transversality conditions derived from necessary complementary slackness conditions associated with the terminal nonnegativity constraints, these optimality conditions are sufficient for the unique utility maximizing state contingent sequence of capital market intermediated household allocations.

Credit Constrained Households

In period t, the representative credit constrained household chooses state contingent sequences for consumption {Ch,i,s}s,t, housing {Hh,i,s}s,t, and labor force participation {{Nh,f,i,s}f=01}s=t to maximize intertemporal utility function (1) subject to dynamic budget constraint (7), and the applicable restrictions on financial asset holdings. In equilibrium, the solutions to this utility maximization problem satisfy household static budget constraint

Pi,tCCh,i,t+ιi,tHHi,t=Πi,t+(1τi,tL)01Wf,i,tLh,f,i,tdf+Th,i,tC,(28)

which equates the sum of consumption and housing expenditures to the sum of profit and disposable labor income net of transfer payments, where profit income Πi,t satisfies Πi,t=Πi,tH+Πi,tS+Πi,tC. The evaluation of this result abstracts from international bank lending. Furthermore, these solutions satisfy intratemporal optimality condition

uH(h,i,t)uC(h,i,t)=ιi,tHPi,tC,(29)

which equates the marginal rate of substitution between housing and consumption to the real rental price of housing. Finally, these solutions satisfy intratemporal optimality condition

uLf(h,f,i,t)uC(h,i,t)=(1τi,tL)Wf,i,tPi,tC,(30)

which equates the marginal rate of substitution between leisure and consumption for the marginal trade specific labor force participant to the corresponding after tax real wage. Provided that the intertemporal utility function is bounded and strictly concave, these optimality conditions are sufficient for the unique utility maximizing state contingent sequence of credit constrained household allocations.

Labor Supply

The unemployment rate ui,tL measures the share of the labor force Ni,t in unemployment Ui,t, that is ui,tL=Ui,t/Ni,t, where unemployment equals the labor force less employment Li,t, that is Ui,t = Ni,tLi,t. The labor force satisfies Ni,t=01Nf,i,tdf.

There exist a large number of perfectly competitive firms which combine differentiated intermediate labor services Lf,i,t supplied by trade unions of workers to produce final labor service Li,t according to constant elasticity of substitution production function

Li,t=[01(Lf,i,t)θi,tL1θi,tLdf]θi,tLθi,tL1,(31)

where serially uncorrelated wage markup shock ϑi,tL satisfies ϑi,tL=θi,tLθi,tL1 with θi,tL>1 and θiL=θL. The representative final labor service firm maximizes profits derived from production of the final labor service with respect to inputs of intermediate labor services, implying demand functions:

Lf,i,t=(Wf,i,tWi,t)θi,tLLi,t.(32)

Since the production function exhibits constant returns to scale, in equilibrium the representative final labor service firm generates zero profit, implying aggregate wage index:

Wi,t=[01(Wf,i,t)1θi,tLdf]11θi,tL.(33)

As the wage elasticity of demand for intermediate labor services θi,tL increases, they become closer substitutes, and individual trade unions have less market power.

In an extension of the model of nominal wage rigidity proposed by Erceg, Henderson and Levin (2000) along the lines of Smets and Wouters (2003), each period a randomly selected fraction 1 – ωL of trade unions adjust their wage optimally, where 0 ≤ ωL < 1. The remaining fraction ωL of trade unions adjust their wage to account for past consumption price inflation and trend productivity growth according to partial indexation rule

Wf,i,t=(Pi,t1CA˜i,t1Pi,t2CA˜i,t2)γL(P¯i,t1CA¯i,t1P¯i,t2CA¯i,t2)1γLWf,i,t1,(34)

where 0 ≤ γL ≤ 1. Under this specification, although trade unions adjust their wage every period, they infrequently do so optimally, and the interval between optimal wage adjustments is a random variable.

If the representative trade union can adjust its wage optimally in period t, then it does so to maximize intertemporal utility function (1) subject to dynamic budget constraint (7), intermediate labor service demand function (32), and the assumed form of nominal wage rigidity. Since all trade unions that adjust their wage optimally in period t solve an identical utility maximization problem, in equilibrium they all choose a common wage Wi,t* given by necessary first order condition:

Wi,t*Wi,t=Ets=t(ωL)stβstuC(h,i,s)uC(h,i,t)θi,sLuLf(h,f,i,s)uC(h,i,s)[(Pi,t1CA˜i,t1Pi,s1CA˜i,s1)γL(P¯i,t1CA¯i,t1P¯i,s1CA¯i,s1)1γLWi,sWi,t]θi,sL(Wi,t*Wi,t)θi,sLLh,i,sEts=t(ωL)stβstuC(h,i,s)uC(h,i,t)(θi,sL1)(1τi,sL)Wi,sPi,sC[(Pi,t1CA˜i,t1Pi,s1CA˜i,s1)γL(P¯i,t1CA¯i,t1P¯i,s1CA¯i,s1)1γLWi,sWi,t]θi,sL1(Wi,t*Wi,t)θi,sLLh,i,s.(35)

This necessary first order condition equates the expected present value of the marginal utility of consumption gained from labor supply to the expected present value of the marginal utility cost incurred from leisure foregone. Aggregate wage index (33) equals an average of the wage set by the fraction 1 – ωL of trade unions that adjust their wage optimally in period t, and the average of the wages set by the remaining fraction ωL of trade unions that adjust their wage according to partial indexation rule (34):

Wi,t={(1ωL)(Wi,t*)1θi,tL+ωL[(Pi,t1CA˜i,t1Pi,t2CA˜i,t2)γL(P¯i,t1CA¯i,t1P¯i,t2CA¯i,t2)1γLWi,t1]1θi,tL}11θi,tL.(36)

Since those trade unions able to adjust their wage optimally in period t are selected randomly from among all trade unions, the average wage set by the remaining trade unions equals the value of the aggregate wage index that prevailed during period t – 1, rescaled to account for past consumption price inflation and trend productivity growth.

B. The Construction Sector

The construction sector supplies housing services to domestic households. In doing so, developers obtain mortgage loans from domestic banks and accumulate the housing stock through residential investment.

Housing Demand

There exist a large number of perfectly competitive developers which combine differentiated intermediate housing services Hi,e,t supplied by intermediate developers to produce final housing service Hi,t according to constant elasticity of substitution production function

Hi,t=[01(Hi,e,t)θi,tH1θi,tHde]θi,tHθi,tH1,(37)

where endogenous rental price of housing markup shifter ϑi,tH satisfies ϑi,tH=θi,tHθi,tH1 with θi,tH>0 and θiH=θH. The representative final developer maximizes profits derived from production of the final housing service with respect to inputs of intermediate housing services, implying demand functions:

Hi,e,t=(ιi,e,tHιi,tH)θi,tHHi,t.(38)

Since the production function exhibits constant returns to scale, in equilibrium the representative final developer generates zero profit, implying aggregate rental price of housing index:

ιi,tH=[01(ιi,e,tH)1θi,tHde]11θi,tH.(39)

As the price elasticity of demand for intermediate housing services θi,tH increases, they become closer substitutes, and individual intermediate developers have less market power.

Residential Investment

There exist continuums of monopolistically competitive intermediate developers indexed by e ∈ [0,1]. Intermediate developers supply differentiated intermediate housing services, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive intermediate construction sector.

The representative intermediate developer sells shares to domestic bank intermediated households at price Vi,e,tH. Acting in the interests of its shareholders, it maximizes its pre-dividend stock market value, which abstracting from the capitalist spirit motive for holding real property balances equals the expected present value of current and future dividend payments

Πi,e,tH+Vi,e,tH=Ets=tβstλi,sBλi,tBΠi,e,sH,(40)

where λi,sB denotes the Lagrange multiplier associated with the period s bank intermediated household dynamic budget constraint. The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to profits Πi,e,sH, defined as the sum of earnings and net borrowing less residential investment expenditures:

Πi,e,sH=ιi,e,sHHi,e,s+(Bi,e,s+1C,D(1δi,sM)(1+ii,s+1M)Bi,e,sC,D)Pi,sIHIi,e,sH.(41)

Earnings are defined as revenues from sales of differentiated intermediate housing service Hi,e,s at rental price ιi,e,sH.

Motivated by the collateralized borrowing variant of the financial accelerator mechanism due to Kiyotaki and Moore (1997), the financial policy of the representative intermediate developer is to maintain debt equal to a fraction of the value of the housing stock,

Bi,e,s+1C,DPi,sIHHi,e,s+1=ϕi,sD,(42)

given by regulatory mortgage loan to value ratio limit ϕi,sD. Net borrowing is defined as the increase in mortgage loans Bi,e,s+1C,D from domestic banks net of writedowns at mortgage loan default rate δi,sM and interest payments at mortgage loan rate ii,s1M.

The representative intermediate developer enters period s in possession of previously accumulated housing stock Hi,e,s, which subsequently evolves according to accumulation function

Hi,e,s+1=(1δH)Hi,e,s+HH(Ii,e,sH,Ii,e,s1H),(43)

where 0 ≤ δH ≤ 1. Effective residential investment function HH(Ii,e,sH,Ii,e,s1H) incorporates convex adjustment costs in the gross growth rate of the ratio of nominal residential investment to aggregate nominal output,

HH(Ii,e,sH,Ii,e,s1H)=νi,sIH[1χH2(Pi,sIHIi,e,sHPi,s1IHIi,e,s1HPi,s1YYi,s1Pi,sYYi,s1)2]Ii,e,sH,(44)

where serially correlated residential investment demand shock νi,sIH satisfies νi,sIH>0, while χH > 0. In steady state equilibrium, these adjustment costs equal zero, and effective residential investment equals actual residential investment.

In period t, the representative intermediate developer chooses state contingent sequences for residential investment {Ii,e,sH}s=t and the housing stock {Hi,e,s+1}s=t to maximize pre-dividend stock market value (40) subject to housing accumulation function (43) and terminal nonnegativity constraint Hi,e,T+1 ≥ 0 for T → ∞. In equilibrium, demand for the final residential investment good satisfies necessary first order condition

Qi,e,tHH1H(Ii,e,tH,Ii,e,t1H)+Etβλi,t+1Bλi,tBQi,e,t+1HH2H(Ii,e,t+1H,Ii,e,tH)=Pi,tIH,(45)

which equates the expected present value of an additional unit of residential investment to its price, where Qi,e,sH denotes the Lagrange multiplier associated with the period s housing accumulation function. In equilibrium, this shadow price of housing satisfies necessary first order condition

Qi,e,tH=Etβλi,t+1Bλi,tB{Pi,t+1IH{ιi,e,t+1HPi,t+1IHϕi,tDPi,tIHPi,t+1IH[(1δi,t+1M)(1+ii,tM)λi,tBβλi,t+1B]}+(1δH)Qi,e,t+1H},(46)

which equates it to the expected present value of the sum of the future marginal revenue product of housing, and the future shadow price of housing net of depreciation, less the product of the loan to value ratio with the spread of the effective cost of bank over capital market funding. Provided that the pre-dividend stock market value is bounded and strictly concave, together with other necessary first order conditions, and a transversality condition derived from the necessary complementary slackness condition associated with the terminal nonnegativity constraint, these necessary first order conditions are sufficient for the unique value maximizing state contingent sequence of intermediate developer allocations.

Housing Supply

In period t, the representative intermediate developer adjusts its rental price of housing to maximize pre-dividend stock market value (40) subject to housing accumulation function (43) and intermediate housing service demand function (38). We consider a symmetric equilibrium under which all developer specific endogenous state variables are restricted to equal their aggregate counterparts. It follows that all intermediate developers solve an identical value maximization problem, which implies that they all choose a common rental price of housing ιi,tH,* given by necessary first order condition:

ιi,tH,*Pi,tIH=θi,tHθi,tH1[ϕi,t1D(1δi,tM)(1+ii,t+1M)Pi,t1IHPi,tIH(1δH)Qi,e,tHPi,tIH].(47)

This necessary first order condition equates the marginal revenue gained from housing supply to the marginal cost incurred from construction. Aggregate rental price of housing index (39) satisfies ιi,tH=ιi,tH,*.

C. The Production Sector

The production sector supplies output goods for domestic and foreign absorption. In doing so, firms demand labor services from domestic households, obtain corporate loans from domestic and foreign banks, and accumulate the private physical capital stock through business investment.

