Policy, Risk and Spillover Analysis in the World Economy: A Panel Dynamic Stochastic General Equilibrium Approach

Francis Vitek; Ulric Eriksson von Allmen

doi:10.5089/9781475592757.001.A001

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Policy, Risk and Spillover Analysis in the World Economy

A Panel Dynamic Stochastic General Equilibrium Approach

Author: Mr. Francis Vitek https://orcid.org/0000-0001-8661-7519 and Ulric Eriksson von Allmen

Contributor Notes

Author’s E-Mail Address: FVitek@imf.org

Publication Date:: 04 Apr 2017

ISBN:: 9781475592757

Language:: English

Keywords:: WP; marginal revenue

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Abstract

I. Introduction

Estimated dynamic stochastic general equilibrium models are widely used by monetary and fiscal authorities for policy analysis and forecasting purposes. This class of structural macroeconometric models has many variants, incorporating a range of nominal and real rigidities, and an expanding array of macrofinancial linkages. Its unifying feature is the derivation of approximate linear equilibrium conditions from constrained optimization problems facing households and firms, which interact with governments in an uncertain environment to determine equilibrium prices and quantities under rational expectations.

Developing and estimating a dynamic stochastic general equilibrium model of the world economy, disaggregated into a large number of national economies, presents unique challenges. Adequately accounting for international business cycle comovement requires sufficient spillover transmission channels, in particular international financial linkages. Coping with the curse of dimensionality, which manifests through explosions of the numbers of variables and parameters as the number of economies increases, requires targeted parameter restrictions.

This paper develops a structural macroeconometric model of the world economy, disaggregated into forty national economies, to facilitate multilaterally consistent macrofinancial policy, risk and spillover analysis. This panel dynamic stochastic general equilibrium model features a range of nominal and real rigidities, extensive macrofinancial linkages, and diverse spillover transmission channels. Following Smets and Wouters (2003), the model features short run nominal price and wage rigidities generated by monopolistic competition, staggered reoptimization, and partial indexation in the output and labor markets. Building on Christiano, Eichenbaum and Evans (2005), the resultant inertia in inflation and persistence in output is enhanced with other features such as habit persistence in consumption, adjustment costs in residential and business investment, and variable capital utilization. Following Galí (2011), the model incorporates involuntary unemployment though a reinterpretation of the labor market. Households are differentiated according to whether they are bank intermediated, capital market intermediated, or credit constrained. Bank intermediated households have access to domestic banks where they accumulate deposits, as well as to domestic property markets where they trade real estate, whereas capital market intermediated households have access to domestic and foreign capital markets where they trade money, bond and stock market securities. Motivated by Tobin (1969), these households solve portfolio balance problems, allocating their financial wealth across alternative assets which are imperfect substitutes. To cope with the curse of dimensionality, targeted parameter restrictions are imposed on the optimality conditions determining the solutions to these portfolio balance problems, avoiding the need to track the evolution of granular asset allocations. Firms are grouped into differentiated industries. The energy and nonenergy commodity industries produce internationally homogeneous goods for foreign absorption, while all other industries produce internationally heterogeneous goods for domestic and foreign absorption. Banks perform global financial intermediation subject to financial frictions and regulatory constraints. Building on Hülsewig, Mayer and Wollmershäuser (2009), they 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. Also building on Gerali, Neri, Sessa and Signoretti (2010), they obtain funding from domestic bank intermediated households via deposits and from the domestic money market via loans, accumulating bank capital out of retained earnings given credit losses to satisfy a regulatory capital requirement. Motivated by Kiyotaki and Moore (1997), the model incorporates financial accelerator mechanisms linked to borrowing by developers and firms collateralized against the values of the housing and physical capital stocks, respectively. Finally, following Monacelli (2005) the model accounts for short run incomplete exchange rate pass through with short run nominal price rigidities generated by monopolistic competition, staggered reoptimization, and partial indexation in the import markets. An approximate linear state space representation of the model is estimated by Bayesian maximum likelihood, conditional on prior information concerning the generally common values of structural parameters across economies.

A variety of monetary policy analysis, fiscal policy analysis, macroprudential policy analysis, spillover analysis, and forecasting applications of this estimated panel dynamic stochastic general equilibrium model of the world economy are demonstrated. These include quantifying the monetary, fiscal and macroprudential transmission mechanisms, accounting for business cycle fluctuations, and generating forecasts of inflation and output growth. The monetary, fiscal and macroprudential transmission mechanisms, as estimated with impulse responses, are broadly in line with the empirical literature, as are the drivers of business cycle fluctuations, as identified with historical decompositions. Sequential unconditional forecasts of inflation and output growth dominate a random walk in terms of predictive accuracy by wide margins, on average across economies and horizons.

This paper is the latest in a series that progressively develops a structural macroeconometric model of the world economy, disaggregated into forty national economies, to facilitate multilaterally consistent macrofinancial policy, risk and spillover analysis. It primarily adds a housing sector to the panel dynamic stochastic general equilibrium model developed in Vitek (2015), which in turn primarily added a banking sector to the model developed in Vitek (2014). These extensions have significantly enhanced the macrofinancial linkages embedded in the model while expanding its macroprudential policy analysis capabilities.

The organization of this paper is as follows. The next section develops a panel dynamic stochastic general equilibrium model of the world economy, while the following section describes an approximate multivariate linear rational expectations representation of it. Estimation of the model based on an approximate linear state space representation of it is the subject of section four. Policy and spillover analysis within the framework of the estimated model is conducted in section five, while forecasting is undertaken in section six. Finally, section seven offers conclusions and outlines future model development plans.

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 ϕ^B have access to domestic banks where they accumulate deposits, and to domestic property markets where they trade real estate, where 0 < ϕ^B < 1. In contrast, credit unconstrained households of type Z = A and measure ϕ^A have access to domestic and foreign capital markets where they trade financial assets, where 0 < ϕ^A < 1. Finally, credit constrained households of type Z = C and measure ϕ^C do not have access to banks or capital markets, where 0 < ϕ^C < 1 and ϕ^B + ϕ^A + ϕ^C = 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 C_h,i,s, housing H_h,i,s, labor supply ${L_{h, f, i, s}}_{f = 0}^{1}$ , real property balances $A_{h, i, s + 1}^{B, H} / P_{i, s}^{C}$ , and real portfolio balances $A_{h, i, s + 1}^{A, H} / P_{i, s}^{C}$ represented by intertemporal utility function

\begin{matrix} U_{h, i, t} = E_{t} Σ_{s = t}^{\infty} β^{s - t} u [C_{h, i, s}, H_{h, i, s}, {L_{h, f, i, s}}_{f = 0}^{1}, \frac{A_{h, i, s + 1}^{B, H}}{P_{i, s}^{C}}, \frac{A_{h, i, s + 1}^{A, H}}{P_{i, s}^{C}}], & (1) \end{matrix}

where E_t 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,

\begin{matrix} u (C_{h, i, s}, H_{h, i, s}, {L_{h, f, i, s}}_{f = 0}^{1}, \frac{A_{h, i, s + 1}^{B, H}}{P_{i, s}^{C}}, \frac{A_{h, i, s + 1}^{A, H}}{P_{i, s}^{C}}) = v_{i, s}^{c} [\frac{1}{1 - 1 / σ} (C_{h, i, s} - α^{c} \frac{C_{i, s - 1}^{z}}{φ^{z}})^{1 - 1 / σ} + \frac{υ_{i, s}^{H, H}}{1 - 1 / ζ} (H_{h, i, s})^{1 - 1 / ζ} \\ \begin{matrix} - υ_{i, s}^{L, H} \int_{0}^{1} \int_{α^{L} \frac{L_{f, i, s - 1}^{z}}{φ^{z}}}^{L_{h, f, i, s}} (g - α^{L} \frac{L_{f, i, s - 1}^{z}}{φ^{z}})^{1 / η} d g d f + \frac{υ_{i, s}^{B, H}}{1 - 1 / μ} {(\frac{A_{h, i, s + 1}^{B, H}}{P_{i, s}^{C}})}^{1 - 1 / μ} + \frac{υ_{i, s}^{B, H}}{1 - 1 / μ} {(\frac{A_{h, i, s + 1}^{A, H}}{P_{i, s}^{C}})}^{1 - 1 / μ}], & (2) \end{matrix} \end{matrix}

where 0 ≤ α^C < 1 and 0 ≤ α^L < 1. Endogenous preference shifters $υ_{i, s}^{H, H}, υ_{i, s}^{L, H}, υ_{i, s}^{B, H}$ and $υ_{i, s}^{A, H}$ depend on aggregate consumption or employment according to intratemporal subutility functions

\begin{matrix} \begin{matrix} υ_{i, s}^{H, H} = ν_{i}^{H, H} (\frac{C_{i, s}^{Z}}{φ^{Z}} - α^{c} \frac{C_{i, s - 1}^{Z}}{φ^{Z}})^{- 1 / σ} & (C_{i, s})^{1 / ζ}, \end{matrix} & (3) \end{matrix}

\begin{matrix} \begin{matrix} υ_{i, s}^{H, H} = A_{i, s} (\frac{C_{i, s}^{Z}}{φ^{Z}} - α^{c} \frac{C_{i, s - 1}^{Z}}{φ^{Z}})^{- 1 / σ} (\frac{L_{i, s} - α^{L} L_{i, s - 1}}{(L_{i, s} / N_{i, s})^{t}})^{- 1 / η} & , \end{matrix} & (4) \end{matrix}

\begin{matrix} \begin{matrix} υ_{i, s}^{B, H} = ν_{i}^{B, H} (\frac{C_{i, s}^{Z}}{φ^{Z}} - α^{c} \frac{C_{i, s - 1}^{Z}}{φ^{Z}})^{- 1 / σ} & (C_{i, s})^{1 / μ}, \end{matrix} & (5) \end{matrix}

\begin{matrix} \begin{matrix} υ_{i, s}^{A, H} = ν_{i}^{A, H} (\frac{C_{i, s}^{Z}}{φ^{Z}} - α^{c} \frac{C_{i, s - 1}^{Z}}{φ^{Z}})^{- 1 / σ} & (C_{i, s})^{1 / μ}, \end{matrix} & (6) \end{matrix}

where t > 0. The intratemporal utility function is strictly increasing with respect to consumption if and only if serially correlated consumption demand shock $V_{i, s}^{C}$ satisfies $V_{i, s}^{C} > 0$ . Given this parameter restriction, this intratemporal utility function is strictly increasing with respect to housing if and only if $V_{i}^{H, H} > 0$ , is strictly decreasing with respect to labor supply if and only if serially correlated labor supply shock $N_{i, s}$ satisfies $N_{i, s} > 0$ , is strictly increasing with respect to real property balances if and only if $V_{i}^{B, H} > 0$ , and is strictly increasing with respect to real portfolio balances if and only if $V_{i}^{A, H} > 0$ . 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, $V_{i}^{B, H}$ equates the marginal rate of substitution between real property balances and consumption to one, while $V_{i}^{A, H}$ 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 $A_{h, i, s}^{B, H}$ which yield return $i_{h, i, s}^{A^{B, H}}$ , and portfolio balances $A_{h, i, s}^{A, H}$ which yield return $i_{h, i, s}^{A^{A, H}}$ . Property balances are distributed across the values of bank deposits $B_{h, i, s}^{D, H}$ which bear interest at deposit rate $i_{i, s - 1}^{D}$ , and a real estate portfolio $S_{h, i, s}^{H, H}$ which yields return $i_{h, i, s}^{S^{H, H}}$ . It follows that $(1 + i_{h, i, s}^{A^{B, H}}) A_{h, i, s}^{B, H} = (1 + i_{i, s - 1}^{D}) B_{h, i, s}^{D, H} + (1 + i_{h, i, s}^{S^{H, H}}) S_{h, i, s}^{H, H} .$ . The value of this real estate portfolio is in turn distributed across the values of developer specific shares ${V_{i, e, s}^{H}, S_{h, i, e, s}^{H, H}}_{e = 0}^{1}$ , where $V_{i, e, s}^{H}$ denotes the price per share. It follows that $(1 + i_{h, i, s}^{S^{H, H}}) S_{h, i, s}^{H, H} = \int_{0}^{1} (Π_{i, e, s}^{H} + V_{i, e, s}^{H}) S_{h, i, e, s}^{H, H} d e$ de, where $Π_{i, e, s}^{H}$ denotes the dividend payment per share. Portfolio balances are distributed across the values of internationally diversified short term bond $B_{h, i, s}^{S, H}$ , long term bond $B_{h, i, s}^{L, H}$ and stock $S_{h, i, s}^{F, H}$ portfolios which yield returns $i_{h, i, s}^{B^{S, H}}, i_{h, i, s}^{B^{L, H}}$ and $i_{h, i, s}^{S^{F, H}}$ , respectively. It follows that $(1 + i_{h, i, s}^{A^{A, H}}) A_{h, i, s}^{A, H} = (1 + i_{h, i, s}^{B^{S, H}}) B_{h, i, s}^{S, H} + (1 + i_{h, i, s}^{B^{L, H}}) B_{h, i, s}^{L, H} + (1 + i_{h, i, s}^{S^{F, H}}) S_{h, i, s}^{F, H}$ . 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 ${ε_{i, j, s} B_{h, i, j, s}^{S, H}}_{j = 1}^{N}$ , long term bond ${ε_{i, j, s} B_{h, i, j, s}^{L, H}}_{j = 1}^{N}$ and stock ${ε_{i, j, s} S_{h, i, j, s}^{F, H}}_{j = 1}^{N}$ portfolios, where nominal bilateral exchange rate ε_i,j,s measures the price of foreign currency in terms of domestic currency. It follows that $(1 + i_{h, i, s}^{B^{S, H}}) B_{h, i, s}^{S, H} = Σ_{j = 1}^{N} E_{i, j, s} (1 + i_{j, s - 1}^{S}) B_{h, i, j, s}^{S, H}$ where $i_{j, s - 1}^{S}$ denotes the economy specific yield to maturity on short term bonds, $(1 + i_{h, i, s}^{B^{L, H}}) B_{h, i, s}^{L, H} = Σ_{j = 1}^{N} ε_{i, j, s} (1 + i_{h, i, j, s}^{B^{L, H}}) B_{h, i, j, s}^{L, H}$ where $i_{h, i, j, s}^{B^{L, H}}$ denotes the economy specific return on long term bonds, and $(1 + i_{h, i, s}^{S^{F, H}}) S_{h, i, s}^{F, H} = Σ_{j = 1}^{N} ε_{i, j, s} (1 + i_{h, i, j, s}^{S^{F, H}}) S_{h, i, j, s}^{F, H}$ where $i_{h, i, j, s}^{S^{F, H}}$ denotes the economy specific return on stocks. The local currency denominated values of economy specific long term bond portfolios ${B_{h, i, j, s}^{L, H}}_{j = 1}^{N}$ are in turn distributed across the values of economy and vintage specific long term bonds, ${{{V_{j, k, s}^{B} B}_{h, i, j, k, s}^{L, H}}_{k = 1}^{s - 1}}_{j = 1}^{N}$ where $V_{j, k, s}^{B}$ denotes the local currency denominated price per long term bond, with $V_{j, k, k}^{B} = 1$ . It follows that $(1 + i_{h, i, j, s}^{B^{L, H}}) B_{h, i, j, s}^{L, H} = Σ_{k = 1}^{s - 1} (Π_{j, k, s}^{B} + V_{j, k, s}^{B}) B_{h, i, j, k, s}^{L, H}$ , where $Π_{j, k, s}^{B} = (1 + i_{j, k}^{L} - ω^{B}) (ω^{B})^{s - k} V_{j, k, k}^{B}$ denotes the local currency denominated coupon payment per long term bond, and $i_{j, k}^{L}$ 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 ${S_{h, i, j, s}^{F, H}}_{j = 1}^{N}$ are distributed across the values of economy, industry and firm specific shares ${{{V_{j, k, l, s}^{S} S_{h, i, j, k, l, s}^{F, H}}_{l = 0}^{1}}_{k = 1}^{M}}_{j = 1}^{N}$ , where $V_{j, k, l, s}^{S}$ denotes the local currency denominated price per share. It follows that $(1 + i_{h, i, j, s}^{S^{F, H}}) S_{h, i, j, s}^{F, H} = Σ_{k = 1}^{M} \int_{0}^{1} (Π_{j, k, l, s}^{S} + V_{j, k, l, s}^{S}) S_{h, i, j, k, l, s}^{F, H} d l$ , where $Π_{j, k, l, s}^{S}$ denotes the local currency denominated dividend payment per share. During period s, the representative household receives profit income from banks $Π_{i, s}^{C}$ , and supplies differentiated intermediate labor services ${L_{h, f, i, s}}_{f = 0}^{1}$ , earning labor income at trade specific nominal wages ${W_{f, i, s}}_{f = 0}^{1}$ . The government levies a tax on household labor income at rate $τ_{i, s}^{L}$ , and remits household type specific lump sum transfer payment $T_{i, s}^{z}$ . These sources of wealth are summed in household dynamic budget constraint:

\begin{matrix} A_{h, i, s + 1}^{B, H} + A_{h, i, s + 1}^{A, H} = (1 + i_{h, i, s}^{A^{B, H}}) A_{h, i, s}^{B, H} + (1 + i_{h, i, s}^{A^{A, H}}) A_{h, i, s}^{A, H} + Π_{i, s}^{c} + (1 - τ_{i, s}^{L}) \int_{0}^{1} W_{f, i, s} L_{h, f, i, s} d f + T_{i, s}^{z} - P_{i, s}^{c} C_{h, i, s} - l_{i, s}^{H} H_{h, i, s} . & (7) \end{matrix}

According to this dynamic budget constraint, at the end of period s, the representative household holds property balances $A_{h, i, s + 1}^{B, H}$ and portfolio balances $A_{h, i, s + 1}^{A, H}$ . Property balances are allocated across the values of bank deposits $B_{h, i, s + 1}^{D, H}$ and the real estate portfolio $S_{h, i, s + 1}^{H, H}$ , that is $A_{h, i, s + 1}^{B, H} {{= B}_{h, i, s + 1}^{D, H} + S}_{h, i, s + 1}^{H, H}$ . The value of this real estate portfolio is in turn allocated across the values of developer specific shares ${V_{i, e, s}^{H} S_{h, i, e, s + 1}^{H, H}}_{e = 0}^{1}$ subject to $S_{h, i, s + 1}^{H, H} = \int_{0}^{1} V_{i, e, s}^{H} S_{h, i, e, s + 1}^{H, H} d e$ . Portfolio balances are allocated across the values of internationally diversified short term bond $B_{h, i, s + 1}^{S, H}$ , long term bond $B_{h, i, s + 1}^{L, H}$ and stock portfolios $S_{h, i, s + 1}^{F, H}$ , that is ${A_{h, i, s + 1}^{A, H} = B_{h, i, s + 1}^{S, H} + B}_{h, i, s + 1}^{L, H} + S_{h, i, s + 1}^{F, H}$ . 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 ${ε_{i, j, s} B_{h, i, j, s + 1}^{S, H}}_{j = 1}^{N}$ , long term bond ${ε_{i, j, s} B_{h, i, j, s + 1}^{L, H}}_{j = 1}^{N}$ and stock ${ε_{i, j, s} S_{h, i, j, s + 1}^{F, H}}_{j = 1}^{N}$ portfolios subject to $B_{h, i, s + 1}^{S, H} = Σ_{j = 1}^{N} ε_{i, j, s} B_{h, i, j, s + 1}^{S, H}, B_{h, i, s + 1}^{L, H} = Σ_{j = 1}^{N} E_{i, j, s} B_{h, i, j, s + 1}^{L, H}$ and $S_{h, i, s + 1}^{F, H} = Σ_{j = 1}^{N} E_{i, j, s} S_{h, i, j, s + 1}^{F, H}$ , respectively. The local currency denominated values of economy specific long term bond portfolios ${B_{h, i, j, s + 1}^{L, H}}_{j = 1}^{N}$ are in turn allocated across the local currency denominated values of economy and vintage specific long term bonds ${{V_{j, k, s}^{B} B_{h, i, j, k, s + 1}^{L, H}}_{k = 1}^{s}}_{j = 1}^{N}$ subject to $B_{h, i, j, s + 1}^{L, H} = Σ_{k = 1}^{s} V_{j, k, s}^{B} B_{h, i, j, k, s + 1}^{L, H}$ . In parallel, the local currency denominated values of economy specific stock portfolios ${S_{h, i, j, s + 1}^{F, H}}_{j = 1}^{N}$ are allocated across the local currency denominated values of economy, industry and firm specific shares ${{{V_{j, k, l, s}^{S} S_{h, i, j, k, l, s + 1}^{F, H}}_{l = 0}^{1}}_{k = 1}^{M}}_{j = 1}^{N}$ subject to $S_{h, i, j, s + 1}^{F, H} = Σ_{k = 1}^{M} \int_{0}^{1} V_{j, k, l, s}^{S} S_{h, i, j, k, l, s + 1}^{F, H} d l$ . Finally, the representative household purchases final private consumption good C_h,i,s at price $P_{i, s}^{C}$ , and rents final housing service H_h,i,s at price $l_{i, s}^{H}$ .

