Banking Frictions and Integrated Financial Markets in a Two Country DSGE Model

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Elena Carniti

^∗

LUISS Guido Carli

February 2012

Abstract

Financial frictions and integrated financial markets matter by spread- ing and amplifying country specific shocks. We develop a two country DSGE world with incomplete markets to address these issues. The main reference for the model’s framework is a work by Gertler and Kiyotaki ¹. In the basic version of the model countries trade in goods but financial markets are closed. Then, we enrich the model by allowing for integrated financial markets but portfolio shares remain exoge- nously set. We end up with the complete model that also allows for portfolio choice by implementing a method developed by Devereux and Sutherland².We find home bias in international portfolios, that under incomplete markets allows for less volatility than under full portfolio diversification. The model provides a simple two country world framework that may also be used for monetary policy issues.

Key Words: two country DSGE model, incomplete markets, endogenous portfolio choice

∗Supervisor Prof. Pierpaolo Benigno. I would like to thank my Supervisor for all the insightful suggestions. All errors are my own responsibility.

1Gertler, M. and Kiyotaki, N.,”Financial Intermediation and Credit Policy in Business Cycle Analysis”, N.Y.U. and Princeton, March 2010.

2See Bibliography for the many papers of this topic. Among the others, Devereux, M. B. and Sutherland, A.,”Solving for Country Portfolios in Open Economy Macro Mod- els”, ”International Capital Flows”, CEPR Discussion Papers 5966, C.E.P.R. Discussion Papers, 2006.

1

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cerca e didattici, con citazione della fonte.

1 INTRODUCTION

1 Introduction

Banking balance sheets and leverage constraints have been crucial in transmitting and amplifying shocks from a country to another one during last financial crises. These two channels can separately act in propagating the contagion, how were pointed out, among the others, by Devereux and Yet- man (2010). What we developed here is a DSGE model that aims to keep into account these real-world features in order to understand better what mechanisms are at work during a modern financial crisis.

By trying to analyze how the current Eurozone crisis developed, we see that both banking system’s integrated financial portfolios and leverage constraints were crucial factors in transforming a single country’s problem in a systemic one. What directly exposes a national banking system to exter- nal shocks, indeed, are not just portfolio’s linkages (by owning Government bonds subject to speculative attack), but also the presence of constraints on the ratio asset/liability in the balance sheets.

When the banks are obliged to maintain (or reach) precise capital requirements (Basel Accords, or last European Banking Authority’s requirements), they may be obliged to sell potentially “toxic” assets, by causing in this way a chain of sells that further exacerbates the crisis.

The basic framework of our model closely follows that contained in Gertler and Kiyotaki(2010). We complicate their model by extending the analysis to a two country world with integrated good and financial markets and endogenous portfolio shares. We simplify, however, the analysis by not allowing for idiosyncratic shock to hit separately banks in the same country.

We know that according to empirical evidence markets in the real world are incomplete. We therefore choose to introduce these feature into our model by considering four independent sources of shocks, two for each country (to the technology and to the quality of capital), while only two assets are available as investment instruments for the banking system: home and foreign equities.

The deposit markets, where depositors and banks meet, are segmented at the national level.

The presence of incomplete markets can cause stationarity problems to

2 2

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2 THE MODEL

the model. In order to avoid these, we followed Schmitt-Groh and Uribe (2002,2004)³ and the literature related by making the discount factor endogenous. We chose this solution among the others proposed by the literature because it is the same that also Devereux and Yetman (2010) used in their work, similar in many ways to the one proposed here.

In order to understand better how the portfolio integration channel works under leverage constraints and incomplete markets, we consider three different versions of the same model: in the first version, the good markets are integrated but financial markets are closed (Model A). In the second version, both good and financial markets are integrated, but portfolio’s shares are ex- ogenously given as parameters (Model BK). In the third and last case, both markets are open and portfolio’s shares are endogenous (Model BH/F). In order to determine the equilibrium portfolio’s shares we followed the method developed by Devereux and Sutherland (2006-2011).

With this baseline model we would like to build a theoretical framework that might be used to two purposes. First of all, for analyzing the implication of coordinated versus uncoordinated unconventional monetary policies, such those used during the last financial turmoils by the main Central Banks.

Secondly, for studying the linkages in balance sheets across Governments and banks, that revealed to be a crucial channel in transmitting the current Eurozone debt’s crisis.

We will deepen the first task in the second part of our PhD dissertation, while we leave the second one for a future research.

Our work is organized in the following sequence: 1) the model 2)the steady state 3) a description of the method developed by Devereux and Sutherland (2006-2011) for finding the equilibrium portfolios 4)calibration 5)results. Theoretical models are in the Appendix.