The production sector consists of a finite set of industries indexed by k ∈ {1, …, M}, of which the first M * produce nonrenewable commodities. In particular, the energy commodity industry labeled k = 1 and the nonenergy commodity industry labeled k = 2 produce internationally homogeneous output goods for foreign absorption, while all other industries produce internationally heterogeneous output goods for domestic and foreign absorption.

Output Demand

There exist a large number of perfectly competitive firms which combine industry specific final output goods {Yi,k,t}k=1M to produce final output good Yi,t according to fixed proportions production function

Yi,t=min{Yi,k,tϕi,kY}k=1M,(48)

where 0ϕi,kY1 and k=1Mϕi,kY=1. The representative final output good firm maximizes profits derived from production of the final output good with respect to inputs of industry specific final output goods, implying demand functions:

Yi,k,t=ϕi,kYYi,t.(49)

Since the production function exhibits constant returns to scale, in equilibrium the representative final output good firm generates zero profit, implying aggregate output price index

Pi,tY=k=1Mϕi,kYPi,k,tY,(50)

where Pi,k,tY=Pi,k,tX for 1 ≤ kM*. This aggregate output price index equals the minimum cost of producing one unit of the final output good, given the prices of industry specific final output goods.

There exist a large number of perfectly competitive firms which combine industry specific differentiated intermediate output goods Yi,k,l,t supplied by industry specific intermediate output good firms to produce industry specific final output good Yi,k,t according to constant elasticity of substitution production function

Yi,k,t=[01(Yi,k,l,t)θi,k,tY1θi,k,tYdl]θi,k,tYθi,k,tY1,(51)

where serially uncorrelated output price markup shock ϑi,k,tY satisfies ϑi,k,tY=θi,k,tYθi,k,tY1 with θi,k,tY>1 and θi,kY=θY, while θi,k,tY=θk,tY for 1 ≤ kM* and θi,k,tY=θi,tY otherwise. The representative industry specific final output good firm maximizes profits derived from production of the industry specific final output good with respect to inputs of industry specific intermediate output goods, implying demand functions:

Yi,k,l,t=(Pi,k,l,tYPi,k,tY)θi,k,tYYi,k,t.(52)

Since the production function exhibits constant returns to scale, in equilibrium the representative industry specific final output good firm generates zero profit, implying industry specific aggregate output price index:

Pi,k,tY=[01(Pi,k,l,tY)1θi,k,tYdl]11θi,k,tY.(53)

As the price elasticity of demand for industry specific intermediate output goods θi,k,tY increases, they become closer substitutes, and individual industry specific intermediate output good firms have less market power.

Labor Demand and Business Investment

There exist continuums of monopolistically competitive industry specific intermediate output good firms indexed by l ∈ [0,1]. Intermediate output good firms supply industry specific differentiated intermediate output goods, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive industry specific intermediate output good sectors.

The representative industry specific intermediate output good firm sells shares to domestic and foreign capital market intermediated households at price Vi,k,l,tS. Acting in the interests of its shareholders, it maximizes its pre-dividend stock market value, which abstracting from the capitalist spirit motive for holding real portfolio balances equals the expected present value of current and future dividend payments

Πi,k,l,tS+Vi,k,l,tS=Ets=tβstλi,sAλi,tAΠi,k,l,sS,(54)

where λi,sA denotes the Lagrange multiplier associated with the period s capital market intermediated household dynamic budget constraint. The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to net profits Πi,k,l,sS, defined as the sum of after tax corporate earnings and net borrowing less business investment expenditures,

Πi,k,l,sS=(1τi,sK)(Pi,k,l,sYYi,k,l,sWi,sLi,k,l,sΦi,k,l,s)+(Bi,k,l,s+1C,F(1δi,sC)(1+ii,sC,E)Bi,k,l,sC,F)Pi,sIKIi,k,l,sK,(55)

where Yi,k,l.s=(ui,k,l,sK,Ki,k,l,s,Ai,sLi,k,l,s). Corporate earnings are defined as revenues from sales of industry specific differentiated intermediate output good Yi,k,l,s at price Pi,k,l,sY less expenditures on final labor service Li,k,l,s, and other variable costs Φi,k,l,s. The government levies a tax on corporate earnings at rate τi,sK.

Motivated by the collateralized borrowing variant of the financial accelerator mechanism due to Kiyotaki and Moore (1997), the financial policy of the representative industry specific intermediate output good firm is to maintain debt equal to a fraction of the value of the private physical capital stock,

Bi,k,l,s+1C,FPi,sIKKi,k,l,s+1=ϕi,sF,(56)

given by regulatory corporate loan to value ratio limit ϕi,sF. Net borrowing is defined as the increase in corporate loans Bi,k,l,s+1C,F from domestic and foreign banks net of writedowns at corporate loan default rate δi,sC and interest payments at effective corporate loan rate ii,sC,E. This corporate loan default rate applies uniformly to all corporate loans received from domestic and foreign banks.

The representative industry specific intermediate output good firm utilizes private physical capital Ki,k,l,s at rate ui,k,l,sK and rents final labor service Li,k,l,s to produce industry specific differentiated intermediate output good Yi,k,l,s according to production function:

(ui,k,l,sKKi,k,l,s,Ai,sLi,k,l,s)=(ui,k,l,sKKi,k,l,s)ϕiK(Ai,sLi,k,l,s)1ϕiK.(57)

This production function exhibits constant returns to scale, with 0ϕiK1. Productivity Ai,s depends on the ratio of the public physical capital stock to the aggregate labor force,

Ai,s=(υi,sA)ϕA(Ki,sGNi,s)1ϕA,(58)

where internationally and serially correlated productivity shock υi,sA satisfies υi,sA>0, while 0<ϕA1. Trend productivity A˜i,s exhibits partial adjustment dynamics A˜i,s=(A˜i,s1)ρA(Ai,s)1ρA, where 0ρA<1.

In utilizing private physical capital to produce output, the representative industry specific intermediate output good firm incurs a cost G(ui,k,l,sK,Ki,k,l,s) denominated in terms of business investment,

Φi,k,l,s=Pi,sIKG(ui,k,l,sK,Ki,k.l,s)+Fi,k,sF,(59)

where industry specific fixed cost Fi,k,sF ensures that Φi,k,s = 0. Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the capital utilization rate at an increasing rate,

G(ui,k,l,sK,Ki,k,l,s)=μiK[eηK(ui,k,l,sK1)1]Ki,k,l,s,(60)

where ηK > 0, while μiK=μK1τi with μK > 0. In steady state equilibrium, the capital utilization rate equals one, and the cost of utilizing private physical capital equals zero.

The representative industry specific intermediate output good firm enters period s in possession of previously accumulated private physical capital stock Ki,k,l,s, which subsequently evolves according to accumulation function

Ki,k,l,s+1=(1δK)Ki,k,l,s+H(Ii,k,l,sK,Ii,k,l,s1K),(61)

where 0 < δK ≤ 1. Building on Christiano, Eichenbaum and Evans (2005), effective business investment function H(Ii,k,l,sK,Ii,k,l,s1K) incorporates convex adjustment costs in the gross growth rate of the ratio of nominal business investment to aggregate nominal output,

H(Ii,k,l,sK,Ii,k,l,s1K)=νi,sIK[1χK2(Pi,sIKIi,k,l,sKPi,s1IKIi,k,l,s1KPi,s1YYi,s1Pi,sYYi,s1)2]Ii,k,l,sK,(62)

where serially correlated business investment demand shock νi,sIK satisfies νi,sIK>0, while χK > 0. In steady state equilibrium, these adjustment costs equal zero, and effective business investment equals actual business investment.

In period t, the representative industry specific intermediate output good firm chooses state contingent sequences for employment {Li,k,l,s}s=t, the capital utilization rate {ui,k,l,sK}s=t, business investment {Ii,k,l,sK}s=t, and the private physical capital stock {Ki,k,l,s+1}st to maximize pre-dividend stock market value (54) subject to production function (57), private physical capital accumulation function (61), and terminal nonnegativity constraint Ki,k,l,T+1 ≥ 0 for T → ∞. In equilibrium, demand for the final labor service satisfies necessary first order condition

AL(ui,k,l,tKKi,k,l,t,Ai,tLi,k,l,t)Ψi,k,l,t=(1τi,tK)Wi,tPi,k,tYAi,t,(63)

where Pi,k,sYΨi,k,l,s denotes the Lagrange multiplier associated with the period s production technology constraint. This necessary first order condition equates real marginal cost Ψi,k,l,t to the ratio of the after tax industry specific real wage to the marginal product of labor. In equilibrium, the capital utilization rate satisfies necessary first order condition

uKK(ui,k,l,tKKi,k,l,t,Ai,tLi,k,l,t)Pi,k,tYΨi,k,l,tPi,tIK=(1τi,tK)GuK(ui,k,l,tK,Ki,k,l,t)Ki,k,l,t,(64)

which equates the marginal revenue product of utilized private physical capital to its marginal cost. In equilibrium, demand for the final business investment good satisfies necessary first order condition

Qi,k,l,tKH1(Ii,k,l,tKIi,k,l,t1K)+Etβλi,t+1Aλi,tAQi,k,l,t+1KH2(Ii,k,l,t+1K,Ii,k,l,tK)=Pi,tIK,(65)

which equates the expected present value of an additional unit of business investment to its price, where Qi,k,l,sK denotes the Lagrange multiplier associated with the period s private physical capital accumulation function. In equilibrium, this shadow price of private physical capital satisfies necessary first order condition

Qi,k,l,tK=Etβλi,t+1Aλi,tA{Pi,t+1IK{ui,k,l,t+1KuKK(ui,k,l,t+1KKi,k,l,t+1,Ai,t+1Li,k,l,t+1)Pi,k,t+1YΨi,k,l,t+1Pi,t+1IK(1τi,t+1K)GK(ui,k,l,t+1K,Ki,k,l,t+1)ϕi,tFPi,tIKPi,t+1IK[(1δi,t+1C)(1+ii,t+1C,E)λi,tAβλi,t+1A]}+(1δK)Qi,k,l,t+1K},(66)

which equates it to the expected present value of the sum of the future marginal revenue product of private physical capital net of its marginal utilization cost, and the future shadow price of private physical capital net of depreciation, less the product of the loan to value ratio with the spread of the effective cost of bank over capital market funding. Provided that the pre-dividend stock market value is bounded and strictly concave, together with other necessary first order conditions, and a transversality condition derived from the necessary complementary slackness condition associated with the terminal nonnegativity constraint, these necessary first order conditions are sufficient for the unique value maximizing state contingent sequence of industry specific intermediate output good firm allocations.

Output Supply

In an extension of the model of nominal output price rigidity proposed by Calvo (1983) along the lines of Smets and Wouters (2003), each period a randomly selected fraction 1ωkY of industry specific intermediate output good firms adjust their price optimally, where 0ωkY<1 with ωkY=ωY for k > M *. The remaining fraction ωkY of intermediate output good firms adjust their price to account for past industry specific output price inflation according to partial indexation rule

Pi,k,l,tY=(Pi,k,t1YPi,k,t2Y)γkY(P¯i,k,t1YP¯i,k,t2Y)1γkYPi,k,l,t1Y,(67)

where 0γkY1 with γkY=0 for 1 ≤ kM* and γkY=γY otherwise. Under this specification, optimal price adjustment opportunities arrive randomly, and the interval between optimal price adjustments is a random variable.