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 $B_{h, i, s + 1}^{D, H} / P_{i, s}^{C}$ and the real estate portfolio $S_{h, i, s + 1}^{H, H} / P_{i, s}^{C}$ are represented by constant elasticity of substitution intratemporal subutility function

\begin{matrix} \frac{A_{h, i, s + 1}^{B, H}}{P_{i, s}^{C}} = [(1 - φ^{H})^{\frac{1}{ψ^{H}}} (\frac{B_{h, i, s + 1}^{D, H}}{P_{i, s}^{C}})^{\frac{ψ^{H} - 1}{ψ^{H}}} + (φ^{H})^{\frac{1}{ψ^{H}}} (\frac{1}{V_{i, s}^{H}} \frac{S_{h, i, s + 1}^{H, H}}{P_{i, s}^{C}})^{\frac{ψ^{H} - 1}{ψ^{H}}}]^{\frac{ψ^{H}}{ψ^{H} - 1}}, & (8) \end{matrix}

where serially correlated housing risk premium shock $V_{i, s}^{H}$ satisfies $V_{i, s}^{H} > 0$ , while 0 ≤ ϕ^H ≤ 1 and ψ^H > 0. Preferences over the real values of developer specific shares ${V_{i, e, s}^{H} S_{h, i, e, s + 1}^{H, H} / P_{i, s}^{C}}_{e = 0}^{1}$ are in turn represented by constant elasticity of substitution intratemporal subutility function:

\begin{matrix} \frac{S_{h, i, s + 1}^{H, H}}{P_{i, s}^{C}} = {[\int_{0}^{1} (\frac{V_{i, e, s}^{H} S_{h, i, e, s + 1}^{H, H}}{P_{i, s}^{C}})^{\frac{ψ^{H} - 1}{ψ^{H}}} d e]}^{\frac{ψ^{H}}{ψ^{H} - 1}} . & (9) \end{matrix}

In the limit as $V_{i}^{B, H} \to 0$ there is no capitalist spirit motive for holding real property balances, while in the limit as ψ^H → ∞ 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 ${C_{h, i, s}}_{s = t}^{\infty}$ , housing ${H_{h, i, s}}_{s = t}^{\infty}$ , labor force participation ${{N_{h, f, i, s}}_{f = 0}^{1}}_{s = t}^{\infty}$ , property balances ${A_{h, i, s + 1}^{B, H}}_{s = t}^{\infty}$ , bank deposit holdings ${B_{h, i, s + 1}^{D, H}}_{s = t}^{\infty}$ , and real estate holdings ${{S_{h, i, e, s + 1}^{H, H}}_{e = 0}^{1}}_{s = t}^{\infty}$ to maximize intertemporal utility function (1) subject to dynamic budget constraint (7), the applicable restrictions on financial asset holdings, and terminal nonnegativity constraints $B_{h, i, T + 1}^{D, H} \geq 0$ and $S_{h,, i, e, T + 1}^{H, H} \geq 0$ 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

\begin{matrix} E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} (1 + i_{h, i, t + 1}^{A^{B, H}}) = 1, & (10) \end{matrix}

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

\begin{matrix} \frac{u_{H} (h, i, t)}{u_{C} (h, i, t)} = \frac{l_{i, t}^{H}}{P_{i, t}^{C}}, & (11) \end{matrix}

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

\begin{matrix} \frac{u_{L_{f}} (h, f, i, t)}{u_{C} (h, i, t)} = (1 - τ_{i, t}^{L}) \frac{W_{f, i, t}}{P_{i, t}^{C}}, & (12) \end{matrix}

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

\begin{matrix} (1 - φ^{H}) {1 + E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{u_{C} (h, i, t)}{u_{A^{B} (h, i, t)}} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{h, i, t + 1}^{A^{B, H}}) - (1 + i_{i, t}^{D})]}^{1 - ψ^{H}} \\ \begin{matrix} + φ^{H} \int_{0}^{1} {1 + E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{u_{C} (h, i, t)}{u_{A^{B} (h, i, t)}} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [1 + i_{h, i, t + 1}^{A^{B, H}} - \frac{Π_{i, e, t + 1}^{H} + V_{i, e, t + 1}^{H}}{V_{i, e, t}^{H}}]}^{1 - ψ^{H}} d e = 1, & (13) \end{matrix} \end{matrix}

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

\begin{matrix} E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{i, t}^{D}) - \frac{Π_{i, e, t + 1}^{H} + V_{i, e, t + 1}^{H}}{V_{i, e, t}^{H}}] = - \frac{u_{A^{B} (h, i, t)}}{u_{C} (h, i, t)} (1 - \frac{1}{V_{i, t}^{H}}), & (14) \end{matrix}

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 ω^B where 0 < ω^B < 1, following Woodford (2001). Preferences over the real values of internationally diversified short term bond $B_{h, i, s + 1}^{S, H} / P_{i, s}^{C}$ , long term bond $B_{h, i, s + 1}^{L, H} / P_{i, s}^{C}$ and stock $S_{h, i, s + 1}^{F, H} / P_{i, s}^{C}$ portfolios are represented by constant elasticity of substitution intratemporal subutility function

\begin{matrix} \frac{A_{h, i, s + 1}^{A, H}}{P_{i, s}^{C}} = {[(φ_{M}^{A})^{\frac{1}{ψ^{A}}} (\frac{B_{h, i, s + 1}^{S, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}} + (φ_{B}^{A})^{\frac{1}{ψ^{A}}} (\frac{1}{υ_{i, s}^{B}} \frac{B_{h, i, s + 1}^{L, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}} + (φ_{S}^{A})^{\frac{1}{ψ^{A}}} (\frac{1}{υ_{i, s}^{S}} \frac{S_{h, i, s + 1}^{F, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}}]}^{\frac{ψ^{A}}{ψ^{A} - 1}}, & (15) \end{matrix}

where internationally and serially correlated duration risk premium shock $υ_{i, s}^{B}$ satisfies $υ_{i, s}^{B} > 0$ , and internationally and serially correlated equity risk premium shock $υ_{i, s}^{S}$ satisfies $υ_{i, s}^{S} > 0$ , while $0 \leq φ_{M}^{A} \leq 1, 0 \leq φ_{B}^{A} \leq 1, 0 \leq φ_{S}^{A} \leq 1, φ_{M}^{A} + φ_{B}^{A} + φ_{S}^{A} = 1$ and ψ^A > 0. Preferences over the real values of economy specific short term bond ${ε_{i, j, s} B_{h, i, j, s + 1}^{S, H} / P_{i, s}^{C}}_{j = 1}^{N}$ , long term bond ${ε_{i, j, s} B_{h, i, j, s + 1}^{L, H} / P_{i, s}^{C}}_{j = 1}^{N}$ and stock ${ε_{i, j, s} S_{h, i, j, s + 1}^{F, H} / P_{i, s}^{C}}_{j = 1}^{N}$ portfolios are in turn represented by constant elasticity of substitution intratemporal subutility functions

\begin{matrix} \frac{B_{h, i, s + 1}^{S, H}}{P_{i, s}^{C}} = [Σ_{j = 1}^{N} (φ_{i, j}^{B})^{\frac{1}{ψ^{A}}} (\frac{1}{V_{j, s}^{\in}} \frac{\in_{i, j, s} B_{h, i, j, s + 1}^{S, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}}]^{\frac{ψ^{A}}{ψ^{A} - 1}}, & (16) \end{matrix}

\begin{matrix} \frac{B_{h, i, s + 1}^{L, H}}{P_{i, s}^{C}} = [Σ_{j = 1}^{N} (φ_{i, j}^{B})^{\frac{1}{ψ^{A}}} (\frac{1}{V_{j, s}^{\in}} \frac{\in_{i, j, s} B_{h, i, j, s + 1}^{L, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}}]^{\frac{ψ^{A}}{ψ^{A} - 1}}, & (17) \end{matrix}

\begin{matrix} \frac{S_{h, i, s + 1}^{F, H}}{P_{i, s}^{C}} = [Σ_{j = 1}^{N} (φ_{i, j}^{S})^{\frac{1}{ψ^{A}}} (\frac{1}{V_{j, s}^{\in}} \frac{\in_{i, j, s} S_{h, i, j, s + 1}^{F, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}}]^{\frac{ψ^{A}}{ψ^{A} - 1}}, & (18) \end{matrix}

where serially correlated currency risk premium shocks $V_{j, s}^{ɛ}$ satisfy $V_{j, s}^{ɛ} > 0$ , while $0 \leq φ_{i, j}^{B} \leq 1$ , $Σ_{j = 1}^{N} φ_{i, j}^{B} = 1, 0 \leq φ_{i, j}^{S} \leq 1$ and $Σ_{j = 1}^{N} φ_{i, j}^{S} = 1$ . Finally, preferences over the real values of economy and vintage specific long term bonds ${{ε_{i, j, s} V_{j, k, s}^{B} B_{h, i, j, k, s + 1}^{L, H} / P_{i, s}^{C}}_{k = 1}^{s}}_{j = 1}^{N}$ and economy, industry and firm specific shares ${{{ε_{i, j, s} V_{j, k, l, s}^{S} S_{h, i, j, k, s + 1}^{F, H} / P_{i, s}^{C}}_{l = 0}^{1}}_{k = 1}^{M}}_{j = 1}^{N}$ are represented by constant elasticity of substitution intratemporal subutility functions

\begin{matrix} \frac{ε_{i, j, s} B_{h, i, j, s + 1}^{L, H}}{P_{i, s}^{C}} = [Σ_{k = 1}^{s} (φ_{i, j, k, s}^{B})^{\frac{1}{ψ^{A}}} (\frac{ε_{i, j, s} V_{j, k, s}^{B} B_{h, i, j, s + 1}^{L, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}}]^{\frac{ψ^{A}}{ψ^{A} - 1}}, & (19) \end{matrix}

\begin{matrix} \frac{ε_{i, j, s} S_{h, i, j, s + 1}^{F, H}}{P_{i, s}^{C}} = {[Σ_{k = 1}^{M} (φ_{i, j, k}^{S})^{\frac{1}{ψ^{A}}} \int_{0}^{1} (\frac{ε_{i, j, s} V_{j, k, l, s}^{S} S_{h, i, j, k, l, s + 1}^{F, H}}{P_{i, s}^{C}})^{\frac{ψ^{A} - 1}{ψ^{A}}} d l]}^{\frac{ψ^{A}}{ψ^{A} - 1}}, & (20) \end{matrix}

where $0 \leq φ_{i, j, k, s}^{B} \leq 1, Σ_{k = 1}^{s} φ_{i, j, k, s}^{B} = 1, 0 \leq φ_{i, j, k}^{S} \leq 1$ and $Σ_{k = 1}^{M} φ_{i, j, k}^{S} = 1$ . In the limit as $V_{i}^{A, H} \to 0$ there is no capitalist spirit motive for holding real portfolio balances, while in the limit as ψ^A → ∞ 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 ${C_{h, i, s}}_{s = t}^{\infty}$ , housing ${H_{h, i, s}}_{s = t}^{\infty}$ , labor force participation ${{N_{h, f, i, s}}_{f = 0}^{1}}_{s = t}^{\infty}$ , portfolio balances ${A_{h, i, s + 1}^{A, H}}_{s = t}^{\infty}$ , short term bond holdings ${{B_{h, i, j, s + 1}^{S, H}}_{j = 1}^{N}}_{s = t}^{\infty}$ , long term bond holdings ${{{B_{h, i, j, k, s + 1}^{L, H}}_{k = 1}^{t}}_{j = 1}^{N}}_{s = t}^{\infty}$ , and stock holdings ${{{{S_{h, i, j, k, l, s + 1}^{H}}_{l = 0}^{1}}_{k = 1}^{M}}_{j = 1}^{N}}_{s = t}^{\infty}$ to maximize intertemporal utility function (1) subject to dynamic budget constraint (7), the applicable restrictions on financial asset holdings, and terminal nonnegativity constraints $B_{h, i, j, T + 1}^{S, H} \geq 0, B_{h, i, j, k, T + 1}^{L, H} \geq 0$ and $S_{h, i, j, k, l, T + 1}^{H} \geq 0$ 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

\begin{matrix} E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} (1 + i_{h, i, t + 1}^{A^{A, H}}) = 1, & (21) \end{matrix}

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

\begin{matrix} \frac{u_{H} (h, i, t)}{u_{C} (h, i, t)} = \frac{l_{i, t}^{H}}{P_{i, t}^{C}}, & (22) \end{matrix}

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

- \begin{matrix} \frac{u_{L_{f}} (h, f, i, t)}{u_{C} (h, i, t)} = (1 - τ_{i, t}^{L}) \frac{W_{f, i, t}}{P_{i, t}^{C}}, & (23) \end{matrix}

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

\begin{matrix} φ_{M}^{A} Σ_{j = 1}^{N} φ_{i, j}^{B} {1 + E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{u_{C} (h, i, t)}{u_{A^{A}} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{h, i, t + 1}^{A^{A, H}}) - (1 + i_{j, t}^{s}) \frac{ε_{i, j, t + 1}}{ε_{i, j, t}}]}^{1 - ψ^{A}} \\ + φ_{M}^{A} Σ_{j = 1}^{N} φ_{i, j}^{B} Σ_{k = 1}^{t} φ_{i, j, k, t}^{B} {1 + E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{u_{C} (h, i, t)}{u_{A^{A}} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{h, i, t + 1}^{A^{A, H}}) - \frac{Π_{j, k, t + 1}^{B} + V_{j, k, t + 1}^{B}}{V_{j, k, t}^{B}} \frac{ε_{i, j, t + 1}}{ε_{i, j, t}}]}^{1 - ψ^{A}} \\ \begin{matrix} + φ_{S}^{A} Σ_{j = 1}^{N} φ_{i, j}^{S} Σ_{k = 1}^{M} φ_{i, j, k}^{S} \int_{0}^{1} {1 + E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{u_{C} (h, i, t)}{u_{A^{A}} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{h, i, t + 1}^{A^{A, H}}) - \frac{Π_{j, k, l, t + 1}^{S} + V_{j, k, l, t + 1}^{S}}{V_{j, k, t}^{S}} \frac{ε_{i, j, t + 1}}{ε_{i, j, t}}]}^{1 - ψ^{A}} d l = 1, & (24) \end{matrix} \end{matrix}

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

\begin{matrix} E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{i, t}^{S}) - (1 + i_{j, t}^{S}) \frac{ε_{i, j, t + 1}}{ε_{i, j, t}}] = - \frac{u_{A^{A}} (h, i, t)}{u_{C} (h, i, t)} (\frac{1}{V_{i, t}^{ε}} - \frac{1}{V_{i, t}^{ε}}), & (25) \end{matrix}

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

\begin{matrix} E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{i, t}^{S}) - \frac{Π_{i, k, t + 1}^{B} + V_{i, k, t + 1}^{B}}{V_{i, k, t}^{B}}] = - \frac{u_{A^{A}} (h, i, t)}{u_{C} (h, i, t)} \frac{1}{V_{i, t}^{ε}} (1 - \frac{1}{υ_{i, t}^{B}}), & (26) \end{matrix}

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

\begin{matrix} E_{t} \frac{β u_{C} (h, i, t + 1)}{u_{C} (h, i, t)} \frac{P_{i, t}^{C}}{P_{i, t + 1}^{C}} [(1 + i_{i, t}^{S}) - \frac{Π_{i, k, l, t + 1}^{S} + V_{i, k, l, t + 1}^{S}}{V_{i, k, l, t}^{S}}] = - \frac{u_{A^{A}} (h, i, t)}{u_{C} (h, i, t)} \frac{1}{V_{i, t}^{ε}} (1 - \frac{1}{υ_{i, t}^{S}}), & (27) \end{matrix}

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 ${C_{h, i, s}}_{s = t}^{\infty}$ , housing ${H_{h, i, s}}_{s = t}^{\infty}$ , and labor force participation ${{N_{h, f, i, s}}_{f = 0}^{1}}_{s = t}^{\infty}$ 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

\begin{matrix} P_{i, t}^{c} C_{h, i, t} + l_{i, t}^{H} H_{i, t} = Π_{i, t} + (1 - τ_{i, t}^{L}) \int_{0}^{1} W_{f, i, t} L_{h, f, i, t} d f + T_{i, t}^{C}, & (28) \end{matrix}

which equates the sum of consumption and housing expenditures to the sum of profit and disposable labor income net of transfers, where profit income Π_i,t satisfies $Π_{i, t} = Π_{i, t}^{H} + Π_{i, t}^{S} + Π_{i, t}^{C}$ . The evaluation of this result abstracts from international bank lending. Furthermore, these solutions satisfy intratemporal optimality condition

\begin{matrix} \frac{u_{H} (h, i, t)}{u_{C} (h, i, t)} = \frac{l_{i, t}^{H}}{P_{i, t}^{C}}, & (29) \end{matrix}

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

- \begin{matrix} \frac{u_{L_{f}} (h, f, i, t)}{u_{C} (h, i, t)} = (1 - τ_{i, t}^{L}) \frac{W_{f, i, t}}{P_{i, t}^{C}}, & (30) \end{matrix}

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 $u_{i, t}^{L}$ measures the share of the labor force N_i,t in unemployment U_i,t, that is $u_{i, t}^{L} = U_{i, t} / N_{i, t}$ , where unemployment equals the labor force less employment L_i,t, that is U_i,t = N_i,t – L_i,t. The labor force satisfies $N_{i, t} \int_{0}^{1} N_{f, i, t} d f$ .