2 The Model

We extend a work by Gertler and Kiyotaky (2010) by analyzing a two country world with international trade and banking frictions. In order to determine endogenous portfolio’s shares in an equilibrium with international asset markets we refer to the approach followed by Devereux and Sutherland (2006-2011). Tille and Van Wincoop (2007) also developed a similar method

3See Schmitt-Groh and Uribe for a more detailed discussion

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2.1 General Hypotheses 2 THE MODEL

to get the same results. It is worth mentioning also the work by Devereux and Yetman (2010) as a main reference for this model.

2.1 General Hypotheses

We assume a two country model, where each country is specialized in the production of a single traded good. In each country there is a continuum of households. Population size is normalized to 1. Every time a fraction 1 − f of the home (foreign) representative household is done by depositors, while the remaining f is done by bankers. This turnover between the two groups is random, keeping the relative proportion of each type fixed. Depositors supply labour and return the wage to the household. Moreover, they deposit funds in a bank different from the one they own. Bankers manage a financial intermediary and transfer earnings back to the household. Within the family, we assume perfect consumption insurance.

We assume that with probability θ a banker will be banker also at the following time. Instead with probability 1 − θ she will become a depositor.

Therefore, each period (1 − θ)f bankers exit and become depositors, θf bankers remain bankers, (1−f )θ depositors become bankers and (1−f )(1−θ) depositors remain depositors. There is one risk-free asset (deposit) D_t for the home country (D_t^∗ abroad) called in composite consumption units. Only household within a country hold the deposit of that country (deposits are not internationally traded). The Cobb-Douglas consumption index for the representative home household will be:

C_t≡ C_h,t^γ C_f,t^(1−γ) 0 < γ < 1 (1)

where C_h,t is the domestic consumption of the home good and C_f,t is the domestic consumption of the foreign good.

Similarly, for the foreign representative agent we have identical preferences:

C_t^∗ ≡ C_f,t^∗γC_h,t^∗(1−γ) 0 < γ < 1 (1^∗)

where γ is the ”economic size” of the home economy (i.e. the share of the home good in the consumption basket, while 1 − γ is the economic size of the foreign country). We assume γ = ¹₂ be the same for both home/foreign agents, meaning that there is no home consumption bias. Consumption- based price indexes will be:

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2.1 General Hypotheses 2 THE MODEL

P_t≡ _γγ(1−γ)¹ ^1−γP_h,t^γ P_f,t^(1−γ) (2) P_t^∗ ≡ _γγ(1−γ)¹ ^1−γP_f,t^∗γP_h,t^∗(1−γ) (2^∗)

where P_h,t and P_f,t are the prices of home/foreign goods in domestic currency, while P_h,t^∗ and P_f,t^∗ are the prices of home and foreign goods in foreign currency. P_t (or P_t^∗) represents the minimum expenditure required to buy 1 unit of the composite consumption good. We assume the law of one price for both home and foreign good, that is:

P_h,t = _tP_h,t^∗ (3a) P_f,t = _tP_f,t^∗ (3b)

Moreover, being households’ preferences identical across borders with no home bias, the consumption baskets will also be identical and then the relative price of consumption (real exchange rate ReRt) will be equal to one (Purchase Power Parity holds), that is:

P_t = _tP_t^∗ (3c)

where we define _t as the nominal exchange rate (the price of the foreign currency in terms of the home one).

Home consumption expenditure will be given by:

P_tC_t = P_h,tC_h,t + P_f,tC_f,t

while the lifetime utility of the home household will be:

E_tP∞

τ =tβ^{τ −t C}_1−ρ^τ^1−ρ −^κ₂L²_τ (4)

where β is the discount rate, that can be exogenous or endogenous. We set ρ , γ and κ equal to the same value both home and abroad.

Generic home household chooses consumption, labour supply and deposits in order to maximize (4) subject to the following flow of funds constraint:

PtCt+ PtDt+1= PtRtDt+ WtLt+ Πt (5) where we define:

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2.2 The Production Side 2 THE MODEL

W_t(W_t^∗): nominal wage rate of the generic household.

Rt(R^∗_t): real gross return on deposit from period t − 1 to period t.

D_t(D^∗_t): quantity of risk-less deposit in composite consumption units.

Π_t(Π^∗_t): net distribution from the ownership of both banks and

non-financial firms (nominal), net the transfers that the household gives to its members that start a new activity by entering at t in the banking system.

Symmetrically, for the foreign household we have:

P_t^∗C_t^∗+ P_t^∗D^∗_t+1= P_t^∗R^∗_tD^∗_t + W_t^∗L^∗_t + Π^∗_t (5^∗)

Consumption based power parity always holds due to the law of one price plus the assumptions on preferences (from the (3)) . Within/across countries preferences and constraints are symmetric. Since agents (households and firms) are equal within countries, in what follows all the variables will be considered in per capita (aggregate) terms.