If the representative industry specific intermediate output good firm can adjust its price optimally in period t, then it does so to maximize pre-dividend stock market value (54) subject to production function (57), industry specific intermediate output good demand function (52), and the assumed form of nominal output price rigidity. We consider a symmetric equilibrium under which all industry and firm specific endogenous state variables are restricted to equal their industry specific aggregate counterparts. It follows that all intermediate output good firms that adjust their price optimally in period t solve an identical value maximization problem, which implies that they all choose a common price Pi,k,tY,* given by necessary first order condition:

Pi,k,tY,*Pi,k,tY=Ets=t(ωkY)stβstλi,sAλi,tAθi,k,sYΨi,k,l,s[(Pi,k,t1YPi,k,s1Y)γkY(P¯i,k,t1YP¯i,k,s1Y)1γkYPi,k,sYPi,k,tY]θi,k,sY(Pi,k,tY,*Pi,k,tY)θi,k,sYPi,k,sYYi,k,sEts=t(ωkY)stβstλi,sAλi,tA(θi,k,sY1)(1τi,sK)[(Pi,k,t1YPi,k,s1Y)γkY(P¯i,k,t1YP¯i,k,s1Y)1γkYPi,k,sYPi,k,tY]θi,k,sY1(Pi,k,tYPi,k,tY)θi,k,sYPi,k,sYYi,k,s.(68)

This necessary first order condition equates the expected present value of the after tax marginal revenue gained from output supply to the expected present value of the marginal cost incurred from production. Aggregate output price index (53) equals an average of the price set by the fraction 1ωkY of intermediate output good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction ωkY of intermediate output good firms that adjust their price according to partial indexation rule (67):

Pi,k,tY={(1ωkY)(Pi,k,tY,*)1θi,k,tY+ωkY[(Pi,k,t1YPi,k,t2Y)γkY(P¯i,k,t1YP¯i,k,t2Y)1γkYPi,k,t1Y]1θi,k,tY}11θi,k,tY.(69)

Since those intermediate output good firms able to adjust their price optimally in period t are selected randomly from among all intermediate output good firms, the average price set by the remaining intermediate output good firms equals the value of the industry specific aggregate output price index that prevailed during period t – 1, rescaled to account for past industry specific output price inflation.

D. The Banking Sector

The banking sector supplies global financial intermediation services subject to financial frictions and regulatory constraints. In particular, banks issue risky mortgage loans to domestic developers at infrequently adjusted predetermined mortgage loan rates, as well as risky domestic currency denominated corporate loans to domestic and foreign firms at infrequently adjusted predetermined corporate loan rates, given regulatory loan to value ratio limits. They obtain funding from domestic bank intermediated households via deposits and from the domestic interbank market via loans, accumulating bank capital out of retained earnings given credit losses to satisfy a regulatory capital requirement.

Credit Demand

There exist a large number of perfectly competitive banks which combine economy specific local currency denominated final corporate loans {Bi,j,tC,F}j=1N to produce domestic currency denominated final corporate loan Bi,tC,F according to fixed proportions portfolio aggregator

Bi,tC,F=min{i,j,t1Bi,j,tC,Fϕi,jF}j=1N,(70)

where 0ϕi,jF1 and j=1Nϕi,jF=1. The representative global final bank maximizes profits derived from intermediation of the domestic currency denominated final corporate loan with respect to inputs of economy specific local currency denominated final corporate loans, implying demand functions:

Bi,j,tC,F=ϕi,jFBi,tC,Fi,j,t1.(71)

Since the portfolio aggregator exhibits constant returns to scale, in equilibrium the representative global final bank generates zero profit, implying aggregate effective gross corporate loan rate index:

1+ii,tC,E=j=1Nϕi,jF(1+ij,t1C)i,j,ti,j,t1.(72)

This aggregate effective gross corporate loan rate index equals the minimum cost of producing one unit of the domestic currency denominated final corporate loan, given the rates on economy specific local currency denominated final corporate loans.

There exist a large number of perfectly competitive banks which combine differentiated intermediate mortgage or corporate loans Bi,m,t+1CZ,B supplied by intermediate banks to produce final mortgage or corporate loan Bi,t+1CZ,B according to constant elasticity of substitution portfolio aggregator

Bi,t+1CZ,B=[01(Bi,m,t+1CZ,B)θi,t+1CZ1θi,t+1CZdm]θi,t+1CZθi,t+1CZ1,(73)

where Z ∈ {D,F}, while serially uncorrelated mortgage or corporate loan rate markup shock ϑi,t+1CZ satisfies ϑi,t+1CZ=θi,t+1CZθi,t+1CZ1 with θi,t+1CZ>1 and θiCZ=θC. The representative domestic final bank maximizes profits derived from intermediation of the final mortgage or corporate loan with respect to inputs of intermediate mortgage or corporate loans, implying demand functions

Bi,m,t+1CZ,B=(1+ii,m,tf(Z)1+ii,tf(Z))θi,t+1CZBi,t+1CZ,B,(74)

where f(D) = M and f (F) = C. Since the portfolio aggregator exhibits constant returns to scale, in equilibrium the representative domestic final bank generates zero profit, implying aggregate gross mortgage or corporate loan rate index:

1+ii,tf(Z)=[01(1+ii,m,tf(Z))1θi,t+1CZdm]11θi,t+1CZ.(75)

As the rate elasticity of demand for intermediate mortgage or corporate loans θi,t+1CZ increases, they become closer substitutes, and individual intermediate banks have less market power.

Funding Demand and Bank Capital Accumulation

There exists a continuum of monopolistically competitive intermediate banks indexed by m ∈ [0,1]. Intermediate banks supply differentiated intermediate mortgage and corporate loans, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive intermediate banking sector.

The representative intermediate bank sells shares to domestic bank intermediated households at price Vi,m,tC. Acting in the interests of its shareholders, it maximizes its pre-dividend stock market value, which equals the expected present value of current and future dividend payments:

Πi,m,tC+Vi,m,tC=Ets=tβstλi,sBλi,tBΠi,m,sC.(76)

The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments Πi,m,sC, defined as profits derived from providing financial intermediation services less retained earnings Ii,m,sB:

Πi,m,sC=(Bi,m,s+1D,B(1+ii,s1D)Bi,m,sD,B)+(Bi,m,s+1B,B(1+ii,s1B)Bi,m,sB,B)(77)(Bi,m,s+1CD,B(1δi,sM)(1+ii,m,s1M)Bi,m,sCD,B)(Bi,m,s+1CF,B(1δi,sC,E)(1+ii,m,s1C)Bi,m,sCF,B)Φi,m,sBIi,m,sB.

Profits are defined as the sum of the increase in deposits Bi,m,s+1D,B from domestic bank intermediated households net of interest payments at the deposit rate and the increase in net loans Bi,m,s+1B,B from the domestic interbank market net of interest payments at the interbank loans rate ii,s1B, less the increase in differentiated intermediate mortgage loans Bi,m,s+1CD,B to domestic developers net of writedowns at mortgage credit loss rate δi,sM and interest receipts at mortgage loan rate ii,m,s1M, less the increase in differentiated intermediate corporate loans Bi,m,s+1CF,B to domestic and foreign firms net of writedowns at corporate credit loss rate δi,sC,E and interest receipts at corporate loan rate ii,m,s1C, less a cost of satisfying the regulatory capital requirement Φi,m,sB.

The representative intermediate bank transforms deposit and money market funding into risky differentiated intermediate mortgage and corporate loans according to balance sheet identity:

Bi,m,s+1CD,B+Bi,m,s+1CF,B=Bi,m,s+1D,B+Bi,m,s+1B,B+Ki,m,s+1B.(78)

The bank credit stock Bi,s+1C,B measures aggregate bank assets, that is Bi,s+1C,B=Bi,s+1CD,B+Bi,s+1CF,B, while the money stock Mi,s+1S measures aggregate bank funding, that is Mi,s+1S=Bi,s+1D,B+Bi,s+1B,B where Bi,s+1B,B=0. The bank capital ratio κi,s+1 equals the ratio of aggregate bank capital to assets, that is κi,s+1=Ki,s+1B/Bi,s+1C,B.

In transforming deposit and money market funding into risky mortgage and corporate loans, the representative intermediate bank incurs a cost of satisfying the regulatory capital requirement,

Φi,m,sB=GB(Bi,m,sCD,B,Bi,m,sCF,B,Ki,m,sB)+Fi,sB,(79)

where fixed cost Fi,sB ensures that Φi,sB=Ii,sB. Motivated by Gerali, Neri, Sessa and Signoretti (2010), this regulation cost is decreasing in the ratio of bank capital to assets at a decreasing rate,

GB(Bi,m,sCD,B,Bi,m,sCF,B,Ki,m,sB)=μC[e(2+ηC)(11κi,sRKi,m,sBBi,m,sCD+Bi,m,sCF,B)1]Ki,m,sB,(80)

given regulatory capital requirement κi,sR, where ηC > 0 and μC > 0. In steady state equilibrium, the bank capital ratio equals its required value, and the cost of regulation is constant.

The financial policy of the representative intermediate bank is to smooth retained earnings intertemporally, given credit losses. It enters period s in possession of previously accumulated bank capital stock Ki,m,sB, which subsequently evolves according to accumulation function

Ki,m,s+1B=(1δi,sB)Ki,m,sB+HB(Ii,m,sB,Ii,m,s1B),(81)

where bank capital destruction rate δi,sB satisfies δi,sB=χC(wiCδi,sM+(1wiC)δi,sC,E) with χC < 0, while mortgage loan weight wiC satisfies 0<wiC<1. Effective retained earnings function HB(Ii,m,sB,Ii,m,s1B) incorporates convex adjustment costs,

HB(Ii,m,sB,Ii,m,s1B)=[1χB2(Ii,m,sBIi,m,s1B1)2]Ii,m,sB,(82)

where χB > 0. In steady state equilibrium, these adjustment costs equal zero, and effective retained earnings equals actual retained earnings.

In period t, the representative intermediate bank chooses state contingent sequences for deposit funding {Bi,m,s+1D,B}s=t, net interbank market funding {Bi,m,s+1B,B}s=t, retained earnings {Ii,m,sB}s=t, and the bank capital stock {Ki,m,s+1B}s=t to maximize pre-dividend stock market value (76) subject to balance sheet identity (78), bank capital accumulation function (81), and terminal nonnegativity constraints Bi,m,T+1D,B0,Bi,m,T+1B,B0 and Ki,m,T+1B0 for T → ∞. In equilibrium, the solutions to this value maximization problem satisfy necessary first order condition

1+ii,tD=1+ii,tB,(83)

which equates the deposit rate to the interbank loans rate. In equilibrium, retained earnings satisfies necessary first order condition

Qi,m,tBH1B(Ii,m,tB,Ii,m,t1B)+Etβλi,t+1Bλi,tBQi,m,t+1BH2B(Ii,m,t+1B,Ii,m,tB)=1,(84)

which equates the expected present value of an additional unit of retained earnings to its marginal cost, where Qi,m,sB denotes the Lagrange multiplier associated with the period s bank capital accumulation function. In equilibrium, this shadow price of bank capital satisfies necessary first order condition

Qi,m,tB=Etβλi,t+1Bλi,tB{(1δi,t+1B)Qi,m,t+1B{G3B(Bi,m,t+1CD,B,Bi,m,t+1CF,B,Ki,m,t+1B)+[λi,tBβλi,t+1B(1+ii,tB)]}},(85)

which equates it to the expected present value of the future shadow price of bank capital net of destruction, less the sum of the marginal utilization cost of bank capital and the spread of the cost of deposit over interbank market funding. The evaluation of this result abstracts from risk premium shocks. Provided that the pre-dividend stock market value is bounded and strictly concave, together with other necessary first order conditions, and transversality conditions derived from the necessary complementary slackness conditions associated with the terminal nonnegativity constraints, these necessary first order conditions are sufficient for the unique value maximizing state contingent sequence of intermediate bank allocations.

Credit Supply

In an adaptation of the model of nominal output price rigidity proposed by Calvo (1983) to the banking sector along the lines of Hülsewig, Mayer and Wollmershäuser (2009), each period a randomly selected fraction 1 – ωC of intermediate banks adjust their gross mortgage and corporate loan rates optimally, where 0 ≤ ωC < 1. The remaining fraction ωC of intermediate banks do not adjust their loan rates,

1+ii,m,tf(Z)=1+ii,m,t1f(Z),(86)

where Z ∈ {D,F}, while f (D) = M and f (F) = C. Under this financial friction, intermediate banks infrequently adjust their loan rates, mimicking the effect of maturity transformation on the spreads between the loan and deposit rates.