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

\begin{matrix} L_{i, t} = {[\int_{0}^{1} (L_{f, i, t})^{\frac{θ_{i, t}^{L} - 1}{θ_{i, t}^{L}}} d f]}^{\frac{θ_{i, t}^{L}}{θ_{i, t}^{L} - 1}}, & (31) \end{matrix}

where serially uncorrelated wage markup shock $θ_{i, t}^{L}$ satisfies $θ_{i, t}^{L} > 1$ with $θ_{i}^{L} = θ^{L}$ . 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:

\begin{matrix} L_{f, i, t} = (\frac{W_{f, i, t}}{W_{i, t}})^{- θ_{i, t}^{L}} L_{i, t} . & (32) \end{matrix}

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

\begin{matrix} W_{i, t} = [\int_{0}^{1} (W_{f, i, t})^{1 - θ_{i, t}^{L}} d f]^{\frac{1}{1 - θ_{i, t}^{L}}} . & (33) \end{matrix}

As the wage elasticity of demand for intermediate labor services $θ_{i, t}^{L}$ 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 -ω^l of trade unions adjust their wage optimally, where 0≤ ω^L < 1. The remaining fraction ω^l of trade unions adjust their wage to account for past consumption price inflation and productivity growth according to partial indexation rule

\begin{matrix} W_{f, i, t} = {(\frac{P_{i, t - 1}^{C} A_{i, t - 1}}{P_{i, t - 2}^{C} A_{i, t - 2}})}^{r^{L}} {(\frac{{\bar{P}}_{i, t - 1}^{C} {\bar{A}}_{i, t - 1}}{{\bar{P}}_{i, t - 2}^{C} {\bar{A}}_{i, t - 2}})}^{{1 - r}^{L}} W_{f, i, t - 1}, & (34) \end{matrix}

where 0 ≤ γ ^L ≤ 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 $W_{i, t}^{*}$ given by necessary first order condition:

\begin{matrix} \frac{W_{i, t}^{*}}{W_{i, t}} = - \frac{E_{t} Σ_{s = t}^{\infty} (ω^{L})^{s - t} \frac{β^{s - t} u_{c} (h, i, s)}{u_{c} (h, i, t)} θ_{i, s}^{L} \frac{u_{L_{f} (h, f, i, s)}}{u_{c} (h, i, s)} [{(\frac{P_{i, s - 1}^{C} A_{i, t - 1}}{P_{i, s - 1}^{C} A_{i, s - 1}})}^{γ^{L}} {(\frac{{\bar{P}}_{i, t - 1}^{C} {\bar{A}}_{i, t - 1}}{{\bar{P}}_{i, s - 1}^{C} {\bar{A}}_{i, s - 1}})}^{1 - γ^{L}} {\frac{W_{i, s}}{W_{i, t}}]}^{θ_{i, s}^{L}} (\frac{W_{i, t}^{*}}{W_{i, t}})^{- θ_{i, s}^{L}} L_{h, i, s}}{E_{t} Σ_{s = t}^{\infty} (ω^{L})^{s - t} \frac{β^{s - t} u_{c} (h, i, s)}{u_{c} (h, i, t)} (θ_{i, s}^{L} - 1) (1 - τ_{i, s}^{L}) \frac{W_{i, s}}{P_{i, s}^{C}} [{(\frac{P_{i, t - 1}^{C} A_{i, t - 1}}{P_{i, s - 1}^{C} A_{i, s - 1}})}^{γ^{L}} {(\frac{{\bar{P}}_{i, t - 1}^{C} {\bar{A}}_{i, t - 1}}{{\bar{P}}_{i, s - 1}^{C} {\bar{A}}_{i, s - 1}})}^{1 - γ^{L}} {\frac{W_{i, s}}{W_{i, t}}]}^{θ_{i, s}^{L} - 1} (\frac{W_{i, t}^{*}}{W_{i, t}})^{- θ_{i, s}^{L}} L_{h, i, s}} . & (35) \end{matrix}

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 – ω^L of trade unions that adjust their wage optimally in period t, and the average of the wages set by the remaining fraction ω^L of trade unions that adjust their wage according to partial indexation rule (34):

\begin{matrix} W_{i, t} = {(1 - ω^{L}) (W_{i, t}^{*})^{1 - θ_{i, t}^{L}} + ω^{L} {[{(\frac{P_{i, t - 1}^{C} A_{i, t - 1}}{P_{i, t - 2}^{C} A_{i, t - 2}})}^{γ^{L}} {(\frac{{\bar{P}}_{i, t - 1}^{C} {\bar{A}}_{i, t - 1}}{{\bar{P}}_{i, t - 2}^{C} {\bar{A}}_{i, t - 2}})}^{1 - γ^{L}} W_{i, t - 1}]}^{1 - θ_{i, t}^{L}}}^{\frac{1}{1 - θ_{i, t}^{L}}} . & (36) \end{matrix}

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 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 H_i,e,t supplied by intermediate developers to produce final housing service H_i,t according to constant elasticity of substitution production function

\begin{matrix} H_{i, t} = [\int_{0}^{1} (H_{i, e, t})^{\frac{θ_{i, t}^{H} - 1}{θ_{i, t}^{H}}} d e]^{\frac{θ_{i, t}^{H}}{θ_{i, t}^{H} - 1}}, & (37) \end{matrix}

where endogenous rental price of housing markup $θ_{i, t}^{H}$ satisfies $θ_{i, t}^{H} > 1$ with $θ_{i}^{H} = θ^{H}$ . 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:

\begin{matrix} H_{i, e, t} = (\frac{l_{i, e, t}^{H}}{l_{i, t}^{H}})^{- θ_{i, t}^{H}} H_{i, t} . & (38) \end{matrix}

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:

\begin{matrix} l_{i, t}^{H} = {[\int_{0}^{1} (l_{i, e, t}^{H})^{1 - θ_{i, t}^{H}} d e]}^{\frac{1}{1 - θ_{i, t}^{H}}} . & (39) \end{matrix}

As the price elasticity of demand for intermediate housing services $θ_{i, t}^{H}$ 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 $V_{i, e, t}^{H}$ . 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

\begin{matrix} Π_{i, e, t}^{H} + V_{i, e, t}^{H} = E_{t} Σ_{s = t}^{\infty} \frac{β^{s - t} λ_{i, s}^{B}}{λ_{i, t}^{B}} Π_{i, e, s}^{H}, & (40) \end{matrix}

where $λ_{i, s}^{B}$ denotes the Lagrange multiplier associated with the period s bank intermediated household dynamic budget constraint. The derivation of this result imposes a transversality condition which rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to profits $Π_{i, e, s}^{H}$ , defined as the sum of earnings and net borrowing less residential investment expenditures:

\begin{matrix} Π_{i, e, s}^{H} = l_{i, e, s}^{H} H_{i, e, s} + (B_{i, e, s + 1}^{C, D} - (1 - δ_{i, s}^{M}) (1 + i_{i, s - 1}^{M}) B_{i, e, s}^{C, D}) - P_{i, s}^{I^{H}} I_{i, e, s}^{H} . & (41) \end{matrix}

Earnings are defined as revenues derived from sales of differentiated intermediate housing service H_i,e,s at rental price $l_{i, e, s}^{H}$ .

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,

\begin{matrix} \frac{B_{i, e, s + 1}^{C, D}}{P_{i, s}^{I^{H}} H_{i, e, s + 1}} = φ_{i, s}^{D}, & (42) \end{matrix}

given by regulatory mortgage loan to value ratio limit $φ_{i, s}^{D}$ . Net borrowing is defined as the increase in mortgage loans $B_{i, e, s + 1}^{C, D}$ from domestic banks net of writedowns at mortgage loan default rate $δ_{i, s}^{M}$ and interest payments at mortgage loan rate $i_{i, s - 1}^{M}$ .

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

\begin{matrix} H_{i, e, s + 1} = (1 - δ^{H}) H_{i, e, s} + H^{H} (I_{i, e, s}^{H}, I_{i, e, s - 1}^{H}), & (43) \end{matrix}

where 0 ≤ δ^H ≤ 1. Effective residential investment function $H^{H} (I_{i, e, s}^{H}, I_{i, e, s - 1}^{H})$ incorporates convex adjustment costs,

\begin{matrix} H^{H} (l_{i, e, s}^{H}, I_{i, e, s - 1}^{H}) = ν_{i, s}^{I^{H}} [1 - \frac{χ^{H}}{2} (\frac{I_{i, e, s}^{H}}{I_{i, e, s - 1}^{H}} - 1)^{2}] I_{i, e, s}^{H}, & (44) \end{matrix}

where serially correlated residential investment demand shock $V_{i, s}^{I^{H}}$ satisfies $V_{i, s}^{I^{H}} > 0$ , while χ^H > 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 ${I_{i, e, s}^{H}}_{s = t}^{\infty}$ and the housing stock ${H_{i, e, s + 1}}_{s = t}^{\infty}$ to maximize pre-dividend stock market value (40) subject to housing accumulation function (43) and terminal nonnegativity constraint H_i,e,T+1 ≥ 0 for T → ∞. In equilibrium, demand for the final residential investment good satisfies necessary first order condition

\begin{matrix} Q_{i, e, t}^{H} H_{1}^{H} (I_{i, e, t}^{H}, I_{i, e, t - 1}^{H}) + E_{t} \frac{β λ_{i, t + 1}^{B}}{λ_{i, t}^{B}} Q_{i, e, t + 1}^{H} H_{2}^{H} (I_{i, e, t + 1}^{H}, I_{i, e, t}^{H}) = P_{i, t}^{I^{H}}, & (45) \end{matrix}

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

\begin{matrix} Q_{i, e, t}^{H} = E_{t} \frac{β λ_{i, t + 1}^{B}}{λ_{i, t}^{B}} {P_{i, t + 1}^{I^{H}} {\frac{l_{i, e, t + 1}^{H}}{P_{i, t + 1}^{I^{H}}} - φ_{i, t}^{D} \frac{P_{i, t}^{I^{H}}}{P_{i, t + 1}^{I^{H}}} [(1 - δ_{i, t + 1}^{M}) (1 + i_{i, t}^{M}) - \frac{λ_{i, t}^{B}}{β λ_{i, t + 1}^{B}}]} + (1 - δ^{H}) Q_{i, e, t + 1}^{H}}, & (46) \end{matrix}

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 $l_{i, t}^{H, *}$ given by necessary first order condition:

\begin{matrix} \frac{l_{i, t}^{H, *}}{P_{i, t}^{I^{H}}} = \frac{θ_{i, t}^{H}}{θ_{i, t}^{H} - 1} [φ_{i, t - 1}^{D} (1 - δ_{i, t}^{M}) (1 + i_{i, t - 1}^{M}) \frac{P_{i, t - 1}^{I^{H}}}{P_{i, t}^{I^{H}}} - (1 - δ^{H}) \frac{Q_{i, e, t}^{H}}{P_{i, t}^{I^{H}}}] . & (47) \end{matrix}

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 $l_{i, t}^{H} {= l}_{i, t}^{H, *}$ .

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 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 ${Y_{i, k, t}}_{k = 1}^{M}$ to produce final output good Y_i,t according to fixed proportions production function

\begin{matrix} Y_{i, t} = \min {\frac{Y_{i, k, t}}{φ_{i, k}^{Y}}}_{k = 1}^{M}, & (48) \end{matrix}

where $0 \leq φ_{i, k}^{Y} \leq 1$ and $Σ_{k = 1}^{M} φ_{i, k}^{Y} = 1$ . 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:

\begin{matrix} Y_{i, k, t} = φ_{i, k}^{Y} Y_{i, t} . & (49) \end{matrix}

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:

\begin{matrix} P_{i, t}^{Y} = Σ_{k = 1}^{M} φ_{i, k}^{Y} P_{i, k, t}^{Y} . & (50) \end{matrix}

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 Y_i,k,l,t supplied by industry specific intermediate output good firms to produce industry specific final output good Y_i,k,t according to constant elasticity of substitution production function

\begin{matrix} Y_{i, k, t} = [\int_{0}^{1} (Y_{i, k, l, t})^{\frac{θ_{i, k, t}^{Y} - 1}{θ_{i, k, t}^{Y}}} d l]^{\frac{θ_{i, k, t}^{Y}}{θ_{i, k, t}^{Y} - 1}}, & (51) \end{matrix}

where serially uncorrelated output price markup shock $θ_{i, k, t}^{Y}$ satisfies $θ_{i, k, t}^{Y} > 1$ with $θ_{i, k}^{Y} = θ^{Y}$ , while $θ_{i, k, t}^{Y} = θ_{k, t}^{Y}$ for 1 ≤ k ≤ M * and $θ_{i, k, t}^{Y} = θ_{i, t}^{Y}$ 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:

\begin{matrix} Y_{i, k, l, t} = (\frac{P_{i, k, l, t}^{Y}}{P_{i, k, t}^{Y}})^{- θ_{i, k, t}^{Y}} Y_{i, k, t} . & (52) \end{matrix}

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:

\begin{matrix} P_{i, k, t}^{Y} = {[\int_{0}^{1} (P_{i, k, l, t}^{Y})^{1 - θ_{i, k, t}^{Y}} d l]}^{\frac{1}{1 - θ_{i, k, t}^{Y}}} . & (53) \end{matrix}

As the price elasticity of demand for industry specific intermediate output goods $θ_{i, k, t}^{Y}$ 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 $V_{i, k, l, t}^{S}$ . 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

\begin{matrix} Π_{i, k, l, t}^{S} + V_{i, k, l, t}^{S} = E_{t} Σ_{s = t}^{\infty} \frac{β^{s - t} λ_{i, s}^{A}}{λ_{i, t}^{A}} Π_{i, k, l, s}^{S}, & (54) \end{matrix}

where $λ_{i, s}^{A}$ 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 which rules out self-fulfilling speculative asset price bubbles.

Shares entitle households to dividend payments equal to net profits $Π_{i, k, l, s}^{S}$ , defined as the sum of after tax corporate earnings and net borrowing less business investment expenditures,

\begin{matrix} Π_{i, k, l, s}^{S} = (1 - τ_{i, s}^{K}) (P_{i, k, l, s}^{Y} Y_{i, k, l, s} - W_{i, s} L_{i, k, l, s} - Φ_{i, k, l, s}) + (B_{i, k, l, s + 1}^{C, F} - (1 - δ_{i, s}^{C}) + (1 + i_{i, s}^{C, E}) B_{i, k, l, s}^{C, F}) - P_{i, s}^{I^{K}} I_{i, k, l, s}^{K}, & (55) \end{matrix}

where $Y_{i, k, l, s} = F (u_{i, k, l, s}^{K} K_{i, k, l, s}, A_{i, s} L_{i, k, l, s})$ . Corporate earnings are defined as revenues derived from sales of industry specific differentiated intermediate output good Y_i,k,l,s at price $P_{i, k, l, s}^{Y}$ less expenditures on final labor service L_i,k,l,s, and other variable costs Φ_i,k,l,s. The government levies a tax on corporate earnings at rate $τ_{i, s}^{K}$ .

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 physical capital stock,

\begin{matrix} \frac{B_{i, k, l, s + 1}^{C, F}}{P_{i, s}^{I^{K}} K_{i, k, l, s + 1}} = φ_{i, s}^{F}, & (56) \end{matrix}

given by regulatory corporate loan to value ratio limit $φ_{i, s}^{F}$ . Net borrowing is defined as the increase in corporate loans $B_{i, k, l, s + 1}^{C, F}$ from domestic and foreign banks net of writedowns at corporate loan default rate $δ_{i, s}^{C}$ and interest payments at effective corporate loan rate $i_{i, s}^{C, E}$ . 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 physical capital K_i,k,l,s at rate $u_{i, k, l, s}^{K}$ and rents final labor service L_i,k,l,s to produce industry specific differentiated intermediate output good Y_i,k,l,s according to production function

\begin{matrix} F (u_{i, k, l, s}^{K} K_{i, k, l, s}, A_{i, s} L_{i, k, l, s}) = (u_{i, k, l, s}^{K} K_{i, k, l, s})^{φ_{i}^{K}} (A_{i, s} L_{i, k, l, s})^{1 - φ_{i}^{K}}, & (57) \end{matrix}

where serially correlated productivity shock $A_{i, s}$ satisfies $A_{i, s} > 0$ > 0. This production function exhibits constant returns to scale, with $0 \leq φ_{i}^{K} \leq 1$ .

In utilizing physical capital to produce output, the representative industry specific intermediate output good firm incurs a cost $G (u_{i, k, l, s}^{K}, K_{i, k, l, s})$ denominated in terms of business investment,

\begin{matrix} Φ_{i, k, l, s} = P_{i, s}^{I^{K}} G (u_{i, k, l, s}^{K}, K_{i, k, l, s}) + F_{i, k, s}^{F}, & (58) \end{matrix}

where industry specific fixed cost $F_{i, k, s}^{F}$ ensures that Φ_i,k,s = 0. Following Christiano, Eichenbaum and Evans (2005), this capital utilization cost is increasing in the capital utilization rate at an increasing rate,

\begin{matrix} G (u_{i, k, l, s}^{K}, K_{i, k, l, s}) = μ_{i}^{K} [e^{η^{K} (u_{i, k, l, s}^{K} - 1)} - 1] K_{i, k, l, s}, & (59) \end{matrix}

where η^k > 0, while $μ_{i}^{K} = \frac{μ^{K}}{1 - τ_{i}}$ with μ^k > 0. In steady state equilibrium, the capital utilization rate equals one, and the cost of utilizing physical capital equals zero.

