## Abstract

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. These macrofinancial linkages encompass bank and capital market based financial intermediation, with financial accelerator mechanisms linked to the values of the housing and physical capital stocks. A variety of monetary policy analysis, fiscal policy analysis, macroprudential policy analysis, spillover analysis, and forecasting applications of the estimated model are demonstrated. These include quantifying the monetary, fiscal and macroprudential transmission mechanisms, accounting for business cycle fluctuations, and generating relatively accurate forecasts of inflation and output growth.

## 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*1 and

^{C}<*ϕ*. All households are originally endowed with one share of each domestic developer, firm and bank.

^{B}+ ϕ^{A}+ ϕ^{C}= 1In 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

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,

where *0 ≤ α ^{C} < 1* and

*0 ≤ α*

^{L}

*< 1*. Endogenous preference shifters

where *t > 0*. The intratemporal utility function is strictly increasing with respect to consumption if and only if serially correlated consumption demand shock *σ > 0, ς > 0, η > 0* and *μ > 0*. In steady state equilibrium,

The representative household enters period *s* in possession of previously accumulated property balances *de*, where *ε _{i,j,s}* measures the price of foreign currency in terms of domestic currency. It follows that

*s*, the representative household receives profit income from banks

According to this dynamic budget constraint, at the end of period *s*, the representative household holds property balances *C*_{h,i,s} at price *H*_{h,i,s} at price

##### 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

where serially correlated housing risk premium shock *0 ≤ ϕ*^{H} ≤ 1 and *ψ*^{H} > 0. Preferences over the real values of developer specific shares

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 *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

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

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

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

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

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

where internationally and serially correlated duration risk premium shock *ψ ^{A}* > 0. Preferences over the real values of economy specific short term bond

where serially correlated currency risk premium shocks

where *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 *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

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

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

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

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

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

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

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

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

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

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 *N*_{i,t} in unemployment *U*_{i,t}, that is *L*_{i,t}, that is *U _{i,t} = N_{i,t} – L_{i,t}*. The labor force satisfies

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

where serially uncorrelated wage markup shock

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

As the wage elasticity of demand for intermediate labor services

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

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

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*of trade unions that adjust their wage according to partial indexation rule (34):

^{L}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

where endogenous rental price of housing markup

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:

As the price elasticity of demand for intermediate housing services

#### 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

where *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

Earnings are defined as revenues derived from sales of differentiated intermediate housing service *H*_{i,e,s} at rental price

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,

given by regulatory mortgage loan to value ratio limit

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

where *0 ≤ δ*^{H} ≤ 1. Effective residential investment function

where serially correlated residential investment demand shock *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 *H*_{i,e,T+1} ≥ 0 for *T → ∞*. In equilibrium, demand for the final residential investment good satisfies necessary first order condition

which equates the expected present value of an additional unit of residential investment to its price, where *s* housing accumulation function. In equilibrium, this shadow price of housing satisfies necessary first order condition

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

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

### 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,t} according to fixed proportions production function

where

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:

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

where serially uncorrelated output price markup shock *k* ≤ *M* * and

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:

As the price elasticity of demand for industry specific intermediate output goods

#### 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

where *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

where *Y*_{i,k,l,s} at price *L*_{i,k,l,s}, and other variable costs *Φ*_{i,k,l,s}. The government levies a tax on corporate earnings at rate

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,

given by regulatory corporate loan to value ratio limit

The representative industry specific intermediate output good firm utilizes physical capital *K*_{i,k,l,s} at rate *L*_{i,k,l,s} to produce industry specific differentiated intermediate output good *Y*_{i,k,l,s} according to production function

where serially correlated productivity shock

In utilizing physical capital to produce output, the representative industry specific intermediate output good firm incurs a cost

where industry specific fixed cost *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,

where *η*^{k} > 0, while *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

where 0 ≤ *δ*^{K} ≤ 1. Following Christiano, Eichenbaum and Evans (2005), effective business investment function

where serially correlated business investment demand shock *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 *K*_{i,k,l,T+1} ≥ 0 for *T* → ∞. In equilibrium, demand for the final labor service satisfies necessary first order condition

where *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

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

which equates the expected present value of an additional unit of business investment to its price, where *s* physical capital accumulation function. In equilibrium, this shadow price of physical capital satisfies necessary first order condition

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 *k* > *M**. The remaining fraction

where *k* ≤ *M** and

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

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 *t*, and the average of the prices set by the remaining fraction

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

where

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:

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

where *Z* ∈ {*D*, *F*}, while serially uncorrelated mortgage or corporate loan rate markup shock

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:

As the rate elasticity of demand for intermediate mortgage or corporate loans

#### 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

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

Shares entitle households to dividend payments

Profits are defined as the sum of the increase in deposits

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

The bank credit stock *κ*_{i,s+1} equals the ratio of aggregate bank capital to assets, that is

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,

where fixed cost

given regulatory capital requirement *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

where bank capital destruction rate *χ*^{C} > 0, while mortgage loan weight

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

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

which equates the expected present value of an additional unit of retained earnings to its marginal cost, where *s* bank capital accumulation function. In equilibrium, this shadow price of bank capital satisfies necessary first order condition