2.2 The Production Side

2.2.1 Final Good Firms

Each firm faces an identical Cobb Douglas, therefore in aggregate we can assume:

Y_t= A_tK_t^(1−α)L^α_t (6) or, for the foreign country:

Y_t^∗ = A^∗_tK_t^∗(1−α)L^∗α_t (6^∗)

Each home/foreign competitive final good firm produces a single (but differentiated among countries) output by using an identical CRTS Cobb Douglas production function with capital and labour as inputs. Labour is provided by the household of the same country. Now we define:

δ: rate of physical depreciation.

Ψ_t(Ψ^∗_t): shock to the quality of capital.

I_t(I_t^∗): aggregate home (foreign) investment in unit of consumption good.

Every period the law of motion for aggregate capital (in units of consumption good) will be given by:

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K_t+1 = Ψ_t+1[I_t+ (1 − δ)K_t] (7)

The generic firm’s maximization function (in real terms) at time t will then be the following:

E_tβΛ_t,t+1n_P

h,t

Pt A_tK_t^(1−α)L^α_t + Q_h,t^K_Ψ^t+1

t+1 − R_hk,tQ_h,t−1^K_Ψ^t

t −^W_P^t

tL_t− Q_h,tI_to

=

=E_tβΛ_t,t+1n_P

h,t

Pt A_tK_t^(1−α)L^α_t + Q_h,t(I_t+ (1 − δ)K_t) − R_hk,tQ_h,t−1^K_Ψ^t

t − ^W_P^t

tL_t− Q_h,tI_to where βΛ_t,t+1 is the firms’ stochastic discount factor ( by assumption,

firms belong to the households), R_hk,t is the real return on home capital required by investing banks and Q_hk,t is the real price of one unit of home capital. We are assuming here that the replacement price of capital that has depreciated is equal to 1. Each period the firm issues new state-contingent securities in order to obtain funds from an intermediary at price Q_hk,t and to buy new capital good at the same price, and then it has to pay back capital returns on the securities issued at the previous period.

From the FOCs w.r.t. L_t and K_t:

Wt/Ph,t = αYt/Lt (8) Rhk,t+1 = Ψt+1

(1−α)^Ph,t+1_Pt+1 _Kt+1^Yt+1+(1−δ)Qh,t+1

Qh,t (9)

where the real gross profit per unit of capital is:

Z_t = (1 − α)^P_P^h,t

t A_t(_K^L^t

t)^α(10)

Therefore, each unit of home equity bought at time t will be a state- contingent claim to the future returns from one unit of investment:

Ψ_t+1Z_t+1, (1 − δ)Ψ_t+1Ψ_t+2Z_t+2, (1 − δ)²Ψ_t+1Ψ_t+2Ψ_t+3Z_t+3 2.2.2 Capital Good Firms

Home competitive capital good firms belong to home households and operate in a national market. They produce capital good using national output as input subject to adjustment costs. They choose I_t (in unit of the composite consumption good) in order to solve the following problem:

max E_tP∞

τ =tβ^{τ −t}Λ_t,τnh

Q_h,τI_τ −

1 + f (_I^I^τ

τ −1)i I_τo

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where f (_I^I^t

t−1) is the physical adjustment cost (in units of consumption good) with f (1) = f⁰(1) = 0 and f⁰⁰(1) > 0. From the FOC:

Q_h,t = 1 + f (_I^I^t

t−1) + f⁰(_I^I^t

t−1)_I^I^t

t−1 − E_tΛ_t,t+1f⁰(^I^t+1_I

t )^I^t+1_I

t

2 (11)

where the real price of the capital good has to be equal to the marginal cost of producing the investment good. Profits arise only outside the steady state, and are lump-sum distributed to the home households.

2.2.3 The Households

The home generic household takes the nominal prices as given in maximizing the following function:

E_tn P∞

τ =tβ^{τ −t}h

Cτ^1−ρ

1−ρ − ^κ₂L²_τi

+P∞

τ =tβ^{τ −t}µ_τ[P_τR_τD_τ + W_τL_τ + Π_τ− P_τC_τ − P_τD_{τ +1}]o where the FOCs with respect to C_t,D_t+1,L_t are:

C_t^−ρ = P_τµ_t (12) E_tn

Pτ +1µt+1

Pτµt R_t+1o

= ¹_β (13) W_t= ^κL_µ^t

t (14)

where µ_t is the Lagrange multiplier. The Euler Equation is therefore given by:

1 = E_t

βR_t+1

Ct+1

Ct

^−ρ (15) while the following is the labour supply equation:

Wt

Pt = κLtC_t^ρ (16)

To complete the household’s problem we now need to find the demands for home and foreign consumption goods. The representative household solves:

min Ph,tCh,t+ P_f,t^∗ tCf,t

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s.t.