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

\begin{matrix} K_{i, k, l, s + 1} = (1 - δ^{K}) K_{i, k, l, s} + H (I_{i, k, l, s}^{K}, I_{i, k, l, s - 1}^{K}), & (60) \end{matrix}

where 0 ≤ δ^K ≤ 1. Following Christiano, Eichenbaum and Evans (2005), effective business investment function $H (I_{i, k, l, s}^{K}, I_{i, k, l, s - 1}^{K})$ incorporates convex adjustment costs,

\begin{matrix} H (I_{i, k, l, s}^{K}, I_{i, k, l, s - 1}^{K}) = ν_{i, s}^{I^{K}} [1 - \frac{χ^{K}}{2} (\frac{I_{i, k, l, s}^{K}}{I_{i, k, l, s - 1}^{K}} - 1)^{2}] I_{i, k, l, s}^{K}, & (61) \end{matrix}

where serially correlated business investment demand shock $v_{i, s}^{I^{K}}$ satisfies $v_{i, s}^{I^{K}} > 0$ > 0, while χ^K > 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 ${L_{i, k, l, s}}_{s = t}^{\infty}$ , the capital utilization rate ${u_{i, k, l, s}^{K}}_{s = t}^{\infty}$ , business investment ${I_{i, k, l, s}^{K}}_{s = t}^{\infty}$ , and the physical capital stock ${K_{i, k, l, s + 1}}_{s = t}^{\infty}$ to maximize pre-dividend stock market value (54) subject to production function (57), physical capital accumulation function (60), and terminal nonnegativity constraint K_i,k,l,T+1 ≥ 0 for T → ∞. In equilibrium, demand for the final labor service satisfies necessary first order condition

\begin{matrix} F_{A L} (u_{i, k, l, t}^{K} K_{i, k, l, t} {, A}_{i, t} L_{i, k, l, t}) Ψ_{i, k, l, t} = (1 - τ_{i, t}^{K}) \frac{W_{i, t}}{P_{i, k, t}^{Y} A_{i, t}}, & (62) \end{matrix}

where $P_{i, k, s}^{Y} ψ_{i, k, l, s}$ denotes the Lagrange multiplier associated with the period s production technology constraint. This necessary first order condition equates real marginal cost Ψ_i,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

\begin{matrix} F_{u^{K} K} (u_{i, k, l, t}^{K} K_{i, k, l, t}, A_{i, t} L_{i, k, l, t}) \frac{P_{i, k, t}^{Y} Ψ_{i, k, l, t}}{P_{i, t}^{I^{K}}} = (1 - τ_{i, t}^{K}) \frac{G_{u} k (u_{i, k, l, t}^{K}, K_{i, k, l, t})}{K_{i, k, l, t}}, & (63) \end{matrix}

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

\begin{matrix} Q_{i, k, l, t}^{K} H_{1} (I_{i, k, l, t}^{K}, I_{i, k, l, t - 1}^{K}) + E_{t} \frac{β λ_{i, t + 1}^{A}}{λ_{i, t}^{A}} & Q_{i, k, l, t}^{K} H_{2} (I_{i, k, l, t + 1}^{K}, I_{i, k, l, t}^{K}) = P_{i, t}^{I^{K}}, & (64) \end{matrix}

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

\begin{matrix} \begin{matrix} Q_{i, k, l, t}^{K} = & E_{t} \frac{β λ_{i, t + 1}^{A}}{λ_{i, t}^{A}} {P_{i, t + 1}^{I^{K}} {u_{i, k, l, t + 1}^{K} F_{u^{K} K} (u_{i, k, l, t + 1}^{K} K_{i, k, l, t + 1}, A_{i, t + 1} L_{i, k, l, t + 1}) \frac{P_{i, k, t + 1}^{Y} Ψ_{i, k, l, t + 1}}{P_{i, t + 1}^{I^{K}}} \end{matrix} \\ - (1 - τ_{i, t + 1}^{K}) G_{K} (u_{i, k, l, t + 1}^{K}, K_{i, k, l, t + 1}) - φ_{i, t}^{F} \frac{P_{i, t}^{I^{K}}}{P_{i, t + 1}^{I^{K}}} [(1 - δ_{i, t + 1}^{C}) (1 + i_{i, t + 1}^{C, E}) - \frac{λ_{i, t}^{A}}{β λ_{i, t + 1}^{A}}]} + (1 - δ^{K}) Q_{i, k, l, t + 1}^{K}}, (65) \end{matrix}

which equates it to the expected present value of the sum of the future marginal revenue product of physical capital net of its marginal utilization cost, and the future shadow price of 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 $1 - ω_{k}^{Y}$ of industry specific intermediate output good firms adjust their price optimally, where $0 \leq ω_{k}^{Y} < 1$ with $ω_{k}^{Y} = ω^{Y}$ for k > M*. The remaining fraction $ω_{k}^{Y}$ of intermediate output good firms adjust their price to account for past industry specific output price inflation according to partial indexation rule

\begin{matrix} P_{i, k, l, t}^{Y} = (\frac{P_{i, k, t - 1}^{Y}}{P_{i, k, t - 2}^{Y}})^{γ_{k}^{Y}} (\frac{{\bar{P}}_{i, k, t - 1}^{Y}}{{\bar{P}}_{i, k, t - 2}^{Y}})^{1 - γ_{k}^{Y}} P_{i, k, l, t - 1}^{Y}, & (66) \end{matrix}

where $0 \leq γ_{k}^{Y} \leq 1$ with $γ_{k}^{Y} = 0$ for 1 ≤ k ≤ M* and $γ_{k}^{Y} = γ^{Y}$ 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 $P_{i, k, t}^{Y, *}$ given by necessary first order condition:

\begin{matrix} \frac{P_{i, k, t}^{Y, *}}{P_{i, k, t}^{Y}} = \frac{E_{t} Σ_{s = t}^{\infty} (ω_{k}^{Y})^{s - t} \frac{β^{s - t} λ_{i, s}^{A}}{λ_{i, t}^{A}} θ_{i, k, s}^{Y} Ψ_{i, k, l, s} {[(\frac{P_{i, k, t - 1}^{Y}}{P_{i, k, s - 1}^{Y}})^{γ_{k}^{Y}} (\frac{{\bar{P}}_{i, k, t - 1}^{Y}}{{\bar{P}}_{i, k, s - 1}^{Y}})^{1 - γ_{k}^{Y}} \frac{P_{i, k, s}^{Y}}{P_{i, k, t}^{Y}}]}^{θ_{i, k, s}^{Y}} {(\frac{P_{i, k, t}^{Y, *}}{P_{i, k, t}^{Y}})}^{- θ_{i, k, s}^{Y}} P_{i, k, s}^{Y} Y_{i, k, s}}{E_{t} Σ_{s = t}^{\infty} (ω_{k}^{Y})^{s - t} \frac{β^{s - t} λ_{i, s}^{A}}{λ_{i, t}^{A}} (θ_{i, k, s}^{Y} - 1) (1 - τ_{i, s}^{K}) [(\frac{P_{i, k, t - 1}^{Y}}{P_{i, k, s - 1}^{Y}})^{γ_{k}^{Y}} (\frac{{\bar{P}}_{i, k, t - 1}^{Y}}{{\bar{P}}_{i, k, s - 1}^{Y}})^{1 - γ_{k}^{Y}} \frac{P_{i, k, s}^{Y}}{P_{i, k, t}^{Y}}]^{θ_{i, k, s}^{Y} - 1} {(\frac{P_{i, k, t}^{Y, *}}{P_{i, k, t}^{Y}})}^{- θ_{i, k, s}^{Y}} P_{i, k, s}^{Y} Y_{i, k, s}} & (67) \end{matrix}

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 $1 - ω_{k}^{Y}$ of intermediate output good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction $ω_{k}^{Y}$ of intermediate output good firms that adjust their price according to partial indexation rule (66):

\begin{matrix} P_{i, k, t}^{Y} = {(1 - ω_{k}^{Y}) (P_{i, k, t}^{Y, *})^{1 - θ_{i, k, t}^{Y}} + ω_{k}^{Y} [(\frac{P_{i, k, t - 1}^{Y}}{P_{i, k, t - 2}^{Y}})^{γ_{k}^{Y}} (\frac{{\bar{P}}_{i, k, t - 1}^{Y}}{{\bar{P}}_{i, k, t - 2}^{Y}})^{1 - γ_{k}^{Y}} P_{i, k, t - 1}^{Y}]^{1 - θ_{i, k, t}^{Y}}}^{\frac{1}{1 - θ_{i, k, t}^{Y}}} . & (68) \end{matrix}

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 money 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 local currency denominated final corporate loans ${B_{i, j, t}^{C, F}}_{j = 1}^{N}$ to produce domestic currency denominated final corporate loan $B_{i, t}^{C, F}$ according to fixed proportions portfolio aggregator

B_{i, t}^{C, F} = \min \begin{matrix} {\frac{ε_{i, j, t - 1} B_{i, j, t}^{C, F}}{φ_{i, j}^{F}}}_{J - 1}^{N}, & (69) \end{matrix}

where $0 \leq ϕ_{i, j}^{F} \leq 1$ and $Σ_{j = 1}^{N} φ_{i, j}^{F} = 1$ . The representative global final bank maximizes profits derived from intermediation of the domestic currency denominated final corporate loan with respect to inputs of local currency denominated final corporate loans, implying demand functions:

B_{i, t}^{C, F} = φ_{i, j}^{F} \frac{B_{i, t}^{C, F}}{\in_{i, j, t - 1}} . (70)

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 + i_{i, t}^{C, E} = Σ_{j = 1}^{N} φ_{i, j}^{F} (1 + i_{j, t - 1}^{C}) \frac{\in_{i, j, t}}{\in_{i, j, t} - 1} . (71)

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 local currency denominated final corporate loans.

There exist a large number of perfectly competitive banks which combine differentiated intermediate mortgage or corporate loans $B_{i, m, t + 1}^{C^{Z}, B}$ supplied by intermediate banks to produce final mortgage or corporate loan $B_{i, t + 1}^{C^{Z}, B}$ according to constant elasticity of substitution portfolio aggregator

B_{i, t + 1}^{C^{Z}, B} = [\int_{0}^{1} (B_{i, m, t + 1}^{C^{Z}, B})^{\frac{θ_{i, t + 1}^{C^{Z}} - 1}{θ_{i, t + 1}^{C^{Z}}}} d m]^{\frac{θ_{i, t + 1}^{C^{Z}}}{θ_{i, t + 1}^{C^{Z}} - 1}}, (72)

where Z ∈ {D, F}, while serially uncorrelated mortgage or corporate loan rate markup shock $θ_{i, t + 1}^{C^{Z}}$ satisfies $θ_{i, t + 1}^{C^{Z}} > 1$ with $θ_{i}^{C^{Z}} = θ^{C}$ . 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

B_{i, m, t + 1}^{C^{Z}, B} = (\frac{1 + i_{i, m, t}^{f (Z)}}{1 + i_{i, t}^{f (Z)}})^{- θ_{i, t + 1}^{C^{Z}}} B_{i, t + 1}^{C^{Z}, B}, (73)

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 + i_{i, t}^{f (Z)} = [\int_{0}^{1} (1 + i_{i, m, t}^{f (Z)})^{1 - θ_{i, t + 1}^{C^{Z}}} d m]^{\frac{1}{1 - θ_{i, t + 1}^{C^{Z}}}} . (74)

As the rate elasticity of demand for intermediate mortgage or corporate loans $θ_{i, t + 1}^{C^{Z}}$ 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 $V_{i, m, t}^{C}$ . 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:

Π_{i, m, t}^{C} + V_{i, m, t}^{C} = E_{t} Σ_{s = t}^{\infty} \frac{β^{s - t} λ_{i, s}^{B}}{λ_{i, t}^{B}} Π_{i, m, s}^{C} . (75)

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

Shares entitle households to dividend payments $Π_{i, m, s}^{C}$ defined as profits derived from providing financial intermediation services less retained earnings $I_{i, m, s}^{B}$ :

\begin{matrix} \begin{matrix} Π_{i, m, s}^{C} = (B_{i, m, s + 1}^{D, B} - (1 + i_{i, s - 1}^{D}) B_{i, m, s}^{D, B}) + (B_{i, m, s + 1}^{S, B} - (1 + i_{i, s - 1}^{S}) B_{i, m, s}^{S, B}) \end{matrix} \\ - (B_{i, m, s + 1}^{C^{D}, B} - (1 - δ_{i, s}^{M}) (1 + i_{i, m, s - 1}^{M}) B_{i, m, s}^{C^{D}, B}) - (B_{i, m, s + 1}^{C^{F}, B} - (1 - δ_{i, s}^{C, E}) (1 + i_{i, m, s - 1}^{C}) B_{i, m, s}^{C^{F}, B}) - Φ_{i, m, s}^{B} - I_{i, m, s}^{B} . (76) \end{matrix}

Profits are defined as the sum of the increase in deposits $B_{i, m, s + 1}^{D, B}$ from domestic bank intermediated households net of interest payments at the deposit rate and the increase in loans $B_{i, m, s + 1}^{S, B}$ from the domestic money market net of interest payments at the yield to maturity on short term bonds, less the increase in differentiated intermediate mortgage loans $B_{i, m, s + 1}^{C^{D}, B}$ to domestic developers net of writedowns at mortgage credit loss rate $δ_{i, s}^{M}$ and interest receipts at mortgage loan rate $i_{i, m, s - 1}^{M}$ , less the increase in differentiated intermediate corporate loans $B_{i, m, s + 1}^{C^{F}, B}$ to domestic and foreign firms net of writedowns at corporate credit loss rate $δ_{i, s}^{C, E}$ and interest receipts at corporate loan rate $i_{i, m, s - 1}^{C}$ , less a cost of satisfying the regulatory capital requirement $Φ_{i, m, s}^{B}$ .

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

B_{i, m, s + 1}^{C^{D}, B} + B_{i, m, s + 1}^{C^{F}, B} = B_{i, m, s + 1}^{D, B} + B_{i, m, s + 1}^{S, B} + K_{i, m, s + 1}^{B} . (77)

The bank credit stock $B_{i, s + 1}^{C, B}$ measures aggregate bank assets, that is $B_{i, s + 1}^{C, B} = B_{i, s + 1}^{C^{D}, B} + B_{i, s + 1}^{C^{F}, B}$ , while the money stock $M_{i, s + 1}^{S}$ measures aggregate bank funding, that is $M_{i, s + 1}^{S} = B_{i, s + 1}^{D, B} + B_{i, s + 1}^{S, B}$ . The bank capital ratio κ_i,s+1 equals the ratio of aggregate bank capital to assets, that is $κ_{i, s + 1} = K_{i, s + 1}^{B} / B_{i, s + 1}^{C, B}$ .

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,

Φ_{i, m, s}^{B} = G^{B} (B_{i, m, s}^{C^{D}, B}, B_{i, m, s}^{C^{F}, B}, K_{i, m, s}^{B}) + F_{i, s}^{B}, (78)

where fixed cost $F_{i, s}^{B}$ ensures that $Φ_{i, s}^{B} = - I_{i, s}^{B}$ . 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,

G^{B} (B_{i, m, s}^{C^{D}, B}, B_{i, m, s}^{C^{F}, B}, K_{i, m, s}^{B}) = μ^{c} [_{e} (2 + η^{c}) (1 - \frac{1 K_{i, m, s}^{B}}{K_{i, s}^{R} B_{i, m, s}^{c^{D}, B} + B_{i, m, s}^{C^{F}, B}})_{- 1}] K_{i, m, s}^{B}, (79)

given regulatory capital requirement $κ_{i, s}^{R}$ , where η^C > 0 and μ^C > 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 $K_{i, m, s}^{B}$ , which subsequently evolves according to accumulation function

\begin{matrix} K_{i, m, s + 1}^{B} = (1 - δ_{i, s}^{B}) K_{i, m, s}^{B} + H^{B} (I_{i, m, s}^{B}, I_{i, m, s - 1}^{B}), & (80) \end{matrix}

where bank capital destruction rate $δ_{i, s}^{B}$ satisfies $δ_{i, s}^{B} = χ^{C} (w_{i}^{C} δ_{i, s}^{M} + (1 - w_{i}^{C}) δ_{i, s}^{C, E})$ with χ^C > 0, while mortgage loan weight $w_{i}^{C}$ satisfies $0 < w_{i}^{C} < 1$ . Effective retained earnings function $H^{B} (I_{i, m, s}^{B}, I_{i, m, s - 1}^{B})$ incorporates convex adjustment costs,

H^{B} (I_{i, m, s}^{B}, I_{i, m, s - 1}^{B}) = [1 - \frac{χ^{B}}{2} (\frac{I_{i, m, s}^{B}}{I_{i, m, s - 1}^{B}} - 1)^{2}] I_{i, m, s}^{B}, (81)

where χ^B > 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 ${B_{i, m, s + 1}^{D, B}}_{s = t}^{\infty}$ , money market funding ${B_{i, m, s + 1}^{S, B}}_{s = t}^{\infty}$ , retained earnings ${I_{i, m, s}^{B}}_{s = t}^{\infty}$ , and the bank capital stock ${K_{i, m, s + 1}^{B}}_{s = t}^{\infty}$ to maximize pre-dividend stock market value (75) subject to balance sheet identity (77), bank capital accumulation function (80), and terminal nonnegativity constraints $B_{i, m, T + 1}^{D, B} \geq 0, B_{i, m, T + 1}^{S, B} \geq 0$ and $K_{i, m, T + 1}^{B} \geq 0 f o r T \to \infty$ . In equilibrium, the solutions to this value maximization problem satisfy necessary first order condition

1 + i_{i, t}^{D} = 1 + i_{i, t}^{S}, (82)

which equates the deposit rate to the yield to maturity on short term bonds. In equilibrium, retained earnings satisfies necessary first order condition

Q_{i, m, t}^{B} H_{1}^{B} (I_{i, m, t}^{B}, I_{i, m, t - 1}^{B}) + E_{t} \frac{β λ_{i, t + 1}^{B}}{λ_{i, t}^{B}} Q_{i, m, t + 1}^{B} H_{2}^{B} (I_{i, m, t + 1}^{B}, I_{i, m, t}^{B}) = 1, (83)

which equates the expected present value of an additional unit of retained earnings to its marginal cost, where $Q_{i, m, s}^{B}$ 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

Q_{i, m, t}^{B} = E_{t} \frac{β λ_{i, t + 1}^{B}}{λ_{i, t}^{B}} {(1 - δ_{i, t + 1}^{B}) Q_{i, m, t + 1}^{B} - {G_{3}^{B} (B_{i, m, t + 1}^{C^{D}, B}, B_{i, m, t + 1}^{C^{F}, B}, K_{i, m, t + 1}^{B}) + [\frac{λ_{i, t}^{B}}{β λ_{i, t + 1}^{B}} - (1 + i_{i, t}^{S})]}}, (84)

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 money 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 − ω^C of intermediate banks adjust their gross mortgage and corporate loan rates optimally, where 0 ≤ ω^C < 1. The remaining fraction ω^C of intermediate banks do not adjust their loan rates,

1 + i_{i, m, t}^{f (Z)} = 1 + i_{i, m, t - 1}^{f (Z)}, (85)