If the representative intermediate bank can adjust its gross mortgage and corporate loan rates in period t, then it does so to maximize pre-dividend stock market value (76) subject to balance sheet identity (78), intermediate loan demand function (74), and the assumed financial friction. We consider a symmetric equilibrium under which all bank specific endogenous state variables are restricted to equal their aggregate counterparts. It follows that all intermediate banks that adjust their loan rates in period t solve an identical value maximization problem, which implies that they all choose common loan rates ii,tf(Z),* given by necessary first order conditions

1+ii,tf(Z),*1+ii,tf(Z)=Ets=t(ωC)stβstλi,sBλi,tBθi,sCZ(1+ii,s+1B)+Gh(Z)B(Bi,m,sCD,B,Bi,m,sCF,B,Ki,m,sB)1+ii,s1f(Z)(1+ii,s1f(Z)1+ii,tf(Z))θi,sCZ(1+ii,tf(Z),*1+ii,tf(Z))θi,sCZ(1+ii,s1f(Z))Bi,sCZ,BEts=t(ωC)stβstλi,sBλi,tB(θi,sCZ1)(1δi,sg(Z))(1+ii,s1f(Z)1+ii,tf(Z))θi,sCZ1(1+ii,tf(Z),*1+ii,tf(Z))θi,sCZ(1+ii,s1f(Z))Bi,sCZ,B,(87)

where g(D) = M and g(F) = C, E, while h(D) = 1 and h(F) = 2. These necessary first order conditions equate the expected present value of the marginal revenue gained from mortgage or corporate loan supply to the expected present value of the marginal cost incurred from intermediation. Aggregate gross mortgage or corporate loan rate index (75) equals an average of the gross mortgage or corporate loan rate set by the fraction 1 – ωC of intermediate banks that adjust their loan rates in period t, and the average of the gross mortgage or corporate loan rates set by the remaining fraction ωC of intermediate banks that do not adjust their loan rates:

1+ii,tf(Z)=[(1ωC)(1+ii,tf(Z),*)1θi,t+1CZ+ωC(1+ii,t1f(Z))1θi,t+1CZ]11θi,t+1CZ.(88)

Since those intermediate banks able to adjust their loan rates in period t are selected randomly from among all intermediate banks, the average gross mortgage or corporate loan rate set by the remaining intermediate banks equals the value of the aggregate gross mortgage or corporate loan rate index that prevailed during period t – 1.

E. The Trade Sector

The nominal effective exchange rate i,t measures the trade weighted average price of foreign currency in terms of domestic currency, while the real effective exchange rate Qi, t measures the trade weighted average price of foreign output in terms of domestic output,

i,t=j=1N(i,j,t)wi,jT,Qi,t=j=1N(Qi,j,t)wi,jT,(89)

where the real bilateral exchange rate Qi,j,t satisfies Qi,j,t=i,j,tPj,tY/Pi,tY, and bilateral trade weight wi,jT satisfies wi,iT=0,0wi,jT1 and j=1Nwi,jT=1. Furthermore, the terms of trade Ti,t equals the ratio of the internal terms of trade to the external terms of trade,

Ti,t=υtTTi,tXTi,tM,Ti,tX=Pi,tXPi,t,Ti,tM=Pi,tMPi,t,(90)

where the internal terms of trade Ti,tX measures the relative price of exports, and the external terms of trade Ti,tM measures the relative price of imports, while Pi,t denotes the price of the final noncommodity output good. Finally, under the law of one price for 1 ≤ kM *,

Pk,tY=i=1NwiYi*,i,tPi,k,tY,(91)

where Pk,tY denotes the quotation currency denominated price of energy or nonenergy commodities, and world output weight wiY satisfies 0<wiY<1 and i=1NwiY=1. Endogenous global terms of trade shifter υtT adjusts to ensure multilateral consistency in nominal trade flows, and in steady state equilibrium satisfies υT=1.

The Export Sector

There exist a large number of perfectly competitive firms which combine industry specific final export goods {Xi,k,t}k=1M to produce final export good Xi,t according to fixed proportions production function

Xi,t=min{Xi,k,tϕi,kX}k=1M,(92)

where Xi,k,t = Yi,k,t for 1 ≤ kM *, while 0ϕi,kX1 and k=1Mϕi,kX=1.. The representative final export good firm maximizes profits derived from production of the final export good with respect to inputs of industry specific final export goods, implying demand functions:

Xi,k,t=ϕi,kXXi,t.(93)

Since the production function exhibits constant returns to scale, in equilibrium the representative final export good firm generates zero profit, implying aggregate export price index

Pi,tX=k=1Mϕi,kXPi,k,tX,(94)

where Pi,k,tX=Pi,k,tY for k > M *. This aggregate export price index equals the minimum cost of producing one unit of the final export good, given the prices of industry specific final export goods.

Export Demand

There exist a large number of perfectly competitive firms which combine industry specific differentiated intermediate export goods Xi,k,n,t supplied by industry specific intermediate export good firms to produce industry specific final export good Xi,k,t according to constant elasticity of substitution production function

Xi,k,t=[01(Xi,k,,n,t)θi,tX1θi,tXdn]θi,tXθi,tX1,(95)

for 1 ≤ kM *, where serially uncorrelated export price markup shock ϑi,tX satisfies ϑi,tX=θi,tXθi,tX1 with θiX>1 and θiX=θX. The representative industry specific final export good firm maximizes profits derived from production of the industry specific final export good with respect to inputs of industry specific intermediate export goods, implying demand functions:

Xi,k,n,t=(Pi,k,n,tXPi,k,tX)θi,tXXi,k,t.(96)

Since the production function exhibits constant returns to scale, in equilibrium the representative industry specific final export good firm generates zero profit, implying industry specific aggregate export price index:

Pi,k,tX=[01(Pi,k,n,tX)1θi,tXdn]11θi,tX.(97)

As the price elasticity of demand for industry specific intermediate export goods θi,tX increases, they become closer substitutes, and individual industry specific intermediate export good firms have less market power.

Export Supply

There exist continuums of monopolistically competitive industry specific intermediate export good firms indexed by n ∈ [0,1]. Intermediate export good firms supply industry specific differentiated intermediate export goods, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive industry specific intermediate export good sectors.

The representative industry specific intermediate export good firm sells shares to domestic capital market intermediated households at price Vi,k,n,tX. Acting in the interests of its shareholders, it maximizes its pre-dividend stock market value, which equals the expected present value of current and future dividend payments:

Πi,k,n,tX+Vi,k,n,tX=Ets=tβstλi,sAλi,tAΠi,k,n,sX.(98)

The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to profits Πi,k,n,sX, defined as earnings less industry specific fixed cost Fi,k,sX:

Πi,k,n,sX=Pi,k,n,sXXi,k,n,si,i*,sPk,sYXi,k,n,sFi,k,sX.(99)

Earnings are defined as revenues from sales of industry specific differentiated intermediate export good Xi,k,n,s at price Pi,k,n,sX less expenditures on energy or nonenergy commodity good Xi,k,n,s. The representative industry specific intermediate export good firm purchases the energy or nonenergy commodity good and differentiates it. Fixed cost Fi,k,sX ensures that i,k,sX=0.

In an adaptation of the model of nominal import price rigidity proposed by Monacelli (2005) to the export sector, each period a randomly selected fraction 1 – ωX of industry specific intermediate export good firms adjust their price optimally, where 0 ≤ ωX < 1. The remaining fraction ωX of intermediate export good firms adjust their price to account for past industry specific export price inflation, as well as the contemporaneous change in the domestic currency denominated price of energy or nonenergy commodities, according to partial indexation rule

Pi,k,n,tX=[(Pi,k,t1XPi,k,t2X)1μX(i,i*,tPk,tYi,i*,t1Pk,t1Y)μX]γX[(P¯i,k,t1XP¯i,k,t2X)1μX(¯i,i*,tP¯k,tY¯i,i*,t1P¯k,t1Y)μX]1γXPi,k,n,t1X,(100)

where 0 ≤ γX ≤ 1 and μX ≥ 0. Under this specification, the probability that an intermediate export good firm has adjusted its price optimally is time dependent but state independent.

If the representative industry specific intermediate export good firm can adjust its price optimally in period t, then it does so to maximize pre-dividend stock market value (113) subject to industry specific intermediate export good demand function (111), and the assumed form of nominal export price rigidity. Since all intermediate export good firms that adjust their price optimally in period t solve an identical value maximization problem, in equilibrium they all choose a common price Pi,k,tX,* given by necessary first order condition:

Pi,k,tX,*Pi,k,tX=Ets=t(ωX)stβstλi,sAλi,tAθi,sXi,i*,sPk,sYPi,k,sX{[(Pi,k,t1XPi,k,s1X)1μX(i,i*,tPk,tYi,i*,sPk,sY)μX]γX[(P¯i,k,t1XP¯i,k,s1X)1μX(¯i,i*,tP¯k,tY¯i,i*,sP¯k,sY)μX]1γXPi,k,sXPi,k,tX}θi,sX(Pi,k,tX,*Pi,k,tX)θi,sXPi,k,sXXi,k,sEts=t(ωX)stβstλi,sAλi,tA(θi,sX1){[(Pi,k,t1XPi,k,s1X)1μX(i,i*,tPk,tYi,i*,sPk,sY)μX]γX[(P¯i,k,t1XP¯i,k,s1X)1μX(¯i,i*,tP¯k,tY¯i,i*,sP¯k,sY)μX]1γXPi,k,sXPi,k,tX}θi,sX1(Pi,k,tX,*Pi,k,tX)θi,sXPi,k,sXXi,k,s.(101)

This necessary first order condition equates the expected present value of the marginal revenue gained from export supply to the expected present value of the marginal cost incurred from production. Aggregate export price index (112) equals an average of the price set by the fraction 1 – ωX of intermediate export good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction ωX of intermediate export good firms that adjust their price according to partial indexation rule (115):

Pi,k,tX={(1ωX)(Pi,k,tX,*)1θi,tX+ωX{[(Pi,k,t1XPi,k,t2X)1μX(i,i*,tPk,tYi,i*,t1Pk,t1Y)μX]γX[(P¯i,k,t1XP¯i,k,t2X)1μX(¯i,i*,tP¯k,tY¯i,i*,t1P¯k,t1Y)μX]1γXPi,k,t1X}1θi,tX}11θi,tX.(102)

Since those intermediate export good firms able to adjust their price optimally in period t are selected randomly from among all intermediate export good firms, the average price set by the remaining intermediate export good firms equals the value of the industry specific aggregate export price index that prevailed during period t – 1, rescaled to account for past industry specific export price inflation, as well as the contemporaneous change in the domestic currency denominated price of energy or nonenergy commodities.

The Import Sector

There exist a large number of perfectly competitive firms which combine the final noncommodity output good Zi,th{Ci,th,Ii,tH,h,Ii,tK,h,Gi,tC,h,Gi,tI,h} with the final import good Zi,tf{Ci,tf,Ii,tH,f,Ii,tK,f,Gi,tC,f,Gi,tI,f} to produce final private consumption, residential investment, business investment, public consumption or public investment good Zi,t{Ci,t,Ii,tH,Ii,tK,Gi,tC,Gi,tI} according to constant elasticity of substitution production function

Zi,t=[(1ϕiM)1ψiM(Zi,th)ψiM1ψiM+(ϕiM)1ψiM(νi,tMZi,tf)ψiM1ψiM]ψiMψiM1,(103)

where serially correlated import demand shock νi,tM satisfies νi,tM>0, while 0ϕiM<1 and ψiM=ψM(1θMθM1M¯i,tY¯i,t)1 with ψM > 0. The representative final absorption good firm maximizes profits derived from production of the final private consumption, residential investment, business investment, public consumption or public investment good, with respect to inputs of the final noncommodity output and import goods, implying demand functions:

Zi,th=(1ϕiM)(Pi,tPi,tZ)ψiMZi,t,Zi,tf=ϕiM(1νiMPi,tMPi,tZ)ψiMZi,tνi,tM.(104)

Since the production function exhibits constant returns to scale, in equilibrium the representative final absorption good firm generates zero profit, implying aggregate private consumption, residential investment, business investment, public consumption or public investment price index:

Pi,tZ=[(1ϕiM)(Pi,t)1ψiM+ϕiM(Pi,tMνiM)1ψiM]11ψiM.(105)

Combination of this aggregate private consumption, residential investment, business investment, public consumption or public investment price index with final noncommodity output and import good demand functions (104) yields:

Zi,th=(1ϕiM)[(1ϕiM)+ϕiM(Ti,tMνiM)1ψiM]ψiM1ψiMZi,t,Zi,tf=ϕiM[ϕiM+(1ϕiM)(Ti,tMνiM)ψiM1]ψiM1ψiMZi,tνi,tM.(106)

These demand functions for the final noncommodity output and import goods are directly proportional to final private consumption, residential investment, business investment, public consumption or public investment good demand, with a proportionality coefficient that varies with the external terms of trade. The derivation of these results selectively abstracts from import demand shocks.