C_t= C_h,t^γ C_f,t^(1−γ) in order to get from the FOCs:

C_h,t = γC_t_P^P^t

h,t (17) C_f,t= (1 − γ)C_t_P^P^t

f,t (18) while for the foreign household we will have:

C_h,t^∗ = (1 − γ)C_t^∗_P^P∗^t^∗ h,t

(17^∗) C_f,t^∗ = γC_t^∗_P^P∗^t^∗

f,t

(18^∗) 2.2.4 The Banks

Each home bank borrows d_t(in composite consumption units) in the national deposit market at the real gross deposit rate Rt+1, and then purchases sh,t

and s_f,t units of financial claims on final goods’ producing firms at the real prices Q_h,t and Q_f,t. These are the prices of the banking system’s claim on the future returns from one unit of present capital of the non-financial firm at the end of period. Firms are able to offer to the banks perfectly state-contingent debt. For an individual bank, the flow of funds constraint (where everything is in units of the composite consumption good) holds (intermediary balance sheet):

Q_h,ts_h,t+ Q_f,ts_f,t= n_t+ d_t (19)

where s_h,t and s_f,t are held by an individual national bank (s^∗_h,t and s^∗_f,t are held abroad) and where Q_f,t = ^Q

∗ f,tPt

tP_t^∗ with _t as the nominal exchange rate. Home and foreign financial claims rispectly pay the real gross returns R_hk,tand R_{f k,t}. On the LHS of the (19) we have the value of the loans funded within a given period, while and on the RHS there are the equity capital (the net worth of the bank) plus the debt. Net worth evolves according to:

nt= Rhk,tQh,t−1sh,t−1+ Rf k,tQf,t−1sf,t−1− Rtdt−1 (20)

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or, given that

nt= Rhk,tQh,t−1sh,t−1+Rf k,tQf,t−1sf,t−1−Rt[Qh,t−1sh,t−1+Qf,t−1sf,t−1−nt−1] we can simplify the net worth as:

n_t= [R_hk,t− R_t] Q_h,t−1s_h,t−1+ [R_{f k,t}− R_t] Q_f,t−1s_f,t−1+ R_tn_t−1 The end of period objective of the bank will be:

V_t= E_tP∞

τ =t+1(1 − θ)θ^{τ −(t+1)}β^{τ −(t+1)}Λ_t,τn_τ

where βΛ_t,t+1is the stochastic discount factor, equal to the marginal rate of substitution of the representative household that owns the bank. In order to solve its maximization’s problem, the bank will have to take into account an endogenous constraint. We limit the bank’s ability to obtain funds in the deposit market in this way. Suppose that after a bank obtains funds, the banker managing the bank may transfer a fraction ζ of ”divertable” assets to her household (0 < ζ < 1). Since creditors recognize this bank’s incentive to divert funds, a borrowing constraint must arise:

V (s_h,t, s_f,t, d_t) ≥ ζ(Q_h,ts_h,t+ Q_f,ts_f,t) (21)

where on the LHS we have the maximized value of the bank’s objective at the end of t. The value of the bank at the end of t − 1 will be then:

V (s_h,t−1, s_f,t−1, d_t−1) =

= E_t−1Λ_t−1,t(1 − θ) n_t+ θ max_s_h,t_,s_f,t_,d_tV (s_h,t, s_f,t, d_t)

where quantities are all in units of composite consumption good. In order to solve it, we guess that the value function is linear (we will verify this guess later):

V (s_h,t, s_f,t, d_t) = v_h,ts_h,t+ v_f,ts_f,t− v_td_t where the parameters are defined as:

v_h,t: the value to the bank at the end of t of an additional unit of home asset v_f,t: the value to the bank at the end of t of an additional unit of foreign asset

v_t: the marginal cost of deposit

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Being now λ_tthe Lagrange multiplier for the incentive constraint, we have that:

max V (sh,t, sf,t, dt) = vh,tsh,t+ vf,tsf,t− vt(Qh,tsh,t+ Qf,tsf,t− nt) − - λ_t[ζ(Q_h,ts_h,t+ Q_f,ts_f,t) − v_h,ts_h,t− v_f,ts_f,t+ v_t(Q_h,ts_h,t+ Q_f,ts_f,t− n_t)]

We derive the FOCs with respect to s_h,t ,s_f,t and λ_t:

_v

h,t

Qh,t − v_t

(1 + λ_t) = ζλ_t (22)

_v

f,t

Qf,t − v_t

(1 + λ_t) = ζλ_t (23) Q_h,ts_h,th

ζ −_v

h,t

Qh,t − v_ti

+ Q_f,ts_f,th

ζ −_v

f,t

Qf,t − v_ti

≤ v_tn_t (24) where it must be that

λt

n

Qh,tsh,t

h ζ −

_v

h,t

Qh,t − vt

i

+ Qf,tsf,t

h ζ −

_v

f,t

Qf,t − vt

i

− vtnt

o

(25) Then, from the above equations, as long as the incentive constraint is binding (λ_t > 0), the marginal value of asset (in terms of home goods) will be greater than the marginal cost of deposit. Now, by defining:

µ_h,t = _Q^v^h,t

h,t − v_t= µ_f,t= _Q^v^f,t

f,t − v_t> 0

at an equilibrium where both assets (home and foreign) are held and the incentive constraint is binding, the last ones becomes:

Q_h,ts_h,t[ζ − µ_h,t] + Q_f,ts_f,t[ζ − µ_f,t] = v_tn_t (26) or, since µ_h,t = µ_f,t= µ_t,

(Q_h,ts_h,t+ Q_f,ts_f,t) [ζ − µ_t] = v_tn_t (Qh,tsh,t+ Qf,tsf,t) = _[ζ−µ^v^tⁿ^t

t]

where

φ_t= _[ζ−µ^v^t

t]

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is the leverage ratio net of interbank borrowing. Note that the leverage ratio is increasing in µt. Note also that the constraint is binding as long as 0 < µ_t < ζ. If, instead, µ_t > ζ the constraint is not binding. What we assume is that at an equilibrium, for reasonable values of parameters, the incentive constraint always binds. Now, by combining the conjectured value function with the Bellman equation, and given that, at the equilibrium, FOCs plus binding incentive constraint imply that:

Q_h,ts_h,t+ Q_f,ts_f,t= φ_tn_t= n_t+ d_t we can write:

V_t−1 = E_t−1βΛ_t−1,t{(1 − θ) n_t+ θV_t}

Therefore, holding everything else constant, the expected discounted marginal gain to the banker of expanding asset Qh,tsh,t or Qf,tsf,t by one unit will be:

∂Vt

∂Qh,tsh,t = E_tβΛ_t,t+1[R_hk,t+1− R_t+1] (1 − θ + θ (v_t+1+ φ_t+1µ_t+1))

∂Vt

∂Qf,tsf,t = E_tβΛ_t,t+1[R_{f k,t+1}− R_t+1] (1 − θ + θ (v_t+1+ φ_t+1µ_t+1)) while the expected marginal cost of expanding deposit by one unit will be:

∂Vt

∂dt = −E_tβΛ_t,t+1R_t+1(1 − θ + θ ((v_t+1+ φ_t+1µ_t+1))) Therefore, if we rewrite the conjectured value function in this way:

V_t=_v

h,t

Qh,t − v_t

s_h,tQ_h,t+_v

f,t

Qf,t − v_t

s_f,tQ_f,t+ v_tn_t=

= µ_h,tQ_h,ts_h,t+ µ_f,tQ_f,ts_f,t+ v_tn_t

we can verify that the value function is linear in (s_h,t, s_f,t, d_t) if µ_h,t , µ_f,t (the marginal value of holding home/foreign assets net marginal costs) and vt (the marginal cost of deposit) satisfy:

∂Vt

∂Q_h,ts_h,t = µ_h,t =

=E_tβΛ_t,t+1[R_hk,t+1− R_t+1] (1 − θ + θ (v_t+1+ φ_t+1µ_t+1)) (27)

∂Vt

∂Qf,tsf,t = µf,t=

=EtβΛt,t+1[Rf k,t+1− Rt+1] (1 − θ + θ (vt+1+ φt+1µt+1)) (28)

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∂Vt

∂nt = −^∂V_∂d^t

t =

= EtβΛt,t+1Rt+1(1 − θ + θ ((vt+1+ φt+1µt+1))) (29)

where using the definition of µh,t, µf,t we can also rewrite the equations above as:

v_h,t = E_tβΛ_t,t+1[Z_t+1+ (1 − δ)Q_h,t+1] (1 − θ + θ ((v_t+1+ φ_t+1µ_t+1))) (30) v_f,t= E_tβΛ_t,t+1Z_t+1^∗ + (1 − δ)Q_f,t+1 (1 − θ + θ ((vt+1+ φ_t+1µ_t+1))) (31)

vt = EtβΛt,t+1Rt+1(1 − θ + θ ((vt+1+ φt+1µt+1))) (32) or, for the foreign country,

v_h,t^∗ = E_tβΛ^∗_t,t+1[Z_t+1+ (1 − δ)Q_h,t+1] 1 − θ + θ v^∗_t+1+ φ^∗_t+1µ^∗_t+1

(30^∗) v_f,t^∗ =

= EtβΛ^∗_t,t+1Z_t+1^∗ + (1 − δ)Qf,t+1

1 − θ + θ v_t+1^∗ + φ^∗_t+1µ^∗_t+1

(31^∗) v_t^∗ = EtβΛ^∗_t,t+1R^∗_t+1 1 − θ + θ v_t+1^∗

(32^∗)