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 (75) subject to balance sheet identity (77), intermediate loan demand function (73), 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 $i_{i, t}^{f (Z), *}$ given by necessary first order conditions

\frac{1 + i_{i, t}^{f (Z), *}}{1 + i_{i, t}^{f (Z)}} = \frac{E_{t} Σ_{s = t}^{\infty} (ω^{C})^{s - t} \frac{β^{s - t} λ_{i, s}^{B}}{λ_{i, t}^{B}} θ_{i, s}^{C^{Z}} \frac{(1 + i_{i, s - 1}^{S}) + G_{h (Z)}^{B} (B_{i, m, s}^{C^{D}, B}, B_{i, m, s}^{C^{F}, B}, K_{i, m, s}^{B})}{1 + i_{i, s - 1}^{f (Z)}} (\frac{1 + i_{i, s - 1}^{f (Z)}}{1 + i_{i, t}^{f (Z)}})^{θ_{i, s}^{C^{Z}}} (\frac{1 + i_{i, t}^{f (Z), *}}{1 + i_{i, t}^{f (Z)}})^{- θ_{i, s}^{C^{Z}}} (1 + i_{i, s - 1}^{f (Z)}) B_{i, s}^{C^{Z}, B}}{E_{t} Σ_{s = t}^{\infty} (ω^{C})^{s - t} \frac{β^{s - t} λ_{i, s}^{B}}{λ_{i, t}^{B}} (θ_{i, s}^{C^{Z}} - 1) (1 - δ_{i, s}^{g (Z)}) (\frac{1 + i_{i, s - 1}^{f (Z)}}{1 + i_{i, t}^{f (Z)}})^{θ_{i, s}^{C^{Z}} - 1} (\frac{1 + i_{i, t}^{f (Z), *}}{1 + i_{i, t}^{f (Z)}})^{- θ_{i, s}^{C^{Z}}} (1 + i_{i, s - 1}^{f (Z)}) B_{i, s}^{C^{Z}, B}}, (86)

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 (74) equals an average of the gross mortgage or corporate loan rate set by the fraction 1 − ω^C 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 ω^C of intermediate banks that do not adjust their loan rates:

1 + i_{i, t}^{f (Z)} = [(1 - ω^{C}) (1 + i_{i, t}^{f (Z), *})^{1 - θ_{i, t + 1}^{C^{Z}}} + ω^{C} (1 + i_{i, t - 1}^{f (Z)})^{1 - θ_{i, t + 1}^{C^{Z}}}]^{\frac{1}{1 - θ_{i, t + 1}^{C^{Z}}}} . (87)

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 ε_i,t measures the trade weighted average price of foreign currency in terms of domestic currency, while the real effective exchange rate $Q_{i, t}$ measures the trade weighted average price of foreign output in terms of domestic output,

ε_{i, t} = Π_{j = 1}^{N} (ε_{i, j, t})^{w_{i, j}^{T}}, Q_{i, t} = Π_{j = 1}^{N} (Q_{i, j, t})^{w_{i, j}^{T}}, (88)

where the real bilateral exchange rate $Q_{i, j, t}$ satisfies $Q_{i, j, t} = ε_{i, j, t} P_{j, t}^{Y} / P_{i, t}^{Y}$ , and bilateral trade weight $w_{i, j}^{T}$ satisfies $w_{i, i}^{T} = 0, 0 \leq w_{i, j}^{T} \leq 1$ and $Σ_{j = 1}^{N} w_{i, j}^{T} = 1$ . Furthermore, the terms of trade $T_{i, t}$ equals the ratio of the internal terms of trade to the external terms of trade,

T_{i, t} = υ_{t}^{T} \frac{T_{i, t}^{X}}{T_{i, t}^{M}}, T_{i, t}^{X} = \frac{P_{i, t}^{X}}{P_{i, t}}, T_{i, t}^{M} = \frac{P_{i, t}^{M}}{P_{i, t}}, (89)

where the internal terms of trade $T_{i, t}^{X}$ measures the relative price of exports, and the external terms of trade $T_{i, t}^{M}$ measures the relative price of imports, while P_i,t denotes the price of the final noncommodity output good. Finally, under the law of one price $ε_{i^{*}, i, t} P_{i, k, t}^{Y} = P_{k, t}^{Y}$ for 1 ≤ k ≤ M *, which implies that

P_{k, t}^{Y} = Σ_{i = 1}^{N} w_{i}^{Y} ε_{i *, i, t} P_{i, k, t}^{Y}, (90)

where $P_{k, t}^{Y}$ denotes the quotation currency denominated price of energy or nonenergy commodities, and world output weight $W_{i}^{Y}$ satisfies $0 < W_{i}^{Y} < 1$ and $Σ_{i = 1}^{N} W_{i}^{Y} = 1$ . Endogenous terms of trade shifter $υ_{t}^{T}$ adjusts to ensure multilateral consistency in nominal trade flows, and in steady state equilibrium satisfies $υ^{T} = 1$ .

The Export Sector

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

X_{i, t} = \min {\frac{X_{i, k, t}}{φ_{i, k}^{X}}}_{k = 1}^{M}, (91)

where X_i,k,t = Y_i,k,t for 1 ≤ k ≤ M *, while $0 \leq φ_{i, k}^{X} \leq 1$ and $Σ_{k = 1}^{M} φ_{i, k}^{X} = 1$ . The representative final export good firm maximizes profits derived from production of the final export good with respect to inputs of industry specific final output goods, implying demand functions:

X_{i, k, t} = φ_{i, k}^{X} X_{i, t} . (92)

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:

P_{i, t}^{X} = Σ_{k = 1}^{M} φ_{i, k}^{X} P_{i, k, t}^{Y} . (93)

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 output goods.

The Import Sector

There exist a large number of perfectly competitive firms which combine the final noncommodity output good $Z_{i, t}^{h} \in {C_{i, t}^{h}, I_{i, t}^{H, h}, I_{i, t}^{K, h}, G_{i, t}^{h}}$ with the final import good $Z_{i, t}^{f} \in {C_{i, t}^{f}, I_{i, t}^{H, f}, I_{i, t}^{K, f}, G_{i, t}^{f}}$ to produce final private consumption, residential investment, business investment or public consumption good $Z_{i, t} \in {C_{i, t}, I_{i, t}^{H}, I_{i, t}^{K}, G_{i, t}}$ according to constant elasticity of substitution production function

Z_{i, t} = [{(1 - φ_{i}^{M})}^{\frac{1}{ψ^{M}}} (Z_{i, t}^{h})^{\frac{ψ^{M} - 1}{ψ^{M}}} + (φ_{i}^{M})^{\frac{1}{ψ^{M}}} (v_{i, t}^{M} Z_{i, t}^{f})^{\frac{ψ^{M} - 1}{ψ^{M}}}]^{\frac{ψ^{M}}{ψ^{M} - 1}}, (94)

where serially correlated import demand shock $V_{i, t}^{M}$ satisfies $V_{i, t}^{M} > 0$ , while $0 \leq φ_{i}^{M} \leq 1$ and ψ^M > 0. The representative final absorption good firm maximizes profits derived from production of the final private consumption, residential investment, business investment or public consumption good, with respect to inputs of the final noncommodity output and import goods, implying demand functions:

Z_{i, t}^{h} = (1 - φ_{i}^{M}) (\frac{P_{i, t}}{P_{i, t}^{Z}})^{- ψ^{M}} Z_{i, t}, Z_{i, t}^{f} = φ_{i}^{M} (\frac{1}{v_{i}^{M}} \frac{P_{i, t}^{M}}{P_{i, t}^{Z}})^{- ψ^{M}} \frac{Z_{i, t}}{V_{i, t}^{M}} . (95)

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 or public consumption price index:

P_{i, t}^{Z} = [(1 - φ_{i}^{M}) (P_{i, t})^{1 - ψ^{M}} + φ_{i}^{M} (\frac{P_{i, t}^{M}}{V_{i}^{M}})^{1 - ψ^{M}}]^{\frac{1}{1 - ψ^{M}}} . (96)

Combination of this aggregate private consumption, residential investment, business investment or public consumption price index with final noncommodity output and import good demand functions (95) yields:

Z_{i, t}^{h} = (1 - φ_{i}^{M}) [(1 - φ_{i}^{M}) + φ_{i}^{M} (\frac{T_{i, t}^{M}}{v_{i}^{M}})^{1 - ψ^{M}}]^{\frac{ψ^{M}}{1 - ψ^{M}}} Z_{i, t}, Z_{i, t}^{f} = φ_{i}^{M} [φ_{i}^{M} + (1 - φ_{i}^{M}) (\frac{T_{i, t}^{M}}{V_{i}^{M}})^{ψ^{M} - 1}]^{\frac{ψ^{M}}{1 - ψ^{M}}} \frac{Z_{i, t}}{V_{i, t}^{M}} . (97)

These demand functions for the final noncommodity output and import goods are directly proportional to final private consumption, residential investment, business investment or public consumption 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 ${M_{i, j, t}}_{j = 1}^{N}$ to produce final import good M_i,t according to fixed proportions production function

M_{i, t} = \min {v_{j, t}^{X} \frac{M_{i, j, t}}{φ_{i, j}^{M}}}_{j = 1}^{N}, (98)

where serially correlated export demand shock $V_{i, t}^{X}$ satisfies $V_{i, t}^{X} > 0$ , while $φ_{i, i}^{M} = 0, 0 \leq φ_{i, j}^{M} \leq 1$ and $Σ_{j = 1}^{N} ϕ_{i, j}^{M} = 1$ . 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:

M_{i, j, t} = φ_{i, j}^{M} \frac{M_{i, t}}{V_{j, t}^{X}} . (99)

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:

P_{i, t}^{M} = Σ_{j = 1}^{N} φ_{i, j}^{M} \frac{P_{i, j, t}^{M}}{V_{j}^{X}} . (100)

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 M_i,j,n,t supplied by economy specific intermediate import good firms to produce economy specific final import good M_i,j,t according to constant elasticity of substitution production function

M_{i, j, t} = [\int_{0}^{1} (M_{i, j, n, t})^{\frac{θ_{i, t}^{M} - 1}{θ_{i, t}^{M}}} d n]^{\frac{θ_{i, t}^{M}}{θ_{i, t}^{M} - 1}}, (101)

where serially uncorrelated import price markup shock $θ_{i, t}^{M}$ satisfies $θ_{i, t}^{M} > 1$ with $θ_{i}^{M} = θ^{M}$ . 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:

M_{i, j, n, t} = (\frac{P_{i, j, n, t}^{M}}{P_{i, j, t}^{M}})^{- θ_{i, t}^{M}} M_{i, j, t} . (102)

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:

P_{i, j, t}^{M} = [\int_{0}^{1} (P_{i, j, n, t}^{M})^{1 - θ_{i, t}^{M}} d n]^{\frac{1}{1 - θ_{i, t}^{M}}} . (103)

As the price elasticity of demand for economy specific intermediate import goods $θ_{i, t}^{M}$ 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 $V_{i, j, n, t}^{M}$ . 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:

Π_{i, j, n, t}^{M} + V_{i, j, n, t}^{M} = E_{t} Σ_{s = t}^{\infty} \frac{β^{s - t} λ_{i, s}^{A}}{λ_{i, t}^{A}} Π_{i, j, n, s}^{M} . (104)

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

Shares entitle households to dividend payments equal to profits $Π_{i, j, n, s}^{M}$ , defined as earnings less economy specific fixed cost $F_{i, j, s}^{M}$ :

Π_{i, j, n, s}^{M} = P_{i, j, n, s}^{M} M_{i, j, n, s} - ε_{i, j, s} P_{j, s}^{X} M_{i, j, n, s} - F_{i, j, s}^{M} . (105)

Earnings are defined as revenues derived from sales of economy specific differentiated intermediate import good M_i,j,n,s at price $P_{i, j, n, s}^{M}$ less expenditures on foreign final export good M_i,j,n,s. The representative economy specific intermediate import good firm purchases the foreign final export good and differentiates it. Fixed cost $F_{i, j, s}^{M}$ ensures that $Π_{i, j, s}^{M} = 0$ .

In an extension of the model of nominal import price rigidity proposed by Monacelli (2005), each period a randomly selected fraction 1 − ω^M of economy specific intermediate import good firms adjust their price optimally, where 0 ≤ ω^M < 1. The remaining fraction ω^M 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

P_{i, j, n, t}^{M} = [(\frac{P_{i, j, t - 1}^{M}}{P_{i, j, t - 2}^{M}})^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{ε_{i, i *, t} P_{k, t}^{Y}}{ε_{i, i *, t - 1} P_{k, t - 1}^{Y}})^{μ_{i, k}^{M}}]^{γ^{M}} [(\frac{{\bar{P}}_{i, j, t - 1}^{M}}{{\bar{P}}_{i, j, t - 2}^{M}})^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{{\bar{ε}}_{i, i *, t} {\bar{P}}_{k, t}^{Y}}{{\bar{ε}}_{i, i *, t - 1} {\bar{P}}_{k, t - 1}^{Y}})^{μ_{i, k}^{M}}]^{1 - γ^{M}} P_{i, j, n, t - 1}^{M}, (106)

where 0 ≤ γ^M ≤ 1, while ${μ_{i}^{M} = Σ}_{k = 1}^{M^{*}} μ_{i, k}^{M}$ with $μ_{i, k}^{M} = μ^{M} \frac{{\bar{M}}_{i, k, t}}{{\bar{M}}_{i, t}}$ and μ^M ≥ 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 (104) subject to economy specific intermediate import good demand function (102), 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 $P_{i, j, t}^{M, *}$ given by necessary first order condition:

\frac{P_{i, j, t}^{M, *}}{P_{i, j, t}^{M}} = \frac{E_{t} Σ_{s = t}^{\infty} (ω^{M})^{s - t} \frac{β^{s - t} λ_{i, s}^{A}}{λ_{i, t}^{A}} θ_{i, s}^{M} \frac{ε_{i, j, s} P_{j, s}^{X}}{P_{i, j, s}^{M}} {[(\frac{P_{i, j, t - 1}^{M}}{P_{i, j, s - 1}^{M}})^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{ε_{i, i *, t} P_{k, t}^{Y}}{ε_{i, i *, s} P_{k, s}^{Y}})^{μ_{i, k}^{M}}]^{γ^{M}} [(\frac{{\bar{P}}_{i, j, t - 1}^{M}}{{\bar{P}}_{i, j, s - 1}^{M}})^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{{\bar{ε}}_{i, i *, t} {\bar{P}}_{k, t}^{Y}}{{\bar{ε}}_{i, i *, s} {\bar{P}}_{k, s}^{Y}})^{μ_{i, k}^{M}}]^{1 - γ^{M}} \frac{P_{i, j, s}^{M}}{P_{i, j, t}^{M}}}^{θ_{i, s}^{M}} (\frac{P_{i, j, t}^{M, *}}{P_{i, j, t}^{M}})^{- θ_{i, s}^{M}} P_{i, j, s}^{M} M_{i, j, s}}{E_{t} Σ_{s = t}^{\infty} (ω^{M})^{s - t} \frac{β^{s - t} λ_{i, s}^{A}}{λ_{i, t}^{A}} (θ_{i, s}^{M} - 1) {[(\frac{P_{i, j, t - 1}^{M}}{P_{i, j, s - 1}^{M}})^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{ε_{i, i *, t} P_{k, t}^{Y}}{ε_{i, i *, s} P_{k, s}^{Y}})^{μ_{i, k}^{M}}]^{γ^{M}} [(\frac{{\bar{P}}_{i, j, t - 1}^{M}}{{\bar{P}}_{i, j, s - 1}^{M}})^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{{\bar{ε}}_{i, i *, t} {\bar{P}}_{k, t}^{Y}}{{\bar{ε}}_{i, i *, s} {\bar{P}}_{k, s}^{Y}})^{μ_{i, k}^{M}}]^{1 - γ^{M}} \frac{P_{i, j, s}^{M}}{P_{i, j, t}^{M}}}^{θ_{i, s}^{M} - 1} (\frac{P_{i, j, t}^{M, *}}{P_{i, j, t}^{M}})^{- θ_{i, s}^{M}} P_{i, j, s}^{M} M_{i, j, s}} . (107)

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 (103) equals an average of the price set by the fraction 1 − ω^M of intermediate import good firms that adjust their price optimally in period t, and the average of the prices set by the remaining fraction ω^M of intermediate import good firms that adjust their price according to partial indexation rule (106):

P_{i, j, t}^{M} = {(1 - ω^{M}) (P_{i, j, t}^{M, *})^{1 - θ_{i, t}^{M}} + ω^{M} \begin{matrix} {[{(\frac{P_{i, j, t - 1}^{M}}{P_{i, j, t - 2}^{M}})}^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{\in_{i, i *, t} P_{k, t}^{Y}}{\in_{i, i *, t - 1} P_{k, t - 1}^{Y}})^{μ_{i, k}^{M}}]^{γ^{M}} [{(\frac{{\bar{P}}_{i, j, t - 1}^{M}}{{\bar{P}}_{i, j, t - 2}^{M}})}^{1 - μ_{i}^{M}} Π_{k = 1}^{M *} (\frac{{\bar{\in}}_{i, i *, t} {\bar{P}}_{k, t}^{Y}}{{\bar{\in}}_{i, i *, t - 1} {\bar{P}}_{k, t - 1}^{Y}})^{μ_{i, k}^{M}}]^{1 - γ^{M}} P_{i, j, t - 1}^{M}}^{1 - θ_{i, t}^{M}}}^{\frac{1}{1 - θ_{i, t}^{M}}} . & (108) \end{matrix}

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 according to a monetary policy rule exhibiting partial adjustment dynamics of the form

\begin{matrix} i_{i, t}^{P} - {\bar{i}}_{i, t}^{P} = ρ_{j}^{i} (i_{i, t - 1}^{P} - i_{i, t - 1}^{\bar{P}}) + (1 - ρ_{j}^{i}) [ζ_{j}^{π} E_{t} (π_{i, t + 1}^{C} - {\bar{π}}_{i, t + 1}^{C}) + ζ_{j}^{Y} (1 n Y_{i, t} - 1 n {\tilde{Y}}_{i, t}) + ζ_{j}^{Δ \in} (Δ 1 n \in_{i, t} - Δ 1 n {\bar{\in}}_{i, t}) \\ \begin{matrix} + ζ_{j}^{i} (i_{k, t}^{s} - {\bar{i}}_{k, t}^{s}) + ζ_{j}^{\in} (1 n \in_{i, k, t} - 1 n {\bar{\in}}_{i, k, t})] + V_{i, t}^{i^{P}}, & (109) \end{matrix} \end{matrix}

where $0 \leq ρ_{j}^{i} < 1, ξ_{j}^{π} \geq 0, ξ_{j}^{Y} \geq 0, ξ_{j}^{Δ \in} \geq 0, ξ_{j}^{\in} \geq 0$ and $ξ_{j}^{\in} \geq 0$ . 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 with $ξ_{j}^{π} > 1$ , as well as the contemporaneous output gap with $ξ_{j}^{Y} > 0$ . 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 physical capital and effective labor, given the physical capital stock and effective labor force. 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 $ξ_{j}^{Δ \in} > 0$ . Under a fixed exchange rate regime j = 2, and the deviation of the nominal policy interest rate from its steady state equilibrium value instead tracks the contemporaneous deviation of the yield to maturity on short term bonds of the economy that issues the anchor currency from its steady state equilibrium value one for one with $ξ_{j}^{i} = 1$ , while responding to any contemporaneous deviation of the corresponding nominal bilateral exchange rate from its target value with $ξ_{j}^{\in} > 0$ . For economies belonging to a monetary union, the target variables entering into their common monetary policy rule are expressed as output weighted averages across union members. Deviations from this monetary policy rule are captured by mean zero and serially uncorrelated monetary policy shock $V_{i, t}^{i^{P}}$ .