Import Demand

There exist a large number of perfectly competitive firms which combine economy specific final import goods {Mi,j,t}j=1N to produce final import good Mi,t according to fixed proportions production function

Mi,t=min{νj,tXMi,j,tϕi,jM}j=1N,(107)

where serially correlated export demand shock νi,tX satisfies νi,tX>0, while ϕi,iM=0,0ϕi,jM1 and j=1Nϕi,jM=1. The representative final import good firm maximizes profits derived from production of the final import good with respect to inputs of economy specific final import goods, implying demand functions:

Mi,j,t=ϕi,jMMi,tνj,tX.(108)

Since the production function exhibits constant returns to scale, in equilibrium the representative final import good firm generates zero profit, implying aggregate import price index:

Pi,tM=j=1Nϕi,jMPi,j,tMνjX.(109)

This aggregate import price index equals the minimum cost of producing one unit of the final import good, given the prices of economy specific final import goods. The derivation of these results selectively abstracts from export demand shocks.

There exist a large number of perfectly competitive firms which combine economy specific differentiated intermediate import goods Mi,j,n,t supplied by economy specific intermediate import good firms to produce economy specific final import good Mi,j,t according to constant elasticity of substitution production function

Mi,j,t=[01(Mi,j,n,t)θi,tM1θi,tMdn]θi,tMθi,tM1,(110)

where serially uncorrelated import price markup shock ϑi,tM satisfies ϑi,tM=θi,tMθi,tM1 with θi,tM>1 and θiM=θM. The representative economy specific final import good firm maximizes profits derived from production of the economy specific final import good with respect to inputs of economy specific intermediate import goods, implying demand functions:

Mi,j,n,t=(Pi,j,n,tMPi,j,tM)θi,tMMi,j,t.(111)

Since the production function exhibits constant returns to scale, in equilibrium the representative economy specific final import good firm generates zero profit, implying economy specific aggregate import price index:

Pi,j,tM=[01(Pi,j,n,tM)1θi,tMdn]11θi,tM.(112)

As the price elasticity of demand for economy specific intermediate import goods θi,tM increases, they become closer substitutes, and individual economy specific intermediate import good firms have less market power.

Import Supply

There exist continuums of monopolistically competitive economy specific intermediate import good firms indexed by n ∈ [0,1]. Intermediate import good firms supply economy specific differentiated intermediate import goods, but are otherwise identical. We rule out entry into and exit out of the monopolistically competitive economy specific intermediate import good sectors.

The representative economy specific intermediate import good firm sells shares to domestic capital market intermediated households at price Vi,j,n,tM. Acting in the interests of its shareholders, it maximizes its pre-dividend stock market value, which equals the expected present value of current and future dividend payments:

Πi,j,n,tM+Vi,j,n,tM=Ets=tβstλi,sAλi,tAΠi,j,n,sM.(113)

The derivation of this result imposes a transversality condition that rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to profits Πi,j,n,sM, defined as earnings less economy specific fixed cost Fi,j,sM:

Πi,j,n,sM=Pi,j,n,sMMi,j,n,si,j,sPj,sXMi,j,n,sFi,j,sM.(114)

Earnings are defined as revenues from sales of economy specific differentiated intermediate import good Mi,j,n,s at price Pi,j,n,sM less expenditures on foreign final export good Mi,j,n,s. The representative economy specific intermediate import good firm purchases the foreign final export good and differentiates it. Fixed cost Fi,j,sM ensures that Πi,j,sM=0.

In an extension of the model of nominal import price rigidity proposed by Monacelli (2005), each period a randomly selected fraction 1 – ωM of economy specific intermediate import good firms adjust their price optimally, where 0 ≤ ωM < 1. The remaining fraction ωM of intermediate import good firms adjust their price to account for past economy specific import price inflation, as well as contemporaneous changes in the domestic currency denominated prices of energy and nonenergy commodities, according to partial indexation rule

Pi,j,n,tM=[(Pi,j,t1MPi,j,t2M)1μiMk=1M*(i,i*,tPk,tYi,i*,t1Pk,t1Y)μi,kM]γM[(P¯i,j,t1MP¯i,j,t2M)1μiMk=1M*(¯i,i*,tP¯k,tY¯i,i*,t1P¯k,t1Y)μi,kM]1γMPi,j,n,t1M,(115)

where 0 ≤ γM ≤ 1, while μiM=k=1M*μi,kM with μi,kM=μMM¯i,k,tM¯i,t and μM ≥ 0. Under this specification, the probability that an intermediate import good firm has adjusted its price optimally is time dependent but state independent.

If the representative economy specific intermediate import good firm can adjust its price optimally in period t, then it does so to maximize pre-dividend stock market value (113) subject to economy specific intermediate import good demand function (111), and the assumed form of nominal import price rigidity. Since all intermediate import good firms that adjust their price optimally in period t solve an identical value maximization problem, in equilibrium they all choose a common price Pi,j,tM,* given by necessary first order condition:

Pi,j,tM,*Pi,j,tM=Ets=t(ωM)stβstλi,sAλi,tAθi,sMi,j,sPj,sXPi,j,s{[(Pi,j,t1MPi,j,s1M)1μiMk=1M*(i,i*,tPk,tYi,i*,sPk,sY)μi,kM]γM[(P¯i,j,t1MP¯i,j,s1M)1μiMk=1M*(¯i,i*,tP¯k,tY¯i,i*,sP¯k,sY)μi,kM]1γMPi,j,sMPi,j,tM}θi,sM(Pi,j,tM,*Pi,j,tM)θi,sMPi,j,sMMi,j,sEts=t(ωM)stβstλi,sAλi,tA(θi,sM1){[(Pi,j,t1MPi,j,s1M)1μiMk=1M*(i,i*,tPk,tYi,i*,sPk,sY)μi,kM]γM[(P¯i,j,t1MP¯i,j,s1M)1μiMk=1M*(¯i,i*,tP¯k,tY¯i,i*,sP¯k,sY)μi,kM]1γMPi,j,sMPi,j,tM}θi,sM1(Pi,j,tM,*Pi,j,tM)θi,sMPi,j,sMMi,j,s.(116)

This necessary first order condition equates the expected present value of the marginal revenue gained from import supply to the expected present value of the marginal cost incurred from production. Aggregate import price index (112) equals an average of the price set by the fraction 1 – ωM of intermediate import good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction ωM of intermediate import good firms that adjust their price according to partial indexation rule (115):

Pi,j,tM={(1ωM)(Pi,j,tM,*)1θi,tM+ωM{[(Pi,j,t1MPi,j,t2M)1μiMk=1M*(i,i*,tPk,t*i,i*,t1Pk,t1Y)μi,kM]γM[(P¯i,j,t1MP¯i,j,t2M)1μiMk=1M*(¯i,i*,tP¯k,t*¯i,i*,t1P¯k,t1Y)μi,kM]1γMPi,j,t1}1θi,tM}11θi,tM.(117)

Since those intermediate import good firms able to adjust their price optimally in period t are selected randomly from among all intermediate import good firms, the average price set by the remaining intermediate import good firms equals the value of the economy specific aggregate import price index that prevailed during period t – 1, rescaled to account for past economy specific import price inflation, as well as contemporaneous changes in the domestic currency denominated prices of energy and nonenergy commodities.

F. Monetary, Fiscal, and Macroprudential Policy

The government consists of a monetary authority, a fiscal authority, and a macroprudential authority. The monetary authority conducts monetary policy, while the fiscal authority conducts fiscal policy, and the macroprudential authority conducts macroprudential policy.

The Monetary Authority

The monetary authority implements monetary policy through control of the nominal policy interest rate. We differentiate between flexible inflation targeting, managed exchange rate, and fixed exchange rate regimes. Under a monetary union, the leader economy follows a modified flexible inflation targeting regime, while all other union members follow fixed exchange rate regimes.

Under a flexible inflation targeting or managed exchange rate regime, the nominal policy interest rate satisfies a monetary policy rule exhibiting partial adjustment dynamics of the form

ii,tPi¯i,tP=ρi(ii,t1Pi¯i,t1P)+(1ρi)[ξπEt(πi,t+1Cπ¯i,t+1C)+ξY(lnYi,tlnY˜i,t)+ξj(Δlni,tΔln¯i,t)]+νi,tiP,(118)

where 0 ≤ ρi < 1, ξπ > 1, ξY > 0 and ξj0. This rule prescribing the conduct of monetary policy is consistent with achieving some combination of inflation control, output stabilization, and exchange rate stabilization objectives. As specified, the deviation of the nominal policy interest rate from its steady state equilibrium value depends on a weighted average of its past deviation and its desired deviation. Under a flexible inflation targeting regime j = 0, and this desired deviation is increasing in the expected future deviation of consumption price inflation from its target value, as well as the contemporaneous output gap. We define the output gap as the deviation of output from its potential value, which we define as that output level consistent with full utilization of private physical capital and effective labor, given the private physical capital stock and effective labor force. For the leader economy of a monetary union, the target variables entering into its monetary policy rule are expressed as output weighted averages across union members. Under a managed exchange rate regime j = 1, and the desired deviation of the nominal policy interest rate from its steady state equilibrium value is also increasing in the contemporaneous deviation of the change in the nominal effective exchange rate from its steady state equilibrium value with ξj=ξ>0. Deviations from this monetary policy rule are captured by mean zero and serially uncorrelated monetary policy shock νi,tiP.

Under a fixed exchange rate regime, the nominal policy interest rate instead satisfies a monetary policy rule exhibiting feedback of the form

ii,tPi¯i,tP=(ik,tPi¯k,tP)+ξk(Δlni,k,tΔln¯i,k,t),(119)

where ξk > 1. As specified, the deviation of the nominal policy interest rate from its steady state equilibrium value tracks the contemporaneous deviation of the nominal policy interest rate of the leader economy from its steady state equilibrium value one for one, and is increasing in the contemporaneous deviation of the change in the corresponding nominal bilateral exchange rate from its target value.

The Fiscal Authority

The fiscal authority implements fiscal policy through control of public consumption and investment, as well as the tax rates applicable to corporate earnings and household labor income. It also operates a budget neutral nondiscretionary lump sum transfer program that redistributes national financial wealth from capital market intermediated households to credit constrained households while equalizing steady state equilibrium consumption across households, as well as a discretionary lump sum transfer program that provides income support to credit constrained households. The fiscal authority can transfer its budgetary resources intertemporally through transactions in the domestic money and bond markets. Considered jointly, the rules prescribing the conduct of this distortionary fiscal policy are countercyclical, representing automatic fiscal stabilizers, and are consistent with achieving public and national financial wealth stabilization objectives.

Public consumption and investment satisfy countercyclical fiscal expenditure rules exhibiting partial adjustment dynamics of the form

lnGi,tZG¯i,tZ=ρGlnGi,t1ZG¯i,t1Z+(1ρG)lnY˜i,tY¯i,t+νi,tGZ,(120)

where Z ∈ {C, I}, while 0 ≤ ρG < 1. As specified, the deviation of public consumption or investment from its steady state equilibrium value depends on a weighted average of its past deviation and its desired deviation, which in turn tracks the contemporaneous deviation of potential output from its steady state equilibrium value one for one. Deviations from these fiscal expenditure rules are captured by mean zero and serially correlated public consumption or investment shock νi,tGZ.

The tax rates applicable to corporate earnings and household labor income satisfy acyclical fiscal revenue rules of the form

τi,tZτi=ρτ(τi,t1Zτi)+νi,tτZ,(121)

where Z ∈ (K, L}, while 0 < τi < 1 and 0 ≤ ρτ < 1. As specified, the deviations of these tax rates from their steady state equilibrium value depend on their past deviations. Deviations from these fiscal revenue rules are captured by mean zero and serially correlated corporate or labor income tax rate shock νi,tτZ.

The ratio of nondiscretionary lump sum transfer payments to nominal output satisfies a nondiscretionary transfer payment rule that stabilizes national financial wealth of the form

Ti,tC,NPi,tYYi,tT¯i,tC,NP¯i,tYY¯i,t=ζTN(Ai,tPi,t1YYi,t1A¯i,tP¯i,t1YY¯i,t1),(122)

where ζTN > 0. As specified, the deviation of the ratio of nondiscretionary lump sum transfer payments to nominal output from its steady state equilibrium value is increasing in the past deviation of the ratio of national financial wealth to nominal output from its target value. The ratio of discretionary lump sum transfer payments to nominal output satisfies a discretionary transfer payment rule that stabilizes public financial wealth of the form

Ti,tC,DPi,tYYi,tT¯i,tD,NP¯i,tYY¯i,t=ζTD(Ai,tGPi,t1YYi,t1A¯i,tGP¯i,t1YY¯i,t1)+νi,tT,(123)

where ζTD > 0. As specified, the deviation of the ratio of discretionary lump sum transfer payments to nominal output from its steady state equilibrium value is increasing in the past deviation of the ratio of public financial wealth to nominal output from its target value. Deviations from this discretionary transfer payment rule are captured by mean zero and serially correlated transfer payment shock νi,tτ.