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,

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

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:

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

where the real bilateral exchange rate

where the internal terms of trade *P*_{i,t} denotes the price of the final noncommodity output good. Finally, under the law of one price *k* ≤ *M* *, which implies that

where

#### The Export Sector

There exist a large number of perfectly competitive firms which combine industry specific final output goods *X*_{i,t} according to fixed proportions production function

where *X*_{i,k,t} = *Y*_{i,k,t} for 1 ≤ *k* ≤ *M* *, while

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:

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

where serially correlated import demand shock *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:

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:

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:

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,t} according to fixed proportions production function

where serially correlated export demand shock

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:

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

where serially uncorrelated import price markup shock

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:

As the price elasticity of demand for economy specific intermediate import goods

##### 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

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

Earnings are defined as revenues derived from sales of economy specific differentiated intermediate import good *M*_{i,j,n,s} at price *M*_{i,j,n,s}. The representative economy specific intermediate import good firm purchases the foreign final export good and differentiates it. Fixed cost

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

where 0 ≤ *γ*^{M} ≤ 1, while *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

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):

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

where *j* = 0, and this desired deviation is increasing in the expected future deviation of consumption price inflation from its target value with *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 *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

#### 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

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

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

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

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

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:

Deviations from this money market relationship are captured by internationally and serially correlated credit risk premium shock

The fiscal authority enters period *t* in possession of previously accumulated financial wealth *t*, the fiscal authority levies taxes on corporate earnings at rate *T*_{i,t}. It also remits household type specific lump sum transfer payments

According to this dynamic budget constraint, at the end of period *t*, the fiscal authority holds financial wealth *G*_{i,t} at price

#### 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

where 0 < *κ*^{R} < 1, 0 ≤ *ρ*_{κ} < 1, *ζ*^{κ,B} > 0, *ζ*^{κ,VH} 0 and *ζ*^{κ,VS} > 0. As specified, the deviation of the regulatory capital ratio requirement from its steady state equilibrium value depends on a weighted average of its past deviation and its desired deviation. This desired deviation is increasing in the contemporaneous deviation of bank credit growth from its steady state equilibrium value, as well as the contemporaneous deviations of the changes in the prices of housing and equity from their steady state equilibrium values. Deviations from this countercyclical capital buffer rule are captured by mean zero and serially uncorrelated capital requirement shock

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

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*_{ϕZ} < 1, *ζ*^{ϕZ,B} > 0 and *ζ*^{ϕZ,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

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

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*^{δZ,Y} > 0 and *ζ*^{δZ,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

### 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,

where *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:

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

where the price of investment satisfies

Clearing of the final bank loan markets requires that mortgage loan supply equals the total demand of domestic developers, that is

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

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

where

where *CA _{i,t}* =

*ε*−

_{i*,i,t}A_{i,t+1}*ε*equals the sum of net international investment income and the trade balance

_{i*,i,t-1}A_{i,t}The derivation of this result abstracts from international financial intermediation except via the money markets and imposes restriction

## 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, *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 *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 *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 *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 *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* = 0 and

### A. Endogenous Variables

Output price inflation

Output price inflation also depends on contemporaneous, past and expected future changes in the internal terms of trade, where polynomial in the lag operator

Consumption price inflation

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

Output ln

Domestic demand ln

Investment ln

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

Consumption ln

Reflecting the existence of credit constraints, consumption also depends on contemporaneous, past and expected future credit constrained consumption, where polynomial in the lag operator

where economy specific auxiliary parameter

Residential investment ln

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

which determines the shadow price of housing ln

which determines the rental price of housing ln

Business investment ln

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

which determines the shadow price of physical capital ln

The capital utilization rate also depends on the contemporaneous deviation of utilized physical capital from effective employment. The physical capital stock ln

Exports ln

The response coefficients of this relationship vary across economies with their trade pattern and the trade openness of their trading partners. Imports ln

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

The nominal property return

Reflecting the existence of a portfolio balance mechanism, the nominal property return also depends on the contemporaneous housing risk premium. The real property return

The price of housing ln

Developer profits ln

where economy specific auxiliary parameter

The nominal portfolio return

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

The nominal money market interest rate

The real money market interest rate *k =* 0 for low debt contagion economies, *k =* 1 for medium debt contagion economies, and *k =* 2 for high debt contagion economies, where

The nominal long term bond yield

The real long term bond yield

The term premium ln

The duration risk premium ln *k =* 0 for low debt contagion economies, *k =* 1 for medium debt contagion economies, and *k =* 2 for high debt contagion economies, where

The price of equity ln

Nonfinancial corporate profits ln

where economy specific auxiliary parameter *k =* 0 for low equity contagion economies, *k =* 1 for medium equity contagion economies, and *k =* 2 for high equity contagion economies, where

The nominal policy interest rate

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

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

which determines the money stock ln

Mortgage debt ln

The nominal effective corporate loan rate

The corporate credit loss rate

The real effective corporate loan rate

The nominal mortgage or corporate loan rate

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

Bank retained earnings ln

The shadow price of bank capital ln

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

The regulatory bank capital ratio requirement

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

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

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