The Fiscal Authority

The fiscal authority implements fiscal policy through control of public consumption and the tax rates applicable to corporate earnings and household labor income. It also operates a budget neutral lump sum transfer program which redistributes national financial wealth from capital market intermediated households to credit constrained households, while equalizing steady state equilibrium consumption across 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 satisfies a countercyclical fiscal expenditure rule exhibiting partial adjustment dynamics of the form

\begin{matrix} \ln \frac{G_{i, t}}{{\bar{G}}_{i, t}} - \ln \frac{{\tilde{Y}}_{i, t}}{{\bar{Y}}_{i, t}} = ρ_{G} (\ln \frac{G_{i, t - 1}}{{\bar{G}}_{i, t - 1}} - \ln \frac{{\tilde{Y}}_{i, t - 1}}{{\bar{Y}}_{i, t - 1}}) + (1 - ρ_{G}) ξ^{G} (\frac{A_{i, t + 1}^{G}}{P_{i, t}^{Y} Y_{i, t}} - \frac{{\bar{A}}_{i, t + 1}^{G}}{{\bar{P}}_{i, t}^{Y} {\bar{Y}}_{i, t}}) + v_{i, t}^{G}, & (110) \end{matrix}

where 0 ≤ ρ_G < 1 and ζ^G > 0. As specified, the deviation of public consumption from its steady state equilibrium value, less the deviation of potential output from its steady state equilibrium value, depends on a weighted average of its past value and its desired value, which in turn is increasing in the contemporaneous deviation of the ratio of public financial wealth to nominal output from its target value. Deviations from this fiscal expenditure rule are captured by mean zero and serially uncorrelated government expenditure shock $V_{i, t}^{G}$ .

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

\begin{matrix} τ_{i, t}^{Z} - τ_{i} = ρ_{τ} (τ_{i, t - 1}^{Z} - τ_{i}) - (1 - ρ_{τ}) ζ^{τ} (\frac{A_{i, t + 1}^{G}}{P_{i, t}^{Y} Y_{i, t}} - \frac{{\bar{A}}_{i, t + 1}^{G}}{{\bar{P}}_{i, t}^{Y} {\bar{Y}}_{i, t}}) + v_{i, t}^{τ^{Z}}, & (111) \end{matrix}

where Z ∈ {K, L}, while 0 < τ_i < 1, 0 ≤ ρ_τ < 1 and ζ^τ > 0. As specified, the deviations of these tax rates from their steady state equilibrium value depend on a weighted average of their past deviations and their desired deviation, which in turn is decreasing in the contemporaneous deviation of the ratio of public financial wealth to nominal output from its target value. Deviations from these fiscal revenue rules are captured by mean zero and serially uncorrelated corporate and labor income tax rate shocks $V_{i, t}^{τ^{Z}}$ .

The real lump sum transfer payment to credit constrained households satisfies a transfer payment rule exhibiting partial adjustment dynamics of the form

\begin{matrix} \ln \frac{T_{i, t}^{C}}{P_{i, t}^{C}} - \ln \frac{{\bar{T}}_{i, t}^{C}}{{\bar{P}}_{i, t}^{C}} = ρ_{T} (\ln \frac{T_{i, t - 1}^{C}}{P_{i, t - 1}^{C}} - \ln \frac{{\bar{T}}_{i, t - 1}^{C}}{{\bar{P}}_{i, t - 1}^{C}}) + (1 - ρ_{T}) ζ^{T} (\frac{A_{i, t + 1}}{P_{i, t}^{Y} Y_{i, t}} - \frac{{\bar{A}}_{i, t + 1}}{{\bar{P}}_{i, t}^{Y} {\bar{Y}}_{i, t}}), & (112) \end{matrix}

where 0 ≤ ρ_T < 1 and ζ^T > 0. As specified, the deviation of the real transfer payment from its steady state equilibrium value depends on a weighted average of its past deviation and its desired deviation, which in turn is increasing in the contemporaneous deviation of the ratio of national financial wealth to nominal output from its target value. For economies belonging to a monetary union, the ratio of national financial wealth to nominal output is expressed as an output weighted average across union members.

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

\begin{matrix} 1 + i_{i, t}^{S} = ν_{i, t}^{i^{s}} (1 + i_{i, t}^{p}) . & (113) \end{matrix}

Deviations from this money market relationship are captured by internationally and serially correlated credit risk premium shock $υ_{i, t}^{i^{S}}$ .

The fiscal authority enters period t in possession of previously accumulated financial wealth $A_{i, t}^{G}$ which yields return $i_{i, t}^{A^{G}}$ . This financial wealth is distributed across the values of domestic short term bond $B_{i, t}^{S, G}$ and long term bond $B_{i, t}^{L, G}$ portfolios which yield returns $i_{i, t - 1}^{S}$ and $i_{i, t}^{B^{L, G}}$ , respectively. It follows that $(1 + i_{i, t}^{A^{G}}) A_{i, t}^{G} = (1 + i_{i, t - 1}^{S}) B_{i, t}^{S, G} + (1 + i_{i, t}^{B^{L, G}}) B_{i, t}^{L, G}$ where $(1 + i_{i, t}^{B^{L, G}}) B_{i, t}^{L, G} = Σ_{k = 1}^{t - 1} (Π_{i, k, t}^{B} + V_{i, k, t}^{B}) B_{i, k, t}^{L, G}$ with $Π_{i, k, t}^{B} = (1 + i_{i, k}^{L} - ω^{B}) (ω^{B})^{t - k} V_{i, k, k}^{B}$ and $V_{i, k, k}^{B} = 1$ . At the end of period t, the fiscal authority levies taxes on corporate earnings at rate $τ_{i, t}^{K}$ and household labor income at rate $τ_{i, t}^{L}$ , generating tax revenues T_i,t. It also remits household type specific lump sum transfer payments $T_{i, t}^{Z}$ , where $\int_{0}^{1} T_{i, t}^{Z} d h = 0$ . These sources of public wealth are summed in government dynamic budget constraint:

\begin{matrix} A_{i, t + 1}^{G} = (1 + i_{i, t}^{A^{G}}) A_{i, t}^{G} + Σ_{k = 1}^{M} \int_{0}^{1} τ_{i, t}^{K} (P_{i, k, l, t}^{Y} Y_{i, k, l, t} - W_{i, t} L_{i, k, l, t}) d l + \int_{0}^{1} τ_{i, t}^{L} \int_{0}^{1} W_{f, i, t} L_{h, f, i, t} d f d h - P_{i, t}^{G} G_{i, t} . & (114) \end{matrix}

According to this dynamic budget constraint, at the end of period t, the fiscal authority holds financial wealth $A_{i, t + 1}^{G}$ , which it allocates across the values of domestic short term bond $B_{i, t + 1}^{S, G}$ and long term bond $B_{i, t + 1}^{L, G}$ portfolios, that is $A_{i, t + 1}^{G} = B_{i, t + 1}^{S, G} + B_{i, t + 1}^{L, G}$ where $B_{i, t + 1}^{L, G} = Σ_{k = 1}^{t} V_{i, k, t}^{B} B_{i, k, t + 1}^{L, G}$ . Finally, the fiscal authority purchases final public consumption good G_i,t at price $P_{i, t}^{G}$ .

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

\begin{matrix} \begin{matrix} κ_{i, t + 1}^{R} - κ^{R} = ρ_{κ} (κ_{i, t}^{R} - κ^{R}) + (1 - ρ_{κ}) [ζ^{κ, B} (Δ \ln B_{i, t + 1}^{C, B} - Δ \ln {\bar{B}}_{i, t + 1}^{C, B}) + ζ^{κ, V^{H}} (Δ \ln V_{i, t}^{H} - Δ \ln {\bar{V}}_{i, t}^{H}) \end{matrix} \\ + ζ^{κ, V^{s}} (Δ \ln V_{i, t}^{S} - Δ \ln {\bar{V}}_{i, t}^{S})] + V_{i, t}^{κ}, (115) \end{matrix}

where 0 < κ^R < 1, 0 ≤ ρ_κ < 1, ζ^κ,B > 0, ζ^{κ,V^H} 0 and ζ^{κ,V^S} > 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 uncorrelated capital requirement shock $V_{i, t}^{κ}$ .

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

\begin{matrix} φ_{i, t}^{Z} - φ^{Z} = ρ_{φ^{z}} (φ_{i, t - 1}^{Z} - φ^{Z}) - (1 - ρ_{φ^{z}}) [ζ^{φ^{z}, B} (Δ \ln B_{i, t + 1}^{C, Z} - Δ \ln {\bar{B}}_{i, t + 1}^{C, Z}) + ζ^{φ^{z}, V} (Δ \ln V_{i, t}^{f (Z)} - Δ \ln {\bar{V}}_{i, t}^{f (Z)})] + V_{i, t}^{φ^{z}}, & (116) \end{matrix}

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 < ϕ^Z < 1, 0 ≤ ρ_ϕ^Z < 1, ζ^{ϕ^Z,B} > 0 and ζ^{ϕ^Z,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 shocks $V_{i, t}^{φ^{z}}$ .

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

\begin{matrix} δ_{i, t}^{z} - δ = ρ_{δ} (δ_{i, t - 1}^{z} - δ) - (1 - ρ_{δ}) [ζ^{δ^{z}, Y} (\ln Y_{i, t} - \ln {\tilde{Y}}_{i, t}) + ζ^{δ^{z}, V} (Δ \ln V_{i, t}^{f (z)} - Δ \ln {\bar{V}}_{i, t}^{f (z)})] + ν_{i, t}^{δ^{z}}, & (117) \end{matrix}

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 ≤ ρ_δ < 1, ζ^{δ^Z,Y} > 0 and ζ^{δ^Z,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 shocks $V_{i, t}^{δ^{Z}}$ .

G. Market Clearing Conditions

A rational expectations equilibrium in this panel dynamic stochastic general equilibrium model of the world economy consists of state contingent sequences of allocations for the households, developers, firms and banks of all economies which solve their constrained optimization problems given prices and policies, together with state contingent sequences of allocations for the governments of all economies which 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 X_i,t equal production of the domestic final output good less the total demand of domestic households, developers, firms and the government,

X_{i, t} = Y_{i, t} - C_{i, t}^{h} - I_{i, t}^{H, h} - \begin{matrix} I_{i, t}^{K, h} - G_{i, t}^{h}, & (118) \end{matrix}

where $X_{i, t} = Σ_{j = 1}^{N} X_{i, j, t}$ and X_i,j,t = M_j,i,t. Clearing of the final import good market requires that imports M_i,t equal the total demand of domestic households, developers, firms and the government:

M_{i, t} = C_{i, t}^{f} + I_{i, t}^{H, f} + \begin{matrix} I_{i, t}^{K, f} + G_{i, t}^{f} . & (119) \end{matrix}

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

\begin{matrix} P_{i, t}^{Y} Y_{i, t} = P_{i, t}^{C} C_{i, t} + P_{i, t}^{I} I_{i, t} + P_{i, t}^{G} G_{i, t} + P_{i, t}^{X} X_{i, t} - P_{i, t}^{M} M_{i, t}, & (120) \end{matrix}

where the price of investment satisfies $P_{i, t}^{I} = P_{i, t}^{I^{H}} = P_{i, t}^{I^{K}}$ , while investment satisfies $I_{i, t} = I_{i, t}^{H} + I_{i, t}^{K}$ . The price of domestic demand satisfies $P_{i, t}^{D} = P_{i, t}^{C} = P_{i, t}^{I} = P_{i, t}^{G}$ , while domestic demand satisfies $\begin{matrix} D_{i, t} = C_{i, t} + I_{i, t} + G_{i, t} \end{matrix}$ .

Clearing of the final bank loan markets requires that mortgage loan supply equals the total demand of domestic developers, that is $B_{i, t + 1}^{C^{D}, B} = B_{i, t + 1}^{C, D}$ while corporate loan supply equals the total demand of domestic and foreign firms:

B_{i, t + 1}^{C^{F}, B} \begin{matrix} = Σ_{j = 1}^{N} B_{j, i, t + 1}^{C, F} . & (121) \end{matrix}

In equilibrium, clearing of the final corporate loan payments system implies that the corporate credit loss rate satisfies:

1 - δ_{i, t}^{C, E} = Σ_{j = 1}^{N} \frac{B_{j, i, t}^{C, F}}{B_{i, t}^{C^{F}, B}} \begin{matrix} (1 - δ_{j, t}^{C}) . & (122) \end{matrix}

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 A_i,t+1 denote the net foreign asset position, which equals the sum of the financial wealth of households $A_{i, t + 1}^{H}$ , developers $A_{i, t + 1}^{D}$ , firms $A_{i, t + 1}^{F}$ , banks $A_{i, t + 1}^{B}$ and the government $A_{i, t + 1}^{G}$ ,

A_{i, t + 1} = A_{i, t + 1}^{H} + A_{i, t + 1}^{D} + A_{i, t + 1}^{F} + A_{i, t + 1}^{B} \begin{matrix} + A_{i, t + 1}^{G}, & (123) \end{matrix}

where $A_{i, t + 1}^{H} = A_{i, t + 1}^{B, H} + A_{i, t + 1}^{A, H} + V_{i, t}^{C}, A_{i, t + 1}^{D} = - B_{i, t + 1}^{C, D} - V_{i, t}^{H}, A_{i, t + 1}^{F} = - B_{i, t + 1}^{C, F} - V_{i, t}^{S}$ and $A_{i, t + 1}^{B} = K_{i, t + 1}^{B} {- V}_{i, t}^{C}$ . Imposing equilibrium conditions on government dynamic budget constraint (114) reveals that the increase in public financial wealth equals public saving, or equivalently that the fiscal balance $F B_{i, t =} A_{i, t + 1}^{G} - A_{i, t}^{G}$ equals the sum of net interest income and the primary fiscal balance $P B_{i, t} = T_{i, t} - P_{i, t}^{G} G_{i, t}$ ,

F B_{i, t} = [\frac{B_{i}^{S, G}}{A_{i}^{G}} i_{i, t - 1}^{s} + \frac{B_{i}^{L, G}}{A_{i}^{G}} i_{i, t - 1}^{L, E}] A_{i, t}^{G} + \begin{matrix} P B_{i, t}, & (124) \end{matrix}

where $T_{i, t} = τ_{i, t}^{K} (P_{i, t}^{Y} Y_{i, t} - W_{i, t} L_{i, t}) + τ_{i, t}^{L} W_{i, t} L_{i, t}$ , while nominal effective long term bond yield $i_{i, t}^{L, E}$ satisfies $i_{i, t}^{L, E} = ω^{B} i_{i, t - 1}^{L, E} + (1 - ω^{B}) [ω^{B} ((1 + i_{i, t}^{L} - ω^{B}) + 1) - 1]$ . The derivation of this result abstracts from valuation gains on long term bond holdings, and imposes restrictions $B_{i, k, t}^{L, G} = B_{i, k - 1, t}^{L, G}$ , $B_{i, t}^{S, G} / A_{i, t}^{G} = B_{i}^{S, G} / A_{i}^{G} and B_{i, t}^{L, G} / A_{i, t}^{G} = B_{i}^{L, G} / A_{i}^{G}$ . Imposing equilibrium conditions on household dynamic budget constraint (7), and combining it with government dynamic budget constraint (124), developer dividend payment definition (41), firm dividend payment definition (55), bank dividend payment definition (76), bank balance sheet identity (77), output expenditure decomposition (120), and final corporate loan payments system clearing condition (122) reveals that the increase in net foreign assets equals national saving less investment expenditures, or equivalently that the current account balance CA_i,t = ε_i*,i,tA_i,t+1 − ε_i*,i,t-1A_i,t equals the sum of net international investment income and the trade balance $T B_{i, t} = ε_{i *, i, t} P_{i, t}^{X} X_{i, t} - ε_{i *, i, t} P_{i, t}^{M} M_{i, t}$ :

\begin{matrix} C A_{i, t} = {Σ_{j = 1}^{N} w_{j}^{M} [(1 + i_{j, t - 1}^{S}) \frac{ɛ_{i *, j, t}}{ɛ_{i *, j, t - 1}} - 1]} ɛ_{i *, i, t - 1} A_{i, t} + T B_{i, t} . & (125) \end{matrix}

The derivation of this result abstracts from international financial intermediation except via the money markets and imposes restriction $ε_{i, j, t - 1} B_{i, j, t}^{S} / A_{i, t} = w_{j}^{M}$ , where world money market capitalization weight $w_{i}^{M}$ satisfies ${0 < w}_{i}^{M} < 1$ and $Σ_{i = 1}^{N} w_{i}^{M} = 1$ . Multilateral consistency in nominal trade flows requires that $Σ_{j = 1}^{N} T B_{j, t} = 0$ .

III. The Empirical Framework

Estimation, inference and forecasting are based on a linear state space representation of an approximate multivariate linear rational expectations representation of this panel dynamic stochastic general equilibrium model of the world economy. This multivariate linear rational expectations representation is derived by analytically linearizing the equilibrium conditions of this panel dynamic stochastic general equilibrium model around its stationary deterministic steady state equilibrium, and consolidating them by substituting out intermediate variables assuming small capital utilization costs. The response coefficients of these consolidated approximate linear equilibrium conditions are functions of behavioral structural 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, ${\hat{x}}_{i, t}$ denotes the deviation of variable x_i,t from its steady state equilibrium value x_i, while E_t x_i,t+s denotes the rational expectation of variable x_i,t+s conditional on information available in period t. Bilateral weights $w_{i, j}^{Z}$ for evaluating the trade weighted average of variable x_i,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 $w_{i, j}^{Z}$ for evaluating the weighted average of variable x_i,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 $w_{i, j}^{Z}$ for evaluating the portfolio weighted average of variable x_i,t across the investment destinations of economy i are based on debt for Z = B and equity for Z = S. Finally, world weights $w_{i}^{Z}$ for evaluating the weighted average of variable x_i,t across all economies are based on output for Z = Y, money market capitalization for Z = M, bond market capitalization for Z = B, and stock market capitalization for Z = S. Auxiliary parameters $λ_{j}^{Z}$ are theoretically predicted to equal one, and satisfy λ = 0 and $λ_{j}^{Z} > 0$ .