The gross yield to maturity on short term bonds depends on the contemporaneous gross nominal policy interest rate according to money market relationship:

1+ii,tS=υi,tiS(1+ii,tP).(124)

Deviations from this money market relationship are captured by internationally and serially correlated credit risk premium shock υi,tiS. In parallel, the gross nominal interbank loans rate depends on the contemporaneous gross yield to maturity on short term bonds according to interbank market relationship:

1+ii,tB=υi,tiB(1+ii,tS).(125)

Deviations from this interbank market relationship are captured by internationally and serially correlated liquidity risk premium shock υi,tiB.

The fiscal authority enters period t in possession of previously accumulated financial wealth Ai,tG which yields return ii,tAG. This financial wealth is distributed across the values of domestic short term bond Bi,tS,G and long term bond Bi,tL,G portfolios which yield returns ii,t1S and ii,tBL,G, respectively. It follows that (1+ii,tAG)Ai,tG=(1+ii,t1S)Bi,tS,G+(1+ii,tBL.G)Bi,tL,G, where (1+ii,tBL,G)Bi,tL,G=k=1t1(Πi,k,tB+Vi,k,tB)Bi,k,tL,G with Πi,k,tB=(1+ii,kLωB)(ωB)tkVi,k,kB and Vi,k,kB=1. At the end of period t, the fiscal authority levies taxes on corporate earnings at rate τi,tK and household labor income at rate τi,tL, generating tax revenues Ti,t. These sources of public wealth are summed in government dynamic budget constraint:

Ai,t+1G=(1+ii,tAG)Ai,tG+k=1M01τi,tK(Pi,k,l,tYYi,k,l,tWi,tLi,k,l,t)dl+01τi,tL01Wf,i,tLh,f,i,tdfdh01Th,i,tZdhPi,tGCGi,tCPi,tGIGi,tI.(126)

According to this dynamic budget constraint, at the end of period t, the fiscal authority holds financial wealth Ai,t+1G, which it allocates across the values of domestic short term bond Bi,t+1S,G and long term bond Bi,t+1L,G portfolios, that is Ai,t+1G=Bi,t+1S,G+Bi,t+1L,G where Bi,t+1L,G=k=1tVi,k,tBBi,k,t+1L,G. It also remits household type specific lump sum transfer payments {Th,i,tZ}h=01, which it allocates across nondiscretionary transfers {Th,i,tZ,N}h=01 and discretionary transfers {Th,i,tZ,D}h=01, that is Th,i,tZ=Th,i,tZ,N+Th,i,tZ,D where 01Th,i,tZ,Ndh=0 and Th,i,tB,D=Th,i,tA,D=Th,iC,D=0. Finally, the fiscal authority purchases final public consumption good Gi,tC at price Pi,tGC, and final public investment good Gi,tI at price Pi,tGI, accumulating the public physical capital stock Ki,t+1G according to Ki,t+1G=(1δG)Ki,tG+Gi,tI where 0 ≤ δG ≤ 1.

The Macroprudential Authority

The macroprudential authority implements macroprudential policy through control of a regulatory capital requirement and loan to value ratio limits. It imposes the regulatory capital requirement on lending by domestic banks, and the regulatory loan to value ratio limits on borrowing by domestic developers and firms.

The regulatory capital ratio requirement applicable to lending by domestic banks to domestic and foreign developers and firms satisfies a countercyclical capital buffer rule exhibiting partial adjustment dynamics of the form

κi,t+1RκR=ρκ(κi,tRκR)+(1ρκ)[ζκ,B(ΔlnBi,t+1C,BΔlnB¯i,t+1C,B)+ζκ,VH(ΔlnVi,tHΔlnV¯i,tH)+ζκ,VS(ΔlnVi,tSΔlnV¯i,tS)]+νi,tκ,(127)

where 0 < κR < 1, 0 ≤ ρκ < 1, ζκ,B > 0, ζκ,VH > 0 and ζκ,VS > 0. As specified, the deviation of the regulatory capital ratio requirement from its steady state equilibrium value depends on a weighted average of its past deviation and its desired deviation. This desired deviation is increasing in the contemporaneous deviation of bank credit growth from its steady state equilibrium value, as well as the contemporaneous deviations of the changes in the prices of housing and equity from their steady state equilibrium values. Deviations from this countercyclical capital buffer rule are captured by mean zero and serially correlated capital requirement shock νi,tκ.

The regulatory loan to value ratio limits applicable to borrowing by domestic developers and firms from domestic and foreign banks satisfy loan to value limit rules exhibiting partial adjustment dynamics of the form

ϕi,tZϕZ=ρϕZ(ϕi,t1ZϕZ)(1ρϕZ)[ζϕZ,B(ΔlnBi,t+1C,ZΔlnB¯i,t+1C,Z)+ζϕZ,V(ΔlnVi,tf(Z)ΔlnV¯i,tf(Z))]+νi,tϕZ,(128)

where Z ∈ {D, F}, while f (D) = H and f (F) = S. As specified, the deviations of the regulatory loan to value ratio limits from their steady state equilibrium values depend on a weighted average of their past deviations and their desired deviations, where 0 < ϕZ < 1, 0 ≤ ρϕZ < 1, ζϕZ,B > 0 and ζϕZ, V > 0. These desired deviations are decreasing in the contemporaneous deviation of mortgage or nonfinancial corporate debt growth from its steady state equilibrium value, as well as the contemporaneous deviation of the change in the price of housing or equity from its steady state equilibrium value, respectively. Deviations from these loan to value limit rules are captured by mean zero and serially uncorrelated mortgage or corporate loan to value limit shock νi,tϕZ.

The loan default rates applicable to borrowing by domestic developers and firms from domestic and foreign banks satisfy default rate relationships exhibiting partial adjustment dynamics of the form

δi,tZδ=ρδ(δi,t1Zδ)(1ρδ)[ζδZ,Y(lnYi,tlnY˜i,t)+ζδZ,V(ΔlnVi,tf(Z)ΔlnV¯i,tf(Z))]+νi,tδZ,(129)

where Z ∈ {M, C}, while f (M) = H and f (C) = S. As specified, the deviations of the mortgage or corporate loan default rates from their steady state equilibrium value depend on a weighted average of their past deviations and their attractor deviations, where 0 < δ < 1, 0 ≤ ρδ < 1, ζδZ,Y > 0 and ζδZ,V > 0. These attractor deviations are decreasing in the contemporaneous deviations of output from its potential value and the change in the price of housing or equity from its steady state equilibrium value, which affect the probability of default and loss given default, respectively. Deviations from these default rate relationships are captured by mean zero and serially uncorrelated mortgage or corporate loan default shock νi,tδZ.

G. Market Clearing Conditions

A rational expectations equilibrium in this DSGE model of the world economy consists of state contingent sequences of allocations for the households, developers, firms and banks of all economies that solve their constrained optimization problems given prices and policies, together with state contingent sequences of allocations for the governments of all economies that satisfy their policy rules and constraints given prices, with supporting prices such that all markets clear.

Clearing of the final output good market requires that exports Xi,t equal production of the domestic final output good less the total demand of domestic households, developers, firms and the government,

Xi,t=Yi,tCi,thIi,tH,hIi,tK,hGi,tC,hGi,tI,h,(130)

where Xi,t=j=1NXi,j,t and Xi,j,t = Mj,i,t. Clearing of the final import good market requires that imports Mi,t equal the total demand of domestic households, developers, firms and the government:

Mi,t=Ci,tf+Ii,tH,f+Ii,tK,f+Gi,tC,f+Gi,tI,f.(131)

In equilibrium, combination of these final output and import good market clearing conditions yields output expenditure decomposition,

Pi,tYYi,t=Pi,tCCi,t+Pi,tIIi,t+Pi,tGGi,t+Pi,tXXi,tPi,tMMi,t,(132)

where the price of investment satisfies Pi,tI=Pi,tIH=Pi,tIK while investment satisfies Ii,t=Ii,tH+Ii,tK, and the price of public domestic demand satisfies Pi,tG=Pi,tGC=Pi,tGI while public domestic demand satisfies Gi,t=Gi,tC+Gi,tI. The price of domestic demand satisfies Pi,tD=Pi,tC=Pi,tI=Pi,tG while domestic demand satisfies Di,t = Ci,t + Ii,t + Gi,t.

Clearing of the final bank loan markets requires that mortgage loan supply equals the total demand of domestic developers, that is Bi,t+1CD,B=Bi,t+1C,D, while corporate loan supply equals the total demand of domestic and foreign firms

Bi,t+1CF,B=j=1NBj,i,t+1C,F,(133)

where Bi,j,t+1CF,B=Bj,i,t+1C,F. In equilibrium, clearing of the final corporate loan payments system implies that the corporate credit loss rate satisfies:

1δi,tC,E=j=1NBj,i,tC,FBi,tCF,B(1δj,tC).(134)

The derivation of this result equates the principal and interest receipts of the banking sector to the total domestic currency denominated principal and interest payments of domestic and foreign firms.

Let Ai,t+1 denote the net foreign asset position, which equals the sum of the financial wealth of households Ai,t+1H, developers Ai,t+1D, firms Ai,t+1F, banks Ai,t+1B and the government Ai,t+1G,

Ai,t+1=Ai,t+1H+Ai,t+1D+Ai,t+1F+Ai,t+1B+Ai,t+1G,(135)

where Ai,t+1H=Ai,t+1B,H+Ai,t+1A,H+Vi,tC,Ai,t+1D=Bi,t+1C,DVi,tH,Ai,t+1F=Bi,t+1C,FVi,tS and Ai,t+1B=Ki,t+1BVi,tC. Imposing equilibrium conditions on government dynamic budget constraint (126) reveals that the increase in public financial wealth equals public saving, or equivalently that the fiscal balance FBi,t=Ai,t+1GAi,tG equals the sum of net interest income and the primary fiscal balance PBi,t=Ti,tTi,tC,DPi,tGGi,t,

FBi,t=[BiS,GAiGii,t1S+BiL,GAiGii,t1L,E]Ai,tG+PBi,t,(136)

where Ti,t=τi,tK(Pi,tYYi,tWi,tLi,t)+τi,tLWi,tLi,t, while nominal effective long term market interest rate ii,tL,E satisfies ii,tL,E=ωBii,t1L,E+(1ωB)[ωB((1+ii,tL)+(1ωB))1]. The derivation of this result abstracts from valuation gains on long term bond holdings, and imposes restrictions Bi,k,tL,G=Bi,k1,tL,G,Bi,tS,G/Ai,tG=BiS,G/AiG and Bi,tL,G/Ai,tG=BiL,G/AiG. Imposing equilibrium conditions on household dynamic budget constraint (7), and combining it with government dynamic budget constraint (136), developer dividend payment definition (41), firm dividend payment definition (55), bank dividend payment definition (77), bank balance sheet identity (78), output expenditure decomposition (132), and final corporate loan payments system clearing condition (134) reveals that the increase in net foreign assets equals national saving less investment expenditures, or equivalently that the current account balance CAi,t = i*,i,tAi,t+1i*,i,t−1Ai,t equals the sum of net international investment income and the trade balance TBi,t=i*,i,tPi,tXXi,ti*,i,tPi,tMMi,t:

CAi,t={j=1NwjA[(1+ij,t1S)i*,j,ti*,j,t11]}i*,i,t1Ai,t+TBi,t.(137)

The derivation of this result abstracts from international financial intermediation except via the money markets and imposes restriction i,j,t1Bi,j,tS/Ai,t=wjA, where world capital market capitalization weight wiA satisfies 0<wiA<1 and i=1NwiA=1. Multilateral consistency in nominal trade flows requires that j=1NTBj,t=0.