To conclude, at the equilibrium, if the incentive constraint is binding:

E_tβΛ_t,t+1R_h,t+1(1 − θ + θ ((v_t+1+ φ_t+1µ_t+1))) =

= E_tβΛ_t,t+1R_f,t+1(1 − θ + θ ((v_t+1+ φ_t+1µ_t+1))) >

¿ E_tβΛ_t,t+1R_t+1(1 − θ + θ ((v_t+1+ φ_t+1µ_t+1)))

Otherwise, the banks will increase the amount of home and foreign assets to the point where:

E_tβΛ_t,t+1R_h,t+1(1 − θ + θ ((v_t+1+ φ_t+1µ_t+1))) =

= EtβΛt,t+1Rf,t+1(1 − θ + θ ((vt+1+ φt+1µt+1))) =

= E_tβΛ_t,t+1R_t+1(1 − θ + θ ((v_t+1+ φ_t+1µ_t+1)))

We sum up across individual banks in order to obtain the demand for total bank assets Q_h,tS_h,t and Q_f,tS_f,t as a function of total net worth N_t as:

Q_h,tS_h,t+ Q_f,tS_f,t = _ζ−µ^v^t

tN_t (33)

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2.3 Equilibrium 2 THE MODEL

2.2.5 Evolution of Bank Net Worth

The total net worth for the home banks is given by the sum of the total net worth of existing bankers (o for old) and of entering bankers (y for young):

N_t= N_o,t+ N_y,t (34) where

N_o,t = θ [R_hk,tQ_h,t−1S_h,t−1+ R_{f k,t}Q_f,t−1S_f,t−1− R_tD_t−1] (35) N_y,t= ^ξ(1−θ)_(1−θ) [R_hk,tQ_h,t−1S_h,t−1+ R_{f k,t}Q_f,t−1S_f,t−1] (36)

with θ being the fraction of bankers that survive from the previous period, (1 − θ) the fraction of ”new” bankers that were workers the previous period,

ξ

(1−θ) is the fraction of total value assets gained by exiting bankers that is transferred from the family to each new banker, and D_t is the aggregate deposit (in real terms). Therefore, from the whole banking sector’s balance sheet, deposits will be given by:

D_t= Q_h,tS_h,t+ Q_f,tS_f,t− N_t (37)

2.3 Equilibrium

Since households are symmetric, we can rewrite the previously defined equations in terms of the following equilibrium conditions.

Optimal intertemporal allocation of consumption will be given by : 1 = βE_tR_t+1

Ct+1

Ct

−ρ

(15) 1 = βE_tR_t+1^∗ _C∗

t+1

C_t^∗

^−ρ

(15^∗) Equilibrium in the goods market is described by:

Yt= Ch,t + C_h,t^∗ + h

1 + f

It

It−1

i It P t

P h,t (38) Y_t^∗ = C_f,t+ C_f,t^∗ +h

1 + f _I∗ t

I_t−1^∗

i I_t^∗_P^P^t

f,t (38^∗)

For the labour markets we get the following equilibrium conditions:

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α^P_P^h,t

t

Yt

Lt = κL_tC_t^ρ (39) α^P

∗ f,t

P_t^∗ Y_t^∗

L^∗_t = κL^∗_tC_t^∗ρ (39^∗) The supply for home securities will be given by:

Sh,t+ S_h,t^∗ = [It+ (1 − δ) Kt] (40) S_f,t^∗ + Sf,t = [I_t^∗+ (1 − δ) K_t^∗] (40^∗)

while the demands are obtained by the (33), giving the equilibrium in the security markets.

Lastly, being the household also the owner of both national banks and firms, it follows that in the equation (5):

C_t= −D_t+1+ R_tD_t+^W_P^t

tL_t+ ^Π_P^t

t (5)

the real dividend will be given by the sum of the following terms:

- real profits from both final and capital good sectors - the difference in a generic bank’s wealth

- the transfers from the exiting banker to the household net from the transfer from the household to the new banker

By summing up all these terms, the representative household’s budget constraint in the home country will be given by:

Ph,t

Pt Y_t+ Q_h,tS_h,t^∗ + R_f,tQ_f,t−1S_f,t−1− R_h,tQ_h,t−1S_h,t−1^∗ − Q_f,tS_f,t=

= C_t+h

1 + f

It

It−1

i

I_t (41) and, symmetrically, for the foreign country:

Pf,t

Pt Y_t^∗+ Q_f,tS_f,t+ R_h,tQ_h,t−1S_h,t−1^∗ − R_f,tQ_f,t−1S_f,t−1− Q_h,tS_h,t^∗ =

= C^∗_t + h

1 + f

I_t^∗ I_t−1^∗

i

I_t^∗ (41^∗)

Lastly, notice that from the equilibrium conditions in the financial markets, we must have:

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vh,t

Qh,t − v_t= _Q^v^f,t

f,t − v_t

v_h,t^∗

Qh,t − v_t^∗ = ^v

∗ f,t

Qf,t − v_t^∗ that can be summarized as:

E_t{R_h,t+1− R_f,t+1} = 0 (42)

Then, given that A_t, A^∗_t, Ψ_t and Ψ^∗_t follow exogenous stochastic processes, we need to find the real prices:

Q_h,t, R_ht, R_t+1,^P_P^h,t

t

Q_f,t, R_{f t}, R^∗_t+1,^P_P^f,t

t

the quantities:

Y_t, C_t, C_h,t, C_f,t, L_t, I_t, K_t+1, Z_t, D_t, N_t, S_h,t, S_f,t, Y u_t Y_t^∗, C_t^∗, C_h,t^∗ , C_f,t^∗ , L^∗_t, I_t^∗, K_t+1^∗ , Z_t^∗,

D^∗_t, N_t^∗, S_h,t^∗ , S_f,t^∗ , Y u^∗_t the shadow prices:

v_t, v_h,t, v_f,t, λ_t, Φ_t v_t^∗, v_h,t^∗ , v_f,t^∗ , λ^∗_t, Φ^∗_t determined as a function of the state variables:

K_t, C_t−1, I_t−1, A_t, Ψ_t, Q_h,t−1, Y u_t−1 K_t^∗, C_t−1^∗ , I_t−1^∗ , A^∗_t, Ψ^∗_t, Q_f,t−1, Y u^∗_t−1 by the sequence of the following 44 equations:

(1) − (6) − (7) − (9) − (10) − (11) − (15) − (17) (1^∗) − (6^∗) − (7^∗) − (9^∗) − (10^∗) − (11^∗) − (15^∗) − (17^∗)

that describe the optimization conditions of households and final good firms (home and abroad),

(22) − (23) − (24) − (30) − (31) − (32) − (34) (22^∗) − (23^∗) − (24^∗) − (30^∗) − (31^∗) − (32^∗) − (34^∗)

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2.4 The Steady State 2 THE MODEL

that describe the optimization conditions for banks, (37) − (38) − (39) − (40) − (41) (37^∗) − (38^∗) − (39^∗) − (40^∗) − (41^∗)

that describe the market clearing conditions for goods, securities and labour.

We also include the definitions of Φ_t and of Y u_t : Φ_t= ^v^t

ζ−

vh,t Qh,t−vt

Y ut= _I^I^t

t−1

2.4 The Steady State

By assumption, at steady state the real capital prices Q_h,t and Q_f,t are equal to unity. Then, being at equilibrium µh,t = µf,t= µ, it must be the case that

Rhk = Rf k = Rk

and also from the demands for securities we have:

S_h+ S_f = φN S^∗_h+ S_f^∗ = φN^∗ where φ is defined as:

φt= _ζ−µ^v^t

t (42) while from the supply side we get:

S_h+ S_f = N + D S^∗_h+ S_f^∗ = N^∗+ D^∗

Lastly,from the definition of bank’s wealth we have:

S_h+ S_h^∗ = K S_f + S_f^∗ = K^∗

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Moreover, from Euler Equation (15) and (15^∗) we know that at steady state:

R = R^∗ = _β¹

Then, from equations (34) − (35) − (36) and (34^∗) − (35^∗) − (36^∗) it follows that at the equilibrium (in the binding case, the only one that we consider):

R_k= ^β+θφ−θ_(θ+ξ)βφ (43)

and therefore from equations (9) − (9^∗) also follows that:

Ph

P Y K = ^P

∗ f

P^∗ Y^∗ K^∗ =

δ − 1 + _Ψ¹^β+θφ−θ_(θ+ξ)βφ

1

1−α (44)

Then, by considering the definition of φ_t and the equations (27) − (28) − (29) and (27^∗) − (28^∗) − (29^∗) we have that:

φ_t= _ζ−µ^v^t

t

where at steady state:

v = [1 − θ + θv + θφµ]

and then:

v = ^{(1−θ)(ζ−µ)}_{(ζ−µ−θζ)} (45) φ = _{(ζ−µ−θζ)}^(1−θ) (46) and therefore we find that at the steady state:

φ = φ^∗ since at equilibrium

v_h = v_f = v and also

µh = µf = µ

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where µ from the (27) − (28) and (27^∗) − (28^∗) is given by:

µ = βh

R_k− _β¹i

(1 − θ + θv + θφµ) =

=(β − θ − φξ) (1 − θ + θφζ) _φ(θ+ξ)¹ (47)

Therefore, by substituting the (47) into the (46) we find a second-order equation in φ:

φ [ζξ + ζθ − ζθξ − ζθ²+ ξ − ξθ − βθζ + ζθ²] + +φ²[θξζ] + [θ − β + βθ − θ²− (1 − θ) (ξ + θ)] = 0

Then, since we know that an equilibrium where the constraints are binding we must have µ < ζ, and since we assume v to be positive (being it a marginal cost), only one root of the above equation will guarantee that φ > 0.