A. Endogenous Variables

Output price inflation ${\hat{π}}_{i, t}^{Y}$ depends on a linear combination of its past and expected future values driven by the contemporaneous labor income share and the internal terms of trade according to output price Phillips curve:

\begin{matrix} {\hat{π}}_{i, t}^{Y} = \frac{γ^{Y}}{1 + γ^{Y} β} {\hat{π}}_{i, t - 1}^{Y} + \frac{β}{1 + γ^{Y} β} E_{t} {\hat{π}}_{i, t + 1}^{Y} \\ \begin{matrix} + \frac{(1 - ω^{Y}) (1 - ω^{Y} β)}{ω^{Y} (1 + γ^{Y} β)} [\ln \frac{{\hat{W}}_{i, t} {\hat{L}}_{i, t}}{{\hat{P}}_{i, t}^{Y} {\hat{Y}}_{i, t}} + \frac{X_{i}}{Y_{i}} \ln {\hat{T}}_{i, t}^{X} - \frac{1}{θ^{Y} - 1} \ln {\hat{θ}}_{i, t}^{Y}] + \frac{X_{i}}{Y_{i}} P_{1} (L) Δ \ln {\hat{T}}_{i, t}^{X} . & (126) \end{matrix} \end{matrix}

Output price inflation also depends on contemporaneous, past and expected future changes in the internal terms of trade, where polynomial in the lag operator $P_{1} (L) = 1 - \frac{γ^{Y}}{1 + γ^{Y} β} L - \frac{β}{1 + γ^{Y} β} E_{t} L^{- 1}$ . The response coefficients of this relationship vary across economies with their trade openness. Output price inflation satisfies ${\hat{π}}_{i, t}^{Y} = \ln {\hat{P}}_{i, t}^{Y} - \ln {\hat{P}}_{i, t - 1}^{Y}$ , which determines the output price level ln ${\hat{P}}_{i, t}^{Y}$ .

Consumption price inflation ${\hat{π}}_{i, t}^{C}$ depends on a linear combination of its past and expected future values driven by the contemporaneous labor income share and the internal terms of trade according to consumption price Phillips curve:

\begin{matrix} {\hat{π}}_{i, t}^{C} = \frac{γ^{Y}}{1 + γ^{Y} β} {\hat{π}}_{i, t - 1}^{C} + \frac{β}{1 + γ^{Y} β} E_{t} {\hat{π}}_{i, t + 1}^{C} \\ \begin{matrix} + \frac{(1 - ω^{Y}) (1 - ω^{Y} β)}{ω^{Y} (1 + γ^{Y} β)} [\ln \frac{{\hat{W}}_{i, t} {\hat{L}}_{i, t}}{{\hat{P}}_{i, t}^{Y} {\hat{Y}}_{i, t}} + \frac{X_{i}}{Y_{i}} \ln {\hat{T}}_{i, t}^{X} - \frac{1}{θ^{Y} - 1} \ln {\hat{θ}}_{i, t}^{Y}] + \frac{M_{i}}{Y_{i}} P_{1} (L) Δ \ln {\hat{T}}_{i, t}^{M} . & (127) \end{matrix} \end{matrix}

Consumption price inflation also depends on contemporaneous, past and expected future changes in the external terms of trade. The response coefficients of this relationship vary across economies with their trade openness. Consumption price inflation satisfies ${\hat{π}}_{i, t}^{C} = \ln {\hat{P}}_{i, t}^{C} - \ln {\hat{P}}_{i, t - 1}^{C}$ , which determines the consumption price level ln ${\hat{P}}_{i, t}^{C}$ .

Output ln ${\hat{Y}}_{i, t}$ depends on contemporaneous domestic demand, exports and imports according to output demand relationship:

\ln {\hat{Y}}_{i, t} = \ln {\hat{D}}_{i, t} + \frac{X_{i}}{Y_{i}} \begin{matrix} \ln \frac{{\hat{X}}_{i, t}}{M_{i, t}} . & (128) \end{matrix}

Domestic demand ln ${\hat{D}}_{i, t}$ depends on a weighted average of contemporaneous consumption, investment and public domestic demand according to domestic demand relationship:

\ln {\hat{D}}_{i, t} = \frac{C_{i}}{Y_{i}} \ln {\hat{C}}_{i, t} + \frac{I_{i}}{Y_{i}} \begin{matrix} \ln {\hat{I}}_{i, t} + \frac{G_{i}}{Y_{i}} \ln {\hat{G}}_{i, t} . & (129) \end{matrix}

Investment ln ${\hat{I}}_{i, t}$ depends on a weighted average of contemporaneous residential and business investment according to investment demand relationship:

\begin{matrix} \ln {\hat{I}}_{i, t} = \frac{I_{i}^{H}}{I_{i}} \ln {\hat{I}}_{i, t}^{H} + \frac{I_{i}^{K}}{I_{i}} \ln {\hat{I}}_{i, t}^{K} . & (130) \end{matrix}

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

Consumption ln ${\hat{C}}_{i, t}$ 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:

\begin{matrix} \ln {\hat{C}}_{i, t} = \frac{α^{c}}{1 + α^{c}} \ln {\hat{C}}_{i, t - 1} + \frac{1}{1 + α^{c}} E_{t} \ln {\hat{C}}_{i, t + 1} \\ - (1 - φ^{c}) σ \frac{1 - α^{c}}{1 + α^{c}} E_{t} [\frac{φ^{B}}{1 - φ^{c}} {\hat{r}}_{i, t + 1}^{A^{B, H}} \begin{matrix} + (1 - \frac{φ^{B}}{1 - φ^{C}}) {\hat{r}}_{i, t + 1}^{A^{A, H}} - \ln \frac{{\hat{v}}_{i, t}^{C}}{v_{i, t + 1}^{c}}] + φ^{c} P_{2} (L) \ln {\hat{C}}_{i, t}^{C} . & (131) \end{matrix} \end{matrix}

Reflecting the existence of credit constraints, consumption also depends on contemporaneous, past and expected future credit constrained consumption, where polynomial in the lag operator $P_{2} (L) = 1 - \frac{α^{C}}{1 + α^{C}} L - \frac{1}{1 + α^{C}} E_{t} L^{- 1}$ . Credit constrained consumption ln ${\hat{C}}_{i, t}^{C}$ depends on contemporaneous output and the terms of trade according to credit constrained consumption demand relationship

\begin{matrix} \ln {\hat{C}}_{i, t}^{C} \begin{matrix} = λ_{i}^{C} (\frac{C_{i}}{Y_{i}})^{- 1} {(1 - τ_{i}) {\ln {\hat{Y}}_{i, t} + \frac{X_{i}}{Y_{i}} (\ln {\hat{τ}}_{i, t} - λ \ln {\hat{v}}_{t}^{τ}) - \frac{1}{1 - τ_{i}} [(1 - \frac{W_{i} L_{i}}{P_{i}^{y} Y_{i}}) {\hat{τ}}_{i, t}^{K} + \frac{W_{i} L_{i}}{P_{i}^{y} Y_{i}} {\hat{τ}}_{i, t}^{L}]} \end{matrix} \\ + λ {\frac{M_{I}^{s}}{P_{I}^{Y} Y_{i}} [\ln \frac{{\hat{M}}_{i, t + 1}^{s}}{{\hat{p}}_{i, t}^{C}} - \frac{1}{β} (({\hat{i}}_{i, t - 1}^{s} - {\hat{π}}_{i, t}^{C}) + \ln \frac{{\hat{M}}_{i, t}^{s}}{{\hat{P}}_{i, t - 1}^{C}})] - \frac{I_{i}}{Y_{i}} \ln {\hat{I}}_{i, t}} + \frac{1 - β}{β} \frac{M_{i}^{s}}{P_{i}^{Y} Y_{i}} \ln \frac{{\hat{T}}_{i, t}^{C}}{{\hat{P}}_{i, t}^{C}}}, (132) \end{matrix}

where economy specific auxiliary parameter $λ_{i}^{C} = \frac{1}{1 - τ_{i}} \frac{C_{i}}{Y_{i}}$ Credit constrained consumption also depends on a weighted average of the contemporaneous corporate and labor income tax rates, as well as the contemporaneous real transfer payment to credit constrained households. The response coefficients of this relationship vary across economies with the size of their government, their trade openness, their labor income share, and their financial depth.

Residential investment ln ${\hat{I}}_{i, t}^{H}$ 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:

\ln {\hat{I}}_{i, t}^{H} = \frac{1}{1 + β} \ln {\hat{I}}_{i, t - 1}^{H} + \frac{β}{1 + β} E_{t} \ln {\hat{I}}_{i, t + 1}^{H} + \frac{1}{x^{H} (1 + β)} \ln \begin{matrix} ({\hat{v}}_{i, t}^{I^{H}} \frac{{\hat{Q}}_{i, t}^{^{H}}}{{\hat{P}}_{i, t}^{C}}) . & (133) \end{matrix}

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

\begin{matrix} \ln \frac{{\hat{Q}}_{i, t}^{H}}{{\hat{P}}_{i, t}^{C}} = E_{t} {β (1 - δ^{H}) \ln \frac{{\hat{Q}}_{i, t + 1}^{H}}{{\hat{P}}_{i, t + 1}^{C}} - [(1 - φ^{D}) ({\hat{r}}_{i, t + 1}^{A^{B, H}} + \frac{{\hat{φ}}_{i, t}^{D}}{φ^{D}}) + φ^{D} β \frac{θ^{C}}{θ^{C} - 1} \frac{1 + K^{R} (1 - β (1 - χ^{C} δ))}{β} ({\hat{r}}_{i, t}^{M} - λ {\hat{δ}}_{i, t + 1}^{M} + λ \frac{{\hat{φ}}_{i, t}^{D}}{{\hat{φ}}^{D}})] \\ + [(1 - β (1 - δ^{H})) + φ^{D} β (\frac{θ^{C}}{θ^{C} - 1} \frac{1 + κ^{R} (1 - β (1 - χ^{C} δ))}{β} - \frac{1}{β})] \ln \frac{{\hat{t}}_{i, t + 1}^{H}}{{\hat{P}}_{i, t + 1}^{C}}} + \frac{{\hat{φ}}_{i, t}^{D}}{φ^{D}} \begin{matrix} , & (134) \end{matrix} \end{matrix}

which determines the shadow price of housing ln ${\hat{Q}}_{i, t}^{H}$ . 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 function

\ln \frac{{\hat{i}}_{i, t}^{H}}{{\hat{P}}_{i, t}^{C}} = \frac{1}{ζ} \ln \begin{matrix} \frac{{\hat{C}}_{i, t}}{H_{i, t}}, & (135) \end{matrix}

which determines the rental price of housing ln ${\hat{ι}}_{i, t}^{H}$ . The housing stock ln ${\hat{H}}_{i, t + 1}$ is accumulated according to ln ${\hat{H}}_{i, t + 1} = (1 - δ^{H}) \ln {\hat{H}}_{i, t} + δ^{H} \ln ({\hat{v}}_{i, t}^{I^{H}} {\hat{I}}_{i, t}^{H})$ .

Business investment ln ${\hat{I}}_{i, t}^{K}$ depends on a weighted average of its past and expected future values driven by the contemporaneous relative shadow price of physical capital according to business investment demand relationship:

\ln {\hat{I}}_{i, t}^{K} = \frac{1}{1 + β} \ln {\hat{I}}_{i, t - 1}^{K} + \frac{β}{1 + β} E_{t} {\ln \hat{I}}_{i, t + 1}^{K} + \frac{1}{χ^{K} (1 + β)} \ln \begin{matrix} ({\hat{v}}_{i, t}^{I^{K}} \frac{{\hat{Q}}_{i, t}^{K}}{{\hat{P}}_{i, t}^{C}}) . & (136) \end{matrix}

Reflecting the existence of a financial accelerator mechanism, the relative shadow price of 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

\begin{matrix} \ln \frac{{\hat{Q}}_{i, t}^{K}}{{\hat{P}}_{i, t}^{C}} = E_{t} {β (1 - δ^{K}) \ln \frac{{\hat{Q}}_{i, t + 1}^{K}}{{\hat{P}}_{i, t + 1}^{C}} - [(1 - φ^{F}) ({\hat{r}}_{i, t + 1}^{A^{A, H}} + \frac{{\hat{φ}}_{i, t}^{F}}{φ^{F}}) + φ^{F} β \frac{θ^{C}}{θ^{C} - 1} \frac{1 + κ^{R} (1 - β (1 - χ^{C} δ))}{β} ({\hat{r}}_{i, t + 1}^{C, E} - λ {\hat{δ}}_{i, t + 1}^{C} + λ \frac{{\hat{φ}}_{i, t}^{F}}{{\hat{φ}}^{F}})] \\ + [(1 - β (1 - δ^{K})) + φ^{F} β (\frac{θ^{C}}{θ^{C} - 1} \frac{1 + κ^{R} (1 - β (1 - χ^{C} δ))}{β} - \frac{1}{β})] (η^{K} \ln {\hat{u}}_{i, t + 1}^{K} - \frac{1}{1 - τ_{i}} {\hat{τ}}_{i, t + 1}^{K})} + \frac{{\hat{φ}}_{i, t}^{F}}{φ^{F}} \begin{matrix} , & (137) \end{matrix} \end{matrix}

which determines the shadow price of physical capital ln ${\hat{Q}}_{i, t}^{K}$ . The relative shadow price of 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 ln ${\hat{u}}_{i, t}^{K}$ depends on the contemporaneous real effective wage according to capital utilization relationship:

\ln {\hat{u}}_{i, t}^{K} = \frac{1}{η^{K}} (\ln \frac{{\hat{W}}_{i, t}}{{\hat{p}}_{i, t}^{C} {\hat{A}}_{i, t}} - \begin{matrix} \ln \frac{{\hat{u}}_{i, t}^{K} {\hat{K}}_{i, t}}{{\hat{A}}_{i, t} {\hat{L}}_{i, t}}) . & (\begin{matrix} 138 \end{matrix}) \end{matrix}

The capital utilization rate also depends on the contemporaneous deviation of utilized physical capital from effective employment. The physical capital stock ln ${\hat{K}}_{i, t + 1}$ is accumulated according to ln ${\hat{K}}_{i, t + 1} = (1 - δ^{K}) \ln {\hat{K}}_{i, t} + δ^{K} \ln ({\hat{v}}_{i, t}^{I^{K}} {\hat{I}}_{i, t}^{K})$ .

Exports ln ${\hat{X}}_{i, t}$ depend on contemporaneous export weighted foreign demand, as well as the export weighted average foreign external terms of trade, according to export demand relationship:

\begin{matrix} \ln {\hat{X}}_{i, t} = Σ_{J = 1}^{N} w_{i, j}^{X} [\ln \frac{{\hat{D}}_{j, t}}{{\hat{V}}_{i, t}^{X} {\hat{V}}_{j, t}^{M}} - ψ^{M} (1 - \frac{M_{j}}{Y_{j}}) \ln {\hat{T}}_{j, t}^{M}] . & (139) \end{matrix}

The response coefficients of this relationship vary across economies with their trade pattern and the trade openness of their trading partners. Imports ln ${\hat{M}}_{i, t}$ depend on contemporaneous domestic demand, as well as the domestic external terms of trade, according to import demand relationship:

\begin{matrix} \ln {\hat{M}}_{i, t} = \ln \frac{{\hat{D}}_{i, t}}{{\hat{ν}}_{i, t}^{M}} - ψ^{M} (1 - \frac{M_{i}}{Y_{i}}) \ln {\hat{T}}_{i, t}^{M} . & (\begin{matrix} 140 \end{matrix}) \end{matrix}

The response coefficients of this relationship vary across economies with their trade openness.

The nominal property return $E_{t} {\hat{i}}_{i, t + 1}^{A^{B, H}}$ depends on the contemporaneous nominal money market interest rate according to property return function:

\begin{matrix} E_{t} {\hat{i}}_{i, t + 1}^{A^{B, H}} = {\hat{i}}_{i, t}^{S} + φ^{H} \ln {\hat{v}}_{i, t}^{H} . & (141) \end{matrix}

Reflecting the existence of a portfolio balance mechanism, the nominal property return also depends on the contemporaneous housing risk premium. The real property return $E_{t} {\hat{r}}_{i, t + 1}^{A^{B, H}}$ satisfies $E_{t} {\hat{r}}_{i, t + 1}^{A^{B, H}} = E_{t} {\hat{i}}_{i, t + 1}^{A^{B, H}} - E_{t} {\hat{π}}_{i, t + 1}^{C}$ .

The price of housing ln ${\hat{V}}_{i, t}^{H}$ depends on its expected future value driven by expected future developer profits, and the contemporaneous nominal money market interest rate adjusted by the housing risk premium, according to housing market relationship:

\ln {\hat{V}}_{i, t}^{H} = β E_{t} \ln {\hat{V}}_{i, t + 1}^{H} + (1 - β) E_{t} \ln {\hat{Π}}_{i, t + 1}^{H} - ({\hat{i}}_{i, t}^{S} + \ln {\hat{v}}_{i, t}^{H}) . (142)

Developer profits ln ${\hat{Π}}_{i, t}^{H}$ depend on contemporaneous housing rental revenues according to developer profit function

\begin{matrix} \begin{matrix} \ln {\hat{Π}}_{i, t}^{H} & = λ_{i}^{H} (\frac{Π_{i}^{H}}{P_{i}^{Y} Y_{i}})^{- 1} {\frac{ι_{i}^{H}}{P_{i}^{Y}} \frac{H_{i}}{Y_{i}} \ln ({\hat{ι}}_{i, t}^{H} {\hat{H}}_{i, t}) \\ + λ \frac{B_{i}^{C, D}}{P_{i}^{Y} Y_{i}} [\ln {\hat{B}}_{i, t + 1}^{C, D} - (1 - δ) (1 + i_{i}^{M}) ({\hat{i}}_{i, t - 1}^{M} - {\hat{δ}}_{i, t}^{M} + \ln {\hat{B}}_{i, t}^{C, D})] - λ \frac{I_{i}^{H}}{Y_{i}} \ln ({\hat{P}}_{i, t}^{C} {\hat{I}}_{i, t}^{H})}, \end{matrix} & (143) \end{matrix}

where economy specific auxiliary parameter $λ_{i}^{H} = \frac{Π_{i}^{H}}{P_{i}^{Y} Y_{i}} (\frac{ι_{i}^{H}}{P_{i}^{Y}} \frac{H_{i}}{Y_{i}})^{- 1}$ .