III. The Empirical Framework

Estimation and inference are based on a linear state space representation of an approximate multivariate linear rational expectations representation of this DSGE model of the world economy, expressed as a function of its potentially heteroskedastic structural shocks. This multivariate linear rational expectations representation is derived by analytically linearizing the equilibrium conditions of the DSGE model around its stationary deterministic steady state equilibrium, and consolidating them by substituting out intermediate variables assuming small capital utilization costs and abstracting from the global terms of trade shifter. The response coefficients of these consolidated approximate linear equilibrium conditions are functions of behavioral parameters that have been restricted to coincide across economies—occasionally within groups sharing a structural characteristic—and economy specific structural characteristics implied by steady state equilibrium relationships. Except where stated otherwise, this steady state equilibrium features zero inflation, productivity and labor force growth, as well as public and national financial wealth.2

In what follows, x^i,t denotes the deviation of variable xi,t from its steady state equilibrium value xi, while Et xi,t+s denotes the rational expectation of variable xi,t+s conditional on information available in period t. Bilateral weights wi,jZ for evaluating the trade weighted average of variable xi,t across the trading partners of economy i are based on exports for Z = X, imports for Z = M, and their average for Z = T. In addition, bilateral weights wi,jZ for evaluating the weighted average of variable xi,t across the lending destinations and borrowing sources of economy i are based on bank lending for Z = C and nonfinancial corporate borrowing for Z = F. Furthermore, bilateral weights wi,jZ for evaluating the portfolio weighted average of variable xi,t across the investment destinations of economy i are based on debt for Z = B and equity for Z = S. Finally, world weights wiZ for evaluating the weighted average of variable xi,t across all economies are based on output for Z = Y and capital market capitalization for Z = A. Auxiliary parameters λZ are theoretically predicted to equal one, and satisfy λ = 0 and λZ > 0.

A. Endogenous Variables

Core inflation depends on a linear combination of its past and expected future values driven by contemporaneous real unit labor cost according to Phillips curve

π^i,t=γY1+γYβπ^i,t1+β1+γYβEtπ^i,t+1+(1ωY)(1ωYβ)ωY(1+γYβ)[lnW^i,tL^i,tP^i,tY^i,t+lnϑ^i,tY],(138)

which determines the core price level lnP^i,t. Core inflation π^i,t satisfies π^i,t=lnP^i,tlnP^i,t1. The output price level lnP^i,tY depends on the contemporaneous core price level and internal terms of trade according to output price relationship:

lnP^i,tY=lnP^i,t+XiYilnT^i,tX.(139)

Output price inflation π^i,tY satisfies π^i,tY=lnP^i,tYlnP^i,t1Y. The consumption price level lnP^i,tC depends on the contemporaneous core price level and external terms of trade according to consumption price relationship:

lnP^i,tC=lnP^i,t+MiYilnT^i,tM.(140)

Consumption price inflation π^i,tC satisfies π^i,tC=lnP^i,tClnP^i,t1C. The response coefficients of these relationships vary across economies with their trade openness.

Output lnY^i,t depends on contemporaneous domestic demand, exports and imports according to output demand relationship:

lnY^i,t=lnD^i,t+XiYilnX^i,tM^i,t.(141)

Domestic demand lnD^i,t depends on a weighted average of contemporaneous consumption, investment and public domestic demand according to domestic demand relationship:

lnD^i,t=CiYilnC^i,t+IiYilnI^i,t+GiYilnG^i,t.(142)

Investment lnI^i,t depends on a weighted average of contemporaneous residential and business investment according to investment demand relationship:

IiYilnI^i,t=IiHYilnI^i,tH+IiKYilnI^i,tK.(143)

Public domestic demand lnG^i,t depends on a weighted average of contemporaneous public consumption and investment according to public domestic demand relationship:

GiYilnG^i,t=GiCYilnG^i,tC+GiIYilnG^i,tI.(144)

The response coefficients of these relationships vary across economies with the composition of their domestic demand or their trade openness.

Consumption lnC^i,t depends on a weighted average of its past and expected future values driven by a weighted average of the contemporaneous real property and portfolio returns according to consumption demand relationship:

lnC^i,t=αC1+αClnC^i,t1+11+αCEtlnC^i,t+1(1ϕC)σ1αC1+αCEt[ϕB1ϕCr^i,t+1AB,H+(1ϕB1ϕC)r^i,t+1AA,Hlnν^i,tCν^i,t+1C]+ϕCP1(L)lnC^i,tC.(145)

Reflecting the existence of credit constraints, consumption also depends on contemporaneous, past and expected future credit constrained consumption, where polynomial in the lag operator P1(L)=1αC1+αCL11+αCEtL1. Credit constrained consumption lnC^i,tC depends on contemporaneous output and the terms of trade according to credit constrained consumption demand relationship

lnC^i,tC=λiC(CiYi)1{(1τi){lnY^i,t+XiYilnT^i,t11τi[(1WiLiPiYYi)τ^i,tK+WiLiPiYYiτ^i,tL]}+λiT1ϕCT^i,tCPi,tYYi,t(146)+λMiSPiYYi[lnM^i,t+1SP^i,tC1β(i^i,t1Bπ^i,tC+lnM^i,tSP^i,t1C)+1ββ(lnY^i,t+XiYilnT^i,t)]λIiYilnI^i,t},

where economy specific auxiliary parameters λiC=11τiCiYi and λiT=1τi. Credit constrained consumption also depends on a weighted average of the contemporaneous corporate and labor income tax rates, as well as the contemporaneous transfer payment ratio. The response coefficients of this relationship vary across economies with the size of their government, their trade openness, and their labor income share.

Residential investment lnI^i,tH depends on a weighted average of its past and expected future values driven by the contemporaneous relative shadow price of housing according to residential investment demand relationship:

lnI^i,tH=11+βlnI^i,t1H+β1+βEtlnI^i,t+1H+1χH(1+β)ln(ν^i,tIHQ^i,tHP^i,tC)+P2(L)(lnY^i,t+XiYilnT^i,t).(147)

Residential investment also depends on contemporaneous, past and expected future output and the terms of trade, where polynomial in the lag operator P2(L)=111+βLβ1βEtL1. Reflecting the existence of a financial accelerator mechanism, the relative shadow price of housing depends on its expected future value, as well as the contemporaneous real property return and mortgage loan rate, according to residential investment Euler equation

lnQ^i,tHP^i,tC=Et{β(1δH)lnQ^i,t+1HP^i,t+1C[(1ϕD)(r^i,t+1AB,H+ϕ^i,tDϕD)+ϕDβθCθC11+κR(1β(1χCδ))β(r^i,tMλδ^i,t+1M+λϕ^i,tDϕD)](148)+[(1β(1δH))+ϕDβ(θCθC11+κR(1β(1χCδ))β1β)]lnι^i,t+1HP^i,t+1C}+ϕ^i,tDϕD,

which determines the shadow price of housing lnQ^i,tH. The relative shadow price of housing also depends on the expected future real rental price of housing and the contemporaneous regulatory mortgage loan to value ratio limit. The real rental price of housing depends on the deviation of contemporaneous consumption from the past housing stock according to rental price of housing relationship

lnι^i,tHP^i,tC=1ςlnC^i,tH^i,t,(149)

which determines the rental price of housing lnι^i,tH. The housing stock lnH^i,t+1 is accumulated according to lnH^i,t+1=(1δH)lnH^i,t+δHln(ν^i,tIHI^i,tH).

Business investment lnI^i,tK depends on a weighted average of its past and expected future values driven by the contemporaneous relative shadow price of private physical capital according to business investment demand relationship:

lnI^i,tK=11+βlnI^i,t1K+β1+βEtlnI^i,t+1K+1χK(1+β)ln(ν^i,tIKQ^i,tKP^i,tC)+P2(L)(lnY^i,t+XiYilnT^i,t).(150)

Business investment also depends on contemporaneous, past and expected future output and the terms of trade. Reflecting the existence of a financial accelerator mechanism, the relative shadow price of private physical capital depends on its expected future value, as well as the contemporaneous real portfolio return and effective corporate loan rate, according to business investment Euler equation

lnQ^i,tKP^i,tC=Et{β(1δK)lnQ^i,t+1KP^i,t+1C[(1ϕF)(r^i,t+1AA,H+ϕ^i,tFϕF)+ϕFβθCθC11+κR(1β(1χCδ))β(r^i,t+1C,Eλδ^i,t+1C+λϕ^i,tFϕF)](151)+[(1β(1δK))+ϕFβ(θCθC11+κR(1β(1χCδ))β1β)](ηKlnu^i,t+1K11τiτ^i,t+1K)}+ϕ^i,tFϕF,

which determines the shadow price of private physical capital lnQ^i,tK. The relative shadow price of private physical capital also depends on the expected future capital utilization and corporate tax rates, as well as the contemporaneous regulatory corporate loan to value ratio limit. The capital utilization rate lnu^i,tK depends on the contemporaneous real wage, as well as the deviation of the past private physical capital stock from contemporaneous employment, according to capital utilization relationship:

lnu^i,tK=11+ηK(lnW^i,tP^i,tClnK^i,tL^i,t).(152)

The private physical capital stock lnK^i,t+1 is accumulated according to lnK^i,t+1=(1δK)lnK^i,t+δKln(ν^i,tIKI^i,tK).

Exports lnX^i,t depend on contemporaneous export weighted foreign imports according to export demand relationship:

lnX^i,t=j=1Nwi,jXlnM^j,tν^i,tX.(153)

Imports lnM^i,t depend on contemporaneous domestic demand, as well as the external terms of trade, according to import demand relationship:

lnM^i,t=lnD^i,tν^i,tMψMlnT^i,tM.(154)

The response coefficients of the former relationship vary across economies with their trade pattern.

The nominal property return Eti^i,t+1AB,H depends on the contemporaneous nominal interbank loans rate according to property return function:

Eti^i,t+1AB,H=i^i,tB+ϕHlnν^i,tH.(155)

Reflecting the existence of a portfolio balance mechanism, the nominal property return also depends on the contemporaneous housing risk premium. The real property return Etr^i,t+1AB,H satisfies Etr^i,t+1AB,H=Eti^i,t+1AB,HEtπ^i,t+1C.

The nominal interbank loans rate i^i,tB depends on the contemporaneous nominal short term bond yield adjusted by the liquidity risk premium according to interbank market relationship:

i^i,tB=i^i,tS+lnυ^i,tiB.(156)

The real interbank loans rate r^i,tB satisfies r^i,tB=i^i,tBEtπ^i,t+1C. The liquidity risk premium lnυ^i,tiB satisfies dynamic factor process lnυ^i,tiB=λkMj=1NwjAlnν^j,tiB+(1λkMwiA)lnν^i,tiB. The intensity of international interbank market contagion varies across economies, with k = 0 for low interbank market contagion economies, k = 1 for medium interbank market contagion economies, and k = 2 for high interbank market contagion economies, where λ0M<λ1M<λ2M.

The price of housing lnV^i,tH depends on its expected future value driven by expected future developer profits, and the contemporaneous nominal interbank loans rate adjusted by the housing risk premium, according to housing market relationship:

lnV^i,tH=βEtlnV^i,t+1H+(1β)EtlnΠ^i,t+1H(i^i,tB+lnν^i,tH).(157)

Developer profits lnΠ^i,tH depends on contemporaneous housing rental revenues according to developer profit function

lnΠ^i,tH=λiH(ΠiHPiYYi)1{ιiHPiYHiYiln(ι^i,tHH^i,t)(158)+λBiC,DPiYYi[lnB^i,t+1C,D(1δ)(1+iiM)(i^i,t1Mδ^i,tM+lnB^i,tC,D)]λIiHYiln(P^i,tCI^i,tH)},

where economy specific auxiliary parameter λiH=ΠiHPiYYi(ιiHPiYHiYi)1.

The nominal portfolio return Eti^i,t+1AA,H depends on the contemporaneous nominal short term bond yield according to portfolio return function:

Eti^i,t+1AA,H=i^i,tS+ϕBAj=1Nwi,jB(lnυ^j,tB+λ1ϕSAϕBAlnν^j,tν^i,t)+ϕSAj=1Nwi,jS(lnυ^j,tS+λlnν^j,tν^i,t).(159)

Reflecting the existence of a portfolio balance mechanism, the nominal portfolio return also depends on contemporaneous domestic and foreign duration and equity risk premia. The response coefficients of this relationship vary across economies with their domestic and foreign bond and stock market exposures. The real portfolio return Etr^i,t+1AA,H satisfies Etr^i,t+1AA,H=Eti^i,t+1AA,HEtπ^i,t+1C.