From (46), we can derive the numerical steady state values for R_k, while from (43) − (44) we can obtain ^P_{P K}^h^Yand ^P

∗ fY^∗ P^∗K^∗. Now, by considering ^P_P^h_K^Y = ^P

∗ f

P^∗ Y^∗

K^∗ into the equilibrium conditions in the Good Markets (38) − (38^∗) it follows that:

K

K^∗ = _(1−γ)^γ (48)

Therefore, from equations (8) − (9) and (8^∗) − (9^∗) it follows that:

Y

Y^∗ = ^Ψ_Ψ^∗^P_P^f

h

K

K^∗ +^(1−δ)(Ψ_Ψ(1−α)^∗^−Ψ)_Y^KP∗Ph

that, since at steady state we set Ψ = Ψ^∗ = 1, can be rewritten as:

Y Y^∗ = ^P_P^f

h

K K^∗ (49) and the labour ratio across countries

L

L^∗ = _K^K∗ (50)

while from the production function (6)−(6^∗) we obtain the relative prices:

Ph

P_f = ^A_A^∗ (51)

Therefore from the equation (49) and (50) we get the output and labour ratios:

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Y

Y^∗ = _(1−γ)¹ _A^A∗

L

L^∗ = _(1−γ)^γ

where the above equations can be simplified since we are assuming A, A^∗ equal to unity at steady state.

Also by considering equation (51) into the definitions of home and foreign price of one unit of the composite consumption good (equations (2) − (2^∗)) we derive the relative prices ^P_P^h and ^P_P^f where:

Ph

P = γ^γ(1 − γ)^(1−γ)

Ph

Pf

1−γ

Pf

P = γ^γ(1 − γ)^(1−γ)_P

f

Ph

1−γ

From the definitions of Zt and Z_t^∗ (equations (10) − (10^∗)) we get now the labour to capital ratios:

L K =

Z 1−α

P APh

_α¹ (52)

L^∗ K^∗ =

Z^∗ 1−α

P^∗ A^∗P_f^∗

_α¹

(52^∗) where

Z = β + θφ − θ

(θ + ξ) βφ + δ − 1

Now from the equations (9) − (9^∗) and knowing the capital to labour ratios we find the real wages:

W

P = α^{P hY}_{P L} (53)

W^∗

P^∗ = α^{P f Y}_P∗L^∗^∗ (53^∗)

Then by considering the above results into the equilibrium conditions in the Good Markets (38) − (38^∗)) it follows that:

(C + C^∗) = ( ˜α − δ) (K + K^∗) =

=(K + K^∗) (54)

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where we set

˜

α = ^P_{P K}^h^Y = ^P

∗ fY^∗ P^∗K^∗

β = ˜˜ α − δ

Therefore, by combining the labour supply equations (16) − (16^∗) with equations (54) and knowing the capital to labour ratios from equations (52) and the capital ratio from equation (48) we obtain:

K^∗ =

W κP ˜γ

¹_ρ

1+˜δ^{− 1}^ρ β˜(^1+˜^δ)

_1+ρ^ρ (54) where we set

˜

γ = _K^L = _K^L^∗∗

and

δ =˜ _K^K∗

From the above results it is easy to obtain K, Y, Y^∗, L, L^∗. Then, from the labour supply equations, we also derive C and C^∗, while from equations (17) − (18) and (17^∗) − (18^∗) we end up with C_h, C_f, C_h^∗, C_f^∗.

Lastly, from equations (40) − (40^∗) we obtain S_h+ S_h^∗ and S_f+ S_f^∗. Then, by combining the (40) − (40^∗) with the home and foreign household’s budget constraint (41) − (41^∗), we find that at steady state:

C−^Ph_P Y +δK

(1−Rk) = (S_h^∗− S_f)

C^∗−^{P ∗}^f

P ∗Y^∗+δK^∗

(1−R_k) = (S_f − S_h^∗)

In a steady state where trade balances are equal to zero, S_h^∗ = Sf. Oth- erwise (if we not constraint the trade balances to be equal to zero at steady state), we can derive a relationship between S_h^∗ and S_f. From the last above equations we can also obtain Sh + Sf and S_h^∗ + S_f^∗ and therefore we get N and N^∗ from the demands for securities. Lastly, from equations (37) − (37^∗) we find D and D^∗ and then N_y, N_o, N_y^∗ and N_o^∗.

There is, however, no way to distinguish across Sh and Sf (in the home financial portfolio) and across S_h^∗ and S_f^∗ (in the foreign financial portfolio).

The steady state is therefore not completely determined yet. We need a method in order to overcome this step.