The nominal portfolio return $E_{t} {\hat{i}}_{i, t + 1}^{A^{A, H}}$ depends on the contemporaneous nominal money market interest rate according to portfolio return function:

E_{t} {\hat{i}}_{i, t + 1}^{A^{A, H}} = {\hat{i}}_{i, t}^{S} + φ_{B}^{A} Σ_{j = 1}^{N} w_{i, j}^{B} (\ln {\hat{υ}}_{j, t}^{B} + λ \frac{1 - φ_{S}^{A}}{φ_{B}^{A}} \ln \frac{{\hat{v}}_{j, t}^{ɛ}}{{\hat{v}}_{i, t}^{ɛ}}) + φ_{S}^{A} Σ_{j = 1}^{N} w_{i, j}^{S} (\ln {\hat{υ}}_{j, t}^{S} + γ \ln \frac{{\hat{v}}_{j, t}^{ɛ}}{{\hat{v}}_{i, t}^{ɛ}}) . (144)

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 $E_{t} {\hat{r}}_{i, t + 1}^{A^{A, H}}$ satisfies $E_{t} {\hat{r}}_{i, t + 1}^{A^{A, H}} = E_{t} {\hat{i}}_{i, t + 1}^{A^{A, H}} - E_{t} {\hat{π}}_{i, t + 1}^{C}$

The nominal money market interest rate ${\hat{i}}_{i, t}^{S}$ depends on the contemporaneous nominal policy interest rate adjusted by the credit risk premium according to money market relationship:

{\hat{i}}_{i, t}^{S} = {\hat{i}}_{i, t}^{P} + \ln {\hat{υ}}_{i, t}^{i^{S}} . (145)

The real money market interest rate ${\hat{r}}_{i, t}^{S}$ satisfies ${\hat{r}}_{i, t}^{S} = {\hat{i}}_{i, t}^{S} - E_{t} {\hat{π}}_{i, t + 1}^{C}$ . The credit risk premium ln ${\hat{υ}}_{i, t}^{i^{S}}$ satisfies dynamic factor process ln ${\hat{υ}}_{i, t}^{i^{S}} = λ_{k}^{M} Σ_{j = 1}^{N} w_{j}^{M} \ln {\hat{v}}_{j, t}^{i^{S}} + (1 - λ_{k}^{M} w_{i}^{M}) \ln {\hat{ν}}_{i, t}^{i^{S}}$ , except under a fixed exchange rate regime where ln ${\hat{υ}}_{i, t}^{i} = \ln {\hat{ν}}_{i, t}^{i^{S}}$ . The intensity of international money market contagion varies across economies, with k = 0 for low debt contagion economies, k = 1 for medium debt contagion economies, and k = 2 for high debt contagion economies, where $λ_{0}^{M} < λ_{1}^{M} < λ_{2}^{M}$

The nominal long term bond yield ${\hat{i}}_{i, t}^{L}$ depends on its expected future value, driven by the contemporaneous nominal money market interest rate adjusted by the duration risk premium, according to bond market relationship:

{\hat{i}}_{i, t}^{L} = ω^{B} β E_{t} {\hat{i}}_{i, t + 1}^{L} + \frac{1 - ω^{B} β}{ω^{B} β} (ω^{B} + \frac{1 - ω^{B} β}{ω^{B} β})^{- 1} ({\hat{i}}_{i, t}^{S} + \ln {\hat{υ}}_{i, t}^{B}) . (146)

The real long term bond yield ${\hat{r}}_{i, t}^{L}$ depends on its expected future value, driven by the contemporaneous real money market interest rate adjusted by the duration risk premium, according to:

{\hat{r}}_{i, t}^{L} = ω^{B} β E_{t} {\hat{r}}_{i, t + 1}^{L} + \frac{1 - ω^{B} β}{ω^{B} β} (ω^{B} + \frac{1 - ω^{B} β}{ω^{B} β})^{- 1} ({\hat{r}}_{i, t}^{S} + \ln {\hat{υ}}_{i, t}^{B}) . (147)

The term premium ln ${\hat{μ}}_{i, t}^{B}$ depends on its expected future value driven by the contemporaneous duration risk premium according to:

\ln {\hat{μ}}_{i, t}^{B} = ω^{B} β E_{t} \ln {\hat{μ}}_{i, t + 1}^{B} + \frac{1 - ω^{B} β}{ω^{B} β} (ω^{B} + \frac{1 - ω^{B} β}{ω^{B} β})^{- 1} \ln {\hat{υ}}_{i, t}^{B} . (148)

The duration risk premium ln ${\hat{υ}}_{i, t}^{B}$ satisfies dynamic factor process ln ${\hat{υ}}_{i, t}^{B} = λ_{k}^{B} Σ_{j = 1}^{N} w_{j}^{B} \ln {\hat{ν}}_{j, t}^{B} + (1 - λ_{k}^{B} w_{i}^{B}) \ln {\hat{ν}}_{i, t}^{B}$ . The intensity of international bond market contagion varies across economies, with k = 0 for low debt contagion economies, k = 1 for medium debt contagion economies, and k = 2 for high debt contagion economies, where $λ_{0}^{B} < λ_{1}^{B} < λ_{2}^{B}$ .

The price of equity ln ${\hat{V}}_{i, t}^{S}$ depends on its expected future value driven by expected future nonfinancial corporate profits, and the contemporaneous nominal money market interest rate adjusted by the equity risk premium, according to stock market relationship:

\ln {\hat{V}}_{i, t}^{S} = β E_{t} \ln {\hat{V}}_{i, t + 1}^{S} + (1 - β) E_{t} \ln {\hat{Π}}_{i, t + 1}^{S} - ({\hat{i}}_{i, t}^{S} + \ln {\hat{υ}}_{i, t}^{S}) . (149)

Nonfinancial corporate profits ln ${\hat{Π}}_{i, t}^{S}$ depend on contemporaneous nominal output and the corporate tax rate according to nonfinancial corporate profit function

\begin{matrix} \begin{matrix} \ln {\hat{Π}}_{i, t}^{S} & = λ_{i}^{S} (\frac{Π_{i}^{S}}{P_{i}^{Y} Y_{i}})^{- 1} {(1 - τ_{i}) [\ln ({\hat{P}}_{i, t}^{Y} {\hat{Y}}_{i, t}) - λ \frac{W_{i} L_{i}}{P_{i}^{Y} Y_{i}} \ln ({\hat{W}}_{i, t} {\hat{L}}_{i, t}) - \frac{1}{1 - τ_{i}} (1 - \frac{W_{i} L_{i}}{P_{i}^{Y} Y_{i}}) {\hat{τ}}_{i, t}^{K}] \\ + λ \frac{B_{i}^{C, F}}{P_{i}^{Y} Y_{i}} [\ln {\hat{B}}_{i, t + 1}^{C, F} - (1 - δ) (1 + i_{i}^{C}) ({\hat{i}}_{i, t}^{C, E} - {\hat{δ}}_{i, t}^{C} + \ln {\hat{B}}_{i, t}^{C, F})] - λ \frac{I_{i}^{K}}{Y_{i}} \ln ({\hat{P}}_{i, t}^{C} {\hat{I}}_{i, t}^{K})}, \end{matrix} & (150) \end{matrix}

where economy specific auxiliary parameter $λ_{i}^{S} = \frac{1}{1 - τ_{i}} \frac{Π_{i}^{S}}{P_{i}^{Y} Y_{i}}$ . The response coefficients of this relationship vary across economies with the size of their government and their labor income share. The equity risk premium ln ${\hat{υ}}_{i, t}^{S}$ satisfies dynamic factor process ln ${\hat{υ}}_{i, t}^{S} = λ_{k}^{S} Σ_{j = 1}^{N} w_{j}^{S} \ln {\hat{ν}}_{j, t}^{S} + (1 - λ_{k}^{S} w_{i}^{S}) \ln {\hat{ν}}_{i, t}^{S}$ . The intensity of international stock market contagion varies across economies, with k = 0 for low equity contagion economies, k = 1 for medium equity contagion economies, and k = 2 for high equity contagion economies, where $λ_{0}^{S} < λ_{1}^{S} < λ_{2}^{S}$ .

The nominal policy interest rate ${\hat{i}}_{i, t}^{P}$ depends on a weighted average of its past and desired values according to monetary policy rule:

{\hat{i}}_{i, t}^{P} = ρ_{j}^{i} {\hat{i}}_{i, t - 1}^{P} + (1 - ρ_{j}^{i}) (ξ_{j}^{π} E_{t} {\hat{π}}_{i, t + 1}^{C} + ξ_{j}^{Y} \ln {\hat{\hat{Y}}}_{i, t} + ξ_{j}^{Δ ɛ} Δ \ln {\hat{ɛ}}_{i, t} + ξ_{j}^{i} {\hat{i}}_{k, t}^{S} + ξ_{j}^{ɛ} 1 n {\hat{ɛ}}_{i, k, t} \begin{matrix} ) + {\hat{v}}_{i, t}^{i^{P}} . & (151) \end{matrix}

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. Under a managed exchange rate regime j = 1, and it also responds to the contemporaneous change in the nominal effective exchange rate. Under a fixed exchange rate regime j = 2, and the nominal policy interest rate instead tracks the contemporaneous nominal money market interest rate of the economy that issues the anchor currency one for one, while responding to the contemporaneous corresponding nominal bilateral exchange rate. For economies belonging to a monetary union, the target variables entering into their common monetary policy rule are expressed as output weighted averages across union members. The real policy interest rate ${\hat{r}}_{i, t}^{P}$ satisfies ${\hat{r}}_{i, t}^{P} = {\hat{i}}_{i, t}^{P} - E_{t} {\hat{π}}_{i, t + 1}^{C}$ .

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

\ln {\hat{B}}_{i, t + 1}^{C, B} = \begin{matrix} (1 - κ^{R}) \ln {\hat{M}}_{i, t + 1}^{S} + κ^{R} \ln {\hat{K}}_{i, t + 1}^{B}, & (152) \end{matrix}

which determines the money stock ln ${\hat{M}}_{i, t + 1}^{S}$ . Bank credit ln ${\hat{B}}_{i, t + 1}^{C, B}$ 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:

\ln {\hat{B}}_{i, t + 1}^{C, B} = w_{i}^{C} \ln {\hat{B}}_{i, t + 1}^{C, D} \begin{matrix} + (1 - w_{i}^{C}) Σ_{j = 1}^{N} w_{i, j}^{C} \ln \frac{{\hat{B}}_{j, t + 1}^{C, F}}{{\hat{ɛ}}_{j, i, t}} . & (153) \end{matrix}

Mortgage debt ln ${\hat{B}}_{i, t + 1}^{C, D}$ satisfies ln ${\hat{B}}_{i, t + 1}^{C, D} = \ln {\hat{P}}_{i, t}^{C} + \ln {\hat{H}}_{i, t + 1} + \frac{{\hat{φ}}_{i, t}^{D}}{φ^{D}}$ , while nonfinancial corporate debt ln ${\hat{B}}_{i, t + 1}^{C, F}$ satisfies ln ${\hat{B}}_{i, t + 1}^{C, F} = \ln {\hat{P}}_{i, t}^{C} + \ln {\hat{K}}_{i, t + 1} + \frac{{\hat{φ}}_{i, t}^{F i, t}}{φ^{F}}$ . The bank capital ratio ${\hat{K}}_{i, t + 1}$ satisfies ${\hat{K}}_{i, t + 1} = K^{R} (\ln {\hat{K}}_{i, t + 1}^{B} - \ln {\hat{B}}_{i, t + 1}^{C, B})$

The nominal effective corporate loan rate ${\hat{i}}_{i, t}^{C, E}$ 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:

{\hat{i}}_{i, t}^{C, E} = \begin{matrix} Σ_{j = 1}^{N} w_{i, j}^{F} ({\hat{i}}_{j, t - 1}^{C} + \ln \frac{{\hat{ɛ}}_{i, j, t}}{{\hat{ɛ}}_{i, j, t - 1}}) . & (154) \end{matrix}

The corporate credit loss rate ${\hat{δ}}_{i, t}^{C, E}$ depends on the bank lending weighted average of contemporaneous domestic and foreign corporate loan default rates according to corporate credit loss rate function:

{\hat{δ}}_{i, t}^{C, E} = \begin{matrix} Σ_{j = 1}^{N} w_{i, j}^{C} {\hat{δ}}_{j, t}^{c} . & (155) \end{matrix}

The real effective corporate loan rate $E_{t} {\hat{r}}_{i, t + 1}^{C, E}$ satisfies $E_{t} {\hat{r}}_{i, t + 1}^{C, E} = E_{t} {\hat{i}}_{i, t + 1}^{C, E} - E_{t} {\hat{π}}_{i, t + 1}^{C}$ .

The nominal mortgage or corporate loan rate ${\hat{i}}_{i, t}^{f (Z)}$ depends on a weighted average of its past and expected future values, driven by the deviation of the past nominal money market interest 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

\begin{matrix} \begin{matrix} {\hat{i}}_{i, t}^{f (z)} & = \frac{1}{1 + β} {\hat{i}}_{i, t - 1}^{f (z)} \begin{matrix} + \frac{β}{1 + β} E_{t} {\hat{i}}_{i, t + 1}^{f (z)} \frac{(1 - ω^{c}) (1 - ω^{c} β)}{ω^{c} (1 + β)} {[{\hat{i}}_{i, t - 1}^{S} - ({\hat{i}}_{i, t}^{f (z)} - {\hat{δ}}_{i, t}^{g^{(z)}})] \end{matrix} \\ - \frac{1 - β (1 - χ^{c} δ)}{1 + κ^{R} (1 - β (1 - χ^{C} δ))} [η^{C} ({\hat{κ}}_{i, t} - {\hat{κ}}_{i, t}^{R}) - ({\hat{κ}}_{i, t}^{R} - κ^{R} {\hat{i}}_{i, t - 1}^{S})] - \frac{1}{θ^{c} - 1} 1 n {\hat{θ}}_{i, t}^{C^{Z}}}, \end{matrix} & (156) \end{matrix}

where Z ∈ {D, F}, while f (D) = M and f (F) = C. The nominal mortgage or corporate loan rate also depends 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 or corporate loan rate ${\hat{r}}_{i, t}^{f (Z)}$ satisfies ${\hat{r}}_{i, t}^{f (Z)} = {\hat{i}}_{i, t}^{f (Z)} - E_{t} {\hat{π}}_{i, t + 1}^{C}$

Bank retained earnings ln ${\hat{I}}_{i, t}^{B}$ 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:

\begin{matrix} \ln {\hat{I}}_{i, t}^{B} = \frac{1}{1 + β} \ln {\hat{I}}_{i, t - 1}^{B} + \frac{β}{1 + β} E_{t} \ln {\hat{I}}_{i, t + 1}^{B} + \frac{1}{χ^{B} (1 + β)} \ln {\hat{Q}}_{i, t}^{B} . & (157) \end{matrix}

The shadow price of bank capital ln ${\hat{Q}}_{i, t}^{B}$ depends on its expected future value net of the expected future bank capital destruction rate, as well as the contemporaneous nominal money market interest rate, according to retained earnings Euler equation:

\ln {\hat{Q}}_{i, t}^{B} = E_{t} {β (1 - χ^{c} δ) (\ln {\hat{Q}}_{i, t + 1}^{B} - {\hat{δ}}_{i, t + 1}^{B}) - [{\hat{i}}_{i, t}^{S} + (1 - β (1 - χ^{c} δ)) {\frac{η}{κ^{R}}}^{c} ({\hat{κ}}_{i, t + 1} - {\hat{κ}}_{i, t + 1}^{R})]} . (158)

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 ln ${\hat{K}}_{i, t + 1}^{B}$ is accumulated according to ln ${\hat{K}}_{i, t + 1}^{B} = (1 - χ^{C} δ) (\ln {\hat{K}}_{i, t}^{B} - {\hat{δ}}_{i, t}^{B}) + χ^{C} δ \ln {\hat{I}}_{i, t}^{B}$ , where the bank capital destruction rate ${\hat{δ}}_{i, t}^{B}$ satisfies ${\hat{δ}}_{i, t}^{B} = χ^{C} (W_{i}^{C} {\hat{δ}}_{i, t}^{M} + (1 - W_{i}^{C}) {\hat{δ}}_{i, t}^{C, E})$

The regulatory bank capital ratio requirement ${\hat{K}}_{i, t + 1}^{R}$ depends on a weighted average of its past and desired values according to countercyclical capital buffer rule:

\begin{matrix} {\hat{κ}}_{i, t + 1}^{R} = ρ_{κ} {\hat{κ}}_{i, t}^{R} + (1 - ρ_{κ}) (ζ^{κ, B} Δ \ln {\hat{B}}_{i, t + 1}^{C, B} + ζ^{κ, V^{H}} Δ \ln {\hat{V}}_{i, t}^{H} + {ζ^{κ, V}}^{S} Δ \ln {\hat{V}}_{i, t}^{S}) + {\hat{v}}_{i, t}^{κ} . & (159) \end{matrix}

The desired regulatory bank capital ratio requirement responds to contemporaneous bank credit growth, as well as contemporaneous changes in the prices of housing and equity. The regulatory mortgage or corporate loan to value ratio limit ${\hat{φ}}_{i, t}^{Z}$ depends on a weighted average of its past and desired values according to loan to value limit rules

\begin{matrix} {\hat{φ}}_{i, t}^{z} = ρ_{φ^{z}} {\hat{φ}}_{i, t - 1}^{z} - (1 - ρ_{φ^{z}}) (ζ^{φ^{z}, B} Δ \ln {\hat{B}}_{i, t + 1}^{c, z} + ζ^{φ^{z}, v} Δ \ln {\hat{V}}_{i, t}^{f (z)}) + {\hat{ν}}_{i, t}^{φ^{z}}, & (160) \end{matrix}

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 the contemporaneous change in the price of housing or equity, respectively. The mortgage or corporate loan default rate ${\hat{δ}}_{i, t}^{Z}$ depends on a weighted average of its past and attractor values according to default rate relationships

\begin{matrix} {\hat{δ}}_{i, t}^{z} = ρ_{δ} {\hat{δ}}_{i, t - 1}^{z} - (1 - ρ_{δ}) (ζ^{δ^{z}, Y} \ln {\hat{\hat{Y}}}_{i, t} + ζ^{δ^{z}, v} Δ \ln {\hat{V}}_{i, t}^{f (z)}) + {\hat{ν}}_{i, t}^{δ^{z}}, & (161) \end{matrix}

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

\begin{matrix} \begin{matrix} \ln \frac{{\hat{W}}_{i, t}}{{\hat{P}}_{i, t}^{c} {\hat{A}}_{i, t}} & = \frac{1}{1 + β} \ln \frac{{\hat{W}}_{i, t - 1}}{{\hat{P}}_{i, t - 1}^{c} {\hat{A}}_{i, t - 1}} + \frac{β}{1 + β} E_{t} \ln \frac{{\hat{W}}_{i, t + 1}}{{\hat{P}}_{i, t + 1}^{c} \hat{A}} \end{matrix} \end{matrix}

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