The nominal short term bond yield i^i,tS depends on the contemporaneous nominal policy interest rate adjusted by the credit risk premium according to money market relationship:

i^i,tS=i^i,tP+lnυ^i,tiS.(160)

The real short term bond yield r^i,tS satisfies r^i,tS=i^i,tSEtπ^i,t+1C. The credit risk premium lnυ^i,tiS satisfies dynamic factor process lnυ^i,tiS=λkBj=1NwjAlnν^j,tiS+(1λkBwiA)lnν^i,tiS. The intensity of international money market contagion varies across economies, with k = 0 for low capital market contagion economies, k = 1 for medium capital market contagion economies, and k = 2 for high capital market contagion economies, where λ0B<λ1B<λ2B.

The nominal long term bond yield i^i,tL depends on its expected future value, driven by the contemporaneous nominal short term bond yield adjusted by the duration risk premium, according to bond market relationship:

i^i,tL=ωBβEti^i,t+1L+1ωBβωBβ(ωB+1ωBβωBβ)1(i^i,tS+lnυ^i,tB).(161)

The real long term bond yield r^i,tL depends on its expected future value, driven by the contemporaneous real short term bond yield adjusted by the duration risk premium, according to:

r^i,tL=ωBβEtr^i,t+1L+1ωBβωBβ(ωB+1ωBβωBβ)1(r^i,tS+lnυ^i,tB).(162)

The term premium lnμ^i,tB depends on its expected future value driven by the contemporaneous duration risk premium according to:

lnμ^i,tB=ωBβlnμ^i,tB+1ωBβωBβ(ωB+1ωBβωBβ)1lnυ^i,tB.(163)

The duration risk premium lnυ^i,tB satisfies dynamic factor process lnυ^i,tB=λkBj=1NwjAlnν^j,tB+(1λkBwiA)lnν^i,tB. The intensity of international bond market contagion varies across economies, with k = 0 for low capital market contagion economies, k = 1 for medium capital market contagion economies, and k = 2 for high capital market contagion economies, where λ0B<λ1B<λ2B.

The price of equity lnV^i,tS depends on its expected future value driven by expected future nonfinancial corporate profits, and the contemporaneous nominal short term bond yield adjusted by the equity risk premium, according to stock market relationship:

lnV^i,tS=βEtlnV^i,t+1S+(1β)EtlnΠ^i,t+1S(i^i,tS+lnυ^i,tS).(164)

Nonfinancial corporate profits lnΠ^i,tS depends on contemporaneous nominal output and the corporate tax rate according to nonfinancial corporate profit function

lnΠ^i,tS=λiS(ΠiSPiYYi)1{(1τi)[ln(P^i,tYY^i,t)λWiLiPiYYiln(W^i,tL^i,t)11τi(1WiLiPiYYi)τ^i,tK](165)+λBiC,FPiYYi[lnB^i,t+1C,F(1δ)(1+iiC)(i^i,tC,Eδ^i,tC+lnB^i,tC,F)]λIiKYiln(P^i,tCI^i,tK)},

where economy specific auxiliary parameter λiS=11τiΠisPiYYi. The response coefficients of this relationship vary across economies with the size of their government and their labor income share. The equity risk premium lnυ^i,tS satisfies dynamic factor process lnυ^i,tS=λkSj=1NwjAlnν^j,tS+(1λkSwiA)lnν^i,tS. The intensity of international stock market contagion varies across economies, with k = 0 for low capital market contagion economies, k = 1 for medium capital market contagion economies, and k = 2 for high capital market contagion economies, where λ0S<λ1S<λ2S.

Under a flexible inflation targeting or managed exchange rate regime, the nominal policy interest rate i^i,tP depends on a weighted average of its past and desired values according to monetary policy rule:

i^i,tP=ρii^i,t1P+(1ρi)(ξπEtπ^i,t+1C+ξYlnY^^i,t+ξjΔln^i,t)+ν^i,tiP.(166)

Under a flexible inflation targeting regime j = 0, and the desired nominal policy interest rate responds to expected future consumption price inflation and the contemporaneous output gap. For the leader economy of a monetary union, the target variables entering into its monetary policy rule are expressed as output weighted averages across union members. Under a managed exchange rate regime j = 1, and the desired nominal policy interest rate also responds to the contemporaneous change in the nominal effective exchange rate. Under a fixed exchange rate regime, the nominal policy interest rate instead tracks the contemporaneous nominal policy interest rate of the economy that issues the anchor currency one for one, while responding to the contemporaneous change in the corresponding nominal bilateral exchange rate, according to monetary policy rule:

i^i,tP=i^k,tP+ξkΔln^i,k,t.(167)

It follows that under a fixed exchange rate regime, ln^i,k,t=0 in equilibrium, in the absence of asymmetric credit and currency risk premium shocks. The real policy interest rate r^i,tP satisfies r^i,tP=i^i,tPEtπ^i,t+1C.

Bank credit depends on a weighted average of the contemporaneous money and bank capital stocks according to bank balance sheet identity

lnB^i,t+1C,B=(1κR)lnM^i,t+1S+κRlnK^i,t+1B,(168)

which determines the money stock lnM^i,t+1S. Bank credit lnB^i,t+1C,B depends on a weighted average of contemporaneous mortgage debt, and the bank lending weighted average of contemporaneous domestic currency denominated domestic and foreign nonfinancial corporate debt, according to bank credit demand function:

lnB^i,t+1C,B=wiClnB^i,t+1C,D+(1wiC)j=1Nwi,jClnB^j,t+1C,F^j,i,t.(169)

Mortgage debt lnB^i,t+1C,D satisfies lnB^i,t+1C,D=lnP^i,tC+lnH^i,t+1+ϕ^i,tDϕD, while nonfinancial corporate debt lnB^i,t+1C,F satisfies lnB^i,t+1C,F=lnP^i,tC+lnK^i,t+1+ϕi,tFϕF. The bank capital ratio κ^i,t+1 satisfies κ^i,t+1=κR(lnK^i,t+1BlnB^i,t+1C,B).

The nominal effective corporate loan rate i^i,tC,E depends on the nonfinancial corporate borrowing weighted average of past domestic and foreign nominal corporate loan rates, adjusted for contemporaneous changes in nominal bilateral exchange rates, according to effective corporate loan rate function:

i^i,tC,E=j=1Nwi,jF(i^j,t1C+ln^i,j,t^i,j,t1).(170)

The corporate credit loss rate δ^i,tC,E depends on the bank lending weighted average of contemporaneous domestic and foreign corporate loan default rates according to corporate credit loss rate function:

δ^i,tC,E=j=1Nwi,jCδ^j,tC.(171)

The real effective corporate loan rate Etr^i,t+1C,E satisfies Etr^i,t+1C,E=Eti^i,t+1C,EEtπ^i,t+1C.

The nominal mortgage and corporate loan rates i^i,tf(Z) depend on a weighted average of their past and expected future values, driven by the deviation of the past nominal interbank loans rate from the contemporaneous nominal mortgage or corporate loan rate net of the contemporaneous mortgage or corporate credit loss rate, according to lending rate Phillips curves

i^i,tf(Z)=11+βi^i,t1f(Z)+β1+βEti^i,t+1f(Z)+(1ωC)(1ωCβ)ωC(1+β){[i^i,t1B(i^i,tf(Z)δ^i,tg(Z))](172)1β(1χCδ)1+κR(1β(1χCδ))[ηC(κ^i,tκ^i,tR)(κ^i,tRκRi^i,t1B)]+lnϑ^i,tCZ},

where Z ∈ {D,F}, while f (D) = M and f (F) = C. The nominal mortgage and corporate loan rates also depend on the past deviation of the bank capital ratio from its required value, as well as the past deviation of the regulatory bank capital ratio requirement from its funding cost, where g(D) = M and g(F) = C,E. The real mortgage and corporate loan rates r^i,tf(Z) satisfy r^i,tf(Z)=i^i,tf(Z)Etπ^i,t+1C.

Bank retained earnings lnI^i,tB depends on a weighted average of its past and expected future values driven by the contemporaneous shadow price of bank capital according to retained earnings relationship:

lnI^i,tB=11+βlnI^i,t1B+β1+βEtlnI^i,t+1B+1χB(1+β)lnQ^i,tB.(173)

The shadow price of bank capital lnQ^i,tB depends on its expected future value net of the expected future bank capital destruction rate, as well as the contemporaneous nominal interbank loans rate, according to retained earnings Euler equation:

lnQ^i,tB=Et{β(1χCδ)(lnQ^i,t+1Bδ^i,t+1B)[i^i,tB+(1β(1χCδ))ηCκR(κ^i,t+1κ^i,t+1R)]}.(174)

The shadow price of bank capital also depends on the contemporaneous deviation of the bank capital ratio from its required value. The bank capital stock lnK^i,t+1B is accumulated according to lnK^i,t+1B=(1χCδ)(lnK^i,tBδ^i,tB)+χCδlnI^i,tB, where the bank capital destruction rate δ^i,tB satisfies δ^i,tB=χC(wiCδ^i,tM+(1wiC)δ^i,tC,E).

The regulatory bank capital ratio requirement κ^i,t+1R depends on a weighted average of its past and desired values according to countercyclical capital buffer rule:

κ^i,t+1R=ρκκ^i,tR+(1ρκ)(ζκ,BΔlnB^i,t+1C,B+ζκ,VHΔlnV^i,tH+ζκ,VSΔlnV^i,tS)+ν^i,tκ.(175)

The desired regulatory bank capital ratio requirement responds to contemporaneous bank credit growth, as well as to contemporaneous changes in the prices of housing and equity. The regulatory mortgage and corporate loan to value ratio limits ϕ^i,tZ depend on a weighted average of their past and desired values according to loan to value limit rules

ϕ^i,tZ=ρϕZϕ^i,t1Z(1ρϕZ)(ζϕZ,BΔlnB^i,t+1C,Z+ζϕZ,VΔlnV^i,tf(Z))+ν^i,tϕZ,(176)

where Z ∈ {D,F}, while f (D) = H and f (F) = S. The desired regulatory mortgage or corporate loan to value ratio limit responds to contemporaneous mortgage or nonfinancial corporate debt growth, as well as to the contemporaneous change in the price of housing or equity, respectively. The mortgage and corporate loan default rates δ^i,tZ depend on a weighted average of their past and attractor values according to default rate relationships

δ^i,tZ=ρδδ^i,t1Z(1ρδ)(ζδZ,YlnY^^i,t+ζδZ,VΔlnV^i,tf(Z))+ν^i,tδZ,(177)

where Z ∈ {M, C}, while f (M) = H and f (C) = S. The attractor loan default rate depends on the contemporaneous output gap, as well as the contemporaneous change in the price of housing or equity, respectively.

The real effective wage depends on a weighted average of its past and expected future values driven by the contemporaneous and past unemployment rates according to wage Phillips curve

lnW^i,tP^i,tCA˜^i,t=11+βlnW^i,t1P^i,t1CA˜^i,t1+β1+βEtlnW^i,t+1P^i,t+1CA˜^i,t+1(178)(1ωL)(1ωLβ)ωL(1+β)[1η11αL(u^i,tLαLu^i,t1L)lnϑi,tL]1+γLβ1+βP3(L)Δln(P^i,tCA˜^i,t),

which determines the nominal wage lnW^i,t. The real effective wage also depends on contemporaneous, past and expected future consumption price inflation and trend productivity growth, where polynomial in the lag operator P3(L)=1γL1+γLβLβ1+γLβEtL1. Wage inflation π^i,tW satisfies π^i,tW=lnW^i,tlnW^i,t1, while the unemployment rate u^i,tL satisfies u^i,tL=lnN^i,tlnL^i,t.

The unemployment rate depends on its past value driven by contemporaneous employment and the real effective wage according to labor supply relationship

u^i,tL=αLu^i,t1L(1αL)[ιlnL^i,tν^i,tNη(lnW^i,tP^i,tCA˜^i,tλ11τiτ^i,tL)],(179)

which determines the labor force lnN^i,t. The response coefficients of this relationship vary across economies with the size of their government.

Output depends on the contemporaneous utilized private physical capital stock and effective employment according to production function

lnY^i,t=(1θYθY1WiLiPiYYi)ln(u^i,tKK^i,t)+θYθY1WiLiPiYYiln(A^i,tL^i,t),(180)

which determines employment lnL^i,t. The output gap lnY^^i,t satisfies lnY^^i,t=lnY^i,tlnY˜^